# Properties

 Label 2175.2.c.p.349.2 Level $2175$ Weight $2$ Character 2175.349 Analytic conductor $17.367$ Analytic rank $0$ Dimension $16$ 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: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 28x^{14} + 308x^{12} + 1671x^{10} + 4568x^{8} + 5616x^{6} + 2105x^{4} + 256x^{2} + 4$$ x^16 + 28*x^14 + 308*x^12 + 1671*x^10 + 4568*x^8 + 5616*x^6 + 2105*x^4 + 256*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.2 Root $$-2.59587i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.p.349.15

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.59587i q^{2} +1.00000i q^{3} -4.73855 q^{4} +2.59587 q^{6} +3.93826i q^{7} +7.10893i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-2.59587i q^{2} +1.00000i q^{3} -4.73855 q^{4} +2.59587 q^{6} +3.93826i q^{7} +7.10893i q^{8} -1.00000 q^{9} +2.24645 q^{11} -4.73855i q^{12} +2.09464i q^{13} +10.2232 q^{14} +8.97678 q^{16} -6.95310i q^{17} +2.59587i q^{18} -4.29591 q^{19} -3.93826 q^{21} -5.83149i q^{22} -5.19955i q^{23} -7.10893 q^{24} +5.43743 q^{26} -1.00000i q^{27} -18.6617i q^{28} -1.00000 q^{29} +2.25035 q^{31} -9.08471i q^{32} +2.24645i q^{33} -18.0494 q^{34} +4.73855 q^{36} +10.5280i q^{37} +11.1516i q^{38} -2.09464 q^{39} -2.66629 q^{41} +10.2232i q^{42} +6.03681i q^{43} -10.6449 q^{44} -13.4974 q^{46} +7.74829i q^{47} +8.97678i q^{48} -8.50991 q^{49} +6.95310 q^{51} -9.92559i q^{52} -1.41309i q^{53} -2.59587 q^{54} -27.9968 q^{56} -4.29591i q^{57} +2.59587i q^{58} -5.76800 q^{59} -10.9192 q^{61} -5.84162i q^{62} -3.93826i q^{63} -5.62918 q^{64} +5.83149 q^{66} +11.7471i q^{67} +32.9476i q^{68} +5.19955 q^{69} -4.55691 q^{71} -7.10893i q^{72} +12.3564i q^{73} +27.3294 q^{74} +20.3564 q^{76} +8.84710i q^{77} +5.43743i q^{78} -14.4098 q^{79} +1.00000 q^{81} +6.92135i q^{82} -3.81272i q^{83} +18.6617 q^{84} +15.6708 q^{86} -1.00000i q^{87} +15.9699i q^{88} +2.07340 q^{89} -8.24926 q^{91} +24.6383i q^{92} +2.25035i q^{93} +20.1136 q^{94} +9.08471 q^{96} -12.0068i q^{97} +22.0906i q^{98} -2.24645 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10})$$ 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9 $$16 q - 24 q^{4} + 4 q^{6} - 16 q^{9} + 12 q^{11} - 18 q^{14} + 64 q^{16} - 4 q^{21} - 6 q^{24} + 36 q^{26} - 16 q^{29} + 16 q^{31} + 26 q^{34} + 24 q^{36} + 12 q^{39} + 4 q^{41} + 30 q^{44} + 48 q^{46} - 76 q^{49} + 24 q^{51} - 4 q^{54} + 116 q^{56} - 36 q^{59} + 24 q^{61} - 42 q^{64} - 6 q^{66} - 28 q^{69} + 48 q^{71} + 44 q^{74} - 20 q^{79} + 16 q^{81} + 28 q^{84} + 16 q^{86} - 68 q^{89} + 52 q^{91} + 86 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100})$$ 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9 + 12 * q^11 - 18 * q^14 + 64 * q^16 - 4 * q^21 - 6 * q^24 + 36 * q^26 - 16 * q^29 + 16 * q^31 + 26 * q^34 + 24 * q^36 + 12 * q^39 + 4 * q^41 + 30 * q^44 + 48 * q^46 - 76 * q^49 + 24 * q^51 - 4 * q^54 + 116 * q^56 - 36 * q^59 + 24 * q^61 - 42 * q^64 - 6 * q^66 - 28 * q^69 + 48 * q^71 + 44 * q^74 - 20 * q^79 + 16 * q^81 + 28 * q^84 + 16 * q^86 - 68 * q^89 + 52 * q^91 + 86 * q^94 - 4 * q^96 - 12 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.59587i − 1.83556i −0.397090 0.917779i $$-0.629980\pi$$
0.397090 0.917779i $$-0.370020\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −4.73855 −2.36928
$$5$$ 0 0
$$6$$ 2.59587 1.05976
$$7$$ 3.93826i 1.48852i 0.667888 + 0.744262i $$0.267198\pi$$
−0.667888 + 0.744262i $$0.732802\pi$$
$$8$$ 7.10893i 2.51339i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.24645 0.677329 0.338665 0.940907i $$-0.390025\pi$$
0.338665 + 0.940907i $$0.390025\pi$$
$$12$$ − 4.73855i − 1.36790i
$$13$$ 2.09464i 0.580950i 0.956883 + 0.290475i $$0.0938133\pi$$
−0.956883 + 0.290475i $$0.906187\pi$$
$$14$$ 10.2232 2.73227
$$15$$ 0 0
$$16$$ 8.97678 2.24420
$$17$$ − 6.95310i − 1.68637i −0.537620 0.843187i $$-0.680676\pi$$
0.537620 0.843187i $$-0.319324\pi$$
$$18$$ 2.59587i 0.611853i
$$19$$ −4.29591 −0.985549 −0.492774 0.870157i $$-0.664017\pi$$
−0.492774 + 0.870157i $$0.664017\pi$$
$$20$$ 0 0
$$21$$ −3.93826 −0.859399
$$22$$ − 5.83149i − 1.24328i
$$23$$ − 5.19955i − 1.08418i −0.840320 0.542090i $$-0.817633\pi$$
0.840320 0.542090i $$-0.182367\pi$$
$$24$$ −7.10893 −1.45111
$$25$$ 0 0
$$26$$ 5.43743 1.06637
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 18.6617i − 3.52672i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 2.25035 0.404175 0.202087 0.979367i $$-0.435227\pi$$
0.202087 + 0.979367i $$0.435227\pi$$
$$32$$ − 9.08471i − 1.60596i
$$33$$ 2.24645i 0.391056i
$$34$$ −18.0494 −3.09544
$$35$$ 0 0
$$36$$ 4.73855 0.789759
$$37$$ 10.5280i 1.73079i 0.501086 + 0.865397i $$0.332934\pi$$
−0.501086 + 0.865397i $$0.667066\pi$$
$$38$$ 11.1516i 1.80903i
$$39$$ −2.09464 −0.335412
$$40$$ 0 0
$$41$$ −2.66629 −0.416405 −0.208202 0.978086i $$-0.566761\pi$$
−0.208202 + 0.978086i $$0.566761\pi$$
$$42$$ 10.2232i 1.57748i
$$43$$ 6.03681i 0.920605i 0.887762 + 0.460302i $$0.152259\pi$$
−0.887762 + 0.460302i $$0.847741\pi$$
$$44$$ −10.6449 −1.60478
$$45$$ 0 0
$$46$$ −13.4974 −1.99008
