LMFDB - Embedded newform 5070.2.b.v.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.v.1351.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.55496i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.55496i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.85086i q^{11} +1.00000 q^{12} -2.55496 q^{14} -1.00000i q^{15} +1.00000 q^{16} +7.40581 q^{17} -1.00000i q^{18} -2.93900i q^{19} -1.00000i q^{20} +2.55496i q^{21} -3.85086 q^{22} +1.80194 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.55496i q^{28} +8.03684 q^{29} -1.00000 q^{30} +1.13706i q^{31} -1.00000i q^{32} +3.85086i q^{33} -7.40581i q^{34} +2.55496 q^{35} -1.00000 q^{36} +3.44504i q^{37} -2.93900 q^{38} -1.00000 q^{40} -1.55496i q^{41} +2.55496 q^{42} +10.4547 q^{43} +3.85086i q^{44} +1.00000i q^{45} -1.80194i q^{46} +9.71379i q^{47} -1.00000 q^{48} +0.472189 q^{49} +1.00000i q^{50} -7.40581 q^{51} -8.93900 q^{53} +1.00000i q^{54} +3.85086 q^{55} +2.55496 q^{56} +2.93900i q^{57} -8.03684i q^{58} +11.7845i q^{59} +1.00000i q^{60} -10.0586 q^{61} +1.13706 q^{62} -2.55496i q^{63} -1.00000 q^{64} +3.85086 q^{66} +1.36227i q^{67} -7.40581 q^{68} -1.80194 q^{69} -2.55496i q^{70} -12.0761i q^{71} +1.00000i q^{72} -2.29590i q^{73} +3.44504 q^{74} +1.00000 q^{75} +2.93900i q^{76} -9.83877 q^{77} +8.81700 q^{79} +1.00000i q^{80} +1.00000 q^{81} -1.55496 q^{82} -15.9051i q^{83} -2.55496i q^{84} +7.40581i q^{85} -10.4547i q^{86} -8.03684 q^{87} +3.85086 q^{88} +15.6799i q^{89} +1.00000 q^{90} -1.80194 q^{92} -1.13706i q^{93} +9.71379 q^{94} +2.93900 q^{95} +1.00000i q^{96} +1.98254i q^{97} -0.472189i q^{98} -3.85086i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} + 6 q^{12} - 16 q^{14} + 6 q^{16} + 18 q^{17} + 4 q^{22} + 2 q^{23} - 6 q^{25} - 6 q^{27} - 8 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 16 q^{42} + 18 q^{43} - 6 q^{48} - 10 q^{49} - 18 q^{51} - 34 q^{53} - 4 q^{55} + 16 q^{56} + 2 q^{61} - 4 q^{62} - 6 q^{64} - 4 q^{66} - 18 q^{68} - 2 q^{69} + 20 q^{74} + 6 q^{75} + 6 q^{77} - 6 q^{79} + 6 q^{81} - 10 q^{82} + 8 q^{87} - 4 q^{88} + 6 q^{90} - 2 q^{92} + 42 q^{94} - 2 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 + 18 * q^17 + 4 * q^22 + 2 * q^23 - 6 * q^25 - 6 * q^27 - 8 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 16 * q^42 + 18 * q^43 - 6 * q^48 - 10 * q^49 - 18 * q^51 - 34 * q^53 - 4 * q^55 + 16 * q^56 + 2 * q^61 - 4 * q^62 - 6 * q^64 - 4 * q^66 - 18 * q^68 - 2 * q^69 + 20 * q^74 + 6 * q^75 + 6 * q^77 - 6 * q^79 + 6 * q^81 - 10 * q^82 + 8 * q^87 - 4 * q^88 + 6 * q^90 - 2 * q^92 + 42 * q^94 - 2 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	1.00000i	0.408248i
$7$	− 2.55496i	− 0.965683i	−0.875708	−	0.482842i	$-0.839605\pi$
$7$	− 2.55496i	− 0.965683i	0.875708	−	0.482842i	$-0.160395\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	− 3.85086i	− 1.16108i	−0.814233	−	0.580538i	$-0.802842\pi$
$11$	− 3.85086i	− 1.16108i	0.814233	−	0.580538i	$-0.197158\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−2.55496	−0.682841
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	7.40581	1.79617	0.898087	−	0.439818i	$-0.144957\pi$
$17$	7.40581	1.79617	0.898087	+	0.439818i	$0.144957\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 2.93900i	− 0.674253i	−0.941459	−	0.337127i	$-0.890545\pi$
$19$	− 2.93900i	− 0.674253i	0.941459	−	0.337127i	$-0.109455\pi$
$20$	− 1.00000i	− 0.223607i
$21$	2.55496i	0.557538i
$22$	−3.85086	−0.821005
$23$	1.80194	0.375730	0.187865	−	0.982195i	$-0.439843\pi$
$23$	1.80194	0.375730	0.187865	+	0.982195i	$0.439843\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.55496i	0.482842i
$29$	8.03684	1.49240	0.746201	−	0.665720i	$-0.231876\pi$
$29$	8.03684	1.49240	0.746201	+	0.665720i	$0.231876\pi$
$30$	−1.00000	−0.182574
$31$	1.13706i	0.204223i	0.994773	+	0.102111i	$0.0325598\pi$
$31$	1.13706i	0.204223i	−0.994773	+	0.102111i	$0.967440\pi$
$32$	− 1.00000i	− 0.176777i
$33$	3.85086i	0.670348i
$34$	− 7.40581i	− 1.27009i
$35$	2.55496	0.431867
$36$	−1.00000	−0.166667
$37$	3.44504i	0.566361i	0.959067	+	0.283181i	$0.0913896\pi$
$37$	3.44504i	0.566361i	−0.959067	+	0.283181i	$0.908610\pi$
$38$	−2.93900	−0.476769
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 1.55496i	− 0.242844i	−0.992601	−	0.121422i	$-0.961255\pi$
$41$	− 1.55496i	− 0.242844i	0.992601	−	0.121422i	$-0.0387454\pi$
$42$	2.55496	0.394239
$43$	10.4547	1.59433	0.797166	−	0.603761i	$-0.206332\pi$
$43$	10.4547	1.59433	0.797166	+	0.603761i	$0.206332\pi$
$44$	3.85086i	0.580538i
$45$	1.00000i	0.149071i
$46$	− 1.80194i	− 0.265681i
$47$	9.71379i	1.41690i	0.705760	+	0.708451i	$0.250606\pi$
$47$	9.71379i	1.41690i	−0.705760	+	0.708451i	$0.749394\pi$
$48$	−1.00000	−0.144338
$49$	0.472189	0.0674556
$50$	1.00000i	0.141421i
$51$	−7.40581	−1.03702
$52$	0	0
$53$	−8.93900	−1.22787	−0.613933	−	0.789358i	$-0.710414\pi$
$53$	−8.93900	−1.22787	−0.613933	+	0.789358i	$0.710414\pi$
$54$	1.00000i	0.136083i
$55$	3.85086	0.519249
$56$	2.55496	0.341421
$57$	2.93900i	0.389280i
$58$	− 8.03684i	− 1.05529i
$59$	11.7845i	1.53421i	0.641522	+	0.767104i	$0.278303\pi$
$59$	11.7845i	1.53421i	−0.641522	+	0.767104i	$0.721697\pi$
$60$	1.00000i	0.129099i
$61$	−10.0586	−1.28787	−0.643936	−	0.765079i	$-0.722700\pi$
$61$	−10.0586	−1.28787	−0.643936	+	0.765079i	$0.722700\pi$
$62$	1.13706	0.144407
$63$	− 2.55496i	− 0.321894i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.85086	0.474007
$67$	1.36227i	0.166428i	0.996532	+	0.0832140i	$0.0265185\pi$
$67$	1.36227i	0.166428i	−0.996532	+	0.0832140i	$0.973481\pi$
$68$	−7.40581	−0.898087
$69$	−1.80194	−0.216928
$70$	− 2.55496i	− 0.305376i
$71$	− 12.0761i	− 1.43317i	−0.697502	−	0.716583i	$-0.745705\pi$
$71$	− 12.0761i	− 1.43317i	0.697502	−	0.716583i	$-0.254295\pi$
$72$	1.00000i	0.117851i
$73$	− 2.29590i	− 0.268714i	−0.990933	−	0.134357i	$-0.957103\pi$
