LMFDB - Embedded newform 3724.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, 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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3724 = 2^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3724.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:	$29.7362897127$
Analytic rank:	$0$
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} - 4x^{7} - 4x^{6} + 24x^{5} + x^{4} - 40x^{3} + 6x^{2} + 16x - 1 $ x^8 - 4x^7 - 4x^6 + 24x^5 + x^4 - 40x^3 + 6x^2 + 16x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.39347$ of defining polynomial
Character	$\chi$	$=$	3724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.39347 q^{3} +3.30592 q^{5} -1.05823 q^{9} -1.10587 q^{11} -2.11454 q^{13} -4.60671 q^{15} -6.05149 q^{17} -1.00000 q^{19} +4.53461 q^{23} +5.92909 q^{25} +5.65504 q^{27} -7.82502 q^{29} +1.85247 q^{31} +1.54100 q^{33} +5.34419 q^{37} +2.94655 q^{39} +6.37802 q^{41} -2.53815 q^{43} -3.49842 q^{45} +9.90863 q^{47} +8.43260 q^{51} +8.26962 q^{53} -3.65592 q^{55} +1.39347 q^{57} +8.35210 q^{59} +7.17111 q^{61} -6.99048 q^{65} +3.78722 q^{67} -6.31886 q^{69} +2.96100 q^{71} +7.20336 q^{73} -8.26203 q^{75} +0.252831 q^{79} -4.70547 q^{81} +13.2004 q^{83} -20.0057 q^{85} +10.9040 q^{87} +12.0299 q^{89} -2.58137 q^{93} -3.30592 q^{95} +5.47652 q^{97} +1.17026 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 8 q^{11} + 8 q^{17} - 8 q^{19} + 4 q^{23} + 4 q^{25} + 16 q^{27} - 4 q^{29} + 24 q^{31} - 12 q^{33} + 12 q^{37} + 4 q^{39} + 20 q^{41} + 12 q^{43} + 24 q^{45} + 24 q^{47} + 12 q^{51} + 4 q^{53}+ \cdots + 20 q^{97}+O(q^{100}) $ 8 * q + 4 * q^3 - 8 * q^11 + 8 * q^17 - 8 * q^19 + 4 * q^23 + 4 * q^25 + 16 * q^27 - 4 * q^29 + 24 * q^31 - 12 * q^33 + 12 * q^37 + 4 * q^39 + 20 * q^41 + 12 * q^43 + 24 * q^45 + 24 * q^47 + 12 * q^51 + 4 * q^53 + 16 * q^55 - 4 * q^57 + 52 * q^59 - 16 * q^61 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 16 * q^71 + 24 * q^73 + 8 * q^75 - 20 * q^79 + 12 * q^81 + 48 * q^83 - 12 * q^85 + 40 * q^87 + 12 * q^89 + 8 * q^93 + 20 * q^97

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
$3$	−1.39347	−0.804523	−0.402261	−	0.915525i	$-0.631776\pi$
$3$	−1.39347	−0.804523	−0.402261	+	0.915525i	$0.631776\pi$
$4$	0	0
$5$	3.30592	1.47845	0.739226	−	0.673458i	$-0.235192\pi$
$5$	3.30592	1.47845	0.739226	+	0.673458i	$0.235192\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.05823	−0.352743
$10$	0	0
$11$	−1.10587	−0.333433	−0.166716	−	0.986005i	$-0.553316\pi$
$11$	−1.10587	−0.333433	−0.166716	+	0.986005i	$0.553316\pi$
$12$	0	0
$13$	−2.11454	−0.586467	−0.293233	−	0.956041i	$-0.594731\pi$
$13$	−2.11454	−0.586467	−0.293233	+	0.956041i	$0.594731\pi$
$14$	0	0
$15$	−4.60671	−1.18945
$16$	0	0
$17$	−6.05149	−1.46770	−0.733851	−	0.679310i	$-0.762279\pi$
$17$	−6.05149	−1.46770	−0.733851	+	0.679310i	$0.762279\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.53461	0.945531	0.472766	−	0.881188i	$-0.343256\pi$
$23$	4.53461	0.945531	0.472766	+	0.881188i	$0.343256\pi$
$24$	0	0
$25$	5.92909	1.18582
$26$	0	0
$27$	5.65504	1.08831
$28$	0	0
$29$	−7.82502	−1.45307	−0.726535	−	0.687130i	$-0.758870\pi$
$29$	−7.82502	−1.45307	−0.726535	+	0.687130i	$0.758870\pi$
$30$	0	0
$31$	1.85247	0.332714	0.166357	−	0.986066i	$-0.446800\pi$
$31$	1.85247	0.332714	0.166357	+	0.986066i	$0.446800\pi$
$32$	0	0
$33$	1.54100	0.268254
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.34419	0.878580	0.439290	−	0.898345i	$-0.355230\pi$
$37$	5.34419	0.878580	0.439290	+	0.898345i	$0.355230\pi$
$38$	0	0
$39$	2.94655	0.471826
$40$	0	0
$41$	6.37802	0.996079	0.498040	−	0.867154i	$-0.334053\pi$
$41$	6.37802	0.996079	0.498040	+	0.867154i	$0.334053\pi$
$42$	0	0
$43$	−2.53815	−0.387064	−0.193532	−	0.981094i	$-0.561994\pi$
$43$	−2.53815	−0.387064	−0.193532	+	0.981094i	$0.561994\pi$
$44$	0	0
$45$	−3.49842	−0.521513
$46$	0	0
$47$	9.90863	1.44532	0.722661	−	0.691202i	$-0.242919\pi$
$47$	9.90863	1.44532	0.722661	+	0.691202i	$0.242919\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.43260	1.18080
$52$	0	0
$53$	8.26962	1.13592	0.567960	−	0.823056i	$-0.307733\pi$
$53$	8.26962	1.13592	0.567960	+	0.823056i	$0.307733\pi$
$54$	0	0
$55$	−3.65592	−0.492964
$56$	0	0
$57$	1.39347	0.184570
$58$	0	0
$59$	8.35210	1.08735	0.543675	−	0.839296i	$-0.317032\pi$
$59$	8.35210	1.08735	0.543675	+	0.839296i	$0.317032\pi$
$60$	0	0
$61$	7.17111	0.918167	0.459083	−	0.888393i	$-0.348178\pi$
$61$	7.17111	0.918167	0.459083	+	0.888393i	$0.348178\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.99048	−0.867062
$66$	0	0
