LMFDB - Embedded newform 8004.2.a.k.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.69179$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.69179 q^{5} +3.04338 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.69179 q^{5} +3.04338 q^{7} +1.00000 q^{9} +5.13887 q^{11} +4.38922 q^{13} +1.69179 q^{15} -7.10930 q^{17} +7.10526 q^{19} +3.04338 q^{21} -1.00000 q^{23} -2.13784 q^{25} +1.00000 q^{27} +1.00000 q^{29} +5.13088 q^{31} +5.13887 q^{33} +5.14878 q^{35} -0.265385 q^{37} +4.38922 q^{39} -7.58673 q^{41} -3.69813 q^{43} +1.69179 q^{45} -1.55160 q^{47} +2.26219 q^{49} -7.10930 q^{51} +6.33821 q^{53} +8.69391 q^{55} +7.10526 q^{57} +10.7809 q^{59} +7.54577 q^{61} +3.04338 q^{63} +7.42565 q^{65} -0.875564 q^{67} -1.00000 q^{69} +1.43673 q^{71} -5.91547 q^{73} -2.13784 q^{75} +15.6396 q^{77} +12.7350 q^{79} +1.00000 q^{81} -9.39850 q^{83} -12.0275 q^{85} +1.00000 q^{87} -12.4868 q^{89} +13.3581 q^{91} +5.13088 q^{93} +12.0206 q^{95} +3.58447 q^{97} +5.13887 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.69179	0.756593	0.378297	−	0.925684i	$-0.376510\pi$
$5$	1.69179	0.756593	0.378297	+	0.925684i	$0.376510\pi$
$6$	0	0
$7$	3.04338	1.15029	0.575146	−	0.818051i	$-0.304945\pi$
$7$	3.04338	1.15029	0.575146	+	0.818051i	$0.304945\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.13887	1.54943	0.774714	−	0.632312i	$-0.217894\pi$
$11$	5.13887	1.54943	0.774714	+	0.632312i	$0.217894\pi$
$12$	0	0
$13$	4.38922	1.21735	0.608675	−	0.793420i	$-0.291701\pi$
$13$	4.38922	1.21735	0.608675	+	0.793420i	$0.291701\pi$
$14$	0	0
$15$	1.69179	0.436819
$16$	0	0
$17$	−7.10930	−1.72426	−0.862130	−	0.506688i	$-0.830870\pi$
$17$	−7.10930	−1.72426	−0.862130	+	0.506688i	$0.830870\pi$
$18$	0	0
$19$	7.10526	1.63006	0.815029	−	0.579420i	$-0.196721\pi$
$19$	7.10526	1.63006	0.815029	+	0.579420i	$0.196721\pi$
$20$	0	0
$21$	3.04338	0.664121
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.13784	−0.427567
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.13088	0.921533	0.460767	−	0.887521i	$-0.347575\pi$
$31$	5.13088	0.921533	0.460767	+	0.887521i	$0.347575\pi$
$32$	0	0
$33$	5.13887	0.894562
$34$	0	0
$35$	5.14878	0.870302
$36$	0	0
$37$	−0.265385	−0.0436290	−0.0218145	−	0.999762i	$-0.506944\pi$
$37$	−0.265385	−0.0436290	−0.0218145	+	0.999762i	$0.506944\pi$
$38$	0	0
$39$	4.38922	0.702837
$40$	0	0
$41$	−7.58673	−1.18485	−0.592424	−	0.805626i	$-0.701829\pi$
$41$	−7.58673	−1.18485	−0.592424	+	0.805626i	$0.701829\pi$
$42$	0	0
$43$	−3.69813	−0.563959	−0.281979	−	0.959420i	$-0.590991\pi$
$43$	−3.69813	−0.563959	−0.281979	+	0.959420i	$0.590991\pi$
$44$	0	0
$45$	1.69179	0.252198
$46$	0	0
$47$	−1.55160	−0.226324	−0.113162	−	0.993577i	$-0.536098\pi$
$47$	−1.55160	−0.226324	−0.113162	+	0.993577i	$0.536098\pi$
$48$	0	0
$49$	2.26219	0.323170
$50$	0	0
$51$	−7.10930	−0.995502
$52$	0	0
$53$	6.33821	0.870620	0.435310	−	0.900281i	$-0.356639\pi$
$53$	6.33821	0.870620	0.435310	+	0.900281i	$0.356639\pi$
$54$	0	0
$55$	8.69391	1.17229
$56$	0	0
$57$	7.10526	0.941114
$58$	0	0
$59$	10.7809	1.40355	0.701777	−	0.712397i	$-0.252390\pi$
$59$	10.7809	1.40355	0.701777	+	0.712397i	$0.252390\pi$
$60$	0	0
$61$	7.54577	0.966136	0.483068	−	0.875583i	$-0.339522\pi$
$61$	7.54577	0.966136	0.483068	+	0.875583i	$0.339522\pi$
$62$	0	0
$63$	3.04338	0.383430
$64$	0	0
$65$	7.42565	0.921038
$66$	0	0
$67$	−0.875564	−0.106967	−0.0534836	−	0.998569i	$-0.517032\pi$
$67$	−0.875564	−0.106967	−0.0534836	+	0.998569i	$0.517032\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	1.43673	0.170509	0.0852544	−	0.996359i	$-0.472830\pi$
$71$	1.43673	0.170509	0.0852544	+	0.996359i	$0.472830\pi$
$72$	0	0
$73$	−5.91547	−0.692354	−0.346177	−	0.938169i	$-0.612520\pi$
$73$	−5.91547	−0.692354	−0.346177	+	0.938169i	$0.612520\pi$
$74$	0	0
$75$	−2.13784	−0.246856
$76$	0	0
$77$	15.6396	1.78229
$78$	0	0
$79$	12.7350	1.43280	0.716401	−	0.697689i	$-0.245788\pi$
$79$	12.7350	1.43280	0.716401	+	0.697689i	$0.245788\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.39850	−1.03162	−0.515810	−	0.856703i	$-0.672509\pi$
$83$	−9.39850	−1.03162	−0.515810	+	0.856703i	$0.672509\pi$
$84$	0	0
$85$	−12.0275	−1.30456
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−12.4868	−1.32360	−0.661802	−	0.749679i	$-0.730208\pi$
$89$	−12.4868	−1.32360	−0.661802	+	0.749679i	$0.730208\pi$
$90$	0	0
$91$	13.3581	1.40031
$92$	0	0
$93$	5.13088	0.532047
