LMFDB - Embedded newform 1075.6.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
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} - 7 x^{19} - 469 x^{18} + 3101 x^{17} + 91700 x^{16} - 569288 x^{15} - 9711682 x^{14} + \cdots + 15620859125760 $ x^20 - 7x^19 - 469x^18 + 3101x^17 + 91700x^16 - 569288x^15 - 9711682x^14 + 56438335x^13 + 604830685x^12 - 3303543877x^11 - 22405860367x^10 + 117036723704x^9 + 471797168660x^8 - 2454443105088x^7 - 4913335707456x^6 + 27842865972800x^5 + 14823414415872x^4 - 133312312389248x^3 + 52513733857792x^2 + 79407200338432*x + 15620859125760
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{5} $
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.47195$ of defining polynomial
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.47195 q^{2} -6.34889 q^{3} -19.9456 q^{4} +22.0430 q^{6} -214.575 q^{7} +180.352 q^{8} -202.692 q^{9} +O(q^{10})$	\(q-3.47195 q^{2} -6.34889 q^{3} -19.9456 q^{4} +22.0430 q^{6} -214.575 q^{7} +180.352 q^{8} -202.692 q^{9} -122.077 q^{11} +126.632 q^{12} -433.738 q^{13} +744.994 q^{14} +12.0856 q^{16} -1349.74 q^{17} +703.734 q^{18} -104.137 q^{19} +1362.31 q^{21} +423.846 q^{22} -544.885 q^{23} -1145.04 q^{24} +1505.92 q^{26} +2829.65 q^{27} +4279.83 q^{28} +3139.82 q^{29} +3797.31 q^{31} -5813.23 q^{32} +775.054 q^{33} +4686.22 q^{34} +4042.81 q^{36} -7868.79 q^{37} +361.559 q^{38} +2753.75 q^{39} +11290.2 q^{41} -4729.88 q^{42} -1849.00 q^{43} +2434.90 q^{44} +1891.81 q^{46} -24147.0 q^{47} -76.7303 q^{48} +29235.6 q^{49} +8569.34 q^{51} +8651.16 q^{52} +26428.7 q^{53} -9824.37 q^{54} -38699.2 q^{56} +661.156 q^{57} -10901.3 q^{58} -11126.8 q^{59} +16645.4 q^{61} -13184.1 q^{62} +43492.6 q^{63} +19796.5 q^{64} -2690.95 q^{66} +13535.5 q^{67} +26921.3 q^{68} +3459.41 q^{69} +973.219 q^{71} -36555.9 q^{72} -12031.3 q^{73} +27320.0 q^{74} +2077.08 q^{76} +26194.8 q^{77} -9560.88 q^{78} +78035.4 q^{79} +31289.0 q^{81} -39199.0 q^{82} +21779.4 q^{83} -27172.2 q^{84} +6419.63 q^{86} -19934.4 q^{87} -22016.9 q^{88} -19302.5 q^{89} +93069.5 q^{91} +10868.1 q^{92} -24108.7 q^{93} +83837.1 q^{94} +36907.6 q^{96} +15335.5 q^{97} -101504. q^{98} +24744.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 7 q^{2} - 16 q^{3} + 347 q^{4} - 128 q^{6} - 372 q^{7} - 441 q^{8} + 1808 q^{9}+O(q^{10}) $ 20 * q - 7 * q^2 - 16 * q^3 + 347 * q^4 - 128 * q^6 - 372 * q^7 - 441 * q^8 + 1808 * q^9	\( 20 q - 7 q^{2} - 16 q^{3} + 347 q^{4} - 128 q^{6} - 372 q^{7} - 441 q^{8} + 1808 q^{9} - 147 q^{11} + 751 q^{12} + 151 q^{13} + 311 q^{14} + 6347 q^{16} - 231 q^{17} - 6125 q^{18} + 4146 q^{19} + 2128 q^{21} - 10263 q^{22} - 5101 q^{23} - 6144 q^{24} + 4007 q^{26} - 10372 q^{27} - 42211 q^{28} - 4792 q^{29} + 18673 q^{31} - 32839 q^{32} - 42332 q^{33} - 1064 q^{34} + 76091 q^{36} - 39718 q^{37} - 34535 q^{38} + 12528 q^{39} + 12889 q^{41} - 8789 q^{42} - 36980 q^{43} + 9156 q^{44} + 77524 q^{46} + 31012 q^{47} - 5074 q^{48} + 24162 q^{49} + 37160 q^{51} + 10943 q^{52} - 1913 q^{53} + 7523 q^{54} + 220786 q^{56} - 91576 q^{57} - 215883 q^{58} - 46908 q^{59} + 58128 q^{61} - 216209 q^{62} - 304976 q^{63} + 186347 q^{64} + 376113 q^{66} - 144469 q^{67} - 164172 q^{68} - 8540 q^{69} + 43954 q^{71} - 467998 q^{72} - 189870 q^{73} + 275403 q^{74} + 318411 q^{76} - 189206 q^{77} - 346786 q^{78} + 149228 q^{79} + 195100 q^{81} - 337685 q^{82} - 177193 q^{83} + 160493 q^{84} + 12943 q^{86} - 185584 q^{87} - 510938 q^{88} + 78564 q^{89} - 33858 q^{91} - 63558 q^{92} - 41492 q^{93} + 133334 q^{94} - 565639 q^{96} - 448697 q^{97} + 161151 q^{98} + 395765 q^{99}+O(q^{100}) \) 20 * q - 7 * q^2 - 16 * q^3 + 347 * q^4 - 128 * q^6 - 372 * q^7 - 441 * q^8 + 1808 * q^9 - 147 * q^11 + 751 * q^12 + 151 * q^13 + 311 * q^14 + 6347 * q^16 - 231 * q^17 - 6125 * q^18 + 4146 * q^19 + 2128 * q^21 - 10263 * q^22 - 5101 * q^23 - 6144 * q^24 + 4007 * q^26 - 10372 * q^27 - 42211 * q^28 - 4792 * q^29 + 18673 * q^31 - 32839 * q^32 - 42332 * q^33 - 1064 * q^34 + 76091 * q^36 - 39718 * q^37 - 34535 * q^38 + 12528 * q^39 + 12889 * q^41 - 8789 * q^42 - 36980 * q^43 + 9156 * q^44 + 77524 * q^46 + 31012 * q^47 - 5074 * q^48 + 24162 * q^49 + 37160 * q^51 + 10943 * q^52 - 1913 * q^53 + 7523 * q^54 + 220786 * q^56 - 91576 * q^57 - 215883 * q^58 - 46908 * q^59 + 58128 * q^61 - 216209 * q^62 - 304976 * q^63 + 186347 * q^64 + 376113 * q^66 - 144469 * q^67 - 164172 * q^68 - 8540 * q^69 + 43954 * q^71 - 467998 * q^72 - 189870 * q^73 + 275403 * q^74 + 318411 * q^76 - 189206 * q^77 - 346786 * q^78 + 149228 * q^79 + 195100 * q^81 - 337685 * q^82 - 177193 * q^83 + 160493 * q^84 + 12943 * q^86 - 185584 * q^87 - 510938 * q^88 + 78564 * q^89 - 33858 * q^91 - 63558 * q^92 - 41492 * q^93 + 133334 * q^94 - 565639 * q^96 - 448697 * q^97 + 161151 * q^98 + 395765 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.47195	−0.613759	−0.306880	−	0.951748i	$-0.599285\pi$
$2$	−3.47195	−0.613759	−0.306880	+	0.951748i	$0.599285\pi$
$3$	−6.34889	−0.407281	−0.203641	−	0.979046i	$-0.565277\pi$
$3$	−6.34889	−0.407281	−0.203641	+	0.979046i	$0.565277\pi$
$4$	−19.9456	−0.623300
$5$	0	0
$5$	0	0
$6$	22.0430	0.249973
$7$	−214.575	−1.65514	−0.827570	−	0.561363i	$-0.810277\pi$
$7$	−214.575	−1.65514	−0.827570	+	0.561363i	$0.810277\pi$
$8$	180.352	0.996315
$9$	−202.692	−0.834122
$10$	0	0
$11$	−122.077	−0.304196	−0.152098	−	0.988365i	$-0.548603\pi$
$11$	−122.077	−0.304196	−0.152098	+	0.988365i	$0.548603\pi$
$12$	126.632	0.253858
$13$	−433.738	−0.711818	−0.355909	−	0.934521i	$-0.615829\pi$
$13$	−433.738	−0.711818	−0.355909	+	0.934521i	$0.615829\pi$
$14$	744.994	1.01586
$15$	0	0
$16$	12.0856	0.0118024
$17$	−1349.74	−1.13273	−0.566366	−	0.824154i	$-0.691651\pi$
$17$	−1349.74	−1.13273	−0.566366	+	0.824154i	$0.691651\pi$
$18$	703.734	0.511950
$19$	−104.137	−0.0661793	−0.0330897	−	0.999452i	$-0.510535\pi$
$19$	−104.137	−0.0661793	−0.0330897	+	0.999452i	$0.510535\pi$
$20$	0	0
$21$	1362.31	0.674107
$22$	423.846	0.186703
