LMFDB - Embedded newform 3856.2.a.j.1.2

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:	$0$
Dimension:	$7$
Coefficient field:	7.7.31056073.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 $ x^7 - 3x^6 - 3x^5 + 11x^4 + x^3 - 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.2
Root			$-1.60363$ 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)

$f(q)$	$=$	$q-0.980039 q^{3} -1.69135 q^{5} +1.30586 q^{7} -2.03952 q^{9} +O(q^{10})$	$q-0.980039 q^{3} -1.69135 q^{5} +1.30586 q^{7} -2.03952 q^{9} +3.27094 q^{11} +4.30649 q^{13} +1.65759 q^{15} -1.02456 q^{17} +7.01250 q^{19} -1.27980 q^{21} -0.835873 q^{23} -2.13934 q^{25} +4.93893 q^{27} -1.11761 q^{29} +3.97344 q^{31} -3.20565 q^{33} -2.20867 q^{35} -11.3098 q^{37} -4.22053 q^{39} +1.22869 q^{41} -10.8406 q^{43} +3.44955 q^{45} +0.151820 q^{47} -5.29473 q^{49} +1.00411 q^{51} +3.02053 q^{53} -5.53231 q^{55} -6.87252 q^{57} +4.15373 q^{59} +5.62714 q^{61} -2.66334 q^{63} -7.28378 q^{65} -12.9934 q^{67} +0.819188 q^{69} +11.2862 q^{71} +11.7148 q^{73} +2.09663 q^{75} +4.27140 q^{77} +1.66517 q^{79} +1.27823 q^{81} +2.34322 q^{83} +1.73290 q^{85} +1.09530 q^{87} +18.1099 q^{89} +5.62368 q^{91} -3.89413 q^{93} -11.8606 q^{95} -7.17873 q^{97} -6.67116 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 - 8 * q^5 + 7 * q^7 - 2 * q^9	$ 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9} + 18 q^{11} - q^{13} + 11 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{21} + 22 q^{23} + 5 q^{25} - 3 q^{27} - 16 q^{29} + 18 q^{31} + 4 q^{33} - 7 q^{35} + 8 q^{37} + 9 q^{39} - 15 q^{41} - 14 q^{43} + 3 q^{45} + 10 q^{47} + 6 q^{49} - 13 q^{51} + 15 q^{53} - 29 q^{55} + 14 q^{57} + 18 q^{59} + 4 q^{61} + 16 q^{63} - 7 q^{65} - 18 q^{67} + 26 q^{69} + 50 q^{71} - 16 q^{75} + 17 q^{77} + 15 q^{79} - 9 q^{81} + 24 q^{83} - 2 q^{85} - 12 q^{87} - 13 q^{89} + 12 q^{91} + 14 q^{93} + 41 q^{95} + q^{97} + 20 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 8 * q^5 + 7 * q^7 - 2 * q^9 + 18 * q^11 - q^13 + 11 * q^15 - 2 * q^17 + 6 * q^19 - 2 * q^21 + 22 * q^23 + 5 * q^25 - 3 * q^27 - 16 * q^29 + 18 * q^31 + 4 * q^33 - 7 * q^35 + 8 * q^37 + 9 * q^39 - 15 * q^41 - 14 * q^43 + 3 * q^45 + 10 * q^47 + 6 * q^49 - 13 * q^51 + 15 * q^53 - 29 * q^55 + 14 * q^57 + 18 * q^59 + 4 * q^61 + 16 * q^63 - 7 * q^65 - 18 * q^67 + 26 * q^69 + 50 * q^71 - 16 * q^75 + 17 * q^77 + 15 * q^79 - 9 * q^81 + 24 * q^83 - 2 * q^85 - 12 * q^87 - 13 * q^89 + 12 * q^91 + 14 * q^93 + 41 * q^95 + q^97 + 20 * 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
$3$	−0.980039	−0.565826	−0.282913	−	0.959146i	$-0.591301\pi$
$3$	−0.980039	−0.565826	−0.282913	+	0.959146i	$0.591301\pi$
$4$	0	0
$5$	−1.69135	−0.756395	−0.378197	−	0.925725i	$-0.623456\pi$
$5$	−1.69135	−0.756395	−0.378197	+	0.925725i	$0.623456\pi$
$6$	0	0
$7$	1.30586	0.493569	0.246785	−	0.969070i	$-0.420626\pi$
$7$	1.30586	0.493569	0.246785	+	0.969070i	$0.420626\pi$
$8$	0	0
$9$	−2.03952	−0.679841
$10$	0	0
$11$	3.27094	0.986226	0.493113	−	0.869965i	$-0.335859\pi$
$11$	3.27094	0.986226	0.493113	+	0.869965i	$0.335859\pi$
$12$	0	0
$13$	4.30649	1.19441	0.597203	−	0.802090i	$-0.296279\pi$
$13$	4.30649	1.19441	0.597203	+	0.802090i	$0.296279\pi$
$14$	0	0
$15$	1.65759	0.427988
$16$	0	0
$17$	−1.02456	−0.248493	−0.124247	−	0.992251i	$-0.539651\pi$
$17$	−1.02456	−0.248493	−0.124247	+	0.992251i	$0.539651\pi$
$18$	0	0
$19$	7.01250	1.60878	0.804388	−	0.594104i	$-0.202493\pi$
$19$	7.01250	1.60878	0.804388	+	0.594104i	$0.202493\pi$
$20$	0	0
$21$	−1.27980	−0.279274
$22$	0	0
$23$	−0.835873	−0.174292	−0.0871458	−	0.996196i	$-0.527775\pi$
$23$	−0.835873	−0.174292	−0.0871458	+	0.996196i	$0.527775\pi$
$24$	0	0
$25$	−2.13934	−0.427867
$26$	0	0
$27$	4.93893	0.950498
$28$	0	0
$29$	−1.11761	−0.207535	−0.103767	−	0.994602i	$-0.533090\pi$
$29$	−1.11761	−0.207535	−0.103767	+	0.994602i	$0.533090\pi$
$30$	0	0
$31$	3.97344	0.713652	0.356826	−	0.934171i	$-0.383859\pi$
$31$	3.97344	0.713652	0.356826	+	0.934171i	$0.383859\pi$
$32$	0	0
$33$	−3.20565	−0.558032
$34$	0	0
$35$	−2.20867	−0.373333
$36$	0	0
$37$	−11.3098	−1.85932	−0.929658	−	0.368423i	$-0.879898\pi$
$37$	−11.3098	−1.85932	−0.929658	+	0.368423i	$0.879898\pi$
$38$	0	0
$39$	−4.22053	−0.675825
$40$	0	0
$41$	1.22869	0.191890	0.0959448	−	0.995387i	$-0.469413\pi$
$41$	1.22869	0.191890	0.0959448	+	0.995387i	$0.469413\pi$
$42$	0	0
$43$	−10.8406	−1.65318	−0.826591	−	0.562804i	$-0.809723\pi$
$43$	−10.8406	−1.65318	−0.826591	+	0.562804i	$0.809723\pi$
$44$	0	0
$45$	3.44955	0.514228
$46$	0	0
$47$	0.151820	0.0221453	0.0110726	−	0.999939i	$-0.496475\pi$
$47$	0.151820	0.0221453	0.0110726	+	0.999939i	$0.496475\pi$
$48$	0	0
$49$	−5.29473	−0.756389
$50$	0	0
$51$	1.00411	0.140604
$52$	0	0
$53$	3.02053	0.414902	0.207451	−	0.978245i	$-0.433483\pi$
