LMFDB - Embedded newform 2672.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2672,2,Mod(1,2672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2672 = 2^{4} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2672.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:	$21.3360274201$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1336)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{3} +2.23607 q^{5} +4.61803 q^{7} -2.85410 q^{9} +O(q^{10})$	$q-0.381966 q^{3} +2.23607 q^{5} +4.61803 q^{7} -2.85410 q^{9} +4.00000 q^{11} +5.61803 q^{13} -0.854102 q^{15} +3.09017 q^{17} -0.472136 q^{19} -1.76393 q^{21} +5.85410 q^{23} +2.23607 q^{27} -6.23607 q^{29} -0.763932 q^{31} -1.52786 q^{33} +10.3262 q^{35} -7.94427 q^{37} -2.14590 q^{39} -2.23607 q^{41} +7.47214 q^{43} -6.38197 q^{45} -0.236068 q^{47} +14.3262 q^{49} -1.18034 q^{51} -9.23607 q^{53} +8.94427 q^{55} +0.180340 q^{57} +1.70820 q^{59} -12.2361 q^{61} -13.1803 q^{63} +12.5623 q^{65} -6.23607 q^{67} -2.23607 q^{69} -8.85410 q^{71} -5.09017 q^{73} +18.4721 q^{77} +3.00000 q^{79} +7.70820 q^{81} -1.94427 q^{83} +6.90983 q^{85} +2.38197 q^{87} +1.23607 q^{89} +25.9443 q^{91} +0.291796 q^{93} -1.05573 q^{95} -13.0344 q^{97} -11.4164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + 7 q^{7} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 + 7 * q^7 + q^9	$ 2 q - 3 q^{3} + 7 q^{7} + q^{9} + 8 q^{11} + 9 q^{13} + 5 q^{15} - 5 q^{17} + 8 q^{19} - 8 q^{21} + 5 q^{23} - 8 q^{29} - 6 q^{31} - 12 q^{33} + 5 q^{35} + 2 q^{37} - 11 q^{39} + 6 q^{43} - 15 q^{45} + 4 q^{47} + 13 q^{49} + 20 q^{51} - 14 q^{53} - 22 q^{57} - 10 q^{59} - 20 q^{61} - 4 q^{63} + 5 q^{65} - 8 q^{67} - 11 q^{71} + q^{73} + 28 q^{77} + 6 q^{79} + 2 q^{81} + 14 q^{83} + 25 q^{85} + 7 q^{87} - 2 q^{89} + 34 q^{91} + 14 q^{93} - 20 q^{95} + 3 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 + 7 * q^7 + q^9 + 8 * q^11 + 9 * q^13 + 5 * q^15 - 5 * q^17 + 8 * q^19 - 8 * q^21 + 5 * q^23 - 8 * q^29 - 6 * q^31 - 12 * q^33 + 5 * q^35 + 2 * q^37 - 11 * q^39 + 6 * q^43 - 15 * q^45 + 4 * q^47 + 13 * q^49 + 20 * q^51 - 14 * q^53 - 22 * q^57 - 10 * q^59 - 20 * q^61 - 4 * q^63 + 5 * q^65 - 8 * q^67 - 11 * q^71 + q^73 + 28 * q^77 + 6 * q^79 + 2 * q^81 + 14 * q^83 + 25 * q^85 + 7 * q^87 - 2 * q^89 + 34 * q^91 + 14 * q^93 - 20 * q^95 + 3 * 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
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	0	0
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	4.61803	1.74545	0.872726	−	0.488210i	$-0.162350\pi$
$7$	4.61803	1.74545	0.872726	+	0.488210i	$0.162350\pi$
$8$	0	0
$9$	−2.85410	−0.951367
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	5.61803	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$13$	5.61803	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$14$	0	0
$15$	−0.854102	−0.220528
$16$	0	0
$17$	3.09017	0.749476	0.374738	−	0.927131i	$-0.377733\pi$
$17$	3.09017	0.749476	0.374738	+	0.927131i	$0.377733\pi$
$18$	0	0
$19$	−0.472136	−0.108315	−0.0541577	−	0.998532i	$-0.517247\pi$
$19$	−0.472136	−0.108315	−0.0541577	+	0.998532i	$0.517247\pi$
$20$	0	0
$21$	−1.76393	−0.384922
$22$	0	0
$23$	5.85410	1.22066	0.610332	−	0.792145i	$-0.291036\pi$
$23$	5.85410	1.22066	0.610332	+	0.792145i	$0.291036\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	−6.23607	−1.15801	−0.579004	−	0.815324i	$-0.696559\pi$
$29$	−6.23607	−1.15801	−0.579004	+	0.815324i	$0.696559\pi$
$30$	0	0
$31$	−0.763932	−0.137206	−0.0686031	−	0.997644i	$-0.521854\pi$
$31$	−0.763932	−0.137206	−0.0686031	+	0.997644i	$0.521854\pi$
$32$	0	0
$33$	−1.52786	−0.265967
$34$	0	0
$35$	10.3262	1.74545
$36$	0	0
$37$	−7.94427	−1.30603	−0.653015	−	0.757345i	$-0.726496\pi$
$37$	−7.94427	−1.30603	−0.653015	+	0.757345i	$0.726496\pi$
$38$	0	0
$39$	−2.14590	−0.343619
$40$	0	0
$41$	−2.23607	−0.349215	−0.174608	−	0.984638i	$-0.555866\pi$
$41$	−2.23607	−0.349215	−0.174608	+	0.984638i	$0.555866\pi$
$42$	0	0
$43$	7.47214	1.13949	0.569745	−	0.821822i	$-0.307042\pi$
$43$	7.47214	1.13949	0.569745	+	0.821822i	$0.307042\pi$
$44$	0	0
$45$	−6.38197	−0.951367
$46$	0	0
$47$	−0.236068	−0.0344341	−0.0172170	−	0.999852i	$-0.505481\pi$
$47$	−0.236068	−0.0344341	−0.0172170	+	0.999852i	$0.505481\pi$
$48$	0	0
$49$	14.3262	2.04661
$50$	0	0
$51$	−1.18034	−0.165281
$52$	0	0
$53$	−9.23607	−1.26867	−0.634336	−	0.773058i	$-0.718726\pi$
$53$	−9.23607	−1.26867	−0.634336	+	0.773058i	$0.718726\pi$
$54$	0	0
$55$	8.94427	1.20605
$56$	0	0
$57$	0.180340	0.0238866
$58$	0	0
$59$	1.70820	0.222389	0.111195	−	0.993799i	$-0.464532\pi$
$59$	1.70820	0.222389	0.111195	+	0.993799i	$0.464532\pi$
$60$	0	0
