LMFDB - Embedded newform 3627.2.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3627,2,Mod(1,3627)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3627, 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("3627.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3627 = 3^{2} \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3627.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:	$28.9617408131$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 37x^{4} - 33x^{3} - 36x^{2} + 33x - 4 $ x^8 - x^7 - 11x^6 + 10x^5 + 37x^4 - 33x^3 - 36x^2 + 33x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.656875$ of defining polynomial
Character	$\chi$	$=$	3627.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.656875 q^{2} -1.56852 q^{4} +1.55873 q^{5} +2.17607 q^{7} -2.34407 q^{8} +O(q^{10})$	\(q+0.656875 q^{2} -1.56852 q^{4} +1.55873 q^{5} +2.17607 q^{7} -2.34407 q^{8} +1.02389 q^{10} -0.877966 q^{11} +1.00000 q^{13} +1.42941 q^{14} +1.59727 q^{16} -4.00763 q^{17} -0.601130 q^{19} -2.44490 q^{20} -0.576714 q^{22} -7.74079 q^{23} -2.57035 q^{25} +0.656875 q^{26} -3.41320 q^{28} -4.35321 q^{29} -1.00000 q^{31} +5.73734 q^{32} -2.63251 q^{34} +3.39192 q^{35} -2.60850 q^{37} -0.394867 q^{38} -3.65378 q^{40} +5.00956 q^{41} -2.25813 q^{43} +1.37710 q^{44} -5.08473 q^{46} +1.99049 q^{47} -2.26472 q^{49} -1.68840 q^{50} -1.56852 q^{52} +3.22380 q^{53} -1.36852 q^{55} -5.10086 q^{56} -2.85951 q^{58} -6.18276 q^{59} +2.46585 q^{61} -0.656875 q^{62} +0.574175 q^{64} +1.55873 q^{65} -6.59746 q^{67} +6.28603 q^{68} +2.22806 q^{70} +0.570087 q^{71} -6.17960 q^{73} -1.71346 q^{74} +0.942882 q^{76} -1.91052 q^{77} +9.85609 q^{79} +2.48972 q^{80} +3.29065 q^{82} -7.25438 q^{83} -6.24683 q^{85} -1.48331 q^{86} +2.05801 q^{88} +16.2492 q^{89} +2.17607 q^{91} +12.1415 q^{92} +1.30750 q^{94} -0.937002 q^{95} -15.9725 q^{97} -1.48764 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 7 q^{4} - 11 q^{5} - 2 q^{7}+O(q^{10}) $ 8 * q + q^2 + 7 * q^4 - 11 * q^5 - 2 * q^7	$ 8 q + q^{2} + 7 q^{4} - 11 q^{5} - 2 q^{7} - 2 q^{10} + 2 q^{11} + 8 q^{13} - 3 q^{14} - 7 q^{16} - 7 q^{17} - 5 q^{19} - 6 q^{20} - 4 q^{22} - 14 q^{23} + 17 q^{25} + q^{26} - 9 q^{28} - 12 q^{29} - 8 q^{31} + 21 q^{32} - 12 q^{34} + q^{35} + 2 q^{37} - 24 q^{38} - 19 q^{40} - 13 q^{41} - 5 q^{43} - 22 q^{44} + 17 q^{46} - 23 q^{47} + 26 q^{49} + 26 q^{50} + 7 q^{52} - 25 q^{53} - 17 q^{55} - 8 q^{56} - 29 q^{58} + 5 q^{59} - 9 q^{61} - q^{62} - 14 q^{64} - 11 q^{65} + 22 q^{67} + 6 q^{68} - 29 q^{70} + 17 q^{71} - 27 q^{73} - 14 q^{74} - 36 q^{76} - 31 q^{77} - 23 q^{79} - 9 q^{80} + 18 q^{82} + 25 q^{83} + 13 q^{85} - 11 q^{86} + 5 q^{88} - 2 q^{89} - 2 q^{91} + 20 q^{92} - 38 q^{94} - 3 q^{95} - 15 q^{97} - 39 q^{98}+O(q^{100}) $ 8 * q + q^2 + 7 * q^4 - 11 * q^5 - 2 * q^7 - 2 * q^10 + 2 * q^11 + 8 * q^13 - 3 * q^14 - 7 * q^16 - 7 * q^17 - 5 * q^19 - 6 * q^20 - 4 * q^22 - 14 * q^23 + 17 * q^25 + q^26 - 9 * q^28 - 12 * q^29 - 8 * q^31 + 21 * q^32 - 12 * q^34 + q^35 + 2 * q^37 - 24 * q^38 - 19 * q^40 - 13 * q^41 - 5 * q^43 - 22 * q^44 + 17 * q^46 - 23 * q^47 + 26 * q^49 + 26 * q^50 + 7 * q^52 - 25 * q^53 - 17 * q^55 - 8 * q^56 - 29 * q^58 + 5 * q^59 - 9 * q^61 - q^62 - 14 * q^64 - 11 * q^65 + 22 * q^67 + 6 * q^68 - 29 * q^70 + 17 * q^71 - 27 * q^73 - 14 * q^74 - 36 * q^76 - 31 * q^77 - 23 * q^79 - 9 * q^80 + 18 * q^82 + 25 * q^83 + 13 * q^85 - 11 * q^86 + 5 * q^88 - 2 * q^89 - 2 * q^91 + 20 * q^92 - 38 * q^94 - 3 * q^95 - 15 * q^97 - 39 * q^98

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 $

$2$	0.656875	0.464481	0.232240	−	0.972658i	$-0.425394\pi$
$2$	0.656875	0.464481	0.232240	+	0.972658i	$0.425394\pi$
$3$	0	0
$3$	0	0
$4$	−1.56852	−0.784258
$5$	1.55873	0.697087	0.348544	−	0.937293i	$-0.386676\pi$
$5$	1.55873	0.697087	0.348544	+	0.937293i	$0.386676\pi$
$6$	0	0
$7$	2.17607	0.822477	0.411239	−	0.911528i	$-0.365096\pi$
$7$	2.17607	0.822477	0.411239	+	0.911528i	$0.365096\pi$
$8$	−2.34407	−0.828753
$9$	0	0
$10$	1.02389	0.323784
$11$	−0.877966	−0.264717	−0.132358	−	0.991202i	$-0.542255\pi$
$11$	−0.877966	−0.264717	−0.132358	+	0.991202i	$0.542255\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.42941	0.382025
$15$	0	0
$16$	1.59727	0.399318
$17$	−4.00763	−0.971993	−0.485996	−	0.873961i	$-0.661543\pi$
$17$	−4.00763	−0.971993	−0.485996	+	0.873961i	$0.661543\pi$
$18$	0	0
$19$	−0.601130	−0.137909	−0.0689543	−	0.997620i	$-0.521966\pi$
$19$	−0.601130	−0.137909	−0.0689543	+	0.997620i	$0.521966\pi$
$20$	−2.44490	−0.546696
$21$	0	0
$22$	−0.576714	−0.122956
$23$	−7.74079	−1.61407	−0.807033	−	0.590507i	$-0.798928\pi$
$23$	−7.74079	−1.61407	−0.807033	+	0.590507i	$0.798928\pi$
$24$	0	0
$25$	−2.57035	−0.514070
$26$	0.656875	0.128824
$27$	0	0
$28$	−3.41320	−0.645034
$29$	−4.35321	−0.808371	−0.404185	−	0.914677i	$-0.632445\pi$
$29$	−4.35321	−0.808371	−0.404185	+	0.914677i	$0.632445\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.73734	1.01423
$33$	0	0
$34$	−2.63251	−0.451472
$35$	3.39192	0.573338
$36$	0	0
$37$	−2.60850	−0.428834	−0.214417	−	0.976742i	$-0.568785\pi$
$37$	−2.60850	−0.428834	−0.214417	+	0.976742i	$0.568785\pi$
$38$	−0.394867	−0.0640559
$39$	0	0
$40$	−3.65378	−0.577713
$41$	5.00956	0.782361	0.391181	−	0.920314i	$-0.372067\pi$
$41$	5.00956	0.782361	0.391181	+	0.920314i	$0.372067\pi$
