LMFDB - Embedded newform 1856.4.a.bl.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + \cdots + 14751072 $ x^12 - 2x^11 - 217x^10 + 520x^9 + 16022x^8 - 37368x^7 - 509640x^6 + 989168x^5 + 7106592x^4 - 7979328x^3 - 38912400x^2 + 1083456*x + 14751072
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$8.22268$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.22268 q^{3} -9.91795 q^{5} +18.1458 q^{7} +58.0578 q^{9} +O(q^{10})$	$q+9.22268 q^{3} -9.91795 q^{5} +18.1458 q^{7} +58.0578 q^{9} +30.3346 q^{11} +38.5792 q^{13} -91.4701 q^{15} +59.0959 q^{17} +50.7598 q^{19} +167.353 q^{21} -155.842 q^{23} -26.6342 q^{25} +286.437 q^{27} -29.0000 q^{29} +241.283 q^{31} +279.767 q^{33} -179.969 q^{35} -371.714 q^{37} +355.804 q^{39} +61.4490 q^{41} +369.530 q^{43} -575.815 q^{45} +281.699 q^{47} -13.7311 q^{49} +545.023 q^{51} +374.175 q^{53} -300.857 q^{55} +468.141 q^{57} -129.028 q^{59} -463.061 q^{61} +1053.50 q^{63} -382.627 q^{65} +970.507 q^{67} -1437.28 q^{69} -241.521 q^{71} +565.154 q^{73} -245.639 q^{75} +550.445 q^{77} -424.422 q^{79} +1074.15 q^{81} -161.180 q^{83} -586.110 q^{85} -267.458 q^{87} -1138.20 q^{89} +700.050 q^{91} +2225.28 q^{93} -503.433 q^{95} -1449.36 q^{97} +1761.16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) $ 12 * q + 14 * q^3 + 10 * q^5 - 44 * q^7 + 130 * q^9	$ 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9} + 46 q^{11} + 34 q^{13} + 50 q^{15} + 36 q^{17} + 148 q^{19} + 92 q^{21} - 328 q^{23} + 486 q^{25} + 326 q^{27} - 348 q^{29} - 374 q^{31} + 710 q^{33} + 204 q^{35} + 340 q^{37} + 122 q^{39} + 32 q^{41} + 462 q^{43} + 1132 q^{45} - 434 q^{47} + 1508 q^{49} + 440 q^{51} - 610 q^{53} - 46 q^{55} - 932 q^{57} + 1240 q^{59} + 1228 q^{61} - 4240 q^{63} + 730 q^{65} + 1672 q^{67} + 528 q^{69} - 3220 q^{71} + 564 q^{73} + 6032 q^{75} - 644 q^{77} - 1862 q^{79} + 3040 q^{81} + 3736 q^{83} + 808 q^{85} - 406 q^{87} + 584 q^{89} + 4844 q^{91} + 3226 q^{93} - 2844 q^{95} + 904 q^{97} + 6832 q^{99}+O(q^{100}) $ 12 * q + 14 * q^3 + 10 * q^5 - 44 * q^7 + 130 * q^9 + 46 * q^11 + 34 * q^13 + 50 * q^15 + 36 * q^17 + 148 * q^19 + 92 * q^21 - 328 * q^23 + 486 * q^25 + 326 * q^27 - 348 * q^29 - 374 * q^31 + 710 * q^33 + 204 * q^35 + 340 * q^37 + 122 * q^39 + 32 * q^41 + 462 * q^43 + 1132 * q^45 - 434 * q^47 + 1508 * q^49 + 440 * q^51 - 610 * q^53 - 46 * q^55 - 932 * q^57 + 1240 * q^59 + 1228 * q^61 - 4240 * q^63 + 730 * q^65 + 1672 * q^67 + 528 * q^69 - 3220 * q^71 + 564 * q^73 + 6032 * q^75 - 644 * q^77 - 1862 * q^79 + 3040 * q^81 + 3736 * q^83 + 808 * q^85 - 406 * q^87 + 584 * q^89 + 4844 * q^91 + 3226 * q^93 - 2844 * q^95 + 904 * q^97 + 6832 * 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
$3$	9.22268	1.77491	0.887453	−	0.460898i	$-0.152473\pi$
$3$	9.22268	1.77491	0.887453	+	0.460898i	$0.152473\pi$
$4$	0	0
$5$	−9.91795	−0.887089	−0.443544	−	0.896252i	$-0.646279\pi$
$5$	−9.91795	−0.887089	−0.443544	+	0.896252i	$0.646279\pi$
$6$	0	0
$7$	18.1458	0.979779	0.489890	−	0.871784i	$-0.337037\pi$
$7$	18.1458	0.979779	0.489890	+	0.871784i	$0.337037\pi$
$8$	0	0
$9$	58.0578	2.15029
$10$	0	0
$11$	30.3346	0.831476	0.415738	−	0.909485i	$-0.363523\pi$
$11$	30.3346	0.831476	0.415738	+	0.909485i	$0.363523\pi$
$12$	0	0
$13$	38.5792	0.823073	0.411536	−	0.911393i	$-0.364992\pi$
$13$	38.5792	0.823073	0.411536	+	0.911393i	$0.364992\pi$
$14$	0	0
$15$	−91.4701	−1.57450
$16$	0	0
$17$	59.0959	0.843109	0.421555	−	0.906803i	$-0.361485\pi$
$17$	59.0959	0.843109	0.421555	+	0.906803i	$0.361485\pi$
$18$	0	0
$19$	50.7598	0.612900	0.306450	−	0.951887i	$-0.400859\pi$
$19$	50.7598	0.612900	0.306450	+	0.951887i	$0.400859\pi$
$20$	0	0
$21$	167.353	1.73902
$22$	0	0
$23$	−155.842	−1.41284	−0.706418	−	0.707795i	$-0.749690\pi$
$23$	−155.842	−1.41284	−0.706418	+	0.707795i	$0.749690\pi$
$24$	0	0
$25$	−26.6342	−0.213073
$26$	0	0
$27$	286.437	2.04166
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	241.283	1.39793	0.698964	−	0.715157i	$-0.253645\pi$
$31$	241.283	1.39793	0.698964	+	0.715157i	$0.253645\pi$
$32$	0	0
$33$	279.767	1.47579
$34$	0	0
$35$	−179.969	−0.869151
$36$	0	0
$37$	−371.714	−1.65160	−0.825802	−	0.563960i	$-0.809277\pi$
$37$	−371.714	−1.65160	−0.825802	+	0.563960i	$0.809277\pi$
$38$	0	0
$39$	355.804	1.46088
$40$	0	0
$41$	61.4490	0.234066	0.117033	−	0.993128i	$-0.462662\pi$
$41$	61.4490	0.234066	0.117033	+	0.993128i	$0.462662\pi$
$42$	0	0
$43$	369.530	1.31053	0.655265	−	0.755399i	$-0.272557\pi$
$43$	369.530	1.31053	0.655265	+	0.755399i	$0.272557\pi$
$44$	0	0
$45$	−575.815	−1.90750
$46$	0	0
$47$	281.699	0.874255	0.437127	−	0.899400i	$-0.355996\pi$
$47$	281.699	0.874255	0.437127	+	0.899400i	$0.355996\pi$
$48$	0	0
$49$	−13.7311	−0.0400322
$50$	0	0
$51$	545.023	1.49644
$52$	0	0
$53$	374.175	0.969753	0.484876	−	0.874583i	$-0.338865\pi$
