LMFDB - Embedded newform 1856.4.a.bl.1.10

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			1.10
Root			$5.28045$ 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)

$f(q)$	$=$	$q+6.28045 q^{3} -18.9942 q^{5} -32.3083 q^{7} +12.4440 q^{9} +O(q^{10})$	$q+6.28045 q^{3} -18.9942 q^{5} -32.3083 q^{7} +12.4440 q^{9} -29.5430 q^{11} +0.0104004 q^{13} -119.292 q^{15} -9.38869 q^{17} -98.7408 q^{19} -202.911 q^{21} -152.527 q^{23} +235.780 q^{25} -91.4181 q^{27} -29.0000 q^{29} -192.050 q^{31} -185.543 q^{33} +613.671 q^{35} -3.01616 q^{37} +0.0653192 q^{39} -140.233 q^{41} +346.260 q^{43} -236.364 q^{45} +282.092 q^{47} +700.829 q^{49} -58.9652 q^{51} -628.255 q^{53} +561.146 q^{55} -620.136 q^{57} -586.318 q^{59} +818.471 q^{61} -402.045 q^{63} -0.197547 q^{65} -463.640 q^{67} -957.937 q^{69} -23.3845 q^{71} -83.8973 q^{73} +1480.80 q^{75} +954.486 q^{77} -141.642 q^{79} -910.135 q^{81} +994.167 q^{83} +178.331 q^{85} -182.133 q^{87} -270.071 q^{89} -0.336020 q^{91} -1206.16 q^{93} +1875.50 q^{95} -1235.11 q^{97} -367.634 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	6.28045	1.20867	0.604336	−	0.796729i	$-0.293438\pi$
$3$	6.28045	1.20867	0.604336	+	0.796729i	$0.293438\pi$
$4$	0	0
$5$	−18.9942	−1.69889	−0.849447	−	0.527675i	$-0.823064\pi$
$5$	−18.9942	−1.69889	−0.849447	+	0.527675i	$0.823064\pi$
$6$	0	0
$7$	−32.3083	−1.74449	−0.872243	−	0.489072i	$-0.837335\pi$
$7$	−32.3083	−1.74449	−0.872243	+	0.489072i	$0.837335\pi$
$8$	0	0
$9$	12.4440	0.460889
$10$	0	0
$11$	−29.5430	−0.809778	−0.404889	−	0.914366i	$-0.632690\pi$
$11$	−29.5430	−0.809778	−0.404889	+	0.914366i	$0.632690\pi$
$12$	0	0
$13$	0.0104004	0.000221889 0	0.000110944	−	1.00000i	$-0.499965\pi$
$13$	0.0104004	0.000221889 0	0.000110944		1.00000i	$0.499965\pi$
$14$	0	0
$15$	−119.292	−2.05341
$16$	0	0
$17$	−9.38869	−0.133947	−0.0669733	−	0.997755i	$-0.521334\pi$
$17$	−9.38869	−0.133947	−0.0669733	+	0.997755i	$0.521334\pi$
$18$	0	0
$19$	−98.7408	−1.19225	−0.596123	−	0.802893i	$-0.703293\pi$
$19$	−98.7408	−1.19225	−0.596123	+	0.802893i	$0.703293\pi$
$20$	0	0
$21$	−202.911	−2.10851
$22$	0	0
$23$	−152.527	−1.38278	−0.691392	−	0.722480i	$-0.743002\pi$
$23$	−152.527	−1.38278	−0.691392	+	0.722480i	$0.743002\pi$
$24$	0	0
$25$	235.780	1.88624
$26$	0	0
$27$	−91.4181	−0.651608
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−192.050	−1.11268	−0.556341	−	0.830954i	$-0.687795\pi$
$31$	−192.050	−1.11268	−0.556341	+	0.830954i	$0.687795\pi$
$32$	0	0
$33$	−185.543	−0.978756
$34$	0	0
$35$	613.671	2.96370
$36$	0	0
$37$	−3.01616	−0.0134014	−0.00670072	−	0.999978i	$-0.502133\pi$
$37$	−3.01616	−0.0134014	−0.00670072	+	0.999978i	$0.502133\pi$
$38$	0	0
$39$	0.0653192	0.000268191 0
$40$	0	0
$41$	−140.233	−0.534164	−0.267082	−	0.963674i	$-0.586059\pi$
$41$	−140.233	−0.534164	−0.267082	+	0.963674i	$0.586059\pi$
$42$	0	0
$43$	346.260	1.22800	0.614002	−	0.789304i	$-0.289559\pi$
$43$	346.260	1.22800	0.614002	+	0.789304i	$0.289559\pi$
$44$	0	0
$45$	−236.364	−0.783001
$46$	0	0
$47$	282.092	0.875477	0.437738	−	0.899102i	$-0.355780\pi$
$47$	282.092	0.875477	0.437738	+	0.899102i	$0.355780\pi$
$48$	0	0
$49$	700.829	2.04323
$50$	0	0
$51$	−58.9652	−0.161898
$52$	0	0
$53$	−628.255	−1.62825	−0.814127	−	0.580687i	$-0.802784\pi$
$53$	−628.255	−1.62825	−0.814127	+	0.580687i	$0.802784\pi$
$54$	0	0
$55$	561.146	1.37573
$56$	0	0
$57$	−620.136	−1.44104
$58$	0	0
$59$	−586.318	−1.29376	−0.646882	−	0.762590i	$-0.723927\pi$
$59$	−586.318	−1.29376	−0.646882	+	0.762590i	$0.723927\pi$
$60$	0	0
$61$	818.471	1.71794	0.858971	−	0.512024i	$-0.171104\pi$
$61$	818.471	1.71794	0.858971	+	0.512024i	$0.171104\pi$
$62$	0	0
$63$	−402.045	−0.804015
$64$	0	0
$65$	−0.197547	−0.000376965 0
$66$	0	0
$67$	−463.640	−0.845413	−0.422707	−	0.906267i	$-0.638920\pi$
$67$	−463.640	−0.845413	−0.422707	+	0.906267i	$0.638920\pi$
$68$	0	0
$69$	−957.937	−1.67133
$70$	0	0
$71$	−23.3845	−0.0390878	−0.0195439	−	0.999809i	$-0.506221\pi$
$71$	−23.3845	−0.0390878	−0.0195439	+	0.999809i	$0.506221\pi$
$72$	0	0
$73$	−83.8973	−0.134513	−0.0672564	−	0.997736i	$-0.521425\pi$
$73$	−83.8973	−0.134513	−0.0672564	+	0.997736i	$0.521425\pi$
$74$	0	0
$75$	1480.80	2.27984
$76$	0	0
$77$	954.486	1.41265
$78$	0	0
$79$	−141.642	−0.201721	−0.100860	−	0.994901i	$-0.532160\pi$
$79$	−141.642	−0.201721	−0.100860	+	0.994901i	$0.532160\pi$
$80$	0	0
$81$	−910.135	−1.24847
$82$	0	0
$83$	994.167	1.31475	0.657373	−	0.753565i	$-0.271667\pi$
$83$	994.167	1.31475	0.657373	+	0.753565i	$0.271667\pi$
