LMFDB - Embedded newform 2368.2.a.bd.1.2

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:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*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			1.2
Root			$-0.523976$ 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)

$f(q)$	$=$	$q-0.523976 q^{3} +1.52398 q^{5} +4.20147 q^{7} -2.72545 q^{9} +O(q^{10})$	$q-0.523976 q^{3} +1.52398 q^{5} +4.20147 q^{7} -2.72545 q^{9} +4.67750 q^{11} +6.72545 q^{13} -0.798528 q^{15} +6.40294 q^{17} +2.00000 q^{19} -2.20147 q^{21} -2.72545 q^{23} -2.67750 q^{25} +3.00000 q^{27} -1.67750 q^{29} -10.9749 q^{31} -2.45090 q^{33} +6.40294 q^{35} +1.00000 q^{37} -3.52398 q^{39} +4.52398 q^{41} -3.45090 q^{43} -4.15352 q^{45} -6.20147 q^{47} +10.6524 q^{49} -3.35499 q^{51} -3.24943 q^{53} +7.12839 q^{55} -1.04795 q^{57} -9.85384 q^{59} +6.87897 q^{61} -11.4509 q^{63} +10.2494 q^{65} +5.52398 q^{67} +1.42807 q^{69} -5.15352 q^{71} -7.62954 q^{73} +1.40294 q^{75} +19.6524 q^{77} -2.32250 q^{79} +6.60442 q^{81} -3.15352 q^{83} +9.75794 q^{85} +0.878968 q^{87} +16.4029 q^{89} +28.2568 q^{91} +5.75057 q^{93} +3.04795 q^{95} -10.8059 q^{97} -12.7483 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{5} + 3 q^{7} + 3 q^{9} + 6 q^{11} + 9 q^{13} - 12 q^{15} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 9 q^{27} + 3 q^{29} - 9 q^{31} + 15 q^{33} + 3 q^{37} - 9 q^{39} + 12 q^{41} + 12 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{51} + 3 q^{53} - 9 q^{55} + 12 q^{59} + 3 q^{61} - 12 q^{63} + 18 q^{65} + 15 q^{67} + 9 q^{69} - 9 q^{71} - 18 q^{73} - 15 q^{75} + 27 q^{77} - 15 q^{79} - 9 q^{81} - 3 q^{83} - 6 q^{85} - 15 q^{87} + 30 q^{89} + 24 q^{91} + 30 q^{93} + 6 q^{95} + 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 3 * q^7 + 3 * q^9 + 6 * q^11 + 9 * q^13 - 12 * q^15 + 6 * q^19 + 3 * q^21 + 3 * q^23 + 9 * q^27 + 3 * q^29 - 9 * q^31 + 15 * q^33 + 3 * q^37 - 9 * q^39 + 12 * q^41 + 12 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^51 + 3 * q^53 - 9 * q^55 + 12 * q^59 + 3 * q^61 - 12 * q^63 + 18 * q^65 + 15 * q^67 + 9 * q^69 - 9 * q^71 - 18 * q^73 - 15 * q^75 + 27 * q^77 - 15 * q^79 - 9 * q^81 - 3 * q^83 - 6 * q^85 - 15 * q^87 + 30 * q^89 + 24 * q^91 + 30 * q^93 + 6 * q^95 + 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.523976	−0.302518	−0.151259	−	0.988494i	$-0.548333\pi$
$3$	−0.523976	−0.302518	−0.151259	+	0.988494i	$0.548333\pi$
$4$	0	0
$5$	1.52398	0.681543	0.340771	−	0.940146i	$-0.389312\pi$
$5$	1.52398	0.681543	0.340771	+	0.940146i	$0.389312\pi$
$6$	0	0
$7$	4.20147	1.58801	0.794004	−	0.607913i	$-0.207993\pi$
$7$	4.20147	1.58801	0.794004	+	0.607913i	$0.207993\pi$
$8$	0	0
$9$	−2.72545	−0.908483
$10$	0	0
$11$	4.67750	1.41032	0.705159	−	0.709049i	$-0.250876\pi$
$11$	4.67750	1.41032	0.705159	+	0.709049i	$0.250876\pi$
$12$	0	0
$13$	6.72545	1.86530	0.932652	−	0.360777i	$-0.117489\pi$
$13$	6.72545	1.86530	0.932652	+	0.360777i	$0.117489\pi$
$14$	0	0
$15$	−0.798528	−0.206179
$16$	0	0
$17$	6.40294	1.55294	0.776471	−	0.630153i	$-0.217008\pi$
$17$	6.40294	1.55294	0.776471	+	0.630153i	$0.217008\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−2.20147	−0.480401
$22$	0	0
$23$	−2.72545	−0.568295	−0.284148	−	0.958781i	$-0.591711\pi$
$23$	−2.72545	−0.568295	−0.284148	+	0.958781i	$0.591711\pi$
$24$	0	0
$25$	−2.67750	−0.535499
$26$	0	0
$27$	3.00000	0.577350
$28$	0	0
$29$	−1.67750	−0.311503	−0.155752	−	0.987796i	$-0.549780\pi$
$29$	−1.67750	−0.311503	−0.155752	+	0.987796i	$0.549780\pi$
$30$	0	0
$31$	−10.9749	−1.97115	−0.985573	−	0.169252i	$-0.945865\pi$
$31$	−10.9749	−1.97115	−0.985573	+	0.169252i	$0.945865\pi$
$32$	0	0
$33$	−2.45090	−0.426646
$34$	0	0
$35$	6.40294	1.08230
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−3.52398	−0.564288
$40$	0	0
$41$	4.52398	0.706526	0.353263	−	0.935524i	$-0.385072\pi$
$41$	4.52398	0.706526	0.353263	+	0.935524i	$0.385072\pi$
$42$	0	0
$43$	−3.45090	−0.526257	−0.263128	−	0.964761i	$-0.584754\pi$
$43$	−3.45090	−0.526257	−0.263128	+	0.964761i	$0.584754\pi$
$44$	0	0
$45$	−4.15352	−0.619170
$46$	0	0
$47$	−6.20147	−0.904578	−0.452289	−	0.891872i	$-0.649392\pi$
$47$	−6.20147	−0.904578	−0.452289	+	0.891872i	$0.649392\pi$
$48$	0	0
$49$	10.6524	1.52177
$50$	0	0
$51$	−3.35499	−0.469793
$52$	0	0
$53$	−3.24943	−0.446343	−0.223171	−	0.974779i	$-0.571641\pi$
$53$	−3.24943	−0.446343	−0.223171	+	0.974779i	$0.571641\pi$
$54$	0	0
$55$	7.12839	0.961192
$56$	0	0
$57$	−1.04795	−0.138805
$58$	0	0
$59$	−9.85384	−1.28286	−0.641430	−	0.767181i	$-0.721659\pi$
$59$	−9.85384	−1.28286	−0.641430	+	0.767181i	$0.721659\pi$
$60$	0	0
$61$	6.87897	0.880762	0.440381	−	0.897811i	$-0.354843\pi$
