LMFDB - Embedded newform 8001.2.a.x.1.3

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.21969 q^{2} +2.92701 q^{4} +0.739015 q^{5} +1.00000 q^{7} -2.05767 q^{8} +O(q^{10})$	\(q-2.21969 q^{2} +2.92701 q^{4} +0.739015 q^{5} +1.00000 q^{7} -2.05767 q^{8} -1.64038 q^{10} -1.02193 q^{11} +1.47543 q^{13} -2.21969 q^{14} -1.28663 q^{16} +3.12624 q^{17} -5.88288 q^{19} +2.16311 q^{20} +2.26836 q^{22} +1.53806 q^{23} -4.45386 q^{25} -3.27499 q^{26} +2.92701 q^{28} +1.52533 q^{29} +1.14314 q^{31} +6.97126 q^{32} -6.93928 q^{34} +0.739015 q^{35} -11.1273 q^{37} +13.0581 q^{38} -1.52065 q^{40} +11.0248 q^{41} +1.60578 q^{43} -2.99119 q^{44} -3.41401 q^{46} -8.29699 q^{47} +1.00000 q^{49} +9.88617 q^{50} +4.31859 q^{52} +11.9210 q^{53} -0.755220 q^{55} -2.05767 q^{56} -3.38574 q^{58} -3.45124 q^{59} +5.01324 q^{61} -2.53741 q^{62} -12.9008 q^{64} +1.09036 q^{65} -8.41905 q^{67} +9.15055 q^{68} -1.64038 q^{70} -14.2836 q^{71} -1.36288 q^{73} +24.6992 q^{74} -17.2192 q^{76} -1.02193 q^{77} +12.8685 q^{79} -0.950838 q^{80} -24.4715 q^{82} -7.52064 q^{83} +2.31034 q^{85} -3.56433 q^{86} +2.10279 q^{88} +1.31226 q^{89} +1.47543 q^{91} +4.50192 q^{92} +18.4167 q^{94} -4.34754 q^{95} -8.68801 q^{97} -2.21969 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) $ 22 * q + 10 * q^4 + 22 * q^7	$ 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) $ 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * 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 $

$2$	−2.21969	−1.56956	−0.784778	−	0.619777i	$-0.787223\pi$
$2$	−2.21969	−1.56956	−0.784778	+	0.619777i	$0.787223\pi$
$3$	0	0
$3$	0	0
$4$	2.92701	1.46351
$5$	0.739015	0.330498	0.165249	−	0.986252i	$-0.447157\pi$
$5$	0.739015	0.330498	0.165249	+	0.986252i	$0.447157\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.05767	−0.727498
$9$	0	0
$10$	−1.64038	−0.518735
$11$	−1.02193	−0.308123	−0.154061	−	0.988061i	$-0.549235\pi$
$11$	−1.02193	−0.308123	−0.154061	+	0.988061i	$0.549235\pi$
$12$	0	0
$13$	1.47543	0.409210	0.204605	−	0.978845i	$-0.434409\pi$
$13$	1.47543	0.409210	0.204605	+	0.978845i	$0.434409\pi$
$14$	−2.21969	−0.593236
$15$	0	0
$16$	−1.28663	−0.321657
$17$	3.12624	0.758225	0.379113	−	0.925351i	$-0.376229\pi$
$17$	3.12624	0.758225	0.379113	+	0.925351i	$0.376229\pi$
$18$	0	0
$19$	−5.88288	−1.34962	−0.674812	−	0.737989i	$-0.735775\pi$
$19$	−5.88288	−1.34962	−0.674812	+	0.737989i	$0.735775\pi$
$20$	2.16311	0.483685
$21$	0	0
$22$	2.26836	0.483616
$23$	1.53806	0.320708	0.160354	−	0.987060i	$-0.448736\pi$
$23$	1.53806	0.320708	0.160354	+	0.987060i	$0.448736\pi$
$24$	0	0
$25$	−4.45386	−0.890771
$26$	−3.27499	−0.642278
$27$	0	0
$28$	2.92701	0.553153
$29$	1.52533	0.283246	0.141623	−	0.989921i	$-0.454768\pi$
$29$	1.52533	0.283246	0.141623	+	0.989921i	$0.454768\pi$
$30$	0	0
$31$	1.14314	0.205313	0.102657	−	0.994717i	$-0.467266\pi$
$31$	1.14314	0.205313	0.102657	+	0.994717i	$0.467266\pi$
$32$	6.97126	1.23236
$33$	0	0
$34$	−6.93928	−1.19008
$35$	0.739015	0.124916
$36$	0	0
$37$	−11.1273	−1.82932	−0.914662	−	0.404220i	$-0.867543\pi$
$37$	−11.1273	−1.82932	−0.914662	+	0.404220i	$0.867543\pi$
$38$	13.0581	2.11831
$39$	0	0
$40$	−1.52065	−0.240436
$41$	11.0248	1.72178	0.860890	−	0.508792i	$-0.169908\pi$
$41$	11.0248	1.72178	0.860890	+	0.508792i	$0.169908\pi$
$42$	0	0
$43$	1.60578	0.244879	0.122440	−	0.992476i	$-0.460928\pi$
$43$	1.60578	0.244879	0.122440	+	0.992476i	$0.460928\pi$
$44$	−2.99119	−0.450939
$45$	0	0
$46$	−3.41401	−0.503368
$47$	−8.29699	−1.21024	−0.605120	−	0.796134i	$-0.706875\pi$
$47$	−8.29699	−1.21024	−0.605120	+	0.796134i	$0.706875\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	9.88617	1.39812
$51$	0	0
$52$	4.31859	0.598881
$53$	11.9210	1.63748	0.818738	−	0.574168i	$-0.194674\pi$
$53$	11.9210	1.63748	0.818738	+	0.574168i	$0.194674\pi$
$54$	0	0
$55$	−0.755220	−0.101834
$56$	−2.05767	−0.274968
$57$	0	0
$58$	−3.38574	−0.444570
$59$	−3.45124	−0.449313	−0.224657	−	0.974438i	$-0.572126\pi$
$59$	−3.45124	−0.449313	−0.224657	+	0.974438i	$0.572126\pi$
$60$	0	0
$61$	5.01324	0.641880	0.320940	−	0.947100i	$-0.396001\pi$
$61$	5.01324	0.641880	0.320940	+	0.947100i	$0.396001\pi$
$62$	−2.53741	−0.322251
$63$	0	0
$64$	−12.9008	−1.61260
$65$	1.09036	0.135243
$66$	0	0
$67$	−8.41905	−1.02855	−0.514275	−	0.857625i	$-0.671939\pi$
$67$	−8.41905	−1.02855	−0.514275	+	0.857625i	$0.671939\pi$
$68$	9.15055	1.10967
$69$	0	0
$70$	−1.64038	−0.196063
$71$	−14.2836	−1.69515	−0.847573	−	0.530678i	$-0.821937\pi$
$71$	−14.2836	−1.69515	−0.847573	+	0.530678i	$0.821937\pi$
$72$	0	0
$73$	−1.36288	−0.159513	−0.0797565	−	0.996814i	$-0.525414\pi$
$73$	−1.36288	−0.159513	−0.0797565	+	0.996814i	$0.525414\pi$
$74$	24.6992	2.87123
$75$	0	0
$76$	−17.2192	−1.97518
$77$	−1.02193	−0.116459
$78$	0	0
$79$	12.8685	1.44782	0.723910	−	0.689894i	$-0.242343\pi$
