LMFDB - Embedded newform 1849.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$109.094531601$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 59x^{8} + 42x^{7} + 1187x^{6} - 541x^{5} - 9389x^{4} + 2180x^{3} + 22676x^{2} - 320x - 768 $ x^10 - x^9 - 59x^8 + 42x^7 + 1187x^6 - 541x^5 - 9389x^4 + 2180x^3 + 22676x^2 - 320x - 768
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.23473$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.23473 q^{2} -8.85858 q^{3} +9.93292 q^{4} +5.96488 q^{5} +37.5137 q^{6} +22.5055 q^{7} -8.18537 q^{8} +51.4744 q^{9} +O(q^{10})$	\(q-4.23473 q^{2} -8.85858 q^{3} +9.93292 q^{4} +5.96488 q^{5} +37.5137 q^{6} +22.5055 q^{7} -8.18537 q^{8} +51.4744 q^{9} -25.2597 q^{10} -46.0555 q^{11} -87.9915 q^{12} +9.08413 q^{13} -95.3047 q^{14} -52.8404 q^{15} -44.8005 q^{16} -80.2232 q^{17} -217.980 q^{18} -144.978 q^{19} +59.2487 q^{20} -199.367 q^{21} +195.033 q^{22} -111.217 q^{23} +72.5108 q^{24} -89.4201 q^{25} -38.4688 q^{26} -216.809 q^{27} +223.545 q^{28} -81.3734 q^{29} +223.765 q^{30} -244.968 q^{31} +255.201 q^{32} +407.987 q^{33} +339.723 q^{34} +134.243 q^{35} +511.291 q^{36} -176.027 q^{37} +613.941 q^{38} -80.4725 q^{39} -48.8248 q^{40} -186.729 q^{41} +844.264 q^{42} -457.466 q^{44} +307.039 q^{45} +470.972 q^{46} +29.1612 q^{47} +396.869 q^{48} +163.498 q^{49} +378.670 q^{50} +710.664 q^{51} +90.2319 q^{52} -547.832 q^{53} +918.126 q^{54} -274.716 q^{55} -184.216 q^{56} +1284.30 q^{57} +344.594 q^{58} -281.600 q^{59} -524.859 q^{60} -699.247 q^{61} +1037.37 q^{62} +1158.46 q^{63} -722.302 q^{64} +54.1858 q^{65} -1727.71 q^{66} +30.6364 q^{67} -796.850 q^{68} +985.221 q^{69} -568.482 q^{70} +344.362 q^{71} -421.337 q^{72} -649.371 q^{73} +745.425 q^{74} +792.136 q^{75} -1440.05 q^{76} -1036.50 q^{77} +340.779 q^{78} -124.517 q^{79} -267.230 q^{80} +530.808 q^{81} +790.745 q^{82} +752.750 q^{83} -1980.29 q^{84} -478.522 q^{85} +720.853 q^{87} +376.981 q^{88} -44.3603 q^{89} -1300.23 q^{90} +204.443 q^{91} -1104.70 q^{92} +2170.07 q^{93} -123.490 q^{94} -864.775 q^{95} -2260.72 q^{96} -1133.55 q^{97} -692.370 q^{98} -2370.68 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 5 q^{3} + 39 q^{4} + 19 q^{5} - 15 q^{6} + 51 q^{7} - 36 q^{8} + 117 q^{9}+O(q^{10}) $ 10 * q - q^2 + 5 * q^3 + 39 * q^4 + 19 * q^5 - 15 * q^6 + 51 * q^7 - 36 * q^8 + 117 * q^9	\( 10 q - q^{2} + 5 q^{3} + 39 q^{4} + 19 q^{5} - 15 q^{6} + 51 q^{7} - 36 q^{8} + 117 q^{9} - 27 q^{10} + 27 q^{11} + 72 q^{12} + 15 q^{13} - 96 q^{14} - 65 q^{15} + 67 q^{16} + 82 q^{17} - 247 q^{18} - 78 q^{19} + 495 q^{20} - 9 q^{21} + 190 q^{22} + 61 q^{23} - 202 q^{24} + 151 q^{25} + 21 q^{26} - 97 q^{27} + 794 q^{28} + 53 q^{29} - 627 q^{30} - 253 q^{31} - 399 q^{32} + 424 q^{33} + 231 q^{34} + 355 q^{35} + 1092 q^{36} + 129 q^{37} + 854 q^{38} + 691 q^{39} - 1345 q^{40} + 391 q^{41} + 31 q^{42} + 377 q^{44} + 944 q^{45} + 40 q^{46} - 334 q^{47} + 2401 q^{48} + 115 q^{49} - 424 q^{50} + 795 q^{51} + 564 q^{52} - 773 q^{53} + 182 q^{54} + 1242 q^{55} + 923 q^{56} + 765 q^{57} - 1328 q^{58} - 1483 q^{59} + 1075 q^{60} - 437 q^{61} - 1509 q^{62} + 2222 q^{63} - 738 q^{64} - 1063 q^{65} - 1483 q^{66} + 642 q^{67} + 1052 q^{68} + 3503 q^{69} - 85 q^{70} + 1545 q^{71} - 3834 q^{72} - 1292 q^{73} + 2232 q^{74} + 82 q^{75} + 252 q^{76} - 1448 q^{77} + 2822 q^{78} + 1405 q^{79} + 3157 q^{80} - 974 q^{81} + 3304 q^{82} - 543 q^{83} + 3652 q^{84} + 973 q^{85} + 1409 q^{87} - 2686 q^{88} + 2196 q^{89} - 742 q^{90} + 3513 q^{91} - 2629 q^{92} + 983 q^{93} + 4939 q^{94} + 149 q^{95} - 3540 q^{96} - 425 q^{97} + 213 q^{98} + 3181 q^{99}+O(q^{100}) \) 10 * q - q^2 + 5 * q^3 + 39 * q^4 + 19 * q^5 - 15 * q^6 + 51 * q^7 - 36 * q^8 + 117 * q^9 - 27 * q^10 + 27 * q^11 + 72 * q^12 + 15 * q^13 - 96 * q^14 - 65 * q^15 + 67 * q^16 + 82 * q^17 - 247 * q^18 - 78 * q^19 + 495 * q^20 - 9 * q^21 + 190 * q^22 + 61 * q^23 - 202 * q^24 + 151 * q^25 + 21 * q^26 - 97 * q^27 + 794 * q^28 + 53 * q^29 - 627 * q^30 - 253 * q^31 - 399 * q^32 + 424 * q^33 + 231 * q^34 + 355 * q^35 + 1092 * q^36 + 129 * q^37 + 854 * q^38 + 691 * q^39 - 1345 * q^40 + 391 * q^41 + 31 * q^42 + 377 * q^44 + 944 * q^45 + 40 * q^46 - 334 * q^47 + 2401 * q^48 + 115 * q^49 - 424 * q^50 + 795 * q^51 + 564 * q^52 - 773 * q^53 + 182 * q^54 + 1242 * q^55 + 923 * q^56 + 765 * q^57 - 1328 * q^58 - 1483 * q^59 + 1075 * q^60 - 437 * q^61 - 1509 * q^62 + 2222 * q^63 - 738 * q^64 - 1063 * q^65 - 1483 * q^66 + 642 * q^67 + 1052 * q^68 + 3503 * q^69 - 85 * q^70 + 1545 * q^71 - 3834 * q^72 - 1292 * q^73 + 2232 * q^74 + 82 * q^75 + 252 * q^76 - 1448 * q^77 + 2822 * q^78 + 1405 * q^79 + 3157 * q^80 - 974 * q^81 + 3304 * q^82 - 543 * q^83 + 3652 * q^84 + 973 * q^85 + 1409 * q^87 - 2686 * q^88 + 2196 * q^89 - 742 * q^90 + 3513 * q^91 - 2629 * q^92 + 983 * q^93 + 4939 * q^94 + 149 * q^95 - 3540 * q^96 - 425 * q^97 + 213 * q^98 + 3181 * 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 $

$2$	−4.23473	−1.49720	−0.748601	−	0.663021i	$-0.769274\pi$
$2$	−4.23473	−1.49720	−0.748601	+	0.663021i	$0.769274\pi$
$3$	−8.85858	−1.70483	−0.852417	−	0.522862i	$-0.824864\pi$
$3$	−8.85858	−1.70483	−0.852417	+	0.522862i	$0.824864\pi$
$4$	9.93292	1.24161
$5$	5.96488	0.533516	0.266758	−	0.963764i	$-0.414048\pi$
$5$	5.96488	0.533516	0.266758	+	0.963764i	$0.414048\pi$
$6$	37.5137	2.55248
$7$	22.5055	1.21518	0.607592	−	0.794249i	$-0.292136\pi$
$7$	22.5055	1.21518	0.607592	+	0.794249i	$0.292136\pi$
$8$	−8.18537	−0.361746
$9$	51.4744	1.90646
$10$	−25.2597	−0.798781
$11$	−46.0555	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$11$	−46.0555	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$12$	−87.9915	−2.11675
$13$	9.08413	0.193806	0.0969032	−	0.995294i	$-0.469106\pi$
$13$	9.08413	0.193806	0.0969032	+	0.995294i	$0.469106\pi$
$14$	−95.3047	−1.81938
$15$	−52.8404	−0.909556
$16$	−44.8005	−0.700008
$17$	−80.2232	−1.14453	−0.572264	−	0.820069i	$-0.693935\pi$
$17$	−80.2232	−1.14453	−0.572264	+	0.820069i	$0.693935\pi$
$18$	−217.980	−2.85436
$19$	−144.978	−1.75054	−0.875268	−	0.483639i	$-0.839315\pi$
$19$	−144.978	−1.75054	−0.875268	+	0.483639i	$0.839315\pi$
$20$	59.2487	0.662421
$21$	−199.367	−2.07169
