LMFDB - Embedded newform 6028.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$2$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} -1.38197 q^{5} -4.61803 q^{7} -0.381966 q^{9} +O(q^{10})$	$q-1.61803 q^{3} -1.38197 q^{5} -4.61803 q^{7} -0.381966 q^{9} +1.00000 q^{11} +0.236068 q^{13} +2.23607 q^{15} -1.76393 q^{17} -0.381966 q^{19} +7.47214 q^{21} +0.854102 q^{23} -3.09017 q^{25} +5.47214 q^{27} -7.23607 q^{29} -9.47214 q^{31} -1.61803 q^{33} +6.38197 q^{35} -5.85410 q^{37} -0.381966 q^{39} -3.00000 q^{41} +2.70820 q^{43} +0.527864 q^{45} -11.9443 q^{47} +14.3262 q^{49} +2.85410 q^{51} -8.23607 q^{53} -1.38197 q^{55} +0.618034 q^{57} -6.09017 q^{59} -9.76393 q^{61} +1.76393 q^{63} -0.326238 q^{65} -10.3262 q^{67} -1.38197 q^{69} -9.85410 q^{71} -1.94427 q^{73} +5.00000 q^{75} -4.61803 q^{77} -1.29180 q^{79} -7.70820 q^{81} +15.5623 q^{83} +2.43769 q^{85} +11.7082 q^{87} -4.09017 q^{89} -1.09017 q^{91} +15.3262 q^{93} +0.527864 q^{95} +13.9443 q^{97} -0.381966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 5 q^{5} - 7 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 5 * q^5 - 7 * q^7 - 3 * q^9	$ 2 q - q^{3} - 5 q^{5} - 7 q^{7} - 3 q^{9} + 2 q^{11} - 4 q^{13} - 8 q^{17} - 3 q^{19} + 6 q^{21} - 5 q^{23} + 5 q^{25} + 2 q^{27} - 10 q^{29} - 10 q^{31} - q^{33} + 15 q^{35} - 5 q^{37} - 3 q^{39} - 6 q^{41} - 8 q^{43} + 10 q^{45} - 6 q^{47} + 13 q^{49} - q^{51} - 12 q^{53} - 5 q^{55} - q^{57} - q^{59} - 24 q^{61} + 8 q^{63} + 15 q^{65} - 5 q^{67} - 5 q^{69} - 13 q^{71} + 14 q^{73} + 10 q^{75} - 7 q^{77} - 16 q^{79} - 2 q^{81} + 11 q^{83} + 25 q^{85} + 10 q^{87} + 3 q^{89} + 9 q^{91} + 15 q^{93} + 10 q^{95} + 10 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q - q^3 - 5 * q^5 - 7 * q^7 - 3 * q^9 + 2 * q^11 - 4 * q^13 - 8 * q^17 - 3 * q^19 + 6 * q^21 - 5 * q^23 + 5 * q^25 + 2 * q^27 - 10 * q^29 - 10 * q^31 - q^33 + 15 * q^35 - 5 * q^37 - 3 * q^39 - 6 * q^41 - 8 * q^43 + 10 * q^45 - 6 * q^47 + 13 * q^49 - q^51 - 12 * q^53 - 5 * q^55 - q^57 - q^59 - 24 * q^61 + 8 * q^63 + 15 * q^65 - 5 * q^67 - 5 * q^69 - 13 * q^71 + 14 * q^73 + 10 * q^75 - 7 * q^77 - 16 * q^79 - 2 * q^81 + 11 * q^83 + 25 * q^85 + 10 * q^87 + 3 * q^89 + 9 * q^91 + 15 * q^93 + 10 * q^95 + 10 * q^97 - 3 * 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	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	−1.38197	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$5$	−1.38197	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$6$	0	0
$7$	−4.61803	−1.74545	−0.872726	−	0.488210i	$-0.837650\pi$
$7$	−4.61803	−1.74545	−0.872726	+	0.488210i	$0.837650\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	0	0
$15$	2.23607	0.577350
$16$	0	0
$17$	−1.76393	−0.427816	−0.213908	−	0.976854i	$-0.568619\pi$
$17$	−1.76393	−0.427816	−0.213908	+	0.976854i	$0.568619\pi$
$18$	0	0
$19$	−0.381966	−0.0876290	−0.0438145	−	0.999040i	$-0.513951\pi$
$19$	−0.381966	−0.0876290	−0.0438145	+	0.999040i	$0.513951\pi$
$20$	0	0
$21$	7.47214	1.63055
$22$	0	0
$23$	0.854102	0.178093	0.0890463	−	0.996027i	$-0.471618\pi$
$23$	0.854102	0.178093	0.0890463	+	0.996027i	$0.471618\pi$
$24$	0	0
$25$	−3.09017	−0.618034
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−7.23607	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$29$	−7.23607	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$30$	0	0
$31$	−9.47214	−1.70125	−0.850623	−	0.525776i	$-0.823775\pi$
$31$	−9.47214	−1.70125	−0.850623	+	0.525776i	$0.823775\pi$
$32$	0	0
$33$	−1.61803	−0.281664
$34$	0	0
$35$	6.38197	1.07875
$36$	0	0
$37$	−5.85410	−0.962408	−0.481204	−	0.876609i	$-0.659800\pi$
$37$	−5.85410	−0.962408	−0.481204	+	0.876609i	$0.659800\pi$
$38$	0	0
$39$	−0.381966	−0.0611635
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	2.70820	0.412997	0.206499	−	0.978447i	$-0.433793\pi$
$43$	2.70820	0.412997	0.206499	+	0.978447i	$0.433793\pi$
$44$	0	0
$45$	0.527864	0.0786893
$46$	0	0
$47$	−11.9443	−1.74225	−0.871126	−	0.491060i	$-0.836609\pi$
$47$	−11.9443	−1.74225	−0.871126	+	0.491060i	$0.836609\pi$
$48$	0	0
$49$	14.3262	2.04661
$50$	0	0
$51$	2.85410	0.399654
$52$	0	0
$53$	−8.23607	−1.13131	−0.565655	−	0.824642i	$-0.691377\pi$
$53$	−8.23607	−1.13131	−0.565655	+	0.824642i	$0.691377\pi$
$54$	0	0
$55$	−1.38197	−0.186344
$56$	0	0
$57$	0.618034	0.0818606
$58$	0	0
$59$	−6.09017	−0.792873	−0.396436	−	0.918062i	$-0.629753\pi$
$59$	−6.09017	−0.792873	−0.396436	+	0.918062i	$0.629753\pi$
$60$	0	0
$61$	−9.76393	−1.25014	−0.625072	−	0.780567i	$-0.714930\pi$
$61$	−9.76393	−1.25014	−0.625072	+	0.780567i	$0.714930\pi$
$62$	0	0
$63$	1.76393	0.222235
$64$	0	0
$65$	−0.326238	−0.0404648
$66$	0	0
$67$	−10.3262	−1.26155	−0.630775	−	0.775965i	$-0.717263\pi$
