LMFDB - Embedded newform 3720.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3720,2,Mod(1,3720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.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:	$29.7043495519$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +3.23607 q^{7} +1.00000 q^{9} +1.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -2.47214 q^{19} -3.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.76393 q^{29} -1.00000 q^{31} -3.23607 q^{35} +9.23607 q^{37} -1.23607 q^{39} -6.00000 q^{41} -2.47214 q^{43} -1.00000 q^{45} -8.94427 q^{47} +3.47214 q^{49} +4.47214 q^{51} -0.472136 q^{53} +2.47214 q^{57} +11.2361 q^{59} -3.52786 q^{61} +3.23607 q^{63} -1.23607 q^{65} -9.70820 q^{67} +4.00000 q^{69} -4.76393 q^{71} -10.1803 q^{73} -1.00000 q^{75} +8.94427 q^{79} +1.00000 q^{81} +12.9443 q^{83} +4.47214 q^{85} +2.76393 q^{87} -2.76393 q^{89} +4.00000 q^{91} +1.00000 q^{93} +2.47214 q^{95} +0.472136 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^39 - 12 * q^41 + 4 * q^43 - 2 * q^45 - 2 * q^49 + 8 * q^53 - 4 * q^57 + 18 * q^59 - 16 * q^61 + 2 * q^63 + 2 * q^65 - 6 * q^67 + 8 * q^69 - 14 * q^71 + 2 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 10 * q^87 - 10 * q^89 + 8 * q^91 + 2 * q^93 - 4 * q^95 - 8 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	−3.23607	−0.706168
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	9.23607	1.51840	0.759200	−	0.650857i	$-0.225590\pi$
$37$	9.23607	1.51840	0.759200	+	0.650857i	$0.225590\pi$
$38$	0	0
$39$	−1.23607	−0.197929
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.94427	−1.30466	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427	−1.30466	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	4.47214	0.626224
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.47214	0.327442
$58$	0	0
$59$	11.2361	1.46281	0.731406	−	0.681943i	$-0.238865\pi$
$59$	11.2361	1.46281	0.731406	+	0.681943i	$0.238865\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	3.23607	0.407706
$64$	0	0
$65$	−1.23607	−0.153315
$66$	0	0
$67$	−9.70820	−1.18605	−0.593023	−	0.805186i	$-0.702066\pi$
$67$	−9.70820	−1.18605	−0.593023	+	0.805186i	$0.702066\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−4.76393	−0.565375	−0.282687	−	0.959212i	$-0.591226\pi$
$71$	−4.76393	−0.565375	−0.282687	+	0.959212i	$0.591226\pi$
$72$	0	0
$73$	−10.1803	−1.19152	−0.595759	−	0.803163i	$-0.703149\pi$
$73$	−10.1803	−1.19152	−0.595759	+	0.803163i	$0.703149\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9443	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$83$	12.9443	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$84$	0	0
$85$	4.47214	0.485071
$86$	0	0
$87$	2.76393	0.296325
$88$	0	0
$89$	−2.76393	−0.292976	−0.146488	−	0.989212i	$-0.546797\pi$
$89$	−2.76393	−0.292976	−0.146488	+	0.989212i	$0.546797\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	2.47214	0.253636
$96$	0	0
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.94427	0.690981	0.345490	−	0.938422i	$-0.387713\pi$
$101$	6.94427	0.690981	0.345490	+	0.938422i	$0.387713\pi$
$102$	0	0
$103$	14.6525	1.44375	0.721876	−	0.692023i	$-0.243280\pi$
$103$	14.6525	1.44375	0.721876	+	0.692023i	$0.243280\pi$
$104$	0	0
$105$	3.23607	0.315808
$106$	0	0
$107$	−4.94427	−0.477981	−0.238990	−	0.971022i	$-0.576816\pi$
$107$	−4.94427	−0.477981	−0.238990	+	0.971022i	$0.576816\pi$
$108$	0	0
$109$	−12.4721	−1.19461	−0.597307	−	0.802013i	$-0.703763\pi$
$109$	−12.4721	−1.19461	−0.597307	+	0.802013i	$0.703763\pi$
$110$	0	0
$111$	−9.23607	−0.876649
$112$	0	0
$113$	−14.9443	−1.40584	−0.702919	−	0.711269i	$-0.748121\pi$
$113$	−14.9443	−1.40584	−0.702919	+	0.711269i	$0.748121\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	1.23607	0.114275
$118$	0	0
$119$	−14.4721	−1.32666
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.4164	1.36798	0.683992	−	0.729489i	$-0.260242\pi$
