LMFDB - Embedded newform 6728.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6728,2,Mod(1,6728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6728, 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("6728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6728 = 2^{3} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6728.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:	$53.7233504799$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{3} +1.38197 q^{5} -1.38197 q^{7} -1.47214 q^{9} +O(q^{10})$	$q-1.23607 q^{3} +1.38197 q^{5} -1.38197 q^{7} -1.47214 q^{9} -2.23607 q^{11} -6.23607 q^{13} -1.70820 q^{15} -2.00000 q^{17} -4.61803 q^{19} +1.70820 q^{21} +1.14590 q^{23} -3.09017 q^{25} +5.52786 q^{27} -1.47214 q^{31} +2.76393 q^{33} -1.90983 q^{35} -6.47214 q^{37} +7.70820 q^{39} -3.47214 q^{41} -5.61803 q^{43} -2.03444 q^{45} -3.38197 q^{47} -5.09017 q^{49} +2.47214 q^{51} +10.9443 q^{53} -3.09017 q^{55} +5.70820 q^{57} +14.0902 q^{59} +1.00000 q^{61} +2.03444 q^{63} -8.61803 q^{65} -9.47214 q^{67} -1.41641 q^{69} +10.0902 q^{71} +10.3262 q^{73} +3.81966 q^{75} +3.09017 q^{77} -1.00000 q^{79} -2.41641 q^{81} +4.52786 q^{83} -2.76393 q^{85} -1.29180 q^{89} +8.61803 q^{91} +1.81966 q^{93} -6.38197 q^{95} -11.4164 q^{97} +3.29180 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 5 q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 5 * q^5 - 5 * q^7 + 6 * q^9	$ 2 q + 2 q^{3} + 5 q^{5} - 5 q^{7} + 6 q^{9} - 8 q^{13} + 10 q^{15} - 4 q^{17} - 7 q^{19} - 10 q^{21} + 9 q^{23} + 5 q^{25} + 20 q^{27} + 6 q^{31} + 10 q^{33} - 15 q^{35} - 4 q^{37} + 2 q^{39} + 2 q^{41} - 9 q^{43} + 25 q^{45} - 9 q^{47} + q^{49} - 4 q^{51} + 4 q^{53} + 5 q^{55} - 2 q^{57} + 17 q^{59} + 2 q^{61} - 25 q^{63} - 15 q^{65} - 10 q^{67} + 24 q^{69} + 9 q^{71} + 5 q^{73} + 30 q^{75} - 5 q^{77} - 2 q^{79} + 22 q^{81} + 18 q^{83} - 10 q^{85} - 16 q^{89} + 15 q^{91} + 26 q^{93} - 15 q^{95} + 4 q^{97} + 20 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 5 * q^5 - 5 * q^7 + 6 * q^9 - 8 * q^13 + 10 * q^15 - 4 * q^17 - 7 * q^19 - 10 * q^21 + 9 * q^23 + 5 * q^25 + 20 * q^27 + 6 * q^31 + 10 * q^33 - 15 * q^35 - 4 * q^37 + 2 * q^39 + 2 * q^41 - 9 * q^43 + 25 * q^45 - 9 * q^47 + q^49 - 4 * q^51 + 4 * q^53 + 5 * q^55 - 2 * q^57 + 17 * q^59 + 2 * q^61 - 25 * q^63 - 15 * q^65 - 10 * q^67 + 24 * q^69 + 9 * q^71 + 5 * q^73 + 30 * q^75 - 5 * q^77 - 2 * q^79 + 22 * q^81 + 18 * q^83 - 10 * q^85 - 16 * q^89 + 15 * q^91 + 26 * q^93 - 15 * q^95 + 4 * q^97 + 20 * 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
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	0	0
$5$	1.38197	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$5$	1.38197	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$6$	0	0
$7$	−1.38197	−0.522334	−0.261167	−	0.965294i	$-0.584107\pi$
$7$	−1.38197	−0.522334	−0.261167	+	0.965294i	$0.584107\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0	0
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	0	0
$15$	−1.70820	−0.441056
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−4.61803	−1.05945	−0.529725	−	0.848170i	$-0.677705\pi$
$19$	−4.61803	−1.05945	−0.529725	+	0.848170i	$0.677705\pi$
$20$	0	0
$21$	1.70820	0.372761
$22$	0	0
$23$	1.14590	0.238936	0.119468	−	0.992838i	$-0.461881\pi$
$23$	1.14590	0.238936	0.119468	+	0.992838i	$0.461881\pi$
$24$	0	0
$25$	−3.09017	−0.618034
$26$	0	0
$27$	5.52786	1.06384
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−1.47214	−0.264403	−0.132202	−	0.991223i	$-0.542205\pi$
$31$	−1.47214	−0.264403	−0.132202	+	0.991223i	$0.542205\pi$
$32$	0	0
$33$	2.76393	0.481139
$34$	0	0
$35$	−1.90983	−0.322820
$36$	0	0
$37$	−6.47214	−1.06401	−0.532006	−	0.846740i	$-0.678562\pi$
$37$	−6.47214	−1.06401	−0.532006	+	0.846740i	$0.678562\pi$
$38$	0	0
$39$	7.70820	1.23430
$40$	0	0
$41$	−3.47214	−0.542257	−0.271128	−	0.962543i	$-0.587397\pi$
$41$	−3.47214	−0.542257	−0.271128	+	0.962543i	$0.587397\pi$
$42$	0	0
$43$	−5.61803	−0.856742	−0.428371	−	0.903603i	$-0.640912\pi$
$43$	−5.61803	−0.856742	−0.428371	+	0.903603i	$0.640912\pi$
$44$	0	0
$45$	−2.03444	−0.303277
$46$	0	0
$47$	−3.38197	−0.493310	−0.246655	−	0.969103i	$-0.579332\pi$
$47$	−3.38197	−0.493310	−0.246655	+	0.969103i	$0.579332\pi$
$48$	0	0
$49$	−5.09017	−0.727167
$50$	0	0
$51$	2.47214	0.346168
$52$	0	0
$53$	10.9443	1.50331	0.751656	−	0.659556i	$-0.229256\pi$
$53$	10.9443	1.50331	0.751656	+	0.659556i	$0.229256\pi$
$54$	0	0
$55$	−3.09017	−0.416678
$56$	0	0
$57$	5.70820	0.756070
$58$	0	0
$59$	14.0902	1.83438	0.917192	−	0.398446i	$-0.130450\pi$
$59$	14.0902	1.83438	0.917192	+	0.398446i	$0.130450\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	2.03444	0.256316
