# Properties

 Label 75.22.b.a.49.2 Level $75$ Weight $22$ Character 75.49 Analytic conductor $209.608$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$209.608008215$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 75.49 Dual form 75.22.b.a.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2844.00i q^{2} -59049.0i q^{3} -5.99118e6 q^{4} +1.67935e8 q^{6} -3.63304e8i q^{7} -1.10746e10i q^{8} -3.48678e9 q^{9} +O(q^{10})$$ $$q+2844.00i q^{2} -59049.0i q^{3} -5.99118e6 q^{4} +1.67935e8 q^{6} -3.63304e8i q^{7} -1.10746e10i q^{8} -3.48678e9 q^{9} +1.45818e10 q^{11} +3.53773e11i q^{12} +1.13351e11i q^{13} +1.03324e12 q^{14} +1.89318e13 q^{16} +8.58939e12i q^{17} -9.91641e12i q^{18} +2.92029e13 q^{19} -2.14527e13 q^{21} +4.14707e13i q^{22} -1.55899e14i q^{23} -6.53946e14 q^{24} -3.22370e14 q^{26} +2.05891e14i q^{27} +2.17662e15i q^{28} -2.40079e15 q^{29} +2.23982e15 q^{31} +3.06169e16i q^{32} -8.61043e14i q^{33} -2.44282e16 q^{34} +2.08900e16 q^{36} +3.07851e16i q^{37} +8.30532e16i q^{38} +6.69325e15 q^{39} -1.03208e17 q^{41} -6.10116e16i q^{42} -1.65557e17i q^{43} -8.73624e16 q^{44} +4.43377e17 q^{46} +6.65872e16i q^{47} -1.11790e18i q^{48} +4.26556e17 q^{49} +5.07195e17 q^{51} -6.79105e17i q^{52} +4.35423e17i q^{53} -5.85554e17 q^{54} -4.02346e18 q^{56} -1.72440e18i q^{57} -6.82784e18i q^{58} -5.53437e18 q^{59} -7.17621e18 q^{61} +6.37005e18i q^{62} +1.26676e18i q^{63} -4.73716e19 q^{64} +2.44881e18 q^{66} +1.57554e19i q^{67} -5.14606e19i q^{68} -9.20569e18 q^{69} +2.64579e19 q^{71} +3.86148e19i q^{72} +1.34712e19i q^{73} -8.75527e19 q^{74} -1.74960e20 q^{76} -5.29764e18i q^{77} +1.90356e19i q^{78} +1.68861e19 q^{79} +1.21577e19 q^{81} -2.93522e20i q^{82} -1.70688e20i q^{83} +1.28527e20 q^{84} +4.70845e20 q^{86} +1.41764e20i q^{87} -1.61488e20i q^{88} +3.12592e20 q^{89} +4.11808e19 q^{91} +9.34021e20i q^{92} -1.32259e20i q^{93} -1.89374e20 q^{94} +1.80790e21 q^{96} -9.49015e20i q^{97} +1.21313e21i q^{98} -5.08437e19 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 11982368 q^{4} + 335870712 q^{6} - 6973568802 q^{9}+O(q^{10})$$ 2 * q - 11982368 * q^4 + 335870712 * q^6 - 6973568802 * q^9 $$2 q - 11982368 q^{4} + 335870712 q^{6} - 6973568802 q^{9} + 29163666312 q^{11} + 2066472696960 q^{14} + 37863631405568 q^{16} + 58405878547592 q^{19} - 42905466344160 q^{21} - 13\!\cdots\!84 q^{24}+ \cdots - 10\!\cdots\!12 q^{99}+O(q^{100})$$ 2 * q - 11982368 * q^4 + 335870712 * q^6 - 6973568802 * q^9 + 29163666312 * q^11 + 2066472696960 * q^14 + 37863631405568 * q^16 + 58405878547592 * q^19 - 42905466344160 * q^21 - 1307891300390784 * q^24 - 644739297512976 * q^26 - 4801577414181516 * q^29 + 4479641353894000 * q^31 - 48856448033321616 * q^34 + 41779933829441568 * q^36 + 13386501680324796 * q^39 - 206415142082562060 * q^41 - 174724890989793408 * q^44 + 886754734659944640 * q^46 + 853112251591835214 * q^49 + 1014389732742478236 * q^51 - 1171108759354363512 * q^54 - 8046910809088542720 * q^56 - 11068731596518162632 * q^59 - 14352410329445922404 * q^61 - 94743180552322555904 * q^64 + 4897610684370927072 * q^66 - 18411385487670559440 * q^69 + 52915709748518752464 * q^71 - 175105474652204086320 * q^74 - 349920365060276428928 * q^76 + 33772250171051973680 * q^79 + 24315330918113857602 * q^81 + 257054543473669885440 * q^84 + 941689755272272449312 * q^86 + 625184973879174087372 * q^89 + 82361573194272303680 * q^91 - 378748085896203496704 * q^94 + 3615795563149709015040 * q^96 - 101687416772650799112 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2844.00i 1.96388i 0.189196 + 0.981939i $$0.439412\pi$$
−0.189196 + 0.981939i $$0.560588\pi$$
$$3$$ − 59049.0i − 0.577350i
$$4$$ −5.99118e6 −2.85682
$$5$$ 0 0
$$6$$ 1.67935e8 1.13385
$$7$$ − 3.63304e8i − 0.486117i −0.970012 0.243058i $$-0.921849\pi$$
0.970012 0.243058i $$-0.0781507\pi$$
$$8$$ − 1.10746e10i − 3.64657i
$$9$$ −3.48678e9 −0.333333
$$10$$ 0 0
$$11$$ 1.45818e10 0.169508 0.0847538 0.996402i $$-0.472990\pi$$
0.0847538 + 0.996402i $$0.472990\pi$$
$$12$$ 3.53773e11i 1.64939i
$$13$$ 1.13351e11i 0.228044i 0.993478 + 0.114022i $$0.0363735\pi$$
−0.993478 + 0.114022i $$0.963627\pi$$
$$14$$ 1.03324e12 0.954674
$$15$$ 0 0
$$16$$ 1.89318e13 4.30460
$$17$$ 8.58939e12i 1.03335i 0.856181 + 0.516676i $$0.172831\pi$$
−0.856181 + 0.516676i $$0.827169\pi$$
$$18$$ − 9.91641e12i − 0.654626i
$$19$$ 2.92029e13 1.09273 0.546366 0.837546i $$-0.316011\pi$$
0.546366 + 0.837546i $$0.316011\pi$$
$$20$$ 0 0
$$21$$ −2.14527e13 −0.280660
$$22$$ 4.14707e13i 0.332892i
$$23$$ − 1.55899e14i − 0.784696i −0.919817 0.392348i $$-0.871663\pi$$
0.919817 0.392348i $$-0.128337\pi$$
$$24$$ −6.53946e14 −2.10535
$$25$$ 0 0
$$26$$ −3.22370e14 −0.447851
$$27$$ 2.05891e14i 0.192450i
$$28$$ 2.17662e15i 1.38875i
$$29$$ −2.40079e15 −1.05968 −0.529840 0.848097i $$-0.677748\pi$$
−0.529840 + 0.848097i $$0.677748\pi$$
$$30$$ 0 0
$$31$$ 2.23982e15 0.490812 0.245406 0.969420i $$-0.421079\pi$$
0.245406 + 0.969420i $$0.421079\pi$$
$$32$$ 3.06169e16i 4.80714i
$$33$$ − 8.61043e14i − 0.0978652i
$$34$$ −2.44282e16 −2.02938
$$35$$ 0 0
$$36$$ 2.08900e16 0.952273
$$37$$ 3.07851e16i 1.05250i 0.850330 + 0.526250i $$0.176402\pi$$
−0.850330 + 0.526250i $$0.823598\pi$$
$$38$$ 8.30532e16i 2.14599i
$$39$$ 6.69325e15 0.131661
$$40$$ 0 0
$$41$$ −1.03208e17 −1.20083 −0.600414 0.799689i $$-0.704998\pi$$
−0.600414 + 0.799689i $$0.704998\pi$$
$$42$$ − 6.10116e16i − 0.551182i