$61$	6.87897	0.880762	0.440381	+	0.897811i	$0.354843\pi$
$62$	0	0
$63$	−11.4509	−1.44268
$64$	0	0
$65$	10.2494	1.27128
$66$	0	0
$67$	5.52398	0.674861	0.337431	−	0.941350i	$-0.390442\pi$
$67$	5.52398	0.674861	0.337431	+	0.941350i	$0.390442\pi$
$68$	0	0
$69$	1.42807	0.171920
$70$	0	0
$71$	−5.15352	−0.611610	−0.305805	−	0.952094i	$-0.598926\pi$
$71$	−5.15352	−0.611610	−0.305805	+	0.952094i	$0.598926\pi$
$72$	0	0
$73$	−7.62954	−0.892970	−0.446485	−	0.894791i	$-0.647324\pi$
$73$	−7.62954	−0.892970	−0.446485	+	0.894791i	$0.647324\pi$
$74$	0	0
$75$	1.40294	0.161998
$76$	0	0
$77$	19.6524	2.23960
$78$	0	0
$79$	−2.32250	−0.261302	−0.130651	−	0.991428i	$-0.541707\pi$
$79$	−2.32250	−0.261302	−0.130651	+	0.991428i	$0.541707\pi$
$80$	0	0
$81$	6.60442	0.733824
$82$	0	0
$83$	−3.15352	−0.346144	−0.173072	−	0.984909i	$-0.555369\pi$
$83$	−3.15352	−0.346144	−0.173072	+	0.984909i	$0.555369\pi$
$84$	0	0
$85$	9.75794	1.05840
$86$	0	0
$87$	0.878968	0.0942353
$88$	0	0
$89$	16.4029	1.73871	0.869354	−	0.494189i	$-0.164535\pi$
$89$	16.4029	1.73871	0.869354	+	0.494189i	$0.164535\pi$
$90$	0	0
$91$	28.2568	2.96212
$92$	0	0
$93$	5.75057	0.596307
$94$	0	0
$95$	3.04795	0.312713
$96$	0	0
$97$	−10.8059	−1.09717	−0.548586	−	0.836094i	$-0.684834\pi$
$97$	−10.8059	−1.09717	−0.548586	+	0.836094i	$0.684834\pi$
$98$	0	0
$99$	−12.7483	−1.28125
$100$	0	0
$101$	−0.942386	−0.0937709	−0.0468855	−	0.998900i	$-0.514930\pi$
$101$	−0.942386	−0.0937709	−0.0468855	+	0.998900i	$0.514930\pi$
$102$	0	0
$103$	−17.2819	−1.70284	−0.851419	−	0.524487i	$-0.824257\pi$
$103$	−17.2819	−1.70284	−0.851419	+	0.524487i	$0.824257\pi$
$104$	0	0
$105$	−3.35499	−0.327414
$106$	0	0
$107$	0.0804406	0.00777649	0.00388824	−	0.999992i	$-0.498762\pi$
$107$	0.0804406	0.00777649	0.00388824	+	0.999992i	$0.498762\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−0.523976	−0.0497336
$112$	0	0
$113$	−15.2088	−1.43073	−0.715363	−	0.698753i	$-0.753739\pi$
$113$	−15.2088	−1.43073	−0.715363	+	0.698753i	$0.753739\pi$
$114$	0	0
$115$	−4.15352	−0.387318
$116$	0	0
$117$	−18.3299	−1.69460
$118$	0	0
$119$	26.9018	2.46608
$120$	0	0
$121$	10.8790	0.988997
$122$	0	0
$123$	−2.37046	−0.213737
$124$	0	0
$125$	−11.7003	−1.04651
$126$	0	0
$127$	−8.05531	−0.714794	−0.357397	−	0.933953i	$-0.616336\pi$
$127$	−8.05531	−0.714794	−0.357397	+	0.933953i	$0.616336\pi$
$128$	0	0
$129$	1.80819	0.159202
$130$	0	0
$131$	19.5468	1.70781	0.853906	−	0.520427i	$-0.174227\pi$
$131$	19.5468	1.70781	0.853906	+	0.520427i	$0.174227\pi$
$132$	0	0
$133$	8.40294	0.728628
$134$	0	0
$135$	4.57193	0.393489
$136$	0	0
$137$	12.3299	1.05341	0.526706	−	0.850048i	$-0.323427\pi$
$137$	12.3299	1.05341	0.526706	+	0.850048i	$0.323427\pi$
$138$	0	0
$139$	−6.17635	−0.523871	−0.261935	−	0.965085i	$-0.584361\pi$
$139$	−6.17635	−0.523871	−0.261935	+	0.965085i	$0.584361\pi$
$140$	0	0
$141$	3.24943	0.273651
$142$	0	0
$143$	31.4583	2.63067
$144$	0	0
$145$	−2.55646	−0.212303
$146$	0	0
$147$	−5.58159	−0.460362
$148$	0	0
$149$	−10.5085	−0.860891	−0.430445	−	0.902617i	$-0.641643\pi$
$149$	−10.5085	−0.860891	−0.430445	+	0.902617i	$0.641643\pi$
$150$	0	0
$151$	1.14386	0.0930859	0.0465429	−	0.998916i	$-0.485180\pi$
$151$	1.14386	0.0930859	0.0465429	+	0.998916i	$0.485180\pi$
$152$	0	0
$153$	−17.4509	−1.41082
$154$	0	0
$155$	−16.7254	−1.34342
$156$	0	0
$157$	−22.3624	−1.78471	−0.892355	−	0.451334i	$-0.850948\pi$
$157$	−22.3624	−1.78471	−0.892355	+	0.451334i	$0.850948\pi$
$158$	0	0
$159$	1.70262	0.135027
$160$	0	0
$161$	−11.4509	−0.902457
$162$	0	0
$163$	−4.30704	−0.337353	−0.168677	−	0.985671i	$-0.553949\pi$
$163$	−4.30704	−0.337353	−0.168677	+	0.985671i	$0.553949\pi$
$164$	0	0
$165$	−3.73511	−0.290778
$166$	0	0
$167$	9.61988	0.744409	0.372204	−	0.928151i	$-0.378602\pi$
$167$	9.61988	0.744409	0.372204	+	0.928151i	$0.378602\pi$
$168$	0	0
$169$	32.2317	2.47936
$170$	0	0
$171$	−5.45090	−0.416841
$172$	0	0
$173$	12.7506	0.969408	0.484704	−	0.874678i	$-0.338927\pi$
$173$	12.7506	0.969408	0.484704	+	0.874678i	$0.338927\pi$
$174$	0	0
$175$	−11.2494	−0.850377
$176$	0	0
$177$	5.16318	0.388088
$178$	0	0
$179$	15.8538	1.18497	0.592486	−	0.805581i	$-0.298147\pi$
$179$	15.8538	1.18497	0.592486	+	0.805581i	$0.298147\pi$
$180$	0	0
$181$	−17.6524	−1.31209	−0.656045	−	0.754722i	$-0.727772\pi$
$181$	−17.6524	−1.31209	−0.656045	+	0.754722i	$0.727772\pi$
$182$	0	0
$183$	−3.60442	−0.266446
$184$	0	0
$185$	1.52398	0.112045
$186$	0	0
$187$	29.9497	2.19014
$188$	0	0
$189$	12.6044	0.916836
$190$	0	0
$191$	18.4258	1.33324	0.666621	−	0.745397i	$-0.267740\pi$
$191$	18.4258	1.33324	0.666621	+	0.745397i	$0.267740\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	−5.37046	−0.384586
