# Properties

 Label 175.4.b.c.99.4 Level $175$ Weight $4$ Character 175.99 Analytic conductor $10.325$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(99,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (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: $$10.3253342510$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.4 Root $$-0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 175.99 Dual form 175.4.b.c.99.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.41421i q^{2} +4.65685i q^{3} -21.3137 q^{4} -25.2132 q^{6} -7.00000i q^{7} -72.0833i q^{8} +5.31371 q^{9} +O(q^{10})$$ $$q+5.41421i q^{2} +4.65685i q^{3} -21.3137 q^{4} -25.2132 q^{6} -7.00000i q^{7} -72.0833i q^{8} +5.31371 q^{9} -52.2548 q^{11} -99.2548i q^{12} -30.6569i q^{13} +37.8995 q^{14} +219.765 q^{16} +37.2254i q^{17} +28.7696i q^{18} -80.2254 q^{19} +32.5980 q^{21} -282.919i q^{22} -25.8335i q^{23} +335.681 q^{24} +165.983 q^{26} +150.480i q^{27} +149.196i q^{28} -20.9411 q^{29} -314.558 q^{31} +613.186i q^{32} -243.343i q^{33} -201.546 q^{34} -113.255 q^{36} +197.147i q^{37} -434.357i q^{38} +142.765 q^{39} +11.3625 q^{41} +176.492i q^{42} +33.8335i q^{43} +1113.74 q^{44} +139.868 q^{46} -361.676i q^{47} +1023.41i q^{48} -49.0000 q^{49} -173.353 q^{51} +653.411i q^{52} -153.019i q^{53} -814.732 q^{54} -504.583 q^{56} -373.598i q^{57} -113.380i q^{58} +616.000 q^{59} +15.2649 q^{61} -1703.09i q^{62} -37.1960i q^{63} -1561.80 q^{64} +1317.51 q^{66} -166.510i q^{67} -793.411i q^{68} +120.303 q^{69} -952.000 q^{71} -383.029i q^{72} +148.489i q^{73} -1067.40 q^{74} +1709.90 q^{76} +365.784i q^{77} +772.958i q^{78} -857.725 q^{79} -557.294 q^{81} +61.5189i q^{82} -660.528i q^{83} -694.784 q^{84} -183.182 q^{86} -97.5198i q^{87} +3766.70i q^{88} +45.7746 q^{89} -214.598 q^{91} +550.607i q^{92} -1464.85i q^{93} +1958.19 q^{94} -2855.52 q^{96} +1682.13i q^{97} -265.296i q^{98} -277.667 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 40 q^{4} - 16 q^{6} - 24 q^{9}+O(q^{10})$$ 4 * q - 40 * q^4 - 16 * q^6 - 24 * q^9 $$4 q - 40 q^{4} - 16 q^{6} - 24 q^{9} - 28 q^{11} + 112 q^{14} + 336 q^{16} - 72 q^{19} - 28 q^{21} + 992 q^{24} + 432 q^{26} + 52 q^{29} - 240 q^{31} + 48 q^{34} - 272 q^{36} + 28 q^{39} - 656 q^{41} + 2328 q^{44} + 1408 q^{46} - 196 q^{49} - 1508 q^{51} - 1296 q^{54} - 672 q^{56} + 2464 q^{59} + 672 q^{61} - 4256 q^{64} + 3952 q^{66} - 2664 q^{69} - 3808 q^{71} - 4032 q^{74} + 3536 q^{76} - 2028 q^{79} - 2908 q^{81} - 1512 q^{84} - 1536 q^{86} + 432 q^{89} - 700 q^{91} + 3856 q^{94} - 4928 q^{96} - 1880 q^{99}+O(q^{100})$$ 4 * q - 40 * q^4 - 16 * q^6 - 24 * q^9 - 28 * q^11 + 112 * q^14 + 336 * q^16 - 72 * q^19 - 28 * q^21 + 992 * q^24 + 432 * q^26 + 52 * q^29 - 240 * q^31 + 48 * q^34 - 272 * q^36 + 28 * q^39 - 656 * q^41 + 2328 * q^44 + 1408 * q^46 - 196 * q^49 - 1508 * q^51 - 1296 * q^54 - 672 * q^56 + 2464 * q^59 + 672 * q^61 - 4256 * q^64 + 3952 * q^66 - 2664 * q^69 - 3808 * q^71 - 4032 * q^74 + 3536 * q^76 - 2028 * q^79 - 2908 * q^81 - 1512 * q^84 - 1536 * q^86 + 432 * q^89 - 700 * q^91 + 3856 * q^94 - 4928 * q^96 - 1880 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.41421i 1.91421i 0.289735 + 0.957107i $$0.406433\pi$$
−0.289735 + 0.957107i $$0.593567\pi$$
$$3$$ 4.65685i 0.896212i 0.893980 + 0.448106i $$0.147901\pi$$
−0.893980 + 0.448106i $$0.852099\pi$$
$$4$$ −21.3137 −2.66421
$$5$$ 0 0
$$6$$ −25.2132 −1.71554
$$7$$ − 7.00000i − 0.377964i
$$8$$ − 72.0833i − 3.18566i
$$9$$ 5.31371 0.196804
$$10$$ 0 0
$$11$$ −52.2548 −1.43231 −0.716156 0.697941i $$-0.754100\pi$$
−0.716156 + 0.697941i $$0.754100\pi$$
$$12$$ − 99.2548i − 2.38770i
$$13$$ − 30.6569i − 0.654052i −0.945015 0.327026i $$-0.893953\pi$$
0.945015 0.327026i $$-0.106047\pi$$
$$14$$ 37.8995 0.723505
$$15$$ 0 0
$$16$$ 219.765 3.43382
$$17$$ 37.2254i 0.531087i 0.964099 + 0.265544i $$0.0855514\pi$$
−0.964099 + 0.265544i $$0.914449\pi$$
$$18$$ 28.7696i 0.376725i
$$19$$ −80.2254 −0.968683 −0.484341 0.874879i $$-0.660941\pi$$
−0.484341 + 0.874879i $$0.660941\pi$$
$$20$$ 0 0
$$21$$ 32.5980 0.338736
$$22$$ − 282.919i − 2.74175i
$$23$$ − 25.8335i − 0.234202i −0.993120 0.117101i $$-0.962640\pi$$
0.993120 0.117101i $$-0.0373602\pi$$
$$24$$ 335.681 2.85503
$$25$$ 0 0
$$26$$ 165.983 1.25200
$$27$$ 150.480i 1.07259i
$$28$$ 149.196i 1.00698i
$$29$$ −20.9411 −0.134092 −0.0670460 0.997750i $$-0.521357\pi$$
−0.0670460 + 0.997750i $$0.521357\pi$$
$$30$$ 0 0
$$31$$ −314.558 −1.82246 −0.911232 0.411894i $$-0.864867\pi$$
−0.911232 + 0.411894i $$0.864867\pi$$
$$32$$ 613.186i 3.38741i
$$33$$ − 243.343i − 1.28365i
$$34$$ −201.546 −1.01661
$$35$$ 0 0
$$36$$ −113.255 −0.524328
$$37$$ 197.147i 0.875968i 0.898983 + 0.437984i $$0.144307\pi$$
−0.898983 + 0.437984i $$0.855693\pi$$
$$38$$ − 434.357i − 1.85427i
$$39$$ 142.765 0.586170
$$40$$ 0 0
$$41$$ 11.3625 0.0432810 0.0216405 0.999766i $$-0.493111\pi$$
0.0216405 + 0.999766i $$0.493111\pi$$
$$42$$ 176.492i 0.648414i
$$43$$ 33.8335i 0.119990i 0.998199 + 0.0599948i $$0.0191084\pi$$
−0.998199 + 0.0599948i $$0.980892\pi$$
$$44$$ 1113.74 3.81598
$$45$$ 0 0
$$46$$ 139.868 0.448313
