LMFDB - Embedded newform 8001.2.a.z.1.2

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:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.49464 q^{2} +4.22324 q^{4} -0.373762 q^{5} -1.00000 q^{7} -5.54618 q^{8} +O(q^{10})$	\(q-2.49464 q^{2} +4.22324 q^{4} -0.373762 q^{5} -1.00000 q^{7} -5.54618 q^{8} +0.932401 q^{10} +6.40807 q^{11} +2.54237 q^{13} +2.49464 q^{14} +5.38926 q^{16} -4.88897 q^{17} -5.83201 q^{19} -1.57848 q^{20} -15.9858 q^{22} +8.35556 q^{23} -4.86030 q^{25} -6.34231 q^{26} -4.22324 q^{28} +1.63505 q^{29} -6.96821 q^{31} -2.35191 q^{32} +12.1962 q^{34} +0.373762 q^{35} +7.46339 q^{37} +14.5488 q^{38} +2.07295 q^{40} +9.02251 q^{41} -7.64100 q^{43} +27.0628 q^{44} -20.8441 q^{46} -6.14673 q^{47} +1.00000 q^{49} +12.1247 q^{50} +10.7370 q^{52} +2.53581 q^{53} -2.39509 q^{55} +5.54618 q^{56} -4.07885 q^{58} -1.55774 q^{59} -11.0709 q^{61} +17.3832 q^{62} -4.91135 q^{64} -0.950241 q^{65} +11.4944 q^{67} -20.6473 q^{68} -0.932401 q^{70} -7.04746 q^{71} +5.49704 q^{73} -18.6185 q^{74} -24.6300 q^{76} -6.40807 q^{77} -7.21739 q^{79} -2.01430 q^{80} -22.5079 q^{82} -4.98747 q^{83} +1.82731 q^{85} +19.0616 q^{86} -35.5403 q^{88} -6.29176 q^{89} -2.54237 q^{91} +35.2875 q^{92} +15.3339 q^{94} +2.17978 q^{95} -15.3484 q^{97} -2.49464 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * 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 $

$2$	−2.49464	−1.76398	−0.881989	−	0.471270i	$-0.843796\pi$
$2$	−2.49464	−1.76398	−0.881989	+	0.471270i	$0.843796\pi$
$3$	0	0
$3$	0	0
$4$	4.22324	2.11162
$5$	−0.373762	−0.167151	−0.0835757	−	0.996501i	$-0.526634\pi$
$5$	−0.373762	−0.167151	−0.0835757	+	0.996501i	$0.526634\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−5.54618	−1.96087
$9$	0	0
$10$	0.932401	0.294851
$11$	6.40807	1.93210	0.966052	−	0.258346i	$-0.0831775\pi$
$11$	6.40807	1.93210	0.966052	+	0.258346i	$0.0831775\pi$
$12$	0	0
$13$	2.54237	0.705127	0.352564	−	0.935788i	$-0.385310\pi$
$13$	2.54237	0.705127	0.352564	+	0.935788i	$0.385310\pi$
$14$	2.49464	0.666721
$15$	0	0
$16$	5.38926	1.34731
$17$	−4.88897	−1.18575	−0.592875	−	0.805295i	$-0.702007\pi$
$17$	−4.88897	−1.18575	−0.592875	+	0.805295i	$0.702007\pi$
$18$	0	0
$19$	−5.83201	−1.33796	−0.668978	−	0.743283i	$-0.733268\pi$
$19$	−5.83201	−1.33796	−0.668978	+	0.743283i	$0.733268\pi$
$20$	−1.57848	−0.352960
$21$	0	0
$22$	−15.9858	−3.40819
$23$	8.35556	1.74225	0.871127	−	0.491058i	$-0.163390\pi$
$23$	8.35556	1.74225	0.871127	+	0.491058i	$0.163390\pi$
$24$	0	0
$25$	−4.86030	−0.972060
$26$	−6.34231	−1.24383
$27$	0	0
$28$	−4.22324	−0.798117
$29$	1.63505	0.303620	0.151810	−	0.988410i	$-0.451490\pi$
$29$	1.63505	0.303620	0.151810	+	0.988410i	$0.451490\pi$
$30$	0	0
$31$	−6.96821	−1.25153	−0.625764	−	0.780013i	$-0.715213\pi$
$31$	−6.96821	−1.25153	−0.625764	+	0.780013i	$0.715213\pi$
$32$	−2.35191	−0.415763
$33$	0	0
$34$	12.1962	2.09164
$35$	0.373762	0.0631773
$36$	0	0
$37$	7.46339	1.22697	0.613487	−	0.789705i	$-0.289766\pi$
$37$	7.46339	1.22697	0.613487	+	0.789705i	$0.289766\pi$
$38$	14.5488	2.36012
$39$	0	0
$40$	2.07295	0.327762
$41$	9.02251	1.40908	0.704540	−	0.709664i	$-0.251153\pi$
$41$	9.02251	1.40908	0.704540	+	0.709664i	$0.251153\pi$
$42$	0	0
$43$	−7.64100	−1.16524	−0.582621	−	0.812744i	$-0.697973\pi$
$43$	−7.64100	−1.16524	−0.582621	+	0.812744i	$0.697973\pi$
$44$	27.0628	4.07987
$45$	0	0
$46$	−20.8441	−3.07330
$47$	−6.14673	−0.896592	−0.448296	−	0.893885i	$-0.647969\pi$
$47$	−6.14673	−0.896592	−0.448296	+	0.893885i	$0.647969\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	12.1247	1.71469
$51$	0	0
$52$	10.7370	1.48896
$53$	2.53581	0.348321	0.174160	−	0.984717i	$-0.444279\pi$
$53$	2.53581	0.348321	0.174160	+	0.984717i	$0.444279\pi$
$54$	0	0
$55$	−2.39509	−0.322954
$56$	5.54618	0.741140
$57$	0	0
$58$	−4.07885	−0.535580
$59$	−1.55774	−0.202800	−0.101400	−	0.994846i	$-0.532332\pi$
$59$	−1.55774	−0.202800	−0.101400	+	0.994846i	$0.532332\pi$
$60$	0	0
$61$	−11.0709	−1.41749	−0.708745	−	0.705465i	$-0.750738\pi$
$61$	−11.0709	−1.41749	−0.708745	+	0.705465i	$0.750738\pi$
$62$	17.3832	2.20767
$63$	0	0
$64$	−4.91135	−0.613918
$65$	−0.950241	−0.117863
$66$	0	0
$67$	11.4944	1.40427	0.702134	−	0.712045i	$-0.252231\pi$
$67$	11.4944	1.40427	0.702134	+	0.712045i	$0.252231\pi$
$68$	−20.6473	−2.50385
$69$	0	0
$70$	−0.932401	−0.111443
$71$	−7.04746	−0.836380	−0.418190	−	0.908360i	$-0.637335\pi$
$71$	−7.04746	−0.836380	−0.418190	+	0.908360i	$0.637335\pi$
$72$	0	0
$73$	5.49704	0.643380	0.321690	−	0.946845i	$-0.395749\pi$
$73$	5.49704	0.643380	0.321690	+	0.946845i	$0.395749\pi$
$74$	−18.6185	−2.16436
$75$	0	0
$76$	−24.6300	−2.82525
$77$	−6.40807	−0.730267
$78$	0	0
$79$	−7.21739	−0.812020	−0.406010	−	0.913869i	$-0.633080\pi$
$79$	−7.21739	−0.812020	−0.406010	+	0.913869i	$0.633080\pi$
