LMFDB - Embedded newform 3856.2.a.n.1.6

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			1.6
Root			$2.01020$ of defining polynomial
Character	$\chi$	$=$	3856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.500591 q^{3} +1.92585 q^{5} +0.852319 q^{7} -2.74941 q^{9} +O(q^{10})$	$q-0.500591 q^{3} +1.92585 q^{5} +0.852319 q^{7} -2.74941 q^{9} -0.719546 q^{11} -1.93309 q^{13} -0.964064 q^{15} +0.439843 q^{17} -5.85432 q^{19} -0.426663 q^{21} +7.09215 q^{23} -1.29109 q^{25} +2.87810 q^{27} +10.0222 q^{29} -5.69622 q^{31} +0.360198 q^{33} +1.64144 q^{35} -3.17197 q^{37} +0.967685 q^{39} -6.39435 q^{41} -4.29669 q^{43} -5.29496 q^{45} +0.642479 q^{47} -6.27355 q^{49} -0.220181 q^{51} +0.729714 q^{53} -1.38574 q^{55} +2.93062 q^{57} -0.348904 q^{59} +1.12656 q^{61} -2.34337 q^{63} -3.72284 q^{65} -12.5549 q^{67} -3.55026 q^{69} -0.552289 q^{71} -10.9882 q^{73} +0.646309 q^{75} -0.613283 q^{77} -10.9569 q^{79} +6.80748 q^{81} -6.62478 q^{83} +0.847072 q^{85} -5.01703 q^{87} +12.5513 q^{89} -1.64760 q^{91} +2.85148 q^{93} -11.2746 q^{95} -5.00536 q^{97} +1.97833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.500591	−0.289016	−0.144508	−	0.989504i	$-0.546160\pi$
$3$	−0.500591	−0.289016	−0.144508	+	0.989504i	$0.546160\pi$
$4$	0	0
$5$	1.92585	0.861267	0.430634	−	0.902527i	$-0.358290\pi$
$5$	1.92585	0.861267	0.430634	+	0.902527i	$0.358290\pi$
$6$	0	0
$7$	0.852319	0.322146	0.161073	−	0.986942i	$-0.448505\pi$
$7$	0.852319	0.322146	0.161073	+	0.986942i	$0.448505\pi$
$8$	0	0
$9$	−2.74941	−0.916470
$10$	0	0
$11$	−0.719546	−0.216951	−0.108476	−	0.994099i	$-0.534597\pi$
$11$	−0.719546	−0.216951	−0.108476	+	0.994099i	$0.534597\pi$
$12$	0	0
$13$	−1.93309	−0.536141	−0.268071	−	0.963399i	$-0.586386\pi$
$13$	−1.93309	−0.536141	−0.268071	+	0.963399i	$0.586386\pi$
$14$	0	0
$15$	−0.964064	−0.248920
$16$	0	0
$17$	0.439843	0.106678	0.0533388	−	0.998576i	$-0.483014\pi$
$17$	0.439843	0.106678	0.0533388	+	0.998576i	$0.483014\pi$
$18$	0	0
$19$	−5.85432	−1.34307	−0.671537	−	0.740971i	$-0.734365\pi$
$19$	−5.85432	−1.34307	−0.671537	+	0.740971i	$0.734365\pi$
$20$	0	0
$21$	−0.426663	−0.0931055
$22$	0	0
$23$	7.09215	1.47881	0.739407	−	0.673258i	$-0.235106\pi$
$23$	7.09215	1.47881	0.739407	+	0.673258i	$0.235106\pi$
$24$	0	0
$25$	−1.29109	−0.258219
$26$	0	0
$27$	2.87810	0.553891
$28$	0	0
$29$	10.0222	1.86108	0.930540	−	0.366190i	$-0.119338\pi$
$29$	10.0222	1.86108	0.930540	+	0.366190i	$0.119338\pi$
$30$	0	0
$31$	−5.69622	−1.02307	−0.511536	−	0.859262i	$-0.670923\pi$
$31$	−5.69622	−1.02307	−0.511536	+	0.859262i	$0.670923\pi$
$32$	0	0
$33$	0.360198	0.0627025
$34$	0	0
$35$	1.64144	0.277454
$36$	0	0
$37$	−3.17197	−0.521468	−0.260734	−	0.965411i	$-0.583965\pi$
$37$	−3.17197	−0.521468	−0.260734	+	0.965411i	$0.583965\pi$
$38$	0	0
$39$	0.967685	0.154954
$40$	0	0
$41$	−6.39435	−0.998630	−0.499315	−	0.866421i	$-0.666415\pi$
$41$	−6.39435	−0.998630	−0.499315	+	0.866421i	$0.666415\pi$
$42$	0	0
$43$	−4.29669	−0.655239	−0.327620	−	0.944810i	$-0.606246\pi$
$43$	−4.29669	−0.655239	−0.327620	+	0.944810i	$0.606246\pi$
$44$	0	0
$45$	−5.29496	−0.789325
$46$	0	0
$47$	0.642479	0.0937152	0.0468576	−	0.998902i	$-0.485079\pi$
$47$	0.642479	0.0937152	0.0468576	+	0.998902i	$0.485079\pi$
$48$	0	0
$49$	−6.27355	−0.896222
$50$	0	0
$51$	−0.220181	−0.0308315
$52$	0	0
$53$	0.729714	0.100234	0.0501170	−	0.998743i	$-0.484041\pi$
$53$	0.729714	0.100234	0.0501170	+	0.998743i	$0.484041\pi$
$54$	0	0
$55$	−1.38574	−0.186853
$56$	0	0
$57$	2.93062	0.388170
$58$	0	0
$59$	−0.348904	−0.0454234	−0.0227117	−	0.999742i	$-0.507230\pi$
$59$	−0.348904	−0.0454234	−0.0227117	+	0.999742i	$0.507230\pi$
$60$	0	0
$61$	1.12656	0.144241	0.0721204	−	0.997396i	$-0.477023\pi$
$61$	1.12656	0.144241	0.0721204	+	0.997396i	$0.477023\pi$
$62$	0	0
$63$	−2.34337	−0.295237
$64$	0	0
$65$	−3.72284	−0.461761
$66$	0	0
$67$	−12.5549	−1.53383	−0.766915	−	0.641749i	$-0.778209\pi$
$67$	−12.5549	−1.53383	−0.766915	+	0.641749i	$0.778209\pi$
$68$	0	0
$69$	−3.55026	−0.427401
$70$	0	0
$71$	−0.552289	−0.0655446	−0.0327723	−	0.999463i	$-0.510434\pi$
$71$	−0.552289	−0.0655446	−0.0327723	+	0.999463i	$0.510434\pi$
$72$	0	0
$73$	−10.9882	−1.28607	−0.643037	−	0.765835i	$-0.722326\pi$
$73$	−10.9882	−1.28607	−0.643037	+	0.765835i	$0.722326\pi$
$74$	0	0
$75$	0.646309	0.0746294
$76$	0	0
$77$	−0.613283	−0.0698901
$78$	0	0
$79$	−10.9569	−1.23275	−0.616375	−	0.787453i	$-0.711400\pi$
$79$	−10.9569	−1.23275	−0.616375	+	0.787453i	$0.711400\pi$
$80$	0	0
