LMFDB - Embedded newform 1210.3.d.c.241.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1210,3,Mod(241,1210)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1210.241"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1210 = 2 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1210.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$32.9701119876$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3078 x^{12} - 12280 x^{11} + 49836 x^{10} - 147672 x^{9} + \cdots + 339856 $ x^16 - 8x^15 + 92x^14 - 504x^13 + 3078x^12 - 12280x^11 + 49836x^10 - 147672x^9 + 423553x^8 - 924384x^7 + 1876224x^6 - 2898272x^5 + 3947928x^4 - 3906048x^3 + 3026408x^2 - 1437952*x + 339856
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.4
Root			$0.500000 + 1.09141i$ of defining polynomial
Character	$\chi$	$=$	1210.241
Dual form			1210.3.d.c.241.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -1.40090 q^{3} -2.00000 q^{4} -2.23607 q^{5} +1.98117i q^{6} -12.9437i q^{7} +2.82843i q^{8} -7.03748 q^{9} +3.16228i q^{10} +2.80180 q^{12} -13.2743i q^{13} -18.3052 q^{14} +3.13251 q^{15} +4.00000 q^{16} -22.4912i q^{17} +9.95250i q^{18} -20.9333i q^{19} +4.47214 q^{20} +18.1329i q^{21} -3.64326 q^{23} -3.96234i q^{24} +5.00000 q^{25} -18.7727 q^{26} +22.4669 q^{27} +25.8875i q^{28} -0.408409i q^{29} -4.43004i q^{30} -14.7219 q^{31} -5.65685i q^{32} -31.8073 q^{34} +28.9431i q^{35} +14.0750 q^{36} +72.6033 q^{37} -29.6041 q^{38} +18.5960i q^{39} -6.32456i q^{40} +15.5499i q^{41} +25.6438 q^{42} -83.0551i q^{43} +15.7363 q^{45} +5.15234i q^{46} -66.4284 q^{47} -5.60360 q^{48} -118.541 q^{49} -7.07107i q^{50} +31.5079i q^{51} +26.5486i q^{52} -52.2639 q^{53} -31.7730i q^{54} +36.6104 q^{56} +29.3254i q^{57} -0.577578 q^{58} +23.6028 q^{59} -6.26502 q^{60} +56.9548i q^{61} +20.8199i q^{62} +91.0913i q^{63} -8.00000 q^{64} +29.6823i q^{65} +65.4437 q^{67} +44.9823i q^{68} +5.10384 q^{69} +40.9317 q^{70} -33.5584 q^{71} -19.9050i q^{72} -17.1197i q^{73} -102.677i q^{74} -7.00450 q^{75} +41.8666i q^{76} +26.2987 q^{78} -70.8035i q^{79} -8.94427 q^{80} +31.8634 q^{81} +21.9908 q^{82} +93.3805i q^{83} -36.2658i q^{84} +50.2918i q^{85} -117.458 q^{86} +0.572141i q^{87} -24.2634 q^{89} -22.2545i q^{90} -171.819 q^{91} +7.28652 q^{92} +20.6239 q^{93} +93.9440i q^{94} +46.8082i q^{95} +7.92469i q^{96} +77.6721 q^{97} +167.642i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} - 32 q^{4} + 40 q^{9} + 32 q^{12} + 16 q^{14} + 40 q^{15} + 64 q^{16} + 108 q^{23} + 80 q^{25} - 292 q^{27} - 268 q^{31} - 16 q^{34} - 80 q^{36} + 44 q^{37} - 280 q^{38} - 16 q^{42} - 476 q^{47}+ \cdots + 296 q^{97}+O(q^{100}) $ 16 * q - 16 * q^3 - 32 * q^4 + 40 * q^9 + 32 * q^12 + 16 * q^14 + 40 * q^15 + 64 * q^16 + 108 * q^23 + 80 * q^25 - 292 * q^27 - 268 * q^31 - 16 * q^34 - 80 * q^36 + 44 * q^37 - 280 * q^38 - 16 * q^42 - 476 * q^47 - 64 * q^48 - 644 * q^49 - 396 * q^53 - 32 * q^56 + 192 * q^58 - 180 * q^59 - 80 * q^60 - 128 * q^64 + 188 * q^67 - 416 * q^69 + 308 * q^71 - 80 * q^75 + 8 * q^78 + 536 * q^81 + 16 * q^82 + 304 * q^86 - 176 * q^89 + 80 * q^91 - 216 * q^92 + 436 * q^93 + 296 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times$.

$n$	$727$	$1091$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	−1.40090	−0.466967	−0.233483	−	0.972361i	$-0.575012\pi$
$3$	−1.40090	−0.466967	−0.233483	+	0.972361i	$0.575012\pi$
$4$	−2.00000	−0.500000
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	1.98117i	0.330195i
$7$	− 12.9437i	− 1.84911i	−0.381053	−	0.924553i	$-0.624438\pi$
$7$	− 12.9437i	− 1.84911i	0.381053	−	0.924553i	$-0.375562\pi$
$8$	2.82843i	0.353553i
$9$	−7.03748	−0.781942
$10$	3.16228i	0.316228i
$11$	0	0
$11$	0	0
$12$	2.80180	0.233483
$13$	− 13.2743i	− 1.02110i	−0.859848	−	0.510551i	$-0.829442\pi$
$13$	− 13.2743i	− 1.02110i	0.859848	−	0.510551i	$-0.170558\pi$
$14$	−18.3052	−1.30752
$15$	3.13251	0.208834
$16$	4.00000	0.250000
$17$	− 22.4912i	− 1.32301i	−0.749941	−	0.661505i	$-0.769918\pi$
$17$	− 22.4912i	− 1.32301i	0.749941	−	0.661505i	$-0.230082\pi$
$18$	9.95250i	0.552917i
$19$	− 20.9333i	− 1.10175i	−0.834587	−	0.550876i	$-0.814294\pi$
$19$	− 20.9333i	− 1.10175i	0.834587	−	0.550876i	$-0.185706\pi$
$20$	4.47214	0.223607
$21$	18.1329i	0.863471i
$22$	0	0
$23$	−3.64326	−0.158403	−0.0792013	−	0.996859i	$-0.525237\pi$
$23$	−3.64326	−0.158403	−0.0792013	+	0.996859i	$0.525237\pi$
$24$	− 3.96234i	− 0.165098i
$25$	5.00000	0.200000
$26$	−18.7727	−0.722028
$27$	22.4669	0.832108
$28$	25.8875i	0.924553i
$29$	− 0.408409i	− 0.0140831i	−0.999975	−	0.00704154i	$-0.997759\pi$
$29$	− 0.408409i	− 0.0140831i	0.999975	−	0.00704154i	$-0.00224141\pi$
$30$	− 4.43004i	− 0.147668i
$31$	−14.7219	−0.474900	−0.237450	−	0.971400i	$-0.576312\pi$
$31$	−14.7219	−0.474900	−0.237450	+	0.971400i	$0.576312\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−31.8073	−0.935509
$35$	28.9431i	0.826946i
$36$	14.0750	0.390971
$37$	72.6033	1.96225	0.981126	−	0.193368i	$-0.0619411\pi$
$37$	72.6033	1.96225	0.981126	+	0.193368i	$0.0619411\pi$
$38$	−29.6041	−0.779056
$39$	18.5960i	0.476820i
$40$	− 6.32456i	− 0.158114i
$41$	15.5499i	0.379265i	0.981855	+	0.189633i	$0.0607297\pi$
$41$	15.5499i	0.379265i	−0.981855	+	0.189633i	$0.939270\pi$
$42$	25.6438	0.610566
$43$	− 83.0551i	− 1.93151i	−0.259449	−	0.965757i	$-0.583541\pi$
$43$	− 83.0551i	− 1.93151i	0.259449	−	0.965757i	$-0.416459\pi$
$44$	0	0
$45$	15.7363	0.349695
$46$	5.15234i	0.112007i
$47$	−66.4284	−1.41337	−0.706686	−	0.707528i	$-0.749810\pi$
$47$	−66.4284	−1.41337	−0.706686	+	0.707528i	$0.749810\pi$
$48$	−5.60360	−0.116742
$49$	−118.541	−2.41919
$50$	− 7.07107i	− 0.141421i
$51$	31.5079i	0.617802i
$52$	26.5486i	0.510551i
$53$	−52.2639	−0.986112	−0.493056	−	0.869998i	$-0.664120\pi$
$53$	−52.2639	−0.986112	−0.493056	+	0.869998i	$0.664120\pi$
$54$	− 31.7730i	− 0.588389i
$55$	0	0
