LMFDB - Embedded newform 1216.4.a.n.1.1

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{93}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 23 $ x^2 - x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.32183$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -17.2873 q^{5} +12.3563 q^{7} -26.0000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -17.2873 q^{5} +12.3563 q^{7} -26.0000 q^{9} -39.2873 q^{11} -18.9310 q^{13} -17.2873 q^{15} +100.149 q^{17} -19.0000 q^{19} +12.3563 q^{21} -158.218 q^{23} +173.851 q^{25} -53.0000 q^{27} -141.229 q^{29} -215.586 q^{31} -39.2873 q^{33} -213.608 q^{35} -209.862 q^{37} -18.9310 q^{39} +276.873 q^{41} +166.713 q^{43} +449.470 q^{45} +60.5524 q^{47} -190.321 q^{49} +100.149 q^{51} +76.2405 q^{53} +679.171 q^{55} -19.0000 q^{57} +575.575 q^{59} +91.0111 q^{61} -321.265 q^{63} +327.265 q^{65} -444.978 q^{67} -158.218 q^{69} +943.287 q^{71} -569.851 q^{73} +173.851 q^{75} -485.448 q^{77} -336.205 q^{79} +649.000 q^{81} +603.929 q^{83} -1731.31 q^{85} -141.229 q^{87} +1635.63 q^{89} -233.917 q^{91} -215.586 q^{93} +328.459 q^{95} +319.470 q^{97} +1021.47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9} - 40 q^{11} + 20 q^{13} + 4 q^{15} + 46 q^{17} - 38 q^{19} + 44 q^{21} - 220 q^{23} + 502 q^{25} - 106 q^{27} + 84 q^{29} - 84 q^{31} - 40 q^{33} + 460 q^{35} - 304 q^{37} + 20 q^{39} + 168 q^{41} + 372 q^{43} - 104 q^{45} + 584 q^{47} + 468 q^{49} + 46 q^{51} - 484 q^{53} + 664 q^{55} - 38 q^{57} + 1074 q^{59} - 88 q^{61} - 1144 q^{63} + 1156 q^{65} - 1430 q^{67} - 220 q^{69} + 1848 q^{71} - 1294 q^{73} + 502 q^{75} - 508 q^{77} + 832 q^{79} + 1298 q^{81} - 528 q^{83} - 2884 q^{85} + 84 q^{87} + 1844 q^{89} + 998 q^{91} - 84 q^{93} - 76 q^{95} - 364 q^{97} + 1040 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9 - 40 * q^11 + 20 * q^13 + 4 * q^15 + 46 * q^17 - 38 * q^19 + 44 * q^21 - 220 * q^23 + 502 * q^25 - 106 * q^27 + 84 * q^29 - 84 * q^31 - 40 * q^33 + 460 * q^35 - 304 * q^37 + 20 * q^39 + 168 * q^41 + 372 * q^43 - 104 * q^45 + 584 * q^47 + 468 * q^49 + 46 * q^51 - 484 * q^53 + 664 * q^55 - 38 * q^57 + 1074 * q^59 - 88 * q^61 - 1144 * q^63 + 1156 * q^65 - 1430 * q^67 - 220 * q^69 + 1848 * q^71 - 1294 * q^73 + 502 * q^75 - 508 * q^77 + 832 * q^79 + 1298 * q^81 - 528 * q^83 - 2884 * q^85 + 84 * q^87 + 1844 * q^89 + 998 * q^91 - 84 * q^93 - 76 * q^95 - 364 * q^97 + 1040 * 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
$3$	1.00000	0.192450	0.0962250	−	0.995360i	$-0.469323\pi$
$3$	1.00000	0.192450	0.0962250	+	0.995360i	$0.469323\pi$
$4$	0	0
$5$	−17.2873	−1.54622	−0.773112	−	0.634270i	$-0.781301\pi$
$5$	−17.2873	−1.54622	−0.773112	+	0.634270i	$0.781301\pi$
$6$	0	0
$7$	12.3563	0.667180	0.333590	−	0.942718i	$-0.391740\pi$
$7$	12.3563	0.667180	0.333590	+	0.942718i	$0.391740\pi$
$8$	0	0
$9$	−26.0000	−0.962963
$10$	0	0
$11$	−39.2873	−1.07687	−0.538435	−	0.842667i	$-0.680984\pi$
$11$	−39.2873	−1.07687	−0.538435	+	0.842667i	$0.680984\pi$
$12$	0	0
$13$	−18.9310	−0.403885	−0.201942	−	0.979397i	$-0.564725\pi$
$13$	−18.9310	−0.403885	−0.201942	+	0.979397i	$0.564725\pi$
$14$	0	0
$15$	−17.2873	−0.297571
$16$	0	0
$17$	100.149	1.42881	0.714404	−	0.699733i	$-0.246698\pi$
$17$	100.149	1.42881	0.714404	+	0.699733i	$0.246698\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	12.3563	0.128399
$22$	0	0
$23$	−158.218	−1.43438	−0.717191	−	0.696877i	$-0.754572\pi$
$23$	−158.218	−1.43438	−0.717191	+	0.696877i	$0.754572\pi$
$24$	0	0
$25$	173.851	1.39081
$26$	0	0
$27$	−53.0000	−0.377772
$28$	0	0
$29$	−141.229	−0.904332	−0.452166	−	0.891934i	$-0.649349\pi$
$29$	−141.229	−0.904332	−0.452166	+	0.891934i	$0.649349\pi$
$30$	0	0
$31$	−215.586	−1.24904	−0.624522	−	0.781008i	$-0.714706\pi$
$31$	−215.586	−1.24904	−0.624522	+	0.781008i	$0.714706\pi$
$32$	0	0
$33$	−39.2873	−0.207244
$34$	0	0
$35$	−213.608	−1.03161
$36$	0	0
$37$	−209.862	−0.932462	−0.466231	−	0.884663i	$-0.654388\pi$
$37$	−209.862	−0.932462	−0.466231	+	0.884663i	$0.654388\pi$
$38$	0	0
$39$	−18.9310	−0.0777277
$40$	0	0
$41$	276.873	1.05464	0.527321	−	0.849666i	$-0.323197\pi$
$41$	276.873	1.05464	0.527321	+	0.849666i	$0.323197\pi$
$42$	0	0
$43$	166.713	0.591243	0.295621	−	0.955305i	$-0.404473\pi$
$43$	166.713	0.591243	0.295621	+	0.955305i	$0.404473\pi$
$44$	0	0
$45$	449.470	1.48896
$46$	0	0
$47$	60.5524	0.187925	0.0939625	−	0.995576i	$-0.470047\pi$
$47$	60.5524	0.187925	0.0939625	+	0.995576i	$0.470047\pi$
$48$	0	0
$49$	−190.321	−0.554871
$50$	0	0
$51$	100.149	0.274974
$52$	0	0
$53$	76.2405	0.197593	0.0987966	−	0.995108i	$-0.468501\pi$
$53$	76.2405	0.197593	0.0987966	+	0.995108i	$0.468501\pi$
$54$	0	0
$55$	679.171	1.66508
$56$	0	0
$57$	−19.0000	−0.0441511
$58$	0	0
$59$	575.575	1.27006	0.635029	−	0.772488i	$-0.280988\pi$
