LMFDB - Embedded newform 8001.2.a.x.1.15

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			1.15
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)

$f(q)$	$=$	$q+0.691771 q^{2} -1.52145 q^{4} +0.236679 q^{5} +1.00000 q^{7} -2.43604 q^{8} +O(q^{10})$	\(q+0.691771 q^{2} -1.52145 q^{4} +0.236679 q^{5} +1.00000 q^{7} -2.43604 q^{8} +0.163728 q^{10} +4.73037 q^{11} -1.51889 q^{13} +0.691771 q^{14} +1.35773 q^{16} -4.88453 q^{17} +2.13777 q^{19} -0.360096 q^{20} +3.27233 q^{22} -1.62008 q^{23} -4.94398 q^{25} -1.05072 q^{26} -1.52145 q^{28} +0.481543 q^{29} -9.10709 q^{31} +5.81131 q^{32} -3.37897 q^{34} +0.236679 q^{35} +10.9544 q^{37} +1.47885 q^{38} -0.576559 q^{40} +4.78089 q^{41} +1.26680 q^{43} -7.19704 q^{44} -1.12073 q^{46} -6.43162 q^{47} +1.00000 q^{49} -3.42010 q^{50} +2.31092 q^{52} -11.2045 q^{53} +1.11958 q^{55} -2.43604 q^{56} +0.333117 q^{58} +7.68081 q^{59} +5.26239 q^{61} -6.30002 q^{62} +1.30465 q^{64} -0.359489 q^{65} -15.1697 q^{67} +7.43158 q^{68} +0.163728 q^{70} -2.66621 q^{71} +10.5894 q^{73} +7.57791 q^{74} -3.25252 q^{76} +4.73037 q^{77} -13.1527 q^{79} +0.321345 q^{80} +3.30728 q^{82} +11.1108 q^{83} -1.15606 q^{85} +0.876336 q^{86} -11.5234 q^{88} -11.0913 q^{89} -1.51889 q^{91} +2.46488 q^{92} -4.44921 q^{94} +0.505966 q^{95} -9.66578 q^{97} +0.691771 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.691771	0.489156	0.244578	−	0.969630i	$-0.421351\pi$
$2$	0.691771	0.489156	0.244578	+	0.969630i	$0.421351\pi$
$3$	0	0
$3$	0	0
$4$	−1.52145	−0.760727
$5$	0.236679	0.105846	0.0529230	−	0.998599i	$-0.483146\pi$
$5$	0.236679	0.105846	0.0529230	+	0.998599i	$0.483146\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.43604	−0.861270
$9$	0	0
$10$	0.163728	0.0517752
$11$	4.73037	1.42626	0.713130	−	0.701032i	$-0.247277\pi$
$11$	4.73037	1.42626	0.713130	+	0.701032i	$0.247277\pi$
$12$	0	0
$13$	−1.51889	−0.421264	−0.210632	−	0.977565i	$-0.567552\pi$
$13$	−1.51889	−0.421264	−0.210632	+	0.977565i	$0.567552\pi$
$14$	0.691771	0.184884
$15$	0	0
$16$	1.35773	0.339431
$17$	−4.88453	−1.18467	−0.592336	−	0.805691i	$-0.701794\pi$
$17$	−4.88453	−1.18467	−0.592336	+	0.805691i	$0.701794\pi$
$18$	0	0
$19$	2.13777	0.490439	0.245219	−	0.969468i	$-0.421140\pi$
$19$	2.13777	0.490439	0.245219	+	0.969468i	$0.421140\pi$
$20$	−0.360096	−0.0805199
$21$	0	0
$22$	3.27233	0.697664
$23$	−1.62008	−0.337811	−0.168905	−	0.985632i	$-0.554023\pi$
$23$	−1.62008	−0.337811	−0.168905	+	0.985632i	$0.554023\pi$
$24$	0	0
$25$	−4.94398	−0.988797
$26$	−1.05072	−0.206064
$27$	0	0
$28$	−1.52145	−0.287528
$29$	0.481543	0.0894203	0.0447101	−	0.999000i	$-0.485764\pi$
$29$	0.481543	0.0894203	0.0447101	+	0.999000i	$0.485764\pi$
$30$	0	0
$31$	−9.10709	−1.63568	−0.817841	−	0.575445i	$-0.804829\pi$
$31$	−9.10709	−1.63568	−0.817841	+	0.575445i	$0.804829\pi$
$32$	5.81131	1.02730
$33$	0	0
$34$	−3.37897	−0.579489
$35$	0.236679	0.0400060
$36$	0	0
$37$	10.9544	1.80089	0.900443	−	0.434974i	$-0.143242\pi$
$37$	10.9544	1.80089	0.900443	+	0.434974i	$0.143242\pi$
$38$	1.47885	0.239901
$39$	0	0
$40$	−0.576559	−0.0911620
$41$	4.78089	0.746650	0.373325	−	0.927701i	$-0.378218\pi$
$41$	4.78089	0.746650	0.373325	+	0.927701i	$0.378218\pi$
$42$	0	0
$43$	1.26680	0.193185	0.0965927	−	0.995324i	$-0.469206\pi$
$43$	1.26680	0.193185	0.0965927	+	0.995324i	$0.469206\pi$
$44$	−7.19704	−1.08499
$45$	0	0
$46$	−1.12073	−0.165242
$47$	−6.43162	−0.938148	−0.469074	−	0.883159i	$-0.655412\pi$
$47$	−6.43162	−0.938148	−0.469074	+	0.883159i	$0.655412\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.42010	−0.483676
$51$	0	0
$52$	2.31092	0.320467
$53$	−11.2045	−1.53906	−0.769531	−	0.638610i	$-0.779510\pi$
$53$	−11.2045	−1.53906	−0.769531	+	0.638610i	$0.779510\pi$
$54$	0	0
$55$	1.11958	0.150964
$56$	−2.43604	−0.325529
$57$	0	0
$58$	0.333117	0.0437405
$59$	7.68081	0.999956	0.499978	−	0.866038i	$-0.333341\pi$
$59$	7.68081	0.999956	0.499978	+	0.866038i	$0.333341\pi$
$60$	0	0
$61$	5.26239	0.673780	0.336890	−	0.941544i	$-0.390625\pi$
$61$	5.26239	0.673780	0.336890	+	0.941544i	$0.390625\pi$
$62$	−6.30002	−0.800103
$63$	0	0
$64$	1.30465	0.163081
$65$	−0.359489	−0.0445891
$66$	0	0
$67$	−15.1697	−1.85327	−0.926635	−	0.375963i	$-0.877312\pi$
$67$	−15.1697	−1.85327	−0.926635	+	0.375963i	$0.877312\pi$
$68$	7.43158	0.901211
$69$	0	0
$70$	0.163728	0.0195692
$71$	−2.66621	−0.316421	−0.158210	−	0.987405i	$-0.550572\pi$
$71$	−2.66621	−0.316421	−0.158210	+	0.987405i	$0.550572\pi$
$72$	0	0
$73$	10.5894	1.23939	0.619695	−	0.784843i	$-0.287256\pi$
$73$	10.5894	1.23939	0.619695	+	0.784843i	$0.287256\pi$
$74$	7.57791	0.880914
$75$	0	0
$76$	−3.25252	−0.373090
$77$	4.73037	0.539076
$78$	0	0
$79$	−13.1527	−1.47979	−0.739895	−	0.672722i	$-0.765125\pi$
$79$	−13.1527	−1.47979	−0.739895	+	0.672722i	$0.765125\pi$
$80$	0.321345	0.0359275
$81$	0	0
$82$	3.30728	0.365228
