LMFDB - Embedded newform 6416.2.a.k.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6416,2,Mod(1,6416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6416, 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("6416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6416 = 2^{4} \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6416.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:	$51.2320179369$
Analytic rank:	$0$
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} - 10 x^{10} + 34 x^{9} + 29 x^{8} - 129 x^{7} - 24 x^{6} + 203 x^{5} + x^{4} + \cdots + 4 $ x^12 - 3x^11 - 10x^10 + 34x^9 + 29x^8 - 129x^7 - 24x^6 + 203x^5 + x^4 - 130x^3 - 5x^2 + 22x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 401)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.72691$ of defining polynomial
Character	$\chi$	$=$	6416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.96155 q^{3} +1.08846 q^{5} +1.51444 q^{7} +5.77079 q^{9} +O(q^{10})$	$q+2.96155 q^{3} +1.08846 q^{5} +1.51444 q^{7} +5.77079 q^{9} +5.06421 q^{11} -2.50035 q^{13} +3.22353 q^{15} -1.60664 q^{17} -0.616854 q^{19} +4.48511 q^{21} +0.286907 q^{23} -3.81526 q^{25} +8.20583 q^{27} +3.32920 q^{29} +7.41721 q^{31} +14.9979 q^{33} +1.64841 q^{35} +6.94840 q^{37} -7.40492 q^{39} -1.08712 q^{41} -6.85484 q^{43} +6.28127 q^{45} -5.85772 q^{47} -4.70646 q^{49} -4.75816 q^{51} +10.8594 q^{53} +5.51219 q^{55} -1.82684 q^{57} +11.0313 q^{59} -5.58465 q^{61} +8.73954 q^{63} -2.72153 q^{65} -13.4085 q^{67} +0.849690 q^{69} +12.5845 q^{71} +2.65532 q^{73} -11.2991 q^{75} +7.66946 q^{77} -2.93442 q^{79} +6.98963 q^{81} -0.372029 q^{83} -1.74877 q^{85} +9.85960 q^{87} -10.6127 q^{89} -3.78664 q^{91} +21.9665 q^{93} -0.671420 q^{95} -4.14292 q^{97} +29.2245 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 5 q^{3} - 7 q^{5} + 20 q^{7} + 3 q^{9}+O(q^{10}) $ 12 * q + 5 * q^3 - 7 * q^5 + 20 * q^7 + 3 * q^9	$ 12 q + 5 q^{3} - 7 q^{5} + 20 q^{7} + 3 q^{9} + 11 q^{11} - 11 q^{13} + 11 q^{15} + q^{17} + 34 q^{19} - 3 q^{21} + 7 q^{23} + 7 q^{25} + 2 q^{27} - 6 q^{29} + 52 q^{31} + 4 q^{33} - 12 q^{35} + 3 q^{37} + 24 q^{39} - 16 q^{41} + 2 q^{43} - 23 q^{45} + 3 q^{47} + 6 q^{49} + 16 q^{51} + 19 q^{53} + 43 q^{55} + 11 q^{57} + q^{59} - 24 q^{61} + 11 q^{63} + 13 q^{65} - 6 q^{67} + 29 q^{69} + 15 q^{71} - 20 q^{73} - 31 q^{75} + 38 q^{77} + 53 q^{79} - 8 q^{81} - 17 q^{83} + 7 q^{85} + 5 q^{87} - q^{89} + 6 q^{91} + 44 q^{93} - 34 q^{95} + 12 q^{97}+O(q^{100}) $ 12 * q + 5 * q^3 - 7 * q^5 + 20 * q^7 + 3 * q^9 + 11 * q^11 - 11 * q^13 + 11 * q^15 + q^17 + 34 * q^19 - 3 * q^21 + 7 * q^23 + 7 * q^25 + 2 * q^27 - 6 * q^29 + 52 * q^31 + 4 * q^33 - 12 * q^35 + 3 * q^37 + 24 * q^39 - 16 * q^41 + 2 * q^43 - 23 * q^45 + 3 * q^47 + 6 * q^49 + 16 * q^51 + 19 * q^53 + 43 * q^55 + 11 * q^57 + q^59 - 24 * q^61 + 11 * q^63 + 13 * q^65 - 6 * q^67 + 29 * q^69 + 15 * q^71 - 20 * q^73 - 31 * q^75 + 38 * q^77 + 53 * q^79 - 8 * q^81 - 17 * q^83 + 7 * q^85 + 5 * q^87 - q^89 + 6 * q^91 + 44 * q^93 - 34 * q^95 + 12 * q^97

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
$3$	2.96155	1.70985	0.854926	−	0.518749i	$-0.173602\pi$
$3$	2.96155	1.70985	0.854926	+	0.518749i	$0.173602\pi$
$4$	0	0
$5$	1.08846	0.486774	0.243387	−	0.969929i	$-0.421742\pi$
$5$	1.08846	0.486774	0.243387	+	0.969929i	$0.421742\pi$
$6$	0	0
$7$	1.51444	0.572406	0.286203	−	0.958169i	$-0.407607\pi$
$7$	1.51444	0.572406	0.286203	+	0.958169i	$0.407607\pi$
$8$	0	0
$9$	5.77079	1.92360
$10$	0	0
$11$	5.06421	1.52692	0.763458	−	0.645857i	$-0.223500\pi$
$11$	5.06421	1.52692	0.763458	+	0.645857i	$0.223500\pi$
$12$	0	0
$13$	−2.50035	−0.693472	−0.346736	−	0.937963i	$-0.612710\pi$
$13$	−2.50035	−0.693472	−0.346736	+	0.937963i	$0.612710\pi$
$14$	0	0
$15$	3.22353	0.832312
$16$	0	0
$17$	−1.60664	−0.389668	−0.194834	−	0.980836i	$-0.562417\pi$
$17$	−1.60664	−0.389668	−0.194834	+	0.980836i	$0.562417\pi$
$18$	0	0
$19$	−0.616854	−0.141516	−0.0707580	−	0.997494i	$-0.522542\pi$
$19$	−0.616854	−0.141516	−0.0707580	+	0.997494i	$0.522542\pi$
$20$	0	0
$21$	4.48511	0.978730
$22$	0	0
$23$	0.286907	0.0598243	0.0299121	−	0.999553i	$-0.490477\pi$
$23$	0.286907	0.0598243	0.0299121	+	0.999553i	$0.490477\pi$
$24$	0	0
$25$	−3.81526	−0.763051
$26$	0	0
$27$	8.20583	1.57921
$28$	0	0
$29$	3.32920	0.618217	0.309108	−	0.951027i	$-0.399969\pi$
$29$	3.32920	0.618217	0.309108	+	0.951027i	$0.399969\pi$
$30$	0	0
$31$	7.41721	1.33217	0.666085	−	0.745875i	$-0.267969\pi$
$31$	7.41721	1.33217	0.666085	+	0.745875i	$0.267969\pi$
$32$	0	0
$33$	14.9979	2.61080
$34$	0	0
$35$	1.64841	0.278632
$36$	0	0
$37$	6.94840	1.14231	0.571155	−	0.820842i	$-0.306495\pi$
$37$	6.94840	1.14231	0.571155	+	0.820842i	$0.306495\pi$
$38$	0	0
$39$	−7.40492	−1.18574
$40$	0	0
$41$	−1.08712	−0.169780	−0.0848901	−	0.996390i	$-0.527054\pi$
$41$	−1.08712	−0.169780	−0.0848901	+	0.996390i	$0.527054\pi$
