LMFDB - Embedded newform 2368.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93543 q^{3} +2.93543 q^{5} +4.68133 q^{7} +0.745898 q^{9} -0.762305 q^{11} -1.76231 q^{13} +5.68133 q^{15} +3.36266 q^{17} -7.36266 q^{19} +9.06040 q^{21} +3.25410 q^{23} +3.61676 q^{25} -4.36266 q^{27} +3.25410 q^{29} +3.06457 q^{31} -1.47539 q^{33} +13.7417 q^{35} +1.00000 q^{37} -3.41082 q^{39} -7.42723 q^{41} +12.2499 q^{43} +2.18953 q^{45} +0.302263 q^{47} +14.9149 q^{49} +6.50820 q^{51} -5.53579 q^{53} -2.23769 q^{55} -14.2499 q^{57} -10.2499 q^{59} -12.2981 q^{61} +3.49180 q^{63} -5.17313 q^{65} +13.1526 q^{67} +6.29809 q^{69} +0.173127 q^{71} -1.23769 q^{73} +7.00000 q^{75} -3.56860 q^{77} +4.61676 q^{79} -10.6813 q^{81} -3.53579 q^{83} +9.87086 q^{85} +6.29809 q^{87} +15.7417 q^{89} -8.24993 q^{91} +5.93126 q^{93} -21.6126 q^{95} -16.1208 q^{97} -0.568602 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9} - 3 q^{13} + 10 q^{15} - 4 q^{17} - 8 q^{19} + 3 q^{21} + 9 q^{23} - 4 q^{25} + q^{27} + 9 q^{29} + 17 q^{31} - 9 q^{33} + 10 q^{35} + 3 q^{37} - 7 q^{39} - 16 q^{41}+ \cdots + 24 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9 - 3 * q^13 + 10 * q^15 - 4 * q^17 - 8 * q^19 + 3 * q^21 + 9 * q^23 - 4 * q^25 + q^27 + 9 * q^29 + 17 * q^31 - 9 * q^33 + 10 * q^35 + 3 * q^37 - 7 * q^39 - 16 * q^41 + 4 * q^43 - 2 * q^45 + 11 * q^47 + 8 * q^49 + 18 * q^51 + 3 * q^53 - 9 * q^55 - 10 * q^57 + 2 * q^59 - 15 * q^61 + 12 * q^63 - 10 * q^65 + 5 * q^67 - 3 * q^69 - 5 * q^71 - 6 * q^73 + 21 * q^75 + 15 * q^77 - q^79 - 25 * q^81 + 9 * q^83 + 14 * q^85 - 3 * q^87 + 16 * q^89 + 8 * q^91 - 22 * q^93 - 18 * q^95 + 24 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.93543	1.11742	0.558711	−	0.829362i	$-0.311296\pi$
$3$	1.93543	1.11742	0.558711	+	0.829362i	$0.311296\pi$
$4$	0	0
$5$	2.93543	1.31277	0.656383	−	0.754428i	$-0.272086\pi$
$5$	2.93543	1.31277	0.656383	+	0.754428i	$0.272086\pi$
$6$	0	0
$7$	4.68133	1.76938	0.884688	−	0.466183i	$-0.154371\pi$
$7$	4.68133	1.76938	0.884688	+	0.466183i	$0.154371\pi$
$8$	0	0
$9$	0.745898	0.248633
$10$	0	0
$11$	−0.762305	−0.229844	−0.114922	−	0.993375i	$-0.536662\pi$
$11$	−0.762305	−0.229844	−0.114922	+	0.993375i	$0.536662\pi$
$12$	0	0
$13$	−1.76231	−0.488775	−0.244388	−	0.969678i	$-0.578587\pi$
$13$	−1.76231	−0.488775	−0.244388	+	0.969678i	$0.578587\pi$
$14$	0	0
$15$	5.68133	1.46691
$16$	0	0
$17$	3.36266	0.815565	0.407783	−	0.913079i	$-0.366302\pi$
$17$	3.36266	0.815565	0.407783	+	0.913079i	$0.366302\pi$
$18$	0	0
$19$	−7.36266	−1.68911	−0.844555	−	0.535469i	$-0.820135\pi$
$19$	−7.36266	−1.68911	−0.844555	+	0.535469i	$0.820135\pi$
$20$	0	0
$21$	9.06040	1.97714
$22$	0	0
$23$	3.25410	0.678527	0.339264	−	0.940691i	$-0.389822\pi$
$23$	3.25410	0.678527	0.339264	+	0.940691i	$0.389822\pi$
$24$	0	0
$25$	3.61676	0.723353
$26$	0	0
$27$	−4.36266	−0.839595
$28$	0	0
$29$	3.25410	0.604272	0.302136	−	0.953265i	$-0.402300\pi$
$29$	3.25410	0.604272	0.302136	+	0.953265i	$0.402300\pi$
$30$	0	0
$31$	3.06457	0.550413	0.275206	−	0.961385i	$-0.411254\pi$
$31$	3.06457	0.550413	0.275206	+	0.961385i	$0.411254\pi$
$32$	0	0
$33$	−1.47539	−0.256832
$34$	0	0
$35$	13.7417	2.32278
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−3.41082	−0.546169
$40$	0	0
$41$	−7.42723	−1.15994	−0.579969	−	0.814638i	$-0.696935\pi$
$41$	−7.42723	−1.15994	−0.579969	+	0.814638i	$0.696935\pi$
$42$	0	0
$43$	12.2499	1.86810	0.934049	−	0.357146i	$-0.116250\pi$
$43$	12.2499	1.86810	0.934049	+	0.357146i	$0.116250\pi$
$44$	0	0
$45$	2.18953	0.326396
$46$	0	0
$47$	0.302263	0.0440895	0.0220448	−	0.999757i	$-0.492982\pi$
$47$	0.302263	0.0440895	0.0220448	+	0.999757i	$0.492982\pi$
$48$	0	0
$49$	14.9149	2.13069
$50$	0	0
$51$	6.50820	0.911331
$52$	0	0
$53$	−5.53579	−0.760399	−0.380200	−	0.924904i	$-0.624145\pi$
$53$	−5.53579	−0.760399	−0.380200	+	0.924904i	$0.624145\pi$
$54$	0	0
$55$	−2.23769	−0.301731
$56$	0	0
$57$	−14.2499	−1.88745
$58$	0	0
$59$	−10.2499	−1.33443	−0.667214	−	0.744866i	$-0.732513\pi$
$59$	−10.2499	−1.33443	−0.667214	+	0.744866i	$0.732513\pi$
$60$	0	0
$61$	−12.2981	−1.57461	−0.787305	−	0.616564i	$-0.788524\pi$
$61$	−12.2981	−1.57461	−0.787305	+	0.616564i	$0.788524\pi$
$62$	0	0
$63$	3.49180	0.439925
$64$	0	0
$65$	−5.17313	−0.641647
$66$	0	0
$67$	13.1526	1.60684	0.803420	−	0.595413i	$-0.203011\pi$
$67$	13.1526	1.60684	0.803420	+	0.595413i	$0.203011\pi$
$68$	0	0
$69$	6.29809	0.758201
$70$	0	0
$71$	0.173127	0.0205464	0.0102732	−	0.999947i	$-0.496730\pi$
$71$	0.173127	0.0205464	0.0102732	+	0.999947i	$0.496730\pi$
$72$	0	0
$73$	−1.23769	−0.144861	−0.0724306	−	0.997373i	$-0.523076\pi$
