LMFDB - Embedded newform 3640.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23724.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 4x + 10 $ x^4 - x^3 - 7x^2 + 4x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.52350$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36807 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.60777 q^{9} +O(q^{10})$	$q-2.36807 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.60777 q^{9} +5.04701 q^{11} +1.00000 q^{13} +2.36807 q^{15} +1.60777 q^{17} -3.29691 q^{19} -2.36807 q^{21} +2.31086 q^{23} +1.00000 q^{25} +0.928833 q^{27} +1.32106 q^{29} -4.36807 q^{31} -11.9517 q^{33} -1.00000 q^{35} +10.6306 q^{37} -2.36807 q^{39} +0.249898 q^{41} +3.68914 q^{43} -2.60777 q^{45} -3.21554 q^{47} +1.00000 q^{49} -3.80731 q^{51} -4.73615 q^{53} -5.04701 q^{55} +7.80731 q^{57} -5.58361 q^{59} -0.142334 q^{61} +2.60777 q^{63} -1.00000 q^{65} -6.34391 q^{67} -5.47229 q^{69} -6.42528 q^{71} +3.35787 q^{73} -2.36807 q^{75} +5.04701 q^{77} -11.0801 q^{79} -10.0228 q^{81} -4.73615 q^{83} -1.60777 q^{85} -3.12838 q^{87} -7.27275 q^{89} +1.00000 q^{91} +10.3439 q^{93} +3.29691 q^{95} +3.52061 q^{97} +13.1614 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9	$ 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - q^{15} - q^{17} - 3 q^{19} + q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} + 13 q^{29} - 7 q^{31} - 12 q^{33} - 4 q^{35} - q^{37} + q^{39} + 9 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 4 q^{49} + 6 q^{51} + 2 q^{53} - 2 q^{55} + 10 q^{57} + 3 q^{59} + 3 q^{63} - 4 q^{65} + 3 q^{67} + 20 q^{69} - 2 q^{71} - 2 q^{73} + q^{75} + 2 q^{77} + 5 q^{79} - 4 q^{81} + 2 q^{83} + q^{85} + q^{87} - q^{89} + 4 q^{91} + 13 q^{93} + 3 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - q^15 - q^17 - 3 * q^19 + q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 + 13 * q^29 - 7 * q^31 - 12 * q^33 - 4 * q^35 - q^37 + q^39 + 9 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 4 * q^49 + 6 * q^51 + 2 * q^53 - 2 * q^55 + 10 * q^57 + 3 * q^59 + 3 * q^63 - 4 * q^65 + 3 * q^67 + 20 * q^69 - 2 * q^71 - 2 * q^73 + q^75 + 2 * q^77 + 5 * q^79 - 4 * q^81 + 2 * q^83 + q^85 + q^87 - q^89 + 4 * q^91 + 13 * q^93 + 3 * q^95 + 8 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.36807	−1.36721	−0.683604	−	0.729853i	$-0.739588\pi$
$3$	−2.36807	−1.36721	−0.683604	+	0.729853i	$0.739588\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.60777	0.869256
$10$	0	0
$11$	5.04701	1.52173	0.760865	−	0.648910i	$-0.224775\pi$
$11$	5.04701	1.52173	0.760865	+	0.648910i	$0.224775\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.36807	0.611434
$16$	0	0
$17$	1.60777	0.389941	0.194971	−	0.980809i	$-0.437539\pi$
$17$	1.60777	0.389941	0.194971	+	0.980809i	$0.437539\pi$
$18$	0	0
$19$	−3.29691	−0.756362	−0.378181	−	0.925732i	$-0.623450\pi$
$19$	−3.29691	−0.756362	−0.378181	+	0.925732i	$0.623450\pi$
$20$	0	0
$21$	−2.36807	−0.516756
$22$	0	0
$23$	2.31086	0.481848	0.240924	−	0.970544i	$-0.422550\pi$
$23$	2.31086	0.481848	0.240924	+	0.970544i	$0.422550\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.928833	0.178754
$28$	0	0
$29$	1.32106	0.245316	0.122658	−	0.992449i	$-0.460858\pi$
$29$	1.32106	0.245316	0.122658	+	0.992449i	$0.460858\pi$
$30$	0	0
$31$	−4.36807	−0.784529	−0.392265	−	0.919852i	$-0.628308\pi$
$31$	−4.36807	−0.784529	−0.392265	+	0.919852i	$0.628308\pi$
$32$	0	0
$33$	−11.9517	−2.08052
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	10.6306	1.74766	0.873831	−	0.486229i	$-0.161628\pi$
$37$	10.6306	1.74766	0.873831	+	0.486229i	$0.161628\pi$
$38$	0	0
$39$	−2.36807	−0.379195
$40$	0	0
$41$	0.249898	0.0390274	0.0195137	−	0.999810i	$-0.493788\pi$
$41$	0.249898	0.0390274	0.0195137	+	0.999810i	$0.493788\pi$
$42$	0	0
$43$	3.68914	0.562588	0.281294	−	0.959622i	$-0.409236\pi$
$43$	3.68914	0.562588	0.281294	+	0.959622i	$0.409236\pi$
$44$	0	0
$45$	−2.60777	−0.388743
$46$	0	0
$47$	−3.21554	−0.469034	−0.234517	−	0.972112i	$-0.575351\pi$
$47$	−3.21554	−0.469034	−0.234517	+	0.972112i	$0.575351\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.80731	−0.533130
$52$	0	0
$53$	−4.73615	−0.650560	−0.325280	−	0.945618i	$-0.605458\pi$
$53$	−4.73615	−0.650560	−0.325280	+	0.945618i	$0.605458\pi$
$54$	0	0
$55$	−5.04701	−0.680538
$56$	0	0
$57$	7.80731	1.03410
$58$	0	0
$59$	−5.58361	−0.726924	−0.363462	−	0.931609i	$-0.618405\pi$
$59$	−5.58361	−0.726924	−0.363462	+	0.931609i	$0.618405\pi$
$60$	0	0
$61$	−0.142334	−0.0182240	−0.00911200	−	0.999958i	$-0.502900\pi$
$61$	−0.142334	−0.0182240	−0.00911200	+	0.999958i	$0.502900\pi$
$62$	0	0
$63$	2.60777	0.328548
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−6.34391	−0.775032	−0.387516	−	0.921863i	$-0.626667\pi$
