LMFDB - Embedded newform 1856.4.a.bj.1.5

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			1.5
Root			$4.35906$ 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)

$f(q)$	$=$	$q-5.35906 q^{3} +20.7045 q^{5} -26.2041 q^{7} +1.71955 q^{9} +O(q^{10})$	$q-5.35906 q^{3} +20.7045 q^{5} -26.2041 q^{7} +1.71955 q^{9} +11.3114 q^{11} +82.2044 q^{13} -110.957 q^{15} +74.7229 q^{17} -49.4062 q^{19} +140.430 q^{21} -126.182 q^{23} +303.676 q^{25} +135.480 q^{27} -29.0000 q^{29} +169.187 q^{31} -60.6184 q^{33} -542.543 q^{35} -333.991 q^{37} -440.538 q^{39} -222.203 q^{41} +474.657 q^{43} +35.6024 q^{45} +49.3028 q^{47} +343.656 q^{49} -400.445 q^{51} -409.259 q^{53} +234.197 q^{55} +264.771 q^{57} -281.484 q^{59} +407.163 q^{61} -45.0592 q^{63} +1702.00 q^{65} +296.724 q^{67} +676.215 q^{69} +483.489 q^{71} +152.749 q^{73} -1627.42 q^{75} -296.405 q^{77} +657.772 q^{79} -772.471 q^{81} -9.66724 q^{83} +1547.10 q^{85} +155.413 q^{87} -647.946 q^{89} -2154.09 q^{91} -906.686 q^{93} -1022.93 q^{95} +540.332 q^{97} +19.4505 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.35906	−1.03135	−0.515676	−	0.856784i	$-0.672459\pi$
$3$	−5.35906	−1.03135	−0.515676	+	0.856784i	$0.672459\pi$
$4$	0	0
$5$	20.7045	1.85187	0.925933	−	0.377687i	$-0.123280\pi$
$5$	20.7045	1.85187	0.925933	+	0.377687i	$0.123280\pi$
$6$	0	0
$7$	−26.2041	−1.41489	−0.707445	−	0.706768i	$-0.750152\pi$
$7$	−26.2041	−1.41489	−0.707445	+	0.706768i	$0.750152\pi$
$8$	0	0
$9$	1.71955	0.0636869
$10$	0	0
$11$	11.3114	0.310047	0.155023	−	0.987911i	$-0.450455\pi$
$11$	11.3114	0.310047	0.155023	+	0.987911i	$0.450455\pi$
$12$	0	0
$13$	82.2044	1.75380	0.876900	−	0.480673i	$-0.159608\pi$
$13$	82.2044	1.75380	0.876900	+	0.480673i	$0.159608\pi$
$14$	0	0
$15$	−110.957	−1.90993
$16$	0	0
$17$	74.7229	1.06606	0.533029	−	0.846097i	$-0.321054\pi$
$17$	74.7229	1.06606	0.533029	+	0.846097i	$0.321054\pi$
$18$	0	0
$19$	−49.4062	−0.596555	−0.298278	−	0.954479i	$-0.596412\pi$
$19$	−49.4062	−0.596555	−0.298278	+	0.954479i	$0.596412\pi$
$20$	0	0
$21$	140.430	1.45925
$22$	0	0
$23$	−126.182	−1.14394	−0.571971	−	0.820274i	$-0.693821\pi$
$23$	−126.182	−1.14394	−0.571971	+	0.820274i	$0.693821\pi$
$24$	0	0
$25$	303.676	2.42941
$26$	0	0
$27$	135.480	0.965668
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	169.187	0.980225	0.490112	−	0.871659i	$-0.336956\pi$
$31$	169.187	0.980225	0.490112	+	0.871659i	$0.336956\pi$
$32$	0	0
$33$	−60.6184	−0.319767
$34$	0	0
$35$	−542.543	−2.62019
$36$	0	0
$37$	−333.991	−1.48399	−0.741997	−	0.670403i	$-0.766121\pi$
$37$	−333.991	−1.48399	−0.741997	+	0.670403i	$0.766121\pi$
$38$	0	0
$39$	−440.538	−1.80878
$40$	0	0
$41$	−222.203	−0.846396	−0.423198	−	0.906037i	$-0.639092\pi$
$41$	−222.203	−0.846396	−0.423198	+	0.906037i	$0.639092\pi$
$42$	0	0
$43$	474.657	1.68336	0.841681	−	0.539975i	$-0.181566\pi$
$43$	474.657	1.68336	0.841681	+	0.539975i	$0.181566\pi$
$44$	0	0
$45$	35.6024	0.117940
$46$	0	0
$47$	49.3028	0.153012	0.0765059	−	0.997069i	$-0.475624\pi$
$47$	49.3028	0.153012	0.0765059	+	0.997069i	$0.475624\pi$
$48$	0	0
$49$	343.656	1.00191
$50$	0	0
$51$	−400.445	−1.09948
$52$	0	0
$53$	−409.259	−1.06068	−0.530340	−	0.847785i	$-0.677936\pi$
$53$	−409.259	−1.06068	−0.530340	+	0.847785i	$0.677936\pi$
$54$	0	0
$55$	234.197	0.574165
$56$	0	0
$57$	264.771	0.615259
$58$	0	0
$59$	−281.484	−0.621120	−0.310560	−	0.950554i	$-0.600517\pi$
$59$	−281.484	−0.621120	−0.310560	+	0.950554i	$0.600517\pi$
$60$	0	0
$61$	407.163	0.854620	0.427310	−	0.904105i	$-0.359461\pi$
$61$	407.163	0.854620	0.427310	+	0.904105i	$0.359461\pi$
$62$	0	0
$63$	−45.0592	−0.0901100
$64$	0	0
$65$	1702.00	3.24780
$66$	0	0
$67$	296.724	0.541054	0.270527	−	0.962712i	$-0.412802\pi$
$67$	296.724	0.541054	0.270527	+	0.962712i	$0.412802\pi$
$68$	0	0
$69$	676.215	1.17981
$70$	0	0
$71$	483.489	0.808163	0.404082	−	0.914723i	$-0.367591\pi$
$71$	483.489	0.808163	0.404082	+	0.914723i	$0.367591\pi$
$72$	0	0
$73$	152.749	0.244902	0.122451	−	0.992475i	$-0.460925\pi$
$73$	152.749	0.244902	0.122451	+	0.992475i	$0.460925\pi$
$74$	0	0
$75$	−1627.42	−2.50558
$76$	0	0
$77$	−296.405	−0.438682
$78$	0	0
$79$	657.772	0.936774	0.468387	−	0.883523i	$-0.344835\pi$
$79$	657.772	0.936774	0.468387	+	0.883523i	$0.344835\pi$
$80$	0	0
$81$	−772.471	−1.05963
$82$	0	0
$83$	−9.66724	−0.0127845	−0.00639227	−	0.999980i	$-0.502035\pi$
$83$	−9.66724	−0.0127845	−0.00639227	+	0.999980i	$0.502035\pi$
$84$	0	0
$85$	1547.10	1.97420
