LMFDB - Embedded newform 5070.2.b.w.1351.5

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.44504i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.44504i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.24698i q^{11} -1.00000 q^{12} -3.44504 q^{14} +1.00000i q^{15} +1.00000 q^{16} -6.78986 q^{17} +1.00000i q^{18} -6.26875i q^{19} -1.00000i q^{20} +3.44504i q^{21} -4.24698 q^{22} +1.30798 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.44504i q^{28} -9.14675 q^{29} -1.00000 q^{30} -3.75302i q^{31} +1.00000i q^{32} +4.24698i q^{33} -6.78986i q^{34} -3.44504 q^{35} -1.00000 q^{36} +6.82908i q^{37} +6.26875 q^{38} +1.00000 q^{40} -4.26875i q^{41} -3.44504 q^{42} -3.07069 q^{43} -4.24698i q^{44} +1.00000i q^{45} +1.30798i q^{46} +7.76809i q^{47} +1.00000 q^{48} -4.86831 q^{49} -1.00000i q^{50} -6.78986 q^{51} -8.93900 q^{53} +1.00000i q^{54} -4.24698 q^{55} +3.44504 q^{56} -6.26875i q^{57} -9.14675i q^{58} -10.3327i q^{59} -1.00000i q^{60} +2.53319 q^{61} +3.75302 q^{62} +3.44504i q^{63} -1.00000 q^{64} -4.24698 q^{66} +0.0760644i q^{67} +6.78986 q^{68} +1.30798 q^{69} -3.44504i q^{70} -0.374354i q^{71} -1.00000i q^{72} -16.7114i q^{73} -6.82908 q^{74} -1.00000 q^{75} +6.26875i q^{76} -14.6310 q^{77} -1.33513 q^{79} +1.00000i q^{80} +1.00000 q^{81} +4.26875 q^{82} +0.740939i q^{83} -3.44504i q^{84} -6.78986i q^{85} -3.07069i q^{86} -9.14675 q^{87} +4.24698 q^{88} +13.3274i q^{89} -1.00000 q^{90} -1.30798 q^{92} -3.75302i q^{93} -7.76809 q^{94} +6.26875 q^{95} +1.00000i q^{96} +13.1903i q^{97} -4.86831i q^{98} +4.24698i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 20 q^{14} + 6 q^{16} + 6 q^{17} - 16 q^{22} + 18 q^{23} - 6 q^{25} + 6 q^{27} - 6 q^{30} - 20 q^{35} - 6 q^{36} + 22 q^{38} + 6 q^{40} - 20 q^{42} + 6 q^{43} + 6 q^{48} - 34 q^{49} + 6 q^{51} - 34 q^{53} - 16 q^{55} + 20 q^{56} + 22 q^{61} + 32 q^{62} - 6 q^{64} - 16 q^{66} - 6 q^{68} + 18 q^{69} - 20 q^{74} - 6 q^{75} - 58 q^{77} - 6 q^{79} + 6 q^{81} + 10 q^{82} + 16 q^{88} - 6 q^{90} - 18 q^{92} - 6 q^{94} + 22 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 - 6 * q^12 - 20 * q^14 + 6 * q^16 + 6 * q^17 - 16 * q^22 + 18 * q^23 - 6 * q^25 + 6 * q^27 - 6 * q^30 - 20 * q^35 - 6 * q^36 + 22 * q^38 + 6 * q^40 - 20 * q^42 + 6 * q^43 + 6 * q^48 - 34 * q^49 + 6 * q^51 - 34 * q^53 - 16 * q^55 + 20 * q^56 + 22 * q^61 + 32 * q^62 - 6 * q^64 - 16 * q^66 - 6 * q^68 + 18 * q^69 - 20 * q^74 - 6 * q^75 - 58 * q^77 - 6 * q^79 + 6 * q^81 + 10 * q^82 + 16 * q^88 - 6 * q^90 - 18 * q^92 - 6 * q^94 + 22 * 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
$7$	3.44504i	1.30210i	0.759033	+	0.651052i	$0.225672\pi$
$7$	3.44504i	1.30210i	−0.759033	+	0.651052i	$0.774328\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.24698i	1.28051i	0.768161	+	0.640256i	$0.221172\pi$
$11$	4.24698i	1.28051i	−0.768161	+	0.640256i	$0.778828\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−3.44504	−0.920726
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−6.78986	−1.64678	−0.823391	−	0.567474i	$-0.807921\pi$
$17$	−6.78986	−1.64678	−0.823391	+	0.567474i	$0.807921\pi$
$18$	1.00000i	0.235702i
$19$	− 6.26875i	− 1.43815i	−0.694933	−	0.719075i	$-0.744566\pi$
$19$	− 6.26875i	− 1.43815i	0.694933	−	0.719075i	$-0.255434\pi$
$20$	− 1.00000i	− 0.223607i
$21$	3.44504i	0.751770i
$22$	−4.24698	−0.905459
$23$	1.30798	0.272732	0.136366	−	0.990658i	$-0.456458\pi$
$23$	1.30798	0.272732	0.136366	+	0.990658i	$0.456458\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 3.44504i	− 0.651052i
$29$	−9.14675	−1.69851	−0.849255	−	0.527984i	$-0.822948\pi$
$29$	−9.14675	−1.69851	−0.849255	+	0.527984i	$0.822948\pi$
$30$	−1.00000	−0.182574
$31$	− 3.75302i	− 0.674062i	−0.941493	−	0.337031i	$-0.890577\pi$
$31$	− 3.75302i	− 0.674062i	0.941493	−	0.337031i	$-0.109423\pi$
$32$	1.00000i	0.176777i
$33$	4.24698i	0.739304i
$34$	− 6.78986i	− 1.16445i
$35$	−3.44504	−0.582318
$36$	−1.00000	−0.166667
$37$	6.82908i	1.12269i	0.827580	+	0.561347i	$0.189717\pi$
$37$	6.82908i	1.12269i	−0.827580	+	0.561347i	$0.810283\pi$
$38$	6.26875	1.01693
$39$	0	0
$40$	1.00000	0.158114
$41$	− 4.26875i	− 0.666667i	−0.942809	−	0.333333i	$-0.891827\pi$
$41$	− 4.26875i	− 0.666667i	0.942809	−	0.333333i	$-0.108173\pi$
$42$	−3.44504	−0.531582
$43$	−3.07069	−0.468275	−0.234138	−	0.972203i	$-0.575227\pi$
$43$	−3.07069	−0.468275	−0.234138	+	0.972203i	$0.575227\pi$
$44$	− 4.24698i	− 0.640256i
$45$	1.00000i	0.149071i
$46$	1.30798i	0.192851i
$47$	7.76809i	1.13309i	0.824030	+	0.566546i	$0.191721\pi$
$47$	7.76809i	1.13309i	−0.824030	+	0.566546i	$0.808279\pi$
$48$	1.00000	0.144338
$49$	−4.86831	−0.695473
$50$	− 1.00000i	− 0.141421i
$51$	−6.78986	−0.950770
$52$	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
$55$	−4.24698	−0.572663
$56$	3.44504	0.460363
$57$	− 6.26875i	− 0.830316i
$58$	− 9.14675i	− 1.20103i
$59$	− 10.3327i	− 1.34521i	−0.740003	−	0.672604i	$-0.765176\pi$
$59$	− 10.3327i	− 1.34521i	0.740003	−	0.672604i	$-0.234824\pi$
$60$	− 1.00000i	− 0.129099i
$61$	2.53319	0.324341	0.162171	−	0.986763i	$-0.448150\pi$
$61$	2.53319	0.324341	0.162171	+	0.986763i	$0.448150\pi$
$62$	3.75302	0.476634
$63$	3.44504i	0.434034i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.24698	−0.522767
$67$	0.0760644i	0.00929275i	0.999989	+	0.00464637i	$0.00147899\pi$
$67$	0.0760644i	0.00929275i	−0.999989	+	0.00464637i	$0.998521\pi$
$68$	6.78986	0.823391
$69$	1.30798	0.157462
$70$	− 3.44504i	− 0.411761i
$71$	− 0.374354i	− 0.0444277i	−0.999753	−	0.0222138i	$-0.992929\pi$
$71$	− 0.374354i	− 0.0444277i	0.999753	−	0.0222138i	$-0.00707147\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 16.7114i	− 1.95592i	−0.208789	−	0.977961i	$-0.566952\pi$
$73$	− 16.7114i	− 1.95592i	0.208789	−	0.977961i	$-0.433048\pi$