$73$	−1.23769	−0.144861	−0.0724306	+	0.997373i	$0.523076\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	−3.56860	−0.406680
$78$	0	0
$79$	4.61676	0.519426	0.259713	−	0.965686i	$-0.416372\pi$
$79$	4.61676	0.519426	0.259713	+	0.965686i	$0.416372\pi$
$80$	0	0
$81$	−10.6813	−1.18681
$82$	0	0
$83$	−3.53579	−0.388103	−0.194052	−	0.980991i	$-0.562163\pi$
$83$	−3.53579	−0.388103	−0.194052	+	0.980991i	$0.562163\pi$
$84$	0	0
$85$	9.87086	1.07065
$86$	0	0
$87$	6.29809	0.675227
$88$	0	0
$89$	15.7417	1.66862	0.834310	−	0.551296i	$-0.185866\pi$
$89$	15.7417	1.66862	0.834310	+	0.551296i	$0.185866\pi$
$90$	0	0
$91$	−8.24993	−0.864828
$92$	0	0
$93$	5.93126	0.615043
$94$	0	0
$95$	−21.6126	−2.21741
$96$	0	0
$97$	−16.1208	−1.63682	−0.818409	−	0.574635i	$-0.805144\pi$
$97$	−16.1208	−1.63682	−0.818409	+	0.574635i	$0.805144\pi$
$98$	0	0
$99$	−0.568602	−0.0571467
$100$	0	0
$101$	9.53579	0.948846	0.474423	−	0.880297i	$-0.342657\pi$
$101$	9.53579	0.948846	0.474423	+	0.880297i	$0.342657\pi$
$102$	0	0
$103$	−3.31450	−0.326587	−0.163294	−	0.986577i	$-0.552212\pi$
$103$	−3.31450	−0.326587	−0.163294	+	0.986577i	$0.552212\pi$
$104$	0	0
$105$	26.5962	2.59552
$106$	0	0
$107$	−17.2130	−1.66404	−0.832019	−	0.554747i	$-0.812815\pi$
$107$	−17.2130	−1.66404	−0.832019	+	0.554747i	$0.812815\pi$
$108$	0	0
$109$	−20.3791	−1.95196	−0.975980	−	0.217859i	$-0.930093\pi$
$109$	−20.3791	−1.95196	−0.975980	+	0.217859i	$0.930093\pi$
$110$	0	0
$111$	1.93543	0.183703
$112$	0	0
$113$	−14.7581	−1.38833	−0.694164	−	0.719817i	$-0.744226\pi$
$113$	−14.7581	−1.38833	−0.694164	+	0.719817i	$0.744226\pi$
$114$	0	0
$115$	9.55220	0.890747
$116$	0	0
$117$	−1.31450	−0.121526
$118$	0	0
$119$	15.7417	1.44304
$120$	0	0
$121$	−10.4189	−0.947172
$122$	0	0
$123$	−14.3749	−1.29614
$124$	0	0
$125$	−4.06040	−0.363173
$126$	0	0
$127$	7.22235	0.640880	0.320440	−	0.947269i	$-0.396169\pi$
$127$	7.22235	0.640880	0.320440	+	0.947269i	$0.396169\pi$
$128$	0	0
$129$	23.7089	2.08745
$130$	0	0
$131$	0.475390	0.0415350	0.0207675	−	0.999784i	$-0.493389\pi$
$131$	0.475390	0.0415350	0.0207675	+	0.999784i	$0.493389\pi$
$132$	0	0
$133$	−34.4671	−2.98867
$134$	0	0
$135$	−12.8063	−1.10219
$136$	0	0
$137$	−11.1526	−0.952827	−0.476413	−	0.879221i	$-0.658063\pi$
$137$	−11.1526	−0.952827	−0.476413	+	0.879221i	$0.658063\pi$
$138$	0	0
$139$	15.5040	1.31504	0.657518	−	0.753439i	$-0.271607\pi$
$139$	15.5040	1.31504	0.657518	+	0.753439i	$0.271607\pi$
$140$	0	0
$141$	0.585009	0.0492666
$142$	0	0
$143$	1.34341	0.112342
$144$	0	0
$145$	9.55220	0.793267
$146$	0	0
$147$	28.8667	2.38088
$148$	0	0
$149$	4.68133	0.383510	0.191755	−	0.981443i	$-0.438582\pi$
$149$	4.68133	0.383510	0.191755	+	0.981443i	$0.438582\pi$
$150$	0	0
$151$	7.83805	0.637852	0.318926	−	0.947780i	$-0.396678\pi$
$151$	7.83805	0.637852	0.318926	+	0.947780i	$0.396678\pi$
$152$	0	0
$153$	2.50820	0.202776
$154$	0	0
$155$	8.99583	0.722563
$156$	0	0
$157$	5.53579	0.441804	0.220902	−	0.975296i	$-0.429100\pi$
$157$	5.53579	0.441804	0.220902	+	0.975296i	$0.429100\pi$
$158$	0	0
$159$	−10.7141	−0.849687
$160$	0	0
$161$	15.2335	1.20057
$162$	0	0
$163$	1.74173	0.136423	0.0682114	−	0.997671i	$-0.478271\pi$
$163$	1.74173	0.136423	0.0682114	+	0.997671i	$0.478271\pi$
$164$	0	0
$165$	−4.33091	−0.337161
$166$	0	0
$167$	23.5644	1.82347	0.911735	−	0.410778i	$-0.134743\pi$
$167$	23.5644	1.82347	0.911735	+	0.410778i	$0.134743\pi$
$168$	0	0
$169$	−9.89428	−0.761099
$170$	0	0
$171$	−5.49180	−0.419968
$172$	0	0
$173$	9.18953	0.698667	0.349334	−	0.936998i	$-0.386408\pi$
$173$	9.18953	0.698667	0.349334	+	0.936998i	$0.386408\pi$
$174$	0	0
$175$	16.9313	1.27988
$176$	0	0
$177$	−19.8381	−1.49112
$178$	0	0
$179$	15.9917	1.19527	0.597636	−	0.801767i	$-0.296107\pi$
$179$	15.9917	1.19527	0.597636	+	0.801767i	$0.296107\pi$
$180$	0	0
$181$	−5.15672	−0.383296	−0.191648	−	0.981464i	$-0.561383\pi$
$181$	−5.15672	−0.383296	−0.191648	+	0.981464i	$0.561383\pi$
$182$	0	0
$183$	−23.8021	−1.75950
$184$	0	0
$185$	2.93543	0.215817
$186$	0	0
$187$	−2.56337	−0.187452
$188$	0	0
$189$	−20.4231	−1.48556
$190$	0	0
$191$	−10.6443	−0.770198	−0.385099	−	0.922875i	$-0.625833\pi$
$191$	−10.6443	−0.770198	−0.385099	+	0.922875i	$0.625833\pi$
$192$	0	0
$193$	0.346255	0.0249239	0.0124620	−	0.999922i	$-0.496033\pi$
$193$	0.346255	0.0249239	0.0124620	+	0.999922i	$0.496033\pi$
$194$	0	0
$195$	−10.0122	−0.716991
$196$	0	0
$197$	17.5358	1.24937	0.624687	−	0.780876i	$-0.285227\pi$
$197$	17.5358	1.24937	0.624687	+	0.780876i	$0.285227\pi$
$198$	0	0
$199$	6.37907	0.452200	0.226100	−	0.974104i	$-0.427402\pi$
$199$	6.37907	0.452200	0.226100	+	0.974104i	$0.427402\pi$
$200$	0	0
$201$	25.4559	1.79552
