LMFDB - Embedded newform 952.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [952,2,Mod(1,952)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(952, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("952.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 952 = 2^{3} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	952.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:	$7.60175827243$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	952.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278 q^{3} +0.302776 q^{5} -1.00000 q^{7} -1.30278 q^{9} +4.60555 q^{11} -4.00000 q^{13} -0.394449 q^{15} +1.00000 q^{17} +4.60555 q^{19} +1.30278 q^{21} -6.00000 q^{23} -4.90833 q^{25} +5.60555 q^{27} -8.60555 q^{29} -4.90833 q^{31} -6.00000 q^{33} -0.302776 q^{35} -4.60555 q^{37} +5.21110 q^{39} +0.697224 q^{41} -2.69722 q^{43} -0.394449 q^{45} +9.21110 q^{47} +1.00000 q^{49} -1.30278 q^{51} -7.69722 q^{53} +1.39445 q^{55} -6.00000 q^{57} -5.21110 q^{59} -4.69722 q^{61} +1.30278 q^{63} -1.21110 q^{65} -13.5139 q^{67} +7.81665 q^{69} +0.605551 q^{71} -5.90833 q^{73} +6.39445 q^{75} -4.60555 q^{77} -1.21110 q^{79} -3.39445 q^{81} -17.8167 q^{83} +0.302776 q^{85} +11.2111 q^{87} +17.2111 q^{89} +4.00000 q^{91} +6.39445 q^{93} +1.39445 q^{95} +4.30278 q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 3 q^{5} - 2 q^{7} + q^{9} + 2 q^{11} - 8 q^{13} - 8 q^{15} + 2 q^{17} + 2 q^{19} - q^{21} - 12 q^{23} + q^{25} + 4 q^{27} - 10 q^{29} + q^{31} - 12 q^{33} + 3 q^{35} - 2 q^{37} - 4 q^{39}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q + q^3 - 3 * q^5 - 2 * q^7 + q^9 + 2 * q^11 - 8 * q^13 - 8 * q^15 + 2 * q^17 + 2 * q^19 - q^21 - 12 * q^23 + q^25 + 4 * q^27 - 10 * q^29 + q^31 - 12 * q^33 + 3 * q^35 - 2 * q^37 - 4 * q^39 + 5 * q^41 - 9 * q^43 - 8 * q^45 + 4 * q^47 + 2 * q^49 + q^51 - 19 * q^53 + 10 * q^55 - 12 * q^57 + 4 * q^59 - 13 * q^61 - q^63 + 12 * q^65 - 9 * q^67 - 6 * q^69 - 6 * q^71 - q^73 + 20 * q^75 - 2 * q^77 + 12 * q^79 - 14 * q^81 - 14 * q^83 - 3 * q^85 + 8 * q^87 + 20 * q^89 + 8 * q^91 + 20 * q^93 + 10 * q^95 + 5 * q^97 - 12 * q^99

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	0
$2$	0	0
$3$	−1.30278	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$3$	−1.30278	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$4$	0	0
$5$	0.302776	0.135405	0.0677027	−	0.997706i	$-0.478433\pi$
$5$	0.302776	0.135405	0.0677027	+	0.997706i	$0.478433\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	4.60555	1.38863	0.694313	−	0.719673i	$-0.255708\pi$
$11$	4.60555	1.38863	0.694313	+	0.719673i	$0.255708\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−0.394449	−0.101846
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.60555	1.05659	0.528293	−	0.849062i	$-0.322832\pi$
$19$	4.60555	1.05659	0.528293	+	0.849062i	$0.322832\pi$
$20$	0	0
$21$	1.30278	0.284289
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−4.90833	−0.981665
$26$	0	0
$27$	5.60555	1.07879
$28$	0	0
$29$	−8.60555	−1.59801	−0.799005	−	0.601324i	$-0.794640\pi$
$29$	−8.60555	−1.59801	−0.799005	+	0.601324i	$0.794640\pi$
$30$	0	0
$31$	−4.90833	−0.881562	−0.440781	−	0.897615i	$-0.645298\pi$
$31$	−4.90833	−0.881562	−0.440781	+	0.897615i	$0.645298\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	−0.302776	−0.0511784
$36$	0	0
$37$	−4.60555	−0.757148	−0.378574	−	0.925571i	$-0.623585\pi$
$37$	−4.60555	−0.757148	−0.378574	+	0.925571i	$0.623585\pi$
$38$	0	0
$39$	5.21110	0.834444
$40$	0	0
$41$	0.697224	0.108888	0.0544441	−	0.998517i	$-0.482661\pi$
$41$	0.697224	0.108888	0.0544441	+	0.998517i	$0.482661\pi$
$42$	0	0
$43$	−2.69722	−0.411323	−0.205661	−	0.978623i	$-0.565935\pi$
$43$	−2.69722	−0.411323	−0.205661	+	0.978623i	$0.565935\pi$
$44$	0	0
$45$	−0.394449	−0.0588009
$46$	0	0
$47$	9.21110	1.34358	0.671789	−	0.740743i	$-0.265526\pi$
$47$	9.21110	1.34358	0.671789	+	0.740743i	$0.265526\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.30278	−0.182425
$52$	0	0
$53$	−7.69722	−1.05730	−0.528648	−	0.848841i	$-0.677301\pi$
$53$	−7.69722	−1.05730	−0.528648	+	0.848841i	$0.677301\pi$
$54$	0	0
$55$	1.39445	0.188027
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−5.21110	−0.678428	−0.339214	−	0.940709i	$-0.610161\pi$
$59$	−5.21110	−0.678428	−0.339214	+	0.940709i	$0.610161\pi$
$60$	0	0
$61$	−4.69722	−0.601418	−0.300709	−	0.953716i	$-0.597223\pi$
$61$	−4.69722	−0.601418	−0.300709	+	0.953716i	$0.597223\pi$
$62$	0	0
$63$	1.30278	0.164134
$64$	0	0
$65$	−1.21110	−0.150219
$66$	0	0
$67$	−13.5139	−1.65098	−0.825491	−	0.564415i	$-0.809102\pi$
$67$	−13.5139	−1.65098	−0.825491	+	0.564415i	$0.809102\pi$
