LMFDB - Embedded newform 6020.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - x^{12} - 28 x^{11} + 26 x^{10} + 286 x^{9} - 235 x^{8} - 1298 x^{7} + 895 x^{6} + 2571 x^{5} + \cdots - 216 $ x^13 - x^12 - 28x^11 + 26x^10 + 286x^9 - 235x^8 - 1298x^7 + 895x^6 + 2571x^5 - 1548x^4 - 1833x^3 + 1119x^2 + 342*x - 216
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.98613$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.98613 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.91695 q^{9} +O(q^{10})$	$q-2.98613 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.91695 q^{9} +3.75528 q^{11} -0.833420 q^{13} +2.98613 q^{15} -5.24848 q^{17} -4.89581 q^{19} +2.98613 q^{21} +1.36777 q^{23} +1.00000 q^{25} -8.71037 q^{27} +4.52944 q^{29} +6.15146 q^{31} -11.2137 q^{33} +1.00000 q^{35} +8.62496 q^{37} +2.48870 q^{39} -5.92710 q^{41} -1.00000 q^{43} -5.91695 q^{45} -0.779212 q^{47} +1.00000 q^{49} +15.6726 q^{51} -4.29816 q^{53} -3.75528 q^{55} +14.6195 q^{57} -0.499700 q^{59} -1.63793 q^{61} -5.91695 q^{63} +0.833420 q^{65} -10.9945 q^{67} -4.08434 q^{69} +8.19000 q^{71} +5.33119 q^{73} -2.98613 q^{75} -3.75528 q^{77} -13.6587 q^{79} +8.25941 q^{81} +1.33071 q^{83} +5.24848 q^{85} -13.5255 q^{87} -0.943603 q^{89} +0.833420 q^{91} -18.3690 q^{93} +4.89581 q^{95} +6.64826 q^{97} +22.2198 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * 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
$3$	−2.98613	−1.72404	−0.862020	−	0.506874i	$-0.830801\pi$
$3$	−2.98613	−1.72404	−0.862020	+	0.506874i	$0.830801\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.91695	1.97232
$10$	0	0
$11$	3.75528	1.13226	0.566130	−	0.824316i	$-0.308440\pi$
$11$	3.75528	1.13226	0.566130	+	0.824316i	$0.308440\pi$
$12$	0	0
$13$	−0.833420	−0.231149	−0.115575	−	0.993299i	$-0.536871\pi$
$13$	−0.833420	−0.231149	−0.115575	+	0.993299i	$0.536871\pi$
$14$	0	0
$15$	2.98613	0.771014
$16$	0	0
$17$	−5.24848	−1.27294	−0.636471	−	0.771300i	$-0.719607\pi$
$17$	−5.24848	−1.27294	−0.636471	+	0.771300i	$0.719607\pi$
$18$	0	0
$19$	−4.89581	−1.12318	−0.561588	−	0.827417i	$-0.689809\pi$
$19$	−4.89581	−1.12318	−0.561588	+	0.827417i	$0.689809\pi$
$20$	0	0
$21$	2.98613	0.651626
$22$	0	0
$23$	1.36777	0.285200	0.142600	−	0.989780i	$-0.454454\pi$
$23$	1.36777	0.285200	0.142600	+	0.989780i	$0.454454\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−8.71037	−1.67631
$28$	0	0
$29$	4.52944	0.841095	0.420548	−	0.907270i	$-0.361838\pi$
$29$	4.52944	0.841095	0.420548	+	0.907270i	$0.361838\pi$
$30$	0	0
$31$	6.15146	1.10484	0.552418	−	0.833568i	$-0.313705\pi$
$31$	6.15146	1.10484	0.552418	+	0.833568i	$0.313705\pi$
$32$	0	0
$33$	−11.2137	−1.95206
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	8.62496	1.41794	0.708968	−	0.705241i	$-0.249161\pi$
$37$	8.62496	1.41794	0.708968	+	0.705241i	$0.249161\pi$
$38$	0	0
$39$	2.48870	0.398510
$40$	0	0
$41$	−5.92710	−0.925657	−0.462829	−	0.886448i	$-0.653165\pi$
$41$	−5.92710	−0.925657	−0.462829	+	0.886448i	$0.653165\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−5.91695	−0.882046
$46$	0	0
$47$	−0.779212	−0.113660	−0.0568299	−	0.998384i	$-0.518099\pi$
$47$	−0.779212	−0.113660	−0.0568299	+	0.998384i	$0.518099\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	15.6726	2.19460
$52$	0	0
$53$	−4.29816	−0.590398	−0.295199	−	0.955436i	$-0.595386\pi$
$53$	−4.29816	−0.590398	−0.295199	+	0.955436i	$0.595386\pi$
$54$	0	0
$55$	−3.75528	−0.506362
$56$	0	0
$57$	14.6195	1.93640
$58$	0	0
$59$	−0.499700	−0.0650554	−0.0325277	−	0.999471i	$-0.510356\pi$
$59$	−0.499700	−0.0650554	−0.0325277	+	0.999471i	$0.510356\pi$
$60$	0	0
$61$	−1.63793	−0.209715	−0.104858	−	0.994487i	$-0.533439\pi$
$61$	−1.63793	−0.209715	−0.104858	+	0.994487i	$0.533439\pi$
$62$	0	0
$63$	−5.91695	−0.745465
$64$	0	0
$65$	0.833420	0.103373
$66$	0	0
$67$	−10.9945	−1.34320	−0.671599	−	0.740915i	$-0.734392\pi$
$67$	−10.9945	−1.34320	−0.671599	+	0.740915i	$0.734392\pi$
$68$	0	0
$69$	−4.08434	−0.491696
$70$	0	0
$71$	8.19000	0.971974	0.485987	−	0.873966i	$-0.338460\pi$
$71$	8.19000	0.971974	0.485987	+	0.873966i	$0.338460\pi$
$72$	0	0
$73$	5.33119	0.623969	0.311984	−	0.950087i	$-0.399006\pi$
$73$	5.33119	0.623969	0.311984	+	0.950087i	$0.399006\pi$
$74$	0	0
$75$	−2.98613	−0.344808
$76$	0	0
$77$	−3.75528	−0.427954
$78$	0	0
$79$	−13.6587	−1.53672	−0.768360	−	0.640018i	$-0.778927\pi$
$79$	−13.6587	−1.53672	−0.768360	+	0.640018i	$0.778927\pi$
$80$	0	0
$81$	8.25941	0.917712