$67$	−10.3262	−1.26155	−0.630775	+	0.775965i	$0.717263\pi$
$68$	0	0
$69$	−1.38197	−0.166369
$70$	0	0
$71$	−9.85410	−1.16947	−0.584733	−	0.811226i	$-0.698801\pi$
$71$	−9.85410	−1.16947	−0.584733	+	0.811226i	$0.698801\pi$
$72$	0	0
$73$	−1.94427	−0.227560	−0.113780	−	0.993506i	$-0.536296\pi$
$73$	−1.94427	−0.227560	−0.113780	+	0.993506i	$0.536296\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	0	0
$77$	−4.61803	−0.526274
$78$	0	0
$79$	−1.29180	−0.145338	−0.0726692	−	0.997356i	$-0.523152\pi$
$79$	−1.29180	−0.145338	−0.0726692	+	0.997356i	$0.523152\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	15.5623	1.70818	0.854092	−	0.520121i	$-0.174113\pi$
$83$	15.5623	1.70818	0.854092	+	0.520121i	$0.174113\pi$
$84$	0	0
$85$	2.43769	0.264405
$86$	0	0
$87$	11.7082	1.25525
$88$	0	0
$89$	−4.09017	−0.433557	−0.216779	−	0.976221i	$-0.569555\pi$
$89$	−4.09017	−0.433557	−0.216779	+	0.976221i	$0.569555\pi$
$90$	0	0
$91$	−1.09017	−0.114281
$92$	0	0
$93$	15.3262	1.58926
$94$	0	0
$95$	0.527864	0.0541577
$96$	0	0
$97$	13.9443	1.41583	0.707913	−	0.706299i	$-0.249637\pi$
$97$	13.9443	1.41583	0.707913	+	0.706299i	$0.249637\pi$
$98$	0	0
$99$	−0.381966	−0.0383890
$100$	0	0
$101$	−15.8541	−1.57754	−0.788771	−	0.614687i	$-0.789282\pi$
$101$	−15.8541	−1.57754	−0.788771	+	0.614687i	$0.789282\pi$
$102$	0	0
$103$	−17.9443	−1.76810	−0.884051	−	0.467391i	$-0.845194\pi$
$103$	−17.9443	−1.76810	−0.884051	+	0.467391i	$0.845194\pi$
$104$	0	0
$105$	−10.3262	−1.00774
$106$	0	0
$107$	13.5623	1.31112	0.655559	−	0.755144i	$-0.272433\pi$
$107$	13.5623	1.31112	0.655559	+	0.755144i	$0.272433\pi$
$108$	0	0
$109$	−2.23607	−0.214176	−0.107088	−	0.994250i	$-0.534153\pi$
$109$	−2.23607	−0.214176	−0.107088	+	0.994250i	$0.534153\pi$
$110$	0	0
$111$	9.47214	0.899055
$112$	0	0
$113$	−10.9443	−1.02955	−0.514775	−	0.857325i	$-0.672125\pi$
$113$	−10.9443	−1.02955	−0.514775	+	0.857325i	$0.672125\pi$
$114$	0	0
$115$	−1.18034	−0.110067
$116$	0	0
$117$	−0.0901699	−0.00833621
$118$	0	0
$119$	8.14590	0.746733
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.85410	0.437680
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	−17.4721	−1.55040	−0.775201	−	0.631715i	$-0.782351\pi$
$127$	−17.4721	−1.55040	−0.775201	+	0.631715i	$0.782351\pi$
$128$	0	0
$129$	−4.38197	−0.385811
$130$	0	0
$131$	15.7082	1.37243	0.686216	−	0.727398i	$-0.259270\pi$
$131$	15.7082	1.37243	0.686216	+	0.727398i	$0.259270\pi$
$132$	0	0
$133$	1.76393	0.152952
$134$	0	0
$135$	−7.56231	−0.650860
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	14.7082	1.24753	0.623767	−	0.781611i	$-0.285601\pi$
$139$	14.7082	1.24753	0.623767	+	0.781611i	$0.285601\pi$
$140$	0	0
$141$	19.3262	1.62756
$142$	0	0
$143$	0.236068	0.0197410
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	−23.1803	−1.91188
$148$	0	0
$149$	−6.32624	−0.518266	−0.259133	−	0.965842i	$-0.583437\pi$
$149$	−6.32624	−0.518266	−0.259133	+	0.965842i	$0.583437\pi$
$150$	0	0
$151$	5.32624	0.433443	0.216722	−	0.976233i	$-0.430464\pi$
$151$	5.32624	0.433443	0.216722	+	0.976233i	$0.430464\pi$
$152$	0	0
$153$	0.673762	0.0544704
$154$	0	0
$155$	13.0902	1.05143
$156$	0	0
$157$	7.85410	0.626826	0.313413	−	0.949617i	$-0.398528\pi$
$157$	7.85410	0.626826	0.313413	+	0.949617i	$0.398528\pi$
$158$	0	0
$159$	13.3262	1.05684
$160$	0	0
$161$	−3.94427	−0.310852
$162$	0	0
$163$	11.4164	0.894202	0.447101	−	0.894483i	$-0.352456\pi$
$163$	11.4164	0.894202	0.447101	+	0.894483i	$0.352456\pi$
$164$	0	0
$165$	2.23607	0.174078
$166$	0	0
$167$	0.0901699	0.00697756	0.00348878	−	0.999994i	$-0.498889\pi$
$167$	0.0901699	0.00697756	0.00348878	+	0.999994i	$0.498889\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	0	0
$171$	0.145898	0.0111571
$172$	0	0
$173$	−15.3262	−1.16523	−0.582616	−	0.812747i	$-0.697971\pi$
$173$	−15.3262	−1.16523	−0.582616	+	0.812747i	$0.697971\pi$
$174$	0	0
$175$	14.2705	1.07875
$176$	0	0
$177$	9.85410	0.740680
$178$	0	0
$179$	10.9443	0.818013	0.409007	−	0.912531i	$-0.365875\pi$
$179$	10.9443	0.818013	0.409007	+	0.912531i	$0.365875\pi$
$180$	0	0
$181$	5.18034	0.385052	0.192526	−	0.981292i	$-0.438332\pi$
$181$	5.18034	0.385052	0.192526	+	0.981292i	$0.438332\pi$
$182$	0	0
$183$	15.7984	1.16785
$184$	0	0
$185$	8.09017	0.594801
$186$	0	0
$187$	−1.76393	−0.128991
$188$	0	0
$189$	−25.2705	−1.83816
$190$	0	0
$191$	12.7082	0.919533	0.459767	−	0.888040i	$-0.347933\pi$
$191$	12.7082	0.919533	0.459767	+	0.888040i	$0.347933\pi$
$192$	0	0
$193$	11.7082	0.842775	0.421387	−	0.906881i	$-0.361543\pi$
$193$	11.7082	0.842775	0.421387	+	0.906881i	$0.361543\pi$
$194$	0	0
$195$	0.527864	0.0378011
$196$	0	0
$197$	2.76393	0.196922	0.0984610	−	0.995141i	$-0.468608\pi$