$73$	− 2.29590i	− 0.268714i	0.990933	−	0.134357i	$-0.0428970\pi$
$74$	3.44504	0.400478
$75$	1.00000	0.115470
$76$	2.93900i	0.337127i
$77$	−9.83877	−1.12123
$78$	0	0
$79$	8.81700	0.991990	0.495995	−	0.868325i	$-0.334803\pi$
$79$	8.81700	0.991990	0.495995	+	0.868325i	$0.334803\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−1.55496	−0.171716
$83$	− 15.9051i	− 1.74582i	−0.487884	−	0.872908i	$-0.662231\pi$
$83$	− 15.9051i	− 1.74582i	0.487884	−	0.872908i	$-0.337769\pi$
$84$	− 2.55496i	− 0.278769i
$85$	7.40581i	0.803273i
$86$	− 10.4547i	− 1.12736i
$87$	−8.03684	−0.861639
$88$	3.85086	0.410503
$89$	15.6799i	1.66207i	0.556220	+	0.831035i	$0.312251\pi$
$89$	15.6799i	1.66207i	−0.556220	+	0.831035i	$0.687749\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−1.80194	−0.187865
$93$	− 1.13706i	− 0.117908i
$94$	9.71379	1.00190
$95$	2.93900	0.301535
$96$	1.00000i	0.102062i
$97$	1.98254i	0.201297i	0.994922	+	0.100648i	$0.0320917\pi$
$97$	1.98254i	0.201297i	−0.994922	+	0.100648i	$0.967908\pi$
$98$	− 0.472189i	− 0.0476983i
$99$	− 3.85086i	− 0.387025i
$100$	1.00000	0.100000
$101$	3.48188	0.346460	0.173230	−	0.984881i	$-0.444580\pi$
$101$	3.48188	0.346460	0.173230	+	0.984881i	$0.444580\pi$
$102$	7.40581i	0.733285i
$103$	−8.65817	−0.853115	−0.426557	−	0.904460i	$-0.640274\pi$
$103$	−8.65817	−0.853115	−0.426557	+	0.904460i	$0.640274\pi$
$104$	0	0
$105$	−2.55496	−0.249338
$106$	8.93900i	0.868233i
$107$	13.9366	1.34730	0.673651	−	0.739049i	$-0.264725\pi$
$107$	13.9366	1.34730	0.673651	+	0.739049i	$0.264725\pi$
$108$	1.00000	0.0962250
$109$	5.39373i	0.516626i	0.966061	+	0.258313i	$0.0831665\pi$
$109$	5.39373i	0.516626i	−0.966061	+	0.258313i	$0.916833\pi$
$110$	− 3.85086i	− 0.367165i
$111$	− 3.44504i	− 0.326989i
$112$	− 2.55496i	− 0.241421i
$113$	9.44265	0.888290	0.444145	−	0.895955i	$-0.353508\pi$
$113$	9.44265	0.888290	0.444145	+	0.895955i	$0.353508\pi$
$114$	2.93900	0.275263
$115$	1.80194i	0.168032i
$116$	−8.03684	−0.746201
$117$	0	0
$118$	11.7845	1.08485
$119$	− 18.9215i	− 1.73453i
$120$	1.00000	0.0912871
$121$	−3.82908	−0.348099
$122$	10.0586i	0.910663i
$123$	1.55496i	0.140206i
$124$	− 1.13706i	− 0.102111i
$125$	− 1.00000i	− 0.0894427i
$126$	−2.55496	−0.227614
$127$	0.499336	0.0443089	0.0221545	−	0.999755i	$-0.492947\pi$
$127$	0.499336	0.0443089	0.0221545	+	0.999755i	$0.492947\pi$
$128$	1.00000i	0.0883883i
$129$	−10.4547	−0.920488
$130$	0	0
$131$	−1.12498	−0.0982901	−0.0491451	−	0.998792i	$-0.515650\pi$
$131$	−1.12498	−0.0982901	−0.0491451	+	0.998792i	$0.515650\pi$
$132$	− 3.85086i	− 0.335174i
$133$	−7.50902	−0.651115
$134$	1.36227	0.117682
$135$	− 1.00000i	− 0.0860663i
$136$	7.40581i	0.635043i
$137$	− 14.1468i	− 1.20864i	−0.796742	−	0.604319i	$-0.793445\pi$
$137$	− 14.1468i	− 1.20864i	0.796742	−	0.604319i	$-0.206555\pi$
$138$	1.80194i	0.153391i
$139$	7.17629	0.608685	0.304343	−	0.952563i	$-0.401563\pi$
$139$	7.17629	0.608685	0.304343	+	0.952563i	$0.401563\pi$
$140$	−2.55496	−0.215933
$141$	− 9.71379i	− 0.818049i
$142$	−12.0761	−1.01340
$143$	0	0
$144$	1.00000	0.0833333
$145$	8.03684i	0.667423i
$146$	−2.29590	−0.190010
$147$	−0.472189	−0.0389455
$148$	− 3.44504i	− 0.283181i
$149$	3.25667i	0.266797i	0.991063	+	0.133398i	$0.0425890\pi$
$149$	3.25667i	0.266797i	−0.991063	+	0.133398i	$0.957411\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 0.137063i	− 0.0111541i	−0.999984	−	0.00557703i	$-0.998225\pi$
$151$	− 0.137063i	− 0.0111541i	0.999984	−	0.00557703i	$-0.00177523\pi$
$152$	2.93900	0.238384
$153$	7.40581	0.598725
$154$	9.83877i	0.792831i
$155$	−1.13706	−0.0913311
$156$	0	0
$157$	4.35152	0.347289	0.173645	−	0.984808i	$-0.444446\pi$
$157$	4.35152	0.347289	0.173645	+	0.984808i	$0.444446\pi$
$158$	− 8.81700i	− 0.701443i
$159$	8.93900	0.708909
$160$	1.00000	0.0790569
$161$	− 4.60388i	− 0.362836i
$162$	− 1.00000i	− 0.0785674i
$163$	− 11.6189i	− 0.910066i	−0.890475	−	0.455033i	$-0.849628\pi$
$163$	− 11.6189i	− 0.910066i	0.890475	−	0.455033i	$-0.150372\pi$
$164$	1.55496i	0.121422i
$165$	−3.85086	−0.299789
$166$	−15.9051	−1.23448
$167$	− 20.8146i	− 1.61068i	−0.592811	−	0.805341i	$-0.701982\pi$
$167$	− 20.8146i	− 1.61068i	0.592811	−	0.805341i	$-0.298018\pi$
$168$	−2.55496	−0.197119
$169$	0	0
$170$	7.40581	0.568000
$171$	− 2.93900i	− 0.224751i
$172$	−10.4547	−0.797166
$173$	8.89440	0.676228	0.338114	−	0.941105i	$-0.390211\pi$
$173$	8.89440	0.676228	0.338114	+	0.941105i	$0.390211\pi$
$174$	8.03684i	0.609271i
$175$	2.55496i	0.193137i
$176$	− 3.85086i	− 0.290269i
$177$	− 11.7845i	− 0.885776i
$178$	15.6799	1.17526
$179$	−0.188374	−0.0140797	−0.00703985	−	0.999975i	$-0.502241\pi$
$179$	−0.188374	−0.0140797	−0.00703985	+	0.999975i	$0.502241\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−18.1588	−1.34974	−0.674868	−	0.737939i	$-0.735799\pi$
$181$	−18.1588	−1.34974	−0.674868	+	0.737939i	$0.735799\pi$
$182$	0	0
$183$	10.0586	0.743554
$184$	1.80194i	0.132841i
$185$	−3.44504	−0.253285
$186$	−1.13706	−0.0833735
$187$	− 28.5187i	− 2.08549i
$188$	− 9.71379i	− 0.708451i
$189$	2.55496i	0.185846i
$190$	− 2.93900i	− 0.213218i
$191$	0.469205	0.0339505	0.0169752	−	0.999856i	$-0.494596\pi$
$191$	0.469205	0.0339505	0.0169752	+	0.999856i	$0.494596\pi$
$192$	1.00000	0.0721688
$193$	17.9312i	1.29072i	0.763879	+	0.645359i	$0.223292\pi$
$193$	17.9312i	1.29072i	−0.763879	+	0.645359i	$0.776708\pi$
$194$	1.98254	0.142338
$195$	0	0
$196$	−0.472189	−0.0337278
$197$	− 8.17390i	− 0.582366i	−0.956667	−	0.291183i	$-0.905951\pi$
$197$	− 8.17390i	− 0.582366i	0.956667	−	0.291183i	$-0.0940489\pi$
$198$	−3.85086	−0.273668
$199$	23.7536	1.68385	0.841924	−	0.539595i	$-0.181423\pi$
$199$	23.7536	1.68385	0.841924	+	0.539595i	$0.181423\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 1.36227i	− 0.0960873i
$202$	− 3.48188i	− 0.244984i
$203$	− 20.5338i	− 1.44119i
