LMFDB - Embedded newform 2220.2.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2220,2,Mod(1,2220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2220.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2220 = 2^{2} \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2220.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.7267892487$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.39605.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 9x - 2 $ x^4 - 2x^3 - 8x^2 + 9*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.57685$ of defining polynomial
Character	$\chi$	$=$	2220.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +2.33529 q^{7} +1.00000 q^{9} +6.01770 q^{11} +2.44085 q^{13} +1.00000 q^{15} +0.800705 q^{17} +0.800705 q^{19} +2.33529 q^{21} -2.81840 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.09374 q^{29} -7.48899 q^{31} +6.01770 q^{33} +2.33529 q^{35} -1.00000 q^{37} +2.44085 q^{39} -9.97210 q^{41} +2.90626 q^{43} +1.00000 q^{45} -13.1468 q^{47} -1.54640 q^{49} +0.800705 q^{51} +11.9298 q^{53} +6.01770 q^{55} +0.800705 q^{57} +13.0658 q^{59} -5.21699 q^{61} +2.33529 q^{63} +2.44085 q^{65} -3.47129 q^{67} -2.81840 q^{69} -9.90527 q^{71} +0.948519 q^{73} +1.00000 q^{75} +14.0531 q^{77} -5.40800 q^{79} +1.00000 q^{81} -5.46443 q^{83} +0.800705 q^{85} +1.09374 q^{87} +9.59454 q^{89} +5.70010 q^{91} -7.48899 q^{93} +0.800705 q^{95} -8.41981 q^{97} +6.01770 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} + 5 q^{11} + 3 q^{13} + 4 q^{15} + 3 q^{17} + 3 q^{19} + 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 6 q^{29} + 5 q^{33} + 4 q^{35} - 4 q^{37} + 3 q^{39} + 4 q^{41}+ \cdots + 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 + 5 * q^11 + 3 * q^13 + 4 * q^15 + 3 * q^17 + 3 * q^19 + 4 * q^21 + 8 * q^23 + 4 * q^25 + 4 * q^27 + 6 * q^29 + 5 * q^33 + 4 * q^35 - 4 * q^37 + 3 * q^39 + 4 * q^41 + 10 * q^43 + 4 * q^45 + 3 * q^47 + 2 * q^49 + 3 * q^51 + 11 * q^53 + 5 * q^55 + 3 * q^57 + 10 * q^59 - 2 * q^61 + 4 * q^63 + 3 * q^65 - 3 * q^67 + 8 * q^69 + 9 * q^71 - 5 * q^73 + 4 * q^75 - q^77 - 5 * q^79 + 4 * q^81 + 20 * q^83 + 3 * q^85 + 6 * q^87 + 7 * q^89 - 10 * q^91 + 3 * q^95 - 14 * q^97 + 5 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.33529	0.882658	0.441329	−	0.897345i	$-0.354507\pi$
$7$	2.33529	0.882658	0.441329	+	0.897345i	$0.354507\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.01770	1.81440	0.907202	−	0.420696i	$-0.138214\pi$
$11$	6.01770	1.81440	0.907202	+	0.420696i	$0.138214\pi$
$12$	0	0
$13$	2.44085	0.676970	0.338485	−	0.940972i	$-0.390086\pi$
$13$	2.44085	0.676970	0.338485	+	0.940972i	$0.390086\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.800705	0.194200	0.0970998	−	0.995275i	$-0.469043\pi$
$17$	0.800705	0.194200	0.0970998	+	0.995275i	$0.469043\pi$
$18$	0	0
$19$	0.800705	0.183694	0.0918472	−	0.995773i	$-0.470723\pi$
$19$	0.800705	0.183694	0.0918472	+	0.995773i	$0.470723\pi$
$20$	0	0
$21$	2.33529	0.509603
$22$	0	0
$23$	−2.81840	−0.587677	−0.293839	−	0.955855i	$-0.594933\pi$
$23$	−2.81840	−0.587677	−0.293839	+	0.955855i	$0.594933\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.09374	0.203102	0.101551	−	0.994830i	$-0.467619\pi$
$29$	1.09374	0.203102	0.101551	+	0.994830i	$0.467619\pi$
$30$	0	0
$31$	−7.48899	−1.34506	−0.672531	−	0.740069i	$-0.734793\pi$
$31$	−7.48899	−1.34506	−0.672531	+	0.740069i	$0.734793\pi$
$32$	0	0
$33$	6.01770	1.04755
$34$	0	0
$35$	2.33529	0.394737
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	2.44085	0.390849
$40$	0	0
$41$	−9.97210	−1.55738	−0.778690	−	0.627409i	$-0.784115\pi$
$41$	−9.97210	−1.55738	−0.778690	+	0.627409i	$0.784115\pi$
$42$	0	0
$43$	2.90626	0.443200	0.221600	−	0.975138i	$-0.428872\pi$
$43$	2.90626	0.443200	0.221600	+	0.975138i	$0.428872\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−13.1468	−1.91766	−0.958831	−	0.283977i	$-0.908346\pi$
$47$	−13.1468	−1.91766	−0.958831	+	0.283977i	$0.908346\pi$
$48$	0	0
$49$	−1.54640	−0.220915
$50$	0	0
$51$	0.800705	0.112121
$52$	0	0
$53$	11.9298	1.63869	0.819345	−	0.573301i	$-0.194338\pi$
$53$	11.9298	1.63869	0.819345	+	0.573301i	$0.194338\pi$
$54$	0	0
$55$	6.01770	0.811426
$56$	0	0
$57$	0.800705	0.106056
$58$	0	0
$59$	13.0658	1.70103	0.850513	−	0.525953i	$-0.176291\pi$
$59$	13.0658	1.70103	0.850513	+	0.525953i	$0.176291\pi$
$60$	0	0
$61$	−5.21699	−0.667967	−0.333984	−	0.942579i	$-0.608393\pi$
$61$	−5.21699	−0.667967	−0.333984	+	0.942579i	$0.608393\pi$
$62$	0	0
$63$	2.33529	0.294219
$64$	0	0
$65$	2.44085	0.302750
$66$	0	0
$67$	−3.47129	−0.424086	−0.212043	−	0.977260i	$-0.568012\pi$
$67$	−3.47129	−0.424086	−0.212043	+	0.977260i	$0.568012\pi$
