LMFDB - Embedded newform 3856.2.a.h.1.5

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:	$6$
Coefficient field:	6.6.131357120.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 26x^{2} - 30x - 13 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 26x^2 - 30*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 482)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.77369$ 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)

$f(q)$	$=$	$q+1.77369 q^{3} +1.52670 q^{5} -4.00586 q^{7} +0.145968 q^{9} +O(q^{10})$	$q+1.77369 q^{3} +1.52670 q^{5} -4.00586 q^{7} +0.145968 q^{9} +1.91966 q^{11} +2.11142 q^{13} +2.70788 q^{15} -3.44635 q^{17} -4.14251 q^{19} -7.10515 q^{21} -0.392960 q^{23} -2.66920 q^{25} -5.06216 q^{27} +4.62754 q^{29} -8.22418 q^{31} +3.40487 q^{33} -6.11573 q^{35} -6.16010 q^{37} +3.74501 q^{39} +3.68457 q^{41} -10.9937 q^{43} +0.222849 q^{45} -10.9246 q^{47} +9.04695 q^{49} -6.11275 q^{51} +6.77004 q^{53} +2.93073 q^{55} -7.34751 q^{57} +11.2518 q^{59} -9.47876 q^{61} -0.584729 q^{63} +3.22350 q^{65} -0.318566 q^{67} -0.696989 q^{69} +0.234085 q^{71} +8.54719 q^{73} -4.73433 q^{75} -7.68988 q^{77} +6.82063 q^{79} -9.41660 q^{81} +1.63035 q^{83} -5.26153 q^{85} +8.20781 q^{87} +1.46703 q^{89} -8.45808 q^{91} -14.5871 q^{93} -6.32434 q^{95} +12.7398 q^{97} +0.280209 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9	$ 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9} + 4 q^{11} + 9 q^{13} - 5 q^{15} + q^{17} - 10 q^{19} + 8 q^{21} - 9 q^{23} + 13 q^{25} - 8 q^{27} - q^{29} - 14 q^{31} + 10 q^{33} + 3 q^{35} + 20 q^{37} - 13 q^{39} - 4 q^{41} - 19 q^{43} - 6 q^{45} - q^{47} + 14 q^{49} - 5 q^{51} - 3 q^{53} - 11 q^{55} + 12 q^{57} + q^{59} + 4 q^{61} - 14 q^{63} + 5 q^{65} - 5 q^{67} - 15 q^{69} - 2 q^{71} + 15 q^{73} + 11 q^{75} - 6 q^{77} - 12 q^{79} - 18 q^{81} + 8 q^{83} - 32 q^{85} - 11 q^{87} - 24 q^{89} - q^{91} - 18 q^{93} - 9 q^{95} - 12 q^{97} + 40 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9 + 4 * q^11 + 9 * q^13 - 5 * q^15 + q^17 - 10 * q^19 + 8 * q^21 - 9 * q^23 + 13 * q^25 - 8 * q^27 - q^29 - 14 * q^31 + 10 * q^33 + 3 * q^35 + 20 * q^37 - 13 * q^39 - 4 * q^41 - 19 * q^43 - 6 * q^45 - q^47 + 14 * q^49 - 5 * q^51 - 3 * q^53 - 11 * q^55 + 12 * q^57 + q^59 + 4 * q^61 - 14 * q^63 + 5 * q^65 - 5 * q^67 - 15 * q^69 - 2 * q^71 + 15 * q^73 + 11 * q^75 - 6 * q^77 - 12 * q^79 - 18 * q^81 + 8 * q^83 - 32 * q^85 - 11 * q^87 - 24 * q^89 - q^91 - 18 * q^93 - 9 * q^95 - 12 * q^97 + 40 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.77369	1.02404	0.512020	−	0.858974i	$-0.328898\pi$
$3$	1.77369	1.02404	0.512020	+	0.858974i	$0.328898\pi$
$4$	0	0
$5$	1.52670	0.682759	0.341380	−	0.939926i	$-0.389106\pi$
$5$	1.52670	0.682759	0.341380	+	0.939926i	$0.389106\pi$
$6$	0	0
$7$	−4.00586	−1.51407	−0.757037	−	0.653372i	$-0.773354\pi$
$7$	−4.00586	−1.51407	−0.757037	+	0.653372i	$0.773354\pi$
$8$	0	0
$9$	0.145968	0.0486561
$10$	0	0
$11$	1.91966	0.578798	0.289399	−	0.957209i	$-0.406545\pi$
$11$	1.91966	0.578798	0.289399	+	0.957209i	$0.406545\pi$
$12$	0	0
$13$	2.11142	0.585604	0.292802	−	0.956173i	$-0.405412\pi$
$13$	2.11142	0.585604	0.292802	+	0.956173i	$0.405412\pi$
$14$	0	0
$15$	2.70788	0.699172
$16$	0	0
$17$	−3.44635	−0.835863	−0.417932	−	0.908479i	$-0.637245\pi$
$17$	−3.44635	−0.835863	−0.417932	+	0.908479i	$0.637245\pi$
$18$	0	0
$19$	−4.14251	−0.950356	−0.475178	−	0.879890i	$-0.657616\pi$
$19$	−4.14251	−0.950356	−0.475178	+	0.879890i	$0.657616\pi$
$20$	0	0
$21$	−7.10515	−1.55047
$22$	0	0
$23$	−0.392960	−0.0819379	−0.0409690	−	0.999160i	$-0.513044\pi$
$23$	−0.392960	−0.0819379	−0.0409690	+	0.999160i	$0.513044\pi$
$24$	0	0
$25$	−2.66920	−0.533840
$26$	0	0
$27$	−5.06216	−0.974213
$28$	0	0
$29$	4.62754	0.859312	0.429656	−	0.902993i	$-0.358635\pi$
$29$	4.62754	0.859312	0.429656	+	0.902993i	$0.358635\pi$
$30$	0	0
$31$	−8.22418	−1.47711	−0.738553	−	0.674196i	$-0.764490\pi$
$31$	−8.22418	−1.47711	−0.738553	+	0.674196i	$0.764490\pi$
$32$	0	0
$33$	3.40487	0.592712
$34$	0	0
$35$	−6.11573	−1.03375
$36$	0	0
$37$	−6.16010	−1.01271	−0.506357	−	0.862324i	$-0.669008\pi$
$37$	−6.16010	−1.01271	−0.506357	+	0.862324i	$0.669008\pi$
$38$	0	0
$39$	3.74501	0.599681
$40$	0	0
$41$	3.68457	0.575434	0.287717	−	0.957716i	$-0.407104\pi$
$41$	3.68457	0.575434	0.287717	+	0.957716i	$0.407104\pi$
$42$	0	0
$43$	−10.9937	−1.67653	−0.838264	−	0.545265i	$-0.816429\pi$
$43$	−10.9937	−1.67653	−0.838264	+	0.545265i	$0.816429\pi$
$44$	0	0
$45$	0.222849	0.0332204
$46$	0	0
$47$	−10.9246	−1.59352	−0.796761	−	0.604294i	$-0.793455\pi$
$47$	−10.9246	−1.59352	−0.796761	+	0.604294i	$0.793455\pi$
$48$	0	0
$49$	9.04695	1.29242
$50$	0	0
$51$	−6.11275	−0.855956
$52$	0	0
$53$	6.77004	0.929937	0.464968	−	0.885327i	$-0.346066\pi$
