# Properties

 Label 275.6.a.b.1.1 Level $275$ Weight $6$ Character 275.1 Self dual yes Analytic conductor $44.106$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,6,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 275.a (trivial)

## 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: yes Analytic conductor: $$44.1055504486$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.54492.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x - 38$$ x^3 - x^2 - 52*x - 38 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-6.29828$$ of defining polynomial Character $$\chi$$ $$=$$ 275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-8.18772 q^{2} +3.48600 q^{3} +35.0388 q^{4} -28.5424 q^{6} -145.071 q^{7} -24.8808 q^{8} -230.848 q^{9} +O(q^{10})$$ $$q-8.18772 q^{2} +3.48600 q^{3} +35.0388 q^{4} -28.5424 q^{6} -145.071 q^{7} -24.8808 q^{8} -230.848 q^{9} +121.000 q^{11} +122.145 q^{12} -615.772 q^{13} +1187.80 q^{14} -917.524 q^{16} -1840.68 q^{17} +1890.12 q^{18} +366.633 q^{19} -505.718 q^{21} -990.714 q^{22} +4516.38 q^{23} -86.7344 q^{24} +5041.77 q^{26} -1651.83 q^{27} -5083.12 q^{28} -1717.00 q^{29} -2650.54 q^{31} +8308.62 q^{32} +421.806 q^{33} +15070.9 q^{34} -8088.63 q^{36} -9660.61 q^{37} -3001.89 q^{38} -2146.58 q^{39} -11154.8 q^{41} +4140.68 q^{42} -8368.48 q^{43} +4239.69 q^{44} -36978.9 q^{46} +2221.22 q^{47} -3198.49 q^{48} +4238.64 q^{49} -6416.60 q^{51} -21575.9 q^{52} -23707.9 q^{53} +13524.8 q^{54} +3609.48 q^{56} +1278.08 q^{57} +14058.3 q^{58} +19517.8 q^{59} +20937.3 q^{61} +21701.9 q^{62} +33489.4 q^{63} -38667.9 q^{64} -3453.63 q^{66} +51707.7 q^{67} -64495.1 q^{68} +15744.1 q^{69} -1398.38 q^{71} +5743.67 q^{72} -72466.6 q^{73} +79098.4 q^{74} +12846.4 q^{76} -17553.6 q^{77} +17575.6 q^{78} +64632.2 q^{79} +50337.7 q^{81} +91332.4 q^{82} +96790.3 q^{83} -17719.8 q^{84} +68518.8 q^{86} -5985.47 q^{87} -3010.57 q^{88} -47614.1 q^{89} +89330.7 q^{91} +158249. q^{92} -9239.79 q^{93} -18186.7 q^{94} +28963.9 q^{96} +38399.6 q^{97} -34704.8 q^{98} -27932.6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9}+O(q^{10})$$ 3 * q - 34 * q^3 + 84 * q^4 - 206 * q^6 - 84 * q^7 + 564 * q^8 - 7 * q^9 $$3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9} + 363 q^{11} - 992 q^{12} - 486 q^{13} - 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 3706 q^{18} + 1380 q^{19} - 908 q^{21} + 3066 q^{23} - 11748 q^{24} + 12132 q^{26} + 2990 q^{27} - 23712 q^{28} - 3426 q^{29} - 4098 q^{31} + 12408 q^{32} - 4114 q^{33} + 25320 q^{34} + 4756 q^{36} - 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 5994 q^{41} + 47828 q^{42} + 26208 q^{43} + 10164 q^{44} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 48531 q^{49} - 22724 q^{51} + 35304 q^{52} - 50586 q^{53} + 18814 q^{54} - 42312 q^{56} - 20160 q^{57} + 29172 q^{58} - 3738 q^{59} + 18486 q^{61} + 19974 q^{62} + 12496 q^{63} - 20352 q^{64} - 24926 q^{66} + 47754 q^{67} + 12600 q^{68} + 35042 q^{69} + 39282 q^{71} + 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 103920 q^{76} - 10164 q^{77} - 124984 q^{78} + 125148 q^{79} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} + 343616 q^{84} + 243060 q^{86} + 19368 q^{87} + 68244 q^{88} - 106824 q^{89} - 109632 q^{91} + 336528 q^{92} + 16622 q^{93} - 74928 q^{94} - 76456 q^{96} - 9684 q^{97} - 3480 q^{98} - 847 q^{99}+O(q^{100})$$ 3 * q - 34 * q^3 + 84 * q^4 - 206 * q^6 - 84 * q^7 + 564 * q^8 - 7 * q^9 + 363 * q^11 - 992 * q^12 - 486 * q^13 - 1020 * q^14 + 1992 * q^16 - 1086 * q^17 + 3706 * q^18 + 1380 * q^19 - 908 * q^21 + 3066 * q^23 - 11748 * q^24 + 12132 * q^26 + 2990 * q^27 - 23712 * q^28 - 3426 * q^29 - 4098 * q^31 + 12408 * q^32 - 4114 * q^33 + 25320 * q^34 + 4756 * q^36 - 17724 * q^37 + 9240 * q^38 - 6560 * q^39 + 5994 * q^41 + 47828 * q^42 + 26208 * q^43 + 10164 * q^44 - 16806 * q^46 + 17232 * q^47 - 61064 * q^48 + 48531 * q^49 - 22724 * q^51 + 35304 * q^52 - 50586 * q^53 + 18814 * q^54 - 42312 * q^56 - 20160 * q^57 + 29172 * q^58 - 3738 * q^59 + 18486 * q^61 + 19974 * q^62 + 12496 * q^63 - 20352 * q^64 - 24926 * q^66 + 47754 * q^67 + 12600 * q^68 + 35042 * q^69 + 39282 * q^71 + 95040 * q^72 - 15426 * q^73 + 153294 * q^74 + 103920 * q^76 - 10164 * q^77 - 124984 * q^78 + 125148 * q^79 - 86917 * q^81 + 255372 * q^82 + 143928 * q^83 + 343616 * q^84 + 243060 * q^86 + 19368 * q^87 + 68244 * q^88 - 106824 * q^89 - 109632 * q^91 + 336528 * q^92 + 16622 * q^93 - 74928 * q^94 - 76456 * q^96 - 9684 * q^97 - 3480 * q^98 - 847 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −8.18772 −1.44740 −0.723699 0.690116i $$-0.757560\pi$$
−0.723699 + 0.690116i $$0.757560\pi$$
$$3$$ 3.48600 0.223627 0.111814 0.993729i $$-0.464334\pi$$
0.111814 + 0.993729i $$0.464334\pi$$
$$4$$ 35.0388 1.09496
$$5$$ 0 0
$$6$$ −28.5424 −0.323678
$$7$$ −145.071 −1.11902 −0.559508 0.828825i $$-0.689010\pi$$
−0.559508 + 0.828825i $$0.689010\pi$$
$$8$$ −24.8808 −0.137448
$$9$$ −230.848 −0.949991
$$10$$ 0 0
$$11$$ 121.000 0.301511
$$12$$ 122.145 0.244863
$$13$$ −615.772 −1.01056 −0.505279 0.862956i $$-0.668610\pi$$
−0.505279 + 0.862956i $$0.668610\pi$$
$$14$$ 1187.80 1.61966
$$15$$ 0 0
$$16$$ −917.524 −0.896020
$$17$$ −1840.68 −1.54474 −0.772369 0.635174i $$-0.780929\pi$$
