LMFDB - Embedded newform 1216.4.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 $ x^7 - 3x^6 - 121x^5 + 402x^4 + 4234x^3 - 14542x^2 - 40996x + 141664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.37394$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.88542 q^{3} -4.69330 q^{5} +3.62949 q^{7} +20.4091 q^{9} +O(q^{10})$	$q-6.88542 q^{3} -4.69330 q^{5} +3.62949 q^{7} +20.4091 q^{9} -43.4975 q^{11} -75.0368 q^{13} +32.3154 q^{15} -13.1508 q^{17} +19.0000 q^{19} -24.9906 q^{21} -96.5591 q^{23} -102.973 q^{25} +45.3813 q^{27} -40.5580 q^{29} +138.828 q^{31} +299.499 q^{33} -17.0343 q^{35} -348.686 q^{37} +516.660 q^{39} -44.3243 q^{41} -214.385 q^{43} -95.7860 q^{45} -334.307 q^{47} -329.827 q^{49} +90.5487 q^{51} -311.122 q^{53} +204.147 q^{55} -130.823 q^{57} -502.875 q^{59} +54.7193 q^{61} +74.0746 q^{63} +352.170 q^{65} -312.959 q^{67} +664.850 q^{69} +642.657 q^{71} -837.348 q^{73} +709.012 q^{75} -157.874 q^{77} +349.595 q^{79} -863.515 q^{81} -521.399 q^{83} +61.7206 q^{85} +279.259 q^{87} +63.4094 q^{89} -272.345 q^{91} -955.887 q^{93} -89.1728 q^{95} -1229.92 q^{97} -887.745 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} + 17 q^{5} + 42 q^{7} + 86 q^{9}+O(q^{10}) $ 7 * q - 3 * q^3 + 17 * q^5 + 42 * q^7 + 86 * q^9	$ 7 q - 3 q^{3} + 17 q^{5} + 42 q^{7} + 86 q^{9} + 33 q^{11} + 35 q^{13} + 120 q^{15} + 66 q^{17} + 133 q^{19} - 33 q^{21} + 389 q^{23} + 44 q^{25} + 39 q^{27} - 233 q^{29} + 158 q^{31} - 206 q^{33} - 123 q^{35} + 436 q^{37} + 807 q^{39} - 94 q^{41} - 645 q^{43} - 103 q^{45} + 1451 q^{47} + 93 q^{49} - 1741 q^{51} - 3 q^{53} + 1971 q^{55} - 57 q^{57} - 297 q^{59} - 93 q^{61} + 2999 q^{63} - 788 q^{65} - 1641 q^{67} - 945 q^{69} + 2392 q^{71} + 324 q^{73} - 1909 q^{75} + 711 q^{77} + 2492 q^{79} + 143 q^{81} - 310 q^{83} - 2353 q^{85} + 4795 q^{87} - 440 q^{89} + 107 q^{91} - 900 q^{93} + 323 q^{95} - 532 q^{97} + 1591 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 + 17 * q^5 + 42 * q^7 + 86 * q^9 + 33 * q^11 + 35 * q^13 + 120 * q^15 + 66 * q^17 + 133 * q^19 - 33 * q^21 + 389 * q^23 + 44 * q^25 + 39 * q^27 - 233 * q^29 + 158 * q^31 - 206 * q^33 - 123 * q^35 + 436 * q^37 + 807 * q^39 - 94 * q^41 - 645 * q^43 - 103 * q^45 + 1451 * q^47 + 93 * q^49 - 1741 * q^51 - 3 * q^53 + 1971 * q^55 - 57 * q^57 - 297 * q^59 - 93 * q^61 + 2999 * q^63 - 788 * q^65 - 1641 * q^67 - 945 * q^69 + 2392 * q^71 + 324 * q^73 - 1909 * q^75 + 711 * q^77 + 2492 * q^79 + 143 * q^81 - 310 * q^83 - 2353 * q^85 + 4795 * q^87 - 440 * q^89 + 107 * q^91 - 900 * q^93 + 323 * q^95 - 532 * q^97 + 1591 * 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
$3$	−6.88542	−1.32510	−0.662550	−	0.749017i	$-0.730526\pi$
$3$	−6.88542	−1.32510	−0.662550	+	0.749017i	$0.730526\pi$
$4$	0	0
$5$	−4.69330	−0.419782	−0.209891	−	0.977725i	$-0.567311\pi$
$5$	−4.69330	−0.419782	−0.209891	+	0.977725i	$0.567311\pi$
$6$	0	0
$7$	3.62949	0.195974	0.0979871	−	0.995188i	$-0.468760\pi$
$7$	3.62949	0.195974	0.0979871	+	0.995188i	$0.468760\pi$
$8$	0	0
$9$	20.4091	0.755892
$10$	0	0
$11$	−43.4975	−1.19227	−0.596137	−	0.802883i	$-0.703298\pi$
$11$	−43.4975	−1.19227	−0.596137	+	0.802883i	$0.703298\pi$
$12$	0	0
$13$	−75.0368	−1.60088	−0.800441	−	0.599412i	$-0.795401\pi$
$13$	−75.0368	−1.60088	−0.800441	+	0.599412i	$0.795401\pi$
$14$	0	0
$15$	32.3154	0.556253
$16$	0	0
$17$	−13.1508	−0.187620	−0.0938098	−	0.995590i	$-0.529905\pi$
$17$	−13.1508	−0.187620	−0.0938098	+	0.995590i	$0.529905\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−24.9906	−0.259686
$22$	0	0
$23$	−96.5591	−0.875390	−0.437695	−	0.899124i	$-0.644205\pi$
$23$	−96.5591	−0.875390	−0.437695	+	0.899124i	$0.644205\pi$
$24$	0	0
$25$	−102.973	−0.823783
$26$	0	0
$27$	45.3813	0.323468
$28$	0	0
$29$	−40.5580	−0.259704	−0.129852	−	0.991533i	$-0.541450\pi$
$29$	−40.5580	−0.259704	−0.129852	+	0.991533i	$0.541450\pi$
$30$	0	0
$31$	138.828	0.804328	0.402164	−	0.915568i	$-0.368258\pi$
$31$	138.828	0.804328	0.402164	+	0.915568i	$0.368258\pi$
$32$	0	0
$33$	299.499	1.57988
$34$	0	0
$35$	−17.0343	−0.0822664
$36$	0	0
$37$	−348.686	−1.54929	−0.774644	−	0.632398i	$-0.782071\pi$
$37$	−348.686	−1.54929	−0.774644	+	0.632398i	$0.782071\pi$
$38$	0	0
$39$	516.660	2.12133
$40$	0	0
$41$	−44.3243	−0.168836	−0.0844182	−	0.996430i	$-0.526903\pi$
$41$	−44.3243	−0.168836	−0.0844182	+	0.996430i	$0.526903\pi$
$42$	0	0
$43$	−214.385	−0.760313	−0.380157	−	0.924922i	$-0.624130\pi$
$43$	−214.385	−0.760313	−0.380157	+	0.924922i	$0.624130\pi$
$44$	0	0
$45$	−95.7860	−0.317310
$46$	0	0
$47$	−334.307	−1.03752	−0.518762	−	0.854919i	$-0.673607\pi$
$47$	−334.307	−1.03752	−0.518762	+	0.854919i	$0.673607\pi$
$48$	0	0
$49$	−329.827	−0.961594
$50$	0	0
$51$	90.5487	0.248615
$52$	0	0
$53$	−311.122	−0.806338	−0.403169	−	0.915126i	$-0.632091\pi$
$53$	−311.122	−0.806338	−0.403169	+	0.915126i	$0.632091\pi$