$196$	0	0
$197$	9.95941	0.709579	0.354789	−	0.934946i	$-0.384553\pi$
$197$	9.95941	0.709579	0.354789	+	0.934946i	$0.384553\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−2.89443	−0.204158
$202$	0	0
$203$	−7.04795	−0.494669
$204$	0	0
$205$	6.89443	0.481528
$206$	0	0
$207$	7.42807	0.516287
$208$	0	0
$209$	9.35499	0.647098
$210$	0	0
$211$	−10.9845	−0.756207	−0.378103	−	0.925763i	$-0.623424\pi$
$211$	−10.9845	−0.756207	−0.378103	+	0.925763i	$0.623424\pi$
$212$	0	0
$213$	2.70032	0.185023
$214$	0	0
$215$	−5.25909	−0.358667
$216$	0	0
$217$	−46.1106	−3.13019
$218$	0	0
$219$	3.99770	0.270140
$220$	0	0
$221$	43.0627	2.89671
$222$	0	0
$223$	19.4103	1.29981	0.649905	−	0.760015i	$-0.274809\pi$
$223$	19.4103	1.29981	0.649905	+	0.760015i	$0.274809\pi$
$224$	0	0
$225$	7.29738	0.486492
$226$	0	0
$227$	−4.14616	−0.275190	−0.137595	−	0.990489i	$-0.543937\pi$
$227$	−4.14616	−0.275190	−0.137595	+	0.990489i	$0.543937\pi$
$228$	0	0
$229$	6.79623	0.449107	0.224554	−	0.974462i	$-0.427908\pi$
$229$	6.79623	0.449107	0.224554	+	0.974462i	$0.427908\pi$
$230$	0	0
$231$	−10.2974	−0.677518
$232$	0	0
$233$	2.41841	0.158435	0.0792176	−	0.996857i	$-0.474758\pi$
$233$	2.41841	0.158435	0.0792176	+	0.996857i	$0.474758\pi$
$234$	0	0
$235$	−9.45090	−0.616509
$236$	0	0
$237$	1.21694	0.0790486
$238$	0	0
$239$	−9.07078	−0.586740	−0.293370	−	0.955999i	$-0.594777\pi$
$239$	−9.07078	−0.586740	−0.293370	+	0.955999i	$0.594777\pi$
$240$	0	0
$241$	16.9977	1.09492	0.547459	−	0.836832i	$-0.315595\pi$
$241$	16.9977	1.09492	0.547459	+	0.836832i	$0.315595\pi$
$242$	0	0
$243$	−12.4606	−0.799345
$244$	0	0
$245$	16.2340	1.03715
$246$	0	0
$247$	13.4509	0.855860
$248$	0	0
$249$	1.65237	0.104715
$250$	0	0
$251$	−21.6118	−1.36412	−0.682062	−	0.731295i	$-0.738916\pi$
$251$	−21.6118	−1.36412	−0.682062	+	0.731295i	$0.738916\pi$
$252$	0	0
$253$	−12.7483	−0.801477
$254$	0	0
$255$	−5.11293	−0.320184
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	4.20147	0.261067
$260$	0	0
$261$	4.57193	0.282995
$262$	0	0
$263$	−5.10327	−0.314681	−0.157340	−	0.987544i	$-0.550292\pi$
$263$	−5.10327	−0.314681	−0.157340	+	0.987544i	$0.550292\pi$
$264$	0	0
$265$	−4.95205	−0.304202
$266$	0	0
$267$	−8.59476	−0.525991
$268$	0	0
$269$	16.5948	1.01180	0.505900	−	0.862592i	$-0.331160\pi$
$269$	16.5948	1.01180	0.505900	+	0.862592i	$0.331160\pi$
$270$	0	0
$271$	10.0553	0.610817	0.305408	−	0.952221i	$-0.401207\pi$
$271$	10.0553	0.610817	0.305408	+	0.952221i	$0.401207\pi$
$272$	0	0
$273$	−14.8059	−0.896093
$274$	0	0
$275$	−12.5240	−0.755224
$276$	0	0
$277$	5.71579	0.343428	0.171714	−	0.985147i	$-0.445069\pi$
$277$	5.71579	0.343428	0.171714	+	0.985147i	$0.445069\pi$
$278$	0	0
$279$	29.9115	1.79075
$280$	0	0
$281$	−14.7100	−0.877524	−0.438762	−	0.898603i	$-0.644583\pi$
$281$	−14.7100	−0.877524	−0.438762	+	0.898603i	$0.644583\pi$
$282$	0	0
$283$	−6.64501	−0.395005	−0.197502	−	0.980302i	$-0.563283\pi$
$283$	−6.64501	−0.395005	−0.197502	+	0.980302i	$0.563283\pi$
$284$	0	0
$285$	−1.59706	−0.0946014
$286$	0	0
$287$	19.0074	1.12197
$288$	0	0
$289$	23.9977	1.41163
$290$	0	0
$291$	5.66203	0.331914
$292$	0	0
$293$	−18.3527	−1.07218	−0.536088	−	0.844162i	$-0.680098\pi$
$293$	−18.3527	−1.07218	−0.536088	+	0.844162i	$0.680098\pi$
$294$	0	0
$295$	−15.0170	−0.874325
$296$	0	0
$297$	14.0325	0.814248
$298$	0	0
$299$	−18.3299	−1.06004
$300$	0	0
$301$	−14.4989	−0.835700
$302$	0	0
$303$	0.493788	0.0283674
$304$	0	0
$305$	10.4834	0.600277
$306$	0	0
$307$	27.4258	1.56527	0.782636	−	0.622480i	$-0.213875\pi$
$307$	27.4258	1.56527	0.782636	+	0.622480i	$0.213875\pi$
$308$	0	0
$309$	9.05531	0.515139
$310$	0	0
$311$	−27.7231	−1.57204	−0.786018	−	0.618204i	$-0.787861\pi$
$311$	−27.7231	−1.57204	−0.786018	+	0.618204i	$0.787861\pi$
$312$	0	0
$313$	−9.23976	−0.522262	−0.261131	−	0.965303i	$-0.584096\pi$
$313$	−9.23976	−0.522262	−0.261131	+	0.965303i	$0.584096\pi$
$314$	0	0
$315$	−17.4509	−0.983247
$316$	0	0
$317$	2.30704	0.129576	0.0647881	−	0.997899i	$-0.479363\pi$
$317$	2.30704	0.129576	0.0647881	+	0.997899i	$0.479363\pi$
$318$	0	0
$319$	−7.84648	−0.439319
$320$	0	0
$321$	−0.0421490	−0.00235253
$322$	0	0
$323$	12.8059	0.712539
$324$	0	0
$325$	−18.0074	−0.998869
$326$	0	0
$327$	−5.23976	−0.289760
$328$	0	0
$329$	−26.0553	−1.43648
$330$	0	0
$331$	−14.4989	−0.796929	−0.398464	−	0.917184i	$-0.630457\pi$
$331$	−14.4989	−0.796929	−0.398464	+	0.917184i	$0.630457\pi$
$332$	0	0
$333$	−2.72545	−0.149354
$334$	0	0
$335$	8.41841	0.459947
$336$	0	0
$337$	−3.82135	−0.208162	−0.104081	−	0.994569i	$-0.533190\pi$
$337$	−3.82135	−0.208162	−0.104081	+	0.994569i	$0.533190\pi$
$338$	0	0
$339$	7.96907	0.432820
$340$	0	0
$341$	−51.3349	−2.77994
