LMFDB - Embedded newform 3856.2.a.n.1.12

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			1.12
Root			$0.115670$ 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)

$f(q)$	$=$	$q+3.28295 q^{3} -1.31091 q^{5} -3.19647 q^{7} +7.77775 q^{9} +O(q^{10})$	$q+3.28295 q^{3} -1.31091 q^{5} -3.19647 q^{7} +7.77775 q^{9} -1.38968 q^{11} -5.87704 q^{13} -4.30365 q^{15} +5.28927 q^{17} -4.99913 q^{19} -10.4938 q^{21} -3.07207 q^{23} -3.28152 q^{25} +15.6851 q^{27} +3.28657 q^{29} -0.672296 q^{31} -4.56225 q^{33} +4.19028 q^{35} -3.79547 q^{37} -19.2940 q^{39} +0.970489 q^{41} -7.93946 q^{43} -10.1959 q^{45} -2.82021 q^{47} +3.21740 q^{49} +17.3644 q^{51} +8.45419 q^{53} +1.82175 q^{55} -16.4119 q^{57} -5.70844 q^{59} +0.717980 q^{61} -24.8613 q^{63} +7.70427 q^{65} -8.81215 q^{67} -10.0854 q^{69} -15.8552 q^{71} -8.75018 q^{73} -10.7730 q^{75} +4.44207 q^{77} -11.4452 q^{79} +28.1601 q^{81} +11.9246 q^{83} -6.93376 q^{85} +10.7896 q^{87} -11.9996 q^{89} +18.7858 q^{91} -2.20711 q^{93} +6.55340 q^{95} +1.18886 q^{97} -10.8086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.28295	1.89541	0.947705	−	0.319146i	$-0.103396\pi$
$3$	3.28295	1.89541	0.947705	+	0.319146i	$0.103396\pi$
$4$	0	0
$5$	−1.31091	−0.586256	−0.293128	−	0.956073i	$-0.594696\pi$
$5$	−1.31091	−0.586256	−0.293128	+	0.956073i	$0.594696\pi$
$6$	0	0
$7$	−3.19647	−1.20815	−0.604076	−	0.796927i	$-0.706457\pi$
$7$	−3.19647	−1.20815	−0.604076	+	0.796927i	$0.706457\pi$
$8$	0	0
$9$	7.77775	2.59258
$10$	0	0
$11$	−1.38968	−0.419005	−0.209502	−	0.977808i	$-0.567184\pi$
$11$	−1.38968	−0.419005	−0.209502	+	0.977808i	$0.567184\pi$
$12$	0	0
$13$	−5.87704	−1.63000	−0.814999	−	0.579463i	$-0.803262\pi$
$13$	−5.87704	−1.63000	−0.814999	+	0.579463i	$0.803262\pi$
$14$	0	0
$15$	−4.30365	−1.11120
$16$	0	0
$17$	5.28927	1.28284	0.641418	−	0.767191i	$-0.278346\pi$
$17$	5.28927	1.28284	0.641418	+	0.767191i	$0.278346\pi$
$18$	0	0
$19$	−4.99913	−1.14688	−0.573439	−	0.819248i	$-0.694391\pi$
$19$	−4.99913	−1.14688	−0.573439	+	0.819248i	$0.694391\pi$
$20$	0	0
$21$	−10.4938	−2.28994
$22$	0	0
$23$	−3.07207	−0.640571	−0.320285	−	0.947321i	$-0.603779\pi$
$23$	−3.07207	−0.640571	−0.320285	+	0.947321i	$0.603779\pi$
$24$	0	0
$25$	−3.28152	−0.656303
$26$	0	0
$27$	15.6851	3.01860
$28$	0	0
$29$	3.28657	0.610301	0.305150	−	0.952304i	$-0.401293\pi$
$29$	3.28657	0.610301	0.305150	+	0.952304i	$0.401293\pi$
$30$	0	0
$31$	−0.672296	−0.120748	−0.0603740	−	0.998176i	$-0.519229\pi$
$31$	−0.672296	−0.120748	−0.0603740	+	0.998176i	$0.519229\pi$
$32$	0	0
$33$	−4.56225	−0.794186
$34$	0	0
$35$	4.19028	0.708286
$36$	0	0
$37$	−3.79547	−0.623971	−0.311986	−	0.950087i	$-0.600994\pi$
$37$	−3.79547	−0.623971	−0.311986	+	0.950087i	$0.600994\pi$
$38$	0	0
$39$	−19.2940	−3.08952
$40$	0	0
$41$	0.970489	0.151565	0.0757824	−	0.997124i	$-0.475855\pi$
$41$	0.970489	0.151565	0.0757824	+	0.997124i	$0.475855\pi$
$42$	0	0
$43$	−7.93946	−1.21076	−0.605378	−	0.795938i	$-0.706978\pi$
$43$	−7.93946	−1.21076	−0.605378	+	0.795938i	$0.706978\pi$
$44$	0	0
$45$	−10.1959	−1.51992
$46$	0	0
$47$	−2.82021	−0.411370	−0.205685	−	0.978618i	$-0.565942\pi$
$47$	−2.82021	−0.411370	−0.205685	+	0.978618i	$0.565942\pi$
$48$	0	0
$49$	3.21740	0.459629
$50$	0	0
$51$	17.3644	2.43150
$52$	0	0
$53$	8.45419	1.16127	0.580636	−	0.814163i	$-0.302804\pi$
$53$	8.45419	1.16127	0.580636	+	0.814163i	$0.302804\pi$
$54$	0	0
$55$	1.82175	0.245644
$56$	0	0
$57$	−16.4119	−2.17381
$58$	0	0
$59$	−5.70844	−0.743176	−0.371588	−	0.928398i	$-0.621187\pi$
$59$	−5.70844	−0.743176	−0.371588	+	0.928398i	$0.621187\pi$
$60$	0	0
$61$	0.717980	0.0919279	0.0459639	−	0.998943i	$-0.485364\pi$
$61$	0.717980	0.0919279	0.0459639	+	0.998943i	$0.485364\pi$
$62$	0	0
$63$	−24.8613	−3.13223
$64$	0	0
$65$	7.70427	0.955597
$66$	0	0
$67$	−8.81215	−1.07658	−0.538288	−	0.842761i	$-0.680929\pi$
$67$	−8.81215	−1.07658	−0.538288	+	0.842761i	$0.680929\pi$
$68$	0	0
$69$	−10.0854	−1.21414
$70$	0	0
$71$	−15.8552	−1.88166	−0.940832	−	0.338874i	$-0.889954\pi$
$71$	−15.8552	−1.88166	−0.940832	+	0.338874i	$0.889954\pi$
$72$	0	0
$73$	−8.75018	−1.02413	−0.512066	−	0.858946i	$-0.671120\pi$
$73$	−8.75018	−1.02413	−0.512066	+	0.858946i	$0.671120\pi$
$74$	0	0
$75$	−10.7730	−1.24396
$76$	0	0
$77$	4.44207	0.506221
$78$	0	0
$79$	−11.4452	−1.28769	−0.643843	−	0.765158i	$-0.722661\pi$
$79$	−11.4452	−1.28769	−0.643843	+	0.765158i	$0.722661\pi$
$80$	0	0
$81$	28.1601	3.12890
$82$	0	0
$83$	11.9246	1.30889	0.654446	−	0.756109i	$-0.272902\pi$