$127$	15.4164	1.36798	0.683992	+	0.729489i	$0.260242\pi$
$128$	0	0
$129$	2.47214	0.217659
$130$	0	0
$131$	−4.76393	−0.416227	−0.208113	−	0.978105i	$-0.566732\pi$
$131$	−4.76393	−0.416227	−0.208113	+	0.978105i	$0.566732\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	8.94427	0.753244
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.76393	0.229532
$146$	0	0
$147$	−3.47214	−0.286377
$148$	0	0
$149$	−1.05573	−0.0864886	−0.0432443	−	0.999065i	$-0.513769\pi$
$149$	−1.05573	−0.0864886	−0.0432443	+	0.999065i	$0.513769\pi$
$150$	0	0
$151$	−21.8885	−1.78126	−0.890632	−	0.454724i	$-0.849738\pi$
$151$	−21.8885	−1.78126	−0.890632	+	0.454724i	$0.849738\pi$
$152$	0	0
$153$	−4.47214	−0.361551
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	21.4164	1.70922	0.854608	−	0.519274i	$-0.173798\pi$
$157$	21.4164	1.70922	0.854608	+	0.519274i	$0.173798\pi$
$158$	0	0
$159$	0.472136	0.0374428
$160$	0	0
$161$	−12.9443	−1.02015
$162$	0	0
$163$	11.2361	0.880077	0.440038	−	0.897979i	$-0.354965\pi$
$163$	11.2361	0.880077	0.440038	+	0.897979i	$0.354965\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	−2.47214	−0.189049
$172$	0	0
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	3.23607	0.244624
$176$	0	0
$177$	−11.2361	−0.844555
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−5.41641	−0.402598	−0.201299	−	0.979530i	$-0.564516\pi$
$181$	−5.41641	−0.402598	−0.201299	+	0.979530i	$0.564516\pi$
$182$	0	0
$183$	3.52786	0.260787
$184$	0	0
$185$	−9.23607	−0.679049
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.23607	−0.235389
$190$	0	0
$191$	−19.2361	−1.39187	−0.695937	−	0.718103i	$-0.745011\pi$
$191$	−19.2361	−1.39187	−0.695937	+	0.718103i	$0.745011\pi$
$192$	0	0
$193$	−17.4164	−1.25366	−0.626830	−	0.779156i	$-0.715648\pi$
$193$	−17.4164	−1.25366	−0.626830	+	0.779156i	$0.715648\pi$
$194$	0	0
$195$	1.23607	0.0885167
$196$	0	0
$197$	−14.9443	−1.06474	−0.532368	−	0.846513i	$-0.678698\pi$
$197$	−14.9443	−1.06474	−0.532368	+	0.846513i	$0.678698\pi$
$198$	0	0
$199$	−21.8885	−1.55164	−0.775819	−	0.630956i	$-0.782663\pi$
$199$	−21.8885	−1.55164	−0.775819	+	0.630956i	$0.782663\pi$
$200$	0	0
$201$	9.70820	0.684764
$202$	0	0
$203$	−8.94427	−0.627765
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−20.3607	−1.40169	−0.700844	−	0.713315i	$-0.747193\pi$
$211$	−20.3607	−1.40169	−0.700844	+	0.713315i	$0.747193\pi$
$212$	0	0
$213$	4.76393	0.326419
$214$	0	0
$215$	2.47214	0.168598
$216$	0	0
$217$	−3.23607	−0.219679
$218$	0	0
$219$	10.1803	0.687924
$220$	0	0
$221$	−5.52786	−0.371844
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	11.4164	0.757734	0.378867	−	0.925451i	$-0.376314\pi$
$227$	11.4164	0.757734	0.378867	+	0.925451i	$0.376314\pi$
$228$	0	0
$229$	1.05573	0.0697645	0.0348822	−	0.999391i	$-0.488894\pi$
$229$	1.05573	0.0697645	0.0348822	+	0.999391i	$0.488894\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.58359	−0.169257	−0.0846284	−	0.996413i	$-0.526970\pi$
$233$	−2.58359	−0.169257	−0.0846284	+	0.996413i	$0.526970\pi$
$234$	0	0
$235$	8.94427	0.583460
$236$	0	0
$237$	−8.94427	−0.580993
$238$	0	0
$239$	19.4164	1.25594	0.627972	−	0.778236i	$-0.283885\pi$
$239$	19.4164	1.25594	0.627972	+	0.778236i	$0.283885\pi$
$240$	0	0
$241$	−18.9443	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$241$	−18.9443	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.47214	−0.221827
$246$	0	0
$247$	−3.05573	−0.194431
$248$	0	0
$249$	−12.9443	−0.820310
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.47214	−0.280056
$256$	0	0
$257$	5.41641	0.337866	0.168933	−	0.985628i	$-0.445968\pi$
$257$	5.41641	0.337866	0.168933	+	0.985628i	$0.445968\pi$
$258$	0	0
$259$	29.8885	1.85718
$260$	0	0
$261$	−2.76393	−0.171083
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	0.472136	0.0290031
$266$	0	0
$267$	2.76393	0.169150
