LMFDB - Embedded newform 2166.4.a.bh.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 350x^{6} + 948x^{5} + 37019x^{4} - 115308x^{3} - 1098530x^{2} + 2724222x + 7883581 $ x^8 - 2x^7 - 350x^6 + 948x^5 + 37019x^4 - 115308x^3 - 1098530x^2 + 2724222*x + 7883581
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 19 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-5.58215$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -3.04160 q^{5} -6.00000 q^{6} +19.3581 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -3.04160 q^{5} -6.00000 q^{6} +19.3581 q^{7} -8.00000 q^{8} +9.00000 q^{9} +6.08320 q^{10} +10.3359 q^{11} +12.0000 q^{12} +9.21828 q^{13} -38.7162 q^{14} -9.12481 q^{15} +16.0000 q^{16} -24.5848 q^{17} -18.0000 q^{18} -12.1664 q^{20} +58.0744 q^{21} -20.6718 q^{22} -109.448 q^{23} -24.0000 q^{24} -115.749 q^{25} -18.4366 q^{26} +27.0000 q^{27} +77.4325 q^{28} +93.5874 q^{29} +18.2496 q^{30} +73.4153 q^{31} -32.0000 q^{32} +31.0076 q^{33} +49.1696 q^{34} -58.8797 q^{35} +36.0000 q^{36} -52.7662 q^{37} +27.6548 q^{39} +24.3328 q^{40} -368.610 q^{41} -116.149 q^{42} -136.522 q^{43} +41.3435 q^{44} -27.3744 q^{45} +218.896 q^{46} -422.115 q^{47} +48.0000 q^{48} +31.7368 q^{49} +231.497 q^{50} -73.7544 q^{51} +36.8731 q^{52} -178.321 q^{53} -54.0000 q^{54} -31.4376 q^{55} -154.865 q^{56} -187.175 q^{58} +243.769 q^{59} -36.4992 q^{60} -571.908 q^{61} -146.831 q^{62} +174.223 q^{63} +64.0000 q^{64} -28.0383 q^{65} -62.0153 q^{66} +489.746 q^{67} -98.3392 q^{68} -328.344 q^{69} +117.759 q^{70} -503.852 q^{71} -72.0000 q^{72} +576.803 q^{73} +105.532 q^{74} -347.246 q^{75} +200.083 q^{77} -55.3097 q^{78} -744.045 q^{79} -48.6656 q^{80} +81.0000 q^{81} +737.219 q^{82} -272.455 q^{83} +232.297 q^{84} +74.7771 q^{85} +273.044 q^{86} +280.762 q^{87} -82.6871 q^{88} -180.777 q^{89} +54.7488 q^{90} +178.449 q^{91} -437.792 q^{92} +220.246 q^{93} +844.230 q^{94} -96.0000 q^{96} -93.2530 q^{97} -63.4736 q^{98} +93.0229 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{2} + 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q - 16 * q^2 + 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} + 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} - 64 q^{8} + 72 q^{9} + 60 q^{10} - 6 q^{11} + 96 q^{12} - 48 q^{13} + 68 q^{14} - 90 q^{15} + 128 q^{16} - 36 q^{17} - 144 q^{18} - 120 q^{20} - 102 q^{21} + 12 q^{22} - 282 q^{23} - 192 q^{24} + 396 q^{25} + 96 q^{26} + 216 q^{27} - 136 q^{28} + 380 q^{29} + 180 q^{30} + 48 q^{31} - 256 q^{32} - 18 q^{33} + 72 q^{34} + 762 q^{35} + 288 q^{36} + 168 q^{37} - 144 q^{39} + 240 q^{40} - 342 q^{41} + 204 q^{42} - 788 q^{43} - 24 q^{44} - 270 q^{45} + 564 q^{46} - 468 q^{47} + 384 q^{48} + 222 q^{49} - 792 q^{50} - 108 q^{51} - 192 q^{52} - 1682 q^{53} - 432 q^{54} - 46 q^{55} + 272 q^{56} - 760 q^{58} - 292 q^{59} - 360 q^{60} + 522 q^{61} - 96 q^{62} - 306 q^{63} + 512 q^{64} + 1120 q^{65} + 36 q^{66} + 2484 q^{67} - 144 q^{68} - 846 q^{69} - 1524 q^{70} - 1182 q^{71} - 576 q^{72} + 182 q^{73} - 336 q^{74} + 1188 q^{75} - 504 q^{77} + 288 q^{78} + 2232 q^{79} - 480 q^{80} + 648 q^{81} + 684 q^{82} - 750 q^{83} - 408 q^{84} - 2238 q^{85} + 1576 q^{86} + 1140 q^{87} + 48 q^{88} - 1304 q^{89} + 540 q^{90} + 624 q^{91} - 1128 q^{92} + 144 q^{93} + 936 q^{94} - 768 q^{96} + 1248 q^{97} - 444 q^{98} - 54 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 + 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 - 64 * q^8 + 72 * q^9 + 60 * q^10 - 6 * q^11 + 96 * q^12 - 48 * q^13 + 68 * q^14 - 90 * q^15 + 128 * q^16 - 36 * q^17 - 144 * q^18 - 120 * q^20 - 102 * q^21 + 12 * q^22 - 282 * q^23 - 192 * q^24 + 396 * q^25 + 96 * q^26 + 216 * q^27 - 136 * q^28 + 380 * q^29 + 180 * q^30 + 48 * q^31 - 256 * q^32 - 18 * q^33 + 72 * q^34 + 762 * q^35 + 288 * q^36 + 168 * q^37 - 144 * q^39 + 240 * q^40 - 342 * q^41 + 204 * q^42 - 788 * q^43 - 24 * q^44 - 270 * q^45 + 564 * q^46 - 468 * q^47 + 384 * q^48 + 222 * q^49 - 792 * q^50 - 108 * q^51 - 192 * q^52 - 1682 * q^53 - 432 * q^54 - 46 * q^55 + 272 * q^56 - 760 * q^58 - 292 * q^59 - 360 * q^60 + 522 * q^61 - 96 * q^62 - 306 * q^63 + 512 * q^64 + 1120 * q^65 + 36 * q^66 + 2484 * q^67 - 144 * q^68 - 846 * q^69 - 1524 * q^70 - 1182 * q^71 - 576 * q^72 + 182 * q^73 - 336 * q^74 + 1188 * q^75 - 504 * q^77 + 288 * q^78 + 2232 * q^79 - 480 * q^80 + 648 * q^81 + 684 * q^82 - 750 * q^83 - 408 * q^84 - 2238 * q^85 + 1576 * q^86 + 1140 * q^87 + 48 * q^88 - 1304 * q^89 + 540 * q^90 + 624 * q^91 - 1128 * q^92 + 144 * q^93 + 936 * q^94 - 768 * q^96 + 1248 * q^97 - 444 * q^98 - 54 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−3.04160	−0.272049	−0.136025	−	0.990705i	$-0.543433\pi$
$5$	−3.04160	−0.272049	−0.136025	+	0.990705i	$0.543433\pi$
$6$	−6.00000	−0.408248
$7$	19.3581	1.04524	0.522620	−	0.852566i	$-0.324955\pi$
$7$	19.3581	1.04524	0.522620	+	0.852566i	$0.324955\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	6.08320	0.192368
$11$	10.3359	0.283308	0.141654	−	0.989916i	$-0.454758\pi$
$11$	10.3359	0.283308	0.141654	+	0.989916i	$0.454758\pi$
$12$	12.0000	0.288675
$13$	9.21828	0.196669	0.0983343	−	0.995153i	$-0.468649\pi$
$13$	9.21828	0.196669	0.0983343	+	0.995153i	$0.468649\pi$
$14$	−38.7162	−0.739096
$15$	−9.12481	−0.157068
$16$	16.0000	0.250000
$17$	−24.5848	−0.350746	−0.175373	−	0.984502i	$-0.556113\pi$
$17$	−24.5848	−0.350746	−0.175373	+	0.984502i	$0.556113\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	−12.1664	−0.136025
$21$	58.0744	0.603470
$22$	−20.6718	−0.200329
$23$	−109.448	−0.992238	−0.496119	−	0.868254i	$-0.665242\pi$
$23$	−109.448	−0.992238	−0.496119	+	0.868254i	$0.665242\pi$
$24$	−24.0000	−0.204124
$25$	−115.749	−0.925989
$26$	−18.4366	−0.139066
$27$	27.0000	0.192450
$28$	77.4325	0.522620
$29$	93.5874	0.599267	0.299633	−	0.954054i	$-0.403136\pi$
$29$	93.5874	0.599267	0.299633	+	0.954054i	$0.403136\pi$
$30$	18.2496	0.111064
$31$	73.4153	0.425347	0.212674	−	0.977123i	$-0.431783\pi$
$31$	73.4153	0.425347	0.212674	+	0.977123i	$0.431783\pi$
$32$	−32.0000	−0.176777
$33$	31.0076	0.163568
$34$	49.1696	0.248015