$$47$$ − 361.676i − 1.12247i −0.827658 0.561233i $$-0.810327\pi$$
0.827658 0.561233i $$-0.189673\pi$$
$$48$$ 1023.41i 3.07743i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ −173.353 −0.475967
$$52$$ 653.411i 1.74254i
$$53$$ − 153.019i − 0.396582i −0.980143 0.198291i $$-0.936461\pi$$
0.980143 0.198291i $$-0.0635390\pi$$
$$54$$ −814.732 −2.05317
$$55$$ 0 0
$$56$$ −504.583 −1.20407
$$57$$ − 373.598i − 0.868145i
$$58$$ − 113.380i − 0.256681i
$$59$$ 616.000 1.35926 0.679630 0.733555i $$-0.262140\pi$$
0.679630 + 0.733555i $$0.262140\pi$$
$$60$$ 0 0
$$61$$ 15.2649 0.0320406 0.0160203 0.999872i $$-0.494900\pi$$
0.0160203 + 0.999872i $$0.494900\pi$$
$$62$$ − 1703.09i − 3.48858i
$$63$$ − 37.1960i − 0.0743849i
$$64$$ −1561.80 −3.05040
$$65$$ 0 0
$$66$$ 1317.51 2.45719
$$67$$ − 166.510i − 0.303618i −0.988410 0.151809i $$-0.951490\pi$$
0.988410 0.151809i $$-0.0485098\pi$$
$$68$$ − 793.411i − 1.41493i
$$69$$ 120.303 0.209895
$$70$$ 0 0
$$71$$ −952.000 −1.59129 −0.795645 0.605763i $$-0.792868\pi$$
−0.795645 + 0.605763i $$0.792868\pi$$
$$72$$ − 383.029i − 0.626951i
$$73$$ 148.489i 0.238074i 0.992890 + 0.119037i $$0.0379807\pi$$
−0.992890 + 0.119037i $$0.962019\pi$$
$$74$$ −1067.40 −1.67679
$$75$$ 0 0
$$76$$ 1709.90 2.58078
$$77$$ 365.784i 0.541363i
$$78$$ 772.958i 1.12205i
$$79$$ −857.725 −1.22154 −0.610770 0.791808i $$-0.709140\pi$$
−0.610770 + 0.791808i $$0.709140\pi$$
$$80$$ 0 0
$$81$$ −557.294 −0.764464
$$82$$ 61.5189i 0.0828491i
$$83$$ − 660.528i − 0.873523i −0.899577 0.436761i $$-0.856125\pi$$
0.899577 0.436761i $$-0.143875\pi$$
$$84$$ −694.784 −0.902466
$$85$$ 0 0
$$86$$ −183.182 −0.229686
$$87$$ − 97.5198i − 0.120175i
$$88$$ 3766.70i 4.56286i
$$89$$ 45.7746 0.0545180 0.0272590 0.999628i $$-0.491322\pi$$
0.0272590 + 0.999628i $$0.491322\pi$$
$$90$$ 0 0
$$91$$ −214.598 −0.247209
$$92$$ 550.607i 0.623965i
$$93$$ − 1464.85i − 1.63331i
$$94$$ 1958.19 2.14864
$$95$$ 0 0
$$96$$ −2855.52 −3.03583
$$97$$ 1682.13i 1.76076i 0.474265 + 0.880382i $$0.342714\pi$$
−0.474265 + 0.880382i $$0.657286\pi$$
$$98$$ − 265.296i − 0.273459i
$$99$$ −277.667 −0.281885
$$100$$ 0 0
$$101$$ −434.167 −0.427734 −0.213867 0.976863i $$-0.568606\pi$$
−0.213867 + 0.976863i $$0.568606\pi$$
$$102$$ − 938.572i − 0.911102i
$$103$$ − 345.577i − 0.330589i −0.986244 0.165295i $$-0.947142\pi$$
0.986244 0.165295i $$-0.0528575\pi$$
$$104$$ −2209.85 −2.08359
$$105$$ 0 0
$$106$$ 828.479 0.759142
$$107$$ 217.119i 0.196165i 0.995178 + 0.0980825i $$0.0312709\pi$$
−0.995178 + 0.0980825i $$0.968729\pi$$
$$108$$ − 3207.29i − 2.85761i
$$109$$ −1734.41 −1.52409 −0.762047 0.647521i $$-0.775806\pi$$
−0.762047 + 0.647521i $$0.775806\pi$$
$$110$$ 0 0
$$111$$ −918.086 −0.785053
$$112$$ − 1538.35i − 1.29786i
$$113$$ 1854.20i 1.54362i 0.635855 + 0.771809i $$0.280648\pi$$
−0.635855 + 0.771809i $$0.719352\pi$$
$$114$$ 2022.74 1.66181
$$115$$ 0 0
$$116$$ 446.333 0.357250
$$117$$ − 162.902i − 0.128720i
$$118$$ 3335.16i 2.60191i
$$119$$ 260.578 0.200732
$$120$$ 0 0
$$121$$ 1399.57 1.05152
$$122$$ 82.6476i 0.0613325i
$$123$$ 52.9134i 0.0387890i
$$124$$ 6704.41 4.85543
$$125$$ 0 0
$$126$$ 201.387 0.142389
$$127$$ 1394.51i 0.974352i 0.873304 + 0.487176i $$0.161973\pi$$
−0.873304 + 0.487176i $$0.838027\pi$$
$$128$$ − 3550.45i − 2.45171i
$$129$$ −157.558 −0.107536
$$130$$ 0 0
$$131$$ 1762.42 1.17544 0.587722 0.809063i $$-0.300025\pi$$
0.587722 + 0.809063i $$0.300025\pi$$
$$132$$ 5186.54i 3.41993i
$$133$$ 561.578i 0.366128i
$$134$$ 901.519 0.581189
$$135$$ 0 0
$$136$$ 2683.33 1.69186
$$137$$ − 922.949i − 0.575568i −0.957695 0.287784i $$-0.907081\pi$$
0.957695 0.287784i $$-0.0929185\pi$$
$$138$$ 651.345i 0.401784i
$$139$$ 196.039 0.119624 0.0598122 0.998210i $$-0.480950\pi$$
0.0598122 + 0.998210i $$0.480950\pi$$
$$140$$ 0 0
$$141$$ 1684.27 1.00597
$$142$$ − 5154.33i − 3.04607i
$$143$$ 1601.97i 0.936807i
$$144$$ 1167.76 0.675790
$$145$$ 0 0
$$146$$ −803.954 −0.455724
$$147$$ − 228.186i − 0.128030i
$$148$$ − 4201.94i − 2.33376i
$$149$$ −780.372 −0.429064 −0.214532 0.976717i $$-0.568823\pi$$
−0.214532 + 0.976717i $$0.568823\pi$$
$$150$$ 0 0
$$151$$ −2319.43 −1.25002 −0.625008 0.780618i $$-0.714904\pi$$
−0.625008 + 0.780618i $$0.714904\pi$$
$$152$$ 5782.91i 3.08589i
$$153$$ 197.805i 0.104520i
$$154$$ −1980.43 −1.03628
$$155$$ 0 0
$$156$$ −3042.84 −1.56168
$$157$$ 1022.90i 0.519977i 0.965612 + 0.259989i $$0.0837188\pi$$
−0.965612 + 0.259989i $$0.916281\pi$$
$$158$$ − 4643.91i − 2.33829i
$$159$$ 712.589 0.355421
$$160$$ 0 0
$$161$$ −180.834 −0.0885201
$$162$$ − 3017.31i − 1.46335i
$$163$$ 1350.63i 0.649013i 0.945883 + 0.324507i $$0.105198\pi$$
−0.945883 + 0.324507i $$0.894802\pi$$
$$164$$ −242.177 −0.115310
$$165$$ 0 0
$$166$$ 3576.24 1.67211
$$167$$ − 1230.58i − 0.570209i −0.958496 0.285105i $$-0.907972\pi$$
0.958496 0.285105i $$-0.0920284\pi$$
$$168$$ − 2349.77i − 1.07910i
$$169$$ 1257.16 0.572215
$$170$$ 0 0
$$171$$ −426.294 −0.190641
$$172$$ − 721.117i − 0.319678i
$$173$$ 2487.65i 1.09325i 0.837377 + 0.546626i $$0.184088\pi$$
−0.837377 + 0.546626i $$0.815912\pi$$
$$174$$ 527.993 0.230040
$$175$$ 0 0
$$176$$ −11483.8 −4.91830
$$177$$ 2868.62i 1.21819i
$$178$$ 247.833i 0.104359i
