# Properties

 Label 6960.2.a.co.1.1 Level $6960$ Weight $2$ Character 6960.1 Self dual yes Analytic conductor $55.576$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6960,2,Mod(1,6960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6960.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: $$55.5758798068$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.13856$$ of defining polynomial Character $$\chi$$ $$=$$ 6960.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} -5.07830 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} -5.07830 q^{7} +1.00000 q^{9} +5.07830 q^{11} -3.67096 q^{13} -1.00000 q^{15} -2.60617 q^{17} +6.74926 q^{19} -5.07830 q^{21} +3.21234 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} +0.787665 q^{31} +5.07830 q^{33} +5.07830 q^{35} -5.13021 q^{37} -3.67096 q^{39} -8.81469 q^{41} +2.61968 q^{43} -1.00000 q^{45} +1.11670 q^{47} +18.7892 q^{49} -2.60617 q^{51} +0.619678 q^{53} -5.07830 q^{55} +6.74926 q^{57} -12.7763 q^{59} +8.90587 q^{61} -5.07830 q^{63} +3.67096 q^{65} +0.524047 q^{67} +3.21234 q^{69} -0.195007 q^{71} +1.13021 q^{73} +1.00000 q^{75} -25.7892 q^{77} -15.3035 q^{79} +1.00000 q^{81} +18.1182 q^{83} +2.60617 q^{85} -1.00000 q^{87} -15.5942 q^{89} +18.6423 q^{91} +0.787665 q^{93} -6.74926 q^{95} -12.6671 q^{97} +5.07830 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^5 - 2 * q^7 + 4 * q^9 $$4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 10 q^{17} + 2 q^{19} - 2 q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 16 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} + 12 q^{47} + 6 q^{49} - 10 q^{51} - 10 q^{53} - 2 q^{55} + 2 q^{57} - 2 q^{59} - 26 q^{61} - 2 q^{63} + 8 q^{65} - 2 q^{67} + 12 q^{69} + 10 q^{71} + 4 q^{75} - 34 q^{77} - 22 q^{79} + 4 q^{81} + 10 q^{83} + 10 q^{85} - 4 q^{87} - 4 q^{89} + 8 q^{91} + 4 q^{93} - 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 4 * q^5 - 2 * q^7 + 4 * q^9 + 2 * q^11 - 8 * q^13 - 4 * q^15 - 10 * q^17 + 2 * q^19 - 2 * q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 - 4 * q^29 + 4 * q^31 + 2 * q^33 + 2 * q^35 - 16 * q^37 - 8 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 + 12 * q^47 + 6 * q^49 - 10 * q^51 - 10 * q^53 - 2 * q^55 + 2 * q^57 - 2 * q^59 - 26 * q^61 - 2 * q^63 + 8 * q^65 - 2 * q^67 + 12 * q^69 + 10 * q^71 + 4 * q^75 - 34 * q^77 - 22 * q^79 + 4 * q^81 + 10 * q^83 + 10 * q^85 - 4 * q^87 - 4 * q^89 + 8 * q^91 + 4 * q^93 - 2 * q^95 - 22 * q^97 + 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −5.07830 −1.91942 −0.959709 0.280995i $$-0.909336\pi$$
−0.959709 + 0.280995i $$0.909336\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.07830 1.53117 0.765583 0.643337i $$-0.222451\pi$$
0.765583 + 0.643337i $$0.222451\pi$$
$$12$$ 0 0
$$13$$ −3.67096 −1.01814 −0.509071 0.860725i $$-0.670011\pi$$
−0.509071 + 0.860725i $$0.670011\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −2.60617 −0.632089 −0.316044 0.948744i $$-0.602355\pi$$
−0.316044 + 0.948744i $$0.602355\pi$$
$$18$$ 0 0
$$19$$ 6.74926 1.54839 0.774194 0.632949i $$-0.218156\pi$$
0.774194 + 0.632949i $$0.218156\pi$$
$$20$$ 0 0
$$21$$ −5.07830 −1.10818
$$22$$ 0 0
$$23$$ 3.21234 0.669818 0.334909 0.942250i $$-0.391294\pi$$
0.334909 + 0.942250i $$0.391294\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 0.787665 0.141469 0.0707344 0.997495i $$-0.477466\pi$$
0.0707344 + 0.997495i $$0.477466\pi$$
$$32$$ 0 0
$$33$$ 5.07830 0.884019
$$34$$ 0 0
$$35$$ 5.07830 0.858390
$$36$$ 0 0
$$37$$ −5.13021 −0.843402 −0.421701 0.906735i $$-0.638567\pi$$
−0.421701 + 0.906735i $$0.638567\pi$$
$$38$$ 0 0
$$39$$ −3.67096 −0.587824
$$40$$ 0 0
$$41$$ −8.81469 −1.37662 −0.688311 0.725415i $$-0.741648\pi$$
−0.688311 + 0.725415i $$0.741648\pi$$
$$42$$ 0 0
$$43$$ 2.61968 0.399497 0.199749 0.979847i $$-0.435987\pi$$
0.199749 + 0.979847i $$0.435987\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 1.11670 0.162888 0.0814440 0.996678i $$-0.474047\pi$$
0.0814440 + 0.996678i $$0.474047\pi$$
$$48$$ 0 0
$$49$$ 18.7892 2.68417
$$50$$ 0 0
$$51$$ −2.60617 −0.364936
$$52$$ 0 0
$$53$$ 0.619678 0.0851194 0.0425597 0.999094i $$-0.486449\pi$$
0.0425597 + 0.999094i $$0.486449\pi$$
$$54$$ 0 0
$$55$$ −5.07830 −0.684758
$$56$$ 0 0
$$57$$ 6.74926 0.893962
$$58$$ 0 0
