# Properties

 Label 175.2.b.c.99.2 Level $175$ Weight $2$ Character 175.99 Analytic conductor $1.397$ 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] = [175,2,Mod(99,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("175.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$1.39738203537$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.2 Root $$-0.618034i$$ of defining polynomial Character $$\chi$$ $$=$$ 175.99 Dual form 175.2.b.c.99.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034i q^{2} +3.23607i q^{3} +1.61803 q^{4} +2.00000 q^{6} +1.00000i q^{7} -2.23607i q^{8} -7.47214 q^{9} +O(q^{10})$$ $$q-0.618034i q^{2} +3.23607i q^{3} +1.61803 q^{4} +2.00000 q^{6} +1.00000i q^{7} -2.23607i q^{8} -7.47214 q^{9} -0.236068 q^{11} +5.23607i q^{12} -1.23607i q^{13} +0.618034 q^{14} +1.85410 q^{16} +2.47214i q^{17} +4.61803i q^{18} +4.47214 q^{19} -3.23607 q^{21} +0.145898i q^{22} -6.23607i q^{23} +7.23607 q^{24} -0.763932 q^{26} -14.4721i q^{27} +1.61803i q^{28} -5.00000 q^{29} +3.70820 q^{31} -5.61803i q^{32} -0.763932i q^{33} +1.52786 q^{34} -12.0902 q^{36} +3.00000i q^{37} -2.76393i q^{38} +4.00000 q^{39} +4.76393 q^{41} +2.00000i q^{42} -1.76393i q^{43} -0.381966 q^{44} -3.85410 q^{46} -2.00000i q^{47} +6.00000i q^{48} -1.00000 q^{49} -8.00000 q^{51} -2.00000i q^{52} -8.47214i q^{53} -8.94427 q^{54} +2.23607 q^{56} +14.4721i q^{57} +3.09017i q^{58} -11.7082 q^{59} -9.70820 q^{61} -2.29180i q^{62} -7.47214i q^{63} +0.236068 q^{64} -0.472136 q^{66} -4.23607i q^{67} +4.00000i q^{68} +20.1803 q^{69} +8.70820 q^{71} +16.7082i q^{72} +8.76393i q^{73} +1.85410 q^{74} +7.23607 q^{76} -0.236068i q^{77} -2.47214i q^{78} +11.1803 q^{79} +24.4164 q^{81} -2.94427i q^{82} +7.70820i q^{83} -5.23607 q^{84} -1.09017 q^{86} -16.1803i q^{87} +0.527864i q^{88} -17.2361 q^{89} +1.23607 q^{91} -10.0902i q^{92} +12.0000i q^{93} -1.23607 q^{94} +18.1803 q^{96} +5.23607i q^{97} +0.618034i q^{98} +1.76393 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 8 q^{6} - 12 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 8 * q^6 - 12 * q^9 $$4 q + 2 q^{4} + 8 q^{6} - 12 q^{9} + 8 q^{11} - 2 q^{14} - 6 q^{16} - 4 q^{21} + 20 q^{24} - 12 q^{26} - 20 q^{29} - 12 q^{31} + 24 q^{34} - 26 q^{36} + 16 q^{39} + 28 q^{41} - 6 q^{44} - 2 q^{46} - 4 q^{49} - 32 q^{51} - 20 q^{59} - 12 q^{61} - 8 q^{64} + 16 q^{66} + 36 q^{69} + 8 q^{71} - 6 q^{74} + 20 q^{76} + 44 q^{81} - 12 q^{84} + 18 q^{86} - 60 q^{89} - 4 q^{91} + 4 q^{94} + 28 q^{96} + 16 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 + 8 * q^6 - 12 * q^9 + 8 * q^11 - 2 * q^14 - 6 * q^16 - 4 * q^21 + 20 * q^24 - 12 * q^26 - 20 * q^29 - 12 * q^31 + 24 * q^34 - 26 * q^36 + 16 * q^39 + 28 * q^41 - 6 * q^44 - 2 * q^46 - 4 * q^49 - 32 * q^51 - 20 * q^59 - 12 * q^61 - 8 * q^64 + 16 * q^66 + 36 * q^69 + 8 * q^71 - 6 * q^74 + 20 * q^76 + 44 * q^81 - 12 * q^84 + 18 * q^86 - 60 * q^89 - 4 * q^91 + 4 * q^94 + 28 * q^96 + 16 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 0.618034i − 0.437016i −0.975835 0.218508i $$-0.929881\pi$$
0.975835 0.218508i $$-0.0701190\pi$$
$$3$$ 3.23607i 1.86834i 0.356822 + 0.934172i $$0.383860\pi$$
−0.356822 + 0.934172i $$0.616140\pi$$
$$4$$ 1.61803 0.809017
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 1.00000i 0.377964i
$$8$$ − 2.23607i − 0.790569i
$$9$$ −7.47214 −2.49071
$$10$$ 0 0
$$11$$ −0.236068 −0.0711772 −0.0355886 0.999367i $$-0.511331\pi$$
−0.0355886 + 0.999367i $$0.511331\pi$$
$$12$$ 5.23607i 1.51152i
$$13$$ − 1.23607i − 0.342824i −0.985199 0.171412i $$-0.945167\pi$$
0.985199 0.171412i $$-0.0548329\pi$$
$$14$$ 0.618034 0.165177
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ 2.47214i 0.599581i 0.954005 + 0.299791i $$0.0969168\pi$$
−0.954005 + 0.299791i $$0.903083\pi$$
$$18$$ 4.61803i 1.08848i
$$19$$ 4.47214 1.02598 0.512989 0.858395i $$-0.328538\pi$$
0.512989 + 0.858395i $$0.328538\pi$$
$$20$$ 0 0
$$21$$ −3.23607 −0.706168
$$22$$ 0.145898i 0.0311056i
$$23$$ − 6.23607i − 1.30031i −0.759802 0.650155i $$-0.774704\pi$$
0.759802 0.650155i $$-0.225296\pi$$
$$24$$ 7.23607 1.47706
$$25$$ 0 0
$$26$$ −0.763932 −0.149819
$$27$$ − 14.4721i − 2.78516i
$$28$$ 1.61803i 0.305780i
$$29$$ −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 $$-0.391937\pi$$
0.333007 + 0.942925i $$0.391937\pi$$
$$32$$ − 5.61803i − 0.993137i
$$33$$ − 0.763932i − 0.132983i
$$34$$ 1.52786 0.262027
$$35$$ 0 0
$$36$$ −12.0902 −2.01503
$$37$$ 3.00000i 0.493197i 0.969118 + 0.246598i $$0.0793129\pi$$
−0.969118 + 0.246598i $$0.920687\pi$$
$$38$$ − 2.76393i − 0.448369i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 4.76393 0.744001 0.372001 0.928232i $$-0.378672\pi$$
0.372001 + 0.928232i $$0.378672\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ − 1.76393i − 0.268997i −0.990914 0.134499i $$-0.957058\pi$$
0.990914 0.134499i $$-0.0429424\pi$$
$$44$$ −0.381966 −0.0575835
$$45$$ 0 0
$$46$$ −3.85410 −0.568256