$$47$$ 7.74829i 1.13020i 0.825021 + 0.565102i $$0.191163\pi$$
−0.825021 + 0.565102i $$0.808837\pi$$
$$48$$ 8.97678i 1.29569i
$$49$$ −8.50991 −1.21570
$$50$$ 0 0
$$51$$ 6.95310 0.973629
$$52$$ − 9.92559i − 1.37643i
$$53$$ − 1.41309i − 0.194103i −0.995279 0.0970513i $$-0.969059\pi$$
0.995279 0.0970513i $$-0.0309411\pi$$
$$54$$ −2.59587 −0.353253
$$55$$ 0 0
$$56$$ −27.9968 −3.74124
$$57$$ − 4.29591i − 0.569007i
$$58$$ 2.59587i 0.340855i
$$59$$ −5.76800 −0.750929 −0.375465 0.926837i $$-0.622517\pi$$
−0.375465 + 0.926837i $$0.622517\pi$$
$$60$$ 0 0
$$61$$ −10.9192 −1.39806 −0.699030 0.715092i $$-0.746385\pi$$
−0.699030 + 0.715092i $$0.746385\pi$$
$$62$$ − 5.84162i − 0.741887i
$$63$$ − 3.93826i − 0.496174i
$$64$$ −5.62918 −0.703647
$$65$$ 0 0
$$66$$ 5.83149 0.717807
$$67$$ 11.7471i 1.43514i 0.696487 + 0.717569i $$0.254745\pi$$
−0.696487 + 0.717569i $$0.745255\pi$$
$$68$$ 32.9476i 3.99549i
$$69$$ 5.19955 0.625952
$$70$$ 0 0
$$71$$ −4.55691 −0.540806 −0.270403 0.962747i $$-0.587157\pi$$
−0.270403 + 0.962747i $$0.587157\pi$$
$$72$$ − 7.10893i − 0.837796i
$$73$$ 12.3564i 1.44621i 0.690740 + 0.723103i $$0.257285\pi$$
−0.690740 + 0.723103i $$0.742715\pi$$
$$74$$ 27.3294 3.17698
$$75$$ 0 0
$$76$$ 20.3564 2.33504
$$77$$ 8.84710i 1.00822i
$$78$$ 5.43743i 0.615668i
$$79$$ −14.4098 −1.62122 −0.810612 0.585584i $$-0.800865\pi$$
−0.810612 + 0.585584i $$0.800865\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.92135i 0.764335i
$$83$$ − 3.81272i − 0.418501i −0.977862 0.209250i $$-0.932898\pi$$
0.977862 0.209250i $$-0.0671024\pi$$
$$84$$ 18.6617 2.03615
$$85$$ 0 0
$$86$$ 15.6708 1.68982
$$87$$ − 1.00000i − 0.107211i
$$88$$ 15.9699i 1.70239i
$$89$$ 2.07340 0.219780 0.109890 0.993944i $$-0.464950\pi$$
0.109890 + 0.993944i $$0.464950\pi$$
$$90$$ 0 0
$$91$$ −8.24926 −0.864757
$$92$$ 24.6383i 2.56872i
$$93$$ 2.25035i 0.233350i
$$94$$ 20.1136 2.07456
$$95$$ 0 0
$$96$$ 9.08471 0.927204
$$97$$ − 12.0068i − 1.21911i −0.792745 0.609553i $$-0.791349\pi$$
0.792745 0.609553i $$-0.208651\pi$$
$$98$$ 22.0906i 2.23149i
$$99$$ −2.24645 −0.225776
$$100$$ 0 0
$$101$$ −15.8642 −1.57855 −0.789276 0.614039i $$-0.789544\pi$$
−0.789276 + 0.614039i $$0.789544\pi$$
$$102$$ − 18.0494i − 1.78715i
$$103$$ − 10.3119i − 1.01606i −0.861339 0.508031i $$-0.830373\pi$$
0.861339 0.508031i $$-0.169627\pi$$
$$104$$ −14.8907 −1.46015
$$105$$ 0 0
$$106$$ −3.66820 −0.356287
$$107$$ − 4.47645i − 0.432755i −0.976310 0.216378i $$-0.930576\pi$$
0.976310 0.216378i $$-0.0694242\pi$$
$$108$$ 4.73855i 0.455967i
$$109$$ −13.4783 −1.29098 −0.645492 0.763767i $$-0.723348\pi$$
−0.645492 + 0.763767i $$0.723348\pi$$
$$110$$ 0 0
$$111$$ −10.5280 −0.999275
$$112$$ 35.3529i 3.34054i
$$113$$ − 11.4525i − 1.07736i −0.842511 0.538679i $$-0.818923\pi$$
0.842511 0.538679i $$-0.181077\pi$$
$$114$$ −11.1516 −1.04445
$$115$$ 0 0
$$116$$ 4.73855 0.439964
$$117$$ − 2.09464i − 0.193650i
$$118$$ 14.9730i 1.37838i
$$119$$ 27.3831 2.51021
$$120$$ 0 0
$$121$$ −5.95347 −0.541225
$$122$$ 28.3449i 2.56622i
$$123$$ − 2.66629i − 0.240411i
$$124$$ −10.6634 −0.957602
$$125$$ 0 0
$$126$$ −10.2232 −0.910757
$$127$$ 3.23499i 0.287059i 0.989646 + 0.143529i $$0.0458451\pi$$
−0.989646 + 0.143529i $$0.954155\pi$$
$$128$$ − 3.55679i − 0.314378i
$$129$$ −6.03681 −0.531511
$$130$$ 0 0
$$131$$ −13.3567 −1.16698 −0.583489 0.812121i $$-0.698313\pi$$
−0.583489 + 0.812121i $$0.698313\pi$$
$$132$$ − 10.6449i − 0.926521i
$$133$$ − 16.9184i − 1.46701i
$$134$$ 30.4940 2.63428
$$135$$ 0 0
$$136$$ 49.4291 4.23851
$$137$$ − 4.35642i − 0.372194i −0.982531 0.186097i $$-0.940416\pi$$
0.982531 0.186097i $$-0.0595839\pi$$
$$138$$ − 13.4974i − 1.14897i
$$139$$ 18.4171 1.56212 0.781061 0.624455i $$-0.214679\pi$$
0.781061 + 0.624455i $$0.214679\pi$$
$$140$$ 0 0
$$141$$ −7.74829 −0.652523
$$142$$ 11.8292i 0.992681i
$$143$$ 4.70551i 0.393495i
$$144$$ −8.97678 −0.748065
$$145$$ 0 0
$$146$$ 32.0756 2.65460
$$147$$ − 8.50991i − 0.701885i
$$148$$ − 49.8876i − 4.10073i
$$149$$ −17.6427 −1.44535 −0.722674 0.691189i $$-0.757087\pi$$
−0.722674 + 0.691189i $$0.757087\pi$$
$$150$$ 0 0
$$151$$ 4.02381 0.327453 0.163726 0.986506i $$-0.447649\pi$$
0.163726 + 0.986506i $$0.447649\pi$$
$$152$$ − 30.5393i − 2.47707i
$$153$$ 6.95310i 0.562125i
$$154$$ 22.9659 1.85065
$$155$$ 0 0
$$156$$ 9.92559 0.794683
$$157$$ 7.70119i 0.614622i 0.951609 + 0.307311i $$0.0994292\pi$$
−0.951609 + 0.307311i $$0.900571\pi$$
$$158$$ 37.4059i 2.97585i
$$159$$ 1.41309 0.112065
$$160$$ 0 0
$$161$$ 20.4772 1.61383
$$162$$ − 2.59587i − 0.203951i
$$163$$ − 13.6829i − 1.07173i −0.844303 0.535866i $$-0.819985\pi$$
0.844303 0.535866i $$-0.180015\pi$$
$$164$$ 12.6344 0.986578
$$165$$ 0 0
$$166$$ −9.89734 −0.768183
$$167$$ 24.9607i 1.93151i 0.259449 + 0.965757i $$0.416459\pi$$
−0.259449 + 0.965757i $$0.583541\pi$$
$$168$$ − 27.9968i − 2.16000i
$$169$$ 8.61246 0.662497
$$170$$ 0 0
$$171$$ 4.29591 0.328516
$$172$$ − 28.6057i − 2.18117i
$$173$$ 19.9187i 1.51439i 0.653191 + 0.757194i $$0.273430\pi$$
−0.653191 + 0.757194i $$0.726570\pi$$
$$174$$ −2.59587 −0.196793