$342$	0	0
$343$	15.3453	0.828570
$344$	0	0
$345$	2.17635	0.117171
$346$	0	0
$347$	13.1439	0.705599	0.352800	−	0.935699i	$-0.385230\pi$
$347$	13.1439	0.705599	0.352800	+	0.935699i	$0.385230\pi$
$348$	0	0
$349$	−34.3527	−1.83886	−0.919429	−	0.393257i	$-0.871348\pi$
$349$	−34.3527	−1.83886	−0.919429	+	0.393257i	$0.871348\pi$
$350$	0	0
$351$	20.1763	1.07693
$352$	0	0
$353$	15.6620	0.833606	0.416803	−	0.908997i	$-0.363151\pi$
$353$	15.6620	0.833606	0.416803	+	0.908997i	$0.363151\pi$
$354$	0	0
$355$	−7.85384	−0.416839
$356$	0	0
$357$	−14.0959	−0.746034
$358$	0	0
$359$	−1.15352	−0.0608804	−0.0304402	−	0.999537i	$-0.509691\pi$
$359$	−1.15352	−0.0608804	−0.0304402	+	0.999537i	$0.509691\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−5.70032	−0.299189
$364$	0	0
$365$	−11.6272	−0.608598
$366$	0	0
$367$	8.35269	0.436007	0.218003	−	0.975948i	$-0.430046\pi$
$367$	8.35269	0.436007	0.218003	+	0.975948i	$0.430046\pi$
$368$	0	0
$369$	−12.3299	−0.641867
$370$	0	0
$371$	−13.6524	−0.708796
$372$	0	0
$373$	25.2951	1.30973	0.654865	−	0.755746i	$-0.272726\pi$
$373$	25.2951	1.30973	0.654865	+	0.755746i	$0.272726\pi$
$374$	0	0
$375$	6.13069	0.316588
$376$	0	0
$377$	−11.2819	−0.581048
$378$	0	0
$379$	10.4281	0.535654	0.267827	−	0.963467i	$-0.413694\pi$
$379$	10.4281	0.535654	0.267827	+	0.963467i	$0.413694\pi$
$380$	0	0
$381$	4.22079	0.216238
$382$	0	0
$383$	−36.9977	−1.89049	−0.945247	−	0.326356i	$-0.894179\pi$
$383$	−36.9977	−1.89049	−0.945247	+	0.326356i	$0.894179\pi$
$384$	0	0
$385$	29.9497	1.52638
$386$	0	0
$387$	9.40524	0.478095
$388$	0	0
$389$	6.03019	0.305743	0.152871	−	0.988246i	$-0.451148\pi$
$389$	6.03019	0.305743	0.152871	+	0.988246i	$0.451148\pi$
$390$	0	0
$391$	−17.4509	−0.882530
$392$	0	0
$393$	−10.2421	−0.516644
$394$	0	0
$395$	−3.53944	−0.178089
$396$	0	0
$397$	0.508511	0.0255215	0.0127607	−	0.999919i	$-0.495938\pi$
$397$	0.508511	0.0255215	0.0127607	+	0.999919i	$0.495938\pi$
$398$	0	0
$399$	−4.40294	−0.220423
$400$	0	0
$401$	−15.5661	−0.777335	−0.388668	−	0.921378i	$-0.627065\pi$
$401$	−15.5661	−0.777335	−0.388668	+	0.921378i	$0.627065\pi$
$402$	0	0
$403$	−73.8110	−3.67679
$404$	0	0
$405$	10.0650	0.500133
$406$	0	0
$407$	4.67750	0.231855
$408$	0	0
$409$	−6.45320	−0.319090	−0.159545	−	0.987191i	$-0.551003\pi$
$409$	−6.45320	−0.319090	−0.159545	+	0.987191i	$0.551003\pi$
$410$	0	0
$411$	−6.46056	−0.318676
$412$	0	0
$413$	−41.4006	−2.03719
$414$	0	0
$415$	−4.80589	−0.235912
$416$	0	0
$417$	3.23626	0.158480
$418$	0	0
$419$	9.78306	0.477934	0.238967	−	0.971028i	$-0.423191\pi$
$419$	9.78306	0.477934	0.238967	+	0.971028i	$0.423191\pi$
$420$	0	0
$421$	−31.9150	−1.55544	−0.777720	−	0.628611i	$-0.783624\pi$
$421$	−31.9150	−1.55544	−0.777720	+	0.628611i	$0.783624\pi$
$422$	0	0
$423$	16.9018	0.821793
$424$	0	0
$425$	−17.1439	−0.831599
$426$	0	0
$427$	28.9018	1.39866
$428$	0	0
$429$	−16.4834	−0.795825
$430$	0	0
$431$	8.21113	0.395516	0.197758	−	0.980251i	$-0.436634\pi$
$431$	8.21113	0.395516	0.197758	+	0.980251i	$0.436634\pi$
$432$	0	0
$433$	−17.1381	−0.823602	−0.411801	−	0.911274i	$-0.635100\pi$
$433$	−17.1381	−0.823602	−0.411801	+	0.911274i	$0.635100\pi$
$434$	0	0
$435$	1.33953	0.0642254
$436$	0	0
$437$	−5.45090	−0.260752
$438$	0	0
$439$	−27.5313	−1.31400	−0.657000	−	0.753891i	$-0.728175\pi$
$439$	−27.5313	−1.31400	−0.657000	+	0.753891i	$0.728175\pi$
$440$	0	0
$441$	−29.0325	−1.38250
$442$	0	0
$443$	−29.2916	−1.39168	−0.695842	−	0.718195i	$-0.744969\pi$
$443$	−29.2916	−1.39168	−0.695842	+	0.718195i	$0.744969\pi$
$444$	0	0
$445$	24.9977	1.18500
$446$	0	0
$447$	5.50621	0.260435
$448$	0	0
$449$	1.80819	0.0853337	0.0426669	−	0.999089i	$-0.486415\pi$
$449$	1.80819	0.0853337	0.0426669	+	0.999089i	$0.486415\pi$
$450$	0	0
$451$	21.1609	0.996427
$452$	0	0
$453$	−0.599355	−0.0281601
$454$	0	0
$455$	43.0627	2.01881
$456$	0	0
$457$	−21.6427	−1.01240	−0.506202	−	0.862415i	$-0.668951\pi$
$457$	−21.6427	−1.01240	−0.506202	+	0.862415i	$0.668951\pi$
$458$	0	0
$459$	19.2088	0.896592
$460$	0	0
$461$	14.9977	0.698513	0.349256	−	0.937027i	$-0.386434\pi$
$461$	14.9977	0.698513	0.349256	+	0.937027i	$0.386434\pi$
$462$	0	0
$463$	9.67750	0.449751	0.224876	−	0.974387i	$-0.427802\pi$
$463$	9.67750	0.449751	0.224876	+	0.974387i	$0.427802\pi$
$464$	0	0
$465$	8.76374	0.406409
$466$	0	0
$467$	32.5948	1.50831	0.754153	−	0.656699i	$-0.228048\pi$
$467$	32.5948	1.50831	0.754153	+	0.656699i	$0.228048\pi$
$468$	0	0
$469$	23.2088	1.07168
$470$	0	0
$471$	11.7173	0.539907
$472$	0	0
$473$	−16.1416	−0.742190
$474$	0	0
$475$	−5.35499	−0.245704
$476$	0	0
$477$	8.85614	0.405495
$478$	0	0
$479$	−9.67750	−0.442176	−0.221088	−	0.975254i	$-0.570961\pi$
