LMFDB - Embedded newform 740.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [740,2,Mod(1,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("740.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 740 = 2^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	740.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.90892974957$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	740.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67513 q^{3} -1.00000 q^{5} -4.63752 q^{7} -0.193937 q^{9} -4.96239 q^{11} +5.76845 q^{13} +1.67513 q^{15} +6.54420 q^{17} +4.32487 q^{19} +7.76845 q^{21} +2.96239 q^{23} +1.00000 q^{25} +5.35026 q^{27} -4.57452 q^{29} -5.02539 q^{31} +8.31265 q^{33} +4.63752 q^{35} -1.00000 q^{37} -9.66291 q^{39} -5.35026 q^{41} -0.574515 q^{43} +0.193937 q^{45} +13.5999 q^{47} +14.5066 q^{49} -10.9624 q^{51} -5.53690 q^{53} +4.96239 q^{55} -7.24472 q^{57} +1.54912 q^{59} +9.47627 q^{61} +0.899385 q^{63} -5.76845 q^{65} +14.7635 q^{67} -4.96239 q^{69} -3.68735 q^{71} -2.96239 q^{73} -1.67513 q^{75} +23.0132 q^{77} +5.98778 q^{79} -8.38058 q^{81} -10.6375 q^{83} -6.54420 q^{85} +7.66291 q^{87} -1.16362 q^{89} -26.7513 q^{91} +8.41819 q^{93} -4.32487 q^{95} +2.64974 q^{97} +0.962389 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7} - q^{9} - 4 q^{11} + 6 q^{13} + 10 q^{17} + 18 q^{19} + 12 q^{21} - 2 q^{23} + 3 q^{25} + 6 q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} - 3 q^{37} + 2 q^{39} - 6 q^{41} + 10 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - q^9 - 4 * q^11 + 6 * q^13 + 10 * q^17 + 18 * q^19 + 12 * q^21 - 2 * q^23 + 3 * q^25 + 6 * q^27 - 2 * q^29 + 4 * q^33 - 2 * q^35 - 3 * q^37 + 2 * q^39 - 6 * q^41 + 10 * q^43 + q^45 + 14 * q^47 + 23 * q^49 - 22 * q^51 + 6 * q^53 + 4 * q^55 + 8 * q^57 + 8 * q^59 + 10 * q^61 - 4 * q^63 - 6 * q^65 + 20 * q^67 - 4 * q^69 - 32 * q^71 + 2 * q^73 + 28 * q^77 - 8 * q^79 - 13 * q^81 - 16 * q^83 - 10 * q^85 - 8 * q^87 - 6 * q^89 - 30 * q^91 + 24 * q^93 - 18 * q^95 + 18 * q^97 - 8 * 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
$3$	−1.67513	−0.967137	−0.483569	−	0.875306i	$-0.660660\pi$
$3$	−1.67513	−0.967137	−0.483569	+	0.875306i	$0.660660\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.63752	−1.75282	−0.876409	−	0.481568i	$-0.840067\pi$
$7$	−4.63752	−1.75282	−0.876409	+	0.481568i	$0.840067\pi$
$8$	0	0
$9$	−0.193937	−0.0646455
$10$	0	0
$11$	−4.96239	−1.49622	−0.748108	−	0.663577i	$-0.769038\pi$
$11$	−4.96239	−1.49622	−0.748108	+	0.663577i	$0.769038\pi$
$12$	0	0
$13$	5.76845	1.59988	0.799940	−	0.600079i	$-0.204864\pi$
$13$	5.76845	1.59988	0.799940	+	0.600079i	$0.204864\pi$
$14$	0	0
$15$	1.67513	0.432517
$16$	0	0
$17$	6.54420	1.58720	0.793601	−	0.608439i	$-0.208204\pi$
$17$	6.54420	1.58720	0.793601	+	0.608439i	$0.208204\pi$
$18$	0	0
$19$	4.32487	0.992193	0.496097	−	0.868267i	$-0.334766\pi$
$19$	4.32487	0.992193	0.496097	+	0.868267i	$0.334766\pi$
$20$	0	0
$21$	7.76845	1.69522
$22$	0	0
$23$	2.96239	0.617701	0.308850	−	0.951111i	$-0.400056\pi$
$23$	2.96239	0.617701	0.308850	+	0.951111i	$0.400056\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.35026	1.02966
$28$	0	0
$29$	−4.57452	−0.849466	−0.424733	−	0.905319i	$-0.639632\pi$
$29$	−4.57452	−0.849466	−0.424733	+	0.905319i	$0.639632\pi$
$30$	0	0
$31$	−5.02539	−0.902587	−0.451294	−	0.892376i	$-0.649037\pi$
$31$	−5.02539	−0.902587	−0.451294	+	0.892376i	$0.649037\pi$
$32$	0	0
$33$	8.31265	1.44705
$34$	0	0
$35$	4.63752	0.783884
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−9.66291	−1.54730
$40$	0	0
$41$	−5.35026	−0.835571	−0.417785	−	0.908546i	$-0.637194\pi$
$41$	−5.35026	−0.835571	−0.417785	+	0.908546i	$0.637194\pi$
$42$	0	0
$43$	−0.574515	−0.0876128	−0.0438064	−	0.999040i	$-0.513948\pi$
$43$	−0.574515	−0.0876128	−0.0438064	+	0.999040i	$0.513948\pi$
$44$	0	0
$45$	0.193937	0.0289104
$46$	0	0
$47$	13.5999	1.98375	0.991875	−	0.127214i	$-0.0406036\pi$
$47$	13.5999	1.98375	0.991875	+	0.127214i	$0.0406036\pi$
$48$	0	0
$49$	14.5066	2.07237
$50$	0	0
$51$	−10.9624	−1.53504
$52$	0	0
$53$	−5.53690	−0.760552	−0.380276	−	0.924873i	$-0.624171\pi$
$53$	−5.53690	−0.760552	−0.380276	+	0.924873i	$0.624171\pi$
$54$	0	0
$55$	4.96239	0.669128
$56$	0	0
$57$	−7.24472	−0.959587
$58$	0	0
$59$	1.54912	0.201679	0.100839	−	0.994903i	$-0.467847\pi$
$59$	1.54912	0.201679	0.100839	+	0.994903i	$0.467847\pi$
$60$	0	0
$61$	9.47627	1.21331	0.606656	−	0.794964i	$-0.292511\pi$
$61$	9.47627	1.21331	0.606656	+	0.794964i	$0.292511\pi$
$62$	0	0
$63$	0.899385	0.113312
$64$	0	0
$65$	−5.76845	−0.715488
$66$	0	0
$67$	14.7635	1.80365	0.901826	−	0.432099i	$-0.142227\pi$
$67$	14.7635	1.80365	0.901826	+	0.432099i	$0.142227\pi$