$82$	0	0
$83$	1.33071	0.146065	0.0730324	−	0.997330i	$-0.476732\pi$
$83$	1.33071	0.146065	0.0730324	+	0.997330i	$0.476732\pi$
$84$	0	0
$85$	5.24848	0.569277
$86$	0	0
$87$	−13.5255	−1.45008
$88$	0	0
$89$	−0.943603	−0.100022	−0.0500109	−	0.998749i	$-0.515926\pi$
$89$	−0.943603	−0.100022	−0.0500109	+	0.998749i	$0.515926\pi$
$90$	0	0
$91$	0.833420	0.0873661
$92$	0	0
$93$	−18.3690	−1.90478
$94$	0	0
$95$	4.89581	0.502299
$96$	0	0
$97$	6.64826	0.675028	0.337514	−	0.941320i	$-0.390414\pi$
$97$	6.64826	0.675028	0.337514	+	0.941320i	$0.390414\pi$
$98$	0	0
$99$	22.2198	2.23317
$100$	0	0
$101$	−6.13106	−0.610063	−0.305032	−	0.952342i	$-0.598667\pi$
$101$	−6.13106	−0.610063	−0.305032	+	0.952342i	$0.598667\pi$
$102$	0	0
$103$	−19.4240	−1.91391	−0.956954	−	0.290239i	$-0.906265\pi$
$103$	−19.4240	−1.91391	−0.956954	+	0.290239i	$0.906265\pi$
$104$	0	0
$105$	−2.98613	−0.291416
$106$	0	0
$107$	−19.5620	−1.89113	−0.945563	−	0.325439i	$-0.894488\pi$
$107$	−19.5620	−1.89113	−0.945563	+	0.325439i	$0.894488\pi$
$108$	0	0
$109$	14.6155	1.39991	0.699954	−	0.714188i	$-0.253204\pi$
$109$	14.6155	1.39991	0.699954	+	0.714188i	$0.253204\pi$
$110$	0	0
$111$	−25.7552	−2.44458
$112$	0	0
$113$	3.58889	0.337614	0.168807	−	0.985649i	$-0.446009\pi$
$113$	3.58889	0.337614	0.168807	+	0.985649i	$0.446009\pi$
$114$	0	0
$115$	−1.36777	−0.127545
$116$	0	0
$117$	−4.93130	−0.455899
$118$	0	0
$119$	5.24848	0.481127
$120$	0	0
$121$	3.10213	0.282011
$122$	0	0
$123$	17.6991	1.59587
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.7629	−1.84241	−0.921204	−	0.389081i	$-0.872793\pi$
$127$	−20.7629	−1.84241	−0.921204	+	0.389081i	$0.872793\pi$
$128$	0	0
$129$	2.98613	0.262914
$130$	0	0
$131$	−10.9347	−0.955371	−0.477686	−	0.878531i	$-0.658524\pi$
$131$	−10.9347	−0.955371	−0.477686	+	0.878531i	$0.658524\pi$
$132$	0	0
$133$	4.89581	0.424520
$134$	0	0
$135$	8.71037	0.749669
$136$	0	0
$137$	−14.7907	−1.26365	−0.631826	−	0.775110i	$-0.717694\pi$
$137$	−14.7907	−1.26365	−0.631826	+	0.775110i	$0.717694\pi$
$138$	0	0
$139$	1.82325	0.154646	0.0773231	−	0.997006i	$-0.475363\pi$
$139$	1.82325	0.154646	0.0773231	+	0.997006i	$0.475363\pi$
$140$	0	0
$141$	2.32682	0.195954
$142$	0	0
$143$	−3.12972	−0.261721
$144$	0	0
$145$	−4.52944	−0.376149
$146$	0	0
$147$	−2.98613	−0.246291
$148$	0	0
$149$	17.0722	1.39861	0.699306	−	0.714822i	$-0.253492\pi$
$149$	17.0722	1.39861	0.699306	+	0.714822i	$0.253492\pi$
$150$	0	0
$151$	9.58076	0.779671	0.389835	−	0.920885i	$-0.372532\pi$
$151$	9.58076	0.779671	0.389835	+	0.920885i	$0.372532\pi$
$152$	0	0
$153$	−31.0549	−2.51064
$154$	0	0
$155$	−6.15146	−0.494097
$156$	0	0
$157$	11.9015	0.949841	0.474920	−	0.880029i	$-0.342477\pi$
$157$	11.9015	0.949841	0.474920	+	0.880029i	$0.342477\pi$
$158$	0	0
$159$	12.8348	1.01787
$160$	0	0
$161$	−1.36777	−0.107795
$162$	0	0
$163$	−7.32780	−0.573958	−0.286979	−	0.957937i	$-0.592651\pi$
$163$	−7.32780	−0.573958	−0.286979	+	0.957937i	$0.592651\pi$
$164$	0	0
$165$	11.2137	0.872988
$166$	0	0
$167$	−3.90495	−0.302174	−0.151087	−	0.988520i	$-0.548277\pi$
$167$	−3.90495	−0.302174	−0.151087	+	0.988520i	$0.548277\pi$
$168$	0	0
$169$	−12.3054	−0.946570
$170$	0	0
$171$	−28.9682	−2.21526
$172$	0	0
$173$	22.5417	1.71382	0.856908	−	0.515470i	$-0.172383\pi$
$173$	22.5417	1.71382	0.856908	+	0.515470i	$0.172383\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	1.49217	0.112158
$178$	0	0
$179$	2.79073	0.208589	0.104295	−	0.994546i	$-0.466742\pi$
$179$	2.79073	0.208589	0.104295	+	0.994546i	$0.466742\pi$
$180$	0	0
$181$	24.5230	1.82278	0.911390	−	0.411544i	$-0.135010\pi$
$181$	24.5230	1.82278	0.911390	+	0.411544i	$0.135010\pi$
$182$	0	0
$183$	4.89106	0.361558
$184$	0	0
$185$	−8.62496	−0.634120
$186$	0	0
$187$	−19.7095	−1.44130
$188$	0	0
$189$	8.71037	0.633586
$190$	0	0
$191$	12.3079	0.890569	0.445284	−	0.895389i	$-0.353103\pi$
$191$	12.3079	0.890569	0.445284	+	0.895389i	$0.353103\pi$
$192$	0	0
$193$	−1.79341	−0.129092	−0.0645461	−	0.997915i	$-0.520560\pi$
$193$	−1.79341	−0.129092	−0.0645461	+	0.997915i	$0.520560\pi$
$194$	0	0
$195$	−2.48870	−0.178219
$196$	0	0
$197$	23.4638	1.67173	0.835864	−	0.548937i	$-0.184967\pi$
$197$	23.4638	1.67173	0.835864	+	0.548937i	$0.184967\pi$
$198$	0	0
$199$	−17.1447	−1.21536	−0.607679	−	0.794183i	$-0.707899\pi$
$199$	−17.1447	−1.21536	−0.607679	+	0.794183i	$0.707899\pi$
$200$	0	0
$201$	32.8311	2.31573
$202$	0	0
$203$	−4.52944	−0.317904
$204$	0	0
$205$	5.92710	0.413966
$206$	0	0