$42$	0	0
$43$	−2.25813	−0.344362	−0.172181	−	0.985065i	$-0.555081\pi$
$43$	−2.25813	−0.344362	−0.172181	+	0.985065i	$0.555081\pi$
$44$	1.37710	0.207606
$45$	0	0
$46$	−5.08473	−0.749702
$47$	1.99049	0.290342	0.145171	−	0.989407i	$-0.453627\pi$
$47$	1.99049	0.290342	0.145171	+	0.989407i	$0.453627\pi$
$48$	0	0
$49$	−2.26472	−0.323531
$50$	−1.68840	−0.238775
$51$	0	0
$52$	−1.56852	−0.217514
$53$	3.22380	0.442823	0.221411	−	0.975180i	$-0.428934\pi$
$53$	3.22380	0.442823	0.221411	+	0.975180i	$0.428934\pi$
$54$	0	0
$55$	−1.36852	−0.184531
$56$	−5.10086	−0.681631
$57$	0	0
$58$	−2.85951	−0.375473
$59$	−6.18276	−0.804927	−0.402464	−	0.915436i	$-0.631846\pi$
$59$	−6.18276	−0.804927	−0.402464	+	0.915436i	$0.631846\pi$
$60$	0	0
$61$	2.46585	0.315720	0.157860	−	0.987462i	$-0.449541\pi$
$61$	2.46585	0.315720	0.157860	+	0.987462i	$0.449541\pi$
$62$	−0.656875	−0.0834232
$63$	0	0
$64$	0.574175	0.0717718
$65$	1.55873	0.193337
$66$	0	0
$67$	−6.59746	−0.806007	−0.403004	−	0.915198i	$-0.632034\pi$
$67$	−6.59746	−0.806007	−0.403004	+	0.915198i	$0.632034\pi$
$68$	6.28603	0.762293
$69$	0	0
$70$	2.22806	0.266305
$71$	0.570087	0.0676569	0.0338284	−	0.999428i	$-0.489230\pi$
$71$	0.570087	0.0676569	0.0338284	+	0.999428i	$0.489230\pi$
$72$	0	0
$73$	−6.17960	−0.723267	−0.361633	−	0.932320i	$-0.617781\pi$
$73$	−6.17960	−0.723267	−0.361633	+	0.932320i	$0.617781\pi$
$74$	−1.71346	−0.199185
$75$	0	0
$76$	0.942882	0.108156
$77$	−1.91052	−0.217723
$78$	0	0
$79$	9.85609	1.10890	0.554448	−	0.832218i	$-0.312929\pi$
$79$	9.85609	1.10890	0.554448	+	0.832218i	$0.312929\pi$
$80$	2.48972	0.278359
$81$	0	0
$82$	3.29065	0.363392
$83$	−7.25438	−0.796272	−0.398136	−	0.917326i	$-0.630343\pi$
$83$	−7.25438	−0.796272	−0.398136	+	0.917326i	$0.630343\pi$
$84$	0	0
$85$	−6.24683	−0.677564
$86$	−1.48331	−0.159949
$87$	0	0
$88$	2.05801	0.219385
$89$	16.2492	1.72242	0.861208	−	0.508252i	$-0.169708\pi$
$89$	16.2492	1.72242	0.861208	+	0.508252i	$0.169708\pi$
$90$	0	0
$91$	2.17607	0.228114
$92$	12.1415	1.26584
$93$	0	0
$94$	1.30750	0.134858
$95$	−0.937002	−0.0961344
$96$	0	0
$97$	−15.9725	−1.62176	−0.810882	−	0.585210i	$-0.801012\pi$
$97$	−15.9725	−1.62176	−0.810882	+	0.585210i	$0.801012\pi$
$98$	−1.48764	−0.150274
$99$	0	0
$100$	4.03163	0.403163
$101$	−0.816483	−0.0812431	−0.0406215	−	0.999175i	$-0.512934\pi$
$101$	−0.816483	−0.0812431	−0.0406215	+	0.999175i	$0.512934\pi$
$102$	0	0
$103$	−0.363514	−0.0358181	−0.0179090	−	0.999840i	$-0.505701\pi$
$103$	−0.363514	−0.0358181	−0.0179090	+	0.999840i	$0.505701\pi$
$104$	−2.34407	−0.229855
$105$	0	0
$106$	2.11763	0.205683
$107$	17.8288	1.72357	0.861787	−	0.507270i	$-0.169345\pi$
$107$	17.8288	1.72357	0.861787	+	0.507270i	$0.169345\pi$
$108$	0	0
$109$	−10.3157	−0.988069	−0.494034	−	0.869442i	$-0.664478\pi$
$109$	−10.3157	−0.988069	−0.494034	+	0.869442i	$0.664478\pi$
$110$	−0.898944	−0.0857109
$111$	0	0
$112$	3.47577	0.328430
$113$	−15.5910	−1.46668	−0.733339	−	0.679863i	$-0.762039\pi$
$113$	−15.5910	−1.46668	−0.733339	+	0.679863i	$0.762039\pi$
$114$	0	0
$115$	−12.0658	−1.12514
$116$	6.82808	0.633971
$117$	0	0
$118$	−4.06130	−0.373873
$119$	−8.72088	−0.799442
$120$	0	0
$121$	−10.2292	−0.929925
$122$	1.61975	0.146646
$123$	0	0
$124$	1.56852	0.140857
$125$	−11.8002	−1.05544
$126$	0	0
$127$	−9.71552	−0.862113	−0.431057	−	0.902325i	$-0.641859\pi$
$127$	−9.71552	−0.862113	−0.431057	+	0.902325i	$0.641859\pi$
$128$	−11.0975	−0.980892
$129$	0	0
$130$	1.02389	0.0898014
$131$	−13.5147	−1.18078	−0.590392	−	0.807117i	$-0.701027\pi$
$131$	−13.5147	−1.18078	−0.590392	+	0.807117i	$0.701027\pi$
$132$	0	0
$133$	−1.30810	−0.113427
$134$	−4.33370	−0.374375
$135$	0	0
$136$	9.39415	0.805542
$137$	5.15662	0.440560	0.220280	−	0.975437i	$-0.429303\pi$
$137$	5.15662	0.440560	0.220280	+	0.975437i	$0.429303\pi$
$138$	0	0
$139$	22.2319	1.88568	0.942842	−	0.333241i	$-0.108142\pi$
$139$	22.2319	1.88568	0.942842	+	0.333241i	$0.108142\pi$
$140$	−5.32027	−0.449645
$141$	0	0
$142$	0.374476	0.0314253
$143$	−0.877966	−0.0734192
$144$	0	0
$145$	−6.78550	−0.563505
$146$	−4.05922	−0.335944
$147$	0	0
$148$	4.09147	0.336317
$149$	−4.62898	−0.379221	−0.189611	−	0.981859i	$-0.560723\pi$
$149$	−4.62898	−0.379221	−0.189611	+	0.981859i	$0.560723\pi$
$150$	0	0
$151$	−4.33901	−0.353104	−0.176552	−	0.984291i	$-0.556494\pi$
$151$	−4.33901	−0.353104	−0.176552	+	0.984291i	$0.556494\pi$
$152$	1.40909	0.114292
$153$	0	0
$154$	−1.25497	−0.101128
$155$	−1.55873	−0.125201
$156$	0	0
$157$	−12.7698	−1.01914	−0.509572	−	0.860428i	$-0.670196\pi$
$157$	−12.7698	−1.01914	−0.509572	+	0.860428i	$0.670196\pi$
$158$	6.47422	0.515061
$159$	0	0
$160$	8.94299	0.707006
$161$	−16.8445	−1.32753
$162$	0	0
$163$	−19.4359	−1.52234	−0.761168	−	0.648554i	$-0.775374\pi$
$163$	−19.4359	−1.52234	−0.761168	+	0.648554i	$0.775374\pi$
$164$	−7.85756	−0.613573
$165$	0	0
$166$	−4.76522	−0.369853
$167$	11.6238	0.899476	0.449738	−	0.893160i	$-0.351517\pi$
$167$	11.6238	0.899476	0.449738	+	0.893160i	$0.351517\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.10338	−0.314715
$171$	0	0
$172$	3.54191	0.270068
$173$	−5.18733	−0.394385	−0.197193	−	0.980365i	$-0.563182\pi$
$173$	−5.18733	−0.394385	−0.197193	+	0.980365i	$0.563182\pi$
$174$	0	0
$175$	−5.59326	−0.422811
$176$	−1.40235	−0.105706
$177$	0	0
$178$	10.6737	0.800029
$179$	−20.0692	−1.50005	−0.750023	−	0.661412i	$-0.769958\pi$
