LMFDB - Embedded newform 8048.2.a.q.1.3

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.44300$ of defining polynomial
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)

$f(q)$	$=$	$q-2.44300 q^{3} -1.37594 q^{5} -4.82526 q^{7} +2.96827 q^{9} +O(q^{10})$	$q-2.44300 q^{3} -1.37594 q^{5} -4.82526 q^{7} +2.96827 q^{9} -1.27457 q^{11} -6.41156 q^{13} +3.36142 q^{15} -0.544926 q^{17} -0.210204 q^{19} +11.7881 q^{21} -4.60756 q^{23} -3.10680 q^{25} +0.0775098 q^{27} +3.80926 q^{29} +6.00668 q^{31} +3.11378 q^{33} +6.63925 q^{35} +4.44816 q^{37} +15.6635 q^{39} -3.21576 q^{41} +4.07065 q^{43} -4.08415 q^{45} +11.8610 q^{47} +16.2831 q^{49} +1.33126 q^{51} +4.73490 q^{53} +1.75373 q^{55} +0.513530 q^{57} +8.45718 q^{59} -1.34178 q^{61} -14.3227 q^{63} +8.82190 q^{65} -6.98601 q^{67} +11.2563 q^{69} -14.8444 q^{71} -7.74673 q^{73} +7.58993 q^{75} +6.15013 q^{77} -4.07857 q^{79} -9.09418 q^{81} -7.46503 q^{83} +0.749783 q^{85} -9.30604 q^{87} +6.41106 q^{89} +30.9375 q^{91} -14.6743 q^{93} +0.289228 q^{95} -5.59721 q^{97} -3.78327 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * 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
$3$	−2.44300	−1.41047	−0.705235	−	0.708974i	$-0.749158\pi$
$3$	−2.44300	−1.41047	−0.705235	+	0.708974i	$0.749158\pi$
$4$	0	0
$5$	−1.37594	−0.615337	−0.307669	−	0.951494i	$-0.599549\pi$
$5$	−1.37594	−0.615337	−0.307669	+	0.951494i	$0.599549\pi$
$6$	0	0
$7$	−4.82526	−1.82378	−0.911889	−	0.410438i	$-0.865376\pi$
$7$	−4.82526	−1.82378	−0.911889	+	0.410438i	$0.865376\pi$
$8$	0	0
$9$	2.96827	0.989424
$10$	0	0
$11$	−1.27457	−0.384297	−0.192149	−	0.981366i	$-0.561546\pi$
$11$	−1.27457	−0.384297	−0.192149	+	0.981366i	$0.561546\pi$
$12$	0	0
$13$	−6.41156	−1.77825	−0.889124	−	0.457667i	$-0.848685\pi$
$13$	−6.41156	−1.77825	−0.889124	+	0.457667i	$0.848685\pi$
$14$	0	0
$15$	3.36142	0.867914
$16$	0	0
$17$	−0.544926	−0.132164	−0.0660820	−	0.997814i	$-0.521050\pi$
$17$	−0.544926	−0.132164	−0.0660820	+	0.997814i	$0.521050\pi$
$18$	0	0
$19$	−0.210204	−0.0482242	−0.0241121	−	0.999709i	$-0.507676\pi$
$19$	−0.210204	−0.0482242	−0.0241121	+	0.999709i	$0.507676\pi$
$20$	0	0
$21$	11.7881	2.57238
$22$	0	0
$23$	−4.60756	−0.960743	−0.480372	−	0.877065i	$-0.659498\pi$
$23$	−4.60756	−0.960743	−0.480372	+	0.877065i	$0.659498\pi$
$24$	0	0
$25$	−3.10680	−0.621360
$26$	0	0
$27$	0.0775098	0.0149168
$28$	0	0
$29$	3.80926	0.707362	0.353681	−	0.935366i	$-0.384930\pi$
$29$	3.80926	0.707362	0.353681	+	0.935366i	$0.384930\pi$
$30$	0	0
$31$	6.00668	1.07883	0.539416	−	0.842040i	$-0.318645\pi$
$31$	6.00668	1.07883	0.539416	+	0.842040i	$0.318645\pi$
$32$	0	0
$33$	3.11378	0.542039
$34$	0	0
$35$	6.63925	1.12224
$36$	0	0
$37$	4.44816	0.731274	0.365637	−	0.930758i	$-0.380851\pi$
$37$	4.44816	0.731274	0.365637	+	0.930758i	$0.380851\pi$
$38$	0	0
$39$	15.6635	2.50816
$40$	0	0
$41$	−3.21576	−0.502217	−0.251109	−	0.967959i	$-0.580795\pi$
$41$	−3.21576	−0.502217	−0.251109	+	0.967959i	$0.580795\pi$
$42$	0	0
$43$	4.07065	0.620768	0.310384	−	0.950611i	$-0.399542\pi$
$43$	4.07065	0.620768	0.310384	+	0.950611i	$0.399542\pi$
$44$	0	0
$45$	−4.08415	−0.608830
$46$	0	0
$47$	11.8610	1.73011	0.865053	−	0.501681i	$-0.167285\pi$
$47$	11.8610	1.73011	0.865053	+	0.501681i	$0.167285\pi$
$48$	0	0
$49$	16.2831	2.32616
$50$	0	0
$51$	1.33126	0.186413
$52$	0	0
$53$	4.73490	0.650388	0.325194	−	0.945647i	$-0.394570\pi$
$53$	4.73490	0.650388	0.325194	+	0.945647i	$0.394570\pi$
$54$	0	0
$55$	1.75373	0.236472
$56$	0	0
$57$	0.513530	0.0680188
$58$	0	0
$59$	8.45718	1.10103	0.550515	−	0.834825i	$-0.314431\pi$
$59$	8.45718	1.10103	0.550515	+	0.834825i	$0.314431\pi$
$60$	0	0
$61$	−1.34178	−0.171798	−0.0858990	−	0.996304i	$-0.527376\pi$
$61$	−1.34178	−0.171798	−0.0858990	+	0.996304i	$0.527376\pi$
$62$	0	0
$63$	−14.3227	−1.80449
$64$	0	0
$65$	8.82190	1.09422
$66$	0	0
$67$	−6.98601	−0.853477	−0.426738	−	0.904375i	$-0.640337\pi$
$67$	−6.98601	−0.853477	−0.426738	+	0.904375i	$0.640337\pi$
$68$	0	0
$69$	11.2563	1.35510
$70$	0	0
$71$	−14.8444	−1.76171	−0.880854	−	0.473387i	$-0.843031\pi$
$71$	−14.8444	−1.76171	−0.880854	+	0.473387i	$0.843031\pi$
$72$	0	0
$73$	−7.74673	−0.906687	−0.453343	−	0.891336i	$-0.649769\pi$
$73$	−7.74673	−0.906687	−0.453343	+	0.891336i	$0.649769\pi$
$74$	0	0
$75$	7.58993	0.876410
$76$	0	0
$77$	6.15013	0.700872
$78$	0	0
$79$	−4.07857	−0.458875	−0.229437	−	0.973323i	$-0.573689\pi$
$79$	−4.07857	−0.458875	−0.229437	+	0.973323i	$0.573689\pi$
$80$	0	0