$$47$$ − 2.00000i − 0.291730i −0.989305 0.145865i $$-0.953403\pi$$
0.989305 0.145865i $$-0.0465965\pi$$
$$48$$ 6.00000i 0.866025i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 8.47214i − 1.16374i −0.813283 0.581869i $$-0.802322\pi$$
0.813283 0.581869i $$-0.197678\pi$$
$$54$$ −8.94427 −1.21716
$$55$$ 0 0
$$56$$ 2.23607 0.298807
$$57$$ 14.4721i 1.91688i
$$58$$ 3.09017i 0.405759i
$$59$$ −11.7082 −1.52428 −0.762139 0.647413i $$-0.775851\pi$$
−0.762139 + 0.647413i $$0.775851\pi$$
$$60$$ 0 0
$$61$$ −9.70820 −1.24301 −0.621504 0.783411i $$-0.713478\pi$$
−0.621504 + 0.783411i $$0.713478\pi$$
$$62$$ − 2.29180i − 0.291058i
$$63$$ − 7.47214i − 0.941401i
$$64$$ 0.236068 0.0295085
$$65$$ 0 0
$$66$$ −0.472136 −0.0581159
$$67$$ − 4.23607i − 0.517518i −0.965942 0.258759i $$-0.916686\pi$$
0.965942 0.258759i $$-0.0833136\pi$$
$$68$$ 4.00000i 0.485071i
$$69$$ 20.1803 2.42943
$$70$$ 0 0
$$71$$ 8.70820 1.03347 0.516737 0.856144i $$-0.327147\pi$$
0.516737 + 0.856144i $$0.327147\pi$$
$$72$$ 16.7082i 1.96908i
$$73$$ 8.76393i 1.02574i 0.858466 + 0.512870i $$0.171418\pi$$
−0.858466 + 0.512870i $$0.828582\pi$$
$$74$$ 1.85410 0.215535
$$75$$ 0 0
$$76$$ 7.23607 0.830034
$$77$$ − 0.236068i − 0.0269024i
$$78$$ − 2.47214i − 0.279914i
$$79$$ 11.1803 1.25789 0.628943 0.777451i $$-0.283488\pi$$
0.628943 + 0.777451i $$0.283488\pi$$
$$80$$ 0 0
$$81$$ 24.4164 2.71293
$$82$$ − 2.94427i − 0.325140i
$$83$$ 7.70820i 0.846085i 0.906110 + 0.423043i $$0.139038\pi$$
−0.906110 + 0.423043i $$0.860962\pi$$
$$84$$ −5.23607 −0.571302
$$85$$ 0 0
$$86$$ −1.09017 −0.117556
$$87$$ − 16.1803i − 1.73471i
$$88$$ 0.527864i 0.0562705i
$$89$$ −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
$$92$$ − 10.0902i − 1.05197i
$$93$$ 12.0000i 1.24434i
$$94$$ −1.23607 −0.127491
$$95$$ 0 0
$$96$$ 18.1803 1.85552
$$97$$ 5.23607i 0.531642i 0.964022 + 0.265821i $$0.0856430\pi$$
−0.964022 + 0.265821i $$0.914357\pi$$
$$98$$ 0.618034i 0.0624309i
$$99$$ 1.76393 0.177282
$$100$$ 0 0
$$101$$ 4.76393 0.474029 0.237014 0.971506i $$-0.423831\pi$$
0.237014 + 0.971506i $$0.423831\pi$$
$$102$$ 4.94427i 0.489556i
$$103$$ − 8.47214i − 0.834784i −0.908726 0.417392i $$-0.862944\pi$$
0.908726 0.417392i $$-0.137056\pi$$
$$104$$ −2.76393 −0.271026
$$105$$ 0 0
$$106$$ −5.23607 −0.508572
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ − 23.4164i − 2.25324i
$$109$$ −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
$$112$$ 1.85410i 0.175196i
$$113$$ 14.4164i 1.35618i 0.734978 + 0.678091i $$0.237192\pi$$
−0.734978 + 0.678091i $$0.762808\pi$$
$$114$$ 8.94427 0.837708
$$115$$ 0 0
$$116$$ −8.09017 −0.751153
$$117$$ 9.23607i 0.853875i
$$118$$ 7.23607i 0.666134i
$$119$$ −2.47214 −0.226620
$$120$$ 0 0
$$121$$ −10.9443 −0.994934
$$122$$ 6.00000i 0.543214i
$$123$$ 15.4164i 1.39005i
$$124$$ 6.00000 0.538816
$$125$$ 0 0
$$126$$ −4.61803 −0.411407
$$127$$ 13.6525i 1.21146i 0.795670 + 0.605731i $$0.207119\pi$$
−0.795670 + 0.605731i $$0.792881\pi$$
$$128$$ − 11.3820i − 1.00603i
$$129$$ 5.70820 0.502579
$$130$$ 0 0
$$131$$ −16.9443 −1.48043 −0.740214 0.672371i $$-0.765276\pi$$
−0.740214 + 0.672371i $$0.765276\pi$$
$$132$$ − 1.23607i − 0.107586i
$$133$$ 4.47214i 0.387783i
$$134$$ −2.61803 −0.226164
$$135$$ 0 0
$$136$$ 5.52786 0.474010
$$137$$ − 10.9443i − 0.935032i −0.883985 0.467516i $$-0.845149\pi$$
0.883985 0.467516i $$-0.154851\pi$$
$$138$$ − 12.4721i − 1.06170i
$$139$$ −10.6525 −0.903531 −0.451766 0.892137i $$-0.649206\pi$$
−0.451766 + 0.892137i $$0.649206\pi$$
$$140$$ 0 0
$$141$$ 6.47214 0.545052
$$142$$ − 5.38197i − 0.451645i
$$143$$ 0.291796i 0.0244012i
$$144$$ −13.8541 −1.15451
$$145$$ 0 0
$$146$$ 5.41641 0.448265
$$147$$ − 3.23607i − 0.266906i
$$148$$ 4.85410i 0.399005i
$$149$$ −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 $$-0.807922\pi$$
−0.823394 + 0.567470i $$0.807922\pi$$
$$152$$ − 10.0000i − 0.811107i
$$153$$ − 18.4721i − 1.49338i
$$154$$ −0.145898 −0.0117568
$$155$$ 0 0
$$156$$ 6.47214 0.518186
$$157$$ 0.763932i 0.0609684i 0.999535 + 0.0304842i $$0.00970493\pi$$
−0.999535 + 0.0304842i $$0.990295\pi$$
$$158$$ − 6.90983i − 0.549717i
$$159$$ 27.4164 2.17426
$$160$$ 0 0
$$161$$ 6.23607 0.491471
$$162$$ − 15.0902i − 1.18560i
$$163$$ 1.52786i 0.119672i 0.998208 + 0.0598358i $$0.0190577\pi$$
−0.998208 + 0.0598358i $$0.980942\pi$$
$$164$$ 7.70820 0.601910
$$165$$ 0 0
$$166$$ 4.76393 0.369753
$$167$$ 5.23607i 0.405179i 0.979264 + 0.202590i $$0.0649357\pi$$
−0.979264 + 0.202590i $$0.935064\pi$$
$$168$$ 7.23607i 0.558275i
$$169$$ 11.4721 0.882472
$$170$$ 0 0
$$171$$ −33.4164 −2.55542
$$172$$ − 2.85410i − 0.217623i
$$173$$ 11.5279i 0.876447i 0.898866 + 0.438224i $$0.144392\pi$$
−0.898866 + 0.438224i $$0.855608\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ −0.437694 −0.0329924
$$177$$ − 37.8885i − 2.84788i
$$178$$ 10.6525i 0.798437i
$$179$$ 23.4164 1.75022 0.875112 0.483920i $$-0.160787\pi$$
0.875112 + 0.483920i $$0.160787\pi$$
$$180$$ 0 0