$86$	0	0
$87$	155.413	0.191517
$88$	0	0
$89$	−647.946	−0.771709	−0.385855	−	0.922560i	$-0.626093\pi$
$89$	−647.946	−0.771709	−0.385855	+	0.922560i	$0.626093\pi$
$90$	0	0
$91$	−2154.09	−2.48143
$92$	0	0
$93$	−906.686	−1.01096
$94$	0	0
$95$	−1022.93	−1.10474
$96$	0	0
$97$	540.332	0.565592	0.282796	−	0.959180i	$-0.408738\pi$
$97$	540.332	0.565592	0.282796	+	0.959180i	$0.408738\pi$
$98$	0	0
$99$	19.4505	0.0197459
$100$	0	0
$101$	105.499	0.103936	0.0519680	−	0.998649i	$-0.483451\pi$
$101$	105.499	0.103936	0.0519680	+	0.998649i	$0.483451\pi$
$102$	0	0
$103$	1190.88	1.13923	0.569614	−	0.821913i	$-0.307093\pi$
$103$	1190.88	1.13923	0.569614	+	0.821913i	$0.307093\pi$
$104$	0	0
$105$	2907.52	2.70234
$106$	0	0
$107$	−1084.82	−0.980123	−0.490062	−	0.871688i	$-0.663026\pi$
$107$	−1084.82	−0.980123	−0.490062	+	0.871688i	$0.663026\pi$
$108$	0	0
$109$	−599.692	−0.526973	−0.263487	−	0.964663i	$-0.584873\pi$
$109$	−599.692	−0.526973	−0.263487	+	0.964663i	$0.584873\pi$
$110$	0	0
$111$	1789.88	1.53052
$112$	0	0
$113$	−2131.00	−1.77405	−0.887023	−	0.461724i	$-0.847231\pi$
$113$	−2131.00	−1.77405	−0.887023	+	0.461724i	$0.847231\pi$
$114$	0	0
$115$	−2612.53	−2.11843
$116$	0	0
$117$	141.354	0.111694
$118$	0	0
$119$	−1958.05	−1.50835
$120$	0	0
$121$	−1203.05	−0.903871
$122$	0	0
$123$	1190.80	0.872932
$124$	0	0
$125$	3699.40	2.64708
$126$	0	0
$127$	753.254	0.526303	0.263152	−	0.964754i	$-0.415238\pi$
$127$	753.254	0.526303	0.263152	+	0.964754i	$0.415238\pi$
$128$	0	0
$129$	−2543.72	−1.73614
$130$	0	0
$131$	−42.0357	−0.0280357	−0.0140179	−	0.999902i	$-0.504462\pi$
$131$	−42.0357	−0.0280357	−0.0140179	+	0.999902i	$0.504462\pi$
$132$	0	0
$133$	1294.65	0.844060
$134$	0	0
$135$	2805.04	1.78829
$136$	0	0
$137$	−1681.41	−1.04856	−0.524280	−	0.851546i	$-0.675666\pi$
$137$	−1681.41	−1.04856	−0.524280	+	0.851546i	$0.675666\pi$
$138$	0	0
$139$	2829.80	1.72676	0.863382	−	0.504550i	$-0.168342\pi$
$139$	2829.80	1.72676	0.863382	+	0.504550i	$0.168342\pi$
$140$	0	0
$141$	−264.217	−0.157809
$142$	0	0
$143$	929.846	0.543759
$144$	0	0
$145$	−600.430	−0.343883
$146$	0	0
$147$	−1841.68	−1.03333
$148$	0	0
$149$	1704.69	0.937273	0.468637	−	0.883391i	$-0.344745\pi$
$149$	1704.69	0.937273	0.468637	+	0.883391i	$0.344745\pi$
$150$	0	0
$151$	−2800.24	−1.50914	−0.754572	−	0.656218i	$-0.772155\pi$
$151$	−2800.24	−1.50914	−0.754572	+	0.656218i	$0.772155\pi$
$152$	0	0
$153$	128.490	0.0678939
$154$	0	0
$155$	3502.94	1.81525
$156$	0	0
$157$	−464.761	−0.236254	−0.118127	−	0.992998i	$-0.537689\pi$
$157$	−464.761	−0.236254	−0.118127	+	0.992998i	$0.537689\pi$
$158$	0	0
$159$	2193.24	1.09393
$160$	0	0
$161$	3306.48	1.61855
$162$	0	0
$163$	2925.54	1.40581	0.702903	−	0.711286i	$-0.251887\pi$
$163$	2925.54	1.40581	0.702903	+	0.711286i	$0.251887\pi$
$164$	0	0
$165$	−1255.07	−0.592166
$166$	0	0
$167$	4050.34	1.87679	0.938397	−	0.345560i	$-0.112311\pi$
$167$	4050.34	1.87679	0.938397	+	0.345560i	$0.112311\pi$
$168$	0	0
$169$	4560.56	2.07581
$170$	0	0
$171$	−84.9563	−0.0379928
$172$	0	0
$173$	−1777.89	−0.781332	−0.390666	−	0.920533i	$-0.627755\pi$
$173$	−1777.89	−0.781332	−0.390666	+	0.920533i	$0.627755\pi$
$174$	0	0
$175$	−7957.57	−3.43735
$176$	0	0
$177$	1508.49	0.640594
$178$	0	0
$179$	614.881	0.256750	0.128375	−	0.991726i	$-0.459024\pi$
$179$	614.881	0.256750	0.128375	+	0.991726i	$0.459024\pi$
$180$	0	0
$181$	1002.39	0.411641	0.205821	−	0.978590i	$-0.434014\pi$
$181$	1002.39	0.411641	0.205821	+	0.978590i	$0.434014\pi$
$182$	0	0
$183$	−2182.01	−0.881414
$184$	0	0
$185$	−6915.12	−2.74816
$186$	0	0
$187$	845.220	0.330527
$188$	0	0
$189$	−3550.12	−1.36631
$190$	0	0
$191$	2012.04	0.762232	0.381116	−	0.924527i	$-0.375540\pi$
$191$	2012.04	0.762232	0.381116	+	0.924527i	$0.375540\pi$
$192$	0	0
$193$	4079.23	1.52139	0.760697	−	0.649107i	$-0.224857\pi$
$193$	4079.23	1.52139	0.760697	+	0.649107i	$0.224857\pi$
$194$	0	0
$195$	−9121.13	−3.34963
$196$	0	0
$197$	241.774	0.0874401	0.0437201	−	0.999044i	$-0.486079\pi$
$197$	241.774	0.0874401	0.0437201	+	0.999044i	$0.486079\pi$
$198$	0	0
$199$	−1692.80	−0.603013	−0.301506	−	0.953464i	$-0.597489\pi$
$199$	−1692.80	−0.603013	−0.301506	+	0.953464i	$0.597489\pi$
$200$	0	0
$201$	−1590.16	−0.558017
$202$	0	0
$203$	759.920	0.262738
$204$	0	0
$205$	−4600.60	−1.56741
$206$	0	0
$207$	−216.975	−0.0728542
$208$	0	0
$209$	−558.852	−0.184960
$210$	0	0
$211$	−5337.17	−1.74136	−0.870678	−	0.491853i	$-0.836320\pi$
$211$	−5337.17	−1.74136	−0.870678	+	0.491853i	$0.836320\pi$
