LMFDB - Embedded newform 5952.2.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5952,2,Mod(1,5952)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5952, 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("5952.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5952 = 2^{6} \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5952.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:	$47.5269592831$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 93)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5952.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.23607 q^{5} +0.236068 q^{7} +1.00000 q^{9} +0.763932 q^{11} -1.23607 q^{13} +4.23607 q^{15} -6.47214 q^{17} +6.23607 q^{19} +0.236068 q^{21} -1.23607 q^{23} +12.9443 q^{25} +1.00000 q^{27} -3.23607 q^{29} -1.00000 q^{31} +0.763932 q^{33} +1.00000 q^{35} +5.70820 q^{37} -1.23607 q^{39} +6.70820 q^{41} +9.70820 q^{43} +4.23607 q^{45} -2.47214 q^{47} -6.94427 q^{49} -6.47214 q^{51} -8.94427 q^{53} +3.23607 q^{55} +6.23607 q^{57} +3.00000 q^{59} -8.00000 q^{61} +0.236068 q^{63} -5.23607 q^{65} +12.0000 q^{67} -1.23607 q^{69} +9.00000 q^{71} +3.23607 q^{73} +12.9443 q^{75} +0.180340 q^{77} +8.47214 q^{79} +1.00000 q^{81} +16.4721 q^{83} -27.4164 q^{85} -3.23607 q^{87} +6.94427 q^{89} -0.291796 q^{91} -1.00000 q^{93} +26.4164 q^{95} +9.00000 q^{97} +0.763932 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{13} + 4 q^{15} - 4 q^{17} + 8 q^{19} - 4 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} - 2 q^{29} - 2 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^13 + 4 * q^15 - 4 * q^17 + 8 * q^19 - 4 * q^21 + 2 * q^23 + 8 * q^25 + 2 * q^27 - 2 * q^29 - 2 * q^31 + 6 * q^33 + 2 * q^35 - 2 * q^37 + 2 * q^39 + 6 * q^43 + 4 * q^45 + 4 * q^47 + 4 * q^49 - 4 * q^51 + 2 * q^55 + 8 * q^57 + 6 * q^59 - 16 * q^61 - 4 * q^63 - 6 * q^65 + 24 * q^67 + 2 * q^69 + 18 * q^71 + 2 * q^73 + 8 * q^75 - 22 * q^77 + 8 * q^79 + 2 * q^81 + 24 * q^83 - 28 * q^85 - 2 * q^87 - 4 * q^89 - 14 * q^91 - 2 * q^93 + 26 * q^95 + 18 * q^97 + 6 * q^99

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
$5$	4.23607	1.89443	0.947214	−	0.320603i	$-0.103886\pi$
$5$	4.23607	1.89443	0.947214	+	0.320603i	$0.103886\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	0	0
$15$	4.23607	1.09375
$16$	0	0
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	0	0
$19$	6.23607	1.43065	0.715326	−	0.698791i	$-0.246278\pi$
$19$	6.23607	1.43065	0.715326	+	0.698791i	$0.246278\pi$
$20$	0	0
$21$	0.236068	0.0515143
$22$	0	0
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	0	0
$25$	12.9443	2.58885
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.23607	−0.600923	−0.300461	−	0.953794i	$-0.597141\pi$
$29$	−3.23607	−0.600923	−0.300461	+	0.953794i	$0.597141\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0.763932	0.132983
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	5.70820	0.938423	0.469211	−	0.883086i	$-0.344538\pi$
$37$	5.70820	0.938423	0.469211	+	0.883086i	$0.344538\pi$
$38$	0	0
$39$	−1.23607	−0.197929
$40$	0	0
$41$	6.70820	1.04765	0.523823	−	0.851827i	$-0.324505\pi$
$41$	6.70820	1.04765	0.523823	+	0.851827i	$0.324505\pi$
$42$	0	0
$43$	9.70820	1.48049	0.740244	−	0.672339i	$-0.234710\pi$
$43$	9.70820	1.48049	0.740244	+	0.672339i	$0.234710\pi$
$44$	0	0
$45$	4.23607	0.631476
$46$	0	0
$47$	−2.47214	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	−6.47214	−0.906280
$52$	0	0
$53$	−8.94427	−1.22859	−0.614295	−	0.789076i	$-0.710560\pi$
$53$	−8.94427	−1.22859	−0.614295	+	0.789076i	$0.710560\pi$
$54$	0	0
$55$	3.23607	0.436351
$56$	0	0
$57$	6.23607	0.825987
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0.236068	0.0297418
$64$	0	0
$65$	−5.23607	−0.649454
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−1.23607	−0.148805
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	3.23607	0.378753	0.189377	−	0.981905i	$-0.439353\pi$
$73$	3.23607	0.378753	0.189377	+	0.981905i	$0.439353\pi$
$74$	0	0
$75$	12.9443	1.49468
$76$	0	0
$77$	0.180340	0.0205516
$78$	0	0
$79$	8.47214	0.953190	0.476595	−	0.879123i	$-0.341871\pi$
$79$	8.47214	0.953190	0.476595	+	0.879123i	$0.341871\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.4721	1.80805	0.904026	−	0.427478i	$-0.140598\pi$
$83$	16.4721	1.80805	0.904026	+	0.427478i	$0.140598\pi$
$84$	0	0
$85$	−27.4164	−2.97373
$86$	0	0
$87$	−3.23607	−0.346943
$88$	0	0
$89$	6.94427	0.736091	0.368046	−	0.929808i	$-0.380027\pi$
$89$	6.94427	0.736091	0.368046	+	0.929808i	$0.380027\pi$
$90$	0	0