$$181$$ 8.18034 0.608040 0.304020 0.952666i $$-0.401671\pi$$
0.304020 + 0.952666i $$0.401671\pi$$
$$182$$ − 0.763932i − 0.0566264i
$$183$$ − 31.4164i − 2.32237i
$$184$$ −13.9443 −1.02799
$$185$$ 0 0
$$186$$ 7.41641 0.543797
$$187$$ − 0.583592i − 0.0426765i
$$188$$ − 3.23607i − 0.236015i
$$189$$ 14.4721 1.05269
$$190$$ 0 0
$$191$$ 6.47214 0.468307 0.234154 0.972200i $$-0.424768\pi$$
0.234154 + 0.972200i $$0.424768\pi$$
$$192$$ 0.763932i 0.0551320i
$$193$$ − 12.4164i − 0.893753i −0.894596 0.446876i $$-0.852536\pi$$
0.894596 0.446876i $$-0.147464\pi$$
$$194$$ 3.23607 0.232336
$$195$$ 0 0
$$196$$ −1.61803 −0.115574
$$197$$ − 1.47214i − 0.104885i −0.998624 0.0524427i $$-0.983299\pi$$
0.998624 0.0524427i $$-0.0167007\pi$$
$$198$$ − 1.09017i − 0.0774750i
$$199$$ −7.23607 −0.512951 −0.256476 0.966551i $$-0.582561\pi$$
−0.256476 + 0.966551i $$0.582561\pi$$
$$200$$ 0 0
$$201$$ 13.7082 0.966902
$$202$$ − 2.94427i − 0.207158i
$$203$$ − 5.00000i − 0.350931i
$$204$$ −12.9443 −0.906280
$$205$$ 0 0
$$206$$ −5.23607 −0.364814
$$207$$ 46.5967i 3.23870i
$$208$$ − 2.29180i − 0.158907i
$$209$$ −1.05573 −0.0730262
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ − 13.7082i − 0.941483i
$$213$$ 28.1803i 1.93089i
$$214$$ 4.94427 0.337983
$$215$$ 0 0
$$216$$ −32.3607 −2.20187
$$217$$ 3.70820i 0.251729i
$$218$$ 5.20163i 0.352299i
$$219$$ −28.3607 −1.91644
$$220$$ 0 0
$$221$$ 3.05573 0.205551
$$222$$ 6.00000i 0.402694i
$$223$$ − 20.1803i − 1.35138i −0.737188 0.675688i $$-0.763847\pi$$
0.737188 0.675688i $$-0.236153\pi$$
$$224$$ 5.61803 0.375371
$$225$$ 0 0
$$226$$ 8.90983 0.592673
$$227$$ 21.4164i 1.42146i 0.703466 + 0.710728i $$0.251635\pi$$
−0.703466 + 0.710728i $$0.748365\pi$$
$$228$$ 23.4164i 1.55079i
$$229$$ 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
$$232$$ 11.1803i 0.734025i
$$233$$ − 7.94427i − 0.520447i −0.965548 0.260223i $$-0.916204\pi$$
0.965548 0.260223i $$-0.0837962\pi$$
$$234$$ 5.70820 0.373157
$$235$$ 0 0
$$236$$ −18.9443 −1.23317
$$237$$ 36.1803i 2.35017i
$$238$$ 1.52786i 0.0990367i
$$239$$ 5.52786 0.357568 0.178784 0.983888i $$-0.442784\pi$$
0.178784 + 0.983888i $$0.442784\pi$$
$$240$$ 0 0
$$241$$ −3.52786 −0.227250 −0.113625 0.993524i $$-0.536246\pi$$
−0.113625 + 0.993524i $$0.536246\pi$$
$$242$$ 6.76393i 0.434802i
$$243$$ 35.5967i 2.28353i
$$244$$ −15.7082 −1.00561
$$245$$ 0 0
$$246$$ 9.52786 0.607474
$$247$$ − 5.52786i − 0.351730i
$$248$$ − 8.29180i − 0.526530i
$$249$$ −24.9443 −1.58078
$$250$$ 0 0
$$251$$ 6.47214 0.408518 0.204259 0.978917i $$-0.434522\pi$$
0.204259 + 0.978917i $$0.434522\pi$$
$$252$$ − 12.0902i − 0.761609i
$$253$$ 1.47214i 0.0925524i
$$254$$ 8.43769 0.529428
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ − 12.6525i − 0.789240i −0.918844 0.394620i $$-0.870876\pi$$
0.918844 0.394620i $$-0.129124\pi$$
$$258$$ − 3.52786i − 0.219635i
$$259$$ −3.00000 −0.186411
$$260$$ 0 0
$$261$$ 37.3607 2.31257
$$262$$ 10.4721i 0.646971i
$$263$$ − 16.2361i − 1.00116i −0.865691 0.500579i $$-0.833120\pi$$
0.865691 0.500579i $$-0.166880\pi$$
$$264$$ −1.70820 −0.105133
$$265$$ 0 0
$$266$$ 2.76393 0.169468
$$267$$ − 55.7771i − 3.41350i
$$268$$ − 6.85410i − 0.418681i
$$269$$ 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 $$-0.244103\pi$$
0.720085 + 0.693885i $$0.244103\pi$$
$$272$$ 4.58359i 0.277921i
$$273$$ 4.00000i 0.242091i
$$274$$ −6.76393 −0.408624
$$275$$ 0 0
$$276$$ 32.6525 1.96545
$$277$$ − 19.8885i − 1.19499i −0.801874 0.597493i $$-0.796163\pi$$
0.801874 0.597493i $$-0.203837\pi$$
$$278$$ 6.58359i 0.394858i
$$279$$ −27.7082 −1.65885
$$280$$ 0 0
$$281$$ −15.3607 −0.916341 −0.458171 0.888864i $$-0.651495\pi$$
−0.458171 + 0.888864i $$0.651495\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ − 17.4164i − 1.03530i −0.855593 0.517649i $$-0.826807\pi$$
0.855593 0.517649i $$-0.173193\pi$$
$$284$$ 14.0902 0.836098
$$285$$ 0 0
$$286$$ 0.180340 0.0106637
$$287$$ 4.76393i 0.281206i
$$288$$ 41.9787i 2.47362i
$$289$$ 10.8885 0.640503
$$290$$ 0 0
$$291$$ −16.9443 −0.993291
$$292$$ 14.1803i 0.829842i
$$293$$ 31.1246i 1.81832i 0.416448 + 0.909160i $$0.363275\pi$$
−0.416448 + 0.909160i $$0.636725\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 6.70820 0.389906
$$297$$ 3.41641i 0.198240i
$$298$$ 2.43769i 0.141212i
$$299$$ −7.70820 −0.445777
$$300$$ 0 0
$$301$$ 1.76393 0.101671
$$302$$ 12.5066i 0.719673i
$$303$$ 15.4164i 0.885649i
$$304$$ 8.29180 0.475567
$$305$$ 0 0
$$306$$ −11.4164 −0.652633
$$307$$ 4.58359i 0.261599i 0.991409 + 0.130800i $$0.0417545\pi$$
−0.991409 + 0.130800i $$0.958246\pi$$
$$308$$ − 0.381966i − 0.0217645i
$$309$$ 27.4164 1.55966
$$310$$ 0 0
$$311$$ 24.3607 1.38137 0.690684 0.723157i $$-0.257310\pi$$
0.690684 + 0.723157i $$0.257310\pi$$
$$312$$ − 8.94427i − 0.506370i
$$313$$ − 19.5279i − 1.10378i −0.833917 0.551890i $$-0.813907\pi$$
0.833917 0.551890i $$-0.186093\pi$$
$$314$$ 0.472136 0.0266442
$$315$$ 0 0
$$316$$ 18.0902 1.01765
