LMFDB - Embedded newform 2368.2.a.v.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ 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)

$f(q)$	$=$	$q-2.30278 q^{3} +1.30278 q^{5} +4.60555 q^{7} +2.30278 q^{9} +O(q^{10})$	$q-2.30278 q^{3} +1.30278 q^{5} +4.60555 q^{7} +2.30278 q^{9} +0.697224 q^{11} -2.30278 q^{13} -3.00000 q^{15} -7.21110 q^{17} -6.00000 q^{19} -10.6056 q^{21} +8.30278 q^{23} -3.30278 q^{25} +1.60555 q^{27} -10.3028 q^{29} -5.30278 q^{31} -1.60555 q^{33} +6.00000 q^{35} +1.00000 q^{37} +5.30278 q^{39} +0.302776 q^{41} -1.39445 q^{43} +3.00000 q^{45} -10.6056 q^{47} +14.2111 q^{49} +16.6056 q^{51} +7.21110 q^{53} +0.908327 q^{55} +13.8167 q^{57} +4.60555 q^{59} -13.3028 q^{61} +10.6056 q^{63} -3.00000 q^{65} +0.697224 q^{67} -19.1194 q^{69} -3.21110 q^{71} -0.697224 q^{73} +7.60555 q^{75} +3.21110 q^{77} +8.30278 q^{79} -10.6056 q^{81} -2.78890 q^{83} -9.39445 q^{85} +23.7250 q^{87} +1.21110 q^{89} -10.6056 q^{91} +12.2111 q^{93} -7.81665 q^{95} +15.2111 q^{97} +1.60555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q - q^3 - q^5 + 2 * q^7 + q^9	$ 2 q - q^{3} - q^{5} + 2 q^{7} + q^{9} + 5 q^{11} - q^{13} - 6 q^{15} - 12 q^{19} - 14 q^{21} + 13 q^{23} - 3 q^{25} - 4 q^{27} - 17 q^{29} - 7 q^{31} + 4 q^{33} + 12 q^{35} + 2 q^{37} + 7 q^{39} - 3 q^{41} - 10 q^{43} + 6 q^{45} - 14 q^{47} + 14 q^{49} + 26 q^{51} - 9 q^{55} + 6 q^{57} + 2 q^{59} - 23 q^{61} + 14 q^{63} - 6 q^{65} + 5 q^{67} - 13 q^{69} + 8 q^{71} - 5 q^{73} + 8 q^{75} - 8 q^{77} + 13 q^{79} - 14 q^{81} - 20 q^{83} - 26 q^{85} + 15 q^{87} - 12 q^{89} - 14 q^{91} + 10 q^{93} + 6 q^{95} + 16 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 + 2 * q^7 + q^9 + 5 * q^11 - q^13 - 6 * q^15 - 12 * q^19 - 14 * q^21 + 13 * q^23 - 3 * q^25 - 4 * q^27 - 17 * q^29 - 7 * q^31 + 4 * q^33 + 12 * q^35 + 2 * q^37 + 7 * q^39 - 3 * q^41 - 10 * q^43 + 6 * q^45 - 14 * q^47 + 14 * q^49 + 26 * q^51 - 9 * q^55 + 6 * q^57 + 2 * q^59 - 23 * q^61 + 14 * q^63 - 6 * q^65 + 5 * q^67 - 13 * q^69 + 8 * q^71 - 5 * q^73 + 8 * q^75 - 8 * q^77 + 13 * q^79 - 14 * q^81 - 20 * q^83 - 26 * q^85 + 15 * q^87 - 12 * q^89 - 14 * q^91 + 10 * q^93 + 6 * q^95 + 16 * q^97 - 4 * 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
$3$	−2.30278	−1.32951	−0.664754	−	0.747062i	$-0.731464\pi$
$3$	−2.30278	−1.32951	−0.664754	+	0.747062i	$0.731464\pi$
$4$	0	0
$5$	1.30278	0.582619	0.291309	−	0.956629i	$-0.405909\pi$
$5$	1.30278	0.582619	0.291309	+	0.956629i	$0.405909\pi$
$6$	0	0
$7$	4.60555	1.74073	0.870367	−	0.492403i	$-0.163881\pi$
$7$	4.60555	1.74073	0.870367	+	0.492403i	$0.163881\pi$
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	0.697224	0.210221	0.105111	−	0.994461i	$-0.466480\pi$
$11$	0.697224	0.210221	0.105111	+	0.994461i	$0.466480\pi$
$12$	0	0
$13$	−2.30278	−0.638675	−0.319338	−	0.947641i	$-0.603460\pi$
$13$	−2.30278	−0.638675	−0.319338	+	0.947641i	$0.603460\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−7.21110	−1.74895	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	−7.21110	−1.74895	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−10.6056	−2.31432
$22$	0	0
$23$	8.30278	1.73125	0.865624	−	0.500694i	$-0.166922\pi$
$23$	8.30278	1.73125	0.865624	+	0.500694i	$0.166922\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	0	0
$27$	1.60555	0.308988
$28$	0	0
$29$	−10.3028	−1.91318	−0.956589	−	0.291441i	$-0.905865\pi$
$29$	−10.3028	−1.91318	−0.956589	+	0.291441i	$0.905865\pi$
$30$	0	0
$31$	−5.30278	−0.952407	−0.476203	−	0.879335i	$-0.657987\pi$
$31$	−5.30278	−0.952407	−0.476203	+	0.879335i	$0.657987\pi$
$32$	0	0
$33$	−1.60555	−0.279491
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	5.30278	0.849124
$40$	0	0
$41$	0.302776	0.0472856	0.0236428	−	0.999720i	$-0.492474\pi$
$41$	0.302776	0.0472856	0.0236428	+	0.999720i	$0.492474\pi$
$42$	0	0
$43$	−1.39445	−0.212651	−0.106326	−	0.994331i	$-0.533909\pi$
$43$	−1.39445	−0.212651	−0.106326	+	0.994331i	$0.533909\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	−10.6056	−1.54698	−0.773489	−	0.633809i	$-0.781490\pi$
$47$	−10.6056	−1.54698	−0.773489	+	0.633809i	$0.781490\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	0	0
$51$	16.6056	2.32524
$52$	0	0
$53$	7.21110	0.990521	0.495261	−	0.868744i	$-0.335073\pi$
$53$	7.21110	0.990521	0.495261	+	0.868744i	$0.335073\pi$
$54$	0	0
$55$	0.908327	0.122479
$56$	0	0
$57$	13.8167	1.83006
$58$	0	0
$59$	4.60555	0.599592	0.299796	−	0.954003i	$-0.403081\pi$
$59$	4.60555	0.599592	0.299796	+	0.954003i	$0.403081\pi$
$60$	0	0
$61$	−13.3028	−1.70325	−0.851623	−	0.524155i	$-0.824381\pi$
$61$	−13.3028	−1.70325	−0.851623	+	0.524155i	$0.824381\pi$
$62$	0	0
