# Properties

 Label 1850.2.b.i.149.3 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.3 Root $$-1.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.i.149.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} -4.60555i q^{7} -1.00000i q^{8} +2.90833 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} -4.60555i q^{7} -1.00000i q^{8} +2.90833 q^{9} +1.30278 q^{11} +0.302776i q^{12} -2.30278i q^{13} +4.60555 q^{14} +1.00000 q^{16} +6.00000i q^{17} +2.90833i q^{18} -2.00000 q^{19} -1.39445 q^{21} +1.30278i q^{22} -6.90833i q^{23} -0.302776 q^{24} +2.30278 q^{26} -1.78890i q^{27} +4.60555i q^{28} -6.90833 q^{29} +3.30278 q^{31} +1.00000i q^{32} -0.394449i q^{33} -6.00000 q^{34} -2.90833 q^{36} -1.00000i q^{37} -2.00000i q^{38} -0.697224 q^{39} -0.908327 q^{41} -1.39445i q^{42} -6.60555i q^{43} -1.30278 q^{44} +6.90833 q^{46} +2.60555i q^{47} -0.302776i q^{48} -14.2111 q^{49} +1.81665 q^{51} +2.30278i q^{52} -6.00000i q^{53} +1.78890 q^{54} -4.60555 q^{56} +0.605551i q^{57} -6.90833i q^{58} -3.39445 q^{59} -10.5139 q^{61} +3.30278i q^{62} -13.3944i q^{63} -1.00000 q^{64} +0.394449 q^{66} -14.5139i q^{67} -6.00000i q^{68} -2.09167 q^{69} +6.00000 q^{71} -2.90833i q^{72} -8.69722i q^{73} +1.00000 q^{74} +2.00000 q^{76} -6.00000i q^{77} -0.697224i q^{78} +16.1194 q^{79} +8.18335 q^{81} -0.908327i q^{82} +17.2111i q^{83} +1.39445 q^{84} +6.60555 q^{86} +2.09167i q^{87} -1.30278i q^{88} -5.21110 q^{89} -10.6056 q^{91} +6.90833i q^{92} -1.00000i q^{93} -2.60555 q^{94} +0.302776 q^{96} -12.4222i q^{97} -14.2111i q^{98} +3.78890 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 6 q^{6} - 10 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9 $$4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46} - 28 q^{49} - 36 q^{51} + 36 q^{54} - 4 q^{56} - 28 q^{59} - 6 q^{61} - 4 q^{64} + 16 q^{66} - 30 q^{69} + 24 q^{71} + 4 q^{74} + 8 q^{76} + 14 q^{79} + 76 q^{81} + 20 q^{84} + 12 q^{86} + 8 q^{89} - 28 q^{91} + 4 q^{94} - 6 q^{96} + 44 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9 - 2 * q^11 + 4 * q^14 + 4 * q^16 - 8 * q^19 - 20 * q^21 + 6 * q^24 + 2 * q^26 - 6 * q^29 + 6 * q^31 - 24 * q^34 + 10 * q^36 - 10 * q^39 + 18 * q^41 + 2 * q^44 + 6 * q^46 - 28 * q^49 - 36 * q^51 + 36 * q^54 - 4 * q^56 - 28 * q^59 - 6 * q^61 - 4 * q^64 + 16 * q^66 - 30 * q^69 + 24 * q^71 + 4 * q^74 + 8 * q^76 + 14 * q^79 + 76 * q^81 + 20 * q^84 + 12 * q^86 + 8 * q^89 - 28 * q^91 + 4 * q^94 - 6 * q^96 + 44 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 0.302776i − 0.174808i −0.996173 0.0874038i $$-0.972143\pi$$
0.996173 0.0874038i $$-0.0278570\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0.302776 0.123608
$$7$$ − 4.60555i − 1.74073i −0.492403 0.870367i $$-0.663881\pi$$
0.492403 0.870367i $$-0.336119\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 2.90833 0.969442
$$10$$ 0 0
$$11$$ 1.30278 0.392802 0.196401 0.980524i $$-0.437075\pi$$
0.196401 + 0.980524i $$0.437075\pi$$
$$12$$ 0.302776i 0.0874038i
$$13$$ − 2.30278i − 0.638675i −0.947641 0.319338i $$-0.896540\pi$$
0.947641 0.319338i $$-0.103460\pi$$
$$14$$ 4.60555 1.23089
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 2.90833i 0.685499i
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −1.39445 −0.304294
$$22$$ 1.30278i 0.277753i
$$23$$ − 6.90833i − 1.44049i −0.693722 0.720243i $$-0.744030\pi$$
0.693722 0.720243i $$-0.255970\pi$$
$$24$$ −0.302776 −0.0618038
$$25$$ 0 0
$$26$$ 2.30278 0.451611
$$27$$ − 1.78890i − 0.344273i
$$28$$ 4.60555i 0.870367i
$$29$$ −6.90833 −1.28284 −0.641422 0.767188i $$-0.721655\pi$$
−0.641422 + 0.767188i $$0.721655\pi$$
$$30$$ 0 0
$$31$$ 3.30278 0.593196 0.296598 0.955002i $$-0.404148\pi$$
0.296598 + 0.955002i $$0.404148\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 0.394449i − 0.0686647i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ −2.90833 −0.484721
$$37$$ − 1.00000i − 0.164399i
$$38$$ − 2.00000i − 0.324443i
$$39$$ −0.697224 −0.111645
$$40$$ 0 0
$$41$$ −0.908327 −0.141857 −0.0709284 0.997481i $$-0.522596\pi$$
−0.0709284 + 0.997481i $$0.522596\pi$$
$$42$$ − 1.39445i − 0.215168i
$$43$$ − 6.60555i − 1.00734i −0.863897 0.503669i $$-0.831983\pi$$
0.863897 0.503669i $$-0.168017\pi$$
$$44$$ −1.30278 −0.196401
$$45$$ 0 0
$$46$$ 6.90833 1.01858
$$47$$ 2.60555i 0.380059i 0.981778 + 0.190029i $$0.0608583\pi$$
−0.981778 + 0.190029i $$0.939142\pi$$
$$48$$ − 0.302776i − 0.0437019i
$$49$$ −14.2111 −2.03016
$$50$$ 0 0
$$51$$ 1.81665 0.254382
$$52$$ 2.30278i 0.319338i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 1.78890 0.243438
$$55$$ 0 0
$$56$$ −4.60555 −0.615443
$$57$$ 0.605551i 0.0802072i
$$58$$ − 6.90833i − 0.907108i
$$59$$ −3.39445 −0.441920 −0.220960 0.975283i $$-0.570919\pi$$
−0.220960 + 0.975283i $$0.570919\pi$$
$$60$$ 0 0
$$61$$ −10.5139 −1.34616 −0.673082 0.739568i $$-0.735030\pi$$
−0.673082 + 0.739568i $$0.735030\pi$$
$$62$$ 3.30278i 0.419453i