$$317$$ 25.3607i 1.42440i 0.701978 + 0.712199i $$0.252301\pi$$
−0.701978 + 0.712199i $$0.747699\pi$$
$$318$$ − 16.9443i − 0.950188i
$$319$$ 1.18034 0.0660863
$$320$$ 0 0
$$321$$ −25.8885 −1.44496
$$322$$ − 3.85410i − 0.214781i
$$323$$ 11.0557i 0.615157i
$$324$$ 39.5066 2.19481
$$325$$ 0 0
$$326$$ 0.944272 0.0522984
$$327$$ − 27.2361i − 1.50616i
$$328$$ − 10.6525i − 0.588185i
$$329$$ 2.00000 0.110264
$$330$$ 0 0
$$331$$ −24.7082 −1.35809 −0.679043 0.734099i $$-0.737605\pi$$
−0.679043 + 0.734099i $$0.737605\pi$$
$$332$$ 12.4721i 0.684497i
$$333$$ − 22.4164i − 1.22841i
$$334$$ 3.23607 0.177070
$$335$$ 0 0
$$336$$ −6.00000 −0.327327
$$337$$ − 16.4721i − 0.897294i −0.893709 0.448647i $$-0.851906\pi$$
0.893709 0.448647i $$-0.148094\pi$$
$$338$$ − 7.09017i − 0.385654i
$$339$$ −46.6525 −2.53381
$$340$$ 0 0
$$341$$ −0.875388 −0.0474049
$$342$$ 20.6525i 1.11676i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −3.94427 −0.212661
$$345$$ 0 0
$$346$$ 7.12461 0.383022
$$347$$ 20.2361i 1.08633i 0.839626 + 0.543165i $$0.182774\pi$$
−0.839626 + 0.543165i $$0.817226\pi$$
$$348$$ − 26.1803i − 1.40341i
$$349$$ −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
$$352$$ 1.32624i 0.0706887i
$$353$$ 2.18034i 0.116048i 0.998315 + 0.0580239i $$0.0184800\pi$$
−0.998315 + 0.0580239i $$0.981520\pi$$
$$354$$ −23.4164 −1.24457
$$355$$ 0 0
$$356$$ −27.8885 −1.47809
$$357$$ − 8.00000i − 0.423405i
$$358$$ − 14.4721i − 0.764876i
$$359$$ 30.1246 1.58992 0.794958 0.606664i $$-0.207493\pi$$
0.794958 + 0.606664i $$0.207493\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 5.05573i − 0.265723i
$$363$$ − 35.4164i − 1.85888i
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ −19.4164 −1.01491
$$367$$ − 37.1246i − 1.93789i −0.247278 0.968944i $$-0.579536\pi$$
0.247278 0.968944i $$-0.420464\pi$$
$$368$$ − 11.5623i − 0.602727i
$$369$$ −35.5967 −1.85309
$$370$$ 0 0
$$371$$ 8.47214 0.439851
$$372$$ 19.4164i 1.00669i
$$373$$ 37.8328i 1.95891i 0.201665 + 0.979454i $$0.435365\pi$$
−0.201665 + 0.979454i $$0.564635\pi$$
$$374$$ −0.360680 −0.0186503
$$375$$ 0 0
$$376$$ −4.47214 −0.230633
$$377$$ 6.18034i 0.318304i
$$378$$ − 8.94427i − 0.460044i
$$379$$ 11.1803 0.574295 0.287148 0.957886i $$-0.407293\pi$$
0.287148 + 0.957886i $$0.407293\pi$$
$$380$$ 0 0
$$381$$ −44.1803 −2.26343
$$382$$ − 4.00000i − 0.204658i
$$383$$ 33.2361i 1.69828i 0.528165 + 0.849142i $$0.322880\pi$$
−0.528165 + 0.849142i $$0.677120\pi$$
$$384$$ 36.8328 1.87962
$$385$$ 0 0
$$386$$ −7.67376 −0.390584
$$387$$ 13.1803i 0.669994i
$$388$$ 8.47214i 0.430108i
$$389$$ −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
$$392$$ 2.23607i 0.112938i
$$393$$ − 54.8328i − 2.76595i
$$394$$ −0.909830 −0.0458366
$$395$$ 0 0
$$396$$ 2.85410 0.143424
$$397$$ 9.05573i 0.454494i 0.973837 + 0.227247i $$0.0729725\pi$$
−0.973837 + 0.227247i $$0.927028\pi$$
$$398$$ 4.47214i 0.224168i
$$399$$ −14.4721 −0.724513
$$400$$ 0 0
$$401$$ 2.52786 0.126236 0.0631178 0.998006i $$-0.479896\pi$$
0.0631178 + 0.998006i $$0.479896\pi$$
$$402$$ − 8.47214i − 0.422552i
$$403$$ − 4.58359i − 0.228325i
$$404$$ 7.70820 0.383497
$$405$$ 0 0
$$406$$ −3.09017 −0.153363
$$407$$ − 0.708204i − 0.0351044i
$$408$$ 17.8885i 0.885615i
$$409$$ −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
$$412$$ − 13.7082i − 0.675355i
$$413$$ − 11.7082i − 0.576123i
$$414$$ 28.7984 1.41536
$$415$$ 0 0
$$416$$ −6.94427 −0.340471
$$417$$ − 34.4721i − 1.68811i
$$418$$ 0.652476i 0.0319136i
$$419$$ 26.1803 1.27899 0.639497 0.768794i $$-0.279143\pi$$
0.639497 + 0.768794i $$0.279143\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ − 7.41641i − 0.361025i
$$423$$ 14.9443i 0.726615i
$$424$$ −18.9443 −0.920015
$$425$$ 0 0
$$426$$ 17.4164 0.843828
$$427$$ − 9.70820i − 0.469813i
$$428$$ 12.9443i 0.625685i
$$429$$ −0.944272 −0.0455899
$$430$$ 0 0
$$431$$ 17.5279 0.844288 0.422144 0.906529i $$-0.361278\pi$$
0.422144 + 0.906529i $$0.361278\pi$$
$$432$$ − 26.8328i − 1.29099i
$$433$$ 28.3607i 1.36293i 0.731852 + 0.681464i $$0.238656\pi$$
−0.731852 + 0.681464i $$0.761344\pi$$
$$434$$ 2.29180 0.110010
$$435$$ 0 0
$$436$$ −13.6180 −0.652186
$$437$$ − 27.8885i − 1.33409i
$$438$$ 17.5279i 0.837514i
$$439$$ 8.29180 0.395746 0.197873 0.980228i $$-0.436597\pi$$
0.197873 + 0.980228i $$0.436597\pi$$
$$440$$ 0 0
$$441$$ 7.47214 0.355816
$$442$$ − 1.88854i − 0.0898289i
$$443$$ 19.4164i 0.922501i 0.887270 + 0.461251i $$0.152599\pi$$
−0.887270 + 0.461251i $$0.847401\pi$$
$$444$$ −15.7082 −0.745478
$$445$$ 0 0
$$446$$ −12.4721 −0.590573
$$447$$ − 12.7639i − 0.603713i
$$448$$ 0.236068i 0.0111532i
$$449$$ −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
$$452$$ 23.3262i 1.09717i
$$453$$ − 65.4853i − 3.07677i
$$454$$ 13.2361 0.621199
$$455$$ 0 0
$$456$$ 32.3607 1.51543
$$457$$ − 12.5279i − 0.586029i −0.956108 0.293014i $$-0.905342\pi$$
0.956108 0.293014i $$-0.0946584\pi$$
$$458$$ − 2.76393i − 0.129150i
$$459$$ 35.7771 1.66993