$53$	6.77004	0.929937	0.464968	+	0.885327i	$0.346066\pi$
$54$	0	0
$55$	2.93073	0.395180
$56$	0	0
$57$	−7.34751	−0.973202
$58$	0	0
$59$	11.2518	1.46486	0.732429	−	0.680843i	$-0.238386\pi$
$59$	11.2518	1.46486	0.732429	+	0.680843i	$0.238386\pi$
$60$	0	0
$61$	−9.47876	−1.21363	−0.606815	−	0.794843i	$-0.707553\pi$
$61$	−9.47876	−1.21363	−0.606815	+	0.794843i	$0.707553\pi$
$62$	0	0
$63$	−0.584729	−0.0736689
$64$	0	0
$65$	3.22350	0.399826
$66$	0	0
$67$	−0.318566	−0.0389190	−0.0194595	−	0.999811i	$-0.506195\pi$
$67$	−0.318566	−0.0389190	−0.0194595	+	0.999811i	$0.506195\pi$
$68$	0	0
$69$	−0.696989	−0.0839076
$70$	0	0
$71$	0.234085	0.0277808	0.0138904	−	0.999904i	$-0.495578\pi$
$71$	0.234085	0.0277808	0.0138904	+	0.999904i	$0.495578\pi$
$72$	0	0
$73$	8.54719	1.00037	0.500187	−	0.865918i	$-0.333265\pi$
$73$	8.54719	1.00037	0.500187	+	0.865918i	$0.333265\pi$
$74$	0	0
$75$	−4.73433	−0.546673
$76$	0	0
$77$	−7.68988	−0.876343
$78$	0	0
$79$	6.82063	0.767381	0.383691	−	0.923462i	$-0.374653\pi$
$79$	6.82063	0.767381	0.383691	+	0.923462i	$0.374653\pi$
$80$	0	0
$81$	−9.41660	−1.04629
$82$	0	0
$83$	1.63035	0.178954	0.0894769	−	0.995989i	$-0.471480\pi$
$83$	1.63035	0.178954	0.0894769	+	0.995989i	$0.471480\pi$
$84$	0	0
$85$	−5.26153	−0.570693
$86$	0	0
$87$	8.20781	0.879969
$88$	0	0
$89$	1.46703	0.155505	0.0777525	−	0.996973i	$-0.475226\pi$
$89$	1.46703	0.155505	0.0777525	+	0.996973i	$0.475226\pi$
$90$	0	0
$91$	−8.45808	−0.886648
$92$	0	0
$93$	−14.5871	−1.51261
$94$	0	0
$95$	−6.32434	−0.648864
$96$	0	0
$97$	12.7398	1.29353	0.646766	−	0.762689i	$-0.276121\pi$
$97$	12.7398	1.29353	0.646766	+	0.762689i	$0.276121\pi$
$98$	0	0
$99$	0.280209	0.0281621
$100$	0	0
$101$	−8.53274	−0.849040	−0.424520	−	0.905419i	$-0.639557\pi$
$101$	−8.53274	−0.849040	−0.424520	+	0.905419i	$0.639557\pi$
$102$	0	0
$103$	−9.47829	−0.933923	−0.466962	−	0.884278i	$-0.654651\pi$
$103$	−9.47829	−0.933923	−0.466962	+	0.884278i	$0.654651\pi$
$104$	0	0
$105$	−10.8474	−1.05860
$106$	0	0
$107$	11.7585	1.13673	0.568367	−	0.822775i	$-0.307575\pi$
$107$	11.7585	1.13673	0.568367	+	0.822775i	$0.307575\pi$
$108$	0	0
$109$	−9.85916	−0.944336	−0.472168	−	0.881509i	$-0.656528\pi$
$109$	−9.85916	−0.944336	−0.472168	+	0.881509i	$0.656528\pi$
$110$	0	0
$111$	−10.9261	−1.03706
$112$	0	0
$113$	15.4274	1.45128	0.725642	−	0.688072i	$-0.241543\pi$
$113$	15.4274	1.45128	0.725642	+	0.688072i	$0.241543\pi$
$114$	0	0
$115$	−0.599931	−0.0559439
$116$	0	0
$117$	0.308201	0.0284932
$118$	0	0
$119$	13.8056	1.26556
$120$	0	0
$121$	−7.31492	−0.664993
$122$	0	0
$123$	6.53528	0.589267
$124$	0	0
$125$	−11.7085	−1.04724
$126$	0	0
$127$	10.5263	0.934057	0.467028	−	0.884242i	$-0.345325\pi$
$127$	10.5263	0.934057	0.467028	+	0.884242i	$0.345325\pi$
$128$	0	0
$129$	−19.4994	−1.71683
$130$	0	0
$131$	−2.79109	−0.243858	−0.121929	−	0.992539i	$-0.538908\pi$
$131$	−2.79109	−0.243858	−0.121929	+	0.992539i	$0.538908\pi$
$132$	0	0
$133$	16.5943	1.43891
$134$	0	0
$135$	−7.72838	−0.665153
$136$	0	0
$137$	−0.220545	−0.0188424	−0.00942121	−	0.999956i	$-0.502999\pi$
$137$	−0.220545	−0.0188424	−0.00942121	+	0.999956i	$0.502999\pi$
$138$	0	0
$139$	−6.27606	−0.532328	−0.266164	−	0.963928i	$-0.585756\pi$
$139$	−6.27606	−0.532328	−0.266164	+	0.963928i	$0.585756\pi$
$140$	0	0
$141$	−19.3769	−1.63183
$142$	0	0
$143$	4.05321	0.338946
$144$	0	0
$145$	7.06484	0.586703
$146$	0	0
$147$	16.0465	1.32349
$148$	0	0
$149$	−15.4892	−1.26892	−0.634461	−	0.772955i	$-0.718778\pi$
$149$	−15.4892	−1.26892	−0.634461	+	0.772955i	$0.718778\pi$
$150$	0	0
$151$	−23.4753	−1.91039	−0.955196	−	0.295974i	$-0.904356\pi$
$151$	−23.4753	−1.91039	−0.955196	+	0.295974i	$0.904356\pi$
$152$	0	0
$153$	−0.503058	−0.0406698
$154$	0	0
$155$	−12.5558	−1.00851
$156$	0	0
$157$	−16.5031	−1.31709	−0.658545	−	0.752541i	$-0.728828\pi$
$157$	−16.5031	−1.31709	−0.658545	+	0.752541i	$0.728828\pi$
$158$	0	0
$159$	12.0079	0.952292
$160$	0	0
$161$	1.57415	0.124060
$162$	0	0
$163$	−16.4102	−1.28534	−0.642672	−	0.766142i	$-0.722174\pi$
$163$	−16.4102	−1.28534	−0.642672	+	0.766142i	$0.722174\pi$
$164$	0	0
$165$	5.19820	0.404679
$166$	0	0
$167$	−0.683727	−0.0529084	−0.0264542	−	0.999650i	$-0.508422\pi$
$167$	−0.683727	−0.0529084	−0.0264542	+	0.999650i	$0.508422\pi$
$168$	0	0
$169$	−8.54189	−0.657068
$170$	0	0
$171$	−0.604674	−0.0462406
$172$	0	0
$173$	−8.56431	−0.651133	−0.325566	−	0.945519i	$-0.605555\pi$
$173$	−8.56431	−0.651133	−0.325566	+	0.945519i	$0.605555\pi$
$174$	0	0
$175$	10.6925	0.808274
$176$	0	0
$177$	19.9572	1.50007
$178$	0	0
$179$	15.1572	1.13290	0.566451	−	0.824096i	$-0.308316\pi$
$179$	15.1572	1.13290	0.566451	+	0.824096i	$0.308316\pi$
$180$	0	0
$181$	−21.5786	−1.60393	−0.801963	−	0.597374i	$-0.796211\pi$
$181$	−21.5786	−1.60393	−0.801963	+	0.597374i	$0.796211\pi$
$182$	0	0