$207$	8.09303	0.562504
$208$	0	0
$209$	−18.3851	−1.27173
$210$	0	0
$211$	21.5604	1.48428	0.742140	−	0.670245i	$-0.233811\pi$
$211$	21.5604	1.48428	0.742140	+	0.670245i	$0.233811\pi$
$212$	0	0
$213$	−24.4564	−1.67572
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−6.15146	−0.417588
$218$	0	0
$219$	−15.9196	−1.07575
$220$	0	0
$221$	4.37418	0.294239
$222$	0	0
$223$	5.34358	0.357832	0.178916	−	0.983864i	$-0.442741\pi$
$223$	5.34358	0.357832	0.178916	+	0.983864i	$0.442741\pi$
$224$	0	0
$225$	5.91695	0.394463
$226$	0	0
$227$	4.82289	0.320106	0.160053	−	0.987108i	$-0.448833\pi$
$227$	4.82289	0.320106	0.160053	+	0.987108i	$0.448833\pi$
$228$	0	0
$229$	−0.419170	−0.0276995	−0.0138498	−	0.999904i	$-0.504409\pi$
$229$	−0.419170	−0.0276995	−0.0138498	+	0.999904i	$0.504409\pi$
$230$	0	0
$231$	11.2137	0.737810
$232$	0	0
$233$	−10.9908	−0.720031	−0.360016	−	0.932946i	$-0.617229\pi$
$233$	−10.9908	−0.720031	−0.360016	+	0.932946i	$0.617229\pi$
$234$	0	0
$235$	0.779212	0.0508302
$236$	0	0
$237$	40.7865	2.64937
$238$	0	0
$239$	−4.41229	−0.285408	−0.142704	−	0.989765i	$-0.545580\pi$
$239$	−4.41229	−0.285408	−0.142704	+	0.989765i	$0.545580\pi$
$240$	0	0
$241$	14.9804	0.964973	0.482486	−	0.875904i	$-0.339734\pi$
$241$	14.9804	0.964973	0.482486	+	0.875904i	$0.339734\pi$
$242$	0	0
$243$	1.46746	0.0941376
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.08026	0.259621
$248$	0	0
$249$	−3.97368	−0.251822
$250$	0	0
$251$	15.9228	1.00504	0.502518	−	0.864566i	$-0.332407\pi$
$251$	15.9228	1.00504	0.502518	+	0.864566i	$0.332407\pi$
$252$	0	0
$253$	5.13636	0.322920
$254$	0	0
$255$	−15.6726	−0.981457
$256$	0	0
$257$	19.3829	1.20907	0.604537	−	0.796577i	$-0.293358\pi$
$257$	19.3829	1.20907	0.604537	+	0.796577i	$0.293358\pi$
$258$	0	0
$259$	−8.62496	−0.535929
$260$	0	0
$261$	26.8004	1.65891
$262$	0	0
$263$	32.1342	1.98148	0.990741	−	0.135767i	$-0.0433500\pi$
$263$	32.1342	1.98148	0.990741	+	0.135767i	$0.0433500\pi$
$264$	0	0
$265$	4.29816	0.264034
$266$	0	0
$267$	2.81772	0.172441
$268$	0	0
$269$	−9.65857	−0.588893	−0.294447	−	0.955668i	$-0.595135\pi$
$269$	−9.65857	−0.588893	−0.294447	+	0.955668i	$0.595135\pi$
$270$	0	0
$271$	27.1444	1.64891	0.824454	−	0.565929i	$-0.191482\pi$
$271$	27.1444	1.64891	0.824454	+	0.565929i	$0.191482\pi$
$272$	0	0
$273$	−2.48870	−0.150623
$274$	0	0
$275$	3.75528	0.226452
$276$	0	0
$277$	−23.4333	−1.40797	−0.703984	−	0.710215i	$-0.748598\pi$
$277$	−23.4333	−1.40797	−0.703984	+	0.710215i	$0.748598\pi$
$278$	0	0
$279$	36.3979	2.17908
$280$	0	0
$281$	−23.9861	−1.43089	−0.715444	−	0.698670i	$-0.753776\pi$
$281$	−23.9861	−1.43089	−0.715444	+	0.698670i	$0.753776\pi$
$282$	0	0
$283$	10.6880	0.635338	0.317669	−	0.948202i	$-0.397100\pi$
$283$	10.6880	0.635338	0.317669	+	0.948202i	$0.397100\pi$
$284$	0	0
$285$	−14.6195	−0.865984
$286$	0	0
$287$	5.92710	0.349865
$288$	0	0
$289$	10.5465	0.620382
$290$	0	0
$291$	−19.8525	−1.16378
$292$	0	0
$293$	6.95513	0.406323	0.203161	−	0.979145i	$-0.434878\pi$
$293$	6.95513	0.406323	0.203161	+	0.979145i	$0.434878\pi$
$294$	0	0
$295$	0.499700	0.0290937
$296$	0	0
$297$	−32.7099	−1.89802
$298$	0	0
$299$	−1.13993	−0.0659237
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	18.3081	1.05177
$304$	0	0
$305$	1.63793	0.0937875
$306$	0	0
$307$	5.83986	0.333299	0.166649	−	0.986016i	$-0.446705\pi$
$307$	5.83986	0.333299	0.166649	+	0.986016i	$0.446705\pi$
$308$	0	0
$309$	58.0026	3.29966
$310$	0	0
$311$	3.47883	0.197266	0.0986332	−	0.995124i	$-0.468553\pi$
$311$	3.47883	0.197266	0.0986332	+	0.995124i	$0.468553\pi$
$312$	0	0
$313$	14.9319	0.843999	0.421999	−	0.906596i	$-0.361328\pi$
$313$	14.9319	0.843999	0.421999	+	0.906596i	$0.361328\pi$
$314$	0	0
$315$	5.91695	0.333382
$316$	0	0
$317$	4.37644	0.245805	0.122903	−	0.992419i	$-0.460780\pi$
$317$	4.37644	0.245805	0.122903	+	0.992419i	$0.460780\pi$
$318$	0	0
$319$	17.0093	0.952338
$320$	0	0
$321$	58.4145	3.26038
$322$	0	0
$323$	25.6955	1.42974
$324$	0	0
$325$	−0.833420	−0.0462298
$326$	0	0
$327$	−43.6437	−2.41350
$328$	0	0
$329$	0.779212	0.0429593
$330$	0	0
$331$	−11.3533	−0.624033	−0.312016	−	0.950077i	$-0.601004\pi$
$331$	−11.3533	−0.624033	−0.312016	+	0.950077i	$0.601004\pi$
$332$	0	0
$333$	51.0334	2.79662
$334$	0	0
$335$	10.9945	0.600696
$336$	0	0
$337$	30.8090	1.67828	0.839138	−	0.543919i	$-0.183060\pi$
$337$	30.8090	1.67828	0.839138	+	0.543919i	$0.183060\pi$
$338$	0	0
$339$	−10.7169	−0.582060
$340$	0	0
$341$	23.1005	1.25096