$59$	575.575	1.27006	0.635029	+	0.772488i	$0.280988\pi$
$60$	0	0
$61$	91.0111	0.191029	0.0955146	−	0.995428i	$-0.469550\pi$
$61$	91.0111	0.191029	0.0955146	+	0.995428i	$0.469550\pi$
$62$	0	0
$63$	−321.265	−0.642470
$64$	0	0
$65$	327.265	0.624496
$66$	0	0
$67$	−444.978	−0.811383	−0.405692	−	0.914010i	$-0.632969\pi$
$67$	−444.978	−0.811383	−0.405692	+	0.914010i	$0.632969\pi$
$68$	0	0
$69$	−158.218	−0.276047
$70$	0	0
$71$	943.287	1.57673	0.788363	−	0.615210i	$-0.210929\pi$
$71$	943.287	1.57673	0.788363	+	0.615210i	$0.210929\pi$
$72$	0	0
$73$	−569.851	−0.913644	−0.456822	−	0.889558i	$-0.651012\pi$
$73$	−569.851	−0.913644	−0.456822	+	0.889558i	$0.651012\pi$
$74$	0	0
$75$	173.851	0.267661
$76$	0	0
$77$	−485.448	−0.718466
$78$	0	0
$79$	−336.205	−0.478810	−0.239405	−	0.970920i	$-0.576952\pi$
$79$	−336.205	−0.478810	−0.239405	+	0.970920i	$0.576952\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	0	0
$83$	603.929	0.798672	0.399336	−	0.916805i	$-0.369241\pi$
$83$	603.929	0.798672	0.399336	+	0.916805i	$0.369241\pi$
$84$	0	0
$85$	−1731.31	−2.20926
$86$	0	0
$87$	−141.229	−0.174039
$88$	0	0
$89$	1635.63	1.94805	0.974025	−	0.226441i	$-0.0727089\pi$
$89$	1635.63	1.94805	0.974025	+	0.226441i	$0.0727089\pi$
$90$	0	0
$91$	−233.917	−0.269464
$92$	0	0
$93$	−215.586	−0.240378
$94$	0	0
$95$	328.459	0.354728
$96$	0	0
$97$	319.470	0.334405	0.167202	−	0.985923i	$-0.446527\pi$
$97$	319.470	0.334405	0.167202	+	0.985923i	$0.446527\pi$
$98$	0	0
$99$	1021.47	1.03699
$100$	0	0
$101$	−1595.19	−1.57156	−0.785781	−	0.618505i	$-0.787739\pi$
$101$	−1595.19	−1.57156	−0.785781	+	0.618505i	$0.787739\pi$
$102$	0	0
$103$	1927.51	1.84392	0.921959	−	0.387287i	$-0.126588\pi$
$103$	1927.51	1.84392	0.921959	+	0.387287i	$0.126588\pi$
$104$	0	0
$105$	−213.608	−0.198533
$106$	0	0
$107$	−717.089	−0.647884	−0.323942	−	0.946077i	$-0.605008\pi$
$107$	−717.089	−0.647884	−0.323942	+	0.946077i	$0.605008\pi$
$108$	0	0
$109$	−383.047	−0.336598	−0.168299	−	0.985736i	$-0.553827\pi$
$109$	−383.047	−0.336598	−0.168299	+	0.985736i	$0.553827\pi$
$110$	0	0
$111$	−209.862	−0.179452
$112$	0	0
$113$	622.917	0.518577	0.259288	−	0.965800i	$-0.416512\pi$
$113$	622.917	0.518577	0.259288	+	0.965800i	$0.416512\pi$
$114$	0	0
$115$	2735.17	2.21787
$116$	0	0
$117$	492.205	0.388926
$118$	0	0
$119$	1237.48	0.953273
$120$	0	0
$121$	212.492	0.159648
$122$	0	0
$123$	276.873	0.202966
$124$	0	0
$125$	−844.498	−0.604274
$126$	0	0
$127$	89.3095	0.0624011	0.0312005	−	0.999513i	$-0.490067\pi$
$127$	89.3095	0.0624011	0.0312005	+	0.999513i	$0.490067\pi$
$128$	0	0
$129$	166.713	0.113785
$130$	0	0
$131$	−348.690	−0.232559	−0.116279	−	0.993217i	$-0.537097\pi$
$131$	−348.690	−0.232559	−0.116279	+	0.993217i	$0.537097\pi$
$132$	0	0
$133$	−234.771	−0.153062
$134$	0	0
$135$	916.227	0.584120
$136$	0	0
$137$	1356.98	0.846237	0.423118	−	0.906074i	$-0.360935\pi$
$137$	1356.98	0.846237	0.423118	+	0.906074i	$0.360935\pi$
$138$	0	0
$139$	−629.287	−0.383996	−0.191998	−	0.981395i	$-0.561497\pi$
$139$	−629.287	−0.383996	−0.191998	+	0.981395i	$0.561497\pi$
$140$	0	0
$141$	60.5524	0.0361662
$142$	0	0
$143$	743.746	0.434931
$144$	0	0
$145$	2441.47	1.39830
$146$	0	0
$147$	−190.321	−0.106785
$148$	0	0
$149$	872.641	0.479796	0.239898	−	0.970798i	$-0.422886\pi$
$149$	872.641	0.479796	0.239898	+	0.970798i	$0.422886\pi$
$150$	0	0
$151$	−1892.71	−1.02004	−0.510021	−	0.860162i	$-0.670362\pi$
$151$	−1892.71	−1.02004	−0.510021	+	0.860162i	$0.670362\pi$
$152$	0	0
$153$	−2603.88	−1.37589
$154$	0	0
$155$	3726.90	1.93130
$156$	0	0
$157$	2939.51	1.49426	0.747129	−	0.664679i	$-0.231432\pi$
$157$	2939.51	1.49426	0.747129	+	0.664679i	$0.231432\pi$
$158$	0	0
$159$	76.2405	0.0380268
$160$	0	0
$161$	−1955.00	−0.956991
$162$	0	0
$163$	3301.38	1.58640	0.793202	−	0.608959i	$-0.208413\pi$
$163$	3301.38	1.58640	0.793202	+	0.608959i	$0.208413\pi$
$164$	0	0
$165$	679.171	0.320445
$166$	0	0
$167$	2630.39	1.21884	0.609418	−	0.792849i	$-0.291403\pi$
$167$	2630.39	1.21884	0.609418	+	0.792849i	$0.291403\pi$
$168$	0	0
$169$	−1838.62	−0.836877
$170$	0	0
$171$	494.000	0.220919
$172$	0	0
$173$	1482.41	0.651479	0.325740	−	0.945460i	$-0.394387\pi$
$173$	1482.41	0.651479	0.325740	+	0.945460i	$0.394387\pi$
$174$	0	0
$175$	2148.16	0.927918
$176$	0	0
$177$	575.575	0.244423
$178$	0	0
$179$	1048.00	0.437604	0.218802	−	0.975769i	$-0.429785\pi$
$179$	1048.00	0.437604	0.218802	+	0.975769i	$0.429785\pi$
$180$	0	0
$181$	761.354	0.312657	0.156329	−	0.987705i	$-0.450034\pi$
$181$	761.354	0.312657	0.156329	+	0.987705i	$0.450034\pi$
$182$	0	0
$183$	91.0111	0.0367636
$184$	0	0
$185$	3627.95	1.44179
$186$	0	0
$187$	−3934.59	−1.53864
$188$	0	0
$189$	−654.887	−0.252042
$190$	0	0
$191$	−4005.98	−1.51760	−0.758802	−	0.651322i	$-0.774215\pi$
$191$	−4005.98	−1.51760	−0.758802	+	0.651322i	$0.774215\pi$