$$63$$ − 13.3944i − 1.68754i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0.394449 0.0485533
$$67$$ − 14.5139i − 1.77315i −0.462583 0.886576i $$-0.653077\pi$$
0.462583 0.886576i $$-0.346923\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ −2.09167 −0.251808
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ − 2.90833i − 0.342750i
$$73$$ − 8.69722i − 1.01793i −0.860786 0.508967i $$-0.830028\pi$$
0.860786 0.508967i $$-0.169972\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ − 6.00000i − 0.683763i
$$78$$ − 0.697224i − 0.0789451i
$$79$$ 16.1194 1.81358 0.906789 0.421585i $$-0.138526\pi$$
0.906789 + 0.421585i $$0.138526\pi$$
$$80$$ 0 0
$$81$$ 8.18335 0.909261
$$82$$ − 0.908327i − 0.100308i
$$83$$ 17.2111i 1.88916i 0.328276 + 0.944582i $$0.393533\pi$$
−0.328276 + 0.944582i $$0.606467\pi$$
$$84$$ 1.39445 0.152147
$$85$$ 0 0
$$86$$ 6.60555 0.712295
$$87$$ 2.09167i 0.224251i
$$88$$ − 1.30278i − 0.138876i
$$89$$ −5.21110 −0.552376 −0.276188 0.961104i $$-0.589071\pi$$
−0.276188 + 0.961104i $$0.589071\pi$$
$$90$$ 0 0
$$91$$ −10.6056 −1.11176
$$92$$ 6.90833i 0.720243i
$$93$$ − 1.00000i − 0.103695i
$$94$$ −2.60555 −0.268742
$$95$$ 0 0
$$96$$ 0.302776 0.0309019
$$97$$ − 12.4222i − 1.26128i −0.776074 0.630642i $$-0.782792\pi$$
0.776074 0.630642i $$-0.217208\pi$$
$$98$$ − 14.2111i − 1.43554i
$$99$$ 3.78890 0.380799
$$100$$ 0 0
$$101$$ 16.4222 1.63407 0.817035 0.576588i $$-0.195616\pi$$
0.817035 + 0.576588i $$0.195616\pi$$
$$102$$ 1.81665i 0.179876i
$$103$$ 3.30278i 0.325432i 0.986673 + 0.162716i $$0.0520255\pi$$
−0.986673 + 0.162716i $$0.947975\pi$$
$$104$$ −2.30278 −0.225806
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 4.30278i − 0.415965i −0.978133 0.207983i $$-0.933310\pi$$
0.978133 0.207983i $$-0.0666897\pi$$
$$108$$ 1.78890i 0.172137i
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −0.302776 −0.0287382
$$112$$ − 4.60555i − 0.435184i
$$113$$ 11.2111i 1.05465i 0.849663 + 0.527326i $$0.176805\pi$$
−0.849663 + 0.527326i $$0.823195\pi$$
$$114$$ −0.605551 −0.0567151
$$115$$ 0 0
$$116$$ 6.90833 0.641422
$$117$$ − 6.69722i − 0.619159i
$$118$$ − 3.39445i − 0.312484i
$$119$$ 27.6333 2.53314
$$120$$ 0 0
$$121$$ −9.30278 −0.845707
$$122$$ − 10.5139i − 0.951882i
$$123$$ 0.275019i 0.0247977i
$$124$$ −3.30278 −0.296598
$$125$$ 0 0
$$126$$ 13.3944 1.19327
$$127$$ 4.78890i 0.424946i 0.977167 + 0.212473i $$0.0681517\pi$$
−0.977167 + 0.212473i $$0.931848\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ 3.39445 0.296574 0.148287 0.988944i $$-0.452624\pi$$
0.148287 + 0.988944i $$0.452624\pi$$
$$132$$ 0.394449i 0.0343324i
$$133$$ 9.21110i 0.798704i
$$134$$ 14.5139 1.25381
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 9.90833i 0.846525i 0.906007 + 0.423263i $$0.139115\pi$$
−0.906007 + 0.423263i $$0.860885\pi$$
$$138$$ − 2.09167i − 0.178055i
$$139$$ −8.90833 −0.755594 −0.377797 0.925888i $$-0.623318\pi$$
−0.377797 + 0.925888i $$0.623318\pi$$
$$140$$ 0 0
$$141$$ 0.788897 0.0664372
$$142$$ 6.00000i 0.503509i
$$143$$ − 3.00000i − 0.250873i
$$144$$ 2.90833 0.242361
$$145$$ 0 0
$$146$$ 8.69722 0.719787
$$147$$ 4.30278i 0.354887i
$$148$$ 1.00000i 0.0821995i
$$149$$ 1.81665 0.148826 0.0744130 0.997228i $$-0.476292\pi$$
0.0744130 + 0.997228i $$0.476292\pi$$
$$150$$ 0 0
$$151$$ −13.3944 −1.09002 −0.545012 0.838428i $$-0.683475\pi$$
−0.545012 + 0.838428i $$0.683475\pi$$
$$152$$ 2.00000i 0.162221i
$$153$$ 17.4500i 1.41075i
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 0.697224 0.0558226
$$157$$ − 7.21110i − 0.575509i −0.957704 0.287754i $$-0.907091\pi$$
0.957704 0.287754i $$-0.0929087\pi$$
$$158$$ 16.1194i 1.28239i
$$159$$ −1.81665 −0.144070
$$160$$ 0 0
$$161$$ −31.8167 −2.50750
$$162$$ 8.18335i 0.642944i
$$163$$ − 20.4222i − 1.59959i −0.600273 0.799795i $$-0.704941\pi$$
0.600273 0.799795i $$-0.295059\pi$$
$$164$$ 0.908327 0.0709284
$$165$$ 0 0
$$166$$ −17.2111 −1.33584
$$167$$ − 12.5139i − 0.968353i −0.874970 0.484176i $$-0.839119\pi$$
0.874970 0.484176i $$-0.160881\pi$$
$$168$$ 1.39445i 0.107584i
$$169$$ 7.69722 0.592094
$$170$$ 0 0
$$171$$ −5.81665 −0.444811
$$172$$ 6.60555i 0.503669i
$$173$$ − 23.2111i − 1.76471i −0.470587 0.882354i $$-0.655958\pi$$
0.470587 0.882354i $$-0.344042\pi$$
$$174$$ −2.09167 −0.158569
$$175$$ 0 0
$$176$$ 1.30278 0.0982004
$$177$$ 1.02776i 0.0772509i
$$178$$ − 5.21110i − 0.390589i
$$179$$ −7.81665 −0.584244 −0.292122 0.956381i $$-0.594361\pi$$
−0.292122 + 0.956381i $$0.594361\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ − 10.6056i − 0.786136i
$$183$$ 3.18335i 0.235320i
$$184$$ −6.90833 −0.509289
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 7.81665i 0.571610i
$$188$$ − 2.60555i − 0.190029i
$$189$$ −8.23886 −0.599289
$$190$$ 0 0
$$191$$ 12.5139 0.905472 0.452736 0.891644i $$-0.350448\pi$$
0.452736 + 0.891644i $$0.350448\pi$$
$$192$$ 0.302776i 0.0218509i
$$193$$ − 4.00000i − 0.287926i −0.989583 0.143963i $$-0.954015\pi$$
0.989583 0.143963i $$-0.0459847\pi$$