$68$	0	0
$69$	7.81665	0.941015
$70$	0	0
$71$	0.605551	0.0718657	0.0359329	−	0.999354i	$-0.488560\pi$
$71$	0.605551	0.0718657	0.0359329	+	0.999354i	$0.488560\pi$
$72$	0	0
$73$	−5.90833	−0.691517	−0.345759	−	0.938323i	$-0.612378\pi$
$73$	−5.90833	−0.691517	−0.345759	+	0.938323i	$0.612378\pi$
$74$	0	0
$75$	6.39445	0.738367
$76$	0	0
$77$	−4.60555	−0.524851
$78$	0	0
$79$	−1.21110	−0.136260	−0.0681298	−	0.997676i	$-0.521703\pi$
$79$	−1.21110	−0.136260	−0.0681298	+	0.997676i	$0.521703\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	−17.8167	−1.95563	−0.977816	−	0.209466i	$-0.932827\pi$
$83$	−17.8167	−1.95563	−0.977816	+	0.209466i	$0.932827\pi$
$84$	0	0
$85$	0.302776	0.0328406
$86$	0	0
$87$	11.2111	1.20196
$88$	0	0
$89$	17.2111	1.82437	0.912187	−	0.409775i	$-0.134393\pi$
$89$	17.2111	1.82437	0.912187	+	0.409775i	$0.134393\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	6.39445	0.663073
$94$	0	0
$95$	1.39445	0.143067
$96$	0	0
$97$	4.30278	0.436881	0.218440	−	0.975850i	$-0.429903\pi$
$97$	4.30278	0.436881	0.218440	+	0.975850i	$0.429903\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−12.6056	−1.25430	−0.627150	−	0.778899i	$-0.715779\pi$
$101$	−12.6056	−1.25430	−0.627150	+	0.778899i	$0.715779\pi$
$102$	0	0
$103$	−5.81665	−0.573132	−0.286566	−	0.958061i	$-0.592514\pi$
$103$	−5.81665	−0.573132	−0.286566	+	0.958061i	$0.592514\pi$
$104$	0	0
$105$	0.394449	0.0384943
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−3.21110	−0.307568	−0.153784	−	0.988105i	$-0.549146\pi$
$109$	−3.21110	−0.307568	−0.153784	+	0.988105i	$0.549146\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	1.21110	0.113931	0.0569655	−	0.998376i	$-0.481858\pi$
$113$	1.21110	0.113931	0.0569655	+	0.998376i	$0.481858\pi$
$114$	0	0
$115$	−1.81665	−0.169404
$116$	0	0
$117$	5.21110	0.481767
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	10.2111	0.928282
$122$	0	0
$123$	−0.908327	−0.0819011
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	18.7250	1.66157	0.830787	−	0.556591i	$-0.187891\pi$
$127$	18.7250	1.66157	0.830787	+	0.556591i	$0.187891\pi$
$128$	0	0
$129$	3.51388	0.309380
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−4.60555	−0.399352
$134$	0	0
$135$	1.69722	0.146074
$136$	0	0
$137$	−21.9083	−1.87175	−0.935877	−	0.352326i	$-0.885391\pi$
$137$	−21.9083	−1.87175	−0.935877	+	0.352326i	$0.885391\pi$
$138$	0	0
$139$	13.5139	1.14623	0.573116	−	0.819474i	$-0.305734\pi$
$139$	13.5139	1.14623	0.573116	+	0.819474i	$0.305734\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	−18.4222	−1.54054
$144$	0	0
$145$	−2.60555	−0.216379
$146$	0	0
$147$	−1.30278	−0.107451
$148$	0	0
$149$	22.9361	1.87900	0.939499	−	0.342553i	$-0.111292\pi$
$149$	22.9361	1.87900	0.939499	+	0.342553i	$0.111292\pi$
$150$	0	0
$151$	8.09167	0.658491	0.329246	−	0.944244i	$-0.393206\pi$
$151$	8.09167	0.658491	0.329246	+	0.944244i	$0.393206\pi$
$152$	0	0
$153$	−1.30278	−0.105323
$154$	0	0
$155$	−1.48612	−0.119368
$156$	0	0
$157$	−9.39445	−0.749759	−0.374879	−	0.927074i	$-0.622316\pi$
$157$	−9.39445	−0.749759	−0.374879	+	0.927074i	$0.622316\pi$
$158$	0	0
$159$	10.0278	0.795253
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	23.2111	1.81803	0.909017	−	0.416759i	$-0.136834\pi$
$163$	23.2111	1.81803	0.909017	+	0.416759i	$0.136834\pi$
$164$	0	0
$165$	−1.81665	−0.141426
$166$	0	0
$167$	1.11943	0.0866241	0.0433120	−	0.999062i	$-0.486209\pi$
$167$	1.11943	0.0866241	0.0433120	+	0.999062i	$0.486209\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	2.09167	0.159027	0.0795135	−	0.996834i	$-0.474663\pi$
$173$	2.09167	0.159027	0.0795135	+	0.996834i	$0.474663\pi$
$174$	0	0
$175$	4.90833	0.371035
$176$	0	0
$177$	6.78890	0.510285
$178$	0	0
$179$	−0.0916731	−0.00685197	−0.00342598	−	0.999994i	$-0.501091\pi$
$179$	−0.0916731	−0.00685197	−0.00342598	+	0.999994i	$0.501091\pi$
$180$	0	0
$181$	25.6333	1.90531	0.952654	−	0.304055i	$-0.0983408\pi$
$181$	25.6333	1.90531	0.952654	+	0.304055i	$0.0983408\pi$
$182$	0	0
$183$	6.11943	0.452361
$184$	0	0
$185$	−1.39445	−0.102522
$186$	0	0
$187$	4.60555	0.336791
$188$	0	0
$189$	−5.60555	−0.407744
$190$	0	0
$191$	−20.3028	−1.46906	−0.734529	−	0.678578i	$-0.762597\pi$
$191$	−20.3028	−1.46906	−0.734529	+	0.678578i	$0.762597\pi$
$192$	0	0
$193$	21.2111	1.52681	0.763404	−	0.645921i	$-0.223526\pi$
$193$	21.2111	1.52681	0.763404	+	0.645921i	$0.223526\pi$
$194$	0	0
$195$	1.57779	0.112988
$196$	0	0
$197$	−4.42221	−0.315069	−0.157535	−	0.987513i	$-0.550355\pi$
$197$	−4.42221	−0.315069	−0.157535	+	0.987513i	$0.550355\pi$
$198$	0	0
$199$	22.6972	1.60896	0.804482	−	0.593977i	$-0.202443\pi$
