# Properties

 Label 2175.2.c.f.349.3 Level $2175$ Weight $2$ Character 2175.349 Analytic conductor $17.367$ 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] = [2175,2,Mod(349,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.3 Root $$1.79129i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.f.349.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.79129i q^{2} -1.00000i q^{3} -1.20871 q^{4} +1.79129 q^{6} +1.00000i q^{7} +1.41742i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.79129i q^{2} -1.00000i q^{3} -1.20871 q^{4} +1.79129 q^{6} +1.00000i q^{7} +1.41742i q^{8} -1.00000 q^{9} +5.00000 q^{11} +1.20871i q^{12} +4.58258i q^{13} -1.79129 q^{14} -4.95644 q^{16} -3.00000i q^{17} -1.79129i q^{18} -3.58258 q^{19} +1.00000 q^{21} +8.95644i q^{22} +4.00000i q^{23} +1.41742 q^{24} -8.20871 q^{26} +1.00000i q^{27} -1.20871i q^{28} -1.00000 q^{29} +4.00000 q^{31} -6.04356i q^{32} -5.00000i q^{33} +5.37386 q^{34} +1.20871 q^{36} -4.00000i q^{37} -6.41742i q^{38} +4.58258 q^{39} -9.16515 q^{41} +1.79129i q^{42} +9.58258i q^{43} -6.04356 q^{44} -7.16515 q^{46} +10.5826i q^{47} +4.95644i q^{48} +6.00000 q^{49} -3.00000 q^{51} -5.53901i q^{52} -0.417424i q^{53} -1.79129 q^{54} -1.41742 q^{56} +3.58258i q^{57} -1.79129i q^{58} +7.58258 q^{59} +12.7477 q^{61} +7.16515i q^{62} -1.00000i q^{63} +0.912878 q^{64} +8.95644 q^{66} -4.16515i q^{67} +3.62614i q^{68} +4.00000 q^{69} -9.58258 q^{71} -1.41742i q^{72} -4.00000i q^{73} +7.16515 q^{74} +4.33030 q^{76} +5.00000i q^{77} +8.20871i q^{78} -7.58258 q^{79} +1.00000 q^{81} -16.4174i q^{82} +11.5826i q^{83} -1.20871 q^{84} -17.1652 q^{86} +1.00000i q^{87} +7.08712i q^{88} -1.41742 q^{89} -4.58258 q^{91} -4.83485i q^{92} -4.00000i q^{93} -18.9564 q^{94} -6.04356 q^{96} +11.5826i q^{97} +10.7477i q^{98} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 14 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 14 * q^4 - 2 * q^6 - 4 * q^9 $$4 q - 14 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} + 2 q^{14} + 26 q^{16} + 4 q^{19} + 4 q^{21} + 24 q^{24} - 42 q^{26} - 4 q^{29} + 16 q^{31} - 6 q^{34} + 14 q^{36} - 70 q^{44} + 8 q^{46} + 24 q^{49} - 12 q^{51} + 2 q^{54} - 24 q^{56} + 12 q^{59} - 4 q^{61} - 88 q^{64} - 10 q^{66} + 16 q^{69} - 20 q^{71} - 8 q^{74} - 56 q^{76} - 12 q^{79} + 4 q^{81} - 14 q^{84} - 32 q^{86} - 24 q^{89} - 30 q^{94} - 70 q^{96} - 20 q^{99}+O(q^{100})$$ 4 * q - 14 * q^4 - 2 * q^6 - 4 * q^9 + 20 * q^11 + 2 * q^14 + 26 * q^16 + 4 * q^19 + 4 * q^21 + 24 * q^24 - 42 * q^26 - 4 * q^29 + 16 * q^31 - 6 * q^34 + 14 * q^36 - 70 * q^44 + 8 * q^46 + 24 * q^49 - 12 * q^51 + 2 * q^54 - 24 * q^56 + 12 * q^59 - 4 * q^61 - 88 * q^64 - 10 * q^66 + 16 * q^69 - 20 * q^71 - 8 * q^74 - 56 * q^76 - 12 * q^79 + 4 * q^81 - 14 * q^84 - 32 * q^86 - 24 * q^89 - 30 * q^94 - 70 * q^96 - 20 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.79129i 1.26663i 0.773893 + 0.633316i $$0.218307\pi$$
−0.773893 + 0.633316i $$0.781693\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.20871 −0.604356
$$5$$ 0 0
$$6$$ 1.79129 0.731290
$$7$$ 1.00000i 0.377964i 0.981981 + 0.188982i $$0.0605189\pi$$
−0.981981 + 0.188982i $$0.939481\pi$$
$$8$$ 1.41742i 0.501135i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 1.20871i 0.348925i
$$13$$ 4.58258i 1.27098i 0.772110 + 0.635489i $$0.219201\pi$$
−0.772110 + 0.635489i $$0.780799\pi$$
$$14$$ −1.79129 −0.478742
$$15$$ 0 0
$$16$$ −4.95644 −1.23911
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ − 1.79129i − 0.422211i
$$19$$ −3.58258 −0.821899 −0.410950 0.911658i $$-0.634803\pi$$
−0.410950 + 0.911658i $$0.634803\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 8.95644i 1.90952i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 1.41742 0.289331
$$25$$ 0 0
$$26$$ −8.20871 −1.60986
$$27$$ 1.00000i 0.192450i
$$28$$ − 1.20871i − 0.228425i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ − 6.04356i − 1.06836i
$$33$$ − 5.00000i − 0.870388i
$$34$$ 5.37386 0.921610
$$35$$ 0 0
$$36$$ 1.20871 0.201452
$$37$$ − 4.00000i − 0.657596i −0.944400 0.328798i $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ − 6.41742i − 1.04104i
$$39$$ 4.58258 0.733799
$$40$$ 0 0
$$41$$ −9.16515 −1.43136 −0.715678 0.698430i $$-0.753882\pi$$
−0.715678 + 0.698430i $$0.753882\pi$$
$$42$$ 1.79129i 0.276402i
$$43$$ 9.58258i 1.46133i 0.682737 + 0.730665i $$0.260790\pi$$
−0.682737 + 0.730665i $$0.739210\pi$$
$$44$$ −6.04356 −0.911101
$$45$$ 0 0
$$46$$ −7.16515 −1.05644
$$47$$ 10.5826i 1.54363i 0.635849 + 0.771814i $$0.280650\pi$$
−0.635849 + 0.771814i $$0.719350\pi$$
$$48$$ 4.95644i 0.715400i
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ − 5.53901i − 0.768123i
$$53$$ − 0.417424i − 0.0573376i −0.999589 0.0286688i $$-0.990873\pi$$
0.999589 0.0286688i $$-0.00912682\pi$$
$$54$$ −1.79129 −0.243763
$$55$$ 0 0
$$56$$ −1.41742 −0.189411
$$57$$ 3.58258i 0.474524i
$$58$$ − 1.79129i − 0.235208i
