LMFDB - Embedded newform 2008.2.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.452441 q^{3} +1.91039 q^{5} +1.96124 q^{7} -2.79530 q^{9} +O(q^{10})$	$q+0.452441 q^{3} +1.91039 q^{5} +1.96124 q^{7} -2.79530 q^{9} +0.787387 q^{11} +5.79638 q^{13} +0.864341 q^{15} -3.13249 q^{17} +4.63676 q^{19} +0.887346 q^{21} +2.51256 q^{23} -1.35039 q^{25} -2.62203 q^{27} +3.62687 q^{29} -6.20008 q^{31} +0.356246 q^{33} +3.74674 q^{35} +9.71962 q^{37} +2.62252 q^{39} -0.892103 q^{41} -5.80229 q^{43} -5.34012 q^{45} +3.65849 q^{47} -3.15353 q^{49} -1.41727 q^{51} +12.6489 q^{53} +1.50422 q^{55} +2.09786 q^{57} +6.08621 q^{59} +6.47835 q^{61} -5.48225 q^{63} +11.0734 q^{65} -8.94464 q^{67} +1.13679 q^{69} +7.14955 q^{71} -11.3319 q^{73} -0.610974 q^{75} +1.54426 q^{77} -11.9354 q^{79} +7.19957 q^{81} +4.55162 q^{83} -5.98429 q^{85} +1.64095 q^{87} +1.55704 q^{89} +11.3681 q^{91} -2.80517 q^{93} +8.85805 q^{95} +5.55502 q^{97} -2.20098 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * 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
$3$	0.452441	0.261217	0.130609	−	0.991434i	$-0.458307\pi$
$3$	0.452441	0.261217	0.130609	+	0.991434i	$0.458307\pi$
$4$	0	0
$5$	1.91039	0.854354	0.427177	−	0.904168i	$-0.359508\pi$
$5$	1.91039	0.854354	0.427177	+	0.904168i	$0.359508\pi$
$6$	0	0
$7$	1.96124	0.741279	0.370640	−	0.928777i	$-0.379138\pi$
$7$	1.96124	0.741279	0.370640	+	0.928777i	$0.379138\pi$
$8$	0	0
$9$	−2.79530	−0.931766
$10$	0	0
$11$	0.787387	0.237406	0.118703	−	0.992930i	$-0.462126\pi$
$11$	0.787387	0.237406	0.118703	+	0.992930i	$0.462126\pi$
$12$	0	0
$13$	5.79638	1.60763	0.803814	−	0.594881i	$-0.202801\pi$
$13$	5.79638	1.60763	0.803814	+	0.594881i	$0.202801\pi$
$14$	0	0
$15$	0.864341	0.223172
$16$	0	0
$17$	−3.13249	−0.759740	−0.379870	−	0.925040i	$-0.624031\pi$
$17$	−3.13249	−0.759740	−0.379870	+	0.925040i	$0.624031\pi$
$18$	0	0
$19$	4.63676	1.06375	0.531873	−	0.846824i	$-0.321488\pi$
$19$	4.63676	1.06375	0.531873	+	0.846824i	$0.321488\pi$
$20$	0	0
$21$	0.887346	0.193635
$22$	0	0
$23$	2.51256	0.523906	0.261953	−	0.965081i	$-0.415633\pi$
$23$	2.51256	0.523906	0.261953	+	0.965081i	$0.415633\pi$
$24$	0	0
$25$	−1.35039	−0.270079
$26$	0	0
$27$	−2.62203	−0.504610
$28$	0	0
$29$	3.62687	0.673493	0.336746	−	0.941595i	$-0.390674\pi$
$29$	3.62687	0.673493	0.336746	+	0.941595i	$0.390674\pi$
$30$	0	0
$31$	−6.20008	−1.11357	−0.556783	−	0.830658i	$-0.687965\pi$
$31$	−6.20008	−1.11357	−0.556783	+	0.830658i	$0.687965\pi$
$32$	0	0
$33$	0.356246	0.0620145
$34$	0	0
$35$	3.74674	0.633315
$36$	0	0
$37$	9.71962	1.59790	0.798948	−	0.601400i	$-0.205390\pi$
$37$	9.71962	1.59790	0.798948	+	0.601400i	$0.205390\pi$
$38$	0	0
$39$	2.62252	0.419940
$40$	0	0
$41$	−0.892103	−0.139323	−0.0696615	−	0.997571i	$-0.522192\pi$
$41$	−0.892103	−0.139323	−0.0696615	+	0.997571i	$0.522192\pi$
$42$	0	0
$43$	−5.80229	−0.884840	−0.442420	−	0.896808i	$-0.645880\pi$
$43$	−5.80229	−0.884840	−0.442420	+	0.896808i	$0.645880\pi$
$44$	0	0
$45$	−5.34012	−0.796058
$46$	0	0
$47$	3.65849	0.533646	0.266823	−	0.963746i	$-0.414026\pi$
$47$	3.65849	0.533646	0.266823	+	0.963746i	$0.414026\pi$
$48$	0	0
$49$	−3.15353	−0.450505
$50$	0	0
$51$	−1.41727	−0.198457
$52$	0	0
$53$	12.6489	1.73746	0.868729	−	0.495288i	$-0.164938\pi$
$53$	12.6489	1.73746	0.868729	+	0.495288i	$0.164938\pi$
$54$	0	0
$55$	1.50422	0.202829
$56$	0	0
$57$	2.09786	0.277869
$58$	0	0
$59$	6.08621	0.792357	0.396178	−	0.918174i	$-0.370336\pi$
$59$	6.08621	0.792357	0.396178	+	0.918174i	$0.370336\pi$
$60$	0	0
$61$	6.47835	0.829467	0.414734	−	0.909943i	$-0.363875\pi$
$61$	6.47835	0.829467	0.414734	+	0.909943i	$0.363875\pi$
$62$	0	0
$63$	−5.48225	−0.690699
$64$	0	0
$65$	11.0734	1.37348
$66$	0	0
$67$	−8.94464	−1.09276	−0.546381	−	0.837537i	$-0.683995\pi$
$67$	−8.94464	−1.09276	−0.546381	+	0.837537i	$0.683995\pi$
$68$	0	0
$69$	1.13679	0.136853
$70$	0	0
$71$	7.14955	0.848496	0.424248	−	0.905546i	$-0.360539\pi$
$71$	7.14955	0.848496	0.424248	+	0.905546i	$0.360539\pi$
$72$	0	0
$73$	−11.3319	−1.32629	−0.663147	−	0.748489i	$-0.730780\pi$
$73$	−11.3319	−1.32629	−0.663147	+	0.748489i	$0.730780\pi$
$74$	0	0
$75$	−0.610974	−0.0705492
$76$	0	0
$77$	1.54426	0.175984
$78$	0	0
$79$	−11.9354	−1.34283	−0.671416	−	0.741080i	$-0.734314\pi$
$79$	−11.9354	−1.34283	−0.671416	+	0.741080i	$0.734314\pi$
$80$	0	0
$81$	7.19957	0.799953
$82$	0	0
$83$	4.55162	0.499606	0.249803	−	0.968297i	$-0.419634\pi$
$83$	4.55162	0.499606	0.249803	+	0.968297i	$0.419634\pi$
