LMFDB - Embedded newform 3312.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,2,Mod(1,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, 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("3312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3312.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:	$26.4464531494$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{5} -4.47214 q^{7} +O(q^{10})$	$q+3.23607 q^{5} -4.47214 q^{7} -0.763932 q^{11} -4.47214 q^{13} +4.00000 q^{17} +7.70820 q^{19} +1.00000 q^{23} +5.47214 q^{25} -4.47214 q^{29} -6.47214 q^{31} -14.4721 q^{35} +6.76393 q^{37} +2.00000 q^{41} +9.23607 q^{43} +4.00000 q^{47} +13.0000 q^{49} -0.763932 q^{53} -2.47214 q^{55} +8.94427 q^{59} +5.23607 q^{61} -14.4721 q^{65} +3.70820 q^{67} -8.94427 q^{71} +4.47214 q^{73} +3.41641 q^{77} +4.47214 q^{79} +8.76393 q^{83} +12.9443 q^{85} +1.52786 q^{89} +20.0000 q^{91} +24.9443 q^{95} -8.47214 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} - 6 q^{11} + 8 q^{17} + 2 q^{19} + 2 q^{23} + 2 q^{25} - 4 q^{31} - 20 q^{35} + 18 q^{37} + 4 q^{41} + 14 q^{43} + 8 q^{47} + 26 q^{49} - 6 q^{53} + 4 q^{55} + 6 q^{61} - 20 q^{65} - 6 q^{67} - 20 q^{77} + 22 q^{83} + 8 q^{85} + 12 q^{89} + 40 q^{91} + 32 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 6 * q^11 + 8 * q^17 + 2 * q^19 + 2 * q^23 + 2 * q^25 - 4 * q^31 - 20 * q^35 + 18 * q^37 + 4 * q^41 + 14 * q^43 + 8 * q^47 + 26 * q^49 - 6 * q^53 + 4 * q^55 + 6 * q^61 - 20 * q^65 - 6 * q^67 - 20 * q^77 + 22 * q^83 + 8 * q^85 + 12 * q^89 + 40 * q^91 + 32 * q^95 - 8 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	−4.47214	−1.69031	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	−4.47214	−1.69031	−0.845154	+	0.534522i	$0.820491\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	7.70820	1.76838	0.884192	−	0.467124i	$-0.154710\pi$
$19$	7.70820	1.76838	0.884192	+	0.467124i	$0.154710\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−14.4721	−2.44624
$36$	0	0
$37$	6.76393	1.11198	0.555992	−	0.831188i	$-0.312339\pi$
$37$	6.76393	1.11198	0.555992	+	0.831188i	$0.312339\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	9.23607	1.40849	0.704244	−	0.709958i	$-0.251286\pi$
$43$	9.23607	1.40849	0.704244	+	0.709958i	$0.251286\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.763932	−0.104934	−0.0524671	−	0.998623i	$-0.516708\pi$
$53$	−0.763932	−0.104934	−0.0524671	+	0.998623i	$0.516708\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	5.23607	0.670410	0.335205	−	0.942145i	$-0.391194\pi$
$61$	5.23607	0.670410	0.335205	+	0.942145i	$0.391194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.4721	−1.79505
$66$	0	0
$67$	3.70820	0.453029	0.226515	−	0.974008i	$-0.427267\pi$
$67$	3.70820	0.453029	0.226515	+	0.974008i	$0.427267\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	0	0
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.41641	0.389336
$78$	0	0
$79$	4.47214	0.503155	0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214	0.503155	0.251577	+	0.967837i	$0.419051\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.76393	0.961967	0.480983	−	0.876730i	$-0.340280\pi$
$83$	8.76393	0.961967	0.480983	+	0.876730i	$0.340280\pi$
$84$	0	0
$85$	12.9443	1.40400
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.52786	0.161953	0.0809766	−	0.996716i	$-0.474196\pi$
$89$	1.52786	0.161953	0.0809766	+	0.996716i	$0.474196\pi$
$90$	0	0
$91$	20.0000	2.09657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	24.9443	2.55923
$96$	0	0
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.6525	1.80320	0.901601	−	0.432568i	$-0.142392\pi$
$107$	18.6525	1.80320	0.901601	+	0.432568i	$0.142392\pi$
$108$	0	0
$109$	9.23607	0.884655	0.442327	−	0.896854i	$-0.354153\pi$
$109$	9.23607	0.884655	0.442327	+	0.896854i	$0.354153\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.4721	1.36142	0.680712	−	0.732551i	$-0.261671\pi$
$113$	14.4721	1.36142	0.680712	+	0.732551i	$0.261671\pi$
$114$	0	0
$115$	3.23607	0.301765
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−17.8885	−1.63984