$212$	0	0
$213$	−2591.05	−0.833501
$214$	0	0
$215$	9827.54	3.11736
$216$	0	0
$217$	−4433.41	−1.38691
$218$	0	0
$219$	−818.589	−0.252580
$220$	0	0
$221$	6142.55	1.86965
$222$	0	0
$223$	2336.46	0.701618	0.350809	−	0.936447i	$-0.385907\pi$
$223$	2336.46	0.701618	0.350809	+	0.936447i	$0.385907\pi$
$224$	0	0
$225$	522.186	0.154722
$226$	0	0
$227$	−5753.24	−1.68219	−0.841093	−	0.540891i	$-0.818087\pi$
$227$	−5753.24	−1.68219	−0.841093	+	0.540891i	$0.818087\pi$
$228$	0	0
$229$	3042.77	0.878043	0.439021	−	0.898477i	$-0.355325\pi$
$229$	3042.77	0.878043	0.439021	+	0.898477i	$0.355325\pi$
$230$	0	0
$231$	1588.45	0.452435
$232$	0	0
$233$	5326.14	1.49754	0.748771	−	0.662829i	$-0.230644\pi$
$233$	5326.14	1.49754	0.748771	+	0.662829i	$0.230644\pi$
$234$	0	0
$235$	1020.79	0.283357
$236$	0	0
$237$	−3525.04	−0.966144
$238$	0	0
$239$	4507.17	1.21985	0.609926	−	0.792459i	$-0.291199\pi$
$239$	4507.17	1.21985	0.609926	+	0.792459i	$0.291199\pi$
$240$	0	0
$241$	6039.87	1.61437	0.807183	−	0.590302i	$-0.200991\pi$
$241$	6039.87	1.61437	0.807183	+	0.590302i	$0.200991\pi$
$242$	0	0
$243$	481.773	0.127184
$244$	0	0
$245$	7115.23	1.85541
$246$	0	0
$247$	−4061.40	−1.04624
$248$	0	0
$249$	51.8073	0.0131854
$250$	0	0
$251$	1340.79	0.337170	0.168585	−	0.985687i	$-0.446080\pi$
$251$	1340.79	0.337170	0.168585	+	0.985687i	$0.446080\pi$
$252$	0	0
$253$	−1427.29	−0.354675
$254$	0	0
$255$	−8291.01	−2.03609
$256$	0	0
$257$	4631.69	1.12419	0.562096	−	0.827072i	$-0.309995\pi$
$257$	4631.69	1.12419	0.562096	+	0.827072i	$0.309995\pi$
$258$	0	0
$259$	8751.95	2.09969
$260$	0	0
$261$	−49.8669	−0.0118264
$262$	0	0
$263$	3210.60	0.752754	0.376377	−	0.926467i	$-0.377170\pi$
$263$	3210.60	0.752754	0.376377	+	0.926467i	$0.377170\pi$
$264$	0	0
$265$	−8473.50	−1.96424
$266$	0	0
$267$	3472.38	0.795904
$268$	0	0
$269$	6363.94	1.44244	0.721219	−	0.692707i	$-0.243582\pi$
$269$	6363.94	1.44244	0.721219	+	0.692707i	$0.243582\pi$
$270$	0	0
$271$	−661.429	−0.148262	−0.0741309	−	0.997249i	$-0.523618\pi$
$271$	−661.429	−0.148262	−0.0741309	+	0.997249i	$0.523618\pi$
$272$	0	0
$273$	11543.9	2.55923
$274$	0	0
$275$	3435.00	0.753230
$276$	0	0
$277$	8160.47	1.77009	0.885045	−	0.465506i	$-0.154127\pi$
$277$	8160.47	1.77009	0.885045	+	0.465506i	$0.154127\pi$
$278$	0	0
$279$	290.926	0.0624275
$280$	0	0
$281$	2437.91	0.517557	0.258779	−	0.965937i	$-0.416680\pi$
$281$	2437.91	0.517557	0.258779	+	0.965937i	$0.416680\pi$
$282$	0	0
$283$	3423.83	0.719172	0.359586	−	0.933112i	$-0.382918\pi$
$283$	3423.83	0.719172	0.359586	+	0.933112i	$0.382918\pi$
$284$	0	0
$285$	5481.95	1.13938
$286$	0	0
$287$	5822.63	1.19756
$288$	0	0
$289$	670.515	0.136478
$290$	0	0
$291$	−2895.67	−0.583324
$292$	0	0
$293$	1420.58	0.283246	0.141623	−	0.989921i	$-0.454768\pi$
$293$	1420.58	0.283246	0.141623	+	0.989921i	$0.454768\pi$
$294$	0	0
$295$	−5827.99	−1.15023
$296$	0	0
$297$	1532.46	0.299402
$298$	0	0
$299$	−10372.7	−2.00625
$300$	0	0
$301$	−12438.0	−2.38177
$302$	0	0
$303$	−565.376	−0.107195
$304$	0	0
$305$	8430.10	1.58264
$306$	0	0
$307$	−9364.30	−1.74088	−0.870438	−	0.492278i	$-0.836164\pi$
$307$	−9364.30	−1.74088	−0.870438	+	0.492278i	$0.836164\pi$
$308$	0	0
$309$	−6381.97	−1.17494
$310$	0	0
$311$	−6223.53	−1.13474	−0.567370	−	0.823463i	$-0.692039\pi$
$311$	−6223.53	−1.13474	−0.567370	+	0.823463i	$0.692039\pi$
$312$	0	0
$313$	6835.20	1.23434	0.617170	−	0.786830i	$-0.288279\pi$
$313$	6835.20	1.23434	0.617170	+	0.786830i	$0.288279\pi$
$314$	0	0
$315$	−932.929	−0.166872
$316$	0	0
$317$	−1822.62	−0.322929	−0.161464	−	0.986879i	$-0.551622\pi$
$317$	−1822.62	−0.322929	−0.161464	+	0.986879i	$0.551622\pi$
$318$	0	0
$319$	−328.030	−0.0575742
$320$	0	0
$321$	5813.60	1.01085
$322$	0	0
$323$	−3691.77	−0.635962
$324$	0	0
$325$	24963.5	4.26070
$326$	0	0
$327$	3213.79	0.543495
$328$	0	0
$329$	−1291.94	−0.216495
$330$	0	0
$331$	7526.92	1.24990	0.624950	−	0.780664i	$-0.285119\pi$
$331$	7526.92	1.24990	0.624950	+	0.780664i	$0.285119\pi$
$332$	0	0
$333$	−574.314	−0.0945111
$334$	0	0
$335$	6143.53	1.00196
$336$	0	0
$337$	−5078.25	−0.820860	−0.410430	−	0.911892i	$-0.634621\pi$
$337$	−5078.25	−0.820860	−0.410430	+	0.911892i	$0.634621\pi$
$338$	0	0
$339$	11420.1	1.82967
$340$	0	0
$341$	1913.74	0.303915
$342$	0	0
$343$	−17.1978	−0.00270727
$344$	0	0
$345$	14000.7	2.18485
$346$	0	0
$347$	5206.74	0.805511	0.402756	−	0.915308i	$-0.368052\pi$
$347$	5206.74	0.805511	0.402756	+	0.915308i	$0.368052\pi$