$53$	374.175	0.969753	0.484876	+	0.874583i	$0.338865\pi$
$54$	0	0
$55$	−300.857	−0.737593
$56$	0	0
$57$	468.141	1.08784
$58$	0	0
$59$	−129.028	−0.284713	−0.142357	−	0.989815i	$-0.545468\pi$
$59$	−129.028	−0.284713	−0.142357	+	0.989815i	$0.545468\pi$
$60$	0	0
$61$	−463.061	−0.971948	−0.485974	−	0.873973i	$-0.661535\pi$
$61$	−463.061	−0.971948	−0.485974	+	0.873973i	$0.661535\pi$
$62$	0	0
$63$	1053.50	2.10681
$64$	0	0
$65$	−382.627	−0.730139
$66$	0	0
$67$	970.507	1.76965	0.884823	−	0.465927i	$-0.154279\pi$
$67$	970.507	1.76965	0.884823	+	0.465927i	$0.154279\pi$
$68$	0	0
$69$	−1437.28	−2.50765
$70$	0	0
$71$	−241.521	−0.403708	−0.201854	−	0.979416i	$-0.564697\pi$
$71$	−241.521	−0.403708	−0.201854	+	0.979416i	$0.564697\pi$
$72$	0	0
$73$	565.154	0.906113	0.453057	−	0.891482i	$-0.350334\pi$
$73$	565.154	0.906113	0.453057	+	0.891482i	$0.350334\pi$
$74$	0	0
$75$	−245.639	−0.378185
$76$	0	0
$77$	550.445	0.814663
$78$	0	0
$79$	−424.422	−0.604446	−0.302223	−	0.953237i	$-0.597729\pi$
$79$	−424.422	−0.604446	−0.302223	+	0.953237i	$0.597729\pi$
$80$	0	0
$81$	1074.15	1.47346
$82$	0	0
$83$	−161.180	−0.213154	−0.106577	−	0.994304i	$-0.533989\pi$
$83$	−161.180	−0.213154	−0.106577	+	0.994304i	$0.533989\pi$
$84$	0	0
$85$	−586.110	−0.747913
$86$	0	0
$87$	−267.458	−0.329592
$88$	0	0
$89$	−1138.20	−1.35561	−0.677806	−	0.735241i	$-0.737069\pi$
$89$	−1138.20	−1.35561	−0.677806	+	0.735241i	$0.737069\pi$
$90$	0	0
$91$	700.050	0.806430
$92$	0	0
$93$	2225.28	2.48119
$94$	0	0
$95$	−503.433	−0.543696
$96$	0	0
$97$	−1449.36	−1.51712	−0.758558	−	0.651606i	$-0.774096\pi$
$97$	−1449.36	−1.51712	−0.758558	+	0.651606i	$0.774096\pi$
$98$	0	0
$99$	1761.16	1.78791
$100$	0	0
$101$	−979.900	−0.965383	−0.482692	−	0.875790i	$-0.660341\pi$
$101$	−979.900	−0.965383	−0.482692	+	0.875790i	$0.660341\pi$
$102$	0	0
$103$	375.174	0.358903	0.179451	−	0.983767i	$-0.442568\pi$
$103$	375.174	0.358903	0.179451	+	0.983767i	$0.442568\pi$
$104$	0	0
$105$	−1659.80	−1.54266
$106$	0	0
$107$	−779.197	−0.703999	−0.351999	−	0.936000i	$-0.614498\pi$
$107$	−779.197	−0.703999	−0.351999	+	0.936000i	$0.614498\pi$
$108$	0	0
$109$	146.498	0.128734	0.0643669	−	0.997926i	$-0.479497\pi$
$109$	146.498	0.128734	0.0643669	+	0.997926i	$0.479497\pi$
$110$	0	0
$111$	−3428.20	−2.93144
$112$	0	0
$113$	−209.441	−0.174359	−0.0871795	−	0.996193i	$-0.527785\pi$
$113$	−209.441	−0.174359	−0.0871795	+	0.996193i	$0.527785\pi$
$114$	0	0
$115$	1545.63	1.25331
$116$	0	0
$117$	2239.83	1.76985
$118$	0	0
$119$	1072.34	0.826061
$120$	0	0
$121$	−410.811	−0.308648
$122$	0	0
$123$	566.725	0.415446
$124$	0	0
$125$	1503.90	1.07610
$126$	0	0
$127$	2328.39	1.62686	0.813429	−	0.581664i	$-0.197598\pi$
$127$	2328.39	1.62686	0.813429	+	0.581664i	$0.197598\pi$
$128$	0	0
$129$	3408.06	2.32607
$130$	0	0
$131$	2293.89	1.52991	0.764955	−	0.644084i	$-0.222761\pi$
$131$	2293.89	1.52991	0.764955	+	0.644084i	$0.222761\pi$
$132$	0	0
$133$	921.075	0.600506
$134$	0	0
$135$	−2840.87	−1.81113
$136$	0	0
$137$	−1579.37	−0.984927	−0.492463	−	0.870333i	$-0.663903\pi$
$137$	−1579.37	−0.984927	−0.492463	+	0.870333i	$0.663903\pi$
$138$	0	0
$139$	−1220.98	−0.745054	−0.372527	−	0.928021i	$-0.621509\pi$
$139$	−1220.98	−0.745054	−0.372527	+	0.928021i	$0.621509\pi$
$140$	0	0
$141$	2598.02	1.55172
$142$	0	0
$143$	1170.29	0.684365
$144$	0	0
$145$	287.621	0.164728
$146$	0	0
$147$	−126.637	−0.0710534
$148$	0	0
$149$	−400.388	−0.220141	−0.110071	−	0.993924i	$-0.535108\pi$
$149$	−400.388	−0.220141	−0.110071	+	0.993924i	$0.535108\pi$
$150$	0	0
$151$	−611.484	−0.329549	−0.164774	−	0.986331i	$-0.552690\pi$
$151$	−611.484	−0.329549	−0.164774	+	0.986331i	$0.552690\pi$
$152$	0	0
$153$	3430.98	1.81293
$154$	0	0
$155$	−2393.04	−1.24009
$156$	0	0
$157$	2219.20	1.12810	0.564048	−	0.825742i	$-0.309243\pi$
$157$	2219.20	1.12810	0.564048	+	0.825742i	$0.309243\pi$
$158$	0	0
$159$	3450.90	1.72122
$160$	0	0
$161$	−2827.87	−1.38427
$162$	0	0
$163$	3384.80	1.62649	0.813245	−	0.581922i	$-0.197699\pi$
$163$	3384.80	1.62649	0.813245	+	0.581922i	$0.197699\pi$
$164$	0	0
$165$	−2774.71	−1.30916
$166$	0	0
$167$	1469.18	0.680769	0.340384	−	0.940286i	$-0.389443\pi$
$167$	1469.18	0.680769	0.340384	+	0.940286i	$0.389443\pi$
$168$	0	0
$169$	−708.644	−0.322551
$170$	0	0
$171$	2947.00	1.31791
$172$	0	0
$173$	2494.19	1.09613	0.548063	−	0.836437i	$-0.315365\pi$
$173$	2494.19	1.09613	0.548063	+	0.836437i	$0.315365\pi$
$174$	0	0
$175$	−483.298	−0.208765
$176$	0	0
$177$	−1189.99	−0.505339
$178$	0	0
$179$	1681.66	0.702197	0.351099	−	0.936339i	$-0.385808\pi$
$179$	1681.66	0.702197	0.351099	+	0.936339i	$0.385808\pi$
$180$	0	0
$181$	3463.28	1.42223	0.711115	−	0.703076i	$-0.248191\pi$
$181$	3463.28	1.42223	0.711115	+	0.703076i	$0.248191\pi$
$182$	0	0
$183$	−4270.66	−1.72512
$184$	0	0
$185$	3686.64	1.46512
$186$	0	0
$187$	1792.65	0.701025
$188$	0	0
$189$	5197.61	2.00037