$81$	6.80748	0.756386
$82$	0	0
$83$	−6.62478	−0.727164	−0.363582	−	0.931562i	$-0.618446\pi$
$83$	−6.62478	−0.727164	−0.363582	+	0.931562i	$0.618446\pi$
$84$	0	0
$85$	0.847072	0.0918779
$86$	0	0
$87$	−5.01703	−0.537882
$88$	0	0
$89$	12.5513	1.33044	0.665219	−	0.746648i	$-0.268338\pi$
$89$	12.5513	1.33044	0.665219	+	0.746648i	$0.268338\pi$
$90$	0	0
$91$	−1.64760	−0.172716
$92$	0	0
$93$	2.85148	0.295684
$94$	0	0
$95$	−11.2746	−1.15675
$96$	0	0
$97$	−5.00536	−0.508217	−0.254108	−	0.967176i	$-0.581782\pi$
$97$	−5.00536	−0.508217	−0.254108	+	0.967176i	$0.581782\pi$
$98$	0	0
$99$	1.97833	0.198829
$100$	0	0
$101$	0.191991	0.0191038	0.00955192	−	0.999954i	$-0.496959\pi$
$101$	0.191991	0.0191038	0.00955192	+	0.999954i	$0.496959\pi$
$102$	0	0
$103$	3.97237	0.391409	0.195705	−	0.980663i	$-0.437301\pi$
$103$	3.97237	0.391409	0.195705	+	0.980663i	$0.437301\pi$
$104$	0	0
$105$	−0.821690	−0.0801887
$106$	0	0
$107$	−6.70922	−0.648605	−0.324303	−	0.945953i	$-0.605130\pi$
$107$	−6.70922	−0.648605	−0.324303	+	0.945953i	$0.605130\pi$
$108$	0	0
$109$	4.03051	0.386053	0.193027	−	0.981194i	$-0.438170\pi$
$109$	4.03051	0.386053	0.193027	+	0.981194i	$0.438170\pi$
$110$	0	0
$111$	1.58786	0.150713
$112$	0	0
$113$	0.430322	0.0404813	0.0202407	−	0.999795i	$-0.493557\pi$
$113$	0.430322	0.0404813	0.0202407	+	0.999795i	$0.493557\pi$
$114$	0	0
$115$	13.6584	1.27365
$116$	0	0
$117$	5.31484	0.491357
$118$	0	0
$119$	0.374886	0.0343658
$120$	0	0
$121$	−10.4823	−0.952932
$122$	0	0
$123$	3.20095	0.288620
$124$	0	0
$125$	−12.1157	−1.08366
$126$	0	0
$127$	0.0290958	0.00258183	0.00129092	−	0.999999i	$-0.499589\pi$
$127$	0.0290958	0.00258183	0.00129092	+	0.999999i	$0.499589\pi$
$128$	0	0
$129$	2.15088	0.189375
$130$	0	0
$131$	7.50804	0.655980	0.327990	−	0.944681i	$-0.393629\pi$
$131$	7.50804	0.655980	0.327990	+	0.944681i	$0.393629\pi$
$132$	0	0
$133$	−4.98975	−0.432666
$134$	0	0
$135$	5.54280	0.477048
$136$	0	0
$137$	−2.79071	−0.238427	−0.119213	−	0.992869i	$-0.538037\pi$
$137$	−2.79071	−0.238427	−0.119213	+	0.992869i	$0.538037\pi$
$138$	0	0
$139$	−5.56312	−0.471858	−0.235929	−	0.971770i	$-0.575813\pi$
$139$	−5.56312	−0.471858	−0.235929	+	0.971770i	$0.575813\pi$
$140$	0	0
$141$	−0.321619	−0.0270852
$142$	0	0
$143$	1.39094	0.116317
$144$	0	0
$145$	19.3013	1.60289
$146$	0	0
$147$	3.14048	0.259023
$148$	0	0
$149$	14.8309	1.21500	0.607499	−	0.794321i	$-0.292173\pi$
$149$	14.8309	1.21500	0.607499	+	0.794321i	$0.292173\pi$
$150$	0	0
$151$	19.6745	1.60109	0.800543	−	0.599276i	$-0.204545\pi$
$151$	19.6745	1.60109	0.800543	+	0.599276i	$0.204545\pi$
$152$	0	0
$153$	−1.20931	−0.0977667
$154$	0	0
$155$	−10.9701	−0.881138
$156$	0	0
$157$	14.3437	1.14475	0.572377	−	0.819991i	$-0.306022\pi$
$157$	14.3437	1.14475	0.572377	+	0.819991i	$0.306022\pi$
$158$	0	0
$159$	−0.365288	−0.0289692
$160$	0	0
$161$	6.04477	0.476395
$162$	0	0
$163$	−4.99007	−0.390853	−0.195426	−	0.980718i	$-0.562609\pi$
$163$	−4.99007	−0.390853	−0.195426	+	0.980718i	$0.562609\pi$
$164$	0	0
$165$	0.693689	0.0540036
$166$	0	0
$167$	−15.6290	−1.20941	−0.604703	−	0.796451i	$-0.706708\pi$
$167$	−15.6290	−1.20941	−0.604703	+	0.796451i	$0.706708\pi$
$168$	0	0
$169$	−9.26318	−0.712552
$170$	0	0
$171$	16.0959	1.23089
$172$	0	0
$173$	0.833365	0.0633596	0.0316798	−	0.999498i	$-0.489914\pi$
$173$	0.833365	0.0633596	0.0316798	+	0.999498i	$0.489914\pi$
$174$	0	0
$175$	−1.10042	−0.0831842
$176$	0	0
$177$	0.174658	0.0131281
$178$	0	0
$179$	−14.9985	−1.12104	−0.560520	−	0.828141i	$-0.689399\pi$
$179$	−14.9985	−1.12104	−0.560520	+	0.828141i	$0.689399\pi$
$180$	0	0
$181$	−14.9752	−1.11310	−0.556550	−	0.830814i	$-0.687875\pi$
$181$	−14.9752	−1.11310	−0.556550	+	0.830814i	$0.687875\pi$
$182$	0	0
$183$	−0.563944	−0.0416879
$184$	0	0
$185$	−6.10874	−0.449123
$186$	0	0
$187$	−0.316487	−0.0231438
$188$	0	0
$189$	2.45306	0.178434
$190$	0	0
$191$	−8.51858	−0.616383	−0.308191	−	0.951324i	$-0.599724\pi$
$191$	−8.51858	−0.616383	−0.308191	+	0.951324i	$0.599724\pi$
$192$	0	0
$193$	−23.5303	−1.69375	−0.846874	−	0.531793i	$-0.821518\pi$
$193$	−23.5303	−1.69375	−0.846874	+	0.531793i	$0.821518\pi$
$194$	0	0
$195$	1.86362	0.133456
$196$	0	0
$197$	−26.7548	−1.90620	−0.953102	−	0.302649i	$-0.902129\pi$
$197$	−26.7548	−1.90620	−0.953102	+	0.302649i	$0.902129\pi$
$198$	0	0
$199$	19.5915	1.38880	0.694401	−	0.719589i	$-0.255670\pi$
$199$	19.5915	1.38880	0.694401	+	0.719589i	$0.255670\pi$
$200$	0	0
$201$	6.28488	0.443301
$202$	0	0
$203$	8.54213	0.599540
$204$	0	0