$81$	−9.09418	−1.01046
$82$	0	0
$83$	−7.46503	−0.819394	−0.409697	−	0.912222i	$-0.634366\pi$
$83$	−7.46503	−0.819394	−0.409697	+	0.912222i	$0.634366\pi$
$84$	0	0
$85$	0.749783	0.0813254
$86$	0	0
$87$	−9.30604	−0.997712
$88$	0	0
$89$	6.41106	0.679571	0.339786	−	0.940503i	$-0.389646\pi$
$89$	6.41106	0.679571	0.339786	+	0.940503i	$0.389646\pi$
$90$	0	0
$91$	30.9375	3.24313
$92$	0	0
$93$	−14.6743	−1.52166
$94$	0	0
$95$	0.289228	0.0296741
$96$	0	0
$97$	−5.59721	−0.568310	−0.284155	−	0.958778i	$-0.591713\pi$
$97$	−5.59721	−0.568310	−0.284155	+	0.958778i	$0.591713\pi$
$98$	0	0
$99$	−3.78327	−0.380233
$100$	0	0
$101$	−0.413217	−0.0411167	−0.0205583	−	0.999789i	$-0.506544\pi$
$101$	−0.413217	−0.0411167	−0.0205583	+	0.999789i	$0.506544\pi$
$102$	0	0
$103$	9.87371	0.972885	0.486443	−	0.873713i	$-0.338294\pi$
$103$	9.87371	0.972885	0.486443	+	0.873713i	$0.338294\pi$
$104$	0	0
$105$	−16.2197	−1.58288
$106$	0	0
$107$	0.246859	0.0238648	0.0119324	−	0.999929i	$-0.496202\pi$
$107$	0.246859	0.0238648	0.0119324	+	0.999929i	$0.496202\pi$
$108$	0	0
$109$	14.2919	1.36892	0.684459	−	0.729052i	$-0.260039\pi$
$109$	14.2919	1.36892	0.684459	+	0.729052i	$0.260039\pi$
$110$	0	0
$111$	−10.8669	−1.03144
$112$	0	0
$113$	−14.9485	−1.40623	−0.703117	−	0.711075i	$-0.748209\pi$
$113$	−14.9485	−1.40623	−0.703117	+	0.711075i	$0.748209\pi$
$114$	0	0
$115$	6.33971	0.591181
$116$	0	0
$117$	−19.0313	−1.75944
$118$	0	0
$119$	2.62941	0.241038
$120$	0	0
$121$	−9.37547	−0.852316
$122$	0	0
$123$	7.85612	0.708362
$124$	0	0
$125$	11.1544	0.997683
$126$	0	0
$127$	14.8031	1.31356	0.656779	−	0.754083i	$-0.271918\pi$
$127$	14.8031	1.31356	0.656779	+	0.754083i	$0.271918\pi$
$128$	0	0
$129$	−9.94461	−0.875574
$130$	0	0
$131$	−7.52164	−0.657169	−0.328584	−	0.944475i	$-0.606571\pi$
$131$	−7.52164	−0.657169	−0.328584	+	0.944475i	$0.606571\pi$
$132$	0	0
$133$	1.01429	0.0879502
$134$	0	0
$135$	−0.106648	−0.00917884
$136$	0	0
$137$	1.23151	0.105215	0.0526075	−	0.998615i	$-0.483247\pi$
$137$	1.23151	0.105215	0.0526075	+	0.998615i	$0.483247\pi$
$138$	0	0
$139$	11.3065	0.959007	0.479504	−	0.877540i	$-0.340817\pi$
$139$	11.3065	0.959007	0.479504	+	0.877540i	$0.340817\pi$
$140$	0	0
$141$	−28.9765	−2.44026
$142$	0	0
$143$	8.17198	0.683375
$144$	0	0
$145$	−5.24130	−0.435266
$146$	0	0
$147$	−39.7798	−3.28098
$148$	0	0
$149$	16.2468	1.33099	0.665497	−	0.746401i	$-0.268220\pi$
$149$	16.2468	1.33099	0.665497	+	0.746401i	$0.268220\pi$
$150$	0	0
$151$	17.6785	1.43866	0.719330	−	0.694669i	$-0.244449\pi$
$151$	17.6785	1.43866	0.719330	+	0.694669i	$0.244449\pi$
$152$	0	0
$153$	−1.61749	−0.130766
$154$	0	0
$155$	−8.26480	−0.663845
$156$	0	0
$157$	4.82367	0.384971	0.192485	−	0.981300i	$-0.438345\pi$
$157$	4.82367	0.384971	0.192485	+	0.981300i	$0.438345\pi$
$158$	0	0
$159$	−11.5674	−0.917353
$160$	0	0
$161$	22.2327	1.75218
$162$	0	0
$163$	11.9340	0.934743	0.467372	−	0.884061i	$-0.345201\pi$
$163$	11.9340	0.934743	0.467372	+	0.884061i	$0.345201\pi$
$164$	0	0
$165$	−4.28436	−0.333537
$166$	0	0
$167$	−1.66056	−0.128498	−0.0642488	−	0.997934i	$-0.520465\pi$
$167$	−1.66056	−0.128498	−0.0642488	+	0.997934i	$0.520465\pi$
$168$	0	0
$169$	28.1081	2.16216
$170$	0	0
$171$	−0.623944	−0.0477142
$172$	0	0
$173$	−9.66903	−0.735123	−0.367561	−	0.929999i	$-0.619807\pi$
$173$	−9.66903	−0.735123	−0.367561	+	0.929999i	$0.619807\pi$
$174$	0	0
$175$	14.9911	1.13322
$176$	0	0
$177$	−20.6609	−1.55297
$178$	0	0
$179$	5.56787	0.416162	0.208081	−	0.978112i	$-0.433278\pi$
$179$	5.56787	0.416162	0.208081	+	0.978112i	$0.433278\pi$
$180$	0	0
$181$	22.1563	1.64686	0.823432	−	0.567415i	$-0.192057\pi$
$181$	22.1563	1.64686	0.823432	+	0.567415i	$0.192057\pi$
$182$	0	0
$183$	3.27799	0.242316
$184$	0	0
$185$	−6.12039	−0.449980
$186$	0	0
$187$	0.694546	0.0507902
$188$	0	0
$189$	−0.374005	−0.0272049
$190$	0	0
$191$	3.84081	0.277912	0.138956	−	0.990299i	$-0.455625\pi$
$191$	3.84081	0.277912	0.138956	+	0.990299i	$0.455625\pi$
$192$	0	0
$193$	−18.5974	−1.33867	−0.669336	−	0.742960i	$-0.733421\pi$
$193$	−18.5974	−1.33867	−0.669336	+	0.742960i	$0.733421\pi$
$194$	0	0
$195$	−21.5519	−1.54337
$196$	0	0
$197$	7.64326	0.544560	0.272280	−	0.962218i	$-0.412222\pi$
$197$	7.64326	0.544560	0.272280	+	0.962218i	$0.412222\pi$
$198$	0	0
$199$	3.22386	0.228533	0.114267	−	0.993450i	$-0.463548\pi$
$199$	3.22386	0.228533	0.114267	+	0.993450i	$0.463548\pi$
$200$	0	0
$201$	17.0668	1.20380
$202$	0	0
$203$	−18.3807	−1.29007
$204$	0	0
$205$	4.42468	0.309033