$91$	−0.291796	−0.0305885
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	26.4164	2.71027
$96$	0	0
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	0	0
$99$	0.763932	0.0767781
$100$	0	0
$101$	5.76393	0.573533	0.286766	−	0.958001i	$-0.407420\pi$
$101$	5.76393	0.573533	0.286766	+	0.958001i	$0.407420\pi$
$102$	0	0
$103$	−13.7639	−1.35620	−0.678100	−	0.734969i	$-0.737196\pi$
$103$	−13.7639	−1.35620	−0.678100	+	0.734969i	$0.737196\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−1.47214	−0.142317	−0.0711584	−	0.997465i	$-0.522670\pi$
$107$	−1.47214	−0.142317	−0.0711584	+	0.997465i	$0.522670\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	5.70820	0.541799
$112$	0	0
$113$	−1.76393	−0.165937	−0.0829684	−	0.996552i	$-0.526440\pi$
$113$	−1.76393	−0.165937	−0.0829684	+	0.996552i	$0.526440\pi$
$114$	0	0
$115$	−5.23607	−0.488266
$116$	0	0
$117$	−1.23607	−0.114275
$118$	0	0
$119$	−1.52786	−0.140059
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	6.70820	0.604858
$124$	0	0
$125$	33.6525	3.00997
$126$	0	0
$127$	7.23607	0.642097	0.321049	−	0.947063i	$-0.395965\pi$
$127$	7.23607	0.642097	0.321049	+	0.947063i	$0.395965\pi$
$128$	0	0
$129$	9.70820	0.854760
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	1.47214	0.127650
$134$	0	0
$135$	4.23607	0.364583
$136$	0	0
$137$	−10.1803	−0.869765	−0.434883	−	0.900487i	$-0.643210\pi$
$137$	−10.1803	−0.869765	−0.434883	+	0.900487i	$0.643210\pi$
$138$	0	0
$139$	−5.70820	−0.484164	−0.242082	−	0.970256i	$-0.577830\pi$
$139$	−5.70820	−0.484164	−0.242082	+	0.970256i	$0.577830\pi$
$140$	0	0
$141$	−2.47214	−0.208191
$142$	0	0
$143$	−0.944272	−0.0789640
$144$	0	0
$145$	−13.7082	−1.13840
$146$	0	0
$147$	−6.94427	−0.572754
$148$	0	0
$149$	13.4164	1.09911	0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164	1.09911	0.549557	+	0.835456i	$0.314796\pi$
$150$	0	0
$151$	15.2361	1.23989	0.619947	−	0.784644i	$-0.287154\pi$
$151$	15.2361	1.23989	0.619947	+	0.784644i	$0.287154\pi$
$152$	0	0
$153$	−6.47214	−0.523241
$154$	0	0
$155$	−4.23607	−0.340249
$156$	0	0
$157$	4.05573	0.323682	0.161841	−	0.986817i	$-0.448257\pi$
$157$	4.05573	0.323682	0.161841	+	0.986817i	$0.448257\pi$
$158$	0	0
$159$	−8.94427	−0.709327
$160$	0	0
$161$	−0.291796	−0.0229968
$162$	0	0
$163$	7.18034	0.562408	0.281204	−	0.959648i	$-0.409266\pi$
$163$	7.18034	0.562408	0.281204	+	0.959648i	$0.409266\pi$
$164$	0	0
$165$	3.23607	0.251928
$166$	0	0
$167$	−16.6525	−1.28861	−0.644304	−	0.764770i	$-0.722853\pi$
$167$	−16.6525	−1.28861	−0.644304	+	0.764770i	$0.722853\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	6.23607	0.476884
$172$	0	0
$173$	−6.94427	−0.527963	−0.263982	−	0.964528i	$-0.585036\pi$
$173$	−6.94427	−0.527963	−0.263982	+	0.964528i	$0.585036\pi$
$174$	0	0
$175$	3.05573	0.230991
$176$	0	0
$177$	3.00000	0.225494
$178$	0	0
$179$	−19.4164	−1.45125	−0.725625	−	0.688090i	$-0.758449\pi$
$179$	−19.4164	−1.45125	−0.725625	+	0.688090i	$0.758449\pi$
$180$	0	0
$181$	−19.4164	−1.44321	−0.721605	−	0.692305i	$-0.756595\pi$
$181$	−19.4164	−1.44321	−0.721605	+	0.692305i	$0.756595\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	24.1803	1.77777
$186$	0	0
$187$	−4.94427	−0.361561
$188$	0	0
$189$	0.236068	0.0171714
$190$	0	0
$191$	−13.9443	−1.00897	−0.504486	−	0.863420i	$-0.668318\pi$
$191$	−13.9443	−1.00897	−0.504486	+	0.863420i	$0.668318\pi$
$192$	0	0
$193$	−3.00000	−0.215945	−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000	−0.215945	−0.107972	+	0.994154i	$0.534436\pi$
$194$	0	0
$195$	−5.23607	−0.374963
$196$	0	0
$197$	−5.23607	−0.373054	−0.186527	−	0.982450i	$-0.559723\pi$
$197$	−5.23607	−0.373054	−0.186527	+	0.982450i	$0.559723\pi$
$198$	0	0
$199$	−12.1803	−0.863441	−0.431721	−	0.902007i	$-0.642093\pi$
$199$	−12.1803	−0.863441	−0.431721	+	0.902007i	$0.642093\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−0.763932	−0.0536175
$204$	0	0
$205$	28.4164	1.98469
$206$	0	0
$207$	−1.23607	−0.0859127
$208$	0	0
$209$	4.76393	0.329528
$210$	0	0
$211$	−0.708204	−0.0487548	−0.0243774	−	0.999703i	$-0.507760\pi$
$211$	−0.708204	−0.0487548	−0.0243774	+	0.999703i	$0.507760\pi$
$212$	0	0
$213$	9.00000	0.616670
$214$	0	0
$215$	41.1246	2.80468
$216$	0	0
$217$	−0.236068	−0.0160253
$218$	0	0
$219$	3.23607	0.218673
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−1.81966	−0.121853	−0.0609267	−	0.998142i	$-0.519406\pi$
$223$	−1.81966	−0.121853	−0.0609267	+	0.998142i	$0.519406\pi$
$224$	0	0
$225$	12.9443	0.862951
$226$	0	0
$227$	−4.94427	−0.328163	−0.164081	−	0.986447i	$-0.552466\pi$