$183$	−16.8124	−1.24281
$184$	0	0
$185$	−9.40459	−0.691439
$186$	0	0
$187$	−6.61581	−0.483796
$188$	0	0
$189$	20.2783	1.47503
$190$	0	0
$191$	−5.31889	−0.384861	−0.192431	−	0.981311i	$-0.561637\pi$
$191$	−5.31889	−0.384861	−0.192431	+	0.981311i	$0.561637\pi$
$192$	0	0
$193$	8.33214	0.599760	0.299880	−	0.953977i	$-0.403053\pi$
$193$	8.33214	0.599760	0.299880	+	0.953977i	$0.403053\pi$
$194$	0	0
$195$	5.71749	0.409438
$196$	0	0
$197$	0.585565	0.0417198	0.0208599	−	0.999782i	$-0.493360\pi$
$197$	0.585565	0.0417198	0.0208599	+	0.999782i	$0.493360\pi$
$198$	0	0
$199$	−7.16064	−0.507605	−0.253802	−	0.967256i	$-0.581681\pi$
$199$	−7.16064	−0.507605	−0.253802	+	0.967256i	$0.581681\pi$
$200$	0	0
$201$	−0.565037	−0.0398546
$202$	0	0
$203$	−18.5373	−1.30106
$204$	0	0
$205$	5.62522	0.392883
$206$	0	0
$207$	−0.0573598	−0.00398678
$208$	0	0
$209$	−7.95218	−0.550064
$210$	0	0
$211$	10.0311	0.690569	0.345284	−	0.938498i	$-0.387782\pi$
$211$	10.0311	0.690569	0.345284	+	0.938498i	$0.387782\pi$
$212$	0	0
$213$	0.415194	0.0284486
$214$	0	0
$215$	−16.7841	−1.14466
$216$	0	0
$217$	32.9449	2.23645
$218$	0	0
$219$	15.1601	1.02442
$220$	0	0
$221$	−7.27671	−0.489485
$222$	0	0
$223$	17.3298	1.16049	0.580245	−	0.814442i	$-0.302957\pi$
$223$	17.3298	1.16049	0.580245	+	0.814442i	$0.302957\pi$
$224$	0	0
$225$	−0.389619	−0.0259746
$226$	0	0
$227$	18.4902	1.22724	0.613618	−	0.789603i	$-0.289713\pi$
$227$	18.4902	1.22724	0.613618	+	0.789603i	$0.289713\pi$
$228$	0	0
$229$	−12.2168	−0.807306	−0.403653	−	0.914912i	$-0.632260\pi$
$229$	−12.2168	−0.807306	−0.403653	+	0.914912i	$0.632260\pi$
$230$	0	0
$231$	−13.6394	−0.897410
$232$	0	0
$233$	−7.64789	−0.501030	−0.250515	−	0.968113i	$-0.580600\pi$
$233$	−7.64789	−0.501030	−0.250515	+	0.968113i	$0.580600\pi$
$234$	0	0
$235$	−16.6786	−1.08799
$236$	0	0
$237$	12.0977	0.785828
$238$	0	0
$239$	2.62790	0.169985	0.0849924	−	0.996382i	$-0.472913\pi$
$239$	2.62790	0.169985	0.0849924	+	0.996382i	$0.472913\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	−1.51562	−0.0972271
$244$	0	0
$245$	13.8119	0.882412
$246$	0	0
$247$	−8.74659	−0.556532
$248$	0	0
$249$	2.89173	0.183256
$250$	0	0
$251$	9.73104	0.614218	0.307109	−	0.951674i	$-0.400638\pi$
$251$	9.73104	0.614218	0.307109	+	0.951674i	$0.400638\pi$
$252$	0	0
$253$	−0.754349	−0.0474255
$254$	0	0
$255$	−9.33231	−0.584412
$256$	0	0
$257$	16.3528	1.02006	0.510030	−	0.860156i	$-0.329634\pi$
$257$	16.3528	1.02006	0.510030	+	0.860156i	$0.329634\pi$
$258$	0	0
$259$	24.6765	1.53332
$260$	0	0
$261$	0.675474	0.0418108
$262$	0	0
$263$	7.90280	0.487308	0.243654	−	0.969862i	$-0.421654\pi$
$263$	7.90280	0.487308	0.243654	+	0.969862i	$0.421654\pi$
$264$	0	0
$265$	10.3358	0.634923
$266$	0	0
$267$	2.60206	0.159243
$268$	0	0
$269$	14.5736	0.888570	0.444285	−	0.895886i	$-0.353458\pi$
$269$	14.5736	0.888570	0.444285	+	0.895886i	$0.353458\pi$
$270$	0	0
$271$	13.5423	0.822634	0.411317	−	0.911492i	$-0.365069\pi$
$271$	13.5423	0.822634	0.411317	+	0.911492i	$0.365069\pi$
$272$	0	0
$273$	−15.0020	−0.907962
$274$	0	0
$275$	−5.12395	−0.308986
$276$	0	0
$277$	14.7375	0.885491	0.442746	−	0.896647i	$-0.354004\pi$
$277$	14.7375	0.885491	0.442746	+	0.896647i	$0.354004\pi$
$278$	0	0
$279$	−1.20047	−0.0718702
$280$	0	0
$281$	17.1139	1.02093	0.510465	−	0.859898i	$-0.329473\pi$
$281$	17.1139	1.02093	0.510465	+	0.859898i	$0.329473\pi$
$282$	0	0
$283$	27.1565	1.61429	0.807145	−	0.590354i	$-0.201012\pi$
$283$	27.1565	1.61429	0.807145	+	0.590354i	$0.201012\pi$
$284$	0	0
$285$	−11.2174	−0.664462
$286$	0	0
$287$	−14.7599	−0.871249
$288$	0	0
$289$	−5.12266	−0.301333
$290$	0	0
$291$	22.5964	1.32463
$292$	0	0
$293$	20.8664	1.21903	0.609514	−	0.792775i	$-0.291365\pi$
$293$	20.8664	1.21903	0.609514	+	0.792775i	$0.291365\pi$
$294$	0	0
$295$	17.1781	1.00015
$296$	0	0
$297$	−9.71761	−0.563873
$298$	0	0
$299$	−0.829706	−0.0479832
$300$	0	0
$301$	44.0394	2.53839
$302$	0	0
$303$	−15.1344	−0.869450
$304$	0	0
$305$	−14.4712	−0.828617
$306$	0	0
$307$	−27.6687	−1.57913	−0.789567	−	0.613664i	$-0.789695\pi$
$307$	−27.6687	−1.57913	−0.789567	+	0.613664i	$0.789695\pi$
$308$	0	0
$309$	−16.8115	−0.956374
$310$	0	0
$311$	13.7820	0.781506	0.390753	−	0.920496i	$-0.372215\pi$
$311$	13.7820	0.781506	0.390753	+	0.920496i	$0.372215\pi$
$312$	0	0
$313$	−15.9327	−0.900571	−0.450286	−	0.892885i	$-0.648678\pi$
$313$	−15.9327	−0.900571	−0.450286	+	0.892885i	$0.648678\pi$
$314$	0	0
$315$	−0.892703	−0.0502981
$316$	0	0
$317$	−7.54834	−0.423957	−0.211978	−	0.977274i	$-0.567991\pi$
$317$	−7.54834	−0.423957	−0.211978	+	0.977274i	$0.567991\pi$
$318$	0	0
$319$	8.88328	0.497368
$320$	0	0
$321$	20.8558	1.16406
$322$	0	0
$323$	14.2765	0.794367
$324$	0	0
$325$	−5.63582	−0.312619
$326$	0	0
$327$	−17.4871	−0.967037
$328$	0	0
$329$	43.7626	2.41271
$330$	0	0
