LMFDB - Embedded newform 8048.2.a.u.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27911 q^{3} +3.79116 q^{5} -4.03115 q^{7} +2.19436 q^{9} +O(q^{10})$	$q-2.27911 q^{3} +3.79116 q^{5} -4.03115 q^{7} +2.19436 q^{9} +4.62241 q^{11} -3.86363 q^{13} -8.64048 q^{15} +1.73504 q^{17} +4.30211 q^{19} +9.18744 q^{21} +7.16718 q^{23} +9.37288 q^{25} +1.83615 q^{27} -3.40299 q^{29} -1.29624 q^{31} -10.5350 q^{33} -15.2827 q^{35} +11.4689 q^{37} +8.80564 q^{39} -6.76299 q^{41} -1.28015 q^{43} +8.31915 q^{45} -8.34021 q^{47} +9.25015 q^{49} -3.95436 q^{51} +9.55893 q^{53} +17.5243 q^{55} -9.80499 q^{57} -3.24630 q^{59} -12.3562 q^{61} -8.84578 q^{63} -14.6476 q^{65} +12.1005 q^{67} -16.3348 q^{69} -7.17544 q^{71} +4.96822 q^{73} -21.3619 q^{75} -18.6336 q^{77} -10.6927 q^{79} -10.7679 q^{81} +5.83442 q^{83} +6.57783 q^{85} +7.75580 q^{87} +6.64931 q^{89} +15.5748 q^{91} +2.95429 q^{93} +16.3100 q^{95} -10.1113 q^{97} +10.1432 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * 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
$3$	−2.27911	−1.31585	−0.657923	−	0.753085i	$-0.728565\pi$
$3$	−2.27911	−1.31585	−0.657923	+	0.753085i	$0.728565\pi$
$4$	0	0
$5$	3.79116	1.69546	0.847729	−	0.530430i	$-0.177970\pi$
$5$	3.79116	1.69546	0.847729	+	0.530430i	$0.177970\pi$
$6$	0	0
$7$	−4.03115	−1.52363	−0.761815	−	0.647794i	$-0.775692\pi$
$7$	−4.03115	−1.52363	−0.761815	+	0.647794i	$0.775692\pi$
$8$	0	0
$9$	2.19436	0.731452
$10$	0	0
$11$	4.62241	1.39371	0.696855	−	0.717212i	$-0.254582\pi$
$11$	4.62241	1.39371	0.696855	+	0.717212i	$0.254582\pi$
$12$	0	0
$13$	−3.86363	−1.07158	−0.535788	−	0.844352i	$-0.679986\pi$
$13$	−3.86363	−1.07158	−0.535788	+	0.844352i	$0.679986\pi$
$14$	0	0
$15$	−8.64048	−2.23096
$16$	0	0
$17$	1.73504	0.420810	0.210405	−	0.977614i	$-0.432522\pi$
$17$	1.73504	0.420810	0.210405	+	0.977614i	$0.432522\pi$
$18$	0	0
$19$	4.30211	0.986971	0.493485	−	0.869754i	$-0.335723\pi$
$19$	4.30211	0.986971	0.493485	+	0.869754i	$0.335723\pi$
$20$	0	0
$21$	9.18744	2.00486
$22$	0	0
$23$	7.16718	1.49446	0.747230	−	0.664566i	$-0.231383\pi$
$23$	7.16718	1.49446	0.747230	+	0.664566i	$0.231383\pi$
$24$	0	0
$25$	9.37288	1.87458
$26$	0	0
$27$	1.83615	0.353368
$28$	0	0
$29$	−3.40299	−0.631919	−0.315960	−	0.948773i	$-0.602326\pi$
$29$	−3.40299	−0.631919	−0.315960	+	0.948773i	$0.602326\pi$
$30$	0	0
$31$	−1.29624	−0.232812	−0.116406	−	0.993202i	$-0.537137\pi$
$31$	−1.29624	−0.232812	−0.116406	+	0.993202i	$0.537137\pi$
$32$	0	0
$33$	−10.5350	−1.83391
$34$	0	0
$35$	−15.2827	−2.58325
$36$	0	0
$37$	11.4689	1.88548	0.942738	−	0.333535i	$-0.108242\pi$
$37$	11.4689	1.88548	0.942738	+	0.333535i	$0.108242\pi$
$38$	0	0
$39$	8.80564	1.41003
$40$	0	0
$41$	−6.76299	−1.05620	−0.528101	−	0.849182i	$-0.677096\pi$
$41$	−6.76299	−1.05620	−0.528101	+	0.849182i	$0.677096\pi$
$42$	0	0
$43$	−1.28015	−0.195221	−0.0976104	−	0.995225i	$-0.531120\pi$
$43$	−1.28015	−0.195221	−0.0976104	+	0.995225i	$0.531120\pi$
$44$	0	0
$45$	8.31915	1.24015
$46$	0	0
$47$	−8.34021	−1.21654	−0.608272	−	0.793728i	$-0.708137\pi$
$47$	−8.34021	−1.21654	−0.608272	+	0.793728i	$0.708137\pi$
$48$	0	0
$49$	9.25015	1.32145
$50$	0	0
$51$	−3.95436	−0.553722
$52$	0	0
$53$	9.55893	1.31302	0.656510	−	0.754317i	$-0.272032\pi$
$53$	9.55893	1.31302	0.656510	+	0.754317i	$0.272032\pi$
$54$	0	0
$55$	17.5243	2.36298
$56$	0	0
$57$	−9.80499	−1.29870
$58$	0	0
$59$	−3.24630	−0.422632	−0.211316	−	0.977418i	$-0.567775\pi$
$59$	−3.24630	−0.422632	−0.211316	+	0.977418i	$0.567775\pi$
$60$	0	0
$61$	−12.3562	−1.58205	−0.791025	−	0.611784i	$-0.790452\pi$
$61$	−12.3562	−1.58205	−0.791025	+	0.611784i	$0.790452\pi$
$62$	0	0
$63$	−8.84578	−1.11446
$64$	0	0
$65$	−14.6476	−1.81681
$66$	0	0
$67$	12.1005	1.47832	0.739158	−	0.673532i	$-0.235223\pi$
$67$	12.1005	1.47832	0.739158	+	0.673532i	$0.235223\pi$
$68$	0	0
$69$	−16.3348	−1.96648
$70$	0	0
$71$	−7.17544	−0.851568	−0.425784	−	0.904825i	$-0.640002\pi$
$71$	−7.17544	−0.851568	−0.425784	+	0.904825i	$0.640002\pi$
$72$	0	0
$73$	4.96822	0.581486	0.290743	−	0.956801i	$-0.406098\pi$
$73$	4.96822	0.581486	0.290743	+	0.956801i	$0.406098\pi$
$74$	0	0
$75$	−21.3619	−2.46665
$76$	0	0
$77$	−18.6336	−2.12350
$78$	0	0
$79$	−10.6927	−1.20303	−0.601514	−	0.798862i	$-0.705436\pi$
$79$	−10.6927	−1.20303	−0.601514	+	0.798862i	$0.705436\pi$
$80$	0	0
$81$	−10.7679	−1.19643
$82$	0	0
$83$	5.83442	0.640411	0.320206	−	0.947348i	$-0.396248\pi$
