# Properties

 Label 2175.2.c.n.349.1 Level $2175$ Weight $2$ Character 2175.349 Analytic conductor $17.367$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.1267360000.3 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 11x^{6} + 37x^{4} + 44x^{2} + 16$$ x^8 + 11*x^6 + 37*x^4 + 44*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 435) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.1 Root $$-1.75660i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.n.349.8

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.75660i q^{2} +1.00000i q^{3} -5.59883 q^{4} +2.75660 q^{6} +0.393832i q^{7} +9.92054i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-2.75660i q^{2} +1.00000i q^{3} -5.59883 q^{4} +2.75660 q^{6} +0.393832i q^{7} +9.92054i q^{8} -1.00000 q^{9} -0.393832 q^{11} -5.59883i q^{12} +2.56511i q^{13} +1.08564 q^{14} +16.1493 q^{16} +2.07830i q^{17} +2.75660i q^{18} +0.958939 q^{19} -0.393832 q^{21} +1.08564i q^{22} -6.15661i q^{23} -9.92054 q^{24} +7.07097 q^{26} -1.00000i q^{27} -2.20500i q^{28} +1.00000 q^{29} -10.1566 q^{31} -24.6760i q^{32} -0.393832i q^{33} +5.72905 q^{34} +5.59883 q^{36} -7.34192i q^{37} -2.64341i q^{38} -2.56511 q^{39} -1.65745 q^{41} +1.08564i q^{42} -10.3279i q^{43} +2.20500 q^{44} -16.9713 q^{46} -11.5915i q^{47} +16.1493i q^{48} +6.84490 q^{49} -2.07830 q^{51} -14.3616i q^{52} +12.3279i q^{53} -2.75660 q^{54} -3.90703 q^{56} +0.958939i q^{57} -2.75660i q^{58} +9.54022 q^{59} -6.25340 q^{61} +27.9977i q^{62} -0.393832i q^{63} -35.7232 q^{64} -1.08564 q^{66} -7.42023i q^{67} -11.6361i q^{68} +6.15661 q^{69} +5.98533 q^{71} -9.92054i q^{72} -3.34192i q^{73} -20.2387 q^{74} -5.36894 q^{76} -0.155104i q^{77} +7.07097i q^{78} +2.06745 q^{79} +1.00000 q^{81} +4.56892i q^{82} -6.41000i q^{83} +2.20500 q^{84} -28.4698 q^{86} +1.00000i q^{87} -3.90703i q^{88} -15.8302 q^{89} -1.01022 q^{91} +34.4698i q^{92} -10.1566i q^{93} -31.9531 q^{94} +24.6760 q^{96} -18.4575i q^{97} -18.8686i q^{98} +0.393832 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 10 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10})$$ 8 * q - 10 * q^4 + 6 * q^6 - 8 * q^9 $$8 q - 10 q^{4} + 6 q^{6} - 8 q^{9} - 4 q^{11} + 6 q^{14} + 22 q^{16} + 4 q^{19} - 4 q^{21} - 24 q^{24} - 14 q^{26} + 8 q^{29} - 8 q^{31} + 2 q^{34} + 10 q^{36} - 16 q^{39} - 24 q^{41} - 18 q^{44} - 16 q^{46} - 12 q^{49} + 20 q^{51} - 6 q^{54} - 4 q^{59} - 52 q^{61} - 68 q^{64} - 6 q^{66} - 24 q^{69} - 20 q^{71} - 96 q^{74} + 32 q^{76} - 44 q^{79} + 8 q^{81} - 18 q^{84} - 8 q^{86} + 8 q^{89} - 16 q^{91} - 78 q^{94} + 34 q^{96} + 4 q^{99}+O(q^{100})$$ 8 * q - 10 * q^4 + 6 * q^6 - 8 * q^9 - 4 * q^11 + 6 * q^14 + 22 * q^16 + 4 * q^19 - 4 * q^21 - 24 * q^24 - 14 * q^26 + 8 * q^29 - 8 * q^31 + 2 * q^34 + 10 * q^36 - 16 * q^39 - 24 * q^41 - 18 * q^44 - 16 * q^46 - 12 * q^49 + 20 * q^51 - 6 * q^54 - 4 * q^59 - 52 * q^61 - 68 * q^64 - 6 * q^66 - 24 * q^69 - 20 * q^71 - 96 * q^74 + 32 * q^76 - 44 * q^79 + 8 * q^81 - 18 * q^84 - 8 * q^86 + 8 * q^89 - 16 * q^91 - 78 * q^94 + 34 * q^96 + 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.75660i − 1.94921i −0.223932 0.974605i $$-0.571890\pi$$
0.223932 0.974605i $$-0.428110\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −5.59883 −2.79942
$$5$$ 0 0
$$6$$ 2.75660 1.12538
$$7$$ 0.393832i 0.148855i 0.997226 + 0.0744273i $$0.0237129\pi$$
−0.997226 + 0.0744273i $$0.976287\pi$$
$$8$$ 9.92054i 3.50744i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −0.393832 −0.118745 −0.0593725 0.998236i $$-0.518910\pi$$
−0.0593725 + 0.998236i $$0.518910\pi$$
$$12$$ − 5.59883i − 1.61624i
$$13$$ 2.56511i 0.711433i 0.934594 + 0.355716i $$0.115763\pi$$
−0.934594 + 0.355716i $$0.884237\pi$$
$$14$$ 1.08564 0.290149
$$15$$ 0 0
$$16$$ 16.1493 4.03732
$$17$$ 2.07830i 0.504063i 0.967719 + 0.252031i $$0.0810986\pi$$
−0.967719 + 0.252031i $$0.918901\pi$$
$$18$$ 2.75660i 0.649736i
$$19$$ 0.958939 0.219996 0.109998 0.993932i $$-0.464916\pi$$
0.109998 + 0.993932i $$0.464916\pi$$
$$20$$ 0 0
$$21$$ −0.393832 −0.0859413
$$22$$ 1.08564i 0.231459i
$$23$$ − 6.15661i − 1.28374i −0.766813 0.641871i $$-0.778159\pi$$
0.766813 0.641871i $$-0.221841\pi$$
$$24$$ −9.92054 −2.02502
$$25$$ 0 0
$$26$$ 7.07097 1.38673
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 2.20500i − 0.416706i
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −10.1566 −1.82418 −0.912090 0.409989i $$-0.865532\pi$$
−0.912090 + 0.409989i $$0.865532\pi$$
$$32$$ − 24.6760i − 4.36214i
$$33$$ − 0.393832i − 0.0685574i
$$34$$ 5.72905 0.982524
$$35$$ 0 0
$$36$$ 5.59883 0.933139
$$37$$ − 7.34192i − 1.20700i −0.797361 0.603502i $$-0.793771\pi$$
0.797361 0.603502i $$-0.206229\pi$$
$$38$$ − 2.64341i − 0.428818i
$$39$$ −2.56511 −0.410746
$$40$$ 0 0
$$41$$ −1.65745 −0.258850 −0.129425 0.991589i $$-0.541313\pi$$
−0.129425 + 0.991589i $$0.541313\pi$$
$$42$$ 1.08564i 0.167517i
$$43$$ − 10.3279i − 1.57499i −0.616323 0.787494i $$-0.711378\pi$$
0.616323 0.787494i $$-0.288622\pi$$
$$44$$ 2.20500 0.332417
$$45$$ 0 0
$$46$$ −16.9713 −2.50228