$83$	11.1108	1.21957	0.609783	−	0.792568i	$-0.291257\pi$
$83$	11.1108	1.21957	0.609783	+	0.792568i	$0.291257\pi$
$84$	0	0
$85$	−1.15606	−0.125393
$86$	0.876336	0.0944978
$87$	0	0
$88$	−11.5234	−1.22839
$89$	−11.0913	−1.17568	−0.587838	−	0.808979i	$-0.700021\pi$
$89$	−11.0913	−1.17568	−0.587838	+	0.808979i	$0.700021\pi$
$90$	0	0
$91$	−1.51889	−0.159223
$92$	2.46488	0.256982
$93$	0	0
$94$	−4.44921	−0.458901
$95$	0.505966	0.0519110
$96$	0	0
$97$	−9.66578	−0.981411	−0.490706	−	0.871325i	$-0.663261\pi$
$97$	−9.66578	−0.981411	−0.490706	+	0.871325i	$0.663261\pi$
$98$	0.691771	0.0698794
$99$	0	0
$100$	7.52204	0.752204
$101$	−13.6540	−1.35863	−0.679314	−	0.733848i	$-0.737723\pi$
$101$	−13.6540	−1.35863	−0.679314	+	0.733848i	$0.737723\pi$
$102$	0	0
$103$	6.04299	0.595434	0.297717	−	0.954654i	$-0.403775\pi$
$103$	6.04299	0.595434	0.297717	+	0.954654i	$0.403775\pi$
$104$	3.70007	0.362822
$105$	0	0
$106$	−7.75097	−0.752841
$107$	6.94638	0.671532	0.335766	−	0.941945i	$-0.391005\pi$
$107$	6.94638	0.671532	0.335766	+	0.941945i	$0.391005\pi$
$108$	0	0
$109$	−7.77730	−0.744930	−0.372465	−	0.928046i	$-0.621487\pi$
$109$	−7.77730	−0.744930	−0.372465	+	0.928046i	$0.621487\pi$
$110$	0.774492	0.0738449
$111$	0	0
$112$	1.35773	0.128293
$113$	−5.69588	−0.535823	−0.267911	−	0.963444i	$-0.586333\pi$
$113$	−5.69588	−0.535823	−0.267911	+	0.963444i	$0.586333\pi$
$114$	0	0
$115$	−0.383440	−0.0357559
$116$	−0.732645	−0.0680244
$117$	0	0
$118$	5.31336	0.489134
$119$	−4.88453	−0.447764
$120$	0	0
$121$	11.3764	1.03422
$122$	3.64037	0.329583
$123$	0	0
$124$	13.8560	1.24431
$125$	−2.35353	−0.210506
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−10.7201	−0.947533
$129$	0	0
$130$	−0.248684	−0.0218110
$131$	2.77660	0.242593	0.121296	−	0.992616i	$-0.461295\pi$
$131$	2.77660	0.242593	0.121296	+	0.992616i	$0.461295\pi$
$132$	0	0
$133$	2.13777	0.185368
$134$	−10.4939	−0.906538
$135$	0	0
$136$	11.8989	1.02032
$137$	5.85349	0.500097	0.250048	−	0.968233i	$-0.419553\pi$
$137$	5.85349	0.500097	0.250048	+	0.968233i	$0.419553\pi$
$138$	0	0
$139$	12.8728	1.09186	0.545929	−	0.837832i	$-0.316177\pi$
$139$	12.8728	1.09186	0.545929	+	0.837832i	$0.316177\pi$
$140$	−0.360096	−0.0304337
$141$	0	0
$142$	−1.84440	−0.154779
$143$	−7.18490	−0.600832
$144$	0	0
$145$	0.113971	0.00946478
$146$	7.32541	0.606255
$147$	0	0
$148$	−16.6665	−1.36998
$149$	−22.2004	−1.81873	−0.909364	−	0.416001i	$-0.863431\pi$
$149$	−22.2004	−1.81873	−0.909364	+	0.416001i	$0.863431\pi$
$150$	0	0
$151$	13.5226	1.10045	0.550225	−	0.835016i	$-0.314542\pi$
$151$	13.5226	1.10045	0.550225	+	0.835016i	$0.314542\pi$
$152$	−5.20770	−0.422400
$153$	0	0
$154$	3.27233	0.263692
$155$	−2.15546	−0.173130
$156$	0	0
$157$	1.03369	0.0824973	0.0412487	−	0.999149i	$-0.486866\pi$
$157$	1.03369	0.0824973	0.0412487	+	0.999149i	$0.486866\pi$
$158$	−9.09863	−0.723848
$159$	0	0
$160$	1.37542	0.108736
$161$	−1.62008	−0.127680
$162$	0	0
$163$	−21.0280	−1.64704	−0.823519	−	0.567289i	$-0.807992\pi$
$163$	−21.0280	−1.64704	−0.823519	+	0.567289i	$0.807992\pi$
$164$	−7.27391	−0.567997
$165$	0	0
$166$	7.68611	0.596558
$167$	18.2548	1.41260	0.706300	−	0.707913i	$-0.250363\pi$
$167$	18.2548	1.41260	0.706300	+	0.707913i	$0.250363\pi$
$168$	0	0
$169$	−10.6930	−0.822537
$170$	−0.799732	−0.0613366
$171$	0	0
$172$	−1.92738	−0.146961
$173$	10.4129	0.791679	0.395840	−	0.918320i	$-0.370454\pi$
$173$	10.4129	0.791679	0.395840	+	0.918320i	$0.370454\pi$
$174$	0	0
$175$	−4.94398	−0.373730
$176$	6.42254	0.484117
$177$	0	0
$178$	−7.67264	−0.575089
$179$	5.46552	0.408513	0.204256	−	0.978917i	$-0.434522\pi$
$179$	5.46552	0.408513	0.204256	+	0.978917i	$0.434522\pi$
$180$	0	0
$181$	2.91504	0.216673	0.108337	−	0.994114i	$-0.465448\pi$
$181$	2.91504	0.216673	0.108337	+	0.994114i	$0.465448\pi$
$182$	−1.05072	−0.0778848
$183$	0	0
$184$	3.94659	0.290946
$185$	2.59267	0.190617
$186$	0	0
$187$	−23.1056	−1.68965
$188$	9.78540	0.713674
$189$	0	0
$190$	0.350012	0.0253926
$191$	−4.76963	−0.345118	−0.172559	−	0.984999i	$-0.555204\pi$
$191$	−4.76963	−0.345118	−0.172559	+	0.984999i	$0.555204\pi$
$192$	0	0
$193$	0.130634	0.00940327	0.00470163	−	0.999989i	$-0.498503\pi$
$193$	0.130634	0.00940327	0.00470163	+	0.999989i	$0.498503\pi$
$194$	−6.68651	−0.480063
$195$	0	0
$196$	−1.52145	−0.108675
$197$	−19.2367	−1.37056	−0.685280	−	0.728279i	$-0.740320\pi$
$197$	−19.2367	−1.37056	−0.685280	+	0.728279i	$0.740320\pi$
$198$	0	0
$199$	−11.8797	−0.842128	−0.421064	−	0.907031i	$-0.638343\pi$
$199$	−11.8797	−0.842128	−0.421064	+	0.907031i	$0.638343\pi$
$200$	12.0437	0.851621
$201$	0	0
$202$	−9.44547	−0.664581
$203$	0.481543	0.0337977
$204$	0	0
$205$	1.13154	0.0790300
$206$	4.18037	0.291260
$207$	0	0
$208$	−2.06223	−0.142990
$209$	10.1125	0.699493
$210$	0	0
$211$	14.3794	0.989915	0.494958	−	0.868917i	$-0.335184\pi$