$22$	195.033	1.89005
$23$	−111.217	−1.00827	−0.504136	−	0.863624i	$-0.668189\pi$
$23$	−111.217	−1.00827	−0.504136	+	0.863624i	$0.668189\pi$
$24$	72.5108	0.616717
$25$	−89.4201	−0.715361
$26$	−38.4688	−0.290167
$27$	−216.809	−1.54537
$28$	223.545	1.50879
$29$	−81.3734	−0.521057	−0.260529	−	0.965466i	$-0.583897\pi$
$29$	−81.3734	−0.521057	−0.260529	+	0.965466i	$0.583897\pi$
$30$	223.765	1.36179
$31$	−244.968	−1.41928	−0.709638	−	0.704566i	$-0.751142\pi$
$31$	−244.968	−1.41928	−0.709638	+	0.704566i	$0.751142\pi$
$32$	255.201	1.40980
$33$	407.987	2.15216
$34$	339.723	1.71359
$35$	134.243	0.648319
$36$	511.291	2.36709
$37$	−176.027	−0.782125	−0.391062	−	0.920364i	$-0.627892\pi$
$37$	−176.027	−0.782125	−0.391062	+	0.920364i	$0.627892\pi$
$38$	613.941	2.62091
$39$	−80.4725	−0.330408
$40$	−48.8248	−0.192997
$41$	−186.729	−0.711271	−0.355636	−	0.934625i	$-0.615736\pi$
$41$	−186.729	−0.711271	−0.355636	+	0.934625i	$0.615736\pi$
$42$	844.264	3.10173
$43$	0	0
$43$	0	0
$44$	−457.466	−1.56740
$45$	307.039	1.01713
$46$	470.972	1.50959
$47$	29.1612	0.0905020	0.0452510	−	0.998976i	$-0.485591\pi$
$47$	29.1612	0.0905020	0.0452510	+	0.998976i	$0.485591\pi$
$48$	396.869	1.19340
$49$	163.498	0.476671
$50$	378.670	1.07104
$51$	710.664	1.95123
$52$	90.2319	0.240633
$53$	−547.832	−1.41982	−0.709910	−	0.704292i	$-0.751265\pi$
$53$	−547.832	−1.41982	−0.709910	+	0.704292i	$0.751265\pi$
$54$	918.126	2.31373
$55$	−274.716	−0.673503
$56$	−184.216	−0.439587
$57$	1284.30	2.98437
$58$	344.594	0.780128
$59$	−281.600	−0.621376	−0.310688	−	0.950512i	$-0.600560\pi$
$59$	−281.600	−0.621376	−0.310688	+	0.950512i	$0.600560\pi$
$60$	−524.859	−1.12932
$61$	−699.247	−1.46770	−0.733848	−	0.679314i	$-0.762277\pi$
$61$	−699.247	−1.46770	−0.733848	+	0.679314i	$0.762277\pi$
$62$	1037.37	2.12494
$63$	1158.46	2.31670
$64$	−722.302	−1.41075
$65$	54.1858	0.103399
$66$	−1727.71	−3.22222
$67$	30.6364	0.0558631	0.0279316	−	0.999610i	$-0.491108\pi$
$67$	30.6364	0.0558631	0.0279316	+	0.999610i	$0.491108\pi$
$68$	−796.850	−1.42106
$69$	985.221	1.71894
$70$	−568.482	−0.970665
$71$	344.362	0.575609	0.287805	−	0.957689i	$-0.407075\pi$
$71$	344.362	0.575609	0.287805	+	0.957689i	$0.407075\pi$
$72$	−421.337	−0.689654
$73$	−649.371	−1.04114	−0.520569	−	0.853819i	$-0.674280\pi$
$73$	−649.371	−1.04114	−0.520569	+	0.853819i	$0.674280\pi$
$74$	745.425	1.17100
$75$	792.136	1.21957
$76$	−1440.05	−2.17349
$77$	−1036.50	−1.53403
$78$	340.779	0.494687
$79$	−124.517	−0.177332	−0.0886662	−	0.996061i	$-0.528260\pi$
$79$	−124.517	−0.177332	−0.0886662	+	0.996061i	$0.528260\pi$
$80$	−267.230	−0.373465
$81$	530.808	0.728132
$82$	790.745	1.06492
$83$	752.750	0.995483	0.497741	−	0.867326i	$-0.334163\pi$
$83$	752.750	0.995483	0.497741	+	0.867326i	$0.334163\pi$
$84$	−1980.29	−2.57224
$85$	−478.522	−0.610624
$86$	0	0
$87$	720.853	0.888317
$88$	376.981	0.456663
$89$	−44.3603	−0.0528335	−0.0264167	−	0.999651i	$-0.508410\pi$
$89$	−44.3603	−0.0528335	−0.0264167	+	0.999651i	$0.508410\pi$
$90$	−1300.23	−1.52284
$91$	204.443	0.235510
$92$	−1104.70	−1.25188
$93$	2170.07	2.41963
$94$	−123.490	−0.135500
$95$	−864.775	−0.933938
$96$	−2260.72	−2.40347
$97$	−1133.55	−1.18654	−0.593271	−	0.805003i	$-0.702164\pi$
$97$	−1133.55	−1.18654	−0.593271	+	0.805003i	$0.702164\pi$
$98$	−692.370	−0.713673
$99$	−2370.68	−2.40669
$100$	−888.203	−0.888203
$101$	941.388	0.927442	0.463721	−	0.885981i	$-0.346514\pi$
$101$	941.388	0.927442	0.463721	+	0.885981i	$0.346514\pi$
$102$	−3009.47	−2.92139
$103$	93.1351	0.0890959	0.0445479	−	0.999007i	$-0.485815\pi$
$103$	93.1351	0.0890959	0.0445479	+	0.999007i	$0.485815\pi$
$104$	−74.3569	−0.0701086
$105$	−1189.20	−1.10528
$106$	2319.92	2.12576
$107$	493.709	0.446062	0.223031	−	0.974811i	$-0.428405\pi$
$107$	493.709	0.446062	0.223031	+	0.974811i	$0.428405\pi$
$108$	−2153.54	−1.91875
$109$	−1345.60	−1.18243	−0.591215	−	0.806514i	$-0.701352\pi$
$109$	−1345.60	−1.18243	−0.591215	+	0.806514i	$0.701352\pi$
$110$	1163.35	1.00837
$111$	1559.35	1.33339
$112$	−1008.26	−0.850638
$113$	851.829	0.709144	0.354572	−	0.935029i	$-0.384627\pi$
$113$	851.829	0.709144	0.354572	+	0.935029i	$0.384627\pi$
$114$	−5438.65	−4.46821
$115$	−663.394	−0.537929
$116$	−808.275	−0.646952
$117$	467.600	0.369484
$118$	1192.50	0.930326
$119$	−1805.46	−1.39081
$120$	432.518	0.329028
$121$	790.111	0.593622
$122$	2961.12	2.19744
$123$	1654.15	1.21260
$124$	−2433.25	−1.76219
$125$	−1278.99	−0.915172
$126$	−4905.76	−3.46857
$127$	887.141	0.619850	0.309925	−	0.950761i	$-0.399696\pi$
$127$	887.141	0.619850	0.309925	+	0.950761i	$0.399696\pi$
$128$	1017.15	0.702374
$129$	0	0
$130$	−229.462	−0.154809
$131$	192.747	0.128553	0.0642764	−	0.997932i	$-0.479526\pi$
$131$	192.747	0.128553	0.0642764	+	0.997932i	$0.479526\pi$
$132$	4052.50	2.67216
$133$	−3262.80	−2.12722
$134$	−129.737	−0.0836384
$135$	−1293.24	−0.824477
$136$	656.657	0.414028
$137$	489.758	0.305423	0.152711	−	0.988271i	$-0.451200\pi$
$137$	489.758	0.305423	0.152711	+	0.988271i	$0.451200\pi$
$138$	−4172.14	−2.57360
$139$	2898.23	1.76852	0.884261	−	0.466993i	$-0.154663\pi$
$139$	2898.23	1.76852	0.884261	+	0.466993i	$0.154663\pi$
$140$	1333.42	0.804963
$141$	−258.326	−0.154291
$142$	−1458.28	−0.861804
$143$	−418.374	−0.244659
$144$	−2306.08	−1.33454
$145$	−485.383	−0.277992
$146$	2749.91	1.55879
$147$	−1448.36	−0.812645
$148$	−1748.46	−0.971098
$149$	960.239	0.527959	0.263979	−	0.964528i	$-0.414965\pi$
$149$	960.239	0.527959	0.263979	+	0.964528i	$0.414965\pi$
$150$	−3354.48	−1.82595
$151$	−819.300	−0.441548	−0.220774	−	0.975325i	$-0.570858\pi$
$151$	−819.300	−0.441548	−0.220774	+	0.975325i	$0.570858\pi$
$152$	1186.70	0.633249
$153$	−4129.44	−2.18200
$154$	4389.31	2.29676
$155$	−1461.21	−0.757206
$156$	−799.326	−0.410239
$157$	−134.800	−0.0685235	−0.0342618	−	0.999413i	$-0.510908\pi$
$157$	−134.800	−0.0685235	−0.0342618	+	0.999413i	$0.510908\pi$
$158$	527.296	0.265502
$159$	4853.01	2.42056
$160$	1522.24	0.752150
$161$	−2502.99	−1.22524
$162$	−2247.83	−1.09016
$163$	−3326.69	−1.59857	−0.799283	−	0.600955i	$-0.794787\pi$
$163$	−3326.69	−1.59857	−0.799283	+	0.600955i	$0.794787\pi$
$164$	−1854.76	−0.883125
$165$	2433.59	1.14821
$166$	−3187.69	−1.49044
