LMFDB - Embedded newform 208.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,8,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	208.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:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+73.0000 q^{3} -295.000 q^{5} -1373.00 q^{7} +3142.00 q^{9} +O(q^{10})$

$q+73.0000 q^{3} -295.000 q^{5} -1373.00 q^{7} +3142.00 q^{9} +7646.00 q^{11} +2197.00 q^{13} -21535.0 q^{15} -4147.00 q^{17} +3186.00 q^{19} -100229. q^{21} +17784.0 q^{23} +8900.00 q^{25} +69715.0 q^{27} -93322.0 q^{29} +124484. q^{31} +558158. q^{33} +405035. q^{35} +273661. q^{37} +160381. q^{39} +585816. q^{41} +533559. q^{43} -926890. q^{45} +530055. q^{47} +1.06159e6 q^{49} -302731. q^{51} -615288. q^{53} -2.25557e6 q^{55} +232578. q^{57} +392514. q^{59} +1.87806e6 q^{61} -4.31397e6 q^{63} -648115. q^{65} +3.97144e6 q^{67} +1.29823e6 q^{69} +3.74660e6 q^{71} +2.48580e6 q^{73} +649700. q^{75} -1.04980e7 q^{77} +1.26446e6 q^{79} -1.78236e6 q^{81} -434308. q^{83} +1.22336e6 q^{85} -6.81251e6 q^{87} +5.83081e6 q^{89} -3.01648e6 q^{91} +9.08733e6 q^{93} -939870. q^{95} -2.04533e6 q^{97} +2.40237e7 q^{99} +O(q^{100})$

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
$3$	73.0000	1.56098	0.780492	−	0.625166i	$-0.214969\pi$
$3$	73.0000	1.56098	0.780492	+	0.625166i	$0.214969\pi$
$4$	0	0
$5$	−295.000	−1.05542	−0.527712	−	0.849423i	$-0.676950\pi$
$5$	−295.000	−1.05542	−0.527712	+	0.849423i	$0.676950\pi$
$6$	0	0
$7$	−1373.00	−1.51296	−0.756480	−	0.654017i	$-0.773082\pi$
$7$	−1373.00	−1.51296	−0.756480	+	0.654017i	$0.773082\pi$
$8$	0	0
$9$	3142.00	1.43667
$10$	0	0
$11$	7646.00	1.73205	0.866024	−	0.500003i	$-0.166668\pi$
$11$	7646.00	1.73205	0.866024	+	0.500003i	$0.166668\pi$
$12$	0	0
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	0	0
$15$	−21535.0	−1.64750
$16$	0	0
$17$	−4147.00	−0.204721	−0.102361	−	0.994747i	$-0.532640\pi$
$17$	−4147.00	−0.204721	−0.102361	+	0.994747i	$0.532640\pi$
$18$	0	0
$19$	3186.00	0.106563	0.0532817	−	0.998580i	$-0.483032\pi$
$19$	3186.00	0.106563	0.0532817	+	0.998580i	$0.483032\pi$
$20$	0	0
$21$	−100229.	−2.36171
$22$	0	0
$23$	17784.0	0.304777	0.152388	−	0.988321i	$-0.451304\pi$
$23$	17784.0	0.304777	0.152388	+	0.988321i	$0.451304\pi$
$24$	0	0
$25$	8900.00	0.113920
$26$	0	0
$27$	69715.0	0.681637
$28$	0	0
$29$	−93322.0	−0.710544	−0.355272	−	0.934763i	$-0.615612\pi$
$29$	−93322.0	−0.710544	−0.355272	+	0.934763i	$0.615612\pi$
$30$	0	0
$31$	124484.	0.750495	0.375247	−	0.926925i	$-0.377558\pi$
$31$	124484.	0.750495	0.375247	+	0.926925i	$0.377558\pi$
$32$	0	0
$33$	558158.	2.70370
$34$	0	0
$35$	405035.	1.59681
$36$	0	0
$37$	273661.	0.888192	0.444096	−	0.895979i	$-0.353525\pi$
$37$	273661.	0.888192	0.444096	+	0.895979i	$0.353525\pi$
$38$	0	0
$39$	160381.	0.432939
$40$	0	0
$41$	585816.	1.32745	0.663724	−	0.747977i	$-0.268975\pi$
$41$	585816.	1.32745	0.663724	+	0.747977i	$0.268975\pi$
$42$	0	0
$43$	533559.	1.02339	0.511697	−	0.859166i	$-0.329017\pi$
$43$	533559.	1.02339	0.511697	+	0.859166i	$0.329017\pi$
$44$	0	0
$45$	−926890.	−1.51630
$46$	0	0
$47$	530055.	0.744695	0.372347	−	0.928093i	$-0.378553\pi$
$47$	530055.	0.744695	0.372347	+	0.928093i	$0.378553\pi$
$48$	0	0
$49$	1.06159e6	1.28905
$50$	0	0
$51$	−302731.	−0.319567
$52$	0	0
$53$	−615288.	−0.567692	−0.283846	−	0.958870i	$-0.591610\pi$
$53$	−615288.	−0.567692	−0.283846	+	0.958870i	$0.591610\pi$
$54$	0	0
$55$	−2.25557e6	−1.82805
$56$	0	0
$57$	232578.	0.166344
$58$	0	0
$59$	392514.	0.248813	0.124407	−	0.992231i	$-0.460297\pi$
$59$	392514.	0.248813	0.124407	+	0.992231i	$0.460297\pi$
$60$	0	0
$61$	1.87806e6	1.05939	0.529695	−	0.848188i	$-0.322306\pi$
$61$	1.87806e6	1.05939	0.529695	+	0.848188i	$0.322306\pi$
$62$	0	0
$63$	−4.31397e6	−2.17363
$64$	0	0
$65$	−648115.	−0.292722
$66$	0	0
$67$	3.97144e6	1.61319	0.806596	−	0.591103i	$-0.201307\pi$
$67$	3.97144e6	1.61319	0.806596	+	0.591103i	$0.201307\pi$
$68$	0	0
$69$	1.29823e6	0.475752
$70$	0	0
$71$	3.74660e6	1.24232	0.621160	−	0.783684i	$-0.286662\pi$
$71$	3.74660e6	1.24232	0.621160	+	0.783684i	$0.286662\pi$
$72$	0	0
$73$	2.48580e6	0.747888	0.373944	−	0.927451i	$-0.378005\pi$
$73$	2.48580e6	0.747888	0.373944	+	0.927451i	$0.378005\pi$
$74$	0	0
$75$	649700.	0.177827
$76$	0	0
$77$	−1.04980e7	−2.62052
$78$	0	0
$79$	1.26446e6	0.288542	0.144271	−	0.989538i	$-0.453916\pi$
$79$	1.26446e6	0.288542	0.144271	+	0.989538i	$0.453916\pi$
$80$	0	0
$81$	−1.78236e6	−0.372647
$82$	0	0
$83$	−434308.	−0.0833728	−0.0416864	−	0.999131i	$-0.513273\pi$
