LMFDB - Embedded newform 8024.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.23607 q^{5} +2.23607 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.23607 q^{5} +2.23607 q^{7} -2.00000 q^{9} +0.763932 q^{11} +1.23607 q^{13} -4.23607 q^{15} -1.00000 q^{17} -5.47214 q^{19} +2.23607 q^{21} +4.76393 q^{23} +12.9443 q^{25} -5.00000 q^{27} +2.70820 q^{29} +9.23607 q^{31} +0.763932 q^{33} -9.47214 q^{35} -8.00000 q^{37} +1.23607 q^{39} +0.527864 q^{41} -5.23607 q^{43} +8.47214 q^{45} +10.4721 q^{47} -2.00000 q^{49} -1.00000 q^{51} +0.708204 q^{53} -3.23607 q^{55} -5.47214 q^{57} -1.00000 q^{59} -11.2361 q^{61} -4.47214 q^{63} -5.23607 q^{65} +7.23607 q^{67} +4.76393 q^{69} -4.00000 q^{71} -3.70820 q^{73} +12.9443 q^{75} +1.70820 q^{77} +8.23607 q^{79} +1.00000 q^{81} +3.23607 q^{83} +4.23607 q^{85} +2.70820 q^{87} +6.94427 q^{89} +2.76393 q^{91} +9.23607 q^{93} +23.1803 q^{95} -2.00000 q^{97} -1.52786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 - 4 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} - 4 q^{9} + 6 q^{11} - 2 q^{13} - 4 q^{15} - 2 q^{17} - 2 q^{19} + 14 q^{23} + 8 q^{25} - 10 q^{27} - 8 q^{29} + 14 q^{31} + 6 q^{33} - 10 q^{35} - 16 q^{37} - 2 q^{39} + 10 q^{41} - 6 q^{43} + 8 q^{45} + 12 q^{47} - 4 q^{49} - 2 q^{51} - 12 q^{53} - 2 q^{55} - 2 q^{57} - 2 q^{59} - 18 q^{61} - 6 q^{65} + 10 q^{67} + 14 q^{69} - 8 q^{71} + 6 q^{73} + 8 q^{75} - 10 q^{77} + 12 q^{79} + 2 q^{81} + 2 q^{83} + 4 q^{85} - 8 q^{87} - 4 q^{89} + 10 q^{91} + 14 q^{93} + 24 q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 - 4 * q^9 + 6 * q^11 - 2 * q^13 - 4 * q^15 - 2 * q^17 - 2 * q^19 + 14 * q^23 + 8 * q^25 - 10 * q^27 - 8 * q^29 + 14 * q^31 + 6 * q^33 - 10 * q^35 - 16 * q^37 - 2 * q^39 + 10 * q^41 - 6 * q^43 + 8 * q^45 + 12 * q^47 - 4 * q^49 - 2 * q^51 - 12 * q^53 - 2 * q^55 - 2 * q^57 - 2 * q^59 - 18 * q^61 - 6 * q^65 + 10 * q^67 + 14 * q^69 - 8 * q^71 + 6 * q^73 + 8 * q^75 - 10 * q^77 + 12 * q^79 + 2 * q^81 + 2 * q^83 + 4 * q^85 - 8 * q^87 - 4 * q^89 + 10 * q^91 + 14 * q^93 + 24 * q^95 - 4 * q^97 - 12 * 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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−4.23607	−1.89443	−0.947214	−	0.320603i	$-0.896114\pi$
$5$	−4.23607	−1.89443	−0.947214	+	0.320603i	$0.896114\pi$
$6$	0	0
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	−4.23607	−1.09375
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.47214	−1.25539	−0.627697	−	0.778458i	$-0.716002\pi$
$19$	−5.47214	−1.25539	−0.627697	+	0.778458i	$0.716002\pi$
$20$	0	0
$21$	2.23607	0.487950
$22$	0	0
$23$	4.76393	0.993348	0.496674	−	0.867937i	$-0.334554\pi$
$23$	4.76393	0.993348	0.496674	+	0.867937i	$0.334554\pi$
$24$	0	0
$25$	12.9443	2.58885
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	2.70820	0.502901	0.251450	−	0.967870i	$-0.419092\pi$
$29$	2.70820	0.502901	0.251450	+	0.967870i	$0.419092\pi$
$30$	0	0
$31$	9.23607	1.65885	0.829423	−	0.558620i	$-0.188669\pi$
$31$	9.23607	1.65885	0.829423	+	0.558620i	$0.188669\pi$
$32$	0	0
$33$	0.763932	0.132983
$34$	0	0
$35$	−9.47214	−1.60108
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	1.23607	0.197929
$40$	0	0
$41$	0.527864	0.0824385	0.0412193	−	0.999150i	$-0.486876\pi$
$41$	0.527864	0.0824385	0.0412193	+	0.999150i	$0.486876\pi$
$42$	0	0
$43$	−5.23607	−0.798493	−0.399246	−	0.916844i	$-0.630728\pi$
$43$	−5.23607	−0.798493	−0.399246	+	0.916844i	$0.630728\pi$
$44$	0	0
$45$	8.47214	1.26295
$46$	0	0
$47$	10.4721	1.52752	0.763759	−	0.645501i	$-0.223352\pi$
$47$	10.4721	1.52752	0.763759	+	0.645501i	$0.223352\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	0.708204	0.0972793	0.0486396	−	0.998816i	$-0.484511\pi$
$53$	0.708204	0.0972793	0.0486396	+	0.998816i	$0.484511\pi$
$54$	0	0
$55$	−3.23607	−0.436351
$56$	0	0
$57$	−5.47214	−0.724802
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−11.2361	−1.43863	−0.719316	−	0.694683i	$-0.755544\pi$
$61$	−11.2361	−1.43863	−0.719316	+	0.694683i	$0.755544\pi$
$62$	0	0
$63$	−4.47214	−0.563436
$64$	0	0
$65$	−5.23607	−0.649454
$66$	0	0
$67$	7.23607	0.884026	0.442013	−	0.897009i	$-0.354264\pi$
$67$	7.23607	0.884026	0.442013	+	0.897009i	$0.354264\pi$
$68$	0	0
$69$	4.76393	0.573510
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−3.70820	−0.434012	−0.217006	−	0.976170i	$-0.569629\pi$
$73$	−3.70820	−0.434012	−0.217006	+	0.976170i	$0.569629\pi$
$74$	0	0
