LMFDB - Embedded newform 8008.2.a.u.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 16x^{8} + 26x^{7} + 93x^{6} - 113x^{5} - 230x^{4} + 197x^{3} + 201x^{2} - 115x + 8 $ x^10 - 2x^9 - 16x^8 + 26x^7 + 93x^6 - 113x^5 - 230x^4 + 197x^3 + 201x^2 - 115*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.97233$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.97233 q^{3} -3.07774 q^{5} -1.00000 q^{7} +5.83476 q^{9} +O(q^{10})$	$q+2.97233 q^{3} -3.07774 q^{5} -1.00000 q^{7} +5.83476 q^{9} -1.00000 q^{11} -1.00000 q^{13} -9.14806 q^{15} +2.23499 q^{17} +0.103324 q^{19} -2.97233 q^{21} -2.91620 q^{23} +4.47248 q^{25} +8.42585 q^{27} +5.61031 q^{29} -0.191615 q^{31} -2.97233 q^{33} +3.07774 q^{35} -6.04946 q^{37} -2.97233 q^{39} -10.4530 q^{41} +4.42785 q^{43} -17.9579 q^{45} -4.29716 q^{47} +1.00000 q^{49} +6.64315 q^{51} -0.962428 q^{53} +3.07774 q^{55} +0.307112 q^{57} +7.41338 q^{59} -6.36668 q^{61} -5.83476 q^{63} +3.07774 q^{65} -10.9683 q^{67} -8.66791 q^{69} +0.646837 q^{71} +2.81140 q^{73} +13.2937 q^{75} +1.00000 q^{77} -13.8459 q^{79} +7.54015 q^{81} +1.79514 q^{83} -6.87873 q^{85} +16.6757 q^{87} +2.92179 q^{89} +1.00000 q^{91} -0.569544 q^{93} -0.318003 q^{95} -1.39715 q^{97} -5.83476 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * 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
$3$	2.97233	1.71608	0.858038	−	0.513585i	$-0.171683\pi$
$3$	2.97233	1.71608	0.858038	+	0.513585i	$0.171683\pi$
$4$	0	0
$5$	−3.07774	−1.37641	−0.688203	−	0.725518i	$-0.741600\pi$
$5$	−3.07774	−1.37641	−0.688203	+	0.725518i	$0.741600\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.83476	1.94492
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−9.14806	−2.36202
$16$	0	0
$17$	2.23499	0.542066	0.271033	−	0.962570i	$-0.412635\pi$
$17$	2.23499	0.542066	0.271033	+	0.962570i	$0.412635\pi$
$18$	0	0
$19$	0.103324	0.0237041	0.0118520	−	0.999930i	$-0.496227\pi$
$19$	0.103324	0.0237041	0.0118520	+	0.999930i	$0.496227\pi$
$20$	0	0
$21$	−2.97233	−0.648616
$22$	0	0
$23$	−2.91620	−0.608069	−0.304035	−	0.952661i	$-0.598334\pi$
$23$	−2.91620	−0.608069	−0.304035	+	0.952661i	$0.598334\pi$
$24$	0	0
$25$	4.47248	0.894496
$26$	0	0
$27$	8.42585	1.62156
$28$	0	0
$29$	5.61031	1.04181	0.520904	−	0.853615i	$-0.325595\pi$
$29$	5.61031	1.04181	0.520904	+	0.853615i	$0.325595\pi$
$30$	0	0
$31$	−0.191615	−0.0344151	−0.0172075	−	0.999852i	$-0.505478\pi$
$31$	−0.191615	−0.0344151	−0.0172075	+	0.999852i	$0.505478\pi$
$32$	0	0
$33$	−2.97233	−0.517417
$34$	0	0
$35$	3.07774	0.520233
$36$	0	0
$37$	−6.04946	−0.994525	−0.497262	−	0.867600i	$-0.665661\pi$
$37$	−6.04946	−0.994525	−0.497262	+	0.867600i	$0.665661\pi$
$38$	0	0
$39$	−2.97233	−0.475954
$40$	0	0
$41$	−10.4530	−1.63248	−0.816240	−	0.577713i	$-0.803945\pi$
$41$	−10.4530	−1.63248	−0.816240	+	0.577713i	$0.803945\pi$
$42$	0	0
$43$	4.42785	0.675241	0.337621	−	0.941282i	$-0.390378\pi$
$43$	4.42785	0.675241	0.337621	+	0.941282i	$0.390378\pi$
$44$	0	0
$45$	−17.9579	−2.67700
$46$	0	0
$47$	−4.29716	−0.626805	−0.313402	−	0.949620i	$-0.601469\pi$
$47$	−4.29716	−0.626805	−0.313402	+	0.949620i	$0.601469\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.64315	0.930226
$52$	0	0
$53$	−0.962428	−0.132200	−0.0660998	−	0.997813i	$-0.521056\pi$
$53$	−0.962428	−0.132200	−0.0660998	+	0.997813i	$0.521056\pi$
$54$	0	0
$55$	3.07774	0.415002
$56$	0	0
$57$	0.307112	0.0406780
$58$	0	0
$59$	7.41338	0.965141	0.482570	−	0.875857i	$-0.339703\pi$
$59$	7.41338	0.965141	0.482570	+	0.875857i	$0.339703\pi$
$60$	0	0
$61$	−6.36668	−0.815169	−0.407585	−	0.913167i	$-0.633629\pi$
$61$	−6.36668	−0.815169	−0.407585	+	0.913167i	$0.633629\pi$
$62$	0	0
$63$	−5.83476	−0.735111
$64$	0	0
$65$	3.07774	0.381747
$66$	0	0
$67$	−10.9683	−1.33999	−0.669994	−	0.742367i	$-0.733703\pi$
$67$	−10.9683	−1.33999	−0.669994	+	0.742367i	$0.733703\pi$
$68$	0	0
$69$	−8.66791	−1.04349
$70$	0	0
$71$	0.646837	0.0767655	0.0383827	−	0.999263i	$-0.487779\pi$
$71$	0.646837	0.0767655	0.0383827	+	0.999263i	$0.487779\pi$
$72$	0	0
$73$	2.81140	0.329049	0.164525	−	0.986373i	$-0.447391\pi$
$73$	2.81140	0.329049	0.164525	+	0.986373i	$0.447391\pi$
$74$	0	0
$75$	13.2937	1.53502
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−13.8459	−1.55779	−0.778895	−	0.627155i	$-0.784219\pi$
$79$	−13.8459	−1.55779	−0.778895	+	0.627155i	$0.784219\pi$
$80$	0	0
$81$	7.54015	0.837795
$82$	0	0
$83$	1.79514	0.197042	0.0985210	−	0.995135i	$-0.468589\pi$