$211$	14.3794	0.989915	0.494958	+	0.868917i	$0.335184\pi$
$212$	17.0472	1.17080
$213$	0	0
$214$	4.80531	0.328484
$215$	0.299825	0.0204479
$216$	0	0
$217$	−9.10709	−0.618229
$218$	−5.38011	−0.364387
$219$	0	0
$220$	−1.70339	−0.114842
$221$	7.41905	0.499059
$222$	0	0
$223$	−8.11238	−0.543245	−0.271623	−	0.962404i	$-0.587560\pi$
$223$	−8.11238	−0.543245	−0.271623	+	0.962404i	$0.587560\pi$
$224$	5.81131	0.388285
$225$	0	0
$226$	−3.94024	−0.262101
$227$	−24.9231	−1.65420	−0.827101	−	0.562054i	$-0.810011\pi$
$227$	−24.9231	−1.65420	−0.827101	+	0.562054i	$0.810011\pi$
$228$	0	0
$229$	−9.85102	−0.650974	−0.325487	−	0.945547i	$-0.605528\pi$
$229$	−9.85102	−0.650974	−0.325487	+	0.945547i	$0.605528\pi$
$230$	−0.265252	−0.0174902
$231$	0	0
$232$	−1.17306	−0.0770150
$233$	27.7502	1.81797	0.908987	−	0.416823i	$-0.136857\pi$
$233$	27.7502	1.81797	0.908987	+	0.416823i	$0.136857\pi$
$234$	0	0
$235$	−1.52223	−0.0992992
$236$	−11.6860	−0.760693
$237$	0	0
$238$	−3.37897	−0.219026
$239$	−0.812448	−0.0525529	−0.0262765	−	0.999655i	$-0.508365\pi$
$239$	−0.812448	−0.0525529	−0.0262765	+	0.999655i	$0.508365\pi$
$240$	0	0
$241$	−5.39668	−0.347631	−0.173815	−	0.984778i	$-0.555610\pi$
$241$	−5.39668	−0.347631	−0.173815	+	0.984778i	$0.555610\pi$
$242$	7.86986	0.505894
$243$	0	0
$244$	−8.00648	−0.512562
$245$	0.236679	0.0151209
$246$	0	0
$247$	−3.24704	−0.206604
$248$	22.1852	1.40876
$249$	0	0
$250$	−1.62810	−0.102970
$251$	13.4081	0.846314	0.423157	−	0.906056i	$-0.360922\pi$
$251$	13.4081	0.846314	0.423157	+	0.906056i	$0.360922\pi$
$252$	0	0
$253$	−7.66359	−0.481806
$254$	0.691771	0.0434056
$255$	0	0
$256$	−10.0252	−0.626572
$257$	23.1837	1.44616	0.723079	−	0.690765i	$-0.242726\pi$
$257$	23.1837	1.44616	0.723079	+	0.690765i	$0.242726\pi$
$258$	0	0
$259$	10.9544	0.680671
$260$	0.546946	0.0339201
$261$	0	0
$262$	1.92077	0.118666
$263$	−29.6493	−1.82825	−0.914126	−	0.405430i	$-0.867122\pi$
$263$	−29.6493	−1.82825	−0.914126	+	0.405430i	$0.867122\pi$
$264$	0	0
$265$	−2.65188	−0.162904
$266$	1.47885	0.0906741
$267$	0	0
$268$	23.0799	1.40983
$269$	−12.4588	−0.759624	−0.379812	−	0.925064i	$-0.624011\pi$
$269$	−12.4588	−0.759624	−0.379812	+	0.925064i	$0.624011\pi$
$270$	0	0
$271$	−7.16036	−0.434961	−0.217481	−	0.976065i	$-0.569784\pi$
$271$	−7.16036	−0.434961	−0.217481	+	0.976065i	$0.569784\pi$
$272$	−6.63185	−0.402115
$273$	0	0
$274$	4.04927	0.244625
$275$	−23.3869	−1.41028
$276$	0	0
$277$	−29.1691	−1.75260	−0.876299	−	0.481767i	$-0.839995\pi$
$277$	−29.1691	−1.75260	−0.876299	+	0.481767i	$0.839995\pi$
$278$	8.90504	0.534089
$279$	0	0
$280$	−0.576559	−0.0344560
$281$	−2.42921	−0.144914	−0.0724572	−	0.997372i	$-0.523084\pi$
$281$	−2.42921	−0.144914	−0.0724572	+	0.997372i	$0.523084\pi$
$282$	0	0
$283$	−13.0816	−0.777621	−0.388810	−	0.921318i	$-0.627114\pi$
$283$	−13.0816	−0.777621	−0.388810	+	0.921318i	$0.627114\pi$
$284$	4.05651	0.240710
$285$	0	0
$286$	−4.97031	−0.293900
$287$	4.78089	0.282207
$288$	0	0
$289$	6.85859	0.403447
$290$	0.0788419	0.00462975
$291$	0	0
$292$	−16.1112	−0.942837
$293$	11.9508	0.698174	0.349087	−	0.937090i	$-0.386492\pi$
$293$	11.9508	0.698174	0.349087	+	0.937090i	$0.386492\pi$
$294$	0	0
$295$	1.81788	0.105841
$296$	−26.6852	−1.55105
$297$	0	0
$298$	−15.3576	−0.889642
$299$	2.46073	0.142307
$300$	0	0
$301$	1.26680	0.0730172
$302$	9.35451	0.538292
$303$	0	0
$304$	2.90251	0.166470
$305$	1.24550	0.0713169
$306$	0	0
$307$	−33.3233	−1.90186	−0.950931	−	0.309402i	$-0.899871\pi$
$307$	−33.3233	−1.90186	−0.950931	+	0.309402i	$0.899871\pi$
$308$	−7.19704	−0.410089
$309$	0	0
$310$	−1.49108	−0.0846877
$311$	−17.5249	−0.993747	−0.496874	−	0.867823i	$-0.665519\pi$
$311$	−17.5249	−0.993747	−0.496874	+	0.867823i	$0.665519\pi$
$312$	0	0
$313$	−10.4399	−0.590099	−0.295050	−	0.955482i	$-0.595336\pi$
$313$	−10.4399	−0.590099	−0.295050	+	0.955482i	$0.595336\pi$
$314$	0.715076	0.0403540
$315$	0	0
$316$	20.0112	1.12572
$317$	−17.5619	−0.986377	−0.493188	−	0.869923i	$-0.664169\pi$
$317$	−17.5619	−0.986377	−0.493188	+	0.869923i	$0.664169\pi$
$318$	0	0
$319$	2.27788	0.127537
$320$	0.308782	0.0172614
$321$	0	0
$322$	−1.12073	−0.0624556
$323$	−10.4420	−0.581009
$324$	0	0
$325$	7.50936	0.416544
$326$	−14.5465	−0.805658
$327$	0	0
$328$	−11.6464	−0.643067
$329$	−6.43162	−0.354587
$330$	0	0
$331$	−23.3554	−1.28373	−0.641865	−	0.766818i	$-0.721839\pi$
$331$	−23.3554	−1.28373	−0.641865	+	0.766818i	$0.721839\pi$
$332$	−16.9045	−0.927756
$333$	0	0
$334$	12.6281	0.690981
$335$	−3.59034	−0.196161
$336$	0	0
$337$	−4.90547	−0.267218	−0.133609	−	0.991034i	$-0.542657\pi$
$337$	−4.90547	−0.267218	−0.133609	+	0.991034i	$0.542657\pi$
$338$	−7.39709	−0.402349
$339$	0	0
$340$	1.75890	0.0953896
$341$	−43.0799	−2.33291
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.08598	−0.166385
$345$	0	0
$346$	7.20335	0.387255