$$59$$ 7.58258 0.987167 0.493584 0.869698i $$-0.335687\pi$$
0.493584 + 0.869698i $$0.335687\pi$$
$$60$$ 0 0
$$61$$ 12.7477 1.63218 0.816090 0.577925i $$-0.196138\pi$$
0.816090 + 0.577925i $$0.196138\pi$$
$$62$$ 7.16515i 0.909975i
$$63$$ − 1.00000i − 0.125988i
$$64$$ 0.912878 0.114110
$$65$$ 0 0
$$66$$ 8.95644 1.10246
$$67$$ − 4.16515i − 0.508854i −0.967092 0.254427i $$-0.918113\pi$$
0.967092 0.254427i $$-0.0818869\pi$$
$$68$$ 3.62614i 0.439734i
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −9.58258 −1.13724 −0.568621 0.822599i $$-0.692523\pi$$
−0.568621 + 0.822599i $$0.692523\pi$$
$$72$$ − 1.41742i − 0.167045i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ 7.16515 0.832932
$$75$$ 0 0
$$76$$ 4.33030 0.496720
$$77$$ 5.00000i 0.569803i
$$78$$ 8.20871i 0.929454i
$$79$$ −7.58258 −0.853106 −0.426553 0.904462i $$-0.640272\pi$$
−0.426553 + 0.904462i $$0.640272\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 16.4174i − 1.81300i
$$83$$ 11.5826i 1.27135i 0.771956 + 0.635676i $$0.219279\pi$$
−0.771956 + 0.635676i $$0.780721\pi$$
$$84$$ −1.20871 −0.131881
$$85$$ 0 0
$$86$$ −17.1652 −1.85097
$$87$$ 1.00000i 0.107211i
$$88$$ 7.08712i 0.755490i
$$89$$ −1.41742 −0.150247 −0.0751233 0.997174i $$-0.523935\pi$$
−0.0751233 + 0.997174i $$0.523935\pi$$
$$90$$ 0 0
$$91$$ −4.58258 −0.480384
$$92$$ − 4.83485i − 0.504068i
$$93$$ − 4.00000i − 0.414781i
$$94$$ −18.9564 −1.95521
$$95$$ 0 0
$$96$$ −6.04356 −0.616818
$$97$$ 11.5826i 1.17603i 0.808849 + 0.588016i $$0.200091\pi$$
−0.808849 + 0.588016i $$0.799909\pi$$
$$98$$ 10.7477i 1.08568i
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ −0.582576 −0.0579684 −0.0289842 0.999580i $$-0.509227\pi$$
−0.0289842 + 0.999580i $$0.509227\pi$$
$$102$$ − 5.37386i − 0.532092i
$$103$$ − 15.1652i − 1.49427i −0.664674 0.747133i $$-0.731430\pi$$
0.664674 0.747133i $$-0.268570\pi$$
$$104$$ −6.49545 −0.636932
$$105$$ 0 0
$$106$$ 0.747727 0.0726257
$$107$$ 5.16515i 0.499334i 0.968332 + 0.249667i $$0.0803212\pi$$
−0.968332 + 0.249667i $$0.919679\pi$$
$$108$$ − 1.20871i − 0.116308i
$$109$$ −14.1652 −1.35678 −0.678388 0.734704i $$-0.737321\pi$$
−0.678388 + 0.734704i $$0.737321\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ − 4.95644i − 0.468339i
$$113$$ 14.1652i 1.33255i 0.745708 + 0.666273i $$0.232111\pi$$
−0.745708 + 0.666273i $$0.767889\pi$$
$$114$$ −6.41742 −0.601047
$$115$$ 0 0
$$116$$ 1.20871 0.112226
$$117$$ − 4.58258i − 0.423659i
$$118$$ 13.5826i 1.25038i
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 22.8348i 2.06737i
$$123$$ 9.16515i 0.826394i
$$124$$ −4.83485 −0.434182
$$125$$ 0 0
$$126$$ 1.79129 0.159581
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ − 10.4519i − 0.923826i
$$129$$ 9.58258 0.843699
$$130$$ 0 0
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ 6.04356i 0.526024i
$$133$$ − 3.58258i − 0.310649i
$$134$$ 7.46099 0.644531
$$135$$ 0 0
$$136$$ 4.25227 0.364629
$$137$$ 16.3303i 1.39519i 0.716491 + 0.697596i $$0.245747\pi$$
−0.716491 + 0.697596i $$0.754253\pi$$
$$138$$ 7.16515i 0.609938i
$$139$$ 9.41742 0.798776 0.399388 0.916782i $$-0.369223\pi$$
0.399388 + 0.916782i $$0.369223\pi$$
$$140$$ 0 0
$$141$$ 10.5826 0.891214
$$142$$ − 17.1652i − 1.44047i
$$143$$ 22.9129i 1.91607i
$$144$$ 4.95644 0.413037
$$145$$ 0 0
$$146$$ 7.16515 0.592992
$$147$$ − 6.00000i − 0.494872i
$$148$$ 4.83485i 0.397422i
$$149$$ −16.7477 −1.37203 −0.686014 0.727589i $$-0.740641\pi$$
−0.686014 + 0.727589i $$0.740641\pi$$
$$150$$ 0 0
$$151$$ 7.16515 0.583092 0.291546 0.956557i $$-0.405830\pi$$
0.291546 + 0.956557i $$0.405830\pi$$
$$152$$ − 5.07803i − 0.411883i
$$153$$ 3.00000i 0.242536i
$$154$$ −8.95644 −0.721730
$$155$$ 0 0
$$156$$ −5.53901 −0.443476
$$157$$ 10.7477i 0.857762i 0.903361 + 0.428881i $$0.141092\pi$$
−0.903361 + 0.428881i $$0.858908\pi$$
$$158$$ − 13.5826i − 1.08057i
$$159$$ −0.417424 −0.0331039
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 1.79129i 0.140737i
$$163$$ − 7.58258i − 0.593913i −0.954891 0.296957i $$-0.904028\pi$$
0.954891 0.296957i $$-0.0959717\pi$$
$$164$$ 11.0780 0.865049
$$165$$ 0 0
$$166$$ −20.7477 −1.61034
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 1.41742i 0.109357i
$$169$$ −8.00000 −0.615385
$$170$$ 0 0
$$171$$ 3.58258 0.273966
$$172$$ − 11.5826i − 0.883163i
$$173$$ 3.16515i 0.240642i 0.992735 + 0.120321i $$0.0383924\pi$$
−0.992735 + 0.120321i $$0.961608\pi$$
$$174$$ −1.79129 −0.135797
$$175$$ 0 0
$$176$$ −24.7822 −1.86803
$$177$$ − 7.58258i − 0.569941i
$$178$$ − 2.53901i − 0.190307i
$$179$$ −4.74773 −0.354862 −0.177431 0.984133i $$-0.556779\pi$$
−0.177431 + 0.984133i $$0.556779\pi$$
$$180$$ 0 0
$$181$$ 16.1652 1.20155 0.600773 0.799420i $$-0.294860\pi$$
0.600773 + 0.799420i $$0.294860\pi$$
$$182$$ − 8.20871i − 0.608470i
$$183$$ − 12.7477i − 0.942339i
$$184$$ −5.66970 −0.417976
$$185$$ 0 0
$$186$$ 7.16515 0.525374
$$187$$ − 15.0000i − 1.09691i
$$188$$ − 12.7913i − 0.932901i