$35$	−58.8797	−0.284357
$36$	36.0000	0.166667
$37$	−52.7662	−0.234451	−0.117226	−	0.993105i	$-0.537400\pi$
$37$	−52.7662	−0.234451	−0.117226	+	0.993105i	$0.537400\pi$
$38$	0	0
$39$	27.6548	0.113547
$40$	24.3328	0.0961839
$41$	−368.610	−1.40408	−0.702039	−	0.712139i	$-0.747727\pi$
$41$	−368.610	−1.40408	−0.702039	+	0.712139i	$0.747727\pi$
$42$	−116.149	−0.426718
$43$	−136.522	−0.484173	−0.242086	−	0.970255i	$-0.577832\pi$
$43$	−136.522	−0.484173	−0.242086	+	0.970255i	$0.577832\pi$
$44$	41.3435	0.141654
$45$	−27.3744	−0.0906830
$46$	218.896	0.701618
$47$	−422.115	−1.31004	−0.655019	−	0.755612i	$-0.727339\pi$
$47$	−422.115	−1.31004	−0.655019	+	0.755612i	$0.727339\pi$
$48$	48.0000	0.144338
$49$	31.7368	0.0925271
$50$	231.497	0.654773
$51$	−73.7544	−0.202503
$52$	36.8731	0.0983343
$53$	−178.321	−0.462157	−0.231079	−	0.972935i	$-0.574225\pi$
$53$	−178.321	−0.462157	−0.231079	+	0.972935i	$0.574225\pi$
$54$	−54.0000	−0.136083
$55$	−31.4376	−0.0770736
$56$	−154.865	−0.369548
$57$	0	0
$58$	−187.175	−0.423746
$59$	243.769	0.537899	0.268949	−	0.963154i	$-0.413324\pi$
$59$	243.769	0.537899	0.268949	+	0.963154i	$0.413324\pi$
$60$	−36.4992	−0.0785338
$61$	−571.908	−1.20041	−0.600207	−	0.799844i	$-0.704915\pi$
$61$	−571.908	−1.20041	−0.600207	+	0.799844i	$0.704915\pi$
$62$	−146.831	−0.300766
$63$	174.223	0.348413
$64$	64.0000	0.125000
$65$	−28.0383	−0.0535035
$66$	−62.0153	−0.115660
$67$	489.746	0.893015	0.446508	−	0.894780i	$-0.352668\pi$
$67$	489.746	0.893015	0.446508	+	0.894780i	$0.352668\pi$
$68$	−98.3392	−0.175373
$69$	−328.344	−0.572869
$70$	117.759	0.201071
$71$	−503.852	−0.842201	−0.421100	−	0.907014i	$-0.638356\pi$
$71$	−503.852	−0.842201	−0.421100	+	0.907014i	$0.638356\pi$
$72$	−72.0000	−0.117851
$73$	576.803	0.924790	0.462395	−	0.886674i	$-0.346990\pi$
$73$	576.803	0.924790	0.462395	+	0.886674i	$0.346990\pi$
$74$	105.532	0.165782
$75$	−347.246	−0.534620
$76$	0	0
$77$	200.083	0.296125
$78$	−55.3097	−0.0802896
$79$	−744.045	−1.05964	−0.529820	−	0.848110i	$-0.677741\pi$
$79$	−744.045	−1.05964	−0.529820	+	0.848110i	$0.677741\pi$
$80$	−48.6656	−0.0680123
$81$	81.0000	0.111111
$82$	737.219	0.992832
$83$	−272.455	−0.360311	−0.180156	−	0.983638i	$-0.557660\pi$
$83$	−272.455	−0.360311	−0.180156	+	0.983638i	$0.557660\pi$
$84$	232.297	0.301735
$85$	74.7771	0.0954202
$86$	273.044	0.342362
$87$	280.762	0.345987
$88$	−82.6871	−0.100164
$89$	−180.777	−0.215307	−0.107654	−	0.994188i	$-0.534334\pi$
$89$	−180.777	−0.215307	−0.107654	+	0.994188i	$0.534334\pi$
$90$	54.7488	0.0641226
$91$	178.449	0.205566
$92$	−437.792	−0.496119
$93$	220.246	0.245574
$94$	844.230	0.926337
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−93.2530	−0.0976125	−0.0488062	−	0.998808i	$-0.515542\pi$
$97$	−93.2530	−0.0976125	−0.0488062	+	0.998808i	$0.515542\pi$
$98$	−63.4736	−0.0654265
$99$	93.0229	0.0944359
$100$	−462.995	−0.462995
$101$	57.6363	0.0567824	0.0283912	−	0.999597i	$-0.490962\pi$
$101$	57.6363	0.0567824	0.0283912	+	0.999597i	$0.490962\pi$
$102$	147.509	0.143192
$103$	544.940	0.521307	0.260653	−	0.965432i	$-0.416062\pi$
$103$	544.940	0.521307	0.260653	+	0.965432i	$0.416062\pi$
$104$	−73.7462	−0.0695328
$105$	−176.639	−0.164173
$106$	356.643	0.326794
$107$	−1487.68	−1.34411	−0.672056	−	0.740501i	$-0.734588\pi$
$107$	−1487.68	−1.34411	−0.672056	+	0.740501i	$0.734588\pi$
$108$	108.000	0.0962250
$109$	1396.27	1.22696	0.613478	−	0.789712i	$-0.289770\pi$
$109$	1396.27	1.22696	0.613478	+	0.789712i	$0.289770\pi$
$110$	62.8753	0.0544993
$111$	−158.298	−0.135361
$112$	309.730	0.261310
$113$	1358.55	1.13099	0.565493	−	0.824753i	$-0.308686\pi$
$113$	1358.55	1.13099	0.565493	+	0.824753i	$0.308686\pi$
$114$	0	0
$115$	332.897	0.269938
$116$	374.349	0.299633
$117$	82.9645	0.0655562
$118$	−487.538	−0.380352
$119$	−475.915	−0.366614
$120$	72.9984	0.0555318
$121$	−1224.17	−0.919737
$122$	1143.82	0.848821
$123$	−1105.83	−0.810644
$124$	293.661	0.212674
$125$	732.262	0.523964
$126$	−348.446	−0.246365
$127$	−975.103	−0.681310	−0.340655	−	0.940188i	$-0.610649\pi$
$127$	−975.103	−0.681310	−0.340655	+	0.940188i	$0.610649\pi$
$128$	−128.000	−0.0883883
$129$	−409.566	−0.279537
$130$	56.0767	0.0378327
$131$	−545.903	−0.364090	−0.182045	−	0.983290i	$-0.558272\pi$
$131$	−545.903	−0.364090	−0.182045	+	0.983290i	$0.558272\pi$
$132$	124.031	0.0817839
$133$	0	0
$134$	−979.492	−0.631457
$135$	−82.1233	−0.0523559
$136$	196.678	0.124008
$137$	−2995.72	−1.86819	−0.934094	−	0.357026i	$-0.883791\pi$
$137$	−2995.72	−1.86819	−0.934094	+	0.357026i	$0.883791\pi$
$138$	656.688	0.405080
$139$	1454.53	0.887566	0.443783	−	0.896134i	$-0.353636\pi$
$139$	1454.53	0.887566	0.443783	+	0.896134i	$0.353636\pi$
$140$	−235.519	−0.142178
$141$	−1266.34	−0.756351
$142$	1007.70	0.595526
$143$	95.2790	0.0557177
$144$	144.000	0.0833333
$145$	−284.655	−0.163030
$146$	−1153.61	−0.653926
$147$	95.2104	0.0534205
$148$	−211.065	−0.117226
$149$	−585.486	−0.321912	−0.160956	−	0.986962i	$-0.551458\pi$
$149$	−585.486	−0.321912	−0.160956	+	0.986962i	$0.551458\pi$
$150$	694.492	0.378034
$151$	3611.02	1.94610	0.973048	−	0.230603i	$-0.0740699\pi$
$151$	3611.02	1.94610	0.973048	+	0.230603i	$0.0740699\pi$
$152$	0	0
$153$	−221.263	−0.116915
$154$	−400.166	−0.209392
$155$	−223.300	−0.115715
$156$	110.619	0.0567733
$157$	−2591.71	−1.31746	−0.658730	−	0.752379i	$-0.728906\pi$
$157$	−2591.71	−1.31746	−0.658730	+	0.752379i	$0.728906\pi$
$158$	1488.09	0.749279
$159$	−534.964	−0.266826
$160$	97.3313	0.0480919
$161$	−2118.71	−1.03713
$162$	−162.000	−0.0785674
$163$	2108.77	1.01332	0.506661	−	0.862146i	$-0.330880\pi$
$163$	2108.77	1.01332	0.506661	+	0.862146i	$0.330880\pi$
$164$	−1474.44	−0.702039
$165$	−94.3129	−0.0444985
$166$	544.910	0.254778
$167$	−956.682	−0.443295	−0.221647	−	0.975127i	$-0.571143\pi$
$167$	−956.682	−0.443295	−0.221647	+	0.975127i	$0.571143\pi$
$168$	−464.595	−0.213359
$169$	−2112.02	−0.961321
$170$	−149.554	−0.0674723
$171$	0	0
$172$	−546.088	−0.242086
$173$	−1197.56	−0.526296	−0.263148	−	0.964756i	$-0.584761\pi$
$173$	−1197.56	−0.526296	−0.263148	+	0.964756i	$0.584761\pi$
$174$	−561.524	−0.244650
