LMFDB - Embedded newform 1216.4.a.p.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:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.77200$ 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)

$f(q)$	$=$	$q+8.77200 q^{3} +17.3160 q^{5} +26.0880 q^{7} +49.9480 q^{9} +O(q^{10})$	$q+8.77200 q^{3} +17.3160 q^{5} +26.0880 q^{7} +49.9480 q^{9} -4.22800 q^{11} -64.0360 q^{13} +151.896 q^{15} -48.5440 q^{17} +19.0000 q^{19} +228.844 q^{21} -92.0360 q^{23} +174.844 q^{25} +201.300 q^{27} +88.2120 q^{29} +81.9681 q^{31} -37.0880 q^{33} +451.740 q^{35} +23.6161 q^{37} -561.724 q^{39} +17.7200 q^{41} +368.404 q^{43} +864.900 q^{45} +497.812 q^{47} +337.584 q^{49} -425.828 q^{51} +536.876 q^{53} -73.2120 q^{55} +166.668 q^{57} -36.7000 q^{59} -630.692 q^{61} +1303.04 q^{63} -1108.85 q^{65} +282.556 q^{67} -807.340 q^{69} -595.552 q^{71} -597.048 q^{73} +1533.73 q^{75} -110.300 q^{77} -427.224 q^{79} +417.208 q^{81} +493.768 q^{83} -840.588 q^{85} +773.796 q^{87} -921.136 q^{89} -1670.57 q^{91} +719.024 q^{93} +329.004 q^{95} +1082.74 q^{97} -211.180 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{3} + 9 q^{5} + 18 q^{7} + 23 q^{9}+O(q^{10}) $ 2 * q + 9 * q^3 + 9 * q^5 + 18 * q^7 + 23 * q^9	$ 2 q + 9 q^{3} + 9 q^{5} + 18 q^{7} + 23 q^{9} - 17 q^{11} - 17 q^{13} + 150 q^{15} - 80 q^{17} + 38 q^{19} + 227 q^{21} - 73 q^{23} + 119 q^{25} + 189 q^{27} - 3 q^{29} - 212 q^{31} - 40 q^{33} + 519 q^{35} - 192 q^{37} - 551 q^{39} - 50 q^{41} + 677 q^{43} + 1089 q^{45} + 389 q^{47} + 60 q^{49} - 433 q^{51} + 1219 q^{53} + 33 q^{55} + 171 q^{57} - 287 q^{59} - 313 q^{61} + 1521 q^{63} - 1500 q^{65} + 1223 q^{67} - 803 q^{69} - 200 q^{71} + 378 q^{73} + 1521 q^{75} - 7 q^{77} - 1350 q^{79} + 1142 q^{81} - 670 q^{83} - 579 q^{85} + 753 q^{87} - 236 q^{89} - 2051 q^{91} + 652 q^{93} + 171 q^{95} + 1294 q^{97} + 133 q^{99}+O(q^{100}) $ 2 * q + 9 * q^3 + 9 * q^5 + 18 * q^7 + 23 * q^9 - 17 * q^11 - 17 * q^13 + 150 * q^15 - 80 * q^17 + 38 * q^19 + 227 * q^21 - 73 * q^23 + 119 * q^25 + 189 * q^27 - 3 * q^29 - 212 * q^31 - 40 * q^33 + 519 * q^35 - 192 * q^37 - 551 * q^39 - 50 * q^41 + 677 * q^43 + 1089 * q^45 + 389 * q^47 + 60 * q^49 - 433 * q^51 + 1219 * q^53 + 33 * q^55 + 171 * q^57 - 287 * q^59 - 313 * q^61 + 1521 * q^63 - 1500 * q^65 + 1223 * q^67 - 803 * q^69 - 200 * q^71 + 378 * q^73 + 1521 * q^75 - 7 * q^77 - 1350 * q^79 + 1142 * q^81 - 670 * q^83 - 579 * q^85 + 753 * q^87 - 236 * q^89 - 2051 * q^91 + 652 * q^93 + 171 * q^95 + 1294 * q^97 + 133 * 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
$3$	8.77200	1.68817	0.844086	−	0.536207i	$-0.180144\pi$
$3$	8.77200	1.68817	0.844086	+	0.536207i	$0.180144\pi$
$4$	0	0
$5$	17.3160	1.54879	0.774395	−	0.632702i	$-0.218054\pi$
$5$	17.3160	1.54879	0.774395	+	0.632702i	$0.218054\pi$
$6$	0	0
$7$	26.0880	1.40862	0.704310	−	0.709893i	$-0.251257\pi$
$7$	26.0880	1.40862	0.704310	+	0.709893i	$0.251257\pi$
$8$	0	0
$9$	49.9480	1.84993
$10$	0	0
$11$	−4.22800	−0.115890	−0.0579450	−	0.998320i	$-0.518455\pi$
$11$	−4.22800	−0.115890	−0.0579450	+	0.998320i	$0.518455\pi$
$12$	0	0
$13$	−64.0360	−1.36618	−0.683092	−	0.730332i	$-0.739365\pi$
$13$	−64.0360	−1.36618	−0.683092	+	0.730332i	$0.739365\pi$
$14$	0	0
$15$	151.896	2.61463
$16$	0	0
$17$	−48.5440	−0.692568	−0.346284	−	0.938130i	$-0.612557\pi$
$17$	−48.5440	−0.692568	−0.346284	+	0.938130i	$0.612557\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	228.844	2.37799
$22$	0	0
$23$	−92.0360	−0.834384	−0.417192	−	0.908818i	$-0.636986\pi$
$23$	−92.0360	−0.834384	−0.417192	+	0.908818i	$0.636986\pi$
$24$	0	0
$25$	174.844	1.39875
$26$	0	0
$27$	201.300	1.43482
$28$	0	0
$29$	88.2120	0.564847	0.282424	−	0.959290i	$-0.408862\pi$
$29$	88.2120	0.564847	0.282424	+	0.959290i	$0.408862\pi$
$30$	0	0
$31$	81.9681	0.474900	0.237450	−	0.971400i	$-0.423688\pi$
$31$	81.9681	0.474900	0.237450	+	0.971400i	$0.423688\pi$
$32$	0	0
$33$	−37.0880	−0.195642
$34$	0	0
$35$	451.740	2.18166
$36$	0	0
$37$	23.6161	0.104931	0.0524656	−	0.998623i	$-0.483292\pi$
$37$	23.6161	0.104931	0.0524656	+	0.998623i	$0.483292\pi$
$38$	0	0
$39$	−561.724	−2.30636
$40$	0	0
$41$	17.7200	0.0674976	0.0337488	−	0.999430i	$-0.489255\pi$
$41$	17.7200	0.0674976	0.0337488	+	0.999430i	$0.489255\pi$
$42$	0	0
$43$	368.404	1.30654	0.653268	−	0.757126i	$-0.273397\pi$
$43$	368.404	1.30654	0.653268	+	0.757126i	$0.273397\pi$
$44$	0	0
$45$	864.900	2.86515
$46$	0	0
$47$	497.812	1.54497	0.772483	−	0.635036i	$-0.219015\pi$
$47$	497.812	1.54497	0.772483	+	0.635036i	$0.219015\pi$
$48$	0	0
$49$	337.584	0.984210
$50$	0	0
$51$	−425.828	−1.16917
$52$	0	0
$53$	536.876	1.39143	0.695713	−	0.718320i	$-0.255089\pi$
$53$	536.876	1.39143	0.695713	+	0.718320i	$0.255089\pi$
$54$	0	0
$55$	−73.2120	−0.179489
$56$	0	0
$57$	166.668	0.387293
$58$	0	0
$59$	−36.7000	−0.0809818	−0.0404909	−	0.999180i	$-0.512892\pi$
