LMFDB - Embedded newform 7872.2.a.bx.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 123)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.10278 q^{5} +2.52444 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.10278 q^{5} +2.52444 q^{7} +1.00000 q^{9} -0.813607 q^{11} -5.10278 q^{13} +1.10278 q^{15} +3.39194 q^{17} -3.10278 q^{19} +2.52444 q^{21} -0.897225 q^{23} -3.78389 q^{25} +1.00000 q^{27} -4.44082 q^{29} -8.96526 q^{31} -0.813607 q^{33} +2.78389 q^{35} -2.08362 q^{37} -5.10278 q^{39} +1.00000 q^{41} -9.07306 q^{43} +1.10278 q^{45} -0.235269 q^{47} -0.627213 q^{49} +3.39194 q^{51} -13.8328 q^{53} -0.897225 q^{55} -3.10278 q^{57} -1.04888 q^{59} -1.91638 q^{61} +2.52444 q^{63} -5.62721 q^{65} +10.0383 q^{67} -0.897225 q^{69} -12.8136 q^{71} +4.75971 q^{73} -3.78389 q^{75} -2.05390 q^{77} -15.8328 q^{79} +1.00000 q^{81} +7.68111 q^{83} +3.74055 q^{85} -4.44082 q^{87} +13.2544 q^{89} -12.8816 q^{91} -8.96526 q^{93} -3.42166 q^{95} +2.72999 q^{97} -0.813607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 4 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 4 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 4 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} + 2 q^{21} - 10 q^{23} + 5 q^{25} + 3 q^{27} + 6 q^{29} - 2 q^{31} + 4 q^{33} - 8 q^{35} - 20 q^{37} - 8 q^{39} + 3 q^{41} - 10 q^{43} - 4 q^{45} + 4 q^{47} + 11 q^{49} + 2 q^{51} - 14 q^{53} - 10 q^{55} - 2 q^{57} + 8 q^{59} + 8 q^{61} + 2 q^{63} - 4 q^{65} - 12 q^{67} - 10 q^{69} - 32 q^{71} + 4 q^{73} + 5 q^{75} - 10 q^{77} - 20 q^{79} + 3 q^{81} + 14 q^{83} + 22 q^{85} + 6 q^{87} + 14 q^{89} - 2 q^{93} - 12 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 4 * q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 - 8 * q^13 - 4 * q^15 + 2 * q^17 - 2 * q^19 + 2 * q^21 - 10 * q^23 + 5 * q^25 + 3 * q^27 + 6 * q^29 - 2 * q^31 + 4 * q^33 - 8 * q^35 - 20 * q^37 - 8 * q^39 + 3 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^47 + 11 * q^49 + 2 * q^51 - 14 * q^53 - 10 * q^55 - 2 * q^57 + 8 * q^59 + 8 * q^61 + 2 * q^63 - 4 * q^65 - 12 * q^67 - 10 * q^69 - 32 * q^71 + 4 * q^73 + 5 * q^75 - 10 * q^77 - 20 * q^79 + 3 * q^81 + 14 * q^83 + 22 * q^85 + 6 * q^87 + 14 * q^89 - 2 * q^93 - 12 * q^95 - 12 * q^97 + 4 * 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
$5$	1.10278	0.493176	0.246588	−	0.969120i	$-0.420691\pi$
$5$	1.10278	0.493176	0.246588	+	0.969120i	$0.420691\pi$
$6$	0	0
$7$	2.52444	0.954148	0.477074	−	0.878863i	$-0.341697\pi$
$7$	2.52444	0.954148	0.477074	+	0.878863i	$0.341697\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.813607	−0.245312	−0.122656	−	0.992449i	$-0.539141\pi$
$11$	−0.813607	−0.245312	−0.122656	+	0.992449i	$0.539141\pi$
$12$	0	0
$13$	−5.10278	−1.41526	−0.707628	−	0.706586i	$-0.750235\pi$
$13$	−5.10278	−1.41526	−0.707628	+	0.706586i	$0.750235\pi$
$14$	0	0
$15$	1.10278	0.284735
$16$	0	0
$17$	3.39194	0.822667	0.411334	−	0.911485i	$-0.365063\pi$
$17$	3.39194	0.822667	0.411334	+	0.911485i	$0.365063\pi$
$18$	0	0
$19$	−3.10278	−0.711825	−0.355913	−	0.934519i	$-0.615830\pi$
$19$	−3.10278	−0.711825	−0.355913	+	0.934519i	$0.615830\pi$
$20$	0	0
$21$	2.52444	0.550878
$22$	0	0
$23$	−0.897225	−0.187084	−0.0935422	−	0.995615i	$-0.529819\pi$
$23$	−0.897225	−0.187084	−0.0935422	+	0.995615i	$0.529819\pi$
$24$	0	0
$25$	−3.78389	−0.756777
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.44082	−0.824639	−0.412320	−	0.911039i	$-0.635281\pi$
$29$	−4.44082	−0.824639	−0.412320	+	0.911039i	$0.635281\pi$
$30$	0	0
$31$	−8.96526	−1.61021	−0.805104	−	0.593134i	$-0.797890\pi$
$31$	−8.96526	−1.61021	−0.805104	+	0.593134i	$0.797890\pi$
$32$	0	0
$33$	−0.813607	−0.141631
$34$	0	0
$35$	2.78389	0.470563
$36$	0	0
$37$	−2.08362	−0.342545	−0.171272	−	0.985224i	$-0.554788\pi$
$37$	−2.08362	−0.342545	−0.171272	+	0.985224i	$0.554788\pi$
$38$	0	0
$39$	−5.10278	−0.817098
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−9.07306	−1.38363	−0.691814	−	0.722076i	$-0.743188\pi$
$43$	−9.07306	−1.38363	−0.691814	+	0.722076i	$0.743188\pi$
$44$	0	0
$45$	1.10278	0.164392
$46$	0	0
$47$	−0.235269	−0.0343176	−0.0171588	−	0.999853i	$-0.505462\pi$
$47$	−0.235269	−0.0343176	−0.0171588	+	0.999853i	$0.505462\pi$
$48$	0	0
$49$	−0.627213	−0.0896019
$50$	0	0
$51$	3.39194	0.474967
$52$	0	0
$53$	−13.8328	−1.90008	−0.950038	−	0.312134i	$-0.898956\pi$
$53$	−13.8328	−1.90008	−0.950038	+	0.312134i	$0.898956\pi$
$54$	0	0
$55$	−0.897225	−0.120982
$56$	0	0
$57$	−3.10278	−0.410973
$58$	0	0
$59$	−1.04888	−0.136552	−0.0682760	−	0.997666i	$-0.521750\pi$
$59$	−1.04888	−0.136552	−0.0682760	+	0.997666i	$0.521750\pi$
$60$	0	0
$61$	−1.91638	−0.245368	−0.122684	−	0.992446i	$-0.539150\pi$
$61$	−1.91638	−0.245368	−0.122684	+	0.992446i	$0.539150\pi$
$62$	0	0
$63$	2.52444	0.318049
$64$	0	0
$65$	−5.62721	−0.697970
$66$	0	0