$$43$$ − 1.65557e17i − 1.16823i −0.811670 0.584117i $$-0.801441\pi$$
0.811670 0.584117i $$-0.198559\pi$$
$$44$$ −8.73624e16 −0.484252
$$45$$ 0 0
$$46$$ 4.43377e17 1.54105
$$47$$ 6.65872e16i 0.184656i 0.995729 + 0.0923280i $$0.0294308\pi$$
−0.995729 + 0.0923280i $$0.970569\pi$$
$$48$$ − 1.11790e18i − 2.48526i
$$49$$ 4.26556e17 0.763690
$$50$$ 0 0
$$51$$ 5.07195e17 0.596607
$$52$$ − 6.79105e17i − 0.651482i
$$53$$ 4.35423e17i 0.341991i 0.985272 + 0.170995i $$0.0546983\pi$$
−0.985272 + 0.170995i $$0.945302\pi$$
$$54$$ −5.85554e17 −0.377949
$$55$$ 0 0
$$56$$ −4.02346e18 −1.77266
$$57$$ − 1.72440e18i − 0.630889i
$$58$$ − 6.82784e18i − 2.08108i
$$59$$ −5.53437e18 −1.40968 −0.704842 0.709364i $$-0.748982\pi$$
−0.704842 + 0.709364i $$0.748982\pi$$
$$60$$ 0 0
$$61$$ −7.17621e18 −1.28805 −0.644023 0.765006i $$-0.722736\pi$$
−0.644023 + 0.765006i $$0.722736\pi$$
$$62$$ 6.37005e18i 0.963896i
$$63$$ 1.26676e18i 0.162039i
$$64$$ −4.73716e19 −5.13604
$$65$$ 0 0
$$66$$ 2.44881e18 0.192195
$$67$$ 1.57554e19i 1.05595i 0.849258 + 0.527977i $$0.177049\pi$$
−0.849258 + 0.527977i $$0.822951\pi$$
$$68$$ − 5.14606e19i − 2.95210i
$$69$$ −9.20569e18 −0.453045
$$70$$ 0 0
$$71$$ 2.64579e19 0.964588 0.482294 0.876009i $$-0.339804\pi$$
0.482294 + 0.876009i $$0.339804\pi$$
$$72$$ 3.86148e19i 1.21552i
$$73$$ 1.34712e19i 0.366875i 0.983031 + 0.183437i $$0.0587225\pi$$
−0.983031 + 0.183437i $$0.941278\pi$$
$$74$$ −8.75527e19 −2.06698
$$75$$ 0 0
$$76$$ −1.74960e20 −3.12174
$$77$$ − 5.29764e18i − 0.0824005i
$$78$$ 1.90356e19i 0.258567i
$$79$$ 1.68861e19 0.200653 0.100326 0.994955i $$-0.468011\pi$$
0.100326 + 0.994955i $$0.468011\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ − 2.93522e20i − 2.35828i
$$83$$ − 1.70688e20i − 1.20749i −0.797178 0.603744i $$-0.793675\pi$$
0.797178 0.603744i $$-0.206325\pi$$
$$84$$ 1.28527e20 0.801794
$$85$$ 0 0
$$86$$ 4.70845e20 2.29427
$$87$$ 1.41764e20i 0.611807i
$$88$$ − 1.61488e20i − 0.618121i
$$89$$ 3.12592e20 1.06263 0.531317 0.847173i $$-0.321697\pi$$
0.531317 + 0.847173i $$0.321697\pi$$
$$90$$ 0 0
$$91$$ 4.11808e19 0.110856
$$92$$ 9.34021e20i 2.24174i
$$93$$ − 1.32259e20i − 0.283371i
$$94$$ −1.89374e20 −0.362642
$$95$$ 0 0
$$96$$ 1.80790e21 2.77540
$$97$$ − 9.49015e20i − 1.30668i −0.757064 0.653341i $$-0.773367\pi$$
0.757064 0.653341i $$-0.226633\pi$$
$$98$$ 1.21313e21i 1.49980i
$$99$$ −5.08437e19 −0.0565025
$$100$$ 0 0
$$101$$ 1.44798e20 0.130433 0.0652166 0.997871i $$-0.479226\pi$$
0.0652166 + 0.997871i $$0.479226\pi$$
$$102$$ 1.44246e21i 1.17166i
$$103$$ 2.19627e21i 1.61025i 0.593102 + 0.805127i $$0.297903\pi$$
−0.593102 + 0.805127i $$0.702097\pi$$
$$104$$ 1.25532e21 0.831579
$$105$$ 0 0
$$106$$ −1.23834e21 −0.671629
$$107$$ 1.63087e20i 0.0801473i 0.999197 + 0.0400737i $$0.0127593\pi$$
−0.999197 + 0.0400737i $$0.987241\pi$$
$$108$$ − 1.23353e21i − 0.549795i
$$109$$ −2.24852e20 −0.0909743 −0.0454871 0.998965i $$-0.514484\pi$$
−0.0454871 + 0.998965i $$0.514484\pi$$
$$110$$ 0 0
$$111$$ 1.81783e21 0.607661
$$112$$ − 6.87800e21i − 2.09254i
$$113$$ − 4.24118e21i − 1.17534i −0.809101 0.587670i $$-0.800045\pi$$
0.809101 0.587670i $$-0.199955\pi$$
$$114$$ 4.90421e21 1.23899
$$115$$ 0 0
$$116$$ 1.43836e22 3.02732
$$117$$ − 3.95230e20i − 0.0760148i
$$118$$ − 1.57397e22i − 2.76845i
$$119$$ 3.12056e21 0.502330
$$120$$ 0 0
$$121$$ −7.18762e21 −0.971267
$$122$$ − 2.04091e22i − 2.52957i
$$123$$ 6.09430e21i 0.693299i
$$124$$ −1.34192e22 −1.40216
$$125$$ 0 0
$$126$$ −3.60267e21 −0.318225
$$127$$ − 1.66312e21i − 0.135202i −0.997712 0.0676012i $$-0.978465\pi$$
0.997712 0.0676012i $$-0.0215346\pi$$
$$128$$ − 7.05165e22i − 5.27942i
$$129$$ −9.77599e21 −0.674480
$$130$$ 0 0
$$131$$ 6.40663e21 0.376081 0.188040 0.982161i $$-0.439786\pi$$
0.188040 + 0.982161i $$0.439786\pi$$
$$132$$ 5.15867e21i 0.279583i
$$133$$ − 1.06095e22i − 0.531196i
$$134$$ −4.48085e22 −2.07377
$$135$$ 0 0
$$136$$ 9.51243e22 3.76819
$$137$$ 1.98314e22i 0.727423i 0.931512 + 0.363711i $$0.118491\pi$$
−0.931512 + 0.363711i $$0.881509\pi$$
$$138$$ − 2.61810e22i − 0.889725i
$$139$$ 5.20143e21 0.163858 0.0819290 0.996638i $$-0.473892\pi$$
0.0819290 + 0.996638i $$0.473892\pi$$
$$140$$ 0 0
$$141$$ 3.93191e21 0.106611
$$142$$ 7.52461e22i 1.89433i
$$143$$ 1.65286e21i 0.0386552i
$$144$$ −6.60112e22 −1.43487
$$145$$ 0 0
$$146$$ −3.83122e22 −0.720498
$$147$$ − 2.51877e22i − 0.440917i
$$148$$ − 1.84439e23i − 3.00680i
$$149$$ 7.47631e22 1.13562 0.567808 0.823161i $$-0.307792\pi$$
0.567808 + 0.823161i $$0.307792\pi$$
$$150$$ 0 0
$$151$$ 1.11044e23 1.46635 0.733174 0.680042i $$-0.238038\pi$$
0.733174 + 0.680042i $$0.238038\pi$$
$$152$$ − 3.23412e23i − 3.98472i
$$153$$ − 2.99493e22i − 0.344451i
$$154$$ 1.50665e22 0.161824
$$155$$ 0 0
$$156$$ −4.01005e22 −0.376133
$$157$$ − 4.36563e22i − 0.382913i −0.981501 0.191457i $$-0.938679\pi$$
0.981501 0.191457i $$-0.0613211\pi$$
$$158$$ 4.80241e22i 0.394058i
$$159$$ 2.57113e22 0.197449
$$160$$ 0 0
$$161$$ −5.66388e22 −0.381454
$$162$$ 3.45764e22i 0.218209i
$$163$$ 2.85661e23i 1.68998i 0.534783 + 0.844989i $$0.320393\pi$$
−0.534783 + 0.844989i $$0.679607\pi$$
$$164$$ 6.18336e23 3.43055
$$165$$ 0 0
$$166$$ 4.85437e23 2.37136
$$167$$ − 2.66950e23i − 1.22435i −0.790720 0.612177i $$-0.790294\pi$$
0.790720 0.612177i $$-0.209706\pi$$
$$168$$ 2.37581e23i 1.02344i
$$169$$ 2.34216e23 0.947996
$$170$$ 0 0
$$171$$ −1.01824e23 −0.364244
$$172$$ 9.91884e23i 3.33743i