$80$	−2.01430	−0.225205
$81$	0	0
$82$	−22.5079	−2.48559
$83$	−4.98747	−0.547445	−0.273723	−	0.961809i	$-0.588255\pi$
$83$	−4.98747	−0.547445	−0.273723	+	0.961809i	$0.588255\pi$
$84$	0	0
$85$	1.82731	0.198200
$86$	19.0616	2.05546
$87$	0	0
$88$	−35.5403	−3.78861
$89$	−6.29176	−0.666925	−0.333463	−	0.942763i	$-0.608217\pi$
$89$	−6.29176	−0.666925	−0.333463	+	0.942763i	$0.608217\pi$
$90$	0	0
$91$	−2.54237	−0.266513
$92$	35.2875	3.67898
$93$	0	0
$94$	15.3339	1.58157
$95$	2.17978	0.223641
$96$	0	0
$97$	−15.3484	−1.55840	−0.779199	−	0.626776i	$-0.784374\pi$
$97$	−15.3484	−1.55840	−0.779199	+	0.626776i	$0.784374\pi$
$98$	−2.49464	−0.251997
$99$	0	0
$100$	−20.5262	−2.05262
$101$	−14.8823	−1.48085	−0.740424	−	0.672140i	$-0.765375\pi$
$101$	−14.8823	−1.48085	−0.740424	+	0.672140i	$0.765375\pi$
$102$	0	0
$103$	10.5754	1.04203	0.521013	−	0.853548i	$-0.325554\pi$
$103$	10.5754	1.04203	0.521013	+	0.853548i	$0.325554\pi$
$104$	−14.1005	−1.38266
$105$	0	0
$106$	−6.32594	−0.614430
$107$	−14.1331	−1.36630	−0.683150	−	0.730278i	$-0.739391\pi$
$107$	−14.1331	−1.36630	−0.683150	+	0.730278i	$0.739391\pi$
$108$	0	0
$109$	7.01182	0.671610	0.335805	−	0.941932i	$-0.390992\pi$
$109$	7.01182	0.671610	0.335805	+	0.941932i	$0.390992\pi$
$110$	5.97489	0.569684
$111$	0	0
$112$	−5.38926	−0.509237
$113$	−14.2300	−1.33865	−0.669325	−	0.742970i	$-0.733417\pi$
$113$	−14.2300	−1.33865	−0.669325	+	0.742970i	$0.733417\pi$
$114$	0	0
$115$	−3.12299	−0.291220
$116$	6.90519	0.641130
$117$	0	0
$118$	3.88600	0.357735
$119$	4.88897	0.448171
$120$	0	0
$121$	30.0633	2.73303
$122$	27.6181	2.50042
$123$	0	0
$124$	−29.4284	−2.64275
$125$	3.68540	0.329632
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	16.9559	1.49870
$129$	0	0
$130$	2.37051	0.207908
$131$	19.2456	1.68150	0.840750	−	0.541424i	$-0.182115\pi$
$131$	19.2456	1.68150	0.840750	+	0.541424i	$0.182115\pi$
$132$	0	0
$133$	5.83201	0.505700
$134$	−28.6745	−2.47710
$135$	0	0
$136$	27.1151	2.32510
$137$	0.417270	0.0356498	0.0178249	−	0.999841i	$-0.494326\pi$
$137$	0.417270	0.0356498	0.0178249	+	0.999841i	$0.494326\pi$
$138$	0	0
$139$	−7.80248	−0.661798	−0.330899	−	0.943666i	$-0.607352\pi$
$139$	−7.80248	−0.661798	−0.330899	+	0.943666i	$0.607352\pi$
$140$	1.57848	0.133406
$141$	0	0
$142$	17.5809	1.47536
$143$	16.2917	1.36238
$144$	0	0
$145$	−0.611117	−0.0507505
$146$	−13.7132	−1.13491
$147$	0	0
$148$	31.5197	2.59090
$149$	−0.799730	−0.0655165	−0.0327582	−	0.999463i	$-0.510429\pi$
$149$	−0.799730	−0.0655165	−0.0327582	+	0.999463i	$0.510429\pi$
$150$	0	0
$151$	−17.4297	−1.41841	−0.709204	−	0.705003i	$-0.750946\pi$
$151$	−17.4297	−1.41841	−0.709204	+	0.705003i	$0.750946\pi$
$152$	32.3454	2.62356
$153$	0	0
$154$	15.9858	1.28817
$155$	2.60445	0.209194
$156$	0	0
$157$	−22.3819	−1.78627	−0.893133	−	0.449792i	$-0.851498\pi$
$157$	−22.3819	−1.78627	−0.893133	+	0.449792i	$0.851498\pi$
$158$	18.0048	1.43238
$159$	0	0
$160$	0.879054	0.0694953
$161$	−8.35556	−0.658510
$162$	0	0
$163$	17.1243	1.34128	0.670638	−	0.741785i	$-0.266021\pi$
$163$	17.1243	1.34128	0.670638	+	0.741785i	$0.266021\pi$
$164$	38.1042	2.97544
$165$	0	0
$166$	12.4419	0.965682
$167$	−8.64194	−0.668734	−0.334367	−	0.942443i	$-0.608522\pi$
$167$	−8.64194	−0.668734	−0.334367	+	0.942443i	$0.608522\pi$
$168$	0	0
$169$	−6.53634	−0.502796
$170$	−4.55848	−0.349620
$171$	0	0
$172$	−32.2697	−2.46055
$173$	25.8065	1.96203	0.981016	−	0.193925i	$-0.0621218\pi$
$173$	25.8065	1.96203	0.981016	+	0.193925i	$0.0621218\pi$
$174$	0	0
$175$	4.86030	0.367404
$176$	34.5347	2.60315
$177$	0	0
$178$	15.6957	1.17644
$179$	−14.0778	−1.05222	−0.526111	−	0.850416i	$-0.676351\pi$
$179$	−14.0778	−1.05222	−0.526111	+	0.850416i	$0.676351\pi$
$180$	0	0
$181$	−16.7679	−1.24635	−0.623174	−	0.782083i	$-0.714157\pi$
$181$	−16.7679	−1.24635	−0.623174	+	0.782083i	$0.714157\pi$
$182$	6.34231	0.470123
$183$	0	0
$184$	−46.3414	−3.41634
$185$	−2.78953	−0.205090
$186$	0	0
$187$	−31.3289	−2.29099
$188$	−25.9591	−1.89326
$189$	0	0
$190$	−5.43778	−0.394498
$191$	22.6982	1.64238	0.821191	−	0.570653i	$-0.193310\pi$
$191$	22.6982	1.64238	0.821191	+	0.570653i	$0.193310\pi$
$192$	0	0
$193$	1.65402	0.119059	0.0595296	−	0.998227i	$-0.481040\pi$
$193$	1.65402	0.119059	0.0595296	+	0.998227i	$0.481040\pi$
$194$	38.2889	2.74898
$195$	0	0
$196$	4.22324	0.301660
$197$	12.5827	0.896481	0.448241	−	0.893913i	$-0.352051\pi$
$197$	12.5827	0.896481	0.448241	+	0.893913i	$0.352051\pi$
$198$	0	0
$199$	10.6610	0.755737	0.377869	−	0.925859i	$-0.376657\pi$
$199$	10.6610	0.755737	0.377869	+	0.925859i	$0.376657\pi$
$200$	26.9561	1.90609
$201$	0	0
$202$	37.1261	2.61218
$203$	−1.63505	−0.114758
$204$	0	0
$205$	−3.37227	−0.235529
$206$	−26.3819	−1.83811
$207$	0	0
$208$	13.7015	0.950028
$209$	−37.3719	−2.58507
$210$	0	0