$84$	0	0
$85$	178.331	0.227561
$86$	0	0
$87$	−182.133	−0.224445
$88$	0	0
$89$	−270.071	−0.321657	−0.160828	−	0.986982i	$-0.551417\pi$
$89$	−270.071	−0.321657	−0.160828	+	0.986982i	$0.551417\pi$
$90$	0	0
$91$	−0.336020	−0.000387082 0
$92$	0	0
$93$	−1206.16	−1.34487
$94$	0	0
$95$	1875.50	2.02550
$96$	0	0
$97$	−1235.11	−1.29284	−0.646422	−	0.762980i	$-0.723736\pi$
$97$	−1235.11	−1.29284	−0.646422	+	0.762980i	$0.723736\pi$
$98$	0	0
$99$	−367.634	−0.373218
$100$	0	0
$101$	−62.3712	−0.0614472	−0.0307236	−	0.999528i	$-0.509781\pi$
$101$	−62.3712	−0.0614472	−0.0307236	+	0.999528i	$0.509781\pi$
$102$	0	0
$103$	−51.1389	−0.0489210	−0.0244605	−	0.999701i	$-0.507787\pi$
$103$	−51.1389	−0.0489210	−0.0244605	+	0.999701i	$0.507787\pi$
$104$	0	0
$105$	3854.13	3.58214
$106$	0	0
$107$	1395.49	1.26081	0.630405	−	0.776266i	$-0.282889\pi$
$107$	1395.49	1.26081	0.630405	+	0.776266i	$0.282889\pi$
$108$	0	0
$109$	1420.08	1.24788	0.623940	−	0.781472i	$-0.285531\pi$
$109$	1420.08	1.24788	0.623940	+	0.781472i	$0.285531\pi$
$110$	0	0
$111$	−18.9428	−0.0161979
$112$	0	0
$113$	−1636.33	−1.36224	−0.681120	−	0.732172i	$-0.738507\pi$
$113$	−1636.33	−1.36224	−0.681120	+	0.732172i	$0.738507\pi$
$114$	0	0
$115$	2897.13	2.34920
$116$	0	0
$117$	0.129423	0.000102266 0
$118$	0	0
$119$	303.333	0.233668
$120$	0	0
$121$	−458.210	−0.344260
$122$	0	0
$123$	−880.726	−0.645629
$124$	0	0
$125$	−2104.17	−1.50562
$126$	0	0
$127$	−2572.09	−1.79713	−0.898566	−	0.438838i	$-0.855390\pi$
$127$	−2572.09	−1.79713	−0.898566	+	0.438838i	$0.855390\pi$
$128$	0	0
$129$	2174.67	1.48426
$130$	0	0
$131$	−1720.18	−1.14728	−0.573638	−	0.819109i	$-0.694468\pi$
$131$	−1720.18	−1.14728	−0.573638	+	0.819109i	$0.694468\pi$
$132$	0	0
$133$	3190.15	2.07986
$134$	0	0
$135$	1736.41	1.10701
$136$	0	0
$137$	2805.40	1.74950	0.874749	−	0.484576i	$-0.161026\pi$
$137$	2805.40	1.74950	0.874749	+	0.484576i	$0.161026\pi$
$138$	0	0
$139$	1674.76	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$139$	1674.76	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$140$	0	0
$141$	1771.67	1.05816
$142$	0	0
$143$	−0.307259	−0.000179681 0
$144$	0	0
$145$	550.832	0.315477
$146$	0	0
$147$	4401.52	2.46960
$148$	0	0
$149$	676.411	0.371905	0.185952	−	0.982559i	$-0.440463\pi$
$149$	676.411	0.371905	0.185952	+	0.982559i	$0.440463\pi$
$150$	0	0
$151$	168.195	0.0906460	0.0453230	−	0.998972i	$-0.485568\pi$
$151$	168.195	0.0906460	0.0453230	+	0.998972i	$0.485568\pi$
$152$	0	0
$153$	−116.833	−0.0617345
$154$	0	0
$155$	3647.83	1.89033
$156$	0	0
$157$	3222.42	1.63807	0.819035	−	0.573744i	$-0.194509\pi$
$157$	3222.42	1.63807	0.819035	+	0.573744i	$0.194509\pi$
$158$	0	0
$159$	−3945.72	−1.96803
$160$	0	0
$161$	4927.89	2.41225
$162$	0	0
$163$	−29.9587	−0.0143960	−0.00719799	−	0.999974i	$-0.502291\pi$
$163$	−29.9587	−0.0143960	−0.00719799	+	0.999974i	$0.502291\pi$
$164$	0	0
$165$	3524.25	1.66280
$166$	0	0
$167$	−1636.27	−0.758194	−0.379097	−	0.925357i	$-0.623765\pi$
$167$	−1636.27	−0.758194	−0.379097	+	0.925357i	$0.623765\pi$
$168$	0	0
$169$	−2197.00	−1.00000
$170$	0	0
$171$	−1228.73	−0.549494
$172$	0	0
$173$	1536.82	0.675388	0.337694	−	0.941256i	$-0.390353\pi$
$173$	1536.82	0.675388	0.337694	+	0.941256i	$0.390353\pi$
$174$	0	0
$175$	−7617.65	−3.29052
$176$	0	0
$177$	−3682.34	−1.56374
$178$	0	0
$179$	−489.179	−0.204262	−0.102131	−	0.994771i	$-0.532566\pi$
$179$	−489.179	−0.204262	−0.102131	+	0.994771i	$0.532566\pi$
$180$	0	0
$181$	221.343	0.0908965	0.0454482	−	0.998967i	$-0.485528\pi$
$181$	221.343	0.0908965	0.0454482	+	0.998967i	$0.485528\pi$
$182$	0	0
$183$	5140.36	2.07643
$184$	0	0
$185$	57.2895	0.0227676
$186$	0	0
$187$	277.370	0.108467
$188$	0	0
$189$	2953.57	1.13672
$190$	0	0
$191$	1867.61	0.707516	0.353758	−	0.935337i	$-0.384904\pi$
$191$	1867.61	0.707516	0.353758	+	0.935337i	$0.384904\pi$
$192$	0	0
$193$	−4503.52	−1.67964	−0.839820	−	0.542864i	$-0.817340\pi$
$193$	−4503.52	−1.67964	−0.839820	+	0.542864i	$0.817340\pi$
$194$	0	0
$195$	−1.24069	−0.000455627 0
$196$	0	0
$197$	3480.19	1.25865	0.629324	−	0.777143i	$-0.283332\pi$
$197$	3480.19	1.25865	0.629324	+	0.777143i	$0.283332\pi$
$198$	0	0
$199$	2586.80	0.921475	0.460737	−	0.887536i	$-0.347585\pi$
$199$	2586.80	0.921475	0.460737	+	0.887536i	$0.347585\pi$
$200$	0	0
$201$	−2911.87	−1.02183
$202$	0	0
$203$	936.942	0.323943
$204$	0	0
$205$	2663.62	0.907488
$206$	0	0
$207$	−1898.05	−0.637310
$208$	0	0
$209$	2917.10	0.965455
$210$	0	0
$211$	3493.45	1.13981	0.569903	−	0.821712i	$-0.306981\pi$
