# Properties

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.79129i q^{2} +1.00000i q^{3} -1.20871 q^{4} +1.79129 q^{6} -1.00000i q^{7} -1.41742i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.79129i q^{2} +1.00000i q^{3} -1.20871 q^{4} +1.79129 q^{6} -1.00000i q^{7} -1.41742i q^{8} -1.00000 q^{9} +5.00000 q^{11} -1.20871i q^{12} -4.58258i q^{13} -1.79129 q^{14} -4.95644 q^{16} +3.00000i q^{17} +1.79129i q^{18} -3.58258 q^{19} +1.00000 q^{21} -8.95644i q^{22} -4.00000i q^{23} +1.41742 q^{24} -8.20871 q^{26} -1.00000i q^{27} +1.20871i q^{28} -1.00000 q^{29} +4.00000 q^{31} +6.04356i q^{32} +5.00000i q^{33} +5.37386 q^{34} +1.20871 q^{36} +4.00000i q^{37} +6.41742i q^{38} +4.58258 q^{39} -9.16515 q^{41} -1.79129i q^{42} -9.58258i q^{43} -6.04356 q^{44} -7.16515 q^{46} -10.5826i q^{47} -4.95644i q^{48} +6.00000 q^{49} -3.00000 q^{51} +5.53901i q^{52} +0.417424i q^{53} -1.79129 q^{54} -1.41742 q^{56} -3.58258i q^{57} +1.79129i q^{58} +7.58258 q^{59} +12.7477 q^{61} -7.16515i q^{62} +1.00000i q^{63} +0.912878 q^{64} +8.95644 q^{66} +4.16515i q^{67} -3.62614i q^{68} +4.00000 q^{69} -9.58258 q^{71} +1.41742i q^{72} +4.00000i q^{73} +7.16515 q^{74} +4.33030 q^{76} -5.00000i q^{77} -8.20871i q^{78} -7.58258 q^{79} +1.00000 q^{81} +16.4174i q^{82} -11.5826i q^{83} -1.20871 q^{84} -17.1652 q^{86} -1.00000i q^{87} -7.08712i q^{88} -1.41742 q^{89} -4.58258 q^{91} +4.83485i q^{92} +4.00000i q^{93} -18.9564 q^{94} -6.04356 q^{96} -11.5826i q^{97} -10.7477i q^{98} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 14 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10})$$ 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 $$-0.781693\pi$$
0.773893 0.633316i $$-0.218307\pi$$
$$3$$ 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 $$-0.939481\pi$$
0.981981 0.188982i $$-0.0605189\pi$$
$$8$$ − 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 $$-0.780799\pi$$
0.772110 0.635489i $$-0.219201\pi$$
$$14$$ −1.79129 −0.478742
$$15$$ 0 0
$$16$$ −4.95644 −1.23911
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ 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 $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 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 $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 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 $$-0.739210\pi$$
0.682737 0.730665i $$-0.260790\pi$$
$$44$$ −6.04356 −0.911101
$$45$$ 0 0
$$46$$ −7.16515 −1.05644
$$47$$ − 10.5826i − 1.54363i −0.635849 0.771814i $$-0.719350\pi$$
0.635849 0.771814i $$-0.280650\pi$$
$$48$$ − 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 $$0.00912682\pi$$
−0.999589 + 0.0286688i $$0.990873\pi$$
$$54$$ −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 $$0.0818869\pi$$
−0.967092 + 0.254427i $$0.918113\pi$$
$$68$$ − 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 $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ 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 $$-0.780721\pi$$
0.771956 0.635676i $$-0.219279\pi$$
$$84$$ −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 $$-0.799909\pi$$
0.808849 0.588016i $$-0.200091\pi$$
$$98$$ − 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 $$0.268570\pi$$
−0.664674 + 0.747133i $$0.731430\pi$$
$$104$$ −6.49545 −0.636932
$$105$$ 0 0
$$106$$ 0.747727 0.0726257
$$107$$ − 5.16515i − 0.499334i −0.968332 0.249667i $$-0.919679\pi$$
0.968332 0.249667i $$-0.0803212\pi$$
$$108$$ 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 $$-0.767889\pi$$
0.745708 0.666273i $$-0.232111\pi$$
$$114$$ −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 $$-0.971717\pi$$
0.996055 0.0887357i $$-0.0282826\pi$$
$$128$$ 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 $$-0.754253\pi$$
0.716491 0.697596i $$-0.245747\pi$$
$$138$$ − 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 $$-0.858908\pi$$
0.903361 0.428881i $$-0.141092\pi$$
$$158$$ 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 $$0.0959717\pi$$
−0.954891 + 0.296957i $$0.904028\pi$$
$$164$$ 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 $$-0.961608\pi$$
0.992735 0.120321i $$-0.0383924\pi$$
$$174$$ −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 $$-0.738726\pi$$
0.681623 0.731704i $$-0.261274\pi$$
