LMFDB - Embedded newform 279.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [279,2,Mod(1,279)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(279, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("279.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 279 = 3^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	279.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:	$2.22782621639$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 93)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	279.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.381966 q^{2} -1.85410 q^{4} +4.23607 q^{5} +0.236068 q^{7} -1.47214 q^{8} +1.61803 q^{10} +0.763932 q^{11} +1.23607 q^{13} +0.0901699 q^{14} +3.14590 q^{16} +6.47214 q^{17} -6.23607 q^{19} -7.85410 q^{20} +0.291796 q^{22} +1.23607 q^{23} +12.9443 q^{25} +0.472136 q^{26} -0.437694 q^{28} -3.23607 q^{29} -1.00000 q^{31} +4.14590 q^{32} +2.47214 q^{34} +1.00000 q^{35} -5.70820 q^{37} -2.38197 q^{38} -6.23607 q^{40} -6.70820 q^{41} -9.70820 q^{43} -1.41641 q^{44} +0.472136 q^{46} +2.47214 q^{47} -6.94427 q^{49} +4.94427 q^{50} -2.29180 q^{52} -8.94427 q^{53} +3.23607 q^{55} -0.347524 q^{56} -1.23607 q^{58} +3.00000 q^{59} +8.00000 q^{61} -0.381966 q^{62} -4.70820 q^{64} +5.23607 q^{65} -12.0000 q^{67} -12.0000 q^{68} +0.381966 q^{70} -9.00000 q^{71} +3.23607 q^{73} -2.18034 q^{74} +11.5623 q^{76} +0.180340 q^{77} +8.47214 q^{79} +13.3262 q^{80} -2.56231 q^{82} +16.4721 q^{83} +27.4164 q^{85} -3.70820 q^{86} -1.12461 q^{88} -6.94427 q^{89} +0.291796 q^{91} -2.29180 q^{92} +0.944272 q^{94} -26.4164 q^{95} +9.00000 q^{97} -2.65248 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} + q^{10} + 6 q^{11} - 2 q^{13} - 11 q^{14} + 13 q^{16} + 4 q^{17} - 8 q^{19} - 9 q^{20} + 14 q^{22} - 2 q^{23} + 8 q^{25} - 8 q^{26} - 21 q^{28} - 2 q^{29}+ \cdots + 26 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8 + q^10 + 6 * q^11 - 2 * q^13 - 11 * q^14 + 13 * q^16 + 4 * q^17 - 8 * q^19 - 9 * q^20 + 14 * q^22 - 2 * q^23 + 8 * q^25 - 8 * q^26 - 21 * q^28 - 2 * q^29 - 2 * q^31 + 15 * q^32 - 4 * q^34 + 2 * q^35 + 2 * q^37 - 7 * q^38 - 8 * q^40 - 6 * q^43 + 24 * q^44 - 8 * q^46 - 4 * q^47 + 4 * q^49 - 8 * q^50 - 18 * q^52 + 2 * q^55 - 32 * q^56 + 2 * q^58 + 6 * q^59 + 16 * q^61 - 3 * q^62 + 4 * q^64 + 6 * q^65 - 24 * q^67 - 24 * q^68 + 3 * q^70 - 18 * q^71 + 2 * q^73 + 18 * q^74 + 3 * q^76 - 22 * q^77 + 8 * q^79 + 11 * q^80 + 15 * q^82 + 24 * q^83 + 28 * q^85 + 6 * q^86 + 38 * q^88 + 4 * q^89 + 14 * q^91 - 18 * q^92 - 16 * q^94 - 26 * q^95 + 18 * q^97 + 26 * q^98

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 $

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	0	0
$3$	0	0
$4$	−1.85410	−0.927051
$5$	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	$0.485795\pi$
$8$	−1.47214	−0.520479
$9$	0	0
$10$	1.61803	0.511667
$11$	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	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0.0901699	0.0240989
$15$	0	0
$16$	3.14590	0.786475
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	−6.23607	−1.43065	−0.715326	−	0.698791i	$-0.753722\pi$
$19$	−6.23607	−1.43065	−0.715326	+	0.698791i	$0.753722\pi$
$20$	−7.85410	−1.75623
$21$	0	0
$22$	0.291796	0.0622111
$23$	1.23607	0.257738	0.128869	−	0.991662i	$-0.458865\pi$
$23$	1.23607	0.257738	0.128869	+	0.991662i	$0.458865\pi$
$24$	0	0
$25$	12.9443	2.58885
$26$	0.472136	0.0925935
$27$	0	0
$28$	−0.437694	−0.0827164
$29$	−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
$32$	4.14590	0.732898
$33$	0	0
$34$	2.47214	0.423968
$35$	1.00000	0.169031
$36$	0	0
$37$	−5.70820	−0.938423	−0.469211	−	0.883086i	$-0.655462\pi$
$37$	−5.70820	−0.938423	−0.469211	+	0.883086i	$0.655462\pi$
$38$	−2.38197	−0.386406
$39$	0	0
$40$	−6.23607	−0.986009
$41$	−6.70820	−1.04765	−0.523823	−	0.851827i	$-0.675495\pi$
$41$	−6.70820	−1.04765	−0.523823	+	0.851827i	$0.675495\pi$
$42$	0	0
$43$	−9.70820	−1.48049	−0.740244	−	0.672339i	$-0.765290\pi$
$43$	−9.70820	−1.48049	−0.740244	+	0.672339i	$0.765290\pi$
$44$	−1.41641	−0.213532
$45$	0	0
$46$	0.472136	0.0696126
$47$	2.47214	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	4.94427	0.699226
$51$	0	0
$52$	−2.29180	−0.317815
$53$	−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
$56$	−0.347524	−0.0464399
$57$	0	0