$74$	−6.82908	−0.793865
$75$	−1.00000	−0.115470
$76$	6.26875i	0.719075i
$77$	−14.6310	−1.66736
$78$	0	0
$79$	−1.33513	−0.150213	−0.0751067	−	0.997176i	$-0.523930\pi$
$79$	−1.33513	−0.150213	−0.0751067	+	0.997176i	$0.523930\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	4.26875	0.471405
$83$	0.740939i	0.0813286i	0.999173	+	0.0406643i	$0.0129474\pi$
$83$	0.740939i	0.0813286i	−0.999173	+	0.0406643i	$0.987053\pi$
$84$	− 3.44504i	− 0.375885i
$85$	− 6.78986i	− 0.736463i
$86$	− 3.07069i	− 0.331121i
$87$	−9.14675	−0.980635
$88$	4.24698	0.452730
$89$	13.3274i	1.41270i	0.707864	+	0.706348i	$0.249659\pi$
$89$	13.3274i	1.41270i	−0.707864	+	0.706348i	$0.750341\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−1.30798	−0.136366
$93$	− 3.75302i	− 0.389170i
$94$	−7.76809	−0.801217
$95$	6.26875	0.643160
$96$	1.00000i	0.102062i
$97$	13.1903i	1.33927i	0.742690	+	0.669636i	$0.233550\pi$
$97$	13.1903i	1.33927i	−0.742690	+	0.669636i	$0.766450\pi$
$98$	− 4.86831i	− 0.491774i
$99$	4.24698i	0.426838i
$100$	1.00000	0.100000
$101$	15.0422	1.49676	0.748378	−	0.663272i	$-0.230833\pi$
$101$	15.0422	1.49676	0.748378	+	0.663272i	$0.230833\pi$
$102$	− 6.78986i	− 0.672296i
$103$	8.92154	0.879066	0.439533	−	0.898227i	$-0.355144\pi$
$103$	8.92154	0.879066	0.439533	+	0.898227i	$0.355144\pi$
$104$	0	0
$105$	−3.44504	−0.336202
$106$	− 8.93900i	− 0.868233i
$107$	14.6746	1.41864	0.709322	−	0.704885i	$-0.249001\pi$
$107$	14.6746	1.41864	0.709322	+	0.704885i	$0.249001\pi$
$108$	−1.00000	−0.0962250
$109$	− 9.66786i	− 0.926013i	−0.886355	−	0.463006i	$-0.846771\pi$
$109$	− 9.66786i	− 0.926013i	0.886355	−	0.463006i	$-0.153229\pi$
$110$	− 4.24698i	− 0.404934i
$111$	6.82908i	0.648188i
$112$	3.44504i	0.325526i
$113$	−11.3448	−1.06723	−0.533615	−	0.845727i	$-0.679167\pi$
$113$	−11.3448	−1.06723	−0.533615	+	0.845727i	$0.679167\pi$
$114$	6.26875	0.587122
$115$	1.30798i	0.121970i
$116$	9.14675	0.849255
$117$	0	0
$118$	10.3327	0.951205
$119$	− 23.3913i	− 2.14428i
$120$	1.00000	0.0912871
$121$	−7.03684	−0.639712
$122$	2.53319i	0.229344i
$123$	− 4.26875i	− 0.384900i
$124$	3.75302i	0.337031i
$125$	− 1.00000i	− 0.0894427i
$126$	−3.44504	−0.306909
$127$	2.54288	0.225644	0.112822	−	0.993615i	$-0.464011\pi$
$127$	2.54288	0.225644	0.112822	+	0.993615i	$0.464011\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−3.07069	−0.270359
$130$	0	0
$131$	−10.8509	−0.948044	−0.474022	−	0.880513i	$-0.657198\pi$
$131$	−10.8509	−0.948044	−0.474022	+	0.880513i	$0.657198\pi$
$132$	− 4.24698i	− 0.369652i
$133$	21.5961	1.87262
$134$	−0.0760644	−0.00657096
$135$	1.00000i	0.0860663i
$136$	6.78986i	0.582225i
$137$	− 18.6407i	− 1.59258i	−0.604913	−	0.796292i	$-0.706792\pi$
$137$	− 18.6407i	− 1.59258i	0.604913	−	0.796292i	$-0.293208\pi$
$138$	1.30798i	0.111343i
$139$	3.53750	0.300047	0.150023	−	0.988682i	$-0.452065\pi$
$139$	3.53750	0.300047	0.150023	+	0.988682i	$0.452065\pi$
$140$	3.44504	0.291159
$141$	7.76809i	0.654191i
$142$	0.374354	0.0314151
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 9.14675i	− 0.759596i
$146$	16.7114	1.38305
$147$	−4.86831	−0.401532
$148$	− 6.82908i	− 0.561347i
$149$	16.6843i	1.36683i	0.730031	+	0.683414i	$0.239505\pi$
$149$	16.6843i	1.36683i	−0.730031	+	0.683414i	$0.760495\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	18.3327i	1.49190i	0.666004	+	0.745948i	$0.268003\pi$
$151$	18.3327i	1.49190i	−0.666004	+	0.745948i	$0.731997\pi$
$152$	−6.26875	−0.508463
$153$	−6.78986	−0.548927
$154$	− 14.6310i	− 1.17900i
$155$	3.75302	0.301450
$156$	0	0
$157$	−18.6015	−1.48456	−0.742280	−	0.670090i	$-0.766256\pi$
$157$	−18.6015	−1.48456	−0.742280	+	0.670090i	$0.766256\pi$
$158$	− 1.33513i	− 0.106217i
$159$	−8.93900	−0.708909
$160$	−1.00000	−0.0790569
$161$	4.50604i	0.355126i
$162$	1.00000i	0.0785674i
$163$	18.8944i	1.47992i	0.672649	+	0.739962i	$0.265156\pi$
$163$	18.8944i	1.47992i	−0.672649	+	0.739962i	$0.734844\pi$
$164$	4.26875i	0.333333i
$165$	−4.24698	−0.330627
$166$	−0.740939	−0.0575080
$167$	− 4.24698i	− 0.328641i	−0.986407	−	0.164321i	$-0.947457\pi$
$167$	− 4.24698i	− 0.328641i	0.986407	−	0.164321i	$-0.0525432\pi$
$168$	3.44504	0.265791
$169$	0	0
$170$	6.78986	0.520758
$171$	− 6.26875i	− 0.479383i
$172$	3.07069	0.234138
$173$	−1.16852	−0.0888411	−0.0444206	−	0.999013i	$-0.514144\pi$
$173$	−1.16852	−0.0888411	−0.0444206	+	0.999013i	$0.514144\pi$
$174$	− 9.14675i	− 0.693413i
$175$	− 3.44504i	− 0.260421i
$176$	4.24698i	0.320128i
$177$	− 10.3327i	− 0.776656i
$178$	−13.3274	−0.998928
$179$	10.1836	0.761157	0.380579	−	0.924749i	$-0.375725\pi$
$179$	10.1836	0.761157	0.380579	+	0.924749i	$0.375725\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−10.3351	−0.768204	−0.384102	−	0.923291i	$-0.625489\pi$
$181$	−10.3351	−0.768204	−0.384102	+	0.923291i	$0.625489\pi$
$182$	0	0
$183$	2.53319	0.187259
$184$	− 1.30798i	− 0.0964255i
$185$	−6.82908	−0.502084
$186$	3.75302	0.275185
$187$	− 28.8364i	− 2.10872i
$188$	− 7.76809i	− 0.566546i
$189$	3.44504i	0.250590i
$190$	6.26875i	0.454783i
$191$	−9.43296	−0.682545	−0.341273	−	0.939964i	$-0.610858\pi$
$191$	−9.43296	−0.682545	−0.341273	+	0.939964i	$0.610858\pi$
$192$	−1.00000	−0.0721688
$193$	22.9191i	1.64976i	0.565310	+	0.824878i	$0.308756\pi$
$193$	22.9191i	1.64976i	−0.565310	+	0.824878i	$0.691244\pi$
$194$	−13.1903	−0.947008
$195$	0	0
$196$	4.86831	0.347737
$197$	14.3502i	1.02241i	0.859459	+	0.511204i	$0.170800\pi$
$197$	14.3502i	1.02241i	−0.859459	+	0.511204i	$0.829200\pi$
$198$	−4.24698	−0.301820
$199$	−14.2524	−1.01032	−0.505161	−	0.863025i	$-0.668567\pi$
$199$	−14.2524	−1.01032	−0.505161	+	0.863025i	$0.668567\pi$
$200$	1.00000i	0.0707107i
$201$	0.0760644i	0.00536517i
$202$	15.0422i	1.05837i
$203$	− 31.5109i	− 2.21163i
$204$	6.78986	0.475385