$175$	−2240.68	−0.967881
$176$	165.374	0.0708269
$177$	731.307	0.310556
$178$	361.555	0.152245
$179$	859.173	0.358758	0.179379	−	0.983780i	$-0.442591\pi$
$179$	859.173	0.358758	0.179379	+	0.983780i	$0.442591\pi$
$180$	−109.498	−0.0453415
$181$	2310.65	0.948890	0.474445	−	0.880285i	$-0.342649\pi$
$181$	2310.65	0.948890	0.474445	+	0.880285i	$0.342649\pi$
$182$	−356.897	−0.145357
$183$	−1715.72	−0.693060
$184$	875.584	0.350809
$185$	160.494	0.0637823
$186$	−440.492	−0.173647
$187$	−254.105	−0.0993692
$188$	−1688.46	−0.655019
$189$	522.669	0.201157
$190$	0	0
$191$	1953.31	0.739984	0.369992	−	0.929035i	$-0.379360\pi$
$191$	1953.31	0.739984	0.369992	+	0.929035i	$0.379360\pi$
$192$	192.000	0.0721688
$193$	3266.99	1.21846	0.609231	−	0.792992i	$-0.291478\pi$
$193$	3266.99	1.21846	0.609231	+	0.792992i	$0.291478\pi$
$194$	186.506	0.0690224
$195$	−84.1150	−0.0308903
$196$	126.947	0.0462635
$197$	3314.64	1.19877	0.599387	−	0.800459i	$-0.295411\pi$
$197$	3314.64	1.19877	0.599387	+	0.800459i	$0.295411\pi$
$198$	−186.046	−0.0667763
$199$	−1170.68	−0.417020	−0.208510	−	0.978020i	$-0.566861\pi$
$199$	−1170.68	−0.417020	−0.208510	+	0.978020i	$0.566861\pi$
$200$	925.989	0.327387
$201$	1469.24	0.515583
$202$	−115.273	−0.0401512
$203$	1811.68	0.626378
$204$	−295.017	−0.101252
$205$	1121.16	0.381978
$206$	−1089.88	−0.368619
$207$	−985.032	−0.330746
$208$	147.492	0.0491671
$209$	0	0
$210$	353.278	0.116088
$211$	−4345.46	−1.41779	−0.708895	−	0.705314i	$-0.750806\pi$
$211$	−4345.46	−1.41779	−0.708895	+	0.705314i	$0.750806\pi$
$212$	−713.285	−0.231079
$213$	−1511.56	−0.486245
$214$	2975.37	0.950430
$215$	415.246	0.131719
$216$	−216.000	−0.0680414
$217$	1421.18	0.444590
$218$	−2792.53	−0.867589
$219$	1730.41	0.533928
$220$	−125.751	−0.0385368
$221$	−226.629	−0.0689808
$222$	316.597	0.0957144
$223$	−941.480	−0.282718	−0.141359	−	0.989958i	$-0.545147\pi$
$223$	−941.480	−0.282718	−0.141359	+	0.989958i	$0.545147\pi$
$224$	−619.460	−0.184774
$225$	−1041.74	−0.308663
$226$	−2717.09	−0.799728
$227$	−873.234	−0.255324	−0.127662	−	0.991818i	$-0.540747\pi$
$227$	−873.234	−0.255324	−0.127662	+	0.991818i	$0.540747\pi$
$228$	0	0
$229$	−3509.71	−1.01279	−0.506394	−	0.862302i	$-0.669022\pi$
$229$	−3509.71	−1.01279	−0.506394	+	0.862302i	$0.669022\pi$
$230$	−665.794	−0.190875
$231$	600.250	0.170968
$232$	−748.699	−0.211873
$233$	1266.16	0.356004	0.178002	−	0.984030i	$-0.443037\pi$
$233$	1266.16	0.356004	0.178002	+	0.984030i	$0.443037\pi$
$234$	−165.929	−0.0463552
$235$	1283.91	0.356395
$236$	975.076	0.268949
$237$	−2232.13	−0.611783
$238$	951.831	0.259235
$239$	−3665.28	−0.991996	−0.495998	−	0.868324i	$-0.665198\pi$
$239$	−3665.28	−0.991996	−0.495998	+	0.868324i	$0.665198\pi$
$240$	−145.997	−0.0392669
$241$	4539.85	1.21343	0.606716	−	0.794919i	$-0.292486\pi$
$241$	4539.85	1.21343	0.606716	+	0.794919i	$0.292486\pi$
$242$	2448.34	0.650352
$243$	243.000	0.0641500
$244$	−2287.63	−0.600207
$245$	−96.5307	−0.0251719
$246$	2211.66	0.573212
$247$	0	0
$248$	−587.322	−0.150383
$249$	−817.365	−0.208026
$250$	−1464.52	−0.370498
$251$	−6367.19	−1.60117	−0.800584	−	0.599220i	$-0.795477\pi$
$251$	−6367.19	−1.60117	−0.800584	+	0.599220i	$0.795477\pi$
$252$	696.892	0.174207
$253$	−1131.24	−0.281109
$254$	1950.21	0.481759
$255$	224.331	0.0550909
$256$	256.000	0.0625000
$257$	3351.70	0.813514	0.406757	−	0.913536i	$-0.366660\pi$
$257$	3351.70	0.813514	0.406757	+	0.913536i	$0.366660\pi$
$258$	819.132	0.197663
$259$	−1021.45	−0.245058
$260$	−112.153	−0.0267517
$261$	842.286	0.199756
$262$	1091.81	0.257451
$263$	−3046.58	−0.714298	−0.357149	−	0.934047i	$-0.616251\pi$
$263$	−3046.58	−0.714298	−0.357149	+	0.934047i	$0.616251\pi$
$264$	−248.061	−0.0578300
$265$	542.383	0.125729
$266$	0	0
$267$	−542.332	−0.124308
$268$	1958.98	0.446508
$269$	−5179.78	−1.17404	−0.587020	−	0.809572i	$-0.699699\pi$
$269$	−5179.78	−1.17404	−0.587020	+	0.809572i	$0.699699\pi$
$270$	164.247	0.0370212
$271$	−8718.39	−1.95426	−0.977131	−	0.212640i	$-0.931794\pi$
$271$	−8718.39	−1.95426	−0.977131	+	0.212640i	$0.931794\pi$
$272$	−393.357	−0.0876866
$273$	535.346	0.118683
$274$	5991.45	1.32101
$275$	−1196.36	−0.262340
$276$	−1313.38	−0.286435
$277$	5574.27	1.20912	0.604559	−	0.796561i	$-0.293350\pi$
$277$	5574.27	1.20912	0.604559	+	0.796561i	$0.293350\pi$
$278$	−2909.06	−0.627604
$279$	660.737	0.141782
$280$	471.038	0.100535
$281$	−1895.74	−0.402456	−0.201228	−	0.979544i	$-0.564493\pi$
$281$	−1895.74	−0.402456	−0.201228	+	0.979544i	$0.564493\pi$
$282$	2532.69	0.534821
$283$	−517.541	−0.108709	−0.0543545	−	0.998522i	$-0.517310\pi$
$283$	−517.541	−0.108709	−0.0543545	+	0.998522i	$0.517310\pi$
$284$	−2015.41	−0.421100
$285$	0	0
$286$	−190.558	−0.0393984
$287$	−7135.59	−1.46760
$288$	−288.000	−0.0589256
$289$	−4308.59	−0.876977
$290$	569.311	0.115280
$291$	−279.759	−0.0563566
$292$	2307.21	0.462395
$293$	2619.19	0.522234	0.261117	−	0.965307i	$-0.415909\pi$
$293$	2619.19	0.522234	0.261117	+	0.965307i	$0.415909\pi$
$294$	−190.421	−0.0377740
$295$	−741.448	−0.146335
$296$	422.129	0.0828911
$297$	279.069	0.0545226
$298$	1170.97	0.227626
$299$	−1008.92	−0.195142
$300$	−1388.98	−0.267310
$301$	−2642.81	−0.506077
$302$	−7222.04	−1.37610
$303$	172.909	0.0327833
$304$	0	0
$305$	1739.52	0.326572
$306$	442.526	0.0826717
$307$	7432.64	1.38177	0.690885	−	0.722965i	$-0.257221\pi$
$307$	7432.64	1.38177	0.690885	+	0.722965i	$0.257221\pi$
$308$	800.333	0.148062
$309$	1634.82	0.300977
$310$	446.600	0.0818232
$311$	8630.63	1.57363	0.786814	−	0.617191i	$-0.211729\pi$
$311$	8630.63	1.57363	0.786814	+	0.617191i	$0.211729\pi$
$312$	−221.239	−0.0401448
$313$	−4005.12	−0.723267	−0.361633	−	0.932320i	$-0.617781\pi$
$313$	−4005.12	−0.723267	−0.361633	+	0.932320i	$0.617781\pi$
$314$	5183.43	0.931586
$315$	−529.917	−0.0947856
$316$	−2976.18	−0.529820
$317$	7064.70	1.25171	0.625857	−	0.779938i	$-0.284749\pi$
$317$	7064.70	1.25171	0.625857	+	0.779938i	$0.284749\pi$
$318$	1069.93	0.188675
$319$	967.308	0.169777
$320$	−194.663	−0.0340061
$321$	−4463.05	−0.776023
$322$	4237.41	0.733360
$323$	0	0
$324$	324.000	0.0555556
