LMFDB - Embedded newform 8008.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{3} -0.618034 q^{5} +1.00000 q^{7} -0.381966 q^{9} +O(q^{10})$	$q+1.61803 q^{3} -0.618034 q^{5} +1.00000 q^{7} -0.381966 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -1.14590 q^{17} +2.61803 q^{19} +1.61803 q^{21} -7.70820 q^{23} -4.61803 q^{25} -5.47214 q^{27} +8.47214 q^{29} -7.23607 q^{31} +1.61803 q^{33} -0.618034 q^{35} -1.70820 q^{37} +1.61803 q^{39} -0.763932 q^{41} -0.381966 q^{43} +0.236068 q^{45} -9.23607 q^{47} +1.00000 q^{49} -1.85410 q^{51} +0.381966 q^{53} -0.618034 q^{55} +4.23607 q^{57} -3.23607 q^{59} -12.8541 q^{61} -0.381966 q^{63} -0.618034 q^{65} +1.90983 q^{67} -12.4721 q^{69} +7.56231 q^{71} -4.47214 q^{73} -7.47214 q^{75} +1.00000 q^{77} +14.0344 q^{79} -7.70820 q^{81} +2.61803 q^{83} +0.708204 q^{85} +13.7082 q^{87} -16.3820 q^{89} +1.00000 q^{91} -11.7082 q^{93} -1.61803 q^{95} -2.00000 q^{97} -0.381966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9	$ 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} - 9 q^{17} + 3 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} - 2 q^{27} + 8 q^{29} - 10 q^{31} + q^{33} + q^{35} + 10 q^{37} + q^{39} - 6 q^{41} - 3 q^{43} - 4 q^{45} - 14 q^{47} + 2 q^{49} + 3 q^{51} + 3 q^{53} + q^{55} + 4 q^{57} - 2 q^{59} - 19 q^{61} - 3 q^{63} + q^{65} + 15 q^{67} - 16 q^{69} - 5 q^{71} - 6 q^{75} + 2 q^{77} - q^{79} - 2 q^{81} + 3 q^{83} - 12 q^{85} + 14 q^{87} - 35 q^{89} + 2 q^{91} - 10 q^{93} - q^{95} - 4 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 - 9 * q^17 + 3 * q^19 + q^21 - 2 * q^23 - 7 * q^25 - 2 * q^27 + 8 * q^29 - 10 * q^31 + q^33 + q^35 + 10 * q^37 + q^39 - 6 * q^41 - 3 * q^43 - 4 * q^45 - 14 * q^47 + 2 * q^49 + 3 * q^51 + 3 * q^53 + q^55 + 4 * q^57 - 2 * q^59 - 19 * q^61 - 3 * q^63 + q^65 + 15 * q^67 - 16 * q^69 - 5 * q^71 - 6 * q^75 + 2 * q^77 - q^79 - 2 * q^81 + 3 * q^83 - 12 * q^85 + 14 * q^87 - 35 * q^89 + 2 * q^91 - 10 * q^93 - q^95 - 4 * q^97 - 3 * 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
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	0	0
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.14590	−0.277921	−0.138961	−	0.990298i	$-0.544376\pi$
$17$	−1.14590	−0.277921	−0.138961	+	0.990298i	$0.544376\pi$
$18$	0	0
$19$	2.61803	0.600618	0.300309	−	0.953842i	$-0.402910\pi$
$19$	2.61803	0.600618	0.300309	+	0.953842i	$0.402910\pi$
$20$	0	0
$21$	1.61803	0.353084
$22$	0	0
$23$	−7.70820	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−7.70820	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	−7.23607	−1.29964	−0.649818	−	0.760090i	$-0.725155\pi$
$31$	−7.23607	−1.29964	−0.649818	+	0.760090i	$0.725155\pi$
$32$	0	0
$33$	1.61803	0.281664
$34$	0	0
$35$	−0.618034	−0.104467
$36$	0	0
$37$	−1.70820	−0.280827	−0.140413	−	0.990093i	$-0.544843\pi$
$37$	−1.70820	−0.280827	−0.140413	+	0.990093i	$0.544843\pi$
$38$	0	0
$39$	1.61803	0.259093
$40$	0	0
$41$	−0.763932	−0.119306	−0.0596531	−	0.998219i	$-0.518999\pi$
$41$	−0.763932	−0.119306	−0.0596531	+	0.998219i	$0.518999\pi$
$42$	0	0
$43$	−0.381966	−0.0582493	−0.0291246	−	0.999576i	$-0.509272\pi$
$43$	−0.381966	−0.0582493	−0.0291246	+	0.999576i	$0.509272\pi$
$44$	0	0
$45$	0.236068	0.0351909
$46$	0	0
$47$	−9.23607	−1.34722	−0.673609	−	0.739087i	$-0.735257\pi$
$47$	−9.23607	−1.34722	−0.673609	+	0.739087i	$0.735257\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.85410	−0.259626
$52$	0	0
$53$	0.381966	0.0524671	0.0262335	−	0.999656i	$-0.491649\pi$
$53$	0.381966	0.0524671	0.0262335	+	0.999656i	$0.491649\pi$
$54$	0	0
$55$	−0.618034	−0.0833357
$56$	0	0
$57$	4.23607	0.561081
$58$	0	0
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	0	0
$61$	−12.8541	−1.64580	−0.822900	−	0.568187i	$-0.807645\pi$
$61$	−12.8541	−1.64580	−0.822900	+	0.568187i	$0.807645\pi$
$62$	0	0
$63$	−0.381966	−0.0481232
$64$	0	0
$65$	−0.618034	−0.0766577
$66$	0	0
$67$	1.90983	0.233323	0.116661	−	0.993172i	$-0.462781\pi$
$67$	1.90983	0.233323	0.116661	+	0.993172i	$0.462781\pi$
$68$	0	0
$69$	−12.4721	−1.50147
$70$	0	0
$71$	7.56231	0.897481	0.448740	−	0.893662i	$-0.351873\pi$
$71$	7.56231	0.897481	0.448740	+	0.893662i	$0.351873\pi$
$72$	0	0
$73$	−4.47214	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$73$	−4.47214	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$74$	0	0
$75$	−7.47214	−0.862808
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	14.0344	1.57900	0.789499	−	0.613752i	$-0.210340\pi$