$205$	−12.3146	−0.860087
$206$	0	0
$207$	−19.4992	−1.35529
$208$	0	0
$209$	4.21246	0.291382
$210$	0	0
$211$	−4.58617	−0.315725	−0.157862	−	0.987461i	$-0.550460\pi$
$211$	−4.58617	−0.315725	−0.157862	+	0.987461i	$0.550460\pi$
$212$	0	0
$213$	0.276471	0.0189435
$214$	0	0
$215$	−8.27479	−0.564336
$216$	0	0
$217$	−4.85500	−0.329579
$218$	0	0
$219$	5.50060	0.371696
$220$	0	0
$221$	−0.850254	−0.0571943
$222$	0	0
$223$	11.8124	0.791018	0.395509	−	0.918462i	$-0.370568\pi$
$223$	11.8124	0.791018	0.395509	+	0.918462i	$0.370568\pi$
$224$	0	0
$225$	3.54974	0.236650
$226$	0	0
$227$	10.4668	0.694703	0.347352	−	0.937735i	$-0.387081\pi$
$227$	10.4668	0.694703	0.347352	+	0.937735i	$0.387081\pi$
$228$	0	0
$229$	−28.8373	−1.90562	−0.952811	−	0.303563i	$-0.901824\pi$
$229$	−28.8373	−1.90562	−0.952811	+	0.303563i	$0.901824\pi$
$230$	0	0
$231$	0.307004	0.0201994
$232$	0	0
$233$	0.483058	0.0316462	0.0158231	−	0.999875i	$-0.494963\pi$
$233$	0.483058	0.0316462	0.0158231	+	0.999875i	$0.494963\pi$
$234$	0	0
$235$	1.23732	0.0807138
$236$	0	0
$237$	5.48494	0.356285
$238$	0	0
$239$	2.08293	0.134734	0.0673668	−	0.997728i	$-0.478540\pi$
$239$	2.08293	0.134734	0.0673668	+	0.997728i	$0.478540\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	−12.0421	−0.772499
$244$	0	0
$245$	−12.0819	−0.771887
$246$	0	0
$247$	11.3169	0.720077
$248$	0	0
$249$	3.31631	0.210162
$250$	0	0
$251$	9.67130	0.610447	0.305223	−	0.952281i	$-0.401269\pi$
$251$	9.67130	0.610447	0.305223	+	0.952281i	$0.401269\pi$
$252$	0	0
$253$	−5.10313	−0.320831
$254$	0	0
$255$	−0.424037	−0.0265542
$256$	0	0
$257$	−16.0352	−1.00025	−0.500123	−	0.865954i	$-0.666712\pi$
$257$	−16.0352	−1.00025	−0.500123	+	0.865954i	$0.666712\pi$
$258$	0	0
$259$	−2.70353	−0.167989
$260$	0	0
$261$	−27.5552	−1.70562
$262$	0	0
$263$	−26.9402	−1.66120	−0.830602	−	0.556867i	$-0.812003\pi$
$263$	−26.9402	−1.66120	−0.830602	+	0.556867i	$0.812003\pi$
$264$	0	0
$265$	1.40532	0.0863282
$266$	0	0
$267$	−6.28308	−0.384518
$268$	0	0
$269$	30.7223	1.87317	0.936585	−	0.350441i	$-0.113968\pi$
$269$	30.7223	1.87317	0.936585	+	0.350441i	$0.113968\pi$
$270$	0	0
$271$	32.2539	1.95928	0.979641	−	0.200757i	$-0.0643403\pi$
$271$	32.2539	1.95928	0.979641	+	0.200757i	$0.0643403\pi$
$272$	0	0
$273$	0.824776	0.0499177
$274$	0	0
$275$	0.929002	0.0560209
$276$	0	0
$277$	9.84003	0.591230	0.295615	−	0.955307i	$-0.404475\pi$
$277$	9.84003	0.591230	0.295615	+	0.955307i	$0.404475\pi$
$278$	0	0
$279$	15.6612	0.937614
$280$	0	0
$281$	0.0623558	0.00371983	0.00185992	−	0.999998i	$-0.499408\pi$
$281$	0.0623558	0.00371983	0.00185992	+	0.999998i	$0.499408\pi$
$282$	0	0
$283$	23.1136	1.37396	0.686979	−	0.726677i	$-0.258936\pi$
$283$	23.1136	1.37396	0.686979	+	0.726677i	$0.258936\pi$
$284$	0	0
$285$	5.64394	0.334318
$286$	0	0
$287$	−5.45002	−0.321705
$288$	0	0
$289$	−16.8065	−0.988620
$290$	0	0
$291$	2.50564	0.146883
$292$	0	0
$293$	−7.59717	−0.443831	−0.221916	−	0.975066i	$-0.571231\pi$
$293$	−7.59717	−0.443831	−0.221916	+	0.975066i	$0.571231\pi$
$294$	0	0
$295$	−0.671938	−0.0391217
$296$	0	0
$297$	−2.07093	−0.120167
$298$	0	0
$299$	−13.7097	−0.792854
$300$	0	0
$301$	−3.66215	−0.211083
$302$	0	0
$303$	−0.0961090	−0.00552132
$304$	0	0
$305$	2.16958	0.124230
$306$	0	0
$307$	15.3566	0.876447	0.438224	−	0.898866i	$-0.355608\pi$
$307$	15.3566	0.876447	0.438224	+	0.898866i	$0.355608\pi$
$308$	0	0
$309$	−1.98853	−0.113124
$310$	0	0
$311$	−25.0513	−1.42053	−0.710263	−	0.703936i	$-0.751424\pi$
$311$	−25.0513	−1.42053	−0.710263	+	0.703936i	$0.751424\pi$
$312$	0	0
$313$	−8.95197	−0.505995	−0.252997	−	0.967467i	$-0.581416\pi$
$313$	−8.95197	−0.505995	−0.252997	+	0.967467i	$0.581416\pi$
$314$	0	0
$315$	−4.51299	−0.254278
$316$	0	0
$317$	20.7326	1.16446	0.582228	−	0.813025i	$-0.302181\pi$
$317$	20.7326	1.16446	0.582228	+	0.813025i	$0.302181\pi$
$318$	0	0
$319$	−7.21146	−0.403764
$320$	0	0
$321$	3.35858	0.187457
$322$	0	0
$323$	−2.57498	−0.143276
$324$	0	0
$325$	2.49579	0.138442
$326$	0	0
$327$	−2.01764	−0.111576
$328$	0	0
$329$	0.547597	0.0301900
$330$	0	0
$331$	−6.99950	−0.384727	−0.192364	−	0.981324i	$-0.561615\pi$
$331$	−6.99950	−0.384727	−0.192364	+	0.981324i	$0.561615\pi$
$332$	0	0
$333$	8.72103	0.477910
$334$	0	0
$335$	−24.1789	−1.32104
$336$	0	0
$337$	1.75653	0.0956843	0.0478421	−	0.998855i	$-0.484766\pi$
$337$	1.75653	0.0956843	0.0478421	+	0.998855i	$0.484766\pi$
$338$	0	0
$339$	−0.215415	−0.0116998
$340$	0	0
$341$	4.09870	0.221957
$342$	0	0
$343$	−11.3133	−0.610861
$344$	0	0