$61$	−12.2361	−1.56667	−0.783334	−	0.621601i	$-0.786483\pi$
$61$	−12.2361	−1.56667	−0.783334	+	0.621601i	$0.786483\pi$
$62$	0	0
$63$	−13.1803	−1.66057
$64$	0	0
$65$	12.5623	1.55816
$66$	0	0
$67$	−6.23607	−0.761857	−0.380928	−	0.924605i	$-0.624396\pi$
$67$	−6.23607	−0.761857	−0.380928	+	0.924605i	$0.624396\pi$
$68$	0	0
$69$	−2.23607	−0.269191
$70$	0	0
$71$	−8.85410	−1.05079	−0.525394	−	0.850859i	$-0.676082\pi$
$71$	−8.85410	−1.05079	−0.525394	+	0.850859i	$0.676082\pi$
$72$	0	0
$73$	−5.09017	−0.595759	−0.297880	−	0.954603i	$-0.596279\pi$
$73$	−5.09017	−0.595759	−0.297880	+	0.954603i	$0.596279\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.4721	2.10510
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	0	0
$83$	−1.94427	−0.213412	−0.106706	−	0.994291i	$-0.534030\pi$
$83$	−1.94427	−0.213412	−0.106706	+	0.994291i	$0.534030\pi$
$84$	0	0
$85$	6.90983	0.749476
$86$	0	0
$87$	2.38197	0.255374
$88$	0	0
$89$	1.23607	0.131023	0.0655115	−	0.997852i	$-0.479132\pi$
$89$	1.23607	0.131023	0.0655115	+	0.997852i	$0.479132\pi$
$90$	0	0
$91$	25.9443	2.71970
$92$	0	0
$93$	0.291796	0.0302578
$94$	0	0
$95$	−1.05573	−0.108315
$96$	0	0
$97$	−13.0344	−1.32345	−0.661724	−	0.749748i	$-0.730175\pi$
$97$	−13.0344	−1.32345	−0.661724	+	0.749748i	$0.730175\pi$
$98$	0	0
$99$	−11.4164	−1.14739
$100$	0	0
$101$	11.9443	1.18850	0.594250	−	0.804281i	$-0.297449\pi$
$101$	11.9443	1.18850	0.594250	+	0.804281i	$0.297449\pi$
$102$	0	0
$103$	−5.32624	−0.524810	−0.262405	−	0.964958i	$-0.584516\pi$
$103$	−5.32624	−0.524810	−0.262405	+	0.964958i	$0.584516\pi$
$104$	0	0
$105$	−3.94427	−0.384922
$106$	0	0
$107$	−17.6525	−1.70653	−0.853265	−	0.521478i	$-0.825381\pi$
$107$	−17.6525	−1.70653	−0.853265	+	0.521478i	$0.825381\pi$
$108$	0	0
$109$	16.4721	1.57774	0.788872	−	0.614557i	$-0.210665\pi$
$109$	16.4721	1.57774	0.788872	+	0.614557i	$0.210665\pi$
$110$	0	0
$111$	3.03444	0.288016
$112$	0	0
$113$	−6.23607	−0.586640	−0.293320	−	0.956014i	$-0.594760\pi$
$113$	−6.23607	−0.586640	−0.293320	+	0.956014i	$0.594760\pi$
$114$	0	0
$115$	13.0902	1.22066
$116$	0	0
$117$	−16.0344	−1.48238
$118$	0	0
$119$	14.2705	1.30818
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0.854102	0.0770118
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−16.8885	−1.49862	−0.749308	−	0.662222i	$-0.769614\pi$
$127$	−16.8885	−1.49862	−0.749308	+	0.662222i	$0.769614\pi$
$128$	0	0
$129$	−2.85410	−0.251290
$130$	0	0
$131$	15.9443	1.39306	0.696529	−	0.717529i	$-0.254727\pi$
$131$	15.9443	1.39306	0.696529	+	0.717529i	$0.254727\pi$
$132$	0	0
$133$	−2.18034	−0.189059
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	−16.9443	−1.44765	−0.723823	−	0.689985i	$-0.757617\pi$
$137$	−16.9443	−1.44765	−0.723823	+	0.689985i	$0.757617\pi$
$138$	0	0
$139$	7.03444	0.596654	0.298327	−	0.954464i	$-0.403571\pi$
$139$	7.03444	0.596654	0.298327	+	0.954464i	$0.403571\pi$
$140$	0	0
$141$	0.0901699	0.00759368
$142$	0	0
$143$	22.4721	1.87921
$144$	0	0
$145$	−13.9443	−1.15801
$146$	0	0
$147$	−5.47214	−0.451334
$148$	0	0
$149$	8.14590	0.667338	0.333669	−	0.942690i	$-0.391713\pi$
$149$	8.14590	0.667338	0.333669	+	0.942690i	$0.391713\pi$
$150$	0	0
$151$	2.14590	0.174631	0.0873154	−	0.996181i	$-0.472171\pi$
$151$	2.14590	0.174631	0.0873154	+	0.996181i	$0.472171\pi$
$152$	0	0
$153$	−8.81966	−0.713027
$154$	0	0
$155$	−1.70820	−0.137206
$156$	0	0
$157$	20.2361	1.61501	0.807507	−	0.589858i	$-0.200816\pi$
$157$	20.2361	1.61501	0.807507	+	0.589858i	$0.200816\pi$
$158$	0	0
$159$	3.52786	0.279778
$160$	0	0
$161$	27.0344	2.13061
$162$	0	0
$163$	16.7639	1.31305	0.656526	−	0.754303i	$-0.272025\pi$
$163$	16.7639	1.31305	0.656526	+	0.754303i	$0.272025\pi$
$164$	0	0
$165$	−3.41641	−0.265967
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	18.5623	1.42787
$170$	0	0
$171$	1.34752	0.103048
$172$	0	0
$173$	−3.47214	−0.263982	−0.131991	−	0.991251i	$-0.542137\pi$
$173$	−3.47214	−0.263982	−0.131991	+	0.991251i	$0.542137\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.652476	−0.0490431
$178$	0	0
$179$	−18.7984	−1.40506	−0.702528	−	0.711656i	$-0.747946\pi$
$179$	−18.7984	−1.40506	−0.702528	+	0.711656i	$0.747946\pi$
$180$	0	0
$181$	−5.05573	−0.375789	−0.187895	−	0.982189i	$-0.560166\pi$
$181$	−5.05573	−0.375789	−0.187895	+	0.982189i	$0.560166\pi$
$182$	0	0
$183$	4.67376	0.345494
$184$	0	0
$185$	−17.7639	−1.30603
$186$	0	0
$187$	12.3607	0.903902
$188$	0	0
$189$	10.3262	0.751123
$190$	0	0
$191$	−3.90983	−0.282905	−0.141453	−	0.989945i	$-0.545177\pi$
$191$	−3.90983	−0.282905	−0.141453	+	0.989945i	$0.545177\pi$
$192$	0	0
$193$	7.61803	0.548358	0.274179	−	0.961679i	$-0.411594\pi$
$193$	7.61803	0.548358	0.274179	+	0.961679i	$0.411594\pi$