$67$	10.0383	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$67$	10.0383	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$68$	0	0
$69$	−0.897225	−0.108013
$70$	0	0
$71$	−12.8136	−1.52070	−0.760348	−	0.649516i	$-0.774971\pi$
$71$	−12.8136	−1.52070	−0.760348	+	0.649516i	$0.774971\pi$
$72$	0	0
$73$	4.75971	0.557082	0.278541	−	0.960424i	$-0.410149\pi$
$73$	4.75971	0.557082	0.278541	+	0.960424i	$0.410149\pi$
$74$	0	0
$75$	−3.78389	−0.436926
$76$	0	0
$77$	−2.05390	−0.234064
$78$	0	0
$79$	−15.8328	−1.78133	−0.890663	−	0.454665i	$-0.849759\pi$
$79$	−15.8328	−1.78133	−0.890663	+	0.454665i	$0.849759\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.68111	0.843112	0.421556	−	0.906802i	$-0.361484\pi$
$83$	7.68111	0.843112	0.421556	+	0.906802i	$0.361484\pi$
$84$	0	0
$85$	3.74055	0.405720
$86$	0	0
$87$	−4.44082	−0.476106
$88$	0	0
$89$	13.2544	1.40497	0.702483	−	0.711700i	$-0.252075\pi$
$89$	13.2544	1.40497	0.702483	+	0.711700i	$0.252075\pi$
$90$	0	0
$91$	−12.8816	−1.35036
$92$	0	0
$93$	−8.96526	−0.929654
$94$	0	0
$95$	−3.42166	−0.351055
$96$	0	0
$97$	2.72999	0.277188	0.138594	−	0.990349i	$-0.455742\pi$
$97$	2.72999	0.277188	0.138594	+	0.990349i	$0.455742\pi$
$98$	0	0
$99$	−0.813607	−0.0817705
$100$	0	0
$101$	11.8625	1.18036	0.590181	−	0.807271i	$-0.299057\pi$
$101$	11.8625	1.18036	0.590181	+	0.807271i	$0.299057\pi$
$102$	0	0
$103$	−0.494719	−0.0487461	−0.0243730	−	0.999703i	$-0.507759\pi$
$103$	−0.494719	−0.0487461	−0.0243730	+	0.999703i	$0.507759\pi$
$104$	0	0
$105$	2.78389	0.271680
$106$	0	0
$107$	15.0872	1.45853	0.729267	−	0.684229i	$-0.239861\pi$
$107$	15.0872	1.45853	0.729267	+	0.684229i	$0.239861\pi$
$108$	0	0
$109$	−5.10278	−0.488757	−0.244379	−	0.969680i	$-0.578584\pi$
$109$	−5.10278	−0.488757	−0.244379	+	0.969680i	$0.578584\pi$
$110$	0	0
$111$	−2.08362	−0.197768
$112$	0	0
$113$	−17.5139	−1.64757	−0.823783	−	0.566905i	$-0.808141\pi$
$113$	−17.5139	−1.64757	−0.823783	+	0.566905i	$0.808141\pi$
$114$	0	0
$115$	−0.989437	−0.0922655
$116$	0	0
$117$	−5.10278	−0.471752
$118$	0	0
$119$	8.56275	0.784946
$120$	0	0
$121$	−10.3380	−0.939822
$122$	0	0
$123$	1.00000	0.0901670
$124$	0	0
$125$	−9.68665	−0.866400
$126$	0	0
$127$	12.8816	1.14306	0.571530	−	0.820581i	$-0.306350\pi$
$127$	12.8816	1.14306	0.571530	+	0.820581i	$0.306350\pi$
$128$	0	0
$129$	−9.07306	−0.798838
$130$	0	0
$131$	6.35720	0.555431	0.277716	−	0.960663i	$-0.410423\pi$
$131$	6.35720	0.555431	0.277716	+	0.960663i	$0.410423\pi$
$132$	0	0
$133$	−7.83276	−0.679187
$134$	0	0
$135$	1.10278	0.0949118
$136$	0	0
$137$	−17.4897	−1.49425	−0.747123	−	0.664686i	$-0.768565\pi$
$137$	−17.4897	−1.49425	−0.747123	+	0.664686i	$0.768565\pi$
$138$	0	0
$139$	−15.7250	−1.33377	−0.666887	−	0.745159i	$-0.732374\pi$
$139$	−15.7250	−1.33377	−0.666887	+	0.745159i	$0.732374\pi$
$140$	0	0
$141$	−0.235269	−0.0198133
$142$	0	0
$143$	4.15165	0.347178
$144$	0	0
$145$	−4.89722	−0.406692
$146$	0	0
$147$	−0.627213	−0.0517317
$148$	0	0
$149$	−11.5678	−0.947669	−0.473835	−	0.880614i	$-0.657131\pi$
$149$	−11.5678	−0.947669	−0.473835	+	0.880614i	$0.657131\pi$
$150$	0	0
$151$	19.2544	1.56690	0.783451	−	0.621453i	$-0.213457\pi$
$151$	19.2544	1.56690	0.783451	+	0.621453i	$0.213457\pi$
$152$	0	0
$153$	3.39194	0.274222
$154$	0	0
$155$	−9.88666	−0.794116
$156$	0	0
$157$	20.9200	1.66959	0.834797	−	0.550558i	$-0.185585\pi$
$157$	20.9200	1.66959	0.834797	+	0.550558i	$0.185585\pi$
$158$	0	0
$159$	−13.8328	−1.09701
$160$	0	0
$161$	−2.26499	−0.178506
$162$	0	0
$163$	12.7980	1.00242	0.501209	−	0.865326i	$-0.332889\pi$
$163$	12.7980	1.00242	0.501209	+	0.865326i	$0.332889\pi$
$164$	0	0
$165$	−0.897225	−0.0698489
$166$	0	0
$167$	9.62721	0.744976	0.372488	−	0.928037i	$-0.378505\pi$
$167$	9.62721	0.744976	0.372488	+	0.928037i	$0.378505\pi$
$168$	0	0
$169$	13.0383	1.00295
$170$	0	0
$171$	−3.10278	−0.237275
$172$	0	0
$173$	−11.9461	−0.908245	−0.454123	−	0.890939i	$-0.650047\pi$
$173$	−11.9461	−0.908245	−0.454123	+	0.890939i	$0.650047\pi$
$174$	0	0
$175$	−9.55219	−0.722078
$176$	0	0
$177$	−1.04888	−0.0788383
$178$	0	0
$179$	2.13752	0.159766	0.0798828	−	0.996804i	$-0.474545\pi$
$179$	2.13752	0.159766	0.0798828	+	0.996804i	$0.474545\pi$
$180$	0	0
$181$	1.83276	0.136228	0.0681141	−	0.997678i	$-0.478302\pi$
$181$	1.83276	0.136228	0.0681141	+	0.997678i	$0.478302\pi$
$182$	0	0
$183$	−1.91638	−0.141663
$184$	0	0
$185$	−2.29776	−0.168935
$186$	0	0
$187$	−2.75971	−0.201810
$188$	0	0
$189$	2.52444	0.183626
$190$	0	0
$191$	−18.6167	−1.34705	−0.673527	−	0.739163i	$-0.735221\pi$
$191$	−18.6167	−1.34705	−0.673527	+	0.739163i	$0.735221\pi$
$192$	0	0
$193$	15.6116	1.12375	0.561875	−	0.827222i	$-0.310080\pi$
$193$	15.6116	1.12375	0.561875	+	0.827222i	$0.310080\pi$
$194$	0	0
$195$	−5.62721	−0.402973
$196$	0	0
$197$	5.21057	0.371238	0.185619	−	0.982622i	$-0.440571\pi$
$197$	5.21057	0.371238	0.185619	+	0.982622i	$0.440571\pi$