$345$	−6.83728	−0.368107
$346$	0	0
$347$	11.0359	0.592440	0.296220	−	0.955120i	$-0.404274\pi$
$347$	11.0359	0.592440	0.296220	+	0.955120i	$0.404274\pi$
$348$	0	0
$349$	−22.8659	−1.22398	−0.611992	−	0.790864i	$-0.709631\pi$
$349$	−22.8659	−1.22398	−0.611992	+	0.790864i	$0.709631\pi$
$350$	0	0
$351$	−5.56361	−0.296964
$352$	0	0
$353$	−19.2684	−1.02556	−0.512778	−	0.858522i	$-0.671383\pi$
$353$	−19.2684	−1.02556	−0.512778	+	0.858522i	$0.671383\pi$
$354$	0	0
$355$	−1.06363	−0.0564514
$356$	0	0
$357$	−0.187665	−0.00993226
$358$	0	0
$359$	−20.1999	−1.06611	−0.533056	−	0.846080i	$-0.678957\pi$
$359$	−20.1999	−1.06611	−0.533056	+	0.846080i	$0.678957\pi$
$360$	0	0
$361$	15.2731	0.803847
$362$	0	0
$363$	5.24732	0.275413
$364$	0	0
$365$	−21.1617	−1.10765
$366$	0	0
$367$	−3.91082	−0.204143	−0.102072	−	0.994777i	$-0.532547\pi$
$367$	−3.91082	−0.204143	−0.102072	+	0.994777i	$0.532547\pi$
$368$	0	0
$369$	17.5807	0.915214
$370$	0	0
$371$	0.621949	0.0322900
$372$	0	0
$373$	9.05376	0.468786	0.234393	−	0.972142i	$-0.424690\pi$
$373$	9.05376	0.468786	0.234393	+	0.972142i	$0.424690\pi$
$374$	0	0
$375$	6.06502	0.313196
$376$	0	0
$377$	−19.3738	−0.997802
$378$	0	0
$379$	−28.2491	−1.45106	−0.725530	−	0.688191i	$-0.758405\pi$
$379$	−28.2491	−1.45106	−0.725530	+	0.688191i	$0.758405\pi$
$380$	0	0
$381$	−0.0145651	−0.000746192 0
$382$	0	0
$383$	−31.6901	−1.61929	−0.809645	−	0.586920i	$-0.800340\pi$
$383$	−31.6901	−1.61929	−0.809645	+	0.586920i	$0.800340\pi$
$384$	0	0
$385$	−1.18109	−0.0601940
$386$	0	0
$387$	11.8134	0.600507
$388$	0	0
$389$	23.1689	1.17471	0.587355	−	0.809330i	$-0.300169\pi$
$389$	23.1689	1.17471	0.587355	+	0.809330i	$0.300169\pi$
$390$	0	0
$391$	3.11943	0.157756
$392$	0	0
$393$	−3.75845	−0.189589
$394$	0	0
$395$	−21.1014	−1.06173
$396$	0	0
$397$	30.6366	1.53761	0.768804	−	0.639484i	$-0.220852\pi$
$397$	30.6366	1.53761	0.768804	+	0.639484i	$0.220852\pi$
$398$	0	0
$399$	2.49782	0.125047
$400$	0	0
$401$	−8.47109	−0.423026	−0.211513	−	0.977375i	$-0.567839\pi$
$401$	−8.47109	−0.423026	−0.211513	+	0.977375i	$0.567839\pi$
$402$	0	0
$403$	11.0113	0.548511
$404$	0	0
$405$	13.1102	0.651451
$406$	0	0
$407$	2.28238	0.113133
$408$	0	0
$409$	23.2857	1.15140	0.575702	−	0.817660i	$-0.304729\pi$
$409$	23.2857	1.15140	0.575702	+	0.817660i	$0.304729\pi$
$410$	0	0
$411$	1.39701	0.0689092
$412$	0	0
$413$	−0.297377	−0.0146330
$414$	0	0
$415$	−12.7584	−0.626283
$416$	0	0
$417$	2.78485	0.136375
$418$	0	0
$419$	9.29059	0.453875	0.226937	−	0.973909i	$-0.427129\pi$
$419$	9.29059	0.453875	0.226937	+	0.973909i	$0.427129\pi$
$420$	0	0
$421$	15.4991	0.755380	0.377690	−	0.925932i	$-0.376719\pi$
$421$	15.4991	0.755380	0.377690	+	0.925932i	$0.376719\pi$
$422$	0	0
$423$	−1.76644	−0.0858871
$424$	0	0
$425$	−0.567878	−0.0275461
$426$	0	0
$427$	0.960185	0.0464666
$428$	0	0
$429$	−0.696294	−0.0336174
$430$	0	0
$431$	19.5308	0.940765	0.470382	−	0.882463i	$-0.344116\pi$
$431$	19.5308	0.940765	0.470382	+	0.882463i	$0.344116\pi$
$432$	0	0
$433$	−5.89329	−0.283213	−0.141607	−	0.989923i	$-0.545227\pi$
$433$	−5.89329	−0.283213	−0.141607	+	0.989923i	$0.545227\pi$
$434$	0	0
$435$	−9.66206	−0.463261
$436$	0	0
$437$	−41.5197	−1.98616
$438$	0	0
$439$	−22.1839	−1.05878	−0.529390	−	0.848379i	$-0.677579\pi$
$439$	−22.1839	−1.05878	−0.529390	+	0.848379i	$0.677579\pi$
$440$	0	0
$441$	17.2486	0.821360
$442$	0	0
$443$	14.6268	0.694940	0.347470	−	0.937691i	$-0.387041\pi$
$443$	14.6268	0.694940	0.347470	+	0.937691i	$0.387041\pi$
$444$	0	0
$445$	24.1720	1.14586
$446$	0	0
$447$	−7.42423	−0.351154
$448$	0	0
$449$	−34.2308	−1.61545	−0.807726	−	0.589558i	$-0.799302\pi$
$449$	−34.2308	−1.61545	−0.807726	+	0.589558i	$0.799302\pi$
$450$	0	0
$451$	4.60103	0.216654
$452$	0	0
$453$	−9.84886	−0.462740
$454$	0	0
$455$	−3.17304	−0.148755
$456$	0	0
$457$	−24.3534	−1.13921	−0.569603	−	0.821920i	$-0.692903\pi$
$457$	−24.3534	−1.13921	−0.569603	+	0.821920i	$0.692903\pi$
$458$	0	0
$459$	1.26591	0.0590877
$460$	0	0
$461$	10.3474	0.481924	0.240962	−	0.970535i	$-0.422537\pi$
$461$	10.3474	0.481924	0.240962	+	0.970535i	$0.422537\pi$
$462$	0	0
$463$	−30.7015	−1.42682	−0.713409	−	0.700748i	$-0.752850\pi$
$463$	−30.7015	−1.42682	−0.713409	+	0.700748i	$0.752850\pi$
$464$	0	0
$465$	5.49152	0.254663
$466$	0	0
$467$	20.5093	0.949056	0.474528	−	0.880240i	$-0.342619\pi$
$467$	20.5093	0.949056	0.474528	+	0.880240i	$0.342619\pi$
$468$	0	0
$469$	−10.7008	−0.494117
$470$	0	0
$471$	−7.18033	−0.330852
$472$	0	0
$473$	3.09167	0.142155