$79$	12.8685	1.44782	0.723910	+	0.689894i	$0.242343\pi$
$80$	−0.950838	−0.106307
$81$	0	0
$82$	−24.4715	−2.70243
$83$	−7.52064	−0.825497	−0.412748	−	0.910845i	$-0.635431\pi$
$83$	−7.52064	−0.825497	−0.412748	+	0.910845i	$0.635431\pi$
$84$	0	0
$85$	2.31034	0.250592
$86$	−3.56433	−0.384352
$87$	0	0
$88$	2.10279	0.224159
$89$	1.31226	0.139100	0.0695498	−	0.997578i	$-0.477844\pi$
$89$	1.31226	0.139100	0.0695498	+	0.997578i	$0.477844\pi$
$90$	0	0
$91$	1.47543	0.154667
$92$	4.50192	0.469357
$93$	0	0
$94$	18.4167	1.89954
$95$	−4.34754	−0.446048
$96$	0	0
$97$	−8.68801	−0.882134	−0.441067	−	0.897474i	$-0.645400\pi$
$97$	−8.68801	−0.882134	−0.441067	+	0.897474i	$0.645400\pi$
$98$	−2.21969	−0.224222
$99$	0	0
$100$	−13.0365	−1.30365
$101$	1.74924	0.174056	0.0870282	−	0.996206i	$-0.472263\pi$
$101$	1.74924	0.174056	0.0870282	+	0.996206i	$0.472263\pi$
$102$	0	0
$103$	1.31728	0.129796	0.0648980	−	0.997892i	$-0.479328\pi$
$103$	1.31728	0.129796	0.0648980	+	0.997892i	$0.479328\pi$
$104$	−3.03595	−0.297699
$105$	0	0
$106$	−26.4609	−2.57011
$107$	−8.82108	−0.852766	−0.426383	−	0.904543i	$-0.640212\pi$
$107$	−8.82108	−0.852766	−0.426383	+	0.904543i	$0.640212\pi$
$108$	0	0
$109$	2.01285	0.192796	0.0963980	−	0.995343i	$-0.469268\pi$
$109$	2.01285	0.192796	0.0963980	+	0.995343i	$0.469268\pi$
$110$	1.67635	0.159834
$111$	0	0
$112$	−1.28663	−0.121575
$113$	9.48735	0.892494	0.446247	−	0.894910i	$-0.352760\pi$
$113$	9.48735	0.892494	0.446247	+	0.894910i	$0.352760\pi$
$114$	0	0
$115$	1.13665	0.105993
$116$	4.46464	0.414532
$117$	0	0
$118$	7.66067	0.705222
$119$	3.12624	0.286582
$120$	0	0
$121$	−9.95567	−0.905060
$122$	−11.1278	−1.00747
$123$	0	0
$124$	3.34597	0.300477
$125$	−6.98655	−0.624896
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	14.6931	1.29870
$129$	0	0
$130$	−2.42027	−0.212271
$131$	−11.7303	−1.02488	−0.512441	−	0.858722i	$-0.671259\pi$
$131$	−11.7303	−1.02488	−0.512441	+	0.858722i	$0.671259\pi$
$132$	0	0
$133$	−5.88288	−0.510110
$134$	18.6877	1.61437
$135$	0	0
$136$	−6.43279	−0.551607
$137$	6.66031	0.569029	0.284514	−	0.958672i	$-0.408168\pi$
$137$	6.66031	0.569029	0.284514	+	0.958672i	$0.408168\pi$
$138$	0	0
$139$	5.24371	0.444766	0.222383	−	0.974959i	$-0.428616\pi$
$139$	5.24371	0.444766	0.222383	+	0.974959i	$0.428616\pi$
$140$	2.16311	0.182816
$141$	0	0
$142$	31.7050	2.66063
$143$	−1.50778	−0.126087
$144$	0	0
$145$	1.12724	0.0936121
$146$	3.02517	0.250365
$147$	0	0
$148$	−32.5699	−2.67723
$149$	4.38002	0.358825	0.179413	−	0.983774i	$-0.442580\pi$
$149$	4.38002	0.358825	0.179413	+	0.983774i	$0.442580\pi$
$150$	0	0
$151$	−1.81200	−0.147459	−0.0737293	−	0.997278i	$-0.523490\pi$
$151$	−1.81200	−0.147459	−0.0737293	+	0.997278i	$0.523490\pi$
$152$	12.1050	0.981849
$153$	0	0
$154$	2.26836	0.182790
$155$	0.844796	0.0678556
$156$	0	0
$157$	−6.36291	−0.507816	−0.253908	−	0.967228i	$-0.581716\pi$
$157$	−6.36291	−0.507816	−0.253908	+	0.967228i	$0.581716\pi$
$158$	−28.5641	−2.27243
$159$	0	0
$160$	5.15187	0.407291
$161$	1.53806	0.121216
$162$	0	0
$163$	−15.1417	−1.18599	−0.592996	−	0.805205i	$-0.702055\pi$
$163$	−15.1417	−1.18599	−0.592996	+	0.805205i	$0.702055\pi$
$164$	32.2696	2.51983
$165$	0	0
$166$	16.6935	1.29566
$167$	−5.73094	−0.443473	−0.221737	−	0.975107i	$-0.571173\pi$
$167$	−5.73094	−0.443473	−0.221737	+	0.975107i	$0.571173\pi$
$168$	0	0
$169$	−10.8231	−0.832547
$170$	−5.12824	−0.393318
$171$	0	0
$172$	4.70014	0.358382
$173$	−12.0392	−0.915321	−0.457660	−	0.889127i	$-0.651312\pi$
$173$	−12.0392	−0.915321	−0.457660	+	0.889127i	$0.651312\pi$
$174$	0	0
$175$	−4.45386	−0.336680
$176$	1.31484	0.0991098
$177$	0	0
$178$	−2.91281	−0.218325
$179$	7.99030	0.597223	0.298612	−	0.954375i	$-0.403476\pi$
$179$	7.99030	0.597223	0.298612	+	0.954375i	$0.403476\pi$
$180$	0	0
$181$	−2.79186	−0.207517	−0.103759	−	0.994603i	$-0.533087\pi$
$181$	−2.79186	−0.207517	−0.103759	+	0.994603i	$0.533087\pi$
$182$	−3.27499	−0.242758
$183$	0	0
$184$	−3.16483	−0.233314
$185$	−8.22328	−0.604587
$186$	0	0
$187$	−3.19479	−0.233626
$188$	−24.2854	−1.77119
$189$	0	0
$190$	9.65017	0.700097
$191$	4.42522	0.320198	0.160099	−	0.987101i	$-0.448819\pi$
$191$	4.42522	0.320198	0.160099	+	0.987101i	$0.448819\pi$
$192$	0	0
$193$	−5.04282	−0.362990	−0.181495	−	0.983392i	$-0.558094\pi$
$193$	−5.04282	−0.362990	−0.181495	+	0.983392i	$0.558094\pi$
$194$	19.2847	1.38456
$195$	0	0
$196$	2.92701	0.209072
$197$	10.3156	0.734958	0.367479	−	0.930032i	$-0.380221\pi$
$197$	10.3156	0.734958	0.367479	+	0.930032i	$0.380221\pi$
$198$	0	0
$199$	−21.7203	−1.53971	−0.769854	−	0.638220i	$-0.779671\pi$
$199$	−21.7203	−1.53971	−0.769854	+	0.638220i	$0.779671\pi$
$200$	9.16459	0.648034
$201$	0	0
$202$	−3.88278	−0.273191
$203$	1.52533	0.107057
$204$	0	0
$205$	8.14747	0.569044
$206$	−2.92396	−0.203722
$207$	0	0
$208$	−1.89833	−0.131625
$209$	6.01187	0.415850
$210$	0	0