$83$	1.79514	0.197042	0.0985210	+	0.995135i	$0.468589\pi$
$84$	0	0
$85$	−6.87873	−0.746103
$86$	0	0
$87$	16.6757	1.78782
$88$	0	0
$89$	2.92179	0.309709	0.154855	−	0.987937i	$-0.450509\pi$
$89$	2.92179	0.309709	0.154855	+	0.987937i	$0.450509\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−0.569544	−0.0590589
$94$	0	0
$95$	−0.318003	−0.0326265
$96$	0	0
$97$	−1.39715	−0.141859	−0.0709296	−	0.997481i	$-0.522597\pi$
$97$	−1.39715	−0.141859	−0.0709296	+	0.997481i	$0.522597\pi$
$98$	0	0
$99$	−5.83476	−0.586416
$100$	0	0
$101$	−13.2388	−1.31731	−0.658656	−	0.752444i	$-0.728875\pi$
$101$	−13.2388	−1.31731	−0.658656	+	0.752444i	$0.728875\pi$
$102$	0	0
$103$	9.76380	0.962056	0.481028	−	0.876705i	$-0.340264\pi$
$103$	9.76380	0.962056	0.481028	+	0.876705i	$0.340264\pi$
$104$	0	0
$105$	9.14806	0.892760
$106$	0	0
$107$	5.03968	0.487204	0.243602	−	0.969875i	$-0.421671\pi$
$107$	5.03968	0.487204	0.243602	+	0.969875i	$0.421671\pi$
$108$	0	0
$109$	−10.5141	−1.00706	−0.503532	−	0.863977i	$-0.667966\pi$
$109$	−10.5141	−1.00706	−0.503532	+	0.863977i	$0.667966\pi$
$110$	0	0
$111$	−17.9810	−1.70668
$112$	0	0
$113$	−0.483443	−0.0454785	−0.0227393	−	0.999741i	$-0.507239\pi$
$113$	−0.483443	−0.0454785	−0.0227393	+	0.999741i	$0.507239\pi$
$114$	0	0
$115$	8.97529	0.836950
$116$	0	0
$117$	−5.83476	−0.539424
$118$	0	0
$119$	−2.23499	−0.204882
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−31.0697	−2.80146
$124$	0	0
$125$	1.62357	0.145217
$126$	0	0
$127$	−4.50838	−0.400054	−0.200027	−	0.979790i	$-0.564103\pi$
$127$	−4.50838	−0.400054	−0.200027	+	0.979790i	$0.564103\pi$
$128$	0	0
$129$	13.1611	1.15877
$130$	0	0
$131$	−10.8575	−0.948627	−0.474314	−	0.880356i	$-0.657304\pi$
$131$	−10.8575	−0.948627	−0.474314	+	0.880356i	$0.657304\pi$
$132$	0	0
$133$	−0.103324	−0.00895930
$134$	0	0
$135$	−25.9326	−2.23192
$136$	0	0
$137$	−11.5054	−0.982971	−0.491486	−	0.870886i	$-0.663546\pi$
$137$	−11.5054	−0.982971	−0.491486	+	0.870886i	$0.663546\pi$
$138$	0	0
$139$	−3.51308	−0.297976	−0.148988	−	0.988839i	$-0.547602\pi$
$139$	−3.51308	−0.297976	−0.148988	+	0.988839i	$0.547602\pi$
$140$	0	0
$141$	−12.7726	−1.07565
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−17.2671	−1.43395
$146$	0	0
$147$	2.97233	0.245154
$148$	0	0
$149$	−8.93487	−0.731973	−0.365986	−	0.930620i	$-0.619268\pi$
$149$	−8.93487	−0.731973	−0.365986	+	0.930620i	$0.619268\pi$
$150$	0	0
$151$	2.73619	0.222668	0.111334	−	0.993783i	$-0.464488\pi$
$151$	2.73619	0.222668	0.111334	+	0.993783i	$0.464488\pi$
$152$	0	0
$153$	13.0407	1.05427
$154$	0	0
$155$	0.589741	0.0473691
$156$	0	0
$157$	6.99619	0.558357	0.279178	−	0.960239i	$-0.409938\pi$
$157$	6.99619	0.558357	0.279178	+	0.960239i	$0.409938\pi$
$158$	0	0
$159$	−2.86066	−0.226865
$160$	0	0
$161$	2.91620	0.229828
$162$	0	0
$163$	−5.19927	−0.407238	−0.203619	−	0.979050i	$-0.565270\pi$
$163$	−5.19927	−0.407238	−0.203619	+	0.979050i	$0.565270\pi$
$164$	0	0
$165$	9.14806	0.712176
$166$	0	0
$167$	−22.5642	−1.74607	−0.873036	−	0.487656i	$-0.837852\pi$
$167$	−22.5642	−1.74607	−0.873036	+	0.487656i	$0.837852\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.602869	0.0461026
$172$	0	0
$173$	−23.1188	−1.75769	−0.878844	−	0.477109i	$-0.841685\pi$
$173$	−23.1188	−1.75769	−0.878844	+	0.477109i	$0.841685\pi$
$174$	0	0
$175$	−4.47248	−0.338088
$176$	0	0
$177$	22.0350	1.65626
$178$	0	0
$179$	−17.0313	−1.27298	−0.636488	−	0.771286i	$-0.719614\pi$
$179$	−17.0313	−1.27298	−0.636488	+	0.771286i	$0.719614\pi$
$180$	0	0
$181$	−2.10710	−0.156619	−0.0783097	−	0.996929i	$-0.524952\pi$
$181$	−2.10710	−0.156619	−0.0783097	+	0.996929i	$0.524952\pi$
$182$	0	0
$183$	−18.9239	−1.39889
$184$	0	0
$185$	18.6187	1.36887
$186$	0	0
$187$	−2.23499	−0.163439
$188$	0	0
$189$	−8.42585	−0.612891
$190$	0	0
$191$	−11.2206	−0.811893	−0.405947	−	0.913897i	$-0.633058\pi$
$191$	−11.2206	−0.811893	−0.405947	+	0.913897i	$0.633058\pi$
$192$	0	0
$193$	14.5364	1.04635	0.523177	−	0.852224i	$-0.324747\pi$
$193$	14.5364	1.04635	0.523177	+	0.852224i	$0.324747\pi$
$194$	0	0
$195$	9.14806	0.655107
$196$	0	0
$197$	10.4063	0.741417	0.370708	−	0.928749i	$-0.379115\pi$
$197$	10.4063	0.741417	0.370708	+	0.928749i	$0.379115\pi$
$198$	0	0
$199$	12.4838	0.884954	0.442477	−	0.896780i	$-0.354100\pi$
$199$	12.4838	0.884954	0.442477	+	0.896780i	$0.354100\pi$
$200$	0	0
$201$	−32.6013	−2.29952
$202$	0	0
$203$	−5.61031	−0.393766
$204$	0	0
$205$	32.1715	2.24696
$206$	0	0
$207$	−17.0153	−1.18265