$268$	0	0
$269$	28.0689	1.71139	0.855695	−	0.517480i	$-0.173130\pi$
$269$	28.0689	1.71139	0.855695	+	0.517480i	$0.173130\pi$
$270$	0	0
$271$	9.88854	0.600686	0.300343	−	0.953831i	$-0.402899\pi$
$271$	9.88854	0.600686	0.300343	+	0.953831i	$0.402899\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−26.1803	−1.57302	−0.786512	−	0.617575i	$-0.788115\pi$
$277$	−26.1803	−1.57302	−0.786512	+	0.617575i	$0.788115\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−7.52786	−0.449075	−0.224537	−	0.974465i	$-0.572087\pi$
$281$	−7.52786	−0.449075	−0.224537	+	0.974465i	$0.572087\pi$
$282$	0	0
$283$	8.18034	0.486271	0.243135	−	0.969992i	$-0.421824\pi$
$283$	8.18034	0.486271	0.243135	+	0.969992i	$0.421824\pi$
$284$	0	0
$285$	−2.47214	−0.146437
$286$	0	0
$287$	−19.4164	−1.14611
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−0.472136	−0.0276771
$292$	0	0
$293$	−13.4164	−0.783795	−0.391897	−	0.920009i	$-0.628181\pi$
$293$	−13.4164	−0.783795	−0.391897	+	0.920009i	$0.628181\pi$
$294$	0	0
$295$	−11.2361	−0.654189
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.94427	−0.285935
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−6.94427	−0.398938
$304$	0	0
$305$	3.52786	0.202005
$306$	0	0
$307$	−11.2361	−0.641276	−0.320638	−	0.947202i	$-0.603897\pi$
$307$	−11.2361	−0.641276	−0.320638	+	0.947202i	$0.603897\pi$
$308$	0	0
$309$	−14.6525	−0.833550
$310$	0	0
$311$	−19.2361	−1.09078	−0.545389	−	0.838183i	$-0.683618\pi$
$311$	−19.2361	−1.09078	−0.545389	+	0.838183i	$0.683618\pi$
$312$	0	0
$313$	20.6525	1.16735	0.583673	−	0.811988i	$-0.301615\pi$
$313$	20.6525	1.16735	0.583673	+	0.811988i	$0.301615\pi$
$314$	0	0
$315$	−3.23607	−0.182332
$316$	0	0
$317$	−19.5279	−1.09679	−0.548397	−	0.836218i	$-0.684762\pi$
$317$	−19.5279	−1.09679	−0.548397	+	0.836218i	$0.684762\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.94427	0.275962
$322$	0	0
$323$	11.0557	0.615157
$324$	0	0
$325$	1.23607	0.0685647
$326$	0	0
$327$	12.4721	0.689711
$328$	0	0
$329$	−28.9443	−1.59575
$330$	0	0
$331$	−11.0557	−0.607678	−0.303839	−	0.952723i	$-0.598268\pi$
$331$	−11.0557	−0.607678	−0.303839	+	0.952723i	$0.598268\pi$
$332$	0	0
$333$	9.23607	0.506133
$334$	0	0
$335$	9.70820	0.530416
$336$	0	0
$337$	−9.81966	−0.534911	−0.267455	−	0.963570i	$-0.586183\pi$
$337$	−9.81966	−0.534911	−0.267455	+	0.963570i	$0.586183\pi$
$338$	0	0
$339$	14.9443	0.811661
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	21.8885	1.17504	0.587519	−	0.809210i	$-0.300105\pi$
$347$	21.8885	1.17504	0.587519	+	0.809210i	$0.300105\pi$
$348$	0	0
$349$	11.5279	0.617072	0.308536	−	0.951213i	$-0.400161\pi$
$349$	11.5279	0.617072	0.308536	+	0.951213i	$0.400161\pi$
$350$	0	0
$351$	−1.23607	−0.0659764
$352$	0	0
$353$	−7.52786	−0.400668	−0.200334	−	0.979728i	$-0.564203\pi$
$353$	−7.52786	−0.400668	−0.200334	+	0.979728i	$0.564203\pi$
$354$	0	0
$355$	4.76393	0.252843
$356$	0	0
$357$	14.4721	0.765947
$358$	0	0
$359$	−13.1246	−0.692691	−0.346345	−	0.938107i	$-0.612577\pi$
$359$	−13.1246	−0.692691	−0.346345	+	0.938107i	$0.612577\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	10.1803	0.532863
$366$	0	0
$367$	24.9443	1.30208	0.651040	−	0.759043i	$-0.274333\pi$
$367$	24.9443	1.30208	0.651040	+	0.759043i	$0.274333\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−1.52786	−0.0793227
$372$	0	0
$373$	−14.3607	−0.743568	−0.371784	−	0.928319i	$-0.621254\pi$
$373$	−14.3607	−0.743568	−0.371784	+	0.928319i	$0.621254\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−3.41641	−0.175954
$378$	0	0
$379$	7.41641	0.380955	0.190478	−	0.981692i	$-0.438996\pi$
$379$	7.41641	0.380955	0.190478	+	0.981692i	$0.438996\pi$
$380$	0	0
$381$	−15.4164	−0.789807
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.47214	−0.125666
$388$	0	0
$389$	−23.7082	−1.20205	−0.601027	−	0.799229i	$-0.705242\pi$
$389$	−23.7082	−1.20205	−0.601027	+	0.799229i	$0.705242\pi$