$94$	0	0
$95$	12.0206	1.23329
$96$	0	0
$97$	3.58447	0.363948	0.181974	−	0.983303i	$-0.441751\pi$
$97$	3.58447	0.363948	0.181974	+	0.983303i	$0.441751\pi$
$98$	0	0
$99$	5.13887	0.516476
$100$	0	0
$101$	−18.3824	−1.82912	−0.914558	−	0.404454i	$-0.867462\pi$
$101$	−18.3824	−1.82912	−0.914558	+	0.404454i	$0.867462\pi$
$102$	0	0
$103$	17.8490	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$103$	17.8490	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$104$	0	0
$105$	5.14878	0.502469
$106$	0	0
$107$	−16.7653	−1.62076	−0.810380	−	0.585905i	$-0.800739\pi$
$107$	−16.7653	−1.62076	−0.810380	+	0.585905i	$0.800739\pi$
$108$	0	0
$109$	−15.0206	−1.43872	−0.719358	−	0.694639i	$-0.755564\pi$
$109$	−15.0206	−1.43872	−0.719358	+	0.694639i	$0.755564\pi$
$110$	0	0
$111$	−0.265385	−0.0251892
$112$	0	0
$113$	−19.2169	−1.80778	−0.903889	−	0.427767i	$-0.859300\pi$
$113$	−19.2169	−1.80778	−0.903889	+	0.427767i	$0.859300\pi$
$114$	0	0
$115$	−1.69179	−0.157761
$116$	0	0
$117$	4.38922	0.405783
$118$	0	0
$119$	−21.6363	−1.98340
$120$	0	0
$121$	15.4080	1.40073
$122$	0	0
$123$	−7.58673	−0.684073
$124$	0	0
$125$	−12.0757	−1.08009
$126$	0	0
$127$	1.41558	0.125613	0.0628064	−	0.998026i	$-0.479995\pi$
$127$	1.41558	0.125613	0.0628064	+	0.998026i	$0.479995\pi$
$128$	0	0
$129$	−3.69813	−0.325602
$130$	0	0
$131$	9.24008	0.807309	0.403655	−	0.914911i	$-0.367740\pi$
$131$	9.24008	0.807309	0.403655	+	0.914911i	$0.367740\pi$
$132$	0	0
$133$	21.6240	1.87504
$134$	0	0
$135$	1.69179	0.145606
$136$	0	0
$137$	3.81449	0.325894	0.162947	−	0.986635i	$-0.447900\pi$
$137$	3.81449	0.325894	0.162947	+	0.986635i	$0.447900\pi$
$138$	0	0
$139$	−19.3336	−1.63985	−0.819926	−	0.572470i	$-0.805985\pi$
$139$	−19.3336	−1.63985	−0.819926	+	0.572470i	$0.805985\pi$
$140$	0	0
$141$	−1.55160	−0.130668
$142$	0	0
$143$	22.5556	1.88620
$144$	0	0
$145$	1.69179	0.140496
$146$	0	0
$147$	2.26219	0.186582
$148$	0	0
$149$	9.98911	0.818340	0.409170	−	0.912458i	$-0.365818\pi$
$149$	9.98911	0.818340	0.409170	+	0.912458i	$0.365818\pi$
$150$	0	0
$151$	7.67288	0.624410	0.312205	−	0.950015i	$-0.398932\pi$
$151$	7.67288	0.624410	0.312205	+	0.950015i	$0.398932\pi$
$152$	0	0
$153$	−7.10930	−0.574753
$154$	0	0
$155$	8.68039	0.697225
$156$	0	0
$157$	1.50593	0.120187	0.0600933	−	0.998193i	$-0.480860\pi$
$157$	1.50593	0.120187	0.0600933	+	0.998193i	$0.480860\pi$
$158$	0	0
$159$	6.33821	0.502652
$160$	0	0
$161$	−3.04338	−0.239852
$162$	0	0
$163$	−11.4718	−0.898537	−0.449269	−	0.893397i	$-0.648315\pi$
$163$	−11.4718	−0.898537	−0.449269	+	0.893397i	$0.648315\pi$
$164$	0	0
$165$	8.69391	0.676820
$166$	0	0
$167$	1.12820	0.0873024	0.0436512	−	0.999047i	$-0.486101\pi$
$167$	1.12820	0.0873024	0.0436512	+	0.999047i	$0.486101\pi$
$168$	0	0
$169$	6.26522	0.481940
$170$	0	0
$171$	7.10526	0.543353
$172$	0	0
$173$	−17.2231	−1.30945	−0.654726	−	0.755867i	$-0.727216\pi$
$173$	−17.2231	−1.30945	−0.654726	+	0.755867i	$0.727216\pi$
$174$	0	0
$175$	−6.50626	−0.491827
$176$	0	0
$177$	10.7809	0.810342
$178$	0	0
$179$	24.0634	1.79858	0.899292	−	0.437349i	$-0.144083\pi$
$179$	24.0634	1.79858	0.899292	+	0.437349i	$0.144083\pi$
$180$	0	0
$181$	−14.1215	−1.04964	−0.524822	−	0.851212i	$-0.675868\pi$
$181$	−14.1215	−1.04964	−0.524822	+	0.851212i	$0.675868\pi$
$182$	0	0
$183$	7.54577	0.557799
$184$	0	0
$185$	−0.448977	−0.0330094
$186$	0	0
$187$	−36.5338	−2.67161
$188$	0	0
$189$	3.04338	0.221374
$190$	0	0
$191$	−1.65646	−0.119857	−0.0599285	−	0.998203i	$-0.519087\pi$
$191$	−1.65646	−0.119857	−0.0599285	+	0.998203i	$0.519087\pi$
$192$	0	0
$193$	−24.5205	−1.76502	−0.882512	−	0.470290i	$-0.844149\pi$
$193$	−24.5205	−1.76502	−0.882512	+	0.470290i	$0.844149\pi$
$194$	0	0
$195$	7.42565	0.531762
$196$	0	0
$197$	12.2904	0.875653	0.437826	−	0.899060i	$-0.355748\pi$
$197$	12.2904	0.875653	0.437826	+	0.899060i	$0.355748\pi$
$198$	0	0
$199$	12.4860	0.885111	0.442555	−	0.896741i	$-0.354072\pi$
$199$	12.4860	0.885111	0.442555	+	0.896741i	$0.354072\pi$
$200$	0	0
$201$	−0.875564	−0.0617575
$202$	0	0
$203$	3.04338	0.213604
$204$	0	0
$205$	−12.8352	−0.896448
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	36.5130	2.52566
$210$	0	0
$211$	1.97134	0.135713	0.0678565	−	0.997695i	$-0.478384\pi$
$211$	1.97134	0.135713	0.0678565	+	0.997695i	$0.478384\pi$
$212$	0	0
$213$	1.43673	0.0984433
$214$	0	0
$215$	−6.25646	−0.426687
$216$	0	0
$217$	15.6152	1.06003
$218$	0	0