$199$	22.6972	1.60896	0.804482	+	0.593977i	$0.202443\pi$
$200$	0	0
$201$	17.6056	1.24180
$202$	0	0
$203$	8.60555	0.603991
$204$	0	0
$205$	0.211103	0.0147440
$206$	0	0
$207$	7.81665	0.543295
$208$	0	0
$209$	21.2111	1.46720
$210$	0	0
$211$	−7.81665	−0.538121	−0.269060	−	0.963123i	$-0.586713\pi$
$211$	−7.81665	−0.538121	−0.269060	+	0.963123i	$0.586713\pi$
$212$	0	0
$213$	−0.788897	−0.0540544
$214$	0	0
$215$	−0.816654	−0.0556953
$216$	0	0
$217$	4.90833	0.333199
$218$	0	0
$219$	7.69722	0.520130
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	25.8167	1.72881	0.864406	−	0.502795i	$-0.167695\pi$
$223$	25.8167	1.72881	0.864406	+	0.502795i	$0.167695\pi$
$224$	0	0
$225$	6.39445	0.426297
$226$	0	0
$227$	−10.4861	−0.695988	−0.347994	−	0.937497i	$-0.613137\pi$
$227$	−10.4861	−0.695988	−0.347994	+	0.937497i	$0.613137\pi$
$228$	0	0
$229$	−9.81665	−0.648703	−0.324351	−	0.945937i	$-0.605146\pi$
$229$	−9.81665	−0.648703	−0.324351	+	0.945937i	$0.605146\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−22.2389	−1.45692	−0.728458	−	0.685090i	$-0.759763\pi$
$233$	−22.2389	−1.45692	−0.728458	+	0.685090i	$0.759763\pi$
$234$	0	0
$235$	2.78890	0.181928
$236$	0	0
$237$	1.57779	0.102489
$238$	0	0
$239$	−30.3305	−1.96192	−0.980960	−	0.194212i	$-0.937785\pi$
$239$	−30.3305	−1.96192	−0.980960	+	0.194212i	$0.937785\pi$
$240$	0	0
$241$	1.51388	0.0975175	0.0487587	−	0.998811i	$-0.484473\pi$
$241$	1.51388	0.0975175	0.0487587	+	0.998811i	$0.484473\pi$
$242$	0	0
$243$	−12.3944	−0.795104
$244$	0	0
$245$	0.302776	0.0193436
$246$	0	0
$247$	−18.4222	−1.17218
$248$	0	0
$249$	23.2111	1.47094
$250$	0	0
$251$	−8.60555	−0.543178	−0.271589	−	0.962413i	$-0.587549\pi$
$251$	−8.60555	−0.543178	−0.271589	+	0.962413i	$0.587549\pi$
$252$	0	0
$253$	−27.6333	−1.73729
$254$	0	0
$255$	−0.394449	−0.0247013
$256$	0	0
$257$	18.4222	1.14915	0.574573	−	0.818453i	$-0.305168\pi$
$257$	18.4222	1.14915	0.574573	+	0.818453i	$0.305168\pi$
$258$	0	0
$259$	4.60555	0.286175
$260$	0	0
$261$	11.2111	0.693950
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−2.33053	−0.143163
$266$	0	0
$267$	−22.4222	−1.37222
$268$	0	0
$269$	4.78890	0.291984	0.145992	−	0.989286i	$-0.453363\pi$
$269$	4.78890	0.291984	0.145992	+	0.989286i	$0.453363\pi$
$270$	0	0
$271$	4.42221	0.268630	0.134315	−	0.990939i	$-0.457117\pi$
$271$	4.42221	0.268630	0.134315	+	0.990939i	$0.457117\pi$
$272$	0	0
$273$	−5.21110	−0.315390
$274$	0	0
$275$	−22.6056	−1.36317
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	6.39445	0.382826
$280$	0	0
$281$	−16.3028	−0.972542	−0.486271	−	0.873808i	$-0.661643\pi$
$281$	−16.3028	−0.972542	−0.486271	+	0.873808i	$0.661643\pi$
$282$	0	0
$283$	−0.302776	−0.0179981	−0.00899907	−	0.999960i	$-0.502865\pi$
$283$	−0.302776	−0.0179981	−0.00899907	+	0.999960i	$0.502865\pi$
$284$	0	0
$285$	−1.81665	−0.107609
$286$	0	0
$287$	−0.697224	−0.0411559
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.60555	−0.328603
$292$	0	0
$293$	−27.6333	−1.61436	−0.807178	−	0.590309i	$-0.799006\pi$
$293$	−27.6333	−1.61436	−0.807178	+	0.590309i	$0.799006\pi$
$294$	0	0
$295$	−1.57779	−0.0918628
$296$	0	0
$297$	25.8167	1.49803
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	2.69722	0.155465
$302$	0	0
$303$	16.4222	0.943431
$304$	0	0
$305$	−1.42221	−0.0814352
$306$	0	0
$307$	13.3944	0.764462	0.382231	−	0.924067i	$-0.375156\pi$
$307$	13.3944	0.764462	0.382231	+	0.924067i	$0.375156\pi$
$308$	0	0
$309$	7.57779	0.431086
$310$	0	0
$311$	15.6972	0.890108	0.445054	−	0.895504i	$-0.353184\pi$
$311$	15.6972	0.890108	0.445054	+	0.895504i	$0.353184\pi$
$312$	0	0
$313$	23.9361	1.35295	0.676474	−	0.736467i	$-0.263507\pi$
$313$	23.9361	1.35295	0.676474	+	0.736467i	$0.263507\pi$
$314$	0	0
$315$	0.394449	0.0222247
$316$	0	0
$317$	22.4222	1.25936	0.629678	−	0.776856i	$-0.283187\pi$
$317$	22.4222	1.25936	0.629678	+	0.776856i	$0.283187\pi$
$318$	0	0
$319$	−39.6333	−2.21904
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.60555	0.256260
$324$	0	0
$325$	19.6333	1.08906
$326$	0	0
$327$	4.18335	0.231340
$328$	0	0
$329$	−9.21110	−0.507825
$330$	0	0
$331$	−16.9083	−0.929366	−0.464683	−	0.885477i	$-0.653832\pi$
$331$	−16.9083	−0.929366	−0.464683	+	0.885477i	$0.653832\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−4.09167	−0.223552
$336$	0	0
$337$	11.6333	0.633707	0.316853	−	0.948475i	$-0.397374\pi$
$337$	11.6333	0.633707	0.316853	+	0.948475i	$0.397374\pi$
$338$	0	0
$339$	−1.57779	−0.0856941
$340$	0	0
$341$	−22.6056	−1.22416
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.36669	0.127418
$346$	0	0
$347$	21.6333	1.16134	0.580668	−	0.814140i	$-0.302791\pi$