$83$	5.83442	0.640411	0.320206	+	0.947348i	$0.396248\pi$
$84$	0	0
$85$	6.57783	0.713466
$86$	0	0
$87$	7.75580	0.831509
$88$	0	0
$89$	6.64931	0.704826	0.352413	−	0.935845i	$-0.385361\pi$
$89$	6.64931	0.704826	0.352413	+	0.935845i	$0.385361\pi$
$90$	0	0
$91$	15.5748	1.63269
$92$	0	0
$93$	2.95429	0.306345
$94$	0	0
$95$	16.3100	1.67337
$96$	0	0
$97$	−10.1113	−1.02665	−0.513323	−	0.858195i	$-0.671586\pi$
$97$	−10.1113	−1.02665	−0.513323	+	0.858195i	$0.671586\pi$
$98$	0	0
$99$	10.1432	1.01943
$100$	0	0
$101$	5.70553	0.567721	0.283861	−	0.958866i	$-0.408385\pi$
$101$	5.70553	0.567721	0.283861	+	0.958866i	$0.408385\pi$
$102$	0	0
$103$	−9.24132	−0.910575	−0.455287	−	0.890345i	$-0.650463\pi$
$103$	−9.24132	−0.910575	−0.455287	+	0.890345i	$0.650463\pi$
$104$	0	0
$105$	34.8310	3.39916
$106$	0	0
$107$	15.0186	1.45190	0.725951	−	0.687746i	$-0.241400\pi$
$107$	15.0186	1.45190	0.725951	+	0.687746i	$0.241400\pi$
$108$	0	0
$109$	−2.99448	−0.286819	−0.143409	−	0.989663i	$-0.545807\pi$
$109$	−2.99448	−0.286819	−0.143409	+	0.989663i	$0.545807\pi$
$110$	0	0
$111$	−26.1389	−2.48100
$112$	0	0
$113$	−2.05064	−0.192908	−0.0964539	−	0.995337i	$-0.530750\pi$
$113$	−2.05064	−0.192908	−0.0964539	+	0.995337i	$0.530750\pi$
$114$	0	0
$115$	27.1719	2.53379
$116$	0	0
$117$	−8.47817	−0.783807
$118$	0	0
$119$	−6.99422	−0.641159
$120$	0	0
$121$	10.3667	0.942427
$122$	0	0
$123$	15.4136	1.38980
$124$	0	0
$125$	16.5783	1.48281
$126$	0	0
$127$	2.63530	0.233845	0.116923	−	0.993141i	$-0.462697\pi$
$127$	2.63530	0.233845	0.116923	+	0.993141i	$0.462697\pi$
$128$	0	0
$129$	2.91760	0.256881
$130$	0	0
$131$	9.66184	0.844159	0.422080	−	0.906559i	$-0.361300\pi$
$131$	9.66184	0.844159	0.422080	+	0.906559i	$0.361300\pi$
$132$	0	0
$133$	−17.3424	−1.50378
$134$	0	0
$135$	6.96114	0.599120
$136$	0	0
$137$	3.82672	0.326939	0.163469	−	0.986548i	$-0.447732\pi$
$137$	3.82672	0.326939	0.163469	+	0.986548i	$0.447732\pi$
$138$	0	0
$139$	−12.8713	−1.09173	−0.545863	−	0.837875i	$-0.683798\pi$
$139$	−12.8713	−1.09173	−0.545863	+	0.837875i	$0.683798\pi$
$140$	0	0
$141$	19.0083	1.60079
$142$	0	0
$143$	−17.8593	−1.49347
$144$	0	0
$145$	−12.9013	−1.07139
$146$	0	0
$147$	−21.0821	−1.73883
$148$	0	0
$149$	15.0108	1.22973	0.614867	−	0.788631i	$-0.289210\pi$
$149$	15.0108	1.22973	0.614867	+	0.788631i	$0.289210\pi$
$150$	0	0
$151$	−7.14463	−0.581422	−0.290711	−	0.956811i	$-0.593892\pi$
$151$	−7.14463	−0.581422	−0.290711	+	0.956811i	$0.593892\pi$
$152$	0	0
$153$	3.80731	0.307803
$154$	0	0
$155$	−4.91427	−0.394724
$156$	0	0
$157$	−3.67900	−0.293616	−0.146808	−	0.989165i	$-0.546900\pi$
$157$	−3.67900	−0.293616	−0.146808	+	0.989165i	$0.546900\pi$
$158$	0	0
$159$	−21.7859	−1.72773
$160$	0	0
$161$	−28.8920	−2.27700
$162$	0	0
$163$	6.36798	0.498779	0.249389	−	0.968403i	$-0.419770\pi$
$163$	6.36798	0.498779	0.249389	+	0.968403i	$0.419770\pi$
$164$	0	0
$165$	−39.9399	−3.10931
$166$	0	0
$167$	1.87848	0.145361	0.0726804	−	0.997355i	$-0.476845\pi$
$167$	1.87848	0.145361	0.0726804	+	0.997355i	$0.476845\pi$
$168$	0	0
$169$	1.92760	0.148277
$170$	0	0
$171$	9.44036	0.721922
$172$	0	0
$173$	−10.0104	−0.761080	−0.380540	−	0.924764i	$-0.624262\pi$
$173$	−10.0104	−0.761080	−0.380540	+	0.924764i	$0.624262\pi$
$174$	0	0
$175$	−37.7835	−2.85616
$176$	0	0
$177$	7.39868	0.556118
$178$	0	0
$179$	9.09399	0.679717	0.339858	−	0.940477i	$-0.389621\pi$
$179$	9.09399	0.679717	0.339858	+	0.940477i	$0.389621\pi$
$180$	0	0
$181$	24.7002	1.83595	0.917976	−	0.396635i	$-0.129822\pi$
$181$	24.7002	1.83595	0.917976	+	0.396635i	$0.129822\pi$
$182$	0	0
$183$	28.1612	2.08173
$184$	0	0
$185$	43.4804	3.19674
$186$	0	0
$187$	8.02009	0.586487
$188$	0	0
$189$	−7.40180	−0.538402
$190$	0	0
$191$	13.2841	0.961204	0.480602	−	0.876939i	$-0.340418\pi$
$191$	13.2841	0.961204	0.480602	+	0.876939i	$0.340418\pi$
$192$	0	0
$193$	26.6666	1.91950	0.959750	−	0.280855i	$-0.0906180\pi$
$193$	26.6666	1.91950	0.959750	+	0.280855i	$0.0906180\pi$
$194$	0	0
$195$	33.3836	2.39065
$196$	0	0
$197$	3.65462	0.260381	0.130190	−	0.991489i	$-0.458441\pi$
$197$	3.65462	0.260381	0.130190	+	0.991489i	$0.458441\pi$
$198$	0	0
$199$	−18.6263	−1.32038	−0.660191	−	0.751098i	$-0.729525\pi$
$199$	−18.6263	−1.32038	−0.660191	+	0.751098i	$0.729525\pi$
$200$	0	0
$201$	−27.5785	−1.94524
$202$	0	0
$203$	13.7180	0.962811
$204$	0	0
$205$	−25.6396	−1.79074
$206$	0	0
$207$	15.7273	1.09313
$208$	0	0
$209$	19.8861	1.37555
$210$	0	0
