LMFDB - Embedded newform 3696.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3696,2,Mod(1,3696)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3696, 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("3696.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3696.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:	$29.5127085871$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	3696.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.39821 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.39821 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.04502 q^{13} -2.39821 q^{15} -6.36842 q^{17} +5.32340 q^{19} -1.00000 q^{21} -4.92520 q^{23} +0.751399 q^{25} -1.00000 q^{27} +5.04502 q^{29} +7.57201 q^{31} -1.00000 q^{33} +2.39821 q^{35} +4.24860 q^{37} -5.04502 q^{39} -0.646809 q^{41} +10.5180 q^{43} +2.39821 q^{45} -0.526989 q^{47} +1.00000 q^{49} +6.36842 q^{51} +3.72161 q^{53} +2.39821 q^{55} -5.32340 q^{57} -7.97021 q^{59} -2.00000 q^{61} +1.00000 q^{63} +12.0990 q^{65} -8.76663 q^{67} +4.92520 q^{69} +11.4432 q^{71} -13.0450 q^{73} -0.751399 q^{75} +1.00000 q^{77} -11.4432 q^{79} +1.00000 q^{81} -13.1648 q^{83} -15.2728 q^{85} -5.04502 q^{87} +11.8504 q^{89} +5.04502 q^{91} -7.57201 q^{93} +12.7666 q^{95} -1.87122 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41} + 14 q^{43} + 4 q^{45} + 3 q^{49} - 8 q^{51} + 4 q^{55} - 8 q^{57} - 6 q^{61} + 3 q^{63} + 14 q^{65} + 4 q^{67} + 10 q^{69} + 12 q^{71} - 20 q^{73} - 15 q^{75} + 3 q^{77} - 12 q^{79} + 3 q^{81} - 6 q^{83} - 6 q^{85} + 4 q^{87} + 26 q^{89} - 4 q^{91} - 2 q^{93} + 8 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 4 * q^15 + 8 * q^17 + 8 * q^19 - 3 * q^21 - 10 * q^23 + 15 * q^25 - 3 * q^27 - 4 * q^29 + 2 * q^31 - 3 * q^33 + 4 * q^35 + 4 * q^39 + 14 * q^41 + 14 * q^43 + 4 * q^45 + 3 * q^49 - 8 * q^51 + 4 * q^55 - 8 * q^57 - 6 * q^61 + 3 * q^63 + 14 * q^65 + 4 * q^67 + 10 * q^69 + 12 * q^71 - 20 * q^73 - 15 * q^75 + 3 * q^77 - 12 * q^79 + 3 * q^81 - 6 * q^83 - 6 * q^85 + 4 * q^87 + 26 * q^89 - 4 * q^91 - 2 * q^93 + 8 * q^95 - 4 * q^97 + 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.39821	1.07251	0.536255	−	0.844056i	$-0.319838\pi$
$5$	2.39821	1.07251	0.536255	+	0.844056i	$0.319838\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.04502	1.39924	0.699618	−	0.714517i	$-0.253354\pi$
$13$	5.04502	1.39924	0.699618	+	0.714517i	$0.253354\pi$
$14$	0	0
$15$	−2.39821	−0.619214
$16$	0	0
$17$	−6.36842	−1.54457	−0.772284	−	0.635277i	$-0.780886\pi$
$17$	−6.36842	−1.54457	−0.772284	+	0.635277i	$0.780886\pi$
$18$	0	0
$19$	5.32340	1.22127	0.610636	−	0.791911i	$-0.290914\pi$
$19$	5.32340	1.22127	0.610636	+	0.791911i	$0.290914\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.92520	−1.02697	−0.513487	−	0.858097i	$-0.671647\pi$
$23$	−4.92520	−1.02697	−0.513487	+	0.858097i	$0.671647\pi$
$24$	0	0
$25$	0.751399	0.150280
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.04502	0.936836	0.468418	−	0.883507i	$-0.344824\pi$
$29$	5.04502	0.936836	0.468418	+	0.883507i	$0.344824\pi$
$30$	0	0
$31$	7.57201	1.35997	0.679986	−	0.733225i	$-0.261986\pi$
$31$	7.57201	1.35997	0.679986	+	0.733225i	$0.261986\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	2.39821	0.405371
$36$	0	0
$37$	4.24860	0.698466	0.349233	−	0.937036i	$-0.386442\pi$
$37$	4.24860	0.698466	0.349233	+	0.937036i	$0.386442\pi$
$38$	0	0
$39$	−5.04502	−0.807849
$40$	0	0
$41$	−0.646809	−0.101015	−0.0505073	−	0.998724i	$-0.516084\pi$
$41$	−0.646809	−0.101015	−0.0505073	+	0.998724i	$0.516084\pi$
$42$	0	0
$43$	10.5180	1.60398	0.801992	−	0.597335i	$-0.203774\pi$
$43$	10.5180	1.60398	0.801992	+	0.597335i	$0.203774\pi$
$44$	0	0
$45$	2.39821	0.357504
$46$	0	0
$47$	−0.526989	−0.0768693	−0.0384347	−	0.999261i	$-0.512237\pi$
$47$	−0.526989	−0.0768693	−0.0384347	+	0.999261i	$0.512237\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.36842	0.891757
$52$	0	0
$53$	3.72161	0.511203	0.255601	−	0.966782i	$-0.417727\pi$
$53$	3.72161	0.511203	0.255601	+	0.966782i	$0.417727\pi$
$54$	0	0
$55$	2.39821	0.323374
$56$	0	0
$57$	−5.32340	−0.705102
$58$	0	0
$59$	−7.97021	−1.03763	−0.518817	−	0.854886i	$-0.673627\pi$
$59$	−7.97021	−1.03763	−0.518817	+	0.854886i	$0.673627\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	12.0990	1.50070
$66$	0	0
$67$	−8.76663	−1.07101	−0.535507	−	0.844531i	$-0.679879\pi$
$67$	−8.76663	−1.07101	−0.535507	+	0.844531i	$0.679879\pi$
$68$	0	0
$69$	4.92520	0.592924
$70$	0	0
$71$	11.4432	1.35806	0.679030	−	0.734110i	$-0.262400\pi$
$71$	11.4432	1.35806	0.679030	+	0.734110i	$0.262400\pi$
$72$	0	0
$73$	−13.0450	−1.52680	−0.763402	−	0.645924i	$-0.776472\pi$