$79$	14.0344	1.57900	0.789499	+	0.613752i	$0.210340\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	2.61803	0.287367	0.143683	−	0.989624i	$-0.454105\pi$
$83$	2.61803	0.287367	0.143683	+	0.989624i	$0.454105\pi$
$84$	0	0
$85$	0.708204	0.0768155
$86$	0	0
$87$	13.7082	1.46967
$88$	0	0
$89$	−16.3820	−1.73648	−0.868242	−	0.496140i	$-0.834750\pi$
$89$	−16.3820	−1.73648	−0.868242	+	0.496140i	$0.834750\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−11.7082	−1.21408
$94$	0	0
$95$	−1.61803	−0.166007
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−0.381966	−0.0383890
$100$	0	0
$101$	3.38197	0.336518	0.168259	−	0.985743i	$-0.446185\pi$
$101$	3.38197	0.336518	0.168259	+	0.985743i	$0.446185\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−8.38197	−0.810315	−0.405158	−	0.914247i	$-0.632783\pi$
$107$	−8.38197	−0.810315	−0.405158	+	0.914247i	$0.632783\pi$
$108$	0	0
$109$	−2.32624	−0.222813	−0.111407	−	0.993775i	$-0.535536\pi$
$109$	−2.32624	−0.222813	−0.111407	+	0.993775i	$0.535536\pi$
$110$	0	0
$111$	−2.76393	−0.262341
$112$	0	0
$113$	−9.85410	−0.926996	−0.463498	−	0.886098i	$-0.653406\pi$
$113$	−9.85410	−0.926996	−0.463498	+	0.886098i	$0.653406\pi$
$114$	0	0
$115$	4.76393	0.444239
$116$	0	0
$117$	−0.381966	−0.0353128
$118$	0	0
$119$	−1.14590	−0.105044
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.23607	−0.111452
$124$	0	0
$125$	5.94427	0.531672
$126$	0	0
$127$	−0.145898	−0.0129464	−0.00647318	−	0.999979i	$-0.502060\pi$
$127$	−0.145898	−0.0129464	−0.00647318	+	0.999979i	$0.502060\pi$
$128$	0	0
$129$	−0.618034	−0.0544149
$130$	0	0
$131$	18.4721	1.61392	0.806959	−	0.590607i	$-0.201112\pi$
$131$	18.4721	1.61392	0.806959	+	0.590607i	$0.201112\pi$
$132$	0	0
$133$	2.61803	0.227012
$134$	0	0
$135$	3.38197	0.291073
$136$	0	0
$137$	−20.6525	−1.76446	−0.882230	−	0.470819i	$-0.843959\pi$
$137$	−20.6525	−1.76446	−0.882230	+	0.470819i	$0.843959\pi$
$138$	0	0
$139$	3.41641	0.289776	0.144888	−	0.989448i	$-0.453718\pi$
$139$	3.41641	0.289776	0.144888	+	0.989448i	$0.453718\pi$
$140$	0	0
$141$	−14.9443	−1.25853
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−5.23607	−0.434832
$146$	0	0
$147$	1.61803	0.133453
$148$	0	0
$149$	19.2705	1.57870	0.789351	−	0.613942i	$-0.210417\pi$
$149$	19.2705	1.57870	0.789351	+	0.613942i	$0.210417\pi$
$150$	0	0
$151$	−8.76393	−0.713199	−0.356599	−	0.934257i	$-0.616064\pi$
$151$	−8.76393	−0.713199	−0.356599	+	0.934257i	$0.616064\pi$
$152$	0	0
$153$	0.437694	0.0353855
$154$	0	0
$155$	4.47214	0.359211
$156$	0	0
$157$	9.23607	0.737118	0.368559	−	0.929604i	$-0.379851\pi$
$157$	9.23607	0.737118	0.368559	+	0.929604i	$0.379851\pi$
$158$	0	0
$159$	0.618034	0.0490133
$160$	0	0
$161$	−7.70820	−0.607492
$162$	0	0
$163$	−5.85410	−0.458529	−0.229264	−	0.973364i	$-0.573632\pi$
$163$	−5.85410	−0.458529	−0.229264	+	0.973364i	$0.573632\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−13.9098	−1.07637	−0.538187	−	0.842825i	$-0.680891\pi$
$167$	−13.9098	−1.07637	−0.538187	+	0.842825i	$0.680891\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	10.7984	0.820985	0.410493	−	0.911864i	$-0.365357\pi$
$173$	10.7984	0.820985	0.410493	+	0.911864i	$0.365357\pi$
$174$	0	0
$175$	−4.61803	−0.349091
$176$	0	0
$177$	−5.23607	−0.393567
$178$	0	0
$179$	5.23607	0.391362	0.195681	−	0.980668i	$-0.437308\pi$
$179$	5.23607	0.391362	0.195681	+	0.980668i	$0.437308\pi$
$180$	0	0
$181$	1.52786	0.113565	0.0567826	−	0.998387i	$-0.481916\pi$
$181$	1.52786	0.113565	0.0567826	+	0.998387i	$0.481916\pi$
$182$	0	0
$183$	−20.7984	−1.53746
$184$	0	0
$185$	1.05573	0.0776187
$186$	0	0
$187$	−1.14590	−0.0837964
$188$	0	0
$189$	−5.47214	−0.398039
$190$	0	0
$191$	−24.6525	−1.78379	−0.891895	−	0.452242i	$-0.850624\pi$
$191$	−24.6525	−1.78379	−0.891895	+	0.452242i	$0.850624\pi$
$192$	0	0
$193$	−16.0902	−1.15820	−0.579098	−	0.815258i	$-0.696595\pi$
$193$	−16.0902	−1.15820	−0.579098	+	0.815258i	$0.696595\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	14.8541	1.05831	0.529155	−	0.848525i	$-0.322509\pi$
$197$	14.8541	1.05831	0.529155	+	0.848525i	$0.322509\pi$
$198$	0	0
$199$	14.7984	1.04903	0.524514	−	0.851402i	$-0.324247\pi$
$199$	14.7984	1.04903	0.524514	+	0.851402i	$0.324247\pi$
$200$	0	0
$201$	3.09017	0.217964
$202$	0	0
$203$	8.47214	0.594627
$204$	0	0
$205$	0.472136	0.0329754
$206$	0	0
$207$	2.94427	0.204641
$208$	0	0
$209$	2.61803	0.181093
$210$	0	0
$211$	5.32624	0.366673	0.183337	−	0.983050i	$-0.441310\pi$
$211$	5.32624	0.366673	0.183337	+	0.983050i	$0.441310\pi$
$212$	0	0
$213$	12.2361	0.838402
$214$	0	0
$215$	0.236068	0.0160997
$216$	0	0
$217$	−7.23607	−0.491216