$83$	11.9246	1.30889	0.654446	+	0.756109i	$0.272902\pi$
$84$	0	0
$85$	−6.93376	−0.752071
$86$	0	0
$87$	10.7896	1.15677
$88$	0	0
$89$	−11.9996	−1.27195	−0.635975	−	0.771709i	$-0.719402\pi$
$89$	−11.9996	−1.27195	−0.635975	+	0.771709i	$0.719402\pi$
$90$	0	0
$91$	18.7858	1.96928
$92$	0	0
$93$	−2.20711	−0.228867
$94$	0	0
$95$	6.55340	0.672365
$96$	0	0
$97$	1.18886	0.120710	0.0603550	−	0.998177i	$-0.480777\pi$
$97$	1.18886	0.120710	0.0603550	+	0.998177i	$0.480777\pi$
$98$	0	0
$99$	−10.8086	−1.08630
$100$	0	0
$101$	19.9051	1.98064	0.990318	−	0.138820i	$-0.0443311\pi$
$101$	19.9051	1.98064	0.990318	+	0.138820i	$0.0443311\pi$
$102$	0	0
$103$	2.74129	0.270108	0.135054	−	0.990838i	$-0.456879\pi$
$103$	2.74129	0.270108	0.135054	+	0.990838i	$0.456879\pi$
$104$	0	0
$105$	13.7565	1.34249
$106$	0	0
$107$	9.95829	0.962704	0.481352	−	0.876527i	$-0.340146\pi$
$107$	9.95829	0.962704	0.481352	+	0.876527i	$0.340146\pi$
$108$	0	0
$109$	−5.65622	−0.541768	−0.270884	−	0.962612i	$-0.587316\pi$
$109$	−5.65622	−0.541768	−0.270884	+	0.962612i	$0.587316\pi$
$110$	0	0
$111$	−12.4603	−1.18268
$112$	0	0
$113$	−12.9604	−1.21921	−0.609604	−	0.792706i	$-0.708671\pi$
$113$	−12.9604	−1.21921	−0.609604	+	0.792706i	$0.708671\pi$
$114$	0	0
$115$	4.02720	0.375539
$116$	0	0
$117$	−45.7101	−4.22590
$118$	0	0
$119$	−16.9070	−1.54986
$120$	0	0
$121$	−9.06879	−0.824435
$122$	0	0
$123$	3.18606	0.287278
$124$	0	0
$125$	10.8563	0.971019
$126$	0	0
$127$	10.4647	0.928593	0.464297	−	0.885680i	$-0.346307\pi$
$127$	10.4647	0.928593	0.464297	+	0.885680i	$0.346307\pi$
$128$	0	0
$129$	−26.0648	−2.29488
$130$	0	0
$131$	−12.9607	−1.13238	−0.566190	−	0.824275i	$-0.691583\pi$
$131$	−12.9607	−1.13238	−0.566190	+	0.824275i	$0.691583\pi$
$132$	0	0
$133$	15.9795	1.38560
$134$	0	0
$135$	−20.5617	−1.76967
$136$	0	0
$137$	14.1420	1.20824	0.604118	−	0.796895i	$-0.293526\pi$
$137$	14.1420	1.20824	0.604118	+	0.796895i	$0.293526\pi$
$138$	0	0
$139$	16.8589	1.42996	0.714978	−	0.699147i	$-0.246437\pi$
$139$	16.8589	1.42996	0.714978	+	0.699147i	$0.246437\pi$
$140$	0	0
$141$	−9.25861	−0.779716
$142$	0	0
$143$	8.16721	0.682976
$144$	0	0
$145$	−4.30840	−0.357793
$146$	0	0
$147$	10.5626	0.871186
$148$	0	0
$149$	16.2523	1.33144	0.665722	−	0.746200i	$-0.268124\pi$
$149$	16.2523	1.33144	0.665722	+	0.746200i	$0.268124\pi$
$150$	0	0
$151$	10.4085	0.847035	0.423517	−	0.905888i	$-0.360795\pi$
$151$	10.4085	0.847035	0.423517	+	0.905888i	$0.360795\pi$
$152$	0	0
$153$	41.1386	3.32586
$154$	0	0
$155$	0.881320	0.0707893
$156$	0	0
$157$	−18.9894	−1.51552	−0.757758	−	0.652535i	$-0.773705\pi$
$157$	−18.9894	−1.51552	−0.757758	+	0.652535i	$0.773705\pi$
$158$	0	0
$159$	27.7547	2.20109
$160$	0	0
$161$	9.81977	0.773906
$162$	0	0
$163$	−7.43947	−0.582704	−0.291352	−	0.956616i	$-0.594105\pi$
$163$	−7.43947	−0.582704	−0.291352	+	0.956616i	$0.594105\pi$
$164$	0	0
$165$	5.98070	0.465596
$166$	0	0
$167$	−13.4159	−1.03815	−0.519076	−	0.854728i	$-0.673724\pi$
$167$	−13.4159	−1.03815	−0.519076	+	0.854728i	$0.673724\pi$
$168$	0	0
$169$	21.5396	1.65689
$170$	0	0
$171$	−38.8819	−2.97338
$172$	0	0
$173$	−3.18442	−0.242107	−0.121053	−	0.992646i	$-0.538627\pi$
$173$	−3.18442	−0.242107	−0.121053	+	0.992646i	$0.538627\pi$
$174$	0	0
$175$	10.4893	0.792914
$176$	0	0
$177$	−18.7405	−1.40862
$178$	0	0
$179$	−11.8290	−0.884142	−0.442071	−	0.896980i	$-0.645756\pi$
$179$	−11.8290	−0.884142	−0.442071	+	0.896980i	$0.645756\pi$
$180$	0	0
$181$	−3.92038	−0.291399	−0.145700	−	0.989329i	$-0.546543\pi$
$181$	−3.92038	−0.291399	−0.145700	+	0.989329i	$0.546543\pi$
$182$	0	0
$183$	2.35709	0.174241
$184$	0	0
$185$	4.97551	0.365807
$186$	0	0
$187$	−7.35040	−0.537514
$188$	0	0
$189$	−50.1369	−3.64692
$190$	0	0
$191$	−1.27094	−0.0919617	−0.0459809	−	0.998942i	$-0.514641\pi$
$191$	−1.27094	−0.0919617	−0.0459809	+	0.998942i	$0.514641\pi$
$192$	0	0
$193$	−7.18460	−0.517159	−0.258580	−	0.965990i	$-0.583254\pi$
$193$	−7.18460	−0.517159	−0.258580	+	0.965990i	$0.583254\pi$
$194$	0	0
$195$	25.2927	1.81125
$196$	0	0
$197$	6.64645	0.473540	0.236770	−	0.971566i	$-0.423911\pi$
$197$	6.64645	0.473540	0.236770	+	0.971566i	$0.423911\pi$
$198$	0	0
$199$	−12.7358	−0.902819	−0.451410	−	0.892317i	$-0.649079\pi$
$199$	−12.7358	−0.902819	−0.451410	+	0.892317i	$0.649079\pi$
$200$	0	0
$201$	−28.9298	−2.04055
$202$	0	0
$203$	−10.5054	−0.737336
$204$	0	0
$205$	−1.27222	−0.0888559
$206$	0	0
$207$	−23.8938	−1.66073
$208$	0	0
$209$	6.94719	0.480547