$390$	0	0
$391$	17.8885	0.904663
$392$	0	0
$393$	4.76393	0.240309
$394$	0	0
$395$	−8.94427	−0.450035
$396$	0	0
$397$	21.0557	1.05676	0.528378	−	0.849009i	$-0.322800\pi$
$397$	21.0557	1.05676	0.528378	+	0.849009i	$0.322800\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−9.23607	−0.461227	−0.230614	−	0.973045i	$-0.574073\pi$
$401$	−9.23607	−0.461227	−0.230614	+	0.973045i	$0.574073\pi$
$402$	0	0
$403$	−1.23607	−0.0615729
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	37.4164	1.85012	0.925061	−	0.379818i	$-0.124013\pi$
$409$	37.4164	1.85012	0.925061	+	0.379818i	$0.124013\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	0	0
$413$	36.3607	1.78919
$414$	0	0
$415$	−12.9443	−0.635409
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	6.65248	0.324995	0.162497	−	0.986709i	$-0.448045\pi$
$419$	6.65248	0.324995	0.162497	+	0.986709i	$0.448045\pi$
$420$	0	0
$421$	10.3607	0.504949	0.252474	−	0.967604i	$-0.418756\pi$
$421$	10.3607	0.504949	0.252474	+	0.967604i	$0.418756\pi$
$422$	0	0
$423$	−8.94427	−0.434885
$424$	0	0
$425$	−4.47214	−0.216930
$426$	0	0
$427$	−11.4164	−0.552479
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.12461	−0.246844	−0.123422	−	0.992354i	$-0.539387\pi$
$431$	−5.12461	−0.246844	−0.123422	+	0.992354i	$0.539387\pi$
$432$	0	0
$433$	18.7639	0.901737	0.450869	−	0.892590i	$-0.351114\pi$
$433$	18.7639	0.901737	0.450869	+	0.892590i	$0.351114\pi$
$434$	0	0
$435$	−2.76393	−0.132520
$436$	0	0
$437$	9.88854	0.473033
$438$	0	0
$439$	−22.8328	−1.08975	−0.544875	−	0.838517i	$-0.683423\pi$
$439$	−22.8328	−1.08975	−0.544875	+	0.838517i	$0.683423\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	30.4721	1.44777	0.723887	−	0.689918i	$-0.242353\pi$
$443$	30.4721	1.44777	0.723887	+	0.689918i	$0.242353\pi$
$444$	0	0
$445$	2.76393	0.131023
$446$	0	0
$447$	1.05573	0.0499342
$448$	0	0
$449$	−12.2918	−0.580086	−0.290043	−	0.957014i	$-0.593670\pi$
$449$	−12.2918	−0.580086	−0.290043	+	0.957014i	$0.593670\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	21.8885	1.02841
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	10.7639	0.503516	0.251758	−	0.967790i	$-0.418991\pi$
$457$	10.7639	0.503516	0.251758	+	0.967790i	$0.418991\pi$
$458$	0	0
$459$	4.47214	0.208741
$460$	0	0
$461$	−16.0689	−0.748403	−0.374201	−	0.927348i	$-0.622083\pi$
$461$	−16.0689	−0.748403	−0.374201	+	0.927348i	$0.622083\pi$
$462$	0	0
$463$	15.4164	0.716461	0.358231	−	0.933633i	$-0.383380\pi$
$463$	15.4164	0.716461	0.358231	+	0.933633i	$0.383380\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−12.9443	−0.598989	−0.299495	−	0.954098i	$-0.596818\pi$
$467$	−12.9443	−0.598989	−0.299495	+	0.954098i	$0.596818\pi$
$468$	0	0
$469$	−31.4164	−1.45067
$470$	0	0
$471$	−21.4164	−0.986816
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.47214	−0.113429
$476$	0	0
$477$	−0.472136	−0.0216176
$478$	0	0
$479$	−25.7082	−1.17464	−0.587319	−	0.809356i	$-0.699817\pi$
$479$	−25.7082	−1.17464	−0.587319	+	0.809356i	$0.699817\pi$
$480$	0	0
$481$	11.4164	0.520543
$482$	0	0
$483$	12.9443	0.588985
$484$	0	0
$485$	−0.472136	−0.0214386
$486$	0	0
$487$	18.4721	0.837052	0.418526	−	0.908205i	$-0.362547\pi$
$487$	18.4721	0.837052	0.418526	+	0.908205i	$0.362547\pi$
$488$	0	0
$489$	−11.2361	−0.508113
$490$	0	0
$491$	27.4164	1.23729	0.618643	−	0.785673i	$-0.287683\pi$
$491$	27.4164	1.23729	0.618643	+	0.785673i	$0.287683\pi$
$492$	0	0
$493$	12.3607	0.556697
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.4164	−0.691520
$498$	0	0
$499$	15.0557	0.673987	0.336993	−	0.941507i	$-0.390590\pi$
$499$	15.0557	0.673987	0.336993	+	0.941507i	$0.390590\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	34.8328	1.55312	0.776559	−	0.630044i	$-0.216963\pi$
$503$	34.8328	1.55312	0.776559	+	0.630044i	$0.216963\pi$
$504$	0	0
$505$	−6.94427	−0.309016
$506$	0	0
