LMFDB - Embedded newform 1560.2.l.c.1249.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(1249,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1560.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1560.l (of order $2$, degree $1$, 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:	$12.4566627153$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.4
Root			$-1.75233 + 1.75233i$ of defining polynomial
Character	$\chi$	$=$	1560.1249
Dual form			1560.2.l.c.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.75233 - 1.38900i) q^{5} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-1.75233 - 1.38900i) q^{5} -1.00000 q^{9} +1.50466 q^{11} +1.00000i q^{13} +(1.38900 - 1.75233i) q^{15} +2.72666i q^{17} -0.726656 q^{19} -4.72666i q^{23} +(1.14134 + 4.86799i) q^{25} -1.00000i q^{27} +7.55602 q^{29} -3.00933 q^{31} +1.50466i q^{33} +5.00933i q^{37} -1.00000 q^{39} +5.78734 q^{41} +2.72666i q^{43} +(1.75233 + 1.38900i) q^{45} +10.2313i q^{47} +7.00000 q^{49} -2.72666 q^{51} -7.55602i q^{53} +(-2.63667 - 2.08998i) q^{55} -0.726656i q^{57} +12.5140 q^{59} +6.28267 q^{61} +(1.38900 - 1.75233i) q^{65} +12.5653i q^{67} +4.72666 q^{69} +4.77801 q^{71} +12.0187i q^{73} +(-4.86799 + 1.14134i) q^{75} -5.27334 q^{79} +1.00000 q^{81} +7.78734i q^{83} +(3.78734 - 4.77801i) q^{85} +7.55602i q^{87} -1.78734 q^{89} -3.00933i q^{93} +(1.27334 + 1.00933i) q^{95} +6.00000i q^{97} -1.50466 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} - 12 q^{11} + 2 q^{15} + 4 q^{19} - 10 q^{25} + 20 q^{29} + 24 q^{31} - 6 q^{39} - 20 q^{41} + 42 q^{49} - 8 q^{51} - 20 q^{55} + 12 q^{59} + 4 q^{61} + 2 q^{65} + 20 q^{69} + 16 q^{71} - 4 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} + 44 q^{89} + 16 q^{95} + 12 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 - 12 * q^11 + 2 * q^15 + 4 * q^19 - 10 * q^25 + 20 * q^29 + 24 * q^31 - 6 * q^39 - 20 * q^41 + 42 * q^49 - 8 * q^51 - 20 * q^55 + 12 * q^59 + 4 * q^61 + 2 * q^65 + 20 * q^69 + 16 * q^71 - 4 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 + 44 * q^89 + 16 * q^95 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$.

$n$	$391$	$521$	$781$	$937$	$1081$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.75233	−	1.38900i	−0.783667	−	0.621181i
$5$	−1.75233	−	1.38900i	−0.783667	−	0.621181i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.50466			0.453673			0.226837	−	0.973933i	$-0.427162\pi$
$11$	1.50466			0.453673			0.226837	+	0.973933i	$0.427162\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	1.38900	−	1.75233i	0.358639	−	0.452450i
$16$		0			0
$17$			2.72666i			0.661311i	0.943751	+	0.330656i	$0.107270\pi$
$17$			2.72666i			0.661311i	−0.943751	+	0.330656i	$0.892730\pi$
$18$		0			0
$19$	−0.726656			−0.166706			−0.0833532	−	0.996520i	$-0.526563\pi$
$19$	−0.726656			−0.166706			−0.0833532	+	0.996520i	$0.526563\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	4.72666i		−	0.985576i	−0.870149	−	0.492788i	$-0.835978\pi$
$23$		−	4.72666i		−	0.985576i	0.870149	−	0.492788i	$-0.164022\pi$
$24$		0			0
$25$	1.14134	+	4.86799i	0.228267	+	0.973599i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	7.55602			1.40312			0.701558	−	0.712612i	$-0.252488\pi$
$29$	7.55602			1.40312			0.701558	+	0.712612i	$0.252488\pi$
$30$		0			0
$31$	−3.00933			−0.540491			−0.270246	−	0.962791i	$-0.587105\pi$
$31$	−3.00933			−0.540491			−0.270246	+	0.962791i	$0.587105\pi$
$32$		0			0
$33$			1.50466i			0.261928i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			5.00933i			0.823529i	0.911290	+	0.411764i	$0.135087\pi$
$37$			5.00933i			0.823529i	−0.911290	+	0.411764i	$0.864913\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	5.78734			0.903830			0.451915	−	0.892061i	$-0.350741\pi$
$41$	5.78734			0.903830			0.451915	+	0.892061i	$0.350741\pi$
$42$		0			0
$43$			2.72666i			0.415811i	0.978149	+	0.207906i	$0.0666647\pi$
$43$			2.72666i			0.415811i	−0.978149	+	0.207906i	$0.933335\pi$
$44$		0			0
$45$	1.75233	+	1.38900i	0.261222	+	0.207060i
$46$		0			0
$47$			10.2313i			1.49239i	0.665727	+	0.746196i	$0.268122\pi$
$47$			10.2313i			1.49239i	−0.665727	+	0.746196i	$0.731878\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−2.72666			−0.381808
$52$		0			0
$53$		−	7.55602i		−	1.03790i	−0.854805	−	0.518949i	$-0.826323\pi$
$53$		−	7.55602i		−	1.03790i	0.854805	−	0.518949i	$-0.173677\pi$