$83$	−434308.	−0.0833728	−0.0416864	+	0.999131i	$0.513273\pi$
$84$	0	0
$85$	1.22336e6	0.216068
$86$	0	0
$87$	−6.81251e6	−1.10915
$88$	0	0
$89$	5.83081e6	0.876726	0.438363	−	0.898798i	$-0.355558\pi$
$89$	5.83081e6	0.876726	0.438363	+	0.898798i	$0.355558\pi$
$90$	0	0
$91$	−3.01648e6	−0.419620
$92$	0	0
$93$	9.08733e6	1.17151
$94$	0	0
$95$	−939870.	−0.112470
$96$	0	0
$97$	−2.04533e6	−0.227542	−0.113771	−	0.993507i	$-0.536293\pi$
$97$	−2.04533e6	−0.227542	−0.113771	+	0.993507i	$0.536293\pi$
$98$	0	0
$99$	2.40237e7	2.48838
$100$	0	0
$101$	−1.55142e6	−0.149832	−0.0749160	−	0.997190i	$-0.523869\pi$
$101$	−1.55142e6	−0.149832	−0.0749160	+	0.997190i	$0.523869\pi$
$102$	0	0
$103$	1.68251e7	1.51714	0.758572	−	0.651590i	$-0.225898\pi$
$103$	1.68251e7	1.51714	0.758572	+	0.651590i	$0.225898\pi$
$104$	0	0
$105$	2.95676e7	2.49260
$106$	0	0
$107$	−2.19295e7	−1.73055	−0.865277	−	0.501294i	$-0.832858\pi$
$107$	−2.19295e7	−1.73055	−0.865277	+	0.501294i	$0.832858\pi$
$108$	0	0
$109$	−1.96595e7	−1.45405	−0.727024	−	0.686612i	$-0.759097\pi$
$109$	−1.96595e7	−1.45405	−0.727024	+	0.686612i	$0.759097\pi$
$110$	0	0
$111$	1.99773e7	1.38645
$112$	0	0
$113$	−2.14963e7	−1.40149	−0.700744	−	0.713412i	$-0.747149\pi$
$113$	−2.14963e7	−1.40149	−0.700744	+	0.713412i	$0.747149\pi$
$114$	0	0
$115$	−5.24628e6	−0.321669
$116$	0	0
$117$	6.90297e6	0.398461
$118$	0	0
$119$	5.69383e6	0.309735
$120$	0	0
$121$	3.89741e7	1.99999
$122$	0	0
$123$	4.27646e7	2.07213
$124$	0	0
$125$	2.04214e7	0.935190
$126$	0	0
$127$	1.77419e7	0.768578	0.384289	−	0.923213i	$-0.374447\pi$
$127$	1.77419e7	0.768578	0.384289	+	0.923213i	$0.374447\pi$
$128$	0	0
$129$	3.89498e7	1.59750
$130$	0	0
$131$	8.61184e6	0.334693	0.167346	−	0.985898i	$-0.446480\pi$
$131$	8.61184e6	0.334693	0.167346	+	0.985898i	$0.446480\pi$
$132$	0	0
$133$	−4.37438e6	−0.161226
$134$	0	0
$135$	−2.05659e7	−0.719416
$136$	0	0
$137$	6.30262e6	0.209411	0.104705	−	0.994503i	$-0.466610\pi$
$137$	6.30262e6	0.209411	0.104705	+	0.994503i	$0.466610\pi$
$138$	0	0
$139$	−1.34997e7	−0.426355	−0.213177	−	0.977014i	$-0.568381\pi$
$139$	−1.34997e7	−0.426355	−0.213177	+	0.977014i	$0.568381\pi$
$140$	0	0
$141$	3.86940e7	1.16246
$142$	0	0
$143$	1.67983e7	0.480384
$144$	0	0
$145$	2.75300e7	0.749925
$146$	0	0
$147$	7.74958e7	2.01218
$148$	0	0
$149$	−1.43791e7	−0.356105	−0.178053	−	0.984021i	$-0.556980\pi$
$149$	−1.43791e7	−0.356105	−0.178053	+	0.984021i	$0.556980\pi$
$150$	0	0
$151$	−8.24764e7	−1.94944	−0.974721	−	0.223424i	$-0.928277\pi$
$151$	−8.24764e7	−1.94944	−0.974721	+	0.223424i	$0.928277\pi$
$152$	0	0
$153$	−1.30299e7	−0.294117
$154$	0	0
$155$	−3.67228e7	−0.792090
$156$	0	0
$157$	8.92107e6	0.183979	0.0919895	−	0.995760i	$-0.470677\pi$
$157$	8.92107e6	0.183979	0.0919895	+	0.995760i	$0.470677\pi$
$158$	0	0
$159$	−4.49160e7	−0.886158
$160$	0	0
$161$	−2.44174e7	−0.461115
$162$	0	0
$163$	−2.09065e7	−0.378116	−0.189058	−	0.981966i	$-0.560543\pi$
$163$	−2.09065e7	−0.378116	−0.189058	+	0.981966i	$0.560543\pi$
$164$	0	0
$165$	−1.64657e8	−2.85355
$166$	0	0
$167$	1.88221e7	0.312724	0.156362	−	0.987700i	$-0.450023\pi$
$167$	1.88221e7	0.312724	0.156362	+	0.987700i	$0.450023\pi$
$168$	0	0
$169$	4.82681e6	0.0769231
$170$	0	0
$171$	1.00104e7	0.153097
$172$	0	0
$173$	−4.78358e6	−0.0702412	−0.0351206	−	0.999383i	$-0.511182\pi$
$173$	−4.78358e6	−0.0702412	−0.0351206	+	0.999383i	$0.511182\pi$
$174$	0	0
$175$	−1.22197e7	−0.172356
$176$	0	0
$177$	2.86535e7	0.388393
$178$	0	0
$179$	−9.09914e7	−1.18581	−0.592904	−	0.805273i	$-0.702019\pi$
$179$	−9.09914e7	−1.18581	−0.592904	+	0.805273i	$0.702019\pi$
$180$	0	0
$181$	−1.72015e7	−0.215622	−0.107811	−	0.994171i	$-0.534384\pi$
$181$	−1.72015e7	−0.215622	−0.107811	+	0.994171i	$0.534384\pi$
$182$	0	0
$183$	1.37099e8	1.65369
$184$	0	0
$185$	−8.07300e7	−0.937419
$186$	0	0
$187$	−3.17080e7	−0.354587
$188$	0	0
$189$	−9.57187e7	−1.03129
$190$	0	0
$191$	6.68698e7	0.694405	0.347203	−	0.937790i	$-0.387132\pi$
$191$	6.68698e7	0.694405	0.347203	+	0.937790i	$0.387132\pi$
$192$	0	0
$193$	−4.86222e7	−0.486838	−0.243419	−	0.969921i	$-0.578269\pi$
$193$	−4.86222e7	−0.486838	−0.243419	+	0.969921i	$0.578269\pi$
$194$	0	0
$195$	−4.73124e7	−0.456934
$196$	0	0
$197$	−8.42682e7	−0.785293	−0.392646	−	0.919689i	$-0.628440\pi$
$197$	−8.42682e7	−0.785293	−0.392646	+	0.919689i	$0.628440\pi$
$198$	0	0
$199$	−1.39905e8	−1.25849	−0.629243	−	0.777208i	$-0.716635\pi$
$199$	−1.39905e8	−1.25849	−0.629243	+	0.777208i	$0.716635\pi$
$200$	0	0
$201$	2.89915e8	2.51817
$202$	0	0
$203$	1.28131e8	1.07502
$204$	0	0
$205$	−1.72816e8	−1.40102
