LMFDB - Embedded newform 8008.2.a.y.1.3

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.52493$ 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)

$f(q)$	$=$	$q-2.52493 q^{3} +3.43935 q^{5} +1.00000 q^{7} +3.37528 q^{9} +O(q^{10})$	$q-2.52493 q^{3} +3.43935 q^{5} +1.00000 q^{7} +3.37528 q^{9} -1.00000 q^{11} -1.00000 q^{13} -8.68412 q^{15} -6.56793 q^{17} +2.74521 q^{19} -2.52493 q^{21} -2.80225 q^{23} +6.82912 q^{25} -0.947546 q^{27} +0.240184 q^{29} +0.281315 q^{31} +2.52493 q^{33} +3.43935 q^{35} +5.95065 q^{37} +2.52493 q^{39} +11.4744 q^{41} -5.59622 q^{43} +11.6088 q^{45} -13.3176 q^{47} +1.00000 q^{49} +16.5836 q^{51} +3.03445 q^{53} -3.43935 q^{55} -6.93147 q^{57} -11.5644 q^{59} -0.183548 q^{61} +3.37528 q^{63} -3.43935 q^{65} +0.658984 q^{67} +7.07550 q^{69} -5.80420 q^{71} -8.78419 q^{73} -17.2431 q^{75} -1.00000 q^{77} -0.408207 q^{79} -7.73334 q^{81} -5.01797 q^{83} -22.5894 q^{85} -0.606448 q^{87} +3.72200 q^{89} -1.00000 q^{91} -0.710301 q^{93} +9.44174 q^{95} +1.91747 q^{97} -3.37528 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	−2.52493	−1.45777	−0.728885	−	0.684636i	$-0.759961\pi$
$3$	−2.52493	−1.45777	−0.728885	+	0.684636i	$0.759961\pi$
$4$	0	0
$5$	3.43935	1.53812	0.769062	−	0.639174i	$-0.220724\pi$
$5$	3.43935	1.53812	0.769062	+	0.639174i	$0.220724\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.37528	1.12509
$10$	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
$15$	−8.68412	−2.24223
$16$	0	0
$17$	−6.56793	−1.59296	−0.796478	−	0.604667i	$-0.793306\pi$
$17$	−6.56793	−1.59296	−0.796478	+	0.604667i	$0.793306\pi$
$18$	0	0
$19$	2.74521	0.629795	0.314897	−	0.949126i	$-0.398030\pi$
$19$	2.74521	0.629795	0.314897	+	0.949126i	$0.398030\pi$
$20$	0	0
$21$	−2.52493	−0.550985
$22$	0	0
$23$	−2.80225	−0.584310	−0.292155	−	0.956371i	$-0.594372\pi$
$23$	−2.80225	−0.584310	−0.292155	+	0.956371i	$0.594372\pi$
$24$	0	0
$25$	6.82912	1.36582
$26$	0	0
$27$	−0.947546	−0.182355
$28$	0	0
$29$	0.240184	0.0446010	0.0223005	−	0.999751i	$-0.492901\pi$
$29$	0.240184	0.0446010	0.0223005	+	0.999751i	$0.492901\pi$
$30$	0	0
$31$	0.281315	0.0505257	0.0252628	−	0.999681i	$-0.491958\pi$
$31$	0.281315	0.0505257	0.0252628	+	0.999681i	$0.491958\pi$
$32$	0	0
$33$	2.52493	0.439534
$34$	0	0
$35$	3.43935	0.581356
$36$	0	0
$37$	5.95065	0.978281	0.489140	−	0.872205i	$-0.337311\pi$
$37$	5.95065	0.978281	0.489140	+	0.872205i	$0.337311\pi$
$38$	0	0
$39$	2.52493	0.404313
$40$	0	0
$41$	11.4744	1.79200	0.895998	−	0.444059i	$-0.146462\pi$
$41$	11.4744	1.79200	0.895998	+	0.444059i	$0.146462\pi$
$42$	0	0
$43$	−5.59622	−0.853416	−0.426708	−	0.904389i	$-0.640327\pi$
$43$	−5.59622	−0.853416	−0.426708	+	0.904389i	$0.640327\pi$
$44$	0	0
$45$	11.6088	1.73053
$46$	0	0
$47$	−13.3176	−1.94257	−0.971287	−	0.237912i	$-0.923537\pi$
$47$	−13.3176	−1.94257	−0.971287	+	0.237912i	$0.923537\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	16.5836	2.32216
$52$	0	0
$53$	3.03445	0.416813	0.208407	−	0.978042i	$-0.433172\pi$
$53$	3.03445	0.416813	0.208407	+	0.978042i	$0.433172\pi$
$54$	0	0
$55$	−3.43935	−0.463762
$56$	0	0
$57$	−6.93147	−0.918095
$58$	0	0
$59$	−11.5644	−1.50555	−0.752776	−	0.658276i	$-0.771286\pi$
$59$	−11.5644	−1.50555	−0.752776	+	0.658276i	$0.771286\pi$
$60$	0	0
$61$	−0.183548	−0.0235009	−0.0117504	−	0.999931i	$-0.503740\pi$
$61$	−0.183548	−0.0235009	−0.0117504	+	0.999931i	$0.503740\pi$
$62$	0	0
$63$	3.37528	0.425245
$64$	0	0
$65$	−3.43935	−0.426599
$66$	0	0
$67$	0.658984	0.0805077	0.0402538	−	0.999189i	$-0.487183\pi$
$67$	0.658984	0.0805077	0.0402538	+	0.999189i	$0.487183\pi$
$68$	0	0
$69$	7.07550	0.851790
$70$	0	0
$71$	−5.80420	−0.688832	−0.344416	−	0.938817i	$-0.611923\pi$
$71$	−5.80420	−0.688832	−0.344416	+	0.938817i	$0.611923\pi$
$72$	0	0
$73$	−8.78419	−1.02811	−0.514056	−	0.857757i	$-0.671858\pi$
$73$	−8.78419	−1.02811	−0.514056	+	0.857757i	$0.671858\pi$
$74$	0	0
$75$	−17.2431	−1.99106
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−0.408207	−0.0459268	−0.0229634	−	0.999736i	$-0.507310\pi$
$79$	−0.408207	−0.0459268	−0.0229634	+	0.999736i	$0.507310\pi$
$80$	0	0
$81$	−7.73334	−0.859260
$82$	0	0
$83$	−5.01797	−0.550794	−0.275397	−	0.961331i	$-0.588809\pi$
$83$	−5.01797	−0.550794	−0.275397	+	0.961331i	$0.588809\pi$
$84$	0	0
$85$	−22.5894	−2.45016
$86$	0	0
$87$	−0.606448	−0.0650180
$88$	0	0
$89$	3.72200	0.394532	0.197266	−	0.980350i	$-0.436794\pi$