$179$	−20.0692	−1.50005	−0.750023	+	0.661412i	$0.769958\pi$
$180$	0	0
$181$	17.1393	1.27395	0.636976	−	0.770883i	$-0.280185\pi$
$181$	17.1393	1.27395	0.636976	+	0.770883i	$0.280185\pi$
$182$	1.42941	0.105955
$183$	0	0
$184$	18.1449	1.33766
$185$	−4.06596	−0.298935
$186$	0	0
$187$	3.51856	0.257303
$188$	−3.12211	−0.227703
$189$	0	0
$190$	−0.615493	−0.0446526
$191$	−0.321161	−0.0232384	−0.0116192	−	0.999932i	$-0.503699\pi$
$191$	−0.321161	−0.0232384	−0.0116192	+	0.999932i	$0.503699\pi$
$192$	0	0
$193$	−4.54684	−0.327289	−0.163644	−	0.986519i	$-0.552325\pi$
$193$	−4.54684	−0.327289	−0.163644	+	0.986519i	$0.552325\pi$
$194$	−10.4920	−0.753278
$195$	0	0
$196$	3.55224	0.253732
$197$	−9.75213	−0.694811	−0.347405	−	0.937715i	$-0.612937\pi$
$197$	−9.75213	−0.694811	−0.347405	+	0.937715i	$0.612937\pi$
$198$	0	0
$199$	−7.30281	−0.517683	−0.258841	−	0.965920i	$-0.583341\pi$
$199$	−7.30281	−0.517683	−0.258841	+	0.965920i	$0.583341\pi$
$200$	6.02507	0.426037
$201$	0	0
$202$	−0.536327	−0.0377358
$203$	−9.47289	−0.664867
$204$	0	0
$205$	7.80856	0.545374
$206$	−0.238783	−0.0166368
$207$	0	0
$208$	1.59727	0.110751
$209$	0.527772	0.0365067
$210$	0	0
$211$	−14.7678	−1.01666	−0.508330	−	0.861162i	$-0.669737\pi$
$211$	−14.7678	−1.01666	−0.508330	+	0.861162i	$0.669737\pi$
$212$	−5.05658	−0.347287
$213$	0	0
$214$	11.7113	0.800567
$215$	−3.51983	−0.240050
$216$	0	0
$217$	−2.17607	−0.147721
$218$	−6.77615	−0.458939
$219$	0	0
$220$	2.14654	0.144720
$221$	−4.00763	−0.269582
$222$	0	0
$223$	24.6628	1.65155	0.825773	−	0.564003i	$-0.190739\pi$
$223$	24.6628	1.65155	0.825773	+	0.564003i	$0.190739\pi$
$224$	12.4849	0.834180
$225$	0	0
$226$	−10.2413	−0.681244
$227$	−1.39159	−0.0923628	−0.0461814	−	0.998933i	$-0.514705\pi$
$227$	−1.39159	−0.0923628	−0.0461814	+	0.998933i	$0.514705\pi$
$228$	0	0
$229$	24.9011	1.64551	0.822756	−	0.568394i	$-0.192435\pi$
$229$	24.9011	1.64551	0.822756	+	0.568394i	$0.192435\pi$
$230$	−7.92574	−0.522608
$231$	0	0
$232$	10.2042	0.669940
$233$	22.4331	1.46964	0.734820	−	0.678263i	$-0.237267\pi$
$233$	22.4331	1.46964	0.734820	+	0.678263i	$0.237267\pi$
$234$	0	0
$235$	3.10264	0.202394
$236$	9.69776	0.631270
$237$	0	0
$238$	−5.72853	−0.371325
$239$	−9.04754	−0.585237	−0.292618	−	0.956229i	$-0.594527\pi$
$239$	−9.04754	−0.585237	−0.292618	+	0.956229i	$0.594527\pi$
$240$	0	0
$241$	4.46843	0.287837	0.143918	−	0.989590i	$-0.454030\pi$
$241$	4.46843	0.287837	0.143918	+	0.989590i	$0.454030\pi$
$242$	−6.71929	−0.431932
$243$	0	0
$244$	−3.86772	−0.247606
$245$	−3.53009	−0.225529
$246$	0	0
$247$	−0.601130	−0.0382490
$248$	2.34407	0.148848
$249$	0	0
$250$	−7.75123	−0.490231
$251$	16.7302	1.05600	0.528001	−	0.849244i	$-0.322942\pi$
$251$	16.7302	1.05600	0.528001	+	0.849244i	$0.322942\pi$
$252$	0	0
$253$	6.79615	0.427270
$254$	−6.38188	−0.400435
$255$	0	0
$256$	−8.43804	−0.527377
$257$	−13.0642	−0.814921	−0.407461	−	0.913223i	$-0.633586\pi$
$257$	−13.0642	−0.814921	−0.407461	+	0.913223i	$0.633586\pi$
$258$	0	0
$259$	−5.67628	−0.352707
$260$	−2.44490	−0.151626
$261$	0	0
$262$	−8.87746	−0.548451
$263$	−18.6866	−1.15227	−0.576134	−	0.817355i	$-0.695439\pi$
$263$	−18.6866	−1.15227	−0.576134	+	0.817355i	$0.695439\pi$
$264$	0	0
$265$	5.02505	0.308686
$266$	−0.859259	−0.0526845
$267$	0	0
$268$	10.3482	0.632118
$269$	3.71975	0.226797	0.113398	−	0.993550i	$-0.463826\pi$
$269$	3.71975	0.226797	0.113398	+	0.993550i	$0.463826\pi$
$270$	0	0
$271$	−25.4488	−1.54591	−0.772953	−	0.634464i	$-0.781221\pi$
$271$	−25.4488	−1.54591	−0.772953	+	0.634464i	$0.781221\pi$
$272$	−6.40127	−0.388134
$273$	0	0
$274$	3.38726	0.204632
$275$	2.25668	0.136083
$276$	0	0
$277$	16.2270	0.974986	0.487493	−	0.873127i	$-0.337911\pi$
$277$	16.2270	0.974986	0.487493	+	0.873127i	$0.337911\pi$
$278$	14.6036	0.875864
$279$	0	0
$280$	−7.95088	−0.475156
$281$	10.9457	0.652966	0.326483	−	0.945203i	$-0.394136\pi$
$281$	10.9457	0.652966	0.326483	+	0.945203i	$0.394136\pi$
$282$	0	0
$283$	−27.2032	−1.61706	−0.808532	−	0.588452i	$-0.799738\pi$
$283$	−27.2032	−1.61706	−0.808532	+	0.588452i	$0.799738\pi$
$284$	−0.894190	−0.0530604
$285$	0	0
$286$	−0.576714	−0.0341018
$287$	10.9011	0.643474
$288$	0	0
$289$	−0.938913	−0.0552302
$290$	−4.45722	−0.261737
$291$	0	0
$292$	9.69279	0.567228
$293$	−32.5419	−1.90112	−0.950559	−	0.310544i	$-0.899489\pi$
$293$	−32.5419	−1.90112	−0.950559	+	0.310544i	$0.899489\pi$
$294$	0	0
$295$	−9.63728	−0.561104
$296$	6.11450	0.355398
$297$	0	0
$298$	−3.04066	−0.176141
$299$	−7.74079	−0.447661
$300$	0	0
$301$	−4.91385	−0.283230
$302$	−2.85019	−0.164010
$303$	0	0
$304$	−0.960168	−0.0550694
$305$	3.84360	0.220084
$306$	0	0
$307$	27.5404	1.57181	0.785906	−	0.618346i	$-0.212197\pi$
$307$	27.5404	1.57181	0.785906	+	0.618346i	$0.212197\pi$
$308$	2.99667	0.170751
$309$	0	0
$310$	−1.02389	−0.0581532
$311$	27.8161	1.57731	0.788653	−	0.614838i	$-0.210779\pi$
$311$	27.8161	1.57731	0.788653	+	0.614838i	$0.210779\pi$
$312$	0	0
$313$	13.7900	0.779458	0.389729	−	0.920930i	$-0.372569\pi$
$313$	13.7900	0.779458	0.389729	+	0.920930i	$0.372569\pi$
$314$	−8.38818	−0.473372
$315$	0	0
$316$	−15.4594	−0.869661
$317$	−5.05640	−0.283995	−0.141998	−	0.989867i	$-0.545353\pi$
$317$	−5.05640	−0.283995	−0.141998	+	0.989867i	$0.545353\pi$
$318$	0	0
$319$	3.82197	0.213989
$320$	0.894986	0.0500312
$321$	0	0
