LMFDB - Embedded newform 8040.2.a.t.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 3x^{4} + 43x^{3} - 6x^{2} - 29x + 6 $ x^7 - x^6 - 15x^5 + 3x^4 + 43x^3 - 6x^2 - 29*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.922407$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +4.52856 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +4.52856 q^{7} +1.00000 q^{9} +3.01746 q^{11} -4.99509 q^{13} -1.00000 q^{15} +4.07157 q^{17} +1.78959 q^{19} +4.52856 q^{21} +4.12194 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.71175 q^{29} +4.66138 q^{31} +3.01746 q^{33} -4.52856 q^{35} +4.96844 q^{37} -4.99509 q^{39} -2.66138 q^{41} -9.11704 q^{43} -1.00000 q^{45} -2.72945 q^{47} +13.5078 q^{49} +4.07157 q^{51} +0.311159 q^{53} -3.01746 q^{55} +1.78959 q^{57} -2.27322 q^{59} -0.999758 q^{61} +4.52856 q^{63} +4.99509 q^{65} +1.00000 q^{67} +4.12194 q^{69} -3.73447 q^{71} +13.6689 q^{73} +1.00000 q^{75} +13.6647 q^{77} -2.72419 q^{79} +1.00000 q^{81} -8.69329 q^{83} -4.07157 q^{85} +2.71175 q^{87} -11.8698 q^{89} -22.6206 q^{91} +4.66138 q^{93} -1.78959 q^{95} +8.55597 q^{97} +3.01746 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * q^97

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.52856	1.71163	0.855817	−	0.517279i	$-0.173055\pi$
$7$	4.52856	1.71163	0.855817	+	0.517279i	$0.173055\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.01746	0.909798	0.454899	−	0.890543i	$-0.349675\pi$
$11$	3.01746	0.909798	0.454899	+	0.890543i	$0.349675\pi$
$12$	0	0
$13$	−4.99509	−1.38539	−0.692695	−	0.721231i	$-0.743577\pi$
$13$	−4.99509	−1.38539	−0.692695	+	0.721231i	$0.743577\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.07157	0.987501	0.493751	−	0.869603i	$-0.335626\pi$
$17$	4.07157	0.987501	0.493751	+	0.869603i	$0.335626\pi$
$18$	0	0
$19$	1.78959	0.410559	0.205280	−	0.978703i	$-0.434190\pi$
$19$	1.78959	0.410559	0.205280	+	0.978703i	$0.434190\pi$
$20$	0	0
$21$	4.52856	0.988212
$22$	0	0
$23$	4.12194	0.859485	0.429742	−	0.902951i	$-0.358604\pi$
$23$	4.12194	0.859485	0.429742	+	0.902951i	$0.358604\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.71175	0.503560	0.251780	−	0.967785i	$-0.418984\pi$
$29$	2.71175	0.503560	0.251780	+	0.967785i	$0.418984\pi$
$30$	0	0
$31$	4.66138	0.837208	0.418604	−	0.908169i	$-0.362519\pi$
$31$	4.66138	0.837208	0.418604	+	0.908169i	$0.362519\pi$
$32$	0	0
$33$	3.01746	0.525272
$34$	0	0
$35$	−4.52856	−0.765466
$36$	0	0
$37$	4.96844	0.816807	0.408403	−	0.912802i	$-0.366086\pi$
$37$	4.96844	0.816807	0.408403	+	0.912802i	$0.366086\pi$
$38$	0	0
$39$	−4.99509	−0.799855
$40$	0	0
$41$	−2.66138	−0.415638	−0.207819	−	0.978167i	$-0.566636\pi$
$41$	−2.66138	−0.415638	−0.207819	+	0.978167i	$0.566636\pi$
$42$	0	0
$43$	−9.11704	−1.39034	−0.695168	−	0.718848i	$-0.744670\pi$
$43$	−9.11704	−1.39034	−0.695168	+	0.718848i	$0.744670\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.72945	−0.398131	−0.199066	−	0.979986i	$-0.563791\pi$
$47$	−2.72945	−0.398131	−0.199066	+	0.979986i	$0.563791\pi$
$48$	0	0
$49$	13.5078	1.92969
$50$	0	0
$51$	4.07157	0.570134
$52$	0	0
$53$	0.311159	0.0427409	0.0213705	−	0.999772i	$-0.493197\pi$
$53$	0.311159	0.0427409	0.0213705	+	0.999772i	$0.493197\pi$
$54$	0	0
$55$	−3.01746	−0.406874
$56$	0	0
$57$	1.78959	0.237036
$58$	0	0
$59$	−2.27322	−0.295948	−0.147974	−	0.988991i	$-0.547275\pi$
$59$	−2.27322	−0.295948	−0.147974	+	0.988991i	$0.547275\pi$
$60$	0	0
$61$	−0.999758	−0.128006	−0.0640030	−	0.997950i	$-0.520387\pi$
$61$	−0.999758	−0.128006	−0.0640030	+	0.997950i	$0.520387\pi$
$62$	0	0
$63$	4.52856	0.570545
$64$	0	0
$65$	4.99509	0.619565
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	4.12194	0.496224
$70$	0	0
$71$	−3.73447	−0.443200	−0.221600	−	0.975138i	$-0.571128\pi$
$71$	−3.73447	−0.443200	−0.221600	+	0.975138i	$0.571128\pi$
$72$	0	0
$73$	13.6689	1.59982	0.799912	−	0.600117i	$-0.204879\pi$
$73$	13.6689	1.59982	0.799912	+	0.600117i	$0.204879\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	13.6647	1.55724
$78$	0	0
$79$	−2.72419	−0.306495	−0.153248	−	0.988188i	$-0.548973\pi$
$79$	−2.72419	−0.306495	−0.153248	+	0.988188i	$0.548973\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.69329	−0.954213	−0.477106	−	0.878846i	$-0.658314\pi$
$83$	−8.69329	−0.954213	−0.477106	+	0.878846i	$0.658314\pi$
$84$	0	0
$85$	−4.07157	−0.441624
$86$	0	0
$87$	2.71175	0.290730
$88$	0	0
$89$	−11.8698	−1.25820	−0.629098	−	0.777326i	$-0.716576\pi$