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.52786	0.136656
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.4721	−1.61392	−0.806959	−	0.590607i	$-0.798888\pi$
$131$	−18.4721	−1.61392	−0.806959	+	0.590607i	$0.798888\pi$
$132$	0	0
$133$	−34.4721	−2.98911
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.9443	−1.78939	−0.894695	−	0.446678i	$-0.852607\pi$
$137$	−20.9443	−1.78939	−0.894695	+	0.446678i	$0.852607\pi$
$138$	0	0
$139$	−0.944272	−0.0800921	−0.0400460	−	0.999198i	$-0.512750\pi$
$139$	−0.944272	−0.0800921	−0.0400460	+	0.999198i	$0.512750\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.41641	0.285694
$144$	0	0
$145$	−14.4721	−1.20185
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.70820	−0.139942	−0.0699708	−	0.997549i	$-0.522291\pi$
$149$	−1.70820	−0.139942	−0.0699708	+	0.997549i	$0.522291\pi$
$150$	0	0
$151$	5.52786	0.449851	0.224926	−	0.974376i	$-0.427786\pi$
$151$	5.52786	0.449851	0.224926	+	0.974376i	$0.427786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−20.9443	−1.68228
$156$	0	0
$157$	24.6525	1.96748	0.983741	−	0.179594i	$-0.0574783\pi$
$157$	24.6525	1.96748	0.983741	+	0.179594i	$0.0574783\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.47214	−0.352454
$162$	0	0
$163$	−6.47214	−0.506937	−0.253468	−	0.967344i	$-0.581571\pi$
$163$	−6.47214	−0.506937	−0.253468	+	0.967344i	$0.581571\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.944272	−0.0730700	−0.0365350	−	0.999332i	$-0.511632\pi$
$167$	−0.944272	−0.0730700	−0.0365350	+	0.999332i	$0.511632\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.41641	−0.715916	−0.357958	−	0.933738i	$-0.616527\pi$
$173$	−9.41641	−0.715916	−0.357958	+	0.933738i	$0.616527\pi$
$174$	0	0
$175$	−24.4721	−1.84992
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.41641	−0.554328	−0.277164	−	0.960823i	$-0.589395\pi$
$179$	−7.41641	−0.554328	−0.277164	+	0.960823i	$0.589395\pi$
$180$	0	0
$181$	−6.76393	−0.502759	−0.251380	−	0.967889i	$-0.580884\pi$
$181$	−6.76393	−0.502759	−0.251380	+	0.967889i	$0.580884\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	21.8885	1.60928
$186$	0	0
$187$	−3.05573	−0.223457
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.47214	−0.178877	−0.0894387	−	0.995992i	$-0.528507\pi$
$191$	−2.47214	−0.178877	−0.0894387	+	0.995992i	$0.528507\pi$
$192$	0	0
$193$	−11.8885	−0.855756	−0.427878	−	0.903836i	$-0.640739\pi$
$193$	−11.8885	−0.855756	−0.427878	+	0.903836i	$0.640739\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.9443	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$197$	14.9443	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$198$	0	0
$199$	9.41641	0.667511	0.333756	−	0.942660i	$-0.391684\pi$
$199$	9.41641	0.667511	0.333756	+	0.942660i	$0.391684\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.0000	1.40372
$204$	0	0
$205$	6.47214	0.452034
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.88854	−0.407319
$210$	0	0
$211$	−3.41641	−0.235195	−0.117598	−	0.993061i	$-0.537519\pi$
$211$	−3.41641	−0.235195	−0.117598	+	0.993061i	$0.537519\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	29.8885	2.03838
$216$	0	0
$217$	28.9443	1.96487
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−17.8885	−1.20331
$222$	0	0
$223$	7.41641	0.496639	0.248320	−	0.968678i	$-0.420122\pi$
$223$	7.41641	0.496639	0.248320	+	0.968678i	$0.420122\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.76393	−0.316193	−0.158097	−	0.987424i	$-0.550536\pi$
$227$	−4.76393	−0.316193	−0.158097	+	0.987424i	$0.550536\pi$
$228$	0	0
$229$	4.29180	0.283610	0.141805	−	0.989895i	$-0.454709\pi$
$229$	4.29180	0.283610	0.141805	+	0.989895i	$0.454709\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.8885	1.04089	0.520447	−	0.853894i	$-0.325765\pi$
$233$	15.8885	1.04089	0.520447	+	0.853894i	$0.325765\pi$
$234$	0	0
$235$	12.9443	0.844391
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.9443	0.837295	0.418648	−	0.908149i	$-0.362504\pi$
$239$	12.9443	0.837295	0.418648	+	0.908149i	$0.362504\pi$
$240$	0	0
