LMFDB - Embedded newform 1216.4.a.ba.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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 106x^{3} - 401x^{2} + 356x + 2112 $ x^5 - 106x^3 - 401x^2 + 356*x + 2112
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.76535$ 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)

$f(q)$	$=$	$q-5.24575 q^{3} -18.7154 q^{5} +28.3947 q^{7} +0.517848 q^{9} +O(q^{10})$	$q-5.24575 q^{3} -18.7154 q^{5} +28.3947 q^{7} +0.517848 q^{9} +70.3030 q^{11} -5.85425 q^{13} +98.1762 q^{15} -50.9448 q^{17} -19.0000 q^{19} -148.951 q^{21} +60.6722 q^{23} +225.266 q^{25} +138.919 q^{27} +16.2560 q^{29} -223.646 q^{31} -368.792 q^{33} -531.417 q^{35} -301.646 q^{37} +30.7099 q^{39} +209.103 q^{41} +225.873 q^{43} -9.69173 q^{45} -299.267 q^{47} +463.256 q^{49} +267.244 q^{51} -40.5816 q^{53} -1315.75 q^{55} +99.6692 q^{57} +11.8126 q^{59} -656.327 q^{61} +14.7041 q^{63} +109.565 q^{65} +500.509 q^{67} -318.271 q^{69} -569.538 q^{71} -1150.82 q^{73} -1181.69 q^{75} +1996.23 q^{77} +141.299 q^{79} -742.714 q^{81} -546.177 q^{83} +953.453 q^{85} -85.2748 q^{87} +1556.66 q^{89} -166.230 q^{91} +1173.19 q^{93} +355.593 q^{95} +485.716 q^{97} +36.4063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 22 q^{7} + 83 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 22 * q^7 + 83 * q^9	$ 5 q - 2 q^{3} + 22 q^{7} + 83 q^{9} - 6 q^{11} - 82 q^{13} + 204 q^{15} + 234 q^{17} - 95 q^{19} - 308 q^{21} - 60 q^{23} + 549 q^{25} + 190 q^{27} - 210 q^{29} - 224 q^{31} + 704 q^{33} - 42 q^{35} - 614 q^{37} - 132 q^{39} + 194 q^{41} + 38 q^{43} + 516 q^{45} - 1066 q^{47} + 489 q^{49} + 1382 q^{51} + 42 q^{53} - 618 q^{55} + 38 q^{57} + 1090 q^{59} + 276 q^{61} - 1790 q^{63} - 184 q^{65} + 314 q^{67} + 268 q^{69} - 1276 q^{71} - 106 q^{73} + 2066 q^{75} + 1226 q^{77} + 52 q^{79} - 283 q^{81} + 580 q^{83} + 3498 q^{85} + 900 q^{87} + 810 q^{89} - 2522 q^{91} + 1332 q^{93} + 2694 q^{97} - 126 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 22 * q^7 + 83 * q^9 - 6 * q^11 - 82 * q^13 + 204 * q^15 + 234 * q^17 - 95 * q^19 - 308 * q^21 - 60 * q^23 + 549 * q^25 + 190 * q^27 - 210 * q^29 - 224 * q^31 + 704 * q^33 - 42 * q^35 - 614 * q^37 - 132 * q^39 + 194 * q^41 + 38 * q^43 + 516 * q^45 - 1066 * q^47 + 489 * q^49 + 1382 * q^51 + 42 * q^53 - 618 * q^55 + 38 * q^57 + 1090 * q^59 + 276 * q^61 - 1790 * q^63 - 184 * q^65 + 314 * q^67 + 268 * q^69 - 1276 * q^71 - 106 * q^73 + 2066 * q^75 + 1226 * q^77 + 52 * q^79 - 283 * q^81 + 580 * q^83 + 3498 * q^85 + 900 * q^87 + 810 * q^89 - 2522 * q^91 + 1332 * q^93 + 2694 * q^97 - 126 * 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
$3$	−5.24575	−1.00954	−0.504772	−	0.863253i	$-0.668423\pi$
$3$	−5.24575	−1.00954	−0.504772	+	0.863253i	$0.668423\pi$
$4$	0	0
$5$	−18.7154	−1.67396	−0.836978	−	0.547236i	$-0.815680\pi$
$5$	−18.7154	−1.67396	−0.836978	+	0.547236i	$0.815680\pi$
$6$	0	0
$7$	28.3947	1.53317	0.766583	−	0.642145i	$-0.221955\pi$
$7$	28.3947	1.53317	0.766583	+	0.642145i	$0.221955\pi$
$8$	0	0
$9$	0.517848	0.0191795
$10$	0	0
$11$	70.3030	1.92701	0.963507	−	0.267682i	$-0.0862575\pi$
$11$	70.3030	1.92701	0.963507	+	0.267682i	$0.0862575\pi$
$12$	0	0
$13$	−5.85425	−0.124898	−0.0624492	−	0.998048i	$-0.519891\pi$
$13$	−5.85425	−0.124898	−0.0624492	+	0.998048i	$0.519891\pi$
$14$	0	0
$15$	98.1762	1.68993
$16$	0	0
$17$	−50.9448	−0.726820	−0.363410	−	0.931629i	$-0.618388\pi$
$17$	−50.9448	−0.726820	−0.363410	+	0.931629i	$0.618388\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−148.951	−1.54780
$22$	0	0
$23$	60.6722	0.550045	0.275022	−	0.961438i	$-0.411315\pi$
$23$	60.6722	0.550045	0.275022	+	0.961438i	$0.411315\pi$
$24$	0	0
$25$	225.266	1.80213
$26$	0	0
$27$	138.919	0.990182
$28$	0	0
$29$	16.2560	0.104092	0.0520459	−	0.998645i	$-0.483426\pi$
$29$	16.2560	0.104092	0.0520459	+	0.998645i	$0.483426\pi$
$30$	0	0
$31$	−223.646	−1.29574	−0.647871	−	0.761750i	$-0.724340\pi$
$31$	−223.646	−1.29574	−0.647871	+	0.761750i	$0.724340\pi$
$32$	0	0
$33$	−368.792	−1.94541
$34$	0	0
$35$	−531.417	−2.56645
$36$	0	0
$37$	−301.646	−1.34028	−0.670139	−	0.742236i	$-0.733765\pi$
$37$	−301.646	−1.34028	−0.670139	+	0.742236i	$0.733765\pi$
$38$	0	0
$39$	30.7099	0.126090
$40$	0	0
$41$	209.103	0.796499	0.398249	−	0.917277i	$-0.369618\pi$
$41$	209.103	0.796499	0.398249	+	0.917277i	$0.369618\pi$
$42$	0	0
$43$	225.873	0.801052	0.400526	−	0.916285i	$-0.368827\pi$
$43$	225.873	0.801052	0.400526	+	0.916285i	$0.368827\pi$
$44$	0	0
$45$	−9.69173	−0.0321057
$46$	0	0
$47$	−299.267	−0.928778	−0.464389	−	0.885631i	$-0.653726\pi$
$47$	−299.267	−0.928778	−0.464389	+	0.885631i	$0.653726\pi$
$48$	0	0
$49$	463.256	1.35060
$50$	0	0
$51$	267.244	0.733757
$52$	0	0
$53$	−40.5816	−0.105176	−0.0525879	−	0.998616i	$-0.516747\pi$