$$460$$ 0 0
$$461$$ −14.1803 −0.660444 −0.330222 0.943903i $$-0.607124\pi$$
−0.330222 + 0.943903i $$0.607124\pi$$
$$462$$ − 0.472136i − 0.0219658i
$$463$$ 13.8885i 0.645455i 0.946492 + 0.322728i $$0.104600\pi$$
−0.946492 + 0.322728i $$0.895400\pi$$
$$464$$ −9.27051 −0.430373
$$465$$ 0 0
$$466$$ −4.90983 −0.227443
$$467$$ 6.94427i 0.321343i 0.987008 + 0.160671i $$0.0513659\pi$$
−0.987008 + 0.160671i $$0.948634\pi$$
$$468$$ 14.9443i 0.690799i
$$469$$ 4.23607 0.195603
$$470$$ 0 0
$$471$$ −2.47214 −0.113910
$$472$$ 26.1803i 1.20505i
$$473$$ 0.416408i 0.0191465i
$$474$$ 22.3607 1.02706
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ 63.3050i 2.89853i
$$478$$ − 3.41641i − 0.156263i
$$479$$ 26.1803 1.19621 0.598105 0.801418i $$-0.295921\pi$$
0.598105 + 0.801418i $$0.295921\pi$$
$$480$$ 0 0
$$481$$ 3.70820 0.169080
$$482$$ 2.18034i 0.0993118i
$$483$$ 20.1803i 0.918237i
$$484$$ −17.7082 −0.804918
$$485$$ 0 0
$$486$$ 22.0000 0.997940
$$487$$ 5.76393i 0.261189i 0.991436 + 0.130594i $$0.0416885\pi$$
−0.991436 + 0.130594i $$0.958311\pi$$
$$488$$ 21.7082i 0.982684i
$$489$$ −4.94427 −0.223588
$$490$$ 0 0
$$491$$ −5.76393 −0.260123 −0.130061 0.991506i $$-0.541517\pi$$
−0.130061 + 0.991506i $$0.541517\pi$$
$$492$$ 24.9443i 1.12457i
$$493$$ − 12.3607i − 0.556697i
$$494$$ −3.41641 −0.153711
$$495$$ 0 0
$$496$$ 6.87539 0.308714
$$497$$ 8.70820i 0.390616i
$$498$$ 15.4164i 0.690826i
$$499$$ −11.0557 −0.494922 −0.247461 0.968898i $$-0.579596\pi$$
−0.247461 + 0.968898i $$0.579596\pi$$
$$500$$ 0 0
$$501$$ −16.9443 −0.757014
$$502$$ − 4.00000i − 0.178529i
$$503$$ 8.11146i 0.361672i 0.983513 + 0.180836i $$0.0578803\pi$$
−0.983513 + 0.180836i $$0.942120\pi$$
$$504$$ −16.7082 −0.744243
$$505$$ 0 0
$$506$$ 0.909830 0.0404469
$$507$$ 37.1246i 1.64876i
$$508$$ 22.0902i 0.980093i
$$509$$ −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
$$512$$ − 18.7082i − 0.826794i
$$513$$ − 64.7214i − 2.85752i
$$514$$ −7.81966 −0.344910
$$515$$ 0 0
$$516$$ 9.23607 0.406595
$$517$$ 0.472136i 0.0207645i
$$518$$ 1.85410i 0.0814646i
$$519$$ −37.3050 −1.63751
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ − 23.0902i − 1.01063i
$$523$$ − 16.3607i − 0.715403i −0.933836 0.357701i $$-0.883561\pi$$
0.933836 0.357701i $$-0.116439\pi$$
$$524$$ −27.4164 −1.19769
$$525$$ 0 0
$$526$$ −10.0344 −0.437522
$$527$$ 9.16718i 0.399329i
$$528$$ − 1.41641i − 0.0616412i
$$529$$ −15.8885 −0.690806
$$530$$ 0 0
$$531$$ 87.4853 3.79654
$$532$$ 7.23607i 0.313723i
$$533$$ − 5.88854i − 0.255061i
$$534$$ −34.4721 −1.49176
$$535$$ 0 0
$$536$$ −9.47214 −0.409134
$$537$$ 75.7771i 3.27002i
$$538$$ − 7.23607i − 0.311969i
$$539$$ 0.236068 0.0101682
$$540$$ 0 0
$$541$$ 15.9443 0.685498 0.342749 0.939427i $$-0.388642\pi$$
0.342749 + 0.939427i $$0.388642\pi$$
$$542$$ − 14.6525i − 0.629378i
$$543$$ 26.4721i 1.13603i
$$544$$ 13.8885 0.595466
$$545$$ 0 0
$$546$$ 2.47214 0.105798
$$547$$ − 9.76393i − 0.417476i −0.977972 0.208738i $$-0.933064\pi$$
0.977972 0.208738i $$-0.0669355\pi$$
$$548$$ − 17.7082i − 0.756457i
$$549$$ 72.5410 3.09598
$$550$$ 0 0
$$551$$ −22.3607 −0.952597
$$552$$ − 45.1246i − 1.92063i
$$553$$ 11.1803i 0.475436i
$$554$$ −12.2918 −0.522228
$$555$$ 0 0
$$556$$ −17.2361 −0.730972
$$557$$ − 9.11146i − 0.386065i −0.981192 0.193032i $$-0.938168\pi$$
0.981192 0.193032i $$-0.0618322\pi$$
$$558$$ 17.1246i 0.724943i
$$559$$ −2.18034 −0.0922186
$$560$$ 0 0
$$561$$ 1.88854 0.0797344
$$562$$ 9.49342i 0.400456i
$$563$$ − 17.4164i − 0.734014i −0.930218 0.367007i $$-0.880382\pi$$
0.930218 0.367007i $$-0.119618\pi$$
$$564$$ 10.4721 0.440956
$$565$$ 0 0
$$566$$ −10.7639 −0.452442
$$567$$ 24.4164i 1.02539i
$$568$$ − 19.4721i − 0.817033i
$$569$$ 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 $$-0.222362\pi$$
0.765763 + 0.643123i $$0.222362\pi$$
$$572$$ 0.472136i 0.0197410i
$$573$$ 20.9443i 0.874960i
$$574$$ 2.94427 0.122892
$$575$$ 0 0
$$576$$ −1.76393 −0.0734972
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ − 6.72949i − 0.279910i
$$579$$ 40.1803 1.66984
$$580$$ 0 0
$$581$$ −7.70820 −0.319790
$$582$$ 10.4721i 0.434084i
$$583$$ 2.00000i 0.0828315i
$$584$$ 19.5967 0.810919
$$585$$ 0 0
$$586$$ 19.2361 0.794635
$$587$$ − 24.7639i − 1.02212i −0.859546 0.511058i $$-0.829254\pi$$
0.859546 0.511058i $$-0.170746\pi$$
$$588$$ − 5.23607i − 0.215932i
$$589$$ 16.5836 0.683315
$$590$$ 0 0
$$591$$ 4.76393 0.195962
$$592$$ 5.56231i 0.228609i
$$593$$ 37.3050i 1.53193i 0.642882 + 0.765965i $$0.277739\pi$$
−0.642882 + 0.765965i $$0.722261\pi$$
$$594$$ 2.11146 0.0866341
$$595$$ 0 0
$$596$$ −6.38197 −0.261416
$$597$$ − 23.4164i − 0.958370i
$$598$$ 4.76393i 0.194812i
$$599$$ 11.1803 0.456816 0.228408 0.973565i $$-0.426648\pi$$
0.228408 + 0.973565i $$0.426648\pi$$
$$600$$ 0 0
$$601$$ −36.9443 −1.50699 −0.753494 0.657455i $$-0.771633\pi$$
−0.753494 + 0.657455i $$0.771633\pi$$
$$602$$ − 1.09017i − 0.0444320i
$$603$$ 31.6525i 1.28899i
$$604$$ −32.7426 −1.33228