$63$	10.6056	1.33617
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	0.697224	0.0851795	0.0425898	−	0.999093i	$-0.486439\pi$
$67$	0.697224	0.0851795	0.0425898	+	0.999093i	$0.486439\pi$
$68$	0	0
$69$	−19.1194	−2.30171
$70$	0	0
$71$	−3.21110	−0.381088	−0.190544	−	0.981679i	$-0.561025\pi$
$71$	−3.21110	−0.381088	−0.190544	+	0.981679i	$0.561025\pi$
$72$	0	0
$73$	−0.697224	−0.0816039	−0.0408020	−	0.999167i	$-0.512991\pi$
$73$	−0.697224	−0.0816039	−0.0408020	+	0.999167i	$0.512991\pi$
$74$	0	0
$75$	7.60555	0.878213
$76$	0	0
$77$	3.21110	0.365939
$78$	0	0
$79$	8.30278	0.934135	0.467068	−	0.884222i	$-0.345310\pi$
$79$	8.30278	0.934135	0.467068	+	0.884222i	$0.345310\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	−2.78890	−0.306121	−0.153061	−	0.988217i	$-0.548913\pi$
$83$	−2.78890	−0.306121	−0.153061	+	0.988217i	$0.548913\pi$
$84$	0	0
$85$	−9.39445	−1.01897
$86$	0	0
$87$	23.7250	2.54358
$88$	0	0
$89$	1.21110	0.128377	0.0641883	−	0.997938i	$-0.479554\pi$
$89$	1.21110	0.128377	0.0641883	+	0.997938i	$0.479554\pi$
$90$	0	0
$91$	−10.6056	−1.11176
$92$	0	0
$93$	12.2111	1.26623
$94$	0	0
$95$	−7.81665	−0.801972
$96$	0	0
$97$	15.2111	1.54445	0.772227	−	0.635347i	$-0.219143\pi$
$97$	15.2111	1.54445	0.772227	+	0.635347i	$0.219143\pi$
$98$	0	0
$99$	1.60555	0.161364
$100$	0	0
$101$	−0.788897	−0.0784982	−0.0392491	−	0.999229i	$-0.512497\pi$
$101$	−0.788897	−0.0784982	−0.0392491	+	0.999229i	$0.512497\pi$
$102$	0	0
$103$	−8.09167	−0.797296	−0.398648	−	0.917104i	$-0.630521\pi$
$103$	−8.09167	−0.797296	−0.398648	+	0.917104i	$0.630521\pi$
$104$	0	0
$105$	−13.8167	−1.34837
$106$	0	0
$107$	5.09167	0.492231	0.246115	−	0.969241i	$-0.420846\pi$
$107$	5.09167	0.492231	0.246115	+	0.969241i	$0.420846\pi$
$108$	0	0
$109$	12.4222	1.18983	0.594916	−	0.803788i	$-0.297185\pi$
$109$	12.4222	1.18983	0.594916	+	0.803788i	$0.297185\pi$
$110$	0	0
$111$	−2.30278	−0.218570
$112$	0	0
$113$	−20.4222	−1.92116	−0.960580	−	0.278005i	$-0.910327\pi$
$113$	−20.4222	−1.92116	−0.960580	+	0.278005i	$0.910327\pi$
$114$	0	0
$115$	10.8167	1.00866
$116$	0	0
$117$	−5.30278	−0.490242
$118$	0	0
$119$	−33.2111	−3.04446
$120$	0	0
$121$	−10.5139	−0.955807
$122$	0	0
$123$	−0.697224	−0.0628666
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	−12.4222	−1.10229	−0.551146	−	0.834409i	$-0.685809\pi$
$127$	−12.4222	−1.10229	−0.551146	+	0.834409i	$0.685809\pi$
$128$	0	0
$129$	3.21110	0.282722
$130$	0	0
$131$	−13.8167	−1.20717	−0.603583	−	0.797300i	$-0.706261\pi$
$131$	−13.8167	−1.20717	−0.603583	+	0.797300i	$0.706261\pi$
$132$	0	0
$133$	−27.6333	−2.39611
$134$	0	0
$135$	2.09167	0.180023
$136$	0	0
$137$	−4.69722	−0.401311	−0.200655	−	0.979662i	$-0.564307\pi$
$137$	−4.69722	−0.401311	−0.200655	+	0.979662i	$0.564307\pi$
$138$	0	0
$139$	−2.30278	−0.195319	−0.0976594	−	0.995220i	$-0.531136\pi$
$139$	−2.30278	−0.195319	−0.0976594	+	0.995220i	$0.531136\pi$
$140$	0	0
$141$	24.4222	2.05672
$142$	0	0
$143$	−1.60555	−0.134263
$144$	0	0
$145$	−13.4222	−1.11465
$146$	0	0
$147$	−32.7250	−2.69911
$148$	0	0
$149$	−0.605551	−0.0496087	−0.0248043	−	0.999692i	$-0.507896\pi$
$149$	−0.605551	−0.0496087	−0.0248043	+	0.999692i	$0.507896\pi$
$150$	0	0
$151$	10.6056	0.863068	0.431534	−	0.902097i	$-0.357973\pi$
$151$	10.6056	0.863068	0.431534	+	0.902097i	$0.357973\pi$
$152$	0	0
$153$	−16.6056	−1.34248
$154$	0	0
$155$	−6.90833	−0.554890
$156$	0	0
$157$	−7.21110	−0.575509	−0.287754	−	0.957704i	$-0.592909\pi$
$157$	−7.21110	−0.575509	−0.287754	+	0.957704i	$0.592909\pi$
$158$	0	0
$159$	−16.6056	−1.31691
$160$	0	0
$161$	38.2389	3.01364
$162$	0	0
$163$	18.0000	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$164$	0	0
$165$	−2.09167	−0.162837
$166$	0	0
$167$	−14.5139	−1.12312	−0.561559	−	0.827437i	$-0.689798\pi$
$167$	−14.5139	−1.12312	−0.561559	+	0.827437i	$0.689798\pi$
$168$	0	0
$169$	−7.69722	−0.592094
$170$	0	0
$171$	−13.8167	−1.05659
$172$	0	0
$173$	−7.21110	−0.548250	−0.274125	−	0.961694i	$-0.588388\pi$
$173$	−7.21110	−0.548250	−0.274125	+	0.961694i	$0.588388\pi$
$174$	0	0
$175$	−15.2111	−1.14985
$176$	0	0
$177$	−10.6056	−0.797162
$178$	0	0
$179$	−4.18335	−0.312678	−0.156339	−	0.987703i	$-0.549969\pi$
$179$	−4.18335	−0.312678	−0.156339	+	0.987703i	$0.549969\pi$
$180$	0	0
$181$	9.21110	0.684656	0.342328	−	0.939581i	$-0.388785\pi$
$181$	9.21110	0.684656	0.342328	+	0.939581i	$0.388785\pi$
$182$	0	0
$183$	30.6333	2.26448
$184$	0	0
$185$	1.30278	0.0957820
$186$	0	0
$187$	−5.02776	−0.367666
$188$	0	0
$189$	7.39445	0.537867
$190$	0	0
$191$	−11.7250	−0.848390	−0.424195	−	0.905571i	$-0.639443\pi$
$191$	−11.7250	−0.848390	−0.424195	+	0.905571i	$0.639443\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	6.90833	0.494716