$75$	12.9443	1.49468
$76$	0	0
$77$	1.70820	0.194668
$78$	0	0
$79$	8.23607	0.926630	0.463315	−	0.886194i	$-0.346660\pi$
$79$	8.23607	0.926630	0.463315	+	0.886194i	$0.346660\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.23607	0.355205	0.177602	−	0.984102i	$-0.443166\pi$
$83$	3.23607	0.355205	0.177602	+	0.984102i	$0.443166\pi$
$84$	0	0
$85$	4.23607	0.459466
$86$	0	0
$87$	2.70820	0.290350
$88$	0	0
$89$	6.94427	0.736091	0.368046	−	0.929808i	$-0.380027\pi$
$89$	6.94427	0.736091	0.368046	+	0.929808i	$0.380027\pi$
$90$	0	0
$91$	2.76393	0.289739
$92$	0	0
$93$	9.23607	0.957736
$94$	0	0
$95$	23.1803	2.37825
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−1.52786	−0.153556
$100$	0	0
$101$	−9.70820	−0.966002	−0.483001	−	0.875620i	$-0.660453\pi$
$101$	−9.70820	−0.966002	−0.483001	+	0.875620i	$0.660453\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	−9.47214	−0.924386
$106$	0	0
$107$	16.4164	1.58703	0.793517	−	0.608548i	$-0.208248\pi$
$107$	16.4164	1.58703	0.793517	+	0.608548i	$0.208248\pi$
$108$	0	0
$109$	−19.7082	−1.88770	−0.943852	−	0.330369i	$-0.892827\pi$
$109$	−19.7082	−1.88770	−0.943852	+	0.330369i	$0.892827\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−20.1803	−1.88183
$116$	0	0
$117$	−2.47214	−0.228549
$118$	0	0
$119$	−2.23607	−0.204980
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	0.527864	0.0475959
$124$	0	0
$125$	−33.6525	−3.00997
$126$	0	0
$127$	−8.70820	−0.772728	−0.386364	−	0.922346i	$-0.626269\pi$
$127$	−8.70820	−0.772728	−0.386364	+	0.922346i	$0.626269\pi$
$128$	0	0
$129$	−5.23607	−0.461010
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−12.2361	−1.06100
$134$	0	0
$135$	21.1803	1.82291
$136$	0	0
$137$	−5.94427	−0.507853	−0.253927	−	0.967223i	$-0.581722\pi$
$137$	−5.94427	−0.507853	−0.253927	+	0.967223i	$0.581722\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	10.4721	0.881913
$142$	0	0
$143$	0.944272	0.0789640
$144$	0	0
$145$	−11.4721	−0.952709
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	14.4721	1.18560	0.592802	−	0.805348i	$-0.298022\pi$
$149$	14.4721	1.18560	0.592802	+	0.805348i	$0.298022\pi$
$150$	0	0
$151$	−0.763932	−0.0621679	−0.0310840	−	0.999517i	$-0.509896\pi$
$151$	−0.763932	−0.0621679	−0.0310840	+	0.999517i	$0.509896\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−39.1246	−3.14256
$156$	0	0
$157$	−21.1246	−1.68593	−0.842964	−	0.537970i	$-0.819191\pi$
$157$	−21.1246	−1.68593	−0.842964	+	0.537970i	$0.819191\pi$
$158$	0	0
$159$	0.708204	0.0561642
$160$	0	0
$161$	10.6525	0.839533
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	−3.23607	−0.251928
$166$	0	0
$167$	−16.2361	−1.25638	−0.628192	−	0.778058i	$-0.716205\pi$
$167$	−16.2361	−1.25638	−0.628192	+	0.778058i	$0.716205\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	10.9443	0.836929
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	28.9443	2.18798
$176$	0	0
$177$	−1.00000	−0.0751646
$178$	0	0
$179$	1.70820	0.127677	0.0638386	−	0.997960i	$-0.479666\pi$
$179$	1.70820	0.127677	0.0638386	+	0.997960i	$0.479666\pi$
$180$	0	0
$181$	−15.7639	−1.17172	−0.585862	−	0.810411i	$-0.699244\pi$
$181$	−15.7639	−1.17172	−0.585862	+	0.810411i	$0.699244\pi$
$182$	0	0
$183$	−11.2361	−0.830594
$184$	0	0
$185$	33.8885	2.49154
$186$	0	0
$187$	−0.763932	−0.0558642
$188$	0	0
$189$	−11.1803	−0.813250
$190$	0	0
$191$	7.41641	0.536632	0.268316	−	0.963331i	$-0.413533\pi$
$191$	7.41641	0.536632	0.268316	+	0.963331i	$0.413533\pi$
$192$	0	0
$193$	−14.8885	−1.07170	−0.535850	−	0.844313i	$-0.680009\pi$
$193$	−14.8885	−1.07170	−0.535850	+	0.844313i	$0.680009\pi$
$194$	0	0
$195$	−5.23607	−0.374963
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	20.7082	1.46797	0.733983	−	0.679168i	$-0.237659\pi$
$199$	20.7082	1.46797	0.733983	+	0.679168i	$0.237659\pi$
$200$	0	0
$201$	7.23607	0.510393
$202$	0	0
$203$	6.05573	0.425029
$204$	0	0
$205$	−2.23607	−0.156174
$206$	0	0
$207$	−9.52786	−0.662232
$208$	0	0
$209$	−4.18034	−0.289160
$210$	0	0
$211$	22.3607	1.53937	0.769686	−	0.638422i	$-0.220413\pi$
$211$	22.3607	1.53937	0.769686	+	0.638422i	$0.220413\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	22.1803	1.51269
$216$	0	0
$217$	20.6525	1.40198
$218$	0	0
$219$	−3.70820	−0.250577
$220$	0	0
$221$	−1.23607	−0.0831469
$222$	0	0
$223$	−13.5279	−0.905893	−0.452946	−	0.891538i	$-0.649627\pi$
$223$	−13.5279	−0.905893	−0.452946	+	0.891538i	$0.649627\pi$
$224$	0	0