$211$	−10.2254	−0.703943	−0.351972	−	0.936011i	$-0.614489\pi$
$211$	−10.2254	−0.703943	−0.351972	+	0.936011i	$0.614489\pi$
$212$	34.8929	2.39645
$213$	0	0
$214$	19.5800	1.33846
$215$	1.18670	0.0809321
$216$	0	0
$217$	1.14314	0.0776012
$218$	−4.46789	−0.302604
$219$	0	0
$220$	−2.21054	−0.149034
$221$	4.61254	0.310273
$222$	0	0
$223$	3.59777	0.240924	0.120462	−	0.992718i	$-0.461562\pi$
$223$	3.59777	0.240924	0.120462	+	0.992718i	$0.461562\pi$
$224$	6.97126	0.465787
$225$	0	0
$226$	−21.0589	−1.40082
$227$	−11.5633	−0.767485	−0.383743	−	0.923440i	$-0.625365\pi$
$227$	−11.5633	−0.767485	−0.383743	+	0.923440i	$0.625365\pi$
$228$	0	0
$229$	−0.00980539	−0.000647959 0	−0.000323979	−	1.00000i	$-0.500103\pi$
$229$	−0.00980539	−0.000647959 0	−0.000323979		1.00000i	$0.500103\pi$
$230$	−2.52301	−0.166362
$231$	0	0
$232$	−3.13862	−0.206061
$233$	4.27471	0.280046	0.140023	−	0.990148i	$-0.455282\pi$
$233$	4.27471	0.280046	0.140023	+	0.990148i	$0.455282\pi$
$234$	0	0
$235$	−6.13161	−0.399982
$236$	−10.1018	−0.657572
$237$	0	0
$238$	−6.93928	−0.449807
$239$	13.6832	0.885094	0.442547	−	0.896745i	$-0.354075\pi$
$239$	13.6832	0.885094	0.442547	+	0.896745i	$0.354075\pi$
$240$	0	0
$241$	−13.2841	−0.855703	−0.427852	−	0.903849i	$-0.640729\pi$
$241$	−13.2841	−0.855703	−0.427852	+	0.903849i	$0.640729\pi$
$242$	22.0985	1.42054
$243$	0	0
$244$	14.6738	0.939395
$245$	0.739015	0.0472140
$246$	0	0
$247$	−8.67975	−0.552280
$248$	−2.35220	−0.149365
$249$	0	0
$250$	15.5079	0.980809
$251$	15.3625	0.969673	0.484837	−	0.874605i	$-0.338879\pi$
$251$	15.3625	0.969673	0.484837	+	0.874605i	$0.338879\pi$
$252$	0	0
$253$	−1.57178	−0.0988172
$254$	−2.21969	−0.139276
$255$	0	0
$256$	−6.81264	−0.425790
$257$	−13.7288	−0.856378	−0.428189	−	0.903689i	$-0.640848\pi$
$257$	−13.7288	−0.856378	−0.428189	+	0.903689i	$0.640848\pi$
$258$	0	0
$259$	−11.1273	−0.691419
$260$	3.19151	0.197929
$261$	0	0
$262$	26.0376	1.60861
$263$	−24.0677	−1.48408	−0.742040	−	0.670356i	$-0.766141\pi$
$263$	−24.0677	−1.48408	−0.742040	+	0.670356i	$0.766141\pi$
$264$	0	0
$265$	8.80980	0.541182
$266$	13.0581	0.800646
$267$	0	0
$268$	−24.6427	−1.50529
$269$	3.85428	0.235000	0.117500	−	0.993073i	$-0.462512\pi$
$269$	3.85428	0.235000	0.117500	+	0.993073i	$0.462512\pi$
$270$	0	0
$271$	−17.5674	−1.06714	−0.533572	−	0.845755i	$-0.679151\pi$
$271$	−17.5674	−1.06714	−0.533572	+	0.845755i	$0.679151\pi$
$272$	−4.02231	−0.243888
$273$	0	0
$274$	−14.7838	−0.893122
$275$	4.55152	0.274467
$276$	0	0
$277$	27.4751	1.65082	0.825408	−	0.564536i	$-0.190945\pi$
$277$	27.4751	1.65082	0.825408	+	0.564536i	$0.190945\pi$
$278$	−11.6394	−0.698085
$279$	0	0
$280$	−1.52065	−0.0908764
$281$	−11.3522	−0.677215	−0.338607	−	0.940928i	$-0.609956\pi$
$281$	−11.3522	−0.677215	−0.338607	+	0.940928i	$0.609956\pi$
$282$	0	0
$283$	23.2577	1.38252	0.691262	−	0.722604i	$-0.257055\pi$
$283$	23.2577	1.38252	0.691262	+	0.722604i	$0.257055\pi$
$284$	−41.8081	−2.48086
$285$	0	0
$286$	3.34680	0.197900
$287$	11.0248	0.650771
$288$	0	0
$289$	−7.22660	−0.425094
$290$	−2.50212	−0.146929
$291$	0	0
$292$	−3.98916	−0.233448
$293$	17.0994	0.998956	0.499478	−	0.866327i	$-0.333525\pi$
$293$	17.0994	0.998956	0.499478	+	0.866327i	$0.333525\pi$
$294$	0	0
$295$	−2.55052	−0.148497
$296$	22.8964	1.33083
$297$	0	0
$298$	−9.72228	−0.563197
$299$	2.26929	0.131237
$300$	0	0
$301$	1.60578	0.0925557
$302$	4.02208	0.231445
$303$	0	0
$304$	7.56907	0.434116
$305$	3.70486	0.212140
$306$	0	0
$307$	−3.63911	−0.207695	−0.103847	−	0.994593i	$-0.533115\pi$
$307$	−3.63911	−0.207695	−0.103847	+	0.994593i	$0.533115\pi$
$308$	−2.99119	−0.170439
$309$	0	0
$310$	−1.87518	−0.106503
$311$	−8.68792	−0.492647	−0.246323	−	0.969188i	$-0.579223\pi$
$311$	−8.68792	−0.492647	−0.246323	+	0.969188i	$0.579223\pi$
$312$	0	0
$313$	4.35518	0.246169	0.123085	−	0.992396i	$-0.460721\pi$
$313$	4.35518	0.246169	0.123085	+	0.992396i	$0.460721\pi$
$314$	14.1237	0.797045
$315$	0	0
$316$	37.6663	2.11889
$317$	−11.9892	−0.673381	−0.336691	−	0.941615i	$-0.609308\pi$
$317$	−11.9892	−0.673381	−0.336691	+	0.941615i	$0.609308\pi$
$318$	0	0
$319$	−1.55877	−0.0872744
$320$	−9.53386	−0.532959
$321$	0	0
$322$	−3.41401	−0.190255
$323$	−18.3913	−1.02332
$324$	0	0
$325$	−6.57134	−0.364512
$326$	33.6099	1.86148
$327$	0	0
$328$	−22.6854	−1.25259
$329$	−8.29699	−0.457428
$330$	0	0
$331$	3.76813	0.207115	0.103558	−	0.994623i	$-0.466977\pi$
$331$	3.76813	0.207115	0.103558	+	0.994623i	$0.466977\pi$
$332$	−22.0130	−1.20812
$333$	0	0
$334$	12.7209	0.696056
$335$	−6.22181	−0.339934
$336$	0	0
$337$	−5.84070	−0.318163	−0.159082	−	0.987265i	$-0.550853\pi$
$337$	−5.84070	−0.318163	−0.159082	+	0.987265i	$0.550853\pi$
$338$	24.0239	1.30673
$339$	0	0
$340$	6.76240	0.366742
$341$	−1.16820	−0.0632617
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.30418	−0.178149
$345$	0	0
$346$	26.7232	1.43665