$205$	4.26875	0.298142
$206$	8.92154i	0.621593i
$207$	1.30798	0.0909108
$208$	0	0
$209$	26.6233	1.84157
$210$	− 3.44504i	− 0.237730i
$211$	0.396125	0.0272703	0.0136352	−	0.999907i	$-0.495660\pi$
$211$	0.396125	0.0272703	0.0136352	+	0.999907i	$0.495660\pi$
$212$	8.93900	0.613933
$213$	− 0.374354i	− 0.0256503i
$214$	14.6746i	1.00313i
$215$	− 3.07069i	− 0.209419i
$216$	− 1.00000i	− 0.0680414i
$217$	12.9293	0.877699
$218$	9.66786	0.654790
$219$	− 16.7114i	− 1.12925i
$220$	4.24698	0.286331
$221$	0	0
$222$	−6.82908	−0.458338
$223$	− 1.49635i	− 0.100203i	−0.998744	−	0.0501016i	$-0.984045\pi$
$223$	− 1.49635i	− 0.100203i	0.998744	−	0.0501016i	$-0.0159545\pi$
$224$	−3.44504	−0.230182
$225$	−1.00000	−0.0666667
$226$	− 11.3448i	− 0.754646i
$227$	− 25.2150i	− 1.67358i	−0.547523	−	0.836791i	$-0.684429\pi$
$227$	− 25.2150i	− 1.67358i	0.547523	−	0.836791i	$-0.315571\pi$
$228$	6.26875i	0.415158i
$229$	− 22.3666i	− 1.47803i	−0.673691	−	0.739013i	$-0.735292\pi$
$229$	− 22.3666i	− 1.47803i	0.673691	−	0.739013i	$-0.264708\pi$
$230$	−1.30798	−0.0862456
$231$	−14.6310	−0.962651
$232$	9.14675i	0.600514i
$233$	−14.0858	−0.922788	−0.461394	−	0.887195i	$-0.652651\pi$
$233$	−14.0858	−0.922788	−0.461394	+	0.887195i	$0.652651\pi$
$234$	0	0
$235$	−7.76809	−0.506734
$236$	10.3327i	0.672604i
$237$	−1.33513	−0.0867257
$238$	23.3913	1.51624
$239$	− 5.38106i	− 0.348072i	−0.984739	−	0.174036i	$-0.944319\pi$
$239$	− 5.38106i	− 0.348072i	0.984739	−	0.174036i	$-0.0556809\pi$
$240$	1.00000i	0.0645497i
$241$	− 29.9420i	− 1.92873i	−0.264570	−	0.964366i	$-0.585230\pi$
$241$	− 29.9420i	− 1.92873i	0.264570	−	0.964366i	$-0.414770\pi$
$242$	− 7.03684i	− 0.452345i
$243$	1.00000	0.0641500
$244$	−2.53319	−0.162171
$245$	− 4.86831i	− 0.311025i
$246$	4.26875	0.272166
$247$	0	0
$248$	−3.75302	−0.238317
$249$	0.740939i	0.0469551i
$250$	1.00000	0.0632456
$251$	8.73556	0.551384	0.275692	−	0.961246i	$-0.411093\pi$
$251$	8.73556	0.551384	0.275692	+	0.961246i	$0.411093\pi$
$252$	− 3.44504i	− 0.217017i
$253$	5.55496i	0.349237i
$254$	2.54288i	0.159554i
$255$	− 6.78986i	− 0.425197i
$256$	1.00000	0.0625000
$257$	−21.2717	−1.32689	−0.663447	−	0.748223i	$-0.730907\pi$
$257$	−21.2717	−1.32689	−0.663447	+	0.748223i	$0.730907\pi$
$258$	− 3.07069i	− 0.191173i
$259$	−23.5265	−1.46186
$260$	0	0
$261$	−9.14675	−0.566170
$262$	− 10.8509i	− 0.670368i
$263$	−22.2325	−1.37091	−0.685457	−	0.728113i	$-0.740398\pi$
$263$	−22.2325	−1.37091	−0.685457	+	0.728113i	$0.740398\pi$
$264$	4.24698	0.261384
$265$	− 8.93900i	− 0.549118i
$266$	21.5961i	1.32414i
$267$	13.3274i	0.815621i
$268$	− 0.0760644i	− 0.00464637i
$269$	−10.2241	−0.623377	−0.311689	−	0.950184i	$-0.600895\pi$
$269$	−10.2241	−0.623377	−0.311689	+	0.950184i	$0.600895\pi$
$270$	−1.00000	−0.0608581
$271$	0.716185i	0.0435051i	0.999763	+	0.0217526i	$0.00692460\pi$
$271$	0.716185i	0.0435051i	−0.999763	+	0.0217526i	$0.993075\pi$
$272$	−6.78986	−0.411696
$273$	0	0
$274$	18.6407	1.12613
$275$	− 4.24698i	− 0.256103i
$276$	−1.30798	−0.0787311
$277$	22.8713	1.37420	0.687102	−	0.726561i	$-0.258883\pi$
$277$	22.8713	1.37420	0.687102	+	0.726561i	$0.258883\pi$
$278$	3.53750i	0.212165i
$279$	− 3.75302i	− 0.224687i
$280$	3.44504i	0.205881i
$281$	4.13036i	0.246397i	0.992382	+	0.123198i	$0.0393151\pi$
$281$	4.13036i	0.246397i	−0.992382	+	0.123198i	$0.960685\pi$
$282$	−7.76809	−0.462583
$283$	−16.2959	−0.968691	−0.484345	−	0.874877i	$-0.660942\pi$
$283$	−16.2959	−0.968691	−0.484345	+	0.874877i	$0.660942\pi$
$284$	0.374354i	0.0222138i
$285$	6.26875	0.371329
$286$	0	0
$287$	14.7060	0.868069
$288$	1.00000i	0.0589256i
$289$	29.1021	1.71189
$290$	9.14675	0.537116
$291$	13.1903i	0.773229i
$292$	16.7114i	0.977961i
$293$	7.72587i	0.451350i	0.974203	+	0.225675i	$0.0724588\pi$
$293$	7.72587i	0.451350i	−0.974203	+	0.225675i	$0.927541\pi$
$294$	− 4.86831i	− 0.283926i
$295$	10.3327	0.601595
$296$	6.82908	0.396932
$297$	4.24698i	0.246435i
$298$	−16.6843	−0.966493
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 10.5786i	− 0.609743i
$302$	−18.3327	−1.05493
$303$	15.0422	0.864153
$304$	− 6.26875i	− 0.359537i
$305$	2.53319i	0.145050i
$306$	− 6.78986i	− 0.388150i
$307$	− 5.13036i	− 0.292805i	−0.989225	−	0.146403i	$-0.953231\pi$
$307$	− 5.13036i	− 0.292805i	0.989225	−	0.146403i	$-0.0467695\pi$
$308$	14.6310	0.833680
$309$	8.92154	0.507529
$310$	3.75302i	0.213157i
$311$	−33.1987	−1.88252	−0.941261	−	0.337679i	$-0.890358\pi$
$311$	−33.1987	−1.88252	−0.941261	+	0.337679i	$0.890358\pi$
$312$	0	0
$313$	0.0814412	0.00460333	0.00230167	−	0.999997i	$-0.499267\pi$
$313$	0.0814412	0.00460333	0.00230167	+	0.999997i	$0.499267\pi$
$314$	− 18.6015i	− 1.04974i
$315$	−3.44504	−0.194106
$316$	1.33513	0.0751067
$317$	19.4969i	1.09506i	0.836787	+	0.547529i	$0.184431\pi$
$317$	19.4969i	1.09506i	−0.836787	+	0.547529i	$0.815569\pi$
$318$	− 8.93900i	− 0.501274i
$319$	− 38.8461i	− 2.17496i
$320$	− 1.00000i	− 0.0559017i
$321$	14.6746	0.819054
$322$	−4.50604	−0.251112
$323$	42.5639i	2.36832i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−18.8944	−1.04646
$327$	− 9.66786i	− 0.534634i
$328$	−4.26875	−0.235702
$329$	−26.7614	−1.47540
$330$	− 4.24698i	− 0.233789i
$331$	− 23.8049i	− 1.30844i	−0.756306	−	0.654218i	$-0.772998\pi$
$331$	− 23.8049i	− 1.30844i	0.756306	−	0.654218i	$-0.227002\pi$
$332$	− 0.740939i	− 0.0406643i
$333$	6.82908i	0.374232i
$334$	4.24698	0.232384
$335$	−0.0760644	−0.00415584
$336$	3.44504i	0.187942i
$337$	−21.2064	−1.15519	−0.577594	−	0.816324i	$-0.696008\pi$
$337$	−21.2064	−1.15519	−0.577594	+	0.816324i	$0.696008\pi$
$338$	0	0
$339$	−11.3448	−0.616166
$340$	6.78986i	0.368232i
$341$	15.9390	0.863145
$342$	6.26875	0.338975
$343$	7.34375i	0.396525i
$344$	3.07069i	0.165560i
$345$	1.30798i	0.0704192i
$346$	− 1.16852i	− 0.0628201i