$202$	0	0
$203$	15.2335	1.06918
$204$	0	0
$205$	−21.8021	−1.52273
$206$	0	0
$207$	2.42723	0.168704
$208$	0	0
$209$	5.61259	0.388231
$210$	0	0
$211$	−21.6168	−1.48816	−0.744080	−	0.668091i	$-0.767112\pi$
$211$	−21.6168	−1.48816	−0.744080	+	0.668091i	$0.767112\pi$
$212$	0	0
$213$	0.335076	0.0229590
$214$	0	0
$215$	35.9588	2.45237
$216$	0	0
$217$	14.3463	0.973887
$218$	0	0
$219$	−2.39547	−0.161871
$220$	0	0
$221$	−5.92604	−0.398628
$222$	0	0
$223$	−9.40665	−0.629916	−0.314958	−	0.949106i	$-0.601990\pi$
$223$	−9.40665	−0.629916	−0.314958	+	0.949106i	$0.601990\pi$
$224$	0	0
$225$	2.69774	0.179849
$226$	0	0
$227$	9.87086	0.655152	0.327576	−	0.944825i	$-0.393768\pi$
$227$	9.87086	0.655152	0.327576	+	0.944825i	$0.393768\pi$
$228$	0	0
$229$	0.560533	0.0370411	0.0185205	−	0.999828i	$-0.494104\pi$
$229$	0.560533	0.0370411	0.0185205	+	0.999828i	$0.494104\pi$
$230$	0	0
$231$	−6.90679	−0.454433
$232$	0	0
$233$	−11.1578	−0.730970	−0.365485	−	0.930817i	$-0.619097\pi$
$233$	−11.1578	−0.730970	−0.365485	+	0.930817i	$0.619097\pi$
$234$	0	0
$235$	0.887271	0.0578792
$236$	0	0
$237$	8.93543	0.580419
$238$	0	0
$239$	0.427229	0.0276351	0.0138176	−	0.999905i	$-0.495602\pi$
$239$	0.427229	0.0276351	0.0138176	+	0.999905i	$0.495602\pi$
$240$	0	0
$241$	−12.0880	−0.778655	−0.389328	−	0.921099i	$-0.627293\pi$
$241$	−12.0880	−0.778655	−0.389328	+	0.921099i	$0.627293\pi$
$242$	0	0
$243$	−7.58501	−0.486579
$244$	0	0
$245$	43.7816	2.79710
$246$	0	0
$247$	12.9753	0.825596
$248$	0	0
$249$	−6.84328	−0.433675
$250$	0	0
$251$	1.62093	0.102312	0.0511562	−	0.998691i	$-0.483709\pi$
$251$	1.62093	0.102312	0.0511562	+	0.998691i	$0.483709\pi$
$252$	0	0
$253$	−2.48062	−0.155955
$254$	0	0
$255$	19.1044	1.19636
$256$	0	0
$257$	−17.1372	−1.06899	−0.534495	−	0.845172i	$-0.679498\pi$
$257$	−17.1372	−1.06899	−0.534495	+	0.845172i	$0.679498\pi$
$258$	0	0
$259$	4.68133	0.290884
$260$	0	0
$261$	2.42723	0.150242
$262$	0	0
$263$	−5.73055	−0.353361	−0.176680	−	0.984268i	$-0.556536\pi$
$263$	−5.73055	−0.353361	−0.176680	+	0.984268i	$0.556536\pi$
$264$	0	0
$265$	−16.2499	−0.998225
$266$	0	0
$267$	30.4671	1.86455
$268$	0	0
$269$	15.8297	0.965155	0.482577	−	0.875853i	$-0.339701\pi$
$269$	15.8297	0.965155	0.482577	+	0.875853i	$0.339701\pi$
$270$	0	0
$271$	−12.2059	−0.741458	−0.370729	−	0.928741i	$-0.620892\pi$
$271$	−12.2059	−0.741458	−0.370729	+	0.928741i	$0.620892\pi$
$272$	0	0
$273$	−15.9672	−0.966378
$274$	0	0
$275$	−2.75708	−0.166258
$276$	0	0
$277$	15.2817	0.918188	0.459094	−	0.888388i	$-0.348174\pi$
$277$	15.2817	0.918188	0.459094	+	0.888388i	$0.348174\pi$
$278$	0	0
$279$	2.28586	0.136851
$280$	0	0
$281$	6.03281	0.359887	0.179944	−	0.983677i	$-0.442408\pi$
$281$	6.03281	0.359887	0.179944	+	0.983677i	$0.442408\pi$
$282$	0	0
$283$	−8.16195	−0.485177	−0.242589	−	0.970129i	$-0.577997\pi$
$283$	−8.16195	−0.485177	−0.242589	+	0.970129i	$0.577997\pi$
$284$	0	0
$285$	−41.8297	−2.47778
$286$	0	0
$287$	−34.7693	−2.05237
$288$	0	0
$289$	−5.69251	−0.334853
$290$	0	0
$291$	−31.2007	−1.82902
$292$	0	0
$293$	11.7501	0.686446	0.343223	−	0.939254i	$-0.388481\pi$
$293$	11.7501	0.686446	0.343223	+	0.939254i	$0.388481\pi$
$294$	0	0
$295$	−30.0880	−1.75179
$296$	0	0
$297$	3.32568	0.192975
$298$	0	0
$299$	−5.73472	−0.331647
$300$	0	0
$301$	57.3460	3.30537
$302$	0	0
$303$	18.4559	1.06026
$304$	0	0
$305$	−36.1002	−2.06709
$306$	0	0
$307$	−1.67716	−0.0957207	−0.0478603	−	0.998854i	$-0.515240\pi$
$307$	−1.67716	−0.0957207	−0.0478603	+	0.998854i	$0.515240\pi$
$308$	0	0
$309$	−6.41499	−0.364936
$310$	0	0
$311$	16.8750	0.956895	0.478448	−	0.878116i	$-0.341200\pi$
$311$	16.8750	0.956895	0.478448	+	0.878116i	$0.341200\pi$
$312$	0	0
$313$	2.59619	0.146745	0.0733726	−	0.997305i	$-0.476624\pi$
$313$	2.59619	0.146745	0.0733726	+	0.997305i	$0.476624\pi$
$314$	0	0
$315$	10.2499	0.577518
$316$	0	0
$317$	−23.0716	−1.29583	−0.647914	−	0.761713i	$-0.724359\pi$
$317$	−23.0716	−1.29583	−0.647914	+	0.761713i	$0.724359\pi$
$318$	0	0
$319$	−2.48062	−0.138888
$320$	0	0
$321$	−33.3145	−1.85943
$322$	0	0
$323$	−24.7581	−1.37758
$324$	0	0
$325$	−6.37384	−0.353557
$326$	0	0
$327$	−39.4423	−2.18116
$328$	0	0
$329$	1.41499	0.0780110
$330$	0	0
$331$	8.34625	0.458752	0.229376	−	0.973338i	$-0.426332\pi$
$331$	8.34625	0.458752	0.229376	+	0.973338i	$0.426332\pi$
$332$	0	0
$333$	0.745898	0.0408750
$334$	0	0
$335$	38.6084	2.10940
$336$	0	0
$337$	−17.2377	−0.938997	−0.469499	−	0.882933i	$-0.655565\pi$
$337$	−17.2377	−0.938997	−0.469499	+	0.882933i	$0.655565\pi$
$338$	0	0
$339$	−28.5634	−1.55135
$340$	0	0
$341$	−2.33614	−0.126509
$342$	0	0
$343$	37.0521	2.00062
$344$	0	0
