LMFDB - Embedded newform 3856.2.a.n.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3856,2,Mod(1,3856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3856, 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("3856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3856 = 2^{4} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3856.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:	$30.7903150194$
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} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.70063$ of defining polynomial
Character	$\chi$	$=$	3856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.50808 q^{3} +0.533570 q^{5} -0.354992 q^{7} +3.29045 q^{9} +O(q^{10})$	$q+2.50808 q^{3} +0.533570 q^{5} -0.354992 q^{7} +3.29045 q^{9} -4.18781 q^{11} -3.72447 q^{13} +1.33823 q^{15} -6.46259 q^{17} +1.31002 q^{19} -0.890347 q^{21} +4.10799 q^{23} -4.71530 q^{25} +0.728479 q^{27} -8.85227 q^{29} -5.11371 q^{31} -10.5034 q^{33} -0.189413 q^{35} +5.41403 q^{37} -9.34127 q^{39} +11.8244 q^{41} -0.673253 q^{43} +1.75569 q^{45} -5.22965 q^{47} -6.87398 q^{49} -16.2087 q^{51} -9.92404 q^{53} -2.23449 q^{55} +3.28564 q^{57} +1.23315 q^{59} +4.04837 q^{61} -1.16808 q^{63} -1.98727 q^{65} +14.0522 q^{67} +10.3032 q^{69} -13.0525 q^{71} +7.76916 q^{73} -11.8263 q^{75} +1.48664 q^{77} +1.17673 q^{79} -8.04428 q^{81} -6.25297 q^{83} -3.44824 q^{85} -22.2022 q^{87} +3.80839 q^{89} +1.32216 q^{91} -12.8256 q^{93} +0.698989 q^{95} -9.91591 q^{97} -13.7798 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * 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
$3$	2.50808	1.44804	0.724020	−	0.689779i	$-0.242292\pi$
$3$	2.50808	1.44804	0.724020	+	0.689779i	$0.242292\pi$
$4$	0	0
$5$	0.533570	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$5$	0.533570	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$6$	0	0
$7$	−0.354992	−0.134174	−0.0670872	−	0.997747i	$-0.521371\pi$
$7$	−0.354992	−0.134174	−0.0670872	+	0.997747i	$0.521371\pi$
$8$	0	0
$9$	3.29045	1.09682
$10$	0	0
$11$	−4.18781	−1.26267	−0.631336	−	0.775509i	$-0.717493\pi$
$11$	−4.18781	−1.26267	−0.631336	+	0.775509i	$0.717493\pi$
$12$	0	0
$13$	−3.72447	−1.03298	−0.516492	−	0.856292i	$-0.672762\pi$
$13$	−3.72447	−1.03298	−0.516492	+	0.856292i	$0.672762\pi$
$14$	0	0
$15$	1.33823	0.345531
$16$	0	0
$17$	−6.46259	−1.56741	−0.783704	−	0.621134i	$-0.786672\pi$
$17$	−6.46259	−1.56741	−0.783704	+	0.621134i	$0.786672\pi$
$18$	0	0
$19$	1.31002	0.300540	0.150270	−	0.988645i	$-0.451986\pi$
$19$	1.31002	0.300540	0.150270	+	0.988645i	$0.451986\pi$
$20$	0	0
$21$	−0.890347	−0.194290
$22$	0	0
$23$	4.10799	0.856576	0.428288	−	0.903642i	$-0.359117\pi$
$23$	4.10799	0.856576	0.428288	+	0.903642i	$0.359117\pi$
$24$	0	0
$25$	−4.71530	−0.943061
$26$	0	0
$27$	0.728479	0.140196
$28$	0	0
$29$	−8.85227	−1.64382	−0.821912	−	0.569614i	$-0.807093\pi$
$29$	−8.85227	−1.64382	−0.821912	+	0.569614i	$0.807093\pi$
$30$	0	0
$31$	−5.11371	−0.918449	−0.459225	−	0.888320i	$-0.651873\pi$
$31$	−5.11371	−0.918449	−0.459225	+	0.888320i	$0.651873\pi$
$32$	0	0
$33$	−10.5034	−1.82840
$34$	0	0
$35$	−0.189413	−0.0320166
$36$	0	0
$37$	5.41403	0.890061	0.445030	−	0.895515i	$-0.353193\pi$
$37$	5.41403	0.890061	0.445030	+	0.895515i	$0.353193\pi$
$38$	0	0
$39$	−9.34127	−1.49580
$40$	0	0
$41$	11.8244	1.84665	0.923327	−	0.384014i	$-0.125459\pi$
$41$	11.8244	1.84665	0.923327	+	0.384014i	$0.125459\pi$
$42$	0	0
$43$	−0.673253	−0.102670	−0.0513350	−	0.998681i	$-0.516348\pi$
$43$	−0.673253	−0.102670	−0.0513350	+	0.998681i	$0.516348\pi$
$44$	0	0
$45$	1.75569	0.261722
$46$	0	0
$47$	−5.22965	−0.762823	−0.381412	−	0.924405i	$-0.624562\pi$
$47$	−5.22965	−0.762823	−0.381412	+	0.924405i	$0.624562\pi$
$48$	0	0
$49$	−6.87398	−0.981997
$50$	0	0
$51$	−16.2087	−2.26967
$52$	0	0
$53$	−9.92404	−1.36317	−0.681586	−	0.731738i	$-0.738709\pi$
$53$	−9.92404	−1.36317	−0.681586	+	0.731738i	$0.738709\pi$
$54$	0	0
$55$	−2.23449	−0.301298
$56$	0	0
$57$	3.28564	0.435194
$58$	0	0
$59$	1.23315	0.160542	0.0802710	−	0.996773i	$-0.474421\pi$
$59$	1.23315	0.160542	0.0802710	+	0.996773i	$0.474421\pi$
$60$	0	0
$61$	4.04837	0.518341	0.259170	−	0.965832i	$-0.416551\pi$
$61$	4.04837	0.518341	0.259170	+	0.965832i	$0.416551\pi$
$62$	0	0
$63$	−1.16808	−0.147165
$64$	0	0
$65$	−1.98727	−0.246490
$66$	0	0
$67$	14.0522	1.71675	0.858375	−	0.513022i	$-0.171474\pi$
$67$	14.0522	1.71675	0.858375	+	0.513022i	$0.171474\pi$
$68$	0	0
$69$	10.3032	1.24036
$70$	0	0
$71$	−13.0525	−1.54905	−0.774526	−	0.632542i	$-0.782011\pi$