$208$	0	0
$209$	−0.103324	−0.00714705
$210$	0	0
$211$	0.779158	0.0536394	0.0268197	−	0.999640i	$-0.491462\pi$
$211$	0.779158	0.0536394	0.0268197	+	0.999640i	$0.491462\pi$
$212$	0	0
$213$	1.92262	0.131735
$214$	0	0
$215$	−13.6278	−0.929407
$216$	0	0
$217$	0.191615	0.0130077
$218$	0	0
$219$	8.35641	0.564674
$220$	0	0
$221$	−2.23499	−0.150342
$222$	0	0
$223$	−16.3416	−1.09432	−0.547158	−	0.837030i	$-0.684290\pi$
$223$	−16.3416	−1.09432	−0.547158	+	0.837030i	$0.684290\pi$
$224$	0	0
$225$	26.0958	1.73972
$226$	0	0
$227$	−11.7555	−0.780238	−0.390119	−	0.920764i	$-0.627566\pi$
$227$	−11.7555	−0.780238	−0.390119	+	0.920764i	$0.627566\pi$
$228$	0	0
$229$	10.0978	0.667283	0.333641	−	0.942700i	$-0.391723\pi$
$229$	10.0978	0.667283	0.333641	+	0.942700i	$0.391723\pi$
$230$	0	0
$231$	2.97233	0.195565
$232$	0	0
$233$	20.7608	1.36009	0.680043	−	0.733172i	$-0.261961\pi$
$233$	20.7608	1.36009	0.680043	+	0.733172i	$0.261961\pi$
$234$	0	0
$235$	13.2255	0.862739
$236$	0	0
$237$	−41.1547	−2.67329
$238$	0	0
$239$	−13.1024	−0.847521	−0.423761	−	0.905774i	$-0.639290\pi$
$239$	−13.1024	−0.847521	−0.423761	+	0.905774i	$0.639290\pi$
$240$	0	0
$241$	5.60987	0.361363	0.180682	−	0.983542i	$-0.442170\pi$
$241$	5.60987	0.361363	0.180682	+	0.983542i	$0.442170\pi$
$242$	0	0
$243$	−2.86572	−0.183836
$244$	0	0
$245$	−3.07774	−0.196630
$246$	0	0
$247$	−0.103324	−0.00657433
$248$	0	0
$249$	5.33575	0.338139
$250$	0	0
$251$	−17.8824	−1.12872	−0.564362	−	0.825527i	$-0.690878\pi$
$251$	−17.8824	−1.12872	−0.564362	+	0.825527i	$0.690878\pi$
$252$	0	0
$253$	2.91620	0.183340
$254$	0	0
$255$	−20.4459	−1.28037
$256$	0	0
$257$	−18.0040	−1.12306	−0.561529	−	0.827457i	$-0.689787\pi$
$257$	−18.0040	−1.12306	−0.561529	+	0.827457i	$0.689787\pi$
$258$	0	0
$259$	6.04946	0.375895
$260$	0	0
$261$	32.7348	2.02623
$262$	0	0
$263$	21.0315	1.29686	0.648429	−	0.761275i	$-0.275427\pi$
$263$	21.0315	1.29686	0.648429	+	0.761275i	$0.275427\pi$
$264$	0	0
$265$	2.96210	0.181961
$266$	0	0
$267$	8.68453	0.531485
$268$	0	0
$269$	21.5117	1.31159	0.655795	−	0.754939i	$-0.272333\pi$
$269$	21.5117	1.31159	0.655795	+	0.754939i	$0.272333\pi$
$270$	0	0
$271$	2.67313	0.162381	0.0811907	−	0.996699i	$-0.474128\pi$
$271$	2.67313	0.162381	0.0811907	+	0.996699i	$0.474128\pi$
$272$	0	0
$273$	2.97233	0.179894
$274$	0	0
$275$	−4.47248	−0.269701
$276$	0	0
$277$	7.37325	0.443016	0.221508	−	0.975159i	$-0.428902\pi$
$277$	7.37325	0.443016	0.221508	+	0.975159i	$0.428902\pi$
$278$	0	0
$279$	−1.11803	−0.0669346
$280$	0	0
$281$	−5.42569	−0.323669	−0.161835	−	0.986818i	$-0.551741\pi$
$281$	−5.42569	−0.323669	−0.161835	+	0.986818i	$0.551741\pi$
$282$	0	0
$283$	13.7443	0.817016	0.408508	−	0.912755i	$-0.366049\pi$
$283$	13.7443	0.817016	0.408508	+	0.912755i	$0.366049\pi$
$284$	0	0
$285$	−0.945212	−0.0559895
$286$	0	0
$287$	10.4530	0.617019
$288$	0	0
$289$	−12.0048	−0.706165
$290$	0	0
$291$	−4.15280	−0.243441
$292$	0	0
$293$	5.02676	0.293667	0.146833	−	0.989161i	$-0.453092\pi$
$293$	5.02676	0.293667	0.146833	+	0.989161i	$0.453092\pi$
$294$	0	0
$295$	−22.8165	−1.32843
$296$	0	0
$297$	−8.42585	−0.488917
$298$	0	0
$299$	2.91620	0.168648
$300$	0	0
$301$	−4.42785	−0.255217
$302$	0	0
$303$	−39.3502	−2.26061
$304$	0	0
$305$	19.5950	1.12200
$306$	0	0
$307$	34.6980	1.98032	0.990161	−	0.139935i	$-0.0446893\pi$
$307$	34.6980	1.98032	0.990161	+	0.139935i	$0.0446893\pi$
$308$	0	0
$309$	29.0213	1.65096
$310$	0	0
$311$	15.7071	0.890670	0.445335	−	0.895364i	$-0.353085\pi$
$311$	15.7071	0.890670	0.445335	+	0.895364i	$0.353085\pi$
$312$	0	0
$313$	−12.3047	−0.695503	−0.347752	−	0.937587i	$-0.613055\pi$
$313$	−12.3047	−0.695503	−0.347752	+	0.937587i	$0.613055\pi$
$314$	0	0
$315$	17.9579	1.01181
$316$	0	0
$317$	−18.4765	−1.03774	−0.518872	−	0.854852i	$-0.673648\pi$
$317$	−18.4765	−1.03774	−0.518872	+	0.854852i	$0.673648\pi$
$318$	0	0
$319$	−5.61031	−0.314117
$320$	0	0
$321$	14.9796	0.836080
$322$	0	0
$323$	0.230928	0.0128492
$324$	0	0
$325$	−4.47248	−0.248088
$326$	0	0
$327$	−31.2513	−1.72820
$328$	0	0
$329$	4.29716	0.236910
$330$	0	0
$331$	−17.5194	−0.962952	−0.481476	−	0.876459i	$-0.659899\pi$
$331$	−17.5194	−0.962952	−0.481476	+	0.876459i	$0.659899\pi$
$332$	0	0
$333$	−35.2971	−1.93427
$334$	0	0
$335$	33.7575	1.84437
$336$	0	0
$337$	15.4363	0.840870	0.420435	−	0.907323i	$-0.361877\pi$
$337$	15.4363	0.840870	0.420435	+	0.907323i	$0.361877\pi$
$338$	0	0
$339$	−1.43695	−0.0780446
$340$	0	0