$54$		0			0
$55$	−2.63667	−	2.08998i	−0.355529	−	0.281813i
$56$		0			0
$57$		−	0.726656i		−	0.0962480i
$58$		0			0
$59$	12.5140			1.62918			0.814592	−	0.580035i	$-0.196961\pi$
$59$	12.5140			1.62918			0.814592	+	0.580035i	$0.196961\pi$
$60$		0			0
$61$	6.28267			0.804414			0.402207	−	0.915549i	$-0.368243\pi$
$61$	6.28267			0.804414			0.402207	+	0.915549i	$0.368243\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.38900	−	1.75233i	0.172285	−	0.217350i
$66$		0			0
$67$			12.5653i			1.53510i	0.640988	+	0.767551i	$0.278525\pi$
$67$			12.5653i			1.53510i	−0.640988	+	0.767551i	$0.721475\pi$
$68$		0			0
$69$	4.72666			0.569023
$70$		0			0
$71$	4.77801			0.567045			0.283523	−	0.958966i	$-0.408497\pi$
$71$	4.77801			0.567045			0.283523	+	0.958966i	$0.408497\pi$
$72$		0			0
$73$			12.0187i			1.40668i	0.710855	+	0.703339i	$0.248308\pi$
$73$			12.0187i			1.40668i	−0.710855	+	0.703339i	$0.751692\pi$
$74$		0			0
$75$	−4.86799	+	1.14134i	−0.562107	+	0.131790i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.27334			−0.593297			−0.296649	−	0.954987i	$-0.595869\pi$
$79$	−5.27334			−0.593297			−0.296649	+	0.954987i	$0.595869\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			7.78734i			0.854771i	0.904069	+	0.427386i	$0.140565\pi$
$83$			7.78734i			0.854771i	−0.904069	+	0.427386i	$0.859435\pi$
$84$		0			0
$85$	3.78734	−	4.77801i	0.410794	−	0.518248i
$86$		0			0
$87$			7.55602i			0.810090i
$88$		0			0
$89$	−1.78734			−0.189457			−0.0947286	−	0.995503i	$-0.530198\pi$
$89$	−1.78734			−0.189457			−0.0947286	+	0.995503i	$0.530198\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	3.00933i		−	0.312053i
$94$		0			0
$95$	1.27334	+	1.00933i	0.130642	+	0.103555i
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$	−1.50466			−0.151224
$100$		0			0
$101$	−2.99067			−0.297583			−0.148791	−	0.988869i	$-0.547538\pi$
$101$	−2.99067			−0.297583			−0.148791	+	0.988869i	$0.547538\pi$
$102$		0			0
$103$		−	0.443984i		−	0.0437471i	−0.999761	−	0.0218735i	$-0.993037\pi$
$103$		−	0.443984i		−	0.0437471i	0.999761	−	0.0218735i	$-0.00696312\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	7.00933i		−	0.677617i	−0.940855	−	0.338809i	$-0.889976\pi$
$107$		−	7.00933i		−	0.677617i	0.940855	−	0.338809i	$-0.110024\pi$
$108$		0			0
$109$	13.8387			1.32551			0.662753	−	0.748838i	$-0.269388\pi$
$109$	13.8387			1.32551			0.662753	+	0.748838i	$0.269388\pi$
$110$		0			0
$111$	−5.00933			−0.475464
$112$		0			0
$113$		−	4.28267i		−	0.402880i	−0.979501	−	0.201440i	$-0.935438\pi$
$113$		−	4.28267i		−	0.402880i	0.979501	−	0.201440i	$-0.0645621\pi$
$114$		0			0
$115$	−6.56534	+	8.28267i	−0.612222	+	0.772363i
$116$		0			0
$117$		−	1.00000i		−	0.0924500i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.73599			−0.794180
$122$		0			0
$123$			5.78734i			0.521827i
$124$		0			0
$125$	4.76166	−	10.1157i	0.425896	−	0.904772i
$126$		0			0
$127$			5.71733i			0.507331i	0.967292	+	0.253665i	$0.0816362\pi$
$127$			5.71733i			0.507331i	−0.967292	+	0.253665i	$0.918364\pi$
$128$		0			0
$129$	−2.72666			−0.240069
$130$		0			0
$131$	5.55602			0.485431			0.242716	−	0.970097i	$-0.421962\pi$
$131$	5.55602			0.485431			0.242716	+	0.970097i	$0.421962\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.38900	+	1.75233i	−0.119546	+	0.150817i
$136$		0			0
$137$			6.67531i			0.570310i	0.958481	+	0.285155i	$0.0920450\pi$
$137$			6.67531i			0.570310i	−0.958481	+	0.285155i	$0.907955\pi$
$138$		0			0
$139$	−19.4720			−1.65159			−0.825795	−	0.563970i	$-0.809273\pi$
$139$	−19.4720			−1.65159			−0.825795	+	0.563970i	$0.809273\pi$
$140$		0			0
$141$	−10.2313			−0.861633
$142$		0			0
$143$			1.50466i			0.125826i
$144$		0			0
$145$	−13.2406	−	10.4953i	−1.09958	−	0.871590i
$146$		0			0
$147$			7.00000i			0.577350i
$148$		0			0
$149$	14.5140			1.18903			0.594516	−	0.804084i	$-0.297344\pi$
$149$	14.5140			1.18903			0.594516	+	0.804084i	$0.297344\pi$
$150$		0			0
$151$	−4.46264			−0.363165			−0.181582	−	0.983376i	$-0.558122\pi$
$151$	−4.46264			−0.363165			−0.181582	+	0.983376i	$0.558122\pi$
$152$		0			0
$153$		−	2.72666i		−	0.220437i
$154$		0			0
$155$	5.27334	+	4.17997i	0.423565	+	0.335743i
$156$		0			0
$157$		−	8.30133i		−	0.662518i	−0.943540	−	0.331259i	$-0.892527\pi$
$157$		−	8.30133i		−	0.662518i	0.943540	−	0.331259i	$-0.107473\pi$
$158$		0			0
$159$	7.55602			0.599231