$347$	−10.4466	−0.560805	−0.280403	−	0.959882i	$-0.590468\pi$
$347$	−10.4466	−0.560805	−0.280403	+	0.959882i	$0.590468\pi$
$348$	0	0
$349$	−19.3334	−1.03489	−0.517446	−	0.855716i	$-0.673117\pi$
$349$	−19.3334	−1.03489	−0.517446	+	0.855716i	$0.673117\pi$
$350$	−3.42010	−0.182812
$351$	0	0
$352$	27.4897	1.46520
$353$	4.03886	0.214967	0.107483	−	0.994207i	$-0.465721\pi$
$353$	4.03886	0.214967	0.107483	+	0.994207i	$0.465721\pi$
$354$	0	0
$355$	−0.631035	−0.0334919
$356$	16.8749	0.894368
$357$	0	0
$358$	3.78089	0.199826
$359$	0.855429	0.0451478	0.0225739	−	0.999745i	$-0.492814\pi$
$359$	0.855429	0.0451478	0.0225739	+	0.999745i	$0.492814\pi$
$360$	0	0
$361$	−14.4299	−0.759470
$362$	2.01654	0.105987
$363$	0	0
$364$	2.31092	0.121125
$365$	2.50628	0.131185
$366$	0	0
$367$	24.7046	1.28957	0.644784	−	0.764365i	$-0.276947\pi$
$367$	24.7046	1.28957	0.644784	+	0.764365i	$0.276947\pi$
$368$	−2.19963	−0.114664
$369$	0	0
$370$	1.79353	0.0932412
$371$	−11.2045	−0.581711
$372$	0	0
$373$	0.282770	0.0146413	0.00732065	−	0.999973i	$-0.497670\pi$
$373$	0.282770	0.0146413	0.00732065	+	0.999973i	$0.497670\pi$
$374$	−15.9838	−0.826502
$375$	0	0
$376$	15.6677	0.807998
$377$	−0.731410	−0.0376695
$378$	0	0
$379$	−12.3555	−0.634657	−0.317329	−	0.948316i	$-0.602786\pi$
$379$	−12.3555	−0.634657	−0.317329	+	0.948316i	$0.602786\pi$
$380$	−0.769803	−0.0394901
$381$	0	0
$382$	−3.29949	−0.168817
$383$	22.1873	1.13372	0.566859	−	0.823815i	$-0.308159\pi$
$383$	22.1873	1.13372	0.566859	+	0.823815i	$0.308159\pi$
$384$	0	0
$385$	1.11958	0.0570590
$386$	0.0903691	0.00459966
$387$	0	0
$388$	14.7060	0.746586
$389$	−27.5859	−1.39866	−0.699331	−	0.714798i	$-0.746519\pi$
$389$	−27.5859	−1.39866	−0.699331	+	0.714798i	$0.746519\pi$
$390$	0	0
$391$	7.91334	0.400195
$392$	−2.43604	−0.123039
$393$	0	0
$394$	−13.3074	−0.670418
$395$	−3.11296	−0.156630
$396$	0	0
$397$	−34.4752	−1.73026	−0.865130	−	0.501547i	$-0.832764\pi$
$397$	−34.4752	−1.73026	−0.865130	+	0.501547i	$0.832764\pi$
$398$	−8.21801	−0.411932
$399$	0	0
$400$	−6.71257	−0.335629
$401$	−18.3464	−0.916174	−0.458087	−	0.888907i	$-0.651465\pi$
$401$	−18.3464	−0.916174	−0.458087	+	0.888907i	$0.651465\pi$
$402$	0	0
$403$	13.8327	0.689053
$404$	20.7740	1.03354
$405$	0	0
$406$	0.333117	0.0165323
$407$	51.8182	2.56853
$408$	0	0
$409$	−11.2920	−0.558355	−0.279178	−	0.960239i	$-0.590062\pi$
$409$	−11.2920	−0.558355	−0.279178	+	0.960239i	$0.590062\pi$
$410$	0.782764	0.0386580
$411$	0	0
$412$	−9.19413	−0.452962
$413$	7.68081	0.377948
$414$	0	0
$415$	2.62969	0.129086
$416$	−8.82674	−0.432766
$417$	0	0
$418$	6.99550	0.342161
$419$	4.64156	0.226755	0.113377	−	0.993552i	$-0.463833\pi$
$419$	4.64156	0.226755	0.113377	+	0.993552i	$0.463833\pi$
$420$	0	0
$421$	−18.1742	−0.885756	−0.442878	−	0.896582i	$-0.646042\pi$
$421$	−18.1742	−0.885756	−0.442878	+	0.896582i	$0.646042\pi$
$422$	9.94722	0.484223
$423$	0	0
$424$	27.2947	1.32555
$425$	24.1490	1.17140
$426$	0	0
$427$	5.26239	0.254665
$428$	−10.5686	−0.510852
$429$	0	0
$430$	0.207410	0.0100022
$431$	13.9712	0.672969	0.336485	−	0.941689i	$-0.390762\pi$
$431$	13.9712	0.672969	0.336485	+	0.941689i	$0.390762\pi$
$432$	0	0
$433$	−1.48040	−0.0711437	−0.0355719	−	0.999367i	$-0.511325\pi$
$433$	−1.48040	−0.0711437	−0.0355719	+	0.999367i	$0.511325\pi$
$434$	−6.30002	−0.302411
$435$	0	0
$436$	11.8328	0.566688
$437$	−3.46337	−0.165675
$438$	0	0
$439$	−40.9084	−1.95245	−0.976225	−	0.216758i	$-0.930452\pi$
$439$	−40.9084	−1.95245	−0.976225	+	0.216758i	$0.930452\pi$
$440$	−2.72734	−0.130021
$441$	0	0
$442$	5.13228	0.244118
$443$	2.15526	0.102400	0.0511998	−	0.998688i	$-0.483695\pi$
$443$	2.15526	0.102400	0.0511998	+	0.998688i	$0.483695\pi$
$444$	0	0
$445$	−2.62508	−0.124441
$446$	−5.61191	−0.265732
$447$	0	0
$448$	1.30465	0.0616387
$449$	1.25094	0.0590356	0.0295178	−	0.999564i	$-0.490603\pi$
$449$	1.25094	0.0590356	0.0295178	+	0.999564i	$0.490603\pi$
$450$	0	0
$451$	22.6154	1.06492
$452$	8.66601	0.407615
$453$	0	0
$454$	−17.2410	−0.809162
$455$	−0.359489	−0.0168531
$456$	0	0
$457$	−9.33789	−0.436808	−0.218404	−	0.975858i	$-0.570085\pi$
$457$	−9.33789	−0.436808	−0.218404	+	0.975858i	$0.570085\pi$
$458$	−6.81465	−0.318428
$459$	0	0
$460$	0.583385	0.0272005
$461$	−9.40350	−0.437965	−0.218982	−	0.975729i	$-0.570274\pi$
$461$	−9.40350	−0.437965	−0.218982	+	0.975729i	$0.570274\pi$
$462$	0	0
$463$	35.8889	1.66790	0.833950	−	0.551840i	$-0.186074\pi$
$463$	35.8889	1.66790	0.833950	+	0.551840i	$0.186074\pi$
$464$	0.653803	0.0303520
$465$	0	0
$466$	19.1968	0.889273
$467$	−9.27065	−0.428994	−0.214497	−	0.976725i	$-0.568811\pi$
$467$	−9.27065	−0.428994	−0.214497	+	0.976725i	$0.568811\pi$
$468$	0	0
$469$	−15.1697	−0.700470
$470$	−1.05303	−0.0485728
$471$	0	0
$472$	−18.7107	−0.861232
$473$	5.99244	0.275533
$474$	0	0
$475$	−10.5691	−0.484944
$476$	7.43158	0.340626
$477$	0	0
$478$	−0.562028	−0.0257066