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ − 0.912878i − 0.0658813i
$$193$$ 20.3303i 1.46341i 0.681623 + 0.731704i $$0.261274\pi$$
−0.681623 + 0.731704i $$0.738726\pi$$
$$194$$ −20.7477 −1.48960
$$195$$ 0 0
$$196$$ −7.25227 −0.518019
$$197$$ − 16.3303i − 1.16349i −0.813373 0.581743i $$-0.802371\pi$$
0.813373 0.581743i $$-0.197629\pi$$
$$198$$ − 8.95644i − 0.636506i
$$199$$ −13.4174 −0.951136 −0.475568 0.879679i $$-0.657757\pi$$
−0.475568 + 0.879679i $$0.657757\pi$$
$$200$$ 0 0
$$201$$ −4.16515 −0.293787
$$202$$ − 1.04356i − 0.0734247i
$$203$$ − 1.00000i − 0.0701862i
$$204$$ 3.62614 0.253880
$$205$$ 0 0
$$206$$ 27.1652 1.89269
$$207$$ − 4.00000i − 0.278019i
$$208$$ − 22.7133i − 1.57488i
$$209$$ −17.9129 −1.23906
$$210$$ 0 0
$$211$$ 20.3303 1.39960 0.699798 0.714341i $$-0.253273\pi$$
0.699798 + 0.714341i $$0.253273\pi$$
$$212$$ 0.504546i 0.0346523i
$$213$$ 9.58258i 0.656587i
$$214$$ −9.25227 −0.632472
$$215$$ 0 0
$$216$$ −1.41742 −0.0964435
$$217$$ 4.00000i 0.271538i
$$218$$ − 25.3739i − 1.71853i
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ 13.7477 0.924772
$$222$$ − 7.16515i − 0.480893i
$$223$$ − 7.00000i − 0.468755i −0.972146 0.234377i $$-0.924695\pi$$
0.972146 0.234377i $$-0.0753051\pi$$
$$224$$ 6.04356 0.403802
$$225$$ 0 0
$$226$$ −25.3739 −1.68784
$$227$$ − 7.58258i − 0.503273i −0.967822 0.251637i $$-0.919031\pi$$
0.967822 0.251637i $$-0.0809688\pi$$
$$228$$ − 4.33030i − 0.286781i
$$229$$ 17.1652 1.13431 0.567153 0.823613i $$-0.308045\pi$$
0.567153 + 0.823613i $$0.308045\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ − 1.41742i − 0.0930585i
$$233$$ 13.1652i 0.862478i 0.902238 + 0.431239i $$0.141923\pi$$
−0.902238 + 0.431239i $$0.858077\pi$$
$$234$$ 8.20871 0.536620
$$235$$ 0 0
$$236$$ −9.16515 −0.596601
$$237$$ 7.58258i 0.492541i
$$238$$ 5.37386i 0.348336i
$$239$$ 26.0000 1.68180 0.840900 0.541190i $$-0.182026\pi$$
0.840900 + 0.541190i $$0.182026\pi$$
$$240$$ 0 0
$$241$$ 29.3303 1.88933 0.944665 0.328035i $$-0.106387\pi$$
0.944665 + 0.328035i $$0.106387\pi$$
$$242$$ 25.0780i 1.61208i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −15.4083 −0.986417
$$245$$ 0 0
$$246$$ −16.4174 −1.04674
$$247$$ − 16.4174i − 1.04462i
$$248$$ 5.66970i 0.360026i
$$249$$ 11.5826 0.734016
$$250$$ 0 0
$$251$$ 16.1652 1.02034 0.510168 0.860075i $$-0.329583\pi$$
0.510168 + 0.860075i $$0.329583\pi$$
$$252$$ 1.20871i 0.0761417i
$$253$$ 20.0000i 1.25739i
$$254$$ −3.58258 −0.224791
$$255$$ 0 0
$$256$$ 20.5481 1.28426
$$257$$ 12.7477i 0.795181i 0.917563 + 0.397591i $$0.130154\pi$$
−0.917563 + 0.397591i $$0.869846\pi$$
$$258$$ 17.1652i 1.06866i
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ − 26.8693i − 1.65999i
$$263$$ − 30.3303i − 1.87025i −0.354322 0.935123i $$-0.615288\pi$$
0.354322 0.935123i $$-0.384712\pi$$
$$264$$ 7.08712 0.436182
$$265$$ 0 0
$$266$$ 6.41742 0.393478
$$267$$ 1.41742i 0.0867450i
$$268$$ 5.03447i 0.307529i
$$269$$ −22.5826 −1.37688 −0.688442 0.725291i $$-0.741705\pi$$
−0.688442 + 0.725291i $$0.741705\pi$$
$$270$$ 0 0
$$271$$ 1.16515 0.0707779 0.0353890 0.999374i $$-0.488733\pi$$
0.0353890 + 0.999374i $$0.488733\pi$$
$$272$$ 14.8693i 0.901585i
$$273$$ 4.58258i 0.277350i
$$274$$ −29.2523 −1.76719
$$275$$ 0 0
$$276$$ −4.83485 −0.291024
$$277$$ − 24.9129i − 1.49687i −0.663208 0.748435i $$-0.730806\pi$$
0.663208 0.748435i $$-0.269194\pi$$
$$278$$ 16.8693i 1.01175i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −0.417424 −0.0249014 −0.0124507 0.999922i $$-0.503963\pi$$
−0.0124507 + 0.999922i $$0.503963\pi$$
$$282$$ 18.9564i 1.12884i
$$283$$ 0.834849i 0.0496266i 0.999692 + 0.0248133i $$0.00789913\pi$$
−0.999692 + 0.0248133i $$0.992101\pi$$
$$284$$ 11.5826 0.687299
$$285$$ 0 0
$$286$$ −41.0436 −2.42696
$$287$$ − 9.16515i − 0.541002i
$$288$$ 6.04356i 0.356120i
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 11.5826 0.678983
$$292$$ 4.83485i 0.282938i
$$293$$ − 30.1652i − 1.76227i −0.472868 0.881133i $$-0.656781\pi$$
0.472868 0.881133i $$-0.343219\pi$$
$$294$$ 10.7477 0.626820
$$295$$ 0 0
$$296$$ 5.66970 0.329544
$$297$$ 5.00000i 0.290129i
$$298$$ − 30.0000i − 1.73785i
$$299$$ −18.3303 −1.06007
$$300$$ 0 0
$$301$$ −9.58258 −0.552330
$$302$$ 12.8348i 0.738563i
$$303$$ 0.582576i 0.0334681i
$$304$$ 17.7568 1.01842
$$305$$ 0 0
$$306$$ −5.37386 −0.307203
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ − 6.04356i − 0.344364i
$$309$$ −15.1652 −0.862715
$$310$$ 0 0
$$311$$ −3.00000 −0.170114 −0.0850572 0.996376i $$-0.527107\pi$$
−0.0850572 + 0.996376i $$0.527107\pi$$
$$312$$ 6.49545i 0.367733i
$$313$$ − 10.5826i − 0.598163i −0.954228 0.299081i $$-0.903320\pi$$
0.954228 0.299081i $$-0.0966802\pi$$
$$314$$ −19.2523 −1.08647
$$315$$ 0 0
$$316$$ 9.16515 0.515580
$$317$$ − 25.0000i − 1.40414i −0.712108 0.702070i $$-0.752259\pi$$
0.712108 0.702070i $$-0.247741\pi$$
$$318$$ − 0.747727i − 0.0419305i
$$319$$ −5.00000 −0.279946
$$320$$ 0 0
$$321$$ 5.16515 0.288291
$$322$$ − 7.16515i − 0.399298i