$474$	0	0
$475$	7.55848	0.346807
$476$	0	0
$477$	−2.00628	−0.0918614
$478$	0	0
$479$	−28.0475	−1.28152	−0.640761	−	0.767740i	$-0.721381\pi$
$479$	−28.0475	−1.28152	−0.640761	+	0.767740i	$0.721381\pi$
$480$	0	0
$481$	6.13168	0.279581
$482$	0	0
$483$	−3.02596	−0.137686
$484$	0	0
$485$	−9.63958	−0.437711
$486$	0	0
$487$	−10.9035	−0.494085	−0.247043	−	0.969005i	$-0.579459\pi$
$487$	−10.9035	−0.494085	−0.247043	+	0.969005i	$0.579459\pi$
$488$	0	0
$489$	2.49798	0.112963
$490$	0	0
$491$	27.6672	1.24860	0.624301	−	0.781184i	$-0.285384\pi$
$491$	27.6672	1.24860	0.624301	+	0.781184i	$0.285384\pi$
$492$	0	0
$493$	4.40820	0.198536
$494$	0	0
$495$	3.80997	0.171245
$496$	0	0
$497$	−0.470726	−0.0211149
$498$	0	0
$499$	17.3522	0.776792	0.388396	−	0.921493i	$-0.373029\pi$
$499$	17.3522	0.776792	0.388396	+	0.921493i	$0.373029\pi$
$500$	0	0
$501$	7.82371	0.349538
$502$	0	0
$503$	26.4875	1.18102	0.590510	−	0.807030i	$-0.298926\pi$
$503$	26.4875	1.18102	0.590510	+	0.807030i	$0.298926\pi$
$504$	0	0
$505$	0.369747	0.0164535
$506$	0	0
$507$	4.63706	0.205939
$508$	0	0
$509$	3.50048	0.155156	0.0775780	−	0.996986i	$-0.475281\pi$
$509$	3.50048	0.155156	0.0775780	+	0.996986i	$0.475281\pi$
$510$	0	0
$511$	−9.36547	−0.414304
$512$	0	0
$513$	−16.8493	−0.743916
$514$	0	0
$515$	7.65019	0.337108
$516$	0	0
$517$	−0.462293	−0.0203316
$518$	0	0
$519$	−0.417175	−0.0183119
$520$	0	0
$521$	8.93651	0.391516	0.195758	−	0.980652i	$-0.437283\pi$
$521$	8.93651	0.391516	0.195758	+	0.980652i	$0.437283\pi$
$522$	0	0
$523$	−6.13438	−0.268238	−0.134119	−	0.990965i	$-0.542820\pi$
$523$	−6.13438	−0.268238	−0.134119	+	0.990965i	$0.542820\pi$
$524$	0	0
$525$	0.550862	0.0240416
$526$	0	0
$527$	−2.50544	−0.109139
$528$	0	0
$529$	27.2986	1.18689
$530$	0	0
$531$	0.959280	0.0416292
$532$	0	0
$533$	12.3608	0.535407
$534$	0	0
$535$	−12.9210	−0.558622
$536$	0	0
$537$	7.50810	0.323999
$538$	0	0
$539$	4.51411	0.194437
$540$	0	0
$541$	32.8484	1.41226	0.706132	−	0.708081i	$-0.250439\pi$
$541$	32.8484	1.41226	0.706132	+	0.708081i	$0.250439\pi$
$542$	0	0
$543$	7.49647	0.321704
$544$	0	0
$545$	7.76217	0.332495
$546$	0	0
$547$	27.5818	1.17931	0.589656	−	0.807655i	$-0.299263\pi$
$547$	27.5818	1.17931	0.589656	+	0.807655i	$0.299263\pi$
$548$	0	0
$549$	−3.09736	−0.132192
$550$	0	0
$551$	−58.6733	−2.49957
$552$	0	0
$553$	−9.33879	−0.397126
$554$	0	0
$555$	3.05798	0.129804
$556$	0	0
$557$	12.3393	0.522834	0.261417	−	0.965226i	$-0.415810\pi$
$557$	12.3393	0.522834	0.261417	+	0.965226i	$0.415810\pi$
$558$	0	0
$559$	8.30587	0.351301
$560$	0	0
$561$	0.158431	0.00668895
$562$	0	0
$563$	35.3473	1.48971	0.744855	−	0.667227i	$-0.232519\pi$
$563$	35.3473	1.48971	0.744855	+	0.667227i	$0.232519\pi$
$564$	0	0
$565$	0.828737	0.0348652
$566$	0	0
$567$	5.80214	0.243667
$568$	0	0
$569$	19.0099	0.796938	0.398469	−	0.917182i	$-0.369542\pi$
$569$	19.0099	0.796938	0.398469	+	0.917182i	$0.369542\pi$
$570$	0	0
$571$	−23.9059	−1.00043	−0.500215	−	0.865901i	$-0.666746\pi$
$571$	−23.9059	−1.00043	−0.500215	+	0.865901i	$0.666746\pi$
$572$	0	0
$573$	4.26432	0.178145
$574$	0	0
$575$	−9.15662	−0.381858
$576$	0	0
$577$	24.7986	1.03238	0.516190	−	0.856474i	$-0.327350\pi$
$577$	24.7986	1.03238	0.516190	+	0.856474i	$0.327350\pi$
$578$	0	0
$579$	11.7791	0.489521
$580$	0	0
$581$	−5.64643	−0.234253
$582$	0	0
$583$	−0.525063	−0.0217459
$584$	0	0
$585$	10.2356	0.423190
$586$	0	0
$587$	45.8195	1.89117	0.945586	−	0.325373i	$-0.105490\pi$
$587$	45.8195	1.89117	0.945586	+	0.325373i	$0.105490\pi$
$588$	0	0
$589$	33.3475	1.37406
$590$	0	0
$591$	13.3932	0.550924
$592$	0	0
$593$	12.9205	0.530581	0.265290	−	0.964169i	$-0.414532\pi$
$593$	12.9205	0.530581	0.265290	+	0.964169i	$0.414532\pi$
$594$	0	0
$595$	0.721976	0.0295981
$596$	0	0
$597$	−9.80730	−0.401386
$598$	0	0
$599$	−26.4741	−1.08170	−0.540851	−	0.841118i	$-0.681898\pi$
$599$	−26.4741	−1.08170	−0.540851	+	0.841118i	$0.681898\pi$
$600$	0	0
$601$	12.3729	0.504702	0.252351	−	0.967636i	$-0.418796\pi$
$601$	12.3729	0.504702	0.252351	+	0.967636i	$0.418796\pi$
$602$	0	0
$603$	34.5186	1.40571
$604$	0	0
$605$	−20.1873	−0.820729
$606$	0	0
$607$	6.58880	0.267431	0.133716	−	0.991020i	$-0.457309\pi$
$607$	6.58880	0.267431	0.133716	+	0.991020i	$0.457309\pi$
$608$	0	0
$609$	−4.27611	−0.173277
$610$	0	0
$611$	−1.24197	−0.0502446
$612$	0	0
$613$	−32.6185	−1.31745	−0.658724	−	0.752384i	$-0.728904\pi$
$613$	−32.6185	−1.31745	−0.658724	+	0.752384i	$0.728904\pi$
$614$	0	0
$615$	6.16456	0.248579
$616$	0	0
