LMFDB - Embedded newform 2672.2.a.e.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:	$1$
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 334)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.61803 q^{3} -1.00000 q^{5} -2.23607 q^{7} -0.381966 q^{9} +O(q^{10})$	$q+1.61803 q^{3} -1.00000 q^{5} -2.23607 q^{7} -0.381966 q^{9} +1.38197 q^{11} +2.23607 q^{13} -1.61803 q^{15} -5.85410 q^{17} +2.23607 q^{19} -3.61803 q^{21} +4.23607 q^{23} -4.00000 q^{25} -5.47214 q^{27} -3.76393 q^{29} +8.70820 q^{31} +2.23607 q^{33} +2.23607 q^{35} -8.56231 q^{37} +3.61803 q^{39} -1.76393 q^{41} -10.0902 q^{43} +0.381966 q^{45} -3.70820 q^{47} -2.00000 q^{49} -9.47214 q^{51} +1.38197 q^{53} -1.38197 q^{55} +3.61803 q^{57} -3.47214 q^{59} -3.61803 q^{61} +0.854102 q^{63} -2.23607 q^{65} -8.85410 q^{67} +6.85410 q^{69} +11.4164 q^{71} +1.61803 q^{73} -6.47214 q^{75} -3.09017 q^{77} -3.76393 q^{79} -7.70820 q^{81} -8.14590 q^{83} +5.85410 q^{85} -6.09017 q^{87} +10.1803 q^{89} -5.00000 q^{91} +14.0902 q^{93} -2.23607 q^{95} -9.70820 q^{97} -0.527864 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{5} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 - 2 * q^5 - 3 * q^9	$ 2 q + q^{3} - 2 q^{5} - 3 q^{9} + 5 q^{11} - q^{15} - 5 q^{17} - 5 q^{21} + 4 q^{23} - 8 q^{25} - 2 q^{27} - 12 q^{29} + 4 q^{31} + 3 q^{37} + 5 q^{39} - 8 q^{41} - 9 q^{43} + 3 q^{45} + 6 q^{47} - 4 q^{49} - 10 q^{51} + 5 q^{53} - 5 q^{55} + 5 q^{57} + 2 q^{59} - 5 q^{61} - 5 q^{63} - 11 q^{67} + 7 q^{69} - 4 q^{71} + q^{73} - 4 q^{75} + 5 q^{77} - 12 q^{79} - 2 q^{81} - 23 q^{83} + 5 q^{85} - q^{87} - 2 q^{89} - 10 q^{91} + 17 q^{93} - 6 q^{97} - 10 q^{99}+O(q^{100}) $ 2 * q + q^3 - 2 * q^5 - 3 * q^9 + 5 * q^11 - q^15 - 5 * q^17 - 5 * q^21 + 4 * q^23 - 8 * q^25 - 2 * q^27 - 12 * q^29 + 4 * q^31 + 3 * q^37 + 5 * q^39 - 8 * q^41 - 9 * q^43 + 3 * q^45 + 6 * q^47 - 4 * q^49 - 10 * q^51 + 5 * q^53 - 5 * q^55 + 5 * q^57 + 2 * q^59 - 5 * q^61 - 5 * q^63 - 11 * q^67 + 7 * q^69 - 4 * q^71 + q^73 - 4 * q^75 + 5 * q^77 - 12 * q^79 - 2 * q^81 - 23 * q^83 + 5 * q^85 - q^87 - 2 * q^89 - 10 * q^91 + 17 * q^93 - 6 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−2.23607	−0.845154	−0.422577	−	0.906327i	$-0.638874\pi$
$7$	−2.23607	−0.845154	−0.422577	+	0.906327i	$0.638874\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	0	0
$13$	2.23607	0.620174	0.310087	−	0.950708i	$-0.399642\pi$
$13$	2.23607	0.620174	0.310087	+	0.950708i	$0.399642\pi$
$14$	0	0
$15$	−1.61803	−0.417775
$16$	0	0
$17$	−5.85410	−1.41983	−0.709914	−	0.704288i	$-0.751266\pi$
$17$	−5.85410	−1.41983	−0.709914	+	0.704288i	$0.751266\pi$
$18$	0	0
$19$	2.23607	0.512989	0.256495	−	0.966546i	$-0.417432\pi$
$19$	2.23607	0.512989	0.256495	+	0.966546i	$0.417432\pi$
$20$	0	0
$21$	−3.61803	−0.789520
$22$	0	0
$23$	4.23607	0.883281	0.441641	−	0.897192i	$-0.354397\pi$
$23$	4.23607	0.883281	0.441641	+	0.897192i	$0.354397\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	−3.76393	−0.698945	−0.349472	−	0.936947i	$-0.613639\pi$
$29$	−3.76393	−0.698945	−0.349472	+	0.936947i	$0.613639\pi$
$30$	0	0
$31$	8.70820	1.56404	0.782020	−	0.623254i	$-0.214190\pi$
$31$	8.70820	1.56404	0.782020	+	0.623254i	$0.214190\pi$
$32$	0	0
$33$	2.23607	0.389249
$34$	0	0
$35$	2.23607	0.377964
$36$	0	0
$37$	−8.56231	−1.40763	−0.703817	−	0.710381i	$-0.748523\pi$
$37$	−8.56231	−1.40763	−0.703817	+	0.710381i	$0.748523\pi$
$38$	0	0
$39$	3.61803	0.579349
$40$	0	0
$41$	−1.76393	−0.275480	−0.137740	−	0.990468i	$-0.543984\pi$
$41$	−1.76393	−0.275480	−0.137740	+	0.990468i	$0.543984\pi$
$42$	0	0
$43$	−10.0902	−1.53874	−0.769368	−	0.638806i	$-0.779429\pi$
$43$	−10.0902	−1.53874	−0.769368	+	0.638806i	$0.779429\pi$
$44$	0	0
$45$	0.381966	0.0569401
$46$	0	0
$47$	−3.70820	−0.540897	−0.270449	−	0.962734i	$-0.587172\pi$
$47$	−3.70820	−0.540897	−0.270449	+	0.962734i	$0.587172\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−9.47214	−1.32636
$52$	0	0
$53$	1.38197	0.189828	0.0949138	−	0.995485i	$-0.469742\pi$
$53$	1.38197	0.189828	0.0949138	+	0.995485i	$0.469742\pi$
$54$	0	0
$55$	−1.38197	−0.186344
$56$	0	0
$57$	3.61803	0.479220
$58$	0	0
$59$	−3.47214	−0.452034	−0.226017	−	0.974123i	$-0.572570\pi$
$59$	−3.47214	−0.452034	−0.226017	+	0.974123i	$0.572570\pi$
$60$	0	0
$61$	−3.61803	−0.463242	−0.231621	−	0.972806i	$-0.574403\pi$
$61$	−3.61803	−0.463242	−0.231621	+	0.972806i	$0.574403\pi$
$62$	0	0
$63$	0.854102	0.107607
$64$	0	0
$65$	−2.23607	−0.277350
$66$	0	0
$67$	−8.85410	−1.08170	−0.540850	−	0.841119i	$-0.681897\pi$
$67$	−8.85410	−1.08170	−0.540850	+	0.841119i	$0.681897\pi$
$68$	0	0
$69$	6.85410	0.825137
$70$	0	0
$71$	11.4164	1.35488	0.677439	−	0.735579i	$-0.263090\pi$