$$175$$ 0 0
$$176$$ 20.1659 1.52006
$$177$$ − 5.76800i − 0.433549i
$$178$$ − 5.38227i − 0.403418i
$$179$$ −10.8816 −0.813329 −0.406664 0.913578i $$-0.633308\pi$$
−0.406664 + 0.913578i $$0.633308\pi$$
$$180$$ 0 0
$$181$$ −13.9372 −1.03594 −0.517972 0.855398i $$-0.673313\pi$$
−0.517972 + 0.855398i $$0.673313\pi$$
$$182$$ 21.4140i 1.58731i
$$183$$ − 10.9192i − 0.807171i
$$184$$ 36.9632 2.72497
$$185$$ 0 0
$$186$$ 5.84162 0.428329
$$187$$ − 15.6198i − 1.14223i
$$188$$ − 36.7157i − 2.67777i
$$189$$ 3.93826 0.286466
$$190$$ 0 0
$$191$$ 13.3458 0.965670 0.482835 0.875711i $$-0.339607\pi$$
0.482835 + 0.875711i $$0.339607\pi$$
$$192$$ − 5.62918i − 0.406251i
$$193$$ 15.6337i 1.12534i 0.826683 + 0.562668i $$0.190225\pi$$
−0.826683 + 0.562668i $$0.809775\pi$$
$$194$$ −31.1681 −2.23774
$$195$$ 0 0
$$196$$ 40.3247 2.88033
$$197$$ 20.1140i 1.43306i 0.697556 + 0.716530i $$0.254271\pi$$
−0.697556 + 0.716530i $$0.745729\pi$$
$$198$$ 5.83149i 0.414426i
$$199$$ 9.27232 0.657297 0.328649 0.944452i $$-0.393407\pi$$
0.328649 + 0.944452i $$0.393407\pi$$
$$200$$ 0 0
$$201$$ −11.7471 −0.828578
$$202$$ 41.1816i 2.89752i
$$203$$ − 3.93826i − 0.276412i
$$204$$ −32.9476 −2.30680
$$205$$ 0 0
$$206$$ −26.7684 −1.86504
$$207$$ 5.19955i 0.361394i
$$208$$ 18.8032i 1.30376i
$$209$$ −9.65053 −0.667541
$$210$$ 0 0
$$211$$ −0.228998 −0.0157649 −0.00788245 0.999969i $$-0.502509\pi$$
−0.00788245 + 0.999969i $$0.502509\pi$$
$$212$$ 6.69600i 0.459883i
$$213$$ − 4.55691i − 0.312235i
$$214$$ −11.6203 −0.794348
$$215$$ 0 0
$$216$$ 7.10893 0.483702
$$217$$ 8.86247i 0.601624i
$$218$$ 34.9879i 2.36968i
$$219$$ −12.3564 −0.834968
$$220$$ 0 0
$$221$$ 14.5643 0.979699
$$222$$ 27.3294i 1.83423i
$$223$$ − 25.1117i − 1.68160i −0.541342 0.840802i $$-0.682084\pi$$
0.541342 0.840802i $$-0.317916\pi$$
$$224$$ 35.7780 2.39052
$$225$$ 0 0
$$226$$ −29.7292 −1.97756
$$227$$ 4.98706i 0.331003i 0.986210 + 0.165501i $$0.0529242\pi$$
−0.986210 + 0.165501i $$0.947076\pi$$
$$228$$ 20.3564i 1.34813i
$$229$$ −1.60591 −0.106122 −0.0530608 0.998591i $$-0.516898\pi$$
−0.0530608 + 0.998591i $$0.516898\pi$$
$$230$$ 0 0
$$231$$ −8.84710 −0.582096
$$232$$ − 7.10893i − 0.466724i
$$233$$ 6.65588i 0.436041i 0.975944 + 0.218021i $$0.0699600\pi$$
−0.975944 + 0.218021i $$0.930040\pi$$
$$234$$ −5.43743 −0.355456
$$235$$ 0 0
$$236$$ 27.3320 1.77916
$$237$$ − 14.4098i − 0.936014i
$$238$$ − 71.0831i − 4.60763i
$$239$$ −2.87227 −0.185792 −0.0928959 0.995676i $$-0.529612\pi$$
−0.0928959 + 0.995676i $$0.529612\pi$$
$$240$$ 0 0
$$241$$ 15.2768 0.984068 0.492034 0.870576i $$-0.336254\pi$$
0.492034 + 0.870576i $$0.336254\pi$$
$$242$$ 15.4545i 0.993450i
$$243$$ 1.00000i 0.0641500i
$$244$$ 51.7412 3.31239
$$245$$ 0 0
$$246$$ −6.92135 −0.441289
$$247$$ − 8.99840i − 0.572554i
$$248$$ 15.9976i 1.01585i
$$249$$ 3.81272 0.241621
$$250$$ 0 0
$$251$$ 10.8782 0.686627 0.343314 0.939221i $$-0.388451\pi$$
0.343314 + 0.939221i $$0.388451\pi$$
$$252$$ 18.6617i 1.17557i
$$253$$ − 11.6805i − 0.734348i
$$254$$ 8.39761 0.526913
$$255$$ 0 0
$$256$$ −20.4913 −1.28071
$$257$$ 18.0748i 1.12748i 0.825953 + 0.563739i $$0.190637\pi$$
−0.825953 + 0.563739i $$0.809363\pi$$
$$258$$ 15.6708i 0.975621i
$$259$$ −41.4621 −2.57633
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 34.6722i 2.14206i
$$263$$ − 18.4454i − 1.13739i −0.822547 0.568697i $$-0.807448\pi$$
0.822547 0.568697i $$-0.192552\pi$$
$$264$$ −15.9699 −0.982876
$$265$$ 0 0
$$266$$ −43.9180 −2.69279
$$267$$ 2.07340i 0.126890i
$$268$$ − 55.6643i − 3.40024i
$$269$$ −11.6702 −0.711544 −0.355772 0.934573i $$-0.615782\pi$$
−0.355772 + 0.934573i $$0.615782\pi$$
$$270$$ 0 0
$$271$$ 20.2501 1.23011 0.615054 0.788485i $$-0.289134\pi$$
0.615054 + 0.788485i $$0.289134\pi$$
$$272$$ − 62.4165i − 3.78455i
$$273$$ − 8.24926i − 0.499268i
$$274$$ −11.3087 −0.683184
$$275$$ 0 0
$$276$$ −24.6383 −1.48305
$$277$$ − 3.66952i − 0.220480i −0.993905 0.110240i $$-0.964838\pi$$
0.993905 0.110240i $$-0.0351620\pi$$
$$278$$ − 47.8085i − 2.86737i
$$279$$ −2.25035 −0.134725
$$280$$ 0 0
$$281$$ 22.8056 1.36047 0.680233 0.732996i $$-0.261879\pi$$
0.680233 + 0.732996i $$0.261879\pi$$
$$282$$ 20.1136i 1.19775i
$$283$$ − 2.38657i − 0.141867i −0.997481 0.0709335i $$-0.977402\pi$$
0.997481 0.0709335i $$-0.0225978\pi$$
$$284$$ 21.5932 1.28132
$$285$$ 0 0
$$286$$ 12.2149 0.722282
$$287$$ − 10.5006i − 0.619828i
$$288$$ 9.08471i 0.535321i
$$289$$ −31.3456 −1.84386
$$290$$ 0 0
$$291$$ 12.0068 0.703852
$$292$$ − 58.5514i − 3.42646i
$$293$$ 19.4843i 1.13828i 0.822239 + 0.569142i $$0.192725\pi$$
−0.822239 + 0.569142i $$0.807275\pi$$
$$294$$ −22.0906 −1.28835
$$295$$ 0 0
$$296$$ −74.8430 −4.35016
$$297$$ − 2.24645i − 0.130352i
$$298$$ 45.7982i 2.65302i
$$299$$ 10.8912 0.629855
$$300$$ 0 0
$$301$$ −23.7745 −1.37034
$$302$$ − 10.4453i − 0.601059i
$$303$$ − 15.8642i − 0.911377i
$$304$$ −38.5634 −2.21176
$$305$$ 0 0
$$306$$ 18.0494 1.03181
$$307$$ 0.527333i 0.0300965i 0.999887 + 0.0150482i $$0.00479018\pi$$
−0.999887 + 0.0150482i $$0.995210\pi$$
$$308$$ − 41.9225i − 2.38875i
$$309$$ 10.3119 0.586624
$$310$$ 0 0