$53$	−40.5816	−0.105176	−0.0525879	+	0.998616i	$0.516747\pi$
$54$	0	0
$55$	−1315.75	−3.22574
$56$	0	0
$57$	99.6692	0.231605
$58$	0	0
$59$	11.8126	0.0260656	0.0130328	−	0.999915i	$-0.495851\pi$
$59$	11.8126	0.0260656	0.0130328	+	0.999915i	$0.495851\pi$
$60$	0	0
$61$	−656.327	−1.37761	−0.688804	−	0.724948i	$-0.741864\pi$
$61$	−656.327	−1.37761	−0.688804	+	0.724948i	$0.741864\pi$
$62$	0	0
$63$	14.7041	0.0294054
$64$	0	0
$65$	109.565	0.209074
$66$	0	0
$67$	500.509	0.912640	0.456320	−	0.889816i	$-0.349167\pi$
$67$	500.509	0.912640	0.456320	+	0.889816i	$0.349167\pi$
$68$	0	0
$69$	−318.271	−0.555294
$70$	0	0
$71$	−569.538	−0.951995	−0.475998	−	0.879447i	$-0.657913\pi$
$71$	−569.538	−0.951995	−0.475998	+	0.879447i	$0.657913\pi$
$72$	0	0
$73$	−1150.82	−1.84512	−0.922558	−	0.385858i	$-0.873905\pi$
$73$	−1150.82	−1.84512	−0.922558	+	0.385858i	$0.873905\pi$
$74$	0	0
$75$	−1181.69	−1.81933
$76$	0	0
$77$	1996.23	2.95444
$78$	0	0
$79$	141.299	0.201233	0.100616	−	0.994925i	$-0.467919\pi$
$79$	141.299	0.201233	0.100616	+	0.994925i	$0.467919\pi$
$80$	0	0
$81$	−742.714	−1.01881
$82$	0	0
$83$	−546.177	−0.722298	−0.361149	−	0.932508i	$-0.617615\pi$
$83$	−546.177	−0.722298	−0.361149	+	0.932508i	$0.617615\pi$
$84$	0	0
$85$	953.453	1.21666
$86$	0	0
$87$	−85.2748	−0.105085
$88$	0	0
$89$	1556.66	1.85399	0.926997	−	0.375070i	$-0.122381\pi$
$89$	1556.66	1.85399	0.926997	+	0.375070i	$0.122381\pi$
$90$	0	0
$91$	−166.230	−0.191490
$92$	0	0
$93$	1173.19	1.30811
$94$	0	0
$95$	355.593	0.384032
$96$	0	0
$97$	485.716	0.508422	0.254211	−	0.967149i	$-0.418184\pi$
$97$	485.716	0.508422	0.254211	+	0.967149i	$0.418184\pi$
$98$	0	0
$99$	36.4063	0.0369593
$100$	0	0
$101$	1822.02	1.79502	0.897512	−	0.440990i	$-0.145373\pi$
$101$	1822.02	1.79502	0.897512	+	0.440990i	$0.145373\pi$
$102$	0	0
$103$	1883.11	1.80144	0.900722	−	0.434397i	$-0.143038\pi$
$103$	1883.11	1.80144	0.900722	+	0.434397i	$0.143038\pi$
$104$	0	0
$105$	2787.68	2.59095
$106$	0	0
$107$	1545.87	1.39668	0.698342	−	0.715764i	$-0.253921\pi$
$107$	1545.87	1.39668	0.698342	+	0.715764i	$0.253921\pi$
$108$	0	0
$109$	−150.811	−0.132524	−0.0662620	−	0.997802i	$-0.521107\pi$
$109$	−150.811	−0.132524	−0.0662620	+	0.997802i	$0.521107\pi$
$110$	0	0
$111$	1582.36	1.35307
$112$	0	0
$113$	−1262.79	−1.05126	−0.525632	−	0.850712i	$-0.676171\pi$
$113$	−1262.79	−1.05126	−0.525632	+	0.850712i	$0.676171\pi$
$114$	0	0
$115$	−1135.50	−0.920750
$116$	0	0
$117$	−3.03161	−0.00239549
$118$	0	0
$119$	−1446.56	−1.11434
$120$	0	0
$121$	3611.52	2.71339
$122$	0	0
$123$	−1096.90	−0.804101
$124$	0	0
$125$	−1876.52	−1.34273
$126$	0	0
$127$	1376.72	0.961925	0.480962	−	0.876741i	$-0.340287\pi$
$127$	1376.72	0.961925	0.480962	+	0.876741i	$0.340287\pi$
$128$	0	0
$129$	−1184.87	−0.808697
$130$	0	0
$131$	−957.917	−0.638882	−0.319441	−	0.947606i	$-0.603495\pi$
$131$	−957.917	−0.638882	−0.319441	+	0.947606i	$0.603495\pi$
$132$	0	0
$133$	−539.498	−0.351733
$134$	0	0
$135$	−2599.92	−1.65752
$136$	0	0
$137$	95.2627	0.0594076	0.0297038	−	0.999559i	$-0.490544\pi$
$137$	95.2627	0.0594076	0.0297038	+	0.999559i	$0.490544\pi$
$138$	0	0
$139$	743.815	0.453882	0.226941	−	0.973909i	$-0.427128\pi$
$139$	743.815	0.453882	0.226941	+	0.973909i	$0.427128\pi$
$140$	0	0
$141$	1569.88	0.937643
$142$	0	0
$143$	−411.572	−0.240681
$144$	0	0
$145$	−304.237	−0.174245
$146$	0	0
$147$	−2430.12	−1.36349
$148$	0	0
$149$	783.730	0.430911	0.215455	−	0.976514i	$-0.430876\pi$
$149$	783.730	0.430911	0.215455	+	0.976514i	$0.430876\pi$
$150$	0	0
$151$	2050.55	1.10511	0.552553	−	0.833477i	$-0.313653\pi$
$151$	2050.55	1.10511	0.552553	+	0.833477i	$0.313653\pi$
$152$	0	0
$153$	−26.3817	−0.0139401
$154$	0	0
$155$	4185.62	2.16901
$156$	0	0
$157$	−675.256	−0.343256	−0.171628	−	0.985162i	$-0.554903\pi$
$157$	−675.256	−0.343256	−0.171628	+	0.985162i	$0.554903\pi$
$158$	0	0
$159$	212.881	0.106180
$160$	0	0
$161$	1722.77	0.843310
$162$	0	0
$163$	1631.88	0.784165	0.392082	−	0.919930i	$-0.371755\pi$
$163$	1631.88	0.784165	0.392082	+	0.919930i	$0.371755\pi$
$164$	0	0
$165$	6902.09	3.25653
$166$	0	0
$167$	1460.33	0.676667	0.338333	−	0.941026i	$-0.390137\pi$
$167$	1460.33	0.676667	0.338333	+	0.941026i	$0.390137\pi$
$168$	0	0
$169$	−2162.73	−0.984400
$170$	0	0
$171$	−9.83911	−0.00440009
$172$	0	0
$173$	462.512	0.203261	0.101631	−	0.994822i	$-0.467594\pi$
$173$	462.512	0.203261	0.101631	+	0.994822i	$0.467594\pi$
$174$	0	0
$175$	6396.35	2.76297
$176$	0	0
$177$	−61.9660	−0.0263144
$178$	0	0
$179$	2463.77	1.02878	0.514388	−	0.857558i	$-0.328019\pi$
$179$	2463.77	1.02878	0.514388	+	0.857558i	$0.328019\pi$
$180$	0	0
$181$	−2767.14	−1.13635	−0.568177	−	0.822906i	$-0.692351\pi$
$181$	−2767.14	−1.13635	−0.568177	+	0.822906i	$0.692351\pi$
$182$	0	0
$183$	3442.92	1.39076
$184$	0	0
$185$	5645.42	2.24357
$186$	0	0
$187$	−3581.58	−1.40059
$188$	0	0
$189$	3944.55	1.51811