−0.772369 + 0.635174i $$0.780929\pi$$
$$18$$ 1890.12 1.37502
$$19$$ 366.633 0.232996 0.116498 0.993191i $$-0.462833\pi$$
0.116498 + 0.993191i $$0.462833\pi$$
$$20$$ 0 0
$$21$$ −505.718 −0.250242
$$22$$ −990.714 −0.436407
$$23$$ 4516.38 1.78021 0.890104 0.455757i $$-0.150631\pi$$
0.890104 + 0.455757i $$0.150631\pi$$
$$24$$ −86.7344 −0.0307371
$$25$$ 0 0
$$26$$ 5041.77 1.46268
$$27$$ −1651.83 −0.436071
$$28$$ −5083.12 −1.22528
$$29$$ −1717.00 −0.379119 −0.189560 0.981869i $$-0.560706\pi$$
−0.189560 + 0.981869i $$0.560706\pi$$
$$30$$ 0 0
$$31$$ −2650.54 −0.495371 −0.247685 0.968841i $$-0.579670\pi$$
−0.247685 + 0.968841i $$0.579670\pi$$
$$32$$ 8308.62 1.43435
$$33$$ 421.806 0.0674261
$$34$$ 15070.9 2.23585
$$35$$ 0 0
$$36$$ −8088.63 −1.04020
$$37$$ −9660.61 −1.16011 −0.580057 0.814576i $$-0.696970\pi$$
−0.580057 + 0.814576i $$0.696970\pi$$
$$38$$ −3001.89 −0.337238
$$39$$ −2146.58 −0.225988
$$40$$ 0 0
$$41$$ −11154.8 −1.03634 −0.518170 0.855278i $$-0.673386\pi$$
−0.518170 + 0.855278i $$0.673386\pi$$
$$42$$ 4140.68 0.362200
$$43$$ −8368.48 −0.690201 −0.345100 0.938566i $$-0.612155\pi$$
−0.345100 + 0.938566i $$0.612155\pi$$
$$44$$ 4239.69 0.330144
$$45$$ 0 0
$$46$$ −36978.9 −2.57667
$$47$$ 2221.22 0.146672 0.0733360 0.997307i $$-0.476635\pi$$
0.0733360 + 0.997307i $$0.476635\pi$$
$$48$$ −3198.49 −0.200374
$$49$$ 4238.64 0.252195
$$50$$ 0 0
$$51$$ −6416.60 −0.345445
$$52$$ −21575.9 −1.10652
$$53$$ −23707.9 −1.15932 −0.579659 0.814859i $$-0.696814\pi$$
−0.579659 + 0.814859i $$0.696814\pi$$
$$54$$ 13524.8 0.631168
$$55$$ 0 0
$$56$$ 3609.48 0.153807
$$57$$ 1278.08 0.0521042
$$58$$ 14058.3 0.548737
$$59$$ 19517.8 0.729964 0.364982 0.931015i $$-0.381075\pi$$
0.364982 + 0.931015i $$0.381075\pi$$
$$60$$ 0 0
$$61$$ 20937.3 0.720436 0.360218 0.932868i $$-0.382702\pi$$
0.360218 + 0.932868i $$0.382702\pi$$
$$62$$ 21701.9 0.716999
$$63$$ 33489.4 1.06305
$$64$$ −38667.9 −1.18005
$$65$$ 0 0
$$66$$ −3453.63 −0.0975924
$$67$$ 51707.7 1.40724 0.703619 0.710577i $$-0.251566\pi$$
0.703619 + 0.710577i $$0.251566\pi$$
$$68$$ −64495.1 −1.69143
$$69$$ 15744.1 0.398103
$$70$$ 0 0
$$71$$ −1398.38 −0.0329216 −0.0164608 0.999865i $$-0.505240\pi$$
−0.0164608 + 0.999865i $$0.505240\pi$$
$$72$$ 5743.67 0.130574
$$73$$ −72466.6 −1.59159 −0.795794 0.605567i $$-0.792946\pi$$
−0.795794 + 0.605567i $$0.792946\pi$$
$$74$$ 79098.4 1.67915
$$75$$ 0 0
$$76$$ 12846.4 0.255122
$$77$$ −17553.6 −0.337396
$$78$$ 17575.6 0.327095
$$79$$ 64632.2 1.16515 0.582574 0.812777i $$-0.302045\pi$$
0.582574 + 0.812777i $$0.302045\pi$$
$$80$$ 0 0
$$81$$ 50337.7 0.852474
$$82$$ 91332.4 1.50000
$$83$$ 96790.3 1.54219 0.771093 0.636723i $$-0.219710\pi$$
0.771093 + 0.636723i $$0.219710\pi$$
$$84$$ −17719.8 −0.274006
$$85$$ 0 0
$$86$$ 68518.8 0.998995
$$87$$ −5985.47 −0.0847814
$$88$$ −3010.57 −0.0414422
$$89$$ −47614.1 −0.637178 −0.318589 0.947893i $$-0.603209\pi$$
−0.318589 + 0.947893i $$0.603209\pi$$
$$90$$ 0 0
$$91$$ 89330.7 1.13083
$$92$$ 158249. 1.94926
$$93$$ −9239.79 −0.110778
$$94$$ −18186.7 −0.212293
$$95$$ 0 0
$$96$$ 28963.9 0.320759
$$97$$ 38399.6 0.414378 0.207189 0.978301i $$-0.433568\pi$$
0.207189 + 0.978301i $$0.433568\pi$$
$$98$$ −34704.8 −0.365027
$$99$$ −27932.6 −0.286433
$$100$$ 0 0
$$101$$ −41011.2 −0.400036 −0.200018 0.979792i $$-0.564100\pi$$
−0.200018 + 0.979792i $$0.564100\pi$$
$$102$$ 52537.3 0.499997
$$103$$ 49634.4 0.460988 0.230494 0.973074i $$-0.425966\pi$$
0.230494 + 0.973074i $$0.425966\pi$$
$$104$$ 15320.9 0.138899
$$105$$ 0 0
$$106$$ 194113. 1.67800
$$107$$ 6791.34 0.0573450 0.0286725 0.999589i $$-0.490872\pi$$
0.0286725 + 0.999589i $$0.490872\pi$$
$$108$$ −57878.3 −0.477481
$$109$$ 96780.7 0.780230 0.390115 0.920766i $$-0.372435\pi$$
0.390115 + 0.920766i $$0.372435\pi$$
$$110$$ 0 0
$$111$$ −33676.9 −0.259433
$$112$$ 133106. 1.00266
$$113$$ 212938. 1.56876 0.784379 0.620281i $$-0.212982\pi$$
0.784379 + 0.620281i $$0.212982\pi$$
$$114$$ −10464.6 −0.0754155
$$115$$ 0 0
$$116$$ −60161.7 −0.415121
$$117$$ 142149. 0.960021
$$118$$ −159806. −1.05655
$$119$$ 267029. 1.72859
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −171428. −1.04276
$$123$$ −38885.6 −0.231754
$$124$$ −92871.8 −0.542412
$$125$$ 0 0
$$126$$ −274202. −1.53866
$$127$$ 90363.9 0.497148 0.248574 0.968613i $$-0.420038\pi$$
0.248574 + 0.968613i $$0.420038\pi$$
$$128$$ 50726.1 0.273657
$$129$$ −29172.5 −0.154348
$$130$$ 0 0
$$131$$ 65299.5 0.332454 0.166227 0.986088i $$-0.446842\pi$$
0.166227 + 0.986088i $$0.446842\pi$$
$$132$$ 14779.6 0.0738290
$$133$$ −53187.9 −0.260726
$$134$$ −423368. −2.03684
$$135$$ 0 0
$$136$$ 45797.4 0.212321
$$137$$ −5322.74 −0.0242289 −0.0121145 0.999927i $$-0.503856\pi$$
−0.0121145 + 0.999927i $$0.503856\pi$$
$$138$$ −128908. −0.576214
$$139$$ 89967.1 0.394954 0.197477 0.980308i $$-0.436725\pi$$
0.197477 + 0.980308i $$0.436725\pi$$
$$140$$ 0 0
$$141$$ 7743.18 0.0327998
$$142$$ 11449.6 0.0476506
$$143$$ −74508.4 −0.304695
$$144$$ 211808. 0.851211
$$145$$ 0 0
$$146$$ 593336. 2.30366
$$147$$ 14775.9 0.0563977
$$148$$ −338496. −1.27028
$$149$$ 66489.8 0.245352 0.122676 0.992447i $$-0.460852\pi$$
0.122676 + 0.992447i $$0.460852\pi$$
$$150$$ 0 0
$$151$$ −130866. −0.467074 −0.233537 0.972348i $$-0.575030\pi$$
−0.233537 + 0.972348i $$0.575030\pi$$