$$179$$ −1621.18 −0.676941 −0.338471 0.940977i $$-0.609910\pi$$
−0.338471 + 0.940977i $$0.609910\pi$$
$$180$$ 0 0
$$181$$ 2593.69 1.06512 0.532561 0.846392i $$-0.321230\pi$$
0.532561 + 0.846392i $$0.321230\pi$$
$$182$$ − 1161.88i − 0.473210i
$$183$$ 71.0866i 0.0287151i
$$184$$ −1862.16 −0.746089
$$185$$ 0 0
$$186$$ 7931.03 3.12651
$$187$$ − 1945.21i − 0.760682i
$$188$$ 7708.66i 2.99049i
$$189$$ 1053.36 0.405401
$$190$$ 0 0
$$191$$ −1823.08 −0.690645 −0.345323 0.938484i $$-0.612231\pi$$
−0.345323 + 0.938484i $$0.612231\pi$$
$$192$$ − 7273.09i − 2.73380i
$$193$$ 1541.03i 0.574744i 0.957819 + 0.287372i $$0.0927816\pi$$
−0.957819 + 0.287372i $$0.907218\pi$$
$$194$$ −9107.39 −3.37048
$$195$$ 0 0
$$196$$ 1044.37 0.380602
$$197$$ 701.243i 0.253612i 0.991928 + 0.126806i $$0.0404725\pi$$
−0.991928 + 0.126806i $$0.959527\pi$$
$$198$$ − 1503.35i − 0.539587i
$$199$$ −3294.96 −1.17374 −0.586868 0.809682i $$-0.699639\pi$$
−0.586868 + 0.809682i $$0.699639\pi$$
$$200$$ 0 0
$$201$$ 775.411 0.272106
$$202$$ − 2350.67i − 0.818775i
$$203$$ 146.588i 0.0506820i
$$204$$ 3694.80 1.26808
$$205$$ 0 0
$$206$$ 1871.03 0.632819
$$207$$ − 137.272i − 0.0460920i
$$208$$ − 6737.29i − 2.24590i
$$209$$ 4192.16 1.38746
$$210$$ 0 0
$$211$$ 4082.35 1.33195 0.665974 0.745975i $$-0.268016\pi$$
0.665974 + 0.745975i $$0.268016\pi$$
$$212$$ 3261.41i 1.05658i
$$213$$ − 4433.33i − 1.42613i
$$214$$ −1175.53 −0.375502
$$215$$ 0 0
$$216$$ 10847.1 3.41691
$$217$$ 2201.91i 0.688826i
$$218$$ − 9390.46i − 2.91744i
$$219$$ −691.494 −0.213364
$$220$$ 0 0
$$221$$ 1141.21 0.347359
$$222$$ − 4970.71i − 1.50276i
$$223$$ − 747.161i − 0.224366i −0.993688 0.112183i $$-0.964216\pi$$
0.993688 0.112183i $$-0.0357843\pi$$
$$224$$ 4292.30 1.28032
$$225$$ 0 0
$$226$$ −10039.1 −2.95481
$$227$$ 1665.67i 0.487025i 0.969898 + 0.243513i $$0.0782997\pi$$
−0.969898 + 0.243513i $$0.921700\pi$$
$$228$$ 7962.76i 2.31292i
$$229$$ 6628.35 1.91272 0.956362 0.292183i $$-0.0943816\pi$$
0.956362 + 0.292183i $$0.0943816\pi$$
$$230$$ 0 0
$$231$$ −1703.40 −0.485176
$$232$$ 1509.50i 0.427172i
$$233$$ 432.431i 0.121586i 0.998150 + 0.0607929i $$0.0193629\pi$$
−0.998150 + 0.0607929i $$0.980637\pi$$
$$234$$ 881.984 0.246398
$$235$$ 0 0
$$236$$ −13129.2 −3.62136
$$237$$ − 3994.30i − 1.09476i
$$238$$ 1410.82i 0.384244i
$$239$$ −5580.44 −1.51033 −0.755165 0.655535i $$-0.772443\pi$$
−0.755165 + 0.655535i $$0.772443\pi$$
$$240$$ 0 0
$$241$$ −6296.87 −1.68306 −0.841529 0.540212i $$-0.818344\pi$$
−0.841529 + 0.540212i $$0.818344\pi$$
$$242$$ 7577.56i 2.01283i
$$243$$ 1467.73i 0.387468i
$$244$$ −325.352 −0.0853629
$$245$$ 0 0
$$246$$ −286.485 −0.0742504
$$247$$ 2459.46i 0.633569i
$$248$$ 22674.4i 5.80575i
$$249$$ 3075.98 0.782862
$$250$$ 0 0
$$251$$ 311.921 0.0784393 0.0392197 0.999231i $$-0.487513\pi$$
0.0392197 + 0.999231i $$0.487513\pi$$
$$252$$ 792.784i 0.198177i
$$253$$ 1349.92i 0.335451i
$$254$$ −7550.17 −1.86512
$$255$$ 0 0
$$256$$ 6728.46 1.64269
$$257$$ − 7861.39i − 1.90809i −0.299659 0.954046i $$-0.596873\pi$$
0.299659 0.954046i $$-0.403127\pi$$
$$258$$ − 853.050i − 0.205847i
$$259$$ 1380.03 0.331085
$$260$$ 0 0
$$261$$ −111.275 −0.0263899
$$262$$ 9542.11i 2.25005i
$$263$$ − 5227.09i − 1.22554i −0.790262 0.612769i $$-0.790056\pi$$
0.790262 0.612769i $$-0.209944\pi$$
$$264$$ −17541.0 −4.08929
$$265$$ 0 0
$$266$$ −3040.50 −0.700846
$$267$$ 213.166i 0.0488596i
$$268$$ 3548.94i 0.808903i
$$269$$ −1281.71 −0.290510 −0.145255 0.989394i $$-0.546400\pi$$
−0.145255 + 0.989394i $$0.546400\pi$$
$$270$$ 0 0
$$271$$ 4704.14 1.05445 0.527226 0.849725i $$-0.323232\pi$$
0.527226 + 0.849725i $$0.323232\pi$$
$$272$$ 8180.82i 1.82366i
$$273$$ − 999.352i − 0.221551i
$$274$$ 4997.04 1.10176
$$275$$ 0 0
$$276$$ −2564.10 −0.559205
$$277$$ 8958.56i 1.94321i 0.236619 + 0.971603i $$0.423961\pi$$
−0.236619 + 0.971603i $$0.576039\pi$$
$$278$$ 1061.40i 0.228987i
$$279$$ −1671.47 −0.358668
$$280$$ 0 0
$$281$$ −370.904 −0.0787412 −0.0393706 0.999225i $$-0.512535\pi$$
−0.0393706 + 0.999225i $$0.512535\pi$$
$$282$$ 9119.02i 1.92564i
$$283$$ 5822.26i 1.22296i 0.791261 + 0.611479i $$0.209425\pi$$
−0.791261 + 0.611479i $$0.790575\pi$$
$$284$$ 20290.7 4.23954
$$285$$ 0 0
$$286$$ −8673.40 −1.79325
$$287$$ − 79.5374i − 0.0163587i
$$288$$ 3258.29i 0.666655i
$$289$$ 3527.27 0.717946
$$290$$ 0 0
$$291$$ −7833.42 −1.57802
$$292$$ − 3164.86i − 0.634279i
$$293$$ − 7443.79i − 1.48420i −0.670289 0.742100i $$-0.733830\pi$$
0.670289 0.742100i $$-0.266170\pi$$
$$294$$ 1235.45 0.245077
$$295$$ 0 0
$$296$$ 14211.0 2.79053
$$297$$ − 7863.32i − 1.53628i
$$298$$ − 4225.10i − 0.821320i
$$299$$ −791.973 −0.153181
$$300$$ 0 0
$$301$$ 236.834 0.0453518
$$302$$ − 12557.9i − 2.39280i
$$303$$ − 2021.85i − 0.383341i
$$304$$ −17630.7 −3.32628
$$305$$ 0 0
$$306$$ −1070.96 −0.200074
$$307$$ − 761.674i − 0.141600i −0.997491 0.0707998i $$-0.977445\pi$$
0.997491 0.0707998i $$-0.0225551\pi$$
$$308$$ − 7796.21i − 1.44231i
$$309$$ 1609.30 0.296278
$$310$$ 0 0
$$311$$ 7718.69 1.40735 0.703677 0.710520i $$-0.251540\pi$$
0.703677 + 0.710520i $$0.251540\pi$$
$$312$$ − 10290.9i − 1.86734i
$$313$$ − 8556.00i − 1.54509i −0.634959 0.772546i $$-0.718983\pi$$