$197$	2.76393	0.196922	0.0984610	+	0.995141i	$0.468608\pi$
$198$	0	0
$199$	5.76393	0.408594	0.204297	−	0.978909i	$-0.434509\pi$
$199$	5.76393	0.408594	0.204297	+	0.978909i	$0.434509\pi$
$200$	0	0
$201$	16.7082	1.17851
$202$	0	0
$203$	33.4164	2.34537
$204$	0	0
$205$	4.14590	0.289562
$206$	0	0
$207$	−0.326238	−0.0226751
$208$	0	0
$209$	−0.381966	−0.0264211
$210$	0	0
$211$	10.0902	0.694636	0.347318	−	0.937747i	$-0.387092\pi$
$211$	10.0902	0.694636	0.347318	+	0.937747i	$0.387092\pi$
$212$	0	0
$213$	15.9443	1.09248
$214$	0	0
$215$	−3.74265	−0.255246
$216$	0	0
$217$	43.7426	2.96944
$218$	0	0
$219$	3.14590	0.212580
$220$	0	0
$221$	−0.416408	−0.0280106
$222$	0	0
$223$	28.8885	1.93452	0.967260	−	0.253788i	$-0.0816764\pi$
$223$	28.8885	1.93452	0.967260	+	0.253788i	$0.0816764\pi$
$224$	0	0
$225$	1.18034	0.0786893
$226$	0	0
$227$	−17.9787	−1.19329	−0.596645	−	0.802506i	$-0.703500\pi$
$227$	−17.9787	−1.19329	−0.596645	+	0.802506i	$0.703500\pi$
$228$	0	0
$229$	−6.94427	−0.458890	−0.229445	−	0.973322i	$-0.573691\pi$
$229$	−6.94427	−0.458890	−0.229445	+	0.973322i	$0.573691\pi$
$230$	0	0
$231$	7.47214	0.491630
$232$	0	0
$233$	−17.6180	−1.15420	−0.577098	−	0.816675i	$-0.695815\pi$
$233$	−17.6180	−1.15420	−0.577098	+	0.816675i	$0.695815\pi$
$234$	0	0
$235$	16.5066	1.07677
$236$	0	0
$237$	2.09017	0.135771
$238$	0	0
$239$	−1.00000	−0.0646846	−0.0323423	−	0.999477i	$-0.510297\pi$
$239$	−1.00000	−0.0646846	−0.0323423	+	0.999477i	$0.510297\pi$
$240$	0	0
$241$	−17.2361	−1.11027	−0.555136	−	0.831759i	$-0.687334\pi$
$241$	−17.2361	−1.11027	−0.555136	+	0.831759i	$0.687334\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	−19.7984	−1.26487
$246$	0	0
$247$	−0.0901699	−0.00573738
$248$	0	0
$249$	−25.1803	−1.59574
$250$	0	0
$251$	−20.7984	−1.31278	−0.656391	−	0.754421i	$-0.727918\pi$
$251$	−20.7984	−1.31278	−0.656391	+	0.754421i	$0.727918\pi$
$252$	0	0
$253$	0.854102	0.0536969
$254$	0	0
$255$	−3.94427	−0.247000
$256$	0	0
$257$	−2.61803	−0.163308	−0.0816542	−	0.996661i	$-0.526020\pi$
$257$	−2.61803	−0.163308	−0.0816542	+	0.996661i	$0.526020\pi$
$258$	0	0
$259$	27.0344	1.67984
$260$	0	0
$261$	2.76393	0.171083
$262$	0	0
$263$	4.09017	0.252211	0.126105	−	0.992017i	$-0.459752\pi$
$263$	4.09017	0.252211	0.126105	+	0.992017i	$0.459752\pi$
$264$	0	0
$265$	11.3820	0.699189
$266$	0	0
$267$	6.61803	0.405017
$268$	0	0
$269$	−11.9443	−0.728255	−0.364128	−	0.931349i	$-0.618633\pi$
$269$	−11.9443	−0.728255	−0.364128	+	0.931349i	$0.618633\pi$
$270$	0	0
$271$	−2.38197	−0.144694	−0.0723471	−	0.997380i	$-0.523049\pi$
$271$	−2.38197	−0.144694	−0.0723471	+	0.997380i	$0.523049\pi$
$272$	0	0
$273$	1.76393	0.106758
$274$	0	0
$275$	−3.09017	−0.186344
$276$	0	0
$277$	−9.65248	−0.579961	−0.289981	−	0.957033i	$-0.593649\pi$
$277$	−9.65248	−0.579961	−0.289981	+	0.957033i	$0.593649\pi$
$278$	0	0
$279$	3.61803	0.216606
$280$	0	0
$281$	−10.2361	−0.610633	−0.305316	−	0.952251i	$-0.598762\pi$
$281$	−10.2361	−0.610633	−0.305316	+	0.952251i	$0.598762\pi$
$282$	0	0
$283$	−6.52786	−0.388041	−0.194021	−	0.980997i	$-0.562153\pi$
$283$	−6.52786	−0.388041	−0.194021	+	0.980997i	$0.562153\pi$
$284$	0	0
$285$	−0.854102	−0.0505926
$286$	0	0
$287$	13.8541	0.817782
$288$	0	0
$289$	−13.8885	−0.816973
$290$	0	0
$291$	−22.5623	−1.32263
$292$	0	0
$293$	−4.70820	−0.275056	−0.137528	−	0.990498i	$-0.543916\pi$
$293$	−4.70820	−0.275056	−0.137528	+	0.990498i	$0.543916\pi$
$294$	0	0
$295$	8.41641	0.490022
$296$	0	0
$297$	5.47214	0.317526
$298$	0	0
$299$	0.201626	0.0116603
$300$	0	0
$301$	−12.5066	−0.720867
$302$	0	0
$303$	25.6525	1.47370
$304$	0	0
$305$	13.4934	0.772631
$306$	0	0
$307$	2.38197	0.135946	0.0679730	−	0.997687i	$-0.478347\pi$
$307$	2.38197	0.135946	0.0679730	+	0.997687i	$0.478347\pi$
$308$	0	0
$309$	29.0344	1.65171
$310$	0	0
$311$	−2.58359	−0.146502	−0.0732510	−	0.997314i	$-0.523337\pi$
$311$	−2.58359	−0.146502	−0.0732510	+	0.997314i	$0.523337\pi$
$312$	0	0
$313$	−12.2705	−0.693570	−0.346785	−	0.937945i	$-0.612727\pi$
$313$	−12.2705	−0.693570	−0.346785	+	0.937945i	$0.612727\pi$
$314$	0	0
$315$	−2.43769	−0.137349
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−7.23607	−0.405142
$320$	0	0
$321$	−21.9443	−1.22481
$322$	0	0
$323$	0.673762	0.0374891
$324$	0	0
$325$	−0.729490	−0.0404648
$326$	0	0
$327$	3.61803	0.200078
$328$	0	0
$329$	55.1591	3.04102
$330$	0	0
$331$	12.0344	0.661473	0.330736	−	0.943723i	$-0.392703\pi$
$331$	12.0344	0.661473	0.330736	+	0.943723i	$0.392703\pi$
$332$	0	0
$333$	2.23607	0.122536
$334$	0	0
$335$	14.2705	0.779681
$336$	0	0
$337$	−11.2361	−0.612068	−0.306034	−	0.952021i	$-0.599002\pi$
$337$	−11.2361	−0.612068	−0.306034	+	0.952021i	$0.599002\pi$
$338$	0	0
$339$	17.7082	0.961778
$340$	0	0
$341$	−9.47214	−0.512945