$198$	0	0
$199$	2.05390	0.145597	0.0727985	−	0.997347i	$-0.476807\pi$
$199$	2.05390	0.145597	0.0727985	+	0.997347i	$0.476807\pi$
$200$	0	0
$201$	10.0383	0.708048
$202$	0	0
$203$	−11.2106	−0.786828
$204$	0	0
$205$	1.10278	0.0770212
$206$	0	0
$207$	−0.897225	−0.0623614
$208$	0	0
$209$	2.52444	0.174619
$210$	0	0
$211$	9.04888	0.622950	0.311475	−	0.950254i	$-0.399177\pi$
$211$	9.04888	0.622950	0.311475	+	0.950254i	$0.399177\pi$
$212$	0	0
$213$	−12.8136	−0.877974
$214$	0	0
$215$	−10.0055	−0.682372
$216$	0	0
$217$	−22.6322	−1.53638
$218$	0	0
$219$	4.75971	0.321631
$220$	0	0
$221$	−17.3083	−1.16428
$222$	0	0
$223$	24.8222	1.66222	0.831109	−	0.556110i	$-0.187707\pi$
$223$	24.8222	1.66222	0.831109	+	0.556110i	$0.187707\pi$
$224$	0	0
$225$	−3.78389	−0.252259
$226$	0	0
$227$	−16.6464	−1.10486	−0.552429	−	0.833560i	$-0.686299\pi$
$227$	−16.6464	−1.10486	−0.552429	+	0.833560i	$0.686299\pi$
$228$	0	0
$229$	15.7789	1.04270	0.521348	−	0.853344i	$-0.325429\pi$
$229$	15.7789	1.04270	0.521348	+	0.853344i	$0.325429\pi$
$230$	0	0
$231$	−2.05390	−0.135137
$232$	0	0
$233$	−24.9200	−1.63256	−0.816280	−	0.577656i	$-0.803967\pi$
$233$	−24.9200	−1.63256	−0.816280	+	0.577656i	$0.803967\pi$
$234$	0	0
$235$	−0.259449	−0.0169246
$236$	0	0
$237$	−15.8328	−1.02845
$238$	0	0
$239$	−2.95112	−0.190892	−0.0954462	−	0.995435i	$-0.530428\pi$
$239$	−2.95112	−0.190892	−0.0954462	+	0.995435i	$0.530428\pi$
$240$	0	0
$241$	−16.5925	−1.06881	−0.534407	−	0.845227i	$-0.679465\pi$
$241$	−16.5925	−1.06881	−0.534407	+	0.845227i	$0.679465\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.691675	−0.0441895
$246$	0	0
$247$	15.8328	1.00741
$248$	0	0
$249$	7.68111	0.486771
$250$	0	0
$251$	20.1361	1.27098	0.635489	−	0.772110i	$-0.280799\pi$
$251$	20.1361	1.27098	0.635489	+	0.772110i	$0.280799\pi$
$252$	0	0
$253$	0.729988	0.0458940
$254$	0	0
$255$	3.74055	0.234242
$256$	0	0
$257$	25.9008	1.61565	0.807824	−	0.589424i	$-0.200645\pi$
$257$	25.9008	1.61565	0.807824	+	0.589424i	$0.200645\pi$
$258$	0	0
$259$	−5.25997	−0.326838
$260$	0	0
$261$	−4.44082	−0.274880
$262$	0	0
$263$	−22.4408	−1.38376	−0.691880	−	0.722012i	$-0.743217\pi$
$263$	−22.4408	−1.38376	−0.691880	+	0.722012i	$0.743217\pi$
$264$	0	0
$265$	−15.2544	−0.937072
$266$	0	0
$267$	13.2544	0.811158
$268$	0	0
$269$	−7.62721	−0.465039	−0.232520	−	0.972592i	$-0.574697\pi$
$269$	−7.62721	−0.465039	−0.232520	+	0.972592i	$0.574697\pi$
$270$	0	0
$271$	19.9164	1.20983	0.604917	−	0.796289i	$-0.293206\pi$
$271$	19.9164	1.20983	0.604917	+	0.796289i	$0.293206\pi$
$272$	0	0
$273$	−12.8816	−0.779632
$274$	0	0
$275$	3.07860	0.185646
$276$	0	0
$277$	17.6413	1.05997	0.529983	−	0.848008i	$-0.322198\pi$
$277$	17.6413	1.05997	0.529983	+	0.848008i	$0.322198\pi$
$278$	0	0
$279$	−8.96526	−0.536736
$280$	0	0
$281$	−0.0297193	−0.00177291	−0.000886453	−	1.00000i	$-0.500282\pi$
$281$	−0.0297193	−0.00177291	−0.000886453		1.00000i	$0.500282\pi$
$282$	0	0
$283$	10.1814	0.605220	0.302610	−	0.953115i	$-0.402142\pi$
$283$	10.1814	0.605220	0.302610	+	0.953115i	$0.402142\pi$
$284$	0	0
$285$	−3.42166	−0.202682
$286$	0	0
$287$	2.52444	0.149013
$288$	0	0
$289$	−5.49472	−0.323219
$290$	0	0
$291$	2.72999	0.160035
$292$	0	0
$293$	−3.14808	−0.183913	−0.0919564	−	0.995763i	$-0.529312\pi$
$293$	−3.14808	−0.183913	−0.0919564	+	0.995763i	$0.529312\pi$
$294$	0	0
$295$	−1.15667	−0.0673442
$296$	0	0
$297$	−0.813607	−0.0472102
$298$	0	0
$299$	4.57834	0.264772
$300$	0	0
$301$	−22.9044	−1.32019
$302$	0	0
$303$	11.8625	0.681482
$304$	0	0
$305$	−2.11334	−0.121009
$306$	0	0
$307$	12.7980	0.730422	0.365211	−	0.930925i	$-0.380997\pi$
$307$	12.7980	0.730422	0.365211	+	0.930925i	$0.380997\pi$
$308$	0	0
$309$	−0.494719	−0.0281436
$310$	0	0
$311$	26.6167	1.50929	0.754646	−	0.656132i	$-0.227809\pi$
$311$	26.6167	1.50929	0.754646	+	0.656132i	$0.227809\pi$
$312$	0	0
$313$	−3.00502	−0.169854	−0.0849270	−	0.996387i	$-0.527066\pi$
$313$	−3.00502	−0.169854	−0.0849270	+	0.996387i	$0.527066\pi$
$314$	0	0
$315$	2.78389	0.156854
$316$	0	0
$317$	4.33302	0.243367	0.121683	−	0.992569i	$-0.461171\pi$
$317$	4.33302	0.243367	0.121683	+	0.992569i	$0.461171\pi$
$318$	0	0
$319$	3.61308	0.202294
$320$	0	0
$321$	15.0872	0.842085
$322$	0	0
$323$	−10.5244	−0.585595
$324$	0	0
$325$	19.3083	1.07103
$326$	0	0
$327$	−5.10278	−0.282184
$328$	0	0
$329$	−0.593923	−0.0327440
$330$	0	0
$331$	−5.53500	−0.304231	−0.152116	−	0.988363i	$-0.548609\pi$
$331$	−5.53500	−0.304231	−0.152116	+	0.988363i	$0.548609\pi$
$332$	0	0
$333$	−2.08362	−0.114182
$334$	0	0
$335$	11.0700	0.604819
$336$	0	0
$337$	−3.71083	−0.202142	−0.101071	−	0.994879i	$-0.532227\pi$
$337$	−3.71083	−0.202142	−0.101071	+	0.994879i	$0.532227\pi$
$338$	0	0
$339$	−17.5139	−0.951223
$340$	0	0
$341$	7.29419	0.395003
$342$	0	0
$343$	−19.2544	−1.03964
$344$	0	0
$345$	−0.989437	−0.0532695