$89$	−11.8698	−1.25820	−0.629098	+	0.777326i	$0.716576\pi$
$90$	0	0
$91$	−22.6206	−2.37128
$92$	0	0
$93$	4.66138	0.483362
$94$	0	0
$95$	−1.78959	−0.183608
$96$	0	0
$97$	8.55597	0.868727	0.434363	−	0.900738i	$-0.356973\pi$
$97$	8.55597	0.868727	0.434363	+	0.900738i	$0.356973\pi$
$98$	0	0
$99$	3.01746	0.303266
$100$	0	0
$101$	10.4184	1.03667	0.518335	−	0.855178i	$-0.326552\pi$
$101$	10.4184	1.03667	0.518335	+	0.855178i	$0.326552\pi$
$102$	0	0
$103$	−7.83575	−0.772079	−0.386040	−	0.922482i	$-0.626157\pi$
$103$	−7.83575	−0.772079	−0.386040	+	0.922482i	$0.626157\pi$
$104$	0	0
$105$	−4.52856	−0.441942
$106$	0	0
$107$	10.7245	1.03678	0.518388	−	0.855146i	$-0.326532\pi$
$107$	10.7245	1.03678	0.518388	+	0.855146i	$0.326532\pi$
$108$	0	0
$109$	−12.1961	−1.16818	−0.584088	−	0.811690i	$-0.698548\pi$
$109$	−12.1961	−1.16818	−0.584088	+	0.811690i	$0.698548\pi$
$110$	0	0
$111$	4.96844	0.471584
$112$	0	0
$113$	3.55298	0.334237	0.167118	−	0.985937i	$-0.446554\pi$
$113$	3.55298	0.334237	0.167118	+	0.985937i	$0.446554\pi$
$114$	0	0
$115$	−4.12194	−0.384373
$116$	0	0
$117$	−4.99509	−0.461796
$118$	0	0
$119$	18.4384	1.69024
$120$	0	0
$121$	−1.89494	−0.172268
$122$	0	0
$123$	−2.66138	−0.239968
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.7682	1.75415	0.877073	−	0.480357i	$-0.159493\pi$
$127$	19.7682	1.75415	0.877073	+	0.480357i	$0.159493\pi$
$128$	0	0
$129$	−9.11704	−0.802710
$130$	0	0
$131$	−3.62342	−0.316580	−0.158290	−	0.987393i	$-0.550598\pi$
$131$	−3.62342	−0.316580	−0.158290	+	0.987393i	$0.550598\pi$
$132$	0	0
$133$	8.10424	0.702727
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−14.4342	−1.23320	−0.616598	−	0.787278i	$-0.711489\pi$
$137$	−14.4342	−1.23320	−0.616598	+	0.787278i	$0.711489\pi$
$138$	0	0
$139$	2.13358	0.180968	0.0904839	−	0.995898i	$-0.471159\pi$
$139$	2.13358	0.180968	0.0904839	+	0.995898i	$0.471159\pi$
$140$	0	0
$141$	−2.72945	−0.229861
$142$	0	0
$143$	−15.0725	−1.26042
$144$	0	0
$145$	−2.71175	−0.225199
$146$	0	0
$147$	13.5078	1.11411
$148$	0	0
$149$	14.0243	1.14892	0.574458	−	0.818534i	$-0.305213\pi$
$149$	14.0243	1.14892	0.574458	+	0.818534i	$0.305213\pi$
$150$	0	0
$151$	14.1469	1.15126	0.575631	−	0.817710i	$-0.304757\pi$
$151$	14.1469	1.15126	0.575631	+	0.817710i	$0.304757\pi$
$152$	0	0
$153$	4.07157	0.329167
$154$	0	0
$155$	−4.66138	−0.374411
$156$	0	0
$157$	20.9743	1.67393	0.836965	−	0.547256i	$-0.184328\pi$
$157$	20.9743	1.67393	0.836965	+	0.547256i	$0.184328\pi$
$158$	0	0
$159$	0.311159	0.0246765
$160$	0	0
$161$	18.6665	1.47112
$162$	0	0
$163$	−20.2727	−1.58788	−0.793941	−	0.607995i	$-0.791974\pi$
$163$	−20.2727	−1.58788	−0.793941	+	0.607995i	$0.791974\pi$
$164$	0	0
$165$	−3.01746	−0.234909
$166$	0	0
$167$	6.04113	0.467476	0.233738	−	0.972300i	$-0.424904\pi$
$167$	6.04113	0.467476	0.233738	+	0.972300i	$0.424904\pi$
$168$	0	0
$169$	11.9509	0.919304
$170$	0	0
$171$	1.78959	0.136853
$172$	0	0
$173$	−3.61707	−0.275001	−0.137500	−	0.990502i	$-0.543907\pi$
$173$	−3.61707	−0.275001	−0.137500	+	0.990502i	$0.543907\pi$
$174$	0	0
$175$	4.52856	0.342327
$176$	0	0
$177$	−2.27322	−0.170866
$178$	0	0
$179$	−2.97723	−0.222529	−0.111264	−	0.993791i	$-0.535490\pi$
$179$	−2.97723	−0.222529	−0.111264	+	0.993791i	$0.535490\pi$
$180$	0	0
$181$	3.47802	0.258519	0.129260	−	0.991611i	$-0.458740\pi$
$181$	3.47802	0.258519	0.129260	+	0.991611i	$0.458740\pi$
$182$	0	0
$183$	−0.999758	−0.0739043
$184$	0	0
$185$	−4.96844	−0.365287
$186$	0	0
$187$	12.2858	0.898427
$188$	0	0
$189$	4.52856	0.329404
$190$	0	0
$191$	9.48662	0.686428	0.343214	−	0.939257i	$-0.388484\pi$
$191$	9.48662	0.686428	0.343214	+	0.939257i	$0.388484\pi$
$192$	0	0
$193$	15.7833	1.13610	0.568052	−	0.822992i	$-0.307697\pi$
$193$	15.7833	1.13610	0.568052	+	0.822992i	$0.307697\pi$
$194$	0	0
$195$	4.99509	0.357706
$196$	0	0
$197$	−6.46333	−0.460493	−0.230247	−	0.973132i	$-0.573953\pi$
$197$	−6.46333	−0.460493	−0.230247	+	0.973132i	$0.573953\pi$
$198$	0	0
$199$	9.52523	0.675226	0.337613	−	0.941285i	$-0.390381\pi$
$199$	9.52523	0.675226	0.337613	+	0.941285i	$0.390381\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	12.2803	0.861910
$204$	0	0
$205$	2.66138	0.185879
$206$	0	0
$207$	4.12194	0.286495
$208$	0	0
$209$	5.40000	0.373526
$210$	0	0
$211$	−18.4972	−1.27340	−0.636702	−	0.771110i	$-0.719702\pi$
$211$	−18.4972	−1.27340	−0.636702	+	0.771110i	$0.719702\pi$
$212$	0	0
$213$	−3.73447	−0.255882