$71$	11.4164	1.35488	0.677439	+	0.735579i	$0.263090\pi$
$72$	0	0
$73$	1.61803	0.189377	0.0946883	−	0.995507i	$-0.469815\pi$
$73$	1.61803	0.189377	0.0946883	+	0.995507i	$0.469815\pi$
$74$	0	0
$75$	−6.47214	−0.747338
$76$	0	0
$77$	−3.09017	−0.352158
$78$	0	0
$79$	−3.76393	−0.423475	−0.211738	−	0.977327i	$-0.567912\pi$
$79$	−3.76393	−0.423475	−0.211738	+	0.977327i	$0.567912\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−8.14590	−0.894128	−0.447064	−	0.894502i	$-0.647530\pi$
$83$	−8.14590	−0.894128	−0.447064	+	0.894502i	$0.647530\pi$
$84$	0	0
$85$	5.85410	0.634967
$86$	0	0
$87$	−6.09017	−0.652935
$88$	0	0
$89$	10.1803	1.07911	0.539557	−	0.841949i	$-0.318592\pi$
$89$	10.1803	1.07911	0.539557	+	0.841949i	$0.318592\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	0	0
$93$	14.0902	1.46108
$94$	0	0
$95$	−2.23607	−0.229416
$96$	0	0
$97$	−9.70820	−0.985719	−0.492859	−	0.870109i	$-0.664048\pi$
$97$	−9.70820	−0.985719	−0.492859	+	0.870109i	$0.664048\pi$
$98$	0	0
$99$	−0.527864	−0.0530523
$100$	0	0
$101$	−12.2361	−1.21753	−0.608767	−	0.793349i	$-0.708336\pi$
$101$	−12.2361	−1.21753	−0.608767	+	0.793349i	$0.708336\pi$
$102$	0	0
$103$	−11.8541	−1.16802	−0.584010	−	0.811747i	$-0.698517\pi$
$103$	−11.8541	−1.16802	−0.584010	+	0.811747i	$0.698517\pi$
$104$	0	0
$105$	3.61803	0.353084
$106$	0	0
$107$	1.61803	0.156421	0.0782106	−	0.996937i	$-0.475079\pi$
$107$	1.61803	0.156421	0.0782106	+	0.996937i	$0.475079\pi$
$108$	0	0
$109$	3.70820	0.355182	0.177591	−	0.984104i	$-0.443170\pi$
$109$	3.70820	0.355182	0.177591	+	0.984104i	$0.443170\pi$
$110$	0	0
$111$	−13.8541	−1.31497
$112$	0	0
$113$	−7.61803	−0.716644	−0.358322	−	0.933598i	$-0.616651\pi$
$113$	−7.61803	−0.716644	−0.358322	+	0.933598i	$0.616651\pi$
$114$	0	0
$115$	−4.23607	−0.395015
$116$	0	0
$117$	−0.854102	−0.0789618
$118$	0	0
$119$	13.0902	1.19997
$120$	0	0
$121$	−9.09017	−0.826379
$122$	0	0
$123$	−2.85410	−0.257346
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−7.85410	−0.696939	−0.348469	−	0.937320i	$-0.613298\pi$
$127$	−7.85410	−0.696939	−0.348469	+	0.937320i	$0.613298\pi$
$128$	0	0
$129$	−16.3262	−1.43745
$130$	0	0
$131$	−7.18034	−0.627349	−0.313675	−	0.949531i	$-0.601560\pi$
$131$	−7.18034	−0.627349	−0.313675	+	0.949531i	$0.601560\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	0	0
$135$	5.47214	0.470966
$136$	0	0
$137$	−16.6180	−1.41977	−0.709887	−	0.704315i	$-0.751254\pi$
$137$	−16.6180	−1.41977	−0.709887	+	0.704315i	$0.751254\pi$
$138$	0	0
$139$	7.23607	0.613755	0.306878	−	0.951749i	$-0.400716\pi$
$139$	7.23607	0.613755	0.306878	+	0.951749i	$0.400716\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	3.09017	0.258413
$144$	0	0
$145$	3.76393	0.312578
$146$	0	0
$147$	−3.23607	−0.266906
$148$	0	0
$149$	14.1803	1.16170	0.580849	−	0.814011i	$-0.302721\pi$
$149$	14.1803	1.16170	0.580849	+	0.814011i	$0.302721\pi$
$150$	0	0
$151$	14.7082	1.19694	0.598468	−	0.801146i	$-0.295776\pi$
$151$	14.7082	1.19694	0.598468	+	0.801146i	$0.295776\pi$
$152$	0	0
$153$	2.23607	0.180775
$154$	0	0
$155$	−8.70820	−0.699460
$156$	0	0
$157$	6.79837	0.542569	0.271285	−	0.962499i	$-0.412552\pi$
$157$	6.79837	0.542569	0.271285	+	0.962499i	$0.412552\pi$
$158$	0	0
$159$	2.23607	0.177332
$160$	0	0
$161$	−9.47214	−0.746509
$162$	0	0
$163$	16.0344	1.25591	0.627957	−	0.778248i	$-0.283891\pi$
$163$	16.0344	1.25591	0.627957	+	0.778248i	$0.283891\pi$
$164$	0	0
$165$	−2.23607	−0.174078
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	−0.854102	−0.0653148
$172$	0	0
$173$	1.29180	0.0982134	0.0491067	−	0.998794i	$-0.484363\pi$
$173$	1.29180	0.0982134	0.0491067	+	0.998794i	$0.484363\pi$
$174$	0	0
$175$	8.94427	0.676123
$176$	0	0
$177$	−5.61803	−0.422277
$178$	0	0
$179$	16.8541	1.25973	0.629867	−	0.776703i	$-0.283109\pi$
$179$	16.8541	1.25973	0.629867	+	0.776703i	$0.283109\pi$
$180$	0	0
$181$	14.4164	1.07156	0.535782	−	0.844357i	$-0.320017\pi$
$181$	14.4164	1.07156	0.535782	+	0.844357i	$0.320017\pi$
$182$	0	0
$183$	−5.85410	−0.432748
$184$	0	0
$185$	8.56231	0.629513
$186$	0	0
$187$	−8.09017	−0.591612
$188$	0	0
$189$	12.2361	0.890043
$190$	0	0
$191$	−10.7639	−0.778851	−0.389425	−	0.921058i	$-0.627326\pi$
$191$	−10.7639	−0.778851	−0.389425	+	0.921058i	$0.627326\pi$
$192$	0	0
$193$	14.7984	1.06521	0.532605	−	0.846364i	$-0.321213\pi$
$193$	14.7984	1.06521	0.532605	+	0.846364i	$0.321213\pi$
$194$	0	0
$195$	−3.61803	−0.259093
$196$	0	0
$197$	7.61803	0.542762	0.271381	−	0.962472i	$-0.412520\pi$
$197$	7.61803	0.542762	0.271381	+	0.962472i	$0.412520\pi$
$198$	0	0
$199$	−10.0344	−0.711323	−0.355661	−	0.934615i	$-0.615744\pi$
$199$	−10.0344	−0.711323	−0.355661	+	0.934615i	$0.615744\pi$