$56$	36.6104	0.653758
$57$	29.3254i	0.514481i
$58$	−0.577578	−0.00995824
$59$	23.6028	0.400048	0.200024	−	0.979791i	$-0.435898\pi$
$59$	23.6028	0.400048	0.200024	+	0.979791i	$0.435898\pi$
$60$	−6.26502	−0.104417
$61$	56.9548i	0.933685i	0.884341	+	0.466842i	$0.154608\pi$
$61$	56.9548i	0.933685i	−0.884341	+	0.466842i	$0.845392\pi$
$62$	20.8199i	0.335805i
$63$	91.0913i	1.44589i
$64$	−8.00000	−0.125000
$65$	29.6823i	0.456650i
$66$	0	0
$67$	65.4437	0.976771	0.488386	−	0.872628i	$-0.337586\pi$
$67$	65.4437	0.976771	0.488386	+	0.872628i	$0.337586\pi$
$68$	44.9823i	0.661505i
$69$	5.10384	0.0739687
$70$	40.9317	0.584739
$71$	−33.5584	−0.472654	−0.236327	−	0.971674i	$-0.575944\pi$
$71$	−33.5584	−0.472654	−0.236327	+	0.971674i	$0.575944\pi$
$72$	− 19.9050i	− 0.276458i
$73$	− 17.1197i	− 0.234517i	−0.993101	−	0.117258i	$-0.962589\pi$
$73$	− 17.1197i	− 0.234517i	0.993101	−	0.117258i	$-0.0374106\pi$
$74$	− 102.677i	− 1.38752i
$75$	−7.00450	−0.0933934
$76$	41.8666i	0.550876i
$77$	0	0
$78$	26.2987	0.337163
$79$	− 70.8035i	− 0.896247i	−0.893972	−	0.448123i	$-0.852092\pi$
$79$	− 70.8035i	− 0.896247i	0.893972	−	0.448123i	$-0.147908\pi$
$80$	−8.94427	−0.111803
$81$	31.8634	0.393375
$82$	21.9908	0.268181
$83$	93.3805i	1.12507i	0.826775	+	0.562533i	$0.190173\pi$
$83$	93.3805i	1.12507i	−0.826775	+	0.562533i	$0.809827\pi$
$84$	− 36.2658i	− 0.431736i
$85$	50.2918i	0.591668i
$86$	−117.458	−1.36579
$87$	0.572141i	0.00657633i
$88$	0	0
$89$	−24.2634	−0.272622	−0.136311	−	0.990666i	$-0.543525\pi$
$89$	−24.2634	−0.272622	−0.136311	+	0.990666i	$0.543525\pi$
$90$	− 22.2545i	− 0.247272i
$91$	−171.819	−1.88812
$92$	7.28652	0.0792013
$93$	20.6239	0.221762
$94$	93.9440i	0.999404i
$95$	46.8082i	0.492718i
$96$	7.92469i	0.0825488i
$97$	77.6721	0.800743	0.400371	−	0.916353i	$-0.368881\pi$
$97$	77.6721	0.800743	0.400371	+	0.916353i	$0.368881\pi$
$98$	167.642i	1.71063i
$99$	0	0
$100$	−10.0000	−0.100000
$101$	63.0549i	0.624306i	0.950032	+	0.312153i	$0.101050\pi$
$101$	63.0549i	0.624306i	−0.950032	+	0.312153i	$0.898950\pi$
$102$	44.5589	0.436852
$103$	33.8999	0.329125	0.164562	−	0.986367i	$-0.447379\pi$
$103$	33.8999	0.329125	0.164562	+	0.986367i	$0.447379\pi$
$104$	37.5454	0.361014
$105$	− 40.5464i	− 0.386156i
$106$	73.9124i	0.697286i
$107$	58.9331i	0.550777i	0.961333	+	0.275388i	$0.0888064\pi$
$107$	58.9331i	0.550777i	−0.961333	+	0.275388i	$0.911194\pi$
$108$	−44.9338	−0.416054
$109$	170.424i	1.56352i	0.623579	+	0.781760i	$0.285678\pi$
$109$	170.424i	1.56352i	−0.623579	+	0.781760i	$0.714322\pi$
$110$	0	0
$111$	−101.710	−0.916307
$112$	− 51.7750i	− 0.462277i
$113$	−22.4704	−0.198853	−0.0994264	−	0.995045i	$-0.531701\pi$
$113$	−22.4704	−0.198853	−0.0994264	+	0.995045i	$0.531701\pi$
$114$	41.4724	0.363793
$115$	8.14657	0.0708398
$116$	0.816818i	0.00704154i
$117$	93.4177i	0.798442i
$118$	− 33.3794i	− 0.282877i
$119$	−291.120	−2.44639
$120$	8.86007i	0.0738339i
$121$	0	0
$122$	80.5462	0.660215
$123$	− 21.7838i	− 0.177104i
$124$	29.4438	0.237450
$125$	−11.1803	−0.0894427
$126$	128.823	1.02240
$127$	− 49.6601i	− 0.391024i	−0.980701	−	0.195512i	$-0.937363\pi$
$127$	− 49.6601i	− 0.391024i	0.980701	−	0.195512i	$-0.0626369\pi$
$128$	11.3137i	0.0883883i
$129$	116.352i	0.901953i
$130$	41.9771	0.322901
$131$	− 120.711i	− 0.921456i	−0.887541	−	0.460728i	$-0.847588\pi$
$131$	− 120.711i	− 0.921456i	0.887541	−	0.460728i	$-0.152412\pi$
$132$	0	0
$133$	−270.955	−2.03726
$134$	− 92.5513i	− 0.690682i
$135$	−50.2375	−0.372130
$136$	63.6146	0.467755
$137$	−164.807	−1.20297	−0.601485	−	0.798884i	$-0.705424\pi$
$137$	−164.807	−1.20297	−0.601485	+	0.798884i	$0.705424\pi$
$138$	− 7.21792i	− 0.0523038i
$139$	114.360i	0.822735i	0.911470	+	0.411368i	$0.134949\pi$
$139$	114.360i	0.822735i	−0.911470	+	0.411368i	$0.865051\pi$
$140$	− 57.8862i	− 0.413473i
$141$	93.0596	0.659997
$142$	47.4588i	0.334217i
$143$	0	0
$144$	−28.1499	−0.195486
$145$	0.913231i	0.00629814i
$146$	−24.2109	−0.165828
$147$	166.064	1.12968
$148$	−145.207	−0.981126
$149$	− 4.26385i	− 0.0286164i	−0.999898	−	0.0143082i	$-0.995445\pi$
$149$	− 4.26385i	− 0.0286164i	0.999898	−	0.0143082i	$-0.00455460\pi$
$150$	9.90586i	0.0660391i
$151$	− 63.7963i	− 0.422492i	−0.977433	−	0.211246i	$-0.932248\pi$
$151$	− 63.7963i	− 0.422492i	0.977433	−	0.211246i	$-0.0677521\pi$
$152$	59.2083	0.389528
$153$	158.281i	1.03452i
$154$	0	0
$155$	32.9192	0.212382
$156$	− 37.1920i	− 0.238410i
$157$	13.8309	0.0880946	0.0440473	−	0.999029i	$-0.485975\pi$
$157$	13.8309	0.0880946	0.0440473	+	0.999029i	$0.485975\pi$
$158$	−100.131	−0.633742
$159$	73.2166	0.460482
$160$	12.6491i	0.0790569i
$161$	47.1574i	0.292903i
$162$	− 45.0617i	− 0.278158i
$163$	151.812	0.931362	0.465681	−	0.884953i	$-0.345809\pi$
$163$	151.812	0.931362	0.465681	+	0.884953i	$0.345809\pi$
$164$	− 31.0997i	− 0.189633i
$165$	0	0
$166$	132.060	0.795542
$167$	104.443i	0.625407i	0.949851	+	0.312703i	$0.101235\pi$
$167$	104.443i	0.625407i	−0.949851	+	0.312703i	$0.898765\pi$
$168$	−51.2876	−0.305283
$169$	−7.20745	−0.0426476
$170$	71.1233	0.418372
$171$	147.317i	0.861506i
$172$	166.110i	0.965757i
$173$	177.042i	1.02336i	0.859176	+	0.511681i	$0.170977\pi$
$173$	177.042i	1.02336i	−0.859176	+	0.511681i	$0.829023\pi$
$174$	0.809129	0.00465017
$175$	− 64.7187i	− 0.369821i
$176$	0	0
$177$	−33.0652	−0.186809
$178$	34.3136i	0.192773i
$179$	−73.9295	−0.413014	−0.206507	−	0.978445i	$-0.566210\pi$
$179$	−73.9295	−0.413014	−0.206507	+	0.978445i	$0.566210\pi$
$180$	−31.4726	−0.174848
$181$	113.155	0.625164	0.312582	−	0.949891i	$-0.398806\pi$
$181$	113.155	0.625164	0.312582	+	0.949891i	$0.398806\pi$
$182$	242.989i	1.33511i
$183$	− 79.7879i	− 0.436000i
$184$	− 10.3047i	− 0.0560037i
$185$	−162.346	−0.877546
$186$	− 29.1666i	− 0.156810i
$187$	0	0
$188$	132.857	0.706686
$189$	− 290.806i	− 1.53866i
$190$	66.1968	0.348404
$191$	330.451	1.73011	0.865055	−	0.501677i	$-0.167283\pi$