$68$	0	0
$69$	−4.96239	−0.597401
$70$	0	0
$71$	−3.68735	−0.437608	−0.218804	−	0.975769i	$-0.570216\pi$
$71$	−3.68735	−0.437608	−0.218804	+	0.975769i	$0.570216\pi$
$72$	0	0
$73$	−2.96239	−0.346721	−0.173361	−	0.984858i	$-0.555463\pi$
$73$	−2.96239	−0.346721	−0.173361	+	0.984858i	$0.555463\pi$
$74$	0	0
$75$	−1.67513	−0.193427
$76$	0	0
$77$	23.0132	2.62259
$78$	0	0
$79$	5.98778	0.673678	0.336839	−	0.941562i	$-0.390642\pi$
$79$	5.98778	0.673678	0.336839	+	0.941562i	$0.390642\pi$
$80$	0	0
$81$	−8.38058	−0.931175
$82$	0	0
$83$	−10.6375	−1.16762	−0.583810	−	0.811891i	$-0.698439\pi$
$83$	−10.6375	−1.16762	−0.583810	+	0.811891i	$0.698439\pi$
$84$	0	0
$85$	−6.54420	−0.709818
$86$	0	0
$87$	7.66291	0.821550
$88$	0	0
$89$	−1.16362	−0.123343	−0.0616717	−	0.998096i	$-0.519643\pi$
$89$	−1.16362	−0.123343	−0.0616717	+	0.998096i	$0.519643\pi$
$90$	0	0
$91$	−26.7513	−2.80430
$92$	0	0
$93$	8.41819	0.872926
$94$	0	0
$95$	−4.32487	−0.443722
$96$	0	0
$97$	2.64974	0.269040	0.134520	−	0.990911i	$-0.457051\pi$
$97$	2.64974	0.269040	0.134520	+	0.990911i	$0.457051\pi$
$98$	0	0
$99$	0.962389	0.0967237
$100$	0	0
$101$	11.0435	1.09887	0.549434	−	0.835537i	$-0.314843\pi$
$101$	11.0435	1.09887	0.549434	+	0.835537i	$0.314843\pi$
$102$	0	0
$103$	−13.5369	−1.33383	−0.666915	−	0.745133i	$-0.732386\pi$
$103$	−13.5369	−1.33383	−0.666915	+	0.745133i	$0.732386\pi$
$104$	0	0
$105$	−7.76845	−0.758123
$106$	0	0
$107$	13.7866	1.33280	0.666398	−	0.745596i	$-0.267835\pi$
$107$	13.7866	1.33280	0.666398	+	0.745596i	$0.267835\pi$
$108$	0	0
$109$	−0.387873	−0.0371515	−0.0185758	−	0.999827i	$-0.505913\pi$
$109$	−0.387873	−0.0371515	−0.0185758	+	0.999827i	$0.505913\pi$
$110$	0	0
$111$	1.67513	0.158996
$112$	0	0
$113$	−0.932071	−0.0876819	−0.0438410	−	0.999039i	$-0.513959\pi$
$113$	−0.932071	−0.0876819	−0.0438410	+	0.999039i	$0.513959\pi$
$114$	0	0
$115$	−2.96239	−0.276244
$116$	0	0
$117$	−1.11871	−0.103425
$118$	0	0
$119$	−30.3488	−2.78207
$120$	0	0
$121$	13.6253	1.23866
$122$	0	0
$123$	8.96239	0.808111
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.6375	−1.29887	−0.649435	−	0.760417i	$-0.724995\pi$
$127$	−14.6375	−1.29887	−0.649435	+	0.760417i	$0.724995\pi$
$128$	0	0
$129$	0.962389	0.0847336
$130$	0	0
$131$	7.33804	0.641128	0.320564	−	0.947227i	$-0.396128\pi$
$131$	7.33804	0.641128	0.320564	+	0.947227i	$0.396128\pi$
$132$	0	0
$133$	−20.0567	−1.73913
$134$	0	0
$135$	−5.35026	−0.460477
$136$	0	0
$137$	−6.18664	−0.528561	−0.264280	−	0.964446i	$-0.585134\pi$
$137$	−6.18664	−0.528561	−0.264280	+	0.964446i	$0.585134\pi$
$138$	0	0
$139$	9.61213	0.815290	0.407645	−	0.913140i	$-0.366350\pi$
$139$	9.61213	0.815290	0.407645	+	0.913140i	$0.366350\pi$
$140$	0	0
$141$	−22.7816	−1.91856
$142$	0	0
$143$	−28.6253	−2.39377
$144$	0	0
$145$	4.57452	0.379893
$146$	0	0
$147$	−24.3004	−2.00427
$148$	0	0
$149$	19.0435	1.56010	0.780052	−	0.625715i	$-0.215193\pi$
$149$	19.0435	1.56010	0.780052	+	0.625715i	$0.215193\pi$
$150$	0	0
$151$	9.58769	0.780235	0.390118	−	0.920765i	$-0.372434\pi$
$151$	9.58769	0.780235	0.390118	+	0.920765i	$0.372434\pi$
$152$	0	0
$153$	−1.26916	−0.102605
$154$	0	0
$155$	5.02539	0.403649
$156$	0	0
$157$	11.7381	0.936805	0.468403	−	0.883515i	$-0.344830\pi$
$157$	11.7381	0.936805	0.468403	+	0.883515i	$0.344830\pi$
$158$	0	0
$159$	9.27504	0.735558
$160$	0	0
$161$	−13.7381	−1.08272
$162$	0	0
$163$	21.3258	1.67037	0.835184	−	0.549971i	$-0.185361\pi$
$163$	21.3258	1.67037	0.835184	+	0.549971i	$0.185361\pi$
$164$	0	0
$165$	−8.31265	−0.647139
$166$	0	0
$167$	−11.9248	−0.922767	−0.461383	−	0.887201i	$-0.652647\pi$
$167$	−11.9248	−0.922767	−0.461383	+	0.887201i	$0.652647\pi$
$168$	0	0
$169$	20.2750	1.55962
$170$	0	0
$171$	−0.838750	−0.0641408
$172$	0	0
$173$	−0.0507852	−0.00386113	−0.00193056	−	0.999998i	$-0.500615\pi$
$173$	−0.0507852	−0.00386113	−0.00193056	+	0.999998i	$0.500615\pi$
$174$	0	0
$175$	−4.63752	−0.350564
$176$	0	0
$177$	−2.59498	−0.195051
$178$	0	0
$179$	−3.02539	−0.226128	−0.113064	−	0.993588i	$-0.536067\pi$
$179$	−3.02539	−0.226128	−0.113064	+	0.993588i	$0.536067\pi$
$180$	0	0
$181$	−7.19394	−0.534721	−0.267361	−	0.963597i	$-0.586151\pi$
$181$	−7.19394	−0.534721	−0.267361	+	0.963597i	$0.586151\pi$
$182$	0	0
$183$	−15.8740	−1.17344
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	−32.4749	−2.37480
$188$	0	0
$189$	−24.8119	−1.80480
$190$	0	0
$191$	−9.98778	−0.722690	−0.361345	−	0.932432i	$-0.617682\pi$
$191$	−9.98778	−0.722690	−0.361345	+	0.932432i	$0.617682\pi$
$192$	0	0
$193$	13.8496	0.996913	0.498456	−	0.866915i	$-0.333900\pi$
$193$	13.8496	0.996913	0.498456	+	0.866915i	$0.333900\pi$
$194$	0	0
$195$	9.66291	0.691976
$196$	0	0
