LMFDB - Embedded newform 8001.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} -0.561553 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-2.00000 q^{4} -0.561553 q^{5} -1.00000 q^{7} +5.12311 q^{11} +0.561553 q^{13} +4.00000 q^{16} -2.43845 q^{17} +4.00000 q^{19} +1.12311 q^{20} -7.68466 q^{23} -4.68466 q^{25} +2.00000 q^{28} +4.12311 q^{29} -7.68466 q^{31} +0.561553 q^{35} -10.1231 q^{37} -9.80776 q^{41} +12.2462 q^{43} -10.2462 q^{44} +13.1231 q^{47} +1.00000 q^{49} -1.12311 q^{52} +11.0000 q^{53} -2.87689 q^{55} +8.56155 q^{59} -5.43845 q^{61} -8.00000 q^{64} -0.315342 q^{65} -2.00000 q^{67} +4.87689 q^{68} +4.87689 q^{71} -8.56155 q^{73} -8.00000 q^{76} -5.12311 q^{77} -12.6847 q^{79} -2.24621 q^{80} +2.80776 q^{83} +1.36932 q^{85} +12.8078 q^{89} -0.561553 q^{91} +15.3693 q^{92} -2.24621 q^{95} +11.8078 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 4 * q^4 + 3 * q^5 - 2 * q^7	$ 2 q - 4 q^{4} + 3 q^{5} - 2 q^{7} + 2 q^{11} - 3 q^{13} + 8 q^{16} - 9 q^{17} + 8 q^{19} - 6 q^{20} - 3 q^{23} + 3 q^{25} + 4 q^{28} - 3 q^{31} - 3 q^{35} - 12 q^{37} + q^{41} + 8 q^{43} - 4 q^{44} + 18 q^{47} + 2 q^{49} + 6 q^{52} + 22 q^{53} - 14 q^{55} + 13 q^{59} - 15 q^{61} - 16 q^{64} - 13 q^{65} - 4 q^{67} + 18 q^{68} + 18 q^{71} - 13 q^{73} - 16 q^{76} - 2 q^{77} - 13 q^{79} + 12 q^{80} - 15 q^{83} - 22 q^{85} + 5 q^{89} + 3 q^{91} + 6 q^{92} + 12 q^{95} + 3 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 3 * q^5 - 2 * q^7 + 2 * q^11 - 3 * q^13 + 8 * q^16 - 9 * q^17 + 8 * q^19 - 6 * q^20 - 3 * q^23 + 3 * q^25 + 4 * q^28 - 3 * q^31 - 3 * q^35 - 12 * q^37 + q^41 + 8 * q^43 - 4 * q^44 + 18 * q^47 + 2 * q^49 + 6 * q^52 + 22 * q^53 - 14 * q^55 + 13 * q^59 - 15 * q^61 - 16 * q^64 - 13 * q^65 - 4 * q^67 + 18 * q^68 + 18 * q^71 - 13 * q^73 - 16 * q^76 - 2 * q^77 - 13 * q^79 + 12 * q^80 - 15 * q^83 - 22 * q^85 + 5 * q^89 + 3 * q^91 + 6 * q^92 + 12 * q^95 + 3 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	−0.561553	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	−0.561553	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	0	0
$13$	0.561553	0.155747	0.0778734	−	0.996963i	$-0.475187\pi$
$13$	0.561553	0.155747	0.0778734	+	0.996963i	$0.475187\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	−2.43845	−0.591410	−0.295705	−	0.955279i	$-0.595555\pi$
$17$	−2.43845	−0.591410	−0.295705	+	0.955279i	$0.595555\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.12311	0.251134
$21$	0	0
$22$	0	0
$23$	−7.68466	−1.60236	−0.801181	−	0.598422i	$-0.795795\pi$
$23$	−7.68466	−1.60236	−0.801181	+	0.598422i	$0.795795\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	0	0
$28$	2.00000	0.377964
$29$	4.12311	0.765641	0.382821	−	0.923823i	$-0.374953\pi$
$29$	4.12311	0.765641	0.382821	+	0.923823i	$0.374953\pi$
$30$	0	0
$31$	−7.68466	−1.38021	−0.690103	−	0.723711i	$-0.742435\pi$
$31$	−7.68466	−1.38021	−0.690103	+	0.723711i	$0.742435\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.561553	0.0949197
$36$	0	0
$37$	−10.1231	−1.66423	−0.832114	−	0.554604i	$-0.812870\pi$
$37$	−10.1231	−1.66423	−0.832114	+	0.554604i	$0.812870\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.80776	−1.53172	−0.765858	−	0.643010i	$-0.777685\pi$
$41$	−9.80776	−1.53172	−0.765858	+	0.643010i	$0.777685\pi$
$42$	0	0
$43$	12.2462	1.86753	0.933765	−	0.357887i	$-0.116503\pi$
$43$	12.2462	1.86753	0.933765	+	0.357887i	$0.116503\pi$
$44$	−10.2462	−1.54467
$45$	0	0
$46$	0	0
$47$	13.1231	1.91420	0.957101	−	0.289755i	$-0.0935738\pi$
$47$	13.1231	1.91420	0.957101	+	0.289755i	$0.0935738\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−1.12311	−0.155747
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	−2.87689	−0.387920
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.56155	1.11462	0.557310	−	0.830305i	$-0.311834\pi$
$59$	8.56155	1.11462	0.557310	+	0.830305i	$0.311834\pi$
$60$	0	0
$61$	−5.43845	−0.696322	−0.348161	−	0.937435i	$-0.613194\pi$
$61$	−5.43845	−0.696322	−0.348161	+	0.937435i	$0.613194\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−0.315342	−0.0391133
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	4.87689	0.591410
$69$	0	0
$70$	0	0
$71$	4.87689	0.578781	0.289390	−	0.957211i	$-0.406547\pi$
$71$	4.87689	0.578781	0.289390	+	0.957211i	$0.406547\pi$
$72$	0	0
$73$	−8.56155	−1.00205	−0.501027	−	0.865432i	$-0.667044\pi$
$73$	−8.56155	−1.00205	−0.501027	+	0.865432i	$0.667044\pi$