$64$	0	0
$65$	−8.61803	−1.06894
$66$	0	0
$67$	−9.47214	−1.15721	−0.578603	−	0.815609i	$-0.696402\pi$
$67$	−9.47214	−1.15721	−0.578603	+	0.815609i	$0.696402\pi$
$68$	0	0
$69$	−1.41641	−0.170515
$70$	0	0
$71$	10.0902	1.19748	0.598741	−	0.800942i	$-0.295668\pi$
$71$	10.0902	1.19748	0.598741	+	0.800942i	$0.295668\pi$
$72$	0	0
$73$	10.3262	1.20859	0.604297	−	0.796759i	$-0.293454\pi$
$73$	10.3262	1.20859	0.604297	+	0.796759i	$0.293454\pi$
$74$	0	0
$75$	3.81966	0.441056
$76$	0	0
$77$	3.09017	0.352158
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	4.52786	0.496998	0.248499	−	0.968632i	$-0.420063\pi$
$83$	4.52786	0.496998	0.248499	+	0.968632i	$0.420063\pi$
$84$	0	0
$85$	−2.76393	−0.299791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.29180	−0.136930	−0.0684651	−	0.997654i	$-0.521810\pi$
$89$	−1.29180	−0.136930	−0.0684651	+	0.997654i	$0.521810\pi$
$90$	0	0
$91$	8.61803	0.903415
$92$	0	0
$93$	1.81966	0.188690
$94$	0	0
$95$	−6.38197	−0.654776
$96$	0	0
$97$	−11.4164	−1.15916	−0.579580	−	0.814915i	$-0.696783\pi$
$97$	−11.4164	−1.15916	−0.579580	+	0.814915i	$0.696783\pi$
$98$	0	0
$99$	3.29180	0.330838
$100$	0	0
$101$	−8.85410	−0.881016	−0.440508	−	0.897749i	$-0.645202\pi$
$101$	−8.85410	−0.881016	−0.440508	+	0.897749i	$0.645202\pi$
$102$	0	0
$103$	14.2361	1.40272	0.701361	−	0.712807i	$-0.252576\pi$
$103$	14.2361	1.40272	0.701361	+	0.712807i	$0.252576\pi$
$104$	0	0
$105$	2.36068	0.230379
$106$	0	0
$107$	7.85410	0.759285	0.379642	−	0.925133i	$-0.376047\pi$
$107$	7.85410	0.759285	0.379642	+	0.925133i	$0.376047\pi$
$108$	0	0
$109$	11.6525	1.11610	0.558052	−	0.829806i	$-0.311549\pi$
$109$	11.6525	1.11610	0.558052	+	0.829806i	$0.311549\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	0.472136	0.0444148	0.0222074	−	0.999753i	$-0.492931\pi$
$113$	0.472136	0.0444148	0.0222074	+	0.999753i	$0.492931\pi$
$114$	0	0
$115$	1.58359	0.147671
$116$	0	0
$117$	9.18034	0.848723
$118$	0	0
$119$	2.76393	0.253369
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	4.29180	0.386978
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	5.94427	0.527469	0.263734	−	0.964595i	$-0.415046\pi$
$127$	5.94427	0.527469	0.263734	+	0.964595i	$0.415046\pi$
$128$	0	0
$129$	6.94427	0.611409
$130$	0	0
$131$	13.5279	1.18193	0.590967	−	0.806695i	$-0.298746\pi$
$131$	13.5279	1.18193	0.590967	+	0.806695i	$0.298746\pi$
$132$	0	0
$133$	6.38197	0.553387
$134$	0	0
$135$	7.63932	0.657488
$136$	0	0
$137$	−1.94427	−0.166110	−0.0830552	−	0.996545i	$-0.526468\pi$
$137$	−1.94427	−0.166110	−0.0830552	+	0.996545i	$0.526468\pi$
$138$	0	0
$139$	−19.4721	−1.65161	−0.825803	−	0.563959i	$-0.809277\pi$
$139$	−19.4721	−1.65161	−0.825803	+	0.563959i	$0.809277\pi$
$140$	0	0
$141$	4.18034	0.352048
$142$	0	0
$143$	13.9443	1.16608
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.29180	0.518939
$148$	0	0
$149$	5.29180	0.433521	0.216760	−	0.976225i	$-0.430451\pi$
$149$	5.29180	0.433521	0.216760	+	0.976225i	$0.430451\pi$
$150$	0	0
$151$	−12.2361	−0.995757	−0.497879	−	0.867247i	$-0.665887\pi$
$151$	−12.2361	−0.995757	−0.497879	+	0.867247i	$0.665887\pi$
$152$	0	0
$153$	2.94427	0.238030
$154$	0	0
$155$	−2.03444	−0.163410
$156$	0	0
$157$	−19.4721	−1.55405	−0.777023	−	0.629472i	$-0.783271\pi$
$157$	−19.4721	−1.55405	−0.777023	+	0.629472i	$0.783271\pi$
$158$	0	0
$159$	−13.5279	−1.07283
$160$	0	0
$161$	−1.58359	−0.124805
$162$	0	0
$163$	22.2705	1.74436	0.872180	−	0.489184i	$-0.162705\pi$
$163$	22.2705	1.74436	0.872180	+	0.489184i	$0.162705\pi$
$164$	0	0
$165$	3.81966	0.297360
$166$	0	0
$167$	20.9443	1.62072	0.810358	−	0.585935i	$-0.199273\pi$
$167$	20.9443	1.62072	0.810358	+	0.585935i	$0.199273\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	0	0
$171$	6.79837	0.519885
$172$	0	0
$173$	18.2361	1.38646	0.693231	−	0.720715i	$-0.256186\pi$
$173$	18.2361	1.38646	0.693231	+	0.720715i	$0.256186\pi$
$174$	0	0
$175$	4.27051	0.322820
$176$	0	0
$177$	−17.4164	−1.30910
$178$	0	0
$179$	21.2361	1.58726	0.793629	−	0.608402i	$-0.208189\pi$
$179$	21.2361	1.58726	0.793629	+	0.608402i	$0.208189\pi$
$180$	0	0
$181$	−13.9098	−1.03391	−0.516955	−	0.856013i	$-0.672934\pi$
$181$	−13.9098	−1.03391	−0.516955	+	0.856013i	$0.672934\pi$
$182$	0	0
$183$	−1.23607	−0.0913728
$184$	0	0
$185$	−8.94427	−0.657596
$186$	0	0
$187$	4.47214	0.327035
$188$	0	0
$189$	−7.63932	−0.555679
$190$	0	0
$191$	−25.4164	−1.83907	−0.919533	−	0.393012i	$-0.871433\pi$
$191$	−25.4164	−1.83907	−0.919533	+	0.393012i	$0.871433\pi$
$192$	0	0
$193$	−22.0902	−1.59009	−0.795043	−	0.606554i	$-0.792552\pi$
$193$	−22.0902	−1.59009	−0.795043	+	0.606554i	$0.792552\pi$
$194$	0	0
$195$	10.6525	0.762840
$196$	0	0
$197$	15.5623	1.10877	0.554384	−	0.832261i	$-0.312954\pi$