$190$	0	0
$191$	−2962.78	−1.12240	−0.561202	−	0.827679i	$-0.689661\pi$
$191$	−2962.78	−1.12240	−0.561202	+	0.827679i	$0.689661\pi$
$192$	0	0
$193$	3212.73	1.19822	0.599112	−	0.800665i	$-0.295520\pi$
$193$	3212.73	1.19822	0.599112	+	0.800665i	$0.295520\pi$
$194$	0	0
$195$	−3528.85	−1.29593
$196$	0	0
$197$	40.3205	0.0145823	0.00729115	−	0.999973i	$-0.497679\pi$
$197$	40.3205	0.0145823	0.00729115	+	0.999973i	$0.497679\pi$
$198$	0	0
$199$	492.569	0.175464	0.0877319	−	0.996144i	$-0.472038\pi$
$199$	492.569	0.175464	0.0877319	+	0.996144i	$0.472038\pi$
$200$	0	0
$201$	8950.68	3.14096
$202$	0	0
$203$	−526.227	−0.181940
$204$	0	0
$205$	−609.449	−0.207638
$206$	0	0
$207$	−9047.83	−3.03801
$208$	0	0
$209$	1539.78	0.509611
$210$	0	0
$211$	−2004.02	−0.653850	−0.326925	−	0.945050i	$-0.606012\pi$
$211$	−2004.02	−0.653850	−0.326925	+	0.945050i	$0.606012\pi$
$212$	0	0
$213$	−2227.47	−0.716544
$214$	0	0
$215$	−3664.98	−1.16256
$216$	0	0
$217$	4378.27	1.36966
$218$	0	0
$219$	5212.23	1.60827
$220$	0	0
$221$	2279.87	0.693941
$222$	0	0
$223$	−1975.53	−0.593234	−0.296617	−	0.954996i	$-0.595859\pi$
$223$	−1975.53	−0.593234	−0.296617	+	0.954996i	$0.595859\pi$
$224$	0	0
$225$	−1546.32	−0.458170
$226$	0	0
$227$	5304.20	1.55089	0.775446	−	0.631414i	$-0.217525\pi$
$227$	5304.20	1.55089	0.775446	+	0.631414i	$0.217525\pi$
$228$	0	0
$229$	2463.70	0.710944	0.355472	−	0.934687i	$-0.384320\pi$
$229$	2463.70	0.710944	0.355472	+	0.934687i	$0.384320\pi$
$230$	0	0
$231$	5076.58	1.44595
$232$	0	0
$233$	−3230.57	−0.908332	−0.454166	−	0.890917i	$-0.650063\pi$
$233$	−3230.57	−0.908332	−0.454166	+	0.890917i	$0.650063\pi$
$234$	0	0
$235$	−2793.87	−0.775541
$236$	0	0
$237$	−3914.31	−1.07283
$238$	0	0
$239$	4280.39	1.15848	0.579238	−	0.815159i	$-0.303350\pi$
$239$	4280.39	1.15848	0.579238	+	0.815159i	$0.303350\pi$
$240$	0	0
$241$	6488.41	1.73425	0.867127	−	0.498087i	$-0.165964\pi$
$241$	6488.41	1.73425	0.867127	+	0.498087i	$0.165964\pi$
$242$	0	0
$243$	2172.77	0.573594
$244$	0	0
$245$	136.184	0.0355121
$246$	0	0
$247$	1958.27	0.504461
$248$	0	0
$249$	−1486.51	−0.378328
$250$	0	0
$251$	−407.291	−0.102422	−0.0512111	−	0.998688i	$-0.516308\pi$
$251$	−407.291	−0.102422	−0.0512111	+	0.998688i	$0.516308\pi$
$252$	0	0
$253$	−4727.40	−1.17474
$254$	0	0
$255$	−5405.51	−1.32748
$256$	0	0
$257$	−5519.50	−1.33968	−0.669839	−	0.742507i	$-0.733637\pi$
$257$	−5519.50	−1.33968	−0.669839	+	0.742507i	$0.733637\pi$
$258$	0	0
$259$	−6745.03	−1.61821
$260$	0	0
$261$	−1683.68	−0.399299
$262$	0	0
$263$	4159.14	0.975146	0.487573	−	0.873082i	$-0.337882\pi$
$263$	4159.14	0.975146	0.487573	+	0.873082i	$0.337882\pi$
$264$	0	0
$265$	−3711.05	−0.860257
$266$	0	0
$267$	−10497.3	−2.40608
$268$	0	0
$269$	−3408.03	−0.772459	−0.386230	−	0.922403i	$-0.626223\pi$
$269$	−3408.03	−0.772459	−0.386230	+	0.922403i	$0.626223\pi$
$270$	0	0
$271$	−7638.65	−1.71223	−0.856116	−	0.516783i	$-0.827129\pi$
$271$	−7638.65	−1.71223	−0.856116	+	0.516783i	$0.827129\pi$
$272$	0	0
$273$	6456.33	1.43134
$274$	0	0
$275$	−807.938	−0.177165
$276$	0	0
$277$	−254.643	−0.0552347	−0.0276173	−	0.999619i	$-0.508792\pi$
$277$	−254.643	−0.0552347	−0.0276173	+	0.999619i	$0.508792\pi$
$278$	0	0
$279$	14008.4	3.00595
$280$	0	0
$281$	−3750.69	−0.796255	−0.398128	−	0.917330i	$-0.630340\pi$
$281$	−3750.69	−0.796255	−0.398128	+	0.917330i	$0.630340\pi$
$282$	0	0
$283$	−7621.54	−1.60090	−0.800448	−	0.599402i	$-0.795405\pi$
$283$	−7621.54	−1.60090	−0.800448	+	0.599402i	$0.795405\pi$
$284$	0	0
$285$	−4643.00	−0.965010
$286$	0	0
$287$	1115.04	0.229334
$288$	0	0
$289$	−1420.67	−0.289166
$290$	0	0
$291$	−13367.0	−2.69274
$292$	0	0
$293$	3420.15	0.681935	0.340968	−	0.940075i	$-0.389245\pi$
$293$	3420.15	0.681935	0.340968	+	0.940075i	$0.389245\pi$
$294$	0	0
$295$	1279.70	0.252566
$296$	0	0
$297$	8688.95	1.69759
$298$	0	0
$299$	−6012.25	−1.16287
$300$	0	0
$301$	6705.40	1.28403
$302$	0	0
$303$	−9037.31	−1.71346
$304$	0	0
$305$	4592.61	0.862204
$306$	0	0
$307$	6813.99	1.26676	0.633380	−	0.773841i	$-0.281667\pi$
$307$	6813.99	1.26676	0.633380	+	0.773841i	$0.281667\pi$
$308$	0	0
$309$	3460.11	0.637018
$310$	0	0
$311$	−6160.82	−1.12331	−0.561653	−	0.827373i	$-0.689834\pi$
$311$	−6160.82	−1.12331	−0.561653	+	0.827373i	$0.689834\pi$
$312$	0	0
$313$	−5355.56	−0.967138	−0.483569	−	0.875306i	$-0.660660\pi$
$313$	−5355.56	−0.967138	−0.483569	+	0.875306i	$0.660660\pi$
$314$	0	0
$315$	−10448.6	−1.86893
$316$	0	0
$317$	−1812.38	−0.321115	−0.160557	−	0.987027i	$-0.551329\pi$
$317$	−1812.38	−0.321115	−0.160557	+	0.987027i	$0.551329\pi$
$318$	0	0
$319$	−879.704	−0.154401
$320$	0	0
$321$	−7186.29	−1.24953
$322$	0	0
$323$	2999.69	0.516741
$324$	0	0
$325$	−1027.53	−0.175375
$326$	0	0
$327$	1351.11	0.228490
$328$	0	0
$329$	5111.64	0.856577
$330$	0	0
$331$	10192.2	1.69249	0.846243	−	0.532797i	$-0.178859\pi$