$210$	0	0
$211$	−5.22527	−0.359722	−0.179861	−	0.983692i	$-0.557565\pi$
$211$	−5.22527	−0.359722	−0.179861	+	0.983692i	$0.557565\pi$
$212$	0	0
$213$	−52.0517	−3.56653
$214$	0	0
$215$	10.4079	0.709814
$216$	0	0
$217$	2.14897	0.145882
$218$	0	0
$219$	−28.7264	−1.94115
$220$	0	0
$221$	−31.0853	−2.09102
$222$	0	0
$223$	1.68503	0.112838	0.0564191	−	0.998407i	$-0.482032\pi$
$223$	1.68503	0.112838	0.0564191	+	0.998407i	$0.482032\pi$
$224$	0	0
$225$	−25.5228	−1.70152
$226$	0	0
$227$	1.27020	0.0843064	0.0421532	−	0.999111i	$-0.486578\pi$
$227$	1.27020	0.0843064	0.0421532	+	0.999111i	$0.486578\pi$
$228$	0	0
$229$	24.4833	1.61790	0.808950	−	0.587877i	$-0.200036\pi$
$229$	24.4833	1.61790	0.808950	+	0.587877i	$0.200036\pi$
$230$	0	0
$231$	14.5831	0.959496
$232$	0	0
$233$	3.58351	0.234764	0.117382	−	0.993087i	$-0.462550\pi$
$233$	3.58351	0.234764	0.117382	+	0.993087i	$0.462550\pi$
$234$	0	0
$235$	3.69704	0.241169
$236$	0	0
$237$	−37.5740	−2.44069
$238$	0	0
$239$	27.0261	1.74817	0.874087	−	0.485769i	$-0.161460\pi$
$239$	27.0261	1.74817	0.874087	+	0.485769i	$0.161460\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	45.3929	2.91195
$244$	0	0
$245$	−4.21772	−0.269460
$246$	0	0
$247$	29.3801	1.86941
$248$	0	0
$249$	39.1478	2.48089
$250$	0	0
$251$	19.0710	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$251$	19.0710	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$252$	0	0
$253$	4.26920	0.268402
$254$	0	0
$255$	−22.7632	−1.42548
$256$	0	0
$257$	−29.8662	−1.86300	−0.931502	−	0.363736i	$-0.881501\pi$
$257$	−29.8662	−1.86300	−0.931502	+	0.363736i	$0.881501\pi$
$258$	0	0
$259$	12.1321	0.753851
$260$	0	0
$261$	25.5621	1.58225
$262$	0	0
$263$	14.4912	0.893563	0.446781	−	0.894643i	$-0.352570\pi$
$263$	14.4912	0.893563	0.446781	+	0.894643i	$0.352570\pi$
$264$	0	0
$265$	−11.0827	−0.680803
$266$	0	0
$267$	−39.3939	−2.41087
$268$	0	0
$269$	9.73320	0.593444	0.296722	−	0.954964i	$-0.404107\pi$
$269$	9.73320	0.593444	0.296722	+	0.954964i	$0.404107\pi$
$270$	0	0
$271$	−17.7984	−1.08117	−0.540587	−	0.841288i	$-0.681798\pi$
$271$	−17.7984	−1.08117	−0.540587	+	0.841288i	$0.681798\pi$
$272$	0	0
$273$	61.6727	3.73260
$274$	0	0
$275$	4.56026	0.274994
$276$	0	0
$277$	2.28565	0.137332	0.0686658	−	0.997640i	$-0.478126\pi$
$277$	2.28565	0.137332	0.0686658	+	0.997640i	$0.478126\pi$
$278$	0	0
$279$	−5.22895	−0.313049
$280$	0	0
$281$	9.34433	0.557436	0.278718	−	0.960373i	$-0.410090\pi$
$281$	9.34433	0.557436	0.278718	+	0.960373i	$0.410090\pi$
$282$	0	0
$283$	−8.86628	−0.527045	−0.263523	−	0.964653i	$-0.584884\pi$
$283$	−8.86628	−0.527045	−0.263523	+	0.964653i	$0.584884\pi$
$284$	0	0
$285$	21.5145	1.27441
$286$	0	0
$287$	−3.10214	−0.183113
$288$	0	0
$289$	10.9764	0.645670
$290$	0	0
$291$	3.90295	0.228795
$292$	0	0
$293$	17.1228	1.00032	0.500161	−	0.865932i	$-0.333274\pi$
$293$	17.1228	1.00032	0.500161	+	0.865932i	$0.333274\pi$
$294$	0	0
$295$	7.48325	0.435692
$296$	0	0
$297$	−21.7973	−1.26481
$298$	0	0
$299$	18.0547	1.04413
$300$	0	0
$301$	25.3782	1.46278
$302$	0	0
$303$	65.3475	3.75412
$304$	0	0
$305$	−0.941206	−0.0538933
$306$	0	0
$307$	−5.03009	−0.287082	−0.143541	−	0.989644i	$-0.545849\pi$
$307$	−5.03009	−0.287082	−0.143541	+	0.989644i	$0.545849\pi$
$308$	0	0
$309$	8.99952	0.511965
$310$	0	0
$311$	−9.99750	−0.566906	−0.283453	−	0.958986i	$-0.591480\pi$
$311$	−9.99750	−0.566906	−0.283453	+	0.958986i	$0.591480\pi$
$312$	0	0
$313$	28.1466	1.59094	0.795471	−	0.605992i	$-0.207224\pi$
$313$	28.1466	1.59094	0.795471	+	0.605992i	$0.207224\pi$
$314$	0	0
$315$	32.5909	1.83629
$316$	0	0
$317$	−5.66871	−0.318387	−0.159193	−	0.987247i	$-0.550889\pi$
$317$	−5.66871	−0.318387	−0.159193	+	0.987247i	$0.550889\pi$
$318$	0	0
$319$	−4.56728	−0.255719
$320$	0	0
$321$	32.6925	1.82472
$322$	0	0
$323$	−26.4417	−1.47126
$324$	0	0
$325$	19.2856	1.06977
$326$	0	0
$327$	−18.5691	−1.02687
$328$	0	0
$329$	9.01472	0.496998
$330$	0	0
$331$	14.8950	0.818701	0.409350	−	0.912377i	$-0.365755\pi$
$331$	14.8950	0.818701	0.409350	+	0.912377i	$0.365755\pi$
$332$	0	0
$333$	−29.5202	−1.61770
$334$	0	0
$335$	11.5519	0.631149
$336$	0	0
$337$	21.4134	1.16646	0.583231	−	0.812306i	$-0.301788\pi$
$337$	21.4134	1.16646	0.583231	+	0.812306i	$0.301788\pi$
$338$	0	0
$339$	−42.5482	−2.31090
$340$	0	0
$341$	0.934278	0.0505940
$342$	0	0
$343$	12.0909	0.652850
$344$	0	0
$345$	13.2211	0.711800
$346$	0	0
$347$	6.12740	0.328936	0.164468	−	0.986382i	$-0.447409\pi$
$347$	6.12740	0.328936	0.164468	+	0.986382i	$0.447409\pi$