$197$	−6.64974	−0.473774	−0.236887	−	0.971537i	$-0.576127\pi$
$197$	−6.64974	−0.473774	−0.236887	+	0.971537i	$0.576127\pi$
$198$	0	0
$199$	10.1890	0.722280	0.361140	−	0.932512i	$-0.382388\pi$
$199$	10.1890	0.722280	0.361140	+	0.932512i	$0.382388\pi$
$200$	0	0
$201$	−24.7308	−1.74438
$202$	0	0
$203$	21.2144	1.48896
$204$	0	0
$205$	5.35026	0.373678
$206$	0	0
$207$	−0.574515	−0.0399316
$208$	0	0
$209$	−21.4617	−1.48454
$210$	0	0
$211$	13.4010	0.922566	0.461283	−	0.887253i	$-0.347389\pi$
$211$	13.4010	0.922566	0.461283	+	0.887253i	$0.347389\pi$
$212$	0	0
$213$	6.17679	0.423227
$214$	0	0
$215$	0.574515	0.0391816
$216$	0	0
$217$	23.3054	1.58207
$218$	0	0
$219$	4.96239	0.335327
$220$	0	0
$221$	37.7499	2.53933
$222$	0	0
$223$	21.7113	1.45390	0.726949	−	0.686691i	$-0.240938\pi$
$223$	21.7113	1.45390	0.726949	+	0.686691i	$0.240938\pi$
$224$	0	0
$225$	−0.193937	−0.0129291
$226$	0	0
$227$	11.4617	0.760739	0.380369	−	0.924835i	$-0.375797\pi$
$227$	11.4617	0.760739	0.380369	+	0.924835i	$0.375797\pi$
$228$	0	0
$229$	16.4241	1.08533	0.542667	−	0.839948i	$-0.317415\pi$
$229$	16.4241	1.08533	0.542667	+	0.839948i	$0.317415\pi$
$230$	0	0
$231$	−38.5501	−2.53641
$232$	0	0
$233$	−10.1866	−0.667349	−0.333675	−	0.942688i	$-0.608289\pi$
$233$	−10.1866	−0.667349	−0.333675	+	0.942688i	$0.608289\pi$
$234$	0	0
$235$	−13.5999	−0.887160
$236$	0	0
$237$	−10.0303	−0.651539
$238$	0	0
$239$	1.53927	0.0995673	0.0497837	−	0.998760i	$-0.484147\pi$
$239$	1.53927	0.0995673	0.0497837	+	0.998760i	$0.484147\pi$
$240$	0	0
$241$	−4.11142	−0.264840	−0.132420	−	0.991194i	$-0.542275\pi$
$241$	−4.11142	−0.264840	−0.132420	+	0.991194i	$0.542275\pi$
$242$	0	0
$243$	−2.01222	−0.129084
$244$	0	0
$245$	−14.5066	−0.926792
$246$	0	0
$247$	24.9478	1.58739
$248$	0	0
$249$	17.8192	1.12925
$250$	0	0
$251$	−1.91256	−0.120720	−0.0603598	−	0.998177i	$-0.519225\pi$
$251$	−1.91256	−0.120720	−0.0603598	+	0.998177i	$0.519225\pi$
$252$	0	0
$253$	−14.7005	−0.924214
$254$	0	0
$255$	10.9624	0.686491
$256$	0	0
$257$	13.7685	0.858852	0.429426	−	0.903102i	$-0.358716\pi$
$257$	13.7685	0.858852	0.429426	+	0.903102i	$0.358716\pi$
$258$	0	0
$259$	4.63752	0.288161
$260$	0	0
$261$	0.887166	0.0549142
$262$	0	0
$263$	8.89938	0.548760	0.274380	−	0.961621i	$-0.411527\pi$
$263$	8.89938	0.548760	0.274380	+	0.961621i	$0.411527\pi$
$264$	0	0
$265$	5.53690	0.340129
$266$	0	0
$267$	1.94921	0.119290
$268$	0	0
$269$	−22.7308	−1.38592	−0.692962	−	0.720974i	$-0.743695\pi$
$269$	−22.7308	−1.38592	−0.692962	+	0.720974i	$0.743695\pi$
$270$	0	0
$271$	−24.3127	−1.47689	−0.738444	−	0.674315i	$-0.764439\pi$
$271$	−24.3127	−1.47689	−0.738444	+	0.674315i	$0.764439\pi$
$272$	0	0
$273$	44.8119	2.71214
$274$	0	0
$275$	−4.96239	−0.299243
$276$	0	0
$277$	25.7685	1.54828	0.774138	−	0.633017i	$-0.218184\pi$
$277$	25.7685	1.54828	0.774138	+	0.633017i	$0.218184\pi$
$278$	0	0
$279$	0.974607	0.0583482
$280$	0	0
$281$	−27.4372	−1.63677	−0.818384	−	0.574671i	$-0.805130\pi$
$281$	−27.4372	−1.63677	−0.818384	+	0.574671i	$0.805130\pi$
$282$	0	0
$283$	25.3865	1.50907	0.754534	−	0.656261i	$-0.227863\pi$
$283$	25.3865	1.50907	0.754534	+	0.656261i	$0.227863\pi$
$284$	0	0
$285$	7.24472	0.429140
$286$	0	0
$287$	24.8119	1.46460
$288$	0	0
$289$	25.8265	1.51921
$290$	0	0
$291$	−4.43866	−0.260199
$292$	0	0
$293$	−7.40105	−0.432374	−0.216187	−	0.976352i	$-0.569362\pi$
$293$	−7.40105	−0.432374	−0.216187	+	0.976352i	$0.569362\pi$
$294$	0	0
$295$	−1.54912	−0.0901934
$296$	0	0
$297$	−26.5501	−1.54059
$298$	0	0
$299$	17.0884	0.988247
$300$	0	0
$301$	2.66433	0.153569
$302$	0	0
$303$	−18.4993	−1.06276
$304$	0	0
$305$	−9.47627	−0.542610
$306$	0	0
$307$	−4.95017	−0.282521	−0.141261	−	0.989972i	$-0.545116\pi$
$307$	−4.95017	−0.282521	−0.141261	+	0.989972i	$0.545116\pi$
$308$	0	0
$309$	22.6761	1.29000
$310$	0	0
$311$	−21.3380	−1.20997	−0.604985	−	0.796237i	$-0.706821\pi$
$311$	−21.3380	−1.20997	−0.604985	+	0.796237i	$0.706821\pi$
$312$	0	0
$313$	15.7988	0.892999	0.446500	−	0.894784i	$-0.352670\pi$
$313$	15.7988	0.892999	0.446500	+	0.894784i	$0.352670\pi$
$314$	0	0
$315$	−0.899385	−0.0506746
$316$	0	0
$317$	−35.4372	−1.99035	−0.995177	−	0.0980961i	$-0.968725\pi$
$317$	−35.4372	−1.99035	−0.995177	+	0.0980961i	$0.968725\pi$
$318$	0	0
$319$	22.7005	1.27099
$320$	0	0
$321$	−23.0943	−1.28900
$322$	0	0
$323$	28.3028	1.57481
$324$	0	0
$325$	5.76845	0.319976
$326$	0	0
$327$	0.649738	0.0359306
$328$	0	0
$329$	−63.0698	−3.47715
$330$	0	0
$331$	−11.8618	−0.651982	−0.325991	−	0.945373i	$-0.605698\pi$
$331$	−11.8618	−0.651982	−0.325991	+	0.945373i	$0.605698\pi$
$332$	0	0
$333$	0.193937	0.0106277
$334$	0	0
$335$	−14.7635	−0.806618
$336$	0	0
$337$	−15.5877	−0.849116	−0.424558	−	0.905401i	$-0.639570\pi$