$71$	−13.0525	−1.54905	−0.774526	+	0.632542i	$0.782011\pi$
$72$	0	0
$73$	7.76916	0.909312	0.454656	−	0.890667i	$-0.349762\pi$
$73$	7.76916	0.909312	0.454656	+	0.890667i	$0.349762\pi$
$74$	0	0
$75$	−11.8263	−1.36559
$76$	0	0
$77$	1.48664	0.169418
$78$	0	0
$79$	1.17673	0.132393	0.0661963	−	0.997807i	$-0.478914\pi$
$79$	1.17673	0.132393	0.0661963	+	0.997807i	$0.478914\pi$
$80$	0	0
$81$	−8.04428	−0.893809
$82$	0	0
$83$	−6.25297	−0.686353	−0.343176	−	0.939271i	$-0.611503\pi$
$83$	−6.25297	−0.686353	−0.343176	+	0.939271i	$0.611503\pi$
$84$	0	0
$85$	−3.44824	−0.374014
$86$	0	0
$87$	−22.2022	−2.38032
$88$	0	0
$89$	3.80839	0.403688	0.201844	−	0.979418i	$-0.435307\pi$
$89$	3.80839	0.403688	0.201844	+	0.979418i	$0.435307\pi$
$90$	0	0
$91$	1.32216	0.138600
$92$	0	0
$93$	−12.8256	−1.32995
$94$	0	0
$95$	0.698989	0.0717147
$96$	0	0
$97$	−9.91591	−1.00681	−0.503404	−	0.864051i	$-0.667919\pi$
$97$	−9.91591	−1.00681	−0.503404	+	0.864051i	$0.667919\pi$
$98$	0	0
$99$	−13.7798	−1.38492
$100$	0	0
$101$	7.46224	0.742521	0.371260	−	0.928529i	$-0.378926\pi$
$101$	7.46224	0.742521	0.371260	+	0.928529i	$0.378926\pi$
$102$	0	0
$103$	−7.83988	−0.772486	−0.386243	−	0.922397i	$-0.626227\pi$
$103$	−7.83988	−0.772486	−0.386243	+	0.922397i	$0.626227\pi$
$104$	0	0
$105$	−0.475062	−0.0463613
$106$	0	0
$107$	0.890206	0.0860595	0.0430297	−	0.999074i	$-0.486299\pi$
$107$	0.890206	0.0860595	0.0430297	+	0.999074i	$0.486299\pi$
$108$	0	0
$109$	3.92555	0.375999	0.188000	−	0.982169i	$-0.439800\pi$
$109$	3.92555	0.375999	0.188000	+	0.982169i	$0.439800\pi$
$110$	0	0
$111$	13.5788	1.28884
$112$	0	0
$113$	17.3995	1.63681	0.818404	−	0.574643i	$-0.194859\pi$
$113$	17.3995	1.63681	0.818404	+	0.574643i	$0.194859\pi$
$114$	0	0
$115$	2.19190	0.204396
$116$	0	0
$117$	−12.2552	−1.13299
$118$	0	0
$119$	2.29417	0.210306
$120$	0	0
$121$	6.53776	0.594342
$122$	0	0
$123$	29.6564	2.67403
$124$	0	0
$125$	−5.18379	−0.463653
$126$	0	0
$127$	−18.7331	−1.66230	−0.831148	−	0.556052i	$-0.812316\pi$
$127$	−18.7331	−1.66230	−0.831148	+	0.556052i	$0.812316\pi$
$128$	0	0
$129$	−1.68857	−0.148670
$130$	0	0
$131$	1.40627	0.122867	0.0614333	−	0.998111i	$-0.480433\pi$
$131$	1.40627	0.122867	0.0614333	+	0.998111i	$0.480433\pi$
$132$	0	0
$133$	−0.465048	−0.0403247
$134$	0	0
$135$	0.388694	0.0334535
$136$	0	0
$137$	5.54974	0.474146	0.237073	−	0.971492i	$-0.423812\pi$
$137$	5.54974	0.474146	0.237073	+	0.971492i	$0.423812\pi$
$138$	0	0
$139$	11.1540	0.946072	0.473036	−	0.881043i	$-0.343158\pi$
$139$	11.1540	0.946072	0.473036	+	0.881043i	$0.343158\pi$
$140$	0	0
$141$	−13.1164	−1.10460
$142$	0	0
$143$	15.5974	1.30432
$144$	0	0
$145$	−4.72330	−0.392249
$146$	0	0
$147$	−17.2405	−1.42197
$148$	0	0
$149$	8.27009	0.677512	0.338756	−	0.940874i	$-0.389994\pi$
$149$	8.27009	0.677512	0.338756	+	0.940874i	$0.389994\pi$
$150$	0	0
$151$	11.8990	0.968323	0.484162	−	0.874979i	$-0.339125\pi$
$151$	11.8990	0.968323	0.484162	+	0.874979i	$0.339125\pi$
$152$	0	0
$153$	−21.2648	−1.71916
$154$	0	0
$155$	−2.72852	−0.219160
$156$	0	0
$157$	−5.88156	−0.469399	−0.234700	−	0.972068i	$-0.575411\pi$
$157$	−5.88156	−0.469399	−0.234700	+	0.972068i	$0.575411\pi$
$158$	0	0
$159$	−24.8903	−1.97393
$160$	0	0
$161$	−1.45830	−0.114930
$162$	0	0
$163$	3.21033	0.251452	0.125726	−	0.992065i	$-0.459874\pi$
$163$	3.21033	0.251452	0.125726	+	0.992065i	$0.459874\pi$
$164$	0	0
$165$	−5.60427	−0.436292
$166$	0	0
$167$	9.62511	0.744813	0.372407	−	0.928070i	$-0.378533\pi$
$167$	9.62511	0.744813	0.372407	+	0.928070i	$0.378533\pi$
$168$	0	0
$169$	0.871711	0.0670547
$170$	0	0
$171$	4.31057	0.329637
$172$	0	0
$173$	−11.7677	−0.894682	−0.447341	−	0.894364i	$-0.647629\pi$
$173$	−11.7677	−0.894682	−0.447341	+	0.894364i	$0.647629\pi$
$174$	0	0
$175$	1.67389	0.126535
$176$	0	0
$177$	3.09283	0.232471
$178$	0	0
$179$	4.74606	0.354737	0.177369	−	0.984144i	$-0.443241\pi$
$179$	4.74606	0.354737	0.177369	+	0.984144i	$0.443241\pi$
$180$	0	0
$181$	−9.56167	−0.710713	−0.355357	−	0.934731i	$-0.615641\pi$
$181$	−9.56167	−0.710713	−0.355357	+	0.934731i	$0.615641\pi$
$182$	0	0
$183$	10.1536	0.750578
$184$	0	0
$185$	2.88876	0.212386
$186$	0	0
$187$	27.0641	1.97912
$188$	0	0
$189$	−0.258604	−0.0188107
$190$	0	0
$191$	−23.1412	−1.67444	−0.837218	−	0.546870i	$-0.815819\pi$
$191$	−23.1412	−1.67444	−0.837218	+	0.546870i	$0.815819\pi$
$192$	0	0
$193$	9.15643	0.659095	0.329547	−	0.944139i	$-0.393104\pi$
$193$	9.15643	0.659095	0.329547	+	0.944139i	$0.393104\pi$
$194$	0	0
$195$	−4.98422	−0.356927
$196$	0	0
$197$	−6.75045	−0.480950	−0.240475	−	0.970655i	$-0.577303\pi$
