LMFDB - Embedded newform 8048.2.a.v.1.9

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:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.59308 q^{3} +2.75433 q^{5} -1.46360 q^{7} -0.462108 q^{9} +O(q^{10})$	$q-1.59308 q^{3} +2.75433 q^{5} -1.46360 q^{7} -0.462108 q^{9} +5.86989 q^{11} -6.69840 q^{13} -4.38785 q^{15} -6.07288 q^{17} +1.48446 q^{19} +2.33163 q^{21} -4.40969 q^{23} +2.58632 q^{25} +5.51540 q^{27} +8.17517 q^{29} +8.84800 q^{31} -9.35119 q^{33} -4.03123 q^{35} -7.89315 q^{37} +10.6711 q^{39} +5.35916 q^{41} +8.36082 q^{43} -1.27280 q^{45} -9.07330 q^{47} -4.85788 q^{49} +9.67456 q^{51} +12.3433 q^{53} +16.1676 q^{55} -2.36485 q^{57} +2.57315 q^{59} +3.03940 q^{61} +0.676342 q^{63} -18.4496 q^{65} -1.68264 q^{67} +7.02497 q^{69} -4.21882 q^{71} -12.5331 q^{73} -4.12021 q^{75} -8.59118 q^{77} -14.2617 q^{79} -7.40013 q^{81} +17.2574 q^{83} -16.7267 q^{85} -13.0237 q^{87} -8.75945 q^{89} +9.80378 q^{91} -14.0955 q^{93} +4.08868 q^{95} +2.54144 q^{97} -2.71253 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	−1.59308	−0.919763	−0.459881	−	0.887980i	$-0.652108\pi$
$3$	−1.59308	−0.919763	−0.459881	+	0.887980i	$0.652108\pi$
$4$	0	0
$5$	2.75433	1.23177	0.615886	−	0.787835i	$-0.288798\pi$
$5$	2.75433	1.23177	0.615886	+	0.787835i	$0.288798\pi$
$6$	0	0
$7$	−1.46360	−0.553189	−0.276594	−	0.960987i	$-0.589206\pi$
$7$	−1.46360	−0.553189	−0.276594	+	0.960987i	$0.589206\pi$
$8$	0	0
$9$	−0.462108	−0.154036
$10$	0	0
$11$	5.86989	1.76984	0.884920	−	0.465743i	$-0.154213\pi$
$11$	5.86989	1.76984	0.884920	+	0.465743i	$0.154213\pi$
$12$	0	0
$13$	−6.69840	−1.85780	−0.928901	−	0.370328i	$-0.879245\pi$
$13$	−6.69840	−1.85780	−0.928901	+	0.370328i	$0.879245\pi$
$14$	0	0
$15$	−4.38785	−1.13294
$16$	0	0
$17$	−6.07288	−1.47289	−0.736445	−	0.676497i	$-0.763497\pi$
$17$	−6.07288	−1.47289	−0.736445	+	0.676497i	$0.763497\pi$
$18$	0	0
$19$	1.48446	0.340558	0.170279	−	0.985396i	$-0.445533\pi$
$19$	1.48446	0.340558	0.170279	+	0.985396i	$0.445533\pi$
$20$	0	0
$21$	2.33163	0.508802
$22$	0	0
$23$	−4.40969	−0.919483	−0.459742	−	0.888053i	$-0.652058\pi$
$23$	−4.40969	−0.919483	−0.459742	+	0.888053i	$0.652058\pi$
$24$	0	0
$25$	2.58632	0.517264
$26$	0	0
$27$	5.51540	1.06144
$28$	0	0
$29$	8.17517	1.51809	0.759045	−	0.651038i	$-0.225666\pi$
$29$	8.17517	1.51809	0.759045	+	0.651038i	$0.225666\pi$
$30$	0	0
$31$	8.84800	1.58915	0.794574	−	0.607168i	$-0.207694\pi$
$31$	8.84800	1.58915	0.794574	+	0.607168i	$0.207694\pi$
$32$	0	0
$33$	−9.35119	−1.62783
$34$	0	0
$35$	−4.03123	−0.681403
$36$	0	0
$37$	−7.89315	−1.29763	−0.648813	−	0.760948i	$-0.724734\pi$
$37$	−7.89315	−1.29763	−0.648813	+	0.760948i	$0.724734\pi$
$38$	0	0
$39$	10.6711	1.70874
$40$	0	0
$41$	5.35916	0.836961	0.418480	−	0.908226i	$-0.362563\pi$
$41$	5.35916	0.836961	0.418480	+	0.908226i	$0.362563\pi$
$42$	0	0
$43$	8.36082	1.27501	0.637507	−	0.770445i	$-0.279966\pi$
$43$	8.36082	1.27501	0.637507	+	0.770445i	$0.279966\pi$
$44$	0	0
$45$	−1.27280	−0.189738
$46$	0	0
$47$	−9.07330	−1.32348	−0.661739	−	0.749735i	$-0.730181\pi$
$47$	−9.07330	−1.32348	−0.661739	+	0.749735i	$0.730181\pi$
$48$	0	0
$49$	−4.85788	−0.693982
$50$	0	0
$51$	9.67456	1.35471
$52$	0	0
$53$	12.3433	1.69549	0.847743	−	0.530406i	$-0.177961\pi$
$53$	12.3433	1.69549	0.847743	+	0.530406i	$0.177961\pi$
$54$	0	0
$55$	16.1676	2.18004
$56$	0	0
$57$	−2.36485	−0.313232
$58$	0	0
$59$	2.57315	0.334996	0.167498	−	0.985872i	$-0.446431\pi$
$59$	2.57315	0.334996	0.167498	+	0.985872i	$0.446431\pi$
$60$	0	0
$61$	3.03940	0.389156	0.194578	−	0.980887i	$-0.437666\pi$
$61$	3.03940	0.389156	0.194578	+	0.980887i	$0.437666\pi$
$62$	0	0
$63$	0.676342	0.0852110
$64$	0	0
$65$	−18.4496	−2.28839
$66$	0	0
$67$	−1.68264	−0.205568	−0.102784	−	0.994704i	$-0.532775\pi$
$67$	−1.68264	−0.205568	−0.102784	+	0.994704i	$0.532775\pi$
$68$	0	0
$69$	7.02497	0.845707
$70$	0	0
$71$	−4.21882	−0.500681	−0.250341	−	0.968158i	$-0.580543\pi$
$71$	−4.21882	−0.500681	−0.250341	+	0.968158i	$0.580543\pi$
$72$	0	0
$73$	−12.5331	−1.46689	−0.733445	−	0.679749i	$-0.762089\pi$
$73$	−12.5331	−1.46689	−0.733445	+	0.679749i	$0.762089\pi$
$74$	0	0
$75$	−4.12021	−0.475761
$76$	0	0
$77$	−8.59118	−0.979055
$78$	0	0
$79$	−14.2617	−1.60457	−0.802284	−	0.596943i	$-0.796382\pi$
$79$	−14.2617	−1.60457	−0.802284	+	0.596943i	$0.796382\pi$
$80$	0	0
$81$	−7.40013	−0.822237
$82$	0	0
$83$	17.2574	1.89425	0.947124	−	0.320868i	$-0.103975\pi$