$$47$$ − 11.5915i − 1.69079i −0.534138 0.845397i $$-0.679364\pi$$
0.534138 0.845397i $$-0.320636\pi$$
$$48$$ 16.1493i 2.33095i
$$49$$ 6.84490 0.977842
$$50$$ 0 0
$$51$$ −2.07830 −0.291021
$$52$$ − 14.3616i − 1.99160i
$$53$$ 12.3279i 1.69336i 0.532099 + 0.846682i $$0.321404\pi$$
−0.532099 + 0.846682i $$0.678596\pi$$
$$54$$ −2.75660 −0.375126
$$55$$ 0 0
$$56$$ −3.90703 −0.522099
$$57$$ 0.958939i 0.127015i
$$58$$ − 2.75660i − 0.361959i
$$59$$ 9.54022 1.24203 0.621015 0.783798i $$-0.286720\pi$$
0.621015 + 0.783798i $$0.286720\pi$$
$$60$$ 0 0
$$61$$ −6.25340 −0.800665 −0.400333 0.916370i $$-0.631105\pi$$
−0.400333 + 0.916370i $$0.631105\pi$$
$$62$$ 27.9977i 3.55571i
$$63$$ − 0.393832i − 0.0496182i
$$64$$ −35.7232 −4.46540
$$65$$ 0 0
$$66$$ −1.08564 −0.133633
$$67$$ − 7.42023i − 0.906525i −0.891377 0.453262i $$-0.850260\pi$$
0.891377 0.453262i $$-0.149740\pi$$
$$68$$ − 11.6361i − 1.41108i
$$69$$ 6.15661 0.741168
$$70$$ 0 0
$$71$$ 5.98533 0.710328 0.355164 0.934804i $$-0.384425\pi$$
0.355164 + 0.934804i $$0.384425\pi$$
$$72$$ − 9.92054i − 1.16915i
$$73$$ − 3.34192i − 0.391142i −0.980690 0.195571i $$-0.937344\pi$$
0.980690 0.195571i $$-0.0626561\pi$$
$$74$$ −20.2387 −2.35270
$$75$$ 0 0
$$76$$ −5.36894 −0.615860
$$77$$ − 0.155104i − 0.0176757i
$$78$$ 7.07097i 0.800630i
$$79$$ 2.06745 0.232607 0.116303 0.993214i $$-0.462896\pi$$
0.116303 + 0.993214i $$0.462896\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 4.56892i 0.504553i
$$83$$ − 6.41000i − 0.703589i −0.936077 0.351795i $$-0.885572\pi$$
0.936077 0.351795i $$-0.114428\pi$$
$$84$$ 2.20500 0.240585
$$85$$ 0 0
$$86$$ −28.4698 −3.06998
$$87$$ 1.00000i 0.107211i
$$88$$ − 3.90703i − 0.416491i
$$89$$ −15.8302 −1.67800 −0.839000 0.544131i $$-0.816860\pi$$
−0.839000 + 0.544131i $$0.816860\pi$$
$$90$$ 0 0
$$91$$ −1.01022 −0.105900
$$92$$ 34.4698i 3.59373i
$$93$$ − 10.1566i − 1.05319i
$$94$$ −31.9531 −3.29571
$$95$$ 0 0
$$96$$ 24.6760 2.51848
$$97$$ − 18.4575i − 1.87407i −0.349233 0.937036i $$-0.613558\pi$$
0.349233 0.937036i $$-0.386442\pi$$
$$98$$ − 18.8686i − 1.90602i
$$99$$ 0.393832 0.0395816
$$100$$ 0 0
$$101$$ −12.8038 −1.27403 −0.637015 0.770852i $$-0.719831\pi$$
−0.637015 + 0.770852i $$0.719831\pi$$
$$102$$ 5.72905i 0.567260i
$$103$$ − 4.86979i − 0.479834i −0.970793 0.239917i $$-0.922880\pi$$
0.970793 0.239917i $$-0.0771203\pi$$
$$104$$ −25.4472 −2.49531
$$105$$ 0 0
$$106$$ 33.9830 3.30072
$$107$$ − 2.34255i − 0.226463i −0.993569 0.113231i $$-0.963880\pi$$
0.993569 0.113231i $$-0.0361201\pi$$
$$108$$ 5.59883i 0.538748i
$$109$$ −8.55044 −0.818984 −0.409492 0.912314i $$-0.634294\pi$$
−0.409492 + 0.912314i $$0.634294\pi$$
$$110$$ 0 0
$$111$$ 7.34192 0.696864
$$112$$ 6.36011i 0.600973i
$$113$$ 11.2085i 1.05441i 0.849739 + 0.527204i $$0.176760\pi$$
−0.849739 + 0.527204i $$0.823240\pi$$
$$114$$ 2.64341 0.247578
$$115$$ 0 0
$$116$$ −5.59883 −0.519839
$$117$$ − 2.56511i − 0.237144i
$$118$$ − 26.2985i − 2.42098i
$$119$$ −0.818503 −0.0750321
$$120$$ 0 0
$$121$$ −10.8449 −0.985900
$$122$$ 17.2381i 1.56066i
$$123$$ − 1.65745i − 0.149447i
$$124$$ 56.8652 5.10664
$$125$$ 0 0
$$126$$ −1.08564 −0.0967163
$$127$$ 20.1566i 1.78861i 0.447458 + 0.894305i $$0.352329\pi$$
−0.447458 + 0.894305i $$0.647671\pi$$
$$128$$ 49.1226i 4.34187i
$$129$$ 10.3279 0.909319
$$130$$ 0 0
$$131$$ −5.50235 −0.480742 −0.240371 0.970681i $$-0.577269\pi$$
−0.240371 + 0.970681i $$0.577269\pi$$
$$132$$ 2.20500i 0.191921i
$$133$$ 0.377661i 0.0327474i
$$134$$ −20.4546 −1.76701
$$135$$ 0 0
$$136$$ −20.6179 −1.76797
$$137$$ 7.49853i 0.640643i 0.947309 + 0.320321i $$0.103791\pi$$
−0.947309 + 0.320321i $$0.896209\pi$$
$$138$$ − 16.9713i − 1.44469i
$$139$$ 9.35277 0.793292 0.396646 0.917972i $$-0.370174\pi$$
0.396646 + 0.917972i $$0.370174\pi$$
$$140$$ 0 0
$$141$$ 11.5915 0.976180
$$142$$ − 16.4992i − 1.38458i
$$143$$ − 1.01022i − 0.0844790i
$$144$$ −16.1493 −1.34577
$$145$$ 0 0
$$146$$ −9.21234 −0.762418
$$147$$ 6.84490i 0.564558i
$$148$$ 41.1062i 3.37891i
$$149$$ 11.5402 0.945411 0.472706 0.881220i $$-0.343277\pi$$
0.472706 + 0.881220i $$0.343277\pi$$
$$150$$ 0 0
$$151$$ 6.08212 0.494956 0.247478 0.968894i $$-0.420398\pi$$
0.247478 + 0.968894i $$0.420398\pi$$
$$152$$ 9.51320i 0.771622i
$$153$$ − 2.07830i − 0.168021i
$$154$$ −0.427559 −0.0344537
$$155$$ 0 0
$$156$$ 14.3616 1.14985
$$157$$ − 11.5953i − 0.925407i −0.886513 0.462704i $$-0.846879\pi$$
0.886513 0.462704i $$-0.153121\pi$$
$$158$$ − 5.69914i − 0.453399i
$$159$$ −12.3279 −0.977665
$$160$$ 0 0
$$161$$ 2.42467 0.191091
$$162$$ − 2.75660i − 0.216579i
$$163$$ 0.855118i 0.0669780i 0.999439 + 0.0334890i $$0.0106619\pi$$
−0.999439 + 0.0334890i $$0.989338\pi$$
$$164$$ 9.27979 0.724630
$$165$$ 0 0
$$166$$ −17.6698 −1.37144
$$167$$ 7.73194i 0.598315i 0.954204 + 0.299158i $$0.0967056\pi$$
−0.954204 + 0.299158i $$0.903294\pi$$
$$168$$ − 3.90703i − 0.301434i
$$169$$ 6.42023 0.493863
$$170$$ 0 0
$$171$$ −0.958939 −0.0733319
$$172$$ 57.8241i 4.40905i
$$173$$ − 8.07449i − 0.613892i −0.951727 0.306946i $$-0.900693\pi$$
0.951727 0.306946i $$-0.0993071\pi$$
$$174$$ 2.75660 0.208977
$$175$$ 0 0