$219$	−5.91547	−0.399731
$220$	0	0
$221$	−31.2043	−2.09903
$222$	0	0
$223$	−15.4769	−1.03641	−0.518206	−	0.855256i	$-0.673400\pi$
$223$	−15.4769	−1.03641	−0.518206	+	0.855256i	$0.673400\pi$
$224$	0	0
$225$	−2.13784	−0.142522
$226$	0	0
$227$	−19.6138	−1.30181	−0.650906	−	0.759158i	$-0.725611\pi$
$227$	−19.6138	−1.30181	−0.650906	+	0.759158i	$0.725611\pi$
$228$	0	0
$229$	−22.1372	−1.46287	−0.731434	−	0.681912i	$-0.761149\pi$
$229$	−22.1372	−1.46287	−0.731434	+	0.681912i	$0.761149\pi$
$230$	0	0
$231$	15.6396	1.02901
$232$	0	0
$233$	15.1677	0.993669	0.496834	−	0.867845i	$-0.334496\pi$
$233$	15.1677	0.993669	0.496834	+	0.867845i	$0.334496\pi$
$234$	0	0
$235$	−2.62498	−0.171235
$236$	0	0
$237$	12.7350	0.827229
$238$	0	0
$239$	2.34086	0.151417	0.0757087	−	0.997130i	$-0.475878\pi$
$239$	2.34086	0.151417	0.0757087	+	0.997130i	$0.475878\pi$
$240$	0	0
$241$	−21.1769	−1.36412	−0.682060	−	0.731296i	$-0.738916\pi$
$241$	−21.1769	−1.36412	−0.682060	+	0.731296i	$0.738916\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.82716	0.244508
$246$	0	0
$247$	31.1865	1.98435
$248$	0	0
$249$	−9.39850	−0.595606
$250$	0	0
$251$	10.6322	0.671097	0.335548	−	0.942023i	$-0.391078\pi$
$251$	10.6322	0.671097	0.335548	+	0.942023i	$0.391078\pi$
$252$	0	0
$253$	−5.13887	−0.323078
$254$	0	0
$255$	−12.0275	−0.753190
$256$	0	0
$257$	5.22502	0.325928	0.162964	−	0.986632i	$-0.447895\pi$
$257$	5.22502	0.325928	0.162964	+	0.986632i	$0.447895\pi$
$258$	0	0
$259$	−0.807669	−0.0501861
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−24.0737	−1.48445	−0.742224	−	0.670152i	$-0.766229\pi$
$263$	−24.0737	−1.48445	−0.742224	+	0.670152i	$0.766229\pi$
$264$	0	0
$265$	10.7229	0.658705
$266$	0	0
$267$	−12.4868	−0.764183
$268$	0	0
$269$	17.9501	1.09443	0.547217	−	0.836991i	$-0.315687\pi$
$269$	17.9501	1.09443	0.547217	+	0.836991i	$0.315687\pi$
$270$	0	0
$271$	−7.66136	−0.465394	−0.232697	−	0.972549i	$-0.574755\pi$
$271$	−7.66136	−0.465394	−0.232697	+	0.972549i	$0.574755\pi$
$272$	0	0
$273$	13.3581	0.808468
$274$	0	0
$275$	−10.9861	−0.662484
$276$	0	0
$277$	−24.1458	−1.45078	−0.725390	−	0.688339i	$-0.758340\pi$
$277$	−24.1458	−1.45078	−0.725390	+	0.688339i	$0.758340\pi$
$278$	0	0
$279$	5.13088	0.307178
$280$	0	0
$281$	−5.55040	−0.331109	−0.165555	−	0.986201i	$-0.552941\pi$
$281$	−5.55040	−0.331109	−0.165555	+	0.986201i	$0.552941\pi$
$282$	0	0
$283$	28.8679	1.71602	0.858010	−	0.513633i	$-0.171701\pi$
$283$	28.8679	1.71602	0.858010	+	0.513633i	$0.171701\pi$
$284$	0	0
$285$	12.0206	0.712041
$286$	0	0
$287$	−23.0894	−1.36292
$288$	0	0
$289$	33.5422	1.97307
$290$	0	0
$291$	3.58447	0.210125
$292$	0	0
$293$	5.59117	0.326640	0.163320	−	0.986573i	$-0.447780\pi$
$293$	5.59117	0.326640	0.163320	+	0.986573i	$0.447780\pi$
$294$	0	0
$295$	18.2391	1.06192
$296$	0	0
$297$	5.13887	0.298187
$298$	0	0
$299$	−4.38922	−0.253835
$300$	0	0
$301$	−11.2548	−0.648717
$302$	0	0
$303$	−18.3824	−1.05604
$304$	0	0
$305$	12.7659	0.730972
$306$	0	0
$307$	−17.2764	−0.986018	−0.493009	−	0.870024i	$-0.664103\pi$
$307$	−17.2764	−0.986018	−0.493009	+	0.870024i	$0.664103\pi$
$308$	0	0
$309$	17.8490	1.01539
$310$	0	0
$311$	−6.41420	−0.363716	−0.181858	−	0.983325i	$-0.558211\pi$
$311$	−6.41420	−0.363716	−0.181858	+	0.983325i	$0.558211\pi$
$312$	0	0
$313$	19.6563	1.11104	0.555520	−	0.831503i	$-0.312519\pi$
$313$	19.6563	1.11104	0.555520	+	0.831503i	$0.312519\pi$
$314$	0	0
$315$	5.14878	0.290101
$316$	0	0
$317$	17.7169	0.995082	0.497541	−	0.867440i	$-0.334236\pi$
$317$	17.7169	0.995082	0.497541	+	0.867440i	$0.334236\pi$
$318$	0	0
$319$	5.13887	0.287721
$320$	0	0
$321$	−16.7653	−0.935746
$322$	0	0
$323$	−50.5134	−2.81064
$324$	0	0
$325$	−9.38342	−0.520499
$326$	0	0
$327$	−15.0206	−0.830643
$328$	0	0
$329$	−4.72211	−0.260338
$330$	0	0
$331$	6.42127	0.352945	0.176473	−	0.984306i	$-0.443531\pi$
$331$	6.42127	0.352945	0.176473	+	0.984306i	$0.443531\pi$
$332$	0	0
$333$	−0.265385	−0.0145430
$334$	0	0
$335$	−1.48127	−0.0809306
$336$	0	0
$337$	−0.545708	−0.0297266	−0.0148633	−	0.999890i	$-0.504731\pi$
$337$	−0.545708	−0.0297266	−0.0148633	+	0.999890i	$0.504731\pi$
$338$	0	0
$339$	−19.2169	−1.04372
$340$	0	0
$341$	26.3669	1.42785
$342$	0	0
$343$	−14.4190	−0.778551
$344$	0	0
$345$	−1.69179	−0.0910831
$346$	0	0
$347$	−5.25389	−0.282044	−0.141022	−	0.990006i	$-0.545039\pi$
$347$	−5.25389	−0.282044	−0.141022	+	0.990006i	$0.545039\pi$
$348$	0	0