$197$	15.5623	1.10877	0.554384	+	0.832261i	$0.312954\pi$
$198$	0	0
$199$	13.1803	0.934330	0.467165	−	0.884170i	$-0.345275\pi$
$199$	13.1803	0.934330	0.467165	+	0.884170i	$0.345275\pi$
$200$	0	0
$201$	11.7082	0.825833
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.79837	−0.335133
$206$	0	0
$207$	−1.68692	−0.117249
$208$	0	0
$209$	10.3262	0.714281
$210$	0	0
$211$	−7.65248	−0.526818	−0.263409	−	0.964684i	$-0.584847\pi$
$211$	−7.65248	−0.526818	−0.263409	+	0.964684i	$0.584847\pi$
$212$	0	0
$213$	−12.4721	−0.854577
$214$	0	0
$215$	−7.76393	−0.529496
$216$	0	0
$217$	2.03444	0.138107
$218$	0	0
$219$	−12.7639	−0.862507
$220$	0	0
$221$	12.4721	0.838967
$222$	0	0
$223$	−9.32624	−0.624531	−0.312266	−	0.949995i	$-0.601088\pi$
$223$	−9.32624	−0.624531	−0.312266	+	0.949995i	$0.601088\pi$
$224$	0	0
$225$	4.54915	0.303277
$226$	0	0
$227$	−20.1246	−1.33572	−0.667859	−	0.744288i	$-0.732789\pi$
$227$	−20.1246	−1.33572	−0.667859	+	0.744288i	$0.732789\pi$
$228$	0	0
$229$	−18.8541	−1.24591	−0.622957	−	0.782256i	$-0.714069\pi$
$229$	−18.8541	−1.24591	−0.622957	+	0.782256i	$0.714069\pi$
$230$	0	0
$231$	−3.81966	−0.251315
$232$	0	0
$233$	−19.1246	−1.25289	−0.626447	−	0.779464i	$-0.715492\pi$
$233$	−19.1246	−1.25289	−0.626447	+	0.779464i	$0.715492\pi$
$234$	0	0
$235$	−4.67376	−0.304883
$236$	0	0
$237$	1.23607	0.0802912
$238$	0	0
$239$	−17.0902	−1.10547	−0.552736	−	0.833357i	$-0.686416\pi$
$239$	−17.0902	−1.10547	−0.552736	+	0.833357i	$0.686416\pi$
$240$	0	0
$241$	−12.4164	−0.799811	−0.399906	−	0.916556i	$-0.630957\pi$
$241$	−12.4164	−0.799811	−0.399906	+	0.916556i	$0.630957\pi$
$242$	0	0
$243$	−13.5967	−0.872232
$244$	0	0
$245$	−7.03444	−0.449414
$246$	0	0
$247$	28.7984	1.83240
$248$	0	0
$249$	−5.59675	−0.354679
$250$	0	0
$251$	6.67376	0.421244	0.210622	−	0.977568i	$-0.432451\pi$
$251$	6.67376	0.421244	0.210622	+	0.977568i	$0.432451\pi$
$252$	0	0
$253$	−2.56231	−0.161091
$254$	0	0
$255$	3.41641	0.213944
$256$	0	0
$257$	13.3262	0.831268	0.415634	−	0.909532i	$-0.363560\pi$
$257$	13.3262	0.831268	0.415634	+	0.909532i	$0.363560\pi$
$258$	0	0
$259$	8.94427	0.555770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.0902	−0.868837	−0.434419	−	0.900711i	$-0.643046\pi$
$263$	−14.0902	−0.868837	−0.434419	+	0.900711i	$0.643046\pi$
$264$	0	0
$265$	15.1246	0.929098
$266$	0	0
$267$	1.59675	0.0977194
$268$	0	0
$269$	8.03444	0.489869	0.244934	−	0.969540i	$-0.421234\pi$
$269$	8.03444	0.489869	0.244934	+	0.969540i	$0.421234\pi$
$270$	0	0
$271$	26.9787	1.63884	0.819420	−	0.573193i	$-0.194296\pi$
$271$	26.9787	1.63884	0.819420	+	0.573193i	$0.194296\pi$
$272$	0	0
$273$	−10.6525	−0.644717
$274$	0	0
$275$	6.90983	0.416678
$276$	0	0
$277$	2.85410	0.171486	0.0857432	−	0.996317i	$-0.472674\pi$
$277$	2.85410	0.171486	0.0857432	+	0.996317i	$0.472674\pi$
$278$	0	0
$279$	2.16718	0.129746
$280$	0	0
$281$	−22.4164	−1.33725	−0.668625	−	0.743599i	$-0.733117\pi$
$281$	−22.4164	−1.33725	−0.668625	+	0.743599i	$0.733117\pi$
$282$	0	0
$283$	3.18034	0.189052	0.0945258	−	0.995522i	$-0.469867\pi$
$283$	3.18034	0.189052	0.0945258	+	0.995522i	$0.469867\pi$
$284$	0	0
$285$	7.88854	0.467277
$286$	0	0
$287$	4.79837	0.283239
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.1115	0.827228
$292$	0	0
$293$	−30.5623	−1.78547	−0.892734	−	0.450583i	$-0.851216\pi$
$293$	−30.5623	−1.78547	−0.892734	+	0.450583i	$0.851216\pi$
$294$	0	0
$295$	19.4721	1.13371
$296$	0	0
$297$	−12.3607	−0.717239
$298$	0	0
$299$	−7.14590	−0.413258
$300$	0	0
$301$	7.76393	0.447506
$302$	0	0
$303$	10.9443	0.628732
$304$	0	0
$305$	1.38197	0.0791311
$306$	0	0
$307$	5.18034	0.295658	0.147829	−	0.989013i	$-0.452772\pi$
$307$	5.18034	0.295658	0.147829	+	0.989013i	$0.452772\pi$
$308$	0	0
$309$	−17.5967	−1.00104
$310$	0	0
$311$	20.7426	1.17621	0.588104	−	0.808785i	$-0.299875\pi$
$311$	20.7426	1.17621	0.588104	+	0.808785i	$0.299875\pi$
$312$	0	0
$313$	−23.3607	−1.32042	−0.660212	−	0.751079i	$-0.729534\pi$
$313$	−23.3607	−1.32042	−0.660212	+	0.751079i	$0.729534\pi$
$314$	0	0
$315$	2.81153	0.158412
$316$	0	0
$317$	27.3607	1.53673	0.768364	−	0.640013i	$-0.221071\pi$
$317$	27.3607	1.53673	0.768364	+	0.640013i	$0.221071\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−9.70820	−0.541859
$322$	0	0
$323$	9.23607	0.513909
$324$	0	0
$325$	19.2705	1.06894
$326$	0	0
$327$	−14.4033	−0.796502
$328$	0	0
$329$	4.67376	0.257673
$330$	0	0
$331$	−10.5623	−0.580557	−0.290278	−	0.956942i	$-0.593748\pi$
$331$	−10.5623	−0.580557	−0.290278	+	0.956942i	$0.593748\pi$
$332$	0	0
$333$	9.52786	0.522124
$334$	0	0
$335$	−13.0902	−0.715192
$336$	0	0
$337$	26.6180	1.44998	0.724988	−	0.688761i	$-0.241845\pi$