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	11.0093i		−	0.862317i	−0.902276	−	0.431159i	$-0.858105\pi$
$163$		−	11.0093i		−	0.862317i	0.902276	−	0.431159i	$-0.141895\pi$
$164$		0			0
$165$	2.08998	−	2.63667i	0.162705	−	0.205265i
$166$		0			0
$167$			5.76868i			0.446394i	0.974773	+	0.223197i	$0.0716493\pi$
$167$			5.76868i			0.446394i	−0.974773	+	0.223197i	$0.928351\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	0.726656			0.0555688
$172$		0			0
$173$			4.90663i			0.373044i	0.982451	+	0.186522i	$0.0597215\pi$
$173$			4.90663i			0.373044i	−0.982451	+	0.186522i	$0.940278\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			12.5140i			0.940609i
$178$		0			0
$179$	−9.45331			−0.706574			−0.353287	−	0.935515i	$-0.614936\pi$
$179$	−9.45331			−0.706574			−0.353287	+	0.935515i	$0.614936\pi$
$180$		0			0
$181$	17.4720			1.29868			0.649341	−	0.760498i	$-0.275045\pi$
$181$	17.4720			1.29868			0.649341	+	0.760498i	$0.275045\pi$
$182$		0			0
$183$			6.28267i			0.464428i
$184$		0			0
$185$	6.95798	−	8.77801i	0.511561	−	0.645372i
$186$		0			0
$187$			4.10270i			0.300019i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.1120			−0.804038			−0.402019	−	0.915631i	$-0.631691\pi$
$191$	−11.1120			−0.804038			−0.402019	+	0.915631i	$0.631691\pi$
$192$		0			0
$193$			6.10270i			0.439282i	0.975581	+	0.219641i	$0.0704886\pi$
$193$			6.10270i			0.439282i	−0.975581	+	0.219641i	$0.929511\pi$
$194$		0			0
$195$	1.75233	+	1.38900i	0.125487	+	0.0994686i
$196$		0			0
$197$		−	21.3620i		−	1.52198i	−0.648764	−	0.760990i	$-0.724714\pi$
$197$		−	21.3620i		−	1.52198i	0.648764	−	0.760990i	$-0.275286\pi$
$198$		0			0
$199$	−8.38538			−0.594423			−0.297212	−	0.954812i	$-0.596057\pi$
$199$	−8.38538			−0.594423			−0.297212	+	0.954812i	$0.596057\pi$
$200$		0			0
$201$	−12.5653			−0.886291
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−10.1413	−	8.03863i	−0.708302	−	0.561443i
$206$		0			0
$207$			4.72666i			0.328525i
$208$		0			0
$209$	−1.09337			−0.0756303
$210$		0			0
$211$	−1.27334			−0.0876606			−0.0438303	−	0.999039i	$-0.513956\pi$
$211$	−1.27334			−0.0876606			−0.0438303	+	0.999039i	$0.513956\pi$
$212$		0			0
$213$			4.77801i			0.327384i
$214$		0			0
$215$	3.78734	−	4.77801i	0.258294	−	0.325857i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−12.0187			−0.812146
$220$		0			0
$221$	−2.72666			−0.183415
$222$		0			0
$223$		−	16.4626i		−	1.10242i	−0.834367	−	0.551210i	$-0.814166\pi$
$223$		−	16.4626i		−	1.10242i	0.834367	−	0.551210i	$-0.185834\pi$
$224$		0			0
$225$	−1.14134	−	4.86799i	−0.0760891	−	0.324533i
$226$		0			0
$227$			12.2500i			0.813060i	0.913638	+	0.406530i	$0.133261\pi$
$227$			12.2500i			0.813060i	−0.913638	+	0.406530i	$0.866739\pi$
$228$		0			0
$229$	1.27334			0.0841449			0.0420725	−	0.999115i	$-0.486604\pi$
$229$	1.27334			0.0841449			0.0420725	+	0.999115i	$0.486604\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18.3013i		−	1.19896i	−0.800390	−	0.599480i	$-0.795374\pi$
$233$		−	18.3013i		−	1.19896i	0.800390	−	0.599480i	$-0.204626\pi$
$234$		0			0
$235$	14.2113	−	17.9287i	0.927046	−	1.16954i
$236$		0			0
$237$		−	5.27334i		−	0.342540i
$238$		0			0
$239$	−6.23132			−0.403071			−0.201535	−	0.979481i	$-0.564593\pi$
$239$	−6.23132			−0.403071			−0.201535	+	0.979481i	$0.564593\pi$
$240$		0			0
$241$	19.5560			1.25971			0.629857	−	0.776711i	$-0.283114\pi$
$241$	19.5560			1.25971			0.629857	+	0.776711i	$0.283114\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−12.2663	−	9.72303i	−0.783667	−	0.621181i
$246$		0			0
$247$		−	0.726656i		−	0.0462360i
$248$		0			0
$249$	−7.78734			−0.493502
$250$		0			0
$251$	26.4813			1.67148			0.835742	−	0.549122i	$-0.185038\pi$
$251$	26.4813			1.67148			0.835742	+	0.549122i	$0.185038\pi$
$252$		0			0
$253$		−	7.11203i		−	0.447130i
$254$		0			0
$255$	4.77801	+	3.78734i	0.299210	+	0.237172i
$256$		0			0
$257$			14.8294i			0.925030i	0.886611	+	0.462515i	$0.153053\pi$
$257$			14.8294i			0.925030i	−0.886611	+	0.462515i	$0.846947\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−7.55602			−0.467706
$262$		0			0
$263$		−	22.9507i		−	1.41520i	−0.706612	−	0.707601i	$-0.749777\pi$
$263$		−	22.9507i		−	1.41520i	0.706612	−	0.707601i	$-0.250223\pi$
$264$		0			0
$265$	−10.4953	+	13.2406i	−0.644723	+	0.813367i
$266$		0			0
$267$		−	1.78734i		−	0.109383i
$268$		0			0