0.634959 0.772546i $$-0.281017\pi$$
$$314$$ −5538.21 −0.995348
$$315$$ 0 0
$$316$$ 18281.3 3.25444
$$317$$ − 7780.95i − 1.37862i −0.724468 0.689309i $$-0.757914\pi$$
0.724468 0.689309i $$-0.242086\pi$$
$$318$$ 3858.11i 0.680352i
$$319$$ 1094.28 0.192062
$$320$$ 0 0
$$321$$ −1011.09 −0.175805
$$322$$ − 979.076i − 0.169446i
$$323$$ − 2986.42i − 0.514455i
$$324$$ 11878.0 2.03670
$$325$$ 0 0
$$326$$ −7312.58 −1.24235
$$327$$ − 8076.89i − 1.36591i
$$328$$ − 819.045i − 0.137879i
$$329$$ −2531.73 −0.424252
$$330$$ 0 0
$$331$$ −4932.12 −0.819015 −0.409507 0.912307i $$-0.634299\pi$$
−0.409507 + 0.912307i $$0.634299\pi$$
$$332$$ 14078.3i 2.32725i
$$333$$ 1047.58i 0.172394i
$$334$$ 6662.61 1.09150
$$335$$ 0 0
$$336$$ 7163.88 1.16316
$$337$$ − 7121.13i − 1.15108i −0.817775 0.575538i $$-0.804793\pi$$
0.817775 0.575538i $$-0.195207\pi$$
$$338$$ 6806.52i 1.09534i
$$339$$ −8634.76 −1.38341
$$340$$ 0 0
$$341$$ 16437.2 2.61034
$$342$$ − 2308.05i − 0.364927i
$$343$$ 343.000i 0.0539949i
$$344$$ 2438.83 0.382246
$$345$$ 0 0
$$346$$ −13468.7 −2.09272
$$347$$ 9540.58i 1.47598i 0.674811 + 0.737991i $$0.264225\pi$$
−0.674811 + 0.737991i $$0.735775\pi$$
$$348$$ 2078.51i 0.320172i
$$349$$ −1281.65 −0.196576 −0.0982880 0.995158i $$-0.531337\pi$$
−0.0982880 + 0.995158i $$0.531337\pi$$
$$350$$ 0 0
$$351$$ 4613.25 0.701530
$$352$$ − 32041.9i − 4.85182i
$$353$$ − 5798.07i − 0.874221i −0.899408 0.437110i $$-0.856002\pi$$
0.899408 0.437110i $$-0.143998\pi$$
$$354$$ −15531.3 −2.33187
$$355$$ 0 0
$$356$$ −975.627 −0.145247
$$357$$ 1213.47i 0.179899i
$$358$$ − 8777.40i − 1.29581i
$$359$$ −2267.29 −0.333323 −0.166662 0.986014i $$-0.553299\pi$$
−0.166662 + 0.986014i $$0.553299\pi$$
$$360$$ 0 0
$$361$$ −422.886 −0.0616541
$$362$$ 14042.8i 2.03887i
$$363$$ 6517.58i 0.942381i
$$364$$ 4573.88 0.658616
$$365$$ 0 0
$$366$$ −384.878 −0.0549669
$$367$$ − 7372.85i − 1.04866i −0.851514 0.524332i $$-0.824315\pi$$
0.851514 0.524332i $$-0.175685\pi$$
$$368$$ − 5677.28i − 0.804209i
$$369$$ 60.3769 0.00851788
$$370$$ 0 0
$$371$$ −1071.14 −0.149894
$$372$$ 31221.4i 4.35150i
$$373$$ − 6447.14i − 0.894961i −0.894294 0.447480i $$-0.852321\pi$$
0.894294 0.447480i $$-0.147679\pi$$
$$374$$ 10531.8 1.45611
$$375$$ 0 0
$$376$$ −26070.8 −3.57579
$$377$$ 641.989i 0.0877032i
$$378$$ 5703.12i 0.776024i
$$379$$ 4247.57 0.575680 0.287840 0.957678i $$-0.407063\pi$$
0.287840 + 0.957678i $$0.407063\pi$$
$$380$$ 0 0
$$381$$ −6494.03 −0.873226
$$382$$ − 9870.53i − 1.32204i
$$383$$ 6681.86i 0.891454i 0.895169 + 0.445727i $$0.147055\pi$$
−0.895169 + 0.445727i $$0.852945\pi$$
$$384$$ 16533.9 2.19725
$$385$$ 0 0
$$386$$ −8343.45 −1.10018
$$387$$ 179.781i 0.0236145i
$$388$$ − 35852.4i − 4.69105i
$$389$$ 6371.78 0.830494 0.415247 0.909709i $$-0.363695\pi$$
0.415247 + 0.909709i $$0.363695\pi$$
$$390$$ 0 0
$$391$$ 961.661 0.124382
$$392$$ 3532.08i 0.455094i
$$393$$ 8207.33i 1.05345i
$$394$$ −3796.68 −0.485467
$$395$$ 0 0
$$396$$ 5918.11 0.751001
$$397$$ 4247.93i 0.537021i 0.963277 + 0.268510i $$0.0865314\pi$$
−0.963277 + 0.268510i $$0.913469\pi$$
$$398$$ − 17839.6i − 2.24678i
$$399$$ −2615.19 −0.328128
$$400$$ 0 0
$$401$$ −8833.62 −1.10008 −0.550038 0.835140i $$-0.685387\pi$$
−0.550038 + 0.835140i $$0.685387\pi$$
$$402$$ 4198.24i 0.520869i
$$403$$ 9643.37i 1.19199i
$$404$$ 9253.70 1.13958
$$405$$ 0 0
$$406$$ −793.658 −0.0970162
$$407$$ − 10301.9i − 1.25466i
$$408$$ 12495.9i 1.51627i
$$409$$ 319.205 0.0385908 0.0192954 0.999814i $$-0.493858\pi$$
0.0192954 + 0.999814i $$0.493858\pi$$
$$410$$ 0 0
$$411$$ 4298.04 0.515831
$$412$$ 7365.53i 0.880761i
$$413$$ − 4312.00i − 0.513752i
$$414$$ 743.218 0.0882298
$$415$$ 0 0
$$416$$ 18798.3 2.21554
$$417$$ 912.924i 0.107209i
$$418$$ 22697.3i 2.65589i
$$419$$ 12789.2 1.49115 0.745577 0.666420i $$-0.232174\pi$$
0.745577 + 0.666420i $$0.232174\pi$$
$$420$$ 0 0
$$421$$ −6747.40 −0.781112 −0.390556 0.920579i $$-0.627717\pi$$
−0.390556 + 0.920579i $$0.627717\pi$$
$$422$$ 22102.7i 2.54963i
$$423$$ − 1921.84i − 0.220906i
$$424$$ −11030.1 −1.26337
$$425$$ 0 0
$$426$$ 24003.0 2.72992
$$427$$ − 106.855i − 0.0121102i
$$428$$ − 4627.60i − 0.522625i
$$429$$ −7460.14 −0.839577
$$430$$ 0 0
$$431$$ −5184.75 −0.579444 −0.289722 0.957111i $$-0.593563\pi$$
−0.289722 + 0.957111i $$0.593563\pi$$
$$432$$ 33070.2i 3.68308i
$$433$$ 4242.03i 0.470806i 0.971898 + 0.235403i $$0.0756410\pi$$
−0.971898 + 0.235403i $$0.924359\pi$$
$$434$$ −11921.6 −1.31856
$$435$$ 0 0
$$436$$ 36966.7 4.06051
$$437$$ 2072.50i 0.226868i
$$438$$ − 3743.90i − 0.408425i
$$439$$ 5434.12 0.590789 0.295394 0.955375i $$-0.404549\pi$$
0.295394 + 0.955375i $$0.404549\pi$$
$$440$$ 0 0
$$441$$ −260.372 −0.0281149
$$442$$ 6178.77i 0.664919i
$$443$$ 11493.8i 1.23270i 0.787472 + 0.616350i $$0.211389\pi$$
−0.787472 + 0.616350i $$0.788611\pi$$
$$444$$ 19567.8 2.09155
$$445$$ 0 0
$$446$$ 4045.29 0.429484
$$447$$ − 3634.08i − 0.384532i
$$448$$ 10932.6i 1.15294i
$$449$$ 16849.3 1.77098 0.885489 0.464661i $$-0.153824\pi$$
0.885489 + 0.464661i $$0.153824\pi$$
$$450$$ 0 0
$$451$$ −593.745 −0.0619919
$$452$$ − 39520.0i − 4.11253i
$$453$$ − 10801.2i − 1.12028i