$190$	0	0
$191$	−216.262	−0.0819275	−0.0409637	−	0.999161i	$-0.513043\pi$
$191$	−216.262	−0.0819275	−0.0409637	+	0.999161i	$0.513043\pi$
$192$	0	0
$193$	396.764	0.147978	0.0739889	−	0.997259i	$-0.476427\pi$
$193$	396.764	0.147978	0.0739889	+	0.997259i	$0.476427\pi$
$194$	0	0
$195$	−574.749	−0.211070
$196$	0	0
$197$	−1454.38	−0.525992	−0.262996	−	0.964797i	$-0.584711\pi$
$197$	−1454.38	−0.525992	−0.262996	+	0.964797i	$0.584711\pi$
$198$	0	0
$199$	−2588.49	−0.922075	−0.461037	−	0.887381i	$-0.652523\pi$
$199$	−2588.49	−0.922075	−0.461037	+	0.887381i	$0.652523\pi$
$200$	0	0
$201$	−2625.54	−0.921350
$202$	0	0
$203$	461.583	0.159590
$204$	0	0
$205$	−3913.45	−1.33330
$206$	0	0
$207$	31.4189	0.0105496
$208$	0	0
$209$	−1335.76	−0.442088
$210$	0	0
$211$	−1712.05	−0.558589	−0.279295	−	0.960205i	$-0.590101\pi$
$211$	−1712.05	−0.558589	−0.279295	+	0.960205i	$0.590101\pi$
$212$	0	0
$213$	2987.65	0.961081
$214$	0	0
$215$	−4227.29	−1.34093
$216$	0	0
$217$	−6350.34	−1.98659
$218$	0	0
$219$	6036.92	1.86273
$220$	0	0
$221$	298.244	0.0907786
$222$	0	0
$223$	−2063.61	−0.619683	−0.309842	−	0.950788i	$-0.600276\pi$
$223$	−2063.61	−0.619683	−0.309842	+	0.950788i	$0.600276\pi$
$224$	0	0
$225$	116.654	0.0345640
$226$	0	0
$227$	3548.28	1.03748	0.518739	−	0.854932i	$-0.326402\pi$
$227$	3548.28	1.03748	0.518739	+	0.854932i	$0.326402\pi$
$228$	0	0
$229$	149.686	0.0431945	0.0215972	−	0.999767i	$-0.493125\pi$
$229$	149.686	0.0431945	0.0215972	+	0.999767i	$0.493125\pi$
$230$	0	0
$231$	−10471.7	−2.98263
$232$	0	0
$233$	−3016.89	−0.848254	−0.424127	−	0.905603i	$-0.639419\pi$
$233$	−3016.89	−0.848254	−0.424127	+	0.905603i	$0.639419\pi$
$234$	0	0
$235$	5600.90	1.55473
$236$	0	0
$237$	−741.219	−0.203153
$238$	0	0
$239$	4959.10	1.34217	0.671083	−	0.741382i	$-0.265829\pi$
$239$	4959.10	1.34217	0.671083	+	0.741382i	$0.265829\pi$
$240$	0	0
$241$	5515.58	1.47423	0.737115	−	0.675767i	$-0.236188\pi$
$241$	5515.58	1.47423	0.737115	+	0.675767i	$0.236188\pi$
$242$	0	0
$243$	145.284	0.0383538
$244$	0	0
$245$	−8670.02	−2.26085
$246$	0	0
$247$	111.231	0.0286536
$248$	0	0
$249$	2865.11	0.729192
$250$	0	0
$251$	−3546.11	−0.891746	−0.445873	−	0.895096i	$-0.647107\pi$
$251$	−3546.11	−0.891746	−0.445873	+	0.895096i	$0.647107\pi$
$252$	0	0
$253$	4265.44	1.05994
$254$	0	0
$255$	−5001.57	−1.22828
$256$	0	0
$257$	917.460	0.222683	0.111342	−	0.993782i	$-0.464485\pi$
$257$	917.460	0.222683	0.111342	+	0.993782i	$0.464485\pi$
$258$	0	0
$259$	−8565.13	−2.05487
$260$	0	0
$261$	8.41813	0.00199643
$262$	0	0
$263$	2479.92	0.581438	0.290719	−	0.956808i	$-0.406106\pi$
$263$	2479.92	0.581438	0.290719	+	0.956808i	$0.406106\pi$
$264$	0	0
$265$	759.502	0.176060
$266$	0	0
$267$	−8165.83	−1.87169
$268$	0	0
$269$	6309.83	1.43018	0.715088	−	0.699034i	$-0.246387\pi$
$269$	6309.83	1.43018	0.715088	+	0.699034i	$0.246387\pi$
$270$	0	0
$271$	316.709	0.0709914	0.0354957	−	0.999370i	$-0.488699\pi$
$271$	316.709	0.0709914	0.0354957	+	0.999370i	$0.488699\pi$
$272$	0	0
$273$	871.998	0.193318
$274$	0	0
$275$	15836.9	3.47273
$276$	0	0
$277$	114.210	0.0247733	0.0123866	−	0.999923i	$-0.496057\pi$
$277$	114.210	0.0247733	0.0123866	+	0.999923i	$0.496057\pi$
$278$	0	0
$279$	−115.814	−0.0248517
$280$	0	0
$281$	5847.06	1.24130	0.620651	−	0.784087i	$-0.286868\pi$
$281$	5847.06	1.24130	0.620651	+	0.784087i	$0.286868\pi$
$282$	0	0
$283$	35.9363	0.00754838	0.00377419	−	0.999993i	$-0.498799\pi$
$283$	35.9363	0.00754838	0.00377419	+	0.999993i	$0.498799\pi$
$284$	0	0
$285$	−1865.35	−0.387697
$286$	0	0
$287$	5937.41	1.22117
$288$	0	0
$289$	−2317.62	−0.471733
$290$	0	0
$291$	−2547.94	−0.513275
$292$	0	0
$293$	−2597.89	−0.517987	−0.258994	−	0.965879i	$-0.583391\pi$
$293$	−2597.89	−0.517987	−0.258994	+	0.965879i	$0.583391\pi$
$294$	0	0
$295$	−221.078	−0.0436327
$296$	0	0
$297$	9766.40	1.90809
$298$	0	0
$299$	−355.190	−0.0686996
$300$	0	0
$301$	6413.57	1.22815
$302$	0	0
$303$	−9557.83	−1.81216
$304$	0	0
$305$	12283.4	2.30605
$306$	0	0
$307$	−7510.29	−1.39620	−0.698102	−	0.715998i	$-0.745972\pi$
$307$	−7510.29	−1.39620	−0.698102	+	0.715998i	$0.745972\pi$
$308$	0	0
$309$	−9878.33	−1.81864
$310$	0	0
$311$	1282.09	0.233765	0.116882	−	0.993146i	$-0.462710\pi$
$311$	1282.09	0.233765	0.116882	+	0.993146i	$0.462710\pi$
$312$	0	0
$313$	−1816.97	−0.328119	−0.164060	−	0.986450i	$-0.552459\pi$
$313$	−1816.97	−0.328119	−0.164060	+	0.986450i	$0.552459\pi$
$314$	0	0
$315$	−275.193	−0.0492234
$316$	0	0
$317$	9574.10	1.69632	0.848162	−	0.529737i	$-0.177709\pi$
$317$	9574.10	1.69632	0.848162	+	0.529737i	$0.177709\pi$
$318$	0	0
$319$	1142.85	0.200586
$320$	0	0
$321$	−8109.26	−1.41002
$322$	0	0
$323$	967.952	0.166744
$324$	0	0
$325$	−1318.77	−0.225083
$326$	0	0
$327$	791.118	0.133789
$328$	0	0
$329$	−8497.58	−1.42397
$330$	0	0
$331$	−11094.8	−1.84237	−0.921186	−	0.389122i	$-0.872778\pi$