$211$	3493.45	1.13981	0.569903	+	0.821712i	$0.306981\pi$
$212$	0	0
$213$	−146.865	−0.0472444
$214$	0	0
$215$	−6576.94	−2.08625
$216$	0	0
$217$	6204.81	1.94106
$218$	0	0
$219$	−526.913	−0.162582
$220$	0	0
$221$	−0.0976462	−2.97212e−5 0
$222$	0	0
$223$	−5067.51	−1.52173	−0.760865	−	0.648910i	$-0.775225\pi$
$223$	−5067.51	−1.52173	−0.760865	+	0.648910i	$0.775225\pi$
$224$	0	0
$225$	2934.04	0.869347
$226$	0	0
$227$	−4721.64	−1.38056	−0.690278	−	0.723544i	$-0.742512\pi$
$227$	−4721.64	−1.38056	−0.690278	+	0.723544i	$0.742512\pi$
$228$	0	0
$229$	−6378.17	−1.84053	−0.920264	−	0.391297i	$-0.872026\pi$
$229$	−6378.17	−1.84053	−0.920264	+	0.391297i	$0.872026\pi$
$230$	0	0
$231$	5994.60	1.70743
$232$	0	0
$233$	1920.67	0.540030	0.270015	−	0.962856i	$-0.412971\pi$
$233$	1920.67	0.540030	0.270015	+	0.962856i	$0.412971\pi$
$234$	0	0
$235$	−5358.12	−1.48734
$236$	0	0
$237$	−889.574	−0.243815
$238$	0	0
$239$	−3128.27	−0.846657	−0.423329	−	0.905976i	$-0.639138\pi$
$239$	−3128.27	−0.846657	−0.423329	+	0.905976i	$0.639138\pi$
$240$	0	0
$241$	5730.59	1.53170	0.765851	−	0.643018i	$-0.222318\pi$
$241$	5730.59	1.53170	0.765851	+	0.643018i	$0.222318\pi$
$242$	0	0
$243$	−3247.76	−0.857383
$244$	0	0
$245$	−13311.7	−3.47124
$246$	0	0
$247$	−1.02694	−0.000264546 0
$248$	0	0
$249$	6243.81	1.58910
$250$	0	0
$251$	−6856.93	−1.72433	−0.862163	−	0.506631i	$-0.830891\pi$
$251$	−6856.93	−1.72433	−0.862163	+	0.506631i	$0.830891\pi$
$252$	0	0
$253$	4506.10	1.11975
$254$	0	0
$255$	1120.00	0.275047
$256$	0	0
$257$	−451.686	−0.109632	−0.0548159	−	0.998496i	$-0.517457\pi$
$257$	−451.686	−0.109632	−0.0548159	+	0.998496i	$0.517457\pi$
$258$	0	0
$259$	97.4470	0.0233786
$260$	0	0
$261$	−360.876	−0.0855850
$262$	0	0
$263$	−2353.81	−0.551871	−0.275936	−	0.961176i	$-0.588988\pi$
$263$	−2353.81	−0.551871	−0.275936	+	0.961176i	$0.588988\pi$
$264$	0	0
$265$	11933.2	2.76623
$266$	0	0
$267$	−1696.17	−0.388778
$268$	0	0
$269$	−303.900	−0.0688814	−0.0344407	−	0.999407i	$-0.510965\pi$
$269$	−303.900	−0.0688814	−0.0344407	+	0.999407i	$0.510965\pi$
$270$	0	0
$271$	−7009.45	−1.57120	−0.785598	−	0.618738i	$-0.787644\pi$
$271$	−7009.45	−1.57120	−0.785598	+	0.618738i	$0.787644\pi$
$272$	0	0
$273$	−2.11035	−0.000467855 0
$274$	0	0
$275$	−6965.65	−1.52743
$276$	0	0
$277$	7463.19	1.61884	0.809421	−	0.587228i	$-0.199781\pi$
$277$	7463.19	1.61884	0.809421	+	0.587228i	$0.199781\pi$
$278$	0	0
$279$	−2389.87	−0.512823
$280$	0	0
$281$	2381.78	0.505642	0.252821	−	0.967513i	$-0.418642\pi$
$281$	2381.78	0.505642	0.252821	+	0.967513i	$0.418642\pi$
$282$	0	0
$283$	1996.49	0.419361	0.209681	−	0.977770i	$-0.432758\pi$
$283$	1996.49	0.419361	0.209681	+	0.977770i	$0.432758\pi$
$284$	0	0
$285$	11779.0	2.44817
$286$	0	0
$287$	4530.70	0.931842
$288$	0	0
$289$	−4824.85	−0.982058
$290$	0	0
$291$	−7757.01	−1.56263
$292$	0	0
$293$	−4475.34	−0.892328	−0.446164	−	0.894951i	$-0.647210\pi$
$293$	−4475.34	−0.892328	−0.446164	+	0.894951i	$0.647210\pi$
$294$	0	0
$295$	11136.6	2.19797
$296$	0	0
$297$	2700.77	0.527658
$298$	0	0
$299$	−1.58634	−0.000306824 0
$300$	0	0
$301$	−11187.1	−2.14224
$302$	0	0
$303$	−391.719	−0.0742696
$304$	0	0
$305$	−15546.2	−2.91860
$306$	0	0
$307$	360.202	0.0669635	0.0334818	−	0.999439i	$-0.489340\pi$
$307$	360.202	0.0669635	0.0334818	+	0.999439i	$0.489340\pi$
$308$	0	0
$309$	−321.175	−0.0591295
$310$	0	0
$311$	−1660.17	−0.302699	−0.151350	−	0.988480i	$-0.548362\pi$
$311$	−1660.17	−0.302699	−0.151350	+	0.988480i	$0.548362\pi$
$312$	0	0
$313$	−6722.41	−1.21397	−0.606986	−	0.794713i	$-0.707621\pi$
$313$	−6722.41	−1.21397	−0.606986	+	0.794713i	$0.707621\pi$
$314$	0	0
$315$	7636.53	1.36594
$316$	0	0
$317$	−4232.40	−0.749890	−0.374945	−	0.927047i	$-0.622338\pi$
$317$	−4232.40	−0.749890	−0.374945	+	0.927047i	$0.622338\pi$
$318$	0	0
$319$	856.748	0.150372
$320$	0	0
$321$	8764.27	1.52391
$322$	0	0
$323$	927.047	0.159697
$324$	0	0
$325$	2.45220	0.000418535 0
$326$	0	0
$327$	8918.74	1.50828
$328$	0	0
$329$	−9113.94	−1.52726
$330$	0	0
$331$	−3018.60	−0.501261	−0.250630	−	0.968083i	$-0.580638\pi$
$331$	−3018.60	−0.501261	−0.250630	+	0.968083i	$0.580638\pi$
$332$	0	0
$333$	−37.5331	−0.00617657
$334$	0	0
$335$	8806.48	1.43627
$336$	0	0
$337$	−782.270	−0.126448	−0.0632240	−	0.997999i	$-0.520138\pi$
$337$	−782.270	−0.126448	−0.0632240	+	0.997999i	$0.520138\pi$
$338$	0	0
$339$	−10276.9	−1.64650
$340$	0	0
$341$	5673.73	0.901026
$342$	0	0
$343$	−11560.9	−1.81991
$344$	0	0
$345$	18195.2	2.83942
$346$	0	0