$348$	0	0
$349$	2063.87	0.316551	0.158276	−	0.987395i	$-0.449407\pi$
$349$	2063.87	0.316551	0.158276	+	0.987395i	$0.449407\pi$
$350$	0	0
$351$	11137.0	1.69359
$352$	0	0
$353$	4301.18	0.648523	0.324262	−	0.945967i	$-0.394884\pi$
$353$	4301.18	0.648523	0.324262	+	0.945967i	$0.394884\pi$
$354$	0	0
$355$	10010.4	1.49661
$356$	0	0
$357$	10493.3	1.55564
$358$	0	0
$359$	13272.3	1.95121	0.975607	−	0.219524i	$-0.0704504\pi$
$359$	13272.3	1.95121	0.975607	+	0.219524i	$0.0704504\pi$
$360$	0	0
$361$	−4418.03	−0.644122
$362$	0	0
$363$	6447.23	0.932209
$364$	0	0
$365$	3162.58	0.453526
$366$	0	0
$367$	−1015.56	−0.144446	−0.0722229	−	0.997389i	$-0.523009\pi$
$367$	−1015.56	−0.144446	−0.0722229	+	0.997389i	$0.523009\pi$
$368$	0	0
$369$	−382.088	−0.0539044
$370$	0	0
$371$	10724.3	1.50074
$372$	0	0
$373$	3313.81	0.460007	0.230003	−	0.973190i	$-0.426126\pi$
$373$	3313.81	0.460007	0.230003	+	0.973190i	$0.426126\pi$
$374$	0	0
$375$	−19825.3	−2.73007
$376$	0	0
$377$	−2383.93	−0.325672
$378$	0	0
$379$	11282.1	1.52908	0.764540	−	0.644576i	$-0.222966\pi$
$379$	11282.1	1.52908	0.764540	+	0.644576i	$0.222966\pi$
$380$	0	0
$381$	−4036.74	−0.542804
$382$	0	0
$383$	−1572.07	−0.209736	−0.104868	−	0.994486i	$-0.533442\pi$
$383$	−1572.07	−0.209736	−0.104868	+	0.994486i	$0.533442\pi$
$384$	0	0
$385$	−6136.92	−0.812380
$386$	0	0
$387$	816.196	0.107208
$388$	0	0
$389$	−5089.44	−0.663354	−0.331677	−	0.943393i	$-0.607614\pi$
$389$	−5089.44	−0.663354	−0.331677	+	0.943393i	$0.607614\pi$
$390$	0	0
$391$	−9428.65	−1.21951
$392$	0	0
$393$	225.272	0.0289147
$394$	0	0
$395$	13618.8	1.73478
$396$	0	0
$397$	−5483.61	−0.693236	−0.346618	−	0.938006i	$-0.612670\pi$
$397$	−5483.61	−0.693236	−0.346618	+	0.938006i	$0.612670\pi$
$398$	0	0
$399$	−6938.09	−0.870523
$400$	0	0
$401$	−12471.4	−1.55309	−0.776547	−	0.630059i	$-0.783031\pi$
$401$	−12471.4	−1.55309	−0.776547	+	0.630059i	$0.783031\pi$
$402$	0	0
$403$	13907.9	1.71912
$404$	0	0
$405$	−15993.6	−1.96230
$406$	0	0
$407$	−3777.90	−0.460107
$408$	0	0
$409$	5453.23	0.659278	0.329639	−	0.944107i	$-0.393073\pi$
$409$	5453.23	0.659278	0.329639	+	0.944107i	$0.393073\pi$
$410$	0	0
$411$	9010.80	1.08144
$412$	0	0
$413$	7376.04	0.878817
$414$	0	0
$415$	−200.155	−0.0236753
$416$	0	0
$417$	−15165.1	−1.78090
$418$	0	0
$419$	−1762.32	−0.205478	−0.102739	−	0.994708i	$-0.532761\pi$
$419$	−1762.32	−0.205478	−0.102739	+	0.994708i	$0.532761\pi$
$420$	0	0
$421$	3653.85	0.422988	0.211494	−	0.977379i	$-0.432167\pi$
$421$	3653.85	0.422988	0.211494	+	0.977379i	$0.432167\pi$
$422$	0	0
$423$	84.7785	0.00974485
$424$	0	0
$425$	22691.6	2.58989
$426$	0	0
$427$	−10669.3	−1.20919
$428$	0	0
$429$	−4983.10	−0.560807
$430$	0	0
$431$	10803.8	1.20743	0.603715	−	0.797201i	$-0.293687\pi$
$431$	10803.8	1.20743	0.603715	+	0.797201i	$0.293687\pi$
$432$	0	0
$433$	2752.14	0.305449	0.152725	−	0.988269i	$-0.451195\pi$
$433$	2752.14	0.305449	0.152725	+	0.988269i	$0.451195\pi$
$434$	0	0
$435$	3217.74	0.354664
$436$	0	0
$437$	6234.15	0.682425
$438$	0	0
$439$	−10403.9	−1.13110	−0.565550	−	0.824714i	$-0.691336\pi$
$439$	−10403.9	−1.13110	−0.565550	+	0.824714i	$0.691336\pi$
$440$	0	0
$441$	590.933	0.0638088
$442$	0	0
$443$	−18622.6	−1.99726	−0.998630	−	0.0523329i	$-0.983334\pi$
$443$	−18622.6	−1.99726	−0.998630	+	0.0523329i	$0.983334\pi$
$444$	0	0
$445$	−13415.4	−1.42910
$446$	0	0
$447$	−9135.54	−0.966658
$448$	0	0
$449$	7200.51	0.756822	0.378411	−	0.925638i	$-0.376471\pi$
$449$	7200.51	0.756822	0.378411	+	0.925638i	$0.376471\pi$
$450$	0	0
$451$	−2513.42	−0.262422
$452$	0	0
$453$	15006.7	1.55646
$454$	0	0
$455$	−44599.4	−4.59528
$456$	0	0
$457$	10835.9	1.10915	0.554574	−	0.832134i	$-0.312881\pi$
$457$	10835.9	1.10915	0.554574	+	0.832134i	$0.312881\pi$
$458$	0	0
$459$	10123.4	1.02946
$460$	0	0
$461$	18532.6	1.87234	0.936171	−	0.351546i	$-0.114344\pi$
$461$	18532.6	1.87234	0.936171	+	0.351546i	$0.114344\pi$
$462$	0	0
$463$	7482.76	0.751087	0.375544	−	0.926805i	$-0.377456\pi$
$463$	7482.76	0.751087	0.375544	+	0.926805i	$0.377456\pi$
$464$	0	0
$465$	−18772.5	−1.87216
$466$	0	0
$467$	−271.992	−0.0269514	−0.0134757	−	0.999909i	$-0.504290\pi$
$467$	−271.992	−0.0269514	−0.0134757	+	0.999909i	$0.504290\pi$
$468$	0	0
$469$	−7775.40	−0.765532
$470$	0	0
$471$	2490.68	0.243661
$472$	0	0
$473$	5369.03	0.521921
$474$	0	0
$475$	−15003.5	−1.44928
$476$	0	0
$477$	−703.740	−0.0675514
$478$	0	0
$479$	3242.63	0.309310	0.154655	−	0.987969i	$-0.450573\pi$