$345$	18.4876	0.995340
$346$	0	0
$347$	28.5082	1.53040	0.765200	−	0.643792i	$-0.222640\pi$
$347$	28.5082	1.53040	0.765200	+	0.643792i	$0.222640\pi$
$348$	0	0
$349$	−16.5082	−0.883664	−0.441832	−	0.897098i	$-0.645671\pi$
$349$	−16.5082	−0.883664	−0.441832	+	0.897098i	$0.645671\pi$
$350$	0	0
$351$	7.68834	0.410373
$352$	0	0
$353$	13.8709	0.738272	0.369136	−	0.929375i	$-0.379654\pi$
$353$	13.8709	0.738272	0.369136	+	0.929375i	$0.379654\pi$
$354$	0	0
$355$	0.508203	0.0269726
$356$	0	0
$357$	30.4671	1.61249
$358$	0	0
$359$	29.9477	1.58058	0.790289	−	0.612735i	$-0.209931\pi$
$359$	29.9477	1.58058	0.790289	+	0.612735i	$0.209931\pi$
$360$	0	0
$361$	35.2088	1.85309
$362$	0	0
$363$	−20.1651	−1.05839
$364$	0	0
$365$	−3.63317	−0.190169
$366$	0	0
$367$	−23.9588	−1.25064	−0.625321	−	0.780368i	$-0.715032\pi$
$367$	−23.9588	−1.25064	−0.625321	+	0.780368i	$0.715032\pi$
$368$	0	0
$369$	−5.53996	−0.288399
$370$	0	0
$371$	−25.9149	−1.34543
$372$	0	0
$373$	−13.7529	−0.712099	−0.356049	−	0.934467i	$-0.615876\pi$
$373$	−13.7529	−0.712099	−0.356049	+	0.934467i	$0.615876\pi$
$374$	0	0
$375$	−7.85863	−0.405818
$376$	0	0
$377$	−5.73472	−0.295353
$378$	0	0
$379$	21.0482	1.08117	0.540586	−	0.841289i	$-0.318203\pi$
$379$	21.0482	1.08117	0.540586	+	0.841289i	$0.318203\pi$
$380$	0	0
$381$	13.9784	0.716133
$382$	0	0
$383$	−8.69251	−0.444166	−0.222083	−	0.975028i	$-0.571286\pi$
$383$	−8.69251	−0.444166	−0.222083	+	0.975028i	$0.571286\pi$
$384$	0	0
$385$	−10.4754	−0.533875
$386$	0	0
$387$	9.13720	0.464470
$388$	0	0
$389$	−15.3832	−0.779961	−0.389981	−	0.920823i	$-0.627518\pi$
$389$	−15.3832	−0.779961	−0.389981	+	0.920823i	$0.627518\pi$
$390$	0	0
$391$	10.9424	0.553383
$392$	0	0
$393$	0.920085	0.0464121
$394$	0	0
$395$	13.5522	0.681885
$396$	0	0
$397$	−0.648517	−0.0325481	−0.0162741	−	0.999868i	$-0.505180\pi$
$397$	−0.648517	−0.0325481	−0.0162741	+	0.999868i	$0.505180\pi$
$398$	0	0
$399$	−66.7086	−3.33961
$400$	0	0
$401$	22.5962	1.12840	0.564200	−	0.825638i	$-0.309185\pi$
$401$	22.5962	1.12840	0.564200	+	0.825638i	$0.309185\pi$
$402$	0	0
$403$	−5.40070	−0.269028
$404$	0	0
$405$	−31.3543	−1.55801
$406$	0	0
$407$	−0.762305	−0.0377861
$408$	0	0
$409$	24.9201	1.23222	0.616109	−	0.787661i	$-0.288708\pi$
$409$	24.9201	1.23222	0.616109	+	0.787661i	$0.288708\pi$
$410$	0	0
$411$	−21.5850	−1.06471
$412$	0	0
$413$	−47.9833	−2.36111
$414$	0	0
$415$	−10.3791	−0.509488
$416$	0	0
$417$	30.0070	1.46945
$418$	0	0
$419$	−6.41082	−0.313189	−0.156595	−	0.987663i	$-0.550052\pi$
$419$	−6.41082	−0.313189	−0.156595	+	0.987663i	$0.550052\pi$
$420$	0	0
$421$	−0.551136	−0.0268607	−0.0134304	−	0.999910i	$-0.504275\pi$
$421$	−0.551136	−0.0268607	−0.0134304	+	0.999910i	$0.504275\pi$
$422$	0	0
$423$	0.225457	0.0109621
$424$	0	0
$425$	12.1619	0.589941
$426$	0	0
$427$	−57.5714	−2.78608
$428$	0	0
$429$	2.60009	0.125533
$430$	0	0
$431$	−30.8789	−1.48739	−0.743693	−	0.668521i	$-0.766928\pi$
$431$	−30.8789	−1.48739	−0.743693	+	0.668521i	$0.766928\pi$
$432$	0	0
$433$	−26.0890	−1.25376	−0.626880	−	0.779116i	$-0.715668\pi$
$433$	−26.0890	−1.25376	−0.626880	+	0.779116i	$0.715668\pi$
$434$	0	0
$435$	18.4876	0.886414
$436$	0	0
$437$	−23.9588	−1.14611
$438$	0	0
$439$	−5.85029	−0.279219	−0.139610	−	0.990207i	$-0.544585\pi$
$439$	−5.85029	−0.279219	−0.139610	+	0.990207i	$0.544585\pi$
$440$	0	0
$441$	11.1250	0.529760
$442$	0	0
$443$	−26.7212	−1.26956	−0.634780	−	0.772693i	$-0.718909\pi$
$443$	−26.7212	−1.26956	−0.634780	+	0.772693i	$0.718909\pi$
$444$	0	0
$445$	46.2088	2.19051
$446$	0	0
$447$	9.06040	0.428542
$448$	0	0
$449$	26.8461	1.26695	0.633473	−	0.773764i	$-0.281629\pi$
$449$	26.8461	1.26695	0.633473	+	0.773764i	$0.281629\pi$
$450$	0	0
$451$	5.66181	0.266604
$452$	0	0
$453$	15.1700	0.712750
$454$	0	0
$455$	−24.2171	−1.13532
$456$	0	0
$457$	13.1455	0.614923	0.307461	−	0.951561i	$-0.400521\pi$
$457$	13.1455	0.614923	0.307461	+	0.951561i	$0.400521\pi$
$458$	0	0
$459$	−14.6702	−0.684744
$460$	0	0
$461$	38.2416	1.78109	0.890544	−	0.454896i	$-0.150324\pi$
$461$	38.2416	1.78109	0.890544	+	0.454896i	$0.150324\pi$
$462$	0	0
$463$	−15.7951	−0.734061	−0.367031	−	0.930209i	$-0.619626\pi$
$463$	−15.7951	−0.734061	−0.367031	+	0.930209i	$0.619626\pi$
$464$	0	0
$465$	17.4108	0.807408
$466$	0	0
$467$	−13.4283	−0.621387	−0.310694	−	0.950510i	$-0.600561\pi$
$467$	−13.4283	−0.621387	−0.310694	+	0.950510i	$0.600561\pi$
$468$	0	0
$469$	61.5714	2.84311
$470$	0	0
$471$	10.7141	0.493682
$472$	0	0
$473$	−9.33819	−0.429370
$474$	0	0
$475$	−26.6290	−1.22182
$476$	0	0
$477$	−4.12914	−0.189060
$478$	0	0
$479$	27.2130	1.24339	0.621696	−	0.783259i	$-0.286444\pi$