$337$	26.6180	1.44998	0.724988	+	0.688761i	$0.241845\pi$
$338$	0	0
$339$	−0.583592	−0.0316964
$340$	0	0
$341$	3.29180	0.178261
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	−1.95743	−0.105384
$346$	0	0
$347$	13.1459	0.705709	0.352854	−	0.935678i	$-0.385211\pi$
$347$	13.1459	0.705709	0.352854	+	0.935678i	$0.385211\pi$
$348$	0	0
$349$	−5.88854	−0.315207	−0.157603	−	0.987503i	$-0.550377\pi$
$349$	−5.88854	−0.315207	−0.157603	+	0.987503i	$0.550377\pi$
$350$	0	0
$351$	−34.4721	−1.83999
$352$	0	0
$353$	16.4164	0.873757	0.436879	−	0.899520i	$-0.356084\pi$
$353$	16.4164	0.873757	0.436879	+	0.899520i	$0.356084\pi$
$354$	0	0
$355$	13.9443	0.740085
$356$	0	0
$357$	−3.41641	−0.180815
$358$	0	0
$359$	−11.1459	−0.588258	−0.294129	−	0.955766i	$-0.595030\pi$
$359$	−11.1459	−0.588258	−0.294129	+	0.955766i	$0.595030\pi$
$360$	0	0
$361$	2.32624	0.122434
$362$	0	0
$363$	7.41641	0.389260
$364$	0	0
$365$	14.2705	0.746953
$366$	0	0
$367$	28.1246	1.46809	0.734046	−	0.679099i	$-0.237629\pi$
$367$	28.1246	1.46809	0.734046	+	0.679099i	$0.237629\pi$
$368$	0	0
$369$	5.11146	0.266092
$370$	0	0
$371$	−15.1246	−0.785231
$372$	0	0
$373$	0.527864	0.0273318	0.0136659	−	0.999907i	$-0.495650\pi$
$373$	0.527864	0.0273318	0.0136659	+	0.999907i	$0.495650\pi$
$374$	0	0
$375$	13.8197	0.713644
$376$	0	0
$377$	0	0
$378$	0	0
$379$	12.2361	0.628525	0.314262	−	0.949336i	$-0.398243\pi$
$379$	12.2361	0.628525	0.314262	+	0.949336i	$0.398243\pi$
$380$	0	0
$381$	−7.34752	−0.376425
$382$	0	0
$383$	−8.03444	−0.410541	−0.205270	−	0.978705i	$-0.565807\pi$
$383$	−8.03444	−0.410541	−0.205270	+	0.978705i	$0.565807\pi$
$384$	0	0
$385$	4.27051	0.217645
$386$	0	0
$387$	8.27051	0.420414
$388$	0	0
$389$	−16.4377	−0.833424	−0.416712	−	0.909039i	$-0.636818\pi$
$389$	−16.4377	−0.833424	−0.416712	+	0.909039i	$0.636818\pi$
$390$	0	0
$391$	−2.29180	−0.115901
$392$	0	0
$393$	−16.7214	−0.843481
$394$	0	0
$395$	−1.38197	−0.0695343
$396$	0	0
$397$	6.74265	0.338404	0.169202	−	0.985581i	$-0.445881\pi$
$397$	6.74265	0.338404	0.169202	+	0.985581i	$0.445881\pi$
$398$	0	0
$399$	−7.88854	−0.394921
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	9.18034	0.457305
$404$	0	0
$405$	−3.33939	−0.165936
$406$	0	0
$407$	14.4721	0.717357
$408$	0	0
$409$	15.1803	0.750619	0.375310	−	0.926899i	$-0.377536\pi$
$409$	15.1803	0.750619	0.375310	+	0.926899i	$0.377536\pi$
$410$	0	0
$411$	2.40325	0.118544
$412$	0	0
$413$	−19.4721	−0.958161
$414$	0	0
$415$	6.25735	0.307161
$416$	0	0
$417$	24.0689	1.17866
$418$	0	0
$419$	30.2361	1.47713	0.738564	−	0.674183i	$-0.235504\pi$
$419$	30.2361	1.47713	0.738564	+	0.674183i	$0.235504\pi$
$420$	0	0
$421$	5.94427	0.289706	0.144853	−	0.989453i	$-0.453729\pi$
$421$	5.94427	0.289706	0.144853	+	0.989453i	$0.453729\pi$
$422$	0	0
$423$	4.97871	0.242073
$424$	0	0
$425$	6.18034	0.299791
$426$	0	0
$427$	−1.38197	−0.0668780
$428$	0	0
$429$	−17.2361	−0.832165
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	−29.6180	−1.42335	−0.711676	−	0.702508i	$-0.752064\pi$
$433$	−29.6180	−1.42335	−0.711676	+	0.702508i	$0.752064\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.29180	−0.253141
$438$	0	0
$439$	−20.6738	−0.986705	−0.493352	−	0.869830i	$-0.664229\pi$
$439$	−20.6738	−0.986705	−0.493352	+	0.869830i	$0.664229\pi$
$440$	0	0
$441$	7.49342	0.356830
$442$	0	0
$443$	32.4508	1.54179	0.770893	−	0.636964i	$-0.219810\pi$
$443$	32.4508	1.54179	0.770893	+	0.636964i	$0.219810\pi$
$444$	0	0
$445$	−1.78522	−0.0846275
$446$	0	0
$447$	−6.54102	−0.309380
$448$	0	0
$449$	−9.76393	−0.460788	−0.230394	−	0.973097i	$-0.574002\pi$
$449$	−9.76393	−0.460788	−0.230394	+	0.973097i	$0.574002\pi$
$450$	0	0
$451$	7.76393	0.365589
$452$	0	0
$453$	15.1246	0.710616
$454$	0	0
$455$	11.9098	0.558341
$456$	0	0
$457$	30.2148	1.41339	0.706694	−	0.707519i	$-0.250186\pi$
$457$	30.2148	1.41339	0.706694	+	0.707519i	$0.250186\pi$
$458$	0	0
$459$	−11.0557	−0.516037
$460$	0	0
$461$	−2.81966	−0.131325	−0.0656623	−	0.997842i	$-0.520916\pi$
$461$	−2.81966	−0.131325	−0.0656623	+	0.997842i	$0.520916\pi$
$462$	0	0
$463$	−29.4164	−1.36710	−0.683548	−	0.729905i	$-0.739564\pi$
$463$	−29.4164	−1.36710	−0.683548	+	0.729905i	$0.739564\pi$
$464$	0	0
$465$	2.51471	0.116617
$466$	0	0
$467$	−14.7082	−0.680615	−0.340307	−	0.940314i	$-0.610531\pi$
$467$	−14.7082	−0.680615	−0.340307	+	0.940314i	$0.610531\pi$
$468$	0	0
$469$	13.0902	0.604448
$470$	0	0
$471$	24.0689	1.10904
$472$	0	0
$473$	12.5623	0.577615
$474$	0	0
$475$	14.2705	0.654776
$476$	0	0
$477$	−16.1115	−0.737693
$478$	0	0
$479$	31.7771	1.45193	0.725966	−	0.687730i	$-0.241393\pi$
$479$	31.7771	1.45193	0.725966	+	0.687730i	$0.241393\pi$
$480$	0	0
$481$	40.3607	1.84029
$482$	0	0
$483$	1.95743	0.0890660