$322$	−11.0647	−0.616613
$323$	2.40911	0.134046
$324$	0	0
$325$	−2.57035	−0.142577
$326$	−12.7670	−0.707096
$327$	0	0
$328$	−11.7427	−0.648384
$329$	4.33144	0.238800
$330$	0	0
$331$	31.5996	1.73687	0.868436	−	0.495801i	$-0.165125\pi$
$331$	31.5996	1.73687	0.868436	+	0.495801i	$0.165125\pi$
$332$	11.3786	0.624482
$333$	0	0
$334$	7.63538	0.417789
$335$	−10.2837	−0.561857
$336$	0	0
$337$	17.6494	0.961425	0.480713	−	0.876878i	$-0.340378\pi$
$337$	17.6494	0.961425	0.480713	+	0.876878i	$0.340378\pi$
$338$	0.656875	0.0357293
$339$	0	0
$340$	9.79825	0.531384
$341$	0.877966	0.0475445
$342$	0	0
$343$	−20.1607	−1.08857
$344$	5.29321	0.285391
$345$	0	0
$346$	−3.40743	−0.183184
$347$	6.21085	0.333416	0.166708	−	0.986006i	$-0.446686\pi$
$347$	6.21085	0.333416	0.166708	+	0.986006i	$0.446686\pi$
$348$	0	0
$349$	−28.3495	−1.51752	−0.758758	−	0.651373i	$-0.774193\pi$
$349$	−28.3495	−1.51752	−0.758758	+	0.651373i	$0.774193\pi$
$350$	−3.67407	−0.196387
$351$	0	0
$352$	−5.03719	−0.268483
$353$	−29.8862	−1.59068	−0.795341	−	0.606162i	$-0.792708\pi$
$353$	−29.8862	−1.59068	−0.795341	+	0.606162i	$0.792708\pi$
$354$	0	0
$355$	0.888614	0.0471627
$356$	−25.4872	−1.35082
$357$	0	0
$358$	−13.1830	−0.696742
$359$	−12.6922	−0.669869	−0.334934	−	0.942241i	$-0.608714\pi$
$359$	−12.6922	−0.669869	−0.334934	+	0.942241i	$0.608714\pi$
$360$	0	0
$361$	−18.6386	−0.980981
$362$	11.2584	0.591727
$363$	0	0
$364$	−3.41320	−0.178900
$365$	−9.63235	−0.504180
$366$	0	0
$367$	6.55762	0.342305	0.171152	−	0.985245i	$-0.445251\pi$
$367$	6.55762	0.342305	0.171152	+	0.985245i	$0.445251\pi$
$368$	−12.3641	−0.644525
$369$	0	0
$370$	−2.67082	−0.138850
$371$	7.01521	0.364212
$372$	0	0
$373$	2.47432	0.128115	0.0640577	−	0.997946i	$-0.479596\pi$
$373$	2.47432	0.128115	0.0640577	+	0.997946i	$0.479596\pi$
$374$	2.31125	0.119512
$375$	0	0
$376$	−4.66584	−0.240622
$377$	−4.35321	−0.224202
$378$	0	0
$379$	19.7360	1.01377	0.506885	−	0.862014i	$-0.330797\pi$
$379$	19.7360	1.01377	0.506885	+	0.862014i	$0.330797\pi$
$380$	1.46970	0.0753941
$381$	0	0
$382$	−0.210962	−0.0107938
$383$	19.3671	0.989614	0.494807	−	0.869003i	$-0.335239\pi$
$383$	19.3671	0.989614	0.494807	+	0.869003i	$0.335239\pi$
$384$	0	0
$385$	−2.97799	−0.151772
$386$	−2.98671	−0.152019
$387$	0	0
$388$	25.0532	1.27188
$389$	31.2291	1.58338	0.791688	−	0.610925i	$-0.209202\pi$
$389$	31.2291	1.58338	0.791688	+	0.610925i	$0.209202\pi$
$390$	0	0
$391$	31.0222	1.56886
$392$	5.30865	0.268127
$393$	0	0
$394$	−6.40593	−0.322726
$395$	15.3630	0.772998
$396$	0	0
$397$	8.14879	0.408976	0.204488	−	0.978869i	$-0.434447\pi$
$397$	8.14879	0.408976	0.204488	+	0.978869i	$0.434447\pi$
$398$	−4.79704	−0.240454
$399$	0	0
$400$	−4.10554	−0.205277
$401$	−6.37211	−0.318208	−0.159104	−	0.987262i	$-0.550861\pi$
$401$	−6.37211	−0.318208	−0.159104	+	0.987262i	$0.550861\pi$
$402$	0	0
$403$	−1.00000	−0.0498135
$404$	1.28067	0.0637155
$405$	0	0
$406$	−6.22250	−0.308818
$407$	2.29017	0.113520
$408$	0	0
$409$	−9.57256	−0.473332	−0.236666	−	0.971591i	$-0.576055\pi$
$409$	−9.57256	−0.473332	−0.236666	+	0.971591i	$0.576055\pi$
$410$	5.12925	0.253316
$411$	0	0
$412$	0.570177	0.0280906
$413$	−13.4541	−0.662034
$414$	0	0
$415$	−11.3077	−0.555071
$416$	5.73734	0.281296
$417$	0	0
$418$	0.346680	0.0169567
$419$	22.4308	1.09582	0.547908	−	0.836538i	$-0.315424\pi$
$419$	22.4308	1.09582	0.547908	+	0.836538i	$0.315424\pi$
$420$	0	0
$421$	27.5293	1.34169	0.670847	−	0.741596i	$-0.265931\pi$
$421$	27.5293	1.34169	0.670847	+	0.741596i	$0.265931\pi$
$422$	−9.70062	−0.472219
$423$	0	0
$424$	−7.55680	−0.366991
$425$	10.3010	0.499672
$426$	0	0
$427$	5.36586	0.259672
$428$	−27.9647	−1.35173
$429$	0	0
$430$	−2.31209	−0.111499
$431$	−14.2990	−0.688759	−0.344379	−	0.938831i	$-0.611911\pi$
$431$	−14.2990	−0.688759	−0.344379	+	0.938831i	$0.611911\pi$
$432$	0	0
$433$	−6.73207	−0.323523	−0.161761	−	0.986830i	$-0.551717\pi$
$433$	−6.73207	−0.323523	−0.161761	+	0.986830i	$0.551717\pi$
$434$	−1.42941	−0.0686137
$435$	0	0
$436$	16.1804	0.774901
$437$	4.65322	0.222594
$438$	0	0
$439$	−20.3619	−0.971819	−0.485910	−	0.874009i	$-0.661512\pi$
$439$	−20.3619	−0.971819	−0.485910	+	0.874009i	$0.661512\pi$
$440$	3.20789	0.152930
$441$	0	0
$442$	−2.63251	−0.125216
$443$	−17.6424	−0.838216	−0.419108	−	0.907936i	$-0.637657\pi$
$443$	−17.6424	−0.838216	−0.419108	+	0.907936i	$0.637657\pi$
$444$	0	0
$445$	25.3283	1.20067
$446$	16.2004	0.767111
$447$	0	0
$448$	1.24944	0.0590307
$449$	−19.1462	−0.903565	−0.451782	−	0.892128i	$-0.649212\pi$
$449$	−19.1462	−0.903565	−0.451782	+	0.892128i	$0.649212\pi$
$450$	0	0
$451$	−4.39822	−0.207104
$452$	24.4547	1.15025
$453$	0	0
$454$	−0.914097	−0.0429007
$455$	3.39192	0.159015
$456$	0	0
$457$	−26.0477	−1.21846	−0.609230	−	0.792994i	$-0.708521\pi$
$457$	−26.0477	−1.21846	−0.609230	+	0.792994i	$0.708521\pi$
$458$	16.3569	0.764309
$459$	0	0
$460$	18.9254	0.882403
$461$	−4.57637	−0.213143	−0.106571	−	0.994305i	$-0.533987\pi$
$461$	−4.57637	−0.213143	−0.106571	+	0.994305i	$0.533987\pi$
$462$	0	0
$463$	10.6466	0.494790	0.247395	−	0.968915i	$-0.420426\pi$
$463$	10.6466	0.494790	0.247395	+	0.968915i	$0.420426\pi$
$464$	−6.95326	−0.322797
$465$	0	0
$466$	14.7357	0.682619
$467$	15.0034	0.694275	0.347138	−	0.937814i	$-0.387154\pi$
$467$	15.0034	0.694275	0.347138	+	0.937814i	$0.387154\pi$
$468$	0	0
