# Properties

 Label 975.2.c.g.274.2 Level $975$ Weight $2$ Character 975.274 Analytic conductor $7.785$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.c (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: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 975.274 Dual form 975.2.c.g.274.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} +3.00000 q^{14} -1.00000 q^{16} -5.00000i q^{17} -1.00000i q^{18} +8.00000 q^{19} -3.00000 q^{21} -1.00000i q^{22} +3.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} -3.00000i q^{28} -1.00000 q^{29} +3.00000 q^{31} +5.00000i q^{32} +1.00000i q^{33} +5.00000 q^{34} -1.00000 q^{36} -8.00000i q^{37} +8.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} -3.00000i q^{42} -8.00000i q^{43} -1.00000 q^{44} -11.0000i q^{47} +1.00000i q^{48} -2.00000 q^{49} -5.00000 q^{51} -1.00000i q^{52} +11.0000i q^{53} -1.00000 q^{54} +9.00000 q^{56} -8.00000i q^{57} -1.00000i q^{58} -5.00000 q^{59} +1.00000 q^{61} +3.00000i q^{62} +3.00000i q^{63} -7.00000 q^{64} -1.00000 q^{66} +3.00000i q^{67} -5.00000i q^{68} +16.0000 q^{71} -3.00000i q^{72} +4.00000i q^{73} +8.00000 q^{74} +8.00000 q^{76} +3.00000i q^{77} -1.00000i q^{78} -12.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +3.00000i q^{83} -3.00000 q^{84} +8.00000 q^{86} +1.00000i q^{87} -3.00000i q^{88} -3.00000 q^{91} -3.00000i q^{93} +11.0000 q^{94} +5.00000 q^{96} -2.00000i q^{97} -2.00000i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 6 q^{14} - 2 q^{16} + 16 q^{19} - 6 q^{21} + 6 q^{24} + 2 q^{26} - 2 q^{29} + 6 q^{31} + 10 q^{34} - 2 q^{36} - 2 q^{39} - 4 q^{41} - 2 q^{44} - 4 q^{49} - 10 q^{51} - 2 q^{54} + 18 q^{56} - 10 q^{59} + 2 q^{61} - 14 q^{64} - 2 q^{66} + 32 q^{71} + 16 q^{74} + 16 q^{76} - 24 q^{79} + 2 q^{81} - 6 q^{84} + 16 q^{86} - 6 q^{91} + 22 q^{94} + 10 q^{96} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 + 6 * q^14 - 2 * q^16 + 16 * q^19 - 6 * q^21 + 6 * q^24 + 2 * q^26 - 2 * q^29 + 6 * q^31 + 10 * q^34 - 2 * q^36 - 2 * q^39 - 4 * q^41 - 2 * q^44 - 4 * q^49 - 10 * q^51 - 2 * q^54 + 18 * q^56 - 10 * q^59 + 2 * q^61 - 14 * q^64 - 2 * q^66 + 32 * q^71 + 16 * q^74 + 16 * q^76 - 24 * q^79 + 2 * q^81 - 6 * q^84 + 16 * q^86 - 6 * q^91 + 22 * q^94 + 10 * q^96 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 1.00000i − 0.277350i
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 5.00000i − 1.21268i −0.795206 0.606339i $$-0.792637\pi$$
0.795206 0.606339i $$-0.207363\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ − 1.00000i − 0.213201i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 1.00000i 0.192450i
$$28$$ − 3.00000i − 0.566947i
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 1.00000i 0.174078i
$$34$$ 5.00000 0.857493
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 8.00000i 1.29777i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ − 3.00000i − 0.462910i
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 11.0000i − 1.60451i −0.596978 0.802257i $$-0.703632\pi$$
0.596978 0.802257i $$-0.296368\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −5.00000 −0.700140
$$52$$ − 1.00000i − 0.138675i
$$53$$ 11.0000i 1.51097i 0.655168 + 0.755483i $$0.272598\pi$$
−0.655168 + 0.755483i $$0.727402\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 9.00000 1.20268
$$57$$ − 8.00000i − 1.05963i
$$58$$ − 1.00000i − 0.131306i
$$59$$ −5.00000 −0.650945 −0.325472 0.945552i $$-0.605523\pi$$
−0.325472 + 0.945552i $$0.605523\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 3.00000i 0.381000i
$$63$$ 3.00000i 0.377964i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ 3.00000i 0.366508i 0.983066 + 0.183254i $$0.0586631\pi$$
−0.983066 + 0.183254i $$0.941337\pi$$
$$68$$ − 5.00000i − 0.606339i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ 3.00000i 0.341882i
$$78$$ − 1.00000i − 0.113228i
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 2.00000i − 0.220863i
$$83$$ 3.00000i 0.329293i 0.986353 + 0.164646i $$0.0526483\pi$$
−0.986353 + 0.164646i $$0.947352\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 1.00000i 0.107211i