$67$	3.78722	0.462683	0.231342	−	0.972873i	$-0.425689\pi$
$67$	3.78722	0.462683	0.231342	+	0.972873i	$0.425689\pi$
$68$	0	0
$69$	−6.31886	−0.760702
$70$	0	0
$71$	2.96100	0.351406	0.175703	−	0.984443i	$-0.443780\pi$
$71$	2.96100	0.351406	0.175703	+	0.984443i	$0.443780\pi$
$72$	0	0
$73$	7.20336	0.843089	0.421545	−	0.906808i	$-0.361488\pi$
$73$	7.20336	0.843089	0.421545	+	0.906808i	$0.361488\pi$
$74$	0	0
$75$	−8.26203	−0.954017
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.252831	0.0284457	0.0142229	−	0.999899i	$-0.495473\pi$
$79$	0.252831	0.0284457	0.0142229	+	0.999899i	$0.495473\pi$
$80$	0	0
$81$	−4.70547	−0.522830
$82$	0	0
$83$	13.2004	1.44893	0.724464	−	0.689313i	$-0.242087\pi$
$83$	13.2004	1.44893	0.724464	+	0.689313i	$0.242087\pi$
$84$	0	0
$85$	−20.0057	−2.16993
$86$	0	0
$87$	10.9040	1.16903
$88$	0	0
$89$	12.0299	1.27516	0.637581	−	0.770383i	$-0.279935\pi$
$89$	12.0299	1.27516	0.637581	+	0.770383i	$0.279935\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.58137	−0.267676
$94$	0	0
$95$	−3.30592	−0.339180
$96$	0	0
$97$	5.47652	0.556056	0.278028	−	0.960573i	$-0.410319\pi$
$97$	5.47652	0.556056	0.278028	+	0.960573i	$0.410319\pi$
$98$	0	0
$99$	1.17026	0.117616
$100$	0	0
$101$	−9.48169	−0.943463	−0.471732	−	0.881742i	$-0.656371\pi$
$101$	−9.48169	−0.943463	−0.471732	+	0.881742i	$0.656371\pi$
$102$	0	0
$103$	5.26893	0.519163	0.259581	−	0.965721i	$-0.416415\pi$
$103$	5.26893	0.519163	0.259581	+	0.965721i	$0.416415\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.45965	−0.431130	−0.215565	−	0.976489i	$-0.569159\pi$
$107$	−4.45965	−0.431130	−0.215565	+	0.976489i	$0.569159\pi$
$108$	0	0
$109$	8.16815	0.782367	0.391183	−	0.920313i	$-0.372066\pi$
$109$	8.16815	0.782367	0.391183	+	0.920313i	$0.372066\pi$
$110$	0	0
$111$	−7.44700	−0.706838
$112$	0	0
$113$	−17.6303	−1.65852	−0.829260	−	0.558863i	$-0.811238\pi$
$113$	−17.6303	−1.65852	−0.829260	+	0.558863i	$0.811238\pi$
$114$	0	0
$115$	14.9910	1.39792
$116$	0	0
$117$	2.23766	0.206872
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.77705	−0.888823
$122$	0	0
$123$	−8.88761	−0.801369
$124$	0	0
$125$	3.07149	0.274722
$126$	0	0
$127$	−3.63795	−0.322815	−0.161408	−	0.986888i	$-0.551603\pi$
$127$	−3.63795	−0.322815	−0.161408	+	0.986888i	$0.551603\pi$
$128$	0	0
$129$	3.53685	0.311402
$130$	0	0
$131$	8.10458	0.708101	0.354050	−	0.935226i	$-0.384804\pi$
$131$	8.10458	0.708101	0.354050	+	0.935226i	$0.384804\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	18.6951	1.60902
$136$	0	0
$137$	−16.6223	−1.42014	−0.710070	−	0.704132i	$-0.751337\pi$
$137$	−16.6223	−1.42014	−0.710070	+	0.704132i	$0.751337\pi$
$138$	0	0
$139$	−20.0563	−1.70115	−0.850576	−	0.525852i	$-0.823747\pi$
$139$	−20.0563	−1.70115	−0.850576	+	0.525852i	$0.823747\pi$
$140$	0	0
$141$	−13.8074	−1.16280
$142$	0	0
$143$	2.33840	0.195547
$144$	0	0
$145$	−25.8689	−2.14829
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.56699	−0.537989	−0.268994	−	0.963142i	$-0.586691\pi$
$149$	−6.56699	−0.537989	−0.268994	+	0.963142i	$0.586691\pi$
$150$	0	0
$151$	9.02020	0.734053	0.367027	−	0.930210i	$-0.380376\pi$
$151$	9.02020	0.734053	0.367027	+	0.930210i	$0.380376\pi$
$152$	0	0
$153$	6.40386	0.517721
$154$	0	0
$155$	6.12412	0.491901
$156$	0	0
$157$	17.2831	1.37934	0.689669	−	0.724124i	$-0.257756\pi$
$157$	17.2831	1.37934	0.689669	+	0.724124i	$0.257756\pi$
$158$	0	0
$159$	−11.5235	−0.913873
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.97122	−0.781007	−0.390503	−	0.920602i	$-0.627699\pi$
$163$	−9.97122	−0.781007	−0.390503	+	0.920602i	$0.627699\pi$
$164$	0	0
$165$	5.09443	0.396601
$166$	0	0
$167$	17.6449	1.36541	0.682703	−	0.730696i	$-0.260804\pi$
$167$	17.6449	1.36541	0.682703	+	0.730696i	$0.260804\pi$
$168$	0	0
$169$	−8.52874	−0.656057
$170$	0	0
$171$	1.05823	0.0809247
$172$	0	0
$173$	−21.2223	−1.61350	−0.806752	−	0.590890i	$-0.798777\pi$
$173$	−21.2223	−1.61350	−0.806752	+	0.590890i	$0.798777\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.6384	−0.874799
$178$	0	0
$179$	12.5697	0.939503	0.469751	−	0.882799i	$-0.344344\pi$
$179$	12.5697	0.939503	0.469751	+	0.882799i	$0.344344\pi$
$180$	0	0
$181$	0.739327	0.0549538	0.0274769	−	0.999622i	$-0.491253\pi$
$181$	0.739327	0.0549538	0.0274769	+	0.999622i	$0.491253\pi$
$182$	0	0
$183$	−9.99276	−0.738686
$184$	0	0
$185$	17.6675	1.29894
$186$	0	0
$187$	6.69217	0.489380
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.3243	1.54297	0.771486	−	0.636246i	$-0.219514\pi$
$191$	21.3243	1.54297	0.771486	+	0.636246i	$0.219514\pi$
$192$	0	0
$193$	−6.18487	−0.445197	−0.222598	−	0.974910i	$-0.571454\pi$
$193$	−6.18487	−0.445197	−0.222598	+	0.974910i	$0.571454\pi$