$342$	0	0
$343$	−33.8328	−1.82680
$344$	0	0
$345$	1.90983	0.102822
$346$	0	0
$347$	14.7082	0.789578	0.394789	−	0.918772i	$-0.370818\pi$
$347$	14.7082	0.789578	0.394789	+	0.918772i	$0.370818\pi$
$348$	0	0
$349$	30.2148	1.61736	0.808680	−	0.588249i	$-0.200182\pi$
$349$	30.2148	1.61736	0.808680	+	0.588249i	$0.200182\pi$
$350$	0	0
$351$	1.29180	0.0689510
$352$	0	0
$353$	−20.1246	−1.07113	−0.535563	−	0.844496i	$-0.679900\pi$
$353$	−20.1246	−1.07113	−0.535563	+	0.844496i	$0.679900\pi$
$354$	0	0
$355$	13.6180	0.722770
$356$	0	0
$357$	−13.1803	−0.697578
$358$	0	0
$359$	0.944272	0.0498368	0.0249184	−	0.999689i	$-0.492067\pi$
$359$	0.944272	0.0498368	0.0249184	+	0.999689i	$0.492067\pi$
$360$	0	0
$361$	−18.8541	−0.992321
$362$	0	0
$363$	−1.61803	−0.0849248
$364$	0	0
$365$	2.68692	0.140640
$366$	0	0
$367$	−5.70820	−0.297966	−0.148983	−	0.988840i	$-0.547600\pi$
$367$	−5.70820	−0.297966	−0.148983	+	0.988840i	$0.547600\pi$
$368$	0	0
$369$	1.14590	0.0596531
$370$	0	0
$371$	38.0344	1.97465
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	−18.0902	−0.934172
$376$	0	0
$377$	−1.70820	−0.0879770
$378$	0	0
$379$	−11.2918	−0.580021	−0.290010	−	0.957024i	$-0.593659\pi$
$379$	−11.2918	−0.580021	−0.290010	+	0.957024i	$0.593659\pi$
$380$	0	0
$381$	28.2705	1.44834
$382$	0	0
$383$	15.2918	0.781374	0.390687	−	0.920524i	$-0.372237\pi$
$383$	15.2918	0.781374	0.390687	+	0.920524i	$0.372237\pi$
$384$	0	0
$385$	6.38197	0.325255
$386$	0	0
$387$	−1.03444	−0.0525836
$388$	0	0
$389$	34.5410	1.75130	0.875650	−	0.482947i	$-0.160434\pi$
$389$	34.5410	1.75130	0.875650	+	0.482947i	$0.160434\pi$
$390$	0	0
$391$	−1.50658	−0.0761909
$392$	0	0
$393$	−25.4164	−1.28209
$394$	0	0
$395$	1.78522	0.0898241
$396$	0	0
$397$	30.5066	1.53108	0.765541	−	0.643388i	$-0.222472\pi$
$397$	30.5066	1.53108	0.765541	+	0.643388i	$0.222472\pi$
$398$	0	0
$399$	−2.85410	−0.142884
$400$	0	0
$401$	1.50658	0.0752349	0.0376175	−	0.999292i	$-0.488023\pi$
$401$	1.50658	0.0752349	0.0376175	+	0.999292i	$0.488023\pi$
$402$	0	0
$403$	−2.23607	−0.111386
$404$	0	0
$405$	10.6525	0.529326
$406$	0	0
$407$	−5.85410	−0.290177
$408$	0	0
$409$	−17.3262	−0.856727	−0.428364	−	0.903606i	$-0.640910\pi$
$409$	−17.3262	−0.856727	−0.428364	+	0.903606i	$0.640910\pi$
$410$	0	0
$411$	−1.61803	−0.0798117
$412$	0	0
$413$	28.1246	1.38392
$414$	0	0
$415$	−21.5066	−1.05572
$416$	0	0
$417$	−23.7984	−1.16541
$418$	0	0
$419$	−16.1803	−0.790461	−0.395231	−	0.918582i	$-0.629335\pi$
$419$	−16.1803	−0.790461	−0.395231	+	0.918582i	$0.629335\pi$
$420$	0	0
$421$	−27.7426	−1.35209	−0.676047	−	0.736859i	$-0.736308\pi$
$421$	−27.7426	−1.35209	−0.676047	+	0.736859i	$0.736308\pi$
$422$	0	0
$423$	4.56231	0.221827
$424$	0	0
$425$	5.45085	0.264405
$426$	0	0
$427$	45.0902	2.18207
$428$	0	0
$429$	−0.381966	−0.0184415
$430$	0	0
$431$	−24.6525	−1.18747	−0.593734	−	0.804661i	$-0.702347\pi$
$431$	−24.6525	−1.18747	−0.593734	+	0.804661i	$0.702347\pi$
$432$	0	0
$433$	26.9443	1.29486	0.647430	−	0.762125i	$-0.275844\pi$
$433$	26.9443	1.29486	0.647430	+	0.762125i	$0.275844\pi$
$434$	0	0
$435$	−16.1803	−0.775788
$436$	0	0
$437$	−0.326238	−0.0156061
$438$	0	0
$439$	−21.2361	−1.01354	−0.506771	−	0.862081i	$-0.669161\pi$
$439$	−21.2361	−1.01354	−0.506771	+	0.862081i	$0.669161\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	0	0
$443$	9.90983	0.470830	0.235415	−	0.971895i	$-0.424355\pi$
$443$	9.90983	0.470830	0.235415	+	0.971895i	$0.424355\pi$
$444$	0	0
$445$	5.65248	0.267953
$446$	0	0
$447$	10.2361	0.484149
$448$	0	0
$449$	10.1803	0.480440	0.240220	−	0.970718i	$-0.422780\pi$
$449$	10.1803	0.480440	0.240220	+	0.970718i	$0.422780\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	0	0
$453$	−8.61803	−0.404911
$454$	0	0
$455$	1.50658	0.0706295
$456$	0	0
$457$	−36.1803	−1.69244	−0.846222	−	0.532830i	$-0.821129\pi$
$457$	−36.1803	−1.69244	−0.846222	+	0.532830i	$0.821129\pi$
$458$	0	0
$459$	−9.65248	−0.450539
$460$	0	0
$461$	−37.7082	−1.75625	−0.878123	−	0.478435i	$-0.841204\pi$
$461$	−37.7082	−1.75625	−0.878123	+	0.478435i	$0.841204\pi$
$462$	0	0
$463$	18.0689	0.839732	0.419866	−	0.907586i	$-0.362077\pi$
$463$	18.0689	0.839732	0.419866	+	0.907586i	$0.362077\pi$
$464$	0	0
$465$	−21.1803	−0.982215
$466$	0	0
$467$	−22.9098	−1.06014	−0.530070	−	0.847954i	$-0.677834\pi$
$467$	−22.9098	−1.06014	−0.530070	+	0.847954i	$0.677834\pi$
$468$	0	0
$469$	47.6869	2.20198
$470$	0	0
$471$	−12.7082	−0.585563
$472$	0	0
$473$	2.70820	0.124523
$474$	0	0
$475$	1.18034	0.0541577
$476$	0	0
$477$	3.14590	0.144041
$478$	0	0
$479$	24.1459	1.10325	0.551627	−	0.834091i	$-0.314007\pi$
$479$	24.1459	1.10325	0.551627	+	0.834091i	$0.314007\pi$
$480$	0	0
$481$	−1.38197	−0.0630122
$482$	0	0