$$194$$ −20.7477 −1.48960
$$195$$ 0 0
$$196$$ −7.25227 −0.518019
$$197$$ 16.3303i 1.16349i 0.813373 + 0.581743i $$0.197629\pi$$
−0.813373 + 0.581743i $$0.802371\pi$$
$$198$$ 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 $$0.0753051\pi$$
−0.972146 + 0.234377i $$0.924695\pi$$
$$224$$ 6.04356 0.403802
$$225$$ 0 0
$$226$$ −25.3739 −1.68784
$$227$$ 7.58258i 0.503273i 0.967822 + 0.251637i $$0.0809688\pi$$
−0.967822 + 0.251637i $$0.919031\pi$$
$$228$$ 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 $$-0.858077\pi$$
0.902238 0.431239i $$-0.141923\pi$$
$$234$$ 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 $$-0.869846\pi$$
0.917563 0.397591i $$-0.130154\pi$$
$$258$$ − 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 $$0.384712\pi$$
−0.354322 + 0.935123i $$0.615288\pi$$
$$264$$ 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 $$0.269194\pi$$
−0.663208 + 0.748435i $$0.730806\pi$$
$$278$$ − 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 $$-0.992101\pi$$
0.999692 0.0248133i $$-0.00789913\pi$$
$$284$$ 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 $$0.343219\pi$$
−0.472868 + 0.881133i $$0.656781\pi$$
$$294$$ 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 $$0.0966802\pi$$
−0.954228 + 0.299081i $$0.903320\pi$$
$$314$$ −19.2523 −1.08647
$$315$$ 0 0
$$316$$ 9.16515 0.515580
$$317$$ 25.0000i 1.40414i 0.712108 + 0.702070i $$0.247741\pi$$
−0.712108 + 0.702070i $$0.752259\pi$$
$$318$$ 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 $$0.195570\pi$$
−0.817119 + 0.576470i $$0.804430\pi$$
$$338$$ 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 $$-0.938398\pi$$
0.981332 0.192323i $$-0.0616020\pi$$
$$348$$ 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 $$-0.870817\pi$$
0.918772 0.394790i $$-0.129183\pi$$
$$354$$ 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 $$0.434066\pi$$
−0.205661 + 0.978623i $$0.565934\pi$$
$$368$$ 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 $$0.262088\pi$$
−0.679750 + 0.733444i $$0.737912\pi$$
$$374$$ 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 $$0.0291759\pi$$
−0.995802 + 0.0915305i $$0.970824\pi$$
$$384$$ −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 $$-0.912350\pi$$
0.962328 0.271893i $$-0.0876496\pi$$
$$398$$ 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 $$0.205863\pi$$
−0.798053 + 0.602587i $$0.794137\pi$$
$$434$$ −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 $$0.00440539\pi$$
−0.999904 + 0.0138395i $$0.995595\pi$$
$$444$$ 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 $$0.0279375\pi$$
−0.996151 + 0.0876556i $$0.972062\pi$$
$$458$$ − 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 $$-0.998778\pi$$
0.999993 0.00383762i $$-0.00122155\pi$$
$$464$$ 4.95644 0.230097
$$465$$ 0 0
$$466$$ −23.5826 −1.09244
$$467$$ 8.00000i 0.370196i 0.982720 + 0.185098i $$0.0592602\pi$$
−0.982720 + 0.185098i $$0.940740\pi$$
$$468$$ − 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 $$-0.983186\pi$$
0.998605 0.0527980i $$-0.0168140\pi$$
$$488$$ − 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 $$-0.829343\pi$$
0.859689 0.510817i $$-0.170657\pi$$
$$504$$ 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 $$-0.740117\pi$$
0.684816 0.728716i $$-0.259883\pi$$
$$524$$ 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 $$-0.971619\pi$$
0.996028 0.0890445i $$-0.0283813\pi$$
$$548$$ 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 $$0.160063\pi$$
−0.876212 + 0.481926i $$0.839937\pi$$
$$558$$ 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 $$-0.900577\pi$$
0.951616 0.307291i $$-0.0994225\pi$$
$$564$$ −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 $$0.130618\pi$$
−0.916982 + 0.398928i $$0.869382\pi$$
$$578$$ − 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 $$0.0768301\pi$$
−0.971012 + 0.239032i $$0.923170\pi$$
$$588$$ − 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 $$0.0552915\pi$$
−0.984951 + 0.172831i $$0.944709\pi$$
$$594$$ −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 $$-0.976836\pi$$
0.997353 0.0727061i $$-0.0231635\pi$$
$$608$$ − 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 $$-0.655196\pi$$
0.468474 0.883477i $$-0.344804\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −7.08712 −0.285548