$59$	−36.7000	−0.0809818	−0.0404909	+	0.999180i	$0.512892\pi$
$60$	0	0
$61$	−630.692	−1.32380	−0.661901	−	0.749592i	$-0.730250\pi$
$61$	−630.692	−1.32380	−0.661901	+	0.749592i	$0.730250\pi$
$62$	0	0
$63$	1303.04	2.60584
$64$	0	0
$65$	−1108.85	−2.11593
$66$	0	0
$67$	282.556	0.515219	0.257610	−	0.966249i	$-0.417065\pi$
$67$	282.556	0.515219	0.257610	+	0.966249i	$0.417065\pi$
$68$	0	0
$69$	−807.340	−1.40858
$70$	0	0
$71$	−595.552	−0.995480	−0.497740	−	0.867326i	$-0.665837\pi$
$71$	−595.552	−0.995480	−0.497740	+	0.867326i	$0.665837\pi$
$72$	0	0
$73$	−597.048	−0.957250	−0.478625	−	0.878020i	$-0.658865\pi$
$73$	−597.048	−0.957250	−0.478625	+	0.878020i	$0.658865\pi$
$74$	0	0
$75$	1533.73	2.36134
$76$	0	0
$77$	−110.300	−0.163245
$78$	0	0
$79$	−427.224	−0.608436	−0.304218	−	0.952602i	$-0.598395\pi$
$79$	−427.224	−0.608436	−0.304218	+	0.952602i	$0.598395\pi$
$80$	0	0
$81$	417.208	0.572302
$82$	0	0
$83$	493.768	0.652989	0.326495	−	0.945199i	$-0.394132\pi$
$83$	493.768	0.652989	0.326495	+	0.945199i	$0.394132\pi$
$84$	0	0
$85$	−840.588	−1.07264
$86$	0	0
$87$	773.796	0.953559
$88$	0	0
$89$	−921.136	−1.09708	−0.548541	−	0.836124i	$-0.684816\pi$
$89$	−921.136	−1.09708	−0.548541	+	0.836124i	$0.684816\pi$
$90$	0	0
$91$	−1670.57	−1.92443
$92$	0	0
$93$	719.024	0.801713
$94$	0	0
$95$	329.004	0.355317
$96$	0	0
$97$	1082.74	1.13336	0.566680	−	0.823938i	$-0.308227\pi$
$97$	1082.74	1.13336	0.566680	+	0.823938i	$0.308227\pi$
$98$	0	0
$99$	−211.180	−0.214388
$100$	0	0
$101$	712.448	0.701893	0.350947	−	0.936395i	$-0.385860\pi$
$101$	712.448	0.701893	0.350947	+	0.936395i	$0.385860\pi$
$102$	0	0
$103$	26.4797	0.0253313	0.0126656	−	0.999920i	$-0.495968\pi$
$103$	26.4797	0.0253313	0.0126656	+	0.999920i	$0.495968\pi$
$104$	0	0
$105$	3962.66	3.68301
$106$	0	0
$107$	−740.996	−0.669484	−0.334742	−	0.942310i	$-0.608649\pi$
$107$	−740.996	−0.669484	−0.334742	+	0.942310i	$0.608649\pi$
$108$	0	0
$109$	1983.08	1.74261	0.871304	−	0.490744i	$-0.163275\pi$
$109$	1983.08	1.74261	0.871304	+	0.490744i	$0.163275\pi$
$110$	0	0
$111$	207.160	0.177142
$112$	0	0
$113$	−718.720	−0.598332	−0.299166	−	0.954201i	$-0.596708\pi$
$113$	−718.720	−0.598332	−0.299166	+	0.954201i	$0.596708\pi$
$114$	0	0
$115$	−1593.70	−1.29229
$116$	0	0
$117$	−3198.47	−2.52734
$118$	0	0
$119$	−1266.42	−0.975565
$120$	0	0
$121$	−1313.12	−0.986570
$122$	0	0
$123$	155.440	0.113948
$124$	0	0
$125$	863.100	0.617584
$126$	0	0
$127$	−2610.72	−1.82413	−0.912063	−	0.410050i	$-0.865511\pi$
$127$	−2610.72	−1.82413	−0.912063	+	0.410050i	$0.865511\pi$
$128$	0	0
$129$	3231.64	2.20566
$130$	0	0
$131$	−1216.69	−0.811472	−0.405736	−	0.913990i	$-0.632985\pi$
$131$	−1216.69	−0.811472	−0.405736	+	0.913990i	$0.632985\pi$
$132$	0	0
$133$	495.672	0.323160
$134$	0	0
$135$	3485.71	2.22224
$136$	0	0
$137$	1170.67	0.730053	0.365026	−	0.930997i	$-0.381060\pi$
$137$	1170.67	0.730053	0.365026	+	0.930997i	$0.381060\pi$
$138$	0	0
$139$	−271.083	−0.165417	−0.0827086	−	0.996574i	$-0.526357\pi$
$139$	−271.083	−0.165417	−0.0827086	+	0.996574i	$0.526357\pi$
$140$	0	0
$141$	4366.81	2.60817
$142$	0	0
$143$	270.744	0.158327
$144$	0	0
$145$	1527.48	0.874830
$146$	0	0
$147$	2961.29	1.66152
$148$	0	0
$149$	−1841.19	−1.01232	−0.506161	−	0.862439i	$-0.668936\pi$
$149$	−1841.19	−1.01232	−0.506161	+	0.862439i	$0.668936\pi$
$150$	0	0
$151$	−3322.32	−1.79051	−0.895254	−	0.445557i	$-0.853006\pi$
$151$	−3322.32	−1.79051	−0.895254	+	0.445557i	$0.853006\pi$
$152$	0	0
$153$	−2424.68	−1.28120
$154$	0	0
$155$	1419.36	0.735521
$156$	0	0
$157$	−243.616	−0.123839	−0.0619194	−	0.998081i	$-0.519722\pi$
$157$	−243.616	−0.123839	−0.0619194	+	0.998081i	$0.519722\pi$
$158$	0	0
$159$	4709.48	2.34897
$160$	0	0
$161$	−2401.04	−1.17533
$162$	0	0
$163$	−2598.11	−1.24847	−0.624233	−	0.781238i	$-0.714588\pi$
$163$	−2598.11	−1.24847	−0.624233	+	0.781238i	$0.714588\pi$
$164$	0	0
$165$	−642.216	−0.303009
$166$	0	0
$167$	491.064	0.227543	0.113772	−	0.993507i	$-0.463707\pi$
$167$	491.064	0.227543	0.113772	+	0.993507i	$0.463707\pi$
$168$	0	0
$169$	1903.61	0.866460
$170$	0	0
$171$	949.012	0.424402
$172$	0	0
$173$	−1648.56	−0.724496	−0.362248	−	0.932082i	$-0.617991\pi$
$173$	−1648.56	−0.724496	−0.362248	+	0.932082i	$0.617991\pi$
$174$	0	0
$175$	4561.33	1.97031
$176$	0	0
$177$	−321.932	−0.136711
$178$	0	0
$179$	2326.81	0.971586	0.485793	−	0.874074i	$-0.338531\pi$
$179$	2326.81	0.971586	0.485793	+	0.874074i	$0.338531\pi$
$180$	0	0
$181$	4637.46	1.90442	0.952208	−	0.305449i	$-0.0988066\pi$
$181$	4637.46	1.90442	0.952208	+	0.305449i	$0.0988066\pi$
$182$	0	0
$183$	−5532.43	−2.23480
$184$	0	0
$185$	408.936	0.162516
$186$	0	0
$187$	205.244	0.0802616
$188$	0	0
$189$	5251.52	2.02112
$190$	0	0
$191$	−5260.38	−1.99281	−0.996407	−	0.0846903i	$-0.973010\pi$
$191$	−5260.38	−1.99281	−0.996407	+	0.0846903i	$0.973010\pi$
$192$	0	0