$484$	0	0
$485$	−15.7771	−0.716401
$486$	0	0
$487$	−7.09017	−0.321286	−0.160643	−	0.987013i	$-0.551357\pi$
$487$	−7.09017	−0.321286	−0.160643	+	0.987013i	$0.551357\pi$
$488$	0	0
$489$	−27.5279	−1.24485
$490$	0	0
$491$	−28.1459	−1.27021	−0.635103	−	0.772427i	$-0.719042\pi$
$491$	−28.1459	−1.27021	−0.635103	+	0.772427i	$0.719042\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.54915	0.204469
$496$	0	0
$497$	−13.9443	−0.625486
$498$	0	0
$499$	−36.4721	−1.63272	−0.816359	−	0.577545i	$-0.804011\pi$
$499$	−36.4721	−1.63272	−0.816359	+	0.577545i	$0.804011\pi$
$500$	0	0
$501$	−25.8885	−1.15661
$502$	0	0
$503$	15.7082	0.700394	0.350197	−	0.936676i	$-0.386115\pi$
$503$	15.7082	0.700394	0.350197	+	0.936676i	$0.386115\pi$
$504$	0	0
$505$	−12.2361	−0.544498
$506$	0	0
$507$	−32.0000	−1.42117
$508$	0	0
$509$	40.6525	1.80189	0.900945	−	0.433934i	$-0.142875\pi$
$509$	40.6525	1.80189	0.900945	+	0.433934i	$0.142875\pi$
$510$	0	0
$511$	−14.2705	−0.631290
$512$	0	0
$513$	−25.5279	−1.12708
$514$	0	0
$515$	19.6738	0.866930
$516$	0	0
$517$	7.56231	0.332590
$518$	0	0
$519$	−22.5410	−0.989441
$520$	0	0
$521$	16.5066	0.723166	0.361583	−	0.932340i	$-0.382236\pi$
$521$	16.5066	0.723166	0.361583	+	0.932340i	$0.382236\pi$
$522$	0	0
$523$	13.7426	0.600924	0.300462	−	0.953794i	$-0.402859\pi$
$523$	13.7426	0.600924	0.300462	+	0.953794i	$0.402859\pi$
$524$	0	0
$525$	−5.27864	−0.230379
$526$	0	0
$527$	2.94427	0.128254
$528$	0	0
$529$	−21.6869	−0.942909
$530$	0	0
$531$	−20.7426	−0.900154
$532$	0	0
$533$	21.6525	0.937873
$534$	0	0
$535$	10.8541	0.469264
$536$	0	0
$537$	−26.2492	−1.13274
$538$	0	0
$539$	11.3820	0.490256
$540$	0	0
$541$	21.1803	0.910614	0.455307	−	0.890335i	$-0.349530\pi$
$541$	21.1803	0.910614	0.455307	+	0.890335i	$0.349530\pi$
$542$	0	0
$543$	17.1935	0.737844
$544$	0	0
$545$	16.1033	0.689791
$546$	0	0
$547$	25.1803	1.07663	0.538317	−	0.842743i	$-0.319060\pi$
$547$	25.1803	1.07663	0.538317	+	0.842743i	$0.319060\pi$
$548$	0	0
$549$	−1.47214	−0.0628292
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.38197	0.0587672
$554$	0	0
$555$	11.0557	0.469290
$556$	0	0
$557$	44.4508	1.88344	0.941721	−	0.336394i	$-0.109207\pi$
$557$	44.4508	1.88344	0.941721	+	0.336394i	$0.109207\pi$
$558$	0	0
$559$	35.0344	1.48180
$560$	0	0
$561$	−5.52786	−0.233387
$562$	0	0
$563$	−30.7984	−1.29800	−0.648998	−	0.760790i	$-0.724812\pi$
$563$	−30.7984	−1.29800	−0.648998	+	0.760790i	$0.724812\pi$
$564$	0	0
$565$	0.652476	0.0274499
$566$	0	0
$567$	3.33939	0.140241
$568$	0	0
$569$	37.2705	1.56246	0.781231	−	0.624243i	$-0.214592\pi$
$569$	37.2705	1.56246	0.781231	+	0.624243i	$0.214592\pi$
$570$	0	0
$571$	−13.2361	−0.553912	−0.276956	−	0.960883i	$-0.589326\pi$
$571$	−13.2361	−0.553912	−0.276956	+	0.960883i	$0.589326\pi$
$572$	0	0
$573$	31.4164	1.31244
$574$	0	0
$575$	−3.54102	−0.147671
$576$	0	0
$577$	−4.05573	−0.168842	−0.0844211	−	0.996430i	$-0.526904\pi$
$577$	−4.05573	−0.168842	−0.0844211	+	0.996430i	$0.526904\pi$
$578$	0	0
$579$	27.3050	1.13476
$580$	0	0
$581$	−6.25735	−0.259599
$582$	0	0
$583$	−24.4721	−1.01353
$584$	0	0
$585$	12.6869	0.524539
$586$	0	0
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	6.79837	0.280122
$590$	0	0
$591$	−19.2361	−0.791266
$592$	0	0
$593$	3.96556	0.162846	0.0814230	−	0.996680i	$-0.474054\pi$
$593$	3.96556	0.162846	0.0814230	+	0.996680i	$0.474054\pi$
$594$	0	0
$595$	3.81966	0.156591
$596$	0	0
$597$	−16.2918	−0.666779
$598$	0	0
$599$	−19.7426	−0.806663	−0.403331	−	0.915054i	$-0.632148\pi$
$599$	−19.7426	−0.806663	−0.403331	+	0.915054i	$0.632148\pi$
$600$	0	0
$601$	8.45085	0.344717	0.172359	−	0.985034i	$-0.444861\pi$
$601$	8.45085	0.344717	0.172359	+	0.985034i	$0.444861\pi$
$602$	0	0
$603$	13.9443	0.567855
$604$	0	0
$605$	−8.29180	−0.337109
$606$	0	0
$607$	0.854102	0.0346669	0.0173335	−	0.999850i	$-0.494482\pi$
$607$	0.854102	0.0346669	0.0173335	+	0.999850i	$0.494482\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21.0902	0.853217
$612$	0	0
$613$	−18.9443	−0.765152	−0.382576	−	0.923924i	$-0.624963\pi$
$613$	−18.9443	−0.765152	−0.382576	+	0.923924i	$0.624963\pi$
$614$	0	0
$615$	5.93112	0.239166
$616$	0	0
$617$	−11.7639	−0.473598	−0.236799	−	0.971559i	$-0.576098\pi$
$617$	−11.7639	−0.473598	−0.236799	+	0.971559i	$0.576098\pi$
$618$	0	0
$619$	7.50658	0.301715	0.150857	−	0.988556i	$-0.451797\pi$
$619$	7.50658	0.301715	0.150857	+	0.988556i	$0.451797\pi$
$620$	0	0
$621$	6.33437	0.254189
$622$	0	0
$623$	1.78522	0.0715233
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−12.7639	−0.509742
$628$	0	0
$629$	12.9443	0.516122
$630$	0	0
$631$	−6.32624	−0.251844	−0.125922	−	0.992040i	$-0.540189\pi$
$631$	−6.32624	−0.251844	−0.125922	+	0.992040i	$0.540189\pi$