$206$	0	0
$207$	5.58773e7	0.437864
$208$	0	0
$209$	2.43602e7	0.184573
$210$	0	0
$211$	−2.26349e8	−1.65878	−0.829391	−	0.558669i	$-0.811312\pi$
$211$	−2.26349e8	−1.65878	−0.829391	+	0.558669i	$0.811312\pi$
$212$	0	0
$213$	2.73502e8	1.93924
$214$	0	0
$215$	−1.57400e8	−1.08011
$216$	0	0
$217$	−1.70917e8	−1.13547
$218$	0	0
$219$	1.81464e8	1.16744
$220$	0	0
$221$	−9.11096e6	−0.0567794
$222$	0	0
$223$	2.19897e8	1.32786	0.663929	−	0.747796i	$-0.268888\pi$
$223$	2.19897e8	1.32786	0.663929	+	0.747796i	$0.268888\pi$
$224$	0	0
$225$	2.79638e7	0.163666
$226$	0	0
$227$	2.30377e8	1.30722	0.653611	−	0.756831i	$-0.273253\pi$
$227$	2.30377e8	1.30722	0.653611	+	0.756831i	$0.273253\pi$
$228$	0	0
$229$	−5.41755e7	−0.298111	−0.149056	−	0.988829i	$-0.547623\pi$
$229$	−5.41755e7	−0.298111	−0.149056	+	0.988829i	$0.547623\pi$
$230$	0	0
$231$	−7.66351e8	−4.09059
$232$	0	0
$233$	1.41580e8	0.733259	0.366629	−	0.930367i	$-0.380512\pi$
$233$	1.41580e8	0.733259	0.366629	+	0.930367i	$0.380512\pi$
$234$	0	0
$235$	−1.56366e8	−0.785969
$236$	0	0
$237$	9.23053e7	0.450409
$238$	0	0
$239$	2.57365e8	1.21943	0.609715	−	0.792621i	$-0.291284\pi$
$239$	2.57365e8	1.21943	0.609715	+	0.792621i	$0.291284\pi$
$240$	0	0
$241$	−2.46818e8	−1.13584	−0.567921	−	0.823083i	$-0.692252\pi$
$241$	−2.46818e8	−1.13584	−0.567921	+	0.823083i	$0.692252\pi$
$242$	0	0
$243$	−2.82579e8	−1.26333
$244$	0	0
$245$	−3.13168e8	−1.36049
$246$	0	0
$247$	6.99964e6	0.0295554
$248$	0	0
$249$	−3.17045e7	−0.130144
$250$	0	0
$251$	−2.39628e7	−0.0956490	−0.0478245	−	0.998856i	$-0.515229\pi$
$251$	−2.39628e7	−0.0956490	−0.0478245	+	0.998856i	$0.515229\pi$
$252$	0	0
$253$	1.35976e8	0.527888
$254$	0	0
$255$	8.93056e7	0.337278
$256$	0	0
$257$	−2.50050e8	−0.918885	−0.459443	−	0.888207i	$-0.651951\pi$
$257$	−2.50050e8	−0.918885	−0.459443	+	0.888207i	$0.651951\pi$
$258$	0	0
$259$	−3.75737e8	−1.34380
$260$	0	0
$261$	−2.93218e8	−1.02082
$262$	0	0
$263$	2.09182e8	0.709055	0.354527	−	0.935046i	$-0.384642\pi$
$263$	2.09182e8	0.709055	0.354527	+	0.935046i	$0.384642\pi$
$264$	0	0
$265$	1.81510e8	0.599156
$266$	0	0
$267$	4.25649e8	1.36856
$268$	0	0
$269$	3.71414e8	1.16339	0.581695	−	0.813407i	$-0.302390\pi$
$269$	3.71414e8	1.16339	0.581695	+	0.813407i	$0.302390\pi$
$270$	0	0
$271$	−3.30225e8	−1.00790	−0.503950	−	0.863733i	$-0.668121\pi$
$271$	−3.30225e8	−1.00790	−0.503950	+	0.863733i	$0.668121\pi$
$272$	0	0
$273$	−2.20203e8	−0.655019
$274$	0	0
$275$	6.80494e7	0.197315
$276$	0	0
$277$	5.06278e8	1.43123	0.715616	−	0.698494i	$-0.246146\pi$
$277$	5.06278e8	1.43123	0.715616	+	0.698494i	$0.246146\pi$
$278$	0	0
$279$	3.91129e8	1.07821
$280$	0	0
$281$	1.06744e8	0.286994	0.143497	−	0.989651i	$-0.454165\pi$
$281$	1.06744e8	0.286994	0.143497	+	0.989651i	$0.454165\pi$
$282$	0	0
$283$	5.56521e8	1.45958	0.729792	−	0.683669i	$-0.239617\pi$
$283$	5.56521e8	1.45958	0.729792	+	0.683669i	$0.239617\pi$
$284$	0	0
$285$	−6.86105e7	−0.175563
$286$	0	0
$287$	−8.04325e8	−2.00838
$288$	0	0
$289$	−3.93141e8	−0.958089
$290$	0	0
$291$	−1.49309e8	−0.355190
$292$	0	0
$293$	2.23708e8	0.519571	0.259786	−	0.965666i	$-0.416348\pi$
$293$	2.23708e8	0.519571	0.259786	+	0.965666i	$0.416348\pi$
$294$	0	0
$295$	−1.15792e8	−0.262603
$296$	0	0
$297$	5.33041e8	1.18063
$298$	0	0
$299$	3.90714e7	0.0845299
$300$	0	0
$301$	−7.32577e8	−1.54835
$302$	0	0
$303$	−1.13254e8	−0.233885
$304$	0	0
$305$	−5.54029e8	−1.11811
$306$	0	0
$307$	9.91919e8	1.95655	0.978277	−	0.207300i	$-0.0664678\pi$
$307$	9.91919e8	1.95655	0.978277	+	0.207300i	$0.0664678\pi$
$308$	0	0
$309$	1.22823e9	2.36824
$310$	0	0
$311$	2.48269e8	0.468016	0.234008	−	0.972235i	$-0.424816\pi$
$311$	2.48269e8	0.468016	0.234008	+	0.972235i	$0.424816\pi$
$312$	0	0
$313$	−2.00737e8	−0.370018	−0.185009	−	0.982737i	$-0.559231\pi$
$313$	−2.00737e8	−0.370018	−0.185009	+	0.982737i	$0.559231\pi$
$314$	0	0
$315$	1.27262e9	2.29410
$316$	0	0
$317$	−1.02635e8	−0.180962	−0.0904808	−	0.995898i	$-0.528840\pi$
$317$	−1.02635e8	−0.180962	−0.0904808	+	0.995898i	$0.528840\pi$
$318$	0	0
$319$	−7.13540e8	−1.23070
$320$	0	0
$321$	−1.60085e9	−2.70137
$322$	0	0
$323$	−1.32123e7	−0.0218158
$324$	0	0
$325$	1.95533e7	0.0315957
$326$	0	0
$327$	−1.43514e9	−2.26975
$328$	0	0
$329$	−7.27766e8	−1.12669
$330$	0	0
$331$	7.60053e8	1.15198	0.575991	−	0.817456i	$-0.304616\pi$
$331$	7.60053e8	1.15198	0.575991	+	0.817456i	$0.304616\pi$
$332$	0	0
$333$	8.59843e8	1.27604
$334$	0	0
$335$	−1.17157e9	−1.70260
$336$	0	0
$337$	−4.36659e7	−0.0621495	−0.0310748	−	0.999517i	$-0.509893\pi$