$227$	−4.94427	−0.328163	−0.164081	+	0.986447i	$0.552466\pi$
$228$	0	0
$229$	−9.70820	−0.641536	−0.320768	−	0.947158i	$-0.603941\pi$
$229$	−9.70820	−0.641536	−0.320768	+	0.947158i	$0.603941\pi$
$230$	0	0
$231$	0.180340	0.0118655
$232$	0	0
$233$	0.708204	0.0463960	0.0231980	−	0.999731i	$-0.492615\pi$
$233$	0.708204	0.0463960	0.0231980	+	0.999731i	$0.492615\pi$
$234$	0	0
$235$	−10.4721	−0.683127
$236$	0	0
$237$	8.47214	0.550324
$238$	0	0
$239$	−11.8885	−0.769006	−0.384503	−	0.923124i	$-0.625627\pi$
$239$	−11.8885	−0.769006	−0.384503	+	0.923124i	$0.625627\pi$
$240$	0	0
$241$	−13.7082	−0.883023	−0.441512	−	0.897256i	$-0.645558\pi$
$241$	−13.7082	−0.883023	−0.441512	+	0.897256i	$0.645558\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−29.4164	−1.87935
$246$	0	0
$247$	−7.70820	−0.490461
$248$	0	0
$249$	16.4721	1.04388
$250$	0	0
$251$	11.2361	0.709214	0.354607	−	0.935015i	$-0.384615\pi$
$251$	11.2361	0.709214	0.354607	+	0.935015i	$0.384615\pi$
$252$	0	0
$253$	−0.944272	−0.0593659
$254$	0	0
$255$	−27.4164	−1.71688
$256$	0	0
$257$	5.29180	0.330093	0.165047	−	0.986286i	$-0.447223\pi$
$257$	5.29180	0.330093	0.165047	+	0.986286i	$0.447223\pi$
$258$	0	0
$259$	1.34752	0.0837311
$260$	0	0
$261$	−3.23607	−0.200308
$262$	0	0
$263$	18.4721	1.13904	0.569520	−	0.821977i	$-0.307129\pi$
$263$	18.4721	1.13904	0.569520	+	0.821977i	$0.307129\pi$
$264$	0	0
$265$	−37.8885	−2.32747
$266$	0	0
$267$	6.94427	0.424983
$268$	0	0
$269$	12.6525	0.771435	0.385718	−	0.922617i	$-0.373954\pi$
$269$	12.6525	0.771435	0.385718	+	0.922617i	$0.373954\pi$
$270$	0	0
$271$	−13.4164	−0.814989	−0.407494	−	0.913208i	$-0.633597\pi$
$271$	−13.4164	−0.814989	−0.407494	+	0.913208i	$0.633597\pi$
$272$	0	0
$273$	−0.291796	−0.0176603
$274$	0	0
$275$	9.88854	0.596302
$276$	0	0
$277$	8.47214	0.509041	0.254521	−	0.967067i	$-0.418082\pi$
$277$	8.47214	0.509041	0.254521	+	0.967067i	$0.418082\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−28.2361	−1.68442	−0.842211	−	0.539148i	$-0.818746\pi$
$281$	−28.2361	−1.68442	−0.842211	+	0.539148i	$0.818746\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	26.4164	1.56477
$286$	0	0
$287$	1.58359	0.0934765
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	9.00000	0.527589
$292$	0	0
$293$	−14.3607	−0.838960	−0.419480	−	0.907765i	$-0.637788\pi$
$293$	−14.3607	−0.838960	−0.419480	+	0.907765i	$0.637788\pi$
$294$	0	0
$295$	12.7082	0.739900
$296$	0	0
$297$	0.763932	0.0443278
$298$	0	0
$299$	1.52786	0.0883587
$300$	0	0
$301$	2.29180	0.132097
$302$	0	0
$303$	5.76393	0.331129
$304$	0	0
$305$	−33.8885	−1.94045
$306$	0	0
$307$	10.2361	0.584203	0.292102	−	0.956387i	$-0.405645\pi$
$307$	10.2361	0.584203	0.292102	+	0.956387i	$0.405645\pi$
$308$	0	0
$309$	−13.7639	−0.783003
$310$	0	0
$311$	−5.47214	−0.310296	−0.155148	−	0.987891i	$-0.549586\pi$
$311$	−5.47214	−0.310296	−0.155148	+	0.987891i	$0.549586\pi$
$312$	0	0
$313$	−6.47214	−0.365827	−0.182913	−	0.983129i	$-0.558553\pi$
$313$	−6.47214	−0.365827	−0.182913	+	0.983129i	$0.558553\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−32.1246	−1.80430	−0.902149	−	0.431425i	$-0.858011\pi$
$317$	−32.1246	−1.80430	−0.902149	+	0.431425i	$0.858011\pi$
$318$	0	0
$319$	−2.47214	−0.138413
$320$	0	0
$321$	−1.47214	−0.0821666
$322$	0	0
$323$	−40.3607	−2.24573
$324$	0	0
$325$	−16.0000	−0.887520
$326$	0	0
$327$	5.00000	0.276501
$328$	0	0
$329$	−0.583592	−0.0321745
$330$	0	0
$331$	−20.8328	−1.14508	−0.572538	−	0.819878i	$-0.694041\pi$
$331$	−20.8328	−1.14508	−0.572538	+	0.819878i	$0.694041\pi$
$332$	0	0
$333$	5.70820	0.312808
$334$	0	0
$335$	50.8328	2.77729
$336$	0	0
$337$	17.0557	0.929085	0.464542	−	0.885551i	$-0.346219\pi$
$337$	17.0557	0.929085	0.464542	+	0.885551i	$0.346219\pi$
$338$	0	0
$339$	−1.76393	−0.0958036
$340$	0	0
$341$	−0.763932	−0.0413692
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	−5.23607	−0.281900
$346$	0	0
$347$	−3.34752	−0.179705	−0.0898523	−	0.995955i	$-0.528639\pi$
$347$	−3.34752	−0.179705	−0.0898523	+	0.995955i	$0.528639\pi$
$348$	0	0
$349$	−27.8885	−1.49284	−0.746420	−	0.665475i	$-0.768229\pi$
$349$	−27.8885	−1.49284	−0.746420	+	0.665475i	$0.768229\pi$
$350$	0	0
$351$	−1.23607	−0.0659764
$352$	0	0
$353$	−17.2361	−0.917383	−0.458692	−	0.888595i	$-0.651682\pi$
$353$	−17.2361	−0.917383	−0.458692	+	0.888595i	$0.651682\pi$
$354$	0	0
$355$	38.1246	2.02344
$356$	0	0
$357$	−1.52786	−0.0808631
$358$	0	0
$359$	−8.88854	−0.469119	−0.234560	−	0.972102i	$-0.575365\pi$
$359$	−8.88854	−0.469119	−0.234560	+	0.972102i	$0.575365\pi$