$469$	−14.3565	−0.662923
$470$	2.03805	0.0940080
$471$	0	0
$472$	14.4928	0.667086
$473$	1.98256	0.0911583
$474$	0	0
$475$	1.54511	0.0708947
$476$	13.6788	0.626968
$477$	0	0
$478$	−5.94310	−0.271831
$479$	37.1477	1.69732	0.848661	−	0.528937i	$-0.177409\pi$
$479$	37.1477	1.69732	0.848661	+	0.528937i	$0.177409\pi$
$480$	0	0
$481$	−2.60850	−0.118937
$482$	2.93520	0.133695
$483$	0	0
$484$	16.0446	0.729301
$485$	−24.8969	−1.13051
$486$	0	0
$487$	−23.1824	−1.05049	−0.525247	−	0.850950i	$-0.676027\pi$
$487$	−23.1824	−1.05049	−0.525247	+	0.850950i	$0.676027\pi$
$488$	−5.78012	−0.261654
$489$	0	0
$490$	−2.31883	−0.104754
$491$	26.1739	1.18121	0.590606	−	0.806960i	$-0.298889\pi$
$491$	26.1739	1.18121	0.590606	+	0.806960i	$0.298889\pi$
$492$	0	0
$493$	17.4460	0.785730
$494$	−0.394867	−0.0177659
$495$	0	0
$496$	−1.59727	−0.0717196
$497$	1.24055	0.0556462
$498$	0	0
$499$	−38.5295	−1.72482	−0.862408	−	0.506215i	$-0.831044\pi$
$499$	−38.5295	−1.72482	−0.862408	+	0.506215i	$0.831044\pi$
$500$	18.5087	0.827736
$501$	0	0
$502$	10.9897	0.490492
$503$	23.8172	1.06195	0.530977	−	0.847386i	$-0.321825\pi$
$503$	23.8172	1.06195	0.530977	+	0.847386i	$0.321825\pi$
$504$	0	0
$505$	−1.27268	−0.0566335
$506$	4.46422	0.198459
$507$	0	0
$508$	15.2389	0.676119
$509$	10.5621	0.468156	0.234078	−	0.972218i	$-0.424793\pi$
$509$	10.5621	0.468156	0.234078	+	0.972218i	$0.424793\pi$
$510$	0	0
$511$	−13.4472	−0.594871
$512$	16.6523	0.735935
$513$	0	0
$514$	−8.58153	−0.378515
$515$	−0.566621	−0.0249683
$516$	0	0
$517$	−1.74758	−0.0768584
$518$	−3.72860	−0.163825
$519$	0	0
$520$	−3.65378	−0.160229
$521$	28.8357	1.26331	0.631657	−	0.775248i	$-0.282375\pi$
$521$	28.8357	1.26331	0.631657	+	0.775248i	$0.282375\pi$
$522$	0	0
$523$	−13.2684	−0.580188	−0.290094	−	0.956998i	$-0.593687\pi$
$523$	−13.2684	−0.580188	−0.290094	+	0.956998i	$0.593687\pi$
$524$	21.1980	0.926039
$525$	0	0
$526$	−12.2748	−0.535206
$527$	4.00763	0.174575
$528$	0	0
$529$	36.9198	1.60521
$530$	3.30083	0.143379
$531$	0	0
$532$	2.05178	0.0889558
$533$	5.00956	0.216988
$534$	0	0
$535$	27.7903	1.20148
$536$	15.4649	0.667981
$537$	0	0
$538$	2.44341	0.105343
$539$	1.98834	0.0856441
$540$	0	0
$541$	−7.22402	−0.310585	−0.155292	−	0.987869i	$-0.549632\pi$
$541$	−7.22402	−0.310585	−0.155292	+	0.987869i	$0.549632\pi$
$542$	−16.7167	−0.718043
$543$	0	0
$544$	−22.9931	−0.985823
$545$	−16.0795	−0.688770
$546$	0	0
$547$	−10.7901	−0.461352	−0.230676	−	0.973031i	$-0.574094\pi$
$547$	−10.7901	−0.461352	−0.230676	+	0.973031i	$0.574094\pi$
$548$	−8.08824	−0.345513
$549$	0	0
$550$	1.48235	0.0632078
$551$	2.61685	0.111481
$552$	0	0
$553$	21.4475	0.912042
$554$	10.6591	0.452862
$555$	0	0
$556$	−34.8710	−1.47886
$557$	12.8194	0.543175	0.271588	−	0.962414i	$-0.412451\pi$
$557$	12.8194	0.543175	0.271588	+	0.962414i	$0.412451\pi$
$558$	0	0
$559$	−2.25813	−0.0955088
$560$	5.41781	0.228944
$561$	0	0
$562$	7.18996	0.303290
$563$	33.1561	1.39736	0.698680	−	0.715434i	$-0.253771\pi$
$563$	33.1561	1.39736	0.698680	+	0.715434i	$0.253771\pi$
$564$	0	0
$565$	−24.3022	−1.02240
$566$	−17.8691	−0.751095
$567$	0	0
$568$	−1.33632	−0.0560709
$569$	14.1575	0.593512	0.296756	−	0.954953i	$-0.404095\pi$
$569$	14.1575	0.593512	0.296756	+	0.954953i	$0.404095\pi$
$570$	0	0
$571$	−5.28503	−0.221172	−0.110586	−	0.993867i	$-0.535273\pi$
$571$	−5.28503	−0.221172	−0.110586	+	0.993867i	$0.535273\pi$
$572$	1.37710	0.0575796
$573$	0	0
$574$	7.16069	0.298881
$575$	19.8965	0.829742
$576$	0	0
$577$	−4.65661	−0.193857	−0.0969286	−	0.995291i	$-0.530902\pi$
$577$	−4.65661	−0.193857	−0.0969286	+	0.995291i	$0.530902\pi$
$578$	−0.616748	−0.0256534
$579$	0	0
$580$	10.6432	0.441933
$581$	−15.7860	−0.654916
$582$	0	0
$583$	−2.83039	−0.117223
$584$	14.4854	0.599410
$585$	0	0
$586$	−21.3760	−0.883033
$587$	7.30288	0.301422	0.150711	−	0.988578i	$-0.451844\pi$
$587$	7.30288	0.301422	0.150711	+	0.988578i	$0.451844\pi$
$588$	0	0
$589$	0.601130	0.0247691
$590$	−6.33049	−0.260622
$591$	0	0
$592$	−4.16648	−0.171241
$593$	−11.8112	−0.485030	−0.242515	−	0.970148i	$-0.577972\pi$
$593$	−11.8112	−0.485030	−0.242515	+	0.970148i	$0.577972\pi$
$594$	0	0
$595$	−13.5935	−0.557281
$596$	7.26063	0.297407
$597$	0	0
$598$	−5.08473	−0.207930
$599$	20.1734	0.824262	0.412131	−	0.911125i	$-0.364785\pi$
$599$	20.1734	0.824262	0.412131	+	0.911125i	$0.364785\pi$
$600$	0	0
$601$	35.7716	1.45916	0.729578	−	0.683898i	$-0.239717\pi$
$601$	35.7716	1.45916	0.729578	+	0.683898i	$0.239717\pi$
$602$	−3.22779	−0.131555
$603$	0	0
$604$	6.80580	0.276924
$605$	−15.9446	−0.648239
$606$	0	0
$607$	29.9711	1.21649	0.608245	−	0.793750i	$-0.291874\pi$
$607$	29.9711	1.21649	0.608245	+	0.793750i	$0.291874\pi$
$608$	−3.44889	−0.139871
$609$	0	0
$610$	2.52477	0.102225
$611$	1.99049	0.0805265
$612$	0	0
$613$	−15.7324	−0.635425	−0.317713	−	0.948187i	$-0.602915\pi$
$613$	−15.7324	−0.635425	−0.317713	+	0.948187i	$0.602915\pi$
$614$	18.0906	0.730076
$615$	0	0
$616$	4.47838	0.180439
$617$	−46.9097	−1.88851	−0.944257	−	0.329209i	$-0.893218\pi$
$617$	−46.9097	−1.88851	−0.944257	+	0.329209i	$0.893218\pi$
$618$	0	0
$619$	23.5719	0.947436	0.473718	−	0.880677i	$-0.342912\pi$
$619$	23.5719	0.947436	0.473718	+	0.880677i	$0.342912\pi$
$620$	2.44490	0.0981895
$621$	0	0
$622$	18.2717	0.732628
$623$	35.3595	1.41665
$624$	0	0
$625$	−5.54157	−0.221663
$626$	9.05832	0.362043
$627$	0	0