$507$	11.4721	0.509495
$508$	0	0
$509$	−3.12461	−0.138496	−0.0692480	−	0.997599i	$-0.522060\pi$
$509$	−3.12461	−0.138496	−0.0692480	+	0.997599i	$0.522060\pi$
$510$	0	0
$511$	−32.9443	−1.45737
$512$	0	0
$513$	2.47214	0.109147
$514$	0	0
$515$	−14.6525	−0.645665
$516$	0	0
$517$	0	0
$518$	0	0
$519$	22.0000	0.965693
$520$	0	0
$521$	−9.05573	−0.396739	−0.198369	−	0.980127i	$-0.563565\pi$
$521$	−9.05573	−0.396739	−0.198369	+	0.980127i	$0.563565\pi$
$522$	0	0
$523$	−4.36068	−0.190679	−0.0953396	−	0.995445i	$-0.530394\pi$
$523$	−4.36068	−0.190679	−0.0953396	+	0.995445i	$0.530394\pi$
$524$	0	0
$525$	−3.23607	−0.141234
$526$	0	0
$527$	4.47214	0.194809
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	11.2361	0.487604
$532$	0	0
$533$	−7.41641	−0.321240
$534$	0	0
$535$	4.94427	0.213760
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23.3050	1.00196	0.500979	−	0.865459i	$-0.332973\pi$
$541$	23.3050	1.00196	0.500979	+	0.865459i	$0.332973\pi$
$542$	0	0
$543$	5.41641	0.232440
$544$	0	0
$545$	12.4721	0.534248
$546$	0	0
$547$	−14.6525	−0.626495	−0.313247	−	0.949672i	$-0.601417\pi$
$547$	−14.6525	−0.626495	−0.313247	+	0.949672i	$0.601417\pi$
$548$	0	0
$549$	−3.52786	−0.150566
$550$	0	0
$551$	6.83282	0.291088
$552$	0	0
$553$	28.9443	1.23084
$554$	0	0
$555$	9.23607	0.392049
$556$	0	0
$557$	9.41641	0.398986	0.199493	−	0.979899i	$-0.436070\pi$
$557$	9.41641	0.398986	0.199493	+	0.979899i	$0.436070\pi$
$558$	0	0
$559$	−3.05573	−0.129244
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	0	0
$565$	14.9443	0.628710
$566$	0	0
$567$	3.23607	0.135902
$568$	0	0
$569$	24.6525	1.03349	0.516743	−	0.856141i	$-0.327144\pi$
$569$	24.6525	1.03349	0.516743	+	0.856141i	$0.327144\pi$
$570$	0	0
$571$	−7.05573	−0.295273	−0.147637	−	0.989042i	$-0.547167\pi$
$571$	−7.05573	−0.295273	−0.147637	+	0.989042i	$0.547167\pi$
$572$	0	0
$573$	19.2361	0.803598
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−26.9443	−1.12170	−0.560852	−	0.827916i	$-0.689526\pi$
$577$	−26.9443	−1.12170	−0.560852	+	0.827916i	$0.689526\pi$
$578$	0	0
$579$	17.4164	0.723801
$580$	0	0
$581$	41.8885	1.73783
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.23607	−0.0511051
$586$	0	0
$587$	−7.05573	−0.291221	−0.145611	−	0.989342i	$-0.546515\pi$
$587$	−7.05573	−0.291221	−0.145611	+	0.989342i	$0.546515\pi$
$588$	0	0
$589$	2.47214	0.101863
$590$	0	0
$591$	14.9443	0.614725
$592$	0	0
$593$	44.8328	1.84106	0.920532	−	0.390668i	$-0.127756\pi$
$593$	44.8328	1.84106	0.920532	+	0.390668i	$0.127756\pi$
$594$	0	0
$595$	14.4721	0.593300
$596$	0	0
$597$	21.8885	0.895838
$598$	0	0
$599$	−21.1246	−0.863128	−0.431564	−	0.902082i	$-0.642038\pi$
$599$	−21.1246	−0.863128	−0.431564	+	0.902082i	$0.642038\pi$
$600$	0	0
$601$	27.5279	1.12288	0.561442	−	0.827516i	$-0.310247\pi$
$601$	27.5279	1.12288	0.561442	+	0.827516i	$0.310247\pi$
$602$	0	0
$603$	−9.70820	−0.395349
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	43.5967	1.76954	0.884769	−	0.466030i	$-0.154316\pi$
$607$	43.5967	1.76954	0.884769	+	0.466030i	$0.154316\pi$
$608$	0	0
$609$	8.94427	0.362440
$610$	0	0
$611$	−11.0557	−0.447267
$612$	0	0
$613$	7.70820	0.311331	0.155666	−	0.987810i	$-0.450248\pi$
$613$	7.70820	0.311331	0.155666	+	0.987810i	$0.450248\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−16.8328	−0.677664	−0.338832	−	0.940847i	$-0.610032\pi$
$617$	−16.8328	−0.677664	−0.338832	+	0.940847i	$0.610032\pi$
$618$	0	0
$619$	−27.7771	−1.11646	−0.558228	−	0.829688i	$-0.688518\pi$
$619$	−27.7771	−1.11646	−0.558228	+	0.829688i	$0.688518\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	−8.94427	−0.358345
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−41.3050	−1.64694
$630$	0	0
$631$	−20.9443	−0.833778	−0.416889	−	0.908957i	$-0.636880\pi$
$631$	−20.9443	−0.833778	−0.416889	+	0.908957i	$0.636880\pi$
$632$	0	0
$633$	20.3607	0.809264
$634$	0	0
