LMFDB - Embedded newform 7872.2.a.cj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7872,2,Mod(1,7872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7872, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7872.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7872 = 2^{6} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7872.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:	$62.8582364712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15188.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 2 $ x^4 - x^3 - 7*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3936)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.09178$ of defining polynomial
Character	$\chi$	$=$	7872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.488785 q^{5} -0.956122 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.488785 q^{5} -0.956122 q^{7} +1.00000 q^{9} -2.60300 q^{11} +2.95612 q^{13} +0.488785 q^{15} +1.58057 q^{17} +3.22744 q^{19} -0.956122 q^{21} +4.40103 q^{23} -4.76109 q^{25} +1.00000 q^{27} -1.02547 q^{29} +6.26537 q^{31} -2.60300 q^{33} -0.467338 q^{35} +6.38553 q^{37} +2.95612 q^{39} -1.00000 q^{41} +0.820462 q^{43} +0.488785 q^{45} +3.15809 q^{47} -6.08583 q^{49} +1.58057 q^{51} -3.62847 q^{53} -1.27231 q^{55} +3.22744 q^{57} +9.20600 q^{59} -9.74269 q^{61} -0.956122 q^{63} +1.44491 q^{65} -12.3671 q^{67} +4.40103 q^{69} +4.40698 q^{71} -10.4489 q^{73} -4.76109 q^{75} +2.48878 q^{77} +2.27132 q^{79} +1.00000 q^{81} +15.1272 q^{83} +0.772557 q^{85} -1.02547 q^{87} -12.3223 q^{89} -2.82641 q^{91} +6.26537 q^{93} +1.57752 q^{95} +5.89080 q^{97} -2.60300 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 + 6 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} + 6 q^{7} + 4 q^{9} + 3 q^{11} + 2 q^{13} + 4 q^{15} - 3 q^{17} + 6 q^{21} + 4 q^{27} + 13 q^{29} + 9 q^{31} + 3 q^{33} + 10 q^{35} + 7 q^{37} + 2 q^{39} - 4 q^{41} - 5 q^{43} + 4 q^{45} + 7 q^{47} - 3 q^{51} + 16 q^{53} + 16 q^{55} + 10 q^{59} + 7 q^{61} + 6 q^{63} - 2 q^{65} - 4 q^{67} + 13 q^{71} - 3 q^{73} + 12 q^{77} + 6 q^{79} + 4 q^{81} + 14 q^{83} + 16 q^{85} + 13 q^{87} - 12 q^{89} - 16 q^{91} + 9 q^{93} + 10 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 6 * q^7 + 4 * q^9 + 3 * q^11 + 2 * q^13 + 4 * q^15 - 3 * q^17 + 6 * q^21 + 4 * q^27 + 13 * q^29 + 9 * q^31 + 3 * q^33 + 10 * q^35 + 7 * q^37 + 2 * q^39 - 4 * q^41 - 5 * q^43 + 4 * q^45 + 7 * q^47 - 3 * q^51 + 16 * q^53 + 16 * q^55 + 10 * q^59 + 7 * q^61 + 6 * q^63 - 2 * q^65 - 4 * q^67 + 13 * q^71 - 3 * q^73 + 12 * q^77 + 6 * q^79 + 4 * q^81 + 14 * q^83 + 16 * q^85 + 13 * q^87 - 12 * q^89 - 16 * q^91 + 9 * q^93 + 10 * q^95 - 10 * q^97 + 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.488785	0.218591	0.109296	−	0.994009i	$-0.465141\pi$
$5$	0.488785	0.218591	0.109296	+	0.994009i	$0.465141\pi$
$6$	0	0
$7$	−0.956122	−0.361380	−0.180690	−	0.983540i	$-0.557833\pi$
$7$	−0.956122	−0.361380	−0.180690	+	0.983540i	$0.557833\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.60300	−0.784833	−0.392417	−	0.919788i	$-0.628361\pi$
$11$	−2.60300	−0.784833	−0.392417	+	0.919788i	$0.628361\pi$
$12$	0	0
$13$	2.95612	0.819881	0.409940	−	0.912112i	$-0.365549\pi$
$13$	2.95612	0.819881	0.409940	+	0.912112i	$0.365549\pi$
$14$	0	0
$15$	0.488785	0.126204
$16$	0	0
$17$	1.58057	0.383344	0.191672	−	0.981459i	$-0.438609\pi$
$17$	1.58057	0.383344	0.191672	+	0.981459i	$0.438609\pi$
$18$	0	0
$19$	3.22744	0.740426	0.370213	−	0.928947i	$-0.379285\pi$
$19$	3.22744	0.740426	0.370213	+	0.928947i	$0.379285\pi$
$20$	0	0
$21$	−0.956122	−0.208643
$22$	0	0
$23$	4.40103	0.917678	0.458839	−	0.888519i	$-0.348265\pi$
$23$	4.40103	0.917678	0.458839	+	0.888519i	$0.348265\pi$
$24$	0	0
$25$	−4.76109	−0.952218
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.02547	−0.190426	−0.0952129	−	0.995457i	$-0.530353\pi$
$29$	−1.02547	−0.190426	−0.0952129	+	0.995457i	$0.530353\pi$
$30$	0	0
$31$	6.26537	1.12529	0.562647	−	0.826697i	$-0.309783\pi$
$31$	6.26537	1.12529	0.562647	+	0.826697i	$0.309783\pi$
$32$	0	0
$33$	−2.60300	−0.453124
$34$	0	0
$35$	−0.467338	−0.0789945
$36$	0	0
$37$	6.38553	1.04978	0.524888	−	0.851171i	$-0.324107\pi$
$37$	6.38553	1.04978	0.524888	+	0.851171i	$0.324107\pi$
$38$	0	0
$39$	2.95612	0.473358
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	0.820462	0.125119	0.0625596	−	0.998041i	$-0.480074\pi$
$43$	0.820462	0.125119	0.0625596	+	0.998041i	$0.480074\pi$
$44$	0	0
$45$	0.488785	0.0728637
$46$	0	0
$47$	3.15809	0.460655	0.230327	−	0.973113i	$-0.426020\pi$
$47$	3.15809	0.460655	0.230327	+	0.973113i	$0.426020\pi$
$48$	0	0
$49$	−6.08583	−0.869404
$50$	0	0
$51$	1.58057	0.221324
$52$	0	0
$53$	−3.62847	−0.498409	−0.249204	−	0.968451i	$-0.580169\pi$
$53$	−3.62847	−0.498409	−0.249204	+	0.968451i	$0.580169\pi$
$54$	0	0