$$311$$ −7.89624 −0.447755 −0.223877 0.974617i $$-0.571871\pi$$
−0.223877 + 0.974617i $$0.571871\pi$$
$$312$$ − 14.8907i − 0.843019i
$$313$$ 10.0064i 0.565597i 0.959179 + 0.282799i $$0.0912628\pi$$
−0.959179 + 0.282799i $$0.908737\pi$$
$$314$$ 19.9913 1.12818
$$315$$ 0 0
$$316$$ 68.2814 3.84113
$$317$$ − 32.5401i − 1.82763i −0.406125 0.913817i $$-0.633120\pi$$
0.406125 0.913817i $$-0.366880\pi$$
$$318$$ − 3.66820i − 0.205702i
$$319$$ −2.24645 −0.125777
$$320$$ 0 0
$$321$$ 4.47645 0.249851
$$322$$ − 53.1562i − 2.96228i
$$323$$ 29.8699i 1.66200i
$$324$$ −4.73855 −0.263253
$$325$$ 0 0
$$326$$ −35.5192 −1.96723
$$327$$ − 13.4783i − 0.745350i
$$328$$ − 18.9545i − 1.04659i
$$329$$ −30.5148 −1.68233
$$330$$ 0 0
$$331$$ 22.9411 1.26096 0.630480 0.776206i $$-0.282858\pi$$
0.630480 + 0.776206i $$0.282858\pi$$
$$332$$ 18.0668i 0.991544i
$$333$$ − 10.5280i − 0.576932i
$$334$$ 64.7947 3.54541
$$335$$ 0 0
$$336$$ −35.3529 −1.92866
$$337$$ − 16.0039i − 0.871789i −0.899998 0.435894i $$-0.856432\pi$$
0.899998 0.435894i $$-0.143568\pi$$
$$338$$ − 22.3569i − 1.21605i
$$339$$ 11.4525 0.622013
$$340$$ 0 0
$$341$$ 5.05529 0.273760
$$342$$ − 11.1516i − 0.603011i
$$343$$ − 5.94642i − 0.321076i
$$344$$ −42.9153 −2.31384
$$345$$ 0 0
$$346$$ 51.7063 2.77975
$$347$$ − 22.9990i − 1.23465i −0.786707 0.617326i $$-0.788216\pi$$
0.786707 0.617326i $$-0.211784\pi$$
$$348$$ 4.73855i 0.254013i
$$349$$ −24.3108 −1.30133 −0.650664 0.759366i $$-0.725509\pi$$
−0.650664 + 0.759366i $$0.725509\pi$$
$$350$$ 0 0
$$351$$ 2.09464 0.111804
$$352$$ − 20.4083i − 1.08777i
$$353$$ − 29.2685i − 1.55780i −0.627146 0.778902i $$-0.715777\pi$$
0.627146 0.778902i $$-0.284223\pi$$
$$354$$ −14.9730 −0.795805
$$355$$ 0 0
$$356$$ −9.82490 −0.520719
$$357$$ 27.3831i 1.44927i
$$358$$ 28.2472i 1.49291i
$$359$$ −18.5090 −0.976868 −0.488434 0.872601i $$-0.662432\pi$$
−0.488434 + 0.872601i $$0.662432\pi$$
$$360$$ 0 0
$$361$$ −0.545179 −0.0286936
$$362$$ 36.1792i 1.90154i
$$363$$ − 5.95347i − 0.312476i
$$364$$ 39.0896 2.04885
$$365$$ 0 0
$$366$$ −28.3449 −1.48161
$$367$$ − 0.896398i − 0.0467916i −0.999726 0.0233958i $$-0.992552\pi$$
0.999726 0.0233958i $$-0.00744779\pi$$
$$368$$ − 46.6752i − 2.43311i
$$369$$ 2.66629 0.138802
$$370$$ 0 0
$$371$$ 5.56511 0.288926
$$372$$ − 10.6634i − 0.552872i
$$373$$ 4.64559i 0.240540i 0.992741 + 0.120270i $$0.0383760\pi$$
−0.992741 + 0.120270i $$0.961624\pi$$
$$374$$ −40.5469 −2.09663
$$375$$ 0 0
$$376$$ −55.0821 −2.84064
$$377$$ − 2.09464i − 0.107880i
$$378$$ − 10.2232i − 0.525826i
$$379$$ 32.4863 1.66871 0.834356 0.551226i $$-0.185840\pi$$
0.834356 + 0.551226i $$0.185840\pi$$
$$380$$ 0 0
$$381$$ −3.23499 −0.165733
$$382$$ − 34.6441i − 1.77254i
$$383$$ 9.48311i 0.484564i 0.970206 + 0.242282i $$0.0778959\pi$$
−0.970206 + 0.242282i $$0.922104\pi$$
$$384$$ 3.55679 0.181507
$$385$$ 0 0
$$386$$ 40.5830 2.06562
$$387$$ − 6.03681i − 0.306868i
$$388$$ 56.8949i 2.88840i
$$389$$ −33.9509 −1.72138 −0.860689 0.509132i $$-0.829967\pi$$
−0.860689 + 0.509132i $$0.829967\pi$$
$$390$$ 0 0
$$391$$ −36.1530 −1.82833
$$392$$ − 60.4964i − 3.05553i
$$393$$ − 13.3567i − 0.673755i
$$394$$ 52.2132 2.63047
$$395$$ 0 0
$$396$$ 10.6449 0.534927
$$397$$ 25.0892i 1.25919i 0.776923 + 0.629596i $$0.216780\pi$$
−0.776923 + 0.629596i $$0.783220\pi$$
$$398$$ − 24.0698i − 1.20651i
$$399$$ 16.9184 0.846980
$$400$$ 0 0
$$401$$ −4.85814 −0.242604 −0.121302 0.992616i $$-0.538707\pi$$
−0.121302 + 0.992616i $$0.538707\pi$$
$$402$$ 30.4940i 1.52090i
$$403$$ 4.71368i 0.234805i
$$404$$ 75.1736 3.74003
$$405$$ 0 0
$$406$$ −10.2232 −0.507370
$$407$$ 23.6506i 1.17232i
$$408$$ 49.4291i 2.44711i
$$409$$ −14.6604 −0.724911 −0.362456 0.932001i $$-0.618062\pi$$
−0.362456 + 0.932001i $$0.618062\pi$$
$$410$$ 0 0
$$411$$ 4.35642 0.214886
$$412$$ 48.8635i 2.40733i
$$413$$ − 22.7159i − 1.11778i
$$414$$ 13.4974 0.663359
$$415$$ 0 0
$$416$$ 19.0292 0.932985
$$417$$ 18.4171i 0.901891i
$$418$$ 25.0515i 1.22531i
$$419$$ 18.4123 0.899502 0.449751 0.893154i $$-0.351513\pi$$
0.449751 + 0.893154i $$0.351513\pi$$
$$420$$ 0 0
$$421$$ 29.1489 1.42063 0.710316 0.703883i $$-0.248552\pi$$
0.710316 + 0.703883i $$0.248552\pi$$
$$422$$ 0.594451i 0.0289374i
$$423$$ − 7.74829i − 0.376735i
$$424$$ 10.0456 0.487855
$$425$$ 0 0
$$426$$ −11.8292 −0.573125
$$427$$ − 43.0027i − 2.08105i
$$428$$ 21.2119i 1.02532i
$$429$$ −4.70551 −0.227184
$$430$$ 0 0
$$431$$ −2.05315 −0.0988966 −0.0494483 0.998777i $$-0.515746\pi$$
−0.0494483 + 0.998777i $$0.515746\pi$$
$$432$$ − 8.97678i − 0.431896i
$$433$$ − 18.7820i − 0.902603i −0.892371 0.451302i $$-0.850960\pi$$
0.892371 0.451302i $$-0.149040\pi$$
$$434$$ 23.0058 1.10432
$$435$$ 0 0
$$436$$ 63.8675 3.05870
$$437$$ 22.3368i 1.06851i
$$438$$ 32.0756i 1.53263i
$$439$$ 33.7889 1.61266 0.806329 0.591467i $$-0.201451\pi$$
0.806329 + 0.591467i $$0.201451\pi$$
$$440$$ 0 0
$$441$$ 8.50991 0.405234
$$442$$ − 37.8070i − 1.79830i
$$443$$ 24.2962i 1.15435i 0.816622 + 0.577173i $$0.195844\pi$$
−0.816622 + 0.577173i $$0.804156\pi$$
$$444$$ 49.8876 2.36756
$$445$$ 0 0
$$446$$ −65.1868 −3.08668