$$59$$ −12.7763 −1.66333 −0.831665 0.555277i $$-0.812612\pi$$
−0.831665 + 0.555277i $$0.812612\pi$$
$$60$$ 0 0
$$61$$ 8.90587 1.14028 0.570140 0.821548i $$-0.306889\pi$$
0.570140 + 0.821548i $$0.306889\pi$$
$$62$$ 0 0
$$63$$ −5.07830 −0.639806
$$64$$ 0 0
$$65$$ 3.67096 0.455327
$$66$$ 0 0
$$67$$ 0.524047 0.0640225 0.0320112 0.999488i $$-0.489809\pi$$
0.0320112 + 0.999488i $$0.489809\pi$$
$$68$$ 0 0
$$69$$ 3.21234 0.386720
$$70$$ 0 0
$$71$$ −0.195007 −0.0231431 −0.0115716 0.999933i $$-0.503683\pi$$
−0.0115716 + 0.999933i $$0.503683\pi$$
$$72$$ 0 0
$$73$$ 1.13021 0.132282 0.0661408 0.997810i $$-0.478931\pi$$
0.0661408 + 0.997810i $$0.478931\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −25.7892 −2.93895
$$78$$ 0 0
$$79$$ −15.3035 −1.72178 −0.860890 0.508791i $$-0.830093\pi$$
−0.860890 + 0.508791i $$0.830093\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 18.1182 1.98873 0.994366 0.106003i $$-0.0338053\pi$$
0.994366 + 0.106003i $$0.0338053\pi$$
$$84$$ 0 0
$$85$$ 2.60617 0.282679
$$86$$ 0 0
$$87$$ −1.00000 −0.107211
$$88$$ 0 0
$$89$$ −15.5942 −1.65298 −0.826489 0.562953i $$-0.809665\pi$$
−0.826489 + 0.562953i $$0.809665\pi$$
$$90$$ 0 0
$$91$$ 18.6423 1.95424
$$92$$ 0 0
$$93$$ 0.787665 0.0816770
$$94$$ 0 0
$$95$$ −6.74926 −0.692460
$$96$$ 0 0
$$97$$ −12.6671 −1.28615 −0.643077 0.765802i $$-0.722342\pi$$
−0.643077 + 0.765802i $$0.722342\pi$$
$$98$$ 0 0
$$99$$ 5.07830 0.510389
$$100$$ 0 0
$$101$$ 7.03990 0.700497 0.350248 0.936657i $$-0.386097\pi$$
0.350248 + 0.936657i $$0.386097\pi$$
$$102$$ 0 0
$$103$$ −2.65808 −0.261908 −0.130954 0.991388i $$-0.541804\pi$$
−0.130954 + 0.991388i $$0.541804\pi$$
$$104$$ 0 0
$$105$$ 5.07830 0.495592
$$106$$ 0 0
$$107$$ −4.81469 −0.465453 −0.232727 0.972542i $$-0.574765\pi$$
−0.232727 + 0.972542i $$0.574765\pi$$
$$108$$ 0 0
$$109$$ 3.86597 0.370293 0.185146 0.982711i $$-0.440724\pi$$
0.185146 + 0.982711i $$0.440724\pi$$
$$110$$ 0 0
$$111$$ −5.13021 −0.486938
$$112$$ 0 0
$$113$$ −8.73575 −0.821791 −0.410895 0.911683i $$-0.634784\pi$$
−0.410895 + 0.911683i $$0.634784\pi$$
$$114$$ 0 0
$$115$$ −3.21234 −0.299552
$$116$$ 0 0
$$117$$ −3.67096 −0.339380
$$118$$ 0 0
$$119$$ 13.2349 1.21324
$$120$$ 0 0
$$121$$ 14.7892 1.34447
$$122$$ 0 0
$$123$$ −8.81469 −0.794793
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −10.7877 −0.957250 −0.478625 0.878019i $$-0.658865\pi$$
−0.478625 + 0.878019i $$0.658865\pi$$
$$128$$ 0 0
$$129$$ 2.61968 0.230650
$$130$$ 0 0
$$131$$ −12.9745 −1.13359 −0.566793 0.823860i $$-0.691816\pi$$
−0.566793 + 0.823860i $$0.691816\pi$$
$$132$$ 0 0
$$133$$ −34.2748 −2.97200
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −4.08212 −0.348759 −0.174380 0.984679i $$-0.555792\pi$$
−0.174380 + 0.984679i $$0.555792\pi$$
$$138$$ 0 0
$$139$$ 19.8276 1.68175 0.840876 0.541228i $$-0.182040\pi$$
0.840876 + 0.541228i $$0.182040\pi$$
$$140$$ 0 0
$$141$$ 1.11670 0.0940434
$$142$$ 0 0
$$143$$ −18.6423 −1.55894
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 0 0
$$147$$ 18.7892 1.54970
$$148$$ 0 0
$$149$$ 10.7763 0.882828 0.441414 0.897304i $$-0.354477\pi$$
0.441414 + 0.897304i $$0.354477\pi$$
$$150$$ 0 0
$$151$$ 5.49853 0.447464 0.223732 0.974651i $$-0.428176\pi$$
0.223732 + 0.974651i $$0.428176\pi$$
$$152$$ 0 0
$$153$$ −2.60617 −0.210696
$$154$$ 0 0
$$155$$ −0.787665 −0.0632667
$$156$$ 0 0
$$157$$ 5.77566 0.460948 0.230474 0.973079i $$-0.425972\pi$$
0.230474 + 0.973079i $$0.425972\pi$$
$$158$$ 0 0
$$159$$ 0.619678 0.0491437
$$160$$ 0 0
$$161$$ −16.3132 −1.28566
$$162$$ 0 0
$$163$$ −7.14691 −0.559790 −0.279895 0.960031i $$-0.590300\pi$$
−0.279895 + 0.960031i $$0.590300\pi$$
$$164$$ 0 0
$$165$$ −5.07830 −0.395345
$$166$$ 0 0
$$167$$ −17.1009 −1.32331 −0.661653 0.749810i $$-0.730145\pi$$
−0.661653 + 0.749810i $$0.730145\pi$$
$$168$$ 0 0
$$169$$ 0.475953 0.0366118
$$170$$ 0 0
$$171$$ 6.74926 0.516129
$$172$$ 0 0
$$173$$ 10.2862 0.782045 0.391022 0.920381i $$-0.372121\pi$$
0.391022 + 0.920381i $$0.372121\pi$$
$$174$$ 0 0
$$175$$ −5.07830 −0.383884
$$176$$ 0 0
$$177$$ −12.7763 −0.960324
$$178$$ 0 0
$$179$$ 12.1182 0.905757 0.452879 0.891572i $$-0.350397\pi$$
0.452879 + 0.891572i $$0.350397\pi$$
$$180$$ 0 0
$$181$$ −20.9745 −1.55902 −0.779510 0.626389i $$-0.784532\pi$$
−0.779510 + 0.626389i $$0.784532\pi$$