$348$	0	0
$349$	−7.15661	−0.383085	−0.191542	−	0.981484i	$-0.561349\pi$
$349$	−7.15661	−0.383085	−0.191542	+	0.981484i	$0.561349\pi$
$350$	0	0
$351$	−92.1819	−4.92031
$352$	0	0
$353$	−9.47379	−0.504239	−0.252120	−	0.967696i	$-0.581128\pi$
$353$	−9.47379	−0.504239	−0.252120	+	0.967696i	$0.581128\pi$
$354$	0	0
$355$	20.7847	1.10314
$356$	0	0
$357$	−55.5047	−2.93762
$358$	0	0
$359$	−7.14694	−0.377201	−0.188601	−	0.982054i	$-0.560395\pi$
$359$	−7.14694	−0.377201	−0.188601	+	0.982054i	$0.560395\pi$
$360$	0	0
$361$	5.99126	0.315330
$362$	0	0
$363$	−29.7724	−1.56264
$364$	0	0
$365$	11.4707	0.600404
$366$	0	0
$367$	−2.63925	−0.137767	−0.0688837	−	0.997625i	$-0.521944\pi$
$367$	−2.63925	−0.137767	−0.0688837	+	0.997625i	$0.521944\pi$
$368$	0	0
$369$	7.54822	0.392944
$370$	0	0
$371$	−27.0235	−1.40299
$372$	0	0
$373$	20.5114	1.06204	0.531020	−	0.847360i	$-0.321809\pi$
$373$	20.5114	1.06204	0.531020	+	0.847360i	$0.321809\pi$
$374$	0	0
$375$	35.6407	1.84048
$376$	0	0
$377$	−19.3153	−0.994789
$378$	0	0
$379$	24.7263	1.27010	0.635051	−	0.772470i	$-0.280979\pi$
$379$	24.7263	1.27010	0.635051	+	0.772470i	$0.280979\pi$
$380$	0	0
$381$	34.3551	1.76007
$382$	0	0
$383$	33.2690	1.69997	0.849983	−	0.526810i	$-0.176612\pi$
$383$	33.2690	1.69997	0.849983	+	0.526810i	$0.176612\pi$
$384$	0	0
$385$	−5.82315	−0.296775
$386$	0	0
$387$	−61.7511	−3.13899
$388$	0	0
$389$	−11.2024	−0.567985	−0.283992	−	0.958827i	$-0.591659\pi$
$389$	−11.2024	−0.567985	−0.283992	+	0.958827i	$0.591659\pi$
$390$	0	0
$391$	−16.2490	−0.821748
$392$	0	0
$393$	−42.5493	−2.14633
$394$	0	0
$395$	15.0036	0.754914
$396$	0	0
$397$	5.20629	0.261296	0.130648	−	0.991429i	$-0.458294\pi$
$397$	5.20629	0.261296	0.130648	+	0.991429i	$0.458294\pi$
$398$	0	0
$399$	52.4600	2.62629
$400$	0	0
$401$	−2.79646	−0.139649	−0.0698243	−	0.997559i	$-0.522244\pi$
$401$	−2.79646	−0.139649	−0.0698243	+	0.997559i	$0.522244\pi$
$402$	0	0
$403$	3.95111	0.196819
$404$	0	0
$405$	−36.9153	−1.83434
$406$	0	0
$407$	5.27449	0.261447
$408$	0	0
$409$	0.743107	0.0367443	0.0183721	−	0.999831i	$-0.494152\pi$
$409$	0.743107	0.0367443	0.0183721	+	0.999831i	$0.494152\pi$
$410$	0	0
$411$	46.4276	2.29010
$412$	0	0
$413$	18.2469	0.897869
$414$	0	0
$415$	−15.6320	−0.767346
$416$	0	0
$417$	55.3470	2.71035
$418$	0	0
$419$	11.8016	0.576544	0.288272	−	0.957549i	$-0.406919\pi$
$419$	11.8016	0.576544	0.288272	+	0.957549i	$0.406919\pi$
$420$	0	0
$421$	−9.89956	−0.482475	−0.241237	−	0.970466i	$-0.577553\pi$
$421$	−9.89956	−0.482475	−0.241237	+	0.970466i	$0.577553\pi$
$422$	0	0
$423$	−21.9349	−1.06651
$424$	0	0
$425$	−17.3568	−0.841930
$426$	0	0
$427$	−2.29500	−0.111063
$428$	0	0
$429$	26.8125	1.29452
$430$	0	0
$431$	−9.64934	−0.464793	−0.232396	−	0.972621i	$-0.574657\pi$
$431$	−9.64934	−0.464793	−0.232396	+	0.972621i	$0.574657\pi$
$432$	0	0
$433$	−38.9496	−1.87180	−0.935900	−	0.352265i	$-0.885412\pi$
$433$	−38.9496	−1.87180	−0.935900	+	0.352265i	$0.885412\pi$
$434$	0	0
$435$	−14.1442	−0.678164
$436$	0	0
$437$	15.3577	0.734657
$438$	0	0
$439$	16.8089	0.802243	0.401121	−	0.916025i	$-0.368621\pi$
$439$	16.8089	0.802243	0.401121	+	0.916025i	$0.368621\pi$
$440$	0	0
$441$	25.0241	1.19163
$442$	0	0
$443$	0.438324	0.0208254	0.0104127	−	0.999946i	$-0.496685\pi$
$443$	0.438324	0.0208254	0.0104127	+	0.999946i	$0.496685\pi$
$444$	0	0
$445$	15.7303	0.745689
$446$	0	0
$447$	53.3556	2.52363
$448$	0	0
$449$	−31.7722	−1.49942	−0.749711	−	0.661766i	$-0.769807\pi$
$449$	−31.7722	−1.49942	−0.749711	+	0.661766i	$0.769807\pi$
$450$	0	0
$451$	−1.34867	−0.0635064
$452$	0	0
$453$	34.1707	1.60548
$454$	0	0
$455$	−24.6264	−1.15451
$456$	0	0
$457$	17.1543	0.802444	0.401222	−	0.915981i	$-0.368586\pi$
$457$	17.1543	0.802444	0.401222	+	0.915981i	$0.368586\pi$
$458$	0	0
$459$	82.9627	3.87237
$460$	0	0
$461$	−30.9242	−1.44028	−0.720141	−	0.693827i	$-0.755923\pi$
$461$	−30.9242	−1.44028	−0.720141	+	0.693827i	$0.755923\pi$
$462$	0	0
$463$	−15.6931	−0.729319	−0.364659	−	0.931141i	$-0.618815\pi$
$463$	−15.6931	−0.729319	−0.364659	+	0.931141i	$0.618815\pi$
$464$	0	0
$465$	2.89333	0.134175
$466$	0	0
$467$	−21.3617	−0.988501	−0.494250	−	0.869320i	$-0.664557\pi$
$467$	−21.3617	−0.988501	−0.494250	+	0.869320i	$0.664557\pi$
$468$	0	0
$469$	28.1678	1.30067
$470$	0	0
$471$	−62.3411	−2.87253
$472$	0	0
$473$	11.0333	0.507313
$474$	0	0
$475$	16.4047	0.752700
$476$	0	0
$477$	65.7545	3.01069
$478$	0	0
$479$	−27.3633	−1.25026	−0.625130	−	0.780521i	$-0.714954\pi$