$$176$$ −6.36011 −0.479411
$$177$$ 9.54022i 0.717087i
$$178$$ 43.6376i 3.27077i
$$179$$ −12.4100 −0.927567 −0.463784 0.885949i $$-0.653508\pi$$
−0.463784 + 0.885949i $$0.653508\pi$$
$$180$$ 0 0
$$181$$ −2.49765 −0.185649 −0.0928246 0.995682i $$-0.529590\pi$$
−0.0928246 + 0.995682i $$0.529590\pi$$
$$182$$ 2.78478i 0.206421i
$$183$$ − 6.25340i − 0.462264i
$$184$$ 61.0769 4.50265
$$185$$ 0 0
$$186$$ −27.9977 −2.05289
$$187$$ − 0.818503i − 0.0598549i
$$188$$ 64.8989i 4.73324i
$$189$$ 0.393832 0.0286471
$$190$$ 0 0
$$191$$ −6.32085 −0.457361 −0.228680 0.973502i $$-0.573441\pi$$
−0.228680 + 0.973502i $$0.573441\pi$$
$$192$$ − 35.7232i − 2.57810i
$$193$$ − 25.2921i − 1.82057i −0.413984 0.910284i $$-0.635863\pi$$
0.413984 0.910284i $$-0.364137\pi$$
$$194$$ −50.8798 −3.65296
$$195$$ 0 0
$$196$$ −38.3234 −2.73739
$$197$$ − 21.2047i − 1.51077i −0.655280 0.755386i $$-0.727449\pi$$
0.655280 0.755386i $$-0.272551\pi$$
$$198$$ − 1.08564i − 0.0771529i
$$199$$ 2.64723 0.187657 0.0938285 0.995588i $$-0.470089\pi$$
0.0938285 + 0.995588i $$0.470089\pi$$
$$200$$ 0 0
$$201$$ 7.42023 0.523382
$$202$$ 35.2950i 2.48335i
$$203$$ 0.393832i 0.0276416i
$$204$$ 11.6361 0.814688
$$205$$ 0 0
$$206$$ −13.4240 −0.935297
$$207$$ 6.15661i 0.427914i
$$208$$ 41.4246i 2.87228i
$$209$$ −0.377661 −0.0261234
$$210$$ 0 0
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ − 69.0218i − 4.74043i
$$213$$ 5.98533i 0.410108i
$$214$$ −6.45747 −0.441423
$$215$$ 0 0
$$216$$ 9.92054 0.675007
$$217$$ − 4.00000i − 0.271538i
$$218$$ 23.5701i 1.59637i
$$219$$ 3.34192 0.225826
$$220$$ 0 0
$$221$$ −5.33107 −0.358607
$$222$$ − 20.2387i − 1.35833i
$$223$$ − 12.5504i − 0.840440i −0.907422 0.420220i $$-0.861953\pi$$
0.907422 0.420220i $$-0.138047\pi$$
$$224$$ 9.71820 0.649324
$$225$$ 0 0
$$226$$ 30.8974 2.05526
$$227$$ 1.30149i 0.0863829i 0.999067 + 0.0431914i $$0.0137525\pi$$
−0.999067 + 0.0431914i $$0.986247\pi$$
$$228$$ − 5.36894i − 0.355567i
$$229$$ −3.28682 −0.217199 −0.108600 0.994086i $$-0.534637\pi$$
−0.108600 + 0.994086i $$0.534637\pi$$
$$230$$ 0 0
$$231$$ 0.155104 0.0102051
$$232$$ 9.92054i 0.651315i
$$233$$ − 17.7115i − 1.16032i −0.814503 0.580159i $$-0.802990\pi$$
0.814503 0.580159i $$-0.197010\pi$$
$$234$$ −7.07097 −0.462244
$$235$$ 0 0
$$236$$ −53.4141 −3.47696
$$237$$ 2.06745i 0.134296i
$$238$$ 2.25628i 0.146253i
$$239$$ −21.2651 −1.37553 −0.687763 0.725935i $$-0.741407\pi$$
−0.687763 + 0.725935i $$0.741407\pi$$
$$240$$ 0 0
$$241$$ 15.3177 0.986697 0.493349 0.869832i $$-0.335773\pi$$
0.493349 + 0.869832i $$0.335773\pi$$
$$242$$ 29.8950i 1.92172i
$$243$$ 1.00000i 0.0641500i
$$244$$ 35.0117 2.24140
$$245$$ 0 0
$$246$$ −4.56892 −0.291304
$$247$$ 2.45978i 0.156512i
$$248$$ − 100.759i − 6.39820i
$$249$$ 6.41000 0.406217
$$250$$ 0 0
$$251$$ 26.8917 1.69739 0.848696 0.528882i $$-0.177388\pi$$
0.848696 + 0.528882i $$0.177388\pi$$
$$252$$ 2.20500i 0.138902i
$$253$$ 2.42467i 0.152438i
$$254$$ 55.5637 3.48637
$$255$$ 0 0
$$256$$ 63.9648 3.99780
$$257$$ − 6.85512i − 0.427611i −0.976876 0.213805i $$-0.931414\pi$$
0.976876 0.213805i $$-0.0685858\pi$$
$$258$$ − 28.4698i − 1.77245i
$$259$$ 2.89149 0.179668
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ 15.1678i 0.937067i
$$263$$ 13.6294i 0.840423i 0.907426 + 0.420212i $$0.138044\pi$$
−0.907426 + 0.420212i $$0.861956\pi$$
$$264$$ 3.90703 0.240461
$$265$$ 0 0
$$266$$ 1.04106 0.0638315
$$267$$ − 15.8302i − 0.968794i
$$268$$ 41.5446i 2.53774i
$$269$$ 8.46129 0.515894 0.257947 0.966159i $$-0.416954\pi$$
0.257947 + 0.966159i $$0.416954\pi$$
$$270$$ 0 0
$$271$$ 6.34255 0.385282 0.192641 0.981269i $$-0.438295\pi$$
0.192641 + 0.981269i $$0.438295\pi$$
$$272$$ 33.5631i 2.03506i
$$273$$ − 1.01022i − 0.0611414i
$$274$$ 20.6704 1.24875
$$275$$ 0 0
$$276$$ −34.4698 −2.07484
$$277$$ − 17.1464i − 1.03023i −0.857122 0.515113i $$-0.827750\pi$$
0.857122 0.515113i $$-0.172250\pi$$
$$278$$ − 25.7818i − 1.54629i
$$279$$ 10.1566 0.608060
$$280$$ 0 0
$$281$$ −12.2985 −0.733670 −0.366835 0.930286i $$-0.619559\pi$$
−0.366835 + 0.930286i $$0.619559\pi$$
$$282$$ − 31.9531i − 1.90278i
$$283$$ − 25.1830i − 1.49697i −0.663149 0.748487i $$-0.730781\pi$$
0.663149 0.748487i $$-0.269219\pi$$
$$284$$ −33.5109 −1.98851
$$285$$ 0 0
$$286$$ −2.78478 −0.164667
$$287$$ − 0.652757i − 0.0385311i
$$288$$ 24.6760i 1.45405i
$$289$$ 12.6807 0.745921
$$290$$ 0 0
$$291$$ 18.4575 1.08200
$$292$$ 18.7109i 1.09497i
$$293$$ − 12.3170i − 0.719569i −0.933035 0.359784i $$-0.882850\pi$$
0.933035 0.359784i $$-0.117150\pi$$
$$294$$ 18.8686 1.10044
$$295$$ 0 0
$$296$$ 72.8358 4.23350
$$297$$ 0.393832i 0.0228525i
$$298$$ − 31.8117i − 1.84280i
$$299$$ 15.7924 0.913296
$$300$$ 0 0
$$301$$ 4.06745 0.234444
$$302$$ − 16.7660i − 0.964773i
$$303$$ − 12.8038i − 0.735561i
$$304$$ 15.4862 0.888193
$$305$$ 0 0
$$306$$ −5.72905 −0.327508
$$307$$ − 22.7379i − 1.29772i −0.760908 0.648860i $$-0.775246\pi$$
0.760908 0.648860i $$-0.224754\pi$$
$$308$$ 0.868401i 0.0494817i
$$309$$ 4.86979 0.277032
$$310$$ 0 0
$$311$$ −5.33810 −0.302696 −0.151348 0.988481i $$-0.548361\pi$$