$196$	0	0
$197$	−0.788897	−0.0562066	−0.0281033	−	0.999605i	$-0.508947\pi$
$197$	−0.788897	−0.0562066	−0.0281033	+	0.999605i	$0.508947\pi$
$198$	0	0
$199$	−6.42221	−0.455258	−0.227629	−	0.973748i	$-0.573097\pi$
$199$	−6.42221	−0.455258	−0.227629	+	0.973748i	$0.573097\pi$
$200$	0	0
$201$	−1.60555	−0.113247
$202$	0	0
$203$	−47.4500	−3.33033
$204$	0	0
$205$	0.394449	0.0275495
$206$	0	0
$207$	19.1194	1.32889
$208$	0	0
$209$	−4.18335	−0.289368
$210$	0	0
$211$	20.5139	1.41223	0.706117	−	0.708095i	$-0.250445\pi$
$211$	20.5139	1.41223	0.706117	+	0.708095i	$0.250445\pi$
$212$	0	0
$213$	7.39445	0.506659
$214$	0	0
$215$	−1.81665	−0.123895
$216$	0	0
$217$	−24.4222	−1.65789
$218$	0	0
$219$	1.60555	0.108493
$220$	0	0
$221$	16.6056	1.11701
$222$	0	0
$223$	10.6056	0.710200	0.355100	−	0.934828i	$-0.384447\pi$
$223$	10.6056	0.710200	0.355100	+	0.934828i	$0.384447\pi$
$224$	0	0
$225$	−7.60555	−0.507037
$226$	0	0
$227$	−7.81665	−0.518810	−0.259405	−	0.965769i	$-0.583526\pi$
$227$	−7.81665	−0.518810	−0.259405	+	0.965769i	$0.583526\pi$
$228$	0	0
$229$	10.6056	0.700835	0.350417	−	0.936594i	$-0.386040\pi$
$229$	10.6056	0.700835	0.350417	+	0.936594i	$0.386040\pi$
$230$	0	0
$231$	−7.39445	−0.486519
$232$	0	0
$233$	−15.0917	−0.988688	−0.494344	−	0.869266i	$-0.664592\pi$
$233$	−15.0917	−0.988688	−0.494344	+	0.869266i	$0.664592\pi$
$234$	0	0
$235$	−13.8167	−0.901299
$236$	0	0
$237$	−19.1194	−1.24194
$238$	0	0
$239$	13.1194	0.848625	0.424313	−	0.905516i	$-0.360516\pi$
$239$	13.1194	0.848625	0.424313	+	0.905516i	$0.360516\pi$
$240$	0	0
$241$	9.21110	0.593339	0.296670	−	0.954980i	$-0.404124\pi$
$241$	9.21110	0.593339	0.296670	+	0.954980i	$0.404124\pi$
$242$	0	0
$243$	19.6056	1.25770
$244$	0	0
$245$	18.5139	1.18281
$246$	0	0
$247$	13.8167	0.879133
$248$	0	0
$249$	6.42221	0.406991
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	5.78890	0.363945
$254$	0	0
$255$	21.6333	1.35473
$256$	0	0
$257$	−16.4222	−1.02439	−0.512195	−	0.858869i	$-0.671167\pi$
$257$	−16.4222	−1.02439	−0.512195	+	0.858869i	$0.671167\pi$
$258$	0	0
$259$	4.60555	0.286175
$260$	0	0
$261$	−23.7250	−1.46854
$262$	0	0
$263$	19.8167	1.22195	0.610974	−	0.791651i	$-0.290778\pi$
$263$	19.8167	1.22195	0.610974	+	0.791651i	$0.290778\pi$
$264$	0	0
$265$	9.39445	0.577096
$266$	0	0
$267$	−2.78890	−0.170678
$268$	0	0
$269$	14.4222	0.879337	0.439669	−	0.898160i	$-0.355096\pi$
$269$	14.4222	0.879337	0.439669	+	0.898160i	$0.355096\pi$
$270$	0	0
$271$	−18.4222	−1.11907	−0.559535	−	0.828807i	$-0.689020\pi$
$271$	−18.4222	−1.11907	−0.559535	+	0.828807i	$0.689020\pi$
$272$	0	0
$273$	24.4222	1.47810
$274$	0	0
$275$	−2.30278	−0.138863
$276$	0	0
$277$	−3.90833	−0.234829	−0.117414	−	0.993083i	$-0.537461\pi$
$277$	−3.90833	−0.234829	−0.117414	+	0.993083i	$0.537461\pi$
$278$	0	0
$279$	−12.2111	−0.731060
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	29.0278	1.72552	0.862761	−	0.505613i	$-0.168734\pi$
$283$	29.0278	1.72552	0.862761	+	0.505613i	$0.168734\pi$
$284$	0	0
$285$	18.0000	1.06623
$286$	0	0
$287$	1.39445	0.0823117
$288$	0	0
$289$	35.0000	2.05882
$290$	0	0
$291$	−35.0278	−2.05336
$292$	0	0
$293$	−13.0278	−0.761090	−0.380545	−	0.924762i	$-0.624264\pi$
$293$	−13.0278	−0.761090	−0.380545	+	0.924762i	$0.624264\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	1.11943	0.0649559
$298$	0	0
$299$	−19.1194	−1.10571
$300$	0	0
$301$	−6.42221	−0.370170
$302$	0	0
$303$	1.81665	0.104364
$304$	0	0
$305$	−17.3305	−0.992343
$306$	0	0
$307$	−21.6972	−1.23833	−0.619163	−	0.785262i	$-0.712528\pi$
$307$	−21.6972	−1.23833	−0.619163	+	0.785262i	$0.712528\pi$
$308$	0	0
$309$	18.6333	1.06001
$310$	0	0
$311$	−12.9083	−0.731964	−0.365982	−	0.930622i	$-0.619267\pi$
$311$	−12.9083	−0.731964	−0.365982	+	0.930622i	$0.619267\pi$
$312$	0	0
$313$	−5.81665	−0.328777	−0.164388	−	0.986396i	$-0.552565\pi$
$313$	−5.81665	−0.328777	−0.164388	+	0.986396i	$0.552565\pi$
$314$	0	0
$315$	13.8167	0.778480
$316$	0	0
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	0	0
$319$	−7.18335	−0.402190
$320$	0	0
$321$	−11.7250	−0.654425
$322$	0	0
$323$	43.2666	2.40742
$324$	0	0
$325$	7.60555	0.421880
$326$	0	0
$327$	−28.6056	−1.58189
$328$	0	0
$329$	−48.8444	−2.69288
$330$	0	0
$331$	6.42221	0.352996	0.176498	−	0.984301i	$-0.443523\pi$
$331$	6.42221	0.352996	0.176498	+	0.984301i	$0.443523\pi$
$332$	0	0
$333$	2.30278	0.126191
$334$	0	0
$335$	0.908327	0.0496272
$336$	0	0
$337$	20.5139	1.11746	0.558731	−	0.829349i	$-0.311288\pi$
$337$	20.5139	1.11746	0.558731	+	0.829349i	$0.311288\pi$
$338$	0	0
$339$	47.0278	2.55420
$340$	0	0
$341$	−3.69722	−0.200216
$342$	0	0
$343$	33.2111	1.79323