$218$	0	0
$219$	−7.23607	−0.488968
$220$	0	0
$221$	−1.14590	−0.0770814
$222$	0	0
$223$	−0.763932	−0.0511567	−0.0255783	−	0.999673i	$-0.508143\pi$
$223$	−0.763932	−0.0511567	−0.0255783	+	0.999673i	$0.508143\pi$
$224$	0	0
$225$	1.76393	0.117595
$226$	0	0
$227$	5.43769	0.360912	0.180456	−	0.983583i	$-0.442243\pi$
$227$	5.43769	0.360912	0.180456	+	0.983583i	$0.442243\pi$
$228$	0	0
$229$	9.56231	0.631895	0.315947	−	0.948777i	$-0.397678\pi$
$229$	9.56231	0.631895	0.315947	+	0.948777i	$0.397678\pi$
$230$	0	0
$231$	1.61803	0.106459
$232$	0	0
$233$	6.29180	0.412189	0.206095	−	0.978532i	$-0.433925\pi$
$233$	6.29180	0.412189	0.206095	+	0.978532i	$0.433925\pi$
$234$	0	0
$235$	5.70820	0.372362
$236$	0	0
$237$	22.7082	1.47506
$238$	0	0
$239$	−24.6525	−1.59464	−0.797318	−	0.603559i	$-0.793749\pi$
$239$	−24.6525	−1.59464	−0.797318	+	0.603559i	$0.793749\pi$
$240$	0	0
$241$	−16.1803	−1.04227	−0.521134	−	0.853475i	$-0.674491\pi$
$241$	−16.1803	−1.04227	−0.521134	+	0.853475i	$0.674491\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	−0.618034	−0.0394847
$246$	0	0
$247$	2.61803	0.166582
$248$	0	0
$249$	4.23607	0.268450
$250$	0	0
$251$	−7.85410	−0.495747	−0.247873	−	0.968792i	$-0.579732\pi$
$251$	−7.85410	−0.495747	−0.247873	+	0.968792i	$0.579732\pi$
$252$	0	0
$253$	−7.70820	−0.484611
$254$	0	0
$255$	1.14590	0.0717589
$256$	0	0
$257$	−21.1246	−1.31772	−0.658859	−	0.752267i	$-0.728960\pi$
$257$	−21.1246	−1.31772	−0.658859	+	0.752267i	$0.728960\pi$
$258$	0	0
$259$	−1.70820	−0.106143
$260$	0	0
$261$	−3.23607	−0.200308
$262$	0	0
$263$	−31.8541	−1.96421	−0.982104	−	0.188339i	$-0.939690\pi$
$263$	−31.8541	−1.96421	−0.982104	+	0.188339i	$0.939690\pi$
$264$	0	0
$265$	−0.236068	−0.0145015
$266$	0	0
$267$	−26.5066	−1.62218
$268$	0	0
$269$	−12.4721	−0.760440	−0.380220	−	0.924896i	$-0.624152\pi$
$269$	−12.4721	−0.760440	−0.380220	+	0.924896i	$0.624152\pi$
$270$	0	0
$271$	15.0344	0.913277	0.456639	−	0.889652i	$-0.349053\pi$
$271$	15.0344	0.913277	0.456639	+	0.889652i	$0.349053\pi$
$272$	0	0
$273$	1.61803	0.0979279
$274$	0	0
$275$	−4.61803	−0.278478
$276$	0	0
$277$	−7.70820	−0.463141	−0.231571	−	0.972818i	$-0.574386\pi$
$277$	−7.70820	−0.463141	−0.231571	+	0.972818i	$0.574386\pi$
$278$	0	0
$279$	2.76393	0.165472
$280$	0	0
$281$	31.6869	1.89028	0.945142	−	0.326661i	$-0.105924\pi$
$281$	31.6869	1.89028	0.945142	+	0.326661i	$0.105924\pi$
$282$	0	0
$283$	−29.8885	−1.77669	−0.888345	−	0.459177i	$-0.848144\pi$
$283$	−29.8885	−1.77669	−0.888345	+	0.459177i	$0.848144\pi$
$284$	0	0
$285$	−2.61803	−0.155079
$286$	0	0
$287$	−0.763932	−0.0450935
$288$	0	0
$289$	−15.6869	−0.922760
$290$	0	0
$291$	−3.23607	−0.189702
$292$	0	0
$293$	−19.1246	−1.11727	−0.558636	−	0.829413i	$-0.688675\pi$
$293$	−19.1246	−1.11727	−0.558636	+	0.829413i	$0.688675\pi$
$294$	0	0
$295$	2.00000	0.116445
$296$	0	0
$297$	−5.47214	−0.317526
$298$	0	0
$299$	−7.70820	−0.445777
$300$	0	0
$301$	−0.381966	−0.0220162
$302$	0	0
$303$	5.47214	0.314366
$304$	0	0
$305$	7.94427	0.454888
$306$	0	0
$307$	3.32624	0.189838	0.0949192	−	0.995485i	$-0.469741\pi$
$307$	3.32624	0.189838	0.0949192	+	0.995485i	$0.469741\pi$
$308$	0	0
$309$	−6.47214	−0.368187
$310$	0	0
$311$	1.85410	0.105136	0.0525682	−	0.998617i	$-0.483259\pi$
$311$	1.85410	0.105136	0.0525682	+	0.998617i	$0.483259\pi$
$312$	0	0
$313$	23.2361	1.31338	0.656690	−	0.754161i	$-0.271956\pi$
$313$	23.2361	1.31338	0.656690	+	0.754161i	$0.271956\pi$
$314$	0	0
$315$	0.236068	0.0133009
$316$	0	0
$317$	−16.7639	−0.941556	−0.470778	−	0.882252i	$-0.656027\pi$
$317$	−16.7639	−0.941556	−0.470778	+	0.882252i	$0.656027\pi$
$318$	0	0
$319$	8.47214	0.474349
$320$	0	0
$321$	−13.5623	−0.756974
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−4.61803	−0.256162
$326$	0	0
$327$	−3.76393	−0.208146
$328$	0	0
$329$	−9.23607	−0.509201
$330$	0	0
$331$	8.32624	0.457651	0.228826	−	0.973467i	$-0.426511\pi$
$331$	8.32624	0.457651	0.228826	+	0.973467i	$0.426511\pi$
$332$	0	0
$333$	0.652476	0.0357555
$334$	0	0
$335$	−1.18034	−0.0644889
$336$	0	0
$337$	14.1803	0.772452	0.386226	−	0.922404i	$-0.373778\pi$
$337$	14.1803	0.772452	0.386226	+	0.922404i	$0.373778\pi$
$338$	0	0
$339$	−15.9443	−0.865974
$340$	0	0
$341$	−7.23607	−0.391855
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	7.70820	0.414996
$346$	0	0
$347$	20.0344	1.07551	0.537753	−	0.843103i	$-0.319273\pi$
$347$	20.0344	1.07551	0.537753	+	0.843103i	$0.319273\pi$
$348$	0	0
$349$	16.2918	0.872080	0.436040	−	0.899927i	$-0.356381\pi$
$349$	16.2918	0.872080	0.436040	+	0.899927i	$0.356381\pi$