$67$	−6.34391	−0.775032	−0.387516	+	0.921863i	$0.626667\pi$
$68$	0	0
$69$	−5.47229	−0.658786
$70$	0	0
$71$	−6.42528	−0.762541	−0.381270	−	0.924464i	$-0.624513\pi$
$71$	−6.42528	−0.762541	−0.381270	+	0.924464i	$0.624513\pi$
$72$	0	0
$73$	3.35787	0.393009	0.196505	−	0.980503i	$-0.437041\pi$
$73$	3.35787	0.393009	0.196505	+	0.980503i	$0.437041\pi$
$74$	0	0
$75$	−2.36807	−0.273441
$76$	0	0
$77$	5.04701	0.575160
$78$	0	0
$79$	−11.0801	−1.24660	−0.623302	−	0.781981i	$-0.714209\pi$
$79$	−11.0801	−1.24660	−0.623302	+	0.781981i	$0.714209\pi$
$80$	0	0
$81$	−10.0228	−1.11365
$82$	0	0
$83$	−4.73615	−0.519859	−0.259930	−	0.965628i	$-0.583699\pi$
$83$	−4.73615	−0.519859	−0.259930	+	0.965628i	$0.583699\pi$
$84$	0	0
$85$	−1.60777	−0.174387
$86$	0	0
$87$	−3.12838	−0.335397
$88$	0	0
$89$	−7.27275	−0.770910	−0.385455	−	0.922727i	$-0.625955\pi$
$89$	−7.27275	−0.770910	−0.385455	+	0.922727i	$0.625955\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	10.3439	1.07261
$94$	0	0
$95$	3.29691	0.338255
$96$	0	0
$97$	3.52061	0.357464	0.178732	−	0.983898i	$-0.442801\pi$
$97$	3.52061	0.357464	0.178732	+	0.983898i	$0.442801\pi$
$98$	0	0
$99$	13.1614	1.32277
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	1.91863	0.189048	0.0945242	−	0.995523i	$-0.469867\pi$
$103$	1.91863	0.189048	0.0945242	+	0.995523i	$0.469867\pi$
$104$	0	0
$105$	2.36807	0.231100
$106$	0	0
$107$	11.6891	1.13003	0.565016	−	0.825080i	$-0.308870\pi$
$107$	11.6891	1.13003	0.565016	+	0.825080i	$0.308870\pi$
$108$	0	0
$109$	18.0940	1.73309	0.866546	−	0.499097i	$-0.166335\pi$
$109$	18.0940	1.73309	0.866546	+	0.499097i	$0.166335\pi$
$110$	0	0
$111$	−25.1741	−2.38942
$112$	0	0
$113$	8.25675	0.776730	0.388365	−	0.921506i	$-0.373040\pi$
$113$	8.25675	0.776730	0.388365	+	0.921506i	$0.373040\pi$
$114$	0	0
$115$	−2.31086	−0.215489
$116$	0	0
$117$	2.60777	0.241088
$118$	0	0
$119$	1.60777	0.147384
$120$	0	0
$121$	14.4723	1.31566
$122$	0	0
$123$	−0.591776	−0.0533586
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.18934	0.460480	0.230240	−	0.973134i	$-0.426049\pi$
$127$	5.18934	0.460480	0.230240	+	0.973134i	$0.426049\pi$
$128$	0	0
$129$	−8.73615	−0.769175
$130$	0	0
$131$	−3.37827	−0.295161	−0.147581	−	0.989050i	$-0.547149\pi$
$131$	−3.37827	−0.295161	−0.147581	+	0.989050i	$0.547149\pi$
$132$	0	0
$133$	−3.29691	−0.285878
$134$	0	0
$135$	−0.928833	−0.0799412
$136$	0	0
$137$	2.84167	0.242781	0.121390	−	0.992605i	$-0.461265\pi$
$137$	2.84167	0.242781	0.121390	+	0.992605i	$0.461265\pi$
$138$	0	0
$139$	7.95168	0.674453	0.337226	−	0.941424i	$-0.390511\pi$
$139$	7.95168	0.674453	0.337226	+	0.941424i	$0.390511\pi$
$140$	0	0
$141$	7.61462	0.641267
$142$	0	0
$143$	5.04701	0.422052
$144$	0	0
$145$	−1.32106	−0.109708
$146$	0	0
$147$	−2.36807	−0.195315
$148$	0	0
$149$	−2.09402	−0.171548	−0.0857742	−	0.996315i	$-0.527336\pi$
$149$	−2.09402	−0.171548	−0.0857742	+	0.996315i	$0.527336\pi$
$150$	0	0
$151$	5.64082	0.459043	0.229522	−	0.973304i	$-0.426284\pi$
$151$	5.64082	0.459043	0.229522	+	0.973304i	$0.426284\pi$
$152$	0	0
$153$	4.19269	0.338959
$154$	0	0
$155$	4.36807	0.350852
$156$	0	0
$157$	22.7450	1.81525	0.907626	−	0.419780i	$-0.137893\pi$
$157$	22.7450	1.81525	0.907626	+	0.419780i	$0.137893\pi$
$158$	0	0
$159$	11.2155	0.889450
$160$	0	0
$161$	2.31086	0.182121
$162$	0	0
$163$	−11.7202	−0.917993	−0.458997	−	0.888438i	$-0.651791\pi$
$163$	−11.7202	−0.917993	−0.458997	+	0.888438i	$0.651791\pi$
$164$	0	0
$165$	11.9517	0.930437
$166$	0	0
$167$	16.2364	1.25641	0.628203	−	0.778049i	$-0.283791\pi$
$167$	16.2364	1.25641	0.628203	+	0.778049i	$0.283791\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−8.59757	−0.657472
$172$	0	0
$173$	21.3872	1.62604	0.813018	−	0.582238i	$-0.197823\pi$
$173$	21.3872	1.62604	0.813018	+	0.582238i	$0.197823\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	13.2224	0.993856
$178$	0	0
$179$	−0.823305	−0.0615367	−0.0307684	−	0.999527i	$-0.509795\pi$
$179$	−0.823305	−0.0615367	−0.0307684	+	0.999527i	$0.509795\pi$
$180$	0	0
$181$	11.3579	0.844224	0.422112	−	0.906544i	$-0.361289\pi$
$181$	11.3579	0.844224	0.422112	+	0.906544i	$0.361289\pi$
$182$	0	0
$183$	0.337057	0.0249160
$184$	0	0
$185$	−10.6306	−0.781579
$186$	0	0
$187$	8.11442	0.593385
$188$	0	0
$189$	0.928833	0.0675627
$190$	0	0
$191$	−18.4400	−1.33427	−0.667135	−	0.744937i	$-0.732479\pi$
$191$	−18.4400	−1.33427	−0.667135	+	0.744937i	$0.732479\pi$
$192$	0	0
$193$	6.93773	0.499388	0.249694	−	0.968325i	$-0.419670\pi$
$193$	6.93773	0.499388	0.249694	+	0.968325i	$0.419670\pi$
$194$	0	0
$195$	2.36807	0.169581
$196$	0	0
$197$	27.3668	1.94980	0.974901	−	0.222641i	$-0.0714677\pi$