$360$	0	0
$361$	19.8885	1.04677
$362$	0	0
$363$	−10.4164	−0.546720
$364$	0	0
$365$	13.7082	0.717520
$366$	0	0
$367$	17.0557	0.890302	0.445151	−	0.895456i	$-0.353150\pi$
$367$	17.0557	0.890302	0.445151	+	0.895456i	$0.353150\pi$
$368$	0	0
$369$	6.70820	0.349215
$370$	0	0
$371$	−2.11146	−0.109621
$372$	0	0
$373$	−7.58359	−0.392664	−0.196332	−	0.980538i	$-0.562903\pi$
$373$	−7.58359	−0.392664	−0.196332	+	0.980538i	$0.562903\pi$
$374$	0	0
$375$	33.6525	1.73781
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−17.5279	−0.900346	−0.450173	−	0.892941i	$-0.648638\pi$
$379$	−17.5279	−0.900346	−0.450173	+	0.892941i	$0.648638\pi$
$380$	0	0
$381$	7.23607	0.370715
$382$	0	0
$383$	2.94427	0.150445	0.0752226	−	0.997167i	$-0.476033\pi$
$383$	2.94427	0.150445	0.0752226	+	0.997167i	$0.476033\pi$
$384$	0	0
$385$	0.763932	0.0389336
$386$	0	0
$387$	9.70820	0.493496
$388$	0	0
$389$	−26.1803	−1.32740	−0.663698	−	0.748001i	$-0.731014\pi$
$389$	−26.1803	−1.32740	−0.663698	+	0.748001i	$0.731014\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	35.8885	1.80575
$396$	0	0
$397$	30.4164	1.52656	0.763278	−	0.646070i	$-0.223589\pi$
$397$	30.4164	1.52656	0.763278	+	0.646070i	$0.223589\pi$
$398$	0	0
$399$	1.47214	0.0736990
$400$	0	0
$401$	0.763932	0.0381489	0.0190745	−	0.999818i	$-0.493928\pi$
$401$	0.763932	0.0381489	0.0190745	+	0.999818i	$0.493928\pi$
$402$	0	0
$403$	1.23607	0.0615729
$404$	0	0
$405$	4.23607	0.210492
$406$	0	0
$407$	4.36068	0.216151
$408$	0	0
$409$	23.1246	1.14344	0.571719	−	0.820449i	$-0.306277\pi$
$409$	23.1246	1.14344	0.571719	+	0.820449i	$0.306277\pi$
$410$	0	0
$411$	−10.1803	−0.502159
$412$	0	0
$413$	0.708204	0.0348484
$414$	0	0
$415$	69.7771	3.42522
$416$	0	0
$417$	−5.70820	−0.279532
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	14.4164	0.702613	0.351306	−	0.936261i	$-0.385738\pi$
$421$	14.4164	0.702613	0.351306	+	0.936261i	$0.385738\pi$
$422$	0	0
$423$	−2.47214	−0.120199
$424$	0	0
$425$	−83.7771	−4.06379
$426$	0	0
$427$	−1.88854	−0.0913930
$428$	0	0
$429$	−0.944272	−0.0455899
$430$	0	0
$431$	−6.47214	−0.311752	−0.155876	−	0.987777i	$-0.549820\pi$
$431$	−6.47214	−0.311752	−0.155876	+	0.987777i	$0.549820\pi$
$432$	0	0
$433$	12.4721	0.599373	0.299686	−	0.954038i	$-0.403118\pi$
$433$	12.4721	0.599373	0.299686	+	0.954038i	$0.403118\pi$
$434$	0	0
$435$	−13.7082	−0.657258
$436$	0	0
$437$	−7.70820	−0.368733
$438$	0	0
$439$	−10.7082	−0.511075	−0.255537	−	0.966799i	$-0.582252\pi$
$439$	−10.7082	−0.511075	−0.255537	+	0.966799i	$0.582252\pi$
$440$	0	0
$441$	−6.94427	−0.330680
$442$	0	0
$443$	25.3607	1.20492	0.602461	−	0.798148i	$-0.294187\pi$
$443$	25.3607	1.20492	0.602461	+	0.798148i	$0.294187\pi$
$444$	0	0
$445$	29.4164	1.39447
$446$	0	0
$447$	13.4164	0.634574
$448$	0	0
$449$	−30.6525	−1.44658	−0.723290	−	0.690545i	$-0.757371\pi$
$449$	−30.6525	−1.44658	−0.723290	+	0.690545i	$0.757371\pi$
$450$	0	0
$451$	5.12461	0.241309
$452$	0	0
$453$	15.2361	0.715853
$454$	0	0
$455$	−1.23607	−0.0579478
$456$	0	0
$457$	11.1246	0.520387	0.260194	−	0.965556i	$-0.416214\pi$
$457$	11.1246	0.520387	0.260194	+	0.965556i	$0.416214\pi$
$458$	0	0
$459$	−6.47214	−0.302093
$460$	0	0
$461$	23.8885	1.11260	0.556300	−	0.830981i	$-0.312220\pi$
$461$	23.8885	1.11260	0.556300	+	0.830981i	$0.312220\pi$
$462$	0	0
$463$	−24.8328	−1.15408	−0.577039	−	0.816716i	$-0.695792\pi$
$463$	−24.8328	−1.15408	−0.577039	+	0.816716i	$0.695792\pi$
$464$	0	0
$465$	−4.23607	−0.196443
$466$	0	0
$467$	−20.0557	−0.928068	−0.464034	−	0.885817i	$-0.653599\pi$
$467$	−20.0557	−0.928068	−0.464034	+	0.885817i	$0.653599\pi$
$468$	0	0
$469$	2.83282	0.130807
$470$	0	0
$471$	4.05573	0.186878
$472$	0	0
$473$	7.41641	0.341007
$474$	0	0
$475$	80.7214	3.70375
$476$	0	0
$477$	−8.94427	−0.409530
$478$	0	0
$479$	−0.0557281	−0.00254628	−0.00127314	−	0.999999i	$-0.500405\pi$
$479$	−0.0557281	−0.00254628	−0.00127314	+	0.999999i	$0.500405\pi$
$480$	0	0
$481$	−7.05573	−0.321714
$482$	0	0
$483$	−0.291796	−0.0132772
$484$	0	0
$485$	38.1246	1.73115
$486$	0	0
$487$	−3.70820	−0.168035	−0.0840174	−	0.996464i	$-0.526775\pi$
$487$	−3.70820	−0.168035	−0.0840174	+	0.996464i	$0.526775\pi$
$488$	0	0
$489$	7.18034	0.324706
$490$	0	0
$491$	13.8197	0.623673	0.311836	−	0.950136i	$-0.399056\pi$
$491$	13.8197	0.623673	0.311836	+	0.950136i	$0.399056\pi$
$492$	0	0
$493$	20.9443	0.943283
$494$	0	0
$495$	3.23607	0.145450
$496$	0	0
$497$	2.12461	0.0953019
$498$	0	0
$499$	−19.8885	−0.890333	−0.445167	−	0.895448i	$-0.646856\pi$
$499$	−19.8885	−0.890333	−0.445167	+	0.895448i	$0.646856\pi$