$337$	−15.5877	−0.849116	−0.424558	+	0.905401i	$0.639570\pi$
$338$	0	0
$339$	1.56134	0.0848004
$340$	0	0
$341$	24.9380	1.35047
$342$	0	0
$343$	−34.8119	−1.87967
$344$	0	0
$345$	4.96239	0.267166
$346$	0	0
$347$	0.412311	0.0221340	0.0110670	−	0.999939i	$-0.496477\pi$
$347$	0.412311	0.0221340	0.0110670	+	0.999939i	$0.496477\pi$
$348$	0	0
$349$	−20.2823	−1.08569	−0.542844	−	0.839833i	$-0.682653\pi$
$349$	−20.2823	−1.08569	−0.542844	+	0.839833i	$0.682653\pi$
$350$	0	0
$351$	30.8627	1.64733
$352$	0	0
$353$	−4.70052	−0.250184	−0.125092	−	0.992145i	$-0.539923\pi$
$353$	−4.70052	−0.250184	−0.125092	+	0.992145i	$0.539923\pi$
$354$	0	0
$355$	3.68735	0.195704
$356$	0	0
$357$	50.8383	2.69065
$358$	0	0
$359$	−4.96239	−0.261905	−0.130952	−	0.991389i	$-0.541804\pi$
$359$	−4.96239	−0.261905	−0.130952	+	0.991389i	$0.541804\pi$
$360$	0	0
$361$	−0.295507	−0.0155530
$362$	0	0
$363$	−22.8242	−1.19796
$364$	0	0
$365$	2.96239	0.155059
$366$	0	0
$367$	−11.7259	−0.612088	−0.306044	−	0.952017i	$-0.599005\pi$
$367$	−11.7259	−0.612088	−0.306044	+	0.952017i	$0.599005\pi$
$368$	0	0
$369$	1.03761	0.0540159
$370$	0	0
$371$	25.6775	1.33311
$372$	0	0
$373$	17.0376	0.882174	0.441087	−	0.897464i	$-0.354593\pi$
$373$	17.0376	0.882174	0.441087	+	0.897464i	$0.354593\pi$
$374$	0	0
$375$	1.67513	0.0865034
$376$	0	0
$377$	−26.3879	−1.35904
$378$	0	0
$379$	27.1998	1.39716	0.698580	−	0.715532i	$-0.253815\pi$
$379$	27.1998	1.39716	0.698580	+	0.715532i	$0.253815\pi$
$380$	0	0
$381$	24.5198	1.25619
$382$	0	0
$383$	27.1246	1.38600	0.693001	−	0.720937i	$-0.256288\pi$
$383$	27.1246	1.38600	0.693001	+	0.720937i	$0.256288\pi$
$384$	0	0
$385$	−23.0132	−1.17286
$386$	0	0
$387$	0.111420	0.00566377
$388$	0	0
$389$	−12.4894	−0.633240	−0.316620	−	0.948552i	$-0.602548\pi$
$389$	−12.4894	−0.633240	−0.316620	+	0.948552i	$0.602548\pi$
$390$	0	0
$391$	19.3865	0.980415
$392$	0	0
$393$	−12.2922	−0.620059
$394$	0	0
$395$	−5.98778	−0.301278
$396$	0	0
$397$	30.4142	1.52645	0.763223	−	0.646135i	$-0.223616\pi$
$397$	30.4142	1.52645	0.763223	+	0.646135i	$0.223616\pi$
$398$	0	0
$399$	33.5975	1.68198
$400$	0	0
$401$	14.7757	0.737866	0.368933	−	0.929456i	$-0.379723\pi$
$401$	14.7757	0.737866	0.368933	+	0.929456i	$0.379723\pi$
$402$	0	0
$403$	−28.9887	−1.44403
$404$	0	0
$405$	8.38058	0.416434
$406$	0	0
$407$	4.96239	0.245976
$408$	0	0
$409$	−4.82653	−0.238657	−0.119328	−	0.992855i	$-0.538074\pi$
$409$	−4.82653	−0.238657	−0.119328	+	0.992855i	$0.538074\pi$
$410$	0	0
$411$	10.3634	0.511191
$412$	0	0
$413$	−7.18409	−0.353506
$414$	0	0
$415$	10.6375	0.522175
$416$	0	0
$417$	−16.1016	−0.788497
$418$	0	0
$419$	31.3620	1.53213	0.766067	−	0.642760i	$-0.222211\pi$
$419$	31.3620	1.53213	0.766067	+	0.642760i	$0.222211\pi$
$420$	0	0
$421$	32.6107	1.58935	0.794674	−	0.607036i	$-0.207642\pi$
$421$	32.6107	1.58935	0.794674	+	0.607036i	$0.207642\pi$
$422$	0	0
$423$	−2.63752	−0.128241
$424$	0	0
$425$	6.54420	0.317440
$426$	0	0
$427$	−43.9464	−2.12671
$428$	0	0
$429$	47.9511	2.31510
$430$	0	0
$431$	−26.4264	−1.27292	−0.636458	−	0.771311i	$-0.719601\pi$
$431$	−26.4264	−1.27292	−0.636458	+	0.771311i	$0.719601\pi$
$432$	0	0
$433$	37.9756	1.82499	0.912495	−	0.409089i	$-0.134153\pi$
$433$	37.9756	1.82499	0.912495	+	0.409089i	$0.134153\pi$
$434$	0	0
$435$	−7.66291	−0.367409
$436$	0	0
$437$	12.8119	0.612878
$438$	0	0
$439$	−19.7767	−0.943890	−0.471945	−	0.881628i	$-0.656448\pi$
$439$	−19.7767	−0.943890	−0.471945	+	0.881628i	$0.656448\pi$
$440$	0	0
$441$	−2.81336	−0.133969
$442$	0	0
$443$	−3.92715	−0.186584	−0.0932922	−	0.995639i	$-0.529739\pi$
$443$	−3.92715	−0.186584	−0.0932922	+	0.995639i	$0.529739\pi$
$444$	0	0
$445$	1.16362	0.0551608
$446$	0	0
$447$	−31.9003	−1.50883
$448$	0	0
$449$	16.6761	0.786993	0.393497	−	0.919326i	$-0.371265\pi$
$449$	16.6761	0.786993	0.393497	+	0.919326i	$0.371265\pi$
$450$	0	0
$451$	26.5501	1.25019
$452$	0	0
$453$	−16.0606	−0.754594
$454$	0	0
$455$	26.7513	1.25412
$456$	0	0
$457$	24.3028	1.13684	0.568419	−	0.822740i	$-0.307555\pi$
$457$	24.3028	1.13684	0.568419	+	0.822740i	$0.307555\pi$
$458$	0	0
$459$	35.0132	1.63427
$460$	0	0
$461$	−30.4387	−1.41767	−0.708835	−	0.705375i	$-0.750779\pi$
$461$	−30.4387	−1.41767	−0.708835	+	0.705375i	$0.750779\pi$
$462$	0	0
$463$	−23.7988	−1.10602	−0.553011	−	0.833174i	$-0.686521\pi$
$463$	−23.7988	−1.10602	−0.553011	+	0.833174i	$0.686521\pi$
$464$	0	0
$465$	−8.41819	−0.390384
$466$	0	0
$467$	25.2652	1.16913	0.584567	−	0.811346i	$-0.301264\pi$
$467$	25.2652	1.16913	0.584567	+	0.811346i	$0.301264\pi$
$468$	0	0
$469$	−68.4661	−3.16147
$470$	0	0
$471$	−19.6629	−0.906019
$472$	0	0
$473$	2.85097	0.131088
$474$	0	0
$475$	4.32487	0.198439
$476$	0	0
$477$	1.07381	0.0491663