$241$	−3.52786	−0.227250	−0.113625	−	0.993524i	$-0.536246\pi$
$241$	−3.52786	−0.227250	−0.113625	+	0.993524i	$0.536246\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	42.0689	2.68768
$246$	0	0
$247$	−34.4721	−2.19341
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.29180	−0.397135	−0.198567	−	0.980087i	$-0.563629\pi$
$251$	−6.29180	−0.397135	−0.198567	+	0.980087i	$0.563629\pi$
$252$	0	0
$253$	−0.763932	−0.0480280
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.8885	−1.49013	−0.745063	−	0.666994i	$-0.767581\pi$
$257$	−23.8885	−1.49013	−0.745063	+	0.666994i	$0.767581\pi$
$258$	0	0
$259$	−30.2492	−1.87960
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.05573	0.435075	0.217537	−	0.976052i	$-0.430198\pi$
$263$	7.05573	0.435075	0.217537	+	0.976052i	$0.430198\pi$
$264$	0	0
$265$	−2.47214	−0.151862
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.9443	1.88671	0.943353	−	0.331791i	$-0.107653\pi$
$269$	30.9443	1.88671	0.943353	+	0.331791i	$0.107653\pi$
$270$	0	0
$271$	−0.944272	−0.0573604	−0.0286802	−	0.999589i	$-0.509130\pi$
$271$	−0.944272	−0.0573604	−0.0286802	+	0.999589i	$0.509130\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.18034	−0.252084
$276$	0	0
$277$	−11.5279	−0.692642	−0.346321	−	0.938116i	$-0.612569\pi$
$277$	−11.5279	−0.692642	−0.346321	+	0.938116i	$0.612569\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.4721	1.34058	0.670288	−	0.742101i	$-0.266171\pi$
$281$	22.4721	1.34058	0.670288	+	0.742101i	$0.266171\pi$
$282$	0	0
$283$	26.1803	1.55626	0.778130	−	0.628103i	$-0.216169\pi$
$283$	26.1803	1.55626	0.778130	+	0.628103i	$0.216169\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.94427	−0.527964
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.7082	−0.800842	−0.400421	−	0.916331i	$-0.631136\pi$
$293$	−13.7082	−0.800842	−0.400421	+	0.916331i	$0.631136\pi$
$294$	0	0
$295$	28.9443	1.68520
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.47214	−0.258630
$300$	0	0
$301$	−41.3050	−2.38078
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16.9443	0.970226
$306$	0	0
$307$	11.4164	0.651569	0.325784	−	0.945444i	$-0.394372\pi$
$307$	11.4164	0.651569	0.325784	+	0.945444i	$0.394372\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.05573	−0.173274	−0.0866372	−	0.996240i	$-0.527612\pi$
$311$	−3.05573	−0.173274	−0.0866372	+	0.996240i	$0.527612\pi$
$312$	0	0
$313$	24.4721	1.38325	0.691623	−	0.722258i	$-0.256896\pi$
$313$	24.4721	1.38325	0.691623	+	0.722258i	$0.256896\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−28.4721	−1.59915	−0.799577	−	0.600563i	$-0.794943\pi$
$317$	−28.4721	−1.59915	−0.799577	+	0.600563i	$0.794943\pi$
$318$	0	0
$319$	3.41641	0.191282
$320$	0	0
$321$	0	0
$322$	0	0
$323$	30.8328	1.71558
$324$	0	0
$325$	−24.4721	−1.35747
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−17.8885	−0.986227
$330$	0	0
$331$	1.52786	0.0839790	0.0419895	−	0.999118i	$-0.486630\pi$
$331$	1.52786	0.0839790	0.0419895	+	0.999118i	$0.486630\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	15.8885	0.865504	0.432752	−	0.901513i	$-0.357543\pi$
$337$	15.8885	0.865504	0.432752	+	0.901513i	$0.357543\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.94427	0.267747
$342$	0	0
$343$	−26.8328	−1.44884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.5279	1.15568	0.577838	−	0.816151i	$-0.303896\pi$
$347$	21.5279	1.15568	0.577838	+	0.816151i	$0.303896\pi$
$348$	0	0
$349$	−31.8885	−1.70695	−0.853477	−	0.521130i	$-0.825511\pi$
$349$	−31.8885	−1.70695	−0.853477	+	0.521130i	$0.825511\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−31.8885	−1.69726	−0.848628	−	0.528990i	$-0.822571\pi$
$353$	−31.8885	−1.69726	−0.848628	+	0.528990i	$0.822571\pi$
$354$	0	0
$355$	−28.9443	−1.53620
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.3050	−1.75777	−0.878884	−	0.477035i	$-0.841711\pi$
$359$	−33.3050	−1.75777	−0.878884	+	0.477035i	$0.841711\pi$
$360$	0	0
$361$	40.4164	2.12718
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.4721	0.757506
$366$	0	0
$367$	17.4164	0.909129	0.454565	−	0.890714i	$-0.349795\pi$