$83$	17.2574	1.89425	0.947124	+	0.320868i	$0.103975\pi$
$84$	0	0
$85$	−16.7267	−1.81427
$86$	0	0
$87$	−13.0237	−1.39628
$88$	0	0
$89$	−8.75945	−0.928500	−0.464250	−	0.885704i	$-0.653676\pi$
$89$	−8.75945	−0.928500	−0.464250	+	0.885704i	$0.653676\pi$
$90$	0	0
$91$	9.80378	1.02772
$92$	0	0
$93$	−14.0955	−1.46164
$94$	0	0
$95$	4.08868	0.419490
$96$	0	0
$97$	2.54144	0.258044	0.129022	−	0.991642i	$-0.458816\pi$
$97$	2.54144	0.258044	0.129022	+	0.991642i	$0.458816\pi$
$98$	0	0
$99$	−2.71253	−0.272619
$100$	0	0
$101$	−11.5172	−1.14600	−0.573001	−	0.819554i	$-0.694221\pi$
$101$	−11.5172	−1.14600	−0.573001	+	0.819554i	$0.694221\pi$
$102$	0	0
$103$	−1.61935	−0.159559	−0.0797797	−	0.996813i	$-0.525422\pi$
$103$	−1.61935	−0.159559	−0.0797797	+	0.996813i	$0.525422\pi$
$104$	0	0
$105$	6.42206	0.626729
$106$	0	0
$107$	−4.13438	−0.399685	−0.199843	−	0.979828i	$-0.564043\pi$
$107$	−4.13438	−0.399685	−0.199843	+	0.979828i	$0.564043\pi$
$108$	0	0
$109$	−18.4290	−1.76518	−0.882588	−	0.470147i	$-0.844201\pi$
$109$	−18.4290	−1.76518	−0.882588	+	0.470147i	$0.844201\pi$
$110$	0	0
$111$	12.5744	1.19351
$112$	0	0
$113$	6.13298	0.576943	0.288471	−	0.957489i	$-0.406853\pi$
$113$	6.13298	0.576943	0.288471	+	0.957489i	$0.406853\pi$
$114$	0	0
$115$	−12.1457	−1.13259
$116$	0	0
$117$	3.09539	0.286169
$118$	0	0
$119$	8.88827	0.814786
$120$	0	0
$121$	23.4557	2.13233
$122$	0	0
$123$	−8.53756	−0.769806
$124$	0	0
$125$	−6.64806	−0.594621
$126$	0	0
$127$	−17.1025	−1.51760	−0.758802	−	0.651322i	$-0.774215\pi$
$127$	−17.1025	−1.51760	−0.758802	+	0.651322i	$0.774215\pi$
$128$	0	0
$129$	−13.3194	−1.17271
$130$	0	0
$131$	−12.3670	−1.08051	−0.540256	−	0.841500i	$-0.681673\pi$
$131$	−12.3670	−1.08051	−0.540256	+	0.841500i	$0.681673\pi$
$132$	0	0
$133$	−2.17265	−0.188393
$134$	0	0
$135$	15.1912	1.30745
$136$	0	0
$137$	1.35637	0.115882	0.0579411	−	0.998320i	$-0.481546\pi$
$137$	1.35637	0.115882	0.0579411	+	0.998320i	$0.481546\pi$
$138$	0	0
$139$	9.42003	0.798996	0.399498	−	0.916734i	$-0.369184\pi$
$139$	9.42003	0.798996	0.399498	+	0.916734i	$0.369184\pi$
$140$	0	0
$141$	14.4545	1.21729
$142$	0	0
$143$	−39.3189	−3.28801
$144$	0	0
$145$	22.5171	1.86994
$146$	0	0
$147$	7.73897	0.638299
$148$	0	0
$149$	−14.0620	−1.15201	−0.576003	−	0.817448i	$-0.695388\pi$
$149$	−14.0620	−1.15201	−0.576003	+	0.817448i	$0.695388\pi$
$150$	0	0
$151$	9.72194	0.791161	0.395580	−	0.918431i	$-0.370543\pi$
$151$	9.72194	0.791161	0.395580	+	0.918431i	$0.370543\pi$
$152$	0	0
$153$	2.80633	0.226878
$154$	0	0
$155$	24.3703	1.95747
$156$	0	0
$157$	20.3959	1.62777	0.813886	−	0.581025i	$-0.197348\pi$
$157$	20.3959	1.62777	0.813886	+	0.581025i	$0.197348\pi$
$158$	0	0
$159$	−19.6639	−1.55945
$160$	0	0
$161$	6.45402	0.508648
$162$	0	0
$163$	−6.02566	−0.471966	−0.235983	−	0.971757i	$-0.575831\pi$
$163$	−6.02566	−0.471966	−0.235983	+	0.971757i	$0.575831\pi$
$164$	0	0
$165$	−25.7562	−2.00512
$166$	0	0
$167$	4.04988	0.313389	0.156695	−	0.987647i	$-0.449916\pi$
$167$	4.04988	0.313389	0.156695	+	0.987647i	$0.449916\pi$
$168$	0	0
$169$	31.8686	2.45143
$170$	0	0
$171$	−0.685979	−0.0524582
$172$	0	0
$173$	9.38104	0.713227	0.356614	−	0.934252i	$-0.383931\pi$
$173$	9.38104	0.713227	0.356614	+	0.934252i	$0.383931\pi$
$174$	0	0
$175$	−3.78534	−0.286145
$176$	0	0
$177$	−4.09923	−0.308117
$178$	0	0
$179$	−9.00076	−0.672748	−0.336374	−	0.941728i	$-0.609201\pi$
$179$	−9.00076	−0.672748	−0.336374	+	0.941728i	$0.609201\pi$
$180$	0	0
$181$	5.30418	0.394257	0.197128	−	0.980378i	$-0.436838\pi$
$181$	5.30418	0.394257	0.197128	+	0.980378i	$0.436838\pi$
$182$	0	0
$183$	−4.84200	−0.357931
$184$	0	0
$185$	−21.7403	−1.59838
$186$	0	0
$187$	−35.6472	−2.60678
$188$	0	0
$189$	−8.07234	−0.587176
$190$	0	0
$191$	−14.5843	−1.05529	−0.527643	−	0.849466i	$-0.676924\pi$
$191$	−14.5843	−1.05529	−0.527643	+	0.849466i	$0.676924\pi$
$192$	0	0
$193$	−1.26930	−0.0913665	−0.0456833	−	0.998956i	$-0.514546\pi$
$193$	−1.26930	−0.0913665	−0.0456833	+	0.998956i	$0.514546\pi$
$194$	0	0
$195$	29.3916	2.10478
$196$	0	0
$197$	−13.7100	−0.976795	−0.488397	−	0.872621i	$-0.662418\pi$
$197$	−13.7100	−0.976795	−0.488397	+	0.872621i	$0.662418\pi$
$198$	0	0
$199$	−18.2227	−1.29177	−0.645887	−	0.763433i	$-0.723512\pi$
$199$	−18.2227	−1.29177	−0.645887	+	0.763433i	$0.723512\pi$
$200$	0	0
$201$	2.68058	0.189074
$202$	0	0
$203$	−11.9652	−0.839790
$204$	0	0
$205$	14.7609	1.03095
$206$	0	0
$207$	2.03775	0.141634
$208$	0	0
$209$	8.71360	0.602732
$210$	0	0