$191$	330.451	1.73011	0.865055	+	0.501677i	$0.167283\pi$
$192$	11.2072	0.0583708
$193$	67.5028i	0.349755i	0.984590	+	0.174878i	$0.0559530\pi$
$193$	67.5028i	0.349755i	−0.984590	+	0.174878i	$0.944047\pi$
$194$	− 109.845i	− 0.566211i
$195$	− 41.5819i	− 0.213241i
$196$	237.081	1.20960
$197$	− 297.157i	− 1.50841i	−0.656637	−	0.754206i	$-0.728022\pi$
$197$	− 297.157i	− 1.50841i	0.656637	−	0.754206i	$-0.271978\pi$
$198$	0	0
$199$	268.728	1.35039	0.675197	−	0.737638i	$-0.264059\pi$
$199$	268.728	1.35039	0.675197	+	0.737638i	$0.264059\pi$
$200$	14.1421i	0.0707107i
$201$	−91.6801	−0.456120
$202$	89.1730	0.441451
$203$	−5.28634	−0.0260411
$204$	− 63.0158i	− 0.308901i
$205$	− 34.7706i	− 0.169613i
$206$	− 47.9417i	− 0.232727i
$207$	25.6393	0.123862
$208$	− 53.0973i	− 0.255275i
$209$	0	0
$210$	−57.3413	−0.273054
$211$	− 214.310i	− 1.01569i	−0.861450	−	0.507843i	$-0.830443\pi$
$211$	− 214.310i	− 1.01569i	0.861450	−	0.507843i	$-0.169557\pi$
$212$	104.528	0.493056
$213$	47.0120	0.220714
$214$	83.3440	0.389458
$215$	185.717i	0.863799i
$216$	63.5460i	0.294195i
$217$	190.556i	0.878140i
$218$	241.016	1.10558
$219$	23.9830i	0.109512i
$220$	0	0
$221$	−298.555	−1.35093
$222$	143.840i	0.647927i
$223$	135.355	0.606975	0.303488	−	0.952835i	$-0.401849\pi$
$223$	135.355	0.606975	0.303488	+	0.952835i	$0.401849\pi$
$224$	−73.2209	−0.326879
$225$	−35.1874	−0.156388
$226$	31.7779i	0.140610i
$227$	− 176.290i	− 0.776609i	−0.921531	−	0.388304i	$-0.873061\pi$
$227$	− 176.290i	− 0.776609i	0.921531	−	0.388304i	$-0.126939\pi$
$228$	− 58.6509i	− 0.257241i
$229$	−190.449	−0.831654	−0.415827	−	0.909444i	$-0.636508\pi$
$229$	−190.449	−0.831654	−0.415827	+	0.909444i	$0.636508\pi$
$230$	− 11.5210i	− 0.0500913i
$231$	0	0
$232$	1.15516	0.00497912
$233$	− 12.7100i	− 0.0545494i	−0.999628	−	0.0272747i	$-0.991317\pi$
$233$	− 12.7100i	− 0.0545494i	0.999628	−	0.0272747i	$-0.00868289\pi$
$234$	132.113	0.564584
$235$	148.539	0.632079
$236$	−47.2057	−0.200024
$237$	99.1887i	0.418518i
$238$	411.706i	1.72986i
$239$	35.7470i	0.149569i	0.997200	+	0.0747845i	$0.0238269\pi$
$239$	35.7470i	0.149569i	−0.997200	+	0.0747845i	$0.976173\pi$
$240$	12.5300	0.0522085
$241$	68.2311i	0.283117i	0.989930	+	0.141558i	$0.0452113\pi$
$241$	68.2311i	0.283117i	−0.989930	+	0.141558i	$0.954789\pi$
$242$	0	0
$243$	−246.840	−1.01580
$244$	− 113.910i	− 0.466842i
$245$	265.065	1.08190
$246$	−30.8070	−0.125232
$247$	−277.875	−1.12500
$248$	− 41.6398i	− 0.167902i
$249$	− 130.817i	− 0.525369i
$250$	15.8114i	0.0632456i
$251$	−18.4384	−0.0734596	−0.0367298	−	0.999325i	$-0.511694\pi$
$251$	−18.4384	−0.0734596	−0.0367298	+	0.999325i	$0.511694\pi$
$252$	− 182.183i	− 0.722947i
$253$	0	0
$254$	−70.2300	−0.276496
$255$	− 70.4538i	− 0.276289i
$256$	16.0000	0.0625000
$257$	20.7937	0.0809095	0.0404548	−	0.999181i	$-0.487119\pi$
$257$	20.7937	0.0809095	0.0404548	+	0.999181i	$0.487119\pi$
$258$	164.546	0.637777
$259$	− 939.759i	− 3.62841i
$260$	− 59.3645i	− 0.228325i
$261$	2.87417i	0.0110121i
$262$	−170.711	−0.651568
$263$	227.802i	0.866167i	0.901354	+	0.433083i	$0.142574\pi$
$263$	227.802i	0.866167i	−0.901354	+	0.433083i	$0.857426\pi$
$264$	0	0
$265$	116.866	0.441003
$266$	383.188i	1.44056i
$267$	33.9906	0.127306
$268$	−130.887	−0.488386
$269$	253.579	0.942672	0.471336	−	0.881954i	$-0.343772\pi$
$269$	253.579	0.942672	0.471336	+	0.881954i	$0.343772\pi$
$270$	71.0466i	0.263136i
$271$	− 103.650i	− 0.382473i	−0.981544	−	0.191237i	$-0.938750\pi$
$271$	− 103.650i	− 0.382473i	0.981544	−	0.191237i	$-0.0612498\pi$
$272$	− 89.9647i	− 0.330752i
$273$	240.702	0.881692
$274$	233.072i	0.850628i
$275$	0	0
$276$	−10.2077	−0.0369844
$277$	− 329.481i	− 1.18946i	−0.803924	−	0.594732i	$-0.797258\pi$
$277$	− 329.481i	− 1.18946i	0.803924	−	0.594732i	$-0.202742\pi$
$278$	161.730	0.581762
$279$	103.605	0.371344
$280$	−81.8634	−0.292369
$281$	− 436.320i	− 1.55274i	−0.630278	−	0.776370i	$-0.717059\pi$
$281$	− 436.320i	− 1.55274i	0.630278	−	0.776370i	$-0.282941\pi$
$282$	− 131.606i	− 0.466689i
$283$	273.197i	0.965361i	0.875796	+	0.482681i	$0.160337\pi$
$283$	273.197i	0.965361i	−0.875796	+	0.482681i	$0.839663\pi$
$284$	67.1168	0.236327
$285$	− 65.5737i	− 0.230083i
$286$	0	0
$287$	201.274	0.701302
$288$	39.8100i	0.138229i
$289$	−216.853	−0.750355
$290$	1.29150	0.00445346
$291$	−108.811	−0.373920
$292$	34.2394i	0.117258i
$293$	− 247.912i	− 0.846116i	−0.906103	−	0.423058i	$-0.860957\pi$
$293$	− 247.912i	− 0.846116i	0.906103	−	0.423058i	$-0.139043\pi$
$294$	− 234.849i	− 0.798807i
$295$	−52.7775	−0.178907
$296$	205.353i	0.693761i
$297$	0	0
$298$	−6.02999	−0.0202349
$299$	48.3618i	0.161745i
$300$	14.0090	0.0466967
$301$	−1075.04	−3.57157
$302$	−90.2216	−0.298747
$303$	− 88.3336i	− 0.291530i
$304$	− 83.7331i	− 0.275438i
$305$	− 127.355i	− 0.417556i
$306$	223.843	0.731514
$307$	− 382.783i	− 1.24685i	−0.781884	−	0.623425i	$-0.785741\pi$
$307$	− 382.783i	− 1.24685i	0.781884	−	0.623425i	$-0.214259\pi$
$308$	0	0
$309$	−47.4903	−0.153690
$310$	− 46.5547i	− 0.150177i
$311$	202.814	0.652134	0.326067	−	0.945347i	$-0.394277\pi$
$311$	202.814	0.652134	0.326067	+	0.945347i	$0.394277\pi$
$312$	−52.5974	−0.168581
$313$	−95.6903	−0.305720	−0.152860	−	0.988248i	$-0.548848\pi$
$313$	−95.6903	−0.305720	−0.152860	+	0.988248i	$0.548848\pi$
$314$	− 19.5598i	− 0.0622923i
$315$	− 203.686i	− 0.646623i
$316$	141.607i	0.448123i
$317$	−187.943	−0.592881	−0.296441	−	0.955051i	$-0.595800\pi$
$317$	−187.943	−0.592881	−0.296441	+	0.955051i	$0.595800\pi$
$318$	− 103.544i	− 0.325610i
$319$	0	0
$320$	17.8885	0.0559017
$321$	− 82.5594i	− 0.257194i
$322$	66.6906	0.207114
$323$	−470.814	−1.45763
$324$	−63.7268	−0.196688
$325$	− 66.3716i	− 0.204220i
$326$	− 214.695i	− 0.658573i
$327$	− 238.747i	− 0.730112i
$328$	−43.9817	−0.134091
$329$	859.833i	2.61347i
$330$	0	0
$331$	−283.782	−0.857349	−0.428674	−	0.903459i	$-0.641019\pi$