$193$	16.1833	0.00603575	0.00301787	−	0.999995i	$-0.499039\pi$
$193$	16.1833	0.00603575	0.00301787	+	0.999995i	$0.499039\pi$
$194$	0	0
$195$	−9726.82	−3.57206
$196$	0	0
$197$	−3784.71	−1.36878	−0.684390	−	0.729116i	$-0.739931\pi$
$197$	−3784.71	−1.36878	−0.684390	+	0.729116i	$0.739931\pi$
$198$	0	0
$199$	−73.2079	−0.0260783	−0.0130391	−	0.999915i	$-0.504151\pi$
$199$	−73.2079	−0.0260783	−0.0130391	+	0.999915i	$0.504151\pi$
$200$	0	0
$201$	2478.58	0.869779
$202$	0	0
$203$	2301.28	0.795655
$204$	0	0
$205$	306.840	0.104540
$206$	0	0
$207$	−4597.02	−1.54355
$208$	0	0
$209$	−80.3320	−0.0265870
$210$	0	0
$211$	−2945.44	−0.961006	−0.480503	−	0.876993i	$-0.659546\pi$
$211$	−2945.44	−0.961006	−0.480503	+	0.876993i	$0.659546\pi$
$212$	0	0
$213$	−5224.19	−1.68054
$214$	0	0
$215$	6379.29	2.02355
$216$	0	0
$217$	2138.38	0.668954
$218$	0	0
$219$	−5237.31	−1.61600
$220$	0	0
$221$	3108.57	0.946175
$222$	0	0
$223$	−3125.30	−0.938499	−0.469250	−	0.883066i	$-0.655476\pi$
$223$	−3125.30	−0.938499	−0.469250	+	0.883066i	$0.655476\pi$
$224$	0	0
$225$	8733.11	2.58759
$226$	0	0
$227$	−3577.80	−1.04611	−0.523055	−	0.852299i	$-0.675208\pi$
$227$	−3577.80	−1.04611	−0.523055	+	0.852299i	$0.675208\pi$
$228$	0	0
$229$	4802.00	1.38570	0.692850	−	0.721082i	$-0.256355\pi$
$229$	4802.00	1.38570	0.692850	+	0.721082i	$0.256355\pi$
$230$	0	0
$231$	−967.552	−0.275586
$232$	0	0
$233$	5829.49	1.63907	0.819534	−	0.573031i	$-0.194232\pi$
$233$	5829.49	1.63907	0.819534	+	0.573031i	$0.194232\pi$
$234$	0	0
$235$	8620.12	2.39283
$236$	0	0
$237$	−3747.61	−1.02714
$238$	0	0
$239$	−1364.33	−0.369251	−0.184625	−	0.982809i	$-0.559107\pi$
$239$	−1364.33	−0.369251	−0.184625	+	0.982809i	$0.559107\pi$
$240$	0	0
$241$	−2647.22	−0.707563	−0.353782	−	0.935328i	$-0.615104\pi$
$241$	−2647.22	−0.707563	−0.353782	+	0.935328i	$0.615104\pi$
$242$	0	0
$243$	−1775.35	−0.468679
$244$	0	0
$245$	5845.61	1.52434
$246$	0	0
$247$	−1216.68	−0.313424
$248$	0	0
$249$	4331.34	1.10236
$250$	0	0
$251$	1970.73	0.495582	0.247791	−	0.968814i	$-0.420295\pi$
$251$	1970.73	0.495582	0.247791	+	0.968814i	$0.420295\pi$
$252$	0	0
$253$	389.128	0.0966967
$254$	0	0
$255$	−7373.64	−1.81081
$256$	0	0
$257$	−7915.82	−1.92131	−0.960653	−	0.277752i	$-0.910411\pi$
$257$	−7915.82	−1.92131	−0.960653	+	0.277752i	$0.910411\pi$
$258$	0	0
$259$	616.096	0.147808
$260$	0	0
$261$	4406.02	1.04493
$262$	0	0
$263$	−3287.96	−0.770892	−0.385446	−	0.922730i	$-0.625952\pi$
$263$	−3287.96	−0.770892	−0.385446	+	0.922730i	$0.625952\pi$
$264$	0	0
$265$	9296.55	2.15503
$266$	0	0
$267$	−8080.21	−1.85206
$268$	0	0
$269$	4749.61	1.07654	0.538269	−	0.842773i	$-0.319078\pi$
$269$	4749.61	1.07654	0.538269	+	0.842773i	$0.319078\pi$
$270$	0	0
$271$	−242.661	−0.0543933	−0.0271967	−	0.999630i	$-0.508658\pi$
$271$	−242.661	−0.0543933	−0.0271967	+	0.999630i	$0.508658\pi$
$272$	0	0
$273$	−14654.3	−3.24878
$274$	0	0
$275$	−739.240	−0.162101
$276$	0	0
$277$	4131.13	0.896086	0.448043	−	0.894012i	$-0.352121\pi$
$277$	4131.13	0.896086	0.448043	+	0.894012i	$0.352121\pi$
$278$	0	0
$279$	4094.14	0.878530
$280$	0	0
$281$	1007.19	0.213822	0.106911	−	0.994269i	$-0.465904\pi$
$281$	1007.19	0.213822	0.106911	+	0.994269i	$0.465904\pi$
$282$	0	0
$283$	2333.63	0.490176	0.245088	−	0.969501i	$-0.421183\pi$
$283$	2333.63	0.490176	0.245088	+	0.969501i	$0.421183\pi$
$284$	0	0
$285$	2886.02	0.599836
$286$	0	0
$287$	462.280	0.0950785
$288$	0	0
$289$	−2556.48	−0.520350
$290$	0	0
$291$	9497.83	1.91331
$292$	0	0
$293$	1588.68	0.316763	0.158381	−	0.987378i	$-0.449372\pi$
$293$	1588.68	0.316763	0.158381	+	0.987378i	$0.449372\pi$
$294$	0	0
$295$	−635.497	−0.125424
$296$	0	0
$297$	−851.096	−0.166282
$298$	0	0
$299$	5893.62	1.13992
$300$	0	0
$301$	9610.93	1.84041
$302$	0	0
$303$	6249.59	1.18492
$304$	0	0
$305$	−10921.1	−2.05029
$306$	0	0
$307$	4057.46	0.754304	0.377152	−	0.926151i	$-0.376903\pi$
$307$	4057.46	0.754304	0.377152	+	0.926151i	$0.376903\pi$
$308$	0	0
$309$	232.280	0.0427636
$310$	0	0
$311$	−2871.92	−0.523638	−0.261819	−	0.965117i	$-0.584322\pi$
$311$	−2871.92	−0.523638	−0.261819	+	0.965117i	$0.584322\pi$
$312$	0	0
$313$	4322.67	0.780612	0.390306	−	0.920685i	$-0.372369\pi$
$313$	4322.67	0.780612	0.390306	+	0.920685i	$0.372369\pi$
$314$	0	0
$315$	22563.5	4.03591
$316$	0	0
$317$	−2513.56	−0.445349	−0.222674	−	0.974893i	$-0.571479\pi$
$317$	−2513.56	−0.445349	−0.222674	+	0.974893i	$0.571479\pi$
$318$	0	0
$319$	−372.960	−0.0654601
$320$	0	0
$321$	−6500.02	−1.13021
$322$	0	0
$323$	−922.336	−0.158886
$324$	0	0
$325$	−11196.3	−1.91095
$326$	0	0
$327$	17395.6	2.94182
$328$	0	0
$329$	12986.9	2.17627
$330$	0	0
$331$	−4573.78	−0.759509	−0.379754	−	0.925087i	$-0.623992\pi$
$331$	−4573.78	−0.759509	−0.379754	+	0.925087i	$0.623992\pi$
$332$	0	0
$333$	1179.57	0.194115
$334$	0	0
$335$	4892.74	0.797967
$336$	0	0
$337$	9001.71	1.45506	0.727529	−	0.686077i	$-0.240669\pi$