$347$	21.6333	1.16134	0.580668	+	0.814140i	$0.302791\pi$
$348$	0	0
$349$	−30.2389	−1.61865	−0.809325	−	0.587362i	$-0.800167\pi$
$349$	−30.2389	−1.61865	−0.809325	+	0.587362i	$0.800167\pi$
$350$	0	0
$351$	−22.4222	−1.19681
$352$	0	0
$353$	−16.6056	−0.883824	−0.441912	−	0.897058i	$-0.645700\pi$
$353$	−16.6056	−0.883824	−0.441912	+	0.897058i	$0.645700\pi$
$354$	0	0
$355$	0.183346	0.00973100
$356$	0	0
$357$	1.30278	0.0689502
$358$	0	0
$359$	−0.908327	−0.0479397	−0.0239698	−	0.999713i	$-0.507631\pi$
$359$	−0.908327	−0.0479397	−0.0239698	+	0.999713i	$0.507631\pi$
$360$	0	0
$361$	2.21110	0.116374
$362$	0	0
$363$	−13.3028	−0.698215
$364$	0	0
$365$	−1.78890	−0.0936352
$366$	0	0
$367$	−2.51388	−0.131223	−0.0656117	−	0.997845i	$-0.520900\pi$
$367$	−2.51388	−0.131223	−0.0656117	+	0.997845i	$0.520900\pi$
$368$	0	0
$369$	−0.908327	−0.0472856
$370$	0	0
$371$	7.69722	0.399620
$372$	0	0
$373$	12.3305	0.638451	0.319225	−	0.947679i	$-0.396577\pi$
$373$	12.3305	0.638451	0.319225	+	0.947679i	$0.396577\pi$
$374$	0	0
$375$	3.90833	0.201825
$376$	0	0
$377$	34.4222	1.77283
$378$	0	0
$379$	−7.39445	−0.379827	−0.189914	−	0.981801i	$-0.560821\pi$
$379$	−7.39445	−0.379827	−0.189914	+	0.981801i	$0.560821\pi$
$380$	0	0
$381$	−24.3944	−1.24977
$382$	0	0
$383$	−12.2389	−0.625376	−0.312688	−	0.949856i	$-0.601230\pi$
$383$	−12.2389	−0.625376	−0.312688	+	0.949856i	$0.601230\pi$
$384$	0	0
$385$	−1.39445	−0.0710677
$386$	0	0
$387$	3.51388	0.178620
$388$	0	0
$389$	6.72498	0.340970	0.170485	−	0.985360i	$-0.445467\pi$
$389$	6.72498	0.340970	0.170485	+	0.985360i	$0.445467\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	−10.4222	−0.525731
$394$	0	0
$395$	−0.366692	−0.0184503
$396$	0	0
$397$	−31.3305	−1.57243	−0.786217	−	0.617950i	$-0.787963\pi$
$397$	−31.3305	−1.57243	−0.786217	+	0.617950i	$0.787963\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	18.2389	0.910805	0.455403	−	0.890286i	$-0.349495\pi$
$401$	18.2389	0.910805	0.455403	+	0.890286i	$0.349495\pi$
$402$	0	0
$403$	19.6333	0.978005
$404$	0	0
$405$	−1.02776	−0.0510696
$406$	0	0
$407$	−21.2111	−1.05140
$408$	0	0
$409$	−22.6056	−1.11777	−0.558886	−	0.829244i	$-0.688771\pi$
$409$	−22.6056	−1.11777	−0.558886	+	0.829244i	$0.688771\pi$
$410$	0	0
$411$	28.5416	1.40786
$412$	0	0
$413$	5.21110	0.256422
$414$	0	0
$415$	−5.39445	−0.264803
$416$	0	0
$417$	−17.6056	−0.862148
$418$	0	0
$419$	7.33053	0.358120	0.179060	−	0.983838i	$-0.442694\pi$
$419$	7.33053	0.358120	0.179060	+	0.983838i	$0.442694\pi$
$420$	0	0
$421$	−4.88057	−0.237864	−0.118932	−	0.992902i	$-0.537947\pi$
$421$	−4.88057	−0.237864	−0.118932	+	0.992902i	$0.537947\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−4.90833	−0.238089
$426$	0	0
$427$	4.69722	0.227315
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	6.60555	0.318178	0.159089	−	0.987264i	$-0.449144\pi$
$431$	6.60555	0.318178	0.159089	+	0.987264i	$0.449144\pi$
$432$	0	0
$433$	16.7889	0.806823	0.403411	−	0.915019i	$-0.367824\pi$
$433$	16.7889	0.806823	0.403411	+	0.915019i	$0.367824\pi$
$434$	0	0
$435$	3.39445	0.162751
$436$	0	0
$437$	−27.6333	−1.32188
$438$	0	0
$439$	−2.33053	−0.111230	−0.0556151	−	0.998452i	$-0.517712\pi$
$439$	−2.33053	−0.111230	−0.0556151	+	0.998452i	$0.517712\pi$
$440$	0	0
$441$	−1.30278	−0.0620369
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	5.21110	0.247030
$446$	0	0
$447$	−29.8806	−1.41330
$448$	0	0
$449$	3.21110	0.151541	0.0757706	−	0.997125i	$-0.475858\pi$
$449$	3.21110	0.151541	0.0757706	+	0.997125i	$0.475858\pi$
$450$	0	0
$451$	3.21110	0.151205
$452$	0	0
$453$	−10.5416	−0.495289
$454$	0	0
$455$	1.21110	0.0567774
$456$	0	0
$457$	18.4861	0.864744	0.432372	−	0.901695i	$-0.357677\pi$
$457$	18.4861	0.864744	0.432372	+	0.901695i	$0.357677\pi$
$458$	0	0
$459$	5.60555	0.261645
$460$	0	0
$461$	11.2111	0.522153	0.261077	−	0.965318i	$-0.415922\pi$
$461$	11.2111	0.522153	0.261077	+	0.965318i	$0.415922\pi$
$462$	0	0
$463$	−14.6972	−0.683038	−0.341519	−	0.939875i	$-0.610941\pi$
$463$	−14.6972	−0.683038	−0.341519	+	0.939875i	$0.610941\pi$
$464$	0	0
$465$	1.93608	0.0897837
$466$	0	0
$467$	−7.21110	−0.333690	−0.166845	−	0.985983i	$-0.553358\pi$
$467$	−7.21110	−0.333690	−0.166845	+	0.985983i	$0.553358\pi$
$468$	0	0
$469$	13.5139	0.624013
$470$	0	0
$471$	12.2389	0.563937
$472$	0	0
$473$	−12.4222	−0.571174
$474$	0	0
$475$	−22.6056	−1.03721
$476$	0	0
$477$	10.0278	0.459139
$478$	0	0
$479$	−21.1194	−0.964971	−0.482486	−	0.875904i	$-0.660266\pi$
$479$	−21.1194	−0.964971	−0.482486	+	0.875904i	$0.660266\pi$
$480$	0	0
$481$	18.4222	0.839980
$482$	0	0
$483$	−7.81665	−0.355670