$$194$$ 12.4222 0.891862
$$195$$ 0 0
$$196$$ 14.2111 1.01508
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 3.78890i 0.269265i
$$199$$ 2.42221 0.171706 0.0858528 0.996308i $$-0.472639\pi$$
0.0858528 + 0.996308i $$0.472639\pi$$
$$200$$ 0 0
$$201$$ −4.39445 −0.309961
$$202$$ 16.4222i 1.15546i
$$203$$ 31.8167i 2.23309i
$$204$$ −1.81665 −0.127191
$$205$$ 0 0
$$206$$ −3.30278 −0.230115
$$207$$ − 20.0917i − 1.39647i
$$208$$ − 2.30278i − 0.159669i
$$209$$ −2.60555 −0.180230
$$210$$ 0 0
$$211$$ 6.69722 0.461056 0.230528 0.973066i $$-0.425955\pi$$
0.230528 + 0.973066i $$0.425955\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ − 1.81665i − 0.124475i
$$214$$ 4.30278 0.294132
$$215$$ 0 0
$$216$$ −1.78890 −0.121719
$$217$$ − 15.2111i − 1.03260i
$$218$$ − 2.00000i − 0.135457i
$$219$$ −2.63331 −0.177942
$$220$$ 0 0
$$221$$ 13.8167 0.929409
$$222$$ − 0.302776i − 0.0203210i
$$223$$ 15.8167i 1.05916i 0.848260 + 0.529581i $$0.177651\pi$$
−0.848260 + 0.529581i $$0.822349\pi$$
$$224$$ 4.60555 0.307721
$$225$$ 0 0
$$226$$ −11.2111 −0.745751
$$227$$ 7.81665i 0.518810i 0.965769 + 0.259405i $$0.0835264\pi$$
−0.965769 + 0.259405i $$0.916474\pi$$
$$228$$ − 0.605551i − 0.0401036i
$$229$$ −17.3944 −1.14946 −0.574729 0.818344i $$-0.694892\pi$$
−0.574729 + 0.818344i $$0.694892\pi$$
$$230$$ 0 0
$$231$$ −1.81665 −0.119527
$$232$$ 6.90833i 0.453554i
$$233$$ − 9.51388i − 0.623275i −0.950201 0.311637i $$-0.899123\pi$$
0.950201 0.311637i $$-0.100877\pi$$
$$234$$ 6.69722 0.437811
$$235$$ 0 0
$$236$$ 3.39445 0.220960
$$237$$ − 4.88057i − 0.317027i
$$238$$ 27.6333i 1.79120i
$$239$$ −0.513878 −0.0332400 −0.0166200 0.999862i $$-0.505291\pi$$
−0.0166200 + 0.999862i $$0.505291\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ − 9.30278i − 0.598005i
$$243$$ − 7.84441i − 0.503219i
$$244$$ 10.5139 0.673082
$$245$$ 0 0
$$246$$ −0.275019 −0.0175346
$$247$$ 4.60555i 0.293044i
$$248$$ − 3.30278i − 0.209726i
$$249$$ 5.21110 0.330240
$$250$$ 0 0
$$251$$ −6.78890 −0.428511 −0.214256 0.976778i $$-0.568733\pi$$
−0.214256 + 0.976778i $$0.568733\pi$$
$$252$$ 13.3944i 0.843771i
$$253$$ − 9.00000i − 0.565825i
$$254$$ −4.78890 −0.300482
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 11.2111i 0.699329i 0.936875 + 0.349665i $$0.113704\pi$$
−0.936875 + 0.349665i $$0.886296\pi$$
$$258$$ − 2.00000i − 0.124515i
$$259$$ −4.60555 −0.286175
$$260$$ 0 0
$$261$$ −20.0917 −1.24364
$$262$$ 3.39445i 0.209710i
$$263$$ − 7.81665i − 0.481996i −0.970526 0.240998i $$-0.922525\pi$$
0.970526 0.240998i $$-0.0774746\pi$$
$$264$$ −0.394449 −0.0242766
$$265$$ 0 0
$$266$$ −9.21110 −0.564769
$$267$$ 1.57779i 0.0965595i
$$268$$ 14.5139i 0.886576i
$$269$$ 6.78890 0.413926 0.206963 0.978349i $$-0.433642\pi$$
0.206963 + 0.978349i $$0.433642\pi$$
$$270$$ 0 0
$$271$$ 6.42221 0.390121 0.195061 0.980791i $$-0.437510\pi$$
0.195061 + 0.980791i $$0.437510\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 3.21110i 0.194345i
$$274$$ −9.90833 −0.598584
$$275$$ 0 0
$$276$$ 2.09167 0.125904
$$277$$ 25.1194i 1.50928i 0.656139 + 0.754640i $$0.272189\pi$$
−0.656139 + 0.754640i $$0.727811\pi$$
$$278$$ − 8.90833i − 0.534286i
$$279$$ 9.60555 0.575069
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 0.788897i 0.0469782i
$$283$$ 17.3944i 1.03399i 0.855988 + 0.516996i $$0.172950\pi$$
−0.855988 + 0.516996i $$0.827050\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 4.18335i 0.246935i
$$288$$ 2.90833i 0.171375i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −3.76114 −0.220482
$$292$$ 8.69722i 0.508967i
$$293$$ − 25.0278i − 1.46214i −0.682304 0.731069i $$-0.739022\pi$$
0.682304 0.731069i $$-0.260978\pi$$
$$294$$ −4.30278 −0.250943
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ − 2.33053i − 0.135231i
$$298$$ 1.81665i 0.105236i
$$299$$ −15.9083 −0.920002
$$300$$ 0 0
$$301$$ −30.4222 −1.75351
$$302$$ − 13.3944i − 0.770764i
$$303$$ − 4.97224i − 0.285648i
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ −17.4500 −0.997548
$$307$$ − 7.09167i − 0.404743i −0.979309 0.202372i $$-0.935135\pi$$
0.979309 0.202372i $$-0.0648649\pi$$
$$308$$ 6.00000i 0.341882i
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ 5.09167 0.288722 0.144361 0.989525i $$-0.453887\pi$$
0.144361 + 0.989525i $$0.453887\pi$$
$$312$$ 0.697224i 0.0394726i
$$313$$ 27.0278i 1.52770i 0.645394 + 0.763850i $$0.276693\pi$$
−0.645394 + 0.763850i $$0.723307\pi$$
$$314$$ 7.21110 0.406946
$$315$$ 0 0
$$316$$ −16.1194 −0.906789
$$317$$ 5.21110i 0.292685i 0.989234 + 0.146342i $$0.0467501\pi$$
−0.989234 + 0.146342i $$0.953250\pi$$
$$318$$ − 1.81665i − 0.101873i
$$319$$ −9.00000 −0.503903
$$320$$ 0 0
$$321$$ −1.30278 −0.0727138
$$322$$ − 31.8167i − 1.77307i
$$323$$ − 12.0000i − 0.667698i
$$324$$ −8.18335 −0.454630
$$325$$ 0 0
$$326$$ 20.4222 1.13108
$$327$$ 0.605551i 0.0334871i
$$328$$ 0.908327i 0.0501540i
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 1.21110 0.0665682 0.0332841 0.999446i $$-0.489403\pi$$