$192$	0	0
$193$	−3855.87	−1.43809	−0.719046	−	0.694962i	$-0.755421\pi$
$193$	−3855.87	−1.43809	−0.719046	+	0.694962i	$0.755421\pi$
$194$	0	0
$195$	327.265	0.120184
$196$	0	0
$197$	2719.72	0.983616	0.491808	−	0.870704i	$-0.336336\pi$
$197$	2719.72	0.983616	0.491808	+	0.870704i	$0.336336\pi$
$198$	0	0
$199$	4454.56	1.58681	0.793405	−	0.608694i	$-0.208306\pi$
$199$	4454.56	1.58681	0.793405	+	0.608694i	$0.208306\pi$
$200$	0	0
$201$	−444.978	−0.156151
$202$	0	0
$203$	−1745.08	−0.603353
$204$	0	0
$205$	−4786.39	−1.63071
$206$	0	0
$207$	4113.67	1.38126
$208$	0	0
$209$	746.459	0.247051
$210$	0	0
$211$	−3481.06	−1.13576	−0.567881	−	0.823111i	$-0.692237\pi$
$211$	−3481.06	−1.13576	−0.567881	+	0.823111i	$0.692237\pi$
$212$	0	0
$213$	943.287	0.303441
$214$	0	0
$215$	−2882.01	−0.914194
$216$	0	0
$217$	−2663.85	−0.833337
$218$	0	0
$219$	−569.851	−0.175831
$220$	0	0
$221$	−1895.92	−0.577074
$222$	0	0
$223$	−3193.81	−0.959074	−0.479537	−	0.877522i	$-0.659195\pi$
$223$	−3193.81	−0.959074	−0.479537	+	0.877522i	$0.659195\pi$
$224$	0	0
$225$	−4520.12	−1.33930
$226$	0	0
$227$	5139.70	1.50279	0.751396	−	0.659852i	$-0.229381\pi$
$227$	5139.70	1.50279	0.751396	+	0.659852i	$0.229381\pi$
$228$	0	0
$229$	−4310.87	−1.24397	−0.621987	−	0.783027i	$-0.713674\pi$
$229$	−4310.87	−1.24397	−0.621987	+	0.783027i	$0.713674\pi$
$230$	0	0
$231$	−485.448	−0.138269
$232$	0	0
$233$	175.879	0.0494517	0.0247258	−	0.999694i	$-0.492129\pi$
$233$	175.879	0.0494517	0.0247258	+	0.999694i	$0.492129\pi$
$234$	0	0
$235$	−1046.79	−0.290574
$236$	0	0
$237$	−336.205	−0.0921470
$238$	0	0
$239$	−3811.82	−1.03166	−0.515829	−	0.856692i	$-0.672516\pi$
$239$	−3811.82	−1.03166	−0.515829	+	0.856692i	$0.672516\pi$
$240$	0	0
$241$	−1356.97	−0.362698	−0.181349	−	0.983419i	$-0.558046\pi$
$241$	−1356.97	−0.362698	−0.181349	+	0.983419i	$0.558046\pi$
$242$	0	0
$243$	2080.00	0.549103
$244$	0	0
$245$	3290.13	0.857954
$246$	0	0
$247$	359.688	0.0926575
$248$	0	0
$249$	603.929	0.153704
$250$	0	0
$251$	5465.11	1.37432	0.687160	−	0.726506i	$-0.258857\pi$
$251$	5465.11	1.37432	0.687160	+	0.726506i	$0.258857\pi$
$252$	0	0
$253$	6215.97	1.54464
$254$	0	0
$255$	−1731.31	−0.425172
$256$	0	0
$257$	1249.60	0.303299	0.151649	−	0.988434i	$-0.451542\pi$
$257$	1249.60	0.303299	0.151649	+	0.988434i	$0.451542\pi$
$258$	0	0
$259$	−2593.13	−0.622120
$260$	0	0
$261$	3671.96	0.870838
$262$	0	0
$263$	8043.06	1.88577	0.942883	−	0.333124i	$-0.108103\pi$
$263$	8043.06	1.88577	0.942883	+	0.333124i	$0.108103\pi$
$264$	0	0
$265$	−1317.99	−0.305523
$266$	0	0
$267$	1635.63	0.374902
$268$	0	0
$269$	−3051.13	−0.691565	−0.345782	−	0.938315i	$-0.612386\pi$
$269$	−3051.13	−0.691565	−0.345782	+	0.938315i	$0.612386\pi$
$270$	0	0
$271$	−1967.96	−0.441127	−0.220563	−	0.975373i	$-0.570790\pi$
$271$	−1967.96	−0.441127	−0.220563	+	0.975373i	$0.570790\pi$
$272$	0	0
$273$	−233.917	−0.0518583
$274$	0	0
$275$	−6830.13	−1.49772
$276$	0	0
$277$	4077.51	0.884454	0.442227	−	0.896903i	$-0.354189\pi$
$277$	4077.51	0.884454	0.442227	+	0.896903i	$0.354189\pi$
$278$	0	0
$279$	5605.23	1.20278
$280$	0	0
$281$	−7982.51	−1.69465	−0.847325	−	0.531075i	$-0.821788\pi$
$281$	−7982.51	−1.69465	−0.847325	+	0.531075i	$0.821788\pi$
$282$	0	0
$283$	4137.36	0.869049	0.434524	−	0.900660i	$-0.356917\pi$
$283$	4137.36	0.869049	0.434524	+	0.900660i	$0.356917\pi$
$284$	0	0
$285$	328.459	0.0682674
$286$	0	0
$287$	3421.14	0.703636
$288$	0	0
$289$	5116.86	1.04149
$290$	0	0
$291$	319.470	0.0643562
$292$	0	0
$293$	−6807.50	−1.35733	−0.678666	−	0.734447i	$-0.737442\pi$
$293$	−6807.50	−1.35733	−0.678666	+	0.734447i	$0.737442\pi$
$294$	0	0
$295$	−9950.13	−1.96379
$296$	0	0
$297$	2082.23	0.406812
$298$	0	0
$299$	2995.22	0.579325
$300$	0	0
$301$	2059.96	0.394466
$302$	0	0
$303$	−1595.19	−0.302447
$304$	0	0
$305$	−1573.34	−0.295374
$306$	0	0
$307$	2822.75	0.524765	0.262383	−	0.964964i	$-0.415492\pi$
$307$	2822.75	0.524765	0.262383	+	0.964964i	$0.415492\pi$
$308$	0	0
$309$	1927.51	0.354862
$310$	0	0
$311$	8221.36	1.49901	0.749503	−	0.662001i	$-0.230293\pi$
$311$	8221.36	1.49901	0.749503	+	0.662001i	$0.230293\pi$
$312$	0	0
$313$	−1896.42	−0.342466	−0.171233	−	0.985231i	$-0.554775\pi$
$313$	−1896.42	−0.342466	−0.171233	+	0.985231i	$0.554775\pi$
$314$	0	0
$315$	5553.81	0.993402
$316$	0	0
$317$	2875.66	0.509505	0.254753	−	0.967006i	$-0.418006\pi$
$317$	2875.66	0.509505	0.254753	+	0.967006i	$0.418006\pi$
$318$	0	0
$319$	5548.52	0.973848
$320$	0	0
$321$	−717.089	−0.124685
$322$	0	0
$323$	−1902.83	−0.327791
$324$	0	0
$325$	−3291.16	−0.561725
$326$	0	0
$327$	−383.047	−0.0647784
$328$	0	0
$329$	748.206	0.125380
$330$	0	0
$331$	−9263.48	−1.53827	−0.769134	−	0.639088i	$-0.779312\pi$
$331$	−9263.48	−1.53827	−0.769134	+	0.639088i	$0.779312\pi$
$332$	0	0