$337$	9001.71	1.45506	0.727529	+	0.686077i	$0.240669\pi$
$338$	0	0
$339$	−6304.62	−1.01009
$340$	0	0
$341$	−346.561	−0.0550361
$342$	0	0
$343$	−141.289	−0.0222417
$344$	0	0
$345$	−13979.9	−2.18160
$346$	0	0
$347$	9358.68	1.44784	0.723920	−	0.689884i	$-0.242339\pi$
$347$	9358.68	1.44784	0.723920	+	0.689884i	$0.242339\pi$
$348$	0	0
$349$	−5787.76	−0.887712	−0.443856	−	0.896098i	$-0.646390\pi$
$349$	−5787.76	−0.887712	−0.443856	+	0.896098i	$0.646390\pi$
$350$	0	0
$351$	−12890.5	−1.96023
$352$	0	0
$353$	5784.59	0.872188	0.436094	−	0.899901i	$-0.356361\pi$
$353$	5784.59	0.872188	0.436094	+	0.899901i	$0.356361\pi$
$354$	0	0
$355$	−10312.6	−1.54179
$356$	0	0
$357$	−11109.0	−1.64692
$358$	0	0
$359$	10132.3	1.48959	0.744796	−	0.667292i	$-0.232547\pi$
$359$	10132.3	1.48959	0.744796	+	0.667292i	$0.232547\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−11518.7	−1.66550
$364$	0	0
$365$	−10338.5	−1.48258
$366$	0	0
$367$	6993.81	0.994752	0.497376	−	0.867535i	$-0.334297\pi$
$367$	6993.81	0.994752	0.497376	+	0.867535i	$0.334297\pi$
$368$	0	0
$369$	885.080	0.124866
$370$	0	0
$371$	14006.0	1.95999
$372$	0	0
$373$	−6523.15	−0.905512	−0.452756	−	0.891634i	$-0.649559\pi$
$373$	−6523.15	−0.905512	−0.452756	+	0.891634i	$0.649559\pi$
$374$	0	0
$375$	7571.11	1.04259
$376$	0	0
$377$	−5648.75	−0.771685
$378$	0	0
$379$	−9782.00	−1.32577	−0.662886	−	0.748720i	$-0.730669\pi$
$379$	−9782.00	−1.32577	−0.662886	+	0.748720i	$0.730669\pi$
$380$	0	0
$381$	−22901.2	−3.07944
$382$	0	0
$383$	−9878.11	−1.31788	−0.658940	−	0.752196i	$-0.728995\pi$
$383$	−9878.11	−1.31788	−0.658940	+	0.752196i	$0.728995\pi$
$384$	0	0
$385$	−1909.96	−0.252832
$386$	0	0
$387$	18401.0	2.41700
$388$	0	0
$389$	7891.25	1.02854	0.514270	−	0.857628i	$-0.328063\pi$
$389$	7891.25	1.02854	0.514270	+	0.857628i	$0.328063\pi$
$390$	0	0
$391$	4467.80	0.577868
$392$	0	0
$393$	−10672.8	−1.36991
$394$	0	0
$395$	−7397.81	−0.942340
$396$	0	0
$397$	2787.84	0.352437	0.176219	−	0.984351i	$-0.443613\pi$
$397$	2787.84	0.352437	0.176219	+	0.984351i	$0.443613\pi$
$398$	0	0
$399$	4348.04	0.545549
$400$	0	0
$401$	1264.42	0.157461	0.0787306	−	0.996896i	$-0.474913\pi$
$401$	1264.42	0.157461	0.0787306	+	0.996896i	$0.474913\pi$
$402$	0	0
$403$	−5248.91	−0.648801
$404$	0	0
$405$	7224.37	0.886375
$406$	0	0
$407$	−99.8486	−0.0121605
$408$	0	0
$409$	−8140.55	−0.984166	−0.492083	−	0.870548i	$-0.663764\pi$
$409$	−8140.55	−0.984166	−0.492083	+	0.870548i	$0.663764\pi$
$410$	0	0
$411$	10269.1	1.23246
$412$	0	0
$413$	−957.429	−0.114073
$414$	0	0
$415$	8550.10	1.01134
$416$	0	0
$417$	−2377.94	−0.279253
$418$	0	0
$419$	−9601.15	−1.11944	−0.559722	−	0.828680i	$-0.689092\pi$
$419$	−9601.15	−1.11944	−0.559722	+	0.828680i	$0.689092\pi$
$420$	0	0
$421$	−5702.48	−0.660147	−0.330074	−	0.943955i	$-0.607074\pi$
$421$	−5702.48	−0.660147	−0.330074	+	0.943955i	$0.607074\pi$
$422$	0	0
$423$	24864.7	2.85807
$424$	0	0
$425$	−8487.63	−0.968731
$426$	0	0
$427$	−16453.5	−1.86473
$428$	0	0
$429$	2374.97	0.267283
$430$	0	0
$431$	4025.72	0.449912	0.224956	−	0.974369i	$-0.427776\pi$
$431$	4025.72	0.449912	0.224956	+	0.974369i	$0.427776\pi$
$432$	0	0
$433$	−1347.10	−0.149510	−0.0747548	−	0.997202i	$-0.523817\pi$
$433$	−1347.10	−0.149510	−0.0747548	+	0.997202i	$0.523817\pi$
$434$	0	0
$435$	13399.1	1.47686
$436$	0	0
$437$	−1748.68	−0.191421
$438$	0	0
$439$	−4109.36	−0.446763	−0.223381	−	0.974731i	$-0.571710\pi$
$439$	−4109.36	−0.446763	−0.223381	+	0.974731i	$0.571710\pi$
$440$	0	0
$441$	16861.7	1.82072
$442$	0	0
$443$	6964.84	0.746974	0.373487	−	0.927635i	$-0.378162\pi$
$443$	6964.84	0.746974	0.373487	+	0.927635i	$0.378162\pi$
$444$	0	0
$445$	−15950.4	−1.69915
$446$	0	0
$447$	−16150.9	−1.70897
$448$	0	0
$449$	3041.21	0.319652	0.159826	−	0.987145i	$-0.448907\pi$
$449$	3041.21	0.319652	0.159826	+	0.987145i	$0.448907\pi$
$450$	0	0
$451$	−74.9202	−0.00782229
$452$	0	0
$453$	−29143.4	−3.02269
$454$	0	0
$455$	−28927.6	−2.98055
$456$	0	0
$457$	11984.3	1.22670	0.613352	−	0.789810i	$-0.289821\pi$
$457$	11984.3	1.22670	0.613352	+	0.789810i	$0.289821\pi$
$458$	0	0
$459$	−9771.91	−0.993712
$460$	0	0
$461$	12126.7	1.22515	0.612577	−	0.790411i	$-0.290133\pi$
$461$	12126.7	1.22515	0.612577	+	0.790411i	$0.290133\pi$
$462$	0	0
$463$	6399.19	0.642323	0.321162	−	0.947024i	$-0.395927\pi$
$463$	6399.19	0.642323	0.321162	+	0.947024i	$0.395927\pi$
$464$	0	0
$465$	12450.6	1.24169
$466$	0	0
$467$	993.366	0.0984315	0.0492157	−	0.998788i	$-0.484328\pi$
$467$	993.366	0.0984315	0.0492157	+	0.998788i	$0.484328\pi$
$468$	0	0
$469$	7371.32	0.725748
$470$	0	0
$471$	−2137.00	−0.209061
$472$	0	0
$473$	−1557.61	−0.151414
$474$	0	0
$475$	3322.04	0.320896
$476$	0	0
$477$	26815.9	2.57404
$478$	0	0
$479$	−6639.36	−0.633320	−0.316660	−	0.948539i	$-0.602561\pi$
$479$	−6639.36	−0.633320	−0.316660	+	0.948539i	$0.602561\pi$
$480$	0	0