$73$	−13.0450	−1.52680	−0.763402	+	0.645924i	$0.776472\pi$
$74$	0	0
$75$	−0.751399	−0.0867641
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−11.4432	−1.28746	−0.643732	−	0.765251i	$-0.722615\pi$
$79$	−11.4432	−1.28746	−0.643732	+	0.765251i	$0.722615\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.1648	−1.44503	−0.722514	−	0.691356i	$-0.757014\pi$
$83$	−13.1648	−1.44503	−0.722514	+	0.691356i	$0.757014\pi$
$84$	0	0
$85$	−15.2728	−1.65657
$86$	0	0
$87$	−5.04502	−0.540882
$88$	0	0
$89$	11.8504	1.25614	0.628070	−	0.778157i	$-0.283845\pi$
$89$	11.8504	1.25614	0.628070	+	0.778157i	$0.283845\pi$
$90$	0	0
$91$	5.04502	0.528861
$92$	0	0
$93$	−7.57201	−0.785180
$94$	0	0
$95$	12.7666	1.30983
$96$	0	0
$97$	−1.87122	−0.189993	−0.0949967	−	0.995478i	$-0.530284\pi$
$97$	−1.87122	−0.189993	−0.0949967	+	0.995478i	$0.530284\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	4.51803	0.449560	0.224780	−	0.974409i	$-0.427834\pi$
$101$	4.51803	0.449560	0.224780	+	0.974409i	$0.427834\pi$
$102$	0	0
$103$	10.6468	1.04906	0.524531	−	0.851392i	$-0.324241\pi$
$103$	10.6468	1.04906	0.524531	+	0.851392i	$0.324241\pi$
$104$	0	0
$105$	−2.39821	−0.234041
$106$	0	0
$107$	−15.9702	−1.54390	−0.771949	−	0.635684i	$-0.780718\pi$
$107$	−15.9702	−1.54390	−0.771949	+	0.635684i	$0.780718\pi$
$108$	0	0
$109$	12.7756	1.22368	0.611840	−	0.790982i	$-0.290430\pi$
$109$	12.7756	1.22368	0.611840	+	0.790982i	$0.290430\pi$
$110$	0	0
$111$	−4.24860	−0.403259
$112$	0	0
$113$	18.7368	1.76261	0.881307	−	0.472544i	$-0.156664\pi$
$113$	18.7368	1.76261	0.881307	+	0.472544i	$0.156664\pi$
$114$	0	0
$115$	−11.8116	−1.10144
$116$	0	0
$117$	5.04502	0.466412
$118$	0	0
$119$	−6.36842	−0.583792
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.646809	0.0583208
$124$	0	0
$125$	−10.1890	−0.911334
$126$	0	0
$127$	2.27839	0.202174	0.101087	−	0.994878i	$-0.467768\pi$
$127$	2.27839	0.202174	0.101087	+	0.994878i	$0.467768\pi$
$128$	0	0
$129$	−10.5180	−0.926061
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	5.32340	0.461598
$134$	0	0
$135$	−2.39821	−0.206405
$136$	0	0
$137$	−4.77559	−0.408006	−0.204003	−	0.978970i	$-0.565395\pi$
$137$	−4.77559	−0.408006	−0.204003	+	0.978970i	$0.565395\pi$
$138$	0	0
$139$	15.4432	1.30988	0.654939	−	0.755682i	$-0.272694\pi$
$139$	15.4432	1.30988	0.654939	+	0.755682i	$0.272694\pi$
$140$	0	0
$141$	0.526989	0.0443805
$142$	0	0
$143$	5.04502	0.421885
$144$	0	0
$145$	12.0990	1.00477
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−9.84143	−0.806241	−0.403121	−	0.915147i	$-0.632075\pi$
$149$	−9.84143	−0.806241	−0.403121	+	0.915147i	$0.632075\pi$
$150$	0	0
$151$	4.12878	0.335996	0.167998	−	0.985787i	$-0.446270\pi$
$151$	4.12878	0.335996	0.167998	+	0.985787i	$0.446270\pi$
$152$	0	0
$153$	−6.36842	−0.514856
$154$	0	0
$155$	18.1592	1.45859
$156$	0	0
$157$	−0.946021	−0.0755007	−0.0377504	−	0.999287i	$-0.512019\pi$
$157$	−0.946021	−0.0755007	−0.0377504	+	0.999287i	$0.512019\pi$
$158$	0	0
$159$	−3.72161	−0.295143
$160$	0	0
$161$	−4.92520	−0.388160
$162$	0	0
$163$	−8.76663	−0.686655	−0.343328	−	0.939216i	$-0.611554\pi$
$163$	−8.76663	−0.686655	−0.343328	+	0.939216i	$0.611554\pi$
$164$	0	0
$165$	−2.39821	−0.186700
$166$	0	0
$167$	24.3684	1.88568	0.942842	−	0.333239i	$-0.108142\pi$
$167$	24.3684	1.88568	0.942842	+	0.333239i	$0.108142\pi$
$168$	0	0
$169$	12.4522	0.957860
$170$	0	0
$171$	5.32340	0.407091
$172$	0	0
$173$	12.3476	0.938770	0.469385	−	0.882994i	$-0.344476\pi$
$173$	12.3476	0.938770	0.469385	+	0.882994i	$0.344476\pi$
$174$	0	0
$175$	0.751399	0.0568004
$176$	0	0
$177$	7.97021	0.599078
$178$	0	0
$179$	5.59283	0.418028	0.209014	−	0.977913i	$-0.432975\pi$
$179$	5.59283	0.418028	0.209014	+	0.977913i	$0.432975\pi$
$180$	0	0
$181$	13.5720	1.00880	0.504400	−	0.863470i	$-0.331714\pi$
$181$	13.5720	1.00880	0.504400	+	0.863470i	$0.331714\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	10.1890	0.749112
$186$	0	0
$187$	−6.36842	−0.465705
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	9.42240	0.681781	0.340890	−	0.940103i	$-0.389271\pi$
$191$	9.42240	0.681781	0.340890	+	0.940103i	$0.389271\pi$
$192$	0	0
$193$	−10.1288	−0.729086	−0.364543	−	0.931187i	$-0.618775\pi$
$193$	−10.1288	−0.729086	−0.364543	+	0.931187i	$0.618775\pi$
$194$	0	0
$195$	−12.0990	−0.866427
$196$	0	0
$197$	−2.25756	−0.160845	−0.0804224	−	0.996761i	$-0.525627\pi$
$197$	−2.25756	−0.160845	−0.0804224	+	0.996761i	$0.525627\pi$
$198$	0	0
$199$	−3.07480	−0.217967	−0.108984	−	0.994044i	$-0.534760\pi$
$199$	−3.07480	−0.217967	−0.108984	+	0.994044i	$0.534760\pi$
$200$	0	0
$201$	8.76663	0.618350
$202$	0	0
$203$	5.04502	0.354091
$204$	0	0