$333$	5456.41	0.897926
$334$	0	0
$335$	7692.47	1.25458
$336$	0	0
$337$	−2071.30	−0.334810	−0.167405	−	0.985888i	$-0.553539\pi$
$337$	−2071.30	−0.334810	−0.167405	+	0.985888i	$0.553539\pi$
$338$	0	0
$339$	622.917	0.0998001
$340$	0	0
$341$	8469.78	1.34506
$342$	0	0
$343$	−6589.90	−1.03738
$344$	0	0
$345$	2735.17	0.426830
$346$	0	0
$347$	8255.77	1.27721	0.638607	−	0.769533i	$-0.279511\pi$
$347$	8255.77	1.27721	0.638607	+	0.769533i	$0.279511\pi$
$348$	0	0
$349$	6600.97	1.01244	0.506220	−	0.862404i	$-0.331042\pi$
$349$	6600.97	1.01244	0.506220	+	0.862404i	$0.331042\pi$
$350$	0	0
$351$	1003.34	0.152577
$352$	0	0
$353$	−6136.89	−0.925308	−0.462654	−	0.886539i	$-0.653103\pi$
$353$	−6136.89	−0.925308	−0.462654	+	0.886539i	$0.653103\pi$
$354$	0	0
$355$	−16306.9	−2.43797
$356$	0	0
$357$	1237.48	0.183457
$358$	0	0
$359$	4314.86	0.634344	0.317172	−	0.948368i	$-0.397267\pi$
$359$	4314.86	0.634344	0.317172	+	0.948368i	$0.397267\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	212.492	0.0307244
$364$	0	0
$365$	9851.18	1.41270
$366$	0	0
$367$	−2643.59	−0.376006	−0.188003	−	0.982168i	$-0.560202\pi$
$367$	−2643.59	−0.376006	−0.188003	+	0.982168i	$0.560202\pi$
$368$	0	0
$369$	−7198.70	−1.01558
$370$	0	0
$371$	942.054	0.131830
$372$	0	0
$373$	−3643.51	−0.505774	−0.252887	−	0.967496i	$-0.581380\pi$
$373$	−3643.51	−0.505774	−0.252887	+	0.967496i	$0.581380\pi$
$374$	0	0
$375$	−844.498	−0.116293
$376$	0	0
$377$	2673.61	0.365246
$378$	0	0
$379$	9731.09	1.31887	0.659436	−	0.751761i	$-0.270795\pi$
$379$	9731.09	1.31887	0.659436	+	0.751761i	$0.270795\pi$
$380$	0	0
$381$	89.3095	0.0120091
$382$	0	0
$383$	4167.84	0.556049	0.278025	−	0.960574i	$-0.410320\pi$
$383$	4167.84	0.556049	0.278025	+	0.960574i	$0.410320\pi$
$384$	0	0
$385$	8392.08	1.11091
$386$	0	0
$387$	−4334.53	−0.569345
$388$	0	0
$389$	4267.52	0.556226	0.278113	−	0.960548i	$-0.410291\pi$
$389$	4267.52	0.556226	0.278113	+	0.960548i	$0.410291\pi$
$390$	0	0
$391$	−15845.4	−2.04946
$392$	0	0
$393$	−348.690	−0.0447560
$394$	0	0
$395$	5812.07	0.740347
$396$	0	0
$397$	−7610.34	−0.962096	−0.481048	−	0.876694i	$-0.659744\pi$
$397$	−7610.34	−0.962096	−0.481048	+	0.876694i	$0.659744\pi$
$398$	0	0
$399$	−234.771	−0.0294567
$400$	0	0
$401$	−15827.1	−1.97099	−0.985495	−	0.169704i	$-0.945719\pi$
$401$	−15827.1	−1.97099	−0.985495	+	0.169704i	$0.945719\pi$
$402$	0	0
$403$	4081.24	0.504469
$404$	0	0
$405$	−11219.5	−1.37654
$406$	0	0
$407$	8244.91	1.00414
$408$	0	0
$409$	5134.26	0.620716	0.310358	−	0.950620i	$-0.399551\pi$
$409$	5134.26	0.620716	0.310358	+	0.950620i	$0.399551\pi$
$410$	0	0
$411$	1356.98	0.162858
$412$	0	0
$413$	7112.00	0.847358
$414$	0	0
$415$	−10440.3	−1.23493
$416$	0	0
$417$	−629.287	−0.0739001
$418$	0	0
$419$	5437.64	0.634001	0.317000	−	0.948425i	$-0.397324\pi$
$419$	5437.64	0.634001	0.317000	+	0.948425i	$0.397324\pi$
$420$	0	0
$421$	1552.15	0.179685	0.0898423	−	0.995956i	$-0.471364\pi$
$421$	1552.15	0.179685	0.0898423	+	0.995956i	$0.471364\pi$
$422$	0	0
$423$	−1574.36	−0.180965
$424$	0	0
$425$	17411.0	1.98720
$426$	0	0
$427$	1124.57	0.127451
$428$	0	0
$429$	743.746	0.0837026
$430$	0	0
$431$	5106.26	0.570673	0.285336	−	0.958427i	$-0.407895\pi$
$431$	5106.26	0.570673	0.285336	+	0.958427i	$0.407895\pi$
$432$	0	0
$433$	17422.7	1.93367	0.966835	−	0.255401i	$-0.0822076\pi$
$433$	17422.7	1.93367	0.966835	+	0.255401i	$0.0822076\pi$
$434$	0	0
$435$	2441.47	0.269103
$436$	0	0
$437$	3006.15	0.329070
$438$	0	0
$439$	−2682.09	−0.291592	−0.145796	−	0.989315i	$-0.546574\pi$
$439$	−2682.09	−0.291592	−0.145796	+	0.989315i	$0.546574\pi$
$440$	0	0
$441$	4948.34	0.534320
$442$	0	0
$443$	7407.64	0.794464	0.397232	−	0.917718i	$-0.369971\pi$
$443$	7407.64	0.794464	0.397232	+	0.917718i	$0.369971\pi$
$444$	0	0
$445$	−28275.6	−3.01212
$446$	0	0
$447$	872.641	0.0923367
$448$	0	0
$449$	−15930.7	−1.67442	−0.837212	−	0.546878i	$-0.815816\pi$
$449$	−15930.7	−1.67442	−0.837212	+	0.546878i	$0.815816\pi$
$450$	0	0
$451$	−10877.6	−1.13571
$452$	0	0
$453$	−1892.71	−0.196307
$454$	0	0
$455$	4043.80	0.416651
$456$	0	0
$457$	−6952.91	−0.711693	−0.355846	−	0.934544i	$-0.615807\pi$
$457$	−6952.91	−0.711693	−0.355846	+	0.934544i	$0.615807\pi$
$458$	0	0
$459$	−5307.91	−0.539765
$460$	0	0
$461$	−13281.0	−1.34177	−0.670886	−	0.741561i	$-0.734086\pi$
$461$	−13281.0	−1.34177	−0.670886	+	0.741561i	$0.734086\pi$
$462$	0	0
$463$	6220.21	0.624358	0.312179	−	0.950023i	$-0.398941\pi$
$463$	6220.21	0.624358	0.312179	+	0.950023i	$0.398941\pi$
$464$	0	0
$465$	3726.90	0.371679
$466$	0	0
$467$	−9875.06	−0.978508	−0.489254	−	0.872141i	$-0.662731\pi$
$467$	−9875.06	−0.978508	−0.489254	+	0.872141i	$0.662731\pi$
$468$	0	0
$469$	−5498.30	−0.541339
$470$	0	0
$471$	2939.51	0.287570
$472$	0	0
$473$	−6549.69	−0.636692
$474$	0	0