$479$	3242.63	0.309310	0.154655	+	0.987969i	$0.450573\pi$
$480$	0	0
$481$	−27455.5	−2.60263
$482$	0	0
$483$	−17719.6	−1.66930
$484$	0	0
$485$	11187.3	1.04740
$486$	0	0
$487$	8115.28	0.755110	0.377555	−	0.925987i	$-0.376765\pi$
$487$	8115.28	0.755110	0.377555	+	0.925987i	$0.376765\pi$
$488$	0	0
$489$	−15678.2	−1.44988
$490$	0	0
$491$	−8573.07	−0.787978	−0.393989	−	0.919115i	$-0.628905\pi$
$491$	−8573.07	−0.787978	−0.393989	+	0.919115i	$0.628905\pi$
$492$	0	0
$493$	−2166.96	−0.197962
$494$	0	0
$495$	402.712	0.0365668
$496$	0	0
$497$	−12669.4	−1.14346
$498$	0	0
$499$	−8118.91	−0.728361	−0.364181	−	0.931328i	$-0.618651\pi$
$499$	−8118.91	−0.728361	−0.364181	+	0.931328i	$0.618651\pi$
$500$	0	0
$501$	−21706.0	−1.93563
$502$	0	0
$503$	−13953.9	−1.23693	−0.618464	−	0.785813i	$-0.712245\pi$
$503$	−13953.9	−1.23693	−0.618464	+	0.785813i	$0.712245\pi$
$504$	0	0
$505$	2184.30	0.192476
$506$	0	0
$507$	−24440.3	−2.14089
$508$	0	0
$509$	−6319.05	−0.550270	−0.275135	−	0.961406i	$-0.588722\pi$
$509$	−6319.05	−0.550270	−0.275135	+	0.961406i	$0.588722\pi$
$510$	0	0
$511$	−4002.64	−0.346510
$512$	0	0
$513$	−6693.52	−0.576075
$514$	0	0
$515$	24656.5	2.10970
$516$	0	0
$517$	557.683	0.0474408
$518$	0	0
$519$	9527.82	0.805828
$520$	0	0
$521$	10989.2	0.924076	0.462038	−	0.886860i	$-0.347118\pi$
$521$	10989.2	0.924076	0.462038	+	0.886860i	$0.347118\pi$
$522$	0	0
$523$	−15274.9	−1.27710	−0.638549	−	0.769581i	$-0.720465\pi$
$523$	−15274.9	−1.27710	−0.638549	+	0.769581i	$0.720465\pi$
$524$	0	0
$525$	42645.1	3.54512
$526$	0	0
$527$	12642.2	1.04498
$528$	0	0
$529$	3754.78	0.308604
$530$	0	0
$531$	−484.025	−0.0395573
$532$	0	0
$533$	−18266.0	−1.48441
$534$	0	0
$535$	−22460.6	−1.81506
$536$	0	0
$537$	−3295.18	−0.264800
$538$	0	0
$539$	3887.23	0.310640
$540$	0	0
$541$	14521.2	1.15401	0.577003	−	0.816742i	$-0.304222\pi$
$541$	14521.2	1.15401	0.577003	+	0.816742i	$0.304222\pi$
$542$	0	0
$543$	−5371.87	−0.424547
$544$	0	0
$545$	−12416.3	−0.975884
$546$	0	0
$547$	15568.3	1.21691	0.608457	−	0.793587i	$-0.291789\pi$
$547$	15568.3	1.21691	0.608457	+	0.793587i	$0.291789\pi$
$548$	0	0
$549$	700.135	0.0544281
$550$	0	0
$551$	1432.78	0.110778
$552$	0	0
$553$	−17236.3	−1.32543
$554$	0	0
$555$	37058.6	2.83432
$556$	0	0
$557$	−80.4192	−0.00611754	−0.00305877	−	0.999995i	$-0.500974\pi$
$557$	−80.4192	−0.00611754	−0.00305877	+	0.999995i	$0.500974\pi$
$558$	0	0
$559$	39018.9	2.95228
$560$	0	0
$561$	−4529.59	−0.340890
$562$	0	0
$563$	−21472.1	−1.60735	−0.803676	−	0.595067i	$-0.797126\pi$
$563$	−21472.1	−1.60735	−0.803676	+	0.595067i	$0.797126\pi$
$564$	0	0
$565$	−44121.2	−3.28530
$566$	0	0
$567$	20241.9	1.49926
$568$	0	0
$569$	10665.8	0.785827	0.392913	−	0.919575i	$-0.371467\pi$
$569$	10665.8	0.785827	0.392913	+	0.919575i	$0.371467\pi$
$570$	0	0
$571$	−6958.52	−0.509991	−0.254996	−	0.966942i	$-0.582074\pi$
$571$	−6958.52	−0.509991	−0.254996	+	0.966942i	$0.582074\pi$
$572$	0	0
$573$	−10782.7	−0.786129
$574$	0	0
$575$	−38318.3	−2.77911
$576$	0	0
$577$	−11529.1	−0.831826	−0.415913	−	0.909404i	$-0.636538\pi$
$577$	−11529.1	−0.831826	−0.415913	+	0.909404i	$0.636538\pi$
$578$	0	0
$579$	−21860.8	−1.56909
$580$	0	0
$581$	253.322	0.0180887
$582$	0	0
$583$	−4629.28	−0.328860
$584$	0	0
$585$	2926.67	0.206843
$586$	0	0
$587$	−14332.3	−1.00776	−0.503880	−	0.863774i	$-0.668095\pi$
$587$	−14332.3	−1.00776	−0.503880	+	0.863774i	$0.668095\pi$
$588$	0	0
$589$	−8358.90	−0.584758
$590$	0	0
$591$	−1295.68	−0.0901815
$592$	0	0
$593$	24797.3	1.71720	0.858602	−	0.512643i	$-0.171333\pi$
$593$	24797.3	1.71720	0.858602	+	0.512643i	$0.171333\pi$
$594$	0	0
$595$	−40540.4	−2.79327
$596$	0	0
$597$	9071.83	0.621918
$598$	0	0
$599$	−8303.79	−0.566416	−0.283208	−	0.959058i	$-0.591399\pi$
$599$	−8303.79	−0.566416	−0.283208	+	0.959058i	$0.591399\pi$
$600$	0	0
$601$	−20553.6	−1.39501	−0.697503	−	0.716582i	$-0.745706\pi$
$601$	−20553.6	−1.39501	−0.697503	+	0.716582i	$0.745706\pi$
$602$	0	0
$603$	510.231	0.0344581
$604$	0	0
$605$	−24908.6	−1.67385
$606$	0	0
$607$	−24901.2	−1.66509	−0.832544	−	0.553959i	$-0.813116\pi$
$607$	−24901.2	−1.66509	−0.832544	+	0.553959i	$0.813116\pi$
$608$	0	0
$609$	−4072.46	−0.270976
$610$	0	0
$611$	4052.91	0.268352
$612$	0	0
$613$	17115.9	1.12774	0.563869	−	0.825864i	$-0.309312\pi$
$613$	17115.9	1.12774	0.563869	+	0.825864i	$0.309312\pi$
$614$	0	0
$615$	24654.9	1.61655
$616$	0	0
$617$	1640.73	0.107056	0.0535278	−	0.998566i	$-0.482953\pi$
$617$	1640.73	0.107056	0.0535278	+	0.998566i	$0.482953\pi$