$478$	0	0
$479$	−16.0122	−0.731617	−0.365808	−	0.930690i	$-0.619207\pi$
$479$	−16.0122	−0.731617	−0.365808	+	0.930690i	$0.619207\pi$
$480$	0	0
$481$	−5.76845	−0.263019
$482$	0	0
$483$	23.0132	1.04714
$484$	0	0
$485$	−2.64974	−0.120318
$486$	0	0
$487$	35.6121	1.61374	0.806870	−	0.590729i	$-0.201160\pi$
$487$	35.6121	1.61374	0.806870	+	0.590729i	$0.201160\pi$
$488$	0	0
$489$	−35.7235	−1.61547
$490$	0	0
$491$	11.2849	0.509280	0.254640	−	0.967036i	$-0.418043\pi$
$491$	11.2849	0.509280	0.254640	+	0.967036i	$0.418043\pi$
$492$	0	0
$493$	−29.9365	−1.34827
$494$	0	0
$495$	−0.962389	−0.0432562
$496$	0	0
$497$	17.1002	0.767047
$498$	0	0
$499$	5.51293	0.246792	0.123396	−	0.992357i	$-0.460621\pi$
$499$	5.51293	0.246792	0.123396	+	0.992357i	$0.460621\pi$
$500$	0	0
$501$	19.9756	0.892442
$502$	0	0
$503$	36.4241	1.62407	0.812035	−	0.583609i	$-0.198360\pi$
$503$	36.4241	1.62407	0.812035	+	0.583609i	$0.198360\pi$
$504$	0	0
$505$	−11.0435	−0.491429
$506$	0	0
$507$	−33.9633	−1.50837
$508$	0	0
$509$	34.4749	1.52807	0.764036	−	0.645174i	$-0.223215\pi$
$509$	34.4749	1.52807	0.764036	+	0.645174i	$0.223215\pi$
$510$	0	0
$511$	13.7381	0.607739
$512$	0	0
$513$	23.1392	1.02162
$514$	0	0
$515$	13.5369	0.596507
$516$	0	0
$517$	−67.4880	−2.96812
$518$	0	0
$519$	0.0850719	0.00373424
$520$	0	0
$521$	14.9829	0.656411	0.328205	−	0.944606i	$-0.393556\pi$
$521$	14.9829	0.656411	0.328205	+	0.944606i	$0.393556\pi$
$522$	0	0
$523$	−26.7513	−1.16975	−0.584877	−	0.811122i	$-0.698857\pi$
$523$	−26.7513	−1.16975	−0.584877	+	0.811122i	$0.698857\pi$
$524$	0	0
$525$	7.76845	0.339043
$526$	0	0
$527$	−32.8872	−1.43259
$528$	0	0
$529$	−14.2243	−0.618446
$530$	0	0
$531$	−0.300432	−0.0130376
$532$	0	0
$533$	−30.8627	−1.33681
$534$	0	0
$535$	−13.7866	−0.596045
$536$	0	0
$537$	5.06793	0.218697
$538$	0	0
$539$	−71.9873	−3.10071
$540$	0	0
$541$	−33.6747	−1.44779	−0.723894	−	0.689912i	$-0.757649\pi$
$541$	−33.6747	−1.44779	−0.723894	+	0.689912i	$0.757649\pi$
$542$	0	0
$543$	12.0508	0.517149
$544$	0	0
$545$	0.387873	0.0166147
$546$	0	0
$547$	30.9135	1.32177	0.660883	−	0.750489i	$-0.270182\pi$
$547$	30.9135	1.32177	0.660883	+	0.750489i	$0.270182\pi$
$548$	0	0
$549$	−1.83780	−0.0784352
$550$	0	0
$551$	−19.7842	−0.842834
$552$	0	0
$553$	−27.7685	−1.18083
$554$	0	0
$555$	−1.67513	−0.0711053
$556$	0	0
$557$	−0.156325	−0.00662371	−0.00331186	−	0.999995i	$-0.501054\pi$
$557$	−0.156325	−0.00662371	−0.00331186	+	0.999995i	$0.501054\pi$
$558$	0	0
$559$	−3.31406	−0.140170
$560$	0	0
$561$	54.3996	2.29675
$562$	0	0
$563$	−22.2276	−0.936781	−0.468390	−	0.883522i	$-0.655166\pi$
$563$	−22.2276	−0.936781	−0.468390	+	0.883522i	$0.655166\pi$
$564$	0	0
$565$	0.932071	0.0392125
$566$	0	0
$567$	38.8651	1.63218
$568$	0	0
$569$	27.0640	1.13458	0.567290	−	0.823518i	$-0.307992\pi$
$569$	27.0640	1.13458	0.567290	+	0.823518i	$0.307992\pi$
$570$	0	0
$571$	34.0870	1.42650	0.713248	−	0.700912i	$-0.247223\pi$
$571$	34.0870	1.42650	0.713248	+	0.700912i	$0.247223\pi$
$572$	0	0
$573$	16.7308	0.698941
$574$	0	0
$575$	2.96239	0.123540
$576$	0	0
$577$	20.8423	0.867675	0.433837	−	0.900991i	$-0.357159\pi$
$577$	20.8423	0.867675	0.433837	+	0.900991i	$0.357159\pi$
$578$	0	0
$579$	−23.1998	−0.964151
$580$	0	0
$581$	49.3317	2.04662
$582$	0	0
$583$	27.4763	1.13795
$584$	0	0
$585$	1.11871	0.0462531
$586$	0	0
$587$	1.56134	0.0644435	0.0322217	−	0.999481i	$-0.489742\pi$
$587$	1.56134	0.0644435	0.0322217	+	0.999481i	$0.489742\pi$
$588$	0	0
$589$	−21.7342	−0.895541
$590$	0	0
$591$	11.1392	0.458205
$592$	0	0
$593$	−27.5633	−1.13189	−0.565944	−	0.824444i	$-0.691488\pi$
$593$	−27.5633	−1.13189	−0.565944	+	0.824444i	$0.691488\pi$
$594$	0	0
$595$	30.3488	1.24418
$596$	0	0
$597$	−17.0679	−0.698544
$598$	0	0
$599$	21.0640	0.860650	0.430325	−	0.902674i	$-0.358399\pi$
$599$	21.0640	0.860650	0.430325	+	0.902674i	$0.358399\pi$
$600$	0	0
$601$	12.1925	0.497343	0.248672	−	0.968588i	$-0.420006\pi$
$601$	12.1925	0.497343	0.248672	+	0.968588i	$0.420006\pi$
$602$	0	0
$603$	−2.86319	−0.116598
$604$	0	0
$605$	−13.6253	−0.553947
$606$	0	0
$607$	−8.72496	−0.354135	−0.177068	−	0.984199i	$-0.556661\pi$
$607$	−8.72496	−0.354135	−0.177068	+	0.984199i	$0.556661\pi$
$608$	0	0
$609$	−35.5369	−1.44003
$610$	0	0
$611$	78.4504	3.17376
$612$	0	0
$613$	−2.25202	−0.0909581	−0.0454790	−	0.998965i	$-0.514481\pi$
$613$	−2.25202	−0.0909581	−0.0454790	+	0.998965i	$0.514481\pi$
$614$	0	0
$615$	−8.96239	−0.361398
$616$	0	0
$617$	20.0263	0.806230	0.403115	−	0.915149i	$-0.367927\pi$
$617$	20.0263	0.806230	0.403115	+	0.915149i	$0.367927\pi$
$618$	0	0
$619$	−19.3258	−0.776770	−0.388385	−	0.921497i	$-0.626967\pi$
$619$	−19.3258	−0.776770	−0.388385	+	0.921497i	$0.626967\pi$
