LMFDB - Embedded newform 8008.2.a.h.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q+3.30278 q^{3} +4.30278 q^{5} +1.00000 q^{7} +7.90833 q^{9} +O(q^{10})$	$q+3.30278 q^{3} +4.30278 q^{5} +1.00000 q^{7} +7.90833 q^{9} -1.00000 q^{11} -1.00000 q^{13} +14.2111 q^{15} -0.697224 q^{17} -4.90833 q^{19} +3.30278 q^{21} -8.00000 q^{23} +13.5139 q^{25} +16.2111 q^{27} +4.00000 q^{29} +5.21110 q^{31} -3.30278 q^{33} +4.30278 q^{35} +8.60555 q^{37} -3.30278 q^{39} -4.60555 q^{41} -5.90833 q^{43} +34.0278 q^{45} -5.21110 q^{47} +1.00000 q^{49} -2.30278 q^{51} +11.9083 q^{53} -4.30278 q^{55} -16.2111 q^{57} +3.21110 q^{59} -13.1194 q^{61} +7.90833 q^{63} -4.30278 q^{65} +9.30278 q^{67} -26.4222 q^{69} -16.1194 q^{71} -1.39445 q^{73} +44.6333 q^{75} -1.00000 q^{77} -8.90833 q^{79} +29.8167 q^{81} +7.09167 q^{83} -3.00000 q^{85} +13.2111 q^{87} +5.30278 q^{89} -1.00000 q^{91} +17.2111 q^{93} -21.1194 q^{95} -10.0000 q^{97} -7.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 5 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 5 * q^5 + 2 * q^7 + 5 * q^9	$ 2 q + 3 q^{3} + 5 q^{5} + 2 q^{7} + 5 q^{9} - 2 q^{11} - 2 q^{13} + 14 q^{15} - 5 q^{17} + q^{19} + 3 q^{21} - 16 q^{23} + 9 q^{25} + 18 q^{27} + 8 q^{29} - 4 q^{31} - 3 q^{33} + 5 q^{35} + 10 q^{37} - 3 q^{39} - 2 q^{41} - q^{43} + 32 q^{45} + 4 q^{47} + 2 q^{49} - q^{51} + 13 q^{53} - 5 q^{55} - 18 q^{57} - 8 q^{59} - q^{61} + 5 q^{63} - 5 q^{65} + 15 q^{67} - 24 q^{69} - 7 q^{71} - 10 q^{73} + 46 q^{75} - 2 q^{77} - 7 q^{79} + 38 q^{81} + 25 q^{83} - 6 q^{85} + 12 q^{87} + 7 q^{89} - 2 q^{91} + 20 q^{93} - 17 q^{95} - 20 q^{97} - 5 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 5 * q^5 + 2 * q^7 + 5 * q^9 - 2 * q^11 - 2 * q^13 + 14 * q^15 - 5 * q^17 + q^19 + 3 * q^21 - 16 * q^23 + 9 * q^25 + 18 * q^27 + 8 * q^29 - 4 * q^31 - 3 * q^33 + 5 * q^35 + 10 * q^37 - 3 * q^39 - 2 * q^41 - q^43 + 32 * q^45 + 4 * q^47 + 2 * q^49 - q^51 + 13 * q^53 - 5 * q^55 - 18 * q^57 - 8 * q^59 - q^61 + 5 * q^63 - 5 * q^65 + 15 * q^67 - 24 * q^69 - 7 * q^71 - 10 * q^73 + 46 * q^75 - 2 * q^77 - 7 * q^79 + 38 * q^81 + 25 * q^83 - 6 * q^85 + 12 * q^87 + 7 * q^89 - 2 * q^91 + 20 * q^93 - 17 * q^95 - 20 * q^97 - 5 * 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
$3$	3.30278	1.90686	0.953429	−	0.301617i	$-0.0975264\pi$
$3$	3.30278	1.90686	0.953429	+	0.301617i	$0.0975264\pi$
$4$	0	0
$5$	4.30278	1.92426	0.962130	−	0.272591i	$-0.0878807\pi$
$5$	4.30278	1.92426	0.962130	+	0.272591i	$0.0878807\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.90833	2.63611
$10$	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
$15$	14.2111	3.66929
$16$	0	0
$17$	−0.697224	−0.169102	−0.0845509	−	0.996419i	$-0.526946\pi$
$17$	−0.697224	−0.169102	−0.0845509	+	0.996419i	$0.526946\pi$
$18$	0	0
$19$	−4.90833	−1.12605	−0.563024	−	0.826441i	$-0.690362\pi$
$19$	−4.90833	−1.12605	−0.563024	+	0.826441i	$0.690362\pi$
$20$	0	0
$21$	3.30278	0.720725
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	13.5139	2.70278
$26$	0	0
$27$	16.2111	3.11983
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	5.21110	0.935942	0.467971	−	0.883744i	$-0.344985\pi$
$31$	5.21110	0.935942	0.467971	+	0.883744i	$0.344985\pi$
$32$	0	0
$33$	−3.30278	−0.574939
$34$	0	0
$35$	4.30278	0.727302
$36$	0	0
$37$	8.60555	1.41474	0.707372	−	0.706842i	$-0.249881\pi$
$37$	8.60555	1.41474	0.707372	+	0.706842i	$0.249881\pi$
$38$	0	0
$39$	−3.30278	−0.528867
$40$	0	0
$41$	−4.60555	−0.719266	−0.359633	−	0.933094i	$-0.617098\pi$
$41$	−4.60555	−0.719266	−0.359633	+	0.933094i	$0.617098\pi$
$42$	0	0
$43$	−5.90833	−0.901011	−0.450506	−	0.892774i	$-0.648756\pi$
$43$	−5.90833	−0.901011	−0.450506	+	0.892774i	$0.648756\pi$
$44$	0	0
$45$	34.0278	5.07256
$46$	0	0
$47$	−5.21110	−0.760117	−0.380059	−	0.924962i	$-0.624096\pi$
$47$	−5.21110	−0.760117	−0.380059	+	0.924962i	$0.624096\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.30278	−0.322453
$52$	0	0
$53$	11.9083	1.63573	0.817867	−	0.575407i	$-0.195156\pi$
$53$	11.9083	1.63573	0.817867	+	0.575407i	$0.195156\pi$
$54$	0	0
$55$	−4.30278	−0.580186
$56$	0	0
$57$	−16.2111	−2.14721
$58$	0	0
$59$	3.21110	0.418050	0.209025	−	0.977910i	$-0.432971\pi$
$59$	3.21110	0.418050	0.209025	+	0.977910i	$0.432971\pi$
$60$	0	0
$61$	−13.1194	−1.67977	−0.839885	−	0.542764i	$-0.817378\pi$
$61$	−13.1194	−1.67977	−0.839885	+	0.542764i	$0.817378\pi$
$62$	0	0
$63$	7.90833	0.996356
$64$	0	0
$65$	−4.30278	−0.533694
$66$	0	0
$67$	9.30278	1.13651	0.568257	−	0.822851i	$-0.307618\pi$