$54$	0	0
$55$	204.147	0.500495
$56$	0	0
$57$	−130.823	−0.303999
$58$	0	0
$59$	−502.875	−1.10964	−0.554820	−	0.831971i	$-0.687213\pi$
$59$	−502.875	−1.10964	−0.554820	+	0.831971i	$0.687213\pi$
$60$	0	0
$61$	54.7193	0.114854	0.0574270	−	0.998350i	$-0.481710\pi$
$61$	54.7193	0.114854	0.0574270	+	0.998350i	$0.481710\pi$
$62$	0	0
$63$	74.0746	0.148135
$64$	0	0
$65$	352.170	0.672021
$66$	0	0
$67$	−312.959	−0.570658	−0.285329	−	0.958430i	$-0.592103\pi$
$67$	−312.959	−0.570658	−0.285329	+	0.958430i	$0.592103\pi$
$68$	0	0
$69$	664.850	1.15998
$70$	0	0
$71$	642.657	1.07422	0.537108	−	0.843513i	$-0.319517\pi$
$71$	642.657	1.07422	0.537108	+	0.843513i	$0.319517\pi$
$72$	0	0
$73$	−837.348	−1.34252	−0.671261	−	0.741221i	$-0.734247\pi$
$73$	−837.348	−1.34252	−0.671261	+	0.741221i	$0.734247\pi$
$74$	0	0
$75$	709.012	1.09160
$76$	0	0
$77$	−157.874	−0.233655
$78$	0	0
$79$	349.595	0.497880	0.248940	−	0.968519i	$-0.419918\pi$
$79$	349.595	0.497880	0.248940	+	0.968519i	$0.419918\pi$
$80$	0	0
$81$	−863.515	−1.18452
$82$	0	0
$83$	−521.399	−0.689530	−0.344765	−	0.938689i	$-0.612041\pi$
$83$	−521.399	−0.689530	−0.344765	+	0.938689i	$0.612041\pi$
$84$	0	0
$85$	61.7206	0.0787593
$86$	0	0
$87$	279.259	0.344135
$88$	0	0
$89$	63.4094	0.0755211	0.0377606	−	0.999287i	$-0.487978\pi$
$89$	63.4094	0.0755211	0.0377606	+	0.999287i	$0.487978\pi$
$90$	0	0
$91$	−272.345	−0.313731
$92$	0	0
$93$	−955.887	−1.06582
$94$	0	0
$95$	−89.1728	−0.0963046
$96$	0	0
$97$	−1229.92	−1.28742	−0.643711	−	0.765269i	$-0.722606\pi$
$97$	−1229.92	−1.28742	−0.643711	+	0.765269i	$0.722606\pi$
$98$	0	0
$99$	−887.745	−0.901229
$100$	0	0
$101$	1077.89	1.06192	0.530962	−	0.847396i	$-0.321831\pi$
$101$	1077.89	1.06192	0.530962	+	0.847396i	$0.321831\pi$
$102$	0	0
$103$	1896.77	1.81451	0.907255	−	0.420580i	$-0.138174\pi$
$103$	1896.77	1.81451	0.907255	+	0.420580i	$0.138174\pi$
$104$	0	0
$105$	117.288	0.109011
$106$	0	0
$107$	480.564	0.434186	0.217093	−	0.976151i	$-0.430343\pi$
$107$	480.564	0.434186	0.217093	+	0.976151i	$0.430343\pi$
$108$	0	0
$109$	−124.669	−0.109552	−0.0547758	−	0.998499i	$-0.517444\pi$
$109$	−124.669	−0.109552	−0.0547758	+	0.998499i	$0.517444\pi$
$110$	0	0
$111$	2400.85	2.05296
$112$	0	0
$113$	−1778.75	−1.48080	−0.740400	−	0.672166i	$-0.765364\pi$
$113$	−1778.75	−1.48080	−0.740400	+	0.672166i	$0.765364\pi$
$114$	0	0
$115$	453.181	0.367473
$116$	0	0
$117$	−1531.43	−1.21009
$118$	0	0
$119$	−47.7307	−0.0367686
$120$	0	0
$121$	561.036	0.421515
$122$	0	0
$123$	305.191	0.223725
$124$	0	0
$125$	1069.95	0.765591
$126$	0	0
$127$	1077.80	0.753068	0.376534	−	0.926403i	$-0.377116\pi$
$127$	1077.80	0.753068	0.376534	+	0.926403i	$0.377116\pi$
$128$	0	0
$129$	1476.13	1.00749
$130$	0	0
$131$	1092.79	0.728835	0.364417	−	0.931236i	$-0.381268\pi$
$131$	1092.79	0.728835	0.364417	+	0.931236i	$0.381268\pi$
$132$	0	0
$133$	68.9604	0.0449596
$134$	0	0
$135$	−212.988	−0.135786
$136$	0	0
$137$	3052.21	1.90341	0.951706	−	0.307010i	$-0.0993286\pi$
$137$	3052.21	1.90341	0.951706	+	0.307010i	$0.0993286\pi$
$138$	0	0
$139$	1428.73	0.871821	0.435911	−	0.899990i	$-0.356426\pi$
$139$	1428.73	0.871821	0.435911	+	0.899990i	$0.356426\pi$
$140$	0	0
$141$	2301.84	1.37482
$142$	0	0
$143$	3263.92	1.90869
$144$	0	0
$145$	190.351	0.109019
$146$	0	0
$147$	2271.00	1.27421
$148$	0	0
$149$	−571.022	−0.313959	−0.156980	−	0.987602i	$-0.550176\pi$
$149$	−571.022	−0.313959	−0.156980	+	0.987602i	$0.550176\pi$
$150$	0	0
$151$	−997.782	−0.537737	−0.268869	−	0.963177i	$-0.586650\pi$
$151$	−997.782	−0.537737	−0.268869	+	0.963177i	$0.586650\pi$
$152$	0	0
$153$	−268.395	−0.141820
$154$	0	0
$155$	−651.560	−0.337642
$156$	0	0
$157$	2171.23	1.10371	0.551857	−	0.833939i	$-0.313920\pi$
$157$	2171.23	1.10371	0.551857	+	0.833939i	$0.313920\pi$
$158$	0	0
$159$	2142.21	1.06848
$160$	0	0
$161$	−350.461	−0.171554
$162$	0	0
$163$	−305.488	−0.146795	−0.0733977	−	0.997303i	$-0.523384\pi$
$163$	−305.488	−0.146795	−0.0733977	+	0.997303i	$0.523384\pi$
$164$	0	0
$165$	−1405.64	−0.663206
$166$	0	0
$167$	−136.673	−0.0633300	−0.0316650	−	0.999499i	$-0.510081\pi$
$167$	−136.673	−0.0633300	−0.0316650	+	0.999499i	$0.510081\pi$
$168$	0	0
$169$	3433.52	1.56282
$170$	0	0
$171$	387.772	0.173413
$172$	0	0
$173$	1508.96	0.663144	0.331572	−	0.943430i	$-0.392421\pi$
$173$	1508.96	0.663144	0.331572	+	0.943430i	$0.392421\pi$
$174$	0	0
$175$	−373.739	−0.161440
$176$	0	0
$177$	3462.51	1.47038
$178$	0	0
$179$	−770.548	−0.321751	−0.160875	−	0.986975i	$-0.551432\pi$
$179$	−770.548	−0.321751	−0.160875	+	0.986975i	$0.551432\pi$
$180$	0	0
$181$	−2292.62	−0.941486	−0.470743	−	0.882270i	$-0.656014\pi$
$181$	−2292.62	−0.941486	−0.470743	+	0.882270i	$0.656014\pi$
$182$	0	0
$183$	−376.766	−0.152193
$184$	0	0
$185$	1636.49	0.650363
$186$	0	0
$187$	572.027	0.223694
$188$	0	0
$189$	164.711	0.0633914
