LMFDB - Embedded newform 2057.4.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2057,4,Mod(1,2057)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2057, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2057.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2057 = 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2057.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:	$121.366928882$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 112 x^{18} + 438 x^{17} + 5176 x^{16} - 19774 x^{15} - 127872 x^{14} + \cdots + 10454400 $ x^20 - 4x^19 - 112x^18 + 438x^17 + 5176x^16 - 19774x^15 - 127872x^14 + 475275x^13 + 1833771x^12 - 6551919x^11 - 15615332x^10 + 52010407x^9 + 78728552x^8 - 226862943x^7 - 231699432x^6 + 488433304x^5 + 362730704x^4 - 405382512x^3 - 187679328x^2 + 109987200*x + 10454400
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.40856$ of defining polynomial
Character	$\chi$	$=$	2057.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.40856 q^{2} +9.44900 q^{3} +11.4354 q^{4} -19.6923 q^{5} -41.6565 q^{6} +12.0675 q^{7} -15.1450 q^{8} +62.2837 q^{9} +O(q^{10})$	\(q-4.40856 q^{2} +9.44900 q^{3} +11.4354 q^{4} -19.6923 q^{5} -41.6565 q^{6} +12.0675 q^{7} -15.1450 q^{8} +62.2837 q^{9} +86.8147 q^{10} +108.053 q^{12} -35.7452 q^{13} -53.2001 q^{14} -186.073 q^{15} -24.7153 q^{16} -17.0000 q^{17} -274.581 q^{18} -83.9471 q^{19} -225.189 q^{20} +114.026 q^{21} +134.253 q^{23} -143.105 q^{24} +262.788 q^{25} +157.585 q^{26} +333.395 q^{27} +137.996 q^{28} -197.344 q^{29} +820.313 q^{30} +175.791 q^{31} +230.119 q^{32} +74.9455 q^{34} -237.637 q^{35} +712.237 q^{36} +18.0835 q^{37} +370.085 q^{38} -337.757 q^{39} +298.240 q^{40} +349.485 q^{41} -502.688 q^{42} -125.513 q^{43} -1226.51 q^{45} -591.861 q^{46} -489.504 q^{47} -233.535 q^{48} -197.376 q^{49} -1158.51 q^{50} -160.633 q^{51} -408.760 q^{52} +577.046 q^{53} -1469.79 q^{54} -182.762 q^{56} -793.216 q^{57} +870.002 q^{58} +736.852 q^{59} -2127.81 q^{60} -707.512 q^{61} -774.986 q^{62} +751.606 q^{63} -816.770 q^{64} +703.906 q^{65} -159.321 q^{67} -194.401 q^{68} +1268.55 q^{69} +1047.63 q^{70} +6.40350 q^{71} -943.287 q^{72} -628.355 q^{73} -79.7220 q^{74} +2483.08 q^{75} -959.965 q^{76} +1489.02 q^{78} -1084.60 q^{79} +486.702 q^{80} +1468.60 q^{81} -1540.72 q^{82} +1142.72 q^{83} +1303.92 q^{84} +334.770 q^{85} +553.331 q^{86} -1864.70 q^{87} -479.421 q^{89} +5407.14 q^{90} -431.354 q^{91} +1535.23 q^{92} +1661.05 q^{93} +2158.01 q^{94} +1653.11 q^{95} +2174.39 q^{96} -317.777 q^{97} +870.144 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{2} + 8 q^{3} + 80 q^{4} + 6 q^{5} - 93 q^{6} - 52 q^{7} - 30 q^{8} + 194 q^{9}+O(q^{10}) $ 20 * q - 4 * q^2 + 8 * q^3 + 80 * q^4 + 6 * q^5 - 93 * q^6 - 52 * q^7 - 30 * q^8 + 194 * q^9	\( 20 q - 4 q^{2} + 8 q^{3} + 80 q^{4} + 6 q^{5} - 93 q^{6} - 52 q^{7} - 30 q^{8} + 194 q^{9} - 66 q^{10} + 143 q^{12} - 200 q^{13} + 85 q^{14} - 70 q^{15} + 320 q^{16} - 340 q^{17} - 160 q^{18} - 188 q^{19} + 21 q^{20} + 64 q^{21} - 54 q^{23} - 664 q^{24} + 830 q^{25} + 59 q^{26} + 302 q^{27} - 227 q^{28} - 166 q^{29} - 409 q^{30} + 26 q^{31} - 534 q^{32} + 68 q^{34} - 540 q^{35} + 1813 q^{36} - 120 q^{37} + 49 q^{38} + 482 q^{39} - 1265 q^{40} - 480 q^{41} - 675 q^{42} - 1158 q^{43} - 142 q^{45} - 581 q^{46} + 372 q^{47} + 700 q^{48} - 666 q^{49} - 168 q^{50} - 136 q^{51} - 152 q^{52} - 200 q^{53} - 2051 q^{54} + 551 q^{56} - 1504 q^{57} + 2806 q^{58} + 146 q^{59} - 2549 q^{60} - 1476 q^{61} - 2844 q^{62} - 1798 q^{63} + 1532 q^{64} - 740 q^{65} - 254 q^{67} - 1360 q^{68} + 2566 q^{69} - 860 q^{70} + 394 q^{71} - 489 q^{72} - 2244 q^{73} + 3056 q^{74} + 868 q^{75} - 3345 q^{76} + 612 q^{78} - 1674 q^{79} + 3533 q^{80} - 2676 q^{81} - 3925 q^{82} + 96 q^{83} - 4468 q^{84} - 102 q^{85} - 569 q^{86} - 5498 q^{87} + 1592 q^{89} + 4486 q^{90} + 3364 q^{91} - 4854 q^{92} - 2540 q^{93} - 836 q^{94} + 136 q^{95} - 4207 q^{96} + 1802 q^{97} - 1078 q^{98}+O(q^{100}) \) 20 * q - 4 * q^2 + 8 * q^3 + 80 * q^4 + 6 * q^5 - 93 * q^6 - 52 * q^7 - 30 * q^8 + 194 * q^9 - 66 * q^10 + 143 * q^12 - 200 * q^13 + 85 * q^14 - 70 * q^15 + 320 * q^16 - 340 * q^17 - 160 * q^18 - 188 * q^19 + 21 * q^20 + 64 * q^21 - 54 * q^23 - 664 * q^24 + 830 * q^25 + 59 * q^26 + 302 * q^27 - 227 * q^28 - 166 * q^29 - 409 * q^30 + 26 * q^31 - 534 * q^32 + 68 * q^34 - 540 * q^35 + 1813 * q^36 - 120 * q^37 + 49 * q^38 + 482 * q^39 - 1265 * q^40 - 480 * q^41 - 675 * q^42 - 1158 * q^43 - 142 * q^45 - 581 * q^46 + 372 * q^47 + 700 * q^48 - 666 * q^49 - 168 * q^50 - 136 * q^51 - 152 * q^52 - 200 * q^53 - 2051 * q^54 + 551 * q^56 - 1504 * q^57 + 2806 * q^58 + 146 * q^59 - 2549 * q^60 - 1476 * q^61 - 2844 * q^62 - 1798 * q^63 + 1532 * q^64 - 740 * q^65 - 254 * q^67 - 1360 * q^68 + 2566 * q^69 - 860 * q^70 + 394 * q^71 - 489 * q^72 - 2244 * q^73 + 3056 * q^74 + 868 * q^75 - 3345 * q^76 + 612 * q^78 - 1674 * q^79 + 3533 * q^80 - 2676 * q^81 - 3925 * q^82 + 96 * q^83 - 4468 * q^84 - 102 * q^85 - 569 * q^86 - 5498 * q^87 + 1592 * q^89 + 4486 * q^90 + 3364 * q^91 - 4854 * q^92 - 2540 * q^93 - 836 * q^94 + 136 * q^95 - 4207 * q^96 + 1802 * q^97 - 1078 * 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 $

$2$	−4.40856	−1.55866	−0.779330	−	0.626614i	$-0.784440\pi$
$2$	−4.40856	−1.55866	−0.779330	+	0.626614i	$0.784440\pi$
$3$	9.44900	1.81846	0.909231	−	0.416292i	$-0.136671\pi$
$3$	9.44900	1.81846	0.909231	+	0.416292i	$0.136671\pi$
$4$	11.4354	1.42942
$5$	−19.6923	−1.76134	−0.880668	−	0.473735i	$-0.842906\pi$
$5$	−19.6923	−1.76134	−0.880668	+	0.473735i	$0.842906\pi$
$6$	−41.6565	−2.83436
$7$	12.0675	0.651582	0.325791	−	0.945442i	$-0.394369\pi$
$7$	12.0675	0.651582	0.325791	+	0.945442i	$0.394369\pi$
$8$	−15.1450	−0.669321
$9$	62.2837	2.30680
$10$	86.8147	2.74532
$11$	0	0
$11$	0	0
$12$	108.053	2.59935
$13$	−35.7452	−0.762610	−0.381305	−	0.924449i	$-0.624525\pi$
$13$	−35.7452	−0.762610	−0.381305	+	0.924449i	$0.624525\pi$
$14$	−53.2001	−1.01560
$15$	−186.073	−3.20292
$16$	−24.7153	−0.386177
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−274.581	−3.59552