$269$	27.9160			1.70207			0.851033	−	0.525112i	$-0.175977\pi$
$269$	27.9160			1.70207			0.851033	+	0.525112i	$0.175977\pi$
$270$		0			0
$271$	−5.65872			−0.343743			−0.171871	−	0.985119i	$-0.554981\pi$
$271$	−5.65872			−0.343743			−0.171871	+	0.985119i	$0.554981\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.71733	+	7.32469i	0.103559	+	0.441696i
$276$		0			0
$277$			7.73599i			0.464810i	0.972619	+	0.232405i	$0.0746595\pi$
$277$			7.73599i			0.464810i	−0.972619	+	0.232405i	$0.925340\pi$
$278$		0			0
$279$	3.00933			0.180164
$280$		0			0
$281$	5.11929			0.305391			0.152696	−	0.988273i	$-0.451205\pi$
$281$	5.11929			0.305391			0.152696	+	0.988273i	$0.451205\pi$
$282$		0			0
$283$		−	6.90663i		−	0.410556i	−0.978704	−	0.205278i	$-0.934190\pi$
$283$		−	6.90663i		−	0.410556i	0.978704	−	0.205278i	$-0.0658099\pi$
$284$		0			0
$285$	−1.00933	+	1.27334i	−0.0597875	+	0.0754264i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	9.56534			0.562667
$290$		0			0
$291$	−6.00000			−0.351726
$292$		0			0
$293$		−	26.3527i		−	1.53954i	−0.638321	−	0.769770i	$-0.720371\pi$
$293$		−	26.3527i		−	1.53954i	0.638321	−	0.769770i	$-0.279629\pi$
$294$		0			0
$295$	−21.9287	−	17.3820i	−1.27674	−	1.01202i
$296$		0			0
$297$		−	1.50466i		−	0.0873095i
$298$		0			0
$299$	4.72666			0.273350
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	2.99067i		−	0.171810i
$304$		0			0
$305$	−11.0093	−	8.72666i	−0.630392	−	0.499687i
$306$		0			0
$307$			20.0000i			1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$			20.0000i			1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$		0			0
$309$	0.443984			0.0252574
$310$		0			0
$311$	−11.8973			−0.674634			−0.337317	−	0.941391i	$-0.609519\pi$
$311$	−11.8973			−0.674634			−0.337317	+	0.941391i	$0.609519\pi$
$312$		0			0
$313$		−	17.7360i		−	1.00250i	−0.865303	−	0.501249i	$-0.832874\pi$
$313$		−	17.7360i		−	1.00250i	0.865303	−	0.501249i	$-0.167126\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1.32469i		−	0.0744023i	−0.999308	−	0.0372011i	$-0.988156\pi$
$317$		−	1.32469i		−	0.0744023i	0.999308	−	0.0372011i	$-0.0118442\pi$
$318$		0			0
$319$	11.3693			0.636557
$320$		0			0
$321$	7.00933			0.391223
$322$		0			0
$323$		−	1.98134i		−	0.110245i
$324$		0			0
$325$	−4.86799	+	1.14134i	−0.270028	+	0.0633099i
$326$		0			0
$327$			13.8387i			0.765281i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.73599			−0.205348			−0.102674	−	0.994715i	$-0.532740\pi$
$331$	−3.73599			−0.205348			−0.102674	+	0.994715i	$0.532740\pi$
$332$		0			0
$333$		−	5.00933i		−	0.274510i
$334$		0			0
$335$	17.4533	−	22.0187i	0.953576	−	1.20301i
$336$		0			0
$337$			23.3947i			1.27439i	0.770702	+	0.637195i	$0.219906\pi$
$337$			23.3947i			1.27439i	−0.770702	+	0.637195i	$0.780094\pi$
$338$		0			0
$339$	4.28267			0.232603
$340$		0			0
$341$	−4.52803			−0.245207
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.28267	−	6.56534i	−0.445924	−	0.353466i
$346$		0			0
$347$		−	30.5840i		−	1.64184i	−0.571047	−	0.820918i	$-0.693462\pi$
$347$		−	30.5840i		−	1.64184i	0.571047	−	0.820918i	$-0.306538\pi$
$348$		0			0
$349$	−1.37605			−0.0736581			−0.0368290	−	0.999322i	$-0.511726\pi$
$349$	−1.37605			−0.0736581			−0.0368290	+	0.999322i	$0.511726\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$			19.9087i			1.05963i	0.848112	+	0.529817i	$0.177739\pi$
$353$			19.9087i			1.05963i	−0.848112	+	0.529817i	$0.822261\pi$
$354$		0			0
$355$	−8.37266	−	6.63667i	−0.444374	−	0.352238i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−26.6940			−1.40885			−0.704427	−	0.709777i	$-0.748796\pi$
$359$	−26.6940			−1.40885			−0.704427	+	0.709777i	$0.748796\pi$
$360$		0			0
$361$	−18.4720			−0.972209
$362$		0			0
$363$		−	8.73599i		−	0.458520i
$364$		0			0
$365$	16.6940	−	21.0607i	0.873802	−	1.10237i
$366$		0			0
$367$		−	8.12136i		−	0.423932i	−0.977277	−	0.211966i	$-0.932013\pi$
$367$		−	8.12136i		−	0.423932i	0.977277	−	0.211966i	$-0.0679865\pi$
$368$		0			0
$369$	−5.78734			−0.301277
$370$		0			0
$371$		0			0
$372$		0			0
$373$			9.47197i			0.490440i	0.969467	+	0.245220i	$0.0788602\pi$
$373$			9.47197i			0.490440i	−0.969467	+	0.245220i	$0.921140\pi$
$374$		0			0
$375$	10.1157	+	4.76166i	0.522370	+	0.245891i
$376$		0			0
$377$			7.55602i			0.389155i
$378$		0			0
$379$	−16.7267			−0.859191			−0.429595	−	0.903022i	$-0.641344\pi$
$379$	−16.7267			−0.859191			−0.429595	+	0.903022i	$0.641344\pi$