$42$	0	0
$43$	−6.85484	−1.04535	−0.522676	−	0.852531i	$-0.675066\pi$
$43$	−6.85484	−1.04535	−0.522676	+	0.852531i	$0.675066\pi$
$44$	0	0
$45$	6.28127	0.936356
$46$	0	0
$47$	−5.85772	−0.854436	−0.427218	−	0.904149i	$-0.640506\pi$
$47$	−5.85772	−0.854436	−0.427218	+	0.904149i	$0.640506\pi$
$48$	0	0
$49$	−4.70646	−0.672351
$50$	0	0
$51$	−4.75816	−0.666275
$52$	0	0
$53$	10.8594	1.49165	0.745827	−	0.666140i	$-0.232055\pi$
$53$	10.8594	1.49165	0.745827	+	0.666140i	$0.232055\pi$
$54$	0	0
$55$	5.51219	0.743263
$56$	0	0
$57$	−1.82684	−0.241971
$58$	0	0
$59$	11.0313	1.43615	0.718074	−	0.695966i	$-0.245024\pi$
$59$	11.0313	1.43615	0.718074	+	0.695966i	$0.245024\pi$
$60$	0	0
$61$	−5.58465	−0.715042	−0.357521	−	0.933905i	$-0.616378\pi$
$61$	−5.58465	−0.715042	−0.357521	+	0.933905i	$0.616378\pi$
$62$	0	0
$63$	8.73954	1.10108
$64$	0	0
$65$	−2.72153	−0.337564
$66$	0	0
$67$	−13.4085	−1.63811	−0.819054	−	0.573716i	$-0.805501\pi$
$67$	−13.4085	−1.63811	−0.819054	+	0.573716i	$0.805501\pi$
$68$	0	0
$69$	0.849690	0.102291
$70$	0	0
$71$	12.5845	1.49351	0.746756	−	0.665099i	$-0.231611\pi$
$71$	12.5845	1.49351	0.746756	+	0.665099i	$0.231611\pi$
$72$	0	0
$73$	2.65532	0.310782	0.155391	−	0.987853i	$-0.450336\pi$
$73$	2.65532	0.310782	0.155391	+	0.987853i	$0.450336\pi$
$74$	0	0
$75$	−11.2991	−1.30471
$76$	0	0
$77$	7.66946	0.874016
$78$	0	0
$79$	−2.93442	−0.330148	−0.165074	−	0.986281i	$-0.552786\pi$
$79$	−2.93442	−0.330148	−0.165074	+	0.986281i	$0.552786\pi$
$80$	0	0
$81$	6.98963	0.776626
$82$	0	0
$83$	−0.372029	−0.0408355	−0.0204177	−	0.999792i	$-0.506500\pi$
$83$	−0.372029	−0.0408355	−0.0204177	+	0.999792i	$0.506500\pi$
$84$	0	0
$85$	−1.74877	−0.189680
$86$	0	0
$87$	9.85960	1.05706
$88$	0	0
$89$	−10.6127	−1.12495	−0.562474	−	0.826815i	$-0.690150\pi$
$89$	−10.6127	−1.12495	−0.562474	+	0.826815i	$0.690150\pi$
$90$	0	0
$91$	−3.78664	−0.396948
$92$	0	0
$93$	21.9665	2.27782
$94$	0	0
$95$	−0.671420	−0.0688863
$96$	0	0
$97$	−4.14292	−0.420650	−0.210325	−	0.977632i	$-0.567452\pi$
$97$	−4.14292	−0.420650	−0.210325	+	0.977632i	$0.567452\pi$
$98$	0	0
$99$	29.2245	2.93717
$100$	0	0
$101$	16.7441	1.66610	0.833051	−	0.553197i	$-0.186592\pi$
$101$	16.7441	1.66610	0.833051	+	0.553197i	$0.186592\pi$
$102$	0	0
$103$	14.3865	1.41754	0.708771	−	0.705439i	$-0.249250\pi$
$103$	14.3865	1.41754	0.708771	+	0.705439i	$0.249250\pi$
$104$	0	0
$105$	4.88186	0.476420
$106$	0	0
$107$	3.72497	0.360107	0.180053	−	0.983657i	$-0.442373\pi$
$107$	3.72497	0.360107	0.180053	+	0.983657i	$0.442373\pi$
$108$	0	0
$109$	−7.89425	−0.756132	−0.378066	−	0.925779i	$-0.623411\pi$
$109$	−7.89425	−0.756132	−0.378066	+	0.925779i	$0.623411\pi$
$110$	0	0
$111$	20.5781	1.95318
$112$	0	0
$113$	−13.6256	−1.28179	−0.640895	−	0.767629i	$-0.721437\pi$
$113$	−13.6256	−1.28179	−0.640895	+	0.767629i	$0.721437\pi$
$114$	0	0
$115$	0.312287	0.0291209
$116$	0	0
$117$	−14.4290	−1.33396
$118$	0	0
$119$	−2.43317	−0.223048
$120$	0	0
$121$	14.6462	1.33147
$122$	0	0
$123$	−3.21957	−0.290299
$124$	0	0
$125$	−9.59505	−0.858207
$126$	0	0
$127$	11.3170	1.00422	0.502109	−	0.864805i	$-0.332558\pi$
$127$	11.3170	1.00422	0.502109	+	0.864805i	$0.332558\pi$
$128$	0	0
$129$	−20.3009	−1.78740
$130$	0	0
$131$	1.42156	0.124203	0.0621013	−	0.998070i	$-0.480220\pi$
$131$	1.42156	0.124203	0.0621013	+	0.998070i	$0.480220\pi$
$132$	0	0
$133$	−0.934190	−0.0810046
$134$	0	0
$135$	8.93172	0.768720
$136$	0	0
$137$	−11.3583	−0.970404	−0.485202	−	0.874402i	$-0.661254\pi$
$137$	−11.3583	−0.970404	−0.485202	+	0.874402i	$0.661254\pi$
$138$	0	0
$139$	19.9073	1.68852	0.844258	−	0.535937i	$-0.180042\pi$
$139$	19.9073	1.68852	0.844258	+	0.535937i	$0.180042\pi$
$140$	0	0
$141$	−17.3479	−1.46096
$142$	0	0
$143$	−12.6623	−1.05887
$144$	0	0
$145$	3.62370	0.300932
$146$	0	0
$147$	−13.9384	−1.14962
$148$	0	0
$149$	6.91007	0.566095	0.283047	−	0.959106i	$-0.408655\pi$
$149$	6.91007	0.566095	0.283047	+	0.959106i	$0.408655\pi$
$150$	0	0
$151$	−2.46122	−0.200291	−0.100146	−	0.994973i	$-0.531931\pi$
$151$	−2.46122	−0.200291	−0.100146	+	0.994973i	$0.531931\pi$
$152$	0	0
$153$	−9.27160	−0.749564
$154$	0	0
$155$	8.07334	0.648466
$156$	0	0
$157$	−19.4896	−1.55544	−0.777720	−	0.628611i	$-0.783624\pi$
$157$	−19.4896	−1.55544	−0.777720	+	0.628611i	$0.783624\pi$
$158$	0	0
$159$	32.1607	2.55051
$160$	0	0
$161$	0.434505	0.0342438
$162$	0	0
$163$	−10.1755	−0.797011	−0.398505	−	0.917166i	$-0.630471\pi$
$163$	−10.1755	−0.797011	−0.398505	+	0.917166i	$0.630471\pi$
$164$	0	0
$165$	16.3246	1.27087
$166$	0	0
$167$	−10.7496	−0.831829	−0.415915	−	0.909404i	$-0.636538\pi$
$167$	−10.7496	−0.831829	−0.415915	+	0.909404i	$0.636538\pi$
$168$	0	0
$169$	−6.74825	−0.519096
$170$	0	0
$171$	−3.55973	−0.272219
$172$	0	0
$173$	15.4920	1.17784	0.588918	−	0.808193i	$-0.299554\pi$
$173$	15.4920	1.17784	0.588918	+	0.808193i	$0.299554\pi$
$174$	0	0
$175$	−5.77799	−0.436775