$214$	0	0
$215$	9.11704	0.621777
$216$	0	0
$217$	21.1093	1.43299
$218$	0	0
$219$	13.6689	0.923659
$220$	0	0
$221$	−20.3379	−1.36807
$222$	0	0
$223$	−14.4186	−0.965542	−0.482771	−	0.875747i	$-0.660370\pi$
$223$	−14.4186	−0.965542	−0.482771	+	0.875747i	$0.660370\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−27.4412	−1.82134	−0.910668	−	0.413139i	$-0.864432\pi$
$227$	−27.4412	−1.82134	−0.910668	+	0.413139i	$0.864432\pi$
$228$	0	0
$229$	3.95040	0.261050	0.130525	−	0.991445i	$-0.458334\pi$
$229$	3.95040	0.261050	0.130525	+	0.991445i	$0.458334\pi$
$230$	0	0
$231$	13.6647	0.899074
$232$	0	0
$233$	−18.2038	−1.19257	−0.596285	−	0.802773i	$-0.703357\pi$
$233$	−18.2038	−1.19257	−0.596285	+	0.802773i	$0.703357\pi$
$234$	0	0
$235$	2.72945	0.178050
$236$	0	0
$237$	−2.72419	−0.176955
$238$	0	0
$239$	−6.17999	−0.399750	−0.199875	−	0.979821i	$-0.564054\pi$
$239$	−6.17999	−0.399750	−0.199875	+	0.979821i	$0.564054\pi$
$240$	0	0
$241$	12.9358	0.833270	0.416635	−	0.909074i	$-0.363209\pi$
$241$	12.9358	0.833270	0.416635	+	0.909074i	$0.363209\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−13.5078	−0.862984
$246$	0	0
$247$	−8.93915	−0.568784
$248$	0	0
$249$	−8.69329	−0.550915
$250$	0	0
$251$	10.4636	0.660457	0.330228	−	0.943901i	$-0.392874\pi$
$251$	10.4636	0.660457	0.330228	+	0.943901i	$0.392874\pi$
$252$	0	0
$253$	12.4378	0.781958
$254$	0	0
$255$	−4.07157	−0.254972
$256$	0	0
$257$	−6.77262	−0.422464	−0.211232	−	0.977436i	$-0.567748\pi$
$257$	−6.77262	−0.422464	−0.211232	+	0.977436i	$0.567748\pi$
$258$	0	0
$259$	22.4999	1.39807
$260$	0	0
$261$	2.71175	0.167853
$262$	0	0
$263$	−26.2916	−1.62121	−0.810606	−	0.585592i	$-0.800862\pi$
$263$	−26.2916	−1.62121	−0.810606	+	0.585592i	$0.800862\pi$
$264$	0	0
$265$	−0.311159	−0.0191143
$266$	0	0
$267$	−11.8698	−0.726420
$268$	0	0
$269$	−27.6254	−1.68435	−0.842175	−	0.539204i	$-0.818725\pi$
$269$	−27.6254	−1.68435	−0.842175	+	0.539204i	$0.818725\pi$
$270$	0	0
$271$	21.4445	1.30266	0.651330	−	0.758795i	$-0.274211\pi$
$271$	21.4445	1.30266	0.651330	+	0.758795i	$0.274211\pi$
$272$	0	0
$273$	−22.6206	−1.36906
$274$	0	0
$275$	3.01746	0.181960
$276$	0	0
$277$	−5.90868	−0.355018	−0.177509	−	0.984119i	$-0.556804\pi$
$277$	−5.90868	−0.355018	−0.177509	+	0.984119i	$0.556804\pi$
$278$	0	0
$279$	4.66138	0.279069
$280$	0	0
$281$	−15.4143	−0.919537	−0.459769	−	0.888039i	$-0.652068\pi$
$281$	−15.4143	−0.919537	−0.459769	+	0.888039i	$0.652068\pi$
$282$	0	0
$283$	15.1098	0.898186	0.449093	−	0.893485i	$-0.351747\pi$
$283$	15.1098	0.898186	0.449093	+	0.893485i	$0.351747\pi$
$284$	0	0
$285$	−1.78959	−0.106006
$286$	0	0
$287$	−12.0522	−0.711419
$288$	0	0
$289$	−0.422296	−0.0248410
$290$	0	0
$291$	8.55597	0.501560
$292$	0	0
$293$	−10.3820	−0.606526	−0.303263	−	0.952907i	$-0.598076\pi$
$293$	−10.3820	−0.606526	−0.303263	+	0.952907i	$0.598076\pi$
$294$	0	0
$295$	2.27322	0.132352
$296$	0	0
$297$	3.01746	0.175091
$298$	0	0
$299$	−20.5895	−1.19072
$300$	0	0
$301$	−41.2870	−2.37975
$302$	0	0
$303$	10.4184	0.598522
$304$	0	0
$305$	0.999758	0.0572460
$306$	0	0
$307$	18.6699	1.06555	0.532773	−	0.846258i	$-0.321150\pi$
$307$	18.6699	1.06555	0.532773	+	0.846258i	$0.321150\pi$
$308$	0	0
$309$	−7.83575	−0.445760
$310$	0	0
$311$	13.1111	0.743461	0.371730	−	0.928341i	$-0.378765\pi$
$311$	13.1111	0.743461	0.371730	+	0.928341i	$0.378765\pi$
$312$	0	0
$313$	−7.29921	−0.412576	−0.206288	−	0.978491i	$-0.566138\pi$
$313$	−7.29921	−0.412576	−0.206288	+	0.978491i	$0.566138\pi$
$314$	0	0
$315$	−4.52856	−0.255155
$316$	0	0
$317$	31.4771	1.76793	0.883964	−	0.467556i	$-0.154865\pi$
$317$	31.4771	1.76793	0.883964	+	0.467556i	$0.154865\pi$
$318$	0	0
$319$	8.18260	0.458137
$320$	0	0
$321$	10.7245	0.598583
$322$	0	0
$323$	7.28643	0.405428
$324$	0	0
$325$	−4.99509	−0.277078
$326$	0	0
$327$	−12.1961	−0.674447
$328$	0	0
$329$	−12.3605	−0.681455
$330$	0	0
$331$	−7.29632	−0.401042	−0.200521	−	0.979689i	$-0.564264\pi$
$331$	−7.29632	−0.401042	−0.200521	+	0.979689i	$0.564264\pi$
$332$	0	0
$333$	4.96844	0.272269
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−4.70904	−0.256518	−0.128259	−	0.991741i	$-0.540939\pi$
$337$	−4.70904	−0.256518	−0.128259	+	0.991741i	$0.540939\pi$
$338$	0	0
$339$	3.55298	0.192972
$340$	0	0
$341$	14.0655	0.761690
$342$	0	0
$343$	29.4711	1.59129
$344$	0	0
$345$	−4.12194	−0.221918
$346$	0	0
$347$	−17.5439	−0.941805	−0.470902	−	0.882185i	$-0.656072\pi$