$$152$$ −9122.12 −0.0320248
$$153$$ 424916. 1.46749
$$154$$ 143724. 0.488346
$$155$$ 0 0
$$156$$ −75213.6 −0.247448
$$157$$ 163297. 0.528723 0.264362 0.964424i $$-0.414839\pi$$
0.264362 + 0.964424i $$0.414839\pi$$
$$158$$ −529191. −1.68643
$$159$$ −82645.6 −0.259255
$$160$$ 0 0
$$161$$ −655197. −1.99208
$$162$$ −412151. −1.23387
$$163$$ 535758. 1.57943 0.789713 0.613477i $$-0.210229\pi$$
0.789713 + 0.613477i $$0.210229\pi$$
$$164$$ −390851. −1.13475
$$165$$ 0 0
$$166$$ −792492. −2.23216
$$167$$ −553587. −1.53601 −0.768005 0.640443i $$-0.778751\pi$$
−0.768005 + 0.640443i $$0.778751\pi$$
$$168$$ 12582.7 0.0343953
$$169$$ 7881.54 0.0212273
$$170$$ 0 0
$$171$$ −84636.5 −0.221344
$$172$$ −293221. −0.755744
$$173$$ −266973. −0.678190 −0.339095 0.940752i $$-0.610121\pi$$
−0.339095 + 0.940752i $$0.610121\pi$$
$$174$$ 49007.4 0.122712
$$175$$ 0 0
$$176$$ −111020. −0.270160
$$177$$ 68039.2 0.163240
$$178$$ 389851. 0.922250
$$179$$ 3030.33 0.00706900 0.00353450 0.999994i $$-0.498875\pi$$
0.00353450 + 0.999994i $$0.498875\pi$$
$$180$$ 0 0
$$181$$ 761242. 1.72714 0.863568 0.504233i $$-0.168225\pi$$
0.863568 + 0.504233i $$0.168225\pi$$
$$182$$ −731415. −1.63676
$$183$$ 72987.3 0.161109
$$184$$ −112371. −0.244686
$$185$$ 0 0
$$186$$ 75652.8 0.160340
$$187$$ −222722. −0.465756
$$188$$ 77828.9 0.160600
$$189$$ 239634. 0.487970
$$190$$ 0 0
$$191$$ −430653. −0.854170 −0.427085 0.904212i $$-0.640459\pi$$
−0.427085 + 0.904212i $$0.640459\pi$$
$$192$$ −134796. −0.263891
$$193$$ 272285. 0.526175 0.263088 0.964772i $$-0.415259\pi$$
0.263088 + 0.964772i $$0.415259\pi$$
$$194$$ −314405. −0.599770
$$195$$ 0 0
$$196$$ 148517. 0.276144
$$197$$ −574550. −1.05478 −0.527390 0.849623i $$-0.676829\pi$$
−0.527390 + 0.849623i $$0.676829\pi$$
$$198$$ 228704. 0.414583
$$199$$ 269926. 0.483183 0.241592 0.970378i $$-0.422331\pi$$
0.241592 + 0.970378i $$0.422331\pi$$
$$200$$ 0 0
$$201$$ 180253. 0.314697
$$202$$ 335788. 0.579011
$$203$$ 249088. 0.424240
$$204$$ −224830. −0.378250
$$205$$ 0 0
$$206$$ −406393. −0.667234
$$207$$ −1.04260e6 −1.69118
$$208$$ 564985. 0.905480
$$209$$ 44362.6 0.0702509
$$210$$ 0 0
$$211$$ −753372. −1.16494 −0.582470 0.812853i $$-0.697913\pi$$
−0.582470 + 0.812853i $$0.697913\pi$$
$$212$$ −830695. −1.26941
$$213$$ −4874.77 −0.00736216
$$214$$ −55605.6 −0.0830011
$$215$$ 0 0
$$216$$ 41098.9 0.0599371
$$217$$ 384517. 0.554327
$$218$$ −792414. −1.12930
$$219$$ −252619. −0.355922
$$220$$ 0 0
$$221$$ 1.13344e6 1.56105
$$222$$ 275737. 0.375503
$$223$$ 997692. 1.34349 0.671745 0.740783i $$-0.265545\pi$$
0.671745 + 0.740783i $$0.265545\pi$$
$$224$$ −1.20534e6 −1.60506
$$225$$ 0 0
$$226$$ −1.74347e6 −2.27062
$$227$$ 495214. 0.637864 0.318932 0.947778i $$-0.396676\pi$$
0.318932 + 0.947778i $$0.396676\pi$$
$$228$$ 44782.5 0.0570521
$$229$$ −221893. −0.279611 −0.139806 0.990179i $$-0.544648\pi$$
−0.139806 + 0.990179i $$0.544648\pi$$
$$230$$ 0 0
$$231$$ −61191.9 −0.0754508
$$232$$ 42720.4 0.0521093
$$233$$ −619425. −0.747479 −0.373739 0.927534i $$-0.621925\pi$$
−0.373739 + 0.927534i $$0.621925\pi$$
$$234$$ −1.16388e6 −1.38953
$$235$$ 0 0
$$236$$ 683881. 0.799283
$$237$$ 225308. 0.260559
$$238$$ −2.18636e6 −2.50195
$$239$$ −295471. −0.334595 −0.167298 0.985906i $$-0.553504\pi$$
−0.167298 + 0.985906i $$0.553504\pi$$
$$240$$ 0 0
$$241$$ 693153. 0.768753 0.384376 0.923176i $$-0.374416\pi$$
0.384376 + 0.923176i $$0.374416\pi$$
$$242$$ −119876. −0.131582
$$243$$ 576873. 0.626707
$$244$$ 733616. 0.788850
$$245$$ 0 0
$$246$$ 318385. 0.335440
$$247$$ −225762. −0.235456
$$248$$ 65947.5 0.0680878
$$249$$ 337411. 0.344875
$$250$$ 0 0
$$251$$ 533816. 0.534820 0.267410 0.963583i $$-0.413832\pi$$
0.267410 + 0.963583i $$0.413832\pi$$
$$252$$ 1.17343e6 1.16400
$$253$$ 546482. 0.536753
$$254$$ −739874. −0.719571
$$255$$ 0 0
$$256$$ 822041. 0.783960
$$257$$ 652296. 0.616044 0.308022 0.951379i $$-0.400333\pi$$
0.308022 + 0.951379i $$0.400333\pi$$
$$258$$ 238857. 0.223402
$$259$$ 1.40148e6 1.29818
$$260$$ 0 0
$$261$$ 396366. 0.360160
$$262$$ −534654. −0.481193
$$263$$ 622045. 0.554540 0.277270 0.960792i $$-0.410570\pi$$
0.277270 + 0.960792i $$0.410570\pi$$
$$264$$ −10494.9 −0.00926759
$$265$$ 0 0
$$266$$ 435488. 0.377374
$$267$$ −165983. −0.142490
$$268$$ 1.81177e6 1.54087
$$269$$ −482862. −0.406858 −0.203429 0.979090i $$-0.565209\pi$$
−0.203429 + 0.979090i $$0.565209\pi$$
$$270$$ 0 0
$$271$$ −1.10678e6 −0.915460 −0.457730 0.889091i $$-0.651337\pi$$
−0.457730 + 0.889091i $$0.651337\pi$$
$$272$$ 1.68887e6 1.38412
$$273$$ 311407. 0.252884
$$274$$ 43581.1 0.0350689
$$275$$ 0 0
$$276$$ 551655. 0.435908
$$277$$ −639062. −0.500430 −0.250215 0.968190i $$-0.580501\pi$$
−0.250215 + 0.968190i $$0.580501\pi$$
$$278$$ −736626. −0.571656
$$279$$ 611872. 0.470598
$$280$$ 0 0
$$281$$ −257984. −0.194907 −0.0974534 0.995240i $$-0.531070\pi$$
−0.0974534 + 0.995240i $$0.531070\pi$$
$$282$$ −63399.0 −0.0474744
$$283$$ −1.02991e6 −0.764425 −0.382213 0.924074i $$-0.624838\pi$$
−0.382213 + 0.924074i $$0.624838\pi$$
$$284$$ −48997.7 −0.0360479
$$285$$ 0 0
$$286$$ 610054. 0.441015
$$287$$ 1.61824e6 1.15968
$$288$$ −1.91803e6 −1.36262
$$289$$ 1.96823e6 1.38622
$$290$$ 0 0
$$291$$ 133861. 0.0926662
$$292$$ −2.53914e6 −1.74273