$200$	0	0
$201$	−14.3262	−1.01049
$202$	0	0
$203$	8.41641	0.590716
$204$	0	0
$205$	1.76393	0.123198
$206$	0	0
$207$	−1.61803	−0.112461
$208$	0	0
$209$	3.09017	0.213752
$210$	0	0
$211$	13.3262	0.917416	0.458708	−	0.888587i	$-0.348312\pi$
$211$	13.3262	0.917416	0.458708	+	0.888587i	$0.348312\pi$
$212$	0	0
$213$	18.4721	1.26569
$214$	0	0
$215$	10.0902	0.688144
$216$	0	0
$217$	−19.4721	−1.32185
$218$	0	0
$219$	2.61803	0.176910
$220$	0	0
$221$	−13.0902	−0.880540
$222$	0	0
$223$	−13.0000	−0.870544	−0.435272	−	0.900299i	$-0.643348\pi$
$223$	−13.0000	−0.870544	−0.435272	+	0.900299i	$0.643348\pi$
$224$	0	0
$225$	1.52786	0.101858
$226$	0	0
$227$	14.1803	0.941182	0.470591	−	0.882351i	$-0.344041\pi$
$227$	14.1803	0.941182	0.470591	+	0.882351i	$0.344041\pi$
$228$	0	0
$229$	−12.4164	−0.820499	−0.410250	−	0.911973i	$-0.634558\pi$
$229$	−12.4164	−0.820499	−0.410250	+	0.911973i	$0.634558\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	24.9787	1.63641	0.818205	−	0.574927i	$-0.194969\pi$
$233$	24.9787	1.63641	0.818205	+	0.574927i	$0.194969\pi$
$234$	0	0
$235$	3.70820	0.241897
$236$	0	0
$237$	−6.09017	−0.395599
$238$	0	0
$239$	8.67376	0.561059	0.280530	−	0.959845i	$-0.409490\pi$
$239$	8.67376	0.561059	0.280530	+	0.959845i	$0.409490\pi$
$240$	0	0
$241$	0.326238	0.0210148	0.0105074	−	0.999945i	$-0.496655\pi$
$241$	0.326238	0.0210148	0.0105074	+	0.999945i	$0.496655\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	−13.1803	−0.835270
$250$	0	0
$251$	−18.4721	−1.16595	−0.582975	−	0.812490i	$-0.698112\pi$
$251$	−18.4721	−1.16595	−0.582975	+	0.812490i	$0.698112\pi$
$252$	0	0
$253$	5.85410	0.368044
$254$	0	0
$255$	9.47214	0.593168
$256$	0	0
$257$	−14.0344	−0.875444	−0.437722	−	0.899110i	$-0.644215\pi$
$257$	−14.0344	−0.875444	−0.437722	+	0.899110i	$0.644215\pi$
$258$	0	0
$259$	19.1459	1.18967
$260$	0	0
$261$	1.43769	0.0889910
$262$	0	0
$263$	−8.41641	−0.518978	−0.259489	−	0.965746i	$-0.583554\pi$
$263$	−8.41641	−0.518978	−0.259489	+	0.965746i	$0.583554\pi$
$264$	0	0
$265$	−1.38197	−0.0848935
$266$	0	0
$267$	16.4721	1.00808
$268$	0	0
$269$	−0.0901699	−0.00549776	−0.00274888	−	0.999996i	$-0.500875\pi$
$269$	−0.0901699	−0.00549776	−0.00274888	+	0.999996i	$0.500875\pi$
$270$	0	0
$271$	25.3607	1.54055	0.770276	−	0.637711i	$-0.220119\pi$
$271$	25.3607	1.54055	0.770276	+	0.637711i	$0.220119\pi$
$272$	0	0
$273$	−8.09017	−0.489639
$274$	0	0
$275$	−5.52786	−0.333343
$276$	0	0
$277$	6.41641	0.385525	0.192762	−	0.981245i	$-0.438255\pi$
$277$	6.41641	0.385525	0.192762	+	0.981245i	$0.438255\pi$
$278$	0	0
$279$	−3.32624	−0.199137
$280$	0	0
$281$	18.6525	1.11271	0.556357	−	0.830944i	$-0.312199\pi$
$281$	18.6525	1.11271	0.556357	+	0.830944i	$0.312199\pi$
$282$	0	0
$283$	−20.7082	−1.23097	−0.615487	−	0.788147i	$-0.711041\pi$
$283$	−20.7082	−1.23097	−0.615487	+	0.788147i	$0.711041\pi$
$284$	0	0
$285$	−3.61803	−0.214314
$286$	0	0
$287$	3.94427	0.232823
$288$	0	0
$289$	17.2705	1.01591
$290$	0	0
$291$	−15.7082	−0.920831
$292$	0	0
$293$	18.2918	1.06862	0.534309	−	0.845289i	$-0.320572\pi$
$293$	18.2918	1.06862	0.534309	+	0.845289i	$0.320572\pi$
$294$	0	0
$295$	3.47214	0.202156
$296$	0	0
$297$	−7.56231	−0.438809
$298$	0	0
$299$	9.47214	0.547788
$300$	0	0
$301$	22.5623	1.30047
$302$	0	0
$303$	−19.7984	−1.13739
$304$	0	0
$305$	3.61803	0.207168
$306$	0	0
$307$	−34.4508	−1.96621	−0.983107	−	0.183032i	$-0.941409\pi$
$307$	−34.4508	−1.96621	−0.983107	+	0.183032i	$0.941409\pi$
$308$	0	0
$309$	−19.1803	−1.09113
$310$	0	0
$311$	−11.5623	−0.655638	−0.327819	−	0.944741i	$-0.606314\pi$
$311$	−11.5623	−0.655638	−0.327819	+	0.944741i	$0.606314\pi$
$312$	0	0
$313$	27.8328	1.57320	0.786602	−	0.617461i	$-0.211838\pi$
$313$	27.8328	1.57320	0.786602	+	0.617461i	$0.211838\pi$
$314$	0	0
$315$	−0.854102	−0.0481232
$316$	0	0
$317$	−17.3607	−0.975073	−0.487536	−	0.873103i	$-0.662104\pi$
$317$	−17.3607	−0.975073	−0.487536	+	0.873103i	$0.662104\pi$
$318$	0	0
$319$	−5.20163	−0.291235
$320$	0	0
$321$	2.61803	0.146124
$322$	0	0
$323$	−13.0902	−0.728357
$324$	0	0
$325$	−8.94427	−0.496139
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	8.29180	0.457142
$330$	0	0
$331$	16.6180	0.913410	0.456705	−	0.889618i	$-0.349030\pi$
$331$	16.6180	0.913410	0.456705	+	0.889618i	$0.349030\pi$
$332$	0	0
$333$	3.27051	0.179223
$334$	0	0
$335$	8.85410	0.483751
$336$	0	0
$337$	−20.3820	−1.11028	−0.555138	−	0.831758i	$-0.687335\pi$
$337$	−20.3820	−1.11028	−0.555138	+	0.831758i	$0.687335\pi$
$338$	0	0
$339$	−12.3262	−0.669469
$340$	0	0
$341$	12.0344	0.651702
$342$	0	0
$343$	20.1246	1.08663
$344$	0	0
$345$	−6.85410	−0.369012
$346$	0	0
$347$	11.1459	0.598343	0.299172	−	0.954199i	$-0.403290\pi$