−0.151348 + 0.988481i $$0.548361\pi$$
$$312$$ − 25.4472i − 1.44067i
$$313$$ 1.96338i 0.110977i 0.998459 + 0.0554885i $$0.0176716\pi$$
−0.998459 + 0.0554885i $$0.982328\pi$$
$$314$$ −31.9636 −1.80381
$$315$$ 0 0
$$316$$ −11.5753 −0.651163
$$317$$ 1.29064i 0.0724895i 0.999343 + 0.0362448i $$0.0115396\pi$$
−0.999343 + 0.0362448i $$0.988460\pi$$
$$318$$ 33.9830i 1.90567i
$$319$$ −0.393832 −0.0220504
$$320$$ 0 0
$$321$$ 2.34255 0.130748
$$322$$ − 6.68384i − 0.372476i
$$323$$ 1.99297i 0.110892i
$$324$$ −5.59883 −0.311046
$$325$$ 0 0
$$326$$ 2.35722 0.130554
$$327$$ − 8.55044i − 0.472840i
$$328$$ − 16.4428i − 0.907902i
$$329$$ 4.56511 0.251683
$$330$$ 0 0
$$331$$ 28.9971 1.59382 0.796911 0.604096i $$-0.206466\pi$$
0.796911 + 0.604096i $$0.206466\pi$$
$$332$$ 35.8885i 1.96964i
$$333$$ 7.34192i 0.402335i
$$334$$ 21.3138 1.16624
$$335$$ 0 0
$$336$$ −6.36011 −0.346972
$$337$$ − 16.7854i − 0.914356i −0.889375 0.457178i $$-0.848860\pi$$
0.889375 0.457178i $$-0.151140\pi$$
$$338$$ − 17.6980i − 0.962643i
$$339$$ −11.2085 −0.608763
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ 2.64341i 0.142939i
$$343$$ 5.45257i 0.294411i
$$344$$ 102.458 5.52417
$$345$$ 0 0
$$346$$ −22.2581 −1.19660
$$347$$ − 17.8681i − 0.959210i −0.877485 0.479605i $$-0.840780\pi$$
0.877485 0.479605i $$-0.159220\pi$$
$$348$$ − 5.59883i − 0.300129i
$$349$$ −8.73789 −0.467728 −0.233864 0.972269i $$-0.575137\pi$$
−0.233864 + 0.972269i $$0.575137\pi$$
$$350$$ 0 0
$$351$$ 2.56511 0.136915
$$352$$ 9.71820i 0.517982i
$$353$$ 22.6017i 1.20297i 0.798885 + 0.601484i $$0.205424\pi$$
−0.798885 + 0.601484i $$0.794576\pi$$
$$354$$ 26.2985 1.39775
$$355$$ 0 0
$$356$$ 88.6308 4.69742
$$357$$ − 0.818503i − 0.0433198i
$$358$$ 34.2094i 1.80802i
$$359$$ 13.1830 0.695772 0.347886 0.937537i $$-0.386900\pi$$
0.347886 + 0.937537i $$0.386900\pi$$
$$360$$ 0 0
$$361$$ −18.0804 −0.951602
$$362$$ 6.88503i 0.361869i
$$363$$ − 10.8449i − 0.569209i
$$364$$ 5.65607 0.296458
$$365$$ 0 0
$$366$$ −17.2381 −0.901050
$$367$$ 17.4511i 0.910938i 0.890252 + 0.455469i $$0.150528\pi$$
−0.890252 + 0.455469i $$0.849472\pi$$
$$368$$ − 99.4247i − 5.18287i
$$369$$ 1.65745 0.0862834
$$370$$ 0 0
$$371$$ −4.85512 −0.252065
$$372$$ 56.8652i 2.94832i
$$373$$ 32.2018i 1.66734i 0.552260 + 0.833672i $$0.313766\pi$$
−0.552260 + 0.833672i $$0.686234\pi$$
$$374$$ −2.25628 −0.116670
$$375$$ 0 0
$$376$$ 114.994 5.93036
$$377$$ 2.56511i 0.132110i
$$378$$ − 1.08564i − 0.0558392i
$$379$$ −32.3660 −1.66253 −0.831265 0.555876i $$-0.812383\pi$$
−0.831265 + 0.555876i $$0.812383\pi$$
$$380$$ 0 0
$$381$$ −20.1566 −1.03265
$$382$$ 17.4240i 0.891492i
$$383$$ 25.8739i 1.32209i 0.750345 + 0.661047i $$0.229887\pi$$
−0.750345 + 0.661047i $$0.770113\pi$$
$$384$$ −49.1226 −2.50678
$$385$$ 0 0
$$386$$ −69.7203 −3.54867
$$387$$ 10.3279i 0.524996i
$$388$$ 103.340i 5.24631i
$$389$$ −32.6103 −1.65341 −0.826703 0.562639i $$-0.809786\pi$$
−0.826703 + 0.562639i $$0.809786\pi$$
$$390$$ 0 0
$$391$$ 12.7953 0.647086
$$392$$ 67.9051i 3.42972i
$$393$$ − 5.50235i − 0.277557i
$$394$$ −58.4528 −2.94481
$$395$$ 0 0
$$396$$ −2.20500 −0.110806
$$397$$ 4.39534i 0.220596i 0.993899 + 0.110298i $$0.0351805\pi$$
−0.993899 + 0.110298i $$0.964820\pi$$
$$398$$ − 7.29734i − 0.365783i
$$399$$ −0.377661 −0.0189067
$$400$$ 0 0
$$401$$ −19.0658 −0.952099 −0.476049 0.879418i $$-0.657932\pi$$
−0.476049 + 0.879418i $$0.657932\pi$$
$$402$$ − 20.4546i − 1.02018i
$$403$$ − 26.0528i − 1.29778i
$$404$$ 71.6865 3.56654
$$405$$ 0 0
$$406$$ 1.08564 0.0538793
$$407$$ 2.89149i 0.143326i
$$408$$ − 20.6179i − 1.02074i
$$409$$ 1.38235 0.0683530 0.0341765 0.999416i $$-0.489119\pi$$
0.0341765 + 0.999416i $$0.489119\pi$$
$$410$$ 0 0
$$411$$ −7.49853 −0.369875
$$412$$ 27.2651i 1.34326i
$$413$$ 3.75725i 0.184882i
$$414$$ 16.9713 0.834094
$$415$$ 0 0
$$416$$ 63.2965 3.10337
$$417$$ 9.35277i 0.458007i
$$418$$ 1.04106i 0.0509199i
$$419$$ −2.86979 −0.140198 −0.0700991 0.997540i $$-0.522332\pi$$
−0.0700991 + 0.997540i $$0.522332\pi$$
$$420$$ 0 0
$$421$$ 13.6440 0.664970 0.332485 0.943109i $$-0.392113\pi$$
0.332485 + 0.943109i $$0.392113\pi$$
$$422$$ − 5.51320i − 0.268378i
$$423$$ 11.5915i 0.563598i
$$424$$ −122.299 −5.93938
$$425$$ 0 0
$$426$$ 16.4992 0.799387
$$427$$ − 2.46279i − 0.119183i
$$428$$ 13.1155i 0.633964i
$$429$$ 1.01022 0.0487740
$$430$$ 0 0
$$431$$ −11.9707 −0.576607 −0.288303 0.957539i $$-0.593091\pi$$
−0.288303 + 0.957539i $$0.593091\pi$$
$$432$$ − 16.1493i − 0.776982i
$$433$$ − 17.9907i − 0.864576i −0.901736 0.432288i $$-0.857706\pi$$
0.901736 0.432288i $$-0.142294\pi$$
$$434$$ −11.0264 −0.529284
$$435$$ 0 0
$$436$$ 47.8725 2.29268
$$437$$ − 5.90381i − 0.282418i
$$438$$ − 9.21234i − 0.440182i
$$439$$ 16.2226 0.774260 0.387130 0.922025i $$-0.373466\pi$$
0.387130 + 0.922025i $$0.373466\pi$$
$$440$$ 0 0
$$441$$ −6.84490 −0.325947
$$442$$ 14.6956i 0.698999i
$$443$$ 35.3762i 1.68078i 0.541986 + 0.840388i $$0.317673\pi$$
−0.541986 + 0.840388i $$0.682327\pi$$
$$444$$ −41.1062 −1.95081
$$445$$ 0 0
$$446$$ −34.5965 −1.63819
$$447$$ 11.5402i 0.545834i
$$448$$ − 14.0690i − 0.664696i