$68$	0	0
$69$	−2.81840	−0.339296
$70$	0	0
$71$	−9.90527	−1.17554	−0.587770	−	0.809028i	$-0.699994\pi$
$71$	−9.90527	−1.17554	−0.587770	+	0.809028i	$0.699994\pi$
$72$	0	0
$73$	0.948519	0.111016	0.0555079	−	0.998458i	$-0.482322\pi$
$73$	0.948519	0.111016	0.0555079	+	0.998458i	$0.482322\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	14.0531	1.60150
$78$	0	0
$79$	−5.40800	−0.608447	−0.304224	−	0.952601i	$-0.598397\pi$
$79$	−5.40800	−0.608447	−0.304224	+	0.952601i	$0.598397\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.46443	−0.599799	−0.299899	−	0.953971i	$-0.596953\pi$
$83$	−5.46443	−0.599799	−0.299899	+	0.953971i	$0.596953\pi$
$84$	0	0
$85$	0.800705	0.0868487
$86$	0	0
$87$	1.09374	0.117261
$88$	0	0
$89$	9.59454	1.01702	0.508510	−	0.861056i	$-0.330197\pi$
$89$	9.59454	1.01702	0.508510	+	0.861056i	$0.330197\pi$
$90$	0	0
$91$	5.70010	0.597533
$92$	0	0
$93$	−7.48899	−0.776572
$94$	0	0
$95$	0.800705	0.0821506
$96$	0	0
$97$	−8.41981	−0.854902	−0.427451	−	0.904038i	$-0.640588\pi$
$97$	−8.41981	−0.854902	−0.427451	+	0.904038i	$0.640588\pi$
$98$	0	0
$99$	6.01770	0.604801
$100$	0	0
$101$	17.6687	1.75810	0.879049	−	0.476731i	$-0.158179\pi$
$101$	17.6687	1.75810	0.879049	+	0.476731i	$0.158179\pi$
$102$	0	0
$103$	19.1468	1.88659	0.943296	−	0.331951i	$-0.107707\pi$
$103$	19.1468	1.88659	0.943296	+	0.331951i	$0.107707\pi$
$104$	0	0
$105$	2.33529	0.227901
$106$	0	0
$107$	2.10555	0.203552	0.101776	−	0.994807i	$-0.467548\pi$
$107$	2.10555	0.203552	0.101776	+	0.994807i	$0.467548\pi$
$108$	0	0
$109$	−16.1164	−1.54367	−0.771835	−	0.635823i	$-0.780661\pi$
$109$	−16.1164	−1.54367	−0.771835	+	0.635823i	$0.780661\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	1.58371	0.148983	0.0744917	−	0.997222i	$-0.476267\pi$
$113$	1.58371	0.148983	0.0744917	+	0.997222i	$0.476267\pi$
$114$	0	0
$115$	−2.81840	−0.262817
$116$	0	0
$117$	2.44085	0.225657
$118$	0	0
$119$	1.86988	0.171412
$120$	0	0
$121$	25.2127	2.29206
$122$	0	0
$123$	−9.97210	−0.899154
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.47717	0.574756	0.287378	−	0.957817i	$-0.407216\pi$
$127$	6.47717	0.574756	0.287378	+	0.957817i	$0.407216\pi$
$128$	0	0
$129$	2.90626	0.255882
$130$	0	0
$131$	18.6427	1.62882	0.814409	−	0.580291i	$-0.197061\pi$
$131$	18.6427	1.62882	0.814409	+	0.580291i	$0.197061\pi$
$132$	0	0
$133$	1.86988	0.162139
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−18.4703	−1.57802	−0.789012	−	0.614377i	$-0.789407\pi$
$137$	−18.4703	−1.57802	−0.789012	+	0.614377i	$0.789407\pi$
$138$	0	0
$139$	−8.73054	−0.740515	−0.370257	−	0.928929i	$-0.620731\pi$
$139$	−8.73054	−0.740515	−0.370257	+	0.928929i	$0.620731\pi$
$140$	0	0
$141$	−13.1468	−1.10716
$142$	0	0
$143$	14.6883	1.22830
$144$	0	0
$145$	1.09374	0.0908302
$146$	0	0
$147$	−1.54640	−0.127545
$148$	0	0
$149$	−9.97210	−0.816946	−0.408473	−	0.912770i	$-0.633939\pi$
$149$	−9.97210	−0.816946	−0.408473	+	0.912770i	$0.633939\pi$
$150$	0	0
$151$	−23.5808	−1.91898	−0.959490	−	0.281744i	$-0.909087\pi$
$151$	−23.5808	−1.91898	−0.959490	+	0.281744i	$0.909087\pi$
$152$	0	0
$153$	0.800705	0.0647332
$154$	0	0
$155$	−7.48899	−0.601530
$156$	0	0
$157$	−14.8184	−1.18264	−0.591319	−	0.806438i	$-0.701392\pi$
$157$	−14.8184	−1.18264	−0.591319	+	0.806438i	$0.701392\pi$
$158$	0	0
$159$	11.9298	0.946098
$160$	0	0
$161$	−6.58179	−0.518718
$162$	0	0
$163$	10.2465	0.802568	0.401284	−	0.915954i	$-0.368564\pi$
$163$	10.2465	0.802568	0.401284	+	0.915954i	$0.368564\pi$
$164$	0	0
$165$	6.01770	0.468477
$166$	0	0
$167$	12.5759	0.973149	0.486575	−	0.873639i	$-0.338246\pi$
$167$	12.5759	0.973149	0.486575	+	0.873639i	$0.338246\pi$
$168$	0	0
$169$	−7.04226	−0.541712
$170$	0	0
$171$	0.800705	0.0612315
$172$	0	0
$173$	−13.4188	−1.02021	−0.510107	−	0.860111i	$-0.670394\pi$
$173$	−13.4188	−1.02021	−0.510107	+	0.860111i	$0.670394\pi$
$174$	0	0
$175$	2.33529	0.176532
$176$	0	0
$177$	13.0658	0.982088
$178$	0	0
$179$	11.8665	0.886947	0.443473	−	0.896287i	$-0.353746\pi$
$179$	11.8665	0.886947	0.443473	+	0.896287i	$0.353746\pi$
$180$	0	0
$181$	5.06584	0.376541	0.188270	−	0.982117i	$-0.439712\pi$
$181$	5.06584	0.376541	0.188270	+	0.982117i	$0.439712\pi$
$182$	0	0
$183$	−5.21699	−0.385651
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	4.81840	0.352356
$188$	0	0
$189$	2.33529	0.169868
$190$	0	0
$191$	17.1891	1.24376	0.621879	−	0.783113i	$-0.286369\pi$
$191$	17.1891	1.24376	0.621879	+	0.783113i	$0.286369\pi$
$192$	0	0
$193$	−12.1409	−0.873924	−0.436962	−	0.899480i	$-0.643946\pi$
$193$	−12.1409	−0.873924	−0.436962	+	0.899480i	$0.643946\pi$
$194$	0	0
$195$	2.44085	0.174793
$196$	0	0