$347$	−16.5773	−0.889917	−0.444959	−	0.895551i	$-0.646782\pi$
$347$	−16.5773	−0.889917	−0.444959	+	0.895551i	$0.646782\pi$
$348$	9.14675	0.490317
$349$	2.51035i	0.134376i	0.997740	+	0.0671880i	$0.0214027\pi$
$349$	2.51035i	0.134376i	−0.997740	+	0.0671880i	$0.978597\pi$
$350$	3.44504	0.184145
$351$	0	0
$352$	−4.24698	−0.226365
$353$	20.8412i	1.10926i	0.832096	+	0.554632i	$0.187141\pi$
$353$	20.8412i	1.10926i	−0.832096	+	0.554632i	$0.812859\pi$
$354$	10.3327	0.549179
$355$	0.374354	0.0198687
$356$	− 13.3274i	− 0.706348i
$357$	− 23.3913i	− 1.23800i
$358$	10.1836i	0.538219i
$359$	2.42566i	0.128022i	0.997949	+	0.0640108i	$0.0203892\pi$
$359$	2.42566i	0.128022i	−0.997949	+	0.0640108i	$0.979611\pi$
$360$	1.00000	0.0527046
$361$	−20.2972	−1.06827
$362$	− 10.3351i	− 0.543202i
$363$	−7.03684	−0.369338
$364$	0	0
$365$	16.7114	0.874715
$366$	2.53319i	0.132412i
$367$	2.66248	0.138980	0.0694902	−	0.997583i	$-0.477863\pi$
$367$	2.66248	0.138980	0.0694902	+	0.997583i	$0.477863\pi$
$368$	1.30798	0.0681831
$369$	− 4.26875i	− 0.222222i
$370$	− 6.82908i	− 0.355027i
$371$	− 30.7952i	− 1.59881i
$372$	3.75302i	0.194585i
$373$	5.87800	0.304351	0.152176	−	0.988353i	$-0.451372\pi$
$373$	5.87800	0.304351	0.152176	+	0.988353i	$0.451372\pi$
$374$	28.8364	1.49109
$375$	− 1.00000i	− 0.0516398i
$376$	7.76809	0.400608
$377$	0	0
$378$	−3.44504	−0.177194
$379$	− 36.1540i	− 1.85711i	−0.371196	−	0.928554i	$-0.621052\pi$
$379$	− 36.1540i	− 1.85711i	0.371196	−	0.928554i	$-0.378948\pi$
$380$	−6.26875	−0.321580
$381$	2.54288	0.130276
$382$	− 9.43296i	− 0.482632i
$383$	8.98361i	0.459041i	0.973304	+	0.229520i	$0.0737158\pi$
$383$	8.98361i	0.459041i	−0.973304	+	0.229520i	$0.926284\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 14.6310i	− 0.745666i
$386$	−22.9191	−1.16655
$387$	−3.07069	−0.156092
$388$	− 13.1903i	− 0.669636i
$389$	−0.655186	−0.0332192	−0.0166096	−	0.999862i	$-0.505287\pi$
$389$	−0.655186	−0.0332192	−0.0166096	+	0.999862i	$0.505287\pi$
$390$	0	0
$391$	−8.88099	−0.449131
$392$	4.86831i	0.245887i
$393$	−10.8509	−0.547353
$394$	−14.3502	−0.722952
$395$	− 1.33513i	− 0.0671775i
$396$	− 4.24698i	− 0.213419i
$397$	0.271143i	0.0136083i	0.999977	+	0.00680413i	$0.00216584\pi$
$397$	0.271143i	0.0136083i	−0.999977	+	0.00680413i	$0.997834\pi$
$398$	− 14.2524i	− 0.714406i
$399$	21.5961	1.08116
$400$	−1.00000	−0.0500000
$401$	0.344814i	0.0172192i	0.999963	+	0.00860960i	$0.00274056\pi$
$401$	0.344814i	0.0172192i	−0.999963	+	0.00860960i	$0.997259\pi$
$402$	−0.0760644	−0.00379375
$403$	0	0
$404$	−15.0422	−0.748378
$405$	1.00000i	0.0496904i
$406$	31.5109	1.56386
$407$	−29.0030	−1.43762
$408$	6.78986i	0.336148i
$409$	− 27.0388i	− 1.33698i	−0.743721	−	0.668490i	$-0.766941\pi$
$409$	− 27.0388i	− 1.33698i	0.743721	−	0.668490i	$-0.233059\pi$
$410$	4.26875i	0.210819i
$411$	− 18.6407i	− 0.919478i
$412$	−8.92154	−0.439533
$413$	35.5967	1.75160
$414$	1.30798i	0.0642836i
$415$	−0.740939	−0.0363713
$416$	0	0
$417$	3.53750	0.173232
$418$	26.6233i	1.30219i
$419$	5.86533	0.286540	0.143270	−	0.989684i	$-0.454238\pi$
$419$	5.86533	0.286540	0.143270	+	0.989684i	$0.454238\pi$
$420$	3.44504	0.168101
$421$	28.9476i	1.41082i	0.708799	+	0.705410i	$0.249237\pi$
$421$	28.9476i	1.41082i	−0.708799	+	0.705410i	$0.750763\pi$
$422$	0.396125i	0.0192830i
$423$	7.76809i	0.377697i
$424$	8.93900i	0.434116i
$425$	6.78986	0.329356
$426$	0.374354	0.0181375
$427$	8.72694i	0.422326i
$428$	−14.6746	−0.709322
$429$	0	0
$430$	3.07069	0.148082
$431$	− 4.87023i	− 0.234591i	−0.993097	−	0.117295i	$-0.962578\pi$
$431$	− 4.87023i	− 0.234591i	0.993097	−	0.117295i	$-0.0374224\pi$
$432$	1.00000	0.0481125
$433$	36.9788	1.77709	0.888544	−	0.458791i	$-0.151717\pi$
$433$	36.9788	1.77709	0.888544	+	0.458791i	$0.151717\pi$
$434$	12.9293i	0.620627i
$435$	− 9.14675i	− 0.438553i
$436$	9.66786i	0.463006i
$437$	− 8.19939i	− 0.392230i
$438$	16.7114	0.798502
$439$	−22.8799	−1.09200	−0.546000	−	0.837785i	$-0.683850\pi$
$439$	−22.8799	−1.09200	−0.546000	+	0.837785i	$0.683850\pi$
$440$	4.24698i	0.202467i
$441$	−4.86831	−0.231824
$442$	0	0
$443$	−10.5942	−0.503345	−0.251673	−	0.967812i	$-0.580981\pi$
$443$	−10.5942	−0.503345	−0.251673	+	0.967812i	$0.580981\pi$
$444$	− 6.82908i	− 0.324094i
$445$	−13.3274	−0.631777
$446$	1.49635	0.0708543
$447$	16.6843i	0.789138i
$448$	− 3.44504i	− 0.162763i
$449$	19.6485i	0.927269i	0.886027	+	0.463635i	$0.153455\pi$
$449$	19.6485i	0.927269i	−0.886027	+	0.463635i	$0.846545\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	18.1293	0.853675
$452$	11.3448	0.533615
$453$	18.3327i	0.861347i
$454$	25.2150	1.18340
$455$	0	0
$456$	−6.26875	−0.293561
$457$	18.4239i	0.861832i	0.902392	+	0.430916i	$0.141809\pi$
$457$	18.4239i	0.861832i	−0.902392	+	0.430916i	$0.858191\pi$
$458$	22.3666	1.04512
$459$	−6.78986	−0.316923
$460$	− 1.30798i	− 0.0609848i
$461$	22.2446i	1.03603i	0.855370	+	0.518017i	$0.173330\pi$
$461$	22.2446i	1.03603i	−0.855370	+	0.518017i	$0.826670\pi$
$462$	− 14.6310i	− 0.680697i
$463$	− 8.68532i	− 0.403641i	−0.979423	−	0.201820i	$-0.935314\pi$
$463$	− 8.68532i	− 0.403641i	0.979423	−	0.201820i	$-0.0646857\pi$
$464$	−9.14675	−0.424627
$465$	3.75302	0.174042
$466$	− 14.0858i	− 0.652510i
$467$	−19.6136	−0.907608	−0.453804	−	0.891102i	$-0.649933\pi$
$467$	−19.6136	−0.907608	−0.453804	+	0.891102i	$0.649933\pi$
$468$	0	0
$469$	−0.262045	−0.0121001
$470$	− 7.76809i	− 0.358315i
$471$	−18.6015	−0.857111
$472$	−10.3327	−0.475603
$473$	− 13.0411i	− 0.599633i
$474$	− 1.33513i	− 0.0613243i
$475$	6.26875i	0.287630i
$476$	23.3913i	1.07214i
$477$	−8.93900	−0.409289
$478$	5.38106	0.246124
$479$	31.4698i	1.43789i	0.695066	+	0.718946i	$0.255375\pi$
$479$	31.4698i	1.43789i	−0.695066	+	0.718946i	$0.744625\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	29.9420	1.36382