$211$	−8.06656	−0.555325	−0.277662	−	0.960679i	$-0.589560\pi$
$211$	−8.06656	−0.555325	−0.277662	+	0.960679i	$0.589560\pi$
$212$	10.7093	0.735520
$213$	0	0
$214$	35.2571	2.41012
$215$	2.85591	0.194772
$216$	0	0
$217$	6.96821	0.473033
$218$	−17.4920	−1.18471
$219$	0	0
$220$	−10.1150	−0.681955
$221$	−12.4296	−0.836104
$222$	0	0
$223$	−3.52281	−0.235905	−0.117952	−	0.993019i	$-0.537633\pi$
$223$	−3.52281	−0.235905	−0.117952	+	0.993019i	$0.537633\pi$
$224$	2.35191	0.157144
$225$	0	0
$226$	35.4989	2.36135
$227$	−11.6455	−0.772938	−0.386469	−	0.922302i	$-0.626305\pi$
$227$	−11.6455	−0.772938	−0.386469	+	0.922302i	$0.626305\pi$
$228$	0	0
$229$	21.3408	1.41024	0.705120	−	0.709088i	$-0.250893\pi$
$229$	21.3408	1.41024	0.705120	+	0.709088i	$0.250893\pi$
$230$	7.79073	0.513706
$231$	0	0
$232$	−9.06826	−0.595360
$233$	−3.63820	−0.238346	−0.119173	−	0.992873i	$-0.538024\pi$
$233$	−3.63820	−0.238346	−0.119173	+	0.992873i	$0.538024\pi$
$234$	0	0
$235$	2.29741	0.149867
$236$	−6.57870	−0.428237
$237$	0	0
$238$	−12.1962	−0.790564
$239$	−3.73257	−0.241440	−0.120720	−	0.992687i	$-0.538520\pi$
$239$	−3.73257	−0.241440	−0.120720	+	0.992687i	$0.538520\pi$
$240$	0	0
$241$	5.73060	0.369141	0.184570	−	0.982819i	$-0.440911\pi$
$241$	5.73060	0.369141	0.184570	+	0.982819i	$0.440911\pi$
$242$	−74.9972	−4.82100
$243$	0	0
$244$	−46.7552	−2.99320
$245$	−0.373762	−0.0238788
$246$	0	0
$247$	−14.8271	−0.943429
$248$	38.6470	2.45408
$249$	0	0
$250$	−9.19376	−0.581464
$251$	−11.4934	−0.725458	−0.362729	−	0.931895i	$-0.618155\pi$
$251$	−11.4934	−0.725458	−0.362729	+	0.931895i	$0.618155\pi$
$252$	0	0
$253$	53.5430	3.36622
$254$	2.49464	0.156528
$255$	0	0
$256$	−32.4761	−2.02976
$257$	24.9218	1.55458	0.777290	−	0.629143i	$-0.216594\pi$
$257$	24.9218	1.55458	0.777290	+	0.629143i	$0.216594\pi$
$258$	0	0
$259$	−7.46339	−0.463753
$260$	−4.01309	−0.248882
$261$	0	0
$262$	−48.0110	−2.96613
$263$	22.7571	1.40326	0.701632	−	0.712539i	$-0.252455\pi$
$263$	22.7571	1.40326	0.701632	+	0.712539i	$0.252455\pi$
$264$	0	0
$265$	−0.947789	−0.0582222
$266$	−14.5488	−0.892043
$267$	0	0
$268$	48.5437	2.96528
$269$	−6.75574	−0.411905	−0.205952	−	0.978562i	$-0.566029\pi$
$269$	−6.75574	−0.411905	−0.205952	+	0.978562i	$0.566029\pi$
$270$	0	0
$271$	−20.8092	−1.26407	−0.632036	−	0.774939i	$-0.717780\pi$
$271$	−20.8092	−1.26407	−0.632036	+	0.774939i	$0.717780\pi$
$272$	−26.3479	−1.59758
$273$	0	0
$274$	−1.04094	−0.0628854
$275$	−31.1451	−1.87812
$276$	0	0
$277$	16.4321	0.987311	0.493655	−	0.869658i	$-0.335660\pi$
$277$	16.4321	0.987311	0.493655	+	0.869658i	$0.335660\pi$
$278$	19.4644	1.16740
$279$	0	0
$280$	−2.07295	−0.123882
$281$	−17.3979	−1.03787	−0.518935	−	0.854814i	$-0.673671\pi$
$281$	−17.3979	−1.03787	−0.518935	+	0.854814i	$0.673671\pi$
$282$	0	0
$283$	−20.8735	−1.24080	−0.620400	−	0.784285i	$-0.713030\pi$
$283$	−20.8735	−1.24080	−0.620400	+	0.784285i	$0.713030\pi$
$284$	−29.7631	−1.76612
$285$	0	0
$286$	−40.6419	−2.40321
$287$	−9.02251	−0.532582
$288$	0	0
$289$	6.90205	0.406003
$290$	1.52452	0.0895228
$291$	0	0
$292$	23.2153	1.35857
$293$	−2.66238	−0.155538	−0.0777690	−	0.996971i	$-0.524780\pi$
$293$	−2.66238	−0.155538	−0.0777690	+	0.996971i	$0.524780\pi$
$294$	0	0
$295$	0.582223	0.0338984
$296$	−41.3933	−2.40594
$297$	0	0
$298$	1.99504	0.115570
$299$	21.2429	1.22851
$300$	0	0
$301$	7.64100	0.440420
$302$	43.4808	2.50204
$303$	0	0
$304$	−31.4302	−1.80265
$305$	4.13790	0.236935
$306$	0	0
$307$	−19.1541	−1.09318	−0.546592	−	0.837399i	$-0.684075\pi$
$307$	−19.1541	−1.09318	−0.546592	+	0.837399i	$0.684075\pi$
$308$	−27.0628	−1.54205
$309$	0	0
$310$	−6.49717	−0.369014
$311$	−17.6665	−1.00177	−0.500887	−	0.865513i	$-0.666993\pi$
$311$	−17.6665	−1.00177	−0.500887	+	0.865513i	$0.666993\pi$
$312$	0	0
$313$	−21.4016	−1.20969	−0.604846	−	0.796343i	$-0.706765\pi$
$313$	−21.4016	−1.20969	−0.604846	+	0.796343i	$0.706765\pi$
$314$	55.8347	3.15094
$315$	0	0
$316$	−30.4807	−1.71468
$317$	11.4317	0.642070	0.321035	−	0.947067i	$-0.395969\pi$
$317$	11.4317	0.642070	0.321035	+	0.947067i	$0.395969\pi$
$318$	0	0
$319$	10.4775	0.586626
$320$	1.83567	0.102617
$321$	0	0
$322$	20.8441	1.16160
$323$	28.5125	1.58648
$324$	0	0
$325$	−12.3567	−0.685426
$326$	−42.7189	−2.36598
$327$	0	0
$328$	−50.0405	−2.76302
$329$	6.14673	0.338880
$330$	0	0
$331$	25.5607	1.40494	0.702472	−	0.711711i	$-0.252079\pi$
$331$	25.5607	1.40494	0.702472	+	0.711711i	$0.252079\pi$
$332$	−21.0633	−1.15600
$333$	0	0
$334$	21.5586	1.17963
$335$	−4.29618	−0.234725
$336$	0	0
$337$	−0.321760	−0.0175274	−0.00876370	−	0.999962i	$-0.502790\pi$
$337$	−0.321760	−0.0175274	−0.00876370	+	0.999962i	$0.502790\pi$
$338$	16.3058	0.886921
$339$	0	0
$340$	7.71717	0.418522
$341$	−44.6528	−2.41808
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	42.3784	2.28489
$345$	0	0
$346$	−64.3780	−3.46098
