LMFDB - Embedded newform 8001.2.a.x.1.10

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			1.10
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)

$f(q)$	$=$	$q-0.446366 q^{2} -1.80076 q^{4} +2.33773 q^{5} +1.00000 q^{7} +1.69653 q^{8} +O(q^{10})$	\(q-0.446366 q^{2} -1.80076 q^{4} +2.33773 q^{5} +1.00000 q^{7} +1.69653 q^{8} -1.04348 q^{10} -4.53166 q^{11} -0.767582 q^{13} -0.446366 q^{14} +2.84424 q^{16} -3.43169 q^{17} -4.98922 q^{19} -4.20969 q^{20} +2.02278 q^{22} +4.14751 q^{23} +0.464998 q^{25} +0.342623 q^{26} -1.80076 q^{28} +9.37650 q^{29} +5.36743 q^{31} -4.66263 q^{32} +1.53179 q^{34} +2.33773 q^{35} -2.44708 q^{37} +2.22702 q^{38} +3.96603 q^{40} +1.04906 q^{41} +2.90570 q^{43} +8.16042 q^{44} -1.85131 q^{46} +1.19217 q^{47} +1.00000 q^{49} -0.207559 q^{50} +1.38223 q^{52} -10.5726 q^{53} -10.5938 q^{55} +1.69653 q^{56} -4.18535 q^{58} -1.68946 q^{59} +4.42071 q^{61} -2.39584 q^{62} -3.60724 q^{64} -1.79440 q^{65} +11.8999 q^{67} +6.17964 q^{68} -1.04348 q^{70} -5.39083 q^{71} +5.28672 q^{73} +1.09229 q^{74} +8.98437 q^{76} -4.53166 q^{77} -5.18146 q^{79} +6.64908 q^{80} -0.468264 q^{82} -6.33884 q^{83} -8.02238 q^{85} -1.29701 q^{86} -7.68809 q^{88} +1.35913 q^{89} -0.767582 q^{91} -7.46866 q^{92} -0.532145 q^{94} -11.6635 q^{95} -8.94618 q^{97} -0.446366 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.446366	−0.315628	−0.157814	−	0.987469i	$-0.550445\pi$
$2$	−0.446366	−0.315628	−0.157814	+	0.987469i	$0.550445\pi$
$3$	0	0
$3$	0	0
$4$	−1.80076	−0.900379
$5$	2.33773	1.04547	0.522733	−	0.852496i	$-0.324912\pi$
$5$	2.33773	1.04547	0.522733	+	0.852496i	$0.324912\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.69653	0.599814
$9$	0	0
$10$	−1.04348	−0.329979
$11$	−4.53166	−1.36635	−0.683174	−	0.730256i	$-0.739401\pi$
$11$	−4.53166	−1.36635	−0.683174	+	0.730256i	$0.739401\pi$
$12$	0	0
$13$	−0.767582	−0.212889	−0.106445	−	0.994319i	$-0.533947\pi$
$13$	−0.767582	−0.212889	−0.106445	+	0.994319i	$0.533947\pi$
$14$	−0.446366	−0.119296
$15$	0	0
$16$	2.84424	0.711061
$17$	−3.43169	−0.832307	−0.416154	−	0.909294i	$-0.636622\pi$
$17$	−3.43169	−0.832307	−0.416154	+	0.909294i	$0.636622\pi$
$18$	0	0
$19$	−4.98922	−1.14460	−0.572302	−	0.820043i	$-0.693950\pi$
$19$	−4.98922	−1.14460	−0.572302	+	0.820043i	$0.693950\pi$
$20$	−4.20969	−0.941316
$21$	0	0
$22$	2.02278	0.431258
$23$	4.14751	0.864816	0.432408	−	0.901678i	$-0.357664\pi$
$23$	4.14751	0.864816	0.432408	+	0.901678i	$0.357664\pi$
$24$	0	0
$25$	0.464998	0.0929996
$26$	0.342623	0.0671938
$27$	0	0
$28$	−1.80076	−0.340311
$29$	9.37650	1.74117	0.870586	−	0.492016i	$-0.163740\pi$
$29$	9.37650	1.74117	0.870586	+	0.492016i	$0.163740\pi$
$30$	0	0
$31$	5.36743	0.964019	0.482009	−	0.876166i	$-0.339907\pi$
$31$	5.36743	0.964019	0.482009	+	0.876166i	$0.339907\pi$
$32$	−4.66263	−0.824244
$33$	0	0
$34$	1.53179	0.262700
$35$	2.33773	0.395149
$36$	0	0
$37$	−2.44708	−0.402297	−0.201148	−	0.979561i	$-0.564467\pi$
$37$	−2.44708	−0.402297	−0.201148	+	0.979561i	$0.564467\pi$
$38$	2.22702	0.361270
$39$	0	0
$40$	3.96603	0.627085
$41$	1.04906	0.163835	0.0819176	−	0.996639i	$-0.473896\pi$
$41$	1.04906	0.163835	0.0819176	+	0.996639i	$0.473896\pi$
$42$	0	0
$43$	2.90570	0.443116	0.221558	−	0.975147i	$-0.428886\pi$
$43$	2.90570	0.443116	0.221558	+	0.975147i	$0.428886\pi$
$44$	8.16042	1.23023
$45$	0	0
$46$	−1.85131	−0.272961
$47$	1.19217	0.173896	0.0869481	−	0.996213i	$-0.472289\pi$
$47$	1.19217	0.173896	0.0869481	+	0.996213i	$0.472289\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.207559	−0.0293533
$51$	0	0
$52$	1.38223	0.191681
$53$	−10.5726	−1.45226	−0.726130	−	0.687557i	$-0.758683\pi$
$53$	−10.5726	−1.45226	−0.726130	+	0.687557i	$0.758683\pi$
$54$	0	0
$55$	−10.5938	−1.42847
$56$	1.69653	0.226708
$57$	0	0
$58$	−4.18535	−0.549563
$59$	−1.68946	−0.219949	−0.109974	−	0.993934i	$-0.535077\pi$
$59$	−1.68946	−0.219949	−0.109974	+	0.993934i	$0.535077\pi$
$60$	0	0
$61$	4.42071	0.566014	0.283007	−	0.959118i	$-0.408668\pi$
$61$	4.42071	0.566014	0.283007	+	0.959118i	$0.408668\pi$
$62$	−2.39584	−0.304272
$63$	0	0
$64$	−3.60724	−0.450906
$65$	−1.79440	−0.222568
$66$	0	0
$67$	11.8999	1.45380	0.726901	−	0.686742i	$-0.240960\pi$
$67$	11.8999	1.45380	0.726901	+	0.686742i	$0.240960\pi$
$68$	6.17964	0.749392
$69$	0	0
$70$	−1.04348	−0.124720
$71$	−5.39083	−0.639773	−0.319887	−	0.947456i	$-0.603645\pi$
$71$	−5.39083	−0.639773	−0.319887	+	0.947456i	$0.603645\pi$
$72$	0	0
$73$	5.28672	0.618764	0.309382	−	0.950938i	$-0.399878\pi$
$73$	5.28672	0.618764	0.309382	+	0.950938i	$0.399878\pi$
$74$	1.09229	0.126976
$75$	0	0
$76$	8.98437	1.03058
$77$	−4.53166	−0.516431
$78$	0	0
$79$	−5.18146	−0.582959	−0.291480	−	0.956577i	$-0.594148\pi$
$79$	−5.18146	−0.582959	−0.291480	+	0.956577i	$0.594148\pi$
$80$	6.64908	0.743390
$81$	0	0
$82$	−0.468264	−0.0517111
