LMFDB - Embedded newform 1040.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.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:	$8.30444181021$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.65685 q^{9} +O(q^{10})$	$q+0.585786 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.65685 q^{9} +4.24264 q^{11} -1.00000 q^{13} +0.585786 q^{15} +0.828427 q^{17} -0.242641 q^{19} +1.17157 q^{21} +9.07107 q^{23} +1.00000 q^{25} -3.31371 q^{27} +1.65685 q^{29} -1.41421 q^{31} +2.48528 q^{33} +2.00000 q^{35} -6.82843 q^{37} -0.585786 q^{39} +4.82843 q^{41} +10.2426 q^{43} -2.65685 q^{45} -2.00000 q^{47} -3.00000 q^{49} +0.485281 q^{51} -8.82843 q^{53} +4.24264 q^{55} -0.142136 q^{57} +2.58579 q^{59} +15.3137 q^{61} -5.31371 q^{63} -1.00000 q^{65} +4.82843 q^{67} +5.31371 q^{69} -9.89949 q^{71} +1.17157 q^{73} +0.585786 q^{75} +8.48528 q^{77} -1.17157 q^{79} +6.02944 q^{81} +2.00000 q^{83} +0.828427 q^{85} +0.970563 q^{87} -10.0000 q^{89} -2.00000 q^{91} -0.828427 q^{93} -0.242641 q^{95} +11.6569 q^{97} -11.2721 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9	$ 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9} - 2 q^{13} + 4 q^{15} - 4 q^{17} + 8 q^{19} + 8 q^{21} + 4 q^{23} + 2 q^{25} + 16 q^{27} - 8 q^{29} - 12 q^{33} + 4 q^{35} - 8 q^{37} - 4 q^{39} + 4 q^{41} + 12 q^{43} + 6 q^{45} - 4 q^{47} - 6 q^{49} - 16 q^{51} - 12 q^{53} + 28 q^{57} + 8 q^{59} + 8 q^{61} + 12 q^{63} - 2 q^{65} + 4 q^{67} - 12 q^{69} + 8 q^{73} + 4 q^{75} - 8 q^{79} + 46 q^{81} + 4 q^{83} - 4 q^{85} - 32 q^{87} - 20 q^{89} - 4 q^{91} + 4 q^{93} + 8 q^{95} + 12 q^{97} - 48 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9 - 2 * q^13 + 4 * q^15 - 4 * q^17 + 8 * q^19 + 8 * q^21 + 4 * q^23 + 2 * q^25 + 16 * q^27 - 8 * q^29 - 12 * q^33 + 4 * q^35 - 8 * q^37 - 4 * q^39 + 4 * q^41 + 12 * q^43 + 6 * q^45 - 4 * q^47 - 6 * q^49 - 16 * q^51 - 12 * q^53 + 28 * q^57 + 8 * q^59 + 8 * q^61 + 12 * q^63 - 2 * q^65 + 4 * q^67 - 12 * q^69 + 8 * q^73 + 4 * q^75 - 8 * q^79 + 46 * q^81 + 4 * q^83 - 4 * q^85 - 32 * q^87 - 20 * q^89 - 4 * q^91 + 4 * q^93 + 8 * q^95 + 12 * q^97 - 48 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.585786	0.338204	0.169102	−	0.985599i	$-0.445913\pi$
$3$	0.585786	0.338204	0.169102	+	0.985599i	$0.445913\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.65685	−0.885618
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.585786	0.151249
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	−0.242641	−0.0556656	−0.0278328	−	0.999613i	$-0.508861\pi$
$19$	−0.242641	−0.0556656	−0.0278328	+	0.999613i	$0.508861\pi$
$20$	0	0
$21$	1.17157	0.255658
$22$	0	0
$23$	9.07107	1.89145	0.945724	−	0.324970i	$-0.105354\pi$
$23$	9.07107	1.89145	0.945724	+	0.324970i	$0.105354\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.31371	−0.637723
$28$	0	0
$29$	1.65685	0.307670	0.153835	−	0.988097i	$-0.450838\pi$
$29$	1.65685	0.307670	0.153835	+	0.988097i	$0.450838\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	0	0
$33$	2.48528	0.432632
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−6.82843	−1.12259	−0.561293	−	0.827617i	$-0.689696\pi$
$37$	−6.82843	−1.12259	−0.561293	+	0.827617i	$0.689696\pi$
$38$	0	0
$39$	−0.585786	−0.0938009
$40$	0	0
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	0	0
$43$	10.2426	1.56199	0.780994	−	0.624538i	$-0.214713\pi$
$43$	10.2426	1.56199	0.780994	+	0.624538i	$0.214713\pi$
$44$	0	0
$45$	−2.65685	−0.396060
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0.485281	0.0679530
$52$	0	0
$53$	−8.82843	−1.21268	−0.606339	−	0.795206i	$-0.707362\pi$
$53$	−8.82843	−1.21268	−0.606339	+	0.795206i	$0.707362\pi$
$54$	0	0
$55$	4.24264	0.572078
$56$	0	0
$57$	−0.142136	−0.0188263
$58$	0	0
$59$	2.58579	0.336641	0.168320	−	0.985732i	$-0.446166\pi$
$59$	2.58579	0.336641	0.168320	+	0.985732i	$0.446166\pi$
$60$	0	0
$61$	15.3137	1.96072	0.980360	−	0.197218i	$-0.0631906\pi$
$61$	15.3137	1.96072	0.980360	+	0.197218i	$0.0631906\pi$
$62$	0	0
$63$	−5.31371	−0.669464
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	4.82843	0.589886	0.294943	−	0.955515i	$-0.404699\pi$
$67$	4.82843	0.589886	0.294943	+	0.955515i	$0.404699\pi$
$68$	0	0
$69$	5.31371	0.639695
$70$	0	0
$71$	−9.89949	−1.17485	−0.587427	−	0.809277i	$-0.699859\pi$
$71$	−9.89949	−1.17485	−0.587427	+	0.809277i	$0.699859\pi$
$72$	0	0
$73$	1.17157	0.137122	0.0685611	−	0.997647i	$-0.478159\pi$
$73$	1.17157	0.137122	0.0685611	+	0.997647i	$0.478159\pi$
$74$	0	0
$75$	0.585786	0.0676408
$76$	0	0
$77$	8.48528	0.966988
$78$	0	0