$325$	−1067.00	−0.182113
$326$	−4217.53	−0.716526
$327$	4188.80	0.708383
$328$	2948.88	0.496416
$329$	−8171.35	−1.36930
$330$	188.626	0.0314652
$331$	5775.72	0.959101	0.479551	−	0.877514i	$-0.340800\pi$
$331$	5775.72	0.959101	0.479551	+	0.877514i	$0.340800\pi$
$332$	−1089.82	−0.180156
$333$	−474.895	−0.0781505
$334$	1913.36	0.313457
$335$	−1489.61	−0.242944
$336$	929.190	0.150867
$337$	−2351.50	−0.380103	−0.190051	−	0.981774i	$-0.560865\pi$
$337$	−2351.50	−0.380103	−0.190051	+	0.981774i	$0.560865\pi$
$338$	4224.05	0.679757
$339$	4075.64	0.652975
$340$	299.109	0.0477101
$341$	758.812	0.120504
$342$	0	0
$343$	−6025.47	−0.948527
$344$	1092.18	0.171181
$345$	998.691	0.155849
$346$	2395.13	0.372147
$347$	−9749.31	−1.50827	−0.754136	−	0.656718i	$-0.771944\pi$
$347$	−9749.31	−1.50827	−0.754136	+	0.656718i	$0.771944\pi$
$348$	1123.05	0.172993
$349$	6444.57	0.988452	0.494226	−	0.869333i	$-0.335451\pi$
$349$	6444.57	0.988452	0.494226	+	0.869333i	$0.335451\pi$
$350$	4481.35	0.684395
$351$	248.894	0.0378489
$352$	−330.748	−0.0500822
$353$	9480.96	1.42952	0.714760	−	0.699370i	$-0.246536\pi$
$353$	9480.96	1.42952	0.714760	+	0.699370i	$0.246536\pi$
$354$	−1462.61	−0.219596
$355$	1532.52	0.229120
$356$	−723.109	−0.107654
$357$	−1427.75	−0.211665
$358$	−1718.35	−0.253680
$359$	−8116.24	−1.19320	−0.596600	−	0.802539i	$-0.703482\pi$
$359$	−8116.24	−1.19320	−0.596600	+	0.802539i	$0.703482\pi$
$360$	218.995	0.0320613
$361$	0	0
$362$	−4621.29	−0.670966
$363$	−3672.51	−0.531010
$364$	713.794	0.102783
$365$	−1754.41	−0.251588
$366$	3431.45	0.490067
$367$	6842.39	0.973214	0.486607	−	0.873621i	$-0.338234\pi$
$367$	6842.39	0.973214	0.486607	+	0.873621i	$0.338234\pi$
$368$	−1751.17	−0.248060
$369$	−3317.49	−0.468026
$370$	−320.987	−0.0451009
$371$	−3451.97	−0.483065
$372$	880.983	0.122787
$373$	−6294.70	−0.873799	−0.436900	−	0.899510i	$-0.643924\pi$
$373$	−6294.70	−0.873799	−0.436900	+	0.899510i	$0.643924\pi$
$374$	508.211	0.0702646
$375$	2196.78	0.302511
$376$	3376.92	0.463168
$377$	862.714	0.117857
$378$	−1045.34	−0.142239
$379$	−2031.49	−0.275331	−0.137666	−	0.990479i	$-0.543960\pi$
$379$	−2031.49	−0.275331	−0.137666	+	0.990479i	$0.543960\pi$
$380$	0	0
$381$	−2925.31	−0.393355
$382$	−3906.63	−0.523247
$383$	−4655.75	−0.621142	−0.310571	−	0.950550i	$-0.600520\pi$
$383$	−4655.75	−0.621142	−0.310571	+	0.950550i	$0.600520\pi$
$384$	−384.000	−0.0510310
$385$	−608.574	−0.0805605
$386$	−6533.99	−0.861583
$387$	−1228.70	−0.161391
$388$	−373.012	−0.0488062
$389$	−4963.85	−0.646985	−0.323492	−	0.946231i	$-0.604857\pi$
$389$	−4963.85	−0.646985	−0.323492	+	0.946231i	$0.604857\pi$
$390$	168.230	0.0218427
$391$	2690.76	0.348024
$392$	−253.894	−0.0327133
$393$	−1637.71	−0.210208
$394$	−6629.29	−0.847661
$395$	2263.09	0.288274
$396$	372.092	0.0472180
$397$	10315.2	1.30404	0.652021	−	0.758201i	$-0.273922\pi$
$397$	10315.2	1.30404	0.652021	+	0.758201i	$0.273922\pi$
$398$	2341.35	0.294878
$399$	0	0
$400$	−1851.98	−0.231497
$401$	−15567.0	−1.93860	−0.969298	−	0.245888i	$-0.920920\pi$
$401$	−15567.0	−1.93860	−0.969298	+	0.245888i	$0.920920\pi$
$402$	−2938.48	−0.364572
$403$	676.762	0.0836525
$404$	230.545	0.0283912
$405$	−246.370	−0.0302277
$406$	−3623.35	−0.442916
$407$	−545.385	−0.0664219
$408$	590.035	0.0715958
$409$	−6020.51	−0.727860	−0.363930	−	0.931426i	$-0.618565\pi$
$409$	−6020.51	−0.727860	−0.363930	+	0.931426i	$0.618565\pi$
$410$	−2242.33	−0.270099
$411$	−8987.17	−1.07860
$412$	2179.76	0.260653
$413$	4718.91	0.562233
$414$	1970.06	0.233873
$415$	828.700	0.0980223
$416$	−294.985	−0.0347664
$417$	4363.59	0.512436
$418$	0	0
$419$	−2778.57	−0.323967	−0.161983	−	0.986793i	$-0.551789\pi$
$419$	−2778.57	−0.323967	−0.161983	+	0.986793i	$0.551789\pi$
$420$	−706.556	−0.0820867
$421$	6262.16	0.724938	0.362469	−	0.931996i	$-0.381934\pi$
$421$	6262.16	0.724938	0.362469	+	0.931996i	$0.381934\pi$
$422$	8690.91	1.00253
$423$	−3799.03	−0.436679
$424$	1426.57	0.163397
$425$	2845.66	0.324787
$426$	3023.11	0.343827
$427$	−11071.1	−1.25472
$428$	−5950.74	−0.672056
$429$	285.837	0.0321686
$430$	−830.492	−0.0931392
$431$	−2507.59	−0.280247	−0.140124	−	0.990134i	$-0.544750\pi$
$431$	−2507.59	−0.280247	−0.140124	+	0.990134i	$0.544750\pi$
$432$	432.000	0.0481125
$433$	−12259.2	−1.36060	−0.680299	−	0.732935i	$-0.738150\pi$
$433$	−12259.2	−1.36060	−0.680299	+	0.732935i	$0.738150\pi$
$434$	−2842.36	−0.314373
$435$	−853.966	−0.0941254
$436$	5585.07	0.613478
$437$	0	0
$438$	−3460.82	−0.377544
$439$	−9130.03	−0.992603	−0.496301	−	0.868150i	$-0.665309\pi$
$439$	−9130.03	−0.992603	−0.496301	+	0.868150i	$0.665309\pi$
$440$	251.501	0.0272496
$441$	285.631	0.0308424
$442$	453.259	0.0487768
$443$	−9392.32	−1.00732	−0.503660	−	0.863902i	$-0.668013\pi$
$443$	−9392.32	−1.00732	−0.503660	+	0.863902i	$0.668013\pi$
$444$	−633.194	−0.0676803
$445$	549.853	0.0585742
$446$	1882.96	0.199912
$447$	−1756.46	−0.185856
$448$	1238.92	0.130655
$449$	−337.179	−0.0354398	−0.0177199	−	0.999843i	$-0.505641\pi$
$449$	−337.179	−0.0354398	−0.0177199	+	0.999843i	$0.505641\pi$
$450$	2083.48	0.218258
$451$	−3809.91	−0.397786
$452$	5434.19	0.565493
$453$	10833.1	1.12358
$454$	1746.47	0.180541
$455$	−542.769	−0.0559240
$456$	0	0
$457$	−17701.1	−1.81187	−0.905933	−	0.423421i	$-0.860829\pi$
$457$	−17701.1	−1.81187	−0.905933	+	0.423421i	$0.860829\pi$
$458$	7019.43	0.716149
$459$	−663.789	−0.0675012
$460$	1331.59	0.134969
$461$	3530.56	0.356691	0.178346	−	0.983968i	$-0.442926\pi$
$461$	3530.56	0.356691	0.178346	+	0.983968i	$0.442926\pi$
$462$	−1200.50	−0.120892
$463$	−1809.78	−0.181658	−0.0908288	−	0.995867i	$-0.528952\pi$
$463$	−1809.78	−0.181658	−0.0908288	+	0.995867i	$0.528952\pi$
$464$	1497.40	0.149817
$465$	−669.900	−0.0668083
$466$	−2532.32	−0.251733
$467$	−3279.71	−0.324983	−0.162491	−	0.986710i	$-0.551953\pi$
$467$	−3279.71	−0.324983	−0.162491	+	0.986710i	$0.551953\pi$
$468$	331.858	0.0327781
$469$	9480.56	0.933415
$470$	−2567.81	−0.252009
$471$	−7775.14	−0.760636
$472$	−1950.15	−0.190176
$473$	−1411.08	−0.137170
$474$	4464.27	0.432596
$475$	0	0
$476$	−1903.66	−0.183307
$477$	−1604.89	−0.154052
