LMFDB - Embedded newform 1815.4.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, 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("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$1$
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} - 3 x^{11} - 69 x^{10} + 157 x^{9} + 1812 x^{8} - 2703 x^{7} - 22379 x^{6} + 16453 x^{5} + \cdots + 196416 $ x^12 - 3x^11 - 69x^10 + 157x^9 + 1812x^8 - 2703x^7 - 22379x^6 + 16453x^5 + 129167x^4 - 22672x^3 - 298084x^2 - 19392*x + 196416
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.61005$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.61005 q^{2} +3.00000 q^{3} +23.4727 q^{4} -5.00000 q^{5} -16.8301 q^{6} -29.7389 q^{7} -86.8023 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.61005 q^{2} +3.00000 q^{3} +23.4727 q^{4} -5.00000 q^{5} -16.8301 q^{6} -29.7389 q^{7} -86.8023 q^{8} +9.00000 q^{9} +28.0502 q^{10} +70.4180 q^{12} +44.6753 q^{13} +166.837 q^{14} -15.0000 q^{15} +299.184 q^{16} +30.3586 q^{17} -50.4904 q^{18} -64.6796 q^{19} -117.363 q^{20} -89.2167 q^{21} -69.8631 q^{23} -260.407 q^{24} +25.0000 q^{25} -250.631 q^{26} +27.0000 q^{27} -698.051 q^{28} -34.5269 q^{29} +84.1507 q^{30} +185.869 q^{31} -984.019 q^{32} -170.313 q^{34} +148.695 q^{35} +211.254 q^{36} -268.378 q^{37} +362.856 q^{38} +134.026 q^{39} +434.012 q^{40} -118.173 q^{41} +500.510 q^{42} +449.777 q^{43} -45.0000 q^{45} +391.935 q^{46} +0.808815 q^{47} +897.552 q^{48} +541.403 q^{49} -140.251 q^{50} +91.0757 q^{51} +1048.65 q^{52} +128.987 q^{53} -151.471 q^{54} +2581.41 q^{56} -194.039 q^{57} +193.698 q^{58} -191.111 q^{59} -352.090 q^{60} +531.573 q^{61} -1042.74 q^{62} -267.650 q^{63} +3126.92 q^{64} -223.376 q^{65} -368.599 q^{67} +712.596 q^{68} -209.589 q^{69} -834.184 q^{70} +252.618 q^{71} -781.221 q^{72} -537.247 q^{73} +1505.62 q^{74} +75.0000 q^{75} -1518.20 q^{76} -751.892 q^{78} +1004.65 q^{79} -1495.92 q^{80} +81.0000 q^{81} +662.957 q^{82} -396.478 q^{83} -2094.15 q^{84} -151.793 q^{85} -2523.27 q^{86} -103.581 q^{87} -206.340 q^{89} +252.452 q^{90} -1328.59 q^{91} -1639.87 q^{92} +557.608 q^{93} -4.53749 q^{94} +323.398 q^{95} -2952.06 q^{96} +1224.91 q^{97} -3037.30 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} - 27 q^{6} - 21 q^{7} - 123 q^{8} + 108 q^{9}+O(q^{10}) $ 12 * q - 9 * q^2 + 36 * q^3 + 57 * q^4 - 60 * q^5 - 27 * q^6 - 21 * q^7 - 123 * q^8 + 108 * q^9	\( 12 q - 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} - 27 q^{6} - 21 q^{7} - 123 q^{8} + 108 q^{9} + 45 q^{10} + 171 q^{12} - 78 q^{13} + 262 q^{14} - 180 q^{15} + 225 q^{16} - 313 q^{17} - 81 q^{18} - 51 q^{19} - 285 q^{20} - 63 q^{21} - 34 q^{23} - 369 q^{24} + 300 q^{25} + 28 q^{26} + 324 q^{27} - 376 q^{28} - 31 q^{29} + 135 q^{30} + 655 q^{31} - 1578 q^{32} - 10 q^{34} + 105 q^{35} + 513 q^{36} + 84 q^{37} - 1076 q^{38} - 234 q^{39} + 615 q^{40} - 1463 q^{41} + 786 q^{42} + 111 q^{43} - 540 q^{45} + q^{46} + 278 q^{47} + 675 q^{48} - 325 q^{49} - 225 q^{50} - 939 q^{51} - 1957 q^{52} + 517 q^{53} - 243 q^{54} + 1543 q^{56} - 153 q^{57} + 442 q^{58} - 308 q^{59} - 855 q^{60} - 604 q^{61} - 1773 q^{62} - 189 q^{63} + 4323 q^{64} + 390 q^{65} - 357 q^{67} - 2192 q^{68} - 102 q^{69} - 1310 q^{70} - 620 q^{71} - 1107 q^{72} - 1892 q^{73} - 581 q^{74} + 900 q^{75} - 378 q^{76} + 84 q^{78} - 415 q^{79} - 1125 q^{80} + 972 q^{81} - 2802 q^{82} - 3158 q^{83} - 1128 q^{84} + 1565 q^{85} + 747 q^{86} - 93 q^{87} + 1563 q^{89} + 405 q^{90} + 1434 q^{91} - 3466 q^{92} + 1965 q^{93} + 3 q^{94} + 255 q^{95} - 4734 q^{96} + 714 q^{97} - 6586 q^{98}+O(q^{100}) \) 12 * q - 9 * q^2 + 36 * q^3 + 57 * q^4 - 60 * q^5 - 27 * q^6 - 21 * q^7 - 123 * q^8 + 108 * q^9 + 45 * q^10 + 171 * q^12 - 78 * q^13 + 262 * q^14 - 180 * q^15 + 225 * q^16 - 313 * q^17 - 81 * q^18 - 51 * q^19 - 285 * q^20 - 63 * q^21 - 34 * q^23 - 369 * q^24 + 300 * q^25 + 28 * q^26 + 324 * q^27 - 376 * q^28 - 31 * q^29 + 135 * q^30 + 655 * q^31 - 1578 * q^32 - 10 * q^34 + 105 * q^35 + 513 * q^36 + 84 * q^37 - 1076 * q^38 - 234 * q^39 + 615 * q^40 - 1463 * q^41 + 786 * q^42 + 111 * q^43 - 540 * q^45 + q^46 + 278 * q^47 + 675 * q^48 - 325 * q^49 - 225 * q^50 - 939 * q^51 - 1957 * q^52 + 517 * q^53 - 243 * q^54 + 1543 * q^56 - 153 * q^57 + 442 * q^58 - 308 * q^59 - 855 * q^60 - 604 * q^61 - 1773 * q^62 - 189 * q^63 + 4323 * q^64 + 390 * q^65 - 357 * q^67 - 2192 * q^68 - 102 * q^69 - 1310 * q^70 - 620 * q^71 - 1107 * q^72 - 1892 * q^73 - 581 * q^74 + 900 * q^75 - 378 * q^76 + 84 * q^78 - 415 * q^79 - 1125 * q^80 + 972 * q^81 - 2802 * q^82 - 3158 * q^83 - 1128 * q^84 + 1565 * q^85 + 747 * q^86 - 93 * q^87 + 1563 * q^89 + 405 * q^90 + 1434 * q^91 - 3466 * q^92 + 1965 * q^93 + 3 * q^94 + 255 * q^95 - 4734 * q^96 + 714 * q^97 - 6586 * q^98

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 $

$2$	−5.61005	−1.98345	−0.991726	−	0.128373i	$-0.959024\pi$
$2$	−5.61005	−1.98345	−0.991726	+	0.128373i	$0.959024\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	23.4727	2.93408
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−16.8301	−1.14515
$7$	−29.7389	−1.60575	−0.802875	−	0.596147i	$-0.796697\pi$
$7$	−29.7389	−1.60575	−0.802875	+	0.596147i	$0.796697\pi$
$8$	−86.8023	−3.83616
$9$	9.00000	0.333333
$10$	28.0502	0.887027
$11$	0	0
$11$	0	0
$12$	70.4180	1.69399
$13$	44.6753	0.953131	0.476565	−	0.879139i	$-0.341882\pi$
$13$	44.6753	0.953131	0.476565	+	0.879139i	$0.341882\pi$
$14$	166.837	3.18493
$15$	−15.0000	−0.258199
$16$	299.184	4.67475
$17$	30.3586	0.433120	0.216560	−	0.976269i	$-0.430516\pi$
$17$	30.3586	0.433120	0.216560	+	0.976269i	$0.430516\pi$
$18$	−50.4904	−0.661151
$19$	−64.6796	−0.780974	−0.390487	−	0.920608i	$-0.627693\pi$
$19$	−64.6796	−0.780974	−0.390487	+	0.920608i	$0.627693\pi$
$20$	−117.363	−1.31216
$21$	−89.2167	−0.927080
$22$	0	0
$23$	−69.8631	−0.633368	−0.316684	−	0.948531i	$-0.602569\pi$
$23$	−69.8631	−0.633368	−0.316684	+	0.948531i	$0.602569\pi$
$24$	−260.407	−2.21481
$25$	25.0000	0.200000
$26$	−250.631	−1.89049
$27$	27.0000	0.192450