$347$	30.8340	1.65526	0.827629	−	0.561275i	$-0.189689\pi$
$347$	30.8340	1.65526	0.827629	+	0.561275i	$0.189689\pi$
$348$	0	0
$349$	33.3522	1.78530	0.892651	−	0.450748i	$-0.148843\pi$
$349$	33.3522	1.78530	0.892651	+	0.450748i	$0.148843\pi$
$350$	9.88617	0.528438
$351$	0	0
$352$	−7.12412	−0.379717
$353$	−12.7771	−0.680058	−0.340029	−	0.940415i	$-0.610437\pi$
$353$	−12.7771	−0.680058	−0.340029	+	0.940415i	$0.610437\pi$
$354$	0	0
$355$	−10.5558	−0.560242
$356$	3.84101	0.203573
$357$	0	0
$358$	−17.7360	−0.937375
$359$	3.62553	0.191348	0.0956740	−	0.995413i	$-0.469499\pi$
$359$	3.62553	0.191348	0.0956740	+	0.995413i	$0.469499\pi$
$360$	0	0
$361$	15.6082	0.821486
$362$	6.19706	0.325710
$363$	0	0
$364$	4.31859	0.226356
$365$	−1.00719	−0.0527187
$366$	0	0
$367$	26.9151	1.40496	0.702479	−	0.711705i	$-0.252077\pi$
$367$	26.9151	1.40496	0.702479	+	0.711705i	$0.252077\pi$
$368$	−1.97891	−0.103158
$369$	0	0
$370$	18.2531	0.948934
$371$	11.9210	0.618907
$372$	0	0
$373$	5.18060	0.268241	0.134121	−	0.990965i	$-0.457179\pi$
$373$	5.18060	0.268241	0.134121	+	0.990965i	$0.457179\pi$
$374$	7.09144	0.366690
$375$	0	0
$376$	17.0725	0.880448
$377$	2.25051	0.115907
$378$	0	0
$379$	−20.2702	−1.04121	−0.520605	−	0.853798i	$-0.674294\pi$
$379$	−20.2702	−1.04121	−0.520605	+	0.853798i	$0.674294\pi$
$380$	−12.7253	−0.652794
$381$	0	0
$382$	−9.82261	−0.502569
$383$	−29.7716	−1.52126	−0.760630	−	0.649186i	$-0.775110\pi$
$383$	−29.7716	−1.52126	−0.760630	+	0.649186i	$0.775110\pi$
$384$	0	0
$385$	−0.755220	−0.0384896
$386$	11.1935	0.569734
$387$	0	0
$388$	−25.4299	−1.29101
$389$	−11.0618	−0.560858	−0.280429	−	0.959875i	$-0.590477\pi$
$389$	−11.0618	−0.560858	−0.280429	+	0.959875i	$0.590477\pi$
$390$	0	0
$391$	4.80835	0.243169
$392$	−2.05767	−0.103928
$393$	0	0
$394$	−22.8975	−1.15356
$395$	9.51003	0.478501
$396$	0	0
$397$	−0.272445	−0.0136736	−0.00683680	−	0.999977i	$-0.502176\pi$
$397$	−0.272445	−0.0136736	−0.00683680	+	0.999977i	$0.502176\pi$
$398$	48.2122	2.41666
$399$	0	0
$400$	5.73046	0.286523
$401$	−3.20151	−0.159876	−0.0799379	−	0.996800i	$-0.525472\pi$
$401$	−3.20151	−0.159876	−0.0799379	+	0.996800i	$0.525472\pi$
$402$	0	0
$403$	1.68661	0.0840162
$404$	5.12006	0.254732
$405$	0	0
$406$	−3.38574	−0.168032
$407$	11.3713	0.563656
$408$	0	0
$409$	−39.5928	−1.95774	−0.978869	−	0.204486i	$-0.934448\pi$
$409$	−39.5928	−1.95774	−0.978869	+	0.204486i	$0.934448\pi$
$410$	−18.0848	−0.893147
$411$	0	0
$412$	3.85571	0.189957
$413$	−3.45124	−0.169824
$414$	0	0
$415$	−5.55787	−0.272825
$416$	10.2856	0.504292
$417$	0	0
$418$	−13.3445	−0.652699
$419$	15.0772	0.736567	0.368284	−	0.929714i	$-0.379946\pi$
$419$	15.0772	0.736567	0.368284	+	0.929714i	$0.379946\pi$
$420$	0	0
$421$	−10.0571	−0.490155	−0.245077	−	0.969504i	$-0.578813\pi$
$421$	−10.0571	−0.490155	−0.245077	+	0.969504i	$0.578813\pi$
$422$	22.6971	1.10488
$423$	0	0
$424$	−24.5295	−1.19126
$425$	−13.9238	−0.675405
$426$	0	0
$427$	5.01324	0.242608
$428$	−25.8194	−1.24803
$429$	0	0
$430$	−2.63410	−0.127027
$431$	8.22144	0.396013	0.198006	−	0.980201i	$-0.436553\pi$
$431$	8.22144	0.396013	0.198006	+	0.980201i	$0.436553\pi$
$432$	0	0
$433$	1.53568	0.0738001	0.0369001	−	0.999319i	$-0.488252\pi$
$433$	1.53568	0.0738001	0.0369001	+	0.999319i	$0.488252\pi$
$434$	−2.53741	−0.121799
$435$	0	0
$436$	5.89163	0.282158
$437$	−9.04821	−0.432835
$438$	0	0
$439$	18.9535	0.904603	0.452302	−	0.891865i	$-0.350603\pi$
$439$	18.9535	0.904603	0.452302	+	0.891865i	$0.350603\pi$
$440$	1.55400	0.0740839
$441$	0	0
$442$	−10.2384	−0.486991
$443$	31.3030	1.48725	0.743625	−	0.668597i	$-0.233105\pi$
$443$	31.3030	1.48725	0.743625	+	0.668597i	$0.233105\pi$
$444$	0	0
$445$	0.969783	0.0459721
$446$	−7.98592	−0.378144
$447$	0	0
$448$	−12.9008	−0.609504
$449$	12.6009	0.594674	0.297337	−	0.954773i	$-0.403901\pi$
$449$	12.6009	0.594674	0.297337	+	0.954773i	$0.403901\pi$
$450$	0	0
$451$	−11.2665	−0.530519
$452$	27.7696	1.30617
$453$	0	0
$454$	25.6670	1.20461
$455$	1.09036	0.0511170
$456$	0	0
$457$	7.03970	0.329303	0.164652	−	0.986352i	$-0.447350\pi$
$457$	7.03970	0.329303	0.164652	+	0.986352i	$0.447350\pi$
$458$	0.0217649	0.00101701
$459$	0	0
$460$	3.32699	0.155122
$461$	22.4366	1.04497	0.522487	−	0.852647i	$-0.325004\pi$
$461$	22.4366	1.04497	0.522487	+	0.852647i	$0.325004\pi$
$462$	0	0
$463$	−32.9587	−1.53172	−0.765860	−	0.643007i	$-0.777686\pi$
$463$	−32.9587	−1.53172	−0.765860	+	0.643007i	$0.777686\pi$
$464$	−1.96253	−0.0911080
$465$	0	0
$466$	−9.48853	−0.439548
$467$	−35.3566	−1.63611	−0.818055	−	0.575140i	$-0.804947\pi$
$467$	−35.3566	−1.63611	−0.818055	+	0.575140i	$0.804947\pi$
$468$	0	0
$469$	−8.41905	−0.388756
$470$	13.6102	0.627794
$471$	0	0
$472$	7.10153	0.326874
$473$	−1.64099	−0.0754529
$474$	0	0
$475$	26.2015	1.20221
$476$	9.15055	0.419415
$477$	0	0
$478$	−30.3725	−1.38920
$479$	−10.4972	−0.479632	−0.239816	−	0.970818i	$-0.577087\pi$
