LMFDB - Embedded newform 2368.2.a.s.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 74)
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)

$f(q)$	$=$	$q-3.30278 q^{3} +2.30278 q^{5} -2.60555 q^{7} +7.90833 q^{9} +O(q^{10})$	$q-3.30278 q^{3} +2.30278 q^{5} -2.60555 q^{7} +7.90833 q^{9} +2.30278 q^{11} -1.30278 q^{13} -7.60555 q^{15} -6.00000 q^{17} -2.00000 q^{19} +8.60555 q^{21} +3.90833 q^{23} +0.302776 q^{25} -16.2111 q^{27} +3.90833 q^{29} -0.302776 q^{31} -7.60555 q^{33} -6.00000 q^{35} -1.00000 q^{37} +4.30278 q^{39} +9.90833 q^{41} -0.605551 q^{43} +18.2111 q^{45} +4.60555 q^{47} -0.211103 q^{49} +19.8167 q^{51} +6.00000 q^{53} +5.30278 q^{55} +6.60555 q^{57} -10.6056 q^{59} -7.51388 q^{61} -20.6056 q^{63} -3.00000 q^{65} +3.51388 q^{67} -12.9083 q^{69} +6.00000 q^{71} -12.3028 q^{73} -1.00000 q^{75} -6.00000 q^{77} +9.11943 q^{79} +29.8167 q^{81} -2.78890 q^{83} -13.8167 q^{85} -12.9083 q^{87} -9.21110 q^{89} +3.39445 q^{91} +1.00000 q^{93} -4.60555 q^{95} -16.4222 q^{97} +18.2111 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 + q^5 + 2 * q^7 + 5 * q^9	$ 2 q - 3 q^{3} + q^{5} + 2 q^{7} + 5 q^{9} + q^{11} + q^{13} - 8 q^{15} - 12 q^{17} - 4 q^{19} + 10 q^{21} - 3 q^{23} - 3 q^{25} - 18 q^{27} - 3 q^{29} + 3 q^{31} - 8 q^{33} - 12 q^{35} - 2 q^{37} + 5 q^{39} + 9 q^{41} + 6 q^{43} + 22 q^{45} + 2 q^{47} + 14 q^{49} + 18 q^{51} + 12 q^{53} + 7 q^{55} + 6 q^{57} - 14 q^{59} + 3 q^{61} - 34 q^{63} - 6 q^{65} - 11 q^{67} - 15 q^{69} + 12 q^{71} - 21 q^{73} - 2 q^{75} - 12 q^{77} - 7 q^{79} + 38 q^{81} - 20 q^{83} - 6 q^{85} - 15 q^{87} - 4 q^{89} + 14 q^{91} + 2 q^{93} - 2 q^{95} - 4 q^{97} + 22 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 + q^5 + 2 * q^7 + 5 * q^9 + q^11 + q^13 - 8 * q^15 - 12 * q^17 - 4 * q^19 + 10 * q^21 - 3 * q^23 - 3 * q^25 - 18 * q^27 - 3 * q^29 + 3 * q^31 - 8 * q^33 - 12 * q^35 - 2 * q^37 + 5 * q^39 + 9 * q^41 + 6 * q^43 + 22 * q^45 + 2 * q^47 + 14 * q^49 + 18 * q^51 + 12 * q^53 + 7 * q^55 + 6 * q^57 - 14 * q^59 + 3 * q^61 - 34 * q^63 - 6 * q^65 - 11 * q^67 - 15 * q^69 + 12 * q^71 - 21 * q^73 - 2 * q^75 - 12 * q^77 - 7 * q^79 + 38 * q^81 - 20 * q^83 - 6 * q^85 - 15 * q^87 - 4 * q^89 + 14 * q^91 + 2 * q^93 - 2 * q^95 - 4 * q^97 + 22 * 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
$3$	−3.30278	−1.90686	−0.953429	−	0.301617i	$-0.902474\pi$
$3$	−3.30278	−1.90686	−0.953429	+	0.301617i	$0.902474\pi$
$4$	0	0
$5$	2.30278	1.02983	0.514916	−	0.857240i	$-0.327823\pi$
$5$	2.30278	1.02983	0.514916	+	0.857240i	$0.327823\pi$
$6$	0	0
$7$	−2.60555	−0.984806	−0.492403	−	0.870367i	$-0.663881\pi$
$7$	−2.60555	−0.984806	−0.492403	+	0.870367i	$0.663881\pi$
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	2.30278	0.694313	0.347156	−	0.937807i	$-0.387147\pi$
$11$	2.30278	0.694313	0.347156	+	0.937807i	$0.387147\pi$
$12$	0	0
$13$	−1.30278	−0.361325	−0.180662	−	0.983545i	$-0.557824\pi$
$13$	−1.30278	−0.361325	−0.180662	+	0.983545i	$0.557824\pi$
$14$	0	0
$15$	−7.60555	−1.96374
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	8.60555	1.87789
$22$	0	0
$23$	3.90833	0.814942	0.407471	−	0.913218i	$-0.366411\pi$
$23$	3.90833	0.814942	0.407471	+	0.913218i	$0.366411\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	−16.2111	−3.11983
$28$	0	0
$29$	3.90833	0.725758	0.362879	−	0.931836i	$-0.381794\pi$
$29$	3.90833	0.725758	0.362879	+	0.931836i	$0.381794\pi$
$30$	0	0
$31$	−0.302776	−0.0543801	−0.0271901	−	0.999630i	$-0.508656\pi$
$31$	−0.302776	−0.0543801	−0.0271901	+	0.999630i	$0.508656\pi$
$32$	0	0
$33$	−7.60555	−1.32396
$34$	0	0
$35$	−6.00000	−1.01419
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	4.30278	0.688996
$40$	0	0
$41$	9.90833	1.54742	0.773710	−	0.633540i	$-0.218399\pi$
$41$	9.90833	1.54742	0.773710	+	0.633540i	$0.218399\pi$
$42$	0	0
$43$	−0.605551	−0.0923457	−0.0461729	−	0.998933i	$-0.514703\pi$
$43$	−0.605551	−0.0923457	−0.0461729	+	0.998933i	$0.514703\pi$
$44$	0	0
$45$	18.2111	2.71475
$46$	0	0
$47$	4.60555	0.671789	0.335894	−	0.941900i	$-0.390961\pi$
$47$	4.60555	0.671789	0.335894	+	0.941900i	$0.390961\pi$
$48$	0	0
$49$	−0.211103	−0.0301575
$50$	0	0
$51$	19.8167	2.77489
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	5.30278	0.715026
$56$	0	0
$57$	6.60555	0.874927
$58$	0	0
$59$	−10.6056	−1.38073	−0.690363	−	0.723464i	$-0.742549\pi$
$59$	−10.6056	−1.38073	−0.690363	+	0.723464i	$0.742549\pi$
$60$	0	0
$61$	−7.51388	−0.962054	−0.481027	−	0.876706i	$-0.659736\pi$
$61$	−7.51388	−0.962054	−0.481027	+	0.876706i	$0.659736\pi$
$62$	0	0
$63$	−20.6056	−2.59606