$344$	0	0
$345$	−24.9083	−1.34102
$346$	0	0
$347$	7.81665	0.419620	0.209810	−	0.977742i	$-0.432715\pi$
$347$	7.81665	0.419620	0.209810	+	0.977742i	$0.432715\pi$
$348$	0	0
$349$	2.60555	0.139472	0.0697360	−	0.997565i	$-0.477784\pi$
$349$	2.60555	0.139472	0.0697360	+	0.997565i	$0.477784\pi$
$350$	0	0
$351$	−3.69722	−0.197343
$352$	0	0
$353$	−21.3944	−1.13871	−0.569356	−	0.822091i	$-0.692807\pi$
$353$	−21.3944	−1.13871	−0.569356	+	0.822091i	$0.692807\pi$
$354$	0	0
$355$	−4.18335	−0.222029
$356$	0	0
$357$	76.4777	4.04763
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	24.2111	1.27075
$364$	0	0
$365$	−0.908327	−0.0475440
$366$	0	0
$367$	−23.0278	−1.20204	−0.601020	−	0.799234i	$-0.705239\pi$
$367$	−23.0278	−1.20204	−0.601020	+	0.799234i	$0.705239\pi$
$368$	0	0
$369$	0.697224	0.0362961
$370$	0	0
$371$	33.2111	1.72423
$372$	0	0
$373$	−27.8167	−1.44029	−0.720146	−	0.693823i	$-0.755925\pi$
$373$	−27.8167	−1.44029	−0.720146	+	0.693823i	$0.755925\pi$
$374$	0	0
$375$	24.9083	1.28626
$376$	0	0
$377$	23.7250	1.22190
$378$	0	0
$379$	32.7250	1.68097	0.840485	−	0.541835i	$-0.182270\pi$
$379$	32.7250	1.68097	0.840485	+	0.541835i	$0.182270\pi$
$380$	0	0
$381$	28.6056	1.46551
$382$	0	0
$383$	24.8444	1.26949	0.634745	−	0.772722i	$-0.281105\pi$
$383$	24.8444	1.26949	0.634745	+	0.772722i	$0.281105\pi$
$384$	0	0
$385$	4.18335	0.213203
$386$	0	0
$387$	−3.21110	−0.163230
$388$	0	0
$389$	10.9083	0.553074	0.276537	−	0.961003i	$-0.410813\pi$
$389$	10.9083	0.553074	0.276537	+	0.961003i	$0.410813\pi$
$390$	0	0
$391$	−59.8722	−3.02787
$392$	0	0
$393$	31.8167	1.60494
$394$	0	0
$395$	10.8167	0.544245
$396$	0	0
$397$	−15.8167	−0.793815	−0.396908	−	0.917859i	$-0.629917\pi$
$397$	−15.8167	−0.793815	−0.396908	+	0.917859i	$0.629917\pi$
$398$	0	0
$399$	63.6333	3.18565
$400$	0	0
$401$	19.0278	0.950201	0.475100	−	0.879932i	$-0.342412\pi$
$401$	19.0278	0.950201	0.475100	+	0.879932i	$0.342412\pi$
$402$	0	0
$403$	12.2111	0.608278
$404$	0	0
$405$	−13.8167	−0.686555
$406$	0	0
$407$	0.697224	0.0345601
$408$	0	0
$409$	13.3944	0.662313	0.331156	−	0.943576i	$-0.392561\pi$
$409$	13.3944	0.662313	0.331156	+	0.943576i	$0.392561\pi$
$410$	0	0
$411$	10.8167	0.533546
$412$	0	0
$413$	21.2111	1.04373
$414$	0	0
$415$	−3.63331	−0.178352
$416$	0	0
$417$	5.30278	0.259678
$418$	0	0
$419$	18.9083	0.923732	0.461866	−	0.886950i	$-0.347180\pi$
$419$	18.9083	0.923732	0.461866	+	0.886950i	$0.347180\pi$
$420$	0	0
$421$	−30.9083	−1.50638	−0.753190	−	0.657803i	$-0.771486\pi$
$421$	−30.9083	−1.50638	−0.753190	+	0.657803i	$0.771486\pi$
$422$	0	0
$423$	−24.4222	−1.18745
$424$	0	0
$425$	23.8167	1.15528
$426$	0	0
$427$	−61.2666	−2.96490
$428$	0	0
$429$	3.69722	0.178504
$430$	0	0
$431$	9.21110	0.443683	0.221842	−	0.975083i	$-0.428793\pi$
$431$	9.21110	0.443683	0.221842	+	0.975083i	$0.428793\pi$
$432$	0	0
$433$	11.3305	0.544511	0.272255	−	0.962225i	$-0.412230\pi$
$433$	11.3305	0.544511	0.272255	+	0.962225i	$0.412230\pi$
$434$	0	0
$435$	30.9083	1.48194
$436$	0	0
$437$	−49.8167	−2.38305
$438$	0	0
$439$	17.5139	0.835892	0.417946	−	0.908472i	$-0.362750\pi$
$439$	17.5139	0.835892	0.417946	+	0.908472i	$0.362750\pi$
$440$	0	0
$441$	32.7250	1.55833
$442$	0	0
$443$	14.0917	0.669516	0.334758	−	0.942304i	$-0.391345\pi$
$443$	14.0917	0.669516	0.334758	+	0.942304i	$0.391345\pi$
$444$	0	0
$445$	1.57779	0.0747947
$446$	0	0
$447$	1.39445	0.0659552
$448$	0	0
$449$	0.788897	0.0372304	0.0186152	−	0.999827i	$-0.494074\pi$
$449$	0.788897	0.0372304	0.0186152	+	0.999827i	$0.494074\pi$
$450$	0	0
$451$	0.211103	0.00994043
$452$	0	0
$453$	−24.4222	−1.14746
$454$	0	0
$455$	−13.8167	−0.647735
$456$	0	0
$457$	−13.8167	−0.646316	−0.323158	−	0.946345i	$-0.604744\pi$
$457$	−13.8167	−0.646316	−0.323158	+	0.946345i	$0.604744\pi$
$458$	0	0
$459$	−11.5778	−0.540405
$460$	0	0
$461$	−8.42221	−0.392261	−0.196131	−	0.980578i	$-0.562838\pi$
$461$	−8.42221	−0.392261	−0.196131	+	0.980578i	$0.562838\pi$
$462$	0	0
$463$	−3.69722	−0.171825	−0.0859123	−	0.996303i	$-0.527380\pi$
$463$	−3.69722	−0.171825	−0.0859123	+	0.996303i	$0.527380\pi$
$464$	0	0
$465$	15.9083	0.737731
$466$	0	0
$467$	−33.2111	−1.53683	−0.768413	−	0.639954i	$-0.778953\pi$
$467$	−33.2111	−1.53683	−0.768413	+	0.639954i	$0.778953\pi$
$468$	0	0
$469$	3.21110	0.148275
$470$	0	0
$471$	16.6056	0.765143
$472$	0	0
$473$	−0.972244	−0.0447038
$474$	0	0
$475$	19.8167	0.909250
$476$	0	0
$477$	16.6056	0.760316
$478$	0	0
$479$	−11.0917	−0.506791	−0.253396	−	0.967363i	$-0.581547\pi$
$479$	−11.0917	−0.506791	−0.253396	+	0.967363i	$0.581547\pi$
$480$	0	0
$481$	−2.30278	−0.104998
$482$	0	0
$483$	−88.0555	−4.00666
$484$	0	0
$485$	19.8167	0.899828
$486$	0	0