$347$	11.1459	0.598343	0.299172	+	0.954199i	$0.403290\pi$
$348$	0	0
$349$	10.7639	0.576180	0.288090	−	0.957603i	$-0.406980\pi$
$349$	10.7639	0.576180	0.288090	+	0.957603i	$0.406980\pi$
$350$	0	0
$351$	−12.2361	−0.653113
$352$	0	0
$353$	−13.5623	−0.721849	−0.360924	−	0.932595i	$-0.617539\pi$
$353$	−13.5623	−0.721849	−0.360924	+	0.932595i	$0.617539\pi$
$354$	0	0
$355$	−11.4164	−0.605920
$356$	0	0
$357$	21.1803	1.12098
$358$	0	0
$359$	−9.65248	−0.509438	−0.254719	−	0.967015i	$-0.581983\pi$
$359$	−9.65248	−0.509438	−0.254719	+	0.967015i	$0.581983\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	0	0
$363$	−14.7082	−0.771980
$364$	0	0
$365$	−1.61803	−0.0846918
$366$	0	0
$367$	12.7082	0.663363	0.331681	−	0.943391i	$-0.392384\pi$
$367$	12.7082	0.663363	0.331681	+	0.943391i	$0.392384\pi$
$368$	0	0
$369$	0.673762	0.0350747
$370$	0	0
$371$	−3.09017	−0.160434
$372$	0	0
$373$	−10.7984	−0.559119	−0.279559	−	0.960128i	$-0.590188\pi$
$373$	−10.7984	−0.559119	−0.279559	+	0.960128i	$0.590188\pi$
$374$	0	0
$375$	14.5623	0.751994
$376$	0	0
$377$	−8.41641	−0.433467
$378$	0	0
$379$	13.2705	0.681660	0.340830	−	0.940125i	$-0.389292\pi$
$379$	13.2705	0.681660	0.340830	+	0.940125i	$0.389292\pi$
$380$	0	0
$381$	−12.7082	−0.651061
$382$	0	0
$383$	−12.3262	−0.629841	−0.314921	−	0.949118i	$-0.601978\pi$
$383$	−12.3262	−0.629841	−0.314921	+	0.949118i	$0.601978\pi$
$384$	0	0
$385$	3.09017	0.157490
$386$	0	0
$387$	3.85410	0.195915
$388$	0	0
$389$	−22.0902	−1.12002	−0.560008	−	0.828487i	$-0.689202\pi$
$389$	−22.0902	−1.12002	−0.560008	+	0.828487i	$0.689202\pi$
$390$	0	0
$391$	−24.7984	−1.25411
$392$	0	0
$393$	−11.6180	−0.586052
$394$	0	0
$395$	3.76393	0.189384
$396$	0	0
$397$	−35.2361	−1.76845	−0.884224	−	0.467063i	$-0.845312\pi$
$397$	−35.2361	−1.76845	−0.884224	+	0.467063i	$0.845312\pi$
$398$	0	0
$399$	−8.09017	−0.405015
$400$	0	0
$401$	3.76393	0.187962	0.0939809	−	0.995574i	$-0.470041\pi$
$401$	3.76393	0.187962	0.0939809	+	0.995574i	$0.470041\pi$
$402$	0	0
$403$	19.4721	0.969976
$404$	0	0
$405$	7.70820	0.383024
$406$	0	0
$407$	−11.8328	−0.586531
$408$	0	0
$409$	−14.3262	−0.708387	−0.354193	−	0.935172i	$-0.615245\pi$
$409$	−14.3262	−0.708387	−0.354193	+	0.935172i	$0.615245\pi$
$410$	0	0
$411$	−26.8885	−1.32631
$412$	0	0
$413$	7.76393	0.382038
$414$	0	0
$415$	8.14590	0.399866
$416$	0	0
$417$	11.7082	0.573353
$418$	0	0
$419$	−11.4164	−0.557728	−0.278864	−	0.960331i	$-0.589958\pi$
$419$	−11.4164	−0.557728	−0.278864	+	0.960331i	$0.589958\pi$
$420$	0	0
$421$	−35.2148	−1.71626	−0.858132	−	0.513430i	$-0.828375\pi$
$421$	−35.2148	−1.71626	−0.858132	+	0.513430i	$0.828375\pi$
$422$	0	0
$423$	1.41641	0.0688681
$424$	0	0
$425$	23.4164	1.13586
$426$	0	0
$427$	8.09017	0.391511
$428$	0	0
$429$	5.00000	0.241402
$430$	0	0
$431$	6.58359	0.317120	0.158560	−	0.987349i	$-0.449315\pi$
$431$	6.58359	0.317120	0.158560	+	0.987349i	$0.449315\pi$
$432$	0	0
$433$	−13.1246	−0.630729	−0.315364	−	0.948971i	$-0.602127\pi$
$433$	−13.1246	−0.630729	−0.315364	+	0.948971i	$0.602127\pi$
$434$	0	0
$435$	6.09017	0.292001
$436$	0	0
$437$	9.47214	0.453114
$438$	0	0
$439$	19.7639	0.943281	0.471641	−	0.881791i	$-0.343662\pi$
$439$	19.7639	0.943281	0.471641	+	0.881791i	$0.343662\pi$
$440$	0	0
$441$	0.763932	0.0363777
$442$	0	0
$443$	39.6869	1.88558	0.942791	−	0.333384i	$-0.108191\pi$
$443$	39.6869	1.88558	0.942791	+	0.333384i	$0.108191\pi$
$444$	0	0
$445$	−10.1803	−0.482594
$446$	0	0
$447$	22.9443	1.08523
$448$	0	0
$449$	14.2361	0.671842	0.335921	−	0.941890i	$-0.390953\pi$
$449$	14.2361	0.671842	0.335921	+	0.941890i	$0.390953\pi$
$450$	0	0
$451$	−2.43769	−0.114787
$452$	0	0
$453$	23.7984	1.11815
$454$	0	0
$455$	5.00000	0.234404
$456$	0	0
$457$	−36.2705	−1.69666	−0.848331	−	0.529466i	$-0.822392\pi$
$457$	−36.2705	−1.69666	−0.848331	+	0.529466i	$0.822392\pi$
$458$	0	0
$459$	32.0344	1.49524
$460$	0	0
$461$	36.4721	1.69868	0.849338	−	0.527849i	$-0.177001\pi$
$461$	36.4721	1.69868	0.849338	+	0.527849i	$0.177001\pi$
$462$	0	0
$463$	−5.70820	−0.265283	−0.132641	−	0.991164i	$-0.542346\pi$
$463$	−5.70820	−0.265283	−0.132641	+	0.991164i	$0.542346\pi$
$464$	0	0
$465$	−14.0902	−0.653416
$466$	0	0
$467$	42.5410	1.96856	0.984282	−	0.176605i	$-0.0565115\pi$
$467$	42.5410	1.96856	0.984282	+	0.176605i	$0.0565115\pi$
$468$	0	0
$469$	19.7984	0.914204
$470$	0	0
$471$	11.0000	0.506853
$472$	0	0
$473$	−13.9443	−0.641158
$474$	0	0
$475$	−8.94427	−0.410391
$476$	0	0
$477$	−0.527864	−0.0241692
$478$	0	0
$479$	−29.8885	−1.36564	−0.682821	−	0.730586i	$-0.739247\pi$
$479$	−29.8885	−1.36564	−0.682821	+	0.730586i	$0.739247\pi$
$480$	0	0
$481$	−19.1459	−0.872978
$482$	0	0
$483$	−15.3262	−0.697368
$484$	0	0
$485$	9.70820	0.440827
$486$	0	0