$58$	−1.23607	−0.162304
$59$	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	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−0.381966	−0.0485097
$63$	0	0
$64$	−4.70820	−0.588525
$65$	5.23607	0.649454
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−12.0000	−1.45521
$69$	0	0
$70$	0.381966	0.0456537
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	3.23607	0.378753	0.189377	−	0.981905i	$-0.439353\pi$
$73$	3.23607	0.378753	0.189377	+	0.981905i	$0.439353\pi$
$74$	−2.18034	−0.253459
$75$	0	0
$76$	11.5623	1.32629
$77$	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	$0.341871\pi$
$80$	13.3262	1.48992
$81$	0	0
$82$	−2.56231	−0.282959
$83$	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
$86$	−3.70820	−0.399866
$87$	0	0
$88$	−1.12461	−0.119884
$89$	−6.94427	−0.736091	−0.368046	−	0.929808i	$-0.619973\pi$
$89$	−6.94427	−0.736091	−0.368046	+	0.929808i	$0.619973\pi$
$90$	0	0
$91$	0.291796	0.0305885
$92$	−2.29180	−0.238936
$93$	0	0
$94$	0.944272	0.0973942
$95$	−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	$0.348958\pi$
$98$	−2.65248	−0.267941
$99$	0	0
$100$	−24.0000	−2.40000
$101$	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	$0.737196\pi$
$104$	−1.81966	−0.178432
$105$	0	0
$106$	−3.41641	−0.331831
$107$	−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	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	1.23607	0.117854
$111$	0	0
$112$	0.742646	0.0701734
$113$	1.76393	0.165937	0.0829684	−	0.996552i	$-0.473560\pi$
$113$	1.76393	0.165937	0.0829684	+	0.996552i	$0.473560\pi$
$114$	0	0
$115$	5.23607	0.488266
$116$	6.00000	0.557086
$117$	0	0
$118$	1.14590	0.105488
$119$	1.52786	0.140059
$120$	0	0
$121$	−10.4164	−0.946946
$122$	3.05573	0.276653
$123$	0	0
$124$	1.85410	0.166503
$125$	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	$0.395965\pi$
$128$	−10.0902	−0.891853
$129$	0	0
$130$	2.00000	0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−1.47214	−0.127650
$134$	−4.58359	−0.395962
$135$	0	0
$136$	−9.52786	−0.817008
$137$	10.1803	0.869765	0.434883	−	0.900487i	$-0.356790\pi$
$137$	10.1803	0.869765	0.434883	+	0.900487i	$0.356790\pi$
$138$	0	0
$139$	5.70820	0.484164	0.242082	−	0.970256i	$-0.422170\pi$
$139$	5.70820	0.484164	0.242082	+	0.970256i	$0.422170\pi$
$140$	−1.85410	−0.156700
$141$	0	0
$142$	−3.43769	−0.288485
$143$	0.944272	0.0789640
$144$	0	0
$145$	−13.7082	−1.13840
$146$	1.23607	0.102298
$147$	0	0
$148$	10.5836	0.869966
$149$	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	$0.287154\pi$
$152$	9.18034	0.744624
$153$	0	0
$154$	0.0688837	0.00555081
$155$	−4.23607	−0.340249
$156$	0	0
$157$	−4.05573	−0.323682	−0.161841	−	0.986817i	$-0.551743\pi$
$157$	−4.05573	−0.323682	−0.161841	+	0.986817i	$0.551743\pi$
$158$	3.23607	0.257448
$159$	0	0
$160$	17.5623	1.38842
$161$	0.291796	0.0229968
$162$	0	0
$163$	−7.18034	−0.562408	−0.281204	−	0.959648i	$-0.590734\pi$
$163$	−7.18034	−0.562408	−0.281204	+	0.959648i	$0.590734\pi$
$164$	12.4377	0.971221
$165$	0	0
$166$	6.29180	0.488338
$167$	16.6525	1.28861	0.644304	−	0.764770i	$-0.277147\pi$
$167$	16.6525	1.28861	0.644304	+	0.764770i	$0.277147\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	10.4721	0.803176
$171$	0	0
$172$	18.0000	1.37249
$173$	−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
$176$	2.40325	0.181152
$177$	0	0
$178$	−2.65248	−0.198811
$179$	−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	$-0.243405\pi$
$181$	19.4164	1.44321	0.721605	+	0.692305i	$0.243405\pi$
$182$	0.111456	0.00826168
$183$	0	0
$184$	−1.81966	−0.134147
$185$	−24.1803	−1.77777
$186$	0	0
$187$	4.94427	0.361561
$188$	−4.58359	−0.334293
$189$	0	0
$190$	−10.0902	−0.732018
$191$	13.9443	1.00897	0.504486	−	0.863420i	$-0.331682\pi$
$191$	13.9443	1.00897	0.504486	+	0.863420i	$0.331682\pi$
$192$	0	0
$193$	−3.00000	−0.215945	−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000	−0.215945	−0.107972	+	0.994154i	$0.534436\pi$
$194$	3.43769	0.246812
$195$	0	0
$196$	12.8754	0.919671
$197$	−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	$0.642093\pi$
$200$	−19.0557	−1.34744
$201$	0	0
$202$	2.20163	0.154906
$203$	−0.763932	−0.0536175
$204$	0	0
$205$	−28.4164	−1.98469
$206$	−5.25735	−0.366297
$207$	0	0
$208$	3.88854	0.269622
$209$	−4.76393	−0.329528
$210$	0	0
$211$	0.708204	0.0487548	0.0243774	−	0.999703i	$-0.492240\pi$