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	3.51388	0.429289	0.214644	−	0.976692i	$-0.431141\pi$
$67$	3.51388	0.429289	0.214644	+	0.976692i	$0.431141\pi$
$68$	0	0
$69$	−12.9083	−1.55398
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−12.3028	−1.43993	−0.719965	−	0.694010i	$-0.755842\pi$
$73$	−12.3028	−1.43993	−0.719965	+	0.694010i	$0.755842\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	9.11943	1.02602	0.513008	−	0.858384i	$-0.328531\pi$
$79$	9.11943	1.02602	0.513008	+	0.858384i	$0.328531\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	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
$85$	−13.8167	−1.49863
$86$	0	0
$87$	−12.9083	−1.38392
$88$	0	0
$89$	−9.21110	−0.976375	−0.488187	−	0.872739i	$-0.662342\pi$
$89$	−9.21110	−0.976375	−0.488187	+	0.872739i	$0.662342\pi$
$90$	0	0
$91$	3.39445	0.355835
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−4.60555	−0.472520
$96$	0	0
$97$	−16.4222	−1.66742	−0.833711	−	0.552201i	$-0.813788\pi$
$97$	−16.4222	−1.66742	−0.833711	+	0.552201i	$0.813788\pi$
$98$	0	0
$99$	18.2111	1.83028
$100$	0	0
$101$	12.4222	1.23606	0.618028	−	0.786156i	$-0.287932\pi$
$101$	12.4222	1.23606	0.618028	+	0.786156i	$0.287932\pi$
$102$	0	0
$103$	−0.302776	−0.0298334	−0.0149167	−	0.999889i	$-0.504748\pi$
$103$	−0.302776	−0.0298334	−0.0149167	+	0.999889i	$0.504748\pi$
$104$	0	0
$105$	19.8167	1.93391
$106$	0	0
$107$	−0.697224	−0.0674032	−0.0337016	−	0.999432i	$-0.510730\pi$
$107$	−0.697224	−0.0674032	−0.0337016	+	0.999432i	$0.510730\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	3.30278	0.313486
$112$	0	0
$113$	−3.21110	−0.302075	−0.151038	−	0.988528i	$-0.548261\pi$
$113$	−3.21110	−0.302075	−0.151038	+	0.988528i	$0.548261\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	0	0
$117$	−10.3028	−0.952492
$118$	0	0
$119$	15.6333	1.43310
$120$	0	0
$121$	−5.69722	−0.517929
$122$	0	0
$123$	−32.7250	−2.95071
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	−19.2111	−1.70471	−0.852355	−	0.522964i	$-0.824826\pi$
$127$	−19.2111	−1.70471	−0.852355	+	0.522964i	$0.824826\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	−10.6056	−0.926611	−0.463306	−	0.886199i	$-0.653337\pi$
$131$	−10.6056	−0.926611	−0.463306	+	0.886199i	$0.653337\pi$
$132$	0	0
$133$	5.21110	0.451860
$134$	0	0
$135$	−37.3305	−3.21290
$136$	0	0
$137$	0.908327	0.0776036	0.0388018	−	0.999247i	$-0.487646\pi$
$137$	0.908327	0.0776036	0.0388018	+	0.999247i	$0.487646\pi$
$138$	0	0
$139$	1.90833	0.161862	0.0809311	−	0.996720i	$-0.474211\pi$
$139$	1.90833	0.161862	0.0809311	+	0.996720i	$0.474211\pi$
$140$	0	0
$141$	−15.2111	−1.28101
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	9.00000	0.747409
$146$	0	0
$147$	0.697224	0.0575061
$148$	0	0
$149$	−19.8167	−1.62344	−0.811722	−	0.584044i	$-0.801469\pi$
$149$	−19.8167	−1.62344	−0.811722	+	0.584044i	$0.801469\pi$
$150$	0	0
$151$	−20.6056	−1.67686	−0.838428	−	0.545012i	$-0.816525\pi$
$151$	−20.6056	−1.67686	−0.838428	+	0.545012i	$0.816525\pi$
$152$	0	0
$153$	−47.4500	−3.83610
$154$	0	0
$155$	−0.697224	−0.0560024
$156$	0	0
$157$	7.21110	0.575509	0.287754	−	0.957704i	$-0.407091\pi$
$157$	7.21110	0.575509	0.287754	+	0.957704i	$0.407091\pi$
$158$	0	0
$159$	−19.8167	−1.57156
$160$	0	0
$161$	−10.1833	−0.802560
$162$	0	0
$163$	−8.42221	−0.659678	−0.329839	−	0.944037i	$-0.606994\pi$
$163$	−8.42221	−0.659678	−0.329839	+	0.944037i	$0.606994\pi$
$164$	0	0
$165$	−17.5139	−1.36345
$166$	0	0
$167$	−5.51388	−0.426677	−0.213338	−	0.976978i	$-0.568434\pi$
$167$	−5.51388	−0.426677	−0.213338	+	0.976978i	$0.568434\pi$
$168$	0	0
$169$	−11.3028	−0.869444
$170$	0	0
$171$	−15.8167	−1.20953
$172$	0	0
$173$	8.78890	0.668207	0.334104	−	0.942536i	$-0.391566\pi$
$173$	8.78890	0.668207	0.334104	+	0.942536i	$0.391566\pi$
$174$	0	0
$175$	−0.788897	−0.0596350
$176$	0	0
$177$	35.0278	2.63285
$178$	0	0
$179$	13.8167	1.03271	0.516353	−	0.856376i	$-0.327289\pi$
$179$	13.8167	1.03271	0.516353	+	0.856376i	$0.327289\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	24.8167	1.83450
$184$	0	0
$185$	−2.30278	−0.169303
$186$	0	0
$187$	−13.8167	−1.01037
$188$	0	0
$189$	42.2389	3.07242
$190$	0	0
$191$	−5.51388	−0.398970	−0.199485	−	0.979901i	$-0.563927\pi$
$191$	−5.51388	−0.398970	−0.199485	+	0.979901i	$0.563927\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	9.90833	0.709550
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	26.4222	1.87302	0.936510	−	0.350640i	$-0.114036\pi$
$199$	26.4222	1.87302	0.936510	+	0.350640i	$0.114036\pi$
$200$	0	0
$201$	−11.6056	−0.818592
$202$	0	0
$203$	−10.1833	−0.714731
$204$	0	0
$205$	22.8167	1.59358