0.0332841 + 0.999446i $$0.489403\pi$$
$$332$$ − 17.2111i − 0.944582i
$$333$$ − 2.90833i − 0.159375i
$$334$$ 12.5139 0.684729
$$335$$ 0 0
$$336$$ −1.39445 −0.0760734
$$337$$ 19.1194i 1.04150i 0.853709 + 0.520751i $$0.174348\pi$$
−0.853709 + 0.520751i $$0.825652\pi$$
$$338$$ 7.69722i 0.418674i
$$339$$ 3.39445 0.184361
$$340$$ 0 0
$$341$$ 4.30278 0.233008
$$342$$ − 5.81665i − 0.314529i
$$343$$ 33.2111i 1.79323i
$$344$$ −6.60555 −0.356147
$$345$$ 0 0
$$346$$ 23.2111 1.24784
$$347$$ − 31.8167i − 1.70801i −0.520267 0.854004i $$-0.674168\pi$$
0.520267 0.854004i $$-0.325832\pi$$
$$348$$ − 2.09167i − 0.112125i
$$349$$ 22.2389 1.19042 0.595209 0.803571i $$-0.297069\pi$$
0.595209 + 0.803571i $$0.297069\pi$$
$$350$$ 0 0
$$351$$ −4.11943 −0.219879
$$352$$ 1.30278i 0.0694382i
$$353$$ 31.8167i 1.69343i 0.532047 + 0.846715i $$0.321423\pi$$
−0.532047 + 0.846715i $$0.678577\pi$$
$$354$$ −1.02776 −0.0546246
$$355$$ 0 0
$$356$$ 5.21110 0.276188
$$357$$ − 8.36669i − 0.442812i
$$358$$ − 7.81665i − 0.413123i
$$359$$ 11.2111 0.591699 0.295850 0.955235i $$-0.404397\pi$$
0.295850 + 0.955235i $$0.404397\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 20.0000i 1.05118i
$$363$$ 2.81665i 0.147836i
$$364$$ 10.6056 0.555882
$$365$$ 0 0
$$366$$ −3.18335 −0.166396
$$367$$ 17.8167i 0.930022i 0.885305 + 0.465011i $$0.153950\pi$$
−0.885305 + 0.465011i $$0.846050\pi$$
$$368$$ − 6.90833i − 0.360121i
$$369$$ −2.64171 −0.137522
$$370$$ 0 0
$$371$$ −27.6333 −1.43465
$$372$$ 1.00000i 0.0518476i
$$373$$ 3.81665i 0.197619i 0.995106 + 0.0988094i $$0.0315034\pi$$
−0.995106 + 0.0988094i $$0.968497\pi$$
$$374$$ −7.81665 −0.404190
$$375$$ 0 0
$$376$$ 2.60555 0.134371
$$377$$ 15.9083i 0.819321i
$$378$$ − 8.23886i − 0.423761i
$$379$$ 15.3305 0.787477 0.393738 0.919223i $$-0.371182\pi$$
0.393738 + 0.919223i $$0.371182\pi$$
$$380$$ 0 0
$$381$$ 1.44996 0.0742838
$$382$$ 12.5139i 0.640266i
$$383$$ 20.8444i 1.06510i 0.846399 + 0.532550i $$0.178766\pi$$
−0.846399 + 0.532550i $$0.821234\pi$$
$$384$$ −0.302776 −0.0154510
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ − 19.2111i − 0.976555i
$$388$$ 12.4222i 0.630642i
$$389$$ 11.8806 0.602369 0.301184 0.953566i $$-0.402618\pi$$
0.301184 + 0.953566i $$0.402618\pi$$
$$390$$ 0 0
$$391$$ 41.4500 2.09621
$$392$$ 14.2111i 0.717769i
$$393$$ − 1.02776i − 0.0518435i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −3.78890 −0.190399
$$397$$ − 27.8167i − 1.39608i −0.716060 0.698039i $$-0.754056\pi$$
0.716060 0.698039i $$-0.245944\pi$$
$$398$$ 2.42221i 0.121414i
$$399$$ 2.78890 0.139620
$$400$$ 0 0
$$401$$ 13.8167 0.689971 0.344985 0.938608i $$-0.387884\pi$$
0.344985 + 0.938608i $$0.387884\pi$$
$$402$$ − 4.39445i − 0.219175i
$$403$$ − 7.60555i − 0.378859i
$$404$$ −16.4222 −0.817035
$$405$$ 0 0
$$406$$ −31.8167 −1.57903
$$407$$ − 1.30278i − 0.0645762i
$$408$$ − 1.81665i − 0.0899378i
$$409$$ 5.02776 0.248607 0.124303 0.992244i $$-0.460330\pi$$
0.124303 + 0.992244i $$0.460330\pi$$
$$410$$ 0 0
$$411$$ 3.00000 0.147979
$$412$$ − 3.30278i − 0.162716i
$$413$$ 15.6333i 0.769265i
$$414$$ 20.0917 0.987452
$$415$$ 0 0
$$416$$ 2.30278 0.112903
$$417$$ 2.69722i 0.132084i
$$418$$ − 2.60555i − 0.127442i
$$419$$ 25.1472 1.22852 0.614260 0.789104i $$-0.289455\pi$$
0.614260 + 0.789104i $$0.289455\pi$$
$$420$$ 0 0
$$421$$ 28.7250 1.39997 0.699985 0.714158i $$-0.253190\pi$$
0.699985 + 0.714158i $$0.253190\pi$$
$$422$$ 6.69722i 0.326016i
$$423$$ 7.57779i 0.368445i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 1.81665 0.0880172
$$427$$ 48.4222i 2.34331i
$$428$$ 4.30278i 0.207983i
$$429$$ −0.908327 −0.0438544
$$430$$ 0 0
$$431$$ −5.21110 −0.251010 −0.125505 0.992093i $$-0.540055\pi$$
−0.125505 + 0.992093i $$0.540055\pi$$
$$432$$ − 1.78890i − 0.0860684i
$$433$$ − 11.9361i − 0.573612i −0.957989 0.286806i $$-0.907407\pi$$
0.957989 0.286806i $$-0.0925934\pi$$
$$434$$ 15.2111 0.730156
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 13.8167i 0.660940i
$$438$$ − 2.63331i − 0.125824i
$$439$$ 9.33053 0.445322 0.222661 0.974896i $$-0.428526\pi$$
0.222661 + 0.974896i $$0.428526\pi$$
$$440$$ 0 0
$$441$$ −41.3305 −1.96812
$$442$$ 13.8167i 0.657191i
$$443$$ 0.275019i 0.0130666i 0.999979 + 0.00653328i $$0.00207962\pi$$
−0.999979 + 0.00653328i $$0.997920\pi$$
$$444$$ 0.302776 0.0143691
$$445$$ 0 0
$$446$$ −15.8167 −0.748940
$$447$$ − 0.550039i − 0.0260159i
$$448$$ 4.60555i 0.217592i
$$449$$ 0.788897 0.0372304 0.0186152 0.999827i $$-0.494074\pi$$
0.0186152 + 0.999827i $$0.494074\pi$$
$$450$$ 0 0
$$451$$ −1.18335 −0.0557216
$$452$$ − 11.2111i − 0.527326i
$$453$$ 4.05551i 0.190545i
$$454$$ −7.81665 −0.366854
$$455$$ 0 0
$$456$$ 0.605551 0.0283575
$$457$$ − 4.60555i − 0.215439i −0.994181 0.107719i $$-0.965645\pi$$
0.994181 0.107719i $$-0.0343548\pi$$
$$458$$ − 17.3944i − 0.812789i
$$459$$ 10.7334 0.500991
$$460$$ 0 0
$$461$$ −16.4222 −0.764858 −0.382429 0.923985i $$-0.624912\pi$$
−0.382429 + 0.923985i $$0.624912\pi$$