$$447$$ − 17.6427i − 0.834472i
$$448$$ − 22.1692i − 1.04740i
$$449$$ 21.0976 0.995656 0.497828 0.867276i $$-0.334131\pi$$
0.497828 + 0.867276i $$0.334131\pi$$
$$450$$ 0 0
$$451$$ −5.98968 −0.282043
$$452$$ 54.2682i 2.55256i
$$453$$ 4.02381i 0.189055i
$$454$$ 12.9458 0.607575
$$455$$ 0 0
$$456$$ 30.5393 1.43013
$$457$$ − 8.70983i − 0.407429i −0.979030 0.203714i $$-0.934699\pi$$
0.979030 0.203714i $$-0.0653014\pi$$
$$458$$ 4.16874i 0.194793i
$$459$$ −6.95310 −0.324543
$$460$$ 0 0
$$461$$ 29.5424 1.37592 0.687962 0.725746i $$-0.258505\pi$$
0.687962 + 0.725746i $$0.258505\pi$$
$$462$$ 22.9659i 1.06847i
$$463$$ 37.3067i 1.73379i 0.498493 + 0.866894i $$0.333887\pi$$
−0.498493 + 0.866894i $$0.666113\pi$$
$$464$$ −8.97678 −0.416737
$$465$$ 0 0
$$466$$ 17.2778 0.800379
$$467$$ − 30.2144i − 1.39816i −0.715045 0.699079i $$-0.753594\pi$$
0.715045 0.699079i $$-0.246406\pi$$
$$468$$ 9.92559i 0.458810i
$$469$$ −46.2632 −2.13624
$$470$$ 0 0
$$471$$ −7.70119 −0.354852
$$472$$ − 41.0043i − 1.88738i
$$473$$ 13.5614i 0.623553i
$$474$$ −37.4059 −1.71811
$$475$$ 0 0
$$476$$ −129.756 −5.94738
$$477$$ 1.41309i 0.0647009i
$$478$$ 7.45605i 0.341032i
$$479$$ −39.1511 −1.78886 −0.894430 0.447209i $$-0.852418\pi$$
−0.894430 + 0.447209i $$0.852418\pi$$
$$480$$ 0 0
$$481$$ −22.0524 −1.00551
$$482$$ − 39.6567i − 1.80632i
$$483$$ 20.4772i 0.931744i
$$484$$ 28.2108 1.28231
$$485$$ 0 0
$$486$$ 2.59587 0.117751
$$487$$ 23.9182i 1.08384i 0.840430 + 0.541919i $$0.182302\pi$$
−0.840430 + 0.541919i $$0.817698\pi$$
$$488$$ − 77.6239i − 3.51387i
$$489$$ 13.6829 0.618764
$$490$$ 0 0
$$491$$ 4.40974 0.199009 0.0995044 0.995037i $$-0.468274\pi$$
0.0995044 + 0.995037i $$0.468274\pi$$
$$492$$ 12.6344i 0.569601i
$$493$$ 6.95310i 0.313152i
$$494$$ −23.3587 −1.05096
$$495$$ 0 0
$$496$$ 20.2009 0.907047
$$497$$ − 17.9463i − 0.805002i
$$498$$ − 9.89734i − 0.443510i
$$499$$ 37.0446 1.65834 0.829171 0.558995i $$-0.188813\pi$$
0.829171 + 0.558995i $$0.188813\pi$$
$$500$$ 0 0
$$501$$ −24.9607 −1.11516
$$502$$ − 28.2385i − 1.26034i
$$503$$ − 12.6691i − 0.564886i −0.959284 0.282443i $$-0.908855\pi$$
0.959284 0.282443i $$-0.0911448\pi$$
$$504$$ 27.9968 1.24708
$$505$$ 0 0
$$506$$ −30.3211 −1.34794
$$507$$ 8.61246i 0.382493i
$$508$$ − 15.3292i − 0.680121i
$$509$$ 11.7078 0.518939 0.259469 0.965751i $$-0.416452\pi$$
0.259469 + 0.965751i $$0.416452\pi$$
$$510$$ 0 0
$$511$$ −48.6627 −2.15271
$$512$$ 46.0793i 2.03644i
$$513$$ 4.29591i 0.189669i
$$514$$ 46.9200 2.06955
$$515$$ 0 0
$$516$$ 28.6057 1.25930
$$517$$ 17.4061i 0.765520i
$$518$$ 107.630i 4.72900i
$$519$$ −19.9187 −0.874332
$$520$$ 0 0
$$521$$ 0.0559550 0.00245143 0.00122572 0.999999i $$-0.499610\pi$$
0.00122572 + 0.999999i $$0.499610\pi$$
$$522$$ − 2.59587i − 0.113618i
$$523$$ − 0.256669i − 0.0112234i −0.999984 0.00561168i $$-0.998214\pi$$
0.999984 0.00561168i $$-0.00178626\pi$$
$$524$$ 63.2913 2.76489
$$525$$ 0 0
$$526$$ −47.8819 −2.08775
$$527$$ − 15.6469i − 0.681590i
$$528$$ 20.1659i 0.877607i
$$529$$ −4.03530 −0.175448
$$530$$ 0 0
$$531$$ 5.76800 0.250310
$$532$$ 80.1688i 3.47576i
$$533$$ − 5.58493i − 0.241910i
$$534$$ 5.38227 0.232914
$$535$$ 0 0
$$536$$ −83.5095 −3.60706
$$537$$ − 10.8816i − 0.469576i
$$538$$ 30.2943i 1.30608i
$$539$$ −19.1171 −0.823430
$$540$$ 0 0
$$541$$ 1.88887 0.0812090 0.0406045 0.999175i $$-0.487072\pi$$
0.0406045 + 0.999175i $$0.487072\pi$$
$$542$$ − 52.5668i − 2.25794i
$$543$$ − 13.9372i − 0.598102i
$$544$$ −63.1669 −2.70826
$$545$$ 0 0
$$546$$ −21.4140 −0.916436
$$547$$ 0.193341i 0.00826668i 0.999991 + 0.00413334i $$0.00131569\pi$$
−0.999991 + 0.00413334i $$0.998684\pi$$
$$548$$ 20.6431i 0.881831i
$$549$$ 10.9192 0.466020
$$550$$ 0 0
$$551$$ 4.29591 0.183012
$$552$$ 36.9632i 1.57326i
$$553$$ − 56.7494i − 2.41323i
$$554$$ −9.52561 −0.404704
$$555$$ 0 0
$$556$$ −87.2706 −3.70110
$$557$$ 26.4282i 1.11980i 0.828561 + 0.559899i $$0.189160\pi$$
−0.828561 + 0.559899i $$0.810840\pi$$
$$558$$ 5.84162i 0.247296i
$$559$$ −12.6450 −0.534825
$$560$$ 0 0
$$561$$ 15.6198 0.659467
$$562$$ − 59.2003i − 2.49722i
$$563$$ − 6.18961i − 0.260861i −0.991457 0.130430i $$-0.958364\pi$$
0.991457 0.130430i $$-0.0416359\pi$$
$$564$$ 36.7157 1.54601
$$565$$ 0 0
$$566$$ −6.19523 −0.260405
$$567$$ 3.93826i 0.165391i
$$568$$ − 32.3948i − 1.35926i
$$569$$ 15.9034 0.666705 0.333353 0.942802i $$-0.391820\pi$$
0.333353 + 0.942802i $$0.391820\pi$$
$$570$$ 0 0
$$571$$ −16.8555 −0.705380 −0.352690 0.935740i $$-0.614733\pi$$
−0.352690 + 0.935740i $$0.614733\pi$$
$$572$$ − 22.2973i − 0.932297i
$$573$$ 13.3458i 0.557530i
$$574$$ −27.2581 −1.13773
$$575$$ 0 0
$$576$$ 5.62918 0.234549
$$577$$ 23.1237i 0.962650i 0.876542 + 0.481325i $$0.159844\pi$$
−0.876542 + 0.481325i $$0.840156\pi$$
$$578$$ 81.3692i 3.38451i
$$579$$ −15.6337 −0.649713
$$580$$ 0 0
$$581$$ 15.0155 0.622948
$$582$$ − 31.1681i − 1.29196i
$$583$$ − 3.17443i − 0.131471i
$$584$$ −87.8408 −3.63488
$$585$$ 0 0
$$586$$ 50.5787 2.08939
$$587$$ 7.80902i 0.322313i 0.986929 + 0.161156i $$0.0515224\pi$$
−0.986929 + 0.161156i $$0.948478\pi$$