$55$	−1.27231	−0.171558
$56$	0	0
$57$	3.22744	0.427485
$58$	0	0
$59$	9.20600	1.19852	0.599259	−	0.800555i	$-0.295462\pi$
$59$	9.20600	1.19852	0.599259	+	0.800555i	$0.295462\pi$
$60$	0	0
$61$	−9.74269	−1.24742	−0.623712	−	0.781655i	$-0.714376\pi$
$61$	−9.74269	−1.24742	−0.623712	+	0.781655i	$0.714376\pi$
$62$	0	0
$63$	−0.956122	−0.120460
$64$	0	0
$65$	1.44491	0.179219
$66$	0	0
$67$	−12.3671	−1.51089	−0.755443	−	0.655215i	$-0.772578\pi$
$67$	−12.3671	−1.51089	−0.755443	+	0.655215i	$0.772578\pi$
$68$	0	0
$69$	4.40103	0.529822
$70$	0	0
$71$	4.40698	0.523012	0.261506	−	0.965202i	$-0.415781\pi$
$71$	4.40698	0.523012	0.261506	+	0.965202i	$0.415781\pi$
$72$	0	0
$73$	−10.4489	−1.22296	−0.611478	−	0.791262i	$-0.709425\pi$
$73$	−10.4489	−1.22296	−0.611478	+	0.791262i	$0.709425\pi$
$74$	0	0
$75$	−4.76109	−0.549763
$76$	0	0
$77$	2.48878	0.283623
$78$	0	0
$79$	2.27132	0.255544	0.127772	−	0.991804i	$-0.459217\pi$
$79$	2.27132	0.255544	0.127772	+	0.991804i	$0.459217\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.1272	1.66043	0.830215	−	0.557443i	$-0.188218\pi$
$83$	15.1272	1.66043	0.830215	+	0.557443i	$0.188218\pi$
$84$	0	0
$85$	0.772557	0.0837955
$86$	0	0
$87$	−1.02547	−0.109942
$88$	0	0
$89$	−12.3223	−1.30616	−0.653079	−	0.757290i	$-0.726523\pi$
$89$	−12.3223	−1.30616	−0.653079	+	0.757290i	$0.726523\pi$
$90$	0	0
$91$	−2.82641	−0.296289
$92$	0	0
$93$	6.26537	0.649688
$94$	0	0
$95$	1.57752	0.161851
$96$	0	0
$97$	5.89080	0.598120	0.299060	−	0.954234i	$-0.403327\pi$
$97$	5.89080	0.598120	0.299060	+	0.954234i	$0.403327\pi$
$98$	0	0
$99$	−2.60300	−0.261611
$100$	0	0
$101$	16.0803	1.60005	0.800026	−	0.599966i	$-0.204819\pi$
$101$	16.0803	1.60005	0.800026	+	0.599966i	$0.204819\pi$
$102$	0	0
$103$	3.17954	0.313289	0.156645	−	0.987655i	$-0.449932\pi$
$103$	3.17954	0.313289	0.156645	+	0.987655i	$0.449932\pi$
$104$	0	0
$105$	−0.467338	−0.0456075
$106$	0	0
$107$	−1.77963	−0.172043	−0.0860215	−	0.996293i	$-0.527415\pi$
$107$	−1.77963	−0.172043	−0.0860215	+	0.996293i	$0.527415\pi$
$108$	0	0
$109$	12.4334	1.19091	0.595454	−	0.803390i	$-0.296972\pi$
$109$	12.4334	1.19091	0.595454	+	0.803390i	$0.296972\pi$
$110$	0	0
$111$	6.38553	0.606088
$112$	0	0
$113$	4.44589	0.418234	0.209117	−	0.977891i	$-0.432941\pi$
$113$	4.44589	0.418234	0.209117	+	0.977891i	$0.432941\pi$
$114$	0	0
$115$	2.15115	0.200596
$116$	0	0
$117$	2.95612	0.273294
$118$	0	0
$119$	−1.51122	−0.138533
$120$	0	0
$121$	−4.22440	−0.384036
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	−4.77107	−0.426737
$126$	0	0
$127$	16.9981	1.50834	0.754168	−	0.656682i	$-0.228041\pi$
$127$	16.9981	1.50834	0.754168	+	0.656682i	$0.228041\pi$
$128$	0	0
$129$	0.820462	0.0722376
$130$	0	0
$131$	0.0338981	0.00296169	0.00148084	−	0.999999i	$-0.499529\pi$
$131$	0.0338981	0.00296169	0.00148084	+	0.999999i	$0.499529\pi$
$132$	0	0
$133$	−3.08583	−0.267575
$134$	0	0
$135$	0.488785	0.0420679
$136$	0	0
$137$	4.09277	0.349669	0.174834	−	0.984598i	$-0.444061\pi$
$137$	4.09277	0.349669	0.174834	+	0.984598i	$0.444061\pi$
$138$	0	0
$139$	11.7182	0.993924	0.496962	−	0.867772i	$-0.334449\pi$
$139$	11.7182	0.993924	0.496962	+	0.867772i	$0.334449\pi$
$140$	0	0
$141$	3.15809	0.265959
$142$	0	0
$143$	−7.69478	−0.643470
$144$	0	0
$145$	−0.501236	−0.0416254
$146$	0	0
$147$	−6.08583	−0.501951
$148$	0	0
$149$	13.3386	1.09274	0.546371	−	0.837543i	$-0.316009\pi$
$149$	13.3386	1.09274	0.546371	+	0.837543i	$0.316009\pi$
$150$	0	0
$151$	−10.0529	−0.818095	−0.409047	−	0.912513i	$-0.634139\pi$
$151$	−10.0529	−0.818095	−0.409047	+	0.912513i	$0.634139\pi$
$152$	0	0
$153$	1.58057	0.127781
$154$	0	0
$155$	3.06242	0.245979
$156$	0	0
$157$	−2.17748	−0.173782	−0.0868909	−	0.996218i	$-0.527693\pi$
$157$	−2.17748	−0.173782	−0.0868909	+	0.996218i	$0.527693\pi$
$158$	0	0
$159$	−3.62847	−0.287757
$160$	0	0
$161$	−4.20792	−0.331631
$162$	0	0
$163$	−1.52671	−0.119581	−0.0597906	−	0.998211i	$-0.519043\pi$
$163$	−1.52671	−0.119581	−0.0597906	+	0.998211i	$0.519043\pi$
$164$	0	0
$165$	−1.27231	−0.0990488
$166$	0	0
$167$	−21.0180	−1.62642	−0.813212	−	0.581967i	$-0.802283\pi$
$167$	−21.0180	−1.62642	−0.813212	+	0.581967i	$0.802283\pi$
$168$	0	0
$169$	−4.26134	−0.327795
$170$	0	0
$171$	3.22744	0.246809
$172$	0	0
$173$	6.80497	0.517372	0.258686	−	0.965961i	$-0.416710\pi$
$173$	6.80497	0.517372	0.258686	+	0.965961i	$0.416710\pi$
$174$	0	0
$175$	4.55218	0.344113
$176$	0	0
$177$	9.20600	0.691965
$178$	0	0
$179$	8.48283	0.634037	0.317018	−	0.948419i	$-0.397318\pi$
$179$	8.48283	0.634037	0.317018	+	0.948419i	$0.397318\pi$
$180$	0	0
$181$	17.5083	1.30138	0.650691	−	0.759343i	$-0.274479\pi$
$181$	17.5083	1.30138	0.650691	+	0.759343i	$0.274479\pi$
$182$	0	0
$183$	−9.74269	−0.720200
$184$	0	0
$185$	3.12115	0.229472
$186$	0	0
$187$	−4.11421	−0.300861
$188$	0	0
$189$	−0.956122	−0.0695477