$194$	0	0
$195$	9.74106	0.697572
$196$	0	0
$197$	7.18658	0.512023	0.256011	−	0.966674i	$-0.417592\pi$
$197$	7.18658	0.512023	0.256011	+	0.966674i	$0.417592\pi$
$198$	0	0
$199$	11.2582	0.798072	0.399036	−	0.916935i	$-0.369345\pi$
$199$	11.2582	0.798072	0.399036	+	0.916935i	$0.369345\pi$
$200$	0	0
$201$	−5.27740	−0.372239
$202$	0	0
$203$	0	0
$204$	0	0
$205$	21.0852	1.47265
$206$	0	0
$207$	−4.79865	−0.333529
$208$	0	0
$209$	1.10587	0.0764947
$210$	0	0
$211$	26.4910	1.82371	0.911856	−	0.410509i	$-0.134649\pi$
$211$	26.4910	1.82371	0.911856	+	0.410509i	$0.134649\pi$
$212$	0	0
$213$	−4.12608	−0.282714
$214$	0	0
$215$	−8.39092	−0.572256
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−10.0377	−0.678285
$220$	0	0
$221$	12.7961	0.860759
$222$	0	0
$223$	6.11655	0.409594	0.204797	−	0.978804i	$-0.434347\pi$
$223$	6.11655	0.409594	0.204797	+	0.978804i	$0.434347\pi$
$224$	0	0
$225$	−6.27433	−0.418289
$226$	0	0
$227$	21.1675	1.40494	0.702469	−	0.711714i	$-0.252081\pi$
$227$	21.1675	1.40494	0.702469	+	0.711714i	$0.252081\pi$
$228$	0	0
$229$	13.0632	0.863243	0.431621	−	0.902055i	$-0.357942\pi$
$229$	13.0632	0.863243	0.431621	+	0.902055i	$0.357942\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.260769	0.0170836	0.00854178	−	0.999964i	$-0.497281\pi$
$233$	0.260769	0.0170836	0.00854178	+	0.999964i	$0.497281\pi$
$234$	0	0
$235$	32.7571	2.13684
$236$	0	0
$237$	−0.352314	−0.0228852
$238$	0	0
$239$	−3.63705	−0.235261	−0.117631	−	0.993057i	$-0.537530\pi$
$239$	−3.63705	−0.235261	−0.117631	+	0.993057i	$0.537530\pi$
$240$	0	0
$241$	8.84444	0.569721	0.284860	−	0.958569i	$-0.408053\pi$
$241$	8.84444	0.569721	0.284860	+	0.958569i	$0.408053\pi$
$242$	0	0
$243$	−10.4082	−0.667684
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.11454	0.134545
$248$	0	0
$249$	−18.3944	−1.16570
$250$	0	0
$251$	−1.96113	−0.123786	−0.0618929	−	0.998083i	$-0.519714\pi$
$251$	−1.96113	−0.123786	−0.0618929	+	0.998083i	$0.519714\pi$
$252$	0	0
$253$	−5.01469	−0.315271
$254$	0	0
$255$	27.8775	1.74576
$256$	0	0
$257$	1.16068	0.0724011	0.0362006	−	0.999345i	$-0.488474\pi$
$257$	1.16068	0.0724011	0.0362006	+	0.999345i	$0.488474\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.28066	0.512560
$262$	0	0
$263$	8.26461	0.509618	0.254809	−	0.966991i	$-0.417987\pi$
$263$	8.26461	0.509618	0.254809	+	0.966991i	$0.417987\pi$
$264$	0	0
$265$	27.3387	1.67940
$266$	0	0
$267$	−16.7633	−1.02590
$268$	0	0
$269$	−2.68700	−0.163829	−0.0819146	−	0.996639i	$-0.526103\pi$
$269$	−2.68700	−0.163829	−0.0819146	+	0.996639i	$0.526103\pi$
$270$	0	0
$271$	14.5909	0.886337	0.443169	−	0.896438i	$-0.353854\pi$
$271$	14.5909	0.886337	0.443169	+	0.896438i	$0.353854\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.55681	−0.395390
$276$	0	0
$277$	5.52475	0.331950	0.165975	−	0.986130i	$-0.446923\pi$
$277$	5.52475	0.331950	0.165975	+	0.986130i	$0.446923\pi$
$278$	0	0
$279$	−1.96034	−0.117362
$280$	0	0
$281$	−32.3126	−1.92761	−0.963804	−	0.266612i	$-0.914096\pi$
$281$	−32.3126	−1.92761	−0.963804	+	0.266612i	$0.914096\pi$
$282$	0	0
$283$	−8.47948	−0.504053	−0.252027	−	0.967720i	$-0.581097\pi$
$283$	−8.47948	−0.504053	−0.252027	+	0.967720i	$0.581097\pi$
$284$	0	0
$285$	4.60671	0.272878
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.6206	1.15415
$290$	0	0
$291$	−7.63139	−0.447360
$292$	0	0
$293$	5.99888	0.350458	0.175229	−	0.984528i	$-0.443933\pi$
$293$	5.99888	0.350458	0.175229	+	0.984528i	$0.443933\pi$
$294$	0	0
$295$	27.6114	1.60760
$296$	0	0
$297$	−6.25374	−0.362879
$298$	0	0
$299$	−9.58859	−0.554523
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.2125	0.759038
$304$	0	0
$305$	23.7071	1.35746
$306$	0	0
$307$	−15.1571	−0.865061	−0.432530	−	0.901619i	$-0.642379\pi$
$307$	−15.1571	−0.865061	−0.432530	+	0.901619i	$0.642379\pi$
$308$	0	0
$309$	−7.34212	−0.417678
$310$	0	0
$311$	−12.5193	−0.709907	−0.354953	−	0.934884i	$-0.615503\pi$
$311$	−12.5193	−0.709907	−0.354953	+	0.934884i	$0.615503\pi$
$312$	0	0
$313$	7.33341	0.414509	0.207254	−	0.978287i	$-0.433547\pi$
$313$	7.33341	0.414509	0.207254	+	0.978287i	$0.433547\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.12002	−0.287569	−0.143784	−	0.989609i	$-0.545927\pi$
$317$	−5.12002	−0.287569	−0.143784	+	0.989609i	$0.545927\pi$
$318$	0	0
$319$	8.65346	0.484501
$320$	0	0
$321$	6.21441	0.346854
$322$	0	0
$323$	6.05149	0.336714
$324$	0	0
$325$	−12.5373	−0.695442
$326$	0	0
$327$	−11.3821	−0.629432
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.8484	1.64061	0.820307	−	0.571923i	$-0.193802\pi$
$331$	29.8484	1.64061	0.820307	+	0.571923i	$0.193802\pi$