$23$	−544.885	−0.214776	−0.107388	−	0.994217i	$-0.534249\pi$
$23$	−544.885	−0.214776	−0.107388	+	0.994217i	$0.534249\pi$
$24$	−1145.04	−0.405780
$25$	0	0
$26$	1505.92	0.436885
$27$	2829.65	0.747003
$28$	4279.83	1.03165
$29$	3139.82	0.693282	0.346641	−	0.937998i	$-0.387322\pi$
$29$	3139.82	0.693282	0.346641	+	0.937998i	$0.387322\pi$
$30$	0	0
$31$	3797.31	0.709695	0.354848	−	0.934924i	$-0.384533\pi$
$31$	3797.31	0.709695	0.354848	+	0.934924i	$0.384533\pi$
$32$	−5813.23	−1.00356
$33$	775.054	0.123893
$34$	4686.22	0.695225
$35$	0	0
$36$	4042.81	0.519908
$37$	−7868.79	−0.944939	−0.472470	−	0.881347i	$-0.656637\pi$
$37$	−7868.79	−0.944939	−0.472470	+	0.881347i	$0.656637\pi$
$38$	361.559	0.0406182
$39$	2753.75	0.289910
$40$	0	0
$41$	11290.2	1.04892	0.524461	−	0.851435i	$-0.324267\pi$
$41$	11290.2	1.04892	0.524461	+	0.851435i	$0.324267\pi$
$42$	−4729.88	−0.413740
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	2434.90	0.189605
$45$	0	0
$46$	1891.81	0.131821
$47$	−24147.0	−1.59448	−0.797240	−	0.603663i	$-0.793707\pi$
$47$	−24147.0	−1.59448	−0.797240	+	0.603663i	$0.793707\pi$
$48$	−76.7303	−0.00480689
$49$	29235.6	1.73949
$50$	0	0
$51$	8569.34	0.461341
$52$	8651.16	0.443676
$53$	26428.7	1.29237	0.646183	−	0.763182i	$-0.276364\pi$
$53$	26428.7	1.29237	0.646183	+	0.763182i	$0.276364\pi$
$54$	−9824.37	−0.458480
$55$	0	0
$56$	−38699.2	−1.64904
$57$	661.156	0.0269536
$58$	−10901.3	−0.425508
$59$	−11126.8	−0.416142	−0.208071	−	0.978114i	$-0.566719\pi$
$59$	−11126.8	−0.416142	−0.208071	+	0.978114i	$0.566719\pi$
$60$	0	0
$61$	16645.4	0.572756	0.286378	−	0.958117i	$-0.407549\pi$
$61$	16645.4	0.572756	0.286378	+	0.958117i	$0.407549\pi$
$62$	−13184.1	−0.435582
$63$	43492.6	1.38059
$64$	19796.5	0.604141
$65$	0	0
$66$	−2690.95	−0.0760406
$67$	13535.5	0.368372	0.184186	−	0.982891i	$-0.441035\pi$
$67$	13535.5	0.368372	0.184186	+	0.982891i	$0.441035\pi$
$68$	26921.3	0.706032
$69$	3459.41	0.0874741
$70$	0	0
$71$	973.219	0.0229121	0.0114560	−	0.999934i	$-0.496353\pi$
$71$	973.219	0.0229121	0.0114560	+	0.999934i	$0.496353\pi$
$72$	−36555.9	−0.831048
$73$	−12031.3	−0.264244	−0.132122	−	0.991233i	$-0.542179\pi$
$73$	−12031.3	−0.264244	−0.132122	+	0.991233i	$0.542179\pi$
$74$	27320.0	0.579965
$75$	0	0
$76$	2077.08	0.0412496
$77$	26194.8	0.503486
$78$	−9560.88	−0.177935
$79$	78035.4	1.40677	0.703387	−	0.710808i	$-0.251670\pi$
$79$	78035.4	1.40677	0.703387	+	0.710808i	$0.251670\pi$
$80$	0	0
$81$	31289.0	0.529882
$82$	−39199.0	−0.643785
$83$	21779.4	0.347016	0.173508	−	0.984832i	$-0.444490\pi$
$83$	21779.4	0.347016	0.173508	+	0.984832i	$0.444490\pi$
$84$	−27172.2	−0.420171
$85$	0	0
$86$	6419.63	0.0935974
$87$	−19934.4	−0.282361
$88$	−22016.9	−0.303075
$89$	−19302.5	−0.258308	−0.129154	−	0.991625i	$-0.541226\pi$
$89$	−19302.5	−0.258308	−0.129154	+	0.991625i	$0.541226\pi$
$90$	0	0
$91$	93069.5	1.17816
$92$	10868.1	0.133870
$93$	−24108.7	−0.289045
$94$	83837.1	0.978626
$95$	0	0
$96$	36907.6	0.408731
$97$	15335.5	0.165489	0.0827445	−	0.996571i	$-0.473631\pi$
$97$	15335.5	0.165489	0.0827445	+	0.996571i	$0.473631\pi$
$98$	−101504.	−1.06763
$99$	24744.0	0.253736
$100$	0	0
$101$	−29453.7	−0.287301	−0.143650	−	0.989629i	$-0.545884\pi$
$101$	−29453.7	−0.287301	−0.143650	+	0.989629i	$0.545884\pi$
$102$	−29752.3	−0.283152
$103$	−94350.4	−0.876295	−0.438148	−	0.898903i	$-0.644365\pi$
$103$	−94350.4	−0.876295	−0.438148	+	0.898903i	$0.644365\pi$
$104$	−78225.7	−0.709195
$105$	0	0
$106$	−91758.9	−0.793201
$107$	217015.	1.83244	0.916222	−	0.400670i	$-0.131223\pi$
$107$	217015.	1.83244	0.916222	+	0.400670i	$0.131223\pi$
$108$	−56438.9	−0.465607
$109$	−62022.5	−0.500015	−0.250008	−	0.968244i	$-0.580433\pi$
$109$	−62022.5	−0.500015	−0.250008	+	0.968244i	$0.580433\pi$
$110$	0	0
$111$	49958.1	0.384856
$112$	−2593.28	−0.0195346
$113$	−39158.9	−0.288492	−0.144246	−	0.989542i	$-0.546076\pi$
$113$	−39158.9	−0.288492	−0.144246	+	0.989542i	$0.546076\pi$
$114$	−2295.50	−0.0165430
$115$	0	0
$116$	−62625.6	−0.432122
$117$	87915.1	0.593743
$118$	38631.8	0.255411
$119$	289621.	1.87483
$120$	0	0
$121$	−146148.	−0.907465
$122$	−57791.9	−0.351534
$123$	−71680.3	−0.427206
$124$	−75739.6	−0.442353
$125$	0	0
$126$	−151004.	−0.847349
$127$	146022.	0.803359	0.401679	−	0.915780i	$-0.368427\pi$
$127$	146022.	0.803359	0.401679	+	0.915780i	$0.368427\pi$
$128$	117291.	0.632762
$129$	11739.1	0.0621098
$130$	0	0
$131$	−27685.6	−0.140953	−0.0704767	−	0.997513i	$-0.522452\pi$
$131$	−27685.6	−0.140953	−0.0704767	+	0.997513i	$0.522452\pi$
$132$	−15458.9	−0.0772226
$133$	22345.3	0.109536
$134$	−46994.4	−0.226092
$135$	0	0
$136$	−243428.	−1.12856
$137$	305149.	1.38903	0.694513	−	0.719480i	$-0.255620\pi$
$137$	305149.	1.38903	0.694513	+	0.719480i	$0.255620\pi$
$138$	−12010.9	−0.0536880
$139$	312325.	1.37110	0.685552	−	0.728024i	$-0.259561\pi$
$139$	312325.	1.37110	0.685552	+	0.728024i	$0.259561\pi$
$140$	0	0
$141$	153307.	0.649401
$142$	−3378.96	−0.0140625
$143$	52949.6	0.216532
$144$	−2449.66	−0.00984463
$145$	0	0
$146$	41771.9	0.162182
$147$	−185613.	−0.708461
$148$	156948.	0.588980
$149$	59736.8	0.220433	0.110216	−	0.993908i	$-0.464846\pi$
$149$	59736.8	0.220433	0.110216	+	0.993908i	$0.464846\pi$
$150$	0	0
$151$	174282.	0.622028	0.311014	−	0.950405i	$-0.399331\pi$
$151$	174282.	0.622028	0.311014	+	0.950405i	$0.399331\pi$
$152$	−18781.4	−0.0659354
$153$	273581.	0.944837
$154$	−90946.8	−0.309019
$155$	0	0
$156$	−54925.2	−0.180701
$157$	−410782.	−1.33003	−0.665017	−	0.746828i	$-0.731576\pi$
$157$	−410782.	−1.33003	−0.665017	+	0.746828i	$0.731576\pi$
$158$	−270935.	−0.863420
$159$	−167793.	−0.526356
$160$	0	0
$161$	116919.	0.355484
$162$	−108634.	−0.325220
$163$	435266.	1.28318	0.641588	−	0.767050i	$-0.278276\pi$
$163$	435266.	1.28318	0.641588	+	0.767050i	$0.278276\pi$
$164$	−225190.	−0.653792
$165$	0	0
$166$	−75616.8	−0.212984
$167$	−642358.	−1.78232	−0.891161	−	0.453687i	$-0.850108\pi$
$167$	−642358.	−1.78232	−0.891161	+	0.453687i	$0.850108\pi$
$168$	245696.	0.671623
$169$	−183164.	−0.493315