$79$	−1.17157	−0.131812	−0.0659061	−	0.997826i	$-0.520994\pi$
$79$	−1.17157	−0.131812	−0.0659061	+	0.997826i	$0.520994\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	0.828427	0.0898555
$86$	0	0
$87$	0.970563	0.104055
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−0.828427	−0.0859039
$94$	0	0
$95$	−0.242641	−0.0248944
$96$	0	0
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	0	0
$99$	−11.2721	−1.13289
$100$	0	0
$101$	−3.65685	−0.363871	−0.181935	−	0.983311i	$-0.558236\pi$
$101$	−3.65685	−0.363871	−0.181935	+	0.983311i	$0.558236\pi$
$102$	0	0
$103$	−11.4142	−1.12468	−0.562338	−	0.826908i	$-0.690098\pi$
$103$	−11.4142	−1.12468	−0.562338	+	0.826908i	$0.690098\pi$
$104$	0	0
$105$	1.17157	0.114334
$106$	0	0
$107$	−5.07107	−0.490239	−0.245119	−	0.969493i	$-0.578827\pi$
$107$	−5.07107	−0.490239	−0.245119	+	0.969493i	$0.578827\pi$
$108$	0	0
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−8.82843	−0.830509	−0.415254	−	0.909705i	$-0.636307\pi$
$113$	−8.82843	−0.830509	−0.415254	+	0.909705i	$0.636307\pi$
$114$	0	0
$115$	9.07107	0.845881
$116$	0	0
$117$	2.65685	0.245626
$118$	0	0
$119$	1.65685	0.151884
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	2.82843	0.255031
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.8995	−1.41085	−0.705426	−	0.708784i	$-0.749244\pi$
$127$	−15.8995	−1.41085	−0.705426	+	0.708784i	$0.749244\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	19.3137	1.68745	0.843723	−	0.536778i	$-0.180359\pi$
$131$	19.3137	1.68745	0.843723	+	0.536778i	$0.180359\pi$
$132$	0	0
$133$	−0.485281	−0.0420792
$134$	0	0
$135$	−3.31371	−0.285199
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	8.48528	0.719712	0.359856	−	0.933008i	$-0.382826\pi$
$139$	8.48528	0.719712	0.359856	+	0.933008i	$0.382826\pi$
$140$	0	0
$141$	−1.17157	−0.0986642
$142$	0	0
$143$	−4.24264	−0.354787
$144$	0	0
$145$	1.65685	0.137594
$146$	0	0
$147$	−1.75736	−0.144945
$148$	0	0
$149$	−22.9706	−1.88182	−0.940911	−	0.338654i	$-0.890028\pi$
$149$	−22.9706	−1.88182	−0.940911	+	0.338654i	$0.890028\pi$
$150$	0	0
$151$	−15.0711	−1.22647	−0.613233	−	0.789902i	$-0.710131\pi$
$151$	−15.0711	−1.22647	−0.613233	+	0.789902i	$0.710131\pi$
$152$	0	0
$153$	−2.20101	−0.177941
$154$	0	0
$155$	−1.41421	−0.113592
$156$	0	0
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	0	0
$159$	−5.17157	−0.410132
$160$	0	0
$161$	18.1421	1.42980
$162$	0	0
$163$	5.51472	0.431946	0.215973	−	0.976399i	$-0.430708\pi$
$163$	5.51472	0.431946	0.215973	+	0.976399i	$0.430708\pi$
$164$	0	0
$165$	2.48528	0.193479
$166$	0	0
$167$	−9.31371	−0.720716	−0.360358	−	0.932814i	$-0.617346\pi$
$167$	−9.31371	−0.720716	−0.360358	+	0.932814i	$0.617346\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.644661	0.0492985
$172$	0	0
$173$	−15.1716	−1.15347	−0.576737	−	0.816930i	$-0.695674\pi$
$173$	−15.1716	−1.15347	−0.576737	+	0.816930i	$0.695674\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	1.51472	0.113853
$178$	0	0
$179$	−11.3137	−0.845626	−0.422813	−	0.906217i	$-0.638957\pi$
$179$	−11.3137	−0.845626	−0.422813	+	0.906217i	$0.638957\pi$
$180$	0	0
$181$	0.686292	0.0510116	0.0255058	−	0.999675i	$-0.491880\pi$
$181$	0.686292	0.0510116	0.0255058	+	0.999675i	$0.491880\pi$
$182$	0	0
$183$	8.97056	0.663123
$184$	0	0
$185$	−6.82843	−0.502036
$186$	0	0
$187$	3.51472	0.257022
$188$	0	0
$189$	−6.62742	−0.482074
$190$	0	0
$191$	10.3431	0.748404	0.374202	−	0.927347i	$-0.377917\pi$
$191$	10.3431	0.748404	0.374202	+	0.927347i	$0.377917\pi$
$192$	0	0
$193$	−19.6569	−1.41493	−0.707466	−	0.706748i	$-0.750162\pi$
$193$	−19.6569	−1.41493	−0.707466	+	0.706748i	$0.750162\pi$
$194$	0	0
$195$	−0.585786	−0.0419490
$196$	0	0
$197$	−11.6569	−0.830516	−0.415258	−	0.909704i	$-0.636309\pi$
$197$	−11.6569	−0.830516	−0.415258	+	0.909704i	$0.636309\pi$
$198$	0	0
$199$	−17.6569	−1.25166	−0.625831	−	0.779959i	$-0.715240\pi$
$199$	−17.6569	−1.25166	−0.625831	+	0.779959i	$0.715240\pi$
$200$	0	0
$201$	2.82843	0.199502
$202$	0	0
$203$	3.31371	0.232577
$204$	0	0
$205$	4.82843	0.337232
$206$	0	0
$207$	−24.1005	−1.67510
$208$	0	0
$209$	−1.02944	−0.0712077
$210$	0	0
$211$	5.65685	0.389434	0.194717	−	0.980859i	$-0.437621\pi$
$211$	5.65685	0.389434	0.194717	+	0.980859i	$0.437621\pi$
$212$	0	0
$213$	−5.79899	−0.397340
$214$	0	0
$215$	10.2426	0.698542
$216$	0	0
$217$	−2.82843	−0.192006
$218$	0	0
$219$	0.686292	0.0463753
$220$	0	0
$221$	−0.828427	−0.0557260
$222$	0	0
$223$	−2.68629	−0.179887	−0.0899437	−	0.995947i	$-0.528669\pi$
$223$	−2.68629	−0.179887	−0.0899437	+	0.995947i	$0.528669\pi$
$224$	0	0
$225$	−2.65685	−0.177124