$211$	10.1444	0.698373	0.349186	−	0.937053i	$-0.386458\pi$
$211$	10.1444	0.698373	0.349186	+	0.937053i	$0.386458\pi$
$212$	0	0
$213$	6.72090	0.460508
$214$	0	0
$215$	23.0284	1.57053
$216$	0	0
$217$	−12.9499	−0.879098
$218$	0	0
$219$	19.9662	1.34919
$220$	0	0
$221$	40.6786	2.73634
$222$	0	0
$223$	−9.35840	−0.626685	−0.313342	−	0.949640i	$-0.601449\pi$
$223$	−9.35840	−0.626685	−0.313342	+	0.949640i	$0.601449\pi$
$224$	0	0
$225$	−1.19516	−0.0796774
$226$	0	0
$227$	−13.4970	−0.895825	−0.447913	−	0.894077i	$-0.647832\pi$
$227$	−13.4970	−0.895825	−0.447913	+	0.894077i	$0.647832\pi$
$228$	0	0
$229$	3.85673	0.254860	0.127430	−	0.991848i	$-0.459327\pi$
$229$	3.85673	0.254860	0.127430	+	0.991848i	$0.459327\pi$
$230$	0	0
$231$	13.6864	0.900499
$232$	0	0
$233$	−26.6606	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$233$	−26.6606	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$234$	0	0
$235$	−24.9908	−1.63022
$236$	0	0
$237$	22.7200	1.47582
$238$	0	0
$239$	10.6255	0.687304	0.343652	−	0.939097i	$-0.388336\pi$
$239$	10.6255	0.687304	0.343652	+	0.939097i	$0.388336\pi$
$240$	0	0
$241$	−21.2402	−1.36820	−0.684102	−	0.729387i	$-0.739806\pi$
$241$	−21.2402	−1.36820	−0.684102	+	0.729387i	$0.739806\pi$
$242$	0	0
$243$	−4.75724	−0.305177
$244$	0	0
$245$	−13.3802	−0.854828
$246$	0	0
$247$	−9.94348	−0.632688
$248$	0	0
$249$	−27.4924	−1.74226
$250$	0	0
$251$	19.6652	1.24126	0.620629	−	0.784105i	$-0.286878\pi$
$251$	19.6652	1.24126	0.620629	+	0.784105i	$0.286878\pi$
$252$	0	0
$253$	−25.8844	−1.62734
$254$	0	0
$255$	26.6469	1.66869
$256$	0	0
$257$	−23.6914	−1.47783	−0.738916	−	0.673798i	$-0.764662\pi$
$257$	−23.6914	−1.47783	−0.738916	+	0.673798i	$0.764662\pi$
$258$	0	0
$259$	11.5524	0.717832
$260$	0	0
$261$	−3.77781	−0.233841
$262$	0	0
$263$	1.08534	0.0669247	0.0334624	−	0.999440i	$-0.489347\pi$
$263$	1.08534	0.0669247	0.0334624	+	0.999440i	$0.489347\pi$
$264$	0	0
$265$	33.9976	2.08845
$266$	0	0
$267$	13.9545	0.854000
$268$	0	0
$269$	8.92937	0.544433	0.272217	−	0.962236i	$-0.412243\pi$
$269$	8.92937	0.544433	0.272217	+	0.962236i	$0.412243\pi$
$270$	0	0
$271$	−8.03612	−0.488160	−0.244080	−	0.969755i	$-0.578486\pi$
$271$	−8.03612	−0.488160	−0.244080	+	0.969755i	$0.578486\pi$
$272$	0	0
$273$	−15.6182	−0.945254
$274$	0	0
$275$	15.1814	0.915475
$276$	0	0
$277$	10.7093	0.643458	0.321729	−	0.946832i	$-0.395736\pi$
$277$	10.7093	0.643458	0.321729	+	0.946832i	$0.395736\pi$
$278$	0	0
$279$	−4.08873	−0.244786
$280$	0	0
$281$	−10.6054	−0.632663	−0.316332	−	0.948649i	$-0.602451\pi$
$281$	−10.6054	−0.632663	−0.316332	+	0.948649i	$0.602451\pi$
$282$	0	0
$283$	−2.63639	−0.156717	−0.0783586	−	0.996925i	$-0.524968\pi$
$283$	−2.63639	−0.156717	−0.0783586	+	0.996925i	$0.524968\pi$
$284$	0	0
$285$	−6.51358	−0.385831
$286$	0	0
$287$	−7.84367	−0.462997
$288$	0	0
$289$	19.8799	1.16941
$290$	0	0
$291$	−4.04870	−0.237339
$292$	0	0
$293$	7.32611	0.427996	0.213998	−	0.976834i	$-0.431351\pi$
$293$	7.32611	0.427996	0.213998	+	0.976834i	$0.431351\pi$
$294$	0	0
$295$	7.08731	0.412639
$296$	0	0
$297$	32.3748	1.87858
$298$	0	0
$299$	29.5379	1.70822
$300$	0	0
$301$	−12.2369	−0.705323
$302$	0	0
$303$	18.3477	1.05405
$304$	0	0
$305$	8.37152	0.479352
$306$	0	0
$307$	13.2865	0.758301	0.379151	−	0.925335i	$-0.376216\pi$
$307$	13.2865	0.758301	0.379151	+	0.925335i	$0.376216\pi$
$308$	0	0
$309$	2.57975	0.146757
$310$	0	0
$311$	−22.6292	−1.28318	−0.641591	−	0.767047i	$-0.721725\pi$
$311$	−22.6292	−1.28318	−0.641591	+	0.767047i	$0.721725\pi$
$312$	0	0
$313$	2.01120	0.113680	0.0568399	−	0.998383i	$-0.481898\pi$
$313$	2.01120	0.113680	0.0568399	+	0.998383i	$0.481898\pi$
$314$	0	0
$315$	1.86287	0.104961
$316$	0	0
$317$	−30.4277	−1.70899	−0.854495	−	0.519460i	$-0.826133\pi$
$317$	−30.4277	−1.70899	−0.854495	+	0.519460i	$0.826133\pi$
$318$	0	0
$319$	47.9874	2.68678
$320$	0	0
$321$	6.58638	0.367616
$322$	0	0
$323$	−9.01493	−0.501604
$324$	0	0
$325$	−17.3242	−0.960975
$326$	0	0
$327$	29.3588	1.62354
$328$	0	0
$329$	13.2797	0.732133
$330$	0	0
$331$	−0.921540	−0.0506524	−0.0253262	−	0.999679i	$-0.508062\pi$
$331$	−0.921540	−0.0506524	−0.0253262	+	0.999679i	$0.508062\pi$
$332$	0	0
$333$	3.64749	0.199881
$334$	0	0
$335$	−4.63455	−0.253213
$336$	0	0
$337$	2.03118	0.110646	0.0553228	−	0.998469i	$-0.482381\pi$
$337$	2.03118	0.110646	0.0553228	+	0.998469i	$0.482381\pi$
$338$	0	0
$339$	−9.77031	−0.530650
$340$	0	0
$341$	51.9368	2.81254
$342$	0	0
$343$	17.3552	0.937092