$350$	0	0
$351$	−5.47214	−0.292081
$352$	0	0
$353$	−17.0557	−0.907785	−0.453892	−	0.891056i	$-0.649965\pi$
$353$	−17.0557	−0.907785	−0.453892	+	0.891056i	$0.649965\pi$
$354$	0	0
$355$	−4.67376	−0.248058
$356$	0	0
$357$	−1.85410	−0.0981295
$358$	0	0
$359$	−20.7639	−1.09588	−0.547939	−	0.836518i	$-0.684587\pi$
$359$	−20.7639	−1.09588	−0.547939	+	0.836518i	$0.684587\pi$
$360$	0	0
$361$	−12.1459	−0.639258
$362$	0	0
$363$	1.61803	0.0849248
$364$	0	0
$365$	2.76393	0.144671
$366$	0	0
$367$	−22.4508	−1.17192	−0.585962	−	0.810338i	$-0.699283\pi$
$367$	−22.4508	−1.17192	−0.585962	+	0.810338i	$0.699283\pi$
$368$	0	0
$369$	0.291796	0.0151903
$370$	0	0
$371$	0.381966	0.0198307
$372$	0	0
$373$	−27.4164	−1.41957	−0.709784	−	0.704419i	$-0.751207\pi$
$373$	−27.4164	−1.41957	−0.709784	+	0.704419i	$0.751207\pi$
$374$	0	0
$375$	9.61803	0.496673
$376$	0	0
$377$	8.47214	0.436337
$378$	0	0
$379$	10.6738	0.548274	0.274137	−	0.961691i	$-0.411608\pi$
$379$	10.6738	0.548274	0.274137	+	0.961691i	$0.411608\pi$
$380$	0	0
$381$	−0.236068	−0.0120941
$382$	0	0
$383$	19.8885	1.01626	0.508129	−	0.861281i	$-0.330337\pi$
$383$	19.8885	1.01626	0.508129	+	0.861281i	$0.330337\pi$
$384$	0	0
$385$	−0.618034	−0.0314979
$386$	0	0
$387$	0.145898	0.00741641
$388$	0	0
$389$	−8.14590	−0.413013	−0.206507	−	0.978445i	$-0.566210\pi$
$389$	−8.14590	−0.413013	−0.206507	+	0.978445i	$0.566210\pi$
$390$	0	0
$391$	8.83282	0.446695
$392$	0	0
$393$	29.8885	1.50768
$394$	0	0
$395$	−8.67376	−0.436424
$396$	0	0
$397$	17.2148	0.863985	0.431993	−	0.901877i	$-0.357811\pi$
$397$	17.2148	0.863985	0.431993	+	0.901877i	$0.357811\pi$
$398$	0	0
$399$	4.23607	0.212069
$400$	0	0
$401$	−0.291796	−0.0145716	−0.00728580	−	0.999973i	$-0.502319\pi$
$401$	−0.291796	−0.0145716	−0.00728580	+	0.999973i	$0.502319\pi$
$402$	0	0
$403$	−7.23607	−0.360454
$404$	0	0
$405$	4.76393	0.236722
$406$	0	0
$407$	−1.70820	−0.0846725
$408$	0	0
$409$	8.65248	0.427837	0.213919	−	0.976851i	$-0.431377\pi$
$409$	8.65248	0.427837	0.213919	+	0.976851i	$0.431377\pi$
$410$	0	0
$411$	−33.4164	−1.64831
$412$	0	0
$413$	−3.23607	−0.159236
$414$	0	0
$415$	−1.61803	−0.0794262
$416$	0	0
$417$	5.52786	0.270701
$418$	0	0
$419$	−14.5623	−0.711415	−0.355708	−	0.934597i	$-0.615760\pi$
$419$	−14.5623	−0.711415	−0.355708	+	0.934597i	$0.615760\pi$
$420$	0	0
$421$	23.5279	1.14668	0.573339	−	0.819318i	$-0.305648\pi$
$421$	23.5279	1.14668	0.573339	+	0.819318i	$0.305648\pi$
$422$	0	0
$423$	3.52786	0.171531
$424$	0	0
$425$	5.29180	0.256690
$426$	0	0
$427$	−12.8541	−0.622054
$428$	0	0
$429$	1.61803	0.0781194
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	8.94427	0.429834	0.214917	−	0.976632i	$-0.431052\pi$
$433$	8.94427	0.429834	0.214917	+	0.976632i	$0.431052\pi$
$434$	0	0
$435$	−8.47214	−0.406208
$436$	0	0
$437$	−20.1803	−0.965357
$438$	0	0
$439$	27.5967	1.31712	0.658560	−	0.752528i	$-0.271166\pi$
$439$	27.5967	1.31712	0.658560	+	0.752528i	$0.271166\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	0	0
$443$	−37.3050	−1.77241	−0.886206	−	0.463292i	$-0.846668\pi$
$443$	−37.3050	−1.77241	−0.886206	+	0.463292i	$0.846668\pi$
$444$	0	0
$445$	10.1246	0.479953
$446$	0	0
$447$	31.1803	1.47478
$448$	0	0
$449$	30.3607	1.43281	0.716405	−	0.697685i	$-0.245787\pi$
$449$	30.3607	1.43281	0.716405	+	0.697685i	$0.245787\pi$
$450$	0	0
$451$	−0.763932	−0.0359722
$452$	0	0
$453$	−14.1803	−0.666250
$454$	0	0
$455$	−0.618034	−0.0289739
$456$	0	0
$457$	19.7426	0.923522	0.461761	−	0.887004i	$-0.347218\pi$
$457$	19.7426	0.923522	0.461761	+	0.887004i	$0.347218\pi$
$458$	0	0
$459$	6.27051	0.292682
$460$	0	0
$461$	7.23607	0.337017	0.168509	−	0.985700i	$-0.446105\pi$
$461$	7.23607	0.337017	0.168509	+	0.985700i	$0.446105\pi$
$462$	0	0
$463$	0.944272	0.0438840	0.0219420	−	0.999759i	$-0.493015\pi$
$463$	0.944272	0.0438840	0.0219420	+	0.999759i	$0.493015\pi$
$464$	0	0
$465$	7.23607	0.335565
$466$	0	0
$467$	−28.0902	−1.29986	−0.649929	−	0.759995i	$-0.725201\pi$
$467$	−28.0902	−1.29986	−0.649929	+	0.759995i	$0.725201\pi$
$468$	0	0
$469$	1.90983	0.0881878
$470$	0	0
$471$	14.9443	0.688596
$472$	0	0
$473$	−0.381966	−0.0175628
$474$	0	0
$475$	−12.0902	−0.554735
$476$	0	0
$477$	−0.145898	−0.00668021
$478$	0	0
$479$	−39.1591	−1.78922	−0.894611	−	0.446845i	$-0.852548\pi$
$479$	−39.1591	−1.78922	−0.894611	+	0.446845i	$0.852548\pi$
$480$	0	0
$481$	−1.70820	−0.0778874
$482$	0	0
$483$	−12.4721	−0.567502
$484$	0	0
$485$	1.23607	0.0561270
$486$	0	0
$487$	−4.94427	−0.224046	−0.112023	−	0.993706i	$-0.535733\pi$