$347$	2509.12	0.388175	0.194087	−	0.980984i	$-0.437825\pi$
$347$	2509.12	0.388175	0.194087	+	0.980984i	$0.437825\pi$
$348$	0	0
$349$	−5618.39	−0.861735	−0.430868	−	0.902415i	$-0.641792\pi$
$349$	−5618.39	−0.861735	−0.430868	+	0.902415i	$0.641792\pi$
$350$	0	0
$351$	−0.950786	−0.000144585 0
$352$	0	0
$353$	−4662.41	−0.702988	−0.351494	−	0.936190i	$-0.614326\pi$
$353$	−4662.41	−0.702988	−0.351494	+	0.936190i	$0.614326\pi$
$354$	0	0
$355$	444.171	0.0664060
$356$	0	0
$357$	1905.07	0.282428
$358$	0	0
$359$	2759.19	0.405639	0.202820	−	0.979216i	$-0.434989\pi$
$359$	2759.19	0.405639	0.202820	+	0.979216i	$0.434989\pi$
$360$	0	0
$361$	2890.74	0.421453
$362$	0	0
$363$	−2877.76	−0.416097
$364$	0	0
$365$	1593.56	0.228523
$366$	0	0
$367$	1121.65	0.159536	0.0797681	−	0.996813i	$-0.474582\pi$
$367$	1121.65	0.159536	0.0797681	+	0.996813i	$0.474582\pi$
$368$	0	0
$369$	−1745.06	−0.246190
$370$	0	0
$371$	20297.9	2.84047
$372$	0	0
$373$	771.423	0.107085	0.0535426	−	0.998566i	$-0.482949\pi$
$373$	771.423	0.107085	0.0535426	+	0.998566i	$0.482949\pi$
$374$	0	0
$375$	−13215.1	−1.81980
$376$	0	0
$377$	−0.301612	−4.12037e−5 0
$378$	0	0
$379$	−11154.3	−1.51177	−0.755884	−	0.654706i	$-0.772792\pi$
$379$	−11154.3	−1.51177	−0.755884	+	0.654706i	$0.772792\pi$
$380$	0	0
$381$	−16153.9	−2.17214
$382$	0	0
$383$	5468.92	0.729631	0.364815	−	0.931080i	$-0.381132\pi$
$383$	5468.92	0.729631	0.364815	+	0.931080i	$0.381132\pi$
$384$	0	0
$385$	−18129.7	−2.39994
$386$	0	0
$387$	4308.87	0.565974
$388$	0	0
$389$	−5320.87	−0.693518	−0.346759	−	0.937954i	$-0.612718\pi$
$389$	−5320.87	−0.693518	−0.346759	+	0.937954i	$0.612718\pi$
$390$	0	0
$391$	1432.03	0.185219
$392$	0	0
$393$	−10803.5	−1.38668
$394$	0	0
$395$	2690.37	0.342702
$396$	0	0
$397$	13910.4	1.75855	0.879275	−	0.476315i	$-0.158028\pi$
$397$	13910.4	1.75855	0.879275	+	0.476315i	$0.158028\pi$
$398$	0	0
$399$	20035.6	2.51387
$400$	0	0
$401$	−2421.00	−0.301494	−0.150747	−	0.988572i	$-0.548168\pi$
$401$	−2421.00	−0.301494	−0.150747	+	0.988572i	$0.548168\pi$
$402$	0	0
$403$	−1.99740	−0.000246892 0
$404$	0	0
$405$	17287.3	2.12102
$406$	0	0
$407$	89.1064	0.0108522
$408$	0	0
$409$	4610.86	0.557439	0.278720	−	0.960373i	$-0.410090\pi$
$409$	4610.86	0.557439	0.278720	+	0.960373i	$0.410090\pi$
$410$	0	0
$411$	17619.1	2.11457
$412$	0	0
$413$	18943.0	2.25695
$414$	0	0
$415$	−18883.4	−2.23361
$416$	0	0
$417$	10518.2	1.23521
$418$	0	0
$419$	10982.6	1.28051	0.640255	−	0.768162i	$-0.278829\pi$
$419$	10982.6	1.28051	0.640255	+	0.768162i	$0.278829\pi$
$420$	0	0
$421$	−2305.18	−0.266859	−0.133429	−	0.991058i	$-0.542599\pi$
$421$	−2305.18	−0.266859	−0.133429	+	0.991058i	$0.542599\pi$
$422$	0	0
$423$	3510.36	0.403498
$424$	0	0
$425$	−2213.66	−0.252655
$426$	0	0
$427$	−26443.4	−2.99693
$428$	0	0
$429$	−1.92973	−0.000217175 0
$430$	0	0
$431$	5218.34	0.583198	0.291599	−	0.956541i	$-0.405813\pi$
$431$	5218.34	0.583198	0.291599	+	0.956541i	$0.405813\pi$
$432$	0	0
$433$	7208.84	0.800080	0.400040	−	0.916498i	$-0.368996\pi$
$433$	7208.84	0.800080	0.400040	+	0.916498i	$0.368996\pi$
$434$	0	0
$435$	3459.47	0.381308
$436$	0	0
$437$	15060.6	1.64862
$438$	0	0
$439$	1177.60	0.128027	0.0640134	−	0.997949i	$-0.479610\pi$
$439$	1177.60	0.128027	0.0640134	+	0.997949i	$0.479610\pi$
$440$	0	0
$441$	8721.12	0.941704
$442$	0	0
$443$	−13707.3	−1.47010	−0.735051	−	0.678012i	$-0.762842\pi$
$443$	−13707.3	−1.47010	−0.735051	+	0.678012i	$0.762842\pi$
$444$	0	0
$445$	5129.78	0.546460
$446$	0	0
$447$	4248.17	0.449511
$448$	0	0
$449$	−12090.4	−1.27078	−0.635392	−	0.772190i	$-0.719161\pi$
$449$	−12090.4	−1.27078	−0.635392	+	0.772190i	$0.719161\pi$
$450$	0	0
$451$	4142.91	0.432554
$452$	0	0
$453$	1056.34	0.109561
$454$	0	0
$455$	6.38243	0.000657611 0
$456$	0	0
$457$	6893.73	0.705635	0.352817	−	0.935692i	$-0.385224\pi$
$457$	6893.73	0.705635	0.352817	+	0.935692i	$0.385224\pi$
$458$	0	0
$459$	858.296	0.0872807
$460$	0	0
$461$	10151.3	1.02558	0.512789	−	0.858515i	$-0.328612\pi$
$461$	10151.3	1.02558	0.512789	+	0.858515i	$0.328612\pi$
$462$	0	0
$463$	17812.6	1.78795	0.893976	−	0.448115i	$-0.147904\pi$
$463$	17812.6	1.78795	0.893976	+	0.448115i	$0.147904\pi$
$464$	0	0
$465$	22910.0	2.28479
$466$	0	0
$467$	94.0626	0.00932055	0.00466028	−	0.999989i	$-0.498517\pi$
$467$	94.0626	0.00932055	0.00466028	+	0.999989i	$0.498517\pi$
$468$	0	0
$469$	14979.5	1.47481
$470$	0	0
$471$	20238.2	1.97989
$472$	0	0
$473$	−10229.6	−0.994411
$474$	0	0
$475$	−23281.1	−2.24886
$476$	0	0
$477$	−7818.01	−0.750445