$89$	3.72200	0.394532	0.197266	+	0.980350i	$0.436794\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.710301	−0.0736548
$94$	0	0
$95$	9.44174	0.968702
$96$	0	0
$97$	1.91747	0.194689	0.0973447	−	0.995251i	$-0.468965\pi$
$97$	1.91747	0.194689	0.0973447	+	0.995251i	$0.468965\pi$
$98$	0	0
$99$	−3.37528	−0.339228
$100$	0	0
$101$	−0.841815	−0.0837637	−0.0418819	−	0.999123i	$-0.513335\pi$
$101$	−0.841815	−0.0837637	−0.0418819	+	0.999123i	$0.513335\pi$
$102$	0	0
$103$	13.4270	1.32301	0.661503	−	0.749943i	$-0.269919\pi$
$103$	13.4270	1.32301	0.661503	+	0.749943i	$0.269919\pi$
$104$	0	0
$105$	−8.68412	−0.847483
$106$	0	0
$107$	−6.29003	−0.608080	−0.304040	−	0.952659i	$-0.598336\pi$
$107$	−6.29003	−0.608080	−0.304040	+	0.952659i	$0.598336\pi$
$108$	0	0
$109$	1.94525	0.186321	0.0931607	−	0.995651i	$-0.470303\pi$
$109$	1.94525	0.186321	0.0931607	+	0.995651i	$0.470303\pi$
$110$	0	0
$111$	−15.0250	−1.42611
$112$	0	0
$113$	−17.3415	−1.63135	−0.815674	−	0.578512i	$-0.803634\pi$
$113$	−17.3415	−1.63135	−0.815674	+	0.578512i	$0.803634\pi$
$114$	0	0
$115$	−9.63793	−0.898741
$116$	0	0
$117$	−3.37528	−0.312044
$118$	0	0
$119$	−6.56793	−0.602081
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−28.9720	−2.61232
$124$	0	0
$125$	6.29100	0.562684
$126$	0	0
$127$	−7.69408	−0.682740	−0.341370	−	0.939929i	$-0.610891\pi$
$127$	−7.69408	−0.682740	−0.341370	+	0.939929i	$0.610891\pi$
$128$	0	0
$129$	14.1301	1.24408
$130$	0	0
$131$	−3.11125	−0.271831	−0.135915	−	0.990720i	$-0.543398\pi$
$131$	−3.11125	−0.271831	−0.135915	+	0.990720i	$0.543398\pi$
$132$	0	0
$133$	2.74521	0.238040
$134$	0	0
$135$	−3.25894	−0.280485
$136$	0	0
$137$	−1.81615	−0.155164	−0.0775822	−	0.996986i	$-0.524720\pi$
$137$	−1.81615	−0.155164	−0.0775822	+	0.996986i	$0.524720\pi$
$138$	0	0
$139$	−5.63326	−0.477807	−0.238904	−	0.971043i	$-0.576788\pi$
$139$	−5.63326	−0.477807	−0.238904	+	0.971043i	$0.576788\pi$
$140$	0	0
$141$	33.6261	2.83182
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.826076	0.0686019
$146$	0	0
$147$	−2.52493	−0.208253
$148$	0	0
$149$	13.2141	1.08254	0.541271	−	0.840848i	$-0.317943\pi$
$149$	13.2141	1.08254	0.541271	+	0.840848i	$0.317943\pi$
$150$	0	0
$151$	13.9976	1.13911	0.569554	−	0.821954i	$-0.307116\pi$
$151$	13.9976	1.13911	0.569554	+	0.821954i	$0.307116\pi$
$152$	0	0
$153$	−22.1686	−1.79222
$154$	0	0
$155$	0.967541	0.0777148
$156$	0	0
$157$	−3.59693	−0.287066	−0.143533	−	0.989645i	$-0.545846\pi$
$157$	−3.59693	−0.287066	−0.143533	+	0.989645i	$0.545846\pi$
$158$	0	0
$159$	−7.66176	−0.607617
$160$	0	0
$161$	−2.80225	−0.220849
$162$	0	0
$163$	2.82244	0.221071	0.110535	−	0.993872i	$-0.464743\pi$
$163$	2.82244	0.221071	0.110535	+	0.993872i	$0.464743\pi$
$164$	0	0
$165$	8.68412	0.676058
$166$	0	0
$167$	−23.5586	−1.82302	−0.911511	−	0.411277i	$-0.865083\pi$
$167$	−23.5586	−1.82302	−0.911511	+	0.411277i	$0.865083\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	9.26584	0.708577
$172$	0	0
$173$	2.55235	0.194052	0.0970259	−	0.995282i	$-0.469067\pi$
$173$	2.55235	0.194052	0.0970259	+	0.995282i	$0.469067\pi$
$174$	0	0
$175$	6.82912	0.516233
$176$	0	0
$177$	29.1992	2.19475
$178$	0	0
$179$	−0.648782	−0.0484923	−0.0242461	−	0.999706i	$-0.507719\pi$
$179$	−0.648782	−0.0484923	−0.0242461	+	0.999706i	$0.507719\pi$
$180$	0	0
$181$	14.5072	1.07831	0.539155	−	0.842207i	$-0.318744\pi$
$181$	14.5072	1.07831	0.539155	+	0.842207i	$0.318744\pi$
$182$	0	0
$183$	0.463445	0.0342589
$184$	0	0
$185$	20.4664	1.50472
$186$	0	0
$187$	6.56793	0.480294
$188$	0	0
$189$	−0.947546	−0.0689238
$190$	0	0
$191$	−15.5571	−1.12567	−0.562837	−	0.826568i	$-0.690290\pi$
$191$	−15.5571	−1.12567	−0.562837	+	0.826568i	$0.690290\pi$
$192$	0	0
$193$	17.9774	1.29404	0.647019	−	0.762474i	$-0.276015\pi$
$193$	17.9774	1.29404	0.647019	+	0.762474i	$0.276015\pi$
$194$	0	0
$195$	8.68412	0.621883
$196$	0	0
$197$	−8.43697	−0.601109	−0.300555	−	0.953765i	$-0.597172\pi$
$197$	−8.43697	−0.601109	−0.300555	+	0.953765i	$0.597172\pi$
$198$	0	0
$199$	14.3769	1.01915	0.509575	−	0.860426i	$-0.329803\pi$
$199$	14.3769	1.01915	0.509575	+	0.860426i	$0.329803\pi$
$200$	0	0
$201$	−1.66389	−0.117362
$202$	0	0
$203$	0.240184	0.0168576
$204$	0	0
$205$	39.4644	2.75631
$206$	0	0
$207$	−9.45838	−0.657403
$208$	0	0
$209$	−2.74521	−0.189890
$210$	0	0
$211$	−5.79465	−0.398920	−0.199460	−	0.979906i	$-0.563919\pi$
$211$	−5.79465	−0.398920	−0.199460	+	0.979906i	$0.563919\pi$
$212$	0	0
$213$	14.6552	1.00416