$478$	7330.56	0.701447
$479$	−14709.8	−1.40315	−0.701575	−	0.712596i	$-0.747520\pi$
$479$	−14709.8	−1.40315	−0.701575	+	0.712596i	$0.747520\pi$
$480$	291.994	0.0277659
$481$	−486.413	−0.0461092
$482$	−9079.69	−0.858026
$483$	−6356.12	−0.598786
$484$	−4896.68	−0.459868
$485$	283.639	0.0265554
$486$	−486.000	−0.0453609
$487$	−7800.10	−0.725783	−0.362891	−	0.931831i	$-0.618210\pi$
$487$	−7800.10	−0.725783	−0.362891	+	0.931831i	$0.618210\pi$
$488$	4575.26	0.424411
$489$	6326.30	0.585041
$490$	193.061	0.0177992
$491$	15703.0	1.44332	0.721658	−	0.692249i	$-0.243380\pi$
$491$	15703.0	1.44332	0.721658	+	0.692249i	$0.243380\pi$
$492$	−4423.32	−0.405322
$493$	−2300.83	−0.210191
$494$	0	0
$495$	−282.939	−0.0256912
$496$	1174.64	0.106337
$497$	−9753.63	−0.880302
$498$	1634.73	0.147096
$499$	−9144.74	−0.820391	−0.410195	−	0.911998i	$-0.634539\pi$
$499$	−9144.74	−0.820391	−0.410195	+	0.911998i	$0.634539\pi$
$500$	2929.05	0.261982
$501$	−2870.05	−0.255936
$502$	12734.4	1.13220
$503$	−1207.06	−0.106998	−0.0534992	−	0.998568i	$-0.517037\pi$
$503$	−1207.06	−0.106998	−0.0534992	+	0.998568i	$0.517037\pi$
$504$	−1393.78	−0.123183
$505$	−175.307	−0.0154476
$506$	2262.48	0.198774
$507$	−6336.07	−0.555019
$508$	−3900.41	−0.340655
$509$	2244.95	0.195493	0.0977463	−	0.995211i	$-0.468837\pi$
$509$	2244.95	0.195493	0.0977463	+	0.995211i	$0.468837\pi$
$510$	−448.663	−0.0389551
$511$	11165.8	0.966628
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−6703.40	−0.575241
$515$	−1657.49	−0.141821
$516$	−1638.26	−0.139769
$517$	−4362.93	−0.371144
$518$	2042.91	0.173282
$519$	−3592.69	−0.303857
$520$	224.307	0.0189163
$521$	−3875.65	−0.325903	−0.162951	−	0.986634i	$-0.552101\pi$
$521$	−3875.65	−0.325903	−0.162951	+	0.986634i	$0.552101\pi$
$522$	−1684.57	−0.141249
$523$	−11456.7	−0.957872	−0.478936	−	0.877850i	$-0.658977\pi$
$523$	−11456.7	−0.957872	−0.478936	+	0.877850i	$0.658977\pi$
$524$	−2183.61	−0.182045
$525$	−6722.03	−0.558806
$526$	6093.16	0.505085
$527$	−1804.90	−0.149189
$528$	496.122	0.0408920
$529$	−188.141	−0.0154632
$530$	−1084.77	−0.0889041
$531$	2193.92	0.179300
$532$	0	0
$533$	−3397.95	−0.276138
$534$	1084.66	0.0878989
$535$	4524.95	0.365664
$536$	−3917.97	−0.315729
$537$	2577.52	0.207129
$538$	10359.6	0.830172
$539$	328.028	0.0262136
$540$	−328.493	−0.0261779
$541$	17221.7	1.36861	0.684304	−	0.729196i	$-0.260106\pi$
$541$	17221.7	1.36861	0.684304	+	0.729196i	$0.260106\pi$
$542$	17436.8	1.38187
$543$	6931.94	0.547842
$544$	786.713	0.0620038
$545$	−4246.89	−0.333792
$546$	−1070.69	−0.0839219
$547$	15563.6	1.21655	0.608275	−	0.793726i	$-0.291862\pi$
$547$	15563.6	1.21655	0.608275	+	0.793726i	$0.291862\pi$
$548$	−11982.9	−0.934094
$549$	−5147.17	−0.400138
$550$	2392.73	0.185502
$551$	0	0
$552$	2626.75	0.202540
$553$	−14403.3	−1.10758
$554$	−11148.5	−0.854975
$555$	481.481	0.0368247
$556$	5818.12	0.443783
$557$	−12661.5	−0.963167	−0.481584	−	0.876400i	$-0.659938\pi$
$557$	−12661.5	−0.963167	−0.481584	+	0.876400i	$0.659938\pi$
$558$	−1321.47	−0.100255
$559$	−1258.50	−0.0952215
$560$	−942.075	−0.0710892
$561$	−762.316	−0.0573708
$562$	3791.47	0.284579
$563$	−10852.4	−0.812389	−0.406194	−	0.913787i	$-0.633144\pi$
$563$	−10852.4	−0.812389	−0.406194	+	0.913787i	$0.633144\pi$
$564$	−5065.38	−0.378175
$565$	−4132.16	−0.307684
$566$	1035.08	0.0768688
$567$	1568.01	0.116138
$568$	4030.82	0.297763
$569$	−21599.6	−1.59139	−0.795696	−	0.605696i	$-0.792895\pi$
$569$	−21599.6	−1.59139	−0.795696	+	0.605696i	$0.792895\pi$
$570$	0	0
$571$	−2528.85	−0.185340	−0.0926699	−	0.995697i	$-0.529540\pi$
$571$	−2528.85	−0.185340	−0.0926699	+	0.995697i	$0.529540\pi$
$572$	381.116	0.0278589
$573$	5859.94	0.427230
$574$	14271.2	1.03775
$575$	12668.5	0.918802
$576$	576.000	0.0416667
$577$	−2299.80	−0.165931	−0.0829653	−	0.996552i	$-0.526439\pi$
$577$	−2299.80	−0.165931	−0.0829653	+	0.996552i	$0.526439\pi$
$578$	8617.18	0.620116
$579$	9800.98	0.703480
$580$	−1138.62	−0.0815150
$581$	−5274.22	−0.376612
$582$	559.518	0.0398501
$583$	−1843.11	−0.130933
$584$	−4614.42	−0.326963
$585$	−252.345	−0.0178345
$586$	−5238.37	−0.369275
$587$	−10687.0	−0.751450	−0.375725	−	0.926731i	$-0.622606\pi$
$587$	−10687.0	−0.751450	−0.375725	+	0.926731i	$0.622606\pi$
$588$	380.841	0.0267103
$589$	0	0
$590$	1482.90	0.103474
$591$	9943.93	0.692113
$592$	−844.258	−0.0586129
$593$	−13089.9	−0.906468	−0.453234	−	0.891391i	$-0.649730\pi$
$593$	−13089.9	−0.906468	−0.453234	+	0.891391i	$0.649730\pi$
$594$	−558.138	−0.0385533
$595$	1447.54	0.0997371
$596$	−2341.95	−0.160956
$597$	−3512.03	−0.240767
$598$	2017.84	0.137986
$599$	2399.07	0.163645	0.0818226	−	0.996647i	$-0.473926\pi$
$599$	2399.07	0.163645	0.0818226	+	0.996647i	$0.473926\pi$
$600$	2777.97	0.189017
$601$	16276.7	1.10473	0.552363	−	0.833604i	$-0.313726\pi$
$601$	16276.7	1.10473	0.552363	+	0.833604i	$0.313726\pi$
$602$	5285.62	0.357850
$603$	4407.72	0.297672
$604$	14444.1	0.973048
$605$	3723.44	0.250214
$606$	−345.818	−0.0231813
$607$	−15629.4	−1.04511	−0.522553	−	0.852607i	$-0.675020\pi$
$607$	−15629.4	−1.04511	−0.522553	+	0.852607i	$0.675020\pi$
$608$	0	0
$609$	5435.03	0.361639
$610$	−3479.03	−0.230921
$611$	−3891.17	−0.257643
$612$	−885.052	−0.0584577
$613$	−13096.9	−0.862933	−0.431467	−	0.902129i	$-0.642004\pi$
$613$	−13096.9	−0.862933	−0.431467	+	0.902129i	$0.642004\pi$
$614$	−14865.3	−0.977059
$615$	3363.49	0.220535
$616$	−1600.67	−0.104696
$617$	9144.24	0.596650	0.298325	−	0.954464i	$-0.403572\pi$
$617$	9144.24	0.596650	0.298325	+	0.954464i	$0.403572\pi$
$618$	−3269.64	−0.212823
$619$	−17372.2	−1.12803	−0.564014	−	0.825765i	$-0.690744\pi$
$619$	−17372.2	−1.12803	−0.564014	+	0.825765i	$0.690744\pi$
$620$	−893.200	−0.0578577
$621$	−2955.10	−0.190956
$622$	−17261.3	−1.11272
$623$	−3499.51	−0.225048
$624$	442.477	0.0283867
$625$	12241.3	0.783445
$626$	8010.23	0.511427
$627$	0	0
$628$	−10366.9	−0.658730
$629$	1297.24	0.0822330
$630$	1059.83	0.0670235
$631$	−10889.7	−0.687022	−0.343511	−	0.939149i	$-0.611616\pi$
$631$	−10889.7	−0.687022	−0.343511	+	0.939149i	$0.611616\pi$
$632$	5952.36	0.374639
$633$	−13036.4	−0.818561