$479$	23.0284	1.05220	0.526098	−	0.850424i	$-0.323655\pi$
$479$	23.0284	1.05220	0.526098	+	0.850424i	$0.323655\pi$
$480$	0	0
$481$	−16.6385	−0.758648
$482$	−3.73327	−0.170046
$483$	0	0
$484$	−17.3087	−0.786757
$485$	−2.28769	−0.103879
$486$	0	0
$487$	−11.9762	−0.542694	−0.271347	−	0.962482i	$-0.587469\pi$
$487$	−11.9762	−0.542694	−0.271347	+	0.962482i	$0.587469\pi$
$488$	−12.8194	−0.580306
$489$	0	0
$490$	0.163728	0.00739646
$491$	42.2730	1.90775	0.953877	−	0.300199i	$-0.0970531\pi$
$491$	42.2730	1.90775	0.953877	+	0.300199i	$0.0970531\pi$
$492$	0	0
$493$	−2.35211	−0.105934
$494$	−2.24621	−0.101062
$495$	0	0
$496$	−12.3649	−0.555201
$497$	−2.66621	−0.119596
$498$	0	0
$499$	3.88448	0.173893	0.0869465	−	0.996213i	$-0.472289\pi$
$499$	3.88448	0.173893	0.0869465	+	0.996213i	$0.472289\pi$
$500$	3.58079	0.160138
$501$	0	0
$502$	9.27536	0.413980
$503$	−7.79134	−0.347399	−0.173699	−	0.984799i	$-0.555572\pi$
$503$	−7.79134	−0.347399	−0.173699	+	0.984799i	$0.555572\pi$
$504$	0	0
$505$	−3.23162	−0.143805
$506$	−5.30145	−0.235678
$507$	0	0
$508$	−1.52145	−0.0675036
$509$	−25.8939	−1.14773	−0.573864	−	0.818951i	$-0.694556\pi$
$509$	−25.8939	−1.14773	−0.573864	+	0.818951i	$0.694556\pi$
$510$	0	0
$511$	10.5894	0.468445
$512$	14.5051	0.641041
$513$	0	0
$514$	16.0378	0.707397
$515$	1.43025	0.0630243
$516$	0	0
$517$	−30.4239	−1.33804
$518$	7.57791	0.332954
$519$	0	0
$520$	0.875729	0.0384033
$521$	7.34696	0.321876	0.160938	−	0.986964i	$-0.448548\pi$
$521$	7.34696	0.321876	0.160938	+	0.986964i	$0.448548\pi$
$522$	0	0
$523$	2.81126	0.122928	0.0614639	−	0.998109i	$-0.480423\pi$
$523$	2.81126	0.122928	0.0614639	+	0.998109i	$0.480423\pi$
$524$	−4.22447	−0.184547
$525$	0	0
$526$	−20.5105	−0.894300
$527$	44.4838	1.93774
$528$	0	0
$529$	−20.3753	−0.885884
$530$	−1.83449	−0.0796852
$531$	0	0
$532$	−3.25252	−0.141015
$533$	−7.26165	−0.314537
$534$	0	0
$535$	1.64406	0.0710790
$536$	36.9539	1.59616
$537$	0	0
$538$	−8.61860	−0.371574
$539$	4.73037	0.203751
$540$	0	0
$541$	4.76871	0.205023	0.102511	−	0.994732i	$-0.467312\pi$
$541$	4.76871	0.205023	0.102511	+	0.994732i	$0.467312\pi$
$542$	−4.95333	−0.212764
$543$	0	0
$544$	−28.3855	−1.21702
$545$	−1.84072	−0.0788479
$546$	0	0
$547$	4.47547	0.191357	0.0956786	−	0.995412i	$-0.469498\pi$
$547$	4.47547	0.191357	0.0956786	+	0.995412i	$0.469498\pi$
$548$	−8.90580	−0.380437
$549$	0	0
$550$	−16.1784	−0.689847
$551$	1.02943	0.0438552
$552$	0	0
$553$	−13.1527	−0.559308
$554$	−20.1783	−0.857294
$555$	0	0
$556$	−19.5854	−0.830605
$557$	30.3570	1.28627	0.643135	−	0.765753i	$-0.277634\pi$
$557$	30.3570	1.28627	0.643135	+	0.765753i	$0.277634\pi$
$558$	0	0
$559$	−1.92413	−0.0813820
$560$	0.321345	0.0135793
$561$	0	0
$562$	−1.68046	−0.0708858
$563$	40.3405	1.70015	0.850076	−	0.526661i	$-0.176556\pi$
$563$	40.3405	1.70015	0.850076	+	0.526661i	$0.176556\pi$
$564$	0	0
$565$	−1.34809	−0.0567147
$566$	−9.04947	−0.380378
$567$	0	0
$568$	6.49499	0.272524
$569$	−1.96447	−0.0823550	−0.0411775	−	0.999152i	$-0.513111\pi$
$569$	−1.96447	−0.0823550	−0.0411775	+	0.999152i	$0.513111\pi$
$570$	0	0
$571$	17.7620	0.743315	0.371658	−	0.928370i	$-0.378790\pi$
$571$	17.7620	0.743315	0.371658	+	0.928370i	$0.378790\pi$
$572$	10.9315	0.457069
$573$	0	0
$574$	3.30728	0.138043
$575$	8.00966	0.334026
$576$	0	0
$577$	8.06360	0.335692	0.167846	−	0.985813i	$-0.446319\pi$
$577$	8.06360	0.335692	0.167846	+	0.985813i	$0.446319\pi$
$578$	4.74457	0.197348
$579$	0	0
$580$	−0.173402	−0.00720011
$581$	11.1108	0.460953
$582$	0	0
$583$	−53.0016	−2.19510
$584$	−25.7961	−1.06745
$585$	0	0
$586$	8.26723	0.341516
$587$	−2.82469	−0.116588	−0.0582938	−	0.998299i	$-0.518566\pi$
$587$	−2.82469	−0.116588	−0.0582938	+	0.998299i	$0.518566\pi$
$588$	0	0
$589$	−19.4689	−0.802202
$590$	1.25756	0.0517729
$591$	0	0
$592$	14.8730	0.611277
$593$	−25.7116	−1.05585	−0.527925	−	0.849291i	$-0.677030\pi$
$593$	−25.7116	−1.05585	−0.527925	+	0.849291i	$0.677030\pi$
$594$	0	0
$595$	−1.15606	−0.0473940
$596$	33.7769	1.38355
$597$	0	0
$598$	1.70226	0.0696105
$599$	−34.6917	−1.41746	−0.708732	−	0.705478i	$-0.750732\pi$
$599$	−34.6917	−1.41746	−0.708732	+	0.705478i	$0.750732\pi$
$600$	0	0
$601$	12.8762	0.525230	0.262615	−	0.964901i	$-0.415415\pi$
$601$	12.8762	0.525230	0.262615	+	0.964901i	$0.415415\pi$
$602$	0.876336	0.0357168
$603$	0	0
$604$	−20.5739	−0.837142
$605$	2.69255	0.109468
$606$	0	0
$607$	25.8987	1.05120	0.525599	−	0.850733i	$-0.323841\pi$
$607$	25.8987	1.05120	0.525599	+	0.850733i	$0.323841\pi$
$608$	12.4233	0.503830
$609$	0	0
$610$	0.861598	0.0348851
$611$	9.76891	0.395208
$612$	0	0
$613$	−36.6547	−1.48047	−0.740234	−	0.672349i	$-0.765285\pi$
$613$	−36.6547	−1.48047	−0.740234	+	0.672349i	$0.765285\pi$
$614$	−23.0521	−0.930307
$615$	0	0
$616$	−11.5234	−0.464290
$617$	49.5324	1.99410	0.997050	−	0.0767545i	$-0.0244558\pi$