$214$	0	0
$215$	−19.2474	−1.31266
$216$	0	0
$217$	0.281315	0.0190969
$218$	0	0
$219$	22.1795	1.49875
$220$	0	0
$221$	6.56793	0.441807
$222$	0	0
$223$	−23.1268	−1.54868	−0.774342	−	0.632767i	$-0.781919\pi$
$223$	−23.1268	−1.54868	−0.774342	+	0.632767i	$0.781919\pi$
$224$	0	0
$225$	23.0502	1.53668
$226$	0	0
$227$	11.8937	0.789410	0.394705	−	0.918808i	$-0.370847\pi$
$227$	11.8937	0.789410	0.394705	+	0.918808i	$0.370847\pi$
$228$	0	0
$229$	−8.61599	−0.569361	−0.284680	−	0.958622i	$-0.591887\pi$
$229$	−8.61599	−0.569361	−0.284680	+	0.958622i	$0.591887\pi$
$230$	0	0
$231$	2.52493	0.166128
$232$	0	0
$233$	−20.5697	−1.34756	−0.673781	−	0.738931i	$-0.735331\pi$
$233$	−20.5697	−1.34756	−0.673781	+	0.738931i	$0.735331\pi$
$234$	0	0
$235$	−45.8039	−2.98792
$236$	0	0
$237$	1.03069	0.0669507
$238$	0	0
$239$	−23.3524	−1.51054	−0.755269	−	0.655415i	$-0.772494\pi$
$239$	−23.3524	−1.51054	−0.755269	+	0.655415i	$0.772494\pi$
$240$	0	0
$241$	−0.0162209	−0.00104488	−0.000522440	−	1.00000i	$-0.500166\pi$
$241$	−0.0162209	−0.00104488	−0.000522440		1.00000i	$0.500166\pi$
$242$	0	0
$243$	22.3688	1.43496
$244$	0	0
$245$	3.43935	0.219732
$246$	0	0
$247$	−2.74521	−0.174674
$248$	0	0
$249$	12.6700	0.802931
$250$	0	0
$251$	−15.2276	−0.961160	−0.480580	−	0.876951i	$-0.659574\pi$
$251$	−15.2276	−0.961160	−0.480580	+	0.876951i	$0.659574\pi$
$252$	0	0
$253$	2.80225	0.176176
$254$	0	0
$255$	57.0367	3.57177
$256$	0	0
$257$	−5.29617	−0.330366	−0.165183	−	0.986263i	$-0.552822\pi$
$257$	−5.29617	−0.330366	−0.165183	+	0.986263i	$0.552822\pi$
$258$	0	0
$259$	5.95065	0.369755
$260$	0	0
$261$	0.810687	0.0501802
$262$	0	0
$263$	31.3030	1.93023	0.965113	−	0.261833i	$-0.0843271\pi$
$263$	31.3030	1.93023	0.965113	+	0.261833i	$0.0843271\pi$
$264$	0	0
$265$	10.4365	0.641110
$266$	0	0
$267$	−9.39780	−0.575136
$268$	0	0
$269$	0.990063	0.0603652	0.0301826	−	0.999544i	$-0.490391\pi$
$269$	0.990063	0.0603652	0.0301826	+	0.999544i	$0.490391\pi$
$270$	0	0
$271$	25.7198	1.56237	0.781183	−	0.624302i	$-0.214616\pi$
$271$	25.7198	1.56237	0.781183	+	0.624302i	$0.214616\pi$
$272$	0	0
$273$	2.52493	0.152816
$274$	0	0
$275$	−6.82912	−0.411812
$276$	0	0
$277$	−17.5474	−1.05432	−0.527159	−	0.849766i	$-0.676743\pi$
$277$	−17.5474	−1.05432	−0.527159	+	0.849766i	$0.676743\pi$
$278$	0	0
$279$	0.949516	0.0568460
$280$	0	0
$281$	5.54813	0.330974	0.165487	−	0.986212i	$-0.447080\pi$
$281$	5.54813	0.330974	0.165487	+	0.986212i	$0.447080\pi$
$282$	0	0
$283$	−7.14592	−0.424781	−0.212390	−	0.977185i	$-0.568125\pi$
$283$	−7.14592	−0.424781	−0.212390	+	0.977185i	$0.568125\pi$
$284$	0	0
$285$	−23.8397	−1.41214
$286$	0	0
$287$	11.4744	0.677311
$288$	0	0
$289$	26.1377	1.53751
$290$	0	0
$291$	−4.84147	−0.283812
$292$	0	0
$293$	−16.9967	−0.992956	−0.496478	−	0.868049i	$-0.665374\pi$
$293$	−16.9967	−0.992956	−0.496478	+	0.868049i	$0.665374\pi$
$294$	0	0
$295$	−39.7739	−2.31573
$296$	0	0
$297$	0.947546	0.0549822
$298$	0	0
$299$	2.80225	0.162059
$300$	0	0
$301$	−5.59622	−0.322561
$302$	0	0
$303$	2.12552	0.122108
$304$	0	0
$305$	−0.631285	−0.0361473
$306$	0	0
$307$	20.7792	1.18593	0.592966	−	0.805228i	$-0.297957\pi$
$307$	20.7792	1.18593	0.592966	+	0.805228i	$0.297957\pi$
$308$	0	0
$309$	−33.9023	−1.92864
$310$	0	0
$311$	0.483400	0.0274111	0.0137055	−	0.999906i	$-0.495637\pi$
$311$	0.483400	0.0274111	0.0137055	+	0.999906i	$0.495637\pi$
$312$	0	0
$313$	−4.43420	−0.250636	−0.125318	−	0.992117i	$-0.539995\pi$
$313$	−4.43420	−0.250636	−0.125318	+	0.992117i	$0.539995\pi$
$314$	0	0
$315$	11.6088	0.654079
$316$	0	0
$317$	17.4308	0.979012	0.489506	−	0.872000i	$-0.337177\pi$
$317$	17.4308	0.979012	0.489506	+	0.872000i	$0.337177\pi$
$318$	0	0
$319$	−0.240184	−0.0134477
$320$	0	0
$321$	15.8819	0.886441
$322$	0	0
$323$	−18.0303	−1.00323
$324$	0	0
$325$	−6.82912	−0.378812
$326$	0	0
$327$	−4.91163	−0.271614
$328$	0	0
$329$	−13.3176	−0.734224
$330$	0	0
$331$	−30.6243	−1.68327	−0.841633	−	0.540050i	$-0.818405\pi$
$331$	−30.6243	−1.68327	−0.841633	+	0.540050i	$0.818405\pi$
$332$	0	0
$333$	20.0851	1.10066
$334$	0	0
$335$	2.26648	0.123831
$336$	0	0
$337$	33.1976	1.80839	0.904195	−	0.427119i	$-0.140471\pi$
$337$	33.1976	1.80839	0.904195	+	0.427119i	$0.140471\pi$
$338$	0	0
$339$	43.7860	2.37813
$340$	0	0
$341$	−0.281315	−0.0152341
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	24.3351	1.31016
$346$	0	0
$347$	2.82249	0.151519	0.0757596	−	0.997126i	$-0.475862\pi$
