# Properties

 Label 2175.2.c.p.349.11 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 349.11 Root $$0.510732i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.p.349.6

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.510732i q^{2} -1.00000i q^{3} +1.73915 q^{4} +0.510732 q^{6} -4.82343i q^{7} +1.90970i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+0.510732i q^{2} -1.00000i q^{3} +1.73915 q^{4} +0.510732 q^{6} -4.82343i q^{7} +1.90970i q^{8} -1.00000 q^{9} +4.88439 q^{11} -1.73915i q^{12} +4.59669i q^{13} +2.46348 q^{14} +2.50296 q^{16} -6.50987i q^{17} -0.510732i q^{18} -3.09205 q^{19} -4.82343 q^{21} +2.49461i q^{22} -5.62549i q^{23} +1.90970 q^{24} -2.34767 q^{26} +1.00000i q^{27} -8.38869i q^{28} -1.00000 q^{29} +9.24375 q^{31} +5.09775i q^{32} -4.88439i q^{33} +3.32480 q^{34} -1.73915 q^{36} +11.1261i q^{37} -1.57921i q^{38} +4.59669 q^{39} -2.84537 q^{41} -2.46348i q^{42} -4.58611i q^{43} +8.49470 q^{44} +2.87311 q^{46} -3.62713i q^{47} -2.50296i q^{48} -16.2655 q^{49} -6.50987 q^{51} +7.99434i q^{52} -0.967845i q^{53} -0.510732 q^{54} +9.21132 q^{56} +3.09205i q^{57} -0.510732i q^{58} +0.298882 q^{59} -0.786908 q^{61} +4.72108i q^{62} +4.82343i q^{63} +2.40234 q^{64} +2.49461 q^{66} -4.86742i q^{67} -11.3217i q^{68} -5.62549 q^{69} +0.741689 q^{71} -1.90970i q^{72} +5.52981i q^{73} -5.68246 q^{74} -5.37755 q^{76} -23.5595i q^{77} +2.34767i q^{78} +2.96278 q^{79} +1.00000 q^{81} -1.45322i q^{82} -13.6633i q^{83} -8.38869 q^{84} +2.34227 q^{86} +1.00000i q^{87} +9.32773i q^{88} -3.67835 q^{89} +22.1718 q^{91} -9.78358i q^{92} -9.24375i q^{93} +1.85249 q^{94} +5.09775 q^{96} +2.87658i q^{97} -8.30730i q^{98} -4.88439 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.510732i 0.361142i 0.983562 + 0.180571i $$0.0577945\pi$$
−0.983562 + 0.180571i $$0.942205\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ 1.73915 0.869577
$$5$$ 0 0
$$6$$ 0.510732 0.208505
$$7$$ − 4.82343i − 1.82309i −0.411205 0.911543i $$-0.634892\pi$$
0.411205 0.911543i $$-0.365108\pi$$
$$8$$ 1.90970i 0.675182i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.88439 1.47270 0.736349 0.676602i $$-0.236548\pi$$
0.736349 + 0.676602i $$0.236548\pi$$
$$12$$ − 1.73915i − 0.502050i
$$13$$ 4.59669i 1.27489i 0.770495 + 0.637446i $$0.220009\pi$$
−0.770495 + 0.637446i $$0.779991\pi$$
$$14$$ 2.46348 0.658392
$$15$$ 0 0
$$16$$ 2.50296 0.625740
$$17$$ − 6.50987i − 1.57888i −0.613830 0.789438i $$-0.710372\pi$$
0.613830 0.789438i $$-0.289628\pi$$
$$18$$ − 0.510732i − 0.120381i
$$19$$ −3.09205 −0.709364 −0.354682 0.934987i $$-0.615411\pi$$
−0.354682 + 0.934987i $$0.615411\pi$$
$$20$$ 0 0
$$21$$ −4.82343 −1.05256
$$22$$ 2.49461i 0.531853i
$$23$$ − 5.62549i − 1.17299i −0.809951 0.586497i $$-0.800506\pi$$
0.809951 0.586497i $$-0.199494\pi$$
$$24$$ 1.90970 0.389817
$$25$$ 0 0
$$26$$ −2.34767 −0.460416
$$27$$ 1.00000i 0.192450i
$$28$$ − 8.38869i − 1.58531i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 9.24375 1.66023 0.830113 0.557595i $$-0.188276\pi$$
0.830113 + 0.557595i $$0.188276\pi$$
$$32$$ 5.09775i 0.901163i
$$33$$ − 4.88439i − 0.850262i
$$34$$ 3.32480 0.570198
$$35$$ 0 0
$$36$$ −1.73915 −0.289859
$$37$$ 11.1261i 1.82912i 0.404448 + 0.914561i $$0.367464\pi$$
−0.404448 + 0.914561i $$0.632536\pi$$
$$38$$ − 1.57921i − 0.256181i
$$39$$ 4.59669 0.736059
$$40$$ 0 0
$$41$$ −2.84537 −0.444372 −0.222186 0.975004i $$-0.571319\pi$$
−0.222186 + 0.975004i $$0.571319\pi$$
$$42$$ − 2.46348i − 0.380123i
$$43$$ − 4.58611i − 0.699375i −0.936866 0.349688i $$-0.886288\pi$$
0.936866 0.349688i $$-0.113712\pi$$
$$44$$ 8.49470 1.28062
$$45$$ 0 0
$$46$$ 2.87311 0.423617
$$47$$ − 3.62713i − 0.529071i −0.964376 0.264536i $$-0.914781\pi$$
0.964376 0.264536i $$-0.0852187\pi$$
$$48$$ − 2.50296i − 0.361271i
$$49$$ −16.2655 −2.32364
$$50$$ 0 0
$$51$$ −6.50987 −0.911564
$$52$$ 7.99434i 1.10862i
$$53$$ − 0.967845i − 0.132944i −0.997788 0.0664719i $$-0.978826\pi$$
0.997788 0.0664719i $$-0.0211743\pi$$
$$54$$ −0.510732 −0.0695018
$$55$$ 0 0
$$56$$ 9.21132 1.23091
$$57$$ 3.09205i 0.409552i
$$58$$ − 0.510732i − 0.0670623i
$$59$$ 0.298882 0.0389112 0.0194556 0.999811i $$-0.493807\pi$$
0.0194556 + 0.999811i $$0.493807\pi$$
$$60$$ 0 0
$$61$$ −0.786908 −0.100753 −0.0503766 0.998730i $$-0.516042\pi$$
−0.0503766 + 0.998730i $$0.516042\pi$$
$$62$$ 4.72108i 0.599577i
$$63$$ 4.82343i 0.607695i
$$64$$ 2.40234 0.300293
$$65$$ 0 0
$$66$$ 2.49461 0.307065
$$67$$ − 4.86742i − 0.594650i −0.954776 0.297325i $$-0.903906\pi$$
0.954776 0.297325i $$-0.0960945\pi$$
$$68$$ − 11.3217i − 1.37295i
$$69$$ −5.62549 −0.677229
$$70$$ 0 0
$$71$$ 0.741689 0.0880223 0.0440112 0.999031i $$-0.485986\pi$$
0.0440112 + 0.999031i $$0.485986\pi$$
$$72$$ − 1.90970i − 0.225061i
$$73$$ 5.52981i 0.647215i 0.946191 + 0.323608i $$0.104896\pi$$
−0.946191 + 0.323608i $$0.895104\pi$$
$$74$$ −5.68246 −0.660572
$$75$$ 0 0
$$76$$ −5.37755 −0.616847