$84$	0	0
$85$	−5.98429	−0.649087
$86$	0	0
$87$	1.64095	0.175928
$88$	0	0
$89$	1.55704	0.165046	0.0825228	−	0.996589i	$-0.473702\pi$
$89$	1.55704	0.165046	0.0825228	+	0.996589i	$0.473702\pi$
$90$	0	0
$91$	11.3681	1.19170
$92$	0	0
$93$	−2.80517	−0.290883
$94$	0	0
$95$	8.85805	0.908816
$96$	0	0
$97$	5.55502	0.564027	0.282013	−	0.959410i	$-0.408998\pi$
$97$	5.55502	0.564027	0.282013	+	0.959410i	$0.408998\pi$
$98$	0	0
$99$	−2.20098	−0.221207
$100$	0	0
$101$	2.50744	0.249500	0.124750	−	0.992188i	$-0.460187\pi$
$101$	2.50744	0.249500	0.124750	+	0.992188i	$0.460187\pi$
$102$	0	0
$103$	−15.7471	−1.55161	−0.775805	−	0.630973i	$-0.782656\pi$
$103$	−15.7471	−1.55161	−0.775805	+	0.630973i	$0.782656\pi$
$104$	0	0
$105$	1.69518	0.165433
$106$	0	0
$107$	−14.5671	−1.40826	−0.704129	−	0.710072i	$-0.748662\pi$
$107$	−14.5671	−1.40826	−0.704129	+	0.710072i	$0.748662\pi$
$108$	0	0
$109$	19.2481	1.84364	0.921818	−	0.387623i	$-0.126704\pi$
$109$	19.2481	1.84364	0.921818	+	0.387623i	$0.126704\pi$
$110$	0	0
$111$	4.39756	0.417398
$112$	0	0
$113$	15.5531	1.46311	0.731555	−	0.681782i	$-0.238795\pi$
$113$	15.5531	1.46311	0.731555	+	0.681782i	$0.238795\pi$
$114$	0	0
$115$	4.79999	0.447601
$116$	0	0
$117$	−16.2026	−1.49793
$118$	0	0
$119$	−6.14356	−0.563179
$120$	0	0
$121$	−10.3800	−0.943638
$122$	0	0
$123$	−0.403624	−0.0363936
$124$	0	0
$125$	−12.1318	−1.08510
$126$	0	0
$127$	2.29660	0.203790	0.101895	−	0.994795i	$-0.467509\pi$
$127$	2.29660	0.203790	0.101895	+	0.994795i	$0.467509\pi$
$128$	0	0
$129$	−2.62519	−0.231135
$130$	0	0
$131$	−11.3850	−0.994711	−0.497355	−	0.867547i	$-0.665695\pi$
$131$	−11.3850	−0.994711	−0.497355	+	0.867547i	$0.665695\pi$
$132$	0	0
$133$	9.09381	0.788533
$134$	0	0
$135$	−5.00912	−0.431116
$136$	0	0
$137$	−20.6460	−1.76390	−0.881951	−	0.471340i	$-0.843770\pi$
$137$	−20.6460	−1.76390	−0.881951	+	0.471340i	$0.843770\pi$
$138$	0	0
$139$	18.4827	1.56769	0.783843	−	0.620959i	$-0.213257\pi$
$139$	18.4827	1.56769	0.783843	+	0.620959i	$0.213257\pi$
$140$	0	0
$141$	1.65525	0.139398
$142$	0	0
$143$	4.56400	0.381661
$144$	0	0
$145$	6.92875	0.575401
$146$	0	0
$147$	−1.42679	−0.117680
$148$	0	0
$149$	5.25249	0.430301	0.215150	−	0.976581i	$-0.430976\pi$
$149$	5.25249	0.430301	0.215150	+	0.976581i	$0.430976\pi$
$150$	0	0
$151$	23.5014	1.91252	0.956258	−	0.292525i	$-0.0944956\pi$
$151$	23.5014	1.91252	0.956258	+	0.292525i	$0.0944956\pi$
$152$	0	0
$153$	8.75623	0.707899
$154$	0	0
$155$	−11.8446	−0.951381
$156$	0	0
$157$	−0.0874349	−0.00697807	−0.00348903	−	0.999994i	$-0.501111\pi$
$157$	−0.0874349	−0.00697807	−0.00348903	+	0.999994i	$0.501111\pi$
$158$	0	0
$159$	5.72288	0.453854
$160$	0	0
$161$	4.92774	0.388361
$162$	0	0
$163$	−19.1980	−1.50371	−0.751853	−	0.659331i	$-0.770840\pi$
$163$	−19.1980	−1.50371	−0.751853	+	0.659331i	$0.770840\pi$
$164$	0	0
$165$	0.680571	0.0529824
$166$	0	0
$167$	1.82664	0.141349	0.0706747	−	0.997499i	$-0.477485\pi$
$167$	1.82664	0.141349	0.0706747	+	0.997499i	$0.477485\pi$
$168$	0	0
$169$	20.5981	1.58447
$170$	0	0
$171$	−12.9611	−0.991162
$172$	0	0
$173$	21.4790	1.63302	0.816509	−	0.577333i	$-0.195906\pi$
$173$	21.4790	1.63302	0.816509	+	0.577333i	$0.195906\pi$
$174$	0	0
$175$	−2.64845	−0.200204
$176$	0	0
$177$	2.75365	0.206977
$178$	0	0
$179$	7.53705	0.563346	0.281673	−	0.959510i	$-0.409111\pi$
$179$	7.53705	0.563346	0.281673	+	0.959510i	$0.409111\pi$
$180$	0	0
$181$	2.63731	0.196029	0.0980147	−	0.995185i	$-0.468751\pi$
$181$	2.63731	0.196029	0.0980147	+	0.995185i	$0.468751\pi$
$182$	0	0
$183$	2.93107	0.216671
$184$	0	0
$185$	18.5683	1.36517
$186$	0	0
$187$	−2.46648	−0.180367
$188$	0	0
$189$	−5.14244	−0.374057
$190$	0	0
$191$	−5.11657	−0.370222	−0.185111	−	0.982718i	$-0.559264\pi$
$191$	−5.11657	−0.370222	−0.185111	+	0.982718i	$0.559264\pi$
$192$	0	0
$193$	−1.28124	−0.0922257	−0.0461128	−	0.998936i	$-0.514683\pi$
$193$	−1.28124	−0.0922257	−0.0461128	+	0.998936i	$0.514683\pi$
$194$	0	0
$195$	5.01006	0.358778
$196$	0	0
$197$	12.8801	0.917667	0.458833	−	0.888522i	$-0.348267\pi$
$197$	12.8801	0.917667	0.458833	+	0.888522i	$0.348267\pi$
$198$	0	0
$199$	−24.0559	−1.70528	−0.852638	−	0.522503i	$-0.824999\pi$
$199$	−24.0559	−1.70528	−0.852638	+	0.522503i	$0.824999\pi$
$200$	0	0
$201$	−4.04693	−0.285448
$202$	0	0
$203$	7.11316	0.499246
$204$	0	0
$205$	−1.70427	−0.119031
$206$	0	0
$207$	−7.02336	−0.488157
$208$	0	0
$209$	3.65093	0.252540
$210$	0	0