$194$	0	0
$195$	−4.79837	−0.343619
$196$	0	0
$197$	7.85410	0.559582	0.279791	−	0.960061i	$-0.409735\pi$
$197$	7.85410	0.559582	0.279791	+	0.960061i	$0.409735\pi$
$198$	0	0
$199$	−22.0902	−1.56593	−0.782965	−	0.622065i	$-0.786294\pi$
$199$	−22.0902	−1.56593	−0.782965	+	0.622065i	$0.786294\pi$
$200$	0	0
$201$	2.38197	0.168011
$202$	0	0
$203$	−28.7984	−2.02125
$204$	0	0
$205$	−5.00000	−0.349215
$206$	0	0
$207$	−16.7082	−1.16130
$208$	0	0
$209$	−1.88854	−0.130633
$210$	0	0
$211$	21.8541	1.50450	0.752249	−	0.658879i	$-0.228969\pi$
$211$	21.8541	1.50450	0.752249	+	0.658879i	$0.228969\pi$
$212$	0	0
$213$	3.38197	0.231728
$214$	0	0
$215$	16.7082	1.13949
$216$	0	0
$217$	−3.52786	−0.239487
$218$	0	0
$219$	1.94427	0.131382
$220$	0	0
$221$	17.3607	1.16781
$222$	0	0
$223$	11.5066	0.770537	0.385269	−	0.922804i	$-0.374109\pi$
$223$	11.5066	0.770537	0.385269	+	0.922804i	$0.374109\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	0	0
$229$	17.6180	1.16423	0.582116	−	0.813106i	$-0.302225\pi$
$229$	17.6180	1.16423	0.582116	+	0.813106i	$0.302225\pi$
$230$	0	0
$231$	−7.05573	−0.464233
$232$	0	0
$233$	−16.7984	−1.10050	−0.550249	−	0.835001i	$-0.685467\pi$
$233$	−16.7984	−1.10050	−0.550249	+	0.835001i	$0.685467\pi$
$234$	0	0
$235$	−0.527864	−0.0344341
$236$	0	0
$237$	−1.14590	−0.0744341
$238$	0	0
$239$	−9.56231	−0.618534	−0.309267	−	0.950975i	$-0.600084\pi$
$239$	−9.56231	−0.618534	−0.309267	+	0.950975i	$0.600084\pi$
$240$	0	0
$241$	17.0344	1.09728	0.548642	−	0.836057i	$-0.315145\pi$
$241$	17.0344	1.09728	0.548642	+	0.836057i	$0.315145\pi$
$242$	0	0
$243$	−9.65248	−0.619207
$244$	0	0
$245$	32.0344	2.04661
$246$	0	0
$247$	−2.65248	−0.168773
$248$	0	0
$249$	0.742646	0.0470633
$250$	0	0
$251$	20.7082	1.30709	0.653545	−	0.756888i	$-0.273281\pi$
$251$	20.7082	1.30709	0.653545	+	0.756888i	$0.273281\pi$
$252$	0	0
$253$	23.4164	1.47218
$254$	0	0
$255$	−2.63932	−0.165281
$256$	0	0
$257$	23.9787	1.49575	0.747876	−	0.663839i	$-0.231074\pi$
$257$	23.9787	1.49575	0.747876	+	0.663839i	$0.231074\pi$
$258$	0	0
$259$	−36.6869	−2.27961
$260$	0	0
$261$	17.7984	1.10169
$262$	0	0
$263$	−12.4164	−0.765629	−0.382814	−	0.923825i	$-0.625045\pi$
$263$	−12.4164	−0.765629	−0.382814	+	0.923825i	$0.625045\pi$
$264$	0	0
$265$	−20.6525	−1.26867
$266$	0	0
$267$	−0.472136	−0.0288943
$268$	0	0
$269$	−15.5066	−0.945453	−0.472726	−	0.881209i	$-0.656730\pi$
$269$	−15.5066	−0.945453	−0.472726	+	0.881209i	$0.656730\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	−9.90983	−0.599770
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.9787	1.62099	0.810497	−	0.585743i	$-0.199197\pi$
$277$	26.9787	1.62099	0.810497	+	0.585743i	$0.199197\pi$
$278$	0	0
$279$	2.18034	0.130534
$280$	0	0
$281$	−30.8885	−1.84266	−0.921328	−	0.388786i	$-0.872894\pi$
$281$	−30.8885	−1.84266	−0.921328	+	0.388786i	$0.872894\pi$
$282$	0	0
$283$	12.9443	0.769457	0.384729	−	0.923030i	$-0.374295\pi$
$283$	12.9443	0.769457	0.384729	+	0.923030i	$0.374295\pi$
$284$	0	0
$285$	0.403252	0.0238866
$286$	0	0
$287$	−10.3262	−0.609539
$288$	0	0
$289$	−7.45085	−0.438285
$290$	0	0
$291$	4.97871	0.291857
$292$	0	0
$293$	−2.94427	−0.172006	−0.0860031	−	0.996295i	$-0.527409\pi$
$293$	−2.94427	−0.172006	−0.0860031	+	0.996295i	$0.527409\pi$
$294$	0	0
$295$	3.81966	0.222389
$296$	0	0
$297$	8.94427	0.518999
$298$	0	0
$299$	32.8885	1.90199
$300$	0	0
$301$	34.5066	1.98893
$302$	0	0
$303$	−4.56231	−0.262098
$304$	0	0
$305$	−27.3607	−1.56667
$306$	0	0
$307$	−16.9098	−0.965095	−0.482547	−	0.875870i	$-0.660288\pi$
$307$	−16.9098	−0.965095	−0.482547	+	0.875870i	$0.660288\pi$
$308$	0	0
$309$	2.03444	0.115735
$310$	0	0
$311$	−2.41641	−0.137022	−0.0685110	−	0.997650i	$-0.521825\pi$
$311$	−2.41641	−0.137022	−0.0685110	+	0.997650i	$0.521825\pi$
$312$	0	0
$313$	29.2148	1.65132	0.825659	−	0.564170i	$-0.190804\pi$
$313$	29.2148	1.65132	0.825659	+	0.564170i	$0.190804\pi$
$314$	0	0
$315$	−29.4721	−1.66057
$316$	0	0
$317$	8.03444	0.451259	0.225630	−	0.974213i	$-0.427556\pi$
$317$	8.03444	0.451259	0.225630	+	0.974213i	$0.427556\pi$
$318$	0	0
$319$	−24.9443	−1.39661
$320$	0	0
$321$	6.74265	0.376338
$322$	0	0
$323$	−1.45898	−0.0811798
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.29180	−0.347937
$328$	0	0
$329$	−1.09017	−0.0601030
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	0	0
$333$	22.6738	1.24251
$334$	0	0
$335$	−13.9443	−0.761857
$336$	0	0
$337$	−5.65248	−0.307910	−0.153955	−	0.988078i	$-0.549201\pi$
$337$	−5.65248	−0.307910	−0.153955	+	0.988078i	$0.549201\pi$
$338$	0	0
$339$	2.38197	0.129371
$340$	0	0
$341$	−3.05573	−0.165477
$342$	0	0
$343$	33.8328	1.82680
$344$	0	0
$345$	−5.00000	−0.269191