$$449$$ −41.6971 −1.96781 −0.983903 0.178702i $$-0.942810\pi$$
−0.983903 + 0.178702i $$0.942810\pi$$
$$450$$ 0 0
$$451$$ 0.652757 0.0307371
$$452$$ − 62.7546i − 2.95173i
$$453$$ 6.08212i 0.285763i
$$454$$ 3.58768 0.168378
$$455$$ 0 0
$$456$$ −9.51320 −0.445496
$$457$$ − 16.1111i − 0.753646i −0.926285 0.376823i $$-0.877017\pi$$
0.926285 0.376823i $$-0.122983\pi$$
$$458$$ 9.06045i 0.423367i
$$459$$ 2.07830 0.0970069
$$460$$ 0 0
$$461$$ 17.3396 0.807586 0.403793 0.914850i $$-0.367692\pi$$
0.403793 + 0.914850i $$0.367692\pi$$
$$462$$ − 0.427559i − 0.0198918i
$$463$$ − 19.3177i − 0.897768i −0.893590 0.448884i $$-0.851822\pi$$
0.893590 0.448884i $$-0.148178\pi$$
$$464$$ 16.1493 0.749711
$$465$$ 0 0
$$466$$ −48.8235 −2.26170
$$467$$ − 8.00000i − 0.370196i −0.982720 0.185098i $$-0.940740\pi$$
0.982720 0.185098i $$-0.0592602\pi$$
$$468$$ 14.3616i 0.663866i
$$469$$ 2.92232 0.134940
$$470$$ 0 0
$$471$$ 11.5953 0.534284
$$472$$ 94.6441i 4.35635i
$$473$$ 4.06745i 0.187022i
$$474$$ 5.69914 0.261770
$$475$$ 0 0
$$476$$ 4.58266 0.210046
$$477$$ − 12.3279i − 0.564455i
$$478$$ 58.6194i 2.68119i
$$479$$ 40.3877 1.84536 0.922681 0.385565i $$-0.125994\pi$$
0.922681 + 0.385565i $$0.125994\pi$$
$$480$$ 0 0
$$481$$ 18.8328 0.858702
$$482$$ − 42.2246i − 1.92328i
$$483$$ 2.42467i 0.110326i
$$484$$ 60.7188 2.75994
$$485$$ 0 0
$$486$$ 2.75660 0.125042
$$487$$ 2.84171i 0.128770i 0.997925 + 0.0643850i $$0.0205086\pi$$
−0.997925 + 0.0643850i $$0.979491\pi$$
$$488$$ − 62.0371i − 2.80829i
$$489$$ −0.855118 −0.0386698
$$490$$ 0 0
$$491$$ 0.157863 0.00712428 0.00356214 0.999994i $$-0.498866\pi$$
0.00356214 + 0.999994i $$0.498866\pi$$
$$492$$ 9.27979i 0.418365i
$$493$$ 2.07830i 0.0936021i
$$494$$ 6.78063 0.305075
$$495$$ 0 0
$$496$$ −164.022 −7.36480
$$497$$ 2.35722i 0.105736i
$$498$$ − 17.6698i − 0.791803i
$$499$$ −2.64723 −0.118506 −0.0592531 0.998243i $$-0.518872\pi$$
−0.0592531 + 0.998243i $$0.518872\pi$$
$$500$$ 0 0
$$501$$ −7.73194 −0.345437
$$502$$ − 74.1297i − 3.30857i
$$503$$ − 13.9545i − 0.622200i −0.950377 0.311100i $$-0.899303\pi$$
0.950377 0.311100i $$-0.100697\pi$$
$$504$$ 3.90703 0.174033
$$505$$ 0 0
$$506$$ 6.68384 0.297133
$$507$$ 6.42023i 0.285132i
$$508$$ − 112.853i − 5.00706i
$$509$$ 16.4921 0.731001 0.365500 0.930811i $$-0.380898\pi$$
0.365500 + 0.930811i $$0.380898\pi$$
$$510$$ 0 0
$$511$$ 1.31616 0.0582233
$$512$$ − 78.0802i − 3.45069i
$$513$$ − 0.958939i − 0.0423382i
$$514$$ −18.8968 −0.833502
$$515$$ 0 0
$$516$$ −57.8241 −2.54556
$$517$$ 4.56511i 0.200773i
$$518$$ − 7.97066i − 0.350211i
$$519$$ 8.07449 0.354431
$$520$$ 0 0
$$521$$ −12.2253 −0.535601 −0.267800 0.963474i $$-0.586297\pi$$
−0.267800 + 0.963474i $$0.586297\pi$$
$$522$$ 2.75660i 0.120653i
$$523$$ − 1.66659i − 0.0728748i −0.999336 0.0364374i $$-0.988399\pi$$
0.999336 0.0364374i $$-0.0116010\pi$$
$$524$$ 30.8067 1.34580
$$525$$ 0 0
$$526$$ 37.5707 1.63816
$$527$$ − 21.1085i − 0.919501i
$$528$$ − 6.36011i − 0.276788i
$$529$$ −14.9038 −0.647992
$$530$$ 0 0
$$531$$ −9.54022 −0.414010
$$532$$ − 2.11446i − 0.0916736i
$$533$$ − 4.25154i − 0.184155i
$$534$$ −43.6376 −1.88838
$$535$$ 0 0
$$536$$ 73.6126 3.17958
$$537$$ − 12.4100i − 0.535531i
$$538$$ − 23.3244i − 1.00558i
$$539$$ −2.69574 −0.116114
$$540$$ 0 0
$$541$$ −34.6558 −1.48997 −0.744984 0.667083i $$-0.767543\pi$$
−0.744984 + 0.667083i $$0.767543\pi$$
$$542$$ − 17.4839i − 0.750996i
$$543$$ − 2.49765i − 0.107185i
$$544$$ 51.2842 2.19879
$$545$$ 0 0
$$546$$ −2.78478 −0.119177
$$547$$ 38.2530i 1.63558i 0.575515 + 0.817791i $$0.304801\pi$$
−0.575515 + 0.817791i $$0.695199\pi$$
$$548$$ − 41.9830i − 1.79343i
$$549$$ 6.25340 0.266888
$$550$$ 0 0
$$551$$ 0.958939 0.0408522
$$552$$ 61.0769i 2.59960i
$$553$$ 0.814230i 0.0346246i
$$554$$ −47.2657 −2.00813
$$555$$ 0 0
$$556$$ −52.3646 −2.22075
$$557$$ − 28.7760i − 1.21928i −0.792679 0.609639i $$-0.791314\pi$$
0.792679 0.609639i $$-0.208686\pi$$
$$558$$ − 27.9977i − 1.18524i
$$559$$ 26.4921 1.12050
$$560$$ 0 0
$$561$$ 0.818503 0.0345572
$$562$$ 33.9022i 1.43008i
$$563$$ 32.9809i 1.38998i 0.719020 + 0.694989i $$0.244591\pi$$
−0.719020 + 0.694989i $$0.755409\pi$$
$$564$$ −64.8989 −2.73274
$$565$$ 0 0
$$566$$ −69.4194 −2.91792
$$567$$ 0.393832i 0.0165394i
$$568$$ 59.3777i 2.49143i
$$569$$ −16.8038 −0.704453 −0.352227 0.935915i $$-0.614575\pi$$
−0.352227 + 0.935915i $$0.614575\pi$$
$$570$$ 0 0
$$571$$ −6.21703 −0.260175 −0.130087 0.991503i $$-0.541526\pi$$
−0.130087 + 0.991503i $$0.541526\pi$$
$$572$$ 5.65607i 0.236492i
$$573$$ − 6.32085i − 0.264057i
$$574$$ −1.79939 −0.0751051
$$575$$ 0 0
$$576$$ 35.7232 1.48847
$$577$$ 11.9977i 0.499470i 0.968314 + 0.249735i $$0.0803435\pi$$
−0.968314 + 0.249735i $$0.919656\pi$$
$$578$$ − 34.9555i − 1.45396i
$$579$$ 25.2921 1.05111
$$580$$ 0 0
$$581$$ 2.52447 0.104733
$$582$$ − 50.8798i − 2.10904i
$$583$$ − 4.85512i − 0.201078i
$$584$$ 33.1537 1.37191
$$585$$ 0 0
$$586$$ −33.9531 −1.40259
$$587$$ − 11.2194i − 0.463073i −0.972826 0.231536i $$-0.925625\pi$$
0.972826 0.231536i $$-0.0743753\pi$$
$$588$$ − 38.3234i − 1.58043i
$$589$$ −9.73957 −0.401312