$211$	−10.7763	−0.741874	−0.370937	−	0.928658i	$-0.620963\pi$
$211$	−10.7763	−0.741874	−0.370937	+	0.928658i	$0.620963\pi$
$212$	0	0
$213$	3.23475	0.221642
$214$	0	0
$215$	−11.0847	−0.755967
$216$	0	0
$217$	−12.1598	−0.825464
$218$	0	0
$219$	−5.12700	−0.346451
$220$	0	0
$221$	−18.1571	−1.22138
$222$	0	0
$223$	15.8073	1.05854	0.529268	−	0.848455i	$-0.322466\pi$
$223$	15.8073	1.05854	0.529268	+	0.848455i	$0.322466\pi$
$224$	0	0
$225$	3.77475	0.251650
$226$	0	0
$227$	17.8386	1.18399	0.591995	−	0.805942i	$-0.298341\pi$
$227$	17.8386	1.18399	0.591995	+	0.805942i	$0.298341\pi$
$228$	0	0
$229$	−24.0567	−1.58971	−0.794856	−	0.606798i	$-0.792454\pi$
$229$	−24.0567	−1.58971	−0.794856	+	0.606798i	$0.792454\pi$
$230$	0	0
$231$	0.698685	0.0459701
$232$	0	0
$233$	15.4663	1.01323	0.506616	−	0.862172i	$-0.330896\pi$
$233$	15.4663	1.01323	0.506616	+	0.862172i	$0.330896\pi$
$234$	0	0
$235$	6.98917	0.455923
$236$	0	0
$237$	−5.40005	−0.350771
$238$	0	0
$239$	4.57069	0.295653	0.147827	−	0.989013i	$-0.452772\pi$
$239$	4.57069	0.295653	0.147827	+	0.989013i	$0.452772\pi$
$240$	0	0
$241$	15.9646	1.02837	0.514185	−	0.857679i	$-0.328095\pi$
$241$	15.9646	1.02837	0.514185	+	0.857679i	$0.328095\pi$
$242$	0	0
$243$	11.1235	0.713572
$244$	0	0
$245$	−6.02449	−0.384891
$246$	0	0
$247$	26.8765	1.71011
$248$	0	0
$249$	2.05934	0.130506
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	1.97836	0.124378
$254$	0	0
$255$	−2.70754	−0.169553
$256$	0	0
$257$	−8.72312	−0.544133	−0.272067	−	0.962278i	$-0.587707\pi$
$257$	−8.72312	−0.544133	−0.272067	+	0.962278i	$0.587707\pi$
$258$	0	0
$259$	19.0625	1.18449
$260$	0	0
$261$	−10.1382	−0.627537
$262$	0	0
$263$	−23.8788	−1.47243	−0.736214	−	0.676749i	$-0.763388\pi$
$263$	−23.8788	−1.47243	−0.736214	+	0.676749i	$0.763388\pi$
$264$	0	0
$265$	24.1643	1.48440
$266$	0	0
$267$	0.704468	0.0431127
$268$	0	0
$269$	13.4422	0.819587	0.409794	−	0.912178i	$-0.365601\pi$
$269$	13.4422	0.819587	0.409794	+	0.912178i	$0.365601\pi$
$270$	0	0
$271$	−11.5155	−0.699518	−0.349759	−	0.936840i	$-0.613737\pi$
$271$	−11.5155	−0.699518	−0.349759	+	0.936840i	$0.613737\pi$
$272$	0	0
$273$	5.14340	0.311293
$274$	0	0
$275$	−1.06328	−0.0641183
$276$	0	0
$277$	−7.37275	−0.442986	−0.221493	−	0.975162i	$-0.571093\pi$
$277$	−7.37275	−0.442986	−0.221493	+	0.975162i	$0.571093\pi$
$278$	0	0
$279$	17.3311	1.03758
$280$	0	0
$281$	−19.9735	−1.19152	−0.595758	−	0.803164i	$-0.703148\pi$
$281$	−19.9735	−1.19152	−0.595758	+	0.803164i	$0.703148\pi$
$282$	0	0
$283$	−28.0411	−1.66687	−0.833434	−	0.552619i	$-0.813629\pi$
$283$	−28.0411	−1.66687	−0.833434	+	0.552619i	$0.813629\pi$
$284$	0	0
$285$	4.00775	0.237398
$286$	0	0
$287$	−1.74963	−0.103277
$288$	0	0
$289$	−7.18753	−0.422796
$290$	0	0
$291$	2.51332	0.147333
$292$	0	0
$293$	−24.2201	−1.41496	−0.707478	−	0.706735i	$-0.750167\pi$
$293$	−24.2201	−1.41496	−0.707478	+	0.706735i	$0.750167\pi$
$294$	0	0
$295$	11.6271	0.676954
$296$	0	0
$297$	−2.06455	−0.119798
$298$	0	0
$299$	14.5638	0.842246
$300$	0	0
$301$	−11.3797	−0.655914
$302$	0	0
$303$	1.13447	0.0651737
$304$	0	0
$305$	12.3762	0.708659
$306$	0	0
$307$	14.9324	0.852236	0.426118	−	0.904667i	$-0.359881\pi$
$307$	14.9324	0.852236	0.426118	+	0.904667i	$0.359881\pi$
$308$	0	0
$309$	−7.12464	−0.405307
$310$	0	0
$311$	−26.2949	−1.49105	−0.745524	−	0.666479i	$-0.767801\pi$
$311$	−26.2949	−1.49105	−0.745524	+	0.666479i	$0.767801\pi$
$312$	0	0
$313$	−3.51404	−0.198625	−0.0993127	−	0.995056i	$-0.531664\pi$
$313$	−3.51404	−0.198625	−0.0993127	+	0.995056i	$0.531664\pi$
$314$	0	0
$315$	−10.4733	−0.590101
$316$	0	0
$317$	−15.1169	−0.849047	−0.424524	−	0.905417i	$-0.639558\pi$
$317$	−15.1169	−0.849047	−0.424524	+	0.905417i	$0.639558\pi$
$318$	0	0
$319$	2.85575	0.159891
$320$	0	0
$321$	−6.59077	−0.367861
$322$	0	0
$323$	−14.5246	−0.808170
$324$	0	0
$325$	−7.82740	−0.434186
$326$	0	0
$327$	8.70865	0.481589
$328$	0	0
$329$	7.17519	0.395581
$330$	0	0
$331$	−34.0881	−1.87365	−0.936825	−	0.349799i	$-0.886250\pi$
$331$	−34.0881	−1.87365	−0.936825	+	0.349799i	$0.886250\pi$
$332$	0	0
$333$	−27.1692	−1.48886
$334$	0	0
$335$	−17.0878	−0.933606
$336$	0	0
$337$	−2.67369	−0.145645	−0.0728227	−	0.997345i	$-0.523201\pi$
$337$	−2.67369	−0.145645	−0.0728227	+	0.997345i	$0.523201\pi$
$338$	0	0
$339$	7.03686	0.382190
$340$	0	0
$341$	−4.88186	−0.264368
$342$	0	0
$343$	−19.9135	−1.07523
$344$	0	0
$345$	2.17171	0.116921
$346$	0	0
$347$	−24.8044	−1.33157	−0.665786	−	0.746143i	$-0.731904\pi$