$479$	−27.3633	−1.25026	−0.625130	+	0.780521i	$0.714954\pi$
$480$	0	0
$481$	22.3061	1.01707
$482$	0	0
$483$	32.2378	1.46687
$484$	0	0
$485$	−1.55848	−0.0707670
$486$	0	0
$487$	9.12015	0.413274	0.206637	−	0.978418i	$-0.433748\pi$
$487$	9.12015	0.413274	0.206637	+	0.978418i	$0.433748\pi$
$488$	0	0
$489$	−24.4234	−1.10446
$490$	0	0
$491$	7.29684	0.329302	0.164651	−	0.986352i	$-0.447350\pi$
$491$	7.29684	0.329302	0.164651	+	0.986352i	$0.447350\pi$
$492$	0	0
$493$	17.3836	0.782916
$494$	0	0
$495$	14.1691	0.636853
$496$	0	0
$497$	50.6806	2.27333
$498$	0	0
$499$	−1.98278	−0.0887613	−0.0443807	−	0.999015i	$-0.514131\pi$
$499$	−1.98278	−0.0887613	−0.0443807	+	0.999015i	$0.514131\pi$
$500$	0	0
$501$	−44.0436	−1.96772
$502$	0	0
$503$	−7.01318	−0.312702	−0.156351	−	0.987702i	$-0.549973\pi$
$503$	−7.01318	−0.312702	−0.156351	+	0.987702i	$0.549973\pi$
$504$	0	0
$505$	−26.0938	−1.16116
$506$	0	0
$507$	70.7134	3.14049
$508$	0	0
$509$	−28.5738	−1.26651	−0.633256	−	0.773943i	$-0.718282\pi$
$509$	−28.5738	−1.26651	−0.633256	+	0.773943i	$0.718282\pi$
$510$	0	0
$511$	27.9697	1.23731
$512$	0	0
$513$	−78.4118	−3.46196
$514$	0	0
$515$	−3.59359	−0.158352
$516$	0	0
$517$	3.91920	0.172366
$518$	0	0
$519$	−10.4543	−0.458892
$520$	0	0
$521$	−13.1589	−0.576501	−0.288250	−	0.957555i	$-0.593074\pi$
$521$	−13.1589	−0.576501	−0.288250	+	0.957555i	$0.593074\pi$
$522$	0	0
$523$	−23.8792	−1.04416	−0.522082	−	0.852895i	$-0.674845\pi$
$523$	−23.8792	−1.04416	−0.522082	+	0.852895i	$0.674845\pi$
$524$	0	0
$525$	34.4357	1.50290
$526$	0	0
$527$	−3.55596	−0.154900
$528$	0	0
$529$	−13.5624	−0.589669
$530$	0	0
$531$	−44.3988	−1.92674
$532$	0	0
$533$	−5.70360	−0.247050
$534$	0	0
$535$	−13.0544	−0.564391
$536$	0	0
$537$	−38.8340	−1.67581
$538$	0	0
$539$	−4.47116	−0.192587
$540$	0	0
$541$	−7.11389	−0.305850	−0.152925	−	0.988238i	$-0.548869\pi$
$541$	−7.11389	−0.305850	−0.152925	+	0.988238i	$0.548869\pi$
$542$	0	0
$543$	−12.8704	−0.552321
$544$	0	0
$545$	7.41480	0.317615
$546$	0	0
$547$	−2.10736	−0.0901044	−0.0450522	−	0.998985i	$-0.514345\pi$
$547$	−2.10736	−0.0901044	−0.0450522	+	0.998985i	$0.514345\pi$
$548$	0	0
$549$	5.58426	0.238331
$550$	0	0
$551$	−16.4300	−0.699941
$552$	0	0
$553$	36.5842	1.55572
$554$	0	0
$555$	16.3344	0.693355
$556$	0	0
$557$	−9.69468	−0.410777	−0.205388	−	0.978681i	$-0.565846\pi$
$557$	−9.69468	−0.410777	−0.205388	+	0.978681i	$0.565846\pi$
$558$	0	0
$559$	46.6605	1.97353
$560$	0	0
$561$	−24.1310	−1.01881
$562$	0	0
$563$	12.6385	0.532647	0.266324	−	0.963884i	$-0.414191\pi$
$563$	12.6385	0.532647	0.266324	+	0.963884i	$0.414191\pi$
$564$	0	0
$565$	16.9899	0.714768
$566$	0	0
$567$	−90.0129	−3.78018
$568$	0	0
$569$	17.6815	0.741245	0.370622	−	0.928784i	$-0.379144\pi$
$569$	17.6815	0.741245	0.370622	+	0.928784i	$0.379144\pi$
$570$	0	0
$571$	24.5248	1.02633	0.513166	−	0.858289i	$-0.328473\pi$
$571$	24.5248	1.02633	0.513166	+	0.858289i	$0.328473\pi$
$572$	0	0
$573$	−4.17242	−0.174305
$574$	0	0
$575$	10.0810	0.420409
$576$	0	0
$577$	−35.7883	−1.48988	−0.744942	−	0.667129i	$-0.767523\pi$
$577$	−35.7883	−1.48988	−0.744942	+	0.667129i	$0.767523\pi$
$578$	0	0
$579$	−23.5867	−0.980229
$580$	0	0
$581$	−38.1165	−1.58134
$582$	0	0
$583$	−11.7486	−0.486578
$584$	0	0
$585$	59.9218	2.47746
$586$	0	0
$587$	42.4486	1.75204	0.876021	−	0.482273i	$-0.160189\pi$
$587$	42.4486	1.75204	0.876021	+	0.482273i	$0.160189\pi$
$588$	0	0
$589$	3.36089	0.138483
$590$	0	0
$591$	21.8200	0.897553
$592$	0	0
$593$	17.8382	0.732527	0.366263	−	0.930511i	$-0.380637\pi$
$593$	17.8382	0.732527	0.366263	+	0.930511i	$0.380637\pi$
$594$	0	0
$595$	22.1635	0.908616
$596$	0	0
$597$	−41.8111	−1.71121
$598$	0	0
$599$	−21.3734	−0.873291	−0.436646	−	0.899634i	$-0.643834\pi$
$599$	−21.3734	−0.873291	−0.436646	+	0.899634i	$0.643834\pi$
$600$	0	0
$601$	−42.3095	−1.72584	−0.862920	−	0.505341i	$-0.831367\pi$
$601$	−42.3095	−1.72584	−0.862920	+	0.505341i	$0.831367\pi$
$602$	0	0
$603$	−68.5387	−2.79111
$604$	0	0
$605$	11.8884	0.483330
$606$	0	0
$607$	−1.51410	−0.0614552	−0.0307276	−	0.999528i	$-0.509782\pi$
$607$	−1.51410	−0.0614552	−0.0307276	+	0.999528i	$0.509782\pi$
$608$	0	0
$609$	−34.4887	−1.39755
$610$	0	0
$611$	16.5745	0.670533
$612$	0	0
$613$	−15.4991	−0.626001	−0.313000	−	0.949753i	$-0.601334\pi$
$613$	−15.4991	−0.626001	−0.313000	+	0.949753i	$0.601334\pi$
$614$	0	0
$615$	−4.17664	−0.168418
$616$	0	0
$617$	22.9695	0.924718	0.462359	−	0.886693i	$-0.347003\pi$