$$588$$ 40.3247i 1.66296i
$$589$$ −9.66730 −0.398334
$$590$$ 0 0
$$591$$ −20.1140 −0.827377
$$592$$ 94.5077i 3.88424i
$$593$$ 19.5804i 0.804069i 0.915625 + 0.402034i $$0.131697\pi$$
−0.915625 + 0.402034i $$0.868303\pi$$
$$594$$ −5.83149 −0.239269
$$595$$ 0 0
$$596$$ 83.6009 3.42443
$$597$$ 9.27232i 0.379491i
$$598$$ − 28.2722i − 1.15614i
$$599$$ −30.2322 −1.23525 −0.617627 0.786471i $$-0.711906\pi$$
−0.617627 + 0.786471i $$0.711906\pi$$
$$600$$ 0 0
$$601$$ 33.1180 1.35091 0.675457 0.737400i $$-0.263947\pi$$
0.675457 + 0.737400i $$0.263947\pi$$
$$602$$ 61.7157i 2.51534i
$$603$$ − 11.7471i − 0.478380i
$$604$$ −19.0670 −0.775826
$$605$$ 0 0
$$606$$ −41.1816 −1.67289
$$607$$ 23.4711i 0.952661i 0.879266 + 0.476330i $$0.158033\pi$$
−0.879266 + 0.476330i $$0.841967\pi$$
$$608$$ 39.0271i 1.58276i
$$609$$ 3.93826 0.159586
$$610$$ 0 0
$$611$$ −16.2299 −0.656592
$$612$$ − 32.9476i − 1.33183i
$$613$$ 24.1560i 0.975650i 0.872941 + 0.487825i $$0.162210\pi$$
−0.872941 + 0.487825i $$0.837790\pi$$
$$614$$ 1.36889 0.0552439
$$615$$ 0 0
$$616$$ −62.8935 −2.53405
$$617$$ 47.7509i 1.92238i 0.275890 + 0.961189i $$0.411027\pi$$
−0.275890 + 0.961189i $$0.588973\pi$$
$$618$$ − 26.7684i − 1.07678i
$$619$$ −0.779735 −0.0313402 −0.0156701 0.999877i $$-0.504988\pi$$
−0.0156701 + 0.999877i $$0.504988\pi$$
$$620$$ 0 0
$$621$$ −5.19955 −0.208651
$$622$$ 20.4976i 0.821880i
$$623$$ 8.16558i 0.327147i
$$624$$ −18.8032 −0.752729
$$625$$ 0 0
$$626$$ 25.9754 1.03819
$$627$$ − 9.65053i − 0.385405i
$$628$$ − 36.4925i − 1.45621i
$$629$$ 73.2023 2.91877
$$630$$ 0 0
$$631$$ 17.8120 0.709083 0.354542 0.935040i $$-0.384637\pi$$
0.354542 + 0.935040i $$0.384637\pi$$
$$632$$ − 102.438i − 4.07476i
$$633$$ − 0.228998i − 0.00910187i
$$634$$ −84.4700 −3.35473
$$635$$ 0 0
$$636$$ −6.69600 −0.265514
$$637$$ − 17.8252i − 0.706262i
$$638$$ 5.83149i 0.230871i
$$639$$ 4.55691 0.180269
$$640$$ 0 0
$$641$$ 29.7647 1.17563 0.587817 0.808994i $$-0.299987\pi$$
0.587817 + 0.808994i $$0.299987\pi$$
$$642$$ − 11.6203i − 0.458617i
$$643$$ − 41.4049i − 1.63285i −0.577451 0.816426i $$-0.695952\pi$$
0.577451 0.816426i $$-0.304048\pi$$
$$644$$ −97.0322 −3.82361
$$645$$ 0 0
$$646$$ 77.5384 3.05071
$$647$$ − 14.6074i − 0.574275i −0.957889 0.287137i $$-0.907296\pi$$
0.957889 0.287137i $$-0.0927036\pi$$
$$648$$ 7.10893i 0.279265i
$$649$$ −12.9575 −0.508627
$$650$$ 0 0
$$651$$ −8.86247 −0.347348
$$652$$ 64.8374i 2.53923i
$$653$$ 30.5647i 1.19609i 0.801462 + 0.598045i $$0.204056\pi$$
−0.801462 + 0.598045i $$0.795944\pi$$
$$654$$ −34.9879 −1.36813
$$655$$ 0 0
$$656$$ −23.9347 −0.934493
$$657$$ − 12.3564i − 0.482069i
$$658$$ 79.2125i 3.08802i
$$659$$ −22.1893 −0.864371 −0.432186 0.901785i $$-0.642257\pi$$
−0.432186 + 0.901785i $$0.642257\pi$$
$$660$$ 0 0
$$661$$ −9.85731 −0.383405 −0.191702 0.981453i $$-0.561401\pi$$
−0.191702 + 0.981453i $$0.561401\pi$$
$$662$$ − 59.5523i − 2.31457i
$$663$$ 14.5643i 0.565630i
$$664$$ 27.1044 1.05185
$$665$$ 0 0
$$666$$ −27.3294 −1.05899
$$667$$ 5.19955i 0.201327i
$$668$$ − 118.277i − 4.57629i
$$669$$ 25.1117 0.970875
$$670$$ 0 0
$$671$$ −24.5294 −0.946948
$$672$$ 35.7780i 1.38016i
$$673$$ − 25.3057i − 0.975463i −0.872994 0.487732i $$-0.837824\pi$$
0.872994 0.487732i $$-0.162176\pi$$
$$674$$ −41.5441 −1.60022
$$675$$ 0 0
$$676$$ −40.8106 −1.56964
$$677$$ 3.67306i 0.141167i 0.997506 + 0.0705836i $$0.0224862\pi$$
−0.997506 + 0.0705836i $$0.977514\pi$$
$$678$$ − 29.7292i − 1.14174i
$$679$$ 47.2860 1.81467
$$680$$ 0 0
$$681$$ −4.98706 −0.191104
$$682$$ − 13.1229i − 0.502502i
$$683$$ 17.3807i 0.665053i 0.943094 + 0.332527i $$0.107901\pi$$
−0.943094 + 0.332527i $$0.892099\pi$$
$$684$$ −20.3564 −0.778346
$$685$$ 0 0
$$686$$ −15.4361 −0.589354
$$687$$ − 1.60591i − 0.0612694i
$$688$$ 54.1911i 2.06602i
$$689$$ 2.95992 0.112764
$$690$$ 0 0
$$691$$ −28.5585 −1.08642 −0.543209 0.839598i $$-0.682791\pi$$
−0.543209 + 0.839598i $$0.682791\pi$$
$$692$$ − 94.3856i − 3.58800i
$$693$$ − 8.84710i − 0.336074i
$$694$$ −59.7025 −2.26628
$$695$$ 0 0
$$696$$ 7.10893 0.269463
$$697$$ 18.5390i 0.702214i
$$698$$ 63.1078i 2.38866i
$$699$$ −6.65588 −0.251749
$$700$$ 0 0
$$701$$ 21.5401 0.813557 0.406779 0.913527i $$-0.366652\pi$$
0.406779 + 0.913527i $$0.366652\pi$$
$$702$$ − 5.43743i − 0.205223i
$$703$$ − 45.2274i − 1.70578i
$$704$$ −12.6457 −0.476601
$$705$$ 0 0
$$706$$ −75.9772 −2.85944
$$707$$ − 62.4776i − 2.34971i
$$708$$ 27.3320i 1.02720i
$$709$$ 2.29948 0.0863587 0.0431793 0.999067i $$-0.486251\pi$$
0.0431793 + 0.999067i $$0.486251\pi$$
$$710$$ 0 0
$$711$$ 14.4098 0.540408
$$712$$ 14.7396i 0.552391i
$$713$$ − 11.7008i − 0.438199i
$$714$$ 71.0831 2.66022
$$715$$ 0 0
$$716$$ 51.5630 1.92700
$$717$$ − 2.87227i − 0.107267i
$$718$$ 48.0470i 1.79310i
$$719$$ −13.6814 −0.510232 −0.255116 0.966910i $$-0.582114\pi$$
−0.255116 + 0.966910i $$0.582114\pi$$
$$720$$ 0 0
$$721$$ 40.6110 1.51243
$$722$$ 1.41521i 0.0526688i
$$723$$ 15.2768i 0.568152i
$$724$$ 66.0422 2.45444
$$725$$ 0 0
$$726$$ −15.4545 −0.573569
$$727$$ − 50.0451i − 1.85607i −0.372492 0.928035i $$-0.621497\pi$$