$347$	−24.8044	−1.33157	−0.665786	+	0.746143i	$0.731904\pi$
$348$	0	0
$349$	−1.31555	−0.0704198	−0.0352099	−	0.999380i	$-0.511210\pi$
$349$	−1.31555	−0.0704198	−0.0352099	+	0.999380i	$0.511210\pi$
$350$	0	0
$351$	−15.1983	−0.811226
$352$	0	0
$353$	−17.7470	−0.944576	−0.472288	−	0.881444i	$-0.656572\pi$
$353$	−17.7470	−0.944576	−0.472288	+	0.881444i	$0.656572\pi$
$354$	0	0
$355$	13.6585	0.724916
$356$	0	0
$357$	−2.77960	−0.147112
$358$	0	0
$359$	−14.9012	−0.786457	−0.393228	−	0.919441i	$-0.628642\pi$
$359$	−14.9012	−0.786457	−0.393228	+	0.919441i	$0.628642\pi$
$360$	0	0
$361$	2.49957	0.131556
$362$	0	0
$363$	−4.69635	−0.246494
$364$	0	0
$365$	−21.6483	−1.13313
$366$	0	0
$367$	−26.1918	−1.36720	−0.683600	−	0.729857i	$-0.739587\pi$
$367$	−26.1918	−1.36720	−0.683600	+	0.729857i	$0.739587\pi$
$368$	0	0
$369$	2.49369	0.129816
$370$	0	0
$371$	24.8075	1.28794
$372$	0	0
$373$	−1.81955	−0.0942130	−0.0471065	−	0.998890i	$-0.515000\pi$
$373$	−1.81955	−0.0942130	−0.0471065	+	0.998890i	$0.515000\pi$
$374$	0	0
$375$	−5.48891	−0.283446
$376$	0	0
$377$	21.0227	1.08273
$378$	0	0
$379$	1.39119	0.0714607	0.0357304	−	0.999361i	$-0.488624\pi$
$379$	1.39119	0.0714607	0.0357304	+	0.999361i	$0.488624\pi$
$380$	0	0
$381$	1.03908	0.0532335
$382$	0	0
$383$	5.65910	0.289167	0.144583	−	0.989493i	$-0.453816\pi$
$383$	5.65910	0.289167	0.144583	+	0.989493i	$0.453816\pi$
$384$	0	0
$385$	2.95014	0.150353
$386$	0	0
$387$	16.2191	0.824464
$388$	0	0
$389$	33.3761	1.69224	0.846118	−	0.532996i	$-0.178934\pi$
$389$	33.3761	1.69224	0.846118	+	0.532996i	$0.178934\pi$
$390$	0	0
$391$	−7.87057	−0.398032
$392$	0	0
$393$	−5.15104	−0.259835
$394$	0	0
$395$	−22.8012	−1.14726
$396$	0	0
$397$	2.80826	0.140943	0.0704713	−	0.997514i	$-0.477550\pi$
$397$	2.80826	0.140943	0.0704713	+	0.997514i	$0.477550\pi$
$398$	0	0
$399$	4.11442	0.205978
$400$	0	0
$401$	20.0554	1.00152	0.500759	−	0.865587i	$-0.333054\pi$
$401$	20.0554	1.00152	0.500759	+	0.865587i	$0.333054\pi$
$402$	0	0
$403$	−35.9380	−1.79020
$404$	0	0
$405$	13.7540	0.683443
$406$	0	0
$407$	7.65310	0.379350
$408$	0	0
$409$	5.60055	0.276929	0.138465	−	0.990367i	$-0.455783\pi$
$409$	5.60055	0.276929	0.138465	+	0.990367i	$0.455783\pi$
$410$	0	0
$411$	−9.34108	−0.460762
$412$	0	0
$413$	11.9365	0.587358
$414$	0	0
$415$	8.69540	0.426840
$416$	0	0
$417$	8.36236	0.409506
$418$	0	0
$419$	11.6046	0.566922	0.283461	−	0.958984i	$-0.408517\pi$
$419$	11.6046	0.566922	0.283461	+	0.958984i	$0.408517\pi$
$420$	0	0
$421$	−32.0502	−1.56203	−0.781014	−	0.624513i	$-0.785298\pi$
$421$	−32.0502	−1.56203	−0.781014	+	0.624513i	$0.785298\pi$
$422$	0	0
$423$	−10.2266	−0.497233
$424$	0	0
$425$	4.23009	0.205189
$426$	0	0
$427$	12.7056	0.614867
$428$	0	0
$429$	2.06494	0.0996963
$430$	0	0
$431$	−33.0806	−1.59344	−0.796719	−	0.604349i	$-0.793433\pi$
$431$	−33.0806	−1.59344	−0.796719	+	0.604349i	$0.793433\pi$
$432$	0	0
$433$	7.51349	0.361075	0.180538	−	0.983568i	$-0.442216\pi$
$433$	7.51349	0.361075	0.180538	+	0.983568i	$0.442216\pi$
$434$	0	0
$435$	3.13485	0.150305
$436$	0	0
$437$	11.6502	0.557303
$438$	0	0
$439$	−1.95743	−0.0934231	−0.0467116	−	0.998908i	$-0.514874\pi$
$439$	−1.95743	−0.0934231	−0.0467116	+	0.998908i	$0.514874\pi$
$440$	0	0
$441$	8.81506	0.419765
$442$	0	0
$443$	0.880392	0.0418287	0.0209143	−	0.999781i	$-0.493342\pi$
$443$	0.880392	0.0418287	0.0209143	+	0.999781i	$0.493342\pi$
$444$	0	0
$445$	2.97456	0.141007
$446$	0	0
$447$	2.37644	0.112402
$448$	0	0
$449$	−33.1514	−1.56451	−0.782255	−	0.622958i	$-0.785931\pi$
$449$	−33.1514	−1.56451	−0.782255	+	0.622958i	$0.785931\pi$
$450$	0	0
$451$	−0.702430	−0.0330761
$452$	0	0
$453$	10.6330	0.499582
$454$	0	0
$455$	21.7176	1.01814
$456$	0	0
$457$	33.5917	1.57136	0.785678	−	0.618636i	$-0.212314\pi$
$457$	33.5917	1.57136	0.785678	+	0.618636i	$0.212314\pi$
$458$	0	0
$459$	8.21348	0.383372
$460$	0	0
$461$	−14.2767	−0.664930	−0.332465	−	0.943116i	$-0.607880\pi$
$461$	−14.2767	−0.664930	−0.332465	+	0.943116i	$0.607880\pi$
$462$	0	0
$463$	−1.99819	−0.0928636	−0.0464318	−	0.998921i	$-0.514785\pi$
$463$	−1.99819	−0.0928636	−0.0464318	+	0.998921i	$0.514785\pi$
$464$	0	0
$465$	−5.35898	−0.248517
$466$	0	0
$467$	−22.7336	−1.05199	−0.525993	−	0.850489i	$-0.676306\pi$
$467$	−22.7336	−1.05199	−0.525993	+	0.850489i	$0.676306\pi$
$468$	0	0
$469$	−17.5426	−0.810042
$470$	0	0
$471$	−0.0395592	−0.00182279
$472$	0	0
$473$	−4.56864	−0.210066
$474$	0	0
$475$	−6.26145	−0.287295
$476$	0	0