$226$	0	0
$227$	−15.1716	−1.00697	−0.503486	−	0.864003i	$-0.667950\pi$
$227$	−15.1716	−1.00697	−0.503486	+	0.864003i	$0.667950\pi$
$228$	0	0
$229$	25.7990	1.70485	0.852423	−	0.522853i	$-0.175132\pi$
$229$	25.7990	1.70485	0.852423	+	0.522853i	$0.175132\pi$
$230$	0	0
$231$	4.97056	0.327039
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	−0.686292	−0.0445794
$238$	0	0
$239$	16.7279	1.08204	0.541020	−	0.841010i	$-0.318038\pi$
$239$	16.7279	1.08204	0.541020	+	0.841010i	$0.318038\pi$
$240$	0	0
$241$	12.8284	0.826352	0.413176	−	0.910651i	$-0.364419\pi$
$241$	12.8284	0.826352	0.413176	+	0.910651i	$0.364419\pi$
$242$	0	0
$243$	13.4731	0.864299
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0.242641	0.0154389
$248$	0	0
$249$	1.17157	0.0742454
$250$	0	0
$251$	18.1421	1.14512	0.572561	−	0.819862i	$-0.305950\pi$
$251$	18.1421	1.14512	0.572561	+	0.819862i	$0.305950\pi$
$252$	0	0
$253$	38.4853	2.41955
$254$	0	0
$255$	0.485281	0.0303895
$256$	0	0
$257$	−18.9706	−1.18335	−0.591676	−	0.806176i	$-0.701533\pi$
$257$	−18.9706	−1.18335	−0.591676	+	0.806176i	$0.701533\pi$
$258$	0	0
$259$	−13.6569	−0.848596
$260$	0	0
$261$	−4.40202	−0.272478
$262$	0	0
$263$	−22.2426	−1.37154	−0.685770	−	0.727818i	$-0.740534\pi$
$263$	−22.2426	−1.37154	−0.685770	+	0.727818i	$0.740534\pi$
$264$	0	0
$265$	−8.82843	−0.542326
$266$	0	0
$267$	−5.85786	−0.358495
$268$	0	0
$269$	−28.6274	−1.74544	−0.872722	−	0.488217i	$-0.837647\pi$
$269$	−28.6274	−1.74544	−0.872722	+	0.488217i	$0.837647\pi$
$270$	0	0
$271$	9.89949	0.601351	0.300676	−	0.953726i	$-0.402788\pi$
$271$	9.89949	0.601351	0.300676	+	0.953726i	$0.402788\pi$
$272$	0	0
$273$	−1.17157	−0.0709068
$274$	0	0
$275$	4.24264	0.255841
$276$	0	0
$277$	−24.1421	−1.45056	−0.725280	−	0.688454i	$-0.758290\pi$
$277$	−24.1421	−1.45056	−0.725280	+	0.688454i	$0.758290\pi$
$278$	0	0
$279$	3.75736	0.224947
$280$	0	0
$281$	−5.51472	−0.328981	−0.164490	−	0.986379i	$-0.552598\pi$
$281$	−5.51472	−0.328981	−0.164490	+	0.986379i	$0.552598\pi$
$282$	0	0
$283$	15.8995	0.945127	0.472563	−	0.881297i	$-0.343329\pi$
$283$	15.8995	0.945127	0.472563	+	0.881297i	$0.343329\pi$
$284$	0	0
$285$	−0.142136	−0.00841939
$286$	0	0
$287$	9.65685	0.570026
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	6.82843	0.400289
$292$	0	0
$293$	−2.82843	−0.165238	−0.0826192	−	0.996581i	$-0.526329\pi$
$293$	−2.82843	−0.165238	−0.0826192	+	0.996581i	$0.526329\pi$
$294$	0	0
$295$	2.58579	0.150550
$296$	0	0
$297$	−14.0589	−0.815779
$298$	0	0
$299$	−9.07107	−0.524593
$300$	0	0
$301$	20.4853	1.18075
$302$	0	0
$303$	−2.14214	−0.123062
$304$	0	0
$305$	15.3137	0.876860
$306$	0	0
$307$	−4.34315	−0.247876	−0.123938	−	0.992290i	$-0.539552\pi$
$307$	−4.34315	−0.247876	−0.123938	+	0.992290i	$0.539552\pi$
$308$	0	0
$309$	−6.68629	−0.380370
$310$	0	0
$311$	−18.1421	−1.02875	−0.514373	−	0.857567i	$-0.671975\pi$
$311$	−18.1421	−1.02875	−0.514373	+	0.857567i	$0.671975\pi$
$312$	0	0
$313$	−12.8284	−0.725106	−0.362553	−	0.931963i	$-0.618095\pi$
$313$	−12.8284	−0.725106	−0.362553	+	0.931963i	$0.618095\pi$
$314$	0	0
$315$	−5.31371	−0.299394
$316$	0	0
$317$	7.51472	0.422069	0.211034	−	0.977479i	$-0.432317\pi$
$317$	7.51472	0.422069	0.211034	+	0.977479i	$0.432317\pi$
$318$	0	0
$319$	7.02944	0.393573
$320$	0	0
$321$	−2.97056	−0.165801
$322$	0	0
$323$	−0.201010	−0.0111845
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	10.1421	0.560861
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	15.0711	0.828381	0.414190	−	0.910190i	$-0.364065\pi$
$331$	15.0711	0.828381	0.414190	+	0.910190i	$0.364065\pi$
$332$	0	0
$333$	18.1421	0.994183
$334$	0	0
$335$	4.82843	0.263805
$336$	0	0
$337$	−7.85786	−0.428045	−0.214023	−	0.976829i	$-0.568657\pi$
$337$	−7.85786	−0.428045	−0.214023	+	0.976829i	$0.568657\pi$
$338$	0	0
$339$	−5.17157	−0.280881
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	5.31371	0.286080
$346$	0	0
$347$	32.3848	1.73851	0.869253	−	0.494368i	$-0.164600\pi$
$347$	32.3848	1.73851	0.869253	+	0.494368i	$0.164600\pi$
$348$	0	0
$349$	−15.4558	−0.827332	−0.413666	−	0.910429i	$-0.635752\pi$
$349$	−15.4558	−0.827332	−0.413666	+	0.910429i	$0.635752\pi$
$350$	0	0
$351$	3.31371	0.176873
$352$	0	0
$353$	10.8284	0.576339	0.288170	−	0.957579i	$-0.406953\pi$
$353$	10.8284	0.576339	0.288170	+	0.957579i	$0.406953\pi$
$354$	0	0
$355$	−9.89949	−0.525411
$356$	0	0
$357$	0.970563	0.0513676
$358$	0	0
$359$	−3.55635	−0.187697	−0.0938485	−	0.995586i	$-0.529917\pi$
$359$	−3.55635	−0.187697	−0.0938485	+	0.995586i	$0.529917\pi$
$360$	0	0
$361$	−18.9411	−0.996901
$362$	0	0