$211$	0.708204	0.0487548	0.0243774	+	0.999703i	$0.492240\pi$
$212$	16.5836	1.13897
$213$	0	0
$214$	−0.562306	−0.0384384
$215$	−41.1246	−2.80468
$216$	0	0
$217$	−0.236068	−0.0160253
$218$	−1.90983	−0.129350
$219$	0	0
$220$	−6.00000	−0.404520
$221$	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	$0.519406\pi$
$224$	0.978714	0.0653931
$225$	0	0
$226$	0.673762	0.0448180
$227$	−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	$-0.396059\pi$
$229$	9.70820	0.641536	0.320768	+	0.947158i	$0.396059\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	4.76393	0.312767
$233$	−0.708204	−0.0463960	−0.0231980	−	0.999731i	$-0.507385\pi$
$233$	−0.708204	−0.0463960	−0.0231980	+	0.999731i	$0.507385\pi$
$234$	0	0
$235$	10.4721	0.683127
$236$	−5.56231	−0.362075
$237$	0	0
$238$	0.583592	0.0378287
$239$	11.8885	0.769006	0.384503	−	0.923124i	$-0.374373\pi$
$239$	11.8885	0.769006	0.384503	+	0.923124i	$0.374373\pi$
$240$	0	0
$241$	−13.7082	−0.883023	−0.441512	−	0.897256i	$-0.645558\pi$
$241$	−13.7082	−0.883023	−0.441512	+	0.897256i	$0.645558\pi$
$242$	−3.97871	−0.255761
$243$	0	0
$244$	−14.8328	−0.949574
$245$	−29.4164	−1.87935
$246$	0	0
$247$	−7.70820	−0.490461
$248$	1.47214	0.0934807
$249$	0	0
$250$	12.8541	0.812965
$251$	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
$254$	2.76393	0.173425
$255$	0	0
$256$	5.56231	0.347644
$257$	−5.29180	−0.330093	−0.165047	−	0.986286i	$-0.552777\pi$
$257$	−5.29180	−0.330093	−0.165047	+	0.986286i	$0.552777\pi$
$258$	0	0
$259$	−1.34752	−0.0837311
$260$	−9.70820	−0.602077
$261$	0	0
$262$	0	0
$263$	−18.4721	−1.13904	−0.569520	−	0.821977i	$-0.692871\pi$
$263$	−18.4721	−1.13904	−0.569520	+	0.821977i	$0.692871\pi$
$264$	0	0
$265$	−37.8885	−2.32747
$266$	−0.562306	−0.0344772
$267$	0	0
$268$	22.2492	1.35909
$269$	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	$0.633597\pi$
$272$	20.3607	1.23455
$273$	0	0
$274$	3.88854	0.234916
$275$	9.88854	0.596302
$276$	0	0
$277$	−8.47214	−0.509041	−0.254521	−	0.967067i	$-0.581918\pi$
$277$	−8.47214	−0.509041	−0.254521	+	0.967067i	$0.581918\pi$
$278$	2.18034	0.130768
$279$	0	0
$280$	−1.47214	−0.0879770
$281$	28.2361	1.68442	0.842211	−	0.539148i	$-0.181254\pi$
$281$	28.2361	1.68442	0.842211	+	0.539148i	$0.181254\pi$
$282$	0	0
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	16.6869	0.990186
$285$	0	0
$286$	0.360680	0.0213274
$287$	−1.58359	−0.0934765
$288$	0	0
$289$	24.8885	1.46403
$290$	−5.23607	−0.307472
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−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
$296$	8.40325	0.488429
$297$	0	0
$298$	5.12461	0.296861
$299$	1.52786	0.0883587
$300$	0	0
$301$	−2.29180	−0.132097
$302$	5.81966	0.334884
$303$	0	0
$304$	−19.6180	−1.12517
$305$	33.8885	1.94045
$306$	0	0
$307$	−10.2361	−0.584203	−0.292102	−	0.956387i	$-0.594355\pi$
$307$	−10.2361	−0.584203	−0.292102	+	0.956387i	$0.594355\pi$
$308$	−0.334369	−0.0190524
$309$	0	0
$310$	−1.61803	−0.0918982
$311$	5.47214	0.310296	0.155148	−	0.987891i	$-0.450414\pi$
$311$	5.47214	0.310296	0.155148	+	0.987891i	$0.450414\pi$
$312$	0	0
$313$	−6.47214	−0.365827	−0.182913	−	0.983129i	$-0.558553\pi$
$313$	−6.47214	−0.365827	−0.182913	+	0.983129i	$0.558553\pi$
$314$	−1.54915	−0.0874236
$315$	0	0
$316$	−15.7082	−0.883656
$317$	−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
$320$	−19.9443	−1.11492
$321$	0	0
$322$	0.111456	0.00621121
$323$	−40.3607	−2.24573
$324$	0	0
$325$	16.0000	0.887520
$326$	−2.74265	−0.151901
$327$	0	0
$328$	9.87539	0.545277
$329$	0.583592	0.0321745
$330$	0	0
$331$	20.8328	1.14508	0.572538	−	0.819878i	$-0.305959\pi$
$331$	20.8328	1.14508	0.572538	+	0.819878i	$0.305959\pi$
$332$	−30.5410	−1.67616
$333$	0	0
$334$	6.36068	0.348041
$335$	−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	$0.346219\pi$
$338$	−4.38197	−0.238348
$339$	0	0
$340$	−50.8328	−2.75680
$341$	−0.763932	−0.0413692
$342$	0	0
$343$	−3.29180	−0.177740
$344$	14.2918	0.770562
$345$	0	0
$346$	−2.65248	−0.142598
$347$	−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	$-0.231771\pi$
$349$	27.8885	1.49284	0.746420	+	0.665475i	$0.231771\pi$
$350$	1.16718	0.0623886
$351$	0	0
$352$	3.16718	0.168811
$353$	17.2361	0.917383	0.458692	−	0.888595i	$-0.348318\pi$
$353$	17.2361	0.917383	0.458692	+	0.888595i	$0.348318\pi$
$354$	0	0