$347$	2.82249	0.151519	0.0757596	+	0.997126i	$0.475862\pi$
$348$	0	0
$349$	−33.9939	−1.81965	−0.909826	−	0.414991i	$-0.863785\pi$
$349$	−33.9939	−1.81965	−0.909826	+	0.414991i	$0.863785\pi$
$350$	0	0
$351$	0.947546	0.0505763
$352$	0	0
$353$	17.9223	0.953909	0.476955	−	0.878928i	$-0.341741\pi$
$353$	17.9223	0.953909	0.476955	+	0.878928i	$0.341741\pi$
$354$	0	0
$355$	−19.9627	−1.05951
$356$	0	0
$357$	16.5836	0.877695
$358$	0	0
$359$	25.1800	1.32895	0.664475	−	0.747311i	$-0.268655\pi$
$359$	25.1800	1.32895	0.664475	+	0.747311i	$0.268655\pi$
$360$	0	0
$361$	−11.4638	−0.603359
$362$	0	0
$363$	−2.52493	−0.132525
$364$	0	0
$365$	−30.2119	−1.58136
$366$	0	0
$367$	−31.8746	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$367$	−31.8746	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$368$	0	0
$369$	38.7292	2.01616
$370$	0	0
$371$	3.03445	0.157541
$372$	0	0
$373$	−15.2256	−0.788352	−0.394176	−	0.919035i	$-0.628970\pi$
$373$	−15.2256	−0.788352	−0.394176	+	0.919035i	$0.628970\pi$
$374$	0	0
$375$	−15.8843	−0.820263
$376$	0	0
$377$	−0.240184	−0.0123701
$378$	0	0
$379$	−10.2654	−0.527298	−0.263649	−	0.964619i	$-0.584926\pi$
$379$	−10.2654	−0.527298	−0.263649	+	0.964619i	$0.584926\pi$
$380$	0	0
$381$	19.4270	0.995277
$382$	0	0
$383$	−12.1745	−0.622087	−0.311043	−	0.950396i	$-0.600678\pi$
$383$	−12.1745	−0.622087	−0.311043	+	0.950396i	$0.600678\pi$
$384$	0	0
$385$	−3.43935	−0.175285
$386$	0	0
$387$	−18.8888	−0.960172
$388$	0	0
$389$	8.85013	0.448720	0.224360	−	0.974506i	$-0.427971\pi$
$389$	8.85013	0.448720	0.224360	+	0.974506i	$0.427971\pi$
$390$	0	0
$391$	18.4050	0.930781
$392$	0	0
$393$	7.85568	0.396267
$394$	0	0
$395$	−1.40397	−0.0706412
$396$	0	0
$397$	−38.2258	−1.91850	−0.959248	−	0.282565i	$-0.908815\pi$
$397$	−38.2258	−1.91850	−0.959248	+	0.282565i	$0.908815\pi$
$398$	0	0
$399$	−6.93147	−0.347007
$400$	0	0
$401$	−1.93458	−0.0966083	−0.0483041	−	0.998833i	$-0.515382\pi$
$401$	−1.93458	−0.0966083	−0.0483041	+	0.998833i	$0.515382\pi$
$402$	0	0
$403$	−0.281315	−0.0140133
$404$	0	0
$405$	−26.5977	−1.32165
$406$	0	0
$407$	−5.95065	−0.294963
$408$	0	0
$409$	−11.0589	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$409$	−11.0589	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$410$	0	0
$411$	4.58566	0.226194
$412$	0	0
$413$	−11.5644	−0.569045
$414$	0	0
$415$	−17.2586	−0.847190
$416$	0	0
$417$	14.2236	0.696533
$418$	0	0
$419$	−28.7518	−1.40462	−0.702310	−	0.711872i	$-0.747848\pi$
$419$	−28.7518	−1.40462	−0.702310	+	0.711872i	$0.747848\pi$
$420$	0	0
$421$	−2.95139	−0.143842	−0.0719209	−	0.997410i	$-0.522913\pi$
$421$	−2.95139	−0.143842	−0.0719209	+	0.997410i	$0.522913\pi$
$422$	0	0
$423$	−44.9506	−2.18557
$424$	0	0
$425$	−44.8532	−2.17570
$426$	0	0
$427$	−0.183548	−0.00888250
$428$	0	0
$429$	−2.52493	−0.121905
$430$	0	0
$431$	−11.3266	−0.545585	−0.272793	−	0.962073i	$-0.587947\pi$
$431$	−11.3266	−0.545585	−0.272793	+	0.962073i	$0.587947\pi$
$432$	0	0
$433$	−32.7738	−1.57501	−0.787504	−	0.616309i	$-0.788627\pi$
$433$	−32.7738	−1.57501	−0.787504	+	0.616309i	$0.788627\pi$
$434$	0	0
$435$	−2.08578	−0.100006
$436$	0	0
$437$	−7.69278	−0.367995
$438$	0	0
$439$	7.05878	0.336897	0.168449	−	0.985710i	$-0.446124\pi$
$439$	7.05878	0.336897	0.168449	+	0.985710i	$0.446124\pi$
$440$	0	0
$441$	3.37528	0.160727
$442$	0	0
$443$	−5.75409	−0.273385	−0.136692	−	0.990614i	$-0.543647\pi$
$443$	−5.75409	−0.273385	−0.136692	+	0.990614i	$0.543647\pi$
$444$	0	0
$445$	12.8013	0.606838
$446$	0	0
$447$	−33.3647	−1.57810
$448$	0	0
$449$	−4.18170	−0.197347	−0.0986734	−	0.995120i	$-0.531460\pi$
$449$	−4.18170	−0.197347	−0.0986734	+	0.995120i	$0.531460\pi$
$450$	0	0
$451$	−11.4744	−0.540307
$452$	0	0
$453$	−35.3429	−1.66056
$454$	0	0
$455$	−3.43935	−0.161239
$456$	0	0
$457$	−25.5675	−1.19600	−0.597999	−	0.801497i	$-0.704037\pi$
$457$	−25.5675	−1.19600	−0.597999	+	0.801497i	$0.704037\pi$
$458$	0	0
$459$	6.22341	0.290484
$460$	0	0
$461$	−11.5020	−0.535699	−0.267850	−	0.963461i	$-0.586313\pi$
$461$	−11.5020	−0.535699	−0.267850	+	0.963461i	$0.586313\pi$
$462$	0	0
$463$	32.1859	1.49580	0.747902	−	0.663809i	$-0.231061\pi$
$463$	32.1859	1.49580	0.747902	+	0.663809i	$0.231061\pi$
$464$	0	0
$465$	−2.44297	−0.113290
$466$	0	0
$467$	34.2239	1.58370	0.791848	−	0.610719i	$-0.209119\pi$
$467$	34.2239	1.58370	0.791848	+	0.610719i	$0.209119\pi$
$468$	0	0
$469$	0.658984	0.0304290
$470$	0	0
$471$	9.08201	0.418477
$472$	0	0
$473$	5.59622	0.257315