$487$	−4.94427	−0.224046	−0.112023	+	0.993706i	$0.535733\pi$
$488$	0	0
$489$	−9.47214	−0.428345
$490$	0	0
$491$	18.4721	0.833636	0.416818	−	0.908990i	$-0.363145\pi$
$491$	18.4721	0.833636	0.416818	+	0.908990i	$0.363145\pi$
$492$	0	0
$493$	−9.70820	−0.437236
$494$	0	0
$495$	0.236068	0.0106105
$496$	0	0
$497$	7.56231	0.339216
$498$	0	0
$499$	24.6869	1.10514	0.552569	−	0.833467i	$-0.313648\pi$
$499$	24.6869	1.10514	0.552569	+	0.833467i	$0.313648\pi$
$500$	0	0
$501$	−22.5066	−1.00552
$502$	0	0
$503$	−30.9443	−1.37974	−0.689868	−	0.723935i	$-0.742332\pi$
$503$	−30.9443	−1.37974	−0.689868	+	0.723935i	$0.742332\pi$
$504$	0	0
$505$	−2.09017	−0.0930113
$506$	0	0
$507$	1.61803	0.0718594
$508$	0	0
$509$	22.3607	0.991120	0.495560	−	0.868574i	$-0.334963\pi$
$509$	22.3607	0.991120	0.495560	+	0.868574i	$0.334963\pi$
$510$	0	0
$511$	−4.47214	−0.197836
$512$	0	0
$513$	−14.3262	−0.632519
$514$	0	0
$515$	2.47214	0.108935
$516$	0	0
$517$	−9.23607	−0.406202
$518$	0	0
$519$	17.4721	0.766942
$520$	0	0
$521$	−29.1246	−1.27597	−0.637986	−	0.770048i	$-0.720232\pi$
$521$	−29.1246	−1.27597	−0.637986	+	0.770048i	$0.720232\pi$
$522$	0	0
$523$	10.7639	0.470674	0.235337	−	0.971914i	$-0.424381\pi$
$523$	10.7639	0.470674	0.235337	+	0.971914i	$0.424381\pi$
$524$	0	0
$525$	−7.47214	−0.326111
$526$	0	0
$527$	8.29180	0.361196
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	1.23607	0.0536408
$532$	0	0
$533$	−0.763932	−0.0330896
$534$	0	0
$535$	5.18034	0.223966
$536$	0	0
$537$	8.47214	0.365600
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	0	0
$543$	2.47214	0.106090
$544$	0	0
$545$	1.43769	0.0615840
$546$	0	0
$547$	−26.4721	−1.13187	−0.565933	−	0.824451i	$-0.691484\pi$
$547$	−26.4721	−1.13187	−0.565933	+	0.824451i	$0.691484\pi$
$548$	0	0
$549$	4.90983	0.209546
$550$	0	0
$551$	22.1803	0.944914
$552$	0	0
$553$	14.0344	0.596805
$554$	0	0
$555$	1.70820	0.0725092
$556$	0	0
$557$	27.2705	1.15549	0.577744	−	0.816218i	$-0.303933\pi$
$557$	27.2705	1.15549	0.577744	+	0.816218i	$0.303933\pi$
$558$	0	0
$559$	−0.381966	−0.0161554
$560$	0	0
$561$	−1.85410	−0.0782802
$562$	0	0
$563$	40.1803	1.69340	0.846700	−	0.532071i	$-0.178586\pi$
$563$	40.1803	1.69340	0.846700	+	0.532071i	$0.178586\pi$
$564$	0	0
$565$	6.09017	0.256215
$566$	0	0
$567$	−7.70820	−0.323714
$568$	0	0
$569$	−41.1246	−1.72403	−0.862017	−	0.506880i	$-0.830799\pi$
$569$	−41.1246	−1.72403	−0.862017	+	0.506880i	$0.830799\pi$
$570$	0	0
$571$	−29.5623	−1.23714	−0.618572	−	0.785728i	$-0.712288\pi$
$571$	−29.5623	−1.23714	−0.618572	+	0.785728i	$0.712288\pi$
$572$	0	0
$573$	−39.8885	−1.66637
$574$	0	0
$575$	35.5967	1.48449
$576$	0	0
$577$	−36.5623	−1.52211	−0.761054	−	0.648688i	$-0.775318\pi$
$577$	−36.5623	−1.52211	−0.761054	+	0.648688i	$0.775318\pi$
$578$	0	0
$579$	−26.0344	−1.08195
$580$	0	0
$581$	2.61803	0.108614
$582$	0	0
$583$	0.381966	0.0158194
$584$	0	0
$585$	0.236068	0.00976021
$586$	0	0
$587$	27.5967	1.13904	0.569520	−	0.821978i	$-0.307129\pi$
$587$	27.5967	1.13904	0.569520	+	0.821978i	$0.307129\pi$
$588$	0	0
$589$	−18.9443	−0.780585
$590$	0	0
$591$	24.0344	0.988645
$592$	0	0
$593$	−36.1803	−1.48575	−0.742874	−	0.669431i	$-0.766538\pi$
$593$	−36.1803	−1.48575	−0.742874	+	0.669431i	$0.766538\pi$
$594$	0	0
$595$	0.708204	0.0290335
$596$	0	0
$597$	23.9443	0.979974
$598$	0	0
$599$	44.5410	1.81990	0.909948	−	0.414722i	$-0.136121\pi$
$599$	44.5410	1.81990	0.909948	+	0.414722i	$0.136121\pi$
$600$	0	0
$601$	−0.381966	−0.0155807	−0.00779036	−	0.999970i	$-0.502480\pi$
$601$	−0.381966	−0.0155807	−0.00779036	+	0.999970i	$0.502480\pi$
$602$	0	0
$603$	−0.729490	−0.0297071
$604$	0	0
$605$	−0.618034	−0.0251267
$606$	0	0
$607$	9.52786	0.386724	0.193362	−	0.981127i	$-0.438061\pi$
$607$	9.52786	0.386724	0.193362	+	0.981127i	$0.438061\pi$
$608$	0	0
$609$	13.7082	0.555484
$610$	0	0
$611$	−9.23607	−0.373651
$612$	0	0
$613$	44.8328	1.81078	0.905390	−	0.424581i	$-0.139578\pi$
$613$	44.8328	1.81078	0.905390	+	0.424581i	$0.139578\pi$
$614$	0	0
$615$	0.763932	0.0308047
$616$	0	0
$617$	−20.3607	−0.819690	−0.409845	−	0.912155i	$-0.634417\pi$
$617$	−20.3607	−0.819690	−0.409845	+	0.912155i	$0.634417\pi$
$618$	0	0
$619$	−24.7639	−0.995346	−0.497673	−	0.867365i	$-0.665812\pi$
$619$	−24.7639	−0.995346	−0.497673	+	0.867365i	$0.665812\pi$
$620$	0	0
$621$	42.1803	1.69264
$622$	0	0
$623$	−16.3820	−0.656330
$624$	0	0
$625$	19.4164	0.776656
$626$	0	0
$627$	4.23607	0.169172
$628$	0	0
$629$	1.95743	0.0780477
$630$	0	0
$631$	6.96556	0.277294	0.138647	−	0.990342i	$-0.455725\pi$