$337$	−4.36659e7	−0.0621495	−0.0310748	+	0.999517i	$0.509893\pi$
$338$	0	0
$339$	−1.56923e9	−2.18770
$340$	0	0
$341$	9.51805e8	1.29989
$342$	0	0
$343$	−3.26833e8	−0.437317
$344$	0	0
$345$	−3.82978e8	−0.502120
$346$	0	0
$347$	1.82053e8	0.233907	0.116954	−	0.993137i	$-0.462687\pi$
$347$	1.82053e8	0.233907	0.116954	+	0.993137i	$0.462687\pi$
$348$	0	0
$349$	−5.80955e8	−0.731566	−0.365783	−	0.930700i	$-0.619199\pi$
$349$	−5.80955e8	−0.731566	−0.365783	+	0.930700i	$0.619199\pi$
$350$	0	0
$351$	1.53164e8	0.189052
$352$	0	0
$353$	5.45624e8	0.660210	0.330105	−	0.943944i	$-0.392916\pi$
$353$	5.45624e8	0.660210	0.330105	+	0.943944i	$0.392916\pi$
$354$	0	0
$355$	−1.10525e9	−1.31117
$356$	0	0
$357$	4.15650e8	0.483491
$358$	0	0
$359$	1.05196e9	1.19996	0.599981	−	0.800014i	$-0.295175\pi$
$359$	1.05196e9	1.19996	0.599981	+	0.800014i	$0.295175\pi$
$360$	0	0
$361$	−8.83721e8	−0.988644
$362$	0	0
$363$	2.84511e9	3.12195
$364$	0	0
$365$	−7.33312e8	−0.789339
$366$	0	0
$367$	−7.29203e8	−0.770047	−0.385024	−	0.922907i	$-0.625807\pi$
$367$	−7.29203e8	−0.770047	−0.385024	+	0.922907i	$0.625807\pi$
$368$	0	0
$369$	1.84063e9	1.90711
$370$	0	0
$371$	8.44790e8	0.858895
$372$	0	0
$373$	1.32385e9	1.32087	0.660434	−	0.750884i	$-0.270372\pi$
$373$	1.32385e9	1.32087	0.660434	+	0.750884i	$0.270372\pi$
$374$	0	0
$375$	1.49076e9	1.45982
$376$	0	0
$377$	−2.05028e8	−0.197069
$378$	0	0
$379$	1.08474e8	0.102350	0.0511752	−	0.998690i	$-0.483703\pi$
$379$	1.08474e8	0.102350	0.0511752	+	0.998690i	$0.483703\pi$
$380$	0	0
$381$	1.29516e9	1.19974
$382$	0	0
$383$	−2.84754e8	−0.258985	−0.129492	−	0.991580i	$-0.541335\pi$
$383$	−2.84754e8	−0.258985	−0.129492	+	0.991580i	$0.541335\pi$
$384$	0	0
$385$	3.09690e9	2.76576
$386$	0	0
$387$	1.67644e9	1.47028
$388$	0	0
$389$	3.22741e8	0.277991	0.138996	−	0.990293i	$-0.455613\pi$
$389$	3.22741e8	0.277991	0.138996	+	0.990293i	$0.455613\pi$
$390$	0	0
$391$	−7.37502e7	−0.0623943
$392$	0	0
$393$	6.28664e8	0.522450
$394$	0	0
$395$	−3.73015e8	−0.304534
$396$	0	0
$397$	−8.64634e8	−0.693531	−0.346765	−	0.937952i	$-0.612720\pi$
$397$	−8.64634e8	−0.693531	−0.346765	+	0.937952i	$0.612720\pi$
$398$	0	0
$399$	−3.19330e8	−0.251671
$400$	0	0
$401$	−1.31166e9	−1.01582	−0.507909	−	0.861411i	$-0.669581\pi$
$401$	−1.31166e9	−1.01582	−0.507909	+	0.861411i	$0.669581\pi$
$402$	0	0
$403$	2.73491e8	0.208150
$404$	0	0
$405$	5.25796e8	0.393301
$406$	0	0
$407$	2.09241e9	1.53839
$408$	0	0
$409$	−4.00024e7	−0.0289104	−0.0144552	−	0.999896i	$-0.504601\pi$
$409$	−4.00024e7	−0.0289104	−0.0144552	+	0.999896i	$0.504601\pi$
$410$	0	0
$411$	4.60091e8	0.326887
$412$	0	0
$413$	−5.38922e8	−0.376444
$414$	0	0
$415$	1.28121e8	0.0879937
$416$	0	0
$417$	−9.85475e8	−0.665533
$418$	0	0
$419$	−2.64978e9	−1.75979	−0.879896	−	0.475166i	$-0.842388\pi$
$419$	−2.64978e9	−1.75979	−0.879896	+	0.475166i	$0.842388\pi$
$420$	0	0
$421$	8.99741e8	0.587665	0.293833	−	0.955857i	$-0.405069\pi$
$421$	8.99741e8	0.587665	0.293833	+	0.955857i	$0.405069\pi$
$422$	0	0
$423$	1.66543e9	1.06988
$424$	0	0
$425$	−3.69083e7	−0.0233218
$426$	0	0
$427$	−2.57858e9	−1.60281
$428$	0	0
$429$	1.22627e9	0.749871
$430$	0	0
$431$	−3.69212e8	−0.222129	−0.111065	−	0.993813i	$-0.535426\pi$
$431$	−3.69212e8	−0.222129	−0.111065	+	0.993813i	$0.535426\pi$
$432$	0	0
$433$	2.63280e9	1.55851	0.779255	−	0.626707i	$-0.215598\pi$
$433$	2.63280e9	1.55851	0.779255	+	0.626707i	$0.215598\pi$
$434$	0	0
$435$	2.00969e9	1.17062
$436$	0	0
$437$	5.66598e7	0.0324781
$438$	0	0
$439$	−3.44814e8	−0.194518	−0.0972588	−	0.995259i	$-0.531007\pi$
$439$	−3.44814e8	−0.194518	−0.0972588	+	0.995259i	$0.531007\pi$
$440$	0	0
$441$	3.33550e9	1.85194
$442$	0	0
$443$	1.68347e9	0.920012	0.460006	−	0.887916i	$-0.347847\pi$
$443$	1.68347e9	0.920012	0.460006	+	0.887916i	$0.347847\pi$
$444$	0	0
$445$	−1.72009e9	−0.925318
$446$	0	0
$447$	−1.04967e9	−0.555875
$448$	0	0
$449$	3.20869e9	1.67288	0.836440	−	0.548058i	$-0.184633\pi$
$449$	3.20869e9	1.67288	0.836440	+	0.548058i	$0.184633\pi$
$450$	0	0
$451$	4.47915e9	2.29920
$452$	0	0
$453$	−6.02078e9	−3.04305
$454$	0	0
$455$	8.89862e8	0.442877
$456$	0	0
$457$	−8.53834e8	−0.418473	−0.209236	−	0.977865i	$-0.567098\pi$
$457$	−8.53834e8	−0.418473	−0.209236	+	0.977865i	$0.567098\pi$
$458$	0	0
$459$	−2.89108e8	−0.139546
$460$	0	0
$461$	−3.91799e9	−1.86256	−0.931279	−	0.364308i	$-0.881306\pi$
$461$	−3.91799e9	−1.86256	−0.931279	+	0.364308i	$0.881306\pi$
$462$	0	0
$463$	9.00831e8	0.421803	0.210902	−	0.977507i	$-0.432360\pi$
$463$	9.00831e8	0.421803	0.210902	+	0.977507i	$0.432360\pi$
$464$	0	0