$197$	−6.75045	−0.480950	−0.240475	+	0.970655i	$0.577303\pi$
$198$	0	0
$199$	−3.26300	−0.231308	−0.115654	−	0.993290i	$-0.536896\pi$
$199$	−3.26300	−0.231308	−0.115654	+	0.993290i	$0.536896\pi$
$200$	0	0
$201$	35.2440	2.48592
$202$	0	0
$203$	3.14248	0.220559
$204$	0	0
$205$	6.30912	0.440648
$206$	0	0
$207$	13.5172	0.939507
$208$	0	0
$209$	−5.48613	−0.379483
$210$	0	0
$211$	23.1474	1.59354	0.796768	−	0.604285i	$-0.206541\pi$
$211$	23.1474	1.59354	0.796768	+	0.604285i	$0.206541\pi$
$212$	0	0
$213$	−32.7368	−2.24309
$214$	0	0
$215$	−0.359227	−0.0244991
$216$	0	0
$217$	1.81532	0.123232
$218$	0	0
$219$	19.4857	1.31672
$220$	0	0
$221$	24.0698	1.61911
$222$	0	0
$223$	18.2948	1.22511	0.612557	−	0.790427i	$-0.290141\pi$
$223$	18.2948	1.22511	0.612557	+	0.790427i	$0.290141\pi$
$224$	0	0
$225$	−15.5155	−1.03437
$226$	0	0
$227$	−10.4809	−0.695642	−0.347821	−	0.937561i	$-0.613078\pi$
$227$	−10.4809	−0.695642	−0.347821	+	0.937561i	$0.613078\pi$
$228$	0	0
$229$	−5.51626	−0.364524	−0.182262	−	0.983250i	$-0.558342\pi$
$229$	−5.51626	−0.364524	−0.182262	+	0.983250i	$0.558342\pi$
$230$	0	0
$231$	3.72861	0.245324
$232$	0	0
$233$	27.7343	1.81693	0.908466	−	0.417959i	$-0.137254\pi$
$233$	27.7343	1.81693	0.908466	+	0.417959i	$0.137254\pi$
$234$	0	0
$235$	−2.79039	−0.182025
$236$	0	0
$237$	2.95133	0.191710
$238$	0	0
$239$	−6.60568	−0.427286	−0.213643	−	0.976912i	$-0.568533\pi$
$239$	−6.60568	−0.427286	−0.213643	+	0.976912i	$0.568533\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	−22.3611	−1.43447
$244$	0	0
$245$	−3.66775	−0.234324
$246$	0	0
$247$	−4.87915	−0.310453
$248$	0	0
$249$	−15.6829	−0.993865
$250$	0	0
$251$	−21.2628	−1.34210	−0.671049	−	0.741413i	$-0.734156\pi$
$251$	−21.2628	−1.34210	−0.671049	+	0.741413i	$0.734156\pi$
$252$	0	0
$253$	−17.2035	−1.08157
$254$	0	0
$255$	−8.64846	−0.541588
$256$	0	0
$257$	7.33227	0.457374	0.228687	−	0.973500i	$-0.426557\pi$
$257$	7.33227	0.457374	0.228687	+	0.973500i	$0.426557\pi$
$258$	0	0
$259$	−1.92194	−0.119423
$260$	0	0
$261$	−29.1280	−1.80298
$262$	0	0
$263$	−20.6204	−1.27151	−0.635753	−	0.771892i	$-0.719310\pi$
$263$	−20.6204	−1.27151	−0.635753	+	0.771892i	$0.719310\pi$
$264$	0	0
$265$	−5.29517	−0.325280
$266$	0	0
$267$	9.55173	0.584557
$268$	0	0
$269$	−0.178772	−0.0108999	−0.00544997	−	0.999985i	$-0.501735\pi$
$269$	−0.178772	−0.0108999	−0.00544997	+	0.999985i	$0.501735\pi$
$270$	0	0
$271$	17.9311	1.08923	0.544617	−	0.838685i	$-0.316675\pi$
$271$	17.9311	1.08923	0.544617	+	0.838685i	$0.316675\pi$
$272$	0	0
$273$	3.31608	0.200698
$274$	0	0
$275$	19.7468	1.19078
$276$	0	0
$277$	2.56354	0.154028	0.0770140	−	0.997030i	$-0.475461\pi$
$277$	2.56354	0.154028	0.0770140	+	0.997030i	$0.475461\pi$
$278$	0	0
$279$	−16.8264	−1.00737
$280$	0	0
$281$	−22.4531	−1.33944	−0.669720	−	0.742614i	$-0.733586\pi$
$281$	−22.4531	−1.33944	−0.669720	+	0.742614i	$0.733586\pi$
$282$	0	0
$283$	−13.1240	−0.780141	−0.390070	−	0.920785i	$-0.627549\pi$
$283$	−13.1240	−0.780141	−0.390070	+	0.920785i	$0.627549\pi$
$284$	0	0
$285$	1.75312	0.103846
$286$	0	0
$287$	−4.19755	−0.247774
$288$	0	0
$289$	24.7651	1.45677
$290$	0	0
$291$	−24.8699	−1.45790
$292$	0	0
$293$	−32.3935	−1.89245	−0.946225	−	0.323509i	$-0.895137\pi$
$293$	−32.3935	−1.89245	−0.946225	+	0.323509i	$0.895137\pi$
$294$	0	0
$295$	0.657970	0.0383085
$296$	0	0
$297$	−3.05073	−0.177021
$298$	0	0
$299$	−15.3001	−0.884829
$300$	0	0
$301$	0.238999	0.0137757
$302$	0	0
$303$	18.7159	1.07520
$304$	0	0
$305$	2.16009	0.123686
$306$	0	0
$307$	−12.6847	−0.723952	−0.361976	−	0.932187i	$-0.617898\pi$
$307$	−12.6847	−0.723952	−0.361976	+	0.932187i	$0.617898\pi$
$308$	0	0
$309$	−19.6630	−1.11859
$310$	0	0
$311$	−22.1050	−1.25346	−0.626730	−	0.779236i	$-0.715607\pi$
$311$	−22.1050	−1.25346	−0.626730	+	0.779236i	$0.715607\pi$
$312$	0	0
$313$	10.8859	0.615305	0.307653	−	0.951499i	$-0.400457\pi$
$313$	10.8859	0.615305	0.307653	+	0.951499i	$0.400457\pi$
$314$	0	0
$315$	−0.623255	−0.0351164
$316$	0	0
$317$	−7.76634	−0.436201	−0.218101	−	0.975926i	$-0.569986\pi$
$317$	−7.76634	−0.436201	−0.218101	+	0.975926i	$0.569986\pi$
$318$	0	0
$319$	37.0716	2.07561
$320$	0	0
$321$	2.23271	0.124618
$322$	0	0
$323$	−8.46614	−0.471069
$324$	0	0
$325$	17.5620	0.974166
$326$	0	0
$327$	9.84558	0.544462
$328$	0	0
$329$	1.85648	0.102351
$330$	0	0
$331$	−0.317941	−0.0174756	−0.00873781	−	0.999962i	$-0.502781\pi$
$331$	−0.317941	−0.0174756	−0.00873781	+	0.999962i	$0.502781\pi$
$332$	0	0
$333$	17.8146	0.976235
$334$	0	0
$335$	7.49784	0.409651
$336$	0	0
$337$	−10.0901	−0.549641	−0.274821	−	0.961496i	$-0.588618\pi$