$363$	4.10051	0.215221
$364$	0	0
$365$	1.17157	0.0613229
$366$	0	0
$367$	−2.72792	−0.142396	−0.0711982	−	0.997462i	$-0.522682\pi$
$367$	−2.72792	−0.142396	−0.0711982	+	0.997462i	$0.522682\pi$
$368$	0	0
$369$	−12.8284	−0.667821
$370$	0	0
$371$	−17.6569	−0.916698
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0.585786	0.0302499
$376$	0	0
$377$	−1.65685	−0.0853323
$378$	0	0
$379$	−7.75736	−0.398469	−0.199234	−	0.979952i	$-0.563846\pi$
$379$	−7.75736	−0.398469	−0.199234	+	0.979952i	$0.563846\pi$
$380$	0	0
$381$	−9.31371	−0.477156
$382$	0	0
$383$	−22.0000	−1.12415	−0.562074	−	0.827087i	$-0.689996\pi$
$383$	−22.0000	−1.12415	−0.562074	+	0.827087i	$0.689996\pi$
$384$	0	0
$385$	8.48528	0.432450
$386$	0	0
$387$	−27.2132	−1.38332
$388$	0	0
$389$	9.31371	0.472224	0.236112	−	0.971726i	$-0.424127\pi$
$389$	9.31371	0.472224	0.236112	+	0.971726i	$0.424127\pi$
$390$	0	0
$391$	7.51472	0.380036
$392$	0	0
$393$	11.3137	0.570701
$394$	0	0
$395$	−1.17157	−0.0589482
$396$	0	0
$397$	6.82843	0.342709	0.171354	−	0.985209i	$-0.445186\pi$
$397$	6.82843	0.342709	0.171354	+	0.985209i	$0.445186\pi$
$398$	0	0
$399$	−0.284271	−0.0142314
$400$	0	0
$401$	−16.6274	−0.830334	−0.415167	−	0.909745i	$-0.636277\pi$
$401$	−16.6274	−0.830334	−0.415167	+	0.909745i	$0.636277\pi$
$402$	0	0
$403$	1.41421	0.0704470
$404$	0	0
$405$	6.02944	0.299605
$406$	0	0
$407$	−28.9706	−1.43602
$408$	0	0
$409$	−0.828427	−0.0409631	−0.0204815	−	0.999790i	$-0.506520\pi$
$409$	−0.828427	−0.0409631	−0.0204815	+	0.999790i	$0.506520\pi$
$410$	0	0
$411$	1.17157	0.0577894
$412$	0	0
$413$	5.17157	0.254476
$414$	0	0
$415$	2.00000	0.0981761
$416$	0	0
$417$	4.97056	0.243410
$418$	0	0
$419$	5.85786	0.286175	0.143088	−	0.989710i	$-0.454297\pi$
$419$	5.85786	0.286175	0.143088	+	0.989710i	$0.454297\pi$
$420$	0	0
$421$	24.3431	1.18641	0.593206	−	0.805051i	$-0.297862\pi$
$421$	24.3431	1.18641	0.593206	+	0.805051i	$0.297862\pi$
$422$	0	0
$423$	5.31371	0.258361
$424$	0	0
$425$	0.828427	0.0401846
$426$	0	0
$427$	30.6274	1.48216
$428$	0	0
$429$	−2.48528	−0.119991
$430$	0	0
$431$	29.2132	1.40715	0.703575	−	0.710621i	$-0.251586\pi$
$431$	29.2132	1.40715	0.703575	+	0.710621i	$0.251586\pi$
$432$	0	0
$433$	26.9706	1.29612	0.648061	−	0.761588i	$-0.275580\pi$
$433$	26.9706	1.29612	0.648061	+	0.761588i	$0.275580\pi$
$434$	0	0
$435$	0.970563	0.0465349
$436$	0	0
$437$	−2.20101	−0.105289
$438$	0	0
$439$	28.2843	1.34993	0.674967	−	0.737848i	$-0.264158\pi$
$439$	28.2843	1.34993	0.674967	+	0.737848i	$0.264158\pi$
$440$	0	0
$441$	7.97056	0.379551
$442$	0	0
$443$	0.585786	0.0278316	0.0139158	−	0.999903i	$-0.495570\pi$
$443$	0.585786	0.0278316	0.0139158	+	0.999903i	$0.495570\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	−13.4558	−0.636440
$448$	0	0
$449$	36.1421	1.70565	0.852826	−	0.522195i	$-0.174886\pi$
$449$	36.1421	1.70565	0.852826	+	0.522195i	$0.174886\pi$
$450$	0	0
$451$	20.4853	0.964614
$452$	0	0
$453$	−8.82843	−0.414796
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	12.6274	0.590686	0.295343	−	0.955391i	$-0.404566\pi$
$457$	12.6274	0.590686	0.295343	+	0.955391i	$0.404566\pi$
$458$	0	0
$459$	−2.74517	−0.128133
$460$	0	0
$461$	8.14214	0.379217	0.189609	−	0.981860i	$-0.439278\pi$
$461$	8.14214	0.379217	0.189609	+	0.981860i	$0.439278\pi$
$462$	0	0
$463$	−5.51472	−0.256291	−0.128145	−	0.991755i	$-0.540902\pi$
$463$	−5.51472	−0.256291	−0.128145	+	0.991755i	$0.540902\pi$
$464$	0	0
$465$	−0.828427	−0.0384174
$466$	0	0
$467$	−5.75736	−0.266419	−0.133209	−	0.991088i	$-0.542528\pi$
$467$	−5.75736	−0.266419	−0.133209	+	0.991088i	$0.542528\pi$
$468$	0	0
$469$	9.65685	0.445912
$470$	0	0
$471$	−10.1421	−0.467325
$472$	0	0
$473$	43.4558	1.99810
$474$	0	0
$475$	−0.242641	−0.0111331
$476$	0	0
$477$	23.4558	1.07397
$478$	0	0
$479$	23.7574	1.08550	0.542751	−	0.839894i	$-0.317383\pi$
$479$	23.7574	1.08550	0.542751	+	0.839894i	$0.317383\pi$
$480$	0	0
$481$	6.82843	0.311349
$482$	0	0
$483$	10.6274	0.483564
$484$	0	0
$485$	11.6569	0.529310
$486$	0	0
$487$	10.4853	0.475133	0.237567	−	0.971371i	$-0.423650\pi$
$487$	10.4853	0.475133	0.237567	+	0.971371i	$0.423650\pi$
$488$	0	0
$489$	3.23045	0.146086
$490$	0	0
$491$	−43.1127	−1.94565	−0.972824	−	0.231544i	$-0.925622\pi$
$491$	−43.1127	−1.94565	−0.972824	+	0.231544i	$0.925622\pi$
$492$	0	0
$493$	1.37258	0.0618180
$494$	0	0
$495$	−11.2721	−0.506642
$496$	0	0
$497$	−19.7990	−0.888106
$498$	0	0
$499$	29.4142	1.31676	0.658381	−	0.752685i	$-0.271242\pi$
$499$	29.4142	1.31676	0.658381	+	0.752685i	$0.271242\pi$
$500$	0	0
$501$	−5.45584	−0.243749
$502$	0	0