$167$	−1180.06	−0.546803	−0.273401	−	0.961900i	$-0.588149\pi$
$167$	−1180.06	−0.546803	−0.273401	+	0.961900i	$0.588149\pi$
$168$	1631.89	0.749424
$169$	−2114.48	−0.962439
$170$	2026.41	0.914227
$171$	−7462.65	−3.33733
$172$	0	0
$173$	−1091.38	−0.479631	−0.239816	−	0.970818i	$-0.577087\pi$
$173$	−1091.38	−0.479631	−0.239816	+	0.970818i	$0.577087\pi$
$174$	−3052.62	−1.32999
$175$	−2012.45	−0.869295
$176$	2063.31	0.883681
$177$	2494.58	1.05934
$178$	187.854	0.0791024
$179$	3759.15	1.56968	0.784838	−	0.619701i	$-0.212746\pi$
$179$	3759.15	1.56968	0.784838	+	0.619701i	$0.212746\pi$
$180$	3049.79	1.26288
$181$	−425.296	−0.174652	−0.0873259	−	0.996180i	$-0.527832\pi$
$181$	−425.296	−0.174652	−0.0873259	+	0.996180i	$0.527832\pi$
$182$	−865.760	−0.352607
$183$	6194.34	2.50218
$184$	910.349	0.364738
$185$	−1049.98	−0.417276
$186$	−9189.66	−3.62268
$187$	3694.72	1.44484
$188$	289.655	0.112369
$189$	−4879.39	−1.87790
$190$	3662.09	1.39829
$191$	3327.75	1.26067	0.630335	−	0.776324i	$-0.282918\pi$
$191$	3327.75	1.26067	0.630335	+	0.776324i	$0.282918\pi$
$192$	6398.57	2.40509
$193$	2513.07	0.937278	0.468639	−	0.883390i	$-0.344744\pi$
$193$	2513.07	0.937278	0.468639	+	0.883390i	$0.344744\pi$
$194$	4800.27	1.77649
$195$	−480.009	−0.176278
$196$	1624.01	0.591842
$197$	−2433.41	−0.880066	−0.440033	−	0.897982i	$-0.645033\pi$
$197$	−2433.41	−0.880066	−0.440033	+	0.897982i	$0.645033\pi$
$198$	10039.2	3.60330
$199$	2829.37	1.00788	0.503942	−	0.863737i	$-0.331882\pi$
$199$	2829.37	1.00788	0.503942	+	0.863737i	$0.331882\pi$
$200$	731.937	0.258779
$201$	−271.395	−0.0952374
$202$	−3986.52	−1.38857
$203$	−1831.35	−0.633180
$204$	7058.96	2.42268
$205$	−1113.82	−0.379474
$206$	−394.402	−0.133395
$207$	−5724.81	−1.92223
$208$	−406.974	−0.135666
$209$	6677.02	2.20985
$210$	5035.94	1.65482
$211$	−892.853	−0.291311	−0.145655	−	0.989335i	$-0.546529\pi$
$211$	−892.853	−0.291311	−0.145655	+	0.989335i	$0.546529\pi$
$212$	−5441.57	−1.76287
$213$	−3050.56	−0.981319
$214$	−2090.72	−0.667845
$215$	0	0
$216$	1774.66	0.559029
$217$	−5513.14	−1.72468
$218$	5698.24	1.77034
$219$	5752.50	1.77497
$220$	−2728.73	−0.836231
$221$	−728.758	−0.221817
$222$	−6603.41	−1.99636
$223$	4867.53	1.46168	0.730838	−	0.682551i	$-0.239129\pi$
$223$	4867.53	1.46168	0.730838	+	0.682551i	$0.239129\pi$
$224$	5743.43	1.71316
$225$	−4602.85	−1.36381
$226$	−3607.26	−1.06173
$227$	−4376.31	−1.27958	−0.639792	−	0.768548i	$-0.720980\pi$
$227$	−4376.31	−1.27958	−0.639792	+	0.768548i	$0.720980\pi$
$228$	12756.8	3.70544
$229$	2818.42	0.813304	0.406652	−	0.913583i	$-0.366696\pi$
$229$	2818.42	0.813304	0.406652	+	0.913583i	$0.366696\pi$
$230$	2809.29	0.805388
$231$	9181.95	2.61527
$232$	666.072	0.188490
$233$	−2809.91	−0.790058	−0.395029	−	0.918669i	$-0.629265\pi$
$233$	−2809.91	−0.790058	−0.395029	+	0.918669i	$0.629265\pi$
$234$	−1980.16	−0.553193
$235$	173.943	0.0482842
$236$	−2797.11	−0.771510
$237$	1103.04	0.302322
$238$	7645.65	2.08233
$239$	496.604	0.134404	0.0672021	−	0.997739i	$-0.478593\pi$
$239$	496.604	0.134404	0.0672021	+	0.997739i	$0.478593\pi$
$240$	2367.28	0.636696
$241$	−1348.00	−0.360300	−0.180150	−	0.983639i	$-0.557658\pi$
$241$	−1348.00	−0.360300	−0.180150	+	0.983639i	$0.557658\pi$
$242$	−3345.90	−0.888772
$243$	1151.63	0.304021
$244$	−6945.56	−1.82231
$245$	975.248	0.254311
$246$	−7004.88	−1.81551
$247$	−1317.00	−0.339265
$248$	2005.16	0.513417
$249$	−6668.30	−1.69713
$250$	5416.18	1.37020
$251$	−2677.99	−0.673439	−0.336719	−	0.941605i	$-0.609317\pi$
$251$	−2677.99	−0.673439	−0.336719	+	0.941605i	$0.609317\pi$
$252$	11506.9	2.87645
$253$	5122.14	1.27283
$254$	−3756.80	−0.928042
$255$	4239.03	1.04101
$256$	1471.08	0.359151
$257$	3557.37	0.863434	0.431717	−	0.902009i	$-0.357908\pi$
$257$	3557.37	0.863434	0.431717	+	0.902009i	$0.357908\pi$
$258$	0	0
$259$	−3961.57	−0.950425
$260$	538.223	0.128381
$261$	−4188.65	−0.993376
$262$	−816.233	−0.192470
$263$	−154.432	−0.0362080	−0.0181040	−	0.999836i	$-0.505763\pi$
$263$	−154.432	−0.0362080	−0.0181040	+	0.999836i	$0.505763\pi$
$264$	−3339.52	−0.778535
$265$	−3267.75	−0.757496
$266$	13817.1	3.18488
$267$	392.969	0.0900723
$268$	304.309	0.0693605
$269$	83.4892	0.0189235	0.00946176	−	0.999955i	$-0.496988\pi$
$269$	83.4892	0.0189235	0.00946176	+	0.999955i	$0.496988\pi$
$270$	5476.52	1.23441
$271$	3239.98	0.726254	0.363127	−	0.931740i	$-0.381709\pi$
$271$	3239.98	0.726254	0.363127	+	0.931740i	$0.381709\pi$
$272$	3594.04	0.801179
$273$	−1811.07	−0.401506
$274$	−2073.99	−0.457279
$275$	4118.29	0.903063
$276$	9786.11	2.13426
$277$	−5068.83	−1.09948	−0.549741	−	0.835335i	$-0.685274\pi$
$277$	−5068.83	−1.09948	−0.549741	+	0.835335i	$0.685274\pi$
$278$	−12273.2	−2.64784
$279$	−12609.6	−2.70580
$280$	−1098.83	−0.234527
$281$	4572.70	0.970762	0.485381	−	0.874303i	$-0.338681\pi$
$281$	4572.70	0.970762	0.485381	+	0.874303i	$0.338681\pi$
$282$	1093.94	0.231005
$283$	−3634.42	−0.763406	−0.381703	−	0.924285i	$-0.624662\pi$
$283$	−3634.42	−0.763406	−0.381703	+	0.924285i	$0.624662\pi$
$284$	3420.52	0.714685
$285$	7660.68	1.59221
$286$	1771.70	0.366304
$287$	−4202.43	−0.864325
$288$	13136.3	2.68773
$289$	1522.76	0.309945
$290$	2055.47	0.416211
$291$	10041.6	2.02286
$292$	−6450.15	−1.29269
$293$	5587.49	1.11408	0.557039	−	0.830486i	$-0.311937\pi$
$293$	5587.49	1.11408	0.557039	+	0.830486i	$0.311937\pi$
$294$	6133.42	1.21669
$295$	−1679.71	−0.331514
$296$	1440.84	0.282930
$297$	9985.24	1.95085
$298$	−4066.35	−0.790461
$299$	−1010.31	−0.195410
$300$	7868.22	1.51424
$301$	0	0
$302$	3469.51	0.661086
$303$	−8339.36	−1.58114
$304$	6495.08	1.22539
$305$	−4170.93	−0.783038
$306$	17487.1	3.26689
$307$	640.632	0.119097	0.0595486	−	0.998225i	$-0.481034\pi$
$307$	640.632	0.119097	0.0595486	+	0.998225i	$0.481034\pi$
$308$	−10295.5	−1.90468
$309$	−825.045	−0.151894
$310$	6187.81	1.13369
$311$	8471.76	1.54466	0.772330	−	0.635221i	$-0.219091\pi$
$311$	8471.76	1.54466	0.772330	+	0.635221i	$0.219091\pi$
$312$	658.697	0.119524
$313$	−7812.40	−1.41081	−0.705404	−	0.708805i	$-0.749234\pi$
$313$	−7812.40	−1.41081	−0.705404	+	0.708805i	$0.749234\pi$
$314$	570.840	0.102594
$315$	6910.07	1.23600
$316$	−1236.82	−0.220178
$317$	−11274.6	−1.99761	−0.998805	−	0.0488747i	$-0.984437\pi$