$$88$$ − 3.00000i − 0.319801i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 0 0
$$93$$ − 3.00000i − 0.311086i
$$94$$ 11.0000 1.13456
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ − 2.00000i − 0.202031i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 9.00000 0.895533 0.447767 0.894150i $$-0.352219\pi$$
0.447767 + 0.894150i $$0.352219\pi$$
$$102$$ − 5.00000i − 0.495074i
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 3.00000 0.294174
$$105$$ 0 0
$$106$$ −11.0000 −1.06841
$$107$$ 18.0000i 1.74013i 0.492941 + 0.870063i $$0.335922\pi$$
−0.492941 + 0.870063i $$0.664078\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 3.00000i 0.283473i
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ 8.00000 0.749269
$$115$$ 0 0
$$116$$ −1.00000 −0.0928477
$$117$$ 1.00000i 0.0924500i
$$118$$ − 5.00000i − 0.460287i
$$119$$ −15.0000 −1.37505
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 1.00000i 0.0905357i
$$123$$ 2.00000i 0.180334i
$$124$$ 3.00000 0.269408
$$125$$ 0 0
$$126$$ −3.00000 −0.267261
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 1.00000i 0.0870388i
$$133$$ − 24.0000i − 2.08106i
$$134$$ −3.00000 −0.259161
$$135$$ 0 0
$$136$$ 15.0000 1.28624
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ −11.0000 −0.926367
$$142$$ 16.0000i 1.34269i
$$143$$ 1.00000i 0.0836242i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 2.00000i 0.164957i
$$148$$ − 8.00000i − 0.657596i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −17.0000 −1.38344 −0.691720 0.722166i $$-0.743147\pi$$
−0.691720 + 0.722166i $$0.743147\pi$$
$$152$$ 24.0000i 1.94666i
$$153$$ 5.00000i 0.404226i
$$154$$ −3.00000 −0.241747
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ − 11.0000i − 0.877896i −0.898513 0.438948i $$-0.855351\pi$$
0.898513 0.438948i $$-0.144649\pi$$
$$158$$ − 12.0000i − 0.954669i
$$159$$ 11.0000 0.872357
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −3.00000 −0.232845
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ − 9.00000i − 0.694365i
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ −8.00000 −0.611775
$$172$$ − 8.00000i − 0.609994i
$$173$$ 21.0000i 1.59660i 0.602260 + 0.798300i $$0.294267\pi$$
−0.602260 + 0.798300i $$0.705733\pi$$
$$174$$ −1.00000 −0.0758098
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 5.00000i 0.375823i
$$178$$ 0 0
$$179$$ −26.0000 −1.94333 −0.971666 0.236360i $$-0.924046\pi$$
−0.971666 + 0.236360i $$0.924046\pi$$
$$180$$ 0 0
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ − 3.00000i − 0.222375i
$$183$$ − 1.00000i − 0.0739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 3.00000 0.219971
$$187$$ 5.00000i 0.365636i
$$188$$ − 11.0000i − 0.802257i
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 7.00000i 0.505181i
$$193$$ − 10.0000i − 0.719816i −0.932988 0.359908i $$-0.882808\pi$$
0.932988 0.359908i $$-0.117192\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −2.00000 −0.142857
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 1.00000i 0.0710669i
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ 9.00000i 0.633238i
$$203$$ 3.00000i 0.210559i
$$204$$ −5.00000 −0.350070
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ 1.00000i 0.0693375i
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 11.0000i 0.755483i
$$213$$ − 16.0000i − 1.09630i
$$214$$ −18.0000 −1.23045
$$215$$ 0 0
$$216$$ −3.00000 −0.204124
$$217$$ − 9.00000i − 0.610960i
$$218$$ 10.0000i 0.677285i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ −5.00000 −0.336336
$$222$$ − 8.00000i − 0.536925i
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 15.0000 1.00223
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 23.0000i 1.52656i 0.646066 + 0.763282i $$0.276413\pi$$
−0.646066 + 0.763282i $$0.723587\pi$$
$$228$$ − 8.00000i − 0.529813i
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ − 3.00000i − 0.196960i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ −5.00000 −0.325472
$$237$$ 12.0000i 0.779484i
$$238$$ − 15.0000i − 0.972306i