$617$	22.9695	0.924718	0.462359	+	0.886693i	$0.347003\pi$
$618$	0	0
$619$	−36.7503	−1.47712	−0.738559	−	0.674189i	$-0.764493\pi$
$619$	−36.7503	−1.47712	−0.738559	+	0.674189i	$0.764493\pi$
$620$	0	0
$621$	−48.1857	−1.93363
$622$	0	0
$623$	38.3562	1.53671
$624$	0	0
$625$	2.17594	0.0870375
$626$	0	0
$627$	22.8073	0.910834
$628$	0	0
$629$	−20.0753	−0.800453
$630$	0	0
$631$	16.5412	0.658494	0.329247	−	0.944244i	$-0.393205\pi$
$631$	16.5412	0.658494	0.329247	+	0.944244i	$0.393205\pi$
$632$	0	0
$633$	−17.1543	−0.681821
$634$	0	0
$635$	−13.7183	−0.544394
$636$	0	0
$637$	−18.9088	−0.749194
$638$	0	0
$639$	−123.318	−4.87837
$640$	0	0
$641$	1.70611	0.0673873	0.0336937	−	0.999432i	$-0.489273\pi$
$641$	1.70611	0.0673873	0.0336937	+	0.999432i	$0.489273\pi$
$642$	0	0
$643$	12.2046	0.481302	0.240651	−	0.970612i	$-0.422639\pi$
$643$	12.2046	0.481302	0.240651	+	0.970612i	$0.422639\pi$
$644$	0	0
$645$	34.1686	1.34539
$646$	0	0
$647$	−28.0548	−1.10295	−0.551473	−	0.834193i	$-0.685934\pi$
$647$	−28.0548	−1.10295	−0.551473	+	0.834193i	$0.685934\pi$
$648$	0	0
$649$	7.93291	0.311394
$650$	0	0
$651$	7.05497	0.276506
$652$	0	0
$653$	5.68400	0.222432	0.111216	−	0.993796i	$-0.464525\pi$
$653$	5.68400	0.222432	0.111216	+	0.993796i	$0.464525\pi$
$654$	0	0
$655$	16.9903	0.663865
$656$	0	0
$657$	−68.0567	−2.65515
$658$	0	0
$659$	30.6811	1.19517	0.597584	−	0.801806i	$-0.296127\pi$
$659$	30.6811	1.19517	0.597584	+	0.801806i	$0.296127\pi$
$660$	0	0
$661$	−13.8208	−0.537568	−0.268784	−	0.963200i	$-0.586622\pi$
$661$	−13.8208	−0.537568	−0.268784	+	0.963200i	$0.586622\pi$
$662$	0	0
$663$	−102.051	−3.96334
$664$	0	0
$665$	−20.9477	−0.812318
$666$	0	0
$667$	−10.0966	−0.390941
$668$	0	0
$669$	5.53188	0.213875
$670$	0	0
$671$	−0.997763	−0.0385182
$672$	0	0
$673$	19.2294	0.741240	0.370620	−	0.928785i	$-0.379145\pi$
$673$	19.2294	0.741240	0.370620	+	0.928785i	$0.379145\pi$
$674$	0	0
$675$	−51.4709	−1.98112
$676$	0	0
$677$	34.7093	1.33399	0.666994	−	0.745063i	$-0.267581\pi$
$677$	34.7093	1.33399	0.666994	+	0.745063i	$0.267581\pi$
$678$	0	0
$679$	−3.80014	−0.145836
$680$	0	0
$681$	4.17001	0.159795
$682$	0	0
$683$	−2.16428	−0.0828139	−0.0414069	−	0.999142i	$-0.513184\pi$
$683$	−2.16428	−0.0828139	−0.0414069	+	0.999142i	$0.513184\pi$
$684$	0	0
$685$	−18.5389	−0.708336
$686$	0	0
$687$	80.3773	3.06659
$688$	0	0
$689$	−49.6856	−1.89287
$690$	0	0
$691$	−22.7388	−0.865025	−0.432513	−	0.901628i	$-0.642373\pi$
$691$	−22.7388	−0.865025	−0.432513	+	0.901628i	$0.642373\pi$
$692$	0	0
$693$	34.5493	1.31242
$694$	0	0
$695$	−22.1005	−0.838321
$696$	0	0
$697$	5.13318	0.194433
$698$	0	0
$699$	11.7645	0.444974
$700$	0	0
$701$	−21.4836	−0.811424	−0.405712	−	0.914001i	$-0.632976\pi$
$701$	−21.4836	−0.811424	−0.405712	+	0.914001i	$0.632976\pi$
$702$	0	0
$703$	18.9740	0.715619
$704$	0	0
$705$	12.1372	0.457113
$706$	0	0
$707$	−63.6261	−2.39291
$708$	0	0
$709$	−4.43026	−0.166382	−0.0831910	−	0.996534i	$-0.526511\pi$
$709$	−4.43026	−0.166382	−0.0831910	+	0.996534i	$0.526511\pi$
$710$	0	0
$711$	−89.0178	−3.33843
$712$	0	0
$713$	2.06534	0.0773476
$714$	0	0
$715$	−10.7065	−0.400399
$716$	0	0
$717$	88.7254	3.31351
$718$	0	0
$719$	−37.8736	−1.41245	−0.706224	−	0.707989i	$-0.749603\pi$
$719$	−37.8736	−1.41245	−0.706224	+	0.707989i	$0.749603\pi$
$720$	0	0
$721$	−8.76245	−0.326331
$722$	0	0
$723$	3.28295	0.122094
$724$	0	0
$725$	−10.7849	−0.400542
$726$	0	0
$727$	0.346692	0.0128581	0.00642904	−	0.999979i	$-0.497954\pi$
$727$	0.346692	0.0128581	0.00642904	+	0.999979i	$0.497954\pi$
$728$	0	0
$729$	64.5421	2.39045
$730$	0	0
$731$	−41.9940	−1.55320
$732$	0	0
$733$	4.28244	0.158176	0.0790878	−	0.996868i	$-0.474799\pi$
$733$	4.28244	0.158176	0.0790878	+	0.996868i	$0.474799\pi$
$734$	0	0
$735$	−13.8466	−0.510738
$736$	0	0
$737$	12.2461	0.451090
$738$	0	0
$739$	−4.52351	−0.166400	−0.0832000	−	0.996533i	$-0.526514\pi$
$739$	−4.52351	−0.166400	−0.0832000	+	0.996533i	$0.526514\pi$
$740$	0	0
$741$	96.4532	3.54330
$742$	0	0
$743$	−14.4360	−0.529606	−0.264803	−	0.964303i	$-0.585307\pi$
$743$	−14.4360	−0.529606	−0.264803	+	0.964303i	$0.585307\pi$
$744$	0	0
$745$	−21.3054	−0.780568
$746$	0	0
$747$	92.7463	3.39341
$748$	0	0
$749$	−31.8313	−1.16309
$750$	0	0
$751$	18.6022	0.678804	0.339402	−	0.940641i	$-0.389775\pi$
$751$	18.6022	0.678804	0.339402	+	0.940641i	$0.389775\pi$
$752$	0	0
$753$	62.6090	2.28160
$754$	0	0
$755$	−13.6446	−0.496580
$756$	0	0
$757$	1.36472	0.0496015	0.0248007	−	0.999692i	$-0.492105\pi$