$$77$$ − 23.5595i − 2.68485i
$$78$$ 2.34767i 0.265822i
$$79$$ 2.96278 0.333339 0.166669 0.986013i $$-0.446699\pi$$
0.166669 + 0.986013i $$0.446699\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 1.45322i − 0.160481i
$$83$$ − 13.6633i − 1.49975i −0.661581 0.749874i $$-0.730114\pi$$
0.661581 0.749874i $$-0.269886\pi$$
$$84$$ −8.38869 −0.915281
$$85$$ 0 0
$$86$$ 2.34227 0.252574
$$87$$ 1.00000i 0.107211i
$$88$$ 9.32773i 0.994339i
$$89$$ −3.67835 −0.389905 −0.194952 0.980813i $$-0.562455\pi$$
−0.194952 + 0.980813i $$0.562455\pi$$
$$90$$ 0 0
$$91$$ 22.1718 2.32424
$$92$$ − 9.78358i − 1.02001i
$$93$$ − 9.24375i − 0.958532i
$$94$$ 1.85249 0.191070
$$95$$ 0 0
$$96$$ 5.09775 0.520287
$$97$$ 2.87658i 0.292072i 0.989279 + 0.146036i $$0.0466515\pi$$
−0.989279 + 0.146036i $$0.953348\pi$$
$$98$$ − 8.30730i − 0.839164i
$$99$$ −4.88439 −0.490899
$$100$$ 0 0
$$101$$ −14.6780 −1.46051 −0.730255 0.683174i $$-0.760599\pi$$
−0.730255 + 0.683174i $$0.760599\pi$$
$$102$$ − 3.32480i − 0.329204i
$$103$$ − 2.48154i − 0.244514i −0.992498 0.122257i $$-0.960987\pi$$
0.992498 0.122257i $$-0.0390132\pi$$
$$104$$ −8.77831 −0.860784
$$105$$ 0 0
$$106$$ 0.494309 0.0480115
$$107$$ 4.48423i 0.433507i 0.976226 + 0.216753i $$0.0695467\pi$$
−0.976226 + 0.216753i $$0.930453\pi$$
$$108$$ 1.73915i 0.167350i
$$109$$ −3.28140 −0.314301 −0.157151 0.987575i $$-0.550231\pi$$
−0.157151 + 0.987575i $$0.550231\pi$$
$$110$$ 0 0
$$111$$ 11.1261 1.05604
$$112$$ − 12.0729i − 1.14078i
$$113$$ − 17.9820i − 1.69160i −0.533500 0.845800i $$-0.679123\pi$$
0.533500 0.845800i $$-0.320877\pi$$
$$114$$ −1.57921 −0.147906
$$115$$ 0 0
$$116$$ −1.73915 −0.161476
$$117$$ − 4.59669i − 0.424964i
$$118$$ 0.152649i 0.0140524i
$$119$$ −31.3999 −2.87843
$$120$$ 0 0
$$121$$ 12.8572 1.16884
$$122$$ − 0.401899i − 0.0363862i
$$123$$ 2.84537i 0.256558i
$$124$$ 16.0763 1.44369
$$125$$ 0 0
$$126$$ −2.46348 −0.219464
$$127$$ 13.5099i 1.19881i 0.800446 + 0.599405i $$0.204596\pi$$
−0.800446 + 0.599405i $$0.795404\pi$$
$$128$$ 11.4224i 1.00961i
$$129$$ −4.58611 −0.403784
$$130$$ 0 0
$$131$$ 12.1450 1.06111 0.530556 0.847650i $$-0.321983\pi$$
0.530556 + 0.847650i $$0.321983\pi$$
$$132$$ − 8.49470i − 0.739368i
$$133$$ 14.9143i 1.29323i
$$134$$ 2.48594 0.214753
$$135$$ 0 0
$$136$$ 12.4319 1.06603
$$137$$ 9.97466i 0.852193i 0.904678 + 0.426096i $$0.140112\pi$$
−0.904678 + 0.426096i $$0.859888\pi$$
$$138$$ − 2.87311i − 0.244576i
$$139$$ −2.82971 −0.240013 −0.120007 0.992773i $$-0.538292\pi$$
−0.120007 + 0.992773i $$0.538292\pi$$
$$140$$ 0 0
$$141$$ −3.62713 −0.305459
$$142$$ 0.378804i 0.0317885i
$$143$$ 22.4520i 1.87753i
$$144$$ −2.50296 −0.208580
$$145$$ 0 0
$$146$$ −2.82425 −0.233736
$$147$$ 16.2655i 1.34155i
$$148$$ 19.3500i 1.59056i
$$149$$ 22.5396 1.84652 0.923259 0.384178i $$-0.125515\pi$$
0.923259 + 0.384178i $$0.125515\pi$$
$$150$$ 0 0
$$151$$ −14.2205 −1.15725 −0.578626 0.815593i $$-0.696411\pi$$
−0.578626 + 0.815593i $$0.696411\pi$$
$$152$$ − 5.90490i − 0.478950i
$$153$$ 6.50987i 0.526292i
$$154$$ 12.0326 0.969613
$$155$$ 0 0
$$156$$ 7.99434 0.640059
$$157$$ − 10.7712i − 0.859632i −0.902917 0.429816i $$-0.858579\pi$$
0.902917 0.429816i $$-0.141421\pi$$
$$158$$ 1.51319i 0.120383i
$$159$$ −0.967845 −0.0767551
$$160$$ 0 0
$$161$$ −27.1341 −2.13847
$$162$$ 0.510732i 0.0401269i
$$163$$ − 10.6908i − 0.837365i −0.908133 0.418682i $$-0.862492\pi$$
0.908133 0.418682i $$-0.137508\pi$$
$$164$$ −4.94853 −0.386415
$$165$$ 0 0
$$166$$ 6.97830 0.541621
$$167$$ − 7.90643i − 0.611818i −0.952061 0.305909i $$-0.901040\pi$$
0.952061 0.305909i $$-0.0989603\pi$$
$$168$$ − 9.21132i − 0.710669i
$$169$$ −8.12952 −0.625347
$$170$$ 0 0
$$171$$ 3.09205 0.236455
$$172$$ − 7.97595i − 0.608160i
$$173$$ 3.76760i 0.286445i 0.989690 + 0.143223i $$0.0457465\pi$$
−0.989690 + 0.143223i $$0.954253\pi$$
$$174$$ −0.510732 −0.0387185
$$175$$ 0 0
$$176$$ 12.2254 0.921526
$$177$$ − 0.298882i − 0.0224654i
$$178$$ − 1.87865i − 0.140811i
$$179$$ −10.3282 −0.771964 −0.385982 0.922506i $$-0.626137\pi$$
−0.385982 + 0.922506i $$0.626137\pi$$
$$180$$ 0 0
$$181$$ 14.4533 1.07431 0.537154 0.843484i $$-0.319500\pi$$
0.537154 + 0.843484i $$0.319500\pi$$
$$182$$ 11.3238i 0.839378i
$$183$$ 0.786908i 0.0581699i
$$184$$ 10.7430 0.791985
$$185$$ 0 0
$$186$$ 4.72108 0.346166
$$187$$ − 31.7967i − 2.32521i
$$188$$ − 6.30813i − 0.460068i
$$189$$ 4.82343 0.350853
$$190$$ 0 0
$$191$$ 8.81550 0.637867 0.318934 0.947777i $$-0.396675\pi$$
0.318934 + 0.947777i $$0.396675\pi$$
$$192$$ − 2.40234i − 0.173374i
$$193$$ 17.2543i 1.24199i 0.783814 + 0.620996i $$0.213272\pi$$
−0.783814 + 0.620996i $$0.786728\pi$$
$$194$$ −1.46916 −0.105479
$$195$$ 0 0
$$196$$ −28.2882 −2.02058
$$197$$ 22.5719i 1.60818i 0.594508 + 0.804090i $$0.297347\pi$$
−0.594508 + 0.804090i $$0.702653\pi$$
$$198$$ − 2.49461i − 0.177284i
$$199$$ 13.7975 0.978078 0.489039 0.872262i $$-0.337348\pi$$
0.489039 + 0.872262i $$0.337348\pi$$