$346$	0	0
$347$	−3.07860	−0.165268	−0.0826338	−	0.996580i	$-0.526333\pi$
$347$	−3.07860	−0.165268	−0.0826338	+	0.996580i	$0.526333\pi$
$348$	0	0
$349$	−14.7980	−0.792120	−0.396060	−	0.918225i	$-0.629623\pi$
$349$	−14.7980	−0.792120	−0.396060	+	0.918225i	$0.629623\pi$
$350$	0	0
$351$	−5.10278	−0.272366
$352$	0	0
$353$	26.8222	1.42760	0.713801	−	0.700349i	$-0.246972\pi$
$353$	26.8222	1.42760	0.713801	+	0.700349i	$0.246972\pi$
$354$	0	0
$355$	−14.1305	−0.749970
$356$	0	0
$357$	8.56275	0.453189
$358$	0	0
$359$	2.35720	0.124408	0.0622042	−	0.998063i	$-0.480187\pi$
$359$	2.35720	0.124408	0.0622042	+	0.998063i	$0.480187\pi$
$360$	0	0
$361$	−9.37279	−0.493305
$362$	0	0
$363$	−10.3380	−0.542607
$364$	0	0
$365$	5.24889	0.274739
$366$	0	0
$367$	23.1708	1.20951	0.604753	−	0.796413i	$-0.293272\pi$
$367$	23.1708	1.20951	0.604753	+	0.796413i	$0.293272\pi$
$368$	0	0
$369$	1.00000	0.0520579
$370$	0	0
$371$	−34.9200	−1.81295
$372$	0	0
$373$	−35.2091	−1.82306	−0.911530	−	0.411235i	$-0.865098\pi$
$373$	−35.2091	−1.82306	−0.911530	+	0.411235i	$0.865098\pi$
$374$	0	0
$375$	−9.68665	−0.500217
$376$	0	0
$377$	22.6605	1.16708
$378$	0	0
$379$	26.0383	1.33750	0.668749	−	0.743488i	$-0.266830\pi$
$379$	26.0383	1.33750	0.668749	+	0.743488i	$0.266830\pi$
$380$	0	0
$381$	12.8816	0.659946
$382$	0	0
$383$	15.3819	0.785978	0.392989	−	0.919543i	$-0.371441\pi$
$383$	15.3819	0.785978	0.392989	+	0.919543i	$0.371441\pi$
$384$	0	0
$385$	−2.26499	−0.115435
$386$	0	0
$387$	−9.07306	−0.461209
$388$	0	0
$389$	−4.46500	−0.226384	−0.113192	−	0.993573i	$-0.536108\pi$
$389$	−4.46500	−0.226384	−0.113192	+	0.993573i	$0.536108\pi$
$390$	0	0
$391$	−3.04334	−0.153908
$392$	0	0
$393$	6.35720	0.320678
$394$	0	0
$395$	−17.4600	−0.878507
$396$	0	0
$397$	10.5628	0.530129	0.265065	−	0.964231i	$-0.414607\pi$
$397$	10.5628	0.530129	0.265065	+	0.964231i	$0.414607\pi$
$398$	0	0
$399$	−7.83276	−0.392129
$400$	0	0
$401$	13.0872	0.653543	0.326772	−	0.945103i	$-0.394039\pi$
$401$	13.0872	0.653543	0.326772	+	0.945103i	$0.394039\pi$
$402$	0	0
$403$	45.7477	2.27885
$404$	0	0
$405$	1.10278	0.0547973
$406$	0	0
$407$	1.69525	0.0840302
$408$	0	0
$409$	−18.5542	−0.917444	−0.458722	−	0.888580i	$-0.651693\pi$
$409$	−18.5542	−0.917444	−0.458722	+	0.888580i	$0.651693\pi$
$410$	0	0
$411$	−17.4897	−0.862703
$412$	0	0
$413$	−2.64782	−0.130291
$414$	0	0
$415$	8.47054	0.415802
$416$	0	0
$417$	−15.7250	−0.770055
$418$	0	0
$419$	25.6116	1.25121	0.625605	−	0.780140i	$-0.284852\pi$
$419$	25.6116	1.25121	0.625605	+	0.780140i	$0.284852\pi$
$420$	0	0
$421$	8.67609	0.422847	0.211423	−	0.977395i	$-0.432190\pi$
$421$	8.67609	0.422847	0.211423	+	0.977395i	$0.432190\pi$
$422$	0	0
$423$	−0.235269	−0.0114392
$424$	0	0
$425$	−12.8347	−0.622576
$426$	0	0
$427$	−4.83779	−0.234117
$428$	0	0
$429$	4.15165	0.200444
$430$	0	0
$431$	7.10278	0.342129	0.171064	−	0.985260i	$-0.445279\pi$
$431$	7.10278	0.342129	0.171064	+	0.985260i	$0.445279\pi$
$432$	0	0
$433$	−11.0247	−0.529813	−0.264907	−	0.964274i	$-0.585341\pi$
$433$	−11.0247	−0.529813	−0.264907	+	0.964274i	$0.585341\pi$
$434$	0	0
$435$	−4.89722	−0.234804
$436$	0	0
$437$	2.78389	0.133171
$438$	0	0
$439$	−1.32391	−0.0631868	−0.0315934	−	0.999501i	$-0.510058\pi$
$439$	−1.32391	−0.0631868	−0.0315934	+	0.999501i	$0.510058\pi$
$440$	0	0
$441$	−0.627213	−0.0298673
$442$	0	0
$443$	−15.5889	−0.740651	−0.370325	−	0.928902i	$-0.620754\pi$
$443$	−15.5889	−0.740651	−0.370325	+	0.928902i	$0.620754\pi$
$444$	0	0
$445$	14.6167	0.692896
$446$	0	0
$447$	−11.5678	−0.547137
$448$	0	0
$449$	−32.7839	−1.54717	−0.773584	−	0.633694i	$-0.781538\pi$
$449$	−32.7839	−1.54717	−0.773584	+	0.633694i	$0.781538\pi$
$450$	0	0
$451$	−0.813607	−0.0383112
$452$	0	0
$453$	19.2544	0.904652
$454$	0	0
$455$	−14.2056	−0.665966
$456$	0	0
$457$	−14.1672	−0.662715	−0.331358	−	0.943505i	$-0.607507\pi$
$457$	−14.1672	−0.662715	−0.331358	+	0.943505i	$0.607507\pi$
$458$	0	0
$459$	3.39194	0.158322
$460$	0	0
$461$	30.0766	1.40081	0.700404	−	0.713747i	$-0.253003\pi$
$461$	30.0766	1.40081	0.700404	+	0.713747i	$0.253003\pi$
$462$	0	0
$463$	−23.8766	−1.10964	−0.554820	−	0.831970i	$-0.687213\pi$
$463$	−23.8766	−1.10964	−0.554820	+	0.831970i	$0.687213\pi$
$464$	0	0
$465$	−9.88666	−0.458483
$466$	0	0
$467$	9.73501	0.450483	0.225241	−	0.974303i	$-0.427683\pi$
$467$	9.73501	0.450483	0.225241	+	0.974303i	$0.427683\pi$
$468$	0	0
$469$	25.3411	1.17014
$470$	0	0
$471$	20.9200	0.963941
$472$	0	0
$473$	7.38190	0.339420
$474$	0	0
$475$	11.7406	0.538693
$476$	0	0
$477$	−13.8328	−0.633359
$478$	0	0
$479$	−1.17635	−0.0537487	−0.0268743	−	0.999639i	$-0.508555\pi$
$479$	−1.17635	−0.0537487	−0.0268743	+	0.999639i	$0.508555\pi$
$480$	0	0
$481$	10.6322	0.484788
$482$	0	0
$483$	−2.26499	−0.103061
$484$	0	0
$485$	3.01056	0.136703
$486$	0	0
$487$	20.6025	0.933589	0.466795	−	0.884366i	$-0.345409\pi$