$484$	0	0
$485$	1.30278	0.0591560
$486$	0	0
$487$	5.39445	0.244446	0.122223	−	0.992503i	$-0.460998\pi$
$487$	5.39445	0.244446	0.122223	+	0.992503i	$0.460998\pi$
$488$	0	0
$489$	−30.2389	−1.36745
$490$	0	0
$491$	−21.5139	−0.970908	−0.485454	−	0.874262i	$-0.661346\pi$
$491$	−21.5139	−0.970908	−0.485454	+	0.874262i	$0.661346\pi$
$492$	0	0
$493$	−8.60555	−0.387575
$494$	0	0
$495$	−1.81665	−0.0816525
$496$	0	0
$497$	−0.605551	−0.0271627
$498$	0	0
$499$	9.02776	0.404138	0.202069	−	0.979371i	$-0.435233\pi$
$499$	9.02776	0.404138	0.202069	+	0.979371i	$0.435233\pi$
$500$	0	0
$501$	−1.45837	−0.0651550
$502$	0	0
$503$	−20.9083	−0.932256	−0.466128	−	0.884717i	$-0.654351\pi$
$503$	−20.9083	−0.932256	−0.466128	+	0.884717i	$0.654351\pi$
$504$	0	0
$505$	−3.81665	−0.169839
$506$	0	0
$507$	−3.90833	−0.173575
$508$	0	0
$509$	8.42221	0.373308	0.186654	−	0.982426i	$-0.440236\pi$
$509$	8.42221	0.373308	0.186654	+	0.982426i	$0.440236\pi$
$510$	0	0
$511$	5.90833	0.261369
$512$	0	0
$513$	25.8167	1.13983
$514$	0	0
$515$	−1.76114	−0.0776051
$516$	0	0
$517$	42.4222	1.86573
$518$	0	0
$519$	−2.72498	−0.119613
$520$	0	0
$521$	−3.11943	−0.136665	−0.0683323	−	0.997663i	$-0.521768\pi$
$521$	−3.11943	−0.136665	−0.0683323	+	0.997663i	$0.521768\pi$
$522$	0	0
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	0	0
$525$	−6.39445	−0.279077
$526$	0	0
$527$	−4.90833	−0.213810
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	6.78890	0.294613
$532$	0	0
$533$	−2.78890	−0.120801
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.119429	0.00515376
$538$	0	0
$539$	4.60555	0.198375
$540$	0	0
$541$	19.2111	0.825950	0.412975	−	0.910742i	$-0.364490\pi$
$541$	19.2111	0.825950	0.412975	+	0.910742i	$0.364490\pi$
$542$	0	0
$543$	−33.3944	−1.43309
$544$	0	0
$545$	−0.972244	−0.0416463
$546$	0	0
$547$	−31.8167	−1.36038	−0.680191	−	0.733035i	$-0.738103\pi$
$547$	−31.8167	−1.36038	−0.680191	+	0.733035i	$0.738103\pi$
$548$	0	0
$549$	6.11943	0.261171
$550$	0	0
$551$	−39.6333	−1.68844
$552$	0	0
$553$	1.21110	0.0515013
$554$	0	0
$555$	1.81665	0.0771127
$556$	0	0
$557$	5.63331	0.238691	0.119345	−	0.992853i	$-0.461920\pi$
$557$	5.63331	0.238691	0.119345	+	0.992853i	$0.461920\pi$
$558$	0	0
$559$	10.7889	0.456322
$560$	0	0
$561$	−6.00000	−0.253320
$562$	0	0
$563$	28.6056	1.20558	0.602790	−	0.797900i	$-0.294056\pi$
$563$	28.6056	1.20558	0.602790	+	0.797900i	$0.294056\pi$
$564$	0	0
$565$	0.366692	0.0154269
$566$	0	0
$567$	3.39445	0.142553
$568$	0	0
$569$	22.0917	0.926131	0.463066	−	0.886324i	$-0.346749\pi$
$569$	22.0917	0.926131	0.463066	+	0.886324i	$0.346749\pi$
$570$	0	0
$571$	7.39445	0.309448	0.154724	−	0.987958i	$-0.450551\pi$
$571$	7.39445	0.309448	0.154724	+	0.987958i	$0.450551\pi$
$572$	0	0
$573$	26.4500	1.10496
$574$	0	0
$575$	29.4500	1.22815
$576$	0	0
$577$	−12.4222	−0.517143	−0.258572	−	0.965992i	$-0.583252\pi$
$577$	−12.4222	−0.517143	−0.258572	+	0.965992i	$0.583252\pi$
$578$	0	0
$579$	−27.6333	−1.14840
$580$	0	0
$581$	17.8167	0.739159
$582$	0	0
$583$	−35.4500	−1.46819
$584$	0	0
$585$	1.57779	0.0652338
$586$	0	0
$587$	21.6333	0.892902	0.446451	−	0.894808i	$-0.352688\pi$
$587$	21.6333	0.892902	0.446451	+	0.894808i	$0.352688\pi$
$588$	0	0
$589$	−22.6056	−0.931446
$590$	0	0
$591$	5.76114	0.236982
$592$	0	0
$593$	−23.3944	−0.960695	−0.480347	−	0.877078i	$-0.659489\pi$
$593$	−23.3944	−0.960695	−0.480347	+	0.877078i	$0.659489\pi$
$594$	0	0
$595$	−0.302776	−0.0124126
$596$	0	0
$597$	−29.5694	−1.21019
$598$	0	0
$599$	33.1194	1.35322	0.676612	−	0.736340i	$-0.263448\pi$
$599$	33.1194	1.35322	0.676612	+	0.736340i	$0.263448\pi$
$600$	0	0
$601$	−4.78890	−0.195343	−0.0976716	−	0.995219i	$-0.531139\pi$
$601$	−4.78890	−0.195343	−0.0976716	+	0.995219i	$0.531139\pi$
$602$	0	0
$603$	17.6056	0.716953
$604$	0	0
$605$	3.09167	0.125694
$606$	0	0
$607$	24.5416	0.996114	0.498057	−	0.867144i	$-0.334047\pi$
$607$	24.5416	0.996114	0.498057	+	0.867144i	$0.334047\pi$
$608$	0	0
$609$	−11.2111	−0.454297
$610$	0	0
$611$	−36.8444	−1.49057
$612$	0	0
$613$	−12.9083	−0.521362	−0.260681	−	0.965425i	$-0.583947\pi$
$613$	−12.9083	−0.521362	−0.260681	+	0.965425i	$0.583947\pi$
$614$	0	0
$615$	−0.275019	−0.0110898
$616$	0	0
$617$	−44.2389	−1.78099	−0.890495	−	0.454994i	$-0.849642\pi$
$617$	−44.2389	−1.78099	−0.890495	+	0.454994i	$0.849642\pi$
$618$	0	0
$619$	14.4222	0.579677	0.289839	−	0.957076i	$-0.406398\pi$
$619$	14.4222	0.579677	0.289839	+	0.957076i	$0.406398\pi$
$620$	0	0
$621$	−33.6333	−1.34966
$622$	0	0
$623$	−17.2111	−0.689548
$624$	0	0
$625$	23.6333	0.945332
$626$	0	0
$627$	−27.6333	−1.10357
$628$	0	0
$629$	−4.60555	−0.183635
$630$	0	0