$347$	11.5029	0.617507	0.308753	−	0.951142i	$-0.400088\pi$
$347$	11.5029	0.617507	0.308753	+	0.951142i	$0.400088\pi$
$348$	0	0
$349$	15.3049	0.819252	0.409626	−	0.912254i	$-0.365659\pi$
$349$	15.3049	0.819252	0.409626	+	0.912254i	$0.365659\pi$
$350$	−12.1247	−0.648093
$351$	0	0
$352$	−15.0712	−0.803298
$353$	7.04701	0.375075	0.187537	−	0.982257i	$-0.439949\pi$
$353$	7.04701	0.375075	0.187537	+	0.982257i	$0.439949\pi$
$354$	0	0
$355$	2.63407	0.139802
$356$	−26.5716	−1.40829
$357$	0	0
$358$	35.1190	1.85610
$359$	−17.9118	−0.945350	−0.472675	−	0.881237i	$-0.656711\pi$
$359$	−17.9118	−0.945350	−0.472675	+	0.881237i	$0.656711\pi$
$360$	0	0
$361$	15.0124	0.790124
$362$	41.8299	2.19853
$363$	0	0
$364$	−10.7370	−0.562774
$365$	−2.05458	−0.107542
$366$	0	0
$367$	2.13230	0.111305	0.0556527	−	0.998450i	$-0.482276\pi$
$367$	2.13230	0.111305	0.0556527	+	0.998450i	$0.482276\pi$
$368$	45.0303	2.34736
$369$	0	0
$370$	6.95888	0.361775
$371$	−2.53581	−0.131653
$372$	0	0
$373$	−29.0911	−1.50628	−0.753140	−	0.657861i	$-0.771462\pi$
$373$	−29.0911	−1.50628	−0.753140	+	0.657861i	$0.771462\pi$
$374$	78.1543	4.04126
$375$	0	0
$376$	34.0909	1.75810
$377$	4.15689	0.214091
$378$	0	0
$379$	−19.7521	−1.01460	−0.507298	−	0.861771i	$-0.669356\pi$
$379$	−19.7521	−1.01460	−0.507298	+	0.861771i	$0.669356\pi$
$380$	9.20574	0.472244
$381$	0	0
$382$	−56.6238	−2.89713
$383$	13.0295	0.665776	0.332888	−	0.942966i	$-0.391977\pi$
$383$	13.0295	0.665776	0.332888	+	0.942966i	$0.391977\pi$
$384$	0	0
$385$	2.39509	0.122065
$386$	−4.12619	−0.210018
$387$	0	0
$388$	−64.8201	−3.29074
$389$	−27.0519	−1.37159	−0.685794	−	0.727796i	$-0.740545\pi$
$389$	−27.0519	−1.37159	−0.685794	+	0.727796i	$0.740545\pi$
$390$	0	0
$391$	−40.8501	−2.06588
$392$	−5.54618	−0.280124
$393$	0	0
$394$	−31.3894	−1.58137
$395$	2.69758	0.135730
$396$	0	0
$397$	−7.86212	−0.394589	−0.197294	−	0.980344i	$-0.563215\pi$
$397$	−7.86212	−0.394589	−0.197294	+	0.980344i	$0.563215\pi$
$398$	−26.5953	−1.33310
$399$	0	0
$400$	−26.1934	−1.30967
$401$	−7.53993	−0.376526	−0.188263	−	0.982119i	$-0.560286\pi$
$401$	−7.53993	−0.376526	−0.188263	+	0.982119i	$0.560286\pi$
$402$	0	0
$403$	−17.7158	−0.882486
$404$	−62.8517	−3.12699
$405$	0	0
$406$	4.07885	0.202430
$407$	47.8259	2.37064
$408$	0	0
$409$	−5.18427	−0.256346	−0.128173	−	0.991752i	$-0.540911\pi$
$409$	−5.18427	−0.256346	−0.128173	+	0.991752i	$0.540911\pi$
$410$	8.41260	0.415469
$411$	0	0
$412$	44.6625	2.20036
$413$	1.55774	0.0766514
$414$	0	0
$415$	1.86412	0.0915062
$416$	−5.97943	−0.293166
$417$	0	0
$418$	93.2296	4.56001
$419$	23.7502	1.16028	0.580138	−	0.814518i	$-0.302999\pi$
$419$	23.7502	1.16028	0.580138	+	0.814518i	$0.302999\pi$
$420$	0	0
$421$	17.3466	0.845419	0.422710	−	0.906265i	$-0.361079\pi$
$421$	17.3466	0.845419	0.422710	+	0.906265i	$0.361079\pi$
$422$	20.1232	0.979581
$423$	0	0
$424$	−14.0641	−0.683012
$425$	23.7619	1.15262
$426$	0	0
$427$	11.0709	0.535761
$428$	−59.6875	−2.88511
$429$	0	0
$430$	−7.12448	−0.343573
$431$	18.6704	0.899324	0.449662	−	0.893199i	$-0.351544\pi$
$431$	18.6704	0.899324	0.449662	+	0.893199i	$0.351544\pi$
$432$	0	0
$433$	13.6587	0.656398	0.328199	−	0.944609i	$-0.393558\pi$
$433$	13.6587	0.656398	0.328199	+	0.944609i	$0.393558\pi$
$434$	−17.3832	−0.834420
$435$	0	0
$436$	29.6126	1.41818
$437$	−48.7297	−2.33106
$438$	0	0
$439$	25.1317	1.19947	0.599736	−	0.800198i	$-0.295272\pi$
$439$	25.1317	1.19947	0.599736	+	0.800198i	$0.295272\pi$
$440$	13.2836	0.633271
$441$	0	0
$442$	31.0074	1.47487
$443$	26.5215	1.26008	0.630038	−	0.776565i	$-0.283039\pi$
$443$	26.5215	1.26008	0.630038	+	0.776565i	$0.283039\pi$
$444$	0	0
$445$	2.35162	0.111477
$446$	8.78815	0.416131
$447$	0	0
$448$	4.91135	0.232039
$449$	−16.1644	−0.762843	−0.381422	−	0.924401i	$-0.624565\pi$
$449$	−16.1644	−0.762843	−0.381422	+	0.924401i	$0.624565\pi$
$450$	0	0
$451$	57.8169	2.72249
$452$	−60.0968	−2.82672
$453$	0	0
$454$	29.0513	1.36345
$455$	0.950241	0.0445480
$456$	0	0
$457$	−3.04113	−0.142258	−0.0711290	−	0.997467i	$-0.522660\pi$
$457$	−3.04113	−0.142258	−0.0711290	+	0.997467i	$0.522660\pi$
$458$	−53.2377	−2.48763
$459$	0	0
$460$	−13.1891	−0.614946
$461$	−21.0282	−0.979379	−0.489689	−	0.871897i	$-0.662890\pi$
$461$	−21.0282	−0.979379	−0.489689	+	0.871897i	$0.662890\pi$
$462$	0	0
$463$	13.6609	0.634877	0.317439	−	0.948279i	$-0.397177\pi$
$463$	13.6609	0.634877	0.317439	+	0.948279i	$0.397177\pi$
$464$	8.81169	0.409072
$465$	0	0
$466$	9.07601	0.420438
$467$	11.9422	0.552619	0.276310	−	0.961069i	$-0.410888\pi$
$467$	11.9422	0.552619	0.276310	+	0.961069i	$0.410888\pi$
$468$	0	0
$469$	−11.4944	−0.530763
$470$	−5.73122	−0.264361
$471$	0	0
$472$	8.63951	0.397665
$473$	−48.9640	−2.25137
$474$	0	0
$475$	28.3453	1.30057
$476$	20.6473	0.946367
$477$	0	0
$478$	9.31144	0.425895