$487$	20.6025	0.933589	0.466795	+	0.884366i	$0.345409\pi$
$488$	0	0
$489$	12.7980	0.578746
$490$	0	0
$491$	−14.1900	−0.640384	−0.320192	−	0.947353i	$-0.603747\pi$
$491$	−14.1900	−0.640384	−0.320192	+	0.947353i	$0.603747\pi$
$492$	0	0
$493$	−15.0630	−0.678404
$494$	0	0
$495$	−0.897225	−0.0403273
$496$	0	0
$497$	−32.3472	−1.45097
$498$	0	0
$499$	21.4600	0.960680	0.480340	−	0.877082i	$-0.340513\pi$
$499$	21.4600	0.960680	0.480340	+	0.877082i	$0.340513\pi$
$500$	0	0
$501$	9.62721	0.430112
$502$	0	0
$503$	−15.5491	−0.693302	−0.346651	−	0.937994i	$-0.612681\pi$
$503$	−15.5491	−0.693302	−0.346651	+	0.937994i	$0.612681\pi$
$504$	0	0
$505$	13.0816	0.582126
$506$	0	0
$507$	13.0383	0.579052
$508$	0	0
$509$	−22.7542	−1.00856	−0.504280	−	0.863540i	$-0.668242\pi$
$509$	−22.7542	−1.00856	−0.504280	+	0.863540i	$0.668242\pi$
$510$	0	0
$511$	12.0156	0.531538
$512$	0	0
$513$	−3.10278	−0.136991
$514$	0	0
$515$	−0.545563	−0.0240404
$516$	0	0
$517$	0.191417	0.00841850
$518$	0	0
$519$	−11.9461	−0.524376
$520$	0	0
$521$	−38.7230	−1.69649	−0.848243	−	0.529608i	$-0.822339\pi$
$521$	−38.7230	−1.69649	−0.848243	+	0.529608i	$0.822339\pi$
$522$	0	0
$523$	−6.56829	−0.287211	−0.143606	−	0.989635i	$-0.545870\pi$
$523$	−6.56829	−0.287211	−0.143606	+	0.989635i	$0.545870\pi$
$524$	0	0
$525$	−9.55219	−0.416892
$526$	0	0
$527$	−30.4096	−1.32467
$528$	0	0
$529$	−22.1950	−0.964999
$530$	0	0
$531$	−1.04888	−0.0455173
$532$	0	0
$533$	−5.10278	−0.221026
$534$	0	0
$535$	16.6378	0.719314
$536$	0	0
$537$	2.13752	0.0922407
$538$	0	0
$539$	0.510305	0.0219804
$540$	0	0
$541$	−42.0766	−1.80902	−0.904508	−	0.426457i	$-0.859761\pi$
$541$	−42.0766	−1.80902	−0.904508	+	0.426457i	$0.859761\pi$
$542$	0	0
$543$	1.83276	0.0786514
$544$	0	0
$545$	−5.62721	−0.241043
$546$	0	0
$547$	−19.9688	−0.853805	−0.426903	−	0.904298i	$-0.640395\pi$
$547$	−19.9688	−0.853805	−0.426903	+	0.904298i	$0.640395\pi$
$548$	0	0
$549$	−1.91638	−0.0817892
$550$	0	0
$551$	13.7789	0.586999
$552$	0	0
$553$	−39.9688	−1.69965
$554$	0	0
$555$	−2.29776	−0.0975346
$556$	0	0
$557$	−13.5491	−0.574095	−0.287048	−	0.957916i	$-0.592674\pi$
$557$	−13.5491	−0.574095	−0.287048	+	0.957916i	$0.592674\pi$
$558$	0	0
$559$	46.2978	1.95819
$560$	0	0
$561$	−2.75971	−0.116515
$562$	0	0
$563$	−9.75468	−0.411111	−0.205555	−	0.978645i	$-0.565900\pi$
$563$	−9.75468	−0.411111	−0.205555	+	0.978645i	$0.565900\pi$
$564$	0	0
$565$	−19.3139	−0.812540
$566$	0	0
$567$	2.52444	0.106016
$568$	0	0
$569$	−21.4544	−0.899417	−0.449708	−	0.893175i	$-0.648472\pi$
$569$	−21.4544	−0.899417	−0.449708	+	0.893175i	$0.648472\pi$
$570$	0	0
$571$	−20.9411	−0.876357	−0.438178	−	0.898888i	$-0.644376\pi$
$571$	−20.9411	−0.876357	−0.438178	+	0.898888i	$0.644376\pi$
$572$	0	0
$573$	−18.6167	−0.777722
$574$	0	0
$575$	3.39500	0.141581
$576$	0	0
$577$	18.4705	0.768939	0.384469	−	0.923138i	$-0.374384\pi$
$577$	18.4705	0.768939	0.384469	+	0.923138i	$0.374384\pi$
$578$	0	0
$579$	15.6116	0.648797
$580$	0	0
$581$	19.3905	0.804453
$582$	0	0
$583$	11.2544	0.466111
$584$	0	0
$585$	−5.62721	−0.232657
$586$	0	0
$587$	0.127471	0.00526130	0.00263065	−	0.999997i	$-0.499163\pi$
$587$	0.127471	0.00526130	0.00263065	+	0.999997i	$0.499163\pi$
$588$	0	0
$589$	27.8172	1.14619
$590$	0	0
$591$	5.21057	0.214334
$592$	0	0
$593$	−27.2841	−1.12043	−0.560213	−	0.828349i	$-0.689281\pi$
$593$	−27.2841	−1.12043	−0.560213	+	0.828349i	$0.689281\pi$
$594$	0	0
$595$	9.44279	0.387117
$596$	0	0
$597$	2.05390	0.0840605
$598$	0	0
$599$	−42.3260	−1.72939	−0.864697	−	0.502293i	$-0.832490\pi$
$599$	−42.3260	−1.72939	−0.864697	+	0.502293i	$0.832490\pi$
$600$	0	0
$601$	−15.6756	−0.639420	−0.319710	−	0.947515i	$-0.603585\pi$
$601$	−15.6756	−0.639420	−0.319710	+	0.947515i	$0.603585\pi$
$602$	0	0
$603$	10.0383	0.408792
$604$	0	0
$605$	−11.4005	−0.463498
$606$	0	0
$607$	−1.93051	−0.0783572	−0.0391786	−	0.999232i	$-0.512474\pi$
$607$	−1.93051	−0.0783572	−0.0391786	+	0.999232i	$0.512474\pi$
$608$	0	0
$609$	−11.2106	−0.454275
$610$	0	0
$611$	1.20053	0.0485681
$612$	0	0
$613$	0.372787	0.0150567	0.00752836	−	0.999972i	$-0.497604\pi$
$613$	0.372787	0.0150567	0.00752836	+	0.999972i	$0.497604\pi$
$614$	0	0
$615$	1.10278	0.0444682
$616$	0	0
$617$	31.3083	1.26043	0.630213	−	0.776422i	$-0.282968\pi$
$617$	31.3083	1.26043	0.630213	+	0.776422i	$0.282968\pi$
$618$	0	0
$619$	−34.3658	−1.38128	−0.690639	−	0.723200i	$-0.742671\pi$
$619$	−34.3658	−1.38128	−0.690639	+	0.723200i	$0.742671\pi$
$620$	0	0
$621$	−0.897225	−0.0360044
$622$	0	0
$623$	33.4600	1.34055
$624$	0	0
$625$	8.23724	0.329490
$626$	0	0
$627$	2.52444	0.100816
$628$	0	0
$629$	−7.06752	−0.281800
$630$	0	0
$631$	7.33804	0.292123	0.146061	−	0.989276i	$-0.453340\pi$
$631$	7.33804	0.292123	0.146061	+	0.989276i	$0.453340\pi$
$632$	0	0
$633$	9.04888	0.359661
$634$	0	0
$635$	14.2056	0.563730
$636$	0	0
$637$	3.20053	0.126809