$478$	0	0
$479$	−2220.20	−0.211782	−0.105891	−	0.994378i	$-0.533769\pi$
$479$	−2220.20	−0.211782	−0.105891	+	0.994378i	$0.533769\pi$
$480$	0	0
$481$	−0.0313692	−2.97363e−6 0
$482$	0	0
$483$	30949.3	2.91562
$484$	0	0
$485$	23459.8	2.19641
$486$	0	0
$487$	−6774.61	−0.630364	−0.315182	−	0.949031i	$-0.602065\pi$
$487$	−6774.61	−0.630364	−0.315182	+	0.949031i	$0.602065\pi$
$488$	0	0
$489$	−188.154	−0.0174000
$490$	0	0
$491$	−21395.0	−1.96649	−0.983243	−	0.182301i	$-0.941645\pi$
$491$	−21395.0	−1.96649	−0.983243	+	0.182301i	$0.941645\pi$
$492$	0	0
$493$	272.272	0.0248733
$494$	0	0
$495$	6982.91	0.634057
$496$	0	0
$497$	755.516	0.0681882
$498$	0	0
$499$	16933.1	1.51910	0.759548	−	0.650451i	$-0.225420\pi$
$499$	16933.1	1.51910	0.759548	+	0.650451i	$0.225420\pi$
$500$	0	0
$501$	−10276.5	−0.916408
$502$	0	0
$503$	9007.46	0.798455	0.399227	−	0.916852i	$-0.369278\pi$
$503$	9007.46	0.798455	0.399227	+	0.916852i	$0.369278\pi$
$504$	0	0
$505$	1184.69	0.104392
$506$	0	0
$507$	−13798.1	−1.20867
$508$	0	0
$509$	−19817.2	−1.72570	−0.862851	−	0.505459i	$-0.831323\pi$
$509$	−19817.2	−1.72570	−0.862851	+	0.505459i	$0.831323\pi$
$510$	0	0
$511$	2710.58	0.234656
$512$	0	0
$513$	9026.70	0.776878
$514$	0	0
$515$	971.342	0.0831115
$516$	0	0
$517$	−8333.86	−0.708942
$518$	0	0
$519$	9651.90	0.816323
$520$	0	0
$521$	1728.53	0.145352	0.0726758	−	0.997356i	$-0.476846\pi$
$521$	1728.53	0.145352	0.0726758	+	0.997356i	$0.476846\pi$
$522$	0	0
$523$	−18434.1	−1.54123	−0.770617	−	0.637298i	$-0.780052\pi$
$523$	−18434.1	−1.54123	−0.770617	+	0.637298i	$0.780052\pi$
$524$	0	0
$525$	−47842.3	−3.97716
$526$	0	0
$527$	1803.10	0.149040
$528$	0	0
$529$	11097.4	0.912093
$530$	0	0
$531$	−7296.15	−0.596282
$532$	0	0
$533$	−1.45848	−0.000118525 0
$534$	0	0
$535$	−26506.1	−2.14198
$536$	0	0
$537$	−3072.26	−0.246886
$538$	0	0
$539$	−20704.6	−1.65457
$540$	0	0
$541$	−4874.90	−0.387409	−0.193704	−	0.981060i	$-0.562050\pi$
$541$	−4874.90	−0.387409	−0.193704	+	0.981060i	$0.562050\pi$
$542$	0	0
$543$	1390.13	0.109864
$544$	0	0
$545$	−26973.3	−2.12002
$546$	0	0
$547$	−1566.21	−0.122424	−0.0612122	−	0.998125i	$-0.519497\pi$
$547$	−1566.21	−0.122424	−0.0612122	+	0.998125i	$0.519497\pi$
$548$	0	0
$549$	10185.1	0.791781
$550$	0	0
$551$	2863.48	0.221395
$552$	0	0
$553$	4576.21	0.351900
$554$	0	0
$555$	359.803	0.0275186
$556$	0	0
$557$	−16787.2	−1.27702	−0.638508	−	0.769615i	$-0.720448\pi$
$557$	−16787.2	−1.27702	−0.638508	+	0.769615i	$0.720448\pi$
$558$	0	0
$559$	3.60125	0.000272480 0
$560$	0	0
$561$	1742.01	0.131101
$562$	0	0
$563$	19592.1	1.46662	0.733311	−	0.679894i	$-0.237974\pi$
$563$	19592.1	1.46662	0.733311	+	0.679894i	$0.237974\pi$
$564$	0	0
$565$	31080.8	2.31430
$566$	0	0
$567$	29405.0	2.17794
$568$	0	0
$569$	−7969.40	−0.587161	−0.293580	−	0.955934i	$-0.594847\pi$
$569$	−7969.40	−0.587161	−0.293580	+	0.955934i	$0.594847\pi$
$570$	0	0
$571$	18393.2	1.34804	0.674020	−	0.738713i	$-0.264566\pi$
$571$	18393.2	1.34804	0.674020	+	0.738713i	$0.264566\pi$
$572$	0	0
$573$	11729.4	0.855155
$574$	0	0
$575$	−35962.7	−2.60826
$576$	0	0
$577$	−3751.49	−0.270670	−0.135335	−	0.990800i	$-0.543211\pi$
$577$	−3751.49	−0.270670	−0.135335	+	0.990800i	$0.543211\pi$
$578$	0	0
$579$	−28284.1	−2.03014
$580$	0	0
$581$	−32119.9	−2.29356
$582$	0	0
$583$	18560.6	1.31852
$584$	0	0
$585$	−2.45828	−0.000173739 0
$586$	0	0
$587$	9300.75	0.653974	0.326987	−	0.945029i	$-0.393967\pi$
$587$	9300.75	0.653974	0.326987	+	0.945029i	$0.393967\pi$
$588$	0	0
$589$	18963.1	1.32659
$590$	0	0
$591$	21857.2	1.52129
$592$	0	0
$593$	17646.8	1.22204	0.611019	−	0.791616i	$-0.290760\pi$
$593$	17646.8	1.22204	0.611019	+	0.791616i	$0.290760\pi$
$594$	0	0
$595$	−5761.57	−0.396977
$596$	0	0
$597$	16246.3	1.11376
$598$	0	0
$599$	−22271.0	−1.51915	−0.759573	−	0.650422i	$-0.774592\pi$
$599$	−22271.0	−1.51915	−0.759573	+	0.650422i	$0.774592\pi$
$600$	0	0
$601$	11801.2	0.800970	0.400485	−	0.916303i	$-0.368842\pi$
$601$	11801.2	0.800970	0.400485	+	0.916303i	$0.368842\pi$
$602$	0	0
$603$	−5769.54	−0.389642
$604$	0	0
$605$	8703.32	0.584860
$606$	0	0
$607$	−4285.28	−0.286547	−0.143274	−	0.989683i	$-0.545763\pi$
$607$	−4285.28	−0.286547	−0.143274	+	0.989683i	$0.545763\pi$
$608$	0	0
$609$	5884.41	0.391541
$610$	0	0
$611$	2.93387	0.000194258 0
$612$	0	0
$613$	1570.64	0.103487	0.0517435	−	0.998660i	$-0.483522\pi$
$613$	1570.64	0.103487	0.0517435	+	0.998660i	$0.483522\pi$
$614$	0	0
$615$	16728.7	1.09686
$616$	0	0
$617$	−18943.1	−1.23602	−0.618008	−	0.786172i	$-0.712060\pi$