$$200$$ 0 0
$$201$$ −4.86742 −0.343321
$$202$$ − 7.49649i − 0.527451i
$$203$$ 4.82343i 0.338538i
$$204$$ −11.3217 −0.792675
$$205$$ 0 0
$$206$$ 1.26740 0.0883041
$$207$$ 5.62549i 0.390998i
$$208$$ 11.5053i 0.797751i
$$209$$ −15.1028 −1.04468
$$210$$ 0 0
$$211$$ 16.4705 1.13387 0.566937 0.823761i $$-0.308128\pi$$
0.566937 + 0.823761i $$0.308128\pi$$
$$212$$ − 1.68323i − 0.115605i
$$213$$ − 0.741689i − 0.0508197i
$$214$$ −2.29024 −0.156557
$$215$$ 0 0
$$216$$ −1.90970 −0.129939
$$217$$ − 44.5866i − 3.02674i
$$218$$ − 1.67592i − 0.113507i
$$219$$ 5.52981 0.373670
$$220$$ 0 0
$$221$$ 29.9238 2.01289
$$222$$ 5.68246i 0.381382i
$$223$$ − 4.58903i − 0.307304i −0.988125 0.153652i $$-0.950896\pi$$
0.988125 0.153652i $$-0.0491035\pi$$
$$224$$ 24.5886 1.64290
$$225$$ 0 0
$$226$$ 9.18396 0.610908
$$227$$ 25.5254i 1.69418i 0.531452 + 0.847089i $$0.321647\pi$$
−0.531452 + 0.847089i $$0.678353\pi$$
$$228$$ 5.37755i 0.356137i
$$229$$ 8.04982 0.531947 0.265974 0.963980i $$-0.414307\pi$$
0.265974 + 0.963980i $$0.414307\pi$$
$$230$$ 0 0
$$231$$ −23.5595 −1.55010
$$232$$ − 1.90970i − 0.125378i
$$233$$ 3.75903i 0.246262i 0.992390 + 0.123131i $$0.0392935\pi$$
−0.992390 + 0.123131i $$0.960706\pi$$
$$234$$ 2.34767 0.153472
$$235$$ 0 0
$$236$$ 0.519802 0.0338362
$$237$$ − 2.96278i − 0.192453i
$$238$$ − 16.0369i − 1.03952i
$$239$$ 3.53192 0.228461 0.114231 0.993454i $$-0.463560\pi$$
0.114231 + 0.993454i $$0.463560\pi$$
$$240$$ 0 0
$$241$$ 3.05553 0.196824 0.0984118 0.995146i $$-0.468624\pi$$
0.0984118 + 0.995146i $$0.468624\pi$$
$$242$$ 6.56659i 0.422117i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −1.36855 −0.0876126
$$245$$ 0 0
$$246$$ −1.45322 −0.0926538
$$247$$ − 14.2132i − 0.904363i
$$248$$ 17.6528i 1.12096i
$$249$$ −13.6633 −0.865880
$$250$$ 0 0
$$251$$ −23.3283 −1.47247 −0.736233 0.676728i $$-0.763397\pi$$
−0.736233 + 0.676728i $$0.763397\pi$$
$$252$$ 8.38869i 0.528437i
$$253$$ − 27.4770i − 1.72747i
$$254$$ −6.89994 −0.432941
$$255$$ 0 0
$$256$$ −1.02912 −0.0643202
$$257$$ − 28.2268i − 1.76074i −0.474289 0.880369i $$-0.657295\pi$$
0.474289 0.880369i $$-0.342705\pi$$
$$258$$ − 2.34227i − 0.145823i
$$259$$ 53.6660 3.33465
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 6.20283i 0.383212i
$$263$$ − 27.6688i − 1.70613i −0.521802 0.853067i $$-0.674740\pi$$
0.521802 0.853067i $$-0.325260\pi$$
$$264$$ 9.32773 0.574082
$$265$$ 0 0
$$266$$ −7.61719 −0.467040
$$267$$ 3.67835i 0.225112i
$$268$$ − 8.46519i − 0.517094i
$$269$$ −0.839052 −0.0511579 −0.0255790 0.999673i $$-0.508143\pi$$
−0.0255790 + 0.999673i $$0.508143\pi$$
$$270$$ 0 0
$$271$$ 1.27423 0.0774037 0.0387018 0.999251i $$-0.487678\pi$$
0.0387018 + 0.999251i $$0.487678\pi$$
$$272$$ − 16.2940i − 0.987966i
$$273$$ − 22.1718i − 1.34190i
$$274$$ −5.09437 −0.307762
$$275$$ 0 0
$$276$$ −9.78358 −0.588902
$$277$$ 26.7387i 1.60657i 0.595592 + 0.803287i $$0.296917\pi$$
−0.595592 + 0.803287i $$0.703083\pi$$
$$278$$ − 1.44522i − 0.0866788i
$$279$$ −9.24375 −0.553409
$$280$$ 0 0
$$281$$ −16.3446 −0.975038 −0.487519 0.873112i $$-0.662098\pi$$
−0.487519 + 0.873112i $$0.662098\pi$$
$$282$$ − 1.85249i − 0.110314i
$$283$$ 23.7207i 1.41005i 0.709183 + 0.705025i $$0.249064\pi$$
−0.709183 + 0.705025i $$0.750936\pi$$
$$284$$ 1.28991 0.0765422
$$285$$ 0 0
$$286$$ −11.4669 −0.678054
$$287$$ 13.7244i 0.810127i
$$288$$ − 5.09775i − 0.300388i
$$289$$ −25.3784 −1.49285
$$290$$ 0 0
$$291$$ 2.87658 0.168628
$$292$$ 9.61718i 0.562803i
$$293$$ − 8.28535i − 0.484035i −0.970272 0.242018i $$-0.922191\pi$$
0.970272 0.242018i $$-0.0778092\pi$$
$$294$$ −8.30730 −0.484491
$$295$$ 0 0
$$296$$ −21.2476 −1.23499
$$297$$ 4.88439i 0.283421i
$$298$$ 11.5117i 0.666855i
$$299$$ 25.8586 1.49544
$$300$$ 0 0
$$301$$ −22.1208 −1.27502
$$302$$ − 7.26288i − 0.417932i
$$303$$ 14.6780i 0.843226i
$$304$$ −7.73927 −0.443878
$$305$$ 0 0
$$306$$ −3.32480 −0.190066
$$307$$ 16.3024i 0.930425i 0.885199 + 0.465213i $$0.154022\pi$$
−0.885199 + 0.465213i $$0.845978\pi$$
$$308$$ − 40.9736i − 2.33469i
$$309$$ −2.48154 −0.141170
$$310$$ 0 0
$$311$$ −7.72085 −0.437809 −0.218905 0.975746i $$-0.570248\pi$$
−0.218905 + 0.975746i $$0.570248\pi$$
$$312$$ 8.77831i 0.496974i
$$313$$ − 6.22393i − 0.351797i −0.984408 0.175899i $$-0.943717\pi$$
0.984408 0.175899i $$-0.0562831\pi$$
$$314$$ 5.50117 0.310449
$$315$$ 0 0
$$316$$ 5.15273 0.289864
$$317$$ 31.0672i 1.74491i 0.488696 + 0.872454i $$0.337473\pi$$
−0.488696 + 0.872454i $$0.662527\pi$$
$$318$$ − 0.494309i − 0.0277195i
$$319$$ −4.88439 −0.273473
$$320$$ 0 0
$$321$$ 4.48423 0.250285
$$322$$ − 13.8583i − 0.772291i
$$323$$ 20.1288i 1.12000i
$$324$$ 1.73915 0.0966196
$$325$$ 0 0
$$326$$ 5.46011 0.302407
$$327$$ 3.28140i 0.181462i
$$328$$ − 5.43381i − 0.300032i
$$329$$ −17.4952 −0.964542
$$330$$ 0 0
$$331$$ 34.8731 1.91680 0.958399 0.285431i $$-0.0921366\pi$$
0.958399 + 0.285431i $$0.0921366\pi$$
$$332$$ − 23.7627i − 1.30415i
$$333$$ − 11.1261i − 0.609707i