$331$	−11094.8	−1.84237	−0.921186	+	0.389122i	$0.872778\pi$
$332$	0	0
$333$	−156.207	−0.0257059
$334$	0	0
$335$	−9367.22	−1.52772
$336$	0	0
$337$	10324.0	1.66880	0.834401	−	0.551158i	$-0.185814\pi$
$337$	10324.0	1.66880	0.834401	+	0.551158i	$0.185814\pi$
$338$	0	0
$339$	6624.25	1.06130
$340$	0	0
$341$	−15723.0	−2.49691
$342$	0	0
$343$	3414.63	0.537530
$344$	0	0
$345$	5956.57	0.929538
$346$	0	0
$347$	1057.95	0.163671	0.0818353	−	0.996646i	$-0.473922\pi$
$347$	1057.95	0.163671	0.0818353	+	0.996646i	$0.473922\pi$
$348$	0	0
$349$	−7982.42	−1.22432	−0.612162	−	0.790732i	$-0.709700\pi$
$349$	−7982.42	−1.22432	−0.612162	+	0.790732i	$0.709700\pi$
$350$	0	0
$351$	−813.265	−0.123672
$352$	0	0
$353$	8838.49	1.33265	0.666325	−	0.745661i	$-0.267866\pi$
$353$	8838.49	1.33265	0.666325	+	0.745661i	$0.267866\pi$
$354$	0	0
$355$	10659.1	1.59360
$356$	0	0
$357$	7588.29	1.12497
$358$	0	0
$359$	1576.47	0.231764	0.115882	−	0.993263i	$-0.463031\pi$
$359$	1576.47	0.231764	0.115882	+	0.993263i	$0.463031\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−18945.1	−2.73928
$364$	0	0
$365$	21538.1	3.08864
$366$	0	0
$367$	−1818.46	−0.258646	−0.129323	−	0.991603i	$-0.541280\pi$
$367$	−1818.46	−0.258646	−0.129323	+	0.991603i	$0.541280\pi$
$368$	0	0
$369$	108.284	0.0152765
$370$	0	0
$371$	−1152.30	−0.161252
$372$	0	0
$373$	−6121.16	−0.849709	−0.424855	−	0.905262i	$-0.639675\pi$
$373$	−6121.16	−0.849709	−0.424855	+	0.905262i	$0.639675\pi$
$374$	0	0
$375$	9843.76	1.35555
$376$	0	0
$377$	−95.1667	−0.0130009
$378$	0	0
$379$	7910.25	1.07209	0.536045	−	0.844189i	$-0.319918\pi$
$379$	7910.25	1.07209	0.536045	+	0.844189i	$0.319918\pi$
$380$	0	0
$381$	−7221.94	−0.971106
$382$	0	0
$383$	13030.0	1.73838	0.869190	−	0.494478i	$-0.164641\pi$
$383$	13030.0	1.73838	0.869190	+	0.494478i	$0.164641\pi$
$384$	0	0
$385$	−37360.2	−4.94560
$386$	0	0
$387$	116.968	0.0153638
$388$	0	0
$389$	12763.4	1.66357	0.831784	−	0.555100i	$-0.187320\pi$
$389$	12763.4	1.66357	0.831784	+	0.555100i	$0.187320\pi$
$390$	0	0
$391$	−3090.93	−0.399783
$392$	0	0
$393$	5024.99	0.644980
$394$	0	0
$395$	−2644.47	−0.336855
$396$	0	0
$397$	5997.17	0.758159	0.379080	−	0.925364i	$-0.376241\pi$
$397$	5997.17	0.758159	0.379080	+	0.925364i	$0.376241\pi$
$398$	0	0
$399$	2830.07	0.355090
$400$	0	0
$401$	5525.92	0.688157	0.344079	−	0.938941i	$-0.388191\pi$
$401$	5525.92	0.688157	0.344079	+	0.938941i	$0.388191\pi$
$402$	0	0
$403$	1309.28	0.161836
$404$	0	0
$405$	13900.2	1.70545
$406$	0	0
$407$	−21206.6	−2.58273
$408$	0	0
$409$	−2339.42	−0.282829	−0.141414	−	0.989950i	$-0.545165\pi$
$409$	−2339.42	−0.282829	−0.141414	+	0.989950i	$0.545165\pi$
$410$	0	0
$411$	−499.724	−0.0599746
$412$	0	0
$413$	335.415	0.0399630
$414$	0	0
$415$	10221.9	1.20909
$416$	0	0
$417$	−3901.87	−0.458214
$418$	0	0
$419$	4794.39	0.559000	0.279500	−	0.960146i	$-0.409831\pi$
$419$	4794.39	0.559000	0.279500	+	0.960146i	$0.409831\pi$
$420$	0	0
$421$	−1743.22	−0.201804	−0.100902	−	0.994896i	$-0.532173\pi$
$421$	−1743.22	−0.201804	−0.100902	+	0.994896i	$0.532173\pi$
$422$	0	0
$423$	−154.975	−0.0178135
$424$	0	0
$425$	−11476.2	−1.30982
$426$	0	0
$427$	−18636.2	−2.11210
$428$	0	0
$429$	2159.00	0.242978
$430$	0	0
$431$	−10128.3	−1.13193	−0.565965	−	0.824429i	$-0.691496\pi$
$431$	−10128.3	−1.13193	−0.565965	+	0.824429i	$0.691496\pi$
$432$	0	0
$433$	2375.38	0.263634	0.131817	−	0.991274i	$-0.457919\pi$
$433$	2375.38	0.263634	0.131817	+	0.991274i	$0.457919\pi$
$434$	0	0
$435$	1595.95	0.175908
$436$	0	0
$437$	−1152.77	−0.126189
$438$	0	0
$439$	10347.4	1.12495	0.562474	−	0.826815i	$-0.309850\pi$
$439$	10347.4	1.12495	0.562474	+	0.826815i	$0.309850\pi$
$440$	0	0
$441$	239.896	0.0259039
$442$	0	0
$443$	3972.45	0.426042	0.213021	−	0.977048i	$-0.431670\pi$
$443$	3972.45	0.426042	0.213021	+	0.977048i	$0.431670\pi$
$444$	0	0
$445$	−29133.5	−3.10350
$446$	0	0
$447$	−4111.25	−0.435023
$448$	0	0
$449$	−5263.35	−0.553213	−0.276606	−	0.960983i	$-0.589210\pi$
$449$	−5263.35	−0.553213	−0.276606	+	0.960983i	$0.589210\pi$
$450$	0	0
$451$	14700.6	1.53486
$452$	0	0
$453$	−10756.6	−1.11565
$454$	0	0
$455$	3111.05	0.320546
$456$	0	0
$457$	−1413.12	−0.144646	−0.0723228	−	0.997381i	$-0.523041\pi$
$457$	−1413.12	−0.144646	−0.0723228	+	0.997381i	$0.523041\pi$
$458$	0	0
$459$	−7077.19	−0.719684
$460$	0	0
$461$	8546.09	0.863408	0.431704	−	0.902015i	$-0.357913\pi$
$461$	8546.09	0.863408	0.431704	+	0.902015i	$0.357913\pi$
$462$	0	0
$463$	5384.63	0.540486	0.270243	−	0.962792i	$-0.412896\pi$
$463$	5384.63	0.540486	0.270243	+	0.962792i	$0.412896\pi$
$464$	0	0
$465$	−21956.7	−2.18972
$466$	0	0
$467$	6787.88	0.672603	0.336302	−	0.941754i	$-0.390824\pi$
$467$	6787.88	0.672603	0.336302	+	0.941754i	$0.390824\pi$
$468$	0	0
$469$	14211.8	1.39923
$470$	0	0
$471$	3542.22	0.346533
$472$	0	0
$473$	15879.5	1.54364