$28$	−698.051	−4.71140
$29$	−34.5269	−0.221086	−0.110543	−	0.993871i	$-0.535259\pi$
$29$	−34.5269	−0.221086	−0.110543	+	0.993871i	$0.535259\pi$
$30$	84.1507	0.512125
$31$	185.869	1.07687	0.538437	−	0.842666i	$-0.319015\pi$
$31$	185.869	1.07687	0.538437	+	0.842666i	$0.319015\pi$
$32$	−984.019	−5.43599
$33$	0	0
$34$	−170.313	−0.859072
$35$	148.695	0.718113
$36$	211.254	0.978027
$37$	−268.378	−1.19246	−0.596232	−	0.802812i	$-0.703336\pi$
$37$	−268.378	−1.19246	−0.596232	+	0.802812i	$0.703336\pi$
$38$	362.856	1.54903
$39$	134.026	0.550290
$40$	434.012	1.71558
$41$	−118.173	−0.450135	−0.225068	−	0.974343i	$-0.572260\pi$
$41$	−118.173	−0.450135	−0.225068	+	0.974343i	$0.572260\pi$
$42$	500.510	1.83882
$43$	449.777	1.59513	0.797563	−	0.603236i	$-0.206122\pi$
$43$	449.777	1.59513	0.797563	+	0.603236i	$0.206122\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	391.935	1.25625
$47$	0.808815	0.00251017	0.00125508	−	0.999999i	$-0.499600\pi$
$47$	0.808815	0.00251017	0.00125508	+	0.999999i	$0.499600\pi$
$48$	897.552	2.69897
$49$	541.403	1.57843
$50$	−140.251	−0.396690
$51$	91.0757	0.250062
$52$	1048.65	2.79656
$53$	128.987	0.334298	0.167149	−	0.985932i	$-0.446544\pi$
$53$	128.987	0.334298	0.167149	+	0.985932i	$0.446544\pi$
$54$	−151.471	−0.381715
$55$	0	0
$56$	2581.41	6.15991
$57$	−194.039	−0.450896
$58$	193.698	0.438513
$59$	−191.111	−0.421705	−0.210852	−	0.977518i	$-0.567624\pi$
$59$	−191.111	−0.421705	−0.210852	+	0.977518i	$0.567624\pi$
$60$	−352.090	−0.757577
$61$	531.573	1.11575	0.557876	−	0.829924i	$-0.311616\pi$
$61$	531.573	1.11575	0.557876	+	0.829924i	$0.311616\pi$
$62$	−1042.74	−2.13593
$63$	−267.650	−0.535250
$64$	3126.92	6.10727
$65$	−223.376	−0.426253
$66$	0	0
$67$	−368.599	−0.672112	−0.336056	−	0.941842i	$-0.609093\pi$
$67$	−368.599	−0.672112	−0.336056	+	0.941842i	$0.609093\pi$
$68$	712.596	1.27081
$69$	−209.589	−0.365675
$70$	−834.184	−1.42434
$71$	252.618	0.422256	0.211128	−	0.977458i	$-0.432286\pi$
$71$	252.618	0.422256	0.211128	+	0.977458i	$0.432286\pi$
$72$	−781.221	−1.27872
$73$	−537.247	−0.861370	−0.430685	−	0.902502i	$-0.641728\pi$
$73$	−537.247	−0.861370	−0.430685	+	0.902502i	$0.641728\pi$
$74$	1505.62	2.36519
$75$	75.0000	0.115470
$76$	−1518.20	−2.29144
$77$	0	0
$78$	−751.892	−1.09147
$79$	1004.65	1.43079	0.715395	−	0.698720i	$-0.246247\pi$
$79$	1004.65	1.43079	0.715395	+	0.698720i	$0.246247\pi$
$80$	−1495.92	−2.09061
$81$	81.0000	0.111111
$82$	662.957	0.892822
$83$	−396.478	−0.524326	−0.262163	−	0.965024i	$-0.584436\pi$
$83$	−396.478	−0.524326	−0.262163	+	0.965024i	$0.584436\pi$
$84$	−2094.15	−2.72013
$85$	−151.793	−0.193697
$86$	−2523.27	−3.16386
$87$	−103.581	−0.127644
$88$	0	0
$89$	−206.340	−0.245753	−0.122877	−	0.992422i	$-0.539212\pi$
$89$	−206.340	−0.245753	−0.122877	+	0.992422i	$0.539212\pi$
$90$	252.452	0.295676
$91$	−1328.59	−1.53049
$92$	−1639.87	−1.85835
$93$	557.608	0.621733
$94$	−4.53749	−0.00497879
$95$	323.398	0.349262
$96$	−2952.06	−3.13847
$97$	1224.91	1.28217	0.641086	−	0.767469i	$-0.278484\pi$
$97$	1224.91	1.28217	0.641086	+	0.767469i	$0.278484\pi$
$98$	−3037.30	−3.13075
$99$	0	0
$100$	586.816	0.586816
$101$	455.409	0.448662	0.224331	−	0.974513i	$-0.427980\pi$
$101$	455.409	0.448662	0.224331	+	0.974513i	$0.427980\pi$
$102$	−510.939	−0.495985
$103$	−24.4597	−0.0233989	−0.0116994	−	0.999932i	$-0.503724\pi$
$103$	−24.4597	−0.0233989	−0.0116994	+	0.999932i	$0.503724\pi$
$104$	−3877.92	−3.65636
$105$	446.084	0.414603
$106$	−723.626	−0.663064
$107$	−1048.63	−0.947426	−0.473713	−	0.880679i	$-0.657087\pi$
$107$	−1048.63	−0.947426	−0.473713	+	0.880679i	$0.657087\pi$
$108$	633.762	0.564664
$109$	1014.75	0.891703	0.445852	−	0.895107i	$-0.352901\pi$
$109$	1014.75	0.891703	0.445852	+	0.895107i	$0.352901\pi$
$110$	0	0
$111$	−805.135	−0.688469
$112$	−8897.41	−7.50648
$113$	770.811	0.641697	0.320849	−	0.947130i	$-0.396032\pi$
$113$	770.811	0.641697	0.320849	+	0.947130i	$0.396032\pi$
$114$	1088.57	0.894330
$115$	349.315	0.283251
$116$	−810.439	−0.648684
$117$	402.078	0.317710
$118$	1072.14	0.836431
$119$	−902.831	−0.695482
$120$	1302.03	0.990491
$121$	0	0
$122$	−2982.15	−2.21304
$123$	−354.520	−0.259886
$124$	4362.84	3.15964
$125$	−125.000	−0.0894427
$126$	1501.53	1.06164
$127$	−270.985	−0.189339	−0.0946695	−	0.995509i	$-0.530179\pi$
$127$	−270.985	−0.189339	−0.0946695	+	0.995509i	$0.530179\pi$
$128$	−9670.03	−6.67748
$129$	1349.33	0.920946
$130$	1253.15	0.845452
$131$	283.925	0.189364	0.0946820	−	0.995508i	$-0.469817\pi$
$131$	283.925	0.189364	0.0946820	+	0.995508i	$0.469817\pi$
$132$	0	0
$133$	1923.50	1.25405
$134$	2067.86	1.33310
$135$	−135.000	−0.0860663
$136$	−2635.19	−1.66151
$137$	−2292.48	−1.42964	−0.714818	−	0.699311i	$-0.753490\pi$
$137$	−2292.48	−1.42964	−0.714818	+	0.699311i	$0.753490\pi$
$138$	1175.81	0.725299
$139$	1850.33	1.12909	0.564544	−	0.825403i	$-0.309052\pi$
$139$	1850.33	1.12909	0.564544	+	0.825403i	$0.309052\pi$
$140$	3490.25	2.10700
$141$	2.42644	0.00144924
$142$	−1417.20	−0.837525
$143$	0	0
$144$	2692.66	1.55825
$145$	172.635	0.0988726
$146$	3013.98	1.70849
$147$	1624.21	0.911309
$148$	−6299.55	−3.49878
$149$	1408.44	0.774390	0.387195	−	0.921998i	$-0.373444\pi$
$149$	1408.44	0.774390	0.387195	+	0.921998i	$0.373444\pi$
$150$	−420.754	−0.229029
$151$	673.838	0.363154	0.181577	−	0.983377i	$-0.441880\pi$
$151$	673.838	0.363154	0.181577	+	0.983377i	$0.441880\pi$
$152$	5614.34	2.99594
$153$	273.227	0.144373
$154$	0	0
$155$	−929.346	−0.481593
$156$	3145.94	1.61460
$157$	−2986.80	−1.51830	−0.759148	−	0.650918i	$-0.774384\pi$
$157$	−2986.80	−1.51830	−0.759148	+	0.650918i	$0.774384\pi$
$158$	−5636.16	−2.83790
$159$	386.962	0.193007
$160$	4920.09	2.43105
$161$	2077.65	1.01703
$162$	−454.414	−0.220384
$163$	112.390	0.0540065	0.0270032	−	0.999635i	$-0.491404\pi$
$163$	112.390	0.0540065	0.0270032	+	0.999635i	$0.491404\pi$
$164$	−2773.84	−1.32073
$165$	0	0
$166$	2224.26	1.03998
$167$	−2196.33	−1.01771	−0.508854	−	0.860853i	$-0.669931\pi$
$167$	−2196.33	−1.01771	−0.508854	+	0.860853i	$0.669931\pi$
$168$	7744.22	3.55643