$332$	0	0
$333$	−5.65538	−0.309913
$334$	0	0
$335$	12.5202	0.684054
$336$	0	0
$337$	30.2473	1.64768	0.823839	−	0.566824i	$-0.191828\pi$
$337$	30.2473	1.64768	0.823839	+	0.566824i	$0.191828\pi$
$338$	0	0
$339$	24.5674	1.33432
$340$	0	0
$341$	−2.04859	−0.110938
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−20.8896	−1.12466
$346$	0	0
$347$	−32.0242	−1.71915	−0.859573	−	0.511012i	$-0.829271\pi$
$347$	−32.0242	−1.71915	−0.859573	+	0.511012i	$0.829271\pi$
$348$	0	0
$349$	−14.6171	−0.782438	−0.391219	−	0.920298i	$-0.627946\pi$
$349$	−14.6171	−0.782438	−0.391219	+	0.920298i	$0.627946\pi$
$350$	0	0
$351$	−11.9578	−0.638259
$352$	0	0
$353$	−37.4179	−1.99156	−0.995778	−	0.0917968i	$-0.970739\pi$
$353$	−37.4179	−1.99156	−0.995778	+	0.0917968i	$0.970739\pi$
$354$	0	0
$355$	9.78882	0.519537
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.80414	−0.306331	−0.153166	−	0.988201i	$-0.548947\pi$
$359$	−5.80414	−0.306331	−0.153166	+	0.988201i	$0.548947\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	13.6241	0.715078
$364$	0	0
$365$	23.8137	1.24647
$366$	0	0
$367$	−37.9239	−1.97961	−0.989805	−	0.142426i	$-0.954510\pi$
$367$	−37.9239	−1.97961	−0.989805	+	0.142426i	$0.954510\pi$
$368$	0	0
$369$	−6.74940	−0.351360
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.18554	−0.0613848	−0.0306924	−	0.999529i	$-0.509771\pi$
$373$	−1.18554	−0.0613848	−0.0306924	+	0.999529i	$0.509771\pi$
$374$	0	0
$375$	−4.28004	−0.221020
$376$	0	0
$377$	16.5463	0.852177
$378$	0	0
$379$	−12.0108	−0.616954	−0.308477	−	0.951232i	$-0.599819\pi$
$379$	−12.0108	−0.616954	−0.308477	+	0.951232i	$0.599819\pi$
$380$	0	0
$381$	5.06938	0.259712
$382$	0	0
$383$	33.4886	1.71119	0.855595	−	0.517646i	$-0.173192\pi$
$383$	33.4886	1.71119	0.855595	+	0.517646i	$0.173192\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.68594	0.136534
$388$	0	0
$389$	14.7766	0.749204	0.374602	−	0.927186i	$-0.377779\pi$
$389$	14.7766	0.749204	0.374602	+	0.927186i	$0.377779\pi$
$390$	0	0
$391$	−27.4411	−1.38776
$392$	0	0
$393$	−11.2935	−0.569683
$394$	0	0
$395$	0.835839	0.0420556
$396$	0	0
$397$	−5.56015	−0.279056	−0.139528	−	0.990218i	$-0.544559\pi$
$397$	−5.56015	−0.279056	−0.139528	+	0.990218i	$0.544559\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.18319	−0.158961	−0.0794805	−	0.996836i	$-0.525326\pi$
$401$	−3.18319	−0.158961	−0.0794805	+	0.996836i	$0.525326\pi$
$402$	0	0
$403$	−3.91712	−0.195126
$404$	0	0
$405$	−15.5559	−0.772978
$406$	0	0
$407$	−5.90999	−0.292947
$408$	0	0
$409$	−25.8865	−1.28001	−0.640003	−	0.768372i	$-0.721067\pi$
$409$	−25.8865	−1.28001	−0.640003	+	0.768372i	$0.721067\pi$
$410$	0	0
$411$	23.1628	1.14253
$412$	0	0
$413$	0	0
$414$	0	0
$415$	43.6393	2.14217
$416$	0	0
$417$	27.9479	1.36862
$418$	0	0
$419$	31.7199	1.54962	0.774809	−	0.632196i	$-0.217846\pi$
$419$	31.7199	1.54962	0.774809	+	0.632196i	$0.217846\pi$
$420$	0	0
$421$	18.5654	0.904821	0.452411	−	0.891810i	$-0.350564\pi$
$421$	18.5654	0.904821	0.452411	+	0.891810i	$0.350564\pi$
$422$	0	0
$423$	−10.4856	−0.509827
$424$	0	0
$425$	−35.8798	−1.74043
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.25851	−0.157322
$430$	0	0
$431$	−4.48598	−0.216082	−0.108041	−	0.994146i	$-0.534458\pi$
$431$	−4.48598	−0.216082	−0.108041	+	0.994146i	$0.534458\pi$
$432$	0	0
$433$	−9.99063	−0.480119	−0.240060	−	0.970758i	$-0.577167\pi$
$433$	−9.99063	−0.480119	−0.240060	+	0.970758i	$0.577167\pi$
$434$	0	0
$435$	36.0476	1.72835
$436$	0	0
$437$	−4.53461	−0.216920
$438$	0	0
$439$	11.9717	0.571377	0.285688	−	0.958323i	$-0.407778\pi$
$439$	11.9717	0.571377	0.285688	+	0.958323i	$0.407778\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.15582	−0.435006	−0.217503	−	0.976060i	$-0.569791\pi$
$443$	−9.15582	−0.435006	−0.217503	+	0.976060i	$0.569791\pi$
$444$	0	0
$445$	39.7697	1.88527
$446$	0	0
$447$	9.15093	0.432824
$448$	0	0
$449$	35.0930	1.65614	0.828071	−	0.560624i	$-0.189439\pi$
$449$	35.0930	1.65614	0.828071	+	0.560624i	$0.189439\pi$
$450$	0	0
$451$	−7.05327	−0.332125
$452$	0	0
$453$	−12.5694	−0.590563
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.21975	−0.103835	−0.0519177	−	0.998651i	$-0.516533\pi$
$457$	−2.21975	−0.103835	−0.0519177	+	0.998651i	$0.516533\pi$
$458$	0	0
$459$	−34.2214	−1.59732
$460$	0	0
$461$	−3.70689	−0.172647	−0.0863236	−	0.996267i	$-0.527512\pi$
$461$	−3.70689	−0.172647	−0.0863236	+	0.996267i	$0.527512\pi$
$462$	0	0
$463$	2.34922	0.109177	0.0545887	−	0.998509i	$-0.482615\pi$
$463$	2.34922	0.109177	0.0545887	+	0.998509i	$0.482615\pi$
$464$	0	0
$465$	−8.53380	−0.395746
$466$	0	0
$467$	16.9497	0.784339	0.392170	−	0.919893i	$-0.371725\pi$