$479$	−9.67750	−0.442176	−0.221088	+	0.975254i	$0.570961\pi$
$480$	0	0
$481$	6.72545	0.306654
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	−16.4679	−0.747770
$486$	0	0
$487$	6.28772	0.284924	0.142462	−	0.989800i	$-0.454498\pi$
$487$	6.28772	0.284924	0.142462	+	0.989800i	$0.454498\pi$
$488$	0	0
$489$	2.25679	0.102055
$490$	0	0
$491$	9.53134	0.430143	0.215072	−	0.976598i	$-0.431002\pi$
$491$	9.53134	0.430143	0.215072	+	0.976598i	$0.431002\pi$
$492$	0	0
$493$	−10.7409	−0.483746
$494$	0	0
$495$	−19.4281	−0.873227
$496$	0	0
$497$	−21.6524	−0.971242
$498$	0	0
$499$	42.4486	1.90026	0.950130	−	0.311854i	$-0.100950\pi$
$499$	42.4486	1.90026	0.950130	+	0.311854i	$0.100950\pi$
$500$	0	0
$501$	−5.04059	−0.225197
$502$	0	0
$503$	−34.6176	−1.54352	−0.771761	−	0.635913i	$-0.780624\pi$
$503$	−34.6176	−1.54352	−0.771761	+	0.635913i	$0.780624\pi$
$504$	0	0
$505$	−1.43617	−0.0639089
$506$	0	0
$507$	−16.8886	−0.750050
$508$	0	0
$509$	9.89443	0.438563	0.219282	−	0.975662i	$-0.429629\pi$
$509$	9.89443	0.438563	0.219282	+	0.975662i	$0.429629\pi$
$510$	0	0
$511$	−32.0553	−1.41804
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	−26.3372	−1.16056
$516$	0	0
$517$	−29.0074	−1.27574
$518$	0	0
$519$	−6.68100	−0.293263
$520$	0	0
$521$	27.4606	1.20307	0.601534	−	0.798847i	$-0.294556\pi$
$521$	27.4606	1.20307	0.601534	+	0.798847i	$0.294556\pi$
$522$	0	0
$523$	7.16318	0.313224	0.156612	−	0.987660i	$-0.449943\pi$
$523$	7.16318	0.313224	0.156612	+	0.987660i	$0.449943\pi$
$524$	0	0
$525$	5.89443	0.257254
$526$	0	0
$527$	−70.2715	−3.06108
$528$	0	0
$529$	−15.5719	−0.677040
$530$	0	0
$531$	26.8561	1.16546
$532$	0	0
$533$	30.4258	1.31789
$534$	0	0
$535$	0.122590	0.00530001
$536$	0	0
$537$	−8.30704	−0.358475
$538$	0	0
$539$	49.8264	2.14618
$540$	0	0
$541$	−6.62218	−0.284710	−0.142355	−	0.989816i	$-0.545467\pi$
$541$	−6.62218	−0.284710	−0.142355	+	0.989816i	$0.545467\pi$
$542$	0	0
$543$	9.24943	0.396931
$544$	0	0
$545$	15.2398	0.652800
$546$	0	0
$547$	−32.6597	−1.39643	−0.698215	−	0.715888i	$-0.746022\pi$
$547$	−32.6597	−1.39643	−0.698215	+	0.715888i	$0.746022\pi$
$548$	0	0
$549$	−18.7483	−0.800157
$550$	0	0
$551$	−3.35499	−0.142927
$552$	0	0
$553$	−9.75794	−0.414950
$554$	0	0
$555$	−0.798528	−0.0338956
$556$	0	0
$557$	28.0685	1.18930	0.594650	−	0.803985i	$-0.297291\pi$
$557$	28.0685	1.18930	0.594650	+	0.803985i	$0.297291\pi$
$558$	0	0
$559$	−23.2088	−0.981629
$560$	0	0
$561$	−15.6930	−0.662557
$562$	0	0
$563$	36.4177	1.53482	0.767411	−	0.641156i	$-0.221545\pi$
$563$	36.4177	1.53482	0.767411	+	0.641156i	$0.221545\pi$
$564$	0	0
$565$	−23.1779	−0.975102
$566$	0	0
$567$	27.7483	1.16532
$568$	0	0
$569$	7.59706	0.318485	0.159243	−	0.987239i	$-0.449095\pi$
$569$	7.59706	0.318485	0.159243	+	0.987239i	$0.449095\pi$
$570$	0	0
$571$	37.9749	1.58920	0.794600	−	0.607134i	$-0.207681\pi$
$571$	37.9749	1.58920	0.794600	+	0.607134i	$0.207681\pi$
$572$	0	0
$573$	−9.65467	−0.403330
$574$	0	0
$575$	7.29738	0.304322
$576$	0	0
$577$	0.856142	0.0356416	0.0178208	−	0.999841i	$-0.494327\pi$
$577$	0.856142	0.0356416	0.0178208	+	0.999841i	$0.494327\pi$
$578$	0	0
$579$	−8.38362	−0.348411
$580$	0	0
$581$	−13.2494	−0.549679
$582$	0	0
$583$	−15.1992	−0.629485
$584$	0	0
$585$	−27.9343	−1.15494
$586$	0	0
$587$	30.9977	1.27941	0.639706	−	0.768620i	$-0.279056\pi$
$587$	30.9977	1.27941	0.639706	+	0.768620i	$0.279056\pi$
$588$	0	0
$589$	−21.9497	−0.904424
$590$	0	0
$591$	−5.21850	−0.214660
$592$	0	0
$593$	23.6945	0.973017	0.486509	−	0.873676i	$-0.338270\pi$
$593$	23.6945	0.973017	0.486509	+	0.873676i	$0.338270\pi$
$594$	0	0
$595$	40.9977	1.68074
$596$	0	0
$597$	8.38362	0.343119
$598$	0	0
$599$	18.5542	0.758103	0.379051	−	0.925376i	$-0.376250\pi$
$599$	18.5542	0.758103	0.379051	+	0.925376i	$0.376250\pi$
$600$	0	0
$601$	−31.5696	−1.28775	−0.643876	−	0.765130i	$-0.722675\pi$
$601$	−31.5696	−1.28775	−0.643876	+	0.765130i	$0.722675\pi$
$602$	0	0
$603$	−15.0553	−0.613100
$604$	0	0
$605$	16.5793	0.674044
$606$	0	0
$607$	−33.2746	−1.35057	−0.675286	−	0.737556i	$-0.735980\pi$
$607$	−33.2746	−1.35057	−0.675286	+	0.737556i	$0.735980\pi$
$608$	0	0
$609$	3.69296	0.149646
$610$	0	0
$611$	−41.7077	−1.68731
$612$	0	0
$613$	−7.49149	−0.302578	−0.151289	−	0.988490i	$-0.548342\pi$
$613$	−7.49149	−0.302578	−0.151289	+	0.988490i	$0.548342\pi$
$614$	0	0
$615$	−3.61252	−0.145671
$616$	0	0
$617$	41.1187	1.65538	0.827689	−	0.561187i	$-0.189655\pi$
$617$	41.1187	1.65538	0.827689	+	0.561187i	$0.189655\pi$
$618$	0	0
$619$	−21.9320	−0.881521	−0.440760	−	0.897625i	$-0.645291\pi$
$619$	−21.9320	−0.881521	−0.440760	+	0.897625i	$0.645291\pi$
$620$	0	0
$621$	−8.17635	−0.328105
$622$	0	0
$623$	68.9165	2.76108
$624$	0	0