$53$	3.02053	0.414902	0.207451	+	0.978245i	$0.433483\pi$
$54$	0	0
$55$	−5.53231	−0.745976
$56$	0	0
$57$	−6.87252	−0.910287
$58$	0	0
$59$	4.15373	0.540769	0.270385	−	0.962752i	$-0.412849\pi$
$59$	4.15373	0.540769	0.270385	+	0.962752i	$0.412849\pi$
$60$	0	0
$61$	5.62714	0.720482	0.360241	−	0.932859i	$-0.382694\pi$
$61$	5.62714	0.720482	0.360241	+	0.932859i	$0.382694\pi$
$62$	0	0
$63$	−2.66334	−0.335549
$64$	0	0
$65$	−7.28378	−0.903442
$66$	0	0
$67$	−12.9934	−1.58740	−0.793700	−	0.608309i	$-0.791848\pi$
$67$	−12.9934	−1.58740	−0.793700	+	0.608309i	$0.791848\pi$
$68$	0	0
$69$	0.819188	0.0986187
$70$	0	0
$71$	11.2862	1.33942	0.669711	−	0.742622i	$-0.266418\pi$
$71$	11.2862	1.33942	0.669711	+	0.742622i	$0.266418\pi$
$72$	0	0
$73$	11.7148	1.37112	0.685560	−	0.728017i	$-0.259558\pi$
$73$	11.7148	1.37112	0.685560	+	0.728017i	$0.259558\pi$
$74$	0	0
$75$	2.09663	0.242098
$76$	0	0
$77$	4.27140	0.486771
$78$	0	0
$79$	1.66517	0.187346	0.0936732	−	0.995603i	$-0.470139\pi$
$79$	1.66517	0.187346	0.0936732	+	0.995603i	$0.470139\pi$
$80$	0	0
$81$	1.27823	0.142025
$82$	0	0
$83$	2.34322	0.257201	0.128601	−	0.991696i	$-0.458951\pi$
$83$	2.34322	0.257201	0.128601	+	0.991696i	$0.458951\pi$
$84$	0	0
$85$	1.73290	0.187959
$86$	0	0
$87$	1.09530	0.117429
$88$	0	0
$89$	18.1099	1.91965	0.959825	−	0.280598i	$-0.0905329\pi$
$89$	18.1099	1.91965	0.959825	+	0.280598i	$0.0905329\pi$
$90$	0	0
$91$	5.62368	0.589522
$92$	0	0
$93$	−3.89413	−0.403802
$94$	0	0
$95$	−11.8606	−1.21687
$96$	0	0
$97$	−7.17873	−0.728890	−0.364445	−	0.931225i	$-0.618741\pi$
$97$	−7.17873	−0.728890	−0.364445	+	0.931225i	$0.618741\pi$
$98$	0	0
$99$	−6.67116	−0.670477
$100$	0	0
$101$	−5.48113	−0.545393	−0.272696	−	0.962100i	$-0.587915\pi$
$101$	−5.48113	−0.545393	−0.272696	+	0.962100i	$0.587915\pi$
$102$	0	0
$103$	15.7371	1.55062	0.775311	−	0.631580i	$-0.217593\pi$
$103$	15.7371	1.55062	0.775311	+	0.631580i	$0.217593\pi$
$104$	0	0
$105$	2.16458	0.211242
$106$	0	0
$107$	11.1816	1.08097	0.540483	−	0.841355i	$-0.318242\pi$
$107$	11.1816	1.08097	0.540483	+	0.841355i	$0.318242\pi$
$108$	0	0
$109$	−0.296424	−0.0283922	−0.0141961	−	0.999899i	$-0.504519\pi$
$109$	−0.296424	−0.0283922	−0.0141961	+	0.999899i	$0.504519\pi$
$110$	0	0
$111$	11.0840	1.05205
$112$	0	0
$113$	10.5881	0.996045	0.498023	−	0.867164i	$-0.334060\pi$
$113$	10.5881	0.996045	0.498023	+	0.867164i	$0.334060\pi$
$114$	0	0
$115$	1.41375	0.131833
$116$	0	0
$117$	−8.78319	−0.812006
$118$	0	0
$119$	−1.33794	−0.122649
$120$	0	0
$121$	−0.300940	−0.0273582
$122$	0	0
$123$	−1.20417	−0.108576
$124$	0	0
$125$	12.0751	1.08003
$126$	0	0
$127$	−14.6989	−1.30432	−0.652160	−	0.758081i	$-0.726137\pi$
$127$	−14.6989	−1.30432	−0.652160	+	0.758081i	$0.726137\pi$
$128$	0	0
$129$	10.6242	0.935413
$130$	0	0
$131$	−9.63853	−0.842122	−0.421061	−	0.907032i	$-0.638342\pi$
$131$	−9.63853	−0.842122	−0.421061	+	0.907032i	$0.638342\pi$
$132$	0	0
$133$	9.15735	0.794043
$134$	0	0
$135$	−8.35346	−0.718951
$136$	0	0
$137$	23.2376	1.98532	0.992661	−	0.120933i	$-0.0385886\pi$
$137$	23.2376	1.98532	0.992661	+	0.120933i	$0.0385886\pi$
$138$	0	0
$139$	1.29376	0.109736	0.0548678	−	0.998494i	$-0.482526\pi$
$139$	1.29376	0.109736	0.0548678	+	0.998494i	$0.482526\pi$
$140$	0	0
$141$	−0.148790	−0.0125304
$142$	0	0
$143$	14.0863	1.17795
$144$	0	0
$145$	1.89027	0.156978
$146$	0	0
$147$	5.18904	0.427985
$148$	0	0
$149$	17.5316	1.43625	0.718123	−	0.695916i	$-0.245002\pi$
$149$	17.5316	1.43625	0.718123	+	0.695916i	$0.245002\pi$
$150$	0	0
$151$	−5.26990	−0.428858	−0.214429	−	0.976740i	$-0.568789\pi$
$151$	−5.26990	−0.428858	−0.214429	+	0.976740i	$0.568789\pi$
$152$	0	0
$153$	2.08962	0.168936
$154$	0	0
$155$	−6.72048	−0.539802
$156$	0	0
$157$	−14.2165	−1.13460	−0.567299	−	0.823512i	$-0.692012\pi$
$157$	−14.2165	−1.13460	−0.567299	+	0.823512i	$0.692012\pi$
$158$	0	0
$159$	−2.96024	−0.234762
$160$	0	0
$161$	−1.09153	−0.0860250
$162$	0	0
$163$	15.9688	1.25077	0.625386	−	0.780315i	$-0.284941\pi$
$163$	15.9688	1.25077	0.625386	+	0.780315i	$0.284941\pi$
$164$	0	0
$165$	5.42188	0.422093
$166$	0	0
$167$	21.5275	1.66585	0.832925	−	0.553386i	$-0.186665\pi$
$167$	21.5275	1.66585	0.832925	+	0.553386i	$0.186665\pi$
$168$	0	0
$169$	5.54585	0.426604
$170$	0	0
$171$	−14.3021	−1.09371
$172$	0	0
$173$	−5.82531	−0.442890	−0.221445	−	0.975173i	$-0.571077\pi$
$173$	−5.82531	−0.442890	−0.221445	+	0.975173i	$0.571077\pi$
$174$	0	0
$175$	−2.79368	−0.211182
$176$	0	0
$177$	−4.07081	−0.305981
$178$	0	0
$179$	6.03932	0.451400	0.225700	−	0.974197i	$-0.427533\pi$
$179$	6.03932	0.451400	0.225700	+	0.974197i	$0.427533\pi$
$180$	0	0
$181$	−4.03706	−0.300073	−0.150036	−	0.988680i	$-0.547939\pi$
$181$	−4.03706	−0.300073	−0.150036	+	0.988680i	$0.547939\pi$
$182$	0	0
$183$	−5.51482	−0.407667
$184$	0	0
$185$	19.1288	1.40638
$186$	0	0