$74$	0	0
$75$	0	0
$76$	−8.00000	−0.917663
$77$	−5.12311	−0.583832
$78$	0	0
$79$	−12.6847	−1.42714	−0.713568	−	0.700586i	$-0.752922\pi$
$79$	−12.6847	−1.42714	−0.713568	+	0.700586i	$0.752922\pi$
$80$	−2.24621	−0.251134
$81$	0	0
$82$	0	0
$83$	2.80776	0.308192	0.154096	−	0.988056i	$-0.450753\pi$
$83$	2.80776	0.308192	0.154096	+	0.988056i	$0.450753\pi$
$84$	0	0
$85$	1.36932	0.148523
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.8078	1.35762	0.678810	−	0.734314i	$-0.262496\pi$
$89$	12.8078	1.35762	0.678810	+	0.734314i	$0.262496\pi$
$90$	0	0
$91$	−0.561553	−0.0588667
$92$	15.3693	1.60236
$93$	0	0
$94$	0	0
$95$	−2.24621	−0.230456
$96$	0	0
$97$	11.8078	1.19890	0.599448	−	0.800413i	$-0.295387\pi$
$97$	11.8078	1.19890	0.599448	+	0.800413i	$0.295387\pi$
$98$	0	0
$99$	0	0
$100$	9.36932	0.936932
$101$	0.807764	0.0803755	0.0401878	−	0.999192i	$-0.487204\pi$
$101$	0.807764	0.0803755	0.0401878	+	0.999192i	$0.487204\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−11.1231	−1.04637	−0.523187	−	0.852218i	$-0.675257\pi$
$113$	−11.1231	−1.04637	−0.523187	+	0.852218i	$0.675257\pi$
$114$	0	0
$115$	4.31534	0.402408
$116$	−8.24621	−0.765641
$117$	0	0
$118$	0	0
$119$	2.43845	0.223532
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	0	0
$124$	15.3693	1.38021
$125$	5.43845	0.486430
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.31534	−0.114922	−0.0574610	−	0.998348i	$-0.518300\pi$
$131$	−1.31534	−0.114922	−0.0574610	+	0.998348i	$0.518300\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.4384	1.31900	0.659498	−	0.751707i	$-0.270769\pi$
$137$	15.4384	1.31900	0.659498	+	0.751707i	$0.270769\pi$
$138$	0	0
$139$	8.43845	0.715740	0.357870	−	0.933771i	$-0.383503\pi$
$139$	8.43845	0.715740	0.357870	+	0.933771i	$0.383503\pi$
$140$	−1.12311	−0.0949197
$141$	0	0
$142$	0	0
$143$	2.87689	0.240578
$144$	0	0
$145$	−2.31534	−0.192279
$146$	0	0
$147$	0	0
$148$	20.2462	1.66423
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	10.4924	0.853861	0.426931	−	0.904284i	$-0.359595\pi$
$151$	10.4924	0.853861	0.426931	+	0.904284i	$0.359595\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.31534	0.346617
$156$	0	0
$157$	−16.2462	−1.29659	−0.648294	−	0.761390i	$-0.724517\pi$
$157$	−16.2462	−1.29659	−0.648294	+	0.761390i	$0.724517\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.68466	0.605636
$162$	0	0
$163$	18.6155	1.45808	0.729040	−	0.684471i	$-0.239967\pi$
$163$	18.6155	1.45808	0.729040	+	0.684471i	$0.239967\pi$
$164$	19.6155	1.53172
$165$	0	0
$166$	0	0
$167$	6.24621	0.483346	0.241673	−	0.970358i	$-0.422304\pi$
$167$	6.24621	0.483346	0.241673	+	0.970358i	$0.422304\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	0	0
$172$	−24.4924	−1.86753
$173$	−23.3693	−1.77674	−0.888368	−	0.459132i	$-0.848161\pi$
$173$	−23.3693	−1.77674	−0.888368	+	0.459132i	$0.848161\pi$
$174$	0	0
$175$	4.68466	0.354127
$176$	20.4924	1.54467
$177$	0	0
$178$	0	0
$179$	11.1231	0.831380	0.415690	−	0.909506i	$-0.363540\pi$
$179$	11.1231	0.831380	0.415690	+	0.909506i	$0.363540\pi$
$180$	0	0
$181$	−23.8078	−1.76962	−0.884809	−	0.465955i	$-0.845711\pi$
$181$	−23.8078	−1.76962	−0.884809	+	0.465955i	$0.845711\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.68466	0.417944
$186$	0	0
$187$	−12.4924	−0.913536
$188$	−26.2462	−1.91420
$189$	0	0
$190$	0	0
$191$	5.36932	0.388510	0.194255	−	0.980951i	$-0.437771\pi$
$191$	5.36932	0.388510	0.194255	+	0.980951i	$0.437771\pi$
$192$	0	0
$193$	−1.12311	−0.0808429	−0.0404215	−	0.999183i	$-0.512870\pi$
$193$	−1.12311	−0.0808429	−0.0404215	+	0.999183i	$0.512870\pi$
$194$	0	0
$195$	0	0
$196$	−2.00000	−0.142857
$197$	2.63068	0.187428	0.0937142	−	0.995599i	$-0.470126\pi$
$197$	2.63068	0.187428	0.0937142	+	0.995599i	$0.470126\pi$
$198$	0	0
$199$	−5.43845	−0.385521	−0.192761	−	0.981246i	$-0.561744\pi$
$199$	−5.43845	−0.385521	−0.192761	+	0.981246i	$0.561744\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.12311	−0.289385
$204$	0	0
$205$	5.50758	0.384666
$206$	0	0
$207$	0	0
$208$	2.24621	0.155747
$209$	20.4924	1.41749
$210$	0	0
$211$	−16.3153	−1.12319	−0.561597	−	0.827411i	$-0.689813\pi$
$211$	−16.3153	−1.12319	−0.561597	+	0.827411i	$0.689813\pi$
$212$	−22.0000	−1.51097
$213$	0	0
$214$	0	0
$215$	−6.87689	−0.469000
$216$	0	0
$217$	7.68466	0.521669
$218$	0	0
$219$	0	0
$220$	5.75379	0.387920
$221$	−1.36932	−0.0921102
$222$	0	0
$223$	9.80776	0.656776	0.328388	−	0.944543i	$-0.393495\pi$