$625$	−4.44354	−0.177741
$626$	0	0
$627$	−4.90179	−0.195759
$628$	0	0
$629$	6.40294	0.255302
$630$	0	0
$631$	0.982236	0.0391022	0.0195511	−	0.999809i	$-0.493776\pi$
$631$	0.982236	0.0391022	0.0195511	+	0.999809i	$0.493776\pi$
$632$	0	0
$633$	5.75564	0.228766
$634$	0	0
$635$	−12.2761	−0.487163
$636$	0	0
$637$	71.6420	2.83856
$638$	0	0
$639$	14.0457	0.555637
$640$	0	0
$641$	9.32987	0.368508	0.184254	−	0.982879i	$-0.441013\pi$
$641$	9.32987	0.368508	0.184254	+	0.982879i	$0.441013\pi$
$642$	0	0
$643$	17.6427	0.695761	0.347880	−	0.937539i	$-0.386902\pi$
$643$	17.6427	0.695761	0.347880	+	0.937539i	$0.386902\pi$
$644$	0	0
$645$	2.75564	0.108503
$646$	0	0
$647$	25.2243	0.991670	0.495835	−	0.868417i	$-0.334862\pi$
$647$	25.2243	0.991670	0.495835	+	0.868417i	$0.334862\pi$
$648$	0	0
$649$	−46.0913	−1.80924
$650$	0	0
$651$	24.1609	0.946940
$652$	0	0
$653$	2.51432	0.0983928	0.0491964	−	0.998789i	$-0.484334\pi$
$653$	2.51432	0.0983928	0.0491964	+	0.998789i	$0.484334\pi$
$654$	0	0
$655$	29.7889	1.16395
$656$	0	0
$657$	20.7939	0.811248
$658$	0	0
$659$	−1.97487	−0.0769302	−0.0384651	−	0.999260i	$-0.512247\pi$
$659$	−1.97487	−0.0769302	−0.0384651	+	0.999260i	$0.512247\pi$
$660$	0	0
$661$	−32.8214	−1.27660	−0.638301	−	0.769787i	$-0.720363\pi$
$661$	−32.8214	−1.27660	−0.638301	+	0.769787i	$0.720363\pi$
$662$	0	0
$663$	−22.5638	−0.876306
$664$	0	0
$665$	12.8059	0.496591
$666$	0	0
$667$	4.57193	0.177026
$668$	0	0
$669$	−10.1705	−0.393216
$670$	0	0
$671$	32.1763	1.24215
$672$	0	0
$673$	23.4258	0.902997	0.451499	−	0.892272i	$-0.350890\pi$
$673$	23.4258	0.902997	0.451499	+	0.892272i	$0.350890\pi$
$674$	0	0
$675$	−8.03249	−0.309171
$676$	0	0
$677$	−26.8110	−1.03043	−0.515214	−	0.857061i	$-0.672288\pi$
$677$	−26.8110	−1.03043	−0.515214	+	0.857061i	$0.672288\pi$
$678$	0	0
$679$	−45.4006	−1.74232
$680$	0	0
$681$	2.17249	0.0832500
$682$	0	0
$683$	−38.3983	−1.46927	−0.734636	−	0.678462i	$-0.762647\pi$
$683$	−38.3983	−1.46927	−0.734636	+	0.678462i	$0.762647\pi$
$684$	0	0
$685$	18.7904	0.717945
$686$	0	0
$687$	−3.56106	−0.135863
$688$	0	0
$689$	−21.8538	−0.832565
$690$	0	0
$691$	12.6141	0.479862	0.239931	−	0.970790i	$-0.422875\pi$
$691$	12.6141	0.479862	0.239931	+	0.970790i	$0.422875\pi$
$692$	0	0
$693$	−53.5615	−2.03463
$694$	0	0
$695$	−9.41261	−0.357040
$696$	0	0
$697$	28.9668	1.09719
$698$	0	0
$699$	−1.26719	−0.0479295
$700$	0	0
$701$	−17.7808	−0.671570	−0.335785	−	0.941939i	$-0.609002\pi$
$701$	−17.7808	−0.671570	−0.335785	+	0.941939i	$0.609002\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	4.95205	0.186505
$706$	0	0
$707$	−3.95941	−0.148909
$708$	0	0
$709$	47.7955	1.79500	0.897499	−	0.441017i	$-0.145382\pi$
$709$	47.7955	1.79500	0.897499	+	0.441017i	$0.145382\pi$
$710$	0	0
$711$	6.32987	0.237389
$712$	0	0
$713$	29.9115	1.12019
$714$	0	0
$715$	47.9416	1.79292
$716$	0	0
$717$	4.75287	0.177499
$718$	0	0
$719$	35.2641	1.31513	0.657565	−	0.753397i	$-0.271586\pi$
$719$	35.2641	1.31513	0.657565	+	0.753397i	$0.271586\pi$
$720$	0	0
$721$	−72.6095	−2.70412
$722$	0	0
$723$	−8.90639	−0.331232
$724$	0	0
$725$	4.49149	0.166810
$726$	0	0
$727$	−29.3469	−1.08842	−0.544208	−	0.838950i	$-0.683170\pi$
$727$	−29.3469	−1.08842	−0.544208	+	0.838950i	$0.683170\pi$
$728$	0	0
$729$	−13.2842	−0.492008
$730$	0	0
$731$	−22.0959	−0.817247
$732$	0	0
$733$	2.44354	0.0902541	0.0451270	−	0.998981i	$-0.485631\pi$
$733$	2.44354	0.0902541	0.0451270	+	0.998981i	$0.485631\pi$
$734$	0	0
$735$	−8.50621	−0.313756
$736$	0	0
$737$	25.8384	0.951769
$738$	0	0
$739$	−34.0325	−1.25191	−0.625953	−	0.779861i	$-0.715290\pi$
$739$	−34.0325	−1.25191	−0.625953	+	0.779861i	$0.715290\pi$
$740$	0	0
$741$	−7.04795	−0.258913
$742$	0	0
$743$	−32.0244	−1.17486	−0.587430	−	0.809275i	$-0.699860\pi$
$743$	−32.0244	−1.17486	−0.587430	+	0.809275i	$0.699860\pi$
$744$	0	0
$745$	−16.0147	−0.586734
$746$	0	0
$747$	8.59476	0.314466
$748$	0	0
$749$	0.337969	0.0123491
$750$	0	0
$751$	−29.1535	−1.06383	−0.531914	−	0.846799i	$-0.678527\pi$
$751$	−29.1535	−1.06383	−0.531914	+	0.846799i	$0.678527\pi$
$752$	0	0
$753$	11.3241	0.412672
$754$	0	0
$755$	1.74321	0.0634420
$756$	0	0
$757$	−27.0901	−0.984606	−0.492303	−	0.870424i	$-0.663845\pi$
$757$	−27.0901	−0.984606	−0.492303	+	0.870424i	$0.663845\pi$
$758$	0	0
$759$	6.67980	0.242461
$760$	0	0
$761$	−40.6539	−1.47370	−0.736852	−	0.676054i	$-0.763688\pi$
$761$	−40.6539	−1.47370	−0.736852	+	0.676054i	$0.763688\pi$
$762$	0	0
$763$	42.0147	1.52104
$764$	0	0
$765$	−26.5948	−0.961535
$766$	0	0
$767$	−66.2715	−2.39293
$768$	0	0
$769$	−15.4006	−0.555361	−0.277681	−	0.960673i	$-0.589566\pi$
$769$	−15.4006	−0.555361	−0.277681	+	0.960673i	$0.589566\pi$
$770$	0	0