$632$	0	0
$633$	9.45898	0.375961
$634$	0	0
$635$	8.21478	0.325994
$636$	0	0
$637$	31.7426	1.25769
$638$	0	0
$639$	−14.8541	−0.587619
$640$	0	0
$641$	20.0344	0.791313	0.395656	−	0.918399i	$-0.370517\pi$
$641$	20.0344	0.791313	0.395656	+	0.918399i	$0.370517\pi$
$642$	0	0
$643$	−15.5279	−0.612359	−0.306180	−	0.951974i	$-0.599051\pi$
$643$	−15.5279	−0.612359	−0.306180	+	0.951974i	$0.599051\pi$
$644$	0	0
$645$	9.59675	0.377872
$646$	0	0
$647$	−2.18034	−0.0857180	−0.0428590	−	0.999081i	$-0.513647\pi$
$647$	−2.18034	−0.0857180	−0.0428590	+	0.999081i	$0.513647\pi$
$648$	0	0
$649$	−31.5066	−1.23674
$650$	0	0
$651$	−2.51471	−0.0985592
$652$	0	0
$653$	−16.8541	−0.659552	−0.329776	−	0.944059i	$-0.606973\pi$
$653$	−16.8541	−0.659552	−0.329776	+	0.944059i	$0.606973\pi$
$654$	0	0
$655$	18.6950	0.730476
$656$	0	0
$657$	−15.2016	−0.593072
$658$	0	0
$659$	27.9443	1.08855	0.544277	−	0.838905i	$-0.316804\pi$
$659$	27.9443	1.08855	0.544277	+	0.838905i	$0.316804\pi$
$660$	0	0
$661$	11.0557	0.430018	0.215009	−	0.976612i	$-0.431022\pi$
$661$	11.0557	0.430018	0.215009	+	0.976612i	$0.431022\pi$
$662$	0	0
$663$	−15.4164	−0.598724
$664$	0	0
$665$	8.81966	0.342012
$666$	0	0
$667$	0	0
$668$	0	0
$669$	11.5279	0.445693
$670$	0	0
$671$	−2.23607	−0.0863224
$672$	0	0
$673$	−1.59675	−0.0615501	−0.0307751	−	0.999526i	$-0.509798\pi$
$673$	−1.59675	−0.0615501	−0.0307751	+	0.999526i	$0.509798\pi$
$674$	0	0
$675$	−17.0820	−0.657488
$676$	0	0
$677$	−9.65248	−0.370975	−0.185487	−	0.982647i	$-0.559386\pi$
$677$	−9.65248	−0.370975	−0.185487	+	0.982647i	$0.559386\pi$
$678$	0	0
$679$	15.7771	0.605469
$680$	0	0
$681$	24.8754	0.953227
$682$	0	0
$683$	−11.3820	−0.435519	−0.217759	−	0.976002i	$-0.569875\pi$
$683$	−11.3820	−0.435519	−0.217759	+	0.976002i	$0.569875\pi$
$684$	0	0
$685$	−2.68692	−0.102662
$686$	0	0
$687$	23.3050	0.889139
$688$	0	0
$689$	−68.2492	−2.60009
$690$	0	0
$691$	−8.45085	−0.321485	−0.160743	−	0.986996i	$-0.551389\pi$
$691$	−8.45085	−0.321485	−0.160743	+	0.986996i	$0.551389\pi$
$692$	0	0
$693$	−4.54915	−0.172808
$694$	0	0
$695$	−26.9098	−1.02075
$696$	0	0
$697$	6.94427	0.263033
$698$	0	0
$699$	23.6393	0.894121
$700$	0	0
$701$	4.70820	0.177826	0.0889132	−	0.996039i	$-0.471661\pi$
$701$	4.70820	0.177826	0.0889132	+	0.996039i	$0.471661\pi$
$702$	0	0
$703$	29.8885	1.12727
$704$	0	0
$705$	5.77709	0.217578
$706$	0	0
$707$	12.2361	0.460185
$708$	0	0
$709$	21.6525	0.813176	0.406588	−	0.913612i	$-0.366718\pi$
$709$	21.6525	0.813176	0.406588	+	0.913612i	$0.366718\pi$
$710$	0	0
$711$	1.47214	0.0552094
$712$	0	0
$713$	−1.68692	−0.0631756
$714$	0	0
$715$	19.2705	0.720676
$716$	0	0
$717$	21.1246	0.788913
$718$	0	0
$719$	−0.506578	−0.0188922	−0.00944608	−	0.999955i	$-0.503007\pi$
$719$	−0.506578	−0.0188922	−0.00944608	+	0.999955i	$0.503007\pi$
$720$	0	0
$721$	−19.6738	−0.732689
$722$	0	0
$723$	15.3475	0.570781
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−11.0000	−0.407967	−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000	−0.407967	−0.203984	+	0.978974i	$0.565389\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	11.2361	0.415581
$732$	0	0
$733$	1.34752	0.0497719	0.0248860	−	0.999690i	$-0.492078\pi$
$733$	1.34752	0.0497719	0.0248860	+	0.999690i	$0.492078\pi$
$734$	0	0
$735$	8.69505	0.320722
$736$	0	0
$737$	21.1803	0.780188
$738$	0	0
$739$	−45.1591	−1.66120	−0.830601	−	0.556868i	$-0.812003\pi$
$739$	−45.1591	−1.66120	−0.830601	+	0.556868i	$0.812003\pi$
$740$	0	0
$741$	−35.5967	−1.30768
$742$	0	0
$743$	−15.6525	−0.574234	−0.287117	−	0.957896i	$-0.592697\pi$
$743$	−15.6525	−0.574234	−0.287117	+	0.957896i	$0.592697\pi$
$744$	0	0
$745$	7.31308	0.267931
$746$	0	0
$747$	−6.66563	−0.243883
$748$	0	0
$749$	−10.8541	−0.396600
$750$	0	0
$751$	15.0000	0.547358	0.273679	−	0.961821i	$-0.411759\pi$
$751$	15.0000	0.547358	0.273679	+	0.961821i	$0.411759\pi$
$752$	0	0
$753$	−8.24922	−0.300618
$754$	0	0
$755$	−16.9098	−0.615412
$756$	0	0
$757$	−7.23607	−0.262999	−0.131500	−	0.991316i	$-0.541979\pi$
$757$	−7.23607	−0.262999	−0.131500	+	0.991316i	$0.541979\pi$
$758$	0	0
$759$	3.16718	0.114962
$760$	0	0
$761$	−19.0344	−0.689998	−0.344999	−	0.938603i	$-0.612121\pi$
$761$	−19.0344	−0.689998	−0.344999	+	0.938603i	$0.612121\pi$
$762$	0	0
$763$	−16.1033	−0.582980
$764$	0	0
$765$	4.06888	0.147111
$766$	0	0
$767$	−87.8673	−3.17270
$768$	0	0
$769$	3.59675	0.129702	0.0648510	−	0.997895i	$-0.479343\pi$
$769$	3.59675	0.129702	0.0648510	+	0.997895i	$0.479343\pi$
$770$	0	0
$771$	−16.4721	−0.593229
$772$	0	0
$773$	−53.4296	−1.92173	−0.960864	−	0.277021i	$-0.910653\pi$
$773$	−53.4296	−1.92173	−0.960864	+	0.277021i	$0.910653\pi$
$774$	0	0
$775$	4.54915	0.163410
$776$	0	0