$380$		0			0
$381$	−5.71733			−0.292908
$382$		0			0
$383$			19.6846i			1.00584i	0.864334	+	0.502919i	$0.167741\pi$
$383$			19.6846i			1.00584i	−0.864334	+	0.502919i	$0.832259\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	2.72666i		−	0.138604i
$388$		0			0
$389$	14.0000			0.709828			0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000			0.709828			0.354914	+	0.934899i	$0.384510\pi$
$390$		0			0
$391$	12.8880			0.651773
$392$		0			0
$393$			5.55602i			0.280264i
$394$		0			0
$395$	9.24065	+	7.32469i	0.464948	+	0.368545i
$396$		0			0
$397$			3.35061i			0.168162i	0.996459	+	0.0840812i	$0.0267955\pi$
$397$			3.35061i			0.168162i	−0.996459	+	0.0840812i	$0.973205\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−36.3713			−1.81630			−0.908149	−	0.418647i	$-0.862504\pi$
$401$	−36.3713			−1.81630			−0.908149	+	0.418647i	$0.862504\pi$
$402$		0			0
$403$		−	3.00933i		−	0.149905i
$404$		0			0
$405$	−1.75233	−	1.38900i	−0.0870741	−	0.0690202i
$406$		0			0
$407$			7.53736i			0.373613i
$408$		0			0
$409$	−35.5933			−1.75998			−0.879988	−	0.474995i	$-0.842450\pi$
$409$	−35.5933			−1.75998			−0.879988	+	0.474995i	$0.842450\pi$
$410$		0			0
$411$	−6.67531			−0.329269
$412$		0			0
$413$		0			0
$414$		0			0
$415$	10.8166	−	13.6460i	0.530968	−	0.669856i
$416$		0			0
$417$		−	19.4720i		−	0.953546i
$418$		0			0
$419$	13.9160			0.679839			0.339919	−	0.940455i	$-0.389600\pi$
$419$	13.9160			0.679839			0.339919	+	0.940455i	$0.389600\pi$
$420$		0			0
$421$	4.70800			0.229454			0.114727	−	0.993397i	$-0.463401\pi$
$421$	4.70800			0.229454			0.114727	+	0.993397i	$0.463401\pi$
$422$		0			0
$423$		−	10.2313i		−	0.497464i
$424$		0			0
$425$	−13.2733	+	3.11203i	−0.643852	+	0.150956i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−1.50466			−0.0726459
$430$		0			0
$431$	7.89004			0.380050			0.190025	−	0.981779i	$-0.439143\pi$
$431$	7.89004			0.380050			0.190025	+	0.981779i	$0.439143\pi$
$432$		0			0
$433$		−	2.30133i		−	0.110595i	−0.998470	−	0.0552974i	$-0.982389\pi$
$433$		−	2.30133i		−	0.110595i	0.998470	−	0.0552974i	$-0.0176107\pi$
$434$		0			0
$435$	10.4953	−	13.2406i	0.503213	−	0.634841i
$436$		0			0
$437$			3.43466i			0.164302i
$438$		0			0
$439$	−41.1307			−1.96306			−0.981530	−	0.191307i	$-0.938728\pi$
$439$	−41.1307			−1.96306			−0.981530	+	0.191307i	$0.938728\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$		−	29.5933i		−	1.40602i	−0.711179	−	0.703011i	$-0.751839\pi$
$443$		−	29.5933i		−	1.40602i	0.711179	−	0.703011i	$-0.248161\pi$
$444$		0			0
$445$	3.13201	+	2.48262i	0.148471	+	0.117687i
$446$		0			0
$447$			14.5140i			0.686488i
$448$		0			0
$449$	11.4461			0.540173			0.270086	−	0.962836i	$-0.412948\pi$
$449$	11.4461			0.540173			0.270086	+	0.962836i	$0.412948\pi$
$450$		0			0
$451$	8.70800			0.410044
$452$		0			0
$453$		−	4.46264i		−	0.209673i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			16.5467i			0.774021i	0.922075	+	0.387011i	$0.126492\pi$
$457$			16.5467i			0.774021i	−0.922075	+	0.387011i	$0.873508\pi$
$458$		0			0
$459$	2.72666			0.127269
$460$		0			0
$461$	3.50466			0.163228			0.0816142	−	0.996664i	$-0.473992\pi$
$461$	3.50466			0.163228			0.0816142	+	0.996664i	$0.473992\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$	−4.17997	+	5.27334i	−0.193841	+	0.244545i
$466$		0			0
$467$			6.44398i			0.298192i	0.988823	+	0.149096i	$0.0476363\pi$
$467$			6.44398i			0.298192i	−0.988823	+	0.149096i	$0.952364\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	8.30133			0.382505
$472$		0			0
$473$			4.10270i			0.188642i
$474$		0			0
$475$	−0.829359	−	3.53736i	−0.0380536	−	0.162305i
$476$		0			0
$477$			7.55602i			0.345966i
$478$		0			0
$479$	−22.2313			−1.01577			−0.507887	−	0.861423i	$-0.669573\pi$
$479$	−22.2313			−1.01577			−0.507887	+	0.861423i	$0.669573\pi$
$480$		0			0
$481$	−5.00933			−0.228406
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.33402	−	10.5140i	0.378429	−	0.477416i
$486$		0			0
$487$		−	13.9160i		−	0.630592i	−0.948993	−	0.315296i	$-0.897896\pi$
$487$		−	13.9160i		−	0.630592i	0.948993	−	0.315296i	$-0.102104\pi$
$488$		0			0
$489$	11.0093			0.497859
$490$		0			0
$491$	−28.1400			−1.26994			−0.634971	−	0.772536i	$-0.718988\pi$
$491$	−28.1400			−1.26994			−0.634971	+	0.772536i	$0.718988\pi$
$492$		0			0
$493$			20.6027i			0.927897i
$494$		0			0