$206$	0	0
$207$	−13.6765	−0.950583
$208$	0	0
$209$	0.267920	0.0185324
$210$	0	0
$211$	−21.1620	−1.45685	−0.728425	−	0.685126i	$-0.759747\pi$
$211$	−21.1620	−1.45685	−0.728425	+	0.685126i	$0.759747\pi$
$212$	0	0
$213$	36.2650	2.48484
$214$	0	0
$215$	−5.60095	−0.381982
$216$	0	0
$217$	−28.9838	−1.96755
$218$	0	0
$219$	18.9253	1.27885
$220$	0	0
$221$	3.49383	0.235020
$222$	0	0
$223$	−9.38848	−0.628699	−0.314350	−	0.949307i	$-0.601786\pi$
$223$	−9.38848	−0.628699	−0.314350	+	0.949307i	$0.601786\pi$
$224$	0	0
$225$	−9.22183	−0.614789
$226$	0	0
$227$	−18.4530	−1.22477	−0.612385	−	0.790560i	$-0.709790\pi$
$227$	−18.4530	−1.22477	−0.612385	+	0.790560i	$0.709790\pi$
$228$	0	0
$229$	−22.3285	−1.47551	−0.737754	−	0.675069i	$-0.764114\pi$
$229$	−22.3285	−1.47551	−0.737754	+	0.675069i	$0.764114\pi$
$230$	0	0
$231$	−15.0248	−0.988559
$232$	0	0
$233$	27.7302	1.81667	0.908333	−	0.418247i	$-0.137355\pi$
$233$	27.7302	1.81667	0.908333	+	0.418247i	$0.137355\pi$
$234$	0	0
$235$	−16.3200	−1.06460
$236$	0	0
$237$	9.96396	0.647229
$238$	0	0
$239$	−1.73094	−0.111965	−0.0559827	−	0.998432i	$-0.517829\pi$
$239$	−1.73094	−0.111965	−0.0559827	+	0.998432i	$0.517829\pi$
$240$	0	0
$241$	−26.0705	−1.67935	−0.839674	−	0.543091i	$-0.817254\pi$
$241$	−26.0705	−1.67935	−0.839674	+	0.543091i	$0.817254\pi$
$242$	0	0
$243$	21.9846	1.41031
$244$	0	0
$245$	−22.4046	−1.43137
$246$	0	0
$247$	1.34774	0.0857545
$248$	0	0
$249$	18.2371	1.15573
$250$	0	0
$251$	−9.76387	−0.616290	−0.308145	−	0.951339i	$-0.599708\pi$
$251$	−9.76387	−0.616290	−0.308145	+	0.951339i	$0.599708\pi$
$252$	0	0
$253$	5.87266	0.369211
$254$	0	0
$255$	−1.83172	−0.114707
$256$	0	0
$257$	−18.4736	−1.15235	−0.576176	−	0.817325i	$-0.695456\pi$
$257$	−18.4736	−1.15235	−0.576176	+	0.817325i	$0.695456\pi$
$258$	0	0
$259$	−21.4636	−1.33368
$260$	0	0
$261$	11.3069	0.699881
$262$	0	0
$263$	−21.4229	−1.32100	−0.660498	−	0.750828i	$-0.729655\pi$
$263$	−21.4229	−1.32100	−0.660498	+	0.750828i	$0.729655\pi$
$264$	0	0
$265$	−6.51491	−0.400208
$266$	0	0
$267$	−15.6623	−0.958514
$268$	0	0
$269$	17.5935	1.07270	0.536348	−	0.843997i	$-0.319803\pi$
$269$	17.5935	1.07270	0.536348	+	0.843997i	$0.319803\pi$
$270$	0	0
$271$	32.3302	1.96392	0.981959	−	0.189092i	$-0.0605546\pi$
$271$	32.3302	1.96392	0.981959	+	0.189092i	$0.0605546\pi$
$272$	0	0
$273$	−75.5804	−4.57433
$274$	0	0
$275$	3.95983	0.238787
$276$	0	0
$277$	5.54885	0.333398	0.166699	−	0.986008i	$-0.446689\pi$
$277$	5.54885	0.333398	0.166699	+	0.986008i	$0.446689\pi$
$278$	0	0
$279$	17.8295	1.06742
$280$	0	0
$281$	−17.5343	−1.04601	−0.523005	−	0.852330i	$-0.675189\pi$
$281$	−17.5343	−1.04601	−0.523005	+	0.852330i	$0.675189\pi$
$282$	0	0
$283$	−8.79786	−0.522979	−0.261489	−	0.965206i	$-0.584214\pi$
$283$	−8.79786	−0.522979	−0.261489	+	0.965206i	$0.584214\pi$
$284$	0	0
$285$	−0.706585	−0.0418545
$286$	0	0
$287$	15.5169	0.915933
$288$	0	0
$289$	−16.7031	−0.982533
$290$	0	0
$291$	13.6740	0.801584
$292$	0	0
$293$	0.622890	0.0363896	0.0181948	−	0.999834i	$-0.494208\pi$
$293$	0.622890	0.0363896	0.0181948	+	0.999834i	$0.494208\pi$
$294$	0	0
$295$	−11.6365	−0.677505
$296$	0	0
$297$	−0.0987916	−0.00573247
$298$	0	0
$299$	29.5417	1.70844
$300$	0	0
$301$	−19.6419	−1.13214
$302$	0	0
$303$	1.00949	0.0579938
$304$	0	0
$305$	1.84621	0.105714
$306$	0	0
$307$	8.40517	0.479708	0.239854	−	0.970809i	$-0.422900\pi$
$307$	8.40517	0.479708	0.239854	+	0.970809i	$0.422900\pi$
$308$	0	0
$309$	−24.1215	−1.37222
$310$	0	0
$311$	5.88571	0.333748	0.166874	−	0.985978i	$-0.446633\pi$
$311$	5.88571	0.333748	0.166874	+	0.985978i	$0.446633\pi$
$312$	0	0
$313$	13.4203	0.758560	0.379280	−	0.925282i	$-0.376172\pi$
$313$	13.4203	0.758560	0.379280	+	0.925282i	$0.376172\pi$
$314$	0	0
$315$	19.7071	1.11037
$316$	0	0
$317$	−10.2950	−0.578222	−0.289111	−	0.957296i	$-0.593360\pi$
$317$	−10.2950	−0.578222	−0.289111	+	0.957296i	$0.593360\pi$
$318$	0	0
$319$	−4.85517	−0.271837
$320$	0	0
$321$	−0.603079	−0.0336606
$322$	0	0
$323$	0.114546	0.00637350
$324$	0	0
$325$	19.9194	1.10493
$326$	0	0
$327$	−34.9152	−1.93082
$328$	0	0
$329$	−57.2325	−3.15533
$330$	0	0
$331$	−18.5387	−1.01898	−0.509490	−	0.860477i	$-0.670166\pi$
$331$	−18.5387	−1.01898	−0.509490	+	0.860477i	$0.670166\pi$
$332$	0	0
$333$	13.2034	0.723540
$334$	0	0
$335$	9.61230	0.525176
$336$	0	0
$337$	−10.2426	−0.557951	−0.278975	−	0.960298i	$-0.589995\pi$
$337$	−10.2426	−0.557951	−0.278975	+	0.960298i	$0.589995\pi$
$338$	0	0
$339$	36.5192	1.98345