$223$	9.80776	0.656776	0.328388	+	0.944543i	$0.393495\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.3693	−0.754608	−0.377304	−	0.926089i	$-0.623149\pi$
$227$	−11.3693	−0.754608	−0.377304	+	0.926089i	$0.623149\pi$
$228$	0	0
$229$	−13.5616	−0.896173	−0.448086	−	0.893990i	$-0.647894\pi$
$229$	−13.5616	−0.896173	−0.448086	+	0.893990i	$0.647894\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.369317	0.0241948	0.0120974	−	0.999927i	$-0.496149\pi$
$233$	0.369317	0.0241948	0.0120974	+	0.999927i	$0.496149\pi$
$234$	0	0
$235$	−7.36932	−0.480721
$236$	−17.1231	−1.11462
$237$	0	0
$238$	0	0
$239$	−28.3693	−1.83506	−0.917529	−	0.397668i	$-0.869820\pi$
$239$	−28.3693	−1.83506	−0.917529	+	0.397668i	$0.869820\pi$
$240$	0	0
$241$	−9.56155	−0.615914	−0.307957	−	0.951400i	$-0.599645\pi$
$241$	−9.56155	−0.615914	−0.307957	+	0.951400i	$0.599645\pi$
$242$	0	0
$243$	0	0
$244$	10.8769	0.696322
$245$	−0.561553	−0.0358763
$246$	0	0
$247$	2.24621	0.142923
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.19224	0.138373	0.0691864	−	0.997604i	$-0.477960\pi$
$251$	2.19224	0.138373	0.0691864	+	0.997604i	$0.477960\pi$
$252$	0	0
$253$	−39.3693	−2.47513
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−27.0540	−1.68758	−0.843790	−	0.536673i	$-0.819681\pi$
$257$	−27.0540	−1.68758	−0.843790	+	0.536673i	$0.819681\pi$
$258$	0	0
$259$	10.1231	0.629019
$260$	0.630683	0.0391133
$261$	0	0
$262$	0	0
$263$	−27.6155	−1.70285	−0.851423	−	0.524479i	$-0.824260\pi$
$263$	−27.6155	−1.70285	−0.851423	+	0.524479i	$0.824260\pi$
$264$	0	0
$265$	−6.17708	−0.379455
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	−15.8078	−0.963816	−0.481908	−	0.876222i	$-0.660056\pi$
$269$	−15.8078	−0.963816	−0.481908	+	0.876222i	$0.660056\pi$
$270$	0	0
$271$	−15.9309	−0.967731	−0.483866	−	0.875142i	$-0.660768\pi$
$271$	−15.9309	−0.967731	−0.483866	+	0.875142i	$0.660768\pi$
$272$	−9.75379	−0.591410
$273$	0	0
$274$	0	0
$275$	−24.0000	−1.44725
$276$	0	0
$277$	−18.2462	−1.09631	−0.548154	−	0.836377i	$-0.684669\pi$
$277$	−18.2462	−1.09631	−0.548154	+	0.836377i	$0.684669\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.43845	−0.264776	−0.132388	−	0.991198i	$-0.542264\pi$
$281$	−4.43845	−0.264776	−0.132388	+	0.991198i	$0.542264\pi$
$282$	0	0
$283$	−3.31534	−0.197077	−0.0985383	−	0.995133i	$-0.531417\pi$
$283$	−3.31534	−0.197077	−0.0985383	+	0.995133i	$0.531417\pi$
$284$	−9.75379	−0.578781
$285$	0	0
$286$	0	0
$287$	9.80776	0.578934
$288$	0	0
$289$	−11.0540	−0.650234
$290$	0	0
$291$	0	0
$292$	17.1231	1.00205
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	−4.80776	−0.279919
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.31534	−0.249563
$300$	0	0
$301$	−12.2462	−0.705860
$302$	0	0
$303$	0	0
$304$	16.0000	0.917663
$305$	3.05398	0.174870
$306$	0	0
$307$	−14.1922	−0.809994	−0.404997	−	0.914318i	$-0.632727\pi$
$307$	−14.1922	−0.809994	−0.404997	+	0.914318i	$0.632727\pi$
$308$	10.2462	0.583832
$309$	0	0
$310$	0	0
$311$	17.0540	0.967042	0.483521	−	0.875333i	$-0.339358\pi$
$311$	17.0540	0.967042	0.483521	+	0.875333i	$0.339358\pi$
$312$	0	0
$313$	−16.9309	−0.956989	−0.478495	−	0.878090i	$-0.658817\pi$
$313$	−16.9309	−0.956989	−0.478495	+	0.878090i	$0.658817\pi$
$314$	0	0
$315$	0	0
$316$	25.3693	1.42714
$317$	24.2462	1.36180	0.680901	−	0.732375i	$-0.261588\pi$
$317$	24.2462	1.36180	0.680901	+	0.732375i	$0.261588\pi$
$318$	0	0
$319$	21.1231	1.18267
$320$	4.49242	0.251134
$321$	0	0
$322$	0	0
$323$	−9.75379	−0.542715
$324$	0	0
$325$	−2.63068	−0.145924
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13.1231	−0.723500
$330$	0	0
$331$	10.2462	0.563183	0.281591	−	0.959534i	$-0.409138\pi$
$331$	10.2462	0.563183	0.281591	+	0.959534i	$0.409138\pi$
$332$	−5.61553	−0.308192
$333$	0	0
$334$	0	0
$335$	1.12311	0.0613618
$336$	0	0
$337$	26.2462	1.42972	0.714861	−	0.699266i	$-0.246490\pi$
$337$	26.2462	1.42972	0.714861	+	0.699266i	$0.246490\pi$
$338$	0	0
$339$	0	0
$340$	−2.73863	−0.148523
$341$	−39.3693	−2.13197
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.49242	−0.402214	−0.201107	−	0.979569i	$-0.564454\pi$
$347$	−7.49242	−0.402214	−0.201107	+	0.979569i	$0.564454\pi$
$348$	0	0
$349$	−16.0540	−0.859350	−0.429675	−	0.902984i	$-0.641372\pi$
$349$	−16.0540	−0.859350	−0.429675	+	0.902984i	$0.641372\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.1231	0.592023	0.296012	−	0.955184i	$-0.404343\pi$