$495$	2.63667	+	2.08998i	0.118510	+	0.0939378i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−38.8480			−1.73908			−0.869538	−	0.493866i	$-0.835583\pi$
$499$	−38.8480			−1.73908			−0.869538	+	0.493866i	$0.835583\pi$
$500$		0			0
$501$	−5.76868			−0.257726
$502$		0			0
$503$		−	8.19863i		−	0.365559i	−0.983154	−	0.182779i	$-0.941491\pi$
$503$		−	8.19863i		−	0.365559i	0.983154	−	0.182779i	$-0.0585094\pi$
$504$		0			0
$505$	5.24065	+	4.15405i	0.233206	+	0.184853i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	−16.0700			−0.712291			−0.356145	−	0.934431i	$-0.615909\pi$
$509$	−16.0700			−0.712291			−0.356145	+	0.934431i	$0.615909\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			0.726656i			0.0320827i
$514$		0			0
$515$	−0.616696	+	0.778008i	−0.0271749	+	0.0342831i
$516$		0			0
$517$			15.3947i			0.677058i
$518$		0			0
$519$	−4.90663			−0.215377
$520$		0			0
$521$	−23.0280			−1.00887			−0.504437	−	0.863448i	$-0.668300\pi$
$521$	−23.0280			−1.00887			−0.504437	+	0.863448i	$0.668300\pi$
$522$		0			0
$523$		−	5.47875i		−	0.239569i	−0.992800	−	0.119784i	$-0.961780\pi$
$523$		−	5.47875i		−	0.239569i	0.992800	−	0.119784i	$-0.0382204\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8.20541i		−	0.357433i
$528$		0			0
$529$	0.658719			0.0286399
$530$		0			0
$531$	−12.5140			−0.543061
$532$		0			0
$533$			5.78734i			0.250677i
$534$		0			0
$535$	−9.73599	+	12.2827i	−0.420923	+	0.531026i
$536$		0			0
$537$		−	9.45331i		−	0.407941i
$538$		0			0
$539$	10.5327			0.453673
$540$		0			0
$541$	17.8387			0.766945			0.383473	−	0.923552i	$-0.374728\pi$
$541$	17.8387			0.766945			0.383473	+	0.923552i	$0.374728\pi$
$542$		0			0
$543$			17.4720i			0.749794i
$544$		0			0
$545$	−24.2500	−	19.2220i	−1.03875	−	0.823380i
$546$		0			0
$547$			29.1053i			1.24445i	0.782838	+	0.622225i	$0.213771\pi$
$547$			29.1053i			1.24445i	−0.782838	+	0.622225i	$0.786229\pi$
$548$		0			0
$549$	−6.28267			−0.268138
$550$		0			0
$551$	−5.49063			−0.233909
$552$		0			0
$553$		0			0
$554$		0			0
$555$	8.77801	+	6.95798i	0.372606	+	0.295350i
$556$		0			0
$557$			44.0114i			1.86482i	0.361399	+	0.932411i	$0.382299\pi$
$557$			44.0114i			1.86482i	−0.361399	+	0.932411i	$0.617701\pi$
$558$		0			0
$559$	−2.72666			−0.115325
$560$		0			0
$561$	−4.10270			−0.173216
$562$		0			0
$563$		−	7.43466i		−	0.313333i	−0.987652	−	0.156667i	$-0.949925\pi$
$563$		−	7.43466i		−	0.313333i	0.987652	−	0.156667i	$-0.0500748\pi$
$564$		0			0
$565$	−5.94865	+	7.50466i	−0.250262	+	0.315724i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.48130			−0.355555			−0.177777	−	0.984071i	$-0.556891\pi$
$569$	−8.48130			−0.355555			−0.177777	+	0.984071i	$0.556891\pi$
$570$		0			0
$571$	40.7826			1.70670			0.853350	−	0.521339i	$-0.174567\pi$
$571$	40.7826			1.70670			0.853350	+	0.521339i	$0.174567\pi$
$572$		0			0
$573$		−	11.1120i		−	0.464212i
$574$		0			0
$575$	23.0093	−	5.39470i	0.959555	−	0.224975i
$576$		0			0
$577$			1.57467i			0.0655545i	0.999463	+	0.0327773i	$0.0104352\pi$
$577$			1.57467i			0.0655545i	−0.999463	+	0.0327773i	$0.989565\pi$
$578$		0			0
$579$	−6.10270			−0.253620
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	11.3693i		−	0.470867i
$584$		0			0
$585$	−1.38900	+	1.75233i	−0.0574282	+	0.0724500i
$586$		0			0
$587$			27.8247i			1.14845i	0.818699	+	0.574223i	$0.194696\pi$
$587$			27.8247i			1.14845i	−0.818699	+	0.574223i	$0.805304\pi$
$588$		0			0
$589$	2.18675			0.0901034
$590$		0			0
$591$	21.3620			0.878716
$592$		0			0
$593$		−	32.6940i		−	1.34258i	−0.741195	−	0.671290i	$-0.765740\pi$
$593$		−	32.6940i		−	1.34258i	0.741195	−	0.671290i	$-0.234260\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	8.38538i		−	0.343191i
$598$		0			0
$599$	32.1400			1.31321			0.656603	−	0.754237i	$-0.271993\pi$
$599$	32.1400			1.31321			0.656603	+	0.754237i	$0.271993\pi$
$600$		0			0
$601$	40.8667			1.66699			0.833493	−	0.552530i	$-0.186337\pi$
$601$	40.8667			1.66699			0.833493	+	0.552530i	$0.186337\pi$
$602$		0			0
$603$		−	12.5653i		−	0.511700i
$604$		0			0
$605$	15.3083	+	12.1343i	0.622373	+	0.493330i
$606$		0			0
$607$			14.9907i			0.608453i	0.952600	+	0.304226i	$0.0983979\pi$
$607$			14.9907i			0.608453i	−0.952600	+	0.304226i	$0.901602\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.2313			−0.413915
$612$		0			0
$613$		−	13.5747i		−	0.548276i	−0.961690	−	0.274138i	$-0.911608\pi$