$628$	20.0297	0.799271
$629$	10.4539	0.416824
$630$	0	0
$631$	35.1311	1.39855	0.699274	−	0.714854i	$-0.253507\pi$
$631$	35.1311	1.39855	0.699274	+	0.714854i	$0.253507\pi$
$632$	−23.1033	−0.919002
$633$	0	0
$634$	−3.32142	−0.131910
$635$	−15.1439	−0.600968
$636$	0	0
$637$	−2.26472	−0.0897314
$638$	2.51056	0.0993939
$639$	0	0
$640$	−17.2981	−0.683767
$641$	−17.3741	−0.686235	−0.343118	−	0.939292i	$-0.611483\pi$
$641$	−17.3741	−0.686235	−0.343118	+	0.939292i	$0.611483\pi$
$642$	0	0
$643$	43.7159	1.72399	0.861993	−	0.506920i	$-0.169216\pi$
$643$	43.7159	1.72399	0.861993	+	0.506920i	$0.169216\pi$
$644$	26.4208	1.04113
$645$	0	0
$646$	1.58248	0.0622619
$647$	−31.6896	−1.24585	−0.622924	−	0.782283i	$-0.714055\pi$
$647$	−31.6896	−1.24585	−0.622924	+	0.782283i	$0.714055\pi$
$648$	0	0
$649$	5.42826	0.213078
$650$	−1.68840	−0.0662244
$651$	0	0
$652$	30.4855	1.19390
$653$	−1.10032	−0.0430587	−0.0215293	−	0.999768i	$-0.506854\pi$
$653$	−1.10032	−0.0430587	−0.0215293	+	0.999768i	$0.506854\pi$
$654$	0	0
$655$	−21.0658	−0.823109
$656$	8.00162	0.312411
$657$	0	0
$658$	2.84521	0.110918
$659$	−26.0015	−1.01288	−0.506438	−	0.862277i	$-0.669038\pi$
$659$	−26.0015	−1.01288	−0.506438	+	0.862277i	$0.669038\pi$
$660$	0	0
$661$	9.05504	0.352200	0.176100	−	0.984372i	$-0.443652\pi$
$661$	9.05504	0.352200	0.176100	+	0.984372i	$0.443652\pi$
$662$	20.7570	0.806744
$663$	0	0
$664$	17.0048	0.659913
$665$	−2.03898	−0.0790683
$666$	0	0
$667$	33.6973	1.30476
$668$	−18.2321	−0.705421
$669$	0	0
$670$	−6.75509	−0.260972
$671$	−2.16493	−0.0835763
$672$	0	0
$673$	32.2884	1.24463	0.622313	−	0.782769i	$-0.286193\pi$
$673$	32.2884	1.24463	0.622313	+	0.782769i	$0.286193\pi$
$674$	11.5935	0.446564
$675$	0	0
$676$	−1.56852	−0.0603275
$677$	45.1998	1.73717	0.868585	−	0.495540i	$-0.165030\pi$
$677$	45.1998	1.73717	0.868585	+	0.495540i	$0.165030\pi$
$678$	0	0
$679$	−34.7573	−1.33386
$680$	14.6430	0.561533
$681$	0	0
$682$	0.576714	0.0220835
$683$	−13.3857	−0.512189	−0.256094	−	0.966652i	$-0.582436\pi$
$683$	−13.3857	−0.512189	−0.256094	+	0.966652i	$0.582436\pi$
$684$	0	0
$685$	8.03780	0.307109
$686$	−13.2430	−0.505622
$687$	0	0
$688$	−3.60685	−0.137510
$689$	3.22380	0.122817
$690$	0	0
$691$	−0.244786	−0.00931211	−0.00465605	−	0.999989i	$-0.501482\pi$
$691$	−0.244786	−0.00931211	−0.00465605	+	0.999989i	$0.501482\pi$
$692$	8.13641	0.309300
$693$	0	0
$694$	4.07975	0.154865
$695$	34.6536	1.31449
$696$	0	0
$697$	−20.0764	−0.760449
$698$	−18.6221	−0.704857
$699$	0	0
$700$	8.77311	0.331592
$701$	−21.6400	−0.817330	−0.408665	−	0.912684i	$-0.634006\pi$
$701$	−21.6400	−0.817330	−0.408665	+	0.912684i	$0.634006\pi$
$702$	0	0
$703$	1.56805	0.0591400
$704$	−0.504106	−0.0189992
$705$	0	0
$706$	−19.6315	−0.738841
$707$	−1.77672	−0.0668206
$708$	0	0
$709$	3.51683	0.132077	0.0660387	−	0.997817i	$-0.478964\pi$
$709$	3.51683	0.132077	0.0660387	+	0.997817i	$0.478964\pi$
$710$	0.583708	0.0219062
$711$	0	0
$712$	−38.0893	−1.42746
$713$	7.74079	0.289895
$714$	0	0
$715$	−1.36852	−0.0511796
$716$	31.4789	1.17642
$717$	0	0
$718$	−8.33719	−0.311141
$719$	−3.98915	−0.148770	−0.0743852	−	0.997230i	$-0.523699\pi$
$719$	−3.98915	−0.148770	−0.0743852	+	0.997230i	$0.523699\pi$
$720$	0	0
$721$	−0.791031	−0.0294596
$722$	−12.2433	−0.455647
$723$	0	0
$724$	−26.8832	−0.999107
$725$	11.1893	0.415559
$726$	0	0
$727$	−2.47671	−0.0918562	−0.0459281	−	0.998945i	$-0.514625\pi$
$727$	−2.47671	−0.0918562	−0.0459281	+	0.998945i	$0.514625\pi$
$728$	−5.10086	−0.189050
$729$	0	0
$730$	−6.32725	−0.234182
$731$	9.04975	0.334717
$732$	0	0
$733$	25.9259	0.957593	0.478797	−	0.877926i	$-0.341073\pi$
$733$	25.9259	0.957593	0.478797	+	0.877926i	$0.341073\pi$
$734$	4.30753	0.158994
$735$	0	0
$736$	−44.4115	−1.63703
$737$	5.79234	0.213364
$738$	0	0
$739$	−43.5409	−1.60168	−0.800840	−	0.598879i	$-0.795613\pi$
$739$	−43.5409	−1.60168	−0.800840	+	0.598879i	$0.795613\pi$
$740$	6.37751	0.234442
$741$	0	0
$742$	4.60812	0.169169
$743$	−16.0866	−0.590161	−0.295081	−	0.955472i	$-0.595347\pi$
$743$	−16.0866	−0.590161	−0.295081	+	0.955472i	$0.595347\pi$
$744$	0	0
$745$	−7.21535	−0.264350
$746$	1.62532	0.0595072
$747$	0	0
$748$	−5.51892	−0.201792
$749$	38.7967	1.41760
$750$	0	0
$751$	37.4826	1.36776	0.683879	−	0.729595i	$-0.260292\pi$
$751$	37.4826	1.36776	0.683879	+	0.729595i	$0.260292\pi$
$752$	3.17935	0.115939
$753$	0	0
$754$	−2.85951	−0.104137
$755$	−6.76336	−0.246144
$756$	0	0
$757$	10.6822	0.388253	0.194126	−	0.980977i	$-0.437813\pi$
$757$	10.6822	0.388253	0.194126	+	0.980977i	$0.437813\pi$
$758$	12.9641	0.470877
$759$	0	0
$760$	2.19640	0.0796717
$761$	−16.4984	−0.598068	−0.299034	−	0.954242i	$-0.596664\pi$
$761$	−16.4984	−0.598068	−0.299034	+	0.954242i	$0.596664\pi$
$762$	0	0
$763$	−22.4478	−0.812664
$764$	0.503745	0.0182249
$765$	0	0
$766$	12.7218	0.459657
$767$	−6.18276	−0.223247
$768$	0	0
$769$	−38.3588	−1.38325	−0.691627	−	0.722255i	$-0.743106\pi$
$769$	−38.3588	−1.38325	−0.691627	+	0.722255i	$0.743106\pi$
$770$	−1.95616	−0.0704953
$771$	0	0
$772$	7.13179	0.256679
$773$	10.2453	0.368498	0.184249	−	0.982880i	$-0.441015\pi$
$773$	10.2453	0.368498	0.184249	+	0.982880i	$0.441015\pi$
$774$	0	0
$775$	2.57035	0.0923296
$776$	37.4407	1.34404
$777$	0	0
$778$	20.5136	0.735448
$779$	−3.01139	−0.107894
$780$	0	0
$781$	−0.500517	−0.0179099
$782$	20.3777	0.728705
$783$	0	0