$337$	−10.0901	−0.549641	−0.274821	+	0.961496i	$0.588618\pi$
$338$	0	0
$339$	43.6393	2.37016
$340$	0	0
$341$	21.4152	1.15970
$342$	0	0
$343$	4.92515	0.265933
$344$	0	0
$345$	5.49746	0.295973
$346$	0	0
$347$	12.2330	0.656700	0.328350	−	0.944556i	$-0.393508\pi$
$347$	12.2330	0.656700	0.328350	+	0.944556i	$0.393508\pi$
$348$	0	0
$349$	0.0471917	0.00252612	0.00126306	−	0.999999i	$-0.499598\pi$
$349$	0.0471917	0.00252612	0.00126306	+	0.999999i	$0.499598\pi$
$350$	0	0
$351$	−2.71320	−0.144820
$352$	0	0
$353$	9.38275	0.499393	0.249697	−	0.968324i	$-0.419669\pi$
$353$	9.38275	0.499393	0.249697	+	0.968324i	$0.419669\pi$
$354$	0	0
$355$	−6.96444	−0.369634
$356$	0	0
$357$	5.75395	0.304531
$358$	0	0
$359$	−4.83057	−0.254948	−0.127474	−	0.991842i	$-0.540687\pi$
$359$	−4.83057	−0.254948	−0.127474	+	0.991842i	$0.540687\pi$
$360$	0	0
$361$	−17.2838	−0.909676
$362$	0	0
$363$	16.3972	0.860630
$364$	0	0
$365$	4.14539	0.216980
$366$	0	0
$367$	26.6792	1.39264	0.696322	−	0.717730i	$-0.254819\pi$
$367$	26.6792	1.39264	0.696322	+	0.717730i	$0.254819\pi$
$368$	0	0
$369$	38.9075	2.02544
$370$	0	0
$371$	3.52295	0.182903
$372$	0	0
$373$	−1.30441	−0.0675400	−0.0337700	−	0.999430i	$-0.510751\pi$
$373$	−1.30441	−0.0675400	−0.0337700	+	0.999430i	$0.510751\pi$
$374$	0	0
$375$	−13.0014	−0.671387
$376$	0	0
$377$	32.9700	1.69804
$378$	0	0
$379$	3.62474	0.186191	0.0930953	−	0.995657i	$-0.470324\pi$
$379$	3.62474	0.186191	0.0930953	+	0.995657i	$0.470324\pi$
$380$	0	0
$381$	−46.9841	−2.40707
$382$	0	0
$383$	28.5501	1.45884	0.729420	−	0.684066i	$-0.239790\pi$
$383$	28.5501	1.45884	0.729420	+	0.684066i	$0.239790\pi$
$384$	0	0
$385$	0.793226	0.0404265
$386$	0	0
$387$	−2.21531	−0.112610
$388$	0	0
$389$	29.3393	1.48756	0.743780	−	0.668424i	$-0.233031\pi$
$389$	29.3393	1.48756	0.743780	+	0.668424i	$0.233031\pi$
$390$	0	0
$391$	−26.5483	−1.34260
$392$	0	0
$393$	3.52704	0.177916
$394$	0	0
$395$	0.627868	0.0315915
$396$	0	0
$397$	3.94029	0.197758	0.0988788	−	0.995099i	$-0.468474\pi$
$397$	3.94029	0.197758	0.0988788	+	0.995099i	$0.468474\pi$
$398$	0	0
$399$	−1.16638	−0.0583918
$400$	0	0
$401$	0.0336542	0.00168061	0.000840304	−	1.00000i	$-0.499733\pi$
$401$	0.0336542	0.00168061	0.000840304		1.00000i	$0.499733\pi$
$402$	0	0
$403$	19.0459	0.948743
$404$	0	0
$405$	−4.29218	−0.213280
$406$	0	0
$407$	−22.6729	−1.12386
$408$	0	0
$409$	−8.89629	−0.439893	−0.219947	−	0.975512i	$-0.570588\pi$
$409$	−8.89629	−0.439893	−0.219947	+	0.975512i	$0.570588\pi$
$410$	0	0
$411$	13.9192	0.686582
$412$	0	0
$413$	−0.437757	−0.0215406
$414$	0	0
$415$	−3.33640	−0.163777
$416$	0	0
$417$	27.9752	1.36995
$418$	0	0
$419$	29.8619	1.45885	0.729425	−	0.684061i	$-0.239788\pi$
$419$	29.8619	1.45885	0.729425	+	0.684061i	$0.239788\pi$
$420$	0	0
$421$	15.9683	0.778247	0.389124	−	0.921186i	$-0.372778\pi$
$421$	15.9683	0.778247	0.389124	+	0.921186i	$0.372778\pi$
$422$	0	0
$423$	−17.2079	−0.836678
$424$	0	0
$425$	30.4731	1.47816
$426$	0	0
$427$	−1.43714	−0.0695480
$428$	0	0
$429$	39.1195	1.88871
$430$	0	0
$431$	−24.8345	−1.19624	−0.598118	−	0.801408i	$-0.704085\pi$
$431$	−24.8345	−1.19624	−0.598118	+	0.801408i	$0.704085\pi$
$432$	0	0
$433$	−31.1730	−1.49808	−0.749040	−	0.662524i	$-0.769485\pi$
$433$	−31.1730	−1.49808	−0.749040	+	0.662524i	$0.769485\pi$
$434$	0	0
$435$	−11.8464	−0.567992
$436$	0	0
$437$	5.38157	0.257435
$438$	0	0
$439$	0.959384	0.0457889	0.0228944	−	0.999738i	$-0.492712\pi$
$439$	0.959384	0.0457889	0.0228944	+	0.999738i	$0.492712\pi$
$440$	0	0
$441$	−22.6185	−1.07707
$442$	0	0
$443$	−6.52446	−0.309986	−0.154993	−	0.987916i	$-0.549536\pi$
$443$	−6.52446	−0.309986	−0.154993	+	0.987916i	$0.549536\pi$
$444$	0	0
$445$	2.03204	0.0963280
$446$	0	0
$447$	20.7420	0.981065
$448$	0	0
$449$	−9.39811	−0.443524	−0.221762	−	0.975101i	$-0.571181\pi$
$449$	−9.39811	−0.443524	−0.221762	+	0.975101i	$0.571181\pi$
$450$	0	0
$451$	−49.5182	−2.33172
$452$	0	0
$453$	29.8435	1.40217
$454$	0	0
$455$	0.705464	0.0330727
$456$	0	0
$457$	−27.0904	−1.26724	−0.633618	−	0.773646i	$-0.718431\pi$
$457$	−27.0904	−1.26724	−0.633618	+	0.773646i	$0.718431\pi$
$458$	0	0
$459$	−4.70786	−0.219744
$460$	0	0
$461$	−32.1068	−1.49536	−0.747682	−	0.664057i	$-0.768833\pi$
$461$	−32.1068	−1.49536	−0.747682	+	0.664057i	$0.768833\pi$
$462$	0	0
$463$	−17.5754	−0.816797	−0.408399	−	0.912804i	$-0.633913\pi$
$463$	−17.5754	−0.816797	−0.408399	+	0.912804i	$0.633913\pi$
$464$	0	0
$465$	−6.84334	−0.317352
$466$	0	0
$467$	−12.9732	−0.600329	−0.300164	−	0.953887i	$-0.597042\pi$