$613$		−	13.5747i		−	0.548276i	0.961690	−	0.274138i	$-0.0883925\pi$
$614$		0			0
$615$	8.03863	−	10.1413i	0.324149	−	0.408938i
$616$		0			0
$617$		−	4.69396i		−	0.188972i	−0.995526	−	0.0944859i	$-0.969879\pi$
$617$		−	4.69396i		−	0.188972i	0.995526	−	0.0944859i	$-0.0301207\pi$
$618$		0			0
$619$	20.8667			0.838702			0.419351	−	0.907824i	$-0.362258\pi$
$619$	20.8667			0.838702			0.419351	+	0.907824i	$0.362258\pi$
$620$		0			0
$621$	−4.72666			−0.189674
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−22.3947	+	11.1120i	−0.895788	+	0.444481i
$626$		0			0
$627$		−	1.09337i		−	0.0436652i
$628$		0			0
$629$	−13.6587			−0.544609
$630$		0			0
$631$	45.9533			1.82937			0.914685	−	0.404167i	$-0.132438\pi$
$631$	45.9533			1.82937			0.914685	+	0.404167i	$0.132438\pi$
$632$		0			0
$633$		−	1.27334i		−	0.0506109i
$634$		0			0
$635$	7.94139	−	10.0187i	0.315144	−	0.397578i
$636$		0			0
$637$			7.00000i			0.277350i
$638$		0			0
$639$	−4.77801			−0.189015
$640$		0			0
$641$	20.0187			0.790689			0.395345	−	0.918533i	$-0.370625\pi$
$641$	20.0187			0.790689			0.395345	+	0.918533i	$0.370625\pi$
$642$		0			0
$643$			26.5840i			1.04837i	0.851604	+	0.524185i	$0.175630\pi$
$643$			26.5840i			1.04837i	−0.851604	+	0.524185i	$0.824370\pi$
$644$		0			0
$645$	4.77801	+	3.78734i	0.188134	+	0.149126i
$646$		0			0
$647$			9.39470i			0.369344i	0.982800	+	0.184672i	$0.0591223\pi$
$647$			9.39470i			0.369344i	−0.982800	+	0.184672i	$0.940878\pi$
$648$		0			0
$649$	18.8294			0.739117
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	2.00000i		−	0.0782660i	−0.999234	−	0.0391330i	$-0.987540\pi$
$653$		−	2.00000i		−	0.0782660i	0.999234	−	0.0391330i	$-0.0124596\pi$
$654$		0			0
$655$	−9.73599	−	7.71733i	−0.380416	−	0.301541i
$656$		0			0
$657$		−	12.0187i		−	0.468892i
$658$		0			0
$659$	−47.6774			−1.85725			−0.928623	−	0.371024i	$-0.879007\pi$
$659$	−47.6774			−1.85725			−0.928623	+	0.371024i	$0.879007\pi$
$660$		0			0
$661$	−22.0959			−0.859432			−0.429716	−	0.902964i	$-0.641386\pi$
$661$	−22.0959			−0.859432			−0.429716	+	0.902964i	$0.641386\pi$
$662$		0			0
$663$		−	2.72666i		−	0.105895i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	35.7147i		−	1.38288i
$668$		0			0
$669$	16.4626			0.636482
$670$		0			0
$671$	9.45331			0.364941
$672$		0			0
$673$			48.6027i			1.87349i	0.350006	+	0.936747i	$0.386180\pi$
$673$			48.6027i			1.87349i	−0.350006	+	0.936747i	$0.613820\pi$
$674$		0			0
$675$	4.86799	−	1.14134i	0.187369	−	0.0439300i
$676$		0			0
$677$			10.4253i			0.400678i	0.979727	+	0.200339i	$0.0642043\pi$
$677$			10.4253i			0.400678i	−0.979727	+	0.200339i	$0.935796\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−12.2500			−0.469420
$682$		0			0
$683$			9.13795i			0.349654i	0.984599	+	0.174827i	$0.0559366\pi$
$683$			9.13795i			0.349654i	−0.984599	+	0.174827i	$0.944063\pi$
$684$		0			0
$685$	9.27203	−	11.6974i	0.354266	−	0.446933i
$686$		0			0
$687$			1.27334i			0.0485811i
$688$		0			0
$689$	7.55602			0.287861
$690$		0			0
$691$	33.7173			1.28267			0.641334	−	0.767262i	$-0.278381\pi$
$691$	33.7173			1.28267			0.641334	+	0.767262i	$0.278381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	34.1214	+	27.0466i	1.29430	+	1.02594i
$696$		0			0
$697$			15.7801i			0.597713i
$698$		0			0
$699$	18.3013			0.692220
$700$		0			0
$701$	−19.9813			−0.754685			−0.377342	−	0.926074i	$-0.623162\pi$
$701$	−19.9813			−0.754685			−0.377342	+	0.926074i	$0.623162\pi$
$702$		0			0
$703$		−	3.64006i		−	0.137288i
$704$		0			0
$705$	17.9287	+	14.2113i	0.675233	+	0.535230i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	43.3293			1.62727			0.813633	−	0.581378i	$-0.197486\pi$
$709$	43.3293			1.62727			0.813633	+	0.581378i	$0.197486\pi$
$710$		0			0
$711$	5.27334			0.197766
$712$		0			0
$713$			14.2241i			0.532695i
$714$		0			0
$715$	2.08998	−	2.63667i	0.0781610	−	0.0986059i
$716$		0			0
$717$		−	6.23132i		−	0.232713i
$718$		0			0
$719$	34.6867			1.29360			0.646798	−	0.762661i	$-0.276108\pi$
$719$	34.6867			1.29360			0.646798	+	0.762661i	$0.276108\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			19.5560i			0.727296i
$724$		0			0
$725$	8.62395	+	36.7826i	0.320286	+	1.36607i
$726$		0			0
$727$		−	5.39470i		−	0.200078i	−0.994983	−	0.100039i	$-0.968103\pi$
$727$		−	5.39470i		−	0.200078i	0.994983	−	0.100039i	$-0.0318968\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−7.43466			−0.274981