$474$	0	0
$475$	18.7474	0.860189
$476$	0	0
$477$	10.2421	0.468953
$478$	0	0
$479$	14.8111	0.676735	0.338368	−	0.941014i	$-0.390125\pi$
$479$	14.8111	0.676735	0.338368	+	0.941014i	$0.390125\pi$
$480$	0	0
$481$	−5.95065	−0.271326
$482$	0	0
$483$	7.07550	0.321946
$484$	0	0
$485$	6.59484	0.299456
$486$	0	0
$487$	10.9171	0.494700	0.247350	−	0.968926i	$-0.420440\pi$
$487$	10.9171	0.494700	0.247350	+	0.968926i	$0.420440\pi$
$488$	0	0
$489$	−7.12648	−0.322270
$490$	0	0
$491$	−32.5095	−1.46713	−0.733566	−	0.679618i	$-0.762145\pi$
$491$	−32.5095	−1.46713	−0.733566	+	0.679618i	$0.762145\pi$
$492$	0	0
$493$	−1.57751	−0.0710475
$494$	0	0
$495$	−11.6088	−0.521775
$496$	0	0
$497$	−5.80420	−0.260354
$498$	0	0
$499$	−4.38207	−0.196168	−0.0980842	−	0.995178i	$-0.531271\pi$
$499$	−4.38207	−0.196168	−0.0980842	+	0.995178i	$0.531271\pi$
$500$	0	0
$501$	59.4839	2.65754
$502$	0	0
$503$	−5.31536	−0.237000	−0.118500	−	0.992954i	$-0.537809\pi$
$503$	−5.31536	−0.237000	−0.118500	+	0.992954i	$0.537809\pi$
$504$	0	0
$505$	−2.89530	−0.128839
$506$	0	0
$507$	−2.52493	−0.112136
$508$	0	0
$509$	−12.7539	−0.565309	−0.282654	−	0.959222i	$-0.591215\pi$
$509$	−12.7539	−0.565309	−0.282654	+	0.959222i	$0.591215\pi$
$510$	0	0
$511$	−8.78419	−0.388590
$512$	0	0
$513$	−2.60121	−0.114846
$514$	0	0
$515$	46.1803	2.03495
$516$	0	0
$517$	13.3176	0.585708
$518$	0	0
$519$	−6.44451	−0.282883
$520$	0	0
$521$	−20.7866	−0.910678	−0.455339	−	0.890318i	$-0.650482\pi$
$521$	−20.7866	−0.910678	−0.455339	+	0.890318i	$0.650482\pi$
$522$	0	0
$523$	17.6386	0.771284	0.385642	−	0.922649i	$-0.373980\pi$
$523$	17.6386	0.771284	0.385642	+	0.922649i	$0.373980\pi$
$524$	0	0
$525$	−17.2431	−0.752549
$526$	0	0
$527$	−1.84766	−0.0804852
$528$	0	0
$529$	−15.1474	−0.658582
$530$	0	0
$531$	−39.0329	−1.69389
$532$	0	0
$533$	−11.4744	−0.497010
$534$	0	0
$535$	−21.6336	−0.935303
$536$	0	0
$537$	1.63813	0.0706905
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−4.81067	−0.206827	−0.103414	−	0.994638i	$-0.532976\pi$
$541$	−4.81067	−0.206827	−0.103414	+	0.994638i	$0.532976\pi$
$542$	0	0
$543$	−36.6296	−1.57193
$544$	0	0
$545$	6.69040	0.286585
$546$	0	0
$547$	−6.81951	−0.291581	−0.145791	−	0.989315i	$-0.546573\pi$
$547$	−6.81951	−0.291581	−0.145791	+	0.989315i	$0.546573\pi$
$548$	0	0
$549$	−0.619524	−0.0264406
$550$	0	0
$551$	0.659355	0.0280895
$552$	0	0
$553$	−0.408207	−0.0173587
$554$	0	0
$555$	−51.6762	−2.19353
$556$	0	0
$557$	−41.9966	−1.77945	−0.889726	−	0.456494i	$-0.849105\pi$
$557$	−41.9966	−1.77945	−0.889726	+	0.456494i	$0.849105\pi$
$558$	0	0
$559$	5.59622	0.236695
$560$	0	0
$561$	−16.5836	−0.700158
$562$	0	0
$563$	−20.5581	−0.866420	−0.433210	−	0.901293i	$-0.642619\pi$
$563$	−20.5581	−0.866420	−0.433210	+	0.901293i	$0.642619\pi$
$564$	0	0
$565$	−59.6434	−2.50921
$566$	0	0
$567$	−7.73334	−0.324770
$568$	0	0
$569$	35.7553	1.49894	0.749471	−	0.662037i	$-0.230308\pi$
$569$	35.7553	1.49894	0.749471	+	0.662037i	$0.230308\pi$
$570$	0	0
$571$	7.24964	0.303388	0.151694	−	0.988428i	$-0.451527\pi$
$571$	7.24964	0.303388	0.151694	+	0.988428i	$0.451527\pi$
$572$	0	0
$573$	39.2806	1.64097
$574$	0	0
$575$	−19.1369	−0.798065
$576$	0	0
$577$	14.7885	0.615652	0.307826	−	0.951443i	$-0.400399\pi$
$577$	14.7885	0.615652	0.307826	+	0.951443i	$0.400399\pi$
$578$	0	0
$579$	−45.3916	−1.88641
$580$	0	0
$581$	−5.01797	−0.208181
$582$	0	0
$583$	−3.03445	−0.125674
$584$	0	0
$585$	−11.6088	−0.479963
$586$	0	0
$587$	−1.33172	−0.0549658	−0.0274829	−	0.999622i	$-0.508749\pi$
$587$	−1.33172	−0.0549658	−0.0274829	+	0.999622i	$0.508749\pi$
$588$	0	0
$589$	0.772269	0.0318208
$590$	0	0
$591$	21.3028	0.876278
$592$	0	0
$593$	11.0893	0.455382	0.227691	−	0.973733i	$-0.426882\pi$
$593$	11.0893	0.455382	0.227691	+	0.973733i	$0.426882\pi$
$594$	0	0
$595$	−22.5894	−0.926075
$596$	0	0
$597$	−36.3006	−1.48569
$598$	0	0
$599$	23.2983	0.951943	0.475971	−	0.879461i	$-0.342097\pi$
$599$	23.2983	0.951943	0.475971	+	0.879461i	$0.342097\pi$
$600$	0	0
$601$	−30.9967	−1.26438	−0.632190	−	0.774813i	$-0.717844\pi$
$601$	−30.9967	−1.26438	−0.632190	+	0.774813i	$0.717844\pi$
$602$	0	0
$603$	2.22425	0.0905786
$604$	0	0
$605$	3.43935	0.139829
$606$	0	0
$607$	−15.8510	−0.643374	−0.321687	−	0.946846i	$-0.604250\pi$
$607$	−15.8510	−0.643374	−0.321687	+	0.946846i	$0.604250\pi$
$608$	0	0
$609$	−0.606448	−0.0245745
$610$	0	0
$611$	13.3176	0.538773
$612$	0	0
$613$	2.70288	0.109168	0.0545841	−	0.998509i	$-0.482617\pi$