$197$	2.81840	0.200803	0.100401	−	0.994947i	$-0.467987\pi$
$197$	2.81840	0.200803	0.100401	+	0.994947i	$0.467987\pi$
$198$	0	0
$199$	5.76099	0.408386	0.204193	−	0.978931i	$-0.434543\pi$
$199$	5.76099	0.408386	0.204193	+	0.978931i	$0.434543\pi$
$200$	0	0
$201$	−3.47129	−0.244846
$202$	0	0
$203$	2.55420	0.179270
$204$	0	0
$205$	−9.97210	−0.696481
$206$	0	0
$207$	−2.81840	−0.195892
$208$	0	0
$209$	4.81840	0.333296
$210$	0	0
$211$	−7.03044	−0.483996	−0.241998	−	0.970277i	$-0.577803\pi$
$211$	−7.03044	−0.483996	−0.241998	+	0.970277i	$0.577803\pi$
$212$	0	0
$213$	−9.90527	−0.678698
$214$	0	0
$215$	2.90626	0.198205
$216$	0	0
$217$	−17.4890	−1.18723
$218$	0	0
$219$	0.948519	0.0640950
$220$	0	0
$221$	1.95440	0.131467
$222$	0	0
$223$	−4.98038	−0.333511	−0.166756	−	0.985998i	$-0.553329\pi$
$223$	−4.98038	−0.333511	−0.166756	+	0.985998i	$0.553329\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.6451	0.839284	0.419642	−	0.907690i	$-0.362156\pi$
$227$	12.6451	0.839284	0.419642	+	0.907690i	$0.362156\pi$
$228$	0	0
$229$	16.1763	1.06896	0.534481	−	0.845180i	$-0.320507\pi$
$229$	16.1763	1.06896	0.534481	+	0.845180i	$0.320507\pi$
$230$	0	0
$231$	14.0531	0.924625
$232$	0	0
$233$	5.85219	0.383389	0.191695	−	0.981455i	$-0.438602\pi$
$233$	5.85219	0.383389	0.191695	+	0.981455i	$0.438602\pi$
$234$	0	0
$235$	−13.1468	−0.857604
$236$	0	0
$237$	−5.40800	−0.351287
$238$	0	0
$239$	−19.7400	−1.27687	−0.638436	−	0.769675i	$-0.720418\pi$
$239$	−19.7400	−1.27687	−0.638436	+	0.769675i	$0.720418\pi$
$240$	0	0
$241$	−8.94852	−0.576425	−0.288212	−	0.957567i	$-0.593061\pi$
$241$	−8.94852	−0.576425	−0.288212	+	0.957567i	$0.593061\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.54640	−0.0987961
$246$	0	0
$247$	1.95440	0.124356
$248$	0	0
$249$	−5.46443	−0.346294
$250$	0	0
$251$	−26.8858	−1.69702	−0.848510	−	0.529180i	$-0.822500\pi$
$251$	−26.8858	−1.69702	−0.848510	+	0.529180i	$0.822500\pi$
$252$	0	0
$253$	−16.9603	−1.06628
$254$	0	0
$255$	0.800705	0.0501421
$256$	0	0
$257$	−3.91901	−0.244461	−0.122231	−	0.992502i	$-0.539005\pi$
$257$	−3.91901	−0.244461	−0.122231	+	0.992502i	$0.539005\pi$
$258$	0	0
$259$	−2.33529	−0.145108
$260$	0	0
$261$	1.09374	0.0677008
$262$	0	0
$263$	26.2828	1.62067	0.810334	−	0.585968i	$-0.199286\pi$
$263$	26.2828	1.62067	0.810334	+	0.585968i	$0.199286\pi$
$264$	0	0
$265$	11.9298	0.732844
$266$	0	0
$267$	9.59454	0.587177
$268$	0	0
$269$	11.1140	0.677634	0.338817	−	0.940852i	$-0.389973\pi$
$269$	11.1140	0.677634	0.338817	+	0.940852i	$0.389973\pi$
$270$	0	0
$271$	−19.8269	−1.20440	−0.602198	−	0.798346i	$-0.705708\pi$
$271$	−19.8269	−1.20440	−0.602198	+	0.798346i	$0.705708\pi$
$272$	0	0
$273$	5.70010	0.344986
$274$	0	0
$275$	6.01770	0.362881
$276$	0	0
$277$	−7.46777	−0.448695	−0.224347	−	0.974509i	$-0.572025\pi$
$277$	−7.46777	−0.448695	−0.224347	+	0.974509i	$0.572025\pi$
$278$	0	0
$279$	−7.48899	−0.448354
$280$	0	0
$281$	5.60048	0.334096	0.167048	−	0.985949i	$-0.446576\pi$
$281$	5.60048	0.334096	0.167048	+	0.985949i	$0.446576\pi$
$282$	0	0
$283$	26.1281	1.55316	0.776579	−	0.630020i	$-0.216954\pi$
$283$	26.1281	1.55316	0.776579	+	0.630020i	$0.216954\pi$
$284$	0	0
$285$	0.800705	0.0474297
$286$	0	0
$287$	−23.2878	−1.37463
$288$	0	0
$289$	−16.3589	−0.962287
$290$	0	0
$291$	−8.41981	−0.493578
$292$	0	0
$293$	24.9921	1.46006	0.730028	−	0.683417i	$-0.239507\pi$
$293$	24.9921	1.46006	0.730028	+	0.683417i	$0.239507\pi$
$294$	0	0
$295$	13.0658	0.760722
$296$	0	0
$297$	6.01770	0.349182
$298$	0	0
$299$	−6.87929	−0.397840
$300$	0	0
$301$	6.78697	0.391194
$302$	0	0
$303$	17.6687	1.01504
$304$	0	0
$305$	−5.21699	−0.298724
$306$	0	0
$307$	19.1612	1.09359	0.546793	−	0.837268i	$-0.315848\pi$
$307$	19.1612	1.09359	0.546793	+	0.837268i	$0.315848\pi$
$308$	0	0
$309$	19.1468	1.08922
$310$	0	0
$311$	12.0144	0.681272	0.340636	−	0.940195i	$-0.389358\pi$
$311$	12.0144	0.681272	0.340636	+	0.940195i	$0.389358\pi$
$312$	0	0
$313$	4.03633	0.228147	0.114073	−	0.993472i	$-0.463610\pi$
$313$	4.03633	0.228147	0.114073	+	0.993472i	$0.463610\pi$
$314$	0	0
$315$	2.33529	0.131579
$316$	0	0
$317$	−18.2439	−1.02468	−0.512340	−	0.858783i	$-0.671221\pi$
$317$	−18.2439	−1.02468	−0.512340	+	0.858783i	$0.671221\pi$
$318$	0	0
$319$	6.58179	0.368510
$320$	0	0
$321$	2.10555	0.117521
$322$	0	0
$323$	0.641129	0.0356734
$324$	0	0
$325$	2.44085	0.135394
$326$	0	0
$327$	−16.1164	−0.891238
$328$	0	0
$329$	−30.7017	−1.69264
$330$	0	0
$331$	12.9560	0.712127	0.356063	−	0.934462i	$-0.384119\pi$
$331$	12.9560	0.712127	0.356063	+	0.934462i	$0.384119\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−3.47129	−0.189657
$336$	0	0
$337$	21.6207	1.17775	0.588876	−	0.808223i	$-0.299571\pi$