$367$	17.4164	0.909129	0.454565	+	0.890714i	$0.349795\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.41641	0.177371
$372$	0	0
$373$	13.5967	0.704013	0.352006	−	0.935998i	$-0.385500\pi$
$373$	13.5967	0.704013	0.352006	+	0.935998i	$0.385500\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.0000	1.03005
$378$	0	0
$379$	−0.291796	−0.0149886	−0.00749428	−	0.999972i	$-0.502386\pi$
$379$	−0.291796	−0.0149886	−0.00749428	+	0.999972i	$0.502386\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.9443	1.27459	0.637296	−	0.770619i	$-0.280053\pi$
$383$	24.9443	1.27459	0.637296	+	0.770619i	$0.280053\pi$
$384$	0	0
$385$	11.0557	0.563452
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.2918	−0.521815	−0.260907	−	0.965364i	$-0.584022\pi$
$389$	−10.2918	−0.521815	−0.260907	+	0.965364i	$0.584022\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.4721	0.728172
$396$	0	0
$397$	9.05573	0.454494	0.227247	−	0.973837i	$-0.427028\pi$
$397$	9.05573	0.454494	0.227247	+	0.973837i	$0.427028\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.94427	0.446656	0.223328	−	0.974743i	$-0.428308\pi$
$401$	8.94427	0.446656	0.223328	+	0.974743i	$0.428308\pi$
$402$	0	0
$403$	28.9443	1.44182
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.16718	−0.256128
$408$	0	0
$409$	−0.111456	−0.00551115	−0.00275558	−	0.999996i	$-0.500877\pi$
$409$	−0.111456	−0.00551115	−0.00275558	+	0.999996i	$0.500877\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−40.0000	−1.96827
$414$	0	0
$415$	28.3607	1.39217
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.0689	−1.46896	−0.734481	−	0.678630i	$-0.762574\pi$
$419$	−30.0689	−1.46896	−0.734481	+	0.678630i	$0.762574\pi$
$420$	0	0
$421$	−5.23607	−0.255190	−0.127595	−	0.991826i	$-0.540726\pi$
$421$	−5.23607	−0.255190	−0.127595	+	0.991826i	$0.540726\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	21.8885	1.06175
$426$	0	0
$427$	−23.4164	−1.13320
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.41641	0.164563	0.0822813	−	0.996609i	$-0.473779\pi$
$431$	3.41641	0.164563	0.0822813	+	0.996609i	$0.473779\pi$
$432$	0	0
$433$	5.41641	0.260296	0.130148	−	0.991495i	$-0.458455\pi$
$433$	5.41641	0.260296	0.130148	+	0.991495i	$0.458455\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.70820	0.368733
$438$	0	0
$439$	−36.9443	−1.76325	−0.881627	−	0.471947i	$-0.843551\pi$
$439$	−36.9443	−1.76325	−0.881627	+	0.471947i	$0.843551\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.9443	−1.56523	−0.782615	−	0.622506i	$-0.786115\pi$
$443$	−32.9443	−1.56523	−0.782615	+	0.622506i	$0.786115\pi$
$444$	0	0
$445$	4.94427	0.234381
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.8885	0.938598	0.469299	−	0.883039i	$-0.344507\pi$
$449$	19.8885	0.938598	0.469299	+	0.883039i	$0.344507\pi$
$450$	0	0
$451$	−1.52786	−0.0719443
$452$	0	0
$453$	0	0
$454$	0	0
$455$	64.7214	3.03418
$456$	0	0
$457$	36.4721	1.70609	0.853047	−	0.521834i	$-0.174752\pi$
$457$	36.4721	1.70609	0.853047	+	0.521834i	$0.174752\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−40.1803	−1.85932	−0.929662	−	0.368413i	$-0.879901\pi$
$467$	−40.1803	−1.85932	−0.929662	+	0.368413i	$0.879901\pi$
$468$	0	0
$469$	−16.5836	−0.765759
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.05573	−0.324423
$474$	0	0
$475$	42.1803	1.93537
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.0557	−0.505149	−0.252575	−	0.967577i	$-0.581277\pi$
$479$	−11.0557	−0.505149	−0.252575	+	0.967577i	$0.581277\pi$
$480$	0	0
$481$	−30.2492	−1.37925
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−27.4164	−1.24491
$486$	0	0
$487$	25.8885	1.17312	0.586561	−	0.809905i	$-0.300481\pi$
$487$	25.8885	1.17312	0.586561	+	0.809905i	$0.300481\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−33.3050	−1.50303	−0.751516	−	0.659715i	$-0.770677\pi$
$491$	−33.3050	−1.50303	−0.751516	+	0.659715i	$0.770677\pi$
$492$	0	0
$493$	−17.8885	−0.805659
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.0000	1.79425
$498$	0	0
$499$	−27.4164	−1.22733	−0.613663	−	0.789568i	$-0.710305\pi$