$631$	6.96556	0.277294	0.138647	+	0.990342i	$0.455725\pi$
$632$	0	0
$633$	8.61803	0.342536
$634$	0	0
$635$	0.0901699	0.00357829
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−2.88854	−0.114269
$640$	0	0
$641$	−1.79837	−0.0710315	−0.0355157	−	0.999369i	$-0.511307\pi$
$641$	−1.79837	−0.0710315	−0.0355157	+	0.999369i	$0.511307\pi$
$642$	0	0
$643$	−0.583592	−0.0230146	−0.0115073	−	0.999934i	$-0.503663\pi$
$643$	−0.583592	−0.0230146	−0.0115073	+	0.999934i	$0.503663\pi$
$644$	0	0
$645$	0.381966	0.0150399
$646$	0	0
$647$	29.8885	1.17504	0.587520	−	0.809210i	$-0.300104\pi$
$647$	29.8885	1.17504	0.587520	+	0.809210i	$0.300104\pi$
$648$	0	0
$649$	−3.23607	−0.127027
$650$	0	0
$651$	−11.7082	−0.458881
$652$	0	0
$653$	−41.8673	−1.63839	−0.819196	−	0.573513i	$-0.805580\pi$
$653$	−41.8673	−1.63839	−0.819196	+	0.573513i	$0.805580\pi$
$654$	0	0
$655$	−11.4164	−0.446076
$656$	0	0
$657$	1.70820	0.0666434
$658$	0	0
$659$	−17.5066	−0.681959	−0.340980	−	0.940071i	$-0.610759\pi$
$659$	−17.5066	−0.681959	−0.340980	+	0.940071i	$0.610759\pi$
$660$	0	0
$661$	−27.1591	−1.05636	−0.528182	−	0.849131i	$-0.677126\pi$
$661$	−27.1591	−1.05636	−0.528182	+	0.849131i	$0.677126\pi$
$662$	0	0
$663$	−1.85410	−0.0720074
$664$	0	0
$665$	−1.61803	−0.0627447
$666$	0	0
$667$	−65.3050	−2.52862
$668$	0	0
$669$	−1.23607	−0.0477891
$670$	0	0
$671$	−12.8541	−0.496227
$672$	0	0
$673$	1.81966	0.0701427	0.0350714	−	0.999385i	$-0.488834\pi$
$673$	1.81966	0.0701427	0.0350714	+	0.999385i	$0.488834\pi$
$674$	0	0
$675$	25.2705	0.972662
$676$	0	0
$677$	−9.03444	−0.347222	−0.173611	−	0.984814i	$-0.555544\pi$
$677$	−9.03444	−0.347222	−0.173611	+	0.984814i	$0.555544\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	8.79837	0.337154
$682$	0	0
$683$	−49.3262	−1.88742	−0.943708	−	0.330780i	$-0.892688\pi$
$683$	−49.3262	−1.88742	−0.943708	+	0.330780i	$0.892688\pi$
$684$	0	0
$685$	12.7639	0.487685
$686$	0	0
$687$	15.4721	0.590299
$688$	0	0
$689$	0.381966	0.0145517
$690$	0	0
$691$	−40.8328	−1.55335	−0.776677	−	0.629899i	$-0.783096\pi$
$691$	−40.8328	−1.55335	−0.776677	+	0.629899i	$0.783096\pi$
$692$	0	0
$693$	−0.381966	−0.0145097
$694$	0	0
$695$	−2.11146	−0.0800921
$696$	0	0
$697$	0.875388	0.0331577
$698$	0	0
$699$	10.1803	0.385056
$700$	0	0
$701$	3.52786	0.133246	0.0666228	−	0.997778i	$-0.478778\pi$
$701$	3.52786	0.133246	0.0666228	+	0.997778i	$0.478778\pi$
$702$	0	0
$703$	−4.47214	−0.168670
$704$	0	0
$705$	9.23607	0.347850
$706$	0	0
$707$	3.38197	0.127192
$708$	0	0
$709$	44.2492	1.66181	0.830907	−	0.556411i	$-0.187822\pi$
$709$	44.2492	1.66181	0.830907	+	0.556411i	$0.187822\pi$
$710$	0	0
$711$	−5.36068	−0.201041
$712$	0	0
$713$	55.7771	2.08887
$714$	0	0
$715$	−0.618034	−0.0231132
$716$	0	0
$717$	−39.8885	−1.48966
$718$	0	0
$719$	5.32624	0.198635	0.0993176	−	0.995056i	$-0.468334\pi$
$719$	5.32624	0.198635	0.0993176	+	0.995056i	$0.468334\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	−26.1803	−0.973657
$724$	0	0
$725$	−39.1246	−1.45305
$726$	0	0
$727$	−7.96556	−0.295426	−0.147713	−	0.989030i	$-0.547191\pi$
$727$	−7.96556	−0.295426	−0.147713	+	0.989030i	$0.547191\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	0.437694	0.0161887
$732$	0	0
$733$	17.1246	0.632512	0.316256	−	0.948674i	$-0.397574\pi$
$733$	17.1246	0.632512	0.316256	+	0.948674i	$0.397574\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	1.90983	0.0703495
$738$	0	0
$739$	−0.763932	−0.0281017	−0.0140508	−	0.999901i	$-0.504473\pi$
$739$	−0.763932	−0.0281017	−0.0140508	+	0.999901i	$0.504473\pi$
$740$	0	0
$741$	4.23607	0.155616
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	−11.9098	−0.436342
$746$	0	0
$747$	−1.00000	−0.0365881
$748$	0	0
$749$	−8.38197	−0.306270
$750$	0	0
$751$	22.3607	0.815953	0.407976	−	0.912992i	$-0.366235\pi$
$751$	22.3607	0.815953	0.407976	+	0.912992i	$0.366235\pi$
$752$	0	0
$753$	−12.7082	−0.463113
$754$	0	0
$755$	5.41641	0.197123
$756$	0	0
$757$	39.2705	1.42731	0.713655	−	0.700497i	$-0.247038\pi$
$757$	39.2705	1.42731	0.713655	+	0.700497i	$0.247038\pi$
$758$	0	0
$759$	−12.4721	−0.452710
$760$	0	0
$761$	24.3607	0.883074	0.441537	−	0.897243i	$-0.354433\pi$
$761$	24.3607	0.883074	0.441537	+	0.897243i	$0.354433\pi$
$762$	0	0
$763$	−2.32624	−0.0842155
$764$	0	0
$765$	−0.270510	−0.00978030
$766$	0	0
$767$	−3.23607	−0.116848
$768$	0	0
$769$	−8.06888	−0.290971	−0.145486	−	0.989360i	$-0.546474\pi$
$769$	−8.06888	−0.290971	−0.145486	+	0.989360i	$0.546474\pi$
$770$	0	0
$771$	−34.1803	−1.23097
$772$	0	0
$773$	37.2148	1.33852	0.669261	−	0.743027i	$-0.266611\pi$