$732$		0			0
$733$			9.11203i			0.336561i	0.985739	+	0.168280i	$0.0538214\pi$
$733$			9.11203i			0.336561i	−0.985739	+	0.168280i	$0.946179\pi$
$734$		0			0
$735$	9.72303	−	12.2663i	0.358639	−	0.452450i
$736$		0			0
$737$			18.9066i			0.696435i
$738$		0			0
$739$	4.82936			0.177651			0.0888254	−	0.996047i	$-0.471689\pi$
$739$	4.82936			0.177651			0.0888254	+	0.996047i	$0.471689\pi$
$740$		0			0
$741$	0.726656			0.0266944
$742$		0			0
$743$			50.4087i			1.84931i	0.380801	+	0.924657i	$0.375648\pi$
$743$			50.4087i			1.84931i	−0.380801	+	0.924657i	$0.624352\pi$
$744$		0			0
$745$	−25.4333	−	20.1600i	−0.931805	−	0.738605i
$746$		0			0
$747$		−	7.78734i		−	0.284924i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.88797			0.178365			0.0891823	−	0.996015i	$-0.471575\pi$
$751$	4.88797			0.178365			0.0891823	+	0.996015i	$0.471575\pi$
$752$		0			0
$753$			26.4813i			0.965032i
$754$		0			0
$755$	7.82003	+	6.19863i	0.284600	+	0.225591i
$756$		0			0
$757$		−	32.2241i		−	1.17120i	−0.810599	−	0.585602i	$-0.800858\pi$
$757$		−	32.2241i		−	1.17120i	0.810599	−	0.585602i	$-0.199142\pi$
$758$		0			0
$759$	7.11203			0.258150
$760$		0			0
$761$	36.4740			1.32218			0.661091	−	0.750305i	$-0.270094\pi$
$761$	36.4740			1.32218			0.661091	+	0.750305i	$0.270094\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−3.78734	+	4.77801i	−0.136931	+	0.172749i
$766$		0			0
$767$			12.5140i			0.451854i
$768$		0			0
$769$	−13.1120			−0.472832			−0.236416	−	0.971652i	$-0.575973\pi$
$769$	−13.1120			−0.472832			−0.236416	+	0.971652i	$0.575973\pi$
$770$		0			0
$771$	−14.8294			−0.534066
$772$		0			0
$773$		−	12.5913i		−	0.452876i	−0.974026	−	0.226438i	$-0.927292\pi$
$773$		−	12.5913i		−	0.452876i	0.974026	−	0.226438i	$-0.0727081\pi$
$774$		0			0
$775$	−3.43466	−	14.6494i	−0.123376	−	0.526222i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.20541			−0.150674
$780$		0			0
$781$	7.18930			0.257253
$782$		0			0
$783$		−	7.55602i		−	0.270030i
$784$		0			0
$785$	−11.5306	+	14.5467i	−0.411544	+	0.519194i
$786$		0			0
$787$		−	36.9253i		−	1.31624i	−0.752911	−	0.658122i	$-0.771351\pi$
$787$		−	36.9253i		−	1.31624i	0.752911	−	0.658122i	$-0.228649\pi$
$788$		0			0
$789$	22.9507			0.817067
$790$		0			0
$791$		0			0
$792$		0			0
$793$			6.28267i			0.223104i
$794$		0			0
$795$	−13.2406	−	10.4953i	−0.469597	−	0.372231i
$796$		0			0
$797$			25.0466i			0.887198i	0.896225	+	0.443599i	$0.146298\pi$
$797$			25.0466i			0.887198i	−0.896225	+	0.443599i	$0.853702\pi$

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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1560.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.75233 - 1.38900i) q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-1.75233 - 1.38900i) q^{5} -1.00000 q^{9} +1.50466 q^{11} +1.00000i q^{13} +(1.38900 - 1.75233i) q^{15} +2.72666i q^{17} -0.726656 q^{19} -4.72666i q^{23} +(1.14134 + 4.86799i) q^{25} -1.00000i q^{27} +7.55602 q^{29} -3.00933 q^{31} +1.50466i q^{33} +5.00933i q^{37} -1.00000 q^{39} +5.78734 q^{41} +2.72666i q^{43} +(1.75233 + 1.38900i) q^{45} +10.2313i q^{47} +7.00000 q^{49} -2.72666 q^{51} -7.55602i q^{53} +(-2.63667 - 2.08998i) q^{55} -0.726656i q^{57} +12.5140 q^{59} +6.28267 q^{61} +(1.38900 - 1.75233i) q^{65} +12.5653i q^{67} +4.72666 q^{69} +4.77801 q^{71} +12.0187i q^{73} +(-4.86799 + 1.14134i) q^{75} -5.27334 q^{79} +1.00000 q^{81} +7.78734i q^{83} +(3.78734 - 4.77801i) q^{85} +7.55602i q^{87} -1.78734 q^{89} -3.00933i q^{93} +(1.27334 + 1.00933i) q^{95} +6.00000i q^{97} -1.50466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} - 12 q^{11} + 2 q^{15} + 4 q^{19} - 10 q^{25} + 20 q^{29} + 24 q^{31} - 6 q^{39} - 20 q^{41} + 42 q^{49} - 8 q^{51} - 20 q^{55} + 12 q^{59} + 4 q^{61} + 2 q^{65} + 20 q^{69} + 16 q^{71} - 4 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} + 44 q^{89} + 16 q^{95} + 12 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 - 12 * q^11 + 2 * q^15 + 4 * q^19 - 10 * q^25 + 20 * q^29 + 24 * q^31 - 6 * q^39 - 20 * q^41 + 42 * q^49 - 8 * q^51 - 20 * q^55 + 12 * q^59 + 4 * q^61 + 2 * q^65 + 20 * q^69 + 16 * q^71 - 4 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 + 44 * q^89 + 16 * q^95 + 12 * q^99

Introduction

L-functions

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