$205$	−1.55118	−0.108339
$206$	0	0
$207$	−4.92520	−0.342325
$208$	0	0
$209$	5.32340	0.368228
$210$	0	0
$211$	14.6468	1.00833	0.504164	−	0.863608i	$-0.331801\pi$
$211$	14.6468	1.00833	0.504164	+	0.863608i	$0.331801\pi$
$212$	0	0
$213$	−11.4432	−0.784077
$214$	0	0
$215$	25.2244	1.72029
$216$	0	0
$217$	7.57201	0.514021
$218$	0	0
$219$	13.0450	0.881500
$220$	0	0
$221$	−32.1288	−2.16122
$222$	0	0
$223$	1.90997	0.127901	0.0639505	−	0.997953i	$-0.479630\pi$
$223$	1.90997	0.127901	0.0639505	+	0.997953i	$0.479630\pi$
$224$	0	0
$225$	0.751399	0.0500933
$226$	0	0
$227$	3.20359	0.212629	0.106315	−	0.994333i	$-0.466095\pi$
$227$	3.20359	0.212629	0.106315	+	0.994333i	$0.466095\pi$
$228$	0	0
$229$	−18.3088	−1.20988	−0.604941	−	0.796270i	$-0.706803\pi$
$229$	−18.3088	−1.20988	−0.604941	+	0.796270i	$0.706803\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−16.5872	−1.08667	−0.543333	−	0.839517i	$-0.682838\pi$
$233$	−16.5872	−1.08667	−0.543333	+	0.839517i	$0.682838\pi$
$234$	0	0
$235$	−1.26383	−0.0824432
$236$	0	0
$237$	11.4432	0.743317
$238$	0	0
$239$	−2.91623	−0.188635	−0.0943177	−	0.995542i	$-0.530067\pi$
$239$	−2.91623	−0.188635	−0.0943177	+	0.995542i	$0.530067\pi$
$240$	0	0
$241$	−6.09899	−0.392871	−0.196435	−	0.980517i	$-0.562937\pi$
$241$	−6.09899	−0.392871	−0.196435	+	0.980517i	$0.562937\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.39821	0.153216
$246$	0	0
$247$	26.8567	1.70885
$248$	0	0
$249$	13.1648	0.834288
$250$	0	0
$251$	−1.62262	−0.102419	−0.0512093	−	0.998688i	$-0.516308\pi$
$251$	−1.62262	−0.102419	−0.0512093	+	0.998688i	$0.516308\pi$
$252$	0	0
$253$	−4.92520	−0.309644
$254$	0	0
$255$	15.2728	0.956419
$256$	0	0
$257$	−6.89541	−0.430124	−0.215062	−	0.976600i	$-0.568995\pi$
$257$	−6.89541	−0.430124	−0.215062	+	0.976600i	$0.568995\pi$
$258$	0	0
$259$	4.24860	0.263995
$260$	0	0
$261$	5.04502	0.312279
$262$	0	0
$263$	−5.08377	−0.313478	−0.156739	−	0.987640i	$-0.550098\pi$
$263$	−5.08377	−0.313478	−0.156739	+	0.987640i	$0.550098\pi$
$264$	0	0
$265$	8.92520	0.548270
$266$	0	0
$267$	−11.8504	−0.725232
$268$	0	0
$269$	−0.886447	−0.0540476	−0.0270238	−	0.999635i	$-0.508603\pi$
$269$	−0.886447	−0.0540476	−0.0270238	+	0.999635i	$0.508603\pi$
$270$	0	0
$271$	25.3234	1.53829	0.769144	−	0.639076i	$-0.220683\pi$
$271$	25.3234	1.53829	0.769144	+	0.639076i	$0.220683\pi$
$272$	0	0
$273$	−5.04502	−0.305338
$274$	0	0
$275$	0.751399	0.0453111
$276$	0	0
$277$	−24.8269	−1.49170	−0.745851	−	0.666113i	$-0.767957\pi$
$277$	−24.8269	−1.49170	−0.745851	+	0.666113i	$0.767957\pi$
$278$	0	0
$279$	7.57201	0.453324
$280$	0	0
$281$	1.90101	0.113404	0.0567022	−	0.998391i	$-0.481941\pi$
$281$	1.90101	0.113404	0.0567022	+	0.998391i	$0.481941\pi$
$282$	0	0
$283$	22.3178	1.32666	0.663328	−	0.748329i	$-0.269143\pi$
$283$	22.3178	1.32666	0.663328	+	0.748329i	$0.269143\pi$
$284$	0	0
$285$	−12.7666	−0.756230
$286$	0	0
$287$	−0.646809	−0.0381799
$288$	0	0
$289$	23.5568	1.38569
$290$	0	0
$291$	1.87122	0.109693
$292$	0	0
$293$	12.0900	0.706307	0.353154	−	0.935565i	$-0.385109\pi$
$293$	12.0900	0.706307	0.353154	+	0.935565i	$0.385109\pi$
$294$	0	0
$295$	−19.1142	−1.11287
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−24.8477	−1.43698
$300$	0	0
$301$	10.5180	0.606249
$302$	0	0
$303$	−4.51803	−0.259554
$304$	0	0
$305$	−4.79641	−0.274642
$306$	0	0
$307$	13.5928	0.775784	0.387892	−	0.921705i	$-0.373203\pi$
$307$	13.5928	0.775784	0.387892	+	0.921705i	$0.373203\pi$
$308$	0	0
$309$	−10.6468	−0.605676
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	14.9252	0.843622	0.421811	−	0.906684i	$-0.361395\pi$
$313$	14.9252	0.843622	0.421811	+	0.906684i	$0.361395\pi$
$314$	0	0
$315$	2.39821	0.135124
$316$	0	0
$317$	3.97918	0.223493	0.111746	−	0.993737i	$-0.464356\pi$
$317$	3.97918	0.223493	0.111746	+	0.993737i	$0.464356\pi$
$318$	0	0
$319$	5.04502	0.282467
$320$	0	0
$321$	15.9702	0.891370
$322$	0	0
$323$	−33.9017	−1.88634
$324$	0	0
$325$	3.79082	0.210277
$326$	0	0
$327$	−12.7756	−0.706492
$328$	0	0
$329$	−0.526989	−0.0290539
$330$	0	0
$331$	−23.4432	−1.28856	−0.644278	−	0.764791i	$-0.722842\pi$
$331$	−23.4432	−1.28856	−0.644278	+	0.764791i	$0.722842\pi$
$332$	0	0
$333$	4.24860	0.232822
$334$	0	0
$335$	−21.0242	−1.14867
$336$	0	0
$337$	11.1648	0.608187	0.304094	−	0.952642i	$-0.401646\pi$
$337$	11.1648	0.608187	0.304094	+	0.952642i	$0.401646\pi$
$338$	0	0
$339$	−18.7368	−1.01765
$340$	0	0
$341$	7.57201	0.410047
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	11.8116	0.635918
$346$	0	0
$347$	22.5872	1.21255	0.606273	−	0.795256i	$-0.292664\pi$
$347$	22.5872	1.21255	0.606273	+	0.795256i	$0.292664\pi$
$348$	0	0