$$454$$ −9018.32 −0.932270
$$455$$ 0 0
$$456$$ −26930.2 −2.76561
$$457$$ 15348.5i 1.57106i 0.618826 + 0.785528i $$0.287609\pi$$
−0.618826 + 0.785528i $$0.712391\pi$$
$$458$$ 35887.3i 3.66136i
$$459$$ −5601.69 −0.569639
$$460$$ 0 0
$$461$$ 14038.4 1.41830 0.709148 0.705059i $$-0.249080\pi$$
0.709148 + 0.705059i $$0.249080\pi$$
$$462$$ − 9222.58i − 0.928730i
$$463$$ 8661.23i 0.869377i 0.900581 + 0.434689i $$0.143142\pi$$
−0.900581 + 0.434689i $$0.856858\pi$$
$$464$$ −4602.12 −0.460448
$$465$$ 0 0
$$466$$ −2341.27 −0.232741
$$467$$ 7014.71i 0.695079i 0.937665 + 0.347539i $$0.112983\pi$$
−0.937665 + 0.347539i $$0.887017\pi$$
$$468$$ 3472.04i 0.342938i
$$469$$ −1165.57 −0.114757
$$470$$ 0 0
$$471$$ −4763.50 −0.466010
$$472$$ − 44403.3i − 4.33014i
$$473$$ − 1767.96i − 0.171863i
$$474$$ 21626.0 2.09560
$$475$$ 0 0
$$476$$ −5553.88 −0.534793
$$477$$ − 813.100i − 0.0780488i
$$478$$ − 30213.7i − 2.89109i
$$479$$ −18134.7 −1.72984 −0.864922 0.501907i $$-0.832632\pi$$
−0.864922 + 0.501907i $$0.832632\pi$$
$$480$$ 0 0
$$481$$ 6043.91 0.572929
$$482$$ − 34092.6i − 3.22173i
$$483$$ − 842.119i − 0.0793328i
$$484$$ −29830.0 −2.80146
$$485$$ 0 0
$$486$$ −7946.59 −0.741697
$$487$$ 16537.8i 1.53881i 0.638761 + 0.769405i $$0.279447\pi$$
−0.638761 + 0.769405i $$0.720553\pi$$
$$488$$ − 1100.35i − 0.102070i
$$489$$ −6289.67 −0.581654
$$490$$ 0 0
$$491$$ 220.608 0.0202768 0.0101384 0.999949i $$-0.496773\pi$$
0.0101384 + 0.999949i $$0.496773\pi$$
$$492$$ − 1127.78i − 0.103342i
$$493$$ − 779.542i − 0.0712146i
$$494$$ −13316.0 −1.21279
$$495$$ 0 0
$$496$$ −69128.8 −6.25801
$$497$$ 6664.00i 0.601451i
$$498$$ 16654.0i 1.49856i
$$499$$ −5939.04 −0.532801 −0.266401 0.963862i $$-0.585834\pi$$
−0.266401 + 0.963862i $$0.585834\pi$$
$$500$$ 0 0
$$501$$ 5730.62 0.511029
$$502$$ 1688.81i 0.150150i
$$503$$ 11604.8i 1.02869i 0.857584 + 0.514345i $$0.171965\pi$$
−0.857584 + 0.514345i $$0.828035\pi$$
$$504$$ −2681.21 −0.236965
$$505$$ 0 0
$$506$$ −7308.78 −0.642124
$$507$$ 5854.40i 0.512826i
$$508$$ − 29722.2i − 2.59588i
$$509$$ 1867.67 0.162639 0.0813193 0.996688i $$-0.474087\pi$$
0.0813193 + 0.996688i $$0.474087\pi$$
$$510$$ 0 0
$$511$$ 1039.43 0.0899834
$$512$$ 8025.75i 0.692757i
$$513$$ − 12072.3i − 1.03900i
$$514$$ 42563.2 3.65250
$$515$$ 0 0
$$516$$ 3358.14 0.286499
$$517$$ 18899.3i 1.60772i
$$518$$ 7471.78i 0.633767i
$$519$$ −11584.6 −0.979786
$$520$$ 0 0
$$521$$ 6117.21 0.514395 0.257197 0.966359i $$-0.417201\pi$$
0.257197 + 0.966359i $$0.417201\pi$$
$$522$$ − 602.467i − 0.0505158i
$$523$$ 16685.6i 1.39505i 0.716561 + 0.697524i $$0.245715\pi$$
−0.716561 + 0.697524i $$0.754285\pi$$
$$524$$ −37563.7 −3.13164
$$525$$ 0 0
$$526$$ 28300.6 2.34594
$$527$$ − 11709.6i − 0.967887i
$$528$$ − 53478.2i − 4.40784i
$$529$$ 11499.6 0.945149
$$530$$ 0 0
$$531$$ 3273.24 0.267508
$$532$$ − 11969.3i − 0.975442i
$$533$$ − 348.338i − 0.0283081i
$$534$$ −1154.12 −0.0935278
$$535$$ 0 0
$$536$$ −12002.6 −0.967223
$$537$$ − 7549.59i − 0.606683i
$$538$$ − 6939.44i − 0.556097i
$$539$$ 2560.49 0.204616
$$540$$ 0 0
$$541$$ 9309.03 0.739790 0.369895 0.929074i $$-0.379394\pi$$
0.369895 + 0.929074i $$0.379394\pi$$
$$542$$ 25469.2i 2.01845i
$$543$$ 12078.4i 0.954575i
$$544$$ −22826.1 −1.79901
$$545$$ 0 0
$$546$$ 5410.70 0.424097
$$547$$ 10894.7i 0.851598i 0.904818 + 0.425799i $$0.140007\pi$$
−0.904818 + 0.425799i $$0.859993\pi$$
$$548$$ 19671.5i 1.53344i
$$549$$ 81.1134 0.00630571
$$550$$ 0 0
$$551$$ 1680.01 0.129893
$$552$$ − 8671.81i − 0.668654i
$$553$$ 6004.07i 0.461698i
$$554$$ −48503.6 −3.71971
$$555$$ 0 0
$$556$$ −4178.31 −0.318705
$$557$$ − 7873.90i − 0.598973i −0.954101 0.299486i $$-0.903185\pi$$
0.954101 0.299486i $$-0.0968153\pi$$
$$558$$ − 9049.71i − 0.686567i
$$559$$ 1037.23 0.0784796
$$560$$ 0 0
$$561$$ 9058.55 0.681733
$$562$$ − 2008.15i − 0.150728i
$$563$$ − 21770.7i − 1.62971i −0.579666 0.814854i $$-0.696817\pi$$
0.579666 0.814854i $$-0.303183\pi$$
$$564$$ −35898.1 −2.68011
$$565$$ 0 0
$$566$$ −31522.9 −2.34100
$$567$$ 3901.06i 0.288940i
$$568$$ 68623.3i 5.06931i
$$569$$ 12381.3 0.912213 0.456106 0.889925i $$-0.349244\pi$$
0.456106 + 0.889925i $$0.349244\pi$$
$$570$$ 0 0
$$571$$ −5768.38 −0.422765 −0.211383 0.977403i $$-0.567797\pi$$
−0.211383 + 0.977403i $$0.567797\pi$$
$$572$$ − 34143.9i − 2.49585i
$$573$$ − 8489.81i − 0.618965i
$$574$$ 430.632 0.0313140
$$575$$ 0 0
$$576$$ −8298.97 −0.600330
$$577$$ 4733.38i 0.341513i 0.985313 + 0.170757i $$0.0546212\pi$$
−0.985313 + 0.170757i $$0.945379\pi$$
$$578$$ 19097.4i 1.37430i
$$579$$ −7176.34 −0.515093
$$580$$ 0 0
$$581$$ −4623.70 −0.330161
$$582$$ − 42411.8i − 3.02066i
$$583$$ 7996.00i 0.568028i
$$584$$ 10703.6 0.758422
$$585$$ 0 0
$$586$$ 40302.3 2.84108
$$587$$ − 8441.67i − 0.593569i −0.954944 0.296785i $$-0.904086\pi$$
0.954944 0.296785i $$-0.0959143\pi$$
$$588$$ 4863.49i 0.341100i
$$589$$ 25235.6 1.76539
$$590$$ 0 0
$$591$$ −3265.59 −0.227290
$$592$$ 43326.0i 3.00792i
$$593$$ − 18939.9i − 1.31158i −0.754943 0.655791i $$-0.772335\pi$$
0.754943 0.655791i $$-0.227665\pi$$
$$594$$ 42573.7 2.94077
$$595$$ 0 0
$$596$$ 16632.6 1.14312
$$597$$ − 15344.2i − 1.05192i
$$598$$ − 4287.91i − 0.293220i