$620$	0	0
$621$	15.8496	0.636021
$622$	0	0
$623$	5.39631	0.216198
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	35.9511	1.43575
$628$	0	0
$629$	−6.54420	−0.260934
$630$	0	0
$631$	−35.6507	−1.41923	−0.709616	−	0.704589i	$-0.751131\pi$
$631$	−35.6507	−1.41923	−0.709616	+	0.704589i	$0.751131\pi$
$632$	0	0
$633$	−22.4485	−0.892248
$634$	0	0
$635$	14.6375	0.580872
$636$	0	0
$637$	83.6806	3.31554
$638$	0	0
$639$	0.715112	0.0282894
$640$	0	0
$641$	−37.9062	−1.49721	−0.748603	−	0.663019i	$-0.769275\pi$
$641$	−37.9062	−1.49721	−0.748603	+	0.663019i	$0.769275\pi$
$642$	0	0
$643$	−43.2506	−1.70564	−0.852819	−	0.522207i	$-0.825109\pi$
$643$	−43.2506	−1.70564	−0.852819	+	0.522207i	$0.825109\pi$
$644$	0	0
$645$	−0.962389	−0.0378940
$646$	0	0
$647$	−32.7367	−1.28701	−0.643507	−	0.765441i	$-0.722521\pi$
$647$	−32.7367	−1.28701	−0.643507	+	0.765441i	$0.722521\pi$
$648$	0	0
$649$	−7.68735	−0.301755
$650$	0	0
$651$	−39.0395	−1.53008
$652$	0	0
$653$	47.3522	1.85303	0.926517	−	0.376253i	$-0.122788\pi$
$653$	47.3522	1.85303	0.926517	+	0.376253i	$0.122788\pi$
$654$	0	0
$655$	−7.33804	−0.286721
$656$	0	0
$657$	0.574515	0.0224140
$658$	0	0
$659$	27.6385	1.07664	0.538321	−	0.842740i	$-0.319059\pi$
$659$	27.6385	1.07664	0.538321	+	0.842740i	$0.319059\pi$
$660$	0	0
$661$	−9.47627	−0.368584	−0.184292	−	0.982872i	$-0.558999\pi$
$661$	−9.47627	−0.368584	−0.184292	+	0.982872i	$0.558999\pi$
$662$	0	0
$663$	−63.2360	−2.45588
$664$	0	0
$665$	20.0567	0.777764
$666$	0	0
$667$	−13.5515	−0.524716
$668$	0	0
$669$	−36.3693	−1.40612
$670$	0	0
$671$	−47.0249	−1.81538
$672$	0	0
$673$	−18.7757	−0.723752	−0.361876	−	0.932226i	$-0.617864\pi$
$673$	−18.7757	−0.723752	−0.361876	+	0.932226i	$0.617864\pi$
$674$	0	0
$675$	5.35026	0.205932
$676$	0	0
$677$	0.513881	0.0197501	0.00987503	−	0.999951i	$-0.496857\pi$
$677$	0.513881	0.0197501	0.00987503	+	0.999951i	$0.496857\pi$
$678$	0	0
$679$	−12.2882	−0.471578
$680$	0	0
$681$	−19.1998	−0.735739
$682$	0	0
$683$	−22.4387	−0.858591	−0.429296	−	0.903164i	$-0.641238\pi$
$683$	−22.4387	−0.858591	−0.429296	+	0.903164i	$0.641238\pi$
$684$	0	0
$685$	6.18664	0.236379
$686$	0	0
$687$	−27.5125	−1.04967
$688$	0	0
$689$	−31.9394	−1.21679
$690$	0	0
$691$	−0.288213	−0.0109641	−0.00548207	−	0.999985i	$-0.501745\pi$
$691$	−0.288213	−0.0109641	−0.00548207	+	0.999985i	$0.501745\pi$
$692$	0	0
$693$	−4.46310	−0.169539
$694$	0	0
$695$	−9.61213	−0.364609
$696$	0	0
$697$	−35.0132	−1.32622
$698$	0	0
$699$	17.0640	0.645418
$700$	0	0
$701$	−0.911603	−0.0344308	−0.0172154	−	0.999852i	$-0.505480\pi$
$701$	−0.911603	−0.0344308	−0.0172154	+	0.999852i	$0.505480\pi$
$702$	0	0
$703$	−4.32487	−0.163116
$704$	0	0
$705$	22.7816	0.858006
$706$	0	0
$707$	−51.2144	−1.92612
$708$	0	0
$709$	−33.3620	−1.25294	−0.626468	−	0.779447i	$-0.715500\pi$
$709$	−33.3620	−1.25294	−0.626468	+	0.779447i	$0.715500\pi$
$710$	0	0
$711$	−1.16125	−0.0435503
$712$	0	0
$713$	−14.8872	−0.557529
$714$	0	0
$715$	28.6253	1.07053
$716$	0	0
$717$	−2.57848	−0.0962953
$718$	0	0
$719$	−6.82653	−0.254587	−0.127293	−	0.991865i	$-0.540629\pi$
$719$	−6.82653	−0.254587	−0.127293	+	0.991865i	$0.540629\pi$
$720$	0	0
$721$	62.7777	2.33796
$722$	0	0
$723$	6.88717	0.256136
$724$	0	0
$725$	−4.57452	−0.169893
$726$	0	0
$727$	−12.4123	−0.460347	−0.230174	−	0.973150i	$-0.573929\pi$
$727$	−12.4123	−0.460347	−0.230174	+	0.973150i	$0.573929\pi$
$728$	0	0
$729$	28.5125	1.05602
$730$	0	0
$731$	−3.75974	−0.139059
$732$	0	0
$733$	−11.0278	−0.407320	−0.203660	−	0.979042i	$-0.565284\pi$
$733$	−11.0278	−0.407320	−0.203660	+	0.979042i	$0.565284\pi$
$734$	0	0
$735$	24.3004	0.896335
$736$	0	0
$737$	−73.2624	−2.69865
$738$	0	0
$739$	−12.2228	−0.449624	−0.224812	−	0.974402i	$-0.572177\pi$
$739$	−12.2228	−0.449624	−0.224812	+	0.974402i	$0.572177\pi$
$740$	0	0
$741$	−41.7908	−1.53522
$742$	0	0
$743$	45.7015	1.67662	0.838312	−	0.545190i	$-0.183543\pi$
$743$	45.7015	1.67662	0.838312	+	0.545190i	$0.183543\pi$
$744$	0	0
$745$	−19.0435	−0.697700
$746$	0	0
$747$	2.06300	0.0754814
$748$	0	0
$749$	−63.9354	−2.33615
$750$	0	0
$751$	−19.8740	−0.725212	−0.362606	−	0.931942i	$-0.618113\pi$
$751$	−19.8740	−0.725212	−0.362606	+	0.931942i	$0.618113\pi$
$752$	0	0
$753$	3.20379	0.116752
$754$	0	0
$755$	−9.58769	−0.348932
$756$	0	0
$757$	20.8021	0.756065	0.378032	−	0.925792i	$-0.376601\pi$
$757$	20.8021	0.756065	0.378032	+	0.925792i	$0.376601\pi$
$758$	0	0
$759$	24.6253	0.893842
$760$	0	0
$761$	−2.15045	−0.0779536	−0.0389768	−	0.999240i	$-0.512410\pi$
$761$	−2.15045	−0.0779536	−0.0389768	+	0.999240i	$0.512410\pi$
$762$	0	0
$763$	1.79877	0.0651198
$764$	0	0
$765$	1.26916	0.0458866
$766$	0	0
$767$	8.93604	0.322662