$169$	−201.118	−0.0915420
$170$	851.565	0.384189
$171$	−582.116	−0.260325
$172$	10557.5	4.68023
$173$	−4316.09	−1.89680	−0.948400	−	0.317075i	$-0.897299\pi$
$173$	−4316.09	−1.89680	−0.948400	+	0.317075i	$0.897299\pi$
$174$	581.094	0.253176
$175$	−743.473	−0.321150
$176$	0	0
$177$	−573.334	−0.243471
$178$	1157.58	0.487440
$179$	2996.13	1.25107	0.625534	−	0.780197i	$-0.284881\pi$
$179$	2996.13	1.25107	0.625534	+	0.780197i	$0.284881\pi$
$180$	−1056.27	−0.437387
$181$	−2599.46	−1.06749	−0.533747	−	0.845644i	$-0.679217\pi$
$181$	−2599.46	−1.06749	−0.533747	+	0.845644i	$0.679217\pi$
$182$	7453.48	3.03565
$183$	1594.72	0.644180
$184$	6064.28	2.42970
$185$	1341.89	0.533286
$186$	−3128.21	−1.23318
$187$	0	0
$188$	18.9850	0.00736503
$189$	−802.950	−0.309027
$190$	−1814.28	−0.692745
$191$	937.163	0.355030	0.177515	−	0.984118i	$-0.443194\pi$
$191$	937.163	0.355030	0.177515	+	0.984118i	$0.443194\pi$
$192$	9380.76	3.52603
$193$	4003.34	1.49309	0.746546	−	0.665334i	$-0.231711\pi$
$193$	4003.34	1.49309	0.746546	+	0.665334i	$0.231711\pi$
$194$	−6871.80	−2.54313
$195$	−670.129	−0.246097
$196$	12708.2	4.63125
$197$	3081.73	1.11454	0.557270	−	0.830331i	$-0.311849\pi$
$197$	3081.73	1.11454	0.557270	+	0.830331i	$0.311849\pi$
$198$	0	0
$199$	−1089.85	−0.388228	−0.194114	−	0.980979i	$-0.562183\pi$
$199$	−1089.85	−0.388228	−0.194114	+	0.980979i	$0.562183\pi$
$200$	−2170.06	−0.767231
$201$	−1105.80	−0.388044
$202$	−2554.87	−0.889900
$203$	1026.79	0.355009
$204$	2137.79	0.733701
$205$	590.866	0.201307
$206$	137.220	0.0464105
$207$	−628.768	−0.211123
$208$	13366.1	4.45565
$209$	0	0
$210$	−2502.55	−0.822345
$211$	−2937.09	−0.958281	−0.479141	−	0.877738i	$-0.659052\pi$
$211$	−2937.09	−0.958281	−0.479141	+	0.877738i	$0.659052\pi$
$212$	3027.68	0.980857
$213$	757.853	0.243790
$214$	5882.85	1.87917
$215$	−2248.89	−0.713362
$216$	−2343.66	−0.738269
$217$	−5527.55	−1.72919
$218$	−5692.81	−1.76865
$219$	−1611.74	−0.497312
$220$	0	0
$221$	1356.28	0.412820
$222$	4516.85	1.36555
$223$	4317.75	1.29658	0.648291	−	0.761392i	$-0.275484\pi$
$223$	4317.75	1.29658	0.648291	+	0.761392i	$0.275484\pi$
$224$	29263.6	8.72884
$225$	225.000	0.0666667
$226$	−4324.29	−1.27278
$227$	4639.05	1.35641	0.678204	−	0.734874i	$-0.262759\pi$
$227$	4639.05	1.35641	0.678204	+	0.734874i	$0.262759\pi$
$228$	−4554.60	−1.32296
$229$	−1278.54	−0.368944	−0.184472	−	0.982838i	$-0.559058\pi$
$229$	−1278.54	−0.368944	−0.184472	+	0.982838i	$0.559058\pi$
$230$	−1959.68	−0.561814
$231$	0	0
$232$	2997.02	0.848120
$233$	−6012.27	−1.69046	−0.845229	−	0.534404i	$-0.820536\pi$
$233$	−6012.27	−1.69046	−0.845229	+	0.534404i	$0.820536\pi$
$234$	−2255.68	−0.630163
$235$	−4.04407	−0.00112258
$236$	−4485.89	−1.23732
$237$	3013.96	0.826068
$238$	5064.92	1.37945
$239$	−833.668	−0.225630	−0.112815	−	0.993616i	$-0.535987\pi$
$239$	−833.668	−0.225630	−0.112815	+	0.993616i	$0.535987\pi$
$240$	−4487.76	−1.20702
$241$	−6844.62	−1.82946	−0.914732	−	0.404062i	$-0.867598\pi$
$241$	−6844.62	−1.82946	−0.914732	+	0.404062i	$0.867598\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	12477.4	3.27371
$245$	−2707.01	−0.705897
$246$	1988.87	0.515471
$247$	−2889.58	−0.744371
$248$	−16133.9	−4.13106
$249$	−1189.43	−0.302720
$250$	701.256	0.177405
$251$	−111.829	−0.0281219	−0.0140610	−	0.999901i	$-0.504476\pi$
$251$	−111.829	−0.0281219	−0.0140610	+	0.999901i	$0.504476\pi$
$252$	−6282.46	−1.57047
$253$	0	0
$254$	1520.24	0.375545
$255$	−455.379	−0.111831
$256$	29234.0	7.13720
$257$	−633.760	−0.153824	−0.0769122	−	0.997038i	$-0.524506\pi$
$257$	−633.760	−0.153824	−0.0769122	+	0.997038i	$0.524506\pi$
$258$	−7569.82	−1.82665
$259$	7981.28	1.91480
$260$	−5243.24	−1.25066
$261$	−310.742	−0.0736953
$262$	−1592.84	−0.375594
$263$	−1755.76	−0.411654	−0.205827	−	0.978588i	$-0.565988\pi$
$263$	−1755.76	−0.411654	−0.205827	+	0.978588i	$0.565988\pi$
$264$	0	0
$265$	−644.937	−0.149503
$266$	−10790.9	−2.48735
$267$	−619.021	−0.141886
$268$	−8651.99	−1.97203
$269$	−1284.70	−0.291188	−0.145594	−	0.989344i	$-0.546509\pi$
$269$	−1284.70	−0.291188	−0.145594	+	0.989344i	$0.546509\pi$
$270$	757.357	0.170708
$271$	−7167.60	−1.60665	−0.803323	−	0.595544i	$-0.796936\pi$
$271$	−7167.60	−1.60665	−0.803323	+	0.595544i	$0.796936\pi$
$272$	9082.80	2.02473
$273$	−3985.78	−0.883629
$274$	12860.9	2.83561
$275$	0	0
$276$	−4919.62	−1.07292
$277$	1768.27	0.383557	0.191779	−	0.981438i	$-0.438574\pi$
$277$	1768.27	0.383557	0.191779	+	0.981438i	$0.438574\pi$
$278$	−10380.5	−2.23949
$279$	1672.82	0.358958
$280$	−12907.0	−2.75480
$281$	−5265.11	−1.11776	−0.558879	−	0.829249i	$-0.688768\pi$
$281$	−5265.11	−1.11776	−0.558879	+	0.829249i	$0.688768\pi$
$282$	−13.6125	−0.00287451
$283$	1587.98	0.333554	0.166777	−	0.985995i	$-0.446664\pi$
$283$	1587.98	0.333554	0.166777	+	0.985995i	$0.446664\pi$
$284$	5929.61	1.23893
$285$	970.194	0.201647
$286$	0	0
$287$	3514.34	0.722805
$288$	−8856.17	−1.81200
$289$	−3991.36	−0.812407
$290$	−968.489	−0.196109
$291$	3674.73	0.740262
$292$	−12610.6	−2.52733
$293$	−6357.75	−1.26766	−0.633829	−	0.773473i	$-0.718518\pi$
$293$	−6357.75	−1.26766	−0.633829	+	0.773473i	$0.718518\pi$
$294$	−9111.89	−1.80754
$295$	955.557	0.188592
$296$	23295.9	4.57448
$297$	0	0
$298$	−7901.43	−1.53597
$299$	−3121.15	−0.603682
$300$	1760.45	0.338799
$301$	−13375.9	−2.56137
$302$	−3780.27	−0.720298
$303$	1366.23	0.259035
$304$	−19351.1	−3.65086
$305$	−2657.86	−0.498980
$306$	−1532.82	−0.286357
$307$	2282.86	0.424397	0.212199	−	0.977227i	$-0.431938\pi$
$307$	2282.86	0.424397	0.212199	+	0.977227i	$0.431938\pi$
$308$	0	0
$309$	−73.3790	−0.0135093
$310$	5213.68	0.955216
$311$	−4088.21	−0.745405	−0.372703	−	0.927951i	$-0.621569\pi$
$311$	−4088.21	−0.745405	−0.372703	+	0.927951i	$0.621569\pi$
$312$	−11633.8	−2.11100
$313$	−4044.94	−0.730459	−0.365230	−	0.930917i	$-0.619010\pi$
$313$	−4044.94	−0.730459	−0.365230	+	0.930917i	$0.619010\pi$
$314$	16756.1	3.01147
$315$	1338.25	0.239371
$316$	23581.9	4.19806
$317$	−4932.62	−0.873954	−0.436977	−	0.899473i	$-0.643951\pi$
$317$	−4932.62	−0.873954	−0.436977	+	0.899473i	$0.643951\pi$