$479$	27.2130	1.24339	0.621696	+	0.783259i	$0.286444\pi$
$480$	0	0
$481$	−1.76231	−0.0803542
$482$	0	0
$483$	29.4835	1.34154
$484$	0	0
$485$	−47.3215	−2.14876
$486$	0	0
$487$	−23.7417	−1.07584	−0.537920	−	0.842996i	$-0.680790\pi$
$487$	−23.7417	−1.07584	−0.537920	+	0.842996i	$0.680790\pi$
$488$	0	0
$489$	3.37100	0.152442
$490$	0	0
$491$	27.2458	1.22958	0.614792	−	0.788689i	$-0.289240\pi$
$491$	27.2458	1.22958	0.614792	+	0.788689i	$0.289240\pi$
$492$	0	0
$493$	10.9424	0.492823
$494$	0	0
$495$	−1.66909	−0.0750201
$496$	0	0
$497$	0.810466	0.0363544
$498$	0	0
$499$	−3.52461	−0.157783	−0.0788916	−	0.996883i	$-0.525138\pi$
$499$	−3.52461	−0.157783	−0.0788916	+	0.996883i	$0.525138\pi$
$500$	0	0
$501$	45.6074	2.03759
$502$	0	0
$503$	13.9518	0.622082	0.311041	−	0.950397i	$-0.399322\pi$
$503$	13.9518	0.622082	0.311041	+	0.950397i	$0.399322\pi$
$504$	0	0
$505$	27.9917	1.24561
$506$	0	0
$507$	−19.1497	−0.850469
$508$	0	0
$509$	24.5110	1.08643	0.543216	−	0.839593i	$-0.317206\pi$
$509$	24.5110	1.08643	0.543216	+	0.839593i	$0.317206\pi$
$510$	0	0
$511$	−5.79406	−0.256314
$512$	0	0
$513$	32.1208	1.41817
$514$	0	0
$515$	−9.72949	−0.428733
$516$	0	0
$517$	−0.230416	−0.0101337
$518$	0	0
$519$	17.7857	0.780707
$520$	0	0
$521$	35.9149	1.57346	0.786729	−	0.617298i	$-0.211773\pi$
$521$	35.9149	1.57346	0.786729	+	0.617298i	$0.211773\pi$
$522$	0	0
$523$	15.1784	0.663703	0.331852	−	0.943332i	$-0.392327\pi$
$523$	15.1784	0.663703	0.331852	+	0.943332i	$0.392327\pi$
$524$	0	0
$525$	32.7693	1.43017
$526$	0	0
$527$	10.3051	0.448897
$528$	0	0
$529$	−12.4108	−0.539601
$530$	0	0
$531$	−7.64541	−0.331782
$532$	0	0
$533$	13.0890	0.566949
$534$	0	0
$535$	−50.5275	−2.18449
$536$	0	0
$537$	30.9508	1.33562
$538$	0	0
$539$	−11.3697	−0.489726
$540$	0	0
$541$	−12.2981	−0.528736	−0.264368	−	0.964422i	$-0.585163\pi$
$541$	−12.2981	−0.528736	−0.264368	+	0.964422i	$0.585163\pi$
$542$	0	0
$543$	−9.98048	−0.428304
$544$	0	0
$545$	−59.8214	−2.56247
$546$	0	0
$547$	2.38741	0.102078	0.0510391	−	0.998697i	$-0.483747\pi$
$547$	2.38741	0.102078	0.0510391	+	0.998697i	$0.483747\pi$
$548$	0	0
$549$	−9.17313	−0.391500
$550$	0	0
$551$	−23.9588	−1.02068
$552$	0	0
$553$	21.6126	0.919061
$554$	0	0
$555$	5.68133	0.241159
$556$	0	0
$557$	−19.4353	−0.823500	−0.411750	−	0.911297i	$-0.635082\pi$
$557$	−19.4353	−0.823500	−0.411750	+	0.911297i	$0.635082\pi$
$558$	0	0
$559$	−21.5881	−0.913080
$560$	0	0
$561$	−4.96124	−0.209464
$562$	0	0
$563$	−28.1432	−1.18609	−0.593046	−	0.805168i	$-0.702075\pi$
$563$	−28.1432	−1.18609	−0.593046	+	0.805168i	$0.702075\pi$
$564$	0	0
$565$	−43.3215	−1.82255
$566$	0	0
$567$	−50.0028	−2.09992
$568$	0	0
$569$	−3.48346	−0.146034	−0.0730171	−	0.997331i	$-0.523263\pi$
$569$	−3.48346	−0.146034	−0.0730171	+	0.997331i	$0.523263\pi$
$570$	0	0
$571$	−32.4436	−1.35772	−0.678862	−	0.734266i	$-0.737527\pi$
$571$	−32.4436	−1.35772	−0.678862	+	0.734266i	$0.737527\pi$
$572$	0	0
$573$	−20.6014	−0.860636
$574$	0	0
$575$	11.7693	0.490814
$576$	0	0
$577$	17.2335	0.717441	0.358721	−	0.933445i	$-0.383213\pi$
$577$	17.2335	0.717441	0.358721	+	0.933445i	$0.383213\pi$
$578$	0	0
$579$	0.670152	0.0278506
$580$	0	0
$581$	−16.5522	−0.686701
$582$	0	0
$583$	4.21996	0.174773
$584$	0	0
$585$	−3.85863	−0.159535
$586$	0	0
$587$	1.07158	0.0442287	0.0221144	−	0.999755i	$-0.492960\pi$
$587$	1.07158	0.0442287	0.0221144	+	0.999755i	$0.492960\pi$
$588$	0	0
$589$	−22.5634	−0.929708
$590$	0	0
$591$	33.9393	1.39608
$592$	0	0
$593$	−39.7047	−1.63048	−0.815239	−	0.579124i	$-0.803395\pi$
$593$	−39.7047	−1.63048	−0.815239	+	0.579124i	$0.803395\pi$
$594$	0	0
$595$	46.2088	1.89438
$596$	0	0
$597$	12.3463	0.505299
$598$	0	0
$599$	−30.9864	−1.26607	−0.633036	−	0.774123i	$-0.718191\pi$
$599$	−30.9864	−1.26607	−0.633036	+	0.774123i	$0.718191\pi$
$600$	0	0
$601$	−25.4436	−1.03787	−0.518934	−	0.854814i	$-0.673671\pi$
$601$	−25.4436	−1.03787	−0.518934	+	0.854814i	$0.673671\pi$
$602$	0	0
$603$	9.81047	0.399513
$604$	0	0
$605$	−30.5839	−1.24341
$606$	0	0
$607$	23.1250	0.938613	0.469307	−	0.883035i	$-0.344504\pi$
$607$	23.1250	0.938613	0.469307	+	0.883035i	$0.344504\pi$
$608$	0	0
$609$	29.4835	1.19473
$610$	0	0
$611$	−0.532679	−0.0215499
$612$	0	0
$613$	−21.2140	−0.856826	−0.428413	−	0.903583i	$-0.640927\pi$
$613$	−21.2140	−0.856826	−0.428413	+	0.903583i	$0.640927\pi$
$614$	0	0
$615$	−42.1965	−1.70153
$616$	0	0
$617$	−20.0318	−0.806448	−0.403224	−	0.915101i	$-0.632110\pi$
$617$	−20.0318	−0.806448	−0.403224	+	0.915101i	$0.632110\pi$
$618$	0	0
$619$	12.7540	0.512625	0.256313	−	0.966594i	$-0.417492\pi$
$619$	12.7540	0.512625	0.256313	+	0.966594i	$0.417492\pi$
$620$	0	0
$621$	−14.1965	−0.569688