$$293$$ −877712. −0.597287 −0.298644 0.954365i $$-0.596534\pi$$
−0.298644 + 0.954365i $$0.596534\pi$$
$$294$$ −120981. −0.0816299
$$295$$ 0 0
$$296$$ 240363. 0.159455
$$297$$ −199872. −0.131480
$$298$$ −544400. −0.355122
$$299$$ −2.78106e6 −1.79900
$$300$$ 0 0
$$301$$ 1.21403e6 0.772345
$$302$$ 1.07150e6 0.676042
$$303$$ −142965. −0.0894589
$$304$$ −336395. −0.208769
$$305$$ 0 0
$$306$$ −3.47909e6 −2.12404
$$307$$ 1.30925e6 0.792826 0.396413 0.918072i $$-0.370255\pi$$
0.396413 + 0.918072i $$0.370255\pi$$
$$308$$ −615057. −0.369436
$$309$$ 173026. 0.103089
$$310$$ 0 0
$$311$$ 3.35930e6 1.96946 0.984731 0.174083i $$-0.0556962\pi$$
0.984731 + 0.174083i $$0.0556962\pi$$
$$312$$ 53408.6 0.0310616
$$313$$ 3.00640e6 1.73454 0.867272 0.497835i $$-0.165871\pi$$
0.867272 + 0.497835i $$0.165871\pi$$
$$314$$ −1.33703e6 −0.765273
$$315$$ 0 0
$$316$$ 2.26464e6 1.27579
$$317$$ −2.10147e6 −1.17456 −0.587279 0.809385i $$-0.699801\pi$$
−0.587279 + 0.809385i $$0.699801\pi$$
$$318$$ 676679. 0.375245
$$319$$ −207757. −0.114309
$$320$$ 0 0
$$321$$ 23674.6 0.0128239
$$322$$ 5.36457e6 2.88333
$$323$$ −674853. −0.359918
$$324$$ 1.76377e6 0.933426
$$325$$ 0 0
$$326$$ −4.38663e6 −2.28606
$$327$$ 337378. 0.174481
$$328$$ 277540. 0.142443
$$329$$ −322235. −0.164128
$$330$$ 0 0
$$331$$ 1.23338e6 0.618766 0.309383 0.950938i $$-0.399878\pi$$
0.309383 + 0.950938i $$0.399878\pi$$
$$332$$ 3.39142e6 1.68864
$$333$$ 2.23013e6 1.10210
$$334$$ 4.53261e6 2.22322
$$335$$ 0 0
$$336$$ 464009. 0.224222
$$337$$ −679319. −0.325836 −0.162918 0.986640i $$-0.552091\pi$$
−0.162918 + 0.986640i $$0.552091\pi$$
$$338$$ −64531.9 −0.0307243
$$339$$ 742301. 0.350817
$$340$$ 0 0
$$341$$ −320715. −0.149360
$$342$$ 692980. 0.320373
$$343$$ 1.82331e6 0.836805
$$344$$ 208214. 0.0948668
$$345$$ 0 0
$$346$$ 2.18590e6 0.981611
$$347$$ −2.67540e6 −1.19279 −0.596397 0.802690i $$-0.703402\pi$$
−0.596397 + 0.802690i $$0.703402\pi$$
$$348$$ −209724. −0.0928324
$$349$$ −2.37636e6 −1.04435 −0.522177 0.852837i $$-0.674880\pi$$
−0.522177 + 0.852837i $$0.674880\pi$$
$$350$$ 0 0
$$351$$ 1.01715e6 0.440675
$$352$$ 1.00534e6 0.432472
$$353$$ −638696. −0.272808 −0.136404 0.990653i $$-0.543555\pi$$
−0.136404 + 0.990653i $$0.543555\pi$$
$$354$$ −557086. −0.236273
$$355$$ 0 0
$$356$$ −1.66834e6 −0.697686
$$357$$ 930864. 0.386559
$$358$$ −24811.5 −0.0102317
$$359$$ 1.50842e6 0.617712 0.308856 0.951109i $$-0.400054\pi$$
0.308856 + 0.951109i $$0.400054\pi$$
$$360$$ 0 0
$$361$$ −2.34168e6 −0.945713
$$362$$ −6.23284e6 −2.49985
$$363$$ 51038.5 0.0203297
$$364$$ 3.13004e6 1.23822
$$365$$ 0 0
$$366$$ −597600. −0.233189
$$367$$ −1.77368e6 −0.687403 −0.343701 0.939079i $$-0.611681\pi$$
−0.343701 + 0.939079i $$0.611681\pi$$
$$368$$ −4.14389e6 −1.59510
$$369$$ 2.57506e6 0.984513
$$370$$ 0 0
$$371$$ 3.43933e6 1.29729
$$372$$ −323751. −0.121298
$$373$$ −2.27176e6 −0.845456 −0.422728 0.906257i $$-0.638927\pi$$
−0.422728 + 0.906257i $$0.638927\pi$$
$$374$$ 1.82358e6 0.674135
$$375$$ 0 0
$$376$$ −55265.7 −0.0201598
$$377$$ 1.05728e6 0.383122
$$378$$ −1.96205e6 −0.706287
$$379$$ −4.42409e6 −1.58207 −0.791035 0.611771i $$-0.790457\pi$$
−0.791035 + 0.611771i $$0.790457\pi$$
$$380$$ 0 0
$$381$$ 315009. 0.111176
$$382$$ 3.52607e6 1.23632
$$383$$ −2.37588e6 −0.827615 −0.413807 0.910364i $$-0.635801\pi$$
−0.413807 + 0.910364i $$0.635801\pi$$
$$384$$ 176831. 0.0611971
$$385$$ 0 0
$$386$$ −2.22939e6 −0.761586
$$387$$ 1.93185e6 0.655684
$$388$$ 1.34547e6 0.453728
$$389$$ −2.65905e6 −0.890949 −0.445474 0.895295i $$-0.646965\pi$$
−0.445474 + 0.895295i $$0.646965\pi$$
$$390$$ 0 0
$$391$$ −8.31319e6 −2.74996
$$392$$ −105461. −0.0346638
$$393$$ 227634. 0.0743457
$$394$$ 4.70425e6 1.52669
$$395$$ 0 0
$$396$$ −978724. −0.313633
$$397$$ −2.15712e6 −0.686907 −0.343453 0.939170i $$-0.611597\pi$$
−0.343453 + 0.939170i $$0.611597\pi$$
$$398$$ −2.21008e6 −0.699359
$$399$$ −185413. −0.0583054
$$400$$ 0 0
$$401$$ −2.43031e6 −0.754744 −0.377372 0.926062i $$-0.623172\pi$$
−0.377372 + 0.926062i $$0.623172\pi$$
$$402$$ −1.47586e6 −0.455492
$$403$$ 1.63213e6 0.500601
$$404$$ −1.43698e6 −0.438024
$$405$$ 0 0
$$406$$ −2.03946e6 −0.614045
$$407$$ −1.16893e6 −0.349787
$$408$$ 159650. 0.0474808
$$409$$ 6.12831e6 1.81148 0.905738 0.423839i $$-0.139318\pi$$
0.905738 + 0.423839i $$0.139318\pi$$
$$410$$ 0 0
$$411$$ −18555.1 −0.00541824
$$412$$ 1.73913e6 0.504765
$$413$$ −2.83147e6 −0.816841
$$414$$ 8.53649e6 2.44781
$$415$$ 0 0
$$416$$ −5.11621e6 −1.44949
$$417$$ 313625. 0.0883225
$$418$$ −363229. −0.101681
$$419$$ −375626. −0.104525 −0.0522626 0.998633i $$-0.516643\pi$$
−0.0522626 + 0.998633i $$0.516643\pi$$
$$420$$ 0 0
$$421$$ 3.52333e6 0.968831 0.484416 0.874838i $$-0.339032\pi$$
0.484416 + 0.874838i $$0.339032\pi$$
$$422$$ 6.16840e6 1.68613
$$423$$ −512764. −0.139337
$$424$$ 589870. 0.159346
$$425$$ 0 0
$$426$$ 39913.3 0.0106560
$$427$$ −3.03739e6 −0.806178
$$428$$ 237960. 0.0627906
$$429$$ −259736. −0.0681380
$$430$$ 0 0
$$431$$ 3.15287e6 0.817548 0.408774 0.912636i $$-0.365956\pi$$
0.408774 + 0.912636i $$0.365956\pi$$
$$432$$ 1.51560e6 0.390728
$$433$$ −1.62168e6 −0.415667 −0.207833 0.978164i $$-0.566641\pi$$
−0.207833 + 0.978164i $$0.566641\pi$$
$$434$$ −3.14832e6 −0.802333
$$435$$ 0 0
$$436$$ 3.39108e6 0.854322
$$437$$ 1.65586e6 0.414781