$$323$$ 10.7477i 0.598020i
$$324$$ −1.20871 −0.0671507
$$325$$ 0 0
$$326$$ 13.5826 0.752269
$$327$$ 14.1652i 0.783335i
$$328$$ − 12.9909i − 0.717303i
$$329$$ −10.5826 −0.583436
$$330$$ 0 0
$$331$$ −8.33030 −0.457875 −0.228937 0.973441i $$-0.573525\pi$$
−0.228937 + 0.973441i $$0.573525\pi$$
$$332$$ − 14.0000i − 0.768350i
$$333$$ 4.00000i 0.219199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −4.95644 −0.270396
$$337$$ − 21.1652i − 1.15294i −0.817119 0.576470i $$-0.804430\pi$$
0.817119 0.576470i $$-0.195570\pi$$
$$338$$ − 14.3303i − 0.779466i
$$339$$ 14.1652 0.769345
$$340$$ 0 0
$$341$$ 20.0000 1.08306
$$342$$ 6.41742i 0.347015i
$$343$$ 13.0000i 0.701934i
$$344$$ −13.5826 −0.732323
$$345$$ 0 0
$$346$$ −5.66970 −0.304805
$$347$$ 7.16515i 0.384645i 0.981332 + 0.192323i $$0.0616020\pi$$
−0.981332 + 0.192323i $$0.938398\pi$$
$$348$$ − 1.20871i − 0.0647938i
$$349$$ 4.33030 0.231796 0.115898 0.993261i $$-0.463025\pi$$
0.115898 + 0.993261i $$0.463025\pi$$
$$350$$ 0 0
$$351$$ −4.58258 −0.244600
$$352$$ − 30.2178i − 1.61061i
$$353$$ 14.8348i 0.789579i 0.918772 + 0.394790i $$0.129183\pi$$
−0.918772 + 0.394790i $$0.870817\pi$$
$$354$$ 13.5826 0.721906
$$355$$ 0 0
$$356$$ 1.71326 0.0908025
$$357$$ − 3.00000i − 0.158777i
$$358$$ − 8.50455i − 0.449479i
$$359$$ 27.1652 1.43372 0.716861 0.697216i $$-0.245578\pi$$
0.716861 + 0.697216i $$0.245578\pi$$
$$360$$ 0 0
$$361$$ −6.16515 −0.324482
$$362$$ 28.9564i 1.52192i
$$363$$ − 14.0000i − 0.734809i
$$364$$ 5.53901 0.290323
$$365$$ 0 0
$$366$$ 22.8348 1.19360
$$367$$ − 37.4955i − 1.95725i −0.205661 0.978623i $$-0.565934\pi$$
0.205661 0.978623i $$-0.434066\pi$$
$$368$$ − 19.8258i − 1.03349i
$$369$$ 9.16515 0.477119
$$370$$ 0 0
$$371$$ 0.417424 0.0216716
$$372$$ 4.83485i 0.250675i
$$373$$ − 28.3303i − 1.46689i −0.679750 0.733444i $$-0.737912\pi$$
0.679750 0.733444i $$-0.262088\pi$$
$$374$$ 26.8693 1.38938
$$375$$ 0 0
$$376$$ −15.0000 −0.773566
$$377$$ − 4.58258i − 0.236015i
$$378$$ − 1.79129i − 0.0921339i
$$379$$ 26.0000 1.33553 0.667765 0.744372i $$-0.267251\pi$$
0.667765 + 0.744372i $$0.267251\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ − 7.16515i − 0.366601i
$$383$$ − 3.58258i − 0.183061i −0.995802 0.0915305i $$-0.970824\pi$$
0.995802 0.0915305i $$-0.0291759\pi$$
$$384$$ −10.4519 −0.533371
$$385$$ 0 0
$$386$$ −36.4174 −1.85360
$$387$$ − 9.58258i − 0.487110i
$$388$$ − 14.0000i − 0.710742i
$$389$$ 6.58258 0.333750 0.166875 0.985978i $$-0.446632\pi$$
0.166875 + 0.985978i $$0.446632\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 8.50455i 0.429544i
$$393$$ 15.0000i 0.756650i
$$394$$ 29.2523 1.47371
$$395$$ 0 0
$$396$$ 6.04356 0.303700
$$397$$ 10.8348i 0.543785i 0.962328 + 0.271893i $$0.0876496\pi$$
−0.962328 + 0.271893i $$0.912350\pi$$
$$398$$ − 24.0345i − 1.20474i
$$399$$ −3.58258 −0.179353
$$400$$ 0 0
$$401$$ 12.4174 0.620097 0.310048 0.950721i $$-0.399655\pi$$
0.310048 + 0.950721i $$0.399655\pi$$
$$402$$ − 7.46099i − 0.372120i
$$403$$ 18.3303i 0.913097i
$$404$$ 0.704166 0.0350336
$$405$$ 0 0
$$406$$ 1.79129 0.0889001
$$407$$ − 20.0000i − 0.991363i
$$408$$ − 4.25227i − 0.210519i
$$409$$ 2.74773 0.135866 0.0679332 0.997690i $$-0.478360\pi$$
0.0679332 + 0.997690i $$0.478360\pi$$
$$410$$ 0 0
$$411$$ 16.3303 0.805514
$$412$$ 18.3303i 0.903069i
$$413$$ 7.58258i 0.373114i
$$414$$ 7.16515 0.352148
$$415$$ 0 0
$$416$$ 27.6951 1.35786
$$417$$ − 9.41742i − 0.461173i
$$418$$ − 32.0871i − 1.56943i
$$419$$ 37.1652 1.81564 0.907818 0.419364i $$-0.137747\pi$$
0.907818 + 0.419364i $$0.137747\pi$$
$$420$$ 0 0
$$421$$ 4.41742 0.215292 0.107646 0.994189i $$-0.465669\pi$$
0.107646 + 0.994189i $$0.465669\pi$$
$$422$$ 36.4174i 1.77277i
$$423$$ − 10.5826i − 0.514542i
$$424$$ 0.591667 0.0287339
$$425$$ 0 0
$$426$$ −17.1652 −0.831654
$$427$$ 12.7477i 0.616906i
$$428$$ − 6.24318i − 0.301776i
$$429$$ 22.9129 1.10624
$$430$$ 0 0
$$431$$ −39.1652 −1.88652 −0.943259 0.332057i $$-0.892258\pi$$
−0.943259 + 0.332057i $$0.892258\pi$$
$$432$$ − 4.95644i − 0.238467i
$$433$$ − 25.0780i − 1.20517i −0.798053 0.602587i $$-0.794137\pi$$
0.798053 0.602587i $$-0.205863\pi$$
$$434$$ −7.16515 −0.343938
$$435$$ 0 0
$$436$$ 17.1216 0.819975
$$437$$ − 14.3303i − 0.685511i
$$438$$ − 7.16515i − 0.342364i
$$439$$ 16.9129 0.807208 0.403604 0.914934i $$-0.367757\pi$$
0.403604 + 0.914934i $$0.367757\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 24.6261i 1.17135i
$$443$$ − 0.582576i − 0.0276790i −0.999904 0.0138395i $$-0.995595\pi$$
0.999904 0.0138395i $$-0.00440539\pi$$
$$444$$ 4.83485 0.229452
$$445$$ 0 0
$$446$$ 12.5390 0.593740
$$447$$ 16.7477i 0.792140i
$$448$$ 0.912878i 0.0431295i
$$449$$ 30.0780 1.41947 0.709735 0.704469i $$-0.248815\pi$$
0.709735 + 0.704469i $$0.248815\pi$$
$$450$$ 0 0
$$451$$ −45.8258 −2.15785
$$452$$ − 17.1216i − 0.805332i
$$453$$ − 7.16515i − 0.336648i
$$454$$ 13.5826 0.637462
$$455$$ 0 0
$$456$$ −5.07803 −0.237801