$318$	−2170.88	−0.382820
$319$	0	0
$320$	−15634.6	−2.73125
$321$	−3145.88	−0.546997
$322$	−11655.7	−2.01723
$323$	−1963.58	−0.338255
$324$	1901.28	0.326009
$325$	1116.88	0.190626
$326$	−630.513	−0.107119
$327$	3044.26	0.514825
$328$	10257.7	1.72679
$329$	−24.0533	−0.00403070
$330$	0	0
$331$	495.718	0.0823176	0.0411588	−	0.999153i	$-0.486895\pi$
$331$	495.718	0.0823176	0.0411588	+	0.999153i	$0.486895\pi$
$332$	−9306.38	−1.53842
$333$	−2415.41	−0.397488
$334$	12321.5	2.01858
$335$	1842.99	0.300578
$336$	−26692.2	−4.33387
$337$	10939.8	1.76833	0.884167	−	0.467171i	$-0.154727\pi$
$337$	10939.8	1.76833	0.884167	+	0.467171i	$0.154727\pi$
$338$	1128.28	0.181569
$339$	2312.43	0.370484
$340$	−3562.98	−0.568323
$341$	0	0
$342$	3265.70	0.516342
$343$	−5900.28	−0.928819
$344$	−39041.7	−6.11915
$345$	1047.95	0.163535
$346$	24213.5	3.76221
$347$	−2729.35	−0.422245	−0.211122	−	0.977460i	$-0.567712\pi$
$347$	−2729.35	−0.422245	−0.211122	+	0.977460i	$0.567712\pi$
$348$	−2431.32	−0.374518
$349$	9896.20	1.51785	0.758927	−	0.651175i	$-0.225724\pi$
$349$	9896.20	1.51785	0.758927	+	0.651175i	$0.225724\pi$
$350$	4170.92	0.636986
$351$	1206.23	0.183430
$352$	0	0
$353$	11024.6	1.66226	0.831132	−	0.556075i	$-0.187693\pi$
$353$	11024.6	1.66226	0.831132	+	0.556075i	$0.187693\pi$
$354$	3216.43	0.482914
$355$	−1263.09	−0.188839
$356$	−4843.35	−0.721060
$357$	−2708.49	−0.401537
$358$	−16808.4	−2.48143
$359$	10183.3	1.49708	0.748542	−	0.663088i	$-0.230754\pi$
$359$	10183.3	1.49708	0.748542	+	0.663088i	$0.230754\pi$
$360$	3906.10	0.571861
$361$	−2675.55	−0.390079
$362$	14583.1	2.11732
$363$	0	0
$364$	−31185.6	−4.49058
$365$	2686.24	0.385217
$366$	−8946.45	−1.27770
$367$	2663.75	0.378873	0.189436	−	0.981893i	$-0.439334\pi$
$367$	2663.75	0.378873	0.189436	+	0.981893i	$0.439334\pi$
$368$	−20901.9	−2.96084
$369$	−1063.56	−0.150045
$370$	−7528.08	−1.05775
$371$	−3835.95	−0.536799
$372$	13088.5	1.82422
$373$	−1173.86	−0.162949	−0.0814746	−	0.996675i	$-0.525963\pi$
$373$	−1173.86	−0.162949	−0.0814746	+	0.996675i	$0.525963\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	−70.2070	−0.00962939
$377$	−1542.50	−0.210724
$378$	4504.59	0.612940
$379$	248.820	0.0337230	0.0168615	−	0.999858i	$-0.494633\pi$
$379$	248.820	0.0337230	0.0168615	+	0.999858i	$0.494633\pi$
$380$	7591.01	1.02476
$381$	−812.956	−0.109315
$382$	−5257.53	−0.704185
$383$	5593.28	0.746223	0.373111	−	0.927787i	$-0.378291\pi$
$383$	5593.28	0.746223	0.373111	+	0.927787i	$0.378291\pi$
$384$	−29010.1	−3.85525
$385$	0	0
$386$	−22458.9	−2.96148
$387$	4048.00	0.531709
$388$	28751.9	3.76200
$389$	−8645.42	−1.12684	−0.563419	−	0.826171i	$-0.690514\pi$
$389$	−8645.42	−1.12684	−0.563419	+	0.826171i	$0.690514\pi$
$390$	3759.46	0.488122
$391$	−2120.94	−0.274324
$392$	−46995.0	−6.05512
$393$	851.776	0.109329
$394$	−17288.7	−2.21064
$395$	−5023.27	−0.639869
$396$	0	0
$397$	−9061.36	−1.14553	−0.572767	−	0.819718i	$-0.694130\pi$
$397$	−9061.36	−1.14553	−0.572767	+	0.819718i	$0.694130\pi$
$398$	6114.11	0.770032
$399$	5770.50	0.724026
$400$	7479.60	0.934950
$401$	−1649.24	−0.205385	−0.102692	−	0.994713i	$-0.532746\pi$
$401$	−1649.24	−0.205385	−0.102692	+	0.994713i	$0.532746\pi$
$402$	6203.57	0.769667
$403$	8303.76	1.02640
$404$	10689.7	1.31641
$405$	−405.000	−0.0496904
$406$	−5760.36	−0.704143
$407$	0	0
$408$	−7905.58	−0.959276
$409$	−1601.01	−0.193557	−0.0967787	−	0.995306i	$-0.530854\pi$
$409$	−1601.01	−0.193557	−0.0967787	+	0.995306i	$0.530854\pi$
$410$	−3314.79	−0.399282
$411$	−6877.45	−0.825401
$412$	−574.133	−0.0686541
$413$	5683.44	0.677152
$414$	3527.42	0.418752
$415$	1982.39	0.234486
$416$	−43961.3	−5.18121
$417$	5551.00	0.651879
$418$	0	0
$419$	−4185.17	−0.487969	−0.243985	−	0.969779i	$-0.578455\pi$
$419$	−4185.17	−0.487969	−0.243985	+	0.969779i	$0.578455\pi$
$420$	10470.8	1.21648
$421$	−11918.6	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$421$	−11918.6	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$422$	16477.2	1.90070
$423$	7.27933	0.000836722 0
$424$	−11196.4	−1.28242
$425$	758.964	0.0866239
$426$	−4251.59	−0.483545
$427$	−15808.4	−1.79162
$428$	−24614.0	−2.77982
$429$	0	0
$430$	12616.4	1.41492
$431$	6163.18	0.688793	0.344397	−	0.938824i	$-0.388083\pi$
$431$	6163.18	0.688793	0.344397	+	0.938824i	$0.388083\pi$
$432$	8077.97	0.899656
$433$	−17991.8	−1.99684	−0.998419	−	0.0562163i	$-0.982096\pi$
$433$	−17991.8	−1.99684	−0.998419	+	0.0562163i	$0.982096\pi$
$434$	31009.8	3.42977
$435$	517.904	0.0570841
$436$	23818.9	2.61633
$437$	4518.72	0.494644
$438$	9041.95	0.986395
$439$	−14959.2	−1.62634	−0.813170	−	0.582026i	$-0.802260\pi$
$439$	−14959.2	−1.62634	−0.813170	+	0.582026i	$0.802260\pi$
$440$	0	0
$441$	4872.62	0.526144
$442$	−7608.79	−0.818808
$443$	−8505.99	−0.912261	−0.456131	−	0.889913i	$-0.650765\pi$
$443$	−8505.99	−0.912261	−0.456131	+	0.889913i	$0.650765\pi$
$444$	−18898.7	−2.02002
$445$	1031.70	0.109904
$446$	−24222.8	−2.57171
$447$	4225.33	0.447094
$448$	−92991.2	−9.80675
$449$	−1904.67	−0.200194	−0.100097	−	0.994978i	$-0.531915\pi$
$449$	−1904.67	−0.200194	−0.100097	+	0.994978i	$0.531915\pi$
$450$	−1262.26	−0.132230
$451$	0	0
$452$	18093.0	1.88279
$453$	2021.51	0.209667
$454$	−26025.3	−2.69037
$455$	6642.97	0.684456
$456$	16843.0	1.72971
$457$	−8981.66	−0.919353	−0.459677	−	0.888086i	$-0.652035\pi$
$457$	−8981.66	−0.919353	−0.459677	+	0.888086i	$0.652035\pi$
$458$	7172.66	0.731783
$459$	819.681	0.0833539
$460$	8199.36	0.831081
$461$	−11982.4	−1.21058	−0.605290	−	0.796005i	$-0.706943\pi$
$461$	−11982.4	−1.21058	−0.605290	+	0.796005i	$0.706943\pi$
$462$	0	0
$463$	−14573.3	−1.46280	−0.731402	−	0.681946i	$-0.761134\pi$
$463$	−14573.3	−1.46280	−0.731402	+	0.681946i	$0.761134\pi$
$464$	−10329.9	−1.03352
$465$	−2788.04	−0.278048
$466$	33729.1	3.35294
$467$	14157.1	1.40281	0.701406	−	0.712762i	$-0.252556\pi$
$467$	14157.1	1.40281	0.701406	+	0.712762i	$0.252556\pi$
$468$	9437.83	0.932188
$469$	10961.7	1.07924
$470$	22.6875	0.00222658
$471$	−8960.40	−0.876589
$472$	16588.9	1.61773
$473$	0	0