$503$	24.5858	1.09623	0.548113	−	0.836404i	$-0.315346\pi$
$503$	24.5858	1.09623	0.548113	+	0.836404i	$0.315346\pi$
$504$	0	0
$505$	−3.65685	−0.162728
$506$	0	0
$507$	0.585786	0.0260157
$508$	0	0
$509$	34.4853	1.52853	0.764267	−	0.644900i	$-0.223101\pi$
$509$	34.4853	1.52853	0.764267	+	0.644900i	$0.223101\pi$
$510$	0	0
$511$	2.34315	0.103655
$512$	0	0
$513$	0.804041	0.0354993
$514$	0	0
$515$	−11.4142	−0.502970
$516$	0	0
$517$	−8.48528	−0.373182
$518$	0	0
$519$	−8.88730	−0.390109
$520$	0	0
$521$	−35.3137	−1.54712	−0.773561	−	0.633722i	$-0.781526\pi$
$521$	−35.3137	−1.54712	−0.773561	+	0.633722i	$0.781526\pi$
$522$	0	0
$523$	−35.6985	−1.56099	−0.780493	−	0.625165i	$-0.785032\pi$
$523$	−35.6985	−1.56099	−0.780493	+	0.625165i	$0.785032\pi$
$524$	0	0
$525$	1.17157	0.0511316
$526$	0	0
$527$	−1.17157	−0.0510345
$528$	0	0
$529$	59.2843	2.57758
$530$	0	0
$531$	−6.87006	−0.298135
$532$	0	0
$533$	−4.82843	−0.209142
$534$	0	0
$535$	−5.07107	−0.219241
$536$	0	0
$537$	−6.62742	−0.285994
$538$	0	0
$539$	−12.7279	−0.548230
$540$	0	0
$541$	−2.48528	−0.106851	−0.0534253	−	0.998572i	$-0.517014\pi$
$541$	−2.48528	−0.106851	−0.0534253	+	0.998572i	$0.517014\pi$
$542$	0	0
$543$	0.402020	0.0172523
$544$	0	0
$545$	17.3137	0.741638
$546$	0	0
$547$	18.9289	0.809343	0.404671	−	0.914462i	$-0.367386\pi$
$547$	18.9289	0.809343	0.404671	+	0.914462i	$0.367386\pi$
$548$	0	0
$549$	−40.6863	−1.73645
$550$	0	0
$551$	−0.402020	−0.0171266
$552$	0	0
$553$	−2.34315	−0.0996407
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	13.8579	0.587177	0.293588	−	0.955932i	$-0.405151\pi$
$557$	13.8579	0.587177	0.293588	+	0.955932i	$0.405151\pi$
$558$	0	0
$559$	−10.2426	−0.433218
$560$	0	0
$561$	2.05887	0.0869257
$562$	0	0
$563$	13.5563	0.571332	0.285666	−	0.958329i	$-0.407785\pi$
$563$	13.5563	0.571332	0.285666	+	0.958329i	$0.407785\pi$
$564$	0	0
$565$	−8.82843	−0.371415
$566$	0	0
$567$	12.0589	0.506425
$568$	0	0
$569$	−8.68629	−0.364148	−0.182074	−	0.983285i	$-0.558281\pi$
$569$	−8.68629	−0.364148	−0.182074	+	0.983285i	$0.558281\pi$
$570$	0	0
$571$	29.1716	1.22079	0.610396	−	0.792096i	$-0.291010\pi$
$571$	29.1716	1.22079	0.610396	+	0.792096i	$0.291010\pi$
$572$	0	0
$573$	6.05887	0.253113
$574$	0	0
$575$	9.07107	0.378290
$576$	0	0
$577$	5.85786	0.243866	0.121933	−	0.992538i	$-0.461091\pi$
$577$	5.85786	0.243866	0.121933	+	0.992538i	$0.461091\pi$
$578$	0	0
$579$	−11.5147	−0.478535
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	−37.4558	−1.55126
$584$	0	0
$585$	2.65685	0.109847
$586$	0	0
$587$	−2.48528	−0.102579	−0.0512893	−	0.998684i	$-0.516333\pi$
$587$	−2.48528	−0.102579	−0.0512893	+	0.998684i	$0.516333\pi$
$588$	0	0
$589$	0.343146	0.0141391
$590$	0	0
$591$	−6.82843	−0.280884
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	1.65685	0.0679244
$596$	0	0
$597$	−10.3431	−0.423317
$598$	0	0
$599$	11.5147	0.470479	0.235239	−	0.971937i	$-0.424413\pi$
$599$	11.5147	0.470479	0.235239	+	0.971937i	$0.424413\pi$
$600$	0	0
$601$	35.9411	1.46607	0.733035	−	0.680191i	$-0.238103\pi$
$601$	35.9411	1.46607	0.733035	+	0.680191i	$0.238103\pi$
$602$	0	0
$603$	−12.8284	−0.522414
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	6.44365	0.261540	0.130770	−	0.991413i	$-0.458255\pi$
$607$	6.44365	0.261540	0.130770	+	0.991413i	$0.458255\pi$
$608$	0	0
$609$	1.94113	0.0786584
$610$	0	0
$611$	2.00000	0.0809113
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	0	0
$615$	2.82843	0.114053
$616$	0	0
$617$	−3.65685	−0.147219	−0.0736097	−	0.997287i	$-0.523452\pi$
$617$	−3.65685	−0.147219	−0.0736097	+	0.997287i	$0.523452\pi$
$618$	0	0
$619$	−36.0416	−1.44864	−0.724318	−	0.689466i	$-0.757845\pi$
$619$	−36.0416	−1.44864	−0.724318	+	0.689466i	$0.757845\pi$
$620$	0	0
$621$	−30.0589	−1.20622
$622$	0	0
$623$	−20.0000	−0.801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.603030	−0.0240827
$628$	0	0
$629$	−5.65685	−0.225554
$630$	0	0
$631$	12.7279	0.506691	0.253345	−	0.967376i	$-0.418469\pi$
$631$	12.7279	0.506691	0.253345	+	0.967376i	$0.418469\pi$
$632$	0	0
$633$	3.31371	0.131708
$634$	0	0
$635$	−15.8995	−0.630952
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	26.3015	1.04047
$640$	0	0
$641$	28.3431	1.11949	0.559743	−	0.828666i	$-0.310900\pi$
$641$	28.3431	1.11949	0.559743	+	0.828666i	$0.310900\pi$
$642$	0	0
$643$	13.3137	0.525041	0.262521	−	0.964926i	$-0.415446\pi$
$643$	13.3137	0.525041	0.262521	+	0.964926i	$0.415446\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	37.3553	1.46859	0.734295	−	0.678831i	$-0.237513\pi$
$647$	37.3553	1.46859	0.734295	+	0.678831i	$0.237513\pi$
$648$	0	0