$$438$$ 2.06837e6 0.515161
$$439$$ −2.48145e6 −0.614533 −0.307266 0.951624i $$-0.599414\pi$$
−0.307266 + 0.951624i $$0.599414\pi$$
$$440$$ 0 0
$$441$$ −978482. −0.239583
$$442$$ −9.28026e6 −2.25946
$$443$$ −3.75466e6 −0.908994 −0.454497 0.890748i $$-0.650181\pi$$
−0.454497 + 0.890748i $$0.650181\pi$$
$$444$$ −1.18000e6 −0.284069
$$445$$ 0 0
$$446$$ −8.16882e6 −1.94456
$$447$$ 231783. 0.0548673
$$448$$ 5.60960e6 1.32049
$$449$$ −4.80916e6 −1.12578 −0.562890 0.826532i $$-0.690311\pi$$
−0.562890 + 0.826532i $$0.690311\pi$$
$$450$$ 0 0
$$451$$ −1.34973e6 −0.312468
$$452$$ 7.46108e6 1.71773
$$453$$ −456200. −0.104450
$$454$$ −4.05467e6 −0.923244
$$455$$ 0 0
$$456$$ −31799.7 −0.00716162
$$457$$ −7.09951e6 −1.59015 −0.795075 0.606512i $$-0.792568\pi$$
−0.795075 + 0.606512i $$0.792568\pi$$
$$458$$ 1.81680e6 0.404709
$$459$$ 3.04049e6 0.673616
$$460$$ 0 0
$$461$$ 8.12745e6 1.78116 0.890578 0.454830i $$-0.150300\pi$$
0.890578 + 0.454830i $$0.150300\pi$$
$$462$$ 501022. 0.109207
$$463$$ 2.67361e6 0.579623 0.289812 0.957084i $$-0.406407\pi$$
0.289812 + 0.957084i $$0.406407\pi$$
$$464$$ 1.57539e6 0.339699
$$465$$ 0 0
$$466$$ 5.07168e6 1.08190
$$467$$ 4.32733e6 0.918180 0.459090 0.888390i $$-0.348175\pi$$
0.459090 + 0.888390i $$0.348175\pi$$
$$468$$ 4.98075e6 1.05119
$$469$$ −7.50129e6 −1.57472
$$470$$ 0 0
$$471$$ 569253. 0.118237
$$472$$ −485618. −0.100332
$$473$$ −1.01259e6 −0.208103
$$474$$ −1.84476e6 −0.377133
$$475$$ 0 0
$$476$$ 9.35637e6 1.89274
$$477$$ 5.47291e6 1.10134
$$478$$ 2.41923e6 0.484293
$$479$$ 1.55878e6 0.310417 0.155208 0.987882i $$-0.450395\pi$$
0.155208 + 0.987882i $$0.450395\pi$$
$$480$$ 0 0
$$481$$ 5.94873e6 1.17236
$$482$$ −5.67535e6 −1.11269
$$483$$ −2.28402e6 −0.445483
$$484$$ 513003. 0.0995420
$$485$$ 0 0
$$486$$ −4.72328e6 −0.907095
$$487$$ 7.63818e6 1.45938 0.729689 0.683779i $$-0.239665\pi$$
0.729689 + 0.683779i $$0.239665\pi$$
$$488$$ −520935. −0.0990225
$$489$$ 1.86765e6 0.353202
$$490$$ 0 0
$$491$$ 3.60872e6 0.675537 0.337768 0.941229i $$-0.390328\pi$$
0.337768 + 0.941229i $$0.390328\pi$$
$$492$$ −1.36251e6 −0.253761
$$493$$ 3.16045e6 0.585640
$$494$$ 1.84848e6 0.340798
$$495$$ 0 0
$$496$$ 2.43194e6 0.443862
$$497$$ 202865. 0.0368397
$$498$$ −2.76263e6 −0.499171
$$499$$ −8.46131e6 −1.52120 −0.760599 0.649221i $$-0.775095\pi$$
−0.760599 + 0.649221i $$0.775095\pi$$
$$500$$ 0 0
$$501$$ −1.92980e6 −0.343494
$$502$$ −4.37074e6 −0.774098
$$503$$ 8.28353e6 1.45981 0.729904 0.683550i $$-0.239565\pi$$
0.729904 + 0.683550i $$0.239565\pi$$
$$504$$ −833241. −0.146115
$$505$$ 0 0
$$506$$ −4.47444e6 −0.776896
$$507$$ 27475.1 0.00474700
$$508$$ 3.16624e6 0.544358
$$509$$ −7.60138e6 −1.30046 −0.650232 0.759736i $$-0.725328\pi$$
−0.650232 + 0.759736i $$0.725328\pi$$
$$510$$ 0 0
$$511$$ 1.05128e7 1.78101
$$512$$ −8.35388e6 −1.40836
$$513$$ −605618. −0.101603
$$514$$ −5.34082e6 −0.891661
$$515$$ 0 0
$$516$$ −1.02217e6 −0.169005
$$517$$ 268768. 0.0442233
$$518$$ −1.14749e7 −1.87899
$$519$$ −930667. −0.151662
$$520$$ 0 0
$$521$$ 9.60432e6 1.55015 0.775073 0.631872i $$-0.217713\pi$$
0.775073 + 0.631872i $$0.217713\pi$$
$$522$$ −3.24534e6 −0.521295
$$523$$ 9.97831e6 1.59515 0.797577 0.603217i $$-0.206115\pi$$
0.797577 + 0.603217i $$0.206115\pi$$
$$524$$ 2.28802e6 0.364025
$$525$$ 0 0
$$526$$ −5.09313e6 −0.802640
$$527$$ 4.87879e6 0.765218
$$528$$ −387018. −0.0604151
$$529$$ 1.39613e7 2.16914
$$530$$ 0 0
$$531$$ −4.50565e6 −0.693459
$$532$$ −1.86364e6 −0.285485
$$533$$ 6.86881e6 1.04728
$$534$$ 1.35902e6 0.206240
$$535$$ 0 0
$$536$$ −1.28653e6 −0.193422
$$537$$ 10563.8 0.00158082
$$538$$ 3.95354e6 0.588885
$$539$$ 512876. 0.0760397
$$540$$ 0 0
$$541$$ −4.34177e6 −0.637784 −0.318892 0.947791i $$-0.603311\pi$$
−0.318892 + 0.947791i $$0.603311\pi$$
$$542$$ 9.06203e6 1.32503
$$543$$ 2.65369e6 0.386234
$$544$$ −1.52935e7 −2.21569
$$545$$ 0 0
$$546$$ −2.54971e6 −0.366024
$$547$$ −1.14668e7 −1.63860 −0.819302 0.573363i $$-0.805639\pi$$
−0.819302 + 0.573363i $$0.805639\pi$$
$$548$$ −186503. −0.0265298
$$549$$ −4.83332e6 −0.684407
$$550$$ 0 0
$$551$$ −629511. −0.0883332
$$552$$ −391726. −0.0547185
$$553$$ −9.37627e6 −1.30382
$$554$$ 5.23246e6 0.724322
$$555$$ 0 0
$$556$$ 3.15234e6 0.432460
$$557$$ 1.57100e6 0.214555 0.107278 0.994229i $$-0.465787\pi$$
0.107278 + 0.994229i $$0.465787\pi$$
$$558$$ −5.00983e6 −0.681142
$$559$$ 5.15307e6 0.697488
$$560$$ 0 0
$$561$$ −776409. −0.104156
$$562$$ 2.11230e6 0.282108
$$563$$ 850908. 0.113139 0.0565694 0.998399i $$-0.481984\pi$$
0.0565694 + 0.998399i $$0.481984\pi$$
$$564$$ 271312. 0.0359146
$$565$$ 0 0
$$566$$ 8.43265e6 1.10643
$$567$$ −7.30255e6 −0.953931
$$568$$ 34792.9 0.00452501
$$569$$ −1.19642e7 −1.54919 −0.774595 0.632458i $$-0.782046\pi$$
−0.774595 + 0.632458i $$0.782046\pi$$
$$570$$ 0 0
$$571$$ −7.97842e6 −1.02406 −0.512032 0.858967i $$-0.671107\pi$$
−0.512032 + 0.858967i $$0.671107\pi$$
$$572$$ −2.61068e6 −0.333629
$$573$$ −1.50126e6 −0.191016
$$574$$ −1.32497e7 −1.67852
$$575$$ 0 0
$$576$$ 8.92640e6 1.12104
$$577$$ −5.90743e6 −0.738685 −0.369342 0.929293i $$-0.620417\pi$$
−0.369342 + 0.929293i $$0.620417\pi$$
$$578$$ −1.61153e7 −2.00641
$$579$$ 949186. 0.117667
$$580$$ 0 0
$$581$$ −1.40415e7 −1.72573
$$582$$ −1.09602e6 −0.134125
$$583$$ −2.86865e6 −0.349548