$170$	0	0
$171$	21107.8	0.0552016
$172$	36879.4	0.0950523
$173$	631119.	1.60323	0.801615	−	0.597840i	$-0.203974\pi$
$173$	631119.	1.60323	0.801615	+	0.597840i	$0.203974\pi$
$174$	69211.0	0.173301
$175$	0	0
$176$	−1475.38	−0.00359023
$177$	70643.0	0.169487
$178$	67017.1	0.158539
$179$	132741.	0.309651	0.154825	−	0.987942i	$-0.450518\pi$
$179$	132741.	0.309651	0.154825	+	0.987942i	$0.450518\pi$
$180$	0	0
$181$	−460476.	−1.04475	−0.522373	−	0.852717i	$-0.674953\pi$
$181$	−460476.	−1.04475	−0.522373	+	0.852717i	$0.674953\pi$
$182$	−323132.	−0.723106
$183$	−105680.	−0.233273
$184$	−98271.3	−0.213984
$185$	0	0
$186$	83704.1	0.177404
$187$	164772.	0.344572
$188$	481627.	0.993839
$189$	−607172.	−1.23640
$190$	0	0
$191$	701481.	1.39134	0.695669	−	0.718362i	$-0.255108\pi$
$191$	701481.	1.39134	0.695669	+	0.718362i	$0.255108\pi$
$192$	−125686.	−0.246055
$193$	617487.	1.19326	0.596629	−	0.802517i	$-0.296506\pi$
$193$	617487.	1.19326	0.596629	+	0.802517i	$0.296506\pi$
$194$	−53244.1	−0.101570
$195$	0	0
$196$	−583121.	−1.08422
$197$	−65006.0	−0.119340	−0.0596702	−	0.998218i	$-0.519005\pi$
$197$	−65006.0	−0.119340	−0.0596702	+	0.998218i	$0.519005\pi$
$198$	−85910.0	−0.155733
$199$	214650.	0.384236	0.192118	−	0.981372i	$-0.438464\pi$
$199$	214650.	0.384236	0.192118	+	0.981372i	$0.438464\pi$
$200$	0	0
$201$	−85935.2	−0.150031
$202$	102262.	0.176333
$203$	−673728.	−1.14748
$204$	−170920.	−0.287554
$205$	0	0
$206$	327579.	0.537834
$207$	110444.	0.179149
$208$	−5242.00	−0.00840115
$209$	12712.8	0.0201315
$210$	0	0
$211$	−909853.	−1.40691	−0.703453	−	0.710742i	$-0.748359\pi$
$211$	−909853.	−1.40691	−0.703453	+	0.710742i	$0.748359\pi$
$212$	−527135.	−0.805532
$213$	−6178.86	−0.00933166
$214$	−753465.	−1.12468
$215$	0	0
$216$	510333.	0.744251
$217$	−814809.	−1.17464
$218$	215339.	0.306889
$219$	76385.2	0.107621
$220$	0	0
$221$	585433.	0.806300
$222$	−173452.	−0.236209
$223$	−827779.	−1.11469	−0.557343	−	0.830282i	$-0.688179\pi$
$223$	−827779.	−1.11469	−0.557343	+	0.830282i	$0.688179\pi$
$224$	1.24738e6	1.66103
$225$	0	0
$226$	135957.	0.177065
$227$	−726099.	−0.935258	−0.467629	−	0.883925i	$-0.654892\pi$
$227$	−726099.	−0.935258	−0.467629	+	0.883925i	$0.654892\pi$
$228$	−13187.1	−0.0168002
$229$	1.21856e6	1.53553	0.767765	−	0.640731i	$-0.221369\pi$
$229$	1.21856e6	1.53553	0.767765	+	0.640731i	$0.221369\pi$
$230$	0	0
$231$	−166308.	−0.205061
$232$	566274.	0.690727
$233$	369589.	0.445994	0.222997	−	0.974819i	$-0.428416\pi$
$233$	369589.	0.445994	0.222997	+	0.974819i	$0.428416\pi$
$234$	−305236.	−0.364415
$235$	0	0
$236$	221931.	0.259381
$237$	−495438.	−0.572952
$238$	−1.00555e6	−1.15069
$239$	−1.28065e6	−1.45023	−0.725116	−	0.688627i	$-0.758214\pi$
$239$	−1.28065e6	−1.45023	−0.725116	+	0.688627i	$0.758214\pi$
$240$	0	0
$241$	−71562.2	−0.0793673	−0.0396836	−	0.999212i	$-0.512635\pi$
$241$	−71562.2	−0.0793673	−0.0396836	+	0.999212i	$0.512635\pi$
$242$	507418.	0.556965
$243$	−886254.	−0.962814
$244$	−332002.	−0.356999
$245$	0	0
$246$	248870.	0.262202
$247$	45168.3	0.0471077
$248$	684854.	0.707080
$249$	−138275.	−0.141333
$250$	0	0
$251$	−1.16845e6	−1.17064	−0.585322	−	0.810801i	$-0.699032\pi$
$251$	−1.16845e6	−1.17064	−0.585322	+	0.810801i	$0.699032\pi$
$252$	−867486.	−0.860521
$253$	66518.1	0.0653339
$254$	−506981.	−0.493069
$255$	0	0
$256$	−1.04072e6	−0.992504
$257$	978056.	0.923700	0.461850	−	0.886958i	$-0.347186\pi$
$257$	978056.	0.923700	0.461850	+	0.886958i	$0.347186\pi$
$258$	−40757.5	−0.0381205
$259$	1.68845e6	1.56401
$260$	0	0
$261$	−636415.	−0.578282
$262$	96122.8	0.0865114
$263$	415914.	0.370778	0.185389	−	0.982665i	$-0.440645\pi$
$263$	415914.	0.370778	0.185389	+	0.982665i	$0.440645\pi$
$264$	139783.	0.123437
$265$	0	0
$266$	−77581.7	−0.0672287
$267$	122549.	0.105204
$268$	−269973.	−0.229606
$269$	2.15490e6	1.81571	0.907856	−	0.419283i	$-0.137719\pi$
$269$	2.15490e6	1.81571	0.907856	+	0.419283i	$0.137719\pi$
$270$	0	0
$271$	−1.03994e6	−0.860171	−0.430085	−	0.902788i	$-0.641517\pi$
$271$	−1.03994e6	−0.860171	−0.430085	+	0.902788i	$0.641517\pi$
$272$	−16312.5	−0.0133689
$273$	−590888.	−0.479842
$274$	−1.05946e6	−0.852528
$275$	0	0
$276$	−69000.0	−0.0545226
$277$	−1.56813e6	−1.22796	−0.613979	−	0.789323i	$-0.710432\pi$
$277$	−1.56813e6	−1.22796	−0.613979	+	0.789323i	$0.710432\pi$
$278$	−1.08438e6	−0.841528
$279$	−769683.	−0.591972
$280$	0	0
$281$	166293.	0.125634	0.0628170	−	0.998025i	$-0.479992\pi$
$281$	166293.	0.125634	0.0628170	+	0.998025i	$0.479992\pi$
$282$	−532272.	−0.398576
$283$	−423919.	−0.314642	−0.157321	−	0.987548i	$-0.550286\pi$
$283$	−423919.	−0.314642	−0.157321	+	0.987548i	$0.550286\pi$
$284$	−19411.4	−0.0142811
$285$	0	0
$286$	−183838.	−0.132899
$287$	−2.42260e6	−1.73611
$288$	1.17829e6	0.837091
$289$	401937.	0.283083
$290$	0	0
$291$	−97363.4	−0.0674005
$292$	239971.	0.164703
$293$	−1.30489e6	−0.887982	−0.443991	−	0.896031i	$-0.646438\pi$
$293$	−1.30489e6	−0.887982	−0.443991	+	0.896031i	$0.646438\pi$
$294$	644440.	0.434824
$295$	0	0
$296$	−1.41915e6	−0.941457
$297$	−345435.	−0.227235
$298$	−207403.	−0.135293
$299$	236337.	0.152881
$300$	0	0
$301$	396750.	0.252406
$302$	−605098.	−0.381776
$303$	186998.	0.117012
$304$	−1258.57	−0.000781073 0
$305$	0	0
$306$	−949858.	−0.579902
$307$	−1.80847e6	−1.09513	−0.547564	−	0.836764i	$-0.684445\pi$
$307$	−1.80847e6	−1.09513	−0.547564	+	0.836764i	$0.684445\pi$
$308$	−522470.	−0.313823
$309$	599020.	0.356899
$310$	0	0
$311$	2.31222e6	1.35559	0.677794	−	0.735252i	$-0.262936\pi$
$311$	2.31222e6	1.35559	0.677794	+	0.735252i	$0.262936\pi$
$312$	496646.	0.288842
$313$	−2.81344e6	−1.62322	−0.811608	−	0.584202i	$-0.801407\pi$
$313$	−2.81344e6	−1.62322	−0.811608	+	0.584202i	$0.801407\pi$
$314$	1.42621e6	0.816320
$315$	0	0
$316$	−1.55646e6	−0.876841
$317$	794420.	0.444019	0.222010	−	0.975044i	$-0.428738\pi$
$317$	794420.	0.444019	0.222010	+	0.975044i	$0.428738\pi$
$318$	582567.	0.323056
$319$	−383301.	−0.210893
$320$	0	0
$321$	−1.37781e6	−0.746320