$349$	27.9315	1.49514	0.747568	−	0.664185i	$-0.231221\pi$
$349$	27.9315	1.49514	0.747568	+	0.664185i	$0.231221\pi$
$350$	0	0
$351$	−5.04502	−0.269283
$352$	0	0
$353$	−16.5478	−0.880751	−0.440376	−	0.897814i	$-0.645155\pi$
$353$	−16.5478	−0.880751	−0.440376	+	0.897814i	$0.645155\pi$
$354$	0	0
$355$	27.4432	1.45654
$356$	0	0
$357$	6.36842	0.337053
$358$	0	0
$359$	−22.0305	−1.16272	−0.581362	−	0.813645i	$-0.697480\pi$
$359$	−22.0305	−1.16272	−0.581362	+	0.813645i	$0.697480\pi$
$360$	0	0
$361$	9.33863	0.491507
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−31.2847	−1.63751
$366$	0	0
$367$	19.3836	1.01182	0.505909	−	0.862587i	$-0.331157\pi$
$367$	19.3836	1.01182	0.505909	+	0.862587i	$0.331157\pi$
$368$	0	0
$369$	−0.646809	−0.0336715
$370$	0	0
$371$	3.72161	0.193216
$372$	0	0
$373$	29.2549	1.51476	0.757380	−	0.652975i	$-0.226479\pi$
$373$	29.2549	1.51476	0.757380	+	0.652975i	$0.226479\pi$
$374$	0	0
$375$	10.1890	0.526159
$376$	0	0
$377$	25.4522	1.31085
$378$	0	0
$379$	−12.5270	−0.643468	−0.321734	−	0.946830i	$-0.604266\pi$
$379$	−12.5270	−0.643468	−0.321734	+	0.946830i	$0.604266\pi$
$380$	0	0
$381$	−2.27839	−0.116725
$382$	0	0
$383$	17.5928	0.898952	0.449476	−	0.893293i	$-0.351611\pi$
$383$	17.5928	0.898952	0.449476	+	0.893293i	$0.351611\pi$
$384$	0	0
$385$	2.39821	0.122224
$386$	0	0
$387$	10.5180	0.534661
$388$	0	0
$389$	20.0900	1.01861	0.509303	−	0.860588i	$-0.329903\pi$
$389$	20.0900	1.01861	0.509303	+	0.860588i	$0.329903\pi$
$390$	0	0
$391$	31.3657	1.58623
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	−27.4432	−1.38082
$396$	0	0
$397$	−35.1053	−1.76188	−0.880941	−	0.473226i	$-0.843090\pi$
$397$	−35.1053	−1.76188	−0.880941	+	0.473226i	$0.843090\pi$
$398$	0	0
$399$	−5.32340	−0.266504
$400$	0	0
$401$	9.57201	0.478003	0.239002	−	0.971019i	$-0.423180\pi$
$401$	9.57201	0.478003	0.239002	+	0.971019i	$0.423180\pi$
$402$	0	0
$403$	38.2009	1.90292
$404$	0	0
$405$	2.39821	0.119168
$406$	0	0
$407$	4.24860	0.210595
$408$	0	0
$409$	38.1801	1.88788	0.943941	−	0.330113i	$-0.107087\pi$
$409$	38.1801	1.88788	0.943941	+	0.330113i	$0.107087\pi$
$410$	0	0
$411$	4.77559	0.235563
$412$	0	0
$413$	−7.97021	−0.392189
$414$	0	0
$415$	−31.5720	−1.54981
$416$	0	0
$417$	−15.4432	−0.756258
$418$	0	0
$419$	−7.17380	−0.350463	−0.175231	−	0.984527i	$-0.556067\pi$
$419$	−7.17380	−0.350463	−0.175231	+	0.984527i	$0.556067\pi$
$420$	0	0
$421$	15.1530	0.738511	0.369255	−	0.929328i	$-0.379613\pi$
$421$	15.1530	0.738511	0.369255	+	0.929328i	$0.379613\pi$
$422$	0	0
$423$	−0.526989	−0.0256231
$424$	0	0
$425$	−4.78522	−0.232117
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−5.04502	−0.243576
$430$	0	0
$431$	−5.56304	−0.267962	−0.133981	−	0.990984i	$-0.542776\pi$
$431$	−5.56304	−0.267962	−0.133981	+	0.990984i	$0.542776\pi$
$432$	0	0
$433$	−25.6412	−1.23224	−0.616119	−	0.787653i	$-0.711296\pi$
$433$	−25.6412	−1.23224	−0.616119	+	0.787653i	$0.711296\pi$
$434$	0	0
$435$	−12.0990	−0.580102
$436$	0	0
$437$	−26.2188	−1.25422
$438$	0	0
$439$	−23.6710	−1.12976	−0.564878	−	0.825175i	$-0.691077\pi$
$439$	−23.6710	−1.12976	−0.564878	+	0.825175i	$0.691077\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.0305	0.856653	0.428326	−	0.903624i	$-0.359103\pi$
$443$	18.0305	0.856653	0.428326	+	0.903624i	$0.359103\pi$
$444$	0	0
$445$	28.4197	1.34722
$446$	0	0
$447$	9.84143	0.465484
$448$	0	0
$449$	−34.9765	−1.65064	−0.825321	−	0.564664i	$-0.809006\pi$
$449$	−34.9765	−1.65064	−0.825321	+	0.564664i	$0.809006\pi$
$450$	0	0
$451$	−0.646809	−0.0304570
$452$	0	0
$453$	−4.12878	−0.193987
$454$	0	0
$455$	12.0990	0.567210
$456$	0	0
$457$	4.53595	0.212183	0.106091	−	0.994356i	$-0.466166\pi$
$457$	4.53595	0.212183	0.106091	+	0.994356i	$0.466166\pi$
$458$	0	0
$459$	6.36842	0.297252
$460$	0	0
$461$	−2.79641	−0.130242	−0.0651210	−	0.997877i	$-0.520743\pi$
$461$	−2.79641	−0.130242	−0.0651210	+	0.997877i	$0.520743\pi$
$462$	0	0
$463$	−38.3595	−1.78272	−0.891358	−	0.453301i	$-0.850246\pi$
$463$	−38.3595	−1.78272	−0.891358	+	0.453301i	$0.850246\pi$
$464$	0	0
$465$	−18.1592	−0.842115
$466$	0	0
$467$	20.4674	0.947119	0.473560	−	0.880762i	$-0.342969\pi$
$467$	20.4674	0.947119	0.473560	+	0.880762i	$0.342969\pi$
$468$	0	0
$469$	−8.76663	−0.404805
$470$	0	0
$471$	0.946021	0.0435904
$472$	0	0
$473$	10.5180	0.483619
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	3.72161	0.170401
$478$	0	0
$479$	11.6137	0.530641	0.265321	−	0.964160i	$-0.414522\pi$
$479$	11.6137	0.530641	0.265321	+	0.964160i	$0.414522\pi$
$480$	0	0
$481$	21.4343	0.977318
$482$	0	0
$483$	4.92520	0.224104
$484$	0	0
$485$	−4.48757	−0.203770
$486$	0	0
$487$	−32.4793	−1.47178	−0.735888	−	0.677103i	$-0.763235\pi$