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.08434	0.219893
$346$	0	0
$347$	15.4664	0.830280	0.415140	−	0.909758i	$-0.363733\pi$
$347$	15.4664	0.830280	0.415140	+	0.909758i	$0.363733\pi$
$348$	0	0
$349$	19.7365	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$349$	19.7365	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$350$	0	0
$351$	7.25939	0.387478
$352$	0	0
$353$	16.0996	0.856897	0.428448	−	0.903566i	$-0.359060\pi$
$353$	16.0996	0.856897	0.428448	+	0.903566i	$0.359060\pi$
$354$	0	0
$355$	−8.19000	−0.434680
$356$	0	0
$357$	−15.6726	−0.829482
$358$	0	0
$359$	−3.83199	−0.202245	−0.101122	−	0.994874i	$-0.532243\pi$
$359$	−3.83199	−0.202245	−0.101122	+	0.994874i	$0.532243\pi$
$360$	0	0
$361$	4.96895	0.261523
$362$	0	0
$363$	−9.26334	−0.486199
$364$	0	0
$365$	−5.33119	−0.279047
$366$	0	0
$367$	2.35536	0.122949	0.0614743	−	0.998109i	$-0.480420\pi$
$367$	2.35536	0.122949	0.0614743	+	0.998109i	$0.480420\pi$
$368$	0	0
$369$	−35.0703	−1.82569
$370$	0	0
$371$	4.29816	0.223149
$372$	0	0
$373$	−30.9920	−1.60470	−0.802352	−	0.596852i	$-0.796418\pi$
$373$	−30.9920	−1.60470	−0.802352	+	0.596852i	$0.796418\pi$
$374$	0	0
$375$	2.98613	0.154203
$376$	0	0
$377$	−3.77492	−0.194418
$378$	0	0
$379$	−18.2993	−0.939970	−0.469985	−	0.882674i	$-0.655741\pi$
$379$	−18.2993	−0.939970	−0.469985	+	0.882674i	$0.655741\pi$
$380$	0	0
$381$	62.0006	3.17638
$382$	0	0
$383$	2.69477	0.137696	0.0688482	−	0.997627i	$-0.478068\pi$
$383$	2.69477	0.137696	0.0688482	+	0.997627i	$0.478068\pi$
$384$	0	0
$385$	3.75528	0.191387
$386$	0	0
$387$	−5.91695	−0.300775
$388$	0	0
$389$	−15.2747	−0.774459	−0.387229	−	0.921983i	$-0.626568\pi$
$389$	−15.2747	−0.774459	−0.387229	+	0.921983i	$0.626568\pi$
$390$	0	0
$391$	−7.17871	−0.363043
$392$	0	0
$393$	32.6525	1.64710
$394$	0	0
$395$	13.6587	0.687242
$396$	0	0
$397$	−3.00151	−0.150641	−0.0753206	−	0.997159i	$-0.523998\pi$
$397$	−3.00151	−0.150641	−0.0753206	+	0.997159i	$0.523998\pi$
$398$	0	0
$399$	−14.6195	−0.731890
$400$	0	0
$401$	3.30789	0.165188	0.0825941	−	0.996583i	$-0.473679\pi$
$401$	3.30789	0.165188	0.0825941	+	0.996583i	$0.473679\pi$
$402$	0	0
$403$	−5.12675	−0.255382
$404$	0	0
$405$	−8.25941	−0.410413
$406$	0	0
$407$	32.3892	1.60547
$408$	0	0
$409$	−3.92532	−0.194095	−0.0970474	−	0.995280i	$-0.530940\pi$
$409$	−3.92532	−0.194095	−0.0970474	+	0.995280i	$0.530940\pi$
$410$	0	0
$411$	44.1668	2.17859
$412$	0	0
$413$	0.499700	0.0245886
$414$	0	0
$415$	−1.33071	−0.0653222
$416$	0	0
$417$	−5.44446	−0.266616
$418$	0	0
$419$	−16.8517	−0.823259	−0.411630	−	0.911351i	$-0.635040\pi$
$419$	−16.8517	−0.823259	−0.411630	+	0.911351i	$0.635040\pi$
$420$	0	0
$421$	21.4343	1.04465	0.522323	−	0.852748i	$-0.325066\pi$
$421$	21.4343	1.04465	0.522323	+	0.852748i	$0.325066\pi$
$422$	0	0
$423$	−4.61055	−0.224173
$424$	0	0
$425$	−5.24848	−0.254588
$426$	0	0
$427$	1.63793	0.0792649
$428$	0	0
$429$	9.34575	0.451217
$430$	0	0
$431$	−15.8838	−0.765094	−0.382547	−	0.923936i	$-0.624953\pi$
$431$	−15.8838	−0.765094	−0.382547	+	0.923936i	$0.624953\pi$
$432$	0	0
$433$	22.9946	1.10505	0.552525	−	0.833496i	$-0.313664\pi$
$433$	22.9946	1.10505	0.552525	+	0.833496i	$0.313664\pi$
$434$	0	0
$435$	13.5255	0.648497
$436$	0	0
$437$	−6.69634	−0.320330
$438$	0	0
$439$	−21.8536	−1.04301	−0.521507	−	0.853247i	$-0.674630\pi$
$439$	−21.8536	−1.04301	−0.521507	+	0.853247i	$0.674630\pi$
$440$	0	0
$441$	5.91695	0.281759
$442$	0	0
$443$	−17.6894	−0.840447	−0.420223	−	0.907421i	$-0.638048\pi$
$443$	−17.6894	−0.840447	−0.420223	+	0.907421i	$0.638048\pi$
$444$	0	0
$445$	0.943603	0.0447311
$446$	0	0
$447$	−50.9798	−2.41126
$448$	0	0
$449$	24.4452	1.15364	0.576819	−	0.816872i	$-0.304294\pi$
$449$	24.4452	1.15364	0.576819	+	0.816872i	$0.304294\pi$
$450$	0	0
$451$	−22.2579	−1.04808
$452$	0	0
$453$	−28.6093	−1.34418
$454$	0	0
$455$	−0.833420	−0.0390713
$456$	0	0
$457$	−23.9783	−1.12166	−0.560828	−	0.827932i	$-0.689517\pi$
$457$	−23.9783	−1.12166	−0.560828	+	0.827932i	$0.689517\pi$
$458$	0	0
$459$	45.7162	2.13385
$460$	0	0
$461$	19.8371	0.923907	0.461953	−	0.886904i	$-0.347149\pi$
$461$	19.8371	0.923907	0.461953	+	0.886904i	$0.347149\pi$
$462$	0	0
$463$	11.1247	0.517011	0.258505	−	0.966010i	$-0.416770\pi$
$463$	11.1247	0.517011	0.258505	+	0.966010i	$0.416770\pi$
$464$	0	0
$465$	18.3690	0.851844
$466$	0	0
$467$	−12.9508	−0.599292	−0.299646	−	0.954050i	$-0.596869\pi$
$467$	−12.9508	−0.599292	−0.299646	+	0.954050i	$0.596869\pi$
$468$	0	0
$469$	10.9945	0.507681