$204$	7.40581	0.518511
$205$	1.55496	0.108603
$206$	8.65817i	0.603243i
$207$	1.80194	0.125243
$208$	0	0
$209$	−11.3177	−0.782859
$210$	2.55496i	0.176309i
$211$	24.2392	1.66870	0.834348	−	0.551238i	$-0.185844\pi$
$211$	24.2392	1.66870	0.834348	+	0.551238i	$0.185844\pi$
$212$	8.93900	0.613933
$213$	12.0761i	0.827438i
$214$	− 13.9366i	− 0.952687i
$215$	10.4547i	0.713007i
$216$	− 1.00000i	− 0.0680414i
$217$	2.90515	0.197214
$218$	5.39373	0.365310
$219$	2.29590i	0.155142i
$220$	−3.85086	−0.259625
$221$	0	0
$222$	−3.44504	−0.231216
$223$	− 22.5579i	− 1.51059i	−0.655384	−	0.755296i	$-0.727493\pi$
$223$	− 22.5579i	− 1.51059i	0.655384	−	0.755296i	$-0.272507\pi$
$224$	−2.55496	−0.170710
$225$	−1.00000	−0.0666667
$226$	− 9.44265i	− 0.628116i
$227$	− 23.7439i	− 1.57594i	−0.615714	−	0.787970i	$-0.711132\pi$
$227$	− 23.7439i	− 1.57594i	0.615714	−	0.787970i	$-0.288868\pi$
$228$	− 2.93900i	− 0.194640i
$229$	3.11529i	0.205864i	0.994688	+	0.102932i	$0.0328225\pi$
$229$	3.11529i	0.205864i	−0.994688	+	0.102932i	$0.967178\pi$
$230$	1.80194	0.118816
$231$	9.83877	0.647344
$232$	8.03684i	0.527644i
$233$	12.5603	0.822855	0.411427	−	0.911443i	$-0.365030\pi$
$233$	12.5603	0.822855	0.411427	+	0.911443i	$0.365030\pi$
$234$	0	0
$235$	−9.71379	−0.633658
$236$	− 11.7845i	− 0.767104i
$237$	−8.81700	−0.572726
$238$	−18.9215	−1.22650
$239$	− 8.52111i	− 0.551185i	−0.961275	−	0.275592i	$-0.911126\pi$
$239$	− 8.52111i	− 0.551185i	0.961275	−	0.275592i	$-0.0888740\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 11.6136i	− 0.748095i	−0.927409	−	0.374048i	$-0.877970\pi$
$241$	− 11.6136i	− 0.748095i	0.927409	−	0.374048i	$-0.122030\pi$
$242$	3.82908i	0.246143i
$243$	−1.00000	−0.0641500
$244$	10.0586	0.643936
$245$	0.472189i	0.0301670i
$246$	1.55496	0.0991405
$247$	0	0
$248$	−1.13706	−0.0722036
$249$	15.9051i	1.00795i
$250$	−1.00000	−0.0632456
$251$	−22.7463	−1.43573	−0.717867	−	0.696180i	$-0.754882\pi$
$251$	−22.7463	−1.43573	−0.717867	+	0.696180i	$0.754882\pi$
$252$	2.55496i	0.160947i
$253$	− 6.93900i	− 0.436251i
$254$	− 0.499336i	− 0.0313311i
$255$	− 7.40581i	− 0.463770i
$256$	1.00000	0.0625000
$257$	−7.39373	−0.461208	−0.230604	−	0.973048i	$-0.574070\pi$
$257$	−7.39373	−0.461208	−0.230604	+	0.973048i	$0.574070\pi$
$258$	10.4547i	0.650883i
$259$	8.80194	0.546926
$260$	0	0
$261$	8.03684	0.497468
$262$	1.12498i	0.0695016i
$263$	−11.7735	−0.725983	−0.362991	−	0.931792i	$-0.618245\pi$
$263$	−11.7735	−0.725983	−0.362991	+	0.931792i	$0.618245\pi$
$264$	−3.85086	−0.237004
$265$	− 8.93900i	− 0.549118i
$266$	7.50902i	0.460408i
$267$	− 15.6799i	− 0.959597i
$268$	− 1.36227i	− 0.0832140i
$269$	−22.8810	−1.39508	−0.697539	−	0.716547i	$-0.745722\pi$
$269$	−22.8810	−1.39508	−0.697539	+	0.716547i	$0.745722\pi$
$270$	−1.00000	−0.0608581
$271$	− 2.13036i	− 0.129410i	−0.997904	−	0.0647050i	$-0.979389\pi$
$271$	− 2.13036i	− 0.129410i	0.997904	−	0.0647050i	$-0.0206107\pi$
$272$	7.40581	0.449043
$273$	0	0
$274$	−14.1468	−0.854637
$275$	3.85086i	0.232215i
$276$	1.80194	0.108464
$277$	14.7084	0.883743	0.441872	−	0.897078i	$-0.354315\pi$
$277$	14.7084	0.883743	0.441872	+	0.897078i	$0.354315\pi$
$278$	− 7.17629i	− 0.430405i
$279$	1.13706i	0.0680742i
$280$	2.55496i	0.152688i
$281$	12.5942i	0.751306i	0.926760	+	0.375653i	$0.122582\pi$
$281$	12.5942i	0.751306i	−0.926760	+	0.375653i	$0.877418\pi$
$282$	−9.71379	−0.578448
$283$	1.62565	0.0966346	0.0483173	−	0.998832i	$-0.484614\pi$
$283$	1.62565	0.0966346	0.0483173	+	0.998832i	$0.484614\pi$
$284$	12.0761i	0.716583i
$285$	−2.93900	−0.174091
$286$	0	0
$287$	−3.97285	−0.234510
$288$	− 1.00000i	− 0.0589256i
$289$	37.8461	2.22624
$290$	8.03684	0.471939
$291$	− 1.98254i	− 0.116219i
$292$	2.29590i	0.134357i
$293$	13.0664i	0.763346i	0.924297	+	0.381673i	$0.124652\pi$
$293$	13.0664i	0.763346i	−0.924297	+	0.381673i	$0.875348\pi$
$294$	0.472189i	0.0275386i
$295$	−11.7845	−0.686119
$296$	−3.44504	−0.200239
$297$	3.85086i	0.223449i
$298$	3.25667	0.188654
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 26.7114i	− 1.53962i
$302$	−0.137063	−0.00788711
$303$	−3.48188	−0.200029
$304$	− 2.93900i	− 0.168563i
$305$	− 10.0586i	− 0.575954i
$306$	− 7.40581i	− 0.423362i
$307$	26.2489i	1.49810i	0.662511	+	0.749052i	$0.269491\pi$
$307$	26.2489i	1.49810i	−0.662511	+	0.749052i	$0.730509\pi$
$308$	9.83877	0.560616
$309$	8.65817	0.492546
$310$	1.13706i	0.0645809i
$311$	−27.1250	−1.53812	−0.769058	−	0.639179i	$-0.779274\pi$
$311$	−27.1250	−1.53812	−0.769058	+	0.639179i	$0.779274\pi$
$312$	0	0
$313$	19.1323	1.08142	0.540710	−	0.841209i	$-0.318156\pi$
$313$	19.1323	1.08142	0.540710	+	0.841209i	$0.318156\pi$
$314$	− 4.35152i	− 0.245570i
$315$	2.55496	0.143956
$316$	−8.81700	−0.495995
$317$	− 6.67456i	− 0.374881i	−0.982276	−	0.187440i	$-0.939981\pi$
$317$	− 6.67456i	− 0.374881i	0.982276	−	0.187440i	$-0.0600191\pi$
$318$	− 8.93900i	− 0.501274i
$319$	− 30.9487i	− 1.73279i
$320$	− 1.00000i	− 0.0559017i
$321$	−13.9366	−0.777866
$322$	−4.60388	−0.256564
$323$	− 21.7657i	− 1.21108i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−11.6189	−0.643514
$327$	− 5.39373i	− 0.298274i
$328$	1.55496	0.0858582
$329$	24.8183	1.36828
$330$	3.85086i	0.211983i
$331$	18.1360i	0.996845i	0.866934	+	0.498422i	$0.166087\pi$
$331$	18.1360i	0.996845i	−0.866934	+	0.498422i	$0.833913\pi$
$332$	15.9051i	0.872908i
$333$	3.44504i	0.188787i
$334$	−20.8146	−1.13892
$335$	−1.36227	−0.0744289
$336$	2.55496i	0.139384i
$337$	−10.8780	−0.592563	−0.296281	−	0.955101i	$-0.595747\pi$
$337$	−10.8780	−0.592563	−0.296281	+	0.955101i	$0.595747\pi$
$338$	0	0
$339$	−9.44265	−0.512854
$340$	− 7.40581i	− 0.401637i
$341$	4.37867	0.237118
$342$	−2.93900	−0.158923
$343$	− 19.0911i	− 1.03082i
$344$	10.4547i	0.563681i
$345$	− 1.80194i	− 0.0970131i
$346$	− 8.89440i	− 0.478166i
$347$	−7.42865	−0.398791	−0.199395	−	0.979919i	$-0.563898\pi$