$631$	26.4861	1.05440	0.527198	−	0.849743i	$-0.323243\pi$
$631$	26.4861	1.05440	0.527198	+	0.849743i	$0.323243\pi$
$632$	0	0
$633$	10.1833	0.404752
$634$	0	0
$635$	5.66947	0.224986
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	−0.788897	−0.0312083
$640$	0	0
$641$	−30.2389	−1.19436	−0.597182	−	0.802106i	$-0.703713\pi$
$641$	−30.2389	−1.19436	−0.597182	+	0.802106i	$0.703713\pi$
$642$	0	0
$643$	−44.7527	−1.76488	−0.882438	−	0.470429i	$-0.844099\pi$
$643$	−44.7527	−1.76488	−0.882438	+	0.470429i	$0.844099\pi$
$644$	0	0
$645$	1.06392	0.0418917
$646$	0	0
$647$	−9.81665	−0.385932	−0.192966	−	0.981205i	$-0.561811\pi$
$647$	−9.81665	−0.385932	−0.192966	+	0.981205i	$0.561811\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−6.39445	−0.250618
$652$	0	0
$653$	−24.4222	−0.955715	−0.477857	−	0.878437i	$-0.658586\pi$
$653$	−24.4222	−0.955715	−0.477857	+	0.878437i	$0.658586\pi$
$654$	0	0
$655$	2.42221	0.0946434
$656$	0	0
$657$	7.69722	0.300297
$658$	0	0
$659$	−16.3028	−0.635066	−0.317533	−	0.948247i	$-0.602854\pi$
$659$	−16.3028	−0.635066	−0.317533	+	0.948247i	$0.602854\pi$
$660$	0	0
$661$	−42.6056	−1.65716	−0.828582	−	0.559868i	$-0.810852\pi$
$661$	−42.6056	−1.65716	−0.828582	+	0.559868i	$0.810852\pi$
$662$	0	0
$663$	5.21110	0.202382
$664$	0	0
$665$	−1.39445	−0.0540744
$666$	0	0
$667$	51.6333	1.99925
$668$	0	0
$669$	−33.6333	−1.30034
$670$	0	0
$671$	−21.6333	−0.835145
$672$	0	0
$673$	42.8444	1.65153	0.825765	−	0.564014i	$-0.190744\pi$
$673$	42.8444	1.65153	0.825765	+	0.564014i	$0.190744\pi$
$674$	0	0
$675$	−27.5139	−1.05901
$676$	0	0
$677$	−0.788897	−0.0303198	−0.0151599	−	0.999885i	$-0.504826\pi$
$677$	−0.788897	−0.0303198	−0.0151599	+	0.999885i	$0.504826\pi$
$678$	0	0
$679$	−4.30278	−0.165125
$680$	0	0
$681$	13.6611	0.523493
$682$	0	0
$683$	−49.2666	−1.88513	−0.942567	−	0.334016i	$-0.891596\pi$
$683$	−49.2666	−1.88513	−0.942567	+	0.334016i	$0.891596\pi$
$684$	0	0
$685$	−6.63331	−0.253446
$686$	0	0
$687$	12.7889	0.487927
$688$	0	0
$689$	30.7889	1.17296
$690$	0	0
$691$	35.5416	1.35207	0.676034	−	0.736871i	$-0.263697\pi$
$691$	35.5416	1.35207	0.676034	+	0.736871i	$0.263697\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	4.09167	0.155206
$696$	0	0
$697$	0.697224	0.0264093
$698$	0	0
$699$	28.9722	1.09583
$700$	0	0
$701$	45.6333	1.72355	0.861773	−	0.507294i	$-0.169354\pi$
$701$	45.6333	1.72355	0.861773	+	0.507294i	$0.169354\pi$
$702$	0	0
$703$	−21.2111	−0.799992
$704$	0	0
$705$	−3.63331	−0.136838
$706$	0	0
$707$	12.6056	0.474081
$708$	0	0
$709$	15.6333	0.587121	0.293561	−	0.955940i	$-0.405160\pi$
$709$	15.6333	0.587121	0.293561	+	0.955940i	$0.405160\pi$
$710$	0	0
$711$	1.57779	0.0591719
$712$	0	0
$713$	29.4500	1.10291
$714$	0	0
$715$	−5.57779	−0.208598
$716$	0	0
$717$	39.5139	1.47567
$718$	0	0
$719$	−43.3583	−1.61699	−0.808496	−	0.588502i	$-0.799718\pi$
$719$	−43.3583	−1.61699	−0.808496	+	0.588502i	$0.799718\pi$
$720$	0	0
$721$	5.81665	0.216624
$722$	0	0
$723$	−1.97224	−0.0733485
$724$	0	0
$725$	42.2389	1.56871
$726$	0	0
$727$	10.2389	0.379738	0.189869	−	0.981809i	$-0.439194\pi$
$727$	10.2389	0.379738	0.189869	+	0.981809i	$0.439194\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	−2.69722	−0.0997604
$732$	0	0
$733$	−37.6333	−1.39002	−0.695009	−	0.719001i	$-0.744600\pi$
$733$	−37.6333	−1.39002	−0.695009	+	0.719001i	$0.744600\pi$
$734$	0	0
$735$	−0.394449	−0.0145495
$736$	0	0
$737$	−62.2389	−2.29260
$738$	0	0
$739$	35.5416	1.30742	0.653710	−	0.756745i	$-0.273212\pi$
$739$	35.5416	1.30742	0.653710	+	0.756745i	$0.273212\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	0	0
$743$	11.3944	0.418022	0.209011	−	0.977913i	$-0.432976\pi$
$743$	11.3944	0.418022	0.209011	+	0.977913i	$0.432976\pi$
$744$	0	0
$745$	6.94449	0.254426
$746$	0	0
$747$	23.2111	0.849250
$748$	0	0
$749$	0	0
$750$	0	0
$751$	11.0278	0.402409	0.201204	−	0.979549i	$-0.435514\pi$
$751$	11.0278	0.402409	0.201204	+	0.979549i	$0.435514\pi$
$752$	0	0
$753$	11.2111	0.408555
$754$	0	0
$755$	2.44996	0.0891632
$756$	0	0
$757$	−27.3028	−0.992336	−0.496168	−	0.868226i	$-0.665260\pi$
$757$	−27.3028	−0.992336	−0.496168	+	0.868226i	$0.665260\pi$
$758$	0	0
$759$	36.0000	1.30672
$760$	0	0
$761$	29.4500	1.06756	0.533780	−	0.845623i	$-0.320771\pi$
$761$	29.4500	1.06756	0.533780	+	0.845623i	$0.320771\pi$
$762$	0	0
$763$	3.21110	0.116250
$764$	0	0
$765$	−0.394449	−0.0142613
$766$	0	0
$767$	20.8444	0.752648
$768$	0	0
$769$	−6.60555	−0.238202	−0.119101	−	0.992882i	$-0.538001\pi$
$769$	−6.60555	−0.238202	−0.119101	+	0.992882i	$0.538001\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	0	0