$355$	−38.1246	−2.02344
$356$	12.8754	0.682394
$357$	0	0
$358$	−7.41641	−0.391969
$359$	8.88854	0.469119	0.234560	−	0.972102i	$-0.424635\pi$
$359$	8.88854	0.469119	0.234560	+	0.972102i	$0.424635\pi$
$360$	0	0
$361$	19.8885	1.04677
$362$	7.41641	0.389798
$363$	0	0
$364$	−0.541020	−0.0283571
$365$	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	$0.353150\pi$
$368$	3.88854	0.202704
$369$	0	0
$370$	−9.23607	−0.480160
$371$	−2.11146	−0.109621
$372$	0	0
$373$	7.58359	0.392664	0.196332	−	0.980538i	$-0.437097\pi$
$373$	7.58359	0.392664	0.196332	+	0.980538i	$0.437097\pi$
$374$	1.88854	0.0976543
$375$	0	0
$376$	−3.63932	−0.187684
$377$	−4.00000	−0.206010
$378$	0	0
$379$	17.5279	0.900346	0.450173	−	0.892941i	$-0.351362\pi$
$379$	17.5279	0.900346	0.450173	+	0.892941i	$0.351362\pi$
$380$	48.9787	2.51256
$381$	0	0
$382$	5.32624	0.272514
$383$	−2.94427	−0.150445	−0.0752226	−	0.997167i	$-0.523967\pi$
$383$	−2.94427	−0.150445	−0.0752226	+	0.997167i	$0.523967\pi$
$384$	0	0
$385$	0.763932	0.0389336
$386$	−1.14590	−0.0583247
$387$	0	0
$388$	−16.6869	−0.847150
$389$	−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
$392$	10.2229	0.516335
$393$	0	0
$394$	−2.00000	−0.100759
$395$	35.8885	1.80575
$396$	0	0
$397$	−30.4164	−1.52656	−0.763278	−	0.646070i	$-0.776411\pi$
$397$	−30.4164	−1.52656	−0.763278	+	0.646070i	$0.776411\pi$
$398$	−4.65248	−0.233208
$399$	0	0
$400$	40.7214	2.03607
$401$	−0.763932	−0.0381489	−0.0190745	−	0.999818i	$-0.506072\pi$
$401$	−0.763932	−0.0381489	−0.0190745	+	0.999818i	$0.506072\pi$
$402$	0	0
$403$	−1.23607	−0.0615729
$404$	−10.6869	−0.531694
$405$	0	0
$406$	−0.291796	−0.0144816
$407$	−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	$0.306277\pi$
$410$	−10.8541	−0.536046
$411$	0	0
$412$	25.5197	1.25727
$413$	0.708204	0.0348484
$414$	0	0
$415$	69.7771	3.42522
$416$	5.12461	0.251255
$417$	0	0
$418$	−1.81966	−0.0890025
$419$	−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	$-0.614262\pi$
$421$	−14.4164	−0.702613	−0.351306	+	0.936261i	$0.614262\pi$
$422$	0.270510	0.0131682
$423$	0	0
$424$	13.1672	0.639455
$425$	83.7771	4.06379
$426$	0	0
$427$	1.88854	0.0913930
$428$	2.72949	0.131935
$429$	0	0
$430$	−15.7082	−0.757517
$431$	6.47214	0.311752	0.155876	−	0.987777i	$-0.450180\pi$
$431$	6.47214	0.311752	0.155876	+	0.987777i	$0.450180\pi$
$432$	0	0
$433$	12.4721	0.599373	0.299686	−	0.954038i	$-0.403118\pi$
$433$	12.4721	0.599373	0.299686	+	0.954038i	$0.403118\pi$
$434$	−0.0901699	−0.00432830
$435$	0	0
$436$	9.27051	0.443977
$437$	−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	$0.582252\pi$
$440$	−4.76393	−0.227112
$441$	0	0
$442$	3.05573	0.145346
$443$	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
$446$	−0.695048	−0.0329115
$447$	0	0
$448$	−1.11146	−0.0525114
$449$	30.6525	1.44658	0.723290	−	0.690545i	$-0.242629\pi$
$449$	30.6525	1.44658	0.723290	+	0.690545i	$0.242629\pi$
$450$	0	0
$451$	−5.12461	−0.241309
$452$	−3.27051	−0.153832
$453$	0	0
$454$	−1.88854	−0.0886338
$455$	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	$0.416214\pi$
$458$	3.70820	0.173273
$459$	0	0
$460$	−9.70820	−0.452647
$461$	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	$0.695792\pi$
$464$	−10.1803	−0.472610
$465$	0	0
$466$	−0.270510	−0.0125311
$467$	−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
$470$	4.00000	0.184506
$471$	0	0
$472$	−4.41641	−0.203282
$473$	−7.41641	−0.341007
$474$	0	0
$475$	−80.7214	−3.70375
$476$	−2.83282	−0.129842
$477$	0	0
$478$	4.54102	0.207701
$479$	0.0557281	0.00254628	0.00127314	−	0.999999i	$-0.499595\pi$
$479$	0.0557281	0.00254628	0.00127314	+	0.999999i	$0.499595\pi$
$480$	0	0
$481$	−7.05573	−0.321714
$482$	−5.23607	−0.238496
$483$	0	0
$484$	19.3131	0.877867
$485$	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	$0.526775\pi$
$488$	−11.7771	−0.533124
$489$	0	0
$490$	−11.2361	−0.507594
$491$	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
$494$	−2.94427	−0.132469
$495$	0	0
$496$	−3.14590	−0.141255
$497$	−2.12461	−0.0953019
$498$	0	0
$499$	19.8885	0.890333	0.445167	−	0.895448i	$-0.353144\pi$
$499$	19.8885	0.890333	0.445167	+	0.895448i	$0.353144\pi$
$500$	−62.3951	−2.79039
$501$	0	0
$502$	4.29180	0.191552
$503$	−13.3607	−0.595723	−0.297862	−	0.954609i	$-0.596273\pi$
$503$	−13.3607	−0.595723	−0.297862	+	0.954609i	$0.596273\pi$