$$590$$ 0 0
$$591$$ 21.2047 0.872245
$$592$$ − 118.567i − 4.87306i
$$593$$ 7.69682i 0.316071i 0.987433 + 0.158035i $$0.0505160\pi$$
−0.987433 + 0.158035i $$0.949484\pi$$
$$594$$ 1.08564 0.0445442
$$595$$ 0 0
$$596$$ −64.6118 −2.64660
$$597$$ 2.64723i 0.108344i
$$598$$ − 43.5332i − 1.78020i
$$599$$ −20.0543 −0.819396 −0.409698 0.912221i $$-0.634366\pi$$
−0.409698 + 0.912221i $$0.634366\pi$$
$$600$$ 0 0
$$601$$ 3.10851 0.126799 0.0633995 0.997988i $$-0.479806\pi$$
0.0633995 + 0.997988i $$0.479806\pi$$
$$602$$ − 11.2123i − 0.456981i
$$603$$ 7.42023i 0.302175i
$$604$$ −34.0528 −1.38559
$$605$$ 0 0
$$606$$ −35.2950 −1.43376
$$607$$ 37.0441i 1.50357i 0.659407 + 0.751786i $$0.270807\pi$$
−0.659407 + 0.751786i $$0.729193\pi$$
$$608$$ − 23.6628i − 0.959652i
$$609$$ −0.393832 −0.0159589
$$610$$ 0 0
$$611$$ 29.7334 1.20289
$$612$$ 11.6361i 0.470361i
$$613$$ 13.3839i 0.540569i 0.962781 + 0.270284i $$0.0871177\pi$$
−0.962781 + 0.270284i $$0.912882\pi$$
$$614$$ −62.6792 −2.52953
$$615$$ 0 0
$$616$$ 1.53871 0.0619966
$$617$$ − 12.6364i − 0.508721i −0.967109 0.254361i $$-0.918135\pi$$
0.967109 0.254361i $$-0.0818650\pi$$
$$618$$ − 13.4240i − 0.539994i
$$619$$ 41.2447 1.65776 0.828882 0.559424i $$-0.188978\pi$$
0.828882 + 0.559424i $$0.188978\pi$$
$$620$$ 0 0
$$621$$ −6.15661 −0.247056
$$622$$ 14.7150i 0.590018i
$$623$$ − 6.23446i − 0.249778i
$$624$$ −41.4246 −1.65831
$$625$$ 0 0
$$626$$ 5.41226 0.216318
$$627$$ − 0.377661i − 0.0150823i
$$628$$ 64.9203i 2.59060i
$$629$$ 15.2587 0.608406
$$630$$ 0 0
$$631$$ 29.8009 1.18635 0.593177 0.805072i $$-0.297873\pi$$
0.593177 + 0.805072i $$0.297873\pi$$
$$632$$ 20.5103i 0.815854i
$$633$$ 2.00000i 0.0794929i
$$634$$ 3.55777 0.141297
$$635$$ 0 0
$$636$$ 69.0218 2.73689
$$637$$ 17.5579i 0.695669i
$$638$$ 1.08564i 0.0429808i
$$639$$ −5.98533 −0.236776
$$640$$ 0 0
$$641$$ 17.7774 0.702167 0.351083 0.936344i $$-0.385813\pi$$
0.351083 + 0.936344i $$0.385813\pi$$
$$642$$ − 6.45747i − 0.254856i
$$643$$ 44.5534i 1.75702i 0.477727 + 0.878508i $$0.341461\pi$$
−0.477727 + 0.878508i $$0.658539\pi$$
$$644$$ −13.5753 −0.534943
$$645$$ 0 0
$$646$$ 5.49381 0.216151
$$647$$ 42.9339i 1.68790i 0.536418 + 0.843952i $$0.319777\pi$$
−0.536418 + 0.843952i $$0.680223\pi$$
$$648$$ 9.92054i 0.389716i
$$649$$ −3.75725 −0.147485
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ − 4.78766i − 0.187499i
$$653$$ 19.8426i 0.776500i 0.921554 + 0.388250i $$0.126920\pi$$
−0.921554 + 0.388250i $$0.873080\pi$$
$$654$$ −23.5701 −0.921665
$$655$$ 0 0
$$656$$ −26.7666 −1.04506
$$657$$ 3.34192i 0.130381i
$$658$$ − 12.5842i − 0.490582i
$$659$$ −23.0483 −0.897836 −0.448918 0.893573i $$-0.648190\pi$$
−0.448918 + 0.893573i $$0.648190\pi$$
$$660$$ 0 0
$$661$$ −36.2079 −1.40832 −0.704162 0.710040i $$-0.748677\pi$$
−0.704162 + 0.710040i $$0.748677\pi$$
$$662$$ − 79.9332i − 3.10669i
$$663$$ − 5.33107i − 0.207042i
$$664$$ 63.5907 2.46780
$$665$$ 0 0
$$666$$ 20.2387 0.784235
$$667$$ − 6.15661i − 0.238385i
$$668$$ − 43.2898i − 1.67493i
$$669$$ 12.5504 0.485228
$$670$$ 0 0
$$671$$ 2.46279 0.0950749
$$672$$ 9.71820i 0.374888i
$$673$$ − 22.9000i − 0.882731i −0.897327 0.441365i $$-0.854494\pi$$
0.897327 0.441365i $$-0.145506\pi$$
$$674$$ −46.2705 −1.78227
$$675$$ 0 0
$$676$$ −35.9458 −1.38253
$$677$$ 36.0068i 1.38385i 0.721967 + 0.691927i $$0.243238\pi$$
−0.721967 + 0.691927i $$0.756762\pi$$
$$678$$ 30.8974i 1.18661i
$$679$$ 7.26915 0.278964
$$680$$ 0 0
$$681$$ −1.30149 −0.0498732
$$682$$ − 11.0264i − 0.422222i
$$683$$ − 9.12896i − 0.349310i −0.984630 0.174655i $$-0.944119\pi$$
0.984630 0.174655i $$-0.0558810\pi$$
$$684$$ 5.36894 0.205287
$$685$$ 0 0
$$686$$ 15.0305 0.573869
$$687$$ − 3.28682i − 0.125400i
$$688$$ − 166.788i − 6.35873i
$$689$$ −31.6223 −1.20472
$$690$$ 0 0
$$691$$ 5.85956 0.222908 0.111454 0.993770i $$-0.464449\pi$$
0.111454 + 0.993770i $$0.464449\pi$$
$$692$$ 45.2077i 1.71854i
$$693$$ 0.155104i 0.00589191i
$$694$$ −49.2552 −1.86970
$$695$$ 0 0
$$696$$ −9.92054 −0.376037
$$697$$ − 3.44469i − 0.130477i
$$698$$ 24.0868i 0.911700i
$$699$$ 17.7115 0.669910
$$700$$ 0 0
$$701$$ −7.56955 −0.285898 −0.142949 0.989730i $$-0.545658\pi$$
−0.142949 + 0.989730i $$0.545658\pi$$
$$702$$ − 7.07097i − 0.266877i
$$703$$ − 7.04046i − 0.265536i
$$704$$ 14.0690 0.530244
$$705$$ 0 0
$$706$$ 62.3039 2.34484
$$707$$ − 5.04256i − 0.189645i
$$708$$ − 53.4141i − 2.00742i
$$709$$ −14.5209 −0.545342 −0.272671 0.962107i $$-0.587907\pi$$
−0.272671 + 0.962107i $$0.587907\pi$$
$$710$$ 0 0
$$711$$ −2.06745 −0.0775356
$$712$$ − 157.044i − 5.88549i
$$713$$ 62.5302i 2.34178i
$$714$$ −2.25628 −0.0844393
$$715$$ 0 0
$$716$$ 69.4815 2.59665
$$717$$ − 21.2651i − 0.794161i
$$718$$ − 36.3402i − 1.35621i
$$719$$ −12.2164 −0.455596 −0.227798 0.973708i $$-0.573153\pi$$
−0.227798 + 0.973708i $$0.573153\pi$$
$$720$$ 0 0
$$721$$ 1.91788 0.0714255
$$722$$ 49.8405i 1.85487i
$$723$$ 15.3177i 0.569670i
$$724$$ 13.9839 0.519709
$$725$$ 0 0
$$726$$ −29.8950 −1.10951
$$727$$ − 40.8856i − 1.51636i −0.652044 0.758182i $$-0.726088\pi$$
0.652044 0.758182i $$-0.273912\pi$$