$$605$$ 0 0
$$606$$ 9.52786 0.387043
$$607$$ − 7.12461i − 0.289179i −0.989492 0.144590i $$-0.953814\pi$$
0.989492 0.144590i $$-0.0461862\pi$$
$$608$$ − 25.1246i − 1.01894i
$$609$$ 16.1803 0.655660
$$610$$ 0 0
$$611$$ −2.47214 −0.100012
$$612$$ − 29.8885i − 1.20817i
$$613$$ 44.4164i 1.79396i 0.442069 + 0.896981i $$0.354245\pi$$
−0.442069 + 0.896981i $$0.645755\pi$$
$$614$$ 2.83282 0.114323
$$615$$ 0 0
$$616$$ −0.527864 −0.0212682
$$617$$ − 5.94427i − 0.239307i −0.992816 0.119654i $$-0.961822\pi$$
0.992816 0.119654i $$-0.0381784\pi$$
$$618$$ − 16.9443i − 0.681599i
$$619$$ 11.7082 0.470592 0.235296 0.971924i $$-0.424394\pi$$
0.235296 + 0.971924i $$0.424394\pi$$
$$620$$ 0 0
$$621$$ −90.2492 −3.62158
$$622$$ − 15.0557i − 0.603680i
$$623$$ − 17.2361i − 0.690548i
$$624$$ 7.41641 0.296894
$$625$$ 0 0
$$626$$ −12.0689 −0.482370
$$627$$ − 3.41641i − 0.136438i
$$628$$ 1.23607i 0.0493245i
$$629$$ −7.41641 −0.295712
$$630$$ 0 0
$$631$$ 27.6525 1.10083 0.550414 0.834892i $$-0.314470\pi$$
0.550414 + 0.834892i $$0.314470\pi$$
$$632$$ − 25.0000i − 0.994447i
$$633$$ 38.8328i 1.54347i
$$634$$ 15.6738 0.622485
$$635$$ 0 0
$$636$$ 44.3607 1.75902
$$637$$ 1.23607i 0.0489748i
$$638$$ − 0.729490i − 0.0288808i
$$639$$ −65.0689 −2.57409
$$640$$ 0 0
$$641$$ 43.8328 1.73129 0.865646 0.500656i $$-0.166908\pi$$
0.865646 + 0.500656i $$0.166908\pi$$
$$642$$ 16.0000i 0.631470i
$$643$$ − 18.4721i − 0.728470i −0.931307 0.364235i $$-0.881331\pi$$
0.931307 0.364235i $$-0.118669\pi$$
$$644$$ 10.0902 0.397608
$$645$$ 0 0
$$646$$ 6.83282 0.268834
$$647$$ − 19.8885i − 0.781899i −0.920412 0.390950i $$-0.872147\pi$$
0.920412 0.390950i $$-0.127853\pi$$
$$648$$ − 54.5967i − 2.14476i
$$649$$ 2.76393 0.108494
$$650$$ 0 0
$$651$$ −12.0000 −0.470317
$$652$$ 2.47214i 0.0968163i
$$653$$ − 25.0557i − 0.980506i −0.871580 0.490253i $$-0.836904\pi$$
0.871580 0.490253i $$-0.163096\pi$$
$$654$$ −16.8328 −0.658215
$$655$$ 0 0
$$656$$ 8.83282 0.344864
$$657$$ − 65.4853i − 2.55482i
$$658$$ − 1.23607i − 0.0481869i
$$659$$ −17.8885 −0.696839 −0.348419 0.937339i $$-0.613281\pi$$
−0.348419 + 0.937339i $$0.613281\pi$$
$$660$$ 0 0
$$661$$ −42.7214 −1.66167 −0.830834 0.556520i $$-0.812136\pi$$
−0.830834 + 0.556520i $$0.812136\pi$$
$$662$$ 15.2705i 0.593505i
$$663$$ 9.88854i 0.384039i
$$664$$ 17.2361 0.668889
$$665$$ 0 0
$$666$$ −13.8541 −0.536836
$$667$$ 31.1803i 1.20731i
$$668$$ 8.47214i 0.327797i
$$669$$ 65.3050 2.52484
$$670$$ 0 0
$$671$$ 2.29180 0.0884738
$$672$$ 18.1803i 0.701322i
$$673$$ − 19.5279i − 0.752744i −0.926469 0.376372i $$-0.877171\pi$$
0.926469 0.376372i $$-0.122829\pi$$
$$674$$ −10.1803 −0.392132
$$675$$ 0 0
$$676$$ 18.5623 0.713935
$$677$$ − 14.3607i − 0.551926i −0.961168 0.275963i $$-0.911003\pi$$
0.961168 0.275963i $$-0.0889967\pi$$
$$678$$ 28.8328i 1.10732i
$$679$$ −5.23607 −0.200942
$$680$$ 0 0
$$681$$ −69.3050 −2.65577
$$682$$ 0.541020i 0.0207167i
$$683$$ − 14.1246i − 0.540463i −0.962795 0.270232i $$-0.912900\pi$$
0.962795 0.270232i $$-0.0871003\pi$$
$$684$$ −54.0689 −2.06738
$$685$$ 0 0
$$686$$ −0.618034 −0.0235966
$$687$$ 14.4721i 0.552146i
$$688$$ − 3.27051i − 0.124687i
$$689$$ −10.4721 −0.398957
$$690$$ 0 0
$$691$$ −4.18034 −0.159028 −0.0795138 0.996834i $$-0.525337\pi$$
−0.0795138 + 0.996834i $$0.525337\pi$$
$$692$$ 18.6525i 0.709061i
$$693$$ 1.76393i 0.0670062i
$$694$$ 12.5066 0.474743
$$695$$ 0 0
$$696$$ −36.1803 −1.37141
$$697$$ 11.7771i 0.446089i
$$698$$ 2.76393i 0.104616i
$$699$$ 25.7082 0.972374
$$700$$ 0 0
$$701$$ −29.0557 −1.09742 −0.548710 0.836013i $$-0.684881\pi$$
−0.548710 + 0.836013i $$0.684881\pi$$
$$702$$ 11.0557i 0.417272i
$$703$$ 13.4164i 0.506009i
$$704$$ −0.0557281 −0.00210033
$$705$$ 0 0
$$706$$ 1.34752 0.0507147
$$707$$ 4.76393i 0.179166i
$$708$$ − 61.3050i − 2.30398i
$$709$$ 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
$$712$$ 38.5410i 1.44439i
$$713$$ − 23.1246i − 0.866024i
$$714$$ −4.94427 −0.185035
$$715$$ 0 0
$$716$$ 37.8885 1.41596
$$717$$ 17.8885i 0.668060i
$$718$$ − 18.6180i − 0.694819i
$$719$$ −16.1803 −0.603425 −0.301712 0.953399i $$-0.597558\pi$$
−0.301712 + 0.953399i $$0.597558\pi$$
$$720$$ 0 0
$$721$$ 8.47214 0.315519
$$722$$ − 0.618034i − 0.0230008i
$$723$$ − 11.4164i − 0.424581i
$$724$$ 13.2361 0.491915
$$725$$ 0 0
$$726$$ −21.8885 −0.812360
$$727$$ − 3.05573i − 0.113331i −0.998393 0.0566653i $$-0.981953\pi$$
0.998393 0.0566653i $$-0.0180468\pi$$
$$728$$ − 2.76393i − 0.102438i
$$729$$ −41.9443 −1.55349
$$730$$ 0 0
$$731$$ 4.36068 0.161286
$$732$$ − 50.8328i − 1.87883i
$$733$$ − 4.00000i − 0.147743i −0.997268 0.0738717i $$-0.976464\pi$$
0.997268 0.0738717i $$-0.0235355\pi$$
$$734$$ −22.9443 −0.846889
$$735$$ 0 0
$$736$$ −35.0344 −1.29139
$$737$$ 1.00000i 0.0368355i
$$738$$ 22.0000i 0.809831i
$$739$$ −25.6525 −0.943642 −0.471821 0.881694i $$-0.656403\pi$$
−0.471821 + 0.881694i $$0.656403\pi$$
$$740$$ 0 0
$$741$$ 17.8885 0.657152
$$742$$ − 5.23607i − 0.192222i