$190$	0	0
$191$	−4.61849	−0.334182	−0.167091	−	0.985941i	$-0.553437\pi$
$191$	−4.61849	−0.334182	−0.167091	+	0.985941i	$0.553437\pi$
$192$	0	0
$193$	14.5292	1.04584	0.522919	−	0.852382i	$-0.324843\pi$
$193$	14.5292	1.04584	0.522919	+	0.852382i	$0.324843\pi$
$194$	0	0
$195$	1.44491	0.103472
$196$	0	0
$197$	15.4744	1.10251	0.551253	−	0.834338i	$-0.314150\pi$
$197$	15.4744	1.10251	0.551253	+	0.834338i	$0.314150\pi$
$198$	0	0
$199$	16.3457	1.15871	0.579357	−	0.815074i	$-0.303304\pi$
$199$	16.3457	1.15871	0.579357	+	0.815074i	$0.303304\pi$
$200$	0	0
$201$	−12.3671	−0.872310
$202$	0	0
$203$	0.980478	0.0688161
$204$	0	0
$205$	−0.488785	−0.0341382
$206$	0	0
$207$	4.40103	0.305893
$208$	0	0
$209$	−8.40103	−0.581111
$210$	0	0
$211$	20.6265	1.41999	0.709995	−	0.704207i	$-0.248697\pi$
$211$	20.6265	1.41999	0.709995	+	0.704207i	$0.248697\pi$
$212$	0	0
$213$	4.40698	0.301961
$214$	0	0
$215$	0.401029	0.0273500
$216$	0	0
$217$	−5.99046	−0.406659
$218$	0	0
$219$	−10.4489	−0.706074
$220$	0	0
$221$	4.67235	0.314296
$222$	0	0
$223$	10.5875	0.708992	0.354496	−	0.935058i	$-0.384653\pi$
$223$	10.5875	0.708992	0.354496	+	0.935058i	$0.384653\pi$
$224$	0	0
$225$	−4.76109	−0.317406
$226$	0	0
$227$	16.5581	1.09900	0.549501	−	0.835493i	$-0.314818\pi$
$227$	16.5581	1.09900	0.549501	+	0.835493i	$0.314818\pi$
$228$	0	0
$229$	2.68480	0.177417	0.0887083	−	0.996058i	$-0.471726\pi$
$229$	2.68480	0.177417	0.0887083	+	0.996058i	$0.471726\pi$
$230$	0	0
$231$	2.48878	0.163750
$232$	0	0
$233$	−15.7367	−1.03095	−0.515474	−	0.856905i	$-0.672384\pi$
$233$	−15.7367	−1.03095	−0.515474	+	0.856905i	$0.672384\pi$
$234$	0	0
$235$	1.54363	0.100695
$236$	0	0
$237$	2.27132	0.147538
$238$	0	0
$239$	3.61602	0.233901	0.116950	−	0.993138i	$-0.462688\pi$
$239$	3.61602	0.233901	0.116950	+	0.993138i	$0.462688\pi$
$240$	0	0
$241$	−4.11225	−0.264893	−0.132447	−	0.991190i	$-0.542283\pi$
$241$	−4.11225	−0.264893	−0.132447	+	0.991190i	$0.542283\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.97466	−0.190044
$246$	0	0
$247$	9.54072	0.607061
$248$	0	0
$249$	15.1272	0.958650
$250$	0	0
$251$	13.2957	0.839218	0.419609	−	0.907705i	$-0.362167\pi$
$251$	13.2957	0.839218	0.419609	+	0.907705i	$0.362167\pi$
$252$	0	0
$253$	−11.4559	−0.720224
$254$	0	0
$255$	0.772557	0.0483794
$256$	0	0
$257$	8.53520	0.532411	0.266206	−	0.963916i	$-0.414230\pi$
$257$	8.53520	0.532411	0.266206	+	0.963916i	$0.414230\pi$
$258$	0	0
$259$	−6.10535	−0.379368
$260$	0	0
$261$	−1.02547	−0.0634752
$262$	0	0
$263$	−1.62346	−0.100107	−0.0500534	−	0.998747i	$-0.515939\pi$
$263$	−1.62346	−0.100107	−0.0500534	+	0.998747i	$0.515939\pi$
$264$	0	0
$265$	−1.77354	−0.108948
$266$	0	0
$267$	−12.3223	−0.754111
$268$	0	0
$269$	19.1692	1.16877	0.584383	−	0.811478i	$-0.301337\pi$
$269$	19.1692	1.16877	0.584383	+	0.811478i	$0.301337\pi$
$270$	0	0
$271$	−13.2629	−0.805664	−0.402832	−	0.915274i	$-0.631974\pi$
$271$	−13.2629	−0.805664	−0.402832	+	0.915274i	$0.631974\pi$
$272$	0	0
$273$	−2.82641	−0.171062
$274$	0	0
$275$	12.3931	0.747333
$276$	0	0
$277$	9.04692	0.543577	0.271788	−	0.962357i	$-0.412385\pi$
$277$	9.04692	0.543577	0.271788	+	0.962357i	$0.412385\pi$
$278$	0	0
$279$	6.26537	0.375098
$280$	0	0
$281$	4.00304	0.238802	0.119401	−	0.992846i	$-0.461903\pi$
$281$	4.00304	0.238802	0.119401	+	0.992846i	$0.461903\pi$
$282$	0	0
$283$	28.4769	1.69278	0.846389	−	0.532565i	$-0.178772\pi$
$283$	28.4769	1.69278	0.846389	+	0.532565i	$0.178772\pi$
$284$	0	0
$285$	1.57752	0.0934445
$286$	0	0
$287$	0.956122	0.0564381
$288$	0	0
$289$	−14.5018	−0.853047
$290$	0	0
$291$	5.89080	0.345325
$292$	0	0
$293$	−17.4679	−1.02049	−0.510243	−	0.860030i	$-0.670445\pi$
$293$	−17.4679	−1.02049	−0.510243	+	0.860030i	$0.670445\pi$
$294$	0	0
$295$	4.49975	0.261985
$296$	0	0
$297$	−2.60300	−0.151041
$298$	0	0
$299$	13.0100	0.752387
$300$	0	0
$301$	−0.784462	−0.0452156
$302$	0	0
$303$	16.0803	0.923790
$304$	0	0
$305$	−4.76207	−0.272676
$306$	0	0
$307$	−21.1832	−1.20899	−0.604494	−	0.796609i	$-0.706625\pi$
$307$	−21.1832	−1.20899	−0.604494	+	0.796609i	$0.706625\pi$
$308$	0	0
$309$	3.17954	0.180878
$310$	0	0
$311$	22.8550	1.29599	0.647993	−	0.761646i	$-0.275608\pi$
$311$	22.8550	1.29599	0.647993	+	0.761646i	$0.275608\pi$
$312$	0	0
$313$	−6.90517	−0.390304	−0.195152	−	0.980773i	$-0.562520\pi$
$313$	−6.90517	−0.390304	−0.195152	+	0.980773i	$0.562520\pi$
$314$	0	0
$315$	−0.467338	−0.0263315
$316$	0	0
$317$	28.1966	1.58368	0.791840	−	0.610728i	$-0.209123\pi$
$317$	28.1966	1.58368	0.791840	+	0.610728i	$0.209123\pi$
$318$	0	0
$319$	2.66931	0.149452
$320$	0	0
$321$	−1.77963	−0.0993291
$322$	0	0
$323$	5.10119	0.283838
$324$	0	0
$325$	−14.0744	−0.780705
$326$	0	0
$327$	12.4334	0.687571
$328$	0	0
$329$	−3.01952	−0.166472
$330$	0	0
$331$	−5.67624	−0.311995	−0.155997	−	0.987757i	$-0.549859\pi$