$337$	21.6207	1.17775	0.588876	+	0.808223i	$0.299571\pi$
$338$	0	0
$339$	1.58371	0.0860156
$340$	0	0
$341$	−45.0665	−2.44049
$342$	0	0
$343$	−19.9584	−1.07765
$344$	0	0
$345$	−2.81840	−0.151738
$346$	0	0
$347$	−30.8613	−1.65672	−0.828360	−	0.560196i	$-0.810726\pi$
$347$	−30.8613	−1.65672	−0.828360	+	0.560196i	$0.810726\pi$
$348$	0	0
$349$	−27.5631	−1.47542	−0.737710	−	0.675118i	$-0.764093\pi$
$349$	−27.5631	−1.47542	−0.737710	+	0.675118i	$0.764093\pi$
$350$	0	0
$351$	2.44085	0.130283
$352$	0	0
$353$	14.6073	0.777468	0.388734	−	0.921350i	$-0.372913\pi$
$353$	14.6073	0.777468	0.388734	+	0.921350i	$0.372913\pi$
$354$	0	0
$355$	−9.90527	−0.525717
$356$	0	0
$357$	1.86988	0.0989647
$358$	0	0
$359$	−16.0142	−0.845196	−0.422598	−	0.906317i	$-0.638882\pi$
$359$	−16.0142	−0.845196	−0.422598	+	0.906317i	$0.638882\pi$
$360$	0	0
$361$	−18.3589	−0.966256
$362$	0	0
$363$	25.2127	1.32332
$364$	0	0
$365$	0.948519	0.0496478
$366$	0	0
$367$	−12.8125	−0.668805	−0.334403	−	0.942430i	$-0.608535\pi$
$367$	−12.8125	−0.668805	−0.334403	+	0.942430i	$0.608535\pi$
$368$	0	0
$369$	−9.97210	−0.519127
$370$	0	0
$371$	27.8597	1.44640
$372$	0	0
$373$	−16.4139	−0.849878	−0.424939	−	0.905222i	$-0.639705\pi$
$373$	−16.4139	−0.849878	−0.424939	+	0.905222i	$0.639705\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	2.66965	0.137494
$378$	0	0
$379$	−13.1528	−0.675612	−0.337806	−	0.941216i	$-0.609685\pi$
$379$	−13.1528	−0.675612	−0.337806	+	0.941216i	$0.609685\pi$
$380$	0	0
$381$	6.47717	0.331836
$382$	0	0
$383$	29.1832	1.49119	0.745595	−	0.666400i	$-0.232166\pi$
$383$	29.1832	1.49119	0.745595	+	0.666400i	$0.232166\pi$
$384$	0	0
$385$	14.0531	0.716212
$386$	0	0
$387$	2.90626	0.147733
$388$	0	0
$389$	3.13266	0.158832	0.0794160	−	0.996842i	$-0.474694\pi$
$389$	3.13266	0.158832	0.0794160	+	0.996842i	$0.474694\pi$
$390$	0	0
$391$	−2.25671	−0.114127
$392$	0	0
$393$	18.6427	0.940399
$394$	0	0
$395$	−5.40800	−0.272106
$396$	0	0
$397$	−10.1124	−0.507528	−0.253764	−	0.967266i	$-0.581669\pi$
$397$	−10.1124	−0.507528	−0.253764	+	0.967266i	$0.581669\pi$
$398$	0	0
$399$	1.86988	0.0936112
$400$	0	0
$401$	−20.8607	−1.04173	−0.520866	−	0.853639i	$-0.674391\pi$
$401$	−20.8607	−1.04173	−0.520866	+	0.853639i	$0.674391\pi$
$402$	0	0
$403$	−18.2795	−0.910566
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−6.01770	−0.298286
$408$	0	0
$409$	−16.1536	−0.798746	−0.399373	−	0.916788i	$-0.630772\pi$
$409$	−16.1536	−0.798746	−0.399373	+	0.916788i	$0.630772\pi$
$410$	0	0
$411$	−18.4703	−0.911073
$412$	0	0
$413$	30.5126	1.50143
$414$	0	0
$415$	−5.46443	−0.268238
$416$	0	0
$417$	−8.73054	−0.427537
$418$	0	0
$419$	−1.98824	−0.0971318	−0.0485659	−	0.998820i	$-0.515465\pi$
$419$	−1.98824	−0.0971318	−0.0485659	+	0.998820i	$0.515465\pi$
$420$	0	0
$421$	−3.80503	−0.185446	−0.0927230	−	0.995692i	$-0.529557\pi$
$421$	−3.80503	−0.185446	−0.0927230	+	0.995692i	$0.529557\pi$
$422$	0	0
$423$	−13.1468	−0.639221
$424$	0	0
$425$	0.800705	0.0388399
$426$	0	0
$427$	−12.1832	−0.589587
$428$	0	0
$429$	14.6883	0.709157
$430$	0	0
$431$	20.5539	0.990046	0.495023	−	0.868880i	$-0.335160\pi$
$431$	20.5539	0.990046	0.495023	+	0.868880i	$0.335160\pi$
$432$	0	0
$433$	−6.48663	−0.311728	−0.155864	−	0.987779i	$-0.549816\pi$
$433$	−6.48663	−0.311728	−0.155864	+	0.987779i	$0.549816\pi$
$434$	0	0
$435$	1.09374	0.0524408
$436$	0	0
$437$	−2.25671	−0.107953
$438$	0	0
$439$	16.3074	0.778309	0.389155	−	0.921172i	$-0.372767\pi$
$439$	16.3074	0.778309	0.389155	+	0.921172i	$0.372767\pi$
$440$	0	0
$441$	−1.54640	−0.0736382
$442$	0	0
$443$	6.25337	0.297106	0.148553	−	0.988904i	$-0.452538\pi$
$443$	6.25337	0.297106	0.148553	+	0.988904i	$0.452538\pi$
$444$	0	0
$445$	9.59454	0.454825
$446$	0	0
$447$	−9.97210	−0.471664
$448$	0	0
$449$	28.3368	1.33730	0.668649	−	0.743578i	$-0.266873\pi$
$449$	28.3368	1.33730	0.668649	+	0.743578i	$0.266873\pi$
$450$	0	0
$451$	−60.0090	−2.82572
$452$	0	0
$453$	−23.5808	−1.10792
$454$	0	0
$455$	5.70010	0.267225
$456$	0	0
$457$	−23.3438	−1.09198	−0.545988	−	0.837793i	$-0.683845\pi$
$457$	−23.3438	−1.09198	−0.545988	+	0.837793i	$0.683845\pi$
$458$	0	0
$459$	0.800705	0.0373737
$460$	0	0
$461$	−20.7903	−0.968301	−0.484151	−	0.874985i	$-0.660871\pi$
$461$	−20.7903	−0.968301	−0.484151	+	0.874985i	$0.660871\pi$
$462$	0	0
$463$	−7.25925	−0.337366	−0.168683	−	0.985670i	$-0.553951\pi$
$463$	−7.25925	−0.337366	−0.168683	+	0.985670i	$0.553951\pi$
$464$	0	0
$465$	−7.48899	−0.347294
$466$	0	0
$467$	28.4280	1.31549	0.657747	−	0.753239i	$-0.271510\pi$
$467$	28.4280	1.31549	0.657747	+	0.753239i	$0.271510\pi$
$468$	0	0
$469$	−8.10649	−0.374323
$470$	0	0