$331$	10192.2	1.69249	0.846243	+	0.532797i	$0.178859\pi$
$332$	0	0
$333$	−21580.9	−3.55143
$334$	0	0
$335$	−9625.45	−1.56983
$336$	0	0
$337$	4600.14	0.743578	0.371789	−	0.928317i	$-0.378745\pi$
$337$	4600.14	0.743578	0.371789	+	0.928317i	$0.378745\pi$
$338$	0	0
$339$	−1931.61	−0.309471
$340$	0	0
$341$	7319.24	1.16234
$342$	0	0
$343$	−6473.16	−1.01900
$344$	0	0
$345$	14254.8	2.22451
$346$	0	0
$347$	−3833.04	−0.592992	−0.296496	−	0.955034i	$-0.595818\pi$
$347$	−3833.04	−0.592992	−0.296496	+	0.955034i	$0.595818\pi$
$348$	0	0
$349$	−3398.43	−0.521243	−0.260622	−	0.965441i	$-0.583928\pi$
$349$	−3398.43	−0.521243	−0.260622	+	0.965441i	$0.583928\pi$
$350$	0	0
$351$	11050.5	1.68043
$352$	0	0
$353$	−8663.33	−1.30624	−0.653119	−	0.757255i	$-0.726540\pi$
$353$	−8663.33	−1.30624	−0.653119	+	0.757255i	$0.726540\pi$
$354$	0	0
$355$	2395.39	0.358125
$356$	0	0
$357$	9889.85	1.46618
$358$	0	0
$359$	12373.5	1.81908	0.909540	−	0.415616i	$-0.136434\pi$
$359$	12373.5	1.81908	0.909540	+	0.415616i	$0.136434\pi$
$360$	0	0
$361$	−4282.44	−0.624354
$362$	0	0
$363$	−3788.78	−0.547822
$364$	0	0
$365$	−5605.17	−0.803803
$366$	0	0
$367$	3507.48	0.498880	0.249440	−	0.968390i	$-0.419754\pi$
$367$	3507.48	0.498880	0.249440	+	0.968390i	$0.419754\pi$
$368$	0	0
$369$	3567.60	0.503311
$370$	0	0
$371$	6789.69	0.950144
$372$	0	0
$373$	12460.9	1.72976	0.864878	−	0.501982i	$-0.167396\pi$
$373$	12460.9	1.72976	0.864878	+	0.501982i	$0.167396\pi$
$374$	0	0
$375$	13870.0	1.90998
$376$	0	0
$377$	−1118.80	−0.152841
$378$	0	0
$379$	−11480.8	−1.55601	−0.778003	−	0.628260i	$-0.783767\pi$
$379$	−11480.8	−1.55601	−0.778003	+	0.628260i	$0.783767\pi$
$380$	0	0
$381$	21474.0	2.88752
$382$	0	0
$383$	−11746.7	−1.56718	−0.783591	−	0.621278i	$-0.786614\pi$
$383$	−11746.7	−1.56718	−0.783591	+	0.621278i	$0.786614\pi$
$384$	0	0
$385$	−5459.29	−0.722678
$386$	0	0
$387$	21454.1	2.81802
$388$	0	0
$389$	−2142.34	−0.279232	−0.139616	−	0.990206i	$-0.544587\pi$
$389$	−2142.34	−0.279232	−0.139616	+	0.990206i	$0.544587\pi$
$390$	0	0
$391$	−9209.60	−1.19117
$392$	0	0
$393$	21155.8	2.71544
$394$	0	0
$395$	4209.40	0.536197
$396$	0	0
$397$	−3330.09	−0.420989	−0.210494	−	0.977595i	$-0.567507\pi$
$397$	−3330.09	−0.420989	−0.210494	+	0.977595i	$0.567507\pi$
$398$	0	0
$399$	8494.78	1.06584
$400$	0	0
$401$	−2449.26	−0.305013	−0.152507	−	0.988302i	$-0.548735\pi$
$401$	−2449.26	−0.305013	−0.152507	+	0.988302i	$0.548735\pi$
$402$	0	0
$403$	9308.52	1.15060
$404$	0	0
$405$	−10653.4	−1.30709
$406$	0	0
$407$	−11275.8	−1.37327
$408$	0	0
$409$	1978.69	0.239217	0.119608	−	0.992821i	$-0.461836\pi$
$409$	1978.69	0.239217	0.119608	+	0.992821i	$0.461836\pi$
$410$	0	0
$411$	−14566.1	−1.74815
$412$	0	0
$413$	−2341.32	−0.278956
$414$	0	0
$415$	1598.57	0.189087
$416$	0	0
$417$	−11260.7	−1.32240
$418$	0	0
$419$	−3715.93	−0.433258	−0.216629	−	0.976254i	$-0.569506\pi$
$419$	−3715.93	−0.433258	−0.216629	+	0.976254i	$0.569506\pi$
$420$	0	0
$421$	16124.9	1.86670	0.933350	−	0.358968i	$-0.116871\pi$
$421$	16124.9	1.86670	0.933350	+	0.358968i	$0.116871\pi$
$422$	0	0
$423$	16354.8	1.87990
$424$	0	0
$425$	−1573.97	−0.179644
$426$	0	0
$427$	−8402.59	−0.952295
$428$	0	0
$429$	10793.2	1.21468
$430$	0	0
$431$	6764.83	0.756034	0.378017	−	0.925799i	$-0.376606\pi$
$431$	6764.83	0.756034	0.378017	+	0.925799i	$0.376606\pi$
$432$	0	0
$433$	−1794.00	−0.199109	−0.0995544	−	0.995032i	$-0.531742\pi$
$433$	−1794.00	−0.199109	−0.0995544	+	0.995032i	$0.531742\pi$
$434$	0	0
$435$	2652.63	0.292377
$436$	0	0
$437$	−7910.48	−0.865926
$438$	0	0
$439$	5513.07	0.599372	0.299686	−	0.954038i	$-0.403118\pi$
$439$	5513.07	0.599372	0.299686	+	0.954038i	$0.403118\pi$
$440$	0	0
$441$	−797.195	−0.0860809
$442$	0	0
$443$	−11355.7	−1.21789	−0.608945	−	0.793213i	$-0.708407\pi$
$443$	−11355.7	−1.21789	−0.608945	+	0.793213i	$0.708407\pi$
$444$	0	0
$445$	11288.7	1.20255
$446$	0	0
$447$	−3692.65	−0.390730
$448$	0	0
$449$	−3049.12	−0.320483	−0.160242	−	0.987078i	$-0.551227\pi$
$449$	−3049.12	−0.320483	−0.160242	+	0.987078i	$0.551227\pi$
$450$	0	0
$451$	1864.03	0.194621
$452$	0	0
$453$	−5639.52	−0.584918
$454$	0	0
$455$	−6943.06	−0.715375
$456$	0	0
$457$	−12033.7	−1.23176	−0.615879	−	0.787841i	$-0.711199\pi$
$457$	−12033.7	−1.23176	−0.615879	+	0.787841i	$0.711199\pi$
$458$	0	0
$459$	16927.2	1.72134
$460$	0	0
$461$	629.093	0.0635570	0.0317785	−	0.999495i	$-0.489883\pi$
$461$	629.093	0.0635570	0.0317785	+	0.999495i	$0.489883\pi$
$462$	0	0
$463$	−4.68659	−0.000470420 0	−0.000235210	−	1.00000i	$-0.500075\pi$
$463$	−4.68659	−0.000470420 0	−0.000235210		1.00000i	$0.500075\pi$
$464$	0	0
$465$	−22070.2	−2.20104
$466$	0	0
$467$	694.076	0.0687752	0.0343876	−	0.999409i	$-0.489052\pi$
$467$	694.076	0.0687752	0.0343876	+	0.999409i	$0.489052\pi$
$468$	0	0
$469$	17610.6	1.73386
$470$	0	0
$471$	20466.9	2.00227
$472$	0	0