$784$	−3.61737	−0.129192
$785$	−19.9048	−0.710432
$786$	0	0
$787$	21.3288	0.760291	0.380145	−	0.924927i	$-0.375874\pi$
$787$	21.3288	0.760291	0.380145	+	0.924927i	$0.375874\pi$
$788$	15.2964	0.544911
$789$	0	0
$790$	10.0916	0.359042
$791$	−33.9271	−1.20631
$792$	0	0
$793$	2.46585	0.0875649
$794$	5.35274	0.189962
$795$	0	0
$796$	11.4546	0.405997
$797$	−52.2408	−1.85046	−0.925231	−	0.379403i	$-0.876129\pi$
$797$	−52.2408	−1.85046	−0.925231	+	0.379403i	$0.876129\pi$
$798$	0	0
$799$	−7.97713	−0.282211
$800$	−14.7470	−0.521384
$801$	0	0
$802$	−4.18568	−0.147802
$803$	5.42547	0.191461
$804$	0	0
$805$	−26.2561	−0.925406
$806$	−0.656875	−0.0231374
$807$	0	0
$808$	1.91389	0.0673305
$809$	−41.0611	−1.44363	−0.721815	−	0.692086i	$-0.756692\pi$
$809$	−41.0611	−1.44363	−0.721815	+	0.692086i	$0.756692\pi$
$810$	0	0
$811$	−12.9575	−0.455001	−0.227500	−	0.973778i	$-0.573055\pi$
$811$	−12.9575	−0.455001	−0.227500	+	0.973778i	$0.573055\pi$
$812$	14.8584	0.521427
$813$	0	0
$814$	1.50436	0.0527277
$815$	−30.2954	−1.06120
$816$	0	0
$817$	1.35743	0.0474905
$818$	−6.28797	−0.219854
$819$	0	0
$820$	−12.2479	−0.427714
$821$	21.9928	0.767554	0.383777	−	0.923426i	$-0.374623\pi$
$821$	21.9928	0.767554	0.383777	+	0.923426i	$0.374623\pi$
$822$	0	0
$823$	2.80755	0.0978650	0.0489325	−	0.998802i	$-0.484418\pi$
$823$	2.80755	0.0978650	0.0489325	+	0.998802i	$0.484418\pi$
$824$	0.852101	0.0296843
$825$	0	0
$826$	−8.83768	−0.307502
$827$	33.6672	1.17072	0.585362	−	0.810772i	$-0.300952\pi$
$827$	33.6672	1.17072	0.585362	+	0.810772i	$0.300952\pi$
$828$	0	0
$829$	31.8293	1.10548	0.552738	−	0.833355i	$-0.313583\pi$
$829$	31.8293	1.10548	0.552738	+	0.833355i	$0.313583\pi$
$830$	−7.42771	−0.257820
$831$	0	0
$832$	0.574175	0.0199059
$833$	9.07615	0.314470
$834$	0	0
$835$	18.1184	0.627013
$836$	−0.827818	−0.0286307
$837$	0	0
$838$	14.7342	0.508986
$839$	−5.70452	−0.196942	−0.0984709	−	0.995140i	$-0.531395\pi$
$839$	−5.70452	−0.196942	−0.0984709	+	0.995140i	$0.531395\pi$
$840$	0	0
$841$	−10.0496	−0.346537
$842$	18.0833	0.623191
$843$	0	0
$844$	23.1636	0.797323
$845$	1.55873	0.0536221
$846$	0	0
$847$	−22.2594	−0.764842
$848$	5.14928	0.176827
$849$	0	0
$850$	6.76647	0.232088
$851$	20.1918	0.692167
$852$	0	0
$853$	−5.83724	−0.199863	−0.0999316	−	0.994994i	$-0.531862\pi$
$853$	−5.83724	−0.199863	−0.0999316	+	0.994994i	$0.531862\pi$
$854$	3.52470	0.120613
$855$	0	0
$856$	−41.7919	−1.42842
$857$	1.10228	0.0376530	0.0188265	−	0.999823i	$-0.494007\pi$
$857$	1.10228	0.0376530	0.0188265	+	0.999823i	$0.494007\pi$
$858$	0	0
$859$	13.0961	0.446834	0.223417	−	0.974723i	$-0.428279\pi$
$859$	13.0961	0.446834	0.223417	+	0.974723i	$0.428279\pi$
$860$	5.52090	0.188261
$861$	0	0
$862$	−9.39266	−0.319915
$863$	0.0939544	0.00319824	0.00159912	−	0.999999i	$-0.499491\pi$
$863$	0.0939544	0.00319824	0.00159912	+	0.999999i	$0.499491\pi$
$864$	0	0
$865$	−8.08567	−0.274921
$866$	−4.42213	−0.150270
$867$	0	0
$868$	3.41320	0.115852
$869$	−8.65331	−0.293543
$870$	0	0
$871$	−6.59746	−0.223546
$872$	24.1808	0.818865
$873$	0	0
$874$	3.05658	0.103390
$875$	−25.6780	−0.868074
$876$	0	0
$877$	52.5489	1.77445	0.887226	−	0.461335i	$-0.152629\pi$
$877$	52.5489	1.77445	0.887226	+	0.461335i	$0.152629\pi$
$878$	−13.3752	−0.451391
$879$	0	0
$880$	−2.18589	−0.0736864
$881$	−58.3414	−1.96557	−0.982786	−	0.184746i	$-0.940854\pi$
$881$	−58.3414	−1.96557	−0.982786	+	0.184746i	$0.940854\pi$
$882$	0	0
$883$	−38.7524	−1.30412	−0.652061	−	0.758166i	$-0.726095\pi$
$883$	−38.7524	−1.30412	−0.652061	+	0.758166i	$0.726095\pi$
$884$	6.28603	0.211422
$885$	0	0
$886$	−11.5888	−0.389335
$887$	16.4399	0.551998	0.275999	−	0.961158i	$-0.410991\pi$
$887$	16.4399	0.551998	0.275999	+	0.961158i	$0.410991\pi$
$888$	0	0
$889$	−21.1417	−0.709069
$890$	16.6375	0.557690
$891$	0	0
$892$	−38.6840	−1.29524
$893$	−1.19654	−0.0400407
$894$	0	0
$895$	−31.2826	−1.04566
$896$	−24.1490	−0.806761
$897$	0	0
$898$	−12.5767	−0.419688
$899$	4.35321	0.145188
$900$	0	0
$901$	−12.9198	−0.430421
$902$	−2.88908	−0.0961958
$903$	0	0
$904$	36.5464	1.21551
$905$	26.7156	0.888056
$906$	0	0
$907$	40.2105	1.33517	0.667584	−	0.744534i	$-0.267328\pi$
$907$	40.2105	1.33517	0.667584	+	0.744534i	$0.267328\pi$
$908$	2.18272	0.0724362
$909$	0	0
$910$	2.22806	0.0738596
$911$	24.4143	0.808881	0.404441	−	0.914564i	$-0.367466\pi$
$911$	24.4143	0.808881	0.404441	+	0.914564i	$0.367466\pi$
$912$	0	0
$913$	6.36910	0.210786
$914$	−17.1101	−0.565951
$915$	0	0
$916$	−39.0578	−1.29051
$917$	−29.4089	−0.971168
$918$	0	0
$919$	−2.91104	−0.0960264	−0.0480132	−	0.998847i	$-0.515289\pi$
$919$	−2.91104	−0.0960264	−0.0480132	+	0.998847i	$0.515289\pi$
$920$	28.2831	0.932467
$921$	0	0
$922$	−3.00610	−0.0990007
$923$	0.570087	0.0187646
$924$	0	0
$925$	6.70475	0.220451
$926$	6.99349	0.229820
$927$	0	0
$928$	−24.9759	−0.819873
$929$	23.9908	0.787112	0.393556	−	0.919301i	$-0.371245\pi$
$929$	23.9908	0.787112	0.393556	+	0.919301i	$0.371245\pi$
$930$	0	0
$931$	1.36139	0.0446177
$932$	−35.1866	−1.15258
$933$	0	0
$934$	9.85537	0.322478
$935$	5.48450	0.179362
$936$	0	0
$937$	31.4017	1.02585	0.512924	−	0.858434i	$-0.328562\pi$
$937$	31.4017	1.02585	0.512924	+	0.858434i	$0.328562\pi$
$938$	−9.43044	−0.307915
$939$	0	0
$940$	−4.86654	−0.158729
$941$	−33.3661	−1.08771	−0.543853	−	0.839181i	$-0.683035\pi$