$340$	0	0
$341$	−7.65593	−0.414592
$342$	0	0
$343$	−44.7936	−2.41863
$344$	0	0
$345$	−15.4879	−0.833843
$346$	0	0
$347$	10.5984	0.568949	0.284475	−	0.958684i	$-0.408181\pi$
$347$	10.5984	0.568949	0.284475	+	0.958684i	$0.408181\pi$
$348$	0	0
$349$	35.0601	1.87672	0.938361	−	0.345656i	$-0.112344\pi$
$349$	35.0601	1.87672	0.938361	+	0.345656i	$0.112344\pi$
$350$	0	0
$351$	−0.496959	−0.0265257
$352$	0	0
$353$	36.2302	1.92834	0.964169	−	0.265288i	$-0.0854672\pi$
$353$	36.2302	1.92834	0.964169	+	0.265288i	$0.0854672\pi$
$354$	0	0
$355$	20.4250	1.08405
$356$	0	0
$357$	−6.42366	−0.339976
$358$	0	0
$359$	−16.8750	−0.890626	−0.445313	−	0.895375i	$-0.646908\pi$
$359$	−16.8750	−0.890626	−0.445313	+	0.895375i	$0.646908\pi$
$360$	0	0
$361$	−18.9558	−0.997674
$362$	0	0
$363$	22.9043	1.20217
$364$	0	0
$365$	10.6590	0.557918
$366$	0	0
$367$	−12.7359	−0.664810	−0.332405	−	0.943137i	$-0.607860\pi$
$367$	−12.7359	−0.664810	−0.332405	+	0.943137i	$0.607860\pi$
$368$	0	0
$369$	−9.54525	−0.496906
$370$	0	0
$371$	−22.8471	−1.18616
$372$	0	0
$373$	27.7052	1.43452	0.717260	−	0.696806i	$-0.245396\pi$
$373$	27.7052	1.43452	0.717260	+	0.696806i	$0.245396\pi$
$374$	0	0
$375$	−27.2503	−1.40720
$376$	0	0
$377$	−24.4233	−1.25786
$378$	0	0
$379$	−0.782502	−0.0401944	−0.0200972	−	0.999798i	$-0.506398\pi$
$379$	−0.782502	−0.0401944	−0.0200972	+	0.999798i	$0.506398\pi$
$380$	0	0
$381$	−36.1639	−1.85273
$382$	0	0
$383$	−8.28256	−0.423219	−0.211610	−	0.977354i	$-0.567871\pi$
$383$	−8.28256	−0.423219	−0.211610	+	0.977354i	$0.567871\pi$
$384$	0	0
$385$	−8.46218	−0.431273
$386$	0	0
$387$	12.0828	0.614203
$388$	0	0
$389$	18.5901	0.942553	0.471277	−	0.881985i	$-0.343793\pi$
$389$	18.5901	0.942553	0.471277	+	0.881985i	$0.343793\pi$
$390$	0	0
$391$	2.51078	0.126976
$392$	0	0
$393$	18.3754	0.926916
$394$	0	0
$395$	5.61184	0.282363
$396$	0	0
$397$	20.8081	1.04433	0.522165	−	0.852844i	$-0.325124\pi$
$397$	20.8081	1.04433	0.522165	+	0.852844i	$0.325124\pi$
$398$	0	0
$399$	−2.47792	−0.124051
$400$	0	0
$401$	11.7505	0.586790	0.293395	−	0.955991i	$-0.405215\pi$
$401$	11.7505	0.586790	0.293395	+	0.955991i	$0.405215\pi$
$402$	0	0
$403$	−38.5122	−1.91843
$404$	0	0
$405$	12.5130	0.621776
$406$	0	0
$407$	−5.66949	−0.281026
$408$	0	0
$409$	28.5036	1.40941	0.704706	−	0.709500i	$-0.251079\pi$
$409$	28.5036	1.40941	0.704706	+	0.709500i	$0.251079\pi$
$410$	0	0
$411$	−3.00859	−0.148403
$412$	0	0
$413$	−40.8081	−2.00803
$414$	0	0
$415$	10.2714	0.504204
$416$	0	0
$417$	−27.6219	−1.35265
$418$	0	0
$419$	−1.17290	−0.0573000	−0.0286500	−	0.999590i	$-0.509121\pi$
$419$	−1.17290	−0.0573000	−0.0286500	+	0.999590i	$0.509121\pi$
$420$	0	0
$421$	−5.06318	−0.246764	−0.123382	−	0.992359i	$-0.539374\pi$
$421$	−5.06318	−0.246764	−0.123382	+	0.992359i	$0.539374\pi$
$422$	0	0
$423$	35.2067	1.71181
$424$	0	0
$425$	1.69298	0.0821214
$426$	0	0
$427$	6.47446	0.313321
$428$	0	0
$429$	−19.9642	−0.963880
$430$	0	0
$431$	20.2125	0.973603	0.486801	−	0.873513i	$-0.338164\pi$
$431$	20.2125	0.973603	0.486801	+	0.873513i	$0.338164\pi$
$432$	0	0
$433$	15.2425	0.732507	0.366253	−	0.930515i	$-0.380640\pi$
$433$	15.2425	0.732507	0.366253	+	0.930515i	$0.380640\pi$
$434$	0	0
$435$	12.8045	0.613930
$436$	0	0
$437$	0.968530	0.0463311
$438$	0	0
$439$	−21.4430	−1.02342	−0.511708	−	0.859159i	$-0.670987\pi$
$439$	−21.4430	−1.02342	−0.511708	+	0.859159i	$0.670987\pi$
$440$	0	0
$441$	48.3328	2.30156
$442$	0	0
$443$	−22.6558	−1.07641	−0.538205	−	0.842814i	$-0.680898\pi$
$443$	−22.6558	−1.07641	−0.538205	+	0.842814i	$0.680898\pi$
$444$	0	0
$445$	−8.82121	−0.418165
$446$	0	0
$447$	−39.6911	−1.87733
$448$	0	0
$449$	−38.1246	−1.79921	−0.899605	−	0.436705i	$-0.856145\pi$
$449$	−38.1246	−1.79921	−0.899605	+	0.436705i	$0.856145\pi$
$450$	0	0
$451$	4.09871	0.193001
$452$	0	0
$453$	−43.1888	−2.02918
$454$	0	0
$455$	−42.5680	−1.99562
$456$	0	0
$457$	14.7600	0.690446	0.345223	−	0.938521i	$-0.387803\pi$
$457$	14.7600	0.690446	0.345223	+	0.938521i	$0.387803\pi$
$458$	0	0
$459$	−0.0422371	−0.00197146
$460$	0	0
$461$	40.2986	1.87689	0.938446	−	0.345425i	$-0.112265\pi$
$461$	40.2986	1.87689	0.938446	+	0.345425i	$0.112265\pi$
$462$	0	0
$463$	−12.0099	−0.558148	−0.279074	−	0.960270i	$-0.590028\pi$
$463$	−12.0099	−0.558148	−0.279074	+	0.960270i	$0.590028\pi$
$464$	0	0
$465$	20.1910	0.936333
$466$	0	0
$467$	−26.6048	−1.23113	−0.615563	−	0.788088i	$-0.711071\pi$
$467$	−26.6048	−1.23113	−0.615563	+	0.788088i	$0.711071\pi$
$468$	0	0
$469$	33.7093	1.55655