$474$	−16908.5	−1.63847
$475$	−1616.99	−0.156195
$476$	−21191.8	−2.04060
$477$	1160.89	0.111433
$478$	4676.92	0.447526
$479$	2855.93	0.272424	0.136212	−	0.990680i	$-0.456507\pi$
$479$	2855.93	0.272424	0.136212	+	0.990680i	$0.456507\pi$
$480$	14760.3	1.40357
$481$	−11989.9	−1.13657
$482$	38398.7	3.62865
$483$	6232.96	0.587183
$484$	0	0
$485$	−6124.55	−0.573405
$486$	−1363.24	−0.127238
$487$	18165.6	1.69027	0.845133	−	0.534555i	$-0.179521\pi$
$487$	18165.6	1.69027	0.845133	+	0.534555i	$0.179521\pi$
$488$	−46141.7	−4.28020
$489$	337.170	0.0311807
$490$	15186.5	1.40011
$491$	10395.6	0.955493	0.477746	−	0.878498i	$-0.341454\pi$
$491$	10395.6	0.955493	0.477746	+	0.878498i	$0.341454\pi$
$492$	−8321.51	−0.762526
$493$	−1048.19	−0.0957567
$494$	16210.7	1.47642
$495$	0	0
$496$	55609.1	5.03412
$497$	−7512.57	−0.678038
$498$	6672.78	0.600431
$499$	−2837.79	−0.254583	−0.127292	−	0.991865i	$-0.540628\pi$
$499$	−2837.79	−0.254583	−0.127292	+	0.991865i	$0.540628\pi$
$500$	−2934.08	−0.262432
$501$	−6589.00	−0.587574
$502$	627.368	0.0557785
$503$	1012.95	0.0897912	0.0448956	−	0.998992i	$-0.485704\pi$
$503$	1012.95	0.0897912	0.0448956	+	0.998992i	$0.485704\pi$
$504$	23232.7	2.05330
$505$	−2277.04	−0.200648
$506$	0	0
$507$	−603.353	−0.0528518
$508$	−6360.75	−0.555536
$509$	743.483	0.0647432	0.0323716	−	0.999476i	$-0.489694\pi$
$509$	743.483	0.0647432	0.0323716	+	0.999476i	$0.489694\pi$
$510$	2554.70	0.221811
$511$	15977.1	1.38315
$512$	−86643.8	−7.47881
$513$	−1746.35	−0.150299
$514$	3555.42	0.305103
$515$	122.298	0.0104643
$516$	31672.4	2.70213
$517$	0	0
$518$	−44775.4	−3.79791
$519$	−12948.3	−1.09512
$520$	19389.6	1.63517
$521$	−9387.31	−0.789377	−0.394688	−	0.918815i	$-0.629147\pi$
$521$	−9387.31	−0.789377	−0.394688	+	0.918815i	$0.629147\pi$
$522$	1743.28	0.146171
$523$	−2803.79	−0.234419	−0.117209	−	0.993107i	$-0.537395\pi$
$523$	−2803.79	−0.234419	−0.117209	+	0.993107i	$0.537395\pi$
$524$	6664.48	0.555609
$525$	−2230.42	−0.185416
$526$	9849.92	0.816496
$527$	5642.72	0.466415
$528$	0	0
$529$	−7286.15	−0.598845
$530$	3618.13	0.296531
$531$	−1720.00	−0.140568
$532$	45149.6	3.67948
$533$	−5279.42	−0.429038
$534$	3472.74	0.281423
$535$	5243.13	0.423702
$536$	31995.2	2.57833
$537$	8988.38	0.722304
$538$	7207.24	0.577558
$539$	0	0
$540$	−3168.81	−0.252526
$541$	3756.12	0.298500	0.149250	−	0.988800i	$-0.452314\pi$
$541$	3756.12	0.298500	0.149250	+	0.988800i	$0.452314\pi$
$542$	40210.6	3.18670
$543$	−7798.38	−0.616318
$544$	−29873.4	−2.35443
$545$	−5073.76	−0.398782
$546$	22360.4	1.75263
$547$	−13621.1	−1.06471	−0.532355	−	0.846521i	$-0.678693\pi$
$547$	−13621.1	−1.06471	−0.532355	+	0.846521i	$0.678693\pi$
$548$	−53810.7	−4.19467
$549$	4784.15	0.371917
$550$	0	0
$551$	2233.19	0.172662
$552$	18192.8	1.40279
$553$	−29877.3	−2.29749
$554$	−9920.11	−0.760767
$555$	4025.68	0.307893
$556$	43432.2	3.31283
$557$	−17476.1	−1.32942	−0.664708	−	0.747103i	$-0.731444\pi$
$557$	−17476.1	−1.32942	−0.664708	+	0.747103i	$0.731444\pi$
$558$	−9384.62	−0.711976
$559$	20093.9	1.52036
$560$	44487.0	3.35700
$561$	0	0
$562$	29537.5	2.21702
$563$	−2974.61	−0.222673	−0.111336	−	0.993783i	$-0.535513\pi$
$563$	−2974.61	−0.222673	−0.111336	+	0.993783i	$0.535513\pi$
$564$	56.9551	0.00425220
$565$	−3854.06	−0.286976
$566$	−8908.65	−0.661587
$567$	−2408.85	−0.178417
$568$	−21927.8	−1.61984
$569$	4923.08	0.362718	0.181359	−	0.983417i	$-0.441950\pi$
$569$	4923.08	0.362718	0.181359	+	0.983417i	$0.441950\pi$
$570$	−5442.83	−0.399957
$571$	−22267.6	−1.63199	−0.815997	−	0.578056i	$-0.803811\pi$
$571$	−22267.6	−1.63199	−0.815997	+	0.578056i	$0.803811\pi$
$572$	0	0
$573$	2811.49	0.204977
$574$	−19715.6	−1.43365
$575$	−1746.58	−0.126674
$576$	28142.3	2.03576
$577$	−3082.13	−0.222376	−0.111188	−	0.993799i	$-0.535466\pi$
$577$	−3082.13	−0.222376	−0.111188	+	0.993799i	$0.535466\pi$
$578$	22391.7	1.61137
$579$	12010.0	0.862037
$580$	4052.19	0.290100
$581$	11790.8	0.841937
$582$	−20615.4	−1.46827
$583$	0	0
$584$	46634.3	3.30435
$585$	−2010.39	−0.142084
$586$	35667.3	2.51434
$587$	−19190.8	−1.34939	−0.674694	−	0.738098i	$-0.735724\pi$
$587$	−19190.8	−1.34939	−0.674694	+	0.738098i	$0.735724\pi$
$588$	38124.5	2.67385
$589$	−12021.9	−0.841011
$590$	−5360.72	−0.374063
$591$	9245.20	0.643480
$592$	−80294.6	−5.57447
$593$	8566.34	0.593217	0.296608	−	0.954999i	$-0.404144\pi$
$593$	8566.34	0.593217	0.296608	+	0.954999i	$0.404144\pi$
$594$	0	0
$595$	4514.15	0.311029
$596$	33059.9	2.27212
$597$	−3269.55	−0.224144
$598$	17509.8	1.19737
$599$	−21405.3	−1.46009	−0.730047	−	0.683397i	$-0.760502\pi$
$599$	−21405.3	−1.46009	−0.730047	+	0.683397i	$0.760502\pi$
$600$	−6510.17	−0.442961
$601$	−13764.8	−0.934237	−0.467118	−	0.884195i	$-0.654708\pi$
$601$	−13764.8	−0.934237	−0.467118	+	0.884195i	$0.654708\pi$
$602$	75039.4	5.08036
$603$	−3317.39	−0.224037
$604$	15816.8	1.06552
$605$	0	0
$606$	−7664.60	−0.513784
$607$	−15840.3	−1.05920	−0.529601	−	0.848247i	$-0.677658\pi$
$607$	−15840.3	−1.05920	−0.529601	+	0.848247i	$0.677658\pi$
$608$	63645.9	4.24537
$609$	3080.38	0.204964
$610$	14910.7	0.989702
$611$	36.1340	0.00239252
$612$	6413.36	0.423603
$613$	7393.70	0.487160	0.243580	−	0.969881i	$-0.421678\pi$
$613$	7393.70	0.487160	0.243580	+	0.969881i	$0.421678\pi$
$614$	−12807.0	−0.841771
$615$	1772.60	0.116224
$616$	0	0
$617$	−1115.13	−0.0727606	−0.0363803	−	0.999338i	$-0.511583\pi$
$617$	−1115.13	−0.0727606	−0.0363803	+	0.999338i	$0.511583\pi$
$618$	411.660	0.0267951
$619$	2926.56	0.190030	0.0950149	−	0.995476i	$-0.469710\pi$
$619$	2926.56	0.190030	0.0950149	+	0.995476i	$0.469710\pi$
$620$	−21814.2	−1.41303
$621$	−1886.30	−0.121892
$622$	22935.1	1.47848
$623$	6136.34	0.394618
$624$	40098.4	2.57247
$625$	625.000	0.0400000
$626$	22692.3	1.44883
$627$	0	0
$628$	−70108.1	−4.45480
$629$	−8147.59	−0.516479
$630$	−7507.65	−0.474781
$631$	4694.39	0.296166	0.148083	−	0.988975i	$-0.452690\pi$
$631$	4694.39	0.296166	0.148083	+	0.988975i	$0.452690\pi$
$632$	−87206.4	−5.48874
$633$	−8811.26	−0.553264
$634$	27672.2	1.73344
$635$	1354.93	0.0846750