$481$	−1512.28	−0.143355
$482$	0	0
$483$	−21061.9	−1.98416
$484$	0	0
$485$	18748.8	1.75534
$486$	0	0
$487$	11088.8	1.03179	0.515894	−	0.856652i	$-0.327460\pi$
$487$	11088.8	1.03179	0.515894	+	0.856652i	$0.327460\pi$
$488$	0	0
$489$	−22790.6	−2.10762
$490$	0	0
$491$	−13215.2	−1.21465	−0.607324	−	0.794454i	$-0.707757\pi$
$491$	−13215.2	−1.21465	−0.607324	+	0.794454i	$0.707757\pi$
$492$	0	0
$493$	−4282.17	−0.391195
$494$	0	0
$495$	−3656.80	−0.332042
$496$	0	0
$497$	−15536.8	−1.40225
$498$	0	0
$499$	410.640	0.0368393	0.0184196	−	0.999830i	$-0.494137\pi$
$499$	410.640	0.0368393	0.0184196	+	0.999830i	$0.494137\pi$
$500$	0	0
$501$	4307.62	0.384132
$502$	0	0
$503$	9407.88	0.833950	0.416975	−	0.908918i	$-0.363090\pi$
$503$	9407.88	0.833950	0.416975	+	0.908918i	$0.363090\pi$
$504$	0	0
$505$	12336.7	1.08709
$506$	0	0
$507$	16698.5	1.46273
$508$	0	0
$509$	−10482.2	−0.912803	−0.456402	−	0.889774i	$-0.650862\pi$
$509$	−10482.2	−0.912803	−0.456402	+	0.889774i	$0.650862\pi$
$510$	0	0
$511$	−15575.8	−1.34840
$512$	0	0
$513$	3824.70	0.329171
$514$	0	0
$515$	458.523	0.0392329
$516$	0	0
$517$	−2104.75	−0.179046
$518$	0	0
$519$	−14461.2	−1.22307
$520$	0	0
$521$	−3181.02	−0.267492	−0.133746	−	0.991016i	$-0.542701\pi$
$521$	−3181.02	−0.267492	−0.133746	+	0.991016i	$0.542701\pi$
$522$	0	0
$523$	−4360.12	−0.364541	−0.182270	−	0.983248i	$-0.558345\pi$
$523$	−4360.12	−0.364541	−0.182270	+	0.983248i	$0.558345\pi$
$524$	0	0
$525$	40012.0	3.32622
$526$	0	0
$527$	−3979.06	−0.328900
$528$	0	0
$529$	−3696.37	−0.303803
$530$	0	0
$531$	−1833.09	−0.149810
$532$	0	0
$533$	−1134.72	−0.0922142
$534$	0	0
$535$	−12831.1	−1.03689
$536$	0	0
$537$	20410.8	1.64020
$538$	0	0
$539$	−1427.31	−0.114060
$540$	0	0
$541$	23681.2	1.88195	0.940973	−	0.338481i	$-0.109913\pi$
$541$	23681.2	1.88195	0.940973	+	0.338481i	$0.109913\pi$
$542$	0	0
$543$	40679.8	3.21498
$544$	0	0
$545$	34339.0	2.69894
$546$	0	0
$547$	−7373.25	−0.576339	−0.288169	−	0.957579i	$-0.593047\pi$
$547$	−7373.25	−0.576339	−0.288169	+	0.957579i	$0.593047\pi$
$548$	0	0
$549$	−31501.8	−2.44893
$550$	0	0
$551$	1676.03	0.129585
$552$	0	0
$553$	−11145.4	−0.857055
$554$	0	0
$555$	3587.18	0.274356
$556$	0	0
$557$	−4772.14	−0.363020	−0.181510	−	0.983389i	$-0.558098\pi$
$557$	−4772.14	−0.363020	−0.181510	+	0.983389i	$0.558098\pi$
$558$	0	0
$559$	−23591.1	−1.78497
$560$	0	0
$561$	1800.40	0.135495
$562$	0	0
$563$	7276.49	0.544702	0.272351	−	0.962198i	$-0.412199\pi$
$563$	7276.49	0.544702	0.272351	+	0.962198i	$0.412199\pi$
$564$	0	0
$565$	−12445.4	−0.926691
$566$	0	0
$567$	10884.1	0.806156
$568$	0	0
$569$	−10685.1	−0.787245	−0.393622	−	0.919272i	$-0.628778\pi$
$569$	−10685.1	−0.787245	−0.393622	+	0.919272i	$0.628778\pi$
$570$	0	0
$571$	14856.1	1.08881	0.544404	−	0.838823i	$-0.316756\pi$
$571$	14856.1	1.08881	0.544404	+	0.838823i	$0.316756\pi$
$572$	0	0
$573$	−46144.0	−3.36421
$574$	0	0
$575$	−16092.0	−1.16710
$576$	0	0
$577$	3212.67	0.231794	0.115897	−	0.993261i	$-0.463026\pi$
$577$	3212.67	0.231794	0.115897	+	0.993261i	$0.463026\pi$
$578$	0	0
$579$	141.960	0.0101894
$580$	0	0
$581$	12881.4	0.919814
$582$	0	0
$583$	−2269.91	−0.161252
$584$	0	0
$585$	−55384.8	−3.91432
$586$	0	0
$587$	−22321.1	−1.56949	−0.784745	−	0.619818i	$-0.787206\pi$
$587$	−22321.1	−1.56949	−0.784745	+	0.619818i	$0.787206\pi$
$588$	0	0
$589$	1557.39	0.108950
$590$	0	0
$591$	−33199.5	−2.31074
$592$	0	0
$593$	−8202.50	−0.568021	−0.284010	−	0.958821i	$-0.591665\pi$
$593$	−8202.50	−0.568021	−0.284010	+	0.958821i	$0.591665\pi$
$594$	0	0
$595$	−21929.3	−1.51095
$596$	0	0
$597$	−642.180	−0.0440246
$598$	0	0
$599$	−10583.3	−0.721906	−0.360953	−	0.932584i	$-0.617548\pi$
$599$	−10583.3	−0.721906	−0.360953	+	0.932584i	$0.617548\pi$
$600$	0	0
$601$	−9051.94	−0.614370	−0.307185	−	0.951650i	$-0.599387\pi$
$601$	−9051.94	−0.614370	−0.307185	+	0.951650i	$0.599387\pi$
$602$	0	0
$603$	14113.1	0.953118
$604$	0	0
$605$	−22738.1	−1.52799
$606$	0	0
$607$	−8123.48	−0.543199	−0.271599	−	0.962410i	$-0.587553\pi$
$607$	−8123.48	−0.543199	−0.271599	+	0.962410i	$0.587553\pi$
$608$	0	0
$609$	20186.8	1.34320
$610$	0	0
$611$	−31877.9	−2.11071
$612$	0	0
$613$	−22384.7	−1.47490	−0.737448	−	0.675404i	$-0.763969\pi$
$613$	−22384.7	−1.47490	−0.737448	+	0.675404i	$0.763969\pi$
$614$	0	0
$615$	2691.60	0.176481
$616$	0	0
$617$	11349.1	0.740517	0.370259	−	0.928929i	$-0.379269\pi$
$617$	11349.1	0.740517	0.370259	+	0.928929i	$0.379269\pi$
$618$	0	0
$619$	−9106.25	−0.591294	−0.295647	−	0.955297i	$-0.595535\pi$
$619$	−9106.25	−0.591294	−0.295647	+	0.955297i	$0.595535\pi$
$620$	0	0
$621$	−18526.9	−1.19719
$622$	0	0
$623$	−24030.6	−1.54537
$624$	0	0
$625$	−6910.06	−0.442244
$626$	0	0
$627$	−704.672	−0.0448834
$628$	0	0
$629$	−1146.42	−0.0726720
$630$	0	0
$631$	27784.2	1.75289	0.876444	−	0.481505i	$-0.159910\pi$
$631$	27784.2	1.75289	0.876444	+	0.481505i	$0.159910\pi$