$$182$$ 0 0
$$183$$ 8.90587 0.658341
$$184$$ 0 0
$$185$$ 5.13021 0.377181
$$186$$ 0 0
$$187$$ −13.2349 −0.967833
$$188$$ 0 0
$$189$$ −5.07830 −0.369392
$$190$$ 0 0
$$191$$ −26.2094 −1.89645 −0.948223 0.317607i $$-0.897121\pi$$
−0.948223 + 0.317607i $$0.897121\pi$$
$$192$$ 0 0
$$193$$ −23.7643 −1.71059 −0.855295 0.518141i $$-0.826624\pi$$
−0.855295 + 0.518141i $$0.826624\pi$$
$$194$$ 0 0
$$195$$ 3.67096 0.262883
$$196$$ 0 0
$$197$$ −25.6281 −1.82593 −0.912964 0.408041i $$-0.866212\pi$$
−0.912964 + 0.408041i $$0.866212\pi$$
$$198$$ 0 0
$$199$$ −7.82757 −0.554882 −0.277441 0.960743i $$-0.589486\pi$$
−0.277441 + 0.960743i $$0.589486\pi$$
$$200$$ 0 0
$$201$$ 0.524047 0.0369634
$$202$$ 0 0
$$203$$ 5.07830 0.356427
$$204$$ 0 0
$$205$$ 8.81469 0.615644
$$206$$ 0 0
$$207$$ 3.21234 0.223273
$$208$$ 0 0
$$209$$ 34.2748 2.37084
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 0 0
$$213$$ −0.195007 −0.0133617
$$214$$ 0 0
$$215$$ −2.61968 −0.178661
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 0 0
$$219$$ 1.13021 0.0763728
$$220$$ 0 0
$$221$$ 9.56714 0.643555
$$222$$ 0 0
$$223$$ −7.86597 −0.526744 −0.263372 0.964694i $$-0.584835\pi$$
−0.263372 + 0.964694i $$0.584835\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 0.0654212 0.00434216 0.00217108 0.999998i $$-0.499309\pi$$
0.00217108 + 0.999998i $$0.499309\pi$$
$$228$$ 0 0
$$229$$ −3.87041 −0.255764 −0.127882 0.991789i $$-0.540818\pi$$
−0.127882 + 0.991789i $$0.540818\pi$$
$$230$$ 0 0
$$231$$ −25.7892 −1.69680
$$232$$ 0 0
$$233$$ −8.18363 −0.536127 −0.268064 0.963401i $$-0.586384\pi$$
−0.268064 + 0.963401i $$0.586384\pi$$
$$234$$ 0 0
$$235$$ −1.11670 −0.0728457
$$236$$ 0 0
$$237$$ −15.3035 −0.994071
$$238$$ 0 0
$$239$$ 11.2651 0.728680 0.364340 0.931266i $$-0.381295\pi$$
0.364340 + 0.931266i $$0.381295\pi$$
$$240$$ 0 0
$$241$$ −15.2619 −0.983107 −0.491554 0.870847i $$-0.663571\pi$$
−0.491554 + 0.870847i $$0.663571\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −18.7892 −1.20040
$$246$$ 0 0
$$247$$ −24.7763 −1.57648
$$248$$ 0 0
$$249$$ 18.1182 1.14819
$$250$$ 0 0
$$251$$ −24.9411 −1.57427 −0.787134 0.616783i $$-0.788436\pi$$
−0.787134 + 0.616783i $$0.788436\pi$$
$$252$$ 0 0
$$253$$ 16.3132 1.02560
$$254$$ 0 0
$$255$$ 2.60617 0.163205
$$256$$ 0 0
$$257$$ 1.14691 0.0715425 0.0357713 0.999360i $$-0.488611\pi$$
0.0357713 + 0.999360i $$0.488611\pi$$
$$258$$ 0 0
$$259$$ 26.0528 1.61884
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ 0 0
$$263$$ −0.685099 −0.0422450 −0.0211225 0.999777i $$-0.506724\pi$$
−0.0211225 + 0.999777i $$0.506724\pi$$
$$264$$ 0 0
$$265$$ −0.619678 −0.0380665
$$266$$ 0 0
$$267$$ −15.5942 −0.954347
$$268$$ 0 0
$$269$$ 4.22522 0.257616 0.128808 0.991670i $$-0.458885\pi$$
0.128808 + 0.991670i $$0.458885\pi$$
$$270$$ 0 0
$$271$$ 0.814686 0.0494886 0.0247443 0.999694i $$-0.492123\pi$$
0.0247443 + 0.999694i $$0.492123\pi$$
$$272$$ 0 0
$$273$$ 18.6423 1.12828
$$274$$ 0 0
$$275$$ 5.07830 0.306233
$$276$$ 0 0
$$277$$ 9.85459 0.592105 0.296052 0.955172i $$-0.404330\pi$$
0.296052 + 0.955172i $$0.404330\pi$$
$$278$$ 0 0
$$279$$ 0.787665 0.0471562
$$280$$ 0 0
$$281$$ 12.2297 0.729561 0.364780 0.931094i $$-0.381144\pi$$
0.364780 + 0.931094i $$0.381144\pi$$
$$282$$ 0 0
$$283$$ −4.23341 −0.251650 −0.125825 0.992052i $$-0.540158\pi$$
−0.125825 + 0.992052i $$0.540158\pi$$
$$284$$ 0 0
$$285$$ −6.74926 −0.399792
$$286$$ 0 0
$$287$$ 44.7637 2.64231
$$288$$ 0 0
$$289$$ −10.2079 −0.600464
$$290$$ 0 0
$$291$$ −12.6671 −0.742561
$$292$$ 0 0
$$293$$ −13.3170 −0.777989 −0.388995 0.921240i $$-0.627178\pi$$
−0.388995 + 0.921240i $$0.627178\pi$$
$$294$$ 0 0
$$295$$ 12.7763 0.743864
$$296$$ 0 0
$$297$$ 5.07830 0.294673
$$298$$ 0 0
$$299$$ −11.7924 −0.681970
$$300$$ 0 0
$$301$$ −13.3035 −0.766802
$$302$$ 0 0
$$303$$ 7.03990 0.404432
$$304$$ 0 0
$$305$$ −8.90587 −0.509949
$$306$$ 0 0
$$307$$ −14.7379 −0.841136 −0.420568 0.907261i $$-0.638169\pi$$
−0.420568 + 0.907261i $$0.638169\pi$$
$$308$$ 0 0
$$309$$ −2.65808 −0.151213
$$310$$ 0 0
$$311$$ 10.0226 0.568328 0.284164 0.958776i $$-0.408284\pi$$
0.284164 + 0.958776i $$0.408284\pi$$
$$312$$ 0 0
$$313$$ 4.08800 0.231067 0.115534 0.993304i $$-0.463142\pi$$
0.115534 + 0.993304i $$0.463142\pi$$
$$314$$ 0 0
$$315$$ 5.07830 0.286130