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−23.0640	−0.830628
$772$	0	0
$773$	−34.4993	−1.24085	−0.620427	−	0.784264i	$-0.713041\pi$
$773$	−34.4993	−1.24085	−0.620427	+	0.784264i	$0.713041\pi$
$774$	0	0
$775$	−5.02539	−0.180517
$776$	0	0
$777$	−7.76845	−0.278692
$778$	0	0
$779$	−23.1392	−0.829047
$780$	0	0
$781$	18.2981	0.654756
$782$	0	0
$783$	−24.4749	−0.874660
$784$	0	0
$785$	−11.7381	−0.418952
$786$	0	0
$787$	4.17442	0.148802	0.0744011	−	0.997228i	$-0.476296\pi$
$787$	4.17442	0.148802	0.0744011	+	0.997228i	$0.476296\pi$
$788$	0	0
$789$	−14.9076	−0.530726
$790$	0	0
$791$	4.32250	0.153690
$792$	0	0
$793$	54.6634	1.94115
$794$	0	0
$795$	−9.27504	−0.328952
$796$	0	0
$797$	−26.9584	−0.954916	−0.477458	−	0.878655i	$-0.658442\pi$
$797$	−26.9584	−0.954916	−0.477458	+	0.878655i	$0.658442\pi$
$798$	0	0
$799$	89.0005	3.14861
$800$	0	0
$801$	0.225668	0.00797360
$802$	0	0
$803$	14.7005	0.518770
$804$	0	0
$805$	13.7381	0.484206
$806$	0	0
$807$	38.0771	1.34038
$808$	0	0
$809$	−6.15045	−0.216238	−0.108119	−	0.994138i	$-0.534483\pi$
$809$	−6.15045	−0.216238	−0.108119	+	0.994138i	$0.534483\pi$
$810$	0	0
$811$	2.70052	0.0948282	0.0474141	−	0.998875i	$-0.484902\pi$
$811$	2.70052	0.0948282	0.0474141	+	0.998875i	$0.484902\pi$
$812$	0	0
$813$	40.7269	1.42835
$814$	0	0
$815$	−21.3258	−0.747011
$816$	0	0
$817$	−2.48470	−0.0869288
$818$	0	0
$819$	5.18806	0.181285
$820$	0	0
$821$	−30.6253	−1.06883	−0.534415	−	0.845222i	$-0.679468\pi$
$821$	−30.6253	−1.06883	−0.534415	+	0.845222i	$0.679468\pi$
$822$	0	0
$823$	16.7997	0.585602	0.292801	−	0.956173i	$-0.405413\pi$
$823$	16.7997	0.585602	0.292801	+	0.956173i	$0.405413\pi$
$824$	0	0
$825$	8.31265	0.289409
$826$	0	0
$827$	−10.3127	−0.358606	−0.179303	−	0.983794i	$-0.557384\pi$
$827$	−10.3127	−0.358606	−0.179303	+	0.983794i	$0.557384\pi$
$828$	0	0
$829$	−27.6531	−0.960431	−0.480215	−	0.877151i	$-0.659441\pi$
$829$	−27.6531	−0.960431	−0.480215	+	0.877151i	$0.659441\pi$
$830$	0	0
$831$	−43.1655	−1.49740
$832$	0	0
$833$	94.9340	3.28927
$834$	0	0
$835$	11.9248	0.412674
$836$	0	0
$837$	−26.8872	−0.929356
$838$	0	0
$839$	8.37328	0.289078	0.144539	−	0.989499i	$-0.453830\pi$
$839$	8.37328	0.289078	0.144539	+	0.989499i	$0.453830\pi$
$840$	0	0
$841$	−8.07381	−0.278407
$842$	0	0
$843$	45.9610	1.58298
$844$	0	0
$845$	−20.2750	−0.697483
$846$	0	0
$847$	−63.1876	−2.17115
$848$	0	0
$849$	−42.5256	−1.45948
$850$	0	0
$851$	−2.96239	−0.101549
$852$	0	0
$853$	−42.8726	−1.46793	−0.733965	−	0.679188i	$-0.762332\pi$
$853$	−42.8726	−1.46793	−0.733965	+	0.679188i	$0.762332\pi$
$854$	0	0
$855$	0.838750	0.0286847
$856$	0	0
$857$	18.0244	0.615703	0.307852	−	0.951434i	$-0.400390\pi$
$857$	18.0244	0.615703	0.307852	+	0.951434i	$0.400390\pi$
$858$	0	0
$859$	50.4382	1.72093	0.860465	−	0.509510i	$-0.170173\pi$
$859$	50.4382	1.72093	0.860465	+	0.509510i	$0.170173\pi$
$860$	0	0
$861$	−41.5633	−1.41647
$862$	0	0
$863$	−29.5247	−1.00503	−0.502516	−	0.864568i	$-0.667592\pi$
$863$	−29.5247	−1.00503	−0.502516	+	0.864568i	$0.667592\pi$
$864$	0	0
$865$	0.0507852	0.00172675
$866$	0	0
$867$	−43.2628	−1.46928
$868$	0	0
$869$	−29.7137	−1.00797
$870$	0	0
$871$	85.1627	2.88563
$872$	0	0
$873$	−0.513881	−0.0173922
$874$	0	0
$875$	4.63752	0.156777
$876$	0	0
$877$	−40.4650	−1.36641	−0.683203	−	0.730228i	$-0.739414\pi$
$877$	−40.4650	−1.36641	−0.683203	+	0.730228i	$0.739414\pi$
$878$	0	0
$879$	12.3977	0.418165
$880$	0	0
$881$	−5.64244	−0.190099	−0.0950494	−	0.995473i	$-0.530301\pi$
$881$	−5.64244	−0.190099	−0.0950494	+	0.995473i	$0.530301\pi$
$882$	0	0
$883$	−21.1344	−0.711231	−0.355615	−	0.934632i	$-0.615729\pi$
$883$	−21.1344	−0.711231	−0.355615	+	0.934632i	$0.615729\pi$
$884$	0	0
$885$	2.59498	0.0872294
$886$	0	0
$887$	16.6375	0.558633	0.279317	−	0.960199i	$-0.409892\pi$
$887$	16.6375	0.558633	0.279317	+	0.960199i	$0.409892\pi$
$888$	0	0
$889$	67.8818	2.27668
$890$	0	0
$891$	41.5877	1.39324
$892$	0	0
$893$	58.8178	1.96826
$894$	0	0
$895$	3.02539	0.101128
$896$	0	0
$897$	−28.6253	−0.955771
$898$	0	0
$899$	22.9887	0.766717
$900$	0	0
$901$	−36.2346	−1.20715
$902$	0	0
$903$	−4.46310	−0.148523
$904$	0	0
$905$	7.19394	0.239135
$906$	0	0
$907$	25.3865	0.842944	0.421472	−	0.906842i	$-0.361514\pi$
$907$	25.3865	0.842944	0.421472	+	0.906842i	$0.361514\pi$
$908$	0	0
$909$	−2.14174	−0.0710369
$910$	0	0
$911$	23.5755	0.781090	0.390545	−	0.920584i	$-0.372287\pi$
$911$	23.5755	0.781090	0.390545	+	0.920584i	$0.372287\pi$
$912$	0	0
$913$	52.7875	1.74701
$914$	0	0
$915$	15.8740	0.524778
$916$	0	0
$917$	−34.0303	−1.12378
$918$	0	0
$919$	−28.4969	−0.940027	−0.470013	−	0.882659i	$-0.655751\pi$
$919$	−28.4969	−0.940027	−0.470013	+	0.882659i	$0.655751\pi$
$920$	0	0
$921$	8.29218	0.273237