$649$	10.9706	0.430632
$650$	0	0
$651$	−1.65685	−0.0649372
$652$	0	0
$653$	−9.02944	−0.353349	−0.176675	−	0.984269i	$-0.556534\pi$
$653$	−9.02944	−0.353349	−0.176675	+	0.984269i	$0.556534\pi$
$654$	0	0
$655$	19.3137	0.754649
$656$	0	0
$657$	−3.11270	−0.121438
$658$	0	0
$659$	−4.48528	−0.174722	−0.0873609	−	0.996177i	$-0.527843\pi$
$659$	−4.48528	−0.174722	−0.0873609	+	0.996177i	$0.527843\pi$
$660$	0	0
$661$	−9.02944	−0.351204	−0.175602	−	0.984461i	$-0.556187\pi$
$661$	−9.02944	−0.351204	−0.175602	+	0.984461i	$0.556187\pi$
$662$	0	0
$663$	−0.485281	−0.0188468
$664$	0	0
$665$	−0.485281	−0.0188184
$666$	0	0
$667$	15.0294	0.581942
$668$	0	0
$669$	−1.57359	−0.0608386
$670$	0	0
$671$	64.9706	2.50816
$672$	0	0
$673$	33.7990	1.30286	0.651428	−	0.758711i	$-0.274170\pi$
$673$	33.7990	1.30286	0.651428	+	0.758711i	$0.274170\pi$
$674$	0	0
$675$	−3.31371	−0.127545
$676$	0	0
$677$	−20.1421	−0.774125	−0.387063	−	0.922053i	$-0.626510\pi$
$677$	−20.1421	−0.774125	−0.387063	+	0.922053i	$0.626510\pi$
$678$	0	0
$679$	23.3137	0.894698
$680$	0	0
$681$	−8.88730	−0.340562
$682$	0	0
$683$	−37.1127	−1.42008	−0.710039	−	0.704162i	$-0.751323\pi$
$683$	−37.1127	−1.42008	−0.710039	+	0.704162i	$0.751323\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	15.1127	0.576585
$688$	0	0
$689$	8.82843	0.336336
$690$	0	0
$691$	49.4142	1.87981	0.939903	−	0.341443i	$-0.110915\pi$
$691$	49.4142	1.87981	0.939903	+	0.341443i	$0.110915\pi$
$692$	0	0
$693$	−22.5442	−0.856382
$694$	0	0
$695$	8.48528	0.321865
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	−12.8873	−0.487443
$700$	0	0
$701$	−20.6274	−0.779087	−0.389543	−	0.921008i	$-0.627367\pi$
$701$	−20.6274	−0.779087	−0.389543	+	0.921008i	$0.627367\pi$
$702$	0	0
$703$	1.65685	0.0624894
$704$	0	0
$705$	−1.17157	−0.0441240
$706$	0	0
$707$	−7.31371	−0.275060
$708$	0	0
$709$	−19.4558	−0.730680	−0.365340	−	0.930874i	$-0.619047\pi$
$709$	−19.4558	−0.730680	−0.365340	+	0.930874i	$0.619047\pi$
$710$	0	0
$711$	3.11270	0.116735
$712$	0	0
$713$	−12.8284	−0.480428
$714$	0	0
$715$	−4.24264	−0.158666
$716$	0	0
$717$	9.79899	0.365950
$718$	0	0
$719$	18.6274	0.694685	0.347343	−	0.937738i	$-0.387084\pi$
$719$	18.6274	0.694685	0.347343	+	0.937738i	$0.387084\pi$
$720$	0	0
$721$	−22.8284	−0.850175
$722$	0	0
$723$	7.51472	0.279475
$724$	0	0
$725$	1.65685	0.0615340
$726$	0	0
$727$	2.24264	0.0831749	0.0415875	−	0.999135i	$-0.486758\pi$
$727$	2.24264	0.0831749	0.0415875	+	0.999135i	$0.486758\pi$
$728$	0	0
$729$	−10.1960	−0.377628
$730$	0	0
$731$	8.48528	0.313839
$732$	0	0
$733$	32.6274	1.20512	0.602561	−	0.798073i	$-0.294147\pi$
$733$	32.6274	1.20512	0.602561	+	0.798073i	$0.294147\pi$
$734$	0	0
$735$	−1.75736	−0.0648212
$736$	0	0
$737$	20.4853	0.754585
$738$	0	0
$739$	−18.5858	−0.683689	−0.341845	−	0.939756i	$-0.611052\pi$
$739$	−18.5858	−0.683689	−0.341845	+	0.939756i	$0.611052\pi$
$740$	0	0
$741$	0.142136	0.00522148
$742$	0	0
$743$	−30.9706	−1.13620	−0.568100	−	0.822960i	$-0.692321\pi$
$743$	−30.9706	−1.13620	−0.568100	+	0.822960i	$0.692321\pi$
$744$	0	0
$745$	−22.9706	−0.841576
$746$	0	0
$747$	−5.31371	−0.194418
$748$	0	0
$749$	−10.1421	−0.370586
$750$	0	0
$751$	−8.48528	−0.309632	−0.154816	−	0.987943i	$-0.549479\pi$
$751$	−8.48528	−0.309632	−0.154816	+	0.987943i	$0.549479\pi$
$752$	0	0
$753$	10.6274	0.387285
$754$	0	0
$755$	−15.0711	−0.548492
$756$	0	0
$757$	−8.14214	−0.295931	−0.147965	−	0.988993i	$-0.547272\pi$
$757$	−8.14214	−0.295931	−0.147965	+	0.988993i	$0.547272\pi$
$758$	0	0
$759$	22.5442	0.818301
$760$	0	0
$761$	−41.3137	−1.49762	−0.748810	−	0.662784i	$-0.769375\pi$
$761$	−41.3137	−1.49762	−0.748810	+	0.662784i	$0.769375\pi$
$762$	0	0
$763$	34.6274	1.25360
$764$	0	0
$765$	−2.20101	−0.0795777
$766$	0	0
$767$	−2.58579	−0.0933673
$768$	0	0
$769$	−19.6569	−0.708844	−0.354422	−	0.935086i	$-0.615322\pi$
$769$	−19.6569	−0.708844	−0.354422	+	0.935086i	$0.615322\pi$
$770$	0	0
$771$	−11.1127	−0.400214
$772$	0	0
$773$	18.1421	0.652527	0.326264	−	0.945279i	$-0.394210\pi$
$773$	18.1421	0.652527	0.326264	+	0.945279i	$0.394210\pi$
$774$	0	0
$775$	−1.41421	−0.0508001
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	−1.17157	−0.0419760
$780$	0	0
$781$	−42.0000	−1.50288
$782$	0	0
$783$	−5.49033	−0.196208
$784$	0	0
$785$	−17.3137	−0.617953
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	0	0
$789$	−13.0294	−0.463860
$790$	0	0
$791$	−17.6569	−0.627805
$792$	0	0
$793$	−15.3137	−0.543806
$794$	0	0
$795$	−5.17157	−0.183417
$796$	0	0
$797$	49.5980	1.75685	0.878425	−	0.477880i	$-0.158595\pi$
$797$	49.5980	1.75685	0.878425	+	0.477880i	$0.158595\pi$