$635$	−15.4164	−0.611781
$636$	0	0
$637$	4.29180	0.170047
$638$	0	0
$639$	−4.76393	−0.188458
$640$	0	0
$641$	−22.1803	−0.876071	−0.438035	−	0.898958i	$-0.644326\pi$
$641$	−22.1803	−0.876071	−0.438035	+	0.898958i	$0.644326\pi$
$642$	0	0
$643$	−0.583592	−0.0230146	−0.0115073	−	0.999934i	$-0.503663\pi$
$643$	−0.583592	−0.0230146	−0.0115073	+	0.999934i	$0.503663\pi$
$644$	0	0
$645$	−2.47214	−0.0973403
$646$	0	0
$647$	−36.9443	−1.45243	−0.726215	−	0.687468i	$-0.758722\pi$
$647$	−36.9443	−1.45243	−0.726215	+	0.687468i	$0.758722\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	3.23607	0.126832
$652$	0	0
$653$	2.58359	0.101104	0.0505519	−	0.998721i	$-0.483902\pi$
$653$	2.58359	0.101104	0.0505519	+	0.998721i	$0.483902\pi$
$654$	0	0
$655$	4.76393	0.186142
$656$	0	0
$657$	−10.1803	−0.397173
$658$	0	0
$659$	20.7639	0.808848	0.404424	−	0.914572i	$-0.367472\pi$
$659$	20.7639	0.808848	0.404424	+	0.914572i	$0.367472\pi$
$660$	0	0
$661$	10.9443	0.425683	0.212841	−	0.977087i	$-0.431728\pi$
$661$	10.9443	0.425683	0.212841	+	0.977087i	$0.431728\pi$
$662$	0	0
$663$	5.52786	0.214684
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	11.0557	0.428080
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	0	0
$672$	0	0
$673$	17.2361	0.664402	0.332201	−	0.943209i	$-0.392209\pi$
$673$	17.2361	0.664402	0.332201	+	0.943209i	$0.392209\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−6.58359	−0.253028	−0.126514	−	0.991965i	$-0.540379\pi$
$677$	−6.58359	−0.253028	−0.126514	+	0.991965i	$0.540379\pi$
$678$	0	0
$679$	1.52786	0.0586340
$680$	0	0
$681$	−11.4164	−0.437478
$682$	0	0
$683$	19.4164	0.742948	0.371474	−	0.928443i	$-0.378852\pi$
$683$	19.4164	0.742948	0.371474	+	0.928443i	$0.378852\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	0	0
$687$	−1.05573	−0.0402785
$688$	0	0
$689$	−0.583592	−0.0222331
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	26.8328	1.01637
$698$	0	0
$699$	2.58359	0.0977204
$700$	0	0
$701$	−44.4721	−1.67969	−0.839845	−	0.542827i	$-0.817354\pi$
$701$	−44.4721	−1.67969	−0.839845	+	0.542827i	$0.817354\pi$
$702$	0	0
$703$	−22.8328	−0.861156
$704$	0	0
$705$	−8.94427	−0.336861
$706$	0	0
$707$	22.4721	0.845152
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	8.94427	0.335436
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.4164	−0.725119
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	47.4164	1.76588
$722$	0	0
$723$	18.9443	0.704545
$724$	0	0
$725$	−2.76393	−0.102650
$726$	0	0
$727$	−51.5967	−1.91362	−0.956809	−	0.290718i	$-0.906106\pi$
$727$	−51.5967	−1.91362	−0.956809	+	0.290718i	$0.906106\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.0557	0.408911
$732$	0	0
$733$	45.7771	1.69082	0.845408	−	0.534122i	$-0.179358\pi$
$733$	45.7771	1.69082	0.845408	+	0.534122i	$0.179358\pi$
$734$	0	0
$735$	3.47214	0.128072
$736$	0	0
$737$	0	0
$738$	0	0
$739$	3.05573	0.112407	0.0562034	−	0.998419i	$-0.482100\pi$
$739$	3.05573	0.112407	0.0562034	+	0.998419i	$0.482100\pi$
$740$	0	0
$741$	3.05573	0.112255
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	1.05573	0.0386789
$746$	0	0
$747$	12.9443	0.473606
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−33.8885	−1.23661	−0.618305	−	0.785938i	$-0.712180\pi$
$751$	−33.8885	−1.23661	−0.618305	+	0.785938i	$0.712180\pi$
$752$	0	0
$753$	8.00000	0.291536
$754$	0	0
$755$	21.8885	0.796606
$756$	0	0
$757$	27.1246	0.985861	0.492930	−	0.870069i	$-0.335926\pi$
$757$	27.1246	0.985861	0.492930	+	0.870069i	$0.335926\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.8197	0.645962	0.322981	−	0.946405i	$-0.395315\pi$
$761$	17.8197	0.645962	0.322981	+	0.946405i	$0.395315\pi$
$762$	0	0
$763$	−40.3607	−1.46115
$764$	0	0
$765$	4.47214	0.161690
$766$	0	0
$767$	13.8885	0.501486
$768$	0	0
$769$	−0.472136	−0.0170257	−0.00851283	−	0.999964i	$-0.502710\pi$
$769$	−0.472136	−0.0170257	−0.00851283	+	0.999964i	$0.502710\pi$