$$334$$ 4.03806 0.220953
$$335$$ 0 0
$$336$$ −12.0729 −0.658628
$$337$$ 16.5875i 0.903579i 0.892124 + 0.451790i $$0.149214\pi$$
−0.892124 + 0.451790i $$0.850786\pi$$
$$338$$ − 4.15200i − 0.225839i
$$339$$ −17.9820 −0.976646
$$340$$ 0 0
$$341$$ 45.1500 2.44501
$$342$$ 1.57921i 0.0853937i
$$343$$ 44.6914i 2.41311i
$$344$$ 8.75811 0.472206
$$345$$ 0 0
$$346$$ −1.92423 −0.103447
$$347$$ − 0.413605i − 0.0222035i −0.999938 0.0111017i $$-0.996466\pi$$
0.999938 0.0111017i $$-0.00353387\pi$$
$$348$$ 1.73915i 0.0932284i
$$349$$ −35.5590 −1.90343 −0.951715 0.306982i $$-0.900681\pi$$
−0.951715 + 0.306982i $$0.900681\pi$$
$$350$$ 0 0
$$351$$ −4.59669 −0.245353
$$352$$ 24.8994i 1.32714i
$$353$$ 3.08030i 0.163948i 0.996634 + 0.0819740i $$0.0261225\pi$$
−0.996634 + 0.0819740i $$0.973878\pi$$
$$354$$ 0.152649 0.00811318
$$355$$ 0 0
$$356$$ −6.39722 −0.339052
$$357$$ 31.3999i 1.66186i
$$358$$ − 5.27492i − 0.278788i
$$359$$ 14.3219 0.755879 0.377940 0.925830i $$-0.376633\pi$$
0.377940 + 0.925830i $$0.376633\pi$$
$$360$$ 0 0
$$361$$ −9.43924 −0.496802
$$362$$ 7.38177i 0.387977i
$$363$$ − 12.8572i − 0.674829i
$$364$$ 38.5601 2.02110
$$365$$ 0 0
$$366$$ −0.401899 −0.0210076
$$367$$ − 17.3009i − 0.903102i −0.892245 0.451551i $$-0.850871\pi$$
0.892245 0.451551i $$-0.149129\pi$$
$$368$$ − 14.0804i − 0.733990i
$$369$$ 2.84537 0.148124
$$370$$ 0 0
$$371$$ −4.66833 −0.242368
$$372$$ − 16.0763i − 0.833517i
$$373$$ 31.8314i 1.64817i 0.566469 + 0.824083i $$0.308309\pi$$
−0.566469 + 0.824083i $$0.691691\pi$$
$$374$$ 16.2396 0.839729
$$375$$ 0 0
$$376$$ 6.92674 0.357219
$$377$$ − 4.59669i − 0.236741i
$$378$$ 2.46348i 0.126708i
$$379$$ 18.1855 0.934127 0.467064 0.884224i $$-0.345312\pi$$
0.467064 + 0.884224i $$0.345312\pi$$
$$380$$ 0 0
$$381$$ 13.5099 0.692134
$$382$$ 4.50236i 0.230361i
$$383$$ 32.3026i 1.65059i 0.564704 + 0.825293i $$0.308990\pi$$
−0.564704 + 0.825293i $$0.691010\pi$$
$$384$$ 11.4224 0.582899
$$385$$ 0 0
$$386$$ −8.81232 −0.448535
$$387$$ 4.58611i 0.233125i
$$388$$ 5.00281i 0.253979i
$$389$$ 26.5696 1.34713 0.673567 0.739126i $$-0.264761\pi$$
0.673567 + 0.739126i $$0.264761\pi$$
$$390$$ 0 0
$$391$$ −36.6212 −1.85201
$$392$$ − 31.0622i − 1.56888i
$$393$$ − 12.1450i − 0.612634i
$$394$$ −11.5282 −0.580781
$$395$$ 0 0
$$396$$ −8.49470 −0.426875
$$397$$ − 8.87832i − 0.445590i −0.974865 0.222795i $$-0.928482\pi$$
0.974865 0.222795i $$-0.0715181\pi$$
$$398$$ 7.04681i 0.353225i
$$399$$ 14.9143 0.746648
$$400$$ 0 0
$$401$$ 25.6502 1.28091 0.640455 0.767995i $$-0.278746\pi$$
0.640455 + 0.767995i $$0.278746\pi$$
$$402$$ − 2.48594i − 0.123988i
$$403$$ 42.4906i 2.11661i
$$404$$ −25.5272 −1.27003
$$405$$ 0 0
$$406$$ −2.46348 −0.122260
$$407$$ 54.3442i 2.69374i
$$408$$ − 12.4319i − 0.615472i
$$409$$ −9.92575 −0.490797 −0.245398 0.969422i $$-0.578919\pi$$
−0.245398 + 0.969422i $$0.578919\pi$$
$$410$$ 0 0
$$411$$ 9.97466 0.492014
$$412$$ − 4.31579i − 0.212624i
$$413$$ − 1.44164i − 0.0709384i
$$414$$ −2.87311 −0.141206
$$415$$ 0 0
$$416$$ −23.4327 −1.14888
$$417$$ 2.82971i 0.138572i
$$418$$ − 7.71345i − 0.377277i
$$419$$ −8.12620 −0.396991 −0.198495 0.980102i $$-0.563606\pi$$
−0.198495 + 0.980102i $$0.563606\pi$$
$$420$$ 0 0
$$421$$ −8.37826 −0.408332 −0.204166 0.978936i $$-0.565448\pi$$
−0.204166 + 0.978936i $$0.565448\pi$$
$$422$$ 8.41198i 0.409489i
$$423$$ 3.62713i 0.176357i
$$424$$ 1.84830 0.0897612
$$425$$ 0 0
$$426$$ 0.378804 0.0183531
$$427$$ 3.79559i 0.183682i
$$428$$ 7.79876i 0.376967i
$$429$$ 22.4520 1.08399
$$430$$ 0 0
$$431$$ −32.3877 −1.56006 −0.780030 0.625742i $$-0.784796\pi$$
−0.780030 + 0.625742i $$0.784796\pi$$
$$432$$ 2.50296i 0.120424i
$$433$$ 9.89328i 0.475441i 0.971334 + 0.237720i $$0.0764002\pi$$
−0.971334 + 0.237720i $$0.923600\pi$$
$$434$$ 22.7718 1.09308
$$435$$ 0 0
$$436$$ −5.70686 −0.273309
$$437$$ 17.3943i 0.832081i
$$438$$ 2.82425i 0.134948i
$$439$$ −5.84726 −0.279075 −0.139537 0.990217i $$-0.544562\pi$$
−0.139537 + 0.990217i $$0.544562\pi$$
$$440$$ 0 0
$$441$$ 16.2655 0.774547
$$442$$ 15.2830i 0.726940i
$$443$$ 3.14176i 0.149270i 0.997211 + 0.0746348i $$0.0237791\pi$$
−0.997211 + 0.0746348i $$0.976221\pi$$
$$444$$ 19.3500 0.918311
$$445$$ 0 0
$$446$$ 2.34376 0.110980
$$447$$ − 22.5396i − 1.06609i
$$448$$ − 11.5875i − 0.547459i
$$449$$ −27.8643 −1.31500 −0.657500 0.753455i $$-0.728386\pi$$
−0.657500 + 0.753455i $$0.728386\pi$$
$$450$$ 0 0
$$451$$ −13.8979 −0.654425
$$452$$ − 31.2734i − 1.47098i
$$453$$ 14.2205i 0.668139i
$$454$$ −13.0366 −0.611838
$$455$$ 0 0
$$456$$ −5.90490 −0.276522
$$457$$ 31.8928i 1.49188i 0.666013 + 0.745940i $$0.268000\pi$$
−0.666013 + 0.745940i $$0.732000\pi$$
$$458$$ 4.11130i 0.192108i
$$459$$ 6.50987 0.303855
$$460$$ 0 0
$$461$$ 5.64660 0.262989 0.131494 0.991317i $$-0.458022\pi$$
0.131494 + 0.991317i $$0.458022\pi$$
$$462$$ − 12.0326i − 0.559806i
$$463$$ 32.7602i 1.52250i 0.648460 + 0.761248i $$0.275413\pi$$