$483$	6.38197	0.290390
$484$	0	0
$485$	−19.2705	−0.875029
$486$	0	0
$487$	25.7082	1.16495	0.582475	−	0.812849i	$-0.302084\pi$
$487$	25.7082	1.16495	0.582475	+	0.812849i	$0.302084\pi$
$488$	0	0
$489$	−18.4721	−0.835339
$490$	0	0
$491$	−38.0344	−1.71647	−0.858235	−	0.513257i	$-0.828439\pi$
$491$	−38.0344	−1.71647	−0.858235	+	0.513257i	$0.828439\pi$
$492$	0	0
$493$	12.7639	0.574859
$494$	0	0
$495$	0.527864	0.0237257
$496$	0	0
$497$	45.5066	2.04125
$498$	0	0
$499$	−17.4164	−0.779665	−0.389833	−	0.920886i	$-0.627467\pi$
$499$	−17.4164	−0.779665	−0.389833	+	0.920886i	$0.627467\pi$
$500$	0	0
$501$	−0.145898	−0.00651824
$502$	0	0
$503$	20.5623	0.916828	0.458414	−	0.888739i	$-0.348418\pi$
$503$	20.5623	0.916828	0.458414	+	0.888739i	$0.348418\pi$
$504$	0	0
$505$	21.9098	0.974975
$506$	0	0
$507$	20.9443	0.930168
$508$	0	0
$509$	2.97871	0.132029	0.0660146	−	0.997819i	$-0.478972\pi$
$509$	2.97871	0.132029	0.0660146	+	0.997819i	$0.478972\pi$
$510$	0	0
$511$	8.97871	0.397195
$512$	0	0
$513$	−2.09017	−0.0922833
$514$	0	0
$515$	24.7984	1.09275
$516$	0	0
$517$	−11.9443	−0.525308
$518$	0	0
$519$	24.7984	1.08853
$520$	0	0
$521$	7.36068	0.322477	0.161239	−	0.986915i	$-0.448451\pi$
$521$	7.36068	0.322477	0.161239	+	0.986915i	$0.448451\pi$
$522$	0	0
$523$	−1.38197	−0.0604292	−0.0302146	−	0.999543i	$-0.509619\pi$
$523$	−1.38197	−0.0604292	−0.0302146	+	0.999543i	$0.509619\pi$
$524$	0	0
$525$	−23.0902	−1.00774
$526$	0	0
$527$	16.7082	0.727821
$528$	0	0
$529$	−22.2705	−0.968283
$530$	0	0
$531$	2.32624	0.100950
$532$	0	0
$533$	−0.708204	−0.0306757
$534$	0	0
$535$	−18.7426	−0.810315
$536$	0	0
$537$	−17.7082	−0.764165
$538$	0	0
$539$	14.3262	0.617075
$540$	0	0
$541$	−37.7639	−1.62360	−0.811799	−	0.583937i	$-0.801512\pi$
$541$	−37.7639	−1.62360	−0.811799	+	0.583937i	$0.801512\pi$
$542$	0	0
$543$	−8.38197	−0.359705
$544$	0	0
$545$	3.09017	0.132368
$546$	0	0
$547$	−6.27051	−0.268108	−0.134054	−	0.990974i	$-0.542800\pi$
$547$	−6.27051	−0.268108	−0.134054	+	0.990974i	$0.542800\pi$
$548$	0	0
$549$	3.72949	0.159171
$550$	0	0
$551$	2.76393	0.117747
$552$	0	0
$553$	5.96556	0.253681
$554$	0	0
$555$	−13.0902	−0.555647
$556$	0	0
$557$	−12.3475	−0.523181	−0.261591	−	0.965179i	$-0.584247\pi$
$557$	−12.3475	−0.523181	−0.261591	+	0.965179i	$0.584247\pi$
$558$	0	0
$559$	0.639320	0.0270404
$560$	0	0
$561$	2.85410	0.120500
$562$	0	0
$563$	−36.5967	−1.54237	−0.771185	−	0.636612i	$-0.780335\pi$
$563$	−36.5967	−1.54237	−0.771185	+	0.636612i	$0.780335\pi$
$564$	0	0
$565$	15.1246	0.636297
$566$	0	0
$567$	35.5967	1.49492
$568$	0	0
$569$	18.9098	0.792741	0.396371	−	0.918091i	$-0.370270\pi$
$569$	18.9098	0.792741	0.396371	+	0.918091i	$0.370270\pi$
$570$	0	0
$571$	−31.7082	−1.32695	−0.663474	−	0.748200i	$-0.730918\pi$
$571$	−31.7082	−1.32695	−0.663474	+	0.748200i	$0.730918\pi$
$572$	0	0
$573$	−20.5623	−0.859003
$574$	0	0
$575$	−2.63932	−0.110067
$576$	0	0
$577$	−21.3262	−0.887823	−0.443911	−	0.896071i	$-0.646410\pi$
$577$	−21.3262	−0.887823	−0.443911	+	0.896071i	$0.646410\pi$
$578$	0	0
$579$	−18.9443	−0.787297
$580$	0	0
$581$	−71.8673	−2.98156
$582$	0	0
$583$	−8.23607	−0.341103
$584$	0	0
$585$	0.124612	0.00515206
$586$	0	0
$587$	20.1246	0.830632	0.415316	−	0.909677i	$-0.363671\pi$
$587$	20.1246	0.830632	0.415316	+	0.909677i	$0.363671\pi$
$588$	0	0
$589$	3.61803	0.149078
$590$	0	0
$591$	−4.47214	−0.183959
$592$	0	0
$593$	33.2705	1.36626	0.683128	−	0.730299i	$-0.260619\pi$
$593$	33.2705	1.36626	0.683128	+	0.730299i	$0.260619\pi$
$594$	0	0
$595$	−11.2574	−0.461507
$596$	0	0
$597$	−9.32624	−0.381698
$598$	0	0
$599$	28.4164	1.16106	0.580531	−	0.814238i	$-0.302845\pi$
$599$	28.4164	1.16106	0.580531	+	0.814238i	$0.302845\pi$
$600$	0	0
$601$	1.12461	0.0458739	0.0229369	−	0.999737i	$-0.492698\pi$
$601$	1.12461	0.0458739	0.0229369	+	0.999737i	$0.492698\pi$
$602$	0	0
$603$	3.94427	0.160623
$604$	0	0
$605$	−1.38197	−0.0561849
$606$	0	0
$607$	42.1033	1.70892	0.854461	−	0.519516i	$-0.173888\pi$
$607$	42.1033	1.70892	0.854461	+	0.519516i	$0.173888\pi$
$608$	0	0
$609$	−54.0689	−2.19098
$610$	0	0
$611$	−2.81966	−0.114071
$612$	0	0
$613$	−22.1803	−0.895855	−0.447928	−	0.894070i	$-0.647838\pi$
$613$	−22.1803	−0.895855	−0.447928	+	0.894070i	$0.647838\pi$
$614$	0	0
$615$	−6.70820	−0.270501
$616$	0	0
$617$	26.4721	1.06573	0.532864	−	0.846201i	$-0.321116\pi$
$617$	26.4721	1.06573	0.532864	+	0.846201i	$0.321116\pi$
$618$	0	0
$619$	7.11146	0.285834	0.142917	−	0.989735i	$-0.454352\pi$
$619$	7.11146	0.285834	0.142917	+	0.989735i	$0.454352\pi$
$620$	0	0
$621$	4.67376	0.187552
$622$	0	0
$623$	18.8885	0.756754
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.618034	0.0246819
$628$	0	0
$629$	10.3262	0.411734
$630$	0	0