$322$	−405936.	−0.218182
$323$	140558.	0.0749635
$324$	−624077.	−0.330275
$325$	0	0
$326$	−1.51122e6	−0.787561
$327$	393774.	0.203647
$328$	2.03622e6	1.04506
$329$	5.18136e6	2.63909
$330$	0	0
$331$	2.12662e6	1.06689	0.533446	−	0.845834i	$-0.320897\pi$
$331$	2.12662e6	1.06689	0.533446	+	0.845834i	$0.320897\pi$
$332$	−434402.	−0.216295
$333$	1.59494e6	0.788195
$334$	2.23023e6	1.09392
$335$	0	0
$336$	16464.4	0.00795607
$337$	−2.83314e6	−1.35892	−0.679460	−	0.733713i	$-0.737786\pi$
$337$	−2.83314e6	−1.35892	−0.679460	+	0.733713i	$0.737786\pi$
$338$	635936.	0.302776
$339$	248615.	0.117497
$340$	0	0
$341$	−463565.	−0.215886
$342$	−73285.0	−0.0338805
$343$	−2.66687e6	−1.22396
$344$	−333471.	−0.151937
$345$	0	0
$346$	−2.19121e6	−0.983997
$347$	2.98932e6	1.33275	0.666374	−	0.745618i	$-0.267845\pi$
$347$	2.98932e6	1.33275	0.666374	+	0.745618i	$0.267845\pi$
$348$	397603.	0.175995
$349$	−310251.	−0.136348	−0.0681742	−	0.997673i	$-0.521717\pi$
$349$	−310251.	−0.136348	−0.0681742	+	0.997673i	$0.521717\pi$
$350$	0	0
$351$	−1.22732e6	−0.531731
$352$	709664.	0.305278
$353$	−1.91574e6	−0.818274	−0.409137	−	0.912473i	$-0.634170\pi$
$353$	−1.91574e6	−0.818274	−0.409137	+	0.912473i	$0.634170\pi$
$354$	−245269.	−0.104024
$355$	0	0
$356$	384999.	0.161003
$357$	−1.83877e6	−0.763583
$358$	−460869.	−0.190051
$359$	−1.06723e6	−0.437042	−0.218521	−	0.975832i	$-0.570123\pi$
$359$	−1.06723e6	−0.437042	−0.218521	+	0.975832i	$0.570123\pi$
$360$	0	0
$361$	−2.46525e6	−0.995620
$362$	1.59875e6	0.641222
$363$	927878.	0.369593
$364$	−1.85633e6	−0.734346
$365$	0	0
$366$	366914.	0.143173
$367$	−4.51684e6	−1.75053	−0.875265	−	0.483644i	$-0.839313\pi$
$367$	−4.51684e6	−1.75053	−0.875265	+	0.483644i	$0.839313\pi$
$368$	−6585.28	−0.00253487
$369$	−2.28843e6	−0.874928
$370$	0	0
$371$	−5.67094e6	−2.13905
$372$	480862.	0.180162
$373$	−3.76940e6	−1.40282	−0.701408	−	0.712760i	$-0.747445\pi$
$373$	−3.76940e6	−1.40282	−0.701408	+	0.712760i	$0.747445\pi$
$374$	−572081.	−0.211484
$375$	0	0
$376$	−4.35497e6	−1.58860
$377$	−1.36186e6	−0.493491
$378$	2.10807e6	0.758849
$379$	−558937.	−0.199878	−0.0999389	−	0.994994i	$-0.531865\pi$
$379$	−558937.	−0.199878	−0.0999389	+	0.994994i	$0.531865\pi$
$380$	0	0
$381$	−927078.	−0.327193
$382$	−2.43550e6	−0.853946
$383$	2.78912e6	0.971560	0.485780	−	0.874081i	$-0.338536\pi$
$383$	2.78912e6	0.971560	0.485780	+	0.874081i	$0.338536\pi$
$384$	−744668.	−0.257712
$385$	0	0
$386$	−2.14388e6	−0.732373
$387$	374777.	0.127202
$388$	−305876.	−0.103149
$389$	−586578.	−0.196540	−0.0982701	−	0.995160i	$-0.531331\pi$
$389$	−586578.	−0.196540	−0.0982701	+	0.995160i	$0.531331\pi$
$390$	0	0
$391$	735452.	0.243283
$392$	5.27270e6	1.73308
$393$	175773.	0.0574076
$394$	225697.	0.0732463
$395$	0	0
$396$	−493535.	−0.158154
$397$	−921553.	−0.293457	−0.146728	−	0.989177i	$-0.546874\pi$
$397$	−921553.	−0.293457	−0.146728	+	0.989177i	$0.546874\pi$
$398$	−745253.	−0.235828
$399$	−141868.	−0.0446120
$400$	0	0
$401$	2.30516e6	0.715879	0.357939	−	0.933745i	$-0.383479\pi$
$401$	2.30516e6	0.715879	0.357939	+	0.933745i	$0.383479\pi$
$402$	298362.	0.0920829
$403$	−1.64704e6	−0.505174
$404$	587472.	0.179074
$405$	0	0
$406$	2.33915e6	0.704275
$407$	960601.	0.287446
$408$	1.54550e6	0.459641
$409$	799607.	0.236357	0.118178	−	0.992992i	$-0.462295\pi$
$409$	799607.	0.236357	0.118178	+	0.992992i	$0.462295\pi$
$410$	0	0
$411$	−1.93736e6	−0.565724
$412$	1.88187e6	0.546195
$413$	2.38755e6	0.688774
$414$	−383454.	−0.109954
$415$	0	0
$416$	2.52142e6	0.714352
$417$	−1.98292e6	−0.558425
$418$	−44138.1	−0.0123559
$419$	−4.24481e6	−1.18120	−0.590600	−	0.806964i	$-0.701109\pi$
$419$	−4.24481e6	−1.18120	−0.590600	+	0.806964i	$0.701109\pi$
$420$	0	0
$421$	−2.21896e6	−0.610162	−0.305081	−	0.952326i	$-0.598684\pi$
$421$	−2.21896e6	−0.610162	−0.305081	+	0.952326i	$0.598684\pi$
$422$	3.15896e6	0.863501
$423$	4.89440e6	1.32999
$424$	4.76647e6	1.28760
$425$	0	0
$426$	21452.7	0.00572739
$427$	−3.57169e6	−0.947991
$428$	−4.32850e6	−1.14216
$429$	−336171.	−0.0881894
$430$	0	0
$431$	−470092.	−0.121896	−0.0609480	−	0.998141i	$-0.519412\pi$
$431$	−470092.	−0.121896	−0.0609480	+	0.998141i	$0.519412\pi$
$432$	34198.1	0.00881642
$433$	6.11670e6	1.56782	0.783912	−	0.620872i	$-0.213221\pi$
$433$	6.11670e6	1.56782	0.783912	+	0.620872i	$0.213221\pi$
$434$	2.82897e6	0.720949
$435$	0	0
$436$	1.23708e6	0.311659
$437$	56742.8	0.0142137
$438$	−265205.	−0.0660536
$439$	−1.29847e6	−0.321567	−0.160783	−	0.986990i	$-0.551402\pi$
$439$	−1.29847e6	−0.321567	−0.160783	+	0.986990i	$0.551402\pi$
$440$	0	0
$441$	−5.92581e6	−1.45095
$442$	−2.03259e6	−0.494874
$443$	2.36688e6	0.573016	0.286508	−	0.958078i	$-0.407505\pi$
$443$	2.36688e6	0.573016	0.286508	+	0.958078i	$0.407505\pi$
$444$	−996443.	−0.239881
$445$	0	0
$446$	2.87400e6	0.684148
$447$	−379262.	−0.0897782
$448$	−4.24784e6	−0.999938
$449$	−679554.	−0.159077	−0.0795386	−	0.996832i	$-0.525345\pi$
$449$	−679554.	−0.159077	−0.0795386	+	0.996832i	$0.525345\pi$
$450$	0	0
$451$	−1.37828e6	−0.319077
$452$	781047.	0.179817
$453$	−1.10650e6	−0.253340
$454$	2.52098e6	0.574023
$455$	0	0
$456$	119241.	0.0268543
$457$	5.67044e6	1.27007	0.635033	−	0.772485i	$-0.280987\pi$
$457$	5.67044e6	1.27007	0.635033	+	0.772485i	$0.280987\pi$
$458$	−4.23078e6	−0.942446
$459$	−3.81928e6	−0.846155
$460$	0	0
$461$	−4.13319e6	−0.905802	−0.452901	−	0.891561i	$-0.649611\pi$
$461$	−4.13319e6	−0.905802	−0.452901	+	0.891561i	$0.649611\pi$
$462$	577411.	0.125858
$463$	7.61045e6	1.64990	0.824950	−	0.565206i	$-0.191203\pi$
$463$	7.61045e6	1.64990	0.824950	+	0.565206i	$0.191203\pi$
$464$	37946.7	0.00818238
$465$	0	0
$466$	−1.28319e6	−0.273733
$467$	3.60271e6	0.764429	0.382215	−	0.924074i	$-0.375161\pi$
$467$	3.60271e6	0.764429	0.382215	+	0.924074i	$0.375161\pi$
$468$	−1.75352e6	−0.370080
$469$	−2.90438e6	−0.609707
$470$	0	0
$471$	2.60801e6	0.541698
$472$	−2.00675e6	−0.414609
$473$	225721.	0.0463894
$474$	1.72013e6	0.351655
$475$	0	0
$476$	−5.77666e6	−1.16858
$477$	−5.35687e6	−1.07799