$$599$$ −22655.3 −1.54536 −0.772681 0.634794i $$-0.781085\pi$$
−0.772681 + 0.634794i $$0.781085\pi$$
$$600$$ 0 0
$$601$$ −15947.4 −1.08237 −0.541187 0.840902i $$-0.682025\pi$$
−0.541187 + 0.840902i $$0.682025\pi$$
$$602$$ 1282.27i 0.0868131i
$$603$$ − 884.784i − 0.0597532i
$$604$$ 49435.6 3.33031
$$605$$ 0 0
$$606$$ 10946.7 0.733796
$$607$$ − 25993.2i − 1.73811i −0.494719 0.869053i $$-0.664729\pi$$
0.494719 0.869053i $$-0.335271\pi$$
$$608$$ − 49193.1i − 3.28132i
$$609$$ −682.638 −0.0454218
$$610$$ 0 0
$$611$$ −11087.9 −0.734152
$$612$$ − 4215.96i − 0.278464i
$$613$$ − 665.408i − 0.0438427i −0.999760 0.0219213i $$-0.993022\pi$$
0.999760 0.0219213i $$-0.00697834\pi$$
$$614$$ 4123.87 0.271052
$$615$$ 0 0
$$616$$ 26366.9 1.72460
$$617$$ 18401.3i 1.20066i 0.799752 + 0.600330i $$0.204964\pi$$
−0.799752 + 0.600330i $$0.795036\pi$$
$$618$$ 8713.10i 0.567140i
$$619$$ 11150.6 0.724040 0.362020 0.932170i $$-0.382087\pi$$
0.362020 + 0.932170i $$0.382087\pi$$
$$620$$ 0 0
$$621$$ 3887.43 0.251203
$$622$$ 41790.6i 2.69397i
$$623$$ − 320.422i − 0.0206059i
$$624$$ 31374.6 2.01280
$$625$$ 0 0
$$626$$ 46324.0 2.95764
$$627$$ 19522.3i 1.24345i
$$628$$ − 21801.8i − 1.38533i
$$629$$ −7338.88 −0.465215
$$630$$ 0 0
$$631$$ 5381.79 0.339534 0.169767 0.985484i $$-0.445699\pi$$
0.169767 + 0.985484i $$0.445699\pi$$
$$632$$ 61827.6i 3.89141i
$$633$$ 19010.9i 1.19371i
$$634$$ 42127.7 2.63897
$$635$$ 0 0
$$636$$ −15187.9 −0.946918
$$637$$ 1502.19i 0.0934361i
$$638$$ 5924.64i 0.367647i
$$639$$ −5058.65 −0.313172
$$640$$ 0 0
$$641$$ −19455.1 −1.19880 −0.599398 0.800451i $$-0.704593\pi$$
−0.599398 + 0.800451i $$0.704593\pi$$
$$642$$ − 5474.26i − 0.336529i
$$643$$ 14695.8i 0.901317i 0.892696 + 0.450658i $$0.148811\pi$$
−0.892696 + 0.450658i $$0.851189\pi$$
$$644$$ 3854.25 0.235837
$$645$$ 0 0
$$646$$ 16169.1 0.984777
$$647$$ − 12694.8i − 0.771383i −0.922628 0.385691i $$-0.873963\pi$$
0.922628 0.385691i $$-0.126037\pi$$
$$648$$ 40171.6i 2.43532i
$$649$$ −32189.0 −1.94688
$$650$$ 0 0
$$651$$ −10254.0 −0.617334
$$652$$ − 28786.8i − 1.72911i
$$653$$ 12385.6i 0.742247i 0.928583 + 0.371124i $$0.121027\pi$$
−0.928583 + 0.371124i $$0.878973\pi$$
$$654$$ 43730.0 2.61465
$$655$$ 0 0
$$656$$ 2497.07 0.148619
$$657$$ 789.030i 0.0468539i
$$658$$ − 13707.3i − 0.812109i
$$659$$ 2072.18 0.122489 0.0612447 0.998123i $$-0.480493\pi$$
0.0612447 + 0.998123i $$0.480493\pi$$
$$660$$ 0 0
$$661$$ 1074.36 0.0632193 0.0316096 0.999500i $$-0.489937\pi$$
0.0316096 + 0.999500i $$0.489937\pi$$
$$662$$ − 26703.6i − 1.56777i
$$663$$ 5314.47i 0.311307i
$$664$$ −47613.0 −2.78275
$$665$$ 0 0
$$666$$ −5671.84 −0.329999
$$667$$ 540.982i 0.0314047i
$$668$$ 26228.2i 1.51916i
$$669$$ 3479.42 0.201080
$$670$$ 0 0
$$671$$ −797.667 −0.0458921
$$672$$ 19988.6i 1.14744i
$$673$$ − 26195.2i − 1.50037i −0.661226 0.750186i $$-0.729964\pi$$
0.661226 0.750186i $$-0.270036\pi$$
$$674$$ 38555.3 2.20341
$$675$$ 0 0
$$676$$ −26794.7 −1.52450
$$677$$ − 4228.44i − 0.240047i −0.992771 0.120024i $$-0.961703\pi$$
0.992771 0.120024i $$-0.0382970\pi$$
$$678$$ − 46750.4i − 2.64814i
$$679$$ 11774.9 0.665506
$$680$$ 0 0
$$681$$ −7756.80 −0.436478
$$682$$ 88994.5i 4.99674i
$$683$$ − 27525.5i − 1.54207i −0.636792 0.771036i $$-0.719739\pi$$
0.636792 0.771036i $$-0.280261\pi$$
$$684$$ 9085.91 0.507907
$$685$$ 0 0
$$686$$ −1857.08 −0.103358
$$687$$ 30867.3i 1.71421i
$$688$$ 7435.40i 0.412023i
$$689$$ −4691.09 −0.259385
$$690$$ 0 0
$$691$$ −33324.4 −1.83462 −0.917309 0.398177i $$-0.869643\pi$$
−0.917309 + 0.398177i $$0.869643\pi$$
$$692$$ − 53021.1i − 2.91266i
$$693$$ 1943.67i 0.106542i
$$694$$ −51654.8 −2.82534
$$695$$ 0 0
$$696$$ −7029.54 −0.382836
$$697$$ 422.973i 0.0229860i
$$698$$ − 6939.12i − 0.376289i
$$699$$ −2013.77 −0.108967
$$700$$ 0 0
$$701$$ −33262.9 −1.79219 −0.896094 0.443864i $$-0.853607\pi$$
−0.896094 + 0.443864i $$0.853607\pi$$
$$702$$ 24977.1i 1.34288i
$$703$$ − 15816.2i − 0.848534i
$$704$$ 81611.8 4.36912
$$705$$ 0 0
$$706$$ 31392.0 1.67345
$$707$$ 3039.17i 0.161668i
$$708$$ − 61141.0i − 3.24551i
$$709$$ −13703.0 −0.725851 −0.362926 0.931818i $$-0.618222\pi$$
−0.362926 + 0.931818i $$0.618222\pi$$
$$710$$ 0 0
$$711$$ −4557.70 −0.240404
$$712$$ − 3299.58i − 0.173676i
$$713$$ 8126.14i 0.426825i
$$714$$ −6570.00 −0.344364
$$715$$ 0 0
$$716$$ 34553.3 1.80352
$$717$$ − 25987.3i − 1.35358i
$$718$$ − 12275.6i − 0.638052i
$$719$$ 8074.93 0.418838 0.209419 0.977826i $$-0.432843\pi$$
0.209419 + 0.977826i $$0.432843\pi$$
$$720$$ 0 0
$$721$$ −2419.04 −0.124951
$$722$$ − 2289.59i − 0.118019i
$$723$$ − 29323.6i − 1.50838i
$$724$$ −55281.1 −2.83771
$$725$$ 0 0
$$726$$ −35287.6 −1.80392
$$727$$ − 3668.70i − 0.187159i −0.995612 0.0935794i $$-0.970169\pi$$
0.995612 0.0935794i $$-0.0298309\pi$$
$$728$$ 15468.9i 0.787523i
$$729$$ −21881.9 −1.11172
$$730$$ 0 0
$$731$$ −1259.46 −0.0637250
$$732$$ − 1515.12i − 0.0765033i
$$733$$ 14980.3i 0.754857i 0.926039 + 0.377428i $$0.123192\pi$$
−0.926039 + 0.377428i $$0.876808\pi$$
$$734$$ 39918.2 2.00737
$$735$$ 0 0
$$736$$ 15840.7 0.793338
$$737$$ 8700.94i 0.434875i
$$738$$ 326.894i 0.0163050i
$$739$$ −6530.59 −0.325077 −0.162538 0.986702i $$-0.551968\pi$$