$353$	11.1231	0.592023	0.296012	+	0.955184i	$0.404343\pi$
$354$	0	0
$355$	−2.73863	−0.145352
$356$	−25.6155	−1.35762
$357$	0	0
$358$	0	0
$359$	6.93087	0.365797	0.182899	−	0.983132i	$-0.441452\pi$
$359$	6.93087	0.365797	0.182899	+	0.983132i	$0.441452\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	1.12311	0.0588667
$365$	4.80776	0.251650
$366$	0	0
$367$	38.1771	1.99283	0.996414	−	0.0846152i	$-0.0269661\pi$
$367$	38.1771	1.99283	0.996414	+	0.0846152i	$0.0269661\pi$
$368$	−30.7386	−1.60236
$369$	0	0
$370$	0	0
$371$	−11.0000	−0.571092
$372$	0	0
$373$	−5.12311	−0.265264	−0.132632	−	0.991165i	$-0.542343\pi$
$373$	−5.12311	−0.265264	−0.132632	+	0.991165i	$0.542343\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.31534	0.119246
$378$	0	0
$379$	−18.8769	−0.969641	−0.484820	−	0.874614i	$-0.661115\pi$
$379$	−18.8769	−0.969641	−0.484820	+	0.874614i	$0.661115\pi$
$380$	4.49242	0.230456
$381$	0	0
$382$	0	0
$383$	5.06913	0.259020	0.129510	−	0.991578i	$-0.458659\pi$
$383$	5.06913	0.259020	0.129510	+	0.991578i	$0.458659\pi$
$384$	0	0
$385$	2.87689	0.146620
$386$	0	0
$387$	0	0
$388$	−23.6155	−1.19890
$389$	−16.4924	−0.836199	−0.418100	−	0.908401i	$-0.637304\pi$
$389$	−16.4924	−0.836199	−0.418100	+	0.908401i	$0.637304\pi$
$390$	0	0
$391$	18.7386	0.947653
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.12311	0.358402
$396$	0	0
$397$	0.946025	0.0474796	0.0237398	−	0.999718i	$-0.492443\pi$
$397$	0.946025	0.0474796	0.0237398	+	0.999718i	$0.492443\pi$
$398$	0	0
$399$	0	0
$400$	−18.7386	−0.936932
$401$	−10.3693	−0.517819	−0.258909	−	0.965902i	$-0.583363\pi$
$401$	−10.3693	−0.517819	−0.258909	+	0.965902i	$0.583363\pi$
$402$	0	0
$403$	−4.31534	−0.214962
$404$	−1.61553	−0.0803755
$405$	0	0
$406$	0	0
$407$	−51.8617	−2.57069
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	24.0000	1.18240
$413$	−8.56155	−0.421286
$414$	0	0
$415$	−1.57671	−0.0773975
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.12311	−0.445693	−0.222846	−	0.974854i	$-0.571535\pi$
$419$	−9.12311	−0.445693	−0.222846	+	0.974854i	$0.571535\pi$
$420$	0	0
$421$	−21.7538	−1.06021	−0.530107	−	0.847931i	$-0.677848\pi$
$421$	−21.7538	−1.06021	−0.530107	+	0.847931i	$0.677848\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11.4233	0.554111
$426$	0	0
$427$	5.43845	0.263185
$428$	8.00000	0.386695
$429$	0	0
$430$	0	0
$431$	−14.4924	−0.698075	−0.349038	−	0.937109i	$-0.613491\pi$
$431$	−14.4924	−0.698075	−0.349038	+	0.937109i	$0.613491\pi$
$432$	0	0
$433$	2.49242	0.119778	0.0598891	−	0.998205i	$-0.480925\pi$
$433$	2.49242	0.119778	0.0598891	+	0.998205i	$0.480925\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−30.7386	−1.47043
$438$	0	0
$439$	−21.8078	−1.04083	−0.520414	−	0.853914i	$-0.674222\pi$
$439$	−21.8078	−1.04083	−0.520414	+	0.853914i	$0.674222\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−7.19224	−0.340945
$446$	0	0
$447$	0	0
$448$	8.00000	0.377964
$449$	17.6155	0.831328	0.415664	−	0.909518i	$-0.363549\pi$
$449$	17.6155	0.831328	0.415664	+	0.909518i	$0.363549\pi$
$450$	0	0
$451$	−50.2462	−2.36600
$452$	22.2462	1.04637
$453$	0	0
$454$	0	0
$455$	0.315342	0.0147834
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	0	0
$459$	0	0
$460$	−8.63068	−0.402408
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−3.68466	−0.171241	−0.0856203	−	0.996328i	$-0.527287\pi$
$463$	−3.68466	−0.171241	−0.0856203	+	0.996328i	$0.527287\pi$
$464$	16.4924	0.765641
$465$	0	0
$466$	0	0
$467$	10.5616	0.488730	0.244365	−	0.969683i	$-0.421420\pi$
$467$	10.5616	0.488730	0.244365	+	0.969683i	$0.421420\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	62.7386	2.88473
$474$	0	0
$475$	−18.7386	−0.859787
$476$	−4.87689	−0.223532
$477$	0	0
$478$	0	0
$479$	−39.5616	−1.80761	−0.903807	−	0.427941i	$-0.859239\pi$
$479$	−39.5616	−1.80761	−0.903807	+	0.427941i	$0.859239\pi$
$480$	0	0
$481$	−5.68466	−0.259198
$482$	0	0
$483$	0	0
$484$	−30.4924	−1.38602
$485$	−6.63068	−0.301084
$486$	0	0
$487$	9.75379	0.441986	0.220993	−	0.975275i	$-0.429070\pi$
$487$	9.75379	0.441986	0.220993	+	0.975275i	$0.429070\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.50758	0.338812	0.169406	−	0.985546i	$-0.445815\pi$
$491$	7.50758	0.338812	0.169406	+	0.985546i	$0.445815\pi$
$492$	0	0
$493$	−10.0540	−0.452808
$494$	0	0
$495$	0	0
$496$	−30.7386	−1.38021
$497$	−4.87689	−0.218759
$498$	0	0
$499$	0.492423	0.0220439	0.0110219	−	0.999939i	$-0.496492\pi$