$$239$$ 19.0000 1.22901 0.614504 0.788914i $$-0.289356\pi$$
0.614504 + 0.788914i $$0.289356\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ − 10.0000i − 0.642824i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 1.00000 0.0640184
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ − 8.00000i − 0.509028i
$$248$$ 9.00000i 0.571501i
$$249$$ 3.00000 0.190117
$$250$$ 0 0
$$251$$ −30.0000 −1.89358 −0.946792 0.321847i $$-0.895696\pi$$
−0.946792 + 0.321847i $$0.895696\pi$$
$$252$$ 3.00000i 0.188982i
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ − 3.00000i − 0.187135i −0.995613 0.0935674i $$-0.970173\pi$$
0.995613 0.0935674i $$-0.0298271\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 6.00000i 0.370681i
$$263$$ − 14.0000i − 0.863277i −0.902047 0.431638i $$-0.857936\pi$$
0.902047 0.431638i $$-0.142064\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 0 0
$$266$$ 24.0000 1.47153
$$267$$ 0 0
$$268$$ 3.00000i 0.183254i
$$269$$ −17.0000 −1.03651 −0.518254 0.855227i $$-0.673418\pi$$
−0.518254 + 0.855227i $$0.673418\pi$$
$$270$$ 0 0
$$271$$ −5.00000 −0.303728 −0.151864 0.988401i $$-0.548528\pi$$
−0.151864 + 0.988401i $$0.548528\pi$$
$$272$$ 5.00000i 0.303170i
$$273$$ 3.00000i 0.181568i
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6.00000i 0.360505i 0.983620 + 0.180253i $$0.0576915\pi$$
−0.983620 + 0.180253i $$0.942309\pi$$
$$278$$ 10.0000i 0.599760i
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ − 11.0000i − 0.655040i
$$283$$ 18.0000i 1.06999i 0.844856 + 0.534994i $$0.179686\pi$$
−0.844856 + 0.534994i $$0.820314\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ −1.00000 −0.0591312
$$287$$ 6.00000i 0.354169i
$$288$$ − 5.00000i − 0.294628i
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 4.00000i 0.234082i
$$293$$ − 30.0000i − 1.75262i −0.481749 0.876309i $$-0.659998\pi$$
0.481749 0.876309i $$-0.340002\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 24.0000 1.39497
$$297$$ − 1.00000i − 0.0580259i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ − 17.0000i − 0.978240i
$$303$$ − 9.00000i − 0.517036i
$$304$$ −8.00000 −0.458831
$$305$$ 0 0
$$306$$ −5.00000 −0.285831
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 3.00000i 0.170941i
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ − 3.00000i − 0.169842i
$$313$$ 29.0000i 1.63918i 0.572953 + 0.819588i $$0.305798\pi$$
−0.572953 + 0.819588i $$0.694202\pi$$
$$314$$ 11.0000 0.620766
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ − 16.0000i − 0.898650i −0.893368 0.449325i $$-0.851665\pi$$
0.893368 0.449325i $$-0.148335\pi$$
$$318$$ 11.0000i 0.616849i
$$319$$ 1.00000 0.0559893
$$320$$ 0 0
$$321$$ 18.0000 1.00466
$$322$$ 0 0
$$323$$ − 40.0000i − 2.22566i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ − 10.0000i − 0.553001i
$$328$$ − 6.00000i − 0.331295i
$$329$$ −33.0000 −1.81935
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 3.00000i 0.164646i
$$333$$ 8.00000i 0.438397i
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 3.00000 0.163663
$$337$$ 5.00000i 0.272367i 0.990684 + 0.136184i $$0.0434837\pi$$
−0.990684 + 0.136184i $$0.956516\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −3.00000 −0.162459
$$342$$ − 8.00000i − 0.432590i
$$343$$ − 15.0000i − 0.809924i
$$344$$ 24.0000 1.29399
$$345$$ 0 0
$$346$$ −21.0000 −1.12897
$$347$$ − 2.00000i − 0.107366i −0.998558 0.0536828i $$-0.982904\pi$$
0.998558 0.0536828i $$-0.0170960\pi$$
$$348$$ 1.00000i 0.0536056i
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ − 5.00000i − 0.266501i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ −5.00000 −0.265747
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 15.0000i 0.793884i
$$358$$ − 26.0000i − 1.37414i
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 5.00000i 0.262794i
$$363$$ 10.0000i 0.524864i
$$364$$ −3.00000 −0.157243
$$365$$ 0 0
$$366$$ 1.00000 0.0522708
$$367$$ − 12.0000i − 0.626395i −0.949688 0.313197i $$-0.898600\pi$$
0.949688 0.313197i $$-0.101400\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 33.0000 1.71327