$617$	44.7802	1.80278	0.901391	−	0.433006i	$-0.142547\pi$
$617$	44.7802	1.80278	0.901391	+	0.433006i	$0.142547\pi$
$618$	0	0
$619$	−36.4386	−1.46459	−0.732296	−	0.680986i	$-0.761551\pi$
$619$	−36.4386	−1.46459	−0.732296	+	0.680986i	$0.761551\pi$
$620$	0	0
$621$	20.4119	0.819102
$622$	0	0
$623$	10.6977	0.428596
$624$	0	0
$625$	−16.8776	−0.675104
$626$	0	0
$627$	−2.10872	−0.0842140
$628$	0	0
$629$	−1.39517	−0.0556290
$630$	0	0
$631$	−19.4629	−0.774804	−0.387402	−	0.921911i	$-0.626627\pi$
$631$	−19.4629	−0.774804	−0.387402	+	0.921911i	$0.626627\pi$
$632$	0	0
$633$	2.29579	0.0912496
$634$	0	0
$635$	0.0560342	0.00222365
$636$	0	0
$637$	12.1273	0.480502
$638$	0	0
$639$	1.51847	0.0600696
$640$	0	0
$641$	27.8560	1.10025	0.550123	−	0.835084i	$-0.314581\pi$
$641$	27.8560	1.10025	0.550123	+	0.835084i	$0.314581\pi$
$642$	0	0
$643$	5.63664	0.222288	0.111144	−	0.993804i	$-0.464549\pi$
$643$	5.63664	0.222288	0.111144	+	0.993804i	$0.464549\pi$
$644$	0	0
$645$	4.14228	0.163102
$646$	0	0
$647$	17.7323	0.697127	0.348563	−	0.937285i	$-0.386670\pi$
$647$	17.7323	0.697127	0.348563	+	0.937285i	$0.386670\pi$
$648$	0	0
$649$	0.251053	0.00985468
$650$	0	0
$651$	2.43037	0.0952536
$652$	0	0
$653$	14.1435	0.553477	0.276739	−	0.960945i	$-0.410746\pi$
$653$	14.1435	0.553477	0.276739	+	0.960945i	$0.410746\pi$
$654$	0	0
$655$	14.4594	0.564974
$656$	0	0
$657$	30.2111	1.17865
$658$	0	0
$659$	−20.0695	−0.781798	−0.390899	−	0.920434i	$-0.627836\pi$
$659$	−20.0695	−0.781798	−0.390899	+	0.920434i	$0.627836\pi$
$660$	0	0
$661$	−6.85193	−0.266509	−0.133255	−	0.991082i	$-0.542543\pi$
$661$	−6.85193	−0.266509	−0.133255	+	0.991082i	$0.542543\pi$
$662$	0	0
$663$	0.425629	0.0165301
$664$	0	0
$665$	−9.60952	−0.372641
$666$	0	0
$667$	71.0791	2.75219
$668$	0	0
$669$	−5.91319	−0.228617
$670$	0	0
$671$	−0.810610	−0.0312932
$672$	0	0
$673$	17.5575	0.676791	0.338396	−	0.941004i	$-0.390116\pi$
$673$	17.5575	0.676791	0.338396	+	0.941004i	$0.390116\pi$
$674$	0	0
$675$	−3.71590	−0.143025
$676$	0	0
$677$	22.0052	0.845728	0.422864	−	0.906193i	$-0.361025\pi$
$677$	22.0052	0.845728	0.422864	+	0.906193i	$0.361025\pi$
$678$	0	0
$679$	−4.26616	−0.163720
$680$	0	0
$681$	−5.23956	−0.200780
$682$	0	0
$683$	−32.7325	−1.25248	−0.626238	−	0.779632i	$-0.715406\pi$
$683$	−32.7325	−1.25248	−0.626238	+	0.779632i	$0.715406\pi$
$684$	0	0
$685$	−5.37450	−0.205349
$686$	0	0
$687$	14.4357	0.550756
$688$	0	0
$689$	−1.41060	−0.0537396
$690$	0	0
$691$	42.8793	1.63121	0.815604	−	0.578611i	$-0.196405\pi$
$691$	42.8793	1.63121	0.815604	+	0.578611i	$0.196405\pi$
$692$	0	0
$693$	1.68617	0.0640521
$694$	0	0
$695$	−10.7138	−0.406396
$696$	0	0
$697$	−2.81251	−0.106531
$698$	0	0
$699$	−0.241814	−0.00914626
$700$	0	0
$701$	−9.59945	−0.362566	−0.181283	−	0.983431i	$-0.558025\pi$
$701$	−9.59945	−0.362566	−0.181283	+	0.983431i	$0.558025\pi$
$702$	0	0
$703$	18.5697	0.700370
$704$	0	0
$705$	−0.619391	−0.0233276
$706$	0	0
$707$	0.163638	0.00615423
$708$	0	0
$709$	−20.6550	−0.775717	−0.387858	−	0.921719i	$-0.626785\pi$
$709$	−20.6550	−0.775717	−0.387858	+	0.921719i	$0.626785\pi$
$710$	0	0
$711$	30.1251	1.12978
$712$	0	0
$713$	−40.3984	−1.51293
$714$	0	0
$715$	2.67875	0.100180
$716$	0	0
$717$	−1.04270	−0.0389402
$718$	0	0
$719$	36.5196	1.36195	0.680975	−	0.732307i	$-0.261556\pi$
$719$	36.5196	1.36195	0.680975	+	0.732307i	$0.261556\pi$
$720$	0	0
$721$	3.38572	0.126091
$722$	0	0
$723$	−0.500591	−0.0186172
$724$	0	0
$725$	−12.9396	−0.480566
$726$	0	0
$727$	−53.4718	−1.98316	−0.991579	−	0.129503i	$-0.958662\pi$
$727$	−53.4718	−1.98316	−0.991579	+	0.129503i	$0.958662\pi$
$728$	0	0
$729$	−14.3943	−0.533122
$730$	0	0
$731$	−1.88987	−0.0698993
$732$	0	0
$733$	−47.4390	−1.75220	−0.876100	−	0.482129i	$-0.839864\pi$
$733$	−47.4390	−1.75220	−0.876100	+	0.482129i	$0.839864\pi$
$734$	0	0
$735$	6.04810	0.223088
$736$	0	0
$737$	9.03386	0.332766
$738$	0	0
$739$	−24.6810	−0.907904	−0.453952	−	0.891026i	$-0.649986\pi$
$739$	−24.6810	−0.907904	−0.453952	+	0.891026i	$0.649986\pi$
$740$	0	0
$741$	−5.66514	−0.208114
$742$	0	0
$743$	41.1925	1.51121	0.755603	−	0.655030i	$-0.227344\pi$
$743$	41.1925	1.51121	0.755603	+	0.655030i	$0.227344\pi$
$744$	0	0
$745$	28.5622	1.04644
$746$	0	0
$747$	18.2142	0.666424
$748$	0	0
$749$	−5.71840	−0.208946
$750$	0	0
$751$	16.0787	0.586719	0.293360	−	0.956002i	$-0.405227\pi$
$751$	16.0787	0.586719	0.293360	+	0.956002i	$0.405227\pi$
$752$	0	0
$753$	−4.84136	−0.176429
$754$	0	0
$755$	37.8901	1.37896
$756$	0	0
$757$	−38.0614	−1.38336	−0.691682	−	0.722202i	$-0.743130\pi$