$225$	−25.8885	−1.72590
$226$	0	0
$227$	2.18034	0.144714	0.0723571	−	0.997379i	$-0.476948\pi$
$227$	2.18034	0.144714	0.0723571	+	0.997379i	$0.476948\pi$
$228$	0	0
$229$	−7.52786	−0.497455	−0.248728	−	0.968573i	$-0.580012\pi$
$229$	−7.52786	−0.497455	−0.248728	+	0.968573i	$0.580012\pi$
$230$	0	0
$231$	1.70820	0.112392
$232$	0	0
$233$	25.1246	1.64597	0.822984	−	0.568065i	$-0.192308\pi$
$233$	25.1246	1.64597	0.822984	+	0.568065i	$0.192308\pi$
$234$	0	0
$235$	−44.3607	−2.89377
$236$	0	0
$237$	8.23607	0.534990
$238$	0	0
$239$	9.65248	0.624367	0.312183	−	0.950022i	$-0.398940\pi$
$239$	9.65248	0.624367	0.312183	+	0.950022i	$0.398940\pi$
$240$	0	0
$241$	−8.52786	−0.549328	−0.274664	−	0.961540i	$-0.588567\pi$
$241$	−8.52786	−0.549328	−0.274664	+	0.961540i	$0.588567\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	8.47214	0.541265
$246$	0	0
$247$	−6.76393	−0.430379
$248$	0	0
$249$	3.23607	0.205077
$250$	0	0
$251$	0.527864	0.0333185	0.0166592	−	0.999861i	$-0.494697\pi$
$251$	0.527864	0.0333185	0.0166592	+	0.999861i	$0.494697\pi$
$252$	0	0
$253$	3.63932	0.228802
$254$	0	0
$255$	4.23607	0.265273
$256$	0	0
$257$	−23.9443	−1.49360	−0.746801	−	0.665047i	$-0.768411\pi$
$257$	−23.9443	−1.49360	−0.746801	+	0.665047i	$0.768411\pi$
$258$	0	0
$259$	−17.8885	−1.11154
$260$	0	0
$261$	−5.41641	−0.335267
$262$	0	0
$263$	−15.6525	−0.965173	−0.482587	−	0.875848i	$-0.660303\pi$
$263$	−15.6525	−0.965173	−0.482587	+	0.875848i	$0.660303\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	0	0
$267$	6.94427	0.424983
$268$	0	0
$269$	−10.9443	−0.667284	−0.333642	−	0.942700i	$-0.608278\pi$
$269$	−10.9443	−0.667284	−0.333642	+	0.942700i	$0.608278\pi$
$270$	0	0
$271$	−29.0689	−1.76581	−0.882904	−	0.469553i	$-0.844415\pi$
$271$	−29.0689	−1.76581	−0.882904	+	0.469553i	$0.844415\pi$
$272$	0	0
$273$	2.76393	0.167281
$274$	0	0
$275$	9.88854	0.596302
$276$	0	0
$277$	−25.7639	−1.54800	−0.774002	−	0.633183i	$-0.781748\pi$
$277$	−25.7639	−1.54800	−0.774002	+	0.633183i	$0.781748\pi$
$278$	0	0
$279$	−18.4721	−1.10590
$280$	0	0
$281$	−21.0000	−1.25275	−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000	−1.25275	−0.626377	+	0.779520i	$0.715463\pi$
$282$	0	0
$283$	−14.6525	−0.870999	−0.435500	−	0.900189i	$-0.643428\pi$
$283$	−14.6525	−0.870999	−0.435500	+	0.900189i	$0.643428\pi$
$284$	0	0
$285$	23.1803	1.37308
$286$	0	0
$287$	1.18034	0.0696733
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	9.18034	0.536321	0.268161	−	0.963374i	$-0.413584\pi$
$293$	9.18034	0.536321	0.268161	+	0.963374i	$0.413584\pi$
$294$	0	0
$295$	4.23607	0.246633
$296$	0	0
$297$	−3.81966	−0.221639
$298$	0	0
$299$	5.88854	0.340543
$300$	0	0
$301$	−11.7082	−0.674850
$302$	0	0
$303$	−9.70820	−0.557722
$304$	0	0
$305$	47.5967	2.72538
$306$	0	0
$307$	−5.94427	−0.339258	−0.169629	−	0.985508i	$-0.554257\pi$
$307$	−5.94427	−0.339258	−0.169629	+	0.985508i	$0.554257\pi$
$308$	0	0
$309$	2.00000	0.113776
$310$	0	0
$311$	−18.7082	−1.06084	−0.530422	−	0.847733i	$-0.677967\pi$
$311$	−18.7082	−1.06084	−0.530422	+	0.847733i	$0.677967\pi$
$312$	0	0
$313$	3.23607	0.182913	0.0914567	−	0.995809i	$-0.470848\pi$
$313$	3.23607	0.182913	0.0914567	+	0.995809i	$0.470848\pi$
$314$	0	0
$315$	18.9443	1.06739
$316$	0	0
$317$	−3.52786	−0.198145	−0.0990723	−	0.995080i	$-0.531588\pi$
$317$	−3.52786	−0.198145	−0.0990723	+	0.995080i	$0.531588\pi$
$318$	0	0
$319$	2.06888	0.115835
$320$	0	0
$321$	16.4164	0.916275
$322$	0	0
$323$	5.47214	0.304478
$324$	0	0
$325$	16.0000	0.887520
$326$	0	0
$327$	−19.7082	−1.08987
$328$	0	0
$329$	23.4164	1.29099
$330$	0	0
$331$	33.3607	1.83367	0.916834	−	0.399268i	$-0.130736\pi$
$331$	33.3607	1.83367	0.916834	+	0.399268i	$0.130736\pi$
$332$	0	0
$333$	16.0000	0.876795
$334$	0	0
$335$	−30.6525	−1.67472
$336$	0	0
$337$	2.18034	0.118771	0.0593853	−	0.998235i	$-0.481086\pi$
$337$	2.18034	0.118771	0.0593853	+	0.998235i	$0.481086\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.05573	0.382089
$342$	0	0
$343$	−20.1246	−1.08663
$344$	0	0
$345$	−20.1803	−1.08647
$346$	0	0
$347$	−18.6525	−1.00132	−0.500659	−	0.865645i	$-0.666909\pi$
$347$	−18.6525	−1.00132	−0.500659	+	0.865645i	$0.666909\pi$
$348$	0	0
$349$	−24.4721	−1.30996	−0.654982	−	0.755645i	$-0.727324\pi$
$349$	−24.4721	−1.30996	−0.654982	+	0.755645i	$0.727324\pi$
$350$	0	0
$351$	−6.18034	−0.329882
$352$	0	0
$353$	32.3607	1.72239	0.861193	−	0.508279i	$-0.169718\pi$