$773$	37.2148	1.33852	0.669261	+	0.743027i	$0.266611\pi$
$774$	0	0
$775$	33.4164	1.20035
$776$	0	0
$777$	−2.76393	−0.0991555
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	7.56231	0.270601
$782$	0	0
$783$	−46.3607	−1.65680
$784$	0	0
$785$	−5.70820	−0.203735
$786$	0	0
$787$	13.9787	0.498287	0.249144	−	0.968467i	$-0.419851\pi$
$787$	13.9787	0.498287	0.249144	+	0.968467i	$0.419851\pi$
$788$	0	0
$789$	−51.5410	−1.83491
$790$	0	0
$791$	−9.85410	−0.350372
$792$	0	0
$793$	−12.8541	−0.456463
$794$	0	0
$795$	−0.381966	−0.0135469
$796$	0	0
$797$	11.4164	0.404390	0.202195	−	0.979345i	$-0.435193\pi$
$797$	11.4164	0.404390	0.202195	+	0.979345i	$0.435193\pi$
$798$	0	0
$799$	10.5836	0.374421
$800$	0	0
$801$	6.25735	0.221093
$802$	0	0
$803$	−4.47214	−0.157818
$804$	0	0
$805$	4.76393	0.167907
$806$	0	0
$807$	−20.1803	−0.710382
$808$	0	0
$809$	21.1246	0.742702	0.371351	−	0.928493i	$-0.378895\pi$
$809$	21.1246	0.742702	0.371351	+	0.928493i	$0.378895\pi$
$810$	0	0
$811$	−8.50658	−0.298706	−0.149353	−	0.988784i	$-0.547719\pi$
$811$	−8.50658	−0.298706	−0.149353	+	0.988784i	$0.547719\pi$
$812$	0	0
$813$	24.3262	0.853158
$814$	0	0
$815$	3.61803	0.126734
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	−0.381966	−0.0133470
$820$	0	0
$821$	21.5066	0.750585	0.375292	−	0.926906i	$-0.377542\pi$
$821$	21.5066	0.750585	0.375292	+	0.926906i	$0.377542\pi$
$822$	0	0
$823$	−36.3607	−1.26745	−0.633727	−	0.773557i	$-0.718476\pi$
$823$	−36.3607	−1.26745	−0.633727	+	0.773557i	$0.718476\pi$
$824$	0	0
$825$	−7.47214	−0.260146
$826$	0	0
$827$	25.2361	0.877544	0.438772	−	0.898598i	$-0.355414\pi$
$827$	25.2361	0.877544	0.438772	+	0.898598i	$0.355414\pi$
$828$	0	0
$829$	53.0132	1.84122	0.920611	−	0.390480i	$-0.127691\pi$
$829$	53.0132	1.84122	0.920611	+	0.390480i	$0.127691\pi$
$830$	0	0
$831$	−12.4721	−0.432654
$832$	0	0
$833$	−1.14590	−0.0397030
$834$	0	0
$835$	8.59675	0.297503
$836$	0	0
$837$	39.5967	1.36866
$838$	0	0
$839$	7.23607	0.249817	0.124908	−	0.992168i	$-0.460136\pi$
$839$	7.23607	0.249817	0.124908	+	0.992168i	$0.460136\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	51.2705	1.76585
$844$	0	0
$845$	−0.618034	−0.0212610
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−48.3607	−1.65973
$850$	0	0
$851$	13.1672	0.451365
$852$	0	0
$853$	19.4164	0.664805	0.332403	−	0.943138i	$-0.392141\pi$
$853$	19.4164	0.664805	0.332403	+	0.943138i	$0.392141\pi$
$854$	0	0
$855$	0.618034	0.0211363
$856$	0	0
$857$	−34.0344	−1.16259	−0.581297	−	0.813691i	$-0.697455\pi$
$857$	−34.0344	−1.16259	−0.581297	+	0.813691i	$0.697455\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	−1.23607	−0.0421251
$862$	0	0
$863$	−11.0557	−0.376341	−0.188171	−	0.982136i	$-0.560256\pi$
$863$	−11.0557	−0.376341	−0.188171	+	0.982136i	$0.560256\pi$
$864$	0	0
$865$	−6.67376	−0.226915
$866$	0	0
$867$	−25.3820	−0.862017
$868$	0	0
$869$	14.0344	0.476086
$870$	0	0
$871$	1.90983	0.0647121
$872$	0	0
$873$	0.763932	0.0258552
$874$	0	0
$875$	5.94427	0.200953
$876$	0	0
$877$	6.09017	0.205650	0.102825	−	0.994699i	$-0.467212\pi$
$877$	6.09017	0.205650	0.102825	+	0.994699i	$0.467212\pi$
$878$	0	0
$879$	−30.9443	−1.04372
$880$	0	0
$881$	35.2361	1.18713	0.593567	−	0.804785i	$-0.297719\pi$
$881$	35.2361	1.18713	0.593567	+	0.804785i	$0.297719\pi$
$882$	0	0
$883$	29.0557	0.977803	0.488902	−	0.872339i	$-0.337398\pi$
$883$	29.0557	0.977803	0.488902	+	0.872339i	$0.337398\pi$
$884$	0	0
$885$	3.23607	0.108779
$886$	0	0
$887$	1.52786	0.0513007	0.0256503	−	0.999671i	$-0.491834\pi$
$887$	1.52786	0.0513007	0.0256503	+	0.999671i	$0.491834\pi$
$888$	0	0
$889$	−0.145898	−0.00489326
$890$	0	0
$891$	−7.70820	−0.258235
$892$	0	0
$893$	−24.1803	−0.809164
$894$	0	0
$895$	−3.23607	−0.108170
$896$	0	0
$897$	−12.4721	−0.416432
$898$	0	0
$899$	−61.3050	−2.04463
$900$	0	0
$901$	−0.437694	−0.0145817
$902$	0	0
$903$	−0.618034	−0.0205669
$904$	0	0
$905$	−0.944272	−0.0313887
$906$	0	0
$907$	−30.5410	−1.01410	−0.507049	−	0.861917i	$-0.669264\pi$
$907$	−30.5410	−1.01410	−0.507049	+	0.861917i	$0.669264\pi$
$908$	0	0
$909$	−1.29180	−0.0428462
$910$	0	0
$911$	52.5410	1.74076	0.870381	−	0.492379i	$-0.163873\pi$
$911$	52.5410	1.74076	0.870381	+	0.492379i	$0.163873\pi$
$912$	0	0
$913$	2.61803	0.0866443
$914$	0	0
$915$	12.8541	0.424944
$916$	0	0
$917$	18.4721	0.610004
$918$	0	0
$919$	19.3262	0.637514	0.318757	−	0.947837i	$-0.396735\pi$
$919$	19.3262	0.637514	0.318757	+	0.947837i	$0.396735\pi$
$920$	0	0
$921$	5.38197	0.177342
$922$	0	0
$923$	7.56231	0.248916
$924$	0	0
$925$	7.88854	0.259374