$331$	−5.67624	−0.311995	−0.155997	+	0.987757i	$0.549859\pi$
$332$	0	0
$333$	6.38553	0.349925
$334$	0	0
$335$	−6.04486	−0.330266
$336$	0	0
$337$	−36.0116	−1.96168	−0.980838	−	0.194826i	$-0.937586\pi$
$337$	−36.0116	−1.96168	−0.980838	+	0.194826i	$0.937586\pi$
$338$	0	0
$339$	4.44589	0.241468
$340$	0	0
$341$	−16.3087	−0.883168
$342$	0	0
$343$	12.5117	0.675566
$344$	0	0
$345$	2.15115	0.115814
$346$	0	0
$347$	11.2339	0.603070	0.301535	−	0.953455i	$-0.402501\pi$
$347$	11.2339	0.603070	0.301535	+	0.953455i	$0.402501\pi$
$348$	0	0
$349$	−7.86143	−0.420813	−0.210406	−	0.977614i	$-0.567479\pi$
$349$	−7.86143	−0.420813	−0.210406	+	0.977614i	$0.567479\pi$
$350$	0	0
$351$	2.95612	0.157786
$352$	0	0
$353$	3.41250	0.181629	0.0908144	−	0.995868i	$-0.471053\pi$
$353$	3.41250	0.181629	0.0908144	+	0.995868i	$0.471053\pi$
$354$	0	0
$355$	2.15406	0.114326
$356$	0	0
$357$	−1.51122	−0.0799820
$358$	0	0
$359$	19.4825	1.02825	0.514123	−	0.857717i	$-0.328118\pi$
$359$	19.4825	1.02825	0.514123	+	0.857717i	$0.328118\pi$
$360$	0	0
$361$	−8.58361	−0.451769
$362$	0	0
$363$	−4.22440	−0.221724
$364$	0	0
$365$	−5.10728	−0.267327
$366$	0	0
$367$	19.0509	0.994447	0.497223	−	0.867623i	$-0.334353\pi$
$367$	19.0509	0.994447	0.497223	+	0.867623i	$0.334353\pi$
$368$	0	0
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	3.46926	0.180115
$372$	0	0
$373$	−6.58155	−0.340780	−0.170390	−	0.985377i	$-0.554503\pi$
$373$	−6.58155	−0.340780	−0.170390	+	0.985377i	$0.554503\pi$
$374$	0	0
$375$	−4.77107	−0.246377
$376$	0	0
$377$	−3.03143	−0.156126
$378$	0	0
$379$	−8.30373	−0.426534	−0.213267	−	0.976994i	$-0.568410\pi$
$379$	−8.30373	−0.426534	−0.213267	+	0.976994i	$0.568410\pi$
$380$	0	0
$381$	16.9981	0.870838
$382$	0	0
$383$	−20.4350	−1.04418	−0.522090	−	0.852891i	$-0.674847\pi$
$383$	−20.4350	−1.04418	−0.522090	+	0.852891i	$0.674847\pi$
$384$	0	0
$385$	1.21648	0.0619975
$386$	0	0
$387$	0.820462	0.0417064
$388$	0	0
$389$	−22.3152	−1.13143	−0.565714	−	0.824602i	$-0.691399\pi$
$389$	−22.3152	−1.13143	−0.565714	+	0.824602i	$0.691399\pi$
$390$	0	0
$391$	6.95612	0.351786
$392$	0	0
$393$	0.0338981	0.00170993
$394$	0	0
$395$	1.11019	0.0558595
$396$	0	0
$397$	−37.2416	−1.86910	−0.934551	−	0.355830i	$-0.884198\pi$
$397$	−37.2416	−1.86910	−0.934551	+	0.355830i	$0.884198\pi$
$398$	0	0
$399$	−3.08583	−0.154485
$400$	0	0
$401$	30.8979	1.54297	0.771483	−	0.636250i	$-0.219515\pi$
$401$	30.8979	1.54297	0.771483	+	0.636250i	$0.219515\pi$
$402$	0	0
$403$	18.5212	0.922606
$404$	0	0
$405$	0.488785	0.0242879
$406$	0	0
$407$	−16.6215	−0.823899
$408$	0	0
$409$	−8.83291	−0.436759	−0.218380	−	0.975864i	$-0.570077\pi$
$409$	−8.83291	−0.436759	−0.218380	+	0.975864i	$0.570077\pi$
$410$	0	0
$411$	4.09277	0.201881
$412$	0	0
$413$	−8.80206	−0.433121
$414$	0	0
$415$	7.39396	0.362955
$416$	0	0
$417$	11.7182	0.573843
$418$	0	0
$419$	30.1068	1.47081	0.735406	−	0.677627i	$-0.236991\pi$
$419$	30.1068	1.47081	0.735406	+	0.677627i	$0.236991\pi$
$420$	0	0
$421$	15.9142	0.775611	0.387806	−	0.921741i	$-0.373233\pi$
$421$	15.9142	0.775611	0.387806	+	0.921741i	$0.373233\pi$
$422$	0	0
$423$	3.15809	0.153552
$424$	0	0
$425$	−7.52522	−0.365027
$426$	0	0
$427$	9.31520	0.450794
$428$	0	0
$429$	−7.69478	−0.371508
$430$	0	0
$431$	−15.7538	−0.758833	−0.379417	−	0.925226i	$-0.623875\pi$
$431$	−15.7538	−0.758833	−0.379417	+	0.925226i	$0.623875\pi$
$432$	0	0
$433$	2.27782	0.109465	0.0547325	−	0.998501i	$-0.482569\pi$
$433$	2.27782	0.109465	0.0547325	+	0.998501i	$0.482569\pi$
$434$	0	0
$435$	−0.501236	−0.0240324
$436$	0	0
$437$	14.2041	0.679473
$438$	0	0
$439$	−31.6379	−1.51000	−0.754998	−	0.655727i	$-0.772362\pi$
$439$	−31.6379	−1.51000	−0.754998	+	0.655727i	$0.772362\pi$
$440$	0	0
$441$	−6.08583	−0.289801
$442$	0	0
$443$	−17.7428	−0.842987	−0.421493	−	0.906831i	$-0.638494\pi$
$443$	−17.7428	−0.842987	−0.421493	+	0.906831i	$0.638494\pi$
$444$	0	0
$445$	−6.02293	−0.285514
$446$	0	0
$447$	13.3386	0.630895
$448$	0	0
$449$	0.932707	0.0440172	0.0220086	−	0.999758i	$-0.492994\pi$
$449$	0.932707	0.0440172	0.0220086	+	0.999758i	$0.492994\pi$
$450$	0	0
$451$	2.60300	0.122570
$452$	0	0
$453$	−10.0529	−0.472327
$454$	0	0
$455$	−1.38151	−0.0647661
$456$	0	0
$457$	27.2650	1.27540	0.637701	−	0.770284i	$-0.279885\pi$
$457$	27.2650	1.27540	0.637701	+	0.770284i	$0.279885\pi$
$458$	0	0
$459$	1.58057	0.0737746
$460$	0	0
$461$	−15.3137	−0.713231	−0.356615	−	0.934251i	$-0.616069\pi$
$461$	−15.3137	−0.713231	−0.356615	+	0.934251i	$0.616069\pi$
$462$	0	0
$463$	−6.83738	−0.317760	−0.158880	−	0.987298i	$-0.550788\pi$
$463$	−6.83738	−0.317760	−0.158880	+	0.987298i	$0.550788\pi$
$464$	0	0
$465$	3.06242	0.142016
$466$	0	0
$467$	−25.3298	−1.17212	−0.586062	−	0.810266i	$-0.699323\pi$
$467$	−25.3298	−1.17212	−0.586062	+	0.810266i	$0.699323\pi$
$468$	0	0
$469$	11.8245	0.546004