$83$	−6.33884	−0.695778	−0.347889	−	0.937536i	$-0.613101\pi$
$83$	−6.33884	−0.695778	−0.347889	+	0.937536i	$0.613101\pi$
$84$	0	0
$85$	−8.02238	−0.870149
$86$	−1.29701	−0.139860
$87$	0	0
$88$	−7.68809	−0.819554
$89$	1.35913	0.144068	0.0720339	−	0.997402i	$-0.477051\pi$
$89$	1.35913	0.144068	0.0720339	+	0.997402i	$0.477051\pi$
$90$	0	0
$91$	−0.767582	−0.0804645
$92$	−7.46866	−0.778662
$93$	0	0
$94$	−0.532145	−0.0548866
$95$	−11.6635	−1.19665
$96$	0	0
$97$	−8.94618	−0.908347	−0.454173	−	0.890913i	$-0.650065\pi$
$97$	−8.94618	−0.908347	−0.454173	+	0.890913i	$0.650065\pi$
$98$	−0.446366	−0.0450898
$99$	0	0
$100$	−0.837349	−0.0837349
$101$	−0.306288	−0.0304768	−0.0152384	−	0.999884i	$-0.504851\pi$
$101$	−0.306288	−0.0304768	−0.0152384	+	0.999884i	$0.504851\pi$
$102$	0	0
$103$	−7.08821	−0.698422	−0.349211	−	0.937044i	$-0.613550\pi$
$103$	−7.08821	−0.698422	−0.349211	+	0.937044i	$0.613550\pi$
$104$	−1.30223	−0.127694
$105$	0	0
$106$	4.71926	0.458375
$107$	9.23685	0.892960	0.446480	−	0.894794i	$-0.352677\pi$
$107$	9.23685	0.892960	0.446480	+	0.894794i	$0.352677\pi$
$108$	0	0
$109$	−10.5036	−1.00606	−0.503031	−	0.864268i	$-0.667782\pi$
$109$	−10.5036	−1.00606	−0.503031	+	0.864268i	$0.667782\pi$
$110$	4.72872	0.450866
$111$	0	0
$112$	2.84424	0.268756
$113$	−8.67620	−0.816188	−0.408094	−	0.912940i	$-0.633807\pi$
$113$	−8.67620	−0.816188	−0.408094	+	0.912940i	$0.633807\pi$
$114$	0	0
$115$	9.69578	0.904136
$116$	−16.8848	−1.56771
$117$	0	0
$118$	0.754117	0.0694221
$119$	−3.43169	−0.314583
$120$	0	0
$121$	9.53595	0.866905
$122$	−1.97325	−0.178650
$123$	0	0
$124$	−9.66544	−0.867982
$125$	−10.6016	−0.948238
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	10.9354	0.966563
$129$	0	0
$130$	0.800960	0.0702489
$131$	−10.2832	−0.898445	−0.449223	−	0.893420i	$-0.648299\pi$
$131$	−10.2832	−0.898445	−0.449223	+	0.893420i	$0.648299\pi$
$132$	0	0
$133$	−4.98922	−0.432620
$134$	−5.31170	−0.458861
$135$	0	0
$136$	−5.82196	−0.499229
$137$	0.222610	0.0190189	0.00950944	−	0.999955i	$-0.496973\pi$
$137$	0.222610	0.0190189	0.00950944	+	0.999955i	$0.496973\pi$
$138$	0	0
$139$	6.73668	0.571398	0.285699	−	0.958319i	$-0.407774\pi$
$139$	6.73668	0.571398	0.285699	+	0.958319i	$0.407774\pi$
$140$	−4.20969	−0.355784
$141$	0	0
$142$	2.40628	0.201931
$143$	3.47842	0.290880
$144$	0	0
$145$	21.9198	1.82034
$146$	−2.35981	−0.195300
$147$	0	0
$148$	4.40659	0.362219
$149$	−4.25733	−0.348774	−0.174387	−	0.984677i	$-0.555794\pi$
$149$	−4.25733	−0.348774	−0.174387	+	0.984677i	$0.555794\pi$
$150$	0	0
$151$	−18.4936	−1.50499	−0.752495	−	0.658598i	$-0.771150\pi$
$151$	−18.4936	−1.50499	−0.752495	+	0.658598i	$0.771150\pi$
$152$	−8.46435	−0.686549
$153$	0	0
$154$	2.02278	0.163000
$155$	12.5476	1.00785
$156$	0	0
$157$	−10.8175	−0.863333	−0.431666	−	0.902033i	$-0.642074\pi$
$157$	−10.8175	−0.863333	−0.431666	+	0.902033i	$0.642074\pi$
$158$	2.31283	0.183999
$159$	0	0
$160$	−10.9000	−0.861720
$161$	4.14751	0.326870
$162$	0	0
$163$	−16.9398	−1.32683	−0.663415	−	0.748251i	$-0.730894\pi$
$163$	−16.9398	−1.32683	−0.663415	+	0.748251i	$0.730894\pi$
$164$	−1.88910	−0.147514
$165$	0	0
$166$	2.82944	0.219607
$167$	8.15215	0.630832	0.315416	−	0.948953i	$-0.397856\pi$
$167$	8.15215	0.630832	0.315416	+	0.948953i	$0.397856\pi$
$168$	0	0
$169$	−12.4108	−0.954678
$170$	3.58092	0.274644
$171$	0	0
$172$	−5.23247	−0.398972
$173$	16.1018	1.22420	0.612098	−	0.790782i	$-0.290326\pi$
$173$	16.1018	1.22420	0.612098	+	0.790782i	$0.290326\pi$
$174$	0	0
$175$	0.464998	0.0351506
$176$	−12.8891	−0.971556
$177$	0	0
$178$	−0.606671	−0.0454719
$179$	−6.75542	−0.504924	−0.252462	−	0.967607i	$-0.581240\pi$
$179$	−6.75542	−0.504924	−0.252462	+	0.967607i	$0.581240\pi$
$180$	0	0
$181$	−16.5449	−1.22977	−0.614885	−	0.788617i	$-0.710798\pi$
$181$	−16.5449	−1.22977	−0.614885	+	0.788617i	$0.710798\pi$
$182$	0.342623	0.0253969
$183$	0	0
$184$	7.03637	0.518728
$185$	−5.72061	−0.420588
$186$	0	0
$187$	15.5513	1.13722
$188$	−2.14681	−0.156572
$189$	0	0
$190$	5.20617	0.377695
$191$	−4.01445	−0.290475	−0.145238	−	0.989397i	$-0.546395\pi$
$191$	−4.01445	−0.290475	−0.145238	+	0.989397i	$0.546395\pi$
$192$	0	0
$193$	−2.60259	−0.187339	−0.0936693	−	0.995603i	$-0.529860\pi$
$193$	−2.60259	−0.187339	−0.0936693	+	0.995603i	$0.529860\pi$
$194$	3.99327	0.286700
$195$	0	0
$196$	−1.80076	−0.128626
$197$	3.22387	0.229691	0.114846	−	0.993383i	$-0.463363\pi$
$197$	3.22387	0.229691	0.114846	+	0.993383i	$0.463363\pi$
$198$	0	0
$199$	−12.7767	−0.905717	−0.452859	−	0.891582i	$-0.649596\pi$
$199$	−12.7767	−0.905717	−0.452859	+	0.891582i	$0.649596\pi$
$200$	0.788883	0.0557824
$201$	0	0
$202$	0.136716	0.00961934
$203$	9.37650	0.658101
$204$	0	0
$205$	2.45242	0.171284
$206$	3.16393	0.220442
$207$	0	0
$208$	−2.18319	−0.151377
$209$	22.6094	1.56393
$210$	0	0