$341$	0.191615	0.0103765
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	26.6776	1.43627
$346$	0	0
$347$	12.6121	0.677055	0.338527	−	0.940956i	$-0.390071\pi$
$347$	12.6121	0.677055	0.338527	+	0.940956i	$0.390071\pi$
$348$	0	0
$349$	21.0583	1.12722	0.563612	−	0.826040i	$-0.309411\pi$
$349$	21.0583	1.12722	0.563612	+	0.826040i	$0.309411\pi$
$350$	0	0
$351$	−8.42585	−0.449739
$352$	0	0
$353$	−3.45477	−0.183879	−0.0919394	−	0.995765i	$-0.529307\pi$
$353$	−3.45477	−0.183879	−0.0919394	+	0.995765i	$0.529307\pi$
$354$	0	0
$355$	−1.99080	−0.105661
$356$	0	0
$357$	−6.64315	−0.351593
$358$	0	0
$359$	−1.30709	−0.0689855	−0.0344927	−	0.999405i	$-0.510982\pi$
$359$	−1.30709	−0.0689855	−0.0344927	+	0.999405i	$0.510982\pi$
$360$	0	0
$361$	−18.9893	−0.999438
$362$	0	0
$363$	2.97233	0.156007
$364$	0	0
$365$	−8.65275	−0.452906
$366$	0	0
$367$	2.43609	0.127163	0.0635813	−	0.997977i	$-0.479748\pi$
$367$	2.43609	0.127163	0.0635813	+	0.997977i	$0.479748\pi$
$368$	0	0
$369$	−60.9906	−3.17504
$370$	0	0
$371$	0.962428	0.0499668
$372$	0	0
$373$	−13.6475	−0.706640	−0.353320	−	0.935502i	$-0.614947\pi$
$373$	−13.6475	−0.706640	−0.353320	+	0.935502i	$0.614947\pi$
$374$	0	0
$375$	4.82580	0.249203
$376$	0	0
$377$	−5.61031	−0.288945
$378$	0	0
$379$	−16.0155	−0.822662	−0.411331	−	0.911486i	$-0.634936\pi$
$379$	−16.0155	−0.822662	−0.411331	+	0.911486i	$0.634936\pi$
$380$	0	0
$381$	−13.4004	−0.686524
$382$	0	0
$383$	−36.6095	−1.87066	−0.935330	−	0.353777i	$-0.884897\pi$
$383$	−36.6095	−1.87066	−0.935330	+	0.353777i	$0.884897\pi$
$384$	0	0
$385$	−3.07774	−0.156856
$386$	0	0
$387$	25.8355	1.31329
$388$	0	0
$389$	−16.3212	−0.827517	−0.413758	−	0.910387i	$-0.635784\pi$
$389$	−16.3212	−0.827517	−0.413758	+	0.910387i	$0.635784\pi$
$390$	0	0
$391$	−6.51768	−0.329613
$392$	0	0
$393$	−32.2722	−1.62792
$394$	0	0
$395$	42.6142	2.14415
$396$	0	0
$397$	−10.2818	−0.516031	−0.258016	−	0.966141i	$-0.583069\pi$
$397$	−10.2818	−0.516031	−0.258016	+	0.966141i	$0.583069\pi$
$398$	0	0
$399$	−0.307112	−0.0153749
$400$	0	0
$401$	−18.5310	−0.925393	−0.462697	−	0.886517i	$-0.653118\pi$
$401$	−18.5310	−0.925393	−0.462697	+	0.886517i	$0.653118\pi$
$402$	0	0
$403$	0.191615	0.00954502
$404$	0	0
$405$	−23.2066	−1.15315
$406$	0	0
$407$	6.04946	0.299860
$408$	0	0
$409$	30.9073	1.52827	0.764135	−	0.645057i	$-0.223166\pi$
$409$	30.9073	1.52827	0.764135	+	0.645057i	$0.223166\pi$
$410$	0	0
$411$	−34.1978	−1.68685
$412$	0	0
$413$	−7.41338	−0.364789
$414$	0	0
$415$	−5.52497	−0.271210
$416$	0	0
$417$	−10.4421	−0.511349
$418$	0	0
$419$	2.04604	0.0999554	0.0499777	−	0.998750i	$-0.484085\pi$
$419$	2.04604	0.0999554	0.0499777	+	0.998750i	$0.484085\pi$
$420$	0	0
$421$	−17.9547	−0.875056	−0.437528	−	0.899205i	$-0.644146\pi$
$421$	−17.9547	−0.875056	−0.437528	+	0.899205i	$0.644146\pi$
$422$	0	0
$423$	−25.0729	−1.21909
$424$	0	0
$425$	9.99596	0.484875
$426$	0	0
$427$	6.36668	0.308105
$428$	0	0
$429$	2.97233	0.143506
$430$	0	0
$431$	−31.2834	−1.50687	−0.753434	−	0.657523i	$-0.771604\pi$
$431$	−31.2834	−1.50687	−0.753434	+	0.657523i	$0.771604\pi$
$432$	0	0
$433$	2.28678	0.109896	0.0549478	−	0.998489i	$-0.482501\pi$
$433$	2.28678	0.109896	0.0549478	+	0.998489i	$0.482501\pi$
$434$	0	0
$435$	−51.3234	−2.46077
$436$	0	0
$437$	−0.301312	−0.0144137
$438$	0	0
$439$	4.92870	0.235234	0.117617	−	0.993059i	$-0.462474\pi$
$439$	4.92870	0.235234	0.117617	+	0.993059i	$0.462474\pi$
$440$	0	0
$441$	5.83476	0.277846
$442$	0	0
$443$	−10.1997	−0.484602	−0.242301	−	0.970201i	$-0.577902\pi$
$443$	−10.1997	−0.484602	−0.242301	+	0.970201i	$0.577902\pi$
$444$	0	0
$445$	−8.99251	−0.426286
$446$	0	0
$447$	−26.5574	−1.25612
$448$	0	0
$449$	−12.7827	−0.603254	−0.301627	−	0.953426i	$-0.597530\pi$
$449$	−12.7827	−0.603254	−0.301627	+	0.953426i	$0.597530\pi$
$450$	0	0
$451$	10.4530	0.492211
$452$	0	0
$453$	8.13287	0.382116
$454$	0	0
$455$	−3.07774	−0.144287
$456$	0	0
$457$	−24.0107	−1.12317	−0.561587	−	0.827418i	$-0.689809\pi$
$457$	−24.0107	−1.12317	−0.561587	+	0.827418i	$0.689809\pi$
$458$	0	0
$459$	18.8317	0.878990
$460$	0	0
$461$	−5.33623	−0.248533	−0.124267	−	0.992249i	$-0.539658\pi$
$461$	−5.33623	−0.248533	−0.124267	+	0.992249i	$0.539658\pi$
$462$	0	0
$463$	−20.3381	−0.945192	−0.472596	−	0.881279i	$-0.656683\pi$
$463$	−20.3381	−0.945192	−0.472596	+	0.881279i	$0.656683\pi$
$464$	0	0
$465$	1.75291	0.0812891
$466$	0	0
$467$	−11.5450	−0.534240	−0.267120	−	0.963663i	$-0.586072\pi$