$$372$$ − 3.00000i − 0.155543i
$$373$$ 31.0000i 1.60512i 0.596572 + 0.802560i $$0.296529\pi$$
−0.596572 + 0.802560i $$0.703471\pi$$
$$374$$ −5.00000 −0.258544
$$375$$ 0 0
$$376$$ 33.0000 1.70185
$$377$$ 1.00000i 0.0515026i
$$378$$ 3.00000i 0.154303i
$$379$$ 35.0000 1.79783 0.898915 0.438124i $$-0.144357\pi$$
0.898915 + 0.438124i $$0.144357\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 10.0000i 0.511645i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 8.00000i 0.406663i
$$388$$ − 2.00000i − 0.101535i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 6.00000i − 0.303046i
$$393$$ − 6.00000i − 0.302660i
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 1.00000 0.0502519
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ − 10.0000i − 0.501255i
$$399$$ −24.0000 −1.20150
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 3.00000i 0.149626i
$$403$$ − 3.00000i − 0.149441i
$$404$$ 9.00000 0.447767
$$405$$ 0 0
$$406$$ −3.00000 −0.148888
$$407$$ 8.00000i 0.396545i
$$408$$ − 15.0000i − 0.742611i
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 14.0000i 0.689730i
$$413$$ 15.0000i 0.738102i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 5.00000 0.245145
$$417$$ − 10.0000i − 0.489702i
$$418$$ − 8.00000i − 0.391293i
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 28.0000 1.36464 0.682318 0.731055i $$-0.260972\pi$$
0.682318 + 0.731055i $$0.260972\pi$$
$$422$$ − 16.0000i − 0.778868i
$$423$$ 11.0000i 0.534838i
$$424$$ −33.0000 −1.60262
$$425$$ 0 0
$$426$$ 16.0000 0.775203
$$427$$ − 3.00000i − 0.145180i
$$428$$ 18.0000i 0.870063i
$$429$$ 1.00000 0.0482805
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 9.00000 0.432014
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 0 0
$$438$$ 4.00000i 0.191127i
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ − 5.00000i − 0.237826i
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 10.0000i 0.472984i
$$448$$ 21.0000i 0.992157i
$$449$$ 28.0000 1.32140 0.660701 0.750649i $$-0.270259\pi$$
0.660701 + 0.750649i $$0.270259\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ 2.00000i 0.0940721i
$$453$$ 17.0000i 0.798730i
$$454$$ −23.0000 −1.07944
$$455$$ 0 0
$$456$$ 24.0000 1.12390
$$457$$ − 34.0000i − 1.59045i −0.606313 0.795226i $$-0.707352\pi$$
0.606313 0.795226i $$-0.292648\pi$$
$$458$$ 18.0000i 0.841085i
$$459$$ 5.00000 0.233380
$$460$$ 0 0
$$461$$ 42.0000 1.95614 0.978068 0.208288i $$-0.0667892\pi$$
0.978068 + 0.208288i $$0.0667892\pi$$
$$462$$ 3.00000i 0.139573i
$$463$$ 9.00000i 0.418265i 0.977887 + 0.209133i $$0.0670641\pi$$
−0.977887 + 0.209133i $$0.932936\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 32.0000i 1.48078i 0.672176 + 0.740392i $$0.265360\pi$$
−0.672176 + 0.740392i $$0.734640\pi$$
$$468$$ 1.00000i 0.0462250i
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ −11.0000 −0.506853
$$472$$ − 15.0000i − 0.690431i
$$473$$ 8.00000i 0.367840i
$$474$$ −12.0000 −0.551178
$$475$$ 0 0
$$476$$ −15.0000 −0.687524
$$477$$ − 11.0000i − 0.503655i
$$478$$ 19.0000i 0.869040i
$$479$$ 15.0000 0.685367 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 10.0000i 0.455488i
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 37.0000i 1.67663i 0.545186 + 0.838315i $$0.316459\pi$$
−0.545186 + 0.838315i $$0.683541\pi$$
$$488$$ 3.00000i 0.135804i
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ −26.0000 −1.17336 −0.586682 0.809818i $$-0.699566\pi$$
−0.586682 + 0.809818i $$0.699566\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 5.00000i 0.225189i
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −3.00000 −0.134704
$$497$$ − 48.0000i − 2.15309i
$$498$$ 3.00000i 0.134433i
$$499$$ 41.0000 1.83541 0.917706 0.397260i $$-0.130039\pi$$
0.917706 + 0.397260i $$0.130039\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ − 30.0000i − 1.33897i
$$503$$ − 26.0000i − 1.15928i −0.814872 0.579641i $$-0.803193\pi$$
0.814872 0.579641i $$-0.196807\pi$$
$$504$$ −9.00000 −0.400892
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000i 0.0444116i
$$508$$ 8.00000i 0.354943i