$471$	−14.8184	−0.682796
$472$	0	0
$473$	17.4890	0.804145
$474$	0	0
$475$	0.800705	0.0367389
$476$	0	0
$477$	11.9298	0.546230
$478$	0	0
$479$	−32.4999	−1.48496	−0.742478	−	0.669870i	$-0.766350\pi$
$479$	−32.4999	−1.48496	−0.742478	+	0.669870i	$0.766350\pi$
$480$	0	0
$481$	−2.44085	−0.111293
$482$	0	0
$483$	−6.58179	−0.299482
$484$	0	0
$485$	−8.41981	−0.382324
$486$	0	0
$487$	11.5910	0.525239	0.262620	−	0.964899i	$-0.415414\pi$
$487$	11.5910	0.525239	0.262620	+	0.964899i	$0.415414\pi$
$488$	0	0
$489$	10.2465	0.463363
$490$	0	0
$491$	10.0751	0.454681	0.227341	−	0.973815i	$-0.426997\pi$
$491$	10.0751	0.454681	0.227341	+	0.973815i	$0.426997\pi$
$492$	0	0
$493$	0.875763	0.0394424
$494$	0	0
$495$	6.01770	0.270475
$496$	0	0
$497$	−23.1317	−1.03760
$498$	0	0
$499$	16.4340	0.735686	0.367843	−	0.929888i	$-0.380096\pi$
$499$	16.4340	0.735686	0.367843	+	0.929888i	$0.380096\pi$
$500$	0	0
$501$	12.5759	0.561848
$502$	0	0
$503$	28.0590	1.25109	0.625545	−	0.780188i	$-0.284877\pi$
$503$	28.0590	1.25109	0.625545	+	0.780188i	$0.284877\pi$
$504$	0	0
$505$	17.6687	0.786245
$506$	0	0
$507$	−7.04226	−0.312758
$508$	0	0
$509$	−10.4595	−0.463608	−0.231804	−	0.972762i	$-0.574463\pi$
$509$	−10.4595	−0.463608	−0.231804	+	0.972762i	$0.574463\pi$
$510$	0	0
$511$	2.21507	0.0979890
$512$	0	0
$513$	0.800705	0.0353520
$514$	0	0
$515$	19.1468	0.843710
$516$	0	0
$517$	−79.1136	−3.47941
$518$	0	0
$519$	−13.4188	−0.589021
$520$	0	0
$521$	28.4847	1.24794	0.623968	−	0.781450i	$-0.285520\pi$
$521$	28.4847	1.24794	0.623968	+	0.781450i	$0.285520\pi$
$522$	0	0
$523$	3.26964	0.142972	0.0714858	−	0.997442i	$-0.477226\pi$
$523$	3.26964	0.142972	0.0714858	+	0.997442i	$0.477226\pi$
$524$	0	0
$525$	2.33529	0.101921
$526$	0	0
$527$	−5.99647	−0.261210
$528$	0	0
$529$	−15.0566	−0.654635
$530$	0	0
$531$	13.0658	0.567009
$532$	0	0
$533$	−24.3404	−1.05430
$534$	0	0
$535$	2.10555	0.0910311
$536$	0	0
$537$	11.8665	0.512079
$538$	0	0
$539$	−9.30578	−0.400828
$540$	0	0
$541$	−30.6848	−1.31924	−0.659620	−	0.751599i	$-0.729283\pi$
$541$	−30.6848	−1.31924	−0.659620	+	0.751599i	$0.729283\pi$
$542$	0	0
$543$	5.06584	0.217396
$544$	0	0
$545$	−16.1164	−0.690350
$546$	0	0
$547$	33.4047	1.42828	0.714140	−	0.700003i	$-0.246818\pi$
$547$	33.4047	1.42828	0.714140	+	0.700003i	$0.246818\pi$
$548$	0	0
$549$	−5.21699	−0.222656
$550$	0	0
$551$	0.875763	0.0373088
$552$	0	0
$553$	−12.6293	−0.537051
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	23.3334	0.988667	0.494333	−	0.869272i	$-0.335412\pi$
$557$	23.3334	0.988667	0.494333	+	0.869272i	$0.335412\pi$
$558$	0	0
$559$	7.09374	0.300033
$560$	0	0
$561$	4.81840	0.203433
$562$	0	0
$563$	−14.7633	−0.622201	−0.311100	−	0.950377i	$-0.600698\pi$
$563$	−14.7633	−0.622201	−0.311100	+	0.950377i	$0.600698\pi$
$564$	0	0
$565$	1.58371	0.0666274
$566$	0	0
$567$	2.33529	0.0980731
$568$	0	0
$569$	−23.6521	−0.991547	−0.495774	−	0.868452i	$-0.665115\pi$
$569$	−23.6521	−0.991547	−0.495774	+	0.868452i	$0.665115\pi$
$570$	0	0
$571$	13.2288	0.553606	0.276803	−	0.960927i	$-0.410725\pi$
$571$	13.2288	0.553606	0.276803	+	0.960927i	$0.410725\pi$
$572$	0	0
$573$	17.1891	0.718084
$574$	0	0
$575$	−2.81840	−0.117535
$576$	0	0
$577$	−5.89351	−0.245350	−0.122675	−	0.992447i	$-0.539147\pi$
$577$	−5.89351	−0.245350	−0.122675	+	0.992447i	$0.539147\pi$
$578$	0	0
$579$	−12.1409	−0.504561
$580$	0	0
$581$	−12.7610	−0.529417
$582$	0	0
$583$	71.7901	2.97324
$584$	0	0
$585$	2.44085	0.100917
$586$	0	0
$587$	13.9230	0.574662	0.287331	−	0.957831i	$-0.407232\pi$
$587$	13.9230	0.574662	0.287331	+	0.957831i	$0.407232\pi$
$588$	0	0
$589$	−5.99647	−0.247080
$590$	0	0
$591$	2.81840	0.115934
$592$	0	0
$593$	9.22626	0.378877	0.189439	−	0.981893i	$-0.439333\pi$
$593$	9.22626	0.378877	0.189439	+	0.981893i	$0.439333\pi$
$594$	0	0
$595$	1.86988	0.0766577
$596$	0	0
$597$	5.76099	0.235782
$598$	0	0
$599$	−28.6848	−1.17203	−0.586014	−	0.810301i	$-0.699304\pi$
$599$	−28.6848	−1.17203	−0.586014	+	0.810301i	$0.699304\pi$
$600$	0	0
$601$	−29.3555	−1.19744	−0.598719	−	0.800960i	$-0.704323\pi$
$601$	−29.3555	−1.19744	−0.598719	+	0.800960i	$0.704323\pi$
$602$	0	0
$603$	−3.47129	−0.141362
$604$	0	0
$605$	25.2127	1.02504
$606$	0	0
$607$	−19.0530	−0.773339	−0.386669	−	0.922218i	$-0.626375\pi$
$607$	−19.0530	−0.773339	−0.386669	+	0.922218i	$0.626375\pi$
$608$	0	0
$609$	2.55420	0.103502
$610$	0	0
$611$	−32.0894	−1.29820
$612$	0	0
$613$	13.6840	0.552690	0.276345	−	0.961059i	$-0.410877\pi$
$613$	13.6840	0.552690	0.276345	+	0.961059i	$0.410877\pi$
$614$	0	0
$615$	−9.97210	−0.402114
$616$	0	0
$617$	−3.98230	−0.160322	−0.0801608	−	0.996782i	$-0.525543\pi$
$617$	−3.98230	−0.160322	−0.0801608	+	0.996782i	$0.525543\pi$