$617$	−18943.1	−1.23602	−0.618008	+	0.786172i	$0.712060\pi$
$618$	0	0
$619$	2628.57	0.170680	0.0853401	−	0.996352i	$-0.472802\pi$
$619$	2628.57	0.170680	0.0853401	+	0.996352i	$0.472802\pi$
$620$	0	0
$621$	13943.7	0.901034
$622$	0	0
$623$	8725.54	0.561126
$624$	0	0
$625$	10494.6	0.671655
$626$	0	0
$627$	18320.7	1.16692
$628$	0	0
$629$	28.3177	0.00179508
$630$	0	0
$631$	−17214.4	−1.08605	−0.543024	−	0.839717i	$-0.682721\pi$
$631$	−17214.4	−1.08605	−0.543024	+	0.839717i	$0.682721\pi$
$632$	0	0
$633$	21940.4	1.37765
$634$	0	0
$635$	48854.7	3.05314
$636$	0	0
$637$	7.28891	0.000453371 0
$638$	0	0
$639$	−290.998	−0.0180152
$640$	0	0
$641$	20389.4	1.25637	0.628185	−	0.778064i	$-0.283798\pi$
$641$	20389.4	1.25637	0.628185	+	0.778064i	$0.283798\pi$
$642$	0	0
$643$	−26098.5	−1.60066	−0.800330	−	0.599560i	$-0.795342\pi$
$643$	−26098.5	−1.60066	−0.800330	+	0.599560i	$0.795342\pi$
$644$	0	0
$645$	−41306.1	−2.52159
$646$	0	0
$647$	−17556.7	−1.06681	−0.533404	−	0.845860i	$-0.679088\pi$
$647$	−17556.7	−1.06681	−0.533404	+	0.845860i	$0.679088\pi$
$648$	0	0
$649$	17321.6	1.04766
$650$	0	0
$651$	38969.0	2.34611
$652$	0	0
$653$	−4709.71	−0.282244	−0.141122	−	0.989992i	$-0.545071\pi$
$653$	−4709.71	−0.282244	−0.141122	+	0.989992i	$0.545071\pi$
$654$	0	0
$655$	32673.5	1.94910
$656$	0	0
$657$	−1044.02	−0.0619955
$658$	0	0
$659$	−9491.86	−0.561078	−0.280539	−	0.959843i	$-0.590513\pi$
$659$	−9491.86	−0.561078	−0.280539	+	0.959843i	$0.590513\pi$
$660$	0	0
$661$	28228.4	1.66106	0.830528	−	0.556977i	$-0.188039\pi$
$661$	28228.4	1.66106	0.830528	+	0.556977i	$0.188039\pi$
$662$	0	0
$663$	−0.613262	−3.59232e−5 0
$664$	0	0
$665$	−60594.4	−3.53346
$666$	0	0
$667$	4423.28	0.256777
$668$	0	0
$669$	−31826.3	−1.83927
$670$	0	0
$671$	−24180.1	−1.39115
$672$	0	0
$673$	−25181.4	−1.44231	−0.721154	−	0.692775i	$-0.756388\pi$
$673$	−25181.4	−1.44231	−0.721154	+	0.692775i	$0.756388\pi$
$674$	0	0
$675$	−21554.5	−1.22909
$676$	0	0
$677$	−4141.90	−0.235135	−0.117567	−	0.993065i	$-0.537510\pi$
$677$	−4141.90	−0.235135	−0.117567	+	0.993065i	$0.537510\pi$
$678$	0	0
$679$	39904.2	2.25535
$680$	0	0
$681$	−29654.0	−1.66864
$682$	0	0
$683$	19362.5	1.08475	0.542377	−	0.840135i	$-0.317525\pi$
$683$	19362.5	1.08475	0.542377	+	0.840135i	$0.317525\pi$
$684$	0	0
$685$	−53286.3	−2.97221
$686$	0	0
$687$	−40057.7	−2.22460
$688$	0	0
$689$	−6.53411	−0.000361291 0
$690$	0	0
$691$	5910.75	0.325406	0.162703	−	0.986675i	$-0.447979\pi$
$691$	5910.75	0.325406	0.162703	+	0.986675i	$0.447979\pi$
$692$	0	0
$693$	11877.6	0.651074
$694$	0	0
$695$	−31810.7	−1.73619
$696$	0	0
$697$	1316.60	0.0715495
$698$	0	0
$699$	12062.6	0.652719
$700$	0	0
$701$	−6569.86	−0.353980	−0.176990	−	0.984213i	$-0.556636\pi$
$701$	−6569.86	−0.353980	−0.176990	+	0.984213i	$0.556636\pi$
$702$	0	0
$703$	297.818	0.0159778
$704$	0	0
$705$	−33651.4	−1.79771
$706$	0	0
$707$	2015.11	0.107194
$708$	0	0
$709$	27095.5	1.43525	0.717624	−	0.696431i	$-0.245230\pi$
$709$	27095.5	1.43525	0.717624	+	0.696431i	$0.245230\pi$
$710$	0	0
$711$	−1762.59	−0.0929710
$712$	0	0
$713$	29292.7	1.53860
$714$	0	0
$715$	5.83615	0.000305258 0
$716$	0	0
$717$	−19646.9	−1.02333
$718$	0	0
$719$	−33157.5	−1.71984	−0.859921	−	0.510428i	$-0.829487\pi$
$719$	−33157.5	−1.71984	−0.859921	+	0.510428i	$0.829487\pi$
$720$	0	0
$721$	1652.21	0.0853420
$722$	0	0
$723$	35990.7	1.85133
$724$	0	0
$725$	−6837.61	−0.350266
$726$	0	0
$727$	9883.51	0.504208	0.252104	−	0.967700i	$-0.418877\pi$
$727$	9883.51	0.504208	0.252104	+	0.967700i	$0.418877\pi$
$728$	0	0
$729$	4176.23	0.212175
$730$	0	0
$731$	−3250.93	−0.164487
$732$	0	0
$733$	27151.7	1.36817	0.684087	−	0.729400i	$-0.260201\pi$
$733$	27151.7	1.36817	0.684087	+	0.729400i	$0.260201\pi$
$734$	0	0
$735$	−83603.4	−4.19559
$736$	0	0
$737$	13697.3	0.684597
$738$	0	0
$739$	−8290.34	−0.412672	−0.206336	−	0.978481i	$-0.566154\pi$
$739$	−8290.34	−0.412672	−0.206336	+	0.978481i	$0.566154\pi$
$740$	0	0
$741$	−6.44967	−0.000319750 0
$742$	0	0
$743$	−20729.5	−1.02354	−0.511770	−	0.859123i	$-0.671010\pi$
$743$	−20729.5	−1.02354	−0.511770	+	0.859123i	$0.671010\pi$
$744$	0	0
$745$	−12847.9	−0.631826
$746$	0	0
$747$	12371.4	0.605953
$748$	0	0
$749$	−45085.8	−2.19947
$750$	0	0
$751$	19118.0	0.928931	0.464465	−	0.885591i	$-0.346247\pi$
$751$	19118.0	0.928931	0.464465	+	0.885591i	$0.346247\pi$
$752$	0	0
$753$	−43064.6	−2.08415
$754$	0	0
$755$	−3194.74	−0.153998
$756$	0	0
$757$	4009.10	0.192488	0.0962440	−	0.995358i	$-0.469317\pi$