−0.648460 + 0.761248i $$0.724587\pi$$
$$464$$ −2.50296 −0.116197
$$465$$ 0 0
$$466$$ −1.91985 −0.0889355
$$467$$ 37.4129i 1.73126i 0.500680 + 0.865632i $$0.333083\pi$$
−0.500680 + 0.865632i $$0.666917\pi$$
$$468$$ − 7.99434i − 0.369539i
$$469$$ −23.4777 −1.08410
$$470$$ 0 0
$$471$$ −10.7712 −0.496308
$$472$$ 0.570777i 0.0262721i
$$473$$ − 22.4003i − 1.02997i
$$474$$ 1.51319 0.0695029
$$475$$ 0 0
$$476$$ −54.6093 −2.50301
$$477$$ 0.967845i 0.0443146i
$$478$$ 1.80387i 0.0825069i
$$479$$ 13.0770 0.597503 0.298751 0.954331i $$-0.403430\pi$$
0.298751 + 0.954331i $$0.403430\pi$$
$$480$$ 0 0
$$481$$ −51.1432 −2.33193
$$482$$ 1.56055i 0.0710813i
$$483$$ 27.1341i 1.23465i
$$484$$ 22.3607 1.01639
$$485$$ 0 0
$$486$$ 0.510732 0.0231673
$$487$$ − 19.4492i − 0.881328i −0.897672 0.440664i $$-0.854743\pi$$
0.897672 0.440664i $$-0.145257\pi$$
$$488$$ − 1.50276i − 0.0680268i
$$489$$ −10.6908 −0.483453
$$490$$ 0 0
$$491$$ 15.8221 0.714041 0.357020 0.934097i $$-0.383793\pi$$
0.357020 + 0.934097i $$0.383793\pi$$
$$492$$ 4.94853i 0.223097i
$$493$$ 6.50987i 0.293190i
$$494$$ 7.25912 0.326603
$$495$$ 0 0
$$496$$ 23.1367 1.03887
$$497$$ − 3.57749i − 0.160472i
$$498$$ − 6.97830i − 0.312705i
$$499$$ −4.88592 −0.218724 −0.109362 0.994002i $$-0.534881\pi$$
−0.109362 + 0.994002i $$0.534881\pi$$
$$500$$ 0 0
$$501$$ −7.90643 −0.353233
$$502$$ − 11.9145i − 0.531769i
$$503$$ − 16.9719i − 0.756742i −0.925654 0.378371i $$-0.876484\pi$$
0.925654 0.378371i $$-0.123516\pi$$
$$504$$ −9.21132 −0.410305
$$505$$ 0 0
$$506$$ 14.0334 0.623860
$$507$$ 8.12952i 0.361045i
$$508$$ 23.4958i 1.04246i
$$509$$ 7.53223 0.333860 0.166930 0.985969i $$-0.446615\pi$$
0.166930 + 0.985969i $$0.446615\pi$$
$$510$$ 0 0
$$511$$ 26.6726 1.17993
$$512$$ 22.3193i 0.986383i
$$513$$ − 3.09205i − 0.136517i
$$514$$ 14.4163 0.635876
$$515$$ 0 0
$$516$$ −7.97595 −0.351122
$$517$$ − 17.7163i − 0.779162i
$$518$$ 27.4089i 1.20428i
$$519$$ 3.76760 0.165379
$$520$$ 0 0
$$521$$ −0.748655 −0.0327992 −0.0163996 0.999866i $$-0.505220\pi$$
−0.0163996 + 0.999866i $$0.505220\pi$$
$$522$$ 0.510732i 0.0223541i
$$523$$ 18.3909i 0.804178i 0.915601 + 0.402089i $$0.131716\pi$$
−0.915601 + 0.402089i $$0.868284\pi$$
$$524$$ 21.1220 0.922719
$$525$$ 0 0
$$526$$ 14.1313 0.616156
$$527$$ − 60.1756i − 2.62129i
$$528$$ − 12.2254i − 0.532043i
$$529$$ −8.64609 −0.375917
$$530$$ 0 0
$$531$$ −0.298882 −0.0129704
$$532$$ 25.9382i 1.12456i
$$533$$ − 13.0793i − 0.566525i
$$534$$ −1.87865 −0.0812972
$$535$$ 0 0
$$536$$ 9.29533 0.401497
$$537$$ 10.3282i 0.445693i
$$538$$ − 0.428531i − 0.0184753i
$$539$$ −79.4469 −3.42202
$$540$$ 0 0
$$541$$ 33.4160 1.43667 0.718334 0.695698i $$-0.244905\pi$$
0.718334 + 0.695698i $$0.244905\pi$$
$$542$$ 0.650787i 0.0279537i
$$543$$ − 14.4533i − 0.620251i
$$544$$ 33.1857 1.42282
$$545$$ 0 0
$$546$$ 11.3238 0.484615
$$547$$ − 3.00020i − 0.128279i −0.997941 0.0641396i $$-0.979570\pi$$
0.997941 0.0641396i $$-0.0204303\pi$$
$$548$$ 17.3475i 0.741047i
$$549$$ 0.786908 0.0335844
$$550$$ 0 0
$$551$$ 3.09205 0.131726
$$552$$ − 10.7430i − 0.457253i
$$553$$ − 14.2908i − 0.607705i
$$554$$ −13.6563 −0.580201
$$555$$ 0 0
$$556$$ −4.92131 −0.208710
$$557$$ − 4.11112i − 0.174194i −0.996200 0.0870969i $$-0.972241\pi$$
0.996200 0.0870969i $$-0.0277590\pi$$
$$558$$ − 4.72108i − 0.199859i
$$559$$ 21.0809 0.891627
$$560$$ 0 0
$$561$$ −31.7967 −1.34246
$$562$$ − 8.34771i − 0.352127i
$$563$$ − 11.5127i − 0.485204i −0.970126 0.242602i $$-0.921999\pi$$
0.970126 0.242602i $$-0.0780009\pi$$
$$564$$ −6.30813 −0.265620
$$565$$ 0 0
$$566$$ −12.1149 −0.509228
$$567$$ − 4.82343i − 0.202565i
$$568$$ 1.41641i 0.0594311i
$$569$$ −32.9470 −1.38121 −0.690605 0.723232i $$-0.742656\pi$$
−0.690605 + 0.723232i $$0.742656\pi$$
$$570$$ 0 0
$$571$$ −22.3213 −0.934117 −0.467059 0.884226i $$-0.654686\pi$$
−0.467059 + 0.884226i $$0.654686\pi$$
$$572$$ 39.0474i 1.63266i
$$573$$ − 8.81550i − 0.368273i
$$574$$ −7.00950 −0.292571
$$575$$ 0 0
$$576$$ −2.40234 −0.100098
$$577$$ 39.8197i 1.65772i 0.559458 + 0.828859i $$0.311009\pi$$
−0.559458 + 0.828859i $$0.688991\pi$$
$$578$$ − 12.9616i − 0.539130i
$$579$$ 17.2543 0.717064
$$580$$ 0 0
$$581$$ −65.9042 −2.73417
$$582$$ 1.46916i 0.0608986i
$$583$$ − 4.72733i − 0.195786i
$$584$$ −10.5603 −0.436988
$$585$$ 0 0
$$586$$ 4.23159 0.174805
$$587$$ 28.7487i 1.18659i 0.804987 + 0.593293i $$0.202172\pi$$
−0.804987 + 0.593293i $$0.797828\pi$$
$$588$$ 28.2882i 1.16658i
$$589$$ −28.5821 −1.17771
$$590$$ 0 0
$$591$$ 22.5719 0.928483
$$592$$ 27.8482i 1.14455i
$$593$$ 27.5400i 1.13093i 0.824771 + 0.565467i $$0.191304\pi$$
−0.824771 + 0.565467i $$0.808696\pi$$
$$594$$ −2.49461 −0.102355
$$595$$ 0 0
$$596$$ 39.1999 1.60569
$$597$$ − 13.7975i − 0.564693i
$$598$$ 13.2068i 0.540066i
$$599$$ 4.11339 0.168069 0.0840343 0.996463i $$-0.473219\pi$$
0.0840343 + 0.996463i $$0.473219\pi$$
$$600$$ 0 0
$$601$$ −0.656799 −0.0267914 −0.0133957 0.999910i $$-0.504264\pi$$