$$457$$ − 3.74773i − 0.175311i −0.996151 0.0876556i $$-0.972062\pi$$
0.996151 0.0876556i $$-0.0279375\pi$$
$$458$$ 30.7477i 1.43675i
$$459$$ 3.00000 0.140028
$$460$$ 0 0
$$461$$ −9.16515 −0.426864 −0.213432 0.976958i $$-0.568464\pi$$
−0.213432 + 0.976958i $$0.568464\pi$$
$$462$$ 8.95644i 0.416691i
$$463$$ 0.165151i 0.00767524i 0.999993 + 0.00383762i $$0.00122155\pi$$
−0.999993 + 0.00383762i $$0.998778\pi$$
$$464$$ 4.95644 0.230097
$$465$$ 0 0
$$466$$ −23.5826 −1.09244
$$467$$ − 8.00000i − 0.370196i −0.982720 0.185098i $$-0.940740\pi$$
0.982720 0.185098i $$-0.0592602\pi$$
$$468$$ 5.53901i 0.256041i
$$469$$ 4.16515 0.192329
$$470$$ 0 0
$$471$$ 10.7477 0.495229
$$472$$ 10.7477i 0.494704i
$$473$$ 47.9129i 2.20304i
$$474$$ −13.5826 −0.623868
$$475$$ 0 0
$$476$$ −3.62614 −0.166204
$$477$$ 0.417424i 0.0191125i
$$478$$ 46.5735i 2.13022i
$$479$$ 19.1652 0.875678 0.437839 0.899053i $$-0.355744\pi$$
0.437839 + 0.899053i $$0.355744\pi$$
$$480$$ 0 0
$$481$$ 18.3303 0.835790
$$482$$ 52.5390i 2.39309i
$$483$$ 4.00000i 0.182006i
$$484$$ −16.9220 −0.769180
$$485$$ 0 0
$$486$$ 1.79129 0.0812545
$$487$$ 2.33030i 0.105596i 0.998605 + 0.0527980i $$0.0168140\pi$$
−0.998605 + 0.0527980i $$0.983186\pi$$
$$488$$ 18.0689i 0.817942i
$$489$$ −7.58258 −0.342896
$$490$$ 0 0
$$491$$ 16.0000 0.722070 0.361035 0.932552i $$-0.382424\pi$$
0.361035 + 0.932552i $$0.382424\pi$$
$$492$$ − 11.0780i − 0.499436i
$$493$$ 3.00000i 0.135113i
$$494$$ 29.4083 1.32314
$$495$$ 0 0
$$496$$ −19.8258 −0.890203
$$497$$ − 9.58258i − 0.429837i
$$498$$ 20.7477i 0.929728i
$$499$$ 13.4174 0.600646 0.300323 0.953837i $$-0.402905\pi$$
0.300323 + 0.953837i $$0.402905\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 28.9564i 1.29239i
$$503$$ 22.9129i 1.02163i 0.859689 + 0.510817i $$0.170657\pi$$
−0.859689 + 0.510817i $$0.829343\pi$$
$$504$$ 1.41742 0.0631371
$$505$$ 0 0
$$506$$ −35.8258 −1.59265
$$507$$ 8.00000i 0.355292i
$$508$$ − 2.41742i − 0.107256i
$$509$$ −26.7477 −1.18557 −0.592786 0.805360i $$-0.701972\pi$$
−0.592786 + 0.805360i $$0.701972\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 15.9038i 0.702855i
$$513$$ − 3.58258i − 0.158175i
$$514$$ −22.8348 −1.00720
$$515$$ 0 0
$$516$$ −11.5826 −0.509894
$$517$$ 52.9129i 2.32711i
$$518$$ 7.16515i 0.314819i
$$519$$ 3.16515 0.138935
$$520$$ 0 0
$$521$$ 43.0780 1.88728 0.943641 0.330970i $$-0.107376\pi$$
0.943641 + 0.330970i $$0.107376\pi$$
$$522$$ 1.79129i 0.0784025i
$$523$$ 33.3303i 1.45743i 0.684816 + 0.728716i $$0.259883\pi$$
−0.684816 + 0.728716i $$0.740117\pi$$
$$524$$ 18.1307 0.792043
$$525$$ 0 0
$$526$$ 54.3303 2.36891
$$527$$ − 12.0000i − 0.522728i
$$528$$ 24.7822i 1.07851i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −7.58258 −0.329056
$$532$$ 4.33030i 0.187742i
$$533$$ − 42.0000i − 1.81922i
$$534$$ −2.53901 −0.109874
$$535$$ 0 0
$$536$$ 5.90379 0.255005
$$537$$ 4.74773i 0.204880i
$$538$$ − 40.4519i − 1.74400i
$$539$$ 30.0000 1.29219
$$540$$ 0 0
$$541$$ −31.4955 −1.35410 −0.677048 0.735939i $$-0.736741\pi$$
−0.677048 + 0.735939i $$0.736741\pi$$
$$542$$ 2.08712i 0.0896495i
$$543$$ − 16.1652i − 0.693713i
$$544$$ −18.1307 −0.777347
$$545$$ 0 0
$$546$$ −8.20871 −0.351300
$$547$$ 4.16515i 0.178089i 0.996028 + 0.0890445i $$0.0283813\pi$$
−0.996028 + 0.0890445i $$0.971619\pi$$
$$548$$ − 19.7386i − 0.843193i
$$549$$ −12.7477 −0.544060
$$550$$ 0 0
$$551$$ 3.58258 0.152623
$$552$$ 5.66970i 0.241318i
$$553$$ − 7.58258i − 0.322444i
$$554$$ 44.6261 1.89598
$$555$$ 0 0
$$556$$ −11.3830 −0.482745
$$557$$ − 22.7477i − 0.963852i −0.876212 0.481926i $$-0.839937\pi$$
0.876212 0.481926i $$-0.160063\pi$$
$$558$$ − 7.16515i − 0.303325i
$$559$$ −43.9129 −1.85732
$$560$$ 0 0
$$561$$ −15.0000 −0.633300
$$562$$ − 0.747727i − 0.0315410i
$$563$$ 14.5826i 0.614582i 0.951616 + 0.307291i $$0.0994225\pi$$
−0.951616 + 0.307291i $$0.900577\pi$$
$$564$$ −12.7913 −0.538610
$$565$$ 0 0
$$566$$ −1.49545 −0.0628586
$$567$$ 1.00000i 0.0419961i
$$568$$ − 13.5826i − 0.569912i
$$569$$ −28.5826 −1.19824 −0.599122 0.800658i $$-0.704484\pi$$
−0.599122 + 0.800658i $$0.704484\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ − 27.6951i − 1.15799i
$$573$$ 4.00000i 0.167102i
$$574$$ 16.4174 0.685250
$$575$$ 0 0
$$576$$ −0.912878 −0.0380366
$$577$$ − 19.1652i − 0.797856i −0.916982 0.398928i $$-0.869382\pi$$
0.916982 0.398928i $$-0.130618\pi$$
$$578$$ 14.3303i 0.596062i
$$579$$ 20.3303 0.844899
$$580$$ 0 0
$$581$$ −11.5826 −0.480526
$$582$$ 20.7477i 0.860021i
$$583$$ − 2.08712i − 0.0864397i
$$584$$ 5.66970 0.234614
$$585$$ 0 0
$$586$$ 54.0345 2.23214
$$587$$ − 11.5826i − 0.478064i −0.971012 0.239032i $$-0.923170\pi$$
0.971012 0.239032i $$-0.0768301\pi$$
$$588$$ 7.25227i 0.299079i
$$589$$ −14.3303 −0.590470
$$590$$ 0 0
$$591$$ −16.3303 −0.671739
$$592$$ 19.8258i 0.814834i
$$593$$ − 8.41742i − 0.345662i −0.984951 0.172831i $$-0.944709\pi$$
0.984951 0.172831i $$-0.0552915\pi$$
$$594$$ −8.95644 −0.367487
$$595$$ 0 0
$$596$$ 20.2432 0.829193