$483$	4.50604i	0.205032i
$484$	7.03684	0.319856
$485$	−13.1903	−0.598940
$486$	1.00000i	0.0453609i
$487$	− 13.2644i	− 0.601069i	−0.953771	−	0.300535i	$-0.902835\pi$
$487$	− 13.2644i	− 0.601069i	0.953771	−	0.300535i	$-0.0971651\pi$
$488$	− 2.53319i	− 0.114672i
$489$	18.8944i	0.854434i
$490$	4.86831	0.219928
$491$	−11.7259	−0.529181	−0.264591	−	0.964361i	$-0.585237\pi$
$491$	−11.7259	−0.529181	−0.264591	+	0.964361i	$0.585237\pi$
$492$	4.26875i	0.192450i
$493$	62.1051	2.79707
$494$	0	0
$495$	−4.24698	−0.190888
$496$	− 3.75302i	− 0.168516i
$497$	1.28967	0.0578494
$498$	−0.740939	−0.0332023
$499$	26.7633i	1.19809i	0.800715	+	0.599045i	$0.204453\pi$
$499$	26.7633i	1.19809i	−0.800715	+	0.599045i	$0.795547\pi$
$500$	1.00000i	0.0447214i
$501$	− 4.24698i	− 0.189741i
$502$	8.73556i	0.389887i
$503$	17.2239	0.767975	0.383987	−	0.923338i	$-0.374551\pi$
$503$	17.2239	0.767975	0.383987	+	0.923338i	$0.374551\pi$
$504$	3.44504	0.153454
$505$	15.0422i	0.669370i
$506$	−5.55496	−0.246948
$507$	0	0
$508$	−2.54288	−0.112822
$509$	28.2737i	1.25321i	0.779338	+	0.626604i	$0.215556\pi$
$509$	28.2737i	1.25321i	−0.779338	+	0.626604i	$0.784444\pi$
$510$	6.78986	0.300660
$511$	57.5715	2.54681
$512$	1.00000i	0.0441942i
$513$	− 6.26875i	− 0.276772i
$514$	− 21.2717i	− 0.938256i
$515$	8.92154i	0.393130i
$516$	3.07069	0.135179
$517$	−32.9909	−1.45094
$518$	− 23.5265i	− 1.03369i
$519$	−1.16852	−0.0512924
$520$	0	0
$521$	1.47889	0.0647915	0.0323958	−	0.999475i	$-0.489686\pi$
$521$	1.47889	0.0647915	0.0323958	+	0.999475i	$0.489686\pi$
$522$	− 9.14675i	− 0.400342i
$523$	−28.2198	−1.23397	−0.616984	−	0.786976i	$-0.711646\pi$
$523$	−28.2198	−1.23397	−0.616984	+	0.786976i	$0.711646\pi$
$524$	10.8509	0.474022
$525$	− 3.44504i	− 0.150354i
$526$	− 22.2325i	− 0.969383i
$527$	25.4825i	1.11003i
$528$	4.24698i	0.184826i
$529$	−21.2892	−0.925617
$530$	8.93900	0.388285
$531$	− 10.3327i	− 0.448402i
$532$	−21.5961	−0.936310
$533$	0	0
$534$	−13.3274	−0.576731
$535$	14.6746i	0.634437i
$536$	0.0760644	0.00328548
$537$	10.1836	0.439454
$538$	− 10.2241i	− 0.440794i
$539$	− 20.6756i	− 0.890562i
$540$	− 1.00000i	− 0.0430331i
$541$	0.245915i	0.0105727i	0.999986	+	0.00528635i	$0.00168270\pi$
$541$	0.245915i	0.0105727i	−0.999986	+	0.00528635i	$0.998317\pi$
$542$	−0.716185	−0.0307628
$543$	−10.3351	−0.443523
$544$	− 6.78986i	− 0.291113i
$545$	9.66786	0.414126
$546$	0	0
$547$	−13.6407	−0.583235	−0.291617	−	0.956535i	$-0.594193\pi$
$547$	−13.6407	−0.583235	−0.291617	+	0.956535i	$0.594193\pi$
$548$	18.6407i	0.796292i
$549$	2.53319	0.108114
$550$	4.24698	0.181092
$551$	57.3387i	2.44271i
$552$	− 1.30798i	− 0.0556713i
$553$	− 4.59956i	− 0.195593i
$554$	22.8713i	0.971708i
$555$	−6.82908	−0.289879
$556$	−3.53750	−0.150023
$557$	14.0248i	0.594248i	0.954839	+	0.297124i	$0.0960275\pi$
$557$	14.0248i	0.594248i	−0.954839	+	0.297124i	$0.903973\pi$
$558$	3.75302	0.158878
$559$	0	0
$560$	−3.44504	−0.145580
$561$	− 28.8364i	− 1.21747i
$562$	−4.13036	−0.174229
$563$	3.26981	0.137806	0.0689031	−	0.997623i	$-0.478050\pi$
$563$	3.26981	0.137806	0.0689031	+	0.997623i	$0.478050\pi$
$564$	− 7.76809i	− 0.327095i
$565$	− 11.3448i	− 0.477280i
$566$	− 16.2959i	− 0.684968i
$567$	3.44504i	0.144678i
$568$	−0.374354	−0.0157076
$569$	−4.10752	−0.172196	−0.0860982	−	0.996287i	$-0.527440\pi$
$569$	−4.10752	−0.172196	−0.0860982	+	0.996287i	$0.527440\pi$
$570$	6.26875i	0.262569i
$571$	−9.18465	−0.384366	−0.192183	−	0.981359i	$-0.561557\pi$
$571$	−9.18465	−0.384366	−0.192183	+	0.981359i	$0.561557\pi$
$572$	0	0
$573$	−9.43296	−0.394068
$574$	14.7060i	0.613817i
$575$	−1.30798	−0.0545465
$576$	−1.00000	−0.0416667
$577$	− 28.0000i	− 1.16566i	−0.812596	−	0.582828i	$-0.801946\pi$
$577$	− 28.0000i	− 1.16566i	0.812596	−	0.582828i	$-0.198054\pi$
$578$	29.1021i	1.21049i
$579$	22.9191i	0.952487i
$580$	9.14675i	0.379798i
$581$	−2.55257	−0.105898
$582$	−13.1903	−0.546755
$583$	− 37.9638i	− 1.57230i
$584$	−16.7114	−0.691523
$585$	0	0
$586$	−7.72587	−0.319153
$587$	47.6765i	1.96782i	0.178668	+	0.983910i	$0.442821\pi$
$587$	47.6765i	1.96782i	−0.178668	+	0.983910i	$0.557179\pi$
$588$	4.86831	0.200766
$589$	−23.5267	−0.969403
$590$	10.3327i	0.425392i
$591$	14.3502i	0.590288i
$592$	6.82908i	0.280674i
$593$	− 16.7681i	− 0.688583i	−0.938863	−	0.344291i	$-0.888119\pi$
$593$	− 16.7681i	− 0.688583i	0.938863	−	0.344291i	$-0.111881\pi$
$594$	−4.24698	−0.174256
$595$	23.3913	0.958951
$596$	− 16.6843i	− 0.683414i
$597$	−14.2524	−0.583310
$598$	0	0
$599$	44.0157	1.79843	0.899215	−	0.437506i	$-0.144138\pi$
$599$	44.0157	1.79843	0.899215	+	0.437506i	$0.144138\pi$
$600$	1.00000i	0.0408248i
$601$	39.1299	1.59614	0.798071	−	0.602564i	$-0.205854\pi$
$601$	39.1299	1.59614	0.798071	+	0.602564i	$0.205854\pi$
$602$	10.5786	0.431153
$603$	0.0760644i	0.00309758i
$604$	− 18.3327i	− 0.745948i
$605$	− 7.03684i	− 0.286088i
$606$	15.0422i	0.611048i
$607$	6.33273	0.257038	0.128519	−	0.991707i	$-0.458978\pi$
$607$	6.33273	0.257038	0.128519	+	0.991707i	$0.458978\pi$
$608$	6.26875	0.254231
$609$	− 31.5109i	− 1.27689i
$610$	−2.53319	−0.102566
$611$	0	0
$612$	6.78986	0.274464
$613$	− 29.1129i	− 1.17586i	−0.808912	−	0.587929i	$-0.799943\pi$
$613$	− 29.1129i	− 1.17586i	0.808912	−	0.587929i	$-0.200057\pi$
$614$	5.13036	0.207044
$615$	4.26875	0.172133
$616$	14.6310i	0.589501i
$617$	27.9933i	1.12697i	0.826127	+	0.563484i	$0.190539\pi$
$617$	27.9933i	1.12697i	−0.826127	+	0.563484i	$0.809461\pi$
$618$	8.92154i	0.358877i
$619$	23.0084i	0.924784i	0.886676	+	0.462392i	$0.153009\pi$
$619$	23.0084i	0.924784i	−0.886676	+	0.462392i	$0.846991\pi$
$620$	−3.75302	−0.150725
$621$	1.30798	0.0524874
$622$	− 33.1987i	− 1.33114i
$623$	−45.9133	−1.83948
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.0814412i	0.00325505i