$622$	0	0
$623$	73.6922	2.95242
$624$	0	0
$625$	−30.0028	−1.20011
$626$	0	0
$627$	10.8628	0.433818
$628$	0	0
$629$	3.36266	0.134078
$630$	0	0
$631$	−30.1086	−1.19860	−0.599301	−	0.800523i	$-0.704555\pi$
$631$	−30.1086	−1.19860	−0.599301	+	0.800523i	$0.704555\pi$
$632$	0	0
$633$	−41.8378	−1.66290
$634$	0	0
$635$	21.2007	0.841325
$636$	0	0
$637$	−26.2845	−1.04143
$638$	0	0
$639$	0.129135	0.00510851
$640$	0	0
$641$	−8.18537	−0.323302	−0.161651	−	0.986848i	$-0.551682\pi$
$641$	−8.18537	−0.323302	−0.161651	+	0.986848i	$0.551682\pi$
$642$	0	0
$643$	6.85446	0.270313	0.135157	−	0.990824i	$-0.456846\pi$
$643$	6.85446	0.270313	0.135157	+	0.990824i	$0.456846\pi$
$644$	0	0
$645$	69.5959	2.74034
$646$	0	0
$647$	2.36683	0.0930497	0.0465248	−	0.998917i	$-0.485185\pi$
$647$	2.36683	0.0930497	0.0465248	+	0.998917i	$0.485185\pi$
$648$	0	0
$649$	7.81358	0.306710
$650$	0	0
$651$	27.7662	1.08824
$652$	0	0
$653$	17.6332	0.690039	0.345020	−	0.938595i	$-0.387872\pi$
$653$	17.6332	0.690039	0.345020	+	0.938595i	$0.387872\pi$
$654$	0	0
$655$	1.39547	0.0545257
$656$	0	0
$657$	−0.923195	−0.0360173
$658$	0	0
$659$	−31.7980	−1.23867	−0.619336	−	0.785126i	$-0.712598\pi$
$659$	−31.7980	−1.23867	−0.619336	+	0.785126i	$0.712598\pi$
$660$	0	0
$661$	−16.4241	−0.638824	−0.319412	−	0.947616i	$-0.603485\pi$
$661$	−16.4241	−0.638824	−0.319412	+	0.947616i	$0.603485\pi$
$662$	0	0
$663$	−11.4694	−0.445436
$664$	0	0
$665$	−101.176	−3.92343
$666$	0	0
$667$	10.5892	0.410015
$668$	0	0
$669$	−18.2059	−0.703882
$670$	0	0
$671$	9.37490	0.361914
$672$	0	0
$673$	7.79796	0.300589	0.150295	−	0.988641i	$-0.451978\pi$
$673$	7.79796	0.300589	0.150295	+	0.988641i	$0.451978\pi$
$674$	0	0
$675$	−15.7787	−0.607323
$676$	0	0
$677$	−48.3408	−1.85789	−0.928943	−	0.370223i	$-0.879281\pi$
$677$	−48.3408	−1.85789	−0.928943	+	0.370223i	$0.879281\pi$
$678$	0	0
$679$	−75.4668	−2.89615
$680$	0	0
$681$	19.1044	0.732082
$682$	0	0
$683$	20.8461	0.797655	0.398827	−	0.917026i	$-0.369417\pi$
$683$	20.8461	0.797655	0.398827	+	0.917026i	$0.369417\pi$
$684$	0	0
$685$	−32.7376	−1.25084
$686$	0	0
$687$	1.08487	0.0413905
$688$	0	0
$689$	9.75575	0.371664
$690$	0	0
$691$	0.692509	0.0263443	0.0131721	−	0.999913i	$-0.495807\pi$
$691$	0.692509	0.0263443	0.0131721	+	0.999913i	$0.495807\pi$
$692$	0	0
$693$	−2.66181	−0.101114
$694$	0	0
$695$	45.5110	1.72633
$696$	0	0
$697$	−24.9753	−0.946005
$698$	0	0
$699$	−21.5951	−0.816803
$700$	0	0
$701$	4.03982	0.152582	0.0762910	−	0.997086i	$-0.475692\pi$
$701$	4.03982	0.152582	0.0762910	+	0.997086i	$0.475692\pi$
$702$	0	0
$703$	−7.36266	−0.277688
$704$	0	0
$705$	1.71725	0.0646755
$706$	0	0
$707$	44.6402	1.67887
$708$	0	0
$709$	18.4272	0.692049	0.346025	−	0.938225i	$-0.387531\pi$
$709$	18.4272	0.692049	0.346025	+	0.938225i	$0.387531\pi$
$710$	0	0
$711$	3.44364	0.129146
$712$	0	0
$713$	9.97241	0.373470
$714$	0	0
$715$	3.94350	0.147479
$716$	0	0
$717$	0.826873	0.0308801
$718$	0	0
$719$	−31.7774	−1.18510	−0.592548	−	0.805535i	$-0.701878\pi$
$719$	−31.7774	−1.18510	−0.592548	+	0.805535i	$0.701878\pi$
$720$	0	0
$721$	−15.5163	−0.577856
$722$	0	0
$723$	−23.3955	−0.870087
$724$	0	0
$725$	11.7693	0.437101
$726$	0	0
$727$	13.8862	0.515011	0.257506	−	0.966277i	$-0.417099\pi$
$727$	13.8862	0.515011	0.257506	+	0.966277i	$0.417099\pi$
$728$	0	0
$729$	17.3637	0.643101
$730$	0	0
$731$	41.1924	1.52356
$732$	0	0
$733$	46.2939	1.70991	0.854953	−	0.518706i	$-0.173586\pi$
$733$	46.2939	1.70991	0.854953	+	0.518706i	$0.173586\pi$
$734$	0	0
$735$	84.7362	3.12554
$736$	0	0
$737$	−10.0263	−0.369322
$738$	0	0
$739$	19.0451	0.700584	0.350292	−	0.936641i	$-0.386082\pi$
$739$	19.0451	0.700584	0.350292	+	0.936641i	$0.386082\pi$
$740$	0	0
$741$	25.1127	0.922539
$742$	0	0
$743$	24.7141	0.906674	0.453337	−	0.891339i	$-0.350233\pi$
$743$	24.7141	0.906674	0.453337	+	0.891339i	$0.350233\pi$
$744$	0	0
$745$	13.7417	0.503458
$746$	0	0
$747$	−2.63734	−0.0964952
$748$	0	0
$749$	−80.5795	−2.94431
$750$	0	0
$751$	16.9313	0.617831	0.308915	−	0.951090i	$-0.400034\pi$
$751$	16.9313	0.617831	0.308915	+	0.951090i	$0.400034\pi$
$752$	0	0
$753$	3.13720	0.114326
$754$	0	0
$755$	23.0081	0.837349
$756$	0	0
$757$	25.0786	0.911497	0.455748	−	0.890109i	$-0.349372\pi$
$757$	25.0786	0.911497	0.455748	+	0.890109i	$0.349372\pi$
$758$	0	0
$759$	−4.80107	−0.174268
$760$	0	0
$761$	−12.2489	−0.444021	−0.222011	−	0.975044i	$-0.571262\pi$
$761$	−12.2489	−0.444021	−0.222011	+	0.975044i	$0.571262\pi$
$762$	0	0
$763$	−95.4012	−3.45375
$764$	0	0
$765$	7.36266	0.266198
$766$	0	0
$767$	18.0635	0.652235
$768$	0	0
$769$	−12.0328	−0.433914	−0.216957	−	0.976181i	$-0.569613\pi$