$331$	−25.0021	−1.37424	−0.687120	−	0.726544i	$-0.741125\pi$
$331$	−25.0021	−1.37424	−0.687120	+	0.726544i	$0.741125\pi$
$332$	0	0
$333$	−0.899179	−0.0492747
$334$	0	0
$335$	−0.486354	−0.0265723
$336$	0	0
$337$	−12.7940	−0.696932	−0.348466	−	0.937321i	$-0.613297\pi$
$337$	−12.7940	−0.696932	−0.348466	+	0.937321i	$0.613297\pi$
$338$	0	0
$339$	27.3633	1.48617
$340$	0	0
$341$	−15.7876	−0.854946
$342$	0	0
$343$	−8.19978	−0.442747
$344$	0	0
$345$	−1.06409	−0.0572887
$346$	0	0
$347$	16.2493	0.872309	0.436154	−	0.899872i	$-0.356340\pi$
$347$	16.2493	0.872309	0.436154	+	0.899872i	$0.356340\pi$
$348$	0	0
$349$	−10.0687	−0.538966	−0.269483	−	0.963005i	$-0.586853\pi$
$349$	−10.0687	−0.538966	−0.269483	+	0.963005i	$0.586853\pi$
$350$	0	0
$351$	−10.6884	−0.570503
$352$	0	0
$353$	−12.6517	−0.673381	−0.336690	−	0.941615i	$-0.609308\pi$
$353$	−12.6517	−0.673381	−0.336690	+	0.941615i	$0.609308\pi$
$354$	0	0
$355$	0.357377	0.0189676
$356$	0	0
$357$	24.4869	1.29598
$358$	0	0
$359$	−1.89540	−0.100035	−0.0500177	−	0.998748i	$-0.515928\pi$
$359$	−1.89540	−0.100035	−0.0500177	+	0.998748i	$0.515928\pi$
$360$	0	0
$361$	−1.83965	−0.0968238
$362$	0	0
$363$	−12.9744	−0.680979
$364$	0	0
$365$	13.0490	0.683014
$366$	0	0
$367$	21.3889	1.11649	0.558245	−	0.829676i	$-0.311475\pi$
$367$	21.3889	1.11649	0.558245	+	0.829676i	$0.311475\pi$
$368$	0	0
$369$	0.537831	0.0279984
$370$	0	0
$371$	−27.1199	−1.40799
$372$	0	0
$373$	1.49893	0.0776117	0.0388058	−	0.999247i	$-0.487645\pi$
$373$	1.49893	0.0776117	0.0388058	+	0.999247i	$0.487645\pi$
$374$	0	0
$375$	−20.7673	−1.07242
$376$	0	0
$377$	9.77070	0.503216
$378$	0	0
$379$	−20.4949	−1.05275	−0.526377	−	0.850251i	$-0.676450\pi$
$379$	−20.4949	−1.05275	−0.526377	+	0.850251i	$0.676450\pi$
$380$	0	0
$381$	18.6703	0.956511
$382$	0	0
$383$	−12.3337	−0.630221	−0.315110	−	0.949055i	$-0.602042\pi$
$383$	−12.3337	−0.630221	−0.315110	+	0.949055i	$0.602042\pi$
$384$	0	0
$385$	−11.7401	−0.598331
$386$	0	0
$387$	−1.60474	−0.0815733
$388$	0	0
$389$	−23.3762	−1.18522	−0.592611	−	0.805489i	$-0.701903\pi$
$389$	−23.3762	−1.18522	−0.592611	+	0.805489i	$0.701903\pi$
$390$	0	0
$391$	1.35428	0.0684889
$392$	0	0
$393$	−4.95052	−0.249721
$394$	0	0
$395$	10.4130	0.523936
$396$	0	0
$397$	2.14990	0.107901	0.0539503	−	0.998544i	$-0.482819\pi$
$397$	2.14990	0.107901	0.0539503	+	0.998544i	$0.482819\pi$
$398$	0	0
$399$	29.4331	1.47350
$400$	0	0
$401$	9.89274	0.494020	0.247010	−	0.969013i	$-0.420552\pi$
$401$	9.89274	0.494020	0.247010	+	0.969013i	$0.420552\pi$
$402$	0	0
$403$	−17.3647	−0.864999
$404$	0	0
$405$	−14.3763	−0.714363
$406$	0	0
$407$	−11.8253	−0.586157
$408$	0	0
$409$	−30.6591	−1.51600	−0.757998	−	0.652256i	$-0.773823\pi$
$409$	−30.6591	−1.51600	−0.757998	+	0.652256i	$0.773823\pi$
$410$	0	0
$411$	−0.391178	−0.0192954
$412$	0	0
$413$	−45.0732	−2.21790
$414$	0	0
$415$	2.48904	0.122182
$416$	0	0
$417$	−11.1318	−0.545125
$418$	0	0
$419$	19.9273	0.973513	0.486757	−	0.873538i	$-0.338180\pi$
$419$	19.9273	0.973513	0.486757	+	0.873538i	$0.338180\pi$
$420$	0	0
$421$	9.56092	0.465971	0.232985	−	0.972480i	$-0.425151\pi$
$421$	9.56092	0.465971	0.232985	+	0.972480i	$0.425151\pi$
$422$	0	0
$423$	−1.59465	−0.0775346
$424$	0	0
$425$	9.19900	0.446217
$426$	0	0
$427$	37.9706	1.83753
$428$	0	0
$429$	7.18913	0.347094
$430$	0	0
$431$	14.2606	0.686911	0.343455	−	0.939169i	$-0.388403\pi$
$431$	14.2606	0.686911	0.343455	+	0.939169i	$0.388403\pi$
$432$	0	0
$433$	−10.6496	−0.511785	−0.255892	−	0.966705i	$-0.582369\pi$
$433$	−10.6496	−0.511785	−0.255892	+	0.966705i	$0.582369\pi$
$434$	0	0
$435$	12.5308	0.600807
$436$	0	0
$437$	1.62784	0.0778702
$438$	0	0
$439$	−13.8041	−0.658833	−0.329416	−	0.944185i	$-0.606852\pi$
$439$	−13.8041	−0.658833	−0.329416	+	0.944185i	$0.606852\pi$
$440$	0	0
$441$	1.32057	0.0628841
$442$	0	0
$443$	−40.1850	−1.90925	−0.954623	−	0.297816i	$-0.903742\pi$
$443$	−40.1850	−1.90925	−0.954623	+	0.297816i	$0.903742\pi$
$444$	0	0
$445$	2.23971	0.106172
$446$	0	0
$447$	−27.4729	−1.29943
$448$	0	0
$449$	−17.6297	−0.831995	−0.415998	−	0.909366i	$-0.636568\pi$
$449$	−17.6297	−0.831995	−0.415998	+	0.909366i	$0.636568\pi$
$450$	0	0
$451$	7.07311	0.333060
$452$	0	0
$453$	−41.6378	−1.95632
$454$	0	0
$455$	−12.9129	−0.605367
$456$	0	0
$457$	2.05781	0.0962603	0.0481302	−	0.998841i	$-0.484674\pi$
$457$	2.05781	0.0962603	0.0481302	+	0.998841i	$0.484674\pi$
$458$	0	0
$459$	17.4460	0.814309
$460$	0	0
$461$	5.53192	0.257647	0.128824	−	0.991668i	$-0.458880\pi$
$461$	5.53192	0.257647	0.128824	+	0.991668i	$0.458880\pi$
$462$	0	0
$463$	33.0928	1.53795	0.768976	−	0.639278i	$-0.220767\pi$
$463$	33.0928	1.53795	0.768976	+	0.639278i	$0.220767\pi$
$464$	0	0
$465$	−22.2701	−1.03275
$466$	0	0
$467$	37.4259	1.73186	0.865932	−	0.500162i	$-0.166726\pi$