$467$	−12.9732	−0.600329	−0.300164	+	0.953887i	$0.597042\pi$
$468$	0	0
$469$	−4.98842	−0.230344
$470$	0	0
$471$	−14.7514	−0.679708
$472$	0	0
$473$	2.81945	0.129639
$474$	0	0
$475$	−6.17716	−0.283427
$476$	0	0
$477$	−32.6546	−1.49515
$478$	0	0
$479$	6.68222	0.305319	0.152659	−	0.988279i	$-0.451216\pi$
$479$	6.68222	0.305319	0.152659	+	0.988279i	$0.451216\pi$
$480$	0	0
$481$	−20.1644	−0.919418
$482$	0	0
$483$	−3.65754	−0.166424
$484$	0	0
$485$	−5.29083	−0.240244
$486$	0	0
$487$	−19.6021	−0.888256	−0.444128	−	0.895963i	$-0.646486\pi$
$487$	−19.6021	−0.888256	−0.444128	+	0.895963i	$0.646486\pi$
$488$	0	0
$489$	8.05175	0.364113
$490$	0	0
$491$	6.61946	0.298732	0.149366	−	0.988782i	$-0.452277\pi$
$491$	6.61946	0.298732	0.149366	+	0.988782i	$0.452277\pi$
$492$	0	0
$493$	57.2086	2.57654
$494$	0	0
$495$	−7.35248	−0.330470
$496$	0	0
$497$	4.63355	0.207843
$498$	0	0
$499$	−39.6828	−1.77645	−0.888223	−	0.459413i	$-0.848060\pi$
$499$	−39.6828	−1.77645	−0.888223	+	0.459413i	$0.848060\pi$
$500$	0	0
$501$	24.1405	1.07852
$502$	0	0
$503$	13.9042	0.619960	0.309980	−	0.950743i	$-0.399678\pi$
$503$	13.9042	0.619960	0.309980	+	0.950743i	$0.399678\pi$
$504$	0	0
$505$	3.98163	0.177180
$506$	0	0
$507$	2.18632	0.0970979
$508$	0	0
$509$	33.7260	1.49488	0.747439	−	0.664331i	$-0.231283\pi$
$509$	33.7260	1.49488	0.747439	+	0.664331i	$0.231283\pi$
$510$	0	0
$511$	−2.75799	−0.122006
$512$	0	0
$513$	0.954324	0.0421344
$514$	0	0
$515$	−4.18312	−0.184330
$516$	0	0
$517$	21.9008	0.963196
$518$	0	0
$519$	−29.5143	−1.29553
$520$	0	0
$521$	−37.2520	−1.63204	−0.816020	−	0.578023i	$-0.803824\pi$
$521$	−37.2520	−1.63204	−0.816020	+	0.578023i	$0.803824\pi$
$522$	0	0
$523$	14.7673	0.645730	0.322865	−	0.946445i	$-0.395354\pi$
$523$	14.7673	0.645730	0.322865	+	0.946445i	$0.395354\pi$
$524$	0	0
$525$	4.19826	0.183227
$526$	0	0
$527$	33.0478	1.43958
$528$	0	0
$529$	−6.12439	−0.266278
$530$	0	0
$531$	4.05761	0.176085
$532$	0	0
$533$	−44.0395	−1.90756
$534$	0	0
$535$	0.474987	0.0205355
$536$	0	0
$537$	11.9035	0.513674
$538$	0	0
$539$	28.7869	1.23994
$540$	0	0
$541$	0.345044	0.0148346	0.00741730	−	0.999972i	$-0.497639\pi$
$541$	0.345044	0.0148346	0.00741730	+	0.999972i	$0.497639\pi$
$542$	0	0
$543$	−23.9814	−1.02914
$544$	0	0
$545$	2.09455	0.0897209
$546$	0	0
$547$	6.16271	0.263498	0.131749	−	0.991283i	$-0.457941\pi$
$547$	6.16271	0.263498	0.131749	+	0.991283i	$0.457941\pi$
$548$	0	0
$549$	13.3210	0.568525
$550$	0	0
$551$	−11.5967	−0.494035
$552$	0	0
$553$	−0.417730	−0.0177637
$554$	0	0
$555$	7.24524	0.307543
$556$	0	0
$557$	7.45145	0.315728	0.157864	−	0.987461i	$-0.449539\pi$
$557$	7.45145	0.315728	0.157864	+	0.987461i	$0.449539\pi$
$558$	0	0
$559$	2.50751	0.106056
$560$	0	0
$561$	67.8789	2.86585
$562$	0	0
$563$	14.5393	0.612760	0.306380	−	0.951909i	$-0.400882\pi$
$563$	14.5393	0.612760	0.306380	+	0.951909i	$0.400882\pi$
$564$	0	0
$565$	9.28385	0.390575
$566$	0	0
$567$	2.85565	0.119926
$568$	0	0
$569$	12.1696	0.510175	0.255088	−	0.966918i	$-0.417896\pi$
$569$	12.1696	0.510175	0.255088	+	0.966918i	$0.417896\pi$
$570$	0	0
$571$	16.0026	0.669687	0.334843	−	0.942274i	$-0.391317\pi$
$571$	16.0026	0.669687	0.334843	+	0.942274i	$0.391317\pi$
$572$	0	0
$573$	−58.0398	−2.42465
$574$	0	0
$575$	−19.3704	−0.807803
$576$	0	0
$577$	0.0727829	0.00302999	0.00151500	−	0.999999i	$-0.499518\pi$
$577$	0.0727829	0.00302999	0.00151500	+	0.999999i	$0.499518\pi$
$578$	0	0
$579$	22.9650	0.954395
$580$	0	0
$581$	2.21975	0.0920909
$582$	0	0
$583$	41.5600	1.72124
$584$	0	0
$585$	−6.53901	−0.270355
$586$	0	0
$587$	−0.902377	−0.0372451	−0.0186225	−	0.999827i	$-0.505928\pi$
$587$	−0.902377	−0.0372451	−0.0186225	+	0.999827i	$0.505928\pi$
$588$	0	0
$589$	−6.69907	−0.276031
$590$	0	0
$591$	−16.9307	−0.696434
$592$	0	0
$593$	−23.7787	−0.976476	−0.488238	−	0.872711i	$-0.662360\pi$
$593$	−23.7787	−0.976476	−0.488238	+	0.872711i	$0.662360\pi$
$594$	0	0
$595$	1.22410	0.0501831
$596$	0	0
$597$	−8.18387	−0.334943
$598$	0	0
$599$	−48.7217	−1.99072	−0.995358	−	0.0962439i	$-0.969317\pi$
$599$	−48.7217	−1.99072	−0.995358	+	0.0962439i	$0.969317\pi$
$600$	0	0
$601$	−3.90565	−0.159315	−0.0796573	−	0.996822i	$-0.525383\pi$
$601$	−3.90565	−0.159315	−0.0796573	+	0.996822i	$0.525383\pi$
$602$	0	0
$603$	46.2381	1.88296
$604$	0	0
$605$	3.48835	0.141822
$606$	0	0
$607$	15.1915	0.616605	0.308302	−	0.951288i	$-0.400239\pi$
$607$	15.1915	0.616605	0.308302	+	0.951288i	$0.400239\pi$
$608$	0	0
$609$	7.88159	0.319378
$610$	0	0
$611$	19.4777	0.787984
$612$	0	0
$613$	−29.1406	−1.17698	−0.588490	−	0.808505i	$-0.700277\pi$