$631$	−21.7639	−0.866408	−0.433204	−	0.901296i	$-0.642617\pi$
$631$	−21.7639	−0.866408	−0.433204	+	0.901296i	$0.642617\pi$
$632$	0	0
$633$	−16.3262	−0.648910
$634$	0	0
$635$	24.1459	0.958201
$636$	0	0
$637$	3.38197	0.133998
$638$	0	0
$639$	3.76393	0.148899
$640$	0	0
$641$	−11.3607	−0.448720	−0.224360	−	0.974506i	$-0.572029\pi$
$641$	−11.3607	−0.448720	−0.224360	+	0.974506i	$0.572029\pi$
$642$	0	0
$643$	22.2918	0.879103	0.439551	−	0.898217i	$-0.355137\pi$
$643$	22.2918	0.879103	0.439551	+	0.898217i	$0.355137\pi$
$644$	0	0
$645$	6.05573	0.238444
$646$	0	0
$647$	34.4164	1.35305	0.676524	−	0.736420i	$-0.263485\pi$
$647$	34.4164	1.35305	0.676524	+	0.736420i	$0.263485\pi$
$648$	0	0
$649$	−6.09017	−0.239060
$650$	0	0
$651$	−70.7771	−2.77397
$652$	0	0
$653$	−4.52786	−0.177189	−0.0885945	−	0.996068i	$-0.528238\pi$
$653$	−4.52786	−0.177189	−0.0885945	+	0.996068i	$0.528238\pi$
$654$	0	0
$655$	−21.7082	−0.848210
$656$	0	0
$657$	0.742646	0.0289734
$658$	0	0
$659$	28.0902	1.09424	0.547119	−	0.837055i	$-0.315725\pi$
$659$	28.0902	1.09424	0.547119	+	0.837055i	$0.315725\pi$
$660$	0	0
$661$	42.9230	1.66951	0.834755	−	0.550621i	$-0.185609\pi$
$661$	42.9230	1.66951	0.834755	+	0.550621i	$0.185609\pi$
$662$	0	0
$663$	0.673762	0.0261668
$664$	0	0
$665$	−2.43769	−0.0945297
$666$	0	0
$667$	−6.18034	−0.239304
$668$	0	0
$669$	−46.7426	−1.80718
$670$	0	0
$671$	−9.76393	−0.376932
$672$	0	0
$673$	5.47214	0.210935	0.105468	−	0.994423i	$-0.466366\pi$
$673$	5.47214	0.210935	0.105468	+	0.994423i	$0.466366\pi$
$674$	0	0
$675$	−16.9098	−0.650860
$676$	0	0
$677$	29.5279	1.13485	0.567424	−	0.823426i	$-0.307940\pi$
$677$	29.5279	1.13485	0.567424	+	0.823426i	$0.307940\pi$
$678$	0	0
$679$	−64.3951	−2.47126
$680$	0	0
$681$	29.0902	1.11474
$682$	0	0
$683$	−32.1803	−1.23135	−0.615673	−	0.788002i	$-0.711116\pi$
$683$	−32.1803	−1.23135	−0.615673	+	0.788002i	$0.711116\pi$
$684$	0	0
$685$	−1.38197	−0.0528022
$686$	0	0
$687$	11.2361	0.428683
$688$	0	0
$689$	−1.94427	−0.0740709
$690$	0	0
$691$	22.2705	0.847210	0.423605	−	0.905847i	$-0.360764\pi$
$691$	22.2705	0.847210	0.423605	+	0.905847i	$0.360764\pi$
$692$	0	0
$693$	1.76393	0.0670062
$694$	0	0
$695$	−20.3262	−0.771018
$696$	0	0
$697$	5.29180	0.200441
$698$	0	0
$699$	28.5066	1.07822
$700$	0	0
$701$	−5.11146	−0.193057	−0.0965285	−	0.995330i	$-0.530774\pi$
$701$	−5.11146	−0.193057	−0.0965285	+	0.995330i	$0.530774\pi$
$702$	0	0
$703$	2.23607	0.0843349
$704$	0	0
$705$	−26.7082	−1.00589
$706$	0	0
$707$	73.2148	2.75353
$708$	0	0
$709$	−7.56231	−0.284008	−0.142004	−	0.989866i	$-0.545355\pi$
$709$	−7.56231	−0.284008	−0.142004	+	0.989866i	$0.545355\pi$
$710$	0	0
$711$	0.493422	0.0185048
$712$	0	0
$713$	−8.09017	−0.302979
$714$	0	0
$715$	−0.326238	−0.0122006
$716$	0	0
$717$	1.61803	0.0604266
$718$	0	0
$719$	−9.20163	−0.343163	−0.171581	−	0.985170i	$-0.554888\pi$
$719$	−9.20163	−0.343163	−0.171581	+	0.985170i	$0.554888\pi$
$720$	0	0
$721$	82.8673	3.08614
$722$	0	0
$723$	27.8885	1.03719
$724$	0	0
$725$	22.3607	0.830455
$726$	0	0
$727$	−18.5623	−0.688438	−0.344219	−	0.938889i	$-0.611856\pi$
$727$	−18.5623	−0.688438	−0.344219	+	0.938889i	$0.611856\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	−4.77709	−0.176687
$732$	0	0
$733$	−13.2148	−0.488099	−0.244050	−	0.969763i	$-0.578476\pi$
$733$	−13.2148	−0.488099	−0.244050	+	0.969763i	$0.578476\pi$
$734$	0	0
$735$	32.0344	1.18161
$736$	0	0
$737$	−10.3262	−0.380372
$738$	0	0
$739$	−20.4377	−0.751813	−0.375906	−	0.926658i	$-0.622669\pi$
$739$	−20.4377	−0.751813	−0.375906	+	0.926658i	$0.622669\pi$
$740$	0	0
$741$	0.145898	0.00535970
$742$	0	0
$743$	−0.832816	−0.0305530	−0.0152765	−	0.999883i	$-0.504863\pi$
$743$	−0.832816	−0.0305530	−0.0152765	+	0.999883i	$0.504863\pi$
$744$	0	0
$745$	8.74265	0.320306
$746$	0	0
$747$	−5.94427	−0.217490
$748$	0	0
$749$	−62.6312	−2.28849
$750$	0	0
$751$	35.3607	1.29033	0.645165	−	0.764043i	$-0.276789\pi$
$751$	35.3607	1.29033	0.645165	+	0.764043i	$0.276789\pi$
$752$	0	0
$753$	33.6525	1.22636
$754$	0	0
$755$	−7.36068	−0.267883
$756$	0	0
$757$	−46.2705	−1.68173	−0.840865	−	0.541245i	$-0.817953\pi$
$757$	−46.2705	−1.68173	−0.840865	+	0.541245i	$0.817953\pi$
$758$	0	0
$759$	−1.38197	−0.0501622
$760$	0	0
$761$	21.1803	0.767787	0.383893	−	0.923377i	$-0.374583\pi$
$761$	21.1803	0.767787	0.383893	+	0.923377i	$0.374583\pi$
$762$	0	0
$763$	10.3262	0.373835
$764$	0	0
$765$	−0.931116	−0.0336646
$766$	0	0
$767$	−1.43769	−0.0519121
$768$	0	0
$769$	7.29180	0.262949	0.131474	−	0.991320i	$-0.458029\pi$
$769$	7.29180	0.262949	0.131474	+	0.991320i	$0.458029\pi$
$770$	0	0
$771$	4.23607	0.152558
$772$	0	0
$773$	23.7426	0.853964	0.426982	−	0.904260i	$-0.359577\pi$
$773$	23.7426	0.853964	0.426982	+	0.904260i	$0.359577\pi$