$$173$$ − 2.28496e23i − 0.723425i −0.932290 0.361713i $$-0.882192\pi$$
0.932290 0.361713i $$-0.117808\pi$$
$$174$$ −4.03177e23 −1.20151
$$175$$ 0 0
$$176$$ 2.76061e23 0.729661
$$177$$ 3.26799e23i 0.813881i
$$178$$ 8.89013e23i 2.08688i
$$179$$ 1.29151e22 0.0285852 0.0142926 0.999898i $$-0.495450\pi$$
0.0142926 + 0.999898i $$0.495450\pi$$
$$180$$ 0 0
$$181$$ −8.75338e23 −1.72405 −0.862026 0.506863i $$-0.830805\pi$$
−0.862026 + 0.506863i $$0.830805\pi$$
$$182$$ 1.17118e23i 0.217708i
$$183$$ 4.23748e23i 0.743654i
$$184$$ −1.72653e24 −2.86145
$$185$$ 0 0
$$186$$ 3.76145e23 0.556506
$$187$$ 1.25249e23i 0.175161i
$$188$$ − 3.98936e23i − 0.527529i
$$189$$ 7.48011e22 0.0935532
$$190$$ 0 0
$$191$$ 1.33961e24 1.50013 0.750066 0.661363i $$-0.230022\pi$$
0.750066 + 0.661363i $$0.230022\pi$$
$$192$$ 2.79725e24i 2.96529i
$$193$$ − 2.13970e23i − 0.214783i −0.994217 0.107392i $$-0.965750\pi$$
0.994217 0.107392i $$-0.0342499\pi$$
$$194$$ 2.69900e24 2.56616
$$195$$ 0 0
$$196$$ −2.55558e24 −2.18173
$$197$$ − 1.42895e24i − 1.15644i −0.815881 0.578220i $$-0.803748\pi$$
0.815881 0.578220i $$-0.196252\pi$$
$$198$$ − 1.44600e23i − 0.110964i
$$199$$ −7.45856e23 −0.542872 −0.271436 0.962456i $$-0.587499\pi$$
−0.271436 + 0.962456i $$0.587499\pi$$
$$200$$ 0 0
$$201$$ 9.30344e23 0.609656
$$202$$ 4.11805e23i 0.256155i
$$203$$ 8.72216e23i 0.515129i
$$204$$ −3.03870e24 −1.70440
$$205$$ 0 0
$$206$$ −6.24619e24 −3.16234
$$207$$ 5.43587e23i 0.261565i
$$208$$ 2.14594e24i 0.981639i
$$209$$ 4.25832e23 0.185226
$$210$$ 0 0
$$211$$ −2.65090e24 −1.04334 −0.521671 0.853147i $$-0.674691\pi$$
−0.521671 + 0.853147i $$0.674691\pi$$
$$212$$ − 2.60870e24i − 0.977006i
$$213$$ − 1.56231e24i − 0.556905i
$$214$$ −4.63819e23 −0.157400
$$215$$ 0 0
$$216$$ 2.28017e24 0.701782
$$217$$ − 8.13736e23i − 0.238592i
$$218$$ − 6.39479e23i − 0.178662i
$$219$$ 7.95464e23 0.211815
$$220$$ 0 0
$$221$$ −9.73614e23 −0.235650
$$222$$ 5.16990e24i 1.19337i
$$223$$ 3.97174e24i 0.874539i 0.899330 + 0.437270i $$0.144054\pi$$
−0.899330 + 0.437270i $$0.855946\pi$$
$$224$$ 1.11232e25 2.33683
$$225$$ 0 0
$$226$$ 1.20619e25 2.30823
$$227$$ 2.14690e24i 0.392229i 0.980581 + 0.196114i $$0.0628324\pi$$
−0.980581 + 0.196114i $$0.937168\pi$$
$$228$$ 1.03312e25i 1.80234i
$$229$$ −3.66024e24 −0.609869 −0.304934 0.952373i $$-0.598635\pi$$
−0.304934 + 0.952373i $$0.598635\pi$$
$$230$$ 0 0
$$231$$ −3.12820e23 −0.0475739
$$232$$ 2.65878e25i 3.86420i
$$233$$ − 2.90921e24i − 0.404145i −0.979371 0.202073i $$-0.935232\pi$$
0.979371 0.202073i $$-0.0647677\pi$$
$$234$$ 1.12403e24 0.149284
$$235$$ 0 0
$$236$$ 3.31574e25 4.02721
$$237$$ − 9.97109e23i − 0.115847i
$$238$$ 8.87487e24i 0.986516i
$$239$$ 4.80224e24 0.510818 0.255409 0.966833i $$-0.417790\pi$$
0.255409 + 0.966833i $$0.417790\pi$$
$$240$$ 0 0
$$241$$ 7.86263e24 0.766282 0.383141 0.923690i $$-0.374842\pi$$
0.383141 + 0.923690i $$0.374842\pi$$
$$242$$ − 2.04416e25i − 1.90745i
$$243$$ − 7.17898e23i − 0.0641500i
$$244$$ 4.29940e25 3.67972
$$245$$ 0 0
$$246$$ −1.73322e25 −1.36155
$$247$$ 3.31018e24i 0.249191i
$$248$$ − 2.48052e25i − 1.78978i
$$249$$ −1.00790e25 −0.697144
$$250$$ 0 0
$$251$$ −1.99525e25 −1.26889 −0.634443 0.772969i $$-0.718771\pi$$
−0.634443 + 0.772969i $$0.718771\pi$$
$$252$$ − 7.58941e24i − 0.462916i
$$253$$ − 2.27330e24i − 0.133012i
$$254$$ 4.72991e24 0.265521
$$255$$ 0 0
$$256$$ 1.01203e26 5.23210
$$257$$ − 1.33580e25i − 0.662894i −0.943474 0.331447i $$-0.892463\pi$$
0.943474 0.331447i $$-0.107537\pi$$
$$258$$ − 2.78029e25i − 1.32460i
$$259$$ 1.11843e25 0.511638
$$260$$ 0 0
$$261$$ 8.37103e24 0.353227
$$262$$ 1.82204e25i 0.738576i
$$263$$ − 5.83637e24i − 0.227304i −0.993521 0.113652i $$-0.963745\pi$$
0.993521 0.113652i $$-0.0362549\pi$$
$$264$$ −9.53573e24 −0.356872
$$265$$ 0 0
$$266$$ 3.01735e25 1.04320
$$267$$ − 1.84583e25i − 0.613512i
$$268$$ − 9.43938e25i − 3.01667i
$$269$$ −5.35297e25 −1.64511 −0.822557 0.568683i $$-0.807453\pi$$
−0.822557 + 0.568683i $$0.807453\pi$$
$$270$$ 0 0
$$271$$ −1.04403e25 −0.296849 −0.148425 0.988924i $$-0.547420\pi$$
−0.148425 + 0.988924i $$0.547420\pi$$
$$272$$ 1.62613e26i 4.44817i
$$273$$ − 2.43168e24i − 0.0640029i
$$274$$ −5.64004e25 −1.42857
$$275$$ 0 0
$$276$$ 5.51530e25 1.29427
$$277$$ − 3.14884e25i − 0.711399i −0.934600 0.355699i $$-0.884243\pi$$
0.934600 0.355699i $$-0.115757\pi$$
$$278$$ 1.47929e25i 0.321797i
$$279$$ −7.80977e24 −0.163604
$$280$$ 0 0
$$281$$ −1.15887e25 −0.225225 −0.112613 0.993639i $$-0.535922\pi$$
−0.112613 + 0.993639i $$0.535922\pi$$
$$282$$ 1.11823e25i 0.209371i
$$283$$ − 4.80399e25i − 0.866652i −0.901237 0.433326i $$-0.857340\pi$$
0.901237 0.433326i $$-0.142660\pi$$
$$284$$ −1.58514e26 −2.75565
$$285$$ 0 0
$$286$$ −4.70074e24 −0.0759142
$$287$$ 3.74957e25i 0.583743i
$$288$$ − 1.06755e26i − 1.60238i
$$289$$ −4.68568e24 −0.0678180
$$290$$ 0 0
$$291$$ −5.60384e25 −0.754413
$$292$$ − 8.07087e25i − 1.04809i
$$293$$ − 7.96714e25i − 0.998142i −0.866561 0.499071i $$-0.833675\pi$$
0.866561 0.499071i $$-0.166325\pi$$
$$294$$ 7.16339e25 0.865907
$$295$$ 0 0
$$296$$ 3.40933e26 3.83801
$$297$$ 3.00227e24i 0.0326217i
$$298$$ 2.12626e26i 2.23021i
$$299$$ 1.76713e25 0.178946
$$300$$ 0 0
$$301$$ −6.01476e25 −0.567898
$$302$$ 3.15809e26i 2.87973i
$$303$$ − 8.55018e24i − 0.0753056i
$$304$$ 5.52865e26 4.70377
$$305$$ 0 0
$$306$$ 8.51760e25 0.676460
$$307$$ − 1.51498e26i − 1.16266i −0.813666 0.581332i $$-0.802532\pi$$