−0.162538 + 0.986702i $$0.551968\pi$$
$$740$$ 0 0
$$741$$ −11453.3 −0.567812
$$742$$ − 5799.36i − 0.286929i
$$743$$ − 25952.0i − 1.28141i −0.767788 0.640704i $$-0.778643\pi$$
0.767788 0.640704i $$-0.221357\pi$$
$$744$$ −105591. −5.20318
$$745$$ 0 0
$$746$$ 34906.2 1.71315
$$747$$ − 3509.85i − 0.171913i
$$748$$ 41459.6i 2.02662i
$$749$$ 1519.83 0.0741434
$$750$$ 0 0
$$751$$ −14093.9 −0.684813 −0.342407 0.939552i $$-0.611242\pi$$
−0.342407 + 0.939552i $$0.611242\pi$$
$$752$$ − 79483.6i − 3.85435i
$$753$$ 1452.57i 0.0702983i
$$754$$ −3475.87 −0.167883
$$755$$ 0 0
$$756$$ −22451.0 −1.08007
$$757$$ − 2554.41i − 0.122644i −0.998118 0.0613220i $$-0.980468\pi$$
0.998118 0.0613220i $$-0.0195316\pi$$
$$758$$ 22997.2i 1.10197i
$$759$$ −6286.40 −0.300635
$$760$$ 0 0
$$761$$ 2219.08 0.105705 0.0528527 0.998602i $$-0.483169\pi$$
0.0528527 + 0.998602i $$0.483169\pi$$
$$762$$ − 35160.1i − 1.67154i
$$763$$ 12140.9i 0.576054i
$$764$$ 38856.5 1.84003
$$765$$ 0 0
$$766$$ −36177.0 −1.70643
$$767$$ − 18884.6i − 0.889028i
$$768$$ 31333.5i 1.47220i
$$769$$ 22466.2 1.05352 0.526758 0.850015i $$-0.323408\pi$$
0.526758 + 0.850015i $$0.323408\pi$$
$$770$$ 0 0
$$771$$ 36609.3 1.71006
$$772$$ − 32845.0i − 1.53124i
$$773$$ − 9674.79i − 0.450165i −0.974340 0.225083i $$-0.927735\pi$$
0.974340 0.225083i $$-0.0722652\pi$$
$$774$$ −973.374 −0.0452031
$$775$$ 0 0
$$776$$ 121253. 5.60920
$$777$$ 6426.60i 0.296722i
$$778$$ 34498.2i 1.58974i
$$779$$ −911.560 −0.0419256
$$780$$ 0 0
$$781$$ 49746.6 2.27922
$$782$$ 5206.64i 0.238093i
$$783$$ − 3151.23i − 0.143826i
$$784$$ −10768.5 −0.490546
$$785$$ 0 0
$$786$$ −44436.2 −2.01652
$$787$$ − 20942.8i − 0.948577i −0.880370 0.474288i $$-0.842705\pi$$
0.880370 0.474288i $$-0.157295\pi$$
$$788$$ − 14946.1i − 0.675676i
$$789$$ 24341.8 1.09834
$$790$$ 0 0
$$791$$ 12979.4 0.583433
$$792$$ 20015.1i 0.897989i
$$793$$ − 467.975i − 0.0209562i
$$794$$ −22999.2 −1.02797
$$795$$ 0 0
$$796$$ 70227.8 3.12708
$$797$$ 23526.6i 1.04561i 0.852451 + 0.522807i $$0.175115\pi$$
−0.852451 + 0.522807i $$0.824885\pi$$
$$798$$ − 14159.2i − 0.628107i
$$799$$ 13463.5 0.596127
$$800$$ 0 0
$$801$$ 243.233 0.0107294
$$802$$ − 47827.1i − 2.10578i
$$803$$ − 7759.29i − 0.340996i
$$804$$ −16526.9 −0.724948
$$805$$ 0 0
$$806$$ −52211.3 −2.28172
$$807$$ − 5968.72i − 0.260358i
$$808$$ 31296.1i 1.36262i
$$809$$ 18202.2 0.791047 0.395523 0.918456i $$-0.370563\pi$$
0.395523 + 0.918456i $$0.370563\pi$$
$$810$$ 0 0
$$811$$ −2510.24 −0.108689 −0.0543443 0.998522i $$-0.517307\pi$$
−0.0543443 + 0.998522i $$0.517307\pi$$
$$812$$ − 3124.33i − 0.135028i
$$813$$ 21906.5i 0.945012i
$$814$$ 55776.7 2.40168
$$815$$ 0 0
$$816$$ −38096.9 −1.63438
$$817$$ − 2714.30i − 0.116232i
$$818$$ 1728.24i 0.0738711i
$$819$$ −1140.31 −0.0486516
$$820$$ 0 0
$$821$$ 17899.6 0.760903 0.380451 0.924801i $$-0.375769\pi$$
0.380451 + 0.924801i $$0.375769\pi$$
$$822$$ 23270.5i 0.987411i
$$823$$ − 14039.5i − 0.594637i −0.954778 0.297318i $$-0.903908\pi$$
0.954778 0.297318i $$-0.0960923\pi$$
$$824$$ −24910.3 −1.05315
$$825$$ 0 0
$$826$$ 23346.1 0.983431
$$827$$ 15127.4i 0.636073i 0.948079 + 0.318036i $$0.103023\pi$$
−0.948079 + 0.318036i $$0.896977\pi$$
$$828$$ 2925.77i 0.122799i
$$829$$ −21986.5 −0.921136 −0.460568 0.887624i $$-0.652354\pi$$
−0.460568 + 0.887624i $$0.652354\pi$$
$$830$$ 0 0
$$831$$ −41718.7 −1.74152
$$832$$ 47880.0i 1.99512i
$$833$$ − 1824.04i − 0.0758696i
$$834$$ −4942.76 −0.205220
$$835$$ 0 0
$$836$$ −89350.6 −3.69648
$$837$$ − 47334.8i − 1.95476i
$$838$$ 69243.5i 2.85439i
$$839$$ 2276.89 0.0936914 0.0468457 0.998902i $$-0.485083\pi$$
0.0468457 + 0.998902i $$0.485083\pi$$
$$840$$ 0 0
$$841$$ −23950.5 −0.982019
$$842$$ − 36531.8i − 1.49521i
$$843$$ − 1727.25i − 0.0705688i
$$844$$ −87010.1 −3.54859
$$845$$ 0 0
$$846$$ 10405.3 0.422861
$$847$$ − 9796.97i − 0.397436i
$$848$$ − 33628.2i − 1.36179i
$$849$$ −27113.4 −1.09603
$$850$$ 0 0
$$851$$ 5093.00 0.205154
$$852$$ 94490.6i 3.79952i
$$853$$ − 13342.6i − 0.535570i −0.963479 0.267785i $$-0.913708\pi$$
0.963479 0.267785i $$-0.0862917\pi$$
$$854$$ 578.533 0.0231815
$$855$$ 0 0
$$856$$ 15650.6 0.624915
$$857$$ 18690.9i 0.745003i 0.928032 + 0.372502i $$0.121500\pi$$
−0.928032 + 0.372502i $$0.878500\pi$$
$$858$$ − 40390.8i − 1.60713i
$$859$$ −18318.9 −0.727628 −0.363814 0.931472i $$-0.618526\pi$$
−0.363814 + 0.931472i $$0.618526\pi$$
$$860$$ 0 0
$$861$$ 370.394 0.0146609
$$862$$ − 28071.3i − 1.10918i
$$863$$ − 38133.1i − 1.50413i −0.659087 0.752067i $$-0.729057\pi$$
0.659087 0.752067i $$-0.270943\pi$$
$$864$$ −92272.3 −3.63330
$$865$$ 0 0
$$866$$ −22967.3 −0.901223
$$867$$ 16426.0i 0.643432i
$$868$$ − 46930.8i − 1.83518i
$$869$$ 44820.3 1.74962
$$870$$ 0 0
$$871$$ −5104.66 −0.198582
$$872$$ 125022.i 4.85525i
$$873$$ 8938.33i 0.346525i
$$874$$ −11221.0 −0.434273
$$875$$ 0 0
$$876$$ 14738.3 0.568449
$$877$$ − 19707.5i − 0.758807i −0.925231 0.379404i $$-0.876129\pi$$
0.925231 0.379404i $$-0.123871\pi$$
$$878$$ 29421.5i 1.13090i
$$879$$ 34664.6 1.33016
$$880$$ 0 0
$$881$$ −14091.5 −0.538883 −0.269441 0.963017i $$-0.586839\pi$$
−0.269441 + 0.963017i $$0.586839\pi$$