$477$	−35.3574	−1.61890
$478$	0	0
$479$	−8.15942	−0.372813	−0.186407	−	0.982473i	$-0.559684\pi$
$479$	−8.15942	−0.372813	−0.186407	+	0.982473i	$0.559684\pi$
$480$	0	0
$481$	56.3387	2.56882
$482$	0	0
$483$	2.22951	0.101446
$484$	0	0
$485$	10.6123	0.481879
$486$	0	0
$487$	2.44996	0.111019	0.0555093	−	0.998458i	$-0.482322\pi$
$487$	2.44996	0.111019	0.0555093	+	0.998458i	$0.482322\pi$
$488$	0	0
$489$	−8.68599	−0.392794
$490$	0	0
$491$	12.7893	0.577171	0.288586	−	0.957454i	$-0.406815\pi$
$491$	12.7893	0.577171	0.288586	+	0.957454i	$0.406815\pi$
$492$	0	0
$493$	−11.3611	−0.511679
$494$	0	0
$495$	−4.20474	−0.188989
$496$	0	0
$497$	14.0220	0.628972
$498$	0	0
$499$	−0.938315	−0.0420048	−0.0210024	−	0.999779i	$-0.506686\pi$
$499$	−0.938315	−0.0420048	−0.0210024	+	0.999779i	$0.506686\pi$
$500$	0	0
$501$	0.826446	0.0369229
$502$	0	0
$503$	29.2146	1.30262	0.651308	−	0.758814i	$-0.274221\pi$
$503$	29.2146	1.30262	0.651308	+	0.758814i	$0.274221\pi$
$504$	0	0
$505$	4.79021	0.213161
$506$	0	0
$507$	9.31942	0.413890
$508$	0	0
$509$	40.9310	1.81423	0.907117	−	0.420879i	$-0.138278\pi$
$509$	40.9310	1.81423	0.907117	+	0.420879i	$0.138278\pi$
$510$	0	0
$511$	−22.2245	−0.983154
$512$	0	0
$513$	−12.1577	−0.536777
$514$	0	0
$515$	−30.0832	−1.32562
$516$	0	0
$517$	2.88065	0.126691
$518$	0	0
$519$	9.71799	0.426572
$520$	0	0
$521$	29.9804	1.31347	0.656733	−	0.754123i	$-0.271938\pi$
$521$	29.9804	1.31347	0.656733	+	0.754123i	$0.271938\pi$
$522$	0	0
$523$	−26.2955	−1.14982	−0.574911	−	0.818216i	$-0.694963\pi$
$523$	−26.2955	−1.14982	−0.574911	+	0.818216i	$0.694963\pi$
$524$	0	0
$525$	−1.19827	−0.0522966
$526$	0	0
$527$	19.4217	0.846021
$528$	0	0
$529$	−16.6870	−0.725523
$530$	0	0
$531$	−17.0128	−0.738291
$532$	0	0
$533$	−5.17097	−0.223980
$534$	0	0
$535$	−27.8290	−1.20315
$536$	0	0
$537$	3.41008	0.147156
$538$	0	0
$539$	−2.48305	−0.106953
$540$	0	0
$541$	−22.9843	−0.988171	−0.494085	−	0.869413i	$-0.664497\pi$
$541$	−22.9843	−0.988171	−0.494085	+	0.869413i	$0.664497\pi$
$542$	0	0
$543$	1.19323	0.0512063
$544$	0	0
$545$	36.7715	1.57512
$546$	0	0
$547$	7.79728	0.333387	0.166694	−	0.986009i	$-0.446691\pi$
$547$	7.79728	0.333387	0.166694	+	0.986009i	$0.446691\pi$
$548$	0	0
$549$	−18.1089	−0.772869
$550$	0	0
$551$	16.8169	0.716425
$552$	0	0
$553$	−23.4081	−0.995414
$554$	0	0
$555$	8.40107	0.356606
$556$	0	0
$557$	−26.6843	−1.13065	−0.565325	−	0.824868i	$-0.691249\pi$
$557$	−26.6843	−1.13065	−0.565325	+	0.824868i	$0.691249\pi$
$558$	0	0
$559$	−33.6323	−1.42249
$560$	0	0
$561$	−1.11594	−0.0471149
$562$	0	0
$563$	15.3602	0.647357	0.323679	−	0.946167i	$-0.395080\pi$
$563$	15.3602	0.647357	0.323679	+	0.946167i	$0.395080\pi$
$564$	0	0
$565$	29.7125	1.25001
$566$	0	0
$567$	14.1201	0.592988
$568$	0	0
$569$	2.53660	0.106340	0.0531700	−	0.998585i	$-0.483067\pi$
$569$	2.53660	0.106340	0.0531700	+	0.998585i	$0.483067\pi$
$570$	0	0
$571$	24.5090	1.02567	0.512836	−	0.858487i	$-0.328595\pi$
$571$	24.5090	1.02567	0.512836	+	0.858487i	$0.328595\pi$
$572$	0	0
$573$	−2.31495	−0.0967083
$574$	0	0
$575$	−3.39295	−0.141496
$576$	0	0
$577$	−37.4112	−1.55745	−0.778725	−	0.627366i	$-0.784133\pi$
$577$	−37.4112	−1.55745	−0.778725	+	0.627366i	$0.784133\pi$
$578$	0	0
$579$	−0.579686	−0.0240909
$580$	0	0
$581$	8.92683	0.370347
$582$	0	0
$583$	9.95956	0.412483
$584$	0	0
$585$	−30.9534	−1.27977
$586$	0	0
$587$	32.3390	1.33477	0.667387	−	0.744711i	$-0.267413\pi$
$587$	32.3390	1.33477	0.667387	+	0.744711i	$0.267413\pi$
$588$	0	0
$589$	−28.7483	−1.18455
$590$	0	0
$591$	5.82748	0.239710
$592$	0	0
$593$	−17.8932	−0.734785	−0.367392	−	0.930066i	$-0.619749\pi$
$593$	−17.8932	−0.734785	−0.367392	+	0.930066i	$0.619749\pi$
$594$	0	0
$595$	−11.7366	−0.481155
$596$	0	0
$597$	−10.8839	−0.445447
$598$	0	0
$599$	19.3881	0.792175	0.396088	−	0.918213i	$-0.370368\pi$
$599$	19.3881	0.792175	0.396088	+	0.918213i	$0.370368\pi$
$600$	0	0
$601$	−11.0687	−0.451503	−0.225751	−	0.974185i	$-0.572484\pi$
$601$	−11.0687	−0.451503	−0.225751	+	0.974185i	$0.572484\pi$
$602$	0	0
$603$	25.0029	1.01820
$604$	0	0
$605$	−19.8299	−0.806202
$606$	0	0
$607$	−30.8771	−1.25326	−0.626632	−	0.779316i	$-0.715567\pi$
$607$	−30.8771	−1.25326	−0.626632	+	0.779316i	$0.715567\pi$
$608$	0	0
$609$	3.21829	0.130412
$610$	0	0
$611$	21.2060	0.857905
$612$	0	0
$613$	9.94691	0.401752	0.200876	−	0.979617i	$-0.435621\pi$
$613$	9.94691	0.401752	0.200876	+	0.979617i	$0.435621\pi$
$614$	0	0
$615$	−0.771081	−0.0310930