$798$	0	0
$799$	−1.65685	−0.0586153
$800$	0	0
$801$	26.5685	0.938753
$802$	0	0
$803$	4.97056	0.175407
$804$	0	0
$805$	18.1421	0.639426
$806$	0	0
$807$	−16.7696	−0.590316
$808$	0	0
$809$	−41.6569	−1.46458	−0.732289	−	0.680995i	$-0.761548\pi$
$809$	−41.6569	−1.46458	−0.732289	+	0.680995i	$0.761548\pi$
$810$	0	0
$811$	−9.89949	−0.347618	−0.173809	−	0.984779i	$-0.555608\pi$
$811$	−9.89949	−0.347618	−0.173809	+	0.984779i	$0.555608\pi$
$812$	0	0
$813$	5.79899	0.203379
$814$	0	0
$815$	5.51472	0.193172
$816$	0	0
$817$	−2.48528	−0.0869490
$818$	0	0
$819$	5.31371	0.185676
$820$	0	0
$821$	6.68629	0.233353	0.116677	−	0.993170i	$-0.462776\pi$
$821$	6.68629	0.233353	0.116677	+	0.993170i	$0.462776\pi$
$822$	0	0
$823$	16.8701	0.588053	0.294027	−	0.955797i	$-0.405005\pi$
$823$	16.8701	0.588053	0.294027	+	0.955797i	$0.405005\pi$
$824$	0	0
$825$	2.48528	0.0865264
$826$	0	0
$827$	−20.6274	−0.717286	−0.358643	−	0.933475i	$-0.616760\pi$
$827$	−20.6274	−0.717286	−0.358643	+	0.933475i	$0.616760\pi$
$828$	0	0
$829$	14.3431	0.498158	0.249079	−	0.968483i	$-0.419872\pi$
$829$	14.3431	0.498158	0.249079	+	0.968483i	$0.419872\pi$
$830$	0	0
$831$	−14.1421	−0.490585
$832$	0	0
$833$	−2.48528	−0.0861099
$834$	0	0
$835$	−9.31371	−0.322314
$836$	0	0
$837$	4.68629	0.161982
$838$	0	0
$839$	37.8995	1.30844	0.654218	−	0.756306i	$-0.272998\pi$
$839$	37.8995	1.30844	0.654218	+	0.756306i	$0.272998\pi$
$840$	0	0
$841$	−26.2548	−0.905339
$842$	0	0
$843$	−3.23045	−0.111263
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	9.31371	0.319646
$850$	0	0
$851$	−61.9411	−2.12331
$852$	0	0
$853$	−47.1127	−1.61311	−0.806554	−	0.591160i	$-0.798670\pi$
$853$	−47.1127	−1.61311	−0.806554	+	0.591160i	$0.798670\pi$
$854$	0	0
$855$	0.644661	0.0220469
$856$	0	0
$857$	−52.9117	−1.80743	−0.903714	−	0.428136i	$-0.859170\pi$
$857$	−52.9117	−1.80743	−0.903714	+	0.428136i	$0.859170\pi$
$858$	0	0
$859$	−54.8284	−1.87072	−0.935361	−	0.353695i	$-0.884925\pi$
$859$	−54.8284	−1.87072	−0.935361	+	0.353695i	$0.884925\pi$
$860$	0	0
$861$	5.65685	0.192785
$862$	0	0
$863$	−53.5980	−1.82450	−0.912248	−	0.409638i	$-0.865655\pi$
$863$	−53.5980	−1.82450	−0.912248	+	0.409638i	$0.865655\pi$
$864$	0	0
$865$	−15.1716	−0.515849
$866$	0	0
$867$	−9.55635	−0.324551
$868$	0	0
$869$	−4.97056	−0.168615
$870$	0	0
$871$	−4.82843	−0.163605
$872$	0	0
$873$	−30.9706	−1.04819
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−29.3137	−0.989854	−0.494927	−	0.868935i	$-0.664805\pi$
$877$	−29.3137	−0.989854	−0.494927	+	0.868935i	$0.664805\pi$
$878$	0	0
$879$	−1.65685	−0.0558843
$880$	0	0
$881$	25.9411	0.873979	0.436989	−	0.899467i	$-0.356045\pi$
$881$	25.9411	0.873979	0.436989	+	0.899467i	$0.356045\pi$
$882$	0	0
$883$	18.2426	0.613914	0.306957	−	0.951723i	$-0.400689\pi$
$883$	18.2426	0.613914	0.306957	+	0.951723i	$0.400689\pi$
$884$	0	0
$885$	1.51472	0.0509167
$886$	0	0
$887$	−30.0416	−1.00870	−0.504350	−	0.863500i	$-0.668268\pi$
$887$	−30.0416	−1.00870	−0.504350	+	0.863500i	$0.668268\pi$
$888$	0	0
$889$	−31.7990	−1.06650
$890$	0	0
$891$	25.5807	0.856987
$892$	0	0
$893$	0.485281	0.0162393
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	−5.31371	−0.177420
$898$	0	0
$899$	−2.34315	−0.0781483
$900$	0	0
$901$	−7.31371	−0.243655
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	0.686292	0.0228131
$906$	0	0
$907$	−1.07107	−0.0355642	−0.0177821	−	0.999842i	$-0.505661\pi$
$907$	−1.07107	−0.0355642	−0.0177821	+	0.999842i	$0.505661\pi$
$908$	0	0
$909$	9.71573	0.322250
$910$	0	0
$911$	−51.5980	−1.70952	−0.854759	−	0.519026i	$-0.826295\pi$
$911$	−51.5980	−1.70952	−0.854759	+	0.519026i	$0.826295\pi$
$912$	0	0
$913$	8.48528	0.280822
$914$	0	0
$915$	8.97056	0.296558
$916$	0	0
$917$	38.6274	1.27559
$918$	0	0
$919$	3.11270	0.102678	0.0513392	−	0.998681i	$-0.483651\pi$
$919$	3.11270	0.102678	0.0513392	+	0.998681i	$0.483651\pi$
$920$	0	0
$921$	−2.54416	−0.0838328
$922$	0	0
$923$	9.89949	0.325846
$924$	0	0
$925$	−6.82843	−0.224517
$926$	0	0
$927$	30.3259	0.996033
$928$	0	0
$929$	−41.1127	−1.34886	−0.674432	−	0.738337i	$-0.735611\pi$
$929$	−41.1127	−1.34886	−0.674432	+	0.738337i	$0.735611\pi$
$930$	0	0
$931$	0.727922	0.0238567
$932$	0	0
$933$	−10.6274	−0.347926
$934$	0	0
$935$	3.51472	0.114944
$936$	0	0
$937$	46.2843	1.51204	0.756021	−	0.654548i	$-0.227141\pi$
$937$	46.2843	1.51204	0.756021	+	0.654548i	$0.227141\pi$
$938$	0	0
$939$	−7.51472	−0.245234
$940$	0	0
$941$	−18.4853	−0.602603	−0.301301	−	0.953529i	$-0.597421\pi$
$941$	−18.4853	−0.602603	−0.301301	+	0.953529i	$0.597421\pi$
$942$	0	0