$634$	−14129.4	−0.885095
$635$	2965.87	0.185350
$636$	−2139.86	−0.133413
$637$	292.559	0.0181972
$638$	−1934.62	−0.120050
$639$	−4534.67	−0.280734
$640$	389.325	0.0240460
$641$	20997.7	1.29385	0.646926	−	0.762553i	$-0.276054\pi$
$641$	20997.7	1.29385	0.646926	+	0.762553i	$0.276054\pi$
$642$	8926.11	0.548731
$643$	3723.67	0.228378	0.114189	−	0.993459i	$-0.463573\pi$
$643$	3723.67	0.228378	0.114189	+	0.993459i	$0.463573\pi$
$644$	−8474.83	−0.518564
$645$	1245.74	0.0760478
$646$	0	0
$647$	−721.297	−0.0438286	−0.0219143	−	0.999760i	$-0.506976\pi$
$647$	−721.297	−0.0438286	−0.0219143	+	0.999760i	$0.506976\pi$
$648$	−648.000	−0.0392837
$649$	2519.57	0.152391
$650$	2134.01	0.128773
$651$	4263.54	0.256684
$652$	8435.07	0.506661
$653$	19452.4	1.16574	0.582872	−	0.812564i	$-0.301929\pi$
$653$	19452.4	1.16574	0.582872	+	0.812564i	$0.301929\pi$
$654$	−8377.60	−0.500903
$655$	1660.42	0.0990504
$656$	−5897.75	−0.351019
$657$	5191.23	0.308263
$658$	16342.7	0.968245
$659$	−15476.7	−0.914848	−0.457424	−	0.889249i	$-0.651228\pi$
$659$	−15476.7	−0.914848	−0.457424	+	0.889249i	$0.651228\pi$
$660$	−377.252	−0.0222492
$661$	−15317.7	−0.901344	−0.450672	−	0.892690i	$-0.648816\pi$
$661$	−15317.7	−0.901344	−0.450672	+	0.892690i	$0.648816\pi$
$662$	−11551.4	−0.678187
$663$	−679.888	−0.0398261
$664$	2179.64	0.127389
$665$	0	0
$666$	949.791	0.0552607
$667$	−10242.9	−0.594615
$668$	−3826.73	−0.221647
$669$	−2824.44	−0.163227
$670$	2979.23	0.171787
$671$	−5911.17	−0.340087
$672$	−1858.38	−0.106679
$673$	−6598.36	−0.377932	−0.188966	−	0.981984i	$-0.560514\pi$
$673$	−6598.36	−0.377932	−0.188966	+	0.981984i	$0.560514\pi$
$674$	4703.01	0.268773
$675$	−3125.21	−0.178207
$676$	−8448.09	−0.480661
$677$	12164.7	0.690586	0.345293	−	0.938495i	$-0.387780\pi$
$677$	12164.7	0.690586	0.345293	+	0.938495i	$0.387780\pi$
$678$	−8151.28	−0.461723
$679$	−1805.20	−0.102028
$680$	−598.217	−0.0337361
$681$	−2619.70	−0.147411
$682$	−1517.62	−0.0852094
$683$	−31792.2	−1.78110	−0.890552	−	0.454882i	$-0.849682\pi$
$683$	−31792.2	−1.78110	−0.890552	+	0.454882i	$0.849682\pi$
$684$	0	0
$685$	9111.80	0.508239
$686$	12050.9	0.670710
$687$	−10529.1	−0.584733
$688$	−2184.35	−0.121043
$689$	−1643.82	−0.0908917
$690$	−1997.38	−0.110202
$691$	−6318.92	−0.347877	−0.173938	−	0.984757i	$-0.555649\pi$
$691$	−6318.92	−0.347877	−0.173938	+	0.984757i	$0.555649\pi$
$692$	−4790.26	−0.263148
$693$	1800.75	0.0987082
$694$	19498.6	1.06651
$695$	−4424.10	−0.241461
$696$	−2246.10	−0.122325
$697$	9062.19	0.492475
$698$	−12889.1	−0.698941
$699$	3798.48	0.205539
$700$	−8962.71	−0.483941
$701$	4622.11	0.249037	0.124518	−	0.992217i	$-0.460261\pi$
$701$	4622.11	0.249037	0.124518	+	0.992217i	$0.460261\pi$
$702$	−497.787	−0.0267632
$703$	0	0
$704$	661.496	0.0354135
$705$	3851.72	0.205765
$706$	−18961.9	−1.01082
$707$	1115.73	0.0593513
$708$	2925.23	0.155278
$709$	36097.0	1.91206	0.956031	−	0.293265i	$-0.0947420\pi$
$709$	36097.0	1.91206	0.956031	+	0.293265i	$0.0947420\pi$
$710$	−3065.04	−0.162012
$711$	−6696.40	−0.353213
$712$	1446.22	0.0761227
$713$	−8035.15	−0.422046
$714$	2855.49	0.149670
$715$	−289.801	−0.0151580
$716$	3436.69	0.179379
$717$	−10995.8	−0.572729
$718$	16232.5	0.843720
$719$	7833.00	0.406289	0.203144	−	0.979149i	$-0.434884\pi$
$719$	7833.00	0.406289	0.203144	+	0.979149i	$0.434884\pi$
$720$	−437.991	−0.0226708
$721$	10549.0	0.544891
$722$	0	0
$723$	13619.5	0.700575
$724$	9242.59	0.474445
$725$	−10832.6	−0.554915
$726$	7345.02	0.375481
$727$	−22964.0	−1.17151	−0.585756	−	0.810488i	$-0.699202\pi$
$727$	−22964.0	−1.17151	−0.585756	+	0.810488i	$0.699202\pi$
$728$	−1427.59	−0.0726785
$729$	729.000	0.0370370
$730$	3508.81	0.177900
$731$	3356.37	0.169822
$732$	−6862.89	−0.346530
$733$	−8647.16	−0.435730	−0.217865	−	0.975979i	$-0.569909\pi$
$733$	−8647.16	−0.435730	−0.217865	+	0.975979i	$0.569909\pi$
$734$	−13684.8	−0.688166
$735$	−289.592	−0.0145330
$736$	3502.34	0.175405
$737$	5061.96	0.252998
$738$	6634.97	0.330944
$739$	31449.3	1.56547	0.782733	−	0.622357i	$-0.213825\pi$
$739$	31449.3	1.56547	0.782733	+	0.622357i	$0.213825\pi$
$740$	641.975	0.0318912
$741$	0	0
$742$	6903.93	0.341579
$743$	26640.7	1.31541	0.657707	−	0.753274i	$-0.271527\pi$
$743$	26640.7	1.31541	0.657707	+	0.753274i	$0.271527\pi$
$744$	−1761.97	−0.0868237
$745$	1780.82	0.0875759
$746$	12589.4	0.617869
$747$	−2452.10	−0.120104
$748$	−1016.42	−0.0496846
$749$	−28798.8	−1.40492
$750$	−4393.57	−0.213907
$751$	31000.9	1.50631	0.753155	−	0.657843i	$-0.228531\pi$
$751$	31000.9	1.50631	0.753155	+	0.657843i	$0.228531\pi$
$752$	−6753.84	−0.327510
$753$	−19101.6	−0.924435
$754$	−1725.43	−0.0833374
$755$	−10983.3	−0.529434
$756$	2090.68	0.100578
$757$	−34.2137	−0.00164269	−0.000821346	−	1.00000i	$-0.500261\pi$
$757$	−34.2137	−0.00164269	−0.000821346		1.00000i	$0.500261\pi$
$758$	4062.98	0.194689
$759$	−3393.72	−0.162298
$760$	0	0
$761$	37299.0	1.77673	0.888363	−	0.459142i	$-0.151843\pi$
$761$	37299.0	1.77673	0.888363	+	0.459142i	$0.151843\pi$
$762$	5850.62	0.278144
$763$	27029.1	1.28246
$764$	7813.26	0.369992
$765$	672.994	0.0318067
$766$	9311.49	0.439214
$767$	2247.13	0.105788
$768$	768.000	0.0360844
$769$	18459.7	0.865635	0.432818	−	0.901482i	$-0.357519\pi$
$769$	18459.7	0.865635	0.432818	+	0.901482i	$0.357519\pi$
$770$	1217.15	0.0569649
$771$	10055.1	0.469683
$772$	13068.0	0.609231
$773$	−25657.5	−1.19383	−0.596917	−	0.802303i	$-0.703608\pi$
$773$	−25657.5	−1.19383	−0.596917	+	0.802303i	$0.703608\pi$
$774$	2457.40	0.114121
$775$	−8497.72	−0.393867
$776$	746.024	0.0345112
$777$	−3064.36	−0.141484
$778$	9927.69	0.457487
$779$	0	0
$780$	−336.460	−0.0154451
$781$	−5207.76	−0.238602
$782$	−5381.51	−0.246090
$783$	2526.86	0.115329
$784$	507.789	0.0231318
$785$	7882.96	0.358414
$786$	3275.42	0.148639
$787$	6941.29	0.314397	0.157198	−	0.987567i	$-0.449754\pi$
$787$	6941.29	0.314397	0.157198	+	0.987567i	$0.449754\pi$
$788$	13258.6	0.599387
$789$	−9139.74	−0.412400
$790$	−4526.17	−0.203841
$791$	26298.9	1.18215
$792$	−744.184	−0.0333881
$793$	−5272.01	−0.236084
$794$	−20630.4	−0.922096
$795$	1627.15	0.0725899
$796$	−4682.70	−0.208510