$475$	−3303.17	−0.319073
$476$	0	0
$477$	−1982.25	−0.190275
$478$	0	0
$479$	−19014.0	−1.81372	−0.906861	−	0.421429i	$-0.861529\pi$
$479$	−19014.0	−1.81372	−0.906861	+	0.421429i	$0.861529\pi$
$480$	0	0
$481$	3972.89	0.376607
$482$	0	0
$483$	−1955.00	−0.184173
$484$	0	0
$485$	−5522.77	−0.517064
$486$	0	0
$487$	−8907.07	−0.828784	−0.414392	−	0.910099i	$-0.636006\pi$
$487$	−8907.07	−0.828784	−0.414392	+	0.910099i	$0.636006\pi$
$488$	0	0
$489$	3301.38	0.305303
$490$	0	0
$491$	−18451.8	−1.69596	−0.847981	−	0.530027i	$-0.822182\pi$
$491$	−18451.8	−1.69596	−0.847981	+	0.530027i	$0.822182\pi$
$492$	0	0
$493$	−14144.0	−1.29212
$494$	0	0
$495$	−17658.5	−1.60341
$496$	0	0
$497$	11655.6	1.05196
$498$	0	0
$499$	−14771.6	−1.32519	−0.662594	−	0.748979i	$-0.730544\pi$
$499$	−14771.6	−1.32519	−0.662594	+	0.748979i	$0.730544\pi$
$500$	0	0
$501$	2630.39	0.234565
$502$	0	0
$503$	−18175.4	−1.61114	−0.805570	−	0.592501i	$-0.798141\pi$
$503$	−18175.4	−1.61114	−0.805570	+	0.592501i	$0.798141\pi$
$504$	0	0
$505$	27576.6	2.42998
$506$	0	0
$507$	−1838.62	−0.161057
$508$	0	0
$509$	5172.14	0.450395	0.225198	−	0.974313i	$-0.427697\pi$
$509$	5172.14	0.450395	0.225198	+	0.974313i	$0.427697\pi$
$510$	0	0
$511$	−7041.28	−0.609565
$512$	0	0
$513$	1007.00	0.0866669
$514$	0	0
$515$	−33321.5	−2.85111
$516$	0	0
$517$	−2378.94	−0.202371
$518$	0	0
$519$	1482.41	0.125377
$520$	0	0
$521$	−12920.1	−1.08645	−0.543225	−	0.839587i	$-0.682797\pi$
$521$	−12920.1	−1.08645	−0.543225	+	0.839587i	$0.682797\pi$
$522$	0	0
$523$	2486.41	0.207884	0.103942	−	0.994583i	$-0.466854\pi$
$523$	2486.41	0.207884	0.103942	+	0.994583i	$0.466854\pi$
$524$	0	0
$525$	2148.16	0.178578
$526$	0	0
$527$	−21590.7	−1.78464
$528$	0	0
$529$	12866.0	1.05745
$530$	0	0
$531$	−14964.9	−1.22302
$532$	0	0
$533$	−5241.47	−0.425954
$534$	0	0
$535$	12396.5	1.00177
$536$	0	0
$537$	1048.00	0.0842170
$538$	0	0
$539$	7477.18	0.597523
$540$	0	0
$541$	−4553.45	−0.361863	−0.180932	−	0.983496i	$-0.557911\pi$
$541$	−4553.45	−0.361863	−0.180932	+	0.983496i	$0.557911\pi$
$542$	0	0
$543$	761.354	0.0601710
$544$	0	0
$545$	6621.85	0.520456
$546$	0	0
$547$	−18289.1	−1.42959	−0.714796	−	0.699333i	$-0.753480\pi$
$547$	−18289.1	−1.42959	−0.714796	+	0.699333i	$0.753480\pi$
$548$	0	0
$549$	−2366.29	−0.183954
$550$	0	0
$551$	2683.36	0.207468
$552$	0	0
$553$	−4154.26	−0.319453
$554$	0	0
$555$	3627.95	0.277473
$556$	0	0
$557$	21732.8	1.65323	0.826613	−	0.562771i	$-0.190265\pi$
$557$	21732.8	1.65323	0.826613	+	0.562771i	$0.190265\pi$
$558$	0	0
$559$	−3156.03	−0.238794
$560$	0	0
$561$	−3934.59	−0.296112
$562$	0	0
$563$	6231.21	0.466455	0.233227	−	0.972422i	$-0.425071\pi$
$563$	6231.21	0.466455	0.233227	+	0.972422i	$0.425071\pi$
$564$	0	0
$565$	−10768.6	−0.801835
$566$	0	0
$567$	8019.27	0.593964
$568$	0	0
$569$	−24670.7	−1.81766	−0.908829	−	0.417168i	$-0.863023\pi$
$569$	−24670.7	−1.81766	−0.908829	+	0.417168i	$0.863023\pi$
$570$	0	0
$571$	20912.5	1.53268	0.766342	−	0.642433i	$-0.222075\pi$
$571$	20912.5	1.53268	0.766342	+	0.642433i	$0.222075\pi$
$572$	0	0
$573$	−4005.98	−0.292063
$574$	0	0
$575$	−27506.4	−1.99495
$576$	0	0
$577$	20604.1	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$577$	20604.1	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$578$	0	0
$579$	−3855.87	−0.276761
$580$	0	0
$581$	7462.35	0.532858
$582$	0	0
$583$	−2995.28	−0.212782
$584$	0	0
$585$	−8508.89	−0.601366
$586$	0	0
$587$	5511.88	0.387563	0.193781	−	0.981045i	$-0.437925\pi$
$587$	5511.88	0.387563	0.193781	+	0.981045i	$0.437925\pi$
$588$	0	0
$589$	4096.13	0.286550
$590$	0	0
$591$	2719.72	0.189297
$592$	0	0
$593$	11836.1	0.819648	0.409824	−	0.912165i	$-0.365590\pi$
$593$	11836.1	0.819648	0.409824	+	0.912165i	$0.365590\pi$
$594$	0	0
$595$	−21392.7	−1.47397
$596$	0	0
$597$	4454.56	0.305382
$598$	0	0
$599$	8407.13	0.573466	0.286733	−	0.958011i	$-0.407431\pi$
$599$	8407.13	0.573466	0.286733	+	0.958011i	$0.407431\pi$
$600$	0	0
$601$	11313.6	0.767872	0.383936	−	0.923360i	$-0.374568\pi$
$601$	11313.6	0.767872	0.383936	+	0.923360i	$0.374568\pi$
$602$	0	0
$603$	11569.4	0.781332
$604$	0	0
$605$	−3673.41	−0.246852
$606$	0	0
$607$	−1581.43	−0.105747	−0.0528733	−	0.998601i	$-0.516838\pi$
$607$	−1581.43	−0.105747	−0.0528733	+	0.998601i	$0.516838\pi$
$608$	0	0
$609$	−1745.08	−0.116115
$610$	0	0
$611$	−1146.31	−0.0759000
$612$	0	0
$613$	2299.35	0.151500	0.0757502	−	0.997127i	$-0.475865\pi$
$613$	2299.35	0.151500	0.0757502	+	0.997127i	$0.475865\pi$
$614$	0	0
$615$	−4786.39	−0.313831
$616$	0	0
$617$	−16144.5	−1.05341	−0.526703	−	0.850049i	$-0.676572\pi$
$617$	−16144.5	−1.05341	−0.526703	+	0.850049i	$0.676572\pi$
$618$	0	0
$619$	−2734.58	−0.177564	−0.0887819	−	0.996051i	$-0.528297\pi$
$619$	−2734.58	−0.177564	−0.0887819	+	0.996051i	$0.528297\pi$
$620$	0	0
$621$	8385.57	0.541870