$347$	−17.5439	−0.941805	−0.470902	+	0.882185i	$0.656072\pi$
$348$	0	0
$349$	19.5530	1.04665	0.523323	−	0.852134i	$-0.324692\pi$
$349$	19.5530	1.04665	0.523323	+	0.852134i	$0.324692\pi$
$350$	0	0
$351$	−4.99509	−0.266618
$352$	0	0
$353$	6.20618	0.330322	0.165161	−	0.986267i	$-0.447186\pi$
$353$	6.20618	0.330322	0.165161	+	0.986267i	$0.447186\pi$
$354$	0	0
$355$	3.73447	0.198205
$356$	0	0
$357$	18.4384	0.975861
$358$	0	0
$359$	10.8531	0.572804	0.286402	−	0.958110i	$-0.407541\pi$
$359$	10.8531	0.572804	0.286402	+	0.958110i	$0.407541\pi$
$360$	0	0
$361$	−15.7974	−0.831441
$362$	0	0
$363$	−1.89494	−0.0994588
$364$	0	0
$365$	−13.6689	−0.715463
$366$	0	0
$367$	−6.31956	−0.329878	−0.164939	−	0.986304i	$-0.552743\pi$
$367$	−6.31956	−0.329878	−0.164939	+	0.986304i	$0.552743\pi$
$368$	0	0
$369$	−2.66138	−0.138546
$370$	0	0
$371$	1.40910	0.0731568
$372$	0	0
$373$	−1.30921	−0.0677885	−0.0338943	−	0.999425i	$-0.510791\pi$
$373$	−1.30921	−0.0677885	−0.0338943	+	0.999425i	$0.510791\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−13.5454	−0.697626
$378$	0	0
$379$	18.6595	0.958472	0.479236	−	0.877686i	$-0.340914\pi$
$379$	18.6595	0.958472	0.479236	+	0.877686i	$0.340914\pi$
$380$	0	0
$381$	19.7682	1.01276
$382$	0	0
$383$	21.1748	1.08198	0.540990	−	0.841029i	$-0.318050\pi$
$383$	21.1748	1.08198	0.540990	+	0.841029i	$0.318050\pi$
$384$	0	0
$385$	−13.6647	−0.696419
$386$	0	0
$387$	−9.11704	−0.463445
$388$	0	0
$389$	−11.6516	−0.590760	−0.295380	−	0.955380i	$-0.595446\pi$
$389$	−11.6516	−0.590760	−0.295380	+	0.955380i	$0.595446\pi$
$390$	0	0
$391$	16.7828	0.848743
$392$	0	0
$393$	−3.62342	−0.182777
$394$	0	0
$395$	2.72419	0.137069
$396$	0	0
$397$	31.4764	1.57976	0.789879	−	0.613263i	$-0.210143\pi$
$397$	31.4764	1.57976	0.789879	+	0.613263i	$0.210143\pi$
$398$	0	0
$399$	8.10424	0.405720
$400$	0	0
$401$	38.6154	1.92836	0.964181	−	0.265246i	$-0.0854532\pi$
$401$	38.6154	1.92836	0.964181	+	0.265246i	$0.0854532\pi$
$402$	0	0
$403$	−23.2840	−1.15986
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	14.9921	0.743129
$408$	0	0
$409$	25.5356	1.26266	0.631328	−	0.775516i	$-0.282510\pi$
$409$	25.5356	1.26266	0.631328	+	0.775516i	$0.282510\pi$
$410$	0	0
$411$	−14.4342	−0.711986
$412$	0	0
$413$	−10.2944	−0.506555
$414$	0	0
$415$	8.69329	0.426737
$416$	0	0
$417$	2.13358	0.104482
$418$	0	0
$419$	−20.7202	−1.01225	−0.506124	−	0.862461i	$-0.668922\pi$
$419$	−20.7202	−1.01225	−0.506124	+	0.862461i	$0.668922\pi$
$420$	0	0
$421$	−15.5622	−0.758454	−0.379227	−	0.925304i	$-0.623810\pi$
$421$	−15.5622	−0.758454	−0.379227	+	0.925304i	$0.623810\pi$
$422$	0	0
$423$	−2.72945	−0.132710
$424$	0	0
$425$	4.07157	0.197500
$426$	0	0
$427$	−4.52746	−0.219099
$428$	0	0
$429$	−15.0725	−0.727706
$430$	0	0
$431$	−31.7281	−1.52829	−0.764144	−	0.645046i	$-0.776838\pi$
$431$	−31.7281	−1.52829	−0.764144	+	0.645046i	$0.776838\pi$
$432$	0	0
$433$	4.49421	0.215978	0.107989	−	0.994152i	$-0.465559\pi$
$433$	4.49421	0.215978	0.107989	+	0.994152i	$0.465559\pi$
$434$	0	0
$435$	−2.71175	−0.130019
$436$	0	0
$437$	7.37658	0.352869
$438$	0	0
$439$	15.8257	0.755321	0.377660	−	0.925944i	$-0.376729\pi$
$439$	15.8257	0.755321	0.377660	+	0.925944i	$0.376729\pi$
$440$	0	0
$441$	13.5078	0.643230
$442$	0	0
$443$	17.8635	0.848719	0.424360	−	0.905494i	$-0.360499\pi$
$443$	17.8635	0.848719	0.424360	+	0.905494i	$0.360499\pi$
$444$	0	0
$445$	11.8698	0.562683
$446$	0	0
$447$	14.0243	0.663327
$448$	0	0
$449$	−0.122541	−0.00578308	−0.00289154	−	0.999996i	$-0.500920\pi$
$449$	−0.122541	−0.00578308	−0.00289154	+	0.999996i	$0.500920\pi$
$450$	0	0
$451$	−8.03060	−0.378146
$452$	0	0
$453$	14.1469	0.664681
$454$	0	0
$455$	22.6206	1.06047
$456$	0	0
$457$	23.4200	1.09554	0.547771	−	0.836628i	$-0.315477\pi$
$457$	23.4200	1.09554	0.547771	+	0.836628i	$0.315477\pi$
$458$	0	0
$459$	4.07157	0.190045
$460$	0	0
$461$	−30.6749	−1.42867	−0.714337	−	0.699802i	$-0.753272\pi$
$461$	−30.6749	−1.42867	−0.714337	+	0.699802i	$0.753272\pi$
$462$	0	0
$463$	10.1660	0.472455	0.236227	−	0.971698i	$-0.424089\pi$
$463$	10.1660	0.472455	0.236227	+	0.971698i	$0.424089\pi$
$464$	0	0
$465$	−4.66138	−0.216166
$466$	0	0
$467$	6.90183	0.319378	0.159689	−	0.987167i	$-0.448951\pi$
$467$	6.90183	0.319378	0.159689	+	0.987167i	$0.448951\pi$
$468$	0	0
$469$	4.52856	0.209109
$470$	0	0
$471$	20.9743	0.966444
$472$	0	0
$473$	−27.5103	−1.26492
$474$	0	0
$475$	1.78959	0.0821118
$476$	0	0
$477$	0.311159	0.0142470