$618$	0	0
$619$	−13582.1	−0.881921	−0.440961	−	0.897526i	$-0.645362\pi$
$619$	−13582.1	−0.881921	−0.440961	+	0.897526i	$0.645362\pi$
$620$	0	0
$621$	−17095.0	−1.10467
$622$	0	0
$623$	16978.9	1.09188
$624$	0	0
$625$	38634.8	2.47262
$626$	0	0
$627$	2994.92	0.190759
$628$	0	0
$629$	−24956.8	−1.58202
$630$	0	0
$631$	−3160.69	−0.199406	−0.0997031	−	0.995017i	$-0.531789\pi$
$631$	−3160.69	−0.199406	−0.0997031	+	0.995017i	$0.531789\pi$
$632$	0	0
$633$	28602.2	1.79595
$634$	0	0
$635$	15595.8	0.974644
$636$	0	0
$637$	28250.1	1.75716
$638$	0	0
$639$	831.382	0.0514694
$640$	0	0
$641$	−4378.96	−0.269826	−0.134913	−	0.990857i	$-0.543075\pi$
$641$	−4378.96	−0.269826	−0.134913	+	0.990857i	$0.543075\pi$
$642$	0	0
$643$	25480.4	1.56275	0.781375	−	0.624062i	$-0.214519\pi$
$643$	25480.4	1.56275	0.781375	+	0.624062i	$0.214519\pi$
$644$	0	0
$645$	−52666.4	−3.21510
$646$	0	0
$647$	6735.91	0.409298	0.204649	−	0.978835i	$-0.434395\pi$
$647$	6735.91	0.409298	0.204649	+	0.978835i	$0.434395\pi$
$648$	0	0
$649$	−3183.97	−0.192576
$650$	0	0
$651$	23758.9	1.43039
$652$	0	0
$653$	−10942.1	−0.655736	−0.327868	−	0.944724i	$-0.606330\pi$
$653$	−10942.1	−0.655736	−0.327868	+	0.944724i	$0.606330\pi$
$654$	0	0
$655$	−870.328	−0.0519184
$656$	0	0
$657$	262.658	0.0155971
$658$	0	0
$659$	3498.45	0.206799	0.103399	−	0.994640i	$-0.467028\pi$
$659$	3498.45	0.206799	0.103399	+	0.994640i	$0.467028\pi$
$660$	0	0
$661$	21301.8	1.25347	0.626736	−	0.779232i	$-0.284390\pi$
$661$	21301.8	1.25347	0.626736	+	0.779232i	$0.284390\pi$
$662$	0	0
$663$	−32918.3	−1.92827
$664$	0	0
$665$	26805.0	1.56309
$666$	0	0
$667$	3659.27	0.212425
$668$	0	0
$669$	−12521.2	−0.723615
$670$	0	0
$671$	4605.57	0.264972
$672$	0	0
$673$	−15144.5	−0.867428	−0.433714	−	0.901051i	$-0.642797\pi$
$673$	−15144.5	−0.867428	−0.433714	+	0.901051i	$0.642797\pi$
$674$	0	0
$675$	41141.9	2.34600
$676$	0	0
$677$	17045.5	0.967669	0.483835	−	0.875159i	$-0.339244\pi$
$677$	17045.5	0.967669	0.483835	+	0.875159i	$0.339244\pi$
$678$	0	0
$679$	−14158.9	−0.800250
$680$	0	0
$681$	30832.0	1.73493
$682$	0	0
$683$	−23538.8	−1.31872	−0.659360	−	0.751827i	$-0.729173\pi$
$683$	−23538.8	−1.31872	−0.659360	+	0.751827i	$0.729173\pi$
$684$	0	0
$685$	−34812.8	−1.94179
$686$	0	0
$687$	−16306.4	−0.905571
$688$	0	0
$689$	−33642.9	−1.86022
$690$	0	0
$691$	8457.60	0.465618	0.232809	−	0.972522i	$-0.425208\pi$
$691$	8457.60	0.465618	0.232809	+	0.972522i	$0.425208\pi$
$692$	0	0
$693$	−509.683	−0.0279383
$694$	0	0
$695$	58589.5	3.19774
$696$	0	0
$697$	−16603.6	−0.902306
$698$	0	0
$699$	−28543.1	−1.54449
$700$	0	0
$701$	30011.0	1.61697	0.808487	−	0.588513i	$-0.200287\pi$
$701$	30011.0	1.61697	0.808487	+	0.588513i	$0.200287\pi$
$702$	0	0
$703$	16501.2	0.885285
$704$	0	0
$705$	−5470.48	−0.292241
$706$	0	0
$707$	−2764.51	−0.147058
$708$	0	0
$709$	7803.17	0.413334	0.206667	−	0.978411i	$-0.433738\pi$
$709$	7803.17	0.413334	0.206667	+	0.978411i	$0.433738\pi$
$710$	0	0
$711$	1131.07	0.0596603
$712$	0	0
$713$	−21348.3	−1.12132
$714$	0	0
$715$	19252.0	1.00697
$716$	0	0
$717$	−24154.2	−1.25810
$718$	0	0
$719$	−4061.71	−0.210676	−0.105338	−	0.994436i	$-0.533592\pi$
$719$	−4061.71	−0.210676	−0.105338	+	0.994436i	$0.533592\pi$
$720$	0	0
$721$	−31205.8	−1.61188
$722$	0	0
$723$	−32368.0	−1.66498
$724$	0	0
$725$	−8806.61	−0.451130
$726$	0	0
$727$	−4677.31	−0.238613	−0.119307	−	0.992857i	$-0.538067\pi$
$727$	−4677.31	−0.238613	−0.119307	+	0.992857i	$0.538067\pi$
$728$	0	0
$729$	18274.9	0.928459
$730$	0	0
$731$	35467.8	1.79456
$732$	0	0
$733$	−23343.0	−1.17625	−0.588126	−	0.808769i	$-0.700134\pi$
$733$	−23343.0	−1.17625	−0.588126	+	0.808769i	$0.700134\pi$
$734$	0	0
$735$	−38131.0	−1.91358
$736$	0	0
$737$	3356.36	0.167752
$738$	0	0
$739$	−14638.3	−0.728659	−0.364330	−	0.931270i	$-0.618702\pi$
$739$	−14638.3	−0.728659	−0.364330	+	0.931270i	$0.618702\pi$
$740$	0	0
$741$	21765.3	1.07904
$742$	0	0
$743$	10881.0	0.537263	0.268631	−	0.963243i	$-0.413429\pi$
$743$	10881.0	0.537263	0.268631	+	0.963243i	$0.413429\pi$
$744$	0	0
$745$	35294.8	1.73570
$746$	0	0
$747$	−16.6233	−0.000814209 0
$748$	0	0
$749$	28426.7	1.38677
$750$	0	0
$751$	16206.8	0.787477	0.393739	−	0.919222i	$-0.371181\pi$
$751$	16206.8	0.787477	0.393739	+	0.919222i	$0.371181\pi$
$752$	0	0
$753$	−7185.37	−0.347741
$754$	0	0
$755$	−57977.6	−2.79473
$756$	0	0
$757$	26758.4	1.28474	0.642372	−	0.766393i	$-0.277950\pi$
$757$	26758.4	1.28474	0.642372	+	0.766393i	$0.277950\pi$