$197$	27.3668	1.94980	0.974901	+	0.222641i	$0.0714677\pi$
$198$	0	0
$199$	11.5663	0.819914	0.409957	−	0.912105i	$-0.365544\pi$
$199$	11.5663	0.819914	0.409957	+	0.912105i	$0.365544\pi$
$200$	0	0
$201$	15.0228	1.05963
$202$	0	0
$203$	1.32106	0.0927206
$204$	0	0
$205$	−0.249898	−0.0174536
$206$	0	0
$207$	6.02619	0.418849
$208$	0	0
$209$	−16.6395	−1.15098
$210$	0	0
$211$	4.48625	0.308846	0.154423	−	0.988005i	$-0.450648\pi$
$211$	4.48625	0.308846	0.154423	+	0.988005i	$0.450648\pi$
$212$	0	0
$213$	15.2155	1.04255
$214$	0	0
$215$	−3.68914	−0.251597
$216$	0	0
$217$	−4.36807	−0.296524
$218$	0	0
$219$	−7.95168	−0.537325
$220$	0	0
$221$	1.60777	0.108150
$222$	0	0
$223$	−23.2816	−1.55905	−0.779527	−	0.626369i	$-0.784540\pi$
$223$	−23.2816	−1.55905	−0.779527	+	0.626369i	$0.784540\pi$
$224$	0	0
$225$	2.60777	0.173851
$226$	0	0
$227$	17.9238	1.18964	0.594821	−	0.803858i	$-0.297223\pi$
$227$	17.9238	1.18964	0.594821	+	0.803858i	$0.297223\pi$
$228$	0	0
$229$	−20.9834	−1.38662	−0.693312	−	0.720638i	$-0.743849\pi$
$229$	−20.9834	−1.38662	−0.693312	+	0.720638i	$0.743849\pi$
$230$	0	0
$231$	−11.9517	−0.786363
$232$	0	0
$233$	−3.35787	−0.219981	−0.109991	−	0.993933i	$-0.535082\pi$
$233$	−3.35787	−0.219981	−0.109991	+	0.993933i	$0.535082\pi$
$234$	0	0
$235$	3.21554	0.209758
$236$	0	0
$237$	26.2384	1.70437
$238$	0	0
$239$	4.16853	0.269640	0.134820	−	0.990870i	$-0.456954\pi$
$239$	4.16853	0.269640	0.134820	+	0.990870i	$0.456954\pi$
$240$	0	0
$241$	8.93773	0.575729	0.287865	−	0.957671i	$-0.407055\pi$
$241$	8.93773	0.575729	0.287865	+	0.957671i	$0.407055\pi$
$242$	0	0
$243$	20.9483	1.34384
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−3.29691	−0.209777
$248$	0	0
$249$	11.2155	0.710756
$250$	0	0
$251$	23.9034	1.50877	0.754383	−	0.656434i	$-0.227936\pi$
$251$	23.9034	1.50877	0.754383	+	0.656434i	$0.227936\pi$
$252$	0	0
$253$	11.6629	0.733243
$254$	0	0
$255$	3.80731	0.238423
$256$	0	0
$257$	−14.0457	−0.876147	−0.438073	−	0.898939i	$-0.644339\pi$
$257$	−14.0457	−0.876147	−0.438073	+	0.898939i	$0.644339\pi$
$258$	0	0
$259$	10.6306	0.660554
$260$	0	0
$261$	3.44503	0.213242
$262$	0	0
$263$	1.68914	0.104157	0.0520783	−	0.998643i	$-0.483415\pi$
$263$	1.68914	0.104157	0.0520783	+	0.998643i	$0.483415\pi$
$264$	0	0
$265$	4.73615	0.290939
$266$	0	0
$267$	17.2224	1.05399
$268$	0	0
$269$	−10.3991	−0.634044	−0.317022	−	0.948418i	$-0.602683\pi$
$269$	−10.3991	−0.634044	−0.317022	+	0.948418i	$0.602683\pi$
$270$	0	0
$271$	−10.7903	−0.655461	−0.327731	−	0.944771i	$-0.606284\pi$
$271$	−10.7903	−0.655461	−0.327731	+	0.944771i	$0.606284\pi$
$272$	0	0
$273$	−2.36807	−0.143322
$274$	0	0
$275$	5.04701	0.304346
$276$	0	0
$277$	11.6629	0.700758	0.350379	−	0.936608i	$-0.386053\pi$
$277$	11.6629	0.700758	0.350379	+	0.936608i	$0.386053\pi$
$278$	0	0
$279$	−11.3909	−0.681957
$280$	0	0
$281$	29.5180	1.76090	0.880448	−	0.474143i	$-0.157242\pi$
$281$	29.5180	1.76090	0.880448	+	0.474143i	$0.157242\pi$
$282$	0	0
$283$	−27.8703	−1.65672	−0.828359	−	0.560197i	$-0.810725\pi$
$283$	−27.8703	−1.65672	−0.828359	+	0.560197i	$0.810725\pi$
$284$	0	0
$285$	−7.80731	−0.462465
$286$	0	0
$287$	0.249898	0.0147510
$288$	0	0
$289$	−14.4151	−0.847946
$290$	0	0
$291$	−8.33706	−0.488727
$292$	0	0
$293$	6.89888	0.403037	0.201519	−	0.979485i	$-0.435412\pi$
$293$	6.89888	0.403037	0.201519	+	0.979485i	$0.435412\pi$
$294$	0	0
$295$	5.58361	0.325090
$296$	0	0
$297$	4.68783	0.272015
$298$	0	0
$299$	2.31086	0.133641
$300$	0	0
$301$	3.68914	0.212638
$302$	0	0
$303$	−14.2084	−0.816253
$304$	0	0
$305$	0.142334	0.00815002
$306$	0	0
$307$	−1.74325	−0.0994923	−0.0497461	−	0.998762i	$-0.515841\pi$
$307$	−1.74325	−0.0994923	−0.0497461	+	0.998762i	$0.515841\pi$
$308$	0	0
$309$	−4.54346	−0.258468
$310$	0	0
$311$	11.1189	0.630495	0.315248	−	0.949009i	$-0.397912\pi$
$311$	11.1189	0.630495	0.315248	+	0.949009i	$0.397912\pi$
$312$	0	0
$313$	−21.5295	−1.21692	−0.608460	−	0.793585i	$-0.708212\pi$
$313$	−21.5295	−1.21692	−0.608460	+	0.793585i	$0.708212\pi$
$314$	0	0
$315$	−2.60777	−0.146931
$316$	0	0
$317$	29.9034	1.67954	0.839770	−	0.542942i	$-0.182690\pi$
$317$	29.9034	1.67954	0.839770	+	0.542942i	$0.182690\pi$
$318$	0	0
$319$	6.66742	0.373304
$320$	0	0
$321$	−27.6807	−1.54499
$322$	0	0
$323$	−5.30066	−0.294937
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−42.8479	−2.36950
$328$	0	0
$329$	−3.21554	−0.177278
$330$	0	0
$331$	−4.93259	−0.271120	−0.135560	−	0.990769i	$-0.543283\pi$
$331$	−4.93259	−0.271120	−0.135560	+	0.990769i	$0.543283\pi$
$332$	0	0
$333$	27.7222	1.51917
$334$	0	0
$335$	6.34391	0.346605
$336$	0	0
$337$	−3.24345	−0.176682	−0.0883410	−	0.996090i	$-0.528157\pi$
$337$	−3.24345	−0.176682	−0.0883410	+	0.996090i	$0.528157\pi$
$338$	0	0