$467$	−11.5450	−0.534240	−0.267120	+	0.963663i	$0.586072\pi$
$468$	0	0
$469$	10.9683	0.506468
$470$	0	0
$471$	20.7950	0.958183
$472$	0	0
$473$	−4.42785	−0.203593
$474$	0	0
$475$	0.462113	0.0212032
$476$	0	0
$477$	−5.61554	−0.257118
$478$	0	0
$479$	29.8689	1.36475	0.682373	−	0.731004i	$-0.260948\pi$
$479$	29.8689	1.36475	0.682373	+	0.731004i	$0.260948\pi$
$480$	0	0
$481$	6.04946	0.275831
$482$	0	0
$483$	8.66791	0.394403
$484$	0	0
$485$	4.30007	0.195256
$486$	0	0
$487$	−17.4372	−0.790155	−0.395077	−	0.918648i	$-0.629282\pi$
$487$	−17.4372	−0.790155	−0.395077	+	0.918648i	$0.629282\pi$
$488$	0	0
$489$	−15.4540	−0.698852
$490$	0	0
$491$	8.31828	0.375399	0.187699	−	0.982227i	$-0.439897\pi$
$491$	8.31828	0.375399	0.187699	+	0.982227i	$0.439897\pi$
$492$	0	0
$493$	12.5390	0.564728
$494$	0	0
$495$	17.9579	0.807146
$496$	0	0
$497$	−0.646837	−0.0290146
$498$	0	0
$499$	25.9215	1.16041	0.580204	−	0.814471i	$-0.302973\pi$
$499$	25.9215	1.16041	0.580204	+	0.814471i	$0.302973\pi$
$500$	0	0
$501$	−67.0684	−2.99639
$502$	0	0
$503$	−25.0915	−1.11878	−0.559388	−	0.828906i	$-0.688964\pi$
$503$	−25.0915	−1.11878	−0.559388	+	0.828906i	$0.688964\pi$
$504$	0	0
$505$	40.7457	1.81316
$506$	0	0
$507$	2.97233	0.132006
$508$	0	0
$509$	15.7067	0.696189	0.348094	−	0.937460i	$-0.386829\pi$
$509$	15.7067	0.696189	0.348094	+	0.937460i	$0.386829\pi$
$510$	0	0
$511$	−2.81140	−0.124369
$512$	0	0
$513$	0.870590	0.0384375
$514$	0	0
$515$	−30.0504	−1.32418
$516$	0	0
$517$	4.29716	0.188989
$518$	0	0
$519$	−68.7167	−3.01633
$520$	0	0
$521$	−5.67235	−0.248510	−0.124255	−	0.992250i	$-0.539654\pi$
$521$	−5.67235	−0.248510	−0.124255	+	0.992250i	$0.539654\pi$
$522$	0	0
$523$	17.4581	0.763388	0.381694	−	0.924289i	$-0.375341\pi$
$523$	17.4581	0.763388	0.381694	+	0.924289i	$0.375341\pi$
$524$	0	0
$525$	−13.2937	−0.580184
$526$	0	0
$527$	−0.428258	−0.0186552
$528$	0	0
$529$	−14.4958	−0.630252
$530$	0	0
$531$	43.2553	1.87712
$532$	0	0
$533$	10.4530	0.452768
$534$	0	0
$535$	−15.5108	−0.670592
$536$	0	0
$537$	−50.6226	−2.18453
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	34.7310	1.49320	0.746602	−	0.665271i	$-0.231684\pi$
$541$	34.7310	1.49320	0.746602	+	0.665271i	$0.231684\pi$
$542$	0	0
$543$	−6.26299	−0.268771
$544$	0	0
$545$	32.3595	1.38613
$546$	0	0
$547$	4.87867	0.208597	0.104298	−	0.994546i	$-0.466740\pi$
$547$	4.87867	0.208597	0.104298	+	0.994546i	$0.466740\pi$
$548$	0	0
$549$	−37.1480	−1.58544
$550$	0	0
$551$	0.579678	0.0246951
$552$	0	0
$553$	13.8459	0.588789
$554$	0	0
$555$	55.3408	2.34909
$556$	0	0
$557$	42.6830	1.80854	0.904269	−	0.426962i	$-0.140416\pi$
$557$	42.6830	1.80854	0.904269	+	0.426962i	$0.140416\pi$
$558$	0	0
$559$	−4.42785	−0.187278
$560$	0	0
$561$	−6.64315	−0.280474
$562$	0	0
$563$	−9.22792	−0.388911	−0.194455	−	0.980911i	$-0.562294\pi$
$563$	−9.22792	−0.388911	−0.194455	+	0.980911i	$0.562294\pi$
$564$	0	0
$565$	1.48791	0.0625969
$566$	0	0
$567$	−7.54015	−0.316657
$568$	0	0
$569$	39.3733	1.65061	0.825307	−	0.564684i	$-0.191002\pi$
$569$	39.3733	1.65061	0.825307	+	0.564684i	$0.191002\pi$
$570$	0	0
$571$	36.5144	1.52808	0.764039	−	0.645170i	$-0.223213\pi$
$571$	36.5144	1.52808	0.764039	+	0.645170i	$0.223213\pi$
$572$	0	0
$573$	−33.3513	−1.39327
$574$	0	0
$575$	−13.0426	−0.543915
$576$	0	0
$577$	7.24618	0.301663	0.150831	−	0.988560i	$-0.451805\pi$
$577$	7.24618	0.301663	0.150831	+	0.988560i	$0.451805\pi$
$578$	0	0
$579$	43.2070	1.79562
$580$	0	0
$581$	−1.79514	−0.0744749
$582$	0	0
$583$	0.962428	0.0398597
$584$	0	0
$585$	17.9579	0.742467
$586$	0	0
$587$	−11.6928	−0.482612	−0.241306	−	0.970449i	$-0.577576\pi$
$587$	−11.6928	−0.482612	−0.241306	+	0.970449i	$0.577576\pi$
$588$	0	0
$589$	−0.0197984	−0.000815778 0
$590$	0	0
$591$	30.9309	1.27233
$592$	0	0
$593$	36.5638	1.50150	0.750748	−	0.660588i	$-0.229693\pi$
$593$	36.5638	1.50150	0.750748	+	0.660588i	$0.229693\pi$
$594$	0	0
$595$	6.87873	0.282000
$596$	0	0
$597$	37.1061	1.51865
$598$	0	0
$599$	36.3081	1.48351	0.741754	−	0.670672i	$-0.233994\pi$
$599$	36.3081	1.48351	0.741754	+	0.670672i	$0.233994\pi$
$600$	0	0
$601$	−0.848823	−0.0346242	−0.0173121	−	0.999850i	$-0.505511\pi$
$601$	−0.848823	−0.0346242	−0.0173121	+	0.999850i	$0.505511\pi$
$602$	0	0
$603$	−63.9972	−2.60617
$604$	0	0
$605$	−3.07774	−0.125128
$606$	0	0
$607$	33.9714	1.37886	0.689429	−	0.724353i	$-0.257862\pi$
$607$	33.9714	1.37886	0.689429	+	0.724353i	$0.257862\pi$
$608$	0	0
$609$	−16.6757	−0.675733
$610$	0	0
$611$	4.29716	0.173844