$478$	4.44636e6	0.890093
$479$	4.84462e6	0.964764	0.482382	−	0.875961i	$-0.339772\pi$
$479$	4.84462e6	0.964764	0.482382	+	0.875961i	$0.339772\pi$
$480$	0	0
$481$	3.41300e6	0.672625
$482$	248460.	0.0487124
$483$	−742305.	−0.144782
$484$	2.91501e6	0.565623
$485$	0	0
$486$	3.07703e6	0.590936
$487$	−5.40249e6	−1.03222	−0.516109	−	0.856523i	$-0.672620\pi$
$487$	−5.40249e6	−1.03222	−0.516109	+	0.856523i	$0.672620\pi$
$488$	3.00204e6	0.570645
$489$	−2.76346e6	−0.522613
$490$	0	0
$491$	2.94704e6	0.551675	0.275837	−	0.961204i	$-0.411045\pi$
$491$	2.94704e6	0.551675	0.275837	+	0.961204i	$0.411045\pi$
$492$	1.42971e6	0.266277
$493$	−4.23794e6	−0.785303
$494$	−156822.	−0.0289128
$495$	0	0
$496$	45892.9	0.00837609
$497$	−208829.	−0.0379227
$498$	480082.	0.0867446
$499$	2.07044e6	0.372230	0.186115	−	0.982528i	$-0.440410\pi$
$499$	2.07044e6	0.372230	0.186115	+	0.982528i	$0.440410\pi$
$500$	0	0
$501$	4.07826e6	0.725906
$502$	4.05679e6	0.718493
$503$	−8.83444e6	−1.55689	−0.778447	−	0.627710i	$-0.783992\pi$
$503$	−8.83444e6	−1.55689	−0.778447	+	0.627710i	$0.783992\pi$
$504$	7.84400e6	1.37550
$505$	0	0
$506$	−230947.	−0.0400992
$507$	1.16289e6	0.200918
$508$	−2.91250e6	−0.500733
$509$	2.86398e6	0.489976	0.244988	−	0.969526i	$-0.421216\pi$
$509$	2.86398e6	0.489976	0.244988	+	0.969526i	$0.421216\pi$
$510$	0	0
$511$	2.58161e6	0.437360
$512$	−140006.	−0.0236033
$513$	−294672.	−0.0494362
$514$	−3.39576e6	−0.566929
$515$	0	0
$516$	−234143.	−0.0387130
$517$	2.94780e6	0.485034
$518$	−5.86220e6	−0.959923
$519$	−4.00690e6	−0.652966
$520$	0	0
$521$	−7.97515e6	−1.28720	−0.643598	−	0.765364i	$-0.722559\pi$
$521$	−7.97515e6	−1.28720	−0.643598	+	0.765364i	$0.722559\pi$
$522$	2.20960e6	0.354926
$523$	−2.32992e6	−0.372466	−0.186233	−	0.982506i	$-0.559628\pi$
$523$	−2.32992e6	−0.372466	−0.186233	+	0.982506i	$0.559628\pi$
$524$	552205.	0.0878562
$525$	0	0
$526$	−1.44403e6	−0.227568
$527$	−5.12538e6	−0.803895
$528$	9367.03	0.00146223
$529$	−6.13944e6	−0.953871
$530$	0	0
$531$	2.25532e6	0.347113
$532$	−445690.	−0.0682738
$533$	−4.89700e6	−0.746642
$534$	−425484.	−0.0645699
$535$	0	0
$536$	2.44115e6	0.367014
$537$	−842757.	−0.126115
$538$	−7.48170e6	−1.11441
$539$	−3.56900e6	−0.529145
$540$	0	0
$541$	−3.64888e6	−0.536002	−0.268001	−	0.963419i	$-0.586363\pi$
$541$	−3.64888e6	−0.536002	−0.268001	+	0.963419i	$0.586363\pi$
$542$	3.61061e6	0.527938
$543$	2.92351e6	0.425505
$544$	7.84635e6	1.13676
$545$	0	0
$546$	2.05153e6	0.294507
$547$	−622513.	−0.0889570	−0.0444785	−	0.999010i	$-0.514163\pi$
$547$	−622513.	−0.0889570	−0.0444785	+	0.999010i	$0.514163\pi$
$548$	−6.08638e6	−0.865780
$549$	−3.37388e6	−0.477748
$550$	0	0
$551$	−326972.	−0.0458809
$552$	623913.	0.0871518
$553$	−1.67445e7	−2.32841
$554$	5.44447e6	0.753670
$555$	0	0
$556$	−6.22952e6	−0.854609
$557$	−7.17711e6	−0.980193	−0.490096	−	0.871668i	$-0.663038\pi$
$557$	−7.17711e6	−0.980193	−0.490096	+	0.871668i	$0.663038\pi$
$558$	2.67230e6	0.363328
$559$	801982.	0.108551
$560$	0	0
$561$	−1.04612e6	−0.140338
$562$	−577359.	−0.0771090
$563$	1.05227e7	1.39913	0.699564	−	0.714570i	$-0.253378\pi$
$563$	1.05227e7	1.39913	0.699564	+	0.714570i	$0.253378\pi$
$564$	−3.05779e6	−0.404772
$565$	0	0
$566$	1.47182e6	0.193114
$567$	−6.71384e6	−0.877028
$568$	175522.	0.0228277
$569$	2.97618e6	0.385371	0.192685	−	0.981261i	$-0.438280\pi$
$569$	2.97618e6	0.385371	0.192685	+	0.981261i	$0.438280\pi$
$570$	0	0
$571$	−4431.96	−0.000568861 0	−0.000284430	−	1.00000i	$-0.500091\pi$
$571$	−4431.96	−0.000568861 0	−0.000284430		1.00000i	$0.500091\pi$
$572$	−1.05611e6	−0.134964
$573$	−4.45362e6	−0.566666
$574$	8.41115e6	1.06555
$575$	0	0
$576$	−4.01258e6	−0.503927
$577$	5.14649e6	0.643534	0.321767	−	0.946819i	$-0.395723\pi$
$577$	5.14649e6	0.643534	0.321767	+	0.946819i	$0.395723\pi$
$578$	−1.39550e6	−0.173745
$579$	−3.92035e6	−0.485992
$580$	0	0
$581$	−4.67331e6	−0.574361
$582$	338040.	0.0413677
$583$	−3.22634e6	−0.393132
$584$	−2.16987e6	−0.263270
$585$	0	0
$586$	4.53050e6	0.545007
$587$	1.01528e7	1.21616	0.608079	−	0.793877i	$-0.291941\pi$
$587$	1.01528e7	1.21616	0.608079	+	0.793877i	$0.291941\pi$
$588$	3.70217e6	0.441584
$589$	−395442.	−0.0469671
$590$	0	0
$591$	412716.	0.0486051
$592$	−95099.4	−0.0111525
$593$	2.77698e6	0.324292	0.162146	−	0.986767i	$-0.448158\pi$
$593$	2.77698e6	0.324292	0.162146	+	0.986767i	$0.448158\pi$
$594$	1.19933e6	0.139468
$595$	0	0
$596$	−1.19149e6	−0.137396
$597$	−1.36279e6	−0.156492
$598$	−820551.	−0.0938323
$599$	−1.00062e7	−1.13947	−0.569737	−	0.821827i	$-0.692955\pi$
$599$	−1.00062e7	−1.13947	−0.569737	+	0.821827i	$0.692955\pi$
$600$	0	0
$601$	−2.79803e6	−0.315985	−0.157992	−	0.987440i	$-0.550502\pi$
$601$	−2.79803e6	−0.315985	−0.157992	+	0.987440i	$0.550502\pi$
$602$	−1.37749e6	−0.154917
$603$	−2.74353e6	−0.307267
$604$	−3.47616e6	−0.387710
$605$	0	0
$606$	−649248.	−0.0718172
$607$	3.74474e6	0.412525	0.206263	−	0.978497i	$-0.433870\pi$
$607$	3.74474e6	0.412525	0.206263	+	0.978497i	$0.433870\pi$
$608$	605374.	0.0664148
$609$	4.27742e6	0.467346
$610$	0	0
$611$	1.04735e7	1.13498
$612$	−5.45673e6	−0.588917
$613$	1.08663e7	1.16797	0.583985	−	0.811764i	$-0.301493\pi$
$613$	1.08663e7	1.16797	0.583985	+	0.811764i	$0.301493\pi$
$614$	6.27890e6	0.672145
$615$	0	0
$616$	4.72429e6	0.501631
$617$	1.85659e7	1.96338	0.981689	−	0.190490i	$-0.0610076\pi$
$617$	1.85659e7	1.96338	0.981689	+	0.190490i	$0.0610076\pi$
$618$	−2.07976e6	−0.219050
$619$	−1.13424e7	−1.18982	−0.594908	−	0.803794i	$-0.702811\pi$
$619$	−1.13424e7	−1.18982	−0.594908	+	0.803794i	$0.702811\pi$
$620$	0	0
$621$	−1.54183e6	−0.160438
$622$	−8.02789e6	−0.832005
$623$	4.14183e6	0.427536
$624$	33280.9	0.00342163
$625$	0	0
$626$	9.76810e6	0.996264
$627$	−80712.1	−0.00819916
$628$	8.19330e6	0.829010
$629$	1.06208e7	1.07036
$630$	0	0
$631$	1.54649e7	1.54623	0.773117	−	0.634264i	$-0.218697\pi$
$631$	1.54649e7	1.54623	0.773117	+	0.634264i	$0.218697\pi$
$632$	1.40739e7	1.40159
$633$	5.77655e6	0.573006
$634$	−2.75818e6	−0.272521
$635$	0	0