$211$	−19.6650	−1.35380	−0.676898	−	0.736077i	$-0.736676\pi$
$211$	−19.6650	−1.35380	−0.676898	+	0.736077i	$0.736676\pi$
$212$	19.0387	1.30758
$213$	0	0
$214$	−4.12302	−0.281844
$215$	6.79276	0.463263
$216$	0	0
$217$	5.36743	0.364365
$218$	4.68845	0.317542
$219$	0	0
$220$	19.0769	1.28616
$221$	2.63411	0.177189
$222$	0	0
$223$	−10.9512	−0.733346	−0.366673	−	0.930350i	$-0.619503\pi$
$223$	−10.9512	−0.733346	−0.366673	+	0.930350i	$0.619503\pi$
$224$	−4.66263	−0.311535
$225$	0	0
$226$	3.87276	0.257612
$227$	6.15209	0.408329	0.204164	−	0.978937i	$-0.434552\pi$
$227$	6.15209	0.408329	0.204164	+	0.978937i	$0.434552\pi$
$228$	0	0
$229$	20.4006	1.34811	0.674054	−	0.738682i	$-0.264552\pi$
$229$	20.4006	1.34811	0.674054	+	0.738682i	$0.264552\pi$
$230$	−4.32787	−0.285371
$231$	0	0
$232$	15.9075	1.04438
$233$	23.9319	1.56783	0.783914	−	0.620869i	$-0.213220\pi$
$233$	23.9319	1.56783	0.783914	+	0.620869i	$0.213220\pi$
$234$	0	0
$235$	2.78698	0.181803
$236$	3.04231	0.198037
$237$	0	0
$238$	1.53179	0.0992912
$239$	−3.47377	−0.224700	−0.112350	−	0.993669i	$-0.535838\pi$
$239$	−3.47377	−0.224700	−0.112350	+	0.993669i	$0.535838\pi$
$240$	0	0
$241$	10.8457	0.698634	0.349317	−	0.937005i	$-0.386414\pi$
$241$	10.8457	0.698634	0.349317	+	0.937005i	$0.386414\pi$
$242$	−4.25653	−0.273620
$243$	0	0
$244$	−7.96063	−0.509627
$245$	2.33773	0.149352
$246$	0	0
$247$	3.82963	0.243674
$248$	9.10600	0.578231
$249$	0	0
$250$	4.73221	0.299291
$251$	−11.2843	−0.712260	−0.356130	−	0.934437i	$-0.615904\pi$
$251$	−11.2843	−0.712260	−0.356130	+	0.934437i	$0.615904\pi$
$252$	0	0
$253$	−18.7951	−1.18164
$254$	−0.446366	−0.0280075
$255$	0	0
$256$	2.33329	0.145831
$257$	1.31385	0.0819559	0.0409779	−	0.999160i	$-0.486953\pi$
$257$	1.31385	0.0819559	0.0409779	+	0.999160i	$0.486953\pi$
$258$	0	0
$259$	−2.44708	−0.152054
$260$	3.23128	0.200396
$261$	0	0
$262$	4.59006	0.283575
$263$	4.13412	0.254921	0.127460	−	0.991844i	$-0.459317\pi$
$263$	4.13412	0.254921	0.127460	+	0.991844i	$0.459317\pi$
$264$	0	0
$265$	−24.7160	−1.51829
$266$	2.22702	0.136547
$267$	0	0
$268$	−21.4288	−1.30897
$269$	19.5802	1.19382	0.596912	−	0.802307i	$-0.296394\pi$
$269$	19.5802	1.19382	0.596912	+	0.802307i	$0.296394\pi$
$270$	0	0
$271$	25.5373	1.55128	0.775641	−	0.631174i	$-0.217427\pi$
$271$	25.5373	1.55128	0.775641	+	0.631174i	$0.217427\pi$
$272$	−9.76056	−0.591821
$273$	0	0
$274$	−0.0993656	−0.00600290
$275$	−2.10721	−0.127070
$276$	0	0
$277$	−8.17171	−0.490991	−0.245495	−	0.969398i	$-0.578951\pi$
$277$	−8.17171	−0.490991	−0.245495	+	0.969398i	$0.578951\pi$
$278$	−3.00703	−0.180349
$279$	0	0
$280$	3.96603	0.237016
$281$	11.2877	0.673366	0.336683	−	0.941618i	$-0.390695\pi$
$281$	11.2877	0.673366	0.336683	+	0.941618i	$0.390695\pi$
$282$	0	0
$283$	−14.5510	−0.864966	−0.432483	−	0.901642i	$-0.642362\pi$
$283$	−14.5510	−0.864966	−0.432483	+	0.901642i	$0.642362\pi$
$284$	9.70757	0.576038
$285$	0	0
$286$	−1.55265	−0.0918101
$287$	1.04906	0.0619239
$288$	0	0
$289$	−5.22350	−0.307265
$290$	−9.78423	−0.574550
$291$	0	0
$292$	−9.52011	−0.557122
$293$	15.8252	0.924518	0.462259	−	0.886745i	$-0.347039\pi$
$293$	15.8252	0.924518	0.462259	+	0.886745i	$0.347039\pi$
$294$	0	0
$295$	−3.94951	−0.229949
$296$	−4.15153	−0.241303
$297$	0	0
$298$	1.90033	0.110083
$299$	−3.18356	−0.184110
$300$	0	0
$301$	2.90570	0.167482
$302$	8.25493	0.475018
$303$	0	0
$304$	−14.1905	−0.813883
$305$	10.3344	0.591749
$306$	0	0
$307$	10.2596	0.585544	0.292772	−	0.956182i	$-0.405422\pi$
$307$	10.2596	0.585544	0.292772	+	0.956182i	$0.405422\pi$
$308$	8.16042	0.464983
$309$	0	0
$310$	−5.60083	−0.318106
$311$	−19.5160	−1.10665	−0.553325	−	0.832965i	$-0.686641\pi$
$311$	−19.5160	−1.10665	−0.553325	+	0.832965i	$0.686641\pi$
$312$	0	0
$313$	−24.6729	−1.39460	−0.697298	−	0.716781i	$-0.745615\pi$
$313$	−24.6729	−1.39460	−0.697298	+	0.716781i	$0.745615\pi$
$314$	4.82858	0.272492
$315$	0	0
$316$	9.33055	0.524884
$317$	21.4107	1.20255	0.601273	−	0.799043i	$-0.294660\pi$
$317$	21.4107	1.20255	0.601273	+	0.799043i	$0.294660\pi$
$318$	0	0
$319$	−42.4911	−2.37905
$320$	−8.43278	−0.471407
$321$	0	0
$322$	−1.85131	−0.103169
$323$	17.1214	0.952663
$324$	0	0
$325$	−0.356924	−0.0197986
$326$	7.56137	0.418785
$327$	0	0
$328$	1.77976	0.0982706
$329$	1.19217	0.0657266
$330$	0	0
$331$	−18.4219	−1.01256	−0.506280	−	0.862369i	$-0.668980\pi$
$331$	−18.4219	−1.01256	−0.506280	+	0.862369i	$0.668980\pi$
$332$	11.4147	0.626463
$333$	0	0
$334$	−3.63884	−0.199109
$335$	27.8188	1.51990
$336$	0	0
$337$	−23.6053	−1.28586	−0.642930	−	0.765925i	$-0.722281\pi$
$337$	−23.6053	−1.28586	−0.642930	+	0.765925i	$0.722281\pi$
$338$	5.53977	0.301324
$339$	0	0
$340$	14.4464	0.783464
$341$	−24.3234	−1.31718
$342$	0	0
$343$	1.00000	0.0539949
$344$	4.92961	0.265787
$345$	0	0
$346$	−7.18728	−0.386391
$347$	−33.5517	−1.80115	−0.900576	−	0.434699i	$-0.856855\pi$