$211$	3.17606	0.218649	0.109325	−	0.994006i	$-0.465131\pi$
$211$	3.17606	0.218649	0.109325	+	0.994006i	$0.465131\pi$
$212$	0	0
$213$	16.3536	1.12053
$214$	0	0
$215$	−4.85324	−0.330988
$216$	0	0
$217$	5.22535	0.354720
$218$	0	0
$219$	−11.3231	−0.765146
$220$	0	0
$221$	−6.70356	−0.450930
$222$	0	0
$223$	12.4850	0.836059	0.418030	−	0.908433i	$-0.362721\pi$
$223$	12.4850	0.836059	0.418030	+	0.908433i	$0.362721\pi$
$224$	0	0
$225$	20.5674	1.37116
$226$	0	0
$227$	16.1748	1.07356	0.536778	−	0.843723i	$-0.319641\pi$
$227$	16.1748	1.07356	0.536778	+	0.843723i	$0.319641\pi$
$228$	0	0
$229$	−22.6012	−1.49353	−0.746765	−	0.665088i	$-0.768394\pi$
$229$	−22.6012	−1.49353	−0.746765	+	0.665088i	$0.768394\pi$
$230$	0	0
$231$	42.4682	2.79420
$232$	0	0
$233$	−3.70258	−0.242564	−0.121282	−	0.992618i	$-0.538701\pi$
$233$	−3.70258	−0.242564	−0.121282	+	0.992618i	$0.538701\pi$
$234$	0	0
$235$	−31.6191	−2.06260
$236$	0	0
$237$	24.3700	1.58300
$238$	0	0
$239$	−5.99222	−0.387604	−0.193802	−	0.981041i	$-0.562082\pi$
$239$	−5.99222	−0.387604	−0.193802	+	0.981041i	$0.562082\pi$
$240$	0	0
$241$	19.9256	1.28352	0.641760	−	0.766905i	$-0.278204\pi$
$241$	19.9256	1.28352	0.641760	+	0.766905i	$0.278204\pi$
$242$	0	0
$243$	19.0327	1.22095
$244$	0	0
$245$	35.0688	2.24046
$246$	0	0
$247$	−16.6217	−1.05762
$248$	0	0
$249$	−13.2973	−0.842683
$250$	0	0
$251$	−5.43776	−0.343228	−0.171614	−	0.985164i	$-0.554898\pi$
$251$	−5.43776	−0.343228	−0.171614	+	0.985164i	$0.554898\pi$
$252$	0	0
$253$	33.1297	2.08284
$254$	0	0
$255$	−14.9916	−0.938811
$256$	0	0
$257$	16.9275	1.05591	0.527954	−	0.849273i	$-0.322959\pi$
$257$	16.9275	1.05591	0.527954	+	0.849273i	$0.322959\pi$
$258$	0	0
$259$	−46.2328	−2.87277
$260$	0	0
$261$	−7.46737	−0.462219
$262$	0	0
$263$	14.2305	0.877489	0.438744	−	0.898612i	$-0.355423\pi$
$263$	14.2305	0.877489	0.438744	+	0.898612i	$0.355423\pi$
$264$	0	0
$265$	36.2394	2.22617
$266$	0	0
$267$	−15.1545	−0.927443
$268$	0	0
$269$	4.78021	0.291455	0.145727	−	0.989325i	$-0.453448\pi$
$269$	4.78021	0.291455	0.145727	+	0.989325i	$0.453448\pi$
$270$	0	0
$271$	−22.8296	−1.38680	−0.693399	−	0.720554i	$-0.743888\pi$
$271$	−22.8296	−1.38680	−0.693399	+	0.720554i	$0.743888\pi$
$272$	0	0
$273$	−35.4968	−2.14837
$274$	0	0
$275$	43.3253	2.61261
$276$	0	0
$277$	2.84440	0.170903	0.0854516	−	0.996342i	$-0.472767\pi$
$277$	2.84440	0.170903	0.0854516	+	0.996342i	$0.472767\pi$
$278$	0	0
$279$	−2.84442	−0.170291
$280$	0	0
$281$	−8.70598	−0.519356	−0.259678	−	0.965695i	$-0.583616\pi$
$281$	−8.70598	−0.519356	−0.259678	+	0.965695i	$0.583616\pi$
$282$	0	0
$283$	−28.5870	−1.69932	−0.849659	−	0.527333i	$-0.823192\pi$
$283$	−28.5870	−1.69932	−0.849659	+	0.527333i	$0.823192\pi$
$284$	0	0
$285$	−37.1723	−2.20189
$286$	0	0
$287$	27.2626	1.60926
$288$	0	0
$289$	−13.9896	−0.822919
$290$	0	0
$291$	23.0448	1.35091
$292$	0	0
$293$	−7.13843	−0.417032	−0.208516	−	0.978019i	$-0.566863\pi$
$293$	−7.13843	−0.417032	−0.208516	+	0.978019i	$0.566863\pi$
$294$	0	0
$295$	−12.3072	−0.716554
$296$	0	0
$297$	8.48745	0.492492
$298$	0	0
$299$	−27.6913	−1.60143
$300$	0	0
$301$	5.16047	0.297444
$302$	0	0
$303$	−13.0035	−0.747034
$304$	0	0
$305$	−46.8443	−2.68230
$306$	0	0
$307$	−4.76214	−0.271790	−0.135895	−	0.990723i	$-0.543391\pi$
$307$	−4.76214	−0.271790	−0.135895	+	0.990723i	$0.543391\pi$
$308$	0	0
$309$	21.0620	1.19818
$310$	0	0
$311$	29.6663	1.68222	0.841111	−	0.540862i	$-0.181902\pi$
$311$	29.6663	1.68222	0.841111	+	0.540862i	$0.181902\pi$
$312$	0	0
$313$	3.14569	0.177805	0.0889025	−	0.996040i	$-0.471664\pi$
$313$	3.14569	0.177805	0.0889025	+	0.996040i	$0.471664\pi$
$314$	0	0
$315$	−33.5357	−1.88952
$316$	0	0
$317$	−4.87154	−0.273613	−0.136806	−	0.990598i	$-0.543684\pi$
$317$	−4.87154	−0.273613	−0.136806	+	0.990598i	$0.543684\pi$
$318$	0	0
$319$	−15.7300	−0.880712
$320$	0	0
$321$	−34.2291	−1.91048
$322$	0	0
$323$	7.46435	0.415327
$324$	0	0
$325$	−36.2133	−2.00875
$326$	0	0
$327$	6.82475	0.377410
$328$	0	0
$329$	33.6206	1.85356
$330$	0	0
$331$	−19.7091	−1.08331	−0.541655	−	0.840601i	$-0.682202\pi$
$331$	−19.7091	−1.08331	−0.541655	+	0.840601i	$0.682202\pi$
$332$	0	0
$333$	25.1669	1.37914
$334$	0	0
$335$	45.8751	2.50642
$336$	0	0
$337$	6.18352	0.336838	0.168419	−	0.985716i	$-0.446134\pi$
$337$	6.18352	0.336838	0.168419	+	0.985716i	$0.446134\pi$
$338$	0	0
$339$	4.67364	0.253837
$340$	0	0
$341$	−5.99178	−0.324473
$342$	0	0
$343$	−9.07070	−0.489772