$627$	26.6233	1.06323
$628$	18.6015	0.742280
$629$	− 46.3685i	− 1.84883i
$630$	− 3.44504i	− 0.137254i
$631$	− 6.18731i	− 0.246313i	−0.992387	−	0.123156i	$-0.960698\pi$
$631$	− 6.18731i	− 0.246313i	0.992387	−	0.123156i	$-0.0393017\pi$
$632$	1.33513i	0.0531084i
$633$	0.396125	0.0157445
$634$	−19.4969	−0.774323
$635$	2.54288i	0.100911i
$636$	8.93900	0.354454
$637$	0	0
$638$	38.8461	1.53793
$639$	− 0.374354i	− 0.0148092i
$640$	1.00000	0.0395285
$641$	−39.1159	−1.54498	−0.772492	−	0.635024i	$-0.780990\pi$
$641$	−39.1159	−1.54498	−0.772492	+	0.635024i	$0.780990\pi$
$642$	14.6746i	0.579159i
$643$	− 8.78209i	− 0.346332i	−0.984893	−	0.173166i	$-0.944600\pi$
$643$	− 8.78209i	− 0.346332i	0.984893	−	0.173166i	$-0.0553997\pi$
$644$	− 4.50604i	− 0.177563i
$645$	− 3.07069i	− 0.120908i
$646$	−42.5639	−1.67465
$647$	30.1333	1.18466	0.592332	−	0.805694i	$-0.298207\pi$
$647$	30.1333	1.18466	0.592332	+	0.805694i	$0.298207\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	43.8829	1.72255
$650$	0	0
$651$	12.9293	0.506740
$652$	− 18.8944i	− 0.739962i
$653$	−27.0858	−1.05995	−0.529974	−	0.848014i	$-0.677798\pi$
$653$	−27.0858	−1.05995	−0.529974	+	0.848014i	$0.677798\pi$
$654$	9.66786	0.378043
$655$	− 10.8509i	− 0.423978i
$656$	− 4.26875i	− 0.166667i
$657$	− 16.7114i	− 0.651974i
$658$	− 26.7614i	− 1.04327i
$659$	5.69681	0.221916	0.110958	−	0.993825i	$-0.464608\pi$
$659$	5.69681	0.221916	0.110958	+	0.993825i	$0.464608\pi$
$660$	4.24698	0.165313
$661$	− 17.1008i	− 0.665145i	−0.943078	−	0.332572i	$-0.892083\pi$
$661$	− 17.1008i	− 0.665145i	0.943078	−	0.332572i	$-0.107917\pi$
$662$	23.8049	0.925205
$663$	0	0
$664$	0.740939	0.0287540
$665$	21.5961i	0.837461i
$666$	−6.82908	−0.264622
$667$	−11.9638	−0.463238
$668$	4.24698i	0.164321i
$669$	− 1.49635i	− 0.0578523i
$670$	− 0.0760644i	− 0.00293862i
$671$	10.7584i	0.415323i
$672$	−3.44504	−0.132895
$673$	43.1957	1.66507	0.832535	−	0.553972i	$-0.186889\pi$
$673$	43.1957	1.66507	0.832535	+	0.553972i	$0.186889\pi$
$674$	− 21.2064i	− 0.816841i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−7.78746	−0.299297	−0.149648	−	0.988739i	$-0.547814\pi$
$677$	−7.78746	−0.299297	−0.149648	+	0.988739i	$0.547814\pi$
$678$	− 11.3448i	− 0.435695i
$679$	−45.4411	−1.74387
$680$	−6.78986	−0.260379
$681$	− 25.2150i	− 0.966243i
$682$	15.9390i	0.610336i
$683$	30.9138i	1.18288i	0.806348	+	0.591441i	$0.201441\pi$
$683$	30.9138i	1.18288i	−0.806348	+	0.591441i	$0.798559\pi$
$684$	6.26875i	0.239692i
$685$	18.6407	0.712225
$686$	−7.34375	−0.280386
$687$	− 22.3666i	− 0.853338i
$688$	−3.07069	−0.117069
$689$	0	0
$690$	−1.30798	−0.0497939
$691$	6.20237i	0.235949i	0.993017	+	0.117975i	$0.0376402\pi$
$691$	6.20237i	0.235949i	−0.993017	+	0.117975i	$0.962360\pi$
$692$	1.16852	0.0444206
$693$	−14.6310	−0.555787
$694$	− 16.5773i	− 0.629266i
$695$	3.53750i	0.134185i
$696$	9.14675i	0.346707i
$697$	28.9842i	1.09785i
$698$	−2.51035	−0.0950182
$699$	−14.0858	−0.532772
$700$	3.44504i	0.130210i
$701$	−26.3220	−0.994167	−0.497084	−	0.867703i	$-0.665596\pi$
$701$	−26.3220	−0.994167	−0.497084	+	0.867703i	$0.665596\pi$
$702$	0	0
$703$	42.8098	1.61460
$704$	− 4.24698i	− 0.160064i
$705$	−7.76809	−0.292563
$706$	−20.8412	−0.784368
$707$	51.8211i	1.94893i
$708$	10.3327i	0.388328i
$709$	33.7429i	1.26724i	0.773645	+	0.633620i	$0.218432\pi$
$709$	33.7429i	1.26724i	−0.773645	+	0.633620i	$0.781568\pi$
$710$	0.374354i	0.0140493i
$711$	−1.33513	−0.0500711
$712$	13.3274	0.499464
$713$	− 4.90887i	− 0.183839i
$714$	23.3913	0.875399
$715$	0	0
$716$	−10.1836	−0.380579
$717$	− 5.38106i	− 0.200959i
$718$	−2.42566	−0.0905250
$719$	9.50173	0.354355	0.177177	−	0.984179i	$-0.443303\pi$
$719$	9.50173	0.354355	0.177177	+	0.984179i	$0.443303\pi$
$720$	1.00000i	0.0372678i
$721$	30.7351i	1.14463i
$722$	− 20.2972i	− 0.755384i
$723$	− 29.9420i	− 1.11355i
$724$	10.3351	0.384102
$725$	9.14675	0.339702
$726$	− 7.03684i	− 0.261161i
$727$	12.2631	0.454814	0.227407	−	0.973800i	$-0.426975\pi$
$727$	12.2631	0.454814	0.227407	+	0.973800i	$0.426975\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	16.7114i	0.618517i
$731$	20.8495	0.771148
$732$	−2.53319	−0.0936293
$733$	− 1.52217i	− 0.0562227i	−0.999605	−	0.0281113i	$-0.991051\pi$
$733$	− 1.52217i	− 0.0562227i	0.999605	−	0.0281113i	$-0.00894930\pi$
$734$	2.66248i	0.0982740i
$735$	− 4.86831i	− 0.179570i
$736$	1.30798i	0.0482127i
$737$	−0.323044	−0.0118995
$738$	4.26875	0.157135
$739$	51.1075i	1.88002i	0.341146	+	0.940010i	$0.389185\pi$
$739$	51.1075i	1.88002i	−0.341146	+	0.940010i	$0.610815\pi$
$740$	6.82908	0.251042
$741$	0	0
$742$	30.7952	1.13053
$743$	9.08144i	0.333166i	0.986027	+	0.166583i	$0.0532733\pi$
$743$	9.08144i	0.333166i	−0.986027	+	0.166583i	$0.946727\pi$
$744$	−3.75302	−0.137592
$745$	−16.6843	−0.611264
$746$	5.87800i	0.215209i
$747$	0.740939i	0.0271095i
$748$	28.8364i	1.05436i
$749$	50.5545i	1.84722i
$750$	1.00000	0.0365148
$751$	−31.7090	−1.15708	−0.578539	−	0.815655i	$-0.696377\pi$
$751$	−31.7090	−1.15708	−0.578539	+	0.815655i	$0.696377\pi$
$752$	7.76809i	0.283273i
$753$	8.73556	0.318342
$754$	0	0
$755$	−18.3327	−0.667196
$756$	− 3.44504i	− 0.125295i
$757$	−8.49934	−0.308914	−0.154457	−	0.988000i	$-0.549363\pi$
$757$	−8.49934	−0.308914	−0.154457	+	0.988000i	$0.549363\pi$
$758$	36.1540	1.31317
$759$	5.55496i	0.201632i
$760$	− 6.26875i	− 0.227391i
$761$	− 16.4198i	− 0.595218i	−0.954688	−	0.297609i	$-0.903811\pi$
$761$	− 16.4198i	− 0.595218i	0.954688	−	0.297609i	$-0.0961891\pi$
$762$	2.54288i	0.0921187i
$763$	33.3062	1.20576
$764$	9.43296	0.341273
$765$	− 6.78986i	− 0.245488i
$766$	−8.98361	−0.324591
$767$	0	0
$768$	1.00000	0.0360844
$769$	27.1011i	0.977290i	0.872483	+	0.488645i	$0.162509\pi$
$769$	27.1011i	0.977290i	−0.872483	+	0.488645i	$0.837491\pi$
