# Properties

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

## $q$-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 $$-0.593567\pi$$
0.289735 0.957107i $$-0.406433\pi$$
$$3$$ − 4.65685i − 0.896212i −0.893980 0.448106i $$-0.852099\pi$$
0.893980 0.448106i $$-0.147901\pi$$
$$4$$ −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 $$0.106047\pi$$
−0.945015 + 0.327026i $$0.893953\pi$$
$$14$$ 37.8995 0.723505
$$15$$ 0 0
$$16$$ 219.765 3.43382
$$17$$ − 37.2254i − 0.531087i −0.964099 0.265544i $$-0.914449\pi$$
0.964099 0.265544i $$-0.0855514\pi$$
$$18$$ − 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 $$0.0373602\pi$$
−0.993120 + 0.117101i $$0.962640\pi$$
$$24$$ 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 $$-0.855693\pi$$
0.898983 0.437984i $$-0.144307\pi$$
$$38$$ 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 $$-0.980892\pi$$
0.998199 0.0599948i $$-0.0191084\pi$$
$$44$$ 1113.74 3.81598
$$45$$ 0 0
$$46$$ 139.868 0.448313
$$47$$ 361.676i 1.12247i 0.827658 + 0.561233i $$0.189673\pi$$
−0.827658 + 0.561233i $$0.810327\pi$$
$$48$$ − 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 $$0.0635390\pi$$
−0.980143 + 0.198291i $$0.936461\pi$$
$$54$$ −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 $$0.0485098\pi$$
−0.988410 + 0.151809i $$0.951490\pi$$
$$68$$ 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 $$-0.962019\pi$$
0.992890 0.119037i $$-0.0379807\pi$$
$$74$$ −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 $$0.143875\pi$$
−0.899577 + 0.436761i $$0.856125\pi$$
$$84$$ −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 $$-0.657286\pi$$
0.474265 0.880382i $$-0.342714\pi$$
$$98$$ 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 $$0.0528575\pi$$
−0.986244 + 0.165295i $$0.947142\pi$$
$$104$$ −2209.85 −2.08359
$$105$$ 0 0
$$106$$ 828.479 0.759142
$$107$$ − 217.119i − 0.196165i −0.995178 0.0980825i $$-0.968729\pi$$
0.995178 0.0980825i $$-0.0312709\pi$$
$$108$$ 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 $$-0.719352\pi$$
0.635855 0.771809i $$-0.280648\pi$$
$$114$$ 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 $$-0.838027\pi$$
0.873304 0.487176i $$-0.161973\pi$$
$$128$$ 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 $$0.0929185\pi$$
−0.957695 + 0.287784i $$0.907081\pi$$
$$138$$ − 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 $$-0.916281\pi$$
0.965612 0.259989i $$-0.0837188\pi$$
$$158$$ 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 $$-0.894802\pi$$
0.945883 0.324507i $$-0.105198\pi$$
$$164$$ −242.177 −0.115310
$$165$$ 0 0
$$166$$ 3576.24 1.67211
$$167$$ 1230.58i 0.570209i 0.958496 + 0.285105i $$0.0920284\pi$$
−0.958496 + 0.285105i $$0.907972\pi$$
$$168$$ 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 $$-0.815912\pi$$
0.837377 0.546626i $$-0.184088\pi$$
$$174$$ 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 $$-0.907218\pi$$
0.957819 0.287372i $$-0.0927816\pi$$
$$194$$ −9107.39 −3.37048
$$195$$ 0 0
$$196$$ 1044.37 0.380602
$$197$$ − 701.243i − 0.253612i −0.991928 0.126806i $$-0.959527\pi$$
0.991928 0.126806i $$-0.0404725\pi$$
$$198$$ 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 $$0.0357843\pi$$
−0.993688 + 0.112183i $$0.964216\pi$$
$$224$$ 4292.30 1.28032
$$225$$ 0 0
$$226$$ −10039.1 −2.95481
$$227$$ − 1665.67i − 0.487025i −0.969898 0.243513i $$-0.921700\pi$$
0.969898 0.243513i $$-0.0782997\pi$$
$$228$$ − 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 $$-0.980637\pi$$
0.998150 0.0607929i $$-0.0193629\pi$$
$$234$$ 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 $$0.403127\pi$$
−0.299659 + 0.954046i $$0.596873\pi$$
$$258$$ 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 $$0.209944\pi$$
−0.790262 + 0.612769i $$0.790056\pi$$
$$264$$ −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 $$-0.576039\pi$$
0.236619 0.971603i $$-0.423961\pi$$
$$278$$ − 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 $$-0.790575\pi$$
0.791261 0.611479i $$-0.209425\pi$$
$$284$$ 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 $$0.266170\pi$$