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ − 11.0000i − 0.486136i
$$513$$ 8.00000i 0.353209i
$$514$$ 3.00000 0.132324
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ 11.0000i 0.483779i
$$518$$ − 24.0000i − 1.05450i
$$519$$ 21.0000 0.921798
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 1.00000i 0.0437688i
$$523$$ − 26.0000i − 1.13690i −0.822718 0.568450i $$-0.807543\pi$$
0.822718 0.568450i $$-0.192457\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ 14.0000 0.610429
$$527$$ − 15.0000i − 0.653410i
$$528$$ − 1.00000i − 0.0435194i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 5.00000 0.216982
$$532$$ − 24.0000i − 1.04053i
$$533$$ 2.00000i 0.0866296i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −9.00000 −0.388741
$$537$$ 26.0000i 1.12198i
$$538$$ − 17.0000i − 0.732922i
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ 12.0000 0.515920 0.257960 0.966156i $$-0.416950\pi$$
0.257960 + 0.966156i $$0.416950\pi$$
$$542$$ − 5.00000i − 0.214768i
$$543$$ − 5.00000i − 0.214571i
$$544$$ 25.0000 1.07187
$$545$$ 0 0
$$546$$ −3.00000 −0.128388
$$547$$ − 22.0000i − 0.940652i −0.882493 0.470326i $$-0.844136\pi$$
0.882493 0.470326i $$-0.155864\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ −1.00000 −0.0426790
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 36.0000i 1.53088i
$$554$$ −6.00000 −0.254916
$$555$$ 0 0
$$556$$ 10.0000 0.424094
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ − 3.00000i − 0.127000i
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 5.00000 0.211100
$$562$$ − 18.0000i − 0.759284i
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ −11.0000 −0.463184
$$565$$ 0 0
$$566$$ −18.0000 −0.756596
$$567$$ − 3.00000i − 0.125988i
$$568$$ 48.0000i 2.01404i
$$569$$ −15.0000 −0.628833 −0.314416 0.949285i $$-0.601809\pi$$
−0.314416 + 0.949285i $$0.601809\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 1.00000i 0.0418121i
$$573$$ − 10.0000i − 0.417756i
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 10.0000i 0.416305i 0.978096 + 0.208153i $$0.0667451\pi$$
−0.978096 + 0.208153i $$0.933255\pi$$
$$578$$ − 8.00000i − 0.332756i
$$579$$ −10.0000 −0.415586
$$580$$ 0 0
$$581$$ 9.00000 0.373383
$$582$$ − 2.00000i − 0.0829027i
$$583$$ − 11.0000i − 0.455573i
$$584$$ −12.0000 −0.496564
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ − 45.0000i − 1.85735i −0.370896 0.928674i $$-0.620949\pi$$
0.370896 0.928674i $$-0.379051\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ 24.0000 0.988903
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 8.00000i 0.328798i
$$593$$ − 26.0000i − 1.06769i −0.845582 0.533846i $$-0.820746\pi$$
0.845582 0.533846i $$-0.179254\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 10.0000i 0.409273i
$$598$$ 0 0
$$599$$ −46.0000 −1.87951 −0.939755 0.341850i $$-0.888947\pi$$
−0.939755 + 0.341850i $$0.888947\pi$$
$$600$$ 0 0
$$601$$ 5.00000 0.203954 0.101977 0.994787i $$-0.467483\pi$$
0.101977 + 0.994787i $$0.467483\pi$$
$$602$$ − 24.0000i − 0.978167i
$$603$$ − 3.00000i − 0.122169i
$$604$$ −17.0000 −0.691720
$$605$$ 0 0
$$606$$ 9.00000 0.365600
$$607$$ 38.0000i 1.54237i 0.636610 + 0.771186i $$0.280336\pi$$
−0.636610 + 0.771186i $$0.719664\pi$$
$$608$$ 40.0000i 1.62221i
$$609$$ 3.00000 0.121566
$$610$$ 0 0
$$611$$ −11.0000 −0.445012
$$612$$ 5.00000i 0.202113i
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ −9.00000 −0.362620
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 14.0000i 0.563163i
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 6.00000i 0.240578i
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ −29.0000 −1.15907
$$627$$ 8.00000i 0.319489i
$$628$$ − 11.0000i − 0.438948i
$$629$$ −40.0000 −1.59490
$$630$$ 0 0
$$631$$ −44.0000 −1.75161 −0.875806 0.482663i $$-0.839670\pi$$
−0.875806 + 0.482663i $$0.839670\pi$$
$$632$$ − 36.0000i − 1.43200i
$$633$$ 16.0000i 0.635943i
$$634$$ 16.0000 0.635441
$$635$$ 0 0
$$636$$ 11.0000 0.436178
$$637$$ 2.00000i 0.0792429i
$$638$$ 1.00000i 0.0395904i