$478$	0	0
$479$	−7.66608	−0.350272	−0.175136	−	0.984544i	$-0.556037\pi$
$479$	−7.66608	−0.350272	−0.175136	+	0.984544i	$0.556037\pi$
$480$	0	0
$481$	−24.8178	−1.13160
$482$	0	0
$483$	18.6665	0.849354
$484$	0	0
$485$	−8.55597	−0.388506
$486$	0	0
$487$	16.8798	0.764895	0.382447	−	0.923977i	$-0.375081\pi$
$487$	16.8798	0.764895	0.382447	+	0.923977i	$0.375081\pi$
$488$	0	0
$489$	−20.2727	−0.916764
$490$	0	0
$491$	−30.1301	−1.35975	−0.679875	−	0.733328i	$-0.737966\pi$
$491$	−30.1301	−1.35975	−0.679875	+	0.733328i	$0.737966\pi$
$492$	0	0
$493$	11.0411	0.497266
$494$	0	0
$495$	−3.01746	−0.135625
$496$	0	0
$497$	−16.9118	−0.758597
$498$	0	0
$499$	41.5501	1.86004	0.930019	−	0.367511i	$-0.119790\pi$
$499$	41.5501	1.86004	0.930019	+	0.367511i	$0.119790\pi$
$500$	0	0
$501$	6.04113	0.269898
$502$	0	0
$503$	36.2858	1.61790	0.808951	−	0.587876i	$-0.200036\pi$
$503$	36.2858	1.61790	0.808951	+	0.587876i	$0.200036\pi$
$504$	0	0
$505$	−10.4184	−0.463613
$506$	0	0
$507$	11.9509	0.530760
$508$	0	0
$509$	−29.7617	−1.31916	−0.659582	−	0.751633i	$-0.729267\pi$
$509$	−29.7617	−1.31916	−0.659582	+	0.751633i	$0.729267\pi$
$510$	0	0
$511$	61.9004	2.73831
$512$	0	0
$513$	1.78959	0.0790122
$514$	0	0
$515$	7.83575	0.345284
$516$	0	0
$517$	−8.23601	−0.362219
$518$	0	0
$519$	−3.61707	−0.158772
$520$	0	0
$521$	10.5559	0.462461	0.231230	−	0.972899i	$-0.425725\pi$
$521$	10.5559	0.462461	0.231230	+	0.972899i	$0.425725\pi$
$522$	0	0
$523$	21.1314	0.924013	0.462006	−	0.886877i	$-0.347130\pi$
$523$	21.1314	0.924013	0.462006	+	0.886877i	$0.347130\pi$
$524$	0	0
$525$	4.52856	0.197642
$526$	0	0
$527$	18.9791	0.826744
$528$	0	0
$529$	−6.00957	−0.261286
$530$	0	0
$531$	−2.27322	−0.0986495
$532$	0	0
$533$	13.2938	0.575820
$534$	0	0
$535$	−10.7245	−0.463660
$536$	0	0
$537$	−2.97723	−0.128477
$538$	0	0
$539$	40.7593	1.75563
$540$	0	0
$541$	9.65064	0.414913	0.207457	−	0.978244i	$-0.433481\pi$
$541$	9.65064	0.414913	0.207457	+	0.978244i	$0.433481\pi$
$542$	0	0
$543$	3.47802	0.149256
$544$	0	0
$545$	12.1961	0.522424
$546$	0	0
$547$	−37.2906	−1.59443	−0.797216	−	0.603694i	$-0.793695\pi$
$547$	−37.2906	−1.59443	−0.797216	+	0.603694i	$0.793695\pi$
$548$	0	0
$549$	−0.999758	−0.0426686
$550$	0	0
$551$	4.85291	0.206741
$552$	0	0
$553$	−12.3366	−0.524607
$554$	0	0
$555$	−4.96844	−0.210899
$556$	0	0
$557$	22.0909	0.936020	0.468010	−	0.883723i	$-0.344971\pi$
$557$	22.0909	0.936020	0.468010	+	0.883723i	$0.344971\pi$
$558$	0	0
$559$	45.5404	1.92616
$560$	0	0
$561$	12.2858	0.518707
$562$	0	0
$563$	21.4094	0.902300	0.451150	−	0.892448i	$-0.351014\pi$
$563$	21.4094	0.902300	0.451150	+	0.892448i	$0.351014\pi$
$564$	0	0
$565$	−3.55298	−0.149475
$566$	0	0
$567$	4.52856	0.190182
$568$	0	0
$569$	21.6142	0.906116	0.453058	−	0.891481i	$-0.350333\pi$
$569$	21.6142	0.906116	0.453058	+	0.891481i	$0.350333\pi$
$570$	0	0
$571$	25.2151	1.05522	0.527609	−	0.849487i	$-0.323089\pi$
$571$	25.2151	1.05522	0.527609	+	0.849487i	$0.323089\pi$
$572$	0	0
$573$	9.48662	0.396309
$574$	0	0
$575$	4.12194	0.171897
$576$	0	0
$577$	−9.50017	−0.395497	−0.197749	−	0.980253i	$-0.563363\pi$
$577$	−9.50017	−0.395497	−0.197749	+	0.980253i	$0.563363\pi$
$578$	0	0
$579$	15.7833	0.655930
$580$	0	0
$581$	−39.3681	−1.63326
$582$	0	0
$583$	0.938908	0.0388856
$584$	0	0
$585$	4.99509	0.206522
$586$	0	0
$587$	−33.1639	−1.36882	−0.684409	−	0.729098i	$-0.739940\pi$
$587$	−33.1639	−1.36882	−0.684409	+	0.729098i	$0.739940\pi$
$588$	0	0
$589$	8.34194	0.343724
$590$	0	0
$591$	−6.46333	−0.265866
$592$	0	0
$593$	27.3144	1.12167	0.560835	−	0.827928i	$-0.310480\pi$
$593$	27.3144	1.12167	0.560835	+	0.827928i	$0.310480\pi$
$594$	0	0
$595$	−18.4384	−0.755899
$596$	0	0
$597$	9.52523	0.389842
$598$	0	0
$599$	9.42693	0.385174	0.192587	−	0.981280i	$-0.438312\pi$
$599$	9.42693	0.385174	0.192587	+	0.981280i	$0.438312\pi$
$600$	0	0
$601$	−33.1898	−1.35384	−0.676920	−	0.736057i	$-0.736686\pi$
$601$	−33.1898	−1.35384	−0.676920	+	0.736057i	$0.736686\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	1.89494	0.0770404
$606$	0	0
$607$	10.1532	0.412106	0.206053	−	0.978541i	$-0.433938\pi$
$607$	10.1532	0.412106	0.206053	+	0.978541i	$0.433938\pi$
$608$	0	0
$609$	12.2803	0.497624
$610$	0	0
$611$	13.6339	0.551567
$612$	0	0
$613$	−23.0912	−0.932646	−0.466323	−	0.884615i	$-0.654421\pi$
$613$	−23.0912	−0.932646	−0.466323	+	0.884615i	$0.654421\pi$
$614$	0	0
$615$	2.66138	0.107317
$616$	0	0