$773$	4.84441	0.174241	0.0871207	−	0.996198i	$-0.472233\pi$
$773$	4.84441	0.174241	0.0871207	+	0.996198i	$0.472233\pi$
$774$	0	0
$775$	24.0917	0.865398
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	3.21110	0.115050
$780$	0	0
$781$	2.78890	0.0997946
$782$	0	0
$783$	−48.2389	−1.72392
$784$	0	0
$785$	−2.84441	−0.101521
$786$	0	0
$787$	25.2111	0.898679	0.449339	−	0.893361i	$-0.351659\pi$
$787$	25.2111	0.898679	0.449339	+	0.893361i	$0.351659\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.21110	−0.0430618
$792$	0	0
$793$	18.7889	0.667213
$794$	0	0
$795$	3.03616	0.107682
$796$	0	0
$797$	−10.1833	−0.360713	−0.180356	−	0.983601i	$-0.557725\pi$
$797$	−10.1833	−0.360713	−0.180356	+	0.983601i	$0.557725\pi$
$798$	0	0
$799$	9.21110	0.325865
$800$	0	0
$801$	−22.4222	−0.792250
$802$	0	0
$803$	−27.2111	−0.960259
$804$	0	0
$805$	1.81665	0.0640286
$806$	0	0
$807$	−6.23886	−0.219618
$808$	0	0
$809$	−15.2111	−0.534794	−0.267397	−	0.963586i	$-0.586163\pi$
$809$	−15.2111	−0.534794	−0.267397	+	0.963586i	$0.586163\pi$
$810$	0	0
$811$	34.5139	1.21195	0.605973	−	0.795485i	$-0.292784\pi$
$811$	34.5139	1.21195	0.605973	+	0.795485i	$0.292784\pi$
$812$	0	0
$813$	−5.76114	−0.202052
$814$	0	0
$815$	7.02776	0.246172
$816$	0	0
$817$	−12.4222	−0.434598
$818$	0	0
$819$	−5.21110	−0.182091
$820$	0	0
$821$	−7.39445	−0.258068	−0.129034	−	0.991640i	$-0.541188\pi$
$821$	−7.39445	−0.258068	−0.129034	+	0.991640i	$0.541188\pi$
$822$	0	0
$823$	31.0278	1.08156	0.540780	−	0.841164i	$-0.318129\pi$
$823$	31.0278	1.08156	0.540780	+	0.841164i	$0.318129\pi$
$824$	0	0
$825$	29.4500	1.02532
$826$	0	0
$827$	1.76114	0.0612409	0.0306204	−	0.999531i	$-0.490252\pi$
$827$	1.76114	0.0612409	0.0306204	+	0.999531i	$0.490252\pi$
$828$	0	0
$829$	−18.0555	−0.627094	−0.313547	−	0.949573i	$-0.601517\pi$
$829$	−18.0555	−0.627094	−0.313547	+	0.949573i	$0.601517\pi$
$830$	0	0
$831$	33.8722	1.17501
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	0.338936	0.0117294
$836$	0	0
$837$	−27.5139	−0.951019
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	45.0555	1.55364
$842$	0	0
$843$	21.2389	0.731505
$844$	0	0
$845$	0.908327	0.0312474
$846$	0	0
$847$	−10.2111	−0.350858
$848$	0	0
$849$	0.394449	0.0135374
$850$	0	0
$851$	27.6333	0.947258
$852$	0	0
$853$	−36.0555	−1.23452	−0.617259	−	0.786760i	$-0.711757\pi$
$853$	−36.0555	−1.23452	−0.617259	+	0.786760i	$0.711757\pi$
$854$	0	0
$855$	−1.81665	−0.0621282
$856$	0	0
$857$	11.3305	0.387044	0.193522	−	0.981096i	$-0.438009\pi$
$857$	11.3305	0.387044	0.193522	+	0.981096i	$0.438009\pi$
$858$	0	0
$859$	40.8444	1.39359	0.696797	−	0.717269i	$-0.254608\pi$
$859$	40.8444	1.39359	0.696797	+	0.717269i	$0.254608\pi$
$860$	0	0
$861$	0.908327	0.0309557
$862$	0	0
$863$	−15.6972	−0.534340	−0.267170	−	0.963649i	$-0.586088\pi$
$863$	−15.6972	−0.534340	−0.267170	+	0.963649i	$0.586088\pi$
$864$	0	0
$865$	0.633308	0.0215331
$866$	0	0
$867$	−1.30278	−0.0442446
$868$	0	0
$869$	−5.57779	−0.189214
$870$	0	0
$871$	54.0555	1.83160
$872$	0	0
$873$	−5.60555	−0.189719
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−23.2111	−0.783783	−0.391892	−	0.920011i	$-0.628179\pi$
$877$	−23.2111	−0.783783	−0.391892	+	0.920011i	$0.628179\pi$
$878$	0	0
$879$	36.0000	1.21425
$880$	0	0
$881$	−2.72498	−0.0918069	−0.0459035	−	0.998946i	$-0.514617\pi$
$881$	−2.72498	−0.0918069	−0.0459035	+	0.998946i	$0.514617\pi$
$882$	0	0
$883$	−48.5416	−1.63356	−0.816778	−	0.576952i	$-0.804242\pi$
$883$	−48.5416	−1.63356	−0.816778	+	0.576952i	$0.804242\pi$
$884$	0	0
$885$	2.05551	0.0690953
$886$	0	0
$887$	33.1472	1.11297	0.556487	−	0.830856i	$-0.312149\pi$
$887$	33.1472	1.11297	0.556487	+	0.830856i	$0.312149\pi$
$888$	0	0
$889$	−18.7250	−0.628016
$890$	0	0
$891$	−15.6333	−0.523736
$892$	0	0
$893$	42.4222	1.41960
$894$	0	0
$895$	−0.0277564	−0.000927793 0
$896$	0	0
$897$	−31.2666	−1.04396
$898$	0	0
$899$	42.2389	1.40874
$900$	0	0
$901$	−7.69722	−0.256432
$902$	0	0
$903$	−3.51388	−0.116935
$904$	0	0
$905$	7.76114	0.257989
$906$	0	0
$907$	−20.4222	−0.678108	−0.339054	−	0.940767i	$-0.610107\pi$
$907$	−20.4222	−0.678108	−0.339054	+	0.940767i	$0.610107\pi$
$908$	0	0
$909$	16.4222	0.544690
$910$	0	0
$911$	10.9722	0.363527	0.181763	−	0.983342i	$-0.441820\pi$
$911$	10.9722	0.363527	0.181763	+	0.983342i	$0.441820\pi$
$912$	0	0
$913$	−82.0555	−2.71564
$914$	0	0
$915$	1.85281	0.0612521
$916$	0	0
$917$	−8.00000	−0.264183
$918$	0	0
$919$	6.69722	0.220921	0.110461	−	0.993881i	$-0.464767\pi$
$919$	6.69722	0.220921	0.110461	+	0.993881i	$0.464767\pi$
$920$	0	0
$921$	−17.4500	−0.574996
$922$	0	0
$923$	−2.42221	−0.0797279
$924$	0	0
$925$	22.6056	0.743266
$926$	0	0