$943$	43.7990	1.42629
$944$	0	0
$945$	−6.62742	−0.215590
$946$	0	0
$947$	−46.9706	−1.52634	−0.763169	−	0.646199i	$-0.776358\pi$
$947$	−46.9706	−1.52634	−0.763169	+	0.646199i	$0.776358\pi$
$948$	0	0
$949$	−1.17157	−0.0380309
$950$	0	0
$951$	4.40202	0.142745
$952$	0	0
$953$	−19.9411	−0.645956	−0.322978	−	0.946406i	$-0.604684\pi$
$953$	−19.9411	−0.645956	−0.322978	+	0.946406i	$0.604684\pi$
$954$	0	0
$955$	10.3431	0.334696
$956$	0	0
$957$	4.11775	0.133108
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	13.4731	0.434164
$964$	0	0
$965$	−19.6569	−0.632777
$966$	0	0
$967$	44.8284	1.44159	0.720793	−	0.693151i	$-0.243778\pi$
$967$	44.8284	1.44159	0.720793	+	0.693151i	$0.243778\pi$
$968$	0	0
$969$	−0.117749	−0.00378264
$970$	0	0
$971$	6.62742	0.212684	0.106342	−	0.994330i	$-0.466086\pi$
$971$	6.62742	0.212684	0.106342	+	0.994330i	$0.466086\pi$
$972$	0	0
$973$	16.9706	0.544051
$974$	0	0
$975$	−0.585786	−0.0187602
$976$	0	0
$977$	62.4264	1.99720	0.998599	−	0.0529182i	$-0.0168523\pi$
$977$	62.4264	1.99720	0.998599	+	0.0529182i	$0.0168523\pi$
$978$	0	0
$979$	−42.4264	−1.35595
$980$	0	0
$981$	−46.0000	−1.46867
$982$	0	0
$983$	34.4853	1.09991	0.549955	−	0.835194i	$-0.314645\pi$
$983$	34.4853	1.09991	0.549955	+	0.835194i	$0.314645\pi$
$984$	0	0
$985$	−11.6569	−0.371418
$986$	0	0
$987$	−2.34315	−0.0745832
$988$	0	0
$989$	92.9117	2.95442
$990$	0	0
$991$	−55.5980	−1.76613	−0.883064	−	0.469253i	$-0.844523\pi$
$991$	−55.5980	−1.76613	−0.883064	+	0.469253i	$0.844523\pi$
$992$	0	0
$993$	8.82843	0.280162
$994$	0	0
$995$	−17.6569	−0.559760
$996$	0	0
$997$	−3.17157	−0.100445	−0.0502224	−	0.998738i	$-0.515993\pi$
$997$	−3.17157	−0.100445	−0.0502224	+	0.998738i	$0.515993\pi$
$998$	0	0
$999$	22.6274	0.715900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.a.n.1.1		2
3.2	odd	2		9360.2.a.ck.1.1		2
4.3	odd	2		520.2.a.c.1.2	✓	2
5.4	even	2		5200.2.a.bl.1.2		2
8.3	odd	2		4160.2.a.bn.1.1		2
8.5	even	2		4160.2.a.u.1.2		2
12.11	even	2		4680.2.a.w.1.2		2
20.3	even	4		2600.2.d.i.1249.2		4
20.7	even	4		2600.2.d.i.1249.3		4
20.19	odd	2		2600.2.a.w.1.1		2
52.51	odd	2		6760.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.c.1.2	✓	2	4.3	odd	2
1040.2.a.n.1.1		2	1.1	even	1	trivial
2600.2.a.w.1.1		2	20.19	odd	2
2600.2.d.i.1249.2		4	20.3	even	4
2600.2.d.i.1249.3		4	20.7	even	4
4160.2.a.u.1.2		2	8.5	even	2
4160.2.a.bn.1.1		2	8.3	odd	2
4680.2.a.w.1.2		2	12.11	even	2
5200.2.a.bl.1.2		2	5.4	even	2
6760.2.a.n.1.2		2	52.51	odd	2
9360.2.a.ck.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q+0.585786 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.65685 q^{9} +O(q^{10})\)	\(q+0.585786 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.65685 q^{9} +4.24264 q^{11} -1.00000 q^{13} +0.585786 q^{15} +0.828427 q^{17} -0.242641 q^{19} +1.17157 q^{21} +9.07107 q^{23} +1.00000 q^{25} -3.31371 q^{27} +1.65685 q^{29} -1.41421 q^{31} +2.48528 q^{33} +2.00000 q^{35} -6.82843 q^{37} -0.585786 q^{39} +4.82843 q^{41} +10.2426 q^{43} -2.65685 q^{45} -2.00000 q^{47} -3.00000 q^{49} +0.485281 q^{51} -8.82843 q^{53} +4.24264 q^{55} -0.142136 q^{57} +2.58579 q^{59} +15.3137 q^{61} -5.31371 q^{63} -1.00000 q^{65} +4.82843 q^{67} +5.31371 q^{69} -9.89949 q^{71} +1.17157 q^{73} +0.585786 q^{75} +8.48528 q^{77} -1.17157 q^{79} +6.02944 q^{81} +2.00000 q^{83} +0.828427 q^{85} +0.970563 q^{87} -10.0000 q^{89} -2.00000 q^{91} -0.828427 q^{93} -0.242641 q^{95} +11.6569 q^{97} -11.2721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9	\( 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9} - 2 q^{13} + 4 q^{15} - 4 q^{17} + 8 q^{19} + 8 q^{21} + 4 q^{23} + 2 q^{25} + 16 q^{27} - 8 q^{29} - 12 q^{33} + 4 q^{35} - 8 q^{37} - 4 q^{39} + 4 q^{41} + 12 q^{43} + 6 q^{45} - 4 q^{47} - 6 q^{49} - 16 q^{51} - 12 q^{53} + 28 q^{57} + 8 q^{59} + 8 q^{61} + 12 q^{63} - 2 q^{65} + 4 q^{67} - 12 q^{69} + 8 q^{73} + 4 q^{75} - 8 q^{79} + 46 q^{81} + 4 q^{83} - 4 q^{85} - 32 q^{87} - 20 q^{89} - 4 q^{91} + 4 q^{93} + 8 q^{95} + 12 q^{97} - 48 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9 - 2 * q^13 + 4 * q^15 - 4 * q^17 + 8 * q^19 + 8 * q^21 + 4 * q^23 + 2 * q^25 + 16 * q^27 - 8 * q^29 - 12 * q^33 + 4 * q^35 - 8 * q^37 - 4 * q^39 + 4 * q^41 + 12 * q^43 + 6 * q^45 - 4 * q^47 - 6 * q^49 - 16 * q^51 - 12 * q^53 + 28 * q^57 + 8 * q^59 + 8 * q^61 + 12 * q^63 - 2 * q^65 + 4 * q^67 - 12 * q^69 + 8 * q^73 + 4 * q^75 - 8 * q^79 + 46 * q^81 + 4 * q^83 - 4 * q^85 - 32 * q^87 - 20 * q^89 - 4 * q^91 + 4 * q^93 + 8 * q^95 + 12 * q^97 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more