$347$	−7.42865	−0.398791	−0.199395	+	0.979919i	$0.563898\pi$
$348$	8.03684	0.430820
$349$	− 28.7275i	− 1.53775i	−0.639399	−	0.768875i	$-0.720817\pi$
$349$	− 28.7275i	− 1.53775i	0.639399	−	0.768875i	$-0.279183\pi$
$350$	2.55496	0.136568
$351$	0	0
$352$	−3.85086	−0.205251
$353$	− 31.2131i	− 1.66131i	−0.556790	−	0.830654i	$-0.687967\pi$
$353$	− 31.2131i	− 1.66131i	0.556790	−	0.830654i	$-0.312033\pi$
$354$	−11.7845	−0.626338
$355$	12.0761	0.640931
$356$	− 15.6799i	− 0.831035i
$357$	18.9215i	1.00143i
$358$	0.188374i	0.00995585i
$359$	− 0.0247542i	− 0.00130648i	−1.00000		0.000653238i	$-0.999792\pi$
$359$	− 0.0247542i	− 0.00130648i	1.00000		0.000653238i	$-0.000207932\pi$
$360$	−1.00000	−0.0527046
$361$	10.3623	0.545383
$362$	18.1588i	0.954407i
$363$	3.82908	0.200975
$364$	0	0
$365$	2.29590	0.120173
$366$	− 10.0586i	− 0.525772i
$367$	17.5284	0.914975	0.457488	−	0.889216i	$-0.348749\pi$
$367$	17.5284	0.914975	0.457488	+	0.889216i	$0.348749\pi$
$368$	1.80194	0.0939325
$369$	− 1.55496i	− 0.0809479i
$370$	3.44504i	0.179099i
$371$	22.8388i	1.18573i
$372$	1.13706i	0.0589540i
$373$	−35.3056	−1.82805	−0.914027	−	0.405653i	$-0.867044\pi$
$373$	−35.3056	−1.82805	−0.914027	+	0.405653i	$0.867044\pi$
$374$	−28.5187	−1.47467
$375$	1.00000i	0.0516398i
$376$	−9.71379	−0.500951
$377$	0	0
$378$	2.55496	0.131413
$379$	18.2567i	0.937782i	0.883256	+	0.468891i	$0.155346\pi$
$379$	18.2567i	0.937782i	−0.883256	+	0.468891i	$0.844654\pi$
$380$	−2.93900	−0.150768
$381$	−0.499336	−0.0255818
$382$	− 0.469205i	− 0.0240066i
$383$	− 2.83148i	− 0.144682i	−0.997380	−	0.0723409i	$-0.976953\pi$
$383$	− 2.83148i	− 0.144682i	0.997380	−	0.0723409i	$-0.0230469\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 9.83877i	− 0.501430i
$386$	17.9312	0.912676
$387$	10.4547	0.531444
$388$	− 1.98254i	− 0.100648i
$389$	−28.8025	−1.46035	−0.730173	−	0.683262i	$-0.760561\pi$
$389$	−28.8025	−1.46035	−0.730173	+	0.683262i	$0.760561\pi$
$390$	0	0
$391$	13.3448	0.674876
$392$	0.472189i	0.0238491i
$393$	1.12498	0.0567478
$394$	−8.17390	−0.411795
$395$	8.81700i	0.443632i
$396$	3.85086i	0.193513i
$397$	− 21.4306i	− 1.07557i	−0.843082	−	0.537785i	$-0.819261\pi$
$397$	− 21.4306i	− 1.07557i	0.843082	−	0.537785i	$-0.180739\pi$
$398$	− 23.7536i	− 1.19066i
$399$	7.50902	0.375921
$400$	−1.00000	−0.0500000
$401$	− 19.5284i	− 0.975202i	−0.873067	−	0.487601i	$-0.837872\pi$
$401$	− 19.5284i	− 0.975202i	0.873067	−	0.487601i	$-0.162128\pi$
$402$	−1.36227	−0.0679440
$403$	0	0
$404$	−3.48188	−0.173230
$405$	1.00000i	0.0496904i
$406$	−20.5338	−1.01907
$407$	13.2664	0.657589
$408$	− 7.40581i	− 0.366642i
$409$	30.2707i	1.49679i	0.663254	+	0.748394i	$0.269175\pi$
$409$	30.2707i	1.49679i	−0.663254	+	0.748394i	$0.730825\pi$
$410$	− 1.55496i	− 0.0767939i
$411$	14.1468i	0.697808i
$412$	8.65817	0.426557
$413$	30.1089	1.48156
$414$	− 1.80194i	− 0.0885604i
$415$	15.9051	0.780753
$416$	0	0
$417$	−7.17629	−0.351425
$418$	11.3177i	0.553565i
$419$	−24.4873	−1.19628	−0.598140	−	0.801391i	$-0.704093\pi$
$419$	−24.4873	−1.19628	−0.598140	+	0.801391i	$0.704093\pi$
$420$	2.55496	0.124669
$421$	− 8.90408i	− 0.433959i	−0.976176	−	0.216979i	$-0.930380\pi$
$421$	− 8.90408i	− 0.433959i	0.976176	−	0.216979i	$-0.0696204\pi$
$422$	− 24.2392i	− 1.17995i
$423$	9.71379i	0.472301i
$424$	− 8.93900i	− 0.434116i
$425$	−7.40581	−0.359235
$426$	12.0761	0.585087
$427$	25.6993i	1.24368i
$428$	−13.9366	−0.673651
$429$	0	0
$430$	10.4547	0.504172
$431$	1.48619i	0.0715872i	0.999359	+	0.0357936i	$0.0113959\pi$
$431$	1.48619i	0.0715872i	−0.999359	+	0.0357936i	$0.988604\pi$
$432$	−1.00000	−0.0481125
$433$	−18.5719	−0.892511	−0.446255	−	0.894906i	$-0.647243\pi$
$433$	−18.5719	−0.892511	−0.446255	+	0.894906i	$0.647243\pi$
$434$	− 2.90515i	− 0.139452i
$435$	− 8.03684i	− 0.385337i
$436$	− 5.39373i	− 0.258313i
$437$	− 5.29590i	− 0.253337i
$438$	2.29590	0.109702
$439$	−0.532123	−0.0253968	−0.0126984	−	0.999919i	$-0.504042\pi$
$439$	−0.532123	−0.0253968	−0.0126984	+	0.999919i	$0.504042\pi$
$440$	3.85086i	0.183582i
$441$	0.472189	0.0224852
$442$	0	0
$443$	25.5230	1.21264	0.606318	−	0.795222i	$-0.292646\pi$
$443$	25.5230	1.21264	0.606318	+	0.795222i	$0.292646\pi$
$444$	3.44504i	0.163494i
$445$	−15.6799	−0.743300
$446$	−22.5579	−1.06815
$447$	− 3.25667i	− 0.154035i
$448$	2.55496i	0.120710i
$449$	40.7730i	1.92420i	0.272703	+	0.962098i	$0.412082\pi$
$449$	40.7730i	1.92420i	−0.272703	+	0.962098i	$0.587918\pi$
$450$	1.00000i	0.0471405i
$451$	−5.98792	−0.281960
$452$	−9.44265	−0.444145
$453$	0.137063i	0.00643980i
$454$	−23.7439	−1.11436
$455$	0	0
$456$	−2.93900	−0.137631
$457$	− 15.1511i	− 0.708737i	−0.935106	−	0.354368i	$-0.884696\pi$
$457$	− 15.1511i	− 0.708737i	0.935106	−	0.354368i	$-0.115304\pi$
$458$	3.11529	0.145568
$459$	−7.40581	−0.345674
$460$	− 1.80194i	− 0.0840158i
$461$	− 19.5743i	− 0.911668i	−0.890065	−	0.455834i	$-0.849341\pi$
$461$	− 19.5743i	− 0.911668i	0.890065	−	0.455834i	$-0.150659\pi$
$462$	− 9.83877i	− 0.457741i
$463$	− 31.2905i	− 1.45419i	−0.686535	−	0.727097i	$-0.740869\pi$
$463$	− 31.2905i	− 1.45419i	0.686535	−	0.727097i	$-0.259131\pi$
$464$	8.03684	0.373101
$465$	1.13706	0.0527300
$466$	− 12.5603i	− 0.581846i
$467$	32.3032	1.49481	0.747407	−	0.664367i	$-0.231299\pi$
$467$	32.3032	1.49481	0.747407	+	0.664367i	$0.231299\pi$
$468$	0	0
$469$	3.48055	0.160717
$470$	9.71379i	0.448064i
$471$	−4.35152	−0.200507
$472$	−11.7845	−0.542425
$473$	− 40.2597i	− 1.85114i
$474$	8.81700i	0.404978i
$475$	2.93900i	0.134851i
$476$	18.9215i	0.867267i
$477$	−8.93900	−0.409289
$478$	−8.52111	−0.389746
$479$	9.32842i	0.426226i	0.977028	+	0.213113i	$0.0683603\pi$
$479$	9.32842i	0.426226i	−0.977028	+	0.213113i	$0.931640\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−11.6136	−0.528983
$483$	4.60388i	0.209484i
$484$	3.82908	0.174049
$485$	−1.98254	−0.0900226