$$462$$ − 1.81665i − 0.0845184i
$$463$$ 30.3028i 1.40829i 0.710056 + 0.704145i $$0.248669\pi$$
−0.710056 + 0.704145i $$0.751331\pi$$
$$464$$ −6.90833 −0.320711
$$465$$ 0 0
$$466$$ 9.51388 0.440722
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 6.69722i 0.309579i
$$469$$ −66.8444 −3.08659
$$470$$ 0 0
$$471$$ −2.18335 −0.100603
$$472$$ 3.39445i 0.156242i
$$473$$ − 8.60555i − 0.395684i
$$474$$ 4.88057 0.224172
$$475$$ 0 0
$$476$$ −27.6333 −1.26657
$$477$$ − 17.4500i − 0.798979i
$$478$$ − 0.513878i − 0.0235042i
$$479$$ −12.1194 −0.553751 −0.276875 0.960906i $$-0.589299\pi$$
−0.276875 + 0.960906i $$0.589299\pi$$
$$480$$ 0 0
$$481$$ −2.30278 −0.104998
$$482$$ 8.00000i 0.364390i
$$483$$ 9.63331i 0.438331i
$$484$$ 9.30278 0.422853
$$485$$ 0 0
$$486$$ 7.84441 0.355830
$$487$$ 22.7889i 1.03266i 0.856389 + 0.516332i $$0.172703\pi$$
−0.856389 + 0.516332i $$0.827297\pi$$
$$488$$ 10.5139i 0.475941i
$$489$$ −6.18335 −0.279621
$$490$$ 0 0
$$491$$ −14.7250 −0.664529 −0.332265 0.943186i $$-0.607813\pi$$
−0.332265 + 0.943186i $$0.607813\pi$$
$$492$$ − 0.275019i − 0.0123988i
$$493$$ − 41.4500i − 1.86681i
$$494$$ −4.60555 −0.207214
$$495$$ 0 0
$$496$$ 3.30278 0.148299
$$497$$ − 27.6333i − 1.23952i
$$498$$ 5.21110i 0.233515i
$$499$$ −8.23886 −0.368822 −0.184411 0.982849i $$-0.559038\pi$$
−0.184411 + 0.982849i $$0.559038\pi$$
$$500$$ 0 0
$$501$$ −3.78890 −0.169275
$$502$$ − 6.78890i − 0.303003i
$$503$$ − 24.5139i − 1.09302i −0.837453 0.546510i $$-0.815956\pi$$
0.837453 0.546510i $$-0.184044\pi$$
$$504$$ −13.3944 −0.596636
$$505$$ 0 0
$$506$$ 9.00000 0.400099
$$507$$ − 2.33053i − 0.103503i
$$508$$ − 4.78890i − 0.212473i
$$509$$ 25.8167 1.14430 0.572152 0.820148i $$-0.306109\pi$$
0.572152 + 0.820148i $$0.306109\pi$$
$$510$$ 0 0
$$511$$ −40.0555 −1.77195
$$512$$ 1.00000i 0.0441942i
$$513$$ 3.57779i 0.157964i
$$514$$ −11.2111 −0.494501
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ 3.39445i 0.149288i
$$518$$ − 4.60555i − 0.202356i
$$519$$ −7.02776 −0.308484
$$520$$ 0 0
$$521$$ −9.63331 −0.422043 −0.211021 0.977481i $$-0.567679\pi$$
−0.211021 + 0.977481i $$0.567679\pi$$
$$522$$ − 20.0917i − 0.879389i
$$523$$ 32.2389i 1.40971i 0.709353 + 0.704853i $$0.248987\pi$$
−0.709353 + 0.704853i $$0.751013\pi$$
$$524$$ −3.39445 −0.148287
$$525$$ 0 0
$$526$$ 7.81665 0.340822
$$527$$ 19.8167i 0.863227i
$$528$$ − 0.394449i − 0.0171662i
$$529$$ −24.7250 −1.07500
$$530$$ 0 0
$$531$$ −9.87217 −0.428416
$$532$$ − 9.21110i − 0.399352i
$$533$$ 2.09167i 0.0906004i
$$534$$ −1.57779 −0.0682779
$$535$$ 0 0
$$536$$ −14.5139 −0.626904
$$537$$ 2.36669i 0.102130i
$$538$$ 6.78890i 0.292690i
$$539$$ −18.5139 −0.797449
$$540$$ 0 0
$$541$$ −20.9361 −0.900113 −0.450056 0.893000i $$-0.648596\pi$$
−0.450056 + 0.893000i $$0.648596\pi$$
$$542$$ 6.42221i 0.275857i
$$543$$ − 6.05551i − 0.259867i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ −3.21110 −0.137423
$$547$$ 13.3944i 0.572705i 0.958124 + 0.286353i $$0.0924429\pi$$
−0.958124 + 0.286353i $$0.907557\pi$$
$$548$$ − 9.90833i − 0.423263i
$$549$$ −30.5778 −1.30503
$$550$$ 0 0
$$551$$ 13.8167 0.588609
$$552$$ 2.09167i 0.0890275i
$$553$$ − 74.2389i − 3.15696i
$$554$$ −25.1194 −1.06722
$$555$$ 0 0
$$556$$ 8.90833 0.377797
$$557$$ 6.51388i 0.276002i 0.990432 + 0.138001i $$0.0440677\pi$$
−0.990432 + 0.138001i $$0.955932\pi$$
$$558$$ 9.60555i 0.406635i
$$559$$ −15.2111 −0.643361
$$560$$ 0 0
$$561$$ 2.36669 0.0999218
$$562$$ − 12.0000i − 0.506189i
$$563$$ 44.0555i 1.85672i 0.371684 + 0.928359i $$0.378780\pi$$
−0.371684 + 0.928359i $$0.621220\pi$$
$$564$$ −0.788897 −0.0332186
$$565$$ 0 0
$$566$$ −17.3944 −0.731143
$$567$$ − 37.6888i − 1.58278i
$$568$$ − 6.00000i − 0.251754i
$$569$$ 10.4222 0.436922 0.218461 0.975846i $$-0.429896\pi$$
0.218461 + 0.975846i $$0.429896\pi$$
$$570$$ 0 0
$$571$$ −20.3028 −0.849645 −0.424822 0.905277i $$-0.639663\pi$$
−0.424822 + 0.905277i $$0.639663\pi$$
$$572$$ 3.00000i 0.125436i
$$573$$ − 3.78890i − 0.158283i
$$574$$ −4.18335 −0.174609
$$575$$ 0 0
$$576$$ −2.90833 −0.121180
$$577$$ 28.2389i 1.17560i 0.809007 + 0.587800i $$0.200006\pi$$
−0.809007 + 0.587800i $$0.799994\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ −1.21110 −0.0503317
$$580$$ 0 0
$$581$$ 79.2666 3.28853
$$582$$ − 3.76114i − 0.155904i
$$583$$ − 7.81665i − 0.323733i
$$584$$ −8.69722 −0.359894
$$585$$ 0 0
$$586$$ 25.0278 1.03389
$$587$$ 2.36669i 0.0976838i 0.998807 + 0.0488419i $$0.0155531\pi$$
−0.998807 + 0.0488419i $$0.984447\pi$$
$$588$$ − 4.30278i − 0.177443i
$$589$$ −6.60555 −0.272177
$$590$$ 0 0
$$591$$ 1.81665 0.0747272
$$592$$ − 1.00000i − 0.0410997i
$$593$$ 36.5139i 1.49945i 0.661752 + 0.749723i $$0.269813\pi$$
−0.661752 + 0.749723i $$0.730187\pi$$
$$594$$ 2.33053 0.0956229
$$595$$ 0 0
$$596$$ −1.81665 −0.0744130
$$597$$ − 0.733385i − 0.0300154i
$$598$$ − 15.9083i − 0.650540i
$$599$$ 35.2111 1.43869 0.719343 0.694655i $$-0.244443\pi$$
0.719343 + 0.694655i $$0.244443\pi$$
$$600$$ 0 0