$618$	0	0
$619$	39.8813	1.60297	0.801483	−	0.598018i	$-0.204045\pi$
$619$	39.8813	1.60297	0.801483	+	0.598018i	$0.204045\pi$
$620$	0	0
$621$	−2.81840	−0.113099
$622$	0	0
$623$	22.4061	0.897681
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.81840	0.192428
$628$	0	0
$629$	−0.800705	−0.0319262
$630$	0	0
$631$	−46.2693	−1.84195	−0.920975	−	0.389621i	$-0.872606\pi$
$631$	−46.2693	−1.84195	−0.920975	+	0.389621i	$0.872606\pi$
$632$	0	0
$633$	−7.03044	−0.279435
$634$	0	0
$635$	6.47717	0.257039
$636$	0	0
$637$	−3.77454	−0.149553
$638$	0	0
$639$	−9.90527	−0.391847
$640$	0	0
$641$	−44.0122	−1.73838	−0.869189	−	0.494481i	$-0.835358\pi$
$641$	−44.0122	−1.73838	−0.869189	+	0.494481i	$0.835358\pi$
$642$	0	0
$643$	−39.4016	−1.55385	−0.776923	−	0.629596i	$-0.783221\pi$
$643$	−39.4016	−1.55385	−0.776923	+	0.629596i	$0.783221\pi$
$644$	0	0
$645$	2.90626	0.114434
$646$	0	0
$647$	9.45600	0.371754	0.185877	−	0.982573i	$-0.440487\pi$
$647$	9.45600	0.371754	0.185877	+	0.982573i	$0.440487\pi$
$648$	0	0
$649$	78.6262	3.08635
$650$	0	0
$651$	−17.4890	−0.685448
$652$	0	0
$653$	18.4174	0.720729	0.360364	−	0.932812i	$-0.382652\pi$
$653$	18.4174	0.720729	0.360364	+	0.932812i	$0.382652\pi$
$654$	0	0
$655$	18.6427	0.728430
$656$	0	0
$657$	0.948519	0.0370053
$658$	0	0
$659$	−36.1695	−1.40896	−0.704481	−	0.709723i	$-0.748820\pi$
$659$	−36.1695	−1.40896	−0.704481	+	0.709723i	$0.748820\pi$
$660$	0	0
$661$	−31.8621	−1.23929	−0.619646	−	0.784882i	$-0.712724\pi$
$661$	−31.8621	−1.23929	−0.619646	+	0.784882i	$0.712724\pi$
$662$	0	0
$663$	1.95440	0.0759026
$664$	0	0
$665$	1.86988	0.0725109
$666$	0	0
$667$	−3.08260	−0.119359
$668$	0	0
$669$	−4.98038	−0.192553
$670$	0	0
$671$	−31.3943	−1.21196
$672$	0	0
$673$	18.3954	0.709092	0.354546	−	0.935039i	$-0.384635\pi$
$673$	18.3954	0.709092	0.354546	+	0.935039i	$0.384635\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−15.2019	−0.584256	−0.292128	−	0.956379i	$-0.594363\pi$
$677$	−15.2019	−0.584256	−0.292128	+	0.956379i	$0.594363\pi$
$678$	0	0
$679$	−19.6627	−0.754586
$680$	0	0
$681$	12.6451	0.484561
$682$	0	0
$683$	11.6435	0.445526	0.222763	−	0.974873i	$-0.428492\pi$
$683$	11.6435	0.445526	0.222763	+	0.974873i	$0.428492\pi$
$684$	0	0
$685$	−18.4703	−0.705714
$686$	0	0
$687$	16.1763	0.617166
$688$	0	0
$689$	29.1189	1.10934
$690$	0	0
$691$	−13.3107	−0.506362	−0.253181	−	0.967419i	$-0.581477\pi$
$691$	−13.3107	−0.506362	−0.253181	+	0.967419i	$0.581477\pi$
$692$	0	0
$693$	14.0531	0.533833
$694$	0	0
$695$	−8.73054	−0.331168
$696$	0	0
$697$	−7.98471	−0.302442
$698$	0	0
$699$	5.85219	0.221350
$700$	0	0
$701$	−29.8623	−1.12788	−0.563941	−	0.825815i	$-0.690716\pi$
$701$	−29.8623	−1.12788	−0.563941	+	0.825815i	$0.690716\pi$
$702$	0	0
$703$	−0.800705	−0.0301992
$704$	0	0
$705$	−13.1468	−0.495138
$706$	0	0
$707$	41.2615	1.55180
$708$	0	0
$709$	2.41703	0.0907736	0.0453868	−	0.998969i	$-0.485548\pi$
$709$	2.41703	0.0907736	0.0453868	+	0.998969i	$0.485548\pi$
$710$	0	0
$711$	−5.40800	−0.202816
$712$	0	0
$713$	21.1070	0.790462
$714$	0	0
$715$	14.6883	0.549311
$716$	0	0
$717$	−19.7400	−0.737202
$718$	0	0
$719$	6.56806	0.244947	0.122474	−	0.992472i	$-0.460917\pi$
$719$	6.56806	0.244947	0.122474	+	0.992472i	$0.460917\pi$
$720$	0	0
$721$	44.7135	1.66522
$722$	0	0
$723$	−8.94852	−0.332799
$724$	0	0
$725$	1.09374	0.0406205
$726$	0	0
$727$	−40.0048	−1.48369	−0.741847	−	0.670569i	$-0.766050\pi$
$727$	−40.0048	−1.48369	−0.741847	+	0.670569i	$0.766050\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.32706	0.0860693
$732$	0	0
$733$	−42.0783	−1.55420	−0.777098	−	0.629379i	$-0.783309\pi$
$733$	−42.0783	−1.55420	−0.777098	+	0.629379i	$0.783309\pi$
$734$	0	0
$735$	−1.54640	−0.0570399
$736$	0	0
$737$	−20.8892	−0.769463
$738$	0	0
$739$	−8.89958	−0.327376	−0.163688	−	0.986512i	$-0.552339\pi$
$739$	−8.89958	−0.327376	−0.163688	+	0.986512i	$0.552339\pi$
$740$	0	0
$741$	1.95440	0.0717967
$742$	0	0
$743$	10.1562	0.372593	0.186297	−	0.982494i	$-0.440351\pi$
$743$	10.1562	0.372593	0.186297	+	0.982494i	$0.440351\pi$
$744$	0	0
$745$	−9.97210	−0.365349
$746$	0	0
$747$	−5.46443	−0.199933
$748$	0	0
$749$	4.91709	0.179667
$750$	0	0
$751$	0.549930	0.0200672	0.0100336	−	0.999950i	$-0.496806\pi$
$751$	0.549930	0.0200672	0.0100336	+	0.999950i	$0.496806\pi$
$752$	0	0
$753$	−26.8858	−0.979775
$754$	0	0
$755$	−23.5808	−0.858194
$756$	0	0
$757$	−9.65147	−0.350789	−0.175394	−	0.984498i	$-0.556120\pi$
$757$	−9.65147	−0.350789	−0.175394	+	0.984498i	$0.556120\pi$
$758$	0	0
$759$	−16.9603	−0.615619
$760$	0	0
$761$	−2.39463	−0.0868052	−0.0434026	−	0.999058i	$-0.513820\pi$
$761$	−2.39463	−0.0868052	−0.0434026	+	0.999058i	$0.513820\pi$