$500$	0	0
$501$	−16.6525	−0.743978
$502$	0	0
$503$	13.3607	0.595723	0.297862	−	0.954609i	$-0.403727\pi$
$503$	13.3607	0.595723	0.297862	+	0.954609i	$0.403727\pi$
$504$	0	0
$505$	24.4164	1.08652
$506$	0	0
$507$	−11.4721	−0.509495
$508$	0	0
$509$	−26.2918	−1.16536	−0.582682	−	0.812700i	$-0.697997\pi$
$509$	−26.2918	−1.16536	−0.582682	+	0.812700i	$0.697997\pi$
$510$	0	0
$511$	0.763932	0.0337944
$512$	0	0
$513$	6.23607	0.275329
$514$	0	0
$515$	−58.3050	−2.56922
$516$	0	0
$517$	−1.88854	−0.0830581
$518$	0	0
$519$	−6.94427	−0.304820
$520$	0	0
$521$	−27.5279	−1.20602	−0.603009	−	0.797735i	$-0.706032\pi$
$521$	−27.5279	−1.20602	−0.603009	+	0.797735i	$0.706032\pi$
$522$	0	0
$523$	−42.5410	−1.86019	−0.930094	−	0.367320i	$-0.880275\pi$
$523$	−42.5410	−1.86019	−0.930094	+	0.367320i	$0.880275\pi$
$524$	0	0
$525$	3.05573	0.133363
$526$	0	0
$527$	6.47214	0.281931
$528$	0	0
$529$	−21.4721	−0.933571
$530$	0	0
$531$	3.00000	0.130189
$532$	0	0
$533$	−8.29180	−0.359158
$534$	0	0
$535$	−6.23607	−0.269609
$536$	0	0
$537$	−19.4164	−0.837880
$538$	0	0
$539$	−5.30495	−0.228500
$540$	0	0
$541$	9.00000	0.386940	0.193470	−	0.981106i	$-0.438026\pi$
$541$	9.00000	0.386940	0.193470	+	0.981106i	$0.438026\pi$
$542$	0	0
$543$	−19.4164	−0.833238
$544$	0	0
$545$	21.1803	0.907266
$546$	0	0
$547$	31.5410	1.34860	0.674298	−	0.738459i	$-0.264446\pi$
$547$	31.5410	1.34860	0.674298	+	0.738459i	$0.264446\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−20.1803	−0.859711
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	24.1803	1.02640
$556$	0	0
$557$	−24.6525	−1.04456	−0.522279	−	0.852774i	$-0.674918\pi$
$557$	−24.6525	−1.04456	−0.522279	+	0.852774i	$0.674918\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	−4.94427	−0.208747
$562$	0	0
$563$	9.11146	0.384002	0.192001	−	0.981395i	$-0.438502\pi$
$563$	9.11146	0.384002	0.192001	+	0.981395i	$0.438502\pi$
$564$	0	0
$565$	−7.47214	−0.314355
$566$	0	0
$567$	0.236068	0.00991392
$568$	0	0
$569$	29.8885	1.25299	0.626496	−	0.779424i	$-0.284488\pi$
$569$	29.8885	1.25299	0.626496	+	0.779424i	$0.284488\pi$
$570$	0	0
$571$	−23.7082	−0.992157	−0.496079	−	0.868278i	$-0.665227\pi$
$571$	−23.7082	−0.992157	−0.496079	+	0.868278i	$0.665227\pi$
$572$	0	0
$573$	−13.9443	−0.582530
$574$	0	0
$575$	−16.0000	−0.667246
$576$	0	0
$577$	30.3607	1.26393	0.631966	−	0.774996i	$-0.282248\pi$
$577$	30.3607	1.26393	0.631966	+	0.774996i	$0.282248\pi$
$578$	0	0
$579$	−3.00000	−0.124676
$580$	0	0
$581$	3.88854	0.161324
$582$	0	0
$583$	−6.83282	−0.282986
$584$	0	0
$585$	−5.23607	−0.216485
$586$	0	0
$587$	34.4721	1.42282	0.711409	−	0.702779i	$-0.248058\pi$
$587$	34.4721	1.42282	0.711409	+	0.702779i	$0.248058\pi$
$588$	0	0
$589$	−6.23607	−0.256953
$590$	0	0
$591$	−5.23607	−0.215383
$592$	0	0
$593$	−5.29180	−0.217308	−0.108654	−	0.994080i	$-0.534654\pi$
$593$	−5.29180	−0.217308	−0.108654	+	0.994080i	$0.534654\pi$
$594$	0	0
$595$	−6.47214	−0.265332
$596$	0	0
$597$	−12.1803	−0.498508
$598$	0	0
$599$	−37.9443	−1.55036	−0.775180	−	0.631740i	$-0.782341\pi$
$599$	−37.9443	−1.55036	−0.775180	+	0.631740i	$0.782341\pi$
$600$	0	0
$601$	−20.5410	−0.837886	−0.418943	−	0.908013i	$-0.637599\pi$
$601$	−20.5410	−0.837886	−0.418943	+	0.908013i	$0.637599\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	−44.1246	−1.79392
$606$	0	0
$607$	26.8328	1.08911	0.544555	−	0.838725i	$-0.316698\pi$
$607$	26.8328	1.08911	0.544555	+	0.838725i	$0.316698\pi$
$608$	0	0
$609$	−0.763932	−0.0309561
$610$	0	0
$611$	3.05573	0.123622
$612$	0	0
$613$	17.4164	0.703442	0.351721	−	0.936105i	$-0.385597\pi$
$613$	17.4164	0.703442	0.351721	+	0.936105i	$0.385597\pi$
$614$	0	0
$615$	28.4164	1.14586
$616$	0	0
$617$	13.4164	0.540124	0.270062	−	0.962843i	$-0.412956\pi$
$617$	13.4164	0.540124	0.270062	+	0.962843i	$0.412956\pi$
$618$	0	0
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	0	0
$621$	−1.23607	−0.0496017
$622$	0	0
$623$	1.63932	0.0656780
$624$	0	0
$625$	77.8328	3.11331
$626$	0	0
$627$	4.76393	0.190253
$628$	0	0
$629$	−36.9443	−1.47306
$630$	0	0
$631$	−24.8328	−0.988579	−0.494289	−	0.869297i	$-0.664572\pi$
$631$	−24.8328	−0.988579	−0.494289	+	0.869297i	$0.664572\pi$
$632$	0	0
$633$	−0.708204	−0.0281486
$634$	0	0
$635$	30.6525	1.21641
$636$	0	0
$637$	8.58359	0.340094
$638$	0	0
$639$	9.00000	0.356034
$640$	0	0
$641$	4.58359	0.181041	0.0905205	−	0.995895i	$-0.471147\pi$
$641$	4.58359	0.181041	0.0905205	+	0.995895i	$0.471147\pi$
$642$	0	0
$643$	44.4721	1.75381	0.876905	−	0.480664i	$-0.159604\pi$