$487$	−26.7889	−1.21392	−0.606960	−	0.794732i	$-0.707611\pi$
$487$	−26.7889	−1.21392	−0.606960	+	0.794732i	$0.707611\pi$
$488$	0	0
$489$	−41.4500	−1.87443
$490$	0	0
$491$	−32.7250	−1.47686	−0.738429	−	0.674331i	$-0.764432\pi$
$491$	−32.7250	−1.47686	−0.738429	+	0.674331i	$0.764432\pi$
$492$	0	0
$493$	74.2944	3.34605
$494$	0	0
$495$	2.09167	0.0940137
$496$	0	0
$497$	−14.7889	−0.663373
$498$	0	0
$499$	−38.6611	−1.73071	−0.865353	−	0.501162i	$-0.832906\pi$
$499$	−38.6611	−1.73071	−0.865353	+	0.501162i	$0.832906\pi$
$500$	0	0
$501$	33.4222	1.49319
$502$	0	0
$503$	−0.275019	−0.0122625	−0.00613125	−	0.999981i	$-0.501952\pi$
$503$	−0.275019	−0.0122625	−0.00613125	+	0.999981i	$0.501952\pi$
$504$	0	0
$505$	−1.02776	−0.0457346
$506$	0	0
$507$	17.7250	0.787194
$508$	0	0
$509$	−19.0278	−0.843390	−0.421695	−	0.906738i	$-0.638565\pi$
$509$	−19.0278	−0.843390	−0.421695	+	0.906738i	$0.638565\pi$
$510$	0	0
$511$	−3.21110	−0.142051
$512$	0	0
$513$	−9.63331	−0.425321
$514$	0	0
$515$	−10.5416	−0.464520
$516$	0	0
$517$	−7.39445	−0.325207
$518$	0	0
$519$	16.6056	0.728903
$520$	0	0
$521$	35.2111	1.54263	0.771313	−	0.636456i	$-0.219600\pi$
$521$	35.2111	1.54263	0.771313	+	0.636456i	$0.219600\pi$
$522$	0	0
$523$	7.39445	0.323337	0.161668	−	0.986845i	$-0.448313\pi$
$523$	7.39445	0.323337	0.161668	+	0.986845i	$0.448313\pi$
$524$	0	0
$525$	35.0278	1.52874
$526$	0	0
$527$	38.2389	1.66571
$528$	0	0
$529$	45.9361	1.99722
$530$	0	0
$531$	10.6056	0.460242
$532$	0	0
$533$	−0.697224	−0.0302001
$534$	0	0
$535$	6.63331	0.286783
$536$	0	0
$537$	9.63331	0.415708
$538$	0	0
$539$	9.90833	0.426782
$540$	0	0
$541$	−26.5139	−1.13992	−0.569960	−	0.821672i	$-0.693041\pi$
$541$	−26.5139	−1.13992	−0.569960	+	0.821672i	$0.693041\pi$
$542$	0	0
$543$	−21.2111	−0.910256
$544$	0	0
$545$	16.1833	0.693218
$546$	0	0
$547$	−26.2389	−1.12189	−0.560946	−	0.827852i	$-0.689563\pi$
$547$	−26.2389	−1.12189	−0.560946	+	0.827852i	$0.689563\pi$
$548$	0	0
$549$	−30.6333	−1.30740
$550$	0	0
$551$	61.8167	2.63348
$552$	0	0
$553$	38.2389	1.62608
$554$	0	0
$555$	−3.00000	−0.127343
$556$	0	0
$557$	−43.7250	−1.85269	−0.926343	−	0.376680i	$-0.877066\pi$
$557$	−43.7250	−1.85269	−0.926343	+	0.376680i	$0.877066\pi$
$558$	0	0
$559$	3.21110	0.135815
$560$	0	0
$561$	11.5778	0.488815
$562$	0	0
$563$	3.21110	0.135332	0.0676659	−	0.997708i	$-0.478445\pi$
$563$	3.21110	0.135332	0.0676659	+	0.997708i	$0.478445\pi$
$564$	0	0
$565$	−26.6056	−1.11930
$566$	0	0
$567$	−48.8444	−2.05127
$568$	0	0
$569$	−22.4222	−0.939988	−0.469994	−	0.882670i	$-0.655744\pi$
$569$	−22.4222	−0.939988	−0.469994	+	0.882670i	$0.655744\pi$
$570$	0	0
$571$	28.1194	1.17676	0.588381	−	0.808584i	$-0.299766\pi$
$571$	28.1194	1.17676	0.588381	+	0.808584i	$0.299766\pi$
$572$	0	0
$573$	27.0000	1.12794
$574$	0	0
$575$	−27.4222	−1.14359
$576$	0	0
$577$	27.3944	1.14045	0.570223	−	0.821490i	$-0.306857\pi$
$577$	27.3944	1.14045	0.570223	+	0.821490i	$0.306857\pi$
$578$	0	0
$579$	−9.21110	−0.382800
$580$	0	0
$581$	−12.8444	−0.532876
$582$	0	0
$583$	5.02776	0.208228
$584$	0	0
$585$	−6.90833	−0.285624
$586$	0	0
$587$	8.78890	0.362757	0.181378	−	0.983413i	$-0.441944\pi$
$587$	8.78890	0.362757	0.181378	+	0.983413i	$0.441944\pi$
$588$	0	0
$589$	31.8167	1.31098
$590$	0	0
$591$	1.81665	0.0747272
$592$	0	0
$593$	−1.06392	−0.0436898	−0.0218449	−	0.999761i	$-0.506954\pi$
$593$	−1.06392	−0.0436898	−0.0218449	+	0.999761i	$0.506954\pi$
$594$	0	0
$595$	−43.2666	−1.77376
$596$	0	0
$597$	14.7889	0.605269
$598$	0	0
$599$	12.4222	0.507558	0.253779	−	0.967262i	$-0.418326\pi$
$599$	12.4222	0.507558	0.253779	+	0.967262i	$0.418326\pi$
$600$	0	0
$601$	30.0917	1.22746	0.613732	−	0.789514i	$-0.289667\pi$
$601$	30.0917	1.22746	0.613732	+	0.789514i	$0.289667\pi$
$602$	0	0
$603$	1.60555	0.0653831
$604$	0	0
$605$	−13.6972	−0.556871
$606$	0	0
$607$	36.9083	1.49806	0.749031	−	0.662535i	$-0.230519\pi$
$607$	36.9083	1.49806	0.749031	+	0.662535i	$0.230519\pi$
$608$	0	0
$609$	109.267	4.42771
$610$	0	0
$611$	24.4222	0.988017
$612$	0	0
$613$	18.2389	0.736661	0.368330	−	0.929695i	$-0.379930\pi$
$613$	18.2389	0.736661	0.368330	+	0.929695i	$0.379930\pi$
$614$	0	0
$615$	−0.908327	−0.0366273
$616$	0	0
$617$	−21.7527	−0.875732	−0.437866	−	0.899040i	$-0.644266\pi$
$617$	−21.7527	−0.875732	−0.437866	+	0.899040i	$0.644266\pi$
$618$	0	0
$619$	12.6972	0.510345	0.255172	−	0.966896i	$-0.417868\pi$
$619$	12.6972	0.510345	0.255172	+	0.966896i	$0.417868\pi$
$620$	0	0
$621$	13.3305	0.534936
$622$	0	0
$623$	5.57779	0.223470
$624$	0	0
$625$	2.42221	0.0968882
$626$	0	0
$627$	9.63331	0.384717
$628$	0	0
$629$	−7.21110	−0.287525
$630$	0	0
$631$	−46.9638	−1.86960	−0.934800	−	0.355173i	$-0.884422\pi$