$499$	−27.4164	−1.22733	−0.613663	+	0.789568i	$0.710305\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.52786	0.0681241	0.0340620	−	0.999420i	$-0.489156\pi$
$503$	1.52786	0.0681241	0.0340620	+	0.999420i	$0.489156\pi$
$504$	0	0
$505$	14.4721	0.644002
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.58359	0.291813	0.145906	−	0.989298i	$-0.453390\pi$
$509$	6.58359	0.291813	0.145906	+	0.989298i	$0.453390\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.4164	0.855589
$516$	0	0
$517$	−3.05573	−0.134391
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.0000	−0.700973	−0.350486	−	0.936568i	$-0.613984\pi$
$521$	−16.0000	−0.700973	−0.350486	+	0.936568i	$0.613984\pi$
$522$	0	0
$523$	−12.2918	−0.537483	−0.268741	−	0.963212i	$-0.586608\pi$
$523$	−12.2918	−0.537483	−0.268741	+	0.963212i	$0.586608\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.8885	−1.12772
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.94427	−0.387419
$534$	0	0
$535$	60.3607	2.60962
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.93112	−0.427763
$540$	0	0
$541$	43.8885	1.88692	0.943458	−	0.331492i	$-0.107552\pi$
$541$	43.8885	1.88692	0.943458	+	0.331492i	$0.107552\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	29.8885	1.28028
$546$	0	0
$547$	21.3050	0.910934	0.455467	−	0.890253i	$-0.349472\pi$
$547$	21.3050	0.910934	0.455467	+	0.890253i	$0.349472\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34.4721	−1.46856
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.1246	−1.06456	−0.532282	−	0.846567i	$-0.678665\pi$
$557$	−25.1246	−1.06456	−0.532282	+	0.846567i	$0.678665\pi$
$558$	0	0
$559$	−41.3050	−1.74701
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.65248	−0.280368	−0.140184	−	0.990125i	$-0.544769\pi$
$563$	−6.65248	−0.280368	−0.140184	+	0.990125i	$0.544769\pi$
$564$	0	0
$565$	46.8328	1.97027
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	27.7082	1.15955	0.579776	−	0.814776i	$-0.303140\pi$
$571$	27.7082	1.15955	0.579776	+	0.814776i	$0.303140\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.47214	0.228204
$576$	0	0
$577$	10.3607	0.431321	0.215660	−	0.976468i	$-0.430810\pi$
$577$	10.3607	0.431321	0.215660	+	0.976468i	$0.430810\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−39.1935	−1.62602
$582$	0	0
$583$	0.583592	0.0241699
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.47214	−0.102036	−0.0510180	−	0.998698i	$-0.516247\pi$
$587$	−2.47214	−0.102036	−0.0510180	+	0.998698i	$0.516247\pi$
$588$	0	0
$589$	−49.8885	−2.05562
$590$	0	0
$591$	0	0
$592$	0	0
$593$	37.7771	1.55132	0.775660	−	0.631152i	$-0.217417\pi$
$593$	37.7771	1.55132	0.775660	+	0.631152i	$0.217417\pi$
$594$	0	0
$595$	−57.8885	−2.37320
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.88854	−0.0771638	−0.0385819	−	0.999255i	$-0.512284\pi$
$599$	−1.88854	−0.0771638	−0.0385819	+	0.999255i	$0.512284\pi$
$600$	0	0
$601$	34.3607	1.40160	0.700801	−	0.713357i	$-0.252826\pi$
$601$	34.3607	1.40160	0.700801	+	0.713357i	$0.252826\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−33.7082	−1.37043
$606$	0	0
$607$	26.4721	1.07447	0.537235	−	0.843432i	$-0.319469\pi$
$607$	26.4721	1.07447	0.537235	+	0.843432i	$0.319469\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17.8885	−0.723693
$612$	0	0
$613$	15.1246	0.610877	0.305439	−	0.952212i	$-0.401197\pi$
$613$	15.1246	0.610877	0.305439	+	0.952212i	$0.401197\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.3607	0.980724	0.490362	−	0.871519i	$-0.336865\pi$
$617$	24.3607	0.980724	0.490362	+	0.871519i	$0.336865\pi$
$618$	0	0
$619$	−31.7082	−1.27446	−0.637230	−	0.770674i	$-0.719920\pi$
$619$	−31.7082	−1.27446	−0.637230	+	0.770674i	$0.719920\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.83282	−0.273751
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	27.0557	1.07878
$630$	0	0
$631$	6.00000	0.238856	0.119428	−	0.992843i	$-0.461894\pi$
$631$	6.00000	0.238856	0.119428	+	0.992843i	$0.461894\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.9443	0.513678
$636$	0	0
$637$	−58.1378	−2.30350