$467$	16.9497	0.784339	0.392170	+	0.919893i	$0.371725\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−24.0835	−1.10971
$472$	0	0
$473$	2.80687	0.129060
$474$	0	0
$475$	−5.92909	−0.272045
$476$	0	0
$477$	−8.75115	−0.400687
$478$	0	0
$479$	9.45654	0.432080	0.216040	−	0.976385i	$-0.430686\pi$
$479$	9.45654	0.432080	0.216040	+	0.976385i	$0.430686\pi$
$480$	0	0
$481$	−11.3005	−0.515258
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.1049	0.822102
$486$	0	0
$487$	−36.4997	−1.65396	−0.826980	−	0.562231i	$-0.809943\pi$
$487$	−36.4997	−1.65396	−0.826980	+	0.562231i	$0.809943\pi$
$488$	0	0
$489$	13.8946	0.628338
$490$	0	0
$491$	−25.8603	−1.16706	−0.583529	−	0.812093i	$-0.698328\pi$
$491$	−25.8603	−1.16706	−0.583529	+	0.812093i	$0.698328\pi$
$492$	0	0
$493$	47.3530	2.13267
$494$	0	0
$495$	3.86880	0.173889
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.0470	1.25555	0.627777	−	0.778393i	$-0.283965\pi$
$499$	28.0470	1.25555	0.627777	+	0.778393i	$0.283965\pi$
$500$	0	0
$501$	−24.5878	−1.09850
$502$	0	0
$503$	15.9737	0.712233	0.356116	−	0.934442i	$-0.384101\pi$
$503$	15.9737	0.712233	0.356116	+	0.934442i	$0.384101\pi$
$504$	0	0
$505$	−31.3457	−1.39486
$506$	0	0
$507$	11.8846	0.527813
$508$	0	0
$509$	13.1937	0.584803	0.292401	−	0.956296i	$-0.405546\pi$
$509$	13.1937	0.584803	0.292401	+	0.956296i	$0.405546\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.65504	−0.249676
$514$	0	0
$515$	17.4186	0.767557
$516$	0	0
$517$	−10.9577	−0.481918
$518$	0	0
$519$	29.5728	1.29810
$520$	0	0
$521$	13.1695	0.576967	0.288483	−	0.957485i	$-0.406849\pi$
$521$	13.1695	0.576967	0.288483	+	0.957485i	$0.406849\pi$
$522$	0	0
$523$	31.6786	1.38521	0.692603	−	0.721319i	$-0.256464\pi$
$523$	31.6786	1.38521	0.692603	+	0.721319i	$0.256464\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.2102	−0.488325
$528$	0	0
$529$	−2.43733	−0.105971
$530$	0	0
$531$	−8.83843	−0.383555
$532$	0	0
$533$	−13.4866	−0.584167
$534$	0	0
$535$	−14.7432	−0.637405
$536$	0	0
$537$	−17.5155	−0.755851
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.70012	0.331054	0.165527	−	0.986205i	$-0.447068\pi$
$541$	7.70012	0.331054	0.165527	+	0.986205i	$0.447068\pi$
$542$	0	0
$543$	−1.03023	−0.0442116
$544$	0	0
$545$	27.0032	1.15669
$546$	0	0
$547$	−44.2782	−1.89320	−0.946600	−	0.322410i	$-0.895507\pi$
$547$	−44.2782	−1.89320	−0.946600	+	0.322410i	$0.895507\pi$
$548$	0	0
$549$	−7.58867	−0.323877
$550$	0	0
$551$	7.82502	0.333357
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−24.6192	−1.04503
$556$	0	0
$557$	12.0573	0.510884	0.255442	−	0.966824i	$-0.417779\pi$
$557$	12.0573	0.510884	0.255442	+	0.966824i	$0.417779\pi$
$558$	0	0
$559$	5.36701	0.227000
$560$	0	0
$561$	−9.32537	−0.393717
$562$	0	0
$563$	−6.86141	−0.289174	−0.144587	−	0.989492i	$-0.546185\pi$
$563$	−6.86141	−0.289174	−0.144587	+	0.989492i	$0.546185\pi$
$564$	0	0
$565$	−58.2843	−2.45204
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.33611	−0.0979346	−0.0489673	−	0.998800i	$-0.515593\pi$
$569$	−2.33611	−0.0979346	−0.0489673	+	0.998800i	$0.515593\pi$
$570$	0	0
$571$	−36.7697	−1.53876	−0.769381	−	0.638790i	$-0.779435\pi$
$571$	−36.7697	−1.53876	−0.769381	+	0.638790i	$0.779435\pi$
$572$	0	0
$573$	−29.7149	−1.24136
$574$	0	0
$575$	26.8861	1.12123
$576$	0	0
$577$	−25.3319	−1.05458	−0.527289	−	0.849686i	$-0.676792\pi$
$577$	−25.3319	−1.05458	−0.527289	+	0.849686i	$0.676792\pi$
$578$	0	0
$579$	8.61846	0.358171
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.14513	−0.378753
$584$	0	0
$585$	7.39752	0.305850
$586$	0	0
$587$	−40.6611	−1.67826	−0.839132	−	0.543928i	$-0.816937\pi$
$587$	−40.6611	−1.67826	−0.839132	+	0.543928i	$0.816937\pi$
$588$	0	0
$589$	−1.85247	−0.0763298
$590$	0	0
$591$	−10.0143	−0.411934
$592$	0	0
$593$	−15.2067	−0.624463	−0.312232	−	0.950006i	$-0.601076\pi$
$593$	−15.2067	−0.624463	−0.312232	+	0.950006i	$0.601076\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.6880	−0.642067
$598$	0	0
$599$	−44.4621	−1.81667	−0.908336	−	0.418242i	$-0.862646\pi$
$599$	−44.4621	−1.81667	−0.908336	+	0.418242i	$0.862646\pi$
$600$	0	0
$601$	−17.7062	−0.722251	−0.361125	−	0.932517i	$-0.617607\pi$
$601$	−17.7062	−0.722251	−0.361125	+	0.932517i	$0.617607\pi$
$602$	0	0
$603$	−4.00775	−0.163208
$604$	0	0
$605$	−32.3221	−1.31408
$606$	0	0
$607$	27.0208	1.09674	0.548370	−	0.836236i	$-0.315248\pi$
$607$	27.0208	1.09674	0.548370	+	0.836236i	$0.315248\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.9522	−0.847633
$612$	0	0
$613$	37.6644	1.52125	0.760626	−	0.649191i	$-0.224892\pi$
$613$	37.6644	1.52125	0.760626	+	0.649191i	$0.224892\pi$