$479$	−10.4972	−0.479632	−0.239816	+	0.970818i	$0.577087\pi$
$480$	0	0
$481$	−16.4176	−0.748577
$482$	29.4865	1.34307
$483$	0	0
$484$	−29.1403	−1.32456
$485$	−6.42057	−0.291543
$486$	0	0
$487$	−24.2897	−1.10067	−0.550335	−	0.834944i	$-0.685500\pi$
$487$	−24.2897	−1.10067	−0.550335	+	0.834944i	$0.685500\pi$
$488$	−10.3156	−0.466966
$489$	0	0
$490$	−1.64038	−0.0741050
$491$	8.55150	0.385924	0.192962	−	0.981206i	$-0.438191\pi$
$491$	8.55150	0.385924	0.192962	+	0.981206i	$0.438191\pi$
$492$	0	0
$493$	4.76854	0.214764
$494$	19.2663	0.866834
$495$	0	0
$496$	−1.47079	−0.0660405
$497$	−14.2836	−0.640705
$498$	0	0
$499$	27.6417	1.23741	0.618707	−	0.785622i	$-0.287657\pi$
$499$	27.6417	1.23741	0.618707	+	0.785622i	$0.287657\pi$
$500$	−20.4497	−0.914538
$501$	0	0
$502$	−34.1000	−1.52196
$503$	−5.20240	−0.231964	−0.115982	−	0.993251i	$-0.537001\pi$
$503$	−5.20240	−0.231964	−0.115982	+	0.993251i	$0.537001\pi$
$504$	0	0
$505$	1.29272	0.0575252
$506$	3.48887	0.155099
$507$	0	0
$508$	2.92701	0.129865
$509$	−17.9034	−0.793554	−0.396777	−	0.917915i	$-0.629871\pi$
$509$	−17.9034	−0.793554	−0.396777	+	0.917915i	$0.629871\pi$
$510$	0	0
$511$	−1.36288	−0.0602902
$512$	−14.2643	−0.630401
$513$	0	0
$514$	30.4736	1.34413
$515$	0.973494	0.0428973
$516$	0	0
$517$	8.47892	0.372902
$518$	24.6992	1.08522
$519$	0	0
$520$	−2.24361	−0.0983889
$521$	0.564594	0.0247353	0.0123677	−	0.999924i	$-0.496063\pi$
$521$	0.564594	0.0247353	0.0123677	+	0.999924i	$0.496063\pi$
$522$	0	0
$523$	24.2775	1.06158	0.530791	−	0.847503i	$-0.321895\pi$
$523$	24.2775	1.06158	0.530791	+	0.847503i	$0.321895\pi$
$524$	−34.3347	−1.49992
$525$	0	0
$526$	53.4228	2.32935
$527$	3.57372	0.155674
$528$	0	0
$529$	−20.6344	−0.897147
$530$	−19.5550	−0.849415
$531$	0	0
$532$	−17.2192	−0.746549
$533$	16.2662	0.704569
$534$	0	0
$535$	−6.51891	−0.281837
$536$	17.3237	0.748269
$537$	0	0
$538$	−8.55530	−0.368845
$539$	−1.02193	−0.0440175
$540$	0	0
$541$	20.5670	0.884244	0.442122	−	0.896955i	$-0.354226\pi$
$541$	20.5670	0.884244	0.442122	+	0.896955i	$0.354226\pi$
$542$	38.9941	1.67494
$543$	0	0
$544$	21.7939	0.934404
$545$	1.48753	0.0637186
$546$	0	0
$547$	−3.09815	−0.132467	−0.0662336	−	0.997804i	$-0.521098\pi$
$547$	−3.09815	−0.132467	−0.0662336	+	0.997804i	$0.521098\pi$
$548$	19.4948	0.832777
$549$	0	0
$550$	−10.1029	−0.430791
$551$	−8.97330	−0.382275
$552$	0	0
$553$	12.8685	0.547225
$554$	−60.9861	−2.59105
$555$	0	0
$556$	15.3484	0.650917
$557$	−40.8819	−1.73222	−0.866111	−	0.499852i	$-0.833388\pi$
$557$	−40.8819	−1.73222	−0.866111	+	0.499852i	$0.833388\pi$
$558$	0	0
$559$	2.36921	0.100207
$560$	−0.950838	−0.0401802
$561$	0	0
$562$	25.1983	1.06293
$563$	8.38214	0.353265	0.176633	−	0.984277i	$-0.443480\pi$
$563$	8.38214	0.353265	0.176633	+	0.984277i	$0.443480\pi$
$564$	0	0
$565$	7.01130	0.294967
$566$	−51.6247	−2.16995
$567$	0	0
$568$	29.3909	1.23322
$569$	−24.0948	−1.01010	−0.505052	−	0.863089i	$-0.668527\pi$
$569$	−24.0948	−1.01010	−0.505052	+	0.863089i	$0.668527\pi$
$570$	0	0
$571$	33.4430	1.39955	0.699773	−	0.714365i	$-0.253284\pi$
$571$	33.4430	1.39955	0.699773	+	0.714365i	$0.253284\pi$
$572$	−4.41328	−0.184529
$573$	0	0
$574$	−24.4715	−1.02142
$575$	−6.85030	−0.285677
$576$	0	0
$577$	−15.0916	−0.628273	−0.314136	−	0.949378i	$-0.601715\pi$
$577$	−15.0916	−0.628273	−0.314136	+	0.949378i	$0.601715\pi$
$578$	16.0408	0.667209
$579$	0	0
$580$	3.29944	0.137002
$581$	−7.52064	−0.312009
$582$	0	0
$583$	−12.1824	−0.504543
$584$	2.80436	0.116045
$585$	0	0
$586$	−37.9553	−1.56792
$587$	−14.0289	−0.579034	−0.289517	−	0.957173i	$-0.593495\pi$
$587$	−14.0289	−0.579034	−0.289517	+	0.957173i	$0.593495\pi$
$588$	0	0
$589$	−6.72493	−0.277096
$590$	5.66136	0.233074
$591$	0	0
$592$	14.3167	0.588415
$593$	33.0407	1.35682	0.678409	−	0.734684i	$-0.262670\pi$
$593$	33.0407	1.35682	0.678409	+	0.734684i	$0.262670\pi$
$594$	0	0
$595$	2.31034	0.0947148
$596$	12.8204	0.525143
$597$	0	0
$598$	−5.03712	−0.205983
$599$	1.41971	0.0580079	0.0290039	−	0.999579i	$-0.490766\pi$
$599$	1.41971	0.0580079	0.0290039	+	0.999579i	$0.490766\pi$
$600$	0	0
$601$	18.2562	0.744687	0.372344	−	0.928095i	$-0.378554\pi$
$601$	18.2562	0.744687	0.372344	+	0.928095i	$0.378554\pi$
$602$	−3.56433	−0.145271
$603$	0	0
$604$	−5.30375	−0.215807
$605$	−7.35739	−0.299120
$606$	0	0
$607$	−15.1214	−0.613759	−0.306880	−	0.951748i	$-0.599285\pi$
$607$	−15.1214	−0.613759	−0.306880	+	0.951748i	$0.599285\pi$
$608$	−41.0111	−1.66322
$609$	0	0
$610$	−8.22364	−0.332965
$611$	−12.2416	−0.495242
$612$	0	0
$613$	44.2262	1.78628	0.893140	−	0.449779i	$-0.148497\pi$
$613$	44.2262	1.78628	0.893140	+	0.449779i	$0.148497\pi$
$614$	8.07768	0.325989
$615$	0	0
$616$	2.10279	0.0847240
$617$	−25.3037	−1.01869	−0.509344	−	0.860563i	$-0.670112\pi$
$617$	−25.3037	−1.01869	−0.509344	+	0.860563i	$0.670112\pi$
$618$	0	0