$470$	0	0
$471$	−2.17748	−0.100333
$472$	0	0
$473$	−2.13566	−0.0981978
$474$	0	0
$475$	−15.3661	−0.705047
$476$	0	0
$477$	−3.62847	−0.166136
$478$	0	0
$479$	3.66224	0.167332	0.0836659	−	0.996494i	$-0.473337\pi$
$479$	3.66224	0.167332	0.0836659	+	0.996494i	$0.473337\pi$
$480$	0	0
$481$	18.8764	0.860691
$482$	0	0
$483$	−4.20792	−0.191467
$484$	0	0
$485$	2.87933	0.130744
$486$	0	0
$487$	3.98599	0.180623	0.0903113	−	0.995914i	$-0.471214\pi$
$487$	3.98599	0.180623	0.0903113	+	0.995914i	$0.471214\pi$
$488$	0	0
$489$	−1.52671	−0.0690402
$490$	0	0
$491$	−12.9805	−0.585801	−0.292900	−	0.956143i	$-0.594620\pi$
$491$	−12.9805	−0.585801	−0.292900	+	0.956143i	$0.594620\pi$
$492$	0	0
$493$	−1.62083	−0.0729985
$494$	0	0
$495$	−1.27231	−0.0571859
$496$	0	0
$497$	−4.21361	−0.189006
$498$	0	0
$499$	19.0346	0.852106	0.426053	−	0.904698i	$-0.359904\pi$
$499$	19.0346	0.852106	0.426053	+	0.904698i	$0.359904\pi$
$500$	0	0
$501$	−21.0180	−0.939017
$502$	0	0
$503$	−20.4804	−0.913176	−0.456588	−	0.889678i	$-0.650929\pi$
$503$	−20.4804	−0.913176	−0.456588	+	0.889678i	$0.650929\pi$
$504$	0	0
$505$	7.85981	0.349757
$506$	0	0
$507$	−4.26134	−0.189253
$508$	0	0
$509$	22.0989	0.979514	0.489757	−	0.871859i	$-0.337085\pi$
$509$	22.0989	0.979514	0.489757	+	0.871859i	$0.337085\pi$
$510$	0	0
$511$	9.99046	0.441952
$512$	0	0
$513$	3.22744	0.142495
$514$	0	0
$515$	1.55411	0.0684822
$516$	0	0
$517$	−8.22051	−0.361537
$518$	0	0
$519$	6.80497	0.298705
$520$	0	0
$521$	−30.2334	−1.32455	−0.662276	−	0.749260i	$-0.730409\pi$
$521$	−30.2334	−1.32455	−0.662276	+	0.749260i	$0.730409\pi$
$522$	0	0
$523$	−35.2404	−1.54095	−0.770477	−	0.637468i	$-0.779982\pi$
$523$	−35.2404	−1.54095	−0.770477	+	0.637468i	$0.779982\pi$
$524$	0	0
$525$	4.55218	0.198674
$526$	0	0
$527$	9.90284	0.431374
$528$	0	0
$529$	−3.63094	−0.157867
$530$	0	0
$531$	9.20600	0.399506
$532$	0	0
$533$	−2.95612	−0.128044
$534$	0	0
$535$	−0.869854	−0.0376071
$536$	0	0
$537$	8.48283	0.366061
$538$	0	0
$539$	15.8414	0.682338
$540$	0	0
$541$	−36.3813	−1.56415	−0.782077	−	0.623182i	$-0.785839\pi$
$541$	−36.3813	−1.56415	−0.782077	+	0.623182i	$0.785839\pi$
$542$	0	0
$543$	17.5083	0.751353
$544$	0	0
$545$	6.07727	0.260322
$546$	0	0
$547$	43.1343	1.84429	0.922146	−	0.386842i	$-0.126434\pi$
$547$	43.1343	1.84429	0.922146	+	0.386842i	$0.126434\pi$
$548$	0	0
$549$	−9.74269	−0.415808
$550$	0	0
$551$	−3.30966	−0.140996
$552$	0	0
$553$	−2.17166	−0.0923484
$554$	0	0
$555$	3.12115	0.132485
$556$	0	0
$557$	−29.4236	−1.24672	−0.623359	−	0.781936i	$-0.714232\pi$
$557$	−29.4236	−1.24672	−0.623359	+	0.781936i	$0.714232\pi$
$558$	0	0
$559$	2.42539	0.102583
$560$	0	0
$561$	−4.11421	−0.173702
$562$	0	0
$563$	33.4394	1.40930	0.704652	−	0.709553i	$-0.251103\pi$
$563$	33.4394	1.40930	0.704652	+	0.709553i	$0.251103\pi$
$564$	0	0
$565$	2.17308	0.0914223
$566$	0	0
$567$	−0.956122	−0.0401534
$568$	0	0
$569$	19.5212	0.818373	0.409186	−	0.912451i	$-0.365813\pi$
$569$	19.5212	0.818373	0.409186	+	0.912451i	$0.365813\pi$
$570$	0	0
$571$	−2.82499	−0.118222	−0.0591111	−	0.998251i	$-0.518827\pi$
$571$	−2.82499	−0.118222	−0.0591111	+	0.998251i	$0.518827\pi$
$572$	0	0
$573$	−4.61849	−0.192940
$574$	0	0
$575$	−20.9537	−0.873829
$576$	0	0
$577$	−10.9098	−0.454180	−0.227090	−	0.973874i	$-0.572921\pi$
$577$	−10.9098	−0.454180	−0.227090	+	0.973874i	$0.572921\pi$
$578$	0	0
$579$	14.5292	0.603815
$580$	0	0
$581$	−14.4635	−0.600047
$582$	0	0
$583$	9.44491	0.391168
$584$	0	0
$585$	1.44491	0.0597395
$586$	0	0
$587$	14.0030	0.577967	0.288984	−	0.957334i	$-0.406683\pi$
$587$	14.0030	0.577967	0.288984	+	0.957334i	$0.406683\pi$
$588$	0	0
$589$	20.2211	0.833197
$590$	0	0
$591$	15.4744	0.636532
$592$	0	0
$593$	−8.29487	−0.340629	−0.170315	−	0.985390i	$-0.554478\pi$
$593$	−8.29487	−0.340629	−0.170315	+	0.985390i	$0.554478\pi$
$594$	0	0
$595$	−0.738659	−0.0302821
$596$	0	0
$597$	16.3457	0.668984
$598$	0	0
$599$	−8.00291	−0.326990	−0.163495	−	0.986544i	$-0.552277\pi$
$599$	−8.00291	−0.326990	−0.163495	+	0.986544i	$0.552277\pi$
$600$	0	0
$601$	−3.72062	−0.151767	−0.0758837	−	0.997117i	$-0.524178\pi$
$601$	−3.72062	−0.151767	−0.0758837	+	0.997117i	$0.524178\pi$
$602$	0	0
$603$	−12.3671	−0.503629
$604$	0	0
$605$	−2.06482	−0.0839469
$606$	0	0
$607$	26.8483	1.08974	0.544870	−	0.838520i	$-0.316579\pi$
$607$	26.8483	1.08974	0.544870	+	0.838520i	$0.316579\pi$
$608$	0	0
$609$	0.980478	0.0397310
$610$	0	0
$611$	9.33570	0.377682
$612$	0	0
$613$	−4.04344	−0.163313	−0.0816565	−	0.996661i	$-0.526021\pi$
$613$	−4.04344	−0.163313	−0.0816565	+	0.996661i	$0.526021\pi$
$614$	0	0
$615$	−0.488785	−0.0197097
$616$	0	0
$617$	−1.57407	−0.0633696	−0.0316848	−	0.999498i	$-0.510087\pi$
$617$	−1.57407	−0.0633696	−0.0316848	+	0.999498i	$0.510087\pi$
$618$	0	0
$619$	−25.2571	−1.01517	−0.507584	−	0.861602i	$-0.669461\pi$