$331$	−283.782	−0.857349	−0.428674	+	0.903459i	$0.641019\pi$
$332$	− 186.761i	− 0.562533i
$333$	−510.944	−1.53437
$334$	147.705	0.442230
$335$	−146.337	−0.436825
$336$	72.5316i	0.215868i
$337$	498.268i	1.47854i	0.673409	+	0.739270i	$0.264829\pi$
$337$	498.268i	1.47854i	−0.673409	+	0.739270i	$0.735171\pi$
$338$	10.1929i	0.0301564i
$339$	31.4787	0.0928576
$340$	− 100.584i	− 0.295834i
$341$	0	0
$342$	208.338	0.609177
$343$	900.115i	2.62424i
$344$	234.915	0.682893
$345$	−11.4125	−0.0330798
$346$	250.375	0.723626
$347$	247.160i	0.712276i	0.934433	+	0.356138i	$0.115907\pi$
$347$	247.160i	0.712276i	−0.934433	+	0.356138i	$0.884093\pi$
$348$	− 1.14428i	− 0.00328816i
$349$	− 566.031i	− 1.62187i	−0.585139	−	0.810933i	$-0.698960\pi$
$349$	− 566.031i	− 1.62187i	0.585139	−	0.810933i	$-0.301040\pi$
$350$	−91.5261	−0.261503
$351$	− 298.233i	− 0.849666i
$352$	0	0
$353$	206.341	0.584536	0.292268	−	0.956336i	$-0.405590\pi$
$353$	206.341	0.584536	0.292268	+	0.956336i	$0.405590\pi$
$354$	46.7613i	0.132094i
$355$	75.0389	0.211377
$356$	48.5268	0.136311
$357$	407.830	1.14238
$358$	104.552i	0.292045i
$359$	− 618.192i	− 1.72198i	−0.508619	−	0.860992i	$-0.669844\pi$
$359$	− 618.192i	− 1.72198i	0.508619	−	0.860992i	$-0.330156\pi$
$360$	44.5089i	0.123636i
$361$	−77.2022	−0.213856
$362$	− 160.025i	− 0.442058i
$363$	0	0
$364$	343.639	0.944062
$365$	38.2809i	0.104879i
$366$	−112.837	−0.308298
$367$	−616.708	−1.68040	−0.840202	−	0.542274i	$-0.817563\pi$
$367$	−616.708	−1.68040	−0.840202	+	0.542274i	$0.817563\pi$
$368$	−14.5730	−0.0396006
$369$	− 109.432i	− 0.296563i
$370$	229.592i	0.620519i
$371$	676.491i	1.82343i
$372$	−41.2478	−0.110881
$373$	− 661.539i	− 1.77356i	−0.462188	−	0.886782i	$-0.652936\pi$
$373$	− 661.539i	− 1.77356i	0.462188	−	0.886782i	$-0.347064\pi$
$374$	0	0
$375$	15.6625	0.0417668
$376$	− 187.888i	− 0.499702i
$377$	−5.42135	−0.0143802
$378$	−411.262	−1.08799
$379$	113.126	0.298486	0.149243	−	0.988801i	$-0.452316\pi$
$379$	113.126	0.298486	0.149243	+	0.988801i	$0.452316\pi$
$380$	− 93.6165i	− 0.246359i
$381$	69.5689i	0.182595i
$382$	− 467.328i	− 1.22337i
$383$	625.977	1.63440	0.817202	−	0.576352i	$-0.195524\pi$
$383$	625.977	1.63440	0.817202	+	0.576352i	$0.195524\pi$
$384$	− 15.8494i	− 0.0412744i
$385$	0	0
$386$	95.4634	0.247314
$387$	584.498i	1.51033i
$388$	−155.344	−0.400371
$389$	242.609	0.623674	0.311837	−	0.950136i	$-0.399056\pi$
$389$	242.609	0.623674	0.311837	+	0.950136i	$0.399056\pi$
$390$	−58.8057	−0.150784
$391$	81.9411i	0.209568i
$392$	− 335.283i	− 0.855315i
$393$	169.104i	0.430290i
$394$	−420.244	−1.06661
$395$	158.321i	0.400814i
$396$	0	0
$397$	−477.057	−1.20165	−0.600827	−	0.799379i	$-0.705162\pi$
$397$	−477.057	−1.20165	−0.600827	+	0.799379i	$0.705162\pi$
$398$	− 380.039i	− 0.954872i
$399$	379.581	0.951331
$400$	20.0000	0.0500000
$401$	−333.453	−0.831553	−0.415777	−	0.909467i	$-0.636490\pi$
$401$	−333.453	−0.831553	−0.415777	+	0.909467i	$0.636490\pi$
$402$	129.655i	0.322525i
$403$	195.423i	0.484921i
$404$	− 126.110i	− 0.312153i
$405$	−71.2487	−0.175923
$406$	7.47602i	0.0184138i
$407$	0	0
$408$	−89.1178	−0.218426
$409$	− 13.6989i	− 0.0334937i	−0.999860	−	0.0167468i	$-0.994669\pi$
$409$	− 13.6989i	− 0.0334937i	0.999860	−	0.0167468i	$-0.00533094\pi$
$410$	−49.1730	−0.119934
$411$	230.878	0.561747
$412$	−67.7997	−0.164562
$413$	− 305.509i	− 0.739731i
$414$	− 36.2595i	− 0.0875834i
$415$	− 208.805i	− 0.503145i
$416$	−75.0909	−0.180507
$417$	− 160.207i	− 0.384190i
$418$	0	0
$419$	−21.0470	−0.0502316	−0.0251158	−	0.999685i	$-0.507995\pi$
$419$	−21.0470	−0.0502316	−0.0251158	+	0.999685i	$0.507995\pi$
$420$	81.0928i	0.193078i
$421$	−357.356	−0.848828	−0.424414	−	0.905468i	$-0.639520\pi$
$421$	−357.356	−0.848828	−0.424414	+	0.905468i	$0.639520\pi$
$422$	−303.080	−0.718198
$423$	467.489	1.10517
$424$	− 147.825i	− 0.348643i
$425$	− 112.456i	− 0.264602i
$426$	− 66.4850i	− 0.156068i
$427$	737.208	1.72648
$428$	− 117.866i	− 0.275388i
$429$	0	0
$430$	262.643	0.610798
$431$	439.263i	1.01917i	0.860420	+	0.509586i	$0.170201\pi$
$431$	439.263i	1.01917i	−0.860420	+	0.509586i	$0.829799\pi$
$432$	89.8676	0.208027
$433$	−259.257	−0.598745	−0.299373	−	0.954136i	$-0.596777\pi$
$433$	−259.257	−0.598745	−0.299373	+	0.954136i	$0.596777\pi$
$434$	269.488	0.620939
$435$	− 1.27935i	− 0.00294102i
$436$	− 340.847i	− 0.781760i
$437$	76.2653i	0.174520i
$438$	33.9171	0.0774364
$439$	− 531.988i	− 1.21182i	−0.795534	−	0.605909i	$-0.792810\pi$
$439$	− 531.988i	− 1.21182i	0.795534	−	0.605909i	$-0.207190\pi$
$440$	0	0
$441$	834.227	1.89167
$442$	422.220i	0.955250i
$443$	405.912	0.916281	0.458140	−	0.888880i	$-0.348516\pi$
$443$	405.912	0.916281	0.458140	+	0.888880i	$0.348516\pi$
$444$	203.420	0.458153
$445$	54.2546	0.121920
$446$	− 191.422i	− 0.429196i
$447$	5.97323i	0.0133629i
$448$	103.550i	0.231138i
$449$	−566.274	−1.26119	−0.630595	−	0.776112i	$-0.717189\pi$
$449$	−566.274	−1.26119	−0.630595	+	0.776112i	$0.717189\pi$
$450$	49.7625i	0.110583i
$451$	0	0
$452$	44.9407	0.0994264
$453$	89.3723i	0.197290i
$454$	−249.312	−0.549145
$455$	384.200	0.844395
$456$	−82.9449	−0.181897
$457$	− 343.900i	− 0.752517i	−0.926515	−	0.376258i	$-0.877211\pi$
$457$	− 343.900i	− 0.752517i	0.926515	−	0.376258i	$-0.122789\pi$
$458$	269.335i	0.588068i
$459$	− 505.307i	− 1.10089i
$460$	−16.2931	−0.0354199
$461$	− 545.981i	− 1.18434i	−0.805813	−	0.592170i	$-0.798271\pi$
$461$	− 545.981i	− 1.18434i	0.805813	−	0.592170i	$-0.201729\pi$
$462$	0	0
$463$	318.490	0.687883	0.343941	−	0.938991i	$-0.388238\pi$
$463$	318.490	0.687883	0.343941	+	0.938991i	$0.388238\pi$
$464$	− 1.63364i	− 0.00352077i
$465$	−46.1165	−0.0991752
$466$	−17.9747	−0.0385723
$467$	10.2324	0.0219110	0.0109555	−	0.999940i	$-0.496513\pi$
$467$	10.2324	0.0219110	0.0109555	+	0.999940i	$0.496513\pi$
$468$	− 186.835i	− 0.399221i