$317$	−11274.6	−1.99761	−0.998805	+	0.0488747i	$0.984437\pi$
$318$	−20551.2	−3.62407
$319$	3747.70	0.657776
$320$	−4308.45	−0.752655
$321$	−4373.56	−0.760462
$322$	10599.5	1.83443
$323$	11630.6	2.00354
$324$	5272.47	0.904059
$325$	−812.304	−0.138642
$326$	14087.6	2.39338
$327$	11920.1	2.01585
$328$	1528.44	0.257299
$329$	656.287	0.109976
$330$	−10305.6	−1.71911
$331$	−1041.60	−0.172966	−0.0864830	−	0.996253i	$-0.527563\pi$
$331$	−1041.60	−0.172966	−0.0864830	+	0.996253i	$0.527563\pi$
$332$	7477.00	1.23601
$333$	−9060.88	−1.49109
$334$	4997.25	0.818675
$335$	182.742	0.0298038
$336$	8931.74	1.45020
$337$	−5173.02	−0.836180	−0.418090	−	0.908406i	$-0.637300\pi$
$337$	−5173.02	−0.836180	−0.418090	+	0.908406i	$0.637300\pi$
$338$	8954.24	1.44097
$339$	−7545.99	−1.20897
$340$	−4753.12	−0.758159
$341$	11282.1	1.79168
$342$	31602.3	4.99665
$343$	−4039.78	−0.635941
$344$	0	0
$345$	5876.73	0.917079
$346$	4621.71	0.718105
$347$	−3932.41	−0.608366	−0.304183	−	0.952614i	$-0.598383\pi$
$347$	−3932.41	−0.608366	−0.304183	+	0.952614i	$0.598383\pi$
$348$	7160.17	1.10295
$349$	−2167.01	−0.332371	−0.166186	−	0.986094i	$-0.553145\pi$
$349$	−2167.01	−0.332371	−0.166186	+	0.986094i	$0.553145\pi$
$350$	8522.16	1.30151
$351$	−1969.52	−0.299502
$352$	−11753.4	−1.77971
$353$	3792.49	0.571824	0.285912	−	0.958256i	$-0.407703\pi$
$353$	3792.49	0.571824	0.285912	+	0.958256i	$0.407703\pi$
$354$	−10563.9	−1.58605
$355$	2054.08	0.307097
$356$	−440.627	−0.0655988
$357$	15993.8	2.37110
$358$	−15919.0	−2.35012
$359$	−6333.06	−0.931047	−0.465524	−	0.885035i	$-0.654134\pi$
$359$	−6333.06	−0.931047	−0.465524	+	0.885035i	$0.654134\pi$
$360$	−2513.23	−0.367941
$361$	14159.5	2.06437
$362$	1801.01	0.261489
$363$	−6999.26	−1.01203
$364$	2030.71	0.292413
$365$	−3873.42	−0.555464
$366$	−26231.3	−3.74627
$367$	−8820.87	−1.25462	−0.627310	−	0.778770i	$-0.715844\pi$
$367$	−8820.87	−1.25462	−0.627310	+	0.778770i	$0.715844\pi$
$368$	4982.56	0.705798
$369$	−9611.76	−1.35601
$370$	4446.38	0.624746
$371$	−12329.2	−1.72534
$372$	21555.1	3.00425
$373$	5560.79	0.771921	0.385961	−	0.922515i	$-0.373870\pi$
$373$	5560.79	0.771921	0.385961	+	0.922515i	$0.373870\pi$
$374$	−15646.1	−2.16321
$375$	11330.0	1.56022
$376$	−238.695	−0.0327387
$377$	−739.207	−0.100984
$378$	20662.9	2.81160
$379$	−8674.03	−1.17561	−0.587804	−	0.809004i	$-0.700007\pi$
$379$	−8674.03	−1.17561	−0.587804	+	0.809004i	$0.700007\pi$
$380$	−8589.74	−1.15959
$381$	−7858.81	−1.05674
$382$	−14092.1	−1.88748
$383$	14344.6	1.91377	0.956885	−	0.290467i	$-0.0938105\pi$
$383$	14344.6	1.91377	0.956885	+	0.290467i	$0.0938105\pi$
$384$	−9010.47	−1.19743
$385$	−6182.62	−0.818430
$386$	−10642.2	−1.40330
$387$	0	0
$388$	−11259.5	−1.47323
$389$	6725.93	0.876654	0.438327	−	0.898816i	$-0.355571\pi$
$389$	6725.93	0.876654	0.438327	+	0.898816i	$0.355571\pi$
$390$	2032.71	0.263923
$391$	8922.15	1.15400
$392$	−1338.29	−0.172434
$393$	−1707.47	−0.219161
$394$	10304.8	1.31764
$395$	−742.730	−0.0946096
$396$	−23547.8	−2.98818
$397$	8910.23	1.12643	0.563214	−	0.826311i	$-0.309565\pi$
$397$	8910.23	1.12643	0.563214	+	0.826311i	$0.309565\pi$
$398$	−11981.6	−1.50901
$399$	28903.8	3.62656
$400$	4006.07	0.500759
$401$	−8835.43	−1.10030	−0.550150	−	0.835066i	$-0.685430\pi$
$401$	−8835.43	−1.10030	−0.550150	+	0.835066i	$0.685430\pi$
$402$	1149.28	0.142590
$403$	−2225.32	−0.275065
$404$	9350.73	1.15153
$405$	3166.21	0.388470
$406$	7755.27	0.947999
$407$	8107.00	0.987345
$408$	−5817.04	−0.705850
$409$	−5924.70	−0.716277	−0.358139	−	0.933668i	$-0.616588\pi$
$409$	−5924.70	−0.716277	−0.358139	+	0.933668i	$0.616588\pi$
$410$	4716.70	0.568150
$411$	−4338.56	−0.520695
$412$	925.103	0.110623
$413$	−6337.55	−0.755086
$414$	24243.0	2.87797
$415$	4490.07	0.531105
$416$	2318.28	0.273228
$417$	−25674.2	−3.01504
$418$	−28275.4	−3.30860
$419$	−8197.03	−0.955730	−0.477865	−	0.878433i	$-0.658589\pi$
$419$	−8197.03	−0.955730	−0.477865	+	0.878433i	$0.658589\pi$
$420$	−11812.2	−1.37233
$421$	−7101.61	−0.822117	−0.411059	−	0.911609i	$-0.634841\pi$
$421$	−7101.61	−0.822117	−0.411059	+	0.911609i	$0.634841\pi$
$422$	3780.99	0.436151
$423$	1501.05	0.172538
$424$	4484.21	0.513614
$425$	7173.57	0.818751
$426$	12918.3	1.46923
$427$	−15736.9	−1.78352
$428$	4903.97	0.553837
$429$	3706.20	0.417103
$430$	0	0
$431$	3042.03	0.339975	0.169987	−	0.985446i	$-0.445627\pi$
$431$	3042.03	0.339975	0.169987	+	0.985446i	$0.445627\pi$
$432$	9713.15	1.08177
$433$	2767.54	0.307158	0.153579	−	0.988136i	$-0.450920\pi$
$433$	2767.54	0.307158	0.153579	+	0.988136i	$0.450920\pi$
$434$	23346.6	2.58220
$435$	4299.81	0.473931
$436$	−13365.7	−1.46812
$437$	16123.9	1.76502
$438$	−24360.3	−2.65749
$439$	7776.24	0.845421	0.422711	−	0.906265i	$-0.361079\pi$
$439$	7776.24	0.845421	0.422711	+	0.906265i	$0.361079\pi$
$440$	2248.65	0.243637
$441$	8415.98	0.908755
$442$	3086.09	0.332105
$443$	−6881.29	−0.738013	−0.369007	−	0.929427i	$-0.620302\pi$
$443$	−6881.29	−0.738013	−0.369007	+	0.929427i	$0.620302\pi$
$444$	15488.9	1.65556
$445$	−264.604	−0.0281875
$446$	−20612.7	−2.18843
$447$	−8506.36	−0.900083
$448$	−16255.8	−1.71432
$449$	−3267.04	−0.343388	−0.171694	−	0.985150i	$-0.554924\pi$
$449$	−3267.04	−0.343388	−0.171694	+	0.985150i	$0.554924\pi$
$450$	19491.8	2.04190
$451$	8599.89	0.897900
$452$	8461.14	0.880484
$453$	7257.84	0.752766
$454$	18532.5	1.91580
$455$	1219.48	0.125648
$456$	−10512.4	−1.07958
$457$	15041.8	1.53966	0.769832	−	0.638246i	$-0.220340\pi$
$457$	15041.8	1.53966	0.769832	+	0.638246i	$0.220340\pi$
$458$	−11935.2	−1.21768
$459$	17393.1	1.76871
$460$	−6589.44	−0.667900
$461$	6945.59	0.701710	0.350855	−	0.936430i	$-0.385891\pi$
$461$	6945.59	0.701710	0.350855	+	0.936430i	$0.385891\pi$
$462$	−38883.0	−3.91559
$463$	16606.8	1.66692	0.833460	−	0.552580i	$-0.186356\pi$
$463$	16606.8	1.66692	0.833460	+	0.552580i	$0.186356\pi$
$464$	3645.57	0.364744
$465$	12944.2	1.29091
$466$	11899.2	1.18288
$467$	−12959.8	−1.28418	−0.642088	−	0.766631i	$-0.721931\pi$
$467$	−12959.8	−1.28418	−0.642088	+	0.766631i	$0.721931\pi$
$468$	4644.64	0.458757
$469$	689.487	0.0678839
$470$	−736.601	−0.0722912
$471$	1194.13	0.116821
$472$	2305.00	0.224780
$473$	0	0
$474$	−4671.09	−0.452638
$475$	12963.9	1.25226