$612$	0	0
$613$	24.9217	1.00658	0.503288	−	0.864119i	$-0.332123\pi$
$613$	24.9217	1.00658	0.503288	+	0.864119i	$0.332123\pi$
$614$	0	0
$615$	95.6245	3.85595
$616$	0	0
$617$	35.9790	1.44846	0.724229	−	0.689559i	$-0.242196\pi$
$617$	35.9790	1.44846	0.724229	+	0.689559i	$0.242196\pi$
$618$	0	0
$619$	31.4024	1.26217	0.631085	−	0.775714i	$-0.282610\pi$
$619$	31.4024	1.26217	0.631085	+	0.775714i	$0.282610\pi$
$620$	0	0
$621$	−24.5714	−0.986018
$622$	0	0
$623$	−2.92179	−0.117059
$624$	0	0
$625$	−27.3593	−1.09437
$626$	0	0
$627$	−0.307112	−0.0122649
$628$	0	0
$629$	−13.5205	−0.539098
$630$	0	0
$631$	11.1345	0.443256	0.221628	−	0.975131i	$-0.428863\pi$
$631$	11.1345	0.443256	0.221628	+	0.975131i	$0.428863\pi$
$632$	0	0
$633$	2.31592	0.0920494
$634$	0	0
$635$	13.8756	0.550637
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	3.77414	0.149303
$640$	0	0
$641$	−9.52274	−0.376126	−0.188063	−	0.982157i	$-0.560221\pi$
$641$	−9.52274	−0.376126	−0.188063	+	0.982157i	$0.560221\pi$
$642$	0	0
$643$	24.4989	0.966142	0.483071	−	0.875581i	$-0.339521\pi$
$643$	24.4989	0.966142	0.483071	+	0.875581i	$0.339521\pi$
$644$	0	0
$645$	−40.5063	−1.59493
$646$	0	0
$647$	27.8307	1.09414	0.547069	−	0.837087i	$-0.315743\pi$
$647$	27.8307	1.09414	0.547069	+	0.837087i	$0.315743\pi$
$648$	0	0
$649$	−7.41338	−0.291001
$650$	0	0
$651$	0.569544	0.0223222
$652$	0	0
$653$	−7.00807	−0.274247	−0.137123	−	0.990554i	$-0.543786\pi$
$653$	−7.00807	−0.274247	−0.137123	+	0.990554i	$0.543786\pi$
$654$	0	0
$655$	33.4167	1.30570
$656$	0	0
$657$	16.4038	0.639975
$658$	0	0
$659$	−21.7284	−0.846419	−0.423209	−	0.906032i	$-0.639097\pi$
$659$	−21.7284	−0.846419	−0.423209	+	0.906032i	$0.639097\pi$
$660$	0	0
$661$	−23.5817	−0.917223	−0.458612	−	0.888637i	$-0.651653\pi$
$661$	−23.5817	−0.917223	−0.458612	+	0.888637i	$0.651653\pi$
$662$	0	0
$663$	−6.64315	−0.257998
$664$	0	0
$665$	0.318003	0.0123316
$666$	0	0
$667$	−16.3608	−0.633491
$668$	0	0
$669$	−48.5727	−1.87793
$670$	0	0
$671$	6.36668	0.245783
$672$	0	0
$673$	−6.49704	−0.250443	−0.125221	−	0.992129i	$-0.539964\pi$
$673$	−6.49704	−0.250443	−0.125221	+	0.992129i	$0.539964\pi$
$674$	0	0
$675$	37.6844	1.45047
$676$	0	0
$677$	−21.8837	−0.841061	−0.420530	−	0.907279i	$-0.638156\pi$
$677$	−21.8837	−0.841061	−0.420530	+	0.907279i	$0.638156\pi$
$678$	0	0
$679$	1.39715	0.0536177
$680$	0	0
$681$	−34.9412	−1.33895
$682$	0	0
$683$	14.9527	0.572148	0.286074	−	0.958207i	$-0.407650\pi$
$683$	14.9527	0.572148	0.286074	+	0.958207i	$0.407650\pi$
$684$	0	0
$685$	35.4106	1.35297
$686$	0	0
$687$	30.0141	1.14511
$688$	0	0
$689$	0.962428	0.0366656
$690$	0	0
$691$	17.8548	0.679229	0.339615	−	0.940565i	$-0.389703\pi$
$691$	17.8548	0.679229	0.339615	+	0.940565i	$0.389703\pi$
$692$	0	0
$693$	5.83476	0.221644
$694$	0	0
$695$	10.8124	0.410136
$696$	0	0
$697$	−23.3623	−0.884911
$698$	0	0
$699$	61.7080	2.33401
$700$	0	0
$701$	−26.1625	−0.988143	−0.494071	−	0.869421i	$-0.664492\pi$
$701$	−26.1625	−0.988143	−0.494071	+	0.869421i	$0.664492\pi$
$702$	0	0
$703$	−0.625052	−0.0235743
$704$	0	0
$705$	39.3107	1.48053
$706$	0	0
$707$	13.2388	0.497897
$708$	0	0
$709$	38.3064	1.43863	0.719314	−	0.694685i	$-0.244456\pi$
$709$	38.3064	1.43863	0.719314	+	0.694685i	$0.244456\pi$
$710$	0	0
$711$	−80.7877	−3.02978
$712$	0	0
$713$	0.558787	0.0209267
$714$	0	0
$715$	−3.07774	−0.115101
$716$	0	0
$717$	−38.9446	−1.45441
$718$	0	0
$719$	23.2718	0.867893	0.433946	−	0.900939i	$-0.357121\pi$
$719$	23.2718	0.867893	0.433946	+	0.900939i	$0.357121\pi$
$720$	0	0
$721$	−9.76380	−0.363623
$722$	0	0
$723$	16.6744	0.620127
$724$	0	0
$725$	25.0920	0.931893
$726$	0	0
$727$	−39.2543	−1.45586	−0.727931	−	0.685650i	$-0.759518\pi$
$727$	−39.2543	−1.45586	−0.727931	+	0.685650i	$0.759518\pi$
$728$	0	0
$729$	−31.1383	−1.15327
$730$	0	0
$731$	9.89623	0.366025
$732$	0	0
$733$	14.0120	0.517544	0.258772	−	0.965938i	$-0.416682\pi$
$733$	14.0120	0.517544	0.258772	+	0.965938i	$0.416682\pi$
$734$	0	0
$735$	−9.14806	−0.337431
$736$	0	0
$737$	10.9683	0.404021
$738$	0	0
$739$	26.3673	0.969935	0.484968	−	0.874532i	$-0.338831\pi$
$739$	26.3673	0.969935	0.484968	+	0.874532i	$0.338831\pi$
$740$	0	0
$741$	−0.307112	−0.0112821
$742$	0	0
$743$	−7.72910	−0.283553	−0.141777	−	0.989899i	$-0.545281\pi$
$743$	−7.72910	−0.283553	−0.141777	+	0.989899i	$0.545281\pi$
$744$	0	0
$745$	27.4992	1.00749
$746$	0	0
$747$	10.4742	0.383231
$748$	0	0
$749$	−5.03968	−0.184146
$750$	0	0