$617$	−1.52338	−0.0613288	−0.0306644	−	0.999530i	$-0.509762\pi$
$617$	−1.52338	−0.0613288	−0.0306644	+	0.999530i	$0.509762\pi$
$618$	0	0
$619$	−40.3184	−1.62053	−0.810266	−	0.586062i	$-0.800677\pi$
$619$	−40.3184	−1.62053	−0.810266	+	0.586062i	$0.800677\pi$
$620$	0	0
$621$	4.12194	0.165408
$622$	0	0
$623$	−53.7531	−2.15357
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.40000	0.215655
$628$	0	0
$629$	20.2294	0.806598
$630$	0	0
$631$	0.363697	0.0144785	0.00723927	−	0.999974i	$-0.497696\pi$
$631$	0.363697	0.0144785	0.00723927	+	0.999974i	$0.497696\pi$
$632$	0	0
$633$	−18.4972	−0.735200
$634$	0	0
$635$	−19.7682	−0.784478
$636$	0	0
$637$	−67.4729	−2.67337
$638$	0	0
$639$	−3.73447	−0.147733
$640$	0	0
$641$	25.1645	0.993937	0.496968	−	0.867769i	$-0.334447\pi$
$641$	25.1645	0.993937	0.496968	+	0.867769i	$0.334447\pi$
$642$	0	0
$643$	−6.18919	−0.244078	−0.122039	−	0.992525i	$-0.538943\pi$
$643$	−6.18919	−0.244078	−0.122039	+	0.992525i	$0.538943\pi$
$644$	0	0
$645$	9.11704	0.358983
$646$	0	0
$647$	−26.3079	−1.03427	−0.517135	−	0.855904i	$-0.673002\pi$
$647$	−26.3079	−1.03427	−0.517135	+	0.855904i	$0.673002\pi$
$648$	0	0
$649$	−6.85936	−0.269253
$650$	0	0
$651$	21.1093	0.827340
$652$	0	0
$653$	24.9711	0.977196	0.488598	−	0.872509i	$-0.337509\pi$
$653$	24.9711	0.977196	0.488598	+	0.872509i	$0.337509\pi$
$654$	0	0
$655$	3.62342	0.141579
$656$	0	0
$657$	13.6689	0.533275
$658$	0	0
$659$	33.8149	1.31724	0.658621	−	0.752474i	$-0.271140\pi$
$659$	33.8149	1.31724	0.658621	+	0.752474i	$0.271140\pi$
$660$	0	0
$661$	−9.02483	−0.351025	−0.175513	−	0.984477i	$-0.556158\pi$
$661$	−9.02483	−0.351025	−0.175513	+	0.984477i	$0.556158\pi$
$662$	0	0
$663$	−20.3379	−0.789858
$664$	0	0
$665$	−8.10424	−0.314269
$666$	0	0
$667$	11.1777	0.432802
$668$	0	0
$669$	−14.4186	−0.557456
$670$	0	0
$671$	−3.01673	−0.116460
$672$	0	0
$673$	−28.0138	−1.07985	−0.539927	−	0.841712i	$-0.681548\pi$
$673$	−28.0138	−1.07985	−0.539927	+	0.841712i	$0.681548\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−6.97519	−0.268078	−0.134039	−	0.990976i	$-0.542795\pi$
$677$	−6.97519	−0.268078	−0.134039	+	0.990976i	$0.542795\pi$
$678$	0	0
$679$	38.7462	1.48694
$680$	0	0
$681$	−27.4412	−1.05155
$682$	0	0
$683$	−34.0960	−1.30465	−0.652324	−	0.757940i	$-0.726206\pi$
$683$	−34.0960	−1.30465	−0.652324	+	0.757940i	$0.726206\pi$
$684$	0	0
$685$	14.4342	0.551502
$686$	0	0
$687$	3.95040	0.150717
$688$	0	0
$689$	−1.55427	−0.0592128
$690$	0	0
$691$	41.2338	1.56861	0.784304	−	0.620377i	$-0.213020\pi$
$691$	41.2338	1.56861	0.784304	+	0.620377i	$0.213020\pi$
$692$	0	0
$693$	13.6647	0.519080
$694$	0	0
$695$	−2.13358	−0.0809312
$696$	0	0
$697$	−10.8360	−0.410443
$698$	0	0
$699$	−18.2038	−0.688530
$700$	0	0
$701$	3.11548	0.117670	0.0588350	−	0.998268i	$-0.481261\pi$
$701$	3.11548	0.117670	0.0588350	+	0.998268i	$0.481261\pi$
$702$	0	0
$703$	8.89145	0.335347
$704$	0	0
$705$	2.72945	0.102797
$706$	0	0
$707$	47.1804	1.77440
$708$	0	0
$709$	9.97845	0.374749	0.187374	−	0.982289i	$-0.440002\pi$
$709$	9.97845	0.374749	0.187374	+	0.982289i	$0.440002\pi$
$710$	0	0
$711$	−2.72419	−0.102165
$712$	0	0
$713$	19.2139	0.719568
$714$	0	0
$715$	15.0725	0.563679
$716$	0	0
$717$	−6.17999	−0.230796
$718$	0	0
$719$	43.0440	1.60527	0.802635	−	0.596471i	$-0.203431\pi$
$719$	43.0440	1.60527	0.802635	+	0.596471i	$0.203431\pi$
$720$	0	0
$721$	−35.4846	−1.32152
$722$	0	0
$723$	12.9358	0.481089
$724$	0	0
$725$	2.71175	0.100712
$726$	0	0
$727$	−27.8439	−1.03267	−0.516336	−	0.856386i	$-0.672704\pi$
$727$	−27.8439	−1.03267	−0.516336	+	0.856386i	$0.672704\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−37.1207	−1.37296
$732$	0	0
$733$	19.8584	0.733486	0.366743	−	0.930322i	$-0.380473\pi$
$733$	19.8584	0.733486	0.366743	+	0.930322i	$0.380473\pi$
$734$	0	0
$735$	−13.5078	−0.498244
$736$	0	0
$737$	3.01746	0.111150
$738$	0	0
$739$	−28.1107	−1.03407	−0.517035	−	0.855964i	$-0.672964\pi$
$739$	−28.1107	−1.03407	−0.517035	+	0.855964i	$0.672964\pi$
$740$	0	0
$741$	−8.93915	−0.328388
$742$	0	0
$743$	6.81336	0.249958	0.124979	−	0.992159i	$-0.460114\pi$
$743$	6.81336	0.249958	0.124979	+	0.992159i	$0.460114\pi$
$744$	0	0
$745$	−14.0243	−0.513811
$746$	0	0
$747$	−8.69329	−0.318071
$748$	0	0
$749$	48.5665	1.77458
$750$	0	0
$751$	8.45304	0.308456	0.154228	−	0.988035i	$-0.450711\pi$
$751$	8.45304	0.308456	0.154228	+	0.988035i	$0.450711\pi$
$752$	0	0
$753$	10.4636	0.381315
$754$	0	0