−0.670289 + 0.742100i $$0.733830\pi$$
$$294$$ 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 $$0.0225551\pi$$
−0.997491 + 0.0707998i $$0.977445\pi$$
$$308$$ 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 $$0.281017\pi$$
−0.634959 + 0.772546i $$0.718983\pi$$
$$314$$ −5538.21 −0.995348
$$315$$ 0 0
$$316$$ 18281.3 3.25444
$$317$$ 7780.95i 1.37862i 0.724468 + 0.689309i $$0.242086\pi$$
−0.724468 + 0.689309i $$0.757914\pi$$
$$318$$ − 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 $$0.195207\pi$$
−0.817775 + 0.575538i $$0.804793\pi$$
$$338$$ − 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 $$-0.735775\pi$$
0.674811 0.737991i $$-0.264225\pi$$
$$348$$ − 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 $$0.143998\pi$$
−0.899408 + 0.437110i $$0.856002\pi$$
$$354$$ −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 $$0.175685\pi$$
−0.851514 + 0.524332i $$0.824315\pi$$
$$368$$ 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 $$0.147679\pi$$
−0.894294 + 0.447480i $$0.852321\pi$$
$$374$$ 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 $$-0.852945\pi$$
0.895169 0.445727i $$-0.147055\pi$$
$$384$$ 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 $$-0.913469\pi$$
0.963277 0.268510i $$-0.0865314\pi$$
$$398$$ 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 $$-0.924359\pi$$
0.971898 0.235403i $$-0.0756410\pi$$
$$434$$ −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 $$-0.788611\pi$$
0.787472 0.616350i $$-0.211389\pi$$
$$444$$ 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 $$-0.712391\pi$$
0.618826 0.785528i $$-0.287609\pi$$
$$458$$ − 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 $$-0.856858\pi$$
0.900581 0.434689i $$-0.143142\pi$$
$$464$$ −4602.12 −0.460448
$$465$$ 0 0
$$466$$ −2341.27 −0.232741
$$467$$ − 7014.71i − 0.695079i −0.937665 0.347539i $$-0.887017\pi$$
0.937665 0.347539i $$-0.112983\pi$$
$$468$$ − 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 $$-0.720553\pi$$
0.638761 0.769405i $$-0.279447\pi$$
$$488$$ 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 $$-0.828035\pi$$
0.857584 0.514345i $$-0.171965\pi$$
$$504$$ −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 $$-0.754285\pi$$
0.716561 0.697524i $$-0.245715\pi$$
$$524$$ −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 $$-0.859993\pi$$
0.904818 0.425799i $$-0.140007\pi$$
$$548$$ − 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 $$0.0968153\pi$$
−0.954101 + 0.299486i $$0.903185\pi$$
$$558$$ 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 $$0.303183\pi$$
−0.579666 + 0.814854i $$0.696817\pi$$
$$564$$ −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 $$-0.945379\pi$$
0.985313 0.170757i $$-0.0546212\pi$$
$$578$$ − 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 $$0.0959143\pi$$
−0.954944 + 0.296785i $$0.904086\pi$$
$$588$$ − 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 $$0.227665\pi$$
−0.754943 + 0.655791i $$0.772335\pi$$
$$594$$ 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 $$0.335271\pi$$
−0.494719 + 0.869053i $$0.664729\pi$$
$$608$$ 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 $$0.00697834\pi$$
−0.999760 + 0.0219213i $$0.993022\pi$$
$$614$$ 4123.87 0.271052
$$615$$ 0 0
$$616$$ 26366.9 1.72460
$$617$$ − 18401.3i − 1.20066i −0.799752 0.600330i $$-0.795036\pi$$
0.799752 0.600330i $$-0.204964\pi$$
$$618$$ − 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 $$-0.851189\pi$$
0.892696 0.450658i $$-0.148811\pi$$
$$644$$ 3854.25 0.235837
$$645$$ 0 0
$$646$$ 16169.1 0.984777
$$647$$ 12694.8i 0.771383i 0.922628 + 0.385691i $$0.126037\pi$$
−0.922628 + 0.385691i $$0.873963\pi$$
$$648$$ − 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 $$-0.878973\pi$$
0.928583 0.371124i $$-0.121027\pi$$
$$654$$ 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 $$0.270036\pi$$
−0.661226 + 0.750186i $$0.729964\pi$$
$$674$$ 38555.3 2.20341
$$675$$ 0 0
$$676$$ −26794.7 −1.52450
$$677$$ 4228.44i 0.240047i 0.992771 + 0.120024i $$0.0382970\pi$$