$470$	0	0
$471$	−11.7843	−0.542990
$472$	0	0
$473$	−5.18832	−0.238559
$474$	0	0
$475$	0.653063	0.0299646
$476$	0	0
$477$	14.0545	0.643510
$478$	0	0
$479$	−10.6675	−0.487412	−0.243706	−	0.969849i	$-0.578363\pi$
$479$	−10.6675	−0.487412	−0.243706	+	0.969849i	$0.578363\pi$
$480$	0	0
$481$	−28.5197	−1.30039
$482$	0	0
$483$	−54.3146	−2.47140
$484$	0	0
$485$	7.70140	0.349702
$486$	0	0
$487$	−27.9475	−1.26642	−0.633211	−	0.773980i	$-0.718263\pi$
$487$	−27.9475	−1.26642	−0.633211	+	0.773980i	$0.718263\pi$
$488$	0	0
$489$	−29.1548	−1.31843
$490$	0	0
$491$	23.5693	1.06367	0.531834	−	0.846849i	$-0.321503\pi$
$491$	23.5693	1.06367	0.531834	+	0.846849i	$0.321503\pi$
$492$	0	0
$493$	−2.07577	−0.0934878
$494$	0	0
$495$	5.20553	0.233971
$496$	0	0
$497$	71.6282	3.21296
$498$	0	0
$499$	31.2230	1.39773	0.698867	−	0.715251i	$-0.253688\pi$
$499$	31.2230	1.39773	0.698867	+	0.715251i	$0.253688\pi$
$500$	0	0
$501$	4.05675	0.181242
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	0.568561	0.0253006
$506$	0	0
$507$	−68.6683	−3.04966
$508$	0	0
$509$	−41.7608	−1.85102	−0.925508	−	0.378729i	$-0.876361\pi$
$509$	−41.7608	−1.85102	−0.925508	+	0.378729i	$0.876361\pi$
$510$	0	0
$511$	37.3800	1.65359
$512$	0	0
$513$	−0.0162929	−0.000719349 0
$514$	0	0
$515$	−13.5856	−0.598652
$516$	0	0
$517$	−15.1177	−0.664875
$518$	0	0
$519$	23.6215	1.03687
$520$	0	0
$521$	39.4587	1.72872	0.864358	−	0.502876i	$-0.167725\pi$
$521$	39.4587	1.72872	0.864358	+	0.502876i	$0.167725\pi$
$522$	0	0
$523$	−20.3259	−0.888788	−0.444394	−	0.895831i	$-0.646581\pi$
$523$	−20.3259	−0.888788	−0.444394	+	0.895831i	$0.646581\pi$
$524$	0	0
$525$	−36.6234	−1.59838
$526$	0	0
$527$	−3.27320	−0.142583
$528$	0	0
$529$	−1.77035	−0.0769719
$530$	0	0
$531$	25.1032	1.08939
$532$	0	0
$533$	20.6180	0.893067
$534$	0	0
$535$	−0.339663	−0.0146849
$536$	0	0
$537$	−13.6023	−0.586984
$538$	0	0
$539$	−20.7540	−0.893938
$540$	0	0
$541$	20.2289	0.869710	0.434855	−	0.900500i	$-0.356800\pi$
$541$	20.2289	0.869710	0.434855	+	0.900500i	$0.356800\pi$
$542$	0	0
$543$	−54.1279	−2.32285
$544$	0	0
$545$	−19.6648	−0.842346
$546$	0	0
$547$	37.6805	1.61110	0.805551	−	0.592526i	$-0.201869\pi$
$547$	37.6805	1.61110	0.805551	+	0.592526i	$0.201869\pi$
$548$	0	0
$549$	−3.98278	−0.169981
$550$	0	0
$551$	−0.800723	−0.0341120
$552$	0	0
$553$	19.6801	0.836885
$554$	0	0
$555$	14.9521	0.634683
$556$	0	0
$557$	−30.1596	−1.27790	−0.638952	−	0.769247i	$-0.720632\pi$
$557$	−30.1596	−1.27790	−0.638952	+	0.769247i	$0.720632\pi$
$558$	0	0
$559$	−26.0992	−1.10388
$560$	0	0
$561$	−1.69678	−0.0716381
$562$	0	0
$563$	−12.6606	−0.533580	−0.266790	−	0.963755i	$-0.585963\pi$
$563$	−12.6606	−0.533580	−0.266790	+	0.963755i	$0.585963\pi$
$564$	0	0
$565$	20.5681	0.865308
$566$	0	0
$567$	43.8818	1.84286
$568$	0	0
$569$	−3.93752	−0.165069	−0.0825347	−	0.996588i	$-0.526302\pi$
$569$	−3.93752	−0.165069	−0.0825347	+	0.996588i	$0.526302\pi$
$570$	0	0
$571$	29.1371	1.21935	0.609675	−	0.792651i	$-0.291300\pi$
$571$	29.1371	1.21935	0.609675	+	0.792651i	$0.291300\pi$
$572$	0	0
$573$	−9.38313	−0.391986
$574$	0	0
$575$	14.3148	0.596968
$576$	0	0
$577$	−39.2948	−1.63586	−0.817931	−	0.575316i	$-0.804879\pi$
$577$	−39.2948	−1.63586	−0.817931	+	0.575316i	$0.804879\pi$
$578$	0	0
$579$	45.4336	1.88816
$580$	0	0
$581$	36.0207	1.49439
$582$	0	0
$583$	−6.03495	−0.249942
$584$	0	0
$585$	26.1858	1.08265
$586$	0	0
$587$	−0.793453	−0.0327493	−0.0163747	−	0.999866i	$-0.505212\pi$
$587$	−0.793453	−0.0327493	−0.0163747	+	0.999866i	$0.505212\pi$
$588$	0	0
$589$	−1.26263	−0.0520258
$590$	0	0
$591$	−18.6725	−0.768085
$592$	0	0
$593$	18.2475	0.749336	0.374668	−	0.927159i	$-0.377757\pi$
$593$	18.2475	0.749336	0.374668	+	0.927159i	$0.377757\pi$
$594$	0	0
$595$	−3.61790	−0.148319
$596$	0	0
$597$	−7.87590	−0.322339
$598$	0	0
$599$	−13.0913	−0.534896	−0.267448	−	0.963572i	$-0.586180\pi$
$599$	−13.0913	−0.534896	−0.267448	+	0.963572i	$0.586180\pi$
$600$	0	0
$601$	−25.5225	−1.04108	−0.520541	−	0.853836i	$-0.674270\pi$
$601$	−25.5225	−1.04108	−0.520541	+	0.853836i	$0.674270\pi$
$602$	0	0
$603$	−20.7364	−0.844450
$604$	0	0
$605$	12.9001	0.524462
$606$	0	0
$607$	11.1307	0.451782	0.225891	−	0.974153i	$-0.427471\pi$
$607$	11.1307	0.451782	0.225891	+	0.974153i	$0.427471\pi$
$608$	0	0
$609$	44.9041	1.81961
$610$	0	0
$611$	−76.0476	−3.07656
$612$	0	0
$613$	6.56388	0.265113	0.132556	−	0.991175i	$-0.457681\pi$
$613$	6.56388	0.265113	0.132556	+	0.991175i	$0.457681\pi$
$614$	0	0