$636$	9083.03	0.566298
$637$	24187.3	1.50445
$638$	0	0
$639$	2273.56	0.140752
$640$	48350.2	2.98626
$641$	−20775.0	−1.28013	−0.640066	−	0.768320i	$-0.721093\pi$
$641$	−20775.0	−1.28013	−0.640066	+	0.768320i	$0.721093\pi$
$642$	17648.5	1.08494
$643$	1868.85	0.114620	0.0573098	−	0.998356i	$-0.481748\pi$
$643$	1868.85	0.114620	0.0573098	+	0.998356i	$0.481748\pi$
$644$	48768.0	2.98405
$645$	−6746.66	−0.411860
$646$	11015.8	0.670913
$647$	8565.33	0.520460	0.260230	−	0.965547i	$-0.416202\pi$
$647$	8565.33	0.520460	0.260230	+	0.965547i	$0.416202\pi$
$648$	−7030.99	−0.426240
$649$	0	0
$650$	−6265.77	−0.378098
$651$	−16582.6	−0.998349
$652$	2638.09	0.158459
$653$	16428.5	0.984530	0.492265	−	0.870445i	$-0.336169\pi$
$653$	16428.5	0.984530	0.492265	+	0.870445i	$0.336169\pi$
$654$	−17078.4	−1.02113
$655$	−1419.63	−0.0846861
$656$	−35355.5	−2.10427
$657$	−4835.23	−0.287123
$658$	134.940	0.00799470
$659$	−27652.8	−1.63460	−0.817298	−	0.576215i	$-0.804529\pi$
$659$	−27652.8	−1.63460	−0.817298	+	0.576215i	$0.804529\pi$
$660$	0	0
$661$	14887.6	0.876040	0.438020	−	0.898965i	$-0.355680\pi$
$661$	14887.6	0.876040	0.438020	+	0.898965i	$0.355680\pi$
$662$	−2781.00	−0.163273
$663$	4068.83	0.238342
$664$	34415.2	2.01140
$665$	−9617.50	−0.560828
$666$	13550.5	0.788398
$667$	2412.16	0.140029
$668$	−51553.8	−2.98604
$669$	12953.3	0.748582
$670$	−10339.3	−0.596181
$671$	0	0
$672$	87790.9	5.03960
$673$	−8157.45	−0.467231	−0.233616	−	0.972329i	$-0.575056\pi$
$673$	−8157.45	−0.467231	−0.233616	+	0.972329i	$0.575056\pi$
$674$	−61372.8	−3.50740
$675$	675.000	0.0384900
$676$	−4720.77	−0.268592
$677$	26020.1	1.47716	0.738579	−	0.674167i	$-0.235497\pi$
$677$	26020.1	1.47716	0.738579	+	0.674167i	$0.235497\pi$
$678$	−12972.9	−0.734838
$679$	−36427.5	−2.05885
$680$	13176.0	0.743052
$681$	13917.2	0.783123
$682$	0	0
$683$	13678.9	0.766338	0.383169	−	0.923678i	$-0.374833\pi$
$683$	13678.9	0.766338	0.383169	+	0.923678i	$0.374833\pi$
$684$	−13663.8	−0.763814
$685$	11462.4	0.639353
$686$	33100.8	1.84227
$687$	−3835.62	−0.213010
$688$	134566.	7.45682
$689$	5762.55	0.318630
$690$	−5879.03	−0.324364
$691$	−9180.50	−0.505416	−0.252708	−	0.967543i	$-0.581321\pi$
$691$	−9180.50	−0.505416	−0.252708	+	0.967543i	$0.581321\pi$
$692$	−101310.	−5.56537
$693$	0	0
$694$	15311.8	0.837502
$695$	−9251.66	−0.504943
$696$	8991.06	0.489663
$697$	−3587.57	−0.194962
$698$	−55518.2	−3.01059
$699$	−18036.8	−0.975986
$700$	−17451.3	−0.942280
$701$	15498.2	0.835032	0.417516	−	0.908670i	$-0.362901\pi$
$701$	15498.2	0.835032	0.417516	+	0.908670i	$0.362901\pi$
$702$	−6767.03	−0.363825
$703$	17358.6	0.931283
$704$	0	0
$705$	−12.1322	−0.000648122 0
$706$	−61848.5	−3.29702
$707$	−13543.4	−0.720439
$708$	−13457.7	−0.714365
$709$	10636.7	0.563426	0.281713	−	0.959499i	$-0.409097\pi$
$709$	10636.7	0.563426	0.281713	+	0.959499i	$0.409097\pi$
$710$	7085.99	0.374553
$711$	9041.89	0.476930
$712$	17910.8	0.942748
$713$	−12985.4	−0.682057
$714$	15194.8	0.796429
$715$	0	0
$716$	70327.0	3.67073
$717$	−2501.00	−0.130267
$718$	−57128.7	−2.96939
$719$	−24592.3	−1.27557	−0.637787	−	0.770213i	$-0.720150\pi$
$719$	−24592.3	−1.27557	−0.637787	+	0.770213i	$0.720150\pi$
$720$	−13463.3	−0.696871
$721$	727.404	0.0375727
$722$	15010.0	0.773703
$723$	−20533.9	−1.05624
$724$	−61016.2	−3.13211
$725$	−863.174	−0.0442172
$726$	0	0
$727$	−30920.8	−1.57742	−0.788712	−	0.614762i	$-0.789252\pi$
$727$	−30920.8	−1.57742	−0.788712	+	0.614762i	$0.789252\pi$
$728$	115325.	5.87120
$729$	729.000	0.0370370
$730$	−15069.9	−0.764058
$731$	13654.6	0.690880
$732$	37432.3	1.89008
$733$	−8665.70	−0.436664	−0.218332	−	0.975874i	$-0.570062\pi$
$733$	−8665.70	−0.436664	−0.218332	+	0.975874i	$0.570062\pi$
$734$	−14943.7	−0.751476
$735$	−8121.04	−0.407550
$736$	68746.6	3.44298
$737$	0	0
$738$	5966.62	0.297607
$739$	25421.1	1.26540	0.632699	−	0.774398i	$-0.281947\pi$
$739$	25421.1	1.26540	0.632699	+	0.774398i	$0.281947\pi$
$740$	31497.8	1.56470
$741$	−8668.74	−0.429763
$742$	21519.8	1.06471
$743$	23704.1	1.17042	0.585209	−	0.810883i	$-0.301013\pi$
$743$	23704.1	1.17042	0.585209	+	0.810883i	$0.301013\pi$
$744$	−48401.6	−2.38507
$745$	−7042.22	−0.346318
$746$	6585.40	0.323202
$747$	−3568.30	−0.174775
$748$	0	0
$749$	31185.0	1.52133
$750$	2103.77	0.102425
$751$	−864.870	−0.0420234	−0.0210117	−	0.999779i	$-0.506689\pi$
$751$	−864.870	−0.0420234	−0.0210117	+	0.999779i	$0.506689\pi$
$752$	241.985	0.0117344
$753$	−335.488	−0.0162362
$754$	8653.51	0.417960
$755$	−3369.19	−0.162407
$756$	−18847.4	−0.906710
$757$	−37558.5	−1.80329	−0.901643	−	0.432481i	$-0.857638\pi$
$757$	−37558.5	−1.80329	−0.901643	+	0.432481i	$0.857638\pi$
$758$	−1395.89	−0.0668879
$759$	0	0
$760$	−28071.7	−1.33983
$761$	−34183.3	−1.62831	−0.814156	−	0.580647i	$-0.802800\pi$
$761$	−34183.3	−1.62831	−0.814156	+	0.580647i	$0.802800\pi$
$762$	4560.72	0.216821
$763$	−30177.6	−1.43185
$764$	21997.7	1.04169
$765$	−1366.14	−0.0645657
$766$	−31378.6	−1.48010
$767$	−8537.96	−0.401940
$768$	87701.9	4.12067
$769$	−9261.40	−0.434297	−0.217149	−	0.976139i	$-0.569676\pi$
$769$	−9261.40	−0.434297	−0.217149	+	0.976139i	$0.569676\pi$
$770$	0	0
$771$	−1901.28	−0.0888105
$772$	93969.0	4.38085
$773$	37167.3	1.72939	0.864693	−	0.502300i	$-0.167513\pi$
$773$	37167.3	1.72939	0.864693	+	0.502300i	$0.167513\pi$
$774$	−22709.5	−1.05462
$775$	4646.73	0.215375
$776$	−106325.	−4.91861
$777$	23943.8	1.10551
$778$	48501.2	2.23503
$779$	7643.39	0.351544
$780$	−15729.7	−0.722069
$781$	0	0
$782$	11898.6	0.544109
$783$	−932.227	−0.0425480
$784$	161979.	7.37878
$785$	14934.0	0.679003
$786$	−4778.51	−0.216849
$787$	−29942.9	−1.35623	−0.678113	−	0.734958i	$-0.737202\pi$
$787$	−29942.9	−1.35623	−0.678113	+	0.734958i	$0.737202\pi$
$788$	72336.5	3.27015
$789$	−5267.29	−0.237669
$790$	28180.8	1.26915
$791$	−22923.1	−1.03041
$792$	0	0
$793$	23748.2	1.06346
$794$	50834.7	2.27211
$795$	−1934.81	−0.0863154
$796$	−25581.7	−1.13909
$797$	25708.1	1.14257	0.571285	−	0.820752i	$-0.306445\pi$
$797$	25708.1	1.14257	0.571285	+	0.820752i	$0.306445\pi$
$798$	−32372.8	−1.43607