$$597$$ 13.4174i 0.549139i
$$598$$ − 32.8348i − 1.34272i
$$599$$ −0.165151 −0.00674790 −0.00337395 0.999994i $$-0.501074\pi$$
−0.00337395 + 0.999994i $$0.501074\pi$$
$$600$$ 0 0
$$601$$ −32.3303 −1.31878 −0.659390 0.751801i $$-0.729185\pi$$
−0.659390 + 0.751801i $$0.729185\pi$$
$$602$$ − 17.1652i − 0.699599i
$$603$$ 4.16515i 0.169618i
$$604$$ −8.66061 −0.352395
$$605$$ 0 0
$$606$$ −1.04356 −0.0423918
$$607$$ 3.58258i 0.145412i 0.997353 + 0.0727061i $$0.0231635\pi$$
−0.997353 + 0.0727061i $$0.976836\pi$$
$$608$$ 21.6515i 0.878085i
$$609$$ −1.00000 −0.0405220
$$610$$ 0 0
$$611$$ −48.4955 −1.96192
$$612$$ − 3.62614i − 0.146578i
$$613$$ 43.7477i 1.76695i 0.468474 + 0.883477i $$0.344804\pi$$
−0.468474 + 0.883477i $$0.655196\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −7.08712 −0.285548
$$617$$ 15.4955i 0.623823i 0.950111 + 0.311912i $$0.100969\pi$$
−0.950111 + 0.311912i $$0.899031\pi$$
$$618$$ − 27.1652i − 1.09274i
$$619$$ −13.1652 −0.529152 −0.264576 0.964365i $$-0.585232\pi$$
−0.264576 + 0.964365i $$0.585232\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ − 5.37386i − 0.215472i
$$623$$ − 1.41742i − 0.0567879i
$$624$$ −22.7133 −0.909258
$$625$$ 0 0
$$626$$ 18.9564 0.757652
$$627$$ 17.9129i 0.715371i
$$628$$ − 12.9909i − 0.518394i
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 10.9129 0.434435 0.217217 0.976123i $$-0.430302\pi$$
0.217217 + 0.976123i $$0.430302\pi$$
$$632$$ − 10.7477i − 0.427522i
$$633$$ − 20.3303i − 0.808057i
$$634$$ 44.7822 1.77853
$$635$$ 0 0
$$636$$ 0.504546 0.0200065
$$637$$ 27.4955i 1.08941i
$$638$$ − 8.95644i − 0.354589i
$$639$$ 9.58258 0.379081
$$640$$ 0 0
$$641$$ −6.58258 −0.259996 −0.129998 0.991514i $$-0.541497\pi$$
−0.129998 + 0.991514i $$0.541497\pi$$
$$642$$ 9.25227i 0.365158i
$$643$$ − 43.6606i − 1.72181i −0.508769 0.860903i $$-0.669899\pi$$
0.508769 0.860903i $$-0.330101\pi$$
$$644$$ 4.83485 0.190520
$$645$$ 0 0
$$646$$ −19.2523 −0.757471
$$647$$ 24.7477i 0.972934i 0.873699 + 0.486467i $$0.161715\pi$$
−0.873699 + 0.486467i $$0.838285\pi$$
$$648$$ 1.41742i 0.0556817i
$$649$$ 37.9129 1.48821
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ 9.16515i 0.358935i
$$653$$ − 26.1652i − 1.02392i −0.859009 0.511961i $$-0.828919\pi$$
0.859009 0.511961i $$-0.171081\pi$$
$$654$$ −25.3739 −0.992197
$$655$$ 0 0
$$656$$ 45.4265 1.77361
$$657$$ 4.00000i 0.156055i
$$658$$ − 18.9564i − 0.738999i
$$659$$ −18.1652 −0.707614 −0.353807 0.935318i $$-0.615113\pi$$
−0.353807 + 0.935318i $$0.615113\pi$$
$$660$$ 0 0
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ − 14.9220i − 0.579959i
$$663$$ − 13.7477i − 0.533917i
$$664$$ −16.4174 −0.637120
$$665$$ 0 0
$$666$$ −7.16515 −0.277644
$$667$$ − 4.00000i − 0.154881i
$$668$$ 0 0
$$669$$ −7.00000 −0.270636
$$670$$ 0 0
$$671$$ 63.7386 2.46060
$$672$$ − 6.04356i − 0.233135i
$$673$$ − 1.74773i − 0.0673699i −0.999433 0.0336850i $$-0.989276\pi$$
0.999433 0.0336850i $$-0.0107243\pi$$
$$674$$ 37.9129 1.46035
$$675$$ 0 0
$$676$$ 9.66970 0.371911
$$677$$ 44.8258i 1.72279i 0.507932 + 0.861397i $$0.330410\pi$$
−0.507932 + 0.861397i $$0.669590\pi$$
$$678$$ 25.3739i 0.974477i
$$679$$ −11.5826 −0.444498
$$680$$ 0 0
$$681$$ −7.58258 −0.290565
$$682$$ 35.8258i 1.37184i
$$683$$ − 7.16515i − 0.274167i −0.990560 0.137083i $$-0.956227\pi$$
0.990560 0.137083i $$-0.0437728\pi$$
$$684$$ −4.33030 −0.165573
$$685$$ 0 0
$$686$$ −23.2867 −0.889092
$$687$$ − 17.1652i − 0.654891i
$$688$$ − 47.4955i − 1.81075i
$$689$$ 1.91288 0.0728749
$$690$$ 0 0
$$691$$ 42.9129 1.63248 0.816241 0.577711i $$-0.196054\pi$$
0.816241 + 0.577711i $$0.196054\pi$$
$$692$$ − 3.82576i − 0.145433i
$$693$$ − 5.00000i − 0.189934i
$$694$$ −12.8348 −0.487204
$$695$$ 0 0
$$696$$ −1.41742 −0.0537273
$$697$$ 27.4955i 1.04146i
$$698$$ 7.75682i 0.293600i
$$699$$ 13.1652 0.497952
$$700$$ 0 0
$$701$$ −23.0780 −0.871645 −0.435823 0.900033i $$-0.643542\pi$$
−0.435823 + 0.900033i $$0.643542\pi$$
$$702$$ − 8.20871i − 0.309818i
$$703$$ 14.3303i 0.540478i
$$704$$ 4.56439 0.172027
$$705$$ 0 0
$$706$$ −26.5735 −1.00011
$$707$$ − 0.582576i − 0.0219100i
$$708$$ 9.16515i 0.344447i
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 7.58258 0.284369
$$712$$ − 2.00909i − 0.0752939i
$$713$$ 16.0000i 0.599205i
$$714$$ 5.37386 0.201112
$$715$$ 0 0
$$716$$ 5.73864 0.214463
$$717$$ − 26.0000i − 0.970988i
$$718$$ 48.6606i 1.81600i
$$719$$ −7.91288 −0.295101 −0.147550 0.989055i $$-0.547139\pi$$
−0.147550 + 0.989055i $$0.547139\pi$$
$$720$$ 0 0
$$721$$ 15.1652 0.564780
$$722$$ − 11.0436i − 0.410999i
$$723$$ − 29.3303i − 1.09081i
$$724$$ −19.5390 −0.726162
$$725$$ 0 0
$$726$$ 25.0780 0.930733
$$727$$ 9.66970i 0.358629i 0.983792 + 0.179315i $$0.0573880\pi$$
−0.983792 + 0.179315i $$0.942612\pi$$
$$728$$ − 6.49545i − 0.240738i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 28.7477 1.06327
$$732$$ 15.4083i 0.569508i
$$733$$ 38.4174i 1.41898i 0.704716 + 0.709490i $$0.251075\pi$$