$615$	−10.8095	−0.435882
$616$	0	0
$617$	−44.7334	−1.80090	−0.900449	−	0.434961i	$-0.856762\pi$
$617$	−44.7334	−1.80090	−0.900449	+	0.434961i	$0.856762\pi$
$618$	0	0
$619$	−20.3953	−0.819758	−0.409879	−	0.912140i	$-0.634429\pi$
$619$	−20.3953	−0.819758	−0.409879	+	0.912140i	$0.634429\pi$
$620$	0	0
$621$	−0.357131	−0.0143312
$622$	0	0
$623$	−30.9350	−1.23939
$624$	0	0
$625$	0.186213	0.00744852
$626$	0	0
$627$	−0.654530	−0.0261394
$628$	0	0
$629$	−2.42392	−0.0966480
$630$	0	0
$631$	−40.8652	−1.62682	−0.813408	−	0.581693i	$-0.802391\pi$
$631$	−40.8652	−1.62682	−0.813408	+	0.581693i	$0.802391\pi$
$632$	0	0
$633$	51.6988	2.05484
$634$	0	0
$635$	−20.3681	−0.808282
$636$	0	0
$637$	−104.400	−4.13649
$638$	0	0
$639$	−44.0623	−1.74308
$640$	0	0
$641$	45.5783	1.80024	0.900118	−	0.435647i	$-0.143480\pi$
$641$	45.5783	1.80024	0.900118	+	0.435647i	$0.143480\pi$
$642$	0	0
$643$	1.52226	0.0600322	0.0300161	−	0.999549i	$-0.490444\pi$
$643$	1.52226	0.0600322	0.0300161	+	0.999549i	$0.490444\pi$
$644$	0	0
$645$	13.6831	0.538773
$646$	0	0
$647$	7.38127	0.290188	0.145094	−	0.989418i	$-0.453652\pi$
$647$	7.38127	0.290188	0.145094	+	0.989418i	$0.453652\pi$
$648$	0	0
$649$	−10.7793	−0.423123
$650$	0	0
$651$	70.8075	2.77517
$652$	0	0
$653$	−25.9500	−1.01550	−0.507752	−	0.861503i	$-0.669523\pi$
$653$	−25.9500	−1.01550	−0.507752	+	0.861503i	$0.669523\pi$
$654$	0	0
$655$	10.3493	0.404380
$656$	0	0
$657$	−22.9944	−0.897098
$658$	0	0
$659$	4.91637	0.191514	0.0957572	−	0.995405i	$-0.469473\pi$
$659$	4.91637	0.191514	0.0957572	+	0.995405i	$0.469473\pi$
$660$	0	0
$661$	10.4635	0.406982	0.203491	−	0.979077i	$-0.434771\pi$
$661$	10.4635	0.406982	0.203491	+	0.979077i	$0.434771\pi$
$662$	0	0
$663$	−8.53543	−0.331489
$664$	0	0
$665$	−1.39560	−0.0541190
$666$	0	0
$667$	−17.5514	−0.679593
$668$	0	0
$669$	22.9361	0.886761
$670$	0	0
$671$	1.71020	0.0660214
$672$	0	0
$673$	−13.8496	−0.533863	−0.266931	−	0.963716i	$-0.586010\pi$
$673$	−13.8496	−0.533863	−0.266931	+	0.963716i	$0.586010\pi$
$674$	0	0
$675$	−0.240807	−0.00926868
$676$	0	0
$677$	−23.7309	−0.912054	−0.456027	−	0.889966i	$-0.650728\pi$
$677$	−23.7309	−0.912054	−0.456027	+	0.889966i	$0.650728\pi$
$678$	0	0
$679$	27.0080	1.03647
$680$	0	0
$681$	45.0808	1.72750
$682$	0	0
$683$	−3.34525	−0.128002	−0.0640012	−	0.997950i	$-0.520386\pi$
$683$	−3.34525	−0.128002	−0.0640012	+	0.997950i	$0.520386\pi$
$684$	0	0
$685$	−1.69448	−0.0647428
$686$	0	0
$687$	54.5486	2.08116
$688$	0	0
$689$	−30.3581	−1.15655
$690$	0	0
$691$	−32.4097	−1.23292	−0.616461	−	0.787385i	$-0.711434\pi$
$691$	−32.4097	−1.23292	−0.616461	+	0.787385i	$0.711434\pi$
$692$	0	0
$693$	18.2553	0.693460
$694$	0	0
$695$	−15.5571	−0.590113
$696$	0	0
$697$	1.75235	0.0663751
$698$	0	0
$699$	−67.7451	−2.56235
$700$	0	0
$701$	−14.6881	−0.554762	−0.277381	−	0.960760i	$-0.589467\pi$
$701$	−14.6881	−0.554762	−0.277381	+	0.960760i	$0.589467\pi$
$702$	0	0
$703$	−0.935024	−0.0352651
$704$	0	0
$705$	39.8698	1.50158
$706$	0	0
$707$	1.99388	0.0749876
$708$	0	0
$709$	−11.5782	−0.434828	−0.217414	−	0.976080i	$-0.569762\pi$
$709$	−11.5782	−0.434828	−0.217414	+	0.976080i	$0.569762\pi$
$710$	0	0
$711$	−12.1063	−0.454022
$712$	0	0
$713$	−27.6762	−1.03648
$714$	0	0
$715$	−11.2441	−0.420506
$716$	0	0
$717$	4.22870	0.157924
$718$	0	0
$719$	−19.8610	−0.740692	−0.370346	−	0.928894i	$-0.620761\pi$
$719$	−19.8610	−0.740692	−0.370346	+	0.928894i	$0.620761\pi$
$720$	0	0
$721$	−47.6432	−1.77433
$722$	0	0
$723$	63.6903	2.36867
$724$	0	0
$725$	−11.8346	−0.439526
$726$	0	0
$727$	19.0565	0.706767	0.353384	−	0.935479i	$-0.385031\pi$
$727$	19.0565	0.706767	0.353384	+	0.935479i	$0.385031\pi$
$728$	0	0
$729$	−26.4259	−0.978738
$730$	0	0
$731$	−2.21820	−0.0820432
$732$	0	0
$733$	−12.2295	−0.451707	−0.225854	−	0.974161i	$-0.572517\pi$
$733$	−12.2295	−0.451707	−0.225854	+	0.974161i	$0.572517\pi$
$734$	0	0
$735$	54.7345	2.01891
$736$	0	0
$737$	8.90415	0.327989
$738$	0	0
$739$	25.8392	0.950510	0.475255	−	0.879848i	$-0.342356\pi$
$739$	25.8392	0.950510	0.475255	+	0.879848i	$0.342356\pi$
$740$	0	0
$741$	−3.29253	−0.120954
$742$	0	0
$743$	10.7963	0.396079	0.198040	−	0.980194i	$-0.436542\pi$
$743$	10.7963	0.396079	0.198040	+	0.980194i	$0.436542\pi$
$744$	0	0
$745$	−22.3546	−0.819010
$746$	0	0
$747$	−22.1583	−0.810728
$748$	0	0
$749$	−1.19116	−0.0435241
$750$	0	0
$751$	36.1711	1.31990	0.659952	−	0.751308i	$-0.270577\pi$
$751$	36.1711	1.31990	0.659952	+	0.751308i	$0.270577\pi$
$752$	0	0
$753$	23.8532	0.869258