$758$	0	0
$759$	7648.93	0.365795
$760$	0	0
$761$	−23388.2	−1.11409	−0.557044	−	0.830483i	$-0.688065\pi$
$761$	−23388.2	−1.11409	−0.557044	+	0.830483i	$0.688065\pi$
$762$	0	0
$763$	15714.4	0.745609
$764$	0	0
$765$	2660.31	0.125730
$766$	0	0
$767$	−23139.2	−1.08932
$768$	0	0
$769$	9689.44	0.454370	0.227185	−	0.973852i	$-0.427048\pi$
$769$	9689.44	0.454370	0.227185	+	0.973852i	$0.427048\pi$
$770$	0	0
$771$	−24821.5	−1.15944
$772$	0	0
$773$	−25615.7	−1.19189	−0.595946	−	0.803024i	$-0.703223\pi$
$773$	−25615.7	−1.19189	−0.595946	+	0.803024i	$0.703223\pi$
$774$	0	0
$775$	51378.2	2.38137
$776$	0	0
$777$	−46902.2	−2.16552
$778$	0	0
$779$	10978.2	0.504922
$780$	0	0
$781$	5468.93	0.250568
$782$	0	0
$783$	−3928.91	−0.179320
$784$	0	0
$785$	−9622.64	−0.437512
$786$	0	0
$787$	−36122.7	−1.63613	−0.818066	−	0.575124i	$-0.804954\pi$
$787$	−36122.7	−1.63613	−0.818066	+	0.575124i	$0.804954\pi$
$788$	0	0
$789$	−17205.8	−0.776354
$790$	0	0
$791$	55840.9	2.51008
$792$	0	0
$793$	33470.5	1.49883
$794$	0	0
$795$	45410.0	2.02582
$796$	0	0
$797$	−30678.6	−1.36348	−0.681739	−	0.731596i	$-0.738776\pi$
$797$	−30678.6	−1.36348	−0.681739	+	0.731596i	$0.738776\pi$
$798$	0	0
$799$	3684.05	0.163119
$800$	0	0
$801$	−1114.17	−0.0491478
$802$	0	0
$803$	1727.80	0.0759311
$804$	0	0
$805$	68459.0	2.99734
$806$	0	0
$807$	−34104.7	−1.48766
$808$	0	0
$809$	−3906.04	−0.169752	−0.0848758	−	0.996392i	$-0.527049\pi$
$809$	−3906.04	−0.169752	−0.0848758	+	0.996392i	$0.527049\pi$
$810$	0	0
$811$	−6485.48	−0.280809	−0.140405	−	0.990094i	$-0.544840\pi$
$811$	−6485.48	−0.280809	−0.140405	+	0.990094i	$0.544840\pi$
$812$	0	0
$813$	3544.64	0.152910
$814$	0	0
$815$	60571.9	2.60337
$816$	0	0
$817$	−23451.0	−1.00422
$818$	0	0
$819$	−3704.07	−0.158035
$820$	0	0
$821$	−19790.5	−0.841284	−0.420642	−	0.907227i	$-0.638195\pi$
$821$	−19790.5	−0.841284	−0.420642	+	0.907227i	$0.638195\pi$
$822$	0	0
$823$	7205.41	0.305182	0.152591	−	0.988289i	$-0.451238\pi$
$823$	7205.41	0.305182	0.152591	+	0.988289i	$0.451238\pi$
$824$	0	0
$825$	−18408.4	−0.776845
$826$	0	0
$827$	8620.08	0.362454	0.181227	−	0.983441i	$-0.441993\pi$
$827$	8620.08	0.362454	0.181227	+	0.983441i	$0.441993\pi$
$828$	0	0
$829$	−29288.3	−1.22705	−0.613524	−	0.789676i	$-0.710249\pi$
$829$	−29288.3	−1.22705	−0.613524	+	0.789676i	$0.710249\pi$
$830$	0	0
$831$	−43732.4	−1.82559
$832$	0	0
$833$	25679.0	1.06810
$834$	0	0
$835$	83860.2	3.47557
$836$	0	0
$837$	22921.4	0.946572
$838$	0	0
$839$	−13354.6	−0.549525	−0.274763	−	0.961512i	$-0.588599\pi$
$839$	−13354.6	−0.549525	−0.274763	+	0.961512i	$0.588599\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−13064.9	−0.533784
$844$	0	0
$845$	94424.1	3.84413
$846$	0	0
$847$	31524.9	1.27888
$848$	0	0
$849$	−18348.5	−0.741719
$850$	0	0
$851$	42143.5	1.69760
$852$	0	0
$853$	−7277.80	−0.292130	−0.146065	−	0.989275i	$-0.546661\pi$
$853$	−7277.80	−0.292130	−0.146065	+	0.989275i	$0.546661\pi$
$854$	0	0
$855$	−1758.98	−0.0703576
$856$	0	0
$857$	25596.7	1.02026	0.510132	−	0.860096i	$-0.329597\pi$
$857$	25596.7	1.02026	0.510132	+	0.860096i	$0.329597\pi$
$858$	0	0
$859$	−17617.2	−0.699758	−0.349879	−	0.936795i	$-0.613777\pi$
$859$	−17617.2	−0.699758	−0.349879	+	0.936795i	$0.613777\pi$
$860$	0	0
$861$	−31203.8	−1.23510
$862$	0	0
$863$	46545.1	1.83594	0.917969	−	0.396651i	$-0.129828\pi$
$863$	46545.1	1.83594	0.917969	+	0.396651i	$0.129828\pi$
$864$	0	0
$865$	−36810.3	−1.44692
$866$	0	0
$867$	−3593.33	−0.140757
$868$	0	0
$869$	7440.32	0.290444
$870$	0	0
$871$	24392.0	0.948901
$872$	0	0
$873$	929.126	0.0360208
$874$	0	0
$875$	−96939.6	−3.74532
$876$	0	0
$877$	7008.29	0.269844	0.134922	−	0.990856i	$-0.456922\pi$
$877$	7008.29	0.269844	0.134922	+	0.990856i	$0.456922\pi$
$878$	0	0
$879$	−7612.97	−0.292126
$880$	0	0
$881$	−9428.76	−0.360571	−0.180285	−	0.983614i	$-0.557702\pi$
$881$	−9428.76	−0.360571	−0.180285	+	0.983614i	$0.557702\pi$
$882$	0	0
$883$	13904.2	0.529913	0.264956	−	0.964260i	$-0.414642\pi$
$883$	13904.2	0.529913	0.264956	+	0.964260i	$0.414642\pi$
$884$	0	0
$885$	31232.5	1.18629
$886$	0	0
$887$	−18819.6	−0.712400	−0.356200	−	0.934410i	$-0.615928\pi$
$887$	−18819.6	−0.712400	−0.356200	+	0.934410i	$0.615928\pi$
$888$	0	0
$889$	−19738.4	−0.744661
$890$	0	0
$891$	−8737.72	−0.328535
$892$	0	0
$893$	−2435.86	−0.0912800
$894$	0	0
$895$	12730.8	0.475468
$896$	0	0
$897$	55587.8	2.06915
$898$	0	0
$899$	−4906.44	−0.182023
$900$	0	0
$901$	−30581.0	−1.13074
$902$	0	0