−0.992771 + 0.120024i $$0.961703\pi$$
$$678$$ 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 $$0.280261\pi$$
−0.636792 + 0.771036i $$0.719739\pi$$
$$684$$ 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 $$0.0298309\pi$$
−0.995612 + 0.0935794i $$0.970169\pi$$
$$728$$ − 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 $$-0.876808\pi$$
0.926039 0.377428i $$-0.123192\pi$$
$$734$$ 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 $$0.221357\pi$$
−0.767788 + 0.640704i $$0.778643\pi$$
$$744$$ −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 $$0.0195316\pi$$
−0.998118 + 0.0613220i $$0.980468\pi$$
$$758$$ − 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 $$0.0722652\pi$$
−0.974340 + 0.225083i $$0.927735\pi$$
$$774$$ −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 $$0.157295\pi$$
−0.880370 + 0.474288i $$0.842705\pi$$
$$788$$ 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 $$-0.824885\pi$$
0.852451 0.522807i $$-0.175115\pi$$
$$798$$ 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 $$0.0960923\pi$$
−0.954778 + 0.297318i $$0.903908\pi$$
$$824$$ −24910.3 −1.05315
$$825$$ 0 0
$$826$$ 23346.1 0.983431
$$827$$ − 15127.4i − 0.636073i −0.948079 0.318036i $$-0.896977\pi$$
0.948079 0.318036i $$-0.103023\pi$$
$$828$$ − 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 $$0.0862917\pi$$
−0.963479 + 0.267785i $$0.913708\pi$$
$$854$$ 578.533 0.0231815
$$855$$ 0 0
$$856$$ 15650.6 0.624915
$$857$$ − 18690.9i − 0.745003i −0.928032 0.372502i $$-0.878500\pi$$
0.928032 0.372502i $$-0.121500\pi$$
$$858$$ 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 $$0.270943\pi$$
−0.659087 + 0.752067i $$0.729057\pi$$
$$864$$ −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 $$0.123871\pi$$
−0.925231 + 0.379404i $$0.876129\pi$$
$$878$$ − 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 $$0.0189110\pi$$
−0.998236 + 0.0593757i $$0.981089\pi$$
$$884$$ −24323.5 −0.925438
$$885$$ 0 0
$$886$$ −62229.8 −2.35965
$$887$$ − 38734.6i − 1.46627i −0.680084 0.733134i $$-0.738057\pi$$
0.680084 0.733134i $$-0.261943\pi$$
$$888$$ − 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 $$-0.885423\pi$$
0.935913 0.352232i $$-0.114577\pi$$
$$908$$ 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 $$-0.794815\pi$$
0.799336 0.600884i $$-0.205185\pi$$
$$938$$ 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 $$0.226724\pi$$
−0.756877 + 0.653557i $$0.773276\pi$$
$$948$$ − 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 $$0.0272847\pi$$
−0.996329 + 0.0856126i $$0.972715\pi$$
$$954$$ 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 $$-0.938782\pi$$
0.981563 0.191139i $$-0.0612181\pi$$
$$968$$ 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 $$-0.924530\pi$$
0.972024 0.234882i $$-0.0754705\pi$$
$$978$$ 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 $$0.188842\pi$$
−0.829120 + 0.559070i $$0.811158\pi$$
$$984$$ 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 $$0.298229\pi$$
−0.592277 + 0.805734i $$0.701771\pi$$
$$998$$ 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.99.1 4
5.2 odd 4 35.4.a.b.1.2 2
5.3 odd 4 175.4.a.c.1.1 2
5.4 even 2 inner 175.4.b.c.99.4 4
15.2 even 4 315.4.a.f.1.1 2
15.8 even 4 1575.4.a.z.1.2 2
20.7 even 4 560.4.a.r.1.2 2
35.2 odd 12 245.4.e.h.116.1 4
35.12 even 12 245.4.e.i.116.1 4
35.13 even 4 1225.4.a.m.1.1 2
35.17 even 12 245.4.e.i.226.1 4
35.27 even 4 245.4.a.k.1.2 2
35.32 odd 12 245.4.e.h.226.1 4
40.27 even 4 2240.4.a.bo.1.1 2
40.37 odd 4 2240.4.a.bn.1.2 2
105.62 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 5.2 odd 4
175.4.a.c.1.1 2 5.3 odd 4
175.4.b.c.99.1 4 1.1 even 1 trivial
175.4.b.c.99.4 4 5.4 even 2 inner
245.4.a.k.1.2 2 35.27 even 4
245.4.e.h.116.1 4 35.2 odd 12
245.4.e.h.226.1 4 35.32 odd 12
245.4.e.i.116.1 4 35.12 even 12
245.4.e.i.226.1 4 35.17 even 12
315.4.a.f.1.1 2 15.2 even 4
560.4.a.r.1.2 2 20.7 even 4
1225.4.a.m.1.1 2 35.13 even 4
1575.4.a.z.1.2 2 15.8 even 4
2205.4.a.u.1.1 2 105.62 odd 4
2240.4.a.bn.1.2 2 40.37 odd 4
2240.4.a.bo.1.1 2 40.27 even 4