$19$	−83.9471	−1.01362	−0.506810	−	0.862058i	$-0.669175\pi$
$19$	−83.9471	−1.01362	−0.506810	+	0.862058i	$0.669175\pi$
$20$	−225.189	−2.51769
$21$	114.026	1.18488
$22$	0	0
$23$	134.253	1.21711	0.608557	−	0.793510i	$-0.291749\pi$
$23$	134.253	1.21711	0.608557	+	0.793510i	$0.291749\pi$
$24$	−143.105	−1.21713
$25$	262.788	2.10230
$26$	157.585	1.18865
$27$	333.395	2.37637
$28$	137.996	0.931385
$29$	−197.344	−1.26365	−0.631825	−	0.775111i	$-0.717694\pi$
$29$	−197.344	−1.26365	−0.631825	+	0.775111i	$0.717694\pi$
$30$	820.313	4.99226
$31$	175.791	1.01849	0.509243	−	0.860623i	$-0.329925\pi$
$31$	175.791	1.01849	0.509243	+	0.860623i	$0.329925\pi$
$32$	230.119	1.27124
$33$	0	0
$34$	74.9455	0.378031
$35$	−237.637	−1.14765
$36$	712.237	3.29739
$37$	18.0835	0.0803487	0.0401744	−	0.999193i	$-0.487209\pi$
$37$	18.0835	0.0803487	0.0401744	+	0.999193i	$0.487209\pi$
$38$	370.085	1.57989
$39$	−337.757	−1.38678
$40$	298.240	1.17890
$41$	349.485	1.33123	0.665615	−	0.746296i	$-0.268169\pi$
$41$	349.485	1.33123	0.665615	+	0.746296i	$0.268169\pi$
$42$	−502.688	−1.84682
$43$	−125.513	−0.445129	−0.222565	−	0.974918i	$-0.571443\pi$
$43$	−125.513	−0.445129	−0.222565	+	0.974918i	$0.571443\pi$
$44$	0	0
$45$	−1226.51	−4.06305
$46$	−591.861	−1.89707
$47$	−489.504	−1.51918	−0.759590	−	0.650402i	$-0.774601\pi$
$47$	−489.504	−1.51918	−0.759590	+	0.650402i	$0.774601\pi$
$48$	−233.535	−0.702248
$49$	−197.376	−0.575441
$50$	−1158.51	−3.27677
$51$	−160.633	−0.441042
$52$	−408.760	−1.09009
$53$	577.046	1.49554	0.747768	−	0.663960i	$-0.231125\pi$
$53$	577.046	1.49554	0.747768	+	0.663960i	$0.231125\pi$
$54$	−1469.79	−3.70395
$55$	0	0
$56$	−182.762	−0.436118
$57$	−793.216	−1.84323
$58$	870.002	1.96960
$59$	736.852	1.62593	0.812965	−	0.582312i	$-0.197852\pi$
$59$	736.852	1.62593	0.812965	+	0.582312i	$0.197852\pi$
$60$	−2127.81	−4.57832
$61$	−707.512	−1.48504	−0.742521	−	0.669822i	$-0.766370\pi$
$61$	−707.512	−1.48504	−0.742521	+	0.669822i	$0.766370\pi$
$62$	−774.986	−1.58747
$63$	751.606	1.50307
$64$	−816.770	−1.59525
$65$	703.906	1.34321
$66$	0	0
$67$	−159.321	−0.290511	−0.145255	−	0.989394i	$-0.546400\pi$
$67$	−159.321	−0.290511	−0.145255	+	0.989394i	$0.546400\pi$
$68$	−194.401	−0.346685
$69$	1268.55	2.21328
$70$	1047.63	1.78880
$71$	6.40350	0.0107036	0.00535180	−	0.999986i	$-0.498296\pi$
$71$	6.40350	0.0107036	0.00535180	+	0.999986i	$0.498296\pi$
$72$	−943.287	−1.54399
$73$	−628.355	−1.00744	−0.503722	−	0.863866i	$-0.668036\pi$
$73$	−628.355	−1.00744	−0.503722	+	0.863866i	$0.668036\pi$
$74$	−79.7220	−0.125236
$75$	2483.08	3.82295
$76$	−959.965	−1.44889
$77$	0	0
$78$	1489.02	2.16152
$79$	−1084.60	−1.54464	−0.772322	−	0.635232i	$-0.780905\pi$
$79$	−1084.60	−1.54464	−0.772322	+	0.635232i	$0.780905\pi$
$80$	486.702	0.680187
$81$	1468.60	2.01453
$82$	−1540.72	−2.07493
$83$	1142.72	1.51121	0.755603	−	0.655030i	$-0.227344\pi$
$83$	1142.72	1.51121	0.755603	+	0.655030i	$0.227344\pi$
$84$	1303.92	1.69369
$85$	334.770	0.427186
$86$	553.331	0.693805
$87$	−1864.70	−2.29790
$88$	0	0
$89$	−479.421	−0.570995	−0.285498	−	0.958379i	$-0.592159\pi$
$89$	−479.421	−0.570995	−0.285498	+	0.958379i	$0.592159\pi$
$90$	5407.14	6.33292
$91$	−431.354	−0.496903
$92$	1535.23	1.73977
$93$	1661.05	1.85208
$94$	2158.01	2.36789
$95$	1653.11	1.78532
$96$	2174.39	2.31170
$97$	−317.777	−0.332633	−0.166316	−	0.986072i	$-0.553187\pi$
$97$	−317.777	−0.332633	−0.166316	+	0.986072i	$0.553187\pi$
$98$	870.144	0.896916
$99$	0	0
$100$	3005.07	3.00507
$101$	−1269.42	−1.25061	−0.625305	−	0.780380i	$-0.715026\pi$
$101$	−1269.42	−1.25061	−0.625305	+	0.780380i	$0.715026\pi$
$102$	708.160	0.687434
$103$	−933.442	−0.892959	−0.446480	−	0.894794i	$-0.647322\pi$
$103$	−933.442	−0.892959	−0.446480	+	0.894794i	$0.647322\pi$
$104$	541.361	0.510431
$105$	−2245.43	−2.08697
$106$	−2543.94	−2.33103
$107$	640.395	0.578592	0.289296	−	0.957240i	$-0.406579\pi$
$107$	640.395	0.578592	0.289296	+	0.957240i	$0.406579\pi$
$108$	3812.50	3.39683
$109$	1056.54	0.928420	0.464210	−	0.885725i	$-0.346338\pi$
$109$	1056.54	0.928420	0.464210	+	0.885725i	$0.346338\pi$
$110$	0	0
$111$	170.871	0.146111
$112$	−298.251	−0.251626
$113$	−1490.30	−1.24067	−0.620336	−	0.784336i	$-0.713004\pi$
$113$	−1490.30	−1.24067	−0.620336	+	0.784336i	$0.713004\pi$
$114$	3496.94	2.87297
$115$	−2643.75	−2.14375
$116$	−2256.70	−1.80629
$117$	−2226.34	−1.75919
$118$	−3248.45	−2.53427
$119$	−205.147	−0.158032
$120$	2818.07	2.14378
$121$	0	0
$122$	3119.11	2.31468
$123$	3302.29	2.42079
$124$	2010.24	1.45585
$125$	−2713.36	−1.94152
$126$	−3313.50	−2.34278
$127$	−19.9050	−0.0139077	−0.00695386	−	0.999976i	$-0.502213\pi$
$127$	−19.9050	−0.0139077	−0.00695386	+	0.999976i	$0.502213\pi$
$128$	1759.82	1.21522
$129$	−1185.97	−0.809450
$130$	−3103.21	−2.09361
$131$	−10.0666	−0.00671395	−0.00335697	−	0.999994i	$-0.501069\pi$
$131$	−10.0666	−0.00671395	−0.00335697	+	0.999994i	$0.501069\pi$
$132$	0	0
$133$	−1013.03	−0.660457
$134$	702.378	0.452807
$135$	−6565.33	−4.18558
$136$	257.465	0.162334
$137$	1869.24	1.16569	0.582846	−	0.812582i	$-0.301939\pi$
$137$	1869.24	1.16569	0.582846	+	0.812582i	$0.301939\pi$
$138$	−5592.50	−3.44975
$139$	−121.562	−0.0741782	−0.0370891	−	0.999312i	$-0.511809\pi$
$139$	−121.562	−0.0741782	−0.0370891	+	0.999312i	$0.511809\pi$
$140$	−2717.46	−1.64048
$141$	−4625.32	−2.76257
$142$	−28.2302	−0.0166833
$143$	0	0
$144$	−1539.36	−0.890834
$145$	3886.16	2.22571
$146$	2770.14	1.57026
$147$	−1865.01	−1.04642
$148$	206.791	0.114852
$149$	1657.47	0.911310	0.455655	−	0.890157i	$-0.349405\pi$
$149$	1657.47	0.911310	0.455655	+	0.890157i	$0.349405\pi$
$150$	−10946.8	−5.95868
$151$	−2833.03	−1.52681	−0.763407	−	0.645917i	$-0.776475\pi$
$151$	−2833.03	−1.52681	−0.763407	+	0.645917i	$0.776475\pi$
$152$	1271.38	0.678437
$153$	−1058.82	−0.559482
$154$	0	0
$155$	−3461.74	−1.79389
$156$	−3862.37	−1.98229
$157$	1796.07	0.913005	0.456502	−	0.889722i	$-0.349102\pi$
$157$	1796.07	0.913005	0.456502	+	0.889722i	$0.349102\pi$
$158$	4781.51	2.40757
$159$	5452.51	2.71957
$160$	−4531.58	−2.23908
$161$	1620.09	0.793050