$353$	32.3607	1.72239	0.861193	+	0.508279i	$0.169718\pi$
$354$	0	0
$355$	16.9443	0.899309
$356$	0	0
$357$	−2.23607	−0.118345
$358$	0	0
$359$	10.7082	0.565157	0.282579	−	0.959244i	$-0.408810\pi$
$359$	10.7082	0.565157	0.282579	+	0.959244i	$0.408810\pi$
$360$	0	0
$361$	10.9443	0.576014
$362$	0	0
$363$	−10.4164	−0.546720
$364$	0	0
$365$	15.7082	0.822205
$366$	0	0
$367$	4.18034	0.218212	0.109106	−	0.994030i	$-0.465201\pi$
$367$	4.18034	0.218212	0.109106	+	0.994030i	$0.465201\pi$
$368$	0	0
$369$	−1.05573	−0.0549590
$370$	0	0
$371$	1.58359	0.0822160
$372$	0	0
$373$	9.05573	0.468888	0.234444	−	0.972130i	$-0.424673\pi$
$373$	9.05573	0.468888	0.234444	+	0.972130i	$0.424673\pi$
$374$	0	0
$375$	−33.6525	−1.73781
$376$	0	0
$377$	3.34752	0.172406
$378$	0	0
$379$	23.9443	1.22993	0.614967	−	0.788553i	$-0.289169\pi$
$379$	23.9443	1.22993	0.614967	+	0.788553i	$0.289169\pi$
$380$	0	0
$381$	−8.70820	−0.446135
$382$	0	0
$383$	21.5279	1.10002	0.550011	−	0.835157i	$-0.314623\pi$
$383$	21.5279	1.10002	0.550011	+	0.835157i	$0.314623\pi$
$384$	0	0
$385$	−7.23607	−0.368784
$386$	0	0
$387$	10.4721	0.532329
$388$	0	0
$389$	−16.4721	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$389$	−16.4721	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$390$	0	0
$391$	−4.76393	−0.240922
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	−34.8885	−1.75543
$396$	0	0
$397$	30.8328	1.54745	0.773727	−	0.633519i	$-0.218390\pi$
$397$	30.8328	1.54745	0.773727	+	0.633519i	$0.218390\pi$
$398$	0	0
$399$	−12.2361	−0.612570
$400$	0	0
$401$	−28.8328	−1.43984	−0.719921	−	0.694056i	$-0.755822\pi$
$401$	−28.8328	−1.43984	−0.719921	+	0.694056i	$0.755822\pi$
$402$	0	0
$403$	11.4164	0.568692
$404$	0	0
$405$	−4.23607	−0.210492
$406$	0	0
$407$	−6.11146	−0.302934
$408$	0	0
$409$	−38.3607	−1.89681	−0.948407	−	0.317056i	$-0.897306\pi$
$409$	−38.3607	−1.89681	−0.948407	+	0.317056i	$0.897306\pi$
$410$	0	0
$411$	−5.94427	−0.293209
$412$	0	0
$413$	−2.23607	−0.110030
$414$	0	0
$415$	−13.7082	−0.672909
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	18.3607	0.894845	0.447422	−	0.894323i	$-0.352342\pi$
$421$	18.3607	0.894845	0.447422	+	0.894323i	$0.352342\pi$
$422$	0	0
$423$	−20.9443	−1.01835
$424$	0	0
$425$	−12.9443	−0.627889
$426$	0	0
$427$	−25.1246	−1.21587
$428$	0	0
$429$	0.944272	0.0455899
$430$	0	0
$431$	−28.7639	−1.38551	−0.692755	−	0.721173i	$-0.743603\pi$
$431$	−28.7639	−1.38551	−0.692755	+	0.721173i	$0.743603\pi$
$432$	0	0
$433$	24.8885	1.19607	0.598034	−	0.801471i	$-0.295949\pi$
$433$	24.8885	1.19607	0.598034	+	0.801471i	$0.295949\pi$
$434$	0	0
$435$	−11.4721	−0.550047
$436$	0	0
$437$	−26.0689	−1.24704
$438$	0	0
$439$	−30.8328	−1.47157	−0.735785	−	0.677215i	$-0.763187\pi$
$439$	−30.8328	−1.47157	−0.735785	+	0.677215i	$0.763187\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	0	0
$443$	−22.3607	−1.06239	−0.531194	−	0.847250i	$-0.678256\pi$
$443$	−22.3607	−1.06239	−0.531194	+	0.847250i	$0.678256\pi$
$444$	0	0
$445$	−29.4164	−1.39447
$446$	0	0
$447$	14.4721	0.684509
$448$	0	0
$449$	30.3050	1.43018	0.715090	−	0.699033i	$-0.246386\pi$
$449$	30.3050	1.43018	0.715090	+	0.699033i	$0.246386\pi$
$450$	0	0
$451$	0.403252	0.0189884
$452$	0	0
$453$	−0.763932	−0.0358927
$454$	0	0
$455$	−11.7082	−0.548889
$456$	0	0
$457$	−38.6525	−1.80809	−0.904043	−	0.427441i	$-0.859415\pi$
$457$	−38.6525	−1.80809	−0.904043	+	0.427441i	$0.859415\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	11.5279	0.536906	0.268453	−	0.963293i	$-0.413488\pi$
$461$	11.5279	0.536906	0.268453	+	0.963293i	$0.413488\pi$
$462$	0	0
$463$	17.7082	0.822970	0.411485	−	0.911417i	$-0.365010\pi$
$463$	17.7082	0.822970	0.411485	+	0.911417i	$0.365010\pi$
$464$	0	0
$465$	−39.1246	−1.81436
$466$	0	0
$467$	8.29180	0.383699	0.191849	−	0.981424i	$-0.438552\pi$
$467$	8.29180	0.383699	0.191849	+	0.981424i	$0.438552\pi$
$468$	0	0
$469$	16.1803	0.747139
$470$	0	0
$471$	−21.1246	−0.973371
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−70.8328	−3.25003
$476$	0	0
$477$	−1.41641	−0.0648529
$478$	0	0
$479$	−32.9443	−1.50526	−0.752631	−	0.658442i	$-0.771216\pi$
$479$	−32.9443	−1.50526	−0.752631	+	0.658442i	$0.771216\pi$
$480$	0	0
$481$	−9.88854	−0.450879
$482$	0	0
$483$	10.6525	0.484704
$484$	0	0
$485$	8.47214	0.384700
$486$	0	0
$487$	−35.6525	−1.61557	−0.807784	−	0.589479i	$-0.799333\pi$