$349$	9.77828	0.523419	0.261710	−	0.965147i	$-0.415714\pi$
$349$	9.77828	0.523419	0.261710	+	0.965147i	$0.415714\pi$
$350$	0	0
$351$	4.38922	0.234279
$352$	0	0
$353$	24.5427	1.30628	0.653139	−	0.757238i	$-0.273452\pi$
$353$	24.5427	1.30628	0.653139	+	0.757238i	$0.273452\pi$
$354$	0	0
$355$	2.43065	0.129006
$356$	0	0
$357$	−21.6363	−1.14512
$358$	0	0
$359$	33.0792	1.74585	0.872927	−	0.487850i	$-0.162219\pi$
$359$	33.0792	1.74585	0.872927	+	0.487850i	$0.162219\pi$
$360$	0	0
$361$	31.4847	1.65709
$362$	0	0
$363$	15.4080	0.808709
$364$	0	0
$365$	−10.0078	−0.523830
$366$	0	0
$367$	26.3765	1.37684	0.688422	−	0.725310i	$-0.258304\pi$
$367$	26.3765	1.37684	0.688422	+	0.725310i	$0.258304\pi$
$368$	0	0
$369$	−7.58673	−0.394950
$370$	0	0
$371$	19.2896	1.00147
$372$	0	0
$373$	4.11468	0.213050	0.106525	−	0.994310i	$-0.466028\pi$
$373$	4.11468	0.213050	0.106525	+	0.994310i	$0.466028\pi$
$374$	0	0
$375$	−12.0757	−0.623589
$376$	0	0
$377$	4.38922	0.226056
$378$	0	0
$379$	−8.36503	−0.429683	−0.214841	−	0.976649i	$-0.568923\pi$
$379$	−8.36503	−0.429683	−0.214841	+	0.976649i	$0.568923\pi$
$380$	0	0
$381$	1.41558	0.0725226
$382$	0	0
$383$	−4.90830	−0.250802	−0.125401	−	0.992106i	$-0.540022\pi$
$383$	−4.90830	−0.250802	−0.125401	+	0.992106i	$0.540022\pi$
$384$	0	0
$385$	26.4589	1.34847
$386$	0	0
$387$	−3.69813	−0.187986
$388$	0	0
$389$	−8.96786	−0.454689	−0.227344	−	0.973814i	$-0.573004\pi$
$389$	−8.96786	−0.454689	−0.227344	+	0.973814i	$0.573004\pi$
$390$	0	0
$391$	7.10930	0.359533
$392$	0	0
$393$	9.24008	0.466100
$394$	0	0
$395$	21.5450	1.08405
$396$	0	0
$397$	8.20097	0.411595	0.205797	−	0.978595i	$-0.434021\pi$
$397$	8.20097	0.411595	0.205797	+	0.978595i	$0.434021\pi$
$398$	0	0
$399$	21.6240	1.08256
$400$	0	0
$401$	17.9438	0.896073	0.448036	−	0.894015i	$-0.352123\pi$
$401$	17.9438	0.896073	0.448036	+	0.894015i	$0.352123\pi$
$402$	0	0
$403$	22.5205	1.12183
$404$	0	0
$405$	1.69179	0.0840659
$406$	0	0
$407$	−1.36378	−0.0676000
$408$	0	0
$409$	9.18489	0.454163	0.227082	−	0.973876i	$-0.427082\pi$
$409$	9.18489	0.454163	0.227082	+	0.973876i	$0.427082\pi$
$410$	0	0
$411$	3.81449	0.188155
$412$	0	0
$413$	32.8104	1.61450
$414$	0	0
$415$	−15.9003	−0.780516
$416$	0	0
$417$	−19.3336	−0.946769
$418$	0	0
$419$	4.14863	0.202674	0.101337	−	0.994852i	$-0.467688\pi$
$419$	4.14863	0.202674	0.101337	+	0.994852i	$0.467688\pi$
$420$	0	0
$421$	13.5453	0.660159	0.330079	−	0.943953i	$-0.392924\pi$
$421$	13.5453	0.660159	0.330079	+	0.943953i	$0.392924\pi$
$422$	0	0
$423$	−1.55160	−0.0754412
$424$	0	0
$425$	15.1985	0.737236
$426$	0	0
$427$	22.9647	1.11134
$428$	0	0
$429$	22.5556	1.08900
$430$	0	0
$431$	9.60843	0.462822	0.231411	−	0.972856i	$-0.425666\pi$
$431$	9.60843	0.462822	0.231411	+	0.972856i	$0.425666\pi$
$432$	0	0
$433$	−26.8686	−1.29122	−0.645610	−	0.763667i	$-0.723397\pi$
$433$	−26.8686	−1.29122	−0.645610	+	0.763667i	$0.723397\pi$
$434$	0	0
$435$	1.69179	0.0811153
$436$	0	0
$437$	−7.10526	−0.339891
$438$	0	0
$439$	27.8765	1.33047	0.665235	−	0.746634i	$-0.268331\pi$
$439$	27.8765	1.33047	0.665235	+	0.746634i	$0.268331\pi$
$440$	0	0
$441$	2.26219	0.107723
$442$	0	0
$443$	−35.7658	−1.69929	−0.849643	−	0.527359i	$-0.823182\pi$
$443$	−35.7658	−1.69929	−0.849643	+	0.527359i	$0.823182\pi$
$444$	0	0
$445$	−21.1252	−1.00143
$446$	0	0
$447$	9.98911	0.472469
$448$	0	0
$449$	11.3485	0.535571	0.267785	−	0.963479i	$-0.413708\pi$
$449$	11.3485	0.535571	0.267785	+	0.963479i	$0.413708\pi$
$450$	0	0
$451$	−38.9872	−1.83584
$452$	0	0
$453$	7.67288	0.360503
$454$	0	0
$455$	22.5991	1.05946
$456$	0	0
$457$	−0.503279	−0.0235424	−0.0117712	−	0.999931i	$-0.503747\pi$
$457$	−0.503279	−0.0235424	−0.0117712	+	0.999931i	$0.503747\pi$
$458$	0	0
$459$	−7.10930	−0.331834
$460$	0	0
$461$	−12.0836	−0.562790	−0.281395	−	0.959592i	$-0.590797\pi$
$461$	−12.0836	−0.562790	−0.281395	+	0.959592i	$0.590797\pi$
$462$	0	0
$463$	29.4597	1.36911	0.684553	−	0.728963i	$-0.259997\pi$
$463$	29.4597	1.36911	0.684553	+	0.728963i	$0.259997\pi$
$464$	0	0
$465$	8.68039	0.402543
$466$	0	0
$467$	−14.7018	−0.680319	−0.340159	−	0.940368i	$-0.610481\pi$
$467$	−14.7018	−0.680319	−0.340159	+	0.940368i	$0.610481\pi$
$468$	0	0
$469$	−2.66468	−0.123043
$470$	0	0
$471$	1.50593	0.0693898
$472$	0	0
$473$	−19.0042	−0.873813
$474$	0	0
$475$	−15.1899	−0.696959
$476$	0	0
$477$	6.33821	0.290207
$478$	0	0
$479$	23.4391	1.07096	0.535479	−	0.844549i	$-0.320131\pi$