$617$	49.5324	1.99410	0.997050	+	0.0767545i	$0.0244558\pi$
$618$	0	0
$619$	−43.3703	−1.74320	−0.871599	−	0.490219i	$-0.836917\pi$
$619$	−43.3703	−1.74320	−0.871599	+	0.490219i	$0.836917\pi$
$620$	3.27942	0.131705
$621$	0	0
$622$	−12.1232	−0.486097
$623$	−11.0913	−0.444364
$624$	0	0
$625$	24.1629	0.966515
$626$	−7.22203	−0.288651
$627$	0	0
$628$	−1.57271	−0.0627579
$629$	−53.5069	−2.13346
$630$	0	0
$631$	−31.7213	−1.26280	−0.631402	−	0.775455i	$-0.717520\pi$
$631$	−31.7213	−1.26280	−0.631402	+	0.775455i	$0.717520\pi$
$632$	32.0404	1.27450
$633$	0	0
$634$	−12.1488	−0.482492
$635$	0.236679	0.00939232
$636$	0	0
$637$	−1.51889	−0.0601806
$638$	1.57577	0.0623853
$639$	0	0
$640$	−2.53722	−0.100293
$641$	−6.12263	−0.241829	−0.120915	−	0.992663i	$-0.538583\pi$
$641$	−6.12263	−0.241829	−0.120915	+	0.992663i	$0.538583\pi$
$642$	0	0
$643$	−1.36807	−0.0539513	−0.0269757	−	0.999636i	$-0.508588\pi$
$643$	−1.36807	−0.0539513	−0.0269757	+	0.999636i	$0.508588\pi$
$644$	2.46488	0.0971299
$645$	0	0
$646$	−7.22348	−0.284204
$647$	23.2875	0.915525	0.457762	−	0.889075i	$-0.348651\pi$
$647$	23.2875	0.915525	0.457762	+	0.889075i	$0.348651\pi$
$648$	0	0
$649$	36.3330	1.42620
$650$	5.19476	0.203755
$651$	0	0
$652$	31.9931	1.25295
$653$	26.1315	1.02260	0.511302	−	0.859401i	$-0.329164\pi$
$653$	26.1315	1.02260	0.511302	+	0.859401i	$0.329164\pi$
$654$	0	0
$655$	0.657163	0.0256775
$656$	6.49114	0.253436
$657$	0	0
$658$	−4.44921	−0.173448
$659$	10.1531	0.395510	0.197755	−	0.980252i	$-0.436635\pi$
$659$	10.1531	0.395510	0.197755	+	0.980252i	$0.436635\pi$
$660$	0	0
$661$	2.42067	0.0941533	0.0470766	−	0.998891i	$-0.485009\pi$
$661$	2.42067	0.0941533	0.0470766	+	0.998891i	$0.485009\pi$
$662$	−16.1566	−0.627944
$663$	0	0
$664$	−27.0663	−1.05038
$665$	0.505966	0.0196205
$666$	0	0
$667$	−0.780140	−0.0302071
$668$	−27.7738	−1.07460
$669$	0	0
$670$	−2.48369	−0.0959534
$671$	24.8930	0.960985
$672$	0	0
$673$	−26.0790	−1.00527	−0.502636	−	0.864498i	$-0.667636\pi$
$673$	−26.0790	−1.00527	−0.502636	+	0.864498i	$0.667636\pi$
$674$	−3.39346	−0.130711
$675$	0	0
$676$	16.2689	0.625725
$677$	45.3731	1.74383	0.871915	−	0.489657i	$-0.162878\pi$
$677$	45.3731	1.74383	0.871915	+	0.489657i	$0.162878\pi$
$678$	0	0
$679$	−9.66578	−0.370939
$680$	2.81622	0.107997
$681$	0	0
$682$	−29.8014	−1.14116
$683$	−11.9507	−0.457282	−0.228641	−	0.973511i	$-0.573428\pi$
$683$	−11.9507	−0.457282	−0.228641	+	0.973511i	$0.573428\pi$
$684$	0	0
$685$	1.38540	0.0529333
$686$	0.691771	0.0264119
$687$	0	0
$688$	1.71997	0.0655732
$689$	17.0184	0.648351
$690$	0	0
$691$	−11.9466	−0.454469	−0.227235	−	0.973840i	$-0.572968\pi$
$691$	−11.9466	−0.454469	−0.227235	+	0.973840i	$0.572968\pi$
$692$	−15.8428	−0.602251
$693$	0	0
$694$	−7.22669	−0.274321
$695$	3.04672	0.115569
$696$	0	0
$697$	−23.3524	−0.884535
$698$	−13.3743	−0.506223
$699$	0	0
$700$	7.52204	0.284306
$701$	11.7682	0.444478	0.222239	−	0.974992i	$-0.428663\pi$
$701$	11.7682	0.444478	0.222239	+	0.974992i	$0.428663\pi$
$702$	0	0
$703$	23.4179	0.883224
$704$	6.17146	0.232596
$705$	0	0
$706$	2.79396	0.105152
$707$	−13.6540	−0.513513
$708$	0	0
$709$	−21.7838	−0.818108	−0.409054	−	0.912510i	$-0.634141\pi$
$709$	−21.7838	−0.818108	−0.409054	+	0.912510i	$0.634141\pi$
$710$	−0.436532	−0.0163827
$711$	0	0
$712$	27.0189	1.01257
$713$	14.7542	0.552551
$714$	0	0
$715$	−1.70052	−0.0635957
$716$	−8.31554	−0.310766
$717$	0	0
$718$	0.591761	0.0220843
$719$	49.2724	1.83755	0.918774	−	0.394783i	$-0.129180\pi$
$719$	49.2724	1.83755	0.918774	+	0.394783i	$0.129180\pi$
$720$	0	0
$721$	6.04299	0.225053
$722$	−9.98220	−0.371499
$723$	0	0
$724$	−4.43510	−0.164829
$725$	−2.38074	−0.0884185
$726$	0	0
$727$	12.7353	0.472327	0.236164	−	0.971713i	$-0.424110\pi$
$727$	12.7353	0.472327	0.236164	+	0.971713i	$0.424110\pi$
$728$	3.70007	0.137134
$729$	0	0
$730$	1.73377	0.0641697
$731$	−6.18772	−0.228861
$732$	0	0
$733$	22.6644	0.837130	0.418565	−	0.908187i	$-0.362533\pi$
$733$	22.6644	0.837130	0.418565	+	0.908187i	$0.362533\pi$
$734$	17.0899	0.630799
$735$	0	0
$736$	−9.41481	−0.347035
$737$	−71.7581	−2.64324
$738$	0	0
$739$	−7.56804	−0.278395	−0.139197	−	0.990265i	$-0.544452\pi$
$739$	−7.56804	−0.278395	−0.139197	+	0.990265i	$0.544452\pi$
$740$	−3.94462	−0.145007
$741$	0	0
$742$	−7.75097	−0.284547
$743$	−2.44813	−0.0898132	−0.0449066	−	0.998991i	$-0.514299\pi$
$743$	−2.44813	−0.0898132	−0.0449066	+	0.998991i	$0.514299\pi$
$744$	0	0
$745$	−5.25437	−0.192505
$746$	0.195612	0.00716188
$747$	0	0
$748$	35.1541	1.28536
$749$	6.94638	0.253815
$750$	0	0
$751$	36.3013	1.32465	0.662327	−	0.749215i	$-0.269569\pi$
$751$	36.3013	1.32465	0.662327	+	0.749215i	$0.269569\pi$
$752$	−8.73237	−0.318437
$753$	0	0
$754$	−0.505968	−0.0184263
$755$	3.20050	0.116478
$756$	0	0
$757$	19.0015	0.690620	0.345310	−	0.938489i	$-0.387774\pi$