$487$	−35.6525	−1.61557	−0.807784	+	0.589479i	$0.799333\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.4164	0.470086	0.235043	−	0.971985i	$-0.424477\pi$
$491$	10.4164	0.470086	0.235043	+	0.971985i	$0.424477\pi$
$492$	0	0
$493$	−2.70820	−0.121971
$494$	0	0
$495$	6.47214	0.290901
$496$	0	0
$497$	−8.94427	−0.401205
$498$	0	0
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	0	0
$501$	−16.2361	−0.725374
$502$	0	0
$503$	−16.3607	−0.729487	−0.364743	−	0.931108i	$-0.618843\pi$
$503$	−16.3607	−0.729487	−0.364743	+	0.931108i	$0.618843\pi$
$504$	0	0
$505$	41.1246	1.83002
$506$	0	0
$507$	−11.4721	−0.509495
$508$	0	0
$509$	4.18034	0.185290	0.0926452	−	0.995699i	$-0.470468\pi$
$509$	4.18034	0.185290	0.0926452	+	0.995699i	$0.470468\pi$
$510$	0	0
$511$	−8.29180	−0.366807
$512$	0	0
$513$	27.3607	1.20800
$514$	0	0
$515$	−8.47214	−0.373327
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	37.8328	1.65431	0.827157	−	0.561971i	$-0.189957\pi$
$523$	37.8328	1.65431	0.827157	+	0.561971i	$0.189957\pi$
$524$	0	0
$525$	28.9443	1.26323
$526$	0	0
$527$	−9.23607	−0.402329
$528$	0	0
$529$	−0.304952	−0.0132588
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	0.652476	0.0282619
$534$	0	0
$535$	−69.5410	−3.00652
$536$	0	0
$537$	1.70820	0.0737144
$538$	0	0
$539$	−1.52786	−0.0658098
$540$	0	0
$541$	−30.2918	−1.30235	−0.651173	−	0.758929i	$-0.725723\pi$
$541$	−30.2918	−1.30235	−0.651173	+	0.758929i	$0.725723\pi$
$542$	0	0
$543$	−15.7639	−0.676495
$544$	0	0
$545$	83.4853	3.57612
$546$	0	0
$547$	−38.8328	−1.66037	−0.830186	−	0.557487i	$-0.811766\pi$
$547$	−38.8328	−1.66037	−0.830186	+	0.557487i	$0.811766\pi$
$548$	0	0
$549$	22.4721	0.959087
$550$	0	0
$551$	−14.8197	−0.631339
$552$	0	0
$553$	18.4164	0.783145
$554$	0	0
$555$	33.8885	1.43849
$556$	0	0
$557$	−43.0689	−1.82489	−0.912444	−	0.409203i	$-0.865807\pi$
$557$	−43.0689	−1.82489	−0.912444	+	0.409203i	$0.865807\pi$
$558$	0	0
$559$	−6.47214	−0.273742
$560$	0	0
$561$	−0.763932	−0.0322532
$562$	0	0
$563$	16.2918	0.686617	0.343309	−	0.939223i	$-0.388452\pi$
$563$	16.2918	0.686617	0.343309	+	0.939223i	$0.388452\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.23607	0.0939060
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	−9.70820	−0.406276	−0.203138	−	0.979150i	$-0.565114\pi$
$571$	−9.70820	−0.406276	−0.203138	+	0.979150i	$0.565114\pi$
$572$	0	0
$573$	7.41641	0.309825
$574$	0	0
$575$	61.6656	2.57163
$576$	0	0
$577$	−26.4164	−1.09973	−0.549865	−	0.835254i	$-0.685321\pi$
$577$	−26.4164	−1.09973	−0.549865	+	0.835254i	$0.685321\pi$
$578$	0	0
$579$	−14.8885	−0.618747
$580$	0	0
$581$	7.23607	0.300203
$582$	0	0
$583$	0.541020	0.0224067
$584$	0	0
$585$	10.4721	0.432970
$586$	0	0
$587$	−26.0000	−1.07313	−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000	−1.07313	−0.536567	+	0.843857i	$0.680279\pi$
$588$	0	0
$589$	−50.5410	−2.08251
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	−7.00000	−0.287456	−0.143728	−	0.989617i	$-0.545909\pi$
$593$	−7.00000	−0.287456	−0.143728	+	0.989617i	$0.545909\pi$
$594$	0	0
$595$	9.47214	0.388320
$596$	0	0
$597$	20.7082	0.847530
$598$	0	0
$599$	−19.1803	−0.783687	−0.391844	−	0.920032i	$-0.628163\pi$
$599$	−19.1803	−0.783687	−0.391844	+	0.920032i	$0.628163\pi$
$600$	0	0
$601$	−45.3050	−1.84803	−0.924014	−	0.382359i	$-0.875112\pi$
$601$	−45.3050	−1.84803	−0.924014	+	0.382359i	$0.875112\pi$
$602$	0	0
$603$	−14.4721	−0.589351
$604$	0	0
$605$	44.1246	1.79392
$606$	0	0
$607$	35.7639	1.45161	0.725807	−	0.687899i	$-0.241467\pi$
$607$	35.7639	1.45161	0.725807	+	0.687899i	$0.241467\pi$
$608$	0	0
$609$	6.05573	0.245390
$610$	0	0
$611$	12.9443	0.523669
$612$	0	0
$613$	−6.36068	−0.256905	−0.128453	−	0.991716i	$-0.541001\pi$
$613$	−6.36068	−0.256905	−0.128453	+	0.991716i	$0.541001\pi$
$614$	0	0
$615$	−2.23607	−0.0901670
$616$	0	0
$617$	−3.94427	−0.158790	−0.0793952	−	0.996843i	$-0.525299\pi$
$617$	−3.94427	−0.158790	−0.0793952	+	0.996843i	$0.525299\pi$
$618$	0	0
$619$	37.8328	1.52063	0.760315	−	0.649555i	$-0.225045\pi$
$619$	37.8328	1.52063	0.760315	+	0.649555i	$0.225045\pi$
$620$	0	0
$621$	−23.8197	−0.955850
$622$	0	0
$623$	15.5279	0.622111
$624$	0	0
$625$	77.8328	3.11331
$626$	0	0
$627$	−4.18034	−0.166947
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	−20.3607	−0.810546	−0.405273	−	0.914196i	$-0.632824\pi$
$631$	−20.3607	−0.810546	−0.405273	+	0.914196i	$0.632824\pi$
$632$	0	0
$633$	22.3607	0.888757
$634$	0	0
$635$	36.8885	1.46388
$636$	0	0
$637$	−2.47214	−0.0979496