$469$	− 847.086i	− 1.80615i
$470$	− 210.065i	− 0.446947i
$471$	−19.3756	−0.0411372
$472$	66.7589i	0.141438i
$473$	0	0
$474$	140.274	0.295937
$475$	− 104.666i	− 0.220350i
$476$	582.240	1.22319
$477$	367.806	0.771082
$478$	50.5539	0.105761
$479$	364.015i	0.759947i	0.924997	+	0.379973i	$0.124067\pi$
$479$	364.015i	0.759947i	−0.924997	+	0.379973i	$0.875933\pi$
$480$	− 17.7201i	− 0.0369170i
$481$	− 963.760i	− 2.00366i
$482$	96.4933	0.200194
$483$	− 66.0628i	− 0.136776i
$484$	0	0
$485$	−173.680	−0.358103
$486$	349.084i	0.718280i
$487$	215.800	0.443121	0.221560	−	0.975147i	$-0.428885\pi$
$487$	215.800	0.443121	0.221560	+	0.975147i	$0.428885\pi$
$488$	−161.092	−0.330107
$489$	−212.674	−0.434915
$490$	− 374.858i	− 0.765017i
$491$	830.488i	1.69142i	0.533641	+	0.845711i	$0.320823\pi$
$491$	830.488i	1.69142i	−0.533641	+	0.845711i	$0.679177\pi$
$492$	43.5676i	0.0885521i
$493$	−9.18560	−0.0186320
$494$	392.975i	0.795495i
$495$	0	0
$496$	−58.8876	−0.118725
$497$	434.371i	0.873987i
$498$	−185.003	−0.371492
$499$	−201.822	−0.404453	−0.202227	−	0.979339i	$-0.564818\pi$
$499$	−201.822	−0.404453	−0.202227	+	0.979339i	$0.564818\pi$
$500$	22.3607	0.0447214
$501$	− 146.314i	− 0.292044i
$502$	26.0758i	0.0519438i
$503$	540.336i	1.07423i	0.843510	+	0.537113i	$0.180485\pi$
$503$	540.336i	1.07423i	−0.843510	+	0.537113i	$0.819515\pi$
$504$	−257.645	−0.511201
$505$	− 140.995i	− 0.279198i
$506$	0	0
$507$	10.0969	0.0199150
$508$	99.3202i	0.195512i
$509$	441.188	0.866773	0.433387	−	0.901208i	$-0.357318\pi$
$509$	441.188	0.866773	0.433387	+	0.901208i	$0.357318\pi$
$510$	−99.6367	−0.195366
$511$	−221.593	−0.433646
$512$	− 22.6274i	− 0.0441942i
$513$	− 470.306i	− 0.916776i
$514$	− 29.4068i	− 0.0572117i
$515$	−75.8024	−0.147189
$516$	− 232.704i	− 0.450976i
$517$	0	0
$518$	−1329.02	−2.56568
$519$	− 248.018i	− 0.477876i
$520$	−83.9541	−0.161450
$521$	−887.995	−1.70441	−0.852203	−	0.523212i	$-0.824734\pi$
$521$	−887.995	−1.70441	−0.852203	+	0.523212i	$0.824734\pi$
$522$	4.06469	0.00778676
$523$	723.040i	1.38249i	0.722622	+	0.691243i	$0.242937\pi$
$523$	723.040i	1.38249i	−0.722622	+	0.691243i	$0.757063\pi$
$524$	241.422i	0.460728i
$525$	90.6645i	0.172694i
$526$	322.161	0.612473
$527$	331.113i	0.628297i
$528$	0	0
$529$	−515.727	−0.974909
$530$	− 165.273i	− 0.311836i
$531$	−166.104	−0.312814
$532$	541.910	1.01863
$533$	206.414	0.387268
$534$	− 48.0700i	− 0.0900187i
$535$	− 131.778i	− 0.246315i
$536$	185.103i	0.345341i
$537$	103.568	0.192864
$538$	− 358.614i	− 0.666570i
$539$	0	0
$540$	100.475	0.186065
$541$	− 156.074i	− 0.288492i	−0.989542	−	0.144246i	$-0.953924\pi$
$541$	− 156.074i	− 0.288492i	0.989542	−	0.144246i	$-0.0460757\pi$
$542$	−146.584	−0.270450
$543$	−158.519	−0.291931
$544$	−127.229	−0.233877
$545$	− 381.079i	− 0.699227i
$546$	− 340.404i	− 0.623450i
$547$	70.4507i	0.128795i	0.997924	+	0.0643973i	$0.0205125\pi$
$547$	70.4507i	0.128795i	−0.997924	+	0.0643973i	$0.979487\pi$
$548$	329.614	0.601485
$549$	− 400.818i	− 0.730087i
$550$	0	0
$551$	−8.54934	−0.0155160
$552$	14.4358i	0.0261519i
$553$	−916.463	−1.65726
$554$	−465.957	−0.841078
$555$	227.431	0.409785
$556$	− 228.720i	− 0.411368i
$557$	− 65.2295i	− 0.117109i	−0.998284	−	0.0585543i	$-0.981351\pi$
$557$	− 65.2295i	− 0.117109i	0.998284	−	0.0585543i	$-0.0186491\pi$
$558$	− 146.520i	− 0.262580i
$559$	−1102.50	−1.97227
$560$	115.772i	0.206736i
$561$	0	0
$562$	−617.049	−1.09795
$563$	83.5605i	0.148420i	0.997243	+	0.0742100i	$0.0236435\pi$
$563$	83.5605i	0.148420i	−0.997243	+	0.0742100i	$0.976356\pi$
$564$	−186.119	−0.329999
$565$	50.2452	0.0889296
$566$	386.359	0.682614
$567$	− 412.432i	− 0.727393i
$568$	− 94.9175i	− 0.167108i
$569$	− 465.539i	− 0.818170i	−0.912496	−	0.409085i	$-0.865848\pi$
$569$	− 465.539i	− 0.818170i	0.912496	−	0.409085i	$-0.134152\pi$
$570$	−92.7352	−0.162693
$571$	523.333i	0.916520i	0.888818	+	0.458260i	$0.151527\pi$
$571$	523.333i	0.916520i	−0.888818	+	0.458260i	$0.848473\pi$
$572$	0	0
$573$	−462.929	−0.807904
$574$	− 284.644i	− 0.495895i
$575$	−18.2163	−0.0316805
$576$	56.2998	0.0977428
$577$	609.950	1.05711	0.528553	−	0.848900i	$-0.322735\pi$
$577$	609.950	1.05711	0.528553	+	0.848900i	$0.322735\pi$
$578$	306.676i	0.530581i
$579$	− 94.5647i	− 0.163324i
$580$	− 1.82646i	− 0.00314907i
$581$	1208.69	2.08037
$582$	153.882i	0.264402i
$583$	0	0
$584$	48.4219	0.0829142
$585$	− 208.888i	− 0.357074i
$586$	−350.601	−0.598295
$587$	102.776	0.175087	0.0875433	−	0.996161i	$-0.472098\pi$
$587$	102.776	0.175087	0.0875433	+	0.996161i	$0.472098\pi$
$588$	−332.127	−0.564842
$589$	308.178i	0.523222i
$590$	74.6387i	0.126506i
$591$	416.288i	0.704379i
$592$	290.413	0.490563
$593$	1035.02i	1.74539i	0.488265	+	0.872695i	$0.337630\pi$
$593$	1035.02i	1.74539i	−0.488265	+	0.872695i	$0.662370\pi$
$594$	0	0
$595$	650.964	1.09406
$596$	8.52769i	0.0143082i
$597$	−376.462	−0.630589
$598$	68.3939	0.114371
$599$	177.980	0.297129	0.148564	−	0.988903i	$-0.452535\pi$
$599$	177.980	0.297129	0.148564	+	0.988903i	$0.452535\pi$
$600$	− 19.8117i	− 0.0330195i
$601$	199.308i	0.331627i	0.986157	+	0.165813i	$0.0530249\pi$
$601$	199.308i	0.331627i	−0.986157	+	0.165813i	$0.946975\pi$
$602$	1520.34i	2.52548i
$603$	−460.559	−0.763779
$604$	127.593i	0.211246i
$605$	0	0
$606$	−124.923	−0.206143
$607$	− 363.011i	− 0.598041i	−0.954247	−	0.299021i	$-0.903340\pi$
$607$	− 363.011i	− 0.598041i	0.954247	−	0.299021i	$-0.0966600\pi$
$608$	−118.417	−0.194764
$609$	7.40564	0.0121603
$610$	−180.107	−0.295257
$611$	881.792i	1.44320i
$612$	− 316.562i	− 0.517258i
$613$	− 257.811i	− 0.420573i	−0.977640	−	0.210286i	$-0.932560\pi$
$613$	− 257.811i	− 0.420573i	0.977640	−	0.210286i	$-0.0674397\pi$
$614$	−541.336	−0.881655
$615$	48.7101i	0.0792034i
$616$	0	0
$617$	516.144	0.836538	0.418269	−	0.908323i	$-0.362637\pi$
$617$	516.144	0.836538	0.418269	+	0.908323i	$0.362637\pi$
$618$	67.1615i	0.108676i