$632$	0	0
$633$	−25837.4	−1.62234
$634$	0	0
$635$	−45207.3	−2.82519
$636$	0	0
$637$	−21617.5	−1.34461
$638$	0	0
$639$	−29746.7	−1.84156
$640$	0	0
$641$	−16958.3	−1.04495	−0.522476	−	0.852654i	$-0.674992\pi$
$641$	−16958.3	−1.04495	−0.522476	+	0.852654i	$0.674992\pi$
$642$	0	0
$643$	4754.37	0.291592	0.145796	−	0.989315i	$-0.453426\pi$
$643$	4754.37	0.291592	0.145796	+	0.989315i	$0.453426\pi$
$644$	0	0
$645$	55959.1	3.41611
$646$	0	0
$647$	11254.0	0.683831	0.341916	−	0.939731i	$-0.388924\pi$
$647$	11254.0	0.683831	0.341916	+	0.939731i	$0.388924\pi$
$648$	0	0
$649$	155.167	0.00938498
$650$	0	0
$651$	18757.9	1.12931
$652$	0	0
$653$	15515.1	0.929793	0.464896	−	0.885365i	$-0.346092\pi$
$653$	15515.1	0.929793	0.464896	+	0.885365i	$0.346092\pi$
$654$	0	0
$655$	−21068.2	−1.25680
$656$	0	0
$657$	−29821.4	−1.77084
$658$	0	0
$659$	17203.2	1.01691	0.508453	−	0.861090i	$-0.330217\pi$
$659$	17203.2	1.01691	0.508453	+	0.861090i	$0.330217\pi$
$660$	0	0
$661$	−2305.65	−0.135672	−0.0678361	−	0.997696i	$-0.521610\pi$
$661$	−2305.65	−0.135672	−0.0678361	+	0.997696i	$0.521610\pi$
$662$	0	0
$663$	27268.3	1.59731
$664$	0	0
$665$	8583.06	0.500507
$666$	0	0
$667$	−8118.69	−0.471299
$668$	0	0
$669$	−27415.1	−1.58435
$670$	0	0
$671$	2666.57	0.153415
$672$	0	0
$673$	−14242.8	−0.815782	−0.407891	−	0.913031i	$-0.633736\pi$
$673$	−14242.8	−0.815782	−0.407891	+	0.913031i	$0.633736\pi$
$674$	0	0
$675$	35196.1	2.00696
$676$	0	0
$677$	13480.0	0.765256	0.382628	−	0.923902i	$-0.375019\pi$
$677$	13480.0	0.765256	0.382628	+	0.923902i	$0.375019\pi$
$678$	0	0
$679$	28246.6	1.59647
$680$	0	0
$681$	−31384.5	−1.76601
$682$	0	0
$683$	27626.1	1.54771	0.773854	−	0.633365i	$-0.218327\pi$
$683$	27626.1	1.54771	0.773854	+	0.633365i	$0.218327\pi$
$684$	0	0
$685$	20271.4	1.13070
$686$	0	0
$687$	42123.2	2.33930
$688$	0	0
$689$	−34379.4	−1.90094
$690$	0	0
$691$	17419.7	0.959009	0.479505	−	0.877539i	$-0.340816\pi$
$691$	17419.7	0.959009	0.479505	+	0.877539i	$0.340816\pi$
$692$	0	0
$693$	−5509.27	−0.301991
$694$	0	0
$695$	−4694.08	−0.256197
$696$	0	0
$697$	−860.201	−0.0467467
$698$	0	0
$699$	51136.3	2.76703
$700$	0	0
$701$	−5069.39	−0.273136	−0.136568	−	0.990631i	$-0.543607\pi$
$701$	−5069.39	−0.273136	−0.136568	+	0.990631i	$0.543607\pi$
$702$	0	0
$703$	448.705	0.0240729
$704$	0	0
$705$	75615.7	4.03951
$706$	0	0
$707$	18586.3	0.988701
$708$	0	0
$709$	16758.9	0.887719	0.443860	−	0.896096i	$-0.353609\pi$
$709$	16758.9	0.887719	0.443860	+	0.896096i	$0.353609\pi$
$710$	0	0
$711$	−21339.0	−1.12556
$712$	0	0
$713$	−7544.02	−0.396249
$714$	0	0
$715$	4688.21	0.245215
$716$	0	0
$717$	−11967.9	−0.623359
$718$	0	0
$719$	3885.84	0.201554	0.100777	−	0.994909i	$-0.467867\pi$
$719$	3885.84	0.201554	0.100777	+	0.994909i	$0.467867\pi$
$720$	0	0
$721$	690.803	0.0356822
$722$	0	0
$723$	−23221.5	−1.19449
$724$	0	0
$725$	15423.4	0.790081
$726$	0	0
$727$	−6468.37	−0.329984	−0.164992	−	0.986295i	$-0.552760\pi$
$727$	−6468.37	−0.329984	−0.164992	+	0.986295i	$0.552760\pi$
$728$	0	0
$729$	−26838.0	−1.36351
$730$	0	0
$731$	−17883.8	−0.904865
$732$	0	0
$733$	−25245.5	−1.27212	−0.636059	−	0.771640i	$-0.719437\pi$
$733$	−25245.5	−1.27212	−0.636059	+	0.771640i	$0.719437\pi$
$734$	0	0
$735$	51277.7	2.57334
$736$	0	0
$737$	−1194.65	−0.0597087
$738$	0	0
$739$	3229.28	0.160746	0.0803728	−	0.996765i	$-0.474389\pi$
$739$	3229.28	0.160746	0.0803728	+	0.996765i	$0.474389\pi$
$740$	0	0
$741$	−10672.8	−0.529114
$742$	0	0
$743$	−18876.2	−0.932033	−0.466016	−	0.884776i	$-0.654311\pi$
$743$	−18876.2	−0.932033	−0.466016	+	0.884776i	$0.654311\pi$
$744$	0	0
$745$	−31882.0	−1.56788
$746$	0	0
$747$	24662.8	1.20798
$748$	0	0
$749$	−19331.1	−0.943049
$750$	0	0
$751$	−24895.8	−1.20967	−0.604833	−	0.796352i	$-0.706760\pi$
$751$	−24895.8	−1.20967	−0.604833	+	0.796352i	$0.706760\pi$
$752$	0	0
$753$	17287.2	0.836628
$754$	0	0
$755$	−57529.3	−2.77312
$756$	0	0
$757$	36203.2	1.73821	0.869107	−	0.494624i	$-0.164694\pi$
$757$	36203.2	1.73821	0.869107	+	0.494624i	$0.164694\pi$
$758$	0	0
$759$	3413.43	0.163241
$760$	0	0
$761$	11417.5	0.543868	0.271934	−	0.962316i	$-0.412337\pi$
$761$	11417.5	0.543868	0.271934	+	0.962316i	$0.412337\pi$
$762$	0	0
$763$	51734.5	2.45467
$764$	0	0
$765$	−41985.7	−1.98431
$766$	0	0
$767$	2350.12	0.110636
$768$	0	0
$769$	39414.5	1.84828	0.924138	−	0.382058i	$-0.124785\pi$
$769$	39414.5	1.84828	0.924138	+	0.382058i	$0.124785\pi$
$770$	0	0
$771$	−69437.6	−3.24350
$772$	0	0
$773$	14268.5	0.663910	0.331955	−	0.943295i	$-0.392292\pi$
$773$	14268.5	0.663910	0.331955	+	0.943295i	$0.392292\pi$
$774$	0	0
$775$	14331.6	0.664268
$776$	0	0
$777$	5404.39	0.249526
$778$	0	0
$779$	336.680	0.0154850
$780$	0	0
$781$	2517.99	0.115366
$782$	0	0
$783$	17757.1	0.810455
$784$	0	0
$785$	−4218.46	−0.191800
$786$	0	0
$787$	−2922.28	−0.132361	−0.0661804	−	0.997808i	$-0.521081\pi$