$619$	−36.6561	−1.47333	−0.736667	−	0.676256i	$-0.763601\pi$
$619$	−36.6561	−1.47333	−0.736667	+	0.676256i	$0.763601\pi$
$620$	2.47273	0.0993071
$621$	0	0
$622$	19.2845	0.773237
$623$	1.31226	0.0525747
$624$	0	0
$625$	17.1061	0.684245
$626$	−9.66713	−0.386376
$627$	0	0
$628$	−18.6243	−0.743191
$629$	−34.7868	−1.38704
$630$	0	0
$631$	11.2461	0.447699	0.223849	−	0.974624i	$-0.428138\pi$
$631$	11.2461	0.447699	0.223849	+	0.974624i	$0.428138\pi$
$632$	−26.4792	−1.05329
$633$	0	0
$634$	26.6123	1.05691
$635$	0.739015	0.0293269
$636$	0	0
$637$	1.47543	0.0584585
$638$	3.45998	0.136982
$639$	0	0
$640$	10.8585	0.429218
$641$	45.1621	1.78380	0.891899	−	0.452236i	$-0.149373\pi$
$641$	45.1621	1.78380	0.891899	+	0.452236i	$0.149373\pi$
$642$	0	0
$643$	−20.4110	−0.804930	−0.402465	−	0.915435i	$-0.631846\pi$
$643$	−20.4110	−0.804930	−0.402465	+	0.915435i	$0.631846\pi$
$644$	4.50192	0.177400
$645$	0	0
$646$	40.8229	1.60616
$647$	4.53171	0.178160	0.0890799	−	0.996024i	$-0.471607\pi$
$647$	4.53171	0.178160	0.0890799	+	0.996024i	$0.471607\pi$
$648$	0	0
$649$	3.52692	0.138444
$650$	14.5863	0.572122
$651$	0	0
$652$	−44.3200	−1.73571
$653$	13.0536	0.510827	0.255414	−	0.966832i	$-0.417788\pi$
$653$	13.0536	0.510827	0.255414	+	0.966832i	$0.417788\pi$
$654$	0	0
$655$	−8.66888	−0.338721
$656$	−14.1848	−0.553822
$657$	0	0
$658$	18.4167	0.717959
$659$	−38.2628	−1.49051	−0.745254	−	0.666780i	$-0.767672\pi$
$659$	−38.2628	−1.49051	−0.745254	+	0.666780i	$0.767672\pi$
$660$	0	0
$661$	−39.5357	−1.53776	−0.768881	−	0.639392i	$-0.779186\pi$
$661$	−39.5357	−1.53776	−0.768881	+	0.639392i	$0.779186\pi$
$662$	−8.36407	−0.325079
$663$	0	0
$664$	15.4750	0.600547
$665$	−4.34754	−0.168590
$666$	0	0
$667$	2.34604	0.0908391
$668$	−16.7745	−0.649026
$669$	0	0
$670$	13.8105	0.533545
$671$	−5.12317	−0.197778
$672$	0	0
$673$	−3.83621	−0.147875	−0.0739375	−	0.997263i	$-0.523557\pi$
$673$	−3.83621	−0.147875	−0.0739375	+	0.997263i	$0.523557\pi$
$674$	12.9645	0.499375
$675$	0	0
$676$	−31.6794	−1.21844
$677$	3.32219	0.127682	0.0638410	−	0.997960i	$-0.479665\pi$
$677$	3.32219	0.127682	0.0638410	+	0.997960i	$0.479665\pi$
$678$	0	0
$679$	−8.68801	−0.333415
$680$	−4.75393	−0.182305
$681$	0	0
$682$	2.59304	0.0992927
$683$	−22.3302	−0.854440	−0.427220	−	0.904148i	$-0.640507\pi$
$683$	−22.3302	−0.854440	−0.427220	+	0.904148i	$0.640507\pi$
$684$	0	0
$685$	4.92207	0.188063
$686$	−2.21969	−0.0847481
$687$	0	0
$688$	−2.06604	−0.0787672
$689$	17.5886	0.670071
$690$	0	0
$691$	−42.0621	−1.60012	−0.800058	−	0.599923i	$-0.795198\pi$
$691$	−42.0621	−1.60012	−0.800058	+	0.599923i	$0.795198\pi$
$692$	−35.2388	−1.33958
$693$	0	0
$694$	−68.4419	−2.59802
$695$	3.87519	0.146994
$696$	0	0
$697$	34.4661	1.30550
$698$	−74.0314	−2.80213
$699$	0	0
$700$	−13.0365	−0.492733
$701$	−40.9185	−1.54547	−0.772736	−	0.634728i	$-0.781112\pi$
$701$	−40.9185	−1.54547	−0.772736	+	0.634728i	$0.781112\pi$
$702$	0	0
$703$	65.4608	2.46890
$704$	13.1836	0.496877
$705$	0	0
$706$	28.3612	1.06739
$707$	1.74924	0.0657871
$708$	0	0
$709$	−1.63697	−0.0614777	−0.0307388	−	0.999527i	$-0.509786\pi$
$709$	−1.63697	−0.0614777	−0.0307388	+	0.999527i	$0.509786\pi$
$710$	23.4305	0.879331
$711$	0	0
$712$	−2.70021	−0.101195
$713$	1.75821	0.0658456
$714$	0	0
$715$	−1.11427	−0.0416714
$716$	23.3877	0.874039
$717$	0	0
$718$	−8.04753	−0.300331
$719$	22.2967	0.831525	0.415763	−	0.909473i	$-0.363515\pi$
$719$	22.2967	0.831525	0.415763	+	0.909473i	$0.363515\pi$
$720$	0	0
$721$	1.31728	0.0490582
$722$	−34.6454	−1.28937
$723$	0	0
$724$	−8.17181	−0.303703
$725$	−6.79358	−0.252307
$726$	0	0
$727$	11.1451	0.413349	0.206675	−	0.978410i	$-0.433736\pi$
$727$	11.1451	0.413349	0.206675	+	0.978410i	$0.433736\pi$
$728$	−3.03595	−0.112520
$729$	0	0
$730$	2.23564	0.0827449
$731$	5.02006	0.185674
$732$	0	0
$733$	−19.3381	−0.714271	−0.357135	−	0.934053i	$-0.616247\pi$
$733$	−19.3381	−0.714271	−0.357135	+	0.934053i	$0.616247\pi$
$734$	−59.7431	−2.20516
$735$	0	0
$736$	10.7222	0.395226
$737$	8.60365	0.316920
$738$	0	0
$739$	−37.7548	−1.38883	−0.694416	−	0.719574i	$-0.744337\pi$
$739$	−37.7548	−1.38883	−0.694416	+	0.719574i	$0.744337\pi$
$740$	−24.0696	−0.884817
$741$	0	0
$742$	−26.4609	−0.971410
$743$	−23.1183	−0.848127	−0.424063	−	0.905633i	$-0.639397\pi$
$743$	−23.1183	−0.848127	−0.424063	+	0.905633i	$0.639397\pi$
$744$	0	0
$745$	3.23690	0.118591
$746$	−11.4993	−0.421019
$747$	0	0
$748$	−9.35119	−0.341913
$749$	−8.82108	−0.322315
$750$	0	0
$751$	9.42007	0.343743	0.171872	−	0.985119i	$-0.445019\pi$
$751$	9.42007	0.343743	0.171872	+	0.985119i	$0.445019\pi$
$752$	10.6751	0.389282
$753$	0	0
$754$	−4.99542	−0.181922
$755$	−1.33910	−0.0487347
$756$	0	0
$757$	−17.4997	−0.636036	−0.318018	−	0.948085i	$-0.603017\pi$
$757$	−17.4997	−0.636036	−0.318018	+	0.948085i	$0.603017\pi$
$758$	44.9935	1.63424
$759$	0	0
$760$	8.94582	0.324499