$347$	−33.5517	−1.80115	−0.900576	+	0.434699i	$0.856855\pi$
$348$	0	0
$349$	−30.7293	−1.64490	−0.822450	−	0.568837i	$-0.807394\pi$
$349$	−30.7293	−1.64490	−0.822450	+	0.568837i	$0.807394\pi$
$350$	−0.207559	−0.0110945
$351$	0	0
$352$	21.1295	1.12620
$353$	−18.2382	−0.970722	−0.485361	−	0.874314i	$-0.661312\pi$
$353$	−18.2382	−0.970722	−0.485361	+	0.874314i	$0.661312\pi$
$354$	0	0
$355$	−12.6023	−0.668862
$356$	−2.44747	−0.129716
$357$	0	0
$358$	3.01539	0.159368
$359$	11.3158	0.597223	0.298611	−	0.954375i	$-0.403477\pi$
$359$	11.3158	0.597223	0.298611	+	0.954375i	$0.403477\pi$
$360$	0	0
$361$	5.89227	0.310119
$362$	7.38507	0.388150
$363$	0	0
$364$	1.38223	0.0724485
$365$	12.3589	0.646897
$366$	0	0
$367$	−27.5352	−1.43732	−0.718662	−	0.695360i	$-0.755245\pi$
$367$	−27.5352	−1.43732	−0.718662	+	0.695360i	$0.755245\pi$
$368$	11.7965	0.614937
$369$	0	0
$370$	2.55349	0.132749
$371$	−10.5726	−0.548903
$372$	0	0
$373$	1.92003	0.0994156	0.0497078	−	0.998764i	$-0.484171\pi$
$373$	1.92003	0.0994156	0.0497078	+	0.998764i	$0.484171\pi$
$374$	−6.94155	−0.358939
$375$	0	0
$376$	2.02255	0.104305
$377$	−7.19724	−0.370677
$378$	0	0
$379$	5.18999	0.266592	0.133296	−	0.991076i	$-0.457444\pi$
$379$	5.18999	0.266592	0.133296	+	0.991076i	$0.457444\pi$
$380$	21.0031	1.07743
$381$	0	0
$382$	1.79191	0.0916823
$383$	−37.3440	−1.90819	−0.954095	−	0.299505i	$-0.903178\pi$
$383$	−37.3440	−1.90819	−0.954095	+	0.299505i	$0.903178\pi$
$384$	0	0
$385$	−10.5938	−0.539911
$386$	1.16171	0.0591294
$387$	0	0
$388$	16.1099	0.817856
$389$	17.3932	0.881871	0.440935	−	0.897539i	$-0.354647\pi$
$389$	17.3932	0.881871	0.440935	+	0.897539i	$0.354647\pi$
$390$	0	0
$391$	−14.2330	−0.719793
$392$	1.69653	0.0856876
$393$	0	0
$394$	−1.43903	−0.0724971
$395$	−12.1129	−0.609464
$396$	0	0
$397$	−4.97781	−0.249829	−0.124915	−	0.992167i	$-0.539866\pi$
$397$	−4.97781	−0.249829	−0.124915	+	0.992167i	$0.539866\pi$
$398$	5.70309	0.285870
$399$	0	0
$400$	1.32257	0.0661284
$401$	−13.6687	−0.682582	−0.341291	−	0.939958i	$-0.610864\pi$
$401$	−13.6687	−0.682582	−0.341291	+	0.939958i	$0.610864\pi$
$402$	0	0
$403$	−4.11994	−0.205229
$404$	0.551550	0.0274406
$405$	0	0
$406$	−4.18535	−0.207715
$407$	11.0893	0.549677
$408$	0	0
$409$	15.1574	0.749485	0.374742	−	0.927129i	$-0.377731\pi$
$409$	15.1574	0.749485	0.374742	+	0.927129i	$0.377731\pi$
$410$	−1.09468	−0.0540622
$411$	0	0
$412$	12.7641	0.628844
$413$	−1.68946	−0.0831329
$414$	0	0
$415$	−14.8185	−0.727412
$416$	3.57895	0.175473
$417$	0	0
$418$	−10.0921	−0.493620
$419$	16.8436	0.822866	0.411433	−	0.911440i	$-0.365028\pi$
$419$	16.8436	0.822866	0.411433	+	0.911440i	$0.365028\pi$
$420$	0	0
$421$	−0.0881303	−0.00429521	−0.00214760	−	0.999998i	$-0.500684\pi$
$421$	−0.0881303	−0.00429521	−0.00214760	+	0.999998i	$0.500684\pi$
$422$	8.77779	0.427296
$423$	0	0
$424$	−17.9367	−0.871085
$425$	−1.59573	−0.0774043
$426$	0	0
$427$	4.42071	0.213933
$428$	−16.6333	−0.804002
$429$	0	0
$430$	−3.03206	−0.146219
$431$	28.1319	1.35506	0.677532	−	0.735493i	$-0.263049\pi$
$431$	28.1319	1.35506	0.677532	+	0.735493i	$0.263049\pi$
$432$	0	0
$433$	23.5326	1.13090	0.565452	−	0.824781i	$-0.308702\pi$
$433$	23.5326	1.13090	0.565452	+	0.824781i	$0.308702\pi$
$434$	−2.39584	−0.115004
$435$	0	0
$436$	18.9144	0.905837
$437$	−20.6928	−0.989872
$438$	0	0
$439$	−1.13019	−0.0539411	−0.0269705	−	0.999636i	$-0.508586\pi$
$439$	−1.13019	−0.0539411	−0.0269705	+	0.999636i	$0.508586\pi$
$440$	−17.9727	−0.856816
$441$	0	0
$442$	−1.17577	−0.0559259
$443$	25.2206	1.19827	0.599134	−	0.800649i	$-0.295512\pi$
$443$	25.2206	1.19827	0.599134	+	0.800649i	$0.295512\pi$
$444$	0	0
$445$	3.17729	0.150618
$446$	4.88824	0.231465
$447$	0	0
$448$	−3.60724	−0.170426
$449$	2.99798	0.141483	0.0707416	−	0.997495i	$-0.477463\pi$
$449$	2.99798	0.141483	0.0707416	+	0.997495i	$0.477463\pi$
$450$	0	0
$451$	−4.75397	−0.223856
$452$	15.6237	0.734879
$453$	0	0
$454$	−2.74608	−0.128880
$455$	−1.79440	−0.0841229
$456$	0	0
$457$	17.3210	0.810242	0.405121	−	0.914263i	$-0.367229\pi$
$457$	17.3210	0.810242	0.405121	+	0.914263i	$0.367229\pi$
$458$	−9.10612	−0.425501
$459$	0	0
$460$	−17.4597	−0.814065
$461$	−6.28572	−0.292755	−0.146377	−	0.989229i	$-0.546761\pi$
$461$	−6.28572	−0.292755	−0.146377	+	0.989229i	$0.546761\pi$
$462$	0	0
$463$	−11.3025	−0.525270	−0.262635	−	0.964895i	$-0.584591\pi$
$463$	−11.3025	−0.525270	−0.262635	+	0.964895i	$0.584591\pi$
$464$	26.6690	1.23808
$465$	0	0
$466$	−10.6824	−0.494851
$467$	0.664225	0.0307367	0.0153683	−	0.999882i	$-0.495108\pi$
$467$	0.664225	0.0307367	0.0153683	+	0.999882i	$0.495108\pi$
$468$	0	0
$469$	11.8999	0.549486
$470$	−1.24401	−0.0573821
$471$	0	0
$472$	−2.86622	−0.131928
$473$	−13.1677	−0.605450
$474$	0	0
$475$	−2.31998	−0.106448
$476$	6.17964	0.283243
$477$	0	0
$478$	1.55057	0.0709216
$479$	20.6417	0.943145	0.471572	−	0.881827i	$-0.343687\pi$