$476$	−17933.5	−1.72685
$477$	−28199.3	−2.70683
$478$	−2102.98	−0.201230
$479$	−18846.7	−1.79777	−0.898883	−	0.438189i	$-0.855620\pi$
$479$	−18846.7	−1.79777	−0.898883	+	0.438189i	$0.855620\pi$
$480$	−13484.9	−1.28229
$481$	−1599.05	−0.151581
$482$	5708.42	0.539443
$483$	22172.9	2.08882
$484$	7848.10	0.737050
$485$	−6761.49	−0.633038
$486$	−4876.84	−0.455181
$487$	−2.43644	−0.000226706 0	−0.000113353	−	1.00000i	$-0.500036\pi$
$487$	−2.43644	−0.000226706 0	−0.000113353		1.00000i	$0.500036\pi$
$488$	5723.60	0.530933
$489$	29469.7	2.72529
$490$	−4129.91	−0.380756
$491$	17349.9	1.59469	0.797344	−	0.603525i	$-0.206238\pi$
$491$	17349.9	1.59469	0.797344	+	0.603525i	$0.206238\pi$
$492$	16430.5	1.50558
$493$	6528.04	0.596365
$494$	5577.12	0.507948
$495$	−14140.8	−1.28401
$496$	10974.7	0.993505
$497$	7750.05	0.699471
$498$	28238.4	2.54095
$499$	−3385.64	−0.303731	−0.151866	−	0.988401i	$-0.548528\pi$
$499$	−3385.64	−0.303731	−0.151866	+	0.988401i	$0.548528\pi$
$500$	−12704.1	−1.13629
$501$	10453.7	0.932209
$502$	11340.6	1.00827
$503$	−4310.15	−0.382068	−0.191034	−	0.981583i	$-0.561184\pi$
$503$	−4310.15	−0.382068	−0.191034	+	0.981583i	$0.561184\pi$
$504$	−9482.41	−0.838056
$505$	5615.27	0.494805
$506$	−21690.8	−1.90568
$507$	18731.3	1.64080
$508$	8811.89	0.769615
$509$	−3107.28	−0.270585	−0.135292	−	0.990806i	$-0.543197\pi$
$509$	−3107.28	−0.270585	−0.135292	+	0.990806i	$0.543197\pi$
$510$	−17951.1	−1.55861
$511$	−14614.4	−1.26517
$512$	−14366.8	−1.24010
$513$	31432.4	2.70522
$514$	−15064.5	−1.29274
$515$	555.540	0.0475340
$516$	0	0
$517$	−1343.03	−0.114249
$518$	16776.2	1.42298
$519$	9668.10	0.817692
$520$	−443.531	−0.0374040
$521$	−6726.12	−0.565598	−0.282799	−	0.959179i	$-0.591263\pi$
$521$	−6726.12	−0.565598	−0.282799	+	0.959179i	$0.591263\pi$
$522$	17737.8	1.48728
$523$	−10484.3	−0.876569	−0.438285	−	0.898836i	$-0.644414\pi$
$523$	−10484.3	−0.876569	−0.438285	+	0.898836i	$0.644414\pi$
$524$	1914.54	0.159613
$525$	17827.4	1.48200
$526$	653.979	0.0542107
$527$	19652.1	1.62440
$528$	−18278.0	−1.50653
$529$	202.121	0.0166123
$530$	13838.0	1.13413
$531$	−14495.2	−1.18463
$532$	−32409.1	−2.64119
$533$	−1696.27	−0.137849
$534$	−1664.12	−0.134856
$535$	2944.92	0.237981
$536$	−250.770	−0.0202082
$537$	−33300.7	−2.67604
$538$	−353.554	−0.0283323
$539$	−7529.99	−0.601743
$540$	−12845.6	−1.02368
$541$	−8277.14	−0.657786	−0.328893	−	0.944367i	$-0.606675\pi$
$541$	−8277.14	−0.657786	−0.328893	+	0.944367i	$0.606675\pi$
$542$	−13720.4	−1.08735
$543$	3767.51	0.297752
$544$	−20473.0	−1.61356
$545$	−8026.34	−0.630845
$546$	7669.41	0.601136
$547$	3179.07	0.248496	0.124248	−	0.992251i	$-0.460348\pi$
$547$	3179.07	0.248496	0.124248	+	0.992251i	$0.460348\pi$
$548$	4864.73	0.379217
$549$	−35993.4	−2.79810
$550$	−17439.8	−1.35207
$551$	11797.3	0.912129
$552$	−8064.40	−0.621818
$553$	−2802.32	−0.215491
$554$	21465.1	1.64615
$555$	9301.33	0.711386
$556$	28787.9	2.19582
$557$	−19197.2	−1.46034	−0.730172	−	0.683263i	$-0.760560\pi$
$557$	−19197.2	−1.46034	−0.730172	+	0.683263i	$0.760560\pi$
$558$	53398.2	4.05112
$559$	0	0
$560$	−6014.15	−0.453829
$561$	−32730.0	−2.46321
$562$	−19364.1	−1.45343
$563$	−21982.0	−1.64552	−0.822762	−	0.568386i	$-0.807568\pi$
$563$	−21982.0	−1.64552	−0.822762	+	0.568386i	$0.807568\pi$
$564$	−2565.93	−0.191570
$565$	5081.06	0.378339
$566$	15390.8	1.14297
$567$	11946.1	0.884814
$568$	−2818.73	−0.208224
$569$	8889.61	0.654959	0.327480	−	0.944858i	$-0.393801\pi$
$569$	8889.61	0.654959	0.327480	+	0.944858i	$0.393801\pi$
$570$	−32440.9	−2.38386
$571$	10927.5	0.800877	0.400439	−	0.916324i	$-0.368858\pi$
$571$	10927.5	0.800877	0.400439	+	0.916324i	$0.368858\pi$
$572$	−4155.68	−0.303772
$573$	−29479.2	−2.14923
$574$	17796.1	1.29407
$575$	9945.00	0.721279
$576$	−37180.1	−2.68953
$577$	−15347.8	−1.10734	−0.553671	−	0.832736i	$-0.686773\pi$
$577$	−15347.8	−1.10734	−0.553671	+	0.832736i	$0.686773\pi$
$578$	−6448.48	−0.464051
$579$	−22262.2	−1.59790
$580$	−4821.27	−0.345159
$581$	16941.0	1.20969
$582$	−42523.6	−3.02863
$583$	25230.7	1.79236
$584$	5315.34	0.376627
$585$	2789.18	0.197126
$586$	−23661.5	−1.66800
$587$	10600.8	0.745388	0.372694	−	0.927954i	$-0.378434\pi$
$587$	10600.8	0.745388	0.372694	+	0.927954i	$0.378434\pi$
$588$	−14386.5	−1.00899
$589$	35514.9	2.48449
$590$	7113.12	0.496343
$591$	21556.5	1.50037
$592$	7886.09	0.547494
$593$	20036.7	1.38754	0.693768	−	0.720198i	$-0.255949\pi$
$593$	20036.7	1.38754	0.693768	+	0.720198i	$0.255949\pi$
$594$	−42284.8	−2.92082
$595$	−10769.4	−0.742020
$596$	9537.98	0.655521
$597$	−25064.2	−1.71828
$598$	4278.37	0.292568
$599$	−13670.5	−0.932490	−0.466245	−	0.884656i	$-0.654393\pi$
$599$	−13670.5	−0.932490	−0.466245	+	0.884656i	$0.654393\pi$
$600$	−6483.92	−0.441175
$601$	5710.81	0.387602	0.193801	−	0.981041i	$-0.437918\pi$
$601$	5710.81	0.387602	0.193801	+	0.981041i	$0.437918\pi$
$602$	0	0
$603$	1576.99	0.106501
$604$	−8138.04	−0.548232
$605$	4712.92	0.316707
$606$	35314.9	2.36728
$607$	1687.69	0.112852	0.0564259	−	0.998407i	$-0.482030\pi$
$607$	1687.69	0.112852	0.0564259	+	0.998407i	$0.482030\pi$
$608$	−36998.4	−2.46790
$609$	16223.2	1.07947
$610$	17662.7	1.17237
$611$	264.904	0.0175399
$612$	−41017.4	−2.70920
$613$	−28929.9	−1.90614	−0.953072	−	0.302743i	$-0.902098\pi$
$613$	−28929.9	−1.90614	−0.953072	+	0.302743i	$0.902098\pi$
$614$	−2712.90	−0.178312
$615$	9866.82	0.646941
$616$	8484.16	0.554930
$617$	−11347.4	−0.740403	−0.370202	−	0.928951i	$-0.620711\pi$
$617$	−11347.4	−0.740403	−0.370202	+	0.928951i	$0.620711\pi$
$618$	3493.84	0.227416
$619$	−11004.0	−0.714520	−0.357260	−	0.934005i	$-0.616289\pi$
$619$	−11004.0	−0.714520	−0.357260	+	0.934005i	$0.616289\pi$
$620$	−14514.0	−0.940158
$621$	24112.7	1.55815
$622$	−35875.6	−2.31267
$623$	−998.350	−0.0642024
$624$	3605.21	0.231288
$625$	3548.48	0.227103
$626$	33083.4	2.11227
$627$	−59149.0	−3.76743
$628$	−1338.95	−0.0850798
$629$	14121.4	0.895164
$630$	−29262.3	−1.85053
$631$	−8104.30	−0.511295	−0.255647	−	0.966770i	$-0.582289\pi$
$631$	−8104.30	−0.511295	−0.255647	+	0.966770i	$0.582289\pi$
$632$	1019.22	0.0641492
$633$	7909.41	0.496637
$634$	47744.7	2.99083
$635$	5291.69	0.330700
$636$	48204.6	3.00540
$637$	1485.24	0.0923819