$467$	37.4259	1.73186	0.865932	+	0.500162i	$0.166726\pi$
$468$	0	0
$469$	1.27613	0.0589263
$470$	0	0
$471$	−29.2713	−1.34875
$472$	0	0
$473$	−21.1042	−0.970371
$474$	0	0
$475$	11.0572	0.507338
$476$	0	0
$477$	0.988211	0.0452471
$478$	0	0
$479$	11.8492	0.541406	0.270703	−	0.962663i	$-0.412744\pi$
$479$	11.8492	0.541406	0.270703	+	0.962663i	$0.412744\pi$
$480$	0	0
$481$	−13.0066	−0.593049
$482$	0	0
$483$	2.79204	0.127042
$484$	0	0
$485$	19.4498	0.883170
$486$	0	0
$487$	−42.4956	−1.92566	−0.962830	−	0.270108i	$-0.912940\pi$
$487$	−42.4956	−1.92566	−0.962830	+	0.270108i	$0.912940\pi$
$488$	0	0
$489$	−29.1065	−1.31624
$490$	0	0
$491$	−17.7837	−0.802569	−0.401285	−	0.915953i	$-0.631436\pi$
$491$	−17.7837	−0.802569	−0.401285	+	0.915953i	$0.631436\pi$
$492$	0	0
$493$	−15.9481	−0.718267
$494$	0	0
$495$	0.427794	0.0192279
$496$	0	0
$497$	−0.937714	−0.0420622
$498$	0	0
$499$	27.4111	1.22709	0.613545	−	0.789660i	$-0.289743\pi$
$499$	27.4111	1.22709	0.613545	+	0.789660i	$0.289743\pi$
$500$	0	0
$501$	−1.21272	−0.0541802
$502$	0	0
$503$	−3.38490	−0.150925	−0.0754627	−	0.997149i	$-0.524043\pi$
$503$	−3.38490	−0.150925	−0.0754627	+	0.997149i	$0.524043\pi$
$504$	0	0
$505$	−13.0269	−0.579689
$506$	0	0
$507$	−15.1506	−0.672864
$508$	0	0
$509$	34.9091	1.54732	0.773658	−	0.633603i	$-0.218425\pi$
$509$	34.9091	1.54732	0.773658	+	0.633603i	$0.218425\pi$
$510$	0	0
$511$	−34.2389	−1.51464
$512$	0	0
$513$	20.9700	0.925849
$514$	0	0
$515$	−14.4705	−0.637645
$516$	0	0
$517$	−20.9716	−0.922328
$518$	0	0
$519$	−15.1904	−0.666785
$520$	0	0
$521$	28.1444	1.23303	0.616513	−	0.787344i	$-0.288545\pi$
$521$	28.1444	1.23303	0.616513	+	0.787344i	$0.288545\pi$
$522$	0	0
$523$	28.7122	1.25550	0.627749	−	0.778415i	$-0.283976\pi$
$523$	28.7122	1.25550	0.627749	+	0.778415i	$0.283976\pi$
$524$	0	0
$525$	18.9651	0.827704
$526$	0	0
$527$	28.3434	1.23466
$528$	0	0
$529$	−22.8456	−0.993286
$530$	0	0
$531$	1.64240	0.0712743
$532$	0	0
$533$	7.77970	0.336976
$534$	0	0
$535$	17.9516	0.776115
$536$	0	0
$537$	26.8841	1.16014
$538$	0	0
$539$	17.3670	0.748051
$540$	0	0
$541$	33.9149	1.45812	0.729059	−	0.684451i	$-0.239958\pi$
$541$	33.9149	1.45812	0.729059	+	0.684451i	$0.239958\pi$
$542$	0	0
$543$	−38.2737	−1.64248
$544$	0	0
$545$	−15.0519	−0.644754
$546$	0	0
$547$	−23.0028	−0.983528	−0.491764	−	0.870728i	$-0.663648\pi$
$547$	−23.0028	−0.983528	−0.491764	+	0.870728i	$0.663648\pi$
$548$	0	0
$549$	−1.38360	−0.0590505
$550$	0	0
$551$	−19.1696	−0.816652
$552$	0	0
$553$	−27.3225	−1.16187
$554$	0	0
$555$	−16.6808	−0.708061
$556$	0	0
$557$	41.9944	1.77936	0.889681	−	0.456583i	$-0.150927\pi$
$557$	41.9944	1.77936	0.889681	+	0.456583i	$0.150927\pi$
$558$	0	0
$559$	−23.2124	−0.981781
$560$	0	0
$561$	−11.7344	−0.495426
$562$	0	0
$563$	−34.8757	−1.46984	−0.734918	−	0.678156i	$-0.762780\pi$
$563$	−34.8757	−1.46984	−0.734918	+	0.678156i	$0.762780\pi$
$564$	0	0
$565$	23.5529	0.990877
$566$	0	0
$567$	37.7216	1.58416
$568$	0	0
$569$	−16.3677	−0.686169	−0.343085	−	0.939305i	$-0.611472\pi$
$569$	−16.3677	−0.686169	−0.343085	+	0.939305i	$0.611472\pi$
$570$	0	0
$571$	24.9867	1.04566	0.522831	−	0.852436i	$-0.324876\pi$
$571$	24.9867	1.04566	0.522831	+	0.852436i	$0.324876\pi$
$572$	0	0
$573$	−9.43405	−0.394113
$574$	0	0
$575$	1.04889	0.0437418
$576$	0	0
$577$	9.37218	0.390169	0.195085	−	0.980786i	$-0.437502\pi$
$577$	9.37218	0.390169	0.195085	+	0.980786i	$0.437502\pi$
$578$	0	0
$579$	14.7786	0.614178
$580$	0	0
$581$	−6.53095	−0.270949
$582$	0	0
$583$	12.9962	0.538246
$584$	0	0
$585$	0.470529	0.0194540
$586$	0	0
$587$	−40.1071	−1.65540	−0.827699	−	0.561172i	$-0.810351\pi$
$587$	−40.1071	−1.65540	−0.827699	+	0.561172i	$0.810351\pi$
$588$	0	0
$589$	34.0687	1.40378
$590$	0	0
$591$	1.03861	0.0427227
$592$	0	0
$593$	42.1891	1.73250	0.866249	−	0.499613i	$-0.166524\pi$
$593$	42.1891	1.73250	0.866249	+	0.499613i	$0.166524\pi$
$594$	0	0
$595$	21.0770	0.864072
$596$	0	0
$597$	−12.7007	−0.519807
$598$	0	0
$599$	−34.4472	−1.40747	−0.703736	−	0.710461i	$-0.748486\pi$
$599$	−34.4472	−1.40747	−0.703736	+	0.710461i	$0.748486\pi$
$600$	0	0
$601$	−8.51761	−0.347441	−0.173720	−	0.984795i	$-0.555579\pi$
$601$	−8.51761	−0.347441	−0.173720	+	0.984795i	$0.555579\pi$
$602$	0	0
$603$	−0.0465005	−0.00189365
$604$	0	0
$605$	−11.1677	−0.454030
$606$	0	0
$607$	39.7684	1.61415	0.807074	−	0.590450i	$-0.201050\pi$
$607$	39.7684	1.61415	0.807074	+	0.590450i	$0.201050\pi$
$608$	0	0
$609$	−32.8794	−1.33234
$610$	0	0
$611$	−23.0666	−0.933173
$612$	0	0
$613$	−6.00614	−0.242586	−0.121293	−	0.992617i	$-0.538704\pi$
$613$	−6.00614	−0.242586	−0.121293	+	0.992617i	$0.538704\pi$
$614$	0	0
$615$	9.97739	0.402327
$616$	0	0
$617$	3.65465	0.147131	0.0735654	−	0.997290i	$-0.476562\pi$
$617$	3.65465	0.147131	0.0735654	+	0.997290i	$0.476562\pi$
$618$	0	0