$479$	23.4391	1.07096	0.535479	+	0.844549i	$0.320131\pi$
$480$	0	0
$481$	−1.16483	−0.0531118
$482$	0	0
$483$	−3.04338	−0.138479
$484$	0	0
$485$	6.06418	0.275360
$486$	0	0
$487$	−39.1805	−1.77544	−0.887719	−	0.460386i	$-0.847711\pi$
$487$	−39.1805	−1.77544	−0.887719	+	0.460386i	$0.847711\pi$
$488$	0	0
$489$	−11.4718	−0.518771
$490$	0	0
$491$	−42.2265	−1.90565	−0.952827	−	0.303515i	$-0.901840\pi$
$491$	−42.2265	−1.90565	−0.952827	+	0.303515i	$0.901840\pi$
$492$	0	0
$493$	−7.10930	−0.320187
$494$	0	0
$495$	8.69391	0.390762
$496$	0	0
$497$	4.37253	0.196135
$498$	0	0
$499$	31.9955	1.43232	0.716158	−	0.697939i	$-0.245899\pi$
$499$	31.9955	1.43232	0.716158	+	0.697939i	$0.245899\pi$
$500$	0	0
$501$	1.12820	0.0504041
$502$	0	0
$503$	14.1584	0.631293	0.315646	−	0.948877i	$-0.397779\pi$
$503$	14.1584	0.631293	0.315646	+	0.948877i	$0.397779\pi$
$504$	0	0
$505$	−31.0992	−1.38390
$506$	0	0
$507$	6.26522	0.278248
$508$	0	0
$509$	−28.2628	−1.25273	−0.626363	−	0.779532i	$-0.715457\pi$
$509$	−28.2628	−1.25273	−0.626363	+	0.779532i	$0.715457\pi$
$510$	0	0
$511$	−18.0031	−0.796409
$512$	0	0
$513$	7.10526	0.313705
$514$	0	0
$515$	30.1968	1.33063
$516$	0	0
$517$	−7.97346	−0.350672
$518$	0	0
$519$	−17.2231	−0.756012
$520$	0	0
$521$	−6.02538	−0.263977	−0.131988	−	0.991251i	$-0.542136\pi$
$521$	−6.02538	−0.263977	−0.131988	+	0.991251i	$0.542136\pi$
$522$	0	0
$523$	−24.2031	−1.05833	−0.529165	−	0.848519i	$-0.677495\pi$
$523$	−24.2031	−1.05833	−0.529165	+	0.848519i	$0.677495\pi$
$524$	0	0
$525$	−6.50626	−0.283956
$526$	0	0
$527$	−36.4770	−1.58896
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	10.7809	0.467851
$532$	0	0
$533$	−33.2998	−1.44238
$534$	0	0
$535$	−28.3634	−1.22626
$536$	0	0
$537$	24.0634	1.03841
$538$	0	0
$539$	11.6251	0.500729
$540$	0	0
$541$	37.3041	1.60383	0.801915	−	0.597438i	$-0.203815\pi$
$541$	37.3041	1.60383	0.801915	+	0.597438i	$0.203815\pi$
$542$	0	0
$543$	−14.1215	−0.606013
$544$	0	0
$545$	−25.4118	−1.08852
$546$	0	0
$547$	−3.42033	−0.146243	−0.0731214	−	0.997323i	$-0.523296\pi$
$547$	−3.42033	−0.146243	−0.0731214	+	0.997323i	$0.523296\pi$
$548$	0	0
$549$	7.54577	0.322045
$550$	0	0
$551$	7.10526	0.302694
$552$	0	0
$553$	38.7576	1.64814
$554$	0	0
$555$	−0.448977	−0.0190580
$556$	0	0
$557$	−26.3931	−1.11831	−0.559157	−	0.829062i	$-0.688875\pi$
$557$	−26.3931	−1.11831	−0.559157	+	0.829062i	$0.688875\pi$
$558$	0	0
$559$	−16.2319	−0.686535
$560$	0	0
$561$	−36.5338	−1.54246
$562$	0	0
$563$	38.7058	1.63126	0.815628	−	0.578576i	$-0.196392\pi$
$563$	38.7058	1.63126	0.815628	+	0.578576i	$0.196392\pi$
$564$	0	0
$565$	−32.5111	−1.36775
$566$	0	0
$567$	3.04338	0.127810
$568$	0	0
$569$	4.37579	0.183443	0.0917214	−	0.995785i	$-0.470763\pi$
$569$	4.37579	0.183443	0.0917214	+	0.995785i	$0.470763\pi$
$570$	0	0
$571$	1.13259	0.0473975	0.0236987	−	0.999719i	$-0.492456\pi$
$571$	1.13259	0.0473975	0.0236987	+	0.999719i	$0.492456\pi$
$572$	0	0
$573$	−1.65646	−0.0691994
$574$	0	0
$575$	2.13784	0.0891539
$576$	0	0
$577$	12.6032	0.524679	0.262340	−	0.964976i	$-0.415506\pi$
$577$	12.6032	0.524679	0.262340	+	0.964976i	$0.415506\pi$
$578$	0	0
$579$	−24.5205	−1.01904
$580$	0	0
$581$	−28.6033	−1.18666
$582$	0	0
$583$	32.5712	1.34896
$584$	0	0
$585$	7.42565	0.307013
$586$	0	0
$587$	28.8156	1.18935	0.594673	−	0.803968i	$-0.297282\pi$
$587$	28.8156	1.18935	0.594673	+	0.803968i	$0.297282\pi$
$588$	0	0
$589$	36.4562	1.50215
$590$	0	0
$591$	12.2904	0.505558
$592$	0	0
$593$	−32.6713	−1.34165	−0.670825	−	0.741615i	$-0.734060\pi$
$593$	−32.6713	−1.34165	−0.670825	+	0.741615i	$0.734060\pi$
$594$	0	0
$595$	−36.6042	−1.50063
$596$	0	0
$597$	12.4860	0.511019
$598$	0	0
$599$	−4.45856	−0.182172	−0.0910859	−	0.995843i	$-0.529034\pi$
$599$	−4.45856	−0.182172	−0.0910859	+	0.995843i	$0.529034\pi$
$600$	0	0
$601$	40.4230	1.64889	0.824445	−	0.565943i	$-0.191488\pi$
$601$	40.4230	1.64889	0.824445	+	0.565943i	$0.191488\pi$
$602$	0	0
$603$	−0.875564	−0.0356557
$604$	0	0
$605$	26.0671	1.05978
$606$	0	0
$607$	41.9563	1.70295	0.851477	−	0.524392i	$-0.175708\pi$
$607$	41.9563	1.70295	0.851477	+	0.524392i	$0.175708\pi$
$608$	0	0
$609$	3.04338	0.123324
$610$	0	0
$611$	−6.81030	−0.275515
$612$	0	0
$613$	24.5660	0.992210	0.496105	−	0.868263i	$-0.334763\pi$
$613$	24.5660	0.992210	0.496105	+	0.868263i	$0.334763\pi$
$614$	0	0
$615$	−12.8352	−0.517565
$616$	0	0