$187$	−3.35129	−0.245071
$188$	0	0
$189$	6.44956	0.469136
$190$	0	0
$191$	16.4803	1.19247	0.596235	−	0.802810i	$-0.296663\pi$
$191$	16.4803	1.19247	0.596235	+	0.802810i	$0.296663\pi$
$192$	0	0
$193$	−23.0631	−1.66012	−0.830060	−	0.557674i	$-0.811694\pi$
$193$	−23.0631	−1.66012	−0.830060	+	0.557674i	$0.811694\pi$
$194$	0	0
$195$	7.13839	0.511191
$196$	0	0
$197$	−13.5540	−0.965680	−0.482840	−	0.875709i	$-0.660395\pi$
$197$	−13.5540	−0.965680	−0.482840	+	0.875709i	$0.660395\pi$
$198$	0	0
$199$	25.6725	1.81988	0.909938	−	0.414745i	$-0.136129\pi$
$199$	25.6725	1.81988	0.909938	+	0.414745i	$0.136129\pi$
$200$	0	0
$201$	12.7341	0.898192
$202$	0	0
$203$	−1.45944	−0.102433
$204$	0	0
$205$	−2.07815	−0.145144
$206$	0	0
$207$	1.70478	0.118491
$208$	0	0
$209$	22.9375	1.58662
$210$	0	0
$211$	−5.18301	−0.356813	−0.178407	−	0.983957i	$-0.557094\pi$
$211$	−5.18301	−0.356813	−0.178407	+	0.983957i	$0.557094\pi$
$212$	0	0
$213$	−11.0609	−0.757879
$214$	0	0
$215$	18.3353	1.25046
$216$	0	0
$217$	5.18877	0.352237
$218$	0	0
$219$	−11.4810	−0.775815
$220$	0	0
$221$	−4.41227	−0.296802
$222$	0	0
$223$	6.50914	0.435884	0.217942	−	0.975962i	$-0.430066\pi$
$223$	6.50914	0.435884	0.217942	+	0.975962i	$0.430066\pi$
$224$	0	0
$225$	4.36323	0.290882
$226$	0	0
$227$	−11.0460	−0.733150	−0.366575	−	0.930389i	$-0.619470\pi$
$227$	−11.0460	−0.733150	−0.366575	+	0.930389i	$0.619470\pi$
$228$	0	0
$229$	8.95679	0.591881	0.295941	−	0.955206i	$-0.404367\pi$
$229$	8.95679	0.591881	0.295941	+	0.955206i	$0.404367\pi$
$230$	0	0
$231$	−4.18614	−0.275428
$232$	0	0
$233$	−6.46647	−0.423632	−0.211816	−	0.977310i	$-0.567938\pi$
$233$	−6.46647	−0.423632	−0.211816	+	0.977310i	$0.567938\pi$
$234$	0	0
$235$	−0.256781	−0.0167506
$236$	0	0
$237$	−1.63193	−0.106005
$238$	0	0
$239$	25.0588	1.62092	0.810460	−	0.585794i	$-0.199217\pi$
$239$	25.0588	1.62092	0.810460	+	0.585794i	$0.199217\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	0	0
$243$	−16.0695	−1.03086
$244$	0	0
$245$	8.95523	0.572129
$246$	0	0
$247$	30.1992	1.92153
$248$	0	0
$249$	−2.29644	−0.145531
$250$	0	0
$251$	12.5611	0.792847	0.396424	−	0.918068i	$-0.370251\pi$
$251$	12.5611	0.792847	0.396424	+	0.918068i	$0.370251\pi$
$252$	0	0
$253$	−2.73409	−0.171891
$254$	0	0
$255$	−1.69831	−0.106352
$256$	0	0
$257$	16.0367	1.00034	0.500171	−	0.865927i	$-0.333270\pi$
$257$	16.0367	1.00034	0.500171	+	0.865927i	$0.333270\pi$
$258$	0	0
$259$	−14.7690	−0.917702
$260$	0	0
$261$	2.27939	0.141091
$262$	0	0
$263$	−3.33210	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$263$	−3.33210	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$264$	0	0
$265$	−5.10877	−0.313829
$266$	0	0
$267$	−17.7485	−1.08619
$268$	0	0
$269$	−7.44101	−0.453686	−0.226843	−	0.973931i	$-0.572840\pi$
$269$	−7.44101	−0.453686	−0.226843	+	0.973931i	$0.572840\pi$
$270$	0	0
$271$	20.6031	1.25155	0.625774	−	0.780005i	$-0.284783\pi$
$271$	20.6031	1.25155	0.625774	+	0.780005i	$0.284783\pi$
$272$	0	0
$273$	−5.51143	−0.333567
$274$	0	0
$275$	−6.99764	−0.421974
$276$	0	0
$277$	5.17667	0.311036	0.155518	−	0.987833i	$-0.450295\pi$
$277$	5.17667	0.311036	0.155518	+	0.987833i	$0.450295\pi$
$278$	0	0
$279$	−8.10393	−0.485170
$280$	0	0
$281$	−14.1821	−0.846031	−0.423015	−	0.906123i	$-0.639028\pi$
$281$	−14.1821	−0.846031	−0.423015	+	0.906123i	$0.639028\pi$
$282$	0	0
$283$	7.05196	0.419196	0.209598	−	0.977788i	$-0.432785\pi$
$283$	7.05196	0.419196	0.209598	+	0.977788i	$0.432785\pi$
$284$	0	0
$285$	11.6238	0.688537
$286$	0	0
$287$	1.60450	0.0947109
$288$	0	0
$289$	−15.9503	−0.938251
$290$	0	0
$291$	7.03544	0.412425
$292$	0	0
$293$	−2.78966	−0.162974	−0.0814869	−	0.996674i	$-0.525967\pi$
$293$	−2.78966	−0.162974	−0.0814869	+	0.996674i	$0.525967\pi$
$294$	0	0
$295$	−7.02540	−0.409035
$296$	0	0
$297$	16.1550	0.937405
$298$	0	0
$299$	−3.59968	−0.208175
$300$	0	0
$301$	−14.1564	−0.815960
$302$	0	0
$303$	5.37172	0.308597
$304$	0	0
$305$	−9.51747	−0.544969
$306$	0	0
$307$	−19.8285	−1.13167	−0.565837	−	0.824517i	$-0.691447\pi$
$307$	−19.8285	−1.13167	−0.565837	+	0.824517i	$0.691447\pi$
$308$	0	0
$309$	−15.4230	−0.877382
$310$	0	0
$311$	−33.5109	−1.90023	−0.950114	−	0.311902i	$-0.899034\pi$
$311$	−33.5109	−1.90023	−0.950114	+	0.311902i	$0.899034\pi$
$312$	0	0
$313$	19.4152	1.09741	0.548705	−	0.836016i	$-0.315121\pi$
$313$	19.4152	1.09741	0.548705	+	0.836016i	$0.315121\pi$
$314$	0	0
$315$	4.50463	0.253807
$316$	0	0
$317$	−11.7255	−0.658569	−0.329284	−	0.944231i	$-0.606807\pi$
$317$	−11.7255	−0.658569	−0.329284	+	0.944231i	$0.606807\pi$
$318$	0	0
$319$	−3.65564	−0.204676
$320$	0	0
$321$	−10.9584	−0.611638
$322$	0	0
$323$	−7.18475	−0.399770
$324$	0	0
$325$	−9.21303	−0.511047
$326$	0	0
$327$	0.290507	0.0160651
$328$	0	0
$329$	0.198256	0.0109302
$330$	0	0
$331$	−4.93072	−0.271017	−0.135509	−	0.990776i	$-0.543267\pi$