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	0	0
$643$	−34.5410	−1.36216	−0.681082	−	0.732207i	$-0.738490\pi$
$643$	−34.5410	−1.36216	−0.681082	+	0.732207i	$0.738490\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.94427	−0.194379	−0.0971897	−	0.995266i	$-0.530985\pi$
$647$	−4.94427	−0.194379	−0.0971897	+	0.995266i	$0.530985\pi$
$648$	0	0
$649$	−6.83282	−0.268211
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.4164	0.681557	0.340778	−	0.940144i	$-0.389309\pi$
$653$	17.4164	0.681557	0.340778	+	0.940144i	$0.389309\pi$
$654$	0	0
$655$	−59.7771	−2.33568
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.34752	−0.0524921	−0.0262460	−	0.999656i	$-0.508355\pi$
$659$	−1.34752	−0.0524921	−0.0262460	+	0.999656i	$0.508355\pi$
$660$	0	0
$661$	26.1803	1.01830	0.509149	−	0.860679i	$-0.329960\pi$
$661$	26.1803	1.01830	0.509149	+	0.860679i	$0.329960\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−111.554	−4.32589
$666$	0	0
$667$	−4.47214	−0.173162
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−4.47214	−0.172388	−0.0861941	−	0.996278i	$-0.527471\pi$
$673$	−4.47214	−0.172388	−0.0861941	+	0.996278i	$0.527471\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.1246	0.811885	0.405942	−	0.913899i	$-0.366943\pi$
$677$	21.1246	0.811885	0.405942	+	0.913899i	$0.366943\pi$
$678$	0	0
$679$	37.8885	1.45403
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.52786	0.211518	0.105759	−	0.994392i	$-0.466273\pi$
$683$	5.52786	0.211518	0.105759	+	0.994392i	$0.466273\pi$
$684$	0	0
$685$	−67.7771	−2.58963
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.41641	0.130155
$690$	0	0
$691$	−43.4164	−1.65164	−0.825819	−	0.563935i	$-0.809287\pi$
$691$	−43.4164	−1.65164	−0.825819	+	0.563935i	$0.809287\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.05573	−0.115910
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−37.1246	−1.40218	−0.701089	−	0.713074i	$-0.747302\pi$
$701$	−37.1246	−1.40218	−0.701089	+	0.713074i	$0.747302\pi$
$702$	0	0
$703$	52.1378	1.96641
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.0000	−0.752177
$708$	0	0
$709$	−41.0132	−1.54028	−0.770141	−	0.637874i	$-0.779814\pi$
$709$	−41.0132	−1.54028	−0.770141	+	0.637874i	$0.779814\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.47214	−0.242383
$714$	0	0
$715$	11.0557	0.413461
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.9443	−0.781090	−0.390545	−	0.920584i	$-0.627713\pi$
$719$	−20.9443	−0.781090	−0.390545	+	0.920584i	$0.627713\pi$
$720$	0	0
$721$	−26.8328	−0.999306
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24.4721	−0.908872
$726$	0	0
$727$	−43.3050	−1.60609	−0.803046	−	0.595917i	$-0.796789\pi$
$727$	−43.3050	−1.60609	−0.803046	+	0.595917i	$0.796789\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	36.9443	1.36643
$732$	0	0
$733$	−8.29180	−0.306264	−0.153132	−	0.988206i	$-0.548936\pi$
$733$	−8.29180	−0.306264	−0.153132	+	0.988206i	$0.548936\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.83282	−0.104348
$738$	0	0
$739$	15.0557	0.553834	0.276917	−	0.960894i	$-0.410687\pi$
$739$	15.0557	0.553834	0.276917	+	0.960894i	$0.410687\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	−5.52786	−0.202525
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−83.4164	−3.04797
$750$	0	0
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.8885	0.651031
$756$	0	0
$757$	−32.6525	−1.18677	−0.593387	−	0.804917i	$-0.702210\pi$
$757$	−32.6525	−1.18677	−0.593387	+	0.804917i	$0.702210\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.9443	0.831729	0.415865	−	0.909427i	$-0.363479\pi$
$761$	22.9443	0.831729	0.415865	+	0.909427i	$0.363479\pi$
$762$	0	0
$763$	−41.3050	−1.49534
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	3.52786	0.127218	0.0636090	−	0.997975i	$-0.479739\pi$
$769$	3.52786	0.127218	0.0636090	+	0.997975i	$0.479739\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.8197	0.568994	0.284497	−	0.958677i	$-0.408173\pi$
$773$	15.8197	0.568994	0.284497	+	0.958677i	$0.408173\pi$
$774$	0	0
$775$	−35.4164	−1.27219