$479$	20.6417	0.943145	0.471572	+	0.881827i	$0.343687\pi$
$480$	0	0
$481$	1.87833	0.0856446
$482$	−4.84116	−0.220509
$483$	0	0
$484$	−17.1719	−0.780543
$485$	−20.9138	−0.949646
$486$	0	0
$487$	−33.0401	−1.49719	−0.748594	−	0.663028i	$-0.769271\pi$
$487$	−33.0401	−1.49719	−0.748594	+	0.663028i	$0.769271\pi$
$488$	7.49986	0.339503
$489$	0	0
$490$	−1.04348	−0.0471398
$491$	−38.1702	−1.72260	−0.861299	−	0.508099i	$-0.830348\pi$
$491$	−38.1702	−1.72260	−0.861299	+	0.508099i	$0.830348\pi$
$492$	0	0
$493$	−32.1772	−1.44919
$494$	−1.70942	−0.0769104
$495$	0	0
$496$	15.2663	0.685476
$497$	−5.39083	−0.241812
$498$	0	0
$499$	6.83156	0.305823	0.152911	−	0.988240i	$-0.451135\pi$
$499$	6.83156	0.305823	0.152911	+	0.988240i	$0.451135\pi$
$500$	19.0910	0.853774
$501$	0	0
$502$	5.03693	0.224809
$503$	−24.9027	−1.11036	−0.555178	−	0.831732i	$-0.687350\pi$
$503$	−24.9027	−1.11036	−0.555178	+	0.831732i	$0.687350\pi$
$504$	0	0
$505$	−0.716019	−0.0318624
$506$	8.38950	0.372959
$507$	0	0
$508$	−1.80076	−0.0798957
$509$	−19.4291	−0.861180	−0.430590	−	0.902548i	$-0.641694\pi$
$509$	−19.4291	−0.861180	−0.430590	+	0.902548i	$0.641694\pi$
$510$	0	0
$511$	5.28672	0.233871
$512$	−22.9123	−1.01259
$513$	0	0
$514$	−0.586459	−0.0258676
$515$	−16.5703	−0.730176
$516$	0	0
$517$	−5.40252	−0.237603
$518$	1.09229	0.0479925
$519$	0	0
$520$	−3.04426	−0.133499
$521$	−23.0002	−1.00766	−0.503828	−	0.863804i	$-0.668075\pi$
$521$	−23.0002	−1.00766	−0.503828	+	0.863804i	$0.668075\pi$
$522$	0	0
$523$	−24.0527	−1.05175	−0.525876	−	0.850561i	$-0.676262\pi$
$523$	−24.0527	−1.05175	−0.525876	+	0.850561i	$0.676262\pi$
$524$	18.5175	0.808941
$525$	0	0
$526$	−1.84533	−0.0804602
$527$	−18.4194	−0.802360
$528$	0	0
$529$	−5.79814	−0.252093
$530$	11.0324	0.479215
$531$	0	0
$532$	8.98437	0.389522
$533$	−0.805238	−0.0348787
$534$	0	0
$535$	21.5933	0.933560
$536$	20.1885	0.872010
$537$	0	0
$538$	−8.73992	−0.376805
$539$	−4.53166	−0.195192
$540$	0	0
$541$	29.8773	1.28452	0.642262	−	0.766485i	$-0.277996\pi$
$541$	29.8773	1.28452	0.642262	+	0.766485i	$0.277996\pi$
$542$	−11.3990	−0.489629
$543$	0	0
$544$	16.0007	0.686025
$545$	−24.5546	−1.05180
$546$	0	0
$547$	−23.4196	−1.00135	−0.500675	−	0.865635i	$-0.666915\pi$
$547$	−23.4196	−1.00135	−0.500675	+	0.865635i	$0.666915\pi$
$548$	−0.400867	−0.0171242
$549$	0	0
$550$	0.940589	0.0401068
$551$	−46.7814	−1.99295
$552$	0	0
$553$	−5.18146	−0.220338
$554$	3.64757	0.154971
$555$	0	0
$556$	−12.1311	−0.514474
$557$	18.7404	0.794055	0.397028	−	0.917807i	$-0.370042\pi$
$557$	18.7404	0.794055	0.397028	+	0.917807i	$0.370042\pi$
$558$	0	0
$559$	−2.23037	−0.0943345
$560$	6.64908	0.280975
$561$	0	0
$562$	−5.03843	−0.212534
$563$	−35.2058	−1.48375	−0.741874	−	0.670539i	$-0.766063\pi$
$563$	−35.2058	−1.48375	−0.741874	+	0.670539i	$0.766063\pi$
$564$	0	0
$565$	−20.2826	−0.853297
$566$	6.49506	0.273008
$567$	0	0
$568$	−9.14569	−0.383745
$569$	13.5337	0.567361	0.283681	−	0.958919i	$-0.408444\pi$
$569$	13.5337	0.567361	0.283681	+	0.958919i	$0.408444\pi$
$570$	0	0
$571$	−11.0291	−0.461552	−0.230776	−	0.973007i	$-0.574126\pi$
$571$	−11.0291	−0.461552	−0.230776	+	0.973007i	$0.574126\pi$
$572$	−6.26380	−0.261902
$573$	0	0
$574$	−0.468264	−0.0195449
$575$	1.92859	0.0804276
$576$	0	0
$577$	−33.1645	−1.38066	−0.690328	−	0.723496i	$-0.742534\pi$
$577$	−33.1645	−1.38066	−0.690328	+	0.723496i	$0.742534\pi$
$578$	2.33159	0.0969815
$579$	0	0
$580$	−39.4722	−1.63899
$581$	−6.33884	−0.262979
$582$	0	0
$583$	47.9115	1.98429
$584$	8.96908	0.371143
$585$	0	0
$586$	−7.06383	−0.291804
$587$	−17.6262	−0.727512	−0.363756	−	0.931494i	$-0.618506\pi$
$587$	−17.6262	−0.727512	−0.363756	+	0.931494i	$0.618506\pi$
$588$	0	0
$589$	−26.7793	−1.10342
$590$	1.76293	0.0725785
$591$	0	0
$592$	−6.96007	−0.286057
$593$	−11.9687	−0.491497	−0.245749	−	0.969334i	$-0.579034\pi$
$593$	−11.9687	−0.491497	−0.245749	+	0.969334i	$0.579034\pi$
$594$	0	0
$595$	−8.02238	−0.328885
$596$	7.66641	0.314028
$597$	0	0
$598$	1.42103	0.0581103
$599$	−19.2882	−0.788096	−0.394048	−	0.919090i	$-0.628926\pi$
$599$	−19.2882	−0.788096	−0.394048	+	0.919090i	$0.628926\pi$
$600$	0	0
$601$	−43.2184	−1.76292	−0.881458	−	0.472262i	$-0.843438\pi$
$601$	−43.2184	−1.76292	−0.881458	+	0.472262i	$0.843438\pi$
$602$	−1.29701	−0.0528621
$603$	0	0
$604$	33.3025	1.35506
$605$	22.2925	0.906320
$606$	0	0
$607$	−21.6332	−0.878064	−0.439032	−	0.898471i	$-0.644679\pi$
$607$	−21.6332	−0.878064	−0.439032	+	0.898471i	$0.644679\pi$
$608$	23.2629	0.943434
$609$	0	0
$610$	−4.61294	−0.186773
$611$	−0.915090	−0.0370206
$612$	0	0
$613$	−6.25536	−0.252651	−0.126326	−	0.991989i	$-0.540318\pi$
$613$	−6.25536	−0.252651	−0.126326	+	0.991989i	$0.540318\pi$
$614$	−4.57952	−0.184814
$615$	0	0
$616$	−7.68809	−0.309762
$617$	28.4663	1.14601	0.573005	−	0.819552i	$-0.305777\pi$