$331$	−4.93072	−0.271017	−0.135509	+	0.990776i	$0.543267\pi$
$332$	0	0
$333$	23.0666	1.26404
$334$	0	0
$335$	21.9764	1.20070
$336$	0	0
$337$	34.4425	1.87620	0.938100	−	0.346364i	$-0.112584\pi$
$337$	34.4425	1.87620	0.938100	+	0.346364i	$0.112584\pi$
$338$	0	0
$339$	−10.3768	−0.563588
$340$	0	0
$341$	12.9969	0.703822
$342$	0	0
$343$	−16.0552	−0.866900
$344$	0	0
$345$	−1.38553	−0.0745946
$346$	0	0
$347$	23.4553	1.25915	0.629574	−	0.776940i	$-0.283229\pi$
$347$	23.4553	1.25915	0.629574	+	0.776940i	$0.283229\pi$
$348$	0	0
$349$	15.8727	0.849648	0.424824	−	0.905276i	$-0.360336\pi$
$349$	15.8727	0.849648	0.424824	+	0.905276i	$0.360336\pi$
$350$	0	0
$351$	21.2694	1.13528
$352$	0	0
$353$	−28.4598	−1.51476	−0.757382	−	0.652972i	$-0.773522\pi$
$353$	−28.4598	−1.51476	−0.757382	+	0.652972i	$0.773522\pi$
$354$	0	0
$355$	−19.0889	−1.01313
$356$	0	0
$357$	1.31123	0.0693978
$358$	0	0
$359$	0.772338	0.0407625	0.0203812	−	0.999792i	$-0.493512\pi$
$359$	0.772338	0.0407625	0.0203812	+	0.999792i	$0.493512\pi$
$360$	0	0
$361$	30.1751	1.58816
$362$	0	0
$363$	0.294933	0.0154800
$364$	0	0
$365$	−19.8139	−1.03711
$366$	0	0
$367$	23.3631	1.21954	0.609772	−	0.792577i	$-0.291261\pi$
$367$	23.3631	1.21954	0.609772	+	0.792577i	$0.291261\pi$
$368$	0	0
$369$	−2.50595	−0.130454
$370$	0	0
$371$	3.94439	0.204783
$372$	0	0
$373$	27.7831	1.43855	0.719277	−	0.694724i	$-0.244473\pi$
$373$	27.7831	1.43855	0.719277	+	0.694724i	$0.244473\pi$
$374$	0	0
$375$	−11.8341	−0.611109
$376$	0	0
$377$	−4.81297	−0.247881
$378$	0	0
$379$	−7.33328	−0.376685	−0.188343	−	0.982103i	$-0.560312\pi$
$379$	−7.33328	−0.376685	−0.188343	+	0.982103i	$0.560312\pi$
$380$	0	0
$381$	14.4055	0.738018
$382$	0	0
$383$	33.4931	1.71142	0.855708	−	0.517459i	$-0.173122\pi$
$383$	33.4931	1.71142	0.855708	+	0.517459i	$0.173122\pi$
$384$	0	0
$385$	−7.22443	−0.368191
$386$	0	0
$387$	22.1097	1.12390
$388$	0	0
$389$	15.6882	0.795425	0.397713	−	0.917510i	$-0.369804\pi$
$389$	15.6882	0.795425	0.397713	+	0.917510i	$0.369804\pi$
$390$	0	0
$391$	0.856405	0.0433103
$392$	0	0
$393$	9.44613	0.476494
$394$	0	0
$395$	−2.81639	−0.141708
$396$	0	0
$397$	8.34483	0.418815	0.209408	−	0.977828i	$-0.432846\pi$
$397$	8.34483	0.418815	0.209408	+	0.977828i	$0.432846\pi$
$398$	0	0
$399$	−8.97456	−0.449290
$400$	0	0
$401$	−38.5766	−1.92642	−0.963211	−	0.268747i	$-0.913391\pi$
$401$	−38.5766	−1.92642	−0.963211	+	0.268747i	$0.913391\pi$
$402$	0	0
$403$	17.1116	0.852389
$404$	0	0
$405$	−2.16193	−0.107427
$406$	0	0
$407$	−36.9936	−1.83371
$408$	0	0
$409$	−19.5415	−0.966266	−0.483133	−	0.875547i	$-0.660501\pi$
$409$	−19.5415	−0.966266	−0.483133	+	0.875547i	$0.660501\pi$
$410$	0	0
$411$	−22.7737	−1.12335
$412$	0	0
$413$	5.42419	0.266907
$414$	0	0
$415$	−3.96320	−0.194546
$416$	0	0
$417$	−1.26794	−0.0620912
$418$	0	0
$419$	29.7748	1.45459	0.727297	−	0.686323i	$-0.240776\pi$
$419$	29.7748	1.45459	0.727297	+	0.686323i	$0.240776\pi$
$420$	0	0
$421$	−4.78920	−0.233411	−0.116706	−	0.993167i	$-0.537233\pi$
$421$	−4.78920	−0.233411	−0.116706	+	0.993167i	$0.537233\pi$
$422$	0	0
$423$	−0.309641	−0.0150553
$424$	0	0
$425$	2.19189	0.106322
$426$	0	0
$427$	7.34827	0.355608
$428$	0	0
$429$	−13.8051	−0.666517
$430$	0	0
$431$	−5.35939	−0.258153	−0.129076	−	0.991635i	$-0.541201\pi$
$431$	−5.35939	−0.258153	−0.129076	+	0.991635i	$0.541201\pi$
$432$	0	0
$433$	21.5028	1.03336	0.516680	−	0.856179i	$-0.327168\pi$
$433$	21.5028	1.03336	0.516680	+	0.856179i	$0.327168\pi$
$434$	0	0
$435$	−1.85254	−0.0888224
$436$	0	0
$437$	−5.86156	−0.280396
$438$	0	0
$439$	−4.22306	−0.201556	−0.100778	−	0.994909i	$-0.532133\pi$
$439$	−4.22306	−0.201556	−0.100778	+	0.994909i	$0.532133\pi$
$440$	0	0
$441$	10.7987	0.514225
$442$	0	0
$443$	−8.34030	−0.396260	−0.198130	−	0.980176i	$-0.563487\pi$
$443$	−8.34030	−0.396260	−0.198130	+	0.980176i	$0.563487\pi$
$444$	0	0
$445$	−30.6303	−1.45201
$446$	0	0
$447$	−17.1817	−0.812665
$448$	0	0
$449$	15.7796	0.744684	0.372342	−	0.928096i	$-0.378555\pi$
$449$	15.7796	0.744684	0.372342	+	0.928096i	$0.378555\pi$
$450$	0	0
$451$	4.01898	0.189247
$452$	0	0
$453$	5.16470	0.242659
$454$	0	0
$455$	−9.51161	−0.445911
$456$	0	0
$457$	−8.71419	−0.407633	−0.203816	−	0.979009i	$-0.565335\pi$
$457$	−8.71419	−0.407633	−0.203816	+	0.979009i	$0.565335\pi$
$458$	0	0
$459$	−5.06025	−0.236192
$460$	0	0
$461$	24.9140	1.16036	0.580181	−	0.814487i	$-0.302982\pi$
$461$	24.9140	1.16036	0.580181	+	0.814487i	$0.302982\pi$
$462$	0	0
$463$	2.87272	0.133507	0.0667534	−	0.997770i	$-0.478736\pi$
$463$	2.87272	0.133507	0.0667534	+	0.997770i	$0.478736\pi$
$464$	0	0
$465$	6.58634	0.305434
$466$	0	0
$467$	−4.87474	−0.225576	−0.112788	−	0.993619i	$-0.535978\pi$
$467$	−4.87474	−0.225576	−0.112788	+	0.993619i	$0.535978\pi$
$468$	0	0
$469$	−16.9676	−0.783492
$470$	0	0
$471$	13.9327	0.641985