$619$	625.277	1.01014	0.505070	−	0.863078i	$-0.331467\pi$
$619$	625.277	1.01014	0.505070	+	0.863078i	$0.331467\pi$
$620$	−65.8383	−0.106191
$621$	−81.8527	−0.131808
$622$	− 286.822i	− 0.461128i
$623$	314.059i	0.504108i
$624$	74.3840i	0.119205i
$625$	25.0000	0.0400000
$626$	135.327i	0.216177i
$627$	0	0
$628$	−27.6617	−0.0440473
$629$	− 1632.93i	− 2.59608i
$630$	−288.056	−0.457232
$631$	−28.5483	−0.0452429	−0.0226214	−	0.999744i	$-0.507201\pi$
$631$	−28.5483	−0.0452429	−0.0226214	+	0.999744i	$0.507201\pi$
$632$	200.263	0.316871
$633$	300.227i	0.474292i
$634$	265.792i	0.419230i
$635$	111.043i	0.174871i
$636$	−146.433	−0.230241
$637$	1573.54i	2.47024i
$638$	0	0
$639$	236.167	0.369588
$640$	− 25.2982i	− 0.0395285i
$641$	1203.45	1.87745	0.938725	−	0.344667i	$-0.112008\pi$
$641$	1203.45	1.87745	0.938725	+	0.344667i	$0.112008\pi$
$642$	−116.757	−0.181864
$643$	−382.227	−0.594444	−0.297222	−	0.954808i	$-0.596060\pi$
$643$	−382.227	−0.594444	−0.297222	+	0.954808i	$0.596060\pi$
$644$	− 94.3148i	− 0.146452i
$645$	− 260.171i	− 0.403366i
$646$	665.831i	1.03070i
$647$	593.833	0.917825	0.458913	−	0.888481i	$-0.348239\pi$
$647$	593.833	0.917825	0.458913	+	0.888481i	$0.348239\pi$
$648$	90.1233i	0.139079i
$649$	0	0
$650$	−93.8636	−0.144406
$651$	− 266.951i	− 0.410062i
$652$	−303.624	−0.465681
$653$	−175.019	−0.268023	−0.134011	−	0.990980i	$-0.542786\pi$
$653$	−175.019	−0.268023	−0.134011	+	0.990980i	$0.542786\pi$
$654$	−337.639	−0.516267
$655$	269.918i	0.412088i
$656$	62.1995i	0.0948163i
$657$	120.480i	0.183379i
$658$	1215.99	1.84801
$659$	− 455.063i	− 0.690536i	−0.938504	−	0.345268i	$-0.887788\pi$
$659$	− 455.063i	− 0.690536i	0.938504	−	0.345268i	$-0.112212\pi$
$660$	0	0
$661$	−803.041	−1.21489	−0.607444	−	0.794362i	$-0.707805\pi$
$661$	−803.041	−1.21489	−0.607444	+	0.794362i	$0.707805\pi$
$662$	401.329i	0.606237i
$663$	418.246	0.630838
$664$	−264.120	−0.397771
$665$	605.874	0.911089
$666$	722.585i	1.08496i
$667$	1.48794i	0.00223079i
$668$	− 208.886i	− 0.312703i
$669$	−189.620	−0.283437
$670$	206.951i	0.308882i
$671$	0	0
$672$	102.575	0.152642
$673$	534.545i	0.794271i	0.917760	+	0.397136i	$0.129996\pi$
$673$	534.545i	0.794271i	−0.917760	+	0.397136i	$0.870004\pi$
$674$	704.658	1.04549
$675$	112.335	0.166422
$676$	14.4149	0.0213238
$677$	− 47.3543i	− 0.0699473i	−0.999388	−	0.0349736i	$-0.988865\pi$
$677$	− 47.3543i	− 0.0699473i	0.999388	−	0.0349736i	$-0.0111347\pi$
$678$	− 44.5176i	− 0.0656602i
$679$	− 1005.37i	− 1.48066i
$680$	−142.247	−0.209186
$681$	246.965i	0.362651i
$682$	0	0
$683$	496.885	0.727504	0.363752	−	0.931496i	$-0.381496\pi$
$683$	496.885	0.727504	0.363752	+	0.931496i	$0.381496\pi$
$684$	− 294.635i	− 0.430753i
$685$	368.519	0.537984
$686$	1272.96	1.85562
$687$	266.800	0.388355
$688$	− 332.220i	− 0.482878i
$689$	693.768i	1.00692i
$690$	16.1398i	0.0233910i
$691$	164.013	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$691$	164.013	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$692$	− 354.083i	− 0.511681i
$693$	0	0
$694$	349.537	0.503655
$695$	− 255.717i	− 0.367938i
$696$	−1.61826	−0.00232508
$697$	349.735	0.501772
$698$	−800.489	−1.14683
$699$	17.8055i	0.0254728i
$700$	129.437i	0.184911i
$701$	− 221.274i	− 0.315654i	−0.987467	−	0.157827i	$-0.949551\pi$
$701$	− 221.274i	− 0.315654i	0.987467	−	0.157827i	$-0.0504489\pi$
$702$	−421.765	−0.600805
$703$	− 1519.83i	− 2.16191i
$704$	0	0
$705$	−208.088	−0.295160
$706$	− 291.810i	− 0.413329i
$707$	816.166	1.15441
$708$	66.1304	0.0934046
$709$	−648.300	−0.914387	−0.457193	−	0.889367i	$-0.651145\pi$
$709$	−648.300	−0.914387	−0.457193	+	0.889367i	$0.651145\pi$
$710$	− 106.121i	− 0.149466i
$711$	498.278i	0.700813i
$712$	− 68.6273i	− 0.0963866i
$713$	53.6357	0.0752253
$714$	− 576.759i	− 0.807785i
$715$	0	0
$716$	147.859	0.206507
$717$	− 50.0780i	− 0.0698437i
$718$	−874.256	−1.21763
$719$	1163.11	1.61768	0.808842	−	0.588026i	$-0.200095\pi$
$719$	1163.11	1.61768	0.808842	+	0.588026i	$0.200095\pi$
$720$	62.9451	0.0874238
$721$	− 438.791i	− 0.608587i
$722$	109.180i	0.151219i
$723$	− 95.5849i	− 0.132206i
$724$	−226.309	−0.312582
$725$	− 2.04205i	− 0.00281661i
$726$	0	0
$727$	574.840	0.790702	0.395351	−	0.918530i	$-0.370623\pi$
$727$	574.840	0.790702	0.395351	+	0.918530i	$0.370623\pi$
$728$	− 485.979i	− 0.667553i
$729$	59.0271	0.0809700
$730$	54.1373	0.0741607
$731$	−1868.01	−2.55541
$732$	159.576i	0.218000i
$733$	901.359i	1.22968i	0.788650	+	0.614842i	$0.210780\pi$
$733$	901.359i	1.22968i	−0.788650	+	0.614842i	$0.789220\pi$
$734$	872.157i	1.18822i
$735$	−371.329	−0.505210
$736$	20.6094i	0.0280019i
$737$	0	0
$738$	−154.760	−0.209702
$739$	− 298.550i	− 0.403992i	−0.979386	−	0.201996i	$-0.935257\pi$
$739$	− 298.550i	− 0.403992i	0.979386	−	0.201996i	$-0.0647427\pi$
$740$	324.692	0.438773
$741$	389.275	0.525338
$742$	956.703	1.28936
$743$	521.760i	0.702234i	0.936332	+	0.351117i	$0.114198\pi$
$743$	521.760i	0.702234i	−0.936332	+	0.351117i	$0.885802\pi$
$744$	58.3332i	0.0784049i
$745$	9.53425i	0.0127977i
$746$	−935.558	−1.25410
$747$	− 657.163i	− 0.879737i
$748$	0	0
$749$	762.815	1.01844
$750$	− 22.1502i	− 0.0295336i
$751$	−591.409	−0.787496	−0.393748	−	0.919219i	$-0.628822\pi$
$751$	−591.409	−0.787496	−0.393748	+	0.919219i	$0.628822\pi$
$752$	−265.714	−0.353343
$753$	25.8303	0.0343032
$754$	7.66695i	0.0101684i
$755$	142.653i	0.188944i
$756$	581.612i	0.769328i
$757$	−744.808	−0.983894	−0.491947	−	0.870625i	$-0.663715\pi$
$757$	−744.808	−0.983894	−0.491947	+	0.870625i	$0.663715\pi$
$758$	− 159.984i	− 0.211061i
$759$	0	0
$760$	−132.394	−0.174202
$761$	1208.91i	1.58858i	0.607542	+	0.794288i	$0.292156\pi$
$761$	1208.91i	1.58858i	−0.607542	+	0.794288i	$0.707844\pi$
$762$	98.3852	0.129114
$763$	2205.92	2.89112
$764$	−660.902	−0.865055
$765$	− 353.927i	− 0.462650i
$766$	− 885.265i	− 1.15570i
$767$	− 313.311i	− 0.408489i