$162$	−6474.39	−3.13997
$163$	872.051	0.419045	0.209522	−	0.977804i	$-0.432809\pi$
$163$	872.051	0.419045	0.209522	+	0.977804i	$0.432809\pi$
$164$	3996.49	1.90289
$165$	0	0
$166$	−5037.75	−2.35545
$167$	−835.668	−0.387221	−0.193611	−	0.981078i	$-0.562020\pi$
$167$	−835.668	−0.387221	−0.193611	+	0.981078i	$0.562020\pi$
$168$	−1726.92	−0.793063
$169$	−919.280	−0.418425
$170$	−1475.85	−0.665838
$171$	−5228.53	−2.33822
$172$	−1435.29	−0.636277
$173$	455.115	0.200010	0.100005	−	0.994987i	$-0.468114\pi$
$173$	455.115	0.200010	0.100005	+	0.994987i	$0.468114\pi$
$174$	8220.65	3.58164
$175$	3171.18	1.36982
$176$	0	0
$177$	6962.51	2.95669
$178$	2113.56	0.889987
$179$	−2479.96	−1.03554	−0.517769	−	0.855521i	$-0.673237\pi$
$179$	−2479.96	−1.03554	−0.517769	+	0.855521i	$0.673237\pi$
$180$	−14025.6	−5.80781
$181$	4470.86	1.83600	0.918000	−	0.396580i	$-0.129803\pi$
$181$	4470.86	1.83600	0.918000	+	0.396580i	$0.129803\pi$
$182$	1901.65	0.774504
$183$	−6685.28	−2.70049
$184$	−2033.26	−0.814641
$185$	−356.105	−0.141521
$186$	−7322.85	−2.88676
$187$	0	0
$188$	−5597.66	−2.17155
$189$	4023.24	1.54840
$190$	−7287.84	−2.78271
$191$	−1992.24	−0.754730	−0.377365	−	0.926065i	$-0.623170\pi$
$191$	−1992.24	−0.754730	−0.377365	+	0.926065i	$0.623170\pi$
$192$	−7717.66	−2.90091
$193$	−637.832	−0.237887	−0.118943	−	0.992901i	$-0.537951\pi$
$193$	−637.832	−0.237887	−0.118943	+	0.992901i	$0.537951\pi$
$194$	1400.94	0.518462
$195$	6651.21	2.44258
$196$	−2257.07	−0.822547
$197$	−785.696	−0.284155	−0.142077	−	0.989856i	$-0.545378\pi$
$197$	−785.696	−0.284155	−0.142077	+	0.989856i	$0.545378\pi$
$198$	0	0
$199$	−3678.96	−1.31053	−0.655263	−	0.755401i	$-0.727442\pi$
$199$	−3678.96	−1.31053	−0.655263	+	0.755401i	$0.727442\pi$
$200$	−3979.92	−1.40711
$201$	−1505.43	−0.528282
$202$	5596.29	1.94928
$203$	−2381.44	−0.823372
$204$	−1836.90	−0.630434
$205$	−6882.17	−2.34474
$206$	4115.13	1.39182
$207$	8361.75	2.80764
$208$	883.454	0.294502
$209$	0	0
$210$	9899.10	3.25287
$211$	−222.325	−0.0725379	−0.0362689	−	0.999342i	$-0.511547\pi$
$211$	−222.325	−0.0725379	−0.0362689	+	0.999342i	$0.511547\pi$
$212$	6598.74	2.13775
$213$	60.5067	0.0194641
$214$	−2823.22	−0.901828
$215$	2471.64	0.784021
$216$	−5049.28	−1.59055
$217$	2121.36	0.663627
$218$	−4657.80	−1.44709
$219$	−5937.33	−1.83200
$220$	0	0
$221$	607.668	0.184960
$222$	−753.293	−0.227737
$223$	−3766.79	−1.13113	−0.565567	−	0.824702i	$-0.691343\pi$
$223$	−3766.79	−1.13113	−0.565567	+	0.824702i	$0.691343\pi$
$224$	2776.95	0.828317
$225$	16367.4	4.84959
$226$	6570.09	1.93379
$227$	523.675	0.153117	0.0765585	−	0.997065i	$-0.475607\pi$
$227$	523.675	0.153117	0.0765585	+	0.997065i	$0.475607\pi$
$228$	−9070.72	−2.63475
$229$	−4471.27	−1.29026	−0.645131	−	0.764072i	$-0.723197\pi$
$229$	−4471.27	−1.29026	−0.645131	+	0.764072i	$0.723197\pi$
$230$	11655.1	3.34137
$231$	0	0
$232$	2988.78	0.845788
$233$	615.334	0.173012	0.0865061	−	0.996251i	$-0.472430\pi$
$233$	615.334	0.173012	0.0865061	+	0.996251i	$0.472430\pi$
$234$	9814.95	2.74198
$235$	9639.47	2.67579
$236$	8426.17	2.32414
$237$	−10248.4	−2.80887
$238$	904.402	0.246318
$239$	−4744.90	−1.28419	−0.642096	−	0.766624i	$-0.721935\pi$
$239$	−4744.90	−1.28419	−0.642096	+	0.766624i	$0.721935\pi$
$240$	4598.85	1.23689
$241$	4255.07	1.13732	0.568658	−	0.822574i	$-0.307463\pi$
$241$	4255.07	1.13732	0.568658	+	0.822574i	$0.307463\pi$
$242$	0	0
$243$	4875.09	1.28698
$244$	−8090.66	−2.12275
$245$	3886.79	1.01354
$246$	−14558.3	−3.77319
$247$	3000.70	0.772997
$248$	−2662.36	−0.681694
$249$	10797.6	2.74807
$250$	11962.0	3.02617
$251$	−3360.75	−0.845135	−0.422567	−	0.906332i	$-0.638871\pi$
$251$	−3360.75	−0.845135	−0.422567	+	0.906332i	$0.638871\pi$
$252$	8594.90	2.14852
$253$	0	0
$254$	87.7522	0.0216774
$255$	3163.24	0.776822
$256$	−1224.12	−0.298858
$257$	−3895.76	−0.945567	−0.472783	−	0.881179i	$-0.656751\pi$
$257$	−3895.76	−0.945567	−0.472783	+	0.881179i	$0.656751\pi$
$258$	5228.43	1.26166
$259$	218.222	0.0523538
$260$	8049.43	1.92002
$261$	−12291.3	−2.91499
$262$	44.3794	0.0104648
$263$	252.177	0.0591251	0.0295626	−	0.999563i	$-0.490589\pi$
$263$	252.177	0.0591251	0.0295626	+	0.999563i	$0.490589\pi$
$264$	0	0
$265$	−11363.4	−2.63414
$266$	4465.99	1.02943
$267$	−4530.05	−1.03833
$268$	−1821.90	−0.415262
$269$	−8195.51	−1.85758	−0.928790	−	0.370606i	$-0.879150\pi$
$269$	−8195.51	−1.85758	−0.928790	+	0.370606i	$0.879150\pi$
$270$	28943.6	6.52390
$271$	6134.41	1.37505	0.687525	−	0.726161i	$-0.258697\pi$
$271$	6134.41	1.37505	0.687525	+	0.726161i	$0.258697\pi$
$272$	420.160	0.0936616
$273$	−4075.87	−0.903600
$274$	−8240.65	−1.81692
$275$	0	0
$276$	14506.4	3.16370
$277$	−703.552	−0.152608	−0.0763039	−	0.997085i	$-0.524312\pi$
$277$	−703.552	−0.152608	−0.0763039	+	0.997085i	$0.524312\pi$
$278$	535.914	0.115619
$279$	10948.9	2.34945
$280$	3599.01	0.768150
$281$	−1146.34	−0.243362	−0.121681	−	0.992569i	$-0.538829\pi$
$281$	−1146.34	−0.243362	−0.121681	+	0.992569i	$0.538829\pi$
$282$	20391.0	4.30591
$283$	−3087.06	−0.648434	−0.324217	−	0.945983i	$-0.605101\pi$
$283$	−3087.06	−0.648434	−0.324217	+	0.945983i	$0.605101\pi$
$284$	73.2264	0.0153000
$285$	15620.3	3.24654
$286$	0	0
$287$	4217.40	0.867405
$288$	14332.6	2.93250
$289$	289.000	0.0588235
$290$	−17132.4	−3.46913
$291$	−3002.68	−0.604880
$292$	−7185.48	−1.44006
$293$	−266.784	−0.0531934	−0.0265967	−	0.999646i	$-0.508467\pi$
$293$	−266.784	−0.0531934	−0.0265967	+	0.999646i	$0.508467\pi$
$294$	8221.99	1.63101
$295$	−14510.3	−2.86381
$296$	−273.874	−0.0537791
$297$	0	0
$298$	−7307.05	−1.42042
$299$	−4798.89	−0.928184
$300$	28394.9	5.46461
$301$	−1514.62	−0.290038
$302$	12489.6	2.37978
$303$	−11994.7	−2.27419
$304$	2074.78	0.391436
$305$	13932.6	2.61566
$306$	4667.88	0.872042
$307$	−2809.14	−0.522235	−0.261117	−	0.965307i	$-0.584091\pi$
$307$	−2809.14	−0.522235	−0.261117	+	0.965307i	$0.584091\pi$
$308$	0	0
$309$	−8820.10	−1.62381
$310$	15261.3	2.79607
$311$	8543.60	1.55776	0.778880	−	0.627173i	$-0.215788\pi$
$311$	8543.60	1.55776	0.778880	+	0.627173i	$0.215788\pi$
$312$	5115.33	0.928200