$619$	28.8741	1.16055	0.580274	−	0.814422i	$-0.302946\pi$
$619$	28.8741	1.16055	0.580274	+	0.814422i	$0.302946\pi$
$620$	0	0
$621$	1.98923	0.0798250
$622$	0	0
$623$	−5.87673	−0.235446
$624$	0	0
$625$	−4.52936	−0.181175
$626$	0	0
$627$	−14.1047	−0.563287
$628$	0	0
$629$	21.2299	0.846490
$630$	0	0
$631$	−26.9232	−1.07180	−0.535898	−	0.844283i	$-0.680027\pi$
$631$	−26.9232	−1.07180	−0.535898	+	0.844283i	$0.680027\pi$
$632$	0	0
$633$	17.7920	0.707169
$634$	0	0
$635$	16.0704	0.637736
$636$	0	0
$637$	19.1019	0.756846
$638$	0	0
$639$	0.0341690	0.00135171
$640$	0	0
$641$	8.58200	0.338969	0.169484	−	0.985533i	$-0.445790\pi$
$641$	8.58200	0.338969	0.169484	+	0.985533i	$0.445790\pi$
$642$	0	0
$643$	23.6655	0.933277	0.466639	−	0.884448i	$-0.345465\pi$
$643$	23.6655	0.933277	0.466639	+	0.884448i	$0.345465\pi$
$644$	0	0
$645$	−29.7697	−1.17218
$646$	0	0
$647$	27.3187	1.07401	0.537005	−	0.843579i	$-0.319556\pi$
$647$	27.3187	1.07401	0.537005	+	0.843579i	$0.319556\pi$
$648$	0	0
$649$	21.5996	0.847857
$650$	0	0
$651$	58.4340	2.29021
$652$	0	0
$653$	26.4323	1.03438	0.517188	−	0.855872i	$-0.326979\pi$
$653$	26.4323	1.03438	0.517188	+	0.855872i	$0.326979\pi$
$654$	0	0
$655$	−4.26114	−0.166497
$656$	0	0
$657$	1.24762	0.0486743
$658$	0	0
$659$	−2.83007	−0.110244	−0.0551218	−	0.998480i	$-0.517555\pi$
$659$	−2.83007	−0.110244	−0.0551218	+	0.998480i	$0.517555\pi$
$660$	0	0
$661$	10.3796	0.403721	0.201860	−	0.979414i	$-0.435301\pi$
$661$	10.3796	0.403721	0.201860	+	0.979414i	$0.435301\pi$
$662$	0	0
$663$	−12.9066	−0.501251
$664$	0	0
$665$	25.3345	0.982428
$666$	0	0
$667$	−1.81844	−0.0704102
$668$	0	0
$669$	30.7377	1.18839
$670$	0	0
$671$	−18.1960	−0.702447
$672$	0	0
$673$	−0.138389	−0.00533449	−0.00266724	−	0.999996i	$-0.500849\pi$
$673$	−0.138389	−0.00533449	−0.00266724	+	0.999996i	$0.500849\pi$
$674$	0	0
$675$	13.5119	0.520074
$676$	0	0
$677$	−2.29908	−0.0883607	−0.0441803	−	0.999024i	$-0.514068\pi$
$677$	−2.29908	−0.0883607	−0.0441803	+	0.999024i	$0.514068\pi$
$678$	0	0
$679$	−51.0339	−1.95850
$680$	0	0
$681$	32.7958	1.25674
$682$	0	0
$683$	−42.9255	−1.64250	−0.821249	−	0.570570i	$-0.806722\pi$
$683$	−42.9255	−1.64250	−0.821249	+	0.570570i	$0.806722\pi$
$684$	0	0
$685$	−0.336705	−0.0128648
$686$	0	0
$687$	−21.6687	−0.826713
$688$	0	0
$689$	14.2944	0.544575
$690$	0	0
$691$	−22.2885	−0.847895	−0.423948	−	0.905687i	$-0.639356\pi$
$691$	−22.2885	−0.847895	−0.423948	+	0.905687i	$0.639356\pi$
$692$	0	0
$693$	−1.12248	−0.0426394
$694$	0	0
$695$	−9.58163	−0.363452
$696$	0	0
$697$	−12.6983	−0.480984
$698$	0	0
$699$	−13.5650	−0.513075
$700$	0	0
$701$	25.7589	0.972899	0.486450	−	0.873709i	$-0.338292\pi$
$701$	25.7589	0.972899	0.486450	+	0.873709i	$0.338292\pi$
$702$	0	0
$703$	25.5182	0.962438
$704$	0	0
$705$	−29.5826	−1.11415
$706$	0	0
$707$	34.1810	1.28551
$708$	0	0
$709$	−25.8524	−0.970906	−0.485453	−	0.874263i	$-0.661345\pi$
$709$	−25.8524	−0.970906	−0.485453	+	0.874263i	$0.661345\pi$
$710$	0	0
$711$	0.995596	0.0373378
$712$	0	0
$713$	3.23178	0.121031
$714$	0	0
$715$	6.18802	0.231419
$716$	0	0
$717$	4.66108	0.174071
$718$	0	0
$719$	−31.8242	−1.18684	−0.593421	−	0.804892i	$-0.702223\pi$
$719$	−31.8242	−1.18684	−0.593421	+	0.804892i	$0.702223\pi$
$720$	0	0
$721$	37.9687	1.41403
$722$	0	0
$723$	1.77369	0.0659642
$724$	0	0
$725$	−12.3518	−0.458735
$726$	0	0
$727$	16.0395	0.594870	0.297435	−	0.954742i	$-0.403869\pi$
$727$	16.0395	0.594870	0.297435	+	0.954742i	$0.403869\pi$
$728$	0	0
$729$	25.5616	0.946724
$730$	0	0
$731$	37.8882	1.40135
$732$	0	0
$733$	36.6027	1.35195	0.675976	−	0.736923i	$-0.263722\pi$
$733$	36.6027	1.35195	0.675976	+	0.736923i	$0.263722\pi$
$734$	0	0
$735$	24.4981	0.903624
$736$	0	0
$737$	−0.611537	−0.0225263
$738$	0	0
$739$	−30.7203	−1.13007	−0.565033	−	0.825068i	$-0.691137\pi$
$739$	−30.7203	−1.13007	−0.565033	+	0.825068i	$0.691137\pi$
$740$	0	0
$741$	−15.5137	−0.569911
$742$	0	0
$743$	45.4477	1.66731	0.833657	−	0.552283i	$-0.186243\pi$
$743$	45.4477	1.66731	0.833657	+	0.552283i	$0.186243\pi$
$744$	0	0
$745$	−23.6472	−0.866368
$746$	0	0
$747$	0.237979	0.00870719
$748$	0	0
$749$	−47.1028	−1.72110
$750$	0	0
$751$	−51.5079	−1.87955	−0.939775	−	0.341794i	$-0.888965\pi$
$751$	−51.5079	−1.87955	−0.939775	+	0.341794i	$0.888965\pi$
$752$	0	0
$753$	17.2598	0.628983
$754$	0	0
$755$	−35.8396	−1.30434
$756$	0	0
$757$	−45.2438	−1.64442	−0.822208	−	0.569188i	$-0.807258\pi$
$757$	−45.2438	−1.64442	−0.822208	+	0.569188i	$0.807258\pi$
$758$	0	0
$759$	−1.33798	−0.0485656
$760$	0	0
$761$	−28.1908	−1.02192	−0.510959	−	0.859605i	$-0.670710\pi$
$761$	−28.1908	−1.02192	−0.510959	+	0.859605i	$0.670710\pi$
$762$	0	0
$763$	39.4944	1.42979
$764$	0	0
$765$	−0.768016	−0.0277677
$766$	0	0
$767$	23.7573	0.857827
$768$	0	0
$769$	38.8523	1.40105	0.700526	−	0.713627i	$-0.252949\pi$