$465$	−2.68076e9	−1.23644
$466$	0	0
$467$	8.14889e7	0.0370245	0.0185123	−	0.999829i	$-0.494107\pi$
$467$	8.14889e7	0.0370245	0.0185123	+	0.999829i	$0.494107\pi$
$468$	0	0
$469$	−5.45278e9	−2.44069
$470$	0	0
$471$	6.51238e8	0.287188
$472$	0	0
$473$	4.07959e9	1.77257
$474$	0	0
$475$	2.83554e7	0.0121397
$476$	0	0
$477$	−1.93323e9	−0.815587
$478$	0	0
$479$	−4.28234e9	−1.78036	−0.890178	−	0.455612i	$-0.849420\pi$
$479$	−4.28234e9	−1.78036	−0.890178	+	0.455612i	$0.849420\pi$
$480$	0	0
$481$	6.01233e8	0.246340
$482$	0	0
$483$	−1.78247e9	−0.719793
$484$	0	0
$485$	6.03372e8	0.240154
$486$	0	0
$487$	−4.24063e9	−1.66371	−0.831857	−	0.554990i	$-0.812722\pi$
$487$	−4.24063e9	−1.66371	−0.831857	+	0.554990i	$0.812722\pi$
$488$	0	0
$489$	−1.52617e9	−0.590233
$490$	0	0
$491$	−2.21387e9	−0.844046	−0.422023	−	0.906585i	$-0.638680\pi$
$491$	−2.21387e9	−0.844046	−0.422023	+	0.906585i	$0.638680\pi$
$492$	0	0
$493$	3.87006e8	0.145463
$494$	0	0
$495$	−7.08700e9	−2.62630
$496$	0	0
$497$	−5.14408e9	−1.87958
$498$	0	0
$499$	2.45975e9	0.886215	0.443108	−	0.896468i	$-0.353876\pi$
$499$	2.45975e9	0.886215	0.443108	+	0.896468i	$0.353876\pi$
$500$	0	0
$501$	1.37401e9	0.488156
$502$	0	0
$503$	−3.72798e9	−1.30613	−0.653063	−	0.757303i	$-0.726516\pi$
$503$	−3.72798e9	−1.30613	−0.653063	+	0.757303i	$0.726516\pi$
$504$	0	0
$505$	4.57669e8	0.158136
$506$	0	0
$507$	3.52357e8	0.120076
$508$	0	0
$509$	1.48553e9	0.499308	0.249654	−	0.968335i	$-0.419683\pi$
$509$	1.48553e9	0.499308	0.249654	+	0.968335i	$0.419683\pi$
$510$	0	0
$511$	−3.41301e9	−1.13152
$512$	0	0
$513$	2.22112e8	0.0726376
$514$	0	0
$515$	−4.96340e9	−1.60123
$516$	0	0
$517$	4.05280e9	1.28985
$518$	0	0
$519$	−3.49202e8	−0.109645
$520$	0	0
$521$	1.06857e9	0.331031	0.165516	−	0.986207i	$-0.447071\pi$
$521$	1.06857e9	0.331031	0.165516	+	0.986207i	$0.447071\pi$
$522$	0	0
$523$	3.85266e8	0.117762	0.0588809	−	0.998265i	$-0.481247\pi$
$523$	3.85266e8	0.117762	0.0588809	+	0.998265i	$0.481247\pi$
$524$	0	0
$525$	−8.92038e8	−0.269046
$526$	0	0
$527$	−5.16235e8	−0.153642
$528$	0	0
$529$	−3.08855e9	−0.907111
$530$	0	0
$531$	1.23328e9	0.357463
$532$	0	0
$533$	1.28704e9	0.368168
$534$	0	0
$535$	6.46920e9	1.82647
$536$	0	0
$537$	−6.64237e9	−1.85103
$538$	0	0
$539$	8.11689e9	2.23269
$540$	0	0
$541$	−2.81334e9	−0.763891	−0.381946	−	0.924185i	$-0.624746\pi$
$541$	−2.81334e9	−0.763891	−0.381946	+	0.924185i	$0.624746\pi$
$542$	0	0
$543$	−1.25571e9	−0.336582
$544$	0	0
$545$	5.79954e9	1.53464
$546$	0	0
$547$	−1.67344e9	−0.437174	−0.218587	−	0.975818i	$-0.570145\pi$
$547$	−1.67344e9	−0.437174	−0.218587	+	0.975818i	$0.570145\pi$
$548$	0	0
$549$	5.90088e9	1.52200
$550$	0	0
$551$	−2.97324e8	−0.0757180
$552$	0	0
$553$	−1.73610e9	−0.436552
$554$	0	0
$555$	−5.89329e9	−1.46330
$556$	0	0
$557$	−4.46631e9	−1.09511	−0.547553	−	0.836771i	$-0.684441\pi$
$557$	−4.46631e9	−1.09511	−0.547553	+	0.836771i	$0.684441\pi$
$558$	0	0
$559$	1.17223e9	0.283838
$560$	0	0
$561$	−2.31468e9	−0.553505
$562$	0	0
$563$	5.06446e9	1.19606	0.598031	−	0.801473i	$-0.295950\pi$
$563$	5.06446e9	1.19606	0.598031	+	0.801473i	$0.295950\pi$
$564$	0	0
$565$	6.34142e9	1.47917
$566$	0	0
$567$	2.44718e9	0.563800
$568$	0	0
$569$	4.07861e9	0.928152	0.464076	−	0.885795i	$-0.346386\pi$
$569$	4.07861e9	0.928152	0.464076	+	0.885795i	$0.346386\pi$
$570$	0	0
$571$	7.82983e9	1.76005	0.880027	−	0.474923i	$-0.157524\pi$
$571$	7.82983e9	1.76005	0.880027	+	0.474923i	$0.157524\pi$
$572$	0	0
$573$	4.88149e9	1.08396
$574$	0	0
$575$	1.58278e8	0.0347202
$576$	0	0
$577$	−7.94179e9	−1.72109	−0.860544	−	0.509376i	$-0.829876\pi$
$577$	−7.94179e9	−1.72109	−0.860544	+	0.509376i	$0.829876\pi$
$578$	0	0
$579$	−3.54942e9	−0.759946
$580$	0	0
$581$	5.96305e8	0.126140
$582$	0	0
$583$	−4.70449e9	−0.983270
$584$	0	0
$585$	−2.03638e9	−0.420545
$586$	0	0
$587$	2.02009e9	0.412227	0.206114	−	0.978528i	$-0.433918\pi$
$587$	2.02009e9	0.412227	0.206114	+	0.978528i	$0.433918\pi$
$588$	0	0
$589$	3.96606e8	0.0799753
$590$	0	0
$591$	−6.15158e9	−1.22583
$592$	0	0
$593$	5.19728e9	1.02349	0.511746	−	0.859137i	$-0.328999\pi$
$593$	5.19728e9	1.02349	0.511746	+	0.859137i	$0.328999\pi$
$594$	0	0
$595$	−1.67968e9	−0.326902
$596$	0	0
$597$	−1.02131e10	−1.96448
$598$	0	0
$599$	3.92347e9	0.745893	0.372946	−	0.927853i	$-0.378348\pi$
$599$	3.92347e9	0.745893	0.372946	+	0.927853i	$0.378348\pi$
$600$	0	0
$601$	−9.51281e8	−0.178751	−0.0893754	−	0.995998i	$-0.528487\pi$
$601$	−9.51281e8	−0.178751	−0.0893754	+	0.995998i	$0.528487\pi$
$602$	0	0
$603$	1.24783e10	2.31763
$604$	0	0