$344$	0	0
$345$	−61.9278	−3.33408
$346$	0	0
$347$	16.8542	0.904783	0.452392	−	0.891819i	$-0.350571\pi$
$347$	16.8542	0.904783	0.452392	+	0.891819i	$0.350571\pi$
$348$	0	0
$349$	0.876469	0.0469163	0.0234581	−	0.999725i	$-0.492532\pi$
$349$	0.876469	0.0469163	0.0234581	+	0.999725i	$0.492532\pi$
$350$	0	0
$351$	−7.09420	−0.378660
$352$	0	0
$353$	−26.5298	−1.41204	−0.706020	−	0.708192i	$-0.749511\pi$
$353$	−26.5298	−1.41204	−0.706020	+	0.708192i	$0.749511\pi$
$354$	0	0
$355$	−27.2032	−1.44380
$356$	0	0
$357$	15.9406	0.843667
$358$	0	0
$359$	−5.36111	−0.282949	−0.141474	−	0.989942i	$-0.545184\pi$
$359$	−5.36111	−0.282949	−0.141474	+	0.989942i	$0.545184\pi$
$360$	0	0
$361$	−0.491879	−0.0258884
$362$	0	0
$363$	−23.6269	−1.24009
$364$	0	0
$365$	18.8353	0.985885
$366$	0	0
$367$	−24.9627	−1.30304	−0.651520	−	0.758632i	$-0.725868\pi$
$367$	−24.9627	−1.30304	−0.651520	+	0.758632i	$0.725868\pi$
$368$	0	0
$369$	−14.8404	−0.772561
$370$	0	0
$371$	−38.5335	−2.00056
$372$	0	0
$373$	−15.0785	−0.780734	−0.390367	−	0.920659i	$-0.627652\pi$
$373$	−15.0785	−0.780734	−0.390367	+	0.920659i	$0.627652\pi$
$374$	0	0
$375$	−37.7838	−1.95114
$376$	0	0
$377$	13.1479	0.677150
$378$	0	0
$379$	14.0283	0.720587	0.360294	−	0.932839i	$-0.382676\pi$
$379$	14.0283	0.720587	0.360294	+	0.932839i	$0.382676\pi$
$380$	0	0
$381$	−6.00616	−0.307705
$382$	0	0
$383$	13.2932	0.679253	0.339626	−	0.940560i	$-0.389699\pi$
$383$	13.2932	0.679253	0.339626	+	0.940560i	$0.389699\pi$
$384$	0	0
$385$	−70.6430	−3.60030
$386$	0	0
$387$	−2.80910	−0.142795
$388$	0	0
$389$	24.9812	1.26660	0.633298	−	0.773908i	$-0.281701\pi$
$389$	24.9812	1.26660	0.633298	+	0.773908i	$0.281701\pi$
$390$	0	0
$391$	12.4354	0.628884
$392$	0	0
$393$	−22.0204	−1.11078
$394$	0	0
$395$	−40.5379	−2.03968
$396$	0	0
$397$	−16.3754	−0.821860	−0.410930	−	0.911667i	$-0.634796\pi$
$397$	−16.3754	−0.821860	−0.410930	+	0.911667i	$0.634796\pi$
$398$	0	0
$399$	39.5254	1.97874
$400$	0	0
$401$	28.3698	1.41672	0.708360	−	0.705851i	$-0.249435\pi$
$401$	28.3698	1.41672	0.708360	+	0.705851i	$0.249435\pi$
$402$	0	0
$403$	5.00820	0.249476
$404$	0	0
$405$	−40.8227	−2.02850
$406$	0	0
$407$	53.0140	2.62781
$408$	0	0
$409$	25.0188	1.23710	0.618549	−	0.785746i	$-0.287721\pi$
$409$	25.0188	1.23710	0.618549	+	0.785746i	$0.287721\pi$
$410$	0	0
$411$	−8.72152	−0.430201
$412$	0	0
$413$	13.0863	0.643935
$414$	0	0
$415$	22.1192	1.08579
$416$	0	0
$417$	29.3350	1.43654
$418$	0	0
$419$	−16.2484	−0.793788	−0.396894	−	0.917865i	$-0.629912\pi$
$419$	−16.2484	−0.793788	−0.396894	+	0.917865i	$0.629912\pi$
$420$	0	0
$421$	27.6483	1.34750	0.673748	−	0.738961i	$-0.264683\pi$
$421$	27.6483	1.34750	0.673748	+	0.738961i	$0.264683\pi$
$422$	0	0
$423$	−18.3014	−0.889844
$424$	0	0
$425$	16.2624	0.788841
$426$	0	0
$427$	49.8097	2.41046
$428$	0	0
$429$	40.7033	1.96517
$430$	0	0
$431$	30.8322	1.48514	0.742568	−	0.669771i	$-0.233608\pi$
$431$	30.8322	1.48514	0.742568	+	0.669771i	$0.233608\pi$
$432$	0	0
$433$	−20.9217	−1.00543	−0.502716	−	0.864452i	$-0.667666\pi$
$433$	−20.9217	−1.00543	−0.502716	+	0.864452i	$0.667666\pi$
$434$	0	0
$435$	29.4034	1.40979
$436$	0	0
$437$	30.8340	1.47499
$438$	0	0
$439$	34.0600	1.62560	0.812798	−	0.582546i	$-0.197943\pi$
$439$	34.0600	1.62560	0.812798	+	0.582546i	$0.197943\pi$
$440$	0	0
$441$	20.2981	0.966578
$442$	0	0
$443$	28.7016	1.36366	0.681828	−	0.731513i	$-0.261185\pi$
$443$	28.7016	1.36366	0.681828	+	0.731513i	$0.261185\pi$
$444$	0	0
$445$	25.2086	1.19500
$446$	0	0
$447$	−34.2113	−1.61814
$448$	0	0
$449$	35.9605	1.69708	0.848541	−	0.529130i	$-0.177482\pi$
$449$	35.9605	1.69708	0.848541	+	0.529130i	$0.177482\pi$
$450$	0	0
$451$	−31.2613	−1.47204
$452$	0	0
$453$	16.2834	0.765062
$454$	0	0
$455$	59.0467	2.76815
$456$	0	0
$457$	8.41951	0.393848	0.196924	−	0.980419i	$-0.436905\pi$
$457$	8.41951	0.393848	0.196924	+	0.980419i	$0.436905\pi$
$458$	0	0
$459$	3.18581	0.148701
$460$	0	0
$461$	16.3496	0.761476	0.380738	−	0.924683i	$-0.375670\pi$
$461$	16.3496	0.761476	0.380738	+	0.924683i	$0.375670\pi$
$462$	0	0
$463$	25.5880	1.18918	0.594589	−	0.804030i	$-0.297315\pi$
$463$	25.5880	1.18918	0.594589	+	0.804030i	$0.297315\pi$
$464$	0	0
$465$	11.2002	0.519396
$466$	0	0
$467$	−36.2732	−1.67852	−0.839261	−	0.543729i	$-0.817012\pi$
$467$	−36.2732	−1.67852	−0.839261	+	0.543729i	$0.817012\pi$
$468$	0	0
$469$	−48.7791	−2.25241
$470$	0	0
$471$	8.38485	0.386354
$472$	0	0
$473$	−5.91737	−0.272081