$757$	1.36472	0.0496015	0.0248007	+	0.999692i	$0.492105\pi$
$758$	0	0
$759$	14.0155	0.508732
$760$	0	0
$761$	−32.0391	−1.16142	−0.580709	−	0.814111i	$-0.697224\pi$
$761$	−32.0391	−1.16142	−0.580709	+	0.814111i	$0.697224\pi$
$762$	0	0
$763$	18.0799	0.654538
$764$	0	0
$765$	−53.9290	−1.94981
$766$	0	0
$767$	33.5487	1.21138
$768$	0	0
$769$	−7.53070	−0.271564	−0.135782	−	0.990739i	$-0.543355\pi$
$769$	−7.53070	−0.271564	−0.135782	+	0.990739i	$0.543355\pi$
$770$	0	0
$771$	−98.0493	−3.53116
$772$	0	0
$773$	−23.1144	−0.831368	−0.415684	−	0.909509i	$-0.636458\pi$
$773$	−23.1144	−0.831368	−0.415684	+	0.909509i	$0.636458\pi$
$774$	0	0
$775$	2.20615	0.0792473
$776$	0	0
$777$	39.8290	1.42886
$778$	0	0
$779$	−4.85160	−0.173826
$780$	0	0
$781$	22.0336	0.788426
$782$	0	0
$783$	51.5502	1.84225
$784$	0	0
$785$	24.8933	0.888481
$786$	0	0
$787$	−14.4891	−0.516479	−0.258240	−	0.966081i	$-0.583142\pi$
$787$	−14.4891	−0.516479	−0.258240	+	0.966081i	$0.583142\pi$
$788$	0	0
$789$	47.5737	1.69367
$790$	0	0
$791$	41.4274	1.47299
$792$	0	0
$793$	−4.21960	−0.149842
$794$	0	0
$795$	−36.3838	−1.29040
$796$	0	0
$797$	−13.1633	−0.466267	−0.233133	−	0.972445i	$-0.574898\pi$
$797$	−13.1633	−0.466267	−0.233133	+	0.972445i	$0.574898\pi$
$798$	0	0
$799$	−14.9169	−0.527721
$800$	0	0
$801$	−93.3296	−3.29764
$802$	0	0
$803$	12.1600	0.429116
$804$	0	0
$805$	−12.8728	−0.453707
$806$	0	0
$807$	31.9536	1.12482
$808$	0	0
$809$	−33.0845	−1.16319	−0.581595	−	0.813479i	$-0.697571\pi$
$809$	−33.0845	−1.16319	−0.581595	+	0.813479i	$0.697571\pi$
$810$	0	0
$811$	4.74569	0.166644	0.0833219	−	0.996523i	$-0.473447\pi$
$811$	4.74569	0.166644	0.0833219	+	0.996523i	$0.473447\pi$
$812$	0	0
$813$	−58.4311	−2.04927
$814$	0	0
$815$	9.75247	0.341614
$816$	0	0
$817$	39.6904	1.38859
$818$	0	0
$819$	146.111	5.10553
$820$	0	0
$821$	−33.6149	−1.17317	−0.586584	−	0.809888i	$-0.699528\pi$
$821$	−33.6149	−1.17317	−0.586584	+	0.809888i	$0.699528\pi$
$822$	0	0
$823$	−45.7636	−1.59522	−0.797609	−	0.603175i	$-0.793902\pi$
$823$	−45.7636	−1.59522	−0.797609	+	0.603175i	$0.793902\pi$
$824$	0	0
$825$	14.9711	0.521227
$826$	0	0
$827$	−29.9837	−1.04264	−0.521318	−	0.853363i	$-0.674559\pi$
$827$	−29.9837	−1.04264	−0.521318	+	0.853363i	$0.674559\pi$
$828$	0	0
$829$	−2.23933	−0.0777753	−0.0388876	−	0.999244i	$-0.512381\pi$
$829$	−2.23933	−0.0777753	−0.0388876	+	0.999244i	$0.512381\pi$
$830$	0	0
$831$	7.50368	0.260300
$832$	0	0
$833$	17.0177	0.589629
$834$	0	0
$835$	17.5870	0.608623
$836$	0	0
$837$	−10.5450	−0.364490
$838$	0	0
$839$	31.0203	1.07094	0.535469	−	0.844555i	$-0.320135\pi$
$839$	31.0203	1.07094	0.535469	+	0.844555i	$0.320135\pi$
$840$	0	0
$841$	−18.1985	−0.627533
$842$	0	0
$843$	30.6770	1.05657
$844$	0	0
$845$	−28.2365	−0.971364
$846$	0	0
$847$	28.9881	0.996042
$848$	0	0
$849$	−29.1075	−0.998968
$850$	0	0
$851$	11.6599	0.399698
$852$	0	0
$853$	−13.7603	−0.471142	−0.235571	−	0.971857i	$-0.575696\pi$
$853$	−13.7603	−0.471142	−0.235571	+	0.971857i	$0.575696\pi$
$854$	0	0
$855$	50.9707	1.74316
$856$	0	0
$857$	31.2228	1.06655	0.533276	−	0.845941i	$-0.320961\pi$
$857$	31.2228	1.06655	0.533276	+	0.845941i	$0.320961\pi$
$858$	0	0
$859$	55.8137	1.90434	0.952170	−	0.305569i	$-0.0988465\pi$
$859$	55.8137	1.90434	0.952170	+	0.305569i	$0.0988465\pi$
$860$	0	0
$861$	−10.1841	−0.347075
$862$	0	0
$863$	0.720056	0.0245110	0.0122555	−	0.999925i	$-0.496099\pi$
$863$	0.720056	0.0245110	0.0122555	+	0.999925i	$0.496099\pi$
$864$	0	0
$865$	4.17448	0.141937
$866$	0	0
$867$	36.0349	1.22381
$868$	0	0
$869$	15.9052	0.539546
$870$	0	0
$871$	51.7894	1.75482
$872$	0	0
$873$	9.24662	0.312951
$874$	0	0
$875$	−34.7019	−1.17314
$876$	0	0
$877$	48.3619	1.63307	0.816533	−	0.577298i	$-0.195893\pi$
$877$	48.3619	1.63307	0.816533	+	0.577298i	$0.195893\pi$
$878$	0	0
$879$	56.2132	1.89602
$880$	0	0
$881$	36.2946	1.22279	0.611397	−	0.791324i	$-0.290608\pi$
$881$	36.2946	1.22279	0.611397	+	0.791324i	$0.290608\pi$
$882$	0	0
$883$	26.8840	0.904718	0.452359	−	0.891836i	$-0.350583\pi$
$883$	26.8840	0.904718	0.452359	+	0.891836i	$0.350583\pi$
$884$	0	0
$885$	24.5671	0.825815
$886$	0	0
$887$	−13.4296	−0.450923	−0.225461	−	0.974252i	$-0.572389\pi$
$887$	−13.4296	−0.450923	−0.225461	+	0.974252i	$0.572389\pi$
$888$	0	0
$889$	−33.4501	−1.12188
$890$	0	0
$891$	−39.1336	−1.31102
$892$	0	0
$893$	14.0986	0.471792
$894$	0	0
$895$	15.5068	0.518334
$896$	0	0
$897$	59.2726	1.97905
$898$	0	0
$899$	−2.20955	−0.0736926
$900$	0	0
$901$	44.7165	1.48972
$902$	0	0