$339$	−19.5526	−1.06195
$340$	0	0
$341$	−22.0457	−1.19384
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	5.47229	0.294618
$346$	0	0
$347$	25.2758	1.35688	0.678439	−	0.734656i	$-0.262657\pi$
$347$	25.2758	1.35688	0.678439	+	0.734656i	$0.262657\pi$
$348$	0	0
$349$	3.62614	0.194103	0.0970513	−	0.995279i	$-0.469059\pi$
$349$	3.62614	0.194103	0.0970513	+	0.995279i	$0.469059\pi$
$350$	0	0
$351$	0.928833	0.0495774
$352$	0	0
$353$	−15.1876	−0.808356	−0.404178	−	0.914680i	$-0.632442\pi$
$353$	−15.1876	−0.808356	−0.404178	+	0.914680i	$0.632442\pi$
$354$	0	0
$355$	6.42528	0.341019
$356$	0	0
$357$	−3.80731	−0.201504
$358$	0	0
$359$	−1.23766	−0.0653212	−0.0326606	−	0.999467i	$-0.510398\pi$
$359$	−1.23766	−0.0653212	−0.0326606	+	0.999467i	$0.510398\pi$
$360$	0	0
$361$	−8.13041	−0.427916
$362$	0	0
$363$	−34.2714	−1.79878
$364$	0	0
$365$	−3.35787	−0.175759
$366$	0	0
$367$	−6.79025	−0.354448	−0.177224	−	0.984171i	$-0.556712\pi$
$367$	−6.79025	−0.354448	−0.177224	+	0.984171i	$0.556712\pi$
$368$	0	0
$369$	0.651675	0.0339248
$370$	0	0
$371$	−4.73615	−0.245888
$372$	0	0
$373$	−15.8550	−0.820943	−0.410472	−	0.911873i	$-0.634636\pi$
$373$	−15.8550	−0.820943	−0.410472	+	0.911873i	$0.634636\pi$
$374$	0	0
$375$	2.36807	0.122287
$376$	0	0
$377$	1.32106	0.0680383
$378$	0	0
$379$	2.40488	0.123530	0.0617652	−	0.998091i	$-0.480327\pi$
$379$	2.40488	0.123530	0.0617652	+	0.998091i	$0.480327\pi$
$380$	0	0
$381$	−12.2887	−0.629571
$382$	0	0
$383$	30.7818	1.57288	0.786439	−	0.617667i	$-0.211922\pi$
$383$	30.7818	1.57288	0.786439	+	0.617667i	$0.211922\pi$
$384$	0	0
$385$	−5.04701	−0.257219
$386$	0	0
$387$	9.62042	0.489033
$388$	0	0
$389$	18.8233	0.954379	0.477190	−	0.878800i	$-0.341655\pi$
$389$	18.8233	0.954379	0.477190	+	0.878800i	$0.341655\pi$
$390$	0	0
$391$	3.71533	0.187892
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	11.0801	0.557498
$396$	0	0
$397$	−35.5459	−1.78400	−0.891999	−	0.452038i	$-0.850697\pi$
$397$	−35.5459	−1.78400	−0.891999	+	0.452038i	$0.850697\pi$
$398$	0	0
$399$	7.80731	0.390854
$400$	0	0
$401$	17.5180	0.874807	0.437403	−	0.899265i	$-0.355898\pi$
$401$	17.5180	0.874807	0.437403	+	0.899265i	$0.355898\pi$
$402$	0	0
$403$	−4.36807	−0.217589
$404$	0	0
$405$	10.0228	0.498039
$406$	0	0
$407$	53.6528	2.65947
$408$	0	0
$409$	35.2333	1.74218	0.871088	−	0.491127i	$-0.163415\pi$
$409$	35.2333	1.74218	0.871088	+	0.491127i	$0.163415\pi$
$410$	0	0
$411$	−6.72929	−0.331931
$412$	0	0
$413$	−5.58361	−0.274751
$414$	0	0
$415$	4.73615	0.232488
$416$	0	0
$417$	−18.8302	−0.922117
$418$	0	0
$419$	1.58051	0.0772129	0.0386064	−	0.999254i	$-0.487708\pi$
$419$	1.58051	0.0772129	0.0386064	+	0.999254i	$0.487708\pi$
$420$	0	0
$421$	−18.9521	−0.923668	−0.461834	−	0.886966i	$-0.652808\pi$
$421$	−18.9521	−0.923668	−0.461834	+	0.886966i	$0.652808\pi$
$422$	0	0
$423$	−8.38538	−0.407711
$424$	0	0
$425$	1.60777	0.0779882
$426$	0	0
$427$	−0.142334	−0.00688803
$428$	0	0
$429$	−11.9517	−0.577033
$430$	0	0
$431$	−16.3566	−0.787868	−0.393934	−	0.919139i	$-0.628886\pi$
$431$	−16.3566	−0.787868	−0.393934	+	0.919139i	$0.628886\pi$
$432$	0	0
$433$	19.0801	0.916929	0.458465	−	0.888713i	$-0.348400\pi$
$433$	19.0801	0.916929	0.458465	+	0.888713i	$0.348400\pi$
$434$	0	0
$435$	3.12838	0.149994
$436$	0	0
$437$	−7.61870	−0.364452
$438$	0	0
$439$	−28.4244	−1.35662	−0.678311	−	0.734775i	$-0.737288\pi$
$439$	−28.4244	−1.35662	−0.678311	+	0.734775i	$0.737288\pi$
$440$	0	0
$441$	2.60777	0.124179
$442$	0	0
$443$	−17.8493	−0.848044	−0.424022	−	0.905652i	$-0.639382\pi$
$443$	−17.8493	−0.848044	−0.424022	+	0.905652i	$0.639382\pi$
$444$	0	0
$445$	7.27275	0.344761
$446$	0	0
$447$	4.95878	0.234542
$448$	0	0
$449$	9.21554	0.434908	0.217454	−	0.976071i	$-0.430225\pi$
$449$	9.21554	0.434908	0.217454	+	0.976071i	$0.430225\pi$
$450$	0	0
$451$	1.26124	0.0593892
$452$	0	0
$453$	−13.3579	−0.627608
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	1.55741	0.0728528	0.0364264	−	0.999336i	$-0.488403\pi$
$457$	1.55741	0.0728528	0.0364264	+	0.999336i	$0.488403\pi$
$458$	0	0
$459$	1.49335	0.0697035
$460$	0	0
$461$	27.8878	1.29886	0.649432	−	0.760420i	$-0.275007\pi$
$461$	27.8878	1.29886	0.649432	+	0.760420i	$0.275007\pi$
$462$	0	0
$463$	10.7913	0.501515	0.250758	−	0.968050i	$-0.419320\pi$
$463$	10.7913	0.501515	0.250758	+	0.968050i	$0.419320\pi$
$464$	0	0
$465$	−10.3439	−0.479688
$466$	0	0
$467$	7.69599	0.356128	0.178064	−	0.984019i	$-0.443017\pi$
$467$	7.69599	0.356128	0.178064	+	0.984019i	$0.443017\pi$
$468$	0	0
$469$	−6.34391	−0.292935
$470$	0	0
$471$	−53.8619	−2.48183
$472$	0	0
$473$	18.6191	0.856107
$474$	0	0
$475$	−3.29691	−0.151272
$476$	0	0
$477$	−12.3508	−0.565503
$478$	0	0
$479$	20.0488	0.916053	0.458027	−	0.888939i	$-0.348556\pi$