$769$	−12.0328	−0.433914	−0.216957	+	0.976181i	$0.569613\pi$
$770$	0	0
$771$	−33.1679	−1.19451
$772$	0	0
$773$	14.4147	0.518462	0.259231	−	0.965815i	$-0.416531\pi$
$773$	14.4147	0.518462	0.259231	+	0.965815i	$0.416531\pi$
$774$	0	0
$775$	11.0838	0.398142
$776$	0	0
$777$	9.06040	0.325040
$778$	0	0
$779$	54.6842	1.95926
$780$	0	0
$781$	−0.131976	−0.00472247
$782$	0	0
$783$	−14.1965	−0.507343
$784$	0	0
$785$	16.2499	0.579985
$786$	0	0
$787$	−44.1236	−1.57284	−0.786419	−	0.617694i	$-0.788067\pi$
$787$	−44.1236	−1.57284	−0.786419	+	0.617694i	$0.788067\pi$
$788$	0	0
$789$	−11.0911	−0.394853
$790$	0	0
$791$	−69.0877	−2.45648
$792$	0	0
$793$	21.6730	0.769631
$794$	0	0
$795$	−31.4506	−1.11544
$796$	0	0
$797$	53.8503	1.90748	0.953738	−	0.300640i	$-0.0972004\pi$
$797$	53.8503	1.90748	0.953738	+	0.300640i	$0.0972004\pi$
$798$	0	0
$799$	1.01641	0.0359579
$800$	0	0
$801$	11.7417	0.414874
$802$	0	0
$803$	0.943501	0.0332954
$804$	0	0
$805$	44.7170	1.57607
$806$	0	0
$807$	30.6373	1.07849
$808$	0	0
$809$	20.3051	0.713889	0.356945	−	0.934126i	$-0.383818\pi$
$809$	20.3051	0.713889	0.356945	+	0.934126i	$0.383818\pi$
$810$	0	0
$811$	9.44648	0.331711	0.165855	−	0.986150i	$-0.446962\pi$
$811$	9.44648	0.331711	0.165855	+	0.986150i	$0.446962\pi$
$812$	0	0
$813$	−23.6238	−0.828522
$814$	0	0
$815$	5.11273	0.179091
$816$	0	0
$817$	−90.1921	−3.15542
$818$	0	0
$819$	−6.15361	−0.215025
$820$	0	0
$821$	−33.7118	−1.17655	−0.588274	−	0.808662i	$-0.700192\pi$
$821$	−33.7118	−1.17655	−0.588274	+	0.808662i	$0.700192\pi$
$822$	0	0
$823$	−13.9672	−0.486866	−0.243433	−	0.969918i	$-0.578274\pi$
$823$	−13.9672	−0.486866	−0.243433	+	0.969918i	$0.578274\pi$
$824$	0	0
$825$	−5.33614	−0.185780
$826$	0	0
$827$	−14.3379	−0.498578	−0.249289	−	0.968429i	$-0.580197\pi$
$827$	−14.3379	−0.498578	−0.249289	+	0.968429i	$0.580197\pi$
$828$	0	0
$829$	−4.61676	−0.160347	−0.0801734	−	0.996781i	$-0.525547\pi$
$829$	−4.61676	−0.160347	−0.0801734	+	0.996781i	$0.525547\pi$
$830$	0	0
$831$	29.5767	1.02600
$832$	0	0
$833$	50.1536	1.73772
$834$	0	0
$835$	69.1718	2.39379
$836$	0	0
$837$	−13.3697	−0.462123
$838$	0	0
$839$	55.3871	1.91218	0.956088	−	0.293079i	$-0.0946800\pi$
$839$	55.3871	1.91218	0.956088	+	0.293079i	$0.0946800\pi$
$840$	0	0
$841$	−18.4108	−0.634856
$842$	0	0
$843$	11.6761	0.402146
$844$	0	0
$845$	−29.0440	−0.999144
$846$	0	0
$847$	−48.7743	−1.67590
$848$	0	0
$849$	−15.7969	−0.542148
$850$	0	0
$851$	3.25410	0.111549
$852$	0	0
$853$	33.7264	1.15477	0.577385	−	0.816472i	$-0.304073\pi$
$853$	33.7264	1.15477	0.577385	+	0.816472i	$0.304073\pi$
$854$	0	0
$855$	−16.1208	−0.551320
$856$	0	0
$857$	−34.5222	−1.17926	−0.589628	−	0.807675i	$-0.700726\pi$
$857$	−34.5222	−1.17926	−0.589628	+	0.807675i	$0.700726\pi$
$858$	0	0
$859$	56.2311	1.91858	0.959291	−	0.282420i	$-0.0911372\pi$
$859$	56.2311	1.91858	0.959291	+	0.282420i	$0.0911372\pi$
$860$	0	0
$861$	−67.2937	−2.29336
$862$	0	0
$863$	−24.4342	−0.831751	−0.415876	−	0.909422i	$-0.636525\pi$
$863$	−24.4342	−0.831751	−0.415876	+	0.909422i	$0.636525\pi$
$864$	0	0
$865$	26.9753	0.917186
$866$	0	0
$867$	−11.0175	−0.374173
$868$	0	0
$869$	−3.51938	−0.119387
$870$	0	0
$871$	−23.1788	−0.785384
$872$	0	0
$873$	−12.0245	−0.406967
$874$	0	0
$875$	−19.0081	−0.642590
$876$	0	0
$877$	−42.2416	−1.42640	−0.713199	−	0.700962i	$-0.752754\pi$
$877$	−42.2416	−1.42640	−0.713199	+	0.700962i	$0.752754\pi$
$878$	0	0
$879$	22.7415	0.767050
$880$	0	0
$881$	43.6524	1.47069	0.735344	−	0.677694i	$-0.237021\pi$
$881$	43.6524	1.47069	0.735344	+	0.677694i	$0.237021\pi$
$882$	0	0
$883$	12.4342	0.418445	0.209223	−	0.977868i	$-0.432907\pi$
$883$	12.4342	0.418445	0.209223	+	0.977868i	$0.432907\pi$
$884$	0	0
$885$	−58.2333	−1.95749
$886$	0	0
$887$	28.8656	0.969213	0.484607	−	0.874732i	$-0.338963\pi$
$887$	28.8656	0.969213	0.484607	+	0.874732i	$0.338963\pi$
$888$	0	0
$889$	33.8102	1.13396
$890$	0	0
$891$	8.14243	0.272782
$892$	0	0
$893$	−2.22546	−0.0744721
$894$	0	0
$895$	46.9424	1.56911
$896$	0	0
$897$	−11.0992	−0.370590
$898$	0	0
$899$	9.97241	0.332599
$900$	0	0
$901$	−18.6150	−0.620155
$902$	0	0
$903$	110.989	3.69349
$904$	0	0
$905$	−15.1372	−0.503178
$906$	0	0
$907$	−25.0716	−0.832488	−0.416244	−	0.909253i	$-0.636654\pi$
$907$	−25.0716	−0.832488	−0.416244	+	0.909253i	$0.636654\pi$
$908$	0	0
$909$	7.11273	0.235914
$910$	0	0
$911$	−12.4119	−0.411224	−0.205612	−	0.978634i	$-0.565918\pi$
$911$	−12.4119	−0.411224	−0.205612	+	0.978634i	$0.565918\pi$
$912$	0	0
$913$	2.69535	0.0892030
$914$	0	0
$915$	−69.8695	−2.30982
$916$	0	0
$917$	2.22546	0.0734911
$918$	0	0
$919$	34.5550	1.13987	0.569933	−	0.821691i	$-0.306969\pi$