$176$	0	0
$177$	32.6697	2.45560
$178$	0	0
$179$	−17.4084	−1.30116	−0.650582	−	0.759436i	$-0.725475\pi$
$179$	−17.4084	−1.30116	−0.650582	+	0.759436i	$0.725475\pi$
$180$	0	0
$181$	−18.2769	−1.35851	−0.679256	−	0.733901i	$-0.737698\pi$
$181$	−18.2769	−1.35851	−0.679256	+	0.733901i	$0.737698\pi$
$182$	0	0
$183$	−16.5392	−1.22262
$184$	0	0
$185$	7.56306	0.556047
$186$	0	0
$187$	−8.13637	−0.594991
$188$	0	0
$189$	12.4273	0.903951
$190$	0	0
$191$	0.569096	0.0411784	0.0205892	−	0.999788i	$-0.493446\pi$
$191$	0.569096	0.0411784	0.0205892	+	0.999788i	$0.493446\pi$
$192$	0	0
$193$	21.4612	1.54481	0.772404	−	0.635132i	$-0.219054\pi$
$193$	21.4612	1.54481	0.772404	+	0.635132i	$0.219054\pi$
$194$	0	0
$195$	−8.05995	−0.577185
$196$	0	0
$197$	−8.08145	−0.575779	−0.287890	−	0.957664i	$-0.592954\pi$
$197$	−8.08145	−0.575779	−0.287890	+	0.957664i	$0.592954\pi$
$198$	0	0
$199$	−8.79056	−0.623146	−0.311573	−	0.950222i	$-0.600856\pi$
$199$	−8.79056	−0.623146	−0.311573	+	0.950222i	$0.600856\pi$
$200$	0	0
$201$	−39.7099	−2.80092
$202$	0	0
$203$	5.04189	0.353871
$204$	0	0
$205$	−1.18329	−0.0826445
$206$	0	0
$207$	1.65568	0.115078
$208$	0	0
$209$	−3.12387	−0.216083
$210$	0	0
$211$	−7.52698	−0.518178	−0.259089	−	0.965853i	$-0.583422\pi$
$211$	−7.52698	−0.518178	−0.259089	+	0.965853i	$0.583422\pi$
$212$	0	0
$213$	37.2698	2.55368
$214$	0	0
$215$	−7.46121	−0.508850
$216$	0	0
$217$	11.2330	0.762543
$218$	0	0
$219$	7.86388	0.531391
$220$	0	0
$221$	4.01717	0.270224
$222$	0	0
$223$	7.93079	0.531085	0.265542	−	0.964099i	$-0.414449\pi$
$223$	7.93079	0.531085	0.265542	+	0.964099i	$0.414449\pi$
$224$	0	0
$225$	−22.0170	−1.46780
$226$	0	0
$227$	7.35332	0.488057	0.244028	−	0.969768i	$-0.421531\pi$
$227$	7.35332	0.488057	0.244028	+	0.969768i	$0.421531\pi$
$228$	0	0
$229$	−18.7846	−1.24132	−0.620661	−	0.784079i	$-0.713136\pi$
$229$	−18.7846	−1.24132	−0.620661	+	0.784079i	$0.713136\pi$
$230$	0	0
$231$	22.7135	1.49444
$232$	0	0
$233$	−21.8253	−1.42982	−0.714911	−	0.699215i	$-0.753533\pi$
$233$	−21.8253	−1.42982	−0.714911	+	0.699215i	$0.753533\pi$
$234$	0	0
$235$	−6.37589	−0.415917
$236$	0	0
$237$	−8.69043	−0.564504
$238$	0	0
$239$	−28.3059	−1.83095	−0.915477	−	0.402371i	$-0.868186\pi$
$239$	−28.3059	−1.83095	−0.915477	+	0.402371i	$0.868186\pi$
$240$	0	0
$241$	−2.28335	−0.147084	−0.0735418	−	0.997292i	$-0.523430\pi$
$241$	−2.28335	−0.147084	−0.0735418	+	0.997292i	$0.523430\pi$
$242$	0	0
$243$	−3.91734	−0.251297
$244$	0	0
$245$	−5.12279	−0.327283
$246$	0	0
$247$	1.54235	0.0981374
$248$	0	0
$249$	−1.10178	−0.0698226
$250$	0	0
$251$	−11.4580	−0.723220	−0.361610	−	0.932329i	$-0.617773\pi$
$251$	−11.4580	−0.723220	−0.361610	+	0.932329i	$0.617773\pi$
$252$	0	0
$253$	1.45296	0.0913466
$254$	0	0
$255$	−5.17906	−0.324325
$256$	0	0
$257$	24.9743	1.55785	0.778926	−	0.627116i	$-0.215765\pi$
$257$	24.9743	1.55785	0.778926	+	0.627116i	$0.215765\pi$
$258$	0	0
$259$	10.5230	0.653866
$260$	0	0
$261$	19.2121	1.18920
$262$	0	0
$263$	5.52110	0.340446	0.170223	−	0.985406i	$-0.445551\pi$
$263$	5.52110	0.340446	0.170223	+	0.985406i	$0.445551\pi$
$264$	0	0
$265$	11.8200	0.726098
$266$	0	0
$267$	−31.4302	−1.92349
$268$	0	0
$269$	−3.80408	−0.231939	−0.115969	−	0.993253i	$-0.536997\pi$
$269$	−3.80408	−0.231939	−0.115969	+	0.993253i	$0.536997\pi$
$270$	0	0
$271$	−10.3194	−0.626858	−0.313429	−	0.949612i	$-0.601478\pi$
$271$	−10.3194	−0.626858	−0.313429	+	0.949612i	$0.601478\pi$
$272$	0	0
$273$	−11.2143	−0.678722
$274$	0	0
$275$	−19.3212	−1.16512
$276$	0	0
$277$	19.2676	1.15768	0.578840	−	0.815441i	$-0.303506\pi$
$277$	19.2676	1.15768	0.578840	+	0.815441i	$0.303506\pi$
$278$	0	0
$279$	42.8032	2.56256
$280$	0	0
$281$	30.0379	1.79191	0.895956	−	0.444143i	$-0.146492\pi$
$281$	30.0379	1.79191	0.895956	+	0.444143i	$0.146492\pi$
$282$	0	0
$283$	27.6057	1.64099	0.820495	−	0.571654i	$-0.193698\pi$
$283$	27.6057	1.64099	0.820495	+	0.571654i	$0.193698\pi$
$284$	0	0
$285$	−1.98845	−0.117785
$286$	0	0
$287$	−1.64639	−0.0971832
$288$	0	0
$289$	−14.4187	−0.848159
$290$	0	0
$291$	−12.2695	−0.719249
$292$	0	0
$293$	−10.7943	−0.630613	−0.315306	−	0.948990i	$-0.602107\pi$
$293$	−10.7943	−0.630613	−0.315306	+	0.948990i	$0.602107\pi$
$294$	0	0
$295$	12.0071	0.699080
$296$	0	0
$297$	41.5560	2.41133
$298$	0	0
$299$	−0.717368	−0.0414865
$300$	0	0
$301$	−10.3813	−0.598366
$302$	0	0
$303$	49.5886	2.84879
$304$	0	0
$305$	−6.07867	−0.348064
$306$	0	0
$307$	25.4033	1.44985	0.724923	−	0.688830i	$-0.241876\pi$
$307$	25.4033	1.44985	0.724923	+	0.688830i	$0.241876\pi$
$308$	0	0
$309$	42.6063	2.42379
$310$	0	0
$311$	12.8773	0.730204	0.365102	−	0.930968i	$-0.381034\pi$
$311$	12.8773	0.730204	0.365102	+	0.930968i	$0.381034\pi$
$312$	0	0
$313$	−10.9510	−0.618985	−0.309493	−	0.950902i	$-0.600159\pi$
$313$	−10.9510	−0.618985	−0.309493	+	0.950902i	$0.600159\pi$
$314$	0	0
$315$	9.51263	0.535976
$316$	0	0