$$316$$ 0 0
$$317$$ −12.7628 −0.716829 −0.358414 0.933563i $$-0.616683\pi$$
−0.358414 + 0.933563i $$0.616683\pi$$
$$318$$ 0 0
$$319$$ −5.07830 −0.284330
$$320$$ 0 0
$$321$$ −4.81469 −0.268730
$$322$$ 0 0
$$323$$ −17.5897 −0.978718
$$324$$ 0 0
$$325$$ −3.67096 −0.203628
$$326$$ 0 0
$$327$$ 3.86597 0.213789
$$328$$ 0 0
$$329$$ −5.67096 −0.312650
$$330$$ 0 0
$$331$$ −5.83576 −0.320762 −0.160381 0.987055i $$-0.551272\pi$$
−0.160381 + 0.987055i $$0.551272\pi$$
$$332$$ 0 0
$$333$$ −5.13021 −0.281134
$$334$$ 0 0
$$335$$ −0.524047 −0.0286317
$$336$$ 0 0
$$337$$ 1.95253 0.106361 0.0531807 0.998585i $$-0.483064\pi$$
0.0531807 + 0.998585i $$0.483064\pi$$
$$338$$ 0 0
$$339$$ −8.73575 −0.474461
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ −59.8690 −3.23262
$$344$$ 0 0
$$345$$ −3.21234 −0.172946
$$346$$ 0 0
$$347$$ −17.3960 −0.933864 −0.466932 0.884293i $$-0.654641\pi$$
−0.466932 + 0.884293i $$0.654641\pi$$
$$348$$ 0 0
$$349$$ −28.7379 −1.53830 −0.769152 0.639066i $$-0.779321\pi$$
−0.769152 + 0.639066i $$0.779321\pi$$
$$350$$ 0 0
$$351$$ −3.67096 −0.195941
$$352$$ 0 0
$$353$$ −29.7590 −1.58391 −0.791955 0.610580i $$-0.790936\pi$$
−0.791955 + 0.610580i $$0.790936\pi$$
$$354$$ 0 0
$$355$$ 0.195007 0.0103499
$$356$$ 0 0
$$357$$ 13.2349 0.700466
$$358$$ 0 0
$$359$$ −7.76659 −0.409905 −0.204953 0.978772i $$-0.565704\pi$$
−0.204953 + 0.978772i $$0.565704\pi$$
$$360$$ 0 0
$$361$$ 26.5526 1.39750
$$362$$ 0 0
$$363$$ 14.7892 0.776230
$$364$$ 0 0
$$365$$ −1.13021 −0.0591581
$$366$$ 0 0
$$367$$ 12.8675 0.671677 0.335838 0.941920i $$-0.390980\pi$$
0.335838 + 0.941920i $$0.390980\pi$$
$$368$$ 0 0
$$369$$ −8.81469 −0.458874
$$370$$ 0 0
$$371$$ −3.14691 −0.163380
$$372$$ 0 0
$$373$$ −13.4639 −0.697133 −0.348566 0.937284i $$-0.613331\pi$$
−0.348566 + 0.937284i $$0.613331\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 3.67096 0.189064
$$378$$ 0 0
$$379$$ 9.53318 0.489687 0.244843 0.969563i $$-0.421263\pi$$
0.244843 + 0.969563i $$0.421263\pi$$
$$380$$ 0 0
$$381$$ −10.7877 −0.552669
$$382$$ 0 0
$$383$$ 20.0836 1.02622 0.513111 0.858322i $$-0.328493\pi$$
0.513111 + 0.858322i $$0.328493\pi$$
$$384$$ 0 0
$$385$$ 25.7892 1.31434
$$386$$ 0 0
$$387$$ 2.61968 0.133166
$$388$$ 0 0
$$389$$ 24.3472 1.23445 0.617225 0.786787i $$-0.288257\pi$$
0.617225 + 0.786787i $$0.288257\pi$$
$$390$$ 0 0
$$391$$ −8.37188 −0.423384
$$392$$ 0 0
$$393$$ −12.9745 −0.654476
$$394$$ 0 0
$$395$$ 15.3035 0.770004
$$396$$ 0 0
$$397$$ −25.9232 −1.30105 −0.650524 0.759486i $$-0.725451\pi$$
−0.650524 + 0.759486i $$0.725451\pi$$
$$398$$ 0 0
$$399$$ −34.2748 −1.71589
$$400$$ 0 0
$$401$$ 31.3576 1.56592 0.782961 0.622071i $$-0.213708\pi$$
0.782961 + 0.622071i $$0.213708\pi$$
$$402$$ 0 0
$$403$$ −2.89149 −0.144035
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −26.0528 −1.29139
$$408$$ 0 0
$$409$$ 1.67415 0.0827814 0.0413907 0.999143i $$-0.486821\pi$$
0.0413907 + 0.999143i $$0.486821\pi$$
$$410$$ 0 0
$$411$$ −4.08212 −0.201356
$$412$$ 0 0
$$413$$ 64.8819 3.19263
$$414$$ 0 0
$$415$$ −18.1182 −0.889388
$$416$$ 0 0
$$417$$ 19.8276 0.970960
$$418$$ 0 0
$$419$$ −0.658078 −0.0321492 −0.0160746 0.999871i $$-0.505117\pi$$
−0.0160746 + 0.999871i $$0.505117\pi$$
$$420$$ 0 0
$$421$$ 5.11989 0.249528 0.124764 0.992186i $$-0.460183\pi$$
0.124764 + 0.992186i $$0.460183\pi$$
$$422$$ 0 0
$$423$$ 1.11670 0.0542960
$$424$$ 0 0
$$425$$ −2.60617 −0.126418
$$426$$ 0 0
$$427$$ −45.2267 −2.18867
$$428$$ 0 0
$$429$$ −18.6423 −0.900056
$$430$$ 0 0
$$431$$ 0.390015 0.0187864 0.00939318 0.999956i $$-0.497010\pi$$
0.00939318 + 0.999956i $$0.497010\pi$$
$$432$$ 0 0
$$433$$ −29.6989 −1.42724 −0.713618 0.700535i $$-0.752945\pi$$
−0.713618 + 0.700535i $$0.752945\pi$$
$$434$$ 0 0
$$435$$ 1.00000 0.0479463
$$436$$ 0 0
$$437$$ 21.6809 1.03714
$$438$$ 0 0
$$439$$ 24.4856 1.16864 0.584318 0.811525i $$-0.301362\pi$$
0.584318 + 0.811525i $$0.301362\pi$$
$$440$$ 0 0
$$441$$ 18.7892 0.894722
$$442$$ 0 0
$$443$$ 11.1091 0.527808 0.263904 0.964549i $$-0.414990\pi$$
0.263904 + 0.964549i $$0.414990\pi$$
$$444$$ 0 0
$$445$$ 15.5942 0.739234
$$446$$ 0 0
$$447$$ 10.7763 0.509701
$$448$$ 0 0
$$449$$ −15.1003 −0.712628 −0.356314 0.934366i $$-0.615967\pi$$
−0.356314 + 0.934366i $$0.615967\pi$$
$$450$$ 0 0
$$451$$ −44.7637 −2.10784