$770$	0	0
$771$	−5.41641	−0.195067
$772$	0	0
$773$	−39.3050	−1.41370	−0.706850	−	0.707363i	$-0.749885\pi$
$773$	−39.3050	−1.41370	−0.706850	+	0.707363i	$0.749885\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−29.8885	−1.07225
$778$	0	0
$779$	14.8328	0.531441
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.76393	0.0987749
$784$	0	0
$785$	−21.4164	−0.764384
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−48.3607	−1.71951
$792$	0	0
$793$	−4.36068	−0.154852
$794$	0	0
$795$	−0.472136	−0.0167449
$796$	0	0
$797$	−24.8328	−0.879623	−0.439812	−	0.898090i	$-0.644955\pi$
$797$	−24.8328	−0.879623	−0.439812	+	0.898090i	$0.644955\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	0	0
$801$	−2.76393	−0.0976587
$802$	0	0
$803$	0	0
$804$	0	0
$805$	12.9443	0.456226
$806$	0	0
$807$	−28.0689	−0.988072
$808$	0	0
$809$	11.7082	0.411639	0.205819	−	0.978590i	$-0.434014\pi$
$809$	11.7082	0.411639	0.205819	+	0.978590i	$0.434014\pi$
$810$	0	0
$811$	12.3607	0.434042	0.217021	−	0.976167i	$-0.430366\pi$
$811$	12.3607	0.434042	0.217021	+	0.976167i	$0.430366\pi$
$812$	0	0
$813$	−9.88854	−0.346806
$814$	0	0
$815$	−11.2361	−0.393582
$816$	0	0
$817$	6.11146	0.213813
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	18.1803	0.634498	0.317249	−	0.948342i	$-0.397241\pi$
$821$	18.1803	0.634498	0.317249	+	0.948342i	$0.397241\pi$
$822$	0	0
$823$	36.3607	1.26745	0.633727	−	0.773557i	$-0.281524\pi$
$823$	36.3607	1.26745	0.633727	+	0.773557i	$0.281524\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0000	0.556375	0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000	0.556375	0.278187	+	0.960527i	$0.410266\pi$
$828$	0	0
$829$	−31.3050	−1.08727	−0.543633	−	0.839323i	$-0.682952\pi$
$829$	−31.3050	−1.08727	−0.543633	+	0.839323i	$0.682952\pi$
$830$	0	0
$831$	26.1803	0.908186
$832$	0	0
$833$	−15.5279	−0.538009
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−3.23607	−0.111721	−0.0558607	−	0.998439i	$-0.517790\pi$
$839$	−3.23607	−0.111721	−0.0558607	+	0.998439i	$0.517790\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	7.52786	0.259273
$844$	0	0
$845$	11.4721	0.394653
$846$	0	0
$847$	−35.5967	−1.22312
$848$	0	0
$849$	−8.18034	−0.280749
$850$	0	0
$851$	−36.9443	−1.26643
$852$	0	0
$853$	−33.7771	−1.15651	−0.578253	−	0.815858i	$-0.696265\pi$
$853$	−33.7771	−1.15651	−0.578253	+	0.815858i	$0.696265\pi$
$854$	0	0
$855$	2.47214	0.0845453
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	18.1115	0.617955	0.308977	−	0.951069i	$-0.400013\pi$
$859$	18.1115	0.617955	0.308977	+	0.951069i	$0.400013\pi$
$860$	0	0
$861$	19.4164	0.661709
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	0.472136	0.0159794
$874$	0	0
$875$	−3.23607	−0.109399
$876$	0	0
$877$	−2.94427	−0.0994210	−0.0497105	−	0.998764i	$-0.515830\pi$
$877$	−2.94427	−0.0994210	−0.0497105	+	0.998764i	$0.515830\pi$
$878$	0	0
$879$	13.4164	0.452524
$880$	0	0
$881$	44.0689	1.48472	0.742359	−	0.670002i	$-0.233707\pi$
$881$	44.0689	1.48472	0.742359	+	0.670002i	$0.233707\pi$
$882$	0	0
$883$	18.8328	0.633775	0.316887	−	0.948463i	$-0.397362\pi$
$883$	18.8328	0.633775	0.316887	+	0.948463i	$0.397362\pi$
$884$	0	0
$885$	11.2361	0.377696
$886$	0	0
$887$	7.41641	0.249019	0.124509	−	0.992218i	$-0.460264\pi$
$887$	7.41641	0.249019	0.124509	+	0.992218i	$0.460264\pi$
$888$	0	0
$889$	49.8885	1.67321
$890$	0	0
$891$	0	0
$892$	0	0
$893$	22.1115	0.739932
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.94427	0.165084
$898$	0	0
$899$	2.76393	0.0921823
$900$	0	0
$901$	2.11146	0.0703428
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	5.41641	0.180047
$906$	0	0
$907$	16.5410	0.549236	0.274618	−	0.961553i	$-0.411449\pi$
$907$	16.5410	0.549236	0.274618	+	0.961553i	$0.411449\pi$
$908$	0	0
$909$	6.94427	0.230327
$910$	0	0
$911$	25.8885	0.857726	0.428863	−	0.903370i	$-0.358914\pi$