$67$	9.30278	1.13651	0.568257	+	0.822851i	$0.307618\pi$
$68$	0	0
$69$	−26.4222	−3.18086
$70$	0	0
$71$	−16.1194	−1.91302	−0.956512	−	0.291692i	$-0.905782\pi$
$71$	−16.1194	−1.91302	−0.956512	+	0.291692i	$0.905782\pi$
$72$	0	0
$73$	−1.39445	−0.163208	−0.0816039	−	0.996665i	$-0.526004\pi$
$73$	−1.39445	−0.163208	−0.0816039	+	0.996665i	$0.526004\pi$
$74$	0	0
$75$	44.6333	5.15381
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−8.90833	−1.00227	−0.501133	−	0.865371i	$-0.667083\pi$
$79$	−8.90833	−1.00227	−0.501133	+	0.865371i	$0.667083\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	7.09167	0.778412	0.389206	−	0.921151i	$-0.372749\pi$
$83$	7.09167	0.778412	0.389206	+	0.921151i	$0.372749\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	13.2111	1.41638
$88$	0	0
$89$	5.30278	0.562093	0.281047	−	0.959694i	$-0.409318\pi$
$89$	5.30278	0.562093	0.281047	+	0.959694i	$0.409318\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	17.2111	1.78471
$94$	0	0
$95$	−21.1194	−2.16681
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−7.90833	−0.794817
$100$	0	0
$101$	15.5139	1.54369	0.771844	−	0.635812i	$-0.219334\pi$
$101$	15.5139	1.54369	0.771844	+	0.635812i	$0.219334\pi$
$102$	0	0
$103$	−13.2111	−1.30173	−0.650864	−	0.759194i	$-0.725593\pi$
$103$	−13.2111	−1.30173	−0.650864	+	0.759194i	$0.725593\pi$
$104$	0	0
$105$	14.2111	1.38686
$106$	0	0
$107$	19.3028	1.86607	0.933035	−	0.359786i	$-0.117150\pi$
$107$	19.3028	1.86607	0.933035	+	0.359786i	$0.117150\pi$
$108$	0	0
$109$	12.3028	1.17839	0.589196	−	0.807990i	$-0.299445\pi$
$109$	12.3028	1.17839	0.589196	+	0.807990i	$0.299445\pi$
$110$	0	0
$111$	28.4222	2.69772
$112$	0	0
$113$	−4.30278	−0.404771	−0.202386	−	0.979306i	$-0.564869\pi$
$113$	−4.30278	−0.404771	−0.202386	+	0.979306i	$0.564869\pi$
$114$	0	0
$115$	−34.4222	−3.20989
$116$	0	0
$117$	−7.90833	−0.731125
$118$	0	0
$119$	−0.697224	−0.0639145
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−15.2111	−1.37154
$124$	0	0
$125$	36.6333	3.27658
$126$	0	0
$127$	−3.09167	−0.274342	−0.137171	−	0.990547i	$-0.543801\pi$
$127$	−3.09167	−0.274342	−0.137171	+	0.990547i	$0.543801\pi$
$128$	0	0
$129$	−19.5139	−1.71810
$130$	0	0
$131$	−11.2111	−0.979519	−0.489759	−	0.871858i	$-0.662915\pi$
$131$	−11.2111	−0.979519	−0.489759	+	0.871858i	$0.662915\pi$
$132$	0	0
$133$	−4.90833	−0.425606
$134$	0	0
$135$	69.7527	6.00336
$136$	0	0
$137$	−3.21110	−0.274343	−0.137172	−	0.990547i	$-0.543801\pi$
$137$	−3.21110	−0.274343	−0.137172	+	0.990547i	$0.543801\pi$
$138$	0	0
$139$	21.2111	1.79910	0.899551	−	0.436816i	$-0.143894\pi$
$139$	21.2111	1.79910	0.899551	+	0.436816i	$0.143894\pi$
$140$	0	0
$141$	−17.2111	−1.44944
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	17.2111	1.42930
$146$	0	0
$147$	3.30278	0.272408
$148$	0	0
$149$	−1.51388	−0.124022	−0.0620109	−	0.998075i	$-0.519751\pi$
$149$	−1.51388	−0.124022	−0.0620109	+	0.998075i	$0.519751\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−5.51388	−0.445771
$154$	0	0
$155$	22.4222	1.80099
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	39.3305	3.11911
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−21.5139	−1.68510	−0.842548	−	0.538620i	$-0.818946\pi$
$163$	−21.5139	−1.68510	−0.842548	+	0.538620i	$0.818946\pi$
$164$	0	0
$165$	−14.2111	−1.10633
$166$	0	0
$167$	16.5139	1.27788	0.638941	−	0.769256i	$-0.279373\pi$
$167$	16.5139	1.27788	0.638941	+	0.769256i	$0.279373\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−38.8167	−2.96838
$172$	0	0
$173$	−6.90833	−0.525230	−0.262615	−	0.964901i	$-0.584585\pi$
$173$	−6.90833	−0.525230	−0.262615	+	0.964901i	$0.584585\pi$
$174$	0	0
$175$	13.5139	1.02155
$176$	0	0
$177$	10.6056	0.797162
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	2.78890	0.207297	0.103649	−	0.994614i	$-0.466948\pi$
$181$	2.78890	0.207297	0.103649	+	0.994614i	$0.466948\pi$
$182$	0	0
$183$	−43.3305	−3.20309
$184$	0	0
$185$	37.0278	2.72233
$186$	0	0
$187$	0.697224	0.0509861
$188$	0	0
$189$	16.2111	1.17918
$190$	0	0
$191$	3.21110	0.232347	0.116174	−	0.993229i	$-0.462937\pi$
$191$	3.21110	0.232347	0.116174	+	0.993229i	$0.462937\pi$
$192$	0	0
$193$	−21.3028	−1.53341	−0.766704	−	0.642001i	$-0.778104\pi$
$193$	−21.3028	−1.53341	−0.766704	+	0.642001i	$0.778104\pi$
$194$	0	0
$195$	−14.2111	−1.01768
$196$	0	0
$197$	−3.11943	−0.222250	−0.111125	−	0.993806i	$-0.535445\pi$
$197$	−3.11943	−0.222250	−0.111125	+	0.993806i	$0.535445\pi$
$198$	0	0
$199$	−3.69722	−0.262089	−0.131045	−	0.991376i	$-0.541833\pi$
$199$	−3.69722	−0.262089	−0.131045	+	0.991376i	$0.541833\pi$