$$452$$ 0 0
$$453$$ 5.49853 0.258343
$$454$$ 0 0
$$455$$ −18.6423 −0.873962
$$456$$ 0 0
$$457$$ −24.3742 −1.14018 −0.570088 0.821583i $$-0.693091\pi$$
−0.570088 + 0.821583i $$0.693091\pi$$
$$458$$ 0 0
$$459$$ −2.60617 −0.121645
$$460$$ 0 0
$$461$$ −12.9789 −0.604489 −0.302244 0.953230i $$-0.597736\pi$$
−0.302244 + 0.953230i $$0.597736\pi$$
$$462$$ 0 0
$$463$$ 11.2619 0.523386 0.261693 0.965151i $$-0.415719\pi$$
0.261693 + 0.965151i $$0.415719\pi$$
$$464$$ 0 0
$$465$$ −0.787665 −0.0365271
$$466$$ 0 0
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ −2.66127 −0.122886
$$470$$ 0 0
$$471$$ 5.77566 0.266128
$$472$$ 0 0
$$473$$ 13.3035 0.611697
$$474$$ 0 0
$$475$$ 6.74926 0.309677
$$476$$ 0 0
$$477$$ 0.619678 0.0283731
$$478$$ 0 0
$$479$$ 23.8615 1.09026 0.545130 0.838351i $$-0.316480\pi$$
0.545130 + 0.838351i $$0.316480\pi$$
$$480$$ 0 0
$$481$$ 18.8328 0.858702
$$482$$ 0 0
$$483$$ −16.3132 −0.742277
$$484$$ 0 0
$$485$$ 12.6671 0.575185
$$486$$ 0 0
$$487$$ 20.8417 0.944428 0.472214 0.881484i $$-0.343455\pi$$
0.472214 + 0.881484i $$0.343455\pi$$
$$488$$ 0 0
$$489$$ −7.14691 −0.323195
$$490$$ 0 0
$$491$$ 19.1021 0.862067 0.431034 0.902336i $$-0.358149\pi$$
0.431034 + 0.902336i $$0.358149\pi$$
$$492$$ 0 0
$$493$$ 2.60617 0.117376
$$494$$ 0 0
$$495$$ −5.07830 −0.228253
$$496$$ 0 0
$$497$$ 0.990307 0.0444213
$$498$$ 0 0
$$499$$ 7.82757 0.350410 0.175205 0.984532i $$-0.443941\pi$$
0.175205 + 0.984532i $$0.443941\pi$$
$$500$$ 0 0
$$501$$ −17.1009 −0.764011
$$502$$ 0 0
$$503$$ −31.5865 −1.40837 −0.704187 0.710015i $$-0.748688\pi$$
−0.704187 + 0.710015i $$0.748688\pi$$
$$504$$ 0 0
$$505$$ −7.03990 −0.313272
$$506$$ 0 0
$$507$$ 0.475953 0.0211378
$$508$$ 0 0
$$509$$ 19.6167 0.869497 0.434748 0.900552i $$-0.356837\pi$$
0.434748 + 0.900552i $$0.356837\pi$$
$$510$$ 0 0
$$511$$ −5.73957 −0.253904
$$512$$ 0 0
$$513$$ 6.74926 0.297987
$$514$$ 0 0
$$515$$ 2.65808 0.117129
$$516$$ 0 0
$$517$$ 5.67096 0.249409
$$518$$ 0 0
$$519$$ 10.2862 0.451514
$$520$$ 0 0
$$521$$ 24.4057 1.06923 0.534616 0.845095i $$-0.320456\pi$$
0.534616 + 0.845095i $$0.320456\pi$$
$$522$$ 0 0
$$523$$ 39.9715 1.74783 0.873917 0.486076i $$-0.161572\pi$$
0.873917 + 0.486076i $$0.161572\pi$$
$$524$$ 0 0
$$525$$ −5.07830 −0.221635
$$526$$ 0 0
$$527$$ −2.05279 −0.0894208
$$528$$ 0 0
$$529$$ −12.6809 −0.551344
$$530$$ 0 0
$$531$$ −12.7763 −0.554444
$$532$$ 0 0
$$533$$ 32.3584 1.40160
$$534$$ 0 0
$$535$$ 4.81469 0.208157
$$536$$ 0 0
$$537$$ 12.1182 0.522939
$$538$$ 0 0
$$539$$ 95.4171 4.10991
$$540$$ 0 0
$$541$$ −8.76064 −0.376649 −0.188325 0.982107i $$-0.560306\pi$$
−0.188325 + 0.982107i $$0.560306\pi$$
$$542$$ 0 0
$$543$$ −20.9745 −0.900101
$$544$$ 0 0
$$545$$ −3.86597 −0.165600
$$546$$ 0 0
$$547$$ −31.3569 −1.34072 −0.670361 0.742035i $$-0.733861\pi$$
−0.670361 + 0.742035i $$0.733861\pi$$
$$548$$ 0 0
$$549$$ 8.90587 0.380093
$$550$$ 0 0
$$551$$ −6.74926 −0.287528
$$552$$ 0 0
$$553$$ 77.7159 3.30482
$$554$$ 0 0
$$555$$ 5.13021 0.217765
$$556$$ 0 0
$$557$$ 37.6514 1.59534 0.797670 0.603094i $$-0.206066\pi$$
0.797670 + 0.603094i $$0.206066\pi$$
$$558$$ 0 0
$$559$$ −9.61674 −0.406745
$$560$$ 0 0
$$561$$ −13.2349 −0.558778
$$562$$ 0 0
$$563$$ 39.0323 1.64501 0.822507 0.568755i $$-0.192575\pi$$
0.822507 + 0.568755i $$0.192575\pi$$
$$564$$ 0 0
$$565$$ 8.73575 0.367516
$$566$$ 0 0
$$567$$ −5.07830 −0.213269
$$568$$ 0 0
$$569$$ −3.03990 −0.127439 −0.0637197 0.997968i $$-0.520296\pi$$
−0.0637197 + 0.997968i $$0.520296\pi$$
$$570$$ 0 0
$$571$$ −40.1056 −1.67837 −0.839183 0.543849i $$-0.816966\pi$$
−0.839183 + 0.543849i $$0.816966\pi$$
$$572$$ 0 0
$$573$$ −26.2094 −1.09491
$$574$$ 0 0
$$575$$ 3.21234 0.133964
$$576$$ 0 0
$$577$$ −16.1091 −0.670632 −0.335316 0.942106i $$-0.608843\pi$$
−0.335316 + 0.942106i $$0.608843\pi$$
$$578$$ 0 0
$$579$$ −23.7643 −0.987610
$$580$$ 0 0
$$581$$ −92.0098 −3.81721
$$582$$ 0 0
$$583$$ 3.14691 0.130332
$$584$$ 0 0
$$585$$ 3.67096 0.151776
$$586$$ 0 0
$$587$$ 21.4331 0.884639 0.442320 0.896858i $$-0.354156\pi$$
0.442320 + 0.896858i $$0.354156\pi$$
$$588$$ 0 0
$$589$$ 5.31616 0.219048
$$590$$ 0 0
$$591$$ −25.6281 −1.05420
$$592$$ 0 0
$$593$$ 23.9886 0.985095 0.492547 0.870286i $$-0.336066\pi$$
0.492547 + 0.870286i $$0.336066\pi$$
$$594$$ 0 0