$757$	19.0015	0.690620	0.345310	+	0.938489i	$0.387774\pi$
$758$	−8.54714	−0.310446
$759$	0	0
$760$	−1.23255	−0.0447094
$761$	11.8346	0.429005	0.214503	−	0.976723i	$-0.431187\pi$
$761$	11.8346	0.429005	0.214503	+	0.976723i	$0.431187\pi$
$762$	0	0
$763$	−7.77730	−0.281557
$764$	7.25677	0.262541
$765$	0	0
$766$	15.3485	0.554564
$767$	−11.6663	−0.421245
$768$	0	0
$769$	12.9124	0.465634	0.232817	−	0.972521i	$-0.425206\pi$
$769$	12.9124	0.465634	0.232817	+	0.972521i	$0.425206\pi$
$770$	0.774492	0.0279108
$771$	0	0
$772$	−0.198754	−0.00715332
$773$	44.8628	1.61360	0.806801	−	0.590823i	$-0.201197\pi$
$773$	44.8628	1.61360	0.806801	+	0.590823i	$0.201197\pi$
$774$	0	0
$775$	45.0253	1.61736
$776$	23.5462	0.845260
$777$	0	0
$778$	−19.0832	−0.684164
$779$	10.2205	0.366186
$780$	0	0
$781$	−12.6121	−0.451298
$782$	5.47422	0.195758
$783$	0	0
$784$	1.35773	0.0484902
$785$	0.244652	0.00873201
$786$	0	0
$787$	25.5191	0.909658	0.454829	−	0.890579i	$-0.349700\pi$
$787$	25.5191	0.909658	0.454829	+	0.890579i	$0.349700\pi$
$788$	29.2678	1.04262
$789$	0	0
$790$	−2.15345	−0.0766165
$791$	−5.69588	−0.202522
$792$	0	0
$793$	−7.99298	−0.283839
$794$	−23.8489	−0.846367
$795$	0	0
$796$	18.0744	0.640629
$797$	−35.4729	−1.25651	−0.628257	−	0.778006i	$-0.716231\pi$
$797$	−35.4729	−1.25651	−0.628257	+	0.778006i	$0.716231\pi$
$798$	0	0
$799$	31.4154	1.11140
$800$	−28.7310	−1.01580
$801$	0	0
$802$	−12.6915	−0.448152
$803$	50.0916	1.76769
$804$	0	0
$805$	−0.383440	−0.0135145
$806$	9.56903	0.337055
$807$	0	0
$808$	33.2618	1.17015
$809$	39.9982	1.40626	0.703131	−	0.711061i	$-0.251785\pi$
$809$	39.9982	1.40626	0.703131	+	0.711061i	$0.251785\pi$
$810$	0	0
$811$	−21.3383	−0.749287	−0.374644	−	0.927169i	$-0.622235\pi$
$811$	−21.3383	−0.749287	−0.374644	+	0.927169i	$0.622235\pi$
$812$	−0.732645	−0.0257108
$813$	0	0
$814$	35.8463	1.25641
$815$	−4.97688	−0.174332
$816$	0	0
$817$	2.70813	0.0947456
$818$	−7.81150	−0.273123
$819$	0	0
$820$	−1.72158	−0.0601202
$821$	0.612120	0.0213631	0.0106816	−	0.999943i	$-0.496600\pi$
$821$	0.612120	0.0213631	0.0106816	+	0.999943i	$0.496600\pi$
$822$	0	0
$823$	−37.7565	−1.31611	−0.658055	−	0.752970i	$-0.728621\pi$
$823$	−37.7565	−1.31611	−0.658055	+	0.752970i	$0.728621\pi$
$824$	−14.7210	−0.512829
$825$	0	0
$826$	5.31336	0.184875
$827$	38.2190	1.32900	0.664502	−	0.747287i	$-0.268644\pi$
$827$	38.2190	1.32900	0.664502	+	0.747287i	$0.268644\pi$
$828$	0	0
$829$	−23.0302	−0.799871	−0.399935	−	0.916543i	$-0.630967\pi$
$829$	−23.0302	−0.799871	−0.399935	+	0.916543i	$0.630967\pi$
$830$	1.81914	0.0631433
$831$	0	0
$832$	−1.98161	−0.0687000
$833$	−4.88453	−0.169239
$834$	0	0
$835$	4.32053	0.149518
$836$	−15.3856	−0.532123
$837$	0	0
$838$	3.21089	0.110919
$839$	−34.5464	−1.19268	−0.596338	−	0.802734i	$-0.703378\pi$
$839$	−34.5464	−1.19268	−0.596338	+	0.802734i	$0.703378\pi$
$840$	0	0
$841$	−28.7681	−0.992004
$842$	−12.5724	−0.433273
$843$	0	0
$844$	−21.8775	−0.753055
$845$	−2.53080	−0.0870623
$846$	0	0
$847$	11.3764	0.390898
$848$	−15.2127	−0.522406
$849$	0	0
$850$	16.7056	0.572997
$851$	−17.7470	−0.608359
$852$	0	0
$853$	34.0512	1.16589	0.582946	−	0.812511i	$-0.301900\pi$
$853$	34.0512	1.16589	0.582946	+	0.812511i	$0.301900\pi$
$854$	3.64037	0.124571
$855$	0	0
$856$	−16.9217	−0.578370
$857$	33.3224	1.13827	0.569136	−	0.822243i	$-0.307278\pi$
$857$	33.3224	1.13827	0.569136	+	0.822243i	$0.307278\pi$
$858$	0	0
$859$	34.3511	1.17204	0.586022	−	0.810295i	$-0.300693\pi$
$859$	34.3511	1.17204	0.586022	+	0.810295i	$0.300693\pi$
$860$	−0.456170	−0.0155553
$861$	0	0
$862$	9.66487	0.329187
$863$	0.252328	0.00858935	0.00429467	−	0.999991i	$-0.498633\pi$
$863$	0.252328	0.00858935	0.00429467	+	0.999991i	$0.498633\pi$
$864$	0	0
$865$	2.46452	0.0837961
$866$	−1.02410	−0.0348004
$867$	0	0
$868$	13.8560	0.470303
$869$	−62.2170	−2.11057
$870$	0	0
$871$	23.0410	0.780715
$872$	18.9458	0.641586
$873$	0	0
$874$	−2.39586	−0.0810411
$875$	−2.35353	−0.0795639
$876$	0	0
$877$	38.6171	1.30401	0.652004	−	0.758216i	$-0.273929\pi$
$877$	38.6171	1.30401	0.652004	+	0.758216i	$0.273929\pi$
$878$	−28.2992	−0.955053
$879$	0	0
$880$	1.52008	0.0512419
$881$	−0.798968	−0.0269179	−0.0134590	−	0.999909i	$-0.504284\pi$
$881$	−0.798968	−0.0269179	−0.0134590	+	0.999909i	$0.504284\pi$
$882$	0	0
$883$	−8.06536	−0.271421	−0.135711	−	0.990749i	$-0.543332\pi$
$883$	−8.06536	−0.271421	−0.135711	+	0.990749i	$0.543332\pi$
$884$	−11.2877	−0.379648
$885$	0	0
$886$	1.49095	0.0500894
$887$	16.3375	0.548559	0.274280	−	0.961650i	$-0.411561\pi$
$887$	16.3375	0.548559	0.274280	+	0.961650i	$0.411561\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−1.81595	−0.0608709
$891$	0	0
$892$	12.3426	0.413261
$893$	−13.7493	−0.460104
$894$	0	0
$895$	1.29357	0.0432394
$896$	−10.7201	−0.358134
$897$	0	0
$898$	0.865365	0.0288776
$899$	−4.38545	−0.146263
$900$	0	0
$901$	54.7289	1.82328