$926$	0	0
$927$	1.52786	0.0501816
$928$	0	0
$929$	−18.2705	−0.599436	−0.299718	−	0.954028i	$-0.596893\pi$
$929$	−18.2705	−0.599436	−0.299718	+	0.954028i	$0.596893\pi$
$930$	0	0
$931$	2.61803	0.0858026
$932$	0	0
$933$	3.00000	0.0982156
$934$	0	0
$935$	0.708204	0.0231607
$936$	0	0
$937$	−39.8541	−1.30198	−0.650988	−	0.759088i	$-0.725645\pi$
$937$	−39.8541	−1.30198	−0.650988	+	0.759088i	$0.725645\pi$
$938$	0	0
$939$	37.5967	1.22692
$940$	0	0
$941$	42.1803	1.37504	0.687520	−	0.726166i	$-0.258699\pi$
$941$	42.1803	1.37504	0.687520	+	0.726166i	$0.258699\pi$
$942$	0	0
$943$	5.88854	0.191757
$944$	0	0
$945$	3.38197	0.110015
$946$	0	0
$947$	−25.5066	−0.828852	−0.414426	−	0.910083i	$-0.636018\pi$
$947$	−25.5066	−0.828852	−0.414426	+	0.910083i	$0.636018\pi$
$948$	0	0
$949$	−4.47214	−0.145172
$950$	0	0
$951$	−27.1246	−0.879576
$952$	0	0
$953$	53.7082	1.73978	0.869890	−	0.493246i	$-0.164190\pi$
$953$	53.7082	1.73978	0.869890	+	0.493246i	$0.164190\pi$
$954$	0	0
$955$	15.2361	0.493028
$956$	0	0
$957$	13.7082	0.443123
$958$	0	0
$959$	−20.6525	−0.666903
$960$	0	0
$961$	21.3607	0.689054
$962$	0	0
$963$	3.20163	0.103171
$964$	0	0
$965$	9.94427	0.320117
$966$	0	0
$967$	13.4164	0.431443	0.215721	−	0.976455i	$-0.430790\pi$
$967$	13.4164	0.431443	0.215721	+	0.976455i	$0.430790\pi$
$968$	0	0
$969$	−4.85410	−0.155936
$970$	0	0
$971$	−29.0902	−0.933548	−0.466774	−	0.884377i	$-0.654584\pi$
$971$	−29.0902	−0.933548	−0.466774	+	0.884377i	$0.654584\pi$
$972$	0	0
$973$	3.41641	0.109525
$974$	0	0
$975$	−7.47214	−0.239300
$976$	0	0
$977$	26.2492	0.839787	0.419894	−	0.907573i	$-0.362067\pi$
$977$	26.2492	0.839787	0.419894	+	0.907573i	$0.362067\pi$
$978$	0	0
$979$	−16.3820	−0.523570
$980$	0	0
$981$	0.888544	0.0283690
$982$	0	0
$983$	30.2492	0.964800	0.482400	−	0.875951i	$-0.339765\pi$
$983$	30.2492	0.964800	0.482400	+	0.875951i	$0.339765\pi$
$984$	0	0
$985$	−9.18034	−0.292510
$986$	0	0
$987$	−14.9443	−0.475681
$988$	0	0
$989$	2.94427	0.0936224
$990$	0	0
$991$	15.6393	0.496799	0.248400	−	0.968658i	$-0.420095\pi$
$991$	15.6393	0.496799	0.248400	+	0.968658i	$0.420095\pi$
$992$	0	0
$993$	13.4721	0.427525
$994$	0	0
$995$	−9.14590	−0.289944
$996$	0	0
$997$	5.56231	0.176160	0.0880800	−	0.996113i	$-0.471927\pi$
$997$	5.56231	0.176160	0.0880800	+	0.996113i	$0.471927\pi$
$998$	0	0
$999$	9.34752	0.295743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.g.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.g.1.2	✓	2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} -0.618034 q^{5} +1.00000 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q+1.61803 q^{3} -0.618034 q^{5} +1.00000 q^{7} -0.381966 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -1.14590 q^{17} +2.61803 q^{19} +1.61803 q^{21} -7.70820 q^{23} -4.61803 q^{25} -5.47214 q^{27} +8.47214 q^{29} -7.23607 q^{31} +1.61803 q^{33} -0.618034 q^{35} -1.70820 q^{37} +1.61803 q^{39} -0.763932 q^{41} -0.381966 q^{43} +0.236068 q^{45} -9.23607 q^{47} +1.00000 q^{49} -1.85410 q^{51} +0.381966 q^{53} -0.618034 q^{55} +4.23607 q^{57} -3.23607 q^{59} -12.8541 q^{61} -0.381966 q^{63} -0.618034 q^{65} +1.90983 q^{67} -12.4721 q^{69} +7.56231 q^{71} -4.47214 q^{73} -7.47214 q^{75} +1.00000 q^{77} +14.0344 q^{79} -7.70820 q^{81} +2.61803 q^{83} +0.708204 q^{85} +13.7082 q^{87} -16.3820 q^{89} +1.00000 q^{91} -11.7082 q^{93} -1.61803 q^{95} -2.00000 q^{97} -0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9	\( 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} - 9 q^{17} + 3 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} - 2 q^{27} + 8 q^{29} - 10 q^{31} + q^{33} + q^{35} + 10 q^{37} + q^{39} - 6 q^{41} - 3 q^{43} - 4 q^{45} - 14 q^{47} + 2 q^{49} + 3 q^{51} + 3 q^{53} + q^{55} + 4 q^{57} - 2 q^{59} - 19 q^{61} - 3 q^{63} + q^{65} + 15 q^{67} - 16 q^{69} - 5 q^{71} - 6 q^{75} + 2 q^{77} - q^{79} - 2 q^{81} + 3 q^{83} - 12 q^{85} + 14 q^{87} - 35 q^{89} + 2 q^{91} - 10 q^{93} - q^{95} - 4 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 - 9 * q^17 + 3 * q^19 + q^21 - 2 * q^23 - 7 * q^25 - 2 * q^27 + 8 * q^29 - 10 * q^31 + q^33 + q^35 + 10 * q^37 + q^39 - 6 * q^41 - 3 * q^43 - 4 * q^45 - 14 * q^47 + 2 * q^49 + 3 * q^51 + 3 * q^53 + q^55 + 4 * q^57 - 2 * q^59 - 19 * q^61 - 3 * q^63 + q^65 + 15 * q^67 - 16 * q^69 - 5 * q^71 - 6 * q^75 + 2 * q^77 - q^79 - 2 * q^81 + 3 * q^83 - 12 * q^85 + 14 * q^87 - 35 * q^89 + 2 * q^91 - 10 * q^93 - q^95 - 4 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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