$622$	0	0
$623$	20210.4	1.29970
$624$	0	0
$625$	−7132.25	−0.456464
$626$	0	0
$627$	746.459	0.0475450
$628$	0	0
$629$	−21017.5	−1.33231
$630$	0	0
$631$	15419.2	0.972787	0.486393	−	0.873740i	$-0.338312\pi$
$631$	15419.2	0.972787	0.486393	+	0.873740i	$0.338312\pi$
$632$	0	0
$633$	−3481.06	−0.218578
$634$	0	0
$635$	−1543.92	−0.0964860
$636$	0	0
$637$	3602.95	0.224104
$638$	0	0
$639$	−24525.5	−1.51833
$640$	0	0
$641$	−11905.5	−0.733600	−0.366800	−	0.930300i	$-0.619547\pi$
$641$	−11905.5	−0.733600	−0.366800	+	0.930300i	$0.619547\pi$
$642$	0	0
$643$	3525.47	0.216223	0.108111	−	0.994139i	$-0.465520\pi$
$643$	3525.47	0.216223	0.108111	+	0.994139i	$0.465520\pi$
$644$	0	0
$645$	−2882.01	−0.175937
$646$	0	0
$647$	7502.61	0.455886	0.227943	−	0.973675i	$-0.426800\pi$
$647$	7502.61	0.455886	0.227943	+	0.973675i	$0.426800\pi$
$648$	0	0
$649$	−22612.8	−1.36769
$650$	0	0
$651$	−2663.85	−0.160376
$652$	0	0
$653$	6539.35	0.391891	0.195945	−	0.980615i	$-0.437222\pi$
$653$	6539.35	0.391891	0.195945	+	0.980615i	$0.437222\pi$
$654$	0	0
$655$	6027.92	0.359588
$656$	0	0
$657$	14816.1	0.879805
$658$	0	0
$659$	22229.4	1.31401	0.657007	−	0.753884i	$-0.271822\pi$
$659$	22229.4	1.31401	0.657007	+	0.753884i	$0.271822\pi$
$660$	0	0
$661$	19497.3	1.14729	0.573643	−	0.819105i	$-0.305530\pi$
$661$	19497.3	1.14729	0.573643	+	0.819105i	$0.305530\pi$
$662$	0	0
$663$	−1895.92	−0.111058
$664$	0	0
$665$	4058.55	0.236667
$666$	0	0
$667$	22345.1	1.29716
$668$	0	0
$669$	−3193.81	−0.184574
$670$	0	0
$671$	−3575.58	−0.205714
$672$	0	0
$673$	−6952.91	−0.398239	−0.199120	−	0.979975i	$-0.563808\pi$
$673$	−6952.91	−0.398239	−0.199120	+	0.979975i	$0.563808\pi$
$674$	0	0
$675$	−9214.09	−0.525408
$676$	0	0
$677$	−2010.38	−0.114129	−0.0570643	−	0.998371i	$-0.518174\pi$
$677$	−2010.38	−0.114129	−0.0570643	+	0.998371i	$0.518174\pi$
$678$	0	0
$679$	3947.48	0.223108
$680$	0	0
$681$	5139.70	0.289212
$682$	0	0
$683$	−16401.7	−0.918877	−0.459439	−	0.888209i	$-0.651949\pi$
$683$	−16401.7	−0.918877	−0.459439	+	0.888209i	$0.651949\pi$
$684$	0	0
$685$	−23458.5	−1.30847
$686$	0	0
$687$	−4310.87	−0.239403
$688$	0	0
$689$	−1443.30	−0.0798048
$690$	0	0
$691$	15967.5	0.879064	0.439532	−	0.898227i	$-0.355144\pi$
$691$	15967.5	0.879064	0.439532	+	0.898227i	$0.355144\pi$
$692$	0	0
$693$	12621.6	0.691856
$694$	0	0
$695$	10878.7	0.593744
$696$	0	0
$697$	27728.6	1.50688
$698$	0	0
$699$	175.879	0.00951698
$700$	0	0
$701$	−22503.7	−1.21248	−0.606242	−	0.795280i	$-0.707324\pi$
$701$	−22503.7	−1.21248	−0.606242	+	0.795280i	$0.707324\pi$
$702$	0	0
$703$	3987.38	0.213921
$704$	0	0
$705$	−1046.79	−0.0559210
$706$	0	0
$707$	−19710.8	−1.04851
$708$	0	0
$709$	−14922.1	−0.790426	−0.395213	−	0.918590i	$-0.629329\pi$
$709$	−14922.1	−0.790426	−0.395213	+	0.918590i	$0.629329\pi$
$710$	0	0
$711$	8741.32	0.461076
$712$	0	0
$713$	34109.6	1.79161
$714$	0	0
$715$	−12857.4	−0.672501
$716$	0	0
$717$	−3811.82	−0.198543
$718$	0	0
$719$	−3483.14	−0.180667	−0.0903333	−	0.995912i	$-0.528793\pi$
$719$	−3483.14	−0.180667	−0.0903333	+	0.995912i	$0.528793\pi$
$720$	0	0
$721$	23817.0	1.23023
$722$	0	0
$723$	−1356.97	−0.0698013
$724$	0	0
$725$	−24552.8	−1.25775
$726$	0	0
$727$	6572.23	0.335283	0.167641	−	0.985848i	$-0.446385\pi$
$727$	6572.23	0.335283	0.167641	+	0.985848i	$0.446385\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	0	0
$731$	16696.1	0.844773
$732$	0	0
$733$	27149.4	1.36806	0.684030	−	0.729454i	$-0.260226\pi$
$733$	27149.4	1.36806	0.684030	+	0.729454i	$0.260226\pi$
$734$	0	0
$735$	3290.13	0.165113
$736$	0	0
$737$	17482.0	0.873754
$738$	0	0
$739$	26251.0	1.30671	0.653355	−	0.757051i	$-0.273361\pi$
$739$	26251.0	1.30671	0.653355	+	0.757051i	$0.273361\pi$
$740$	0	0
$741$	359.688	0.0178319
$742$	0	0
$743$	10583.0	0.522545	0.261273	−	0.965265i	$-0.415858\pi$
$743$	10583.0	0.522545	0.261273	+	0.965265i	$0.415858\pi$
$744$	0	0
$745$	−15085.6	−0.741871
$746$	0	0
$747$	−15702.1	−0.769092
$748$	0	0
$749$	−8860.60	−0.432255
$750$	0	0
$751$	9812.03	0.476759	0.238380	−	0.971172i	$-0.423384\pi$
$751$	9812.03	0.476759	0.238380	+	0.971172i	$0.423384\pi$
$752$	0	0
$753$	5465.11	0.264488
$754$	0	0
$755$	32719.8	1.57721
$756$	0	0
$757$	−10954.1	−0.525938	−0.262969	−	0.964804i	$-0.584702\pi$
$757$	−10954.1	−0.525938	−0.262969	+	0.964804i	$0.584702\pi$
$758$	0	0
$759$	6215.97	0.297267
$760$	0	0
$761$	−19350.2	−0.921739	−0.460869	−	0.887468i	$-0.652462\pi$
$761$	−19350.2	−0.921739	−0.460869	+	0.887468i	$0.652462\pi$
$762$	0	0
$763$	−4733.06	−0.224572
$764$	0	0
$765$	45014.0	2.12743
$766$	0	0
$767$	−10896.2	−0.512957
$768$	0	0
$769$	17844.5	0.836786	0.418393	−	0.908266i	$-0.362593\pi$
$769$	17844.5	0.836786	0.418393	+	0.908266i	$0.362593\pi$
$770$	0	0
$771$	1249.60	0.0583699
$772$	0	0
$773$	38019.3	1.76903	0.884514	−	0.466513i	$-0.154490\pi$