$911$	25.8885	0.857726	0.428863	+	0.903370i	$0.358914\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−3.52786	−0.116628
$916$	0	0
$917$	−15.4164	−0.509095
$918$	0	0
$919$	29.3050	0.966681	0.483341	−	0.875432i	$-0.339423\pi$
$919$	29.3050	0.966681	0.483341	+	0.875432i	$0.339423\pi$
$920$	0	0
$921$	11.2361	0.370241
$922$	0	0
$923$	−5.88854	−0.193824
$924$	0	0
$925$	9.23607	0.303680
$926$	0	0
$927$	14.6525	0.481250
$928$	0	0
$929$	−25.5967	−0.839802	−0.419901	−	0.907570i	$-0.637935\pi$
$929$	−25.5967	−0.839802	−0.419901	+	0.907570i	$0.637935\pi$
$930$	0	0
$931$	−8.58359	−0.281316
$932$	0	0
$933$	19.2361	0.629761
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.4164	0.960992	0.480496	−	0.876997i	$-0.340457\pi$
$937$	29.4164	0.960992	0.480496	+	0.876997i	$0.340457\pi$
$938$	0	0
$939$	−20.6525	−0.673968
$940$	0	0
$941$	−51.4853	−1.67837	−0.839186	−	0.543844i	$-0.816968\pi$
$941$	−51.4853	−1.67837	−0.839186	+	0.543844i	$0.816968\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	3.23607	0.105269
$946$	0	0
$947$	12.9443	0.420632	0.210316	−	0.977633i	$-0.432551\pi$
$947$	12.9443	0.420632	0.210316	+	0.977633i	$0.432551\pi$
$948$	0	0
$949$	−12.5836	−0.408481
$950$	0	0
$951$	19.5279	0.633234
$952$	0	0
$953$	−9.05573	−0.293344	−0.146672	−	0.989185i	$-0.546856\pi$
$953$	−9.05573	−0.293344	−0.146672	+	0.989185i	$0.546856\pi$
$954$	0	0
$955$	19.2361	0.622465
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−45.3050	−1.46297
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−4.94427	−0.159327
$964$	0	0
$965$	17.4164	0.560654
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	0	0
$969$	−11.0557	−0.355161
$970$	0	0
$971$	−6.29180	−0.201913	−0.100957	−	0.994891i	$-0.532190\pi$
$971$	−6.29180	−0.201913	−0.100957	+	0.994891i	$0.532190\pi$
$972$	0	0
$973$	12.9443	0.414974
$974$	0	0
$975$	−1.23607	−0.0395859
$976$	0	0
$977$	−14.9443	−0.478110	−0.239055	−	0.971006i	$-0.576838\pi$
$977$	−14.9443	−0.478110	−0.239055	+	0.971006i	$0.576838\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−12.4721	−0.398205
$982$	0	0
$983$	59.7771	1.90659	0.953296	−	0.302036i	$-0.0976663\pi$
$983$	59.7771	1.90659	0.953296	+	0.302036i	$0.0976663\pi$
$984$	0	0
$985$	14.9443	0.476164
$986$	0	0
$987$	28.9443	0.921306
$988$	0	0
$989$	9.88854	0.314437
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	11.0557	0.350843
$994$	0	0
$995$	21.8885	0.693913
$996$	0	0
$997$	−20.4721	−0.648359	−0.324180	−	0.945996i	$-0.605088\pi$
$997$	−20.4721	−0.648359	−0.324180	+	0.945996i	$0.605088\pi$
$998$	0	0
$999$	−9.23607	−0.292216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.i.1.2	✓	2
4.3	odd	2		7440.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.i.1.2	✓	2	1.1	even	1	trivial
7440.2.a.bi.1.1		2	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.23607 q^{7} +1.00000 q^{9} +1.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -2.47214 q^{19} -3.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.76393 q^{29} -1.00000 q^{31} -3.23607 q^{35} +9.23607 q^{37} -1.23607 q^{39} -6.00000 q^{41} -2.47214 q^{43} -1.00000 q^{45} -8.94427 q^{47} +3.47214 q^{49} +4.47214 q^{51} -0.472136 q^{53} +2.47214 q^{57} +11.2361 q^{59} -3.52786 q^{61} +3.23607 q^{63} -1.23607 q^{65} -9.70820 q^{67} +4.00000 q^{69} -4.76393 q^{71} -10.1803 q^{73} -1.00000 q^{75} +8.94427 q^{79} +1.00000 q^{81} +12.9443 q^{83} +4.47214 q^{85} +2.76393 q^{87} -2.76393 q^{89} +4.00000 q^{91} +1.00000 q^{93} +2.47214 q^{95} +0.472136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^39 - 12 * q^41 + 4 * q^43 - 2 * q^45 - 2 * q^49 + 8 * q^53 - 4 * q^57 + 18 * q^59 - 16 * q^61 + 2 * q^63 + 2 * q^65 - 6 * q^67 + 8 * q^69 - 14 * q^71 + 2 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 10 * q^87 - 10 * q^89 + 8 * q^91 + 2 * q^93 - 4 * q^95 - 8 * q^97

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