$470$	0	0
$471$	−35.5393	−1.63756
$472$	0	0
$473$	−3.75528	−0.172668
$474$	0	0
$475$	−4.89581	−0.224635
$476$	0	0
$477$	−25.4320	−1.16445
$478$	0	0
$479$	17.9975	0.822326	0.411163	−	0.911562i	$-0.365123\pi$
$479$	17.9975	0.822326	0.411163	+	0.911562i	$0.365123\pi$
$480$	0	0
$481$	−7.18821	−0.327754
$482$	0	0
$483$	4.08434	0.185844
$484$	0	0
$485$	−6.64826	−0.301882
$486$	0	0
$487$	16.7340	0.758292	0.379146	−	0.925337i	$-0.376218\pi$
$487$	16.7340	0.758292	0.379146	+	0.925337i	$0.376218\pi$
$488$	0	0
$489$	21.8817	0.989527
$490$	0	0
$491$	21.5057	0.970540	0.485270	−	0.874364i	$-0.338721\pi$
$491$	21.5057	0.970540	0.485270	+	0.874364i	$0.338721\pi$
$492$	0	0
$493$	−23.7726	−1.07067
$494$	0	0
$495$	−22.2198	−0.998705
$496$	0	0
$497$	−8.19000	−0.367372
$498$	0	0
$499$	4.54449	0.203440	0.101720	−	0.994813i	$-0.467565\pi$
$499$	4.54449	0.203440	0.101720	+	0.994813i	$0.467565\pi$
$500$	0	0
$501$	11.6607	0.520961
$502$	0	0
$503$	−24.3022	−1.08358	−0.541792	−	0.840513i	$-0.682254\pi$
$503$	−24.3022	−1.08358	−0.541792	+	0.840513i	$0.682254\pi$
$504$	0	0
$505$	6.13106	0.272829
$506$	0	0
$507$	36.7455	1.63193
$508$	0	0
$509$	0.00619337	0.000274516 0	0.000137258	−	1.00000i	$-0.499956\pi$
$509$	0.00619337	0.000274516 0	0.000137258		1.00000i	$0.499956\pi$
$510$	0	0
$511$	−5.33119	−0.235838
$512$	0	0
$513$	42.6443	1.88279
$514$	0	0
$515$	19.4240	0.855926
$516$	0	0
$517$	−2.92616	−0.128692
$518$	0	0
$519$	−67.3124	−2.95469
$520$	0	0
$521$	8.88374	0.389204	0.194602	−	0.980882i	$-0.437659\pi$
$521$	8.88374	0.389204	0.194602	+	0.980882i	$0.437659\pi$
$522$	0	0
$523$	42.1733	1.84411	0.922055	−	0.387059i	$-0.126509\pi$
$523$	42.1733	1.84411	0.922055	+	0.387059i	$0.126509\pi$
$524$	0	0
$525$	2.98613	0.130325
$526$	0	0
$527$	−32.2858	−1.40639
$528$	0	0
$529$	−21.1292	−0.918661
$530$	0	0
$531$	−2.95670	−0.128310
$532$	0	0
$533$	4.93976	0.213965
$534$	0	0
$535$	19.5620	0.845737
$536$	0	0
$537$	−8.33348	−0.359616
$538$	0	0
$539$	3.75528	0.161751
$540$	0	0
$541$	29.0953	1.25090	0.625451	−	0.780263i	$-0.284915\pi$
$541$	29.0953	1.25090	0.625451	+	0.780263i	$0.284915\pi$
$542$	0	0
$543$	−73.2287	−3.14255
$544$	0	0
$545$	−14.6155	−0.626058
$546$	0	0
$547$	6.15989	0.263378	0.131689	−	0.991291i	$-0.457960\pi$
$547$	6.15989	0.263378	0.131689	+	0.991291i	$0.457960\pi$
$548$	0	0
$549$	−9.69154	−0.413625
$550$	0	0
$551$	−22.1753	−0.944698
$552$	0	0
$553$	13.6587	0.580825
$554$	0	0
$555$	25.7552	1.09325
$556$	0	0
$557$	10.2081	0.432529	0.216265	−	0.976335i	$-0.430613\pi$
$557$	10.2081	0.432529	0.216265	+	0.976335i	$0.430613\pi$
$558$	0	0
$559$	0.833420	0.0352499
$560$	0	0
$561$	58.8550	2.48486
$562$	0	0
$563$	−33.4948	−1.41164	−0.705818	−	0.708393i	$-0.749420\pi$
$563$	−33.4948	−1.41164	−0.705818	+	0.708393i	$0.749420\pi$
$564$	0	0
$565$	−3.58889	−0.150986
$566$	0	0
$567$	−8.25941	−0.346863
$568$	0	0
$569$	5.36595	0.224952	0.112476	−	0.993654i	$-0.464122\pi$
$569$	5.36595	0.224952	0.112476	+	0.993654i	$0.464122\pi$
$570$	0	0
$571$	24.3523	1.01911	0.509556	−	0.860437i	$-0.329810\pi$
$571$	24.3523	1.01911	0.509556	+	0.860437i	$0.329810\pi$
$572$	0	0
$573$	−36.7529	−1.53538
$574$	0	0
$575$	1.36777	0.0570400
$576$	0	0
$577$	13.0334	0.542586	0.271293	−	0.962497i	$-0.412549\pi$
$577$	13.0334	0.542586	0.271293	+	0.962497i	$0.412549\pi$
$578$	0	0
$579$	5.35533	0.222560
$580$	0	0
$581$	−1.33071	−0.0552073
$582$	0	0
$583$	−16.1408	−0.668484
$584$	0	0
$585$	4.93130	0.203884
$586$	0	0
$587$	17.9143	0.739402	0.369701	−	0.929151i	$-0.379460\pi$
$587$	17.9143	0.739402	0.369701	+	0.929151i	$0.379460\pi$
$588$	0	0
$589$	−30.1164	−1.24092
$590$	0	0
$591$	−70.0659	−2.88213
$592$	0	0
$593$	17.1177	0.702941	0.351471	−	0.936199i	$-0.385682\pi$
$593$	17.1177	0.702941	0.351471	+	0.936199i	$0.385682\pi$
$594$	0	0
$595$	−5.24848	−0.215167
$596$	0	0
$597$	51.1963	2.09533
$598$	0	0
$599$	−35.3847	−1.44578	−0.722891	−	0.690963i	$-0.757187\pi$
$599$	−35.3847	−1.44578	−0.722891	+	0.690963i	$0.757187\pi$
$600$	0	0
$601$	25.1458	1.02572	0.512859	−	0.858473i	$-0.328586\pi$
$601$	25.1458	1.02572	0.512859	+	0.858473i	$0.328586\pi$
$602$	0	0
$603$	−65.0541	−2.64921
$604$	0	0
$605$	−3.10213	−0.126119
$606$	0	0
$607$	8.80524	0.357394	0.178697	−	0.983904i	$-0.442812\pi$
$607$	8.80524	0.357394	0.178697	+	0.983904i	$0.442812\pi$
$608$	0	0
$609$	13.5255	0.548080
$610$	0	0
$611$	0.649410	0.0262723
$612$	0	0
$613$	−4.08466	−0.164978	−0.0824890	−	0.996592i	$-0.526287\pi$