$313$	−3617.79	−0.653322	−0.326661	−	0.945142i	$-0.605924\pi$
$313$	−3617.79	−0.653322	−0.326661	+	0.945142i	$0.605924\pi$
$314$	−7918.06	−1.42306
$315$	−14800.9	−2.64741
$316$	−12402.8	−2.20795
$317$	2663.40	0.471897	0.235948	−	0.971766i	$-0.424180\pi$
$317$	2663.40	0.471897	0.235948	+	0.971766i	$0.424180\pi$
$318$	−24037.7	−4.23889
$319$	0	0
$320$	16084.1	2.80978
$321$	6051.09	1.05215
$322$	−7142.27	−1.23610
$323$	1427.10	0.245839
$324$	16793.9	2.87962
$325$	−9393.40	−1.60324
$326$	−3844.49	−0.653148
$327$	9983.21	1.68830
$328$	−5292.95	−0.891020
$329$	−5907.08	−0.989871
$330$	0	0
$331$	5935.96	0.985710	0.492855	−	0.870112i	$-0.335953\pi$
$331$	5935.96	0.985710	0.492855	+	0.870112i	$0.335953\pi$
$332$	13067.4	2.16015
$333$	1126.30	0.185349
$334$	3684.09	0.603546
$335$	3137.41	0.511687
$336$	−2818.18	−0.457572
$337$	−66.0390	−0.0106747	−0.00533735	−	0.999986i	$-0.501699\pi$
$337$	−66.0390	−0.0106747	−0.00533735	+	0.999986i	$0.501699\pi$
$338$	4052.70	0.652183
$339$	−14081.9	−2.25612
$340$	3828.21	0.610629
$341$	0	0
$342$	23050.3	3.64449
$343$	−6520.97	−1.02653
$344$	1900.90	0.297934
$345$	−24980.8	−3.89832
$346$	−2006.40	−0.311748
$347$	−9428.28	−1.45861	−0.729303	−	0.684191i	$-0.760156\pi$
$347$	−9428.28	−1.45861	−0.729303	+	0.684191i	$0.760156\pi$
$348$	−21323.6	−3.28467
$349$	−9989.45	−1.53216	−0.766079	−	0.642747i	$-0.777795\pi$
$349$	−9989.45	−1.53216	−0.766079	+	0.642747i	$0.777795\pi$
$350$	−13980.3	−2.13509
$351$	−11917.3	−1.81224
$352$	0	0
$353$	2016.69	0.304073	0.152037	−	0.988375i	$-0.451417\pi$
$353$	2016.69	0.304073	0.152037	+	0.988375i	$0.451417\pi$
$354$	−30694.6	−4.60848
$355$	−126.100	−0.0188526
$356$	−5482.36	−0.816192
$357$	−1938.44	−0.287375
$358$	10933.1	1.61405
$359$	2203.58	0.323957	0.161979	−	0.986794i	$-0.448212\pi$
$359$	2203.58	0.323957	0.161979	+	0.986794i	$0.448212\pi$
$360$	18575.5	2.71949
$361$	188.108	0.0274250
$362$	−19710.0	−2.86170
$363$	0	0
$364$	−4932.69	−0.710284
$365$	12373.8	1.77445
$366$	29472.4	4.20915
$367$	−2643.25	−0.375958	−0.187979	−	0.982173i	$-0.560194\pi$
$367$	−2643.25	−0.375958	−0.187979	+	0.982173i	$0.560194\pi$
$368$	−3318.10	−0.470021
$369$	21767.2	3.07088
$370$	1569.91	0.220583
$371$	6963.49	0.974465
$372$	18994.8	2.64740
$373$	8298.79	1.15200	0.575999	−	0.817451i	$-0.304613\pi$
$373$	8298.79	1.15200	0.575999	+	0.817451i	$0.304613\pi$
$374$	0	0
$375$	−25638.5	−3.53058
$376$	7413.54	1.01682
$377$	7054.10	0.963673
$378$	−17736.7	−2.41343
$379$	−10793.8	−1.46291	−0.731454	−	0.681891i	$-0.761158\pi$
$379$	−10793.8	−1.46291	−0.731454	+	0.681891i	$0.761158\pi$
$380$	18904.0	2.55198
$381$	−188.082	−0.0252907
$382$	8782.90	1.17637
$383$	−5453.74	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$383$	−5453.74	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$384$	16628.6	2.20983
$385$	0	0
$386$	2811.92	0.370785
$387$	−7817.41	−1.02682
$388$	−3633.90	−0.475472
$389$	−4906.03	−0.639449	−0.319724	−	0.947511i	$-0.603590\pi$
$389$	−4906.03	−0.639449	−0.319724	+	0.947511i	$0.603590\pi$
$390$	−29322.2	−3.80715
$391$	−2282.30	−0.295194
$392$	2989.26	0.385155
$393$	−95.1198	−0.0122091
$394$	3463.78	0.442901
$395$	21358.3	2.72063
$396$	0	0
$397$	14944.0	1.88921	0.944606	−	0.328207i	$-0.106444\pi$
$397$	14944.0	1.88921	0.944606	+	0.328207i	$0.106444\pi$
$398$	16218.9	2.04266
$399$	−9572.11	−1.20102
$400$	−6494.88	−0.811860
$401$	−8861.46	−1.10354	−0.551771	−	0.833996i	$-0.686048\pi$
$401$	−8861.46	−1.10354	−0.551771	+	0.833996i	$0.686048\pi$
$402$	6636.77	0.823413
$403$	−6283.70	−0.776708
$404$	−14516.2	−1.78765
$405$	−28920.1	−3.54827
$406$	10498.7	1.28336
$407$	0	0
$408$	2432.79	0.295199
$409$	9404.03	1.13692	0.568459	−	0.822712i	$-0.307540\pi$
$409$	9404.03	1.13692	0.568459	+	0.822712i	$0.307540\pi$
$410$	30340.4	3.65465
$411$	17662.5	2.11977
$412$	−10674.3	−1.27641
$413$	8891.94	1.05943
$414$	−36863.3	−4.37616
$415$	−22502.8	−2.66174
$416$	−8225.65	−0.969461
$417$	−1148.64	−0.134890
$418$	0	0
$419$	263.902	0.0307696	0.0153848	−	0.999882i	$-0.495103\pi$
$419$	263.902	0.0307696	0.0153848	+	0.999882i	$0.495103\pi$
$420$	−25677.3	−2.98315
$421$	−12428.7	−1.43880	−0.719402	−	0.694593i	$-0.755584\pi$
$421$	−12428.7	−1.43880	−0.719402	+	0.694593i	$0.755584\pi$
$422$	980.133	0.113062
$423$	−30488.1	−3.50445
$424$	−8739.37	−1.00099
$425$	−4467.39	−0.509883
$426$	−266.747	−0.0303379
$427$	−8537.88	−0.967628
$428$	7323.15	0.827051
$429$	0	0
$430$	−10896.4	−1.22202
$431$	−16397.6	−1.83259	−0.916295	−	0.400505i	$-0.868835\pi$
$431$	−16397.6	−1.83259	−0.916295	+	0.400505i	$0.868835\pi$
$432$	−8239.97	−0.917699
$433$	−11118.2	−1.23397	−0.616984	−	0.786975i	$-0.711646\pi$
$433$	−11118.2	−1.23397	−0.616984	+	0.786975i	$0.711646\pi$
$434$	−9352.12	−1.03437
$435$	36720.4	4.04737
$436$	12081.9	1.32710
$437$	−11270.1	−1.23369
$438$	26175.1	2.85546
$439$	−10285.7	−1.11825	−0.559123	−	0.829085i	$-0.688862\pi$
$439$	−10285.7	−1.11825	−0.559123	+	0.829085i	$0.688862\pi$
$440$	0	0
$441$	−12293.3	−1.32743
$442$	−2678.94	−0.288290
$443$	−5551.38	−0.595382	−0.297691	−	0.954662i	$-0.596216\pi$
$443$	−5551.38	−0.595382	−0.297691	+	0.954662i	$0.596216\pi$
$444$	1953.97	0.208854
$445$	9440.92	1.00571
$446$	16606.1	1.76305
$447$	15661.4	1.65718
$448$	−9856.35	−1.03944
$449$	3085.61	0.324318	0.162159	−	0.986765i	$-0.448154\pi$
$449$	3085.61	0.324318	0.162159	+	0.986765i	$0.448154\pi$
$450$	−72156.5	−7.55887
$451$	0	0
$452$	−17042.2	−1.77344
$453$	−26769.3	−2.77645
$454$	−2308.65	−0.238657
$455$	8494.37	0.875214
$456$	12013.3	1.23371
$457$	12110.8	1.23965	0.619826	−	0.784740i	$-0.287203\pi$
$457$	12110.8	1.23965	0.619826	+	0.784740i	$0.287203\pi$
$458$	19711.9	2.01108
$459$	−5667.72	−0.576354
$460$	−30232.2	−3.06432
$461$	12222.6	1.23485	0.617423	−	0.786631i	$-0.288177\pi$
$461$	12222.6	1.23485	0.617423	+	0.786631i	$0.288177\pi$
$462$	0	0
$463$	−9048.79	−0.908278	−0.454139	−	0.890931i	$-0.650053\pi$
$463$	−9048.79	−0.908278	−0.454139	+	0.890931i	$0.650053\pi$
$464$	4877.42	0.487993
$465$	−32710.0	−3.26213
$466$	−2712.73	−0.269667
$467$	7252.95	0.718687	0.359343	−	0.933205i	$-0.383001\pi$