−0.0133957 + 0.999910i $$0.504264\pi$$
$$602$$ − 11.2978i − 0.460463i
$$603$$ 4.86742i 0.198217i
$$604$$ −24.7317 −1.00632
$$605$$ 0 0
$$606$$ −7.49649 −0.304524
$$607$$ 18.4210i 0.747684i 0.927492 + 0.373842i $$0.121960\pi$$
−0.927492 + 0.373842i $$0.878040\pi$$
$$608$$ − 15.7625i − 0.639253i
$$609$$ 4.82343 0.195455
$$610$$ 0 0
$$611$$ 16.6728 0.674508
$$612$$ 11.3217i 0.457651i
$$613$$ 21.3356i 0.861737i 0.902415 + 0.430869i $$0.141793\pi$$
−0.902415 + 0.430869i $$0.858207\pi$$
$$614$$ −8.32613 −0.336015
$$615$$ 0 0
$$616$$ 44.9917 1.81277
$$617$$ 23.2463i 0.935861i 0.883765 + 0.467931i $$0.155000\pi$$
−0.883765 + 0.467931i $$0.845000\pi$$
$$618$$ − 1.26740i − 0.0509824i
$$619$$ 4.11413 0.165361 0.0826804 0.996576i $$-0.473652\pi$$
0.0826804 + 0.996576i $$0.473652\pi$$
$$620$$ 0 0
$$621$$ 5.62549 0.225743
$$622$$ − 3.94328i − 0.158111i
$$623$$ 17.7423i 0.710830i
$$624$$ 11.5053 0.460582
$$625$$ 0 0
$$626$$ 3.17876 0.127049
$$627$$ 15.1028i 0.603146i
$$628$$ − 18.7327i − 0.747515i
$$629$$ 72.4296 2.88796
$$630$$ 0 0
$$631$$ −45.7221 −1.82017 −0.910083 0.414426i $$-0.863982\pi$$
−0.910083 + 0.414426i $$0.863982\pi$$
$$632$$ 5.65803i 0.225065i
$$633$$ − 16.4705i − 0.654642i
$$634$$ −15.8670 −0.630159
$$635$$ 0 0
$$636$$ −1.68323 −0.0667444
$$637$$ − 74.7673i − 2.96239i
$$638$$ − 2.49461i − 0.0987626i
$$639$$ −0.741689 −0.0293408
$$640$$ 0 0
$$641$$ 28.3443 1.11953 0.559767 0.828650i $$-0.310891\pi$$
0.559767 + 0.828650i $$0.310891\pi$$
$$642$$ 2.29024i 0.0903884i
$$643$$ − 17.4545i − 0.688337i −0.938908 0.344168i $$-0.888161\pi$$
0.938908 0.344168i $$-0.111839\pi$$
$$644$$ −47.1904 −1.85956
$$645$$ 0 0
$$646$$ −10.2804 −0.404478
$$647$$ − 17.3874i − 0.683569i −0.939778 0.341784i $$-0.888969\pi$$
0.939778 0.341784i $$-0.111031\pi$$
$$648$$ 1.90970i 0.0750202i
$$649$$ 1.45986 0.0573044
$$650$$ 0 0
$$651$$ −44.5866 −1.74749
$$652$$ − 18.5929i − 0.728153i
$$653$$ − 30.2052i − 1.18202i −0.806664 0.591010i $$-0.798729\pi$$
0.806664 0.591010i $$-0.201271\pi$$
$$654$$ −1.67592 −0.0655335
$$655$$ 0 0
$$656$$ −7.12184 −0.278061
$$657$$ − 5.52981i − 0.215738i
$$658$$ − 8.93535i − 0.348336i
$$659$$ −26.3659 −1.02707 −0.513535 0.858069i $$-0.671664\pi$$
−0.513535 + 0.858069i $$0.671664\pi$$
$$660$$ 0 0
$$661$$ 15.8187 0.615274 0.307637 0.951504i $$-0.400462\pi$$
0.307637 + 0.951504i $$0.400462\pi$$
$$662$$ 17.8108i 0.692236i
$$663$$ − 29.9238i − 1.16215i
$$664$$ 26.0929 1.01260
$$665$$ 0 0
$$666$$ 5.68246 0.220191
$$667$$ 5.62549i 0.217820i
$$668$$ − 13.7505i − 0.532023i
$$669$$ −4.58903 −0.177422
$$670$$ 0 0
$$671$$ −3.84356 −0.148379
$$672$$ − 24.5886i − 0.948527i
$$673$$ 43.1070i 1.66165i 0.556530 + 0.830827i $$0.312132\pi$$
−0.556530 + 0.830827i $$0.687868\pi$$
$$674$$ −8.47177 −0.326320
$$675$$ 0 0
$$676$$ −14.1385 −0.543788
$$677$$ − 40.6978i − 1.56415i −0.623187 0.782073i $$-0.714163\pi$$
0.623187 0.782073i $$-0.285837\pi$$
$$678$$ − 9.18396i − 0.352708i
$$679$$ 13.8750 0.532472
$$680$$ 0 0
$$681$$ 25.5254 0.978134
$$682$$ 23.0596i 0.882996i
$$683$$ − 21.0217i − 0.804372i −0.915558 0.402186i $$-0.868250\pi$$
0.915558 0.402186i $$-0.131750\pi$$
$$684$$ 5.37755 0.205616
$$685$$ 0 0
$$686$$ −22.8253 −0.871475
$$687$$ − 8.04982i − 0.307120i
$$688$$ − 11.4789i − 0.437627i
$$689$$ 4.44888 0.169489
$$690$$ 0 0
$$691$$ 6.98613 0.265765 0.132882 0.991132i $$-0.457577\pi$$
0.132882 + 0.991132i $$0.457577\pi$$
$$692$$ 6.55244i 0.249086i
$$693$$ 23.5595i 0.894951i
$$694$$ 0.211241 0.00801860
$$695$$ 0 0
$$696$$ −1.90970 −0.0723871
$$697$$ 18.5230i 0.701608i
$$698$$ − 18.1611i − 0.687408i
$$699$$ 3.75903 0.142179
$$700$$ 0 0
$$701$$ 1.40572 0.0530934 0.0265467 0.999648i $$-0.491549\pi$$
0.0265467 + 0.999648i $$0.491549\pi$$
$$702$$ − 2.34767i − 0.0886072i
$$703$$ − 34.4025i − 1.29751i
$$704$$ 11.7340 0.442240
$$705$$ 0 0
$$706$$ −1.57321 −0.0592085
$$707$$ 70.7981i 2.66264i
$$708$$ − 0.519802i − 0.0195354i
$$709$$ −1.27110 −0.0477372 −0.0238686 0.999715i $$-0.507598\pi$$
−0.0238686 + 0.999715i $$0.507598\pi$$
$$710$$ 0 0
$$711$$ −2.96278 −0.111113
$$712$$ − 7.02457i − 0.263257i
$$713$$ − 52.0006i − 1.94744i
$$714$$ −16.0369 −0.600167
$$715$$ 0 0
$$716$$ −17.9623 −0.671282
$$717$$ − 3.53192i − 0.131902i
$$718$$ 7.31463i 0.272979i
$$719$$ −27.8439 −1.03840 −0.519201 0.854652i $$-0.673770\pi$$
−0.519201 + 0.854652i $$0.673770\pi$$
$$720$$ 0 0
$$721$$ −11.9696 −0.445770
$$722$$ − 4.82092i − 0.179416i
$$723$$ − 3.05553i − 0.113636i
$$724$$ 25.1365 0.934192
$$725$$ 0 0
$$726$$ 6.56659 0.243709
$$727$$ 8.97787i 0.332971i 0.986044 + 0.166485i $$0.0532419\pi$$
−0.986044 + 0.166485i $$0.946758\pi$$
$$728$$ 42.3416i 1.56928i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −29.8550 −1.10423
$$732$$ 1.36855i 0.0505832i
$$733$$ − 34.7466i − 1.28339i −0.766958 0.641697i $$-0.778231\pi$$
0.766958 0.641697i $$-0.221769\pi$$
$$734$$ 8.83614 0.326148
$$735$$ 0 0
$$736$$ 28.6773 1.05706