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ −15.0000 −0.592464 −0.296232 0.955116i $$-0.595730\pi$$
−0.296232 + 0.955116i $$0.595730\pi$$
$$642$$ 18.0000i 0.710403i
$$643$$ 12.0000i 0.473234i 0.971603 + 0.236617i $$0.0760386\pi$$
−0.971603 + 0.236617i $$0.923961\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 40.0000 1.57378
$$647$$ 16.0000i 0.629025i 0.949253 + 0.314512i $$0.101841\pi$$
−0.949253 + 0.314512i $$0.898159\pi$$
$$648$$ 3.00000i 0.117851i
$$649$$ 5.00000 0.196267
$$650$$ 0 0
$$651$$ −9.00000 −0.352738
$$652$$ 20.0000i 0.783260i
$$653$$ − 35.0000i − 1.36966i −0.728705 0.684828i $$-0.759877\pi$$
0.728705 0.684828i $$-0.240123\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ − 4.00000i − 0.156055i
$$658$$ − 33.0000i − 1.28647i
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ − 12.0000i − 0.466393i
$$663$$ 5.00000i 0.194184i
$$664$$ −9.00000 −0.349268
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ 0 0
$$668$$ 12.0000i 0.464294i
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ −1.00000 −0.0386046
$$672$$ − 15.0000i − 0.578638i
$$673$$ − 45.0000i − 1.73462i −0.497766 0.867311i $$-0.665846\pi$$
0.497766 0.867311i $$-0.334154\pi$$
$$674$$ −5.00000 −0.192593
$$675$$ 0 0
$$676$$ −1.00000 −0.0384615
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 2.00000i 0.0768095i
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ 23.0000 0.881362
$$682$$ − 3.00000i − 0.114876i
$$683$$ − 9.00000i − 0.344375i −0.985064 0.172188i $$-0.944916\pi$$
0.985064 0.172188i $$-0.0550836\pi$$
$$684$$ −8.00000 −0.305888
$$685$$ 0 0
$$686$$ 15.0000 0.572703
$$687$$ − 18.0000i − 0.686743i
$$688$$ 8.00000i 0.304997i
$$689$$ 11.0000 0.419067
$$690$$ 0 0
$$691$$ −29.0000 −1.10321 −0.551606 0.834105i $$-0.685985\pi$$
−0.551606 + 0.834105i $$0.685985\pi$$
$$692$$ 21.0000i 0.798300i
$$693$$ − 3.00000i − 0.113961i
$$694$$ 2.00000 0.0759190
$$695$$ 0 0
$$696$$ −3.00000 −0.113715
$$697$$ 10.0000i 0.378777i
$$698$$ − 16.0000i − 0.605609i
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 31.0000 1.17085 0.585427 0.810725i $$-0.300927\pi$$
0.585427 + 0.810725i $$0.300927\pi$$
$$702$$ 1.00000i 0.0377426i
$$703$$ − 64.0000i − 2.41381i
$$704$$ 7.00000 0.263822
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ − 27.0000i − 1.01544i
$$708$$ 5.00000i 0.187912i
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 0 0
$$713$$ 0 0
$$714$$ −15.0000 −0.561361
$$715$$ 0 0
$$716$$ −26.0000 −0.971666
$$717$$ − 19.0000i − 0.709568i
$$718$$ − 15.0000i − 0.559795i
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ 42.0000 1.56416
$$722$$ 45.0000i 1.67473i
$$723$$ − 10.0000i − 0.371904i
$$724$$ 5.00000 0.185824
$$725$$ 0 0
$$726$$ −10.0000 −0.371135
$$727$$ − 2.00000i − 0.0741759i −0.999312 0.0370879i $$-0.988192\pi$$
0.999312 0.0370879i $$-0.0118082\pi$$
$$728$$ − 9.00000i − 0.333562i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −40.0000 −1.47945
$$732$$ − 1.00000i − 0.0369611i
$$733$$ 20.0000i 0.738717i 0.929287 + 0.369358i $$0.120423\pi$$
−0.929287 + 0.369358i $$0.879577\pi$$
$$734$$ 12.0000 0.442928
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 3.00000i − 0.110506i
$$738$$ 2.00000i 0.0736210i
$$739$$ −7.00000 −0.257499 −0.128750 0.991677i $$-0.541096\pi$$
−0.128750 + 0.991677i $$0.541096\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 33.0000i 1.21147i
$$743$$ − 13.0000i − 0.476924i −0.971152 0.238462i $$-0.923357\pi$$
0.971152 0.238462i $$-0.0766432\pi$$
$$744$$ 9.00000 0.329956
$$745$$ 0 0
$$746$$ −31.0000 −1.13499
$$747$$ − 3.00000i − 0.109764i
$$748$$ 5.00000i 0.182818i
$$749$$ 54.0000 1.97312
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 11.0000i 0.401129i
$$753$$ 30.0000i 1.09326i
$$754$$ −1.00000 −0.0364179
$$755$$ 0 0
$$756$$ 3.00000 0.109109
$$757$$ 23.0000i 0.835949i 0.908459 + 0.417975i $$0.137260\pi$$
−0.908459 + 0.417975i $$0.862740\pi$$
$$758$$ 35.0000i 1.27126i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −24.0000 −0.869999 −0.435000 0.900431i $$-0.643252\pi$$
−0.435000 + 0.900431i $$0.643252\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ − 30.0000i − 1.08607i