$317$	7.09629	0.398567	0.199284	−	0.979942i	$-0.436138\pi$
$317$	7.09629	0.398567	0.199284	+	0.979942i	$0.436138\pi$
$318$	0	0
$319$	16.8598	0.943965
$320$	0	0
$321$	11.0317	0.615729
$322$	0	0
$323$	0.991063	0.0551442
$324$	0	0
$325$	9.53948	0.529155
$326$	0	0
$327$	−23.3792	−1.29287
$328$	0	0
$329$	−8.87119	−0.489085
$330$	0	0
$331$	10.6649	0.586197	0.293099	−	0.956082i	$-0.405314\pi$
$331$	10.6649	0.586197	0.293099	+	0.956082i	$0.405314\pi$
$332$	0	0
$333$	40.0978	2.19734
$334$	0	0
$335$	−14.5946	−0.797388
$336$	0	0
$337$	−17.1144	−0.932281	−0.466141	−	0.884711i	$-0.654356\pi$
$337$	−17.1144	−0.932281	−0.466141	+	0.884711i	$0.654356\pi$
$338$	0	0
$339$	−40.3529	−2.19167
$340$	0	0
$341$	37.5623	2.03411
$342$	0	0
$343$	−17.7288	−0.957264
$344$	0	0
$345$	0.924853	0.0497924
$346$	0	0
$347$	16.9839	0.911745	0.455873	−	0.890045i	$-0.349327\pi$
$347$	16.9839	0.911745	0.455873	+	0.890045i	$0.349327\pi$
$348$	0	0
$349$	25.7982	1.38094	0.690472	−	0.723359i	$-0.257403\pi$
$349$	25.7982	1.38094	0.690472	+	0.723359i	$0.257403\pi$
$350$	0	0
$351$	−20.5175	−1.09514
$352$	0	0
$353$	−17.9535	−0.955568	−0.477784	−	0.878477i	$-0.658560\pi$
$353$	−17.9535	−0.955568	−0.477784	+	0.878477i	$0.658560\pi$
$354$	0	0
$355$	13.6978	0.727002
$356$	0	0
$357$	−7.20596	−0.381380
$358$	0	0
$359$	18.7493	0.989549	0.494774	−	0.869022i	$-0.335251\pi$
$359$	18.7493	0.989549	0.494774	+	0.869022i	$0.335251\pi$
$360$	0	0
$361$	−18.6195	−0.979973
$362$	0	0
$363$	43.3755	2.27662
$364$	0	0
$365$	2.89021	0.151281
$366$	0	0
$367$	−30.7285	−1.60401	−0.802007	−	0.597314i	$-0.796234\pi$
$367$	−30.7285	−1.60401	−0.802007	+	0.597314i	$0.796234\pi$
$368$	0	0
$369$	−6.27356	−0.326588
$370$	0	0
$371$	16.4460	0.853832
$372$	0	0
$373$	−8.58613	−0.444573	−0.222286	−	0.974981i	$-0.571352\pi$
$373$	−8.58613	−0.444573	−0.222286	+	0.974981i	$0.571352\pi$
$374$	0	0
$375$	−28.4162	−1.46741
$376$	0	0
$377$	−8.32417	−0.428716
$378$	0	0
$379$	13.0968	0.672737	0.336369	−	0.941730i	$-0.390801\pi$
$379$	13.0968	0.672737	0.336369	+	0.941730i	$0.390801\pi$
$380$	0	0
$381$	33.5157	1.71706
$382$	0	0
$383$	−36.3181	−1.85577	−0.927885	−	0.372867i	$-0.878375\pi$
$383$	−36.3181	−1.85577	−0.927885	+	0.372867i	$0.878375\pi$
$384$	0	0
$385$	8.34790	0.425448
$386$	0	0
$387$	−39.5578	−2.01084
$388$	0	0
$389$	18.8570	0.956087	0.478044	−	0.878336i	$-0.341346\pi$
$389$	18.8570	0.956087	0.478044	+	0.878336i	$0.341346\pi$
$390$	0	0
$391$	−0.460957	−0.0233116
$392$	0	0
$393$	4.21003	0.212368
$394$	0	0
$395$	−3.19399	−0.160707
$396$	0	0
$397$	−29.9892	−1.50512	−0.752558	−	0.658526i	$-0.771180\pi$
$397$	−29.9892	−1.50512	−0.752558	+	0.658526i	$0.771180\pi$
$398$	0	0
$399$	−2.76665	−0.138506
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−18.5456	−0.923824
$404$	0	0
$405$	7.60793	0.378041
$406$	0	0
$407$	35.1882	1.74421
$408$	0	0
$409$	−38.1520	−1.88649	−0.943247	−	0.332092i	$-0.892246\pi$
$409$	−38.1520	−1.88649	−0.943247	+	0.332092i	$0.892246\pi$
$410$	0	0
$411$	−33.6382	−1.65925
$412$	0	0
$413$	16.7062	0.822060
$414$	0	0
$415$	−0.404938	−0.0198776
$416$	0	0
$417$	58.9565	2.88711
$418$	0	0
$419$	−21.0626	−1.02897	−0.514487	−	0.857498i	$-0.672018\pi$
$419$	−21.0626	−1.02897	−0.514487	+	0.857498i	$0.672018\pi$
$420$	0	0
$421$	−22.1264	−1.07837	−0.539187	−	0.842186i	$-0.681268\pi$
$421$	−22.1264	−1.07837	−0.539187	+	0.842186i	$0.681268\pi$
$422$	0	0
$423$	−33.8037	−1.64359
$424$	0	0
$425$	6.12975	0.297337
$426$	0	0
$427$	−8.45765	−0.409294
$428$	0	0
$429$	−37.5000	−1.81052
$430$	0	0
$431$	30.8688	1.48690	0.743450	−	0.668792i	$-0.233188\pi$
$431$	30.8688	1.48690	0.743450	+	0.668792i	$0.233188\pi$
$432$	0	0
$433$	18.6379	0.895679	0.447840	−	0.894114i	$-0.352194\pi$
$433$	18.6379	0.895679	0.447840	+	0.894114i	$0.352194\pi$
$434$	0	0
$435$	10.7318	0.514549
$436$	0	0
$437$	−0.176980	−0.00846608
$438$	0	0
$439$	−1.09195	−0.0521159	−0.0260579	−	0.999660i	$-0.508295\pi$
$439$	−1.09195	−0.0521159	−0.0260579	+	0.999660i	$0.508295\pi$
$440$	0	0
$441$	−27.1600	−1.29333
$442$	0	0
$443$	−25.9327	−1.23210	−0.616049	−	0.787708i	$-0.711268\pi$
$443$	−25.9327	−1.23210	−0.616049	+	0.787708i	$0.711268\pi$
$444$	0	0
$445$	−11.5515	−0.547595
$446$	0	0
$447$	20.4645	0.967939
$448$	0	0
$449$	31.3438	1.47921	0.739604	−	0.673042i	$-0.235013\pi$
$449$	31.3438	1.47921	0.739604	+	0.673042i	$0.235013\pi$
$450$	0	0
$451$	−5.50542	−0.259240
$452$	0	0
$453$	−7.28902	−0.342468
$454$	0	0
$455$	−4.12161	−0.193224
$456$	0	0
$457$	−8.93926	−0.418161	−0.209081	−	0.977898i	$-0.567047\pi$
$457$	−8.93926	−0.418161	−0.209081	+	0.977898i	$0.567047\pi$
$458$	0	0
$459$	−13.1838	−0.615369
$460$	0	0
$461$	−0.328419	−0.0152960	−0.00764801	−	0.999971i	$-0.502434\pi$
$461$	−0.328419	−0.0152960	−0.00764801	+	0.999971i	$0.502434\pi$
$462$	0	0
$463$	−25.5393	−1.18691	−0.593457	−	0.804866i	$-0.702237\pi$
$463$	−25.5393	−1.18691	−0.593457	+	0.804866i	$0.702237\pi$