$190$	0	0
$191$	−1670.39	−0.632803	−0.316402	−	0.948625i	$-0.602475\pi$
$191$	−1670.39	−0.632803	−0.316402	+	0.948625i	$0.602475\pi$
$192$	0	0
$193$	−663.849	−0.247590	−0.123795	−	0.992308i	$-0.539507\pi$
$193$	−663.849	−0.247590	−0.123795	+	0.992308i	$0.539507\pi$
$194$	0	0
$195$	−2424.84	−0.890495
$196$	0	0
$197$	2470.86	0.893611	0.446806	−	0.894631i	$-0.352562\pi$
$197$	2470.86	0.893611	0.446806	+	0.894631i	$0.352562\pi$
$198$	0	0
$199$	−4348.26	−1.54894	−0.774472	−	0.632608i	$-0.781984\pi$
$199$	−4348.26	−1.54894	−0.774472	+	0.632608i	$0.781984\pi$
$200$	0	0
$201$	2154.86	0.756179
$202$	0	0
$203$	−147.205	−0.0508954
$204$	0	0
$205$	208.027	0.0708744
$206$	0	0
$207$	−1970.68	−0.661700
$208$	0	0
$209$	−826.453	−0.273526
$210$	0	0
$211$	3910.79	1.27597	0.637986	−	0.770048i	$-0.279768\pi$
$211$	3910.79	1.27597	0.637986	+	0.770048i	$0.279768\pi$
$212$	0	0
$213$	−4424.97	−1.42344
$214$	0	0
$215$	1006.18	0.319166
$216$	0	0
$217$	503.874	0.157628
$218$	0	0
$219$	5765.50	1.77898
$220$	0	0
$221$	986.792	0.300357
$222$	0	0
$223$	−70.6739	−0.0212228	−0.0106114	−	0.999944i	$-0.503378\pi$
$223$	−70.6739	−0.0212228	−0.0106114	+	0.999944i	$0.503378\pi$
$224$	0	0
$225$	−2101.58	−0.622691
$226$	0	0
$227$	1132.37	0.331091	0.165546	−	0.986202i	$-0.447061\pi$
$227$	1132.37	0.331091	0.165546	+	0.986202i	$0.447061\pi$
$228$	0	0
$229$	−2922.00	−0.843194	−0.421597	−	0.906783i	$-0.638530\pi$
$229$	−2922.00	−0.843194	−0.421597	+	0.906783i	$0.638530\pi$
$230$	0	0
$231$	1087.03	0.309616
$232$	0	0
$233$	−926.052	−0.260376	−0.130188	−	0.991489i	$-0.541558\pi$
$233$	−926.052	−0.260376	−0.130188	+	0.991489i	$0.541558\pi$
$234$	0	0
$235$	1569.00	0.435534
$236$	0	0
$237$	−2407.11	−0.659740
$238$	0	0
$239$	4376.11	1.18438	0.592190	−	0.805798i	$-0.298263\pi$
$239$	4376.11	1.18438	0.592190	+	0.805798i	$0.298263\pi$
$240$	0	0
$241$	−3964.69	−1.05970	−0.529850	−	0.848091i	$-0.677752\pi$
$241$	−3964.69	−1.05970	−0.529850	+	0.848091i	$0.677752\pi$
$242$	0	0
$243$	4720.37	1.24614
$244$	0	0
$245$	1547.98	0.403660
$246$	0	0
$247$	−1425.70	−0.367267
$248$	0	0
$249$	3590.05	0.913697
$250$	0	0
$251$	219.854	0.0552870	0.0276435	−	0.999618i	$-0.491200\pi$
$251$	219.854	0.0552870	0.0276435	+	0.999618i	$0.491200\pi$
$252$	0	0
$253$	4200.08	1.04370
$254$	0	0
$255$	−424.973	−0.104364
$256$	0	0
$257$	−5582.44	−1.35495	−0.677477	−	0.735544i	$-0.736926\pi$
$257$	−5582.44	−1.35495	−0.677477	+	0.735544i	$0.736926\pi$
$258$	0	0
$259$	−1265.55	−0.303620
$260$	0	0
$261$	−827.751	−0.196308
$262$	0	0
$263$	−2971.76	−0.696755	−0.348378	−	0.937354i	$-0.613267\pi$
$263$	−2971.76	−0.696755	−0.348378	+	0.937354i	$0.613267\pi$
$264$	0	0
$265$	1460.19	0.338486
$266$	0	0
$267$	−436.601	−0.100073
$268$	0	0
$269$	5969.51	1.35304	0.676519	−	0.736425i	$-0.263487\pi$
$269$	5969.51	1.35304	0.676519	+	0.736425i	$0.263487\pi$
$270$	0	0
$271$	6355.51	1.42461	0.712306	−	0.701869i	$-0.247651\pi$
$271$	6355.51	1.42461	0.712306	+	0.701869i	$0.247651\pi$
$272$	0	0
$273$	1875.21	0.415726
$274$	0	0
$275$	4479.07	0.982174
$276$	0	0
$277$	8038.80	1.74370	0.871849	−	0.489774i	$-0.162921\pi$
$277$	8038.80	1.74370	0.871849	+	0.489774i	$0.162921\pi$
$278$	0	0
$279$	2833.34	0.607985
$280$	0	0
$281$	−4478.81	−0.950831	−0.475416	−	0.879761i	$-0.657702\pi$
$281$	−4478.81	−0.950831	−0.475416	+	0.879761i	$0.657702\pi$
$282$	0	0
$283$	−8812.45	−1.85105	−0.925523	−	0.378691i	$-0.876374\pi$
$283$	−8812.45	−1.85105	−0.925523	+	0.378691i	$0.876374\pi$
$284$	0	0
$285$	613.992	0.127613
$286$	0	0
$287$	−160.875	−0.0330876
$288$	0	0
$289$	−4740.06	−0.964799
$290$	0	0
$291$	8468.55	1.70596
$292$	0	0
$293$	−942.016	−0.187826	−0.0939132	−	0.995580i	$-0.529938\pi$
$293$	−942.016	−0.187826	−0.0939132	+	0.995580i	$0.529938\pi$
$294$	0	0
$295$	2360.14	0.465807
$296$	0	0
$297$	−1973.98	−0.385662
$298$	0	0
$299$	7245.48	1.40140
$300$	0	0
$301$	−778.110	−0.149002
$302$	0	0
$303$	−7421.74	−1.40715
$304$	0	0
$305$	−256.815	−0.0482136
$306$	0	0
$307$	5371.60	0.998610	0.499305	−	0.866426i	$-0.333589\pi$
$307$	5371.60	0.998610	0.499305	+	0.866426i	$0.333589\pi$
$308$	0	0
$309$	−13060.1	−2.40441
$310$	0	0
$311$	−8180.21	−1.49150	−0.745751	−	0.666225i	$-0.767909\pi$
$311$	−8180.21	−1.49150	−0.745751	+	0.666225i	$0.767909\pi$
$312$	0	0
$313$	−621.565	−0.112246	−0.0561229	−	0.998424i	$-0.517874\pi$
$313$	−621.565	−0.112246	−0.0561229	+	0.998424i	$0.517874\pi$
$314$	0	0
$315$	−347.655	−0.0621845
$316$	0	0
$317$	2641.47	0.468012	0.234006	−	0.972235i	$-0.424816\pi$
$317$	2641.47	0.468012	0.234006	+	0.972235i	$0.424816\pi$
$318$	0	0
$319$	1764.17	0.309639
$320$	0	0
$321$	−3308.89	−0.575340
$322$	0	0
$323$	−249.865	−0.0430429
$324$	0	0
$325$	7726.76	1.31878
$326$	0	0
$327$	858.398	0.145167
$328$	0	0
$329$	−1213.36	−0.203328
$330$	0	0
$331$	−7450.56	−1.23722	−0.618610	−	0.785698i	$-0.712304\pi$