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15.4164	0.552350
$780$	0	0
$781$	6.83282	0.244497
$782$	0	0
$783$	0	0
$784$	0	0
$785$	79.7771	2.84737
$786$	0	0
$787$	−30.7639	−1.09662	−0.548308	−	0.836277i	$-0.684728\pi$
$787$	−30.7639	−1.09662	−0.548308	+	0.836277i	$0.684728\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−64.7214	−2.30123
$792$	0	0
$793$	−23.4164	−0.831541
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.59675	0.269091	0.134545	−	0.990907i	$-0.457043\pi$
$797$	7.59675	0.269091	0.134545	+	0.990907i	$0.457043\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.41641	−0.120562
$804$	0	0
$805$	−14.4721	−0.510076
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−25.0557	−0.880912	−0.440456	−	0.897774i	$-0.645183\pi$
$809$	−25.0557	−0.880912	−0.440456	+	0.897774i	$0.645183\pi$
$810$	0	0
$811$	21.3050	0.748118	0.374059	−	0.927405i	$-0.377966\pi$
$811$	21.3050	0.748118	0.374059	+	0.927405i	$0.377966\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.9443	−0.733646
$816$	0	0
$817$	71.1935	2.49075
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.4721	−1.69169	−0.845845	−	0.533429i	$-0.820903\pi$
$821$	−48.4721	−1.69169	−0.845845	+	0.533429i	$0.820903\pi$
$822$	0	0
$823$	7.63932	0.266290	0.133145	−	0.991097i	$-0.457492\pi$
$823$	7.63932	0.266290	0.133145	+	0.991097i	$0.457492\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.5967	0.959633	0.479816	−	0.877369i	$-0.340703\pi$
$827$	27.5967	0.959633	0.479816	+	0.877369i	$0.340703\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	52.0000	1.80169
$834$	0	0
$835$	−3.05573	−0.105748
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.1115	−0.349086	−0.174543	−	0.984650i	$-0.555845\pi$
$839$	−10.1115	−0.349086	−0.174543	+	0.984650i	$0.555845\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	0	0
$844$	0	0
$845$	22.6525	0.779269
$846$	0	0
$847$	46.5836	1.60063
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.76393	0.231865
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.8328	−1.32188	−0.660942	−	0.750437i	$-0.729843\pi$
$863$	−38.8328	−1.32188	−0.660942	+	0.750437i	$0.729843\pi$
$864$	0	0
$865$	−30.4721	−1.03608
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.41641	−0.115894
$870$	0	0
$871$	−16.5836	−0.561914
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.83282	−0.230991
$876$	0	0
$877$	32.2492	1.08898	0.544489	−	0.838768i	$-0.316723\pi$
$877$	32.2492	1.08898	0.544489	+	0.838768i	$0.316723\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.5836	−0.423952	−0.211976	−	0.977275i	$-0.567990\pi$
$881$	−12.5836	−0.423952	−0.211976	+	0.977275i	$0.567990\pi$
$882$	0	0
$883$	−22.4721	−0.756248	−0.378124	−	0.925755i	$-0.623431\pi$
$883$	−22.4721	−0.756248	−0.378124	+	0.925755i	$0.623431\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.944272	0.0317055	0.0158528	−	0.999874i	$-0.494954\pi$
$887$	0.944272	0.0317055	0.0158528	+	0.999874i	$0.494954\pi$
$888$	0	0
$889$	−17.8885	−0.599963
$890$	0	0
$891$	0	0
$892$	0	0
$893$	30.8328	1.03178
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	28.9443	0.965346
$900$	0	0
$901$	−3.05573	−0.101801
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.8885	−0.727600
$906$	0	0
$907$	18.1803	0.603668	0.301834	−	0.953360i	$-0.402401\pi$
$907$	18.1803	0.603668	0.301834	+	0.953360i	$0.402401\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43.4164	1.43845	0.719225	−	0.694777i	$-0.244497\pi$
$911$	43.4164	1.43845	0.719225	+	0.694777i	$0.244497\pi$
$912$	0	0
$913$	−6.69505	−0.221574
$914$	0	0
$915$	0	0
$916$	0	0
$917$	82.6099	2.72802
$918$	0	0
$919$	7.52786	0.248321	0.124161	−	0.992262i	$-0.460376\pi$
$919$	7.52786	0.248321	0.124161	+	0.992262i	$0.460376\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.0000	1.31662
$924$	0	0
$925$	37.0132	1.21699
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.8328	−0.945974	−0.472987	−	0.881069i	$-0.656824\pi$
$929$	−28.8328	−0.945974	−0.472987	+	0.881069i	$0.656824\pi$
$930$	0	0
$931$	100.207	3.28414
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.88854	−0.323390