$467$	7252.95	0.718687	0.359343	+	0.933205i	$0.383001\pi$
$468$	−25459.0	−2.51463
$469$	−1922.61	−0.189292
$470$	−42496.1	−4.17064
$471$	16971.0	1.66026
$472$	−11159.6	−1.08827
$473$	0	0
$474$	45180.5	4.37808
$475$	−22060.2	−2.13093
$476$	−2345.93	−0.225894
$477$	35940.6	3.44991
$478$	20918.1	2.00162
$479$	3027.54	0.288793	0.144396	−	0.989520i	$-0.453876\pi$
$479$	3027.54	0.288793	0.144396	+	0.989520i	$0.453876\pi$
$480$	−42818.9	−4.07168
$481$	−646.397	−0.0612748
$482$	−18758.7	−1.77269
$483$	15308.3	1.44213
$484$	0	0
$485$	6257.77	0.585878
$486$	−21492.1	−2.00597
$487$	−17446.4	−1.62335	−0.811676	−	0.584108i	$-0.801445\pi$
$487$	−17446.4	−1.62335	−0.811676	+	0.584108i	$0.801445\pi$
$488$	10715.3	0.993971
$489$	8240.01	0.762017
$490$	−17135.1	−1.57977
$491$	−13661.0	−1.25563	−0.627815	−	0.778363i	$-0.716050\pi$
$491$	−13661.0	−1.25563	−0.627815	+	0.778363i	$0.716050\pi$
$492$	37762.8	3.46033
$493$	3354.85	0.306480
$494$	−13228.8	−1.20484
$495$	0	0
$496$	−4344.74	−0.393316
$497$	77.2741	0.00697428
$498$	−47601.7	−4.28330
$499$	3446.59	0.309199	0.154600	−	0.987977i	$-0.450591\pi$
$499$	3446.59	0.309199	0.154600	+	0.987977i	$0.450591\pi$
$500$	−31028.3	−2.77525
$501$	−7896.23	−0.704147
$502$	14816.1	1.31728
$503$	−7395.29	−0.655546	−0.327773	−	0.944757i	$-0.606298\pi$
$503$	−7395.29	−0.655546	−0.327773	+	0.944757i	$0.606298\pi$
$504$	−11383.1	−1.00604
$505$	24997.8	2.20274
$506$	0	0
$507$	−8686.28	−0.760890
$508$	−227.621	−0.0198800
$509$	5842.85	0.508801	0.254401	−	0.967099i	$-0.418122\pi$
$509$	5842.85	0.508801	0.254401	+	0.967099i	$0.418122\pi$
$510$	−13945.3	−1.21080
$511$	−7582.66	−0.656433
$512$	−8681.97	−0.749400
$513$	−27987.6	−2.40874
$514$	17174.7	1.47382
$515$	18381.7	1.57280
$516$	−13562.0	−1.15704
$517$	0	0
$518$	−962.043	−0.0816018
$519$	4300.38	0.363711
$520$	−10660.7	−0.899041
$521$	−3375.87	−0.283877	−0.141938	−	0.989876i	$-0.545333\pi$
$521$	−3375.87	−0.283877	−0.141938	+	0.989876i	$0.545333\pi$
$522$	54186.9	4.54348
$523$	−3244.10	−0.271233	−0.135616	−	0.990761i	$-0.543301\pi$
$523$	−3244.10	−0.271233	−0.135616	+	0.990761i	$0.543301\pi$
$524$	−115.116	−0.00959705
$525$	29964.5	2.49097
$526$	−1111.74	−0.0921560
$527$	−2988.45	−0.247019
$528$	0	0
$529$	5856.81	0.481368
$530$	50096.1	4.10573
$531$	45893.8	3.75070
$532$	−11584.4	−0.944071
$533$	−12492.4	−1.01521
$534$	19971.0	1.61841
$535$	−12610.9	−1.01909
$536$	2412.92	0.194445
$537$	−23433.2	−1.88308
$538$	36130.4	2.89534
$539$	0	0
$540$	−75077.0	−5.98296
$541$	−18926.1	−1.50406	−0.752031	−	0.659128i	$-0.770925\pi$
$541$	−18926.1	−1.50406	−0.752031	+	0.659128i	$0.770925\pi$
$542$	−27043.9	−2.14324
$543$	42245.1	3.33870
$544$	−3912.02	−0.308321
$545$	−20805.6	−1.63526
$546$	17968.7	1.40840
$547$	−1913.04	−0.149535	−0.0747676	−	0.997201i	$-0.523821\pi$
$547$	−1913.04	−0.149535	−0.0747676	+	0.997201i	$0.523821\pi$
$548$	21375.4	1.66627
$549$	−44066.4	−3.42570
$550$	0	0
$551$	16566.5	1.28086
$552$	−19212.3	−1.48139
$553$	−13088.4	−1.00646
$554$	3101.65	0.237864
$555$	−3364.84	−0.257351
$556$	−1390.11	−0.106032
$557$	25577.7	1.94571	0.972855	−	0.231416i	$-0.0743358\pi$
$557$	25577.7	1.94571	0.972855	+	0.231416i	$0.0743358\pi$
$558$	−48269.0	−3.66199
$559$	4486.49	0.339460
$560$	5873.26	0.443198
$561$	0	0
$562$	5053.70	0.379319
$563$	15637.2	1.17057	0.585284	−	0.810829i	$-0.300983\pi$
$563$	15637.2	1.17057	0.585284	+	0.810829i	$0.300983\pi$
$564$	−52892.3	−3.94888
$565$	29347.5	2.18524
$566$	13609.5	1.01069
$567$	17722.2	1.31263
$568$	−96.9811	−0.00716415
$569$	−17414.1	−1.28302	−0.641511	−	0.767114i	$-0.721692\pi$
$569$	−17414.1	−1.28302	−0.641511	+	0.767114i	$0.721692\pi$
$570$	−68862.8	−5.06026
$571$	−10447.5	−0.765697	−0.382848	−	0.923811i	$-0.625057\pi$
$571$	−10447.5	−0.765697	−0.382848	+	0.923811i	$0.625057\pi$
$572$	0	0
$573$	−18824.7	−1.37245
$574$	−18592.7	−1.35199
$575$	35280.0	2.55874
$576$	−50871.4	−3.67993
$577$	14205.7	1.02494	0.512470	−	0.858705i	$-0.328730\pi$
$577$	14205.7	1.02494	0.512470	+	0.858705i	$0.328730\pi$
$578$	−1274.07	−0.0916859
$579$	−6026.88	−0.432588
$580$	44439.7	3.18148
$581$	13789.8	0.984674
$582$	13237.5	0.942802
$583$	0	0
$584$	9516.45	0.674304
$585$	43841.9	3.09853
$586$	1176.13	0.0829105
$587$	5783.99	0.406696	0.203348	−	0.979107i	$-0.434818\pi$
$587$	5783.99	0.406696	0.203348	+	0.979107i	$0.434818\pi$
$588$	−21327.0	−1.49577
$589$	−14757.2	−1.03236
$590$	63969.6	4.46370
$591$	−7424.04	−0.516725
$592$	−446.938	−0.0310288
$593$	−5826.57	−0.403488	−0.201744	−	0.979438i	$-0.564661\pi$
$593$	−5826.57	−0.403488	−0.201744	+	0.979438i	$0.564661\pi$
$594$	0	0
$595$	4039.82	0.278347
$596$	18953.8	1.30265
$597$	−34762.5	−2.38314
$598$	21156.2	1.44672
$599$	−5814.65	−0.396628	−0.198314	−	0.980139i	$-0.563547\pi$
$599$	−5814.65	−0.396628	−0.198314	+	0.980139i	$0.563547\pi$
$600$	−37606.3	−2.55878
$601$	14961.0	1.01542	0.507712	−	0.861527i	$-0.330491\pi$
$601$	14961.0	1.01542	0.507712	+	0.861527i	$0.330491\pi$
$602$	6677.31	0.452071
$603$	−9923.12	−0.670151
$604$	−32396.8	−2.18246
$605$	0	0
$606$	52879.4	3.54468
$607$	−4197.41	−0.280671	−0.140336	−	0.990104i	$-0.544818\pi$
$607$	−4197.41	−0.280671	−0.140336	+	0.990104i	$0.544818\pi$
$608$	−19317.8	−1.28855
$609$	−22502.3	−1.49727
$610$	−61422.4	−4.07692
$611$	17497.4	1.15854
$612$	−12108.0	−0.799735
$613$	−25711.5	−1.69409	−0.847047	−	0.531518i	$-0.821622\pi$
$613$	−25711.5	−1.69409	−0.847047	+	0.531518i	$0.821622\pi$
$614$	12384.3	0.813987
$615$	−65029.7	−4.26382
$616$	0	0
$617$	553.010	0.0360832	0.0180416	−	0.999837i	$-0.494257\pi$
$617$	553.010	0.0360832	0.0180416	+	0.999837i	$0.494257\pi$
$618$	38883.9	2.53097
$619$	12788.3	0.830380	0.415190	−	0.909735i	$-0.363715\pi$
$619$	12788.3	0.830380	0.415190	+	0.909735i	$0.363715\pi$
$620$	−39586.3	−2.56423
$621$	44759.3	2.89231
$622$	−37664.9	−2.42802
$623$	−5785.40	−0.372050
$624$	8347.76	0.535541
$625$	20583.9	1.31737
$626$	15949.2	1.01831
$627$	0	0
$628$	20538.7	1.30507
$629$	−307.419	−0.0194874
$630$	65250.5	4.12642
$631$	−30712.3	−1.93762	−0.968808	−	0.247811i	$-0.920289\pi$