$903$	83.3154	2.77256
$904$	0	0
$905$	5.13926	0.170835
$906$	0	0
$907$	−7.62089	−0.253047	−0.126524	−	0.991964i	$-0.540382\pi$
$907$	−7.62089	−0.253047	−0.126524	+	0.991964i	$0.540382\pi$
$908$	0	0
$909$	154.817	5.13496
$910$	0	0
$911$	14.0113	0.464214	0.232107	−	0.972690i	$-0.425438\pi$
$911$	14.0113	0.464214	0.232107	+	0.972690i	$0.425438\pi$
$912$	0	0
$913$	−16.5714	−0.548432
$914$	0	0
$915$	−3.08993	−0.102150
$916$	0	0
$917$	41.4284	1.36809
$918$	0	0
$919$	−12.5233	−0.413107	−0.206553	−	0.978435i	$-0.566225\pi$
$919$	−12.5233	−0.413107	−0.206553	+	0.978435i	$0.566225\pi$
$920$	0	0
$921$	−16.5135	−0.544139
$922$	0	0
$923$	93.1815	3.06711
$924$	0	0
$925$	12.4549	0.409514
$926$	0	0
$927$	21.3211	0.700276
$928$	0	0
$929$	14.3668	0.471359	0.235680	−	0.971831i	$-0.424268\pi$
$929$	14.3668	0.471359	0.235680	+	0.971831i	$0.424268\pi$
$930$	0	0
$931$	−16.0842	−0.527138
$932$	0	0
$933$	−32.8213	−1.07452
$934$	0	0
$935$	9.63571	0.315121
$936$	0	0
$937$	57.2739	1.87106	0.935528	−	0.353253i	$-0.114925\pi$
$937$	57.2739	1.87106	0.935528	+	0.353253i	$0.114925\pi$
$938$	0	0
$939$	92.4039	3.01549
$940$	0	0
$941$	−2.50683	−0.0817202	−0.0408601	−	0.999165i	$-0.513010\pi$
$941$	−2.50683	−0.0817202	−0.0408601	+	0.999165i	$0.513010\pi$
$942$	0	0
$943$	−2.98141	−0.0970880
$944$	0	0
$945$	65.7249	2.13803
$946$	0	0
$947$	53.8904	1.75120	0.875601	−	0.483036i	$-0.160466\pi$
$947$	53.8904	1.75120	0.875601	+	0.483036i	$0.160466\pi$
$948$	0	0
$949$	51.4252	1.66933
$950$	0	0
$951$	−18.6101	−0.603473
$952$	0	0
$953$	−20.7256	−0.671369	−0.335684	−	0.941975i	$-0.608968\pi$
$953$	−20.7256	−0.671369	−0.335684	+	0.941975i	$0.608968\pi$
$954$	0	0
$955$	1.66608	0.0539131
$956$	0	0
$957$	−14.9942	−0.484692
$958$	0	0
$959$	−45.2046	−1.45973
$960$	0	0
$961$	−30.5480	−0.985420
$962$	0	0
$963$	77.4530	2.49589
$964$	0	0
$965$	9.41836	0.303188
$966$	0	0
$967$	−12.2723	−0.394652	−0.197326	−	0.980338i	$-0.563226\pi$
$967$	−12.2723	−0.394652	−0.197326	+	0.980338i	$0.563226\pi$
$968$	0	0
$969$	−86.8068	−2.78864
$970$	0	0
$971$	−48.8599	−1.56799	−0.783994	−	0.620768i	$-0.786821\pi$
$971$	−48.8599	−1.56799	−0.783994	+	0.620768i	$0.786821\pi$
$972$	0	0
$973$	−53.8890	−1.72760
$974$	0	0
$975$	63.3136	2.02766
$976$	0	0
$977$	−29.7335	−0.951260	−0.475630	−	0.879645i	$-0.657780\pi$
$977$	−29.7335	−0.951260	−0.475630	+	0.879645i	$0.657780\pi$
$978$	0	0
$979$	16.6756	0.532953
$980$	0	0
$981$	−43.9927	−1.40458
$982$	0	0
$983$	−26.3255	−0.839655	−0.419827	−	0.907604i	$-0.637909\pi$
$983$	−26.3255	−0.839655	−0.419827	+	0.907604i	$0.637909\pi$
$984$	0	0
$985$	−8.71290	−0.277616
$986$	0	0
$987$	29.5949	0.942015
$988$	0	0
$989$	24.3906	0.775575
$990$	0	0
$991$	−44.7423	−1.42129	−0.710643	−	0.703552i	$-0.751596\pi$
$991$	−44.7423	−1.42129	−0.710643	+	0.703552i	$0.751596\pi$
$992$	0	0
$993$	48.8994	1.55177
$994$	0	0
$995$	16.6955	0.529284
$996$	0	0
$997$	−39.5818	−1.25357	−0.626784	−	0.779193i	$-0.715629\pi$
$997$	−39.5818	−1.25357	−0.626784	+	0.779193i	$0.715629\pi$
$998$	0	0
$999$	−59.5323	−1.88352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.n.1.12		12
4.3	odd	2		241.2.a.b.1.7	✓	12
12.11	even	2		2169.2.a.h.1.6		12
20.19	odd	2		6025.2.a.h.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.7	✓	12	4.3	odd	2
2169.2.a.h.1.6		12	12.11	even	2
3856.2.a.n.1.12		12	1.1	even	1	trivial
6025.2.a.h.1.6		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)

\(f(q)\)	\(=\)	\(q+3.28295 q^{3} -1.31091 q^{5} -3.19647 q^{7} +7.77775 q^{9} +O(q^{10})\)	\(q+3.28295 q^{3} -1.31091 q^{5} -3.19647 q^{7} +7.77775 q^{9} -1.38968 q^{11} -5.87704 q^{13} -4.30365 q^{15} +5.28927 q^{17} -4.99913 q^{19} -10.4938 q^{21} -3.07207 q^{23} -3.28152 q^{25} +15.6851 q^{27} +3.28657 q^{29} -0.672296 q^{31} -4.56225 q^{33} +4.19028 q^{35} -3.79547 q^{37} -19.2940 q^{39} +0.970489 q^{41} -7.93946 q^{43} -10.1959 q^{45} -2.82021 q^{47} +3.21740 q^{49} +17.3644 q^{51} +8.45419 q^{53} +1.82175 q^{55} -16.4119 q^{57} -5.70844 q^{59} +0.717980 q^{61} -24.8613 q^{63} +7.70427 q^{65} -8.81215 q^{67} -10.0854 q^{69} -15.8552 q^{71} -8.75018 q^{73} -10.7730 q^{75} +4.44207 q^{77} -11.4452 q^{79} +28.1601 q^{81} +11.9246 q^{83} -6.93376 q^{85} +10.7896 q^{87} -11.9996 q^{89} +18.7858 q^{91} -2.20711 q^{93} +6.55340 q^{95} +1.18886 q^{97} -10.8086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) 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

Introduction

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