$206$	0	0
$207$	30.9083	2.14828
$208$	0	0
$209$	−4.60555	−0.318573
$210$	0	0
$211$	−10.3028	−0.709272	−0.354636	−	0.935004i	$-0.615395\pi$
$211$	−10.3028	−0.709272	−0.354636	+	0.935004i	$0.615395\pi$
$212$	0	0
$213$	−19.8167	−1.35781
$214$	0	0
$215$	−1.39445	−0.0951006
$216$	0	0
$217$	0.788897	0.0535538
$218$	0	0
$219$	40.6333	2.74574
$220$	0	0
$221$	7.81665	0.525805
$222$	0	0
$223$	−5.81665	−0.389512	−0.194756	−	0.980852i	$-0.562391\pi$
$223$	−5.81665	−0.389512	−0.194756	+	0.980852i	$0.562391\pi$
$224$	0	0
$225$	2.39445	0.159630
$226$	0	0
$227$	−13.8167	−0.917044	−0.458522	−	0.888683i	$-0.651621\pi$
$227$	−13.8167	−0.917044	−0.458522	+	0.888683i	$0.651621\pi$
$228$	0	0
$229$	−24.6056	−1.62598	−0.812990	−	0.582277i	$-0.802162\pi$
$229$	−24.6056	−1.62598	−0.812990	+	0.582277i	$0.802162\pi$
$230$	0	0
$231$	19.8167	1.30384
$232$	0	0
$233$	8.51388	0.557763	0.278881	−	0.960326i	$-0.410036\pi$
$233$	8.51388	0.557763	0.278881	+	0.960326i	$0.410036\pi$
$234$	0	0
$235$	10.6056	0.691830
$236$	0	0
$237$	−30.1194	−1.95647
$238$	0	0
$239$	−17.5139	−1.13288	−0.566439	−	0.824103i	$-0.691679\pi$
$239$	−17.5139	−1.13288	−0.566439	+	0.824103i	$0.691679\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	−49.8444	−3.19752
$244$	0	0
$245$	−0.486122	−0.0310572
$246$	0	0
$247$	2.60555	0.165787
$248$	0	0
$249$	9.21110	0.583730
$250$	0	0
$251$	21.2111	1.33883	0.669416	−	0.742887i	$-0.266544\pi$
$251$	21.2111	1.33883	0.669416	+	0.742887i	$0.266544\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	45.6333	2.85767
$256$	0	0
$257$	3.21110	0.200303	0.100152	−	0.994972i	$-0.468067\pi$
$257$	3.21110	0.200303	0.100152	+	0.994972i	$0.468067\pi$
$258$	0	0
$259$	2.60555	0.161901
$260$	0	0
$261$	30.9083	1.91318
$262$	0	0
$263$	13.8167	0.851971	0.425986	−	0.904730i	$-0.359927\pi$
$263$	13.8167	0.851971	0.425986	+	0.904730i	$0.359927\pi$
$264$	0	0
$265$	13.8167	0.848750
$266$	0	0
$267$	30.4222	1.86181
$268$	0	0
$269$	21.2111	1.29326	0.646632	−	0.762802i	$-0.276177\pi$
$269$	21.2111	1.29326	0.646632	+	0.762802i	$0.276177\pi$
$270$	0	0
$271$	−22.4222	−1.36205	−0.681026	−	0.732259i	$-0.738466\pi$
$271$	−22.4222	−1.36205	−0.681026	+	0.732259i	$0.738466\pi$
$272$	0	0
$273$	−11.2111	−0.678527
$274$	0	0
$275$	0.697224	0.0420442
$276$	0	0
$277$	−0.119429	−0.00717582	−0.00358791	−	0.999994i	$-0.501142\pi$
$277$	−0.119429	−0.00717582	−0.00358791	+	0.999994i	$0.501142\pi$
$278$	0	0
$279$	−2.39445	−0.143352
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−24.6056	−1.46265	−0.731324	−	0.682030i	$-0.761097\pi$
$283$	−24.6056	−1.46265	−0.731324	+	0.682030i	$0.761097\pi$
$284$	0	0
$285$	15.2111	0.901028
$286$	0	0
$287$	−25.8167	−1.52391
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	54.2389	3.17954
$292$	0	0
$293$	−11.0278	−0.644248	−0.322124	−	0.946697i	$-0.604397\pi$
$293$	−11.0278	−0.644248	−0.322124	+	0.946697i	$0.604397\pi$
$294$	0	0
$295$	−24.4222	−1.42192
$296$	0	0
$297$	−37.3305	−2.16614
$298$	0	0
$299$	−5.09167	−0.294459
$300$	0	0
$301$	1.57779	0.0909426
$302$	0	0
$303$	−41.0278	−2.35698
$304$	0	0
$305$	−17.3028	−0.990754
$306$	0	0
$307$	−17.9083	−1.02208	−0.511041	−	0.859556i	$-0.670740\pi$
$307$	−17.9083	−1.02208	−0.511041	+	0.859556i	$0.670740\pi$
$308$	0	0
$309$	1.00000	0.0568880
$310$	0	0
$311$	15.9083	0.902078	0.451039	−	0.892504i	$-0.351053\pi$
$311$	15.9083	0.902078	0.451039	+	0.892504i	$0.351053\pi$
$312$	0	0
$313$	−9.02776	−0.510279	−0.255139	−	0.966904i	$-0.582121\pi$
$313$	−9.02776	−0.510279	−0.255139	+	0.966904i	$0.582121\pi$
$314$	0	0
$315$	−47.4500	−2.67350
$316$	0	0
$317$	−9.21110	−0.517347	−0.258674	−	0.965965i	$-0.583285\pi$
$317$	−9.21110	−0.517347	−0.258674	+	0.965965i	$0.583285\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	2.30278	0.128528
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−0.394449	−0.0218801
$326$	0	0
$327$	6.60555	0.365288
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	13.2111	0.726148	0.363074	−	0.931760i	$-0.381727\pi$
$331$	13.2111	0.726148	0.363074	+	0.931760i	$0.381727\pi$
$332$	0	0
$333$	−7.90833	−0.433374
$334$	0	0
$335$	8.09167	0.442095
$336$	0	0
$337$	6.11943	0.333347	0.166673	−	0.986012i	$-0.446697\pi$
$337$	6.11943	0.333347	0.166673	+	0.986012i	$0.446697\pi$
$338$	0	0
$339$	10.6056	0.576014
$340$	0	0
$341$	−0.697224	−0.0377568
$342$	0	0
$343$	18.7889	1.01451
$344$	0	0
$345$	−29.7250	−1.60034
$346$	0	0
$347$	−10.1833	−0.546671	−0.273335	−	0.961919i	$-0.588127\pi$
$347$	−10.1833	−0.546671	−0.273335	+	0.961919i	$0.588127\pi$
$348$	0	0
$349$	−28.2389	−1.51159	−0.755796	−	0.654807i	$-0.772750\pi$