$605$	−1.14974e10	−2.11084
$606$	0	0
$607$	−8.27679e9	−1.50211	−0.751055	−	0.660240i	$-0.770455\pi$
$607$	−8.27679e9	−1.50211	−0.751055	+	0.660240i	$0.770455\pi$
$608$	0	0
$609$	9.35357e9	1.67810
$610$	0	0
$611$	1.16453e9	0.206541
$612$	0	0
$613$	2.92674e9	0.513183	0.256591	−	0.966520i	$-0.417401\pi$
$613$	2.92674e9	0.513183	0.256591	+	0.966520i	$0.417401\pi$
$614$	0	0
$615$	−1.26155e10	−2.18697
$616$	0	0
$617$	8.88587e9	1.52301	0.761503	−	0.648161i	$-0.224462\pi$
$617$	8.88587e9	1.52301	0.761503	+	0.648161i	$0.224462\pi$
$618$	0	0
$619$	4.16163e9	0.705255	0.352627	−	0.935764i	$-0.385288\pi$
$619$	4.16163e9	0.705255	0.352627	+	0.935764i	$0.385288\pi$
$620$	0	0
$621$	1.23981e9	0.207747
$622$	0	0
$623$	−8.00570e9	−1.32645
$624$	0	0
$625$	−6.71962e9	−1.10094
$626$	0	0
$627$	1.77829e9	0.288115
$628$	0	0
$629$	−1.13487e9	−0.181832
$630$	0	0
$631$	7.30070e9	1.15681	0.578405	−	0.815750i	$-0.303676\pi$
$631$	7.30070e9	1.15681	0.578405	+	0.815750i	$0.303676\pi$
$632$	0	0
$633$	−1.65234e10	−2.58933
$634$	0	0
$635$	−5.23387e9	−0.811176
$636$	0	0
$637$	2.33230e9	0.357517
$638$	0	0
$639$	1.17718e10	1.78480
$640$	0	0
$641$	3.46867e9	0.520187	0.260094	−	0.965583i	$-0.416247\pi$
$641$	3.46867e9	0.520187	0.260094	+	0.965583i	$0.416247\pi$
$642$	0	0
$643$	3.72175e9	0.552089	0.276045	−	0.961145i	$-0.410976\pi$
$643$	3.72175e9	0.552089	0.276045	+	0.961145i	$0.410976\pi$
$644$	0	0
$645$	−1.14902e10	−1.68604
$646$	0	0
$647$	8.03095e9	1.16574	0.582870	−	0.812565i	$-0.301930\pi$
$647$	8.03095e9	1.16574	0.582870	+	0.812565i	$0.301930\pi$
$648$	0	0
$649$	3.00116e9	0.430956
$650$	0	0
$651$	−1.24769e10	−1.77245
$652$	0	0
$653$	8.22151e9	1.15546	0.577731	−	0.816227i	$-0.303938\pi$
$653$	8.22151e9	1.15546	0.577731	+	0.816227i	$0.303938\pi$
$654$	0	0
$655$	−2.54049e9	−0.353243
$656$	0	0
$657$	7.81039e9	1.07447
$658$	0	0
$659$	−6.47061e9	−0.880737	−0.440369	−	0.897817i	$-0.645152\pi$
$659$	−6.47061e9	−0.880737	−0.440369	+	0.897817i	$0.645152\pi$
$660$	0	0
$661$	−3.64380e9	−0.490738	−0.245369	−	0.969430i	$-0.578909\pi$
$661$	−3.64380e9	−0.490738	−0.245369	+	0.969430i	$0.578909\pi$
$662$	0	0
$663$	−6.65100e8	−0.0886318
$664$	0	0
$665$	1.29044e9	0.170162
$666$	0	0
$667$	−1.65964e9	−0.216557
$668$	0	0
$669$	1.60524e10	2.07276
$670$	0	0
$671$	1.43597e10	1.83491
$672$	0	0
$673$	2.32463e9	0.293968	0.146984	−	0.989139i	$-0.453043\pi$
$673$	2.32463e9	0.293968	0.146984	+	0.989139i	$0.453043\pi$
$674$	0	0
$675$	6.20464e8	0.0776521
$676$	0	0
$677$	2.19098e9	0.271380	0.135690	−	0.990751i	$-0.456675\pi$
$677$	2.19098e9	0.271380	0.135690	+	0.990751i	$0.456675\pi$
$678$	0	0
$679$	2.80824e9	0.344262
$680$	0	0
$681$	1.68175e10	2.04055
$682$	0	0
$683$	1.70757e9	0.205072	0.102536	−	0.994729i	$-0.467304\pi$
$683$	1.70757e9	0.205072	0.102536	+	0.994729i	$0.467304\pi$
$684$	0	0
$685$	−1.85927e9	−0.221017
$686$	0	0
$687$	−3.95481e9	−0.465347
$688$	0	0
$689$	−1.35179e9	−0.157449
$690$	0	0
$691$	−1.48657e10	−1.71400	−0.857000	−	0.515316i	$-0.827675\pi$
$691$	−1.48657e10	−1.71400	−0.857000	+	0.515316i	$0.827675\pi$
$692$	0	0
$693$	−3.29846e10	−3.76482
$694$	0	0
$695$	3.98240e9	0.449985
$696$	0	0
$697$	−2.42938e9	−0.271757
$698$	0	0
$699$	1.03354e10	1.14460
$700$	0	0
$701$	8.96793e9	0.983284	0.491642	−	0.870797i	$-0.336397\pi$
$701$	8.96793e9	0.983284	0.491642	+	0.870797i	$0.336397\pi$
$702$	0	0
$703$	8.71884e8	0.0946488
$704$	0	0
$705$	−1.14147e10	−1.22689
$706$	0	0
$707$	2.13010e9	0.226690
$708$	0	0
$709$	1.31197e10	1.38249	0.691247	−	0.722619i	$-0.257062\pi$
$709$	1.31197e10	1.38249	0.691247	+	0.722619i	$0.257062\pi$
$710$	0	0
$711$	3.97292e9	0.414540
$712$	0	0
$713$	2.21382e9	0.228733
$714$	0	0
$715$	−4.95549e9	−0.507008
$716$	0	0
$717$	1.87876e10	1.90351
$718$	0	0
$719$	1.16702e10	1.17092	0.585462	−	0.810700i	$-0.300913\pi$
$719$	1.16702e10	1.17092	0.585462	+	0.810700i	$0.300913\pi$
$720$	0	0
$721$	−2.31008e10	−2.29538
$722$	0	0
$723$	−1.80177e10	−1.77303
$724$	0	0
$725$	−8.30566e8	−0.0809452
$726$	0	0
$727$	−1.03092e10	−0.995068	−0.497534	−	0.867444i	$-0.665761\pi$
$727$	−1.03092e10	−0.995068	−0.497534	+	0.867444i	$0.665761\pi$
$728$	0	0
$729$	−1.67302e10	−1.59940
$730$	0	0
$731$	−2.21267e9	−0.209510
$732$	0	0
$733$	−1.00112e10	−0.938910	−0.469455	−	0.882956i	$-0.655550\pi$
$733$	−1.00112e10	−0.938910	−0.469455	+	0.882956i	$0.655550\pi$
$734$	0	0
$735$	−2.28613e10	−2.12371
$736$	0	0
$737$	3.03656e10	2.79413
$738$	0	0
$739$	−3.30781e9	−0.301498	−0.150749	−	0.988572i	$-0.548169\pi$
$739$	−3.30781e9	−0.301498	−0.150749	+	0.988572i	$0.548169\pi$