$638$	−15870.5	−0.984824
$639$	17725.9	1.09738
$640$	6067.16	0.374727
$641$	1914.29	0.117956	0.0589781	−	0.998259i	$-0.481216\pi$
$641$	1914.29	0.117956	0.0589781	+	0.998259i	$0.481216\pi$
$642$	18520.8	1.13856
$643$	−24289.5	−1.48971	−0.744857	−	0.667224i	$-0.767482\pi$
$643$	−24289.5	−1.48971	−0.744857	+	0.667224i	$0.767482\pi$
$644$	−24861.9	−1.52127
$645$	0	0
$646$	−49252.3	−2.99970
$647$	16984.1	1.03202	0.516008	−	0.856584i	$-0.327417\pi$
$647$	16984.1	1.03202	0.516008	+	0.856584i	$0.327417\pi$
$648$	−4344.86	−0.263399
$649$	12969.2	0.784418
$650$	3439.89	0.207574
$651$	48838.5	2.94030
$652$	−33043.7	−1.98480
$653$	−1182.53	−0.0708666	−0.0354333	−	0.999372i	$-0.511281\pi$
$653$	−1182.53	−0.0708666	−0.0354333	+	0.999372i	$0.511281\pi$
$654$	−50478.3	−3.01813
$655$	1149.72	0.0685849
$656$	8365.54	0.497896
$657$	−33426.0	−1.98489
$658$	−2779.20	−0.164657
$659$	21972.6	1.29883	0.649417	−	0.760432i	$-0.275013\pi$
$659$	21972.6	1.29883	0.649417	+	0.760432i	$0.275013\pi$
$660$	24172.7	1.42564
$661$	−913.201	−0.0537358	−0.0268679	−	0.999639i	$-0.508553\pi$
$661$	−913.201	−0.0537358	−0.0268679	+	0.999639i	$0.508553\pi$
$662$	4410.91	0.258965
$663$	6455.76	0.378161
$664$	−6161.54	−0.360112
$665$	−19462.2	−1.13491
$666$	38370.4	2.23246
$667$	9050.07	0.525368
$668$	−11721.5	−0.678918
$669$	−43119.4	−2.49192
$670$	−773.864	−0.0446224
$671$	32204.2	1.85280
$672$	−50878.6	−2.92066
$673$	−18798.4	−1.07671	−0.538355	−	0.842718i	$-0.680954\pi$
$673$	−18798.4	−1.07671	−0.538355	+	0.842718i	$0.680954\pi$
$674$	21906.3	1.25193
$675$	19387.1	1.10549
$676$	−21002.9	−1.19498
$677$	10811.5	0.613768	0.306884	−	0.951747i	$-0.400714\pi$
$677$	10811.5	0.613768	0.306884	+	0.951747i	$0.400714\pi$
$678$	31955.2	1.81008
$679$	−25511.1	−1.44187
$680$	3916.88	0.220890
$681$	38767.9	2.18148
$682$	−47776.8	−2.68250
$683$	16685.1	0.934754	0.467377	−	0.884058i	$-0.345199\pi$
$683$	16685.1	0.934754	0.467377	+	0.884058i	$0.345199\pi$
$684$	−74125.8	−4.14367
$685$	2921.35	0.162948
$686$	17107.4	0.952132
$687$	−24967.2	−1.38655
$688$	0	0
$689$	−4976.57	−0.275170
$690$	−24886.3	−1.37305
$691$	−2810.44	−0.154724	−0.0773619	−	0.997003i	$-0.524650\pi$
$691$	−2810.44	−0.154724	−0.0773619	+	0.997003i	$0.524650\pi$
$692$	−10840.6	−0.595517
$693$	−53353.4	−2.92457
$694$	16652.7	0.910847
$695$	17287.6	0.943534
$696$	−5900.45	−0.321345
$697$	14980.0	0.814070
$698$	9176.71	0.497627
$699$	24891.8	1.34692
$700$	−19989.5	−1.07933
$701$	−25250.1	−1.36046	−0.680231	−	0.732998i	$-0.738121\pi$
$701$	−25250.1	−1.36046	−0.680231	+	0.732998i	$0.738121\pi$
$702$	8340.38	0.448415
$703$	25520.0	1.36914
$704$	33266.0	1.78091
$705$	−1540.89	−0.0823166
$706$	−16060.2	−0.856136
$707$	21186.4	1.12701
$708$	24778.4	1.31530
$709$	−21224.1	−1.12424	−0.562121	−	0.827055i	$-0.690014\pi$
$709$	−21224.1	−1.12424	−0.562121	+	0.827055i	$0.690014\pi$
$710$	−8698.47	−0.459786
$711$	−6409.44	−0.338077
$712$	363.105	0.0191123
$713$	27244.5	1.43102
$714$	−67729.6	−3.55002
$715$	−2495.55	−0.130529
$716$	37339.3	1.94893
$717$	−4399.20	−0.229137
$718$	26818.8	1.39397
$719$	3199.39	0.165949	0.0829744	−	0.996552i	$-0.473558\pi$
$719$	3199.39	0.165949	0.0829744	+	0.996552i	$0.473558\pi$
$720$	−13755.5	−0.711997
$721$	2096.05	0.108268
$722$	−59961.8	−3.09079
$723$	11941.4	0.614253
$724$	−4224.43	−0.216850
$725$	7276.42	0.372744
$726$	29640.0	1.51521
$727$	−8946.03	−0.456382	−0.228191	−	0.973616i	$-0.573281\pi$
$727$	−8946.03	−0.456382	−0.228191	+	0.973616i	$0.573281\pi$
$728$	−1673.44	−0.0851949
$729$	−24533.6	−1.24644
$730$	16402.9	0.831641
$731$	0	0
$732$	61527.8	3.10674
$733$	769.362	0.0387681	0.0193841	−	0.999812i	$-0.493829\pi$
$733$	769.362	0.0387681	0.0193841	+	0.999812i	$0.493829\pi$
$734$	37354.0	1.87842
$735$	−8639.31	−0.433559
$736$	−28382.6	−1.42146
$737$	−1410.97	−0.0705209
$738$	40703.2	2.03022
$739$	11278.3	0.561405	0.280703	−	0.959795i	$-0.409433\pi$
$739$	11278.3	0.561405	0.280703	+	0.959795i	$0.409433\pi$
$740$	−10429.4	−0.518096
$741$	11666.7	0.578391
$742$	52211.0	2.58319
$743$	−11316.7	−0.558775	−0.279387	−	0.960178i	$-0.590131\pi$
$743$	−11316.7	−0.558775	−0.279387	+	0.960178i	$0.590131\pi$
$744$	−17762.8	−0.875292
$745$	5727.72	0.281674
$746$	−23548.4	−1.15572
$747$	38747.4	1.89785
$748$	36699.3	1.79393
$749$	11111.2	0.542047
$750$	−47979.7	−2.33596
$751$	15324.5	0.744606	0.372303	−	0.928111i	$-0.378568\pi$
$751$	15324.5	0.744606	0.372303	+	0.928111i	$0.378568\pi$
$752$	−1306.43	−0.0633521
$753$	23723.2	1.14810
$754$	3130.34	0.151194
$755$	−4887.03	−0.235573
$756$	−48466.6	−2.33163
$757$	35157.0	1.68798	0.843992	−	0.536356i	$-0.180200\pi$
$757$	35157.0	1.68798	0.843992	+	0.536356i	$0.180200\pi$
$758$	36732.2	1.76012
$759$	−45374.9	−2.16996
$760$	7078.51	0.337848
$761$	−31368.0	−1.49421	−0.747103	−	0.664708i	$-0.768556\pi$
$761$	−31368.0	−1.49421	−0.747103	+	0.664708i	$0.768556\pi$
$762$	33279.9	1.58216
$763$	−30283.4	−1.43687
$764$	33054.3	1.56527
$765$	−24631.7	−1.16413
$766$	−60745.4	−2.86530
$767$	−2558.09	−0.120427
$768$	−13031.7	−0.612293
$769$	5835.79	0.273659	0.136830	−	0.990595i	$-0.456309\pi$
$769$	5835.79	0.273659	0.136830	+	0.990595i	$0.456309\pi$
$770$	26181.7	1.22536
$771$	−31513.2	−1.47201
$772$	24962.1	1.16374
$773$	−1446.63	−0.0673114	−0.0336557	−	0.999433i	$-0.510715\pi$
$773$	−1446.63	−0.0673114	−0.0336557	+	0.999433i	$0.510715\pi$
$774$	0	0
$775$	21905.1	1.01530
$776$	9278.53	0.429226
$777$	35093.9	1.62032
$778$	−28482.5	−1.31253
$779$	27071.5	1.24511
$780$	−4767.89	−0.218869
$781$	−15859.8	−0.726642
$782$	−37782.9	−1.72777
$783$	17642.5	0.805224
$784$	−7324.80	−0.333674
$785$	−804.065	−0.0365584
$786$	7230.66	0.328129
$787$	−39451.7	−1.78691	−0.893457	−	0.449149i	$-0.851727\pi$
$787$	−39451.7	−1.78691	−0.893457	+	0.449149i	$0.851727\pi$
$788$	−24170.8	−1.09270
$789$	1368.05	0.0617287
$790$	3145.26	0.141650
$791$	19170.8	0.861740
$792$	19404.9	0.870611
$793$	−6352.05	−0.284449
$794$	−37732.4	−1.68649
$795$	28947.7	1.29141
$796$	28103.9	1.25140
$797$	34014.5	1.51174	0.755868	−	0.654723i	$-0.227215\pi$
$797$	34014.5	1.51174	0.755868	+	0.654723i	$0.227215\pi$
$798$	−122400.	−5.42970
$799$	−2339.40	−0.103582
$800$	−22820.1	−1.00852
$801$	−2283.42	−0.100725