$631$	−30712.3	−1.93762	−0.968808	+	0.247811i	$0.920289\pi$
$632$	16426.2	1.03386
$633$	−2100.75	−0.131907
$634$	−11741.7	−0.735527
$635$	391.975	0.0244962
$636$	62351.5	3.88742
$637$	7055.25	0.438837
$638$	0	0
$639$	398.834	0.0246911
$640$	−34655.0	−2.14041
$641$	−12642.8	−0.779036	−0.389518	−	0.921019i	$-0.627358\pi$
$641$	−12642.8	−0.779036	−0.389518	+	0.921019i	$0.627358\pi$
$642$	−26676.6	−1.63994
$643$	−6220.29	−0.381499	−0.190750	−	0.981639i	$-0.561092\pi$
$643$	−6220.29	−0.381499	−0.190750	+	0.981639i	$0.561092\pi$
$644$	18526.3	1.13360
$645$	23354.6	1.42571
$646$	−6291.45	−0.383179
$647$	−2031.73	−0.123455	−0.0617276	−	0.998093i	$-0.519661\pi$
$647$	−2031.73	−0.123455	−0.0617276	+	0.998093i	$0.519661\pi$
$648$	−22241.9	−1.34837
$649$	0	0
$650$	41411.3	2.49890
$651$	20044.7	1.20678
$652$	9972.22	0.598991
$653$	3860.47	0.231351	0.115675	−	0.993287i	$-0.463097\pi$
$653$	3860.47	0.231351	0.115675	+	0.993287i	$0.463097\pi$
$654$	−44011.5	−2.63148
$655$	198.236	0.0118255
$656$	−8637.63	−0.514090
$657$	−39136.3	−2.32398
$658$	26041.7	1.54287
$659$	7263.86	0.429378	0.214689	−	0.976683i	$-0.431126\pi$
$659$	7263.86	0.429378	0.214689	+	0.976683i	$0.431126\pi$
$660$	0	0
$661$	−1939.24	−0.114111	−0.0570557	−	0.998371i	$-0.518171\pi$
$661$	−1939.24	−0.114111	−0.0570557	+	0.998371i	$0.518171\pi$
$662$	−26169.0	−1.53639
$663$	5741.86	0.336343
$664$	−17306.5	−1.01148
$665$	19948.9	1.16329
$666$	−4965.38	−0.288895
$667$	−26494.0	−1.53801
$668$	−9556.17	−0.553502
$669$	−35592.4	−2.05692
$670$	−13831.4	−0.797545
$671$	0	0
$672$	26239.4	1.50626
$673$	−14386.9	−0.824031	−0.412016	−	0.911177i	$-0.635175\pi$
$673$	−14386.9	−0.824031	−0.412016	+	0.911177i	$0.635175\pi$
$674$	291.136	0.0166382
$675$	87612.2	4.99584
$676$	−10512.3	−0.598106
$677$	−11350.5	−0.644367	−0.322184	−	0.946677i	$-0.604417\pi$
$677$	−11350.5	−0.644367	−0.322184	+	0.946677i	$0.604417\pi$
$678$	62080.8	3.51652
$679$	−3834.77	−0.216738
$680$	−5070.09	−0.285925
$681$	4948.21	0.278437
$682$	0	0
$683$	−23341.6	−1.30767	−0.653837	−	0.756636i	$-0.726842\pi$
$683$	−23341.6	−1.30767	−0.653837	+	0.756636i	$0.726842\pi$
$684$	−59790.2	−3.34230
$685$	−36809.7	−2.05318
$686$	28748.1	1.60001
$687$	−42249.1	−2.34629
$688$	3102.09	0.171899
$689$	−20626.6	−1.14051
$690$	110129.	6.07616
$691$	14140.2	0.778464	0.389232	−	0.921140i	$-0.372740\pi$
$691$	14140.2	0.778464	0.389232	+	0.921140i	$0.372740\pi$
$692$	5204.41	0.285899
$693$	0	0
$694$	41565.1	2.27347
$695$	2393.84	0.130653
$696$	28241.0	1.53803
$697$	−5941.25	−0.322871
$698$	44039.0	2.38811
$699$	5814.29	0.314616
$700$	36263.6	1.95805
$701$	−16368.4	−0.881919	−0.440959	−	0.897527i	$-0.645362\pi$
$701$	−16368.4	−0.881919	−0.440959	+	0.897527i	$0.645362\pi$
$702$	52538.0	2.82467
$703$	−1518.05	−0.0814431
$704$	0	0
$705$	91083.4	4.86581
$706$	−8890.71	−0.473947
$707$	−15318.7	−0.814876
$708$	79618.9	4.22636
$709$	11421.7	0.605007	0.302503	−	0.953148i	$-0.402178\pi$
$709$	11421.7	0.605007	0.302503	+	0.953148i	$0.402178\pi$
$710$	555.918	0.0293848
$711$	−67552.7	−3.56319
$712$	7260.84	0.382179
$713$	23600.5	1.23961
$714$	8545.70	0.447920
$715$	0	0
$716$	−28359.3	−1.48022
$717$	−44834.5	−2.33525
$718$	−9714.62	−0.504939
$719$	−2273.05	−0.117901	−0.0589503	−	0.998261i	$-0.518775\pi$
$719$	−2273.05	−0.117901	−0.0589503	+	0.998261i	$0.518775\pi$
$720$	30313.6	1.56906
$721$	−11264.3	−0.581837
$722$	−829.286	−0.0427463
$723$	40206.2	2.06817
$724$	51125.9	2.62442
$725$	−51859.6	−2.65657
$726$	0	0
$727$	7823.41	0.399112	0.199556	−	0.979886i	$-0.436050\pi$
$727$	7823.41	0.399112	0.199556	+	0.979886i	$0.436050\pi$
$728$	6532.86	0.332588
$729$	6412.64	0.325796
$730$	−54550.5	−2.76576
$731$	2133.72	0.107960
$732$	−76448.6	−3.86014
$733$	−17449.2	−0.879267	−0.439633	−	0.898177i	$-0.644892\pi$
$733$	−17449.2	−0.879267	−0.439633	+	0.898177i	$0.644892\pi$
$734$	11652.9	0.585990
$735$	36726.3	1.84309
$736$	30894.1	1.54724
$737$	0	0
$738$	−95962.0	−4.78646
$739$	−20014.8	−0.996289	−0.498145	−	0.867094i	$-0.665985\pi$
$739$	−20014.8	−0.996289	−0.498145	+	0.867094i	$0.665985\pi$
$740$	−4072.20	−0.202293
$741$	28353.7	1.40567
$742$	−30698.9	−1.51886
$743$	9846.02	0.486158	0.243079	−	0.970007i	$-0.421843\pi$
$743$	9846.02	0.486158	0.243079	+	0.970007i	$0.421843\pi$
$744$	−25156.7	−1.23963
$745$	−32639.4	−1.60512
$746$	−36585.7	−1.79557
$747$	71172.9	3.48605
$748$	0	0
$749$	7727.95	0.377000
$750$	113029.	5.50298
$751$	40570.2	1.97128	0.985638	−	0.168875i	$-0.0540134\pi$
$751$	40570.2	1.97128	0.985638	+	0.168875i	$0.0540134\pi$
$752$	12098.2	0.586672
$753$	−31755.8	−1.53684
$754$	−31098.4	−1.50204
$755$	55789.0	2.68923
$756$	46007.2	2.21332
$757$	10306.7	0.494852	0.247426	−	0.968907i	$-0.420415\pi$
$757$	10306.7	0.494852	0.247426	+	0.968907i	$0.420415\pi$
$758$	47585.2	2.28018
$759$	0	0
$760$	−25036.4	−1.19496
$761$	39929.8	1.90204	0.951020	−	0.309128i	$-0.100037\pi$
$761$	39929.8	1.90204	0.951020	+	0.309128i	$0.100037\pi$
$762$	829.170	0.0394195
$763$	12749.7	0.604942
$764$	−22782.0	−1.07883
$765$	20850.7	0.985435
$766$	24043.1	1.13409
$767$	−26338.9	−1.23995
$768$	−11566.7	−0.543462
$769$	−2066.43	−0.0969017	−0.0484508	−	0.998826i	$-0.515428\pi$
$769$	−2066.43	−0.0969017	−0.0484508	+	0.998826i	$0.515428\pi$
$770$	0	0
$771$	−36811.0	−1.71948
$772$	−7293.84	−0.340040
$773$	23881.4	1.11120	0.555598	−	0.831451i	$-0.312489\pi$
$773$	23881.4	1.11120	0.555598	+	0.831451i	$0.312489\pi$
$774$	34463.5	1.60047
$775$	46195.8	2.14116
$776$	4812.74	0.222638
$777$	2061.98	0.0952034
$778$	21628.5	0.996683
$779$	−29338.2	−1.34936
$780$	76059.0	3.49148
$781$	0	0
$782$	10061.6	0.460107
$783$	−65793.6	−3.00290
$784$	4878.21	0.222222
$785$	−35368.7	−1.60811
$786$	419.341	0.0190298
$787$	6788.48	0.307475	0.153738	−	0.988112i	$-0.450869\pi$
$787$	6788.48	0.307475	0.153738	+	0.988112i	$0.450869\pi$
$788$	−8984.72	−0.406177
$789$	2382.82	0.107517
$790$	−94159.1	−4.24054
$791$	−17984.2	−0.808400
$792$	0	0
$793$	25290.2	1.13251
$794$	−65881.4	−2.94464
$795$	−107373.	−4.79008
$796$	−42070.3	−1.87329
$797$	6782.05	0.301421	0.150710	−	0.988578i	$-0.451844\pi$