$499$	0.492423	0.0220439	0.0110219	+	0.999939i	$0.496492\pi$
$500$	−10.8769	−0.486430
$501$	0	0
$502$	0	0
$503$	−8.68466	−0.387230	−0.193615	−	0.981078i	$-0.562021\pi$
$503$	−8.68466	−0.387230	−0.193615	+	0.981078i	$0.562021\pi$
$504$	0	0
$505$	−0.453602	−0.0201850
$506$	0	0
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	3.31534	0.146950	0.0734750	−	0.997297i	$-0.476591\pi$
$509$	3.31534	0.146950	0.0734750	+	0.997297i	$0.476591\pi$
$510$	0	0
$511$	8.56155	0.378741
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.73863	0.296940
$516$	0	0
$517$	67.2311	2.95682
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.12311	−0.136826	−0.0684129	−	0.997657i	$-0.521794\pi$
$521$	−3.12311	−0.136826	−0.0684129	+	0.997657i	$0.521794\pi$
$522$	0	0
$523$	30.1771	1.31955	0.659776	−	0.751462i	$-0.270651\pi$
$523$	30.1771	1.31955	0.659776	+	0.751462i	$0.270651\pi$
$524$	2.63068	0.114922
$525$	0	0
$526$	0	0
$527$	18.7386	0.816268
$528$	0	0
$529$	36.0540	1.56756
$530$	0	0
$531$	0	0
$532$	8.00000	0.346844
$533$	−5.50758	−0.238560
$534$	0	0
$535$	2.24621	0.0971122
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.12311	0.220668
$540$	0	0
$541$	14.2462	0.612492	0.306246	−	0.951952i	$-0.400927\pi$
$541$	14.2462	0.612492	0.306246	+	0.951952i	$0.400927\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.12311	−0.0481086
$546$	0	0
$547$	28.4924	1.21825	0.609124	−	0.793075i	$-0.291521\pi$
$547$	28.4924	1.21825	0.609124	+	0.793075i	$0.291521\pi$
$548$	−30.8769	−1.31900
$549$	0	0
$550$	0	0
$551$	16.4924	0.702601
$552$	0	0
$553$	12.6847	0.539407
$554$	0	0
$555$	0	0
$556$	−16.8769	−0.715740
$557$	17.1231	0.725529	0.362765	−	0.931881i	$-0.381833\pi$
$557$	17.1231	0.725529	0.362765	+	0.931881i	$0.381833\pi$
$558$	0	0
$559$	6.87689	0.290862
$560$	2.24621	0.0949197
$561$	0	0
$562$	0	0
$563$	−38.7386	−1.63264	−0.816319	−	0.577601i	$-0.803989\pi$
$563$	−38.7386	−1.63264	−0.816319	+	0.577601i	$0.803989\pi$
$564$	0	0
$565$	6.24621	0.262780
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−37.3693	−1.56660	−0.783302	−	0.621642i	$-0.786466\pi$
$569$	−37.3693	−1.56660	−0.783302	+	0.621642i	$0.786466\pi$
$570$	0	0
$571$	−21.1231	−0.883974	−0.441987	−	0.897021i	$-0.645726\pi$
$571$	−21.1231	−0.883974	−0.441987	+	0.897021i	$0.645726\pi$
$572$	−5.75379	−0.240578
$573$	0	0
$574$	0	0
$575$	36.0000	1.50130
$576$	0	0
$577$	10.8078	0.449933	0.224967	−	0.974366i	$-0.427773\pi$
$577$	10.8078	0.449933	0.224967	+	0.974366i	$0.427773\pi$
$578$	0	0
$579$	0	0
$580$	4.63068	0.192279
$581$	−2.80776	−0.116486
$582$	0	0
$583$	56.3542	2.33395
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.3002	−0.920427	−0.460214	−	0.887808i	$-0.652227\pi$
$587$	−22.3002	−0.920427	−0.460214	+	0.887808i	$0.652227\pi$
$588$	0	0
$589$	−30.7386	−1.26656
$590$	0	0
$591$	0	0
$592$	−40.4924	−1.66423
$593$	−12.3153	−0.505730	−0.252865	−	0.967502i	$-0.581373\pi$
$593$	−12.3153	−0.505730	−0.252865	+	0.967502i	$0.581373\pi$
$594$	0	0
$595$	−1.36932	−0.0561365
$596$	28.0000	1.14692
$597$	0	0
$598$	0	0
$599$	−5.38447	−0.220004	−0.110002	−	0.993931i	$-0.535086\pi$
$599$	−5.38447	−0.220004	−0.110002	+	0.993931i	$0.535086\pi$
$600$	0	0
$601$	−4.93087	−0.201134	−0.100567	−	0.994930i	$-0.532066\pi$
$601$	−4.93087	−0.201134	−0.100567	+	0.994930i	$0.532066\pi$
$602$	0	0
$603$	0	0
$604$	−20.9848	−0.853861
$605$	−8.56155	−0.348077
$606$	0	0
$607$	16.8078	0.682206	0.341103	−	0.940026i	$-0.389199\pi$
$607$	16.8078	0.682206	0.341103	+	0.940026i	$0.389199\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.36932	0.298131
$612$	0	0
$613$	6.63068	0.267811	0.133905	−	0.990994i	$-0.457248\pi$
$613$	6.63068	0.267811	0.133905	+	0.990994i	$0.457248\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.1080	1.65494	0.827472	−	0.561507i	$-0.189778\pi$
$617$	41.1080	1.65494	0.827472	+	0.561507i	$0.189778\pi$
$618$	0	0
$619$	34.6847	1.39409	0.697047	−	0.717025i	$-0.254497\pi$
$619$	34.6847	1.39409	0.697047	+	0.717025i	$0.254497\pi$
$620$	−8.63068	−0.346617
$621$	0	0
$622$	0	0
$623$	−12.8078	−0.513132
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	0	0
$628$	32.4924	1.29659
$629$	24.6847	0.984242
$630$	0	0
$631$	22.4924	0.895409	0.447705	−	0.894182i	$-0.352242\pi$
$631$	22.4924	0.895409	0.447705	+	0.894182i	$0.352242\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.561553	−0.0222845
$636$	0	0
$637$	0.561553	0.0222495