$$584$$ 1.80303e6 0.218761
$$585$$ 0 0
$$586$$ 7.18646e6 0.864512
$$587$$ −1.34766e6 −0.161430 −0.0807151 0.996737i $$-0.525720\pi$$
−0.0807151 + 0.996737i $$0.525720\pi$$
$$588$$ 517730. 0.0617533
$$589$$ −971777. −0.115419
$$590$$ 0 0
$$591$$ −2.00288e6 −0.235877
$$592$$ 8.86385e6 1.03948
$$593$$ −1.05883e7 −1.23649 −0.618243 0.785987i $$-0.712155\pi$$
−0.618243 + 0.785987i $$0.712155\pi$$
$$594$$ 1.63650e6 0.190304
$$595$$ 0 0
$$596$$ 2.32972e6 0.268651
$$597$$ 940962. 0.108053
$$598$$ 2.27705e7 2.60388
$$599$$ 3.48377e6 0.396718 0.198359 0.980129i $$-0.436439\pi$$
0.198359 + 0.980129i $$0.436439\pi$$
$$600$$ 0 0
$$601$$ −6.41433e6 −0.724378 −0.362189 0.932105i $$-0.617970\pi$$
−0.362189 + 0.932105i $$0.617970\pi$$
$$602$$ −9.94010e6 −1.11789
$$603$$ −1.19366e7 −1.33686
$$604$$ −4.58540e6 −0.511428
$$605$$ 0 0
$$606$$ 1.17056e6 0.129483
$$607$$ 700912. 0.0772132 0.0386066 0.999254i $$-0.487708\pi$$
0.0386066 + 0.999254i $$0.487708\pi$$
$$608$$ 3.04622e6 0.334197
$$609$$ 868320. 0.0948717
$$610$$ 0 0
$$611$$ −1.36776e6 −0.148221
$$612$$ 1.48885e7 1.60684
$$613$$ 1.17591e7 1.26393 0.631966 0.774996i $$-0.282248\pi$$
0.631966 + 0.774996i $$0.282248\pi$$
$$614$$ −1.07198e7 −1.14753
$$615$$ 0 0
$$616$$ 436747. 0.0463744
$$617$$ 1.00683e6 0.106474 0.0532371 0.998582i $$-0.483046\pi$$
0.0532371 + 0.998582i $$0.483046\pi$$
$$618$$ −1.41669e6 −0.149212
$$619$$ 1.27458e7 1.33703 0.668513 0.743700i $$-0.266931\pi$$
0.668513 + 0.743700i $$0.266931\pi$$
$$620$$ 0 0
$$621$$ −7.46031e6 −0.776297
$$622$$ −2.75050e7 −2.85060
$$623$$ 6.90744e6 0.713012
$$624$$ 1.96954e6 0.202490
$$625$$ 0 0
$$626$$ −2.46155e7 −2.51058
$$627$$ 154648. 0.0157100
$$628$$ 5.72172e6 0.578932
$$629$$ 1.77821e7 1.79207
$$630$$ 0 0
$$631$$ 1.41284e7 1.41260 0.706299 0.707913i $$-0.250363\pi$$
0.706299 + 0.707913i $$0.250363\pi$$
$$632$$ −1.60810e6 −0.160148
$$633$$ −2.62626e6 −0.260512
$$634$$ 1.72062e7 1.70005
$$635$$ 0 0
$$636$$ −2.89580e6 −0.283874
$$637$$ −2.61004e6 −0.254858
$$638$$ 1.70106e6 0.165450
$$639$$ 322814. 0.0312752
$$640$$ 0 0
$$641$$ −4.36680e6 −0.419777 −0.209888 0.977725i $$-0.567310\pi$$
−0.209888 + 0.977725i $$0.567310\pi$$
$$642$$ −193841. −0.0185613
$$643$$ −7.81597e6 −0.745513 −0.372757 0.927929i $$-0.621587\pi$$
−0.372757 + 0.927929i $$0.621587\pi$$
$$644$$ −2.29573e7 −2.18125
$$645$$ 0 0
$$646$$ 5.52551e6 0.520944
$$647$$ −2.01624e7 −1.89357 −0.946786 0.321863i $$-0.895691\pi$$
−0.946786 + 0.321863i $$0.895691\pi$$
$$648$$ −1.25244e6 −0.117171
$$649$$ 2.36166e6 0.220092
$$650$$ 0 0
$$651$$ 1.34043e6 0.123963
$$652$$ 1.87723e7 1.72941
$$653$$ −324619. −0.0297914 −0.0148957 0.999889i $$-0.504742\pi$$
−0.0148957 + 0.999889i $$0.504742\pi$$
$$654$$ −2.76235e6 −0.252543
$$655$$ 0 0
$$656$$ 1.02348e7 0.928581
$$657$$ 1.67288e7 1.51199
$$658$$ 2.63837e6 0.237559
$$659$$ −1.07107e7 −0.960740 −0.480370 0.877066i $$-0.659498\pi$$
−0.480370 + 0.877066i $$0.659498\pi$$
$$660$$ 0 0
$$661$$ −1.11064e7 −0.988712 −0.494356 0.869260i $$-0.664596\pi$$
−0.494356 + 0.869260i $$0.664596\pi$$
$$662$$ −1.00986e7 −0.895601
$$663$$ 3.95116e6 0.349093
$$664$$ −2.40822e6 −0.211971
$$665$$ 0 0
$$666$$ −1.82597e7 −1.59517
$$667$$ −7.75464e6 −0.674912
$$668$$ −1.93970e7 −1.68187
$$669$$ 3.47795e6 0.300441
$$670$$ 0 0
$$671$$ 2.53341e6 0.217219
$$672$$ −4.20182e6 −0.358934
$$673$$ 1.38137e7 1.17564 0.587818 0.808993i $$-0.299987\pi$$
0.587818 + 0.808993i $$0.299987\pi$$
$$674$$ 5.56208e6 0.471615
$$675$$ 0 0
$$676$$ 276160. 0.0232431
$$677$$ −2.29090e6 −0.192103 −0.0960514 0.995376i $$-0.530621\pi$$
−0.0960514 + 0.995376i $$0.530621\pi$$
$$678$$ −6.07775e6 −0.507772
$$679$$ −5.57067e6 −0.463695
$$680$$ 0 0
$$681$$ 1.72632e6 0.142644
$$682$$ 2.62593e6 0.216183
$$683$$ −4.40512e6 −0.361332 −0.180666 0.983545i $$-0.557825\pi$$
−0.180666 + 0.983545i $$0.557825\pi$$
$$684$$ −2.96556e6 −0.242363
$$685$$ 0 0
$$686$$ −1.49287e7 −1.21119
$$687$$ −773519. −0.0625287
$$688$$ 7.67828e6 0.618434
$$689$$ 1.45986e7 1.17156
$$690$$ 0 0
$$691$$ −5.86199e6 −0.467035 −0.233518 0.972353i $$-0.575024\pi$$
−0.233518 + 0.972353i $$0.575024\pi$$
$$692$$ −9.35440e6 −0.742592
$$693$$ 4.05221e6 0.320523
$$694$$ 2.19054e7 1.72645
$$695$$ 0 0
$$696$$ 148923. 0.0116530
$$697$$ 2.05324e7 1.60087
$$698$$ 1.94569e7 1.51160
$$699$$ −2.15932e6 −0.167157
$$700$$ 0 0
$$701$$ −8.02106e6 −0.616505 −0.308253 0.951305i $$-0.599744\pi$$
−0.308253 + 0.951305i $$0.599744\pi$$
$$702$$ −8.32816e6 −0.637832
$$703$$ −3.54190e6 −0.270301
$$704$$ −4.67881e6 −0.355799
$$705$$ 0 0
$$706$$ 5.22946e6 0.394862
$$707$$ 5.94954e6 0.447646
$$708$$ 2.38401e6 0.178741
$$709$$ 2.17891e7 1.62788 0.813941 0.580948i $$-0.197318\pi$$
0.813941 + 0.580948i $$0.197318\pi$$
$$710$$ 0 0
$$711$$ −1.49202e7 −1.10688
$$712$$ 1.18468e6 0.0875789
$$713$$ −1.19709e7 −0.881863
$$714$$ −7.62165e6 −0.559505
$$715$$ 0 0
$$716$$ 106179. 0.00774029
$$717$$ −1.03001e6 −0.0748246
$$718$$ −1.23505e7 −0.894076
$$719$$ 1.03483e7 0.746531 0.373266 0.927724i $$-0.378238\pi$$
0.373266 + 0.927724i $$0.378238\pi$$
$$720$$ 0 0
$$721$$ −7.20052e6 −0.515853
$$722$$ 1.91730e7 1.36882
$$723$$ 2.41633e6 0.171914
$$724$$ 2.66730e7 1.89115
$$725$$ 0 0
$$726$$ −417889. −0.0294252
$$727$$ 2.03348e7 1.42693 0.713466 0.700690i $$-0.247124\pi$$