$771$	−3.14386	−0.113223
$772$	0	0
$773$	5.26875	0.189504	0.0947518	−	0.995501i	$-0.469794\pi$
$773$	5.26875	0.189504	0.0947518	+	0.995501i	$0.469794\pi$
$774$	0	0
$775$	29.3852	1.05555
$776$	0	0
$777$	−2.20147	−0.0789774
$778$	0	0
$779$	9.04795	0.324177
$780$	0	0
$781$	−24.1056	−0.862565
$782$	0	0
$783$	−5.03249	−0.179846
$784$	0	0
$785$	−34.0797	−1.21636
$786$	0	0
$787$	−39.2641	−1.39962	−0.699808	−	0.714331i	$-0.746731\pi$
$787$	−39.2641	−1.39962	−0.699808	+	0.714331i	$0.746731\pi$
$788$	0	0
$789$	2.67399	0.0951966
$790$	0	0
$791$	−63.8995	−2.27200
$792$	0	0
$793$	46.2641	1.64289
$794$	0	0
$795$	2.59476	0.0920265
$796$	0	0
$797$	11.3705	0.402762	0.201381	−	0.979513i	$-0.435457\pi$
$797$	11.3705	0.402762	0.201381	+	0.979513i	$0.435457\pi$
$798$	0	0
$799$	−39.7077	−1.40476
$800$	0	0
$801$	−44.7054	−1.57959
$802$	0	0
$803$	−35.6872	−1.25937
$804$	0	0
$805$	−17.4509	−0.615063
$806$	0	0
$807$	−8.69526	−0.306088
$808$	0	0
$809$	30.9520	1.08822	0.544108	−	0.839015i	$-0.316868\pi$
$809$	30.9520	1.08822	0.544108	+	0.839015i	$0.316868\pi$
$810$	0	0
$811$	45.8670	1.61061	0.805304	−	0.592862i	$-0.202002\pi$
$811$	45.8670	1.61061	0.805304	+	0.592862i	$0.202002\pi$
$812$	0	0
$813$	−5.26875	−0.184783
$814$	0	0
$815$	−6.56383	−0.229921
$816$	0	0
$817$	−6.90179	−0.241463
$818$	0	0
$819$	−77.0124	−2.69103
$820$	0	0
$821$	36.4583	1.27240	0.636201	−	0.771523i	$-0.280505\pi$
$821$	36.4583	1.27240	0.636201	+	0.771523i	$0.280505\pi$
$822$	0	0
$823$	36.5136	1.27278	0.636392	−	0.771366i	$-0.280426\pi$
$823$	36.5136	1.27278	0.636392	+	0.771366i	$0.280426\pi$
$824$	0	0
$825$	6.56227	0.228469
$826$	0	0
$827$	13.0480	0.453722	0.226861	−	0.973927i	$-0.427154\pi$
$827$	13.0480	0.453722	0.226861	+	0.973927i	$0.427154\pi$
$828$	0	0
$829$	34.0804	1.18366	0.591831	−	0.806062i	$-0.298405\pi$
$829$	34.0804	1.18366	0.591831	+	0.806062i	$0.298405\pi$
$830$	0	0
$831$	−2.99494	−0.103893
$832$	0	0
$833$	68.2065	2.36322
$834$	0	0
$835$	14.6605	0.507347
$836$	0	0
$837$	−32.9246	−1.13804
$838$	0	0
$839$	−8.85154	−0.305589	−0.152795	−	0.988258i	$-0.548827\pi$
$839$	−8.85154	−0.305589	−0.152795	+	0.988258i	$0.548827\pi$
$840$	0	0
$841$	−26.1860	−0.902966
$842$	0	0
$843$	7.70768	0.265467
$844$	0	0
$845$	49.1203	1.68979
$846$	0	0
$847$	45.7077	1.57053
$848$	0	0
$849$	3.48183	0.119496
$850$	0	0
$851$	−2.72545	−0.0934272
$852$	0	0
$853$	15.0251	0.514451	0.257225	−	0.966351i	$-0.417192\pi$
$853$	15.0251	0.514451	0.257225	+	0.966351i	$0.417192\pi$
$854$	0	0
$855$	−8.30704	−0.284095
$856$	0	0
$857$	25.4966	0.870946	0.435473	−	0.900202i	$-0.356581\pi$
$857$	25.4966	0.870946	0.435473	+	0.900202i	$0.356581\pi$
$858$	0	0
$859$	5.98068	0.204058	0.102029	−	0.994781i	$-0.467467\pi$
$859$	5.98068	0.204058	0.102029	+	0.994781i	$0.467467\pi$
$860$	0	0
$861$	−9.95941	−0.339416
$862$	0	0
$863$	−27.8995	−0.949710	−0.474855	−	0.880064i	$-0.657499\pi$
$863$	−27.8995	−0.949710	−0.474855	+	0.880064i	$0.657499\pi$
$864$	0	0
$865$	19.4316	0.660693
$866$	0	0
$867$	−12.5742	−0.427043
$868$	0	0
$869$	−10.8635	−0.368519
$870$	0	0
$871$	37.1512	1.25882
$872$	0	0
$873$	29.4509	0.996762
$874$	0	0
$875$	−49.1586	−1.66186
$876$	0	0
$877$	2.61408	0.0882711	0.0441356	−	0.999026i	$-0.485947\pi$
$877$	2.61408	0.0882711	0.0441356	+	0.999026i	$0.485947\pi$
$878$	0	0
$879$	9.61638	0.324352
$880$	0	0
$881$	−8.81399	−0.296951	−0.148475	−	0.988916i	$-0.547437\pi$
$881$	−8.81399	−0.296951	−0.148475	+	0.988916i	$0.547437\pi$
$882$	0	0
$883$	18.9211	0.636746	0.318373	−	0.947965i	$-0.396864\pi$
$883$	18.9211	0.636746	0.318373	+	0.947965i	$0.396864\pi$
$884$	0	0
$885$	7.86857	0.264499
$886$	0	0
$887$	5.74828	0.193008	0.0965041	−	0.995333i	$-0.469234\pi$
$887$	5.74828	0.193008	0.0965041	+	0.995333i	$0.469234\pi$
$888$	0	0
$889$	−33.8442	−1.13510
$890$	0	0
$891$	30.8921	1.03493
$892$	0	0
$893$	−12.4029	−0.415049
$894$	0	0
$895$	24.1609	0.807609
$896$	0	0
$897$	9.60442	0.320682
$898$	0	0
$899$	18.4103	0.614018
$900$	0	0
$901$	−20.8059	−0.693145
$902$	0	0
$903$	7.59706	0.252814
$904$	0	0
$905$	−26.9018	−0.894246
$906$	0	0
$907$	−3.29002	−0.109243	−0.0546216	−	0.998507i	$-0.517395\pi$
$907$	−3.29002	−0.109243	−0.0546216	+	0.998507i	$0.517395\pi$
$908$	0	0
$909$	2.56842	0.0851893
$910$	0	0
$911$	−26.3070	−0.871591	−0.435796	−	0.900046i	$-0.643533\pi$
$911$	−26.3070	−0.871591	−0.435796	+	0.900046i	$0.643533\pi$
$912$	0	0
$913$	−14.7506	−0.488173
$914$	0	0
$915$	−5.49305	−0.181595
$916$	0	0
$917$	82.1254	2.71202
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−14.3705	−0.473523
$922$	0	0
$923$	−34.6597	−1.14084
$924$	0	0
$925$	−2.67750	−0.0880355
$926$	0	0
$927$	47.1010	1.54700
$928$	0	0