0.813666 0.581332i $$-0.197468\pi$$
$$308$$ 3.17391e25i 0.235403i
$$309$$ 1.29687e26 0.929681
$$310$$ 0 0
$$311$$ 1.41406e26 0.947292 0.473646 0.880715i $$-0.342938\pi$$
0.473646 + 0.880715i $$0.342938\pi$$
$$312$$ − 7.41253e25i − 0.480112i
$$313$$ 4.99598e25i 0.312899i 0.987686 + 0.156450i $$0.0500049\pi$$
−0.987686 + 0.156450i $$0.949995\pi$$
$$314$$ 1.24158e26 0.751995
$$315$$ 0 0
$$316$$ −1.01168e26 −0.573229
$$317$$ − 1.81535e26i − 0.995033i −0.867454 0.497517i $$-0.834245\pi$$
0.867454 0.497517i $$-0.165755\pi$$
$$318$$ 7.31229e25i 0.387765i
$$319$$ −3.50079e25 −0.179624
$$320$$ 0 0
$$321$$ 9.63011e24 0.0462731
$$322$$ − 1.61081e26i − 0.749130i
$$323$$ 2.50835e26i 1.12918i
$$324$$ −7.28388e25 −0.317424
$$325$$ 0 0
$$326$$ −8.12420e26 −3.31891
$$327$$ 1.32773e25i 0.0525240i
$$328$$ 1.14299e27i 4.37890i
$$329$$ 2.41914e25 0.0897644
$$330$$ 0 0
$$331$$ 1.44090e26 0.501695 0.250848 0.968027i $$-0.419291\pi$$
0.250848 + 0.968027i $$0.419291\pi$$
$$332$$ 1.02262e27i 3.44958i
$$333$$ − 1.07341e26i − 0.350833i
$$334$$ 7.59206e26 2.40448
$$335$$ 0 0
$$336$$ −4.06139e26 −1.20813
$$337$$ 3.63051e26i 1.04678i 0.852095 + 0.523388i $$0.175332\pi$$
−0.852095 + 0.523388i $$0.824668\pi$$
$$338$$ 6.66111e26i 1.86175i
$$339$$ −2.50438e26 −0.678583
$$340$$ 0 0
$$341$$ 3.26607e25 0.0831964
$$342$$ − 2.89588e26i − 0.715331i
$$343$$ − 3.57891e26i − 0.857360i
$$344$$ −1.83349e27 −4.26004
$$345$$ 0 0
$$346$$ 6.49841e26 1.42072
$$347$$ 7.09622e25i 0.150511i 0.997164 + 0.0752554i $$0.0239772\pi$$
−0.997164 + 0.0752554i $$0.976023\pi$$
$$348$$ − 8.49335e26i − 1.74782i
$$349$$ 7.03939e26 1.40562 0.702810 0.711378i $$-0.251928\pi$$
0.702810 + 0.711378i $$0.251928\pi$$
$$350$$ 0 0
$$351$$ −2.33379e25 −0.0438872
$$352$$ 4.46451e26i 0.814846i
$$353$$ − 1.08085e26i − 0.191483i −0.995406 0.0957414i $$-0.969478\pi$$
0.995406 0.0957414i $$-0.0305222\pi$$
$$354$$ −9.29416e26 −1.59836
$$355$$ 0 0
$$356$$ −1.87280e27 −3.03575
$$357$$ − 1.84266e26i − 0.290020i
$$358$$ 3.67305e25i 0.0561378i
$$359$$ 1.67492e26 0.248601 0.124301 0.992245i $$-0.460331\pi$$
0.124301 + 0.992245i $$0.460331\pi$$
$$360$$ 0 0
$$361$$ 1.38602e26 0.194064
$$362$$ − 2.48946e27i − 3.38583i
$$363$$ 4.24422e26i 0.560761i
$$364$$ −2.46722e26 −0.316696
$$365$$ 0 0
$$366$$ −1.20514e27 −1.46045
$$367$$ − 9.83667e26i − 1.15839i −0.815190 0.579194i $$-0.803367\pi$$
0.815190 0.579194i $$-0.196633\pi$$
$$368$$ − 2.95146e27i − 3.37780i
$$369$$ 3.59863e26 0.400276
$$370$$ 0 0
$$371$$ 1.58191e26 0.166248
$$372$$ 7.92389e26i 0.809539i
$$373$$ 1.00058e26i 0.0993824i 0.998765 + 0.0496912i $$0.0158237\pi$$
−0.998765 + 0.0496912i $$0.984176\pi$$
$$374$$ −3.56208e26 −0.343995
$$375$$ 0 0
$$376$$ 7.37429e26 0.673360
$$377$$ − 2.72131e26i − 0.241654i
$$378$$ 2.12734e26i 0.183727i
$$379$$ −9.23905e25 −0.0776096 −0.0388048 0.999247i $$-0.512355\pi$$
−0.0388048 + 0.999247i $$0.512355\pi$$
$$380$$ 0 0
$$381$$ −9.82055e25 −0.0780591
$$382$$ 3.80986e27i 2.94608i
$$383$$ − 2.13677e27i − 1.60757i −0.594919 0.803786i $$-0.702816\pi$$
0.594919 0.803786i $$-0.297184\pi$$
$$384$$ −4.16393e27 −3.04807
$$385$$ 0 0
$$386$$ 6.08530e26 0.421809
$$387$$ 5.77263e26i 0.389411i
$$388$$ 5.68572e27i 3.73295i
$$389$$ 1.25581e27 0.802516 0.401258 0.915965i $$-0.368573\pi$$
0.401258 + 0.915965i $$0.368573\pi$$
$$390$$ 0 0
$$391$$ 1.33908e27 0.810868
$$392$$ − 4.72395e27i − 2.78485i
$$393$$ − 3.78305e26i − 0.217130i
$$394$$ 4.06394e27 2.27111
$$395$$ 0 0
$$396$$ 3.04614e26 0.161417
$$397$$ − 1.78165e27i − 0.919439i −0.888064 0.459719i $$-0.847950\pi$$
0.888064 0.459719i $$-0.152050\pi$$
$$398$$ − 2.12121e27i − 1.06614i
$$399$$ −6.26483e26 −0.306686
$$400$$ 0 0
$$401$$ 1.40181e27 0.651141 0.325570 0.945518i $$-0.394444\pi$$
0.325570 + 0.945518i $$0.394444\pi$$
$$402$$ 2.64590e27i 1.19729i
$$403$$ 2.53885e26i 0.111927i
$$404$$ −8.67511e26 −0.372624
$$405$$ 0 0
$$406$$ −2.48058e27 −1.01165
$$407$$ 4.48903e26i 0.178407i
$$408$$ − 5.61699e27i − 2.17557i
$$409$$ −2.17493e27 −0.821015 −0.410507 0.911857i $$-0.634648\pi$$
−0.410507 + 0.911857i $$0.634648\pi$$
$$410$$ 0 0
$$411$$ 1.17102e27 0.419978
$$412$$ − 1.31582e28i − 4.60021i
$$413$$ 2.01066e27i 0.685271i
$$414$$ −1.54596e27 −0.513683
$$415$$ 0 0
$$416$$ −3.47045e27 −1.09624
$$417$$ − 3.07139e26i − 0.0946034i
$$418$$ 1.21107e27i 0.363762i
$$419$$ 6.07636e27 1.77990 0.889952 0.456055i $$-0.150738\pi$$
0.889952 + 0.456055i $$0.150738\pi$$
$$420$$ 0 0
$$421$$ −1.89993e27 −0.529389 −0.264695 0.964332i $$-0.585271\pi$$
−0.264695 + 0.964332i $$0.585271\pi$$
$$422$$ − 7.53915e27i − 2.04900i
$$423$$ − 2.32175e26i − 0.0615520i
$$424$$ 4.82214e27 1.24709
$$425$$ 0 0
$$426$$ 4.44321e27 1.09369
$$427$$ 2.60714e27i 0.626141i
$$428$$ − 9.77083e26i − 0.228966i
$$429$$ 9.75999e25 0.0223176
$$430$$ 0 0
$$431$$ −8.08572e24 −0.00176079 −0.000880395 1.00000i $$-0.500280\pi$$
−0.000880395 1.00000i $$0.500280\pi$$
$$432$$ 3.89789e27i 0.828420i
$$433$$ − 5.60439e27i − 1.16253i −0.813713 0.581267i $$-0.802557\pi$$
0.813713 0.581267i $$-0.197443\pi$$
$$434$$ 2.31426e27 0.468566
$$435$$ 0 0
$$436$$ 1.34713e27 0.259897
$$437$$ − 4.55272e27i − 0.857463i
$$438$$ 2.26230e27i 0.415979i
$$439$$ 8.51110e27 1.52795 0.763973 0.645248i $$-0.223246\pi$$
0.763973 + 0.645248i $$0.223246\pi$$
$$440$$ 0 0
$$441$$ −1.48731e27 −0.254563
$$442$$ − 2.76896e27i − 0.462789i
$$443$$ − 6.63134e27i − 1.08234i −0.840915 0.541168i $$-0.817982\pi$$