$638$	0	0
$639$	−12.8136	−0.506898
$640$	0	0
$641$	−31.5764	−1.24719	−0.623596	−	0.781747i	$-0.714329\pi$
$641$	−31.5764	−1.24719	−0.623596	+	0.781747i	$0.714329\pi$
$642$	0	0
$643$	4.51890	0.178208	0.0891040	−	0.996022i	$-0.471600\pi$
$643$	4.51890	0.178208	0.0891040	+	0.996022i	$0.471600\pi$
$644$	0	0
$645$	−10.0055	−0.393968
$646$	0	0
$647$	−20.6705	−0.812643	−0.406322	−	0.913730i	$-0.633189\pi$
$647$	−20.6705	−0.812643	−0.406322	+	0.913730i	$0.633189\pi$
$648$	0	0
$649$	0.853372	0.0334978
$650$	0	0
$651$	−22.6322	−0.887027
$652$	0	0
$653$	−0.381381	−0.0149246	−0.00746229	−	0.999972i	$-0.502375\pi$
$653$	−0.381381	−0.0149246	−0.00746229	+	0.999972i	$0.502375\pi$
$654$	0	0
$655$	7.01056	0.273925
$656$	0	0
$657$	4.75971	0.185694
$658$	0	0
$659$	−13.4600	−0.524326	−0.262163	−	0.965024i	$-0.584436\pi$
$659$	−13.4600	−0.524326	−0.262163	+	0.965024i	$0.584436\pi$
$660$	0	0
$661$	−30.7738	−1.19696	−0.598482	−	0.801136i	$-0.704229\pi$
$661$	−30.7738	−1.19696	−0.598482	+	0.801136i	$0.704229\pi$
$662$	0	0
$663$	−17.3083	−0.672200
$664$	0	0
$665$	−8.63778	−0.334959
$666$	0	0
$667$	3.98441	0.154277
$668$	0	0
$669$	24.8222	0.959682
$670$	0	0
$671$	1.55918	0.0601915
$672$	0	0
$673$	1.48110	0.0570923	0.0285461	−	0.999592i	$-0.490912\pi$
$673$	1.48110	0.0570923	0.0285461	+	0.999592i	$0.490912\pi$
$674$	0	0
$675$	−3.78389	−0.145642
$676$	0	0
$677$	−45.3749	−1.74390	−0.871950	−	0.489596i	$-0.837144\pi$
$677$	−45.3749	−1.74390	−0.871950	+	0.489596i	$0.837144\pi$
$678$	0	0
$679$	6.89169	0.264479
$680$	0	0
$681$	−16.6464	−0.637890
$682$	0	0
$683$	0.0594386	0.00227436	0.00113718	−	0.999999i	$-0.499638\pi$
$683$	0.0594386	0.00227436	0.00113718	+	0.999999i	$0.499638\pi$
$684$	0	0
$685$	−19.2872	−0.736926
$686$	0	0
$687$	15.7789	0.602001
$688$	0	0
$689$	70.5855	2.68909
$690$	0	0
$691$	18.9355	0.720342	0.360171	−	0.932886i	$-0.382718\pi$
$691$	18.9355	0.720342	0.360171	+	0.932886i	$0.382718\pi$
$692$	0	0
$693$	−2.05390	−0.0780212
$694$	0	0
$695$	−17.3411	−0.657785
$696$	0	0
$697$	3.39194	0.128479
$698$	0	0
$699$	−24.9200	−0.942559
$700$	0	0
$701$	0.540024	0.0203964	0.0101982	−	0.999948i	$-0.496754\pi$
$701$	0.540024	0.0203964	0.0101982	+	0.999948i	$0.496754\pi$
$702$	0	0
$703$	6.46500	0.243832
$704$	0	0
$705$	−0.259449	−0.00977142
$706$	0	0
$707$	29.9461	1.12624
$708$	0	0
$709$	−28.0867	−1.05482	−0.527409	−	0.849612i	$-0.676836\pi$
$709$	−28.0867	−1.05482	−0.527409	+	0.849612i	$0.676836\pi$
$710$	0	0
$711$	−15.8328	−0.593775
$712$	0	0
$713$	8.04385	0.301245
$714$	0	0
$715$	4.57834	0.171220
$716$	0	0
$717$	−2.95112	−0.110212
$718$	0	0
$719$	9.86248	0.367809	0.183904	−	0.982944i	$-0.441126\pi$
$719$	9.86248	0.367809	0.183904	+	0.982944i	$0.441126\pi$
$720$	0	0
$721$	−1.24889	−0.0465110
$722$	0	0
$723$	−16.5925	−0.617081
$724$	0	0
$725$	16.8036	0.624069
$726$	0	0
$727$	−30.4650	−1.12988	−0.564942	−	0.825131i	$-0.691102\pi$
$727$	−30.4650	−1.12988	−0.564942	+	0.825131i	$0.691102\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.7753	−1.13827
$732$	0	0
$733$	−27.5436	−1.01735	−0.508673	−	0.860960i	$-0.669864\pi$
$733$	−27.5436	−1.01735	−0.508673	+	0.860960i	$0.669864\pi$
$734$	0	0
$735$	−0.691675	−0.0255128
$736$	0	0
$737$	−8.16724	−0.300844
$738$	0	0
$739$	−13.1013	−0.481940	−0.240970	−	0.970533i	$-0.577466\pi$
$739$	−13.1013	−0.481940	−0.240970	+	0.970533i	$0.577466\pi$
$740$	0	0
$741$	15.8328	0.581631
$742$	0	0
$743$	34.7527	1.27495	0.637477	−	0.770470i	$-0.279978\pi$
$743$	34.7527	1.27495	0.637477	+	0.770470i	$0.279978\pi$
$744$	0	0
$745$	−12.7567	−0.467368
$746$	0	0
$747$	7.68111	0.281037
$748$	0	0
$749$	38.0867	1.39166
$750$	0	0
$751$	32.7738	1.19593	0.597967	−	0.801521i	$-0.295975\pi$
$751$	32.7738	1.19593	0.597967	+	0.801521i	$0.295975\pi$
$752$	0	0
$753$	20.1361	0.733799
$754$	0	0
$755$	21.2333	0.772759
$756$	0	0
$757$	32.2978	1.17388	0.586941	−	0.809630i	$-0.300332\pi$
$757$	32.2978	1.17388	0.586941	+	0.809630i	$0.300332\pi$
$758$	0	0
$759$	0.729988	0.0264969
$760$	0	0
$761$	44.4494	1.61129	0.805645	−	0.592399i	$-0.201819\pi$
$761$	44.4494	1.61129	0.805645	+	0.592399i	$0.201819\pi$
$762$	0	0
$763$	−12.8816	−0.466347
$764$	0	0
$765$	3.74055	0.135240
$766$	0	0
$767$	5.35218	0.193256
$768$	0	0
$769$	−19.7108	−0.710791	−0.355395	−	0.934716i	$-0.615654\pi$
$769$	−19.7108	−0.710791	−0.355395	+	0.934716i	$0.615654\pi$
$770$	0	0
$771$	25.9008	0.932794
$772$	0	0
$773$	−43.2530	−1.55570	−0.777851	−	0.628449i	$-0.783690\pi$
$773$	−43.2530	−1.55570	−0.777851	+	0.628449i	$0.783690\pi$
$774$	0	0
$775$	33.9235	1.21857
$776$	0	0
$777$	−5.25997	−0.188700
$778$	0	0
$779$	−3.10278	−0.111168
$780$	0	0
$781$	10.4252	0.373044
$782$	0	0
$783$	−4.44082	−0.158702
$784$	0	0
$785$	23.0700	0.823404
$786$	0	0
$787$	21.4358	0.764104	0.382052	−	0.924141i	$-0.375218\pi$
$787$	21.4358	0.764104	0.382052	+	0.924141i	$0.375218\pi$