$757$	4009.10	0.192488	0.0962440	+	0.995358i	$0.469317\pi$
$758$	0	0
$759$	28300.3	1.35341
$760$	0	0
$761$	39807.5	1.89622	0.948109	−	0.317945i	$-0.102993\pi$
$761$	39807.5	1.89622	0.948109	+	0.317945i	$0.102993\pi$
$762$	0	0
$763$	−45880.4	−2.17691
$764$	0	0
$765$	2219.15	0.104880
$766$	0	0
$767$	−6.09795	−0.000287072 0
$768$	0	0
$769$	35921.0	1.68445	0.842226	−	0.539125i	$-0.181245\pi$
$769$	35921.0	1.68445	0.842226	+	0.539125i	$0.181245\pi$
$770$	0	0
$771$	−2836.79	−0.132509
$772$	0	0
$773$	−42533.4	−1.97907	−0.989533	−	0.144305i	$-0.953905\pi$
$773$	−42533.4	−1.97907	−0.989533	+	0.144305i	$0.953905\pi$
$774$	0	0
$775$	−45281.4	−2.09878
$776$	0	0
$777$	612.011	0.0282571
$778$	0	0
$779$	13846.7	0.636856
$780$	0	0
$781$	690.850	0.0316525
$782$	0	0
$783$	2651.13	0.121001
$784$	0	0
$785$	−61207.3	−2.78291
$786$	0	0
$787$	6977.64	0.316043	0.158022	−	0.987436i	$-0.449488\pi$
$787$	6977.64	0.316043	0.158022	+	0.987436i	$0.449488\pi$
$788$	0	0
$789$	−14783.0	−0.667032
$790$	0	0
$791$	52867.2	2.37641
$792$	0	0
$793$	8.51243	0.000381192 0
$794$	0	0
$795$	74945.8	3.34347
$796$	0	0
$797$	−20011.4	−0.889383	−0.444692	−	0.895684i	$-0.646687\pi$
$797$	−20011.4	−0.889383	−0.444692	+	0.895684i	$0.646687\pi$
$798$	0	0
$799$	−2648.48	−0.117267
$800$	0	0
$801$	−3360.76	−0.148248
$802$	0	0
$803$	2478.58	0.108926
$804$	0	0
$805$	−93601.3	−4.09815
$806$	0	0
$807$	−1908.63	−0.0832550
$808$	0	0
$809$	−11122.4	−0.483367	−0.241683	−	0.970355i	$-0.577700\pi$
$809$	−11122.4	−0.483367	−0.241683	+	0.970355i	$0.577700\pi$
$810$	0	0
$811$	−34651.6	−1.50035	−0.750175	−	0.661240i	$-0.770031\pi$
$811$	−34651.6	−1.50035	−0.750175	+	0.661240i	$0.770031\pi$
$812$	0	0
$813$	−44022.5	−1.89906
$814$	0	0
$815$	569.041	0.0244572
$816$	0	0
$817$	−34190.0	−1.46408
$818$	0	0
$819$	−4.18143	−0.000178402 0
$820$	0	0
$821$	28124.9	1.19557	0.597786	−	0.801656i	$-0.296047\pi$
$821$	28124.9	1.19557	0.597786	+	0.801656i	$0.296047\pi$
$822$	0	0
$823$	27914.0	1.18229	0.591143	−	0.806567i	$-0.298677\pi$
$823$	27914.0	1.18229	0.591143	+	0.806567i	$0.298677\pi$
$824$	0	0
$825$	−43747.4	−1.84617
$826$	0	0
$827$	−33125.6	−1.39285	−0.696426	−	0.717628i	$-0.745228\pi$
$827$	−33125.6	−1.39285	−0.696426	+	0.717628i	$0.745228\pi$
$828$	0	0
$829$	15009.7	0.628840	0.314420	−	0.949284i	$-0.398190\pi$
$829$	15009.7	0.628840	0.314420	+	0.949284i	$0.398190\pi$
$830$	0	0
$831$	46872.2	1.95665
$832$	0	0
$833$	−6579.87	−0.273684
$834$	0	0
$835$	31079.6	1.28809
$836$	0	0
$837$	17556.8	0.725033
$838$	0	0
$839$	−43018.1	−1.77014	−0.885072	−	0.465455i	$-0.845891\pi$
$839$	−43018.1	−1.77014	−0.885072	+	0.465455i	$0.845891\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	14958.7	0.611155
$844$	0	0
$845$	41730.3	1.69889
$846$	0	0
$847$	14804.0	0.600556
$848$	0	0
$849$	12538.9	0.506871
$850$	0	0
$851$	460.045	0.0185313
$852$	0	0
$853$	−327.428	−0.0131429	−0.00657146	−	0.999978i	$-0.502092\pi$
$853$	−327.428	−0.0131429	−0.00657146	+	0.999978i	$0.502092\pi$
$854$	0	0
$855$	23338.8	0.933531
$856$	0	0
$857$	−33969.4	−1.35400	−0.676998	−	0.735985i	$-0.736719\pi$
$857$	−33969.4	−1.35400	−0.676998	+	0.735985i	$0.736719\pi$
$858$	0	0
$859$	−7542.55	−0.299591	−0.149795	−	0.988717i	$-0.547862\pi$
$859$	−7542.55	−0.299591	−0.149795	+	0.988717i	$0.547862\pi$
$860$	0	0
$861$	28454.8	1.12629
$862$	0	0
$863$	−985.584	−0.0388756	−0.0194378	−	0.999811i	$-0.506188\pi$
$863$	−985.584	−0.0388756	−0.0194378	+	0.999811i	$0.506188\pi$
$864$	0	0
$865$	−29190.6	−1.14741
$866$	0	0
$867$	−30302.2	−1.18699
$868$	0	0
$869$	4184.53	0.163349
$870$	0	0
$871$	−4.82205	−0.000187588 0
$872$	0	0
$873$	−15369.7	−0.595858
$874$	0	0
$875$	67982.3	2.62654
$876$	0	0
$877$	−35444.0	−1.36472	−0.682360	−	0.731016i	$-0.739046\pi$
$877$	−35444.0	−1.36472	−0.682360	+	0.731016i	$0.739046\pi$
$878$	0	0
$879$	−28107.1	−1.07853
$880$	0	0
$881$	19387.5	0.741409	0.370704	−	0.928751i	$-0.379116\pi$
$881$	19387.5	0.741409	0.370704	+	0.928751i	$0.379116\pi$
$882$	0	0
$883$	6973.44	0.265770	0.132885	−	0.991131i	$-0.457576\pi$
$883$	6973.44	0.265770	0.132885	+	0.991131i	$0.457576\pi$
$884$	0	0
$885$	69943.1	2.65662
$886$	0	0
$887$	−35453.1	−1.34205	−0.671024	−	0.741435i	$-0.734145\pi$
$887$	−35453.1	−1.34205	−0.671024	+	0.741435i	$0.734145\pi$
$888$	0	0
$889$	83099.9	3.13507
$890$	0	0
$891$	26888.1	1.01098
$892$	0	0
$893$	−27854.0	−1.04378
$894$	0	0
$895$	9291.56	0.347019
$896$	0	0
$897$	−9.96293	−0.000370850 0
$898$	0	0
$899$	5569.44	0.206620
$900$	0	0
$901$	5898.49	0.218099