$768$	−22.4144	−0.0291854
$769$	897.903i	1.16762i	0.811889	+	0.583812i	$0.198439\pi$
$769$	897.903i	1.16762i	−0.811889	+	0.583812i	$0.801561\pi$
$770$	0	0
$771$	−29.1300	−0.0377821
$772$	− 135.006i	− 0.174878i
$773$	349.472	0.452098	0.226049	−	0.974116i	$-0.427419\pi$
$773$	349.472	0.452098	0.226049	+	0.974116i	$0.427419\pi$
$774$	826.606	1.06797
$775$	−73.6095	−0.0949800
$776$	219.690i	0.283105i
$777$	1316.51i	1.69435i
$778$	− 343.101i	− 0.441004i
$779$	325.510	0.417856
$780$	83.1638i	0.106620i
$781$	0	0
$782$	115.882	0.148187
$783$	− 9.17569i	− 0.0117186i
$784$	−474.162	−0.604799
$785$	−30.9267	−0.0393971
$786$	239.149	0.304261
$787$	− 34.9579i	− 0.0444192i	−0.999753	−	0.0222096i	$-0.992930\pi$
$787$	− 34.9579i	− 0.0444192i	0.999753	−	0.0222096i	$-0.00707012\pi$
$788$	594.315i	0.754206i
$789$	− 319.128i	− 0.404471i
$790$	223.900	0.283418
$791$	290.851i	0.367700i
$792$	0	0
$793$	756.035	0.953386
$794$	674.660i	0.849698i
$795$	−163.717	−0.205934
$796$	−537.457	−0.675197
$797$	−1210.15	−1.51839	−0.759194	−	0.650865i	$-0.774406\pi$
$797$	−1210.15	−1.51839	−0.759194	+	0.650865i	$0.774406\pi$
$798$	− 536.809i	− 0.672692i
$799$	1494.05i	1.86990i
$800$	− 28.2843i	− 0.0353553i
$801$	170.753	0.213175
$802$	471.574i	0.587997i
$803$	0	0
$804$	183.360	0.228060
$805$	− 105.447i	− 0.130990i
$806$	276.370	0.342891
$807$	−355.238	−0.440196
$808$	−178.346	−0.220725
$809$	302.846i	0.374346i	0.982327	+	0.187173i	$0.0599326\pi$
$809$	302.846i	0.374346i	−0.982327	+	0.187173i	$0.940067\pi$
$810$	100.761i	0.124396i
$811$	327.385i	0.403681i	0.979418	+	0.201841i	$0.0646923\pi$
$811$	327.385i	0.403681i	−0.979418	+	0.201841i	$0.935308\pi$
$812$	10.5727	0.0130206
$813$	145.204i	0.178602i
$814$	0	0
$815$	−339.462	−0.416518
$816$	126.032i	0.154450i
$817$	−1738.62	−2.12805
$818$	−19.3732	−0.0236836
$819$	1209.18	1.47640
$820$	69.5412i	0.0848063i
$821$	− 924.430i	− 1.12598i	−0.826464	−	0.562990i	$-0.809651\pi$
$821$	− 924.430i	− 1.12598i	0.826464	−	0.562990i	$-0.190349\pi$
$822$	− 326.511i	− 0.397215i
$823$	1423.92	1.73015	0.865077	−	0.501639i	$-0.167269\pi$
$823$	1423.92	1.73015	0.865077	+	0.501639i	$0.167269\pi$
$824$	95.8833i	0.116363i
$825$	0	0
$826$	−432.055	−0.523069
$827$	− 699.822i	− 0.846218i	−0.906079	−	0.423109i	$-0.860939\pi$
$827$	− 699.822i	− 0.846218i	0.906079	−	0.423109i	$-0.139061\pi$
$828$	−51.2787	−0.0619308
$829$	924.763	1.11552	0.557758	−	0.830004i	$-0.311662\pi$
$829$	924.763	1.11552	0.557758	+	0.830004i	$0.311662\pi$
$830$	−295.295	−0.355777
$831$	461.571i	0.555440i
$832$	106.195i	0.127638i
$833$	2666.12i	3.20062i
$834$	−226.567	−0.271663
$835$	− 233.542i	− 0.279690i
$836$	0	0
$837$	−330.755	−0.395168
$838$	29.7650i	0.0355191i
$839$	−280.474	−0.334295	−0.167148	−	0.985932i	$-0.553456\pi$
$839$	−280.474	−0.334295	−0.167148	+	0.985932i	$0.553456\pi$
$840$	114.683	0.136527
$841$	840.833	0.999802
$842$	505.378i	0.600212i
$843$	611.241i	0.725078i
$844$	428.619i	0.507843i
$845$	16.1164	0.0190726
$846$	− 661.129i	− 0.781476i
$847$	0	0
$848$	−209.056	−0.246528
$849$	− 382.722i	− 0.450792i
$850$	−159.037	−0.187102
$851$	−264.513	−0.310826
$852$	−94.0240	−0.110357
$853$	1430.22i	1.67669i	0.545138	+	0.838346i	$0.316477\pi$
$853$	1430.22i	1.67669i	−0.545138	+	0.838346i	$0.683523\pi$
$854$	− 1042.57i	− 1.22081i
$855$	− 329.412i	− 0.385277i
$856$	−166.688	−0.194729
$857$	1486.03i	1.73399i	0.498321	+	0.866993i	$0.333950\pi$
$857$	1486.03i	1.73399i	−0.498321	+	0.866993i	$0.666050\pi$
$858$	0	0
$859$	488.310	0.568463	0.284232	−	0.958756i	$-0.408262\pi$
$859$	488.310	0.568463	0.284232	+	0.958756i	$0.408262\pi$
$860$	− 371.434i	− 0.431900i
$861$	−281.964	−0.327485
$862$	621.212	0.720664
$863$	1604.50	1.85921	0.929607	−	0.368552i	$-0.120146\pi$
$863$	1604.50	1.85921	0.929607	+	0.368552i	$0.120146\pi$
$864$	− 127.092i	− 0.147097i
$865$	− 395.877i	− 0.457661i
$866$	366.644i	0.423377i
$867$	303.789	0.350391
$868$	− 381.113i	− 0.439070i
$869$	0	0
$870$	−1.80927	−0.00207962
$871$	− 868.720i	− 0.997382i
$872$	−482.031	−0.552788
$873$	−546.615	−0.626134
$874$	107.855	0.123404
$875$	144.715i	0.165389i
$876$	− 47.9661i	− 0.0547558i
$877$	− 64.7531i	− 0.0738348i	−0.999318	−	0.0369174i	$-0.988246\pi$
$877$	− 64.7531i	− 0.0738348i	0.999318	−	0.0369174i	$-0.0117538\pi$
$878$	−752.345	−0.856885
$879$	347.300i	0.395108i
$880$	0	0
$881$	−1367.56	−1.55228	−0.776139	−	0.630562i	$-0.782825\pi$
$881$	−1367.56	−1.55228	−0.776139	+	0.630562i	$0.782825\pi$
$882$	− 1179.77i	− 1.33761i
$883$	232.780	0.263624	0.131812	−	0.991275i	$-0.457921\pi$
$883$	232.780	0.263624	0.131812	+	0.991275i	$0.457921\pi$
$884$	597.110	0.675463
$885$	73.9361	0.0835436
$886$	− 574.047i	− 0.647908i
$887$	− 601.378i	− 0.677991i	−0.940788	−	0.338995i	$-0.889913\pi$
$887$	− 601.378i	− 0.677991i	0.940788	−	0.338995i	$-0.110087\pi$
$888$	− 287.679i	− 0.323963i
$889$	−642.788	−0.723046
$890$	− 76.7276i	− 0.0862108i
$891$	0	0
$892$	−270.711	−0.303488
$893$	1390.57i	1.55718i
$894$	8.44742	0.00944901
$895$	165.311	0.184706
$896$	146.442	0.163439
$897$	− 67.7500i	− 0.0755295i
$898$	800.833i	0.891796i
$899$	6.01256i	0.00668805i
$900$	70.3748	0.0781942
$901$	1175.48i	1.30464i
$902$	0	0
$903$	1506.03	1.66781
$904$	− 63.5558i	− 0.0703051i
$905$	−253.022	−0.279582
$906$	126.391	0.139505
$907$	−1463.63	−1.61371	−0.806853	−	0.590753i	$-0.798831\pi$
$907$	−1463.63	−1.61371	−0.806853	+	0.590753i	$0.798831\pi$
$908$	352.580i	0.388304i
$909$	− 443.747i	− 0.488171i
$910$	− 543.341i	− 0.597078i
$911$	771.742	0.847137	0.423569	−	0.905864i	$-0.360777\pi$
$911$	771.742	0.847137	0.423569	+	0.905864i	$0.360777\pi$
$912$	117.302i	0.128620i
$913$	0	0
$914$	−486.348	−0.532110
$915$	178.411i	0.194985i
$916$	380.897	0.415827
$917$	−1562.45	−1.70387
$918$	−714.612	−0.778444
$919$	162.755i	0.177100i	0.996072	+	0.0885500i	$0.0282233\pi$
$919$	162.755i	0.177100i	−0.996072	+	0.0885500i	$0.971777\pi$