$797$	6782.05	0.301421	0.150710	+	0.988578i	$0.451844\pi$
$798$	42199.2	1.87197
$799$	8321.57	0.368455
$800$	60472.4	2.67253
$801$	−29860.1	−1.31717
$802$	39066.3	1.72005
$803$	0	0
$804$	−17215.1	−0.755138
$805$	−31903.4	−1.39683
$806$	27702.0	1.21062
$807$	−77439.4	−3.37794
$808$	19225.3	0.837060
$809$	−11593.9	−0.503855	−0.251928	−	0.967746i	$-0.581064\pi$
$809$	−11593.9	−0.503855	−0.251928	+	0.967746i	$0.581064\pi$
$810$	127496.	5.53055
$811$	23836.5	1.03208	0.516038	−	0.856566i	$-0.327406\pi$
$811$	23836.5	1.03208	0.516038	+	0.856566i	$0.327406\pi$
$812$	−27232.7	−1.17695
$813$	57964.0	2.50048
$814$	0	0
$815$	−17172.7	−0.738078
$816$	3970.10	0.170320
$817$	10536.4	0.451192
$818$	−41458.2	−1.77207
$819$	−26866.3	−1.14626
$820$	−78700.2	−3.35162
$821$	−5666.33	−0.240872	−0.120436	−	0.992721i	$-0.538429\pi$
$821$	−5666.33	−0.240872	−0.120436	+	0.992721i	$0.538429\pi$
$822$	−77865.9	−3.30400
$823$	−748.135	−0.0316869	−0.0158435	−	0.999874i	$-0.505043\pi$
$823$	−748.135	−0.0316869	−0.0158435	+	0.999874i	$0.505043\pi$
$824$	14137.0	0.597677
$825$	0	0
$826$	−39200.6	−1.65129
$827$	22707.9	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$827$	22707.9	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$828$	95619.7	4.01330
$829$	−4286.43	−0.179583	−0.0897913	−	0.995961i	$-0.528620\pi$
$829$	−4286.43	−0.179583	−0.0897913	+	0.995961i	$0.528620\pi$
$830$	99205.1	4.14874
$831$	−6647.87	−0.277511
$832$	29195.6	1.21656
$833$	3355.39	0.139565
$834$	5063.85	0.210248
$835$	16456.2	0.682026
$836$	0	0
$837$	58608.0	2.42030
$838$	−1163.43	−0.0479593
$839$	−19930.6	−0.820119	−0.410059	−	0.912059i	$-0.634492\pi$
$839$	−19930.6	−0.820119	−0.410059	+	0.912059i	$0.634492\pi$
$840$	34007.0	1.39685
$841$	14555.7	0.596813
$842$	54792.5	2.24261
$843$	−10831.8	−0.442545
$844$	−2542.37	−0.103687
$845$	18102.8	0.736987
$846$	134408.	5.46225
$847$	0	0
$848$	−14261.9	−0.577541
$849$	−29169.7	−1.17915
$850$	19694.7	0.794734
$851$	2427.76	0.0977936
$852$	691.916	0.0278224
$853$	10563.7	0.424026	0.212013	−	0.977267i	$-0.431998\pi$
$853$	10563.7	0.424026	0.212013	+	0.977267i	$0.431998\pi$
$854$	37639.7	1.50820
$855$	102962.	4.11839
$856$	−9698.79	−0.387264
$857$	4069.61	0.162211	0.0811057	−	0.996706i	$-0.474155\pi$
$857$	4069.61	0.162211	0.0811057	+	0.996706i	$0.474155\pi$
$858$	0	0
$859$	−3498.48	−0.138960	−0.0694799	−	0.997583i	$-0.522134\pi$
$859$	−3498.48	−0.138960	−0.0694799	+	0.997583i	$0.522134\pi$
$860$	28264.1	1.12070
$861$	39850.2	1.57734
$862$	72289.9	2.85638
$863$	−31340.0	−1.23618	−0.618091	−	0.786107i	$-0.712094\pi$
$863$	−31340.0	−1.23618	−0.618091	+	0.786107i	$0.712094\pi$
$864$	76720.6	3.02093
$865$	−8962.27	−0.352285
$866$	49015.4	1.92334
$867$	2730.76	0.106968
$868$	24258.5	0.948603
$869$	0	0
$870$	−161884.	−6.30848
$871$	5694.98	0.221546
$872$	−16001.2	−0.621411
$873$	−19792.3	−0.767318
$874$	49685.0	1.92291
$875$	−32743.4	−1.26506
$876$	−67895.6	−2.61870
$877$	−32586.8	−1.25471	−0.627354	−	0.778734i	$-0.715862\pi$
$877$	−32586.8	−1.25471	−0.627354	+	0.778734i	$0.715862\pi$
$878$	45345.1	1.74296
$879$	−2520.84	−0.0967302
$880$	0	0
$881$	−19805.0	−0.757375	−0.378687	−	0.925525i	$-0.623624\pi$
$881$	−19805.0	−0.757375	−0.378687	+	0.925525i	$0.623624\pi$
$882$	54195.7	2.06901
$883$	15061.4	0.574015	0.287008	−	0.957928i	$-0.407340\pi$
$883$	15061.4	0.574015	0.287008	+	0.957928i	$0.407340\pi$
$884$	6948.91	0.264386
$885$	−137108.	−5.20773
$886$	24473.6	0.927997
$887$	19900.3	0.753310	0.376655	−	0.926354i	$-0.377074\pi$
$887$	19900.3	0.753310	0.376655	+	0.926354i	$0.377074\pi$
$888$	−2587.84	−0.0977952
$889$	−240.203	−0.00906202
$890$	−41620.8	−1.56757
$891$	0	0
$892$	−43074.6	−1.61687
$893$	41092.4	1.53987
$894$	−69044.3	−2.58298
$895$	48836.2	1.82393
$896$	21236.6	0.791814
$897$	−45344.8	−1.68787
$898$	−13603.1	−0.505501
$899$	−34691.4	−1.28701
$900$	187167.	6.93211
$901$	−9809.79	−0.362721
$902$	0	0
$903$	−14311.7	−0.527423
$904$	22570.7	0.830408
$905$	−88041.5	−3.23381
$906$	118014.	4.32755
$907$	49173.0	1.80018	0.900090	−	0.435703i	$-0.143500\pi$
$907$	49173.0	1.80018	0.900090	+	0.435703i	$0.143500\pi$
$908$	5988.42	0.218869
$909$	−79063.9	−2.88491
$910$	−37447.9	−1.36416
$911$	15336.1	0.557746	0.278873	−	0.960328i	$-0.410039\pi$
$911$	15336.1	0.557746	0.278873	+	0.960328i	$0.410039\pi$
$912$	19604.6	0.711812
$913$	0	0
$914$	−53391.3	−1.93220
$915$	131649.	4.75647
$916$	−51130.7	−1.84433
$917$	−121.479	−0.00437469
$918$	24986.5	0.898340
$919$	14152.5	0.507997	0.253998	−	0.967205i	$-0.418254\pi$
$919$	14152.5	0.507997	0.253998	+	0.967205i	$0.418254\pi$
$920$	40039.6	1.43486
$921$	−26543.6	−0.949664
$922$	−53884.1	−1.92471
$923$	−228.894	−0.00816268
$924$	0	0
$925$	4752.11	0.168917
$926$	39892.1	1.41570
$927$	−58138.2	−2.05988
$928$	−45412.6	−1.60640
$929$	−30367.8	−1.07248	−0.536241	−	0.844065i	$-0.680156\pi$
$929$	−30367.8	−1.07248	−0.536241	+	0.844065i	$0.680156\pi$
$930$	144204.	5.08455
$931$	16569.1	0.583278
$932$	7036.57	0.247307
$933$	80728.5	2.83273
$934$	−31975.0	−1.12019
$935$	0	0
$936$	33718.0	1.17746
$937$	−17058.9	−0.594758	−0.297379	−	0.954759i	$-0.596113\pi$
$937$	−17058.9	−0.594758	−0.297379	+	0.954759i	$0.596113\pi$
$938$	8475.92	0.295041
$939$	−34184.5	−1.18804
$940$	110231.	3.82482
$941$	−40737.9	−1.41128	−0.705641	−	0.708569i	$-0.749341\pi$
$941$	−40737.9	−1.41128	−0.705641	+	0.708569i	$0.749341\pi$
$942$	−74817.8	−2.58779
$943$	46919.3	1.62026
$944$	−18211.5	−0.627897
$945$	−79227.0	−2.72725
$946$	0	0
$947$	2981.51	0.102308	0.0511541	−	0.998691i	$-0.483710\pi$
$947$	2981.51	0.102308	0.0511541	+	0.998691i	$0.483710\pi$
$948$	−117194.	−4.01506
$949$	22460.7	0.768288
$950$	97253.8	3.32140
$951$	25166.4	0.858126
$952$	3106.95	0.105774
$953$	18194.7	0.618453	0.309226	−	0.950988i	$-0.399930\pi$
$953$	18194.7	0.618453	0.309226	+	0.950988i	$0.399930\pi$
$954$	−158446.	−5.37723
$955$	39231.8	1.32933
$956$	−54259.6	−1.83565
$957$	0	0
$958$	−13347.1	−0.450129
$959$	22557.0	0.759545
$960$	151979.	5.10947
$961$	1111.61	0.0373136
$962$	2849.68	0.0955066
$963$	39886.1	1.33470