$616$	0	0
$617$	−28.8807	−1.16269	−0.581346	−	0.813656i	$-0.697474\pi$
$617$	−28.8807	−1.16269	−0.581346	+	0.813656i	$0.697474\pi$
$618$	0	0
$619$	9.25598	0.372029	0.186015	−	0.982547i	$-0.440443\pi$
$619$	9.25598	0.372029	0.186015	+	0.982547i	$0.440443\pi$
$620$	0	0
$621$	−6.58802	−0.264368
$622$	0	0
$623$	3.05373	0.122345
$624$	0	0
$625$	−16.4245	−0.656979
$626$	0	0
$627$	1.65183	0.0659678
$628$	0	0
$629$	−30.4466	−1.21398
$630$	0	0
$631$	−7.76256	−0.309023	−0.154511	−	0.987991i	$-0.549380\pi$
$631$	−7.76256	−0.309023	−0.154511	+	0.987991i	$0.549380\pi$
$632$	0	0
$633$	−4.87566	−0.193790
$634$	0	0
$635$	4.38741	0.174109
$636$	0	0
$637$	−18.2791	−0.724244
$638$	0	0
$639$	−19.9851	−0.790599
$640$	0	0
$641$	−39.4109	−1.55664	−0.778319	−	0.627869i	$-0.783927\pi$
$641$	−39.4109	−1.55664	−0.778319	+	0.627869i	$0.783927\pi$
$642$	0	0
$643$	−36.5538	−1.44154	−0.720771	−	0.693174i	$-0.756212\pi$
$643$	−36.5538	−1.44154	−0.720771	+	0.693174i	$0.756212\pi$
$644$	0	0
$645$	−5.01516	−0.197472
$646$	0	0
$647$	−8.94030	−0.351479	−0.175740	−	0.984437i	$-0.556232\pi$
$647$	−8.94030	−0.351479	−0.175740	+	0.984437i	$0.556232\pi$
$648$	0	0
$649$	4.79220	0.188110
$650$	0	0
$651$	−5.50162	−0.215625
$652$	0	0
$653$	−12.1726	−0.476352	−0.238176	−	0.971222i	$-0.576549\pi$
$653$	−12.1726	−0.476352	−0.238176	+	0.971222i	$0.576549\pi$
$654$	0	0
$655$	−21.7498	−0.849835
$656$	0	0
$657$	31.6759	1.23580
$658$	0	0
$659$	29.4761	1.14823	0.574114	−	0.818776i	$-0.305347\pi$
$659$	29.4761	1.14823	0.574114	+	0.818776i	$0.305347\pi$
$660$	0	0
$661$	−10.5472	−0.410239	−0.205120	−	0.978737i	$-0.565758\pi$
$661$	−10.5472	−0.410239	−0.205120	+	0.978737i	$0.565758\pi$
$662$	0	0
$663$	−8.21502	−0.319045
$664$	0	0
$665$	17.3728	0.673687
$666$	0	0
$667$	9.11274	0.352847
$668$	0	0
$669$	7.15189	0.276508
$670$	0	0
$671$	5.10097	0.196921
$672$	0	0
$673$	−28.4861	−1.09806	−0.549029	−	0.835803i	$-0.685002\pi$
$673$	−28.4861	−1.09806	−0.549029	+	0.835803i	$0.685002\pi$
$674$	0	0
$675$	3.54077	0.136284
$676$	0	0
$677$	3.92813	0.150970	0.0754851	−	0.997147i	$-0.475949\pi$
$677$	3.92813	0.150970	0.0754851	+	0.997147i	$0.475949\pi$
$678$	0	0
$679$	10.8947	0.418101
$680$	0	0
$681$	8.07092	0.309278
$682$	0	0
$683$	−42.9694	−1.64418	−0.822089	−	0.569359i	$-0.807192\pi$
$683$	−42.9694	−1.64418	−0.822089	+	0.569359i	$0.807192\pi$
$684$	0	0
$685$	−39.4419	−1.50700
$686$	0	0
$687$	−10.8843	−0.415260
$688$	0	0
$689$	73.3178	2.79318
$690$	0	0
$691$	13.4562	0.511897	0.255949	−	0.966690i	$-0.417612\pi$
$691$	13.4562	0.511897	0.255949	+	0.966690i	$0.417612\pi$
$692$	0	0
$693$	−4.31665	−0.163976
$694$	0	0
$695$	35.3093	1.33936
$696$	0	0
$697$	2.79450	0.105849
$698$	0	0
$699$	6.99760	0.264673
$700$	0	0
$701$	19.9131	0.752107	0.376054	−	0.926598i	$-0.377281\pi$
$701$	19.9131	0.752107	0.376054	+	0.926598i	$0.377281\pi$
$702$	0	0
$703$	45.0676	1.69976
$704$	0	0
$705$	3.16219	0.119095
$706$	0	0
$707$	4.91770	0.184949
$708$	0	0
$709$	−12.1399	−0.455924	−0.227962	−	0.973670i	$-0.573206\pi$
$709$	−12.1399	−0.455924	−0.227962	+	0.973670i	$0.573206\pi$
$710$	0	0
$711$	33.3629	1.25121
$712$	0	0
$713$	−15.5781	−0.583404
$714$	0	0
$715$	8.71904	0.326074
$716$	0	0
$717$	2.06797	0.0772297
$718$	0	0
$719$	28.7871	1.07358	0.536789	−	0.843716i	$-0.319637\pi$
$719$	28.7871	1.07358	0.536789	+	0.843716i	$0.319637\pi$
$720$	0	0
$721$	−30.8839	−1.15018
$722$	0	0
$723$	7.22304	0.268628
$724$	0	0
$725$	−4.89770	−0.181896
$726$	0	0
$727$	36.9370	1.36992	0.684960	−	0.728581i	$-0.259820\pi$
$727$	36.9370	1.36992	0.684960	+	0.728581i	$0.259820\pi$
$728$	0	0
$729$	−16.5660	−0.613556
$730$	0	0
$731$	18.1756	0.672248
$732$	0	0
$733$	−11.7116	−0.432579	−0.216289	−	0.976329i	$-0.569395\pi$
$733$	−11.7116	−0.432579	−0.216289	+	0.976329i	$0.569395\pi$
$734$	0	0
$735$	−2.72573	−0.100540
$736$	0	0
$737$	−7.04289	−0.259428
$738$	0	0
$739$	−28.6602	−1.05428	−0.527141	−	0.849778i	$-0.676736\pi$
$739$	−28.6602	−1.05428	−0.527141	+	0.849778i	$0.676736\pi$
$740$	0	0
$741$	12.1600	0.446710
$742$	0	0
$743$	9.65811	0.354322	0.177161	−	0.984182i	$-0.443309\pi$
$743$	9.65811	0.354322	0.177161	+	0.984182i	$0.443309\pi$
$744$	0	0
$745$	10.0343	0.367629
$746$	0	0
$747$	−12.7231	−0.465515
$748$	0	0
$749$	−28.5696	−1.04391
$750$	0	0
$751$	−25.1095	−0.916260	−0.458130	−	0.888885i	$-0.651481\pi$
$751$	−25.1095	−0.916260	−0.458130	+	0.888885i	$0.651481\pi$
$752$	0	0
$753$	−0.452441	−0.0164879
$754$	0	0