$754$	0	0
$755$	−24.3245	−0.885261
$756$	0	0
$757$	−52.7401	−1.91687	−0.958436	−	0.285308i	$-0.907904\pi$
$757$	−52.7401	−1.91687	−0.958436	+	0.285308i	$0.907904\pi$
$758$	0	0
$759$	−14.3469	−0.520761
$760$	0	0
$761$	32.6400	1.18320	0.591599	−	0.806233i	$-0.298497\pi$
$761$	32.6400	1.18320	0.591599	+	0.806233i	$0.298497\pi$
$762$	0	0
$763$	−68.9622	−2.49660
$764$	0	0
$765$	2.22556	0.0804653
$766$	0	0
$767$	−54.2237	−1.95790
$768$	0	0
$769$	47.7808	1.72302	0.861510	−	0.507740i	$-0.169519\pi$
$769$	47.7808	1.72302	0.861510	+	0.507740i	$0.169519\pi$
$770$	0	0
$771$	45.1311	1.62536
$772$	0	0
$773$	34.4620	1.23951	0.619756	−	0.784795i	$-0.287232\pi$
$773$	34.4620	1.23951	0.619756	+	0.784795i	$0.287232\pi$
$774$	0	0
$775$	−18.6616	−0.670343
$776$	0	0
$777$	52.4356	1.88112
$778$	0	0
$779$	0.675967	0.0242190
$780$	0	0
$781$	18.9202	0.677019
$782$	0	0
$783$	0.295255	0.0105516
$784$	0	0
$785$	−6.63706	−0.236887
$786$	0	0
$787$	29.8493	1.06401	0.532006	−	0.846741i	$-0.321438\pi$
$787$	29.8493	1.06401	0.532006	+	0.846741i	$0.321438\pi$
$788$	0	0
$789$	52.3364	1.86322
$790$	0	0
$791$	72.1302	2.56466
$792$	0	0
$793$	8.60293	0.305499
$794$	0	0
$795$	15.9160	0.564481
$796$	0	0
$797$	43.0957	1.52653	0.763263	−	0.646088i	$-0.223596\pi$
$797$	43.0957	1.52653	0.763263	+	0.646088i	$0.223596\pi$
$798$	0	0
$799$	−6.46337	−0.228658
$800$	0	0
$801$	19.0298	0.672384
$802$	0	0
$803$	9.87375	0.348437
$804$	0	0
$805$	−30.5908	−1.07818
$806$	0	0
$807$	−42.9811	−1.51301
$808$	0	0
$809$	−10.5898	−0.372319	−0.186160	−	0.982520i	$-0.559604\pi$
$809$	−10.5898	−0.372319	−0.186160	+	0.982520i	$0.559604\pi$
$810$	0	0
$811$	−12.4021	−0.435496	−0.217748	−	0.976005i	$-0.569871\pi$
$811$	−12.4021	−0.435496	−0.217748	+	0.976005i	$0.569871\pi$
$812$	0	0
$813$	−78.9828	−2.77005
$814$	0	0
$815$	−16.4204	−0.575182
$816$	0	0
$817$	−0.855668	−0.0299360
$818$	0	0
$819$	91.8308	3.20883
$820$	0	0
$821$	−5.96957	−0.208339	−0.104170	−	0.994560i	$-0.533219\pi$
$821$	−5.96957	−0.208339	−0.104170	+	0.994560i	$0.533219\pi$
$822$	0	0
$823$	−32.1013	−1.11898	−0.559491	−	0.828837i	$-0.689003\pi$
$823$	−32.1013	−1.11898	−0.559491	+	0.828837i	$0.689003\pi$
$824$	0	0
$825$	−9.67389	−0.336802
$826$	0	0
$827$	29.9333	1.04088	0.520441	−	0.853898i	$-0.325768\pi$
$827$	29.9333	1.04088	0.520441	+	0.853898i	$0.325768\pi$
$828$	0	0
$829$	−7.38716	−0.256567	−0.128283	−	0.991738i	$-0.540947\pi$
$829$	−7.38716	−0.256567	−0.128283	+	0.991738i	$0.540947\pi$
$830$	0	0
$831$	−13.5559	−0.470248
$832$	0	0
$833$	−8.87311	−0.307435
$834$	0	0
$835$	2.28482	0.0790694
$836$	0	0
$837$	0.465576	0.0160927
$838$	0	0
$839$	−34.9100	−1.20523	−0.602614	−	0.798033i	$-0.705874\pi$
$839$	−34.9100	−1.20523	−0.602614	+	0.798033i	$0.705874\pi$
$840$	0	0
$841$	−14.4895	−0.499639
$842$	0	0
$843$	42.8364	1.47537
$844$	0	0
$845$	−38.6750	−1.33046
$846$	0	0
$847$	45.2391	1.55443
$848$	0	0
$849$	21.4932	0.737645
$850$	0	0
$851$	−20.4952	−0.702567
$852$	0	0
$853$	−2.19636	−0.0752019	−0.0376010	−	0.999293i	$-0.511972\pi$
$853$	−2.19636	−0.0752019	−0.0376010	+	0.999293i	$0.511972\pi$
$854$	0	0
$855$	0.858507	0.0293603
$856$	0	0
$857$	49.4888	1.69050	0.845252	−	0.534368i	$-0.179450\pi$
$857$	49.4888	1.69050	0.845252	+	0.534368i	$0.179450\pi$
$858$	0	0
$859$	42.4309	1.44773	0.723863	−	0.689944i	$-0.242365\pi$
$859$	42.4309	1.44773	0.723863	+	0.689944i	$0.242365\pi$
$860$	0	0
$861$	−37.9078	−1.29190
$862$	0	0
$863$	−19.4484	−0.662032	−0.331016	−	0.943625i	$-0.607391\pi$
$863$	−19.4484	−0.662032	−0.331016	+	0.943625i	$0.607391\pi$
$864$	0	0
$865$	13.3040	0.452348
$866$	0	0
$867$	40.8056	1.38583
$868$	0	0
$869$	5.19841	0.176344
$870$	0	0
$871$	44.7912	1.51769
$872$	0	0
$873$	−16.6140	−0.562300
$874$	0	0
$875$	−53.8231	−1.81955
$876$	0	0
$877$	36.1955	1.22223	0.611117	−	0.791540i	$-0.290720\pi$
$877$	36.1955	1.22223	0.611117	+	0.791540i	$0.290720\pi$
$878$	0	0
$879$	−1.52172	−0.0513264
$880$	0	0
$881$	−15.5404	−0.523570	−0.261785	−	0.965126i	$-0.584311\pi$
$881$	−15.5404	−0.523570	−0.261785	+	0.965126i	$0.584311\pi$
$882$	0	0
$883$	−20.1170	−0.676992	−0.338496	−	0.940968i	$-0.609918\pi$
$883$	−20.1170	−0.676992	−0.338496	+	0.940968i	$0.609918\pi$
$884$	0	0
$885$	28.4281	0.955600
$886$	0	0
$887$	−18.1902	−0.610766	−0.305383	−	0.952230i	$-0.598785\pi$
$887$	−18.1902	−0.610766	−0.305383	+	0.952230i	$0.598785\pi$
$888$	0	0
$889$	−71.4286	−2.39564
$890$	0	0
$891$	11.5912	0.388318
$892$	0	0
$893$	−2.49324	−0.0834330
$894$	0	0