$755$	−14.1469	−0.514860
$756$	0	0
$757$	9.50624	0.345510	0.172755	−	0.984965i	$-0.444733\pi$
$757$	9.50624	0.345510	0.172755	+	0.984965i	$0.444733\pi$
$758$	0	0
$759$	12.4378	0.451463
$760$	0	0
$761$	−17.3448	−0.628747	−0.314374	−	0.949299i	$-0.601794\pi$
$761$	−17.3448	−0.628747	−0.314374	+	0.949299i	$0.601794\pi$
$762$	0	0
$763$	−55.2308	−1.99949
$764$	0	0
$765$	−4.07157	−0.147208
$766$	0	0
$767$	11.3550	0.410004
$768$	0	0
$769$	36.8263	1.32799	0.663995	−	0.747737i	$-0.268860\pi$
$769$	36.8263	1.32799	0.663995	+	0.747737i	$0.268860\pi$
$770$	0	0
$771$	−6.77262	−0.243910
$772$	0	0
$773$	−5.48691	−0.197351	−0.0986753	−	0.995120i	$-0.531461\pi$
$773$	−5.48691	−0.197351	−0.0986753	+	0.995120i	$0.531461\pi$
$774$	0	0
$775$	4.66138	0.167442
$776$	0	0
$777$	22.4999	0.807178
$778$	0	0
$779$	−4.76277	−0.170644
$780$	0	0
$781$	−11.2686	−0.403223
$782$	0	0
$783$	2.71175	0.0969101
$784$	0	0
$785$	−20.9743	−0.748604
$786$	0	0
$787$	41.7969	1.48990	0.744949	−	0.667121i	$-0.232474\pi$
$787$	41.7969	1.48990	0.744949	+	0.667121i	$0.232474\pi$
$788$	0	0
$789$	−26.2916	−0.936007
$790$	0	0
$791$	16.0899	0.572091
$792$	0	0
$793$	4.99389	0.177338
$794$	0	0
$795$	−0.311159	−0.0110357
$796$	0	0
$797$	−6.99894	−0.247915	−0.123958	−	0.992288i	$-0.539559\pi$
$797$	−6.99894	−0.247915	−0.123958	+	0.992288i	$0.539559\pi$
$798$	0	0
$799$	−11.1132	−0.393155
$800$	0	0
$801$	−11.8698	−0.419399
$802$	0	0
$803$	41.2454	1.45552
$804$	0	0
$805$	−18.6665	−0.657907
$806$	0	0
$807$	−27.6254	−0.972460
$808$	0	0
$809$	22.7908	0.801282	0.400641	−	0.916235i	$-0.368787\pi$
$809$	22.7908	0.801282	0.400641	+	0.916235i	$0.368787\pi$
$810$	0	0
$811$	−26.2680	−0.922395	−0.461197	−	0.887298i	$-0.652580\pi$
$811$	−26.2680	−0.922395	−0.461197	+	0.887298i	$0.652580\pi$
$812$	0	0
$813$	21.4445	0.752091
$814$	0	0
$815$	20.2727	0.710123
$816$	0	0
$817$	−16.3157	−0.570815
$818$	0	0
$819$	−22.6206	−0.790427
$820$	0	0
$821$	−29.8917	−1.04323	−0.521614	−	0.853182i	$-0.674670\pi$
$821$	−29.8917	−1.04323	−0.521614	+	0.853182i	$0.674670\pi$
$822$	0	0
$823$	45.2087	1.57587	0.787937	−	0.615755i	$-0.211149\pi$
$823$	45.2087	1.57587	0.787937	+	0.615755i	$0.211149\pi$
$824$	0	0
$825$	3.01746	0.105054
$826$	0	0
$827$	−24.5552	−0.853868	−0.426934	−	0.904283i	$-0.640406\pi$
$827$	−24.5552	−0.853868	−0.426934	+	0.904283i	$0.640406\pi$
$828$	0	0
$829$	−26.0885	−0.906090	−0.453045	−	0.891488i	$-0.649662\pi$
$829$	−26.0885	−0.906090	−0.453045	+	0.891488i	$0.649662\pi$
$830$	0	0
$831$	−5.90868	−0.204970
$832$	0	0
$833$	54.9982	1.90557
$834$	0	0
$835$	−6.04113	−0.209062
$836$	0	0
$837$	4.66138	0.161121
$838$	0	0
$839$	−25.3367	−0.874720	−0.437360	−	0.899286i	$-0.644086\pi$
$839$	−25.3367	−0.874720	−0.437360	+	0.899286i	$0.644086\pi$
$840$	0	0
$841$	−21.6464	−0.746428
$842$	0	0
$843$	−15.4143	−0.530895
$844$	0	0
$845$	−11.9509	−0.411125
$846$	0	0
$847$	−8.58136	−0.294859
$848$	0	0
$849$	15.1098	0.518568
$850$	0	0
$851$	20.4796	0.702033
$852$	0	0
$853$	−4.49921	−0.154050	−0.0770250	−	0.997029i	$-0.524542\pi$
$853$	−4.49921	−0.154050	−0.0770250	+	0.997029i	$0.524542\pi$
$854$	0	0
$855$	−1.78959	−0.0612026
$856$	0	0
$857$	−31.5055	−1.07621	−0.538103	−	0.842879i	$-0.680859\pi$
$857$	−31.5055	−1.07621	−0.538103	+	0.842879i	$0.680859\pi$
$858$	0	0
$859$	14.2301	0.485524	0.242762	−	0.970086i	$-0.421947\pi$
$859$	14.2301	0.485524	0.242762	+	0.970086i	$0.421947\pi$
$860$	0	0
$861$	−12.0522	−0.410738
$862$	0	0
$863$	−37.2066	−1.26653	−0.633263	−	0.773936i	$-0.718285\pi$
$863$	−37.2066	−1.26653	−0.633263	+	0.773936i	$0.718285\pi$
$864$	0	0
$865$	3.61707	0.122984
$866$	0	0
$867$	−0.422296	−0.0143419
$868$	0	0
$869$	−8.22012	−0.278849
$870$	0	0
$871$	−4.99509	−0.169252
$872$	0	0
$873$	8.55597	0.289576
$874$	0	0
$875$	−4.52856	−0.153093
$876$	0	0
$877$	−21.2708	−0.718262	−0.359131	−	0.933287i	$-0.616927\pi$
$877$	−21.2708	−0.718262	−0.359131	+	0.933287i	$0.616927\pi$
$878$	0	0
$879$	−10.3820	−0.350178
$880$	0	0
$881$	−40.8417	−1.37599	−0.687996	−	0.725714i	$-0.741509\pi$
$881$	−40.8417	−1.37599	−0.687996	+	0.725714i	$0.741509\pi$
$882$	0	0
$883$	16.0760	0.541000	0.270500	−	0.962720i	$-0.412811\pi$
$883$	16.0760	0.541000	0.270500	+	0.962720i	$0.412811\pi$
$884$	0	0
$885$	2.27322	0.0764136
$886$	0	0
$887$	−23.5466	−0.790617	−0.395308	−	0.918548i	$-0.629362\pi$
$887$	−23.5466	−0.790617	−0.395308	+	0.918548i	$0.629362\pi$
$888$	0	0
$889$	89.5216	3.00246
$890$	0	0