$774$	0	0
$775$	29.2705	1.05143
$776$	0	0
$777$	−43.7426	−1.56926
$778$	0	0
$779$	1.14590	0.0410561
$780$	0	0
$781$	−9.85410	−0.352607
$782$	0	0
$783$	−39.5967	−1.41507
$784$	0	0
$785$	−10.8541	−0.387400
$786$	0	0
$787$	6.41641	0.228720	0.114360	−	0.993439i	$-0.463518\pi$
$787$	6.41641	0.228720	0.114360	+	0.993439i	$0.463518\pi$
$788$	0	0
$789$	−6.61803	−0.235608
$790$	0	0
$791$	50.5410	1.79703
$792$	0	0
$793$	−2.30495	−0.0818512
$794$	0	0
$795$	−18.4164	−0.653163
$796$	0	0
$797$	23.3050	0.825504	0.412752	−	0.910844i	$-0.364568\pi$
$797$	23.3050	0.825504	0.412752	+	0.910844i	$0.364568\pi$
$798$	0	0
$799$	21.0689	0.745364
$800$	0	0
$801$	1.56231	0.0552014
$802$	0	0
$803$	−1.94427	−0.0686119
$804$	0	0
$805$	5.45085	0.192117
$806$	0	0
$807$	19.3262	0.680316
$808$	0	0
$809$	−39.1033	−1.37480	−0.687400	−	0.726279i	$-0.741248\pi$
$809$	−39.1033	−1.37480	−0.687400	+	0.726279i	$0.741248\pi$
$810$	0	0
$811$	−4.25735	−0.149496	−0.0747480	−	0.997202i	$-0.523815\pi$
$811$	−4.25735	−0.149496	−0.0747480	+	0.997202i	$0.523815\pi$
$812$	0	0
$813$	3.85410	0.135169
$814$	0	0
$815$	−15.7771	−0.552647
$816$	0	0
$817$	−1.03444	−0.0361905
$818$	0	0
$819$	0.416408	0.0145505
$820$	0	0
$821$	−56.1246	−1.95876	−0.979381	−	0.202021i	$-0.935249\pi$
$821$	−56.1246	−1.95876	−0.979381	+	0.202021i	$0.935249\pi$
$822$	0	0
$823$	27.7984	0.968990	0.484495	−	0.874794i	$-0.339003\pi$
$823$	27.7984	0.968990	0.484495	+	0.874794i	$0.339003\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	0	0
$827$	23.5066	0.817404	0.408702	−	0.912668i	$-0.365982\pi$
$827$	23.5066	0.817404	0.408702	+	0.912668i	$0.365982\pi$
$828$	0	0
$829$	24.7771	0.860544	0.430272	−	0.902699i	$-0.358418\pi$
$829$	24.7771	0.860544	0.430272	+	0.902699i	$0.358418\pi$
$830$	0	0
$831$	15.6180	0.541784
$832$	0	0
$833$	−25.2705	−0.875571
$834$	0	0
$835$	−0.124612	−0.00431237
$836$	0	0
$837$	−51.8328	−1.79160
$838$	0	0
$839$	−30.3050	−1.04624	−0.523122	−	0.852258i	$-0.675233\pi$
$839$	−30.3050	−1.04624	−0.523122	+	0.852258i	$0.675233\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	0	0
$843$	16.5623	0.570436
$844$	0	0
$845$	17.8885	0.615385
$846$	0	0
$847$	−4.61803	−0.158678
$848$	0	0
$849$	10.5623	0.362497
$850$	0	0
$851$	−5.00000	−0.171398
$852$	0	0
$853$	−46.3050	−1.58545	−0.792726	−	0.609579i	$-0.791339\pi$
$853$	−46.3050	−1.58545	−0.792726	+	0.609579i	$0.791339\pi$
$854$	0	0
$855$	−0.201626	−0.00689547
$856$	0	0
$857$	22.1246	0.755762	0.377881	−	0.925854i	$-0.376653\pi$
$857$	22.1246	0.755762	0.377881	+	0.925854i	$0.376653\pi$
$858$	0	0
$859$	−32.5623	−1.11101	−0.555506	−	0.831513i	$-0.687475\pi$
$859$	−32.5623	−1.11101	−0.555506	+	0.831513i	$0.687475\pi$
$860$	0	0
$861$	−22.4164	−0.763949
$862$	0	0
$863$	32.0689	1.09164	0.545819	−	0.837903i	$-0.316219\pi$
$863$	32.0689	1.09164	0.545819	+	0.837903i	$0.316219\pi$
$864$	0	0
$865$	21.1803	0.720153
$866$	0	0
$867$	22.4721	0.763194
$868$	0	0
$869$	−1.29180	−0.0438212
$870$	0	0
$871$	−2.43769	−0.0825981
$872$	0	0
$873$	−5.32624	−0.180266
$874$	0	0
$875$	−51.6312	−1.74545
$876$	0	0
$877$	−25.8197	−0.871868	−0.435934	−	0.899979i	$-0.643582\pi$
$877$	−25.8197	−0.871868	−0.435934	+	0.899979i	$0.643582\pi$
$878$	0	0
$879$	7.61803	0.256950
$880$	0	0
$881$	30.5623	1.02967	0.514835	−	0.857289i	$-0.327853\pi$
$881$	30.5623	1.02967	0.514835	+	0.857289i	$0.327853\pi$
$882$	0	0
$883$	−33.6525	−1.13250	−0.566248	−	0.824235i	$-0.691606\pi$
$883$	−33.6525	−1.13250	−0.566248	+	0.824235i	$0.691606\pi$
$884$	0	0
$885$	−13.6180	−0.457765
$886$	0	0
$887$	44.9787	1.51024	0.755119	−	0.655588i	$-0.227579\pi$
$887$	44.9787	1.51024	0.755119	+	0.655588i	$0.227579\pi$
$888$	0	0
$889$	80.6869	2.70615
$890$	0	0
$891$	−7.70820	−0.258235
$892$	0	0
$893$	4.56231	0.152672
$894$	0	0
$895$	−15.1246	−0.505560
$896$	0	0
$897$	−0.326238	−0.0108928
$898$	0	0
$899$	68.5410	2.28597
$900$	0	0
$901$	14.5279	0.483993
$902$	0	0
$903$	20.2361	0.673414
$904$	0	0
$905$	−7.15905	−0.237975
$906$	0	0
$907$	−13.2148	−0.438790	−0.219395	−	0.975636i	$-0.570408\pi$
$907$	−13.2148	−0.438790	−0.219395	+	0.975636i	$0.570408\pi$
$908$	0	0
$909$	6.05573	0.200856
$910$	0	0
$911$	−13.3050	−0.440813	−0.220406	−	0.975408i	$-0.570738\pi$
$911$	−13.3050	−0.440813	−0.220406	+	0.975408i	$0.570738\pi$
$912$	0	0
$913$	15.5623	0.515037
$914$	0	0
$915$	−21.8328	−0.721771
$916$	0	0
$917$	−72.5410	−2.39552
$918$	0	0
$919$	−13.8885	−0.458141	−0.229070	−	0.973410i	$-0.573569\pi$
$919$	−13.8885	−0.458141	−0.229070	+	0.973410i	$0.573569\pi$
$920$	0	0
$921$	−3.85410	−0.126997
$922$	0	0
$923$	−2.32624	−0.0765691
$924$	0	0
$925$	18.0902	0.594801
$926$	0	0
$927$	6.85410	0.225118
$928$	0	0