0.713466 + 0.700690i $$0.247124\pi$$
$$728$$ −2.22262e6 −0.155430
$$729$$ −1.02211e7 −0.712325
$$730$$ 0 0
$$731$$ 1.54037e7 1.06618
$$732$$ 2.55739e6 0.176408
$$733$$ −4.78280e6 −0.328793 −0.164396 0.986394i $$-0.552568\pi$$
−0.164396 + 0.986394i $$0.552568\pi$$
$$734$$ 1.45224e7 0.994946
$$735$$ 0 0
$$736$$ 3.75249e7 2.55344
$$737$$ 6.25663e6 0.424298
$$738$$ −2.10839e7 −1.42498
$$739$$ −1.08737e7 −0.732429 −0.366215 0.930530i $$-0.619346\pi$$
−0.366215 + 0.930530i $$0.619346\pi$$
$$740$$ 0 0
$$741$$ −787008. −0.0526543
$$742$$ −2.81603e7 −1.87770
$$743$$ 1.01036e7 0.671434 0.335717 0.941963i $$-0.391021\pi$$
0.335717 + 0.941963i $$0.391021\pi$$
$$744$$ 229893. 0.0152263
$$745$$ 0 0
$$746$$ 1.86006e7 1.22371
$$747$$ −2.23438e7 −1.46506
$$748$$ −7.80390e6 −0.509986
$$749$$ −985227. −0.0641700
$$750$$ 0 0
$$751$$ 9.91947e6 0.641784 0.320892 0.947116i $$-0.396017\pi$$
0.320892 + 0.947116i $$0.396017\pi$$
$$752$$ −2.03802e6 −0.131421
$$753$$ 1.86088e6 0.119600
$$754$$ −8.65673e6 −0.554530
$$755$$ 0 0
$$756$$ 8.39647e6 0.534309
$$757$$ 1.33506e7 0.846759 0.423380 0.905952i $$-0.360844\pi$$
0.423380 + 0.905952i $$0.360844\pi$$
$$758$$ 3.62232e7 2.28988
$$759$$ 1.90504e6 0.120033
$$760$$ 0 0
$$761$$ 1.28109e7 0.801898 0.400949 0.916100i $$-0.368680\pi$$
0.400949 + 0.916100i $$0.368680\pi$$
$$762$$ −2.57920e6 −0.160916
$$763$$ −1.40401e7 −0.873089
$$764$$ −1.50896e7 −0.935284
$$765$$ 0 0
$$766$$ 1.94531e7 1.19789
$$767$$ −1.20185e7 −0.737671
$$768$$ 2.86564e6 0.175315
$$769$$ 1.90629e7 1.16245 0.581224 0.813743i $$-0.302574\pi$$
0.581224 + 0.813743i $$0.302574\pi$$
$$770$$ 0 0
$$771$$ 2.27390e6 0.137764
$$772$$ 9.54054e6 0.576142
$$773$$ 2.34434e6 0.141115 0.0705574 0.997508i $$-0.477522\pi$$
0.0705574 + 0.997508i $$0.477522\pi$$
$$774$$ −1.58174e7 −0.949037
$$775$$ 0 0
$$776$$ −955410. −0.0569555
$$777$$ 4.88555e6 0.290309
$$778$$ 2.17716e7 1.28956
$$779$$ −4.08972e6 −0.241463
$$780$$ 0 0
$$781$$ −169204. −0.00992623
$$782$$ 6.80661e7 3.98028
$$783$$ 2.83620e6 0.165323
$$784$$ −3.88906e6 −0.225972
$$785$$ 0 0
$$786$$ −1.86381e6 −0.107608
$$787$$ 1.52283e7 0.876425 0.438212 0.898871i $$-0.355612\pi$$
0.438212 + 0.898871i $$0.355612\pi$$
$$788$$ −2.01315e7 −1.15494
$$789$$ 2.16845e6 0.124010
$$790$$ 0 0
$$791$$ −3.08911e7 −1.75547
$$792$$ 694984. 0.0393697
$$793$$ −1.28926e7 −0.728042
$$794$$ 1.76619e7 0.994228
$$795$$ 0 0
$$796$$ 9.45788e6 0.529068
$$797$$ 2.70618e7 1.50907 0.754537 0.656257i $$-0.227861\pi$$
0.754537 + 0.656257i $$0.227861\pi$$
$$798$$ 1.51811e6 0.0843911
$$799$$ −4.08855e6 −0.226570
$$800$$ 0 0
$$801$$ 1.09916e7 0.605313
$$802$$ 1.98987e7 1.09242
$$803$$ −8.76846e6 −0.479882
$$804$$ 6.31585e6 0.344581
$$805$$ 0 0
$$806$$ −1.33634e7 −0.724569
$$807$$ −1.68326e6 −0.0909844
$$808$$ 1.02039e6 0.0549842
$$809$$ −2.25663e6 −0.121224 −0.0606121 0.998161i $$-0.519305\pi$$
−0.0606121 + 0.998161i $$0.519305\pi$$
$$810$$ 0 0
$$811$$ 4.49121e6 0.239779 0.119890 0.992787i $$-0.461746\pi$$
0.119890 + 0.992787i $$0.461746\pi$$
$$812$$ 8.72773e6 0.464527
$$813$$ −3.85825e6 −0.204722
$$814$$ 9.57091e6 0.506282
$$815$$ 0 0
$$816$$ 5.88739e6 0.309526
$$817$$ −3.06816e6 −0.160814
$$818$$ −5.01769e7 −2.62193
$$819$$ −2.06218e7 −1.07428
$$820$$ 0 0
$$821$$ 592581. 0.0306825 0.0153412 0.999882i $$-0.495117\pi$$
0.0153412 + 0.999882i $$0.495117\pi$$
$$822$$ 151924. 0.00784236
$$823$$ 1.14748e7 0.590533 0.295266 0.955415i $$-0.404592\pi$$
0.295266 + 0.955415i $$0.404592\pi$$
$$824$$ −1.23494e6 −0.0633620
$$825$$ 0 0
$$826$$ 2.31833e7 1.18229
$$827$$ 8.47060e6 0.430676 0.215338 0.976540i $$-0.430915\pi$$
0.215338 + 0.976540i $$0.430915\pi$$
$$828$$ −3.65313e7 −1.85178
$$829$$ −1.58876e7 −0.802919 −0.401460 0.915877i $$-0.631497\pi$$
−0.401460 + 0.915877i $$0.631497\pi$$
$$830$$ 0 0
$$831$$ −2.22777e6 −0.111910
$$832$$ 2.38106e7 1.19251
$$833$$ −7.80197e6 −0.389576
$$834$$ −2.56788e6 −0.127838
$$835$$ 0 0
$$836$$ 1.55441e6 0.0769221
$$837$$ 4.37825e6 0.216017
$$838$$ 3.07552e6 0.151290
$$839$$ 2.66963e7 1.30932 0.654659 0.755924i $$-0.272812\pi$$
0.654659 + 0.755924i $$0.272812\pi$$
$$840$$ 0 0
$$841$$ −1.75631e7 −0.856268
$$842$$ −2.88480e7 −1.40228
$$843$$ −899333. −0.0435864
$$844$$ −2.63972e7 −1.27556
$$845$$ 0 0
$$846$$ 4.19837e6 0.201676
$$847$$ −2.12399e6 −0.101729
$$848$$ 2.17525e7 1.03877
$$849$$ −3.59028e6 −0.170946
$$850$$ 0 0
$$851$$ −4.36310e7 −2.06524
$$852$$ −170806. −0.00806128
$$853$$ −3.58773e6 −0.168829 −0.0844146 0.996431i $$-0.526902\pi$$
−0.0844146 + 0.996431i $$0.526902\pi$$
$$854$$ 2.48693e7 1.16686
$$855$$ 0 0
$$856$$ −168974. −0.00788197
$$857$$ 6.00941e6 0.279499 0.139749 0.990187i $$-0.455370\pi$$
0.139749 + 0.990187i $$0.455370\pi$$
$$858$$ 2.12665e6 0.0986228
$$859$$ −1.74629e7 −0.807484 −0.403742 0.914873i $$-0.632291\pi$$
−0.403742 + 0.914873i $$0.632291\pi$$
$$860$$ 0 0
$$861$$ 5.64118e6 0.259336
$$862$$ −2.58149e7 −1.18332
$$863$$ −2.34431e6 −0.107149 −0.0535746 0.998564i $$-0.517061\pi$$
−0.0535746 + 0.998564i $$0.517061\pi$$
$$864$$ −1.37245e7 −0.625476
$$865$$ 0 0
$$866$$ 1.32779e7 0.601635
$$867$$ 6.86126e6 0.309996
$$868$$ 1.34730e7 0.606968
$$869$$ 7.82050e6 0.351306
$$870$$ 0 0
$$871$$ −3.18401e7 −1.42210
$$872$$ −2.40798e6 −0.107241