$479$	−2.43999	−0.111486	−0.0557430	−	0.998445i	$-0.517753\pi$
$479$	−2.43999	−0.111486	−0.0557430	+	0.998445i	$0.517753\pi$
$480$	0	0
$481$	18.9747	0.865173
$482$	−14.2958	−0.651156
$483$	0	0
$484$	126.965	5.77112
$485$	5.73666	0.260488
$486$	0	0
$487$	33.2175	1.50523	0.752613	−	0.658463i	$-0.228793\pi$
$487$	33.2175	1.50523	0.752613	+	0.658463i	$0.228793\pi$
$488$	61.4015	2.77951
$489$	0	0
$490$	0.932401	0.0421216
$491$	31.9094	1.44005	0.720025	−	0.693948i	$-0.244130\pi$
$491$	31.9094	1.44005	0.720025	+	0.693948i	$0.244130\pi$
$492$	0	0
$493$	−7.99369	−0.360018
$494$	36.9884	1.66419
$495$	0	0
$496$	−37.5535	−1.68620
$497$	7.04746	0.316122
$498$	0	0
$499$	39.0736	1.74917	0.874586	−	0.484870i	$-0.161133\pi$
$499$	39.0736	1.74917	0.874586	+	0.484870i	$0.161133\pi$
$500$	15.5643	0.696058
$501$	0	0
$502$	28.6719	1.27969
$503$	−1.41048	−0.0628903	−0.0314451	−	0.999505i	$-0.510011\pi$
$503$	−1.41048	−0.0628903	−0.0314451	+	0.999505i	$0.510011\pi$
$504$	0	0
$505$	5.56245	0.247526
$506$	−133.571	−5.93793
$507$	0	0
$508$	−4.22324	−0.187376
$509$	−21.3887	−0.948036	−0.474018	−	0.880515i	$-0.657197\pi$
$509$	−21.3887	−0.948036	−0.474018	+	0.880515i	$0.657197\pi$
$510$	0	0
$511$	−5.49704	−0.243175
$512$	47.1046	2.08175
$513$	0	0
$514$	−62.1710	−2.74224
$515$	−3.95269	−0.174176
$516$	0	0
$517$	−39.3886	−1.73231
$518$	18.6185	0.818049
$519$	0	0
$520$	5.27021	0.231114
$521$	−8.11999	−0.355743	−0.177872	−	0.984054i	$-0.556921\pi$
$521$	−8.11999	−0.355743	−0.177872	+	0.984054i	$0.556921\pi$
$522$	0	0
$523$	−19.7403	−0.863184	−0.431592	−	0.902069i	$-0.642048\pi$
$523$	−19.7403	−0.863184	−0.431592	+	0.902069i	$0.642048\pi$
$524$	81.2789	3.55069
$525$	0	0
$526$	−56.7709	−2.47533
$527$	34.0674	1.48400
$528$	0	0
$529$	46.8153	2.03545
$530$	2.36440	0.102703
$531$	0	0
$532$	24.6300	1.06784
$533$	22.9386	0.993580
$534$	0	0
$535$	5.28242	0.228379
$536$	−63.7502	−2.75359
$537$	0	0
$538$	16.8532	0.726591
$539$	6.40807	0.276015
$540$	0	0
$541$	−7.35211	−0.316092	−0.158046	−	0.987432i	$-0.550519\pi$
$541$	−7.35211	−0.316092	−0.158046	+	0.987432i	$0.550519\pi$
$542$	51.9116	2.22979
$543$	0	0
$544$	11.4984	0.492991
$545$	−2.62075	−0.112261
$546$	0	0
$547$	−17.8879	−0.764831	−0.382416	−	0.923990i	$-0.624908\pi$
$547$	−17.8879	−0.764831	−0.382416	+	0.923990i	$0.624908\pi$
$548$	1.76223	0.0752787
$549$	0	0
$550$	77.6960	3.31297
$551$	−9.53561	−0.406230
$552$	0	0
$553$	7.21739	0.306915
$554$	−40.9923	−1.74159
$555$	0	0
$556$	−32.9517	−1.39746
$557$	−11.6752	−0.494693	−0.247346	−	0.968927i	$-0.579559\pi$
$557$	−11.6752	−0.494693	−0.247346	+	0.968927i	$0.579559\pi$
$558$	0	0
$559$	−19.4263	−0.821643
$560$	2.01430	0.0851197
$561$	0	0
$562$	43.4014	1.83078
$563$	23.7886	1.00257	0.501285	−	0.865282i	$-0.332861\pi$
$563$	23.7886	1.00257	0.501285	+	0.865282i	$0.332861\pi$
$564$	0	0
$565$	5.31864	0.223757
$566$	52.0719	2.18874
$567$	0	0
$568$	39.0865	1.64003
$569$	3.67726	0.154159	0.0770794	−	0.997025i	$-0.475440\pi$
$569$	3.67726	0.154159	0.0770794	+	0.997025i	$0.475440\pi$
$570$	0	0
$571$	−41.7874	−1.74875	−0.874375	−	0.485251i	$-0.838728\pi$
$571$	−41.7874	−1.74875	−0.874375	+	0.485251i	$0.838728\pi$
$572$	68.8037	2.87683
$573$	0	0
$574$	22.5079	0.939463
$575$	−40.6105	−1.69358
$576$	0	0
$577$	35.4826	1.47716	0.738579	−	0.674167i	$-0.235497\pi$
$577$	35.4826	1.47716	0.738579	+	0.674167i	$0.235497\pi$
$578$	−17.2181	−0.716180
$579$	0	0
$580$	−2.58089	−0.107166
$581$	4.98747	0.206915
$582$	0	0
$583$	16.2497	0.672992
$584$	−30.4876	−1.26159
$585$	0	0
$586$	6.64169	0.274366
$587$	−18.7445	−0.773669	−0.386834	−	0.922149i	$-0.626431\pi$
$587$	−18.7445	−0.773669	−0.386834	+	0.922149i	$0.626431\pi$
$588$	0	0
$589$	40.6387	1.67449
$590$	−1.45244	−0.0597960
$591$	0	0
$592$	40.2222	1.65312
$593$	15.6452	0.642470	0.321235	−	0.947000i	$-0.395902\pi$
$593$	15.6452	0.642470	0.321235	+	0.947000i	$0.395902\pi$
$594$	0	0
$595$	−1.82731	−0.0749124
$596$	−3.37745	−0.138346
$597$	0	0
$598$	−52.9935	−2.16707
$599$	−21.2278	−0.867343	−0.433672	−	0.901071i	$-0.642782\pi$
$599$	−21.2278	−0.867343	−0.433672	+	0.901071i	$0.642782\pi$
$600$	0	0
$601$	−11.5015	−0.469154	−0.234577	−	0.972098i	$-0.575371\pi$
$601$	−11.5015	−0.469154	−0.234577	+	0.972098i	$0.575371\pi$
$602$	−19.0616	−0.776891
$603$	0	0
$604$	−73.6097	−2.99514
$605$	−11.2365	−0.456829
$606$	0	0
$607$	−29.5860	−1.20086	−0.600430	−	0.799677i	$-0.705004\pi$
$607$	−29.5860	−1.20086	−0.600430	+	0.799677i	$0.705004\pi$
$608$	13.7164	0.556272
$609$	0	0
$610$	−10.3226	−0.417949
$611$	−15.6273	−0.632212
$612$	0	0
$613$	16.0162	0.646888	0.323444	−	0.946247i	$-0.395159\pi$
$613$	16.0162	0.646888	0.323444	+	0.946247i	$0.395159\pi$
$614$	47.7827	1.92835
$615$	0	0
$616$	35.5403	1.43196
$617$	−32.3833	−1.30370	−0.651851	−	0.758347i	$-0.726007\pi$