$770$	14.6310	0.527265
$771$	−21.2717	−0.766083
$772$	− 22.9191i	− 0.824878i
$773$	− 12.6297i	− 0.454259i	−0.973865	−	0.227129i	$-0.927066\pi$
$773$	− 12.6297i	− 0.454259i	0.973865	−	0.227129i	$-0.0729340\pi$
$774$	− 3.07069i	− 0.110374i
$775$	3.75302i	0.134812i
$776$	13.1903	0.473504
$777$	−23.5265	−0.844008
$778$	− 0.655186i	− 0.0234895i
$779$	−26.7597	−0.958767
$780$	0	0
$781$	1.58987	0.0568902
$782$	− 8.88099i	− 0.317583i
$783$	−9.14675	−0.326878
$784$	−4.86831	−0.173868
$785$	− 18.6015i	− 0.663915i
$786$	− 10.8509i	− 0.387037i
$787$	3.02954i	0.107991i	0.998541	+	0.0539957i	$0.0171957\pi$
$787$	3.02954i	0.107991i	−0.998541	+	0.0539957i	$0.982804\pi$
$788$	− 14.3502i	− 0.511204i
$789$	−22.2325	−0.791498
$790$	1.33513	0.0475016
$791$	− 39.0834i	− 1.38964i
$792$	4.24698	0.150910
$793$	0	0
$794$	−0.271143	−0.00962250
$795$	− 8.93900i	− 0.317034i
$796$	14.2524	0.505161
$797$	−40.2218	−1.42473	−0.712364	−	0.701810i	$-0.752375\pi$
$797$	−40.2218	−1.42473	−0.712364	+	0.701810i	$0.752375\pi$
$798$	21.5961i	0.764494i
$799$	− 52.7442i	− 1.86596i
$800$	− 1.00000i	− 0.0353553i
$801$	13.3274i	0.470899i
$802$	−0.344814	−0.0121758
$803$	70.9730	2.50458
$804$	− 0.0760644i	− 0.00268259i
$805$	−4.50604	−0.158817
$806$	0	0
$807$	−10.2241	−0.359907
$808$	− 15.0422i	− 0.529183i
$809$	33.7918	1.18806	0.594028	−	0.804445i	$-0.297537\pi$
$809$	33.7918	1.18806	0.594028	+	0.804445i	$0.297537\pi$
$810$	−1.00000	−0.0351364
$811$	− 33.6534i	− 1.18173i	−0.806770	−	0.590865i	$-0.798787\pi$
$811$	− 33.6534i	− 1.18173i	0.806770	−	0.590865i	$-0.201213\pi$
$812$	31.5109i	1.10582i
$813$	0.716185i	0.0251177i
$814$	− 29.0030i	− 1.01655i
$815$	−18.8944	−0.661842
$816$	−6.78986	−0.237693
$817$	19.2494i	0.673450i
$818$	27.0388	0.945388
$819$	0	0
$820$	−4.26875	−0.149071
$821$	− 12.6939i	− 0.443022i	−0.975158	−	0.221511i	$-0.928901\pi$
$821$	− 12.6939i	− 0.443022i	0.975158	−	0.221511i	$-0.0710988\pi$
$822$	18.6407	0.650169
$823$	8.79523	0.306583	0.153291	−	0.988181i	$-0.451013\pi$
$823$	8.79523	0.306583	0.153291	+	0.988181i	$0.451013\pi$
$824$	− 8.92154i	− 0.310797i
$825$	− 4.24698i	− 0.147861i
$826$	35.5967i	1.23857i
$827$	32.1758i	1.11886i	0.828877	+	0.559431i	$0.188980\pi$
$827$	32.1758i	1.11886i	−0.828877	+	0.559431i	$0.811020\pi$
$828$	−1.30798	−0.0454554
$829$	−21.1812	−0.735653	−0.367827	−	0.929894i	$-0.619898\pi$
$829$	−21.1812	−0.735653	−0.367827	+	0.929894i	$0.619898\pi$
$830$	− 0.740939i	− 0.0257184i
$831$	22.8713	0.793397
$832$	0	0
$833$	33.0551	1.14529
$834$	3.53750i	0.122494i
$835$	4.24698	0.146973
$836$	−26.6233	−0.920784
$837$	− 3.75302i	− 0.129723i
$838$	5.86533i	0.202614i
$839$	26.5931i	0.918097i	0.888411	+	0.459048i	$0.151810\pi$
$839$	26.5931i	0.918097i	−0.888411	+	0.459048i	$0.848190\pi$
$840$	3.44504i	0.118865i
$841$	54.6631	1.88493
$842$	−28.9476	−0.997601
$843$	4.13036i	0.142257i
$844$	−0.396125	−0.0136352
$845$	0	0
$846$	−7.76809	−0.267072
$847$	− 24.2422i	− 0.832972i
$848$	−8.93900	−0.306967
$849$	−16.2959	−0.559274
$850$	6.78986i	0.232890i
$851$	8.93230i	0.306195i
$852$	0.374354i	0.0128252i
$853$	− 32.2457i	− 1.10407i	−0.833821	−	0.552035i	$-0.813851\pi$
$853$	− 32.2457i	− 1.10407i	0.833821	−	0.552035i	$-0.186149\pi$
$854$	−8.72694	−0.298630
$855$	6.26875	0.214387
$856$	− 14.6746i	− 0.501566i
$857$	43.0176	1.46945	0.734726	−	0.678364i	$-0.237311\pi$
$857$	43.0176	1.46945	0.734726	+	0.678364i	$0.237311\pi$
$858$	0	0
$859$	−31.1997	−1.06452	−0.532260	−	0.846581i	$-0.678657\pi$
$859$	−31.1997	−1.06452	−0.532260	+	0.846581i	$0.678657\pi$
$860$	3.07069i	0.104710i
$861$	14.7060	0.501180
$862$	4.87023	0.165881
$863$	34.5424i	1.17584i	0.808920	+	0.587919i	$0.200052\pi$
$863$	34.5424i	1.17584i	−0.808920	+	0.587919i	$0.799948\pi$
$864$	1.00000i	0.0340207i
$865$	− 1.16852i	− 0.0397309i
$866$	36.9788i	1.25659i
$867$	29.1021	0.988361
$868$	−12.9293	−0.438849
$869$	− 5.67025i	− 0.192350i
$870$	9.14675	0.310104
$871$	0	0
$872$	−9.66786	−0.327395
$873$	13.1903i	0.446424i
$874$	8.19939	0.277349
$875$	3.44504	0.116464
$876$	16.7114i	0.564626i
$877$	− 2.12093i	− 0.0716188i	−0.999359	−	0.0358094i	$-0.988599\pi$
$877$	− 2.12093i	− 0.0716188i	0.999359	−	0.0358094i	$-0.0114009\pi$
$878$	− 22.8799i	− 0.772160i
$879$	7.72587i	0.260587i
$880$	−4.24698	−0.143166
$881$	−27.3489	−0.921407	−0.460703	−	0.887554i	$-0.652403\pi$
$881$	−27.3489	−0.921407	−0.460703	+	0.887554i	$0.652403\pi$
$882$	− 4.86831i	− 0.163925i
$883$	48.2959	1.62529	0.812643	−	0.582762i	$-0.198028\pi$
$883$	48.2959	1.62529	0.812643	+	0.582762i	$0.198028\pi$
$884$	0	0
$885$	10.3327	0.347331
$886$	− 10.5942i	− 0.355919i
$887$	25.1943	0.845943	0.422972	−	0.906143i	$-0.360987\pi$
$887$	25.1943	0.845943	0.422972	+	0.906143i	$0.360987\pi$
$888$	6.82908	0.229169
$889$	8.76032i	0.293812i
$890$	− 13.3274i	− 0.446734i
$891$	4.24698i	0.142279i
$892$	1.49635i	0.0501016i
$893$	48.6962	1.62956
$894$	−16.6843	−0.558005
$895$	10.1836i	0.340400i
$896$	3.44504	0.115091
$897$	0	0
$898$	−19.6485	−0.655678
$899$	34.3279i	1.14490i
$900$	1.00000	0.0333333
$901$	60.6945	2.02203
$902$	18.1293i	0.603639i
$903$	− 10.5786i	− 0.352035i
$904$	11.3448i	0.377323i
$905$	− 10.3351i	− 0.343551i
$906$	−18.3327	−0.609064
$907$	20.0949	0.667239	0.333619	−	0.942708i	$-0.391730\pi$
$907$	20.0949	0.667239	0.333619	+	0.942708i	$0.391730\pi$
$908$	25.2150i	0.836791i
$909$	15.0422	0.498919
$910$	0	0
$911$	−7.44563	−0.246685	−0.123342	−	0.992364i	$-0.539361\pi$
$911$	−7.44563	−0.246685	−0.123342	+	0.992364i	$0.539361\pi$
$912$	− 6.26875i	− 0.207579i
$913$	−3.14675	−0.104142
$914$	−18.4239	−0.609407
$915$	2.53319i	0.0837446i
$916$	22.3666i	0.739013i
$917$	− 37.3817i	− 1.23445i
$918$	− 6.78986i	− 0.224099i
$919$	−14.2765	−0.470939	−0.235469	−	0.971882i	$-0.575663\pi$