$617$	28.4663	1.14601	0.573005	+	0.819552i	$0.305777\pi$
$618$	0	0
$619$	11.3680	0.456920	0.228460	−	0.973553i	$-0.426631\pi$
$619$	11.3680	0.456920	0.228460	+	0.973553i	$0.426631\pi$
$620$	−22.5952	−0.907446
$621$	0	0
$622$	8.71128	0.349290
$623$	1.35913	0.0544525
$624$	0	0
$625$	−27.1088	−1.08435
$626$	11.0132	0.440174
$627$	0	0
$628$	19.4797	0.777326
$629$	8.39761	0.334834
$630$	0	0
$631$	25.9887	1.03459	0.517297	−	0.855806i	$-0.326938\pi$
$631$	25.9887	1.03459	0.517297	+	0.855806i	$0.326938\pi$
$632$	−8.79049	−0.349667
$633$	0	0
$634$	−9.55702	−0.379558
$635$	2.33773	0.0927701
$636$	0	0
$637$	−0.767582	−0.0304127
$638$	18.9666	0.750895
$639$	0	0
$640$	25.5641	1.01051
$641$	18.8497	0.744517	0.372258	−	0.928129i	$-0.378584\pi$
$641$	18.8497	0.744517	0.372258	+	0.928129i	$0.378584\pi$
$642$	0	0
$643$	9.58722	0.378083	0.189042	−	0.981969i	$-0.439462\pi$
$643$	9.58722	0.378083	0.189042	+	0.981969i	$0.439462\pi$
$644$	−7.46866	−0.294307
$645$	0	0
$646$	−7.64243	−0.300687
$647$	7.25215	0.285111	0.142556	−	0.989787i	$-0.454468\pi$
$647$	7.25215	0.285111	0.142556	+	0.989787i	$0.454468\pi$
$648$	0	0
$649$	7.65606	0.300527
$650$	0.159319	0.00624900
$651$	0	0
$652$	30.5045	1.19465
$653$	−15.6142	−0.611030	−0.305515	−	0.952187i	$-0.598829\pi$
$653$	−15.6142	−0.611030	−0.305515	+	0.952187i	$0.598829\pi$
$654$	0	0
$655$	−24.0393	−0.939294
$656$	2.98377	0.116497
$657$	0	0
$658$	−0.532145	−0.0207452
$659$	−19.6781	−0.766550	−0.383275	−	0.923634i	$-0.625204\pi$
$659$	−19.6781	−0.766550	−0.383275	+	0.923634i	$0.625204\pi$
$660$	0	0
$661$	16.6371	0.647110	0.323555	−	0.946209i	$-0.395122\pi$
$661$	16.6371	0.647110	0.323555	+	0.946209i	$0.395122\pi$
$662$	8.22291	0.319593
$663$	0	0
$664$	−10.7540	−0.417337
$665$	−11.6635	−0.452289
$666$	0	0
$667$	38.8891	1.50579
$668$	−14.6800	−0.567988
$669$	0	0
$670$	−12.4173	−0.479724
$671$	−20.0332	−0.773372
$672$	0	0
$673$	5.85137	0.225554	0.112777	−	0.993620i	$-0.464025\pi$
$673$	5.85137	0.225554	0.112777	+	0.993620i	$0.464025\pi$
$674$	10.5366	0.405854
$675$	0	0
$676$	22.3489	0.859572
$677$	−6.32236	−0.242988	−0.121494	−	0.992592i	$-0.538769\pi$
$677$	−6.32236	−0.242988	−0.121494	+	0.992592i	$0.538769\pi$
$678$	0	0
$679$	−8.94618	−0.343323
$680$	−13.6102	−0.521927
$681$	0	0
$682$	10.8571	0.415741
$683$	−47.3971	−1.81360	−0.906799	−	0.421563i	$-0.861482\pi$
$683$	−47.3971	−1.81360	−0.906799	+	0.421563i	$0.861482\pi$
$684$	0	0
$685$	0.520403	0.0198836
$686$	−0.446366	−0.0170423
$687$	0	0
$688$	8.26453	0.315082
$689$	8.11535	0.309170
$690$	0	0
$691$	−6.46620	−0.245986	−0.122993	−	0.992408i	$-0.539249\pi$
$691$	−6.46620	−0.245986	−0.122993	+	0.992408i	$0.539249\pi$
$692$	−28.9954	−1.10224
$693$	0	0
$694$	14.9764	0.568495
$695$	15.7486	0.597377
$696$	0	0
$697$	−3.60004	−0.136361
$698$	13.7165	0.519177
$699$	0	0
$700$	−0.837349	−0.0316488
$701$	18.8826	0.713188	0.356594	−	0.934260i	$-0.383938\pi$
$701$	18.8826	0.713188	0.356594	+	0.934260i	$0.383938\pi$
$702$	0	0
$703$	12.2090	0.460471
$704$	16.3468	0.616094
$705$	0	0
$706$	8.14091	0.306387
$707$	−0.306288	−0.0115191
$708$	0	0
$709$	5.01518	0.188349	0.0941746	−	0.995556i	$-0.469979\pi$
$709$	5.01518	0.188349	0.0941746	+	0.995556i	$0.469979\pi$
$710$	5.62525	0.211112
$711$	0	0
$712$	2.30581	0.0864138
$713$	22.2615	0.833699
$714$	0	0
$715$	8.13163	0.304106
$716$	12.1649	0.454623
$717$	0	0
$718$	−5.05097	−0.188500
$719$	48.7874	1.81946	0.909731	−	0.415199i	$-0.136288\pi$
$719$	48.7874	1.81946	0.909731	+	0.415199i	$0.136288\pi$
$720$	0	0
$721$	−7.08821	−0.263979
$722$	−2.63011	−0.0978825
$723$	0	0
$724$	29.7933	1.10726
$725$	4.36006	0.161928
$726$	0	0
$727$	17.0214	0.631288	0.315644	−	0.948878i	$-0.397780\pi$
$727$	17.0214	0.631288	0.315644	+	0.948878i	$0.397780\pi$
$728$	−1.30223	−0.0482637
$729$	0	0
$730$	−5.51661	−0.204179
$731$	−9.97148	−0.368808
$732$	0	0
$733$	8.82439	0.325936	0.162968	−	0.986631i	$-0.447893\pi$
$733$	8.82439	0.325936	0.162968	+	0.986631i	$0.447893\pi$
$734$	12.2908	0.453660
$735$	0	0
$736$	−19.3383	−0.712820
$737$	−53.9262	−1.98640
$738$	0	0
$739$	−44.0616	−1.62083	−0.810417	−	0.585854i	$-0.800759\pi$
$739$	−44.0616	−1.62083	−0.810417	+	0.585854i	$0.800759\pi$
$740$	10.3014	0.378688
$741$	0	0
$742$	4.71926	0.173249
$743$	8.57658	0.314644	0.157322	−	0.987547i	$-0.449714\pi$
$743$	8.57658	0.314644	0.157322	+	0.987547i	$0.449714\pi$
$744$	0	0
$745$	−9.95250	−0.364631
$746$	−0.857038	−0.0313784
$747$	0	0
$748$	−28.0040	−1.02393
$749$	9.23685	0.337507
$750$	0	0
$751$	−26.2586	−0.958191	−0.479095	−	0.877763i	$-0.659035\pi$
$751$	−26.2586	−0.958191	−0.479095	+	0.877763i	$0.659035\pi$
$752$	3.39083	0.123651
$753$	0	0
$754$	3.21260	0.116996
$755$	−43.2332	−1.57342
$756$	0	0
$757$	7.31126	0.265732	0.132866	−	0.991134i	$-0.457582\pi$
$757$	7.31126	0.265732	0.132866	+	0.991134i	$0.457582\pi$