$773$	38019.3	1.76903	0.884514	+	0.466513i	$0.154490\pi$
$774$	0	0
$775$	−37479.7	−1.73718
$776$	0	0
$777$	−2593.13	−0.119727
$778$	0	0
$779$	−5260.59	−0.241951
$780$	0	0
$781$	−37059.2	−1.69793
$782$	0	0
$783$	7485.16	0.341632
$784$	0	0
$785$	−50816.2	−2.31046
$786$	0	0
$787$	−32884.2	−1.48945	−0.744724	−	0.667373i	$-0.767419\pi$
$787$	−32884.2	−1.48945	−0.744724	+	0.667373i	$0.767419\pi$
$788$	0	0
$789$	8043.06	0.362916
$790$	0	0
$791$	7696.99	0.345984
$792$	0	0
$793$	−1722.93	−0.0771538
$794$	0	0
$795$	−1317.99	−0.0587979
$796$	0	0
$797$	−4141.59	−0.184069	−0.0920344	−	0.995756i	$-0.529337\pi$
$797$	−4141.59	−0.184069	−0.0920344	+	0.995756i	$0.529337\pi$
$798$	0	0
$799$	6064.27	0.268509
$800$	0	0
$801$	−42526.4	−1.87590
$802$	0	0
$803$	22387.9	0.983875
$804$	0	0
$805$	33796.7	1.47972
$806$	0	0
$807$	−3051.13	−0.133092
$808$	0	0
$809$	17007.2	0.739111	0.369555	−	0.929209i	$-0.379510\pi$
$809$	17007.2	0.739111	0.369555	+	0.929209i	$0.379510\pi$
$810$	0	0
$811$	18814.0	0.814611	0.407306	−	0.913292i	$-0.366468\pi$
$811$	18814.0	0.814611	0.407306	+	0.913292i	$0.366468\pi$
$812$	0	0
$813$	−1967.96	−0.0848949
$814$	0	0
$815$	−57071.9	−2.45293
$816$	0	0
$817$	−3167.54	−0.135640
$818$	0	0
$819$	6081.85	0.259484
$820$	0	0
$821$	30735.2	1.30653	0.653267	−	0.757128i	$-0.273398\pi$
$821$	30735.2	1.30653	0.653267	+	0.757128i	$0.273398\pi$
$822$	0	0
$823$	11638.0	0.492923	0.246461	−	0.969153i	$-0.420732\pi$
$823$	11638.0	0.492923	0.246461	+	0.969153i	$0.420732\pi$
$824$	0	0
$825$	−6830.13	−0.288236
$826$	0	0
$827$	−4691.30	−0.197258	−0.0986291	−	0.995124i	$-0.531446\pi$
$827$	−4691.30	−0.197258	−0.0986291	+	0.995124i	$0.531446\pi$
$828$	0	0
$829$	12990.2	0.544232	0.272116	−	0.962264i	$-0.412276\pi$
$829$	12990.2	0.544232	0.272116	+	0.962264i	$0.412276\pi$
$830$	0	0
$831$	4077.51	0.170213
$832$	0	0
$833$	−19060.5	−0.792804
$834$	0	0
$835$	−45472.3	−1.88459
$836$	0	0
$837$	11426.0	0.471854
$838$	0	0
$839$	22152.7	0.911558	0.455779	−	0.890093i	$-0.349361\pi$
$839$	22152.7	0.911558	0.455779	+	0.890093i	$0.349361\pi$
$840$	0	0
$841$	−4443.27	−0.182183
$842$	0	0
$843$	−7982.51	−0.326136
$844$	0	0
$845$	31784.8	1.29400
$846$	0	0
$847$	2625.63	0.106514
$848$	0	0
$849$	4137.36	0.167248
$850$	0	0
$851$	33204.0	1.33751
$852$	0	0
$853$	−27556.6	−1.10612	−0.553060	−	0.833141i	$-0.686540\pi$
$853$	−27556.6	−1.10612	−0.553060	+	0.833141i	$0.686540\pi$
$854$	0	0
$855$	−8539.93	−0.341590
$856$	0	0
$857$	−2728.49	−0.108755	−0.0543777	−	0.998520i	$-0.517318\pi$
$857$	−2728.49	−0.108755	−0.0543777	+	0.998520i	$0.517318\pi$
$858$	0	0
$859$	−24012.7	−0.953786	−0.476893	−	0.878961i	$-0.658237\pi$
$859$	−24012.7	−0.953786	−0.476893	+	0.878961i	$0.658237\pi$
$860$	0	0
$861$	3421.14	0.135415
$862$	0	0
$863$	23821.0	0.939603	0.469802	−	0.882772i	$-0.344325\pi$
$863$	23821.0	0.939603	0.469802	+	0.882772i	$0.344325\pi$
$864$	0	0
$865$	−25626.9	−1.00733
$866$	0	0
$867$	5116.86	0.200436
$868$	0	0
$869$	13208.6	0.515616
$870$	0	0
$871$	8423.85	0.327705
$872$	0	0
$873$	−8306.22	−0.322019
$874$	0	0
$875$	−10434.9	−0.403160
$876$	0	0
$877$	−19.8882	−0.000765765 0	−0.000382882	−	1.00000i	$-0.500122\pi$
$877$	−19.8882	−0.000765765 0	−0.000382882		1.00000i	$0.500122\pi$
$878$	0	0
$879$	−6807.50	−0.261219
$880$	0	0
$881$	−846.013	−0.0323529	−0.0161764	−	0.999869i	$-0.505149\pi$
$881$	−846.013	−0.0323529	−0.0161764	+	0.999869i	$0.505149\pi$
$882$	0	0
$883$	−14028.9	−0.534666	−0.267333	−	0.963604i	$-0.586142\pi$
$883$	−14028.9	−0.534666	−0.267333	+	0.963604i	$0.586142\pi$
$884$	0	0
$885$	−9950.13	−0.377932
$886$	0	0
$887$	−6993.33	−0.264727	−0.132364	−	0.991201i	$-0.542257\pi$
$887$	−6993.33	−0.264727	−0.132364	+	0.991201i	$0.542257\pi$
$888$	0	0
$889$	1103.54	0.0416328
$890$	0	0
$891$	−25497.5	−0.958695
$892$	0	0
$893$	−1150.50	−0.0431129
$894$	0	0
$895$	−18117.1	−0.676634
$896$	0	0
$897$	2995.22	0.111491
$898$	0	0
$899$	30447.0	1.12955
$900$	0	0
$901$	7635.42	0.282323
$902$	0	0
$903$	2059.96	0.0759149
$904$	0	0
$905$	−13161.8	−0.483438
$906$	0	0
$907$	26569.7	0.972693	0.486347	−	0.873766i	$-0.338329\pi$
$907$	26569.7	0.972693	0.486347	+	0.873766i	$0.338329\pi$
$908$	0	0
$909$	41475.0	1.51336
$910$	0	0
$911$	3037.05	0.110452	0.0552260	−	0.998474i	$-0.482412\pi$
$911$	3037.05	0.110452	0.0552260	+	0.998474i	$0.482412\pi$
$912$	0	0
$913$	−23726.7	−0.860066
$914$	0	0
$915$	−1573.34	−0.0568447
$916$	0	0
$917$	−4308.54	−0.155159
$918$	0	0
$919$	30480.3	1.09407	0.547036	−	0.837109i	$-0.315756\pi$
$919$	30480.3	1.09407	0.547036	+	0.837109i	$0.315756\pi$
$920$	0	0
$921$	2822.75	0.100991
$922$	0	0
$923$	−17857.3	−0.636816
$924$	0	0
$925$	−36484.7	−1.29687
$926$	0	0
$927$	−50115.4	−1.77563
$928$	0	0
$929$	−12218.1	−0.431498	−0.215749	−	0.976449i	$-0.569219\pi$