$$601$$ −20.6972 −0.844257 −0.422129 0.906536i $$-0.638717\pi$$
−0.422129 + 0.906536i $$0.638717\pi$$
$$602$$ − 30.4222i − 1.23992i
$$603$$ − 42.2111i − 1.71897i
$$604$$ 13.3944 0.545012
$$605$$ 0 0
$$606$$ 4.97224 0.201984
$$607$$ 31.5139i 1.27911i 0.768746 + 0.639554i $$0.220881\pi$$
−0.768746 + 0.639554i $$0.779119\pi$$
$$608$$ − 2.00000i − 0.0811107i
$$609$$ 9.63331 0.390361
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ − 17.4500i − 0.705373i
$$613$$ − 8.18335i − 0.330522i −0.986250 0.165261i $$-0.947153\pi$$
0.986250 0.165261i $$-0.0528467\pi$$
$$614$$ 7.09167 0.286197
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ − 47.5694i − 1.91507i −0.288314 0.957536i $$-0.593095\pi$$
0.288314 0.957536i $$-0.406905\pi$$
$$618$$ 1.00000i 0.0402259i
$$619$$ 2.69722 0.108411 0.0542053 0.998530i $$-0.482737\pi$$
0.0542053 + 0.998530i $$0.482737\pi$$
$$620$$ 0 0
$$621$$ −12.3583 −0.495921
$$622$$ 5.09167i 0.204157i
$$623$$ 24.0000i 0.961540i
$$624$$ −0.697224 −0.0279113
$$625$$ 0 0
$$626$$ −27.0278 −1.08025
$$627$$ 0.788897i 0.0315055i
$$628$$ 7.21110i 0.287754i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 18.3028 0.728622 0.364311 0.931277i $$-0.381305\pi$$
0.364311 + 0.931277i $$0.381305\pi$$
$$632$$ − 16.1194i − 0.641196i
$$633$$ − 2.02776i − 0.0805961i
$$634$$ −5.21110 −0.206959
$$635$$ 0 0
$$636$$ 1.81665 0.0720350
$$637$$ 32.7250i 1.29661i
$$638$$ − 9.00000i − 0.356313i
$$639$$ 17.4500 0.690310
$$640$$ 0 0
$$641$$ −2.48612 −0.0981959 −0.0490980 0.998794i $$-0.515635\pi$$
−0.0490980 + 0.998794i $$0.515635\pi$$
$$642$$ − 1.30278i − 0.0514165i
$$643$$ − 29.8167i − 1.17585i −0.808914 0.587927i $$-0.799944\pi$$
0.808914 0.587927i $$-0.200056\pi$$
$$644$$ 31.8167 1.25375
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ 25.9361i 1.01965i 0.860277 + 0.509826i $$0.170290\pi$$
−0.860277 + 0.509826i $$0.829710\pi$$
$$648$$ − 8.18335i − 0.321472i
$$649$$ −4.42221 −0.173587
$$650$$ 0 0
$$651$$ −4.60555 −0.180506
$$652$$ 20.4222i 0.799795i
$$653$$ 6.90833i 0.270344i 0.990822 + 0.135172i $$0.0431587\pi$$
−0.990822 + 0.135172i $$0.956841\pi$$
$$654$$ −0.605551 −0.0236789
$$655$$ 0 0
$$656$$ −0.908327 −0.0354642
$$657$$ − 25.2944i − 0.986827i
$$658$$ 12.0000i 0.467809i
$$659$$ 42.1194 1.64074 0.820370 0.571833i $$-0.193767\pi$$
0.820370 + 0.571833i $$0.193767\pi$$
$$660$$ 0 0
$$661$$ −12.4861 −0.485654 −0.242827 0.970070i $$-0.578075\pi$$
−0.242827 + 0.970070i $$0.578075\pi$$
$$662$$ 1.21110i 0.0470708i
$$663$$ − 4.18335i − 0.162468i
$$664$$ 17.2111 0.667920
$$665$$ 0 0
$$666$$ 2.90833 0.112695
$$667$$ 47.7250i 1.84792i
$$668$$ 12.5139i 0.484176i
$$669$$ 4.78890 0.185149
$$670$$ 0 0
$$671$$ −13.6972 −0.528775
$$672$$ − 1.39445i − 0.0537920i
$$673$$ 24.3028i 0.936803i 0.883516 + 0.468402i $$0.155170\pi$$
−0.883516 + 0.468402i $$0.844830\pi$$
$$674$$ −19.1194 −0.736453
$$675$$ 0 0
$$676$$ −7.69722 −0.296047
$$677$$ 36.2389i 1.39277i 0.717667 + 0.696386i $$0.245210\pi$$
−0.717667 + 0.696386i $$0.754790\pi$$
$$678$$ 3.39445i 0.130363i
$$679$$ −57.2111 −2.19556
$$680$$ 0 0
$$681$$ 2.36669 0.0906918
$$682$$ 4.30278i 0.164762i
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 5.81665 0.222405
$$685$$ 0 0
$$686$$ −33.2111 −1.26801
$$687$$ 5.26662i 0.200934i
$$688$$ − 6.60555i − 0.251834i
$$689$$ −13.8167 −0.526373
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 23.2111i 0.882354i
$$693$$ − 17.4500i − 0.662869i
$$694$$ 31.8167 1.20774
$$695$$ 0 0
$$696$$ 2.09167 0.0792847
$$697$$ − 5.44996i − 0.206432i
$$698$$ 22.2389i 0.841753i
$$699$$ −2.88057 −0.108953
$$700$$ 0 0
$$701$$ 14.8806 0.562031 0.281016 0.959703i $$-0.409329\pi$$
0.281016 + 0.959703i $$0.409329\pi$$
$$702$$ − 4.11943i − 0.155478i
$$703$$ 2.00000i 0.0754314i
$$704$$ −1.30278 −0.0491002
$$705$$ 0 0
$$706$$ −31.8167 −1.19744
$$707$$ − 75.6333i − 2.84448i
$$708$$ − 1.02776i − 0.0386254i
$$709$$ 1.66947 0.0626982 0.0313491 0.999508i $$-0.490020\pi$$
0.0313491 + 0.999508i $$0.490020\pi$$
$$710$$ 0 0
$$711$$ 46.8806 1.75816
$$712$$ 5.21110i 0.195294i
$$713$$ − 22.8167i − 0.854490i
$$714$$ 8.36669 0.313116
$$715$$ 0 0
$$716$$ 7.81665 0.292122
$$717$$ 0.155590i 0.00581061i
$$718$$ 11.2111i 0.418395i
$$719$$ 8.36669 0.312025 0.156012 0.987755i $$-0.450136\pi$$
0.156012 + 0.987755i $$0.450136\pi$$
$$720$$ 0 0
$$721$$ 15.2111 0.566491
$$722$$ − 15.0000i − 0.558242i
$$723$$ − 2.42221i − 0.0900828i
$$724$$ −20.0000 −0.743294
$$725$$ 0 0
$$726$$ −2.81665 −0.104536
$$727$$ − 29.9083i − 1.10924i −0.832104 0.554619i $$-0.812864\pi$$
0.832104 0.554619i $$-0.187136\pi$$
$$728$$ 10.6056i 0.393068i
$$729$$ 22.1749 0.821294
$$730$$ 0 0
$$731$$ 39.6333 1.46589
$$732$$ − 3.18335i − 0.117660i
$$733$$ 29.6333i 1.09453i 0.836959 + 0.547266i $$0.184331\pi$$
−0.836959 + 0.547266i $$0.815669\pi$$
$$734$$ −17.8167 −0.657625
$$735$$ 0 0
$$736$$ 6.90833 0.254644
$$737$$ − 18.9083i − 0.696497i
$$738$$ − 2.64171i − 0.0972427i