0.840915 0.541168i $$-0.182018\pi$$
$$444$$ −1.08909e28 −1.73598
$$445$$ 0 0
$$446$$ −1.12956e28 −1.71749
$$447$$ − 4.41468e27i − 0.655648i
$$448$$ 1.72103e28i 2.49671i
$$449$$ −1.30394e28 −1.84787 −0.923933 0.382555i $$-0.875044\pi$$
−0.923933 + 0.382555i $$0.875044\pi$$
$$450$$ 0 0
$$451$$ −1.50496e27 −0.203549
$$452$$ 2.54097e28i 3.35773i
$$453$$ − 6.55703e27i − 0.846596i
$$454$$ −6.10577e27 −0.770289
$$455$$ 0 0
$$456$$ −1.90971e28 −2.30058
$$457$$ − 4.72949e27i − 0.556793i −0.960466 0.278397i $$-0.910197\pi$$
0.960466 0.278397i $$-0.0898030\pi$$
$$458$$ − 1.04097e28i − 1.19771i
$$459$$ −1.76848e27 −0.198869
$$460$$ 0 0
$$461$$ 4.92722e27 0.529349 0.264675 0.964338i $$-0.414735\pi$$
0.264675 + 0.964338i $$0.414735\pi$$
$$462$$ − 8.89661e26i − 0.0934294i
$$463$$ 1.20207e28i 1.23404i 0.786947 + 0.617021i $$0.211661\pi$$
−0.786947 + 0.617021i $$0.788339\pi$$
$$464$$ −4.54513e28 −4.56150
$$465$$ 0 0
$$466$$ 8.27379e27 0.793693
$$467$$ − 1.09969e28i − 1.03144i −0.856758 0.515719i $$-0.827525\pi$$
0.856758 0.515719i $$-0.172475\pi$$
$$468$$ 2.36789e27i 0.217161i
$$469$$ 5.72402e27 0.513317
$$470$$ 0 0
$$471$$ −2.57786e27 −0.221075
$$472$$ 6.12910e28i 5.14051i
$$473$$ − 2.41413e27i − 0.198024i
$$474$$ 2.83578e27 0.227510
$$475$$ 0 0
$$476$$ −1.86958e28 −1.43507
$$477$$ − 1.51823e27i − 0.113997i
$$478$$ 1.36576e28i 1.00318i
$$479$$ 1.32717e28 0.953684 0.476842 0.878989i $$-0.341781\pi$$
0.476842 + 0.878989i $$0.341781\pi$$
$$480$$ 0 0
$$481$$ −3.48951e27 −0.240017
$$482$$ 2.23613e28i 1.50489i
$$483$$ 3.34446e27i 0.220233i
$$484$$ 4.30624e28 2.77473
$$485$$ 0 0
$$486$$ 2.04170e27 0.125983
$$487$$ − 2.62576e28i − 1.58563i −0.609466 0.792813i $$-0.708616\pi$$
0.609466 0.792813i $$-0.291384\pi$$
$$488$$ 7.94738e28i 4.69695i
$$489$$ 1.68680e28 0.975710
$$490$$ 0 0
$$491$$ −2.54066e28 −1.40796 −0.703982 0.710218i $$-0.748596\pi$$
−0.703982 + 0.710218i $$0.748596\pi$$
$$492$$ − 3.65121e28i − 1.98063i
$$493$$ − 2.06213e28i − 1.09502i
$$494$$ −9.41414e27 −0.489382
$$495$$ 0 0
$$496$$ 4.24039e28 2.11275
$$497$$ − 9.61224e27i − 0.468903i
$$498$$ − 2.86645e28i − 1.36911i
$$499$$ 1.30048e28 0.608204 0.304102 0.952640i $$-0.401644\pi$$
0.304102 + 0.952640i $$0.401644\pi$$
$$500$$ 0 0
$$501$$ −1.57631e28 −0.706882
$$502$$ − 5.67449e28i − 2.49194i
$$503$$ 1.34993e27i 0.0580559i 0.999579 + 0.0290280i $$0.00924119\pi$$
−0.999579 + 0.0290280i $$0.990759\pi$$
$$504$$ 1.40289e28 0.590886
$$505$$ 0 0
$$506$$ 6.46525e27 0.261219
$$507$$ − 1.38302e28i − 0.547326i
$$508$$ 9.96406e27i 0.386249i
$$509$$ 4.04902e28 1.53749 0.768746 0.639554i $$-0.220881\pi$$
0.768746 + 0.639554i $$0.220881\pi$$
$$510$$ 0 0
$$511$$ 4.89416e27 0.178344
$$512$$ 1.39939e29i 4.99579i
$$513$$ 6.01263e27i 0.210296i
$$514$$ 3.79902e28 1.30184
$$515$$ 0 0
$$516$$ 5.85698e28 1.92687
$$517$$ 9.70964e26i 0.0313006i
$$518$$ 3.18083e28i 1.00479i
$$519$$ −1.34924e28 −0.417670
$$520$$ 0 0
$$521$$ 5.40378e28 1.60658 0.803289 0.595590i $$-0.203082\pi$$
0.803289 + 0.595590i $$0.203082\pi$$
$$522$$ 2.38072e28i 0.693695i
$$523$$ 1.54066e28i 0.439988i 0.975501 + 0.219994i $$0.0706037\pi$$
−0.975501 + 0.219994i $$0.929396\pi$$
$$524$$ −3.83833e28 −1.07439
$$525$$ 0 0
$$526$$ 1.65986e28 0.446398
$$527$$ 1.92387e28i 0.507182i
$$528$$ − 1.63011e28i − 0.421270i
$$529$$ 1.51670e28 0.384252
$$530$$ 0 0
$$531$$ 1.92971e28 0.469895
$$532$$ 6.35637e28i 1.51753i
$$533$$ − 1.16987e28i − 0.273842i
$$534$$ 5.24953e28 1.20486
$$535$$ 0 0
$$536$$ 1.74486e29 3.85061
$$537$$ − 7.62623e26i − 0.0165037i
$$538$$ − 1.52238e29i − 3.23080i
$$539$$ 6.21997e27 0.129451
$$540$$ 0 0
$$541$$ −7.54478e28 −1.51034 −0.755171 0.655528i $$-0.772446\pi$$
−0.755171 + 0.655528i $$0.772446\pi$$
$$542$$ − 2.96923e28i − 0.582976i
$$543$$ 5.16878e28i 0.995382i
$$544$$ −2.62981e29 −4.96747
$$545$$ 0 0
$$546$$ 6.91571e27 0.125694
$$547$$ − 7.90524e28i − 1.40944i −0.709483 0.704722i $$-0.751072\pi$$
0.709483 0.704722i $$-0.248928\pi$$
$$548$$ − 1.18813e29i − 2.07812i
$$549$$ 2.50219e28 0.429349
$$550$$ 0 0
$$551$$ −7.01101e28 −1.15795
$$552$$ 1.01950e29i 1.65206i
$$553$$ − 6.13480e27i − 0.0975408i
$$554$$ 8.95531e28 1.39710
$$555$$ 0 0
$$556$$ −3.11627e28 −0.468113
$$557$$ − 1.17729e29i − 1.73541i −0.497075 0.867707i $$-0.665593\pi$$
0.497075 0.867707i $$-0.334407\pi$$
$$558$$ − 2.22110e28i − 0.321299i
$$559$$ 1.87660e28 0.266409
$$560$$ 0 0
$$561$$ 7.39583e27 0.101129
$$562$$ − 3.29582e28i − 0.442315i
$$563$$ − 9.20807e28i − 1.21291i −0.795116 0.606457i $$-0.792590\pi$$
0.795116 0.606457i $$-0.207410\pi$$
$$564$$ −2.35568e28 −0.304569
$$565$$ 0 0
$$566$$ 1.36626e29 1.70200
$$567$$ − 4.41693e27i − 0.0540130i
$$568$$ − 2.93011e29i − 3.51744i
$$569$$ 2.71795e27 0.0320304 0.0160152 0.999872i $$-0.494902\pi$$
0.0160152 + 0.999872i $$0.494902\pi$$
$$570$$ 0 0
$$571$$ 1.28086e28 0.145487 0.0727434 0.997351i $$-0.476825\pi$$
0.0727434 + 0.997351i $$0.476825\pi$$
$$572$$ − 9.90260e27i − 0.110431i
$$573$$ − 7.91029e28i − 0.866102i
$$574$$ −1.06638e29 −1.14640
$$575$$ 0 0
$$576$$ 1.65175e29 1.71201
$$577$$ − 1.49329e29i − 1.51984i −0.650015 0.759922i $$-0.725237\pi$$
0.650015 0.759922i $$-0.274763\pi$$
$$578$$ − 1.33261e28i − 0.133186i
$$579$$ −1.26347e28 −0.124005
$$580$$ 0 0
$$581$$ −6.20116e28 −0.586980
$$582$$ − 1.59373e29i − 1.48158i
$$583$$ 6.34926e27i 0.0579700i
$$584$$ 1.49189e29 1.33783
$$585$$ 0 0
$$586$$ 2.26586e29 1.96023