$922$	0	0
$923$	−21.2703	−0.700120
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	2.62530	0.0862262
$928$	0	0
$929$	42.8529	1.40596	0.702979	−	0.711211i	$-0.251853\pi$
$929$	42.8529	1.40596	0.702979	+	0.711211i	$0.251853\pi$
$930$	0	0
$931$	62.7391	2.05619
$932$	0	0
$933$	35.7440	1.17021
$934$	0	0
$935$	32.4749	1.06204
$936$	0	0
$937$	29.3211	0.957878	0.478939	−	0.877848i	$-0.341021\pi$
$937$	29.3211	0.957878	0.478939	+	0.877848i	$0.341021\pi$
$938$	0	0
$939$	−26.4650	−0.863653
$940$	0	0
$941$	−38.8930	−1.26788	−0.633939	−	0.773383i	$-0.718563\pi$
$941$	−38.8930	−1.26788	−0.633939	+	0.773383i	$0.718563\pi$
$942$	0	0
$943$	−15.8496	−0.516133
$944$	0	0
$945$	24.8119	0.807133
$946$	0	0
$947$	18.4142	0.598382	0.299191	−	0.954193i	$-0.403283\pi$
$947$	18.4142	0.598382	0.299191	+	0.954193i	$0.403283\pi$
$948$	0	0
$949$	−17.0884	−0.554713
$950$	0	0
$951$	59.3620	1.92495
$952$	0	0
$953$	−36.5012	−1.18239	−0.591195	−	0.806529i	$-0.701344\pi$
$953$	−36.5012	−1.18239	−0.591195	+	0.806529i	$0.701344\pi$
$954$	0	0
$955$	9.98778	0.323197
$956$	0	0
$957$	−38.0263	−1.22922
$958$	0	0
$959$	28.6907	0.926470
$960$	0	0
$961$	−5.74543	−0.185336
$962$	0	0
$963$	−2.67372	−0.0861593
$964$	0	0
$965$	−13.8496	−0.445833
$966$	0	0
$967$	3.56325	0.114586	0.0572932	−	0.998357i	$-0.481753\pi$
$967$	3.56325	0.114586	0.0572932	+	0.998357i	$0.481753\pi$
$968$	0	0
$969$	−47.4109	−1.52306
$970$	0	0
$971$	27.2046	0.873036	0.436518	−	0.899696i	$-0.356212\pi$
$971$	27.2046	0.873036	0.436518	+	0.899696i	$0.356212\pi$
$972$	0	0
$973$	−44.5764	−1.42905
$974$	0	0
$975$	−9.66291	−0.309461
$976$	0	0
$977$	−18.7513	−0.599908	−0.299954	−	0.953954i	$-0.596971\pi$
$977$	−18.7513	−0.599908	−0.299954	+	0.953954i	$0.596971\pi$
$978$	0	0
$979$	5.77433	0.184548
$980$	0	0
$981$	0.0752228	0.00240168
$982$	0	0
$983$	26.8242	0.855558	0.427779	−	0.903883i	$-0.359296\pi$
$983$	26.8242	0.855558	0.427779	+	0.903883i	$0.359296\pi$
$984$	0	0
$985$	6.64974	0.211878
$986$	0	0
$987$	105.650	3.36288
$988$	0	0
$989$	−1.70194	−0.0541185
$990$	0	0
$991$	−1.53453	−0.0487461	−0.0243730	−	0.999703i	$-0.507759\pi$
$991$	−1.53453	−0.0487461	−0.0243730	+	0.999703i	$0.507759\pi$
$992$	0	0
$993$	19.8700	0.630556
$994$	0	0
$995$	−10.1890	−0.323013
$996$	0	0
$997$	−31.2506	−0.989716	−0.494858	−	0.868974i	$-0.664780\pi$
$997$	−31.2506	−0.989716	−0.494858	+	0.868974i	$0.664780\pi$
$998$	0	0
$999$	−5.35026	−0.169275

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	740.2.a.e.1.1	✓	3
3.2	odd	2		6660.2.a.q.1.1		3
4.3	odd	2		2960.2.a.t.1.3		3
5.2	odd	4		3700.2.d.h.149.5		6
5.3	odd	4		3700.2.d.h.149.2		6
5.4	even	2		3700.2.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.e.1.1	✓	3	1.1	even	1	trivial
2960.2.a.t.1.3		3	4.3	odd	2
3700.2.a.i.1.3		3	5.4	even	2
3700.2.d.h.149.2		6	5.3	odd	4
3700.2.d.h.149.5		6	5.2	odd	4
6660.2.a.q.1.1		3	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:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	740.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67513 q^{3} -1.00000 q^{5} -4.63752 q^{7} -0.193937 q^{9} -4.96239 q^{11} +5.76845 q^{13} +1.67513 q^{15} +6.54420 q^{17} +4.32487 q^{19} +7.76845 q^{21} +2.96239 q^{23} +1.00000 q^{25} +5.35026 q^{27} -4.57452 q^{29} -5.02539 q^{31} +8.31265 q^{33} +4.63752 q^{35} -1.00000 q^{37} -9.66291 q^{39} -5.35026 q^{41} -0.574515 q^{43} +0.193937 q^{45} +13.5999 q^{47} +14.5066 q^{49} -10.9624 q^{51} -5.53690 q^{53} +4.96239 q^{55} -7.24472 q^{57} +1.54912 q^{59} +9.47627 q^{61} +0.899385 q^{63} -5.76845 q^{65} +14.7635 q^{67} -4.96239 q^{69} -3.68735 q^{71} -2.96239 q^{73} -1.67513 q^{75} +23.0132 q^{77} +5.98778 q^{79} -8.38058 q^{81} -10.6375 q^{83} -6.54420 q^{85} +7.66291 q^{87} -1.16362 q^{89} -26.7513 q^{91} +8.41819 q^{93} -4.32487 q^{95} +2.64974 q^{97} +0.962389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7} - q^{9} - 4 q^{11} + 6 q^{13} + 10 q^{17} + 18 q^{19} + 12 q^{21} - 2 q^{23} + 3 q^{25} + 6 q^{27} - 2 q^{29} + 4 q^{33} - 2 q^{35} - 3 q^{37} + 2 q^{39} - 6 q^{41} + 10 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - q^9 - 4 * q^11 + 6 * q^13 + 10 * q^17 + 18 * q^19 + 12 * q^21 - 2 * q^23 + 3 * q^25 + 6 * q^27 - 2 * q^29 + 4 * q^33 - 2 * q^35 - 3 * q^37 + 2 * q^39 - 6 * q^41 + 10 * q^43 + q^45 + 14 * q^47 + 23 * q^49 - 22 * q^51 + 6 * q^53 + 4 * q^55 + 8 * q^57 + 8 * q^59 + 10 * q^61 - 4 * q^63 - 6 * q^65 + 20 * q^67 - 4 * q^69 - 32 * q^71 + 2 * q^73 + 28 * q^77 - 8 * q^79 - 13 * q^81 - 16 * q^83 - 10 * q^85 - 8 * q^87 - 6 * q^89 - 30 * q^91 + 24 * q^93 - 18 * q^95 + 18 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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