$797$	−9719.58	−0.431976	−0.215988	−	0.976396i	$-0.569297\pi$
$797$	−9719.58	−0.431976	−0.215988	+	0.976396i	$0.569297\pi$
$798$	0	0
$799$	10377.6	0.459491
$800$	3703.96	0.163693
$801$	−1627.00	−0.0717691
$802$	31133.9	1.37079
$803$	5961.77	0.262000
$804$	5876.95	0.257791
$805$	6444.26	0.282150
$806$	−1353.52	−0.0591512
$807$	−15539.3	−0.677833
$808$	−461.090	−0.0200756
$809$	9857.21	0.428382	0.214191	−	0.976792i	$-0.431289\pi$
$809$	9857.21	0.428382	0.214191	+	0.976792i	$0.431289\pi$
$810$	492.740	0.0213742
$811$	28971.8	1.25442	0.627211	−	0.778849i	$-0.284196\pi$
$811$	28971.8	1.25442	0.627211	+	0.778849i	$0.284196\pi$
$812$	7246.70	0.313189
$813$	−26155.2	−1.12829
$814$	1090.77	0.0469674
$815$	−6414.03	−0.275673
$816$	−1180.07	−0.0506259
$817$	0	0
$818$	12041.0	0.514675
$819$	1606.04	0.0685219
$820$	4484.66	0.190989
$821$	−16800.8	−0.714191	−0.357095	−	0.934068i	$-0.616233\pi$
$821$	−16800.8	−0.714191	−0.357095	+	0.934068i	$0.616233\pi$
$822$	17974.3	0.762685
$823$	39357.2	1.66696	0.833478	−	0.552553i	$-0.186346\pi$
$823$	39357.2	1.66696	0.833478	+	0.552553i	$0.186346\pi$
$824$	−4359.52	−0.184310
$825$	−3589.09	−0.151462
$826$	−9437.82	−0.397559
$827$	9253.49	0.389088	0.194544	−	0.980894i	$-0.437677\pi$
$827$	9253.49	0.389088	0.194544	+	0.980894i	$0.437677\pi$
$828$	−3940.13	−0.165373
$829$	35558.3	1.48973	0.744867	−	0.667213i	$-0.232513\pi$
$829$	35558.3	1.48973	0.744867	+	0.667213i	$0.232513\pi$
$830$	−1657.40	−0.0693123
$831$	16722.8	0.698084
$832$	589.970	0.0245836
$833$	−780.242	−0.0324535
$834$	−8727.18	−0.362347
$835$	2909.85	0.120598
$836$	0	0
$837$	1982.21	0.0818582
$838$	5557.14	0.229079
$839$	42225.4	1.73753	0.868763	−	0.495229i	$-0.164916\pi$
$839$	42225.4	1.73753	0.868763	+	0.495229i	$0.164916\pi$
$840$	1413.11	0.0580441
$841$	−15630.4	−0.640879
$842$	−12524.3	−0.512609
$843$	−5687.21	−0.232358
$844$	−17381.8	−0.708895
$845$	6423.93	0.261527
$846$	7598.07	0.308779
$847$	−23697.6	−0.961346
$848$	−2853.14	−0.115539
$849$	−1552.62	−0.0627631
$850$	−5691.31	−0.229659
$851$	5775.15	0.232632
$852$	−6046.23	−0.243122
$853$	33206.0	1.33289	0.666443	−	0.745556i	$-0.267816\pi$
$853$	33206.0	1.33289	0.666443	+	0.745556i	$0.267816\pi$
$854$	22142.1	0.887222
$855$	0	0
$856$	11901.5	0.475215
$857$	4015.16	0.160041	0.0800205	−	0.996793i	$-0.474501\pi$
$857$	4015.16	0.160041	0.0800205	+	0.996793i	$0.474501\pi$
$858$	−571.674	−0.0227467
$859$	34644.8	1.37609	0.688047	−	0.725666i	$-0.258468\pi$
$859$	34644.8	1.37609	0.688047	+	0.725666i	$0.258468\pi$
$860$	1660.98	0.0658594
$861$	−21406.8	−0.847318
$862$	5015.19	0.198165
$863$	−17205.8	−0.678671	−0.339335	−	0.940665i	$-0.610202\pi$
$863$	−17205.8	−0.678671	−0.339335	+	0.940665i	$0.610202\pi$
$864$	−864.000	−0.0340207
$865$	3642.51	0.143178
$866$	24518.4	0.962088
$867$	−12925.8	−0.506323
$868$	5684.73	0.222295
$869$	−7690.36	−0.300204
$870$	1707.93	0.0665567
$871$	4514.62	0.175628
$872$	−11170.1	−0.433794
$873$	−839.277	−0.0325375
$874$	0	0
$875$	14175.2	0.547668
$876$	6921.64	0.266964
$877$	40344.8	1.55342	0.776710	−	0.629859i	$-0.216887\pi$
$877$	40344.8	1.55342	0.776710	+	0.629859i	$0.216887\pi$
$878$	18260.1	0.701876
$879$	7857.56	0.301512
$880$	−503.002	−0.0192684
$881$	9145.01	0.349720	0.174860	−	0.984593i	$-0.444053\pi$
$881$	9145.01	0.349720	0.174860	+	0.984593i	$0.444053\pi$
$882$	−571.262	−0.0218088
$883$	30990.9	1.18112	0.590558	−	0.806995i	$-0.298908\pi$
$883$	30990.9	1.18112	0.590558	+	0.806995i	$0.298908\pi$
$884$	−906.518	−0.0344904
$885$	−2224.34	−0.0844865
$886$	18784.6	0.712282
$887$	−52746.4	−1.99668	−0.998338	−	0.0576362i	$-0.981644\pi$
$887$	−52746.4	−1.99668	−0.998338	+	0.0576362i	$0.981644\pi$
$888$	1266.39	0.0478572
$889$	−18876.2	−0.712133
$890$	−1099.71	−0.0414182
$891$	837.206	0.0314786
$892$	−3765.92	−0.141359
$893$	0	0
$894$	3512.92	0.131420
$895$	−2613.26	−0.0975997
$896$	−2477.84	−0.0923871
$897$	−3026.77	−0.112665
$898$	674.357	0.0250597
$899$	6870.74	0.254897
$900$	−4166.95	−0.154332
$901$	4383.99	0.162100
$902$	7619.81	0.281277
$903$	−7928.43	−0.292183
$904$	−10868.4	−0.399864
$905$	−7028.07	−0.258145
$906$	−21666.1	−0.794490
$907$	28386.4	1.03920	0.519601	−	0.854409i	$-0.326081\pi$
$907$	28386.4	1.03920	0.519601	+	0.854409i	$0.326081\pi$
$908$	−3492.94	−0.127662
$909$	518.726	0.0189275
$910$	1085.54	0.0395442
$911$	−5809.42	−0.211278	−0.105639	−	0.994405i	$-0.533689\pi$
$911$	−5809.42	−0.211278	−0.105639	+	0.994405i	$0.533689\pi$
$912$	0	0
$913$	−2816.06	−0.102079
$914$	35402.2	1.28118
$915$	5218.55	0.188546
$916$	−14038.9	−0.506394
$917$	−10567.7	−0.380562
$918$	1327.58	0.0477305
$919$	−37489.2	−1.34565	−0.672827	−	0.739800i	$-0.734920\pi$
$919$	−37489.2	−1.34565	−0.672827	+	0.739800i	$0.734920\pi$
$920$	−2663.18	−0.0954373
$921$	22297.9	0.797765
$922$	−7061.12	−0.252219
$923$	−4644.65	−0.165634
$924$	2401.00	0.0854838
$925$	6107.61	0.217099
$926$	3619.55	0.128451
$927$	4904.46	0.173769
$928$	−2994.80	−0.105936
$929$	22731.2	0.802783	0.401392	−	0.915907i	$-0.368527\pi$
$929$	22731.2	0.802783	0.401392	+	0.915907i	$0.368527\pi$
$930$	1339.80	0.0472406
$931$	0	0
$932$	5064.64	0.178002
$933$	25891.9	0.908534
$934$	6559.43	0.229798
$935$	772.888	0.0270333
$936$	−663.716	−0.0231776
$937$	20439.4	0.712623	0.356311	−	0.934367i	$-0.384034\pi$
$937$	20439.4	0.712623	0.356311	+	0.934367i	$0.384034\pi$
$938$	−18961.1	−0.660024
$939$	−12015.3	−0.417578
$940$	5135.62	0.178197
$941$	−53001.2	−1.83612	−0.918060	−	0.396442i	$-0.870245\pi$
$941$	−53001.2	−1.83612	−0.918060	+	0.396442i	$0.870245\pi$
$942$	15550.3	0.537851
$943$	40343.6	1.39318
$944$	3900.30	0.134475
$945$	−1589.75	−0.0547245
$946$	2822.15	0.0969937
$947$	47807.0	1.64046	0.820231	−	0.572032i	$-0.193845\pi$
$947$	47807.0	1.64046	0.820231	+	0.572032i	$0.193845\pi$
$948$	−8928.53	−0.305892
$949$	5317.13	0.181877
$950$	0	0
$951$	21194.1	0.722677
$952$	3807.32	0.129618
$953$	856.874	0.0291258	0.0145629	−	0.999894i	$-0.495364\pi$
$953$	856.874	0.0291258	0.0145629	+	0.999894i	$0.495364\pi$
$954$	3209.78	0.108931
$955$	−5941.21	−0.201312
$956$	−14661.1	−0.495998