$473$	11209.5	1.08967
$474$	0	0
$475$	−1351.95	−0.130593
$476$	0	0
$477$	21723.8	2.08525
$478$	0	0
$479$	−10669.7	−1.01777	−0.508883	−	0.860836i	$-0.669941\pi$
$479$	−10669.7	−1.01777	−0.508883	+	0.860836i	$0.669941\pi$
$480$	0	0
$481$	−14340.4	−1.35939
$482$	0	0
$483$	−26080.5	−2.45694
$484$	0	0
$485$	14374.7	1.34582
$486$	0	0
$487$	1565.26	0.145644	0.0728221	−	0.997345i	$-0.476799\pi$
$487$	1565.26	0.145644	0.0728221	+	0.997345i	$0.476799\pi$
$488$	0	0
$489$	31216.9	2.88687
$490$	0	0
$491$	−15547.9	−1.42906	−0.714529	−	0.699606i	$-0.753359\pi$
$491$	−15547.9	−1.42906	−0.714529	+	0.699606i	$0.753359\pi$
$492$	0	0
$493$	−1713.78	−0.156561
$494$	0	0
$495$	−17467.1	−1.58604
$496$	0	0
$497$	−4382.58	−0.395545
$498$	0	0
$499$	−15465.1	−1.38740	−0.693700	−	0.720264i	$-0.744021\pi$
$499$	−15465.1	−1.38740	−0.693700	+	0.720264i	$0.744021\pi$
$500$	0	0
$501$	13549.8	1.20830
$502$	0	0
$503$	−6096.03	−0.540375	−0.270187	−	0.962808i	$-0.587086\pi$
$503$	−6096.03	−0.540375	−0.270187	+	0.962808i	$0.587086\pi$
$504$	0	0
$505$	9718.60	0.856381
$506$	0	0
$507$	−6535.60	−0.572497
$508$	0	0
$509$	−18928.0	−1.64827	−0.824133	−	0.566397i	$-0.808337\pi$
$509$	−18928.0	−1.64827	−0.824133	+	0.566397i	$0.808337\pi$
$510$	0	0
$511$	10255.2	0.887791
$512$	0	0
$513$	14539.5	1.25133
$514$	0	0
$515$	−3720.96	−0.318379
$516$	0	0
$517$	8545.22	0.726921
$518$	0	0
$519$	23003.1	1.94552
$520$	0	0
$521$	−20910.7	−1.75838	−0.879189	−	0.476474i	$-0.841915\pi$
$521$	−20910.7	−1.75838	−0.879189	+	0.476474i	$0.841915\pi$
$522$	0	0
$523$	16510.9	1.38044	0.690220	−	0.723600i	$-0.257514\pi$
$523$	16510.9	1.38044	0.690220	+	0.723600i	$0.257514\pi$
$524$	0	0
$525$	−4457.30	−0.370538
$526$	0	0
$527$	14258.9	1.17861
$528$	0	0
$529$	12119.6	0.996104
$530$	0	0
$531$	−7491.12	−0.612216
$532$	0	0
$533$	2370.66	0.192654
$534$	0	0
$535$	7728.04	0.624509
$536$	0	0
$537$	15509.4	1.24633
$538$	0	0
$539$	−416.526	−0.0332858
$540$	0	0
$541$	−24189.4	−1.92234	−0.961169	−	0.275960i	$-0.911004\pi$
$541$	−24189.4	−1.92234	−0.961169	+	0.275960i	$0.911004\pi$
$542$	0	0
$543$	31940.7	2.52432
$544$	0	0
$545$	−1452.96	−0.114198
$546$	0	0
$547$	8110.28	0.633950	0.316975	−	0.948434i	$-0.397333\pi$
$547$	8110.28	0.633950	0.316975	+	0.948434i	$0.397333\pi$
$548$	0	0
$549$	−26884.3	−2.08997
$550$	0	0
$551$	−1472.03	−0.113813
$552$	0	0
$553$	−7701.47	−0.592224
$554$	0	0
$555$	34000.7	2.60045
$556$	0	0
$557$	22133.0	1.68367	0.841835	−	0.539735i	$-0.181475\pi$
$557$	22133.0	1.68367	0.841835	+	0.539735i	$0.181475\pi$
$558$	0	0
$559$	14256.2	1.07866
$560$	0	0
$561$	16533.1	1.24425
$562$	0	0
$563$	18464.4	1.38220	0.691102	−	0.722757i	$-0.257126\pi$
$563$	18464.4	1.38220	0.691102	+	0.722757i	$0.257126\pi$
$564$	0	0
$565$	2077.23	0.154672
$566$	0	0
$567$	19491.3	1.44366
$568$	0	0
$569$	−5538.92	−0.408091	−0.204045	−	0.978961i	$-0.565409\pi$
$569$	−5538.92	−0.408091	−0.204045	+	0.978961i	$0.565409\pi$
$570$	0	0
$571$	−11219.3	−0.822262	−0.411131	−	0.911576i	$-0.634866\pi$
$571$	−11219.3	−0.822262	−0.411131	+	0.911576i	$0.634866\pi$
$572$	0	0
$573$	−27324.8	−1.99216
$574$	0	0
$575$	4150.71	0.301038
$576$	0	0
$577$	−8239.06	−0.594448	−0.297224	−	0.954808i	$-0.596061\pi$
$577$	−8239.06	−0.594448	−0.297224	+	0.954808i	$0.596061\pi$
$578$	0	0
$579$	29630.0	2.12674
$580$	0	0
$581$	−2924.73	−0.208844
$582$	0	0
$583$	11350.5	0.806326
$584$	0	0
$585$	−22214.5	−1.57001
$586$	0	0
$587$	−6452.46	−0.453699	−0.226850	−	0.973930i	$-0.572843\pi$
$587$	−6452.46	−0.453699	−0.226850	+	0.973930i	$0.572843\pi$
$588$	0	0
$589$	12247.5	0.856790
$590$	0	0
$591$	371.863	0.0258822
$592$	0	0
$593$	9937.52	0.688171	0.344085	−	0.938938i	$-0.388189\pi$
$593$	9937.52	0.688171	0.344085	+	0.938938i	$0.388189\pi$
$594$	0	0
$595$	−10635.4	−0.732790
$596$	0	0
$597$	4542.81	0.311432
$598$	0	0
$599$	−5754.72	−0.392540	−0.196270	−	0.980550i	$-0.562883\pi$
$599$	−5754.72	−0.392540	−0.196270	+	0.980550i	$0.562883\pi$
$600$	0	0
$601$	17242.3	1.17026	0.585132	−	0.810938i	$-0.301043\pi$
$601$	17242.3	1.17026	0.585132	+	0.810938i	$0.301043\pi$
$602$	0	0
$603$	56345.6	3.80525
$604$	0	0
$605$	4074.40	0.273798
$606$	0	0
$607$	11857.0	0.792854	0.396427	−	0.918066i	$-0.370250\pi$
$607$	11857.0	0.792854	0.396427	+	0.918066i	$0.370250\pi$
$608$	0	0
$609$	−4853.23	−0.322927
$610$	0	0
$611$	10867.7	0.719575
$612$	0	0
$613$	6818.16	0.449238	0.224619	−	0.974447i	$-0.427886\pi$
$613$	6818.16	0.449238	0.224619	+	0.974447i	$0.427886\pi$
$614$	0	0
$615$	−5620.75	−0.368537
$616$	0	0
$617$	−30340.0	−1.97965	−0.989825	−	0.142289i	$-0.954554\pi$
$617$	−30340.0	−1.97965	−0.989825	+	0.142289i	$0.954554\pi$
$618$	0	0
$619$	8090.29	0.525325	0.262663	−	0.964888i	$-0.415399\pi$
$619$	8090.29	0.525325	0.262663	+	0.964888i	$0.415399\pi$
$620$	0	0
$621$	−44638.7	−2.88453
$622$	0	0
$623$	−20653.6	−1.32820
$624$	0	0