$$737$$ − 23.7744i − 0.875740i
$$738$$ 1.45322i 0.0534937i
$$739$$ 3.32518 0.122319 0.0611594 0.998128i $$-0.480520\pi$$
0.0611594 + 0.998128i $$0.480520\pi$$
$$740$$ 0 0
$$741$$ −14.2132 −0.522134
$$742$$ − 2.38426i − 0.0875291i
$$743$$ − 1.06314i − 0.0390029i −0.999810 0.0195015i $$-0.993792\pi$$
0.999810 0.0195015i $$-0.00620790\pi$$
$$744$$ 17.6528 0.647184
$$745$$ 0 0
$$746$$ −16.2573 −0.595222
$$747$$ 13.6633i 0.499916i
$$748$$ − 55.2994i − 2.02195i
$$749$$ 21.6294 0.790320
$$750$$ 0 0
$$751$$ −38.7756 −1.41494 −0.707472 0.706742i $$-0.750164\pi$$
−0.707472 + 0.706742i $$0.750164\pi$$
$$752$$ − 9.07856i − 0.331061i
$$753$$ 23.3283i 0.850129i
$$754$$ 2.34767 0.0854972
$$755$$ 0 0
$$756$$ 8.38869 0.305094
$$757$$ − 40.9695i − 1.48906i −0.667588 0.744531i $$-0.732673\pi$$
0.667588 0.744531i $$-0.267327\pi$$
$$758$$ 9.28792i 0.337352i
$$759$$ −27.4770 −0.997354
$$760$$ 0 0
$$761$$ 13.8561 0.502283 0.251142 0.967950i $$-0.419194\pi$$
0.251142 + 0.967950i $$0.419194\pi$$
$$762$$ 6.89994i 0.249958i
$$763$$ 15.8276i 0.572998i
$$764$$ 15.3315 0.554675
$$765$$ 0 0
$$766$$ −16.4980 −0.596096
$$767$$ 1.37387i 0.0496075i
$$768$$ 1.02912i 0.0371353i
$$769$$ 6.73919 0.243021 0.121511 0.992590i $$-0.461226\pi$$
0.121511 + 0.992590i $$0.461226\pi$$
$$770$$ 0 0
$$771$$ −28.2268 −1.01656
$$772$$ 30.0079i 1.08001i
$$773$$ 7.45730i 0.268220i 0.990966 + 0.134110i $$0.0428176\pi$$
−0.990966 + 0.134110i $$0.957182\pi$$
$$774$$ −2.34227 −0.0841912
$$775$$ 0 0
$$776$$ −5.49341 −0.197202
$$777$$ − 53.6660i − 1.92526i
$$778$$ 13.5700i 0.486506i
$$779$$ 8.79801 0.315221
$$780$$ 0 0
$$781$$ 3.62270 0.129630
$$782$$ − 18.7036i − 0.668839i
$$783$$ − 1.00000i − 0.0357371i
$$784$$ −40.7119 −1.45400
$$785$$ 0 0
$$786$$ 6.20283 0.221248
$$787$$ 7.81608i 0.278613i 0.990249 + 0.139307i $$0.0444873\pi$$
−0.990249 + 0.139307i $$0.955513\pi$$
$$788$$ 39.2559i 1.39844i
$$789$$ −27.6688 −0.985037
$$790$$ 0 0
$$791$$ −86.7347 −3.08393
$$792$$ − 9.32773i − 0.331446i
$$793$$ − 3.61717i − 0.128449i
$$794$$ 4.53444 0.160921
$$795$$ 0 0
$$796$$ 23.9959 0.850514
$$797$$ − 21.8313i − 0.773305i −0.922225 0.386653i $$-0.873631\pi$$
0.922225 0.386653i $$-0.126369\pi$$
$$798$$ 7.61719i 0.269646i
$$799$$ −23.6121 −0.835338
$$800$$ 0 0
$$801$$ 3.67835 0.129968
$$802$$ 13.1004i 0.462590i
$$803$$ 27.0097i 0.953152i
$$804$$ −8.46519 −0.298544
$$805$$ 0 0
$$806$$ −21.7013 −0.764396
$$807$$ 0.839052i 0.0295360i
$$808$$ − 28.0305i − 0.986111i
$$809$$ −5.71715 −0.201004 −0.100502 0.994937i $$-0.532045\pi$$
−0.100502 + 0.994937i $$0.532045\pi$$
$$810$$ 0 0
$$811$$ −44.8661 −1.57546 −0.787732 0.616019i $$-0.788745\pi$$
−0.787732 + 0.616019i $$0.788745\pi$$
$$812$$ 8.38869i 0.294385i
$$813$$ − 1.27423i − 0.0446890i
$$814$$ −27.7553 −0.972823
$$815$$ 0 0
$$816$$ −16.2940 −0.570402
$$817$$ 14.1805i 0.496112i
$$818$$ − 5.06939i − 0.177247i
$$819$$ −22.1718 −0.774745
$$820$$ 0 0
$$821$$ 37.3699 1.30422 0.652110 0.758125i $$-0.273884\pi$$
0.652110 + 0.758125i $$0.273884\pi$$
$$822$$ 5.09437i 0.177687i
$$823$$ − 31.6982i − 1.10493i −0.833537 0.552464i $$-0.813688\pi$$
0.833537 0.552464i $$-0.186312\pi$$
$$824$$ 4.73901 0.165091
$$825$$ 0 0
$$826$$ 0.736290 0.0256188
$$827$$ 14.4089i 0.501046i 0.968111 + 0.250523i $$0.0806025\pi$$
−0.968111 + 0.250523i $$0.919397\pi$$
$$828$$ 9.78358i 0.340003i
$$829$$ 20.6622 0.717630 0.358815 0.933409i $$-0.383181\pi$$
0.358815 + 0.933409i $$0.383181\pi$$
$$830$$ 0 0
$$831$$ 26.7387 0.927556
$$832$$ 11.0428i 0.382840i
$$833$$ 105.886i 3.66874i
$$834$$ −1.44522 −0.0500440
$$835$$ 0 0
$$836$$ −26.2660 −0.908429
$$837$$ 9.24375i 0.319511i
$$838$$ − 4.15031i − 0.143370i
$$839$$ −1.64970 −0.0569541 −0.0284771 0.999594i $$-0.509066\pi$$
−0.0284771 + 0.999594i $$0.509066\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ − 4.27904i − 0.147466i
$$843$$ 16.3446i 0.562938i
$$844$$ 28.6447 0.985990
$$845$$ 0 0
$$846$$ −1.85249 −0.0636899
$$847$$ − 62.0159i − 2.13089i
$$848$$ − 2.42248i − 0.0831882i
$$849$$ 23.7207 0.814092
$$850$$ 0 0
$$851$$ 62.5898 2.14555
$$852$$ − 1.28991i − 0.0441916i
$$853$$ − 23.4066i − 0.801428i −0.916203 0.400714i $$-0.868762\pi$$
0.916203 0.400714i $$-0.131238\pi$$
$$854$$ −1.93853 −0.0663351
$$855$$ 0 0
$$856$$ −8.56355 −0.292696
$$857$$ − 18.7479i − 0.640415i −0.947347 0.320207i $$-0.896247\pi$$
0.947347 0.320207i $$-0.103753\pi$$
$$858$$ 11.4669i 0.391475i
$$859$$ 29.1326 0.993990 0.496995 0.867753i $$-0.334437\pi$$
0.496995 + 0.867753i $$0.334437\pi$$
$$860$$ 0 0
$$861$$ 13.7244 0.467727
$$862$$ − 16.5414i − 0.563403i
$$863$$ 25.5033i 0.868142i 0.900879 + 0.434071i $$0.142923\pi$$
−0.900879 + 0.434071i $$0.857077\pi$$
$$864$$ −5.09775 −0.173429
$$865$$ 0 0
$$866$$ −5.05281 −0.171701
$$867$$ 25.3784i 0.861897i
$$868$$ − 77.5429i − 2.63198i
$$869$$ 14.4714 0.490908
$$870$$ 0 0
$$871$$ 22.3740 0.758114
$$872$$ − 6.26651i − 0.212211i
$$873$$ − 2.87658i − 0.0973573i