$346$	0	0
$347$	11.6180	0.623689	0.311844	−	0.950133i	$-0.399053\pi$
$347$	11.6180	0.623689	0.311844	+	0.950133i	$0.399053\pi$
$348$	0	0
$349$	−31.0000	−1.65939	−0.829696	−	0.558216i	$-0.811486\pi$
$349$	−31.0000	−1.65939	−0.829696	+	0.558216i	$0.811486\pi$
$350$	0	0
$351$	12.5623	0.670526
$352$	0	0
$353$	11.2361	0.598036	0.299018	−	0.954248i	$-0.403341\pi$
$353$	11.2361	0.598036	0.299018	+	0.954248i	$0.403341\pi$
$354$	0	0
$355$	−19.7984	−1.05079
$356$	0	0
$357$	−5.45085	−0.288490
$358$	0	0
$359$	29.1803	1.54008	0.770040	−	0.637996i	$-0.220236\pi$
$359$	29.1803	1.54008	0.770040	+	0.637996i	$0.220236\pi$
$360$	0	0
$361$	−18.7771	−0.988268
$362$	0	0
$363$	−1.90983	−0.100240
$364$	0	0
$365$	−11.3820	−0.595759
$366$	0	0
$367$	−7.29180	−0.380629	−0.190314	−	0.981723i	$-0.560951\pi$
$367$	−7.29180	−0.380629	−0.190314	+	0.981723i	$0.560951\pi$
$368$	0	0
$369$	6.38197	0.332232
$370$	0	0
$371$	−42.6525	−2.21441
$372$	0	0
$373$	1.20163	0.0622178	0.0311089	−	0.999516i	$-0.490096\pi$
$373$	1.20163	0.0622178	0.0311089	+	0.999516i	$0.490096\pi$
$374$	0	0
$375$	4.27051	0.220528
$376$	0	0
$377$	−35.0344	−1.80437
$378$	0	0
$379$	−13.3607	−0.686292	−0.343146	−	0.939282i	$-0.611493\pi$
$379$	−13.3607	−0.686292	−0.343146	+	0.939282i	$0.611493\pi$
$380$	0	0
$381$	6.45085	0.330487
$382$	0	0
$383$	9.94427	0.508129	0.254064	−	0.967187i	$-0.418233\pi$
$383$	9.94427	0.508129	0.254064	+	0.967187i	$0.418233\pi$
$384$	0	0
$385$	41.3050	2.10510
$386$	0	0
$387$	−21.3262	−1.08407
$388$	0	0
$389$	−28.8885	−1.46471	−0.732354	−	0.680924i	$-0.761578\pi$
$389$	−28.8885	−1.46471	−0.732354	+	0.680924i	$0.761578\pi$
$390$	0	0
$391$	18.0902	0.914859
$392$	0	0
$393$	−6.09017	−0.307208
$394$	0	0
$395$	6.70820	0.337526
$396$	0	0
$397$	−13.7984	−0.692521	−0.346260	−	0.938138i	$-0.612549\pi$
$397$	−13.7984	−0.692521	−0.346260	+	0.938138i	$0.612549\pi$
$398$	0	0
$399$	0.832816	0.0416929
$400$	0	0
$401$	−27.3820	−1.36739	−0.683695	−	0.729768i	$-0.739628\pi$
$401$	−27.3820	−1.36739	−0.683695	+	0.729768i	$0.739628\pi$
$402$	0	0
$403$	−4.29180	−0.213790
$404$	0	0
$405$	17.2361	0.856467
$406$	0	0
$407$	−31.7771	−1.57513
$408$	0	0
$409$	3.38197	0.167227	0.0836137	−	0.996498i	$-0.473354\pi$
$409$	3.38197	0.167227	0.0836137	+	0.996498i	$0.473354\pi$
$410$	0	0
$411$	6.47214	0.319247
$412$	0	0
$413$	7.88854	0.388170
$414$	0	0
$415$	−4.34752	−0.213412
$416$	0	0
$417$	−2.68692	−0.131579
$418$	0	0
$419$	−4.52786	−0.221201	−0.110600	−	0.993865i	$-0.535277\pi$
$419$	−4.52786	−0.221201	−0.110600	+	0.993865i	$0.535277\pi$
$420$	0	0
$421$	−0.652476	−0.0317997	−0.0158999	−	0.999874i	$-0.505061\pi$
$421$	−0.652476	−0.0317997	−0.0158999	+	0.999874i	$0.505061\pi$
$422$	0	0
$423$	0.673762	0.0327594
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−56.5066	−2.73454
$428$	0	0
$429$	−8.58359	−0.414420
$430$	0	0
$431$	28.7639	1.38551	0.692755	−	0.721173i	$-0.256397\pi$
$431$	28.7639	1.38551	0.692755	+	0.721173i	$0.256397\pi$
$432$	0	0
$433$	−3.20163	−0.153860	−0.0769302	−	0.997036i	$-0.524512\pi$
$433$	−3.20163	−0.153860	−0.0769302	+	0.997036i	$0.524512\pi$
$434$	0	0
$435$	5.32624	0.255374
$436$	0	0
$437$	−2.76393	−0.132217
$438$	0	0
$439$	−40.1246	−1.91504	−0.957522	−	0.288361i	$-0.906890\pi$
$439$	−40.1246	−1.91504	−0.957522	+	0.288361i	$0.906890\pi$
$440$	0	0
$441$	−40.8885	−1.94707
$442$	0	0
$443$	−20.2705	−0.963081	−0.481540	−	0.876424i	$-0.659923\pi$
$443$	−20.2705	−0.963081	−0.481540	+	0.876424i	$0.659923\pi$
$444$	0	0
$445$	2.76393	0.131023
$446$	0	0
$447$	−3.11146	−0.147167
$448$	0	0
$449$	−29.0689	−1.37185	−0.685923	−	0.727674i	$-0.740601\pi$
$449$	−29.0689	−1.37185	−0.685923	+	0.727674i	$0.740601\pi$
$450$	0	0
$451$	−8.94427	−0.421169
$452$	0	0
$453$	−0.819660	−0.0385110
$454$	0	0
$455$	58.0132	2.71970
$456$	0	0
$457$	−13.6180	−0.637025	−0.318512	−	0.947919i	$-0.603183\pi$
$457$	−13.6180	−0.637025	−0.318512	+	0.947919i	$0.603183\pi$
$458$	0	0
$459$	6.90983	0.322523
$460$	0	0
$461$	−21.0902	−0.982267	−0.491134	−	0.871084i	$-0.663417\pi$
$461$	−21.0902	−0.982267	−0.491134	+	0.871084i	$0.663417\pi$
$462$	0	0
$463$	7.29180	0.338879	0.169439	−	0.985541i	$-0.445804\pi$
$463$	7.29180	0.338879	0.169439	+	0.985541i	$0.445804\pi$
$464$	0	0
$465$	0.652476	0.0302578
$466$	0	0
$467$	−36.5623	−1.69190	−0.845951	−	0.533261i	$-0.820966\pi$
$467$	−36.5623	−1.69190	−0.845951	+	0.533261i	$0.820966\pi$
$468$	0	0
$469$	−28.7984	−1.32979
$470$	0	0
$471$	−7.72949	−0.356156
$472$	0	0
$473$	29.8885	1.37428
$474$	0	0
$475$	0	0
$476$	0	0
$477$	26.3607	1.20697
$478$	0	0
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	−44.6312	−2.03501
$482$	0	0
$483$	−10.3262	−0.469860