$929$	−4.52168	−0.148351	−0.0741757	−	0.997245i	$-0.523633\pi$
$929$	−4.52168	−0.148351	−0.0741757	+	0.997245i	$0.523633\pi$
$930$	0	0
$931$	21.3047	0.698235
$932$	0	0
$933$	14.5263	0.475569
$934$	0	0
$935$	45.6427	1.49268
$936$	0	0
$937$	−39.3948	−1.28697	−0.643487	−	0.765457i	$-0.722513\pi$
$937$	−39.3948	−1.28697	−0.643487	+	0.765457i	$0.722513\pi$
$938$	0	0
$939$	4.84142	0.157994
$940$	0	0
$941$	29.1632	0.950693	0.475346	−	0.879799i	$-0.342323\pi$
$941$	29.1632	0.950693	0.475346	+	0.879799i	$0.342323\pi$
$942$	0	0
$943$	−12.3299	−0.401516
$944$	0	0
$945$	19.2088	0.624863
$946$	0	0
$947$	−10.7912	−0.350666	−0.175333	−	0.984509i	$-0.556100\pi$
$947$	−10.7912	−0.350666	−0.175333	+	0.984509i	$0.556100\pi$
$948$	0	0
$949$	−51.3121	−1.66566
$950$	0	0
$951$	−1.20883	−0.0391991
$952$	0	0
$953$	−23.8407	−0.772275	−0.386138	−	0.922441i	$-0.626191\pi$
$953$	−23.8407	−0.772275	−0.386138	+	0.922441i	$0.626191\pi$
$954$	0	0
$955$	28.0804	0.908662
$956$	0	0
$957$	4.11137	0.132902
$958$	0	0
$959$	51.8036	1.67283
$960$	0	0
$961$	89.4479	2.88541
$962$	0	0
$963$	−0.219237	−0.00706481
$964$	0	0
$965$	24.3836	0.784937
$966$	0	0
$967$	35.0592	1.12743	0.563713	−	0.825970i	$-0.309372\pi$
$967$	35.0592	1.12743	0.563713	+	0.825970i	$0.309372\pi$
$968$	0	0
$969$	−6.70998	−0.215556
$970$	0	0
$971$	52.5719	1.68711	0.843557	−	0.537040i	$-0.180458\pi$
$971$	52.5719	1.68711	0.843557	+	0.537040i	$0.180458\pi$
$972$	0	0
$973$	−25.9497	−0.831911
$974$	0	0
$975$	9.43543	0.302176
$976$	0	0
$977$	−21.4966	−0.687736	−0.343868	−	0.939018i	$-0.611737\pi$
$977$	−21.4966	−0.687736	−0.343868	+	0.939018i	$0.611737\pi$
$978$	0	0
$979$	76.7247	2.45213
$980$	0	0
$981$	−27.2545	−0.870169
$982$	0	0
$983$	−0.170542	−0.00543946	−0.00271973	−	0.999996i	$-0.500866\pi$
$983$	−0.170542	−0.00543946	−0.00271973	+	0.999996i	$0.500866\pi$
$984$	0	0
$985$	15.1779	0.483608
$986$	0	0
$987$	13.6524	0.434560
$988$	0	0
$989$	9.40524	0.299069
$990$	0	0
$991$	39.0975	1.24197	0.620986	−	0.783822i	$-0.286732\pi$
$991$	39.0975	1.24197	0.620986	+	0.783822i	$0.286732\pi$
$992$	0	0
$993$	7.59706	0.241085
$994$	0	0
$995$	−24.3836	−0.773013
$996$	0	0
$997$	24.4989	0.775886	0.387943	−	0.921683i	$-0.373186\pi$
$997$	24.4989	0.775886	0.387943	+	0.921683i	$0.373186\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bd.1.2		3
4.3	odd	2		2368.2.a.bc.1.2		3
8.3	odd	2		1184.2.a.l.1.2	✓	3
8.5	even	2		1184.2.a.m.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.l.1.2	✓	3	8.3	odd	2
1184.2.a.m.1.2	yes	3	8.5	even	2
2368.2.a.bc.1.2		3	4.3	odd	2
2368.2.a.bd.1.2		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-0.523976 q^{3} +1.52398 q^{5} +4.20147 q^{7} -2.72545 q^{9} +O(q^{10})\)	\(q-0.523976 q^{3} +1.52398 q^{5} +4.20147 q^{7} -2.72545 q^{9} +4.67750 q^{11} +6.72545 q^{13} -0.798528 q^{15} +6.40294 q^{17} +2.00000 q^{19} -2.20147 q^{21} -2.72545 q^{23} -2.67750 q^{25} +3.00000 q^{27} -1.67750 q^{29} -10.9749 q^{31} -2.45090 q^{33} +6.40294 q^{35} +1.00000 q^{37} -3.52398 q^{39} +4.52398 q^{41} -3.45090 q^{43} -4.15352 q^{45} -6.20147 q^{47} +10.6524 q^{49} -3.35499 q^{51} -3.24943 q^{53} +7.12839 q^{55} -1.04795 q^{57} -9.85384 q^{59} +6.87897 q^{61} -11.4509 q^{63} +10.2494 q^{65} +5.52398 q^{67} +1.42807 q^{69} -5.15352 q^{71} -7.62954 q^{73} +1.40294 q^{75} +19.6524 q^{77} -2.32250 q^{79} +6.60442 q^{81} -3.15352 q^{83} +9.75794 q^{85} +0.878968 q^{87} +16.4029 q^{89} +28.2568 q^{91} +5.75057 q^{93} +3.04795 q^{95} -10.8059 q^{97} -12.7483 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{5} + 3 q^{7} + 3 q^{9} + 6 q^{11} + 9 q^{13} - 12 q^{15} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 9 q^{27} + 3 q^{29} - 9 q^{31} + 15 q^{33} + 3 q^{37} - 9 q^{39} + 12 q^{41} + 12 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{51} + 3 q^{53} - 9 q^{55} + 12 q^{59} + 3 q^{61} - 12 q^{63} + 18 q^{65} + 15 q^{67} + 9 q^{69} - 9 q^{71} - 18 q^{73} - 15 q^{75} + 27 q^{77} - 15 q^{79} - 9 q^{81} - 3 q^{83} - 6 q^{85} - 15 q^{87} + 30 q^{89} + 24 q^{91} + 30 q^{93} + 6 q^{95} + 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 3 * q^7 + 3 * q^9 + 6 * q^11 + 9 * q^13 - 12 * q^15 + 6 * q^19 + 3 * q^21 + 3 * q^23 + 9 * q^27 + 3 * q^29 - 9 * q^31 + 15 * q^33 + 3 * q^37 - 9 * q^39 + 12 * q^41 + 12 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^51 + 3 * q^53 - 9 * q^55 + 12 * q^59 + 3 * q^61 - 12 * q^63 + 18 * q^65 + 15 * q^67 + 9 * q^69 - 9 * q^71 - 18 * q^73 - 15 * q^75 + 27 * q^77 - 15 * q^79 - 9 * q^81 - 3 * q^83 - 6 * q^85 - 15 * q^87 + 30 * q^89 + 24 * q^91 + 30 * q^93 + 6 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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