$617$	−32.3833	−1.30370	−0.651851	+	0.758347i	$0.726007\pi$
$618$	0	0
$619$	−6.74750	−0.271205	−0.135602	−	0.990763i	$-0.543297\pi$
$619$	−6.74750	−0.271205	−0.135602	+	0.990763i	$0.543297\pi$
$620$	10.9992	0.441739
$621$	0	0
$622$	44.0715	1.76711
$623$	6.29176	0.252074
$624$	0	0
$625$	22.9240	0.916962
$626$	53.3894	2.13387
$627$	0	0
$628$	−94.5239	−3.77191
$629$	−36.4883	−1.45488
$630$	0	0
$631$	−27.4499	−1.09276	−0.546381	−	0.837537i	$-0.683995\pi$
$631$	−27.4499	−1.09276	−0.546381	+	0.837537i	$0.683995\pi$
$632$	40.0289	1.59227
$633$	0	0
$634$	−28.5181	−1.13260
$635$	0.373762	0.0148323
$636$	0	0
$637$	2.54237	0.100732
$638$	−26.1376	−1.03480
$639$	0	0
$640$	−6.33745	−0.250510
$641$	−25.6205	−1.01195	−0.505975	−	0.862548i	$-0.668867\pi$
$641$	−25.6205	−1.01195	−0.505975	+	0.862548i	$0.668867\pi$
$642$	0	0
$643$	−25.0718	−0.988735	−0.494368	−	0.869253i	$-0.664600\pi$
$643$	−25.0718	−0.988735	−0.494368	+	0.869253i	$0.664600\pi$
$644$	−35.2875	−1.39052
$645$	0	0
$646$	−71.1286	−2.79852
$647$	26.5116	1.04228	0.521139	−	0.853472i	$-0.325507\pi$
$647$	26.5116	1.04228	0.521139	+	0.853472i	$0.325507\pi$
$648$	0	0
$649$	−9.98210	−0.391832
$650$	30.8255	1.20908
$651$	0	0
$652$	72.3198	2.83226
$653$	−0.745988	−0.0291928	−0.0145964	−	0.999893i	$-0.504646\pi$
$653$	−0.745988	−0.0291928	−0.0145964	+	0.999893i	$0.504646\pi$
$654$	0	0
$655$	−7.19328	−0.281065
$656$	48.6247	1.89847
$657$	0	0
$658$	−15.3339	−0.597777
$659$	−50.1415	−1.95324	−0.976619	−	0.214979i	$-0.931032\pi$
$659$	−50.1415	−1.95324	−0.976619	+	0.214979i	$0.931032\pi$
$660$	0	0
$661$	−11.0074	−0.428138	−0.214069	−	0.976819i	$-0.568672\pi$
$661$	−11.0074	−0.428138	−0.214069	+	0.976819i	$0.568672\pi$
$662$	−63.7649	−2.47829
$663$	0	0
$664$	27.6614	1.07347
$665$	−2.17978	−0.0845283
$666$	0	0
$667$	13.6617	0.528984
$668$	−36.4970	−1.41211
$669$	0	0
$670$	10.7174	0.414050
$671$	−70.9434	−2.73874
$672$	0	0
$673$	−30.4045	−1.17201	−0.586003	−	0.810309i	$-0.699299\pi$
$673$	−30.4045	−1.17201	−0.586003	+	0.810309i	$0.699299\pi$
$674$	0.802676	0.0309179
$675$	0	0
$676$	−27.6045	−1.06171
$677$	−27.3555	−1.05136	−0.525679	−	0.850683i	$-0.676189\pi$
$677$	−27.3555	−1.05136	−0.525679	+	0.850683i	$0.676189\pi$
$678$	0	0
$679$	15.3484	0.589019
$680$	−10.1346	−0.388644
$681$	0	0
$682$	111.393	4.26544
$683$	17.0233	0.651379	0.325689	−	0.945477i	$-0.394404\pi$
$683$	17.0233	0.651379	0.325689	+	0.945477i	$0.394404\pi$
$684$	0	0
$685$	−0.155960	−0.00595891
$686$	2.49464	0.0952459
$687$	0	0
$688$	−41.1793	−1.56995
$689$	6.44698	0.245610
$690$	0	0
$691$	−48.1416	−1.83139	−0.915696	−	0.401871i	$-0.868360\pi$
$691$	−48.1416	−1.83139	−0.915696	+	0.401871i	$0.868360\pi$
$692$	108.987	4.14307
$693$	0	0
$694$	−28.6956	−1.08927
$695$	2.91627	0.110620
$696$	0	0
$697$	−44.1108	−1.67082
$698$	−38.1802	−1.44514
$699$	0	0
$700$	20.5262	0.775818
$701$	−36.9208	−1.39448	−0.697240	−	0.716837i	$-0.745589\pi$
$701$	−36.9208	−1.39448	−0.697240	+	0.716837i	$0.745589\pi$
$702$	0	0
$703$	−43.5266	−1.64164
$704$	−31.4722	−1.18615
$705$	0	0
$706$	−17.5798	−0.661623
$707$	14.8823	0.559708
$708$	0	0
$709$	−10.1340	−0.380592	−0.190296	−	0.981727i	$-0.560945\pi$
$709$	−10.1340	−0.380592	−0.190296	+	0.981727i	$0.560945\pi$
$710$	−6.57107	−0.246608
$711$	0	0
$712$	34.8952	1.30775
$713$	−58.2233	−2.18048
$714$	0	0
$715$	−6.08921	−0.227724
$716$	−59.4538	−2.22189
$717$	0	0
$718$	44.6836	1.66758
$719$	−6.11067	−0.227890	−0.113945	−	0.993487i	$-0.536349\pi$
$719$	−6.11067	−0.227890	−0.113945	+	0.993487i	$0.536349\pi$
$720$	0	0
$721$	−10.5754	−0.393849
$722$	−37.4505	−1.39376
$723$	0	0
$724$	−70.8148	−2.63181
$725$	−7.94682	−0.295137
$726$	0	0
$727$	32.9229	1.22104	0.610521	−	0.792000i	$-0.290960\pi$
$727$	32.9229	1.22104	0.610521	+	0.792000i	$0.290960\pi$
$728$	14.1005	0.522598
$729$	0	0
$730$	5.12545	0.189701
$731$	37.3566	1.38168
$732$	0	0
$733$	11.5829	0.427823	0.213912	−	0.976853i	$-0.431380\pi$
$733$	11.5829	0.427823	0.213912	+	0.976853i	$0.431380\pi$
$734$	−5.31933	−0.196340
$735$	0	0
$736$	−19.6515	−0.724365
$737$	73.6571	2.71319
$738$	0	0
$739$	−32.8534	−1.20853	−0.604267	−	0.796782i	$-0.706534\pi$
$739$	−32.8534	−1.20853	−0.604267	+	0.796782i	$0.706534\pi$
$740$	−11.7808	−0.433073
$741$	0	0
$742$	6.32594	0.232233
$743$	−16.3217	−0.598783	−0.299392	−	0.954130i	$-0.596784\pi$
$743$	−16.3217	−0.598783	−0.299392	+	0.954130i	$0.596784\pi$
$744$	0	0
$745$	0.298909	0.0109512
$746$	72.5718	2.65704
$747$	0	0
$748$	−132.309	−4.83770
$749$	14.1331	0.516413
$750$	0	0
$751$	−6.51082	−0.237583	−0.118792	−	0.992919i	$-0.537902\pi$
$751$	−6.51082	−0.237583	−0.118792	+	0.992919i	$0.537902\pi$
$752$	−33.1263	−1.20799
$753$	0	0
$754$	−10.3700	−0.377652
$755$	6.51455	0.237089
$756$	0	0
$757$	28.1014	1.02136	0.510682	−	0.859770i	$-0.329393\pi$