$474$	0	0
$475$	−4280.06	−0.413437
$476$	0	0
$477$	−21.0151	−0.00201722
$478$	0	0
$479$	10302.1	0.982707	0.491353	−	0.870960i	$-0.336502\pi$
$479$	10302.1	0.982707	0.491353	+	0.870960i	$0.336502\pi$
$480$	0	0
$481$	1765.91	0.167398
$482$	0	0
$483$	−9037.19	−0.851359
$484$	0	0
$485$	−9090.36	−0.851077
$486$	0	0
$487$	2979.51	0.277237	0.138618	−	0.990346i	$-0.455734\pi$
$487$	2979.51	0.277237	0.138618	+	0.990346i	$0.455734\pi$
$488$	0	0
$489$	−8560.44	−0.791649
$490$	0	0
$491$	4095.83	0.376461	0.188231	−	0.982125i	$-0.439725\pi$
$491$	4095.83	0.376461	0.188231	+	0.982125i	$0.439725\pi$
$492$	0	0
$493$	−828.159	−0.0756560
$494$	0	0
$495$	−681.358	−0.0618682
$496$	0	0
$497$	−16171.8	−1.45957
$498$	0	0
$499$	20923.6	1.87709	0.938546	−	0.345153i	$-0.112173\pi$
$499$	20923.6	1.87709	0.938546	+	0.345153i	$0.112173\pi$
$500$	0	0
$501$	−7660.50	−0.683125
$502$	0	0
$503$	1625.48	0.144089	0.0720445	−	0.997401i	$-0.477048\pi$
$503$	1625.48	0.144089	0.0720445	+	0.997401i	$0.477048\pi$
$504$	0	0
$505$	−34099.8	−3.00479
$506$	0	0
$507$	11345.1	0.993796
$508$	0	0
$509$	3892.33	0.338948	0.169474	−	0.985535i	$-0.445793\pi$
$509$	3892.33	0.338948	0.169474	+	0.985535i	$0.445793\pi$
$510$	0	0
$511$	−32677.2	−2.82887
$512$	0	0
$513$	−2639.45	−0.227163
$514$	0	0
$515$	−35243.2	−3.01554
$516$	0	0
$517$	−21039.4	−1.78977
$518$	0	0
$519$	−2426.22	−0.205201
$520$	0	0
$521$	18590.2	1.56325	0.781623	−	0.623751i	$-0.214392\pi$
$521$	18590.2	1.56325	0.781623	+	0.623751i	$0.214392\pi$
$522$	0	0
$523$	−17808.6	−1.48894	−0.744470	−	0.667655i	$-0.767298\pi$
$523$	−17808.6	−1.48894	−0.744470	+	0.667655i	$0.767298\pi$
$524$	0	0
$525$	−33553.7	−2.78934
$526$	0	0
$527$	11393.6	0.941770
$528$	0	0
$529$	−8485.89	−0.697451
$530$	0	0
$531$	6.11714	0.000499927 0
$532$	0	0
$533$	−1224.14	−0.0994813
$534$	0	0
$535$	−28931.7	−2.33799
$536$	0	0
$537$	−12924.3	−1.03859
$538$	0	0
$539$	32568.3	2.60263
$540$	0	0
$541$	−5395.14	−0.428753	−0.214376	−	0.976751i	$-0.568772\pi$
$541$	−5395.14	−0.428753	−0.214376	+	0.976751i	$0.568772\pi$
$542$	0	0
$543$	14515.7	1.14720
$544$	0	0
$545$	2822.49	0.221839
$546$	0	0
$547$	9917.46	0.775210	0.387605	−	0.921826i	$-0.373302\pi$
$547$	9917.46	0.775210	0.387605	+	0.921826i	$0.373302\pi$
$548$	0	0
$549$	−339.877	−0.0264219
$550$	0	0
$551$	−308.864	−0.0238803
$552$	0	0
$553$	4012.14	0.308524
$554$	0	0
$555$	−29614.4	−2.26498
$556$	0	0
$557$	−14839.3	−1.12884	−0.564418	−	0.825489i	$-0.690899\pi$
$557$	−14839.3	−1.12884	−0.564418	+	0.825489i	$0.690899\pi$
$558$	0	0
$559$	−1322.32	−0.100050
$560$	0	0
$561$	18788.0	1.41396
$562$	0	0
$563$	12211.8	0.914152	0.457076	−	0.889428i	$-0.348897\pi$
$563$	12211.8	0.914152	0.457076	+	0.889428i	$0.348897\pi$
$564$	0	0
$565$	23633.5	1.75977
$566$	0	0
$567$	−21089.1	−1.56201
$568$	0	0
$569$	5877.66	0.433048	0.216524	−	0.976277i	$-0.430528\pi$
$569$	5877.66	0.433048	0.216524	+	0.976277i	$0.430528\pi$
$570$	0	0
$571$	−5167.65	−0.378738	−0.189369	−	0.981906i	$-0.560644\pi$
$571$	−5167.65	−0.378738	−0.189369	+	0.981906i	$0.560644\pi$
$572$	0	0
$573$	1134.45	0.0827094
$574$	0	0
$575$	13667.4	0.991252
$576$	0	0
$577$	6348.30	0.458030	0.229015	−	0.973423i	$-0.426450\pi$
$577$	6348.30	0.458030	0.229015	+	0.973423i	$0.426450\pi$
$578$	0	0
$579$	−2081.33	−0.149390
$580$	0	0
$581$	−15508.5	−1.10740
$582$	0	0
$583$	−2853.01	−0.202675
$584$	0	0
$585$	56.7378	0.00400995
$586$	0	0
$587$	−24451.8	−1.71931	−0.859655	−	0.510875i	$-0.829321\pi$
$587$	−24451.8	−1.71931	−0.859655	+	0.510875i	$0.829321\pi$
$588$	0	0
$589$	4249.27	0.297263
$590$	0	0
$591$	7629.32	0.531012
$592$	0	0
$593$	8842.92	0.612369	0.306185	−	0.951972i	$-0.400947\pi$
$593$	8842.92	0.612369	0.306185	+	0.951972i	$0.400947\pi$
$594$	0	0
$595$	27073.0	1.86535
$596$	0	0
$597$	13578.5	0.930875
$598$	0	0
$599$	10999.9	0.750325	0.375163	−	0.926959i	$-0.377587\pi$
$599$	10999.9	0.750325	0.375163	+	0.926959i	$0.377587\pi$
$600$	0	0
$601$	−8695.13	−0.590153	−0.295076	−	0.955474i	$-0.595345\pi$
$601$	−8695.13	−0.590153	−0.295076	+	0.955474i	$0.595345\pi$
$602$	0	0
$603$	259.187	0.0175040
$604$	0	0
$605$	−67591.0	−4.54209
$606$	0	0
$607$	−4256.02	−0.284591	−0.142295	−	0.989824i	$-0.545448\pi$
$607$	−4256.02	−0.284591	−0.142295	+	0.989824i	$0.545448\pi$
$608$	0	0
$609$	−2421.35	−0.161113
$610$	0	0
$611$	1751.99	0.116003
$612$	0	0
$613$	3583.05	0.236082	0.118041	−	0.993009i	$-0.462339\pi$
$613$	3583.05	0.236082	0.118041	+	0.993009i	$0.462339\pi$
$614$	0	0
$615$	20529.0	1.34603
$616$	0	0
$617$	−19236.1	−1.25513	−0.627566	−	0.778563i	$-0.715949\pi$
$617$	−19236.1	−1.25513	−0.627566	+	0.778563i	$0.715949\pi$
$618$	0	0
$619$	−3730.08	−0.242204	−0.121102	−	0.992640i	$-0.538643\pi$
$619$	−3730.08	−0.242204	−0.121102	+	0.992640i	$0.538643\pi$
$620$	0	0
$621$	8428.50	0.544644
$622$	0	0
$623$	44200.8	2.84248