$613$	−4.08466	−0.164978	−0.0824890	+	0.996592i	$0.526287\pi$
$614$	0	0
$615$	−17.6991	−0.713695
$616$	0	0
$617$	0.378584	0.0152412	0.00762061	−	0.999971i	$-0.497574\pi$
$617$	0.378584	0.0152412	0.00762061	+	0.999971i	$0.497574\pi$
$618$	0	0
$619$	−25.8845	−1.04039	−0.520194	−	0.854048i	$-0.674140\pi$
$619$	−25.8845	−1.04039	−0.520194	+	0.854048i	$0.674140\pi$
$620$	0	0
$621$	−11.9138	−0.478084
$622$	0	0
$623$	0.943603	0.0378046
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	54.9003	2.19251
$628$	0	0
$629$	−45.2679	−1.80495
$630$	0	0
$631$	22.5373	0.897195	0.448598	−	0.893734i	$-0.351924\pi$
$631$	22.5373	0.897195	0.448598	+	0.893734i	$0.351924\pi$
$632$	0	0
$633$	−64.3821	−2.55896
$634$	0	0
$635$	20.7629	0.823950
$636$	0	0
$637$	−0.833420	−0.0330213
$638$	0	0
$639$	48.4598	1.91704
$640$	0	0
$641$	−19.0072	−0.750741	−0.375370	−	0.926875i	$-0.622484\pi$
$641$	−19.0072	−0.750741	−0.375370	+	0.926875i	$0.622484\pi$
$642$	0	0
$643$	31.5390	1.24378	0.621889	−	0.783105i	$-0.286365\pi$
$643$	31.5390	1.24378	0.621889	+	0.783105i	$0.286365\pi$
$644$	0	0
$645$	−2.98613	−0.117579
$646$	0	0
$647$	−44.4709	−1.74833	−0.874166	−	0.485627i	$-0.838591\pi$
$647$	−44.4709	−1.74833	−0.874166	+	0.485627i	$0.838591\pi$
$648$	0	0
$649$	−1.87651	−0.0736596
$650$	0	0
$651$	18.3690	0.719939
$652$	0	0
$653$	1.77076	0.0692952	0.0346476	−	0.999400i	$-0.488969\pi$
$653$	1.77076	0.0692952	0.0346476	+	0.999400i	$0.488969\pi$
$654$	0	0
$655$	10.9347	0.427255
$656$	0	0
$657$	31.5444	1.23066
$658$	0	0
$659$	−24.5990	−0.958240	−0.479120	−	0.877749i	$-0.659044\pi$
$659$	−24.5990	−0.958240	−0.479120	+	0.877749i	$0.659044\pi$
$660$	0	0
$661$	−32.4940	−1.26387	−0.631935	−	0.775021i	$-0.717739\pi$
$661$	−32.4940	−1.26387	−0.631935	+	0.775021i	$0.717739\pi$
$662$	0	0
$663$	−13.0619	−0.507281
$664$	0	0
$665$	−4.89581	−0.189851
$666$	0	0
$667$	6.19523	0.239880
$668$	0	0
$669$	−15.9566	−0.616918
$670$	0	0
$671$	−6.15088	−0.237452
$672$	0	0
$673$	−13.9375	−0.537250	−0.268625	−	0.963245i	$-0.586569\pi$
$673$	−13.9375	−0.537250	−0.268625	+	0.963245i	$0.586569\pi$
$674$	0	0
$675$	−8.71037	−0.335262
$676$	0	0
$677$	−32.9065	−1.26470	−0.632349	−	0.774684i	$-0.717909\pi$
$677$	−32.9065	−1.26470	−0.632349	+	0.774684i	$0.717909\pi$
$678$	0	0
$679$	−6.64826	−0.255137
$680$	0	0
$681$	−14.4018	−0.551877
$682$	0	0
$683$	−5.26672	−0.201526	−0.100763	−	0.994910i	$-0.532128\pi$
$683$	−5.26672	−0.201526	−0.100763	+	0.994910i	$0.532128\pi$
$684$	0	0
$685$	14.7907	0.565123
$686$	0	0
$687$	1.25169	0.0477551
$688$	0	0
$689$	3.58217	0.136470
$690$	0	0
$691$	47.6915	1.81427	0.907135	−	0.420839i	$-0.138264\pi$
$691$	47.6915	1.81427	0.907135	+	0.420839i	$0.138264\pi$
$692$	0	0
$693$	−22.2198	−0.844060
$694$	0	0
$695$	−1.82325	−0.0691599
$696$	0	0
$697$	31.1082	1.17831
$698$	0	0
$699$	32.8199	1.24136
$700$	0	0
$701$	42.6505	1.61089	0.805443	−	0.592674i	$-0.201928\pi$
$701$	42.6505	1.61089	0.805443	+	0.592674i	$0.201928\pi$
$702$	0	0
$703$	−42.2262	−1.59259
$704$	0	0
$705$	−2.32682	−0.0876333
$706$	0	0
$707$	6.13106	0.230582
$708$	0	0
$709$	34.7652	1.30564	0.652818	−	0.757515i	$-0.273587\pi$
$709$	34.7652	1.30564	0.652818	+	0.757515i	$0.273587\pi$
$710$	0	0
$711$	−80.8176	−3.03090
$712$	0	0
$713$	8.41379	0.315099
$714$	0	0
$715$	3.12972	0.117045
$716$	0	0
$717$	13.1757	0.492054
$718$	0	0
$719$	−22.9103	−0.854408	−0.427204	−	0.904155i	$-0.640501\pi$
$719$	−22.9103	−0.854408	−0.427204	+	0.904155i	$0.640501\pi$
$720$	0	0
$721$	19.4240	0.723389
$722$	0	0
$723$	−44.7334	−1.66365
$724$	0	0
$725$	4.52944	0.168219
$726$	0	0
$727$	7.74321	0.287180	0.143590	−	0.989637i	$-0.454135\pi$
$727$	7.74321	0.287180	0.143590	+	0.989637i	$0.454135\pi$
$728$	0	0
$729$	−29.1603	−1.08001
$730$	0	0
$731$	5.24848	0.194122
$732$	0	0
$733$	25.2655	0.933203	0.466602	−	0.884468i	$-0.345478\pi$
$733$	25.2655	0.933203	0.466602	+	0.884468i	$0.345478\pi$
$734$	0	0
$735$	2.98613	0.110145
$736$	0	0
$737$	−41.2876	−1.52085
$738$	0	0
$739$	40.3297	1.48355	0.741775	−	0.670648i	$-0.233984\pi$
$739$	40.3297	1.48355	0.741775	+	0.670648i	$0.233984\pi$
$740$	0	0
$741$	−12.1842	−0.447597
$742$	0	0
$743$	47.7778	1.75280	0.876400	−	0.481585i	$-0.159939\pi$
$743$	47.7778	1.75280	0.876400	+	0.481585i	$0.159939\pi$
$744$	0	0
$745$	−17.0722	−0.625478
$746$	0	0
$747$	7.87376	0.288086
$748$	0	0
$749$	19.5620	0.714778
$750$	0	0
$751$	−21.0181	−0.766961	−0.383480	−	0.923549i	$-0.625275\pi$
$751$	−21.0181	−0.766961	−0.383480	+	0.923549i	$0.625275\pi$