$757$	28.1014	1.02136	0.510682	+	0.859770i	$0.329393\pi$
$758$	49.2744	1.78973
$759$	0	0
$760$	−12.0895	−0.438531
$761$	17.9070	0.649130	0.324565	−	0.945863i	$-0.394782\pi$
$761$	17.9070	0.649130	0.324565	+	0.945863i	$0.394782\pi$
$762$	0	0
$763$	−7.01182	−0.253845
$764$	95.8598	3.46809
$765$	0	0
$766$	−32.5039	−1.17441
$767$	−3.96035	−0.143000
$768$	0	0
$769$	−31.1782	−1.12432	−0.562158	−	0.827030i	$-0.690029\pi$
$769$	−31.1782	−1.12432	−0.562158	+	0.827030i	$0.690029\pi$
$770$	−5.97489	−0.215320
$771$	0	0
$772$	6.98533	0.251408
$773$	−28.7394	−1.03368	−0.516842	−	0.856081i	$-0.672892\pi$
$773$	−28.7394	−1.03368	−0.516842	+	0.856081i	$0.672892\pi$
$774$	0	0
$775$	33.8676	1.21656
$776$	85.1252	3.05582
$777$	0	0
$778$	67.4849	2.41945
$779$	−52.6194	−1.88529
$780$	0	0
$781$	−45.1606	−1.61597
$782$	101.906	3.64416
$783$	0	0
$784$	5.38926	0.192474
$785$	8.36548	0.298577
$786$	0	0
$787$	31.8508	1.13536	0.567680	−	0.823249i	$-0.307841\pi$
$787$	31.8508	1.13536	0.567680	+	0.823249i	$0.307841\pi$
$788$	53.1398	1.89303
$789$	0	0
$790$	−6.72950	−0.239425
$791$	14.2300	0.505962
$792$	0	0
$793$	−28.1465	−0.999511
$794$	19.6132	0.696046
$795$	0	0
$796$	45.0239	1.59583
$797$	7.35905	0.260671	0.130335	−	0.991470i	$-0.458395\pi$
$797$	7.35905	0.260671	0.130335	+	0.991470i	$0.458395\pi$
$798$	0	0
$799$	30.0512	1.06313
$800$	11.4310	0.404147
$801$	0	0
$802$	18.8094	0.664184
$803$	35.2254	1.24308
$804$	0	0
$805$	3.12299	0.110071
$806$	44.1945	1.55669
$807$	0	0
$808$	82.5402	2.90375
$809$	−37.0239	−1.30169	−0.650846	−	0.759210i	$-0.725586\pi$
$809$	−37.0239	−1.30169	−0.650846	+	0.759210i	$0.725586\pi$
$810$	0	0
$811$	−54.7240	−1.92162	−0.960809	−	0.277212i	$-0.910590\pi$
$811$	−54.7240	−1.92162	−0.960809	+	0.277212i	$0.910590\pi$
$812$	−6.90519	−0.242325
$813$	0	0
$814$	−119.309	−4.18176
$815$	−6.40039	−0.224196
$816$	0	0
$817$	44.5624	1.55904
$818$	12.9329	0.452188
$819$	0	0
$820$	−14.2419	−0.497348
$821$	−9.67818	−0.337771	−0.168885	−	0.985636i	$-0.554017\pi$
$821$	−9.67818	−0.337771	−0.168885	+	0.985636i	$0.554017\pi$
$822$	0	0
$823$	−11.7237	−0.408663	−0.204331	−	0.978902i	$-0.565502\pi$
$823$	−11.7237	−0.408663	−0.204331	+	0.978902i	$0.565502\pi$
$824$	−58.6532	−2.04328
$825$	0	0
$826$	−3.88600	−0.135211
$827$	26.6769	0.927646	0.463823	−	0.885928i	$-0.346477\pi$
$827$	26.6769	0.927646	0.463823	+	0.885928i	$0.346477\pi$
$828$	0	0
$829$	8.20281	0.284895	0.142448	−	0.989802i	$-0.454503\pi$
$829$	8.20281	0.284895	0.142448	+	0.989802i	$0.454503\pi$
$830$	−4.65032	−0.161415
$831$	0	0
$832$	−12.4865	−0.432890
$833$	−4.88897	−0.169393
$834$	0	0
$835$	3.23003	0.111780
$836$	−157.830	−5.45868
$837$	0	0
$838$	−59.2484	−2.04670
$839$	32.8245	1.13323	0.566614	−	0.823983i	$-0.308253\pi$
$839$	32.8245	1.13323	0.566614	+	0.823983i	$0.308253\pi$
$840$	0	0
$841$	−26.3266	−0.907815
$842$	−43.2734	−1.49130
$843$	0	0
$844$	−34.0670	−1.17263
$845$	2.44303	0.0840430
$846$	0	0
$847$	−30.0633	−1.03299
$848$	13.6661	0.469298
$849$	0	0
$850$	−59.2774	−2.03320
$851$	62.3608	2.13770
$852$	0	0
$853$	−32.2937	−1.10571	−0.552857	−	0.833276i	$-0.686462\pi$
$853$	−32.2937	−1.10571	−0.552857	+	0.833276i	$0.686462\pi$
$854$	−27.6181	−0.945070
$855$	0	0
$856$	78.3849	2.67914
$857$	−27.3034	−0.932668	−0.466334	−	0.884609i	$-0.654425\pi$
$857$	−27.3034	−0.932668	−0.466334	+	0.884609i	$0.654425\pi$
$858$	0	0
$859$	−51.1565	−1.74544	−0.872718	−	0.488225i	$-0.837645\pi$
$859$	−51.1565	−1.74544	−0.872718	+	0.488225i	$0.837645\pi$
$860$	12.0612	0.411283
$861$	0	0
$862$	−46.5761	−1.58639
$863$	−38.4182	−1.30777	−0.653885	−	0.756594i	$-0.726862\pi$
$863$	−38.4182	−1.30777	−0.653885	+	0.756594i	$0.726862\pi$
$864$	0	0
$865$	−9.64549	−0.327956
$866$	−34.0737	−1.15787
$867$	0	0
$868$	29.4284	0.998865
$869$	−46.2495	−1.56891
$870$	0	0
$871$	29.2231	0.990187
$872$	−38.8888	−1.31694
$873$	0	0
$874$	121.563	4.11194
$875$	−3.68540	−0.124589
$876$	0	0
$877$	−53.3936	−1.80298	−0.901488	−	0.432805i	$-0.857524\pi$
$877$	−53.3936	−1.80298	−0.901488	+	0.432805i	$0.857524\pi$
$878$	−62.6947	−2.11584
$879$	0	0
$880$	−12.9078	−0.435121
$881$	3.49548	0.117766	0.0588828	−	0.998265i	$-0.481246\pi$
$881$	3.49548	0.117766	0.0588828	+	0.998265i	$0.481246\pi$
$882$	0	0
$883$	−2.75544	−0.0927281	−0.0463640	−	0.998925i	$-0.514763\pi$
$883$	−2.75544	−0.0927281	−0.0463640	+	0.998925i	$0.514763\pi$
$884$	−52.4931	−1.76553
$885$	0	0
$886$	−66.1617	−2.22275
$887$	−53.5674	−1.79862	−0.899308	−	0.437315i	$-0.855929\pi$
$887$	−53.5674	−1.79862	−0.899308	+	0.437315i	$0.855929\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−5.86644	−0.196644
$891$	0	0
$892$	−14.8777	−0.498141
$893$	35.8478	1.19960
$894$	0	0
$895$	5.26174	0.175880
$896$	−16.9559	−0.566456
$897$	0	0
$898$	40.3243	1.34564
$899$	−11.3933	−0.379989
$900$	0	0