$$728$$ − 10.0219i − 0.371438i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 21.4645 0.793892
$$732$$ 35.0117i 1.29407i
$$733$$ − 25.4915i − 0.941550i −0.882253 0.470775i $$-0.843974\pi$$
0.882253 0.470775i $$-0.156026\pi$$
$$734$$ 48.1056 1.77561
$$735$$ 0 0
$$736$$ −151.920 −5.59986
$$737$$ 2.92232i 0.107645i
$$738$$ − 4.56892i − 0.168184i
$$739$$ −9.56830 −0.351975 −0.175988 0.984392i $$-0.556312\pi$$
−0.175988 + 0.984392i $$0.556312\pi$$
$$740$$ 0 0
$$741$$ −2.45978 −0.0903624
$$742$$ 13.3836i 0.491328i
$$743$$ 18.0455i 0.662025i 0.943626 + 0.331013i $$0.107390\pi$$
−0.943626 + 0.331013i $$0.892610\pi$$
$$744$$ 100.759 3.69400
$$745$$ 0 0
$$746$$ 88.7673 3.25000
$$747$$ 6.41000i 0.234530i
$$748$$ 4.58266i 0.167559i
$$749$$ 0.922572 0.0337100
$$750$$ 0 0
$$751$$ 11.9255 0.435168 0.217584 0.976042i $$-0.430182\pi$$
0.217584 + 0.976042i $$0.430182\pi$$
$$752$$ − 187.194i − 6.82627i
$$753$$ 26.8917i 0.979989i
$$754$$ 7.07097 0.257510
$$755$$ 0 0
$$756$$ −2.20500 −0.0801951
$$757$$ 5.86152i 0.213041i 0.994311 + 0.106520i $$0.0339709\pi$$
−0.994311 + 0.106520i $$0.966029\pi$$
$$758$$ 89.2201i 3.24062i
$$759$$ −2.42467 −0.0880100
$$760$$ 0 0
$$761$$ 49.8739 1.80793 0.903963 0.427610i $$-0.140644\pi$$
0.903963 + 0.427610i $$0.140644\pi$$
$$762$$ 55.5637i 2.01286i
$$763$$ − 3.36744i − 0.121909i
$$764$$ 35.3894 1.28034
$$765$$ 0 0
$$766$$ 71.3239 2.57704
$$767$$ 24.4717i 0.883621i
$$768$$ 63.9648i 2.30813i
$$769$$ −29.5121 −1.06423 −0.532117 0.846671i $$-0.678604\pi$$
−0.532117 + 0.846671i $$0.678604\pi$$
$$770$$ 0 0
$$771$$ 6.85512 0.246881
$$772$$ 141.607i 5.09653i
$$773$$ 2.41935i 0.0870180i 0.999053 + 0.0435090i $$0.0138537\pi$$
−0.999053 + 0.0435090i $$0.986146\pi$$
$$774$$ 28.4698 1.02333
$$775$$ 0 0
$$776$$ 183.108 6.57320
$$777$$ 2.89149i 0.103731i
$$778$$ 89.8934i 3.22283i
$$779$$ −1.58939 −0.0569460
$$780$$ 0 0
$$781$$ −2.35722 −0.0843479
$$782$$ − 35.2715i − 1.26131i
$$783$$ − 1.00000i − 0.0357371i
$$784$$ 110.540 3.94786
$$785$$ 0 0
$$786$$ −15.1678 −0.541016
$$787$$ − 51.1549i − 1.82348i −0.410772 0.911738i $$-0.634741\pi$$
0.410772 0.911738i $$-0.365259\pi$$
$$788$$ 118.722i 4.22928i
$$789$$ −13.6294 −0.485218
$$790$$ 0 0
$$791$$ −4.41428 −0.156954
$$792$$ 3.90703i 0.138830i
$$793$$ − 16.0406i − 0.569619i
$$794$$ 12.1162 0.429987
$$795$$ 0 0
$$796$$ −14.8214 −0.525330
$$797$$ 28.8662i 1.02249i 0.859434 + 0.511247i $$0.170816\pi$$
−0.859434 + 0.511247i $$0.829184\pi$$
$$798$$ 1.04106i 0.0368531i
$$799$$ 24.0907 0.852266
$$800$$ 0 0
$$801$$ 15.8302 0.559334
$$802$$ 52.5567i 1.85584i
$$803$$ 1.31616i 0.0464462i
$$804$$ −41.5446 −1.46517
$$805$$ 0 0
$$806$$ −71.8171 −2.52965
$$807$$ 8.46129i 0.297851i
$$808$$ − 127.021i − 4.46858i
$$809$$ −19.0426 −0.669501 −0.334750 0.942307i $$-0.608652\pi$$
−0.334750 + 0.942307i $$0.608652\pi$$
$$810$$ 0 0
$$811$$ −20.8783 −0.733137 −0.366569 0.930391i $$-0.619467\pi$$
−0.366569 + 0.930391i $$0.619467\pi$$
$$812$$ − 2.20500i − 0.0773804i
$$813$$ 6.34255i 0.222443i
$$814$$ 7.97066 0.279372
$$815$$ 0 0
$$816$$ −33.5631 −1.17494
$$817$$ − 9.90381i − 0.346491i
$$818$$ − 3.81060i − 0.133234i
$$819$$ 1.01022 0.0353000
$$820$$ 0 0
$$821$$ 3.60466 0.125804 0.0629018 0.998020i $$-0.479965\pi$$
0.0629018 + 0.998020i $$0.479965\pi$$
$$822$$ 20.6704i 0.720964i
$$823$$ − 27.2288i − 0.949135i −0.880219 0.474567i $$-0.842605\pi$$
0.880219 0.474567i $$-0.157395\pi$$
$$824$$ 48.3109 1.68299
$$825$$ 0 0
$$826$$ 10.3572 0.360374
$$827$$ 47.3936i 1.64804i 0.566562 + 0.824019i $$0.308273\pi$$
−0.566562 + 0.824019i $$0.691727\pi$$
$$828$$ − 34.4698i − 1.19791i
$$829$$ 41.8388 1.45312 0.726560 0.687103i $$-0.241118\pi$$
0.726560 + 0.687103i $$0.241118\pi$$
$$830$$ 0 0
$$831$$ 17.1464 0.594802
$$832$$ − 91.6339i − 3.17683i
$$833$$ 14.2258i 0.492894i
$$834$$ 25.7818 0.892752
$$835$$ 0 0
$$836$$ 2.11446 0.0731302
$$837$$ 10.1566i 0.351064i
$$838$$ 7.91085i 0.273276i
$$839$$ 31.6385 1.09228 0.546141 0.837693i $$-0.316096\pi$$
0.546141 + 0.837693i $$0.316096\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ − 37.6111i − 1.29617i
$$843$$ − 12.2985i − 0.423584i
$$844$$ −11.1977 −0.385440
$$845$$ 0 0
$$846$$ 31.9531 1.09857
$$847$$ − 4.27107i − 0.146756i
$$848$$ 199.086i 6.83665i
$$849$$ 25.1830 0.864278
$$850$$ 0 0
$$851$$ −45.2013 −1.54948
$$852$$ − 33.5109i − 1.14806i
$$853$$ − 44.2000i − 1.51338i −0.653773 0.756690i $$-0.726815\pi$$
0.653773 0.756690i $$-0.273185\pi$$
$$854$$ −6.78892 −0.232312
$$855$$ 0 0
$$856$$ 23.2394 0.794305
$$857$$ 14.4775i 0.494541i 0.968947 + 0.247270i $$0.0795335\pi$$
−0.968947 + 0.247270i $$0.920466\pi$$
$$858$$ − 2.78478i − 0.0950707i
$$859$$ −22.5883 −0.770703 −0.385352 0.922770i $$-0.625920\pi$$
−0.385352 + 0.922770i $$0.625920\pi$$
$$860$$ 0 0
$$861$$ 0.652757 0.0222459
$$862$$ 32.9983i 1.12393i
$$863$$ 22.9900i 0.782590i 0.920265 + 0.391295i $$0.127973\pi$$
−0.920265 + 0.391295i $$0.872027\pi$$
$$864$$ −24.6760 −0.839494
$$865$$ 0 0
$$866$$ −49.5930 −1.68524
$$867$$ 12.6807i 0.430658i
$$868$$ 22.3953i 0.760147i
$$869$$ −0.814230 −0.0276209