$788$	0	0
$789$	−22.4408	−0.798914
$790$	0	0
$791$	−44.2127	−1.57202
$792$	0	0
$793$	9.77886	0.347258
$794$	0	0
$795$	−15.2544	−0.541019
$796$	0	0
$797$	4.40105	0.155893	0.0779467	−	0.996958i	$-0.475164\pi$
$797$	4.40105	0.155893	0.0779467	+	0.996958i	$0.475164\pi$
$798$	0	0
$799$	−0.798021	−0.0282319
$800$	0	0
$801$	13.2544	0.468322
$802$	0	0
$803$	−3.87253	−0.136659
$804$	0	0
$805$	−2.49777	−0.0880349
$806$	0	0
$807$	−7.62721	−0.268491
$808$	0	0
$809$	−26.3799	−0.927469	−0.463734	−	0.885974i	$-0.653491\pi$
$809$	−26.3799	−0.927469	−0.463734	+	0.885974i	$0.653491\pi$
$810$	0	0
$811$	−4.96526	−0.174354	−0.0871769	−	0.996193i	$-0.527785\pi$
$811$	−4.96526	−0.174354	−0.0871769	+	0.996193i	$0.527785\pi$
$812$	0	0
$813$	19.9164	0.698498
$814$	0	0
$815$	14.1133	0.494369
$816$	0	0
$817$	28.1517	0.984902
$818$	0	0
$819$	−12.8816	−0.450121
$820$	0	0
$821$	15.7789	0.550686	0.275343	−	0.961346i	$-0.411209\pi$
$821$	15.7789	0.550686	0.275343	+	0.961346i	$0.411209\pi$
$822$	0	0
$823$	−8.35166	−0.291121	−0.145560	−	0.989349i	$-0.546498\pi$
$823$	−8.35166	−0.291121	−0.145560	+	0.989349i	$0.546498\pi$
$824$	0	0
$825$	3.07860	0.107183
$826$	0	0
$827$	18.3925	0.639568	0.319784	−	0.947490i	$-0.396390\pi$
$827$	18.3925	0.639568	0.319784	+	0.947490i	$0.396390\pi$
$828$	0	0
$829$	29.4147	1.02161	0.510807	−	0.859695i	$-0.329347\pi$
$829$	29.4147	1.02161	0.510807	+	0.859695i	$0.329347\pi$
$830$	0	0
$831$	17.6413	0.611972
$832$	0	0
$833$	−2.12747	−0.0737125
$834$	0	0
$835$	10.6167	0.367404
$836$	0	0
$837$	−8.96526	−0.309885
$838$	0	0
$839$	−33.4600	−1.15517	−0.577583	−	0.816332i	$-0.696004\pi$
$839$	−33.4600	−1.15517	−0.577583	+	0.816332i	$0.696004\pi$
$840$	0	0
$841$	−9.27912	−0.319970
$842$	0	0
$843$	−0.0297193	−0.00102359
$844$	0	0
$845$	14.3783	0.494629
$846$	0	0
$847$	−26.0978	−0.896729
$848$	0	0
$849$	10.1814	0.349424
$850$	0	0
$851$	1.86947	0.0640848
$852$	0	0
$853$	−40.4494	−1.38496	−0.692481	−	0.721436i	$-0.743482\pi$
$853$	−40.4494	−1.38496	−0.692481	+	0.721436i	$0.743482\pi$
$854$	0	0
$855$	−3.42166	−0.117018
$856$	0	0
$857$	12.4494	0.425264	0.212632	−	0.977132i	$-0.431796\pi$
$857$	12.4494	0.425264	0.212632	+	0.977132i	$0.431796\pi$
$858$	0	0
$859$	21.6514	0.738736	0.369368	−	0.929283i	$-0.379574\pi$
$859$	21.6514	0.738736	0.369368	+	0.929283i	$0.379574\pi$
$860$	0	0
$861$	2.52444	0.0860326
$862$	0	0
$863$	19.3778	0.659628	0.329814	−	0.944046i	$-0.393014\pi$
$863$	19.3778	0.659628	0.329814	+	0.944046i	$0.393014\pi$
$864$	0	0
$865$	−13.1739	−0.447925
$866$	0	0
$867$	−5.49472	−0.186610
$868$	0	0
$869$	12.8816	0.436980
$870$	0	0
$871$	−51.2233	−1.73563
$872$	0	0
$873$	2.72999	0.0923961
$874$	0	0
$875$	−24.4534	−0.826674
$876$	0	0
$877$	3.08413	0.104144	0.0520719	−	0.998643i	$-0.483417\pi$
$877$	3.08413	0.104144	0.0520719	+	0.998643i	$0.483417\pi$
$878$	0	0
$879$	−3.14808	−0.106182
$880$	0	0
$881$	−34.5783	−1.16497	−0.582487	−	0.812840i	$-0.697920\pi$
$881$	−34.5783	−1.16497	−0.582487	+	0.812840i	$0.697920\pi$
$882$	0	0
$883$	9.64280	0.324506	0.162253	−	0.986749i	$-0.448124\pi$
$883$	9.64280	0.324506	0.162253	+	0.986749i	$0.448124\pi$
$884$	0	0
$885$	−1.15667	−0.0388812
$886$	0	0
$887$	32.2041	1.08131	0.540654	−	0.841245i	$-0.318177\pi$
$887$	32.2041	1.08131	0.540654	+	0.841245i	$0.318177\pi$
$888$	0	0
$889$	32.5189	1.09065
$890$	0	0
$891$	−0.813607	−0.0272568
$892$	0	0
$893$	0.729988	0.0244281
$894$	0	0
$895$	2.35720	0.0787925
$896$	0	0
$897$	4.57834	0.152866
$898$	0	0
$899$	39.8131	1.32784
$900$	0	0
$901$	−46.9200	−1.56313
$902$	0	0
$903$	−22.9044	−0.762210
$904$	0	0
$905$	2.02113	0.0671845
$906$	0	0
$907$	17.8227	0.591794	0.295897	−	0.955220i	$-0.404382\pi$
$907$	17.8227	0.591794	0.295897	+	0.955220i	$0.404382\pi$
$908$	0	0
$909$	11.8625	0.393454
$910$	0	0
$911$	−20.7894	−0.688784	−0.344392	−	0.938826i	$-0.611915\pi$
$911$	−20.7894	−0.688784	−0.344392	+	0.938826i	$0.611915\pi$
$912$	0	0
$913$	−6.24940	−0.206825
$914$	0	0
$915$	−2.11334	−0.0698648
$916$	0	0
$917$	16.0484	0.529964
$918$	0	0
$919$	33.8555	1.11679	0.558395	−	0.829575i	$-0.311417\pi$
$919$	33.8555	1.11679	0.558395	+	0.829575i	$0.311417\pi$
$920$	0	0
$921$	12.7980	0.421709
$922$	0	0
$923$	65.3850	2.15217
$924$	0	0
$925$	7.88418	0.259230
$926$	0	0
$927$	−0.494719	−0.0162487
$928$	0	0
$929$	−4.10635	−0.134725	−0.0673624	−	0.997729i	$-0.521458\pi$
$929$	−4.10635	−0.134725	−0.0673624	+	0.997729i	$0.521458\pi$
$930$	0	0
$931$	1.94610	0.0637809
$932$	0	0
$933$	26.6167	0.871390
$934$	0	0
$935$	−3.04334	−0.0995277
$936$	0	0
$937$	13.3028	0.434583	0.217292	−	0.976107i	$-0.430278\pi$
$937$	13.3028	0.434583	0.217292	+	0.976107i	$0.430278\pi$
$938$	0	0
$939$	−3.00502	−0.0980652
$940$	0	0
$941$	11.7633	0.383472	0.191736	−	0.981447i	$-0.438588\pi$
$941$	11.7633	0.383472	0.191736	+	0.981447i	$0.438588\pi$
$942$	0	0
$943$	−0.897225	−0.0292177