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ 5.00000i 0.180540i
$$768$$ 17.0000i 0.613435i
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ −3.00000 −0.108042
$$772$$ − 10.0000i − 0.359908i
$$773$$ − 18.0000i − 0.647415i −0.946157 0.323708i $$-0.895071\pi$$
0.946157 0.323708i $$-0.104929\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 24.0000i 0.860995i
$$778$$ 30.0000i 1.07555i
$$779$$ −16.0000 −0.573259
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 0 0
$$783$$ − 1.00000i − 0.0357371i
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 6.00000 0.214013
$$787$$ 17.0000i 0.605985i 0.952993 + 0.302992i $$0.0979856\pi$$
−0.952993 + 0.302992i $$0.902014\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ −14.0000 −0.498413
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 3.00000i 0.106600i
$$793$$ − 1.00000i − 0.0355110i
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ − 43.0000i − 1.52314i −0.648084 0.761569i $$-0.724429\pi$$
0.648084 0.761569i $$-0.275571\pi$$
$$798$$ − 24.0000i − 0.849591i
$$799$$ −55.0000 −1.94576
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 4.00000i − 0.141157i
$$804$$ 3.00000 0.105802
$$805$$ 0 0
$$806$$ 3.00000 0.105670
$$807$$ 17.0000i 0.598428i
$$808$$ 27.0000i 0.949857i
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ −33.0000 −1.15879 −0.579393 0.815048i $$-0.696710\pi$$
−0.579393 + 0.815048i $$0.696710\pi$$
$$812$$ 3.00000i 0.105279i
$$813$$ 5.00000i 0.175358i
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 5.00000 0.175035
$$817$$ − 64.0000i − 2.23908i
$$818$$ − 2.00000i − 0.0699284i
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ − 12.0000i − 0.418548i
$$823$$ − 28.0000i − 0.976019i −0.872838 0.488009i $$-0.837723\pi$$
0.872838 0.488009i $$-0.162277\pi$$
$$824$$ −42.0000 −1.46314
$$825$$ 0 0
$$826$$ −15.0000 −0.521917
$$827$$ 1.00000i 0.0347734i 0.999849 + 0.0173867i $$0.00553464\pi$$
−0.999849 + 0.0173867i $$0.994465\pi$$
$$828$$ 0 0
$$829$$ −51.0000 −1.77130 −0.885652 0.464350i $$-0.846288\pi$$
−0.885652 + 0.464350i $$0.846288\pi$$
$$830$$ 0 0
$$831$$ 6.00000 0.208138
$$832$$ 7.00000i 0.242681i
$$833$$ 10.0000i 0.346479i
$$834$$ 10.0000 0.346272
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ 3.00000i 0.103695i
$$838$$ 26.0000i 0.898155i
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 28.0000i 0.964944i
$$843$$ 18.0000i 0.619953i
$$844$$ −16.0000 −0.550743
$$845$$ 0 0
$$846$$ −11.0000 −0.378188
$$847$$ 30.0000i 1.03081i
$$848$$ − 11.0000i − 0.377742i
$$849$$ 18.0000 0.617758
$$850$$ 0 0
$$851$$ 0 0
$$852$$ − 16.0000i − 0.548151i
$$853$$ − 24.0000i − 0.821744i −0.911693 0.410872i $$-0.865224\pi$$
0.911693 0.410872i $$-0.134776\pi$$
$$854$$ 3.00000 0.102658
$$855$$ 0 0
$$856$$ −54.0000 −1.84568
$$857$$ − 38.0000i − 1.29806i −0.760765 0.649028i $$-0.775176\pi$$
0.760765 0.649028i $$-0.224824\pi$$
$$858$$ 1.00000i 0.0341394i
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ − 12.0000i − 0.408722i
$$863$$ 7.00000i 0.238283i 0.992877 + 0.119141i $$0.0380142\pi$$
−0.992877 + 0.119141i $$0.961986\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ 8.00000i 0.271694i
$$868$$ − 9.00000i − 0.305480i
$$869$$ 12.0000 0.407072
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 30.0000i 1.01593i
$$873$$ 2.00000i 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 4.00000 0.135147
$$877$$ 24.0000i 0.810422i 0.914223 + 0.405211i $$0.132802\pi$$
−0.914223 + 0.405211i $$0.867198\pi$$
$$878$$ − 4.00000i − 0.134993i
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ −25.0000 −0.842271 −0.421136 0.906998i $$-0.638368\pi$$
−0.421136 + 0.906998i $$0.638368\pi$$
$$882$$ 2.00000i 0.0673435i
$$883$$ − 16.0000i − 0.538443i −0.963078 0.269221i $$-0.913234\pi$$
0.963078 0.269221i $$-0.0867663\pi$$
$$884$$ −5.00000 −0.168168
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ − 24.0000i − 0.805387i
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 8.00000i 0.267860i
$$893$$ − 88.0000i − 2.94481i