−0.704716 + 0.709490i $$0.748925\pi$$
$$734$$ 67.1652 2.47911
$$735$$ 0 0
$$736$$ 24.1742 0.891074
$$737$$ − 20.8258i − 0.767127i
$$738$$ 16.4174i 0.604334i
$$739$$ −19.2523 −0.708206 −0.354103 0.935206i $$-0.615214\pi$$
−0.354103 + 0.935206i $$0.615214\pi$$
$$740$$ 0 0
$$741$$ −16.4174 −0.603109
$$742$$ 0.747727i 0.0274499i
$$743$$ − 29.7477i − 1.09134i −0.838001 0.545669i $$-0.816276\pi$$
0.838001 0.545669i $$-0.183724\pi$$
$$744$$ 5.66970 0.207861
$$745$$ 0 0
$$746$$ 50.7477 1.85801
$$747$$ − 11.5826i − 0.423784i
$$748$$ 18.1307i 0.662923i
$$749$$ −5.16515 −0.188731
$$750$$ 0 0
$$751$$ −17.4955 −0.638418 −0.319209 0.947684i $$-0.603417\pi$$
−0.319209 + 0.947684i $$0.603417\pi$$
$$752$$ − 52.4519i − 1.91272i
$$753$$ − 16.1652i − 0.589091i
$$754$$ 8.20871 0.298944
$$755$$ 0 0
$$756$$ 1.20871 0.0439604
$$757$$ − 2.33030i − 0.0846963i −0.999103 0.0423481i $$-0.986516\pi$$
0.999103 0.0423481i $$-0.0134839\pi$$
$$758$$ 46.5735i 1.69163i
$$759$$ 20.0000 0.725954
$$760$$ 0 0
$$761$$ 36.4174 1.32013 0.660065 0.751208i $$-0.270529\pi$$
0.660065 + 0.751208i $$0.270529\pi$$
$$762$$ 3.58258i 0.129783i
$$763$$ − 14.1652i − 0.512813i
$$764$$ 4.83485 0.174919
$$765$$ 0 0
$$766$$ 6.41742 0.231871
$$767$$ 34.7477i 1.25467i
$$768$$ − 20.5481i − 0.741466i
$$769$$ −25.5826 −0.922531 −0.461266 0.887262i $$-0.652604\pi$$
−0.461266 + 0.887262i $$0.652604\pi$$
$$770$$ 0 0
$$771$$ 12.7477 0.459098
$$772$$ − 24.5735i − 0.884419i
$$773$$ 5.16515i 0.185778i 0.995676 + 0.0928888i $$0.0296101\pi$$
−0.995676 + 0.0928888i $$0.970390\pi$$
$$774$$ 17.1652 0.616989
$$775$$ 0 0
$$776$$ −16.4174 −0.589351
$$777$$ − 4.00000i − 0.143499i
$$778$$ 11.7913i 0.422738i
$$779$$ 32.8348 1.17643
$$780$$ 0 0
$$781$$ −47.9129 −1.71446
$$782$$ 21.4955i 0.768676i
$$783$$ − 1.00000i − 0.0357371i
$$784$$ −29.7386 −1.06209
$$785$$ 0 0
$$786$$ −26.8693 −0.958397
$$787$$ − 24.0000i − 0.855508i −0.903895 0.427754i $$-0.859305\pi$$
0.903895 0.427754i $$-0.140695\pi$$
$$788$$ 19.7386i 0.703160i
$$789$$ −30.3303 −1.07979
$$790$$ 0 0
$$791$$ −14.1652 −0.503655
$$792$$ − 7.08712i − 0.251830i
$$793$$ 58.4174i 2.07446i
$$794$$ −19.4083 −0.688776
$$795$$ 0 0
$$796$$ 16.2178 0.574825
$$797$$ 32.3303i 1.14520i 0.819836 + 0.572599i $$0.194065\pi$$
−0.819836 + 0.572599i $$0.805935\pi$$
$$798$$ − 6.41742i − 0.227174i
$$799$$ 31.7477 1.12315
$$800$$ 0 0
$$801$$ 1.41742 0.0500822
$$802$$ 22.2432i 0.785434i
$$803$$ − 20.0000i − 0.705785i
$$804$$ 5.03447 0.177552
$$805$$ 0 0
$$806$$ −32.8348 −1.15656
$$807$$ 22.5826i 0.794944i
$$808$$ − 0.825757i − 0.0290500i
$$809$$ −44.0780 −1.54970 −0.774851 0.632145i $$-0.782175\pi$$
−0.774851 + 0.632145i $$0.782175\pi$$
$$810$$ 0 0
$$811$$ 32.5826 1.14413 0.572064 0.820209i $$-0.306143\pi$$
0.572064 + 0.820209i $$0.306143\pi$$
$$812$$ 1.20871i 0.0424175i
$$813$$ − 1.16515i − 0.0408636i
$$814$$ 35.8258 1.25569
$$815$$ 0 0
$$816$$ 14.8693 0.520530
$$817$$ − 34.3303i − 1.20107i
$$818$$ 4.92197i 0.172093i
$$819$$ 4.58258 0.160128
$$820$$ 0 0
$$821$$ −31.4955 −1.09920 −0.549599 0.835428i $$-0.685220\pi$$
−0.549599 + 0.835428i $$0.685220\pi$$
$$822$$ 29.2523i 1.02029i
$$823$$ − 51.0780i − 1.78047i −0.455503 0.890234i $$-0.650541\pi$$
0.455503 0.890234i $$-0.349459\pi$$
$$824$$ 21.4955 0.748830
$$825$$ 0 0
$$826$$ −13.5826 −0.472598
$$827$$ − 43.8258i − 1.52397i −0.647594 0.761985i $$-0.724225\pi$$
0.647594 0.761985i $$-0.275775\pi$$
$$828$$ 4.83485i 0.168023i
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ −24.9129 −0.864218
$$832$$ 4.18333i 0.145031i
$$833$$ − 18.0000i − 0.623663i
$$834$$ 16.8693 0.584137
$$835$$ 0 0
$$836$$ 21.6515 0.748833
$$837$$ 4.00000i 0.138260i
$$838$$ 66.5735i 2.29974i
$$839$$ 48.4955 1.67425 0.837125 0.547012i $$-0.184235\pi$$
0.837125 + 0.547012i $$0.184235\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 7.91288i 0.272696i
$$843$$ 0.417424i 0.0143769i
$$844$$ −24.5735 −0.845854
$$845$$ 0 0
$$846$$ 18.9564 0.651736
$$847$$ 14.0000i 0.481046i
$$848$$ 2.06894i 0.0710476i
$$849$$ 0.834849 0.0286519
$$850$$ 0 0
$$851$$ 16.0000 0.548473
$$852$$ − 11.5826i − 0.396813i
$$853$$ − 20.7477i − 0.710389i −0.934792 0.355194i $$-0.884415\pi$$
0.934792 0.355194i $$-0.115585\pi$$
$$854$$ −22.8348 −0.781392
$$855$$ 0 0
$$856$$ −7.32121 −0.250234
$$857$$ − 46.6606i − 1.59390i −0.604048 0.796948i $$-0.706446\pi$$
0.604048 0.796948i $$-0.293554\pi$$
$$858$$ 41.0436i 1.40120i
$$859$$ 47.5826 1.62350 0.811748 0.584007i $$-0.198516\pi$$
0.811748 + 0.584007i $$0.198516\pi$$
$$860$$ 0 0
$$861$$ −9.16515 −0.312348
$$862$$ − 70.1561i − 2.38952i
$$863$$ 6.41742i 0.218452i 0.994017 + 0.109226i $$0.0348372\pi$$
−0.994017 + 0.109226i $$0.965163\pi$$
$$864$$ 6.04356 0.205606
$$865$$ 0 0
$$866$$ 44.9220 1.52651
$$867$$ − 8.00000i − 0.271694i
$$868$$ − 4.83485i − 0.164105i
$$869$$ −37.9129 −1.28611
$$870$$ 0 0
$$871$$ 19.0871 0.646742
$$872$$ − 20.0780i − 0.679928i
$$873$$ − 11.5826i − 0.392011i