0.372492 0.928035i $$-0.378503\pi$$
$$728$$ − 58.6435i − 2.17347i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 41.9745 1.55248
$$732$$ 51.7412i 1.91241i
$$733$$ 38.7734i 1.43213i 0.698034 + 0.716065i $$0.254058\pi$$
−0.698034 + 0.716065i $$0.745942\pi$$
$$734$$ −2.32694 −0.0858887
$$735$$ 0 0
$$736$$ −47.2364 −1.74116
$$737$$ 26.3893i 0.972062i
$$738$$ − 6.92135i − 0.254778i
$$739$$ 18.0707 0.664742 0.332371 0.943149i $$-0.392151\pi$$
0.332371 + 0.943149i $$0.392151\pi$$
$$740$$ 0 0
$$741$$ 8.99840 0.330564
$$742$$ − 14.4463i − 0.530341i
$$743$$ − 20.9864i − 0.769917i −0.922934 0.384959i $$-0.874216\pi$$
0.922934 0.384959i $$-0.125784\pi$$
$$744$$ −15.9976 −0.586500
$$745$$ 0 0
$$746$$ 12.0594 0.441525
$$747$$ 3.81272i 0.139500i
$$748$$ 74.0151i 2.70626i
$$749$$ 17.6294 0.644166
$$750$$ 0 0
$$751$$ −20.3274 −0.741758 −0.370879 0.928681i $$-0.620944\pi$$
−0.370879 + 0.928681i $$0.620944\pi$$
$$752$$ 69.5547i 2.53640i
$$753$$ 10.8782i 0.396424i
$$754$$ −5.43743 −0.198020
$$755$$ 0 0
$$756$$ −18.6617 −0.678718
$$757$$ − 0.0654846i − 0.00238008i −0.999999 0.00119004i $$-0.999621\pi$$
0.999999 0.00119004i $$-0.000378801\pi$$
$$758$$ − 84.3304i − 3.06302i
$$759$$ 11.6805 0.423976
$$760$$ 0 0
$$761$$ −28.8872 −1.04716 −0.523580 0.851976i $$-0.675404\pi$$
−0.523580 + 0.851976i $$0.675404\pi$$
$$762$$ 8.39761i 0.304213i
$$763$$ − 53.0810i − 1.92166i
$$764$$ −63.2399 −2.28794
$$765$$ 0 0
$$766$$ 24.6169 0.889446
$$767$$ − 12.0819i − 0.436252i
$$768$$ − 20.4913i − 0.739417i
$$769$$ 10.0813 0.363541 0.181771 0.983341i $$-0.441817\pi$$
0.181771 + 0.983341i $$0.441817\pi$$
$$770$$ 0 0
$$771$$ −18.0748 −0.650949
$$772$$ − 74.0809i − 2.66623i
$$773$$ − 20.5353i − 0.738602i −0.929310 0.369301i $$-0.879597\pi$$
0.929310 0.369301i $$-0.120403\pi$$
$$774$$ −15.6708 −0.563275
$$775$$ 0 0
$$776$$ 85.3556 3.06409
$$777$$ − 41.4621i − 1.48744i
$$778$$ 88.1321i 3.15969i
$$779$$ 11.4541 0.410387
$$780$$ 0 0
$$781$$ −10.2369 −0.366304
$$782$$ 93.8485i 3.35602i
$$783$$ 1.00000i 0.0357371i
$$784$$ −76.3916 −2.72827
$$785$$ 0 0
$$786$$ −34.6722 −1.23672
$$787$$ 46.0601i 1.64186i 0.571026 + 0.820932i $$0.306545\pi$$
−0.571026 + 0.820932i $$0.693455\pi$$
$$788$$ − 95.3110i − 3.39531i
$$789$$ 18.4454 0.656674
$$790$$ 0 0
$$791$$ 45.1029 1.60367
$$792$$ − 15.9699i − 0.567464i
$$793$$ − 22.8718i − 0.812203i
$$794$$ 65.1284 2.31132
$$795$$ 0 0
$$796$$ −43.9374 −1.55732
$$797$$ − 37.4745i − 1.32742i −0.747992 0.663708i $$-0.768982\pi$$
0.747992 0.663708i $$-0.231018\pi$$
$$798$$ − 43.9180i − 1.55468i
$$799$$ 53.8746 1.90595
$$800$$ 0 0
$$801$$ −2.07340 −0.0732599
$$802$$ 12.6111i 0.445314i
$$803$$ 27.7580i 0.979558i
$$804$$ 55.6643 1.96313
$$805$$ 0 0
$$806$$ 12.2361 0.430999
$$807$$ − 11.6702i − 0.410810i
$$808$$ − 112.778i − 3.96751i
$$809$$ −29.9284 −1.05223 −0.526113 0.850414i $$-0.676351\pi$$
−0.526113 + 0.850414i $$0.676351\pi$$
$$810$$ 0 0
$$811$$ −49.1738 −1.72673 −0.863363 0.504583i $$-0.831646\pi$$
−0.863363 + 0.504583i $$0.831646\pi$$
$$812$$ 18.6617i 0.654896i
$$813$$ 20.2501i 0.710204i
$$814$$ 61.3940 2.15186
$$815$$ 0 0
$$816$$ 62.4165 2.18501
$$817$$ − 25.9336i − 0.907301i
$$818$$ 38.0566i 1.33062i
$$819$$ 8.24926 0.288252
$$820$$ 0 0
$$821$$ 33.9715 1.18561 0.592807 0.805345i $$-0.298020\pi$$
0.592807 + 0.805345i $$0.298020\pi$$
$$822$$ − 11.3087i − 0.394437i
$$823$$ 33.8272i 1.17914i 0.807716 + 0.589572i $$0.200703\pi$$
−0.807716 + 0.589572i $$0.799297\pi$$
$$824$$ 73.3067 2.55376
$$825$$ 0 0
$$826$$ −58.9676 −2.05174
$$827$$ − 35.1083i − 1.22083i −0.792080 0.610417i $$-0.791002\pi$$
0.792080 0.610417i $$-0.208998\pi$$
$$828$$ − 24.6383i − 0.856241i
$$829$$ −15.6701 −0.544245 −0.272122 0.962263i $$-0.587726\pi$$
−0.272122 + 0.962263i $$0.587726\pi$$
$$830$$ 0 0
$$831$$ 3.66952 0.127294
$$832$$ − 11.7911i − 0.408784i
$$833$$ 59.1702i 2.05013i
$$834$$ 47.8085 1.65547
$$835$$ 0 0
$$836$$ 45.7296 1.58159
$$837$$ − 2.25035i − 0.0777835i
$$838$$ − 47.7961i − 1.65109i
$$839$$ 16.5131 0.570095 0.285047 0.958513i $$-0.407991\pi$$
0.285047 + 0.958513i $$0.407991\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ − 75.6669i − 2.60765i
$$843$$ 22.8056i 0.785466i
$$844$$ 1.08512 0.0373514
$$845$$ 0 0
$$846$$ −20.1136 −0.691518
$$847$$ − 23.4463i − 0.805626i
$$848$$ − 12.6850i − 0.435604i
$$849$$ 2.38657 0.0819069
$$850$$ 0 0
$$851$$ 54.7409 1.87649
$$852$$ 21.5932i 0.739770i
$$853$$ − 16.3926i − 0.561271i −0.959814 0.280636i $$-0.909455\pi$$
0.959814 0.280636i $$-0.0905452\pi$$
$$854$$ −111.629 −3.81988
$$855$$ 0 0
$$856$$ 31.8228 1.08768
$$857$$ − 9.83745i − 0.336041i −0.985784 0.168020i $$-0.946263\pi$$
0.985784 0.168020i $$-0.0537375\pi$$
$$858$$ 12.2149i 0.417010i
$$859$$ −39.8257 −1.35883 −0.679417 0.733752i $$-0.737767\pi$$
−0.679417 + 0.733752i $$0.737767\pi$$
$$860$$ 0 0
$$861$$ 10.5006 0.357858
$$862$$ 5.32971i 0.181530i
$$863$$ 15.6707i 0.533438i 0.963774 + 0.266719i $$0.0859395\pi$$
−0.963774 + 0.266719i $$0.914060\pi$$
$$864$$ −9.08471 −0.309068
$$865$$ 0 0
$$866$$ −48.7556 −1.65678
$$867$$ − 31.3456i − 1.06455i