$799$	24.5545	0.00108720
$800$	−24600.5	−1.08720
$801$	−1857.06	−0.0819177
$802$	9252.33	0.407371
$803$	0	0
$804$	−25956.0	−1.13855
$805$	−10388.3	−0.454830
$806$	−46584.5	−2.03582
$807$	−3854.11	−0.168118
$808$	−39530.5	−1.72114
$809$	27643.2	1.20134	0.600668	−	0.799498i	$-0.294901\pi$
$809$	27643.2	1.20134	0.600668	+	0.799498i	$0.294901\pi$
$810$	2272.07	0.0985585
$811$	−31017.7	−1.34301	−0.671503	−	0.741002i	$-0.734351\pi$
$811$	−31017.7	−1.34301	−0.671503	+	0.741002i	$0.734351\pi$
$812$	24101.6	1.04162
$813$	−21502.8	−0.927597
$814$	0	0
$815$	−561.949	−0.0241524
$816$	27248.4	1.16898
$817$	−29091.4	−1.24575
$818$	8981.76	0.383912
$819$	−11957.4	−0.510163
$820$	13869.2	0.590650
$821$	35876.5	1.52509	0.762544	−	0.646936i	$-0.223950\pi$
$821$	35876.5	1.52509	0.762544	+	0.646936i	$0.223950\pi$
$822$	38582.8	1.63714
$823$	37311.3	1.58030	0.790152	−	0.612911i	$-0.210001\pi$
$823$	37311.3	1.58030	0.790152	+	0.612911i	$0.210001\pi$
$824$	2123.16	0.0897617
$825$	0	0
$826$	−31884.4	−1.34310
$827$	−46140.4	−1.94009	−0.970047	−	0.242917i	$-0.921896\pi$
$827$	−46140.4	−1.94009	−0.970047	+	0.242917i	$0.921896\pi$
$828$	−14758.8	−0.619451
$829$	−1850.93	−0.0775459	−0.0387730	−	0.999248i	$-0.512345\pi$
$829$	−1850.93	−0.0775459	−0.0387730	+	0.999248i	$0.512345\pi$
$830$	−11121.3	−0.465091
$831$	5304.82	0.221447
$832$	139696.	5.82102
$833$	16436.2	0.683650
$834$	−31141.4	−1.29297
$835$	10981.7	0.455133
$836$	0	0
$837$	5018.47	0.207244
$838$	23479.0	0.967863
$839$	43588.3	1.79360	0.896802	−	0.442431i	$-0.145884\pi$
$839$	43588.3	1.79360	0.896802	+	0.442431i	$0.145884\pi$
$840$	−38721.1	−1.59048
$841$	−23196.9	−0.951121
$842$	66864.2	2.73669
$843$	−15795.3	−0.645338
$844$	−68941.2	−2.81168
$845$	1005.59	0.0409388
$846$	−40.8374	−0.00165960
$847$	0	0
$848$	38591.0	1.56276
$849$	4763.94	0.192577
$850$	−4257.83	−0.171814
$851$	18749.8	0.755268
$852$	17788.8	0.715299
$853$	−25084.1	−1.00688	−0.503438	−	0.864032i	$-0.667932\pi$
$853$	−25084.1	−1.00688	−0.503438	+	0.864032i	$0.667932\pi$
$854$	88685.8	3.55359
$855$	2910.58	0.116421
$856$	91023.2	3.63447
$857$	9469.64	0.377452	0.188726	−	0.982030i	$-0.439564\pi$
$857$	9469.64	0.377452	0.188726	+	0.982030i	$0.439564\pi$
$858$	0	0
$859$	4359.10	0.173144	0.0865719	−	0.996246i	$-0.472409\pi$
$859$	4359.10	0.173144	0.0865719	+	0.996246i	$0.472409\pi$
$860$	−52787.3	−2.09306
$861$	10543.0	0.417312
$862$	−34575.7	−1.36619
$863$	7447.71	0.293770	0.146885	−	0.989154i	$-0.453075\pi$
$863$	7447.71	0.293770	0.146885	+	0.989154i	$0.453075\pi$
$864$	−26568.5	−1.04616
$865$	21580.5	0.848275
$866$	100935.	3.96063
$867$	−11974.1	−0.469044
$868$	−129746.	−5.07359
$869$	0	0
$870$	−2905.47	−0.113224
$871$	−16467.3	−0.640610
$872$	−88082.9	−3.42071
$873$	11024.2	0.427391
$874$	−25350.2	−0.981103
$875$	3717.36	0.143623
$876$	−37831.9	−1.45916
$877$	−12798.1	−0.492771	−0.246386	−	0.969172i	$-0.579243\pi$
$877$	−12798.1	−0.492771	−0.246386	+	0.969172i	$0.579243\pi$
$878$	83921.8	3.22577
$879$	−19073.3	−0.731883
$880$	0	0
$881$	−26987.4	−1.03204	−0.516020	−	0.856577i	$-0.672587\pi$
$881$	−26987.4	−1.03204	−0.516020	+	0.856577i	$0.672587\pi$
$882$	−27335.7	−1.04358
$883$	1287.10	0.0490537	0.0245269	−	0.999699i	$-0.492192\pi$
$883$	1287.10	0.0490537	0.0245269	+	0.999699i	$0.492192\pi$
$884$	31835.4	1.21125
$885$	2866.67	0.108884
$886$	47719.0	1.80943
$887$	−44736.0	−1.69345	−0.846725	−	0.532031i	$-0.821429\pi$
$887$	−44736.0	−1.69345	−0.846725	+	0.532031i	$0.821429\pi$
$888$	69887.6	2.64108
$889$	8058.81	0.304031
$890$	−5787.90	−0.217990
$891$	0	0
$892$	101349.	3.80428
$893$	−52.3138	−0.00196037
$894$	−23704.3	−0.886790
$895$	−14980.6	−0.559494
$896$	287576.	10.7224
$897$	−9363.46	−0.348536
$898$	10685.3	0.397075
$899$	−6417.49	−0.238082
$900$	5281.35	0.195605
$901$	3915.87	0.144791
$902$	0	0
$903$	−40127.7	−1.47881
$904$	−66908.2	−2.46165
$905$	12997.3	0.477398
$906$	−11340.8	−0.415864
$907$	43026.2	1.57515	0.787576	−	0.616217i	$-0.211336\pi$
$907$	43026.2	1.57515	0.787576	+	0.616217i	$0.211336\pi$
$908$	108891.	3.97981
$909$	4098.68	0.149554
$910$	−37267.4	−1.35759
$911$	53890.4	1.95990	0.979951	−	0.199240i	$-0.0638475\pi$
$911$	53890.4	1.95990	0.979951	+	0.199240i	$0.0638475\pi$
$912$	−58053.3	−2.10783
$913$	0	0
$914$	50387.6	1.82349
$915$	−7973.59	−0.288086
$916$	−30010.7	−1.08251
$917$	−8443.63	−0.304071
$918$	−4598.45	−0.165328
$919$	−29558.1	−1.06097	−0.530485	−	0.847695i	$-0.677990\pi$
$919$	−29558.1	−1.06097	−0.530485	+	0.847695i	$0.677990\pi$
$920$	−30321.4	−1.08659
$921$	6848.59	0.245026
$922$	67222.0	2.40113
$923$	11285.8	0.402465
$924$	0	0
$925$	−6709.46	−0.238493
$926$	81756.9	2.90140
$927$	−220.137	−0.00779962
$928$	33975.2	1.20182
$929$	15038.5	0.531105	0.265552	−	0.964096i	$-0.414446\pi$
$929$	15038.5	0.531105	0.265552	+	0.964096i	$0.414446\pi$
$930$	15641.0	0.551494
$931$	−35017.7	−1.23272
$932$	−141124.	−4.95994
$933$	−12264.6	−0.430360
$934$	−79422.1	−2.78241
$935$	0	0
$936$	−34901.3	−1.21879
$937$	−34789.9	−1.21295	−0.606476	−	0.795102i	$-0.707417\pi$
$937$	−34789.9	−1.21295	−0.606476	+	0.795102i	$0.707417\pi$
$938$	−61495.8	−2.14063
$939$	−12134.8	−0.421731
$940$	−94.9251	−0.00329374
$941$	−18534.7	−0.642099	−0.321049	−	0.947062i	$-0.604036\pi$
$941$	−18534.7	−0.642099	−0.321049	+	0.947062i	$0.604036\pi$
$942$	50268.3	1.73867
$943$	8255.94	0.285101
$944$	−57177.5	−1.97136
$945$	4014.75	0.138201
$946$	0	0
$947$	−51097.5	−1.75337	−0.876687	−	0.481062i	$-0.840251\pi$
$947$	−51097.5	−1.75337	−0.876687	+	0.481062i	$0.840251\pi$
$948$	70745.7	2.42375
$949$	−24001.7	−0.820999
$950$	9071.39	0.309805
$951$	−14797.8	−0.504577
$952$	78367.8	2.66798
$953$	−28438.2	−0.966637	−0.483318	−	0.875445i	$-0.660569\pi$
$953$	−28438.2	−0.966637	−0.483318	+	0.875445i	$0.660569\pi$
$954$	−6512.63	−0.221021
$955$	−4685.82	−0.158774
$956$	−19568.4	−0.662016
$957$	0	0
$958$	−16021.9	−0.540339
$959$	68176.0	2.29564
$960$	−46903.8	−1.57689
$961$	4756.36	0.159657
$962$	67263.9	2.25434
$963$	−9437.64	−0.315809
$964$	−160661.	−5.36779