$625$	−11586.3	−0.741526
$626$	0	0
$627$	14200.9	0.904512
$628$	0	0
$629$	−21966.8	−1.39248
$630$	0	0
$631$	12014.9	0.758010	0.379005	−	0.925395i	$-0.376266\pi$
$631$	12014.9	0.758010	0.379005	+	0.925395i	$0.376266\pi$
$632$	0	0
$633$	−18482.4	−1.16052
$634$	0	0
$635$	−23092.8	−1.44317
$636$	0	0
$637$	−529.733	−0.0329494
$638$	0	0
$639$	−14022.2	−0.868089
$640$	0	0
$641$	2106.54	0.129802	0.0649011	−	0.997892i	$-0.479327\pi$
$641$	2106.54	0.129802	0.0649011	+	0.997892i	$0.479327\pi$
$642$	0	0
$643$	−9074.16	−0.556532	−0.278266	−	0.960504i	$-0.589760\pi$
$643$	−9074.16	−0.556532	−0.278266	+	0.960504i	$0.589760\pi$
$644$	0	0
$645$	−33800.9	−2.06343
$646$	0	0
$647$	−25416.1	−1.54437	−0.772187	−	0.635395i	$-0.780837\pi$
$647$	−25416.1	−1.54437	−0.772187	+	0.635395i	$0.780837\pi$
$648$	0	0
$649$	−3914.03	−0.236732
$650$	0	0
$651$	40379.4	2.43102
$652$	0	0
$653$	−18241.8	−1.09319	−0.546597	−	0.837396i	$-0.684077\pi$
$653$	−18241.8	−1.09319	−0.546597	+	0.837396i	$0.684077\pi$
$654$	0	0
$655$	−22750.7	−1.35717
$656$	0	0
$657$	32811.6	1.94841
$658$	0	0
$659$	−6878.96	−0.406626	−0.203313	−	0.979114i	$-0.565171\pi$
$659$	−6878.96	−0.406626	−0.203313	+	0.979114i	$0.565171\pi$
$660$	0	0
$661$	−26498.4	−1.55925	−0.779627	−	0.626244i	$-0.784592\pi$
$661$	−26498.4	−1.55925	−0.779627	+	0.626244i	$0.784592\pi$
$662$	0	0
$663$	21026.5	1.23168
$664$	0	0
$665$	−9135.18	−0.532703
$666$	0	0
$667$	4519.41	0.262357
$668$	0	0
$669$	−18219.7	−1.05293
$670$	0	0
$671$	−14046.8	−0.808151
$672$	0	0
$673$	1232.11	0.0705710	0.0352855	−	0.999377i	$-0.488766\pi$
$673$	1232.11	0.0705710	0.0352855	+	0.999377i	$0.488766\pi$
$674$	0	0
$675$	−7629.00	−0.435023
$676$	0	0
$677$	−6769.54	−0.384305	−0.192153	−	0.981365i	$-0.561547\pi$
$677$	−6769.54	−0.384305	−0.192153	+	0.981365i	$0.561547\pi$
$678$	0	0
$679$	−26299.8	−1.48644
$680$	0	0
$681$	48919.0	2.75269
$682$	0	0
$683$	14965.1	0.838393	0.419196	−	0.907896i	$-0.362312\pi$
$683$	14965.1	0.838393	0.419196	+	0.907896i	$0.362312\pi$
$684$	0	0
$685$	15664.2	0.873717
$686$	0	0
$687$	22722.0	1.26186
$688$	0	0
$689$	14435.4	0.798177
$690$	0	0
$691$	−26158.4	−1.44011	−0.720053	−	0.693919i	$-0.755883\pi$
$691$	−26158.4	−1.44011	−0.720053	+	0.693919i	$0.755883\pi$
$692$	0	0
$693$	31957.7	1.75176
$694$	0	0
$695$	12109.7	0.660929
$696$	0	0
$697$	3631.39	0.197344
$698$	0	0
$699$	−29794.5	−1.61220
$700$	0	0
$701$	−1249.14	−0.0673031	−0.0336515	−	0.999434i	$-0.510714\pi$
$701$	−1249.14	−0.0673031	−0.0336515	+	0.999434i	$0.510714\pi$
$702$	0	0
$703$	−18868.1	−1.01227
$704$	0	0
$705$	−25767.0	−1.37651
$706$	0	0
$707$	−17781.0	−0.945863
$708$	0	0
$709$	−16036.0	−0.849427	−0.424713	−	0.905328i	$-0.639625\pi$
$709$	−16036.0	−0.849427	−0.424713	+	0.905328i	$0.639625\pi$
$710$	0	0
$711$	−24641.0	−1.29973
$712$	0	0
$713$	−37602.0	−1.97504
$714$	0	0
$715$	−11606.8	−0.607093
$716$	0	0
$717$	39476.7	2.05618
$718$	0	0
$719$	−17662.1	−0.916111	−0.458056	−	0.888923i	$-0.651454\pi$
$719$	−17662.1	−0.916111	−0.458056	+	0.888923i	$0.651454\pi$
$720$	0	0
$721$	6807.82	0.351645
$722$	0	0
$723$	59840.5	3.07814
$724$	0	0
$725$	772.391	0.0395667
$726$	0	0
$727$	700.740	0.0357483	0.0178741	−	0.999840i	$-0.494310\pi$
$727$	700.740	0.0357483	0.0178741	+	0.999840i	$0.494310\pi$
$728$	0	0
$729$	−8963.33	−0.455385
$730$	0	0
$731$	21837.7	1.10492
$732$	0	0
$733$	−31338.0	−1.57912	−0.789561	−	0.613672i	$-0.789692\pi$
$733$	−31338.0	−1.57912	−0.789561	+	0.613672i	$0.789692\pi$
$734$	0	0
$735$	1255.98	0.0630307
$736$	0	0
$737$	29440.0	1.47142
$738$	0	0
$739$	−34066.8	−1.69576	−0.847880	−	0.530188i	$-0.822121\pi$
$739$	−34066.8	−1.69576	−0.847880	+	0.530188i	$0.822121\pi$
$740$	0	0
$741$	18060.5	0.895371
$742$	0	0
$743$	−6009.39	−0.296720	−0.148360	−	0.988933i	$-0.547399\pi$
$743$	−6009.39	−0.296720	−0.148360	+	0.988933i	$0.547399\pi$
$744$	0	0
$745$	3971.03	0.195285
$746$	0	0
$747$	−9357.75	−0.458343
$748$	0	0
$749$	−14139.1	−0.689763
$750$	0	0
$751$	−4761.05	−0.231336	−0.115668	−	0.993288i	$-0.536901\pi$
$751$	−4761.05	−0.231336	−0.115668	+	0.993288i	$0.536901\pi$
$752$	0	0
$753$	−3756.32	−0.181790
$754$	0	0
$755$	6064.67	0.292339
$756$	0	0
$757$	−28880.1	−1.38661	−0.693307	−	0.720643i	$-0.743847\pi$
$757$	−28880.1	−1.38661	−0.693307	+	0.720643i	$0.743847\pi$
$758$	0	0
$759$	−43599.3	−2.08505
$760$	0	0
$761$	14423.5	0.687059	0.343530	−	0.939142i	$-0.388377\pi$
$761$	14423.5	0.687059	0.343530	+	0.939142i	$0.388377\pi$
$762$	0	0
$763$	2658.32	0.126131
$764$	0	0
$765$	−34028.3	−1.60823
$766$	0	0
$767$	−4977.82	−0.234340
$768$	0	0
$769$	38830.7	1.82090	0.910449	−	0.413621i	$-0.135736\pi$
$769$	38830.7	1.82090	0.910449	+	0.413621i	$0.135736\pi$
$770$	0	0
$771$	−50904.6	−2.37780
$772$	0	0
$773$	−15417.7	−0.717380	−0.358690	−	0.933457i	$-0.616776\pi$
$773$	−15417.7	−0.717380	−0.358690	+	0.933457i	$0.616776\pi$