$484$	0	0
$485$	−29.1459	−1.32345
$486$	0	0
$487$	−29.8328	−1.35185	−0.675927	−	0.736969i	$-0.736257\pi$
$487$	−29.8328	−1.35185	−0.675927	+	0.736969i	$0.736257\pi$
$488$	0	0
$489$	−6.40325	−0.289565
$490$	0	0
$491$	22.8885	1.03295	0.516473	−	0.856304i	$-0.327245\pi$
$491$	22.8885	1.03295	0.516473	+	0.856304i	$0.327245\pi$
$492$	0	0
$493$	−19.2705	−0.867900
$494$	0	0
$495$	−25.5279	−1.14739
$496$	0	0
$497$	−40.8885	−1.83410
$498$	0	0
$499$	−20.7984	−0.931063	−0.465532	−	0.885031i	$-0.654137\pi$
$499$	−20.7984	−0.931063	−0.465532	+	0.885031i	$0.654137\pi$
$500$	0	0
$501$	−0.381966	−0.0170650
$502$	0	0
$503$	22.9098	1.02150	0.510749	−	0.859730i	$-0.329368\pi$
$503$	22.9098	1.02150	0.510749	+	0.859730i	$0.329368\pi$
$504$	0	0
$505$	26.7082	1.18850
$506$	0	0
$507$	−7.09017	−0.314886
$508$	0	0
$509$	21.5623	0.955732	0.477866	−	0.878433i	$-0.341410\pi$
$509$	21.5623	0.955732	0.477866	+	0.878433i	$0.341410\pi$
$510$	0	0
$511$	−23.5066	−1.03987
$512$	0	0
$513$	−1.05573	−0.0466115
$514$	0	0
$515$	−11.9098	−0.524810
$516$	0	0
$517$	−0.944272	−0.0415290
$518$	0	0
$519$	1.32624	0.0582154
$520$	0	0
$521$	7.23607	0.317018	0.158509	−	0.987358i	$-0.449331\pi$
$521$	7.23607	0.317018	0.158509	+	0.987358i	$0.449331\pi$
$522$	0	0
$523$	23.7639	1.03912	0.519562	−	0.854433i	$-0.326095\pi$
$523$	23.7639	1.03912	0.519562	+	0.854433i	$0.326095\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.36068	−0.102833
$528$	0	0
$529$	11.2705	0.490022
$530$	0	0
$531$	−4.87539	−0.211574
$532$	0	0
$533$	−12.5623	−0.544134
$534$	0	0
$535$	−39.4721	−1.70653
$536$	0	0
$537$	7.18034	0.309855
$538$	0	0
$539$	57.3050	2.46830
$540$	0	0
$541$	4.05573	0.174369	0.0871847	−	0.996192i	$-0.472213\pi$
$541$	4.05573	0.174369	0.0871847	+	0.996192i	$0.472213\pi$
$542$	0	0
$543$	1.93112	0.0828721
$544$	0	0
$545$	36.8328	1.57774
$546$	0	0
$547$	−6.61803	−0.282967	−0.141483	−	0.989941i	$-0.545187\pi$
$547$	−6.61803	−0.282967	−0.141483	+	0.989941i	$0.545187\pi$
$548$	0	0
$549$	34.9230	1.49048
$550$	0	0
$551$	2.94427	0.125430
$552$	0	0
$553$	13.8541	0.589136
$554$	0	0
$555$	6.78522	0.288016
$556$	0	0
$557$	5.29180	0.224221	0.112110	−	0.993696i	$-0.464239\pi$
$557$	5.29180	0.224221	0.112110	+	0.993696i	$0.464239\pi$
$558$	0	0
$559$	41.9787	1.77551
$560$	0	0
$561$	−4.72136	−0.199336
$562$	0	0
$563$	−9.09017	−0.383105	−0.191552	−	0.981482i	$-0.561352\pi$
$563$	−9.09017	−0.383105	−0.191552	+	0.981482i	$0.561352\pi$
$564$	0	0
$565$	−13.9443	−0.586640
$566$	0	0
$567$	35.5967	1.49492
$568$	0	0
$569$	−23.7639	−0.996236	−0.498118	−	0.867109i	$-0.665975\pi$
$569$	−23.7639	−0.996236	−0.498118	+	0.867109i	$0.665975\pi$
$570$	0	0
$571$	−36.3262	−1.52021	−0.760103	−	0.649803i	$-0.774851\pi$
$571$	−36.3262	−1.52021	−0.760103	+	0.649803i	$0.774851\pi$
$572$	0	0
$573$	1.49342	0.0623886
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−19.9098	−0.828857	−0.414429	−	0.910082i	$-0.636019\pi$
$577$	−19.9098	−0.828857	−0.414429	+	0.910082i	$0.636019\pi$
$578$	0	0
$579$	−2.90983	−0.120928
$580$	0	0
$581$	−8.97871	−0.372500
$582$	0	0
$583$	−36.9443	−1.53008
$584$	0	0
$585$	−35.8541	−1.48238
$586$	0	0
$587$	−18.7984	−0.775892	−0.387946	−	0.921682i	$-0.626815\pi$
$587$	−18.7984	−0.775892	−0.387946	+	0.921682i	$0.626815\pi$
$588$	0	0
$589$	0.360680	0.0148616
$590$	0	0
$591$	−3.00000	−0.123404
$592$	0	0
$593$	−1.00000	−0.0410651	−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000	−0.0410651	−0.0205325	+	0.999789i	$0.506536\pi$
$594$	0	0
$595$	31.9098	1.30818
$596$	0	0
$597$	8.43769	0.345332
$598$	0	0
$599$	−13.2148	−0.539941	−0.269971	−	0.962869i	$-0.587014\pi$
$599$	−13.2148	−0.539941	−0.269971	+	0.962869i	$0.587014\pi$
$600$	0	0
$601$	32.0344	1.30671	0.653356	−	0.757051i	$-0.273360\pi$
$601$	32.0344	1.30671	0.653356	+	0.757051i	$0.273360\pi$
$602$	0	0
$603$	17.7984	0.724806
$604$	0	0
$605$	11.1803	0.454545
$606$	0	0
$607$	−6.09017	−0.247192	−0.123596	−	0.992333i	$-0.539443\pi$
$607$	−6.09017	−0.247192	−0.123596	+	0.992333i	$0.539443\pi$
$608$	0	0
$609$	11.0000	0.445742
$610$	0	0
$611$	−1.32624	−0.0536538
$612$	0	0
$613$	25.3262	1.02292	0.511459	−	0.859308i	$-0.329105\pi$
$613$	25.3262	1.02292	0.511459	+	0.859308i	$0.329105\pi$
$614$	0	0
$615$	1.90983	0.0770118
$616$	0	0
$617$	−1.87539	−0.0755003	−0.0377501	−	0.999287i	$-0.512019\pi$
$617$	−1.87539	−0.0755003	−0.0377501	+	0.999287i	$0.512019\pi$
$618$	0	0
$619$	48.5410	1.95103	0.975514	−	0.219937i	$-0.0705851\pi$
$619$	48.5410	1.95103	0.975514	+	0.219937i	$0.0705851\pi$
$620$	0	0
$621$	13.0902	0.525290
$622$	0	0
$623$	5.70820	0.228694
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0.721360	0.0288083
$628$	0	0