$$868$$ − 41.9953i − 1.42541i
$$869$$ −32.3708 −1.09810
$$870$$ 0 0
$$871$$ −24.6060 −0.833744
$$872$$ − 95.8162i − 3.24474i
$$873$$ 12.0068i 0.406369i
$$874$$ 57.9834 1.96132
$$875$$ 0 0
$$876$$ 58.5514 1.97827
$$877$$ 21.2819i 0.718640i 0.933214 + 0.359320i $$0.116991\pi$$
−0.933214 + 0.359320i $$0.883009\pi$$
$$878$$ − 87.7118i − 2.96013i
$$879$$ −19.4843 −0.657189
$$880$$ 0 0
$$881$$ 29.1037 0.980529 0.490265 0.871574i $$-0.336900\pi$$
0.490265 + 0.871574i $$0.336900\pi$$
$$882$$ − 22.0906i − 0.743830i
$$883$$ − 4.03715i − 0.135861i −0.997690 0.0679305i $$-0.978360\pi$$
0.997690 0.0679305i $$-0.0216396\pi$$
$$884$$ −69.0136 −2.32118
$$885$$ 0 0
$$886$$ 63.0698 2.11887
$$887$$ 2.85498i 0.0958608i 0.998851 + 0.0479304i $$0.0152626\pi$$
−0.998851 + 0.0479304i $$0.984737\pi$$
$$888$$ − 74.8430i − 2.51157i
$$889$$ −12.7402 −0.427293
$$890$$ 0 0
$$891$$ 2.24645 0.0752588
$$892$$ 118.993i 3.98419i
$$893$$ − 33.2859i − 1.11387i
$$894$$ −45.7982 −1.53172
$$895$$ 0 0
$$896$$ 14.0076 0.467960
$$897$$ 10.8912i 0.363647i
$$898$$ − 54.7666i − 1.82758i
$$899$$ −2.25035 −0.0750534
$$900$$ 0 0
$$901$$ −9.82535 −0.327330
$$902$$ 15.5485i 0.517707i
$$903$$ − 23.7745i − 0.791167i
$$904$$ 81.4150 2.70782
$$905$$ 0 0
$$906$$ 10.4453 0.347022
$$907$$ 3.08757i 0.102521i 0.998685 + 0.0512605i $$0.0163239\pi$$
−0.998685 + 0.0512605i $$0.983676\pi$$
$$908$$ − 23.6314i − 0.784237i
$$909$$ 15.8642 0.526184
$$910$$ 0 0
$$911$$ 34.1306 1.13080 0.565398 0.824818i $$-0.308723\pi$$
0.565398 + 0.824818i $$0.308723\pi$$
$$912$$ − 38.5634i − 1.27696i
$$913$$ − 8.56508i − 0.283463i
$$914$$ −22.6096 −0.747860
$$915$$ 0 0
$$916$$ 7.60970 0.251432
$$917$$ − 52.6021i − 1.73707i
$$918$$ 18.0494i 0.595718i
$$919$$ −16.9603 −0.559468 −0.279734 0.960078i $$-0.590246\pi$$
−0.279734 + 0.960078i $$0.590246\pi$$
$$920$$ 0 0
$$921$$ −0.527333 −0.0173762
$$922$$ − 76.6882i − 2.52559i
$$923$$ − 9.54511i − 0.314181i
$$924$$ 41.9225 1.37915
$$925$$ 0 0
$$926$$ 96.8433 3.18247
$$927$$ 10.3119i 0.338688i
$$928$$ 9.08471i 0.298220i
$$929$$ −5.63831 −0.184987 −0.0924935 0.995713i $$-0.529484\pi$$
−0.0924935 + 0.995713i $$0.529484\pi$$
$$930$$ 0 0
$$931$$ 36.5578 1.19813
$$932$$ − 31.5392i − 1.03310i
$$933$$ − 7.89624i − 0.258511i
$$934$$ −78.4328 −2.56640
$$935$$ 0 0
$$936$$ 14.8907 0.486718
$$937$$ − 32.0790i − 1.04798i −0.851726 0.523988i $$-0.824444\pi$$
0.851726 0.523988i $$-0.175556\pi$$
$$938$$ 120.093i 3.92119i
$$939$$ −10.0064 −0.326548
$$940$$ 0 0
$$941$$ 31.2028 1.01718 0.508592 0.861008i $$-0.330166\pi$$
0.508592 + 0.861008i $$0.330166\pi$$
$$942$$ 19.9913i 0.651352i
$$943$$ 13.8635i 0.451458i
$$944$$ −51.7781 −1.68523
$$945$$ 0 0
$$946$$ 35.2036 1.14457
$$947$$ 27.7302i 0.901110i 0.892749 + 0.450555i $$0.148774\pi$$
−0.892749 + 0.450555i $$0.851226\pi$$
$$948$$ 68.2814i 2.21768i
$$949$$ −25.8822 −0.840173
$$950$$ 0 0
$$951$$ 32.5401 1.05519
$$952$$ 194.665i 6.30913i
$$953$$ 23.9481i 0.775755i 0.921711 + 0.387877i $$0.126792\pi$$
−0.921711 + 0.387877i $$0.873208\pi$$
$$954$$ 3.66820 0.118762
$$955$$ 0 0
$$956$$ 13.6104 0.440192
$$957$$ − 2.24645i − 0.0726173i
$$958$$ 101.631i 3.28356i
$$959$$ 17.1567 0.554020
$$960$$ 0 0
$$961$$ −25.9359 −0.836643
$$962$$ 57.2453i 1.84566i
$$963$$ 4.47645i 0.144252i
$$964$$ −72.3902 −2.33153
$$965$$ 0 0
$$966$$ 53.1562 1.71027
$$967$$ 43.6821i 1.40472i 0.711821 + 0.702361i $$0.247871\pi$$
−0.711821 + 0.702361i $$0.752129\pi$$
$$968$$ − 42.3228i − 1.36031i
$$969$$ −29.8699 −0.959559
$$970$$ 0 0
$$971$$ 1.12406 0.0360729 0.0180365 0.999837i $$-0.494259\pi$$
0.0180365 + 0.999837i $$0.494259\pi$$
$$972$$ − 4.73855i − 0.151989i
$$973$$ 72.5315i 2.32525i
$$974$$ 62.0887 1.98945
$$975$$ 0 0
$$976$$ −98.0193 −3.13752
$$977$$ 18.8819i 0.604086i 0.953294 + 0.302043i $$0.0976686\pi$$
−0.953294 + 0.302043i $$0.902331\pi$$
$$978$$ − 35.5192i − 1.13578i
$$979$$ 4.65778 0.148863
$$980$$ 0 0
$$981$$ 13.4783 0.430328
$$982$$ − 11.4471i − 0.365292i
$$983$$ − 23.5807i − 0.752109i −0.926598 0.376054i $$-0.877281\pi$$
0.926598 0.376054i $$-0.122719\pi$$
$$984$$ 18.9545 0.604247
$$985$$ 0 0
$$986$$ 18.0494 0.574809
$$987$$ − 30.5148i − 0.971296i
$$988$$ 42.6394i 1.35654i
$$989$$ 31.3887 0.998102
$$990$$ 0 0
$$991$$ −31.3818 −0.996877 −0.498438 0.866925i $$-0.666093\pi$$
−0.498438 + 0.866925i $$0.666093\pi$$
$$992$$ − 20.4438i − 0.649090i
$$993$$ 22.9411i 0.728015i
$$994$$ −46.5864 −1.47763
$$995$$ 0 0
$$996$$ −18.0668 −0.572468
$$997$$ − 45.3442i − 1.43606i −0.696010 0.718032i $$-0.745043\pi$$
0.696010 0.718032i $$-0.254957\pi$$
$$998$$ − 96.1629i − 3.04398i
$$999$$ 10.5280 0.333092
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.p.349.2 16
5.2 odd 4 2175.2.a.bd.1.8 yes 8
5.3 odd 4 2175.2.a.bc.1.1 8
5.4 even 2 inner 2175.2.c.p.349.15 16
15.2 even 4 6525.2.a.by.1.1 8
15.8 even 4 6525.2.a.bz.1.8 8

By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.bc.1.1 8 5.3 odd 4
2175.2.a.bd.1.8 yes 8 5.2 odd 4
2175.2.c.p.349.2 16 1.1 even 1 trivial
2175.2.c.p.349.15 16 5.4 even 2 inner
6525.2.a.by.1.1 8 15.2 even 4
6525.2.a.bz.1.8 8 15.8 even 4