$895$	−7.66103	−0.256080
$896$	0	0
$897$	−72.1705	−2.40970
$898$	0	0
$899$	22.8810	0.763124
$900$	0	0
$901$	−2.58017	−0.0859579
$902$	0	0
$903$	47.9854	1.59685
$904$	0	0
$905$	−30.4856	−1.01338
$906$	0	0
$907$	−36.9801	−1.22790	−0.613952	−	0.789344i	$-0.710421\pi$
$907$	−36.9801	−1.22790	−0.613952	+	0.789344i	$0.710421\pi$
$908$	0	0
$909$	−1.22654	−0.0406818
$910$	0	0
$911$	59.1242	1.95887	0.979436	−	0.201755i	$-0.0646645\pi$
$911$	59.1242	1.95887	0.979436	+	0.201755i	$0.0646645\pi$
$912$	0	0
$913$	9.51470	0.314891
$914$	0	0
$915$	−4.51030	−0.149106
$916$	0	0
$917$	36.2939	1.19853
$918$	0	0
$919$	1.01028	0.0333261	0.0166631	−	0.999861i	$-0.494696\pi$
$919$	1.01028	0.0333261	0.0166631	+	0.999861i	$0.494696\pi$
$920$	0	0
$921$	−20.5339	−0.676614
$922$	0	0
$923$	95.1759	3.13275
$924$	0	0
$925$	−13.8196	−0.454384
$926$	0	0
$927$	29.3079	0.962596
$928$	0	0
$929$	35.7252	1.17211	0.586053	−	0.810273i	$-0.300681\pi$
$929$	35.7252	1.17211	0.586053	+	0.810273i	$0.300681\pi$
$930$	0	0
$931$	−3.42279	−0.112177
$932$	0	0
$933$	−14.3788	−0.470741
$934$	0	0
$935$	−0.955651	−0.0312531
$936$	0	0
$937$	19.2920	0.630243	0.315122	−	0.949051i	$-0.397955\pi$
$937$	19.2920	0.630243	0.315122	+	0.949051i	$0.397955\pi$
$938$	0	0
$939$	−32.7859	−1.06993
$940$	0	0
$941$	7.61881	0.248366	0.124183	−	0.992259i	$-0.460369\pi$
$941$	7.61881	0.248366	0.124183	+	0.992259i	$0.460369\pi$
$942$	0	0
$943$	14.8168	0.482502
$944$	0	0
$945$	0.514607	0.0167402
$946$	0	0
$947$	58.8962	1.91387	0.956934	−	0.290304i	$-0.0937565\pi$
$947$	58.8962	1.91387	0.956934	+	0.290304i	$0.0937565\pi$
$948$	0	0
$949$	49.6687	1.61231
$950$	0	0
$951$	25.1506	0.815565
$952$	0	0
$953$	41.8881	1.35689	0.678444	−	0.734652i	$-0.262655\pi$
$953$	41.8881	1.35689	0.678444	+	0.734652i	$0.262655\pi$
$954$	0	0
$955$	−5.28471	−0.171009
$956$	0	0
$957$	11.8612	0.383418
$958$	0	0
$959$	−5.94236	−0.191889
$960$	0	0
$961$	5.08018	0.163877
$962$	0	0
$963$	0.732746	0.0236124
$964$	0	0
$965$	25.5889	0.823734
$966$	0	0
$967$	−18.1860	−0.584824	−0.292412	−	0.956292i	$-0.594458\pi$
$967$	−18.1860	−0.584824	−0.292412	+	0.956292i	$0.594458\pi$
$968$	0	0
$969$	−0.279836	−0.00898963
$970$	0	0
$971$	39.3654	1.26330	0.631648	−	0.775256i	$-0.282379\pi$
$971$	39.3654	1.26330	0.631648	+	0.775256i	$0.282379\pi$
$972$	0	0
$973$	−54.5569	−1.74902
$974$	0	0
$975$	−48.6633	−1.55847
$976$	0	0
$977$	−27.0324	−0.864844	−0.432422	−	0.901671i	$-0.642341\pi$
$977$	−27.0324	−0.864844	−0.432422	+	0.901671i	$0.642341\pi$
$978$	0	0
$979$	−8.17134	−0.261157
$980$	0	0
$981$	42.4223	1.35444
$982$	0	0
$983$	46.9893	1.49873	0.749363	−	0.662160i	$-0.230360\pi$
$983$	46.9893	1.49873	0.749363	+	0.662160i	$0.230360\pi$
$984$	0	0
$985$	−10.5166	−0.335088
$986$	0	0
$987$	139.819	4.45049
$988$	0	0
$989$	−18.7558	−0.596399
$990$	0	0
$991$	5.63218	0.178912	0.0894561	−	0.995991i	$-0.471487\pi$
$991$	5.63218	0.178912	0.0894561	+	0.995991i	$0.471487\pi$
$992$	0	0
$993$	45.2901	1.43724
$994$	0	0
$995$	−4.43582	−0.140625
$996$	0	0
$997$	−13.1041	−0.415011	−0.207506	−	0.978234i	$-0.566535\pi$
$997$	−13.1041	−0.415011	−0.207506	+	0.978234i	$0.566535\pi$
$998$	0	0
$999$	0.344776	0.0109082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.q.1.3		12
4.3	odd	2		1006.2.a.j.1.10	✓	12
12.11	even	2		9054.2.a.bi.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.10	✓	12	4.3	odd	2
8048.2.a.q.1.3		12	1.1	even	1	trivial
9054.2.a.bi.1.9		12	12.11	even	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44300 q^{3} -1.37594 q^{5} -4.82526 q^{7} +2.96827 q^{9} +O(q^{10})\)	\(q-2.44300 q^{3} -1.37594 q^{5} -4.82526 q^{7} +2.96827 q^{9} -1.27457 q^{11} -6.41156 q^{13} +3.36142 q^{15} -0.544926 q^{17} -0.210204 q^{19} +11.7881 q^{21} -4.60756 q^{23} -3.10680 q^{25} +0.0775098 q^{27} +3.80926 q^{29} +6.00668 q^{31} +3.11378 q^{33} +6.63925 q^{35} +4.44816 q^{37} +15.6635 q^{39} -3.21576 q^{41} +4.07065 q^{43} -4.08415 q^{45} +11.8610 q^{47} +16.2831 q^{49} +1.33126 q^{51} +4.73490 q^{53} +1.75373 q^{55} +0.513530 q^{57} +8.45718 q^{59} -1.34178 q^{61} -14.3227 q^{63} +8.82190 q^{65} -6.98601 q^{67} +11.2563 q^{69} -14.8444 q^{71} -7.74673 q^{73} +7.58993 q^{75} +6.15013 q^{77} -4.07857 q^{79} -9.09418 q^{81} -7.46503 q^{83} +0.749783 q^{85} -9.30604 q^{87} +6.41106 q^{89} +30.9375 q^{91} -14.6743 q^{93} +0.289228 q^{95} -5.59721 q^{97} -3.78327 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

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