$629$	−24.5492	−0.978839
$630$	0	0
$631$	−19.8541	−0.790379	−0.395190	−	0.918600i	$-0.629321\pi$
$631$	−19.8541	−0.790379	−0.395190	+	0.918600i	$0.629321\pi$
$632$	0	0
$633$	−8.34752	−0.331784
$634$	0	0
$635$	−37.7639	−1.49862
$636$	0	0
$637$	80.4853	3.18894
$638$	0	0
$639$	25.2705	0.999686
$640$	0	0
$641$	−27.4377	−1.08372	−0.541862	−	0.840468i	$-0.682280\pi$
$641$	−27.4377	−1.08372	−0.541862	+	0.840468i	$0.682280\pi$
$642$	0	0
$643$	38.0902	1.50213	0.751065	−	0.660228i	$-0.229541\pi$
$643$	38.0902	1.50213	0.751065	+	0.660228i	$0.229541\pi$
$644$	0	0
$645$	−6.38197	−0.251290
$646$	0	0
$647$	−14.0344	−0.551751	−0.275875	−	0.961193i	$-0.588968\pi$
$647$	−14.0344	−0.551751	−0.275875	+	0.961193i	$0.588968\pi$
$648$	0	0
$649$	6.83282	0.268211
$650$	0	0
$651$	1.34752	0.0528136
$652$	0	0
$653$	−10.6738	−0.417697	−0.208848	−	0.977948i	$-0.566971\pi$
$653$	−10.6738	−0.417697	−0.208848	+	0.977948i	$0.566971\pi$
$654$	0	0
$655$	35.6525	1.39306
$656$	0	0
$657$	14.5279	0.566786
$658$	0	0
$659$	43.0132	1.67555	0.837777	−	0.546012i	$-0.183855\pi$
$659$	43.0132	1.67555	0.837777	+	0.546012i	$0.183855\pi$
$660$	0	0
$661$	47.5623	1.84996	0.924980	−	0.380017i	$-0.124082\pi$
$661$	47.5623	1.84996	0.924980	+	0.380017i	$0.124082\pi$
$662$	0	0
$663$	−6.63119	−0.257534
$664$	0	0
$665$	−4.87539	−0.189059
$666$	0	0
$667$	−36.5066	−1.41354
$668$	0	0
$669$	−4.39512	−0.169925
$670$	0	0
$671$	−48.9443	−1.88947
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.6525	−0.486274	−0.243137	−	0.969992i	$-0.578176\pi$
$677$	−12.6525	−0.486274	−0.243137	+	0.969992i	$0.578176\pi$
$678$	0	0
$679$	−60.1935	−2.31001
$680$	0	0
$681$	−4.20163	−0.161007
$682$	0	0
$683$	−12.5623	−0.480683	−0.240342	−	0.970688i	$-0.577259\pi$
$683$	−12.5623	−0.480683	−0.240342	+	0.970688i	$0.577259\pi$
$684$	0	0
$685$	−37.8885	−1.44765
$686$	0	0
$687$	−6.72949	−0.256746
$688$	0	0
$689$	−51.8885	−1.97680
$690$	0	0
$691$	−14.8885	−0.566387	−0.283193	−	0.959063i	$-0.591394\pi$
$691$	−14.8885	−0.566387	−0.283193	+	0.959063i	$0.591394\pi$
$692$	0	0
$693$	−52.7214	−2.00272
$694$	0	0
$695$	15.7295	0.596654
$696$	0	0
$697$	−6.90983	−0.261728
$698$	0	0
$699$	6.41641	0.242691
$700$	0	0
$701$	−26.1246	−0.986713	−0.493356	−	0.869827i	$-0.664230\pi$
$701$	−26.1246	−0.986713	−0.493356	+	0.869827i	$0.664230\pi$
$702$	0	0
$703$	3.75078	0.141463
$704$	0	0
$705$	0.201626	0.00759368
$706$	0	0
$707$	55.1591	2.07447
$708$	0	0
$709$	37.7984	1.41955	0.709774	−	0.704430i	$-0.248797\pi$
$709$	37.7984	1.41955	0.709774	+	0.704430i	$0.248797\pi$
$710$	0	0
$711$	−8.56231	−0.321112
$712$	0	0
$713$	−4.47214	−0.167483
$714$	0	0
$715$	50.2492	1.87921
$716$	0	0
$717$	3.65248	0.136404
$718$	0	0
$719$	30.5623	1.13978	0.569891	−	0.821720i	$-0.306985\pi$
$719$	30.5623	1.13978	0.569891	+	0.821720i	$0.306985\pi$
$720$	0	0
$721$	−24.5967	−0.916031
$722$	0	0
$723$	−6.50658	−0.241982
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.0902	0.856367	0.428184	−	0.903692i	$-0.359154\pi$
$727$	23.0902	0.856367	0.428184	+	0.903692i	$0.359154\pi$
$728$	0	0
$729$	−19.4377	−0.719915
$730$	0	0
$731$	23.0902	0.854021
$732$	0	0
$733$	−1.05573	−0.0389942	−0.0194971	−	0.999810i	$-0.506207\pi$
$733$	−1.05573	−0.0389942	−0.0194971	+	0.999810i	$0.506207\pi$
$734$	0	0
$735$	−12.2361	−0.451334
$736$	0	0
$737$	−24.9443	−0.918834
$738$	0	0
$739$	34.6869	1.27598	0.637989	−	0.770045i	$-0.279766\pi$
$739$	34.6869	1.27598	0.637989	+	0.770045i	$0.279766\pi$
$740$	0	0
$741$	1.01316	0.0372192
$742$	0	0
$743$	34.6180	1.27001	0.635006	−	0.772507i	$-0.280998\pi$
$743$	34.6180	1.27001	0.635006	+	0.772507i	$0.280998\pi$
$744$	0	0
$745$	18.2148	0.667338
$746$	0	0
$747$	5.54915	0.203033
$748$	0	0
$749$	−81.5197	−2.97867
$750$	0	0
$751$	−6.41641	−0.234138	−0.117069	−	0.993124i	$-0.537350\pi$
$751$	−6.41641	−0.234138	−0.117069	+	0.993124i	$0.537350\pi$
$752$	0	0
$753$	−7.90983	−0.288250
$754$	0	0
$755$	4.79837	0.174631
$756$	0	0
$757$	−6.50658	−0.236486	−0.118243	−	0.992985i	$-0.537726\pi$
$757$	−6.50658	−0.236486	−0.118243	+	0.992985i	$0.537726\pi$
$758$	0	0
$759$	−8.94427	−0.324657
$760$	0	0
$761$	44.6869	1.61990	0.809950	−	0.586499i	$-0.199494\pi$
$761$	44.6869	1.61990	0.809950	+	0.586499i	$0.199494\pi$
$762$	0	0
$763$	76.0689	2.75388
$764$	0	0
$765$	−19.7214	−0.713027
$766$	0	0
$767$	9.59675	0.346518
$768$	0	0
$769$	−51.8328	−1.86914	−0.934570	−	0.355780i	$-0.884215\pi$
$769$	−51.8328	−1.86914	−0.934570	+	0.355780i	$0.884215\pi$
$770$	0	0
$771$	−9.15905	−0.329855
$772$	0	0
$773$	−6.06888	−0.218283	−0.109141	−	0.994026i	$-0.534810\pi$
$773$	−6.06888	−0.218283	−0.109141	+	0.994026i	$0.534810\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14.0132	0.502719