$755$	44.8969	1.63397
$756$	0	0
$757$	5.68729	0.206708	0.103354	−	0.994645i	$-0.467043\pi$
$757$	5.68729	0.206708	0.103354	+	0.994645i	$0.467043\pi$
$758$	0	0
$759$	0.895092	0.0324898
$760$	0	0
$761$	9.65366	0.349945	0.174972	−	0.984573i	$-0.444016\pi$
$761$	9.65366	0.349945	0.174972	+	0.984573i	$0.444016\pi$
$762$	0	0
$763$	37.7502	1.36665
$764$	0	0
$765$	16.7279	0.604797
$766$	0	0
$767$	35.2780	1.27382
$768$	0	0
$769$	−6.79824	−0.245151	−0.122575	−	0.992459i	$-0.539115\pi$
$769$	−6.79824	−0.245151	−0.122575	+	0.992459i	$0.539115\pi$
$770$	0	0
$771$	−3.94670	−0.142137
$772$	0	0
$773$	−51.3076	−1.84541	−0.922704	−	0.385510i	$-0.874025\pi$
$773$	−51.3076	−1.84541	−0.922704	+	0.385510i	$0.874025\pi$
$774$	0	0
$775$	8.37254	0.300751
$776$	0	0
$777$	8.62467	0.309408
$778$	0	0
$779$	−4.13647	−0.148204
$780$	0	0
$781$	5.62947	0.201438
$782$	0	0
$783$	−9.50976	−0.339851
$784$	0	0
$785$	−0.167035	−0.00596174
$786$	0	0
$787$	36.4549	1.29948	0.649738	−	0.760158i	$-0.274879\pi$
$787$	36.4549	1.29948	0.649738	+	0.760158i	$0.274879\pi$
$788$	0	0
$789$	−10.8037	−0.384623
$790$	0	0
$791$	30.5033	1.08457
$792$	0	0
$793$	37.5510	1.33348
$794$	0	0
$795$	10.9330	0.387752
$796$	0	0
$797$	49.1111	1.73961	0.869803	−	0.493399i	$-0.164246\pi$
$797$	49.1111	1.73961	0.869803	+	0.493399i	$0.164246\pi$
$798$	0	0
$799$	−11.4602	−0.405432
$800$	0	0
$801$	−4.35238	−0.153784
$802$	0	0
$803$	−8.92256	−0.314870
$804$	0	0
$805$	9.41393	0.331798
$806$	0	0
$807$	6.08182	0.214090
$808$	0	0
$809$	−3.96420	−0.139374	−0.0696870	−	0.997569i	$-0.522200\pi$
$809$	−3.96420	−0.139374	−0.0696870	+	0.997569i	$0.522200\pi$
$810$	0	0
$811$	38.6956	1.35879	0.679393	−	0.733774i	$-0.262243\pi$
$811$	38.6956	1.35879	0.679393	+	0.733774i	$0.262243\pi$
$812$	0	0
$813$	−5.21010	−0.182726
$814$	0	0
$815$	−36.6758	−1.28470
$816$	0	0
$817$	−26.9038	−0.941246
$818$	0	0
$819$	−31.7772	−1.11039
$820$	0	0
$821$	−18.4011	−0.642202	−0.321101	−	0.947045i	$-0.604053\pi$
$821$	−18.4011	−0.642202	−0.321101	+	0.947045i	$0.604053\pi$
$822$	0	0
$823$	39.2854	1.36940	0.684701	−	0.728824i	$-0.259933\pi$
$823$	39.2854	1.36940	0.684701	+	0.728824i	$0.259933\pi$
$824$	0	0
$825$	−0.481073	−0.0167488
$826$	0	0
$827$	24.4883	0.851541	0.425771	−	0.904831i	$-0.360003\pi$
$827$	24.4883	0.851541	0.425771	+	0.904831i	$0.360003\pi$
$828$	0	0
$829$	3.40002	0.118088	0.0590438	−	0.998255i	$-0.481195\pi$
$829$	3.40002	0.118088	0.0590438	+	0.998255i	$0.481195\pi$
$830$	0	0
$831$	−3.33574	−0.115715
$832$	0	0
$833$	9.87841	0.342266
$834$	0	0
$835$	3.48960	0.120763
$836$	0	0
$837$	16.2568	0.561917
$838$	0	0
$839$	29.8387	1.03015	0.515074	−	0.857146i	$-0.327765\pi$
$839$	29.8387	1.03015	0.515074	+	0.857146i	$0.327765\pi$
$840$	0	0
$841$	−15.8458	−0.546408
$842$	0	0
$843$	−9.03681	−0.311244
$844$	0	0
$845$	39.3505	1.35370
$846$	0	0
$847$	−20.3577	−0.699500
$848$	0	0
$849$	−12.6869	−0.435415
$850$	0	0
$851$	24.4212	0.837147
$852$	0	0
$853$	2.73462	0.0936315	0.0468157	−	0.998904i	$-0.485093\pi$
$853$	2.73462	0.0936315	0.0468157	+	0.998904i	$0.485093\pi$
$854$	0	0
$855$	−24.7609	−0.846804
$856$	0	0
$857$	−16.3691	−0.559158	−0.279579	−	0.960123i	$-0.590195\pi$
$857$	−16.3691	−0.559158	−0.279579	+	0.960123i	$0.590195\pi$
$858$	0	0
$859$	−3.18932	−0.108818	−0.0544092	−	0.998519i	$-0.517328\pi$
$859$	−3.18932	−0.108818	−0.0544092	+	0.998519i	$0.517328\pi$
$860$	0	0
$861$	−0.791604	−0.0269778
$862$	0	0
$863$	−42.0992	−1.43307	−0.716537	−	0.697549i	$-0.754274\pi$
$863$	−42.0992	−1.43307	−0.716537	+	0.697549i	$0.754274\pi$
$864$	0	0
$865$	41.0334	1.39518
$866$	0	0
$867$	−3.25193	−0.110441
$868$	0	0
$869$	−9.39775	−0.318797
$870$	0	0
$871$	−51.8466	−1.75675
$872$	0	0
$873$	−15.5279	−0.525541
$874$	0	0
$875$	−23.7933	−0.804360
$876$	0	0
$877$	−9.14103	−0.308671	−0.154335	−	0.988019i	$-0.549324\pi$
$877$	−9.14103	−0.308671	−0.154335	+	0.988019i	$0.549324\pi$
$878$	0	0
$879$	−10.9582	−0.369611
$880$	0	0
$881$	3.14380	0.105917	0.0529586	−	0.998597i	$-0.483135\pi$
$881$	3.14380	0.105917	0.0529586	+	0.998597i	$0.483135\pi$
$882$	0	0
$883$	−30.6344	−1.03093	−0.515464	−	0.856911i	$-0.672380\pi$
$883$	−30.6344	−1.03093	−0.515464	+	0.856911i	$0.672380\pi$
$884$	0	0
$885$	5.26056	0.176832
$886$	0	0
$887$	17.3622	0.582965	0.291483	−	0.956576i	$-0.405851\pi$
$887$	17.3622	0.582965	0.291483	+	0.956576i	$0.405851\pi$
$888$	0	0
$889$	4.50418	0.151065
$890$	0	0
$891$	5.66885	0.189914
$892$	0	0
$893$	16.9636	0.567664