$472$	0	0
$473$	−35.4591	−1.63041
$474$	0	0
$475$	−15.0021	−0.688343
$476$	0	0
$477$	−6.16044	−0.282067
$478$	0	0
$479$	−0.194055	−0.00886661	−0.00443331	−	0.999990i	$-0.501411\pi$
$479$	−0.194055	−0.00886661	−0.00443331	+	0.999990i	$0.501411\pi$
$480$	0	0
$481$	−48.7055	−2.22078
$482$	0	0
$483$	1.06975	0.0486752
$484$	0	0
$485$	12.1417	0.551328
$486$	0	0
$487$	30.7592	1.39383	0.696915	−	0.717154i	$-0.254555\pi$
$487$	30.7592	1.39383	0.696915	+	0.717154i	$0.254555\pi$
$488$	0	0
$489$	−15.6500	−0.707719
$490$	0	0
$491$	21.5528	0.972663	0.486332	−	0.873774i	$-0.338335\pi$
$491$	21.5528	0.972663	0.486332	+	0.873774i	$0.338335\pi$
$492$	0	0
$493$	1.14506	0.0515710
$494$	0	0
$495$	11.2833	0.507145
$496$	0	0
$497$	14.7382	0.661097
$498$	0	0
$499$	−21.8992	−0.980341	−0.490170	−	0.871627i	$-0.663065\pi$
$499$	−21.8992	−0.980341	−0.490170	+	0.871627i	$0.663065\pi$
$500$	0	0
$501$	−21.0978	−0.942581
$502$	0	0
$503$	−4.44114	−0.198021	−0.0990104	−	0.995086i	$-0.531568\pi$
$503$	−4.44114	−0.198021	−0.0990104	+	0.995086i	$0.531568\pi$
$504$	0	0
$505$	9.27050	0.412532
$506$	0	0
$507$	−5.43515	−0.241383
$508$	0	0
$509$	29.6966	1.31628	0.658140	−	0.752895i	$-0.271343\pi$
$509$	29.6966	1.31628	0.658140	+	0.752895i	$0.271343\pi$
$510$	0	0
$511$	15.2980	0.676742
$512$	0	0
$513$	34.6342	1.52914
$514$	0	0
$515$	−26.6169	−1.17288
$516$	0	0
$517$	0.496595	0.0218402
$518$	0	0
$519$	5.70903	0.250598
$520$	0	0
$521$	−32.5336	−1.42532	−0.712661	−	0.701509i	$-0.752510\pi$
$521$	−32.5336	−1.42532	−0.712661	+	0.701509i	$0.752510\pi$
$522$	0	0
$523$	21.6812	0.948053	0.474027	−	0.880510i	$-0.342800\pi$
$523$	21.6812	0.948053	0.474027	+	0.880510i	$0.342800\pi$
$524$	0	0
$525$	2.73791	0.119492
$526$	0	0
$527$	−4.07105	−0.177338
$528$	0	0
$529$	−22.3013	−0.969622
$530$	0	0
$531$	−8.47162	−0.367637
$532$	0	0
$533$	5.29135	0.229194
$534$	0	0
$535$	−18.9120	−0.817637
$536$	0	0
$537$	−5.91877	−0.255414
$538$	0	0
$539$	−17.3187	−0.745971
$540$	0	0
$541$	6.08311	0.261533	0.130767	−	0.991413i	$-0.458256\pi$
$541$	6.08311	0.261533	0.130767	+	0.991413i	$0.458256\pi$
$542$	0	0
$543$	3.95648	0.169789
$544$	0	0
$545$	0.501356	0.0214757
$546$	0	0
$547$	12.1956	0.521447	0.260723	−	0.965414i	$-0.416039\pi$
$547$	12.1956	0.521447	0.260723	+	0.965414i	$0.416039\pi$
$548$	0	0
$549$	−11.4767	−0.489813
$550$	0	0
$551$	−7.83723	−0.333877
$552$	0	0
$553$	2.17448	0.0924684
$554$	0	0
$555$	−18.7470	−0.795765
$556$	0	0
$557$	−37.5982	−1.59309	−0.796544	−	0.604581i	$-0.793341\pi$
$557$	−37.5982	−1.59309	−0.796544	+	0.604581i	$0.793341\pi$
$558$	0	0
$559$	−46.6851	−1.97457
$560$	0	0
$561$	3.28439	0.138667
$562$	0	0
$563$	−20.6037	−0.868342	−0.434171	−	0.900830i	$-0.642959\pi$
$563$	−20.6037	−0.868342	−0.434171	+	0.900830i	$0.642959\pi$
$564$	0	0
$565$	−17.9082	−0.753403
$566$	0	0
$567$	1.66919	0.0700992
$568$	0	0
$569$	20.5762	0.862601	0.431301	−	0.902208i	$-0.358055\pi$
$569$	20.5762	0.862601	0.431301	+	0.902208i	$0.358055\pi$
$570$	0	0
$571$	−37.1320	−1.55392	−0.776962	−	0.629547i	$-0.783240\pi$
$571$	−37.1320	−1.55392	−0.776962	+	0.629547i	$0.783240\pi$
$572$	0	0
$573$	−16.1513	−0.674730
$574$	0	0
$575$	1.78821	0.0745736
$576$	0	0
$577$	−34.7064	−1.44484	−0.722422	−	0.691452i	$-0.756971\pi$
$577$	−34.7064	−1.44484	−0.722422	+	0.691452i	$0.756971\pi$
$578$	0	0
$579$	22.6028	0.939339
$580$	0	0
$581$	3.05992	0.126947
$582$	0	0
$583$	9.87998	0.409187
$584$	0	0
$585$	14.8554	0.614197
$586$	0	0
$587$	1.14455	0.0472405	0.0236203	−	0.999721i	$-0.492481\pi$
$587$	1.14455	0.0472405	0.0236203	+	0.999721i	$0.492481\pi$
$588$	0	0
$589$	27.8638	1.14811
$590$	0	0
$591$	13.2834	0.546407
$592$	0	0
$593$	3.60735	0.148136	0.0740680	−	0.997253i	$-0.476402\pi$
$593$	3.60735	0.148136	0.0740680	+	0.997253i	$0.476402\pi$
$594$	0	0
$595$	2.26292	0.0927708
$596$	0	0
$597$	−25.1601	−1.02973
$598$	0	0
$599$	31.6723	1.29410	0.647048	−	0.762449i	$-0.276003\pi$
$599$	31.6723	1.29410	0.647048	+	0.762449i	$0.276003\pi$
$600$	0	0
$601$	6.55665	0.267451	0.133726	−	0.991018i	$-0.457306\pi$
$601$	6.55665	0.267451	0.133726	+	0.991018i	$0.457306\pi$
$602$	0	0
$603$	26.5004	1.07918
$604$	0	0
$605$	0.508996	0.0206936
$606$	0	0
$607$	9.16648	0.372056	0.186028	−	0.982544i	$-0.440438\pi$
$607$	9.16648	0.372056	0.186028	+	0.982544i	$0.440438\pi$
$608$	0	0
$609$	1.43031	0.0579592
$610$	0	0
$611$	0.653812	0.0264504
$612$	0	0
$613$	7.92345	0.320025	0.160012	−	0.987115i	$-0.448847\pi$
$613$	7.92345	0.320025	0.160012	+	0.987115i	$0.448847\pi$
$614$	0	0
$615$	2.03667	0.0821264
$616$	0	0
$617$	16.5959	0.668126	0.334063	−	0.942551i	$-0.391580\pi$
$617$	16.5959	0.668126	0.334063	+	0.942551i	$0.391580\pi$
$618$	0	0
$619$	−4.49086	−0.180503	−0.0902514	−	0.995919i	$-0.528767\pi$
$619$	−4.49086	−0.180503	−0.0902514	+	0.995919i	$0.528767\pi$
$620$	0	0