$617$	−33.9324	−1.36607	−0.683034	−	0.730386i	$-0.739340\pi$
$617$	−33.9324	−1.36607	−0.683034	+	0.730386i	$0.739340\pi$
$618$	0	0
$619$	15.7842	0.634421	0.317210	−	0.948355i	$-0.397254\pi$
$619$	15.7842	0.634421	0.317210	+	0.948355i	$0.397254\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−38.0023	−1.52253
$624$	0	0
$625$	−9.74049	−0.389619
$626$	0	0
$627$	36.5130	1.45819
$628$	0	0
$629$	1.88670	0.0752277
$630$	0	0
$631$	12.7903	0.509174	0.254587	−	0.967050i	$-0.418060\pi$
$631$	12.7903	0.509174	0.254587	+	0.967050i	$0.418060\pi$
$632$	0	0
$633$	1.97134	0.0783539
$634$	0	0
$635$	2.39488	0.0950378
$636$	0	0
$637$	9.92925	0.393411
$638$	0	0
$639$	1.43673	0.0568363
$640$	0	0
$641$	−28.6594	−1.13198	−0.565990	−	0.824412i	$-0.691506\pi$
$641$	−28.6594	−1.13198	−0.565990	+	0.824412i	$0.691506\pi$
$642$	0	0
$643$	−24.8350	−0.979398	−0.489699	−	0.871892i	$-0.662893\pi$
$643$	−24.8350	−0.979398	−0.489699	+	0.871892i	$0.662893\pi$
$644$	0	0
$645$	−6.25646	−0.246348
$646$	0	0
$647$	33.1872	1.30472	0.652362	−	0.757908i	$-0.273778\pi$
$647$	33.1872	1.30472	0.652362	+	0.757908i	$0.273778\pi$
$648$	0	0
$649$	55.4016	2.17470
$650$	0	0
$651$	15.6152	0.612009
$652$	0	0
$653$	5.82473	0.227939	0.113970	−	0.993484i	$-0.463643\pi$
$653$	5.82473	0.227939	0.113970	+	0.993484i	$0.463643\pi$
$654$	0	0
$655$	15.6323	0.610804
$656$	0	0
$657$	−5.91547	−0.230785
$658$	0	0
$659$	−18.4262	−0.717785	−0.358892	−	0.933379i	$-0.616845\pi$
$659$	−18.4262	−0.717785	−0.358892	+	0.933379i	$0.616845\pi$
$660$	0	0
$661$	−14.4073	−0.560379	−0.280190	−	0.959945i	$-0.590397\pi$
$661$	−14.4073	−0.560379	−0.280190	+	0.959945i	$0.590397\pi$
$662$	0	0
$663$	−31.2043	−1.21187
$664$	0	0
$665$	36.5834	1.41864
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−15.4769	−0.598372
$670$	0	0
$671$	38.7767	1.49696
$672$	0	0
$673$	−8.50110	−0.327693	−0.163847	−	0.986486i	$-0.552390\pi$
$673$	−8.50110	−0.327693	−0.163847	+	0.986486i	$0.552390\pi$
$674$	0	0
$675$	−2.13784	−0.0822853
$676$	0	0
$677$	−16.3149	−0.627033	−0.313516	−	0.949583i	$-0.601507\pi$
$677$	−16.3149	−0.627033	−0.313516	+	0.949583i	$0.601507\pi$
$678$	0	0
$679$	10.9089	0.418646
$680$	0	0
$681$	−19.6138	−0.751602
$682$	0	0
$683$	−35.1542	−1.34514	−0.672569	−	0.740034i	$-0.734809\pi$
$683$	−35.1542	−1.34514	−0.672569	+	0.740034i	$0.734809\pi$
$684$	0	0
$685$	6.45332	0.246569
$686$	0	0
$687$	−22.1372	−0.844588
$688$	0	0
$689$	27.8198	1.05985
$690$	0	0
$691$	−17.0872	−0.650028	−0.325014	−	0.945709i	$-0.605369\pi$
$691$	−17.0872	−0.650028	−0.325014	+	0.945709i	$0.605369\pi$
$692$	0	0
$693$	15.6396	0.594098
$694$	0	0
$695$	−32.7084	−1.24070
$696$	0	0
$697$	53.9364	2.04299
$698$	0	0
$699$	15.1677	0.573695
$700$	0	0
$701$	27.6847	1.04564	0.522818	−	0.852445i	$-0.324881\pi$
$701$	27.6847	1.04564	0.522818	+	0.852445i	$0.324881\pi$
$702$	0	0
$703$	−1.88563	−0.0711178
$704$	0	0
$705$	−2.62498	−0.0988625
$706$	0	0
$707$	−55.9447	−2.10402
$708$	0	0
$709$	−10.8733	−0.408355	−0.204178	−	0.978934i	$-0.565452\pi$
$709$	−10.8733	−0.408355	−0.204178	+	0.978934i	$0.565452\pi$
$710$	0	0
$711$	12.7350	0.477601
$712$	0	0
$713$	−5.13088	−0.192153
$714$	0	0
$715$	38.1594	1.42708
$716$	0	0
$717$	2.34086	0.0874209
$718$	0	0
$719$	−25.2174	−0.940450	−0.470225	−	0.882546i	$-0.655827\pi$
$719$	−25.2174	−0.940450	−0.470225	+	0.882546i	$0.655827\pi$
$720$	0	0
$721$	54.3214	2.02304
$722$	0	0
$723$	−21.1769	−0.787576
$724$	0	0
$725$	−2.13784	−0.0793972
$726$	0	0
$727$	12.4026	0.459985	0.229993	−	0.973192i	$-0.426130\pi$
$727$	12.4026	0.459985	0.229993	+	0.973192i	$0.426130\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	26.2911	0.972411
$732$	0	0
$733$	−7.54181	−0.278563	−0.139282	−	0.990253i	$-0.544479\pi$
$733$	−7.54181	−0.278563	−0.139282	+	0.990253i	$0.544479\pi$
$734$	0	0
$735$	3.82716	0.141167
$736$	0	0
$737$	−4.49941	−0.165738
$738$	0	0
$739$	42.9562	1.58017	0.790085	−	0.612998i	$-0.210037\pi$
$739$	42.9562	1.58017	0.790085	+	0.612998i	$0.210037\pi$
$740$	0	0
$741$	31.1865	1.14567
$742$	0	0
$743$	−25.0483	−0.918932	−0.459466	−	0.888195i	$-0.651959\pi$
$743$	−25.0483	−0.918932	−0.459466	+	0.888195i	$0.651959\pi$
$744$	0	0
$745$	16.8995	0.619150
$746$	0	0
$747$	−9.39850	−0.343873
$748$	0	0
$749$	−51.0232	−1.86435
$750$	0	0
$751$	−39.4124	−1.43818	−0.719090	−	0.694917i	$-0.755441\pi$
$751$	−39.4124	−1.43818	−0.719090	+	0.694917i	$0.755441\pi$
$752$	0	0
$753$	10.6322	0.387458