$778$	0	0
$779$	1.05573	0.0378254
$780$	0	0
$781$	−35.4164	−1.26730
$782$	0	0
$783$	−13.9443	−0.498328
$784$	0	0
$785$	45.2492	1.61501
$786$	0	0
$787$	8.18034	0.291598	0.145799	−	0.989314i	$-0.453425\pi$
$787$	8.18034	0.291598	0.145799	+	0.989314i	$0.453425\pi$
$788$	0	0
$789$	4.74265	0.168843
$790$	0	0
$791$	−28.7984	−1.02395
$792$	0	0
$793$	−68.7426	−2.44112
$794$	0	0
$795$	7.88854	0.279778
$796$	0	0
$797$	−17.7639	−0.629231	−0.314615	−	0.949219i	$-0.601876\pi$
$797$	−17.7639	−0.629231	−0.314615	+	0.949219i	$0.601876\pi$
$798$	0	0
$799$	−0.729490	−0.0258075
$800$	0	0
$801$	−3.52786	−0.124651
$802$	0	0
$803$	−20.3607	−0.718513
$804$	0	0
$805$	60.4508	2.13061
$806$	0	0
$807$	5.92299	0.208499
$808$	0	0
$809$	6.49342	0.228297	0.114148	−	0.993464i	$-0.463586\pi$
$809$	6.49342	0.228297	0.114148	+	0.993464i	$0.463586\pi$
$810$	0	0
$811$	33.1803	1.16512	0.582560	−	0.812788i	$-0.302051\pi$
$811$	33.1803	1.16512	0.582560	+	0.812788i	$0.302051\pi$
$812$	0	0
$813$	−4.58359	−0.160754
$814$	0	0
$815$	37.4853	1.31305
$816$	0	0
$817$	−3.52786	−0.123424
$818$	0	0
$819$	−74.0476	−2.58743
$820$	0	0
$821$	−15.8541	−0.553312	−0.276656	−	0.960969i	$-0.589226\pi$
$821$	−15.8541	−0.553312	−0.276656	+	0.960969i	$0.589226\pi$
$822$	0	0
$823$	53.0344	1.84866	0.924332	−	0.381589i	$-0.124623\pi$
$823$	53.0344	1.84866	0.924332	+	0.381589i	$0.124623\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.4377	−0.502048	−0.251024	−	0.967981i	$-0.580767\pi$
$827$	−14.4377	−0.502048	−0.251024	+	0.967981i	$0.580767\pi$
$828$	0	0
$829$	11.2361	0.390245	0.195122	−	0.980779i	$-0.437490\pi$
$829$	11.2361	0.390245	0.195122	+	0.980779i	$0.437490\pi$
$830$	0	0
$831$	−10.3050	−0.357475
$832$	0	0
$833$	44.2705	1.53388
$834$	0	0
$835$	2.23607	0.0773823
$836$	0	0
$837$	−1.70820	−0.0590442
$838$	0	0
$839$	−54.7082	−1.88874	−0.944368	−	0.328889i	$-0.893326\pi$
$839$	−54.7082	−1.88874	−0.944368	+	0.328889i	$0.893326\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	0	0
$843$	11.7984	0.406358
$844$	0	0
$845$	41.5066	1.42787
$846$	0	0
$847$	23.0902	0.793388
$848$	0	0
$849$	−4.94427	−0.169687
$850$	0	0
$851$	−46.5066	−1.59422
$852$	0	0
$853$	−1.45085	−0.0496761	−0.0248381	−	0.999691i	$-0.507907\pi$
$853$	−1.45085	−0.0496761	−0.0248381	+	0.999691i	$0.507907\pi$
$854$	0	0
$855$	3.01316	0.103048
$856$	0	0
$857$	−45.2148	−1.54451	−0.772254	−	0.635314i	$-0.780871\pi$
$857$	−45.2148	−1.54451	−0.772254	+	0.635314i	$0.780871\pi$
$858$	0	0
$859$	10.2918	0.351152	0.175576	−	0.984466i	$-0.443821\pi$
$859$	10.2918	0.351152	0.175576	+	0.984466i	$0.443821\pi$
$860$	0	0
$861$	3.94427	0.134420
$862$	0	0
$863$	−44.5755	−1.51737	−0.758683	−	0.651460i	$-0.774157\pi$
$863$	−44.5755	−1.51737	−0.758683	+	0.651460i	$0.774157\pi$
$864$	0	0
$865$	−7.76393	−0.263982
$866$	0	0
$867$	2.84597	0.0966543
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	−35.0344	−1.18710
$872$	0	0
$873$	37.2016	1.25908
$874$	0	0
$875$	−51.6312	−1.74545
$876$	0	0
$877$	55.6312	1.87853	0.939266	−	0.343190i	$-0.111508\pi$
$877$	55.6312	1.87853	0.939266	+	0.343190i	$0.111508\pi$
$878$	0	0
$879$	1.12461	0.0379322
$880$	0	0
$881$	−30.8328	−1.03878	−0.519392	−	0.854536i	$-0.673842\pi$
$881$	−30.8328	−1.03878	−0.519392	+	0.854536i	$0.673842\pi$
$882$	0	0
$883$	−34.7984	−1.17106	−0.585529	−	0.810651i	$-0.699113\pi$
$883$	−34.7984	−1.17106	−0.585529	+	0.810651i	$0.699113\pi$
$884$	0	0
$885$	−1.45898	−0.0490431
$886$	0	0
$887$	48.7771	1.63777	0.818887	−	0.573955i	$-0.194591\pi$
$887$	48.7771	1.63777	0.818887	+	0.573955i	$0.194591\pi$
$888$	0	0
$889$	−77.9919	−2.61576
$890$	0	0
$891$	30.8328	1.03294
$892$	0	0
$893$	0.111456	0.00372974
$894$	0	0
$895$	−42.0344	−1.40506
$896$	0	0
$897$	−12.5623	−0.419443
$898$	0	0
$899$	4.76393	0.158886
$900$	0	0
$901$	−28.5410	−0.950839
$902$	0	0
$903$	−13.1803	−0.438614
$904$	0	0
$905$	−11.3050	−0.375789
$906$	0	0
$907$	24.3050	0.807033	0.403516	−	0.914972i	$-0.367788\pi$
$907$	24.3050	0.807033	0.403516	+	0.914972i	$0.367788\pi$
$908$	0	0
$909$	−34.0902	−1.13070
$910$	0	0
$911$	2.59675	0.0860341	0.0430170	−	0.999074i	$-0.486303\pi$
$911$	2.59675	0.0860341	0.0430170	+	0.999074i	$0.486303\pi$
$912$	0	0
$913$	−7.77709	−0.257384
$914$	0	0
$915$	10.4508	0.345494
$916$	0	0
$917$	73.6312	2.43152
$918$	0	0
$919$	35.6180	1.17493	0.587465	−	0.809249i	$-0.300126\pi$
$919$	35.6180	1.17493	0.587465	+	0.809249i	$0.300126\pi$
$920$	0	0
$921$	6.45898	0.212831
$922$	0	0
$923$	−49.7426	−1.63730
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.2016	0.499287
$928$	0	0
$929$	−42.1591	−1.38319	−0.691597	−	0.722284i	$-0.743092\pi$