$504$	0	0
$505$	24.4164	1.08652
$506$	0.360680	0.0160342
$507$	0	0
$508$	−13.4164	−0.595257
$509$	−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
$512$	22.3050	0.985749
$513$	0	0
$514$	−2.02129	−0.0891551
$515$	−58.3050	−2.56922
$516$	0	0
$517$	1.88854	0.0830581
$518$	−0.514708	−0.0226150
$519$	0	0
$520$	−7.70820	−0.338027
$521$	27.5279	1.20602	0.603009	−	0.797735i	$-0.293968\pi$
$521$	27.5279	1.20602	0.603009	+	0.797735i	$0.293968\pi$
$522$	0	0
$523$	42.5410	1.86019	0.930094	−	0.367320i	$-0.119725\pi$
$523$	42.5410	1.86019	0.930094	+	0.367320i	$0.119725\pi$
$524$	0	0
$525$	0	0
$526$	−7.05573	−0.307644
$527$	−6.47214	−0.281931
$528$	0	0
$529$	−21.4721	−0.933571
$530$	−14.4721	−0.628629
$531$	0	0
$532$	2.72949	0.118338
$533$	−8.29180	−0.359158
$534$	0	0
$535$	−6.23607	−0.269609
$536$	17.6656	0.763039
$537$	0	0
$538$	4.83282	0.208357
$539$	−5.30495	−0.228500
$540$	0	0
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	−5.12461	−0.220121
$543$	0	0
$544$	26.8328	1.15045
$545$	−21.1803	−0.907266
$546$	0	0
$547$	−31.5410	−1.34860	−0.674298	−	0.738459i	$-0.735554\pi$
$547$	−31.5410	−1.34860	−0.674298	+	0.738459i	$0.735554\pi$
$548$	−18.8754	−0.806317
$549$	0	0
$550$	3.77709	0.161056
$551$	20.1803	0.859711
$552$	0	0
$553$	2.00000	0.0850487
$554$	−3.23607	−0.137487
$555$	0	0
$556$	−10.5836	−0.448844
$557$	−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
$560$	3.14590	0.132938
$561$	0	0
$562$	10.7852	0.454947
$563$	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
$566$	−3.05573	−0.128442
$567$	0	0
$568$	13.2492	0.555925
$569$	−29.8885	−1.25299	−0.626496	−	0.779424i	$-0.715512\pi$
$569$	−29.8885	−1.25299	−0.626496	+	0.779424i	$0.715512\pi$
$570$	0	0
$571$	23.7082	0.992157	0.496079	−	0.868278i	$-0.334773\pi$
$571$	23.7082	0.992157	0.496079	+	0.868278i	$0.334773\pi$
$572$	−1.75078	−0.0732036
$573$	0	0
$574$	−0.604878	−0.0252471
$575$	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	$0.282248\pi$
$578$	9.50658	0.395422
$579$	0	0
$580$	25.4164	1.05536
$581$	3.88854	0.161324
$582$	0	0
$583$	−6.83282	−0.282986
$584$	−4.76393	−0.197133
$585$	0	0
$586$	−5.48529	−0.226595
$587$	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
$590$	4.85410	0.199840
$591$	0	0
$592$	−17.9574	−0.738046
$593$	5.29180	0.217308	0.108654	−	0.994080i	$-0.465346\pi$
$593$	5.29180	0.217308	0.108654	+	0.994080i	$0.465346\pi$
$594$	0	0
$595$	6.47214	0.265332
$596$	−24.8754	−1.01894
$597$	0	0
$598$	0.583592	0.0238649
$599$	37.9443	1.55036	0.775180	−	0.631740i	$-0.217659\pi$
$599$	37.9443	1.55036	0.775180	+	0.631740i	$0.217659\pi$
$600$	0	0
$601$	−20.5410	−0.837886	−0.418943	−	0.908013i	$-0.637599\pi$
$601$	−20.5410	−0.837886	−0.418943	+	0.908013i	$0.637599\pi$
$602$	−0.875388	−0.0356782
$603$	0	0
$604$	−28.2492	−1.14944
$605$	−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	$0.316698\pi$
$608$	−25.8541	−1.04852
$609$	0	0
$610$	12.9443	0.524098
$611$	3.05573	0.123622
$612$	0	0
$613$	−17.4164	−0.703442	−0.351721	−	0.936105i	$-0.614403\pi$
$613$	−17.4164	−0.703442	−0.351721	+	0.936105i	$0.614403\pi$
$614$	−3.90983	−0.157788
$615$	0	0
$616$	−0.265485	−0.0106967
$617$	−13.4164	−0.540124	−0.270062	−	0.962843i	$-0.587044\pi$
$617$	−13.4164	−0.540124	−0.270062	+	0.962843i	$0.587044\pi$
$618$	0	0
$619$	6.00000	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$619$	6.00000	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$620$	7.85410	0.315428
$621$	0	0
$622$	2.09017	0.0838082
$623$	−1.63932	−0.0656780
$624$	0	0
$625$	77.8328	3.11331
$626$	−2.47214	−0.0988064
$627$	0	0
$628$	7.51973	0.300070
$629$	−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	$0.664572\pi$
$632$	−12.4721	−0.496115
$633$	0	0
$634$	−12.2705	−0.487324
$635$	30.6525	1.21641
$636$	0	0
$637$	−8.58359	−0.340094
$638$	−0.944272	−0.0373841
$639$	0	0
$640$	−42.7426	−1.68955
$641$	−4.58359	−0.181041	−0.0905205	−	0.995895i	$-0.528853\pi$
$641$	−4.58359	−0.181041	−0.0905205	+	0.995895i	$0.528853\pi$
$642$	0	0
$643$	−44.4721	−1.75381	−0.876905	−	0.480664i	$-0.840396\pi$
$643$	−44.4721	−1.75381	−0.876905	+	0.480664i	$0.840396\pi$
$644$	−0.541020	−0.0213192
$645$	0	0
$646$	−15.4164	−0.606550
$647$	−9.59675	−0.377287	−0.188644	−	0.982046i	$-0.560409\pi$
$647$	−9.59675	−0.377287	−0.188644	+	0.982046i	$0.560409\pi$
$648$	0	0