$$874$$ −8.88380 −0.300499
$$875$$ 0 0
$$876$$ 9.61718 0.324935
$$877$$ − 24.3134i − 0.821005i −0.911860 0.410502i $$-0.865353\pi$$
0.911860 0.410502i $$-0.134647\pi$$
$$878$$ − 2.98638i − 0.100786i
$$879$$ −8.28535 −0.279458
$$880$$ 0 0
$$881$$ 51.3387 1.72965 0.864823 0.502077i $$-0.167431\pi$$
0.864823 + 0.502077i $$0.167431\pi$$
$$882$$ 8.30730i 0.279721i
$$883$$ 17.1764i 0.578033i 0.957324 + 0.289016i $$0.0933282\pi$$
−0.957324 + 0.289016i $$0.906672\pi$$
$$884$$ 52.0421 1.75037
$$885$$ 0 0
$$886$$ −1.60460 −0.0539075
$$887$$ − 38.2392i − 1.28395i −0.766727 0.641974i $$-0.778116\pi$$
0.766727 0.641974i $$-0.221884\pi$$
$$888$$ 21.2476i 0.713022i
$$889$$ 65.1641 2.18553
$$890$$ 0 0
$$891$$ 4.88439 0.163633
$$892$$ − 7.98103i − 0.267225i
$$893$$ 11.2153i 0.375304i
$$894$$ 11.5117 0.385009
$$895$$ 0 0
$$896$$ 55.0954 1.84061
$$897$$ − 25.8586i − 0.863393i
$$898$$ − 14.2312i − 0.474901i
$$899$$ −9.24375 −0.308296
$$900$$ 0 0
$$901$$ −6.30055 −0.209902
$$902$$ − 7.09808i − 0.236340i
$$903$$ 22.1208i 0.736134i
$$904$$ 34.3402 1.14214
$$905$$ 0 0
$$906$$ −7.26288 −0.241293
$$907$$ − 9.37007i − 0.311128i −0.987826 0.155564i $$-0.950281\pi$$
0.987826 0.155564i $$-0.0497195\pi$$
$$908$$ 44.3925i 1.47322i
$$909$$ 14.6780 0.486837
$$910$$ 0 0
$$911$$ −10.5220 −0.348611 −0.174305 0.984692i $$-0.555768\pi$$
−0.174305 + 0.984692i $$0.555768\pi$$
$$912$$ 7.73927i 0.256273i
$$913$$ − 66.7371i − 2.20867i
$$914$$ −16.2886 −0.538780
$$915$$ 0 0
$$916$$ 13.9999 0.462569
$$917$$ − 58.5805i − 1.93450i
$$918$$ 3.32480i 0.109735i
$$919$$ 47.9043 1.58022 0.790109 0.612967i $$-0.210024\pi$$
0.790109 + 0.612967i $$0.210024\pi$$
$$920$$ 0 0
$$921$$ 16.3024 0.537181
$$922$$ 2.88390i 0.0949762i
$$923$$ 3.40931i 0.112219i
$$924$$ −40.9736 −1.34793
$$925$$ 0 0
$$926$$ −16.7317 −0.549837
$$927$$ 2.48154i 0.0815046i
$$928$$ − 5.09775i − 0.167342i
$$929$$ −32.0568 −1.05175 −0.525875 0.850562i $$-0.676262\pi$$
−0.525875 + 0.850562i $$0.676262\pi$$
$$930$$ 0 0
$$931$$ 50.2937 1.64831
$$932$$ 6.53752i 0.214144i
$$933$$ 7.72085i 0.252769i
$$934$$ −19.1080 −0.625232
$$935$$ 0 0
$$936$$ 8.77831 0.286928
$$937$$ − 23.8183i − 0.778109i −0.921215 0.389054i $$-0.872802\pi$$
0.921215 0.389054i $$-0.127198\pi$$
$$938$$ − 11.9908i − 0.391513i
$$939$$ −6.22393 −0.203110
$$940$$ 0 0
$$941$$ 45.7836 1.49250 0.746251 0.665664i $$-0.231852\pi$$
0.746251 + 0.665664i $$0.231852\pi$$
$$942$$ − 5.50117i − 0.179238i
$$943$$ 16.0066i 0.521246i
$$944$$ 0.748091 0.0243483
$$945$$ 0 0
$$946$$ 11.4406 0.371965
$$947$$ 4.69130i 0.152447i 0.997091 + 0.0762233i $$0.0242862\pi$$
−0.997091 + 0.0762233i $$0.975714\pi$$
$$948$$ − 5.15273i − 0.167353i
$$949$$ −25.4188 −0.825129
$$950$$ 0 0
$$951$$ 31.0672 1.00742
$$952$$ − 59.9645i − 1.94346i
$$953$$ 56.4381i 1.82821i 0.405479 + 0.914104i $$0.367105\pi$$
−0.405479 + 0.914104i $$0.632895\pi$$
$$954$$ −0.494309 −0.0160038
$$955$$ 0 0
$$956$$ 6.14256 0.198665
$$957$$ 4.88439i 0.157890i
$$958$$ 6.67883i 0.215783i
$$959$$ 48.1121 1.55362
$$960$$ 0 0
$$961$$ 54.4469 1.75635
$$962$$ − 26.1205i − 0.842158i
$$963$$ − 4.48423i − 0.144502i
$$964$$ 5.31403 0.171153
$$965$$ 0 0
$$966$$ −13.8583 −0.445882
$$967$$ 42.2611i 1.35902i 0.733664 + 0.679512i $$0.237808\pi$$
−0.733664 + 0.679512i $$0.762192\pi$$
$$968$$ 24.5535i 0.789179i
$$969$$ 20.1288 0.646631
$$970$$ 0 0
$$971$$ 25.6613 0.823510 0.411755 0.911295i $$-0.364916\pi$$
0.411755 + 0.911295i $$0.364916\pi$$
$$972$$ − 1.73915i − 0.0557834i
$$973$$ 13.6489i 0.437565i
$$974$$ 9.93333 0.318284
$$975$$ 0 0
$$976$$ −1.96960 −0.0630453
$$977$$ − 7.35242i − 0.235225i −0.993060 0.117612i $$-0.962476\pi$$
0.993060 0.117612i $$-0.0375240\pi$$
$$978$$ − 5.46011i − 0.174595i
$$979$$ −17.9665 −0.574212
$$980$$ 0 0
$$981$$ 3.28140 0.104767
$$982$$ 8.08084i 0.257870i
$$983$$ 3.23911i 0.103312i 0.998665 + 0.0516558i $$0.0164499\pi$$
−0.998665 + 0.0516558i $$0.983550\pi$$
$$984$$ −5.43381 −0.173223
$$985$$ 0 0
$$986$$ −3.32480 −0.105883
$$987$$ 17.4952i 0.556879i
$$988$$ − 24.7189i − 0.786413i
$$989$$ −25.7991 −0.820364
$$990$$ 0 0
$$991$$ 3.19230 0.101407 0.0507034 0.998714i $$-0.483854\pi$$
0.0507034 + 0.998714i $$0.483854\pi$$
$$992$$ 47.1223i 1.49614i
$$993$$ − 34.8731i − 1.10666i
$$994$$ 1.82714 0.0579532
$$995$$ 0 0
$$996$$ −23.7627 −0.752949
$$997$$ 49.0253i 1.55265i 0.630335 + 0.776323i $$0.282917\pi$$
−0.630335 + 0.776323i $$0.717083\pi$$
$$998$$ − 2.49539i − 0.0789903i
$$999$$ −11.1261 −0.352015
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.349.11 16
5.2 odd 4 2175.2.a.bc.1.4 8
5.3 odd 4 2175.2.a.bd.1.5 yes 8
5.4 even 2 inner 2175.2.c.p.349.6 16
15.2 even 4 6525.2.a.bz.1.5 8
15.8 even 4 6525.2.a.by.1.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.bc.1.4 8 5.2 odd 4
2175.2.a.bd.1.5 yes 8 5.3 odd 4
2175.2.c.p.349.6 16 5.4 even 2 inner
2175.2.c.p.349.11 16 1.1 even 1 trivial
6525.2.a.by.1.4 8 15.8 even 4
6525.2.a.bz.1.5 8 15.2 even 4