$944$	0	0
$945$	2.78389	0.0905599
$946$	0	0
$947$	−1.40054	−0.0455114	−0.0227557	−	0.999741i	$-0.507244\pi$
$947$	−1.40054	−0.0455114	−0.0227557	+	0.999741i	$0.507244\pi$
$948$	0	0
$949$	−24.2877	−0.788413
$950$	0	0
$951$	4.33302	0.140508
$952$	0	0
$953$	33.7577	1.09352	0.546760	−	0.837289i	$-0.315861\pi$
$953$	33.7577	1.09352	0.546760	+	0.837289i	$0.315861\pi$
$954$	0	0
$955$	−20.5300	−0.664334
$956$	0	0
$957$	3.61308	0.116794
$958$	0	0
$959$	−44.1517	−1.42573
$960$	0	0
$961$	49.3758	1.59277
$962$	0	0
$963$	15.0872	0.486178
$964$	0	0
$965$	17.2161	0.554206
$966$	0	0
$967$	−10.4806	−0.337033	−0.168516	−	0.985699i	$-0.553898\pi$
$967$	−10.4806	−0.337033	−0.168516	+	0.985699i	$0.553898\pi$
$968$	0	0
$969$	−10.5244	−0.338094
$970$	0	0
$971$	−57.6358	−1.84962	−0.924811	−	0.380428i	$-0.875777\pi$
$971$	−57.6358	−1.84962	−0.924811	+	0.380428i	$0.875777\pi$
$972$	0	0
$973$	−39.6967	−1.27262
$974$	0	0
$975$	19.3083	0.618361
$976$	0	0
$977$	41.9008	1.34053	0.670263	−	0.742124i	$-0.266181\pi$
$977$	41.9008	1.34053	0.670263	+	0.742124i	$0.266181\pi$
$978$	0	0
$979$	−10.7839	−0.344655
$980$	0	0
$981$	−5.10278	−0.162919
$982$	0	0
$983$	−18.4056	−0.587046	−0.293523	−	0.955952i	$-0.594828\pi$
$983$	−18.4056	−0.587046	−0.293523	+	0.955952i	$0.594828\pi$
$984$	0	0
$985$	5.74609	0.183086
$986$	0	0
$987$	−0.593923	−0.0189048
$988$	0	0
$989$	8.14057	0.258855
$990$	0	0
$991$	26.0666	0.828032	0.414016	−	0.910270i	$-0.364126\pi$
$991$	26.0666	0.828032	0.414016	+	0.910270i	$0.364126\pi$
$992$	0	0
$993$	−5.53500	−0.175648
$994$	0	0
$995$	2.26499	0.0718050
$996$	0	0
$997$	29.8483	0.945307	0.472653	−	0.881248i	$-0.343296\pi$
$997$	29.8483	0.945307	0.472653	+	0.881248i	$0.343296\pi$
$998$	0	0
$999$	−2.08362	−0.0659228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7872.2.a.bx.1.3		3
4.3	odd	2		7872.2.a.bs.1.3		3
8.3	odd	2		1968.2.a.w.1.1		3
8.5	even	2		123.2.a.d.1.1	✓	3
24.5	odd	2		369.2.a.e.1.3		3
24.11	even	2		5904.2.a.bd.1.3		3
40.29	even	2		3075.2.a.t.1.3		3
56.13	odd	2		6027.2.a.s.1.1		3
120.29	odd	2		9225.2.a.bx.1.1		3
328.245	even	2		5043.2.a.n.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.d.1.1	✓	3	8.5	even	2
369.2.a.e.1.3		3	24.5	odd	2
1968.2.a.w.1.1		3	8.3	odd	2
3075.2.a.t.1.3		3	40.29	even	2
5043.2.a.n.1.1		3	328.245	even	2
5904.2.a.bd.1.3		3	24.11	even	2
6027.2.a.s.1.1		3	56.13	odd	2
7872.2.a.bs.1.3		3	4.3	odd	2
7872.2.a.bx.1.3		3	1.1	even	1	trivial
9225.2.a.bx.1.1		3	120.29	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:	\( N \)	\(=\)	\( 7872 = 2^{6} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7872.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.10278 q^{5} +2.52444 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.10278 q^{5} +2.52444 q^{7} +1.00000 q^{9} -0.813607 q^{11} -5.10278 q^{13} +1.10278 q^{15} +3.39194 q^{17} -3.10278 q^{19} +2.52444 q^{21} -0.897225 q^{23} -3.78389 q^{25} +1.00000 q^{27} -4.44082 q^{29} -8.96526 q^{31} -0.813607 q^{33} +2.78389 q^{35} -2.08362 q^{37} -5.10278 q^{39} +1.00000 q^{41} -9.07306 q^{43} +1.10278 q^{45} -0.235269 q^{47} -0.627213 q^{49} +3.39194 q^{51} -13.8328 q^{53} -0.897225 q^{55} -3.10278 q^{57} -1.04888 q^{59} -1.91638 q^{61} +2.52444 q^{63} -5.62721 q^{65} +10.0383 q^{67} -0.897225 q^{69} -12.8136 q^{71} +4.75971 q^{73} -3.78389 q^{75} -2.05390 q^{77} -15.8328 q^{79} +1.00000 q^{81} +7.68111 q^{83} +3.74055 q^{85} -4.44082 q^{87} +13.2544 q^{89} -12.8816 q^{91} -8.96526 q^{93} -3.42166 q^{95} +2.72999 q^{97} -0.813607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 4 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 4 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 4 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} + 2 q^{21} - 10 q^{23} + 5 q^{25} + 3 q^{27} + 6 q^{29} - 2 q^{31} + 4 q^{33} - 8 q^{35} - 20 q^{37} - 8 q^{39} + 3 q^{41} - 10 q^{43} - 4 q^{45} + 4 q^{47} + 11 q^{49} + 2 q^{51} - 14 q^{53} - 10 q^{55} - 2 q^{57} + 8 q^{59} + 8 q^{61} + 2 q^{63} - 4 q^{65} - 12 q^{67} - 10 q^{69} - 32 q^{71} + 4 q^{73} + 5 q^{75} - 10 q^{77} - 20 q^{79} + 3 q^{81} + 14 q^{83} + 22 q^{85} + 6 q^{87} + 14 q^{89} - 2 q^{93} - 12 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 4 * q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 - 8 * q^13 - 4 * q^15 + 2 * q^17 - 2 * q^19 + 2 * q^21 - 10 * q^23 + 5 * q^25 + 3 * q^27 + 6 * q^29 - 2 * q^31 + 4 * q^33 - 8 * q^35 - 20 * q^37 - 8 * q^39 + 3 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^47 + 11 * q^49 + 2 * q^51 - 14 * q^53 - 10 * q^55 - 2 * q^57 + 8 * q^59 + 8 * q^61 + 2 * q^63 - 4 * q^65 - 12 * q^67 - 10 * q^69 - 32 * q^71 + 4 * q^73 + 5 * q^75 - 10 * q^77 - 20 * q^79 + 3 * q^81 + 14 * q^83 + 22 * q^85 + 6 * q^87 + 14 * q^89 - 2 * q^93 - 12 * q^95 - 12 * q^97 + 4 * q^99

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