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	23.5279	0.929295	0.464647	−	0.885496i	$-0.346181\pi$
$641$	23.5279	0.929295	0.464647	+	0.885496i	$0.346181\pi$
$642$	0	0
$643$	−10.4164	−0.410783	−0.205392	−	0.978680i	$-0.565847\pi$
$643$	−10.4164	−0.410783	−0.205392	+	0.978680i	$0.565847\pi$
$644$	0	0
$645$	22.1803	0.873350
$646$	0	0
$647$	12.3475	0.485431	0.242716	−	0.970097i	$-0.421962\pi$
$647$	12.3475	0.485431	0.242716	+	0.970097i	$0.421962\pi$
$648$	0	0
$649$	−0.763932	−0.0299870
$650$	0	0
$651$	20.6525	0.809434
$652$	0	0
$653$	9.29180	0.363616	0.181808	−	0.983334i	$-0.441805\pi$
$653$	9.29180	0.363616	0.181808	+	0.983334i	$0.441805\pi$
$654$	0	0
$655$	25.4164	0.993101
$656$	0	0
$657$	7.41641	0.289342
$658$	0	0
$659$	14.4721	0.563754	0.281877	−	0.959450i	$-0.409043\pi$
$659$	14.4721	0.563754	0.281877	+	0.959450i	$0.409043\pi$
$660$	0	0
$661$	27.5410	1.07122	0.535611	−	0.844465i	$-0.320081\pi$
$661$	27.5410	1.07122	0.535611	+	0.844465i	$0.320081\pi$
$662$	0	0
$663$	−1.23607	−0.0480049
$664$	0	0
$665$	51.8328	2.00999
$666$	0	0
$667$	12.9017	0.499556
$668$	0	0
$669$	−13.5279	−0.523017
$670$	0	0
$671$	−8.58359	−0.331366
$672$	0	0
$673$	11.8885	0.458270	0.229135	−	0.973395i	$-0.426410\pi$
$673$	11.8885	0.458270	0.229135	+	0.973395i	$0.426410\pi$
$674$	0	0
$675$	−64.7214	−2.49113
$676$	0	0
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	−4.47214	−0.171625
$680$	0	0
$681$	2.18034	0.0835508
$682$	0	0
$683$	35.7082	1.36634	0.683168	−	0.730261i	$-0.260602\pi$
$683$	35.7082	1.36634	0.683168	+	0.730261i	$0.260602\pi$
$684$	0	0
$685$	25.1803	0.962091
$686$	0	0
$687$	−7.52786	−0.287206
$688$	0	0
$689$	0.875388	0.0333496
$690$	0	0
$691$	36.1803	1.37636	0.688182	−	0.725538i	$-0.258409\pi$
$691$	36.1803	1.37636	0.688182	+	0.725538i	$0.258409\pi$
$692$	0	0
$693$	−3.41641	−0.129779
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.527864	−0.0199943
$698$	0	0
$699$	25.1246	0.950300
$700$	0	0
$701$	16.4721	0.622144	0.311072	−	0.950386i	$-0.399312\pi$
$701$	16.4721	0.622144	0.311072	+	0.950386i	$0.399312\pi$
$702$	0	0
$703$	43.7771	1.65108
$704$	0	0
$705$	−44.3607	−1.67072
$706$	0	0
$707$	−21.7082	−0.816421
$708$	0	0
$709$	22.1246	0.830907	0.415454	−	0.909614i	$-0.363623\pi$
$709$	22.1246	0.830907	0.415454	+	0.909614i	$0.363623\pi$
$710$	0	0
$711$	−16.4721	−0.617753
$712$	0	0
$713$	44.0000	1.64781
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	9.65248	0.360478
$718$	0	0
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	4.47214	0.166551
$722$	0	0
$723$	−8.52786	−0.317155
$724$	0	0
$725$	35.0557	1.30194
$726$	0	0
$727$	49.8885	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$727$	49.8885	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	5.23607	0.193663
$732$	0	0
$733$	32.4721	1.19939	0.599693	−	0.800230i	$-0.295289\pi$
$733$	32.4721	1.19939	0.599693	+	0.800230i	$0.295289\pi$
$734$	0	0
$735$	8.47214	0.312499
$736$	0	0
$737$	5.52786	0.203621
$738$	0	0
$739$	−23.3050	−0.857287	−0.428643	−	0.903474i	$-0.641008\pi$
$739$	−23.3050	−0.857287	−0.428643	+	0.903474i	$0.641008\pi$
$740$	0	0
$741$	−6.76393	−0.248479
$742$	0	0
$743$	−12.5836	−0.461647	−0.230824	−	0.972996i	$-0.574142\pi$
$743$	−12.5836	−0.461647	−0.230824	+	0.972996i	$0.574142\pi$
$744$	0	0
$745$	−61.3050	−2.24604
$746$	0	0
$747$	−6.47214	−0.236803
$748$	0	0
$749$	36.7082	1.34129
$750$	0	0
$751$	25.0557	0.914297	0.457148	−	0.889391i	$-0.348871\pi$
$751$	25.0557	0.914297	0.457148	+	0.889391i	$0.348871\pi$
$752$	0	0
$753$	0.527864	0.0192364
$754$	0	0
$755$	3.23607	0.117773
$756$	0	0
$757$	−16.1246	−0.586059	−0.293029	−	0.956103i	$-0.594663\pi$
$757$	−16.1246	−0.586059	−0.293029	+	0.956103i	$0.594663\pi$
$758$	0	0
$759$	3.63932	0.132099
$760$	0	0
$761$	33.9443	1.23048	0.615239	−	0.788340i	$-0.289059\pi$
$761$	33.9443	1.23048	0.615239	+	0.788340i	$0.289059\pi$
$762$	0	0
$763$	−44.0689	−1.59540
$764$	0	0
$765$	−8.47214	−0.306311
$766$	0	0
$767$	−1.23607	−0.0446318
$768$	0	0
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	−23.9443	−0.862332
$772$	0	0
$773$	−12.2918	−0.442105	−0.221053	−	0.975262i	$-0.570949\pi$
$773$	−12.2918	−0.442105	−0.221053	+	0.975262i	$0.570949\pi$
$774$	0	0
$775$	119.554	4.29451
$776$	0	0
$777$	−17.8885	−0.641748
$778$	0	0
$779$	−2.88854	−0.103493
$780$	0	0
$781$	−3.05573	−0.109343
$782$	0	0
$783$	−13.5410	−0.483917
$784$	0	0
$785$	89.4853	3.19387