$621$	−4.12832	−0.165664
$622$	0	0
$623$	23.6491	0.947481
$624$	0	0
$625$	−9.72656	−0.389063
$626$	0	0
$627$	−22.4796	−0.897749
$628$	0	0
$629$	11.5876	0.462028
$630$	0	0
$631$	5.96553	0.237484	0.118742	−	0.992925i	$-0.462114\pi$
$631$	5.96553	0.237484	0.118742	+	0.992925i	$0.462114\pi$
$632$	0	0
$633$	5.07955	0.201894
$634$	0	0
$635$	24.8611	0.986581
$636$	0	0
$637$	−22.8017	−0.903435
$638$	0	0
$639$	−23.0184	−0.910594
$640$	0	0
$641$	−1.73375	−0.0684789	−0.0342395	−	0.999414i	$-0.510901\pi$
$641$	−1.73375	−0.0684789	−0.0342395	+	0.999414i	$0.510901\pi$
$642$	0	0
$643$	−38.3312	−1.51163	−0.755817	−	0.654783i	$-0.772760\pi$
$643$	−38.3312	−1.51163	−0.755817	+	0.654783i	$0.772760\pi$
$644$	0	0
$645$	−17.9693	−0.707541
$646$	0	0
$647$	−29.2098	−1.14836	−0.574178	−	0.818730i	$-0.694678\pi$
$647$	−29.2098	−1.14836	−0.574178	+	0.818730i	$0.694678\pi$
$648$	0	0
$649$	13.5866	0.533321
$650$	0	0
$651$	−5.08520	−0.199305
$652$	0	0
$653$	−42.7952	−1.67471	−0.837353	−	0.546662i	$-0.815898\pi$
$653$	−42.7952	−1.67471	−0.837353	+	0.546662i	$0.815898\pi$
$654$	0	0
$655$	16.3021	0.636976
$656$	0	0
$657$	−23.8927	−0.932143
$658$	0	0
$659$	23.7004	0.923238	0.461619	−	0.887078i	$-0.347269\pi$
$659$	23.7004	0.923238	0.461619	+	0.887078i	$0.347269\pi$
$660$	0	0
$661$	24.1228	0.938268	0.469134	−	0.883127i	$-0.344566\pi$
$661$	24.1228	0.938268	0.469134	+	0.883127i	$0.344566\pi$
$662$	0	0
$663$	4.32420	0.167938
$664$	0	0
$665$	−15.4883	−0.600610
$666$	0	0
$667$	0.934180	0.0361716
$668$	0	0
$669$	−6.37921	−0.246634
$670$	0	0
$671$	18.4061	0.710558
$672$	0	0
$673$	−3.67522	−0.141669	−0.0708346	−	0.997488i	$-0.522566\pi$
$673$	−3.67522	−0.141669	−0.0708346	+	0.997488i	$0.522566\pi$
$674$	0	0
$675$	−10.5660	−0.406687
$676$	0	0
$677$	45.7769	1.75935	0.879675	−	0.475575i	$-0.157760\pi$
$677$	45.7769	1.75935	0.879675	+	0.475575i	$0.157760\pi$
$678$	0	0
$679$	−9.37443	−0.359758
$680$	0	0
$681$	10.8255	0.414835
$682$	0	0
$683$	1.52419	0.0583215	0.0291608	−	0.999575i	$-0.490717\pi$
$683$	1.52419	0.0583215	0.0291608	+	0.999575i	$0.490717\pi$
$684$	0	0
$685$	−39.3029	−1.50169
$686$	0	0
$687$	−8.77800	−0.334902
$688$	0	0
$689$	13.0079	0.495561
$690$	0	0
$691$	−20.3036	−0.772386	−0.386193	−	0.922418i	$-0.626210\pi$
$691$	−20.3036	−0.772386	−0.386193	+	0.922418i	$0.626210\pi$
$692$	0	0
$693$	−8.71162	−0.330927
$694$	0	0
$695$	−2.18821	−0.0830034
$696$	0	0
$697$	−1.25888	−0.0476833
$698$	0	0
$699$	6.33739	0.239702
$700$	0	0
$701$	−10.3863	−0.392285	−0.196143	−	0.980575i	$-0.562842\pi$
$701$	−10.3863	−0.392285	−0.196143	+	0.980575i	$0.562842\pi$
$702$	0	0
$703$	−79.3098	−2.99123
$704$	0	0
$705$	0.251656	0.00947790
$706$	0	0
$707$	−7.15759	−0.269189
$708$	0	0
$709$	34.9053	1.31090	0.655448	−	0.755240i	$-0.272480\pi$
$709$	34.9053	1.31090	0.655448	+	0.755240i	$0.272480\pi$
$710$	0	0
$711$	−3.39615	−0.127366
$712$	0	0
$713$	−3.32129	−0.124383
$714$	0	0
$715$	−23.8248	−0.890998
$716$	0	0
$717$	−24.5586	−0.917159
$718$	0	0
$719$	−5.17696	−0.193068	−0.0965341	−	0.995330i	$-0.530776\pi$
$719$	−5.17696	−0.193068	−0.0965341	+	0.995330i	$0.530776\pi$
$720$	0	0
$721$	20.5505	0.765339
$722$	0	0
$723$	0.980039	0.0364480
$724$	0	0
$725$	2.39094	0.0887974
$726$	0	0
$727$	7.00384	0.259758	0.129879	−	0.991530i	$-0.458541\pi$
$727$	7.00384	0.259758	0.129879	+	0.991530i	$0.458541\pi$
$728$	0	0
$729$	11.9141	0.441262
$730$	0	0
$731$	11.1069	0.410804
$732$	0	0
$733$	−1.22518	−0.0452530	−0.0226265	−	0.999744i	$-0.507203\pi$
$733$	−1.22518	−0.0452530	−0.0226265	+	0.999744i	$0.507203\pi$
$734$	0	0
$735$	−8.77648	−0.323725
$736$	0	0
$737$	−42.5008	−1.56554
$738$	0	0
$739$	31.4569	1.15716	0.578580	−	0.815625i	$-0.303607\pi$
$739$	31.4569	1.15716	0.578580	+	0.815625i	$0.303607\pi$
$740$	0	0
$741$	−29.5964	−1.08725
$742$	0	0
$743$	−17.5176	−0.642659	−0.321330	−	0.946967i	$-0.604130\pi$
$743$	−17.5176	−0.642659	−0.321330	+	0.946967i	$0.604130\pi$
$744$	0	0
$745$	−29.6521	−1.08637
$746$	0	0
$747$	−4.77904	−0.174856
$748$	0	0
$749$	14.6016	0.533532
$750$	0	0
$751$	35.2893	1.28773	0.643863	−	0.765141i	$-0.277331\pi$
$751$	35.2893	1.28773	0.643863	+	0.765141i	$0.277331\pi$
$752$	0	0
$753$	−12.3103	−0.448614
$754$	0	0
$755$	8.91324	0.324386
$756$	0	0
$757$	−17.2974	−0.628686	−0.314343	−	0.949309i	$-0.601784\pi$
$757$	−17.2974	−0.628686	−0.314343	+	0.949309i	$0.601784\pi$
$758$	0	0
$759$	2.67952	0.0972603
$760$	0	0
$761$	9.53250	0.345553	0.172776	−	0.984961i	$-0.444726\pi$
$761$	9.53250	0.345553	0.172776	+	0.984961i	$0.444726\pi$
$762$	0	0
$763$	−0.387088	−0.0140135
$764$	0	0
$765$	−3.53428	−0.127782
$766$	0	0
$767$	17.8880	0.645897
$768$	0	0
$769$	−37.0399	−1.33569	−0.667847	−	0.744299i	$-0.732784\pi$
$769$	−37.0399	−1.33569	−0.667847	+	0.744299i	$0.732784\pi$
$770$	0	0
$771$	−15.7166	−0.566019
$772$	0	0