$927$	7.57779	0.248887
$928$	0	0
$929$	18.4861	0.606510	0.303255	−	0.952909i	$-0.401927\pi$
$929$	18.4861	0.606510	0.303255	+	0.952909i	$0.401927\pi$
$930$	0	0
$931$	4.60555	0.150941
$932$	0	0
$933$	−20.4500	−0.669502
$934$	0	0
$935$	1.39445	0.0456033
$936$	0	0
$937$	−30.4222	−0.993850	−0.496925	−	0.867793i	$-0.665538\pi$
$937$	−30.4222	−0.993850	−0.496925	+	0.867793i	$0.665538\pi$
$938$	0	0
$939$	−31.1833	−1.01763
$940$	0	0
$941$	−18.7527	−0.611322	−0.305661	−	0.952140i	$-0.598877\pi$
$941$	−18.7527	−0.611322	−0.305661	+	0.952140i	$0.598877\pi$
$942$	0	0
$943$	−4.18335	−0.136228
$944$	0	0
$945$	−1.69722	−0.0552107
$946$	0	0
$947$	12.6056	0.409625	0.204813	−	0.978801i	$-0.434341\pi$
$947$	12.6056	0.409625	0.204813	+	0.978801i	$0.434341\pi$
$948$	0	0
$949$	23.6333	0.767170
$950$	0	0
$951$	−29.2111	−0.947235
$952$	0	0
$953$	30.3583	0.983401	0.491701	−	0.870764i	$-0.336375\pi$
$953$	30.3583	0.983401	0.491701	+	0.870764i	$0.336375\pi$
$954$	0	0
$955$	−6.14719	−0.198918
$956$	0	0
$957$	51.6333	1.66907
$958$	0	0
$959$	21.9083	0.707457
$960$	0	0
$961$	−6.90833	−0.222849
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.42221	0.206738
$966$	0	0
$967$	45.3028	1.45684	0.728420	−	0.685131i	$-0.240255\pi$
$967$	45.3028	1.45684	0.728420	+	0.685131i	$0.240255\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	32.6056	1.04636	0.523181	−	0.852222i	$-0.324745\pi$
$971$	32.6056	1.04636	0.523181	+	0.852222i	$0.324745\pi$
$972$	0	0
$973$	−13.5139	−0.433235
$974$	0	0
$975$	−25.5778	−0.819145
$976$	0	0
$977$	15.6972	0.502199	0.251099	−	0.967961i	$-0.419208\pi$
$977$	15.6972	0.502199	0.251099	+	0.967961i	$0.419208\pi$
$978$	0	0
$979$	79.2666	2.53337
$980$	0	0
$981$	4.18335	0.133564
$982$	0	0
$983$	59.5694	1.89997	0.949984	−	0.312298i	$-0.101099\pi$
$983$	59.5694	1.89997	0.949984	+	0.312298i	$0.101099\pi$
$984$	0	0
$985$	−1.33894	−0.0426620
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	16.1833	0.514600
$990$	0	0
$991$	−44.4222	−1.41112	−0.705559	−	0.708651i	$-0.749304\pi$
$991$	−44.4222	−1.41112	−0.705559	+	0.708651i	$0.749304\pi$
$992$	0	0
$993$	22.0278	0.699030
$994$	0	0
$995$	6.87217	0.217862
$996$	0	0
$997$	0.880571	0.0278879	0.0139440	−	0.999903i	$-0.495561\pi$
$997$	0.880571	0.0278879	0.0139440	+	0.999903i	$0.495561\pi$
$998$	0	0
$999$	−25.8167	−0.816803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	952.2.a.b.1.1	✓	2
3.2	odd	2		8568.2.a.t.1.1		2
4.3	odd	2		1904.2.a.g.1.2		2
7.6	odd	2		6664.2.a.g.1.2		2
8.3	odd	2		7616.2.a.x.1.1		2
8.5	even	2		7616.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.b.1.1	✓	2	1.1	even	1	trivial
1904.2.a.g.1.2		2	4.3	odd	2
6664.2.a.g.1.2		2	7.6	odd	2
7616.2.a.s.1.2		2	8.5	even	2
7616.2.a.x.1.1		2	8.3	odd	2
8568.2.a.t.1.1		2	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 952 = 2^{3} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	952.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{3} +0.302776 q^{5} -1.00000 q^{7} -1.30278 q^{9} +4.60555 q^{11} -4.00000 q^{13} -0.394449 q^{15} +1.00000 q^{17} +4.60555 q^{19} +1.30278 q^{21} -6.00000 q^{23} -4.90833 q^{25} +5.60555 q^{27} -8.60555 q^{29} -4.90833 q^{31} -6.00000 q^{33} -0.302776 q^{35} -4.60555 q^{37} +5.21110 q^{39} +0.697224 q^{41} -2.69722 q^{43} -0.394449 q^{45} +9.21110 q^{47} +1.00000 q^{49} -1.30278 q^{51} -7.69722 q^{53} +1.39445 q^{55} -6.00000 q^{57} -5.21110 q^{59} -4.69722 q^{61} +1.30278 q^{63} -1.21110 q^{65} -13.5139 q^{67} +7.81665 q^{69} +0.605551 q^{71} -5.90833 q^{73} +6.39445 q^{75} -4.60555 q^{77} -1.21110 q^{79} -3.39445 q^{81} -17.8167 q^{83} +0.302776 q^{85} +11.2111 q^{87} +17.2111 q^{89} +4.00000 q^{91} +6.39445 q^{93} +1.39445 q^{95} +4.30278 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 3 q^{5} - 2 q^{7} + q^{9} + 2 q^{11} - 8 q^{13} - 8 q^{15} + 2 q^{17} + 2 q^{19} - q^{21} - 12 q^{23} + q^{25} + 4 q^{27} - 10 q^{29} + q^{31} - 12 q^{33} + 3 q^{35} - 2 q^{37} - 4 q^{39}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q + q^3 - 3 * q^5 - 2 * q^7 + q^9 + 2 * q^11 - 8 * q^13 - 8 * q^15 + 2 * q^17 + 2 * q^19 - q^21 - 12 * q^23 + q^25 + 4 * q^27 - 10 * q^29 + q^31 - 12 * q^33 + 3 * q^35 - 2 * q^37 - 4 * q^39 + 5 * q^41 - 9 * q^43 - 8 * q^45 + 4 * q^47 + 2 * q^49 + q^51 - 19 * q^53 + 10 * q^55 - 12 * q^57 + 4 * q^59 - 13 * q^61 - q^63 + 12 * q^65 - 9 * q^67 - 6 * q^69 - 6 * q^71 - q^73 + 20 * q^75 - 2 * q^77 + 12 * q^79 - 14 * q^81 - 14 * q^83 - 3 * q^85 + 8 * q^87 + 20 * q^89 + 8 * q^91 + 20 * q^93 + 10 * q^95 + 5 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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