$200$	0	0
$201$	30.7250	2.16717
$202$	0	0
$203$	4.00000	0.280745
$204$	0	0
$205$	−19.8167	−1.38406
$206$	0	0
$207$	−63.2666	−4.39733
$208$	0	0
$209$	4.90833	0.339516
$210$	0	0
$211$	−17.7250	−1.22024	−0.610119	−	0.792310i	$-0.708878\pi$
$211$	−17.7250	−1.22024	−0.610119	+	0.792310i	$0.708878\pi$
$212$	0	0
$213$	−53.2389	−3.64787
$214$	0	0
$215$	−25.4222	−1.73378
$216$	0	0
$217$	5.21110	0.353753
$218$	0	0
$219$	−4.60555	−0.311214
$220$	0	0
$221$	0.697224	0.0469004
$222$	0	0
$223$	−11.3944	−0.763029	−0.381514	−	0.924363i	$-0.624597\pi$
$223$	−11.3944	−0.763029	−0.381514	+	0.924363i	$0.624597\pi$
$224$	0	0
$225$	106.872	7.12481
$226$	0	0
$227$	8.09167	0.537063	0.268532	−	0.963271i	$-0.413462\pi$
$227$	8.09167	0.537063	0.268532	+	0.963271i	$0.413462\pi$
$228$	0	0
$229$	2.11943	0.140056	0.0700279	−	0.997545i	$-0.477691\pi$
$229$	2.11943	0.140056	0.0700279	+	0.997545i	$0.477691\pi$
$230$	0	0
$231$	−3.30278	−0.217307
$232$	0	0
$233$	−21.6333	−1.41725	−0.708623	−	0.705588i	$-0.750683\pi$
$233$	−21.6333	−1.41725	−0.708623	+	0.705588i	$0.750683\pi$
$234$	0	0
$235$	−22.4222	−1.46266
$236$	0	0
$237$	−29.4222	−1.91118
$238$	0	0
$239$	−12.7889	−0.827245	−0.413623	−	0.910448i	$-0.635737\pi$
$239$	−12.7889	−0.827245	−0.413623	+	0.910448i	$0.635737\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	49.8444	3.19752
$244$	0	0
$245$	4.30278	0.274894
$246$	0	0
$247$	4.90833	0.312309
$248$	0	0
$249$	23.4222	1.48432
$250$	0	0
$251$	−30.9083	−1.95092	−0.975458	−	0.220185i	$-0.929334\pi$
$251$	−30.9083	−1.95092	−0.975458	+	0.220185i	$0.929334\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	−9.90833	−0.620483
$256$	0	0
$257$	−7.02776	−0.438379	−0.219190	−	0.975682i	$-0.570341\pi$
$257$	−7.02776	−0.438379	−0.219190	+	0.975682i	$0.570341\pi$
$258$	0	0
$259$	8.60555	0.534723
$260$	0	0
$261$	31.6333	1.95805
$262$	0	0
$263$	−11.3305	−0.698671	−0.349335	−	0.936998i	$-0.613593\pi$
$263$	−11.3305	−0.698671	−0.349335	+	0.936998i	$0.613593\pi$
$264$	0	0
$265$	51.2389	3.14758
$266$	0	0
$267$	17.5139	1.07183
$268$	0	0
$269$	2.78890	0.170042	0.0850210	−	0.996379i	$-0.472904\pi$
$269$	2.78890	0.170042	0.0850210	+	0.996379i	$0.472904\pi$
$270$	0	0
$271$	0.513878	0.0312159	0.0156079	−	0.999878i	$-0.495032\pi$
$271$	0.513878	0.0312159	0.0156079	+	0.999878i	$0.495032\pi$
$272$	0	0
$273$	−3.30278	−0.199893
$274$	0	0
$275$	−13.5139	−0.814918
$276$	0	0
$277$	1.39445	0.0837843	0.0418922	−	0.999122i	$-0.486661\pi$
$277$	1.39445	0.0837843	0.0418922	+	0.999122i	$0.486661\pi$
$278$	0	0
$279$	41.2111	2.46724
$280$	0	0
$281$	−7.72498	−0.460834	−0.230417	−	0.973092i	$-0.574009\pi$
$281$	−7.72498	−0.460834	−0.230417	+	0.973092i	$0.574009\pi$
$282$	0	0
$283$	28.6056	1.70042	0.850212	−	0.526441i	$-0.176474\pi$
$283$	28.6056	1.70042	0.850212	+	0.526441i	$0.176474\pi$
$284$	0	0
$285$	−69.7527	−4.13180
$286$	0	0
$287$	−4.60555	−0.271857
$288$	0	0
$289$	−16.5139	−0.971405
$290$	0	0
$291$	−33.0278	−1.93612
$292$	0	0
$293$	2.78890	0.162929	0.0814646	−	0.996676i	$-0.474040\pi$
$293$	2.78890	0.162929	0.0814646	+	0.996676i	$0.474040\pi$
$294$	0	0
$295$	13.8167	0.804437
$296$	0	0
$297$	−16.2111	−0.940664
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−5.90833	−0.340550
$302$	0	0
$303$	51.2389	2.94360
$304$	0	0
$305$	−56.4500	−3.23232
$306$	0	0
$307$	−10.6972	−0.610523	−0.305261	−	0.952269i	$-0.598744\pi$
$307$	−10.6972	−0.610523	−0.305261	+	0.952269i	$0.598744\pi$
$308$	0	0
$309$	−43.6333	−2.48221
$310$	0	0
$311$	−7.69722	−0.436470	−0.218235	−	0.975896i	$-0.570030\pi$
$311$	−7.69722	−0.436470	−0.218235	+	0.975896i	$0.570030\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	34.0278	1.91725
$316$	0	0
$317$	−6.60555	−0.371005	−0.185502	−	0.982644i	$-0.559391\pi$
$317$	−6.60555	−0.371005	−0.185502	+	0.982644i	$0.559391\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	63.7527	3.55833
$322$	0	0
$323$	3.42221	0.190417
$324$	0	0
$325$	−13.5139	−0.749615
$326$	0	0
$327$	40.6333	2.24703
$328$	0	0
$329$	−5.21110	−0.287297
$330$	0	0
$331$	31.9361	1.75537	0.877683	−	0.479242i	$-0.159088\pi$
$331$	31.9361	1.75537	0.877683	+	0.479242i	$0.159088\pi$
$332$	0	0
$333$	68.0555	3.72942
$334$	0	0
$335$	40.0278	2.18695
$336$	0	0
$337$	−21.3944	−1.16543	−0.582715	−	0.812677i	$-0.698010\pi$
$337$	−21.3944	−1.16543	−0.582715	+	0.812677i	$0.698010\pi$
$338$	0	0
$339$	−14.2111	−0.771841
$340$	0	0
$341$	−5.21110	−0.282197
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−113.689	−6.12080
$346$	0	0
$347$	8.11943	0.435874	0.217937	−	0.975963i	$-0.430067\pi$