$344$	0	0
$345$	19.3491	1.04172
$346$	0	0
$347$	−11.1452	−0.598305	−0.299152	−	0.954205i	$-0.596704\pi$
$347$	−11.1452	−0.598305	−0.299152	+	0.954205i	$0.596704\pi$
$348$	0	0
$349$	29.5827	1.58352	0.791762	−	0.610829i	$-0.209164\pi$
$349$	29.5827	1.58352	0.791762	+	0.610829i	$0.209164\pi$
$350$	0	0
$351$	−36.9444	−1.97194
$352$	0	0
$353$	−0.527017	−0.0280503	−0.0140251	−	0.999902i	$-0.504464\pi$
$353$	−0.527017	−0.0280503	−0.0140251	+	0.999902i	$0.504464\pi$
$354$	0	0
$355$	−11.6200	−0.616726
$356$	0	0
$357$	−14.1597	−0.749410
$358$	0	0
$359$	0.171064	0.00902843	0.00451422	−	0.999990i	$-0.498563\pi$
$359$	0.171064	0.00902843	0.00451422	+	0.999990i	$0.498563\pi$
$360$	0	0
$361$	−16.7964	−0.884021
$362$	0	0
$363$	−37.3667	−1.96124
$364$	0	0
$365$	−34.5203	−1.80687
$366$	0	0
$367$	−3.35445	−0.175101	−0.0875506	−	0.996160i	$-0.527904\pi$
$367$	−3.35445	−0.175101	−0.0875506	+	0.996160i	$0.527904\pi$
$368$	0	0
$369$	−2.47651	−0.128922
$370$	0	0
$371$	−18.0657	−0.937924
$372$	0	0
$373$	−23.2102	−1.20178	−0.600890	−	0.799332i	$-0.705187\pi$
$373$	−23.2102	−1.20178	−0.600890	+	0.799332i	$0.705187\pi$
$374$	0	0
$375$	10.5909	0.546910
$376$	0	0
$377$	−54.7605	−2.82031
$378$	0	0
$379$	25.0589	1.28719	0.643594	−	0.765367i	$-0.277442\pi$
$379$	25.0589	1.28719	0.643594	+	0.765367i	$0.277442\pi$
$380$	0	0
$381$	27.2456	1.39584
$382$	0	0
$383$	13.5652	0.693148	0.346574	−	0.938023i	$-0.387345\pi$
$383$	13.5652	0.693148	0.346574	+	0.938023i	$0.387345\pi$
$384$	0	0
$385$	−23.6629	−1.20597
$386$	0	0
$387$	−3.86361	−0.196398
$388$	0	0
$389$	−24.9073	−1.26285	−0.631424	−	0.775438i	$-0.717529\pi$
$389$	−24.9073	−1.26285	−0.631424	+	0.775438i	$0.717529\pi$
$390$	0	0
$391$	26.7795	1.35430
$392$	0	0
$393$	19.7016	0.993816
$394$	0	0
$395$	−39.2814	−1.97646
$396$	0	0
$397$	−23.8957	−1.19929	−0.599647	−	0.800265i	$-0.704692\pi$
$397$	−23.8957	−1.19929	−0.599647	+	0.800265i	$0.704692\pi$
$398$	0	0
$399$	3.46120	0.173277
$400$	0	0
$401$	−26.2689	−1.31181	−0.655903	−	0.754845i	$-0.727712\pi$
$401$	−26.2689	−1.31181	−0.655903	+	0.754845i	$0.727712\pi$
$402$	0	0
$403$	−59.2674	−2.95232
$404$	0	0
$405$	−20.3824	−1.01281
$406$	0	0
$407$	−46.3320	−2.29659
$408$	0	0
$409$	15.9584	0.789093	0.394546	−	0.918876i	$-0.370902\pi$
$409$	15.9584	0.789093	0.394546	+	0.918876i	$0.370902\pi$
$410$	0	0
$411$	−2.16079	−0.106584
$412$	0	0
$413$	−3.76607	−0.185316
$414$	0	0
$415$	47.5326	2.33328
$416$	0	0
$417$	−15.0068	−0.734887
$418$	0	0
$419$	5.96030	0.291180	0.145590	−	0.989345i	$-0.453492\pi$
$419$	5.96030	0.291180	0.145590	+	0.989345i	$0.453492\pi$
$420$	0	0
$421$	−21.1846	−1.03247	−0.516236	−	0.856446i	$-0.672667\pi$
$421$	−21.1846	−1.03247	−0.516236	+	0.856446i	$0.672667\pi$
$422$	0	0
$423$	4.19285	0.203863
$424$	0	0
$425$	−15.7064	−0.761874
$426$	0	0
$427$	−4.44847	−0.215277
$428$	0	0
$429$	62.6380	3.02419
$430$	0	0
$431$	−2.38874	−0.115062	−0.0575309	−	0.998344i	$-0.518323\pi$
$431$	−2.38874	−0.115062	−0.0575309	+	0.998344i	$0.518323\pi$
$432$	0	0
$433$	3.90143	0.187491	0.0937455	−	0.995596i	$-0.470116\pi$
$433$	3.90143	0.187491	0.0937455	+	0.995596i	$0.470116\pi$
$434$	0	0
$435$	−35.8714	−1.71990
$436$	0	0
$437$	−6.54599	−0.313137
$438$	0	0
$439$	5.34917	0.255302	0.127651	−	0.991819i	$-0.459256\pi$
$439$	5.34917	0.255302	0.127651	+	0.991819i	$0.459256\pi$
$440$	0	0
$441$	2.24487	0.106898
$442$	0	0
$443$	−22.8298	−1.08467	−0.542337	−	0.840161i	$-0.682461\pi$
$443$	−22.8298	−1.08467	−0.542337	+	0.840161i	$0.682461\pi$
$444$	0	0
$445$	−24.1264	−1.14370
$446$	0	0
$447$	22.4019	1.05957
$448$	0	0
$449$	−31.5053	−1.48683	−0.743413	−	0.668833i	$-0.766794\pi$
$449$	−31.5053	−1.48683	−0.743413	+	0.668833i	$0.766794\pi$
$450$	0	0
$451$	31.4577	1.48129
$452$	0	0
$453$	−15.4878	−0.727680
$454$	0	0
$455$	27.0028	1.26591
$456$	0	0
$457$	16.1928	0.757469	0.378734	−	0.925505i	$-0.376359\pi$
$457$	16.1928	0.757469	0.378734	+	0.925505i	$0.376359\pi$
$458$	0	0
$459$	−33.4944	−1.56338
$460$	0	0
$461$	−25.9691	−1.20950	−0.604751	−	0.796415i	$-0.706727\pi$
$461$	−25.9691	−1.20950	−0.604751	+	0.796415i	$0.706727\pi$
$462$	0	0
$463$	22.1706	1.03036	0.515179	−	0.857083i	$-0.327726\pi$
$463$	22.1706	1.03036	0.515179	+	0.857083i	$0.327726\pi$
$464$	0	0
$465$	−38.8237	−1.80041
$466$	0	0
$467$	−29.5908	−1.36930	−0.684649	−	0.728873i	$-0.740044\pi$
$467$	−29.5908	−1.36930	−0.684649	+	0.728873i	$0.740044\pi$
$468$	0	0
$469$	2.46272	0.113718
$470$	0	0