$773$	−6.97450	−0.250855	−0.125428	−	0.992103i	$-0.540030\pi$
$773$	−6.97450	−0.250855	−0.125428	+	0.992103i	$0.540030\pi$
$774$	0	0
$775$	−8.50053	−0.305348
$776$	0	0
$777$	14.4742	0.519259
$778$	0	0
$779$	8.61621	0.308708
$780$	0	0
$781$	36.9164	1.32097
$782$	0	0
$783$	−5.51980	−0.197261
$784$	0	0
$785$	24.0450	0.858204
$786$	0	0
$787$	−29.9359	−1.06710	−0.533550	−	0.845768i	$-0.679143\pi$
$787$	−29.9359	−1.06710	−0.533550	+	0.845768i	$0.679143\pi$
$788$	0	0
$789$	3.26559	0.116258
$790$	0	0
$791$	13.8266	0.491617
$792$	0	0
$793$	24.2332	0.860547
$794$	0	0
$795$	5.00680	0.177573
$796$	0	0
$797$	−10.7103	−0.379380	−0.189690	−	0.981844i	$-0.560748\pi$
$797$	−10.7103	−0.379380	−0.189690	+	0.981844i	$0.560748\pi$
$798$	0	0
$799$	−0.155550	−0.00550295
$800$	0	0
$801$	−36.9357	−1.30506
$802$	0	0
$803$	38.3186	1.35223
$804$	0	0
$805$	1.84617	0.0650688
$806$	0	0
$807$	7.29248	0.256707
$808$	0	0
$809$	−30.4625	−1.07100	−0.535502	−	0.844534i	$-0.679878\pi$
$809$	−30.4625	−1.07100	−0.535502	+	0.844534i	$0.679878\pi$
$810$	0	0
$811$	6.38159	0.224088	0.112044	−	0.993703i	$-0.464260\pi$
$811$	6.38159	0.224088	0.112044	+	0.993703i	$0.464260\pi$
$812$	0	0
$813$	−20.1918	−0.708158
$814$	0	0
$815$	−27.0088	−0.946077
$816$	0	0
$817$	−76.0199	−2.65960
$818$	0	0
$819$	−11.4696	−0.400781
$820$	0	0
$821$	−31.3814	−1.09522	−0.547609	−	0.836734i	$-0.684462\pi$
$821$	−31.3814	−1.09522	−0.547609	+	0.836734i	$0.684462\pi$
$822$	0	0
$823$	8.61720	0.300377	0.150188	−	0.988657i	$-0.452012\pi$
$823$	8.61720	0.300377	0.150188	+	0.988657i	$0.452012\pi$
$824$	0	0
$825$	6.85796	0.238764
$826$	0	0
$827$	18.2047	0.633039	0.316520	−	0.948586i	$-0.397486\pi$
$827$	18.2047	0.633039	0.316520	+	0.948586i	$0.397486\pi$
$828$	0	0
$829$	38.0162	1.32036	0.660179	−	0.751108i	$-0.270480\pi$
$829$	38.0162	1.32036	0.660179	+	0.751108i	$0.270480\pi$
$830$	0	0
$831$	−5.07334	−0.175992
$832$	0	0
$833$	5.42479	0.187958
$834$	0	0
$835$	−36.4106	−1.26004
$836$	0	0
$837$	19.6246	0.678324
$838$	0	0
$839$	−45.6228	−1.57507	−0.787537	−	0.616267i	$-0.788644\pi$
$839$	−45.6228	−1.57507	−0.787537	+	0.616267i	$0.788644\pi$
$840$	0	0
$841$	−27.7509	−0.956929
$842$	0	0
$843$	13.8990	0.478706
$844$	0	0
$845$	−9.37997	−0.322681
$846$	0	0
$847$	−0.392987	−0.0135032
$848$	0	0
$849$	−6.91120	−0.237192
$850$	0	0
$851$	9.45354	0.324063
$852$	0	0
$853$	17.2406	0.590307	0.295153	−	0.955450i	$-0.404629\pi$
$853$	17.2406	0.590307	0.295153	+	0.955450i	$0.404629\pi$
$854$	0	0
$855$	24.1899	0.827278
$856$	0	0
$857$	9.87638	0.337371	0.168685	−	0.985670i	$-0.446048\pi$
$857$	9.87638	0.337371	0.168685	+	0.985670i	$0.446048\pi$
$858$	0	0
$859$	−25.8904	−0.883369	−0.441685	−	0.897170i	$-0.645619\pi$
$859$	−25.8904	−0.883369	−0.441685	+	0.897170i	$0.645619\pi$
$860$	0	0
$861$	−1.57248	−0.0535899
$862$	0	0
$863$	14.8445	0.505311	0.252656	−	0.967556i	$-0.418696\pi$
$863$	14.8445	0.505311	0.252656	+	0.967556i	$0.418696\pi$
$864$	0	0
$865$	9.85263	0.334999
$866$	0	0
$867$	15.6319	0.530887
$868$	0	0
$869$	5.44668	0.184766
$870$	0	0
$871$	−55.9561	−1.89600
$872$	0	0
$873$	14.6412	0.495529
$874$	0	0
$875$	15.7684	0.533070
$876$	0	0
$877$	42.0781	1.42088	0.710439	−	0.703759i	$-0.248497\pi$
$877$	42.0781	1.42088	0.710439	+	0.703759i	$0.248497\pi$
$878$	0	0
$879$	2.73398	0.0922148
$880$	0	0
$881$	−19.0418	−0.641534	−0.320767	−	0.947158i	$-0.603941\pi$
$881$	−19.0418	−0.641534	−0.320767	+	0.947158i	$0.603941\pi$
$882$	0	0
$883$	−6.51087	−0.219108	−0.109554	−	0.993981i	$-0.534942\pi$
$883$	−6.51087	−0.219108	−0.109554	+	0.993981i	$0.534942\pi$
$884$	0	0
$885$	6.88517	0.231442
$886$	0	0
$887$	−5.59140	−0.187741	−0.0938704	−	0.995584i	$-0.529924\pi$
$887$	−5.59140	−0.187741	−0.0938704	+	0.995584i	$0.529924\pi$
$888$	0	0
$889$	−19.1948	−0.643773
$890$	0	0
$891$	4.18100	0.140069
$892$	0	0
$893$	1.06464	0.0356268
$894$	0	0
$895$	−10.2146	−0.341437
$896$	0	0
$897$	3.52783	0.117791
$898$	0	0
$899$	−4.44076	−0.148108
$900$	0	0
$901$	−3.09473	−0.103100
$902$	0	0
$903$	13.8738	0.461691
$904$	0	0
$905$	6.82809	0.226973
$906$	0	0
$907$	−18.3915	−0.610681	−0.305340	−	0.952243i	$-0.598770\pi$
$907$	−18.3915	−0.610681	−0.305340	+	0.952243i	$0.598770\pi$
$908$	0	0
$909$	11.1789	0.370780
$910$	0	0
$911$	38.2665	1.26783	0.633913	−	0.773404i	$-0.281448\pi$
$911$	38.2665	1.26783	0.633913	+	0.773404i	$0.281448\pi$
$912$	0	0
$913$	7.66452	0.253659
$914$	0	0
$915$	9.32749	0.308357
$916$	0	0
$917$	−12.5866	−0.415646
$918$	0	0
$919$	4.89962	0.161624	0.0808118	−	0.996729i	$-0.474249\pi$
$919$	4.89962	0.161624	0.0808118	+	0.996729i	$0.474249\pi$
$920$	0	0
$921$	19.4327	0.640330
$922$	0	0
$923$	48.6038	1.59981
$924$	0	0
$925$	24.1954	0.795541
$926$	0	0
$927$	−32.0962	−1.05418
$928$	0	0
$929$	17.9381	0.588528	0.294264	−	0.955724i	$-0.404925\pi$