$757$	−38.0614	−1.38336	−0.691682	+	0.722202i	$0.743130\pi$
$758$	0	0
$759$	2.55458	0.0927253
$760$	0	0
$761$	29.7253	1.07754	0.538770	−	0.842453i	$-0.318889\pi$
$761$	29.7253	1.07754	0.538770	+	0.842453i	$0.318889\pi$
$762$	0	0
$763$	3.43528	0.124366
$764$	0	0
$765$	−2.32895	−0.0842033
$766$	0	0
$767$	0.674461	0.0243534
$768$	0	0
$769$	−50.1093	−1.80699	−0.903494	−	0.428602i	$-0.859006\pi$
$769$	−50.1093	−1.80699	−0.903494	+	0.428602i	$0.859006\pi$
$770$	0	0
$771$	8.02705	0.289087
$772$	0	0
$773$	1.44897	0.0521159	0.0260579	−	0.999660i	$-0.491705\pi$
$773$	1.44897	0.0521159	0.0260579	+	0.999660i	$0.491705\pi$
$774$	0	0
$775$	7.35435	0.264176
$776$	0	0
$777$	1.35336	0.0485515
$778$	0	0
$779$	37.4346	1.34123
$780$	0	0
$781$	0.397397	0.0142200
$782$	0	0
$783$	28.8450	1.03084
$784$	0	0
$785$	27.6239	0.985938
$786$	0	0
$787$	−10.5396	−0.375695	−0.187848	−	0.982198i	$-0.560151\pi$
$787$	−10.5396	−0.375695	−0.187848	+	0.982198i	$0.560151\pi$
$788$	0	0
$789$	13.4860	0.480115
$790$	0	0
$791$	0.366772	0.0130409
$792$	0	0
$793$	−2.17773	−0.0773335
$794$	0	0
$795$	−0.703491	−0.0249503
$796$	0	0
$797$	31.2275	1.10614	0.553068	−	0.833136i	$-0.313457\pi$
$797$	31.2275	1.10614	0.553068	+	0.833136i	$0.313457\pi$
$798$	0	0
$799$	0.282590	0.00999731
$800$	0	0
$801$	−34.5087	−1.21931
$802$	0	0
$803$	7.90654	0.279016
$804$	0	0
$805$	11.6413	0.410303
$806$	0	0
$807$	−15.3793	−0.541376
$808$	0	0
$809$	−38.9547	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$809$	−38.9547	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$810$	0	0
$811$	−3.89413	−0.136741	−0.0683707	−	0.997660i	$-0.521780\pi$
$811$	−3.89413	−0.136741	−0.0683707	+	0.997660i	$0.521780\pi$
$812$	0	0
$813$	−16.1460	−0.566264
$814$	0	0
$815$	−9.61014	−0.336629
$816$	0	0
$817$	25.1542	0.880035
$818$	0	0
$819$	4.52994	0.158289
$820$	0	0
$821$	−14.5536	−0.507923	−0.253962	−	0.967214i	$-0.581734\pi$
$821$	−14.5536	−0.507923	−0.253962	+	0.967214i	$0.581734\pi$
$822$	0	0
$823$	−37.0978	−1.29315	−0.646575	−	0.762851i	$-0.723799\pi$
$823$	−37.0978	−1.29315	−0.646575	+	0.762851i	$0.723799\pi$
$824$	0	0
$825$	−0.465050	−0.0161909
$826$	0	0
$827$	46.8558	1.62934	0.814668	−	0.579928i	$-0.196919\pi$
$827$	46.8558	1.62934	0.814668	+	0.579928i	$0.196919\pi$
$828$	0	0
$829$	43.4487	1.50904	0.754518	−	0.656280i	$-0.227871\pi$
$829$	43.4487	1.50904	0.754518	+	0.656280i	$0.227871\pi$
$830$	0	0
$831$	−4.92583	−0.170875
$832$	0	0
$833$	−2.75938	−0.0956068
$834$	0	0
$835$	−30.0991	−1.04162
$836$	0	0
$837$	−16.3943	−0.566670
$838$	0	0
$839$	−45.0114	−1.55397	−0.776983	−	0.629521i	$-0.783251\pi$
$839$	−45.0114	−1.55397	−0.776983	+	0.629521i	$0.783251\pi$
$840$	0	0
$841$	71.4450	2.46362
$842$	0	0
$843$	−0.0312147	−0.00107509
$844$	0	0
$845$	−17.8395	−0.613698
$846$	0	0
$847$	−8.93422	−0.306983
$848$	0	0
$849$	−11.5704	−0.397096
$850$	0	0
$851$	−22.4961	−0.771155
$852$	0	0
$853$	5.62487	0.192592	0.0962959	−	0.995353i	$-0.469300\pi$
$853$	5.62487	0.192592	0.0962959	+	0.995353i	$0.469300\pi$
$854$	0	0
$855$	30.9984	1.06012
$856$	0	0
$857$	−1.42125	−0.0485491	−0.0242746	−	0.999705i	$-0.507728\pi$
$857$	−1.42125	−0.0485491	−0.0242746	+	0.999705i	$0.507728\pi$
$858$	0	0
$859$	2.58506	0.0882010	0.0441005	−	0.999027i	$-0.485958\pi$
$859$	2.58506	0.0882010	0.0441005	+	0.999027i	$0.485958\pi$
$860$	0	0
$861$	2.72823	0.0929779
$862$	0	0
$863$	46.3980	1.57941	0.789703	−	0.613490i	$-0.210235\pi$
$863$	46.3980	1.57941	0.789703	+	0.613490i	$0.210235\pi$
$864$	0	0
$865$	1.60494	0.0545695
$866$	0	0
$867$	8.41320	0.285727
$868$	0	0
$869$	7.88402	0.267447
$870$	0	0
$871$	24.2698	0.822349
$872$	0	0
$873$	13.7618	0.465765
$874$	0	0
$875$	−10.3265	−0.349098
$876$	0	0
$877$	−9.47791	−0.320046	−0.160023	−	0.987113i	$-0.551157\pi$
$877$	−9.47791	−0.320046	−0.160023	+	0.987113i	$0.551157\pi$
$878$	0	0
$879$	3.80307	0.128274
$880$	0	0
$881$	50.5681	1.70368	0.851841	−	0.523800i	$-0.175486\pi$
$881$	50.5681	1.70368	0.851841	+	0.523800i	$0.175486\pi$
$882$	0	0
$883$	18.3321	0.616925	0.308462	−	0.951237i	$-0.400186\pi$
$883$	18.3321	0.616925	0.308462	+	0.951237i	$0.400186\pi$
$884$	0	0
$885$	0.336366	0.0113068
$886$	0	0
$887$	−22.4382	−0.753401	−0.376701	−	0.926335i	$-0.622941\pi$
$887$	−22.4382	−0.753401	−0.376701	+	0.926335i	$0.622941\pi$
$888$	0	0
$889$	0.0247989	0.000831728 0
$890$	0	0
$891$	−4.89829	−0.164099
$892$	0	0
$893$	−3.76128	−0.125866
$894$	0	0
$895$	−28.8849	−0.965515
$896$	0	0
$897$	6.86296	0.229148