$774$	0	0
$775$	−6426.39	−0.297861
$776$	0	0
$777$	−62207.3	−2.87217
$778$	0	0
$779$	3119.14	0.143459
$780$	0	0
$781$	−7326.45	−0.335673
$782$	0	0
$783$	−8306.66	−0.379126
$784$	0	0
$785$	−22009.9	−1.00072
$786$	0	0
$787$	−1122.93	−0.0508618	−0.0254309	−	0.999677i	$-0.508096\pi$
$787$	−1122.93	−0.0508618	−0.0254309	+	0.999677i	$0.508096\pi$
$788$	0	0
$789$	38358.4	1.73079
$790$	0	0
$791$	−3800.47	−0.170833
$792$	0	0
$793$	−17864.5	−0.799984
$794$	0	0
$795$	−34225.8	−1.52687
$796$	0	0
$797$	17273.1	0.767684	0.383842	−	0.923399i	$-0.374601\pi$
$797$	17273.1	0.767684	0.383842	+	0.923399i	$0.374601\pi$
$798$	0	0
$799$	16647.2	0.737092
$800$	0	0
$801$	−66081.7	−2.91496
$802$	0	0
$803$	17143.7	0.753411
$804$	0	0
$805$	28046.6	1.22797
$806$	0	0
$807$	−31431.2	−1.37104
$808$	0	0
$809$	31287.2	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$809$	31287.2	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$810$	0	0
$811$	−20399.1	−0.883244	−0.441622	−	0.897201i	$-0.645597\pi$
$811$	−20399.1	−0.883244	−0.441622	+	0.897201i	$0.645597\pi$
$812$	0	0
$813$	−70448.8	−3.03905
$814$	0	0
$815$	−33570.3	−1.44284
$816$	0	0
$817$	18757.3	0.803223
$818$	0	0
$819$	40643.4	1.73406
$820$	0	0
$821$	9158.81	0.389336	0.194668	−	0.980869i	$-0.437637\pi$
$821$	9158.81	0.389336	0.194668	+	0.980869i	$0.437637\pi$
$822$	0	0
$823$	20147.5	0.853340	0.426670	−	0.904407i	$-0.359687\pi$
$823$	20147.5	0.853340	0.426670	+	0.904407i	$0.359687\pi$
$824$	0	0
$825$	−7451.35	−0.314452
$826$	0	0
$827$	−15491.3	−0.651372	−0.325686	−	0.945478i	$-0.605595\pi$
$827$	−15491.3	−0.651372	−0.325686	+	0.945478i	$0.605595\pi$
$828$	0	0
$829$	36020.1	1.50908	0.754542	−	0.656252i	$-0.227859\pi$
$829$	36020.1	1.50908	0.754542	+	0.656252i	$0.227859\pi$
$830$	0	0
$831$	−2348.49	−0.0980363
$832$	0	0
$833$	−811.449	−0.0337515
$834$	0	0
$835$	−14571.2	−0.603902
$836$	0	0
$837$	69112.4	2.85409
$838$	0	0
$839$	−32919.5	−1.35460	−0.677298	−	0.735709i	$-0.736849\pi$
$839$	−32919.5	−1.35460	−0.677298	+	0.735709i	$0.736849\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−34591.5	−1.41328
$844$	0	0
$845$	7028.30	0.286131
$846$	0	0
$847$	−7454.48	−0.302407
$848$	0	0
$849$	−70291.0	−2.84144
$850$	0	0
$851$	57928.5	2.33345
$852$	0	0
$853$	−15790.8	−0.633842	−0.316921	−	0.948452i	$-0.602649\pi$
$853$	−15790.8	−0.633842	−0.316921	+	0.948452i	$0.602649\pi$
$854$	0	0
$855$	−29228.2	−1.16911
$856$	0	0
$857$	31142.7	1.24132	0.620661	−	0.784079i	$-0.286864\pi$
$857$	31142.7	1.24132	0.620661	+	0.784079i	$0.286864\pi$
$858$	0	0
$859$	15038.5	0.597331	0.298665	−	0.954358i	$-0.403459\pi$
$859$	15038.5	0.597331	0.298665	+	0.954358i	$0.403459\pi$
$860$	0	0
$861$	10283.7	0.407045
$862$	0	0
$863$	−28642.7	−1.12979	−0.564894	−	0.825163i	$-0.691083\pi$
$863$	−28642.7	−1.12979	−0.564894	+	0.825163i	$0.691083\pi$
$864$	0	0
$865$	−24737.3	−0.972361
$866$	0	0
$867$	−13102.4	−0.513243
$868$	0	0
$869$	−12874.7	−0.502582
$870$	0	0
$871$	37441.4	1.45655
$872$	0	0
$873$	−84146.7	−3.26224
$874$	0	0
$875$	27289.4	1.05434
$876$	0	0
$877$	23442.1	0.902603	0.451302	−	0.892371i	$-0.350960\pi$
$877$	23442.1	0.902603	0.451302	+	0.892371i	$0.350960\pi$
$878$	0	0
$879$	31542.9	1.21037
$880$	0	0
$881$	48730.1	1.86352	0.931758	−	0.363079i	$-0.118275\pi$
$881$	48730.1	1.86352	0.931758	+	0.363079i	$0.118275\pi$
$882$	0	0
$883$	−37501.4	−1.42924	−0.714622	−	0.699511i	$-0.753401\pi$
$883$	−37501.4	−1.42924	−0.714622	+	0.699511i	$0.753401\pi$
$884$	0	0
$885$	11802.3	0.448281
$886$	0	0
$887$	−23885.6	−0.904170	−0.452085	−	0.891975i	$-0.649320\pi$
$887$	−23885.6	−0.904170	−0.452085	+	0.891975i	$0.649320\pi$
$888$	0	0
$889$	42250.4	1.59396
$890$	0	0
$891$	32584.0	1.22515
$892$	0	0
$893$	14299.0	0.535830
$894$	0	0
$895$	−16678.6	−0.622911
$896$	0	0
$897$	−55449.0	−2.06398
$898$	0	0
$899$	−6997.22	−0.259589
$900$	0	0
$901$	22112.2	0.817608
$902$	0	0
$903$	61841.8	2.27903
$904$	0	0
$905$	−34348.7	−1.26164
$906$	0	0
$907$	10117.1	0.370380	0.185190	−	0.982703i	$-0.440710\pi$
$907$	10117.1	0.370380	0.185190	+	0.982703i	$0.440710\pi$
$908$	0	0
$909$	−56890.9	−2.07585
$910$	0	0
$911$	−7387.19	−0.268659	−0.134330	−	0.990937i	$-0.542888\pi$
$911$	−7387.19	−0.268659	−0.134330	+	0.990937i	$0.542888\pi$
$912$	0	0
$913$	−4889.33	−0.177232
$914$	0	0
$915$	42356.2	1.53033
$916$	0	0
$917$	41624.4	1.49897
$918$	0	0
$919$	−41196.3	−1.47872	−0.739359	−	0.673312i	$-0.764871\pi$
$919$	−41196.3	−1.47872	−0.739359	+	0.673312i	$0.764871\pi$
$920$	0	0
$921$	62843.3	2.24838
$922$	0	0
$923$	−9317.69	−0.332281
$924$	0	0
$925$	9900.29	0.351913
$926$	0	0
$927$	21781.8	0.771745
$928$	0	0
$929$	−42590.3	−1.50414	−0.752068	−	0.659085i	$-0.770944\pi$
$929$	−42590.3	−1.50414	−0.752068	+	0.659085i	$0.770944\pi$
$930$	0	0
$931$	−696.985	−0.0245357
$932$	0	0
$933$	−56819.3	−1.99376
$934$	0	0