$479$	20.0488	0.916053	0.458027	+	0.888939i	$0.348556\pi$
$480$	0	0
$481$	10.6306	0.484714
$482$	0	0
$483$	−5.47229	−0.248998
$484$	0	0
$485$	−3.52061	−0.159863
$486$	0	0
$487$	3.26868	0.148118	0.0740589	−	0.997254i	$-0.476405\pi$
$487$	3.26868	0.148118	0.0740589	+	0.997254i	$0.476405\pi$
$488$	0	0
$489$	27.7542	1.25509
$490$	0	0
$491$	36.3761	1.64163	0.820814	−	0.571195i	$-0.193520\pi$
$491$	36.3761	1.64163	0.820814	+	0.571195i	$0.193520\pi$
$492$	0	0
$493$	2.12397	0.0956586
$494$	0	0
$495$	−13.1614	−0.591562
$496$	0	0
$497$	−6.42528	−0.288213
$498$	0	0
$499$	3.68914	0.165148	0.0825742	−	0.996585i	$-0.473686\pi$
$499$	3.68914	0.165148	0.0825742	+	0.996585i	$0.473686\pi$
$500$	0	0
$501$	−38.4489	−1.71777
$502$	0	0
$503$	5.50600	0.245500	0.122750	−	0.992438i	$-0.460829\pi$
$503$	5.50600	0.245500	0.122750	+	0.992438i	$0.460829\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−2.36807	−0.105170
$508$	0	0
$509$	2.59447	0.114998	0.0574988	−	0.998346i	$-0.481687\pi$
$509$	2.59447	0.114998	0.0574988	+	0.998346i	$0.481687\pi$
$510$	0	0
$511$	3.35787	0.148543
$512$	0	0
$513$	−3.06227	−0.135203
$514$	0	0
$515$	−1.91863	−0.0845450
$516$	0	0
$517$	−16.2288	−0.713744
$518$	0	0
$519$	−50.6464	−2.22313
$520$	0	0
$521$	−2.83278	−0.124106	−0.0620532	−	0.998073i	$-0.519765\pi$
$521$	−2.83278	−0.124106	−0.0620532	+	0.998073i	$0.519765\pi$
$522$	0	0
$523$	8.12021	0.355072	0.177536	−	0.984114i	$-0.443187\pi$
$523$	8.12021	0.355072	0.177536	+	0.984114i	$0.443187\pi$
$524$	0	0
$525$	−2.36807	−0.103351
$526$	0	0
$527$	−7.02285	−0.305920
$528$	0	0
$529$	−17.6599	−0.767822
$530$	0	0
$531$	−14.5608	−0.631883
$532$	0	0
$533$	0.249898	0.0108243
$534$	0	0
$535$	−11.6891	−0.505365
$536$	0	0
$537$	1.94965	0.0841335
$538$	0	0
$539$	5.04701	0.217390
$540$	0	0
$541$	−24.5251	−1.05442	−0.527208	−	0.849736i	$-0.676761\pi$
$541$	−24.5251	−1.05442	−0.527208	+	0.849736i	$0.676761\pi$
$542$	0	0
$543$	−26.8963	−1.15423
$544$	0	0
$545$	−18.0940	−0.775062
$546$	0	0
$547$	14.2168	0.607868	0.303934	−	0.952693i	$-0.401700\pi$
$547$	14.2168	0.607868	0.303934	+	0.952693i	$0.401700\pi$
$548$	0	0
$549$	−0.371174	−0.0158413
$550$	0	0
$551$	−4.35543	−0.185547
$552$	0	0
$553$	−11.0801	−0.471172
$554$	0	0
$555$	25.1741	1.06858
$556$	0	0
$557$	−4.10349	−0.173871	−0.0869353	−	0.996214i	$-0.527707\pi$
$557$	−4.10349	−0.173871	−0.0869353	+	0.996214i	$0.527707\pi$
$558$	0	0
$559$	3.68914	0.156034
$560$	0	0
$561$	−19.2155	−0.811281
$562$	0	0
$563$	−12.0446	−0.507621	−0.253810	−	0.967254i	$-0.581684\pi$
$563$	−12.0446	−0.507621	−0.253810	+	0.967254i	$0.581684\pi$
$564$	0	0
$565$	−8.25675	−0.347364
$566$	0	0
$567$	−10.0228	−0.420920
$568$	0	0
$569$	−9.17204	−0.384512	−0.192256	−	0.981345i	$-0.561580\pi$
$569$	−9.17204	−0.384512	−0.192256	+	0.981345i	$0.561580\pi$
$570$	0	0
$571$	11.9565	0.500364	0.250182	−	0.968199i	$-0.419510\pi$
$571$	11.9565	0.500364	0.250182	+	0.968199i	$0.419510\pi$
$572$	0	0
$573$	43.6672	1.82422
$574$	0	0
$575$	2.31086	0.0963696
$576$	0	0
$577$	−35.5459	−1.47980	−0.739898	−	0.672719i	$-0.765126\pi$
$577$	−35.5459	−1.47980	−0.739898	+	0.672719i	$0.765126\pi$
$578$	0	0
$579$	−16.4290	−0.682768
$580$	0	0
$581$	−4.73615	−0.196488
$582$	0	0
$583$	−23.9034	−0.989976
$584$	0	0
$585$	−2.60777	−0.107818
$586$	0	0
$587$	−30.4373	−1.25628	−0.628140	−	0.778100i	$-0.716184\pi$
$587$	−30.4373	−1.25628	−0.628140	+	0.778100i	$0.716184\pi$
$588$	0	0
$589$	14.4011	0.593388
$590$	0	0
$591$	−64.8065	−2.66578
$592$	0	0
$593$	−11.9197	−0.489483	−0.244742	−	0.969588i	$-0.578703\pi$
$593$	−11.9197	−0.489483	−0.244742	+	0.969588i	$0.578703\pi$
$594$	0	0
$595$	−1.60777	−0.0659121
$596$	0	0
$597$	−27.3899	−1.12099
$598$	0	0
$599$	25.7011	1.05012	0.525060	−	0.851065i	$-0.324043\pi$
$599$	25.7011	1.05012	0.525060	+	0.851065i	$0.324043\pi$
$600$	0	0
$601$	−20.3712	−0.830958	−0.415479	−	0.909603i	$-0.636386\pi$
$601$	−20.3712	−0.830958	−0.415479	+	0.909603i	$0.636386\pi$
$602$	0	0
$603$	−16.5435	−0.673702
$604$	0	0
$605$	−14.4723	−0.588382
$606$	0	0
$607$	44.6746	1.81329	0.906643	−	0.421899i	$-0.138636\pi$
$607$	44.6746	1.81329	0.906643	+	0.421899i	$0.138636\pi$
$608$	0	0
$609$	−3.12838	−0.126768
$610$	0	0
$611$	−3.21554	−0.130087
$612$	0	0
$613$	2.43369	0.0982959	0.0491480	−	0.998792i	$-0.484349\pi$
$613$	2.43369	0.0982959	0.0491480	+	0.998792i	$0.484349\pi$
$614$	0	0
$615$	0.591776	0.0238627
$616$	0	0
$617$	35.9102	1.44569	0.722846	−	0.691010i	$-0.242834\pi$
$617$	35.9102	1.44569	0.722846	+	0.691010i	$0.242834\pi$
$618$	0	0
$619$	33.1677	1.33312	0.666561	−	0.745450i	$-0.267765\pi$
$619$	33.1677	1.33312	0.666561	+	0.745450i	$0.267765\pi$
$620$	0	0
$621$	2.14641	0.0861323
$622$	0	0