$486$	1.00000i	0.0453609i
$487$	10.6571i	0.482920i	0.970411	+	0.241460i	$0.0776262\pi$
$487$	10.6571i	0.482920i	−0.970411	+	0.241460i	$0.922374\pi$
$488$	− 10.0586i	− 0.455332i
$489$	11.6189i	0.525427i
$490$	0.472189	0.0213313
$491$	−40.8611	−1.84404	−0.922019	−	0.387146i	$-0.873461\pi$
$491$	−40.8611	−1.84404	−0.922019	+	0.387146i	$0.873461\pi$
$492$	− 1.55496i	− 0.0701029i
$493$	59.5193	2.68061
$494$	0	0
$495$	3.85086	0.173083
$496$	1.13706i	0.0510557i
$497$	−30.8538	−1.38398
$498$	15.9051	0.712727
$499$	19.0556i	0.853047i	0.904477	+	0.426523i	$0.140262\pi$
$499$	19.0556i	0.853047i	−0.904477	+	0.426523i	$0.859738\pi$
$500$	1.00000i	0.0447214i
$501$	20.8146i	0.929928i
$502$	22.7463i	1.01522i
$503$	−16.9825	−0.757214	−0.378607	−	0.925558i	$-0.623597\pi$
$503$	−16.9825	−0.757214	−0.378607	+	0.925558i	$0.623597\pi$
$504$	2.55496	0.113807
$505$	3.48188i	0.154942i
$506$	−6.93900	−0.308476
$507$	0	0
$508$	−0.499336	−0.0221545
$509$	− 43.0116i	− 1.90646i	−0.302252	−	0.953228i	$-0.597738\pi$
$509$	− 43.0116i	− 1.90646i	0.302252	−	0.953228i	$-0.402262\pi$
$510$	−7.40581	−0.327935
$511$	−5.86592	−0.259493
$512$	− 1.00000i	− 0.0441942i
$513$	2.93900i	0.129760i
$514$	7.39373i	0.326124i
$515$	− 8.65817i	− 0.381525i
$516$	10.4547	0.460244
$517$	37.4064	1.64513
$518$	− 8.80194i	− 0.386735i
$519$	−8.89440	−0.390421
$520$	0	0
$521$	−25.1021	−1.09975	−0.549873	−	0.835249i	$-0.685324\pi$
$521$	−25.1021	−1.09975	−0.549873	+	0.835249i	$0.685324\pi$
$522$	− 8.03684i	− 0.351763i
$523$	−3.86725	−0.169103	−0.0845515	−	0.996419i	$-0.526946\pi$
$523$	−3.86725	−0.169103	−0.0845515	+	0.996419i	$0.526946\pi$
$524$	1.12498	0.0491451
$525$	− 2.55496i	− 0.111508i
$526$	11.7735i	0.513347i
$527$	8.42088i	0.366819i
$528$	3.85086i	0.167587i
$529$	−19.7530	−0.858827
$530$	−8.93900	−0.388285
$531$	11.7845i	0.511403i
$532$	7.50902	0.325558
$533$	0	0
$534$	−15.6799	−0.678537
$535$	13.9366i	0.602532i
$536$	−1.36227	−0.0588412
$537$	0.188374	0.00812892
$538$	22.8810i	0.986469i
$539$	− 1.81833i	− 0.0783211i
$540$	1.00000i	0.0430331i
$541$	42.6152i	1.83217i	0.400983	+	0.916086i	$0.368669\pi$
$541$	42.6152i	1.83217i	−0.400983	+	0.916086i	$0.631331\pi$
$542$	−2.13036	−0.0915067
$543$	18.1588	0.779270
$544$	− 7.40581i	− 0.317522i
$545$	−5.39373	−0.231042
$546$	0	0
$547$	−29.7928	−1.27385	−0.636925	−	0.770926i	$-0.719794\pi$
$547$	−29.7928	−1.27385	−0.636925	+	0.770926i	$0.719794\pi$
$548$	14.1468i	0.604319i
$549$	−10.0586	−0.429291
$550$	3.85086	0.164201
$551$	− 23.6203i	− 1.00626i
$552$	− 1.80194i	− 0.0766956i
$553$	− 22.5271i	− 0.957949i
$554$	− 14.7084i	− 0.624901i
$555$	3.44504	0.146234
$556$	−7.17629	−0.304343
$557$	− 37.7700i	− 1.60037i	−0.599756	−	0.800183i	$-0.704736\pi$
$557$	− 37.7700i	− 1.60037i	0.599756	−	0.800183i	$-0.295264\pi$
$558$	1.13706	0.0481357
$559$	0	0
$560$	2.55496	0.107967
$561$	28.5187i	1.20406i
$562$	12.5942	0.531254
$563$	−25.0237	−1.05462	−0.527311	−	0.849672i	$-0.676800\pi$
$563$	−25.0237	−1.05462	−0.527311	+	0.849672i	$0.676800\pi$
$564$	9.71379i	0.409024i
$565$	9.44265i	0.397255i
$566$	− 1.62565i	− 0.0683310i
$567$	− 2.55496i	− 0.107298i
$568$	12.0761	0.506700
$569$	−3.81402	−0.159892	−0.0799460	−	0.996799i	$-0.525475\pi$
$569$	−3.81402	−0.159892	−0.0799460	+	0.996799i	$0.525475\pi$
$570$	2.93900i	0.123101i
$571$	4.76702	0.199494	0.0997468	−	0.995013i	$-0.468197\pi$
$571$	4.76702	0.199494	0.0997468	+	0.995013i	$0.468197\pi$
$572$	0	0
$573$	−0.469205	−0.0196013
$574$	3.97285i	0.165824i
$575$	−1.80194	−0.0751460
$576$	−1.00000	−0.0416667
$577$	20.9879i	0.873738i	0.899525	+	0.436869i	$0.143913\pi$
$577$	20.9879i	0.873738i	−0.899525	+	0.436869i	$0.856087\pi$
$578$	− 37.8461i	− 1.57419i
$579$	− 17.9312i	− 0.745197i
$580$	− 8.03684i	− 0.333711i
$581$	−40.6370	−1.68591
$582$	−1.98254	−0.0821790
$583$	34.4228i	1.42565i
$584$	2.29590	0.0950049
$585$	0	0
$586$	13.0664	0.539767
$587$	13.9312i	0.575003i	0.957780	+	0.287502i	$0.0928247\pi$
$587$	13.9312i	0.575003i	−0.957780	+	0.287502i	$0.907175\pi$
$588$	0.472189	0.0194727
$589$	3.34183	0.137698
$590$	11.7845i	0.485159i
$591$	8.17390i	0.336229i
$592$	3.44504i	0.141590i
$593$	− 9.75600i	− 0.400631i	−0.979731	−	0.200316i	$-0.935803\pi$
$593$	− 9.75600i	− 0.400631i	0.979731	−	0.200316i	$-0.0641967\pi$
$594$	3.85086	0.158002
$595$	18.9215	0.775708
$596$	− 3.25667i	− 0.133398i
$597$	−23.7536	−0.972171
$598$	0	0
$599$	−9.88769	−0.404000	−0.202000	−	0.979386i	$-0.564744\pi$
$599$	−9.88769	−0.404000	−0.202000	+	0.979386i	$0.564744\pi$
$600$	1.00000i	0.0408248i
$601$	4.32304	0.176341	0.0881703	−	0.996105i	$-0.471898\pi$
$601$	4.32304	0.176341	0.0881703	+	0.996105i	$0.471898\pi$
$602$	−26.7114	−1.08868
$603$	1.36227i	0.0554760i
$604$	0.137063i	0.00557703i
$605$	− 3.82908i	− 0.155674i
$606$	3.48188i	0.141442i
$607$	37.5948	1.52592	0.762962	−	0.646443i	$-0.223744\pi$
$607$	37.5948	1.52592	0.762962	+	0.646443i	$0.223744\pi$
$608$	−2.93900	−0.119192
$609$	20.5338i	0.832071i
$610$	−10.0586	−0.407261
$611$	0	0
$612$	−7.40581	−0.299362
$613$	33.8595i	1.36757i	0.729683	+	0.683786i	$0.239668\pi$
$613$	33.8595i	1.36757i	−0.729683	+	0.683786i	$0.760332\pi$
$614$	26.2489	1.05932
$615$	−1.55496	−0.0627020
$616$	− 9.83877i	− 0.396415i
$617$	43.1333i	1.73648i	0.496142	+	0.868241i	$0.334750\pi$
$617$	43.1333i	1.73648i	−0.496142	+	0.868241i	$0.665250\pi$
$618$	− 8.65817i	− 0.348283i
$619$	− 14.6606i	− 0.589258i	−0.955612	−	0.294629i	$-0.904804\pi$
$619$	− 14.6606i	− 0.589258i	0.955612	−	0.294629i	$-0.0951960\pi$
$620$	1.13706	0.0456656
$621$	−1.80194	−0.0723093
$622$	27.1250i	1.08761i
$623$	40.0616	1.60503
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 19.1323i	− 0.764680i
$627$	11.3177	0.451984
$628$	−4.35152	−0.173645
$629$	25.5133i	1.01728i
$630$	− 2.55496i	− 0.101792i
$631$	10.5265i	0.419053i	0.977803	+	0.209526i	$0.0671922\pi$