$349$	−28.2389	−1.51159	−0.755796	+	0.654807i	$0.772750\pi$
$350$	0	0
$351$	21.1194	1.12727
$352$	0	0
$353$	10.1833	0.542005	0.271002	−	0.962579i	$-0.412645\pi$
$353$	10.1833	0.542005	0.271002	+	0.962579i	$0.412645\pi$
$354$	0	0
$355$	13.8167	0.733312
$356$	0	0
$357$	−51.6333	−2.73272
$358$	0	0
$359$	3.21110	0.169476	0.0847378	−	0.996403i	$-0.472995\pi$
$359$	3.21110	0.169476	0.0847378	+	0.996403i	$0.472995\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	18.8167	0.987618
$364$	0	0
$365$	−28.3305	−1.48289
$366$	0	0
$367$	3.81665	0.199228	0.0996139	−	0.995026i	$-0.468239\pi$
$367$	3.81665	0.199228	0.0996139	+	0.995026i	$0.468239\pi$
$368$	0	0
$369$	78.3583	4.07917
$370$	0	0
$371$	−15.6333	−0.811641
$372$	0	0
$373$	17.8167	0.922511	0.461256	−	0.887267i	$-0.347399\pi$
$373$	17.8167	0.922511	0.461256	+	0.887267i	$0.347399\pi$
$374$	0	0
$375$	35.7250	1.84483
$376$	0	0
$377$	−5.09167	−0.262235
$378$	0	0
$379$	−24.3305	−1.24978	−0.624888	−	0.780715i	$-0.714855\pi$
$379$	−24.3305	−1.24978	−0.624888	+	0.780715i	$0.714855\pi$
$380$	0	0
$381$	63.4500	3.25064
$382$	0	0
$383$	−36.8444	−1.88266	−0.941331	−	0.337486i	$-0.890424\pi$
$383$	−36.8444	−1.88266	−0.941331	+	0.337486i	$0.890424\pi$
$384$	0	0
$385$	−13.8167	−0.704162
$386$	0	0
$387$	−4.78890	−0.243433
$388$	0	0
$389$	37.1194	1.88203	0.941015	−	0.338365i	$-0.109874\pi$
$389$	37.1194	1.88203	0.941015	+	0.338365i	$0.109874\pi$
$390$	0	0
$391$	−23.4500	−1.18592
$392$	0	0
$393$	35.0278	1.76692
$394$	0	0
$395$	21.0000	1.05662
$396$	0	0
$397$	−6.18335	−0.310333	−0.155167	−	0.987888i	$-0.549591\pi$
$397$	−6.18335	−0.310333	−0.155167	+	0.987888i	$0.549591\pi$
$398$	0	0
$399$	−17.2111	−0.861633
$400$	0	0
$401$	−7.81665	−0.390345	−0.195173	−	0.980769i	$-0.562527\pi$
$401$	−7.81665	−0.390345	−0.195173	+	0.980769i	$0.562527\pi$
$402$	0	0
$403$	0.394449	0.0196489
$404$	0	0
$405$	68.6611	3.41180
$406$	0	0
$407$	−2.30278	−0.114144
$408$	0	0
$409$	31.0278	1.53422	0.767112	−	0.641513i	$-0.221693\pi$
$409$	31.0278	1.53422	0.767112	+	0.641513i	$0.221693\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	0	0
$413$	27.6333	1.35975
$414$	0	0
$415$	−6.42221	−0.315254
$416$	0	0
$417$	−6.30278	−0.308648
$418$	0	0
$419$	−36.1472	−1.76591	−0.882953	−	0.469462i	$-0.844448\pi$
$419$	−36.1472	−1.76591	−0.882953	+	0.469462i	$0.844448\pi$
$420$	0	0
$421$	3.72498	0.181544	0.0907722	−	0.995872i	$-0.471066\pi$
$421$	3.72498	0.181544	0.0907722	+	0.995872i	$0.471066\pi$
$422$	0	0
$423$	36.4222	1.77091
$424$	0	0
$425$	−1.81665	−0.0881207
$426$	0	0
$427$	19.5778	0.947436
$428$	0	0
$429$	9.90833	0.478379
$430$	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
$433$	34.9361	1.67892	0.839461	−	0.543421i	$-0.182871\pi$
$433$	34.9361	1.67892	0.839461	+	0.543421i	$0.182871\pi$
$434$	0	0
$435$	−29.7250	−1.42520
$436$	0	0
$437$	−7.81665	−0.373921
$438$	0	0
$439$	30.3305	1.44760	0.723799	−	0.690011i	$-0.242394\pi$
$439$	30.3305	1.44760	0.723799	+	0.690011i	$0.242394\pi$
$440$	0	0
$441$	−1.66947	−0.0794985
$442$	0	0
$443$	−32.7250	−1.55481	−0.777405	−	0.629000i	$-0.783465\pi$
$443$	−32.7250	−1.55481	−0.777405	+	0.629000i	$0.783465\pi$
$444$	0	0
$445$	−21.2111	−1.00550
$446$	0	0
$447$	65.4500	3.09568
$448$	0	0
$449$	−15.2111	−0.717856	−0.358928	−	0.933365i	$-0.616858\pi$
$449$	−15.2111	−0.717856	−0.358928	+	0.933365i	$0.616858\pi$
$450$	0	0
$451$	22.8167	1.07439
$452$	0	0
$453$	68.0555	3.19753
$454$	0	0
$455$	7.81665	0.366450
$456$	0	0
$457$	−2.60555	−0.121883	−0.0609413	−	0.998141i	$-0.519410\pi$
$457$	−2.60555	−0.121883	−0.0609413	+	0.998141i	$0.519410\pi$
$458$	0	0
$459$	97.2666	4.54002
$460$	0	0
$461$	−12.4222	−0.578560	−0.289280	−	0.957245i	$-0.593416\pi$
$461$	−12.4222	−0.578560	−0.289280	+	0.957245i	$0.593416\pi$
$462$	0	0
$463$	26.6972	1.24073	0.620363	−	0.784315i	$-0.286985\pi$
$463$	26.6972	1.24073	0.620363	+	0.784315i	$0.286985\pi$
$464$	0	0
$465$	2.30278	0.106789
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−9.15559	−0.422766
$470$	0	0
$471$	−23.8167	−1.09741
$472$	0	0
$473$	−1.39445	−0.0641168
$474$	0	0
$475$	−0.605551	−0.0277846
$476$	0	0
$477$	47.4500	2.17258
$478$	0	0
$479$	−13.1194	−0.599442	−0.299721	−	0.954027i	$-0.596894\pi$
$479$	−13.1194	−0.599442	−0.299721	+	0.954027i	$0.596894\pi$
$480$	0	0
$481$	1.30278	0.0594015
$482$	0	0
$483$	33.6333	1.53037
$484$	0	0
$485$	−37.8167	−1.71717
$486$	0	0
$487$	−37.2111	−1.68620	−0.843098	−	0.537760i	$-0.819271\pi$
$487$	−37.2111	−1.68620	−0.843098	+	0.537760i	$0.819271\pi$
$488$	0	0
$489$	27.8167	1.25791
$490$	0	0
$491$	−17.7250	−0.799917	−0.399959	−	0.916533i	$-0.630975\pi$