$631$	−46.9638	−1.86960	−0.934800	+	0.355173i	$0.884422\pi$
$632$	0	0
$633$	−47.2389	−1.87758
$634$	0	0
$635$	−16.1833	−0.642217
$636$	0	0
$637$	−32.7250	−1.29661
$638$	0	0
$639$	−7.39445	−0.292520
$640$	0	0
$641$	46.7250	1.84553	0.922763	−	0.385368i	$-0.125926\pi$
$641$	46.7250	1.84553	0.922763	+	0.385368i	$0.125926\pi$
$642$	0	0
$643$	−0.972244	−0.0383415	−0.0191708	−	0.999816i	$-0.506103\pi$
$643$	−0.972244	−0.0383415	−0.0191708	+	0.999816i	$0.506103\pi$
$644$	0	0
$645$	4.18335	0.164719
$646$	0	0
$647$	9.27502	0.364639	0.182319	−	0.983239i	$-0.441640\pi$
$647$	9.27502	0.364639	0.182319	+	0.983239i	$0.441640\pi$
$648$	0	0
$649$	3.21110	0.126047
$650$	0	0
$651$	56.2389	2.20417
$652$	0	0
$653$	−31.1472	−1.21888	−0.609442	−	0.792831i	$-0.708606\pi$
$653$	−31.1472	−1.21888	−0.609442	+	0.792831i	$0.708606\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	0	0
$657$	−1.60555	−0.0626385
$658$	0	0
$659$	−25.3305	−0.986737	−0.493369	−	0.869820i	$-0.664235\pi$
$659$	−25.3305	−0.986737	−0.493369	+	0.869820i	$0.664235\pi$
$660$	0	0
$661$	23.1472	0.900321	0.450161	−	0.892948i	$-0.351367\pi$
$661$	23.1472	0.900321	0.450161	+	0.892948i	$0.351367\pi$
$662$	0	0
$663$	−38.2389	−1.48507
$664$	0	0
$665$	−36.0000	−1.39602
$666$	0	0
$667$	−85.5416	−3.31219
$668$	0	0
$669$	−24.4222	−0.944217
$670$	0	0
$671$	−9.27502	−0.358058
$672$	0	0
$673$	−29.7527	−1.14688	−0.573442	−	0.819246i	$-0.694392\pi$
$673$	−29.7527	−1.14688	−0.573442	+	0.819246i	$0.694392\pi$
$674$	0	0
$675$	−5.30278	−0.204104
$676$	0	0
$677$	−11.3944	−0.437924	−0.218962	−	0.975733i	$-0.570267\pi$
$677$	−11.3944	−0.437924	−0.218962	+	0.975733i	$0.570267\pi$
$678$	0	0
$679$	70.0555	2.68848
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	2.78890	0.106714	0.0533571	−	0.998575i	$-0.483008\pi$
$683$	2.78890	0.106714	0.0533571	+	0.998575i	$0.483008\pi$
$684$	0	0
$685$	−6.11943	−0.233811
$686$	0	0
$687$	−24.4222	−0.931765
$688$	0	0
$689$	−16.6056	−0.632621
$690$	0	0
$691$	6.42221	0.244312	0.122156	−	0.992511i	$-0.461019\pi$
$691$	6.42221	0.244312	0.122156	+	0.992511i	$0.461019\pi$
$692$	0	0
$693$	7.39445	0.280892
$694$	0	0
$695$	−3.00000	−0.113796
$696$	0	0
$697$	−2.18335	−0.0827001
$698$	0	0
$699$	34.7527	1.31447
$700$	0	0
$701$	−17.4861	−0.660442	−0.330221	−	0.943904i	$-0.607123\pi$
$701$	−17.4861	−0.660442	−0.330221	+	0.943904i	$0.607123\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	31.8167	1.19828
$706$	0	0
$707$	−3.63331	−0.136645
$708$	0	0
$709$	37.9638	1.42576	0.712881	−	0.701285i	$-0.247390\pi$
$709$	37.9638	1.42576	0.712881	+	0.701285i	$0.247390\pi$
$710$	0	0
$711$	19.1194	0.717035
$712$	0	0
$713$	−44.0278	−1.64885
$714$	0	0
$715$	−2.09167	−0.0782241
$716$	0	0
$717$	−30.2111	−1.12825
$718$	0	0
$719$	33.2111	1.23857	0.619283	−	0.785168i	$-0.287424\pi$
$719$	33.2111	1.23857	0.619283	+	0.785168i	$0.287424\pi$
$720$	0	0
$721$	−37.2666	−1.38788
$722$	0	0
$723$	−21.2111	−0.788849
$724$	0	0
$725$	34.0278	1.26376
$726$	0	0
$727$	8.93608	0.331421	0.165710	−	0.986174i	$-0.447008\pi$
$727$	8.93608	0.331421	0.165710	+	0.986174i	$0.447008\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	10.0555	0.371917
$732$	0	0
$733$	−3.21110	−0.118605	−0.0593024	−	0.998240i	$-0.518888\pi$
$733$	−3.21110	−0.118605	−0.0593024	+	0.998240i	$0.518888\pi$
$734$	0	0
$735$	−42.6333	−1.57255
$736$	0	0
$737$	0.486122	0.0179065
$738$	0	0
$739$	−5.72498	−0.210597	−0.105298	−	0.994441i	$-0.533580\pi$
$739$	−5.72498	−0.210597	−0.105298	+	0.994441i	$0.533580\pi$
$740$	0	0
$741$	−31.8167	−1.16881
$742$	0	0
$743$	7.81665	0.286765	0.143383	−	0.989667i	$-0.454202\pi$
$743$	7.81665	0.286765	0.143383	+	0.989667i	$0.454202\pi$
$744$	0	0
$745$	−0.788897	−0.0289030
$746$	0	0
$747$	−6.42221	−0.234976
$748$	0	0
$749$	23.4500	0.856843
$750$	0	0
$751$	−28.0555	−1.02376	−0.511880	−	0.859057i	$-0.671051\pi$
$751$	−28.0555	−1.02376	−0.511880	+	0.859057i	$0.671051\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.8167	0.502840
$756$	0	0
$757$	−49.1194	−1.78528	−0.892638	−	0.450774i	$-0.851148\pi$
$757$	−49.1194	−1.78528	−0.892638	+	0.450774i	$0.851148\pi$
$758$	0	0
$759$	−13.3305	−0.483868
$760$	0	0
$761$	31.6972	1.14902	0.574512	−	0.818496i	$-0.305192\pi$
$761$	31.6972	1.14902	0.574512	+	0.818496i	$0.305192\pi$
$762$	0	0
$763$	57.2111	2.07118
$764$	0	0
$765$	−21.6333	−0.782154
$766$	0	0
$767$	−10.6056	−0.382944
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	37.8167	1.36193
$772$	0	0
$773$	34.4222	1.23808	0.619040	−	0.785359i	$-0.287522\pi$
$773$	34.4222	1.23808	0.619040	+	0.785359i	$0.287522\pi$
$774$	0	0
$775$	17.5139	0.629117
$776$	0	0
$777$	−10.6056	−0.380472
$778$	0	0