$802$	37415.6	1.64737
$803$	29907.1	1.31432
$804$	−2695.74	−0.118248
$805$	−14930.0	−0.653682
$806$	9423.63	0.411828
$807$	−739.596	−0.0322615
$808$	−7705.61	−0.335498
$809$	1131.67	0.0491811	0.0245905	−	0.999698i	$-0.492172\pi$
$809$	1131.67	0.0491811	0.0245905	+	0.999698i	$0.492172\pi$
$810$	−13408.0	−0.581618
$811$	−6246.47	−0.270460	−0.135230	−	0.990814i	$-0.543177\pi$
$811$	−6246.47	−0.270460	−0.135230	+	0.990814i	$0.543177\pi$
$812$	−18190.7	−0.786166
$813$	−28701.6	−1.23814
$814$	−34331.0	−1.47825
$815$	−19843.3	−0.852860
$816$	−31838.1	−1.36588
$817$	0	0
$818$	25089.5	1.07241
$819$	10523.6	0.448991
$820$	−11063.4	−0.471161
$821$	12261.7	0.521238	0.260619	−	0.965442i	$-0.416073\pi$
$821$	12261.7	0.521238	0.260619	+	0.965442i	$0.416073\pi$
$822$	18372.6	0.779586
$823$	5771.00	0.244428	0.122214	−	0.992504i	$-0.461001\pi$
$823$	5771.00	0.244428	0.122214	+	0.992504i	$0.461001\pi$
$824$	−762.345	−0.0322300
$825$	−36482.2	−1.53957
$826$	26837.8	1.13052
$827$	−36608.4	−1.53930	−0.769650	−	0.638466i	$-0.779569\pi$
$827$	−36608.4	−1.53930	−0.769650	+	0.638466i	$0.779569\pi$
$828$	−56864.1	−2.38667
$829$	26614.0	1.11501	0.557505	−	0.830173i	$-0.311759\pi$
$829$	26614.0	1.11501	0.557505	+	0.830173i	$0.311759\pi$
$830$	−19014.2	−0.795172
$831$	44902.6	1.87443
$832$	−6561.48	−0.273412
$833$	−13116.3	−0.545563
$834$	108723.	4.51412
$835$	−7038.95	−0.291728
$836$	66322.3	2.74379
$837$	53111.3	2.19330
$838$	34712.2	1.43092
$839$	2825.56	0.116268	0.0581342	−	0.998309i	$-0.481485\pi$
$839$	2825.56	0.116268	0.0581342	+	0.998309i	$0.481485\pi$
$840$	9734.05	0.399829
$841$	−17767.4	−0.728499
$842$	30073.4	1.23088
$843$	−40507.6	−1.65499
$844$	−8868.64	−0.361696
$845$	−12612.6	−0.513476
$846$	−6356.56	−0.258325
$847$	17781.9	0.721360
$848$	24543.2	0.993886
$849$	32195.8	1.30148
$850$	−30378.1	−1.22584
$851$	19577.1	0.788595
$852$	−30301.0	−1.21842
$853$	−13804.1	−0.554094	−0.277047	−	0.960856i	$-0.589356\pi$
$853$	−13804.1	−0.554094	−0.277047	+	0.960856i	$0.589356\pi$
$854$	66641.6	2.67029
$855$	−44513.8	−1.78052
$856$	−4041.19	−0.161361
$857$	18861.4	0.751799	0.375899	−	0.926660i	$-0.377334\pi$
$857$	18861.4	0.751799	0.375899	+	0.926660i	$0.377334\pi$
$858$	−15694.8	−0.624487
$859$	−41726.1	−1.65737	−0.828683	−	0.559718i	$-0.810909\pi$
$859$	−41726.1	−1.65737	−0.828683	+	0.559718i	$0.810909\pi$
$860$	0	0
$861$	37227.5	1.47353
$862$	−12882.1	−0.509011
$863$	−39816.9	−1.57055	−0.785274	−	0.619149i	$-0.787478\pi$
$863$	−39816.9	−1.57055	−0.785274	+	0.619149i	$0.787478\pi$
$864$	−55329.8	−2.17866
$865$	−6509.97	−0.255891
$866$	−11719.8	−0.459877
$867$	−13489.5	−0.528405
$868$	−54761.5	−2.14139
$869$	5734.69	0.223862
$870$	−18208.5	−0.709570
$871$	278.305	0.0108266
$872$	11014.2	0.427739
$873$	−58348.8	−2.26210
$874$	−68280.4	−2.64259
$875$	−28784.4	−1.11210
$876$	57139.1	2.20383
$877$	6115.27	0.235459	0.117730	−	0.993046i	$-0.462438\pi$
$877$	6115.27	0.235459	0.117730	+	0.993046i	$0.462438\pi$
$878$	−32930.3	−1.26577
$879$	−49497.3	−1.89932
$880$	12307.4	0.471458
$881$	38448.0	1.47031	0.735156	−	0.677898i	$-0.237109\pi$
$881$	38448.0	1.47031	0.735156	+	0.677898i	$0.237109\pi$
$882$	−35639.4	−1.36059
$883$	37344.5	1.42326	0.711631	−	0.702553i	$-0.247957\pi$
$883$	37344.5	1.42326	0.711631	+	0.702553i	$0.247957\pi$
$884$	−7238.69	−0.275411
$885$	14879.9	0.565176
$886$	29140.4	1.10495
$887$	−38270.3	−1.44869	−0.724347	−	0.689436i	$-0.757859\pi$
$887$	−38270.3	−1.44869	−0.724347	+	0.689436i	$0.757859\pi$
$888$	−12763.8	−0.482349
$889$	19965.6	0.753232
$890$	1120.53	0.0422023
$891$	−24446.7	−0.919185
$892$	48348.8	1.81484
$893$	−4227.72	−0.158427
$894$	36022.1	1.34761
$895$	22422.9	0.837447
$896$	22891.4	0.853513
$897$	8949.87	0.333141
$898$	13835.0	0.514121
$899$	19933.9	0.739525
$900$	−45719.7	−1.69332
$901$	43948.8	1.62503
$902$	−36418.2	−1.34434
$903$	0	0
$904$	−6972.53	−0.256530
$905$	−2536.84	−0.0931794
$906$	−30735.0	−1.12704
$907$	−26916.5	−0.985390	−0.492695	−	0.870202i	$-0.663988\pi$
$907$	−26916.5	−0.985390	−0.492695	+	0.870202i	$0.663988\pi$
$908$	−43469.5	−1.58875
$909$	48457.4	1.76813
$910$	−5164.16	−0.188121
$911$	39639.9	1.44163	0.720816	−	0.693126i	$-0.243767\pi$
$911$	39639.9	1.44163	0.720816	+	0.693126i	$0.243767\pi$
$912$	−57537.2	−2.08909
$913$	−34668.3	−1.25668
$914$	−63698.0	−2.30519
$915$	36948.5	1.33495
$916$	27995.1	1.00981
$917$	4337.88	0.156215
$918$	−73655.0	−2.64812
$919$	−32819.6	−1.17804	−0.589020	−	0.808119i	$-0.700486\pi$
$919$	−32819.6	−1.17804	−0.589020	+	0.808119i	$0.700486\pi$
$920$	5430.13	0.194593
$921$	−5675.09	−0.203041
$922$	−29412.7	−1.05060
$923$	3128.23	0.111557
$924$	91203.5	3.24716
$925$	15740.3	0.559502
$926$	−70325.3	−2.49572
$927$	4794.08	0.169858
$928$	−20766.6	−0.734586
$929$	−27270.5	−0.963095	−0.481547	−	0.876420i	$-0.659925\pi$
$929$	−27270.5	−0.963095	−0.481547	+	0.876420i	$0.659925\pi$
$930$	−54815.2	−1.93276
$931$	−23703.6	−0.834429
$932$	−27910.6	−0.980947
$933$	−75047.8	−2.63339
$934$	54881.4	1.92267
$935$	22038.6	0.770844
$936$	−3827.48	−0.133659
$937$	12201.1	0.425393	0.212696	−	0.977118i	$-0.431775\pi$
$937$	12201.1	0.425393	0.212696	+	0.977118i	$0.431775\pi$
$938$	−2919.79	−0.101636
$939$	69206.8	2.40519
$940$	1727.76	0.0599504
$941$	−10590.9	−0.366902	−0.183451	−	0.983029i	$-0.558727\pi$
$941$	−10590.9	−0.366902	−0.183451	+	0.983029i	$0.558727\pi$
$942$	−5056.83	−0.174905
$943$	20767.3	0.717155
$944$	12615.8	0.434968
$945$	−29105.0	−1.00189
$946$	0	0
$947$	−8630.28	−0.296142	−0.148071	−	0.988977i	$-0.547306\pi$
$947$	−8630.28	−0.296142	−0.148071	+	0.988977i	$0.547306\pi$
$948$	10956.4	0.375368
$949$	−5898.97	−0.201779
$950$	−54898.7	−1.87489
$951$	99876.6	3.40559
$952$	14778.4	0.503120
$953$	−10184.4	−0.346176	−0.173088	−	0.984906i	$-0.555374\pi$
$953$	−10184.4	−0.346176	−0.173088	+	0.984906i	$0.555374\pi$
$954$	119417.	4.05268
$955$	19849.7	0.672587
$956$	4932.72	0.166878
$957$	−33199.3	−1.12140
$958$	79810.8	2.69162
$959$	11022.3	0.371144
$960$	38166.7	1.28315
$961$	30218.4	1.01435
$962$	6771.54	0.226947
$963$	25413.4	0.850399
$964$	−13389.6	−0.447354
$965$	14990.2	0.500053
$966$	−93896.2	−3.12739
$967$	32879.0	1.09340	0.546700	−	0.837329i	$-0.315884\pi$