$491$	−17.7250	−0.799917	−0.399959	+	0.916533i	$0.630975\pi$
$492$	0	0
$493$	−23.4500	−1.05613
$494$	0	0
$495$	41.9361	1.88489
$496$	0	0
$497$	−15.6333	−0.701250
$498$	0	0
$499$	42.2389	1.89087	0.945436	−	0.325809i	$-0.105637\pi$
$499$	42.2389	1.89087	0.945436	+	0.325809i	$0.105637\pi$
$500$	0	0
$501$	18.2111	0.813612
$502$	0	0
$503$	−6.48612	−0.289202	−0.144601	−	0.989490i	$-0.546190\pi$
$503$	−6.48612	−0.289202	−0.144601	+	0.989490i	$0.546190\pi$
$504$	0	0
$505$	28.6056	1.27293
$506$	0	0
$507$	37.3305	1.65791
$508$	0	0
$509$	4.18335	0.185424	0.0927118	−	0.995693i	$-0.470446\pi$
$509$	4.18335	0.185424	0.0927118	+	0.995693i	$0.470446\pi$
$510$	0	0
$511$	32.0555	1.41805
$512$	0	0
$513$	32.4222	1.43148
$514$	0	0
$515$	−0.697224	−0.0307234
$516$	0	0
$517$	10.6056	0.466432
$518$	0	0
$519$	−29.0278	−1.27418
$520$	0	0
$521$	33.6333	1.47350	0.736751	−	0.676164i	$-0.236359\pi$
$521$	33.6333	1.47350	0.736751	+	0.676164i	$0.236359\pi$
$522$	0	0
$523$	18.2389	0.797530	0.398765	−	0.917053i	$-0.369439\pi$
$523$	18.2389	0.797530	0.398765	+	0.917053i	$0.369439\pi$
$524$	0	0
$525$	2.60555	0.113716
$526$	0	0
$527$	1.81665	0.0791347
$528$	0	0
$529$	−7.72498	−0.335869
$530$	0	0
$531$	−83.8722	−3.63974
$532$	0	0
$533$	−12.9083	−0.559122
$534$	0	0
$535$	−1.60555	−0.0694140
$536$	0	0
$537$	−45.6333	−1.96922
$538$	0	0
$539$	−0.486122	−0.0209387
$540$	0	0
$541$	−25.9361	−1.11508	−0.557540	−	0.830150i	$-0.688255\pi$
$541$	−25.9361	−1.11508	−0.557540	+	0.830150i	$0.688255\pi$
$542$	0	0
$543$	66.0555	2.83471
$544$	0	0
$545$	−4.60555	−0.197280
$546$	0	0
$547$	20.6056	0.881030	0.440515	−	0.897745i	$-0.354796\pi$
$547$	20.6056	0.881030	0.440515	+	0.897745i	$0.354796\pi$
$548$	0	0
$549$	−59.4222	−2.53608
$550$	0	0
$551$	−7.81665	−0.333001
$552$	0	0
$553$	−23.7611	−1.01043
$554$	0	0
$555$	7.60555	0.322838
$556$	0	0
$557$	−11.5139	−0.487859	−0.243929	−	0.969793i	$-0.578436\pi$
$557$	−11.5139	−0.487859	−0.243929	+	0.969793i	$0.578436\pi$
$558$	0	0
$559$	0.788897	0.0333668
$560$	0	0
$561$	45.6333	1.92664
$562$	0	0
$563$	28.0555	1.18240	0.591199	−	0.806525i	$-0.298655\pi$
$563$	28.0555	1.18240	0.591199	+	0.806525i	$0.298655\pi$
$564$	0	0
$565$	−7.39445	−0.311087
$566$	0	0
$567$	−77.6888	−3.26262
$568$	0	0
$569$	18.4222	0.772299	0.386150	−	0.922436i	$-0.373805\pi$
$569$	18.4222	0.772299	0.386150	+	0.922436i	$0.373805\pi$
$570$	0	0
$571$	16.6972	0.698757	0.349379	−	0.936982i	$-0.386393\pi$
$571$	16.6972	0.698757	0.349379	+	0.936982i	$0.386393\pi$
$572$	0	0
$573$	18.2111	0.760780
$574$	0	0
$575$	1.18335	0.0493489
$576$	0	0
$577$	22.2389	0.925816	0.462908	−	0.886406i	$-0.346806\pi$
$577$	22.2389	0.925816	0.462908	+	0.886406i	$0.346806\pi$
$578$	0	0
$579$	13.2111	0.549035
$580$	0	0
$581$	7.26662	0.301470
$582$	0	0
$583$	13.8167	0.572227
$584$	0	0
$585$	−23.7250	−0.980907
$586$	0	0
$587$	45.6333	1.88349	0.941744	−	0.336330i	$-0.109186\pi$
$587$	45.6333	1.88349	0.941744	+	0.336330i	$0.109186\pi$
$588$	0	0
$589$	0.605551	0.0249513
$590$	0	0
$591$	−19.8167	−0.815148
$592$	0	0
$593$	18.4861	0.759134	0.379567	−	0.925164i	$-0.376073\pi$
$593$	18.4861	0.759134	0.379567	+	0.925164i	$0.376073\pi$
$594$	0	0
$595$	36.0000	1.47586
$596$	0	0
$597$	−87.2666	−3.57158
$598$	0	0
$599$	−20.7889	−0.849411	−0.424706	−	0.905331i	$-0.639622\pi$
$599$	−20.7889	−0.849411	−0.424706	+	0.905331i	$0.639622\pi$
$600$	0	0
$601$	−24.3028	−0.991331	−0.495665	−	0.868514i	$-0.665076\pi$
$601$	−24.3028	−0.991331	−0.495665	+	0.868514i	$0.665076\pi$
$602$	0	0
$603$	27.7889	1.13165
$604$	0	0
$605$	−13.1194	−0.533381
$606$	0	0
$607$	−13.4861	−0.547385	−0.273692	−	0.961817i	$-0.588245\pi$
$607$	−13.4861	−0.547385	−0.273692	+	0.961817i	$0.588245\pi$
$608$	0	0
$609$	33.6333	1.36289
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	29.8167	1.20428	0.602142	−	0.798389i	$-0.294314\pi$
$613$	29.8167	1.20428	0.602142	+	0.798389i	$0.294314\pi$
$614$	0	0
$615$	−75.3583	−3.03874
$616$	0	0
$617$	−42.5694	−1.71378	−0.856890	−	0.515500i	$-0.827606\pi$
$617$	−42.5694	−1.71378	−0.856890	+	0.515500i	$0.827606\pi$
$618$	0	0
$619$	6.30278	0.253330	0.126665	−	0.991946i	$-0.459573\pi$
$619$	6.30278	0.253330	0.126665	+	0.991946i	$0.459573\pi$
$620$	0	0
$621$	−63.3583	−2.54248
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	15.2111	0.607473
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	14.6972	0.585087	0.292544	−	0.956252i	$-0.405498\pi$
$631$	14.6972	0.585087	0.292544	+	0.956252i	$0.405498\pi$
$632$	0	0
$633$	34.0278	1.35248
$634$	0	0
$635$	−44.2389	−1.75557
$636$	0	0
$637$	0.275019	0.0108967