$331$	−7450.56	−1.23722	−0.618610	+	0.785698i	$0.712304\pi$
$332$	0	0
$333$	−7116.36	−1.17109
$334$	0	0
$335$	1468.81	0.239552
$336$	0	0
$337$	121.084	0.0195722	0.00978612	−	0.999952i	$-0.496885\pi$
$337$	121.084	0.0195722	0.00978612	+	0.999952i	$0.496885\pi$
$338$	0	0
$339$	12247.4	1.96221
$340$	0	0
$341$	−6038.66	−0.958978
$342$	0	0
$343$	−2442.02	−0.384422
$344$	0	0
$345$	−3120.35	−0.486938
$346$	0	0
$347$	9376.43	1.45059	0.725293	−	0.688441i	$-0.241704\pi$
$347$	9376.43	1.45059	0.725293	+	0.688441i	$0.241704\pi$
$348$	0	0
$349$	4225.67	0.648124	0.324062	−	0.946036i	$-0.394951\pi$
$349$	4225.67	0.648124	0.324062	+	0.946036i	$0.394951\pi$
$350$	0	0
$351$	−3405.27	−0.517834
$352$	0	0
$353$	10833.3	1.63343	0.816714	−	0.577043i	$-0.195794\pi$
$353$	10833.3	1.63343	0.816714	+	0.577043i	$0.195794\pi$
$354$	0	0
$355$	−3016.19	−0.450937
$356$	0	0
$357$	328.646	0.0487221
$358$	0	0
$359$	−1002.24	−0.147343	−0.0736715	−	0.997283i	$-0.523472\pi$
$359$	−1002.24	−0.147343	−0.0736715	+	0.997283i	$0.523472\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−3862.97	−0.558550
$364$	0	0
$365$	3929.93	0.563567
$366$	0	0
$367$	−4868.42	−0.692451	−0.346225	−	0.938151i	$-0.612537\pi$
$367$	−4868.42	−0.692451	−0.346225	+	0.938151i	$0.612537\pi$
$368$	0	0
$369$	−904.617	−0.127622
$370$	0	0
$371$	−1129.22	−0.158021
$372$	0	0
$373$	−10466.0	−1.45284	−0.726421	−	0.687250i	$-0.758818\pi$
$373$	−10466.0	−1.45284	−0.726421	+	0.687250i	$0.758818\pi$
$374$	0	0
$375$	−7367.03	−1.01449
$376$	0	0
$377$	3043.34	0.415756
$378$	0	0
$379$	−11355.1	−1.53898	−0.769490	−	0.638659i	$-0.779490\pi$
$379$	−11355.1	−1.53898	−0.769490	+	0.638659i	$0.779490\pi$
$380$	0	0
$381$	−7421.14	−0.997891
$382$	0	0
$383$	−8723.90	−1.16389	−0.581946	−	0.813228i	$-0.697708\pi$
$383$	−8723.90	−1.16389	−0.581946	+	0.813228i	$0.697708\pi$
$384$	0	0
$385$	740.951	0.0980840
$386$	0	0
$387$	−4375.41	−0.574714
$388$	0	0
$389$	−2155.33	−0.280924	−0.140462	−	0.990086i	$-0.544859\pi$
$389$	−2155.33	−0.280924	−0.140462	+	0.990086i	$0.544859\pi$
$390$	0	0
$391$	1269.83	0.164240
$392$	0	0
$393$	−7524.31	−0.965780
$394$	0	0
$395$	−1640.75	−0.209001
$396$	0	0
$397$	4708.96	0.595305	0.297652	−	0.954674i	$-0.403796\pi$
$397$	4708.96	0.595305	0.297652	+	0.954674i	$0.403796\pi$
$398$	0	0
$399$	−474.821	−0.0595759
$400$	0	0
$401$	−1051.63	−0.130962	−0.0654812	−	0.997854i	$-0.520858\pi$
$401$	−1051.63	−0.130962	−0.0654812	+	0.997854i	$0.520858\pi$
$402$	0	0
$403$	−10417.2	−1.28763
$404$	0	0
$405$	4052.74	0.497240
$406$	0	0
$407$	15167.0	1.84717
$408$	0	0
$409$	3346.05	0.404527	0.202263	−	0.979331i	$-0.435170\pi$
$409$	3346.05	0.404527	0.202263	+	0.979331i	$0.435170\pi$
$410$	0	0
$411$	−21015.7	−2.52221
$412$	0	0
$413$	−1825.18	−0.217461
$414$	0	0
$415$	2447.08	0.289452
$416$	0	0
$417$	−9837.40	−1.15525
$418$	0	0
$419$	15743.9	1.83565	0.917827	−	0.396982i	$-0.129942\pi$
$419$	15743.9	1.83565	0.917827	+	0.396982i	$0.129942\pi$
$420$	0	0
$421$	−3755.30	−0.434732	−0.217366	−	0.976090i	$-0.569746\pi$
$421$	−3755.30	−0.434732	−0.217366	+	0.976090i	$0.569746\pi$
$422$	0	0
$423$	−6822.89	−0.784255
$424$	0	0
$425$	1354.17	0.154558
$426$	0	0
$427$	198.603	0.0225084
$428$	0	0
$429$	−22473.4	−2.52920
$430$	0	0
$431$	9670.09	1.08072	0.540362	−	0.841433i	$-0.318287\pi$
$431$	9670.09	1.08072	0.540362	+	0.841433i	$0.318287\pi$
$432$	0	0
$433$	−10656.7	−1.18274	−0.591370	−	0.806401i	$-0.701413\pi$
$433$	−10656.7	−1.18274	−0.591370	+	0.806401i	$0.701413\pi$
$434$	0	0
$435$	−1310.65	−0.144461
$436$	0	0
$437$	−1834.62	−0.200828
$438$	0	0
$439$	5113.47	0.555929	0.277964	−	0.960591i	$-0.410340\pi$
$439$	5113.47	0.555929	0.277964	+	0.960591i	$0.410340\pi$
$440$	0	0
$441$	−6731.46	−0.726861
$442$	0	0
$443$	−2552.54	−0.273758	−0.136879	−	0.990588i	$-0.543707\pi$
$443$	−2552.54	−0.273758	−0.136879	+	0.990588i	$0.543707\pi$
$444$	0	0
$445$	−297.600	−0.0317024
$446$	0	0
$447$	3931.73	0.416028
$448$	0	0
$449$	4504.62	0.473466	0.236733	−	0.971575i	$-0.423923\pi$
$449$	4504.62	0.473466	0.236733	+	0.971575i	$0.423923\pi$
$450$	0	0
$451$	1928.00	0.201299
$452$	0	0
$453$	6870.15	0.712556
$454$	0	0
$455$	1278.20	0.131699
$456$	0	0
$457$	9278.63	0.949751	0.474875	−	0.880053i	$-0.342493\pi$
$457$	9278.63	0.949751	0.474875	+	0.880053i	$0.342493\pi$
$458$	0	0
$459$	−596.800	−0.0606890
$460$	0	0
$461$	−3861.47	−0.390123	−0.195062	−	0.980791i	$-0.562491\pi$
$461$	−3861.47	−0.390123	−0.195062	+	0.980791i	$0.562491\pi$
$462$	0	0
$463$	−154.016	−0.0154595	−0.00772975	−	0.999970i	$-0.502460\pi$
$463$	−154.016	−0.0154595	−0.00772975	+	0.999970i	$0.502460\pi$
$464$	0	0
$465$	4486.27	0.447410
$466$	0	0
$467$	−16010.0	−1.58641	−0.793205	−	0.608954i	$-0.791589\pi$
$467$	−16010.0	−1.58641	−0.793205	+	0.608954i	$0.791589\pi$
$468$	0	0
$469$	−1135.88	−0.111834
$470$	0	0
$471$	−14949.8	−1.46253
$472$	0	0