$613$	−29.1406	−1.17698	−0.588490	+	0.808505i	$0.700277\pi$
$614$	0	0
$615$	15.8238	0.638076
$616$	0	0
$617$	−31.1143	−1.25261	−0.626307	−	0.779576i	$-0.715435\pi$
$617$	−31.1143	−1.25261	−0.626307	+	0.779576i	$0.715435\pi$
$618$	0	0
$619$	−40.3950	−1.62361	−0.811806	−	0.583927i	$-0.801516\pi$
$619$	−40.3950	−1.62361	−0.811806	+	0.583927i	$0.801516\pi$
$620$	0	0
$621$	2.99259	0.120088
$622$	0	0
$623$	−1.35195	−0.0541646
$624$	0	0
$625$	20.8106	0.832424
$626$	0	0
$627$	−13.7596	−0.549507
$628$	0	0
$629$	−34.9886	−1.39509
$630$	0	0
$631$	−7.99234	−0.318170	−0.159085	−	0.987265i	$-0.550854\pi$
$631$	−7.99234	−0.318170	−0.159085	+	0.987265i	$0.550854\pi$
$632$	0	0
$633$	58.0556	2.30750
$634$	0	0
$635$	−9.99542	−0.396656
$636$	0	0
$637$	25.6020	1.01439
$638$	0	0
$639$	−42.9488	−1.69903
$640$	0	0
$641$	−25.0212	−0.988277	−0.494139	−	0.869383i	$-0.664516\pi$
$641$	−25.0212	−0.988277	−0.494139	+	0.869383i	$0.664516\pi$
$642$	0	0
$643$	9.88861	0.389969	0.194984	−	0.980806i	$-0.437534\pi$
$643$	9.88861	0.389969	0.194984	+	0.980806i	$0.437534\pi$
$644$	0	0
$645$	−0.900970	−0.0354757
$646$	0	0
$647$	−43.3237	−1.70323	−0.851615	−	0.524168i	$-0.824376\pi$
$647$	−43.3237	−1.70323	−0.851615	+	0.524168i	$0.824376\pi$
$648$	0	0
$649$	−5.16419	−0.202712
$650$	0	0
$651$	4.55298	0.178445
$652$	0	0
$653$	−13.6441	−0.533936	−0.266968	−	0.963705i	$-0.586022\pi$
$653$	−13.6441	−0.533936	−0.266968	+	0.963705i	$0.586022\pi$
$654$	0	0
$655$	0.750344	0.0293184
$656$	0	0
$657$	25.5641	0.997349
$658$	0	0
$659$	18.6052	0.724756	0.362378	−	0.932031i	$-0.381965\pi$
$659$	18.6052	0.724756	0.362378	+	0.932031i	$0.381965\pi$
$660$	0	0
$661$	−39.9082	−1.55225	−0.776124	−	0.630580i	$-0.782817\pi$
$661$	−39.9082	−1.55225	−0.776124	+	0.630580i	$0.782817\pi$
$662$	0	0
$663$	60.3688	2.34453
$664$	0	0
$665$	−0.248135	−0.00962228
$666$	0	0
$667$	−36.3650	−1.40806
$668$	0	0
$669$	45.8849	1.77401
$670$	0	0
$671$	−16.9538	−0.654494
$672$	0	0
$673$	−0.931169	−0.0358939	−0.0179470	−	0.999839i	$-0.505713\pi$
$673$	−0.931169	−0.0358939	−0.0179470	+	0.999839i	$0.505713\pi$
$674$	0	0
$675$	−3.43500	−0.132213
$676$	0	0
$677$	35.1023	1.34909	0.674546	−	0.738233i	$-0.264339\pi$
$677$	35.1023	1.34909	0.674546	+	0.738233i	$0.264339\pi$
$678$	0	0
$679$	3.52007	0.135088
$680$	0	0
$681$	−26.2869	−1.00732
$682$	0	0
$683$	17.5204	0.670399	0.335199	−	0.942147i	$-0.391196\pi$
$683$	17.5204	0.670399	0.335199	+	0.942147i	$0.391196\pi$
$684$	0	0
$685$	2.96117	0.113141
$686$	0	0
$687$	−13.8352	−0.527846
$688$	0	0
$689$	36.9618	1.40813
$690$	0	0
$691$	−33.6421	−1.27981	−0.639903	−	0.768455i	$-0.721026\pi$
$691$	−33.6421	−1.27981	−0.639903	+	0.768455i	$0.721026\pi$
$692$	0	0
$693$	4.89172	0.185821
$694$	0	0
$695$	5.95145	0.225751
$696$	0	0
$697$	−76.4160	−2.89446
$698$	0	0
$699$	69.5597	2.63099
$700$	0	0
$701$	−20.5783	−0.777232	−0.388616	−	0.921400i	$-0.627047\pi$
$701$	−20.5783	−0.777232	−0.388616	+	0.921400i	$0.627047\pi$
$702$	0	0
$703$	7.09250	0.267499
$704$	0	0
$705$	−6.99850	−0.263579
$706$	0	0
$707$	−2.64904	−0.0996272
$708$	0	0
$709$	−16.3548	−0.614216	−0.307108	−	0.951675i	$-0.599361\pi$
$709$	−16.3548	−0.614216	−0.307108	+	0.951675i	$0.599361\pi$
$710$	0	0
$711$	3.87198	0.145210
$712$	0	0
$713$	−21.0071	−0.786721
$714$	0	0
$715$	8.32230	0.311236
$716$	0	0
$717$	−16.5675	−0.618726
$718$	0	0
$719$	−21.1731	−0.789624	−0.394812	−	0.918762i	$-0.629190\pi$
$719$	−21.1731	−0.789624	−0.394812	+	0.918762i	$0.629190\pi$
$720$	0	0
$721$	2.78309	0.103648
$722$	0	0
$723$	2.50808	0.0932764
$724$	0	0
$725$	41.7411	1.55023
$726$	0	0
$727$	15.8288	0.587059	0.293530	−	0.955950i	$-0.405170\pi$
$727$	15.8288	0.587059	0.293530	+	0.955950i	$0.405170\pi$
$728$	0	0
$729$	−31.9506	−1.18335
$730$	0	0
$731$	4.35096	0.160926
$732$	0	0
$733$	−8.66500	−0.320049	−0.160025	−	0.987113i	$-0.551157\pi$
$733$	−8.66500	−0.320049	−0.160025	+	0.987113i	$0.551157\pi$
$734$	0	0
$735$	−9.19900	−0.339310
$736$	0	0
$737$	−58.8480	−2.16769
$738$	0	0
$739$	51.3157	1.88768	0.943840	−	0.330403i	$-0.107185\pi$
$739$	51.3157	1.88768	0.943840	+	0.330403i	$0.107185\pi$
$740$	0	0
$741$	−12.2373	−0.449548
$742$	0	0
$743$	−19.9955	−0.733563	−0.366781	−	0.930307i	$-0.619540\pi$
$743$	−19.9955	−0.733563	−0.366781	+	0.930307i	$0.619540\pi$
$744$	0	0
$745$	4.41267	0.161668
$746$	0	0
$747$	−20.5751	−0.752804
$748$	0	0
$749$	−0.316016	−0.0115470
$750$	0	0
$751$	30.4912	1.11264	0.556320	−	0.830968i	$-0.312213\pi$
$751$	30.4912	1.11264	0.556320	+	0.830968i	$0.312213\pi$
$752$	0	0
$753$	−53.3288	−1.94341
$754$	0	0