$614$	0	0
$615$	−29.3817	−1.18478
$616$	0	0
$617$	−3.46803	−0.139618	−0.0698089	−	0.997560i	$-0.522239\pi$
$617$	−3.46803	−0.139618	−0.0698089	+	0.997560i	$0.522239\pi$
$618$	0	0
$619$	−33.6070	−1.35078	−0.675390	−	0.737461i	$-0.736025\pi$
$619$	−33.6070	−1.35078	−0.675390	+	0.737461i	$0.736025\pi$
$620$	0	0
$621$	25.6434	1.02903
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.4914	−0.779654
$626$	0	0
$627$	−1.54100	−0.0615417
$628$	0	0
$629$	−32.3404	−1.28949
$630$	0	0
$631$	44.3149	1.76415	0.882074	−	0.471112i	$-0.156147\pi$
$631$	44.3149	1.76415	0.882074	+	0.471112i	$0.156147\pi$
$632$	0	0
$633$	−36.9145	−1.46722
$634$	0	0
$635$	−12.0267	−0.477267
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.13342	−0.123956
$640$	0	0
$641$	5.54793	0.219130	0.109565	−	0.993980i	$-0.465054\pi$
$641$	5.54793	0.219130	0.109565	+	0.993980i	$0.465054\pi$
$642$	0	0
$643$	−43.2591	−1.70597	−0.852985	−	0.521935i	$-0.825210\pi$
$643$	−43.2591	−1.70597	−0.852985	+	0.521935i	$0.825210\pi$
$644$	0	0
$645$	11.6925	0.460393
$646$	0	0
$647$	15.5566	0.611594	0.305797	−	0.952097i	$-0.401077\pi$
$647$	15.5566	0.611594	0.305797	+	0.952097i	$0.401077\pi$
$648$	0	0
$649$	−9.23634	−0.362558
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.263998	−0.0103310	−0.00516551	−	0.999987i	$-0.501644\pi$
$653$	−0.263998	−0.0103310	−0.00516551	+	0.999987i	$0.501644\pi$
$654$	0	0
$655$	26.7931	1.04689
$656$	0	0
$657$	−7.62280	−0.297394
$658$	0	0
$659$	−5.92918	−0.230968	−0.115484	−	0.993309i	$-0.536842\pi$
$659$	−5.92918	−0.230968	−0.115484	+	0.993309i	$0.536842\pi$
$660$	0	0
$661$	43.9414	1.70912	0.854562	−	0.519349i	$-0.173826\pi$
$661$	43.9414	1.70912	0.854562	+	0.519349i	$0.173826\pi$
$662$	0	0
$663$	−17.8310	−0.692500
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−35.4834	−1.37392
$668$	0	0
$669$	−8.52325	−0.329528
$670$	0	0
$671$	−7.93032	−0.306147
$672$	0	0
$673$	−34.5137	−1.33040	−0.665202	−	0.746663i	$-0.731655\pi$
$673$	−34.5137	−1.33040	−0.665202	+	0.746663i	$0.731655\pi$
$674$	0	0
$675$	33.5292	1.29054
$676$	0	0
$677$	32.3753	1.24428	0.622142	−	0.782904i	$-0.286263\pi$
$677$	32.3753	1.24428	0.622142	+	0.782904i	$0.286263\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−29.4964	−1.13031
$682$	0	0
$683$	−7.82676	−0.299483	−0.149741	−	0.988725i	$-0.547844\pi$
$683$	−7.82676	−0.299483	−0.149741	+	0.988725i	$0.547844\pi$
$684$	0	0
$685$	−54.9520	−2.09961
$686$	0	0
$687$	−18.2033	−0.694498
$688$	0	0
$689$	−17.4864	−0.666179
$690$	0	0
$691$	10.4415	0.397213	0.198607	−	0.980079i	$-0.436358\pi$
$691$	10.4415	0.397213	0.198607	+	0.980079i	$0.436358\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−66.3044	−2.51507
$696$	0	0
$697$	−38.5965	−1.46195
$698$	0	0
$699$	−0.363375	−0.0137441
$700$	0	0
$701$	49.5542	1.87164	0.935818	−	0.352482i	$-0.114662\pi$
$701$	49.5542	1.87164	0.935818	+	0.352482i	$0.114662\pi$
$702$	0	0
$703$	−5.34419	−0.201560
$704$	0	0
$705$	−45.6462	−1.71914
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−33.5339	−1.25939	−0.629696	−	0.776842i	$-0.716820\pi$
$709$	−33.5339	−1.25939	−0.629696	+	0.776842i	$0.716820\pi$
$710$	0	0
$711$	−0.267553	−0.0100340
$712$	0	0
$713$	8.40023	0.314591
$714$	0	0
$715$	7.73057	0.289107
$716$	0	0
$717$	5.06814	0.189273
$718$	0	0
$719$	42.1990	1.57376	0.786879	−	0.617108i	$-0.211696\pi$
$719$	42.1990	1.57376	0.786879	+	0.617108i	$0.211696\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−12.3245	−0.458353
$724$	0	0
$725$	−46.3952	−1.72308
$726$	0	0
$727$	36.5947	1.35722	0.678611	−	0.734498i	$-0.262582\pi$
$727$	36.5947	1.35722	0.678611	+	0.734498i	$0.262582\pi$
$728$	0	0
$729$	28.6199	1.06000
$730$	0	0
$731$	15.3596	0.568095
$732$	0	0
$733$	−17.1700	−0.634190	−0.317095	−	0.948394i	$-0.602707\pi$
$733$	−17.1700	−0.634190	−0.317095	+	0.948394i	$0.602707\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.18818	−0.154274
$738$	0	0
$739$	−25.9736	−0.955453	−0.477726	−	0.878509i	$-0.658539\pi$
$739$	−25.9736	−0.955453	−0.477726	+	0.878509i	$0.658539\pi$
$740$	0	0
$741$	−2.94655	−0.108244
$742$	0	0
$743$	27.5002	1.00888	0.504442	−	0.863445i	$-0.331698\pi$
$743$	27.5002	1.00888	0.504442	+	0.863445i	$0.331698\pi$
$744$	0	0
$745$	−21.7099	−0.795390
$746$	0	0
$747$	−13.9690	−0.511099
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.9167	0.617298	0.308649	−	0.951176i	$-0.400123\pi$
$751$	16.9167	0.617298	0.308649	+	0.951176i	$0.400123\pi$
$752$	0	0
$753$	2.73279	0.0995885
$754$	0	0
$755$	29.8200	1.08526
$756$	0	0
$757$	18.8931	0.686683	0.343342	−	0.939211i	$-0.388441\pi$
$757$	18.8931	0.686683	0.343342	+	0.939211i	$0.388441\pi$