$624$	0	0
$625$	6961.58	0.445541
$626$	0	0
$627$	7007.05	0.446307
$628$	0	0
$629$	15367.3	0.974140
$630$	0	0
$631$	20068.5	1.26611	0.633053	−	0.774108i	$-0.281801\pi$
$631$	20068.5	1.26611	0.633053	+	0.774108i	$0.281801\pi$
$632$	0	0
$633$	8980.98	0.563921
$634$	0	0
$635$	−25765.9	−1.61022
$636$	0	0
$637$	−2712.02	−0.168688
$638$	0	0
$639$	−294.934	−0.0182588
$640$	0	0
$641$	19620.2	1.20897	0.604487	−	0.796615i	$-0.293378\pi$
$641$	19620.2	1.20897	0.604487	+	0.796615i	$0.293378\pi$
$642$	0	0
$643$	−12790.7	−0.784471	−0.392236	−	0.919865i	$-0.628298\pi$
$643$	−12790.7	−0.784471	−0.392236	+	0.919865i	$0.628298\pi$
$644$	0	0
$645$	22175.3	1.35372
$646$	0	0
$647$	−14999.1	−0.911398	−0.455699	−	0.890134i	$-0.650611\pi$
$647$	−14999.1	−0.911398	−0.455699	+	0.890134i	$0.650611\pi$
$648$	0	0
$649$	830.463	0.0502289
$650$	0	0
$651$	33312.3	2.00555
$652$	0	0
$653$	−16307.6	−0.977283	−0.488641	−	0.872485i	$-0.662507\pi$
$653$	−16307.6	−0.977283	−0.488641	+	0.872485i	$0.662507\pi$
$654$	0	0
$655$	17927.8	1.06946
$656$	0	0
$657$	−595.950	−0.0353885
$658$	0	0
$659$	9338.88	0.552035	0.276017	−	0.961153i	$-0.410985\pi$
$659$	9338.88	0.552035	0.276017	+	0.961153i	$0.410985\pi$
$660$	0	0
$661$	16696.2	0.982460	0.491230	−	0.871030i	$-0.336547\pi$
$661$	16696.2	0.982460	0.491230	+	0.871030i	$0.336547\pi$
$662$	0	0
$663$	−1564.51	−0.0916450
$664$	0	0
$665$	10096.9	0.588785
$666$	0	0
$667$	986.287	0.0572551
$668$	0	0
$669$	10825.2	0.625598
$670$	0	0
$671$	−46141.8	−2.65467
$672$	0	0
$673$	−24837.0	−1.42258	−0.711291	−	0.702898i	$-0.751889\pi$
$673$	−24837.0	−1.42258	−0.711291	+	0.702898i	$0.751889\pi$
$674$	0	0
$675$	31293.7	1.78444
$676$	0	0
$677$	−24773.0	−1.40636	−0.703178	−	0.711014i	$-0.748236\pi$
$677$	−24773.0	−1.40636	−0.703178	+	0.711014i	$0.748236\pi$
$678$	0	0
$679$	13791.7	0.779496
$680$	0	0
$681$	−18613.4	−1.04738
$682$	0	0
$683$	29954.0	1.67812	0.839060	−	0.544038i	$-0.183105\pi$
$683$	29954.0	1.67812	0.839060	+	0.544038i	$0.183105\pi$
$684$	0	0
$685$	−1782.88	−0.0994458
$686$	0	0
$687$	−785.215	−0.0436067
$688$	0	0
$689$	237.575	0.0131363
$690$	0	0
$691$	19009.7	1.04655	0.523273	−	0.852165i	$-0.324711\pi$
$691$	19009.7	1.04655	0.523273	+	0.852165i	$0.324711\pi$
$692$	0	0
$693$	1033.74	0.0566647
$694$	0	0
$695$	−13920.8	−0.759778
$696$	0	0
$697$	−10652.7	−0.578911
$698$	0	0
$699$	15825.8	0.856350
$700$	0	0
$701$	11960.9	0.644448	0.322224	−	0.946663i	$-0.395570\pi$
$701$	11960.9	0.644448	0.322224	+	0.946663i	$0.395570\pi$
$702$	0	0
$703$	5731.27	0.307481
$704$	0	0
$705$	−29380.9	−1.56957
$706$	0	0
$707$	51735.5	2.75207
$708$	0	0
$709$	−5684.44	−0.301105	−0.150553	−	0.988602i	$-0.548105\pi$
$709$	−5684.44	−0.301105	−0.150553	+	0.988602i	$0.548105\pi$
$710$	0	0
$711$	73.1714	0.00385955
$712$	0	0
$713$	−13569.1	−0.712715
$714$	0	0
$715$	7702.73	0.402889
$716$	0	0
$717$	−26014.2	−1.35498
$718$	0	0
$719$	−23183.6	−1.20251	−0.601254	−	0.799058i	$-0.705332\pi$
$719$	−23183.6	−1.20251	−0.601254	+	0.799058i	$0.705332\pi$
$720$	0	0
$721$	53470.3	2.76191
$722$	0	0
$723$	−28933.3	−1.48830
$724$	0	0
$725$	3661.93	0.187587
$726$	0	0
$727$	31616.0	1.61289	0.806446	−	0.591308i	$-0.201388\pi$
$727$	31616.0	1.61289	0.806446	+	0.591308i	$0.201388\pi$
$728$	0	0
$729$	19291.1	0.980092
$730$	0	0
$731$	−11507.0	−0.582221
$732$	0	0
$733$	20331.4	1.02450	0.512250	−	0.858837i	$-0.328812\pi$
$733$	20331.4	1.02450	0.512250	+	0.858837i	$0.328812\pi$
$734$	0	0
$735$	45480.7	2.28243
$736$	0	0
$737$	35187.3	1.75867
$738$	0	0
$739$	25290.9	1.25892	0.629460	−	0.777033i	$-0.283276\pi$
$739$	25290.9	1.25892	0.629460	+	0.777033i	$0.283276\pi$
$740$	0	0
$741$	−583.489	−0.0289271
$742$	0	0
$743$	−32088.0	−1.58438	−0.792190	−	0.610275i	$-0.791059\pi$
$743$	−32088.0	−1.58438	−0.792190	+	0.610275i	$0.791059\pi$
$744$	0	0
$745$	−14667.8	−0.721325
$746$	0	0
$747$	−282.836	−0.0138533
$748$	0	0
$749$	43894.6	2.14135
$750$	0	0
$751$	−8013.25	−0.389358	−0.194679	−	0.980867i	$-0.562366\pi$
$751$	−8013.25	−0.389358	−0.194679	+	0.980867i	$0.562366\pi$
$752$	0	0
$753$	18602.0	0.900257
$754$	0	0
$755$	−38376.8	−1.84990
$756$	0	0
$757$	21148.8	1.01541	0.507705	−	0.861531i	$-0.330494\pi$
$757$	21148.8	1.01541	0.507705	+	0.861531i	$0.330494\pi$
$758$	0	0
$759$	−22375.4	−1.07006
$760$	0	0
$761$	−19780.0	−0.942215	−0.471108	−	0.882076i	$-0.656146\pi$
$761$	−19780.0	−0.942215	−0.471108	+	0.882076i	$0.656146\pi$
$762$	0	0
$763$	−4282.23	−0.203181
$764$	0	0
$765$	493.743	0.0233351
$766$	0	0
$767$	−69.1541	−0.00325555
$768$	0	0
$769$	3498.63	0.164062	0.0820310	−	0.996630i	$-0.473859\pi$
$769$	3498.63	0.164062	0.0820310	+	0.996630i	$0.473859\pi$
$770$	0	0
$771$	−4812.76	−0.224809
$772$	0	0
$773$	24747.0	1.15147	0.575736	−	0.817635i	$-0.304715\pi$
$773$	24747.0	1.15147	0.575736	+	0.817635i	$0.304715\pi$
$774$	0	0