$638$	0	0
$639$	47.4500	1.87709
$640$	0	0
$641$	−20.5139	−0.810249	−0.405125	−	0.914261i	$-0.632772\pi$
$641$	−20.5139	−0.810249	−0.405125	+	0.914261i	$0.632772\pi$
$642$	0	0
$643$	8.18335	0.322720	0.161360	−	0.986896i	$-0.448412\pi$
$643$	8.18335	0.322720	0.161360	+	0.986896i	$0.448412\pi$
$644$	0	0
$645$	4.60555	0.181343
$646$	0	0
$647$	20.9361	0.823082	0.411541	−	0.911391i	$-0.364991\pi$
$647$	20.9361	0.823082	0.411541	+	0.911391i	$0.364991\pi$
$648$	0	0
$649$	−24.4222	−0.958655
$650$	0	0
$651$	−2.60555	−0.102120
$652$	0	0
$653$	3.90833	0.152945	0.0764723	−	0.997072i	$-0.475634\pi$
$653$	3.90833	0.152945	0.0764723	+	0.997072i	$0.475634\pi$
$654$	0	0
$655$	−24.4222	−0.954255
$656$	0	0
$657$	−97.2944	−3.79581
$658$	0	0
$659$	16.8806	0.657574	0.328787	−	0.944404i	$-0.393360\pi$
$659$	16.8806	0.657574	0.328787	+	0.944404i	$0.393360\pi$
$660$	0	0
$661$	30.5139	1.18685	0.593426	−	0.804888i	$-0.297775\pi$
$661$	30.5139	1.18685	0.593426	+	0.804888i	$0.297775\pi$
$662$	0	0
$663$	−25.8167	−1.00264
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	15.2750	0.591451
$668$	0	0
$669$	19.2111	0.742744
$670$	0	0
$671$	−17.3028	−0.667966
$672$	0	0
$673$	20.6972	0.797819	0.398910	−	0.916990i	$-0.369389\pi$
$673$	20.6972	0.797819	0.398910	+	0.916990i	$0.369389\pi$
$674$	0	0
$675$	−4.90833	−0.188922
$676$	0	0
$677$	−14.2389	−0.547244	−0.273622	−	0.961837i	$-0.588222\pi$
$677$	−14.2389	−0.547244	−0.273622	+	0.961837i	$0.588222\pi$
$678$	0	0
$679$	42.7889	1.64209
$680$	0	0
$681$	45.6333	1.74867
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	2.09167	0.0799187
$686$	0	0
$687$	81.2666	3.10051
$688$	0	0
$689$	−7.81665	−0.297791
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	−47.4500	−1.80247
$694$	0	0
$695$	4.39445	0.166691
$696$	0	0
$697$	−59.4500	−2.25183
$698$	0	0
$699$	−28.1194	−1.06357
$700$	0	0
$701$	−40.1194	−1.51529	−0.757645	−	0.652667i	$-0.773650\pi$
$701$	−40.1194	−1.51529	−0.757645	+	0.652667i	$0.773650\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	−35.0278	−1.31922
$706$	0	0
$707$	−32.3667	−1.21727
$708$	0	0
$709$	41.3305	1.55220	0.776100	−	0.630609i	$-0.217195\pi$
$709$	41.3305	1.55220	0.776100	+	0.630609i	$0.217195\pi$
$710$	0	0
$711$	72.1194	2.70469
$712$	0	0
$713$	−1.18335	−0.0443167
$714$	0	0
$715$	−6.90833	−0.258357
$716$	0	0
$717$	57.8444	2.16024
$718$	0	0
$719$	−51.6333	−1.92560	−0.962799	−	0.270220i	$-0.912904\pi$
$719$	−51.6333	−1.92560	−0.962799	+	0.270220i	$0.912904\pi$
$720$	0	0
$721$	0.788897	0.0293801
$722$	0	0
$723$	−26.4222	−0.982652
$724$	0	0
$725$	1.18335	0.0439484
$726$	0	0
$727$	19.0917	0.708071	0.354035	−	0.935232i	$-0.384809\pi$
$727$	19.0917	0.708071	0.354035	+	0.935232i	$0.384809\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	3.63331	0.134383
$732$	0	0
$733$	13.6333	0.503558	0.251779	−	0.967785i	$-0.418984\pi$
$733$	13.6333	0.503558	0.251779	+	0.967785i	$0.418984\pi$
$734$	0	0
$735$	1.60555	0.0592217
$736$	0	0
$737$	8.09167	0.298061
$738$	0	0
$739$	2.66947	0.0981980	0.0490990	−	0.998794i	$-0.484365\pi$
$739$	2.66947	0.0981980	0.0490990	+	0.998794i	$0.484365\pi$
$740$	0	0
$741$	−8.60555	−0.316133
$742$	0	0
$743$	−29.4500	−1.08041	−0.540207	−	0.841532i	$-0.681654\pi$
$743$	−29.4500	−1.08041	−0.540207	+	0.841532i	$0.681654\pi$
$744$	0	0
$745$	−45.6333	−1.67188
$746$	0	0
$747$	−22.0555	−0.806969
$748$	0	0
$749$	1.81665	0.0663791
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	0	0
$753$	−70.0555	−2.55296
$754$	0	0
$755$	−47.4500	−1.72688
$756$	0	0
$757$	−5.69722	−0.207069	−0.103535	−	0.994626i	$-0.533015\pi$
$757$	−5.69722	−0.207069	−0.103535	+	0.994626i	$0.533015\pi$
$758$	0	0
$759$	−29.7250	−1.07895
$760$	0	0
$761$	16.8806	0.611920	0.305960	−	0.952044i	$-0.401023\pi$
$761$	16.8806	0.611920	0.305960	+	0.952044i	$0.401023\pi$
$762$	0	0
$763$	5.21110	0.188655
$764$	0	0
$765$	−109.267	−3.95054
$766$	0	0
$767$	13.8167	0.498890
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−10.6056	−0.381950
$772$	0	0
$773$	−22.0555	−0.793282	−0.396641	−	0.917974i	$-0.629824\pi$
$773$	−22.0555	−0.793282	−0.396641	+	0.917974i	$0.629824\pi$
$774$	0	0
$775$	−0.0916731	−0.00329299
$776$	0	0
$777$	−8.60555	−0.308722
$778$	0	0
$779$	−19.8167	−0.710005
$780$	0	0
$781$	13.8167	0.494399
$782$	0	0
$783$	−63.3583	−2.26424
$784$	0	0
$785$	16.6056	0.592678
$786$	0	0
$787$	−10.7889	−0.384583	−0.192291	−	0.981338i	$-0.561592\pi$
$787$	−10.7889	−0.384583	−0.192291	+	0.981338i	$0.561592\pi$
$788$	0	0
$789$	−45.6333	−1.62459
$790$	0	0
$791$	8.36669	0.297485
$792$	0	0
$793$	9.78890	0.347614
$794$	0	0
$795$	−45.6333	−1.61845