$894$	0	0
$895$	14.3987	0.481297
$896$	0	0
$897$	6.58926	0.220009
$898$	0	0
$899$	−22.4869	−0.749979
$900$	0	0
$901$	−39.6224	−1.32002
$902$	0	0
$903$	−5.14864	−0.171336
$904$	0	0
$905$	5.03830	0.167479
$906$	0	0
$907$	13.0635	0.433767	0.216883	−	0.976198i	$-0.430411\pi$
$907$	13.0635	0.433767	0.216883	+	0.976198i	$0.430411\pi$
$908$	0	0
$909$	−7.00905	−0.232475
$910$	0	0
$911$	23.2857	0.771491	0.385746	−	0.922605i	$-0.373944\pi$
$911$	23.2857	0.771491	0.385746	+	0.922605i	$0.373944\pi$
$912$	0	0
$913$	3.58389	0.118609
$914$	0	0
$915$	5.59950	0.185114
$916$	0	0
$917$	−22.3287	−0.737358
$918$	0	0
$919$	49.1025	1.61974	0.809871	−	0.586608i	$-0.199537\pi$
$919$	49.1025	1.61974	0.809871	+	0.586608i	$0.199537\pi$
$920$	0	0
$921$	6.75603	0.222619
$922$	0	0
$923$	41.4416	1.36407
$924$	0	0
$925$	−13.1253	−0.431557
$926$	0	0
$927$	44.0179	1.44574
$928$	0	0
$929$	40.9252	1.34271	0.671356	−	0.741135i	$-0.265712\pi$
$929$	40.9252	1.34271	0.671356	+	0.741135i	$0.265712\pi$
$930$	0	0
$931$	−14.6222	−0.479223
$932$	0	0
$933$	−11.8969	−0.389487
$934$	0	0
$935$	−4.71195	−0.154097
$936$	0	0
$937$	43.3581	1.41645	0.708224	−	0.705988i	$-0.249497\pi$
$937$	43.3581	1.41645	0.708224	+	0.705988i	$0.249497\pi$
$938$	0	0
$939$	−1.58990	−0.0518844
$940$	0	0
$941$	35.9586	1.17222	0.586108	−	0.810233i	$-0.300659\pi$
$941$	35.9586	1.17222	0.586108	+	0.810233i	$0.300659\pi$
$942$	0	0
$943$	−2.24146	−0.0729921
$944$	0	0
$945$	−9.82408	−0.319577
$946$	0	0
$947$	40.6909	1.32228	0.661138	−	0.750265i	$-0.270074\pi$
$947$	40.6909	1.32228	0.661138	+	0.750265i	$0.270074\pi$
$948$	0	0
$949$	−65.6838	−2.13219
$950$	0	0
$951$	−6.83949	−0.221786
$952$	0	0
$953$	2.43322	0.0788199	0.0394099	−	0.999223i	$-0.487452\pi$
$953$	2.43322	0.0788199	0.0394099	+	0.999223i	$0.487452\pi$
$954$	0	0
$955$	−9.77467	−0.316301
$956$	0	0
$957$	1.29206	0.0417663
$958$	0	0
$959$	−40.4917	−1.30754
$960$	0	0
$961$	7.44096	0.240031
$962$	0	0
$963$	40.7194	1.31217
$964$	0	0
$965$	−2.44767	−0.0787934
$966$	0	0
$967$	54.8236	1.76301	0.881504	−	0.472177i	$-0.156532\pi$
$967$	54.8236	1.76301	0.881504	+	0.472177i	$0.156532\pi$
$968$	0	0
$969$	−6.57153	−0.211108
$970$	0	0
$971$	−9.42580	−0.302488	−0.151244	−	0.988496i	$-0.548328\pi$
$971$	−9.42580	−0.302488	−0.151244	+	0.988496i	$0.548328\pi$
$972$	0	0
$973$	36.2491	1.16209
$974$	0	0
$975$	−3.54144	−0.113417
$976$	0	0
$977$	14.3979	0.460630	0.230315	−	0.973116i	$-0.426024\pi$
$977$	14.3979	0.460630	0.230315	+	0.973116i	$0.426024\pi$
$978$	0	0
$979$	1.22599	0.0391828
$980$	0	0
$981$	−53.8042	−1.71784
$982$	0	0
$983$	18.7587	0.598311	0.299156	−	0.954204i	$-0.403295\pi$
$983$	18.7587	0.598311	0.299156	+	0.954204i	$0.403295\pi$
$984$	0	0
$985$	24.6060	0.784013
$986$	0	0
$987$	3.24635	0.103333
$988$	0	0
$989$	−14.5786	−0.463573
$990$	0	0
$991$	−31.3172	−0.994824	−0.497412	−	0.867514i	$-0.665716\pi$
$991$	−31.3172	−0.994824	−0.497412	+	0.867514i	$0.665716\pi$
$992$	0	0
$993$	−15.4228	−0.489429
$994$	0	0
$995$	−45.9562	−1.45691
$996$	0	0
$997$	39.2848	1.24416	0.622082	−	0.782952i	$-0.286287\pi$
$997$	39.2848	1.24416	0.622082	+	0.782952i	$0.286287\pi$
$998$	0	0
$999$	−25.4852	−0.806315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.14	✓	23
4.3	odd	2		4016.2.a.m.1.10		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.14	✓	23	1.1	even	1	trivial
4016.2.a.m.1.10		23	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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.452441 q^{3} +1.91039 q^{5} +1.96124 q^{7} -2.79530 q^{9} +O(q^{10})\)	\(q+0.452441 q^{3} +1.91039 q^{5} +1.96124 q^{7} -2.79530 q^{9} +0.787387 q^{11} +5.79638 q^{13} +0.864341 q^{15} -3.13249 q^{17} +4.63676 q^{19} +0.887346 q^{21} +2.51256 q^{23} -1.35039 q^{25} -2.62203 q^{27} +3.62687 q^{29} -6.20008 q^{31} +0.356246 q^{33} +3.74674 q^{35} +9.71962 q^{37} +2.62252 q^{39} -0.892103 q^{41} -5.80229 q^{43} -5.34012 q^{45} +3.65849 q^{47} -3.15353 q^{49} -1.41727 q^{51} +12.6489 q^{53} +1.50422 q^{55} +2.09786 q^{57} +6.08621 q^{59} +6.47835 q^{61} -5.48225 q^{63} +11.0734 q^{65} -8.94464 q^{67} +1.13679 q^{69} +7.14955 q^{71} -11.3319 q^{73} -0.610974 q^{75} +1.54426 q^{77} -11.9354 q^{79} +7.19957 q^{81} +4.55162 q^{83} -5.98429 q^{85} +1.64095 q^{87} +1.55704 q^{89} +11.3681 q^{91} -2.80517 q^{93} +8.85805 q^{95} +5.55502 q^{97} -2.20098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * q^99

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