$643$	44.4721	1.75381	0.876905	+	0.480664i	$0.159604\pi$
$644$	0	0
$645$	41.1246	1.61928
$646$	0	0
$647$	9.59675	0.377287	0.188644	−	0.982046i	$-0.439591\pi$
$647$	9.59675	0.377287	0.188644	+	0.982046i	$0.439591\pi$
$648$	0	0
$649$	2.29180	0.0899609
$650$	0	0
$651$	−0.236068	−0.00925223
$652$	0	0
$653$	23.8885	0.934831	0.467415	−	0.884038i	$-0.345185\pi$
$653$	23.8885	0.934831	0.467415	+	0.884038i	$0.345185\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.23607	0.126251
$658$	0	0
$659$	44.3050	1.72588	0.862938	−	0.505310i	$-0.168622\pi$
$659$	44.3050	1.72588	0.862938	+	0.505310i	$0.168622\pi$
$660$	0	0
$661$	−9.47214	−0.368423	−0.184212	−	0.982887i	$-0.558973\pi$
$661$	−9.47214	−0.368423	−0.184212	+	0.982887i	$0.558973\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	6.23607	0.241824
$666$	0	0
$667$	4.00000	0.154881
$668$	0	0
$669$	−1.81966	−0.0703521
$670$	0	0
$671$	−6.11146	−0.235930
$672$	0	0
$673$	−33.1246	−1.27686	−0.638430	−	0.769680i	$-0.720416\pi$
$673$	−33.1246	−1.27686	−0.638430	+	0.769680i	$0.720416\pi$
$674$	0	0
$675$	12.9443	0.498225
$676$	0	0
$677$	10.6525	0.409408	0.204704	−	0.978824i	$-0.434377\pi$
$677$	10.6525	0.409408	0.204704	+	0.978824i	$0.434377\pi$
$678$	0	0
$679$	2.12461	0.0815351
$680$	0	0
$681$	−4.94427	−0.189465
$682$	0	0
$683$	26.8885	1.02886	0.514431	−	0.857532i	$-0.328003\pi$
$683$	26.8885	1.02886	0.514431	+	0.857532i	$0.328003\pi$
$684$	0	0
$685$	−43.1246	−1.64771
$686$	0	0
$687$	−9.70820	−0.370391
$688$	0	0
$689$	11.0557	0.421190
$690$	0	0
$691$	22.7082	0.863861	0.431930	−	0.901907i	$-0.357833\pi$
$691$	22.7082	0.863861	0.431930	+	0.901907i	$0.357833\pi$
$692$	0	0
$693$	0.180340	0.00685055
$694$	0	0
$695$	−24.1803	−0.917213
$696$	0	0
$697$	−43.4164	−1.64451
$698$	0	0
$699$	0.708204	0.0267867
$700$	0	0
$701$	24.7082	0.933216	0.466608	−	0.884464i	$-0.345476\pi$
$701$	24.7082	0.933216	0.466608	+	0.884464i	$0.345476\pi$
$702$	0	0
$703$	35.5967	1.34256
$704$	0	0
$705$	−10.4721	−0.394403
$706$	0	0
$707$	1.36068	0.0511736
$708$	0	0
$709$	38.8328	1.45840	0.729199	−	0.684302i	$-0.239893\pi$
$709$	38.8328	1.45840	0.729199	+	0.684302i	$0.239893\pi$
$710$	0	0
$711$	8.47214	0.317730
$712$	0	0
$713$	1.23607	0.0462911
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−11.8885	−0.443986
$718$	0	0
$719$	53.1935	1.98378	0.991891	−	0.127089i	$-0.0405634\pi$
$719$	53.1935	1.98378	0.991891	+	0.127089i	$0.0405634\pi$
$720$	0	0
$721$	−3.24922	−0.121007
$722$	0	0
$723$	−13.7082	−0.509814
$724$	0	0
$725$	−41.8885	−1.55570
$726$	0	0
$727$	24.1246	0.894732	0.447366	−	0.894351i	$-0.352362\pi$
$727$	24.1246	0.894732	0.447366	+	0.894351i	$0.352362\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−62.8328	−2.32396
$732$	0	0
$733$	7.00000	0.258551	0.129275	−	0.991609i	$-0.458735\pi$
$733$	7.00000	0.258551	0.129275	+	0.991609i	$0.458735\pi$
$734$	0	0
$735$	−29.4164	−1.08504
$736$	0	0
$737$	9.16718	0.337678
$738$	0	0
$739$	−43.2361	−1.59046	−0.795232	−	0.606305i	$-0.792651\pi$
$739$	−43.2361	−1.59046	−0.795232	+	0.606305i	$0.792651\pi$
$740$	0	0
$741$	−7.70820	−0.283168
$742$	0	0
$743$	−39.2361	−1.43943	−0.719716	−	0.694269i	$-0.755728\pi$
$743$	−39.2361	−1.43943	−0.719716	+	0.694269i	$0.755728\pi$
$744$	0	0
$745$	56.8328	2.08219
$746$	0	0
$747$	16.4721	0.602684
$748$	0	0
$749$	−0.347524	−0.0126983
$750$	0	0
$751$	0.708204	0.0258427	0.0129214	−	0.999917i	$-0.495887\pi$
$751$	0.708204	0.0258427	0.0129214	+	0.999917i	$0.495887\pi$
$752$	0	0
$753$	11.2361	0.409465
$754$	0	0
$755$	64.5410	2.34889
$756$	0	0
$757$	−7.59675	−0.276108	−0.138054	−	0.990425i	$-0.544085\pi$
$757$	−7.59675	−0.276108	−0.138054	+	0.990425i	$0.544085\pi$
$758$	0	0
$759$	−0.944272	−0.0342749
$760$	0	0
$761$	−32.7639	−1.18769	−0.593846	−	0.804579i	$-0.702391\pi$
$761$	−32.7639	−1.18769	−0.593846	+	0.804579i	$0.702391\pi$
$762$	0	0
$763$	1.18034	0.0427312
$764$	0	0
$765$	−27.4164	−0.991242
$766$	0	0
$767$	−3.70820	−0.133895
$768$	0	0
$769$	25.5836	0.922568	0.461284	−	0.887253i	$-0.347389\pi$
$769$	25.5836	0.922568	0.461284	+	0.887253i	$0.347389\pi$
$770$	0	0
$771$	5.29180	0.190579
$772$	0	0
$773$	−28.7639	−1.03457	−0.517283	−	0.855814i	$-0.673057\pi$
$773$	−28.7639	−1.03457	−0.517283	+	0.855814i	$0.673057\pi$
$774$	0	0
$775$	−12.9443	−0.464972
$776$	0	0
$777$	1.34752	0.0483422
$778$	0	0
$779$	41.8328	1.49882
$780$	0	0
$781$	6.87539	0.246021
$782$	0	0
$783$	−3.23607	−0.115648
$784$	0	0
$785$	17.1803	0.613193
$786$	0	0
$787$	40.8328	1.45553	0.727766	−	0.685825i	$-0.240559\pi$