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.80776	−0.229393	−0.114696	−	0.993401i	$-0.536590\pi$
$641$	−5.80776	−0.229393	−0.114696	+	0.993401i	$0.536590\pi$
$642$	0	0
$643$	−18.7386	−0.738980	−0.369490	−	0.929235i	$-0.620468\pi$
$643$	−18.7386	−0.738980	−0.369490	+	0.929235i	$0.620468\pi$
$644$	−15.3693	−0.605636
$645$	0	0
$646$	0	0
$647$	21.0540	0.827717	0.413859	−	0.910341i	$-0.364181\pi$
$647$	21.0540	0.827717	0.413859	+	0.910341i	$0.364181\pi$
$648$	0	0
$649$	43.8617	1.72172
$650$	0	0
$651$	0	0
$652$	−37.2311	−1.45808
$653$	−35.8617	−1.40338	−0.701689	−	0.712483i	$-0.747570\pi$
$653$	−35.8617	−1.40338	−0.701689	+	0.712483i	$0.747570\pi$
$654$	0	0
$655$	0.738634	0.0288608
$656$	−39.2311	−1.53172
$657$	0	0
$658$	0	0
$659$	−34.5616	−1.34633	−0.673163	−	0.739494i	$-0.735065\pi$
$659$	−34.5616	−1.34633	−0.673163	+	0.739494i	$0.735065\pi$
$660$	0	0
$661$	15.4384	0.600486	0.300243	−	0.953863i	$-0.402932\pi$
$661$	15.4384	0.600486	0.300243	+	0.953863i	$0.402932\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.24621	0.0871043
$666$	0	0
$667$	−31.6847	−1.22683
$668$	−12.4924	−0.483346
$669$	0	0
$670$	0	0
$671$	−27.8617	−1.07559
$672$	0	0
$673$	−49.1080	−1.89297	−0.946486	−	0.322744i	$-0.895395\pi$
$673$	−49.1080	−1.89297	−0.946486	+	0.322744i	$0.895395\pi$
$674$	0	0
$675$	0	0
$676$	25.3693	0.975743
$677$	6.93087	0.266375	0.133187	−	0.991091i	$-0.457479\pi$
$677$	6.93087	0.266375	0.133187	+	0.991091i	$0.457479\pi$
$678$	0	0
$679$	−11.8078	−0.453140
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.7386	0.717014	0.358507	−	0.933527i	$-0.383286\pi$
$683$	18.7386	0.717014	0.358507	+	0.933527i	$0.383286\pi$
$684$	0	0
$685$	−8.66950	−0.331245
$686$	0	0
$687$	0	0
$688$	48.9848	1.86753
$689$	6.17708	0.235328
$690$	0	0
$691$	−36.3002	−1.38092	−0.690462	−	0.723369i	$-0.742593\pi$
$691$	−36.3002	−1.38092	−0.690462	+	0.723369i	$0.742593\pi$
$692$	46.7386	1.77674
$693$	0	0
$694$	0	0
$695$	−4.73863	−0.179747
$696$	0	0
$697$	23.9157	0.905872
$698$	0	0
$699$	0	0
$700$	−9.36932	−0.354127
$701$	7.93087	0.299545	0.149772	−	0.988720i	$-0.452146\pi$
$701$	7.93087	0.299545	0.149772	+	0.988720i	$0.452146\pi$
$702$	0	0
$703$	−40.4924	−1.52720
$704$	−40.9848	−1.54467
$705$	0	0
$706$	0	0
$707$	−0.807764	−0.0303791
$708$	0	0
$709$	−36.9309	−1.38697	−0.693484	−	0.720472i	$-0.743925\pi$
$709$	−36.9309	−1.38697	−0.693484	+	0.720472i	$0.743925\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	59.0540	2.21159
$714$	0	0
$715$	−1.61553	−0.0604173
$716$	−22.2462	−0.831380
$717$	0	0
$718$	0	0
$719$	−33.1771	−1.23730	−0.618648	−	0.785668i	$-0.712319\pi$
$719$	−33.1771	−1.23730	−0.618648	+	0.785668i	$0.712319\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	0	0
$724$	47.6155	1.76962
$725$	−19.3153	−0.717354
$726$	0	0
$727$	31.1771	1.15629	0.578147	−	0.815933i	$-0.303776\pi$
$727$	31.1771	1.15629	0.578147	+	0.815933i	$0.303776\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.8617	−1.10448
$732$	0	0
$733$	3.68466	0.136096	0.0680480	−	0.997682i	$-0.478323\pi$
$733$	3.68466	0.136096	0.0680480	+	0.997682i	$0.478323\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.2462	−0.377424
$738$	0	0
$739$	12.1231	0.445956	0.222978	−	0.974824i	$-0.428422\pi$
$739$	12.1231	0.445956	0.222978	+	0.974824i	$0.428422\pi$
$740$	−11.3693	−0.417944
$741$	0	0
$742$	0	0
$743$	47.1080	1.72822	0.864112	−	0.503300i	$-0.167881\pi$
$743$	47.1080	1.72822	0.864112	+	0.503300i	$0.167881\pi$
$744$	0	0
$745$	7.86174	0.288032
$746$	0	0
$747$	0	0
$748$	24.9848	0.913536
$749$	4.00000	0.146157
$750$	0	0
$751$	−28.8769	−1.05373	−0.526866	−	0.849948i	$-0.676633\pi$
$751$	−28.8769	−1.05373	−0.526866	+	0.849948i	$0.676633\pi$
$752$	52.4924	1.91420
$753$	0	0
$754$	0	0
$755$	−5.89205	−0.214434
$756$	0	0
$757$	41.1080	1.49409	0.747047	−	0.664771i	$-0.231471\pi$
$757$	41.1080	1.49409	0.747047	+	0.664771i	$0.231471\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.5464	0.926056	0.463028	−	0.886344i	$-0.346763\pi$
$761$	25.5464	0.926056	0.463028	+	0.886344i	$0.346763\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	−10.7386	−0.388510
$765$	0	0
$766$	0	0
$767$	4.80776	0.173598
$768$	0	0
$769$	32.0540	1.15590	0.577948	−	0.816074i	$-0.303854\pi$
$769$	32.0540	1.15590	0.577948	+	0.816074i	$0.303854\pi$
$770$	0	0
$771$	0	0
$772$	2.24621	0.0808429
$773$	−30.3002	−1.08982	−0.544911	−	0.838494i	$-0.683437\pi$
$773$	−30.3002	−1.08982	−0.544911	+	0.838494i	$0.683437\pi$