$487$	−32.4793	−1.47178	−0.735888	+	0.677103i	$0.763235\pi$
$488$	0	0
$489$	8.76663	0.396441
$490$	0	0
$491$	−26.6766	−1.20390	−0.601949	−	0.798535i	$-0.705609\pi$
$491$	−26.6766	−1.20390	−0.601949	+	0.798535i	$0.705609\pi$
$492$	0	0
$493$	−32.1288	−1.44701
$494$	0	0
$495$	2.39821	0.107791
$496$	0	0
$497$	11.4432	0.513299
$498$	0	0
$499$	41.2459	1.84642	0.923210	−	0.384296i	$-0.125556\pi$
$499$	41.2459	1.84642	0.923210	+	0.384296i	$0.125556\pi$
$500$	0	0
$501$	−24.3684	−1.08870
$502$	0	0
$503$	−30.5180	−1.36073	−0.680366	−	0.732873i	$-0.738179\pi$
$503$	−30.5180	−1.36073	−0.680366	+	0.732873i	$0.738179\pi$
$504$	0	0
$505$	10.8352	0.482159
$506$	0	0
$507$	−12.4522	−0.553021
$508$	0	0
$509$	−18.9944	−0.841912	−0.420956	−	0.907081i	$-0.638305\pi$
$509$	−18.9944	−0.841912	−0.420956	+	0.907081i	$0.638305\pi$
$510$	0	0
$511$	−13.0450	−0.577078
$512$	0	0
$513$	−5.32340	−0.235034
$514$	0	0
$515$	25.5333	1.12513
$516$	0	0
$517$	−0.526989	−0.0231770
$518$	0	0
$519$	−12.3476	−0.541999
$520$	0	0
$521$	25.2430	1.10592	0.552958	−	0.833209i	$-0.313499\pi$
$521$	25.2430	1.10592	0.552958	+	0.833209i	$0.313499\pi$
$522$	0	0
$523$	2.93416	0.128302	0.0641509	−	0.997940i	$-0.479566\pi$
$523$	2.93416	0.128302	0.0641509	+	0.997940i	$0.479566\pi$
$524$	0	0
$525$	−0.751399	−0.0327937
$526$	0	0
$527$	−48.2217	−2.10057
$528$	0	0
$529$	1.25756	0.0546767
$530$	0	0
$531$	−7.97021	−0.345878
$532$	0	0
$533$	−3.26316	−0.141343
$534$	0	0
$535$	−38.2999	−1.65585
$536$	0	0
$537$	−5.59283	−0.241348
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	8.90437	0.382829	0.191414	−	0.981509i	$-0.438693\pi$
$541$	8.90437	0.382829	0.191414	+	0.981509i	$0.438693\pi$
$542$	0	0
$543$	−13.5720	−0.582430
$544$	0	0
$545$	30.6385	1.31241
$546$	0	0
$547$	29.4737	1.26020	0.630102	−	0.776513i	$-0.283013\pi$
$547$	29.4737	1.26020	0.630102	+	0.776513i	$0.283013\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	26.8567	1.14413
$552$	0	0
$553$	−11.4432	−0.486615
$554$	0	0
$555$	−10.1890	−0.432500
$556$	0	0
$557$	14.8954	0.631139	0.315569	−	0.948903i	$-0.397804\pi$
$557$	14.8954	0.631139	0.315569	+	0.948903i	$0.397804\pi$
$558$	0	0
$559$	53.0636	2.24435
$560$	0	0
$561$	6.36842	0.268875
$562$	0	0
$563$	7.81164	0.329222	0.164611	−	0.986359i	$-0.447363\pi$
$563$	7.81164	0.329222	0.164611	+	0.986359i	$0.447363\pi$
$564$	0	0
$565$	44.9348	1.89042
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−25.5512	−1.06928	−0.534642	−	0.845079i	$-0.679553\pi$
$571$	−25.5512	−1.06928	−0.534642	+	0.845079i	$0.679553\pi$
$572$	0	0
$573$	−9.42240	−0.393626
$574$	0	0
$575$	−3.70079	−0.154334
$576$	0	0
$577$	−29.5124	−1.22862	−0.614309	−	0.789065i	$-0.710565\pi$
$577$	−29.5124	−1.22862	−0.614309	+	0.789065i	$0.710565\pi$
$578$	0	0
$579$	10.1288	0.420938
$580$	0	0
$581$	−13.1648	−0.546169
$582$	0	0
$583$	3.72161	0.154133
$584$	0	0
$585$	12.0990	0.500232
$586$	0	0
$587$	−31.3955	−1.29583	−0.647916	−	0.761712i	$-0.724359\pi$
$587$	−31.3955	−1.29583	−0.647916	+	0.761712i	$0.724359\pi$
$588$	0	0
$589$	40.3088	1.66090
$590$	0	0
$591$	2.25756	0.0928638
$592$	0	0
$593$	−7.90997	−0.324823	−0.162412	−	0.986723i	$-0.551927\pi$
$593$	−7.90997	−0.324823	−0.162412	+	0.986723i	$0.551927\pi$
$594$	0	0
$595$	−15.2728	−0.626123
$596$	0	0
$597$	3.07480	0.125843
$598$	0	0
$599$	−27.4432	−1.12130	−0.560650	−	0.828053i	$-0.689449\pi$
$599$	−27.4432	−1.12130	−0.560650	+	0.828053i	$0.689449\pi$
$600$	0	0
$601$	31.9910	1.30494	0.652471	−	0.757814i	$-0.273732\pi$
$601$	31.9910	1.30494	0.652471	+	0.757814i	$0.273732\pi$
$602$	0	0
$603$	−8.76663	−0.357005
$604$	0	0
$605$	2.39821	0.0975010
$606$	0	0
$607$	−7.41344	−0.300902	−0.150451	−	0.988617i	$-0.548073\pi$
$607$	−7.41344	−0.300902	−0.150451	+	0.988617i	$0.548073\pi$
$608$	0	0
$609$	−5.04502	−0.204434
$610$	0	0
$611$	−2.65867	−0.107558
$612$	0	0
$613$	−33.9917	−1.37291	−0.686456	−	0.727171i	$-0.740835\pi$
$613$	−33.9917	−1.37291	−0.686456	+	0.727171i	$0.740835\pi$
$614$	0	0
$615$	1.55118	0.0625497
$616$	0	0
$617$	−44.0305	−1.77260	−0.886300	−	0.463112i	$-0.846733\pi$
$617$	−44.0305	−1.77260	−0.886300	+	0.463112i	$0.846733\pi$
$618$	0	0
$619$	−40.0096	−1.60812	−0.804061	−	0.594546i	$-0.797332\pi$
$619$	−40.0096	−1.60812	−0.804061	+	0.594546i	$0.797332\pi$
$620$	0	0
$621$	4.92520	0.197641
$622$	0	0
$623$	11.8504	0.474776
$624$	0	0
$625$	−28.1924	−1.12770
$626$	0	0
$627$	−5.32340	−0.212596
$628$	0	0
$629$	−27.0569	−1.07883
$630$	0	0
$631$	−28.5568	−1.13683	−0.568414	−	0.822743i	$-0.692443\pi$
$631$	−28.5568	−1.13683	−0.568414	+	0.822743i	$0.692443\pi$
$632$	0	0
$633$	−14.6468	−0.582158
$634$	0	0
$635$	5.46405	0.216834
$636$	0	0