$$743$$ 10.4721i 0.384185i 0.981377 + 0.192093i $$0.0615274\pi$$
−0.981377 + 0.192093i $$0.938473\pi$$
$$744$$ 26.8328 0.983739
$$745$$ 0 0
$$746$$ 23.3820 0.856075
$$747$$ − 57.5967i − 2.10735i
$$748$$ − 0.944272i − 0.0345260i
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 3.05573 0.111505 0.0557526 0.998445i $$-0.482244\pi$$
0.0557526 + 0.998445i $$0.482244\pi$$
$$752$$ − 3.70820i − 0.135224i
$$753$$ 20.9443i 0.763252i
$$754$$ 3.81966 0.139104
$$755$$ 0 0
$$756$$ 23.4164 0.851647
$$757$$ 19.5836i 0.711778i 0.934528 + 0.355889i $$0.115822\pi$$
−0.934528 + 0.355889i $$0.884178\pi$$
$$758$$ − 6.90983i − 0.250976i
$$759$$ −4.76393 −0.172920
$$760$$ 0 0
$$761$$ 27.7771 1.00692 0.503459 0.864019i $$-0.332060\pi$$
0.503459 + 0.864019i $$0.332060\pi$$
$$762$$ 27.3050i 0.989154i
$$763$$ − 8.41641i − 0.304694i
$$764$$ 10.4721 0.378869
$$765$$ 0 0
$$766$$ 20.5410 0.742177
$$767$$ 14.4721i 0.522559i
$$768$$ − 21.2361i − 0.766291i
$$769$$ 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
$$772$$ − 20.0902i − 0.723061i
$$773$$ − 50.1803i − 1.80486i −0.430835 0.902431i $$-0.641781\pi$$
0.430835 0.902431i $$-0.358219\pi$$
$$774$$ 8.14590 0.292798
$$775$$ 0 0
$$776$$ 11.7082 0.420300
$$777$$ − 9.70820i − 0.348280i
$$778$$ 1.78522i 0.0640032i
$$779$$ 21.3050 0.763329
$$780$$ 0 0
$$781$$ −2.05573 −0.0735597
$$782$$ − 9.52786i − 0.340716i
$$783$$ 72.3607i 2.58596i
$$784$$ −1.85410 −0.0662179
$$785$$ 0 0
$$786$$ −33.8885 −1.20876
$$787$$ 40.7639i 1.45308i 0.687126 + 0.726539i $$0.258872\pi$$
−0.687126 + 0.726539i $$0.741128\pi$$
$$788$$ − 2.38197i − 0.0848540i
$$789$$ 52.5410 1.87051
$$790$$ 0 0
$$791$$ −14.4164 −0.512588
$$792$$ − 3.94427i − 0.140154i
$$793$$ 12.0000i 0.426132i
$$794$$ 5.59675 0.198621
$$795$$ 0 0
$$796$$ −11.7082 −0.414986
$$797$$ − 35.4164i − 1.25451i −0.778813 0.627257i $$-0.784178\pi$$
0.778813 0.627257i $$-0.215822\pi$$
$$798$$ 8.94427i 0.316624i
$$799$$ 4.94427 0.174916
$$800$$ 0 0
$$801$$ 128.790 4.55058
$$802$$ − 1.56231i − 0.0551669i
$$803$$ − 2.06888i − 0.0730093i
$$804$$ 22.1803 0.782240
$$805$$ 0 0
$$806$$ −2.83282 −0.0997817
$$807$$ 37.8885i 1.33374i
$$808$$ − 10.6525i − 0.374753i
$$809$$ −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 $$-0.769983\pi$$
−0.750075 + 0.661353i $$0.769983\pi$$
$$812$$ − 8.09017i − 0.283909i
$$813$$ 76.7214i 2.69074i
$$814$$ −0.437694 −0.0153412
$$815$$ 0 0
$$816$$ −14.8328 −0.519252
$$817$$ − 7.88854i − 0.275985i
$$818$$ 15.1246i 0.528820i
$$819$$ −9.23607 −0.322734
$$820$$ 0 0
$$821$$ 28.8328 1.00627 0.503136 0.864207i $$-0.332179\pi$$
0.503136 + 0.864207i $$0.332179\pi$$
$$822$$ − 21.8885i − 0.763451i
$$823$$ 31.6525i 1.10334i 0.834064 + 0.551668i $$0.186008\pi$$
−0.834064 + 0.551668i $$0.813992\pi$$
$$824$$ −18.9443 −0.659955
$$825$$ 0 0
$$826$$ −7.23607 −0.251775
$$827$$ 41.5410i 1.44452i 0.691620 + 0.722261i $$0.256897\pi$$
−0.691620 + 0.722261i $$0.743103\pi$$
$$828$$ 75.3951i 2.62016i
$$829$$ 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
$$832$$ − 0.291796i − 0.0101162i
$$833$$ − 2.47214i − 0.0856544i
$$834$$ −21.3050 −0.737730
$$835$$ 0 0
$$836$$ −1.70820 −0.0590795
$$837$$ − 53.6656i − 1.85496i
$$838$$ − 16.1803i − 0.558941i
$$839$$ 30.6525 1.05824 0.529120 0.848547i $$-0.322522\pi$$
0.529120 + 0.848547i $$0.322522\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 8.03444i 0.276885i
$$843$$ − 49.7082i − 1.71204i
$$844$$ 19.4164 0.668340
$$845$$ 0 0
$$846$$ 9.23607 0.317543
$$847$$ − 10.9443i − 0.376050i
$$848$$ − 15.7082i − 0.539422i
$$849$$ 56.3607 1.93429
$$850$$ 0 0
$$851$$ 18.7082 0.641309
$$852$$ 45.5967i 1.56212i
$$853$$ − 27.4164i − 0.938720i −0.883007 0.469360i $$-0.844485\pi$$
0.883007 0.469360i $$-0.155515\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ 17.8885 0.611418
$$857$$ − 15.8197i − 0.540389i −0.962806 0.270195i $$-0.912912\pi$$
0.962806 0.270195i $$-0.0870881\pi$$
$$858$$ 0.583592i 0.0199235i
$$859$$ −22.3607 −0.762937 −0.381468 0.924382i $$-0.624581\pi$$
−0.381468 + 0.924382i $$0.624581\pi$$
$$860$$ 0 0
$$861$$ −15.4164 −0.525390
$$862$$ − 10.8328i − 0.368967i
$$863$$ − 18.3475i − 0.624557i −0.949991 0.312278i $$-0.898908\pi$$
0.949991 0.312278i $$-0.101092\pi$$
$$864$$ −81.3050 −2.76605
$$865$$ 0 0
$$866$$ 17.5279 0.595621
$$867$$ 35.2361i 1.19668i
$$868$$ 6.00000i 0.203653i
$$869$$ −2.63932 −0.0895328
$$870$$ 0 0
$$871$$ −5.23607 −0.177417
$$872$$ 18.8197i 0.637314i
$$873$$ − 39.1246i − 1.32417i
$$874$$ −17.2361 −0.583019
$$875$$ 0 0
$$876$$ −45.8885 −1.55043
$$877$$ 30.3607i 1.02521i 0.858625 + 0.512604i $$0.171319\pi$$
−0.858625 + 0.512604i $$0.828681\pi$$
$$878$$ − 5.12461i − 0.172947i
$$879$$ −100.721 −3.39725
$$880$$ 0 0
$$881$$ 5.81966 0.196069 0.0980347 0.995183i $$-0.468744\pi$$
0.0980347 + 0.995183i $$0.468744\pi$$
$$882$$ − 4.61803i − 0.155497i
$$883$$ 1.40325i 0.0472232i 0.999721 + 0.0236116i $$0.00751650\pi$$
−0.999721 + 0.0236116i $$0.992483\pi$$