$474$	0	0
$475$	40.3231	1.85015
$476$	0	0
$477$	20.9757	0.960412
$478$	0	0
$479$	20.8433	0.952357	0.476178	−	0.879349i	$-0.342022\pi$
$479$	20.8433	0.952357	0.476178	+	0.879349i	$0.342022\pi$
$480$	0	0
$481$	−44.3115	−2.02043
$482$	0	0
$483$	65.8480	2.99619
$484$	0	0
$485$	−38.3335	−1.74064
$486$	0	0
$487$	1.61935	0.0733798	0.0366899	−	0.999327i	$-0.488319\pi$
$487$	1.61935	0.0733798	0.0366899	+	0.999327i	$0.488319\pi$
$488$	0	0
$489$	−14.5133	−0.656316
$490$	0	0
$491$	18.4080	0.830741	0.415371	−	0.909652i	$-0.363652\pi$
$491$	18.4080	0.830741	0.415371	+	0.909652i	$0.363652\pi$
$492$	0	0
$493$	−5.90434	−0.265918
$494$	0	0
$495$	38.4546	1.72840
$496$	0	0
$497$	28.9253	1.29748
$498$	0	0
$499$	8.70198	0.389554	0.194777	−	0.980848i	$-0.437602\pi$
$499$	8.70198	0.389554	0.194777	+	0.980848i	$0.437602\pi$
$500$	0	0
$501$	−4.28126	−0.191273
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	21.6306	0.962547
$506$	0	0
$507$	−4.39322	−0.195110
$508$	0	0
$509$	−6.56992	−0.291206	−0.145603	−	0.989343i	$-0.546512\pi$
$509$	−6.56992	−0.291206	−0.145603	+	0.989343i	$0.546512\pi$
$510$	0	0
$511$	−20.0276	−0.885970
$512$	0	0
$513$	7.89932	0.348764
$514$	0	0
$515$	−35.0353	−1.54384
$516$	0	0
$517$	−38.5519	−1.69551
$518$	0	0
$519$	22.8149	1.00146
$520$	0	0
$521$	−25.8155	−1.13100	−0.565499	−	0.824749i	$-0.691316\pi$
$521$	−25.8155	−1.13100	−0.565499	+	0.824749i	$0.691316\pi$
$522$	0	0
$523$	0.711015	0.0310905	0.0155453	−	0.999879i	$-0.495052\pi$
$523$	0.711015	0.0310905	0.0155453	+	0.999879i	$0.495052\pi$
$524$	0	0
$525$	86.1128	3.75827
$526$	0	0
$527$	−2.24904	−0.0979698
$528$	0	0
$529$	28.3684	1.23341
$530$	0	0
$531$	−7.12353	−0.309135
$532$	0	0
$533$	26.1297	1.13180
$534$	0	0
$535$	56.9379	2.46164
$536$	0	0
$537$	−20.7262	−0.894403
$538$	0	0
$539$	42.7580	1.84172
$540$	0	0
$541$	12.4754	0.536358	0.268179	−	0.963369i	$-0.413578\pi$
$541$	12.4754	0.536358	0.268179	+	0.963369i	$0.413578\pi$
$542$	0	0
$543$	−56.2946	−2.41583
$544$	0	0
$545$	−11.3525	−0.486289
$546$	0	0
$547$	2.52452	0.107941	0.0539704	−	0.998543i	$-0.482812\pi$
$547$	2.52452	0.107941	0.0539704	+	0.998543i	$0.482812\pi$
$548$	0	0
$549$	−27.1139	−1.15719
$550$	0	0
$551$	−14.6400	−0.623686
$552$	0	0
$553$	43.1041	1.83297
$554$	0	0
$555$	−99.0968	−4.20642
$556$	0	0
$557$	−18.5546	−0.786184	−0.393092	−	0.919499i	$-0.628595\pi$
$557$	−18.5546	−0.786184	−0.393092	+	0.919499i	$0.628595\pi$
$558$	0	0
$559$	4.94601	0.209194
$560$	0	0
$561$	−18.2787	−0.771727
$562$	0	0
$563$	33.9002	1.42872	0.714362	−	0.699777i	$-0.246717\pi$
$563$	33.9002	1.42872	0.714362	+	0.699777i	$0.246717\pi$
$564$	0	0
$565$	−7.77430	−0.327067
$566$	0	0
$567$	43.4069	1.82292
$568$	0	0
$569$	−12.6955	−0.532223	−0.266111	−	0.963942i	$-0.585739\pi$
$569$	−12.6955	−0.532223	−0.266111	+	0.963942i	$0.585739\pi$
$570$	0	0
$571$	12.3436	0.516562	0.258281	−	0.966070i	$-0.416844\pi$
$571$	12.3436	0.516562	0.258281	+	0.966070i	$0.416844\pi$
$572$	0	0
$573$	−30.2760	−1.26480
$574$	0	0
$575$	67.1771	2.80148
$576$	0	0
$577$	32.8671	1.36828	0.684138	−	0.729353i	$-0.260179\pi$
$577$	32.8671	1.36828	0.684138	+	0.729353i	$0.260179\pi$
$578$	0	0
$579$	−60.7761	−2.52577
$580$	0	0
$581$	−23.5194	−0.975750
$582$	0	0
$583$	44.1853	1.82997
$584$	0	0
$585$	−32.1421	−1.32891
$586$	0	0
$587$	29.7918	1.22964	0.614819	−	0.788668i	$-0.289229\pi$
$587$	29.7918	1.22964	0.614819	+	0.788668i	$0.289229\pi$
$588$	0	0
$589$	−5.57658	−0.229779
$590$	0	0
$591$	−8.32929	−0.342621
$592$	0	0
$593$	45.0621	1.85048	0.925240	−	0.379383i	$-0.123864\pi$
$593$	45.0621	1.85048	0.925240	+	0.379383i	$0.123864\pi$
$594$	0	0
$595$	−26.5162	−1.08706
$596$	0	0
$597$	42.4514	1.73742
$598$	0	0
$599$	41.5406	1.69730	0.848652	−	0.528952i	$-0.177415\pi$
$599$	41.5406	1.69730	0.848652	+	0.528952i	$0.177415\pi$
$600$	0	0
$601$	14.4075	0.587694	0.293847	−	0.955852i	$-0.405064\pi$
$601$	14.4075	0.587694	0.293847	+	0.955852i	$0.405064\pi$
$602$	0	0
$603$	26.5529	1.08132
$604$	0	0
$605$	39.3018	1.59785
$606$	0	0
$607$	18.5714	0.753790	0.376895	−	0.926256i	$-0.376992\pi$
$607$	18.5714	0.753790	0.376895	+	0.926256i	$0.376992\pi$
$608$	0	0
$609$	−31.2648	−1.26691
$610$	0	0
$611$	32.2234	1.30362
$612$	0	0
$613$	−19.5200	−0.788404	−0.394202	−	0.919024i	$-0.628979\pi$
$613$	−19.5200	−0.788404	−0.394202	+	0.919024i	$0.628979\pi$
$614$	0	0
$615$	58.4355	2.35635
$616$	0	0
$617$	43.6158	1.75591	0.877954	−	0.478745i	$-0.158908\pi$