$631$	10.5265i	0.419053i	−0.977803	+	0.209526i	$0.932808\pi$
$632$	8.81700i	0.350722i
$633$	−24.2392	−0.963422
$634$	−6.67456	−0.265081
$635$	0.499336i	0.0198155i
$636$	−8.93900	−0.354454
$637$	0	0
$638$	−30.9487	−1.22527
$639$	− 12.0761i	− 0.477722i
$640$	−1.00000	−0.0395285
$641$	16.5435	0.653428	0.326714	−	0.945123i	$-0.394059\pi$
$641$	16.5435	0.653428	0.326714	+	0.945123i	$0.394059\pi$
$642$	13.9366i	0.550034i
$643$	− 43.4282i	− 1.71264i	−0.516446	−	0.856320i	$-0.672745\pi$
$643$	− 43.4282i	− 1.71264i	0.516446	−	0.856320i	$-0.327255\pi$
$644$	4.60388i	0.181418i
$645$	− 10.4547i	− 0.411655i
$646$	−21.7657	−0.856360
$647$	−28.7687	−1.13101	−0.565507	−	0.824744i	$-0.691319\pi$
$647$	−28.7687	−1.13101	−0.565507	+	0.824744i	$0.691319\pi$
$648$	1.00000i	0.0392837i
$649$	45.3803	1.78133
$650$	0	0
$651$	−2.90515	−0.113862
$652$	11.6189i	0.455033i
$653$	−30.0978	−1.17782	−0.588910	−	0.808199i	$-0.700443\pi$
$653$	−30.0978	−1.17782	−0.588910	+	0.808199i	$0.700443\pi$
$654$	−5.39373	−0.210912
$655$	− 1.12498i	− 0.0439567i
$656$	− 1.55496i	− 0.0607109i
$657$	− 2.29590i	− 0.0895715i
$658$	− 24.8183i	− 0.967519i
$659$	46.6015	1.81534	0.907668	−	0.419689i	$-0.137861\pi$
$659$	46.6015	1.81534	0.907668	+	0.419689i	$0.137861\pi$
$660$	3.85086	0.149894
$661$	40.9788i	1.59389i	0.604051	+	0.796946i	$0.293552\pi$
$661$	40.9788i	1.59389i	−0.604051	+	0.796946i	$0.706448\pi$
$662$	18.1360	0.704876
$663$	0	0
$664$	15.9051	0.617239
$665$	− 7.50902i	− 0.291187i
$666$	3.44504	0.133493
$667$	14.4819	0.560741
$668$	20.8146i	0.805341i
$669$	22.5579i	0.872140i
$670$	1.36227i	0.0526292i
$671$	38.7342i	1.49532i
$672$	2.55496	0.0985596
$673$	49.2586	1.89878	0.949389	−	0.314101i	$-0.101703\pi$
$673$	49.2586	1.89878	0.949389	+	0.314101i	$0.101703\pi$
$674$	10.8780i	0.419005i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−31.1909	−1.19876	−0.599382	−	0.800463i	$-0.704587\pi$
$677$	−31.1909	−1.19876	−0.599382	+	0.800463i	$0.704587\pi$
$678$	9.44265i	0.362643i
$679$	5.06531	0.194389
$680$	−7.40581	−0.284000
$681$	23.7439i	0.909869i
$682$	− 4.37867i	− 0.167668i
$683$	27.0164i	1.03375i	0.856060	+	0.516877i	$0.172905\pi$
$683$	27.0164i	1.03375i	−0.856060	+	0.516877i	$0.827095\pi$
$684$	2.93900i	0.112376i
$685$	14.1468	0.540520
$686$	−19.0911	−0.728903
$687$	− 3.11529i	− 0.118856i
$688$	10.4547	0.398583
$689$	0	0
$690$	−1.80194	−0.0685986
$691$	9.03551i	0.343727i	0.985121	+	0.171863i	$0.0549788\pi$
$691$	9.03551i	0.343727i	−0.985121	+	0.171863i	$0.945021\pi$
$692$	−8.89440	−0.338114
$693$	−9.83877	−0.373744
$694$	7.42865i	0.281988i
$695$	7.17629i	0.272212i
$696$	− 8.03684i	− 0.304635i
$697$	− 11.5157i	− 0.436189i
$698$	−28.7275	−1.08735
$699$	−12.5603	−0.475075
$700$	− 2.55496i	− 0.0965683i
$701$	16.5332	0.624450	0.312225	−	0.950008i	$-0.398926\pi$
$701$	16.5332	0.624450	0.312225	+	0.950008i	$0.398926\pi$
$702$	0	0
$703$	10.1250	0.381871
$704$	3.85086i	0.145135i
$705$	9.71379	0.365843
$706$	−31.2131	−1.17472
$707$	− 8.89605i	− 0.334570i
$708$	11.7845i	0.442888i
$709$	28.4868i	1.06984i	0.844902	+	0.534922i	$0.179659\pi$
$709$	28.4868i	1.06984i	−0.844902	+	0.534922i	$0.820341\pi$
$710$	− 12.0761i	− 0.453207i
$711$	8.81700	0.330663
$712$	−15.6799	−0.587631
$713$	2.04892i	0.0767326i
$714$	18.9215	0.708121
$715$	0	0
$716$	0.188374	0.00703985
$717$	8.52111i	0.318227i
$718$	−0.0247542	−0.000923817 0
$719$	36.4064	1.35773	0.678865	−	0.734263i	$-0.262472\pi$
$719$	36.4064	1.35773	0.678865	+	0.734263i	$0.262472\pi$
$720$	1.00000i	0.0372678i
$721$	22.1213i	0.823839i
$722$	− 10.3623i	− 0.385644i
$723$	11.6136i	0.431913i
$724$	18.1588	0.674868
$725$	−8.03684	−0.298481
$726$	− 3.82908i	− 0.142111i
$727$	−36.1196	−1.33960	−0.669801	−	0.742541i	$-0.733621\pi$
$727$	−36.1196	−1.33960	−0.669801	+	0.742541i	$0.733621\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 2.29590i	− 0.0849750i
$731$	77.4258	2.86370
$732$	−10.0586	−0.371777
$733$	17.1648i	0.633996i	0.948426	+	0.316998i	$0.102675\pi$
$733$	17.1648i	0.633996i	−0.948426	+	0.316998i	$0.897325\pi$
$734$	− 17.5284i	− 0.646985i
$735$	− 0.472189i	− 0.0174170i
$736$	− 1.80194i	− 0.0664203i
$737$	5.24591	0.193236
$738$	−1.55496	−0.0572388
$739$	− 21.5821i	− 0.793911i	−0.917838	−	0.396955i	$-0.870067\pi$
$739$	− 21.5821i	− 0.793911i	0.917838	−	0.396955i	$-0.129933\pi$
$740$	3.44504	0.126642
$741$	0	0
$742$	22.8388	0.838438
$743$	16.6698i	0.611555i	0.952103	+	0.305777i	$0.0989163\pi$
$743$	16.6698i	0.611555i	−0.952103	+	0.305777i	$0.901084\pi$
$744$	1.13706	0.0416868
$745$	−3.25667	−0.119315
$746$	35.3056i	1.29263i
$747$	− 15.9051i	− 0.581939i
$748$	28.5187i	1.04275i
$749$	− 35.6075i	− 1.30107i
$750$	1.00000	0.0365148
$751$	−11.8009	−0.430620	−0.215310	−	0.976546i	$-0.569076\pi$
$751$	−11.8009	−0.430620	−0.215310	+	0.976546i	$0.569076\pi$
$752$	9.71379i	0.354226i
$753$	22.7463	0.828922
$754$	0	0
$755$	0.137063	0.00498825
$756$	− 2.55496i	− 0.0929229i
$757$	5.44371	0.197855	0.0989276	−	0.995095i	$-0.468459\pi$
$757$	5.44371	0.197855	0.0989276	+	0.995095i	$0.468459\pi$
$758$	18.2567	0.663112
$759$	6.93900i	0.251870i
$760$	2.93900i	0.106609i
$761$	20.4064i	0.739732i	0.929085	+	0.369866i	$0.120596\pi$
$761$	20.4064i	0.739732i	−0.929085	+	0.369866i	$0.879404\pi$
$762$	0.499336i	0.0180890i
$763$	13.7808	0.498897
$764$	−0.469205	−0.0169752
$765$	7.40581i	0.267758i
$766$	−2.83148	−0.102305
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 28.4282i	− 1.02515i	−0.858644	−	0.512573i	$-0.828692\pi$
$769$	− 28.4282i	− 1.02515i	0.858644	−	0.512573i	$-0.171308\pi$
$770$	−9.83877	−0.354565
$771$	7.39373	0.266279
$772$	− 17.9312i	− 0.645359i
$773$	− 41.9874i	− 1.51018i	−0.655619	−	0.755092i	$-0.727592\pi$
$773$	− 41.9874i	− 1.51018i	0.655619	−	0.755092i	$-0.272408\pi$
$774$	− 10.4547i	− 0.375788i
$775$	− 1.13706i	− 0.0408445i