$613$	2.70288	0.109168	0.0545841	+	0.998509i	$0.482617\pi$
$614$	0	0
$615$	−99.6448	−4.01807
$616$	0	0
$617$	40.3787	1.62558	0.812792	−	0.582554i	$-0.197947\pi$
$617$	40.3787	1.62558	0.812792	+	0.582554i	$0.197947\pi$
$618$	0	0
$619$	1.98989	0.0799806	0.0399903	−	0.999200i	$-0.487267\pi$
$619$	1.98989	0.0799806	0.0399903	+	0.999200i	$0.487267\pi$
$620$	0	0
$621$	2.65526	0.106552
$622$	0	0
$623$	3.72200	0.149119
$624$	0	0
$625$	−12.5087	−0.500347
$626$	0	0
$627$	6.93147	0.276816
$628$	0	0
$629$	−39.0834	−1.55836
$630$	0	0
$631$	−15.4417	−0.614726	−0.307363	−	0.951592i	$-0.599447\pi$
$631$	−15.4417	−0.614726	−0.307363	+	0.951592i	$0.599447\pi$
$632$	0	0
$633$	14.6311	0.581534
$634$	0	0
$635$	−26.4626	−1.05014
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−19.5908	−0.775000
$640$	0	0
$641$	−2.74295	−0.108340	−0.0541700	−	0.998532i	$-0.517251\pi$
$641$	−2.74295	−0.108340	−0.0541700	+	0.998532i	$0.517251\pi$
$642$	0	0
$643$	−7.68302	−0.302989	−0.151494	−	0.988458i	$-0.548409\pi$
$643$	−7.68302	−0.302989	−0.151494	+	0.988458i	$0.548409\pi$
$644$	0	0
$645$	48.5983	1.91356
$646$	0	0
$647$	17.4155	0.684674	0.342337	−	0.939577i	$-0.388782\pi$
$647$	17.4155	0.684674	0.342337	+	0.939577i	$0.388782\pi$
$648$	0	0
$649$	11.5644	0.453941
$650$	0	0
$651$	−0.710301	−0.0278389
$652$	0	0
$653$	−20.2655	−0.793052	−0.396526	−	0.918024i	$-0.629784\pi$
$653$	−20.2655	−0.793052	−0.396526	+	0.918024i	$0.629784\pi$
$654$	0	0
$655$	−10.7007	−0.418109
$656$	0	0
$657$	−29.6491	−1.15672
$658$	0	0
$659$	34.4566	1.34224	0.671119	−	0.741349i	$-0.265814\pi$
$659$	34.4566	1.34224	0.671119	+	0.741349i	$0.265814\pi$
$660$	0	0
$661$	−33.4705	−1.30185	−0.650925	−	0.759142i	$-0.725619\pi$
$661$	−33.4705	−1.30185	−0.650925	+	0.759142i	$0.725619\pi$
$662$	0	0
$663$	−16.5836	−0.644052
$664$	0	0
$665$	9.44174	0.366135
$666$	0	0
$667$	−0.673056	−0.0260608
$668$	0	0
$669$	58.3936	2.25763
$670$	0	0
$671$	0.183548	0.00708578
$672$	0	0
$673$	35.8896	1.38344	0.691721	−	0.722165i	$-0.256853\pi$
$673$	35.8896	1.38344	0.691721	+	0.722165i	$0.256853\pi$
$674$	0	0
$675$	−6.47091	−0.249065
$676$	0	0
$677$	−30.0190	−1.15372	−0.576862	−	0.816842i	$-0.695723\pi$
$677$	−30.0190	−1.15372	−0.576862	+	0.816842i	$0.695723\pi$
$678$	0	0
$679$	1.91747	0.0735856
$680$	0	0
$681$	−30.0307	−1.15078
$682$	0	0
$683$	−30.4667	−1.16577	−0.582887	−	0.812553i	$-0.698077\pi$
$683$	−30.4667	−1.16577	−0.582887	+	0.812553i	$0.698077\pi$
$684$	0	0
$685$	−6.24638	−0.238662
$686$	0	0
$687$	21.7548	0.829997
$688$	0	0
$689$	−3.03445	−0.115603
$690$	0	0
$691$	23.9026	0.909296	0.454648	−	0.890671i	$-0.349765\pi$
$691$	23.9026	0.909296	0.454648	+	0.890671i	$0.349765\pi$
$692$	0	0
$693$	−3.37528	−0.128216
$694$	0	0
$695$	−19.3748	−0.734927
$696$	0	0
$697$	−75.3628	−2.85457
$698$	0	0
$699$	51.9370	1.96444
$700$	0	0
$701$	6.97966	0.263618	0.131809	−	0.991275i	$-0.457921\pi$
$701$	6.97966	0.263618	0.131809	+	0.991275i	$0.457921\pi$
$702$	0	0
$703$	16.3358	0.616116
$704$	0	0
$705$	115.652	4.35570
$706$	0	0
$707$	−0.841815	−0.0316597
$708$	0	0
$709$	35.2603	1.32423	0.662115	−	0.749402i	$-0.269659\pi$
$709$	35.2603	1.32423	0.662115	+	0.749402i	$0.269659\pi$
$710$	0	0
$711$	−1.37781	−0.0516719
$712$	0	0
$713$	−0.788316	−0.0295227
$714$	0	0
$715$	3.43935	0.128624
$716$	0	0
$717$	58.9631	2.20202
$718$	0	0
$719$	−4.35967	−0.162588	−0.0812941	−	0.996690i	$-0.525905\pi$
$719$	−4.35967	−0.162588	−0.0812941	+	0.996690i	$0.525905\pi$
$720$	0	0
$721$	13.4270	0.500049
$722$	0	0
$723$	0.0409567	0.00152320
$724$	0	0
$725$	1.64025	0.0609172
$726$	0	0
$727$	−19.6893	−0.730234	−0.365117	−	0.930962i	$-0.618971\pi$
$727$	−19.6893	−0.730234	−0.365117	+	0.930962i	$0.618971\pi$
$728$	0	0
$729$	−33.2796	−1.23258
$730$	0	0
$731$	36.7556	1.35945
$732$	0	0
$733$	−27.6532	−1.02140	−0.510698	−	0.859760i	$-0.670613\pi$
$733$	−27.6532	−1.02140	−0.510698	+	0.859760i	$0.670613\pi$
$734$	0	0
$735$	−8.68412	−0.320319
$736$	0	0
$737$	−0.658984	−0.0242740
$738$	0	0
$739$	53.4502	1.96620	0.983099	−	0.183077i	$-0.0586056\pi$
$739$	53.4502	1.96620	0.983099	+	0.183077i	$0.0586056\pi$
$740$	0	0
$741$	6.93147	0.254634
$742$	0	0
$743$	−18.0130	−0.660832	−0.330416	−	0.943835i	$-0.607189\pi$
$743$	−18.0130	−0.660832	−0.330416	+	0.943835i	$0.607189\pi$
$744$	0	0
$745$	45.4480	1.66509
$746$	0	0
$747$	−16.9370	−0.619694
$748$	0	0
$749$	−6.29003	−0.229833
$750$	0	0
$751$	16.7010	0.609429	0.304715	−	0.952444i	$-0.401439\pi$