$769$	38.8523	1.40105	0.700526	+	0.713627i	$0.252949\pi$
$770$	0	0
$771$	29.0048	1.04458
$772$	0	0
$773$	−34.9213	−1.25603	−0.628016	−	0.778200i	$-0.716133\pi$
$773$	−34.9213	−1.25603	−0.628016	+	0.778200i	$0.716133\pi$
$774$	0	0
$775$	21.9520	0.788538
$776$	0	0
$777$	43.7684	1.57018
$778$	0	0
$779$	−15.2634	−0.546867
$780$	0	0
$781$	0.449363	0.0160795
$782$	0	0
$783$	−23.4253	−0.837153
$784$	0	0
$785$	−25.1952	−0.899255
$786$	0	0
$787$	17.5722	0.626381	0.313190	−	0.949690i	$-0.398602\pi$
$787$	17.5722	0.626381	0.313190	+	0.949690i	$0.398602\pi$
$788$	0	0
$789$	14.0171	0.499022
$790$	0	0
$791$	−61.7999	−2.19735
$792$	0	0
$793$	−20.0137	−0.710707
$794$	0	0
$795$	18.3325	0.650186
$796$	0	0
$797$	−30.0300	−1.06372	−0.531859	−	0.846833i	$-0.678506\pi$
$797$	−30.0300	−1.06372	−0.531859	+	0.846833i	$0.678506\pi$
$798$	0	0
$799$	37.6501	1.33197
$800$	0	0
$801$	0.214140	0.00756627
$802$	0	0
$803$	16.4077	0.579014
$804$	0	0
$805$	2.40324	0.0847031
$806$	0	0
$807$	25.8491	0.909930
$808$	0	0
$809$	48.4560	1.70362	0.851811	−	0.523849i	$-0.175505\pi$
$809$	48.4560	1.70362	0.851811	+	0.523849i	$0.175505\pi$
$810$	0	0
$811$	−4.86264	−0.170750	−0.0853751	−	0.996349i	$-0.527209\pi$
$811$	−4.86264	−0.170750	−0.0853751	+	0.996349i	$0.527209\pi$
$812$	0	0
$813$	24.0198	0.842410
$814$	0	0
$815$	−25.0533	−0.877580
$816$	0	0
$817$	45.5416	1.59330
$818$	0	0
$819$	−1.23461	−0.0431408
$820$	0	0
$821$	−39.1894	−1.36772	−0.683860	−	0.729614i	$-0.739700\pi$
$821$	−39.1894	−1.36772	−0.683860	+	0.729614i	$0.739700\pi$
$822$	0	0
$823$	39.4980	1.37681	0.688407	−	0.725324i	$-0.258310\pi$
$823$	39.4980	1.37681	0.688407	+	0.725324i	$0.258310\pi$
$824$	0	0
$825$	−9.08828	−0.316413
$826$	0	0
$827$	28.9936	1.00820	0.504102	−	0.863644i	$-0.331823\pi$
$827$	28.9936	1.00820	0.504102	+	0.863644i	$0.331823\pi$
$828$	0	0
$829$	5.96444	0.207153	0.103577	−	0.994621i	$-0.466971\pi$
$829$	5.96444	0.207153	0.103577	+	0.994621i	$0.466971\pi$
$830$	0	0
$831$	26.1397	0.906778
$832$	0	0
$833$	−31.1790	−1.08029
$834$	0	0
$835$	−1.04384	−0.0361237
$836$	0	0
$837$	41.6321	1.43902
$838$	0	0
$839$	7.82525	0.270158	0.135079	−	0.990835i	$-0.456871\pi$
$839$	7.82525	0.270158	0.135079	+	0.990835i	$0.456871\pi$
$840$	0	0
$841$	−7.58590	−0.261583
$842$	0	0
$843$	30.3547	1.04547
$844$	0	0
$845$	−13.0409	−0.448619
$846$	0	0
$847$	29.3026	1.00685
$848$	0	0
$849$	48.1672	1.65310
$850$	0	0
$851$	2.42067	0.0829797
$852$	0	0
$853$	−11.3189	−0.387552	−0.193776	−	0.981046i	$-0.562074\pi$
$853$	−11.3189	−0.387552	−0.193776	+	0.981046i	$0.562074\pi$
$854$	0	0
$855$	−0.923154	−0.0315712
$856$	0	0
$857$	−38.6810	−1.32132	−0.660659	−	0.750687i	$-0.729723\pi$
$857$	−38.6810	−1.32132	−0.660659	+	0.750687i	$0.729723\pi$
$858$	0	0
$859$	39.1473	1.33569	0.667845	−	0.744301i	$-0.267217\pi$
$859$	39.1473	1.33569	0.667845	+	0.744301i	$0.267217\pi$
$860$	0	0
$861$	−26.1795	−0.892193
$862$	0	0
$863$	−21.6263	−0.736167	−0.368084	−	0.929793i	$-0.619986\pi$
$863$	−21.6263	−0.736167	−0.368084	+	0.929793i	$0.619986\pi$
$864$	0	0
$865$	−13.0751	−0.444567
$866$	0	0
$867$	−9.08600	−0.308577
$868$	0	0
$869$	13.0933	0.444159
$870$	0	0
$871$	−0.672628	−0.0227911
$872$	0	0
$873$	1.85961	0.0629382
$874$	0	0
$875$	46.9028	1.58560
$876$	0	0
$877$	12.6547	0.427319	0.213660	−	0.976908i	$-0.431462\pi$
$877$	12.6547	0.427319	0.213660	+	0.976908i	$0.431462\pi$
$878$	0	0
$879$	37.0105	1.24833
$880$	0	0
$881$	33.7759	1.13794	0.568969	−	0.822359i	$-0.307342\pi$
$881$	33.7759	1.13794	0.568969	+	0.822359i	$0.307342\pi$
$882$	0	0
$883$	−40.5833	−1.36574	−0.682868	−	0.730542i	$-0.739268\pi$
$883$	−40.5833	−1.36574	−0.682868	+	0.730542i	$0.739268\pi$
$884$	0	0
$885$	30.4685	1.02419
$886$	0	0
$887$	−36.0623	−1.21085	−0.605426	−	0.795902i	$-0.706997\pi$
$887$	−36.0623	−1.21085	−0.605426	+	0.795902i	$0.706997\pi$
$888$	0	0
$889$	−42.1669	−1.41423
$890$	0	0
$891$	−18.0766	−0.605590
$892$	0	0
$893$	45.2554	1.51441
$894$	0	0
$895$	23.1404	0.773499
$896$	0	0
$897$	−1.47164	−0.0491366
$898$	0	0
$899$	−38.0577	−1.26929
$900$	0	0
$901$	−23.3319	−0.777300
$902$	0	0
$903$	78.1121	2.59941
$904$	0	0
$905$	−32.9440	−1.09509
$906$	0	0
$907$	35.5570	1.18065	0.590326	−	0.807165i	$-0.298999\pi$
$907$	35.5570	1.18065	0.590326	+	0.807165i	$0.298999\pi$
$908$	0	0
$909$	−1.24551	−0.0413109
$910$	0	0
$911$	14.2809	0.473148	0.236574	−	0.971614i	$-0.423975\pi$
$911$	14.2809	0.473148	0.236574	+	0.971614i	$0.423975\pi$
$912$	0	0
$913$	3.12971	0.103578
$914$	0	0
$915$	−25.6674	−0.848537
$916$	0	0
$917$	11.1807	0.369220
$918$	0	0
$919$	−13.3134	−0.439170	−0.219585	−	0.975593i	$-0.570470\pi$
$919$	−13.3134	−0.439170	−0.219585	+	0.975593i	$0.570470\pi$
$920$	0	0
$921$	−49.0756	−1.61709
$922$	0	0
$923$	0.494254	0.0162686
$924$	0	0
$925$	16.4425	0.540627
$926$	0	0