$774$	0	0
$775$	36.0000	1.29316
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−39.2311	−1.40560
$780$	0	0
$781$	24.9848	0.894028
$782$	0	0
$783$	0	0
$784$	4.00000	0.142857
$785$	9.12311	0.325618
$786$	0	0
$787$	−45.7926	−1.63233	−0.816165	−	0.577819i	$-0.803904\pi$
$787$	−45.7926	−1.63233	−0.816165	+	0.577819i	$0.803904\pi$
$788$	−5.26137	−0.187428
$789$	0	0
$790$	0	0
$791$	11.1231	0.395492
$792$	0	0
$793$	−3.05398	−0.108450
$794$	0	0
$795$	0	0
$796$	10.8769	0.385521
$797$	28.2462	1.00053	0.500266	−	0.865872i	$-0.333236\pi$
$797$	28.2462	1.00053	0.500266	+	0.865872i	$0.333236\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−43.8617	−1.54785
$804$	0	0
$805$	−4.31534	−0.152096
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−15.1231	−0.531700	−0.265850	−	0.964014i	$-0.585653\pi$
$809$	−15.1231	−0.531700	−0.265850	+	0.964014i	$0.585653\pi$
$810$	0	0
$811$	−2.56155	−0.0899483	−0.0449741	−	0.998988i	$-0.514321\pi$
$811$	−2.56155	−0.0899483	−0.0449741	+	0.998988i	$0.514321\pi$
$812$	8.24621	0.289385
$813$	0	0
$814$	0	0
$815$	−10.4536	−0.366174
$816$	0	0
$817$	48.9848	1.71376
$818$	0	0
$819$	0	0
$820$	−11.0152	−0.384666
$821$	47.0000	1.64031	0.820156	−	0.572140i	$-0.193887\pi$
$821$	47.0000	1.64031	0.820156	+	0.572140i	$0.193887\pi$
$822$	0	0
$823$	−35.4233	−1.23478	−0.617389	−	0.786658i	$-0.711810\pi$
$823$	−35.4233	−1.23478	−0.617389	+	0.786658i	$0.711810\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.0000	1.07798	0.538988	−	0.842314i	$-0.318807\pi$
$827$	31.0000	1.07798	0.538988	+	0.842314i	$0.318807\pi$
$828$	0	0
$829$	−53.2311	−1.84879	−0.924396	−	0.381435i	$-0.875430\pi$
$829$	−53.2311	−1.84879	−0.924396	+	0.381435i	$0.875430\pi$
$830$	0	0
$831$	0	0
$832$	−4.49242	−0.155747
$833$	−2.43845	−0.0844872
$834$	0	0
$835$	−3.50758	−0.121385
$836$	−40.9848	−1.41749
$837$	0	0
$838$	0	0
$839$	10.4233	0.359852	0.179926	−	0.983680i	$-0.442414\pi$
$839$	10.4233	0.359852	0.179926	+	0.983680i	$0.442414\pi$
$840$	0	0
$841$	−12.0000	−0.413793
$842$	0	0
$843$	0	0
$844$	32.6307	1.12319
$845$	7.12311	0.245042
$846$	0	0
$847$	−15.2462	−0.523866
$848$	44.0000	1.51097
$849$	0	0
$850$	0	0
$851$	77.7926	2.66670
$852$	0	0
$853$	−11.5616	−0.395860	−0.197930	−	0.980216i	$-0.563422\pi$
$853$	−11.5616	−0.395860	−0.197930	+	0.980216i	$0.563422\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.9460	−0.647184	−0.323592	−	0.946197i	$-0.604890\pi$
$857$	−18.9460	−0.647184	−0.323592	+	0.946197i	$0.604890\pi$
$858$	0	0
$859$	−55.9157	−1.90782	−0.953910	−	0.300094i	$-0.902982\pi$
$859$	−55.9157	−1.90782	−0.953910	+	0.300094i	$0.902982\pi$
$860$	13.7538	0.469000
$861$	0	0
$862$	0	0
$863$	−29.8769	−1.01702	−0.508511	−	0.861056i	$-0.669804\pi$
$863$	−29.8769	−1.01702	−0.508511	+	0.861056i	$0.669804\pi$
$864$	0	0
$865$	13.1231	0.446199
$866$	0	0
$867$	0	0
$868$	−15.3693	−0.521669
$869$	−64.9848	−2.20446
$870$	0	0
$871$	−1.12311	−0.0380550
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.43845	−0.183853
$876$	0	0
$877$	−20.5464	−0.693803	−0.346901	−	0.937902i	$-0.612766\pi$
$877$	−20.5464	−0.693803	−0.346901	+	0.937902i	$0.612766\pi$
$878$	0	0
$879$	0	0
$880$	−11.5076	−0.387920
$881$	−18.0691	−0.608764	−0.304382	−	0.952550i	$-0.598450\pi$
$881$	−18.0691	−0.608764	−0.304382	+	0.952550i	$0.598450\pi$
$882$	0	0
$883$	−5.93087	−0.199590	−0.0997948	−	0.995008i	$-0.531819\pi$
$883$	−5.93087	−0.199590	−0.0997948	+	0.995008i	$0.531819\pi$
$884$	2.73863	0.0921102
$885$	0	0
$886$	0	0
$887$	−30.5616	−1.02616	−0.513078	−	0.858342i	$-0.671495\pi$
$887$	−30.5616	−1.02616	−0.513078	+	0.858342i	$0.671495\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	0	0
$892$	−19.6155	−0.656776
$893$	52.4924	1.75659
$894$	0	0
$895$	−6.24621	−0.208788
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−31.6847	−1.05674
$900$	0	0
$901$	−26.8229	−0.893601
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.3693	0.444411
$906$	0	0
$907$	−4.50758	−0.149672	−0.0748358	−	0.997196i	$-0.523843\pi$
$907$	−4.50758	−0.149672	−0.0748358	+	0.997196i	$0.523843\pi$
$908$	22.7386	0.754608
$909$	0	0
$910$	0	0
$911$	25.3693	0.840523	0.420261	−	0.907403i	$-0.361938\pi$
$911$	25.3693	0.840523	0.420261	+	0.907403i	$0.361938\pi$
$912$	0	0
$913$	14.3845	0.476057
$914$	0	0
$915$	0	0
$916$	27.1231	0.896173
$917$	1.31534	0.0434364
$918$	0	0
$919$	−52.8617	−1.74375	−0.871874	−	0.489730i	$-0.837095\pi$