$473$	9325.24	0.906501
$474$	0	0
$475$	−1956.49	−0.188989
$476$	0	0
$477$	−6349.71	−0.609504
$478$	0	0
$479$	10467.1	0.998441	0.499220	−	0.866475i	$-0.333620\pi$
$479$	10467.1	0.998441	0.499220	+	0.866475i	$0.333620\pi$
$480$	0	0
$481$	26164.3	2.48023
$482$	0	0
$483$	2413.07	0.227326
$484$	0	0
$485$	5772.41	0.540436
$486$	0	0
$487$	5868.52	0.546054	0.273027	−	0.962006i	$-0.411975\pi$
$487$	5868.52	0.546054	0.273027	+	0.962006i	$0.411975\pi$
$488$	0	0
$489$	2103.41	0.194519
$490$	0	0
$491$	20259.1	1.86208	0.931041	−	0.364915i	$-0.118902\pi$
$491$	20259.1	1.86208	0.931041	+	0.364915i	$0.118902\pi$
$492$	0	0
$493$	533.369	0.0487257
$494$	0	0
$495$	4166.46	0.378320
$496$	0	0
$497$	2332.52	0.210519
$498$	0	0
$499$	−6557.19	−0.588256	−0.294128	−	0.955766i	$-0.595029\pi$
$499$	−6557.19	−0.588256	−0.294128	+	0.955766i	$0.595029\pi$
$500$	0	0
$501$	941.054	0.0839186
$502$	0	0
$503$	−20542.5	−1.82097	−0.910484	−	0.413545i	$-0.864291\pi$
$503$	−20542.5	−1.82097	−0.910484	+	0.413545i	$0.864291\pi$
$504$	0	0
$505$	−5058.87	−0.445776
$506$	0	0
$507$	−23641.2	−2.07090
$508$	0	0
$509$	5464.15	0.475823	0.237912	−	0.971287i	$-0.423537\pi$
$509$	5464.15	0.475823	0.237912	+	0.971287i	$0.423537\pi$
$510$	0	0
$511$	−3039.15	−0.263100
$512$	0	0
$513$	862.245	0.0742087
$514$	0	0
$515$	−8902.13	−0.761699
$516$	0	0
$517$	14541.5	1.23701
$518$	0	0
$519$	−10389.8	−0.878733
$520$	0	0
$521$	−12113.9	−1.01865	−0.509327	−	0.860573i	$-0.670106\pi$
$521$	−12113.9	−1.01865	−0.509327	+	0.860573i	$0.670106\pi$
$522$	0	0
$523$	−12706.6	−1.06237	−0.531185	−	0.847256i	$-0.678253\pi$
$523$	−12706.6	−1.06237	−0.531185	+	0.847256i	$0.678253\pi$
$524$	0	0
$525$	2573.35	0.213925
$526$	0	0
$527$	−1825.69	−0.150908
$528$	0	0
$529$	−2843.34	−0.233693
$530$	0	0
$531$	−10263.2	−0.838767
$532$	0	0
$533$	3325.95	0.270287
$534$	0	0
$535$	−2255.43	−0.182263
$536$	0	0
$537$	5305.55	0.426352
$538$	0	0
$539$	14346.7	1.14648
$540$	0	0
$541$	20011.9	1.59035	0.795173	−	0.606383i	$-0.207380\pi$
$541$	20011.9	1.59035	0.795173	+	0.606383i	$0.207380\pi$
$542$	0	0
$543$	15785.6	1.24756
$544$	0	0
$545$	585.109	0.0459877
$546$	0	0
$547$	−14732.5	−1.15158	−0.575790	−	0.817597i	$-0.695306\pi$
$547$	−14732.5	−1.15158	−0.575790	+	0.817597i	$0.695306\pi$
$548$	0	0
$549$	1116.77	0.0868172
$550$	0	0
$551$	−770.602	−0.0595803
$552$	0	0
$553$	1268.85	0.0975715
$554$	0	0
$555$	−11267.9	−0.861796
$556$	0	0
$557$	2202.91	0.167577	0.0837883	−	0.996484i	$-0.473298\pi$
$557$	2202.91	0.167577	0.0837883	+	0.996484i	$0.473298\pi$
$558$	0	0
$559$	16086.8	1.21717
$560$	0	0
$561$	−3938.65	−0.296417
$562$	0	0
$563$	−14661.2	−1.09751	−0.548755	−	0.835983i	$-0.684898\pi$
$563$	−14661.2	−1.09751	−0.548755	+	0.835983i	$0.684898\pi$
$564$	0	0
$565$	8348.20	0.621613
$566$	0	0
$567$	−3134.12	−0.232135
$568$	0	0
$569$	2252.50	0.165957	0.0829785	−	0.996551i	$-0.473557\pi$
$569$	2252.50	0.165957	0.0829785	+	0.996551i	$0.473557\pi$
$570$	0	0
$571$	−13749.4	−1.00770	−0.503848	−	0.863793i	$-0.668083\pi$
$571$	−13749.4	−1.00770	−0.503848	+	0.863793i	$0.668083\pi$
$572$	0	0
$573$	11501.4	0.838528
$574$	0	0
$575$	9942.97	0.721131
$576$	0	0
$577$	−16355.8	−1.18007	−0.590037	−	0.807376i	$-0.700887\pi$
$577$	−16355.8	−1.18007	−0.590037	+	0.807376i	$0.700887\pi$
$578$	0	0
$579$	4570.88	0.328082
$580$	0	0
$581$	−1892.41	−0.135130
$582$	0	0
$583$	13533.0	0.961375
$584$	0	0
$585$	7187.47	0.507975
$586$	0	0
$587$	−12704.1	−0.893279	−0.446640	−	0.894714i	$-0.647379\pi$
$587$	−12704.1	−0.893279	−0.446640	+	0.894714i	$0.647379\pi$
$588$	0	0
$589$	2637.72	0.184525
$590$	0	0
$591$	−17012.9	−1.18412
$592$	0	0
$593$	7442.67	0.515403	0.257701	−	0.966225i	$-0.417035\pi$
$593$	7442.67	0.515403	0.257701	+	0.966225i	$0.417035\pi$
$594$	0	0
$595$	224.015	0.0154348
$596$	0	0
$597$	29939.6	2.05251
$598$	0	0
$599$	−17522.7	−1.19525	−0.597627	−	0.801774i	$-0.703890\pi$
$599$	−17522.7	−1.19525	−0.597627	+	0.801774i	$0.703890\pi$
$600$	0	0
$601$	14983.9	1.01698	0.508490	−	0.861068i	$-0.330204\pi$
$601$	14983.9	1.01698	0.508490	+	0.861068i	$0.330204\pi$
$602$	0	0
$603$	−6387.21	−0.431355
$604$	0	0
$605$	−2633.11	−0.176944
$606$	0	0
$607$	−18937.9	−1.26634	−0.633169	−	0.774013i	$-0.718246\pi$
$607$	−18937.9	−1.26634	−0.633169	+	0.774013i	$0.718246\pi$
$608$	0	0
$609$	1013.57	0.0674415
$610$	0	0
$611$	25085.3	1.66095
$612$	0	0
$613$	−7209.14	−0.474999	−0.237500	−	0.971388i	$-0.576328\pi$
$613$	−7209.14	−0.474999	−0.237500	+	0.971388i	$0.576328\pi$
$614$	0	0
$615$	−1432.36	−0.0939158
$616$	0	0
$617$	−25907.8	−1.69045	−0.845226	−	0.534408i	$-0.820534\pi$
$617$	−25907.8	−1.69045	−0.845226	+	0.534408i	$0.820534\pi$
$618$	0	0
$619$	−26596.0	−1.72695	−0.863475	−	0.504391i	$-0.831717\pi$
$619$	−26596.0	−1.72695	−0.863475	+	0.504391i	$0.831717\pi$
$620$	0	0
$621$	−4381.98	−0.283161
$622$	0	0
$623$	230.144	0.0148002