$752$	0	0
$753$	−47.5474	−1.73272
$754$	0	0
$755$	−9.58076	−0.348679
$756$	0	0
$757$	14.4118	0.523807	0.261904	−	0.965094i	$-0.415650\pi$
$757$	14.4118	0.523807	0.261904	+	0.965094i	$0.415650\pi$
$758$	0	0
$759$	−15.3378	−0.556728
$760$	0	0
$761$	−34.0794	−1.23538	−0.617689	−	0.786423i	$-0.711931\pi$
$761$	−34.0794	−1.23538	−0.617689	+	0.786423i	$0.711931\pi$
$762$	0	0
$763$	−14.6155	−0.529116
$764$	0	0
$765$	31.0549	1.12279
$766$	0	0
$767$	0.416460	0.0150375
$768$	0	0
$769$	40.4484	1.45861	0.729303	−	0.684190i	$-0.239844\pi$
$769$	40.4484	1.45861	0.729303	+	0.684190i	$0.239844\pi$
$770$	0	0
$771$	−57.8799	−2.08449
$772$	0	0
$773$	18.6518	0.670859	0.335430	−	0.942065i	$-0.391119\pi$
$773$	18.6518	0.670859	0.335430	+	0.942065i	$0.391119\pi$
$774$	0	0
$775$	6.15146	0.220967
$776$	0	0
$777$	25.7552	0.923964
$778$	0	0
$779$	29.0179	1.03968
$780$	0	0
$781$	30.7558	1.10053
$782$	0	0
$783$	−39.4531	−1.40994
$784$	0	0
$785$	−11.9015	−0.424782
$786$	0	0
$787$	46.1657	1.64563	0.822815	−	0.568309i	$-0.192402\pi$
$787$	46.1657	1.64563	0.822815	+	0.568309i	$0.192402\pi$
$788$	0	0
$789$	−95.9568	−3.41615
$790$	0	0
$791$	−3.58889	−0.127606
$792$	0	0
$793$	1.36508	0.0484755
$794$	0	0
$795$	−12.8348	−0.455205
$796$	0	0
$797$	30.7955	1.09083	0.545416	−	0.838165i	$-0.316372\pi$
$797$	30.7955	1.09083	0.545416	+	0.838165i	$0.316372\pi$
$798$	0	0
$799$	4.08967	0.144682
$800$	0	0
$801$	−5.58325	−0.197274
$802$	0	0
$803$	20.0201	0.706494
$804$	0	0
$805$	1.36777	0.0482076
$806$	0	0
$807$	28.8417	1.01528
$808$	0	0
$809$	−14.7445	−0.518389	−0.259194	−	0.965825i	$-0.583457\pi$
$809$	−14.7445	−0.518389	−0.259194	+	0.965825i	$0.583457\pi$
$810$	0	0
$811$	46.6139	1.63684	0.818419	−	0.574623i	$-0.194851\pi$
$811$	46.6139	1.63684	0.818419	+	0.574623i	$0.194851\pi$
$812$	0	0
$813$	−81.0567	−2.84278
$814$	0	0
$815$	7.32780	0.256682
$816$	0	0
$817$	4.89581	0.171283
$818$	0	0
$819$	4.93130	0.172314
$820$	0	0
$821$	−2.00386	−0.0699353	−0.0349676	−	0.999388i	$-0.511133\pi$
$821$	−2.00386	−0.0699353	−0.0349676	+	0.999388i	$0.511133\pi$
$822$	0	0
$823$	3.09839	0.108003	0.0540016	−	0.998541i	$-0.482802\pi$
$823$	3.09839	0.108003	0.0540016	+	0.998541i	$0.482802\pi$
$824$	0	0
$825$	−11.2137	−0.390412
$826$	0	0
$827$	20.0398	0.696852	0.348426	−	0.937336i	$-0.386716\pi$
$827$	20.0398	0.696852	0.348426	+	0.937336i	$0.386716\pi$
$828$	0	0
$829$	14.4055	0.500325	0.250162	−	0.968204i	$-0.419516\pi$
$829$	14.4055	0.500325	0.250162	+	0.968204i	$0.419516\pi$
$830$	0	0
$831$	69.9747	2.42740
$832$	0	0
$833$	−5.24848	−0.181849
$834$	0	0
$835$	3.90495	0.135137
$836$	0	0
$837$	−53.5815	−1.85205
$838$	0	0
$839$	47.0301	1.62366	0.811829	−	0.583895i	$-0.198472\pi$
$839$	47.0301	1.62366	0.811829	+	0.583895i	$0.198472\pi$
$840$	0	0
$841$	−8.48420	−0.292559
$842$	0	0
$843$	71.6254	2.46691
$844$	0	0
$845$	12.3054	0.423319
$846$	0	0
$847$	−3.10213	−0.106590
$848$	0	0
$849$	−31.9158	−1.09535
$850$	0	0
$851$	11.7970	0.404395
$852$	0	0
$853$	32.1112	1.09947	0.549733	−	0.835340i	$-0.314729\pi$
$853$	32.1112	1.09947	0.549733	+	0.835340i	$0.314729\pi$
$854$	0	0
$855$	28.9682	0.990693
$856$	0	0
$857$	−11.8076	−0.403341	−0.201671	−	0.979453i	$-0.564637\pi$
$857$	−11.8076	−0.403341	−0.201671	+	0.979453i	$0.564637\pi$
$858$	0	0
$859$	−28.0331	−0.956476	−0.478238	−	0.878230i	$-0.658724\pi$
$859$	−28.0331	−0.956476	−0.478238	+	0.878230i	$0.658724\pi$
$860$	0	0
$861$	−17.6991	−0.603182
$862$	0	0
$863$	1.51350	0.0515203	0.0257601	−	0.999668i	$-0.491799\pi$
$863$	1.51350	0.0515203	0.0257601	+	0.999668i	$0.491799\pi$
$864$	0	0
$865$	−22.5417	−0.766442
$866$	0	0
$867$	−31.4932	−1.06956
$868$	0	0
$869$	−51.2921	−1.73997
$870$	0	0
$871$	9.16307	0.310479
$872$	0	0
$873$	39.3374	1.33137
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	38.5778	1.30268	0.651340	−	0.758786i	$-0.274207\pi$
$877$	38.5778	1.30268	0.651340	+	0.758786i	$0.274207\pi$
$878$	0	0
$879$	−20.7689	−0.700517
$880$	0	0
$881$	3.53407	0.119066	0.0595329	−	0.998226i	$-0.481039\pi$
$881$	3.53407	0.119066	0.0595329	+	0.998226i	$0.481039\pi$
$882$	0	0
$883$	14.4216	0.485326	0.242663	−	0.970111i	$-0.421979\pi$
$883$	14.4216	0.485326	0.242663	+	0.970111i	$0.421979\pi$
$884$	0	0
$885$	−1.49217	−0.0501587
$886$	0	0
$887$	18.3972	0.617719	0.308859	−	0.951108i	$-0.400053\pi$
$887$	18.3972	0.617719	0.308859	+	0.951108i	$0.400053\pi$
$888$	0	0
$889$	20.7629	0.696365
$890$	0	0
$891$	31.0164	1.03909
$892$	0	0