$487$	30.3262	1.37421	0.687107	−	0.726557i	$-0.258881\pi$
$487$	30.3262	1.37421	0.687107	+	0.726557i	$0.258881\pi$
$488$	0	0
$489$	25.9443	1.17324
$490$	0	0
$491$	19.6180	0.885349	0.442675	−	0.896682i	$-0.354030\pi$
$491$	19.6180	0.885349	0.442675	+	0.896682i	$0.354030\pi$
$492$	0	0
$493$	22.0344	0.992381
$494$	0	0
$495$	0.527864	0.0237257
$496$	0	0
$497$	−25.5279	−1.14508
$498$	0	0
$499$	−40.3262	−1.80525	−0.902625	−	0.430427i	$-0.858363\pi$
$499$	−40.3262	−1.80525	−0.902625	+	0.430427i	$0.858363\pi$
$500$	0	0
$501$	1.61803	0.0722884
$502$	0	0
$503$	26.8885	1.19890	0.599450	−	0.800412i	$-0.295386\pi$
$503$	26.8885	1.19890	0.599450	+	0.800412i	$0.295386\pi$
$504$	0	0
$505$	12.2361	0.544498
$506$	0	0
$507$	−12.9443	−0.574875
$508$	0	0
$509$	−5.03444	−0.223148	−0.111574	−	0.993756i	$-0.535589\pi$
$509$	−5.03444	−0.223148	−0.111574	+	0.993756i	$0.535589\pi$
$510$	0	0
$511$	−3.61803	−0.160052
$512$	0	0
$513$	−12.2361	−0.540236
$514$	0	0
$515$	11.8541	0.522354
$516$	0	0
$517$	−5.12461	−0.225380
$518$	0	0
$519$	2.09017	0.0917483
$520$	0	0
$521$	35.4721	1.55406	0.777031	−	0.629462i	$-0.216725\pi$
$521$	35.4721	1.55406	0.777031	+	0.629462i	$0.216725\pi$
$522$	0	0
$523$	20.1246	0.879988	0.439994	−	0.898001i	$-0.354981\pi$
$523$	20.1246	0.879988	0.439994	+	0.898001i	$0.354981\pi$
$524$	0	0
$525$	14.4721	0.631616
$526$	0	0
$527$	−50.9787	−2.22067
$528$	0	0
$529$	−5.05573	−0.219814
$530$	0	0
$531$	1.32624	0.0575538
$532$	0	0
$533$	−3.94427	−0.170845
$534$	0	0
$535$	−1.61803	−0.0699537
$536$	0	0
$537$	27.2705	1.17681
$538$	0	0
$539$	−2.76393	−0.119051
$540$	0	0
$541$	−6.32624	−0.271986	−0.135993	−	0.990710i	$-0.543423\pi$
$541$	−6.32624	−0.271986	−0.135993	+	0.990710i	$0.543423\pi$
$542$	0	0
$543$	23.3262	1.00102
$544$	0	0
$545$	−3.70820	−0.158842
$546$	0	0
$547$	14.4721	0.618784	0.309392	−	0.950935i	$-0.399875\pi$
$547$	14.4721	0.618784	0.309392	+	0.950935i	$0.399875\pi$
$548$	0	0
$549$	1.38197	0.0589809
$550$	0	0
$551$	−8.41641	−0.358551
$552$	0	0
$553$	8.41641	0.357902
$554$	0	0
$555$	13.8541	0.588074
$556$	0	0
$557$	42.9787	1.82107	0.910533	−	0.413436i	$-0.135671\pi$
$557$	42.9787	1.82107	0.910533	+	0.413436i	$0.135671\pi$
$558$	0	0
$559$	−22.5623	−0.954284
$560$	0	0
$561$	−13.0902	−0.552667
$562$	0	0
$563$	−18.2361	−0.768559	−0.384279	−	0.923217i	$-0.625550\pi$
$563$	−18.2361	−0.768559	−0.384279	+	0.923217i	$0.625550\pi$
$564$	0	0
$565$	7.61803	0.320493
$566$	0	0
$567$	17.2361	0.723847
$568$	0	0
$569$	−34.2705	−1.43669	−0.718347	−	0.695685i	$-0.755101\pi$
$569$	−34.2705	−1.43669	−0.718347	+	0.695685i	$0.755101\pi$
$570$	0	0
$571$	−3.59675	−0.150519	−0.0752596	−	0.997164i	$-0.523979\pi$
$571$	−3.59675	−0.150519	−0.0752596	+	0.997164i	$0.523979\pi$
$572$	0	0
$573$	−17.4164	−0.727581
$574$	0	0
$575$	−16.9443	−0.706625
$576$	0	0
$577$	24.8885	1.03612	0.518062	−	0.855343i	$-0.326654\pi$
$577$	24.8885	1.03612	0.518062	+	0.855343i	$0.326654\pi$
$578$	0	0
$579$	23.9443	0.995090
$580$	0	0
$581$	18.2148	0.755676
$582$	0	0
$583$	1.90983	0.0790971
$584$	0	0
$585$	0.854102	0.0353128
$586$	0	0
$587$	−7.67376	−0.316730	−0.158365	−	0.987381i	$-0.550622\pi$
$587$	−7.67376	−0.316730	−0.158365	+	0.987381i	$0.550622\pi$
$588$	0	0
$589$	19.4721	0.802335
$590$	0	0
$591$	12.3262	0.507034
$592$	0	0
$593$	−18.0344	−0.740586	−0.370293	−	0.928915i	$-0.620743\pi$
$593$	−18.0344	−0.740586	−0.370293	+	0.928915i	$0.620743\pi$
$594$	0	0
$595$	−13.0902	−0.536645
$596$	0	0
$597$	−16.2361	−0.664498
$598$	0	0
$599$	17.5066	0.715299	0.357650	−	0.933856i	$-0.383578\pi$
$599$	17.5066	0.715299	0.357650	+	0.933856i	$0.383578\pi$
$600$	0	0
$601$	−18.2016	−0.742460	−0.371230	−	0.928541i	$-0.621064\pi$
$601$	−18.2016	−0.742460	−0.371230	+	0.928541i	$0.621064\pi$
$602$	0	0
$603$	3.38197	0.137724
$604$	0	0
$605$	9.09017	0.369568
$606$	0	0
$607$	−12.5967	−0.511286	−0.255643	−	0.966771i	$-0.582287\pi$
$607$	−12.5967	−0.511286	−0.255643	+	0.966771i	$0.582287\pi$
$608$	0	0
$609$	13.6180	0.551831
$610$	0	0
$611$	−8.29180	−0.335450
$612$	0	0
$613$	−35.4721	−1.43271	−0.716353	−	0.697738i	$-0.754190\pi$
$613$	−35.4721	−1.43271	−0.716353	+	0.697738i	$0.754190\pi$
$614$	0	0
$615$	2.85410	0.115088
$616$	0	0
$617$	2.20163	0.0886341	0.0443171	−	0.999018i	$-0.485889\pi$
$617$	2.20163	0.0886341	0.0443171	+	0.999018i	$0.485889\pi$
$618$	0	0
$619$	28.4164	1.14215	0.571076	−	0.820897i	$-0.306526\pi$
$619$	28.4164	1.14215	0.571076	+	0.820897i	$0.306526\pi$
$620$	0	0
$621$	−23.1803	−0.930195
$622$	0	0
$623$	−22.7639	−0.912018
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	5.00000	0.199681
$628$	0	0
$629$	50.1246	1.99860
$630$	0	0
$631$	−46.1803	−1.83841	−0.919205	−	0.393779i	$-0.871168\pi$