$347$	8.11943	0.435874	0.217937	+	0.975963i	$0.430067\pi$
$348$	0	0
$349$	−4.60555	−0.246530	−0.123265	−	0.992374i	$-0.539336\pi$
$349$	−4.60555	−0.246530	−0.123265	+	0.992374i	$0.539336\pi$
$350$	0	0
$351$	−16.2111	−0.865285
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	−69.3583	−3.68116
$356$	0	0
$357$	−2.30278	−0.121876
$358$	0	0
$359$	−2.18335	−0.115233	−0.0576163	−	0.998339i	$-0.518350\pi$
$359$	−2.18335	−0.115233	−0.0576163	+	0.998339i	$0.518350\pi$
$360$	0	0
$361$	5.09167	0.267983
$362$	0	0
$363$	3.30278	0.173351
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−9.11943	−0.476030	−0.238015	−	0.971261i	$-0.576497\pi$
$367$	−9.11943	−0.476030	−0.238015	+	0.971261i	$0.576497\pi$
$368$	0	0
$369$	−36.4222	−1.89606
$370$	0	0
$371$	11.9083	0.618250
$372$	0	0
$373$	7.39445	0.382870	0.191435	−	0.981505i	$-0.438686\pi$
$373$	7.39445	0.382870	0.191435	+	0.981505i	$0.438686\pi$
$374$	0	0
$375$	120.992	6.24798
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	13.1194	0.673900	0.336950	−	0.941523i	$-0.390605\pi$
$379$	13.1194	0.673900	0.336950	+	0.941523i	$0.390605\pi$
$380$	0	0
$381$	−10.2111	−0.523131
$382$	0	0
$383$	−15.2111	−0.777251	−0.388626	−	0.921396i	$-0.627050\pi$
$383$	−15.2111	−0.777251	−0.388626	+	0.921396i	$0.627050\pi$
$384$	0	0
$385$	−4.30278	−0.219290
$386$	0	0
$387$	−46.7250	−2.37516
$388$	0	0
$389$	−6.30278	−0.319563	−0.159782	−	0.987152i	$-0.551079\pi$
$389$	−6.30278	−0.319563	−0.159782	+	0.987152i	$0.551079\pi$
$390$	0	0
$391$	5.57779	0.282081
$392$	0	0
$393$	−37.0278	−1.86780
$394$	0	0
$395$	−38.3305	−1.92862
$396$	0	0
$397$	−35.1194	−1.76259	−0.881297	−	0.472563i	$-0.843329\pi$
$397$	−35.1194	−1.76259	−0.881297	+	0.472563i	$0.843329\pi$
$398$	0	0
$399$	−16.2111	−0.811570
$400$	0	0
$401$	−18.2389	−0.910805	−0.455403	−	0.890286i	$-0.650505\pi$
$401$	−18.2389	−0.910805	−0.455403	+	0.890286i	$0.650505\pi$
$402$	0	0
$403$	−5.21110	−0.259584
$404$	0	0
$405$	128.294	6.37500
$406$	0	0
$407$	−8.60555	−0.426561
$408$	0	0
$409$	−9.81665	−0.485402	−0.242701	−	0.970101i	$-0.578033\pi$
$409$	−9.81665	−0.485402	−0.242701	+	0.970101i	$0.578033\pi$
$410$	0	0
$411$	−10.6056	−0.523133
$412$	0	0
$413$	3.21110	0.158008
$414$	0	0
$415$	30.5139	1.49787
$416$	0	0
$417$	70.0555	3.43063
$418$	0	0
$419$	15.1194	0.738632	0.369316	−	0.929304i	$-0.379592\pi$
$419$	15.1194	0.738632	0.369316	+	0.929304i	$0.379592\pi$
$420$	0	0
$421$	−27.0278	−1.31725	−0.658626	−	0.752470i	$-0.728862\pi$
$421$	−27.0278	−1.31725	−0.658626	+	0.752470i	$0.728862\pi$
$422$	0	0
$423$	−41.2111	−2.00375
$424$	0	0
$425$	−9.42221	−0.457044
$426$	0	0
$427$	−13.1194	−0.634894
$428$	0	0
$429$	3.30278	0.159460
$430$	0	0
$431$	14.2389	0.685862	0.342931	−	0.939361i	$-0.388580\pi$
$431$	14.2389	0.685862	0.342931	+	0.939361i	$0.388580\pi$
$432$	0	0
$433$	−27.8167	−1.33678	−0.668392	−	0.743810i	$-0.733017\pi$
$433$	−27.8167	−1.33678	−0.668392	+	0.743810i	$0.733017\pi$
$434$	0	0
$435$	56.8444	2.72548
$436$	0	0
$437$	39.2666	1.87838
$438$	0	0
$439$	19.2111	0.916896	0.458448	−	0.888721i	$-0.348406\pi$
$439$	19.2111	0.916896	0.458448	+	0.888721i	$0.348406\pi$
$440$	0	0
$441$	7.90833	0.376587
$442$	0	0
$443$	−1.39445	−0.0662523	−0.0331261	−	0.999451i	$-0.510546\pi$
$443$	−1.39445	−0.0662523	−0.0331261	+	0.999451i	$0.510546\pi$
$444$	0	0
$445$	22.8167	1.08161
$446$	0	0
$447$	−5.00000	−0.236492
$448$	0	0
$449$	−9.21110	−0.434699	−0.217349	−	0.976094i	$-0.569741\pi$
$449$	−9.21110	−0.434699	−0.217349	+	0.976094i	$0.569741\pi$
$450$	0	0
$451$	4.60555	0.216867
$452$	0	0
$453$	26.4222	1.24142
$454$	0	0
$455$	−4.30278	−0.201717
$456$	0	0
$457$	−24.1194	−1.12826	−0.564130	−	0.825686i	$-0.690788\pi$
$457$	−24.1194	−1.12826	−0.564130	+	0.825686i	$0.690788\pi$
$458$	0	0
$459$	−11.3028	−0.527568
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	5.21110	0.242180	0.121090	−	0.992642i	$-0.461361\pi$
$463$	5.21110	0.242180	0.121090	+	0.992642i	$0.461361\pi$
$464$	0	0
$465$	74.0555	3.43424
$466$	0	0
$467$	−36.1472	−1.67269	−0.836346	−	0.548202i	$-0.815313\pi$
$467$	−36.1472	−1.67269	−0.836346	+	0.548202i	$0.815313\pi$
$468$	0	0
$469$	9.30278	0.429562
$470$	0	0
$471$	6.60555	0.304368
$472$	0	0
$473$	5.90833	0.271665
$474$	0	0
$475$	−66.3305	−3.04345
$476$	0	0
$477$	94.1749	4.31197
$478$	0	0
$479$	0.486122	0.0222115	0.0111057	−	0.999938i	$-0.496465\pi$
$479$	0.486122	0.0222115	0.0111057	+	0.999938i	$0.496465\pi$
$480$	0	0
$481$	−8.60555	−0.392379
$482$	0	0
$483$	−26.4222	−1.20225
$484$	0	0
$485$	−43.0278	−1.95379
$486$	0	0