$957$	2901.92	0.0980208
$958$	29419.6	0.992177
$959$	−57991.6	−1.95271
$960$	−583.988	−0.0196335
$961$	−24401.2	−0.819080
$962$	972.826	0.0326041
$963$	−13389.2	−0.448037
$964$	18159.4	0.606716
$965$	−9936.89	−0.331482
$966$	12712.2	0.423405
$967$	30746.8	1.02249	0.511246	−	0.859434i	$-0.329184\pi$
$967$	30746.8	1.02249	0.511246	+	0.859434i	$0.329184\pi$
$968$	9793.36	0.325176
$969$	0	0
$970$	−567.277	−0.0187775
$971$	20123.8	0.665091	0.332545	−	0.943087i	$-0.392093\pi$
$971$	20123.8	0.665091	0.332545	+	0.943087i	$0.392093\pi$
$972$	972.000	0.0320750
$973$	28157.0	0.927719
$974$	15600.2	0.513206
$975$	−3201.01	−0.105143
$976$	−9150.53	−0.300104
$977$	−11372.5	−0.372405	−0.186203	−	0.982511i	$-0.559618\pi$
$977$	−11372.5	−0.372405	−0.186203	+	0.982511i	$0.559618\pi$
$978$	−12652.6	−0.413687
$979$	−1868.49	−0.0609983
$980$	−386.123	−0.0125860
$981$	12566.4	0.408985
$982$	−31406.1	−1.02058
$983$	48284.8	1.56668	0.783339	−	0.621595i	$-0.213515\pi$
$983$	48284.8	1.56668	0.783339	+	0.621595i	$0.213515\pi$
$984$	8846.63	0.286606
$985$	−10081.8	−0.326126
$986$	4601.65	0.148627
$987$	−24514.1	−0.790568
$988$	0	0
$989$	14942.1	0.480415
$990$	565.877	0.0181664
$991$	14879.8	0.476965	0.238482	−	0.971147i	$-0.423350\pi$
$991$	14879.8	0.476965	0.238482	+	0.971147i	$0.423350\pi$
$992$	−2349.29	−0.0751915
$993$	17327.2	0.553737
$994$	19507.3	0.622468
$995$	3560.73	0.113450
$996$	−3269.46	−0.104013
$997$	6580.67	0.209039	0.104519	−	0.994523i	$-0.466670\pi$
$997$	6580.67	0.209039	0.104519	+	0.994523i	$0.466670\pi$
$998$	18289.5	0.580104
$999$	−1424.69	−0.0451202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.bh.1.5	✓	8
19.18	odd	2		2166.4.a.bi.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.bh.1.5	✓	8	1.1	even	1	trivial
2166.4.a.bi.1.5	yes	8	19.18	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -3.04160 q^{5} -6.00000 q^{6} +19.3581 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -3.04160 q^{5} -6.00000 q^{6} +19.3581 q^{7} -8.00000 q^{8} +9.00000 q^{9} +6.08320 q^{10} +10.3359 q^{11} +12.0000 q^{12} +9.21828 q^{13} -38.7162 q^{14} -9.12481 q^{15} +16.0000 q^{16} -24.5848 q^{17} -18.0000 q^{18} -12.1664 q^{20} +58.0744 q^{21} -20.6718 q^{22} -109.448 q^{23} -24.0000 q^{24} -115.749 q^{25} -18.4366 q^{26} +27.0000 q^{27} +77.4325 q^{28} +93.5874 q^{29} +18.2496 q^{30} +73.4153 q^{31} -32.0000 q^{32} +31.0076 q^{33} +49.1696 q^{34} -58.8797 q^{35} +36.0000 q^{36} -52.7662 q^{37} +27.6548 q^{39} +24.3328 q^{40} -368.610 q^{41} -116.149 q^{42} -136.522 q^{43} +41.3435 q^{44} -27.3744 q^{45} +218.896 q^{46} -422.115 q^{47} +48.0000 q^{48} +31.7368 q^{49} +231.497 q^{50} -73.7544 q^{51} +36.8731 q^{52} -178.321 q^{53} -54.0000 q^{54} -31.4376 q^{55} -154.865 q^{56} -187.175 q^{58} +243.769 q^{59} -36.4992 q^{60} -571.908 q^{61} -146.831 q^{62} +174.223 q^{63} +64.0000 q^{64} -28.0383 q^{65} -62.0153 q^{66} +489.746 q^{67} -98.3392 q^{68} -328.344 q^{69} +117.759 q^{70} -503.852 q^{71} -72.0000 q^{72} +576.803 q^{73} +105.532 q^{74} -347.246 q^{75} +200.083 q^{77} -55.3097 q^{78} -744.045 q^{79} -48.6656 q^{80} +81.0000 q^{81} +737.219 q^{82} -272.455 q^{83} +232.297 q^{84} +74.7771 q^{85} +273.044 q^{86} +280.762 q^{87} -82.6871 q^{88} -180.777 q^{89} +54.7488 q^{90} +178.449 q^{91} -437.792 q^{92} +220.246 q^{93} +844.230 q^{94} -96.0000 q^{96} -93.2530 q^{97} -63.4736 q^{98} +93.0229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{2} + 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q - 16 * q^2 + 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} + 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} - 64 q^{8} + 72 q^{9} + 60 q^{10} - 6 q^{11} + 96 q^{12} - 48 q^{13} + 68 q^{14} - 90 q^{15} + 128 q^{16} - 36 q^{17} - 144 q^{18} - 120 q^{20} - 102 q^{21} + 12 q^{22} - 282 q^{23} - 192 q^{24} + 396 q^{25} + 96 q^{26} + 216 q^{27} - 136 q^{28} + 380 q^{29} + 180 q^{30} + 48 q^{31} - 256 q^{32} - 18 q^{33} + 72 q^{34} + 762 q^{35} + 288 q^{36} + 168 q^{37} - 144 q^{39} + 240 q^{40} - 342 q^{41} + 204 q^{42} - 788 q^{43} - 24 q^{44} - 270 q^{45} + 564 q^{46} - 468 q^{47} + 384 q^{48} + 222 q^{49} - 792 q^{50} - 108 q^{51} - 192 q^{52} - 1682 q^{53} - 432 q^{54} - 46 q^{55} + 272 q^{56} - 760 q^{58} - 292 q^{59} - 360 q^{60} + 522 q^{61} - 96 q^{62} - 306 q^{63} + 512 q^{64} + 1120 q^{65} + 36 q^{66} + 2484 q^{67} - 144 q^{68} - 846 q^{69} - 1524 q^{70} - 1182 q^{71} - 576 q^{72} + 182 q^{73} - 336 q^{74} + 1188 q^{75} - 504 q^{77} + 288 q^{78} + 2232 q^{79} - 480 q^{80} + 648 q^{81} + 684 q^{82} - 750 q^{83} - 408 q^{84} - 2238 q^{85} + 1576 q^{86} + 1140 q^{87} + 48 q^{88} - 1304 q^{89} + 540 q^{90} + 624 q^{91} - 1128 q^{92} + 144 q^{93} + 936 q^{94} - 768 q^{96} + 1248 q^{97} - 444 q^{98} - 54 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 + 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 - 64 * q^8 + 72 * q^9 + 60 * q^10 - 6 * q^11 + 96 * q^12 - 48 * q^13 + 68 * q^14 - 90 * q^15 + 128 * q^16 - 36 * q^17 - 144 * q^18 - 120 * q^20 - 102 * q^21 + 12 * q^22 - 282 * q^23 - 192 * q^24 + 396 * q^25 + 96 * q^26 + 216 * q^27 - 136 * q^28 + 380 * q^29 + 180 * q^30 + 48 * q^31 - 256 * q^32 - 18 * q^33 + 72 * q^34 + 762 * q^35 + 288 * q^36 + 168 * q^37 - 144 * q^39 + 240 * q^40 - 342 * q^41 + 204 * q^42 - 788 * q^43 - 24 * q^44 - 270 * q^45 + 564 * q^46 - 468 * q^47 + 384 * q^48 + 222 * q^49 - 792 * q^50 - 108 * q^51 - 192 * q^52 - 1682 * q^53 - 432 * q^54 - 46 * q^55 + 272 * q^56 - 760 * q^58 - 292 * q^59 - 360 * q^60 + 522 * q^61 - 96 * q^62 - 306 * q^63 + 512 * q^64 + 1120 * q^65 + 36 * q^66 + 2484 * q^67 - 144 * q^68 - 846 * q^69 - 1524 * q^70 - 1182 * q^71 - 576 * q^72 + 182 * q^73 - 336 * q^74 + 1188 * q^75 - 504 * q^77 + 288 * q^78 + 2232 * q^79 - 480 * q^80 + 648 * q^81 + 684 * q^82 - 750 * q^83 - 408 * q^84 - 2238 * q^85 + 1576 * q^86 + 1140 * q^87 + 48 * q^88 - 1304 * q^89 + 540 * q^90 + 624 * q^91 - 1128 * q^92 + 144 * q^93 + 936 * q^94 - 768 * q^96 + 1248 * q^97 - 444 * q^98 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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