$649$	2.29180	0.0899609
$650$	6.11146	0.239711
$651$	0	0
$652$	13.3131	0.521381
$653$	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
$656$	−21.1033	−0.823946
$657$	0	0
$658$	0.222912	0.00869003
$659$	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	$-0.441027\pi$
$661$	9.47214	0.368423	0.184212	+	0.982887i	$0.441027\pi$
$662$	7.95743	0.309274
$663$	0	0
$664$	−24.2492	−0.941052
$665$	−6.23607	−0.241824
$666$	0	0
$667$	−4.00000	−0.154881
$668$	−30.8754	−1.19460
$669$	0	0
$670$	−19.4164	−0.750121
$671$	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	$0.720416\pi$
$674$	6.51471	0.250937
$675$	0	0
$676$	21.2705	0.818097
$677$	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
$680$	−40.3607	−1.54776
$681$	0	0
$682$	−0.291796	−0.0111734
$683$	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
$686$	−1.25735	−0.0480060
$687$	0	0
$688$	−30.5410	−1.16437
$689$	−11.0557	−0.421190
$690$	0	0
$691$	−22.7082	−0.863861	−0.431930	−	0.901907i	$-0.642167\pi$
$691$	−22.7082	−0.863861	−0.431930	+	0.901907i	$0.642167\pi$
$692$	12.8754	0.489449
$693$	0	0
$694$	−1.27864	−0.0485365
$695$	24.1803	0.917213
$696$	0	0
$697$	−43.4164	−1.64451
$698$	10.6525	0.403202
$699$	0	0
$700$	−5.66563	−0.214141
$701$	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
$704$	−3.59675	−0.135558
$705$	0	0
$706$	6.58359	0.247777
$707$	1.36068	0.0511736
$708$	0	0
$709$	−38.8328	−1.45840	−0.729199	−	0.684302i	$-0.760107\pi$
$709$	−38.8328	−1.45840	−0.729199	+	0.684302i	$0.760107\pi$
$710$	−14.5623	−0.546514
$711$	0	0
$712$	10.2229	0.383120
$713$	−1.23607	−0.0462911
$714$	0	0
$715$	4.00000	0.149592
$716$	36.0000	1.34538
$717$	0	0
$718$	3.39512	0.126705
$719$	−53.1935	−1.98378	−0.991891	−	0.127089i	$-0.959437\pi$
$719$	−53.1935	−1.98378	−0.991891	+	0.127089i	$0.959437\pi$
$720$	0	0
$721$	−3.24922	−0.121007
$722$	7.59675	0.282722
$723$	0	0
$724$	−36.0000	−1.33793
$725$	−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	$0.352362\pi$
$728$	−0.429563	−0.0159207
$729$	0	0
$730$	5.23607	0.193796
$731$	−62.8328	−2.32396
$732$	0	0
$733$	−7.00000	−0.258551	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000	−0.258551	−0.129275	+	0.991609i	$0.541265\pi$
$734$	6.51471	0.240462
$735$	0	0
$736$	5.12461	0.188896
$737$	−9.16718	−0.337678
$738$	0	0
$739$	43.2361	1.59046	0.795232	−	0.606305i	$-0.207349\pi$
$739$	43.2361	1.59046	0.795232	+	0.606305i	$0.207349\pi$
$740$	44.8328	1.64809
$741$	0	0
$742$	−0.806504	−0.0296077
$743$	39.2361	1.43943	0.719716	−	0.694269i	$-0.244272\pi$
$743$	39.2361	1.43943	0.719716	+	0.694269i	$0.244272\pi$
$744$	0	0
$745$	56.8328	2.08219
$746$	2.89667	0.106055
$747$	0	0
$748$	−9.16718	−0.335185
$749$	−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	$0.495887\pi$
$752$	7.77709	0.283601
$753$	0	0
$754$	−1.52786	−0.0556415
$755$	64.5410	2.34889
$756$	0	0
$757$	7.59675	0.276108	0.138054	−	0.990425i	$-0.455915\pi$
$757$	7.59675	0.276108	0.138054	+	0.990425i	$0.455915\pi$
$758$	6.69505	0.243175
$759$	0	0
$760$	38.8885	1.41064
$761$	32.7639	1.18769	0.593846	−	0.804579i	$-0.297609\pi$
$761$	32.7639	1.18769	0.593846	+	0.804579i	$0.297609\pi$
$762$	0	0
$763$	−1.18034	−0.0427312
$764$	−25.8541	−0.935369
$765$	0	0
$766$	−1.12461	−0.0406339
$767$	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	$0.347389\pi$
$770$	0.291796	0.0105156
$771$	0	0
$772$	5.56231	0.200192
$773$	−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
$776$	−13.2492	−0.475619
$777$	0	0
$778$	−10.0000	−0.358517
$779$	41.8328	1.49882
$780$	0	0
$781$	−6.87539	−0.246021
$782$	3.05573	0.109273
$783$	0	0
$784$	−21.8460	−0.780213
$785$	−17.1803	−0.613193
$786$	0	0
$787$	−40.8328	−1.45553	−0.727766	−	0.685825i	$-0.759441\pi$
$787$	−40.8328	−1.45553	−0.727766	+	0.685825i	$0.759441\pi$
$788$	9.70820	0.345840
$789$	0	0
$790$	13.7082	0.487716
$791$	0.416408	0.0148058
$792$	0	0
$793$	9.88854	0.351152
$794$	−11.6180	−0.412309
$795$	0	0
$796$	22.5836	0.800454
$797$	−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
$800$	53.6656	1.89737
$801$	0	0
$802$	−0.291796	−0.0103037
$803$	2.47214	0.0872398
$804$	0	0
$805$	1.23607	0.0435657
$806$	−0.472136	−0.0166303
$807$	0	0
$808$	−8.48529	−0.298512
$809$	53.0132	1.86384	0.931922	−	0.362660i	$-0.118131\pi$
$809$	53.0132	1.86384	0.931922	+	0.362660i	$0.118131\pi$
$810$	0	0