$487$	25.2111	1.14242	0.571212	−	0.820803i	$-0.306473\pi$
$487$	25.2111	1.14242	0.571212	+	0.820803i	$0.306473\pi$
$488$	0	0
$489$	−71.0555	−3.21324
$490$	0	0
$491$	5.57779	0.251722	0.125861	−	0.992048i	$-0.459831\pi$
$491$	5.57779	0.251722	0.125861	+	0.992048i	$0.459831\pi$
$492$	0	0
$493$	−2.78890	−0.125606
$494$	0	0
$495$	−34.0278	−1.52943
$496$	0	0
$497$	−16.1194	−0.723055
$498$	0	0
$499$	39.9361	1.78778	0.893892	−	0.448282i	$-0.147964\pi$
$499$	39.9361	1.78778	0.893892	+	0.448282i	$0.147964\pi$
$500$	0	0
$501$	54.5416	2.43674
$502$	0	0
$503$	21.6333	0.964582	0.482291	−	0.876011i	$-0.339805\pi$
$503$	21.6333	0.964582	0.482291	+	0.876011i	$0.339805\pi$
$504$	0	0
$505$	66.7527	2.97046
$506$	0	0
$507$	3.30278	0.146681
$508$	0	0
$509$	−27.2111	−1.20611	−0.603055	−	0.797699i	$-0.706050\pi$
$509$	−27.2111	−1.20611	−0.603055	+	0.797699i	$0.706050\pi$
$510$	0	0
$511$	−1.39445	−0.0616868
$512$	0	0
$513$	−79.5694	−3.51307
$514$	0	0
$515$	−56.8444	−2.50486
$516$	0	0
$517$	5.21110	0.229184
$518$	0	0
$519$	−22.8167	−1.00154
$520$	0	0
$521$	−18.8444	−0.825589	−0.412794	−	0.910824i	$-0.635447\pi$
$521$	−18.8444	−0.825589	−0.412794	+	0.910824i	$0.635447\pi$
$522$	0	0
$523$	−43.0278	−1.88147	−0.940736	−	0.339139i	$-0.889864\pi$
$523$	−43.0278	−1.88147	−0.940736	+	0.339139i	$0.889864\pi$
$524$	0	0
$525$	44.6333	1.94796
$526$	0	0
$527$	−3.63331	−0.158269
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	25.3944	1.10203
$532$	0	0
$533$	4.60555	0.199489
$534$	0	0
$535$	83.0555	3.59080
$536$	0	0
$537$	−52.8444	−2.28040
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	26.8444	1.15413	0.577066	−	0.816698i	$-0.304198\pi$
$541$	26.8444	1.15413	0.577066	+	0.816698i	$0.304198\pi$
$542$	0	0
$543$	9.21110	0.395286
$544$	0	0
$545$	52.9361	2.26753
$546$	0	0
$547$	9.57779	0.409517	0.204758	−	0.978813i	$-0.434359\pi$
$547$	9.57779	0.409517	0.204758	+	0.978813i	$0.434359\pi$
$548$	0	0
$549$	−103.753	−4.42806
$550$	0	0
$551$	−19.6333	−0.836407
$552$	0	0
$553$	−8.90833	−0.378821
$554$	0	0
$555$	122.294	5.19111
$556$	0	0
$557$	−1.51388	−0.0641451	−0.0320725	−	0.999486i	$-0.510211\pi$
$557$	−1.51388	−0.0641451	−0.0320725	+	0.999486i	$0.510211\pi$
$558$	0	0
$559$	5.90833	0.249896
$560$	0	0
$561$	2.30278	0.0972233
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−18.5139	−0.778885
$566$	0	0
$567$	29.8167	1.25218
$568$	0	0
$569$	−27.6333	−1.15845	−0.579224	−	0.815168i	$-0.696644\pi$
$569$	−27.6333	−1.15845	−0.579224	+	0.815168i	$0.696644\pi$
$570$	0	0
$571$	−32.1194	−1.34416	−0.672078	−	0.740480i	$-0.734598\pi$
$571$	−32.1194	−1.34416	−0.672078	+	0.740480i	$0.734598\pi$
$572$	0	0
$573$	10.6056	0.443053
$574$	0	0
$575$	−108.111	−4.50854
$576$	0	0
$577$	26.3305	1.09615	0.548077	−	0.836428i	$-0.315360\pi$
$577$	26.3305	1.09615	0.548077	+	0.836428i	$0.315360\pi$
$578$	0	0
$579$	−70.3583	−2.92399
$580$	0	0
$581$	7.09167	0.294212
$582$	0	0
$583$	−11.9083	−0.493193
$584$	0	0
$585$	−34.0278	−1.40687
$586$	0	0
$587$	44.6056	1.84107	0.920534	−	0.390662i	$-0.127754\pi$
$587$	44.6056	1.84107	0.920534	+	0.390662i	$0.127754\pi$
$588$	0	0
$589$	−25.5778	−1.05391
$590$	0	0
$591$	−10.3028	−0.423800
$592$	0	0
$593$	39.6333	1.62754	0.813772	−	0.581184i	$-0.197410\pi$
$593$	39.6333	1.62754	0.813772	+	0.581184i	$0.197410\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	−12.2111	−0.499767
$598$	0	0
$599$	−25.3944	−1.03759	−0.518795	−	0.854899i	$-0.673619\pi$
$599$	−25.3944	−1.03759	−0.518795	+	0.854899i	$0.673619\pi$
$600$	0	0
$601$	−14.9361	−0.609256	−0.304628	−	0.952471i	$-0.598532\pi$
$601$	−14.9361	−0.609256	−0.304628	+	0.952471i	$0.598532\pi$
$602$	0	0
$603$	73.5694	2.99598
$604$	0	0
$605$	4.30278	0.174933
$606$	0	0
$607$	23.8167	0.966688	0.483344	−	0.875430i	$-0.339422\pi$
$607$	23.8167	0.966688	0.483344	+	0.875430i	$0.339422\pi$
$608$	0	0
$609$	13.2111	0.535341
$610$	0	0
$611$	5.21110	0.210819
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	−65.4500	−2.63920
$616$	0	0
$617$	−14.6056	−0.587997	−0.293999	−	0.955806i	$-0.594986\pi$
$617$	−14.6056	−0.587997	−0.293999	+	0.955806i	$0.594986\pi$
$618$	0	0
$619$	−15.4500	−0.620986	−0.310493	−	0.950576i	$-0.600494\pi$
$619$	−15.4500	−0.620986	−0.310493	+	0.950576i	$0.600494\pi$
$620$	0	0
$621$	−129.689	−5.20423
$622$	0	0
$623$	5.30278	0.212451
$624$	0	0
$625$	90.0555	3.60222
$626$	0	0
$627$	16.2111	0.647409
$628$	0	0
$629$	−6.00000	−0.239236
$630$	0	0
$631$	8.69722	0.346231	0.173116	−	0.984902i	$-0.444617\pi$
$631$	8.69722	0.346231	0.173116	+	0.984902i	$0.444617\pi$
$632$	0	0
$633$	−58.5416	−2.32682