$$873$$ −8.86445e6 −0.393655
$$874$$ −1.35577e7 −0.600354
$$875$$ 0 0
$$876$$ −8.85145e6 −0.389721
$$877$$ −1.98979e7 −0.873591 −0.436796 0.899561i $$-0.643887\pi$$
−0.436796 + 0.899561i $$0.643887\pi$$
$$878$$ 2.03175e7 0.889474
$$879$$ −3.05971e6 −0.133570
$$880$$ 0 0
$$881$$ 2.32718e7 1.01016 0.505081 0.863072i $$-0.331463\pi$$
0.505081 + 0.863072i $$0.331463\pi$$
$$882$$ 8.01154e6 0.346772
$$883$$ 2.71777e7 1.17304 0.586518 0.809936i $$-0.300498\pi$$
0.586518 + 0.809936i $$0.300498\pi$$
$$884$$ 3.97142e7 1.70929
$$885$$ 0 0
$$886$$ 3.07421e7 1.31568
$$887$$ 1.39671e7 0.596069 0.298034 0.954555i $$-0.403669\pi$$
0.298034 + 0.954555i $$0.403669\pi$$
$$888$$ 837907. 0.0356585
$$889$$ −1.31092e7 −0.556316
$$890$$ 0 0
$$891$$ 6.09086e6 0.257030
$$892$$ 3.49579e7 1.47107
$$893$$ 814374. 0.0341740
$$894$$ −1.89778e6 −0.0794149
$$895$$ 0 0
$$896$$ −7.35889e6 −0.306226
$$897$$ −9.69477e6 −0.402306
$$898$$ 3.93761e7 1.62945
$$899$$ 4.55099e6 0.187805
$$900$$ 0 0
$$901$$ 4.36385e7 1.79084
$$902$$ 1.10512e7 0.452266
$$903$$ 4.23209e6 0.172717
$$904$$ −5.29805e6 −0.215623
$$905$$ 0 0
$$906$$ 3.73524e6 0.151181
$$907$$ −1.77875e7 −0.717954 −0.358977 0.933346i $$-0.616874\pi$$
−0.358977 + 0.933346i $$0.616874\pi$$
$$908$$ 1.73517e7 0.698437
$$909$$ 9.46734e6 0.380031
$$910$$ 0 0
$$911$$ 30398.8 0.00121356 0.000606780 1.00000i $$-0.499807\pi$$
0.000606780 1.00000i $$0.499807\pi$$
$$912$$ −1.17267e6 −0.0466864
$$913$$ 1.17116e7 0.464987
$$914$$ 5.81288e7 2.30158
$$915$$ 0 0
$$916$$ −7.77486e6 −0.306164
$$917$$ −9.47308e6 −0.372021
$$918$$ −2.48947e7 −0.974990
$$919$$ −4.10055e6 −0.160160 −0.0800798 0.996788i $$-0.525518\pi$$
−0.0800798 + 0.996788i $$0.525518\pi$$
$$920$$ 0 0
$$921$$ 4.56406e6 0.177297
$$922$$ −6.65453e7 −2.57804
$$923$$ 861085. 0.0332692
$$924$$ −2.14409e6 −0.0826158
$$925$$ 0 0
$$926$$ −2.18908e7 −0.838946
$$927$$ −1.14580e7 −0.437935
$$928$$ −1.42659e7 −0.543788
$$929$$ −1.46532e7 −0.557048 −0.278524 0.960429i $$-0.589845\pi$$
−0.278524 + 0.960429i $$0.589845\pi$$
$$930$$ 0 0
$$931$$ 1.55403e6 0.0587604
$$932$$ −2.17039e7 −0.818461
$$933$$ 1.17105e7 0.440425
$$934$$ −3.54310e7 −1.32897
$$935$$ 0 0
$$936$$ −3.53679e6 −0.131953
$$937$$ 3.97538e7 1.47921 0.739604 0.673042i $$-0.235013\pi$$
0.739604 + 0.673042i $$0.235013\pi$$
$$938$$ 6.14185e7 2.27925
$$939$$ 1.04803e7 0.387891
$$940$$ 0 0
$$941$$ 5.32850e6 0.196169 0.0980847 0.995178i $$-0.468728\pi$$
0.0980847 + 0.995178i $$0.468728\pi$$
$$942$$ −4.66088e6 −0.171136
$$943$$ −5.03793e7 −1.84490
$$944$$ −1.79081e7 −0.654062
$$945$$ 0 0
$$946$$ 8.29077e6 0.301208
$$947$$ −3.11430e7 −1.12846 −0.564230 0.825618i $$-0.690827\pi$$
−0.564230 + 0.825618i $$0.690827\pi$$
$$948$$ 7.89452e6 0.285302
$$949$$ 4.46229e7 1.60839
$$950$$ 0 0
$$951$$ −7.32571e6 −0.262663
$$952$$ −6.64389e6 −0.237591
$$953$$ −4.87227e7 −1.73780 −0.868899 0.494990i $$-0.835172\pi$$
−0.868899 + 0.494990i $$0.835172\pi$$
$$954$$ −4.48106e7 −1.59408
$$955$$ 0 0
$$956$$ −1.03529e7 −0.366369
$$957$$ −724242. −0.0255625
$$958$$ −1.27628e7 −0.449297
$$959$$ 772177. 0.0271125
$$960$$ 0 0
$$961$$ −2.16038e7 −0.754608
$$962$$ −4.87065e7 −1.69687
$$963$$ −1.56776e6 −0.0544773
$$964$$ 2.42873e7 0.841755
$$965$$ 0 0
$$966$$ 1.87009e7 0.644792
$$967$$ −4.85436e7 −1.66942 −0.834711 0.550688i $$-0.814365\pi$$
−0.834711 + 0.550688i $$0.814365\pi$$
$$968$$ −364279. −0.0124953
$$969$$ −2.35254e6 −0.0804873
$$970$$ 0 0
$$971$$ −3.15035e7 −1.07229 −0.536143 0.844127i $$-0.680119\pi$$
−0.536143 + 0.844127i $$0.680119\pi$$
$$972$$ 2.02129e7 0.686221
$$973$$ −1.30516e7 −0.441960
$$974$$ −6.25393e7 −2.11230
$$975$$ 0 0
$$976$$ −1.92104e7 −0.645525
$$977$$ 5.35354e7 1.79434 0.897170 0.441684i $$-0.145619\pi$$
0.897170 + 0.441684i $$0.145619\pi$$
$$978$$ −1.52918e7 −0.511225
$$979$$ −5.76131e6 −0.192116
$$980$$ 0 0
$$981$$ −2.23416e7 −0.741211
$$982$$ −2.95472e7 −0.977771
$$983$$ 5.40925e7 1.78547 0.892736 0.450580i $$-0.148783\pi$$
0.892736 + 0.450580i $$0.148783\pi$$
$$984$$ 967505. 0.0318541
$$985$$ 0 0
$$986$$ −2.58769e7 −0.847655
$$987$$ −1.12331e6 −0.0367035
$$988$$ −7.91044e6 −0.257815
$$989$$ −3.77952e7 −1.22870
$$990$$ 0 0
$$991$$ 2.31007e7 0.747208 0.373604 0.927588i $$-0.378122\pi$$
0.373604 + 0.927588i $$0.378122\pi$$
$$992$$ −2.20223e7 −0.710533
$$993$$ 4.29956e6 0.138373
$$994$$ −1.66100e6 −0.0533218
$$995$$ 0 0
$$996$$ 1.18225e7 0.377625
$$997$$ 4.54061e7 1.44669 0.723346 0.690486i $$-0.242603\pi$$
0.723346 + 0.690486i $$0.242603\pi$$
$$998$$ 6.92788e7 2.20178
$$999$$ 1.59577e7 0.505891
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 275.6.a.b.1.1 3
5.2 odd 4 275.6.b.b.199.2 6
5.3 odd 4 275.6.b.b.199.5 6
5.4 even 2 11.6.a.b.1.3 3
15.14 odd 2 99.6.a.g.1.1 3
20.19 odd 2 176.6.a.i.1.3 3
35.34 odd 2 539.6.a.e.1.3 3
40.19 odd 2 704.6.a.t.1.1 3
40.29 even 2 704.6.a.q.1.3 3
55.54 odd 2 121.6.a.d.1.1 3
165.164 even 2 1089.6.a.r.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.3 3 5.4 even 2
99.6.a.g.1.1 3 15.14 odd 2
121.6.a.d.1.1 3 55.54 odd 2
176.6.a.i.1.3 3 20.19 odd 2
275.6.a.b.1.1 3 1.1 even 1 trivial
275.6.b.b.199.2 6 5.2 odd 4
275.6.b.b.199.5 6 5.3 odd 4
539.6.a.e.1.3 3 35.34 odd 2
704.6.a.q.1.3 3 40.29 even 2
704.6.a.t.1.1 3 40.19 odd 2
1089.6.a.r.1.3 3 165.164 even 2