$941$	−33.3661	−1.08771	−0.543853	+	0.839181i	$0.683035\pi$
$942$	0	0
$943$	−38.7779	−1.26278
$944$	−9.87555	−0.321422
$945$	0	0
$946$	1.30230	0.0423413
$947$	47.4687	1.54253	0.771263	−	0.636517i	$-0.219625\pi$
$947$	47.4687	1.54253	0.771263	+	0.636517i	$0.219625\pi$
$948$	0	0
$949$	−6.17960	−0.200598
$950$	1.01495	0.0329292
$951$	0	0
$952$	20.4423	0.662540
$953$	22.1379	0.717118	0.358559	−	0.933507i	$-0.383268\pi$
$953$	22.1379	0.717118	0.358559	+	0.933507i	$0.383268\pi$
$954$	0	0
$955$	−0.500604	−0.0161992
$956$	14.1912	0.458976
$957$	0	0
$958$	24.4014	0.788373
$959$	11.2212	0.362351
$960$	0	0
$961$	1.00000	0.0322581
$962$	−1.71346	−0.0552441
$963$	0	0
$964$	−7.00880	−0.225738
$965$	−7.08732	−0.228149
$966$	0	0
$967$	25.1227	0.807891	0.403946	−	0.914783i	$-0.367638\pi$
$967$	25.1227	0.807891	0.403946	+	0.914783i	$0.367638\pi$
$968$	23.9779	0.770678
$969$	0	0
$970$	−16.3542	−0.525101
$971$	−51.4528	−1.65120	−0.825600	−	0.564256i	$-0.809163\pi$
$971$	−51.4528	−1.65120	−0.825600	+	0.564256i	$0.809163\pi$
$972$	0	0
$973$	48.3781	1.55093
$974$	−15.2279	−0.487934
$975$	0	0
$976$	3.93863	0.126072
$977$	14.7721	0.472601	0.236300	−	0.971680i	$-0.424065\pi$
$977$	14.7721	0.472601	0.236300	+	0.971680i	$0.424065\pi$
$978$	0	0
$979$	−14.2663	−0.455952
$980$	5.53700	0.176873
$981$	0	0
$982$	17.1930	0.548650
$983$	−47.9404	−1.52906	−0.764530	−	0.644588i	$-0.777029\pi$
$983$	−47.9404	−1.52906	−0.764530	+	0.644588i	$0.777029\pi$
$984$	0	0
$985$	−15.2010	−0.484344
$986$	11.4599	0.364957
$987$	0	0
$988$	0.942882	0.0299971
$989$	17.4797	0.555822
$990$	0	0
$991$	−9.92275	−0.315206	−0.157603	−	0.987503i	$-0.550377\pi$
$991$	−9.92275	−0.315206	−0.157603	+	0.987503i	$0.550377\pi$
$992$	−5.73734	−0.182161
$993$	0	0
$994$	0.814886	0.0258466
$995$	−11.3831	−0.360870
$996$	0	0
$997$	8.72610	0.276358	0.138179	−	0.990407i	$-0.455875\pi$
$997$	8.72610	0.276358	0.138179	+	0.990407i	$0.455875\pi$
$998$	−25.3090	−0.801143
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3627.2.a.p.1.5		8
3.2	odd	2		403.2.a.e.1.4	✓	8
12.11	even	2		6448.2.a.bd.1.2		8
39.38	odd	2		5239.2.a.i.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.e.1.4	✓	8	3.2	odd	2
3627.2.a.p.1.5		8	1.1	even	1	trivial
5239.2.a.i.1.5		8	39.38	odd	2
6448.2.a.bd.1.2		8	12.11	even	2

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:	\( N \)	\(=\)	\( 3627 = 3^{2} \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3627.a (trivial)

\(f(q)\)	\(=\)	\(q+0.656875 q^{2} -1.56852 q^{4} +1.55873 q^{5} +2.17607 q^{7} -2.34407 q^{8} +O(q^{10})\)	\(q+0.656875 q^{2} -1.56852 q^{4} +1.55873 q^{5} +2.17607 q^{7} -2.34407 q^{8} +1.02389 q^{10} -0.877966 q^{11} +1.00000 q^{13} +1.42941 q^{14} +1.59727 q^{16} -4.00763 q^{17} -0.601130 q^{19} -2.44490 q^{20} -0.576714 q^{22} -7.74079 q^{23} -2.57035 q^{25} +0.656875 q^{26} -3.41320 q^{28} -4.35321 q^{29} -1.00000 q^{31} +5.73734 q^{32} -2.63251 q^{34} +3.39192 q^{35} -2.60850 q^{37} -0.394867 q^{38} -3.65378 q^{40} +5.00956 q^{41} -2.25813 q^{43} +1.37710 q^{44} -5.08473 q^{46} +1.99049 q^{47} -2.26472 q^{49} -1.68840 q^{50} -1.56852 q^{52} +3.22380 q^{53} -1.36852 q^{55} -5.10086 q^{56} -2.85951 q^{58} -6.18276 q^{59} +2.46585 q^{61} -0.656875 q^{62} +0.574175 q^{64} +1.55873 q^{65} -6.59746 q^{67} +6.28603 q^{68} +2.22806 q^{70} +0.570087 q^{71} -6.17960 q^{73} -1.71346 q^{74} +0.942882 q^{76} -1.91052 q^{77} +9.85609 q^{79} +2.48972 q^{80} +3.29065 q^{82} -7.25438 q^{83} -6.24683 q^{85} -1.48331 q^{86} +2.05801 q^{88} +16.2492 q^{89} +2.17607 q^{91} +12.1415 q^{92} +1.30750 q^{94} -0.937002 q^{95} -15.9725 q^{97} -1.48764 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 7 q^{4} - 11 q^{5} - 2 q^{7}+O(q^{10}) \) 8 * q + q^2 + 7 * q^4 - 11 * q^5 - 2 * q^7	\( 8 q + q^{2} + 7 q^{4} - 11 q^{5} - 2 q^{7} - 2 q^{10} + 2 q^{11} + 8 q^{13} - 3 q^{14} - 7 q^{16} - 7 q^{17} - 5 q^{19} - 6 q^{20} - 4 q^{22} - 14 q^{23} + 17 q^{25} + q^{26} - 9 q^{28} - 12 q^{29} - 8 q^{31} + 21 q^{32} - 12 q^{34} + q^{35} + 2 q^{37} - 24 q^{38} - 19 q^{40} - 13 q^{41} - 5 q^{43} - 22 q^{44} + 17 q^{46} - 23 q^{47} + 26 q^{49} + 26 q^{50} + 7 q^{52} - 25 q^{53} - 17 q^{55} - 8 q^{56} - 29 q^{58} + 5 q^{59} - 9 q^{61} - q^{62} - 14 q^{64} - 11 q^{65} + 22 q^{67} + 6 q^{68} - 29 q^{70} + 17 q^{71} - 27 q^{73} - 14 q^{74} - 36 q^{76} - 31 q^{77} - 23 q^{79} - 9 q^{80} + 18 q^{82} + 25 q^{83} + 13 q^{85} - 11 q^{86} + 5 q^{88} - 2 q^{89} - 2 q^{91} + 20 q^{92} - 38 q^{94} - 3 q^{95} - 15 q^{97} - 39 q^{98}+O(q^{100}) \) 8 * q + q^2 + 7 * q^4 - 11 * q^5 - 2 * q^7 - 2 * q^10 + 2 * q^11 + 8 * q^13 - 3 * q^14 - 7 * q^16 - 7 * q^17 - 5 * q^19 - 6 * q^20 - 4 * q^22 - 14 * q^23 + 17 * q^25 + q^26 - 9 * q^28 - 12 * q^29 - 8 * q^31 + 21 * q^32 - 12 * q^34 + q^35 + 2 * q^37 - 24 * q^38 - 19 * q^40 - 13 * q^41 - 5 * q^43 - 22 * q^44 + 17 * q^46 - 23 * q^47 + 26 * q^49 + 26 * q^50 + 7 * q^52 - 25 * q^53 - 17 * q^55 - 8 * q^56 - 29 * q^58 + 5 * q^59 - 9 * q^61 - q^62 - 14 * q^64 - 11 * q^65 + 22 * q^67 + 6 * q^68 - 29 * q^70 + 17 * q^71 - 27 * q^73 - 14 * q^74 - 36 * q^76 - 31 * q^77 - 23 * q^79 - 9 * q^80 + 18 * q^82 + 25 * q^83 + 13 * q^85 - 11 * q^86 + 5 * q^88 - 2 * q^89 - 2 * q^91 + 20 * q^92 - 38 * q^94 - 3 * q^95 - 15 * q^97 - 39 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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