$891$	3.01746	0.101089
$892$	0	0
$893$	−4.88459	−0.163457
$894$	0	0
$895$	2.97723	0.0995179
$896$	0	0
$897$	−20.5895	−0.687463
$898$	0	0
$899$	12.6405	0.421584
$900$	0	0
$901$	1.26690	0.0422067
$902$	0	0
$903$	−41.2870	−1.37395
$904$	0	0
$905$	−3.47802	−0.115613
$906$	0	0
$907$	−42.8507	−1.42283	−0.711416	−	0.702771i	$-0.751946\pi$
$907$	−42.8507	−1.42283	−0.711416	+	0.702771i	$0.751946\pi$
$908$	0	0
$909$	10.4184	0.345557
$910$	0	0
$911$	−29.3201	−0.971419	−0.485709	−	0.874120i	$-0.661439\pi$
$911$	−29.3201	−0.971419	−0.485709	+	0.874120i	$0.661439\pi$
$912$	0	0
$913$	−26.2317	−0.868141
$914$	0	0
$915$	0.999758	0.0330510
$916$	0	0
$917$	−16.4089	−0.541869
$918$	0	0
$919$	8.77825	0.289568	0.144784	−	0.989463i	$-0.453751\pi$
$919$	8.77825	0.289568	0.144784	+	0.989463i	$0.453751\pi$
$920$	0	0
$921$	18.6699	0.615193
$922$	0	0
$923$	18.6540	0.614005
$924$	0	0
$925$	4.96844	0.163361
$926$	0	0
$927$	−7.83575	−0.257360
$928$	0	0
$929$	23.6432	0.775709	0.387855	−	0.921721i	$-0.373216\pi$
$929$	23.6432	0.775709	0.387855	+	0.921721i	$0.373216\pi$
$930$	0	0
$931$	24.1734	0.792253
$932$	0	0
$933$	13.1111	0.429237
$934$	0	0
$935$	−12.2858	−0.401789
$936$	0	0
$937$	3.80530	0.124314	0.0621568	−	0.998066i	$-0.480202\pi$
$937$	3.80530	0.124314	0.0621568	+	0.998066i	$0.480202\pi$
$938$	0	0
$939$	−7.29921	−0.238201
$940$	0	0
$941$	10.8919	0.355067	0.177534	−	0.984115i	$-0.443188\pi$
$941$	10.8919	0.355067	0.177534	+	0.984115i	$0.443188\pi$
$942$	0	0
$943$	−10.9701	−0.357234
$944$	0	0
$945$	−4.52856	−0.147314
$946$	0	0
$947$	−53.9819	−1.75418	−0.877089	−	0.480329i	$-0.840517\pi$
$947$	−53.9819	−1.75418	−0.877089	+	0.480329i	$0.840517\pi$
$948$	0	0
$949$	−68.2775	−2.21638
$950$	0	0
$951$	31.4771	1.02071
$952$	0	0
$953$	−38.1294	−1.23513	−0.617566	−	0.786519i	$-0.711881\pi$
$953$	−38.1294	−1.23513	−0.617566	+	0.786519i	$0.711881\pi$
$954$	0	0
$955$	−9.48662	−0.306980
$956$	0	0
$957$	8.18260	0.264506
$958$	0	0
$959$	−65.3660	−2.11078
$960$	0	0
$961$	−9.27155	−0.299082
$962$	0	0
$963$	10.7245	0.345592
$964$	0	0
$965$	−15.7833	−0.508081
$966$	0	0
$967$	−9.32514	−0.299876	−0.149938	−	0.988695i	$-0.547907\pi$
$967$	−9.32514	−0.299876	−0.149938	+	0.988695i	$0.547907\pi$
$968$	0	0
$969$	7.28643	0.234074
$970$	0	0
$971$	24.4385	0.784268	0.392134	−	0.919908i	$-0.371737\pi$
$971$	24.4385	0.784268	0.392134	+	0.919908i	$0.371737\pi$
$972$	0	0
$973$	9.66203	0.309751
$974$	0	0
$975$	−4.99509	−0.159971
$976$	0	0
$977$	−26.1127	−0.835419	−0.417710	−	0.908581i	$-0.637167\pi$
$977$	−26.1127	−0.835419	−0.417710	+	0.908581i	$0.637167\pi$
$978$	0	0
$979$	−35.8166	−1.14470
$980$	0	0
$981$	−12.1961	−0.389392
$982$	0	0
$983$	27.5111	0.877468	0.438734	−	0.898617i	$-0.355427\pi$
$983$	27.5111	0.877468	0.438734	+	0.898617i	$0.355427\pi$
$984$	0	0
$985$	6.46333	0.205939
$986$	0	0
$987$	−12.3605	−0.393438
$988$	0	0
$989$	−37.5799	−1.19497
$990$	0	0
$991$	−46.5897	−1.47997	−0.739985	−	0.672623i	$-0.765167\pi$
$991$	−46.5897	−1.47997	−0.739985	+	0.672623i	$0.765167\pi$
$992$	0	0
$993$	−7.29632	−0.231542
$994$	0	0
$995$	−9.52523	−0.301970
$996$	0	0
$997$	−48.9541	−1.55039	−0.775196	−	0.631721i	$-0.782349\pi$
$997$	−48.9541	−1.55039	−0.775196	+	0.631721i	$0.782349\pi$
$998$	0	0
$999$	4.96844	0.157195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.t.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.t.1.7	✓	7	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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.52856 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.52856 q^{7} +1.00000 q^{9} +3.01746 q^{11} -4.99509 q^{13} -1.00000 q^{15} +4.07157 q^{17} +1.78959 q^{19} +4.52856 q^{21} +4.12194 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.71175 q^{29} +4.66138 q^{31} +3.01746 q^{33} -4.52856 q^{35} +4.96844 q^{37} -4.99509 q^{39} -2.66138 q^{41} -9.11704 q^{43} -1.00000 q^{45} -2.72945 q^{47} +13.5078 q^{49} +4.07157 q^{51} +0.311159 q^{53} -3.01746 q^{55} +1.78959 q^{57} -2.27322 q^{59} -0.999758 q^{61} +4.52856 q^{63} +4.99509 q^{65} +1.00000 q^{67} +4.12194 q^{69} -3.73447 q^{71} +13.6689 q^{73} +1.00000 q^{75} +13.6647 q^{77} -2.72419 q^{79} +1.00000 q^{81} -8.69329 q^{83} -4.07157 q^{85} +2.71175 q^{87} -11.8698 q^{89} -22.6206 q^{91} +4.66138 q^{93} -1.78959 q^{95} +8.55597 q^{97} +3.01746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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