$754$	0	0
$755$	12.9809	0.472424
$756$	0	0
$757$	29.7315	1.08061	0.540306	−	0.841469i	$-0.318309\pi$
$757$	29.7315	1.08061	0.540306	+	0.841469i	$0.318309\pi$
$758$	0	0
$759$	−5.13887	−0.186529
$760$	0	0
$761$	7.46878	0.270743	0.135372	−	0.990795i	$-0.456777\pi$
$761$	7.46878	0.270743	0.135372	+	0.990795i	$0.456777\pi$
$762$	0	0
$763$	−45.7136	−1.65494
$764$	0	0
$765$	−12.0275	−0.434854
$766$	0	0
$767$	47.3197	1.70862
$768$	0	0
$769$	−28.9561	−1.04418	−0.522092	−	0.852889i	$-0.674848\pi$
$769$	−28.9561	−1.04418	−0.522092	+	0.852889i	$0.674848\pi$
$770$	0	0
$771$	5.22502	0.188175
$772$	0	0
$773$	25.0825	0.902154	0.451077	−	0.892485i	$-0.351040\pi$
$773$	25.0825	0.902154	0.451077	+	0.892485i	$0.351040\pi$
$774$	0	0
$775$	−10.9690	−0.394017
$776$	0	0
$777$	−0.807669	−0.0289749
$778$	0	0
$779$	−53.9057	−1.93137
$780$	0	0
$781$	7.38318	0.264191
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	2.54773	0.0909324
$786$	0	0
$787$	−24.4100	−0.870123	−0.435061	−	0.900401i	$-0.643273\pi$
$787$	−24.4100	−0.870123	−0.435061	+	0.900401i	$0.643273\pi$
$788$	0	0
$789$	−24.0737	−0.857046
$790$	0	0
$791$	−58.4846	−2.07947
$792$	0	0
$793$	33.1200	1.17613
$794$	0	0
$795$	10.7229	0.380303
$796$	0	0
$797$	22.7875	0.807174	0.403587	−	0.914941i	$-0.367763\pi$
$797$	22.7875	0.807174	0.403587	+	0.914941i	$0.367763\pi$
$798$	0	0
$799$	11.0308	0.390241
$800$	0	0
$801$	−12.4868	−0.441201
$802$	0	0
$803$	−30.3988	−1.07275
$804$	0	0
$805$	−5.14878	−0.181471
$806$	0	0
$807$	17.9501	0.631872
$808$	0	0
$809$	−15.1845	−0.533857	−0.266929	−	0.963716i	$-0.586009\pi$
$809$	−15.1845	−0.533857	−0.266929	+	0.963716i	$0.586009\pi$
$810$	0	0
$811$	10.0999	0.354657	0.177328	−	0.984152i	$-0.443255\pi$
$811$	10.0999	0.354657	0.177328	+	0.984152i	$0.443255\pi$
$812$	0	0
$813$	−7.66136	−0.268696
$814$	0	0
$815$	−19.4078	−0.679827
$816$	0	0
$817$	−26.2761	−0.919285
$818$	0	0
$819$	13.3581	0.466769
$820$	0	0
$821$	22.6174	0.789352	0.394676	−	0.918820i	$-0.370857\pi$
$821$	22.6174	0.789352	0.394676	+	0.918820i	$0.370857\pi$
$822$	0	0
$823$	−49.1637	−1.71374	−0.856870	−	0.515533i	$-0.827594\pi$
$823$	−49.1637	−1.71374	−0.856870	+	0.515533i	$0.827594\pi$
$824$	0	0
$825$	−10.9861	−0.382485
$826$	0	0
$827$	55.5965	1.93328	0.966640	−	0.256138i	$-0.0824503\pi$
$827$	55.5965	1.93328	0.966640	+	0.256138i	$0.0824503\pi$
$828$	0	0
$829$	5.71246	0.198402	0.0992009	−	0.995067i	$-0.468371\pi$
$829$	5.71246	0.198402	0.0992009	+	0.995067i	$0.468371\pi$
$830$	0	0
$831$	−24.1458	−0.837608
$832$	0	0
$833$	−16.0826	−0.557229
$834$	0	0
$835$	1.90867	0.0660524
$836$	0	0
$837$	5.13088	0.177349
$838$	0	0
$839$	−34.4156	−1.18816	−0.594079	−	0.804407i	$-0.702483\pi$
$839$	−34.4156	−1.18816	−0.594079	+	0.804407i	$0.702483\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−5.55040	−0.191166
$844$	0	0
$845$	10.5995	0.364633
$846$	0	0
$847$	46.8924	1.61124
$848$	0	0
$849$	28.8679	0.990745
$850$	0	0
$851$	0.265385	0.00909728
$852$	0	0
$853$	29.2991	1.00318	0.501591	−	0.865105i	$-0.332748\pi$
$853$	29.2991	1.00318	0.501591	+	0.865105i	$0.332748\pi$
$854$	0	0
$855$	12.0206	0.411097
$856$	0	0
$857$	−5.09933	−0.174190	−0.0870949	−	0.996200i	$-0.527758\pi$
$857$	−5.09933	−0.174190	−0.0870949	+	0.996200i	$0.527758\pi$
$858$	0	0
$859$	15.8478	0.540720	0.270360	−	0.962759i	$-0.412857\pi$
$859$	15.8478	0.540720	0.270360	+	0.962759i	$0.412857\pi$
$860$	0	0
$861$	−23.0894	−0.786883
$862$	0	0
$863$	−44.1814	−1.50395	−0.751976	−	0.659191i	$-0.770899\pi$
$863$	−44.1814	−1.50395	−0.751976	+	0.659191i	$0.770899\pi$
$864$	0	0
$865$	−29.1380	−0.990722
$866$	0	0
$867$	33.5422	1.13915
$868$	0	0
$869$	65.4436	2.22002
$870$	0	0
$871$	−3.84304	−0.130216
$872$	0	0
$873$	3.58447	0.121316
$874$	0	0
$875$	−36.7511	−1.24242
$876$	0	0
$877$	2.87077	0.0969391	0.0484695	−	0.998825i	$-0.484566\pi$
$877$	2.87077	0.0969391	0.0484695	+	0.998825i	$0.484566\pi$
$878$	0	0
$879$	5.59117	0.188585
$880$	0	0
$881$	18.5802	0.625981	0.312991	−	0.949756i	$-0.398669\pi$
$881$	18.5802	0.625981	0.312991	+	0.949756i	$0.398669\pi$
$882$	0	0
$883$	−1.40204	−0.0471825	−0.0235912	−	0.999722i	$-0.507510\pi$
$883$	−1.40204	−0.0471825	−0.0235912	+	0.999722i	$0.507510\pi$
$884$	0	0
$885$	18.2391	0.613099
$886$	0	0
$887$	27.0673	0.908832	0.454416	−	0.890790i	$-0.349848\pi$
$887$	27.0673	0.908832	0.454416	+	0.890790i	$0.349848\pi$
$888$	0	0
$889$	4.30817	0.144491