$623$	−7.27275	−0.291376
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	39.4036	1.57363
$628$	0	0
$629$	17.0916	0.681486
$630$	0	0
$631$	−31.8452	−1.26774	−0.633868	−	0.773441i	$-0.718534\pi$
$631$	−31.8452	−1.26774	−0.633868	+	0.773441i	$0.718534\pi$
$632$	0	0
$633$	−10.6238	−0.422257
$634$	0	0
$635$	−5.18934	−0.205933
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−16.7556	−0.662843
$640$	0	0
$641$	−24.5619	−0.970137	−0.485068	−	0.874476i	$-0.661205\pi$
$641$	−24.5619	−0.970137	−0.485068	+	0.874476i	$0.661205\pi$
$642$	0	0
$643$	−22.9521	−0.905142	−0.452571	−	0.891728i	$-0.649493\pi$
$643$	−22.9521	−0.905142	−0.452571	+	0.891728i	$0.649493\pi$
$644$	0	0
$645$	8.73615	0.343985
$646$	0	0
$647$	−22.4138	−0.881176	−0.440588	−	0.897709i	$-0.645230\pi$
$647$	−22.4138	−0.881176	−0.440588	+	0.897709i	$0.645230\pi$
$648$	0	0
$649$	−28.1805	−1.10618
$650$	0	0
$651$	10.3439	0.405410
$652$	0	0
$653$	−24.7539	−0.968696	−0.484348	−	0.874875i	$-0.660943\pi$
$653$	−24.7539	−0.968696	−0.484348	+	0.874875i	$0.660943\pi$
$654$	0	0
$655$	3.37827	0.132000
$656$	0	0
$657$	8.75655	0.341626
$658$	0	0
$659$	−14.1329	−0.550538	−0.275269	−	0.961367i	$-0.588767\pi$
$659$	−14.1329	−0.550538	−0.275269	+	0.961367i	$0.588767\pi$
$660$	0	0
$661$	19.4145	0.755136	0.377568	−	0.925982i	$-0.376760\pi$
$661$	19.4145	0.755136	0.377568	+	0.925982i	$0.376760\pi$
$662$	0	0
$663$	−3.80731	−0.147864
$664$	0	0
$665$	3.29691	0.127849
$666$	0	0
$667$	3.05280	0.118205
$668$	0	0
$669$	55.1326	2.13155
$670$	0	0
$671$	−0.718361	−0.0277320
$672$	0	0
$673$	49.9211	1.92432	0.962159	−	0.272487i	$-0.0878462\pi$
$673$	49.9211	1.92432	0.962159	+	0.272487i	$0.0878462\pi$
$674$	0	0
$675$	0.928833	0.0357508
$676$	0	0
$677$	18.1880	0.699023	0.349511	−	0.936932i	$-0.386348\pi$
$677$	18.1880	0.699023	0.349511	+	0.936932i	$0.386348\pi$
$678$	0	0
$679$	3.52061	0.135109
$680$	0	0
$681$	−42.4448	−1.62649
$682$	0	0
$683$	49.9879	1.91273	0.956367	−	0.292168i	$-0.0943766\pi$
$683$	49.9879	1.91273	0.956367	+	0.292168i	$0.0943766\pi$
$684$	0	0
$685$	−2.84167	−0.108575
$686$	0	0
$687$	49.6903	1.89580
$688$	0	0
$689$	−4.73615	−0.180433
$690$	0	0
$691$	−9.75386	−0.371054	−0.185527	−	0.982639i	$-0.559399\pi$
$691$	−9.75386	−0.371054	−0.185527	+	0.982639i	$0.559399\pi$
$692$	0	0
$693$	13.1614	0.499961
$694$	0	0
$695$	−7.95168	−0.301624
$696$	0	0
$697$	0.401777	0.0152184
$698$	0	0
$699$	7.95168	0.300760
$700$	0	0
$701$	−8.10756	−0.306218	−0.153109	−	0.988209i	$-0.548929\pi$
$701$	−8.10756	−0.306218	−0.153109	+	0.988209i	$0.548929\pi$
$702$	0	0
$703$	−35.0481	−1.32187
$704$	0	0
$705$	−7.61462	−0.286783
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	5.13523	0.192858	0.0964289	−	0.995340i	$-0.469258\pi$
$709$	5.13523	0.192858	0.0964289	+	0.995340i	$0.469258\pi$
$710$	0	0
$711$	−28.8942	−1.08362
$712$	0	0
$713$	−10.0940	−0.378024
$714$	0	0
$715$	−5.04701	−0.188747
$716$	0	0
$717$	−9.87138	−0.368653
$718$	0	0
$719$	31.9034	1.18979	0.594897	−	0.803802i	$-0.297193\pi$
$719$	31.9034	1.18979	0.594897	+	0.803802i	$0.297193\pi$
$720$	0	0
$721$	1.91863	0.0714536
$722$	0	0
$723$	−21.1652	−0.787142
$724$	0	0
$725$	1.32106	0.0490631
$726$	0	0
$727$	−26.3402	−0.976902	−0.488451	−	0.872591i	$-0.662438\pi$
$727$	−26.3402	−0.976902	−0.488451	+	0.872591i	$0.662438\pi$
$728$	0	0
$729$	−19.5386	−0.723653
$730$	0	0
$731$	5.93128	0.219376
$732$	0	0
$733$	5.89929	0.217895	0.108948	−	0.994047i	$-0.465252\pi$
$733$	5.89929	0.217895	0.108948	+	0.994047i	$0.465252\pi$
$734$	0	0
$735$	2.36807	0.0873477
$736$	0	0
$737$	−32.0178	−1.17939
$738$	0	0
$739$	15.8697	0.583775	0.291887	−	0.956453i	$-0.405717\pi$
$739$	15.8697	0.583775	0.291887	+	0.956453i	$0.405717\pi$
$740$	0	0
$741$	7.80731	0.286809
$742$	0	0
$743$	9.55945	0.350702	0.175351	−	0.984506i	$-0.443894\pi$
$743$	9.55945	0.350702	0.175351	+	0.984506i	$0.443894\pi$
$744$	0	0
$745$	2.09402	0.0767188
$746$	0	0
$747$	−12.3508	−0.451891
$748$	0	0
$749$	11.6891	0.427112
$750$	0	0
$751$	−19.7863	−0.722011	−0.361005	−	0.932564i	$-0.617566\pi$
$751$	−19.7863	−0.722011	−0.361005	+	0.932564i	$0.617566\pi$
$752$	0	0
$753$	−56.6049	−2.06280
$754$	0	0
$755$	−5.64082	−0.205290
$756$	0	0
$757$	7.64923	0.278016	0.139008	−	0.990291i	$-0.455609\pi$
$757$	7.64923	0.278016	0.139008	+	0.990291i	$0.455609\pi$
$758$	0	0
$759$	−27.6187	−1.00250
$760$	0	0
$761$	−8.55701	−0.310191	−0.155096	−	0.987899i	$-0.549569\pi$
$761$	−8.55701	−0.310191	−0.155096	+	0.987899i	$0.549569\pi$
$762$	0	0
$763$	18.0940	0.655047
$764$	0	0
$765$	−4.19269	−0.151587
$766$	0	0
$767$	−5.58361	−0.201612
$768$	0	0
$769$	−42.4626	−1.53124	−0.765620	−	0.643293i	$-0.777568\pi$
$769$	−42.4626	−1.53124	−0.765620	+	0.643293i	$0.777568\pi$