$631$	−46.1803	−1.83841	−0.919205	+	0.393779i	$0.871168\pi$
$632$	0	0
$633$	21.5623	0.857025
$634$	0	0
$635$	7.85410	0.311681
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	−4.36068	−0.172506
$640$	0	0
$641$	−10.3607	−0.409222	−0.204611	−	0.978843i	$-0.565593\pi$
$641$	−10.3607	−0.409222	−0.204611	+	0.978843i	$0.565593\pi$
$642$	0	0
$643$	−16.5410	−0.652314	−0.326157	−	0.945316i	$-0.605754\pi$
$643$	−16.5410	−0.652314	−0.326157	+	0.945316i	$0.605754\pi$
$644$	0	0
$645$	16.3262	0.642845
$646$	0	0
$647$	19.2705	0.757602	0.378801	−	0.925478i	$-0.376336\pi$
$647$	19.2705	0.757602	0.378801	+	0.925478i	$0.376336\pi$
$648$	0	0
$649$	−4.79837	−0.188353
$650$	0	0
$651$	−31.5066	−1.23484
$652$	0	0
$653$	33.6525	1.31692	0.658462	−	0.752614i	$-0.271208\pi$
$653$	33.6525	1.31692	0.658462	+	0.752614i	$0.271208\pi$
$654$	0	0
$655$	7.18034	0.280559
$656$	0	0
$657$	−0.618034	−0.0241118
$658$	0	0
$659$	12.7082	0.495041	0.247521	−	0.968883i	$-0.420384\pi$
$659$	12.7082	0.495041	0.247521	+	0.968883i	$0.420384\pi$
$660$	0	0
$661$	−28.7771	−1.11930	−0.559649	−	0.828729i	$-0.689064\pi$
$661$	−28.7771	−1.11930	−0.559649	+	0.828729i	$0.689064\pi$
$662$	0	0
$663$	−21.1803	−0.822576
$664$	0	0
$665$	5.00000	0.193892
$666$	0	0
$667$	−15.9443	−0.617365
$668$	0	0
$669$	−21.0344	−0.813239
$670$	0	0
$671$	−5.00000	−0.193023
$672$	0	0
$673$	−48.4853	−1.86897	−0.934485	−	0.356002i	$-0.884140\pi$
$673$	−48.4853	−1.86897	−0.934485	+	0.356002i	$0.884140\pi$
$674$	0	0
$675$	21.8885	0.842490
$676$	0	0
$677$	−36.5623	−1.40520	−0.702602	−	0.711583i	$-0.747978\pi$
$677$	−36.5623	−1.40520	−0.702602	+	0.711583i	$0.747978\pi$
$678$	0	0
$679$	21.7082	0.833084
$680$	0	0
$681$	22.9443	0.879226
$682$	0	0
$683$	−19.8197	−0.758378	−0.379189	−	0.925319i	$-0.623797\pi$
$683$	−19.8197	−0.758378	−0.379189	+	0.925319i	$0.623797\pi$
$684$	0	0
$685$	16.6180	0.634942
$686$	0	0
$687$	−20.0902	−0.766488
$688$	0	0
$689$	3.09017	0.117726
$690$	0	0
$691$	−0.236068	−0.00898045	−0.00449022	−	0.999990i	$-0.501429\pi$
$691$	−0.236068	−0.00898045	−0.00449022	+	0.999990i	$0.501429\pi$
$692$	0	0
$693$	1.18034	0.0448374
$694$	0	0
$695$	−7.23607	−0.274480
$696$	0	0
$697$	10.3262	0.391134
$698$	0	0
$699$	40.4164	1.52869
$700$	0	0
$701$	34.6525	1.30881	0.654403	−	0.756146i	$-0.272920\pi$
$701$	34.6525	1.30881	0.654403	+	0.756146i	$0.272920\pi$
$702$	0	0
$703$	−19.1459	−0.722101
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	27.3607	1.02900
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	1.43769	0.0539177
$712$	0	0
$713$	36.8885	1.38149
$714$	0	0
$715$	−3.09017	−0.115566
$716$	0	0
$717$	14.0344	0.524126
$718$	0	0
$719$	−23.2918	−0.868637	−0.434319	−	0.900759i	$-0.643011\pi$
$719$	−23.2918	−0.868637	−0.434319	+	0.900759i	$0.643011\pi$
$720$	0	0
$721$	26.5066	0.987157
$722$	0	0
$723$	0.527864	0.0196315
$724$	0	0
$725$	15.0557	0.559156
$726$	0	0
$727$	23.6525	0.877222	0.438611	−	0.898677i	$-0.355471\pi$
$727$	23.6525	0.877222	0.438611	+	0.898677i	$0.355471\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	59.0689	2.18474
$732$	0	0
$733$	−0.742646	−0.0274302	−0.0137151	−	0.999906i	$-0.504366\pi$
$733$	−0.742646	−0.0274302	−0.0137151	+	0.999906i	$0.504366\pi$
$734$	0	0
$735$	3.23607	0.119364
$736$	0	0
$737$	−12.2361	−0.450721
$738$	0	0
$739$	−34.2361	−1.25939	−0.629697	−	0.776841i	$-0.716821\pi$
$739$	−34.2361	−1.25939	−0.629697	+	0.776841i	$0.716821\pi$
$740$	0	0
$741$	8.09017	0.297200
$742$	0	0
$743$	39.9443	1.46541	0.732707	−	0.680545i	$-0.238257\pi$
$743$	39.9443	1.46541	0.732707	+	0.680545i	$0.238257\pi$
$744$	0	0
$745$	−14.1803	−0.519527
$746$	0	0
$747$	3.11146	0.113842
$748$	0	0
$749$	−3.61803	−0.132200
$750$	0	0
$751$	18.6869	0.681895	0.340948	−	0.940082i	$-0.389252\pi$
$751$	18.6869	0.681895	0.340948	+	0.940082i	$0.389252\pi$
$752$	0	0
$753$	−29.8885	−1.08920
$754$	0	0
$755$	−14.7082	−0.535286
$756$	0	0
$757$	52.3951	1.90433	0.952167	−	0.305580i	$-0.0988502\pi$
$757$	52.3951	1.90433	0.952167	+	0.305580i	$0.0988502\pi$
$758$	0	0
$759$	9.47214	0.343817
$760$	0	0
$761$	50.1591	1.81826	0.909132	−	0.416508i	$-0.136746\pi$
$761$	50.1591	1.81826	0.909132	+	0.416508i	$0.136746\pi$
$762$	0	0
$763$	−8.29180	−0.300183
$764$	0	0
$765$	−2.23607	−0.0808452
$766$	0	0
$767$	−7.76393	−0.280339
$768$	0	0
$769$	−19.7984	−0.713948	−0.356974	−	0.934114i	$-0.616191\pi$
$769$	−19.7984	−0.713948	−0.356974	+	0.934114i	$0.616191\pi$
$770$	0	0
$771$	−22.7082	−0.817816
$772$	0	0
$773$	49.5410	1.78187	0.890933	−	0.454134i	$-0.150051\pi$
$773$	49.5410	1.78187	0.890933	+	0.454134i	$0.150051\pi$
$774$	0	0
$775$	−34.8328	−1.25123
$776$	0	0
$777$	30.9787	1.11136
$778$	0	0
$779$	−3.94427	−0.141318