$617$	43.6158	1.75591	0.877954	+	0.478745i	$0.158908\pi$
$618$	0	0
$619$	15.7737	0.633999	0.317000	−	0.948426i	$-0.397325\pi$
$619$	15.7737	0.633999	0.317000	+	0.948426i	$0.397325\pi$
$620$	0	0
$621$	13.1600	0.528094
$622$	0	0
$623$	−26.8044	−1.07389
$624$	0	0
$625$	15.9865	0.639458
$626$	0	0
$627$	−45.3227	−1.81001
$628$	0	0
$629$	19.8991	0.793427
$630$	0	0
$631$	−20.1102	−0.800575	−0.400288	−	0.916390i	$-0.631090\pi$
$631$	−20.1102	−0.800575	−0.400288	+	0.916390i	$0.631090\pi$
$632$	0	0
$633$	−7.23861	−0.287709
$634$	0	0
$635$	9.99085	0.396475
$636$	0	0
$637$	−35.7391	−1.41604
$638$	0	0
$639$	−15.7455	−0.622882
$640$	0	0
$641$	−16.7023	−0.659701	−0.329850	−	0.944033i	$-0.606998\pi$
$641$	−16.7023	−0.659701	−0.329850	+	0.944033i	$0.606998\pi$
$642$	0	0
$643$	−11.8118	−0.465811	−0.232905	−	0.972499i	$-0.574823\pi$
$643$	−11.8118	−0.465811	−0.232905	+	0.972499i	$0.574823\pi$
$644$	0	0
$645$	11.0611	0.435530
$646$	0	0
$647$	5.72612	0.225117	0.112558	−	0.993645i	$-0.464095\pi$
$647$	5.72612	0.225117	0.112558	+	0.993645i	$0.464095\pi$
$648$	0	0
$649$	−15.0057	−0.589026
$650$	0	0
$651$	−11.9092	−0.466757
$652$	0	0
$653$	25.6377	1.00328	0.501640	−	0.865076i	$-0.332730\pi$
$653$	25.6377	1.00328	0.501640	+	0.865076i	$0.332730\pi$
$654$	0	0
$655$	36.6296	1.43124
$656$	0	0
$657$	10.9020	0.425329
$658$	0	0
$659$	19.6150	0.764091	0.382045	−	0.924144i	$-0.375220\pi$
$659$	19.6150	0.764091	0.382045	+	0.924144i	$0.375220\pi$
$660$	0	0
$661$	6.71840	0.261316	0.130658	−	0.991428i	$-0.458291\pi$
$661$	6.71840	0.261316	0.130658	+	0.991428i	$0.458291\pi$
$662$	0	0
$663$	15.2782	0.593355
$664$	0	0
$665$	−65.7479	−2.54959
$666$	0	0
$667$	−24.3898	−0.944378
$668$	0	0
$669$	−28.4548	−1.10013
$670$	0	0
$671$	−57.1155	−2.20492
$672$	0	0
$673$	0.657038	0.0253270	0.0126635	−	0.999920i	$-0.495969\pi$
$673$	0.657038	0.0253270	0.0126635	+	0.999920i	$0.495969\pi$
$674$	0	0
$675$	17.2100	0.662414
$676$	0	0
$677$	7.87250	0.302565	0.151282	−	0.988491i	$-0.451660\pi$
$677$	7.87250	0.302565	0.151282	+	0.988491i	$0.451660\pi$
$678$	0	0
$679$	40.7601	1.56423
$680$	0	0
$681$	−36.8641	−1.41264
$682$	0	0
$683$	−16.5842	−0.634576	−0.317288	−	0.948329i	$-0.602772\pi$
$683$	−16.5842	−0.634576	−0.317288	+	0.948329i	$0.602772\pi$
$684$	0	0
$685$	14.5077	0.554310
$686$	0	0
$687$	51.5107	1.96526
$688$	0	0
$689$	−36.9321	−1.40700
$690$	0	0
$691$	−22.9396	−0.872664	−0.436332	−	0.899786i	$-0.643723\pi$
$691$	−22.9396	−0.872664	−0.436332	+	0.899786i	$0.643723\pi$
$692$	0	0
$693$	−40.8888	−1.55324
$694$	0	0
$695$	−48.7970	−1.85097
$696$	0	0
$697$	−11.7341	−0.444460
$698$	0	0
$699$	8.43859	0.319177
$700$	0	0
$701$	−38.2990	−1.44653	−0.723266	−	0.690569i	$-0.757360\pi$
$701$	−38.2990	−1.44653	−0.723266	+	0.690569i	$0.757360\pi$
$702$	0	0
$703$	49.3404	1.86091
$704$	0	0
$705$	72.0634	2.71406
$706$	0	0
$707$	−22.9998	−0.864998
$708$	0	0
$709$	13.9055	0.522232	0.261116	−	0.965307i	$-0.415909\pi$
$709$	13.9055	0.522232	0.261116	+	0.965307i	$0.415909\pi$
$710$	0	0
$711$	−23.4637	−0.879958
$712$	0	0
$713$	−9.29042	−0.347929
$714$	0	0
$715$	−67.7073	−2.53211
$716$	0	0
$717$	13.6569	0.510028
$718$	0	0
$719$	1.36261	0.0508166	0.0254083	−	0.999677i	$-0.491911\pi$
$719$	1.36261	0.0508166	0.0254083	+	0.999677i	$0.491911\pi$
$720$	0	0
$721$	37.2531	1.38738
$722$	0	0
$723$	−45.4127	−1.68892
$724$	0	0
$725$	−31.8958	−1.18458
$726$	0	0
$727$	−23.4016	−0.867918	−0.433959	−	0.900933i	$-0.642884\pi$
$727$	−23.4016	−0.867918	−0.433959	+	0.900933i	$0.642884\pi$
$728$	0	0
$729$	−11.0742	−0.410154
$730$	0	0
$731$	−2.22111	−0.0821509
$732$	0	0
$733$	−9.99331	−0.369111	−0.184556	−	0.982822i	$-0.559085\pi$
$733$	−9.99331	−0.369111	−0.184556	+	0.982822i	$0.559085\pi$
$734$	0	0
$735$	−79.9257	−2.94811
$736$	0	0
$737$	55.9337	2.06034
$738$	0	0
$739$	−23.1704	−0.852335	−0.426168	−	0.904644i	$-0.640137\pi$
$739$	−23.1704	−0.852335	−0.426168	+	0.904644i	$0.640137\pi$
$740$	0	0
$741$	37.8828	1.39166
$742$	0	0
$743$	−20.2086	−0.741383	−0.370691	−	0.928756i	$-0.620879\pi$
$743$	−20.2086	−0.741383	−0.370691	+	0.928756i	$0.620879\pi$
$744$	0	0
$745$	56.9083	2.08496
$746$	0	0
$747$	12.8028	0.468430
$748$	0	0
$749$	−60.5422	−2.21216
$750$	0	0
$751$	−11.5168	−0.420253	−0.210127	−	0.977674i	$-0.567388\pi$
$751$	−11.5168	−0.420253	−0.210127	+	0.977674i	$0.567388\pi$
$752$	0	0
$753$	12.3933	0.451636
$754$	0	0
$755$	−27.0864	−0.985776
$756$	0	0