$758$	−2.31663	−0.0841439
$759$	0	0
$760$	−19.7874	−0.717764
$761$	−10.9355	−0.396410	−0.198205	−	0.980161i	$-0.563511\pi$
$761$	−10.9355	−0.396410	−0.198205	+	0.980161i	$0.563511\pi$
$762$	0	0
$763$	−10.5036	−0.380256
$764$	7.22905	0.261538
$765$	0	0
$766$	16.6691	0.602279
$767$	1.29680	0.0468247
$768$	0	0
$769$	−52.1535	−1.88070	−0.940352	−	0.340203i	$-0.889504\pi$
$769$	−52.1535	−1.88070	−0.940352	+	0.340203i	$0.889504\pi$
$770$	4.72872	0.170411
$771$	0	0
$772$	4.68664	0.168676
$773$	5.87993	0.211486	0.105743	−	0.994393i	$-0.466278\pi$
$773$	5.87993	0.211486	0.105743	+	0.994393i	$0.466278\pi$
$774$	0	0
$775$	2.49584	0.0896534
$776$	−15.1774	−0.544839
$777$	0	0
$778$	−7.76374	−0.278343
$779$	−5.23397	−0.187527
$780$	0	0
$781$	24.4294	0.874153
$782$	6.35312	0.227187
$783$	0	0
$784$	2.84424	0.101580
$785$	−25.2885	−0.902585
$786$	0	0
$787$	−33.5039	−1.19428	−0.597142	−	0.802135i	$-0.703697\pi$
$787$	−33.5039	−1.19428	−0.597142	+	0.802135i	$0.703697\pi$
$788$	−5.80541	−0.206809
$789$	0	0
$790$	5.40677	0.192364
$791$	−8.67620	−0.308490
$792$	0	0
$793$	−3.39326	−0.120498
$794$	2.22193	0.0788532
$795$	0	0
$796$	23.0078	0.815488
$797$	40.6842	1.44111	0.720554	−	0.693399i	$-0.243888\pi$
$797$	40.6842	1.44111	0.720554	+	0.693399i	$0.243888\pi$
$798$	0	0
$799$	−4.09117	−0.144735
$800$	−2.16811	−0.0766544
$801$	0	0
$802$	6.10124	0.215442
$803$	−23.9576	−0.845447
$804$	0	0
$805$	9.69578	0.341731
$806$	1.83900	0.0647761
$807$	0	0
$808$	−0.519626	−0.0182804
$809$	−0.118941	−0.00418173	−0.00209087	−	0.999998i	$-0.500666\pi$
$809$	−0.118941	−0.00418173	−0.00209087	+	0.999998i	$0.500666\pi$
$810$	0	0
$811$	−19.5686	−0.687148	−0.343574	−	0.939126i	$-0.611638\pi$
$811$	−19.5686	−0.687148	−0.343574	+	0.939126i	$0.611638\pi$
$812$	−16.8848	−0.592540
$813$	0	0
$814$	−4.94989	−0.173494
$815$	−39.6008	−1.38716
$816$	0	0
$817$	−14.4972	−0.507192
$818$	−6.76574	−0.236559
$819$	0	0
$820$	−4.41621	−0.154221
$821$	48.0569	1.67720	0.838599	−	0.544749i	$-0.183375\pi$
$821$	48.0569	1.67720	0.838599	+	0.544749i	$0.183375\pi$
$822$	0	0
$823$	−48.8400	−1.70246	−0.851228	−	0.524797i	$-0.824141\pi$
$823$	−48.8400	−1.70246	−0.851228	+	0.524797i	$0.824141\pi$
$824$	−12.0253	−0.418923
$825$	0	0
$826$	0.754117	0.0262391
$827$	18.4660	0.642125	0.321063	−	0.947058i	$-0.395960\pi$
$827$	18.4660	0.642125	0.321063	+	0.947058i	$0.395960\pi$
$828$	0	0
$829$	31.9964	1.11128	0.555641	−	0.831423i	$-0.312473\pi$
$829$	31.9964	1.11128	0.555641	+	0.831423i	$0.312473\pi$
$830$	6.61448	0.229592
$831$	0	0
$832$	2.76886	0.0959929
$833$	−3.43169	−0.118901
$834$	0	0
$835$	19.0576	0.659514
$836$	−40.7141	−1.40813
$837$	0	0
$838$	−7.51843	−0.259720
$839$	−49.5290	−1.70993	−0.854965	−	0.518685i	$-0.826422\pi$
$839$	−49.5290	−1.70993	−0.854965	+	0.518685i	$0.826422\pi$
$840$	0	0
$841$	58.9188	2.03168
$842$	0.0393384	0.00135569
$843$	0	0
$844$	35.4119	1.21893
$845$	−29.0132	−0.998084
$846$	0	0
$847$	9.53595	0.327659
$848$	−30.0711	−1.03264
$849$	0	0
$850$	0.712279	0.0244310
$851$	−10.1493	−0.347913
$852$	0	0
$853$	−44.7512	−1.53225	−0.766126	−	0.642691i	$-0.777818\pi$
$853$	−44.7512	−1.53225	−0.766126	+	0.642691i	$0.777818\pi$
$854$	−1.97325	−0.0675234
$855$	0	0
$856$	15.6706	0.535609
$857$	−36.3915	−1.24311	−0.621555	−	0.783371i	$-0.713499\pi$
$857$	−36.3915	−1.24311	−0.621555	+	0.783371i	$0.713499\pi$
$858$	0	0
$859$	41.6322	1.42047	0.710237	−	0.703963i	$-0.248588\pi$
$859$	41.6322	1.42047	0.710237	+	0.703963i	$0.248588\pi$
$860$	−12.2321	−0.417112
$861$	0	0
$862$	−12.5571	−0.427697
$863$	27.8245	0.947158	0.473579	−	0.880751i	$-0.342962\pi$
$863$	27.8245	0.947158	0.473579	+	0.880751i	$0.342962\pi$
$864$	0	0
$865$	37.6417	1.27985
$866$	−10.5042	−0.356946
$867$	0	0
$868$	−9.66544	−0.328066
$869$	23.4806	0.796525
$870$	0	0
$871$	−9.13414	−0.309499
$872$	−17.8197	−0.603450
$873$	0	0
$874$	9.23658	0.312432
$875$	−10.6016	−0.358400
$876$	0	0
$877$	17.1138	0.577893	0.288946	−	0.957345i	$-0.406695\pi$
$877$	17.1138	0.577893	0.288946	+	0.957345i	$0.406695\pi$
$878$	0.504479	0.0170253
$879$	0	0
$880$	−30.1314	−1.01573
$881$	26.9903	0.909327	0.454663	−	0.890663i	$-0.349760\pi$
$881$	26.9903	0.909327	0.454663	+	0.890663i	$0.349760\pi$
$882$	0	0
$883$	−35.5601	−1.19669	−0.598346	−	0.801238i	$-0.704175\pi$
$883$	−35.5601	−1.19669	−0.598346	+	0.801238i	$0.704175\pi$
$884$	−4.74338	−0.159537
$885$	0	0
$886$	−11.2576	−0.378207
$887$	−7.42653	−0.249359	−0.124679	−	0.992197i	$-0.539790\pi$
$887$	−7.42653	−0.249359	−0.124679	+	0.992197i	$0.539790\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−1.41823	−0.0475393
$891$	0	0
$892$	19.7204	0.660289
$893$	−5.94800	−0.199042
$894$	0	0
$895$	−15.7924	−0.527881
$896$	10.9354	0.365326
$897$	0	0
$898$	−1.33819	−0.0446561
$899$	50.3277	1.67852
$900$	0	0
$901$	36.2819	1.20873
$902$	2.12201	0.0706553