$787$	40.8328	1.45553	0.727766	+	0.685825i	$0.240559\pi$
$788$	0	0
$789$	18.4721	0.657625
$790$	0	0
$791$	−0.416408	−0.0148058
$792$	0	0
$793$	9.88854	0.351152
$794$	0	0
$795$	−37.8885	−1.34377
$796$	0	0
$797$	−23.7771	−0.842228	−0.421114	−	0.907008i	$-0.638361\pi$
$797$	−23.7771	−0.842228	−0.421114	+	0.907008i	$0.638361\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	6.94427	0.245364
$802$	0	0
$803$	2.47214	0.0872398
$804$	0	0
$805$	−1.23607	−0.0435657
$806$	0	0
$807$	12.6525	0.445388
$808$	0	0
$809$	−53.0132	−1.86384	−0.931922	−	0.362660i	$-0.881869\pi$
$809$	−53.0132	−1.86384	−0.931922	+	0.362660i	$0.881869\pi$
$810$	0	0
$811$	46.8328	1.64452	0.822261	−	0.569110i	$-0.192712\pi$
$811$	46.8328	1.64452	0.822261	+	0.569110i	$0.192712\pi$
$812$	0	0
$813$	−13.4164	−0.470534
$814$	0	0
$815$	30.4164	1.06544
$816$	0	0
$817$	60.5410	2.11806
$818$	0	0
$819$	−0.291796	−0.0101962
$820$	0	0
$821$	−55.3050	−1.93016	−0.965078	−	0.261962i	$-0.915630\pi$
$821$	−55.3050	−1.93016	−0.965078	+	0.261962i	$0.915630\pi$
$822$	0	0
$823$	18.8328	0.656471	0.328235	−	0.944596i	$-0.393546\pi$
$823$	18.8328	0.656471	0.328235	+	0.944596i	$0.393546\pi$
$824$	0	0
$825$	9.88854	0.344275
$826$	0	0
$827$	0.875388	0.0304402	0.0152201	−	0.999884i	$-0.495155\pi$
$827$	0.875388	0.0304402	0.0152201	+	0.999884i	$0.495155\pi$
$828$	0	0
$829$	5.41641	0.188120	0.0940598	−	0.995567i	$-0.470016\pi$
$829$	5.41641	0.188120	0.0940598	+	0.995567i	$0.470016\pi$
$830$	0	0
$831$	8.47214	0.293895
$832$	0	0
$833$	44.9443	1.55723
$834$	0	0
$835$	−70.5410	−2.44117
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−20.9443	−0.723077	−0.361538	−	0.932357i	$-0.617748\pi$
$839$	−20.9443	−0.723077	−0.361538	+	0.932357i	$0.617748\pi$
$840$	0	0
$841$	−18.5279	−0.638892
$842$	0	0
$843$	−28.2361	−0.972502
$844$	0	0
$845$	−48.5967	−1.67178
$846$	0	0
$847$	−2.45898	−0.0844916
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	−7.05573	−0.241867
$852$	0	0
$853$	23.5279	0.805579	0.402789	−	0.915293i	$-0.368041\pi$
$853$	23.5279	0.805579	0.402789	+	0.915293i	$0.368041\pi$
$854$	0	0
$855$	26.4164	0.903422
$856$	0	0
$857$	−7.30495	−0.249532	−0.124766	−	0.992186i	$-0.539818\pi$
$857$	−7.30495	−0.249532	−0.124766	+	0.992186i	$0.539818\pi$
$858$	0	0
$859$	−22.1803	−0.756783	−0.378392	−	0.925646i	$-0.623523\pi$
$859$	−22.1803	−0.756783	−0.378392	+	0.925646i	$0.623523\pi$
$860$	0	0
$861$	1.58359	0.0539687
$862$	0	0
$863$	−11.3050	−0.384825	−0.192413	−	0.981314i	$-0.561631\pi$
$863$	−11.3050	−0.384825	−0.192413	+	0.981314i	$0.561631\pi$
$864$	0	0
$865$	−29.4164	−1.00019
$866$	0	0
$867$	24.8885	0.845259
$868$	0	0
$869$	6.47214	0.219552
$870$	0	0
$871$	−14.8328	−0.502591
$872$	0	0
$873$	9.00000	0.304604
$874$	0	0
$875$	7.94427	0.268565
$876$	0	0
$877$	29.8328	1.00738	0.503691	−	0.863884i	$-0.331975\pi$
$877$	29.8328	1.00738	0.503691	+	0.863884i	$0.331975\pi$
$878$	0	0
$879$	−14.3607	−0.484374
$880$	0	0
$881$	45.3050	1.52636	0.763181	−	0.646184i	$-0.223636\pi$
$881$	45.3050	1.52636	0.763181	+	0.646184i	$0.223636\pi$
$882$	0	0
$883$	−26.8328	−0.902996	−0.451498	−	0.892272i	$-0.649110\pi$
$883$	−26.8328	−0.902996	−0.451498	+	0.892272i	$0.649110\pi$
$884$	0	0
$885$	12.7082	0.427182
$886$	0	0
$887$	19.5836	0.657553	0.328776	−	0.944408i	$-0.393364\pi$
$887$	19.5836	0.657553	0.328776	+	0.944408i	$0.393364\pi$
$888$	0	0
$889$	1.70820	0.0572913
$890$	0	0
$891$	0.763932	0.0255927
$892$	0	0
$893$	−15.4164	−0.515890
$894$	0	0
$895$	−82.2492	−2.74929
$896$	0	0
$897$	1.52786	0.0510139
$898$	0	0
$899$	3.23607	0.107929
$900$	0	0
$901$	57.8885	1.92855
$902$	0	0
$903$	2.29180	0.0762662
$904$	0	0
$905$	−82.2492	−2.73406
$906$	0	0
$907$	−42.9574	−1.42638	−0.713189	−	0.700972i	$-0.752750\pi$
$907$	−42.9574	−1.42638	−0.713189	+	0.700972i	$0.752750\pi$
$908$	0	0
$909$	5.76393	0.191178
$910$	0	0
$911$	−20.1803	−0.668604	−0.334302	−	0.942466i	$-0.608501\pi$
$911$	−20.1803	−0.668604	−0.334302	+	0.942466i	$0.608501\pi$
$912$	0	0
$913$	12.5836	0.416456
$914$	0	0
$915$	−33.8885	−1.12032
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	10.2361	0.337290
$922$	0	0
$923$	−11.1246	−0.366171
$924$	0	0
$925$	73.8885	2.42944
$926$	0	0
$927$	−13.7639	−0.452067
$928$	0	0
$929$	−17.2361	−0.565497	−0.282749	−	0.959194i	$-0.591246\pi$
$929$	−17.2361	−0.565497	−0.282749	+	0.959194i	$0.591246\pi$
$930$	0	0
$931$	−43.3050	−1.41926
$932$	0	0
$933$	−5.47214	−0.179150
$934$	0	0
$935$	−20.9443	−0.684951
$936$	0	0
$937$	−32.8328	−1.07260	−0.536301	−	0.844027i	$-0.680179\pi$