$929$	−42.1591	−1.38319	−0.691597	+	0.722284i	$0.743092\pi$
$930$	0	0
$931$	−6.76393	−0.221679
$932$	0	0
$933$	0.922986	0.0302172
$934$	0	0
$935$	27.6393	0.903902
$936$	0	0
$937$	−13.7082	−0.447828	−0.223914	−	0.974609i	$-0.571883\pi$
$937$	−13.7082	−0.447828	−0.223914	+	0.974609i	$0.571883\pi$
$938$	0	0
$939$	−11.1591	−0.364162
$940$	0	0
$941$	32.8885	1.07214	0.536068	−	0.844175i	$-0.319909\pi$
$941$	32.8885	1.07214	0.536068	+	0.844175i	$0.319909\pi$
$942$	0	0
$943$	−13.0902	−0.426275
$944$	0	0
$945$	23.0902	0.751123
$946$	0	0
$947$	−0.0901699	−0.00293013	−0.00146506	−	0.999999i	$-0.500466\pi$
$947$	−0.0901699	−0.00293013	−0.00146506	+	0.999999i	$0.500466\pi$
$948$	0	0
$949$	−28.5967	−0.928290
$950$	0	0
$951$	−3.06888	−0.0995154
$952$	0	0
$953$	56.5066	1.83043	0.915214	−	0.402969i	$-0.132022\pi$
$953$	56.5066	1.83043	0.915214	+	0.402969i	$0.132022\pi$
$954$	0	0
$955$	−8.74265	−0.282905
$956$	0	0
$957$	9.52786	0.307992
$958$	0	0
$959$	−78.2492	−2.52680
$960$	0	0
$961$	−30.4164	−0.981174
$962$	0	0
$963$	50.3820	1.62354
$964$	0	0
$965$	17.0344	0.548358
$966$	0	0
$967$	−49.1459	−1.58043	−0.790213	−	0.612833i	$-0.790030\pi$
$967$	−49.1459	−1.58043	−0.790213	+	0.612833i	$0.790030\pi$
$968$	0	0
$969$	0.557281	0.0179024
$970$	0	0
$971$	−1.40325	−0.0450325	−0.0225163	−	0.999746i	$-0.507168\pi$
$971$	−1.40325	−0.0450325	−0.0225163	+	0.999746i	$0.507168\pi$
$972$	0	0
$973$	32.4853	1.04143
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.9787	0.511204	0.255602	−	0.966782i	$-0.417726\pi$
$977$	15.9787	0.511204	0.255602	+	0.966782i	$0.417726\pi$
$978$	0	0
$979$	4.94427	0.158020
$980$	0	0
$981$	−47.0132	−1.50101
$982$	0	0
$983$	−58.8328	−1.87648	−0.938238	−	0.345991i	$-0.887543\pi$
$983$	−58.8328	−1.87648	−0.938238	+	0.345991i	$0.887543\pi$
$984$	0	0
$985$	17.5623	0.559582
$986$	0	0
$987$	0.416408	0.0132544
$988$	0	0
$989$	43.7426	1.39094
$990$	0	0
$991$	−0.944272	−0.0299958	−0.0149979	−	0.999888i	$-0.504774\pi$
$991$	−0.944272	−0.0299958	−0.0149979	+	0.999888i	$0.504774\pi$
$992$	0	0
$993$	−2.67376	−0.0848493
$994$	0	0
$995$	−49.3951	−1.56593
$996$	0	0
$997$	−55.6656	−1.76295	−0.881474	−	0.472232i	$-0.843448\pi$
$997$	−55.6656	−1.76295	−0.881474	+	0.472232i	$0.843448\pi$
$998$	0	0
$999$	−17.7639	−0.562026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2672.2.a.b.1.2		2
4.3	odd	2		1336.2.a.a.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.a.1.1	✓	2	4.3	odd	2
2672.2.a.b.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2672 = 2^{4} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2672.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{3} +2.23607 q^{5} +4.61803 q^{7} -2.85410 q^{9} +O(q^{10})\)	\(q-0.381966 q^{3} +2.23607 q^{5} +4.61803 q^{7} -2.85410 q^{9} +4.00000 q^{11} +5.61803 q^{13} -0.854102 q^{15} +3.09017 q^{17} -0.472136 q^{19} -1.76393 q^{21} +5.85410 q^{23} +2.23607 q^{27} -6.23607 q^{29} -0.763932 q^{31} -1.52786 q^{33} +10.3262 q^{35} -7.94427 q^{37} -2.14590 q^{39} -2.23607 q^{41} +7.47214 q^{43} -6.38197 q^{45} -0.236068 q^{47} +14.3262 q^{49} -1.18034 q^{51} -9.23607 q^{53} +8.94427 q^{55} +0.180340 q^{57} +1.70820 q^{59} -12.2361 q^{61} -13.1803 q^{63} +12.5623 q^{65} -6.23607 q^{67} -2.23607 q^{69} -8.85410 q^{71} -5.09017 q^{73} +18.4721 q^{77} +3.00000 q^{79} +7.70820 q^{81} -1.94427 q^{83} +6.90983 q^{85} +2.38197 q^{87} +1.23607 q^{89} +25.9443 q^{91} +0.291796 q^{93} -1.05573 q^{95} -13.0344 q^{97} -11.4164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 7 q^{7} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 + 7 * q^7 + q^9	\( 2 q - 3 q^{3} + 7 q^{7} + q^{9} + 8 q^{11} + 9 q^{13} + 5 q^{15} - 5 q^{17} + 8 q^{19} - 8 q^{21} + 5 q^{23} - 8 q^{29} - 6 q^{31} - 12 q^{33} + 5 q^{35} + 2 q^{37} - 11 q^{39} + 6 q^{43} - 15 q^{45} + 4 q^{47} + 13 q^{49} + 20 q^{51} - 14 q^{53} - 22 q^{57} - 10 q^{59} - 20 q^{61} - 4 q^{63} + 5 q^{65} - 8 q^{67} - 11 q^{71} + q^{73} + 28 q^{77} + 6 q^{79} + 2 q^{81} + 14 q^{83} + 25 q^{85} + 7 q^{87} - 2 q^{89} + 34 q^{91} + 14 q^{93} - 20 q^{95} + 3 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 + 7 * q^7 + q^9 + 8 * q^11 + 9 * q^13 + 5 * q^15 - 5 * q^17 + 8 * q^19 - 8 * q^21 + 5 * q^23 - 8 * q^29 - 6 * q^31 - 12 * q^33 + 5 * q^35 + 2 * q^37 - 11 * q^39 + 6 * q^43 - 15 * q^45 + 4 * q^47 + 13 * q^49 + 20 * q^51 - 14 * q^53 - 22 * q^57 - 10 * q^59 - 20 * q^61 - 4 * q^63 + 5 * q^65 - 8 * q^67 - 11 * q^71 + q^73 + 28 * q^77 + 6 * q^79 + 2 * q^81 + 14 * q^83 + 25 * q^85 + 7 * q^87 - 2 * q^89 + 34 * q^91 + 14 * q^93 - 20 * q^95 + 3 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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