$634$	0	0
$635$	−13.3028	−0.527905
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−127.478	−5.04294
$640$	0	0
$641$	−0.513878	−0.0202970	−0.0101485	−	0.999949i	$-0.503230\pi$
$641$	−0.513878	−0.0202970	−0.0101485	+	0.999949i	$0.503230\pi$
$642$	0	0
$643$	−12.1833	−0.480464	−0.240232	−	0.970715i	$-0.577224\pi$
$643$	−12.1833	−0.480464	−0.240232	+	0.970715i	$0.577224\pi$
$644$	0	0
$645$	−83.9638	−3.30607
$646$	0	0
$647$	5.57779	0.219286	0.109643	−	0.993971i	$-0.465029\pi$
$647$	5.57779	0.219286	0.109643	+	0.993971i	$0.465029\pi$
$648$	0	0
$649$	−3.21110	−0.126047
$650$	0	0
$651$	17.2111	0.674556
$652$	0	0
$653$	19.9083	0.779073	0.389537	−	0.921011i	$-0.372635\pi$
$653$	19.9083	0.779073	0.389537	+	0.921011i	$0.372635\pi$
$654$	0	0
$655$	−48.2389	−1.88485
$656$	0	0
$657$	−11.0278	−0.430234
$658$	0	0
$659$	−39.3028	−1.53102	−0.765509	−	0.643425i	$-0.777513\pi$
$659$	−39.3028	−1.53102	−0.765509	+	0.643425i	$0.777513\pi$
$660$	0	0
$661$	6.48612	0.252281	0.126140	−	0.992012i	$-0.459741\pi$
$661$	6.48612	0.252281	0.126140	+	0.992012i	$0.459741\pi$
$662$	0	0
$663$	2.30278	0.0894324
$664$	0	0
$665$	−21.1194	−0.818976
$666$	0	0
$667$	−32.0000	−1.23904
$668$	0	0
$669$	−37.6333	−1.45499
$670$	0	0
$671$	13.1194	0.506470
$672$	0	0
$673$	24.7889	0.955542	0.477771	−	0.878484i	$-0.341445\pi$
$673$	24.7889	0.955542	0.477771	+	0.878484i	$0.341445\pi$
$674$	0	0
$675$	219.075	8.43220
$676$	0	0
$677$	6.69722	0.257395	0.128698	−	0.991684i	$-0.458920\pi$
$677$	6.69722	0.257395	0.128698	+	0.991684i	$0.458920\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	26.7250	1.02410
$682$	0	0
$683$	19.9083	0.761771	0.380885	−	0.924622i	$-0.375619\pi$
$683$	19.9083	0.761771	0.380885	+	0.924622i	$0.375619\pi$
$684$	0	0
$685$	−13.8167	−0.527907
$686$	0	0
$687$	7.00000	0.267067
$688$	0	0
$689$	−11.9083	−0.453671
$690$	0	0
$691$	22.0555	0.839031	0.419516	−	0.907748i	$-0.362200\pi$
$691$	22.0555	0.839031	0.419516	+	0.907748i	$0.362200\pi$
$692$	0	0
$693$	−7.90833	−0.300412
$694$	0	0
$695$	91.2666	3.46194
$696$	0	0
$697$	3.21110	0.121629
$698$	0	0
$699$	−71.4500	−2.70249
$700$	0	0
$701$	43.8722	1.65703	0.828514	−	0.559968i	$-0.189186\pi$
$701$	43.8722	1.65703	0.828514	+	0.559968i	$0.189186\pi$
$702$	0	0
$703$	−42.2389	−1.59307
$704$	0	0
$705$	−74.0555	−2.78909
$706$	0	0
$707$	15.5139	0.583459
$708$	0	0
$709$	6.78890	0.254962	0.127481	−	0.991841i	$-0.459311\pi$
$709$	6.78890	0.254962	0.127481	+	0.991841i	$0.459311\pi$
$710$	0	0
$711$	−70.4500	−2.64208
$712$	0	0
$713$	−41.6888	−1.56126
$714$	0	0
$715$	4.30278	0.160915
$716$	0	0
$717$	−42.2389	−1.57744
$718$	0	0
$719$	−28.7527	−1.07230	−0.536148	−	0.844124i	$-0.680121\pi$
$719$	−28.7527	−1.07230	−0.536148	+	0.844124i	$0.680121\pi$
$720$	0	0
$721$	−13.2111	−0.492007
$722$	0	0
$723$	26.4222	0.982652
$724$	0	0
$725$	54.0555	2.00757
$726$	0	0
$727$	11.3305	0.420226	0.210113	−	0.977677i	$-0.432617\pi$
$727$	11.3305	0.420226	0.210113	+	0.977677i	$0.432617\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	4.11943	0.152363
$732$	0	0
$733$	15.3944	0.568607	0.284303	−	0.958734i	$-0.408238\pi$
$733$	15.3944	0.568607	0.284303	+	0.958734i	$0.408238\pi$
$734$	0	0
$735$	14.2111	0.524184
$736$	0	0
$737$	−9.30278	−0.342672
$738$	0	0
$739$	−21.2111	−0.780263	−0.390132	−	0.920759i	$-0.627570\pi$
$739$	−21.2111	−0.780263	−0.390132	+	0.920759i	$0.627570\pi$
$740$	0	0
$741$	16.2111	0.595530
$742$	0	0
$743$	4.18335	0.153472	0.0767360	−	0.997051i	$-0.475550\pi$
$743$	4.18335	0.153472	0.0767360	+	0.997051i	$0.475550\pi$
$744$	0	0
$745$	−6.51388	−0.238650
$746$	0	0
$747$	56.0833	2.05198
$748$	0	0
$749$	19.3028	0.705308
$750$	0	0
$751$	17.5778	0.641423	0.320711	−	0.947177i	$-0.396078\pi$
$751$	17.5778	0.641423	0.320711	+	0.947177i	$0.396078\pi$
$752$	0	0
$753$	−102.083	−3.72012
$754$	0	0
$755$	34.4222	1.25275
$756$	0	0
$757$	26.9083	0.978000	0.489000	−	0.872284i	$-0.337362\pi$
$757$	26.9083	0.978000	0.489000	+	0.872284i	$0.337362\pi$
$758$	0	0
$759$	26.4222	0.959065
$760$	0	0
$761$	24.4222	0.885304	0.442652	−	0.896693i	$-0.354038\pi$
$761$	24.4222	0.885304	0.442652	+	0.896693i	$0.354038\pi$
$762$	0	0
$763$	12.3028	0.445390
$764$	0	0
$765$	−23.7250	−0.857778
$766$	0	0
$767$	−3.21110	−0.115946
$768$	0	0
$769$	−34.2389	−1.23468	−0.617342	−	0.786695i	$-0.711791\pi$
$769$	−34.2389	−1.23468	−0.617342	+	0.786695i	$0.711791\pi$
$770$	0	0
$771$	−23.2111	−0.835927
$772$	0	0
$773$	3.66947	0.131982	0.0659908	−	0.997820i	$-0.478979\pi$
$773$	3.66947	0.131982	0.0659908	+	0.997820i	$0.478979\pi$