$$739$$ 42.3305 1.55715 0.778577 0.627549i $$-0.215942\pi$$
0.778577 + 0.627549i $$0.215942\pi$$
$$740$$ 0 0
$$741$$ 1.39445 0.0512264
$$742$$ − 27.6333i − 1.01445i
$$743$$ 35.4500i 1.30053i 0.759706 + 0.650266i $$0.225343\pi$$
−0.759706 + 0.650266i $$0.774657\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ −3.81665 −0.139738
$$747$$ 50.0555i 1.83144i
$$748$$ − 7.81665i − 0.285805i
$$749$$ −19.8167 −0.724085
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 2.60555i 0.0950147i
$$753$$ 2.05551i 0.0749070i
$$754$$ −15.9083 −0.579347
$$755$$ 0 0
$$756$$ 8.23886 0.299644
$$757$$ − 9.30278i − 0.338115i −0.985606 0.169058i $$-0.945928\pi$$
0.985606 0.169058i $$-0.0540724\pi$$
$$758$$ 15.3305i 0.556830i
$$759$$ −2.72498 −0.0989105
$$760$$ 0 0
$$761$$ 42.1194 1.52683 0.763414 0.645909i $$-0.223522\pi$$
0.763414 + 0.645909i $$0.223522\pi$$
$$762$$ 1.44996i 0.0525266i
$$763$$ 9.21110i 0.333464i
$$764$$ −12.5139 −0.452736
$$765$$ 0 0
$$766$$ −20.8444 −0.753139
$$767$$ 7.81665i 0.282243i
$$768$$ − 0.302776i − 0.0109255i
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ 3.39445 0.122248
$$772$$ 4.00000i 0.143963i
$$773$$ − 50.0555i − 1.80037i −0.435506 0.900186i $$-0.643431\pi$$
0.435506 0.900186i $$-0.356569\pi$$
$$774$$ 19.2111 0.690529
$$775$$ 0 0
$$776$$ −12.4222 −0.445931
$$777$$ 1.39445i 0.0500256i
$$778$$ 11.8806i 0.425939i
$$779$$ 1.81665 0.0650884
$$780$$ 0 0
$$781$$ 7.81665 0.279702
$$782$$ 41.4500i 1.48225i
$$783$$ 12.3583i 0.441649i
$$784$$ −14.2111 −0.507539
$$785$$ 0 0
$$786$$ 1.02776 0.0366589
$$787$$ − 25.2111i − 0.898679i −0.893361 0.449339i $$-0.851659\pi$$
0.893361 0.449339i $$-0.148341\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ −2.36669 −0.0842565
$$790$$ 0 0
$$791$$ 51.6333 1.83587
$$792$$ − 3.78890i − 0.134633i
$$793$$ 24.2111i 0.859761i
$$794$$ 27.8167 0.987176
$$795$$ 0 0
$$796$$ −2.42221 −0.0858528
$$797$$ 17.3305i 0.613879i 0.951729 + 0.306939i $$0.0993049\pi$$
−0.951729 + 0.306939i $$0.900695\pi$$
$$798$$ 2.78890i 0.0987259i
$$799$$ −15.6333 −0.553067
$$800$$ 0 0
$$801$$ −15.1556 −0.535496
$$802$$ 13.8167i 0.487883i
$$803$$ − 11.3305i − 0.399846i
$$804$$ 4.39445 0.154980
$$805$$ 0 0
$$806$$ 7.60555 0.267894
$$807$$ − 2.05551i − 0.0723575i
$$808$$ − 16.4222i − 0.577731i
$$809$$ −29.4500 −1.03541 −0.517703 0.855561i $$-0.673213\pi$$
−0.517703 + 0.855561i $$0.673213\pi$$
$$810$$ 0 0
$$811$$ 54.1472 1.90136 0.950682 0.310166i $$-0.100385\pi$$
0.950682 + 0.310166i $$0.100385\pi$$
$$812$$ − 31.8167i − 1.11655i
$$813$$ − 1.94449i − 0.0681961i
$$814$$ 1.30278 0.0456623
$$815$$ 0 0
$$816$$ 1.81665 0.0635956
$$817$$ 13.2111i 0.462198i
$$818$$ 5.02776i 0.175791i
$$819$$ −30.8444 −1.07779
$$820$$ 0 0
$$821$$ 11.2111 0.391270 0.195635 0.980677i $$-0.437323\pi$$
0.195635 + 0.980677i $$0.437323\pi$$
$$822$$ 3.00000i 0.104637i
$$823$$ − 12.8444i − 0.447728i −0.974620 0.223864i $$-0.928133\pi$$
0.974620 0.223864i $$-0.0718671\pi$$
$$824$$ 3.30278 0.115058
$$825$$ 0 0
$$826$$ −15.6333 −0.543952
$$827$$ − 27.3944i − 0.952598i −0.879283 0.476299i $$-0.841978\pi$$
0.879283 0.476299i $$-0.158022\pi$$
$$828$$ 20.0917i 0.698234i
$$829$$ −4.72498 −0.164105 −0.0820527 0.996628i $$-0.526148\pi$$
−0.0820527 + 0.996628i $$0.526148\pi$$
$$830$$ 0 0
$$831$$ 7.60555 0.263834
$$832$$ 2.30278i 0.0798344i
$$833$$ − 85.2666i − 2.95431i
$$834$$ −2.69722 −0.0933972
$$835$$ 0 0
$$836$$ 2.60555 0.0901149
$$837$$ − 5.90833i − 0.204222i
$$838$$ 25.1472i 0.868695i
$$839$$ 49.0278 1.69263 0.846313 0.532686i $$-0.178817\pi$$
0.846313 + 0.532686i $$0.178817\pi$$
$$840$$ 0 0
$$841$$ 18.7250 0.645689
$$842$$ 28.7250i 0.989928i
$$843$$ 3.63331i 0.125138i
$$844$$ −6.69722 −0.230528
$$845$$ 0 0
$$846$$ −7.57779 −0.260530
$$847$$ 42.8444i 1.47215i
$$848$$ − 6.00000i − 0.206041i
$$849$$ 5.26662 0.180750
$$850$$ 0 0
$$851$$ −6.90833 −0.236814
$$852$$ 1.81665i 0.0622375i
$$853$$ − 11.5416i − 0.395178i −0.980285 0.197589i $$-0.936689\pi$$
0.980285 0.197589i $$-0.0633111\pi$$
$$854$$ −48.4222 −1.65697
$$855$$ 0 0
$$856$$ −4.30278 −0.147066
$$857$$ 14.8444i 0.507075i 0.967326 + 0.253538i $$0.0815942\pi$$
−0.967326 + 0.253538i $$0.918406\pi$$
$$858$$ − 0.908327i − 0.0310098i
$$859$$ 24.0555 0.820764 0.410382 0.911914i $$-0.365395\pi$$
0.410382 + 0.911914i $$0.365395\pi$$
$$860$$ 0 0
$$861$$ 1.26662 0.0431661
$$862$$ − 5.21110i − 0.177491i
$$863$$ 12.0000i 0.408485i 0.978920 + 0.204242i $$0.0654731\pi$$
−0.978920 + 0.204242i $$0.934527\pi$$
$$864$$ 1.78890 0.0608595
$$865$$ 0 0
$$866$$ 11.9361 0.405605
$$867$$ 5.75274i 0.195373i
$$868$$ 15.2111i 0.516298i
$$869$$ 21.0000 0.712376
$$870$$ 0 0
$$871$$ −33.4222 −1.13247
$$872$$ 2.00000i 0.0677285i
$$873$$ − 36.1278i − 1.22274i
$$874$$ −13.8167 −0.467355
$$875$$ 0 0
$$876$$ 2.63331 0.0889712
$$877$$ − 7.21110i − 0.243502i −0.992561 0.121751i $$-0.961149\pi$$
0.992561 0.121751i $$-0.0388509\pi$$
$$878$$ 9.33053i 0.314890i
$$879$$ −7.57779 −0.255593
$$880$$ 0 0