$751$	28.5477	1.04172	0.520860	−	0.853642i	$-0.325611\pi$
$751$	28.5477	1.04172	0.520860	+	0.853642i	$0.325611\pi$
$752$	0	0
$753$	−53.1523	−1.93698
$754$	0	0
$755$	−8.42129	−0.306482
$756$	0	0
$757$	12.4235	0.451541	0.225770	−	0.974181i	$-0.427510\pi$
$757$	12.4235	0.451541	0.225770	+	0.974181i	$0.427510\pi$
$758$	0	0
$759$	8.66791	0.314625
$760$	0	0
$761$	−40.9895	−1.48587	−0.742934	−	0.669365i	$-0.766566\pi$
$761$	−40.9895	−1.48587	−0.742934	+	0.669365i	$0.766566\pi$
$762$	0	0
$763$	10.5141	0.380635
$764$	0	0
$765$	−40.1357	−1.45111
$766$	0	0
$767$	−7.41338	−0.267682
$768$	0	0
$769$	11.9596	0.431276	0.215638	−	0.976473i	$-0.430817\pi$
$769$	11.9596	0.431276	0.215638	+	0.976473i	$0.430817\pi$
$770$	0	0
$771$	−53.5139	−1.92726
$772$	0	0
$773$	30.1629	1.08488	0.542442	−	0.840093i	$-0.317500\pi$
$773$	30.1629	1.08488	0.542442	+	0.840093i	$0.317500\pi$
$774$	0	0
$775$	−0.856994	−0.0307841
$776$	0	0
$777$	17.9810	0.645065
$778$	0	0
$779$	−1.08004	−0.0386964
$780$	0	0
$781$	−0.646837	−0.0231457
$782$	0	0
$783$	47.2716	1.68935
$784$	0	0
$785$	−21.5325	−0.768526
$786$	0	0
$787$	−25.0804	−0.894018	−0.447009	−	0.894529i	$-0.647511\pi$
$787$	−25.0804	−0.894018	−0.447009	+	0.894529i	$0.647511\pi$
$788$	0	0
$789$	62.5126	2.22551
$790$	0	0
$791$	0.483443	0.0171893
$792$	0	0
$793$	6.36668	0.226087
$794$	0	0
$795$	8.80436	0.312258
$796$	0	0
$797$	8.08469	0.286374	0.143187	−	0.989696i	$-0.454265\pi$
$797$	8.08469	0.286374	0.143187	+	0.989696i	$0.454265\pi$
$798$	0	0
$799$	−9.60412	−0.339769
$800$	0	0
$801$	17.0479	0.602360
$802$	0	0
$803$	−2.81140	−0.0992121
$804$	0	0
$805$	−8.97529	−0.316337
$806$	0	0
$807$	63.9399	2.25079
$808$	0	0
$809$	−21.0463	−0.739948	−0.369974	−	0.929042i	$-0.620633\pi$
$809$	−21.0463	−0.739948	−0.369974	+	0.929042i	$0.620633\pi$
$810$	0	0
$811$	44.2012	1.55211	0.776057	−	0.630663i	$-0.217217\pi$
$811$	44.2012	1.55211	0.776057	+	0.630663i	$0.217217\pi$
$812$	0	0
$813$	7.94545	0.278659
$814$	0	0
$815$	16.0020	0.560525
$816$	0	0
$817$	0.457502	0.0160060
$818$	0	0
$819$	5.83476	0.203883
$820$	0	0
$821$	19.4852	0.680039	0.340020	−	0.940418i	$-0.389566\pi$
$821$	19.4852	0.680039	0.340020	+	0.940418i	$0.389566\pi$
$822$	0	0
$823$	−8.93217	−0.311356	−0.155678	−	0.987808i	$-0.549756\pi$
$823$	−8.93217	−0.311356	−0.155678	+	0.987808i	$0.549756\pi$
$824$	0	0
$825$	−13.2937	−0.462827
$826$	0	0
$827$	−13.5672	−0.471779	−0.235890	−	0.971780i	$-0.575800\pi$
$827$	−13.5672	−0.471779	−0.235890	+	0.971780i	$0.575800\pi$
$828$	0	0
$829$	−44.8746	−1.55856	−0.779279	−	0.626677i	$-0.784415\pi$
$829$	−44.8746	−1.55856	−0.779279	+	0.626677i	$0.784415\pi$
$830$	0	0
$831$	21.9158	0.760249
$832$	0	0
$833$	2.23499	0.0774380
$834$	0	0
$835$	69.4468	2.40331
$836$	0	0
$837$	−1.61452	−0.0558060
$838$	0	0
$839$	11.8501	0.409111	0.204556	−	0.978855i	$-0.434425\pi$
$839$	11.8501	0.409111	0.204556	+	0.978855i	$0.434425\pi$
$840$	0	0
$841$	2.47554	0.0853635
$842$	0	0
$843$	−16.1270	−0.555442
$844$	0	0
$845$	−3.07774	−0.105877
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	40.8527	1.40206
$850$	0	0
$851$	17.6414	0.604740
$852$	0	0
$853$	14.0146	0.479850	0.239925	−	0.970791i	$-0.422877\pi$
$853$	14.0146	0.479850	0.239925	+	0.970791i	$0.422877\pi$
$854$	0	0
$855$	−1.85547	−0.0634559
$856$	0	0
$857$	−44.1020	−1.50650	−0.753248	−	0.657737i	$-0.771514\pi$
$857$	−44.1020	−1.50650	−0.753248	+	0.657737i	$0.771514\pi$
$858$	0	0
$859$	46.0462	1.57108	0.785539	−	0.618812i	$-0.212386\pi$
$859$	46.0462	1.57108	0.785539	+	0.618812i	$0.212386\pi$
$860$	0	0
$861$	31.0697	1.05885
$862$	0	0
$863$	19.6060	0.667396	0.333698	−	0.942680i	$-0.391703\pi$
$863$	19.6060	0.667396	0.333698	+	0.942680i	$0.391703\pi$
$864$	0	0
$865$	71.1536	2.41929
$866$	0	0
$867$	−35.6823	−1.21183
$868$	0	0
$869$	13.8459	0.469691
$870$	0	0
$871$	10.9683	0.371646
$872$	0	0
$873$	−8.15204	−0.275905
$874$	0	0
$875$	−1.62357	−0.0548868
$876$	0	0
$877$	−31.2707	−1.05594	−0.527968	−	0.849264i	$-0.677046\pi$
$877$	−31.2707	−1.05594	−0.527968	+	0.849264i	$0.677046\pi$
$878$	0	0
$879$	14.9412	0.503954
$880$	0	0
$881$	−44.1429	−1.48721	−0.743606	−	0.668618i	$-0.766886\pi$
$881$	−44.1429	−1.48721	−0.743606	+	0.668618i	$0.766886\pi$
$882$	0	0
$883$	5.06210	0.170353	0.0851767	−	0.996366i	$-0.472855\pi$
$883$	5.06210	0.170353	0.0851767	+	0.996366i	$0.472855\pi$
$884$	0	0
$885$	−67.8181	−2.27968
$886$	0	0
$887$	8.15920	0.273959	0.136980	−	0.990574i	$-0.456261\pi$