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−15.8197	−0.515706	−0.257853	−	0.966184i	$-0.583015\pi$
$941$	−15.8197	−0.515706	−0.257853	+	0.966184i	$0.583015\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.1935	1.66357	0.831783	−	0.555102i	$-0.187321\pi$
$947$	51.1935	1.66357	0.831783	+	0.555102i	$0.187321\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23.7771	−0.770215	−0.385108	−	0.922872i	$-0.625836\pi$
$953$	−23.7771	−0.770215	−0.385108	+	0.922872i	$0.625836\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	93.6656	3.02462
$960$	0	0
$961$	10.8885	0.351243
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−38.4721	−1.23846
$966$	0	0
$967$	11.4164	0.367127	0.183563	−	0.983008i	$-0.441237\pi$
$967$	11.4164	0.367127	0.183563	+	0.983008i	$0.441237\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21.1246	0.677921	0.338961	−	0.940801i	$-0.389925\pi$
$971$	21.1246	0.677921	0.338961	+	0.940801i	$0.389925\pi$
$972$	0	0
$973$	4.22291	0.135380
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−54.8328	−1.75426	−0.877129	−	0.480256i	$-0.840544\pi$
$977$	−54.8328	−1.75426	−0.877129	+	0.480256i	$0.840544\pi$
$978$	0	0
$979$	−1.16718	−0.0373034
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.4164	0.874448	0.437224	−	0.899353i	$-0.355962\pi$
$983$	27.4164	0.874448	0.437224	+	0.899353i	$0.355962\pi$
$984$	0	0
$985$	48.3607	1.54090
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.23607	0.293690
$990$	0	0
$991$	1.52786	0.0485342	0.0242671	−	0.999706i	$-0.492275\pi$
$991$	1.52786	0.0485342	0.0242671	+	0.999706i	$0.492275\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	30.4721	0.966032
$996$	0	0
$997$	22.9443	0.726652	0.363326	−	0.931662i	$-0.381641\pi$
$997$	22.9443	0.726652	0.363326	+	0.931662i	$0.381641\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.bc.1.2		2
3.2	odd	2		1104.2.a.j.1.1		2
4.3	odd	2		414.2.a.f.1.2		2
12.11	even	2		138.2.a.d.1.1	✓	2
24.5	odd	2		4416.2.a.bl.1.2		2
24.11	even	2		4416.2.a.bh.1.2		2
60.23	odd	4		3450.2.d.x.2899.1		4
60.47	odd	4		3450.2.d.x.2899.4		4
60.59	even	2		3450.2.a.be.1.1		2
84.83	odd	2		6762.2.a.cb.1.2		2
92.91	even	2		9522.2.a.q.1.1		2
276.275	odd	2		3174.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.a.d.1.1	✓	2	12.11	even	2
414.2.a.f.1.2		2	4.3	odd	2
1104.2.a.j.1.1		2	3.2	odd	2
3174.2.a.s.1.2		2	276.275	odd	2
3312.2.a.bc.1.2		2	1.1	even	1	trivial
3450.2.a.be.1.1		2	60.59	even	2
3450.2.d.x.2899.1		4	60.23	odd	4
3450.2.d.x.2899.4		4	60.47	odd	4
4416.2.a.bh.1.2		2	24.11	even	2
4416.2.a.bl.1.2		2	24.5	odd	2
6762.2.a.cb.1.2		2	84.83	odd	2
9522.2.a.q.1.1		2	92.91	even	2

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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:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23607 q^{5} -4.47214 q^{7} +O(q^{10})\)	\(q+3.23607 q^{5} -4.47214 q^{7} -0.763932 q^{11} -4.47214 q^{13} +4.00000 q^{17} +7.70820 q^{19} +1.00000 q^{23} +5.47214 q^{25} -4.47214 q^{29} -6.47214 q^{31} -14.4721 q^{35} +6.76393 q^{37} +2.00000 q^{41} +9.23607 q^{43} +4.00000 q^{47} +13.0000 q^{49} -0.763932 q^{53} -2.47214 q^{55} +8.94427 q^{59} +5.23607 q^{61} -14.4721 q^{65} +3.70820 q^{67} -8.94427 q^{71} +4.47214 q^{73} +3.41641 q^{77} +4.47214 q^{79} +8.76393 q^{83} +12.9443 q^{85} +1.52786 q^{89} +20.0000 q^{91} +24.9443 q^{95} -8.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} - 6 q^{11} + 8 q^{17} + 2 q^{19} + 2 q^{23} + 2 q^{25} - 4 q^{31} - 20 q^{35} + 18 q^{37} + 4 q^{41} + 14 q^{43} + 8 q^{47} + 26 q^{49} - 6 q^{53} + 4 q^{55} + 6 q^{61} - 20 q^{65} - 6 q^{67} - 20 q^{77} + 22 q^{83} + 8 q^{85} + 12 q^{89} + 40 q^{91} + 32 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 6 * q^11 + 8 * q^17 + 2 * q^19 + 2 * q^23 + 2 * q^25 - 4 * q^31 - 20 * q^35 + 18 * q^37 + 4 * q^41 + 14 * q^43 + 8 * q^47 + 26 * q^49 - 6 * q^53 + 4 * q^55 + 6 * q^61 - 20 * q^65 - 6 * q^67 - 20 * q^77 + 22 * q^83 + 8 * q^85 + 12 * q^89 + 40 * q^91 + 32 * q^95 - 8 * q^97

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