$779$	−1.81665	−0.0650884
$780$	0	0
$781$	−2.23886	−0.0801127
$782$	0	0
$783$	−16.5416	−0.591150
$784$	0	0
$785$	−9.39445	−0.335302
$786$	0	0
$787$	−34.0555	−1.21395	−0.606974	−	0.794722i	$-0.707617\pi$
$787$	−34.0555	−1.21395	−0.606974	+	0.794722i	$0.707617\pi$
$788$	0	0
$789$	−45.6333	−1.62459
$790$	0	0
$791$	−94.0555	−3.34423
$792$	0	0
$793$	30.6333	1.08782
$794$	0	0
$795$	−21.6333	−0.767254
$796$	0	0
$797$	−4.48612	−0.158907	−0.0794533	−	0.996839i	$-0.525317\pi$
$797$	−4.48612	−0.158907	−0.0794533	+	0.996839i	$0.525317\pi$
$798$	0	0
$799$	76.4777	2.70559
$800$	0	0
$801$	2.78890	0.0985408
$802$	0	0
$803$	−0.486122	−0.0171549
$804$	0	0
$805$	49.8167	1.75581
$806$	0	0
$807$	−33.2111	−1.16909
$808$	0	0
$809$	12.2389	0.430295	0.215148	−	0.976582i	$-0.430977\pi$
$809$	12.2389	0.430295	0.215148	+	0.976582i	$0.430977\pi$
$810$	0	0
$811$	−2.09167	−0.0734486	−0.0367243	−	0.999325i	$-0.511692\pi$
$811$	−2.09167	−0.0734486	−0.0367243	+	0.999325i	$0.511692\pi$
$812$	0	0
$813$	42.4222	1.48781
$814$	0	0
$815$	23.4500	0.821416
$816$	0	0
$817$	8.36669	0.292714
$818$	0	0
$819$	−24.4222	−0.853381
$820$	0	0
$821$	−20.0555	−0.699942	−0.349971	−	0.936760i	$-0.613809\pi$
$821$	−20.0555	−0.699942	−0.349971	+	0.936760i	$0.613809\pi$
$822$	0	0
$823$	18.4222	0.642158	0.321079	−	0.947052i	$-0.395955\pi$
$823$	18.4222	0.642158	0.321079	+	0.947052i	$0.395955\pi$
$824$	0	0
$825$	5.30278	0.184619
$826$	0	0
$827$	32.2389	1.12105	0.560527	−	0.828136i	$-0.310598\pi$
$827$	32.2389	1.12105	0.560527	+	0.828136i	$0.310598\pi$
$828$	0	0
$829$	−21.6972	−0.753576	−0.376788	−	0.926300i	$-0.622971\pi$
$829$	−21.6972	−0.753576	−0.376788	+	0.926300i	$0.622971\pi$
$830$	0	0
$831$	9.00000	0.312207
$832$	0	0
$833$	−102.478	−3.55064
$834$	0	0
$835$	−18.9083	−0.654350
$836$	0	0
$837$	−8.51388	−0.294283
$838$	0	0
$839$	4.18335	0.144425	0.0722126	−	0.997389i	$-0.476994\pi$
$839$	4.18335	0.144425	0.0722126	+	0.997389i	$0.476994\pi$
$840$	0	0
$841$	77.1472	2.66025
$842$	0	0
$843$	46.0555	1.58624
$844$	0	0
$845$	−10.0278	−0.344965
$846$	0	0
$847$	−48.4222	−1.66381
$848$	0	0
$849$	−66.8444	−2.29409
$850$	0	0
$851$	8.30278	0.284615
$852$	0	0
$853$	40.9361	1.40162	0.700812	−	0.713346i	$-0.252821\pi$
$853$	40.9361	1.40162	0.700812	+	0.713346i	$0.252821\pi$
$854$	0	0
$855$	−18.0000	−0.615587
$856$	0	0
$857$	−19.5778	−0.668765	−0.334382	−	0.942437i	$-0.608528\pi$
$857$	−19.5778	−0.668765	−0.334382	+	0.942437i	$0.608528\pi$
$858$	0	0
$859$	−12.4222	−0.423840	−0.211920	−	0.977287i	$-0.567972\pi$
$859$	−12.4222	−0.423840	−0.211920	+	0.977287i	$0.567972\pi$
$860$	0	0
$861$	−3.21110	−0.109434
$862$	0	0
$863$	−9.21110	−0.313550	−0.156775	−	0.987634i	$-0.550110\pi$
$863$	−9.21110	−0.313550	−0.156775	+	0.987634i	$0.550110\pi$
$864$	0	0
$865$	−9.39445	−0.319421
$866$	0	0
$867$	−80.5971	−2.73722
$868$	0	0
$869$	5.78890	0.196375
$870$	0	0
$871$	−1.60555	−0.0544020
$872$	0	0
$873$	35.0278	1.18551
$874$	0	0
$875$	−49.8167	−1.68411
$876$	0	0
$877$	15.5778	0.526025	0.263012	−	0.964792i	$-0.415284\pi$
$877$	15.5778	0.526025	0.263012	+	0.964792i	$0.415284\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	−34.5694	−1.16467	−0.582336	−	0.812948i	$-0.697861\pi$
$881$	−34.5694	−1.16467	−0.582336	+	0.812948i	$0.697861\pi$
$882$	0	0
$883$	39.6333	1.33377	0.666883	−	0.745162i	$-0.267628\pi$
$883$	39.6333	1.33377	0.666883	+	0.745162i	$0.267628\pi$
$884$	0	0
$885$	−13.8167	−0.464442
$886$	0	0
$887$	14.3667	0.482386	0.241193	−	0.970477i	$-0.422461\pi$
$887$	14.3667	0.482386	0.241193	+	0.970477i	$0.422461\pi$
$888$	0	0
$889$	−57.2111	−1.91880
$890$	0	0
$891$	−7.39445	−0.247723
$892$	0	0
$893$	63.6333	2.12941
$894$	0	0
$895$	−5.44996	−0.182172
$896$	0	0
$897$	44.0278	1.47004
$898$	0	0
$899$	54.6333	1.82212
$900$	0	0
$901$	−52.0000	−1.73237
$902$	0	0
$903$	14.7889	0.492144
$904$	0	0
$905$	12.0000	0.398893
$906$	0	0
$907$	20.7889	0.690284	0.345142	−	0.938550i	$-0.387831\pi$
$907$	20.7889	0.690284	0.345142	+	0.938550i	$0.387831\pi$
$908$	0	0
$909$	−1.81665	−0.0602546
$910$	0	0
$911$	18.4222	0.610355	0.305177	−	0.952296i	$-0.401284\pi$
$911$	18.4222	0.610355	0.305177	+	0.952296i	$0.401284\pi$
$912$	0	0
$913$	−1.94449	−0.0643531
$914$	0	0
$915$	39.9083	1.31933
$916$	0	0
$917$	−63.6333	−2.10136
$918$	0	0
$919$	29.5778	0.975681	0.487841	−	0.872933i	$-0.337785\pi$
$919$	29.5778	0.975681	0.487841	+	0.872933i	$0.337785\pi$
$920$	0	0
$921$	49.9638	1.64636
$922$	0	0
$923$	7.39445	0.243391
$924$	0	0
$925$	−3.30278	−0.108595
$926$	0	0
$927$	−18.6333	−0.611998
$928$	0	0
$929$	42.7527	1.40267	0.701336	−	0.712831i	$-0.252587\pi$
$929$	42.7527	1.40267	0.701336	+	0.712831i	$0.252587\pi$
$930$	0	0