$965$	−20016.7	−0.667731
$966$	−34967.2	−1.16465
$967$	−22430.9	−0.745946	−0.372973	−	0.927842i	$-0.621662\pi$
$967$	−22430.9	−0.745946	−0.372973	+	0.927842i	$0.621662\pi$
$968$	0	0
$969$	−5890.74	−0.195292
$970$	34359.0	1.13732
$971$	39552.3	1.30720	0.653602	−	0.756838i	$-0.273257\pi$
$971$	39552.3	1.30720	0.653602	+	0.756838i	$0.273257\pi$
$972$	5703.85	0.188221
$973$	−55026.9	−1.81303
$974$	−101910.	−3.35256
$975$	3350.65	0.110058
$976$	159038.	5.21587
$977$	26623.9	0.871827	0.435913	−	0.899989i	$-0.356425\pi$
$977$	26623.9	0.871827	0.435913	+	0.899989i	$0.356425\pi$
$978$	−1891.54	−0.0618453
$979$	0	0
$980$	−63540.8	−2.07116
$981$	9132.77	0.297234
$982$	−58319.8	−1.89517
$983$	−3949.49	−0.128148	−0.0640739	−	0.997945i	$-0.520409\pi$
$983$	−3949.49	−0.128148	−0.0640739	+	0.997945i	$0.520409\pi$
$984$	30773.1	0.996963
$985$	−15408.7	−0.498438
$986$	5880.39	0.189929
$987$	−72.1598	−0.00232712
$988$	−67826.1	−2.18404
$989$	−31422.8	−1.01030
$990$	0	0
$991$	315.723	0.0101203	0.00506017	−	0.999987i	$-0.498389\pi$
$991$	315.723	0.0101203	0.00506017	+	0.999987i	$0.498389\pi$
$992$	−182899.	−5.85387
$993$	1487.15	0.0475261
$994$	42145.9	1.34486
$995$	5449.25	0.173621
$996$	−27919.2	−0.888205
$997$	11380.2	0.361498	0.180749	−	0.983529i	$-0.442148\pi$
$997$	11380.2	0.361498	0.180749	+	0.983529i	$0.442148\pi$
$998$	15920.2	0.504954
$999$	−7246.22	−0.229490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bg.1.1		12
11.3	even	5		165.4.m.d.31.1	yes	24
11.4	even	5		165.4.m.d.16.1	✓	24
11.10	odd	2		1815.4.a.bo.1.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.m.d.16.1	✓	24	11.4	even	5
165.4.m.d.31.1	yes	24	11.3	even	5
1815.4.a.bg.1.1		12	1.1	even	1	trivial
1815.4.a.bo.1.12		12	11.10	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-5.61005 q^{2} +3.00000 q^{3} +23.4727 q^{4} -5.00000 q^{5} -16.8301 q^{6} -29.7389 q^{7} -86.8023 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.61005 q^{2} +3.00000 q^{3} +23.4727 q^{4} -5.00000 q^{5} -16.8301 q^{6} -29.7389 q^{7} -86.8023 q^{8} +9.00000 q^{9} +28.0502 q^{10} +70.4180 q^{12} +44.6753 q^{13} +166.837 q^{14} -15.0000 q^{15} +299.184 q^{16} +30.3586 q^{17} -50.4904 q^{18} -64.6796 q^{19} -117.363 q^{20} -89.2167 q^{21} -69.8631 q^{23} -260.407 q^{24} +25.0000 q^{25} -250.631 q^{26} +27.0000 q^{27} -698.051 q^{28} -34.5269 q^{29} +84.1507 q^{30} +185.869 q^{31} -984.019 q^{32} -170.313 q^{34} +148.695 q^{35} +211.254 q^{36} -268.378 q^{37} +362.856 q^{38} +134.026 q^{39} +434.012 q^{40} -118.173 q^{41} +500.510 q^{42} +449.777 q^{43} -45.0000 q^{45} +391.935 q^{46} +0.808815 q^{47} +897.552 q^{48} +541.403 q^{49} -140.251 q^{50} +91.0757 q^{51} +1048.65 q^{52} +128.987 q^{53} -151.471 q^{54} +2581.41 q^{56} -194.039 q^{57} +193.698 q^{58} -191.111 q^{59} -352.090 q^{60} +531.573 q^{61} -1042.74 q^{62} -267.650 q^{63} +3126.92 q^{64} -223.376 q^{65} -368.599 q^{67} +712.596 q^{68} -209.589 q^{69} -834.184 q^{70} +252.618 q^{71} -781.221 q^{72} -537.247 q^{73} +1505.62 q^{74} +75.0000 q^{75} -1518.20 q^{76} -751.892 q^{78} +1004.65 q^{79} -1495.92 q^{80} +81.0000 q^{81} +662.957 q^{82} -396.478 q^{83} -2094.15 q^{84} -151.793 q^{85} -2523.27 q^{86} -103.581 q^{87} -206.340 q^{89} +252.452 q^{90} -1328.59 q^{91} -1639.87 q^{92} +557.608 q^{93} -4.53749 q^{94} +323.398 q^{95} -2952.06 q^{96} +1224.91 q^{97} -3037.30 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} - 27 q^{6} - 21 q^{7} - 123 q^{8} + 108 q^{9}+O(q^{10}) \) 12 * q - 9 * q^2 + 36 * q^3 + 57 * q^4 - 60 * q^5 - 27 * q^6 - 21 * q^7 - 123 * q^8 + 108 * q^9	\( 12 q - 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} - 27 q^{6} - 21 q^{7} - 123 q^{8} + 108 q^{9} + 45 q^{10} + 171 q^{12} - 78 q^{13} + 262 q^{14} - 180 q^{15} + 225 q^{16} - 313 q^{17} - 81 q^{18} - 51 q^{19} - 285 q^{20} - 63 q^{21} - 34 q^{23} - 369 q^{24} + 300 q^{25} + 28 q^{26} + 324 q^{27} - 376 q^{28} - 31 q^{29} + 135 q^{30} + 655 q^{31} - 1578 q^{32} - 10 q^{34} + 105 q^{35} + 513 q^{36} + 84 q^{37} - 1076 q^{38} - 234 q^{39} + 615 q^{40} - 1463 q^{41} + 786 q^{42} + 111 q^{43} - 540 q^{45} + q^{46} + 278 q^{47} + 675 q^{48} - 325 q^{49} - 225 q^{50} - 939 q^{51} - 1957 q^{52} + 517 q^{53} - 243 q^{54} + 1543 q^{56} - 153 q^{57} + 442 q^{58} - 308 q^{59} - 855 q^{60} - 604 q^{61} - 1773 q^{62} - 189 q^{63} + 4323 q^{64} + 390 q^{65} - 357 q^{67} - 2192 q^{68} - 102 q^{69} - 1310 q^{70} - 620 q^{71} - 1107 q^{72} - 1892 q^{73} - 581 q^{74} + 900 q^{75} - 378 q^{76} + 84 q^{78} - 415 q^{79} - 1125 q^{80} + 972 q^{81} - 2802 q^{82} - 3158 q^{83} - 1128 q^{84} + 1565 q^{85} + 747 q^{86} - 93 q^{87} + 1563 q^{89} + 405 q^{90} + 1434 q^{91} - 3466 q^{92} + 1965 q^{93} + 3 q^{94} + 255 q^{95} - 4734 q^{96} + 714 q^{97} - 6586 q^{98}+O(q^{100}) \) 12 * q - 9 * q^2 + 36 * q^3 + 57 * q^4 - 60 * q^5 - 27 * q^6 - 21 * q^7 - 123 * q^8 + 108 * q^9 + 45 * q^10 + 171 * q^12 - 78 * q^13 + 262 * q^14 - 180 * q^15 + 225 * q^16 - 313 * q^17 - 81 * q^18 - 51 * q^19 - 285 * q^20 - 63 * q^21 - 34 * q^23 - 369 * q^24 + 300 * q^25 + 28 * q^26 + 324 * q^27 - 376 * q^28 - 31 * q^29 + 135 * q^30 + 655 * q^31 - 1578 * q^32 - 10 * q^34 + 105 * q^35 + 513 * q^36 + 84 * q^37 - 1076 * q^38 - 234 * q^39 + 615 * q^40 - 1463 * q^41 + 786 * q^42 + 111 * q^43 - 540 * q^45 + q^46 + 278 * q^47 + 675 * q^48 - 325 * q^49 - 225 * q^50 - 939 * q^51 - 1957 * q^52 + 517 * q^53 - 243 * q^54 + 1543 * q^56 - 153 * q^57 + 442 * q^58 - 308 * q^59 - 855 * q^60 - 604 * q^61 - 1773 * q^62 - 189 * q^63 + 4323 * q^64 + 390 * q^65 - 357 * q^67 - 2192 * q^68 - 102 * q^69 - 1310 * q^70 - 620 * q^71 - 1107 * q^72 - 1892 * q^73 - 581 * q^74 + 900 * q^75 - 378 * q^76 + 84 * q^78 - 415 * q^79 - 1125 * q^80 + 972 * q^81 - 2802 * q^82 - 3158 * q^83 - 1128 * q^84 + 1565 * q^85 + 747 * q^86 - 93 * q^87 + 1563 * q^89 + 405 * q^90 + 1434 * q^91 - 3466 * q^92 + 1965 * q^93 + 3 * q^94 + 255 * q^95 - 4734 * q^96 + 714 * q^97 - 6586 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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