$619$	−25.2571	−1.01517	−0.507584	+	0.861602i	$0.669461\pi$
$620$	0	0
$621$	4.40103	0.176607
$622$	0	0
$623$	11.7816	0.472020
$624$	0	0
$625$	21.4734	0.858937
$626$	0	0
$627$	−8.40103	−0.335505
$628$	0	0
$629$	10.0928	0.402425
$630$	0	0
$631$	11.9431	0.475447	0.237724	−	0.971333i	$-0.423599\pi$
$631$	11.9431	0.475447	0.237724	+	0.971333i	$0.423599\pi$
$632$	0	0
$633$	20.6265	0.819832
$634$	0	0
$635$	8.30840	0.329709
$636$	0	0
$637$	−17.9905	−0.712808
$638$	0	0
$639$	4.40698	0.174337
$640$	0	0
$641$	−5.16615	−0.204050	−0.102025	−	0.994782i	$-0.532532\pi$
$641$	−5.16615	−0.204050	−0.102025	+	0.994782i	$0.532532\pi$
$642$	0	0
$643$	−24.0280	−0.947572	−0.473786	−	0.880640i	$-0.657113\pi$
$643$	−24.0280	−0.947572	−0.473786	+	0.880640i	$0.657113\pi$
$644$	0	0
$645$	0.401029	0.0157905
$646$	0	0
$647$	−20.5437	−0.807655	−0.403827	−	0.914835i	$-0.632320\pi$
$647$	−20.5437	−0.807655	−0.403827	+	0.914835i	$0.632320\pi$
$648$	0	0
$649$	−23.9632	−0.940638
$650$	0	0
$651$	−5.99046	−0.234785
$652$	0	0
$653$	26.6296	1.04210	0.521048	−	0.853527i	$-0.325541\pi$
$653$	26.6296	1.04210	0.521048	+	0.853527i	$0.325541\pi$
$654$	0	0
$655$	0.0165688	0.000647398 0
$656$	0	0
$657$	−10.4489	−0.407652
$658$	0	0
$659$	1.43049	0.0557239	0.0278619	−	0.999612i	$-0.491130\pi$
$659$	1.43049	0.0557239	0.0278619	+	0.999612i	$0.491130\pi$
$660$	0	0
$661$	−2.69577	−0.104853	−0.0524266	−	0.998625i	$-0.516696\pi$
$661$	−2.69577	−0.104853	−0.0524266	+	0.998625i	$0.516696\pi$
$662$	0	0
$663$	4.67235	0.181459
$664$	0	0
$665$	−1.50831	−0.0584896
$666$	0	0
$667$	−4.51314	−0.174750
$668$	0	0
$669$	10.5875	0.409337
$670$	0	0
$671$	25.3602	0.979019
$672$	0	0
$673$	−29.3956	−1.13312	−0.566559	−	0.824021i	$-0.691726\pi$
$673$	−29.3956	−1.13312	−0.566559	+	0.824021i	$0.691726\pi$
$674$	0	0
$675$	−4.76109	−0.183254
$676$	0	0
$677$	16.3194	0.627204	0.313602	−	0.949555i	$-0.398464\pi$
$677$	16.3194	0.627204	0.313602	+	0.949555i	$0.398464\pi$
$678$	0	0
$679$	−5.63232	−0.216149
$680$	0	0
$681$	16.5581	0.634509
$682$	0	0
$683$	−24.1163	−0.922785	−0.461393	−	0.887196i	$-0.652650\pi$
$683$	−24.1163	−0.922785	−0.461393	+	0.887196i	$0.652650\pi$
$684$	0	0
$685$	2.00048	0.0764345
$686$	0	0
$687$	2.68480	0.102432
$688$	0	0
$689$	−10.7262	−0.408636
$690$	0	0
$691$	24.7357	0.940992	0.470496	−	0.882402i	$-0.344075\pi$
$691$	24.7357	0.940992	0.470496	+	0.882402i	$0.344075\pi$
$692$	0	0
$693$	2.48878	0.0945411
$694$	0	0
$695$	5.72767	0.217263
$696$	0	0
$697$	−1.58057	−0.0598683
$698$	0	0
$699$	−15.7367	−0.595218
$700$	0	0
$701$	−9.32868	−0.352339	−0.176170	−	0.984360i	$-0.556371\pi$
$701$	−9.32868	−0.352339	−0.176170	+	0.984360i	$0.556371\pi$
$702$	0	0
$703$	20.6090	0.777281
$704$	0	0
$705$	1.54363	0.0581363
$706$	0	0
$707$	−15.3747	−0.578227
$708$	0	0
$709$	4.57972	0.171995	0.0859974	−	0.996295i	$-0.472592\pi$
$709$	4.57972	0.171995	0.0859974	+	0.996295i	$0.472592\pi$
$710$	0	0
$711$	2.27132	0.0851812
$712$	0	0
$713$	27.5741	1.03266
$714$	0	0
$715$	−3.76109	−0.140657
$716$	0	0
$717$	3.61602	0.135043
$718$	0	0
$719$	−5.14564	−0.191900	−0.0959500	−	0.995386i	$-0.530589\pi$
$719$	−5.14564	−0.191900	−0.0959500	+	0.995386i	$0.530589\pi$
$720$	0	0
$721$	−3.04003	−0.113217
$722$	0	0
$723$	−4.11225	−0.152936
$724$	0	0
$725$	4.88237	0.181327
$726$	0	0
$727$	14.5542	0.539784	0.269892	−	0.962891i	$-0.413012\pi$
$727$	14.5542	0.539784	0.269892	+	0.962891i	$0.413012\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.29679	0.0479637
$732$	0	0
$733$	−12.5187	−0.462389	−0.231194	−	0.972908i	$-0.574263\pi$
$733$	−12.5187	−0.462389	−0.231194	+	0.972908i	$0.574263\pi$
$734$	0	0
$735$	−2.97466	−0.109722
$736$	0	0
$737$	32.1916	1.18579
$738$	0	0
$739$	−0.145202	−0.00534135	−0.00267068	−	0.999996i	$-0.500850\pi$
$739$	−0.145202	−0.00534135	−0.00267068	+	0.999996i	$0.500850\pi$
$740$	0	0
$741$	9.54072	0.350487
$742$	0	0
$743$	17.8444	0.654649	0.327325	−	0.944912i	$-0.393853\pi$
$743$	17.8444	0.654649	0.327325	+	0.944912i	$0.393853\pi$
$744$	0	0
$745$	6.51971	0.238864
$746$	0	0
$747$	15.1272	0.553477
$748$	0	0
$749$	1.70154	0.0621730
$750$	0	0
$751$	10.6263	0.387758	0.193879	−	0.981025i	$-0.437893\pi$
$751$	10.6263	0.387758	0.193879	+	0.981025i	$0.437893\pi$
$752$	0	0
$753$	13.2957	0.484523
$754$	0	0
$755$	−4.91371	−0.178828
$756$	0	0
$757$	−29.0658	−1.05641	−0.528207	−	0.849115i	$-0.677136\pi$
$757$	−29.0658	−1.05641	−0.528207	+	0.849115i	$0.677136\pi$
$758$	0	0
$759$	−11.4559	−0.415822
$760$	0	0
$761$	45.2302	1.63959	0.819796	−	0.572656i	$-0.194087\pi$
$761$	45.2302	1.63959	0.819796	+	0.572656i	$0.194087\pi$
$762$	0	0
$763$	−11.8879	−0.430370
$764$	0	0
$765$	0.772557	0.0279318
$766$	0	0
$767$	27.2141	0.982642
$768$	0	0
$769$	1.51426	0.0546056	0.0273028	−	0.999627i	$-0.491308\pi$
$769$	1.51426	0.0546056	0.0273028	+	0.999627i	$0.491308\pi$