$471$	−32.4923	−1.49716
$472$	0	0
$473$	49.0772	2.25657
$474$	0	0
$475$	3.83928	0.176158
$476$	0	0
$477$	−5.70396	−0.261166
$478$	0	0
$479$	−25.2331	−1.15293	−0.576465	−	0.817122i	$-0.695568\pi$
$479$	−25.2331	−1.15293	−0.576465	+	0.817122i	$0.695568\pi$
$480$	0	0
$481$	52.8715	2.41073
$482$	0	0
$483$	−10.2817	−0.467835
$484$	0	0
$485$	6.99995	0.317851
$486$	0	0
$487$	19.4539	0.881542	0.440771	−	0.897620i	$-0.354705\pi$
$487$	19.4539	0.881542	0.440771	+	0.897620i	$0.354705\pi$
$488$	0	0
$489$	9.59934	0.434097
$490$	0	0
$491$	4.65848	0.210234	0.105117	−	0.994460i	$-0.466478\pi$
$491$	4.65848	0.210234	0.105117	+	0.994460i	$0.466478\pi$
$492$	0	0
$493$	−49.6468	−2.23598
$494$	0	0
$495$	−7.47119	−0.335805
$496$	0	0
$497$	6.17466	0.276971
$498$	0	0
$499$	9.16907	0.410464	0.205232	−	0.978713i	$-0.434205\pi$
$499$	9.16907	0.410464	0.205232	+	0.978713i	$0.434205\pi$
$500$	0	0
$501$	−6.45177	−0.288244
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−31.7221	−1.41161
$506$	0	0
$507$	−50.7691	−2.25473
$508$	0	0
$509$	11.6477	0.516275	0.258137	−	0.966108i	$-0.416891\pi$
$509$	11.6477	0.516275	0.258137	+	0.966108i	$0.416891\pi$
$510$	0	0
$511$	18.3435	0.811467
$512$	0	0
$513$	8.18737	0.361481
$514$	0	0
$515$	−4.46023	−0.196541
$516$	0	0
$517$	−53.2593	−2.34234
$518$	0	0
$519$	−14.9447	−0.656000
$520$	0	0
$521$	16.4805	0.722024	0.361012	−	0.932561i	$-0.382431\pi$
$521$	16.4805	0.722024	0.361012	+	0.932561i	$0.382431\pi$
$522$	0	0
$523$	−8.75352	−0.382765	−0.191382	−	0.981516i	$-0.561297\pi$
$523$	−8.75352	−0.382765	−0.191382	+	0.981516i	$0.561297\pi$
$524$	0	0
$525$	6.03033	0.263185
$526$	0	0
$527$	−53.7328	−2.34064
$528$	0	0
$529$	−3.55465	−0.154550
$530$	0	0
$531$	−1.18908	−0.0516015
$532$	0	0
$533$	−35.8978	−1.55491
$534$	0	0
$535$	−11.3874	−0.492321
$536$	0	0
$537$	14.3389	0.618769
$538$	0	0
$539$	−28.5152	−1.22824
$540$	0	0
$541$	22.3348	0.960248	0.480124	−	0.877201i	$-0.340592\pi$
$541$	22.3348	0.960248	0.480124	+	0.877201i	$0.340592\pi$
$542$	0	0
$543$	−8.44997	−0.362623
$544$	0	0
$545$	−50.7595	−2.17430
$546$	0	0
$547$	−12.9067	−0.551849	−0.275925	−	0.961179i	$-0.588984\pi$
$547$	−12.9067	−0.551849	−0.275925	+	0.961179i	$0.588984\pi$
$548$	0	0
$549$	−1.40453	−0.0599441
$550$	0	0
$551$	12.1357	0.516997
$552$	0	0
$553$	20.8734	0.887629
$554$	0	0
$555$	34.6340	1.47013
$556$	0	0
$557$	−3.56576	−0.151086	−0.0755430	−	0.997143i	$-0.524069\pi$
$557$	−3.56576	−0.151086	−0.0755430	+	0.997143i	$0.524069\pi$
$558$	0	0
$559$	−56.0041	−2.36872
$560$	0	0
$561$	56.7887	2.39762
$562$	0	0
$563$	35.6331	1.50176	0.750878	−	0.660441i	$-0.229631\pi$
$563$	35.6331	1.50176	0.750878	+	0.660441i	$0.229631\pi$
$564$	0	0
$565$	16.8922	0.710662
$566$	0	0
$567$	10.8308	0.454852
$568$	0	0
$569$	−11.2523	−0.471721	−0.235861	−	0.971787i	$-0.575791\pi$
$569$	−11.2523	−0.471721	−0.235861	+	0.971787i	$0.575791\pi$
$570$	0	0
$571$	12.2326	0.511918	0.255959	−	0.966688i	$-0.417609\pi$
$571$	12.2326	0.511918	0.255959	+	0.966688i	$0.417609\pi$
$572$	0	0
$573$	23.2340	0.970613
$574$	0	0
$575$	−11.4049	−0.475616
$576$	0	0
$577$	−14.9930	−0.624167	−0.312083	−	0.950055i	$-0.601027\pi$
$577$	−14.9930	−0.624167	−0.312083	+	0.950055i	$0.601027\pi$
$578$	0	0
$579$	2.02210	0.0840355
$580$	0	0
$581$	−25.2580	−1.04788
$582$	0	0
$583$	72.4541	3.00074
$584$	0	0
$585$	8.52571	0.352495
$586$	0	0
$587$	−45.7141	−1.88682	−0.943412	−	0.331622i	$-0.892404\pi$
$587$	−45.7141	−1.88682	−0.943412	+	0.331622i	$0.892404\pi$
$588$	0	0
$589$	13.1345	0.541196
$590$	0	0
$591$	21.8410	0.898419
$592$	0	0
$593$	26.8233	1.10150	0.550750	−	0.834670i	$-0.314342\pi$
$593$	26.8233	1.10150	0.550750	+	0.834670i	$0.314342\pi$
$594$	0	0
$595$	24.4812	1.00363
$596$	0	0
$597$	29.0302	1.18813
$598$	0	0
$599$	13.0654	0.533837	0.266918	−	0.963719i	$-0.413995\pi$
$599$	13.0654	0.533837	0.266918	+	0.963719i	$0.413995\pi$
$600$	0	0
$601$	8.45943	0.345068	0.172534	−	0.985004i	$-0.444805\pi$
$601$	8.45943	0.345068	0.172534	+	0.985004i	$0.444805\pi$
$602$	0	0
$603$	0.777564	0.0316648
$604$	0	0
$605$	64.6046	2.62655
$606$	0	0
$607$	−31.3103	−1.27085	−0.635423	−	0.772164i	$-0.719174\pi$
$607$	−31.3103	−1.27085	−0.635423	+	0.772164i	$0.719174\pi$
$608$	0	0
$609$	19.0614	0.772408
$610$	0	0
$611$	60.7766	2.45876
$612$	0	0
$613$	−13.9304	−0.562642	−0.281321	−	0.959614i	$-0.590773\pi$
$613$	−13.9304	−0.562642	−0.281321	+	0.959614i	$0.590773\pi$
$614$	0	0
$615$	−23.5152	−0.948226