$786$	0	0
$787$	−16.3607	−0.583195	−0.291598	−	0.956541i	$-0.594187\pi$
$787$	−16.3607	−0.583195	−0.291598	+	0.956541i	$0.594187\pi$
$788$	0	0
$789$	−15.6525	−0.557243
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−13.8885	−0.493197
$794$	0	0
$795$	−3.00000	−0.106399
$796$	0	0
$797$	−29.0557	−1.02921	−0.514603	−	0.857428i	$-0.672061\pi$
$797$	−29.0557	−1.02921	−0.514603	+	0.857428i	$0.672061\pi$
$798$	0	0
$799$	−10.4721	−0.370478
$800$	0	0
$801$	−13.8885	−0.490728
$802$	0	0
$803$	−2.83282	−0.0999679
$804$	0	0
$805$	−45.1246	−1.59043
$806$	0	0
$807$	−10.9443	−0.385257
$808$	0	0
$809$	40.8328	1.43561	0.717803	−	0.696247i	$-0.245148\pi$
$809$	40.8328	1.43561	0.717803	+	0.696247i	$0.245148\pi$
$810$	0	0
$811$	−17.1246	−0.601326	−0.300663	−	0.953730i	$-0.597208\pi$
$811$	−17.1246	−0.601326	−0.300663	+	0.953730i	$0.597208\pi$
$812$	0	0
$813$	−29.0689	−1.01949
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.6525	1.00242
$818$	0	0
$819$	−5.52786	−0.193159
$820$	0	0
$821$	−19.7082	−0.687821	−0.343911	−	0.939002i	$-0.611752\pi$
$821$	−19.7082	−0.687821	−0.343911	+	0.939002i	$0.611752\pi$
$822$	0	0
$823$	−26.6525	−0.929047	−0.464523	−	0.885561i	$-0.653774\pi$
$823$	−26.6525	−0.929047	−0.464523	+	0.885561i	$0.653774\pi$
$824$	0	0
$825$	9.88854	0.344275
$826$	0	0
$827$	−46.8328	−1.62854	−0.814268	−	0.580489i	$-0.802862\pi$
$827$	−46.8328	−1.62854	−0.814268	+	0.580489i	$0.802862\pi$
$828$	0	0
$829$	8.59675	0.298577	0.149289	−	0.988794i	$-0.452302\pi$
$829$	8.59675	0.298577	0.149289	+	0.988794i	$0.452302\pi$
$830$	0	0
$831$	−25.7639	−0.893741
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	68.7771	2.38013
$836$	0	0
$837$	−46.1803	−1.59623
$838$	0	0
$839$	16.2918	0.562455	0.281228	−	0.959641i	$-0.409258\pi$
$839$	16.2918	0.562455	0.281228	+	0.959641i	$0.409258\pi$
$840$	0	0
$841$	−21.6656	−0.747091
$842$	0	0
$843$	−21.0000	−0.723278
$844$	0	0
$845$	48.5967	1.67178
$846$	0	0
$847$	−23.2918	−0.800316
$848$	0	0
$849$	−14.6525	−0.502872
$850$	0	0
$851$	−38.1115	−1.30644
$852$	0	0
$853$	19.0689	0.652906	0.326453	−	0.945213i	$-0.394146\pi$
$853$	19.0689	0.652906	0.326453	+	0.945213i	$0.394146\pi$
$854$	0	0
$855$	−46.3607	−1.58550
$856$	0	0
$857$	−56.5410	−1.93140	−0.965702	−	0.259652i	$-0.916392\pi$
$857$	−56.5410	−1.93140	−0.965702	+	0.259652i	$0.916392\pi$
$858$	0	0
$859$	−23.0132	−0.785199	−0.392599	−	0.919710i	$-0.628424\pi$
$859$	−23.0132	−0.785199	−0.392599	+	0.919710i	$0.628424\pi$
$860$	0	0
$861$	1.18034	0.0402259
$862$	0	0
$863$	14.7639	0.502570	0.251285	−	0.967913i	$-0.419147\pi$
$863$	14.7639	0.502570	0.251285	+	0.967913i	$0.419147\pi$
$864$	0	0
$865$	42.3607	1.44031
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	6.29180	0.213435
$870$	0	0
$871$	8.94427	0.303065
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	−75.2492	−2.54389
$876$	0	0
$877$	12.5967	0.425362	0.212681	−	0.977122i	$-0.431780\pi$
$877$	12.5967	0.425362	0.212681	+	0.977122i	$0.431780\pi$
$878$	0	0
$879$	9.18034	0.309645
$880$	0	0
$881$	46.0689	1.55210	0.776050	−	0.630671i	$-0.217220\pi$
$881$	46.0689	1.55210	0.776050	+	0.630671i	$0.217220\pi$
$882$	0	0
$883$	28.3050	0.952537	0.476269	−	0.879300i	$-0.341989\pi$
$883$	28.3050	0.952537	0.476269	+	0.879300i	$0.341989\pi$
$884$	0	0
$885$	4.23607	0.142394
$886$	0	0
$887$	4.65248	0.156215	0.0781074	−	0.996945i	$-0.475112\pi$
$887$	4.65248	0.156215	0.0781074	+	0.996945i	$0.475112\pi$
$888$	0	0
$889$	−19.4721	−0.653074
$890$	0	0
$891$	0.763932	0.0255927
$892$	0	0
$893$	−57.3050	−1.91764
$894$	0	0
$895$	−7.23607	−0.241875
$896$	0	0
$897$	5.88854	0.196613
$898$	0	0
$899$	25.0132	0.834235
$900$	0	0
$901$	−0.708204	−0.0235937
$902$	0	0
$903$	−11.7082	−0.389625
$904$	0	0
$905$	66.7771	2.21975
$906$	0	0
$907$	21.0000	0.697294	0.348647	−	0.937254i	$-0.386641\pi$
$907$	21.0000	0.697294	0.348647	+	0.937254i	$0.386641\pi$
$908$	0	0
$909$	19.4164	0.644002
$910$	0	0
$911$	56.1246	1.85949	0.929746	−	0.368202i	$-0.120027\pi$
$911$	56.1246	1.85949	0.929746	+	0.368202i	$0.120027\pi$
$912$	0	0
$913$	2.47214	0.0818158
$914$	0	0
$915$	47.5967	1.57350
$916$	0	0
$917$	−13.4164	−0.443049
$918$	0	0
$919$	39.5967	1.30618	0.653088	−	0.757282i	$-0.273473\pi$
$919$	39.5967	1.30618	0.653088	+	0.757282i	$0.273473\pi$
$920$	0	0
$921$	−5.94427	−0.195870
$922$	0	0
$923$	−4.94427	−0.162743
$924$	0	0
$925$	−103.554	−3.40484