$927$	−1.38353	−0.0454411
$928$	0	0
$929$	11.0313	0.361924	0.180962	−	0.983490i	$-0.442079\pi$
$929$	11.0313	0.361924	0.180962	+	0.983490i	$0.442079\pi$
$930$	0	0
$931$	−37.4770	−1.22826
$932$	0	0
$933$	24.4450	0.800292
$934$	0	0
$935$	−10.1003	−0.330316
$936$	0	0
$937$	18.1481	0.592873	0.296437	−	0.955052i	$-0.404202\pi$
$937$	18.1481	0.592873	0.296437	+	0.955052i	$0.404202\pi$
$938$	0	0
$939$	−28.2597	−0.922220
$940$	0	0
$941$	−43.0588	−1.40368	−0.701838	−	0.712337i	$-0.747637\pi$
$941$	−43.0588	−1.40368	−0.701838	+	0.712337i	$0.747637\pi$
$942$	0	0
$943$	−1.44789	−0.0471498
$944$	0	0
$945$	30.9588	1.00709
$946$	0	0
$947$	53.0639	1.72434	0.862172	−	0.506616i	$-0.169104\pi$
$947$	53.0639	1.72434	0.862172	+	0.506616i	$0.169104\pi$
$948$	0	0
$949$	18.0468	0.585822
$950$	0	0
$951$	−13.3884	−0.434149
$952$	0	0
$953$	20.1929	0.654111	0.327056	−	0.945005i	$-0.393944\pi$
$953$	20.1929	0.654111	0.327056	+	0.945005i	$0.393944\pi$
$954$	0	0
$955$	−8.12032	−0.262768
$956$	0	0
$957$	15.7562	0.509324
$958$	0	0
$959$	0.883473	0.0285288
$960$	0	0
$961$	36.6371	1.18184
$962$	0	0
$963$	1.71636	0.0553090
$964$	0	0
$965$	12.7206	0.409492
$966$	0	0
$967$	1.63585	0.0526053	0.0263026	−	0.999654i	$-0.491627\pi$
$967$	1.63585	0.0526053	0.0263026	+	0.999654i	$0.491627\pi$
$968$	0	0
$969$	25.3221	0.813463
$970$	0	0
$971$	37.9501	1.21788	0.608938	−	0.793218i	$-0.291596\pi$
$971$	37.9501	1.21788	0.608938	+	0.793218i	$0.291596\pi$
$972$	0	0
$973$	25.1410	0.805985
$974$	0	0
$975$	−9.99618	−0.320134
$976$	0	0
$977$	−22.4456	−0.718097	−0.359049	−	0.933319i	$-0.616899\pi$
$977$	−22.4456	−0.718097	−0.359049	+	0.933319i	$0.616899\pi$
$978$	0	0
$979$	2.81620	0.0900060
$980$	0	0
$981$	−1.43912	−0.0459477
$982$	0	0
$983$	−44.3101	−1.41327	−0.706637	−	0.707577i	$-0.749788\pi$
$983$	−44.3101	−1.41327	−0.706637	+	0.707577i	$0.749788\pi$
$984$	0	0
$985$	0.893979	0.0284845
$986$	0	0
$987$	77.6212	2.47071
$988$	0	0
$989$	4.32010	0.137371
$990$	0	0
$991$	−8.15757	−0.259134	−0.129567	−	0.991571i	$-0.541359\pi$
$991$	−8.15757	−0.259134	−0.129567	+	0.991571i	$0.541359\pi$
$992$	0	0
$993$	−44.3459	−1.40727
$994$	0	0
$995$	−10.9321	−0.346572
$996$	0	0
$997$	41.4720	1.31343	0.656716	−	0.754138i	$-0.271945\pi$
$997$	41.4720	1.31343	0.656716	+	0.754138i	$0.271945\pi$
$998$	0	0
$999$	31.1834	0.986599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.h.1.5		6
4.3	odd	2		482.2.a.d.1.2	✓	6
12.11	even	2		4338.2.a.u.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
482.2.a.d.1.2	✓	6	4.3	odd	2
3856.2.a.h.1.5		6	1.1	even	1	trivial
4338.2.a.u.1.2		6	12.11	even	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)

\(f(q)\)	\(=\)	\(q+1.77369 q^{3} +1.52670 q^{5} -4.00586 q^{7} +0.145968 q^{9} +O(q^{10})\)	\(q+1.77369 q^{3} +1.52670 q^{5} -4.00586 q^{7} +0.145968 q^{9} +1.91966 q^{11} +2.11142 q^{13} +2.70788 q^{15} -3.44635 q^{17} -4.14251 q^{19} -7.10515 q^{21} -0.392960 q^{23} -2.66920 q^{25} -5.06216 q^{27} +4.62754 q^{29} -8.22418 q^{31} +3.40487 q^{33} -6.11573 q^{35} -6.16010 q^{37} +3.74501 q^{39} +3.68457 q^{41} -10.9937 q^{43} +0.222849 q^{45} -10.9246 q^{47} +9.04695 q^{49} -6.11275 q^{51} +6.77004 q^{53} +2.93073 q^{55} -7.34751 q^{57} +11.2518 q^{59} -9.47876 q^{61} -0.584729 q^{63} +3.22350 q^{65} -0.318566 q^{67} -0.696989 q^{69} +0.234085 q^{71} +8.54719 q^{73} -4.73433 q^{75} -7.68988 q^{77} +6.82063 q^{79} -9.41660 q^{81} +1.63035 q^{83} -5.26153 q^{85} +8.20781 q^{87} +1.46703 q^{89} -8.45808 q^{91} -14.5871 q^{93} -6.32434 q^{95} +12.7398 q^{97} +0.280209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9	\( 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9} + 4 q^{11} + 9 q^{13} - 5 q^{15} + q^{17} - 10 q^{19} + 8 q^{21} - 9 q^{23} + 13 q^{25} - 8 q^{27} - q^{29} - 14 q^{31} + 10 q^{33} + 3 q^{35} + 20 q^{37} - 13 q^{39} - 4 q^{41} - 19 q^{43} - 6 q^{45} - q^{47} + 14 q^{49} - 5 q^{51} - 3 q^{53} - 11 q^{55} + 12 q^{57} + q^{59} + 4 q^{61} - 14 q^{63} + 5 q^{65} - 5 q^{67} - 15 q^{69} - 2 q^{71} + 15 q^{73} + 11 q^{75} - 6 q^{77} - 12 q^{79} - 18 q^{81} + 8 q^{83} - 32 q^{85} - 11 q^{87} - 24 q^{89} - q^{91} - 18 q^{93} - 9 q^{95} - 12 q^{97} + 40 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9 + 4 * q^11 + 9 * q^13 - 5 * q^15 + q^17 - 10 * q^19 + 8 * q^21 - 9 * q^23 + 13 * q^25 - 8 * q^27 - q^29 - 14 * q^31 + 10 * q^33 + 3 * q^35 + 20 * q^37 - 13 * q^39 - 4 * q^41 - 19 * q^43 - 6 * q^45 - q^47 + 14 * q^49 - 5 * q^51 - 3 * q^53 - 11 * q^55 + 12 * q^57 + q^59 + 4 * q^61 - 14 * q^63 + 5 * q^65 - 5 * q^67 - 15 * q^69 - 2 * q^71 + 15 * q^73 + 11 * q^75 - 6 * q^77 - 12 * q^79 - 18 * q^81 + 8 * q^83 - 32 * q^85 - 11 * q^87 - 24 * q^89 - q^91 - 18 * q^93 - 9 * q^95 - 12 * q^97 + 40 * q^99

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