$920$	23.0420i	0.0250456i
$921$	536.240i	0.582237i
$922$	−772.134	−0.837455
$923$	445.465i	0.482627i
$924$	0	0
$925$	363.017	0.392451
$926$	− 450.413i	− 0.486407i
$927$	−238.570	−0.257357
$928$	−2.31031	−0.00248956
$929$	246.591	0.265437	0.132719	−	0.991154i	$-0.457629\pi$
$929$	246.591	0.265437	0.132719	+	0.991154i	$0.457629\pi$
$930$	65.2185i	0.0701274i
$931$	2481.44i	2.66535i
$932$	25.4200i	0.0272747i
$933$	−284.122	−0.304525
$934$	− 14.4709i	− 0.0154934i
$935$	0	0
$936$	−264.225	−0.282292
$937$	− 1215.07i	− 1.29677i	−0.761314	−	0.648383i	$-0.775446\pi$
$937$	− 1215.07i	− 1.29677i	0.761314	−	0.648383i	$-0.224554\pi$
$938$	−1197.96	−1.27714
$939$	134.053	0.142761
$940$	−297.077	−0.316039
$941$	1228.67i	1.30571i	0.757482	+	0.652856i	$0.226429\pi$
$941$	1228.67i	1.30571i	−0.757482	+	0.652856i	$0.773571\pi$
$942$	27.4013i	0.0290884i
$943$	− 56.6522i	− 0.0600766i
$944$	94.4113	0.100012
$945$	650.262i	0.688108i
$946$	0	0
$947$	−1697.66	−1.79267	−0.896336	−	0.443376i	$-0.853781\pi$
$947$	−1697.66	−1.79267	−0.896336	+	0.443376i	$0.853781\pi$
$948$	− 198.377i	− 0.209259i
$949$	−227.253	−0.239465
$950$	−148.021	−0.155811
$951$	263.290	0.276856
$952$	− 823.412i	− 0.864928i
$953$	− 1470.02i	− 1.54252i	−0.636521	−	0.771260i	$-0.719627\pi$
$953$	− 1470.02i	− 1.54252i	0.636521	−	0.771260i	$-0.280373\pi$
$954$	− 520.157i	− 0.545238i
$955$	−738.911	−0.773729
$956$	− 71.4940i	− 0.0747845i
$957$	0	0
$958$	514.794	0.537364
$959$	2133.22i	2.22442i
$960$	−25.0601	−0.0261042
$961$	−744.266	−0.774470
$962$	−1362.96	−1.41680
$963$	− 414.741i	− 0.430675i
$964$	− 136.462i	− 0.141558i
$965$	− 150.941i	− 0.156415i
$966$	−93.4269	−0.0967153
$967$	− 1224.68i	− 1.26648i	−0.773956	−	0.633239i	$-0.781725\pi$
$967$	− 1224.68i	− 1.26648i	0.773956	−	0.633239i	$-0.218275\pi$
$968$	0	0
$969$	659.563	0.680664
$970$	245.621i	0.253217i
$971$	−1724.54	−1.77605	−0.888025	−	0.459795i	$-0.847923\pi$
$971$	−1724.54	−1.77605	−0.888025	+	0.459795i	$0.847923\pi$
$972$	493.679	0.507900
$973$	1480.25	1.52132
$974$	− 305.187i	− 0.313334i
$975$	92.9800i	0.0953641i
$976$	227.819i	0.233421i
$977$	−1501.88	−1.53724	−0.768620	−	0.639705i	$-0.779056\pi$
$977$	−1501.88	−1.53724	−0.768620	+	0.639705i	$0.779056\pi$
$978$	300.766i	0.307532i
$979$	0	0
$980$	−530.129	−0.540948
$981$	− 1199.35i	− 1.22258i
$982$	1174.49	1.19602
$983$	−459.772	−0.467723	−0.233862	−	0.972270i	$-0.575136\pi$
$983$	−459.772	−0.467723	−0.233862	+	0.972270i	$0.575136\pi$
$984$	61.6140	0.0626158
$985$	664.464i	0.674583i
$986$	12.9904i	0.0131748i
$987$	− 1204.54i	− 1.22041i
$988$	555.750	0.562500
$989$	302.591i	0.305957i
$990$	0	0
$991$	112.519	0.113541	0.0567703	−	0.998387i	$-0.481920\pi$
$991$	112.519	0.113541	0.0567703	+	0.998387i	$0.481920\pi$
$992$	83.2796i	0.0839512i
$993$	397.551	0.400353
$994$	614.294	0.618002
$995$	−600.895	−0.603914
$996$	261.634i	0.262684i
$997$	− 227.895i	− 0.228580i	−0.993447	−	0.114290i	$-0.963541\pi$
$997$	− 227.895i	− 0.228580i	0.993447	−	0.114290i	$-0.0364594\pi$
$998$	285.419i	0.285991i
$999$	1631.17	1.63281

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1210.3.d.c.241.4		16
11.4	even	5		110.3.h.b.61.3	✓	16
11.8	odd	10		110.3.h.b.101.3	yes	16
11.10	odd	2	inner	1210.3.d.c.241.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.3.h.b.61.3	✓	16	11.4	even	5
110.3.h.b.101.3	yes	16	11.8	odd	10
1210.3.d.c.241.4		16	1.1	even	1	trivial
1210.3.d.c.241.12		16	11.10	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1210.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -1.40090 q^{3} -2.00000 q^{4} -2.23607 q^{5} +1.98117i q^{6} -12.9437i q^{7} +2.82843i q^{8} -7.03748 q^{9} +3.16228i q^{10} +2.80180 q^{12} -13.2743i q^{13} -18.3052 q^{14} +3.13251 q^{15} +4.00000 q^{16} -22.4912i q^{17} +9.95250i q^{18} -20.9333i q^{19} +4.47214 q^{20} +18.1329i q^{21} -3.64326 q^{23} -3.96234i q^{24} +5.00000 q^{25} -18.7727 q^{26} +22.4669 q^{27} +25.8875i q^{28} -0.408409i q^{29} -4.43004i q^{30} -14.7219 q^{31} -5.65685i q^{32} -31.8073 q^{34} +28.9431i q^{35} +14.0750 q^{36} +72.6033 q^{37} -29.6041 q^{38} +18.5960i q^{39} -6.32456i q^{40} +15.5499i q^{41} +25.6438 q^{42} -83.0551i q^{43} +15.7363 q^{45} +5.15234i q^{46} -66.4284 q^{47} -5.60360 q^{48} -118.541 q^{49} -7.07107i q^{50} +31.5079i q^{51} +26.5486i q^{52} -52.2639 q^{53} -31.7730i q^{54} +36.6104 q^{56} +29.3254i q^{57} -0.577578 q^{58} +23.6028 q^{59} -6.26502 q^{60} +56.9548i q^{61} +20.8199i q^{62} +91.0913i q^{63} -8.00000 q^{64} +29.6823i q^{65} +65.4437 q^{67} +44.9823i q^{68} +5.10384 q^{69} +40.9317 q^{70} -33.5584 q^{71} -19.9050i q^{72} -17.1197i q^{73} -102.677i q^{74} -7.00450 q^{75} +41.8666i q^{76} +26.2987 q^{78} -70.8035i q^{79} -8.94427 q^{80} +31.8634 q^{81} +21.9908 q^{82} +93.3805i q^{83} -36.2658i q^{84} +50.2918i q^{85} -117.458 q^{86} +0.572141i q^{87} -24.2634 q^{89} -22.2545i q^{90} -171.819 q^{91} +7.28652 q^{92} +20.6239 q^{93} +93.9440i q^{94} +46.8082i q^{95} +7.92469i q^{96} +77.6721 q^{97} +167.642i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} - 32 q^{4} + 40 q^{9} + 32 q^{12} + 16 q^{14} + 40 q^{15} + 64 q^{16} + 108 q^{23} + 80 q^{25} - 292 q^{27} - 268 q^{31} - 16 q^{34} - 80 q^{36} + 44 q^{37} - 280 q^{38} - 16 q^{42} - 476 q^{47}+ \cdots + 296 q^{97}+O(q^{100}) \) 16 * q - 16 * q^3 - 32 * q^4 + 40 * q^9 + 32 * q^12 + 16 * q^14 + 40 * q^15 + 64 * q^16 + 108 * q^23 + 80 * q^25 - 292 * q^27 - 268 * q^31 - 16 * q^34 - 80 * q^36 + 44 * q^37 - 280 * q^38 - 16 * q^42 - 476 * q^47 - 64 * q^48 - 644 * q^49 - 396 * q^53 - 32 * q^56 + 192 * q^58 - 180 * q^59 - 80 * q^60 - 128 * q^64 + 188 * q^67 - 416 * q^69 + 308 * q^71 - 80 * q^75 + 8 * q^78 + 536 * q^81 + 16 * q^82 + 304 * q^86 - 176 * q^89 + 80 * q^91 - 216 * q^92 + 436 * q^93 + 296 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more