$464$	0	0
$465$	23.9096	1.10878
$466$	0	0
$467$	−12.2274	−0.565814	−0.282907	−	0.959147i	$-0.591299\pi$
$467$	−12.2274	−0.565814	−0.282907	+	0.959147i	$0.591299\pi$
$468$	0	0
$469$	−20.3064	−0.937663
$470$	0	0
$471$	−57.7195	−2.65957
$472$	0	0
$473$	−34.7143	−1.59617
$474$	0	0
$475$	2.35345	0.107984
$476$	0	0
$477$	62.6673	2.86934
$478$	0	0
$479$	21.0321	0.960982	0.480491	−	0.877000i	$-0.340458\pi$
$479$	21.0321	0.960982	0.480491	+	0.877000i	$0.340458\pi$
$480$	0	0
$481$	−17.3734	−0.792161
$482$	0	0
$483$	1.28681	0.0585518
$484$	0	0
$485$	−4.50940	−0.204761
$486$	0	0
$487$	20.6376	0.935180	0.467590	−	0.883945i	$-0.345122\pi$
$487$	20.6376	0.935180	0.467590	+	0.883945i	$0.345122\pi$
$488$	0	0
$489$	−30.1354	−1.36277
$490$	0	0
$491$	−2.46382	−0.111191	−0.0555954	−	0.998453i	$-0.517706\pi$
$491$	−2.46382	−0.111191	−0.0555954	+	0.998453i	$0.517706\pi$
$492$	0	0
$493$	−5.34884	−0.240899
$494$	0	0
$495$	31.8097	1.42974
$496$	0	0
$497$	19.0586	0.854895
$498$	0	0
$499$	30.8562	1.38131	0.690656	−	0.723183i	$-0.257322\pi$
$499$	30.8562	1.38131	0.690656	+	0.723183i	$0.257322\pi$
$500$	0	0
$501$	−31.8355	−1.42231
$502$	0	0
$503$	−6.55532	−0.292287	−0.146144	−	0.989263i	$-0.546686\pi$
$503$	−6.55532	−0.292287	−0.146144	+	0.989263i	$0.546686\pi$
$504$	0	0
$505$	18.2253	0.811015
$506$	0	0
$507$	−19.9853	−0.887578
$508$	0	0
$509$	40.4340	1.79221	0.896104	−	0.443845i	$-0.146386\pi$
$509$	40.4340	1.79221	0.896104	+	0.443845i	$0.146386\pi$
$510$	0	0
$511$	4.02134	0.177894
$512$	0	0
$513$	−5.06180	−0.223484
$514$	0	0
$515$	15.6591	0.690023
$516$	0	0
$517$	−29.6647	−1.30465
$518$	0	0
$519$	45.8804	2.01393
$520$	0	0
$521$	−3.05286	−0.133748	−0.0668742	−	0.997761i	$-0.521303\pi$
$521$	−3.05286	−0.133748	−0.0668742	+	0.997761i	$0.521303\pi$
$522$	0	0
$523$	−21.8863	−0.957021	−0.478510	−	0.878082i	$-0.658823\pi$
$523$	−21.8863	−0.957021	−0.478510	+	0.878082i	$0.658823\pi$
$524$	0	0
$525$	−17.1118	−0.746821
$526$	0	0
$527$	−11.9168	−0.519105
$528$	0	0
$529$	−22.9177	−0.996421
$530$	0	0
$531$	63.6591	2.76257
$532$	0	0
$533$	2.71819	0.117738
$534$	0	0
$535$	4.05448	0.175290
$536$	0	0
$537$	−51.5558	−2.22480
$538$	0	0
$539$	−23.8345	−1.02662
$540$	0	0
$541$	25.6055	1.10086	0.550432	−	0.834880i	$-0.314463\pi$
$541$	25.6055	1.10086	0.550432	+	0.834880i	$0.314463\pi$
$542$	0	0
$543$	−54.1280	−2.32286
$544$	0	0
$545$	−8.59258	−0.368065
$546$	0	0
$547$	41.6448	1.78060	0.890302	−	0.455371i	$-0.150493\pi$
$547$	41.6448	1.78060	0.890302	+	0.455371i	$0.150493\pi$
$548$	0	0
$549$	−32.2279	−1.37545
$550$	0	0
$551$	−2.05363	−0.0874875
$552$	0	0
$553$	−4.44401	−0.188979
$554$	0	0
$555$	22.3984	0.950758
$556$	0	0
$557$	−15.2172	−0.644775	−0.322388	−	0.946608i	$-0.604485\pi$
$557$	−15.2172	−0.644775	−0.322388	+	0.946608i	$0.604485\pi$
$558$	0	0
$559$	17.1395	0.724923
$560$	0	0
$561$	−24.0963	−1.01735
$562$	0	0
$563$	32.1711	1.35585	0.677925	−	0.735131i	$-0.262879\pi$
$563$	32.1711	1.35585	0.677925	+	0.735131i	$0.262879\pi$
$564$	0	0
$565$	−14.8309	−0.623942
$566$	0	0
$567$	10.5854	0.444546
$568$	0	0
$569$	29.5267	1.23782	0.618911	−	0.785461i	$-0.287574\pi$
$569$	29.5267	1.23782	0.618911	+	0.785461i	$0.287574\pi$
$570$	0	0
$571$	9.85847	0.412564	0.206282	−	0.978493i	$-0.433864\pi$
$571$	9.85847	0.412564	0.206282	+	0.978493i	$0.433864\pi$
$572$	0	0
$573$	1.68541	0.0704089
$574$	0	0
$575$	−1.09462	−0.0456490
$576$	0	0
$577$	−5.21817	−0.217235	−0.108618	−	0.994084i	$-0.534642\pi$
$577$	−5.21817	−0.217235	−0.108618	+	0.994084i	$0.534642\pi$
$578$	0	0
$579$	63.5583	2.64139
$580$	0	0
$581$	−0.563417	−0.0233745
$582$	0	0
$583$	54.9943	2.27763
$584$	0	0
$585$	−15.7054	−0.649337
$586$	0	0
$587$	−39.8103	−1.64315	−0.821574	−	0.570102i	$-0.806904\pi$
$587$	−39.8103	−1.64315	−0.821574	+	0.570102i	$0.806904\pi$
$588$	0	0
$589$	−4.57533	−0.188523
$590$	0	0
$591$	−23.9336	−0.984498
$592$	0	0
$593$	22.3199	0.916567	0.458283	−	0.888806i	$-0.348464\pi$
$593$	22.3199	0.916567	0.458283	+	0.888806i	$0.348464\pi$
$594$	0	0
$595$	−2.64841	−0.108574
$596$	0	0
$597$	−26.0337	−1.06549
$598$	0	0
$599$	−25.8326	−1.05549	−0.527747	−	0.849402i	$-0.676963\pi$
$599$	−25.8326	−1.05549	−0.527747	+	0.849402i	$0.676963\pi$
$600$	0	0
$601$	12.6739	0.516979	0.258489	−	0.966014i	$-0.416775\pi$
$601$	12.6739	0.516979	0.258489	+	0.966014i	$0.416775\pi$
$602$	0	0
$603$	−77.3776	−3.15106
$604$	0	0
$605$	15.9418	0.648126
$606$	0	0
$607$	10.4701	0.424968	0.212484	−	0.977165i	$-0.431845\pi$
$607$	10.4701	0.424968	0.212484	+	0.977165i	$0.431845\pi$
$608$	0	0
$609$	14.9318	0.605068
$610$	0	0
$611$	14.6464	0.592528
$612$	0	0
$613$	−25.8178	−1.04277	−0.521385	−	0.853322i	$-0.674584\pi$
$613$	−25.8178	−1.04277	−0.521385	+	0.853322i	$0.674584\pi$
$614$	0	0
$615$	−3.50437	−0.141310
$616$	0	0
$617$	−33.2086	−1.33693	−0.668463	−	0.743745i	$-0.733048\pi$
$617$	−33.2086	−1.33693	−0.668463	+	0.743745i	$0.733048\pi$