$929$	−15.3951	−0.505098	−0.252549	−	0.967584i	$-0.581269\pi$
$929$	−15.3951	−0.505098	−0.252549	+	0.967584i	$0.581269\pi$
$930$	0	0
$931$	−5.47214	−0.179342
$932$	0	0
$933$	4.18034	0.136858
$934$	0	0
$935$	2.43769	0.0797211
$936$	0	0
$937$	7.52786	0.245925	0.122962	−	0.992411i	$-0.460761\pi$
$937$	7.52786	0.245925	0.122962	+	0.992411i	$0.460761\pi$
$938$	0	0
$939$	19.8541	0.647914
$940$	0	0
$941$	22.4377	0.731448	0.365724	−	0.930723i	$-0.380821\pi$
$941$	22.4377	0.731448	0.365724	+	0.930723i	$0.380821\pi$
$942$	0	0
$943$	−2.56231	−0.0834402
$944$	0	0
$945$	34.9230	1.13604
$946$	0	0
$947$	−8.58359	−0.278929	−0.139465	−	0.990227i	$-0.544538\pi$
$947$	−8.58359	−0.278929	−0.139465	+	0.990227i	$0.544538\pi$
$948$	0	0
$949$	−0.458980	−0.0148991
$950$	0	0
$951$	−29.1246	−0.944430
$952$	0	0
$953$	−44.5410	−1.44283	−0.721413	−	0.692506i	$-0.756507\pi$
$953$	−44.5410	−1.44283	−0.721413	+	0.692506i	$0.756507\pi$
$954$	0	0
$955$	−17.5623	−0.568303
$956$	0	0
$957$	11.7082	0.378472
$958$	0	0
$959$	−4.61803	−0.149124
$960$	0	0
$961$	58.7214	1.89424
$962$	0	0
$963$	−5.18034	−0.166934
$964$	0	0
$965$	−16.1803	−0.520864
$966$	0	0
$967$	−4.11146	−0.132216	−0.0661078	−	0.997812i	$-0.521058\pi$
$967$	−4.11146	−0.132216	−0.0661078	+	0.997812i	$0.521058\pi$
$968$	0	0
$969$	−1.09017	−0.0350213
$970$	0	0
$971$	−21.9443	−0.704225	−0.352113	−	0.935958i	$-0.614537\pi$
$971$	−21.9443	−0.704225	−0.352113	+	0.935958i	$0.614537\pi$
$972$	0	0
$973$	−67.9230	−2.17751
$974$	0	0
$975$	1.18034	0.0378011
$976$	0	0
$977$	15.0000	0.479893	0.239946	−	0.970786i	$-0.422870\pi$
$977$	15.0000	0.479893	0.239946	+	0.970786i	$0.422870\pi$
$978$	0	0
$979$	−4.09017	−0.130722
$980$	0	0
$981$	0.854102	0.0272694
$982$	0	0
$983$	−35.1803	−1.12208	−0.561039	−	0.827789i	$-0.689598\pi$
$983$	−35.1803	−1.12208	−0.561039	+	0.827789i	$0.689598\pi$
$984$	0	0
$985$	−3.81966	−0.121704
$986$	0	0
$987$	−89.2492	−2.84083
$988$	0	0
$989$	2.31308	0.0735517
$990$	0	0
$991$	−5.40325	−0.171640	−0.0858200	−	0.996311i	$-0.527351\pi$
$991$	−5.40325	−0.171640	−0.0858200	+	0.996311i	$0.527351\pi$
$992$	0	0
$993$	−19.4721	−0.617930
$994$	0	0
$995$	−7.96556	−0.252525
$996$	0	0
$997$	−47.0689	−1.49069	−0.745343	−	0.666681i	$-0.767714\pi$
$997$	−47.0689	−1.49069	−0.745343	+	0.666681i	$0.767714\pi$
$998$	0	0
$999$	−32.0344	−1.01352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.a.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.a.1.1	✓	2	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} -1.38197 q^{5} -4.61803 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q-1.61803 q^{3} -1.38197 q^{5} -4.61803 q^{7} -0.381966 q^{9} +1.00000 q^{11} +0.236068 q^{13} +2.23607 q^{15} -1.76393 q^{17} -0.381966 q^{19} +7.47214 q^{21} +0.854102 q^{23} -3.09017 q^{25} +5.47214 q^{27} -7.23607 q^{29} -9.47214 q^{31} -1.61803 q^{33} +6.38197 q^{35} -5.85410 q^{37} -0.381966 q^{39} -3.00000 q^{41} +2.70820 q^{43} +0.527864 q^{45} -11.9443 q^{47} +14.3262 q^{49} +2.85410 q^{51} -8.23607 q^{53} -1.38197 q^{55} +0.618034 q^{57} -6.09017 q^{59} -9.76393 q^{61} +1.76393 q^{63} -0.326238 q^{65} -10.3262 q^{67} -1.38197 q^{69} -9.85410 q^{71} -1.94427 q^{73} +5.00000 q^{75} -4.61803 q^{77} -1.29180 q^{79} -7.70820 q^{81} +15.5623 q^{83} +2.43769 q^{85} +11.7082 q^{87} -4.09017 q^{89} -1.09017 q^{91} +15.3262 q^{93} +0.527864 q^{95} +13.9443 q^{97} -0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 5 q^{5} - 7 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 5 * q^5 - 7 * q^7 - 3 * q^9	\( 2 q - q^{3} - 5 q^{5} - 7 q^{7} - 3 q^{9} + 2 q^{11} - 4 q^{13} - 8 q^{17} - 3 q^{19} + 6 q^{21} - 5 q^{23} + 5 q^{25} + 2 q^{27} - 10 q^{29} - 10 q^{31} - q^{33} + 15 q^{35} - 5 q^{37} - 3 q^{39} - 6 q^{41} - 8 q^{43} + 10 q^{45} - 6 q^{47} + 13 q^{49} - q^{51} - 12 q^{53} - 5 q^{55} - q^{57} - q^{59} - 24 q^{61} + 8 q^{63} + 15 q^{65} - 5 q^{67} - 5 q^{69} - 13 q^{71} + 14 q^{73} + 10 q^{75} - 7 q^{77} - 16 q^{79} - 2 q^{81} + 11 q^{83} + 25 q^{85} + 10 q^{87} + 3 q^{89} + 9 q^{91} + 15 q^{93} + 10 q^{95} + 10 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q - q^3 - 5 * q^5 - 7 * q^7 - 3 * q^9 + 2 * q^11 - 4 * q^13 - 8 * q^17 - 3 * q^19 + 6 * q^21 - 5 * q^23 + 5 * q^25 + 2 * q^27 - 10 * q^29 - 10 * q^31 - q^33 + 15 * q^35 - 5 * q^37 - 3 * q^39 - 6 * q^41 - 8 * q^43 + 10 * q^45 - 6 * q^47 + 13 * q^49 - q^51 - 12 * q^53 - 5 * q^55 - q^57 - q^59 - 24 * q^61 + 8 * q^63 + 15 * q^65 - 5 * q^67 - 5 * q^69 - 13 * q^71 + 14 * q^73 + 10 * q^75 - 7 * q^77 - 16 * q^79 - 2 * q^81 + 11 * q^83 + 25 * q^85 + 10 * q^87 + 3 * q^89 + 9 * q^91 + 15 * q^93 + 10 * q^95 + 10 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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