$755$	6.34892	0.231061
$756$	0	0
$757$	10.7919	0.392238	0.196119	−	0.980580i	$-0.437166\pi$
$757$	10.7919	0.392238	0.196119	+	0.980580i	$0.437166\pi$
$758$	0	0
$759$	−43.1477	−1.56616
$760$	0	0
$761$	14.3354	0.519657	0.259828	−	0.965655i	$-0.416334\pi$
$761$	14.3354	0.519657	0.259828	+	0.965655i	$0.416334\pi$
$762$	0	0
$763$	−1.39354	−0.0504495
$764$	0	0
$765$	−11.3463	−0.410226
$766$	0	0
$767$	−4.59282	−0.165837
$768$	0	0
$769$	20.7301	0.747547	0.373773	−	0.927520i	$-0.378064\pi$
$769$	20.7301	0.747547	0.373773	+	0.927520i	$0.378064\pi$
$770$	0	0
$771$	18.3899	0.662296
$772$	0	0
$773$	40.3752	1.45220	0.726098	−	0.687591i	$-0.241332\pi$
$773$	40.3752	1.45220	0.726098	+	0.687591i	$0.241332\pi$
$774$	0	0
$775$	24.1127	0.866153
$776$	0	0
$777$	−4.82037	−0.172930
$778$	0	0
$779$	15.4902	0.554993
$780$	0	0
$781$	54.6616	1.95594
$782$	0	0
$783$	−6.44869	−0.230457
$784$	0	0
$785$	−3.13822	−0.112008
$786$	0	0
$787$	18.3264	0.653266	0.326633	−	0.945151i	$-0.394086\pi$
$787$	18.3264	0.653266	0.326633	+	0.945151i	$0.394086\pi$
$788$	0	0
$789$	−51.7175	−1.84119
$790$	0	0
$791$	−6.17668	−0.219618
$792$	0	0
$793$	−15.0780	−0.535437
$794$	0	0
$795$	−13.2807	−0.471018
$796$	0	0
$797$	−36.1893	−1.28189	−0.640945	−	0.767587i	$-0.721457\pi$
$797$	−36.1893	−1.28189	−0.640945	+	0.767587i	$0.721457\pi$
$798$	0	0
$799$	33.7971	1.19566
$800$	0	0
$801$	12.5313	0.442773
$802$	0	0
$803$	−32.5358	−1.14816
$804$	0	0
$805$	−0.778107	−0.0274247
$806$	0	0
$807$	−0.448375	−0.0157835
$808$	0	0
$809$	−0.129006	−0.00453561	−0.00226781	−	0.999997i	$-0.500722\pi$
$809$	−0.129006	−0.00453561	−0.00226781	+	0.999997i	$0.500722\pi$
$810$	0	0
$811$	22.3392	0.784434	0.392217	−	0.919873i	$-0.371708\pi$
$811$	22.3392	0.784434	0.392217	+	0.919873i	$0.371708\pi$
$812$	0	0
$813$	44.9725	1.57725
$814$	0	0
$815$	1.71293	0.0600015
$816$	0	0
$817$	−0.881977	−0.0308565
$818$	0	0
$819$	4.35050	0.152019
$820$	0	0
$821$	27.9841	0.976653	0.488327	−	0.872661i	$-0.337607\pi$
$821$	27.9841	0.976653	0.488327	+	0.872661i	$0.337607\pi$
$822$	0	0
$823$	−4.52041	−0.157572	−0.0787859	−	0.996892i	$-0.525104\pi$
$823$	−4.52041	−0.157572	−0.0787859	+	0.996892i	$0.525104\pi$
$824$	0	0
$825$	49.5265	1.72429
$826$	0	0
$827$	10.2432	0.356189	0.178095	−	0.984013i	$-0.443007\pi$
$827$	10.2432	0.356189	0.178095	+	0.984013i	$0.443007\pi$
$828$	0	0
$829$	15.8251	0.549630	0.274815	−	0.961497i	$-0.411383\pi$
$829$	15.8251	0.549630	0.274815	+	0.961497i	$0.411383\pi$
$830$	0	0
$831$	6.42955	0.223039
$832$	0	0
$833$	44.4237	1.53919
$834$	0	0
$835$	5.13567	0.177727
$836$	0	0
$837$	−3.72523	−0.128763
$838$	0	0
$839$	−33.9655	−1.17262	−0.586309	−	0.810087i	$-0.699420\pi$
$839$	−33.9655	−1.17262	−0.586309	+	0.810087i	$0.699420\pi$
$840$	0	0
$841$	49.3626	1.70216
$842$	0	0
$843$	−56.3141	−1.93956
$844$	0	0
$845$	0.465119	0.0160006
$846$	0	0
$847$	−2.32085	−0.0797454
$848$	0	0
$849$	−32.9160	−1.12967
$850$	0	0
$851$	22.2408	0.762405
$852$	0	0
$853$	51.3501	1.75819	0.879097	−	0.476643i	$-0.158147\pi$
$853$	51.3501	1.75819	0.879097	+	0.476643i	$0.158147\pi$
$854$	0	0
$855$	2.29999	0.0786580
$856$	0	0
$857$	−27.2848	−0.932030	−0.466015	−	0.884777i	$-0.654311\pi$
$857$	−27.2848	−0.932030	−0.466015	+	0.884777i	$0.654311\pi$
$858$	0	0
$859$	6.65604	0.227101	0.113551	−	0.993532i	$-0.463778\pi$
$859$	6.65604	0.227101	0.113551	+	0.993532i	$0.463778\pi$
$860$	0	0
$861$	−10.5278	−0.358786
$862$	0	0
$863$	−43.1149	−1.46765	−0.733824	−	0.679340i	$-0.762266\pi$
$863$	−43.1149	−1.46765	−0.733824	+	0.679340i	$0.762266\pi$
$864$	0	0
$865$	−6.27889	−0.213489
$866$	0	0
$867$	62.1127	2.10946
$868$	0	0
$869$	−4.92793	−0.167168
$870$	0	0
$871$	−52.3371	−1.77338
$872$	0	0
$873$	−32.6278	−1.10429
$874$	0	0
$875$	1.84020	0.0622103
$876$	0	0
$877$	30.7736	1.03915	0.519575	−	0.854425i	$-0.326090\pi$
$877$	30.7736	1.03915	0.519575	+	0.854425i	$0.326090\pi$
$878$	0	0
$879$	−81.2455	−2.74034
$880$	0	0
$881$	−35.5730	−1.19849	−0.599243	−	0.800567i	$-0.704532\pi$
$881$	−35.5730	−1.19849	−0.599243	+	0.800567i	$0.704532\pi$
$882$	0	0
$883$	39.0447	1.31396	0.656979	−	0.753909i	$-0.271834\pi$
$883$	39.0447	1.31396	0.656979	+	0.753909i	$0.271834\pi$
$884$	0	0
$885$	1.65024	0.0554722
$886$	0	0
$887$	35.4372	1.18987	0.594933	−	0.803775i	$-0.297179\pi$
$887$	35.4372	1.18987	0.594933	+	0.803775i	$0.297179\pi$
$888$	0	0
$889$	6.65010	0.223037
$890$	0	0
$891$	33.6879	1.12859
$892$	0	0
$893$	−6.85097	−0.229259
$894$	0	0
$895$	2.53236	0.0846473
$896$	0	0
$897$	−38.3739	−1.28127
$898$	0	0
$899$	45.2679	1.50977
$900$	0	0