$967$	32879.0	1.09340	0.546700	+	0.837329i	$0.315884\pi$
$968$	−6467.35	−0.214740
$969$	−103030.	−3.41570
$970$	28633.1	0.947787
$971$	29420.4	0.972345	0.486172	−	0.873863i	$-0.338393\pi$
$971$	29420.4	0.972345	0.486172	+	0.873863i	$0.338393\pi$
$972$	11439.0	0.377477
$973$	65226.1	2.14908
$974$	10.3177	0.000339424 0
$975$	7195.86	0.236361
$976$	31326.6	1.02740
$977$	31685.1	1.03756	0.518780	−	0.854908i	$-0.326387\pi$
$977$	31685.1	1.03756	0.518780	+	0.854908i	$0.326387\pi$
$978$	−124796.	−4.08031
$979$	2043.03	0.0666963
$980$	9687.05	0.315757
$981$	−69263.9	−2.25426
$982$	−73472.3	−2.38757
$983$	20915.3	0.678631	0.339315	−	0.940673i	$-0.389805\pi$
$983$	20915.3	0.678631	0.339315	+	0.940673i	$0.389805\pi$
$984$	−13539.8	−0.438653
$985$	−14515.0	−0.469529
$986$	−27644.5	−0.892879
$987$	−5813.77	−0.187492
$988$	−13081.6	−0.421236
$989$	0	0
$990$	59882.6	1.92242
$991$	−34852.4	−1.11718	−0.558589	−	0.829445i	$-0.688657\pi$
$991$	−34852.4	−1.11718	−0.558589	+	0.829445i	$0.688657\pi$
$992$	−62516.1	−2.00090
$993$	9227.14	0.294878
$994$	−32819.3	−1.04725
$995$	16876.9	0.537722
$996$	−66235.6	−2.10718
$997$	3913.94	0.124329	0.0621644	−	0.998066i	$-0.480200\pi$
$997$	3913.94	0.124329	0.0621644	+	0.998066i	$0.480200\pi$
$998$	14337.3	0.454747
$999$	38164.2	1.20867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.d.1.2		10
43.6	even	3		43.4.c.a.36.2	yes	20
43.36	even	3		43.4.c.a.6.2	✓	20
43.42	odd	2		1849.4.a.f.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.c.a.6.2	✓	20	43.36	even	3
43.4.c.a.36.2	yes	20	43.6	even	3
1849.4.a.d.1.2		10	1.1	even	1	trivial
1849.4.a.f.1.9		10	43.42	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-4.23473 q^{2} -8.85858 q^{3} +9.93292 q^{4} +5.96488 q^{5} +37.5137 q^{6} +22.5055 q^{7} -8.18537 q^{8} +51.4744 q^{9} +O(q^{10})\)	\(q-4.23473 q^{2} -8.85858 q^{3} +9.93292 q^{4} +5.96488 q^{5} +37.5137 q^{6} +22.5055 q^{7} -8.18537 q^{8} +51.4744 q^{9} -25.2597 q^{10} -46.0555 q^{11} -87.9915 q^{12} +9.08413 q^{13} -95.3047 q^{14} -52.8404 q^{15} -44.8005 q^{16} -80.2232 q^{17} -217.980 q^{18} -144.978 q^{19} +59.2487 q^{20} -199.367 q^{21} +195.033 q^{22} -111.217 q^{23} +72.5108 q^{24} -89.4201 q^{25} -38.4688 q^{26} -216.809 q^{27} +223.545 q^{28} -81.3734 q^{29} +223.765 q^{30} -244.968 q^{31} +255.201 q^{32} +407.987 q^{33} +339.723 q^{34} +134.243 q^{35} +511.291 q^{36} -176.027 q^{37} +613.941 q^{38} -80.4725 q^{39} -48.8248 q^{40} -186.729 q^{41} +844.264 q^{42} -457.466 q^{44} +307.039 q^{45} +470.972 q^{46} +29.1612 q^{47} +396.869 q^{48} +163.498 q^{49} +378.670 q^{50} +710.664 q^{51} +90.2319 q^{52} -547.832 q^{53} +918.126 q^{54} -274.716 q^{55} -184.216 q^{56} +1284.30 q^{57} +344.594 q^{58} -281.600 q^{59} -524.859 q^{60} -699.247 q^{61} +1037.37 q^{62} +1158.46 q^{63} -722.302 q^{64} +54.1858 q^{65} -1727.71 q^{66} +30.6364 q^{67} -796.850 q^{68} +985.221 q^{69} -568.482 q^{70} +344.362 q^{71} -421.337 q^{72} -649.371 q^{73} +745.425 q^{74} +792.136 q^{75} -1440.05 q^{76} -1036.50 q^{77} +340.779 q^{78} -124.517 q^{79} -267.230 q^{80} +530.808 q^{81} +790.745 q^{82} +752.750 q^{83} -1980.29 q^{84} -478.522 q^{85} +720.853 q^{87} +376.981 q^{88} -44.3603 q^{89} -1300.23 q^{90} +204.443 q^{91} -1104.70 q^{92} +2170.07 q^{93} -123.490 q^{94} -864.775 q^{95} -2260.72 q^{96} -1133.55 q^{97} -692.370 q^{98} -2370.68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 5 q^{3} + 39 q^{4} + 19 q^{5} - 15 q^{6} + 51 q^{7} - 36 q^{8} + 117 q^{9}+O(q^{10}) \) 10 * q - q^2 + 5 * q^3 + 39 * q^4 + 19 * q^5 - 15 * q^6 + 51 * q^7 - 36 * q^8 + 117 * q^9	\( 10 q - q^{2} + 5 q^{3} + 39 q^{4} + 19 q^{5} - 15 q^{6} + 51 q^{7} - 36 q^{8} + 117 q^{9} - 27 q^{10} + 27 q^{11} + 72 q^{12} + 15 q^{13} - 96 q^{14} - 65 q^{15} + 67 q^{16} + 82 q^{17} - 247 q^{18} - 78 q^{19} + 495 q^{20} - 9 q^{21} + 190 q^{22} + 61 q^{23} - 202 q^{24} + 151 q^{25} + 21 q^{26} - 97 q^{27} + 794 q^{28} + 53 q^{29} - 627 q^{30} - 253 q^{31} - 399 q^{32} + 424 q^{33} + 231 q^{34} + 355 q^{35} + 1092 q^{36} + 129 q^{37} + 854 q^{38} + 691 q^{39} - 1345 q^{40} + 391 q^{41} + 31 q^{42} + 377 q^{44} + 944 q^{45} + 40 q^{46} - 334 q^{47} + 2401 q^{48} + 115 q^{49} - 424 q^{50} + 795 q^{51} + 564 q^{52} - 773 q^{53} + 182 q^{54} + 1242 q^{55} + 923 q^{56} + 765 q^{57} - 1328 q^{58} - 1483 q^{59} + 1075 q^{60} - 437 q^{61} - 1509 q^{62} + 2222 q^{63} - 738 q^{64} - 1063 q^{65} - 1483 q^{66} + 642 q^{67} + 1052 q^{68} + 3503 q^{69} - 85 q^{70} + 1545 q^{71} - 3834 q^{72} - 1292 q^{73} + 2232 q^{74} + 82 q^{75} + 252 q^{76} - 1448 q^{77} + 2822 q^{78} + 1405 q^{79} + 3157 q^{80} - 974 q^{81} + 3304 q^{82} - 543 q^{83} + 3652 q^{84} + 973 q^{85} + 1409 q^{87} - 2686 q^{88} + 2196 q^{89} - 742 q^{90} + 3513 q^{91} - 2629 q^{92} + 983 q^{93} + 4939 q^{94} + 149 q^{95} - 3540 q^{96} - 425 q^{97} + 213 q^{98} + 3181 q^{99}+O(q^{100}) \) 10 * q - q^2 + 5 * q^3 + 39 * q^4 + 19 * q^5 - 15 * q^6 + 51 * q^7 - 36 * q^8 + 117 * q^9 - 27 * q^10 + 27 * q^11 + 72 * q^12 + 15 * q^13 - 96 * q^14 - 65 * q^15 + 67 * q^16 + 82 * q^17 - 247 * q^18 - 78 * q^19 + 495 * q^20 - 9 * q^21 + 190 * q^22 + 61 * q^23 - 202 * q^24 + 151 * q^25 + 21 * q^26 - 97 * q^27 + 794 * q^28 + 53 * q^29 - 627 * q^30 - 253 * q^31 - 399 * q^32 + 424 * q^33 + 231 * q^34 + 355 * q^35 + 1092 * q^36 + 129 * q^37 + 854 * q^38 + 691 * q^39 - 1345 * q^40 + 391 * q^41 + 31 * q^42 + 377 * q^44 + 944 * q^45 + 40 * q^46 - 334 * q^47 + 2401 * q^48 + 115 * q^49 - 424 * q^50 + 795 * q^51 + 564 * q^52 - 773 * q^53 + 182 * q^54 + 1242 * q^55 + 923 * q^56 + 765 * q^57 - 1328 * q^58 - 1483 * q^59 + 1075 * q^60 - 437 * q^61 - 1509 * q^62 + 2222 * q^63 - 738 * q^64 - 1063 * q^65 - 1483 * q^66 + 642 * q^67 + 1052 * q^68 + 3503 * q^69 - 85 * q^70 + 1545 * q^71 - 3834 * q^72 - 1292 * q^73 + 2232 * q^74 + 82 * q^75 + 252 * q^76 - 1448 * q^77 + 2822 * q^78 + 1405 * q^79 + 3157 * q^80 - 974 * q^81 + 3304 * q^82 - 543 * q^83 + 3652 * q^84 + 973 * q^85 + 1409 * q^87 - 2686 * q^88 + 2196 * q^89 - 742 * q^90 + 3513 * q^91 - 2629 * q^92 + 983 * q^93 + 4939 * q^94 + 149 * q^95 - 3540 * q^96 - 425 * q^97 + 213 * q^98 + 3181 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more