$787$	−2922.28	−0.132361	−0.0661804	+	0.997808i	$0.521081\pi$
$788$	0	0
$789$	−28842.0	−1.30140
$790$	0	0
$791$	−18750.0	−0.842823
$792$	0	0
$793$	40387.0	1.80856
$794$	0	0
$795$	81549.3	3.63806
$796$	0	0
$797$	7724.25	0.343296	0.171648	−	0.985158i	$-0.445091\pi$
$797$	7724.25	0.343296	0.171648	+	0.985158i	$0.445091\pi$
$798$	0	0
$799$	−24165.8	−1.06999
$800$	0	0
$801$	−46008.9	−2.02952
$802$	0	0
$803$	2524.32	0.110936
$804$	0	0
$805$	−41576.4	−1.82034
$806$	0	0
$807$	41663.6	1.81738
$808$	0	0
$809$	42980.8	1.86789	0.933947	−	0.357412i	$-0.116341\pi$
$809$	42980.8	1.86789	0.933947	+	0.357412i	$0.116341\pi$
$810$	0	0
$811$	28749.5	1.24480	0.622398	−	0.782701i	$-0.286158\pi$
$811$	28749.5	1.24480	0.622398	+	0.782701i	$0.286158\pi$
$812$	0	0
$813$	−2128.62	−0.0918253
$814$	0	0
$815$	−44988.9	−1.93361
$816$	0	0
$817$	6999.68	0.299740
$818$	0	0
$819$	−83441.8	−3.56006
$820$	0	0
$821$	30274.8	1.28696	0.643482	−	0.765461i	$-0.277489\pi$
$821$	30274.8	1.28696	0.643482	+	0.765461i	$0.277489\pi$
$822$	0	0
$823$	−17296.1	−0.732568	−0.366284	−	0.930503i	$-0.619370\pi$
$823$	−17296.1	−0.732568	−0.366284	+	0.930503i	$0.619370\pi$
$824$	0	0
$825$	−6484.62	−0.273655
$826$	0	0
$827$	−2022.80	−0.0850541	−0.0425271	−	0.999095i	$-0.513541\pi$
$827$	−2022.80	−0.0850541	−0.0425271	+	0.999095i	$0.513541\pi$
$828$	0	0
$829$	43239.0	1.81152	0.905762	−	0.423786i	$-0.139299\pi$
$829$	43239.0	1.81152	0.905762	+	0.423786i	$0.139299\pi$
$830$	0	0
$831$	36238.3	1.51275
$832$	0	0
$833$	−16387.7	−0.681632
$834$	0	0
$835$	8503.27	0.352417
$836$	0	0
$837$	16500.2	0.681397
$838$	0	0
$839$	−27435.9	−1.12895	−0.564477	−	0.825449i	$-0.690922\pi$
$839$	−27435.9	−1.12895	−0.564477	+	0.825449i	$0.690922\pi$
$840$	0	0
$841$	−16607.6	−0.680948
$842$	0	0
$843$	8835.09	0.360969
$844$	0	0
$845$	32963.0	1.34196
$846$	0	0
$847$	−34256.8	−1.38970
$848$	0	0
$849$	20470.6	0.827502
$850$	0	0
$851$	−2173.53	−0.0875530
$852$	0	0
$853$	20978.4	0.842071	0.421035	−	0.907044i	$-0.361667\pi$
$853$	20978.4	0.842071	0.421035	+	0.907044i	$0.361667\pi$
$854$	0	0
$855$	16433.1	0.657310
$856$	0	0
$857$	−30822.4	−1.22856	−0.614279	−	0.789089i	$-0.710553\pi$
$857$	−30822.4	−1.22856	−0.614279	+	0.789089i	$0.710553\pi$
$858$	0	0
$859$	−39267.6	−1.55971	−0.779856	−	0.625959i	$-0.784708\pi$
$859$	−39267.6	−1.55971	−0.779856	+	0.625959i	$0.784708\pi$
$860$	0	0
$861$	4055.12	0.160509
$862$	0	0
$863$	24131.3	0.951842	0.475921	−	0.879488i	$-0.342115\pi$
$863$	24131.3	0.951842	0.475921	+	0.879488i	$0.342115\pi$
$864$	0	0
$865$	−28546.5	−1.12209
$866$	0	0
$867$	−22425.4	−0.878441
$868$	0	0
$869$	1806.30	0.0705116
$870$	0	0
$871$	−18093.8	−0.703885
$872$	0	0
$873$	54080.9	2.09663
$874$	0	0
$875$	22516.6	0.869941
$876$	0	0
$877$	−39380.6	−1.51629	−0.758147	−	0.652084i	$-0.773895\pi$
$877$	−39380.6	−1.51629	−0.758147	+	0.652084i	$0.773895\pi$
$878$	0	0
$879$	13935.9	0.534750
$880$	0	0
$881$	30887.5	1.18119	0.590595	−	0.806968i	$-0.298893\pi$
$881$	30887.5	1.18119	0.590595	+	0.806968i	$0.298893\pi$
$882$	0	0
$883$	28191.9	1.07444	0.537221	−	0.843441i	$-0.319474\pi$
$883$	28191.9	1.07444	0.537221	+	0.843441i	$0.319474\pi$
$884$	0	0
$885$	−5574.58	−0.211737
$886$	0	0
$887$	−2760.58	−0.104500	−0.0522498	−	0.998634i	$-0.516639\pi$
$887$	−2760.58	−0.104500	−0.0522498	+	0.998634i	$0.516639\pi$
$888$	0	0
$889$	−68108.5	−2.56950
$890$	0	0
$891$	−1763.95	−0.0663240
$892$	0	0
$893$	9458.43	0.354439
$894$	0	0
$895$	40291.0	1.50478
$896$	0	0
$897$	51698.9	1.92439
$898$	0	0
$899$	7230.57	0.268246
$900$	0	0
$901$	−26062.1	−0.963657
$902$	0	0
$903$	84307.1	3.10694
$904$	0	0
$905$	80302.2	2.94954
$906$	0	0
$907$	−18969.1	−0.694443	−0.347222	−	0.937783i	$-0.612875\pi$
$907$	−18969.1	−0.694443	−0.347222	+	0.937783i	$0.612875\pi$
$908$	0	0
$909$	35585.4	1.29845
$910$	0	0
$911$	48732.9	1.77233	0.886164	−	0.463371i	$-0.153360\pi$
$911$	48732.9	1.77233	0.886164	+	0.463371i	$0.153360\pi$
$912$	0	0
$913$	−2087.65	−0.0756749
$914$	0	0
$915$	−95799.6	−3.46124
$916$	0	0
$917$	−31741.1	−1.14306
$918$	0	0
$919$	35850.4	1.28683	0.643414	−	0.765518i	$-0.277517\pi$
$919$	35850.4	1.28683	0.643414	+	0.765518i	$0.277517\pi$
$920$	0	0
$921$	35592.0	1.27340
$922$	0	0
$923$	38136.8	1.36001
$924$	0	0
$925$	4129.13	0.146773
$926$	0	0
$927$	1322.61	0.0468610
$928$	0	0
$929$	22936.8	0.810044	0.405022	−	0.914307i	$-0.367264\pi$
$929$	22936.8	0.810044	0.405022	+	0.914307i	$0.367264\pi$
$930$	0	0
$931$	6414.10	0.225793
$932$	0	0
$933$	−25192.5	−0.883992
$934$	0	0
$935$	3554.01	0.124308
$936$	0	0
$937$	47925.4	1.67092	0.835462	−	0.549548i	$-0.185200\pi$
$937$	47925.4	1.67092	0.835462	+	0.549548i	$0.185200\pi$
$938$	0	0
$939$	37918.5	1.31781
$940$	0	0
$941$	25842.2	0.895251	0.447626	−	0.894221i	$-0.352270\pi$
$941$	25842.2	0.895251	0.447626	+	0.894221i	$0.352270\pi$
$942$	0	0
$943$	−1630.88	−0.0563189
$944$	0	0