$887$	8.15920	0.273959	0.136980	+	0.990574i	$0.456261\pi$
$888$	0	0
$889$	4.50838	0.151206
$890$	0	0
$891$	−7.54015	−0.252605
$892$	0	0
$893$	−0.443998	−0.0148578
$894$	0	0
$895$	52.4178	1.75213
$896$	0	0
$897$	8.66791	0.289413
$898$	0	0
$899$	−1.07502	−0.0358539
$900$	0	0
$901$	−2.15102	−0.0716609
$902$	0	0
$903$	−13.1611	−0.437972
$904$	0	0
$905$	6.48510	0.215572
$906$	0	0
$907$	33.9941	1.12876	0.564378	−	0.825517i	$-0.309116\pi$
$907$	33.9941	1.12876	0.564378	+	0.825517i	$0.309116\pi$
$908$	0	0
$909$	−77.2454	−2.56207
$910$	0	0
$911$	−55.5974	−1.84202	−0.921012	−	0.389534i	$-0.872636\pi$
$911$	−55.5974	−1.84202	−0.921012	+	0.389534i	$0.872636\pi$
$912$	0	0
$913$	−1.79514	−0.0594104
$914$	0	0
$915$	58.2428	1.92545
$916$	0	0
$917$	10.8575	0.358547
$918$	0	0
$919$	−11.5725	−0.381741	−0.190870	−	0.981615i	$-0.561131\pi$
$919$	−11.5725	−0.381741	−0.190870	+	0.981615i	$0.561131\pi$
$920$	0	0
$921$	103.134	3.39838
$922$	0	0
$923$	−0.646837	−0.0212909
$924$	0	0
$925$	−27.0561	−0.889598
$926$	0	0
$927$	56.9694	1.87112
$928$	0	0
$929$	21.2354	0.696712	0.348356	−	0.937362i	$-0.386740\pi$
$929$	21.2354	0.696712	0.348356	+	0.937362i	$0.386740\pi$
$930$	0	0
$931$	0.103324	0.00338630
$932$	0	0
$933$	46.6868	1.52846
$934$	0	0
$935$	6.87873	0.224958
$936$	0	0
$937$	−17.8920	−0.584505	−0.292253	−	0.956341i	$-0.594405\pi$
$937$	−17.8920	−0.584505	−0.292253	+	0.956341i	$0.594405\pi$
$938$	0	0
$939$	−36.5737	−1.19354
$940$	0	0
$941$	−16.7344	−0.545526	−0.272763	−	0.962081i	$-0.587937\pi$
$941$	−16.7344	−0.545526	−0.272763	+	0.962081i	$0.587937\pi$
$942$	0	0
$943$	30.4829	0.992660
$944$	0	0
$945$	25.9326	0.843587
$946$	0	0
$947$	−6.43614	−0.209147	−0.104573	−	0.994517i	$-0.533348\pi$
$947$	−6.43614	−0.209147	−0.104573	+	0.994517i	$0.533348\pi$
$948$	0	0
$949$	−2.81140	−0.0912619
$950$	0	0
$951$	−54.9183	−1.78085
$952$	0	0
$953$	47.0679	1.52468	0.762339	−	0.647177i	$-0.224051\pi$
$953$	47.0679	1.52468	0.762339	+	0.647177i	$0.224051\pi$
$954$	0	0
$955$	34.5340	1.11750
$956$	0	0
$957$	−16.6757	−0.539049
$958$	0	0
$959$	11.5054	0.371528
$960$	0	0
$961$	−30.9633	−0.998816
$962$	0	0
$963$	29.4053	0.947574
$964$	0	0
$965$	−44.7393	−1.44021
$966$	0	0
$967$	34.9496	1.12390	0.561952	−	0.827170i	$-0.310051\pi$
$967$	34.9496	1.12390	0.561952	+	0.827170i	$0.310051\pi$
$968$	0	0
$969$	0.686395	0.0220502
$970$	0	0
$971$	22.8109	0.732037	0.366018	−	0.930608i	$-0.380721\pi$
$971$	22.8109	0.732037	0.366018	+	0.930608i	$0.380721\pi$
$972$	0	0
$973$	3.51308	0.112624
$974$	0	0
$975$	−13.2937	−0.425739
$976$	0	0
$977$	−34.7413	−1.11147	−0.555736	−	0.831359i	$-0.687564\pi$
$977$	−34.7413	−1.11147	−0.555736	+	0.831359i	$0.687564\pi$
$978$	0	0
$979$	−2.92179	−0.0933808
$980$	0	0
$981$	−61.3470	−1.95866
$982$	0	0
$983$	−3.76311	−0.120025	−0.0600123	−	0.998198i	$-0.519114\pi$
$983$	−3.76311	−0.120025	−0.0600123	+	0.998198i	$0.519114\pi$
$984$	0	0
$985$	−32.0278	−1.02049
$986$	0	0
$987$	12.7726	0.406556
$988$	0	0
$989$	−12.9125	−0.410593
$990$	0	0
$991$	36.6661	1.16474	0.582368	−	0.812925i	$-0.302126\pi$
$991$	36.6661	1.16474	0.582368	+	0.812925i	$0.302126\pi$
$992$	0	0
$993$	−52.0734	−1.65250
$994$	0	0
$995$	−38.4219	−1.21806
$996$	0	0
$997$	12.3542	0.391261	0.195631	−	0.980678i	$-0.437325\pi$
$997$	12.3542	0.391261	0.195631	+	0.980678i	$0.437325\pi$
$998$	0	0
$999$	−50.9718	−1.61268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.u.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.u.1.10	✓	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.97233 q^{3} -3.07774 q^{5} -1.00000 q^{7} +5.83476 q^{9} +O(q^{10})\)	\(q+2.97233 q^{3} -3.07774 q^{5} -1.00000 q^{7} +5.83476 q^{9} -1.00000 q^{11} -1.00000 q^{13} -9.14806 q^{15} +2.23499 q^{17} +0.103324 q^{19} -2.97233 q^{21} -2.91620 q^{23} +4.47248 q^{25} +8.42585 q^{27} +5.61031 q^{29} -0.191615 q^{31} -2.97233 q^{33} +3.07774 q^{35} -6.04946 q^{37} -2.97233 q^{39} -10.4530 q^{41} +4.42785 q^{43} -17.9579 q^{45} -4.29716 q^{47} +1.00000 q^{49} +6.64315 q^{51} -0.962428 q^{53} +3.07774 q^{55} +0.307112 q^{57} +7.41338 q^{59} -6.36668 q^{61} -5.83476 q^{63} +3.07774 q^{65} -10.9683 q^{67} -8.66791 q^{69} +0.646837 q^{71} +2.81140 q^{73} +13.2937 q^{75} +1.00000 q^{77} -13.8459 q^{79} +7.54015 q^{81} +1.79514 q^{83} -6.87873 q^{85} +16.6757 q^{87} +2.92179 q^{89} +1.00000 q^{91} -0.569544 q^{93} -0.318003 q^{95} -1.39715 q^{97} -5.83476 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more