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 9.00000 0.300669
$$897$$ 0 0
$$898$$ 28.0000i 0.934372i
$$899$$ −3.00000 −0.100056
$$900$$ 0 0
$$901$$ 55.0000 1.83232
$$902$$ 2.00000i 0.0665927i
$$903$$ 24.0000i 0.798670i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −17.0000 −0.564787
$$907$$ 24.0000i 0.796907i 0.917189 + 0.398453i $$0.130453\pi$$
−0.917189 + 0.398453i $$0.869547\pi$$
$$908$$ 23.0000i 0.763282i
$$909$$ −9.00000 −0.298511
$$910$$ 0 0
$$911$$ 56.0000 1.85536 0.927681 0.373373i $$-0.121799\pi$$
0.927681 + 0.373373i $$0.121799\pi$$
$$912$$ 8.00000i 0.264906i
$$913$$ − 3.00000i − 0.0992855i
$$914$$ 34.0000 1.12462
$$915$$ 0 0
$$916$$ 18.0000 0.594737
$$917$$ − 18.0000i − 0.594412i
$$918$$ 5.00000i 0.165025i
$$919$$ −22.0000 −0.725713 −0.362857 0.931845i $$-0.618198\pi$$
−0.362857 + 0.931845i $$0.618198\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 42.0000i 1.38320i
$$923$$ − 16.0000i − 0.526646i
$$924$$ 3.00000 0.0986928
$$925$$ 0 0
$$926$$ −9.00000 −0.295758
$$927$$ − 14.0000i − 0.459820i
$$928$$ − 5.00000i − 0.164133i
$$929$$ 26.0000 0.853032 0.426516 0.904480i $$-0.359741\pi$$
0.426516 + 0.904480i $$0.359741\pi$$
$$930$$ 0 0
$$931$$ −16.0000 −0.524379
$$932$$ 6.00000i 0.196537i
$$933$$ − 6.00000i − 0.196431i
$$934$$ −32.0000 −1.04707
$$935$$ 0 0
$$936$$ −3.00000 −0.0980581
$$937$$ 45.0000i 1.47009i 0.678021 + 0.735043i $$0.262838\pi$$
−0.678021 + 0.735043i $$0.737162\pi$$
$$938$$ 9.00000i 0.293860i
$$939$$ 29.0000 0.946379
$$940$$ 0 0
$$941$$ −56.0000 −1.82555 −0.912774 0.408465i $$-0.866064\pi$$
−0.912774 + 0.408465i $$0.866064\pi$$
$$942$$ − 11.0000i − 0.358399i
$$943$$ 0 0
$$944$$ 5.00000 0.162736
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ 21.0000i 0.682408i 0.939989 + 0.341204i $$0.110835\pi$$
−0.939989 + 0.341204i $$0.889165\pi$$
$$948$$ 12.0000i 0.389742i
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ −16.0000 −0.518836
$$952$$ − 45.0000i − 1.45846i
$$953$$ − 37.0000i − 1.19855i −0.800544 0.599274i $$-0.795456\pi$$
0.800544 0.599274i $$-0.204544\pi$$
$$954$$ 11.0000 0.356138
$$955$$ 0 0
$$956$$ 19.0000 0.614504
$$957$$ − 1.00000i − 0.0323254i
$$958$$ 15.0000i 0.484628i
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ − 8.00000i − 0.257930i
$$963$$ − 18.0000i − 0.580042i
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 37.0000i 1.18984i 0.803785 + 0.594920i $$0.202816\pi$$
−0.803785 + 0.594920i $$0.797184\pi$$
$$968$$ − 30.0000i − 0.964237i
$$969$$ −40.0000 −1.28499
$$970$$ 0 0
$$971$$ 54.0000 1.73294 0.866471 0.499227i $$-0.166383\pi$$
0.866471 + 0.499227i $$0.166383\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ − 30.0000i − 0.961756i
$$974$$ −37.0000 −1.18556
$$975$$ 0 0
$$976$$ −1.00000 −0.0320092
$$977$$ 28.0000i 0.895799i 0.894084 + 0.447900i $$0.147828\pi$$
−0.894084 + 0.447900i $$0.852172\pi$$
$$978$$ 20.0000i 0.639529i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ − 26.0000i − 0.829693i
$$983$$ 21.0000i 0.669796i 0.942254 + 0.334898i $$0.108702\pi$$
−0.942254 + 0.334898i $$0.891298\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −5.00000 −0.159232
$$987$$ 33.0000i 1.05040i
$$988$$ − 8.00000i − 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 15.0000i 0.476250i
$$993$$ 12.0000i 0.380808i
$$994$$ 48.0000 1.52247
$$995$$ 0 0
$$996$$ 3.00000 0.0950586
$$997$$ 3.00000i 0.0950110i 0.998871 + 0.0475055i $$0.0151272\pi$$
−0.998871 + 0.0475055i $$0.984873\pi$$
$$998$$ 41.0000i 1.29783i
$$999$$ 8.00000 0.253109
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 975.2.c.g.274.2 2
3.2 odd 2 2925.2.c.i.2224.1 2
5.2 odd 4 975.2.a.d.1.1 1
5.3 odd 4 975.2.a.k.1.1 yes 1
5.4 even 2 inner 975.2.c.g.274.1 2
15.2 even 4 2925.2.a.o.1.1 1
15.8 even 4 2925.2.a.b.1.1 1
15.14 odd 2 2925.2.c.i.2224.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
975.2.a.d.1.1 1 5.2 odd 4
975.2.a.k.1.1 yes 1 5.3 odd 4
975.2.c.g.274.1 2 5.4 even 2 inner
975.2.c.g.274.2 2 1.1 even 1 trivial
2925.2.a.b.1.1 1 15.8 even 4
2925.2.a.o.1.1 1 15.2 even 4
2925.2.c.i.2224.1 2 3.2 odd 2
2925.2.c.i.2224.2 2 15.14 odd 2