$618$	0	0
$619$	−13.6054	−0.546846	−0.273423	−	0.961894i	$-0.588156\pi$
$619$	−13.6054	−0.546846	−0.273423	+	0.961894i	$0.588156\pi$
$620$	0	0
$621$	2.35431	0.0944753
$622$	0	0
$623$	−16.0724	−0.643927
$624$	0	0
$625$	8.63245	0.345298
$626$	0	0
$627$	−9.25152	−0.369470
$628$	0	0
$629$	−11.1636	−0.445122
$630$	0	0
$631$	−2.20131	−0.0876326	−0.0438163	−	0.999040i	$-0.513952\pi$
$631$	−2.20131	−0.0876326	−0.0438163	+	0.999040i	$0.513952\pi$
$632$	0	0
$633$	−22.2915	−0.886009
$634$	0	0
$635$	12.3180	0.488827
$636$	0	0
$637$	11.7678	0.466257
$638$	0	0
$639$	72.6228	2.87291
$640$	0	0
$641$	−10.6624	−0.421138	−0.210569	−	0.977579i	$-0.567532\pi$
$641$	−10.6624	−0.421138	−0.210569	+	0.977579i	$0.567532\pi$
$642$	0	0
$643$	−18.8318	−0.742652	−0.371326	−	0.928502i	$-0.621097\pi$
$643$	−18.8318	−0.742652	−0.371326	+	0.928502i	$0.621097\pi$
$644$	0	0
$645$	−22.0968	−0.870059
$646$	0	0
$647$	−15.7799	−0.620371	−0.310186	−	0.950676i	$-0.600391\pi$
$647$	−15.7799	−0.620371	−0.310186	+	0.950676i	$0.600391\pi$
$648$	0	0
$649$	55.8646	2.19288
$650$	0	0
$651$	33.2670	1.30384
$652$	0	0
$653$	−5.88892	−0.230451	−0.115226	−	0.993339i	$-0.536759\pi$
$653$	−5.88892	−0.230451	−0.115226	+	0.993339i	$0.536759\pi$
$654$	0	0
$655$	1.54731	0.0604586
$656$	0	0
$657$	15.3233	0.597819
$658$	0	0
$659$	−21.3246	−0.830689	−0.415344	−	0.909664i	$-0.636339\pi$
$659$	−21.3246	−0.830689	−0.415344	+	0.909664i	$0.636339\pi$
$660$	0	0
$661$	40.4226	1.57226	0.786128	−	0.618064i	$-0.212083\pi$
$661$	40.4226	1.57226	0.786128	+	0.618064i	$0.212083\pi$
$662$	0	0
$663$	11.8971	0.462043
$664$	0	0
$665$	−1.01683	−0.0394309
$666$	0	0
$667$	0.955171	0.0369844
$668$	0	0
$669$	23.4874	0.908077
$670$	0	0
$671$	−28.2818	−1.09181
$672$	0	0
$673$	39.3982	1.51869	0.759345	−	0.650688i	$-0.225519\pi$
$673$	39.3982	1.51869	0.759345	+	0.650688i	$0.225519\pi$
$674$	0	0
$675$	−31.3074	−1.20502
$676$	0	0
$677$	7.04746	0.270856	0.135428	−	0.990787i	$-0.456759\pi$
$677$	7.04746	0.270856	0.135428	+	0.990787i	$0.456759\pi$
$678$	0	0
$679$	−6.27422	−0.240783
$680$	0	0
$681$	21.7772	0.834505
$682$	0	0
$683$	2.67855	0.102492	0.0512459	−	0.998686i	$-0.483681\pi$
$683$	2.67855	0.102492	0.0512459	+	0.998686i	$0.483681\pi$
$684$	0	0
$685$	−12.3630	−0.472367
$686$	0	0
$687$	−55.6316	−2.12248
$688$	0	0
$689$	−27.1523	−1.03442
$690$	0	0
$691$	20.1086	0.764966	0.382483	−	0.923963i	$-0.375069\pi$
$691$	20.1086	0.764966	0.382483	+	0.923963i	$0.375069\pi$
$692$	0	0
$693$	44.2588	1.68125
$694$	0	0
$695$	21.6683	0.821926
$696$	0	0
$697$	1.74662	0.0661579
$698$	0	0
$699$	−64.6367	−2.44479
$700$	0	0
$701$	−13.3827	−0.505459	−0.252730	−	0.967537i	$-0.581328\pi$
$701$	−13.3827	−0.505459	−0.252730	+	0.967537i	$0.581328\pi$
$702$	0	0
$703$	−4.28615	−0.161655
$704$	0	0
$705$	−18.8825	−0.711157
$706$	0	0
$707$	25.3580	0.953687
$708$	0	0
$709$	28.8273	1.08263	0.541315	−	0.840820i	$-0.317927\pi$
$709$	28.8273	1.08263	0.541315	+	0.840820i	$0.317927\pi$
$710$	0	0
$711$	−16.9339	−0.635071
$712$	0	0
$713$	2.12805	0.0796961
$714$	0	0
$715$	−13.7824	−0.515432
$716$	0	0
$717$	−83.8293	−3.13066
$718$	0	0
$719$	−0.300627	−0.0112115	−0.00560574	−	0.999984i	$-0.501784\pi$
$719$	−0.300627	−0.0112115	−0.00560574	+	0.999984i	$0.501784\pi$
$720$	0	0
$721$	21.7875	0.811410
$722$	0	0
$723$	−6.76226	−0.251491
$724$	0	0
$725$	−12.7017	−0.471731
$726$	0	0
$727$	−44.4033	−1.64683	−0.823414	−	0.567441i	$-0.807933\pi$
$727$	−44.4033	−1.64683	−0.823414	+	0.567441i	$0.807933\pi$
$728$	0	0
$729$	−32.5703	−1.20631
$730$	0	0
$731$	11.0133	0.407341
$732$	0	0
$733$	3.12471	0.115414	0.0577069	−	0.998334i	$-0.481621\pi$
$733$	3.12471	0.115414	0.0577069	+	0.998334i	$0.481621\pi$
$734$	0	0
$735$	−15.1714	−0.559606
$736$	0	0
$737$	−67.9034	−2.50125
$738$	0	0
$739$	−27.5191	−1.01231	−0.506154	−	0.862443i	$-0.668933\pi$
$739$	−27.5191	−1.01231	−0.506154	+	0.862443i	$0.668933\pi$
$740$	0	0
$741$	4.56775	0.167800
$742$	0	0
$743$	−25.7083	−0.943146	−0.471573	−	0.881827i	$-0.656314\pi$
$743$	−25.7083	−0.943146	−0.471573	+	0.881827i	$0.656314\pi$
$744$	0	0
$745$	7.52133	0.275560
$746$	0	0
$747$	−2.14690	−0.0785509
$748$	0	0
$749$	5.64126	0.206127
$750$	0	0
$751$	28.6094	1.04397	0.521986	−	0.852954i	$-0.325191\pi$
$751$	28.6094	1.04397	0.521986	+	0.852954i	$0.325191\pi$
$752$	0	0
$753$	−33.9333	−1.23660
$754$	0	0
$755$	−2.67894	−0.0974965
$756$	0	0
$757$	4.03174	0.146536	0.0732680	−	0.997312i	$-0.476657\pi$
$757$	4.03174	0.146536	0.0732680	+	0.997312i	$0.476657\pi$
$758$	0	0
$759$	4.30301	0.156189
$760$	0	0
$761$	−30.8437	−1.11808	−0.559042	−	0.829139i	$-0.688831\pi$
$761$	−30.8437	−1.11808	−0.559042	+	0.829139i	$0.688831\pi$
$762$	0	0
$763$	−11.9554	−0.432815
$764$	0	0
$765$	−10.0918	−0.364868
$766$	0	0
$767$	−27.5820	−0.995930
$768$	0	0
$769$	−42.7342	−1.54104	−0.770518	−	0.637418i	$-0.780002\pi$
$769$	−42.7342	−1.54104	−0.770518	+	0.637418i	$0.780002\pi$