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−9.70820	−0.318516	−0.159258	−	0.987237i	$-0.550910\pi$
$929$	−9.70820	−0.318516	−0.159258	+	0.987237i	$0.550910\pi$
$930$	0	0
$931$	10.9443	0.358684
$932$	0	0
$933$	−18.7082	−0.612479
$934$	0	0
$935$	3.23607	0.105831
$936$	0	0
$937$	−23.8885	−0.780405	−0.390202	−	0.920729i	$-0.627595\pi$
$937$	−23.8885	−0.780405	−0.390202	+	0.920729i	$0.627595\pi$
$938$	0	0
$939$	3.23607	0.105605
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	2.51471	0.0818902
$944$	0	0
$945$	47.3607	1.54064
$946$	0	0
$947$	4.05573	0.131793	0.0658967	−	0.997826i	$-0.479009\pi$
$947$	4.05573	0.131793	0.0658967	+	0.997826i	$0.479009\pi$
$948$	0	0
$949$	−4.58359	−0.148790
$950$	0	0
$951$	−3.52786	−0.114399
$952$	0	0
$953$	25.7771	0.835002	0.417501	−	0.908677i	$-0.362906\pi$
$953$	25.7771	0.835002	0.417501	+	0.908677i	$0.362906\pi$
$954$	0	0
$955$	−31.4164	−1.01661
$956$	0	0
$957$	2.06888	0.0668775
$958$	0	0
$959$	−13.2918	−0.429214
$960$	0	0
$961$	54.3050	1.75177
$962$	0	0
$963$	−32.8328	−1.05802
$964$	0	0
$965$	63.0689	2.03026
$966$	0	0
$967$	−52.2492	−1.68022	−0.840111	−	0.542415i	$-0.817510\pi$
$967$	−52.2492	−1.68022	−0.840111	+	0.542415i	$0.817510\pi$
$968$	0	0
$969$	5.47214	0.175790
$970$	0	0
$971$	−53.7214	−1.72400	−0.862000	−	0.506908i	$-0.830788\pi$
$971$	−53.7214	−1.72400	−0.862000	+	0.506908i	$0.830788\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	16.0000	0.512410
$976$	0	0
$977$	45.8885	1.46810	0.734052	−	0.679093i	$-0.237627\pi$
$977$	45.8885	1.46810	0.734052	+	0.679093i	$0.237627\pi$
$978$	0	0
$979$	5.30495	0.169547
$980$	0	0
$981$	39.4164	1.25847
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	42.3607	1.34972
$986$	0	0
$987$	23.4164	0.745352
$988$	0	0
$989$	−24.9443	−0.793182
$990$	0	0
$991$	−4.76393	−0.151331	−0.0756656	−	0.997133i	$-0.524108\pi$
$991$	−4.76393	−0.151331	−0.0756656	+	0.997133i	$0.524108\pi$
$992$	0	0
$993$	33.3607	1.05867
$994$	0	0
$995$	−87.7214	−2.78095
$996$	0	0
$997$	4.23607	0.134158	0.0670788	−	0.997748i	$-0.478632\pi$
$997$	4.23607	0.134158	0.0670788	+	0.997748i	$0.478632\pi$
$998$	0	0
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.r.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.r.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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.23607 q^{5} +2.23607 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.23607 q^{5} +2.23607 q^{7} -2.00000 q^{9} +0.763932 q^{11} +1.23607 q^{13} -4.23607 q^{15} -1.00000 q^{17} -5.47214 q^{19} +2.23607 q^{21} +4.76393 q^{23} +12.9443 q^{25} -5.00000 q^{27} +2.70820 q^{29} +9.23607 q^{31} +0.763932 q^{33} -9.47214 q^{35} -8.00000 q^{37} +1.23607 q^{39} +0.527864 q^{41} -5.23607 q^{43} +8.47214 q^{45} +10.4721 q^{47} -2.00000 q^{49} -1.00000 q^{51} +0.708204 q^{53} -3.23607 q^{55} -5.47214 q^{57} -1.00000 q^{59} -11.2361 q^{61} -4.47214 q^{63} -5.23607 q^{65} +7.23607 q^{67} +4.76393 q^{69} -4.00000 q^{71} -3.70820 q^{73} +12.9443 q^{75} +1.70820 q^{77} +8.23607 q^{79} +1.00000 q^{81} +3.23607 q^{83} +4.23607 q^{85} +2.70820 q^{87} +6.94427 q^{89} +2.76393 q^{91} +9.23607 q^{93} +23.1803 q^{95} -2.00000 q^{97} -1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 - 4 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} - 4 q^{9} + 6 q^{11} - 2 q^{13} - 4 q^{15} - 2 q^{17} - 2 q^{19} + 14 q^{23} + 8 q^{25} - 10 q^{27} - 8 q^{29} + 14 q^{31} + 6 q^{33} - 10 q^{35} - 16 q^{37} - 2 q^{39} + 10 q^{41} - 6 q^{43} + 8 q^{45} + 12 q^{47} - 4 q^{49} - 2 q^{51} - 12 q^{53} - 2 q^{55} - 2 q^{57} - 2 q^{59} - 18 q^{61} - 6 q^{65} + 10 q^{67} + 14 q^{69} - 8 q^{71} + 6 q^{73} + 8 q^{75} - 10 q^{77} + 12 q^{79} + 2 q^{81} + 2 q^{83} + 4 q^{85} - 8 q^{87} - 4 q^{89} + 10 q^{91} + 14 q^{93} + 24 q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 - 4 * q^9 + 6 * q^11 - 2 * q^13 - 4 * q^15 - 2 * q^17 - 2 * q^19 + 14 * q^23 + 8 * q^25 - 10 * q^27 - 8 * q^29 + 14 * q^31 + 6 * q^33 - 10 * q^35 - 16 * q^37 - 2 * q^39 + 10 * q^41 - 6 * q^43 + 8 * q^45 + 12 * q^47 - 4 * q^49 - 2 * q^51 - 12 * q^53 - 2 * q^55 - 2 * q^57 - 2 * q^59 - 18 * q^61 - 6 * q^65 + 10 * q^67 + 14 * q^69 - 8 * q^71 + 6 * q^73 + 8 * q^75 - 10 * q^77 + 12 * q^79 + 2 * q^81 + 2 * q^83 + 4 * q^85 - 8 * q^87 - 4 * q^89 + 10 * q^91 + 14 * q^93 + 24 * q^95 - 4 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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