$757$	39.7268	1.44390	0.721948	−	0.691947i	$-0.243247\pi$
$757$	39.7268	1.44390	0.721948	+	0.691947i	$0.243247\pi$
$758$	0	0
$759$	−75.5062	−2.74070
$760$	0	0
$761$	−22.6597	−0.821415	−0.410708	−	0.911767i	$-0.634718\pi$
$761$	−22.6597	−0.821415	−0.410708	+	0.911767i	$0.634718\pi$
$762$	0	0
$763$	12.0712	0.437006
$764$	0	0
$765$	14.4341	0.521866
$766$	0	0
$767$	12.5425	0.452882
$768$	0	0
$769$	17.6004	0.634688	0.317344	−	0.948311i	$-0.397209\pi$
$769$	17.6004	0.634688	0.317344	+	0.948311i	$0.397209\pi$
$770$	0	0
$771$	−38.5797	−1.38941
$772$	0	0
$773$	45.8392	1.64872	0.824361	−	0.566064i	$-0.191535\pi$
$773$	45.8392	1.64872	0.824361	+	0.566064i	$0.191535\pi$
$774$	0	0
$775$	−12.1495	−0.436424
$776$	0	0
$777$	105.370	3.78012
$778$	0	0
$779$	−29.0951	−1.04244
$780$	0	0
$781$	−33.1678	−1.18684
$782$	0	0
$783$	−6.24840	−0.223300
$784$	0	0
$785$	−13.9477	−0.497813
$786$	0	0
$787$	17.4603	0.622391	0.311196	−	0.950346i	$-0.399271\pi$
$787$	17.4603	0.622391	0.311196	+	0.950346i	$0.399271\pi$
$788$	0	0
$789$	−32.4329	−1.15464
$790$	0	0
$791$	8.26643	0.293920
$792$	0	0
$793$	47.7397	1.69529
$794$	0	0
$795$	−82.5937	−2.92930
$796$	0	0
$797$	7.37081	0.261087	0.130544	−	0.991443i	$-0.458328\pi$
$797$	7.37081	0.261087	0.130544	+	0.991443i	$0.458328\pi$
$798$	0	0
$799$	−14.4706	−0.511934
$800$	0	0
$801$	14.5910	0.515546
$802$	0	0
$803$	22.9652	0.810423
$804$	0	0
$805$	−109.534	−3.86056
$806$	0	0
$807$	−10.8946	−0.383510
$808$	0	0
$809$	−7.04747	−0.247776	−0.123888	−	0.992296i	$-0.539536\pi$
$809$	−7.04747	−0.247776	−0.123888	+	0.992296i	$0.539536\pi$
$810$	0	0
$811$	52.9606	1.85970	0.929849	−	0.367941i	$-0.119937\pi$
$811$	52.9606	1.85970	0.929849	+	0.367941i	$0.119937\pi$
$812$	0	0
$813$	52.0312	1.82481
$814$	0	0
$815$	24.1420	0.845658
$816$	0	0
$817$	−5.50733	−0.192677
$818$	0	0
$819$	34.1768	1.19423
$820$	0	0
$821$	−32.9925	−1.15145	−0.575724	−	0.817644i	$-0.695280\pi$
$821$	−32.9925	−1.15145	−0.575724	+	0.817644i	$0.695280\pi$
$822$	0	0
$823$	−9.98952	−0.348213	−0.174106	−	0.984727i	$-0.555704\pi$
$823$	−9.98952	−0.348213	−0.174106	+	0.984727i	$0.555704\pi$
$824$	0	0
$825$	−98.7433	−3.43780
$826$	0	0
$827$	−18.2097	−0.633213	−0.316607	−	0.948557i	$-0.602544\pi$
$827$	−18.2097	−0.633213	−0.316607	+	0.948557i	$0.602544\pi$
$828$	0	0
$829$	−9.44705	−0.328110	−0.164055	−	0.986451i	$-0.552457\pi$
$829$	−9.44705	−0.328110	−0.164055	+	0.986451i	$0.552457\pi$
$830$	0	0
$831$	−6.48270	−0.224882
$832$	0	0
$833$	16.0494	0.556080
$834$	0	0
$835$	7.12160	0.246453
$836$	0	0
$837$	−2.38010	−0.0822684
$838$	0	0
$839$	−23.5644	−0.813533	−0.406766	−	0.913532i	$-0.633344\pi$
$839$	−23.5644	−0.813533	−0.406766	+	0.913532i	$0.633344\pi$
$840$	0	0
$841$	−17.4197	−0.600678
$842$	0	0
$843$	19.8419	0.683392
$844$	0	0
$845$	7.30783	0.251397
$846$	0	0
$847$	−41.7897	−1.43591
$848$	0	0
$849$	65.1529	2.23604
$850$	0	0
$851$	82.1996	2.81777
$852$	0	0
$853$	−52.2667	−1.78958	−0.894789	−	0.446490i	$-0.852674\pi$
$853$	−52.2667	−1.78958	−0.894789	+	0.446490i	$0.852674\pi$
$854$	0	0
$855$	35.7899	1.22399
$856$	0	0
$857$	−23.7734	−0.812083	−0.406041	−	0.913855i	$-0.633091\pi$
$857$	−23.7734	−0.812083	−0.406041	+	0.913855i	$0.633091\pi$
$858$	0	0
$859$	−16.6060	−0.566588	−0.283294	−	0.959033i	$-0.591427\pi$
$859$	−16.6060	−0.566588	−0.283294	+	0.959033i	$0.591427\pi$
$860$	0	0
$861$	−62.1346	−2.11754
$862$	0	0
$863$	−30.5826	−1.04104	−0.520521	−	0.853849i	$-0.674262\pi$
$863$	−30.5826	−1.04104	−0.520521	+	0.853849i	$0.674262\pi$
$864$	0	0
$865$	−37.9512	−1.29038
$866$	0	0
$867$	31.8839	1.08283
$868$	0	0
$869$	−49.4263	−1.67667
$870$	0	0
$871$	−46.7520	−1.58413
$872$	0	0
$873$	−22.1878	−0.750943
$874$	0	0
$875$	−66.8295	−2.25925
$876$	0	0
$877$	45.7382	1.54447	0.772235	−	0.635337i	$-0.219139\pi$
$877$	45.7382	1.54447	0.772235	+	0.635337i	$0.219139\pi$
$878$	0	0
$879$	16.2693	0.548750
$880$	0	0
$881$	−9.93887	−0.334849	−0.167425	−	0.985885i	$-0.553545\pi$
$881$	−9.93887	−0.334849	−0.167425	+	0.985885i	$0.553545\pi$
$882$	0	0
$883$	29.0582	0.977886	0.488943	−	0.872316i	$-0.337383\pi$
$883$	29.0582	0.977886	0.488943	+	0.872316i	$0.337383\pi$
$884$	0	0
$885$	28.0495	0.942875
$886$	0	0
$887$	41.7824	1.40291	0.701457	−	0.712712i	$-0.252533\pi$
$887$	41.7824	1.40291	0.701457	+	0.712712i	$0.252533\pi$
$888$	0	0
$889$	−10.6233	−0.356294
$890$	0	0
$891$	−49.7735	−1.66748
$892$	0	0
$893$	−35.8805	−1.20069
$894$	0	0