$775$	−50379.8	−2.33509
$776$	0	0
$777$	44930.5	2.07448
$778$	0	0
$779$	−3972.96	−0.182729
$780$	0	0
$781$	−40040.2	−1.83451
$782$	0	0
$783$	2258.26	0.103070
$784$	0	0
$785$	12637.7	0.574596
$786$	0	0
$787$	−2784.10	−0.126102	−0.0630512	−	0.998010i	$-0.520083\pi$
$787$	−2784.10	−0.126102	−0.0630512	+	0.998010i	$0.520083\pi$
$788$	0	0
$789$	−13009.0	−0.586987
$790$	0	0
$791$	−35856.4	−1.61176
$792$	0	0
$793$	3842.31	0.172061
$794$	0	0
$795$	−3984.15	−0.177740
$796$	0	0
$797$	15288.7	0.679488	0.339744	−	0.940518i	$-0.389660\pi$
$797$	15288.7	0.679488	0.339744	+	0.940518i	$0.389660\pi$
$798$	0	0
$799$	15246.1	0.675055
$800$	0	0
$801$	806.112	0.0355587
$802$	0	0
$803$	−80906.3	−3.55557
$804$	0	0
$805$	−32242.2	−1.41166
$806$	0	0
$807$	−33099.8	−1.44383
$808$	0	0
$809$	21793.1	0.947099	0.473549	−	0.880767i	$-0.342973\pi$
$809$	21793.1	0.947099	0.473549	+	0.880767i	$0.342973\pi$
$810$	0	0
$811$	−24630.6	−1.06646	−0.533230	−	0.845970i	$-0.679022\pi$
$811$	−24630.6	−1.06646	−0.533230	+	0.845970i	$0.679022\pi$
$812$	0	0
$813$	−1661.37	−0.0716690
$814$	0	0
$815$	−30541.3	−1.31266
$816$	0	0
$817$	−4291.58	−0.183774
$818$	0	0
$819$	−86.0816	−0.00367269
$820$	0	0
$821$	23773.7	1.01061	0.505304	−	0.862942i	$-0.331380\pi$
$821$	23773.7	1.01061	0.505304	+	0.862942i	$0.331380\pi$
$822$	0	0
$823$	−21124.1	−0.894704	−0.447352	−	0.894358i	$-0.647633\pi$
$823$	−21124.1	−0.894704	−0.447352	+	0.894358i	$0.647633\pi$
$824$	0	0
$825$	−83076.3	−3.50588
$826$	0	0
$827$	−2775.09	−0.116686	−0.0583430	−	0.998297i	$-0.518582\pi$
$827$	−2775.09	−0.116686	−0.0583430	+	0.998297i	$0.518582\pi$
$828$	0	0
$829$	23973.1	1.00437	0.502183	−	0.864762i	$-0.332531\pi$
$829$	23973.1	1.00437	0.502183	+	0.864762i	$0.332531\pi$
$830$	0	0
$831$	−599.115	−0.0250097
$832$	0	0
$833$	−23600.5	−0.981644
$834$	0	0
$835$	−27330.6	−1.13271
$836$	0	0
$837$	−31068.6	−1.28302
$838$	0	0
$839$	−30637.3	−1.26069	−0.630344	−	0.776316i	$-0.717086\pi$
$839$	−30637.3	−1.26069	−0.630344	+	0.776316i	$0.717086\pi$
$840$	0	0
$841$	−24124.7	−0.989165
$842$	0	0
$843$	−30672.2	−1.25315
$844$	0	0
$845$	40476.3	1.64784
$846$	0	0
$847$	102548.	4.16007
$848$	0	0
$849$	−188.513	−0.00762043
$850$	0	0
$851$	−18301.5	−0.737212
$852$	0	0
$853$	37084.2	1.48856	0.744279	−	0.667869i	$-0.232793\pi$
$853$	37084.2	1.48856	0.744279	+	0.667869i	$0.232793\pi$
$854$	0	0
$855$	184.143	0.00736556
$856$	0	0
$857$	−25346.5	−1.01029	−0.505145	−	0.863034i	$-0.668561\pi$
$857$	−25346.5	−1.01029	−0.505145	+	0.863034i	$0.668561\pi$
$858$	0	0
$859$	21350.6	0.848048	0.424024	−	0.905651i	$-0.360617\pi$
$859$	21350.6	0.848048	0.424024	+	0.905651i	$0.360617\pi$
$860$	0	0
$861$	−31146.2	−1.23282
$862$	0	0
$863$	13873.6	0.547235	0.273618	−	0.961839i	$-0.411780\pi$
$863$	13873.6	0.547235	0.273618	+	0.961839i	$0.411780\pi$
$864$	0	0
$865$	−8656.10	−0.340250
$866$	0	0
$867$	12157.7	0.476235
$868$	0	0
$869$	9933.76	0.387779
$870$	0	0
$871$	−2930.11	−0.113987
$872$	0	0
$873$	251.527	0.00975131
$874$	0	0
$875$	−53283.2	−2.05863
$876$	0	0
$877$	24283.4	0.934998	0.467499	−	0.883994i	$-0.345155\pi$
$877$	24283.4	0.934998	0.467499	+	0.883994i	$0.345155\pi$
$878$	0	0
$879$	13627.9	0.522931
$880$	0	0
$881$	−48539.1	−1.85622	−0.928108	−	0.372312i	$-0.878565\pi$
$881$	−48539.1	−1.85622	−0.928108	+	0.372312i	$0.878565\pi$
$882$	0	0
$883$	15961.7	0.608327	0.304163	−	0.952620i	$-0.401623\pi$
$883$	15961.7	0.608327	0.304163	+	0.952620i	$0.401623\pi$
$884$	0	0
$885$	1159.72	0.0440492
$886$	0	0
$887$	34162.3	1.29319	0.646594	−	0.762835i	$-0.276193\pi$
$887$	34162.3	1.29319	0.646594	+	0.762835i	$0.276193\pi$
$888$	0	0
$889$	39091.6	1.47479
$890$	0	0
$891$	−52215.0	−1.96327
$892$	0	0
$893$	5686.07	0.213076
$894$	0	0
$895$	−46110.4	−1.72212
$896$	0	0
$897$	1863.24	0.0693553
$898$	0	0
$899$	−3635.58	−0.134876
$900$	0	0
$901$	2067.43	0.0764439
$902$	0	0
$903$	−33644.0	−1.23987
$904$	0	0
$905$	51788.2	1.90221
$906$	0	0
$907$	15032.7	0.550335	0.275168	−	0.961396i	$-0.411267\pi$
$907$	15032.7	0.550335	0.275168	+	0.961396i	$0.411267\pi$
$908$	0	0
$909$	943.527	0.0344277
$910$	0	0
$911$	33788.2	1.22882	0.614410	−	0.788987i	$-0.289394\pi$
$911$	33788.2	1.22882	0.614410	+	0.788987i	$0.289394\pi$
$912$	0	0
$913$	−38397.9	−1.39188
$914$	0	0
$915$	−64435.7	−2.32806
$916$	0	0
$917$	−27199.7	−0.979513
$918$	0	0
$919$	16010.9	0.574700	0.287350	−	0.957826i	$-0.407226\pi$
$919$	16010.9	0.574700	0.287350	+	0.957826i	$0.407226\pi$
$920$	0	0
$921$	39397.0	1.40953
$922$	0	0
$923$	3334.22	0.118903
$924$	0	0
$925$	−67950.6	−2.41535
$926$	0	0
$927$	975.166	0.0345509
$928$	0	0
$929$	9722.13	0.343351	0.171675	−	0.985154i	$-0.445082\pi$
$929$	9722.13	0.343351	0.171675	+	0.985154i	$0.445082\pi$
$930$	0	0
$931$	−8801.87	−0.309849
$932$	0	0
$933$	−6725.54	−0.235996
$934$	0	0
$935$	67030.6	2.34453
$936$	0	0