$624$	0	0
$625$	7850.03	0.502402
$626$	0	0
$627$	5690.48	0.362450
$628$	0	0
$629$	4585.50	0.290677
$630$	0	0
$631$	−231.639	−0.0146140	−0.00730699	−	0.999973i	$-0.502326\pi$
$631$	−231.639	−0.0146140	−0.00730699	+	0.999973i	$0.502326\pi$
$632$	0	0
$633$	−26927.5	−1.69079
$634$	0	0
$635$	−5058.46	−0.316124
$636$	0	0
$637$	24749.1	1.53940
$638$	0	0
$639$	13116.0	0.811991
$640$	0	0
$641$	9194.84	0.566574	0.283287	−	0.959035i	$-0.408575\pi$
$641$	9194.84	0.566574	0.283287	+	0.959035i	$0.408575\pi$
$642$	0	0
$643$	12339.1	0.756775	0.378387	−	0.925647i	$-0.376479\pi$
$643$	12339.1	0.756775	0.378387	+	0.925647i	$0.376479\pi$
$644$	0	0
$645$	−6927.95	−0.422927
$646$	0	0
$647$	−10852.4	−0.659428	−0.329714	−	0.944081i	$-0.606952\pi$
$647$	−10852.4	−0.659428	−0.329714	+	0.944081i	$0.606952\pi$
$648$	0	0
$649$	21873.8	1.32299
$650$	0	0
$651$	−3469.38	−0.208872
$652$	0	0
$653$	−23310.7	−1.39696	−0.698482	−	0.715627i	$-0.746141\pi$
$653$	−23310.7	−1.39696	−0.698482	+	0.715627i	$0.746141\pi$
$654$	0	0
$655$	−5128.79	−0.305952
$656$	0	0
$657$	−17089.5	−1.01480
$658$	0	0
$659$	−21318.0	−1.26014	−0.630069	−	0.776539i	$-0.716973\pi$
$659$	−21318.0	−1.26014	−0.630069	+	0.776539i	$0.716973\pi$
$660$	0	0
$661$	−26645.4	−1.56791	−0.783954	−	0.620818i	$-0.786800\pi$
$661$	−26645.4	−1.56791	−0.783954	+	0.620818i	$0.786800\pi$
$662$	0	0
$663$	−6794.49	−0.398003
$664$	0	0
$665$	−323.652	−0.0188732
$666$	0	0
$667$	3916.24	0.227343
$668$	0	0
$669$	486.620	0.0281223
$670$	0	0
$671$	−2380.16	−0.136937
$672$	0	0
$673$	12426.1	0.711728	0.355864	−	0.934538i	$-0.384187\pi$
$673$	12426.1	0.711728	0.355864	+	0.934538i	$0.384187\pi$
$674$	0	0
$675$	−4673.05	−0.266468
$676$	0	0
$677$	28485.7	1.61712	0.808562	−	0.588411i	$-0.200246\pi$
$677$	28485.7	1.61712	0.808562	+	0.588411i	$0.200246\pi$
$678$	0	0
$679$	−4464.00	−0.252301
$680$	0	0
$681$	−7796.81	−0.438729
$682$	0	0
$683$	1899.14	0.106396	0.0531981	−	0.998584i	$-0.483059\pi$
$683$	1899.14	0.106396	0.0531981	+	0.998584i	$0.483059\pi$
$684$	0	0
$685$	−14324.9	−0.799018
$686$	0	0
$687$	20119.2	1.11732
$688$	0	0
$689$	23345.6	1.29085
$690$	0	0
$691$	29568.6	1.62785	0.813923	−	0.580973i	$-0.197328\pi$
$691$	29568.6	1.62785	0.813923	+	0.580973i	$0.197328\pi$
$692$	0	0
$693$	−3222.06	−0.176618
$694$	0	0
$695$	−6705.46	−0.365975
$696$	0	0
$697$	582.899	0.0316770
$698$	0	0
$699$	6376.26	0.345025
$700$	0	0
$701$	6906.35	0.372110	0.186055	−	0.982539i	$-0.440430\pi$
$701$	6906.35	0.372110	0.186055	+	0.982539i	$0.440430\pi$
$702$	0	0
$703$	−6625.04	−0.355431
$704$	0	0
$705$	−10803.2	−0.577126
$706$	0	0
$707$	3912.20	0.208110
$708$	0	0
$709$	−14410.0	−0.763301	−0.381650	−	0.924307i	$-0.624644\pi$
$709$	−14410.0	−0.763301	−0.381650	+	0.924307i	$0.624644\pi$
$710$	0	0
$711$	7134.90	0.376343
$712$	0	0
$713$	−13405.1	−0.704100
$714$	0	0
$715$	−15318.6	−0.801233
$716$	0	0
$717$	−30131.4	−1.56942
$718$	0	0
$719$	−11170.9	−0.579419	−0.289710	−	0.957115i	$-0.593559\pi$
$719$	−11170.9	−0.579419	−0.289710	+	0.957115i	$0.593559\pi$
$720$	0	0
$721$	6884.32	0.355597
$722$	0	0
$723$	27298.6	1.40421
$724$	0	0
$725$	4176.37	0.213940
$726$	0	0
$727$	29676.9	1.51397	0.756985	−	0.653433i	$-0.226672\pi$
$727$	29676.9	1.51397	0.756985	+	0.653433i	$0.226672\pi$
$728$	0	0
$729$	−9186.85	−0.466740
$730$	0	0
$731$	2819.34	0.142650
$732$	0	0
$733$	252.315	0.0127142	0.00635708	−	0.999980i	$-0.497976\pi$
$733$	252.315	0.0127142	0.00635708	+	0.999980i	$0.497976\pi$
$734$	0	0
$735$	−10658.5	−0.534890
$736$	0	0
$737$	13613.0	0.680380
$738$	0	0
$739$	−1702.08	−0.0847252	−0.0423626	−	0.999102i	$-0.513488\pi$
$739$	−1702.08	−0.0847252	−0.0423626	+	0.999102i	$0.513488\pi$
$740$	0	0
$741$	9816.54	0.486666
$742$	0	0
$743$	−11210.7	−0.553539	−0.276769	−	0.960936i	$-0.589264\pi$
$743$	−11210.7	−0.553539	−0.276769	+	0.960936i	$0.589264\pi$
$744$	0	0
$745$	2679.98	0.131794
$746$	0	0
$747$	−10641.3	−0.521210
$748$	0	0
$749$	1744.20	0.0850892
$750$	0	0
$751$	37318.2	1.81326	0.906631	−	0.421925i	$-0.138646\pi$
$751$	37318.2	1.81326	0.906631	+	0.421925i	$0.138646\pi$
$752$	0	0
$753$	−1513.79	−0.0732609
$754$	0	0
$755$	4682.90	0.225732
$756$	0	0
$757$	13356.3	0.641274	0.320637	−	0.947202i	$-0.396103\pi$
$757$	13356.3	0.641274	0.320637	+	0.947202i	$0.396103\pi$
$758$	0	0
$759$	−28919.4	−1.38301
$760$	0	0
$761$	37538.5	1.78814	0.894068	−	0.447932i	$-0.147839\pi$
$761$	37538.5	1.78814	0.894068	+	0.447932i	$0.147839\pi$
$762$	0	0
$763$	−452.485	−0.0214693
$764$	0	0
$765$	1259.66	0.0595335
$766$	0	0
$767$	37734.1	1.77640
$768$	0	0
$769$	422.829	0.0198278	0.00991391	−	0.999951i	$-0.496844\pi$
$769$	422.829	0.0198278	0.00991391	+	0.999951i	$0.496844\pi$
$770$	0	0
$771$	38437.5	1.79545
$772$	0	0
$773$	−23033.6	−1.07175	−0.535873	−	0.844298i	$-0.680018\pi$
$773$	−23033.6	−1.07175	−0.535873	+	0.844298i	$0.680018\pi$