$890$	0	0
$891$	5.13887	0.172159
$892$	0	0
$893$	−11.0245	−0.368921
$894$	0	0
$895$	40.7103	1.36080
$896$	0	0
$897$	−4.38922	−0.146552
$898$	0	0
$899$	5.13088	0.171124
$900$	0	0
$901$	−45.0602	−1.50117
$902$	0	0
$903$	−11.2548	−0.374537
$904$	0	0
$905$	−23.8907	−0.794154
$906$	0	0
$907$	8.39310	0.278688	0.139344	−	0.990244i	$-0.455501\pi$
$907$	8.39310	0.278688	0.139344	+	0.990244i	$0.455501\pi$
$908$	0	0
$909$	−18.3824	−0.609706
$910$	0	0
$911$	−31.9753	−1.05939	−0.529694	−	0.848189i	$-0.677693\pi$
$911$	−31.9753	−1.05939	−0.529694	+	0.848189i	$0.677693\pi$
$912$	0	0
$913$	−48.2977	−1.59842
$914$	0	0
$915$	12.7659	0.422027
$916$	0	0
$917$	28.1211	0.928641
$918$	0	0
$919$	35.1594	1.15980	0.579900	−	0.814687i	$-0.303091\pi$
$919$	35.1594	1.15980	0.579900	+	0.814687i	$0.303091\pi$
$920$	0	0
$921$	−17.2764	−0.569278
$922$	0	0
$923$	6.30613	0.207569
$924$	0	0
$925$	0.567349	0.0186543
$926$	0	0
$927$	17.8490	0.586239
$928$	0	0
$929$	−27.2074	−0.892645	−0.446323	−	0.894872i	$-0.647267\pi$
$929$	−27.2074	−0.892645	−0.446323	+	0.894872i	$0.647267\pi$
$930$	0	0
$931$	16.0735	0.526786
$932$	0	0
$933$	−6.41420	−0.209991
$934$	0	0
$935$	−61.8076	−2.02133
$936$	0	0
$937$	−46.3558	−1.51438	−0.757189	−	0.653196i	$-0.773428\pi$
$937$	−46.3558	−1.51438	−0.757189	+	0.653196i	$0.773428\pi$
$938$	0	0
$939$	19.6563	0.641459
$940$	0	0
$941$	−3.71707	−0.121173	−0.0605865	−	0.998163i	$-0.519297\pi$
$941$	−3.71707	−0.121173	−0.0605865	+	0.998163i	$0.519297\pi$
$942$	0	0
$943$	7.58673	0.247058
$944$	0	0
$945$	5.14878	0.167490
$946$	0	0
$947$	−4.42009	−0.143634	−0.0718168	−	0.997418i	$-0.522880\pi$
$947$	−4.42009	−0.143634	−0.0718168	+	0.997418i	$0.522880\pi$
$948$	0	0
$949$	−25.9643	−0.842837
$950$	0	0
$951$	17.7169	0.574511
$952$	0	0
$953$	−19.8298	−0.642350	−0.321175	−	0.947020i	$-0.604078\pi$
$953$	−19.8298	−0.642350	−0.321175	+	0.947020i	$0.604078\pi$
$954$	0	0
$955$	−2.80238	−0.0906829
$956$	0	0
$957$	5.13887	0.166116
$958$	0	0
$959$	11.6089	0.374872
$960$	0	0
$961$	−4.67408	−0.150777
$962$	0	0
$963$	−16.7653	−0.540253
$964$	0	0
$965$	−41.4836	−1.33540
$966$	0	0
$967$	−35.7294	−1.14898	−0.574490	−	0.818512i	$-0.694800\pi$
$967$	−35.7294	−1.14898	−0.574490	+	0.818512i	$0.694800\pi$
$968$	0	0
$969$	−50.5134	−1.62273
$970$	0	0
$971$	31.3718	1.00677	0.503384	−	0.864063i	$-0.332088\pi$
$971$	31.3718	1.00677	0.503384	+	0.864063i	$0.332088\pi$
$972$	0	0
$973$	−58.8395	−1.88631
$974$	0	0
$975$	−9.38342	−0.300510
$976$	0	0
$977$	53.4531	1.71012	0.855058	−	0.518533i	$-0.173522\pi$
$977$	53.4531	1.71012	0.855058	+	0.518533i	$0.173522\pi$
$978$	0	0
$979$	−64.1683	−2.05083
$980$	0	0
$981$	−15.0206	−0.479572
$982$	0	0
$983$	−34.7975	−1.10987	−0.554934	−	0.831894i	$-0.687257\pi$
$983$	−34.7975	−1.10987	−0.554934	+	0.831894i	$0.687257\pi$
$984$	0	0
$985$	20.7928	0.662513
$986$	0	0
$987$	−4.72211	−0.150306
$988$	0	0
$989$	3.69813	0.117594
$990$	0	0
$991$	36.5028	1.15955	0.579775	−	0.814776i	$-0.303140\pi$
$991$	36.5028	1.15955	0.579775	+	0.814776i	$0.303140\pi$
$992$	0	0
$993$	6.42127	0.203773
$994$	0	0
$995$	21.1238	0.669668
$996$	0	0
$997$	38.1887	1.20945	0.604724	−	0.796435i	$-0.293283\pi$
$997$	38.1887	1.20945	0.604724	+	0.796435i	$0.293283\pi$
$998$	0	0
$999$	−0.265385	−0.00839641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.11	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.11	✓	18	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.69179 q^{5} +3.04338 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.69179 q^{5} +3.04338 q^{7} +1.00000 q^{9} +5.13887 q^{11} +4.38922 q^{13} +1.69179 q^{15} -7.10930 q^{17} +7.10526 q^{19} +3.04338 q^{21} -1.00000 q^{23} -2.13784 q^{25} +1.00000 q^{27} +1.00000 q^{29} +5.13088 q^{31} +5.13887 q^{33} +5.14878 q^{35} -0.265385 q^{37} +4.38922 q^{39} -7.58673 q^{41} -3.69813 q^{43} +1.69179 q^{45} -1.55160 q^{47} +2.26219 q^{49} -7.10930 q^{51} +6.33821 q^{53} +8.69391 q^{55} +7.10526 q^{57} +10.7809 q^{59} +7.54577 q^{61} +3.04338 q^{63} +7.42565 q^{65} -0.875564 q^{67} -1.00000 q^{69} +1.43673 q^{71} -5.91547 q^{73} -2.13784 q^{75} +15.6396 q^{77} +12.7350 q^{79} +1.00000 q^{81} -9.39850 q^{83} -12.0275 q^{85} +1.00000 q^{87} -12.4868 q^{89} +13.3581 q^{91} +5.13088 q^{93} +12.0206 q^{95} +3.58447 q^{97} +5.13887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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