$$874$$ 25.6697 0.868290
$$875$$ 0 0
$$876$$ 4.83485 0.163354
$$877$$ 19.6697i 0.664198i 0.943244 + 0.332099i $$0.107757\pi$$
−0.943244 + 0.332099i $$0.892243\pi$$
$$878$$ 30.2958i 1.02243i
$$879$$ −30.1652 −1.01745
$$880$$ 0 0
$$881$$ −32.0780 −1.08074 −0.540368 0.841429i $$-0.681715\pi$$
−0.540368 + 0.841429i $$0.681715\pi$$
$$882$$ − 10.7477i − 0.361895i
$$883$$ − 16.0000i − 0.538443i −0.963078 0.269221i $$-0.913234\pi$$
0.963078 0.269221i $$-0.0867663\pi$$
$$884$$ −16.6170 −0.558892
$$885$$ 0 0
$$886$$ 1.04356 0.0350591
$$887$$ − 44.9129i − 1.50803i −0.656859 0.754013i $$-0.728115\pi$$
0.656859 0.754013i $$-0.271885\pi$$
$$888$$ − 5.66970i − 0.190263i
$$889$$ −2.00000 −0.0670778
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ 8.46099i 0.283295i
$$893$$ − 37.9129i − 1.26871i
$$894$$ −30.0000 −1.00335
$$895$$ 0 0
$$896$$ 10.4519 0.349173
$$897$$ 18.3303i 0.612031i
$$898$$ 53.8784i 1.79795i
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ −1.25227 −0.0417193
$$902$$ − 82.0871i − 2.73320i
$$903$$ 9.58258i 0.318888i
$$904$$ −20.0780 −0.667785
$$905$$ 0 0
$$906$$ 12.8348 0.426409
$$907$$ 23.5826i 0.783047i 0.920168 + 0.391523i $$0.128052\pi$$
−0.920168 + 0.391523i $$0.871948\pi$$
$$908$$ 9.16515i 0.304156i
$$909$$ 0.582576 0.0193228
$$910$$ 0 0
$$911$$ −50.8258 −1.68393 −0.841966 0.539530i $$-0.818602\pi$$
−0.841966 + 0.539530i $$0.818602\pi$$
$$912$$ − 17.7568i − 0.587987i
$$913$$ 57.9129i 1.91664i
$$914$$ 6.71326 0.222055
$$915$$ 0 0
$$916$$ −20.7477 −0.685524
$$917$$ − 15.0000i − 0.495344i
$$918$$ 5.37386i 0.177364i
$$919$$ −18.9129 −0.623878 −0.311939 0.950102i $$-0.600979\pi$$
−0.311939 + 0.950102i $$0.600979\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 16.4174i − 0.540679i
$$923$$ − 43.9129i − 1.44541i
$$924$$ −6.04356 −0.198819
$$925$$ 0 0
$$926$$ −0.295834 −0.00972170
$$927$$ 15.1652i 0.498089i
$$928$$ 6.04356i 0.198390i
$$929$$ −15.6697 −0.514106 −0.257053 0.966397i $$-0.582752\pi$$
−0.257053 + 0.966397i $$0.582752\pi$$
$$930$$ 0 0
$$931$$ −21.4955 −0.704485
$$932$$ − 15.9129i − 0.521244i
$$933$$ 3.00000i 0.0982156i
$$934$$ 14.3303 0.468902
$$935$$ 0 0
$$936$$ 6.49545 0.212311
$$937$$ − 18.5826i − 0.607066i −0.952821 0.303533i $$-0.901834\pi$$
0.952821 0.303533i $$-0.0981663\pi$$
$$938$$ 7.46099i 0.243610i
$$939$$ −10.5826 −0.345349
$$940$$ 0 0
$$941$$ 19.9129 0.649141 0.324571 0.945861i $$-0.394780\pi$$
0.324571 + 0.945861i $$0.394780\pi$$
$$942$$ 19.2523i 0.627273i
$$943$$ − 36.6606i − 1.19383i
$$944$$ −37.5826 −1.22321
$$945$$ 0 0
$$946$$ −85.8258 −2.79044
$$947$$ − 54.9129i − 1.78443i −0.451612 0.892214i $$-0.649151\pi$$
0.451612 0.892214i $$-0.350849\pi$$
$$948$$ − 9.16515i − 0.297670i
$$949$$ 18.3303 0.595027
$$950$$ 0 0
$$951$$ −25.0000 −0.810681
$$952$$ 4.25227i 0.137817i
$$953$$ 37.5826i 1.21742i 0.793393 + 0.608710i $$0.208313\pi$$
−0.793393 + 0.608710i $$0.791687\pi$$
$$954$$ −0.747727 −0.0242086
$$955$$ 0 0
$$956$$ −31.4265 −1.01641
$$957$$ 5.00000i 0.161627i
$$958$$ 34.3303i 1.10916i
$$959$$ −16.3303 −0.527333
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 32.8348i 1.05864i
$$963$$ − 5.16515i − 0.166445i
$$964$$ −35.4519 −1.14183
$$965$$ 0 0
$$966$$ −7.16515 −0.230535
$$967$$ 9.25227i 0.297533i 0.988872 + 0.148767i $$0.0475303\pi$$
−0.988872 + 0.148767i $$0.952470\pi$$
$$968$$ 19.8439i 0.637808i
$$969$$ 10.7477 0.345267
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 1.20871i 0.0387695i
$$973$$ 9.41742i 0.301909i
$$974$$ −4.17424 −0.133751
$$975$$ 0 0
$$976$$ −63.1833 −2.02245
$$977$$ − 2.50455i − 0.0801275i −0.999197 0.0400638i $$-0.987244\pi$$
0.999197 0.0400638i $$-0.0127561\pi$$
$$978$$ − 13.5826i − 0.434323i
$$979$$ −7.08712 −0.226505
$$980$$ 0 0
$$981$$ 14.1652 0.452258
$$982$$ 28.6606i 0.914597i
$$983$$ − 55.1652i − 1.75950i −0.475441 0.879748i $$-0.657712\pi$$
0.475441 0.879748i $$-0.342288\pi$$
$$984$$ −12.9909 −0.414135
$$985$$ 0 0
$$986$$ −5.37386 −0.171139
$$987$$ 10.5826i 0.336847i
$$988$$ 19.8439i 0.631320i
$$989$$ −38.3303 −1.21883
$$990$$ 0 0
$$991$$ 16.0780 0.510735 0.255368 0.966844i $$-0.417803\pi$$
0.255368 + 0.966844i $$0.417803\pi$$
$$992$$ − 24.1742i − 0.767533i
$$993$$ 8.33030i 0.264354i
$$994$$ 17.1652 0.544446
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ − 0.834849i − 0.0264399i −0.999913 0.0132200i $$-0.995792\pi$$
0.999913 0.0132200i $$-0.00420817\pi$$
$$998$$ 24.0345i 0.760798i
$$999$$ 4.00000 0.126554
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 2175.2.c.f.349.3 4
5.2 odd 4 2175.2.a.r.1.1 2
5.3 odd 4 435.2.a.f.1.2 2
5.4 even 2 inner 2175.2.c.f.349.2 4
15.2 even 4 6525.2.a.t.1.2 2
15.8 even 4 1305.2.a.m.1.1 2
20.3 even 4 6960.2.a.bw.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.2 2 5.3 odd 4
1305.2.a.m.1.1 2 15.8 even 4
2175.2.a.r.1.1 2 5.2 odd 4
2175.2.c.f.349.2 4 5.4 even 2 inner
2175.2.c.f.349.3 4 1.1 even 1 trivial
6525.2.a.t.1.2 2 15.2 even 4
6960.2.a.bw.1.1 2 20.3 even 4