$740$	0	0
$741$	5.10974e8	0.0461355
$742$	0	0
$743$	2.17089e10	1.94168	0.970838	−	0.239735i	$-0.0770607\pi$
$743$	2.17089e10	1.94168	0.970838	+	0.239735i	$0.0770607\pi$
$744$	0	0
$745$	4.24182e9	0.375842
$746$	0	0
$747$	−1.36460e9	−0.119779
$748$	0	0
$749$	3.01092e10	2.61826
$750$	0	0
$751$	−9.19095e9	−0.791809	−0.395904	−	0.918292i	$-0.629569\pi$
$751$	−9.19095e9	−0.791809	−0.395904	+	0.918292i	$0.629569\pi$
$752$	0	0
$753$	−1.74929e9	−0.149307
$754$	0	0
$755$	2.43305e10	2.05749
$756$	0	0
$757$	9.78965e9	0.820222	0.410111	−	0.912036i	$-0.365490\pi$
$757$	9.78965e9	0.820222	0.410111	+	0.912036i	$0.365490\pi$
$758$	0	0
$759$	9.92628e9	0.824025
$760$	0	0
$761$	−2.03733e10	−1.67577	−0.837886	−	0.545845i	$-0.816209\pi$
$761$	−2.03733e10	−1.67577	−0.837886	+	0.545845i	$0.816209\pi$
$762$	0	0
$763$	2.69924e10	2.19992
$764$	0	0
$765$	3.84381e9	0.310418
$766$	0	0
$767$	8.62353e8	0.0690083
$768$	0	0
$769$	−8.96000e9	−0.710503	−0.355251	−	0.934771i	$-0.615605\pi$
$769$	−8.96000e9	−0.710503	−0.355251	+	0.934771i	$0.615605\pi$
$770$	0	0
$771$	−1.82537e10	−1.43436
$772$	0	0
$773$	7.61579e9	0.593044	0.296522	−	0.955026i	$-0.404173\pi$
$773$	7.61579e9	0.593044	0.296522	+	0.955026i	$0.404173\pi$
$774$	0	0
$775$	1.10791e9	0.0854964
$776$	0	0
$777$	−2.74288e10	−2.09765
$778$	0	0
$779$	1.86641e9	0.141457
$780$	0	0
$781$	2.86465e10	2.15176
$782$	0	0
$783$	−6.50594e9	−0.484333
$784$	0	0
$785$	−2.63172e9	−0.194176
$786$	0	0
$787$	−2.15840e10	−1.57841	−0.789205	−	0.614130i	$-0.789507\pi$
$787$	−2.15840e10	−1.57841	−0.789205	+	0.614130i	$0.789507\pi$
$788$	0	0
$789$	1.52703e10	1.10682
$790$	0	0
$791$	2.95145e10	2.12040
$792$	0	0
$793$	4.12611e9	0.293822
$794$	0	0
$795$	1.32502e10	0.935273
$796$	0	0
$797$	−1.58880e10	−1.11164	−0.555822	−	0.831301i	$-0.687597\pi$
$797$	−1.58880e10	−1.11164	−0.555822	+	0.831301i	$0.687597\pi$
$798$	0	0
$799$	−2.19814e9	−0.152455
$800$	0	0
$801$	1.83204e10	1.25957
$802$	0	0
$803$	1.90064e10	1.29538
$804$	0	0
$805$	7.20314e9	0.486672
$806$	0	0
$807$	2.71132e10	1.81603
$808$	0	0
$809$	8.28899e9	0.550404	0.275202	−	0.961386i	$-0.411255\pi$
$809$	8.28899e9	0.550404	0.275202	+	0.961386i	$0.411255\pi$
$810$	0	0
$811$	6.46851e9	0.425825	0.212913	−	0.977071i	$-0.431705\pi$
$811$	6.46851e9	0.425825	0.212913	+	0.977071i	$0.431705\pi$
$812$	0	0
$813$	−2.41064e10	−1.57332
$814$	0	0
$815$	6.16742e9	0.399072
$816$	0	0
$817$	1.69992e9	0.109056
$818$	0	0
$819$	−9.47778e9	−0.602855
$820$	0	0
$821$	1.65268e10	1.04228	0.521142	−	0.853470i	$-0.325506\pi$
$821$	1.65268e10	1.04228	0.521142	+	0.853470i	$0.325506\pi$
$822$	0	0
$823$	2.05119e10	1.28264	0.641322	−	0.767272i	$-0.278386\pi$
$823$	2.05119e10	1.28264	0.641322	+	0.767272i	$0.278386\pi$
$824$	0	0
$825$	4.96761e9	0.308005
$826$	0	0
$827$	−8.63679e9	−0.530986	−0.265493	−	0.964113i	$-0.585535\pi$
$827$	−8.63679e9	−0.530986	−0.265493	+	0.964113i	$0.585535\pi$
$828$	0	0
$829$	8.81187e9	0.537189	0.268595	−	0.963253i	$-0.413441\pi$
$829$	8.81187e9	0.537189	0.268595	+	0.963253i	$0.413441\pi$
$830$	0	0
$831$	3.69583e10	2.23413
$832$	0	0
$833$	−4.40240e9	−0.263895
$834$	0	0
$835$	−5.55252e9	−0.330056
$836$	0	0
$837$	8.67840e9	0.511565
$838$	0	0
$839$	−1.73321e10	−1.01317	−0.506586	−	0.862189i	$-0.669093\pi$
$839$	−1.73321e10	−1.01317	−0.506586	+	0.862189i	$0.669093\pi$
$840$	0	0
$841$	−8.54088e9	−0.495127
$842$	0	0
$843$	7.79234e9	0.447993
$844$	0	0
$845$	−1.42391e9	−0.0811865
$846$	0	0
$847$	−5.35115e10	−3.02590
$848$	0	0
$849$	4.06260e10	2.27839
$850$	0	0
$851$	4.86679e9	0.270700
$852$	0	0
$853$	−1.62475e10	−0.896322	−0.448161	−	0.893953i	$-0.647921\pi$
$853$	−1.62475e10	−0.896322	−0.448161	+	0.893953i	$0.647921\pi$
$854$	0	0
$855$	−2.95307e9	−0.161582
$856$	0	0
$857$	−2.45158e10	−1.33049	−0.665247	−	0.746623i	$-0.731674\pi$
$857$	−2.45158e10	−1.33049	−0.665247	+	0.746623i	$0.731674\pi$
$858$	0	0
$859$	−9.28369e9	−0.499741	−0.249870	−	0.968279i	$-0.580388\pi$
$859$	−9.28369e9	−0.499741	−0.249870	+	0.968279i	$0.580388\pi$
$860$	0	0
$861$	−5.87158e10	−3.13504
$862$	0	0
$863$	6.17565e9	0.327073	0.163536	−	0.986537i	$-0.447710\pi$
$863$	6.17565e9	0.327073	0.163536	+	0.986537i	$0.447710\pi$
$864$	0	0
$865$	1.41116e9	0.0741343
$866$	0	0
$867$	−2.86993e10	−1.49556
$868$	0	0
$869$	9.66803e9	0.499768
$870$	0	0
$871$	8.72525e9	0.447419
$872$	0	0
$873$	−6.42643e9	−0.326904
$874$	0	0
$875$	−2.80385e10	−1.41491
$876$	0	0
$877$	−1.34392e10	−0.672782	−0.336391	−	0.941722i	$-0.609206\pi$
$877$	−1.34392e10	−0.672782	−0.336391	+	0.941722i	$0.609206\pi$
$878$	0	0