$895$	34.4767	1.15243
$896$	0	0
$897$	63.1116	2.10723
$898$	0	0
$899$	4.41111	0.147119
$900$	0	0
$901$	16.5852	0.552532
$902$	0	0
$903$	−11.7613	−0.391391
$904$	0	0
$905$	93.6424	3.11278
$906$	0	0
$907$	15.3714	0.510397	0.255199	−	0.966889i	$-0.417859\pi$
$907$	15.3714	0.510397	0.255199	+	0.966889i	$0.417859\pi$
$908$	0	0
$909$	12.5200	0.415261
$910$	0	0
$911$	−10.3340	−0.342382	−0.171191	−	0.985238i	$-0.554762\pi$
$911$	−10.3340	−0.342382	−0.171191	+	0.985238i	$0.554762\pi$
$912$	0	0
$913$	26.9691	0.892547
$914$	0	0
$915$	106.764	3.52949
$916$	0	0
$917$	−38.9483	−1.28619
$918$	0	0
$919$	−45.0421	−1.48580	−0.742901	−	0.669401i	$-0.766551\pi$
$919$	−45.0421	−1.48580	−0.742901	+	0.669401i	$0.766551\pi$
$920$	0	0
$921$	10.8535	0.357634
$922$	0	0
$923$	27.7232	0.912521
$924$	0	0
$925$	107.497	3.53447
$926$	0	0
$927$	−20.2788	−0.666042
$928$	0	0
$929$	−14.7997	−0.485562	−0.242781	−	0.970081i	$-0.578060\pi$
$929$	−14.7997	−0.485562	−0.242781	+	0.970081i	$0.578060\pi$
$930$	0	0
$931$	39.7951	1.30423
$932$	0	0
$933$	−67.6129	−2.21355
$934$	0	0
$935$	30.4054	0.994364
$936$	0	0
$937$	3.42126	0.111768	0.0558839	−	0.998437i	$-0.482202\pi$
$937$	3.42126	0.111768	0.0558839	+	0.998437i	$0.482202\pi$
$938$	0	0
$939$	−7.16939	−0.233964
$940$	0	0
$941$	17.9750	0.585968	0.292984	−	0.956117i	$-0.405352\pi$
$941$	17.9750	0.585968	0.292984	+	0.956117i	$0.405352\pi$
$942$	0	0
$943$	−48.4715	−1.57845
$944$	0	0
$945$	−28.0614	−0.912837
$946$	0	0
$947$	9.49653	0.308596	0.154298	−	0.988024i	$-0.450688\pi$
$947$	9.49653	0.308596	0.154298	+	0.988024i	$0.450688\pi$
$948$	0	0
$949$	−19.1953	−0.623107
$950$	0	0
$951$	11.1028	0.360033
$952$	0	0
$953$	−48.6172	−1.57487	−0.787433	−	0.616401i	$-0.788590\pi$
$953$	−48.6172	−1.57487	−0.787433	+	0.616401i	$0.788590\pi$
$954$	0	0
$955$	50.3621	1.62968
$956$	0	0
$957$	35.8505	1.15888
$958$	0	0
$959$	−15.4261	−0.498134
$960$	0	0
$961$	−29.3197	−0.945798
$962$	0	0
$963$	32.9562	1.06200
$964$	0	0
$965$	101.097	3.25443
$966$	0	0
$967$	−36.9048	−1.18678	−0.593389	−	0.804916i	$-0.702211\pi$
$967$	−36.9048	−1.18678	−0.593389	+	0.804916i	$0.702211\pi$
$968$	0	0
$969$	−17.0121	−0.546507
$970$	0	0
$971$	−39.9682	−1.28264	−0.641321	−	0.767273i	$-0.721613\pi$
$971$	−39.9682	−1.28264	−0.641321	+	0.767273i	$0.721613\pi$
$972$	0	0
$973$	51.8859	1.66339
$974$	0	0
$975$	82.5342	2.64321
$976$	0	0
$977$	54.4292	1.74134	0.870672	−	0.491864i	$-0.163684\pi$
$977$	54.4292	1.74134	0.870672	+	0.491864i	$0.163684\pi$
$978$	0	0
$979$	30.7359	0.982323
$980$	0	0
$981$	−6.57095	−0.209794
$982$	0	0
$983$	47.7647	1.52346	0.761728	−	0.647897i	$-0.224351\pi$
$983$	47.7647	1.52346	0.761728	+	0.647897i	$0.224351\pi$
$984$	0	0
$985$	13.8552	0.441464
$986$	0	0
$987$	−76.6252	−2.43901
$988$	0	0
$989$	−9.17505	−0.291750
$990$	0	0
$991$	−18.5959	−0.590717	−0.295359	−	0.955386i	$-0.595439\pi$
$991$	−18.5959	−0.590717	−0.295359	+	0.955386i	$0.595439\pi$
$992$	0	0
$993$	44.9193	1.42547
$994$	0	0
$995$	−70.6152	−2.23865
$996$	0	0
$997$	3.78354	0.119826	0.0599129	−	0.998204i	$-0.480918\pi$
$997$	3.78354	0.119826	0.0599129	+	0.998204i	$0.480918\pi$
$998$	0	0
$999$	21.0586	0.666266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.7		26
4.3	odd	2		503.2.a.f.1.16	✓	26
12.11	even	2		4527.2.a.o.1.11		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.16	✓	26	4.3	odd	2
4527.2.a.o.1.11		26	12.11	even	2
8048.2.a.u.1.7		26	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27911 q^{3} +3.79116 q^{5} -4.03115 q^{7} +2.19436 q^{9} +O(q^{10})\)	\(q-2.27911 q^{3} +3.79116 q^{5} -4.03115 q^{7} +2.19436 q^{9} +4.62241 q^{11} -3.86363 q^{13} -8.64048 q^{15} +1.73504 q^{17} +4.30211 q^{19} +9.18744 q^{21} +7.16718 q^{23} +9.37288 q^{25} +1.83615 q^{27} -3.40299 q^{29} -1.29624 q^{31} -10.5350 q^{33} -15.2827 q^{35} +11.4689 q^{37} +8.80564 q^{39} -6.76299 q^{41} -1.28015 q^{43} +8.31915 q^{45} -8.34021 q^{47} +9.25015 q^{49} -3.95436 q^{51} +9.55893 q^{53} +17.5243 q^{55} -9.80499 q^{57} -3.24630 q^{59} -12.3562 q^{61} -8.84578 q^{63} -14.6476 q^{65} +12.1005 q^{67} -16.3348 q^{69} -7.17544 q^{71} +4.96822 q^{73} -21.3619 q^{75} -18.6336 q^{77} -10.6927 q^{79} -10.7679 q^{81} +5.83442 q^{83} +6.57783 q^{85} +7.75580 q^{87} +6.64931 q^{89} +15.5748 q^{91} +2.95429 q^{93} +16.3100 q^{95} -10.1113 q^{97} +10.1432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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