$774$	0	0
$775$	70.4222	2.52964
$776$	0	0
$777$	28.4222	1.01964
$778$	0	0
$779$	22.6056	0.809928
$780$	0	0
$781$	16.1194	0.576799
$782$	0	0
$783$	64.8444	2.31735
$784$	0	0
$785$	8.60555	0.307145
$786$	0	0
$787$	54.3305	1.93667	0.968337	−	0.249646i	$-0.0803143\pi$
$787$	54.3305	1.93667	0.968337	+	0.249646i	$0.0803143\pi$
$788$	0	0
$789$	−37.4222	−1.33227
$790$	0	0
$791$	−4.30278	−0.152989
$792$	0	0
$793$	13.1194	0.465885
$794$	0	0
$795$	169.230	6.00199
$796$	0	0
$797$	37.0278	1.31159	0.655795	−	0.754939i	$-0.272333\pi$
$797$	37.0278	1.31159	0.655795	+	0.754939i	$0.272333\pi$
$798$	0	0
$799$	3.63331	0.128537
$800$	0	0
$801$	41.9361	1.48174
$802$	0	0
$803$	1.39445	0.0492090
$804$	0	0
$805$	−34.4222	−1.21322
$806$	0	0
$807$	9.21110	0.324246
$808$	0	0
$809$	55.4500	1.94952	0.974758	−	0.223262i	$-0.0716706\pi$
$809$	55.4500	1.94952	0.974758	+	0.223262i	$0.0716706\pi$
$810$	0	0
$811$	−29.1472	−1.02350	−0.511748	−	0.859136i	$-0.671002\pi$
$811$	−29.1472	−1.02350	−0.511748	+	0.859136i	$0.671002\pi$
$812$	0	0
$813$	1.69722	0.0595243
$814$	0	0
$815$	−92.5694	−3.24256
$816$	0	0
$817$	29.0000	1.01458
$818$	0	0
$819$	−7.90833	−0.276339
$820$	0	0
$821$	39.1194	1.36528	0.682639	−	0.730756i	$-0.260832\pi$
$821$	39.1194	1.36528	0.682639	+	0.730756i	$0.260832\pi$
$822$	0	0
$823$	18.4222	0.642158	0.321079	−	0.947052i	$-0.395955\pi$
$823$	18.4222	0.642158	0.321079	+	0.947052i	$0.395955\pi$
$824$	0	0
$825$	−44.6333	−1.55393
$826$	0	0
$827$	−25.5778	−0.889427	−0.444714	−	0.895673i	$-0.646695\pi$
$827$	−25.5778	−0.889427	−0.444714	+	0.895673i	$0.646695\pi$
$828$	0	0
$829$	35.2111	1.22293	0.611466	−	0.791271i	$-0.290580\pi$
$829$	35.2111	1.22293	0.611466	+	0.791271i	$0.290580\pi$
$830$	0	0
$831$	4.60555	0.159765
$832$	0	0
$833$	−0.697224	−0.0241574
$834$	0	0
$835$	71.0555	2.45898
$836$	0	0
$837$	84.4777	2.91998
$838$	0	0
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−25.5139	−0.878745
$844$	0	0
$845$	4.30278	0.148020
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	94.4777	3.24247
$850$	0	0
$851$	−68.8444	−2.35996
$852$	0	0
$853$	0.972244	0.0332890	0.0166445	−	0.999861i	$-0.494702\pi$
$853$	0.972244	0.0332890	0.0166445	+	0.999861i	$0.494702\pi$
$854$	0	0
$855$	−167.019	−5.71194
$856$	0	0
$857$	−44.3028	−1.51335	−0.756677	−	0.653789i	$-0.773178\pi$
$857$	−44.3028	−1.51335	−0.756677	+	0.653789i	$0.773178\pi$
$858$	0	0
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	0	0
$861$	−15.2111	−0.518393
$862$	0	0
$863$	27.6333	0.940649	0.470324	−	0.882494i	$-0.344137\pi$
$863$	27.6333	0.940649	0.470324	+	0.882494i	$0.344137\pi$
$864$	0	0
$865$	−29.7250	−1.01068
$866$	0	0
$867$	−54.5416	−1.85233
$868$	0	0
$869$	8.90833	0.302194
$870$	0	0
$871$	−9.30278	−0.315213
$872$	0	0
$873$	−79.0833	−2.67656
$874$	0	0
$875$	36.6333	1.23843
$876$	0	0
$877$	−8.51388	−0.287493	−0.143747	−	0.989615i	$-0.545915\pi$
$877$	−8.51388	−0.287493	−0.143747	+	0.989615i	$0.545915\pi$
$878$	0	0
$879$	9.21110	0.310683
$880$	0	0
$881$	39.6333	1.33528	0.667640	−	0.744484i	$-0.267305\pi$
$881$	39.6333	1.33528	0.667640	+	0.744484i	$0.267305\pi$
$882$	0	0
$883$	18.0000	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$883$	18.0000	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$884$	0	0
$885$	45.6333	1.53395
$886$	0	0
$887$	−55.8722	−1.87600	−0.938002	−	0.346630i	$-0.887326\pi$
$887$	−55.8722	−1.87600	−0.938002	+	0.346630i	$0.887326\pi$
$888$	0	0
$889$	−3.09167	−0.103691
$890$	0	0
$891$	−29.8167	−0.998895
$892$	0	0
$893$	25.5778	0.855928
$894$	0	0
$895$	−68.8444	−2.30121
$896$	0	0
$897$	26.4222	0.882212
$898$	0	0
$899$	20.8444	0.695200
$900$	0	0
$901$	−8.30278	−0.276606
$902$	0	0
$903$	−19.5139	−0.649381
$904$	0	0
$905$	12.0000	0.398893
$906$	0	0
$907$	37.8722	1.25752	0.628762	−	0.777598i	$-0.283562\pi$
$907$	37.8722	1.25752	0.628762	+	0.777598i	$0.283562\pi$
$908$	0	0
$909$	122.689	4.06933
$910$	0	0
$911$	31.6333	1.04806	0.524029	−	0.851700i	$-0.324428\pi$
$911$	31.6333	1.04806	0.524029	+	0.851700i	$0.324428\pi$
$912$	0	0
$913$	−7.09167	−0.234700
$914$	0	0
$915$	−186.442	−6.16357
$916$	0	0
$917$	−11.2111	−0.370223
$918$	0	0
$919$	−1.11943	−0.0369266	−0.0184633	−	0.999830i	$-0.505877\pi$
$919$	−1.11943	−0.0369266	−0.0184633	+	0.999830i	$0.505877\pi$
$920$	0	0
$921$	−35.3305	−1.16418
$922$	0	0
$923$	16.1194	0.530577
$924$	0	0
$925$	116.294	3.82374
$926$	0	0
$927$	−104.478	−3.43150
$928$	0	0
$929$	42.5139	1.39484	0.697418	−	0.716665i	$-0.254332\pi$
$929$	42.5139	1.39484	0.697418	+	0.716665i	$0.254332\pi$
$930$	0	0
$931$	−4.90833	−0.160864