$796$	0	0
$797$	−22.3305	−0.790988	−0.395494	−	0.918469i	$-0.629427\pi$
$797$	−22.3305	−0.790988	−0.395494	+	0.918469i	$0.629427\pi$
$798$	0	0
$799$	−27.6333	−0.977596
$800$	0	0
$801$	−72.8444	−2.57383
$802$	0	0
$803$	−28.3305	−0.999763
$804$	0	0
$805$	−23.4500	−0.826503
$806$	0	0
$807$	−70.0555	−2.46607
$808$	0	0
$809$	−35.4500	−1.24635	−0.623177	−	0.782081i	$-0.714158\pi$
$809$	−35.4500	−1.24635	−0.623177	+	0.782081i	$0.714158\pi$
$810$	0	0
$811$	7.14719	0.250972	0.125486	−	0.992095i	$-0.459951\pi$
$811$	7.14719	0.250972	0.125486	+	0.992095i	$0.459951\pi$
$812$	0	0
$813$	74.0555	2.59724
$814$	0	0
$815$	−19.3944	−0.679358
$816$	0	0
$817$	1.21110	0.0423711
$818$	0	0
$819$	26.8444	0.938020
$820$	0	0
$821$	3.21110	0.112068	0.0560341	−	0.998429i	$-0.482154\pi$
$821$	3.21110	0.112068	0.0560341	+	0.998429i	$0.482154\pi$
$822$	0	0
$823$	44.8444	1.56318	0.781589	−	0.623794i	$-0.214410\pi$
$823$	44.8444	1.56318	0.781589	+	0.623794i	$0.214410\pi$
$824$	0	0
$825$	−2.30278	−0.0801724
$826$	0	0
$827$	−34.6056	−1.20335	−0.601676	−	0.798740i	$-0.705500\pi$
$827$	−34.6056	−1.20335	−0.601676	+	0.798740i	$0.705500\pi$
$828$	0	0
$829$	27.7250	0.962928	0.481464	−	0.876466i	$-0.340105\pi$
$829$	27.7250	0.962928	0.481464	+	0.876466i	$0.340105\pi$
$830$	0	0
$831$	0.394449	0.0136833
$832$	0	0
$833$	1.26662	0.0438856
$834$	0	0
$835$	−12.6972	−0.439406
$836$	0	0
$837$	4.90833	0.169657
$838$	0	0
$839$	−12.9722	−0.447852	−0.223926	−	0.974606i	$-0.571887\pi$
$839$	−12.9722	−0.447852	−0.223926	+	0.974606i	$0.571887\pi$
$840$	0	0
$841$	−13.7250	−0.473275
$842$	0	0
$843$	39.6333	1.36504
$844$	0	0
$845$	−26.0278	−0.895382
$846$	0	0
$847$	14.8444	0.510060
$848$	0	0
$849$	81.2666	2.78906
$850$	0	0
$851$	−3.90833	−0.133976
$852$	0	0
$853$	−42.5416	−1.45660	−0.728299	−	0.685260i	$-0.759689\pi$
$853$	−42.5416	−1.45660	−0.728299	+	0.685260i	$0.759689\pi$
$854$	0	0
$855$	−36.4222	−1.24561
$856$	0	0
$857$	42.8444	1.46354	0.731769	−	0.681553i	$-0.238695\pi$
$857$	42.8444	1.46354	0.731769	+	0.681553i	$0.238695\pi$
$858$	0	0
$859$	−48.0555	−1.63963	−0.819816	−	0.572626i	$-0.805925\pi$
$859$	−48.0555	−1.63963	−0.819816	+	0.572626i	$0.805925\pi$
$860$	0	0
$861$	85.2666	2.90588
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	20.2389	0.688142
$866$	0	0
$867$	−62.7527	−2.13119
$868$	0	0
$869$	21.0000	0.712376
$870$	0	0
$871$	−4.57779	−0.155113
$872$	0	0
$873$	−129.872	−4.39551
$874$	0	0
$875$	28.1833	0.952771
$876$	0	0
$877$	7.21110	0.243502	0.121751	−	0.992561i	$-0.461149\pi$
$877$	7.21110	0.243502	0.121751	+	0.992561i	$0.461149\pi$
$878$	0	0
$879$	36.4222	1.22849
$880$	0	0
$881$	−28.5416	−0.961592	−0.480796	−	0.876832i	$-0.659652\pi$
$881$	−28.5416	−0.961592	−0.480796	+	0.876832i	$0.659652\pi$
$882$	0	0
$883$	−26.4222	−0.889178	−0.444589	−	0.895735i	$-0.646650\pi$
$883$	−26.4222	−0.889178	−0.444589	+	0.895735i	$0.646650\pi$
$884$	0	0
$885$	80.6611	2.71139
$886$	0	0
$887$	−0.422205	−0.0141763	−0.00708813	−	0.999975i	$-0.502256\pi$
$887$	−0.422205	−0.0141763	−0.00708813	+	0.999975i	$0.502256\pi$
$888$	0	0
$889$	50.0555	1.67881
$890$	0	0
$891$	68.6611	2.30023
$892$	0	0
$893$	−9.21110	−0.308238
$894$	0	0
$895$	31.8167	1.06351
$896$	0	0
$897$	16.8167	0.561492
$898$	0	0
$899$	−1.18335	−0.0394668
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	−5.21110	−0.173415
$904$	0	0
$905$	−46.0555	−1.53094
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	0	0
$909$	98.2389	3.25838
$910$	0	0
$911$	−17.5778	−0.582378	−0.291189	−	0.956665i	$-0.594051\pi$
$911$	−17.5778	−0.582378	−0.291189	+	0.956665i	$0.594051\pi$
$912$	0	0
$913$	−6.42221	−0.212544
$914$	0	0
$915$	57.1472	1.88923
$916$	0	0
$917$	27.6333	0.912532
$918$	0	0
$919$	−9.57779	−0.315942	−0.157971	−	0.987444i	$-0.550495\pi$
$919$	−9.57779	−0.315942	−0.157971	+	0.987444i	$0.550495\pi$
$920$	0	0
$921$	59.1472	1.94897
$922$	0	0
$923$	−7.81665	−0.257288
$924$	0	0
$925$	−0.302776	−0.00995520
$926$	0	0
$927$	−2.39445	−0.0786440
$928$	0	0
$929$	−18.4861	−0.606510	−0.303255	−	0.952909i	$-0.598073\pi$
$929$	−18.4861	−0.606510	−0.303255	+	0.952909i	$0.598073\pi$
$930$	0	0
$931$	0.422205	0.0138372
$932$	0	0
$933$	−52.5416	−1.72014
$934$	0	0
$935$	−31.8167	−1.04052
$936$	0	0
$937$	−18.0917	−0.591029	−0.295515	−	0.955338i	$-0.595491\pi$
$937$	−18.0917	−0.591029	−0.295515	+	0.955338i	$0.595491\pi$
$938$	0	0
$939$	29.8167	0.973030
$940$	0	0
$941$	−13.8167	−0.450410	−0.225205	−	0.974311i	$-0.572305\pi$
$941$	−13.8167	−0.450410	−0.225205	+	0.974311i	$0.572305\pi$
$942$	0	0
$943$	38.7250	1.26106
$944$	0	0
$945$	97.2666	3.16408