$776$	−1.98254	−0.0711691
$777$	−8.80194	−0.315768
$778$	28.8025i	1.03262i
$779$	−4.57002	−0.163738
$780$	0	0
$781$	−46.5032	−1.66401
$782$	− 13.3448i	− 0.477210i
$783$	−8.03684	−0.287213
$784$	0.472189	0.0168639
$785$	4.35152i	0.155312i
$786$	− 1.12498i	− 0.0401268i
$787$	− 11.6950i	− 0.416882i	−0.978035	−	0.208441i	$-0.933161\pi$
$787$	− 11.6950i	− 0.416882i	0.978035	−	0.208441i	$-0.0668389\pi$
$788$	8.17390i	0.291183i
$789$	11.7735	0.419146
$790$	8.81700	0.313695
$791$	− 24.1256i	− 0.857807i
$792$	3.85086	0.136834
$793$	0	0
$794$	−21.4306	−0.760542
$795$	8.93900i	0.317034i
$796$	−23.7536	−0.841924
$797$	−50.5502	−1.79058	−0.895289	−	0.445485i	$-0.853031\pi$
$797$	−50.5502	−1.79058	−0.895289	+	0.445485i	$0.853031\pi$
$798$	− 7.50902i	− 0.265817i
$799$	71.9385i	2.54500i
$800$	1.00000i	0.0353553i
$801$	15.6799i	0.554023i
$802$	−19.5284	−0.689572
$803$	−8.84117	−0.311998
$804$	1.36227i	0.0480437i
$805$	4.60388	0.162265
$806$	0	0
$807$	22.8810	0.805449
$808$	3.48188i	0.122492i
$809$	−19.3134	−0.679021	−0.339511	−	0.940602i	$-0.610261\pi$
$809$	−19.3134	−0.679021	−0.339511	+	0.940602i	$0.610261\pi$
$810$	1.00000	0.0351364
$811$	1.06638i	0.0374455i	0.999825	+	0.0187228i	$0.00595999\pi$
$811$	1.06638i	0.0374455i	−0.999825	+	0.0187228i	$0.994040\pi$
$812$	20.5338i	0.720594i
$813$	2.13036i	0.0747149i
$814$	− 13.2664i	− 0.464986i
$815$	11.6189	0.406994
$816$	−7.40581	−0.259255
$817$	− 30.7265i	− 1.07498i
$818$	30.2707	1.05839
$819$	0	0
$820$	−1.55496	−0.0543015
$821$	− 3.22760i	− 0.112644i	−0.998413	−	0.0563220i	$-0.982063\pi$
$821$	− 3.22760i	− 0.112644i	0.998413	−	0.0563220i	$-0.0179374\pi$
$822$	14.1468	0.493425
$823$	29.0017	1.01093	0.505467	−	0.862846i	$-0.331320\pi$
$823$	29.0017	1.01093	0.505467	+	0.862846i	$0.331320\pi$
$824$	− 8.65817i	− 0.301622i
$825$	− 3.85086i	− 0.134070i
$826$	− 30.1089i	− 1.04762i
$827$	− 17.9414i	− 0.623883i	−0.950101	−	0.311942i	$-0.899021\pi$
$827$	− 17.9414i	− 0.623883i	0.950101	−	0.311942i	$-0.100979\pi$
$828$	−1.80194	−0.0626217
$829$	18.9903	0.659561	0.329780	−	0.944058i	$-0.393025\pi$
$829$	18.9903	0.659561	0.329780	+	0.944058i	$0.393025\pi$
$830$	− 15.9051i	− 0.552076i
$831$	−14.7084	−0.510229
$832$	0	0
$833$	3.49694	0.121162
$834$	7.17629i	0.248495i
$835$	20.8146	0.720319
$836$	11.3177	0.391430
$837$	− 1.13706i	− 0.0393027i
$838$	24.4873i	0.845898i
$839$	− 36.8189i	− 1.27113i	−0.772047	−	0.635565i	$-0.780767\pi$
$839$	− 36.8189i	− 1.27113i	0.772047	−	0.635565i	$-0.219233\pi$
$840$	− 2.55496i	− 0.0881544i
$841$	35.5907	1.22727
$842$	−8.90408	−0.306855
$843$	− 12.5942i	− 0.433767i
$844$	−24.2392	−0.834348
$845$	0	0
$846$	9.71379	0.333967
$847$	9.78315i	0.336153i
$848$	−8.93900	−0.306967
$849$	−1.62565	−0.0557920
$850$	7.40581i	0.254017i
$851$	6.20775i	0.212799i
$852$	− 12.0761i	− 0.413719i
$853$	14.6746i	0.502447i	0.967929	+	0.251224i	$0.0808330\pi$
$853$	14.6746i	0.502447i	−0.967929	+	0.251224i	$0.919167\pi$
$854$	25.6993	0.879413
$855$	2.93900	0.100512
$856$	13.9366i	0.476343i
$857$	14.2784	0.487742	0.243871	−	0.969808i	$-0.421583\pi$
$857$	14.2784	0.487742	0.243871	+	0.969808i	$0.421583\pi$
$858$	0	0
$859$	11.3593	0.387574	0.193787	−	0.981044i	$-0.437923\pi$
$859$	11.3593	0.387574	0.193787	+	0.981044i	$0.437923\pi$
$860$	− 10.4547i	− 0.356503i
$861$	3.97285	0.135394
$862$	1.48619	0.0506198
$863$	− 26.7041i	− 0.909018i	−0.890742	−	0.454509i	$-0.849815\pi$
$863$	− 26.7041i	− 0.909018i	0.890742	−	0.454509i	$-0.150185\pi$
$864$	1.00000i	0.0340207i
$865$	8.89440i	0.302419i
$866$	18.5719i	0.631100i
$867$	−37.8461	−1.28532
$868$	−2.90515	−0.0986072
$869$	− 33.9530i	− 1.15178i
$870$	−8.03684	−0.272474
$871$	0	0
$872$	−5.39373	−0.182655
$873$	1.98254i	0.0670989i
$874$	−5.29590	−0.179136
$875$	−2.55496	−0.0863733
$876$	− 2.29590i	− 0.0775712i
$877$	− 12.7041i	− 0.428987i	−0.976725	−	0.214494i	$-0.931190\pi$
$877$	− 12.7041i	− 0.428987i	0.976725	−	0.214494i	$-0.0688101\pi$
$878$	0.532123i	0.0179583i
$879$	− 13.0664i	− 0.440718i
$880$	3.85086	0.129812
$881$	−52.1342	−1.75645	−0.878223	−	0.478252i	$-0.841271\pi$
$881$	−52.1342	−1.75645	−0.878223	+	0.478252i	$0.841271\pi$
$882$	− 0.472189i	− 0.0158994i
$883$	−6.17390	−0.207768	−0.103884	−	0.994589i	$-0.533127\pi$
$883$	−6.17390	−0.207768	−0.103884	+	0.994589i	$0.533127\pi$
$884$	0	0
$885$	11.7845	0.396131
$886$	− 25.5230i	− 0.857463i
$887$	0.0978347	0.00328497	0.00164248	−	0.999999i	$-0.499477\pi$
$887$	0.0978347	0.00328497	0.00164248	+	0.999999i	$0.499477\pi$
$888$	3.44504	0.115608
$889$	− 1.27578i	− 0.0427884i
$890$	15.6799i	0.525593i
$891$	− 3.85086i	− 0.129008i
$892$	22.5579i	0.755296i
$893$	28.5488	0.955351
$894$	−3.25667	−0.108919
$895$	− 0.188374i	− 0.00629663i
$896$	2.55496	0.0853552
$897$	0	0
$898$	40.7730	1.36061
$899$	9.13839i	0.304782i
$900$	1.00000	0.0333333
$901$	−66.2006	−2.20546
$902$	5.98792i	0.199376i
$903$	26.7114i	0.888900i
$904$	9.44265i	0.314058i
$905$	− 18.1588i	− 0.603620i
$906$	0.137063	0.00455362
$907$	6.44265	0.213925	0.106962	−	0.994263i	$-0.465888\pi$
$907$	6.44265	0.213925	0.106962	+	0.994263i	$0.465888\pi$
$908$	23.7439i	0.787970i
$909$	3.48188	0.115487
$910$	0	0
$911$	32.0965	1.06340	0.531702	−	0.846931i	$-0.321553\pi$
$911$	32.0965	1.06340	0.531702	+	0.846931i	$0.321553\pi$
$912$	2.93900i	0.0973201i
$913$	−61.2484	−2.02703
$914$	−15.1511	−0.501153
$915$	10.0586i	0.332527i
$916$	− 3.11529i	− 0.102932i
$917$	2.87428i	0.0949171i
$918$	7.40581i	0.244428i
$919$	14.1062	0.465320	0.232660	−	0.972558i	$-0.425257\pi$
$919$	14.1062	0.465320	0.232660	+	0.972558i	$0.425257\pi$
$920$	−1.80194	−0.0594081
$921$	− 26.2489i	− 0.864931i
$922$	−19.5743	−0.644646
$923$	0	0
$924$	−9.83877	−0.323672
$925$	− 3.44504i	− 0.113272i
$926$	−31.2905	−1.02827
$927$	−8.65817	−0.284372
$928$	− 8.03684i	− 0.263822i
$929$	− 31.4959i	− 1.03335i	−0.856183	−	0.516673i	$-0.827170\pi$