$903$	0	0
$904$	−14.7194	−0.489561
$905$	−38.6775	−1.28568
$906$	0	0
$907$	36.3294	1.20630	0.603149	−	0.797628i	$-0.293912\pi$
$907$	36.3294	1.20630	0.603149	+	0.797628i	$0.293912\pi$
$908$	−11.0784	−0.367651
$909$	0	0
$910$	0.800960	0.0265516
$911$	5.05197	0.167379	0.0836896	−	0.996492i	$-0.473330\pi$
$911$	5.05197	0.167379	0.0836896	+	0.996492i	$0.473330\pi$
$912$	0	0
$913$	28.7255	0.950674
$914$	−7.73150	−0.255735
$915$	0	0
$916$	−36.7365	−1.21381
$917$	−10.2832	−0.339580
$918$	0	0
$919$	41.5970	1.37216	0.686079	−	0.727527i	$-0.259330\pi$
$919$	41.5970	1.37216	0.686079	+	0.727527i	$0.259330\pi$
$920$	16.4492	0.542313
$921$	0	0
$922$	2.80573	0.0924018
$923$	4.13790	0.136201
$924$	0	0
$925$	−1.13789	−0.0374134
$926$	5.04503	0.165790
$927$	0	0
$928$	−43.7192	−1.43515
$929$	20.6724	0.678239	0.339120	−	0.940743i	$-0.389871\pi$
$929$	20.6724	0.678239	0.339120	+	0.940743i	$0.389871\pi$
$930$	0	0
$931$	−4.98922	−0.163515
$932$	−43.0955	−1.41164
$933$	0	0
$934$	−0.296487	−0.00970136
$935$	36.3547	1.18893
$936$	0	0
$937$	−2.93386	−0.0958452	−0.0479226	−	0.998851i	$-0.515260\pi$
$937$	−2.93386	−0.0958452	−0.0479226	+	0.998851i	$0.515260\pi$
$938$	−5.31170	−0.173433
$939$	0	0
$940$	−5.01868	−0.163691
$941$	10.4399	0.340329	0.170165	−	0.985416i	$-0.445570\pi$
$941$	10.4399	0.340329	0.170165	+	0.985416i	$0.445570\pi$
$942$	0	0
$943$	4.35098	0.141687
$944$	−4.80523	−0.156397
$945$	0	0
$946$	5.87760	0.191097
$947$	−25.7647	−0.837240	−0.418620	−	0.908161i	$-0.637486\pi$
$947$	−25.7647	−0.837240	−0.418620	+	0.908161i	$0.637486\pi$
$948$	0	0
$949$	−4.05800	−0.131728
$950$	1.03556	0.0335979
$951$	0	0
$952$	−5.82196	−0.188691
$953$	−46.8318	−1.51703	−0.758516	−	0.651655i	$-0.774075\pi$
$953$	−46.8318	−1.51703	−0.758516	+	0.651655i	$0.774075\pi$
$954$	0	0
$955$	−9.38471	−0.303682
$956$	6.25542	0.202315
$957$	0	0
$958$	−9.21377	−0.297683
$959$	0.222610	0.00718846
$960$	0	0
$961$	−2.19070	−0.0706678
$962$	−0.838423	−0.0270319
$963$	0	0
$964$	−19.5305	−0.629036
$965$	−6.08416	−0.195856
$966$	0	0
$967$	16.2505	0.522581	0.261290	−	0.965260i	$-0.415852\pi$
$967$	16.2505	0.522581	0.261290	+	0.965260i	$0.415852\pi$
$968$	16.1780	0.519981
$969$	0	0
$970$	9.33520	0.299735
$971$	−28.9833	−0.930118	−0.465059	−	0.885280i	$-0.653967\pi$
$971$	−28.9833	−0.930118	−0.465059	+	0.885280i	$0.653967\pi$
$972$	0	0
$973$	6.73668	0.215968
$974$	14.7480	0.472555
$975$	0	0
$976$	12.5736	0.402470
$977$	41.7762	1.33654	0.668269	−	0.743919i	$-0.267035\pi$
$977$	41.7762	1.33654	0.668269	+	0.743919i	$0.267035\pi$
$978$	0	0
$979$	−6.15913	−0.196847
$980$	−4.20969	−0.134474
$981$	0	0
$982$	17.0379	0.543701
$983$	5.70314	0.181902	0.0909509	−	0.995855i	$-0.471009\pi$
$983$	5.70314	0.181902	0.0909509	+	0.995855i	$0.471009\pi$
$984$	0	0
$985$	7.53655	0.240134
$986$	14.3628	0.457406
$987$	0	0
$988$	−6.89624	−0.219399
$989$	12.0514	0.383214
$990$	0	0
$991$	10.9263	0.347087	0.173543	−	0.984826i	$-0.444478\pi$
$991$	10.9263	0.347087	0.173543	+	0.984826i	$0.444478\pi$
$992$	−25.0263	−0.794587
$993$	0	0
$994$	2.40628	0.0763226
$995$	−29.8685	−0.946897
$996$	0	0
$997$	4.34051	0.137465	0.0687327	−	0.997635i	$-0.478104\pi$
$997$	4.34051	0.137465	0.0687327	+	0.997635i	$0.478104\pi$
$998$	−3.04938	−0.0965263
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.10	✓	22
3.2	odd	2	inner	8001.2.a.x.1.13	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.10	✓	22	1.1	even	1	trivial
8001.2.a.x.1.13	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)

\(f(q)\)	\(=\)	\(q-0.446366 q^{2} -1.80076 q^{4} +2.33773 q^{5} +1.00000 q^{7} +1.69653 q^{8} +O(q^{10})\)	\(q-0.446366 q^{2} -1.80076 q^{4} +2.33773 q^{5} +1.00000 q^{7} +1.69653 q^{8} -1.04348 q^{10} -4.53166 q^{11} -0.767582 q^{13} -0.446366 q^{14} +2.84424 q^{16} -3.43169 q^{17} -4.98922 q^{19} -4.20969 q^{20} +2.02278 q^{22} +4.14751 q^{23} +0.464998 q^{25} +0.342623 q^{26} -1.80076 q^{28} +9.37650 q^{29} +5.36743 q^{31} -4.66263 q^{32} +1.53179 q^{34} +2.33773 q^{35} -2.44708 q^{37} +2.22702 q^{38} +3.96603 q^{40} +1.04906 q^{41} +2.90570 q^{43} +8.16042 q^{44} -1.85131 q^{46} +1.19217 q^{47} +1.00000 q^{49} -0.207559 q^{50} +1.38223 q^{52} -10.5726 q^{53} -10.5938 q^{55} +1.69653 q^{56} -4.18535 q^{58} -1.68946 q^{59} +4.42071 q^{61} -2.39584 q^{62} -3.60724 q^{64} -1.79440 q^{65} +11.8999 q^{67} +6.17964 q^{68} -1.04348 q^{70} -5.39083 q^{71} +5.28672 q^{73} +1.09229 q^{74} +8.98437 q^{76} -4.53166 q^{77} -5.18146 q^{79} +6.64908 q^{80} -0.468264 q^{82} -6.33884 q^{83} -8.02238 q^{85} -1.29701 q^{86} -7.68809 q^{88} +1.35913 q^{89} -0.767582 q^{91} -7.46866 q^{92} -0.532145 q^{94} -11.6635 q^{95} -8.94618 q^{97} -0.446366 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) 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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