$616$	0	0
$617$	−47.0125	−1.89265	−0.946325	−	0.323215i	$-0.895236\pi$
$617$	−47.0125	−1.89265	−0.946325	+	0.323215i	$0.895236\pi$
$618$	0	0
$619$	−33.3762	−1.34150	−0.670751	−	0.741682i	$-0.734028\pi$
$619$	−33.3762	−1.34150	−0.670751	+	0.741682i	$0.734028\pi$
$620$	0	0
$621$	−24.3212	−0.975976
$622$	0	0
$623$	12.8203	0.513636
$624$	0	0
$625$	−31.2425	−1.24970
$626$	0	0
$627$	−13.8814	−0.554371
$628$	0	0
$629$	47.9342	1.91126
$630$	0	0
$631$	−3.74341	−0.149023	−0.0745114	−	0.997220i	$-0.523740\pi$
$631$	−3.74341	−0.149023	−0.0745114	+	0.997220i	$0.523740\pi$
$632$	0	0
$633$	−16.1609	−0.642337
$634$	0	0
$635$	−47.1059	−1.86934
$636$	0	0
$637$	32.5400	1.28928
$638$	0	0
$639$	1.94955	0.0771230
$640$	0	0
$641$	−1.81336	−0.0716234	−0.0358117	−	0.999359i	$-0.511402\pi$
$641$	−1.81336	−0.0716234	−0.0358117	+	0.999359i	$0.511402\pi$
$642$	0	0
$643$	13.6210	0.537160	0.268580	−	0.963257i	$-0.413446\pi$
$643$	13.6210	0.537160	0.268580	+	0.963257i	$0.413446\pi$
$644$	0	0
$645$	−36.6861	−1.44451
$646$	0	0
$647$	16.6018	0.652685	0.326342	−	0.945252i	$-0.394184\pi$
$647$	16.6018	0.652685	0.326342	+	0.945252i	$0.394184\pi$
$648$	0	0
$649$	15.1041	0.592889
$650$	0	0
$651$	20.6302	0.808562
$652$	0	0
$653$	−5.76542	−0.225618	−0.112809	−	0.993617i	$-0.535985\pi$
$653$	−5.76542	−0.225618	−0.112809	+	0.993617i	$0.535985\pi$
$654$	0	0
$655$	−34.0629	−1.33095
$656$	0	0
$657$	5.79166	0.225954
$658$	0	0
$659$	19.3602	0.754168	0.377084	−	0.926179i	$-0.376927\pi$
$659$	19.3602	0.754168	0.377084	+	0.926179i	$0.376927\pi$
$660$	0	0
$661$	−5.17780	−0.201393	−0.100697	−	0.994917i	$-0.532107\pi$
$661$	−5.17780	−0.201393	−0.100697	+	0.994917i	$0.532107\pi$
$662$	0	0
$663$	−64.8041	−2.51678
$664$	0	0
$665$	−5.98419	−0.232057
$666$	0	0
$667$	−36.0499	−1.39586
$668$	0	0
$669$	14.9086	0.576401
$670$	0	0
$671$	17.8410	0.688744
$672$	0	0
$673$	2.58114	0.0994958	0.0497479	−	0.998762i	$-0.484158\pi$
$673$	2.58114	0.0994958	0.0497479	+	0.998762i	$0.484158\pi$
$674$	0	0
$675$	14.2646	0.549045
$676$	0	0
$677$	43.2830	1.66350	0.831751	−	0.555149i	$-0.187339\pi$
$677$	43.2830	1.66350	0.831751	+	0.555149i	$0.187339\pi$
$678$	0	0
$679$	−3.71965	−0.142747
$680$	0	0
$681$	21.5017	0.823947
$682$	0	0
$683$	14.8597	0.568590	0.284295	−	0.958737i	$-0.408240\pi$
$683$	14.8597	0.568590	0.284295	+	0.958737i	$0.408240\pi$
$684$	0	0
$685$	3.73588	0.142741
$686$	0	0
$687$	−6.14406	−0.234411
$688$	0	0
$689$	−82.6806	−3.14988
$690$	0	0
$691$	−34.7070	−1.32032	−0.660158	−	0.751127i	$-0.729511\pi$
$691$	−34.7070	−1.32032	−0.660158	+	0.751127i	$0.729511\pi$
$692$	0	0
$693$	3.97005	0.150810
$694$	0	0
$695$	25.9458	0.984182
$696$	0	0
$697$	−32.5456	−1.23275
$698$	0	0
$699$	42.4723	1.60645
$700$	0	0
$701$	12.6514	0.477838	0.238919	−	0.971040i	$-0.423207\pi$
$701$	12.6514	0.477838	0.238919	+	0.971040i	$0.423207\pi$
$702$	0	0
$703$	−11.7170	−0.441917
$704$	0	0
$705$	39.8123	1.49942
$706$	0	0
$707$	16.8565	0.633956
$708$	0	0
$709$	−0.358275	−0.0134553	−0.00672765	−	0.999977i	$-0.502141\pi$
$709$	−0.358275	−0.0134553	−0.00672765	+	0.999977i	$0.502141\pi$
$710$	0	0
$711$	6.59046	0.247161
$712$	0	0
$713$	−39.0169	−1.46119
$714$	0	0
$715$	−108.297	−4.05008
$716$	0	0
$717$	−16.9272	−0.632157
$718$	0	0
$719$	23.9993	0.895024	0.447512	−	0.894278i	$-0.352310\pi$
$719$	23.9993	0.895024	0.447512	+	0.894278i	$0.352310\pi$
$720$	0	0
$721$	2.37008	0.0882665
$722$	0	0
$723$	33.8373	1.25842
$724$	0	0
$725$	21.1436	0.785254
$726$	0	0
$727$	38.0953	1.41288	0.706438	−	0.707775i	$-0.250301\pi$
$727$	38.0953	1.41288	0.706438	+	0.707775i	$0.250301\pi$
$728$	0	0
$729$	29.7790	1.10293
$730$	0	0
$731$	−50.7743	−1.87796
$732$	0	0
$733$	50.7757	1.87544	0.937721	−	0.347388i	$-0.112931\pi$
$733$	50.7757	1.87544	0.937721	+	0.347388i	$0.112931\pi$
$734$	0	0
$735$	21.3156	0.786240
$736$	0	0
$737$	−9.87694	−0.363822
$738$	0	0
$739$	−30.7456	−1.13099	−0.565497	−	0.824750i	$-0.691316\pi$
$739$	−30.7456	−1.13099	−0.565497	+	0.824750i	$0.691316\pi$
$740$	0	0
$741$	15.8407	0.581923
$742$	0	0
$743$	17.3272	0.635673	0.317837	−	0.948146i	$-0.397044\pi$
$743$	17.3272	0.635673	0.317837	+	0.948146i	$0.397044\pi$
$744$	0	0
$745$	−38.7314	−1.41901
$746$	0	0
$747$	−7.97480	−0.291783
$748$	0	0
$749$	6.05107	0.221101
$750$	0	0
$751$	−33.6584	−1.22821	−0.614106	−	0.789224i	$-0.710483\pi$
$751$	−33.6584	−1.22821	−0.614106	+	0.789224i	$0.710483\pi$
$752$	0	0
$753$	−31.3282	−1.14166
$754$	0	0
$755$	26.7774	0.974530