$$587$$ − 2.62975e28i − 0.223468i −0.993738 0.111734i $$-0.964360\pi$$
0.993738 0.111734i $$-0.0356404\pi$$
$$588$$ 1.50904e29i 1.25962i
$$589$$ 6.54093e28 0.536326
$$590$$ 0 0
$$591$$ −8.43783e28 −0.667671
$$592$$ 5.82817e29i 4.53059i
$$593$$ 1.92294e29i 1.46856i 0.678849 + 0.734278i $$0.262479\pi$$
−0.678849 + 0.734278i $$0.737521\pi$$
$$594$$ −8.53846e27 −0.0640651
$$595$$ 0 0
$$596$$ −4.47919e29 −3.24425
$$597$$ 4.40421e28i 0.313428i
$$598$$ 5.02572e28i 0.351427i
$$599$$ 2.45874e29 1.68940 0.844699 0.535242i $$-0.179780\pi$$
0.844699 + 0.535242i $$0.179780\pi$$
$$600$$ 0 0
$$601$$ 7.47252e28 0.495776 0.247888 0.968789i $$-0.420264\pi$$
0.247888 + 0.968789i $$0.420264\pi$$
$$602$$ − 1.71060e29i − 1.11528i
$$603$$ − 5.49359e28i − 0.351985i
$$604$$ −6.65285e29 −4.18909
$$605$$ 0 0
$$606$$ 2.43167e28 0.147891
$$607$$ 1.23466e29i 0.738016i 0.929426 + 0.369008i $$0.120302\pi$$
−0.929426 + 0.369008i $$0.879698\pi$$
$$608$$ 8.94104e29i 5.25291i
$$609$$ 5.15035e28 0.297410
$$610$$ 0 0
$$611$$ −7.54771e27 −0.0421098
$$612$$ 1.79432e29i 0.984034i
$$613$$ − 1.59244e29i − 0.858476i −0.903191 0.429238i $$-0.858782\pi$$
0.903191 0.429238i $$-0.141218\pi$$
$$614$$ 4.30861e29 2.28333
$$615$$ 0 0
$$616$$ −5.86694e28 −0.300479
$$617$$ − 1.23256e29i − 0.620602i −0.950638 0.310301i $$-0.899570\pi$$
0.950638 0.310301i $$-0.100430\pi$$
$$618$$ 3.68831e29i 1.82578i
$$619$$ 5.18990e28 0.252585 0.126292 0.991993i $$-0.459692\pi$$
0.126292 + 0.991993i $$0.459692\pi$$
$$620$$ 0 0
$$621$$ 3.20983e28 0.151015
$$622$$ 4.02159e29i 1.86037i
$$623$$ − 1.13566e29i − 0.516564i
$$624$$ 1.26715e29 0.566750
$$625$$ 0 0
$$626$$ −1.42086e29 −0.614497
$$627$$ − 2.51450e28i − 0.106940i
$$628$$ 2.61553e29i 1.09391i
$$629$$ −2.64425e29 −1.08760
$$630$$ 0 0
$$631$$ −2.05208e28 −0.0816366 −0.0408183 0.999167i $$-0.512996\pi$$
−0.0408183 + 0.999167i $$0.512996\pi$$
$$632$$ − 1.87008e29i − 0.731694i
$$633$$ 1.56533e29i 0.602374i
$$634$$ 5.16285e29 1.95412
$$635$$ 0 0
$$636$$ −1.54041e29 −0.564075
$$637$$ 4.83505e28i 0.174155i
$$638$$ − 9.95625e28i − 0.352759i
$$639$$ −9.22528e28 −0.321529
$$640$$ 0 0
$$641$$ 5.14342e29 1.73477 0.867386 0.497636i $$-0.165798\pi$$
0.867386 + 0.497636i $$0.165798\pi$$
$$642$$ 2.73880e28i 0.0908747i
$$643$$ − 8.85766e28i − 0.289137i −0.989495 0.144569i $$-0.953821\pi$$
0.989495 0.144569i $$-0.0461794\pi$$
$$644$$ 3.39333e29 1.08975
$$645$$ 0 0
$$646$$ −7.13376e29 −2.21757
$$647$$ − 5.46916e29i − 1.67273i −0.548174 0.836364i $$-0.684677\pi$$
0.548174 0.836364i $$-0.315323\pi$$
$$648$$ − 1.34642e29i − 0.405174i
$$649$$ −8.07012e28 −0.238952
$$650$$ 0 0
$$651$$ −4.80503e28 −0.137751
$$652$$ − 1.71145e30i − 4.82796i
$$653$$ − 5.66153e29i − 1.57161i −0.618473 0.785806i $$-0.712248\pi$$
0.618473 0.785806i $$-0.287752\pi$$
$$654$$ −3.77606e28 −0.103151
$$655$$ 0 0
$$656$$ −1.95391e30 −5.16908
$$657$$ − 4.69713e28i − 0.122292i
$$658$$ 6.88003e28i 0.176286i
$$659$$ 1.48653e29 0.374867 0.187434 0.982277i $$-0.439983\pi$$
0.187434 + 0.982277i $$0.439983\pi$$
$$660$$ 0 0
$$661$$ −4.04669e29 −0.988517 −0.494259 0.869315i $$-0.664560\pi$$
−0.494259 + 0.869315i $$0.664560\pi$$
$$662$$ 4.09792e29i 0.985268i
$$663$$ 5.74909e28i 0.136053i
$$664$$ −1.89031e30 −4.40319
$$665$$ 0 0
$$666$$ 3.05278e29 0.688994
$$667$$ 3.74281e29i 0.831527i
$$668$$ 1.59935e30i 3.49776i
$$669$$ 2.34527e29 0.504916
$$670$$ 0 0
$$671$$ −1.04642e29 −0.218334
$$672$$ − 6.56816e29i − 1.34917i
$$673$$ − 1.54590e29i − 0.312625i −0.987708 0.156313i $$-0.950039\pi$$
0.987708 0.156313i $$-0.0499608\pi$$
$$674$$ −1.03252e30 −2.05574
$$675$$ 0 0
$$676$$ −1.40323e30 −2.70825
$$677$$ 9.88421e29i 1.87828i 0.343530 + 0.939142i $$0.388377\pi$$
−0.343530 + 0.939142i $$0.611623\pi$$
$$678$$ − 7.12245e29i − 1.33265i
$$679$$ −3.44781e29 −0.635200
$$680$$ 0 0
$$681$$ 1.26772e29 0.226453
$$682$$ 9.28870e28i 0.163388i
$$683$$ − 1.41169e29i − 0.244524i −0.992498 0.122262i $$-0.960985\pi$$
0.992498 0.122262i $$-0.0390147\pi$$
$$684$$ 6.10048e29 1.04058
$$685$$ 0 0
$$686$$ 1.01784e30 1.68375
$$687$$ 2.16133e29i 0.352108i
$$688$$ − 3.13430e30i − 5.02877i
$$689$$ −4.93555e28 −0.0779891
$$690$$ 0 0
$$691$$ 7.11585e29 1.09070 0.545352 0.838207i $$-0.316396\pi$$
0.545352 + 0.838207i $$0.316396\pi$$
$$692$$ 1.36896e30i 2.06669i
$$693$$ 1.84717e28i 0.0274668i
$$694$$ −2.01817e29 −0.295585
$$695$$ 0 0
$$696$$ 1.56999e30 2.23099
$$697$$ − 8.86490e29i − 1.24088i
$$698$$ 2.00200e30i 2.76047i
$$699$$ −1.71786e29 −0.233334
$$700$$ 0 0
$$701$$ −8.80754e29 −1.16096 −0.580478 0.814276i $$-0.697134\pi$$
−0.580478 + 0.814276i $$0.697134\pi$$
$$702$$ − 6.63731e28i − 0.0861891i
$$703$$ 8.99015e29i 1.15010i
$$704$$ −6.90765e29 −0.870597
$$705$$ 0 0
$$706$$ 3.07393e29 0.376049
$$707$$ − 5.26057e28i − 0.0634058i
$$708$$ − 1.95791e30i − 2.32511i
$$709$$ 2.79000e29 0.326452 0.163226 0.986589i $$-0.447810\pi$$
0.163226 + 0.986589i $$0.447810\pi$$
$$710$$ 0 0
$$711$$ −5.88783e28 −0.0668843
$$712$$ − 3.46185e30i − 3.87496i
$$713$$ − 3.49186e29i − 0.385139i
$$714$$ 5.24052e29 0.569565
$$715$$ 0 0
$$716$$ −7.73767e28 −0.0816626
$$717$$ − 2.83568e29i − 0.294921i
$$718$$ 4.76349e29i 0.488223i
$$719$$ −1.21404e30 −1.22625 −0.613124 0.789987i $$-0.710087\pi$$
−0.613124 + 0.789987i $$0.710087\pi$$
$$720$$ 0 0
$$721$$ 7.97913e29 0.782772
$$722$$ 3.94185e29i 0.381118i
$$723$$ − 4.64280e29i − 0.442413i
$$724$$ 5.24431e30 4.92531
$$725$$ 0 0
$$726$$ −1.20706e30 −1.10127