$758$	0	0
$759$	6.98784	0.253643
$760$	0	0
$761$	53.7492	1.94841	0.974203	−	0.225673i	$-0.0724580\pi$
$761$	53.7492	1.94841	0.974203	+	0.225673i	$0.0724580\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	21.1706	0.765426
$766$	0	0
$767$	−17.6608	−0.637695
$768$	0	0
$769$	13.1844	0.475442	0.237721	−	0.971333i	$-0.423600\pi$
$769$	13.1844	0.475442	0.237721	+	0.971333i	$0.423600\pi$
$770$	0	0
$771$	−1.61738	−0.0582484
$772$	0	0
$773$	12.6052	0.453377	0.226689	−	0.973967i	$-0.427210\pi$
$773$	12.6052	0.453377	0.226689	+	0.973967i	$0.427210\pi$
$774$	0	0
$775$	10.9835	0.394538
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.37802	−0.228516
$780$	0	0
$781$	−3.27448	−0.117170
$782$	0	0
$783$	−44.2508	−1.58139
$784$	0	0
$785$	57.1364	2.03928
$786$	0	0
$787$	−47.0573	−1.67741	−0.838705	−	0.544586i	$-0.816687\pi$
$787$	−47.0573	−1.67741	−0.838705	+	0.544586i	$0.816687\pi$
$788$	0	0
$789$	−11.5165	−0.409999
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−15.1636	−0.538474
$794$	0	0
$795$	−38.0957	−1.35112
$796$	0	0
$797$	39.5629	1.40139	0.700694	−	0.713462i	$-0.252874\pi$
$797$	39.5629	1.40139	0.700694	+	0.713462i	$0.252874\pi$
$798$	0	0
$799$	−59.9620	−2.12130
$800$	0	0
$801$	−12.7303	−0.449804
$802$	0	0
$803$	−7.96598	−0.281113
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.74426	0.131804
$808$	0	0
$809$	52.4636	1.84452	0.922261	−	0.386567i	$-0.126339\pi$
$809$	52.4636	1.84452	0.922261	+	0.386567i	$0.126339\pi$
$810$	0	0
$811$	−11.3982	−0.400246	−0.200123	−	0.979771i	$-0.564134\pi$
$811$	−11.3982	−0.400246	−0.200123	+	0.979771i	$0.564134\pi$
$812$	0	0
$813$	−20.3321	−0.713079
$814$	0	0
$815$	−32.9640	−1.15468
$816$	0	0
$817$	2.53815	0.0887986
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.3729	−1.68823	−0.844113	−	0.536166i	$-0.819872\pi$
$821$	−48.3729	−1.68823	−0.844113	+	0.536166i	$0.819872\pi$
$822$	0	0
$823$	−30.5538	−1.06504	−0.532520	−	0.846418i	$-0.678755\pi$
$823$	−30.5538	−1.06504	−0.532520	+	0.846418i	$0.678755\pi$
$824$	0	0
$825$	9.13674	0.318101
$826$	0	0
$827$	10.8367	0.376827	0.188414	−	0.982090i	$-0.439665\pi$
$827$	10.8367	0.376827	0.188414	+	0.982090i	$0.439665\pi$
$828$	0	0
$829$	3.98821	0.138516	0.0692582	−	0.997599i	$-0.477937\pi$
$829$	3.98821	0.138516	0.0692582	+	0.997599i	$0.477937\pi$
$830$	0	0
$831$	−7.69861	−0.267062
$832$	0	0
$833$	0	0
$834$	0	0
$835$	58.3327	2.01868
$836$	0	0
$837$	10.4758	0.362097
$838$	0	0
$839$	−25.7895	−0.890353	−0.445176	−	0.895443i	$-0.646859\pi$
$839$	−25.7895	−0.890353	−0.445176	+	0.895443i	$0.646859\pi$
$840$	0	0
$841$	32.2309	1.11141
$842$	0	0
$843$	45.0268	1.55080
$844$	0	0
$845$	−28.1953	−0.969948
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.8159	0.405522
$850$	0	0
$851$	24.2338	0.830725
$852$	0	0
$853$	−22.1041	−0.756830	−0.378415	−	0.925636i	$-0.623531\pi$
$853$	−22.1041	−0.756830	−0.378415	+	0.925636i	$0.623531\pi$
$854$	0	0
$855$	3.49842	0.119643
$856$	0	0
$857$	−4.18342	−0.142903	−0.0714514	−	0.997444i	$-0.522763\pi$
$857$	−4.18342	−0.142903	−0.0714514	+	0.997444i	$0.522763\pi$
$858$	0	0
$859$	−13.1003	−0.446976	−0.223488	−	0.974707i	$-0.571744\pi$
$859$	−13.1003	−0.446976	−0.223488	+	0.974707i	$0.571744\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.83219	0.232570	0.116285	−	0.993216i	$-0.462901\pi$
$863$	6.83219	0.232570	0.116285	+	0.993216i	$0.462901\pi$
$864$	0	0
$865$	−70.1593	−2.38549
$866$	0	0
$867$	−27.3408	−0.928541
$868$	0	0
$869$	−0.279599	−0.00948474
$870$	0	0
$871$	−8.00822	−0.271348
$872$	0	0
$873$	−5.79541	−0.196145
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.0873	1.38742	0.693709	−	0.720255i	$-0.255975\pi$
$877$	41.0873	1.38742	0.693709	+	0.720255i	$0.255975\pi$
$878$	0	0
$879$	−8.35929	−0.281952
$880$	0	0
$881$	20.8335	0.701900	0.350950	−	0.936394i	$-0.385859\pi$
$881$	20.8335	0.701900	0.350950	+	0.936394i	$0.385859\pi$
$882$	0	0
$883$	−24.0758	−0.810215	−0.405108	−	0.914269i	$-0.632766\pi$
$883$	−24.0758	−0.810215	−0.405108	+	0.914269i	$0.632766\pi$
$884$	0	0
$885$	−38.4757	−1.29335
$886$	0	0
$887$	−43.4396	−1.45856	−0.729279	−	0.684217i	$-0.760144\pi$
$887$	−43.4396	−1.45856	−0.729279	+	0.684217i	$0.760144\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.20364	0.174328
$892$	0	0
$893$	−9.90863	−0.331580
$894$	0	0
$895$	41.5543	1.38901
$896$	0	0
$897$	13.3615	0.446126
$898$	0	0
$899$	−14.4956	−0.483456
$900$	0	0
$901$	−50.0435	−1.66719
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.44415	0.0812465
$906$	0	0
$907$	−13.7357	−0.456087	−0.228043	−	0.973651i	$-0.573233\pi$
$907$	−13.7357	−0.456087	−0.228043	+	0.973651i	$0.573233\pi$