$$617$$ − 15.4955i − 0.623823i −0.950111 0.311912i $$-0.899031\pi$$
0.950111 0.311912i $$-0.100969\pi$$
$$618$$ 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 $$0.330101\pi$$
−0.508769 + 0.860903i $$0.669899\pi$$
$$644$$ 4.83485 0.190520
$$645$$ 0 0
$$646$$ −19.2523 −0.757471
$$647$$ − 24.7477i − 0.972934i −0.873699 0.486467i $$-0.838285\pi$$
0.873699 0.486467i $$-0.161715\pi$$
$$648$$ − 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 $$0.171081\pi$$
−0.859009 + 0.511961i $$0.828919\pi$$
$$654$$ −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 $$0.0107243\pi$$
−0.999433 + 0.0336850i $$0.989276\pi$$
$$674$$ 37.9129 1.46035
$$675$$ 0 0
$$676$$ 9.66970 0.371911
$$677$$ − 44.8258i − 1.72279i −0.507932 0.861397i $$-0.669590\pi$$
0.507932 0.861397i $$-0.330410\pi$$
$$678$$ − 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 $$0.0437728\pi$$
−0.990560 + 0.137083i $$0.956227\pi$$
$$684$$ −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 $$-0.942612\pi$$
0.983792 0.179315i $$-0.0573880\pi$$
$$728$$ 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 $$-0.748925\pi$$
0.704716 0.709490i $$-0.251075\pi$$
$$734$$ 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 $$0.183724\pi$$
−0.838001 + 0.545669i $$0.816276\pi$$
$$744$$ 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 $$0.0134839\pi$$
−0.999103 + 0.0423481i $$0.986516\pi$$
$$758$$ − 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 $$-0.970390\pi$$
0.995676 0.0928888i $$-0.0296101\pi$$
$$774$$ 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 $$0.140695\pi$$
−0.903895 + 0.427754i $$0.859305\pi$$
$$788$$ − 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 $$-0.805935\pi$$
0.819836 0.572599i $$-0.194065\pi$$
$$798$$ 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 $$0.349459\pi$$
−0.455503 + 0.890234i $$0.650541\pi$$
$$824$$ 21.4955 0.748830
$$825$$ 0 0
$$826$$ −13.5826 −0.472598
$$827$$ 43.8258i 1.52397i 0.647594 + 0.761985i $$0.275775\pi$$
−0.647594 + 0.761985i $$0.724225\pi$$
$$828$$ − 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 $$0.115585\pi$$
−0.934792 + 0.355194i $$0.884415\pi$$
$$854$$ −22.8348 −0.781392
$$855$$ 0 0
$$856$$ −7.32121 −0.250234
$$857$$ 46.6606i 1.59390i 0.604048 + 0.796948i $$0.293554\pi$$
−0.604048 + 0.796948i $$0.706446\pi$$
$$858$$ − 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 $$-0.965163\pi$$
0.994017 0.109226i $$-0.0348372\pi$$
$$864$$ 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 $$-0.892243\pi$$
0.943244 0.332099i $$-0.107757\pi$$
$$878$$ − 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 $$0.0867663\pi$$
−0.963078 + 0.269221i $$0.913234\pi$$
$$884$$ −16.6170 −0.558892
$$885$$ 0 0
$$886$$ 1.04356 0.0350591
$$887$$ 44.9129i 1.50803i 0.656859 + 0.754013i $$0.271885\pi$$
−0.656859 + 0.754013i $$0.728115\pi$$
$$888$$ 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 $$-0.871948\pi$$
0.920168 0.391523i $$-0.128052\pi$$
$$908$$ − 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 $$0.0981663\pi$$
−0.952821 + 0.303533i $$0.901834\pi$$
$$938$$ − 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 $$0.350849\pi$$
−0.451612 + 0.892214i $$0.649151\pi$$
$$948$$ 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 $$-0.791687\pi$$
0.793393 0.608710i $$-0.208313\pi$$
$$954$$ −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 $$-0.952470\pi$$
0.988872 0.148767i $$-0.0475303\pi$$
$$968$$ − 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 $$0.0127561\pi$$
−0.999197 + 0.0400638i $$0.987244\pi$$
$$978$$ 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 $$0.342288\pi$$
−0.475441 + 0.879748i $$0.657712\pi$$
$$984$$ −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 $$0.00420817\pi$$
−0.999913 + 0.0132200i $$0.995792\pi$$
$$998$$ − 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.349.2 4
5.2 odd 4 435.2.a.f.1.2 2
5.3 odd 4 2175.2.a.r.1.1 2
5.4 even 2 inner 2175.2.c.f.349.3 4
15.2 even 4 1305.2.a.m.1.1 2
15.8 even 4 6525.2.a.t.1.2 2
20.7 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 5.2 odd 4
1305.2.a.m.1.1 2 15.2 even 4
2175.2.a.r.1.1 2 5.3 odd 4
2175.2.c.f.349.2 4 1.1 even 1 trivial
2175.2.c.f.349.3 4 5.4 even 2 inner
6525.2.a.t.1.2 2 15.8 even 4
6960.2.a.bw.1.1 2 20.7 even 4