$929$	− 31.4959i	− 1.03335i	0.856183	−	0.516673i	$-0.172830\pi$
$930$	− 1.13706i	− 0.0372858i
$931$	− 1.38776i	− 0.0454821i
$932$	−12.5603	−0.411427
$933$	27.1250	0.888032
$934$	− 32.3032i	− 1.05699i
$935$	28.5187	0.932662
$936$	0	0
$937$	−10.0592	−0.328620	−0.164310	−	0.986409i	$-0.552540\pi$
$937$	−10.0592	−0.328620	−0.164310	+	0.986409i	$0.552540\pi$
$938$	− 3.48055i	− 0.113644i
$939$	−19.1323	−0.624358
$940$	9.71379	0.316829
$941$	20.7802i	0.677414i	0.940892	+	0.338707i	$0.109990\pi$
$941$	20.7802i	0.677414i	−0.940892	+	0.338707i	$0.890010\pi$
$942$	4.35152i	0.141780i
$943$	− 2.80194i	− 0.0912436i
$944$	11.7845i	0.383552i
$945$	−2.55496	−0.0831128
$946$	−40.2597	−1.30895
$947$	− 41.0307i	− 1.33332i	−0.745362	−	0.666660i	$-0.767723\pi$
$947$	− 41.0307i	− 1.33332i	0.745362	−	0.666660i	$-0.232277\pi$
$948$	8.81700	0.286363
$949$	0	0
$950$	2.93900	0.0953538
$951$	6.67456i	0.216438i
$952$	18.9215	0.613251
$953$	56.3484	1.82530	0.912652	−	0.408738i	$-0.134031\pi$
$953$	56.3484	1.82530	0.912652	+	0.408738i	$0.134031\pi$
$954$	8.93900i	0.289411i
$955$	0.469205i	0.0151831i
$956$	8.52111i	0.275592i
$957$	30.9487i	1.00043i
$958$	9.32842	0.301388
$959$	−36.1444	−1.16716
$960$	1.00000i	0.0322749i
$961$	29.7071	0.958293
$962$	0	0
$963$	13.9366	0.449101
$964$	11.6136i	0.374048i
$965$	−17.9312	−0.577227
$966$	4.60388	0.148127
$967$	9.63102i	0.309713i	0.987937	+	0.154856i	$0.0494915\pi$
$967$	9.63102i	0.309713i	−0.987937	+	0.154856i	$0.950509\pi$
$968$	− 3.82908i	− 0.123071i
$969$	21.7657i	0.699215i
$970$	1.98254i	0.0636556i
$971$	4.72050	0.151488	0.0757440	−	0.997127i	$-0.475867\pi$
$971$	4.72050	0.151488	0.0757440	+	0.997127i	$0.475867\pi$
$972$	1.00000	0.0320750
$973$	− 18.3351i	− 0.587797i
$974$	10.6571	0.341476
$975$	0	0
$976$	−10.0586	−0.321968
$977$	− 2.75063i	− 0.0880004i	−0.999032	−	0.0440002i	$-0.985990\pi$
$977$	− 2.75063i	− 0.0880004i	0.999032	−	0.0440002i	$-0.0140102\pi$
$978$	11.6189	0.371533
$979$	60.3812	1.92979
$980$	− 0.472189i	− 0.0150835i
$981$	5.39373i	0.172209i
$982$	40.8611i	1.30393i
$983$	− 7.87907i	− 0.251303i	−0.992074	−	0.125652i	$-0.959898\pi$
$983$	− 7.87907i	− 0.251303i	0.992074	−	0.125652i	$-0.0401021\pi$
$984$	−1.55496	−0.0495703
$985$	8.17390	0.260442
$986$	− 59.5193i	− 1.89548i
$987$	−24.8183	−0.789976
$988$	0	0
$989$	18.8388	0.599038
$990$	− 3.85086i	− 0.122388i
$991$	13.3609	0.424424	0.212212	−	0.977224i	$-0.431933\pi$
$991$	13.3609	0.424424	0.212212	+	0.977224i	$0.431933\pi$
$992$	1.13706	0.0361018
$993$	− 18.1360i	− 0.575529i
$994$	30.8538i	0.978624i
$995$	23.7536i	0.753040i
$996$	− 15.9051i	− 0.503974i
$997$	47.1554	1.49343	0.746713	−	0.665147i	$-0.231631\pi$
$997$	47.1554	1.49343	0.746713	+	0.665147i	$0.231631\pi$
$998$	19.0556	0.603195
$999$	− 3.44504i	− 0.108996i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.v.1351.2		6
13.5	odd	4		5070.2.a.bl.1.2	✓	3
13.8	odd	4		5070.2.a.bs.1.2	yes	3
13.12	even	2	inner	5070.2.b.v.1351.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bl.1.2	✓	3	13.5	odd	4
5070.2.a.bs.1.2	yes	3	13.8	odd	4
5070.2.b.v.1351.2		6	1.1	even	1	trivial
5070.2.b.v.1351.5		6	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.55496i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.55496i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.85086i q^{11} +1.00000 q^{12} -2.55496 q^{14} -1.00000i q^{15} +1.00000 q^{16} +7.40581 q^{17} -1.00000i q^{18} -2.93900i q^{19} -1.00000i q^{20} +2.55496i q^{21} -3.85086 q^{22} +1.80194 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.55496i q^{28} +8.03684 q^{29} -1.00000 q^{30} +1.13706i q^{31} -1.00000i q^{32} +3.85086i q^{33} -7.40581i q^{34} +2.55496 q^{35} -1.00000 q^{36} +3.44504i q^{37} -2.93900 q^{38} -1.00000 q^{40} -1.55496i q^{41} +2.55496 q^{42} +10.4547 q^{43} +3.85086i q^{44} +1.00000i q^{45} -1.80194i q^{46} +9.71379i q^{47} -1.00000 q^{48} +0.472189 q^{49} +1.00000i q^{50} -7.40581 q^{51} -8.93900 q^{53} +1.00000i q^{54} +3.85086 q^{55} +2.55496 q^{56} +2.93900i q^{57} -8.03684i q^{58} +11.7845i q^{59} +1.00000i q^{60} -10.0586 q^{61} +1.13706 q^{62} -2.55496i q^{63} -1.00000 q^{64} +3.85086 q^{66} +1.36227i q^{67} -7.40581 q^{68} -1.80194 q^{69} -2.55496i q^{70} -12.0761i q^{71} +1.00000i q^{72} -2.29590i q^{73} +3.44504 q^{74} +1.00000 q^{75} +2.93900i q^{76} -9.83877 q^{77} +8.81700 q^{79} +1.00000i q^{80} +1.00000 q^{81} -1.55496 q^{82} -15.9051i q^{83} -2.55496i q^{84} +7.40581i q^{85} -10.4547i q^{86} -8.03684 q^{87} +3.85086 q^{88} +15.6799i q^{89} +1.00000 q^{90} -1.80194 q^{92} -1.13706i q^{93} +9.71379 q^{94} +2.93900 q^{95} +1.00000i q^{96} +1.98254i q^{97} -0.472189i q^{98} -3.85086i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} + 6 q^{12} - 16 q^{14} + 6 q^{16} + 18 q^{17} + 4 q^{22} + 2 q^{23} - 6 q^{25} - 6 q^{27} - 8 q^{29} - 6 q^{30} + 16 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 16 q^{42} + 18 q^{43} - 6 q^{48} - 10 q^{49} - 18 q^{51} - 34 q^{53} - 4 q^{55} + 16 q^{56} + 2 q^{61} - 4 q^{62} - 6 q^{64} - 4 q^{66} - 18 q^{68} - 2 q^{69} + 20 q^{74} + 6 q^{75} + 6 q^{77} - 6 q^{79} + 6 q^{81} - 10 q^{82} + 8 q^{87} - 4 q^{88} + 6 q^{90} - 2 q^{92} + 42 q^{94} - 2 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 + 6 * q^12 - 16 * q^14 + 6 * q^16 + 18 * q^17 + 4 * q^22 + 2 * q^23 - 6 * q^25 - 6 * q^27 - 8 * q^29 - 6 * q^30 + 16 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 16 * q^42 + 18 * q^43 - 6 * q^48 - 10 * q^49 - 18 * q^51 - 34 * q^53 - 4 * q^55 + 16 * q^56 + 2 * q^61 - 4 * q^62 - 6 * q^64 - 4 * q^66 - 18 * q^68 - 2 * q^69 + 20 * q^74 + 6 * q^75 + 6 * q^77 - 6 * q^79 + 6 * q^81 - 10 * q^82 + 8 * q^87 - 4 * q^88 + 6 * q^90 - 2 * q^92 + 42 * q^94 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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