$637$	5.04502	0.199891
$638$	0	0
$639$	11.4432	0.452687
$640$	0	0
$641$	−31.1053	−1.22858	−0.614292	−	0.789079i	$-0.710558\pi$
$641$	−31.1053	−1.22858	−0.614292	+	0.789079i	$0.710558\pi$
$642$	0	0
$643$	5.48197	0.216188	0.108094	−	0.994141i	$-0.465525\pi$
$643$	5.48197	0.216188	0.108094	+	0.994141i	$0.465525\pi$
$644$	0	0
$645$	−25.2244	−0.993210
$646$	0	0
$647$	9.26383	0.364199	0.182099	−	0.983280i	$-0.441711\pi$
$647$	9.26383	0.364199	0.182099	+	0.983280i	$0.441711\pi$
$648$	0	0
$649$	−7.97021	−0.312858
$650$	0	0
$651$	−7.57201	−0.296770
$652$	0	0
$653$	−29.9821	−1.17329	−0.586645	−	0.809844i	$-0.699551\pi$
$653$	−29.9821	−1.17329	−0.586645	+	0.809844i	$0.699551\pi$
$654$	0	0
$655$	−9.59283	−0.374823
$656$	0	0
$657$	−13.0450	−0.508935
$658$	0	0
$659$	−23.9702	−0.933747	−0.466873	−	0.884324i	$-0.654620\pi$
$659$	−23.9702	−0.933747	−0.466873	+	0.884324i	$0.654620\pi$
$660$	0	0
$661$	−40.4585	−1.57365	−0.786826	−	0.617175i	$-0.788277\pi$
$661$	−40.4585	−1.57365	−0.786826	+	0.617175i	$0.788277\pi$
$662$	0	0
$663$	32.1288	1.24778
$664$	0	0
$665$	12.7666	0.495069
$666$	0	0
$667$	−24.8477	−0.962107
$668$	0	0
$669$	−1.90997	−0.0738436
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	21.8712	0.843073	0.421537	−	0.906811i	$-0.361491\pi$
$673$	21.8712	0.843073	0.421537	+	0.906811i	$0.361491\pi$
$674$	0	0
$675$	−0.751399	−0.0289214
$676$	0	0
$677$	−1.26316	−0.0485472	−0.0242736	−	0.999705i	$-0.507727\pi$
$677$	−1.26316	−0.0485472	−0.0242736	+	0.999705i	$0.507727\pi$
$678$	0	0
$679$	−1.87122	−0.0718108
$680$	0	0
$681$	−3.20359	−0.122762
$682$	0	0
$683$	−37.6441	−1.44041	−0.720206	−	0.693760i	$-0.755953\pi$
$683$	−37.6441	−1.44041	−0.720206	+	0.693760i	$0.755953\pi$
$684$	0	0
$685$	−11.4529	−0.437591
$686$	0	0
$687$	18.3088	0.698526
$688$	0	0
$689$	18.7756	0.715293
$690$	0	0
$691$	−14.3892	−0.547393	−0.273696	−	0.961816i	$-0.588246\pi$
$691$	−14.3892	−0.547393	−0.273696	+	0.961816i	$0.588246\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	37.0361	1.40486
$696$	0	0
$697$	4.11915	0.156024
$698$	0	0
$699$	16.5872	0.627387
$700$	0	0
$701$	−39.2936	−1.48410	−0.742050	−	0.670345i	$-0.766146\pi$
$701$	−39.2936	−1.48410	−0.742050	+	0.670345i	$0.766146\pi$
$702$	0	0
$703$	22.6170	0.853017
$704$	0	0
$705$	1.26383	0.0475986
$706$	0	0
$707$	4.51803	0.169918
$708$	0	0
$709$	−49.2430	−1.84936	−0.924680	−	0.380745i	$-0.875667\pi$
$709$	−49.2430	−1.84936	−0.924680	+	0.380745i	$0.875667\pi$
$710$	0	0
$711$	−11.4432	−0.429154
$712$	0	0
$713$	−37.2936	−1.39666
$714$	0	0
$715$	12.0990	0.452477
$716$	0	0
$717$	2.91623	0.108909
$718$	0	0
$719$	7.41344	0.276475	0.138237	−	0.990399i	$-0.455856\pi$
$719$	7.41344	0.276475	0.138237	+	0.990399i	$0.455856\pi$
$720$	0	0
$721$	10.6468	0.396508
$722$	0	0
$723$	6.09899	0.226824
$724$	0	0
$725$	3.79082	0.140787
$726$	0	0
$727$	−18.9557	−0.703026	−0.351513	−	0.936183i	$-0.614333\pi$
$727$	−18.9557	−0.703026	−0.351513	+	0.936183i	$0.614333\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−66.9832	−2.47746
$732$	0	0
$733$	−3.59283	−0.132704	−0.0663521	−	0.997796i	$-0.521136\pi$
$733$	−3.59283	−0.132704	−0.0663521	+	0.997796i	$0.521136\pi$
$734$	0	0
$735$	−2.39821	−0.0884592
$736$	0	0
$737$	−8.76663	−0.322923
$738$	0	0
$739$	26.7756	0.984956	0.492478	−	0.870325i	$-0.336091\pi$
$739$	26.7756	0.984956	0.492478	+	0.870325i	$0.336091\pi$
$740$	0	0
$741$	−26.8567	−0.986604
$742$	0	0
$743$	33.8027	1.24010	0.620050	−	0.784562i	$-0.287112\pi$
$743$	33.8027	1.24010	0.620050	+	0.784562i	$0.287112\pi$
$744$	0	0
$745$	−23.6018	−0.864703
$746$	0	0
$747$	−13.1648	−0.481676
$748$	0	0
$749$	−15.9702	−0.583539
$750$	0	0
$751$	−35.3955	−1.29160	−0.645800	−	0.763506i	$-0.723476\pi$
$751$	−35.3955	−1.29160	−0.645800	+	0.763506i	$0.723476\pi$
$752$	0	0
$753$	1.62262	0.0591314
$754$	0	0
$755$	9.90168	0.360359
$756$	0	0
$757$	−29.3442	−1.06653	−0.533267	−	0.845947i	$-0.679036\pi$
$757$	−29.3442	−1.06653	−0.533267	+	0.845947i	$0.679036\pi$
$758$	0	0
$759$	4.92520	0.178773
$760$	0	0
$761$	12.9044	0.467783	0.233892	−	0.972263i	$-0.424854\pi$
$761$	12.9044	0.467783	0.233892	+	0.972263i	$0.424854\pi$
$762$	0	0
$763$	12.7756	0.462507
$764$	0	0
$765$	−15.2728	−0.552189
$766$	0	0
$767$	−40.2099	−1.45189
$768$	0	0
$769$	9.78186	0.352743	0.176371	−	0.984324i	$-0.443564\pi$
$769$	9.78186	0.352743	0.176371	+	0.984324i	$0.443564\pi$
$770$	0	0
$771$	6.89541	0.248332
$772$	0	0
$773$	29.7223	1.06904	0.534518	−	0.845157i	$-0.320493\pi$
$773$	29.7223	1.06904	0.534518	+	0.845157i	$0.320493\pi$
$774$	0	0
$775$	5.68960	0.204376
$776$	0	0
$777$	−4.24860	−0.152418
$778$	0	0
$779$	−3.44322	−0.123366
$780$	0	0
$781$	11.4432	0.409471