$$884$$ 4.94427 0.166294
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ − 21.3475i − 0.716780i −0.933572 0.358390i $$-0.883326\pi$$
0.933572 0.358390i $$-0.116674\pi$$
$$888$$ 21.7082i 0.728480i
$$889$$ −13.6525 −0.457889
$$890$$ 0 0
$$891$$ −5.76393 −0.193099
$$892$$ − 32.6525i − 1.09329i
$$893$$ − 8.94427i − 0.299309i
$$894$$ −7.88854 −0.263832
$$895$$ 0 0
$$896$$ 11.3820 0.380245
$$897$$ − 24.9443i − 0.832865i
$$898$$ 12.6869i 0.423368i
$$899$$ −18.5410 −0.618378
$$900$$ 0 0
$$901$$ 20.9443 0.697755
$$902$$ 0.695048i 0.0231426i
$$903$$ 5.70820i 0.189957i
$$904$$ 32.2361 1.07216
$$905$$ 0 0
$$906$$ −40.4721 −1.34460
$$907$$ 34.8328i 1.15660i 0.815823 + 0.578302i $$0.196285\pi$$
−0.815823 + 0.578302i $$0.803715\pi$$
$$908$$ 34.6525i 1.14998i
$$909$$ −35.5967 −1.18067
$$910$$ 0 0
$$911$$ 0.819660 0.0271566 0.0135783 0.999908i $$-0.495678\pi$$
0.0135783 + 0.999908i $$0.495678\pi$$
$$912$$ 26.8328i 0.888523i
$$913$$ − 1.81966i − 0.0602220i
$$914$$ −7.74265 −0.256104
$$915$$ 0 0
$$916$$ 7.23607 0.239086
$$917$$ − 16.9443i − 0.559549i
$$918$$ − 22.1115i − 0.729787i
$$919$$ 27.7639 0.915848 0.457924 0.888991i $$-0.348593\pi$$
0.457924 + 0.888991i $$0.348593\pi$$
$$920$$ 0 0
$$921$$ −14.8328 −0.488758
$$922$$ 8.76393i 0.288625i
$$923$$ − 10.7639i − 0.354299i
$$924$$ 1.23607 0.0406637
$$925$$ 0 0
$$926$$ 8.58359 0.282074
$$927$$ 63.3050i 2.07921i
$$928$$ 28.0902i 0.922105i
$$929$$ −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
$$932$$ − 12.8541i − 0.421050i
$$933$$ 78.8328i 2.58087i
$$934$$ 4.29180 0.140432
$$935$$ 0 0
$$936$$ 20.6525 0.675047
$$937$$ 35.2361i 1.15111i 0.817762 + 0.575556i $$0.195214\pi$$
−0.817762 + 0.575556i $$0.804786\pi$$
$$938$$ − 2.61803i − 0.0854818i
$$939$$ 63.1935 2.06224
$$940$$ 0 0
$$941$$ −5.23607 −0.170691 −0.0853455 0.996351i $$-0.527199\pi$$
−0.0853455 + 0.996351i $$0.527199\pi$$
$$942$$ 1.52786i 0.0497805i
$$943$$ − 29.7082i − 0.967432i
$$944$$ −21.7082 −0.706542
$$945$$ 0 0
$$946$$ 0.257354 0.00836731
$$947$$ 34.8328i 1.13191i 0.824435 + 0.565957i $$0.191493\pi$$
−0.824435 + 0.565957i $$0.808507\pi$$
$$948$$ 58.5410i 1.90132i
$$949$$ 10.8328 0.351648
$$950$$ 0 0
$$951$$ −82.0689 −2.66127
$$952$$ 5.52786i 0.179159i
$$953$$ − 3.47214i − 0.112474i −0.998417 0.0562368i $$-0.982090\pi$$
0.998417 0.0562368i $$-0.0179102\pi$$
$$954$$ 39.1246 1.26671
$$955$$ 0 0
$$956$$ 8.94427 0.289278
$$957$$ 3.81966i 0.123472i
$$958$$ − 16.1803i − 0.522763i
$$959$$ 10.9443 0.353409
$$960$$ 0 0
$$961$$ −17.2492 −0.556427
$$962$$ − 2.29180i − 0.0738905i
$$963$$ − 59.7771i − 1.92629i
$$964$$ −5.70820 −0.183849
$$965$$ 0 0
$$966$$ 12.4721 0.401284
$$967$$ − 14.1115i − 0.453794i −0.973919 0.226897i $$-0.927142\pi$$
0.973919 0.226897i $$-0.0728580\pi$$
$$968$$ 24.4721i 0.786564i
$$969$$ −35.7771 −1.14933
$$970$$ 0 0
$$971$$ −18.0000 −0.577647 −0.288824 0.957382i $$-0.593264\pi$$
−0.288824 + 0.957382i $$0.593264\pi$$
$$972$$ 57.5967i 1.84742i
$$973$$ − 10.6525i − 0.341503i
$$974$$ 3.56231 0.114144
$$975$$ 0 0
$$976$$ −18.0000 −0.576166
$$977$$ − 11.4721i − 0.367026i −0.983017 0.183513i $$-0.941253\pi$$
0.983017 0.183513i $$-0.0587470\pi$$
$$978$$ 3.05573i 0.0977114i
$$979$$ 4.06888 0.130042
$$980$$ 0 0
$$981$$ 62.8885 2.00788
$$982$$ 3.56231i 0.113678i
$$983$$ 34.5410i 1.10169i 0.834608 + 0.550844i $$0.185694\pi$$
−0.834608 + 0.550844i $$0.814306\pi$$
$$984$$ 34.4721 1.09893
$$985$$ 0 0
$$986$$ −7.63932 −0.243286
$$987$$ 6.47214i 0.206010i
$$988$$ − 8.94427i − 0.284555i
$$989$$ −11.0000 −0.349780
$$990$$ 0 0
$$991$$ 13.1803 0.418687 0.209344 0.977842i $$-0.432867\pi$$
0.209344 + 0.977842i $$0.432867\pi$$
$$992$$ − 20.8328i − 0.661443i
$$993$$ − 79.9574i − 2.53737i
$$994$$ 5.38197 0.170706
$$995$$ 0 0
$$996$$ −40.3607 −1.27888
$$997$$ − 45.4164i − 1.43835i −0.694828 0.719176i $$-0.744519\pi$$
0.694828 0.719176i $$-0.255481\pi$$
$$998$$ 6.83282i 0.216289i
$$999$$ 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 175.2.b.c.99.2 4
3.2 odd 2 1575.2.d.k.1324.3 4
4.3 odd 2 2800.2.g.s.449.1 4
5.2 odd 4 175.2.a.d.1.2 2
5.3 odd 4 175.2.a.e.1.1 yes 2
5.4 even 2 inner 175.2.b.c.99.3 4
7.6 odd 2 1225.2.b.k.99.2 4
15.2 even 4 1575.2.a.s.1.1 2
15.8 even 4 1575.2.a.n.1.2 2
15.14 odd 2 1575.2.d.k.1324.2 4
20.3 even 4 2800.2.a.bp.1.2 2
20.7 even 4 2800.2.a.bh.1.1 2
20.19 odd 2 2800.2.g.s.449.4 4
35.13 even 4 1225.2.a.u.1.1 2
35.27 even 4 1225.2.a.n.1.2 2
35.34 odd 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 5.2 odd 4
175.2.a.e.1.1 yes 2 5.3 odd 4
175.2.b.c.99.2 4 1.1 even 1 trivial
175.2.b.c.99.3 4 5.4 even 2 inner
1225.2.a.n.1.2 2 35.27 even 4
1225.2.a.u.1.1 2 35.13 even 4
1225.2.b.k.99.2 4 7.6 odd 2
1225.2.b.k.99.3 4 35.34 odd 2
1575.2.a.n.1.2 2 15.8 even 4
1575.2.a.s.1.1 2 15.2 even 4
1575.2.d.k.1324.2 4 15.14 odd 2
1575.2.d.k.1324.3 4 3.2 odd 2
2800.2.a.bh.1.1 2 20.7 even 4
2800.2.a.bp.1.2 2 20.3 even 4
2800.2.g.s.449.1 4 4.3 odd 2
2800.2.g.s.449.4 4 20.19 odd 2