$946$	0	0
$947$	3.63331	0.118067	0.0590333	−	0.998256i	$-0.481198\pi$
$947$	3.63331	0.118067	0.0590333	+	0.998256i	$0.481198\pi$
$948$	0	0
$949$	16.0278	0.520283
$950$	0	0
$951$	30.4222	0.986508
$952$	0	0
$953$	49.7527	1.61165	0.805825	−	0.592154i	$-0.201722\pi$
$953$	49.7527	1.61165	0.805825	+	0.592154i	$0.201722\pi$
$954$	0	0
$955$	−12.6972	−0.410873
$956$	0	0
$957$	−29.7250	−0.960872
$958$	0	0
$959$	−2.36669	−0.0764245
$960$	0	0
$961$	−30.9083	−0.997043
$962$	0	0
$963$	−5.51388	−0.177682
$964$	0	0
$965$	−9.21110	−0.296516
$966$	0	0
$967$	−6.72498	−0.216261	−0.108130	−	0.994137i	$-0.534486\pi$
$967$	−6.72498	−0.216261	−0.108130	+	0.994137i	$0.534486\pi$
$968$	0	0
$969$	−39.6333	−1.27321
$970$	0	0
$971$	22.5416	0.723395	0.361698	−	0.932295i	$-0.382197\pi$
$971$	22.5416	0.723395	0.361698	+	0.932295i	$0.382197\pi$
$972$	0	0
$973$	−4.97224	−0.159403
$974$	0	0
$975$	1.30278	0.0417222
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−21.2111	−0.677910
$980$	0	0
$981$	−15.8167	−0.504987
$982$	0	0
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	0	0
$985$	13.8167	0.440235
$986$	0	0
$987$	39.6333	1.26154
$988$	0	0
$989$	−2.36669	−0.0752564
$990$	0	0
$991$	50.6972	1.61045	0.805225	−	0.592969i	$-0.202044\pi$
$991$	50.6972	1.61045	0.805225	+	0.592969i	$0.202044\pi$
$992$	0	0
$993$	−43.6333	−1.38466
$994$	0	0
$995$	60.8444	1.92890
$996$	0	0
$997$	52.4222	1.66023	0.830114	−	0.557594i	$-0.188275\pi$
$997$	52.4222	1.66023	0.830114	+	0.557594i	$0.188275\pi$
$998$	0	0
$999$	16.2111	0.512897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.s.1.1		2
4.3	odd	2		2368.2.a.ba.1.2		2
8.3	odd	2		592.2.a.f.1.1		2
8.5	even	2		74.2.a.a.1.2	✓	2
24.5	odd	2		666.2.a.j.1.2		2
24.11	even	2		5328.2.a.bf.1.2		2
40.13	odd	4		1850.2.b.i.149.4		4
40.29	even	2		1850.2.a.u.1.1		2
40.37	odd	4		1850.2.b.i.149.1		4
56.13	odd	2		3626.2.a.a.1.1		2
88.21	odd	2		8954.2.a.p.1.2		2
296.221	even	2		2738.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.a.1.2	✓	2	8.5	even	2
592.2.a.f.1.1		2	8.3	odd	2
666.2.a.j.1.2		2	24.5	odd	2
1850.2.a.u.1.1		2	40.29	even	2
1850.2.b.i.149.1		4	40.37	odd	4
1850.2.b.i.149.4		4	40.13	odd	4
2368.2.a.s.1.1		2	1.1	even	1	trivial
2368.2.a.ba.1.2		2	4.3	odd	2
2738.2.a.l.1.2		2	296.221	even	2
3626.2.a.a.1.1		2	56.13	odd	2
5328.2.a.bf.1.2		2	24.11	even	2
8954.2.a.p.1.2		2	88.21	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)

\(f(q)\)	\(=\)	\(q-3.30278 q^{3} +2.30278 q^{5} -2.60555 q^{7} +7.90833 q^{9} +O(q^{10})\)	\(q-3.30278 q^{3} +2.30278 q^{5} -2.60555 q^{7} +7.90833 q^{9} +2.30278 q^{11} -1.30278 q^{13} -7.60555 q^{15} -6.00000 q^{17} -2.00000 q^{19} +8.60555 q^{21} +3.90833 q^{23} +0.302776 q^{25} -16.2111 q^{27} +3.90833 q^{29} -0.302776 q^{31} -7.60555 q^{33} -6.00000 q^{35} -1.00000 q^{37} +4.30278 q^{39} +9.90833 q^{41} -0.605551 q^{43} +18.2111 q^{45} +4.60555 q^{47} -0.211103 q^{49} +19.8167 q^{51} +6.00000 q^{53} +5.30278 q^{55} +6.60555 q^{57} -10.6056 q^{59} -7.51388 q^{61} -20.6056 q^{63} -3.00000 q^{65} +3.51388 q^{67} -12.9083 q^{69} +6.00000 q^{71} -12.3028 q^{73} -1.00000 q^{75} -6.00000 q^{77} +9.11943 q^{79} +29.8167 q^{81} -2.78890 q^{83} -13.8167 q^{85} -12.9083 q^{87} -9.21110 q^{89} +3.39445 q^{91} +1.00000 q^{93} -4.60555 q^{95} -16.4222 q^{97} +18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 + q^5 + 2 * q^7 + 5 * q^9	\( 2 q - 3 q^{3} + q^{5} + 2 q^{7} + 5 q^{9} + q^{11} + q^{13} - 8 q^{15} - 12 q^{17} - 4 q^{19} + 10 q^{21} - 3 q^{23} - 3 q^{25} - 18 q^{27} - 3 q^{29} + 3 q^{31} - 8 q^{33} - 12 q^{35} - 2 q^{37} + 5 q^{39} + 9 q^{41} + 6 q^{43} + 22 q^{45} + 2 q^{47} + 14 q^{49} + 18 q^{51} + 12 q^{53} + 7 q^{55} + 6 q^{57} - 14 q^{59} + 3 q^{61} - 34 q^{63} - 6 q^{65} - 11 q^{67} - 15 q^{69} + 12 q^{71} - 21 q^{73} - 2 q^{75} - 12 q^{77} - 7 q^{79} + 38 q^{81} - 20 q^{83} - 6 q^{85} - 15 q^{87} - 4 q^{89} + 14 q^{91} + 2 q^{93} - 2 q^{95} - 4 q^{97} + 22 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 + q^5 + 2 * q^7 + 5 * q^9 + q^11 + q^13 - 8 * q^15 - 12 * q^17 - 4 * q^19 + 10 * q^21 - 3 * q^23 - 3 * q^25 - 18 * q^27 - 3 * q^29 + 3 * q^31 - 8 * q^33 - 12 * q^35 - 2 * q^37 + 5 * q^39 + 9 * q^41 + 6 * q^43 + 22 * q^45 + 2 * q^47 + 14 * q^49 + 18 * q^51 + 12 * q^53 + 7 * q^55 + 6 * q^57 - 14 * q^59 + 3 * q^61 - 34 * q^63 - 6 * q^65 - 11 * q^67 - 15 * q^69 + 12 * q^71 - 21 * q^73 - 2 * q^75 - 12 * q^77 - 7 * q^79 + 38 * q^81 - 20 * q^83 - 6 * q^85 - 15 * q^87 - 4 * q^89 + 14 * q^91 + 2 * q^93 - 2 * q^95 - 4 * q^97 + 22 * q^99

Introduction

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