$780$	0	0
$781$	15.7771	0.564549
$782$	0	0
$783$	20.5967	0.736068
$784$	0	0
$785$	−6.79837	−0.242644
$786$	0	0
$787$	−4.05573	−0.144571	−0.0722855	−	0.997384i	$-0.523029\pi$
$787$	−4.05573	−0.144571	−0.0722855	+	0.997384i	$0.523029\pi$
$788$	0	0
$789$	−13.6180	−0.484815
$790$	0	0
$791$	17.0344	0.605675
$792$	0	0
$793$	−8.09017	−0.287290
$794$	0	0
$795$	−2.23607	−0.0793052
$796$	0	0
$797$	6.43769	0.228035	0.114017	−	0.993479i	$-0.463628\pi$
$797$	6.43769	0.228035	0.114017	+	0.993479i	$0.463628\pi$
$798$	0	0
$799$	21.7082	0.767981
$800$	0	0
$801$	−3.88854	−0.137395
$802$	0	0
$803$	2.23607	0.0789091
$804$	0	0
$805$	9.47214	0.333849
$806$	0	0
$807$	−0.145898	−0.00513585
$808$	0	0
$809$	−24.6738	−0.867483	−0.433742	−	0.901037i	$-0.642807\pi$
$809$	−24.6738	−0.867483	−0.433742	+	0.901037i	$0.642807\pi$
$810$	0	0
$811$	40.5279	1.42313	0.711563	−	0.702622i	$-0.247988\pi$
$811$	40.5279	1.42313	0.711563	+	0.702622i	$0.247988\pi$
$812$	0	0
$813$	41.0344	1.43914
$814$	0	0
$815$	−16.0344	−0.561662
$816$	0	0
$817$	−22.5623	−0.789355
$818$	0	0
$819$	1.90983	0.0667349
$820$	0	0
$821$	−43.3050	−1.51135	−0.755677	−	0.654945i	$-0.772692\pi$
$821$	−43.3050	−1.51135	−0.755677	+	0.654945i	$0.772692\pi$
$822$	0	0
$823$	−48.3820	−1.68649	−0.843245	−	0.537530i	$-0.819358\pi$
$823$	−48.3820	−1.68649	−0.843245	+	0.537530i	$0.819358\pi$
$824$	0	0
$825$	−8.94427	−0.311400
$826$	0	0
$827$	14.5066	0.504443	0.252222	−	0.967670i	$-0.418839\pi$
$827$	14.5066	0.504443	0.252222	+	0.967670i	$0.418839\pi$
$828$	0	0
$829$	23.5967	0.819549	0.409774	−	0.912187i	$-0.365607\pi$
$829$	23.5967	0.819549	0.409774	+	0.912187i	$0.365607\pi$
$830$	0	0
$831$	10.3820	0.360146
$832$	0	0
$833$	11.7082	0.405665
$834$	0	0
$835$	−1.00000	−0.0346064
$836$	0	0
$837$	−47.6525	−1.64711
$838$	0	0
$839$	−5.74265	−0.198258	−0.0991291	−	0.995075i	$-0.531606\pi$
$839$	−5.74265	−0.198258	−0.0991291	+	0.995075i	$0.531606\pi$
$840$	0	0
$841$	−14.8328	−0.511476
$842$	0	0
$843$	30.1803	1.03947
$844$	0	0
$845$	8.00000	0.275208
$846$	0	0
$847$	20.3262	0.698418
$848$	0	0
$849$	−33.5066	−1.14994
$850$	0	0
$851$	−36.2705	−1.24334
$852$	0	0
$853$	−25.9230	−0.887586	−0.443793	−	0.896129i	$-0.646367\pi$
$853$	−25.9230	−0.887586	−0.443793	+	0.896129i	$0.646367\pi$
$854$	0	0
$855$	0.854102	0.0292097
$856$	0	0
$857$	24.8197	0.847823	0.423912	−	0.905704i	$-0.360657\pi$
$857$	24.8197	0.847823	0.423912	+	0.905704i	$0.360657\pi$
$858$	0	0
$859$	−57.3607	−1.95712	−0.978561	−	0.205959i	$-0.933969\pi$
$859$	−57.3607	−1.95712	−0.978561	+	0.205959i	$0.933969\pi$
$860$	0	0
$861$	6.38197	0.217497
$862$	0	0
$863$	−19.8328	−0.675117	−0.337558	−	0.941305i	$-0.609601\pi$
$863$	−19.8328	−0.675117	−0.337558	+	0.941305i	$0.609601\pi$
$864$	0	0
$865$	−1.29180	−0.0439224
$866$	0	0
$867$	27.9443	0.949037
$868$	0	0
$869$	−5.20163	−0.176453
$870$	0	0
$871$	−19.7984	−0.670842
$872$	0	0
$873$	3.70820	0.125504
$874$	0	0
$875$	−20.1246	−0.680336
$876$	0	0
$877$	6.96556	0.235210	0.117605	−	0.993060i	$-0.462478\pi$
$877$	6.96556	0.235210	0.117605	+	0.993060i	$0.462478\pi$
$878$	0	0
$879$	29.5967	0.998274
$880$	0	0
$881$	11.3820	0.383468	0.191734	−	0.981447i	$-0.438589\pi$
$881$	11.3820	0.383468	0.191734	+	0.981447i	$0.438589\pi$
$882$	0	0
$883$	−5.56231	−0.187186	−0.0935932	−	0.995611i	$-0.529835\pi$
$883$	−5.56231	−0.187186	−0.0935932	+	0.995611i	$0.529835\pi$
$884$	0	0
$885$	5.61803	0.188848
$886$	0	0
$887$	54.3262	1.82410	0.912048	−	0.410083i	$-0.134500\pi$
$887$	54.3262	1.82410	0.912048	+	0.410083i	$0.134500\pi$
$888$	0	0
$889$	17.5623	0.589021
$890$	0	0
$891$	−10.6525	−0.356871
$892$	0	0
$893$	−8.29180	−0.277474
$894$	0	0
$895$	−16.8541	−0.563370
$896$	0	0
$897$	15.3262	0.511728
$898$	0	0
$899$	−32.7771	−1.09318
$900$	0	0
$901$	−8.09017	−0.269523
$902$	0	0
$903$	36.5066	1.21486
$904$	0	0
$905$	−14.4164	−0.479218
$906$	0	0
$907$	18.7639	0.623046	0.311523	−	0.950239i	$-0.399161\pi$
$907$	18.7639	0.623046	0.311523	+	0.950239i	$0.399161\pi$
$908$	0	0
$909$	4.67376	0.155019
$910$	0	0
$911$	26.2705	0.870381	0.435190	−	0.900338i	$-0.356681\pi$
$911$	26.2705	0.870381	0.435190	+	0.900338i	$0.356681\pi$
$912$	0	0
$913$	−11.2574	−0.372564
$914$	0	0
$915$	5.85410	0.193531
$916$	0	0
$917$	16.0557	0.530207
$918$	0	0
$919$	14.7639	0.487017	0.243509	−	0.969899i	$-0.421702\pi$
$919$	14.7639	0.487017	0.243509	+	0.969899i	$0.421702\pi$
$920$	0	0
$921$	−55.7426	−1.83678
$922$	0	0
$923$	25.5279	0.840260
$924$	0	0
$925$	34.2492	1.12611
$926$	0	0
$927$	4.52786	0.148715
$928$	0	0
$929$	47.8328	1.56934	0.784672	−	0.619911i	$-0.212831\pi$
$929$	47.8328	1.56934	0.784672	+	0.619911i	$0.212831\pi$
$930$	0	0
$931$	−4.47214	−0.146568
$932$	0	0