$$595$$ −13.2349 −0.542578
$$596$$ 0 0
$$597$$ −7.82757 −0.320361
$$598$$ 0 0
$$599$$ 38.1100 1.55713 0.778567 0.627562i $$-0.215947\pi$$
0.778567 + 0.627562i $$0.215947\pi$$
$$600$$ 0 0
$$601$$ −20.0528 −0.817970 −0.408985 0.912541i $$-0.634117\pi$$
−0.408985 + 0.912541i $$0.634117\pi$$
$$602$$ 0 0
$$603$$ 0.524047 0.0213408
$$604$$ 0 0
$$605$$ −14.7892 −0.601265
$$606$$ 0 0
$$607$$ 38.7523 1.57291 0.786453 0.617650i $$-0.211915\pi$$
0.786453 + 0.617650i $$0.211915\pi$$
$$608$$ 0 0
$$609$$ 5.07830 0.205783
$$610$$ 0 0
$$611$$ −4.09938 −0.165843
$$612$$ 0 0
$$613$$ 24.6757 0.996640 0.498320 0.866993i $$-0.333950\pi$$
0.498320 + 0.866993i $$0.333950\pi$$
$$614$$ 0 0
$$615$$ 8.81469 0.355442
$$616$$ 0 0
$$617$$ 10.5249 0.423717 0.211859 0.977300i $$-0.432048\pi$$
0.211859 + 0.977300i $$0.432048\pi$$
$$618$$ 0 0
$$619$$ −26.5496 −1.06712 −0.533560 0.845762i $$-0.679146\pi$$
−0.533560 + 0.845762i $$0.679146\pi$$
$$620$$ 0 0
$$621$$ 3.21234 0.128907
$$622$$ 0 0
$$623$$ 79.1919 3.17276
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 34.2748 1.36880
$$628$$ 0 0
$$629$$ 13.3702 0.533105
$$630$$ 0 0
$$631$$ 13.2041 0.525649 0.262824 0.964844i $$-0.415346\pi$$
0.262824 + 0.964844i $$0.415346\pi$$
$$632$$ 0 0
$$633$$ −2.00000 −0.0794929
$$634$$ 0 0
$$635$$ 10.7877 0.428095
$$636$$ 0 0
$$637$$ −68.9743 −2.73286
$$638$$ 0 0
$$639$$ −0.195007 −0.00771437
$$640$$ 0 0
$$641$$ 9.51435 0.375794 0.187897 0.982189i $$-0.439833\pi$$
0.187897 + 0.982189i $$0.439833\pi$$
$$642$$ 0 0
$$643$$ −30.1370 −1.18849 −0.594244 0.804285i $$-0.702549\pi$$
−0.594244 + 0.804285i $$0.702549\pi$$
$$644$$ 0 0
$$645$$ −2.61968 −0.103150
$$646$$ 0 0
$$647$$ 42.7535 1.68081 0.840407 0.541955i $$-0.182316\pi$$
0.840407 + 0.541955i $$0.182316\pi$$
$$648$$ 0 0
$$649$$ −64.8819 −2.54684
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ 0 0
$$653$$ 33.8983 1.32654 0.663272 0.748379i $$-0.269167\pi$$
0.663272 + 0.748379i $$0.269167\pi$$
$$654$$ 0 0
$$655$$ 12.9745 0.506955
$$656$$ 0 0
$$657$$ 1.13021 0.0440939
$$658$$ 0 0
$$659$$ −11.7287 −0.456887 −0.228444 0.973557i $$-0.573364\pi$$
−0.228444 + 0.973557i $$0.573364\pi$$
$$660$$ 0 0
$$661$$ −38.6807 −1.50450 −0.752252 0.658876i $$-0.771032\pi$$
−0.752252 + 0.658876i $$0.771032\pi$$
$$662$$ 0 0
$$663$$ 9.56714 0.371557
$$664$$ 0 0
$$665$$ 34.2748 1.32912
$$666$$ 0 0
$$667$$ −3.21234 −0.124382
$$668$$ 0 0
$$669$$ −7.86597 −0.304116
$$670$$ 0 0
$$671$$ 45.2267 1.74596
$$672$$ 0 0
$$673$$ 30.6410 1.18112 0.590562 0.806992i $$-0.298906\pi$$
0.590562 + 0.806992i $$0.298906\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −40.8954 −1.57174 −0.785868 0.618394i $$-0.787784\pi$$
−0.785868 + 0.618394i $$0.787784\pi$$
$$678$$ 0 0
$$679$$ 64.3276 2.46867
$$680$$ 0 0
$$681$$ 0.0654212 0.00250694
$$682$$ 0 0
$$683$$ −21.2317 −0.812409 −0.406205 0.913782i $$-0.633148\pi$$
−0.406205 + 0.913782i $$0.633148\pi$$
$$684$$ 0 0
$$685$$ 4.08212 0.155970
$$686$$ 0 0
$$687$$ −3.87041 −0.147665
$$688$$ 0 0
$$689$$ −2.27481 −0.0866635
$$690$$ 0 0
$$691$$ 13.9842 0.531983 0.265992 0.963975i $$-0.414301\pi$$
0.265992 + 0.963975i $$0.414301\pi$$
$$692$$ 0 0
$$693$$ −25.7892 −0.979649
$$694$$ 0 0
$$695$$ −19.8276 −0.752103
$$696$$ 0 0
$$697$$ 22.9725 0.870147
$$698$$ 0 0
$$699$$ −8.18363 −0.309533
$$700$$ 0 0
$$701$$ 3.16630 0.119590 0.0597948 0.998211i $$-0.480955\pi$$
0.0597948 + 0.998211i $$0.480955\pi$$
$$702$$ 0 0
$$703$$ −34.6252 −1.30591
$$704$$ 0 0
$$705$$ −1.11670 −0.0420575
$$706$$ 0 0
$$707$$ −35.7508 −1.34455
$$708$$ 0 0
$$709$$ 23.3677 0.877592 0.438796 0.898587i $$-0.355405\pi$$
0.438796 + 0.898587i $$0.355405\pi$$
$$710$$ 0 0
$$711$$ −15.3035 −0.573927
$$712$$ 0 0
$$713$$ 2.53024 0.0947583
$$714$$ 0 0
$$715$$ 18.6423 0.697181
$$716$$ 0 0
$$717$$ 11.2651 0.420704
$$718$$ 0 0
$$719$$ 0.731134 0.0272667 0.0136334 0.999907i $$-0.495660\pi$$
0.0136334 + 0.999907i $$0.495660\pi$$
$$720$$ 0 0
$$721$$ 13.4985 0.502711
$$722$$ 0 0
$$723$$ −15.2619 −0.567597
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 17.7243 0.657358 0.328679 0.944442i $$-0.393397\pi$$
0.328679 + 0.944442i $$0.393397\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −6.82732 −0.252518
$$732$$ 0 0
$$733$$ −5.67184 −0.209494 −0.104747 0.994499i $$-0.533403\pi$$