$770$	0	0
$771$	33.2612	1.19787
$772$	0	0
$773$	−25.7290	−0.925409	−0.462705	−	0.886512i	$-0.653121\pi$
$773$	−25.7290	−0.925409	−0.462705	+	0.886512i	$0.653121\pi$
$774$	0	0
$775$	−4.36807	−0.156906
$776$	0	0
$777$	−25.1741	−0.903115
$778$	0	0
$779$	−0.823889	−0.0295189
$780$	0	0
$781$	−32.4285	−1.16038
$782$	0	0
$783$	1.22705	0.0438511
$784$	0	0
$785$	−22.7450	−0.811805
$786$	0	0
$787$	30.9038	1.10160	0.550800	−	0.834637i	$-0.314323\pi$
$787$	30.9038	1.10160	0.550800	+	0.834637i	$0.314323\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	8.25675	0.293576
$792$	0	0
$793$	−0.142334	−0.00505443
$794$	0	0
$795$	−11.2155	−0.397774
$796$	0	0
$797$	51.2585	1.81567	0.907835	−	0.419327i	$-0.137734\pi$
$797$	51.2585	1.81567	0.907835	+	0.419327i	$0.137734\pi$
$798$	0	0
$799$	−5.16984	−0.182896
$800$	0	0
$801$	−18.9656	−0.670118
$802$	0	0
$803$	16.9472	0.598054
$804$	0	0
$805$	−2.31086	−0.0814472
$806$	0	0
$807$	24.6258	0.866869
$808$	0	0
$809$	−19.9701	−0.702112	−0.351056	−	0.936354i	$-0.614177\pi$
$809$	−19.9701	−0.702112	−0.351056	+	0.936354i	$0.614177\pi$
$810$	0	0
$811$	20.8218	0.731151	0.365575	−	0.930782i	$-0.380872\pi$
$811$	20.8218	0.731151	0.365575	+	0.930782i	$0.380872\pi$
$812$	0	0
$813$	25.5521	0.896151
$814$	0	0
$815$	11.7202	0.410539
$816$	0	0
$817$	−12.1627	−0.425520
$818$	0	0
$819$	2.60777	0.0911228
$820$	0	0
$821$	−14.7641	−0.515269	−0.257635	−	0.966242i	$-0.582943\pi$
$821$	−14.7641	−0.515269	−0.257635	+	0.966242i	$0.582943\pi$
$822$	0	0
$823$	9.81107	0.341992	0.170996	−	0.985272i	$-0.445301\pi$
$823$	9.81107	0.341992	0.170996	+	0.985272i	$0.445301\pi$
$824$	0	0
$825$	−11.9517	−0.416104
$826$	0	0
$827$	−26.3344	−0.915738	−0.457869	−	0.889020i	$-0.651387\pi$
$827$	−26.3344	−0.915738	−0.457869	+	0.889020i	$0.651387\pi$
$828$	0	0
$829$	−13.0453	−0.453082	−0.226541	−	0.974002i	$-0.572742\pi$
$829$	−13.0453	−0.453082	−0.226541	+	0.974002i	$0.572742\pi$
$830$	0	0
$831$	−27.6187	−0.958082
$832$	0	0
$833$	1.60777	0.0557059
$834$	0	0
$835$	−16.2364	−0.561882
$836$	0	0
$837$	−4.05721	−0.140238
$838$	0	0
$839$	−42.0719	−1.45248	−0.726242	−	0.687440i	$-0.758735\pi$
$839$	−42.0719	−1.45248	−0.726242	+	0.687440i	$0.758735\pi$
$840$	0	0
$841$	−27.2548	−0.939820
$842$	0	0
$843$	−69.9007	−2.40751
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	14.4723	0.497274
$848$	0	0
$849$	65.9989	2.26508
$850$	0	0
$851$	24.5659	0.842108
$852$	0	0
$853$	49.4444	1.69294	0.846472	−	0.532434i	$-0.178722\pi$
$853$	49.4444	1.69294	0.846472	+	0.532434i	$0.178722\pi$
$854$	0	0
$855$	8.59757	0.294031
$856$	0	0
$857$	−32.9194	−1.12450	−0.562252	−	0.826966i	$-0.690065\pi$
$857$	−32.9194	−1.12450	−0.562252	+	0.826966i	$0.690065\pi$
$858$	0	0
$859$	−22.6599	−0.773146	−0.386573	−	0.922259i	$-0.626341\pi$
$859$	−22.6599	−0.773146	−0.386573	+	0.922259i	$0.626341\pi$
$860$	0	0
$861$	−0.591776	−0.0201677
$862$	0	0
$863$	17.3967	0.592191	0.296095	−	0.955158i	$-0.404315\pi$
$863$	17.3967	0.592191	0.296095	+	0.955158i	$0.404315\pi$
$864$	0	0
$865$	−21.3872	−0.727185
$866$	0	0
$867$	34.1360	1.15932
$868$	0	0
$869$	−55.9211	−1.89700
$870$	0	0
$871$	−6.34391	−0.214955
$872$	0	0
$873$	9.18093	0.310727
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−20.8029	−0.702464	−0.351232	−	0.936288i	$-0.614237\pi$
$877$	−20.8029	−0.702464	−0.351232	+	0.936288i	$0.614237\pi$
$878$	0	0
$879$	−16.3371	−0.551035
$880$	0	0
$881$	22.4515	0.756409	0.378205	−	0.925722i	$-0.376542\pi$
$881$	22.4515	0.756409	0.378205	+	0.925722i	$0.376542\pi$
$882$	0	0
$883$	−22.5309	−0.758224	−0.379112	−	0.925351i	$-0.623771\pi$
$883$	−22.5309	−0.758224	−0.379112	+	0.925351i	$0.623771\pi$
$884$	0	0
$885$	−13.2224	−0.444466
$886$	0	0
$887$	58.7162	1.97150	0.985749	−	0.168223i	$-0.0538030\pi$
$887$	58.7162	1.97150	0.985749	+	0.168223i	$0.0538030\pi$
$888$	0	0
$889$	5.18934	0.174045
$890$	0	0
$891$	−50.5854	−1.69467
$892$	0	0
$893$	10.6013	0.354760
$894$	0	0
$895$	0.823305	0.0275201
$896$	0	0
$897$	−5.47229	−0.182714
$898$	0	0
$899$	−5.77051	−0.192457
$900$	0	0
$901$	−7.61462	−0.253680
$902$	0	0
$903$	−8.73615	−0.290721
$904$	0	0
$905$	−11.3579	−0.377548
$906$	0	0
$907$	27.5264	0.913999	0.457000	−	0.889467i	$-0.348924\pi$
$907$	27.5264	0.913999	0.457000	+	0.889467i	$0.348924\pi$
$908$	0	0
$909$	15.6466	0.518965
$910$	0	0
$911$	−19.9559	−0.661169	−0.330585	−	0.943776i	$-0.607246\pi$
$911$	−19.9559	−0.661169	−0.330585	+	0.943776i	$0.607246\pi$
$912$	0	0
$913$	−23.9034	−0.791086
$914$	0	0
$915$	−0.337057	−0.0111428
$916$	0	0
$917$	−3.37827	−0.111560
$918$	0	0
$919$	−5.92646	−0.195496	−0.0977479	−	0.995211i	$-0.531164\pi$
$919$	−5.92646	−0.195496	−0.0977479	+	0.995211i	$0.531164\pi$
$920$	0	0
$921$	4.12813	0.136027