$$882$$ − 1409.71i − 0.0538178i
$$883$$ − 3115.87i − 0.118751i −0.998236 0.0593757i $$-0.981089\pi$$
0.998236 0.0593757i $$-0.0189110\pi$$
$$884$$ −24323.5 −0.925438
$$885$$ 0 0
$$886$$ −62229.8 −2.35965
$$887$$ 38734.6i 1.46627i 0.680084 + 0.733134i $$0.261943\pi$$
−0.680084 + 0.733134i $$0.738057\pi$$
$$888$$ 66178.6i 2.50091i
$$889$$ 9761.57 0.368270
$$890$$ 0 0
$$891$$ 29121.3 1.09495
$$892$$ 15924.8i 0.597759i
$$893$$ 29015.6i 1.08731i
$$894$$ 19675.7 0.736077
$$895$$ 0 0
$$896$$ −24853.1 −0.926658
$$897$$ − 3688.10i − 0.137282i
$$898$$ 91225.8i 3.39003i
$$899$$ 6587.21 0.244378
$$900$$ 0 0
$$901$$ 5696.21 0.210619
$$902$$ − 3214.66i − 0.118666i
$$903$$ 1102.90i 0.0406449i
$$904$$ 133657. 4.91744
$$905$$ 0 0
$$906$$ 58480.2 2.14445
$$907$$ 19242.9i 0.704464i 0.935913 + 0.352232i $$0.114577\pi$$
−0.935913 + 0.352232i $$0.885423\pi$$
$$908$$ − 35501.7i − 1.29754i
$$909$$ −2307.03 −0.0841799
$$910$$ 0 0
$$911$$ 34613.3 1.25882 0.629412 0.777072i $$-0.283296\pi$$
0.629412 + 0.777072i $$0.283296\pi$$
$$912$$ − 82103.6i − 2.98105i
$$913$$ 34515.8i 1.25116i
$$914$$ −83100.1 −3.00734
$$915$$ 0 0
$$916$$ −141275. −5.09591
$$917$$ − 12336.9i − 0.444276i
$$918$$ − 30328.7i − 1.09041i
$$919$$ −25826.4 −0.927022 −0.463511 0.886091i $$-0.653411\pi$$
−0.463511 + 0.886091i $$0.653411\pi$$
$$920$$ 0 0
$$921$$ 3547.01 0.126903
$$922$$ 76007.0i 2.71492i
$$923$$ 29185.3i 1.04079i
$$924$$ 36305.8 1.29261
$$925$$ 0 0
$$926$$ −46893.8 −1.66417
$$927$$ − 1836.29i − 0.0650613i
$$928$$ − 12840.8i − 0.454224i
$$929$$ −19451.6 −0.686960 −0.343480 0.939160i $$-0.611606\pi$$
−0.343480 + 0.939160i $$0.611606\pi$$
$$930$$ 0 0
$$931$$ 3931.04 0.138383
$$932$$ − 9216.70i − 0.323930i
$$933$$ 35944.8i 1.26129i
$$934$$ −37979.1 −1.33053
$$935$$ 0 0
$$936$$ −11742.5 −0.410059
$$937$$ 34469.1i 1.20177i 0.799336 + 0.600884i $$0.205185\pi$$
−0.799336 + 0.600884i $$0.794815\pi$$
$$938$$ − 6310.63i − 0.219669i
$$939$$ 39844.1 1.38473
$$940$$ 0 0
$$941$$ 14156.4 0.490419 0.245209 0.969470i $$-0.421143\pi$$
0.245209 + 0.969470i $$0.421143\pi$$
$$942$$ − 25790.6i − 0.892042i
$$943$$ − 293.532i − 0.0101365i
$$944$$ 135375. 4.66746
$$945$$ 0 0
$$946$$ 9572.13 0.328982
$$947$$ − 38092.4i − 1.30711i −0.756877 0.653557i $$-0.773276\pi$$
0.756877 0.653557i $$-0.226724\pi$$
$$948$$ 85133.3i 2.91667i
$$949$$ 4552.22 0.155713
$$950$$ 0 0
$$951$$ 36234.8 1.23553
$$952$$ − 18783.3i − 0.639464i
$$953$$ − 5037.40i − 0.171225i −0.996329 0.0856126i $$-0.972715\pi$$
0.996329 0.0856126i $$-0.0272847\pi$$
$$954$$ 4402.30 0.149402
$$955$$ 0 0
$$956$$ 118940. 4.02384
$$957$$ 5095.88i 0.172128i
$$958$$ − 98185.1i − 3.31129i
$$959$$ −6460.64 −0.217544
$$960$$ 0 0
$$961$$ 69156.0 2.32137
$$962$$ 32723.0i 1.09671i
$$963$$ 1153.71i 0.0386060i
$$964$$ 134210. 4.48403
$$965$$ 0 0
$$966$$ 4559.41 0.151860
$$967$$ 11495.3i 0.382278i 0.981563 + 0.191139i $$0.0612181\pi$$
−0.981563 + 0.191139i $$0.938782\pi$$
$$968$$ − 100885.i − 3.34977i
$$969$$ 13907.3 0.461061
$$970$$ 0 0
$$971$$ 22352.7 0.738757 0.369379 0.929279i $$-0.379571\pi$$
0.369379 + 0.929279i $$0.379571\pi$$
$$972$$ − 31282.7i − 1.03230i
$$973$$ − 1372.27i − 0.0452138i
$$974$$ −89539.4 −2.94561
$$975$$ 0 0
$$976$$ 3354.69 0.110022
$$977$$ 14345.7i 0.469765i 0.972024 + 0.234882i $$0.0754705\pi$$
−0.972024 + 0.234882i $$0.924530\pi$$
$$978$$ − 34053.6i − 1.11341i
$$979$$ −2391.94 −0.0780867
$$980$$ 0 0
$$981$$ −9216.15 −0.299948
$$982$$ 1194.42i 0.0388141i
$$983$$ − 34460.9i − 1.11814i −0.829120 0.559070i $$-0.811158\pi$$
0.829120 0.559070i $$-0.188842\pi$$
$$984$$ 3814.17 0.123568
$$985$$ 0 0
$$986$$ 4220.61 0.136320
$$987$$ − 11789.9i − 0.380220i
$$988$$ − 52420.2i − 1.68796i
$$989$$ 874.036 0.0281019
$$990$$ 0 0
$$991$$ −35189.6 −1.12799 −0.563993 0.825780i $$-0.690735\pi$$
−0.563993 + 0.825780i $$0.690735\pi$$
$$992$$ − 192883.i − 6.17342i
$$993$$ − 22968.2i − 0.734011i
$$994$$ −36080.3 −1.15131
$$995$$ 0 0
$$996$$ −65560.6 −2.08571
$$997$$ − 50730.0i − 1.61147i −0.592277 0.805734i $$-0.701771\pi$$
0.592277 0.805734i $$-0.298229\pi$$
$$998$$ − 32155.2i − 1.01990i
$$999$$ −29666.8 −0.939554
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 175.4.b.c.99.4 4
5.2 odd 4 175.4.a.c.1.1 2
5.3 odd 4 35.4.a.b.1.2 2
5.4 even 2 inner 175.4.b.c.99.1 4
15.2 even 4 1575.4.a.z.1.2 2
15.8 even 4 315.4.a.f.1.1 2
20.3 even 4 560.4.a.r.1.2 2
35.3 even 12 245.4.e.i.226.1 4
35.13 even 4 245.4.a.k.1.2 2
35.18 odd 12 245.4.e.h.226.1 4
35.23 odd 12 245.4.e.h.116.1 4
35.27 even 4 1225.4.a.m.1.1 2
35.33 even 12 245.4.e.i.116.1 4
40.3 even 4 2240.4.a.bo.1.1 2
40.13 odd 4 2240.4.a.bn.1.2 2
105.83 odd 4 2205.4.a.u.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 5.3 odd 4
175.4.a.c.1.1 2 5.2 odd 4
175.4.b.c.99.1 4 5.4 even 2 inner
175.4.b.c.99.4 4 1.1 even 1 trivial
245.4.a.k.1.2 2 35.13 even 4
245.4.e.h.116.1 4 35.23 odd 12
245.4.e.h.226.1 4 35.18 odd 12
245.4.e.i.116.1 4 35.33 even 12
245.4.e.i.226.1 4 35.3 even 12
315.4.a.f.1.1 2 15.8 even 4
560.4.a.r.1.2 2 20.3 even 4
1225.4.a.m.1.1 2 35.27 even 4
1575.4.a.z.1.2 2 15.2 even 4
2205.4.a.u.1.1 2 105.83 odd 4
2240.4.a.bn.1.2 2 40.13 odd 4
2240.4.a.bo.1.1 2 40.3 even 4