$756$	0	0
$757$	−34.1538	−1.24134	−0.620671	−	0.784071i	$-0.713140\pi$
$757$	−34.1538	−1.24134	−0.620671	+	0.784071i	$0.713140\pi$
$758$	0	0
$759$	41.2358	1.49677
$760$	0	0
$761$	14.3681	0.520842	0.260421	−	0.965495i	$-0.416139\pi$
$761$	14.3681	0.520842	0.260421	+	0.965495i	$0.416139\pi$
$762$	0	0
$763$	26.9727	0.976476
$764$	0	0
$765$	7.72955	0.279463
$766$	0	0
$767$	−17.2360	−0.622356
$768$	0	0
$769$	27.4962	0.991537	0.495768	−	0.868455i	$-0.334887\pi$
$769$	27.4962	0.991537	0.495768	+	0.868455i	$0.334887\pi$
$770$	0	0
$771$	37.7423	1.35925
$772$	0	0
$773$	−7.75525	−0.278937	−0.139469	−	0.990226i	$-0.544539\pi$
$773$	−7.75525	−0.278937	−0.139469	+	0.990226i	$0.544539\pi$
$774$	0	0
$775$	22.8838	0.822009
$776$	0	0
$777$	−18.4039	−0.660236
$778$	0	0
$779$	7.95544	0.285033
$780$	0	0
$781$	−24.7640	−0.886126
$782$	0	0
$783$	45.0893	1.61136
$784$	0	0
$785$	56.1770	2.00504
$786$	0	0
$787$	19.6903	0.701885	0.350942	−	0.936397i	$-0.385861\pi$
$787$	19.6903	0.701885	0.350942	+	0.936397i	$0.385861\pi$
$788$	0	0
$789$	−1.72902	−0.0615549
$790$	0	0
$791$	−8.97623	−0.319158
$792$	0	0
$793$	−20.3591	−0.722975
$794$	0	0
$795$	−54.1607	−1.92088
$796$	0	0
$797$	−21.9546	−0.777671	−0.388835	−	0.921307i	$-0.627123\pi$
$797$	−21.9546	−0.777671	−0.388835	+	0.921307i	$0.627123\pi$
$798$	0	0
$799$	55.1011	1.94934
$800$	0	0
$801$	4.04782	0.143023
$802$	0	0
$803$	−73.5680	−2.59616
$804$	0	0
$805$	17.7765	0.626539
$806$	0	0
$807$	−14.2252	−0.500750
$808$	0	0
$809$	−10.3431	−0.363644	−0.181822	−	0.983331i	$-0.558199\pi$
$809$	−10.3431	−0.363644	−0.181822	+	0.983331i	$0.558199\pi$
$810$	0	0
$811$	32.4712	1.14022	0.570109	−	0.821569i	$-0.306901\pi$
$811$	32.4712	1.14022	0.570109	+	0.821569i	$0.306901\pi$
$812$	0	0
$813$	12.8022	0.448991
$814$	0	0
$815$	−16.5967	−0.581355
$816$	0	0
$817$	12.4113	0.434215
$818$	0	0
$819$	−4.53041	−0.158305
$820$	0	0
$821$	−54.2464	−1.89321	−0.946606	−	0.322394i	$-0.895513\pi$
$821$	−54.2464	−1.89321	−0.946606	+	0.322394i	$0.895513\pi$
$822$	0	0
$823$	−49.1896	−1.71464	−0.857322	−	0.514781i	$-0.827873\pi$
$823$	−49.1896	−1.71464	−0.857322	+	0.514781i	$0.827873\pi$
$824$	0	0
$825$	−24.1852	−0.842020
$826$	0	0
$827$	−42.9413	−1.49322	−0.746608	−	0.665265i	$-0.768319\pi$
$827$	−42.9413	−1.49322	−0.746608	+	0.665265i	$0.768319\pi$
$828$	0	0
$829$	−21.5437	−0.748245	−0.374122	−	0.927379i	$-0.622056\pi$
$829$	−21.5437	−0.748245	−0.374122	+	0.927379i	$0.622056\pi$
$830$	0	0
$831$	−17.0607	−0.591829
$832$	0	0
$833$	29.5013	1.02216
$834$	0	0
$835$	11.1547	0.386025
$836$	0	0
$837$	48.8003	1.68678
$838$	0	0
$839$	−18.6657	−0.644410	−0.322205	−	0.946670i	$-0.604424\pi$
$839$	−18.6657	−0.644410	−0.322205	+	0.946670i	$0.604424\pi$
$840$	0	0
$841$	37.8333	1.30460
$842$	0	0
$843$	16.8952	0.581900
$844$	0	0
$845$	87.7765	3.01960
$846$	0	0
$847$	−34.3297	−1.17958
$848$	0	0
$849$	4.19997	0.144143
$850$	0	0
$851$	34.8063	1.19315
$852$	0	0
$853$	−29.7303	−1.01795	−0.508974	−	0.860782i	$-0.669975\pi$
$853$	−29.7303	−1.01795	−0.508974	+	0.860782i	$0.669975\pi$
$854$	0	0
$855$	−1.88941	−0.0646165
$856$	0	0
$857$	−22.9356	−0.783465	−0.391733	−	0.920079i	$-0.628124\pi$
$857$	−22.9356	−0.783465	−0.391733	+	0.920079i	$0.628124\pi$
$858$	0	0
$859$	−21.3777	−0.729397	−0.364698	−	0.931126i	$-0.618828\pi$
$859$	−21.3777	−0.729397	−0.364698	+	0.931126i	$0.618828\pi$
$860$	0	0
$861$	12.4956	0.425848
$862$	0	0
$863$	54.1528	1.84338	0.921692	−	0.387922i	$-0.126807\pi$
$863$	54.1528	1.84338	0.921692	+	0.387922i	$0.126807\pi$
$864$	0	0
$865$	25.8385	0.878534
$866$	0	0
$867$	−31.6702	−1.07558
$868$	0	0
$869$	−83.7147	−2.83983
$870$	0	0
$871$	11.2710	0.381904
$872$	0	0
$873$	−1.17442	−0.0397481
$874$	0	0
$875$	9.73010	0.328937
$876$	0	0
$877$	−30.5204	−1.03060	−0.515300	−	0.857010i	$-0.672319\pi$
$877$	−30.5204	−1.03060	−0.515300	+	0.857010i	$0.672319\pi$
$878$	0	0
$879$	−11.6711	−0.393655
$880$	0	0
$881$	3.55114	0.119641	0.0598205	−	0.998209i	$-0.480947\pi$
$881$	3.55114	0.119641	0.0598205	+	0.998209i	$0.480947\pi$
$882$	0	0
$883$	26.5969	0.895056	0.447528	−	0.894270i	$-0.352305\pi$
$883$	26.5969	0.895056	0.447528	+	0.894270i	$0.352305\pi$
$884$	0	0
$885$	−11.2906	−0.379530
$886$	0	0
$887$	47.2877	1.58776	0.793882	−	0.608071i	$-0.208057\pi$
$887$	47.2877	1.58776	0.793882	+	0.608071i	$0.208057\pi$
$888$	0	0
$889$	25.0312	0.839521
$890$	0	0
$891$	−43.4380	−1.45523
$892$	0	0
$893$	−13.4689	−0.450720
$894$	0	0