$751$	16.7010	0.609429	0.304715	+	0.952444i	$0.401439\pi$
$752$	0	0
$753$	38.4487	1.40115
$754$	0	0
$755$	48.1426	1.75209
$756$	0	0
$757$	−46.5971	−1.69360	−0.846801	−	0.531910i	$-0.821475\pi$
$757$	−46.5971	−1.69360	−0.846801	+	0.531910i	$0.821475\pi$
$758$	0	0
$759$	−7.07550	−0.256824
$760$	0	0
$761$	26.9618	0.977364	0.488682	−	0.872462i	$-0.337478\pi$
$761$	26.9618	0.977364	0.488682	+	0.872462i	$0.337478\pi$
$762$	0	0
$763$	1.94525	0.0704228
$764$	0	0
$765$	−76.2454	−2.75666
$766$	0	0
$767$	11.5644	0.417565
$768$	0	0
$769$	31.3803	1.13160	0.565801	−	0.824542i	$-0.308567\pi$
$769$	31.3803	1.13160	0.565801	+	0.824542i	$0.308567\pi$
$770$	0	0
$771$	13.3725	0.481598
$772$	0	0
$773$	3.02622	0.108846	0.0544228	−	0.998518i	$-0.482668\pi$
$773$	3.02622	0.108846	0.0544228	+	0.998518i	$0.482668\pi$
$774$	0	0
$775$	1.92114	0.0690092
$776$	0	0
$777$	−15.0250	−0.539018
$778$	0	0
$779$	31.4996	1.12859
$780$	0	0
$781$	5.80420	0.207691
$782$	0	0
$783$	−0.227585	−0.00813323
$784$	0	0
$785$	−12.3711	−0.441544
$786$	0	0
$787$	−32.3970	−1.15483	−0.577414	−	0.816452i	$-0.695938\pi$
$787$	−32.3970	−1.15483	−0.577414	+	0.816452i	$0.695938\pi$
$788$	0	0
$789$	−79.0379	−2.81382
$790$	0	0
$791$	−17.3415	−0.616592
$792$	0	0
$793$	0.183548	0.00651797
$794$	0	0
$795$	−26.3515	−0.934591
$796$	0	0
$797$	2.99600	0.106124	0.0530618	−	0.998591i	$-0.483102\pi$
$797$	2.99600	0.106124	0.0530618	+	0.998591i	$0.483102\pi$
$798$	0	0
$799$	87.4691	3.09443
$800$	0	0
$801$	12.5628	0.443884
$802$	0	0
$803$	8.78419	0.309987
$804$	0	0
$805$	−9.63793	−0.339692
$806$	0	0
$807$	−2.49984	−0.0879986
$808$	0	0
$809$	2.05449	0.0722320	0.0361160	−	0.999348i	$-0.488501\pi$
$809$	2.05449	0.0722320	0.0361160	+	0.999348i	$0.488501\pi$
$810$	0	0
$811$	−14.9259	−0.524118	−0.262059	−	0.965052i	$-0.584402\pi$
$811$	−14.9259	−0.524118	−0.262059	+	0.965052i	$0.584402\pi$
$812$	0	0
$813$	−64.9407	−2.27757
$814$	0	0
$815$	9.70737	0.340034
$816$	0	0
$817$	−15.3628	−0.537477
$818$	0	0
$819$	−3.37528	−0.117942
$820$	0	0
$821$	29.4503	1.02782	0.513911	−	0.857843i	$-0.328196\pi$
$821$	29.4503	1.02782	0.513911	+	0.857843i	$0.328196\pi$
$822$	0	0
$823$	−42.3023	−1.47457	−0.737283	−	0.675584i	$-0.763892\pi$
$823$	−42.3023	−1.47457	−0.737283	+	0.675584i	$0.763892\pi$
$824$	0	0
$825$	17.2431	0.600327
$826$	0	0
$827$	15.8903	0.552559	0.276279	−	0.961077i	$-0.410898\pi$
$827$	15.8903	0.552559	0.276279	+	0.961077i	$0.410898\pi$
$828$	0	0
$829$	−45.4270	−1.57774	−0.788872	−	0.614558i	$-0.789335\pi$
$829$	−45.4270	−1.57774	−0.788872	+	0.614558i	$0.789335\pi$
$830$	0	0
$831$	44.3059	1.53695
$832$	0	0
$833$	−6.56793	−0.227565
$834$	0	0
$835$	−81.0263	−2.80403
$836$	0	0
$837$	−0.266559	−0.00921363
$838$	0	0
$839$	−15.1543	−0.523186	−0.261593	−	0.965178i	$-0.584248\pi$
$839$	−15.1543	−0.523186	−0.261593	+	0.965178i	$0.584248\pi$
$840$	0	0
$841$	−28.9423	−0.998011
$842$	0	0
$843$	−14.0086	−0.482483
$844$	0	0
$845$	3.43935	0.118317
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	18.0429	0.619232
$850$	0	0
$851$	−16.6752	−0.571620
$852$	0	0
$853$	45.4014	1.55452	0.777258	−	0.629182i	$-0.216610\pi$
$853$	45.4014	1.55452	0.777258	+	0.629182i	$0.216610\pi$
$854$	0	0
$855$	31.8685	1.08988
$856$	0	0
$857$	−7.01882	−0.239758	−0.119879	−	0.992788i	$-0.538251\pi$
$857$	−7.01882	−0.239758	−0.119879	+	0.992788i	$0.538251\pi$
$858$	0	0
$859$	30.1710	1.02942	0.514711	−	0.857363i	$-0.327899\pi$
$859$	30.1710	1.02942	0.514711	+	0.857363i	$0.327899\pi$
$860$	0	0
$861$	−28.9720	−0.987363
$862$	0	0
$863$	41.9524	1.42808	0.714038	−	0.700107i	$-0.246865\pi$
$863$	41.9524	1.42808	0.714038	+	0.700107i	$0.246865\pi$
$864$	0	0
$865$	8.77843	0.298476
$866$	0	0
$867$	−65.9958	−2.24133
$868$	0	0
$869$	0.408207	0.0138475
$870$	0	0
$871$	−0.658984	−0.0223288
$872$	0	0
$873$	6.47198	0.219043
$874$	0	0
$875$	6.29100	0.212675
$876$	0	0
$877$	3.08406	0.104141	0.0520707	−	0.998643i	$-0.483418\pi$
$877$	3.08406	0.104141	0.0520707	+	0.998643i	$0.483418\pi$
$878$	0	0
$879$	42.9154	1.44750
$880$	0	0
$881$	−19.7931	−0.666848	−0.333424	−	0.942777i	$-0.608204\pi$
$881$	−19.7931	−0.666848	−0.333424	+	0.942777i	$0.608204\pi$
$882$	0	0
$883$	−7.24420	−0.243787	−0.121893	−	0.992543i	$-0.538897\pi$
$883$	−7.24420	−0.243787	−0.121893	+	0.992543i	$0.538897\pi$
$884$	0	0
$885$	100.426	3.37580
$886$	0	0
$887$	−57.8221	−1.94148	−0.970738	−	0.240139i	$-0.922807\pi$