$$881$$ 25.5416 0.860520 0.430260 0.902705i $$-0.358422\pi$$
0.430260 + 0.902705i $$0.358422\pi$$
$$882$$ − 41.3305i − 1.39167i
$$883$$ − 2.42221i − 0.0815137i −0.999169 0.0407568i $$-0.987023\pi$$
0.999169 0.0407568i $$-0.0129769\pi$$
$$884$$ −13.8167 −0.464704
$$885$$ 0 0
$$886$$ −0.275019 −0.00923945
$$887$$ − 28.4222i − 0.954324i −0.878815 0.477162i $$-0.841665\pi$$
0.878815 0.477162i $$-0.158335\pi$$
$$888$$ 0.302776i 0.0101605i
$$889$$ 22.0555 0.739718
$$890$$ 0 0
$$891$$ 10.6611 0.357159
$$892$$ − 15.8167i − 0.529581i
$$893$$ − 5.21110i − 0.174383i
$$894$$ 0.550039 0.0183960
$$895$$ 0 0
$$896$$ −4.60555 −0.153861
$$897$$ 4.81665i 0.160823i
$$898$$ 0.788897i 0.0263258i
$$899$$ −22.8167 −0.760978
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ − 1.18335i − 0.0394011i
$$903$$ 9.21110i 0.306526i
$$904$$ 11.2111 0.372876
$$905$$ 0 0
$$906$$ −4.05551 −0.134735
$$907$$ − 26.0000i − 0.863316i −0.902037 0.431658i $$-0.857929\pi$$
0.902037 0.431658i $$-0.142071\pi$$
$$908$$ − 7.81665i − 0.259405i
$$909$$ 47.7611 1.58414
$$910$$ 0 0
$$911$$ −46.4222 −1.53804 −0.769018 0.639227i $$-0.779254\pi$$
−0.769018 + 0.639227i $$0.779254\pi$$
$$912$$ 0.605551i 0.0200518i
$$913$$ 22.4222i 0.742067i
$$914$$ 4.60555 0.152338
$$915$$ 0 0
$$916$$ 17.3944 0.574729
$$917$$ − 15.6333i − 0.516257i
$$918$$ 10.7334i 0.354254i
$$919$$ 38.4222 1.26743 0.633716 0.773566i $$-0.281529\pi$$
0.633716 + 0.773566i $$0.281529\pi$$
$$920$$ 0 0
$$921$$ −2.14719 −0.0707522
$$922$$ − 16.4222i − 0.540837i
$$923$$ − 13.8167i − 0.454781i
$$924$$ 1.81665 0.0597635
$$925$$ 0 0
$$926$$ −30.3028 −0.995811
$$927$$ 9.60555i 0.315488i
$$928$$ − 6.90833i − 0.226777i
$$929$$ 36.5139 1.19798 0.598991 0.800756i $$-0.295569\pi$$
0.598991 + 0.800756i $$0.295569\pi$$
$$930$$ 0 0
$$931$$ 28.4222 0.931500
$$932$$ 9.51388i 0.311637i
$$933$$ − 1.54163i − 0.0504708i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ −6.69722 −0.218906
$$937$$ 28.9083i 0.944394i 0.881493 + 0.472197i $$0.156539\pi$$
−0.881493 + 0.472197i $$0.843461\pi$$
$$938$$ − 66.8444i − 2.18255i
$$939$$ 8.18335 0.267053
$$940$$ 0 0
$$941$$ −7.81665 −0.254816 −0.127408 0.991850i $$-0.540666\pi$$
−0.127408 + 0.991850i $$0.540666\pi$$
$$942$$ − 2.18335i − 0.0711373i
$$943$$ 6.27502i 0.204343i
$$944$$ −3.39445 −0.110480
$$945$$ 0 0
$$946$$ 8.60555 0.279791
$$947$$ − 39.6333i − 1.28791i −0.765064 0.643955i $$-0.777293\pi$$
0.765064 0.643955i $$-0.222707\pi$$
$$948$$ 4.88057i 0.158514i
$$949$$ −20.0278 −0.650128
$$950$$ 0 0
$$951$$ 1.57779 0.0511635
$$952$$ − 27.6333i − 0.895601i
$$953$$ − 18.7527i − 0.607461i −0.952758 0.303730i $$-0.901768\pi$$
0.952758 0.303730i $$-0.0982322\pi$$
$$954$$ 17.4500 0.564963
$$955$$ 0 0
$$956$$ 0.513878 0.0166200
$$957$$ 2.72498i 0.0880861i
$$958$$ − 12.1194i − 0.391561i
$$959$$ 45.6333 1.47358
$$960$$ 0 0
$$961$$ −20.0917 −0.648118
$$962$$ − 2.30278i − 0.0742445i
$$963$$ − 12.5139i − 0.403254i
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ −9.63331 −0.309947
$$967$$ − 25.7250i − 0.827260i −0.910445 0.413630i $$-0.864261\pi$$
0.910445 0.413630i $$-0.135739\pi$$
$$968$$ 9.30278i 0.299003i
$$969$$ −3.63331 −0.116719
$$970$$ 0 0
$$971$$ 31.5416 1.01222 0.506110 0.862469i $$-0.331083\pi$$
0.506110 + 0.862469i $$0.331083\pi$$
$$972$$ 7.84441i 0.251610i
$$973$$ 41.0278i 1.31529i
$$974$$ −22.7889 −0.730203
$$975$$ 0 0
$$976$$ −10.5139 −0.336541
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 6.18335i − 0.197722i
$$979$$ −6.78890 −0.216974
$$980$$ 0 0
$$981$$ −5.81665 −0.185711
$$982$$ − 14.7250i − 0.469893i
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ 0.275019 0.00876729
$$985$$ 0 0
$$986$$ 41.4500 1.32004
$$987$$ − 3.63331i − 0.115649i
$$988$$ − 4.60555i − 0.146522i
$$989$$ −45.6333 −1.45105
$$990$$ 0 0
$$991$$ 54.3028 1.72498 0.862492 0.506070i $$-0.168902\pi$$
0.862492 + 0.506070i $$0.168902\pi$$
$$992$$ 3.30278i 0.104863i
$$993$$ − 0.366692i − 0.0116366i
$$994$$ 27.6333 0.876475
$$995$$ 0 0
$$996$$ −5.21110 −0.165120
$$997$$ 23.5778i 0.746716i 0.927687 + 0.373358i $$0.121794\pi$$
−0.927687 + 0.373358i $$0.878206\pi$$
$$998$$ − 8.23886i − 0.260797i
$$999$$ −1.78890 −0.0565982
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 1850.2.b.i.149.3 4
5.2 odd 4 74.2.a.a.1.1 2
5.3 odd 4 1850.2.a.u.1.2 2
5.4 even 2 inner 1850.2.b.i.149.2 4
15.2 even 4 666.2.a.j.1.1 2
20.7 even 4 592.2.a.f.1.2 2
35.27 even 4 3626.2.a.a.1.2 2
40.27 even 4 2368.2.a.ba.1.1 2
40.37 odd 4 2368.2.a.s.1.2 2
55.32 even 4 8954.2.a.p.1.1 2
60.47 odd 4 5328.2.a.bf.1.1 2
185.147 odd 4 2738.2.a.l.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 5.2 odd 4
592.2.a.f.1.2 2 20.7 even 4
666.2.a.j.1.1 2 15.2 even 4
1850.2.a.u.1.2 2 5.3 odd 4
1850.2.b.i.149.2 4 5.4 even 2 inner
1850.2.b.i.149.3 4 1.1 even 1 trivial
2368.2.a.s.1.2 2 40.37 odd 4
2368.2.a.ba.1.1 2 40.27 even 4
2738.2.a.l.1.1 2 185.147 odd 4
3626.2.a.a.1.2 2 35.27 even 4
5328.2.a.bf.1.1 2 60.47 odd 4
8954.2.a.p.1.1 2 55.32 even 4