$$870$$ 0 0
$$871$$ 19.0337 0.644931
$$872$$ − 84.8250i − 2.87254i
$$873$$ 18.4575i 0.624691i
$$874$$ −16.2744 −0.550491
$$875$$ 0 0
$$876$$ −18.7109 −0.632182
$$877$$ 13.5741i 0.458364i 0.973384 + 0.229182i $$0.0736051\pi$$
−0.973384 + 0.229182i $$0.926395\pi$$
$$878$$ − 44.7191i − 1.50920i
$$879$$ 12.3170 0.415443
$$880$$ 0 0
$$881$$ 9.91235 0.333956 0.166978 0.985961i $$-0.446599\pi$$
0.166978 + 0.985961i $$0.446599\pi$$
$$882$$ 18.8686i 0.635340i
$$883$$ − 39.3192i − 1.32320i −0.749859 0.661598i $$-0.769879\pi$$
0.749859 0.661598i $$-0.230121\pi$$
$$884$$ 29.8478 1.00389
$$885$$ 0 0
$$886$$ 97.5180 3.27618
$$887$$ − 59.2520i − 1.98949i −0.102403 0.994743i $$-0.532653\pi$$
0.102403 0.994743i $$-0.467347\pi$$
$$888$$ 72.8358i 2.44421i
$$889$$ −7.93832 −0.266243
$$890$$ 0 0
$$891$$ −0.393832 −0.0131939
$$892$$ 70.2678i 2.35274i
$$893$$ − 11.1155i − 0.371968i
$$894$$ 31.8117 1.06394
$$895$$ 0 0
$$896$$ −19.3461 −0.646307
$$897$$ 15.7924i 0.527291i
$$898$$ 114.942i 3.83567i
$$899$$ −10.1566 −0.338742
$$900$$ 0 0
$$901$$ −25.6211 −0.853562
$$902$$ − 1.79939i − 0.0599131i
$$903$$ 4.06745i 0.135356i
$$904$$ −111.195 −3.69828
$$905$$ 0 0
$$906$$ 16.7660 0.557012
$$907$$ − 5.91210i − 0.196308i −0.995171 0.0981541i $$-0.968706\pi$$
0.995171 0.0981541i $$-0.0312938\pi$$
$$908$$ − 7.28682i − 0.241822i
$$909$$ 12.8038 0.424676
$$910$$ 0 0
$$911$$ −12.4185 −0.411445 −0.205722 0.978610i $$-0.565954\pi$$
−0.205722 + 0.978610i $$0.565954\pi$$
$$912$$ 15.4862i 0.512799i
$$913$$ 2.52447i 0.0835476i
$$914$$ −44.4118 −1.46901
$$915$$ 0 0
$$916$$ 18.4024 0.608031
$$917$$ − 2.16700i − 0.0715607i
$$918$$ − 5.72905i − 0.189087i
$$919$$ −16.4332 −0.542081 −0.271041 0.962568i $$-0.587368\pi$$
−0.271041 + 0.962568i $$0.587368\pi$$
$$920$$ 0 0
$$921$$ 22.7379 0.749239
$$922$$ − 47.7983i − 1.57415i
$$923$$ 15.3530i 0.505351i
$$924$$ −0.868401 −0.0285683
$$925$$ 0 0
$$926$$ −53.2510 −1.74994
$$927$$ 4.86979i 0.159945i
$$928$$ − 24.6760i − 0.810029i
$$929$$ −13.8358 −0.453936 −0.226968 0.973902i $$-0.572881\pi$$
−0.226968 + 0.973902i $$0.572881\pi$$
$$930$$ 0 0
$$931$$ 6.56384 0.215121
$$932$$ 99.1637i 3.24822i
$$933$$ − 5.33810i − 0.174762i
$$934$$ −22.0528 −0.721589
$$935$$ 0 0
$$936$$ 25.4472 0.831769
$$937$$ 20.7528i 0.677964i 0.940793 + 0.338982i $$0.110083\pi$$
−0.940793 + 0.338982i $$0.889917\pi$$
$$938$$ − 8.05567i − 0.263027i
$$939$$ −1.96338 −0.0640726
$$940$$ 0 0
$$941$$ 40.1666 1.30940 0.654698 0.755891i $$-0.272796\pi$$
0.654698 + 0.755891i $$0.272796\pi$$
$$942$$ − 31.9636i − 1.04143i
$$943$$ 10.2043i 0.332297i
$$944$$ 154.068 5.01447
$$945$$ 0 0
$$946$$ 11.2123 0.364544
$$947$$ − 17.4144i − 0.565894i −0.959136 0.282947i $$-0.908688\pi$$
0.959136 0.282947i $$-0.0913120\pi$$
$$948$$ − 11.5753i − 0.375949i
$$949$$ 8.57239 0.278271
$$950$$ 0 0
$$951$$ −1.29064 −0.0418518
$$952$$ − 8.11999i − 0.263170i
$$953$$ − 13.5121i − 0.437701i −0.975758 0.218851i $$-0.929769\pi$$
0.975758 0.218851i $$-0.0702307\pi$$
$$954$$ −33.9830 −1.10024
$$955$$ 0 0
$$956$$ 119.060 3.85067
$$957$$ − 0.393832i − 0.0127308i
$$958$$ − 111.333i − 3.59700i
$$959$$ −2.95316 −0.0953626
$$960$$ 0 0
$$961$$ 72.1567 2.32763
$$962$$ − 51.9145i − 1.67379i
$$963$$ 2.34255i 0.0754876i
$$964$$ −85.7610 −2.76218
$$965$$ 0 0
$$966$$ 6.68384 0.215049
$$967$$ − 24.0598i − 0.773712i −0.922140 0.386856i $$-0.873561\pi$$
0.922140 0.386856i $$-0.126439\pi$$
$$968$$ − 107.587i − 3.45798i
$$969$$ −1.99297 −0.0640233
$$970$$ 0 0
$$971$$ 34.7877 1.11639 0.558195 0.829710i $$-0.311494\pi$$
0.558195 + 0.829710i $$0.311494\pi$$
$$972$$ − 5.59883i − 0.179583i
$$973$$ 3.68342i 0.118085i
$$974$$ 7.83344 0.251000
$$975$$ 0 0
$$976$$ −100.988 −3.23254
$$977$$ 35.7566i 1.14396i 0.820269 + 0.571978i $$0.193824\pi$$
−0.820269 + 0.571978i $$0.806176\pi$$
$$978$$ 2.35722i 0.0753755i
$$979$$ 6.23446 0.199254
$$980$$ 0 0
$$981$$ 8.55044 0.272995
$$982$$ − 0.435166i − 0.0138867i
$$983$$ 7.19064i 0.229346i 0.993403 + 0.114673i $$0.0365820\pi$$
−0.993403 + 0.114673i $$0.963418\pi$$
$$984$$ 16.4428 0.524177
$$985$$ 0 0
$$986$$ 5.72905 0.182450
$$987$$ 4.56511i 0.145309i
$$988$$ − 13.7719i − 0.438143i
$$989$$ −63.5847 −2.02188
$$990$$ 0 0
$$991$$ −19.9123 −0.632537 −0.316268 0.948670i $$-0.602430\pi$$
−0.316268 + 0.948670i $$0.602430\pi$$
$$992$$ 250.624i 7.95733i
$$993$$ 28.9971i 0.920194i
$$994$$ 6.49790 0.206101
$$995$$ 0 0
$$996$$ −35.8885 −1.13717
$$997$$ − 1.57302i − 0.0498179i −0.999690 0.0249089i $$-0.992070\pi$$
0.999690 0.0249089i $$-0.00792958\pi$$
$$998$$ 7.29734i 0.230993i
$$999$$ −7.34192 −0.232288
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.n.349.1 8
5.2 odd 4 2175.2.a.v.1.4 4
5.3 odd 4 435.2.a.j.1.1 4
5.4 even 2 inner 2175.2.c.n.349.8 8
15.2 even 4 6525.2.a.bi.1.1 4
15.8 even 4 1305.2.a.r.1.4 4
20.3 even 4 6960.2.a.co.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.1 4 5.3 odd 4
1305.2.a.r.1.4 4 15.8 even 4
2175.2.a.v.1.4 4 5.2 odd 4
2175.2.c.n.349.1 8 1.1 even 1 trivial
2175.2.c.n.349.8 8 5.4 even 2 inner
6525.2.a.bi.1.1 4 15.2 even 4
6960.2.a.co.1.2 4 20.3 even 4