$879$	1.63307e10	0.811043
$880$	0	0
$881$	3.85616e9	0.189994	0.0949968	−	0.995478i	$-0.469716\pi$
$881$	3.85616e9	0.189994	0.0949968	+	0.995478i	$0.469716\pi$
$882$	0	0
$883$	−2.27185e10	−1.11050	−0.555248	−	0.831685i	$-0.687377\pi$
$883$	−2.27185e10	−1.11050	−0.555248	+	0.831685i	$0.687377\pi$
$884$	0	0
$885$	−8.45279e9	−0.409920
$886$	0	0
$887$	−2.21671e10	−1.06654	−0.533270	−	0.845945i	$-0.679037\pi$
$887$	−2.21671e10	−1.06654	−0.533270	+	0.845945i	$0.679037\pi$
$888$	0	0
$889$	−2.43597e10	−1.16283
$890$	0	0
$891$	−1.36279e10	−0.645442
$892$	0	0
$893$	1.68876e9	0.0793572
$894$	0	0
$895$	2.68425e10	1.25153
$896$	0	0
$897$	2.85222e9	0.131950
$898$	0	0
$899$	−1.16171e10	−0.533260
$900$	0	0
$901$	2.55160e9	0.116219
$902$	0	0
$903$	−5.34781e10	−2.41696
$904$	0	0
$905$	5.07446e9	0.227572
$906$	0	0
$907$	3.55279e10	1.58104	0.790522	−	0.612434i	$-0.209809\pi$
$907$	3.55279e10	1.58104	0.790522	+	0.612434i	$0.209809\pi$
$908$	0	0
$909$	−4.87456e9	−0.215259
$910$	0	0
$911$	4.10088e9	0.179706	0.0898530	−	0.995955i	$-0.471360\pi$
$911$	4.10088e9	0.179706	0.0898530	+	0.995955i	$0.471360\pi$
$912$	0	0
$913$	−3.32072e9	−0.144406
$914$	0	0
$915$	−4.04441e10	−1.74535
$916$	0	0
$917$	−1.18240e10	−0.506377
$918$	0	0
$919$	−9.41768e8	−0.0400258	−0.0200129	−	0.999800i	$-0.506371\pi$
$919$	−9.41768e8	−0.0400258	−0.0200129	+	0.999800i	$0.506371\pi$
$920$	0	0
$921$	7.24101e10	3.05415
$922$	0	0
$923$	8.23128e9	0.344557
$924$	0	0
$925$	2.43558e9	0.101183
$926$	0	0
$927$	5.28644e10	2.17964
$928$	0	0
$929$	−1.39001e10	−0.568803	−0.284402	−	0.958705i	$-0.591795\pi$
$929$	−1.39001e10	−0.568803	−0.284402	+	0.958705i	$0.591795\pi$
$930$	0	0
$931$	3.38221e9	0.137365
$932$	0	0
$933$	1.81236e10	0.730565
$934$	0	0
$935$	9.35385e9	0.374240
$936$	0	0
$937$	−1.57057e10	−0.623689	−0.311844	−	0.950133i	$-0.600947\pi$
$937$	−1.57057e10	−0.623689	−0.311844	+	0.950133i	$0.600947\pi$
$938$	0	0
$939$	−1.46538e10	−0.577592
$940$	0	0
$941$	−3.84758e10	−1.50530	−0.752651	−	0.658419i	$-0.771225\pi$
$941$	−3.84758e10	−1.50530	−0.752651	+	0.658419i	$0.771225\pi$
$942$	0	0
$943$	1.04182e10	0.404576
$944$	0	0
$945$	2.82370e10	1.08845
$946$	0	0
$947$	1.18417e10	0.453095	0.226548	−	0.974000i	$-0.427256\pi$
$947$	1.18417e10	0.453095	0.226548	+	0.974000i	$0.427256\pi$
$948$	0	0
$949$	5.46131e9	0.207427
$950$	0	0
$951$	−7.49233e9	−0.282478
$952$	0	0
$953$	8.06299e9	0.301767	0.150883	−	0.988552i	$-0.451788\pi$
$953$	8.06299e9	0.301767	0.150883	+	0.988552i	$0.451788\pi$
$954$	0	0
$955$	−1.97266e10	−0.732892
$956$	0	0
$957$	−5.20884e10	−1.92110
$958$	0	0
$959$	−8.65349e9	−0.316830
$960$	0	0
$961$	−1.20163e10	−0.436758
$962$	0	0
$963$	−6.89024e10	−2.48624
$964$	0	0
$965$	1.43436e10	0.513820
$966$	0	0
$967$	−2.18672e10	−0.777679	−0.388840	−	0.921305i	$-0.627124\pi$
$967$	−2.18672e10	−0.777679	−0.388840	+	0.921305i	$0.627124\pi$
$968$	0	0
$969$	−9.64501e8	−0.0340541
$970$	0	0
$971$	−1.48206e10	−0.519516	−0.259758	−	0.965674i	$-0.583643\pi$
$971$	−1.48206e10	−0.519516	−0.259758	+	0.965674i	$0.583643\pi$
$972$	0	0
$973$	1.85350e10	0.645058
$974$	0	0
$975$	1.42739e9	0.0493204
$976$	0	0
$977$	−4.40385e10	−1.51078	−0.755391	−	0.655274i	$-0.772553\pi$
$977$	−4.40385e10	−1.51078	−0.755391	+	0.655274i	$0.772553\pi$
$978$	0	0
$979$	4.45824e10	1.51853
$980$	0	0
$981$	−6.17700e10	−2.08899
$982$	0	0
$983$	−2.32688e10	−0.781333	−0.390666	−	0.920532i	$-0.627755\pi$
$983$	−2.32688e10	−0.781333	−0.390666	+	0.920532i	$0.627755\pi$
$984$	0	0
$985$	2.48591e10	0.828817
$986$	0	0
$987$	−5.31269e10	−1.75875
$988$	0	0
$989$	9.48881e9	0.311907
$990$	0	0
$991$	−1.25560e10	−0.409821	−0.204911	−	0.978781i	$-0.565690\pi$
$991$	−1.25560e10	−0.409821	−0.204911	+	0.978781i	$0.565690\pi$
$992$	0	0
$993$	5.54839e10	1.79823
$994$	0	0
$995$	4.12721e10	1.32824
$996$	0	0
$997$	3.50176e10	1.11906	0.559530	−	0.828810i	$-0.310982\pi$
$997$	3.50176e10	1.11906	0.559530	+	0.828810i	$0.310982\pi$
$998$	0	0
$999$	1.90783e10	0.605424

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.a.d.1.1		1
4.3	odd	2		13.8.a.a.1.1	✓	1
12.11	even	2		117.8.a.a.1.1		1
20.19	odd	2		325.8.a.a.1.1		1
52.31	even	4		169.8.b.a.168.1		2
52.47	even	4		169.8.b.a.168.2		2
52.51	odd	2		169.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.a.a.1.1	✓	1	4.3	odd	2
117.8.a.a.1.1		1	12.11	even	2
169.8.a.a.1.1		1	52.51	odd	2
169.8.b.a.168.1		2	52.31	even	4
169.8.b.a.168.2		2	52.47	even	4
208.8.a.d.1.1		1	1.1	even	1	trivial
325.8.a.a.1.1		1	20.19	odd	2

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more