$919$	−52.8617	−1.74375	−0.871874	+	0.489730i	$0.837095\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.73863	0.0901432
$924$	0	0
$925$	47.4233	1.55927
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−37.6155	−1.23413	−0.617063	−	0.786914i	$-0.711678\pi$
$929$	−37.6155	−1.23413	−0.617063	+	0.786914i	$0.711678\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	−0.738634	−0.0241948
$933$	0	0
$934$	0	0
$935$	7.01515	0.229420
$936$	0	0
$937$	7.12311	0.232702	0.116351	−	0.993208i	$-0.462880\pi$
$937$	7.12311	0.232702	0.116351	+	0.993208i	$0.462880\pi$
$938$	0	0
$939$	0	0
$940$	14.7386	0.480721
$941$	−0.0539753	−0.00175954	−0.000879772	−	1.00000i	$-0.500280\pi$
$941$	−0.0539753	−0.00175954	−0.000879772		1.00000i	$0.500280\pi$
$942$	0	0
$943$	75.3693	2.45436
$944$	34.2462	1.11462
$945$	0	0
$946$	0	0
$947$	−23.4233	−0.761155	−0.380577	−	0.924749i	$-0.624275\pi$
$947$	−23.4233	−0.761155	−0.380577	+	0.924749i	$0.624275\pi$
$948$	0	0
$949$	−4.80776	−0.156067
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.7386	0.801363	0.400681	−	0.916217i	$-0.368773\pi$
$953$	24.7386	0.801363	0.400681	+	0.916217i	$0.368773\pi$
$954$	0	0
$955$	−3.01515	−0.0975681
$956$	56.7386	1.83506
$957$	0	0
$958$	0	0
$959$	−15.4384	−0.498533
$960$	0	0
$961$	28.0540	0.904967
$962$	0	0
$963$	0	0
$964$	19.1231	0.615914
$965$	0.630683	0.0203024
$966$	0	0
$967$	40.3542	1.29770	0.648851	−	0.760915i	$-0.275250\pi$
$967$	40.3542	1.29770	0.648851	+	0.760915i	$0.275250\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.63068	−0.148606	−0.0743028	−	0.997236i	$-0.523673\pi$
$971$	−4.63068	−0.148606	−0.0743028	+	0.997236i	$0.523673\pi$
$972$	0	0
$973$	−8.43845	−0.270524
$974$	0	0
$975$	0	0
$976$	−21.7538	−0.696322
$977$	−12.9848	−0.415422	−0.207711	−	0.978190i	$-0.566601\pi$
$977$	−12.9848	−0.415422	−0.207711	+	0.978190i	$0.566601\pi$
$978$	0	0
$979$	65.6155	2.09708
$980$	1.12311	0.0358763
$981$	0	0
$982$	0	0
$983$	−9.80776	−0.312819	−0.156410	−	0.987692i	$-0.549992\pi$
$983$	−9.80776	−0.312819	−0.156410	+	0.987692i	$0.549992\pi$
$984$	0	0
$985$	−1.47727	−0.0470697
$986$	0	0
$987$	0	0
$988$	−4.49242	−0.142923
$989$	−94.1080	−2.99246
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.05398	0.0968175
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.i.1.1		2
3.2	odd	2		2667.2.a.f.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.f.1.2	✓	2	3.2	odd	2
8001.2.a.i.1.1		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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -0.561553 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} -0.561553 q^{5} -1.00000 q^{7} +5.12311 q^{11} +0.561553 q^{13} +4.00000 q^{16} -2.43845 q^{17} +4.00000 q^{19} +1.12311 q^{20} -7.68466 q^{23} -4.68466 q^{25} +2.00000 q^{28} +4.12311 q^{29} -7.68466 q^{31} +0.561553 q^{35} -10.1231 q^{37} -9.80776 q^{41} +12.2462 q^{43} -10.2462 q^{44} +13.1231 q^{47} +1.00000 q^{49} -1.12311 q^{52} +11.0000 q^{53} -2.87689 q^{55} +8.56155 q^{59} -5.43845 q^{61} -8.00000 q^{64} -0.315342 q^{65} -2.00000 q^{67} +4.87689 q^{68} +4.87689 q^{71} -8.56155 q^{73} -8.00000 q^{76} -5.12311 q^{77} -12.6847 q^{79} -2.24621 q^{80} +2.80776 q^{83} +1.36932 q^{85} +12.8078 q^{89} -0.561553 q^{91} +15.3693 q^{92} -2.24621 q^{95} +11.8078 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 4 * q^4 + 3 * q^5 - 2 * q^7	\( 2 q - 4 q^{4} + 3 q^{5} - 2 q^{7} + 2 q^{11} - 3 q^{13} + 8 q^{16} - 9 q^{17} + 8 q^{19} - 6 q^{20} - 3 q^{23} + 3 q^{25} + 4 q^{28} - 3 q^{31} - 3 q^{35} - 12 q^{37} + q^{41} + 8 q^{43} - 4 q^{44} + 18 q^{47} + 2 q^{49} + 6 q^{52} + 22 q^{53} - 14 q^{55} + 13 q^{59} - 15 q^{61} - 16 q^{64} - 13 q^{65} - 4 q^{67} + 18 q^{68} + 18 q^{71} - 13 q^{73} - 16 q^{76} - 2 q^{77} - 13 q^{79} + 12 q^{80} - 15 q^{83} - 22 q^{85} + 5 q^{89} + 3 q^{91} + 6 q^{92} + 12 q^{95} + 3 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 3 * q^5 - 2 * q^7 + 2 * q^11 - 3 * q^13 + 8 * q^16 - 9 * q^17 + 8 * q^19 - 6 * q^20 - 3 * q^23 + 3 * q^25 + 4 * q^28 - 3 * q^31 - 3 * q^35 - 12 * q^37 + q^41 + 8 * q^43 - 4 * q^44 + 18 * q^47 + 2 * q^49 + 6 * q^52 + 22 * q^53 - 14 * q^55 + 13 * q^59 - 15 * q^61 - 16 * q^64 - 13 * q^65 - 4 * q^67 + 18 * q^68 + 18 * q^71 - 13 * q^73 - 16 * q^76 - 2 * q^77 - 13 * q^79 + 12 * q^80 - 15 * q^83 - 22 * q^85 + 5 * q^89 + 3 * q^91 + 6 * q^92 + 12 * q^95 + 3 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more