$945$	90935.3	3.13029
$946$	0	0
$947$	36562.8	1.25463	0.627314	−	0.778766i	$-0.284154\pi$
$947$	36562.8	1.25463	0.627314	+	0.778766i	$0.284154\pi$
$948$	0	0
$949$	38232.6	1.30778
$950$	0	0
$951$	−22048.9	−0.751825
$952$	0	0
$953$	29813.1	1.01337	0.506684	−	0.862132i	$-0.330871\pi$
$953$	29813.1	1.01337	0.506684	+	0.862132i	$0.330871\pi$
$954$	0	0
$955$	−91088.7	−3.08645
$956$	0	0
$957$	−3271.61	−0.110508
$958$	0	0
$959$	30540.5	1.02837
$960$	0	0
$961$	−23072.2	−0.774470
$962$	0	0
$963$	−37011.3	−1.23850
$964$	0	0
$965$	280.230	0.00934811
$966$	0	0
$967$	−30315.5	−1.00815	−0.504075	−	0.863660i	$-0.668167\pi$
$967$	−30315.5	−1.00815	−0.504075	+	0.863660i	$0.668167\pi$
$968$	0	0
$969$	−8090.73	−0.268227
$970$	0	0
$971$	26455.6	0.874357	0.437178	−	0.899375i	$-0.355978\pi$
$971$	26455.6	0.874357	0.437178	+	0.899375i	$0.355978\pi$
$972$	0	0
$973$	−7072.03	−0.233010
$974$	0	0
$975$	−98214.1	−3.22602
$976$	0	0
$977$	30207.7	0.989183	0.494591	−	0.869126i	$-0.335318\pi$
$977$	30207.7	0.989183	0.494591	+	0.869126i	$0.335318\pi$
$978$	0	0
$979$	3894.56	0.127141
$980$	0	0
$981$	99050.7	3.22370
$982$	0	0
$983$	5878.48	0.190737	0.0953685	−	0.995442i	$-0.469597\pi$
$983$	5878.48	0.190737	0.0953685	+	0.995442i	$0.469597\pi$
$984$	0	0
$985$	−65536.1	−2.11995
$986$	0	0
$987$	113921.	3.67392
$988$	0	0
$989$	−33906.4	−1.09015
$990$	0	0
$991$	42532.3	1.36335	0.681676	−	0.731654i	$-0.261251\pi$
$991$	42532.3	1.36335	0.681676	+	0.731654i	$0.261251\pi$
$992$	0	0
$993$	−40121.2	−1.28218
$994$	0	0
$995$	−1267.67	−0.0403898
$996$	0	0
$997$	−6320.28	−0.200767	−0.100384	−	0.994949i	$-0.532007\pi$
$997$	−6320.28	−0.200767	−0.100384	+	0.994949i	$0.532007\pi$
$998$	0	0
$999$	4753.91	0.150558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.p.1.2		2
4.3	odd	2		1216.4.a.g.1.1		2
8.3	odd	2		38.4.a.c.1.2	✓	2
8.5	even	2		304.4.a.c.1.1		2
24.11	even	2		342.4.a.h.1.2		2
40.3	even	4		950.4.b.i.799.2		4
40.19	odd	2		950.4.a.e.1.1		2
40.27	even	4		950.4.b.i.799.3		4
56.27	even	2		1862.4.a.e.1.1		2
152.75	even	2		722.4.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.c.1.2	✓	2	8.3	odd	2
304.4.a.c.1.1		2	8.5	even	2
342.4.a.h.1.2		2	24.11	even	2
722.4.a.f.1.1		2	152.75	even	2
950.4.a.e.1.1		2	40.19	odd	2
950.4.b.i.799.2		4	40.3	even	4
950.4.b.i.799.3		4	40.27	even	4
1216.4.a.g.1.1		2	4.3	odd	2
1216.4.a.p.1.2		2	1.1	even	1	trivial
1862.4.a.e.1.1		2	56.27	even	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)

\(f(q)\)	\(=\)	\(q+8.77200 q^{3} +17.3160 q^{5} +26.0880 q^{7} +49.9480 q^{9} +O(q^{10})\)	\(q+8.77200 q^{3} +17.3160 q^{5} +26.0880 q^{7} +49.9480 q^{9} -4.22800 q^{11} -64.0360 q^{13} +151.896 q^{15} -48.5440 q^{17} +19.0000 q^{19} +228.844 q^{21} -92.0360 q^{23} +174.844 q^{25} +201.300 q^{27} +88.2120 q^{29} +81.9681 q^{31} -37.0880 q^{33} +451.740 q^{35} +23.6161 q^{37} -561.724 q^{39} +17.7200 q^{41} +368.404 q^{43} +864.900 q^{45} +497.812 q^{47} +337.584 q^{49} -425.828 q^{51} +536.876 q^{53} -73.2120 q^{55} +166.668 q^{57} -36.7000 q^{59} -630.692 q^{61} +1303.04 q^{63} -1108.85 q^{65} +282.556 q^{67} -807.340 q^{69} -595.552 q^{71} -597.048 q^{73} +1533.73 q^{75} -110.300 q^{77} -427.224 q^{79} +417.208 q^{81} +493.768 q^{83} -840.588 q^{85} +773.796 q^{87} -921.136 q^{89} -1670.57 q^{91} +719.024 q^{93} +329.004 q^{95} +1082.74 q^{97} -211.180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{3} + 9 q^{5} + 18 q^{7} + 23 q^{9}+O(q^{10}) \) 2 * q + 9 * q^3 + 9 * q^5 + 18 * q^7 + 23 * q^9	\( 2 q + 9 q^{3} + 9 q^{5} + 18 q^{7} + 23 q^{9} - 17 q^{11} - 17 q^{13} + 150 q^{15} - 80 q^{17} + 38 q^{19} + 227 q^{21} - 73 q^{23} + 119 q^{25} + 189 q^{27} - 3 q^{29} - 212 q^{31} - 40 q^{33} + 519 q^{35} - 192 q^{37} - 551 q^{39} - 50 q^{41} + 677 q^{43} + 1089 q^{45} + 389 q^{47} + 60 q^{49} - 433 q^{51} + 1219 q^{53} + 33 q^{55} + 171 q^{57} - 287 q^{59} - 313 q^{61} + 1521 q^{63} - 1500 q^{65} + 1223 q^{67} - 803 q^{69} - 200 q^{71} + 378 q^{73} + 1521 q^{75} - 7 q^{77} - 1350 q^{79} + 1142 q^{81} - 670 q^{83} - 579 q^{85} + 753 q^{87} - 236 q^{89} - 2051 q^{91} + 652 q^{93} + 171 q^{95} + 1294 q^{97} + 133 q^{99}+O(q^{100}) \) 2 * q + 9 * q^3 + 9 * q^5 + 18 * q^7 + 23 * q^9 - 17 * q^11 - 17 * q^13 + 150 * q^15 - 80 * q^17 + 38 * q^19 + 227 * q^21 - 73 * q^23 + 119 * q^25 + 189 * q^27 - 3 * q^29 - 212 * q^31 - 40 * q^33 + 519 * q^35 - 192 * q^37 - 551 * q^39 - 50 * q^41 + 677 * q^43 + 1089 * q^45 + 389 * q^47 + 60 * q^49 - 433 * q^51 + 1219 * q^53 + 33 * q^55 + 171 * q^57 - 287 * q^59 - 313 * q^61 + 1521 * q^63 - 1500 * q^65 + 1223 * q^67 - 803 * q^69 - 200 * q^71 + 378 * q^73 + 1521 * q^75 - 7 * q^77 - 1350 * q^79 + 1142 * q^81 - 670 * q^83 - 579 * q^85 + 753 * q^87 - 236 * q^89 - 2051 * q^91 + 652 * q^93 + 171 * q^95 + 1294 * q^97 + 133 * q^99

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