$937$	−32.8328	−1.07260	−0.536301	+	0.844027i	$0.680179\pi$
$938$	0	0
$939$	−6.47214	−0.211210
$940$	0	0
$941$	26.9443	0.878358	0.439179	−	0.898400i	$-0.355269\pi$
$941$	26.9443	0.878358	0.439179	+	0.898400i	$0.355269\pi$
$942$	0	0
$943$	−8.29180	−0.270018
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	7.81966	0.254105	0.127052	−	0.991896i	$-0.459448\pi$
$947$	7.81966	0.254105	0.127052	+	0.991896i	$0.459448\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−32.1246	−1.04171
$952$	0	0
$953$	−0.944272	−0.0305880	−0.0152940	−	0.999883i	$-0.504868\pi$
$953$	−0.944272	−0.0305880	−0.0152940	+	0.999883i	$0.504868\pi$
$954$	0	0
$955$	−59.0689	−1.91142
$956$	0	0
$957$	−2.47214	−0.0799128
$958$	0	0
$959$	−2.40325	−0.0776051
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−1.47214	−0.0474389
$964$	0	0
$965$	−12.7082	−0.409092
$966$	0	0
$967$	−50.5410	−1.62529	−0.812645	−	0.582759i	$-0.801973\pi$
$967$	−50.5410	−1.62529	−0.812645	+	0.582759i	$0.801973\pi$
$968$	0	0
$969$	−40.3607	−1.29657
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	−1.34752	−0.0431996
$974$	0	0
$975$	−16.0000	−0.512410
$976$	0	0
$977$	11.2918	0.361257	0.180628	−	0.983551i	$-0.442187\pi$
$977$	11.2918	0.361257	0.180628	+	0.983551i	$0.442187\pi$
$978$	0	0
$979$	5.30495	0.169547
$980$	0	0
$981$	5.00000	0.159638
$982$	0	0
$983$	−28.7639	−0.917427	−0.458713	−	0.888584i	$-0.651690\pi$
$983$	−28.7639	−0.917427	−0.458713	+	0.888584i	$0.651690\pi$
$984$	0	0
$985$	−22.1803	−0.706724
$986$	0	0
$987$	−0.583592	−0.0185759
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	36.7214	1.16649	0.583246	−	0.812295i	$-0.301782\pi$
$991$	36.7214	1.16649	0.583246	+	0.812295i	$0.301782\pi$
$992$	0	0
$993$	−20.8328	−0.661109
$994$	0	0
$995$	−51.5967	−1.63573
$996$	0	0
$997$	4.41641	0.139869	0.0699345	−	0.997552i	$-0.477721\pi$
$997$	4.41641	0.139869	0.0699345	+	0.997552i	$0.477721\pi$
$998$	0	0
$999$	5.70820	0.180600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5952.2.a.bv.1.2		2
4.3	odd	2		5952.2.a.bo.1.2		2
8.3	odd	2		1488.2.a.q.1.1		2
8.5	even	2		93.2.a.a.1.2	✓	2
24.5	odd	2		279.2.a.b.1.1		2
24.11	even	2		4464.2.a.bn.1.2		2
40.13	odd	4		2325.2.c.h.1024.3		4
40.29	even	2		2325.2.a.o.1.1		2
40.37	odd	4		2325.2.c.h.1024.2		4
56.13	odd	2		4557.2.a.p.1.2		2
120.29	odd	2		6975.2.a.t.1.2		2
248.61	odd	2		2883.2.a.a.1.2		2
744.557	even	2		8649.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.a.a.1.2	✓	2	8.5	even	2
279.2.a.b.1.1		2	24.5	odd	2
1488.2.a.q.1.1		2	8.3	odd	2
2325.2.a.o.1.1		2	40.29	even	2
2325.2.c.h.1024.2		4	40.37	odd	4
2325.2.c.h.1024.3		4	40.13	odd	4
2883.2.a.a.1.2		2	248.61	odd	2
4464.2.a.bn.1.2		2	24.11	even	2
4557.2.a.p.1.2		2	56.13	odd	2
5952.2.a.bo.1.2		2	4.3	odd	2
5952.2.a.bv.1.2		2	1.1	even	1	trivial
6975.2.a.t.1.2		2	120.29	odd	2
8649.2.a.m.1.1		2	744.557	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 5952 = 2^{6} \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5952.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.23607 q^{5} +0.236068 q^{7} +1.00000 q^{9} +0.763932 q^{11} -1.23607 q^{13} +4.23607 q^{15} -6.47214 q^{17} +6.23607 q^{19} +0.236068 q^{21} -1.23607 q^{23} +12.9443 q^{25} +1.00000 q^{27} -3.23607 q^{29} -1.00000 q^{31} +0.763932 q^{33} +1.00000 q^{35} +5.70820 q^{37} -1.23607 q^{39} +6.70820 q^{41} +9.70820 q^{43} +4.23607 q^{45} -2.47214 q^{47} -6.94427 q^{49} -6.47214 q^{51} -8.94427 q^{53} +3.23607 q^{55} +6.23607 q^{57} +3.00000 q^{59} -8.00000 q^{61} +0.236068 q^{63} -5.23607 q^{65} +12.0000 q^{67} -1.23607 q^{69} +9.00000 q^{71} +3.23607 q^{73} +12.9443 q^{75} +0.180340 q^{77} +8.47214 q^{79} +1.00000 q^{81} +16.4721 q^{83} -27.4164 q^{85} -3.23607 q^{87} +6.94427 q^{89} -0.291796 q^{91} -1.00000 q^{93} +26.4164 q^{95} +9.00000 q^{97} +0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{13} + 4 q^{15} - 4 q^{17} + 8 q^{19} - 4 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} - 2 q^{29} - 2 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^13 + 4 * q^15 - 4 * q^17 + 8 * q^19 - 4 * q^21 + 2 * q^23 + 8 * q^25 + 2 * q^27 - 2 * q^29 - 2 * q^31 + 6 * q^33 + 2 * q^35 - 2 * q^37 + 2 * q^39 + 6 * q^43 + 4 * q^45 + 4 * q^47 + 4 * q^49 - 4 * q^51 + 2 * q^55 + 8 * q^57 + 6 * q^59 - 16 * q^61 - 4 * q^63 - 6 * q^65 + 24 * q^67 + 2 * q^69 + 18 * q^71 + 2 * q^73 + 8 * q^75 - 22 * q^77 + 8 * q^79 + 2 * q^81 + 24 * q^83 - 28 * q^85 - 2 * q^87 - 4 * q^89 - 14 * q^91 - 2 * q^93 + 26 * q^95 + 18 * q^97 + 6 * q^99

Introduction

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