$811$	−46.8328	−1.64452	−0.822261	−	0.569110i	$-0.807288\pi$
$811$	−46.8328	−1.64452	−0.822261	+	0.569110i	$0.807288\pi$
$812$	1.41641	0.0497062
$813$	0	0
$814$	−1.66563	−0.0583804
$815$	−30.4164	−1.06544
$816$	0	0
$817$	60.5410	2.11806
$818$	8.83282	0.308832
$819$	0	0
$820$	52.6869	1.83991
$821$	−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	$0.393546\pi$
$824$	20.2624	0.705873
$825$	0	0
$826$	0.270510	0.00941224
$827$	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	$-0.529984\pi$
$829$	−5.41641	−0.188120	−0.0940598	+	0.995567i	$0.529984\pi$
$830$	26.6525	0.925121
$831$	0	0
$832$	−5.81966	−0.201760
$833$	−44.9443	−1.55723
$834$	0	0
$835$	70.5410	2.44117
$836$	8.83282	0.305489
$837$	0	0
$838$	−3.43769	−0.118753
$839$	20.9443	0.723077	0.361538	−	0.932357i	$-0.382252\pi$
$839$	20.9443	0.723077	0.361538	+	0.932357i	$0.382252\pi$
$840$	0	0
$841$	−18.5279	−0.638892
$842$	−5.50658	−0.189769
$843$	0	0
$844$	−1.31308	−0.0451982
$845$	−48.5967	−1.67178
$846$	0	0
$847$	−2.45898	−0.0844916
$848$	−28.1378	−0.966255
$849$	0	0
$850$	32.0000	1.09759
$851$	−7.05573	−0.241867
$852$	0	0
$853$	−23.5279	−0.805579	−0.402789	−	0.915293i	$-0.631959\pi$
$853$	−23.5279	−0.805579	−0.402789	+	0.915293i	$0.631959\pi$
$854$	0.721360	0.0246844
$855$	0	0
$856$	2.16718	0.0740728
$857$	7.30495	0.249532	0.124766	−	0.992186i	$-0.460182\pi$
$857$	7.30495	0.249532	0.124766	+	0.992186i	$0.460182\pi$
$858$	0	0
$859$	22.1803	0.756783	0.378392	−	0.925646i	$-0.376477\pi$
$859$	22.1803	0.756783	0.378392	+	0.925646i	$0.376477\pi$
$860$	76.2492	2.60008
$861$	0	0
$862$	2.47214	0.0842013
$863$	11.3050	0.384825	0.192413	−	0.981314i	$-0.438369\pi$
$863$	11.3050	0.384825	0.192413	+	0.981314i	$0.438369\pi$
$864$	0	0
$865$	−29.4164	−1.00019
$866$	4.76393	0.161885
$867$	0	0
$868$	0.437694	0.0148563
$869$	6.47214	0.219552
$870$	0	0
$871$	−14.8328	−0.502591
$872$	7.36068	0.249264
$873$	0	0
$874$	−2.94427	−0.0995915
$875$	7.94427	0.268565
$876$	0	0
$877$	−29.8328	−1.00738	−0.503691	−	0.863884i	$-0.668025\pi$
$877$	−29.8328	−1.00738	−0.503691	+	0.863884i	$0.668025\pi$
$878$	−4.09017	−0.138037
$879$	0	0
$880$	10.1803	0.343179
$881$	−45.3050	−1.52636	−0.763181	−	0.646184i	$-0.776364\pi$
$881$	−45.3050	−1.52636	−0.763181	+	0.646184i	$0.776364\pi$
$882$	0	0
$883$	26.8328	0.902996	0.451498	−	0.892272i	$-0.350890\pi$
$883$	26.8328	0.902996	0.451498	+	0.892272i	$0.350890\pi$
$884$	−14.8328	−0.498882
$885$	0	0
$886$	9.68692	0.325438
$887$	−19.5836	−0.657553	−0.328776	−	0.944408i	$-0.606636\pi$
$887$	−19.5836	−0.657553	−0.328776	+	0.944408i	$0.606636\pi$
$888$	0	0
$889$	1.70820	0.0572913
$890$	−11.2361	−0.376634
$891$	0	0
$892$	3.37384	0.112964
$893$	−15.4164	−0.515890
$894$	0	0
$895$	−82.2492	−2.74929
$896$	−2.38197	−0.0795759
$897$	0	0
$898$	11.7082	0.390708
$899$	3.23607	0.107929
$900$	0	0
$901$	−57.8885	−1.92855
$902$	−1.95743	−0.0651752
$903$	0	0
$904$	−2.59675	−0.0863665
$905$	82.2492	2.73406
$906$	0	0
$907$	42.9574	1.42638	0.713189	−	0.700972i	$-0.247250\pi$
$907$	42.9574	1.42638	0.713189	+	0.700972i	$0.247250\pi$
$908$	9.16718	0.304224
$909$	0	0
$910$	0.472136	0.0156512
$911$	20.1803	0.668604	0.334302	−	0.942466i	$-0.391499\pi$
$911$	20.1803	0.668604	0.334302	+	0.942466i	$0.391499\pi$
$912$	0	0
$913$	12.5836	0.416456
$914$	4.24922	0.140552
$915$	0	0
$916$	−18.0000	−0.594737
$917$	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	$0.629548\pi$
$920$	−7.70820	−0.254132
$921$	0	0
$922$	9.12461	0.300503
$923$	−11.1246	−0.366171
$924$	0	0
$925$	−73.8885	−2.42944
$926$	−9.48529	−0.311706
$927$	0	0
$928$	−13.4164	−0.440415
$929$	17.2361	0.565497	0.282749	−	0.959194i	$-0.408754\pi$
$929$	17.2361	0.565497	0.282749	+	0.959194i	$0.408754\pi$
$930$	0	0
$931$	43.3050	1.41926
$932$	1.31308	0.0430114
$933$	0	0
$934$	−7.66061	−0.250663
$935$	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	$0.680179\pi$
$938$	−1.08204	−0.0353298
$939$	0	0
$940$	−19.4164	−0.633293
$941$	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
$944$	9.43769	0.307171
$945$	0	0
$946$	−2.83282	−0.0921028
$947$	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
$950$	−30.8328	−1.00035
$951$	0	0
$952$	−2.24922	−0.0728978
$953$	0.944272	0.0305880	0.0152940	−	0.999883i	$-0.495132\pi$
$953$	0.944272	0.0305880	0.0152940	+	0.999883i	$0.495132\pi$
$954$	0	0
$955$	59.0689	1.91142