$$727$$ 6.54831e29i 0.588868i 0.955672 + 0.294434i $$0.0951311\pi$$
−0.955672 + 0.294434i $$0.904869\pi$$
$$728$$ − 4.56062e29i − 0.404245i
$$729$$ −4.23912e28 −0.0370370
$$730$$ 0 0
$$731$$ 1.42204e30 1.20720
$$732$$ − 2.53875e30i − 2.12449i
$$733$$ 2.20665e29i 0.182029i 0.995850 + 0.0910147i $$0.0290110\pi$$
−0.995850 + 0.0910147i $$0.970989\pi$$
$$734$$ 2.79755e30 2.27493
$$735$$ 0 0
$$736$$ 4.77315e30 3.77214
$$737$$ 2.29743e29i 0.178992i
$$738$$ 1.02345e30i 0.786094i
$$739$$ 4.07297e29 0.308422 0.154211 0.988038i $$-0.450717\pi$$
0.154211 + 0.988038i $$0.450717\pi$$
$$740$$ 0 0
$$741$$ 1.95463e29 0.143871
$$742$$ 4.49895e29i 0.326490i
$$743$$ 3.97218e29i 0.284214i 0.989851 + 0.142107i $$0.0453878\pi$$
−0.989851 + 0.142107i $$0.954612\pi$$
$$744$$ −1.46472e30 −1.03333
$$745$$ 0 0
$$746$$ −2.84565e29 −0.195175
$$747$$ 5.95152e29i 0.402496i
$$748$$ − 7.50390e29i − 0.500403i
$$749$$ 5.92500e28 0.0389610
$$750$$ 0 0
$$751$$ −1.86890e30 −1.19500 −0.597498 0.801871i $$-0.703838\pi$$
−0.597498 + 0.801871i $$0.703838\pi$$
$$752$$ 1.26062e30i 0.794869i
$$753$$ 1.17817e30i 0.732592i
$$754$$ 7.73941e29 0.474579
$$755$$ 0 0
$$756$$ −4.48147e29 −0.267265
$$757$$ 2.99103e30i 1.75919i 0.475720 + 0.879597i $$0.342188\pi$$
−0.475720 + 0.879597i $$0.657812\pi$$
$$758$$ − 2.62759e29i − 0.152416i
$$759$$ −1.34236e29 −0.0767945
$$760$$ 0 0
$$761$$ 1.51341e30 0.842206 0.421103 0.907013i $$-0.361643\pi$$
0.421103 + 0.907013i $$0.361643\pi$$
$$762$$ − 2.79297e29i − 0.153299i
$$763$$ 8.16896e28i 0.0442241i
$$764$$ −8.02588e30 −4.28561
$$765$$ 0 0
$$766$$ 6.07697e30 3.15707
$$767$$ − 6.27325e29i − 0.321470i
$$768$$ − 5.97596e30i − 3.02075i
$$769$$ −2.53401e30 −1.26352 −0.631759 0.775165i $$-0.717667\pi$$
−0.631759 + 0.775165i $$0.717667\pi$$
$$770$$ 0 0
$$771$$ −7.88777e29 −0.382722
$$772$$ 1.28193e30i 0.613598i
$$773$$ − 1.69545e30i − 0.800571i −0.916390 0.400285i $$-0.868911\pi$$
0.916390 0.400285i $$-0.131089\pi$$
$$774$$ −1.64173e30 −0.764756
$$775$$ 0 0
$$776$$ −1.05100e31 −4.76490
$$777$$ − 6.60424e29i − 0.295394i
$$778$$ 3.57153e30i 1.57604i
$$779$$ −3.01396e30 −1.31218
$$780$$ 0 0
$$781$$ 3.85804e29 0.163505
$$782$$ 3.80834e30i 1.59245i
$$783$$ − 4.94301e29i − 0.203936i
$$784$$ 8.07548e30 3.28738
$$785$$ 0 0
$$786$$ 1.07590e30 0.426417
$$787$$ 1.87332e30i 0.732616i 0.930494 + 0.366308i $$0.119378\pi$$
−0.930494 + 0.366308i $$0.880622\pi$$
$$788$$ 8.56113e30i 3.30374i
$$789$$ −3.44632e29 −0.131234
$$790$$ 0 0
$$791$$ −1.54084e30 −0.571353
$$792$$ 5.63075e29i 0.206040i
$$793$$ − 8.13429e29i − 0.293732i
$$794$$ 5.06702e30 1.80567
$$795$$ 0 0
$$796$$ 4.46856e30 1.55089
$$797$$ 7.79664e29i 0.267051i 0.991045 + 0.133526i $$0.0426299\pi$$
−0.991045 + 0.133526i $$0.957370\pi$$
$$798$$ − 1.78172e30i − 0.602294i
$$799$$ −5.71944e29 −0.190815
$$800$$ 0 0
$$801$$ −1.08994e30 −0.354211
$$802$$ 3.98676e30i 1.27876i
$$803$$ 1.96436e29i 0.0621880i
$$804$$ −5.57386e30 −1.74168
$$805$$ 0 0
$$806$$ −7.22050e29 −0.219811
$$807$$ 3.16088e30i 0.949807i
$$808$$ − 1.60358e30i − 0.475633i
$$809$$ −8.82262e29 −0.258308 −0.129154 0.991625i $$-0.541226\pi$$
−0.129154 + 0.991625i $$0.541226\pi$$
$$810$$ 0 0
$$811$$ −2.06044e30 −0.587815 −0.293907 0.955834i $$-0.594956\pi$$
−0.293907 + 0.955834i $$0.594956\pi$$
$$812$$ − 5.22561e30i − 1.47163i
$$813$$ 6.16490e29i 0.171386i
$$814$$ −1.27668e30 −0.350369
$$815$$ 0 0
$$816$$ 9.60212e30 2.56815
$$817$$ − 4.83476e30i − 1.27657i
$$818$$ − 6.18551e30i − 1.61237i
$$819$$ −1.43589e29 −0.0369521
$$820$$ 0 0
$$821$$ −1.83846e30 −0.461160 −0.230580 0.973053i $$-0.574062\pi$$
−0.230580 + 0.973053i $$0.574062\pi$$
$$822$$ 3.33039e30i 0.824785i
$$823$$ 7.73766e29i 0.189196i 0.995516 + 0.0945979i $$0.0301565\pi$$
−0.995516 + 0.0945979i $$0.969843\pi$$
$$824$$ 2.43229e31 5.87190
$$825$$ 0 0
$$826$$ −5.71831e30 −1.34579
$$827$$ 4.59989e30i 1.06891i 0.845198 + 0.534453i $$0.179482\pi$$
−0.845198 + 0.534453i $$0.820518\pi$$
$$828$$ − 3.25673e30i − 0.747245i
$$829$$ −7.93000e30 −1.79660 −0.898298 0.439386i $$-0.855196\pi$$
−0.898298 + 0.439386i $$0.855196\pi$$
$$830$$ 0 0
$$831$$ −1.85936e30 −0.410726
$$832$$ − 5.36961e30i − 1.17124i
$$833$$ 3.66386e30i 0.789162i
$$834$$ 8.73504e29 0.185790
$$835$$ 0 0
$$836$$ −2.55124e30 −0.529158
$$837$$ 4.61159e29i 0.0944569i
$$838$$ 1.72812e31i 3.49551i
$$839$$ −4.84033e30 −0.966884 −0.483442 0.875376i $$-0.660614\pi$$
−0.483442 + 0.875376i $$0.660614\pi$$
$$840$$ 0 0
$$841$$ 6.30944e29 0.122923
$$842$$ − 5.40340e30i − 1.03966i
$$843$$ 6.84300e29i 0.130034i
$$844$$ 1.58820e31 2.98064
$$845$$ 0 0
$$846$$ 6.60306e29 0.120881
$$847$$ 2.61129e30i 0.472149i
$$848$$ 8.24334e30i 1.47213i
$$849$$ −2.83671e30 −0.500362
$$850$$ 0 0
$$851$$ 4.79937e30 0.825893
$$852$$ 9.36009e30i 1.59098i
$$853$$ 2.96903e30i 0.498483i 0.968441 + 0.249242i $$0.0801813\pi$$
−0.968441 + 0.249242i $$0.919819\pi$$
$$854$$ −7.41472e30 −1.22967
$$855$$ 0 0
$$856$$ 1.80612e30 0.292263
$$857$$ 3.70009e30i 0.591444i 0.955274 + 0.295722i $$0.0955602\pi$$
−0.955274 + 0.295722i $$0.904440\pi$$
$$858$$ 2.77574e29i 0.0438291i
$$859$$ −7.75385e29 −0.120945 −0.0604726 0.998170i $$-0.519261\pi$$
−0.0604726 + 0.998170i $$0.519261\pi$$
$$860$$ 0 0
$$861$$ 2.21408e30 0.337024
$$862$$ − 2.29958e28i − 0.00345798i
$$863$$ − 1.29544e31i − 1.92443i −0.272283 0.962217i $$-0.587779\pi$$
0.272283 0.962217i $$-0.412221\pi$$
$$864$$ −6.30375e30 −0.925134
$$865$$ 0 0