$762$	0	0
$763$	−37.6365	−1.36253
$764$	0	0
$765$	0.800705	0.0289496
$766$	0	0
$767$	31.8917	1.15154
$768$	0	0
$769$	48.5621	1.75120	0.875598	−	0.483041i	$-0.160468\pi$
$769$	48.5621	1.75120	0.875598	+	0.483041i	$0.160468\pi$
$770$	0	0
$771$	−3.91901	−0.141140
$772$	0	0
$773$	−2.42222	−0.0871211	−0.0435606	−	0.999051i	$-0.513870\pi$
$773$	−2.42222	−0.0871211	−0.0435606	+	0.999051i	$0.513870\pi$
$774$	0	0
$775$	−7.48899	−0.269012
$776$	0	0
$777$	−2.33529	−0.0837782
$778$	0	0
$779$	−7.98471	−0.286082
$780$	0	0
$781$	−59.6069	−2.13290
$782$	0	0
$783$	1.09374	0.0390871
$784$	0	0
$785$	−14.8184	−0.528891
$786$	0	0
$787$	39.9312	1.42339	0.711697	−	0.702487i	$-0.247927\pi$
$787$	39.9312	1.42339	0.711697	+	0.702487i	$0.247927\pi$
$788$	0	0
$789$	26.2828	0.935693
$790$	0	0
$791$	3.69844	0.131501
$792$	0	0
$793$	−12.7339	−0.452193
$794$	0	0
$795$	11.9298	0.423108
$796$	0	0
$797$	−42.1628	−1.49348	−0.746741	−	0.665115i	$-0.768383\pi$
$797$	−42.1628	−1.49348	−0.746741	+	0.665115i	$0.768383\pi$
$798$	0	0
$799$	−10.5267	−0.372409
$800$	0	0
$801$	9.59454	0.339007
$802$	0	0
$803$	5.70790	0.201427
$804$	0	0
$805$	−6.58179	−0.231978
$806$	0	0
$807$	11.1140	0.391232
$808$	0	0
$809$	−30.1822	−1.06115	−0.530574	−	0.847638i	$-0.678024\pi$
$809$	−30.1822	−1.06115	−0.530574	+	0.847638i	$0.678024\pi$
$810$	0	0
$811$	42.2952	1.48519	0.742593	−	0.669743i	$-0.233596\pi$
$811$	42.2952	1.48519	0.742593	+	0.669743i	$0.233596\pi$
$812$	0	0
$813$	−19.8269	−0.695359
$814$	0	0
$815$	10.2465	0.358919
$816$	0	0
$817$	2.32706	0.0814134
$818$	0	0
$819$	5.70010	0.199178
$820$	0	0
$821$	21.7826	0.760220	0.380110	−	0.924941i	$-0.375886\pi$
$821$	21.7826	0.760220	0.380110	+	0.924941i	$0.375886\pi$
$822$	0	0
$823$	51.3746	1.79081	0.895403	−	0.445257i	$-0.146888\pi$
$823$	51.3746	1.79081	0.895403	+	0.445257i	$0.146888\pi$
$824$	0	0
$825$	6.01770	0.209509
$826$	0	0
$827$	23.4960	0.817037	0.408519	−	0.912750i	$-0.366045\pi$
$827$	23.4960	0.817037	0.408519	+	0.912750i	$0.366045\pi$
$828$	0	0
$829$	14.0235	0.487057	0.243529	−	0.969894i	$-0.421695\pi$
$829$	14.0235	0.487057	0.243529	+	0.969894i	$0.421695\pi$
$830$	0	0
$831$	−7.46777	−0.259054
$832$	0	0
$833$	−1.23821	−0.0429015
$834$	0	0
$835$	12.5759	0.435206
$836$	0	0
$837$	−7.48899	−0.258857
$838$	0	0
$839$	−32.9804	−1.13861	−0.569305	−	0.822127i	$-0.692788\pi$
$839$	−32.9804	−1.13861	−0.569305	+	0.822127i	$0.692788\pi$
$840$	0	0
$841$	−27.8037	−0.958749
$842$	0	0
$843$	5.60048	0.192891
$844$	0	0
$845$	−7.04226	−0.242261
$846$	0	0
$847$	58.8790	2.02311
$848$	0	0
$849$	26.1281	0.896716
$850$	0	0
$851$	2.81840	0.0966136
$852$	0	0
$853$	33.3116	1.14057	0.570284	−	0.821448i	$-0.306833\pi$
$853$	33.3116	1.14057	0.570284	+	0.821448i	$0.306833\pi$
$854$	0	0
$855$	0.800705	0.0273835
$856$	0	0
$857$	−37.6136	−1.28486	−0.642428	−	0.766346i	$-0.722073\pi$
$857$	−37.6136	−1.28486	−0.642428	+	0.766346i	$0.722073\pi$
$858$	0	0
$859$	8.02550	0.273826	0.136913	−	0.990583i	$-0.456282\pi$
$859$	8.02550	0.273826	0.136913	+	0.990583i	$0.456282\pi$
$860$	0	0
$861$	−23.2878	−0.793645
$862$	0	0
$863$	10.7281	0.365190	0.182595	−	0.983188i	$-0.441550\pi$
$863$	10.7281	0.365190	0.182595	+	0.983188i	$0.441550\pi$
$864$	0	0
$865$	−13.4188	−0.456254
$866$	0	0
$867$	−16.3589	−0.555576
$868$	0	0
$869$	−32.5437	−1.10397
$870$	0	0
$871$	−8.47290	−0.287093
$872$	0	0
$873$	−8.41981	−0.284967
$874$	0	0
$875$	2.33529	0.0789473
$876$	0	0
$877$	17.1459	0.578976	0.289488	−	0.957182i	$-0.406515\pi$
$877$	17.1459	0.578976	0.289488	+	0.957182i	$0.406515\pi$
$878$	0	0
$879$	24.9921	0.842964
$880$	0	0
$881$	16.7988	0.565967	0.282984	−	0.959125i	$-0.408676\pi$
$881$	16.7988	0.565967	0.282984	+	0.959125i	$0.408676\pi$
$882$	0	0
$883$	−25.8809	−0.870962	−0.435481	−	0.900198i	$-0.643422\pi$
$883$	−25.8809	−0.870962	−0.435481	+	0.900198i	$0.643422\pi$
$884$	0	0
$885$	13.0658	0.439203
$886$	0	0
$887$	32.5024	1.09132	0.545661	−	0.838006i	$-0.316279\pi$
$887$	32.5024	1.09132	0.545661	+	0.838006i	$0.316279\pi$
$888$	0	0
$889$	15.1261	0.507313
$890$	0	0
$891$	6.01770	0.201600
$892$	0	0
$893$	−10.5267	−0.352264
$894$	0	0
$895$	11.8665	0.396655
$896$	0	0
$897$	−6.87929	−0.229693
$898$	0	0
$899$	−8.19101	−0.273185
$900$	0	0
$901$	9.55228	0.318233
$902$	0	0
$903$	6.78697	0.225856
$904$	0	0
$905$	5.06584	0.168394
$906$	0	0
$907$	54.7794	1.81892	0.909459	−	0.415793i	$-0.136496\pi$
$907$	54.7794	1.81892	0.909459	+	0.415793i	$0.136496\pi$
$908$	0	0
$909$	17.6687	0.586033
$910$	0	0
$911$	−43.7616	−1.44989	−0.724943	−	0.688808i	$-0.758134\pi$
$911$	−43.7616	−1.44989	−0.724943	+	0.688808i	$0.758134\pi$
$912$	0	0