$937$	11218.3	0.391126	0.195563	−	0.980691i	$-0.437347\pi$
$937$	11218.3	0.391126	0.195563	+	0.980691i	$0.437347\pi$
$938$	0	0
$939$	9531.37	0.331251
$940$	0	0
$941$	−29507.1	−1.02221	−0.511107	−	0.859517i	$-0.670765\pi$
$941$	−29507.1	−1.02221	−0.511107	+	0.859517i	$0.670765\pi$
$942$	0	0
$943$	12686.8	0.438110
$944$	0	0
$945$	−73823.8	−2.54126
$946$	0	0
$947$	30755.3	1.05535	0.527674	−	0.849447i	$-0.323064\pi$
$947$	30755.3	1.05535	0.527674	+	0.849447i	$0.323064\pi$
$948$	0	0
$949$	6737.20	0.230452
$950$	0	0
$951$	−50223.3	−1.71251
$952$	0	0
$953$	−11292.1	−0.383827	−0.191913	−	0.981412i	$-0.561469\pi$
$953$	−11292.1	−0.383827	−0.191913	+	0.981412i	$0.561469\pi$
$954$	0	0
$955$	4047.43	0.137143
$956$	0	0
$957$	−5995.08	−0.202501
$958$	0	0
$959$	2704.95	0.0910818
$960$	0	0
$961$	20226.4	0.678945
$962$	0	0
$963$	800.527	0.0267878
$964$	0	0
$965$	−7425.61	−0.247708
$966$	0	0
$967$	−2304.70	−0.0766434	−0.0383217	−	0.999265i	$-0.512201\pi$
$967$	−2304.70	−0.0766434	−0.0383217	+	0.999265i	$0.512201\pi$
$968$	0	0
$969$	−5077.63	−0.168335
$970$	0	0
$971$	31345.3	1.03596	0.517981	−	0.855392i	$-0.326684\pi$
$971$	31345.3	1.03596	0.517981	+	0.855392i	$0.326684\pi$
$972$	0	0
$973$	21120.4	0.695877
$974$	0	0
$975$	6917.91	0.227231
$976$	0	0
$977$	33205.9	1.08736	0.543680	−	0.839293i	$-0.317031\pi$
$977$	33205.9	1.08736	0.543680	+	0.839293i	$0.317031\pi$
$978$	0	0
$979$	109438.	3.57267
$980$	0	0
$981$	−78.0973	−0.00254175
$982$	0	0
$983$	34788.3	1.12876	0.564382	−	0.825514i	$-0.309115\pi$
$983$	34788.3	1.12876	0.564382	+	0.825514i	$0.309115\pi$
$984$	0	0
$985$	27219.3	0.880488
$986$	0	0
$987$	44576.2	1.43756
$988$	0	0
$989$	13704.2	0.440614
$990$	0	0
$991$	7196.76	0.230689	0.115344	−	0.993326i	$-0.463203\pi$
$991$	7196.76	0.230689	0.115344	+	0.993326i	$0.463203\pi$
$992$	0	0
$993$	58200.5	1.85996
$994$	0	0
$995$	48444.6	1.54351
$996$	0	0
$997$	43544.2	1.38321	0.691604	−	0.722277i	$-0.256905\pi$
$997$	43544.2	1.38321	0.691604	+	0.722277i	$0.256905\pi$
$998$	0	0
$999$	−41904.2	−1.32712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.ba.1.2		5
4.3	odd	2		1216.4.a.bb.1.4		5
8.3	odd	2		304.4.a.k.1.2		5
8.5	even	2		152.4.a.d.1.4	✓	5
24.5	odd	2		1368.4.a.m.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.d.1.4	✓	5	8.5	even	2
304.4.a.k.1.2		5	8.3	odd	2
1216.4.a.ba.1.2		5	1.1	even	1	trivial
1216.4.a.bb.1.4		5	4.3	odd	2
1368.4.a.m.1.1		5	24.5	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)

\(f(q)\)	\(=\)	\(q-5.24575 q^{3} -18.7154 q^{5} +28.3947 q^{7} +0.517848 q^{9} +O(q^{10})\)	\(q-5.24575 q^{3} -18.7154 q^{5} +28.3947 q^{7} +0.517848 q^{9} +70.3030 q^{11} -5.85425 q^{13} +98.1762 q^{15} -50.9448 q^{17} -19.0000 q^{19} -148.951 q^{21} +60.6722 q^{23} +225.266 q^{25} +138.919 q^{27} +16.2560 q^{29} -223.646 q^{31} -368.792 q^{33} -531.417 q^{35} -301.646 q^{37} +30.7099 q^{39} +209.103 q^{41} +225.873 q^{43} -9.69173 q^{45} -299.267 q^{47} +463.256 q^{49} +267.244 q^{51} -40.5816 q^{53} -1315.75 q^{55} +99.6692 q^{57} +11.8126 q^{59} -656.327 q^{61} +14.7041 q^{63} +109.565 q^{65} +500.509 q^{67} -318.271 q^{69} -569.538 q^{71} -1150.82 q^{73} -1181.69 q^{75} +1996.23 q^{77} +141.299 q^{79} -742.714 q^{81} -546.177 q^{83} +953.453 q^{85} -85.2748 q^{87} +1556.66 q^{89} -166.230 q^{91} +1173.19 q^{93} +355.593 q^{95} +485.716 q^{97} +36.4063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 22 q^{7} + 83 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 22 * q^7 + 83 * q^9	\( 5 q - 2 q^{3} + 22 q^{7} + 83 q^{9} - 6 q^{11} - 82 q^{13} + 204 q^{15} + 234 q^{17} - 95 q^{19} - 308 q^{21} - 60 q^{23} + 549 q^{25} + 190 q^{27} - 210 q^{29} - 224 q^{31} + 704 q^{33} - 42 q^{35} - 614 q^{37} - 132 q^{39} + 194 q^{41} + 38 q^{43} + 516 q^{45} - 1066 q^{47} + 489 q^{49} + 1382 q^{51} + 42 q^{53} - 618 q^{55} + 38 q^{57} + 1090 q^{59} + 276 q^{61} - 1790 q^{63} - 184 q^{65} + 314 q^{67} + 268 q^{69} - 1276 q^{71} - 106 q^{73} + 2066 q^{75} + 1226 q^{77} + 52 q^{79} - 283 q^{81} + 580 q^{83} + 3498 q^{85} + 900 q^{87} + 810 q^{89} - 2522 q^{91} + 1332 q^{93} + 2694 q^{97} - 126 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 22 * q^7 + 83 * q^9 - 6 * q^11 - 82 * q^13 + 204 * q^15 + 234 * q^17 - 95 * q^19 - 308 * q^21 - 60 * q^23 + 549 * q^25 + 190 * q^27 - 210 * q^29 - 224 * q^31 + 704 * q^33 - 42 * q^35 - 614 * q^37 - 132 * q^39 + 194 * q^41 + 38 * q^43 + 516 * q^45 - 1066 * q^47 + 489 * q^49 + 1382 * q^51 + 42 * q^53 - 618 * q^55 + 38 * q^57 + 1090 * q^59 + 276 * q^61 - 1790 * q^63 - 184 * q^65 + 314 * q^67 + 268 * q^69 - 1276 * q^71 - 106 * q^73 + 2066 * q^75 + 1226 * q^77 + 52 * q^79 - 283 * q^81 + 580 * q^83 + 3498 * q^85 + 900 * q^87 + 810 * q^89 - 2522 * q^91 + 1332 * q^93 + 2694 * q^97 - 126 * q^99

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