$919$	34.5550	1.13987	0.569933	+	0.821691i	$0.306969\pi$
$920$	0	0
$921$	−3.24603	−0.106960
$922$	0	0
$923$	−0.305103	−0.0100426
$924$	0	0
$925$	3.61676	0.118918
$926$	0	0
$927$	−2.47228	−0.0812003
$928$	0	0
$929$	25.1854	0.826305	0.413153	−	0.910662i	$-0.364428\pi$
$929$	25.1854	0.826305	0.413153	+	0.910662i	$0.364428\pi$
$930$	0	0
$931$	−109.813	−3.59898
$932$	0	0
$933$	32.6605	1.06926
$934$	0	0
$935$	−7.52461	−0.246081
$936$	0	0
$937$	26.9518	0.880478	0.440239	−	0.897881i	$-0.354894\pi$
$937$	26.9518	0.880478	0.440239	+	0.897881i	$0.354894\pi$
$938$	0	0
$939$	5.02474	0.163976
$940$	0	0
$941$	10.9096	0.355644	0.177822	−	0.984063i	$-0.443095\pi$
$941$	10.9096	0.355644	0.177822	+	0.984063i	$0.443095\pi$
$942$	0	0
$943$	−24.1690	−0.787050
$944$	0	0
$945$	−59.9505	−1.95019
$946$	0	0
$947$	9.45065	0.307105	0.153552	−	0.988141i	$-0.450929\pi$
$947$	9.45065	0.307105	0.153552	+	0.988141i	$0.450929\pi$
$948$	0	0
$949$	2.18120	0.0708046
$950$	0	0
$951$	−44.6535	−1.44799
$952$	0	0
$953$	44.9934	1.45748	0.728740	−	0.684790i	$-0.240106\pi$
$953$	44.9934	1.45748	0.728740	+	0.684790i	$0.240106\pi$
$954$	0	0
$955$	−31.2458	−1.01109
$956$	0	0
$957$	−4.80107	−0.155197
$958$	0	0
$959$	−52.2088	−1.68591
$960$	0	0
$961$	−21.6084	−0.697046
$962$	0	0
$963$	−12.8391	−0.413735
$964$	0	0
$965$	1.01641	0.0327193
$966$	0	0
$967$	−8.52355	−0.274099	−0.137049	−	0.990564i	$-0.543762\pi$
$967$	−8.52355	−0.274099	−0.137049	+	0.990564i	$0.543762\pi$
$968$	0	0
$969$	−47.9177	−1.53934
$970$	0	0
$971$	22.8308	0.732674	0.366337	−	0.930482i	$-0.380612\pi$
$971$	22.8308	0.732674	0.366337	+	0.930482i	$0.380612\pi$
$972$	0	0
$973$	72.5795	2.32679
$974$	0	0
$975$	−12.3361	−0.395073
$976$	0	0
$977$	−10.6045	−0.339269	−0.169634	−	0.985507i	$-0.554259\pi$
$977$	−10.6045	−0.339269	−0.169634	+	0.985507i	$0.554259\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−15.2007	−0.485321
$982$	0	0
$983$	9.13436	0.291341	0.145670	−	0.989333i	$-0.453466\pi$
$983$	9.13436	0.291341	0.145670	+	0.989333i	$0.453466\pi$
$984$	0	0
$985$	51.4751	1.64013
$986$	0	0
$987$	2.73862	0.0871712
$988$	0	0
$989$	39.8625	1.26755
$990$	0	0
$991$	31.8196	1.01078	0.505391	−	0.862890i	$-0.331348\pi$
$991$	31.8196	1.01078	0.505391	+	0.862890i	$0.331348\pi$
$992$	0	0
$993$	16.1536	0.512619
$994$	0	0
$995$	18.7253	0.593633
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	0	0
$999$	−4.36266	−0.138028

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bb.1.3		3
4.3	odd	2		2368.2.a.be.1.1		3
8.3	odd	2		592.2.a.i.1.3		3
8.5	even	2		296.2.a.c.1.1	✓	3
24.5	odd	2		2664.2.a.p.1.3		3
24.11	even	2		5328.2.a.bn.1.3		3
40.29	even	2		7400.2.a.k.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.c.1.1	✓	3	8.5	even	2
592.2.a.i.1.3		3	8.3	odd	2
2368.2.a.bb.1.3		3	1.1	even	1	trivial
2368.2.a.be.1.1		3	4.3	odd	2
2664.2.a.p.1.3		3	24.5	odd	2
5328.2.a.bn.1.3		3	24.11	even	2
7400.2.a.k.1.3		3	40.29	even	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93543 q^{3} +2.93543 q^{5} +4.68133 q^{7} +0.745898 q^{9} -0.762305 q^{11} -1.76231 q^{13} +5.68133 q^{15} +3.36266 q^{17} -7.36266 q^{19} +9.06040 q^{21} +3.25410 q^{23} +3.61676 q^{25} -4.36266 q^{27} +3.25410 q^{29} +3.06457 q^{31} -1.47539 q^{33} +13.7417 q^{35} +1.00000 q^{37} -3.41082 q^{39} -7.42723 q^{41} +12.2499 q^{43} +2.18953 q^{45} +0.302263 q^{47} +14.9149 q^{49} +6.50820 q^{51} -5.53579 q^{53} -2.23769 q^{55} -14.2499 q^{57} -10.2499 q^{59} -12.2981 q^{61} +3.49180 q^{63} -5.17313 q^{65} +13.1526 q^{67} +6.29809 q^{69} +0.173127 q^{71} -1.23769 q^{73} +7.00000 q^{75} -3.56860 q^{77} +4.61676 q^{79} -10.6813 q^{81} -3.53579 q^{83} +9.87086 q^{85} +6.29809 q^{87} +15.7417 q^{89} -8.24993 q^{91} +5.93126 q^{93} -21.6126 q^{95} -16.1208 q^{97} -0.568602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9} - 3 q^{13} + 10 q^{15} - 4 q^{17} - 8 q^{19} + 3 q^{21} + 9 q^{23} - 4 q^{25} + q^{27} + 9 q^{29} + 17 q^{31} - 9 q^{33} + 10 q^{35} + 3 q^{37} - 7 q^{39} - 16 q^{41}+ \cdots + 24 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9 - 3 * q^13 + 10 * q^15 - 4 * q^17 - 8 * q^19 + 3 * q^21 + 9 * q^23 - 4 * q^25 + q^27 + 9 * q^29 + 17 * q^31 - 9 * q^33 + 10 * q^35 + 3 * q^37 - 7 * q^39 - 16 * q^41 + 4 * q^43 - 2 * q^45 + 11 * q^47 + 8 * q^49 + 18 * q^51 + 3 * q^53 - 9 * q^55 - 10 * q^57 + 2 * q^59 - 15 * q^61 + 12 * q^63 - 10 * q^65 + 5 * q^67 - 3 * q^69 - 5 * q^71 - 6 * q^73 + 21 * q^75 + 15 * q^77 - q^79 - 25 * q^81 + 9 * q^83 + 14 * q^85 - 3 * q^87 + 16 * q^89 + 8 * q^91 - 22 * q^93 - 18 * q^95 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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