$901$	64.1350	2.13665
$902$	0	0
$903$	0.599429	0.0199477
$904$	0	0
$905$	−5.10182	−0.169590
$906$	0	0
$907$	6.19074	0.205560	0.102780	−	0.994704i	$-0.467226\pi$
$907$	6.19074	0.205560	0.102780	+	0.994704i	$0.467226\pi$
$908$	0	0
$909$	24.5542	0.814410
$910$	0	0
$911$	−0.700721	−0.0232159	−0.0116080	−	0.999933i	$-0.503695\pi$
$911$	−0.700721	−0.0232159	−0.0116080	+	0.999933i	$0.503695\pi$
$912$	0	0
$913$	26.1863	0.866638
$914$	0	0
$915$	5.41767	0.179103
$916$	0	0
$917$	−0.499215	−0.0164855
$918$	0	0
$919$	29.3520	0.968233	0.484117	−	0.875004i	$-0.339141\pi$
$919$	29.3520	0.968233	0.484117	+	0.875004i	$0.339141\pi$
$920$	0	0
$921$	−31.8141	−1.04831
$922$	0	0
$923$	48.6139	1.60014
$924$	0	0
$925$	−25.5288	−0.839381
$926$	0	0
$927$	−25.7967	−0.847276
$928$	0	0
$929$	52.4498	1.72082	0.860411	−	0.509601i	$-0.170207\pi$
$929$	52.4498	1.72082	0.860411	+	0.509601i	$0.170207\pi$
$930$	0	0
$931$	−9.00507	−0.295129
$932$	0	0
$933$	−55.4411	−1.81506
$934$	0	0
$935$	14.4406	0.472258
$936$	0	0
$937$	−25.6384	−0.837569	−0.418784	−	0.908086i	$-0.637544\pi$
$937$	−25.6384	−0.837569	−0.418784	+	0.908086i	$0.637544\pi$
$938$	0	0
$939$	27.3026	0.890986
$940$	0	0
$941$	−12.3906	−0.403921	−0.201960	−	0.979394i	$-0.564731\pi$
$941$	−12.3906	−0.403921	−0.201960	+	0.979394i	$0.564731\pi$
$942$	0	0
$943$	48.5744	1.58180
$944$	0	0
$945$	−0.137983	−0.00448860
$946$	0	0
$947$	7.84757	0.255012	0.127506	−	0.991838i	$-0.459303\pi$
$947$	7.84757	0.255012	0.127506	+	0.991838i	$0.459303\pi$
$948$	0	0
$949$	−28.9361	−0.939304
$950$	0	0
$951$	−19.4786	−0.631637
$952$	0	0
$953$	−31.9352	−1.03448	−0.517242	−	0.855839i	$-0.673041\pi$
$953$	−31.9352	−1.03448	−0.517242	+	0.855839i	$0.673041\pi$
$954$	0	0
$955$	−12.3474	−0.399553
$956$	0	0
$957$	92.9785	3.00557
$958$	0	0
$959$	−1.97011	−0.0636182
$960$	0	0
$961$	−4.84999	−0.156451
$962$	0	0
$963$	2.92918	0.0943916
$964$	0	0
$965$	4.88560	0.157273
$966$	0	0
$967$	−8.42368	−0.270887	−0.135444	−	0.990785i	$-0.543246\pi$
$967$	−8.42368	−0.270887	−0.135444	+	0.990785i	$0.543246\pi$
$968$	0	0
$969$	−21.2337	−0.682126
$970$	0	0
$971$	44.1565	1.41705	0.708525	−	0.705686i	$-0.249361\pi$
$971$	44.1565	1.41705	0.708525	+	0.705686i	$0.249361\pi$
$972$	0	0
$973$	−3.95959	−0.126939
$974$	0	0
$975$	44.0469	1.41063
$976$	0	0
$977$	−5.28247	−0.169001	−0.0845006	−	0.996423i	$-0.526929\pi$
$977$	−5.28247	−0.169001	−0.0845006	+	0.996423i	$0.526929\pi$
$978$	0	0
$979$	−15.9488	−0.509726
$980$	0	0
$981$	12.9168	0.412403
$982$	0	0
$983$	3.33673	0.106425	0.0532125	−	0.998583i	$-0.483054\pi$
$983$	3.33673	0.106425	0.0532125	+	0.998583i	$0.483054\pi$
$984$	0	0
$985$	−3.60184	−0.114764
$986$	0	0
$987$	4.65621	0.148209
$988$	0	0
$989$	−2.76572	−0.0879447
$990$	0	0
$991$	21.6985	0.689274	0.344637	−	0.938736i	$-0.388002\pi$
$991$	21.6985	0.689274	0.344637	+	0.938736i	$0.388002\pi$
$992$	0	0
$993$	−0.797421	−0.0253054
$994$	0	0
$995$	−1.74104	−0.0551947
$996$	0	0
$997$	−10.6834	−0.338347	−0.169174	−	0.985586i	$-0.554110\pi$
$997$	−10.6834	−0.338347	−0.169174	+	0.985586i	$0.554110\pi$
$998$	0	0
$999$	3.94401	0.124783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.n.1.11		12
4.3	odd	2		241.2.a.b.1.12	✓	12
12.11	even	2		2169.2.a.h.1.1		12
20.19	odd	2		6025.2.a.h.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.12	✓	12	4.3	odd	2
2169.2.a.h.1.1		12	12.11	even	2
3856.2.a.n.1.11		12	1.1	even	1	trivial
6025.2.a.h.1.1		12	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3856 = 2^{4} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3856.a (trivial)

\(f(q)\)	\(=\)	\(q+2.50808 q^{3} +0.533570 q^{5} -0.354992 q^{7} +3.29045 q^{9} +O(q^{10})\)	\(q+2.50808 q^{3} +0.533570 q^{5} -0.354992 q^{7} +3.29045 q^{9} -4.18781 q^{11} -3.72447 q^{13} +1.33823 q^{15} -6.46259 q^{17} +1.31002 q^{19} -0.890347 q^{21} +4.10799 q^{23} -4.71530 q^{25} +0.728479 q^{27} -8.85227 q^{29} -5.11371 q^{31} -10.5034 q^{33} -0.189413 q^{35} +5.41403 q^{37} -9.34127 q^{39} +11.8244 q^{41} -0.673253 q^{43} +1.75569 q^{45} -5.22965 q^{47} -6.87398 q^{49} -16.2087 q^{51} -9.92404 q^{53} -2.23449 q^{55} +3.28564 q^{57} +1.23315 q^{59} +4.04837 q^{61} -1.16808 q^{63} -1.98727 q^{65} +14.0522 q^{67} +10.3032 q^{69} -13.0525 q^{71} +7.76916 q^{73} -11.8263 q^{75} +1.48664 q^{77} +1.17673 q^{79} -8.04428 q^{81} -6.25297 q^{83} -3.44824 q^{85} -22.2022 q^{87} +3.80839 q^{89} +1.32216 q^{91} -12.8256 q^{93} +0.698989 q^{95} -9.91591 q^{97} -13.7798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * q^99

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