$902$	15.6447	0.520911
$903$	0	0
$904$	13.8754	0.461488
$905$	0.689928	0.0229340
$906$	0	0
$907$	−9.06870	−0.301121	−0.150561	−	0.988601i	$-0.548108\pi$
$907$	−9.06870	−0.301121	−0.150561	+	0.988601i	$0.548108\pi$
$908$	37.9193	1.25839
$909$	0	0
$910$	−0.248684	−0.00824379
$911$	10.9544	0.362936	0.181468	−	0.983397i	$-0.441915\pi$
$911$	10.9544	0.362936	0.181468	+	0.983397i	$0.441915\pi$
$912$	0	0
$913$	52.5581	1.73942
$914$	−6.45968	−0.213667
$915$	0	0
$916$	14.9879	0.495213
$917$	2.77660	0.0916914
$918$	0	0
$919$	−3.67443	−0.121208	−0.0606041	−	0.998162i	$-0.519303\pi$
$919$	−3.67443	−0.121208	−0.0606041	+	0.998162i	$0.519303\pi$
$920$	0.934074	0.0307955
$921$	0	0
$922$	−6.50507	−0.214233
$923$	4.04967	0.133297
$924$	0	0
$925$	−54.1582	−1.78071
$926$	24.8269	0.815863
$927$	0	0
$928$	2.79840	0.0918619
$929$	44.8141	1.47030	0.735151	−	0.677903i	$-0.237111\pi$
$929$	44.8141	1.47030	0.735151	+	0.677903i	$0.237111\pi$
$930$	0	0
$931$	2.13777	0.0700627
$932$	−42.2206	−1.38298
$933$	0	0
$934$	−6.41316	−0.209845
$935$	−5.46861	−0.178843
$936$	0	0
$937$	7.40715	0.241981	0.120990	−	0.992654i	$-0.461393\pi$
$937$	7.40715	0.241981	0.120990	+	0.992654i	$0.461393\pi$
$938$	−10.4939	−0.342639
$939$	0	0
$940$	2.31600	0.0755396
$941$	50.7499	1.65440	0.827199	−	0.561909i	$-0.189933\pi$
$941$	50.7499	1.65440	0.827199	+	0.561909i	$0.189933\pi$
$942$	0	0
$943$	−7.74545	−0.252226
$944$	10.4284	0.339416
$945$	0	0
$946$	4.14539	0.134778
$947$	40.3285	1.31050	0.655250	−	0.755412i	$-0.272563\pi$
$947$	40.3285	1.31050	0.655250	+	0.755412i	$0.272563\pi$
$948$	0	0
$949$	−16.0840	−0.522110
$950$	−7.31141	−0.237213
$951$	0	0
$952$	11.8989	0.385645
$953$	−3.51760	−0.113946	−0.0569731	−	0.998376i	$-0.518145\pi$
$953$	−3.51760	−0.113946	−0.0569731	+	0.998376i	$0.518145\pi$
$954$	0	0
$955$	−1.12887	−0.0365294
$956$	1.23610	0.0399784
$957$	0	0
$958$	15.9304	0.514688
$959$	5.85349	0.189019
$960$	0	0
$961$	51.9390	1.67545
$962$	−11.5100	−0.371097
$963$	0	0
$964$	8.21080	0.264452
$965$	0.0309184	0.000995299 0
$966$	0	0
$967$	−45.1545	−1.45207	−0.726035	−	0.687658i	$-0.758639\pi$
$967$	−45.1545	−1.45207	−0.726035	+	0.687658i	$0.758639\pi$
$968$	−27.7133	−0.890741
$969$	0	0
$970$	−1.58256	−0.0508128
$971$	−38.2249	−1.22669	−0.613347	−	0.789813i	$-0.710177\pi$
$971$	−38.2249	−1.22669	−0.613347	+	0.789813i	$0.710177\pi$
$972$	0	0
$973$	12.8728	0.412683
$974$	−8.28480	−0.265462
$975$	0	0
$976$	7.14488	0.228702
$977$	−32.0622	−1.02576	−0.512880	−	0.858460i	$-0.671421\pi$
$977$	−32.0622	−1.02576	−0.512880	+	0.858460i	$0.671421\pi$
$978$	0	0
$979$	−52.4660	−1.67682
$980$	−0.360096	−0.0115028
$981$	0	0
$982$	29.2432	0.933189
$983$	31.8590	1.01614	0.508072	−	0.861315i	$-0.330358\pi$
$983$	31.8590	1.01614	0.508072	+	0.861315i	$0.330358\pi$
$984$	0	0
$985$	−4.55293	−0.145068
$986$	−1.62712	−0.0518181
$987$	0	0
$988$	4.94022	0.157169
$989$	−2.05232	−0.0652601
$990$	0	0
$991$	24.9795	0.793500	0.396750	−	0.917927i	$-0.370138\pi$
$991$	24.9795	0.793500	0.396750	+	0.917927i	$0.370138\pi$
$992$	−52.9241	−1.68034
$993$	0	0
$994$	−1.84440	−0.0585010
$995$	−2.81167	−0.0891359
$996$	0	0
$997$	34.0534	1.07848	0.539241	−	0.842151i	$-0.318711\pi$
$997$	34.0534	1.07848	0.539241	+	0.842151i	$0.318711\pi$
$998$	2.68717	0.0850608
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.15	yes	22
3.2	odd	2	inner	8001.2.a.x.1.8	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.8	✓	22	3.2	odd	2	inner
8001.2.a.x.1.15	yes	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.691771 q^{2} -1.52145 q^{4} +0.236679 q^{5} +1.00000 q^{7} -2.43604 q^{8} +O(q^{10})\)	\(q+0.691771 q^{2} -1.52145 q^{4} +0.236679 q^{5} +1.00000 q^{7} -2.43604 q^{8} +0.163728 q^{10} +4.73037 q^{11} -1.51889 q^{13} +0.691771 q^{14} +1.35773 q^{16} -4.88453 q^{17} +2.13777 q^{19} -0.360096 q^{20} +3.27233 q^{22} -1.62008 q^{23} -4.94398 q^{25} -1.05072 q^{26} -1.52145 q^{28} +0.481543 q^{29} -9.10709 q^{31} +5.81131 q^{32} -3.37897 q^{34} +0.236679 q^{35} +10.9544 q^{37} +1.47885 q^{38} -0.576559 q^{40} +4.78089 q^{41} +1.26680 q^{43} -7.19704 q^{44} -1.12073 q^{46} -6.43162 q^{47} +1.00000 q^{49} -3.42010 q^{50} +2.31092 q^{52} -11.2045 q^{53} +1.11958 q^{55} -2.43604 q^{56} +0.333117 q^{58} +7.68081 q^{59} +5.26239 q^{61} -6.30002 q^{62} +1.30465 q^{64} -0.359489 q^{65} -15.1697 q^{67} +7.43158 q^{68} +0.163728 q^{70} -2.66621 q^{71} +10.5894 q^{73} +7.57791 q^{74} -3.25252 q^{76} +4.73037 q^{77} -13.1527 q^{79} +0.321345 q^{80} +3.30728 q^{82} +11.1108 q^{83} -1.15606 q^{85} +0.876336 q^{86} -11.5234 q^{88} -11.0913 q^{89} -1.51889 q^{91} +2.46488 q^{92} -4.44921 q^{94} +0.505966 q^{95} -9.66578 q^{97} +0.691771 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) 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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