$931$	−85.2666	−2.79450
$932$	0	0
$933$	29.7250	0.973152
$934$	0	0
$935$	−6.55004	−0.214209
$936$	0	0
$937$	39.9361	1.30465	0.652327	−	0.757937i	$-0.273793\pi$
$937$	39.9361	1.30465	0.652327	+	0.757937i	$0.273793\pi$
$938$	0	0
$939$	13.3944	0.437111
$940$	0	0
$941$	−2.97224	−0.0968924	−0.0484462	−	0.998826i	$-0.515427\pi$
$941$	−2.97224	−0.0968924	−0.0484462	+	0.998826i	$0.515427\pi$
$942$	0	0
$943$	2.51388	0.0818631
$944$	0	0
$945$	9.63331	0.313372
$946$	0	0
$947$	43.2666	1.40598	0.702988	−	0.711202i	$-0.251849\pi$
$947$	43.2666	1.40598	0.702988	+	0.711202i	$0.251849\pi$
$948$	0	0
$949$	1.60555	0.0521184
$950$	0	0
$951$	−18.4222	−0.597381
$952$	0	0
$953$	−32.3305	−1.04729	−0.523644	−	0.851937i	$-0.675428\pi$
$953$	−32.3305	−1.04729	−0.523644	+	0.851937i	$0.675428\pi$
$954$	0	0
$955$	−15.2750	−0.494288
$956$	0	0
$957$	16.5416	0.534715
$958$	0	0
$959$	−21.6333	−0.698576
$960$	0	0
$961$	−2.88057	−0.0929216
$962$	0	0
$963$	11.7250	0.377832
$964$	0	0
$965$	5.21110	0.167751
$966$	0	0
$967$	−17.3028	−0.556420	−0.278210	−	0.960520i	$-0.589741\pi$
$967$	−17.3028	−0.556420	−0.278210	+	0.960520i	$0.589741\pi$
$968$	0	0
$969$	−99.6333	−3.20068
$970$	0	0
$971$	−43.9638	−1.41087	−0.705433	−	0.708776i	$-0.749248\pi$
$971$	−43.9638	−1.41087	−0.705433	+	0.708776i	$0.749248\pi$
$972$	0	0
$973$	−10.6056	−0.339998
$974$	0	0
$975$	−17.5139	−0.560893
$976$	0	0
$977$	−47.2111	−1.51042	−0.755208	−	0.655485i	$-0.772464\pi$
$977$	−47.2111	−1.51042	−0.755208	+	0.655485i	$0.772464\pi$
$978$	0	0
$979$	0.844410	0.0269875
$980$	0	0
$981$	28.6056	0.913305
$982$	0	0
$983$	−0.844410	−0.0269325	−0.0134663	−	0.999909i	$-0.504287\pi$
$983$	−0.844410	−0.0269325	−0.0134663	+	0.999909i	$0.504287\pi$
$984$	0	0
$985$	−1.02776	−0.0327470
$986$	0	0
$987$	112.478	3.58021
$988$	0	0
$989$	−11.5778	−0.368152
$990$	0	0
$991$	60.7805	1.93076	0.965378	−	0.260855i	$-0.0840044\pi$
$991$	60.7805	1.93076	0.965378	+	0.260855i	$0.0840044\pi$
$992$	0	0
$993$	−14.7889	−0.469311
$994$	0	0
$995$	−8.36669	−0.265242
$996$	0	0
$997$	−38.8444	−1.23021	−0.615107	−	0.788443i	$-0.710887\pi$
$997$	−38.8444	−1.23021	−0.615107	+	0.788443i	$0.710887\pi$
$998$	0	0
$999$	1.60555	0.0507974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.v.1.1		2
4.3	odd	2		2368.2.a.z.1.2		2
8.3	odd	2		1184.2.a.i.1.1	✓	2
8.5	even	2		1184.2.a.k.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.i.1.1	✓	2	8.3	odd	2
1184.2.a.k.1.2	yes	2	8.5	even	2
2368.2.a.v.1.1		2	1.1	even	1	trivial
2368.2.a.z.1.2		2	4.3	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{3} +1.30278 q^{5} +4.60555 q^{7} +2.30278 q^{9} +O(q^{10})\)	\(q-2.30278 q^{3} +1.30278 q^{5} +4.60555 q^{7} +2.30278 q^{9} +0.697224 q^{11} -2.30278 q^{13} -3.00000 q^{15} -7.21110 q^{17} -6.00000 q^{19} -10.6056 q^{21} +8.30278 q^{23} -3.30278 q^{25} +1.60555 q^{27} -10.3028 q^{29} -5.30278 q^{31} -1.60555 q^{33} +6.00000 q^{35} +1.00000 q^{37} +5.30278 q^{39} +0.302776 q^{41} -1.39445 q^{43} +3.00000 q^{45} -10.6056 q^{47} +14.2111 q^{49} +16.6056 q^{51} +7.21110 q^{53} +0.908327 q^{55} +13.8167 q^{57} +4.60555 q^{59} -13.3028 q^{61} +10.6056 q^{63} -3.00000 q^{65} +0.697224 q^{67} -19.1194 q^{69} -3.21110 q^{71} -0.697224 q^{73} +7.60555 q^{75} +3.21110 q^{77} +8.30278 q^{79} -10.6056 q^{81} -2.78890 q^{83} -9.39445 q^{85} +23.7250 q^{87} +1.21110 q^{89} -10.6056 q^{91} +12.2111 q^{93} -7.81665 q^{95} +15.2111 q^{97} +1.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q - q^3 - q^5 + 2 * q^7 + q^9	\( 2 q - q^{3} - q^{5} + 2 q^{7} + q^{9} + 5 q^{11} - q^{13} - 6 q^{15} - 12 q^{19} - 14 q^{21} + 13 q^{23} - 3 q^{25} - 4 q^{27} - 17 q^{29} - 7 q^{31} + 4 q^{33} + 12 q^{35} + 2 q^{37} + 7 q^{39} - 3 q^{41} - 10 q^{43} + 6 q^{45} - 14 q^{47} + 14 q^{49} + 26 q^{51} - 9 q^{55} + 6 q^{57} + 2 q^{59} - 23 q^{61} + 14 q^{63} - 6 q^{65} + 5 q^{67} - 13 q^{69} + 8 q^{71} - 5 q^{73} + 8 q^{75} - 8 q^{77} + 13 q^{79} - 14 q^{81} - 20 q^{83} - 26 q^{85} + 15 q^{87} - 12 q^{89} - 14 q^{91} + 10 q^{93} + 6 q^{95} + 16 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 + 2 * q^7 + q^9 + 5 * q^11 - q^13 - 6 * q^15 - 12 * q^19 - 14 * q^21 + 13 * q^23 - 3 * q^25 - 4 * q^27 - 17 * q^29 - 7 * q^31 + 4 * q^33 + 12 * q^35 + 2 * q^37 + 7 * q^39 - 3 * q^41 - 10 * q^43 + 6 * q^45 - 14 * q^47 + 14 * q^49 + 26 * q^51 - 9 * q^55 + 6 * q^57 + 2 * q^59 - 23 * q^61 + 14 * q^63 - 6 * q^65 + 5 * q^67 - 13 * q^69 + 8 * q^71 - 5 * q^73 + 8 * q^75 - 8 * q^77 + 13 * q^79 - 14 * q^81 - 20 * q^83 - 26 * q^85 + 15 * q^87 - 12 * q^89 - 14 * q^91 + 10 * q^93 + 6 * q^95 + 16 * q^97 - 4 * q^99

Introduction

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