$887$	−57.8221	−1.94148	−0.970738	+	0.240139i	$0.922807\pi$
$888$	0	0
$889$	−7.69408	−0.258051
$890$	0	0
$891$	7.73334	0.259077
$892$	0	0
$893$	−36.5597	−1.22342
$894$	0	0
$895$	−2.23139	−0.0745871
$896$	0	0
$897$	−7.07550	−0.236244
$898$	0	0
$899$	0.0675673	0.00225350
$900$	0	0
$901$	−19.9300	−0.663965
$902$	0	0
$903$	14.1301	0.470220
$904$	0	0
$905$	49.8952	1.65857
$906$	0	0
$907$	10.5234	0.349423	0.174711	−	0.984620i	$-0.444101\pi$
$907$	10.5234	0.349423	0.174711	+	0.984620i	$0.444101\pi$
$908$	0	0
$909$	−2.84136	−0.0942419
$910$	0	0
$911$	−0.294005	−0.00974081	−0.00487041	−	0.999988i	$-0.501550\pi$
$911$	−0.294005	−0.00974081	−0.00487041	+	0.999988i	$0.501550\pi$
$912$	0	0
$913$	5.01797	0.166071
$914$	0	0
$915$	1.59395	0.0526944
$916$	0	0
$917$	−3.11125	−0.102742
$918$	0	0
$919$	37.9983	1.25345	0.626725	−	0.779241i	$-0.284395\pi$
$919$	37.9983	1.25345	0.626725	+	0.779241i	$0.284395\pi$
$920$	0	0
$921$	−52.4660	−1.72881
$922$	0	0
$923$	5.80420	0.191048
$924$	0	0
$925$	40.6377	1.33616
$926$	0	0
$927$	45.3200	1.48850
$928$	0	0
$929$	−30.6251	−1.00478	−0.502388	−	0.864642i	$-0.667545\pi$
$929$	−30.6251	−1.00478	−0.502388	+	0.864642i	$0.667545\pi$
$930$	0	0
$931$	2.74521	0.0899706
$932$	0	0
$933$	−1.22055	−0.0399591
$934$	0	0
$935$	22.5894	0.738752
$936$	0	0
$937$	11.0781	0.361906	0.180953	−	0.983492i	$-0.442082\pi$
$937$	11.0781	0.361906	0.180953	+	0.983492i	$0.442082\pi$
$938$	0	0
$939$	11.1960	0.365369
$940$	0	0
$941$	20.9511	0.682986	0.341493	−	0.939884i	$-0.389067\pi$
$941$	20.9511	0.682986	0.341493	+	0.939884i	$0.389067\pi$
$942$	0	0
$943$	−32.1541	−1.04708
$944$	0	0
$945$	−3.25894	−0.106013
$946$	0	0
$947$	52.2717	1.69860	0.849300	−	0.527910i	$-0.177024\pi$
$947$	52.2717	1.69860	0.849300	+	0.527910i	$0.177024\pi$
$948$	0	0
$949$	8.78419	0.285147
$950$	0	0
$951$	−44.0116	−1.42717
$952$	0	0
$953$	29.4013	0.952402	0.476201	−	0.879336i	$-0.342013\pi$
$953$	29.4013	0.952402	0.476201	+	0.879336i	$0.342013\pi$
$954$	0	0
$955$	−53.5063	−1.73142
$956$	0	0
$957$	0.606448	0.0196037
$958$	0	0
$959$	−1.81615	−0.0586466
$960$	0	0
$961$	−30.9209	−0.997447
$962$	0	0
$963$	−21.2306	−0.684146
$964$	0	0
$965$	61.8304	1.99039
$966$	0	0
$967$	−7.32741	−0.235634	−0.117817	−	0.993035i	$-0.537590\pi$
$967$	−7.32741	−0.235634	−0.117817	+	0.993035i	$0.537590\pi$
$968$	0	0
$969$	45.5254	1.46249
$970$	0	0
$971$	−48.1549	−1.54536	−0.772682	−	0.634793i	$-0.781085\pi$
$971$	−48.1549	−1.54536	−0.772682	+	0.634793i	$0.781085\pi$
$972$	0	0
$973$	−5.63326	−0.180594
$974$	0	0
$975$	17.2431	0.552220
$976$	0	0
$977$	38.6859	1.23767	0.618835	−	0.785521i	$-0.287605\pi$
$977$	38.6859	1.23767	0.618835	+	0.785521i	$0.287605\pi$
$978$	0	0
$979$	−3.72200	−0.118956
$980$	0	0
$981$	6.56576	0.209629
$982$	0	0
$983$	−47.0645	−1.50113	−0.750563	−	0.660799i	$-0.770218\pi$
$983$	−47.0645	−1.50113	−0.750563	+	0.660799i	$0.770218\pi$
$984$	0	0
$985$	−29.0177	−0.924580
$986$	0	0
$987$	33.6261	1.07033
$988$	0	0
$989$	15.6820	0.498660
$990$	0	0
$991$	−26.6427	−0.846335	−0.423167	−	0.906052i	$-0.639082\pi$
$991$	−26.6427	−0.846335	−0.423167	+	0.906052i	$0.639082\pi$
$992$	0	0
$993$	77.3244	2.45381
$994$	0	0
$995$	49.4471	1.56758
$996$	0	0
$997$	−15.4075	−0.487959	−0.243980	−	0.969780i	$-0.578453\pi$
$997$	−15.4075	−0.487959	−0.243980	+	0.969780i	$0.578453\pi$
$998$	0	0
$999$	−5.63852	−0.178395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.3	✓	14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.52493 q^{3} +3.43935 q^{5} +1.00000 q^{7} +3.37528 q^{9} +O(q^{10})\)	\(q-2.52493 q^{3} +3.43935 q^{5} +1.00000 q^{7} +3.37528 q^{9} -1.00000 q^{11} -1.00000 q^{13} -8.68412 q^{15} -6.56793 q^{17} +2.74521 q^{19} -2.52493 q^{21} -2.80225 q^{23} +6.82912 q^{25} -0.947546 q^{27} +0.240184 q^{29} +0.281315 q^{31} +2.52493 q^{33} +3.43935 q^{35} +5.95065 q^{37} +2.52493 q^{39} +11.4744 q^{41} -5.59622 q^{43} +11.6088 q^{45} -13.3176 q^{47} +1.00000 q^{49} +16.5836 q^{51} +3.03445 q^{53} -3.43935 q^{55} -6.93147 q^{57} -11.5644 q^{59} -0.183548 q^{61} +3.37528 q^{63} -3.43935 q^{65} +0.658984 q^{67} +7.07550 q^{69} -5.80420 q^{71} -8.78419 q^{73} -17.2431 q^{75} -1.00000 q^{77} -0.408207 q^{79} -7.73334 q^{81} -5.01797 q^{83} -22.5894 q^{85} -0.606448 q^{87} +3.72200 q^{89} -1.00000 q^{91} -0.710301 q^{93} +9.44174 q^{95} +1.91747 q^{97} -3.37528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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