$929$	−12218.1	−0.431498	−0.215749	+	0.976449i	$0.569219\pi$
$930$	0	0
$931$	3616.09	0.127296
$932$	0	0
$933$	8221.36	0.288484
$934$	0	0
$935$	68018.5	2.37908
$936$	0	0
$937$	29036.4	1.01236	0.506178	−	0.862429i	$-0.331058\pi$
$937$	29036.4	1.01236	0.506178	+	0.862429i	$0.331058\pi$
$938$	0	0
$939$	−1896.42	−0.0659076
$940$	0	0
$941$	15188.6	0.526179	0.263090	−	0.964771i	$-0.415259\pi$
$941$	15188.6	0.526179	0.263090	+	0.964771i	$0.415259\pi$
$942$	0	0
$943$	−43806.4	−1.51276
$944$	0	0
$945$	11321.2	0.389714
$946$	0	0
$947$	38267.7	1.31313	0.656565	−	0.754270i	$-0.272009\pi$
$947$	38267.7	1.31313	0.656565	+	0.754270i	$0.272009\pi$
$948$	0	0
$949$	10787.8	0.369007
$950$	0	0
$951$	2875.66	0.0980544
$952$	0	0
$953$	−30092.6	−1.02287	−0.511434	−	0.859322i	$-0.670886\pi$
$953$	−30092.6	−1.02287	−0.511434	+	0.859322i	$0.670886\pi$
$954$	0	0
$955$	69252.5	2.34655
$956$	0	0
$957$	5548.52	0.187417
$958$	0	0
$959$	16767.3	0.564592
$960$	0	0
$961$	16686.2	0.560109
$962$	0	0
$963$	18644.3	0.623888
$964$	0	0
$965$	66657.7	2.22361
$966$	0	0
$967$	−22521.1	−0.748943	−0.374472	−	0.927238i	$-0.622176\pi$
$967$	−22521.1	−0.748943	−0.374472	+	0.927238i	$0.622176\pi$
$968$	0	0
$969$	−1902.83	−0.0630835
$970$	0	0
$971$	16791.8	0.554969	0.277485	−	0.960730i	$-0.410499\pi$
$971$	16791.8	0.554969	0.277485	+	0.960730i	$0.410499\pi$
$972$	0	0
$973$	−7775.69	−0.256195
$974$	0	0
$975$	−3291.16	−0.108104
$976$	0	0
$977$	−28654.3	−0.938312	−0.469156	−	0.883115i	$-0.655442\pi$
$977$	−28654.3	−0.938312	−0.469156	+	0.883115i	$0.655442\pi$
$978$	0	0
$979$	−64259.5	−2.09780
$980$	0	0
$981$	9959.22	0.324132
$982$	0	0
$983$	43981.3	1.42704	0.713522	−	0.700633i	$-0.247099\pi$
$983$	43981.3	1.42704	0.713522	+	0.700633i	$0.247099\pi$
$984$	0	0
$985$	−47016.7	−1.52089
$986$	0	0
$987$	748.206	0.0241294
$988$	0	0
$989$	−26377.0	−0.848068
$990$	0	0
$991$	10308.7	0.330442	0.165221	−	0.986257i	$-0.447166\pi$
$991$	10308.7	0.330442	0.165221	+	0.986257i	$0.447166\pi$
$992$	0	0
$993$	−9263.48	−0.296040
$994$	0	0
$995$	−77007.3	−2.45356
$996$	0	0
$997$	−56141.7	−1.78337	−0.891687	−	0.452652i	$-0.850478\pi$
$997$	−56141.7	−1.78337	−0.891687	+	0.452652i	$0.850478\pi$
$998$	0	0
$999$	11122.7	0.352258

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.n.1.1		2
4.3	odd	2		1216.4.a.i.1.1		2
8.3	odd	2		608.4.a.d.1.2	yes	2
8.5	even	2		608.4.a.c.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.c.1.2	✓	2	8.5	even	2
608.4.a.d.1.2	yes	2	8.3	odd	2
1216.4.a.i.1.1		2	4.3	odd	2
1216.4.a.n.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -17.2873 q^{5} +12.3563 q^{7} -26.0000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -17.2873 q^{5} +12.3563 q^{7} -26.0000 q^{9} -39.2873 q^{11} -18.9310 q^{13} -17.2873 q^{15} +100.149 q^{17} -19.0000 q^{19} +12.3563 q^{21} -158.218 q^{23} +173.851 q^{25} -53.0000 q^{27} -141.229 q^{29} -215.586 q^{31} -39.2873 q^{33} -213.608 q^{35} -209.862 q^{37} -18.9310 q^{39} +276.873 q^{41} +166.713 q^{43} +449.470 q^{45} +60.5524 q^{47} -190.321 q^{49} +100.149 q^{51} +76.2405 q^{53} +679.171 q^{55} -19.0000 q^{57} +575.575 q^{59} +91.0111 q^{61} -321.265 q^{63} +327.265 q^{65} -444.978 q^{67} -158.218 q^{69} +943.287 q^{71} -569.851 q^{73} +173.851 q^{75} -485.448 q^{77} -336.205 q^{79} +649.000 q^{81} +603.929 q^{83} -1731.31 q^{85} -141.229 q^{87} +1635.63 q^{89} -233.917 q^{91} -215.586 q^{93} +328.459 q^{95} +319.470 q^{97} +1021.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9} - 40 q^{11} + 20 q^{13} + 4 q^{15} + 46 q^{17} - 38 q^{19} + 44 q^{21} - 220 q^{23} + 502 q^{25} - 106 q^{27} + 84 q^{29} - 84 q^{31} - 40 q^{33} + 460 q^{35} - 304 q^{37} + 20 q^{39} + 168 q^{41} + 372 q^{43} - 104 q^{45} + 584 q^{47} + 468 q^{49} + 46 q^{51} - 484 q^{53} + 664 q^{55} - 38 q^{57} + 1074 q^{59} - 88 q^{61} - 1144 q^{63} + 1156 q^{65} - 1430 q^{67} - 220 q^{69} + 1848 q^{71} - 1294 q^{73} + 502 q^{75} - 508 q^{77} + 832 q^{79} + 1298 q^{81} - 528 q^{83} - 2884 q^{85} + 84 q^{87} + 1844 q^{89} + 998 q^{91} - 84 q^{93} - 76 q^{95} - 364 q^{97} + 1040 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9 - 40 * q^11 + 20 * q^13 + 4 * q^15 + 46 * q^17 - 38 * q^19 + 44 * q^21 - 220 * q^23 + 502 * q^25 - 106 * q^27 + 84 * q^29 - 84 * q^31 - 40 * q^33 + 460 * q^35 - 304 * q^37 + 20 * q^39 + 168 * q^41 + 372 * q^43 - 104 * q^45 + 584 * q^47 + 468 * q^49 + 46 * q^51 - 484 * q^53 + 664 * q^55 - 38 * q^57 + 1074 * q^59 - 88 * q^61 - 1144 * q^63 + 1156 * q^65 - 1430 * q^67 - 220 * q^69 + 1848 * q^71 - 1294 * q^73 + 502 * q^75 - 508 * q^77 + 832 * q^79 + 1298 * q^81 - 528 * q^83 - 2884 * q^85 + 84 * q^87 + 1844 * q^89 + 998 * q^91 - 84 * q^93 - 76 * q^95 - 364 * q^97 + 1040 * q^99

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