$893$	3.81487	0.127660
$894$	0	0
$895$	−2.79073	−0.0932839
$896$	0	0
$897$	3.40397	0.113655
$898$	0	0
$899$	27.8627	0.929272
$900$	0	0
$901$	22.5588	0.751542
$902$	0	0
$903$	−2.98613	−0.0993720
$904$	0	0
$905$	−24.5230	−0.815172
$906$	0	0
$907$	17.6641	0.586526	0.293263	−	0.956032i	$-0.405259\pi$
$907$	17.6641	0.586526	0.293263	+	0.956032i	$0.405259\pi$
$908$	0	0
$909$	−36.2771	−1.20324
$910$	0	0
$911$	−21.0526	−0.697503	−0.348751	−	0.937215i	$-0.613394\pi$
$911$	−21.0526	−0.697503	−0.348751	+	0.937215i	$0.613394\pi$
$912$	0	0
$913$	4.99720	0.165383
$914$	0	0
$915$	−4.89106	−0.161693
$916$	0	0
$917$	10.9347	0.361096
$918$	0	0
$919$	30.9640	1.02141	0.510705	−	0.859756i	$-0.329385\pi$
$919$	30.9640	1.02141	0.510705	+	0.859756i	$0.329385\pi$
$920$	0	0
$921$	−17.4386	−0.574620
$922$	0	0
$923$	−6.82571	−0.224671
$924$	0	0
$925$	8.62496	0.283587
$926$	0	0
$927$	−114.931	−3.77483
$928$	0	0
$929$	−3.62025	−0.118777	−0.0593883	−	0.998235i	$-0.518915\pi$
$929$	−3.62025	−0.118777	−0.0593883	+	0.998235i	$0.518915\pi$
$930$	0	0
$931$	−4.89581	−0.160454
$932$	0	0
$933$	−10.3882	−0.340095
$934$	0	0
$935$	19.7095	0.644569
$936$	0	0
$937$	−45.6216	−1.49039	−0.745197	−	0.666845i	$-0.767644\pi$
$937$	−45.6216	−1.49039	−0.745197	+	0.666845i	$0.767644\pi$
$938$	0	0
$939$	−44.5884	−1.45509
$940$	0	0
$941$	−20.5062	−0.668483	−0.334241	−	0.942487i	$-0.608480\pi$
$941$	−20.5062	−0.668483	−0.334241	+	0.942487i	$0.608480\pi$
$942$	0	0
$943$	−8.10691	−0.263997
$944$	0	0
$945$	−8.71037	−0.283348
$946$	0	0
$947$	−7.63075	−0.247966	−0.123983	−	0.992284i	$-0.539567\pi$
$947$	−7.63075	−0.247966	−0.123983	+	0.992284i	$0.539567\pi$
$948$	0	0
$949$	−4.44312	−0.144230
$950$	0	0
$951$	−13.0686	−0.423778
$952$	0	0
$953$	28.8310	0.933927	0.466964	−	0.884277i	$-0.345348\pi$
$953$	28.8310	0.933927	0.466964	+	0.884277i	$0.345348\pi$
$954$	0	0
$955$	−12.3079	−0.398274
$956$	0	0
$957$	−50.7919	−1.64187
$958$	0	0
$959$	14.7907	0.477616
$960$	0	0
$961$	6.84048	0.220661
$962$	0	0
$963$	−115.747	−3.72990
$964$	0	0
$965$	1.79341	0.0577318
$966$	0	0
$967$	−16.0447	−0.515963	−0.257981	−	0.966150i	$-0.583057\pi$
$967$	−16.0447	−0.515963	−0.257981	+	0.966150i	$0.583057\pi$
$968$	0	0
$969$	−76.7301	−2.46493
$970$	0	0
$971$	2.11424	0.0678492	0.0339246	−	0.999424i	$-0.489199\pi$
$971$	2.11424	0.0678492	0.0339246	+	0.999424i	$0.489199\pi$
$972$	0	0
$973$	−1.82325	−0.0584508
$974$	0	0
$975$	2.48870	0.0797021
$976$	0	0
$977$	43.7887	1.40093	0.700463	−	0.713689i	$-0.252977\pi$
$977$	43.7887	1.40093	0.700463	+	0.713689i	$0.252977\pi$
$978$	0	0
$979$	−3.54349	−0.113251
$980$	0	0
$981$	86.4790	2.76106
$982$	0	0
$983$	−54.3134	−1.73233	−0.866163	−	0.499761i	$-0.833421\pi$
$983$	−54.3134	−1.73233	−0.866163	+	0.499761i	$0.833421\pi$
$984$	0	0
$985$	−23.4638	−0.747619
$986$	0	0
$987$	−2.32682	−0.0740636
$988$	0	0
$989$	−1.36777	−0.0434926
$990$	0	0
$991$	44.7313	1.42094	0.710468	−	0.703730i	$-0.248483\pi$
$991$	44.7313	1.42094	0.710468	+	0.703730i	$0.248483\pi$
$992$	0	0
$993$	33.9023	1.07586
$994$	0	0
$995$	17.1447	0.543524
$996$	0	0
$997$	40.7110	1.28933	0.644666	−	0.764464i	$-0.276996\pi$
$997$	40.7110	1.28933	0.644666	+	0.764464i	$0.276996\pi$
$998$	0	0
$999$	−75.1266	−2.37690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.j.1.1	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.j.1.1	✓	13	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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q-2.98613 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.91695 q^{9} +O(q^{10})\)	\(q-2.98613 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.91695 q^{9} +3.75528 q^{11} -0.833420 q^{13} +2.98613 q^{15} -5.24848 q^{17} -4.89581 q^{19} +2.98613 q^{21} +1.36777 q^{23} +1.00000 q^{25} -8.71037 q^{27} +4.52944 q^{29} +6.15146 q^{31} -11.2137 q^{33} +1.00000 q^{35} +8.62496 q^{37} +2.48870 q^{39} -5.92710 q^{41} -1.00000 q^{43} -5.91695 q^{45} -0.779212 q^{47} +1.00000 q^{49} +15.6726 q^{51} -4.29816 q^{53} -3.75528 q^{55} +14.6195 q^{57} -0.499700 q^{59} -1.63793 q^{61} -5.91695 q^{63} +0.833420 q^{65} -10.9945 q^{67} -4.08434 q^{69} +8.19000 q^{71} +5.33119 q^{73} -2.98613 q^{75} -3.75528 q^{77} -13.6587 q^{79} +8.25941 q^{81} +1.33071 q^{83} +5.24848 q^{85} -13.5255 q^{87} -0.943603 q^{89} +0.833420 q^{91} -18.3690 q^{93} +4.89581 q^{95} +6.64826 q^{97} +22.2198 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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