$774$	0	0
$775$	−14295.5	−0.662592
$776$	0	0
$777$	8713.88	0.402328
$778$	0	0
$779$	−842.161	−0.0387337
$780$	0	0
$781$	−27954.0	−1.28076
$782$	0	0
$783$	−1840.58	−0.0840061
$784$	0	0
$785$	−10190.2	−0.463319
$786$	0	0
$787$	−37994.2	−1.72090	−0.860448	−	0.509538i	$-0.829816\pi$
$787$	−37994.2	−1.72090	−0.860448	+	0.509538i	$0.829816\pi$
$788$	0	0
$789$	20461.8	0.923271
$790$	0	0
$791$	−6455.95	−0.290199
$792$	0	0
$793$	−4105.96	−0.183868
$794$	0	0
$795$	−10054.0	−0.448528
$796$	0	0
$797$	23765.7	1.05624	0.528122	−	0.849169i	$-0.322897\pi$
$797$	23765.7	1.05624	0.528122	+	0.849169i	$0.322897\pi$
$798$	0	0
$799$	4396.39	0.194660
$800$	0	0
$801$	1294.13	0.0570858
$802$	0	0
$803$	36422.6	1.60065
$804$	0	0
$805$	1644.82	0.0720152
$806$	0	0
$807$	−41102.6	−1.79291
$808$	0	0
$809$	−7019.58	−0.305062	−0.152531	−	0.988299i	$-0.548742\pi$
$809$	−7019.58	−0.305062	−0.152531	+	0.988299i	$0.548742\pi$
$810$	0	0
$811$	−35061.1	−1.51808	−0.759039	−	0.651045i	$-0.774331\pi$
$811$	−35061.1	−1.51808	−0.759039	+	0.651045i	$0.774331\pi$
$812$	0	0
$813$	−43760.4	−1.88775
$814$	0	0
$815$	1433.75	0.0616220
$816$	0	0
$817$	−4073.32	−0.174428
$818$	0	0
$819$	−5558.32	−0.237147
$820$	0	0
$821$	306.783	0.0130412	0.00652059	−	0.999979i	$-0.497924\pi$
$821$	306.783	0.0130412	0.00652059	+	0.999979i	$0.497924\pi$
$822$	0	0
$823$	−23239.1	−0.984281	−0.492140	−	0.870516i	$-0.663785\pi$
$823$	−23239.1	−0.984281	−0.492140	+	0.870516i	$0.663785\pi$
$824$	0	0
$825$	−30840.3	−1.30148
$826$	0	0
$827$	24374.9	1.02491	0.512455	−	0.858714i	$-0.328736\pi$
$827$	24374.9	1.02491	0.512455	+	0.858714i	$0.328736\pi$
$828$	0	0
$829$	−27129.6	−1.13661	−0.568306	−	0.822817i	$-0.692401\pi$
$829$	−27129.6	−1.13661	−0.568306	+	0.822817i	$0.692401\pi$
$830$	0	0
$831$	−55350.5	−2.31058
$832$	0	0
$833$	4337.48	0.180414
$834$	0	0
$835$	641.450	0.0265848
$836$	0	0
$837$	6300.18	0.260174
$838$	0	0
$839$	−4376.25	−0.180077	−0.0900386	−	0.995938i	$-0.528699\pi$
$839$	−4376.25	−0.180077	−0.0900386	+	0.995938i	$0.528699\pi$
$840$	0	0
$841$	−22744.0	−0.932554
$842$	0	0
$843$	30838.5	1.25995
$844$	0	0
$845$	−16114.6	−0.656044
$846$	0	0
$847$	2036.28	0.0826061
$848$	0	0
$849$	60677.5	2.45282
$850$	0	0
$851$	33668.8	1.35623
$852$	0	0
$853$	−46278.7	−1.85762	−0.928812	−	0.370550i	$-0.879169\pi$
$853$	−46278.7	−1.85762	−0.928812	+	0.370550i	$0.879169\pi$
$854$	0	0
$855$	−1819.93	−0.0727958
$856$	0	0
$857$	−5108.69	−0.203628	−0.101814	−	0.994803i	$-0.532465\pi$
$857$	−5108.69	−0.203628	−0.101814	+	0.994803i	$0.532465\pi$
$858$	0	0
$859$	−12440.5	−0.494137	−0.247069	−	0.968998i	$-0.579467\pi$
$859$	−12440.5	−0.494137	−0.247069	+	0.968998i	$0.579467\pi$
$860$	0	0
$861$	1107.69	0.0438444
$862$	0	0
$863$	33694.7	1.32906	0.664531	−	0.747260i	$-0.268631\pi$
$863$	33694.7	1.32906	0.664531	+	0.747260i	$0.268631\pi$
$864$	0	0
$865$	−7082.00	−0.278376
$866$	0	0
$867$	32637.3	1.27846
$868$	0	0
$869$	−15206.5	−0.593608
$870$	0	0
$871$	23483.5	0.913555
$872$	0	0
$873$	−25101.6	−0.973151
$874$	0	0
$875$	3883.36	0.150036
$876$	0	0
$877$	−6266.75	−0.241292	−0.120646	−	0.992696i	$-0.538497\pi$
$877$	−6266.75	−0.241292	−0.120646	+	0.992696i	$0.538497\pi$
$878$	0	0
$879$	6486.18	0.248889
$880$	0	0
$881$	−18875.2	−0.721818	−0.360909	−	0.932601i	$-0.617534\pi$
$881$	−18875.2	−0.721818	−0.360909	+	0.932601i	$0.617534\pi$
$882$	0	0
$883$	24369.5	0.928765	0.464383	−	0.885635i	$-0.346276\pi$
$883$	24369.5	0.928765	0.464383	+	0.885635i	$0.346276\pi$
$884$	0	0
$885$	−16250.6	−0.617241
$886$	0	0
$887$	−46836.8	−1.77297	−0.886486	−	0.462754i	$-0.846861\pi$
$887$	−46836.8	−1.77297	−0.886486	+	0.462754i	$0.846861\pi$
$888$	0	0
$889$	3911.88	0.147582
$890$	0	0
$891$	37560.8	1.41227
$892$	0	0
$893$	−6351.82	−0.238024
$894$	0	0
$895$	3616.41	0.135065
$896$	0	0
$897$	−49888.2	−1.85699
$898$	0	0
$899$	−5630.57	−0.208888
$900$	0	0
$901$	4091.50	0.151285
$902$	0	0
$903$	5357.62	0.197442
$904$	0	0
$905$	10760.0	0.395219
$906$	0	0
$907$	19015.8	0.696150	0.348075	−	0.937467i	$-0.386836\pi$
$907$	19015.8	0.696150	0.348075	+	0.937467i	$0.386836\pi$
$908$	0	0
$909$	21998.8	0.802699
$910$	0	0
$911$	−30625.3	−1.11379	−0.556894	−	0.830583i	$-0.688007\pi$
$911$	−30625.3	−1.11379	−0.556894	+	0.830583i	$0.688007\pi$
$912$	0	0
$913$	22679.6	0.822108
$914$	0	0
$915$	1768.28	0.0638879
$916$	0	0
$917$	3966.27	0.142833
$918$	0	0
$919$	48642.7	1.74600	0.873001	−	0.487719i	$-0.162171\pi$
$919$	48642.7	1.74600	0.873001	+	0.487719i	$0.162171\pi$
$920$	0	0
$921$	−36985.7	−1.32326
$922$	0	0
$923$	−48222.9	−1.71969
$924$	0	0
$925$	35905.2	1.27628
$926$	0	0
$927$	38711.4	1.37157
$928$	0	0
$929$	42194.6	1.49016	0.745080	−	0.666975i	$-0.232411\pi$
$929$	42194.6	1.49016	0.745080	+	0.666975i	$0.232411\pi$
$930$	0	0
$931$	−6266.71	−0.220605
$932$	0	0
$933$	56324.2	1.97639
$934$	0	0
$935$	−2684.70	−0.0939026