$901$	−12.3975	−0.413021
$902$	−144.232	−4.80241
$903$	0	0
$904$	78.9224	2.62492
$905$	6.26720	0.208329
$906$	0	0
$907$	11.2461	0.373422	0.186711	−	0.982415i	$-0.440217\pi$
$907$	11.2461	0.373422	0.186711	+	0.982415i	$0.440217\pi$
$908$	−49.1816	−1.63215
$909$	0	0
$910$	−2.37051	−0.0785817
$911$	42.0363	1.39272	0.696362	−	0.717690i	$-0.254801\pi$
$911$	42.0363	1.39272	0.696362	+	0.717690i	$0.254801\pi$
$912$	0	0
$913$	−31.9600	−1.05772
$914$	7.58653	0.250940
$915$	0	0
$916$	90.1273	2.97789
$917$	−19.2456	−0.635547
$918$	0	0
$919$	−20.6524	−0.681261	−0.340630	−	0.940197i	$-0.610641\pi$
$919$	−20.6524	−0.681261	−0.340630	+	0.940197i	$0.610641\pi$
$920$	17.3207	0.571045
$921$	0	0
$922$	52.4577	1.72760
$923$	−17.9173	−0.589754
$924$	0	0
$925$	−36.2743	−1.19269
$926$	−34.0791	−1.11991
$927$	0	0
$928$	−3.84548	−0.126234
$929$	8.98955	0.294937	0.147469	−	0.989067i	$-0.452887\pi$
$929$	8.98955	0.294937	0.147469	+	0.989067i	$0.452887\pi$
$930$	0	0
$931$	−5.83201	−0.191136
$932$	−15.3650	−0.503297
$933$	0	0
$934$	−29.7915	−0.974808
$935$	11.7095	0.382942
$936$	0	0
$937$	−18.7903	−0.613853	−0.306927	−	0.951733i	$-0.599301\pi$
$937$	−18.7903	−0.613853	−0.306927	+	0.951733i	$0.599301\pi$
$938$	28.6745	0.936255
$939$	0	0
$940$	9.70251	0.316461
$941$	34.8758	1.13692	0.568459	−	0.822712i	$-0.307540\pi$
$941$	34.8758	1.13692	0.568459	+	0.822712i	$0.307540\pi$
$942$	0	0
$943$	75.3881	2.45497
$944$	−8.39506	−0.273236
$945$	0	0
$946$	122.148	3.97136
$947$	−57.7138	−1.87545	−0.937723	−	0.347383i	$-0.887071\pi$
$947$	−57.7138	−1.87545	−0.937723	+	0.347383i	$0.887071\pi$
$948$	0	0
$949$	13.9755	0.453665
$950$	−70.7115	−2.29418
$951$	0	0
$952$	−27.1151	−0.878806
$953$	0.231947	0.00751349	0.00375675	−	0.999993i	$-0.498804\pi$
$953$	0.231947	0.00751349	0.00375675	+	0.999993i	$0.498804\pi$
$954$	0	0
$955$	−8.48371	−0.274526
$956$	−15.7635	−0.509830
$957$	0	0
$958$	6.08690	0.196659
$959$	−0.417270	−0.0134744
$960$	0	0
$961$	17.5559	0.566321
$962$	−47.3351	−1.52615
$963$	0	0
$964$	24.2017	0.779484
$965$	−0.618210	−0.0199009
$966$	0	0
$967$	−20.3332	−0.653871	−0.326935	−	0.945047i	$-0.606016\pi$
$967$	−20.3332	−0.653871	−0.326935	+	0.945047i	$0.606016\pi$
$968$	−166.737	−5.35912
$969$	0	0
$970$	−14.3109	−0.459496
$971$	−18.7408	−0.601421	−0.300711	−	0.953715i	$-0.597224\pi$
$971$	−18.7408	−0.601421	−0.300711	+	0.953715i	$0.597224\pi$
$972$	0	0
$973$	7.80248	0.250136
$974$	−82.8657	−2.65519
$975$	0	0
$976$	−59.6642	−1.90981
$977$	17.7823	0.568906	0.284453	−	0.958690i	$-0.408188\pi$
$977$	17.7823	0.568906	0.284453	+	0.958690i	$0.408188\pi$
$978$	0	0
$979$	−40.3180	−1.28857
$980$	−1.57848	−0.0504228
$981$	0	0
$982$	−79.6025	−2.54022
$983$	−54.0319	−1.72335	−0.861675	−	0.507460i	$-0.830584\pi$
$983$	−54.0319	−1.72335	−0.861675	+	0.507460i	$0.830584\pi$
$984$	0	0
$985$	−4.70294	−0.149848
$986$	19.9414	0.635064
$987$	0	0
$988$	−62.6185	−1.99216
$989$	−63.8448	−2.03015
$990$	0	0
$991$	−12.9862	−0.412519	−0.206259	−	0.978497i	$-0.566129\pi$
$991$	−12.9862	−0.412519	−0.206259	+	0.978497i	$0.566129\pi$
$992$	16.3886	0.520339
$993$	0	0
$994$	−17.5809	−0.557632
$995$	−3.98467	−0.126323
$996$	0	0
$997$	9.65694	0.305838	0.152919	−	0.988239i	$-0.451133\pi$
$997$	9.65694	0.305838	0.152919	+	0.988239i	$0.451133\pi$
$998$	−97.4745	−3.08550
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.2	✓	32
3.2	odd	2	inner	8001.2.a.z.1.31	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.2	✓	32	1.1	even	1	trivial
8001.2.a.z.1.31	yes	32	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.49464 q^{2} +4.22324 q^{4} -0.373762 q^{5} -1.00000 q^{7} -5.54618 q^{8} +O(q^{10})\)	\(q-2.49464 q^{2} +4.22324 q^{4} -0.373762 q^{5} -1.00000 q^{7} -5.54618 q^{8} +0.932401 q^{10} +6.40807 q^{11} +2.54237 q^{13} +2.49464 q^{14} +5.38926 q^{16} -4.88897 q^{17} -5.83201 q^{19} -1.57848 q^{20} -15.9858 q^{22} +8.35556 q^{23} -4.86030 q^{25} -6.34231 q^{26} -4.22324 q^{28} +1.63505 q^{29} -6.96821 q^{31} -2.35191 q^{32} +12.1962 q^{34} +0.373762 q^{35} +7.46339 q^{37} +14.5488 q^{38} +2.07295 q^{40} +9.02251 q^{41} -7.64100 q^{43} +27.0628 q^{44} -20.8441 q^{46} -6.14673 q^{47} +1.00000 q^{49} +12.1247 q^{50} +10.7370 q^{52} +2.53581 q^{53} -2.39509 q^{55} +5.54618 q^{56} -4.07885 q^{58} -1.55774 q^{59} -11.0709 q^{61} +17.3832 q^{62} -4.91135 q^{64} -0.950241 q^{65} +11.4944 q^{67} -20.6473 q^{68} -0.932401 q^{70} -7.04746 q^{71} +5.49704 q^{73} -18.6185 q^{74} -24.6300 q^{76} -6.40807 q^{77} -7.21739 q^{79} -2.01430 q^{80} -22.5079 q^{82} -4.98747 q^{83} +1.82731 q^{85} +19.0616 q^{86} -35.5403 q^{88} -6.29176 q^{89} -2.54237 q^{91} +35.2875 q^{92} +15.3339 q^{94} +2.17978 q^{95} -15.3484 q^{97} -2.49464 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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