−0.104747 + 0.994499i $$0.533403\pi$$
$$734$$ 0 0
$$735$$ −18.7892 −0.693049
$$736$$ 0 0
$$737$$ 2.66127 0.0980291
$$738$$ 0 0
$$739$$ −8.72350 −0.320899 −0.160450 0.987044i $$-0.551294\pi$$
−0.160450 + 0.987044i $$0.551294\pi$$
$$740$$ 0 0
$$741$$ −24.7763 −0.910180
$$742$$ 0 0
$$743$$ 0.413474 0.0151689 0.00758445 0.999971i $$-0.497586\pi$$
0.00758445 + 0.999971i $$0.497586\pi$$
$$744$$ 0 0
$$745$$ −10.7763 −0.394813
$$746$$ 0 0
$$747$$ 18.1182 0.662911
$$748$$ 0 0
$$749$$ 24.4504 0.893399
$$750$$ 0 0
$$751$$ −9.71381 −0.354462 −0.177231 0.984169i $$-0.556714\pi$$
−0.177231 + 0.984169i $$0.556714\pi$$
$$752$$ 0 0
$$753$$ −24.9411 −0.908904
$$754$$ 0 0
$$755$$ −5.49853 −0.200112
$$756$$ 0 0
$$757$$ 22.3877 0.813695 0.406847 0.913496i $$-0.366628\pi$$
0.406847 + 0.913496i $$0.366628\pi$$
$$758$$ 0 0
$$759$$ 16.3132 0.592132
$$760$$ 0 0
$$761$$ 44.0836 1.59803 0.799014 0.601313i $$-0.205355\pi$$
0.799014 + 0.601313i $$0.205355\pi$$
$$762$$ 0 0
$$763$$ −19.6326 −0.710746
$$764$$ 0 0
$$765$$ 2.60617 0.0942262
$$766$$ 0 0
$$767$$ 46.9012 1.69351
$$768$$ 0 0
$$769$$ −14.2761 −0.514808 −0.257404 0.966304i $$-0.582867\pi$$
−0.257404 + 0.966304i $$0.582867\pi$$
$$770$$ 0 0
$$771$$ 1.14691 0.0413051
$$772$$ 0 0
$$773$$ −25.5807 −0.920072 −0.460036 0.887900i $$-0.652164\pi$$
−0.460036 + 0.887900i $$0.652164\pi$$
$$774$$ 0 0
$$775$$ 0.787665 0.0282937
$$776$$ 0 0
$$777$$ 26.0528 0.934638
$$778$$ 0 0
$$779$$ −59.4926 −2.13155
$$780$$ 0 0
$$781$$ −0.990307 −0.0354360
$$782$$ 0 0
$$783$$ −1.00000 −0.0357371
$$784$$ 0 0
$$785$$ −5.77566 −0.206142
$$786$$ 0 0
$$787$$ 8.73362 0.311320 0.155660 0.987811i $$-0.450250\pi$$
0.155660 + 0.987811i $$0.450250\pi$$
$$788$$ 0 0
$$789$$ −0.685099 −0.0243902
$$790$$ 0 0
$$791$$ 44.3628 1.57736
$$792$$ 0 0
$$793$$ −32.6931 −1.16097
$$794$$ 0 0
$$795$$ −0.619678 −0.0219777
$$796$$ 0 0
$$797$$ 8.43874 0.298915 0.149458 0.988768i $$-0.452247\pi$$
0.149458 + 0.988768i $$0.452247\pi$$
$$798$$ 0 0
$$799$$ −2.91032 −0.102960
$$800$$ 0 0
$$801$$ −15.5942 −0.550993
$$802$$ 0 0
$$803$$ 5.73957 0.202545
$$804$$ 0 0
$$805$$ 16.3132 0.574965
$$806$$ 0 0
$$807$$ 4.22522 0.148735
$$808$$ 0 0
$$809$$ −21.7508 −0.764716 −0.382358 0.924014i $$-0.624888\pi$$
−0.382358 + 0.924014i $$0.624888\pi$$
$$810$$ 0 0
$$811$$ 3.24629 0.113993 0.0569963 0.998374i $$-0.481848\pi$$
0.0569963 + 0.998374i $$0.481848\pi$$
$$812$$ 0 0
$$813$$ 0.814686 0.0285723
$$814$$ 0 0
$$815$$ 7.14691 0.250345
$$816$$ 0 0
$$817$$ 17.6809 0.618576
$$818$$ 0 0
$$819$$ 18.6423 0.651413
$$820$$ 0 0
$$821$$ 33.9232 1.18393 0.591964 0.805964i $$-0.298353\pi$$
0.591964 + 0.805964i $$0.298353\pi$$
$$822$$ 0 0
$$823$$ 36.4648 1.27108 0.635542 0.772066i $$-0.280777\pi$$
0.635542 + 0.772066i $$0.280777\pi$$
$$824$$ 0 0
$$825$$ 5.07830 0.176804
$$826$$ 0 0
$$827$$ 15.9772 0.555583 0.277792 0.960641i $$-0.410398\pi$$
0.277792 + 0.960641i $$0.410398\pi$$
$$828$$ 0 0
$$829$$ 5.00595 0.173864 0.0869319 0.996214i $$-0.472294\pi$$
0.0869319 + 0.996214i $$0.472294\pi$$
$$830$$ 0 0
$$831$$ 9.85459 0.341852
$$832$$ 0 0
$$833$$ −48.9677 −1.69663
$$834$$ 0 0
$$835$$ 17.1009 0.591800
$$836$$ 0 0
$$837$$ 0.787665 0.0272257
$$838$$ 0 0
$$839$$ −31.4713 −1.08651 −0.543255 0.839567i $$-0.682808\pi$$
−0.543255 + 0.839567i $$0.682808\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 12.2297 0.421212
$$844$$ 0 0
$$845$$ −0.475953 −0.0163733
$$846$$ 0 0
$$847$$ −75.1039 −2.58060
$$848$$ 0 0
$$849$$ −4.23341 −0.145290
$$850$$ 0 0
$$851$$ −16.4800 −0.564926
$$852$$ 0 0
$$853$$ −36.0197 −1.23329 −0.616646 0.787241i $$-0.711509\pi$$
−0.616646 + 0.787241i $$0.711509\pi$$
$$854$$ 0 0
$$855$$ −6.74926 −0.230820
$$856$$ 0 0
$$857$$ −27.4217 −0.936708 −0.468354 0.883541i $$-0.655153\pi$$
−0.468354 + 0.883541i $$0.655153\pi$$
$$858$$ 0 0
$$859$$ −14.0642 −0.479863 −0.239932 0.970790i $$-0.577125\pi$$
−0.239932 + 0.970790i $$0.577125\pi$$
$$860$$ 0 0
$$861$$ 44.7637 1.52554
$$862$$ 0 0
$$863$$ −19.7540 −0.672433 −0.336216 0.941785i $$-0.609147\pi$$
−0.336216 + 0.941785i $$0.609147\pi$$
$$864$$ 0 0
$$865$$ −10.2862 −0.349741
$$866$$ 0 0
$$867$$ −10.2079 −0.346678
$$868$$ 0 0
$$869$$ −77.7159 −2.63633
$$870$$ 0 0
$$871$$ −1.92375 −0.0651839
$$872$$ 0 0
$$873$$ −12.6671 −0.428718
$$874$$ 0 0
$$875$$ 5.07830 0.171678