$932$	0	0
$933$	−25.4222	−0.832286
$934$	0	0
$935$	3.00000	0.0981105
$936$	0	0
$937$	−38.4861	−1.25729	−0.628643	−	0.777694i	$-0.716389\pi$
$937$	−38.4861	−1.25729	−0.628643	+	0.777694i	$0.716389\pi$
$938$	0	0
$939$	33.0278	1.07782
$940$	0	0
$941$	40.6611	1.32551	0.662756	−	0.748835i	$-0.269386\pi$
$941$	40.6611	1.32551	0.662756	+	0.748835i	$0.269386\pi$
$942$	0	0
$943$	36.8444	1.19982
$944$	0	0
$945$	69.7527	2.26906
$946$	0	0
$947$	−16.7527	−0.544391	−0.272195	−	0.962242i	$-0.587750\pi$
$947$	−16.7527	−0.544391	−0.272195	+	0.962242i	$0.587750\pi$
$948$	0	0
$949$	1.39445	0.0452657
$950$	0	0
$951$	−21.8167	−0.707453
$952$	0	0
$953$	24.6056	0.797052	0.398526	−	0.917157i	$-0.369522\pi$
$953$	24.6056	0.797052	0.398526	+	0.917157i	$0.369522\pi$
$954$	0	0
$955$	13.8167	0.447096
$956$	0	0
$957$	−13.2111	−0.427054
$958$	0	0
$959$	−3.21110	−0.103692
$960$	0	0
$961$	−3.84441	−0.124013
$962$	0	0
$963$	152.653	4.91916
$964$	0	0
$965$	−91.6611	−2.95067
$966$	0	0
$967$	−30.6611	−0.985993	−0.492997	−	0.870031i	$-0.664099\pi$
$967$	−30.6611	−0.985993	−0.492997	+	0.870031i	$0.664099\pi$
$968$	0	0
$969$	11.3028	0.363097
$970$	0	0
$971$	−8.72498	−0.279998	−0.139999	−	0.990152i	$-0.544710\pi$
$971$	−8.72498	−0.279998	−0.139999	+	0.990152i	$0.544710\pi$
$972$	0	0
$973$	21.2111	0.679997
$974$	0	0
$975$	−44.6333	−1.42941
$976$	0	0
$977$	4.84441	0.154986	0.0774932	−	0.996993i	$-0.475308\pi$
$977$	4.84441	0.154986	0.0774932	+	0.996993i	$0.475308\pi$
$978$	0	0
$979$	−5.30278	−0.169477
$980$	0	0
$981$	97.2944	3.10637
$982$	0	0
$983$	−51.3944	−1.63923	−0.819614	−	0.572916i	$-0.805812\pi$
$983$	−51.3944	−1.63923	−0.819614	+	0.572916i	$0.805812\pi$
$984$	0	0
$985$	−13.4222	−0.427667
$986$	0	0
$987$	−17.2111	−0.547835
$988$	0	0
$989$	47.2666	1.50299
$990$	0	0
$991$	−18.0000	−0.571789	−0.285894	−	0.958261i	$-0.592291\pi$
$991$	−18.0000	−0.571789	−0.285894	+	0.958261i	$0.592291\pi$
$992$	0	0
$993$	105.478	3.34723
$994$	0	0
$995$	−15.9083	−0.504328
$996$	0	0
$997$	7.27502	0.230402	0.115201	−	0.993342i	$-0.463249\pi$
$997$	7.27502	0.230402	0.115201	+	0.993342i	$0.463249\pi$
$998$	0	0
$999$	139.505	4.41376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.h.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.h.1.2	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.30278 q^{3} +4.30278 q^{5} +1.00000 q^{7} +7.90833 q^{9} +O(q^{10})\)	\(q+3.30278 q^{3} +4.30278 q^{5} +1.00000 q^{7} +7.90833 q^{9} -1.00000 q^{11} -1.00000 q^{13} +14.2111 q^{15} -0.697224 q^{17} -4.90833 q^{19} +3.30278 q^{21} -8.00000 q^{23} +13.5139 q^{25} +16.2111 q^{27} +4.00000 q^{29} +5.21110 q^{31} -3.30278 q^{33} +4.30278 q^{35} +8.60555 q^{37} -3.30278 q^{39} -4.60555 q^{41} -5.90833 q^{43} +34.0278 q^{45} -5.21110 q^{47} +1.00000 q^{49} -2.30278 q^{51} +11.9083 q^{53} -4.30278 q^{55} -16.2111 q^{57} +3.21110 q^{59} -13.1194 q^{61} +7.90833 q^{63} -4.30278 q^{65} +9.30278 q^{67} -26.4222 q^{69} -16.1194 q^{71} -1.39445 q^{73} +44.6333 q^{75} -1.00000 q^{77} -8.90833 q^{79} +29.8167 q^{81} +7.09167 q^{83} -3.00000 q^{85} +13.2111 q^{87} +5.30278 q^{89} -1.00000 q^{91} +17.2111 q^{93} -21.1194 q^{95} -10.0000 q^{97} -7.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 5 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 5 * q^5 + 2 * q^7 + 5 * q^9	\( 2 q + 3 q^{3} + 5 q^{5} + 2 q^{7} + 5 q^{9} - 2 q^{11} - 2 q^{13} + 14 q^{15} - 5 q^{17} + q^{19} + 3 q^{21} - 16 q^{23} + 9 q^{25} + 18 q^{27} + 8 q^{29} - 4 q^{31} - 3 q^{33} + 5 q^{35} + 10 q^{37} - 3 q^{39} - 2 q^{41} - q^{43} + 32 q^{45} + 4 q^{47} + 2 q^{49} - q^{51} + 13 q^{53} - 5 q^{55} - 18 q^{57} - 8 q^{59} - q^{61} + 5 q^{63} - 5 q^{65} + 15 q^{67} - 24 q^{69} - 7 q^{71} - 10 q^{73} + 46 q^{75} - 2 q^{77} - 7 q^{79} + 38 q^{81} + 25 q^{83} - 6 q^{85} + 12 q^{87} + 7 q^{89} - 2 q^{91} + 20 q^{93} - 17 q^{95} - 20 q^{97} - 5 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 5 * q^5 + 2 * q^7 + 5 * q^9 - 2 * q^11 - 2 * q^13 + 14 * q^15 - 5 * q^17 + q^19 + 3 * q^21 - 16 * q^23 + 9 * q^25 + 18 * q^27 + 8 * q^29 - 4 * q^31 - 3 * q^33 + 5 * q^35 + 10 * q^37 - 3 * q^39 - 2 * q^41 - q^43 + 32 * q^45 + 4 * q^47 + 2 * q^49 - q^51 + 13 * q^53 - 5 * q^55 - 18 * q^57 - 8 * q^59 - q^61 + 5 * q^63 - 5 * q^65 + 15 * q^67 - 24 * q^69 - 7 * q^71 - 10 * q^73 + 46 * q^75 - 2 * q^77 - 7 * q^79 + 38 * q^81 + 25 * q^83 - 6 * q^85 + 12 * q^87 + 7 * q^89 - 2 * q^91 + 20 * q^93 - 17 * q^95 - 20 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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