$636$	3.34672e6	0.328078
$637$	−1.26806e7	−1.23820
$638$	1.33080e6	0.129438
$639$	−197263.	−0.0191115
$640$	0	0
$641$	−3.65974e6	−0.351807	−0.175904	−	0.984407i	$-0.556285\pi$
$641$	−3.65974e6	−0.351807	−0.175904	+	0.984407i	$0.556285\pi$
$642$	4.78367e6	0.458061
$643$	8.21054e6	0.783148	0.391574	−	0.920147i	$-0.371931\pi$
$643$	8.21054e6	0.783148	0.391574	+	0.920147i	$0.371931\pi$
$644$	−2.33202e6	−0.221573
$645$	0	0
$646$	−488010.	−0.0460095
$647$	−1.45434e7	−1.36586	−0.682929	−	0.730484i	$-0.739294\pi$
$647$	−1.45434e7	−1.36586	−0.682929	+	0.730484i	$0.739294\pi$
$648$	5.64304e6	0.527929
$649$	1.35833e6	0.126589
$650$	0	0
$651$	5.17313e6	0.478411
$652$	−8.68165e6	−0.799803
$653$	5.64255e6	0.517836	0.258918	−	0.965899i	$-0.416634\pi$
$653$	5.64255e6	0.517836	0.258918	+	0.965899i	$0.416634\pi$
$654$	−1.36716e6	−0.124990
$655$	0	0
$656$	136450.	0.0123798
$657$	2.43864e6	0.220411
$658$	−1.79894e7	−1.61976
$659$	−1.13560e7	−1.01862	−0.509311	−	0.860583i	$-0.670100\pi$
$659$	−1.13560e7	−1.01862	−0.509311	+	0.860583i	$0.670100\pi$
$660$	0	0
$661$	1.47131e7	1.30979	0.654895	−	0.755720i	$-0.272713\pi$
$661$	1.47131e7	1.30979	0.654895	+	0.755720i	$0.272713\pi$
$662$	−7.38351e6	−0.654814
$663$	−3.71685e6	−0.328391
$664$	3.92796e6	0.345738
$665$	0	0
$666$	−5.53754e6	−0.483762
$667$	−1.71084e6	−0.148900
$668$	1.28122e7	1.11092
$669$	5.25548e6	0.453990
$670$	0	0
$671$	−2.03202e6	−0.174230
$672$	−7.91945e6	−0.676506
$673$	8.44709e6	0.718902	0.359451	−	0.933164i	$-0.382964\pi$
$673$	8.44709e6	0.718902	0.359451	+	0.933164i	$0.382964\pi$
$674$	9.83652e6	0.834049
$675$	0	0
$676$	3.65332e6	0.307483
$677$	1.58638e7	1.33026	0.665130	−	0.746728i	$-0.268376\pi$
$677$	1.58638e7	1.33026	0.665130	+	0.746728i	$0.268376\pi$
$678$	−863178.	−0.0721151
$679$	−3.29062e6	−0.273907
$680$	0	0
$681$	4.60992e6	0.380913
$682$	1.60947e6	0.132502
$683$	785895.	0.0644633	0.0322317	−	0.999480i	$-0.489739\pi$
$683$	785895.	0.0644633	0.0322317	+	0.999480i	$0.489739\pi$
$684$	−421007.	−0.0344072
$685$	0	0
$686$	9.25922e6	0.751215
$687$	−7.73650e6	−0.625393
$688$	−22346.3	−0.00179985
$689$	−1.14631e7	−0.919930
$690$	0	0
$691$	6.70664e6	0.534330	0.267165	−	0.963651i	$-0.413913\pi$
$691$	6.70664e6	0.534330	0.267165	+	0.963651i	$0.413913\pi$
$692$	−1.25880e7	−0.999293
$693$	−5.30946e6	−0.419969
$694$	−1.03787e7	−0.817986
$695$	0	0
$696$	−3.59521e6	−0.281320
$697$	−1.52389e7	−1.18815
$698$	1.07718e6	0.0836850
$699$	−2.34648e6	−0.181645
$700$	0	0
$701$	3.56250e6	0.273817	0.136908	−	0.990584i	$-0.456283\pi$
$701$	3.56250e6	0.273817	0.136908	+	0.990584i	$0.456283\pi$
$702$	4.26121e6	0.326355
$703$	819435.	0.0625354
$704$	−2.41670e6	−0.183777
$705$	0	0
$706$	6.65133e6	0.502223
$707$	6.32004e6	0.475523
$708$	−1.40902e6	−0.105641
$709$	−308306.	−0.0230339	−0.0115169	−	0.999934i	$-0.503666\pi$
$709$	−308306.	−0.0230339	−0.0115169	+	0.999934i	$0.503666\pi$
$710$	0	0
$711$	−1.58171e7	−1.17342
$712$	−3.48124e6	−0.257356
$713$	−2.06910e6	−0.152425
$714$	6.38410e6	0.468656
$715$	0	0
$716$	−2.64760e6	−0.193005
$717$	8.13073e6	0.590652
$718$	3.70537e6	0.268238
$719$	−1.43460e7	−1.03492	−0.517461	−	0.855707i	$-0.673123\pi$
$719$	−1.43460e7	−1.03492	−0.517461	+	0.855707i	$0.673123\pi$
$720$	0	0
$721$	2.02453e7	1.45039
$722$	8.55923e6	0.611071
$723$	454341.	0.0323248
$724$	9.18447e6	0.651190
$725$	0	0
$726$	−3.22154e6	−0.226841
$727$	2.16482e7	1.51910	0.759548	−	0.650451i	$-0.225420\pi$
$727$	2.16482e7	1.51910	0.759548	+	0.650451i	$0.225420\pi$
$728$	1.67853e7	1.17382
$729$	−1.97650e6	−0.137746
$730$	0	0
$731$	2.49567e6	0.172740
$732$	2.10784e6	0.145399
$733$	−2.14306e6	−0.147324	−0.0736621	−	0.997283i	$-0.523469\pi$
$733$	−2.14306e6	−0.147324	−0.0736621	+	0.997283i	$0.523469\pi$
$734$	1.56822e7	1.07440
$735$	0	0
$736$	3.16754e6	0.215540
$737$	−1.65237e6	−0.112057
$738$	7.94532e6	0.536995
$739$	567933.	0.0382548	0.0191274	−	0.999817i	$-0.493911\pi$
$739$	567933.	0.0382548	0.0191274	+	0.999817i	$0.493911\pi$
$740$	0	0
$741$	−286768.	−0.0191861
$742$	1.96892e7	1.31286
$743$	2.80738e7	1.86565	0.932824	−	0.360332i	$-0.117336\pi$
$743$	2.80738e7	1.86565	0.932824	+	0.360332i	$0.117336\pi$
$744$	−4.34806e6	−0.287980
$745$	0	0
$746$	1.30872e7	0.860991
$747$	−4.41450e6	−0.289454
$748$	−3.28648e6	−0.214772
$749$	−4.65661e7	−3.03295
$750$	0	0
$751$	2.00180e7	1.29515	0.647577	−	0.762000i	$-0.275782\pi$
$751$	2.00180e7	1.29515	0.647577	+	0.762000i	$0.275782\pi$
$752$	−291832.	−0.0188187
$753$	7.41834e6	0.476781
$754$	4.72830e6	0.302884
$755$	0	0
$756$	1.21104e7	0.770645
$757$	1.32045e7	0.837493	0.418747	−	0.908103i	$-0.362470\pi$
$757$	1.32045e7	0.837493	0.418747	+	0.908103i	$0.362470\pi$
$758$	1.94060e6	0.122677
$759$	−422316.	−0.0266093
$760$	0	0
$761$	7.06673e6	0.442341	0.221170	−	0.975235i	$-0.429012\pi$
$761$	7.06673e6	0.442341	0.221170	+	0.975235i	$0.429012\pi$
$762$	3.21877e6	0.200818
$763$	1.33085e7	0.827595
$764$	−1.39915e7	−0.867221
$765$	0	0
$766$	−9.68366e6	−0.596304
$767$	4.82613e6	0.296218
$768$	6.60739e6	0.404228
$769$	3.11206e7	1.89772	0.948859	−	0.315702i	$-0.102240\pi$
$769$	3.11206e7	1.89772	0.948859	+	0.315702i	$0.102240\pi$
$770$	0	0
$771$	−6.20957e6	−0.376206
$772$	−1.23161e7	−0.743758
$773$	8.48847e6	0.510953	0.255476	−	0.966815i	$-0.417768\pi$
$773$	8.48847e6	0.510953	0.255476	+	0.966815i	$0.417768\pi$
$774$	−1.30120e6	−0.0780716
$775$	0	0
$776$	2.76579e6	0.164879
$777$	−1.07198e7	−0.636990
$778$	2.03657e6	0.120628
$779$	−1.17573e6	−0.0694169
$780$	0	0
$781$	−118808.	−0.00696976
$782$	−2.55345e6	−0.149317
$783$	8.88458e6	0.517884
$784$	353331.	0.0205301
$785$	0	0
$786$	−610273.	−0.0352345
$787$	3.85122e6	0.221647	0.110823	−	0.993840i	$-0.464651\pi$
$787$	3.85122e6	0.221647	0.110823	+	0.993840i	$0.464651\pi$
$788$	1.29658e6	0.0743849
$789$	−2.64059e6	−0.151011
$790$	0	0
$791$	8.40253e6	0.477495
$792$	4.46264e6	0.252801
$793$	−7.21974e6	−0.407698
$794$	3.19958e6	0.180112
$795$	0	0
$796$	−4.28132e6	−0.239494
$797$	1.84609e7	1.02946	0.514728	−	0.857354i	$-0.327893\pi$