$929$	17.9381	0.588528	0.294264	+	0.955724i	$0.404925\pi$
$930$	0	0
$931$	−37.1292	−1.21686
$932$	0	0
$933$	32.8420	1.07520
$934$	0	0
$935$	5.66820	0.185370
$936$	0	0
$937$	29.8369	0.974730	0.487365	−	0.873198i	$-0.337958\pi$
$937$	29.8369	0.974730	0.487365	+	0.873198i	$0.337958\pi$
$938$	0	0
$939$	−19.0276	−0.620943
$940$	0	0
$941$	26.8047	0.873809	0.436904	−	0.899508i	$-0.356075\pi$
$941$	26.8047	0.873809	0.436904	+	0.899508i	$0.356075\pi$
$942$	0	0
$943$	−1.02703	−0.0334448
$944$	0	0
$945$	−10.9085	−0.354852
$946$	0	0
$947$	−2.94622	−0.0957393	−0.0478697	−	0.998854i	$-0.515243\pi$
$947$	−2.94622	−0.0957393	−0.0478697	+	0.998854i	$0.515243\pi$
$948$	0	0
$949$	50.4499	1.63767
$950$	0	0
$951$	11.4914	0.372635
$952$	0	0
$953$	−4.62861	−0.149935	−0.0749676	−	0.997186i	$-0.523885\pi$
$953$	−4.62861	−0.149935	−0.0749676	+	0.997186i	$0.523885\pi$
$954$	0	0
$955$	−27.8739	−0.901978
$956$	0	0
$957$	3.58267	0.115811
$958$	0	0
$959$	30.3451	0.979894
$960$	0	0
$961$	−15.2117	−0.490702
$962$	0	0
$963$	−22.8051	−0.734885
$964$	0	0
$965$	39.0078	1.25571
$966$	0	0
$967$	−20.1721	−0.648689	−0.324345	−	0.945939i	$-0.605144\pi$
$967$	−20.1721	−0.648689	−0.324345	+	0.945939i	$0.605144\pi$
$968$	0	0
$969$	7.04134	0.226200
$970$	0	0
$971$	−30.0251	−0.963550	−0.481775	−	0.876295i	$-0.660008\pi$
$971$	−30.0251	−0.963550	−0.481775	+	0.876295i	$0.660008\pi$
$972$	0	0
$973$	1.68948	0.0541621
$974$	0	0
$975$	9.02913	0.289163
$976$	0	0
$977$	57.7096	1.84629	0.923147	−	0.384448i	$-0.125608\pi$
$977$	57.7096	1.84629	0.923147	+	0.384448i	$0.125608\pi$
$978$	0	0
$979$	59.2366	1.89321
$980$	0	0
$981$	0.604563	0.0193022
$982$	0	0
$983$	3.33817	0.106471	0.0532356	−	0.998582i	$-0.483047\pi$
$983$	3.33817	0.106471	0.0532356	+	0.998582i	$0.483047\pi$
$984$	0	0
$985$	22.9245	0.730435
$986$	0	0
$987$	−0.194299	−0.00618460
$988$	0	0
$989$	9.06139	0.288136
$990$	0	0
$991$	−4.48757	−0.142552	−0.0712762	−	0.997457i	$-0.522707\pi$
$991$	−4.48757	−0.142552	−0.0712762	+	0.997457i	$0.522707\pi$
$992$	0	0
$993$	4.83230	0.153349
$994$	0	0
$995$	−43.4212	−1.37654
$996$	0	0
$997$	−47.4116	−1.50154	−0.750770	−	0.660564i	$-0.770317\pi$
$997$	−47.4116	−1.50154	−0.750770	+	0.660564i	$0.770317\pi$
$998$	0	0
$999$	−55.8582	−1.76728

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.j.1.2		7
4.3	odd	2		241.2.a.a.1.1	✓	7
12.11	even	2		2169.2.a.e.1.7		7
20.19	odd	2		6025.2.a.f.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.1	✓	7	4.3	odd	2
2169.2.a.e.1.7		7	12.11	even	2
3856.2.a.j.1.2		7	1.1	even	1	trivial
6025.2.a.f.1.7		7	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)

\(f(q)\)	\(=\)	\(q-0.980039 q^{3} -1.69135 q^{5} +1.30586 q^{7} -2.03952 q^{9} +O(q^{10})\)	\(q-0.980039 q^{3} -1.69135 q^{5} +1.30586 q^{7} -2.03952 q^{9} +3.27094 q^{11} +4.30649 q^{13} +1.65759 q^{15} -1.02456 q^{17} +7.01250 q^{19} -1.27980 q^{21} -0.835873 q^{23} -2.13934 q^{25} +4.93893 q^{27} -1.11761 q^{29} +3.97344 q^{31} -3.20565 q^{33} -2.20867 q^{35} -11.3098 q^{37} -4.22053 q^{39} +1.22869 q^{41} -10.8406 q^{43} +3.44955 q^{45} +0.151820 q^{47} -5.29473 q^{49} +1.00411 q^{51} +3.02053 q^{53} -5.53231 q^{55} -6.87252 q^{57} +4.15373 q^{59} +5.62714 q^{61} -2.66334 q^{63} -7.28378 q^{65} -12.9934 q^{67} +0.819188 q^{69} +11.2862 q^{71} +11.7148 q^{73} +2.09663 q^{75} +4.27140 q^{77} +1.66517 q^{79} +1.27823 q^{81} +2.34322 q^{83} +1.73290 q^{85} +1.09530 q^{87} +18.1099 q^{89} +5.62368 q^{91} -3.89413 q^{93} -11.8606 q^{95} -7.17873 q^{97} -6.67116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 - 8 * q^5 + 7 * q^7 - 2 * q^9	\( 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9} + 18 q^{11} - q^{13} + 11 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{21} + 22 q^{23} + 5 q^{25} - 3 q^{27} - 16 q^{29} + 18 q^{31} + 4 q^{33} - 7 q^{35} + 8 q^{37} + 9 q^{39} - 15 q^{41} - 14 q^{43} + 3 q^{45} + 10 q^{47} + 6 q^{49} - 13 q^{51} + 15 q^{53} - 29 q^{55} + 14 q^{57} + 18 q^{59} + 4 q^{61} + 16 q^{63} - 7 q^{65} - 18 q^{67} + 26 q^{69} + 50 q^{71} - 16 q^{75} + 17 q^{77} + 15 q^{79} - 9 q^{81} + 24 q^{83} - 2 q^{85} - 12 q^{87} - 13 q^{89} + 12 q^{91} + 14 q^{93} + 41 q^{95} + q^{97} + 20 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 8 * q^5 + 7 * q^7 - 2 * q^9 + 18 * q^11 - q^13 + 11 * q^15 - 2 * q^17 + 6 * q^19 - 2 * q^21 + 22 * q^23 + 5 * q^25 - 3 * q^27 - 16 * q^29 + 18 * q^31 + 4 * q^33 - 7 * q^35 + 8 * q^37 + 9 * q^39 - 15 * q^41 - 14 * q^43 + 3 * q^45 + 10 * q^47 + 6 * q^49 - 13 * q^51 + 15 * q^53 - 29 * q^55 + 14 * q^57 + 18 * q^59 + 4 * q^61 + 16 * q^63 - 7 * q^65 - 18 * q^67 + 26 * q^69 + 50 * q^71 - 16 * q^75 + 17 * q^77 + 15 * q^79 - 9 * q^81 + 24 * q^83 - 2 * q^85 - 12 * q^87 - 13 * q^89 + 12 * q^91 + 14 * q^93 + 41 * q^95 + q^97 + 20 * q^99

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