$908$	0	0
$909$	10.0338	0.332800
$910$	0	0
$911$	34.0715	1.12884	0.564420	−	0.825488i	$-0.309100\pi$
$911$	34.0715	1.12884	0.564420	+	0.825488i	$0.309100\pi$
$912$	0	0
$913$	−14.5979	−0.483120
$914$	0	0
$915$	−33.0352	−1.09211
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−6.85550	−0.226142	−0.113071	−	0.993587i	$-0.536069\pi$
$919$	−6.85550	−0.226142	−0.113071	+	0.993587i	$0.536069\pi$
$920$	0	0
$921$	21.1210	0.695961
$922$	0	0
$923$	−6.26114	−0.206088
$924$	0	0
$925$	31.6862	1.04184
$926$	0	0
$927$	−5.57573	−0.183131
$928$	0	0
$929$	41.7650	1.37027	0.685133	−	0.728418i	$-0.259744\pi$
$929$	41.7650	1.37027	0.685133	+	0.728418i	$0.259744\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	17.4454	0.571136
$934$	0	0
$935$	22.1238	0.723524
$936$	0	0
$937$	17.7567	0.580086	0.290043	−	0.957014i	$-0.406330\pi$
$937$	17.7567	0.580086	0.290043	+	0.957014i	$0.406330\pi$
$938$	0	0
$939$	−10.2189	−0.333482
$940$	0	0
$941$	−4.22123	−0.137608	−0.0688041	−	0.997630i	$-0.521918\pi$
$941$	−4.22123	−0.137608	−0.0688041	+	0.997630i	$0.521918\pi$
$942$	0	0
$943$	28.9218	0.941824
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.4986	−0.633619	−0.316809	−	0.948489i	$-0.602612\pi$
$947$	−19.4986	−0.633619	−0.316809	+	0.948489i	$0.602612\pi$
$948$	0	0
$949$	−15.2318	−0.494444
$950$	0	0
$951$	7.13462	0.231356
$952$	0	0
$953$	58.9734	1.91034	0.955168	−	0.296064i	$-0.0956743\pi$
$953$	58.9734	1.91034	0.955168	+	0.296064i	$0.0956743\pi$
$954$	0	0
$955$	70.4964	2.28121
$956$	0	0
$957$	−12.0584	−0.389792
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.5683	−0.889302
$962$	0	0
$963$	4.71933	0.152078
$964$	0	0
$965$	−20.4467	−0.658202
$966$	0	0
$967$	−30.8592	−0.992366	−0.496183	−	0.868218i	$-0.665265\pi$
$967$	−30.8592	−0.992366	−0.496183	+	0.868218i	$0.665265\pi$
$968$	0	0
$969$	−8.43260	−0.270894
$970$	0	0
$971$	−5.55130	−0.178150	−0.0890749	−	0.996025i	$-0.528391\pi$
$971$	−5.55130	−0.178150	−0.0890749	+	0.996025i	$0.528391\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	17.4704	0.559499
$976$	0	0
$977$	36.4094	1.16484	0.582420	−	0.812888i	$-0.302106\pi$
$977$	36.4094	1.16484	0.582420	+	0.812888i	$0.302106\pi$
$978$	0	0
$979$	−13.3035	−0.425181
$980$	0	0
$981$	−8.64377	−0.275974
$982$	0	0
$983$	−36.4131	−1.16140	−0.580699	−	0.814119i	$-0.697221\pi$
$983$	−36.4131	−1.16140	−0.580699	+	0.814119i	$0.697221\pi$
$984$	0	0
$985$	23.7582	0.757000
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.5095	−0.365981
$990$	0	0
$991$	32.1785	1.02218	0.511091	−	0.859526i	$-0.329241\pi$
$991$	32.1785	1.02218	0.511091	+	0.859526i	$0.329241\pi$
$992$	0	0
$993$	−41.5929	−1.31991
$994$	0	0
$995$	37.2186	1.17991
$996$	0	0
$997$	−22.7328	−0.719955	−0.359978	−	0.932961i	$-0.617216\pi$
$997$	−22.7328	−0.719955	−0.359978	+	0.932961i	$0.617216\pi$
$998$	0	0
$999$	30.2216	0.956170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3724.2.a.p.1.2	yes	8
7.6	odd	2		3724.2.a.o.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3724.2.a.o.1.7	✓	8	7.6	odd	2
3724.2.a.p.1.2	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3724.a (trivial)

\(f(q)\)	\(=\)	\(q-1.39347 q^{3} +3.30592 q^{5} -1.05823 q^{9} -1.10587 q^{11} -2.11454 q^{13} -4.60671 q^{15} -6.05149 q^{17} -1.00000 q^{19} +4.53461 q^{23} +5.92909 q^{25} +5.65504 q^{27} -7.82502 q^{29} +1.85247 q^{31} +1.54100 q^{33} +5.34419 q^{37} +2.94655 q^{39} +6.37802 q^{41} -2.53815 q^{43} -3.49842 q^{45} +9.90863 q^{47} +8.43260 q^{51} +8.26962 q^{53} -3.65592 q^{55} +1.39347 q^{57} +8.35210 q^{59} +7.17111 q^{61} -6.99048 q^{65} +3.78722 q^{67} -6.31886 q^{69} +2.96100 q^{71} +7.20336 q^{73} -8.26203 q^{75} +0.252831 q^{79} -4.70547 q^{81} +13.2004 q^{83} -20.0057 q^{85} +10.9040 q^{87} +12.0299 q^{89} -2.58137 q^{93} -3.30592 q^{95} +5.47652 q^{97} +1.17026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{11} + 8 q^{17} - 8 q^{19} + 4 q^{23} + 4 q^{25} + 16 q^{27} - 4 q^{29} + 24 q^{31} - 12 q^{33} + 12 q^{37} + 4 q^{39} + 20 q^{41} + 12 q^{43} + 24 q^{45} + 24 q^{47} + 12 q^{51} + 4 q^{53}+ \cdots + 20 q^{97}+O(q^{100}) \) 8 * q + 4 * q^3 - 8 * q^11 + 8 * q^17 - 8 * q^19 + 4 * q^23 + 4 * q^25 + 16 * q^27 - 4 * q^29 + 24 * q^31 - 12 * q^33 + 12 * q^37 + 4 * q^39 + 20 * q^41 + 12 * q^43 + 24 * q^45 + 24 * q^47 + 12 * q^51 + 4 * q^53 + 16 * q^55 - 4 * q^57 + 52 * q^59 - 16 * q^61 - 4 * q^65 + 8 * q^67 + 8 * q^69 + 16 * q^71 + 24 * q^73 + 8 * q^75 - 20 * q^79 + 12 * q^81 + 48 * q^83 - 12 * q^85 + 40 * q^87 + 12 * q^89 + 8 * q^93 + 20 * q^97

Introduction

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