$933$	−18.7082	−0.612479
$934$	0	0
$935$	8.09017	0.264577
$936$	0	0
$937$	−43.7082	−1.42788	−0.713942	−	0.700204i	$-0.753092\pi$
$937$	−43.7082	−1.42788	−0.713942	+	0.700204i	$0.753092\pi$
$938$	0	0
$939$	45.0344	1.46964
$940$	0	0
$941$	−41.2918	−1.34607	−0.673037	−	0.739609i	$-0.735011\pi$
$941$	−41.2918	−1.34607	−0.673037	+	0.739609i	$0.735011\pi$
$942$	0	0
$943$	−7.47214	−0.243326
$944$	0	0
$945$	−12.2361	−0.398039
$946$	0	0
$947$	−9.92299	−0.322454	−0.161227	−	0.986917i	$-0.551545\pi$
$947$	−9.92299	−0.322454	−0.161227	+	0.986917i	$0.551545\pi$
$948$	0	0
$949$	3.61803	0.117446
$950$	0	0
$951$	−28.0902	−0.910886
$952$	0	0
$953$	−27.7426	−0.898672	−0.449336	−	0.893363i	$-0.648339\pi$
$953$	−27.7426	−0.898672	−0.449336	+	0.893363i	$0.648339\pi$
$954$	0	0
$955$	10.7639	0.348313
$956$	0	0
$957$	−8.41641	−0.272064
$958$	0	0
$959$	37.1591	1.19993
$960$	0	0
$961$	44.8328	1.44622
$962$	0	0
$963$	−0.618034	−0.0199159
$964$	0	0
$965$	−14.7984	−0.476377
$966$	0	0
$967$	−50.2361	−1.61548	−0.807742	−	0.589537i	$-0.799310\pi$
$967$	−50.2361	−1.61548	−0.807742	+	0.589537i	$0.799310\pi$
$968$	0	0
$969$	−21.1803	−0.680411
$970$	0	0
$971$	38.9230	1.24910	0.624549	−	0.780986i	$-0.285283\pi$
$971$	38.9230	1.24910	0.624549	+	0.780986i	$0.285283\pi$
$972$	0	0
$973$	−16.1803	−0.518718
$974$	0	0
$975$	−14.4721	−0.463479
$976$	0	0
$977$	−56.7426	−1.81536	−0.907679	−	0.419665i	$-0.862148\pi$
$977$	−56.7426	−1.81536	−0.907679	+	0.419665i	$0.862148\pi$
$978$	0	0
$979$	14.0689	0.449643
$980$	0	0
$981$	−1.41641	−0.0452224
$982$	0	0
$983$	19.2918	0.615313	0.307656	−	0.951498i	$-0.400455\pi$
$983$	19.2918	0.615313	0.307656	+	0.951498i	$0.400455\pi$
$984$	0	0
$985$	−7.61803	−0.242731
$986$	0	0
$987$	13.4164	0.427049
$988$	0	0
$989$	−42.7426	−1.35914
$990$	0	0
$991$	−14.2705	−0.453318	−0.226659	−	0.973974i	$-0.572780\pi$
$991$	−14.2705	−0.453318	−0.226659	+	0.973974i	$0.572780\pi$
$992$	0	0
$993$	26.8885	0.853282
$994$	0	0
$995$	10.0344	0.318113
$996$	0	0
$997$	−27.2148	−0.861901	−0.430950	−	0.902376i	$-0.641822\pi$
$997$	−27.2148	−0.861901	−0.430950	+	0.902376i	$0.641822\pi$
$998$	0	0
$999$	46.8541	1.48240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2672.2.a.e.1.2		2
4.3	odd	2		334.2.a.b.1.1	✓	2
12.11	even	2		3006.2.a.m.1.2		2
20.19	odd	2		8350.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
334.2.a.b.1.1	✓	2	4.3	odd	2
2672.2.a.e.1.2		2	1.1	even	1	trivial
3006.2.a.m.1.2		2	12.11	even	2
8350.2.a.m.1.2		2	20.19	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:	\( N \)	\(=\)	\( 2672 = 2^{4} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2672.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} -1.00000 q^{5} -2.23607 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q+1.61803 q^{3} -1.00000 q^{5} -2.23607 q^{7} -0.381966 q^{9} +1.38197 q^{11} +2.23607 q^{13} -1.61803 q^{15} -5.85410 q^{17} +2.23607 q^{19} -3.61803 q^{21} +4.23607 q^{23} -4.00000 q^{25} -5.47214 q^{27} -3.76393 q^{29} +8.70820 q^{31} +2.23607 q^{33} +2.23607 q^{35} -8.56231 q^{37} +3.61803 q^{39} -1.76393 q^{41} -10.0902 q^{43} +0.381966 q^{45} -3.70820 q^{47} -2.00000 q^{49} -9.47214 q^{51} +1.38197 q^{53} -1.38197 q^{55} +3.61803 q^{57} -3.47214 q^{59} -3.61803 q^{61} +0.854102 q^{63} -2.23607 q^{65} -8.85410 q^{67} +6.85410 q^{69} +11.4164 q^{71} +1.61803 q^{73} -6.47214 q^{75} -3.09017 q^{77} -3.76393 q^{79} -7.70820 q^{81} -8.14590 q^{83} +5.85410 q^{85} -6.09017 q^{87} +10.1803 q^{89} -5.00000 q^{91} +14.0902 q^{93} -2.23607 q^{95} -9.70820 q^{97} -0.527864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 2 q^{5} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 - 2 * q^5 - 3 * q^9	\( 2 q + q^{3} - 2 q^{5} - 3 q^{9} + 5 q^{11} - q^{15} - 5 q^{17} - 5 q^{21} + 4 q^{23} - 8 q^{25} - 2 q^{27} - 12 q^{29} + 4 q^{31} + 3 q^{37} + 5 q^{39} - 8 q^{41} - 9 q^{43} + 3 q^{45} + 6 q^{47} - 4 q^{49} - 10 q^{51} + 5 q^{53} - 5 q^{55} + 5 q^{57} + 2 q^{59} - 5 q^{61} - 5 q^{63} - 11 q^{67} + 7 q^{69} - 4 q^{71} + q^{73} - 4 q^{75} + 5 q^{77} - 12 q^{79} - 2 q^{81} - 23 q^{83} + 5 q^{85} - q^{87} - 2 q^{89} - 10 q^{91} + 17 q^{93} - 6 q^{97} - 10 q^{99}+O(q^{100}) \) 2 * q + q^3 - 2 * q^5 - 3 * q^9 + 5 * q^11 - q^15 - 5 * q^17 - 5 * q^21 + 4 * q^23 - 8 * q^25 - 2 * q^27 - 12 * q^29 + 4 * q^31 + 3 * q^37 + 5 * q^39 - 8 * q^41 - 9 * q^43 + 3 * q^45 + 6 * q^47 - 4 * q^49 - 10 * q^51 + 5 * q^53 - 5 * q^55 + 5 * q^57 + 2 * q^59 - 5 * q^61 - 5 * q^63 - 11 * q^67 + 7 * q^69 - 4 * q^71 + q^73 - 4 * q^75 + 5 * q^77 - 12 * q^79 - 2 * q^81 - 23 * q^83 + 5 * q^85 - q^87 - 2 * q^89 - 10 * q^91 + 17 * q^93 - 6 * q^97 - 10 * q^99

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