$922$	0	0
$923$	−6.42528	−0.211491
$924$	0	0
$925$	10.6306	0.349533
$926$	0	0
$927$	5.00335	0.164331
$928$	0	0
$929$	0.213745	0.00701275	0.00350637	−	0.999994i	$-0.498884\pi$
$929$	0.213745	0.00701275	0.00350637	+	0.999994i	$0.498884\pi$
$930$	0	0
$931$	−3.29691	−0.108052
$932$	0	0
$933$	−26.3304	−0.862017
$934$	0	0
$935$	−8.11442	−0.265370
$936$	0	0
$937$	25.6992	0.839555	0.419778	−	0.907627i	$-0.362108\pi$
$937$	25.6992	0.839555	0.419778	+	0.907627i	$0.362108\pi$
$938$	0	0
$939$	50.9834	1.66378
$940$	0	0
$941$	−16.5257	−0.538724	−0.269362	−	0.963039i	$-0.586813\pi$
$941$	−16.5257	−0.538724	−0.269362	+	0.963039i	$0.586813\pi$
$942$	0	0
$943$	0.577479	0.0188053
$944$	0	0
$945$	−0.928833	−0.0302149
$946$	0	0
$947$	−1.26320	−0.0410485	−0.0205243	−	0.999789i	$-0.506534\pi$
$947$	−1.26320	−0.0410485	−0.0205243	+	0.999789i	$0.506534\pi$
$948$	0	0
$949$	3.35787	0.109001
$950$	0	0
$951$	−70.8133	−2.29628
$952$	0	0
$953$	57.7238	1.86986	0.934929	−	0.354836i	$-0.115463\pi$
$953$	57.7238	1.86986	0.934929	+	0.354836i	$0.115463\pi$
$954$	0	0
$955$	18.4400	0.596703
$956$	0	0
$957$	−15.7889	−0.510384
$958$	0	0
$959$	2.84167	0.0917624
$960$	0	0
$961$	−11.9199	−0.384514
$962$	0	0
$963$	30.4826	0.982287
$964$	0	0
$965$	−6.93773	−0.223333
$966$	0	0
$967$	−13.3184	−0.428292	−0.214146	−	0.976802i	$-0.568697\pi$
$967$	−13.3184	−0.428292	−0.214146	+	0.976802i	$0.568697\pi$
$968$	0	0
$969$	12.5524	0.403240
$970$	0	0
$971$	−23.1747	−0.743712	−0.371856	−	0.928290i	$-0.621279\pi$
$971$	−23.1747	−0.743712	−0.371856	+	0.928290i	$0.621279\pi$
$972$	0	0
$973$	7.95168	0.254919
$974$	0	0
$975$	−2.36807	−0.0758390
$976$	0	0
$977$	−17.1537	−0.548795	−0.274397	−	0.961616i	$-0.588478\pi$
$977$	−17.1537	−0.548795	−0.274397	+	0.961616i	$0.588478\pi$
$978$	0	0
$979$	−36.7056	−1.17312
$980$	0	0
$981$	47.1850	1.50650
$982$	0	0
$983$	49.0156	1.56335	0.781677	−	0.623683i	$-0.214365\pi$
$983$	49.0156	1.56335	0.781677	+	0.623683i	$0.214365\pi$
$984$	0	0
$985$	−27.3668	−0.871978
$986$	0	0
$987$	7.61462	0.242376
$988$	0	0
$989$	8.52509	0.271082
$990$	0	0
$991$	22.7683	0.723259	0.361629	−	0.932322i	$-0.382221\pi$
$991$	22.7683	0.723259	0.361629	+	0.932322i	$0.382221\pi$
$992$	0	0
$993$	11.6807	0.370677
$994$	0	0
$995$	−11.5663	−0.366677
$996$	0	0
$997$	−36.4796	−1.15532	−0.577660	−	0.816278i	$-0.696034\pi$
$997$	−36.4796	−1.15532	−0.577660	+	0.816278i	$0.696034\pi$
$998$	0	0
$999$	9.87407	0.312402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.w.1.1	✓	4
4.3	odd	2		7280.2.a.bs.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.w.1.1	✓	4	1.1	even	1	trivial
7280.2.a.bs.1.4		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36807 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.60777 q^{9} +O(q^{10})\)	\(q-2.36807 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.60777 q^{9} +5.04701 q^{11} +1.00000 q^{13} +2.36807 q^{15} +1.60777 q^{17} -3.29691 q^{19} -2.36807 q^{21} +2.31086 q^{23} +1.00000 q^{25} +0.928833 q^{27} +1.32106 q^{29} -4.36807 q^{31} -11.9517 q^{33} -1.00000 q^{35} +10.6306 q^{37} -2.36807 q^{39} +0.249898 q^{41} +3.68914 q^{43} -2.60777 q^{45} -3.21554 q^{47} +1.00000 q^{49} -3.80731 q^{51} -4.73615 q^{53} -5.04701 q^{55} +7.80731 q^{57} -5.58361 q^{59} -0.142334 q^{61} +2.60777 q^{63} -1.00000 q^{65} -6.34391 q^{67} -5.47229 q^{69} -6.42528 q^{71} +3.35787 q^{73} -2.36807 q^{75} +5.04701 q^{77} -11.0801 q^{79} -10.0228 q^{81} -4.73615 q^{83} -1.60777 q^{85} -3.12838 q^{87} -7.27275 q^{89} +1.00000 q^{91} +10.3439 q^{93} +3.29691 q^{95} +3.52061 q^{97} +13.1614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9	\( 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - q^{15} - q^{17} - 3 q^{19} + q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} + 13 q^{29} - 7 q^{31} - 12 q^{33} - 4 q^{35} - q^{37} + q^{39} + 9 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 4 q^{49} + 6 q^{51} + 2 q^{53} - 2 q^{55} + 10 q^{57} + 3 q^{59} + 3 q^{63} - 4 q^{65} + 3 q^{67} + 20 q^{69} - 2 q^{71} - 2 q^{73} + q^{75} + 2 q^{77} + 5 q^{79} - 4 q^{81} + 2 q^{83} + q^{85} + q^{87} - q^{89} + 4 q^{91} + 13 q^{93} + 3 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - q^15 - q^17 - 3 * q^19 + q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 + 13 * q^29 - 7 * q^31 - 12 * q^33 - 4 * q^35 - q^37 + q^39 + 9 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 4 * q^49 + 6 * q^51 + 2 * q^53 - 2 * q^55 + 10 * q^57 + 3 * q^59 + 3 * q^63 - 4 * q^65 + 3 * q^67 + 20 * q^69 - 2 * q^71 - 2 * q^73 + q^75 + 2 * q^77 + 5 * q^79 - 4 * q^81 + 2 * q^83 + q^85 + q^87 - q^89 + 4 * q^91 + 13 * q^93 + 3 * q^95 + 8 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more