$770$	0	0
$771$	8.53520	0.307388
$772$	0	0
$773$	−44.0449	−1.58419	−0.792093	−	0.610400i	$-0.791009\pi$
$773$	−44.0449	−1.58419	−0.792093	+	0.610400i	$0.791009\pi$
$774$	0	0
$775$	−29.8300	−1.07152
$776$	0	0
$777$	−6.10535	−0.219028
$778$	0	0
$779$	−3.22744	−0.115635
$780$	0	0
$781$	−11.4714	−0.410478
$782$	0	0
$783$	−1.02547	−0.0366475
$784$	0	0
$785$	−1.06432	−0.0379871
$786$	0	0
$787$	28.3855	1.01184	0.505918	−	0.862582i	$-0.331154\pi$
$787$	28.3855	1.01184	0.505918	+	0.862582i	$0.331154\pi$
$788$	0	0
$789$	−1.62346	−0.0577967
$790$	0	0
$791$	−4.25082	−0.151142
$792$	0	0
$793$	−28.8006	−1.02274
$794$	0	0
$795$	−1.77354	−0.0629010
$796$	0	0
$797$	−46.5688	−1.64955	−0.824775	−	0.565461i	$-0.808698\pi$
$797$	−46.5688	−1.64955	−0.824775	+	0.565461i	$0.808698\pi$
$798$	0	0
$799$	4.99158	0.176589
$800$	0	0
$801$	−12.3223	−0.435386
$802$	0	0
$803$	27.1986	0.959816
$804$	0	0
$805$	−2.05677	−0.0724915
$806$	0	0
$807$	19.1692	0.674787
$808$	0	0
$809$	44.1687	1.55289	0.776444	−	0.630186i	$-0.217021\pi$
$809$	44.1687	1.55289	0.776444	+	0.630186i	$0.217021\pi$
$810$	0	0
$811$	−29.6488	−1.04111	−0.520556	−	0.853828i	$-0.674275\pi$
$811$	−29.6488	−1.04111	−0.520556	+	0.853828i	$0.674275\pi$
$812$	0	0
$813$	−13.2629	−0.465150
$814$	0	0
$815$	−0.746232	−0.0261394
$816$	0	0
$817$	2.64799	0.0926416
$818$	0	0
$819$	−2.82641	−0.0987629
$820$	0	0
$821$	17.5780	0.613477	0.306738	−	0.951794i	$-0.400762\pi$
$821$	17.5780	0.613477	0.306738	+	0.951794i	$0.400762\pi$
$822$	0	0
$823$	12.3652	0.431022	0.215511	−	0.976501i	$-0.430858\pi$
$823$	12.3652	0.431022	0.215511	+	0.976501i	$0.430858\pi$
$824$	0	0
$825$	12.3931	0.431473
$826$	0	0
$827$	7.47427	0.259906	0.129953	−	0.991520i	$-0.458517\pi$
$827$	7.47427	0.259906	0.129953	+	0.991520i	$0.458517\pi$
$828$	0	0
$829$	−4.41598	−0.153373	−0.0766866	−	0.997055i	$-0.524434\pi$
$829$	−4.41598	−0.153373	−0.0766866	+	0.997055i	$0.524434\pi$
$830$	0	0
$831$	9.04692	0.313834
$832$	0	0
$833$	−9.61906	−0.333281
$834$	0	0
$835$	−10.2733	−0.355522
$836$	0	0
$837$	6.26537	0.216563
$838$	0	0
$839$	−53.5931	−1.85024	−0.925119	−	0.379676i	$-0.876035\pi$
$839$	−53.5931	−1.85024	−0.925119	+	0.379676i	$0.876035\pi$
$840$	0	0
$841$	−27.9484	−0.963738
$842$	0	0
$843$	4.00304	0.137872
$844$	0	0
$845$	−2.08288	−0.0716532
$846$	0	0
$847$	4.03904	0.138783
$848$	0	0
$849$	28.4769	0.977326
$850$	0	0
$851$	28.1029	0.963356
$852$	0	0
$853$	−44.6050	−1.52725	−0.763624	−	0.645662i	$-0.776582\pi$
$853$	−44.6050	−1.52725	−0.763624	+	0.645662i	$0.776582\pi$
$854$	0	0
$855$	1.57752	0.0539502
$856$	0	0
$857$	23.3486	0.797574	0.398787	−	0.917044i	$-0.369431\pi$
$857$	23.3486	0.797574	0.398787	+	0.917044i	$0.369431\pi$
$858$	0	0
$859$	−31.9786	−1.09110	−0.545548	−	0.838079i	$-0.683679\pi$
$859$	−31.9786	−1.09110	−0.545548	+	0.838079i	$0.683679\pi$
$860$	0	0
$861$	0.956122	0.0325846
$862$	0	0
$863$	1.28887	0.0438738	0.0219369	−	0.999759i	$-0.493017\pi$
$863$	1.28887	0.0438738	0.0219369	+	0.999759i	$0.493017\pi$
$864$	0	0
$865$	3.32616	0.113093
$866$	0	0
$867$	−14.5018	−0.492507
$868$	0	0
$869$	−5.91224	−0.200559
$870$	0	0
$871$	−36.5588	−1.23875
$872$	0	0
$873$	5.89080	0.199373
$874$	0	0
$875$	4.56173	0.154214
$876$	0	0
$877$	−24.2125	−0.817596	−0.408798	−	0.912625i	$-0.634052\pi$
$877$	−24.2125	−0.817596	−0.408798	+	0.912625i	$0.634052\pi$
$878$	0	0
$879$	−17.4679	−0.589178
$880$	0	0
$881$	38.8979	1.31050	0.655251	−	0.755411i	$-0.272563\pi$
$881$	38.8979	1.31050	0.655251	+	0.755411i	$0.272563\pi$
$882$	0	0
$883$	−43.8330	−1.47510	−0.737549	−	0.675293i	$-0.764017\pi$
$883$	−43.8330	−1.47510	−0.737549	+	0.675293i	$0.764017\pi$
$884$	0	0
$885$	4.49975	0.151257
$886$	0	0
$887$	−10.9358	−0.367187	−0.183593	−	0.983002i	$-0.558773\pi$
$887$	−10.9358	−0.367187	−0.183593	+	0.983002i	$0.558773\pi$
$888$	0	0
$889$	−16.2522	−0.545083
$890$	0	0
$891$	−2.60300	−0.0872037
$892$	0	0
$893$	10.1926	0.341081
$894$	0	0
$895$	4.14628	0.138595
$896$	0	0
$897$	13.0100	0.434391
$898$	0	0
$899$	−6.42497	−0.214285
$900$	0	0
$901$	−5.73504	−0.191062
$902$	0	0
$903$	−0.784462	−0.0261053
$904$	0	0
$905$	8.55779	0.284471
$906$	0	0
$907$	14.4934	0.481245	0.240622	−	0.970619i	$-0.422648\pi$
$907$	14.4934	0.481245	0.240622	+	0.970619i	$0.422648\pi$
$908$	0	0
$909$	16.0803	0.533350
$910$	0	0
$911$	28.5987	0.947518	0.473759	−	0.880654i	$-0.342897\pi$
$911$	28.5987	0.947518	0.473759	+	0.880654i	$0.342897\pi$
$912$	0	0
$913$	−39.3762	−1.30316
$914$	0	0
$915$	−4.76207	−0.157429
$916$	0	0
$917$	−0.0324107	−0.00107030
$918$	0	0
$919$	−10.7297	−0.353939	−0.176969	−	0.984216i	$-0.556629\pi$
$919$	−10.7297	−0.353939	−0.176969	+	0.984216i	$0.556629\pi$
$920$	0	0
$921$	−21.1832	−0.698010
$922$	0	0
$923$	13.0276	0.428808
$924$	0	0
$925$	−30.4021	−0.999615
$926$	0	0
$927$	3.17954	0.104430