$919$	−14.2765	−0.470939	−0.235469	+	0.971882i	$0.575663\pi$
$920$	1.30798	0.0431228
$921$	− 5.13036i	− 0.169051i
$922$	−22.2446	−0.732586
$923$	0	0
$924$	14.6310	0.481325
$925$	− 6.82908i	− 0.224539i
$926$	8.68532	0.285417
$927$	8.92154	0.293022
$928$	− 9.14675i	− 0.300257i
$929$	− 30.9148i	− 1.01428i	−0.861863	−	0.507141i	$-0.830702\pi$
$929$	− 30.9148i	− 1.01428i	0.861863	−	0.507141i	$-0.169298\pi$
$930$	3.75302i	0.123066i
$931$	30.5182i	1.00019i
$932$	14.0858	0.461394
$933$	−33.1987	−1.08688
$934$	− 19.6136i	− 0.641775i
$935$	28.8364	0.943050
$936$	0	0
$937$	−16.7187	−0.546176	−0.273088	−	0.961989i	$-0.588045\pi$
$937$	−16.7187	−0.546176	−0.273088	+	0.961989i	$0.588045\pi$
$938$	− 0.262045i	− 0.00855608i
$939$	0.0814412	0.00265773
$940$	7.76809	0.253367
$941$	− 36.9033i	− 1.20301i	−0.798867	−	0.601507i	$-0.794567\pi$
$941$	− 36.9033i	− 1.20301i	0.798867	−	0.601507i	$-0.205433\pi$
$942$	− 18.6015i	− 0.606069i
$943$	− 5.58343i	− 0.181822i
$944$	− 10.3327i	− 0.336302i
$945$	−3.44504	−0.112067
$946$	13.0411	0.424004
$947$	− 39.1661i	− 1.27273i	−0.771389	−	0.636364i	$-0.780438\pi$
$947$	− 39.1661i	− 1.27273i	0.771389	−	0.636364i	$-0.219562\pi$
$948$	1.33513	0.0433629
$949$	0	0
$950$	−6.26875	−0.203385
$951$	19.4969i	0.632232i
$952$	−23.3913	−0.758118
$953$	19.6200	0.635554	0.317777	−	0.948165i	$-0.397064\pi$
$953$	19.6200	0.635554	0.317777	+	0.948165i	$0.397064\pi$
$954$	− 8.93900i	− 0.289411i
$955$	− 9.43296i	− 0.305243i
$956$	5.38106i	0.174036i
$957$	− 38.8461i	− 1.25572i
$958$	−31.4698	−1.01674
$959$	64.2180	2.07371
$960$	− 1.00000i	− 0.0322749i
$961$	16.9148	0.545640
$962$	0	0
$963$	14.6746	0.472881
$964$	29.9420i	0.964366i
$965$	−22.9191	−0.737794
$966$	−4.50604	−0.144979
$967$	43.1564i	1.38782i	0.720063	+	0.693909i	$0.244113\pi$
$967$	43.1564i	1.38782i	−0.720063	+	0.693909i	$0.755887\pi$
$968$	7.03684i	0.226172i
$969$	42.5639i	1.36735i
$970$	− 13.1903i	− 0.423515i
$971$	39.5032	1.26772	0.633859	−	0.773449i	$-0.281470\pi$
$971$	39.5032	1.26772	0.633859	+	0.773449i	$0.281470\pi$
$972$	−1.00000	−0.0320750
$973$	12.1868i	0.390692i
$974$	13.2644	0.425020
$975$	0	0
$976$	2.53319	0.0810854
$977$	3.84117i	0.122890i	0.998110	+	0.0614449i	$0.0195708\pi$
$977$	3.84117i	0.122890i	−0.998110	+	0.0614449i	$0.980429\pi$
$978$	−18.8944	−0.604176
$979$	−56.6010	−1.80898
$980$	4.86831i	0.155513i
$981$	− 9.66786i	− 0.308671i
$982$	− 11.7259i	− 0.374188i
$983$	− 32.4300i	− 1.03436i	−0.855878	−	0.517178i	$-0.826983\pi$
$983$	− 32.4300i	− 1.03436i	0.855878	−	0.517178i	$-0.173017\pi$
$984$	−4.26875	−0.136083
$985$	−14.3502	−0.457235
$986$	62.1051i	1.97783i
$987$	−26.7614	−0.851824
$988$	0	0
$989$	−4.01639	−0.127714
$990$	− 4.24698i	− 0.134978i
$991$	−61.9807	−1.96888	−0.984442	−	0.175712i	$-0.943777\pi$
$991$	−61.9807	−1.96888	−0.984442	+	0.175712i	$0.943777\pi$
$992$	3.75302	0.119159
$993$	− 23.8049i	− 0.755426i
$994$	1.28967i	0.0409057i
$995$	− 14.2524i	− 0.451830i
$996$	− 0.740939i	− 0.0234775i
$997$	28.9162	0.915784	0.457892	−	0.889008i	$-0.348605\pi$
$997$	28.9162	0.915784	0.457892	+	0.889008i	$0.348605\pi$
$998$	−26.7633	−0.847177
$999$	6.82908i	0.216063i

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.44504i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +3.44504i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.24698i q^{11} -1.00000 q^{12} -3.44504 q^{14} +1.00000i q^{15} +1.00000 q^{16} -6.78986 q^{17} +1.00000i q^{18} -6.26875i q^{19} -1.00000i q^{20} +3.44504i q^{21} -4.24698 q^{22} +1.30798 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.44504i q^{28} -9.14675 q^{29} -1.00000 q^{30} -3.75302i q^{31} +1.00000i q^{32} +4.24698i q^{33} -6.78986i q^{34} -3.44504 q^{35} -1.00000 q^{36} +6.82908i q^{37} +6.26875 q^{38} +1.00000 q^{40} -4.26875i q^{41} -3.44504 q^{42} -3.07069 q^{43} -4.24698i q^{44} +1.00000i q^{45} +1.30798i q^{46} +7.76809i q^{47} +1.00000 q^{48} -4.86831 q^{49} -1.00000i q^{50} -6.78986 q^{51} -8.93900 q^{53} +1.00000i q^{54} -4.24698 q^{55} +3.44504 q^{56} -6.26875i q^{57} -9.14675i q^{58} -10.3327i q^{59} -1.00000i q^{60} +2.53319 q^{61} +3.75302 q^{62} +3.44504i q^{63} -1.00000 q^{64} -4.24698 q^{66} +0.0760644i q^{67} +6.78986 q^{68} +1.30798 q^{69} -3.44504i q^{70} -0.374354i q^{71} -1.00000i q^{72} -16.7114i q^{73} -6.82908 q^{74} -1.00000 q^{75} +6.26875i q^{76} -14.6310 q^{77} -1.33513 q^{79} +1.00000i q^{80} +1.00000 q^{81} +4.26875 q^{82} +0.740939i q^{83} -3.44504i q^{84} -6.78986i q^{85} -3.07069i q^{86} -9.14675 q^{87} +4.24698 q^{88} +13.3274i q^{89} -1.00000 q^{90} -1.30798 q^{92} -3.75302i q^{93} -7.76809 q^{94} +6.26875 q^{95} +1.00000i q^{96} +13.1903i q^{97} -4.86831i q^{98} +4.24698i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} - 6 q^{12} - 20 q^{14} + 6 q^{16} + 6 q^{17} - 16 q^{22} + 18 q^{23} - 6 q^{25} + 6 q^{27} - 6 q^{30} - 20 q^{35} - 6 q^{36} + 22 q^{38} + 6 q^{40} - 20 q^{42} + 6 q^{43} + 6 q^{48} - 34 q^{49} + 6 q^{51} - 34 q^{53} - 16 q^{55} + 20 q^{56} + 22 q^{61} + 32 q^{62} - 6 q^{64} - 16 q^{66} - 6 q^{68} + 18 q^{69} - 20 q^{74} - 6 q^{75} - 58 q^{77} - 6 q^{79} + 6 q^{81} + 10 q^{82} + 16 q^{88} - 6 q^{90} - 18 q^{92} - 6 q^{94} + 22 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 - 6 * q^12 - 20 * q^14 + 6 * q^16 + 6 * q^17 - 16 * q^22 + 18 * q^23 - 6 * q^25 + 6 * q^27 - 6 * q^30 - 20 * q^35 - 6 * q^36 + 22 * q^38 + 6 * q^40 - 20 * q^42 + 6 * q^43 + 6 * q^48 - 34 * q^49 + 6 * q^51 - 34 * q^53 - 16 * q^55 + 20 * q^56 + 22 * q^61 + 32 * q^62 - 6 * q^64 - 16 * q^66 - 6 * q^68 + 18 * q^69 - 20 * q^74 - 6 * q^75 - 58 * q^77 - 6 * q^79 + 6 * q^81 + 10 * q^82 + 16 * q^88 - 6 * q^90 - 18 * q^92 - 6 * q^94 + 22 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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