$761$	−43.4638	−1.57556	−0.787781	−	0.615955i	$-0.788770\pi$
$761$	−43.4638	−1.57556	−0.787781	+	0.615955i	$0.788770\pi$
$762$	0	0
$763$	2.01285	0.0728700
$764$	12.9527	0.468612
$765$	0	0
$766$	66.0837	2.38770
$767$	−5.09205	−0.183863
$768$	0	0
$769$	−4.03307	−0.145436	−0.0727181	−	0.997353i	$-0.523167\pi$
$769$	−4.03307	−0.145436	−0.0727181	+	0.997353i	$0.523167\pi$
$770$	1.67635	0.0604115
$771$	0	0
$772$	−14.7604	−0.531238
$773$	21.5633	0.775576	0.387788	−	0.921749i	$-0.373239\pi$
$773$	21.5633	0.775576	0.387788	+	0.921749i	$0.373239\pi$
$774$	0	0
$775$	−5.09137	−0.182887
$776$	17.8771	0.641750
$777$	0	0
$778$	24.5538	0.880298
$779$	−64.8573	−2.32376
$780$	0	0
$781$	14.5968	0.522313
$782$	−10.6730	−0.381667
$783$	0	0
$784$	−1.28663	−0.0459510
$785$	−4.70229	−0.167832
$786$	0	0
$787$	22.0639	0.786495	0.393247	−	0.919433i	$-0.371352\pi$
$787$	22.0639	0.786495	0.393247	+	0.919433i	$0.371352\pi$
$788$	30.1940	1.07562
$789$	0	0
$790$	−21.1093	−0.751035
$791$	9.48735	0.337331
$792$	0	0
$793$	7.39667	0.262664
$794$	0.604742	0.0214615
$795$	0	0
$796$	−63.5754	−2.25337
$797$	−3.27403	−0.115972	−0.0579861	−	0.998317i	$-0.518468\pi$
$797$	−3.27403	−0.115972	−0.0579861	+	0.998317i	$0.518468\pi$
$798$	0	0
$799$	−25.9384	−0.917635
$800$	−31.0490	−1.09775
$801$	0	0
$802$	7.10636	0.250934
$803$	1.39276	0.0491496
$804$	0	0
$805$	1.13665	0.0400616
$806$	−3.74376	−0.131868
$807$	0	0
$808$	−3.59938	−0.126626
$809$	−6.05140	−0.212756	−0.106378	−	0.994326i	$-0.533925\pi$
$809$	−6.05140	−0.212756	−0.106378	+	0.994326i	$0.533925\pi$
$810$	0	0
$811$	−24.2878	−0.852861	−0.426430	−	0.904520i	$-0.640229\pi$
$811$	−24.2878	−0.852861	−0.426430	+	0.904520i	$0.640229\pi$
$812$	4.46464	0.156678
$813$	0	0
$814$	−25.2408	−0.884689
$815$	−11.1900	−0.391968
$816$	0	0
$817$	−9.44662	−0.330495
$818$	87.8837	3.07278
$819$	0	0
$820$	23.8477	0.832799
$821$	−39.4820	−1.37793	−0.688965	−	0.724795i	$-0.741935\pi$
$821$	−39.4820	−1.37793	−0.688965	+	0.724795i	$0.741935\pi$
$822$	0	0
$823$	−16.4296	−0.572702	−0.286351	−	0.958125i	$-0.592442\pi$
$823$	−16.4296	−0.572702	−0.286351	+	0.958125i	$0.592442\pi$
$824$	−2.71054	−0.0944262
$825$	0	0
$826$	7.66067	0.266549
$827$	−8.05879	−0.280231	−0.140116	−	0.990135i	$-0.544747\pi$
$827$	−8.05879	−0.280231	−0.140116	+	0.990135i	$0.544747\pi$
$828$	0	0
$829$	5.33531	0.185303	0.0926515	−	0.995699i	$-0.470466\pi$
$829$	5.33531	0.185303	0.0926515	+	0.995699i	$0.470466\pi$
$830$	12.3367	0.428214
$831$	0	0
$832$	−19.0341	−0.659890
$833$	3.12624	0.108318
$834$	0	0
$835$	−4.23525	−0.146567
$836$	17.5968	0.608598
$837$	0	0
$838$	−33.4666	−1.15608
$839$	−16.3489	−0.564427	−0.282214	−	0.959352i	$-0.591069\pi$
$839$	−16.3489	−0.564427	−0.282214	+	0.959352i	$0.591069\pi$
$840$	0	0
$841$	−26.6734	−0.919772
$842$	22.3237	0.769325
$843$	0	0
$844$	−29.9298	−1.03022
$845$	−7.99845	−0.275155
$846$	0	0
$847$	−9.95567	−0.342081
$848$	−15.3379	−0.526705
$849$	0	0
$850$	30.9066	1.06009
$851$	−17.1145	−0.586678
$852$	0	0
$853$	13.5673	0.464537	0.232268	−	0.972652i	$-0.425385\pi$
$853$	13.5673	0.464537	0.232268	+	0.972652i	$0.425385\pi$
$854$	−11.1278	−0.380787
$855$	0	0
$856$	18.1509	0.620385
$857$	17.2147	0.588044	0.294022	−	0.955799i	$-0.405006\pi$
$857$	17.2147	0.588044	0.294022	+	0.955799i	$0.405006\pi$
$858$	0	0
$859$	9.09943	0.310469	0.155234	−	0.987878i	$-0.450387\pi$
$859$	9.09943	0.310469	0.155234	+	0.987878i	$0.450387\pi$
$860$	3.47348	0.118445
$861$	0	0
$862$	−18.2490	−0.621564
$863$	−16.5214	−0.562395	−0.281198	−	0.959650i	$-0.590732\pi$
$863$	−16.5214	−0.562395	−0.281198	+	0.959650i	$0.590732\pi$
$864$	0	0
$865$	−8.89713	−0.302511
$866$	−3.40873	−0.115833
$867$	0	0
$868$	3.34597	0.113570
$869$	−13.1507	−0.446106
$870$	0	0
$871$	−12.4217	−0.420893
$872$	−4.14179	−0.140259
$873$	0	0
$874$	20.0842	0.679358
$875$	−6.98655	−0.236188
$876$	0	0
$877$	−34.1895	−1.15450	−0.577249	−	0.816568i	$-0.695874\pi$
$877$	−34.1895	−1.15450	−0.577249	+	0.816568i	$0.695874\pi$
$878$	−42.0709	−1.41983
$879$	0	0
$880$	0.971687	0.0327556
$881$	−42.3220	−1.42587	−0.712933	−	0.701233i	$-0.752633\pi$
$881$	−42.3220	−1.42587	−0.712933	+	0.701233i	$0.752633\pi$
$882$	0	0
$883$	−25.2040	−0.848184	−0.424092	−	0.905619i	$-0.639407\pi$
$883$	−25.2040	−0.848184	−0.424092	+	0.905619i	$0.639407\pi$
$884$	13.5010	0.454087
$885$	0	0
$886$	−69.4828	−2.33432
$887$	−1.96049	−0.0658269	−0.0329135	−	0.999458i	$-0.510479\pi$
$887$	−1.96049	−0.0658269	−0.0329135	+	0.999458i	$0.510479\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−2.15261	−0.0721558
$891$	0	0
$892$	10.5307	0.352594
$893$	48.8102	1.63337
$894$	0	0
$895$	5.90496	0.197381
$896$	14.6931	0.490863
$897$	0	0
$898$	−27.9701	−0.933374
$899$	1.74365	0.0581541
$900$	0	0
$901$	37.2679	1.24158
$902$	25.0081	0.832679
$903$	0	0
$904$	−19.5219	−0.649288
$905$	−2.06323	−0.0685840
$906$	0	0