$935$	−17779.4	−0.621871
$936$	0	0
$937$	−19764.2	−0.689079	−0.344539	−	0.938772i	$-0.611965\pi$
$937$	−19764.2	−0.689079	−0.344539	+	0.938772i	$0.611965\pi$
$938$	0	0
$939$	−49392.6	−1.71658
$940$	0	0
$941$	19163.0	0.663866	0.331933	−	0.943303i	$-0.392299\pi$
$941$	19163.0	0.663866	0.331933	+	0.943303i	$0.392299\pi$
$942$	0	0
$943$	−9576.31	−0.330697
$944$	0	0
$945$	−51549.7	−1.77451
$946$	0	0
$947$	29334.5	1.00659	0.503297	−	0.864114i	$-0.332120\pi$
$947$	29334.5	1.00659	0.503297	+	0.864114i	$0.332120\pi$
$948$	0	0
$949$	21803.2	0.745797
$950$	0	0
$951$	−16715.0	−0.569948
$952$	0	0
$953$	−26975.1	−0.916904	−0.458452	−	0.888719i	$-0.651596\pi$
$953$	−26975.1	−0.916904	−0.458452	+	0.888719i	$0.651596\pi$
$954$	0	0
$955$	29384.7	0.995672
$956$	0	0
$957$	−8113.23	−0.274048
$958$	0	0
$959$	−28658.9	−0.965011
$960$	0	0
$961$	28426.7	0.954204
$962$	0	0
$963$	−45238.5	−1.51380
$964$	0	0
$965$	−31863.7	−1.06293
$966$	0	0
$967$	12036.6	0.400280	0.200140	−	0.979767i	$-0.435860\pi$
$967$	12036.6	0.400280	0.200140	+	0.979767i	$0.435860\pi$
$968$	0	0
$969$	27665.2	0.917167
$970$	0	0
$971$	−43608.2	−1.44125	−0.720625	−	0.693325i	$-0.756145\pi$
$971$	−43608.2	−1.44125	−0.720625	+	0.693325i	$0.756145\pi$
$972$	0	0
$973$	−22155.7	−0.729989
$974$	0	0
$975$	−9476.54	−0.311274
$976$	0	0
$977$	−48826.4	−1.59887	−0.799434	−	0.600754i	$-0.794867\pi$
$977$	−48826.4	−1.59887	−0.799434	+	0.600754i	$0.794867\pi$
$978$	0	0
$979$	−34527.0	−1.12716
$980$	0	0
$981$	8505.37	0.276815
$982$	0	0
$983$	−23279.6	−0.755345	−0.377672	−	0.925939i	$-0.623275\pi$
$983$	−23279.6	−0.755345	−0.377672	+	0.925939i	$0.623275\pi$
$984$	0	0
$985$	−399.897	−0.0129358
$986$	0	0
$987$	47143.0	1.52034
$988$	0	0
$989$	−57588.1	−1.85156
$990$	0	0
$991$	39511.7	1.26653	0.633264	−	0.773936i	$-0.281715\pi$
$991$	39511.7	1.26653	0.633264	+	0.773936i	$0.281715\pi$
$992$	0	0
$993$	93999.3	3.00400
$994$	0	0
$995$	−4885.28	−0.155652
$996$	0	0
$997$	−38690.6	−1.22903	−0.614515	−	0.788905i	$-0.710648\pi$
$997$	−38690.6	−1.22903	−0.614515	+	0.788905i	$0.710648\pi$
$998$	0	0
$999$	−106472.	−3.37201

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bl.1.11		12
4.3	odd	2		1856.4.a.bj.1.2		12
8.3	odd	2		928.4.a.j.1.11	yes	12
8.5	even	2		928.4.a.h.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.2	✓	12	8.5	even	2
928.4.a.j.1.11	yes	12	8.3	odd	2
1856.4.a.bj.1.2		12	4.3	odd	2
1856.4.a.bl.1.11		12	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+9.22268 q^{3} -9.91795 q^{5} +18.1458 q^{7} +58.0578 q^{9} +O(q^{10})\)	\(q+9.22268 q^{3} -9.91795 q^{5} +18.1458 q^{7} +58.0578 q^{9} +30.3346 q^{11} +38.5792 q^{13} -91.4701 q^{15} +59.0959 q^{17} +50.7598 q^{19} +167.353 q^{21} -155.842 q^{23} -26.6342 q^{25} +286.437 q^{27} -29.0000 q^{29} +241.283 q^{31} +279.767 q^{33} -179.969 q^{35} -371.714 q^{37} +355.804 q^{39} +61.4490 q^{41} +369.530 q^{43} -575.815 q^{45} +281.699 q^{47} -13.7311 q^{49} +545.023 q^{51} +374.175 q^{53} -300.857 q^{55} +468.141 q^{57} -129.028 q^{59} -463.061 q^{61} +1053.50 q^{63} -382.627 q^{65} +970.507 q^{67} -1437.28 q^{69} -241.521 q^{71} +565.154 q^{73} -245.639 q^{75} +550.445 q^{77} -424.422 q^{79} +1074.15 q^{81} -161.180 q^{83} -586.110 q^{85} -267.458 q^{87} -1138.20 q^{89} +700.050 q^{91} +2225.28 q^{93} -503.433 q^{95} -1449.36 q^{97} +1761.16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) \) 12 * q + 14 * q^3 + 10 * q^5 - 44 * q^7 + 130 * q^9	\( 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9} + 46 q^{11} + 34 q^{13} + 50 q^{15} + 36 q^{17} + 148 q^{19} + 92 q^{21} - 328 q^{23} + 486 q^{25} + 326 q^{27} - 348 q^{29} - 374 q^{31} + 710 q^{33} + 204 q^{35} + 340 q^{37} + 122 q^{39} + 32 q^{41} + 462 q^{43} + 1132 q^{45} - 434 q^{47} + 1508 q^{49} + 440 q^{51} - 610 q^{53} - 46 q^{55} - 932 q^{57} + 1240 q^{59} + 1228 q^{61} - 4240 q^{63} + 730 q^{65} + 1672 q^{67} + 528 q^{69} - 3220 q^{71} + 564 q^{73} + 6032 q^{75} - 644 q^{77} - 1862 q^{79} + 3040 q^{81} + 3736 q^{83} + 808 q^{85} - 406 q^{87} + 584 q^{89} + 4844 q^{91} + 3226 q^{93} - 2844 q^{95} + 904 q^{97} + 6832 q^{99}+O(q^{100}) \) 12 * q + 14 * q^3 + 10 * q^5 - 44 * q^7 + 130 * q^9 + 46 * q^11 + 34 * q^13 + 50 * q^15 + 36 * q^17 + 148 * q^19 + 92 * q^21 - 328 * q^23 + 486 * q^25 + 326 * q^27 - 348 * q^29 - 374 * q^31 + 710 * q^33 + 204 * q^35 + 340 * q^37 + 122 * q^39 + 32 * q^41 + 462 * q^43 + 1132 * q^45 - 434 * q^47 + 1508 * q^49 + 440 * q^51 - 610 * q^53 - 46 * q^55 - 932 * q^57 + 1240 * q^59 + 1228 * q^61 - 4240 * q^63 + 730 * q^65 + 1672 * q^67 + 528 * q^69 - 3220 * q^71 + 564 * q^73 + 6032 * q^75 - 644 * q^77 - 1862 * q^79 + 3040 * q^81 + 3736 * q^83 + 808 * q^85 - 406 * q^87 + 584 * q^89 + 4844 * q^91 + 3226 * q^93 - 2844 * q^95 + 904 * q^97 + 6832 * q^99

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