$770$	0	0
$771$	73.9626	2.66370
$772$	0	0
$773$	−25.8673	−0.930380	−0.465190	−	0.885211i	$-0.654014\pi$
$773$	−25.8673	−0.930380	−0.465190	+	0.885211i	$0.654014\pi$
$774$	0	0
$775$	−28.2986	−1.01651
$776$	0	0
$777$	31.1643	1.11801
$778$	0	0
$779$	0.670596	0.0240266
$780$	0	0
$781$	63.7308	2.28047
$782$	0	0
$783$	27.3189	0.976296
$784$	0	0
$785$	−21.2136	−0.757147
$786$	0	0
$787$	23.2476	0.828688	0.414344	−	0.910120i	$-0.364011\pi$
$787$	23.2476	0.828688	0.414344	+	0.910120i	$0.364011\pi$
$788$	0	0
$789$	16.3510	0.582112
$790$	0	0
$791$	−20.6352	−0.733704
$792$	0	0
$793$	13.9636	0.495862
$794$	0	0
$795$	35.0056	1.24152
$796$	0	0
$797$	−28.6632	−1.01530	−0.507652	−	0.861562i	$-0.669486\pi$
$797$	−28.6632	−1.01530	−0.507652	+	0.861562i	$0.669486\pi$
$798$	0	0
$799$	9.41127	0.332947
$800$	0	0
$801$	−61.2438	−2.16394
$802$	0	0
$803$	13.4471	0.474538
$804$	0	0
$805$	0.472941	0.0166690
$806$	0	0
$807$	−11.2660	−0.396582
$808$	0	0
$809$	53.0838	1.86633	0.933164	−	0.359451i	$-0.117036\pi$
$809$	53.0838	1.86633	0.933164	+	0.359451i	$0.117036\pi$
$810$	0	0
$811$	−9.52060	−0.334314	−0.167157	−	0.985930i	$-0.553459\pi$
$811$	−9.52060	−0.334314	−0.167157	+	0.985930i	$0.553459\pi$
$812$	0	0
$813$	−30.5614	−1.07184
$814$	0	0
$815$	−11.0757	−0.387964
$816$	0	0
$817$	4.22843	0.147934
$818$	0	0
$819$	−21.8519	−0.763568
$820$	0	0
$821$	−55.7535	−1.94581	−0.972905	−	0.231204i	$-0.925734\pi$
$821$	−55.7535	−1.94581	−0.972905	+	0.231204i	$0.925734\pi$
$822$	0	0
$823$	−13.9314	−0.485618	−0.242809	−	0.970074i	$-0.578069\pi$
$823$	−13.9314	−0.485618	−0.242809	+	0.970074i	$0.578069\pi$
$824$	0	0
$825$	−57.2209	−1.99218
$826$	0	0
$827$	2.83651	0.0986350	0.0493175	−	0.998783i	$-0.484295\pi$
$827$	2.83651	0.0986350	0.0493175	+	0.998783i	$0.484295\pi$
$828$	0	0
$829$	43.9663	1.52701	0.763506	−	0.645801i	$-0.223476\pi$
$829$	43.9663	1.52701	0.763506	+	0.645801i	$0.223476\pi$
$830$	0	0
$831$	57.0621	1.97946
$832$	0	0
$833$	7.56160	0.261994
$834$	0	0
$835$	−11.7005	−0.404913
$836$	0	0
$837$	60.8644	2.10378
$838$	0	0
$839$	27.3679	0.944844	0.472422	−	0.881372i	$-0.343380\pi$
$839$	27.3679	0.944844	0.472422	+	0.881372i	$0.343380\pi$
$840$	0	0
$841$	−17.9164	−0.617808
$842$	0	0
$843$	88.9588	3.06391
$844$	0	0
$845$	−7.34519	−0.252682
$846$	0	0
$847$	22.1809	0.762143
$848$	0	0
$849$	81.7558	2.80585
$850$	0	0
$851$	1.99355	0.0683379
$852$	0	0
$853$	4.54076	0.155473	0.0777363	−	0.996974i	$-0.475231\pi$
$853$	4.54076	0.155473	0.0777363	+	0.996974i	$0.475231\pi$
$854$	0	0
$855$	−3.87462	−0.132509
$856$	0	0
$857$	1.96044	0.0669673	0.0334836	−	0.999439i	$-0.489340\pi$
$857$	1.96044	0.0669673	0.0334836	+	0.999439i	$0.489340\pi$
$858$	0	0
$859$	19.6657	0.670985	0.335492	−	0.942043i	$-0.391097\pi$
$859$	19.6657	0.670985	0.335492	+	0.942043i	$0.391097\pi$
$860$	0	0
$861$	−4.87586	−0.166169
$862$	0	0
$863$	−4.45060	−0.151500	−0.0757501	−	0.997127i	$-0.524135\pi$
$863$	−4.45060	−0.151500	−0.0757501	+	0.997127i	$0.524135\pi$
$864$	0	0
$865$	16.8624	0.573340
$866$	0	0
$867$	−42.7017	−1.45023
$868$	0	0
$869$	−14.8605	−0.504108
$870$	0	0
$871$	33.5259	1.13598
$872$	0	0
$873$	−23.9079	−0.809160
$874$	0	0
$875$	−14.5312	−0.491243
$876$	0	0
$877$	−42.1305	−1.42265	−0.711323	−	0.702865i	$-0.751904\pi$
$877$	−42.1305	−1.42265	−0.711323	+	0.702865i	$0.751904\pi$
$878$	0	0
$879$	−31.9680	−1.07825
$880$	0	0
$881$	0.878824	0.0296083	0.0148042	−	0.999890i	$-0.495288\pi$
$881$	0.878824	0.0296083	0.0148042	+	0.999890i	$0.495288\pi$
$882$	0	0
$883$	17.8313	0.600071	0.300036	−	0.953928i	$-0.403001\pi$
$883$	17.8313	0.600071	0.300036	+	0.953928i	$0.403001\pi$
$884$	0	0
$885$	35.5596	1.19532
$886$	0	0
$887$	30.9292	1.03850	0.519250	−	0.854622i	$-0.326211\pi$
$887$	30.9292	1.03850	0.519250	+	0.854622i	$0.326211\pi$
$888$	0	0
$889$	17.1389	0.574820
$890$	0	0
$891$	35.3970	1.18584
$892$	0	0
$893$	3.61336	0.120916
$894$	0	0
$895$	−18.9483	−0.633373
$896$	0	0
$897$	−2.12452	−0.0709358
$898$	0	0
$899$	24.6934	0.823571
$900$	0	0
$901$	−17.4472	−0.581250
$902$	0	0
$903$	−30.7447	−1.02312
$904$	0	0
$905$	−19.8937	−0.661289
$906$	0	0
$907$	−40.8217	−1.35546	−0.677731	−	0.735310i	$-0.737037\pi$
$907$	−40.8217	−1.35546	−0.677731	+	0.735310i	$0.737037\pi$
$908$	0	0
$909$	96.6267	3.20491
$910$	0	0
$911$	−8.80690	−0.291786	−0.145893	−	0.989300i	$-0.546605\pi$
$911$	−8.80690	−0.291786	−0.145893	+	0.989300i	$0.546605\pi$
$912$	0	0
$913$	−1.88403	−0.0623523
$914$	0	0
$915$	−18.0023	−0.595138
$916$	0	0
$917$	2.15288	0.0710943
$918$	0	0
$919$	19.4416	0.641319	0.320660	−	0.947195i	$-0.396095\pi$
$919$	19.4416	0.641319	0.320660	+	0.947195i	$0.396095\pi$
$920$	0	0
$921$	75.2333	2.47902
$922$	0	0
$923$	−31.4658	−1.03571
$924$	0	0
$925$	−26.5099	−0.871641
$926$	0	0
$927$	83.0213	2.72678
$928$	0	0
$929$	2.28047	0.0748196	0.0374098	−	0.999300i	$-0.488089\pi$