$907$	−4.81708	−0.159948	−0.0799742	−	0.996797i	$-0.525484\pi$
$907$	−4.81708	−0.159948	−0.0799742	+	0.996797i	$0.525484\pi$
$908$	−33.8460	−1.12322
$909$	0	0
$910$	−2.42027	−0.0802310
$911$	−9.61550	−0.318576	−0.159288	−	0.987232i	$-0.550920\pi$
$911$	−9.61550	−0.318576	−0.159288	+	0.987232i	$0.550920\pi$
$912$	0	0
$913$	7.68554	0.254354
$914$	−15.6259	−0.516860
$915$	0	0
$916$	−0.0287005	−0.000948291 0
$917$	−11.7303	−0.387369
$918$	0	0
$919$	−2.91835	−0.0962674	−0.0481337	−	0.998841i	$-0.515327\pi$
$919$	−2.91835	−0.0962674	−0.0481337	+	0.998841i	$0.515327\pi$
$920$	−2.33886	−0.0771098
$921$	0	0
$922$	−49.8021	−1.64015
$923$	−21.0743	−0.693671
$924$	0	0
$925$	49.5596	1.62951
$926$	73.1580	2.40412
$927$	0	0
$928$	10.6334	0.349060
$929$	−2.62004	−0.0859606	−0.0429803	−	0.999076i	$-0.513685\pi$
$929$	−2.62004	−0.0859606	−0.0429803	+	0.999076i	$0.513685\pi$
$930$	0	0
$931$	−5.88288	−0.192803
$932$	12.5121	0.409849
$933$	0	0
$934$	78.4807	2.56797
$935$	−2.36100	−0.0772130
$936$	0	0
$937$	−41.5944	−1.35883	−0.679415	−	0.733754i	$-0.737766\pi$
$937$	−41.5944	−1.35883	−0.679415	+	0.733754i	$0.737766\pi$
$938$	18.6877	0.610174
$939$	0	0
$940$	−17.9473	−0.585376
$941$	−45.9996	−1.49954	−0.749771	−	0.661697i	$-0.769837\pi$
$941$	−45.9996	−1.49954	−0.749771	+	0.661697i	$0.769837\pi$
$942$	0	0
$943$	16.9567	0.552188
$944$	4.44046	0.144525
$945$	0	0
$946$	3.64249	0.118427
$947$	54.4686	1.76999	0.884996	−	0.465599i	$-0.154161\pi$
$947$	54.4686	1.76999	0.884996	+	0.465599i	$0.154161\pi$
$948$	0	0
$949$	−2.01083	−0.0652743
$950$	−58.1591	−1.88693
$951$	0	0
$952$	−6.43279	−0.208488
$953$	8.27052	0.267909	0.133954	−	0.990988i	$-0.457232\pi$
$953$	8.27052	0.267909	0.133954	+	0.990988i	$0.457232\pi$
$954$	0	0
$955$	3.27031	0.105825
$956$	40.0509	1.29534
$957$	0	0
$958$	23.3006	0.752808
$959$	6.66031	0.215073
$960$	0	0
$961$	−29.6932	−0.957846
$962$	36.4419	1.17493
$963$	0	0
$964$	−38.8827	−1.25233
$965$	−3.72672	−0.119967
$966$	0	0
$967$	−54.6520	−1.75749	−0.878745	−	0.477291i	$-0.841619\pi$
$967$	−54.6520	−1.75749	−0.878745	+	0.477291i	$0.841619\pi$
$968$	20.4855	0.658430
$969$	0	0
$970$	14.2517	0.457593
$971$	−50.6336	−1.62491	−0.812455	−	0.583023i	$-0.801870\pi$
$971$	−50.6336	−1.62491	−0.812455	+	0.583023i	$0.801870\pi$
$972$	0	0
$973$	5.24371	0.168106
$974$	53.9154	1.72756
$975$	0	0
$976$	−6.45018	−0.206465
$977$	49.0318	1.56867	0.784334	−	0.620339i	$-0.213005\pi$
$977$	49.0318	1.56867	0.784334	+	0.620339i	$0.213005\pi$
$978$	0	0
$979$	−1.34104	−0.0428597
$980$	2.16311	0.0690979
$981$	0	0
$982$	−18.9817	−0.605729
$983$	53.0129	1.69085	0.845425	−	0.534094i	$-0.179347\pi$
$983$	53.0129	1.69085	0.845425	+	0.534094i	$0.179347\pi$
$984$	0	0
$985$	7.62341	0.242902
$986$	−10.5847	−0.337084
$987$	0	0
$988$	−25.4057	−0.808264
$989$	2.46979	0.0785347
$990$	0	0
$991$	−51.3451	−1.63103	−0.815516	−	0.578735i	$-0.803547\pi$
$991$	−51.3451	−1.63103	−0.815516	+	0.578735i	$0.803547\pi$
$992$	7.96910	0.253019
$993$	0	0
$994$	31.7050	1.00562
$995$	−16.0516	−0.508870
$996$	0	0
$997$	46.0459	1.45829	0.729144	−	0.684360i	$-0.239919\pi$
$997$	46.0459	1.45829	0.729144	+	0.684360i	$0.239919\pi$
$998$	−61.3560	−1.94219
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.3	✓	22
3.2	odd	2	inner	8001.2.a.x.1.20	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.3	✓	22	1.1	even	1	trivial
8001.2.a.x.1.20	yes	22	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)

\(f(q)\)	\(=\)	\(q-2.21969 q^{2} +2.92701 q^{4} +0.739015 q^{5} +1.00000 q^{7} -2.05767 q^{8} +O(q^{10})\)	\(q-2.21969 q^{2} +2.92701 q^{4} +0.739015 q^{5} +1.00000 q^{7} -2.05767 q^{8} -1.64038 q^{10} -1.02193 q^{11} +1.47543 q^{13} -2.21969 q^{14} -1.28663 q^{16} +3.12624 q^{17} -5.88288 q^{19} +2.16311 q^{20} +2.26836 q^{22} +1.53806 q^{23} -4.45386 q^{25} -3.27499 q^{26} +2.92701 q^{28} +1.52533 q^{29} +1.14314 q^{31} +6.97126 q^{32} -6.93928 q^{34} +0.739015 q^{35} -11.1273 q^{37} +13.0581 q^{38} -1.52065 q^{40} +11.0248 q^{41} +1.60578 q^{43} -2.99119 q^{44} -3.41401 q^{46} -8.29699 q^{47} +1.00000 q^{49} +9.88617 q^{50} +4.31859 q^{52} +11.9210 q^{53} -0.755220 q^{55} -2.05767 q^{56} -3.38574 q^{58} -3.45124 q^{59} +5.01324 q^{61} -2.53741 q^{62} -12.9008 q^{64} +1.09036 q^{65} -8.41905 q^{67} +9.15055 q^{68} -1.64038 q^{70} -14.2836 q^{71} -1.36288 q^{73} +24.6992 q^{74} -17.2192 q^{76} -1.02193 q^{77} +12.8685 q^{79} -0.950838 q^{80} -24.4715 q^{82} -7.52064 q^{83} +2.31034 q^{85} -3.56433 q^{86} +2.10279 q^{88} +1.31226 q^{89} +1.47543 q^{91} +4.50192 q^{92} +18.4167 q^{94} -4.34754 q^{95} -8.68801 q^{97} -2.21969 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) 22 * q + 10 * q^4 + 22 * q^7	\( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) \) 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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