$928$	0	0
$929$	−47.1623	−1.54734	−0.773672	−	0.633586i	$-0.781582\pi$
$929$	−47.1623	−1.54734	−0.773672	+	0.633586i	$0.781582\pi$
$930$	0	0
$931$	−19.6417	−0.643730
$932$	0	0
$933$	22.8550	0.748238
$934$	0	0
$935$	−2.01096	−0.0657656
$936$	0	0
$937$	27.2063	0.888790	0.444395	−	0.895831i	$-0.353419\pi$
$937$	27.2063	0.888790	0.444395	+	0.895831i	$0.353419\pi$
$938$	0	0
$939$	−6.90517	−0.225342
$940$	0	0
$941$	−0.398995	−0.0130069	−0.00650344	−	0.999979i	$-0.502070\pi$
$941$	−0.398995	−0.0130069	−0.00650344	+	0.999979i	$0.502070\pi$
$942$	0	0
$943$	−4.40103	−0.143317
$944$	0	0
$945$	−0.467338	−0.0152025
$946$	0	0
$947$	26.4859	0.860675	0.430338	−	0.902668i	$-0.358395\pi$
$947$	26.4859	0.860675	0.430338	+	0.902668i	$0.358395\pi$
$948$	0	0
$949$	−30.8883	−1.00268
$950$	0	0
$951$	28.1966	0.914338
$952$	0	0
$953$	−0.584594	−0.0189369	−0.00946844	−	0.999955i	$-0.503014\pi$
$953$	−0.584594	−0.0189369	−0.00946844	+	0.999955i	$0.503014\pi$
$954$	0	0
$955$	−2.25745	−0.0730493
$956$	0	0
$957$	2.66931	0.0862864
$958$	0	0
$959$	−3.91319	−0.126363
$960$	0	0
$961$	8.25484	0.266285
$962$	0	0
$963$	−1.77963	−0.0573477
$964$	0	0
$965$	7.10167	0.228611
$966$	0	0
$967$	−0.944612	−0.0303767	−0.0151883	−	0.999885i	$-0.504835\pi$
$967$	−0.944612	−0.0303767	−0.0151883	+	0.999885i	$0.504835\pi$
$968$	0	0
$969$	5.10119	0.163874
$970$	0	0
$971$	11.7149	0.375948	0.187974	−	0.982174i	$-0.439808\pi$
$971$	11.7149	0.375948	0.187974	+	0.982174i	$0.439808\pi$
$972$	0	0
$973$	−11.2040	−0.359185
$974$	0	0
$975$	−14.0744	−0.450740
$976$	0	0
$977$	44.7946	1.43311	0.716553	−	0.697532i	$-0.245719\pi$
$977$	44.7946	1.43311	0.716553	+	0.697532i	$0.245719\pi$
$978$	0	0
$979$	32.0748	1.02512
$980$	0	0
$981$	12.4334	0.396969
$982$	0	0
$983$	−17.5234	−0.558909	−0.279455	−	0.960159i	$-0.590154\pi$
$983$	−17.5234	−0.558909	−0.279455	+	0.960159i	$0.590154\pi$
$984$	0	0
$985$	7.56365	0.240998
$986$	0	0
$987$	−3.01952	−0.0961124
$988$	0	0
$989$	3.61088	0.114819
$990$	0	0
$991$	49.6302	1.57655	0.788277	−	0.615321i	$-0.210973\pi$
$991$	49.6302	1.57655	0.788277	+	0.615321i	$0.210973\pi$
$992$	0	0
$993$	−5.67624	−0.180130
$994$	0	0
$995$	7.98952	0.253285
$996$	0	0
$997$	−11.4339	−0.362117	−0.181058	−	0.983472i	$-0.557952\pi$
$997$	−11.4339	−0.362117	−0.181058	+	0.983472i	$0.557952\pi$
$998$	0	0
$999$	6.38553	0.202029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7872.2.a.cj.1.2		4
4.3	odd	2		7872.2.a.cf.1.2		4
8.3	odd	2		3936.2.a.m.1.3	yes	4
8.5	even	2		3936.2.a.i.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3936.2.a.i.1.3	✓	4	8.5	even	2
3936.2.a.m.1.3	yes	4	8.3	odd	2
7872.2.a.cf.1.2		4	4.3	odd	2
7872.2.a.cj.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7872 = 2^{6} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7872.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.488785 q^{5} -0.956122 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.488785 q^{5} -0.956122 q^{7} +1.00000 q^{9} -2.60300 q^{11} +2.95612 q^{13} +0.488785 q^{15} +1.58057 q^{17} +3.22744 q^{19} -0.956122 q^{21} +4.40103 q^{23} -4.76109 q^{25} +1.00000 q^{27} -1.02547 q^{29} +6.26537 q^{31} -2.60300 q^{33} -0.467338 q^{35} +6.38553 q^{37} +2.95612 q^{39} -1.00000 q^{41} +0.820462 q^{43} +0.488785 q^{45} +3.15809 q^{47} -6.08583 q^{49} +1.58057 q^{51} -3.62847 q^{53} -1.27231 q^{55} +3.22744 q^{57} +9.20600 q^{59} -9.74269 q^{61} -0.956122 q^{63} +1.44491 q^{65} -12.3671 q^{67} +4.40103 q^{69} +4.40698 q^{71} -10.4489 q^{73} -4.76109 q^{75} +2.48878 q^{77} +2.27132 q^{79} +1.00000 q^{81} +15.1272 q^{83} +0.772557 q^{85} -1.02547 q^{87} -12.3223 q^{89} -2.82641 q^{91} +6.26537 q^{93} +1.57752 q^{95} +5.89080 q^{97} -2.60300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 + 6 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} + 6 q^{7} + 4 q^{9} + 3 q^{11} + 2 q^{13} + 4 q^{15} - 3 q^{17} + 6 q^{21} + 4 q^{27} + 13 q^{29} + 9 q^{31} + 3 q^{33} + 10 q^{35} + 7 q^{37} + 2 q^{39} - 4 q^{41} - 5 q^{43} + 4 q^{45} + 7 q^{47} - 3 q^{51} + 16 q^{53} + 16 q^{55} + 10 q^{59} + 7 q^{61} + 6 q^{63} - 2 q^{65} - 4 q^{67} + 13 q^{71} - 3 q^{73} + 12 q^{77} + 6 q^{79} + 4 q^{81} + 14 q^{83} + 16 q^{85} + 13 q^{87} - 12 q^{89} - 16 q^{91} + 9 q^{93} + 10 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 6 * q^7 + 4 * q^9 + 3 * q^11 + 2 * q^13 + 4 * q^15 - 3 * q^17 + 6 * q^21 + 4 * q^27 + 13 * q^29 + 9 * q^31 + 3 * q^33 + 10 * q^35 + 7 * q^37 + 2 * q^39 - 4 * q^41 - 5 * q^43 + 4 * q^45 + 7 * q^47 - 3 * q^51 + 16 * q^53 + 16 * q^55 + 10 * q^59 + 7 * q^61 + 6 * q^63 - 2 * q^65 - 4 * q^67 + 13 * q^71 - 3 * q^73 + 12 * q^77 + 6 * q^79 + 4 * q^81 + 14 * q^83 + 16 * q^85 + 13 * q^87 - 12 * q^89 - 16 * q^91 + 9 * q^93 + 10 * q^95 - 10 * q^97 + 3 * q^99

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