$929$	2.28047	0.0748196	0.0374098	+	0.999300i	$0.488089\pi$
$930$	0	0
$931$	2.90320	0.0951484
$932$	0	0
$933$	38.1367	1.24854
$934$	0	0
$935$	−8.85611	−0.289626
$936$	0	0
$937$	31.4892	1.02871	0.514353	−	0.857579i	$-0.328032\pi$
$937$	31.4892	1.02871	0.514353	+	0.857579i	$0.328032\pi$
$938$	0	0
$939$	−32.4318	−1.05837
$940$	0	0
$941$	−41.2715	−1.34541	−0.672706	−	0.739910i	$-0.734868\pi$
$941$	−41.2715	−1.34541	−0.672706	+	0.739910i	$0.734868\pi$
$942$	0	0
$943$	−0.311903	−0.0101570
$944$	0	0
$945$	13.5266	0.440020
$946$	0	0
$947$	18.2225	0.592152	0.296076	−	0.955164i	$-0.404322\pi$
$947$	18.2225	0.592152	0.296076	+	0.955164i	$0.404322\pi$
$948$	0	0
$949$	−6.63924	−0.215519
$950$	0	0
$951$	21.0160	0.681491
$952$	0	0
$953$	−11.1493	−0.361162	−0.180581	−	0.983560i	$-0.557798\pi$
$953$	−11.1493	−0.361162	−0.180581	+	0.983560i	$0.557798\pi$
$954$	0	0
$955$	0.619438	0.0200446
$956$	0	0
$957$	49.9311	1.61404
$958$	0	0
$959$	−17.2015	−0.555465
$960$	0	0
$961$	24.0151	0.774679
$962$	0	0
$963$	21.4960	0.692700
$964$	0	0
$965$	23.3596	0.751972
$966$	0	0
$967$	26.7881	0.861448	0.430724	−	0.902484i	$-0.358258\pi$
$967$	26.7881	0.861448	0.430724	+	0.902484i	$0.358258\pi$
$968$	0	0
$969$	2.93509	0.0942885
$970$	0	0
$971$	40.2602	1.29201	0.646006	−	0.763333i	$-0.276438\pi$
$971$	40.2602	1.29201	0.646006	+	0.763333i	$0.276438\pi$
$972$	0	0
$973$	30.1485	0.966517
$974$	0	0
$975$	28.2517	0.904777
$976$	0	0
$977$	−6.97147	−0.223037	−0.111519	−	0.993762i	$-0.535571\pi$
$977$	−6.97147	−0.223037	−0.111519	+	0.993762i	$0.535571\pi$
$978$	0	0
$979$	−53.7451	−1.71770
$980$	0	0
$981$	−45.5561	−1.45449
$982$	0	0
$983$	−15.5529	−0.496059	−0.248030	−	0.968752i	$-0.579783\pi$
$983$	−15.5529	−0.496059	−0.248030	+	0.968752i	$0.579783\pi$
$984$	0	0
$985$	−8.79633	−0.280274
$986$	0	0
$987$	−26.2725	−0.836263
$988$	0	0
$989$	−1.96670	−0.0625374
$990$	0	0
$991$	3.86885	0.122898	0.0614490	−	0.998110i	$-0.480428\pi$
$991$	3.86885	0.122898	0.0614490	+	0.998110i	$0.480428\pi$
$992$	0	0
$993$	31.5847	1.00231
$994$	0	0
$995$	−9.56817	−0.303331
$996$	0	0
$997$	−3.71255	−0.117578	−0.0587888	−	0.998270i	$-0.518724\pi$
$997$	−3.71255	−0.117578	−0.0587888	+	0.998270i	$0.518724\pi$
$998$	0	0
$999$	57.0174	1.80395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6416.2.a.k.1.12		12
4.3	odd	2		401.2.a.a.1.11	✓	12
12.11	even	2		3609.2.a.b.1.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
401.2.a.a.1.11	✓	12	4.3	odd	2
3609.2.a.b.1.2		12	12.11	even	2
6416.2.a.k.1.12		12	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:	\( N \)	\(=\)	\( 6416 = 2^{4} \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6416.a (trivial)

\(f(q)\)	\(=\)	\(q+2.96155 q^{3} +1.08846 q^{5} +1.51444 q^{7} +5.77079 q^{9} +O(q^{10})\)	\(q+2.96155 q^{3} +1.08846 q^{5} +1.51444 q^{7} +5.77079 q^{9} +5.06421 q^{11} -2.50035 q^{13} +3.22353 q^{15} -1.60664 q^{17} -0.616854 q^{19} +4.48511 q^{21} +0.286907 q^{23} -3.81526 q^{25} +8.20583 q^{27} +3.32920 q^{29} +7.41721 q^{31} +14.9979 q^{33} +1.64841 q^{35} +6.94840 q^{37} -7.40492 q^{39} -1.08712 q^{41} -6.85484 q^{43} +6.28127 q^{45} -5.85772 q^{47} -4.70646 q^{49} -4.75816 q^{51} +10.8594 q^{53} +5.51219 q^{55} -1.82684 q^{57} +11.0313 q^{59} -5.58465 q^{61} +8.73954 q^{63} -2.72153 q^{65} -13.4085 q^{67} +0.849690 q^{69} +12.5845 q^{71} +2.65532 q^{73} -11.2991 q^{75} +7.66946 q^{77} -2.93442 q^{79} +6.98963 q^{81} -0.372029 q^{83} -1.74877 q^{85} +9.85960 q^{87} -10.6127 q^{89} -3.78664 q^{91} +21.9665 q^{93} -0.671420 q^{95} -4.14292 q^{97} +29.2245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{3} - 7 q^{5} + 20 q^{7} + 3 q^{9}+O(q^{10}) \) 12 * q + 5 * q^3 - 7 * q^5 + 20 * q^7 + 3 * q^9	\( 12 q + 5 q^{3} - 7 q^{5} + 20 q^{7} + 3 q^{9} + 11 q^{11} - 11 q^{13} + 11 q^{15} + q^{17} + 34 q^{19} - 3 q^{21} + 7 q^{23} + 7 q^{25} + 2 q^{27} - 6 q^{29} + 52 q^{31} + 4 q^{33} - 12 q^{35} + 3 q^{37} + 24 q^{39} - 16 q^{41} + 2 q^{43} - 23 q^{45} + 3 q^{47} + 6 q^{49} + 16 q^{51} + 19 q^{53} + 43 q^{55} + 11 q^{57} + q^{59} - 24 q^{61} + 11 q^{63} + 13 q^{65} - 6 q^{67} + 29 q^{69} + 15 q^{71} - 20 q^{73} - 31 q^{75} + 38 q^{77} + 53 q^{79} - 8 q^{81} - 17 q^{83} + 7 q^{85} + 5 q^{87} - q^{89} + 6 q^{91} + 44 q^{93} - 34 q^{95} + 12 q^{97}+O(q^{100}) \) 12 * q + 5 * q^3 - 7 * q^5 + 20 * q^7 + 3 * q^9 + 11 * q^11 - 11 * q^13 + 11 * q^15 + q^17 + 34 * q^19 - 3 * q^21 + 7 * q^23 + 7 * q^25 + 2 * q^27 - 6 * q^29 + 52 * q^31 + 4 * q^33 - 12 * q^35 + 3 * q^37 + 24 * q^39 - 16 * q^41 + 2 * q^43 - 23 * q^45 + 3 * q^47 + 6 * q^49 + 16 * q^51 + 19 * q^53 + 43 * q^55 + 11 * q^57 + q^59 - 24 * q^61 + 11 * q^63 + 13 * q^65 - 6 * q^67 + 29 * q^69 + 15 * q^71 - 20 * q^73 - 31 * q^75 + 38 * q^77 + 53 * q^79 - 8 * q^81 - 17 * q^83 + 7 * q^85 + 5 * q^87 - q^89 + 6 * q^91 + 44 * q^93 - 34 * q^95 + 12 * q^97

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