$$866$$ 1.59389e31 2.28307
$$867$$ 2.76685e29i 0.0391548i
$$868$$ 4.87524e30i 0.681615i
$$869$$ 2.46231e29 0.0340122
$$870$$ 0 0
$$871$$ −1.78589e30 −0.240805
$$872$$ 2.49015e30i 0.331744i
$$873$$ 3.30901e30i 0.435560i
$$874$$ 1.29479e31 1.68395
$$875$$ 0 0
$$876$$ −4.76577e30 −0.605118
$$877$$ 1.57355e31i 1.97417i 0.160207 + 0.987083i $$0.448784\pi$$
−0.160207 + 0.987083i $$0.551216\pi$$
$$878$$ 2.42056e31i 3.00070i
$$879$$ −4.70452e30 −0.576278
$$880$$ 0 0
$$881$$ −1.47526e31 −1.76450 −0.882252 0.470778i $$-0.843973\pi$$
−0.882252 + 0.470778i $$0.843973\pi$$
$$882$$ − 4.22991e30i − 0.499932i
$$883$$ − 5.64453e30i − 0.659235i −0.944115 0.329617i $$-0.893080\pi$$
0.944115 0.329617i $$-0.106920\pi$$
$$884$$ 5.83310e30 0.673210
$$885$$ 0 0
$$886$$ 1.88595e31 2.12558
$$887$$ − 5.89300e30i − 0.656354i −0.944616 0.328177i $$-0.893566\pi$$
0.944616 0.328177i $$-0.106434\pi$$
$$888$$ − 2.01318e31i − 2.21588i
$$889$$ −6.04218e29 −0.0657242
$$890$$ 0 0
$$891$$ 1.77281e29 0.0188342
$$892$$ − 2.37954e31i − 2.49840i
$$893$$ 1.94454e30i 0.201780i
$$894$$ 1.25554e31 1.28761
$$895$$ 0 0
$$896$$ −2.56189e31 −2.56641
$$897$$ − 1.04347e30i − 0.103314i
$$898$$ − 3.70840e31i − 3.62898i
$$899$$ −5.37734e30 −0.520104
$$900$$ 0 0
$$901$$ −3.74002e30 −0.353397
$$902$$ − 4.28009e30i − 0.399746i
$$903$$ 3.55166e30i 0.327876i
$$904$$ −4.69695e31 −4.28596
$$905$$ 0 0
$$906$$ 1.86482e31 1.66261
$$907$$ − 8.32782e30i − 0.733931i −0.930235 0.366965i $$-0.880397\pi$$
0.930235 0.366965i $$-0.119603\pi$$
$$908$$ − 1.28624e31i − 1.12053i
$$909$$ −5.04879e29 −0.0434777
$$910$$ 0 0
$$911$$ 1.08086e31 0.909548 0.454774 0.890607i $$-0.349720\pi$$
0.454774 + 0.890607i $$0.349720\pi$$
$$912$$ − 3.26461e31i − 2.71572i
$$913$$ − 2.48894e30i − 0.204678i
$$914$$ 1.34507e31 1.09347
$$915$$ 0 0
$$916$$ 2.19291e31 1.74229
$$917$$ − 2.32755e30i − 0.182819i
$$918$$ − 5.02955e30i − 0.390554i
$$919$$ −6.55205e30 −0.502996 −0.251498 0.967858i $$-0.580923\pi$$
−0.251498 + 0.967858i $$0.580923\pi$$
$$920$$ 0 0
$$921$$ −8.94582e30 −0.671265
$$922$$ 1.40130e31i 1.03958i
$$923$$ 2.99902e30i 0.219969i
$$924$$ 1.87416e30 0.135910
$$925$$ 0 0
$$926$$ −3.41869e31 −2.42351
$$927$$ − 7.65791e30i − 0.536752i
$$928$$ − 7.35047e31i − 5.09403i
$$929$$ −1.04036e31 −0.712883 −0.356442 0.934318i $$-0.616010\pi$$
−0.356442 + 0.934318i $$0.616010\pi$$
$$930$$ 0 0
$$931$$ 1.24567e31 0.834509
$$932$$ 1.74296e31i 1.15457i
$$933$$ − 8.34989e30i − 0.546919i
$$934$$ 3.12752e31 2.02562
$$935$$ 0 0
$$936$$ −4.37702e30 −0.277193
$$937$$ 2.09831e31i 1.31402i 0.753880 + 0.657012i $$0.228180\pi$$
−0.753880 + 0.657012i $$0.771820\pi$$
$$938$$ 1.62791e31i 1.00809i
$$939$$ 2.95007e30 0.180653
$$940$$ 0 0
$$941$$ 2.20967e31 1.32323 0.661617 0.749842i $$-0.269870\pi$$
0.661617 + 0.749842i $$0.269870\pi$$
$$942$$ − 7.33143e30i − 0.434165i
$$943$$ 1.60900e31i 0.942286i
$$944$$ −1.04776e32 −6.06812
$$945$$ 0 0
$$946$$ 6.86578e30 0.388896
$$947$$ 2.59604e31i 1.45424i 0.686509 + 0.727121i $$0.259142\pi$$
−0.686509 + 0.727121i $$0.740858\pi$$
$$948$$ 5.97386e30i 0.330954i
$$949$$ −1.52698e30 −0.0836637
$$950$$ 0 0
$$951$$ −1.07194e31 −0.574483
$$952$$ − 3.45590e31i − 1.83178i
$$953$$ − 2.94729e31i − 1.54507i −0.634973 0.772535i $$-0.718989\pi$$
0.634973 0.772535i $$-0.281011\pi$$
$$954$$ 4.31783e30 0.223876
$$955$$ 0 0
$$956$$ −2.87711e31 −1.45931
$$957$$ 2.06718e30i 0.103706i
$$958$$ 3.77448e31i 1.87292i
$$959$$ 7.20482e30 0.353612
$$960$$ 0 0
$$961$$ −1.58087e31 −0.759103
$$962$$ − 9.92417e30i − 0.471364i
$$963$$ − 5.68648e29i − 0.0267158i
$$964$$ −4.71064e31 −2.18913
$$965$$ 0 0
$$966$$ −9.51166e30 −0.432510
$$967$$ − 6.98077e30i − 0.313997i −0.987599 0.156998i $$-0.949818\pi$$
0.987599 0.156998i $$-0.0501818\pi$$
$$968$$ 7.96002e31i 3.54179i
$$969$$ 1.48116e31 0.651931
$$970$$ 0 0
$$971$$ −1.30234e30 −0.0560946 −0.0280473 0.999607i $$-0.508929\pi$$
−0.0280473 + 0.999607i $$0.508929\pi$$
$$972$$ 4.30106e30i 0.183265i
$$973$$ − 1.88970e30i − 0.0796541i
$$974$$ 7.46765e31 3.11398
$$975$$ 0 0
$$976$$ −1.35859e32 −5.54452
$$977$$ − 3.06599e31i − 1.23788i −0.785439 0.618939i $$-0.787563\pi$$
0.785439 0.618939i $$-0.212437\pi$$
$$978$$ 4.79726e31i 1.91618i
$$979$$ 4.55817e30 0.180124
$$980$$ 0 0
$$981$$ 7.84011e29 0.0303248
$$982$$ − 7.22564e31i − 2.76507i
$$983$$ 8.31325e30i 0.314745i 0.987539 + 0.157373i $$0.0503023\pi$$
−0.987539 + 0.157373i $$0.949698\pi$$
$$984$$ 6.74921e31 2.52816
$$985$$ 0 0
$$986$$ 5.86470e31 2.15049
$$987$$ − 1.42848e30i − 0.0518255i
$$988$$ − 1.98319e31i − 0.711895i
$$989$$ −2.58102e31 −0.916708
$$990$$ 0 0
$$991$$ −1.32568e31 −0.460962 −0.230481 0.973077i $$-0.574030\pi$$
−0.230481 + 0.973077i $$0.574030\pi$$
$$992$$ 6.85764e31i 2.35940i
$$993$$ − 8.50837e30i − 0.289654i
$$994$$ 2.73372e31 0.920868
$$995$$ 0 0
$$996$$ 6.03849e31 1.99161
$$997$$ − 3.42048e31i − 1.11632i −0.829734 0.558159i $$-0.811508\pi$$
0.829734 0.558159i $$-0.188492\pi$$
$$998$$ 3.69857e31i 1.19444i
$$999$$ −6.33837e30 −0.202554
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 75.22.b.a.49.2 2
5.2 odd 4 3.22.a.a.1.1 1
5.3 odd 4 75.22.a.c.1.1 1
5.4 even 2 inner 75.22.b.a.49.1 2
15.2 even 4 9.22.a.d.1.1 1
20.7 even 4 48.22.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 1 5.2 odd 4
9.22.a.d.1.1 1 15.2 even 4
48.22.a.e.1.1 1 20.7 even 4
75.22.a.c.1.1 1 5.3 odd 4
75.22.b.a.49.1 2 5.4 even 2 inner
75.22.b.a.49.2 2 1.1 even 1 trivial