$913$	−32.8833	−1.08828
$914$	0	0
$915$	−5.21699	−0.172468
$916$	0	0
$917$	43.5361	1.43769
$918$	0	0
$919$	28.4694	0.939118	0.469559	−	0.882901i	$-0.344413\pi$
$919$	28.4694	0.939118	0.469559	+	0.882901i	$0.344413\pi$
$920$	0	0
$921$	19.1612	0.631382
$922$	0	0
$923$	−24.1773	−0.795805
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	19.1468	0.628864
$928$	0	0
$929$	28.1808	0.924582	0.462291	−	0.886728i	$-0.347028\pi$
$929$	28.1808	0.924582	0.462291	+	0.886728i	$0.347028\pi$
$930$	0	0
$931$	−1.23821	−0.0405808
$932$	0	0
$933$	12.0144	0.393332
$934$	0	0
$935$	4.81840	0.157579
$936$	0	0
$937$	40.1805	1.31264	0.656320	−	0.754483i	$-0.272112\pi$
$937$	40.1805	1.31264	0.656320	+	0.754483i	$0.272112\pi$
$938$	0	0
$939$	4.03633	0.131721
$940$	0	0
$941$	−3.43349	−0.111929	−0.0559644	−	0.998433i	$-0.517823\pi$
$941$	−3.43349	−0.111929	−0.0559644	+	0.998433i	$0.517823\pi$
$942$	0	0
$943$	28.1054	0.915237
$944$	0	0
$945$	2.33529	0.0759671
$946$	0	0
$947$	7.01016	0.227799	0.113900	−	0.993492i	$-0.463666\pi$
$947$	7.01016	0.227799	0.113900	+	0.993492i	$0.463666\pi$
$948$	0	0
$949$	2.31519	0.0751543
$950$	0	0
$951$	−18.2439	−0.591599
$952$	0	0
$953$	12.2012	0.395236	0.197618	−	0.980279i	$-0.436679\pi$
$953$	12.2012	0.395236	0.197618	+	0.980279i	$0.436679\pi$
$954$	0	0
$955$	17.1891	0.556226
$956$	0	0
$957$	6.58179	0.212759
$958$	0	0
$959$	−43.1336	−1.39286
$960$	0	0
$961$	25.0850	0.809192
$962$	0	0
$963$	2.10555	0.0678505
$964$	0	0
$965$	−12.1409	−0.390831
$966$	0	0
$967$	22.6274	0.727648	0.363824	−	0.931468i	$-0.381471\pi$
$967$	22.6274	0.727648	0.363824	+	0.931468i	$0.381471\pi$
$968$	0	0
$969$	0.641129	0.0205960
$970$	0	0
$971$	29.4871	0.946287	0.473143	−	0.880985i	$-0.343119\pi$
$971$	29.4871	0.946287	0.473143	+	0.880985i	$0.343119\pi$
$972$	0	0
$973$	−20.3884	−0.653622
$974$	0	0
$975$	2.44085	0.0781697
$976$	0	0
$977$	0.279799	0.00895158	0.00447579	−	0.999990i	$-0.498575\pi$
$977$	0.279799	0.00895158	0.00447579	+	0.999990i	$0.498575\pi$
$978$	0	0
$979$	57.7370	1.84528
$980$	0	0
$981$	−16.1164	−0.514557
$982$	0	0
$983$	−50.6891	−1.61673	−0.808365	−	0.588681i	$-0.799647\pi$
$983$	−50.6891	−1.61673	−0.808365	+	0.588681i	$0.799647\pi$
$984$	0	0
$985$	2.81840	0.0898017
$986$	0	0
$987$	−30.7017	−0.977246
$988$	0	0
$989$	−8.19101	−0.260459
$990$	0	0
$991$	33.7763	1.07294	0.536469	−	0.843920i	$-0.319758\pi$
$991$	33.7763	1.07294	0.536469	+	0.843920i	$0.319758\pi$
$992$	0	0
$993$	12.9560	0.411146
$994$	0	0
$995$	5.76099	0.182636
$996$	0	0
$997$	−45.5633	−1.44300	−0.721502	−	0.692412i	$-0.756548\pi$
$997$	−45.5633	−1.44300	−0.721502	+	0.692412i	$0.756548\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2220.2.a.k.1.3	✓	4
3.2	odd	2		6660.2.a.s.1.3		4
4.3	odd	2		8880.2.a.cf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2220.2.a.k.1.3	✓	4	1.1	even	1	trivial
6660.2.a.s.1.3		4	3.2	odd	2
8880.2.a.cf.1.2		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2220 = 2^{2} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.33529 q^{7} +1.00000 q^{9} +6.01770 q^{11} +2.44085 q^{13} +1.00000 q^{15} +0.800705 q^{17} +0.800705 q^{19} +2.33529 q^{21} -2.81840 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.09374 q^{29} -7.48899 q^{31} +6.01770 q^{33} +2.33529 q^{35} -1.00000 q^{37} +2.44085 q^{39} -9.97210 q^{41} +2.90626 q^{43} +1.00000 q^{45} -13.1468 q^{47} -1.54640 q^{49} +0.800705 q^{51} +11.9298 q^{53} +6.01770 q^{55} +0.800705 q^{57} +13.0658 q^{59} -5.21699 q^{61} +2.33529 q^{63} +2.44085 q^{65} -3.47129 q^{67} -2.81840 q^{69} -9.90527 q^{71} +0.948519 q^{73} +1.00000 q^{75} +14.0531 q^{77} -5.40800 q^{79} +1.00000 q^{81} -5.46443 q^{83} +0.800705 q^{85} +1.09374 q^{87} +9.59454 q^{89} +5.70010 q^{91} -7.48899 q^{93} +0.800705 q^{95} -8.41981 q^{97} +6.01770 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} + 5 q^{11} + 3 q^{13} + 4 q^{15} + 3 q^{17} + 3 q^{19} + 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 6 q^{29} + 5 q^{33} + 4 q^{35} - 4 q^{37} + 3 q^{39} + 4 q^{41}+ \cdots + 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 + 5 * q^11 + 3 * q^13 + 4 * q^15 + 3 * q^17 + 3 * q^19 + 4 * q^21 + 8 * q^23 + 4 * q^25 + 4 * q^27 + 6 * q^29 + 5 * q^33 + 4 * q^35 - 4 * q^37 + 3 * q^39 + 4 * q^41 + 10 * q^43 + 4 * q^45 + 3 * q^47 + 2 * q^49 + 3 * q^51 + 11 * q^53 + 5 * q^55 + 3 * q^57 + 10 * q^59 - 2 * q^61 + 4 * q^63 + 3 * q^65 - 3 * q^67 + 8 * q^69 + 9 * q^71 - 5 * q^73 + 4 * q^75 - q^77 - 5 * q^79 + 4 * q^81 + 20 * q^83 + 3 * q^85 + 6 * q^87 + 7 * q^89 - 10 * q^91 + 3 * q^95 - 14 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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