$964$	48658.3	1.62570
$965$	12560.4	0.418998
$966$	−67487.3	−2.24779
$967$	52558.6	1.74785	0.873924	−	0.486062i	$-0.161567\pi$
$967$	52558.6	1.74785	0.873924	+	0.486062i	$0.161567\pi$
$968$	0	0
$969$	13484.7	0.447049
$970$	−27587.7	−0.913184
$971$	−4184.08	−0.138284	−0.0691419	−	0.997607i	$-0.522026\pi$
$971$	−4184.08	−0.138284	−0.0691419	+	0.997607i	$0.522026\pi$
$972$	55748.4	1.83964
$973$	−1466.95	−0.0483332
$974$	76913.5	2.53025
$975$	−88758.2	−2.91542
$976$	17486.4	0.573489
$977$	47245.3	1.54709	0.773547	−	0.633739i	$-0.218481\pi$
$977$	47245.3	1.54709	0.773547	+	0.633739i	$0.218481\pi$
$978$	−36326.6	−1.18773
$979$	0	0
$980$	44446.9	1.44878
$981$	65804.9	2.14168
$982$	60225.5	1.95710
$983$	−15829.5	−0.513614	−0.256807	−	0.966463i	$-0.582670\pi$
$983$	−15829.5	−0.513614	−0.256807	+	0.966463i	$0.582670\pi$
$984$	−50013.1	−1.62029
$985$	15472.2	0.500492
$986$	−14790.0	−0.477699
$987$	−55816.0	−1.80004
$988$	34314.2	1.10494
$989$	−16850.5	−0.541773
$990$	0	0
$991$	14386.2	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$991$	14386.2	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$992$	40452.9	1.29474
$993$	56088.9	1.79248
$994$	−340.667	−0.0108705
$995$	72447.3	2.30827
$996$	123474.	3.92815
$997$	−3311.55	−0.105193	−0.0525967	−	0.998616i	$-0.516750\pi$
$997$	−3311.55	−0.105193	−0.0525967	+	0.998616i	$0.516750\pi$
$998$	−15194.5	−0.481937
$999$	6028.94	0.190938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.n.1.3	✓	20
11.10	odd	2		2057.4.a.p.1.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2057.4.a.n.1.3	✓	20	1.1	even	1	trivial
2057.4.a.p.1.18	yes	20	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:	\( N \)	\(=\)	\( 2057 = 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2057.a (trivial)

\(f(q)\)	\(=\)	\(q-4.40856 q^{2} +9.44900 q^{3} +11.4354 q^{4} -19.6923 q^{5} -41.6565 q^{6} +12.0675 q^{7} -15.1450 q^{8} +62.2837 q^{9} +O(q^{10})\)	\(q-4.40856 q^{2} +9.44900 q^{3} +11.4354 q^{4} -19.6923 q^{5} -41.6565 q^{6} +12.0675 q^{7} -15.1450 q^{8} +62.2837 q^{9} +86.8147 q^{10} +108.053 q^{12} -35.7452 q^{13} -53.2001 q^{14} -186.073 q^{15} -24.7153 q^{16} -17.0000 q^{17} -274.581 q^{18} -83.9471 q^{19} -225.189 q^{20} +114.026 q^{21} +134.253 q^{23} -143.105 q^{24} +262.788 q^{25} +157.585 q^{26} +333.395 q^{27} +137.996 q^{28} -197.344 q^{29} +820.313 q^{30} +175.791 q^{31} +230.119 q^{32} +74.9455 q^{34} -237.637 q^{35} +712.237 q^{36} +18.0835 q^{37} +370.085 q^{38} -337.757 q^{39} +298.240 q^{40} +349.485 q^{41} -502.688 q^{42} -125.513 q^{43} -1226.51 q^{45} -591.861 q^{46} -489.504 q^{47} -233.535 q^{48} -197.376 q^{49} -1158.51 q^{50} -160.633 q^{51} -408.760 q^{52} +577.046 q^{53} -1469.79 q^{54} -182.762 q^{56} -793.216 q^{57} +870.002 q^{58} +736.852 q^{59} -2127.81 q^{60} -707.512 q^{61} -774.986 q^{62} +751.606 q^{63} -816.770 q^{64} +703.906 q^{65} -159.321 q^{67} -194.401 q^{68} +1268.55 q^{69} +1047.63 q^{70} +6.40350 q^{71} -943.287 q^{72} -628.355 q^{73} -79.7220 q^{74} +2483.08 q^{75} -959.965 q^{76} +1489.02 q^{78} -1084.60 q^{79} +486.702 q^{80} +1468.60 q^{81} -1540.72 q^{82} +1142.72 q^{83} +1303.92 q^{84} +334.770 q^{85} +553.331 q^{86} -1864.70 q^{87} -479.421 q^{89} +5407.14 q^{90} -431.354 q^{91} +1535.23 q^{92} +1661.05 q^{93} +2158.01 q^{94} +1653.11 q^{95} +2174.39 q^{96} -317.777 q^{97} +870.144 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} + 8 q^{3} + 80 q^{4} + 6 q^{5} - 93 q^{6} - 52 q^{7} - 30 q^{8} + 194 q^{9}+O(q^{10}) \) 20 * q - 4 * q^2 + 8 * q^3 + 80 * q^4 + 6 * q^5 - 93 * q^6 - 52 * q^7 - 30 * q^8 + 194 * q^9	\( 20 q - 4 q^{2} + 8 q^{3} + 80 q^{4} + 6 q^{5} - 93 q^{6} - 52 q^{7} - 30 q^{8} + 194 q^{9} - 66 q^{10} + 143 q^{12} - 200 q^{13} + 85 q^{14} - 70 q^{15} + 320 q^{16} - 340 q^{17} - 160 q^{18} - 188 q^{19} + 21 q^{20} + 64 q^{21} - 54 q^{23} - 664 q^{24} + 830 q^{25} + 59 q^{26} + 302 q^{27} - 227 q^{28} - 166 q^{29} - 409 q^{30} + 26 q^{31} - 534 q^{32} + 68 q^{34} - 540 q^{35} + 1813 q^{36} - 120 q^{37} + 49 q^{38} + 482 q^{39} - 1265 q^{40} - 480 q^{41} - 675 q^{42} - 1158 q^{43} - 142 q^{45} - 581 q^{46} + 372 q^{47} + 700 q^{48} - 666 q^{49} - 168 q^{50} - 136 q^{51} - 152 q^{52} - 200 q^{53} - 2051 q^{54} + 551 q^{56} - 1504 q^{57} + 2806 q^{58} + 146 q^{59} - 2549 q^{60} - 1476 q^{61} - 2844 q^{62} - 1798 q^{63} + 1532 q^{64} - 740 q^{65} - 254 q^{67} - 1360 q^{68} + 2566 q^{69} - 860 q^{70} + 394 q^{71} - 489 q^{72} - 2244 q^{73} + 3056 q^{74} + 868 q^{75} - 3345 q^{76} + 612 q^{78} - 1674 q^{79} + 3533 q^{80} - 2676 q^{81} - 3925 q^{82} + 96 q^{83} - 4468 q^{84} - 102 q^{85} - 569 q^{86} - 5498 q^{87} + 1592 q^{89} + 4486 q^{90} + 3364 q^{91} - 4854 q^{92} - 2540 q^{93} - 836 q^{94} + 136 q^{95} - 4207 q^{96} + 1802 q^{97} - 1078 q^{98}+O(q^{100}) \) 20 * q - 4 * q^2 + 8 * q^3 + 80 * q^4 + 6 * q^5 - 93 * q^6 - 52 * q^7 - 30 * q^8 + 194 * q^9 - 66 * q^10 + 143 * q^12 - 200 * q^13 + 85 * q^14 - 70 * q^15 + 320 * q^16 - 340 * q^17 - 160 * q^18 - 188 * q^19 + 21 * q^20 + 64 * q^21 - 54 * q^23 - 664 * q^24 + 830 * q^25 + 59 * q^26 + 302 * q^27 - 227 * q^28 - 166 * q^29 - 409 * q^30 + 26 * q^31 - 534 * q^32 + 68 * q^34 - 540 * q^35 + 1813 * q^36 - 120 * q^37 + 49 * q^38 + 482 * q^39 - 1265 * q^40 - 480 * q^41 - 675 * q^42 - 1158 * q^43 - 142 * q^45 - 581 * q^46 + 372 * q^47 + 700 * q^48 - 666 * q^49 - 168 * q^50 - 136 * q^51 - 152 * q^52 - 200 * q^53 - 2051 * q^54 + 551 * q^56 - 1504 * q^57 + 2806 * q^58 + 146 * q^59 - 2549 * q^60 - 1476 * q^61 - 2844 * q^62 - 1798 * q^63 + 1532 * q^64 - 740 * q^65 - 254 * q^67 - 1360 * q^68 + 2566 * q^69 - 860 * q^70 + 394 * q^71 - 489 * q^72 - 2244 * q^73 + 3056 * q^74 + 868 * q^75 - 3345 * q^76 + 612 * q^78 - 1674 * q^79 + 3533 * q^80 - 2676 * q^81 - 3925 * q^82 + 96 * q^83 - 4468 * q^84 - 102 * q^85 - 569 * q^86 - 5498 * q^87 + 1592 * q^89 + 4486 * q^90 + 3364 * q^91 - 4854 * q^92 - 2540 * q^93 - 836 * q^94 + 136 * q^95 - 4207 * q^96 + 1802 * q^97 - 1078 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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