$898$	0	0
$899$	−57.0888	−1.90402
$900$	0	0
$901$	0.320960	0.0106927
$902$	0	0
$903$	1.83324	0.0610064
$904$	0	0
$905$	−28.8401	−0.958677
$906$	0	0
$907$	44.1063	1.46452	0.732262	−	0.681023i	$-0.238465\pi$
$907$	44.1063	1.46452	0.732262	+	0.681023i	$0.238465\pi$
$908$	0	0
$909$	−0.527862	−0.0175081
$910$	0	0
$911$	−36.5594	−1.21127	−0.605633	−	0.795744i	$-0.707080\pi$
$911$	−36.5594	−1.21127	−0.605633	+	0.795744i	$0.707080\pi$
$912$	0	0
$913$	4.76684	0.157759
$914$	0	0
$915$	−1.08607	−0.0359044
$916$	0	0
$917$	6.39924	0.211322
$918$	0	0
$919$	47.3887	1.56321	0.781605	−	0.623774i	$-0.214401\pi$
$919$	47.3887	1.56321	0.781605	+	0.623774i	$0.214401\pi$
$920$	0	0
$921$	−7.68737	−0.253307
$922$	0	0
$923$	1.06762	0.0351412
$924$	0	0
$925$	4.09531	0.134653
$926$	0	0
$927$	−10.9217	−0.358715
$928$	0	0
$929$	23.8472	0.782400	0.391200	−	0.920306i	$-0.372060\pi$
$929$	23.8472	0.782400	0.391200	+	0.920306i	$0.372060\pi$
$930$	0	0
$931$	36.7274	1.20369
$932$	0	0
$933$	12.5404	0.410555
$934$	0	0
$935$	−0.609508	−0.0199330
$936$	0	0
$937$	31.9624	1.04417	0.522083	−	0.852895i	$-0.325155\pi$
$937$	31.9624	1.04417	0.522083	+	0.852895i	$0.325155\pi$
$938$	0	0
$939$	4.48127	0.146241
$940$	0	0
$941$	23.8740	0.778271	0.389136	−	0.921180i	$-0.372774\pi$
$941$	23.8740	0.778271	0.389136	+	0.921180i	$0.372774\pi$
$942$	0	0
$943$	−45.3497	−1.47679
$944$	0	0
$945$	4.72423	0.153679
$946$	0	0
$947$	−44.4722	−1.44515	−0.722575	−	0.691292i	$-0.757042\pi$
$947$	−44.4722	−1.44515	−0.722575	+	0.691292i	$0.757042\pi$
$948$	0	0
$949$	21.2412	0.689518
$950$	0	0
$951$	−10.3785	−0.336547
$952$	0	0
$953$	−54.1645	−1.75456	−0.877280	−	0.479980i	$-0.840644\pi$
$953$	−54.1645	−1.75456	−0.877280	+	0.479980i	$0.840644\pi$
$954$	0	0
$955$	−16.4055	−0.530870
$956$	0	0
$957$	3.60999	0.116694
$958$	0	0
$959$	−2.37858	−0.0768083
$960$	0	0
$961$	1.44694	0.0466756
$962$	0	0
$963$	18.4464	0.594427
$964$	0	0
$965$	−45.3159	−1.45877
$966$	0	0
$967$	35.3826	1.13783	0.568914	−	0.822397i	$-0.307364\pi$
$967$	35.3826	1.13783	0.568914	+	0.822397i	$0.307364\pi$
$968$	0	0
$969$	1.28901	0.0414090
$970$	0	0
$971$	8.91001	0.285936	0.142968	−	0.989727i	$-0.454335\pi$
$971$	8.91001	0.285936	0.142968	+	0.989727i	$0.454335\pi$
$972$	0	0
$973$	−4.74155	−0.152007
$974$	0	0
$975$	−1.24937	−0.0400119
$976$	0	0
$977$	21.0250	0.672650	0.336325	−	0.941746i	$-0.390816\pi$
$977$	21.0250	0.672650	0.336325	+	0.941746i	$0.390816\pi$
$978$	0	0
$979$	−9.03126	−0.288640
$980$	0	0
$981$	−11.0815	−0.353806
$982$	0	0
$983$	33.5818	1.07109	0.535546	−	0.844506i	$-0.320106\pi$
$983$	33.5818	1.07109	0.535546	+	0.844506i	$0.320106\pi$
$984$	0	0
$985$	−51.5259	−1.64175
$986$	0	0
$987$	−0.274122	−0.00872540
$988$	0	0
$989$	−30.4728	−0.968978
$990$	0	0
$991$	32.0529	1.01819	0.509097	−	0.860709i	$-0.329979\pi$
$991$	32.0529	1.01819	0.509097	+	0.860709i	$0.329979\pi$
$992$	0	0
$993$	3.50389	0.111192
$994$	0	0
$995$	37.7302	1.19613
$996$	0	0
$997$	37.4198	1.18510	0.592548	−	0.805535i	$-0.298122\pi$
$997$	37.4198	1.18510	0.592548	+	0.805535i	$0.298122\pi$
$998$	0	0
$999$	−9.12924	−0.288836

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.n.1.6		12
4.3	odd	2		241.2.a.b.1.10	✓	12
12.11	even	2		2169.2.a.h.1.3		12
20.19	odd	2		6025.2.a.h.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.10	✓	12	4.3	odd	2
2169.2.a.h.1.3		12	12.11	even	2
3856.2.a.n.1.6		12	1.1	even	1	trivial
6025.2.a.h.1.3		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)

\(f(q)\)	\(=\)	\(q-0.500591 q^{3} +1.92585 q^{5} +0.852319 q^{7} -2.74941 q^{9} +O(q^{10})\)	\(q-0.500591 q^{3} +1.92585 q^{5} +0.852319 q^{7} -2.74941 q^{9} -0.719546 q^{11} -1.93309 q^{13} -0.964064 q^{15} +0.439843 q^{17} -5.85432 q^{19} -0.426663 q^{21} +7.09215 q^{23} -1.29109 q^{25} +2.87810 q^{27} +10.0222 q^{29} -5.69622 q^{31} +0.360198 q^{33} +1.64144 q^{35} -3.17197 q^{37} +0.967685 q^{39} -6.39435 q^{41} -4.29669 q^{43} -5.29496 q^{45} +0.642479 q^{47} -6.27355 q^{49} -0.220181 q^{51} +0.729714 q^{53} -1.38574 q^{55} +2.93062 q^{57} -0.348904 q^{59} +1.12656 q^{61} -2.34337 q^{63} -3.72284 q^{65} -12.5549 q^{67} -3.55026 q^{69} -0.552289 q^{71} -10.9882 q^{73} +0.646309 q^{75} -0.613283 q^{77} -10.9569 q^{79} +6.80748 q^{81} -6.62478 q^{83} +0.847072 q^{85} -5.01703 q^{87} +12.5513 q^{89} -1.64760 q^{91} +2.85148 q^{93} -11.2746 q^{95} -5.00536 q^{97} +1.97833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * q^99

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