$902$	0	0
$903$	−70260.0	−2.58926
$904$	0	0
$905$	−4204.23	−0.154423
$906$	0	0
$907$	35537.4	1.30099	0.650496	−	0.759510i	$-0.274561\pi$
$907$	35537.4	1.30099	0.650496	+	0.759510i	$0.274561\pi$
$908$	0	0
$909$	−776.148	−0.0283204
$910$	0	0
$911$	43991.8	1.59990	0.799952	−	0.600063i	$-0.204858\pi$
$911$	43991.8	1.59990	0.799952	+	0.600063i	$0.204858\pi$
$912$	0	0
$913$	−29370.7	−1.06465
$914$	0	0
$915$	−97637.1	−3.52763
$916$	0	0
$917$	55576.3	2.00141
$918$	0	0
$919$	−37227.9	−1.33627	−0.668136	−	0.744039i	$-0.732908\pi$
$919$	−37227.9	−1.33627	−0.668136	+	0.744039i	$0.732908\pi$
$920$	0	0
$921$	2262.23	0.0809370
$922$	0	0
$923$	−0.243209	−8.67315e−6 0
$924$	0	0
$925$	−711.148	−0.0252783
$926$	0	0
$927$	−636.373	−0.0225472
$928$	0	0
$929$	8831.43	0.311894	0.155947	−	0.987765i	$-0.450157\pi$
$929$	8831.43	0.311894	0.155947	+	0.987765i	$0.450157\pi$
$930$	0	0
$931$	−69200.4	−2.43604
$932$	0	0
$933$	−10426.6	−0.365864
$934$	0	0
$935$	−5268.43	−0.184274
$936$	0	0
$937$	−9645.52	−0.336292	−0.168146	−	0.985762i	$-0.553778\pi$
$937$	−9645.52	−0.336292	−0.168146	+	0.985762i	$0.553778\pi$
$938$	0	0
$939$	−42219.7	−1.46729
$940$	0	0
$941$	−42094.4	−1.45828	−0.729138	−	0.684367i	$-0.760079\pi$
$941$	−42094.4	−1.45828	−0.729138	+	0.684367i	$0.760079\pi$
$942$	0	0
$943$	21389.3	0.738634
$944$	0	0
$945$	−56100.7	−1.93117
$946$	0	0
$947$	13532.9	0.464373	0.232187	−	0.972671i	$-0.425412\pi$
$947$	13532.9	0.464373	0.232187	+	0.972671i	$0.425412\pi$
$948$	0	0
$949$	−0.872566	−2.98469e−5 0
$950$	0	0
$951$	−26581.3	−0.906371
$952$	0	0
$953$	25603.5	0.870282	0.435141	−	0.900362i	$-0.356699\pi$
$953$	25603.5	0.870282	0.435141	+	0.900362i	$0.356699\pi$
$954$	0	0
$955$	−35473.8	−1.20199
$956$	0	0
$957$	5380.76	0.181751
$958$	0	0
$959$	−90637.7	−3.05198
$960$	0	0
$961$	7092.12	0.238063
$962$	0	0
$963$	17365.4	0.581094
$964$	0	0
$965$	85540.8	2.85353
$966$	0	0
$967$	456.522	0.0151818	0.00759088	−	0.999971i	$-0.497584\pi$
$967$	456.522	0.0151818	0.00759088	+	0.999971i	$0.497584\pi$
$968$	0	0
$969$	5822.27	0.193022
$970$	0	0
$971$	34824.2	1.15094	0.575469	−	0.817824i	$-0.304819\pi$
$971$	34824.2	1.15094	0.575469	+	0.817824i	$0.304819\pi$
$972$	0	0
$973$	−54108.7	−1.78278
$974$	0	0
$975$	15.4009	0.000505872 0
$976$	0	0
$977$	−5051.26	−0.165409	−0.0827043	−	0.996574i	$-0.526356\pi$
$977$	−5051.26	−0.165409	−0.0827043	+	0.996574i	$0.526356\pi$
$978$	0	0
$979$	7978.71	0.260471
$980$	0	0
$981$	17671.5	0.575135
$982$	0	0
$983$	22474.8	0.729232	0.364616	−	0.931158i	$-0.381200\pi$
$983$	22474.8	0.729232	0.364616	+	0.931158i	$0.381200\pi$
$984$	0	0
$985$	−66103.5	−2.13831
$986$	0	0
$987$	−57239.6	−1.84595
$988$	0	0
$989$	−52814.0	−1.69807
$990$	0	0
$991$	19345.4	0.620108	0.310054	−	0.950719i	$-0.399653\pi$
$991$	19345.4	0.620108	0.310054	+	0.950719i	$0.399653\pi$
$992$	0	0
$993$	−18958.2	−0.605860
$994$	0	0
$995$	−49134.2	−1.56549
$996$	0	0
$997$	34370.6	1.09180	0.545901	−	0.837850i	$-0.316187\pi$
$997$	34370.6	1.09180	0.545901	+	0.837850i	$0.316187\pi$
$998$	0	0
$999$	275.731	0.00873248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bl.1.10		12
4.3	odd	2		1856.4.a.bj.1.3		12
8.3	odd	2		928.4.a.j.1.10	yes	12
8.5	even	2		928.4.a.h.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.3	✓	12	8.5	even	2
928.4.a.j.1.10	yes	12	8.3	odd	2
1856.4.a.bj.1.3		12	4.3	odd	2
1856.4.a.bl.1.10		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)

\(f(q)\)	\(=\)	\(q+6.28045 q^{3} -18.9942 q^{5} -32.3083 q^{7} +12.4440 q^{9} +O(q^{10})\)	\(q+6.28045 q^{3} -18.9942 q^{5} -32.3083 q^{7} +12.4440 q^{9} -29.5430 q^{11} +0.0104004 q^{13} -119.292 q^{15} -9.38869 q^{17} -98.7408 q^{19} -202.911 q^{21} -152.527 q^{23} +235.780 q^{25} -91.4181 q^{27} -29.0000 q^{29} -192.050 q^{31} -185.543 q^{33} +613.671 q^{35} -3.01616 q^{37} +0.0653192 q^{39} -140.233 q^{41} +346.260 q^{43} -236.364 q^{45} +282.092 q^{47} +700.829 q^{49} -58.9652 q^{51} -628.255 q^{53} +561.146 q^{55} -620.136 q^{57} -586.318 q^{59} +818.471 q^{61} -402.045 q^{63} -0.197547 q^{65} -463.640 q^{67} -957.937 q^{69} -23.3845 q^{71} -83.8973 q^{73} +1480.80 q^{75} +954.486 q^{77} -141.642 q^{79} -910.135 q^{81} +994.167 q^{83} +178.331 q^{85} -182.133 q^{87} -270.071 q^{89} -0.336020 q^{91} -1206.16 q^{93} +1875.50 q^{95} -1235.11 q^{97} -367.634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) \) 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

Introduction

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