$777$	−11.0557	−0.396622
$778$	0	0
$779$	16.0344	0.574493
$780$	0	0
$781$	−22.5623	−0.807343
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−26.9098	−0.960453
$786$	0	0
$787$	−10.1459	−0.361662	−0.180831	−	0.983514i	$-0.557879\pi$
$787$	−10.1459	−0.361662	−0.180831	+	0.983514i	$0.557879\pi$
$788$	0	0
$789$	17.4164	0.620041
$790$	0	0
$791$	−0.652476	−0.0231994
$792$	0	0
$793$	−6.23607	−0.221449
$794$	0	0
$795$	−18.6950	−0.663045
$796$	0	0
$797$	54.2705	1.92236	0.961180	−	0.275922i	$-0.0889831\pi$
$797$	54.2705	1.92236	0.961180	+	0.275922i	$0.0889831\pi$
$798$	0	0
$799$	6.76393	0.239291
$800$	0	0
$801$	1.90170	0.0671932
$802$	0	0
$803$	−23.0902	−0.814834
$804$	0	0
$805$	−2.18847	−0.0771335
$806$	0	0
$807$	−9.93112	−0.349592
$808$	0	0
$809$	−31.8541	−1.11993	−0.559965	−	0.828516i	$-0.689186\pi$
$809$	−31.8541	−1.11993	−0.559965	+	0.828516i	$0.689186\pi$
$810$	0	0
$811$	4.79837	0.168494	0.0842468	−	0.996445i	$-0.473152\pi$
$811$	4.79837	0.168494	0.0842468	+	0.996445i	$0.473152\pi$
$812$	0	0
$813$	−33.3475	−1.16955
$814$	0	0
$815$	30.7771	1.07807
$816$	0	0
$817$	25.9443	0.907675
$818$	0	0
$819$	−12.6869	−0.443317
$820$	0	0
$821$	−37.8885	−1.32232	−0.661160	−	0.750245i	$-0.729935\pi$
$821$	−37.8885	−1.32232	−0.661160	+	0.750245i	$0.729935\pi$
$822$	0	0
$823$	−33.5967	−1.17111	−0.585555	−	0.810633i	$-0.699123\pi$
$823$	−33.5967	−1.17111	−0.585555	+	0.810633i	$0.699123\pi$
$824$	0	0
$825$	−8.54102	−0.297360
$826$	0	0
$827$	16.5279	0.574730	0.287365	−	0.957821i	$-0.407221\pi$
$827$	16.5279	0.574730	0.287365	+	0.957821i	$0.407221\pi$
$828$	0	0
$829$	14.3262	0.497571	0.248785	−	0.968559i	$-0.419969\pi$
$829$	14.3262	0.497571	0.248785	+	0.968559i	$0.419969\pi$
$830$	0	0
$831$	−3.52786	−0.122380
$832$	0	0
$833$	10.1803	0.352728
$834$	0	0
$835$	28.9443	1.00166
$836$	0	0
$837$	−8.13777	−0.281282
$838$	0	0
$839$	51.7771	1.78754	0.893772	−	0.448522i	$-0.148049\pi$
$839$	51.7771	1.78754	0.893772	+	0.448522i	$0.148049\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	27.7082	0.954321
$844$	0	0
$845$	35.7771	1.23077
$846$	0	0
$847$	8.29180	0.284909
$848$	0	0
$849$	−3.93112	−0.134916
$850$	0	0
$851$	−7.41641	−0.254231
$852$	0	0
$853$	33.6738	1.15297	0.576484	−	0.817109i	$-0.304424\pi$
$853$	33.6738	1.15297	0.576484	+	0.817109i	$0.304424\pi$
$854$	0	0
$855$	9.39512	0.321306
$856$	0	0
$857$	33.2148	1.13460	0.567298	−	0.823513i	$-0.307989\pi$
$857$	33.2148	1.13460	0.567298	+	0.823513i	$0.307989\pi$
$858$	0	0
$859$	−30.8328	−1.05200	−0.526001	−	0.850484i	$-0.676309\pi$
$859$	−30.8328	−1.05200	−0.526001	+	0.850484i	$0.676309\pi$
$860$	0	0
$861$	−5.93112	−0.202132
$862$	0	0
$863$	−16.7426	−0.569926	−0.284963	−	0.958538i	$-0.591981\pi$
$863$	−16.7426	−0.569926	−0.284963	+	0.958538i	$0.591981\pi$
$864$	0	0
$865$	25.2016	0.856881
$866$	0	0
$867$	16.0689	0.545728
$868$	0	0
$869$	2.23607	0.0758534
$870$	0	0
$871$	59.0689	2.00147
$872$	0	0
$873$	16.8065	0.568814
$874$	0	0
$875$	15.4508	0.522334
$876$	0	0
$877$	−10.7639	−0.363472	−0.181736	−	0.983347i	$-0.558172\pi$
$877$	−10.7639	−0.363472	−0.181736	+	0.983347i	$0.558172\pi$
$878$	0	0
$879$	37.7771	1.27419
$880$	0	0
$881$	−28.3820	−0.956213	−0.478106	−	0.878302i	$-0.658677\pi$
$881$	−28.3820	−0.956213	−0.478106	+	0.878302i	$0.658677\pi$
$882$	0	0
$883$	57.4164	1.93222	0.966108	−	0.258138i	$-0.0831090\pi$
$883$	57.4164	1.93222	0.966108	+	0.258138i	$0.0831090\pi$
$884$	0	0
$885$	−24.0689	−0.809067
$886$	0	0
$887$	−58.0689	−1.94976	−0.974881	−	0.222726i	$-0.928505\pi$
$887$	−58.0689	−1.94976	−0.974881	+	0.222726i	$0.928505\pi$
$888$	0	0
$889$	−8.21478	−0.275515
$890$	0	0
$891$	5.40325	0.181016
$892$	0	0
$893$	15.6180	0.522638
$894$	0	0
$895$	29.3475	0.980980
$896$	0	0
$897$	8.83282	0.294919
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−21.8885	−0.729213
$902$	0	0
$903$	−9.59675	−0.319360
$904$	0	0
$905$	−19.2229	−0.638991
$906$	0	0
$907$	26.1246	0.867453	0.433727	−	0.901044i	$-0.357198\pi$
$907$	26.1246	0.867453	0.433727	+	0.901044i	$0.357198\pi$
$908$	0	0
$909$	13.0344	0.432325
$910$	0	0
$911$	41.5623	1.37702	0.688510	−	0.725227i	$-0.258265\pi$
$911$	41.5623	1.37702	0.688510	+	0.725227i	$0.258265\pi$
$912$	0	0
$913$	−10.1246	−0.335076
$914$	0	0
$915$	−1.70820	−0.0564715
$916$	0	0
$917$	−18.6950	−0.617365
$918$	0	0
$919$	1.47214	0.0485613	0.0242806	−	0.999705i	$-0.492270\pi$
$919$	1.47214	0.0485613	0.0242806	+	0.999705i	$0.492270\pi$
$920$	0	0
$921$	−6.40325	−0.210994
$922$	0	0
$923$	−62.9230	−2.07114
$924$	0	0
$925$	20.0000	0.657596
$926$	0	0
$927$	−20.9574	−0.688332
$928$	0	0
$929$	−28.5066	−0.935270	−0.467635	−	0.883922i	$-0.654894\pi$
$929$	−28.5066	−0.935270	−0.467635	+	0.883922i	$0.654894\pi$