$797$	1.84609e7	1.02946	0.514728	+	0.857354i	$0.327893\pi$
$798$	492557.	0.0273810
$799$	3.25922e7	1.80612
$800$	0	0
$801$	3.91245e6	0.215460
$802$	−8.00338e6	−0.439377
$803$	1.46874e6	0.0803817
$804$	1.71403e6	0.0935143
$805$	0	0
$806$	5.71843e6	0.310055
$807$	−1.36812e7	−0.739505
$808$	−5.31204e6	−0.286242
$809$	−3.93437e6	−0.211351	−0.105675	−	0.994401i	$-0.533700\pi$
$809$	−3.93437e6	−0.211351	−0.105675	+	0.994401i	$0.533700\pi$
$810$	0	0
$811$	5.97142e6	0.318805	0.159403	−	0.987214i	$-0.449043\pi$
$811$	5.97142e6	0.318805	0.159403	+	0.987214i	$0.449043\pi$
$812$	1.34379e7	0.715223
$813$	6.60245e6	0.350331
$814$	−3.33515e6	−0.176423
$815$	0	0
$816$	103566.	0.00544492
$817$	192550.	0.0100923
$818$	−2.77619e6	−0.145066
$819$	−1.88644e7	−0.982729
$820$	0	0
$821$	8.21540e6	0.425374	0.212687	−	0.977120i	$-0.431779\pi$
$821$	8.21540e6	0.425374	0.212687	+	0.977120i	$0.431779\pi$
$822$	6.72639e6	0.347218
$823$	−3.73555e6	−0.192245	−0.0961225	−	0.995370i	$-0.530644\pi$
$823$	−3.73555e6	−0.192245	−0.0961225	+	0.995370i	$0.530644\pi$
$824$	−1.70163e7	−0.873066
$825$	0	0
$826$	−8.28943e6	−0.422741
$827$	3.30193e7	1.67882	0.839409	−	0.543500i	$-0.182901\pi$
$827$	3.30193e7	1.67882	0.839409	+	0.543500i	$0.182901\pi$
$828$	−2.20286e6	−0.111664
$829$	2.24340e7	1.13376	0.566878	−	0.823802i	$-0.308151\pi$
$829$	2.24340e7	1.13376	0.566878	+	0.823802i	$0.308151\pi$
$830$	0	0
$831$	9.95589e6	0.500124
$832$	−8.58649e6	−0.430039
$833$	−3.94604e7	−1.97038
$834$	6.88459e6	0.342738
$835$	0	0
$836$	−253564.	−0.0125479
$837$	1.07450e7	0.530145
$838$	1.47378e7	0.724972
$839$	−1.71137e7	−0.839344	−0.419672	−	0.907676i	$-0.637855\pi$
$839$	−1.71137e7	−0.839344	−0.419672	+	0.907676i	$0.637855\pi$
$840$	0	0
$841$	−1.06527e7	−0.519360
$842$	7.70412e6	0.374492
$843$	−1.05577e6	−0.0511684
$844$	1.81476e7	0.876924
$845$	0	0
$846$	−1.69931e7	−0.816294
$847$	3.13598e7	1.50198
$848$	319407.	0.0152530
$849$	2.69141e6	0.128148
$850$	0	0
$851$	4.28759e6	0.202950
$852$	123241.	0.00581642
$853$	−4.09076e7	−1.92500	−0.962501	−	0.271277i	$-0.912554\pi$
$853$	−4.09076e7	−1.92500	−0.962501	+	0.271277i	$0.912554\pi$
$854$	1.24007e7	0.581838
$855$	0	0
$856$	3.91392e7	1.82569
$857$	−2.88043e7	−1.33969	−0.669847	−	0.742499i	$-0.733640\pi$
$857$	−2.88043e7	−1.33969	−0.669847	+	0.742499i	$0.733640\pi$
$858$	1.16717e6	0.0541271
$859$	−812540.	−0.0375718	−0.0187859	−	0.999824i	$-0.505980\pi$
$859$	−812540.	−0.0375718	−0.0187859	+	0.999824i	$0.505980\pi$
$860$	0	0
$861$	1.53808e7	0.707086
$862$	1.63213e6	0.0748148
$863$	−9.56742e6	−0.437288	−0.218644	−	0.975805i	$-0.570163\pi$
$863$	−9.56742e6	−0.437288	−0.218644	+	0.975805i	$0.570163\pi$
$864$	−1.64494e7	−0.749662
$865$	0	0
$866$	−2.12368e7	−0.962266
$867$	−2.55185e6	−0.115294
$868$	1.62519e7	0.732156
$869$	−9.52635e6	−0.427934
$870$	0	0
$871$	−5.87085e6	−0.262214
$872$	−1.11859e7	−0.498173
$873$	−3.10838e6	−0.138038
$874$	−197008.	−0.00872380
$875$	0	0
$876$	−1.52355e6	−0.0670804
$877$	1.83349e7	0.804968	0.402484	−	0.915427i	$-0.368147\pi$
$877$	1.83349e7	0.804968	0.402484	+	0.915427i	$0.368147\pi$
$878$	4.50822e6	0.197365
$879$	8.28458e6	0.361658
$880$	0	0
$881$	3.54341e7	1.53809	0.769045	−	0.639194i	$-0.220732\pi$
$881$	3.54341e7	1.53809	0.769045	+	0.639194i	$0.220732\pi$
$882$	2.05741e7	0.890531
$883$	4.34651e7	1.87603	0.938013	−	0.346600i	$-0.112664\pi$
$883$	4.34651e7	1.87603	0.938013	+	0.346600i	$0.112664\pi$
$884$	−1.16768e7	−0.502567
$885$	0	0
$886$	−8.21768e6	−0.351694
$887$	−6.99795e6	−0.298649	−0.149325	−	0.988788i	$-0.547710\pi$
$887$	−6.99795e6	−0.298649	−0.149325	+	0.988788i	$0.547710\pi$
$888$	9.01005e6	0.383438
$889$	−3.13328e7	−1.32967
$890$	0	0
$891$	−3.81967e6	−0.161188
$892$	1.65105e7	0.694783
$893$	2.51461e6	0.105522
$894$	1.31678e6	0.0551022
$895$	0	0
$896$	−2.51678e7	−1.04731
$897$	−1.50048e6	−0.0622657
$898$	2.35937e6	0.0976351
$899$	1.19229e7	0.492019
$900$	0	0
$901$	−3.56718e7	−1.46391
$902$	4.78531e6	0.195837
$903$	−2.51892e6	−0.102800
$904$	−7.06239e6	−0.287429
$905$	0	0
$906$	3.84170e6	0.155490
$907$	3.40510e6	0.137440	0.0687198	−	0.997636i	$-0.478109\pi$
$907$	3.40510e6	0.137440	0.0687198	+	0.997636i	$0.478109\pi$
$908$	1.44825e7	0.582946
$909$	5.97002e6	0.239644
$910$	0	0
$911$	−2.49420e7	−0.995716	−0.497858	−	0.867259i	$-0.665880\pi$
$911$	−2.49420e7	−0.995716	−0.497858	+	0.867259i	$0.665880\pi$
$912$	7990.49	0.000318117 0
$913$	−2.65876e6	−0.105561
$914$	−1.96875e7	−0.779514
$915$	0	0
$916$	−2.43049e7	−0.957096
$917$	5.94064e6	0.233297
$918$	1.32603e7	0.519335
$919$	4.08231e7	1.59447	0.797236	−	0.603667i	$-0.206294\pi$
$919$	4.08231e7	1.59447	0.797236	+	0.603667i	$0.206294\pi$
$920$	0	0
$921$	1.14818e7	0.446025
$922$	1.43502e7	0.555944
$923$	−422122.	−0.0163092
$924$	3.31710e6	0.127814
$925$	0	0
$926$	−2.64231e7	−1.01264
$927$	1.91240e7	0.730937
$928$	−1.82525e7	−0.695749
$929$	−3.99080e6	−0.151712	−0.0758560	−	0.997119i	$-0.524169\pi$
$929$	−3.99080e6	−0.151712	−0.0758560	+	0.997119i	$0.524169\pi$
$930$	0	0
$931$	−3.04451e6	−0.115118
$932$	−7.37168e6	−0.277988
$933$	−1.46800e7	−0.552105
$934$	−1.25084e7	−0.469175
$935$	0	0
$936$	1.58557e7	0.591555
$937$	−4.87906e7	−1.81546	−0.907732	−	0.419551i	$-0.862187\pi$
$937$	−4.87906e7	−1.81546	−0.907732	+	0.419551i	$0.862187\pi$
$938$	1.00838e7	0.374213
$939$	1.78622e7	0.661105
$940$	0	0
$941$	−1.29288e7	−0.475975	−0.237987	−	0.971268i	$-0.576488\pi$
$941$	−1.29288e7	−0.475975	−0.237987	+	0.971268i	$0.576488\pi$
$942$	−9.05487e6	−0.332472
$943$	−6.15187e6	−0.225283
$944$	−134475.	−0.00491147
$945$	0	0
$946$	−783690.	−0.0284719
$947$	3.59668e7	1.30325	0.651624	−	0.758542i	$-0.274088\pi$
$947$	3.59668e7	1.30325	0.651624	+	0.758542i	$0.274088\pi$
$948$	9.88180e6	0.357121
$949$	5.21842e6	0.188093
$950$	0	0
$951$	−5.04368e6	−0.180841
$952$	5.22337e7	1.86792
$953$	−3.33117e7	−1.18813	−0.594065	−	0.804417i	$-0.702478\pi$
$953$	−3.33117e7	−1.18813	−0.594065	+	0.804417i	$0.702478\pi$