$903$	66655.9	2.45645
$904$	0	0
$905$	20754.0	0.762305
$906$	0	0
$907$	−18083.2	−0.662010	−0.331005	−	0.943629i	$-0.607388\pi$
$907$	−18083.2	−0.662010	−0.331005	+	0.943629i	$0.607388\pi$
$908$	0	0
$909$	181.411	0.00661937
$910$	0	0
$911$	3424.94	0.124559	0.0622796	−	0.998059i	$-0.480163\pi$
$911$	3424.94	0.124559	0.0622796	+	0.998059i	$0.480163\pi$
$912$	0	0
$913$	−109.350	−0.00396380
$914$	0	0
$915$	−45177.4	−1.63226
$916$	0	0
$917$	1101.51	0.0396674
$918$	0	0
$919$	37465.4	1.34480	0.672398	−	0.740190i	$-0.265264\pi$
$919$	37465.4	1.34480	0.672398	+	0.740190i	$0.265264\pi$
$920$	0	0
$921$	50183.9	1.79546
$922$	0	0
$923$	39744.9	1.41736
$924$	0	0
$925$	−101425.	−3.60523
$926$	0	0
$927$	2047.77	0.0725539
$928$	0	0
$929$	−22247.8	−0.785713	−0.392856	−	0.919600i	$-0.628513\pi$
$929$	−22247.8	−0.785713	−0.392856	+	0.919600i	$0.628513\pi$
$930$	0	0
$931$	−16978.7	−0.597697
$932$	0	0
$933$	33352.3	1.17032
$934$	0	0
$935$	17499.9	0.612092
$936$	0	0
$937$	−12937.5	−0.451067	−0.225534	−	0.974235i	$-0.572413\pi$
$937$	−12937.5	−0.451067	−0.225534	+	0.974235i	$0.572413\pi$
$938$	0	0
$939$	−36630.3	−1.27304
$940$	0	0
$941$	−7579.26	−0.262568	−0.131284	−	0.991345i	$-0.541910\pi$
$941$	−7579.26	−0.262568	−0.131284	+	0.991345i	$0.541910\pi$
$942$	0	0
$943$	28037.9	0.968228
$944$	0	0
$945$	−73503.5	−2.53023
$946$	0	0
$947$	14864.2	0.510053	0.255027	−	0.966934i	$-0.417916\pi$
$947$	14864.2	0.510053	0.255027	+	0.966934i	$0.417916\pi$
$948$	0	0
$949$	12556.6	0.429509
$950$	0	0
$951$	9767.52	0.333053
$952$	0	0
$953$	13121.7	0.446017	0.223009	−	0.974816i	$-0.428412\pi$
$953$	13121.7	0.446017	0.223009	+	0.974816i	$0.428412\pi$
$954$	0	0
$955$	41658.3	1.41155
$956$	0	0
$957$	1757.93	0.0593793
$958$	0	0
$959$	44060.0	1.48360
$960$	0	0
$961$	−1166.61	−0.0391597
$962$	0	0
$963$	−1865.39	−0.0624211
$964$	0	0
$965$	84458.3	2.81742
$966$	0	0
$967$	−32017.3	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$967$	−32017.3	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$968$	0	0
$969$	19784.4	0.655901
$970$	0	0
$971$	−7412.86	−0.244995	−0.122497	−	0.992469i	$-0.539090\pi$
$971$	−7412.86	−0.244995	−0.122497	+	0.992469i	$0.539090\pi$
$972$	0	0
$973$	−74152.4	−2.44318
$974$	0	0
$975$	−133781.	−4.39428
$976$	0	0
$977$	−42349.7	−1.38678	−0.693391	−	0.720562i	$-0.743884\pi$
$977$	−42349.7	−1.38678	−0.693391	+	0.720562i	$0.743884\pi$
$978$	0	0
$979$	−7329.17	−0.239266
$980$	0	0
$981$	−1031.20	−0.0335613
$982$	0	0
$983$	31320.6	1.01625	0.508124	−	0.861284i	$-0.330339\pi$
$983$	31320.6	1.01625	0.508124	+	0.861284i	$0.330339\pi$
$984$	0	0
$985$	5005.81	0.161927
$986$	0	0
$987$	6923.57	0.223282
$988$	0	0
$989$	−59893.0	−1.92567
$990$	0	0
$991$	42147.1	1.35100	0.675502	−	0.737358i	$-0.263927\pi$
$991$	42147.1	1.35100	0.675502	+	0.737358i	$0.263927\pi$
$992$	0	0
$993$	−40337.3	−1.28909
$994$	0	0
$995$	−35048.6	−1.11670
$996$	0	0
$997$	−12454.9	−0.395638	−0.197819	−	0.980239i	$-0.563386\pi$
$997$	−12454.9	−0.395638	−0.197819	+	0.980239i	$0.563386\pi$
$998$	0	0
$999$	−45249.0	−1.43305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bj.1.5		12
4.3	odd	2		1856.4.a.bl.1.8		12
8.3	odd	2		928.4.a.h.1.5	✓	12
8.5	even	2		928.4.a.j.1.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.5	✓	12	8.3	odd	2
928.4.a.j.1.8	yes	12	8.5	even	2
1856.4.a.bj.1.5		12	1.1	even	1	trivial
1856.4.a.bl.1.8		12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-5.35906 q^{3} +20.7045 q^{5} -26.2041 q^{7} +1.71955 q^{9} +O(q^{10})\)	\(q-5.35906 q^{3} +20.7045 q^{5} -26.2041 q^{7} +1.71955 q^{9} +11.3114 q^{11} +82.2044 q^{13} -110.957 q^{15} +74.7229 q^{17} -49.4062 q^{19} +140.430 q^{21} -126.182 q^{23} +303.676 q^{25} +135.480 q^{27} -29.0000 q^{29} +169.187 q^{31} -60.6184 q^{33} -542.543 q^{35} -333.991 q^{37} -440.538 q^{39} -222.203 q^{41} +474.657 q^{43} +35.6024 q^{45} +49.3028 q^{47} +343.656 q^{49} -400.445 q^{51} -409.259 q^{53} +234.197 q^{55} +264.771 q^{57} -281.484 q^{59} +407.163 q^{61} -45.0592 q^{63} +1702.00 q^{65} +296.724 q^{67} +676.215 q^{69} +483.489 q^{71} +152.749 q^{73} -1627.42 q^{75} -296.405 q^{77} +657.772 q^{79} -772.471 q^{81} -9.66724 q^{83} +1547.10 q^{85} +155.413 q^{87} -647.946 q^{89} -2154.09 q^{91} -906.686 q^{93} -1022.93 q^{95} +540.332 q^{97} +19.4505 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) \) 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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