$782$	0	0
$783$	−5.04502	−0.180294
$784$	0	0
$785$	−2.26875	−0.0809753
$786$	0	0
$787$	−17.0242	−0.606847	−0.303423	−	0.952856i	$-0.598130\pi$
$787$	−17.0242	−0.606847	−0.303423	+	0.952856i	$0.598130\pi$
$788$	0	0
$789$	5.08377	0.180987
$790$	0	0
$791$	18.7368	0.666205
$792$	0	0
$793$	−10.0900	−0.358308
$794$	0	0
$795$	−8.92520	−0.316544
$796$	0	0
$797$	−36.4287	−1.29037	−0.645185	−	0.764027i	$-0.723220\pi$
$797$	−36.4287	−1.29037	−0.645185	+	0.764027i	$0.723220\pi$
$798$	0	0
$799$	3.35609	0.118730
$800$	0	0
$801$	11.8504	0.418713
$802$	0	0
$803$	−13.0450	−0.460349
$804$	0	0
$805$	−11.8116	−0.416306
$806$	0	0
$807$	0.886447	0.0312044
$808$	0	0
$809$	44.4882	1.56412	0.782062	−	0.623201i	$-0.214168\pi$
$809$	44.4882	1.56412	0.782062	+	0.623201i	$0.214168\pi$
$810$	0	0
$811$	−7.65307	−0.268736	−0.134368	−	0.990932i	$-0.542900\pi$
$811$	−7.65307	−0.268736	−0.134368	+	0.990932i	$0.542900\pi$
$812$	0	0
$813$	−25.3234	−0.888131
$814$	0	0
$815$	−21.0242	−0.736445
$816$	0	0
$817$	55.9917	1.95890
$818$	0	0
$819$	5.04502	0.176287
$820$	0	0
$821$	−44.3691	−1.54849	−0.774246	−	0.632885i	$-0.781871\pi$
$821$	−44.3691	−1.54849	−0.774246	+	0.632885i	$0.781871\pi$
$822$	0	0
$823$	−6.61702	−0.230655	−0.115327	−	0.993328i	$-0.536792\pi$
$823$	−6.61702	−0.230655	−0.115327	+	0.993328i	$0.536792\pi$
$824$	0	0
$825$	−0.751399	−0.0261604
$826$	0	0
$827$	−39.7126	−1.38094	−0.690472	−	0.723359i	$-0.742597\pi$
$827$	−39.7126	−1.38094	−0.690472	+	0.723359i	$0.742597\pi$
$828$	0	0
$829$	−3.90997	−0.135799	−0.0678994	−	0.997692i	$-0.521630\pi$
$829$	−3.90997	−0.135799	−0.0678994	+	0.997692i	$0.521630\pi$
$830$	0	0
$831$	24.8269	0.861235
$832$	0	0
$833$	−6.36842	−0.220653
$834$	0	0
$835$	58.4405	2.02242
$836$	0	0
$837$	−7.57201	−0.261727
$838$	0	0
$839$	9.58097	0.330772	0.165386	−	0.986229i	$-0.447113\pi$
$839$	9.58097	0.330772	0.165386	+	0.986229i	$0.447113\pi$
$840$	0	0
$841$	−3.54781	−0.122338
$842$	0	0
$843$	−1.90101	−0.0654741
$844$	0	0
$845$	29.8629	1.02732
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−22.3178	−0.765945
$850$	0	0
$851$	−20.9252	−0.717307
$852$	0	0
$853$	−14.5568	−0.498415	−0.249207	−	0.968450i	$-0.580170\pi$
$853$	−14.5568	−0.498415	−0.249207	+	0.968450i	$0.580170\pi$
$854$	0	0
$855$	12.7666	0.436609
$856$	0	0
$857$	10.4793	0.357965	0.178983	−	0.983852i	$-0.442719\pi$
$857$	10.4793	0.357965	0.178983	+	0.983852i	$0.442719\pi$
$858$	0	0
$859$	−2.88645	−0.0984843	−0.0492421	−	0.998787i	$-0.515681\pi$
$859$	−2.88645	−0.0984843	−0.0492421	+	0.998787i	$0.515681\pi$
$860$	0	0
$861$	0.646809	0.0220432
$862$	0	0
$863$	20.5485	0.699479	0.349739	−	0.936847i	$-0.386270\pi$
$863$	20.5485	0.699479	0.349739	+	0.936847i	$0.386270\pi$
$864$	0	0
$865$	29.6121	1.00684
$866$	0	0
$867$	−23.5568	−0.800030
$868$	0	0
$869$	−11.4432	−0.388185
$870$	0	0
$871$	−44.2278	−1.49860
$872$	0	0
$873$	−1.87122	−0.0633311
$874$	0	0
$875$	−10.1890	−0.344452
$876$	0	0
$877$	59.1149	1.99617	0.998084	−	0.0618724i	$-0.0197072\pi$
$877$	59.1149	1.99617	0.998084	+	0.0618724i	$0.0197072\pi$
$878$	0	0
$879$	−12.0900	−0.407787
$880$	0	0
$881$	30.3982	1.02414	0.512071	−	0.858943i	$-0.328879\pi$
$881$	30.3982	1.02414	0.512071	+	0.858943i	$0.328879\pi$
$882$	0	0
$883$	35.6114	1.19842	0.599210	−	0.800592i	$-0.295481\pi$
$883$	35.6114	1.19842	0.599210	+	0.800592i	$0.295481\pi$
$884$	0	0
$885$	19.1142	0.642518
$886$	0	0
$887$	−22.9736	−0.771377	−0.385689	−	0.922629i	$-0.626036\pi$
$887$	−22.9736	−0.771377	−0.385689	+	0.922629i	$0.626036\pi$
$888$	0	0
$889$	2.27839	0.0764147
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−2.80538	−0.0938784
$894$	0	0
$895$	13.4128	0.448339
$896$	0	0
$897$	24.8477	0.829640
$898$	0	0
$899$	38.2009	1.27407
$900$	0	0
$901$	−23.7008	−0.789588
$902$	0	0
$903$	−10.5180	−0.350018
$904$	0	0
$905$	32.5485	1.08195
$906$	0	0
$907$	57.1745	1.89845	0.949224	−	0.314602i	$-0.101871\pi$
$907$	57.1745	1.89845	0.949224	+	0.314602i	$0.101871\pi$
$908$	0	0
$909$	4.51803	0.149853
$910$	0	0
$911$	−6.82687	−0.226184	−0.113092	−	0.993584i	$-0.536076\pi$
$911$	−6.82687	−0.226184	−0.113092	+	0.993584i	$0.536076\pi$
$912$	0	0
$913$	−13.1648	−0.435692
$914$	0	0
$915$	4.79641	0.158565
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−12.0692	−0.398126	−0.199063	−	0.979987i	$-0.563790\pi$
$919$	−12.0692	−0.398126	−0.199063	+	0.979987i	$0.563790\pi$
$920$	0	0
$921$	−13.5928	−0.447899
$922$	0	0
$923$	57.7312	1.90025
$924$	0	0
$925$	3.19239	0.104965
$926$	0	0
$927$	10.6468	0.349687
$928$	0	0
$929$	26.8954	0.882410	0.441205	−	0.897406i	$-0.354551\pi$
$929$	26.8954	0.882410	0.441205	+	0.897406i	$0.354551\pi$
$930$	0	0
$931$	5.32340	0.174468