$954$	1.85988e7	0.661627
$955$	0	0
$956$	2.55434e7	0.903929
$957$	2.43353e6	0.0858929
$958$	−1.68203e7	−0.592133
$959$	−6.54775e7	−2.29903
$960$	0	0
$961$	−1.42096e7	−0.496333
$962$	−1.18497e7	−0.412830
$963$	−4.39872e7	−1.52848
$964$	1.42735e6	0.0494696
$965$	0	0
$966$	2.57724e6	0.0888612
$967$	1.06247e7	0.365385	0.182692	−	0.983170i	$-0.441519\pi$
$967$	1.06247e7	0.365385	0.182692	+	0.983170i	$0.441519\pi$
$968$	−2.63582e7	−0.904121
$969$	−892387.	−0.0305312
$970$	0	0
$971$	2.88905e7	0.983346	0.491673	−	0.870780i	$-0.336386\pi$
$971$	2.88905e7	0.983346	0.491673	+	0.870780i	$0.336386\pi$
$972$	1.76769e7	0.600122
$973$	−6.70174e7	−2.26937
$974$	1.87572e7	0.633534
$975$	0	0
$976$	201170.	0.00675988
$977$	−2.62926e7	−0.881246	−0.440623	−	0.897692i	$-0.645242\pi$
$977$	−2.62926e7	−0.881246	−0.440623	+	0.897692i	$0.645242\pi$
$978$	9.59457e6	0.320759
$979$	2.35639e6	0.0785761
$980$	0	0
$981$	1.25714e7	0.417074
$982$	−1.02320e7	−0.338595
$983$	4.52063e7	1.49216	0.746080	−	0.665856i	$-0.231933\pi$
$983$	4.52063e7	1.49216	0.746080	+	0.665856i	$0.231933\pi$
$984$	−1.29277e7	−0.425632
$985$	0	0
$986$	1.47139e7	0.481987
$987$	−3.28958e7	−1.07485
$988$	−900909.	−0.0293622
$989$	1.00749e6	0.0327530
$990$	0	0
$991$	1.12500e6	0.0363890	0.0181945	−	0.999834i	$-0.494208\pi$
$991$	1.12500e6	0.0363890	0.0181945	+	0.999834i	$0.494208\pi$
$992$	−2.20747e7	−0.712221
$993$	−1.35017e7	−0.434525
$994$	725042.	0.0232754
$995$	0	0
$996$	2.75797e6	0.0880930
$997$	3.60034e7	1.14711	0.573556	−	0.819166i	$-0.305563\pi$
$997$	3.60034e7	1.14711	0.573556	+	0.819166i	$0.305563\pi$
$998$	−7.18845e6	−0.228459
$999$	−2.22659e7	−0.705873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.e.1.8		20
5.4	even	2		215.6.a.c.1.13	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.6.a.c.1.13	✓	20	5.4	even	2
1075.6.a.e.1.8		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q-3.47195 q^{2} -6.34889 q^{3} -19.9456 q^{4} +22.0430 q^{6} -214.575 q^{7} +180.352 q^{8} -202.692 q^{9} +O(q^{10})\)	\(q-3.47195 q^{2} -6.34889 q^{3} -19.9456 q^{4} +22.0430 q^{6} -214.575 q^{7} +180.352 q^{8} -202.692 q^{9} -122.077 q^{11} +126.632 q^{12} -433.738 q^{13} +744.994 q^{14} +12.0856 q^{16} -1349.74 q^{17} +703.734 q^{18} -104.137 q^{19} +1362.31 q^{21} +423.846 q^{22} -544.885 q^{23} -1145.04 q^{24} +1505.92 q^{26} +2829.65 q^{27} +4279.83 q^{28} +3139.82 q^{29} +3797.31 q^{31} -5813.23 q^{32} +775.054 q^{33} +4686.22 q^{34} +4042.81 q^{36} -7868.79 q^{37} +361.559 q^{38} +2753.75 q^{39} +11290.2 q^{41} -4729.88 q^{42} -1849.00 q^{43} +2434.90 q^{44} +1891.81 q^{46} -24147.0 q^{47} -76.7303 q^{48} +29235.6 q^{49} +8569.34 q^{51} +8651.16 q^{52} +26428.7 q^{53} -9824.37 q^{54} -38699.2 q^{56} +661.156 q^{57} -10901.3 q^{58} -11126.8 q^{59} +16645.4 q^{61} -13184.1 q^{62} +43492.6 q^{63} +19796.5 q^{64} -2690.95 q^{66} +13535.5 q^{67} +26921.3 q^{68} +3459.41 q^{69} +973.219 q^{71} -36555.9 q^{72} -12031.3 q^{73} +27320.0 q^{74} +2077.08 q^{76} +26194.8 q^{77} -9560.88 q^{78} +78035.4 q^{79} +31289.0 q^{81} -39199.0 q^{82} +21779.4 q^{83} -27172.2 q^{84} +6419.63 q^{86} -19934.4 q^{87} -22016.9 q^{88} -19302.5 q^{89} +93069.5 q^{91} +10868.1 q^{92} -24108.7 q^{93} +83837.1 q^{94} +36907.6 q^{96} +15335.5 q^{97} -101504. q^{98} +24744.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 7 q^{2} - 16 q^{3} + 347 q^{4} - 128 q^{6} - 372 q^{7} - 441 q^{8} + 1808 q^{9}+O(q^{10}) \) 20 * q - 7 * q^2 - 16 * q^3 + 347 * q^4 - 128 * q^6 - 372 * q^7 - 441 * q^8 + 1808 * q^9	\( 20 q - 7 q^{2} - 16 q^{3} + 347 q^{4} - 128 q^{6} - 372 q^{7} - 441 q^{8} + 1808 q^{9} - 147 q^{11} + 751 q^{12} + 151 q^{13} + 311 q^{14} + 6347 q^{16} - 231 q^{17} - 6125 q^{18} + 4146 q^{19} + 2128 q^{21} - 10263 q^{22} - 5101 q^{23} - 6144 q^{24} + 4007 q^{26} - 10372 q^{27} - 42211 q^{28} - 4792 q^{29} + 18673 q^{31} - 32839 q^{32} - 42332 q^{33} - 1064 q^{34} + 76091 q^{36} - 39718 q^{37} - 34535 q^{38} + 12528 q^{39} + 12889 q^{41} - 8789 q^{42} - 36980 q^{43} + 9156 q^{44} + 77524 q^{46} + 31012 q^{47} - 5074 q^{48} + 24162 q^{49} + 37160 q^{51} + 10943 q^{52} - 1913 q^{53} + 7523 q^{54} + 220786 q^{56} - 91576 q^{57} - 215883 q^{58} - 46908 q^{59} + 58128 q^{61} - 216209 q^{62} - 304976 q^{63} + 186347 q^{64} + 376113 q^{66} - 144469 q^{67} - 164172 q^{68} - 8540 q^{69} + 43954 q^{71} - 467998 q^{72} - 189870 q^{73} + 275403 q^{74} + 318411 q^{76} - 189206 q^{77} - 346786 q^{78} + 149228 q^{79} + 195100 q^{81} - 337685 q^{82} - 177193 q^{83} + 160493 q^{84} + 12943 q^{86} - 185584 q^{87} - 510938 q^{88} + 78564 q^{89} - 33858 q^{91} - 63558 q^{92} - 41492 q^{93} + 133334 q^{94} - 565639 q^{96} - 448697 q^{97} + 161151 q^{98} + 395765 q^{99}+O(q^{100}) \) 20 * q - 7 * q^2 - 16 * q^3 + 347 * q^4 - 128 * q^6 - 372 * q^7 - 441 * q^8 + 1808 * q^9 - 147 * q^11 + 751 * q^12 + 151 * q^13 + 311 * q^14 + 6347 * q^16 - 231 * q^17 - 6125 * q^18 + 4146 * q^19 + 2128 * q^21 - 10263 * q^22 - 5101 * q^23 - 6144 * q^24 + 4007 * q^26 - 10372 * q^27 - 42211 * q^28 - 4792 * q^29 + 18673 * q^31 - 32839 * q^32 - 42332 * q^33 - 1064 * q^34 + 76091 * q^36 - 39718 * q^37 - 34535 * q^38 + 12528 * q^39 + 12889 * q^41 - 8789 * q^42 - 36980 * q^43 + 9156 * q^44 + 77524 * q^46 + 31012 * q^47 - 5074 * q^48 + 24162 * q^49 + 37160 * q^51 + 10943 * q^52 - 1913 * q^53 + 7523 * q^54 + 220786 * q^56 - 91576 * q^57 - 215883 * q^58 - 46908 * q^59 + 58128 * q^61 - 216209 * q^62 - 304976 * q^63 + 186347 * q^64 + 376113 * q^66 - 144469 * q^67 - 164172 * q^68 - 8540 * q^69 + 43954 * q^71 - 467998 * q^72 - 189870 * q^73 + 275403 * q^74 + 318411 * q^76 - 189206 * q^77 - 346786 * q^78 + 149228 * q^79 + 195100 * q^81 - 337685 * q^82 - 177193 * q^83 + 160493 * q^84 + 12943 * q^86 - 185584 * q^87 - 510938 * q^88 + 78564 * q^89 - 33858 * q^91 - 63558 * q^92 - 41492 * q^93 + 133334 * q^94 - 565639 * q^96 - 448697 * q^97 + 161151 * q^98 + 395765 * q^99

Introduction

L-functions

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