$936$	0	0
$937$	−28438.1	−0.991498	−0.495749	−	0.868466i	$-0.665106\pi$
$937$	−28438.1	−0.991498	−0.495749	+	0.868466i	$0.665106\pi$
$938$	0	0
$939$	4279.74	0.148737
$940$	0	0
$941$	−29883.2	−1.03524	−0.517622	−	0.855610i	$-0.673182\pi$
$941$	−29883.2	−1.03524	−0.517622	+	0.855610i	$0.673182\pi$
$942$	0	0
$943$	4279.91	0.147798
$944$	0	0
$945$	−773.040	−0.0266106
$946$	0	0
$947$	46897.4	1.60925	0.804626	−	0.593782i	$-0.202366\pi$
$947$	46897.4	1.60925	0.804626	+	0.593782i	$0.202366\pi$
$948$	0	0
$949$	62831.9	2.14922
$950$	0	0
$951$	−18187.7	−0.620163
$952$	0	0
$953$	−20401.1	−0.693449	−0.346724	−	0.937967i	$-0.612706\pi$
$953$	−20401.1	−0.693449	−0.346724	+	0.937967i	$0.612706\pi$
$954$	0	0
$955$	7839.66	0.265639
$956$	0	0
$957$	−12147.1	−0.410302
$958$	0	0
$959$	11078.0	0.373020
$960$	0	0
$961$	−10517.9	−0.353057
$962$	0	0
$963$	9807.87	0.328197
$964$	0	0
$965$	3115.65	0.103934
$966$	0	0
$967$	−23598.5	−0.784775	−0.392388	−	0.919800i	$-0.628351\pi$
$967$	−23598.5	−0.784775	−0.392388	+	0.919800i	$0.628351\pi$
$968$	0	0
$969$	1720.43	0.0570362
$970$	0	0
$971$	17149.1	0.566776	0.283388	−	0.959005i	$-0.408542\pi$
$971$	17149.1	0.566776	0.283388	+	0.959005i	$0.408542\pi$
$972$	0	0
$973$	5185.56	0.170854
$974$	0	0
$975$	−53202.0	−1.74752
$976$	0	0
$977$	−56653.2	−1.85517	−0.927583	−	0.373618i	$-0.878117\pi$
$977$	−56653.2	−1.85517	−0.927583	+	0.373618i	$0.878117\pi$
$978$	0	0
$979$	−2758.15	−0.0900418
$980$	0	0
$981$	−2544.38	−0.0828091
$982$	0	0
$983$	−7895.00	−0.256166	−0.128083	−	0.991763i	$-0.540882\pi$
$983$	−7895.00	−0.256166	−0.128083	+	0.991763i	$0.540882\pi$
$984$	0	0
$985$	−11596.5	−0.375122
$986$	0	0
$987$	8354.52	0.269430
$988$	0	0
$989$	20700.9	0.665570
$990$	0	0
$991$	48458.9	1.55333	0.776664	−	0.629915i	$-0.216910\pi$
$991$	48458.9	1.55333	0.776664	+	0.629915i	$0.216910\pi$
$992$	0	0
$993$	51300.3	1.63944
$994$	0	0
$995$	20407.7	0.650219
$996$	0	0
$997$	−40325.2	−1.28095	−0.640477	−	0.767977i	$-0.721263\pi$
$997$	−40325.2	−1.28095	−0.640477	+	0.767977i	$0.721263\pi$
$998$	0	0
$999$	−15823.8	−0.501145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.bf.1.2		7
4.3	odd	2		1216.4.a.bg.1.6		7
8.3	odd	2		608.4.a.j.1.2	✓	7
8.5	even	2		608.4.a.k.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.j.1.2	✓	7	8.3	odd	2
608.4.a.k.1.6	yes	7	8.5	even	2
1216.4.a.bf.1.2		7	1.1	even	1	trivial
1216.4.a.bg.1.6		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-6.88542 q^{3} -4.69330 q^{5} +3.62949 q^{7} +20.4091 q^{9} +O(q^{10})\)	\(q-6.88542 q^{3} -4.69330 q^{5} +3.62949 q^{7} +20.4091 q^{9} -43.4975 q^{11} -75.0368 q^{13} +32.3154 q^{15} -13.1508 q^{17} +19.0000 q^{19} -24.9906 q^{21} -96.5591 q^{23} -102.973 q^{25} +45.3813 q^{27} -40.5580 q^{29} +138.828 q^{31} +299.499 q^{33} -17.0343 q^{35} -348.686 q^{37} +516.660 q^{39} -44.3243 q^{41} -214.385 q^{43} -95.7860 q^{45} -334.307 q^{47} -329.827 q^{49} +90.5487 q^{51} -311.122 q^{53} +204.147 q^{55} -130.823 q^{57} -502.875 q^{59} +54.7193 q^{61} +74.0746 q^{63} +352.170 q^{65} -312.959 q^{67} +664.850 q^{69} +642.657 q^{71} -837.348 q^{73} +709.012 q^{75} -157.874 q^{77} +349.595 q^{79} -863.515 q^{81} -521.399 q^{83} +61.7206 q^{85} +279.259 q^{87} +63.4094 q^{89} -272.345 q^{91} -955.887 q^{93} -89.1728 q^{95} -1229.92 q^{97} -887.745 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} + 17 q^{5} + 42 q^{7} + 86 q^{9}+O(q^{10}) \) 7 * q - 3 * q^3 + 17 * q^5 + 42 * q^7 + 86 * q^9	\( 7 q - 3 q^{3} + 17 q^{5} + 42 q^{7} + 86 q^{9} + 33 q^{11} + 35 q^{13} + 120 q^{15} + 66 q^{17} + 133 q^{19} - 33 q^{21} + 389 q^{23} + 44 q^{25} + 39 q^{27} - 233 q^{29} + 158 q^{31} - 206 q^{33} - 123 q^{35} + 436 q^{37} + 807 q^{39} - 94 q^{41} - 645 q^{43} - 103 q^{45} + 1451 q^{47} + 93 q^{49} - 1741 q^{51} - 3 q^{53} + 1971 q^{55} - 57 q^{57} - 297 q^{59} - 93 q^{61} + 2999 q^{63} - 788 q^{65} - 1641 q^{67} - 945 q^{69} + 2392 q^{71} + 324 q^{73} - 1909 q^{75} + 711 q^{77} + 2492 q^{79} + 143 q^{81} - 310 q^{83} - 2353 q^{85} + 4795 q^{87} - 440 q^{89} + 107 q^{91} - 900 q^{93} + 323 q^{95} - 532 q^{97} + 1591 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 + 17 * q^5 + 42 * q^7 + 86 * q^9 + 33 * q^11 + 35 * q^13 + 120 * q^15 + 66 * q^17 + 133 * q^19 - 33 * q^21 + 389 * q^23 + 44 * q^25 + 39 * q^27 - 233 * q^29 + 158 * q^31 - 206 * q^33 - 123 * q^35 + 436 * q^37 + 807 * q^39 - 94 * q^41 - 645 * q^43 - 103 * q^45 + 1451 * q^47 + 93 * q^49 - 1741 * q^51 - 3 * q^53 + 1971 * q^55 - 57 * q^57 - 297 * q^59 - 93 * q^61 + 2999 * q^63 - 788 * q^65 - 1641 * q^67 - 945 * q^69 + 2392 * q^71 + 324 * q^73 - 1909 * q^75 + 711 * q^77 + 2492 * q^79 + 143 * q^81 - 310 * q^83 - 2353 * q^85 + 4795 * q^87 - 440 * q^89 + 107 * q^91 - 900 * q^93 + 323 * q^95 - 532 * q^97 + 1591 * q^99

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