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−15.2728	−0.499474
$936$	0	0
$937$	−14.9944	−0.489846	−0.244923	−	0.969543i	$-0.578763\pi$
$937$	−14.9944	−0.489846	−0.244923	+	0.969543i	$0.578763\pi$
$938$	0	0
$939$	−14.9252	−0.487065
$940$	0	0
$941$	−30.1205	−0.981900	−0.490950	−	0.871188i	$-0.663350\pi$
$941$	−30.1205	−0.981900	−0.490950	+	0.871188i	$0.663350\pi$
$942$	0	0
$943$	3.18566	0.103739
$944$	0	0
$945$	−2.39821	−0.0780137
$946$	0	0
$947$	17.3532	0.563903	0.281951	−	0.959429i	$-0.409018\pi$
$947$	17.3532	0.563903	0.281951	+	0.959429i	$0.409018\pi$
$948$	0	0
$949$	−65.8123	−2.13636
$950$	0	0
$951$	−3.97918	−0.129034
$952$	0	0
$953$	−2.14064	−0.0693422	−0.0346711	−	0.999399i	$-0.511038\pi$
$953$	−2.14064	−0.0693422	−0.0346711	+	0.999399i	$0.511038\pi$
$954$	0	0
$955$	22.5969	0.731217
$956$	0	0
$957$	−5.04502	−0.163082
$958$	0	0
$959$	−4.77559	−0.154212
$960$	0	0
$961$	26.3353	0.849525
$962$	0	0
$963$	−15.9702	−0.514633
$964$	0	0
$965$	−24.2909	−0.781952
$966$	0	0
$967$	1.53326	0.0493062	0.0246531	−	0.999696i	$-0.492152\pi$
$967$	1.53326	0.0493062	0.0246531	+	0.999696i	$0.492152\pi$
$968$	0	0
$969$	33.9017	1.08908
$970$	0	0
$971$	26.5574	0.852269	0.426135	−	0.904660i	$-0.359875\pi$
$971$	26.5574	0.852269	0.426135	+	0.904660i	$0.359875\pi$
$972$	0	0
$973$	15.4432	0.495087
$974$	0	0
$975$	−3.79082	−0.121403
$976$	0	0
$977$	−55.9017	−1.78845	−0.894227	−	0.447615i	$-0.852274\pi$
$977$	−55.9017	−1.78845	−0.894227	+	0.447615i	$0.852274\pi$
$978$	0	0
$979$	11.8504	0.378740
$980$	0	0
$981$	12.7756	0.407893
$982$	0	0
$983$	53.0361	1.69159	0.845794	−	0.533510i	$-0.179127\pi$
$983$	53.0361	1.69159	0.845794	+	0.533510i	$0.179127\pi$
$984$	0	0
$985$	−5.41411	−0.172508
$986$	0	0
$987$	0.526989	0.0167743
$988$	0	0
$989$	−51.8034	−1.64725
$990$	0	0
$991$	−14.7362	−0.468110	−0.234055	−	0.972223i	$-0.575200\pi$
$991$	−14.7362	−0.468110	−0.234055	+	0.972223i	$0.575200\pi$
$992$	0	0
$993$	23.4432	0.743948
$994$	0	0
$995$	−7.37402	−0.233772
$996$	0	0
$997$	−45.0665	−1.42727	−0.713635	−	0.700517i	$-0.752953\pi$
$997$	−45.0665	−1.42727	−0.713635	+	0.700517i	$0.752953\pi$
$998$	0	0
$999$	−4.24860	−0.134420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bo.1.2		3
4.3	odd	2		231.2.a.e.1.2	✓	3
12.11	even	2		693.2.a.l.1.2		3
20.19	odd	2		5775.2.a.bp.1.2		3
28.27	even	2		1617.2.a.t.1.2		3
44.43	even	2		2541.2.a.bg.1.2		3
84.83	odd	2		4851.2.a.bi.1.2		3
132.131	odd	2		7623.2.a.cd.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.2	✓	3	4.3	odd	2
693.2.a.l.1.2		3	12.11	even	2
1617.2.a.t.1.2		3	28.27	even	2
2541.2.a.bg.1.2		3	44.43	even	2
3696.2.a.bo.1.2		3	1.1	even	1	trivial
4851.2.a.bi.1.2		3	84.83	odd	2
5775.2.a.bp.1.2		3	20.19	odd	2
7623.2.a.cd.1.2		3	132.131	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:	\( N \)	\(=\)	\( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3696.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.39821 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.39821 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.04502 q^{13} -2.39821 q^{15} -6.36842 q^{17} +5.32340 q^{19} -1.00000 q^{21} -4.92520 q^{23} +0.751399 q^{25} -1.00000 q^{27} +5.04502 q^{29} +7.57201 q^{31} -1.00000 q^{33} +2.39821 q^{35} +4.24860 q^{37} -5.04502 q^{39} -0.646809 q^{41} +10.5180 q^{43} +2.39821 q^{45} -0.526989 q^{47} +1.00000 q^{49} +6.36842 q^{51} +3.72161 q^{53} +2.39821 q^{55} -5.32340 q^{57} -7.97021 q^{59} -2.00000 q^{61} +1.00000 q^{63} +12.0990 q^{65} -8.76663 q^{67} +4.92520 q^{69} +11.4432 q^{71} -13.0450 q^{73} -0.751399 q^{75} +1.00000 q^{77} -11.4432 q^{79} +1.00000 q^{81} -13.1648 q^{83} -15.2728 q^{85} -5.04502 q^{87} +11.8504 q^{89} +5.04502 q^{91} -7.57201 q^{93} +12.7666 q^{95} -1.87122 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41} + 14 q^{43} + 4 q^{45} + 3 q^{49} - 8 q^{51} + 4 q^{55} - 8 q^{57} - 6 q^{61} + 3 q^{63} + 14 q^{65} + 4 q^{67} + 10 q^{69} + 12 q^{71} - 20 q^{73} - 15 q^{75} + 3 q^{77} - 12 q^{79} + 3 q^{81} - 6 q^{83} - 6 q^{85} + 4 q^{87} + 26 q^{89} - 4 q^{91} - 2 q^{93} + 8 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 4 * q^15 + 8 * q^17 + 8 * q^19 - 3 * q^21 - 10 * q^23 + 15 * q^25 - 3 * q^27 - 4 * q^29 + 2 * q^31 - 3 * q^33 + 4 * q^35 + 4 * q^39 + 14 * q^41 + 14 * q^43 + 4 * q^45 + 3 * q^49 - 8 * q^51 + 4 * q^55 - 8 * q^57 - 6 * q^61 + 3 * q^63 + 14 * q^65 + 4 * q^67 + 10 * q^69 + 12 * q^71 - 20 * q^73 - 15 * q^75 + 3 * q^77 - 12 * q^79 + 3 * q^81 - 6 * q^83 - 6 * q^85 + 4 * q^87 + 26 * q^89 - 4 * q^91 - 2 * q^93 + 8 * q^95 - 4 * q^97 + 3 * q^99

Introduction

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