$$876$$ 0 0
$$877$$ 42.2030 1.42509 0.712547 0.701624i $$-0.247541\pi$$
0.712547 + 0.701624i $$0.247541\pi$$
$$878$$ 0 0
$$879$$ −13.3170 −0.449172
$$880$$ 0 0
$$881$$ −33.0927 −1.11492 −0.557461 0.830203i $$-0.688224\pi$$
−0.557461 + 0.830203i $$0.688224\pi$$
$$882$$ 0 0
$$883$$ 26.2634 0.883835 0.441917 0.897056i $$-0.354298\pi$$
0.441917 + 0.897056i $$0.354298\pi$$
$$884$$ 0 0
$$885$$ 12.7763 0.429470
$$886$$ 0 0
$$887$$ −14.0716 −0.472479 −0.236239 0.971695i $$-0.575915\pi$$
−0.236239 + 0.971695i $$0.575915\pi$$
$$888$$ 0 0
$$889$$ 54.7830 1.83736
$$890$$ 0 0
$$891$$ 5.07830 0.170130
$$892$$ 0 0
$$893$$ 7.53693 0.252214
$$894$$ 0 0
$$895$$ −12.1182 −0.405067
$$896$$ 0 0
$$897$$ −11.7924 −0.393735
$$898$$ 0 0
$$899$$ −0.787665 −0.0262701
$$900$$ 0 0
$$901$$ −1.61499 −0.0538030
$$902$$ 0 0
$$903$$ −13.3035 −0.442713
$$904$$ 0 0
$$905$$ 20.9745 0.697215
$$906$$ 0 0
$$907$$ −11.9810 −0.397822 −0.198911 0.980018i $$-0.563740\pi$$
−0.198911 + 0.980018i $$0.563740\pi$$
$$908$$ 0 0
$$909$$ 7.03990 0.233499
$$910$$ 0 0
$$911$$ −27.5300 −0.912109 −0.456055 0.889952i $$-0.650738\pi$$
−0.456055 + 0.889952i $$0.650738\pi$$
$$912$$ 0 0
$$913$$ 92.0098 3.04508
$$914$$ 0 0
$$915$$ −8.90587 −0.294419
$$916$$ 0 0
$$917$$ 65.8884 2.17583
$$918$$ 0 0
$$919$$ 17.7250 0.584694 0.292347 0.956312i $$-0.405564\pi$$
0.292347 + 0.956312i $$0.405564\pi$$
$$920$$ 0 0
$$921$$ −14.7379 −0.485630
$$922$$ 0 0
$$923$$ 0.715865 0.0235630
$$924$$ 0 0
$$925$$ −5.13021 −0.168680
$$926$$ 0 0
$$927$$ −2.65808 −0.0873027
$$928$$ 0 0
$$929$$ 36.9971 1.21383 0.606917 0.794765i $$-0.292406\pi$$
0.606917 + 0.794765i $$0.292406\pi$$
$$930$$ 0 0
$$931$$ 126.813 4.15613
$$932$$ 0 0
$$933$$ 10.0226 0.328124
$$934$$ 0 0
$$935$$ 13.2349 0.432828
$$936$$ 0 0
$$937$$ −36.0446 −1.17753 −0.588763 0.808306i $$-0.700385\pi$$
−0.588763 + 0.808306i $$0.700385\pi$$
$$938$$ 0 0
$$939$$ 4.08800 0.133407
$$940$$ 0 0
$$941$$ −19.6256 −0.639777 −0.319889 0.947455i $$-0.603645\pi$$
−0.319889 + 0.947455i $$0.603645\pi$$
$$942$$ 0 0
$$943$$ −28.3157 −0.922087
$$944$$ 0 0
$$945$$ 5.07830 0.165197
$$946$$ 0 0
$$947$$ −18.9555 −0.615970 −0.307985 0.951391i $$-0.599655\pi$$
−0.307985 + 0.951391i $$0.599655\pi$$
$$948$$ 0 0
$$949$$ −4.14897 −0.134681
$$950$$ 0 0
$$951$$ −12.7628 −0.413861
$$952$$ 0 0
$$953$$ −30.2761 −0.980738 −0.490369 0.871515i $$-0.663138\pi$$
−0.490369 + 0.871515i $$0.663138\pi$$
$$954$$ 0 0
$$955$$ 26.2094 0.848116
$$956$$ 0 0
$$957$$ −5.07830 −0.164158
$$958$$ 0 0
$$959$$ 20.7303 0.669415
$$960$$ 0 0
$$961$$ −30.3796 −0.979987
$$962$$ 0 0
$$963$$ −4.81469 −0.155151
$$964$$ 0 0
$$965$$ 23.7643 0.764999
$$966$$ 0 0
$$967$$ 20.4812 0.658631 0.329316 0.944220i $$-0.393182\pi$$
0.329316 + 0.944220i $$0.393182\pi$$
$$968$$ 0 0
$$969$$ −17.5897 −0.565063
$$970$$ 0 0
$$971$$ −44.1566 −1.41705 −0.708526 0.705684i $$-0.750640\pi$$
−0.708526 + 0.705684i $$0.750640\pi$$
$$972$$ 0 0
$$973$$ −100.690 −3.22799
$$974$$ 0 0
$$975$$ −3.67096 −0.117565
$$976$$ 0 0
$$977$$ 0.492580 0.0157590 0.00787952 0.999969i $$-0.497492\pi$$
0.00787952 + 0.999969i $$0.497492\pi$$
$$978$$ 0 0
$$979$$ −79.1919 −2.53098
$$980$$ 0 0
$$981$$ 3.86597 0.123431
$$982$$ 0 0
$$983$$ −27.5513 −0.878750 −0.439375 0.898304i $$-0.644800\pi$$
−0.439375 + 0.898304i $$0.644800\pi$$
$$984$$ 0 0
$$985$$ 25.6281 0.816580
$$986$$ 0 0
$$987$$ −5.67096 −0.180509
$$988$$ 0 0
$$989$$ 8.41529 0.267590
$$990$$ 0 0
$$991$$ −23.0927 −0.733563 −0.366782 0.930307i $$-0.619540\pi$$
−0.366782 + 0.930307i $$0.619540\pi$$
$$992$$ 0 0
$$993$$ −5.83576 −0.185192
$$994$$ 0 0
$$995$$ 7.82757 0.248151
$$996$$ 0 0
$$997$$ 7.79593 0.246900 0.123450 0.992351i $$-0.460604\pi$$
0.123450 + 0.992351i $$0.460604\pi$$
$$998$$ 0 0
$$999$$ −5.13021 −0.162313
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 6960.2.a.co.1.1 4
4.3 odd 2 435.2.a.j.1.3 4
12.11 even 2 1305.2.a.r.1.2 4
20.3 even 4 2175.2.c.n.349.4 8
20.7 even 4 2175.2.c.n.349.5 8
20.19 odd 2 2175.2.a.v.1.2 4
60.59 even 2 6525.2.a.bi.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.3 4 4.3 odd 2
1305.2.a.r.1.2 4 12.11 even 2
2175.2.a.v.1.2 4 20.19 odd 2
2175.2.c.n.349.4 8 20.3 even 4
2175.2.c.n.349.5 8 20.7 even 4
6525.2.a.bi.1.3 4 60.59 even 2
6960.2.a.co.1.1 4 1.1 even 1 trivial