$956$	−22.0426	−0.712908
$957$	0	0
$958$	0.0212862	0.000687727 0
$959$	2.40325	0.0776051
$960$	0	0
$961$	1.00000	0.0322581
$962$	−2.69505	−0.0868918
$963$	0	0
$964$	25.4164	0.818607
$965$	−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	$0.801973\pi$
$968$	15.3344	0.492865
$969$	0	0
$970$	14.5623	0.467567
$971$	−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
$974$	−1.41641	−0.0453846
$975$	0	0
$976$	25.1672	0.805582
$977$	−11.2918	−0.361257	−0.180628	−	0.983551i	$-0.557813\pi$
$977$	−11.2918	−0.361257	−0.180628	+	0.983551i	$0.557813\pi$
$978$	0	0
$979$	−5.30495	−0.169547
$980$	54.5410	1.74225
$981$	0	0
$982$	5.27864	0.168448
$983$	28.7639	0.917427	0.458713	−	0.888584i	$-0.348310\pi$
$983$	28.7639	0.917427	0.458713	+	0.888584i	$0.348310\pi$
$984$	0	0
$985$	−22.1803	−0.706724
$986$	−8.00000	−0.254772
$987$	0	0
$988$	14.2918	0.454683
$989$	−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	$0.301782\pi$
$992$	−4.14590	−0.131632
$993$	0	0
$994$	−0.811529	−0.0257402
$995$	−51.5967	−1.63573
$996$	0	0
$997$	−4.41641	−0.139869	−0.0699345	−	0.997552i	$-0.522279\pi$
$997$	−4.41641	−0.139869	−0.0699345	+	0.997552i	$0.522279\pi$
$998$	7.59675	0.240471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	279.2.a.b.1.1		2
3.2	odd	2		93.2.a.a.1.2	✓	2
4.3	odd	2		4464.2.a.bn.1.2		2
5.4	even	2		6975.2.a.t.1.2		2
12.11	even	2		1488.2.a.q.1.1		2
15.2	even	4		2325.2.c.h.1024.2		4
15.8	even	4		2325.2.c.h.1024.3		4
15.14	odd	2		2325.2.a.o.1.1		2
21.20	even	2		4557.2.a.p.1.2		2
24.5	odd	2		5952.2.a.bv.1.2		2
24.11	even	2		5952.2.a.bo.1.2		2
31.30	odd	2		8649.2.a.m.1.1		2
93.92	even	2		2883.2.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.a.a.1.2	✓	2	3.2	odd	2
279.2.a.b.1.1		2	1.1	even	1	trivial
1488.2.a.q.1.1		2	12.11	even	2
2325.2.a.o.1.1		2	15.14	odd	2
2325.2.c.h.1024.2		4	15.2	even	4
2325.2.c.h.1024.3		4	15.8	even	4
2883.2.a.a.1.2		2	93.92	even	2
4464.2.a.bn.1.2		2	4.3	odd	2
4557.2.a.p.1.2		2	21.20	even	2
5952.2.a.bo.1.2		2	24.11	even	2
5952.2.a.bv.1.2		2	24.5	odd	2
6975.2.a.t.1.2		2	5.4	even	2
8649.2.a.m.1.1		2	31.30	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:	\( N \)	\(=\)	\( 279 = 3^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	279.a (trivial)

\(f(q)\)	\(=\)	\(q+0.381966 q^{2} -1.85410 q^{4} +4.23607 q^{5} +0.236068 q^{7} -1.47214 q^{8} +1.61803 q^{10} +0.763932 q^{11} +1.23607 q^{13} +0.0901699 q^{14} +3.14590 q^{16} +6.47214 q^{17} -6.23607 q^{19} -7.85410 q^{20} +0.291796 q^{22} +1.23607 q^{23} +12.9443 q^{25} +0.472136 q^{26} -0.437694 q^{28} -3.23607 q^{29} -1.00000 q^{31} +4.14590 q^{32} +2.47214 q^{34} +1.00000 q^{35} -5.70820 q^{37} -2.38197 q^{38} -6.23607 q^{40} -6.70820 q^{41} -9.70820 q^{43} -1.41641 q^{44} +0.472136 q^{46} +2.47214 q^{47} -6.94427 q^{49} +4.94427 q^{50} -2.29180 q^{52} -8.94427 q^{53} +3.23607 q^{55} -0.347524 q^{56} -1.23607 q^{58} +3.00000 q^{59} +8.00000 q^{61} -0.381966 q^{62} -4.70820 q^{64} +5.23607 q^{65} -12.0000 q^{67} -12.0000 q^{68} +0.381966 q^{70} -9.00000 q^{71} +3.23607 q^{73} -2.18034 q^{74} +11.5623 q^{76} +0.180340 q^{77} +8.47214 q^{79} +13.3262 q^{80} -2.56231 q^{82} +16.4721 q^{83} +27.4164 q^{85} -3.70820 q^{86} -1.12461 q^{88} -6.94427 q^{89} +0.291796 q^{91} -2.29180 q^{92} +0.944272 q^{94} -26.4164 q^{95} +9.00000 q^{97} -2.65248 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} + q^{10} + 6 q^{11} - 2 q^{13} - 11 q^{14} + 13 q^{16} + 4 q^{17} - 8 q^{19} - 9 q^{20} + 14 q^{22} - 2 q^{23} + 8 q^{25} - 8 q^{26} - 21 q^{28} - 2 q^{29}+ \cdots + 26 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8 + q^10 + 6 * q^11 - 2 * q^13 - 11 * q^14 + 13 * q^16 + 4 * q^17 - 8 * q^19 - 9 * q^20 + 14 * q^22 - 2 * q^23 + 8 * q^25 - 8 * q^26 - 21 * q^28 - 2 * q^29 - 2 * q^31 + 15 * q^32 - 4 * q^34 + 2 * q^35 + 2 * q^37 - 7 * q^38 - 8 * q^40 - 6 * q^43 + 24 * q^44 - 8 * q^46 - 4 * q^47 + 4 * q^49 - 8 * q^50 - 18 * q^52 + 2 * q^55 - 32 * q^56 + 2 * q^58 + 6 * q^59 + 16 * q^61 - 3 * q^62 + 4 * q^64 + 6 * q^65 - 24 * q^67 - 24 * q^68 + 3 * q^70 - 18 * q^71 + 2 * q^73 + 18 * q^74 + 3 * q^76 - 22 * q^77 + 8 * q^79 + 11 * q^80 + 15 * q^82 + 24 * q^83 + 28 * q^85 + 6 * q^86 + 38 * q^88 + 4 * q^89 + 14 * q^91 - 18 * q^92 - 16 * q^94 - 26 * q^95 + 18 * q^97 + 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more