$895$	−24.7910	−0.828673
$896$	0	0
$897$	−47.0561	−1.57116
$898$	0	0
$899$	72.3339	2.41247
$900$	0	0
$901$	−74.9596	−2.49727
$902$	0	0
$903$	19.4943	0.648730
$904$	0	0
$905$	14.6095	0.485635
$906$	0	0
$907$	−5.29369	−0.175774	−0.0878870	−	0.996130i	$-0.528011\pi$
$907$	−5.29369	−0.175774	−0.0878870	+	0.996130i	$0.528011\pi$
$908$	0	0
$909$	5.32219	0.176526
$910$	0	0
$911$	14.4595	0.479064	0.239532	−	0.970888i	$-0.423006\pi$
$911$	14.4595	0.479064	0.239532	+	0.970888i	$0.423006\pi$
$912$	0	0
$913$	101.299	3.35252
$914$	0	0
$915$	−13.3365	−0.440890
$916$	0	0
$917$	18.1004	0.597727
$918$	0	0
$919$	−3.00395	−0.0990911	−0.0495456	−	0.998772i	$-0.515777\pi$
$919$	−3.00395	−0.0990911	−0.0495456	+	0.998772i	$0.515777\pi$
$920$	0	0
$921$	−21.1664	−0.697457
$922$	0	0
$923$	28.2593	0.930167
$924$	0	0
$925$	−20.4142	−0.671216
$926$	0	0
$927$	0.748316	0.0245779
$928$	0	0
$929$	13.1686	0.432047	0.216023	−	0.976388i	$-0.430691\pi$
$929$	13.1686	0.432047	0.216023	+	0.976388i	$0.430691\pi$
$930$	0	0
$931$	−7.21130	−0.236341
$932$	0	0
$933$	36.0500	1.18022
$934$	0	0
$935$	−98.1840	−3.21096
$936$	0	0
$937$	7.66519	0.250411	0.125205	−	0.992131i	$-0.460041\pi$
$937$	7.66519	0.250411	0.125205	+	0.992131i	$0.460041\pi$
$938$	0	0
$939$	−3.20400	−0.104559
$940$	0	0
$941$	−31.8108	−1.03700	−0.518501	−	0.855077i	$-0.673510\pi$
$941$	−31.8108	−1.03700	−0.518501	+	0.855077i	$0.673510\pi$
$942$	0	0
$943$	−23.6322	−0.769572
$944$	0	0
$945$	−22.2339	−0.723268
$946$	0	0
$947$	20.1008	0.653189	0.326595	−	0.945165i	$-0.394099\pi$
$947$	20.1008	0.653189	0.326595	+	0.945165i	$0.394099\pi$
$948$	0	0
$949$	83.9518	2.72519
$950$	0	0
$951$	48.4736	1.57187
$952$	0	0
$953$	2.24603	0.0727560	0.0363780	−	0.999338i	$-0.488418\pi$
$953$	2.24603	0.0727560	0.0363780	+	0.999338i	$0.488418\pi$
$954$	0	0
$955$	−40.1701	−1.29987
$956$	0	0
$957$	−76.4475	−2.47120
$958$	0	0
$959$	−1.98518	−0.0641047
$960$	0	0
$961$	47.2871	1.52539
$962$	0	0
$963$	1.91053	0.0615660
$964$	0	0
$965$	−3.49608	−0.112543
$966$	0	0
$967$	25.4322	0.817843	0.408921	−	0.912570i	$-0.365905\pi$
$967$	25.4322	0.817843	0.408921	+	0.912570i	$0.365905\pi$
$968$	0	0
$969$	14.3615	0.461357
$970$	0	0
$971$	−10.2958	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$971$	−10.2958	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$972$	0	0
$973$	−13.7872	−0.441996
$974$	0	0
$975$	27.5988	0.883869
$976$	0	0
$977$	−2.26590	−0.0724925	−0.0362462	−	0.999343i	$-0.511540\pi$
$977$	−2.26590	−0.0724925	−0.0362462	+	0.999343i	$0.511540\pi$
$978$	0	0
$979$	−51.4170	−1.64330
$980$	0	0
$981$	8.51619	0.271901
$982$	0	0
$983$	−19.6737	−0.627492	−0.313746	−	0.949507i	$-0.601584\pi$
$983$	−19.6737	−0.627492	−0.313746	+	0.949507i	$0.601584\pi$
$984$	0	0
$985$	−37.7617	−1.20319
$986$	0	0
$987$	−21.1555	−0.673388
$988$	0	0
$989$	−36.8686	−1.17235
$990$	0	0
$991$	−48.1094	−1.52825	−0.764123	−	0.645070i	$-0.776828\pi$
$991$	−48.1094	−1.52825	−0.764123	+	0.645070i	$0.776828\pi$
$992$	0	0
$993$	1.46808	0.0465882
$994$	0	0
$995$	−50.1914	−1.59117
$996$	0	0
$997$	−1.90282	−0.0602629	−0.0301315	−	0.999546i	$-0.509593\pi$
$997$	−1.90282	−0.0602629	−0.0301315	+	0.999546i	$0.509593\pi$
$998$	0	0
$999$	−43.5339	−1.37735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.9		28
4.3	odd	2		4024.2.a.d.1.20	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.20	✓	28	4.3	odd	2
8048.2.a.v.1.9		28	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)

\(f(q)\)	\(=\)	\(q-1.59308 q^{3} +2.75433 q^{5} -1.46360 q^{7} -0.462108 q^{9} +O(q^{10})\)	\(q-1.59308 q^{3} +2.75433 q^{5} -1.46360 q^{7} -0.462108 q^{9} +5.86989 q^{11} -6.69840 q^{13} -4.38785 q^{15} -6.07288 q^{17} +1.48446 q^{19} +2.33163 q^{21} -4.40969 q^{23} +2.58632 q^{25} +5.51540 q^{27} +8.17517 q^{29} +8.84800 q^{31} -9.35119 q^{33} -4.03123 q^{35} -7.89315 q^{37} +10.6711 q^{39} +5.35916 q^{41} +8.36082 q^{43} -1.27280 q^{45} -9.07330 q^{47} -4.85788 q^{49} +9.67456 q^{51} +12.3433 q^{53} +16.1676 q^{55} -2.36485 q^{57} +2.57315 q^{59} +3.03940 q^{61} +0.676342 q^{63} -18.4496 q^{65} -1.68264 q^{67} +7.02497 q^{69} -4.21882 q^{71} -12.5331 q^{73} -4.12021 q^{75} -8.59118 q^{77} -14.2617 q^{79} -7.40013 q^{81} +17.2574 q^{83} -16.7267 q^{85} -13.0237 q^{87} -8.75945 q^{89} +9.80378 q^{91} -14.0955 q^{93} +4.08868 q^{95} +2.54144 q^{97} -2.71253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

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