$930$	0	0
$931$	23.5066	0.770397
$932$	0	0
$933$	−25.6393	−0.839394
$934$	0	0
$935$	6.18034	0.202119
$936$	0	0
$937$	51.2492	1.67424	0.837120	−	0.547020i	$-0.184238\pi$
$937$	51.2492	1.67424	0.837120	+	0.547020i	$0.184238\pi$
$938$	0	0
$939$	28.8754	0.942313
$940$	0	0
$941$	−12.5623	−0.409519	−0.204760	−	0.978812i	$-0.565641\pi$
$941$	−12.5623	−0.409519	−0.204760	+	0.978812i	$0.565641\pi$
$942$	0	0
$943$	−3.97871	−0.129565
$944$	0	0
$945$	−10.5573	−0.343428
$946$	0	0
$947$	53.5967	1.74166	0.870830	−	0.491584i	$-0.163582\pi$
$947$	53.5967	1.74166	0.870830	+	0.491584i	$0.163582\pi$
$948$	0	0
$949$	−64.3951	−2.09035
$950$	0	0
$951$	−33.8197	−1.09668
$952$	0	0
$953$	2.25735	0.0731229	0.0365614	−	0.999331i	$-0.488360\pi$
$953$	2.25735	0.0731229	0.0365614	+	0.999331i	$0.488360\pi$
$954$	0	0
$955$	−35.1246	−1.13661
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.68692	0.0867651
$960$	0	0
$961$	−28.8328	−0.930091
$962$	0	0
$963$	−11.5623	−0.372590
$964$	0	0
$965$	−30.5279	−0.982727
$966$	0	0
$967$	−30.2705	−0.973434	−0.486717	−	0.873560i	$-0.661806\pi$
$967$	−30.2705	−0.973434	−0.486717	+	0.873560i	$0.661806\pi$
$968$	0	0
$969$	−11.4164	−0.366748
$970$	0	0
$971$	−10.7082	−0.343643	−0.171821	−	0.985128i	$-0.554965\pi$
$971$	−10.7082	−0.343643	−0.171821	+	0.985128i	$0.554965\pi$
$972$	0	0
$973$	26.9098	0.862690
$974$	0	0
$975$	−23.8197	−0.762840
$976$	0	0
$977$	46.7639	1.49611	0.748055	−	0.663636i	$-0.230988\pi$
$977$	46.7639	1.49611	0.748055	+	0.663636i	$0.230988\pi$
$978$	0	0
$979$	2.88854	0.0923183
$980$	0	0
$981$	−17.1540	−0.547686
$982$	0	0
$983$	−6.47214	−0.206429	−0.103215	−	0.994659i	$-0.532913\pi$
$983$	−6.47214	−0.206429	−0.103215	+	0.994659i	$0.532913\pi$
$984$	0	0
$985$	21.5066	0.685257
$986$	0	0
$987$	−5.77709	−0.183887
$988$	0	0
$989$	−6.43769	−0.204707
$990$	0	0
$991$	−58.3262	−1.85279	−0.926397	−	0.376548i	$-0.877111\pi$
$991$	−58.3262	−1.85279	−0.926397	+	0.376548i	$0.877111\pi$
$992$	0	0
$993$	13.0557	0.414311
$994$	0	0
$995$	18.2148	0.577447
$996$	0	0
$997$	23.1246	0.732364	0.366182	−	0.930543i	$-0.380665\pi$
$997$	23.1246	0.732364	0.366182	+	0.930543i	$0.380665\pi$
$998$	0	0
$999$	−35.7771	−1.13194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6728.2.a.g.1.1	yes	2
29.28	even	2		6728.2.a.e.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6728.2.a.e.1.2	✓	2	29.28	even	2
6728.2.a.g.1.1	yes	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:	\( N \)	\(=\)	\( 6728 = 2^{3} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{3} +1.38197 q^{5} -1.38197 q^{7} -1.47214 q^{9} +O(q^{10})\)	\(q-1.23607 q^{3} +1.38197 q^{5} -1.38197 q^{7} -1.47214 q^{9} -2.23607 q^{11} -6.23607 q^{13} -1.70820 q^{15} -2.00000 q^{17} -4.61803 q^{19} +1.70820 q^{21} +1.14590 q^{23} -3.09017 q^{25} +5.52786 q^{27} -1.47214 q^{31} +2.76393 q^{33} -1.90983 q^{35} -6.47214 q^{37} +7.70820 q^{39} -3.47214 q^{41} -5.61803 q^{43} -2.03444 q^{45} -3.38197 q^{47} -5.09017 q^{49} +2.47214 q^{51} +10.9443 q^{53} -3.09017 q^{55} +5.70820 q^{57} +14.0902 q^{59} +1.00000 q^{61} +2.03444 q^{63} -8.61803 q^{65} -9.47214 q^{67} -1.41641 q^{69} +10.0902 q^{71} +10.3262 q^{73} +3.81966 q^{75} +3.09017 q^{77} -1.00000 q^{79} -2.41641 q^{81} +4.52786 q^{83} -2.76393 q^{85} -1.29180 q^{89} +8.61803 q^{91} +1.81966 q^{93} -6.38197 q^{95} -11.4164 q^{97} +3.29180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 5 q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 5 * q^5 - 5 * q^7 + 6 * q^9	\( 2 q + 2 q^{3} + 5 q^{5} - 5 q^{7} + 6 q^{9} - 8 q^{13} + 10 q^{15} - 4 q^{17} - 7 q^{19} - 10 q^{21} + 9 q^{23} + 5 q^{25} + 20 q^{27} + 6 q^{31} + 10 q^{33} - 15 q^{35} - 4 q^{37} + 2 q^{39} + 2 q^{41} - 9 q^{43} + 25 q^{45} - 9 q^{47} + q^{49} - 4 q^{51} + 4 q^{53} + 5 q^{55} - 2 q^{57} + 17 q^{59} + 2 q^{61} - 25 q^{63} - 15 q^{65} - 10 q^{67} + 24 q^{69} + 9 q^{71} + 5 q^{73} + 30 q^{75} - 5 q^{77} - 2 q^{79} + 22 q^{81} + 18 q^{83} - 10 q^{85} - 16 q^{89} + 15 q^{91} + 26 q^{93} - 15 q^{95} + 4 q^{97} + 20 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 5 * q^5 - 5 * q^7 + 6 * q^9 - 8 * q^13 + 10 * q^15 - 4 * q^17 - 7 * q^19 - 10 * q^21 + 9 * q^23 + 5 * q^25 + 20 * q^27 + 6 * q^31 + 10 * q^33 - 15 * q^35 - 4 * q^37 + 2 * q^39 + 2 * q^41 - 9 * q^43 + 25 * q^45 - 9 * q^47 + q^49 - 4 * q^51 + 4 * q^53 + 5 * q^55 - 2 * q^57 + 17 * q^59 + 2 * q^61 - 25 * q^63 - 15 * q^65 - 10 * q^67 + 24 * q^69 + 9 * q^71 + 5 * q^73 + 30 * q^75 - 5 * q^77 - 2 * q^79 + 22 * q^81 + 18 * q^83 - 10 * q^85 - 16 * q^89 + 15 * q^91 + 26 * q^93 - 15 * q^95 + 4 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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