LMFDB - Embedded newform 650.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, 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("650.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{145}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 36 $ x^2 - x - 36
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.52080$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} -15.0416 q^{3} +16.0000 q^{4} -60.1664 q^{6} -52.9584 q^{7} +64.0000 q^{8} -16.7504 q^{9} +O(q^{10})$	\(q+4.00000 q^{2} -15.0416 q^{3} +16.0000 q^{4} -60.1664 q^{6} -52.9584 q^{7} +64.0000 q^{8} -16.7504 q^{9} -259.248 q^{11} -240.666 q^{12} +169.000 q^{13} -211.834 q^{14} +256.000 q^{16} +2273.15 q^{17} -67.0017 q^{18} +730.329 q^{19} +796.579 q^{21} -1036.99 q^{22} -1973.36 q^{23} -962.662 q^{24} +676.000 q^{26} +3907.06 q^{27} -847.334 q^{28} -949.515 q^{29} +225.331 q^{31} +1024.00 q^{32} +3899.50 q^{33} +9092.62 q^{34} -268.007 q^{36} +954.367 q^{37} +2921.32 q^{38} -2542.03 q^{39} -17371.4 q^{41} +3186.32 q^{42} +10088.2 q^{43} -4147.97 q^{44} -7893.46 q^{46} +6069.90 q^{47} -3850.65 q^{48} -14002.4 q^{49} -34191.9 q^{51} +2704.00 q^{52} +16177.7 q^{53} +15628.2 q^{54} -3389.34 q^{56} -10985.3 q^{57} -3798.06 q^{58} +326.933 q^{59} -46814.4 q^{61} +901.324 q^{62} +887.076 q^{63} +4096.00 q^{64} +15598.0 q^{66} -43583.3 q^{67} +36370.5 q^{68} +29682.5 q^{69} +51745.5 q^{71} -1072.03 q^{72} +13295.3 q^{73} +3817.47 q^{74} +11685.3 q^{76} +13729.4 q^{77} -10168.1 q^{78} +76268.5 q^{79} -54698.1 q^{81} -69485.7 q^{82} -30147.4 q^{83} +12745.3 q^{84} +40352.9 q^{86} +14282.2 q^{87} -16591.9 q^{88} -66771.0 q^{89} -8949.97 q^{91} -31573.8 q^{92} -3389.34 q^{93} +24279.6 q^{94} -15402.6 q^{96} -151425. q^{97} -56009.6 q^{98} +4342.51 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} - 6 q^{3} + 32 q^{4} - 24 q^{6} - 130 q^{7} + 128 q^{8} - 178 q^{9}+O(q^{10}) $ 2 * q + 8 * q^2 - 6 * q^3 + 32 * q^4 - 24 * q^6 - 130 * q^7 + 128 * q^8 - 178 * q^9	\( 2 q + 8 q^{2} - 6 q^{3} + 32 q^{4} - 24 q^{6} - 130 q^{7} + 128 q^{8} - 178 q^{9} + 204 q^{11} - 96 q^{12} + 338 q^{13} - 520 q^{14} + 512 q^{16} + 404 q^{17} - 712 q^{18} + 112 q^{19} + 100 q^{21} + 816 q^{22} - 262 q^{23} - 384 q^{24} + 1352 q^{26} + 252 q^{27} - 2080 q^{28} - 6812 q^{29} - 320 q^{31} + 2048 q^{32} + 8088 q^{33} + 1616 q^{34} - 2848 q^{36} + 13276 q^{37} + 448 q^{38} - 1014 q^{39} - 35080 q^{41} + 400 q^{42} - 2534 q^{43} + 3264 q^{44} - 1048 q^{46} + 22038 q^{47} - 1536 q^{48} - 24874 q^{49} - 51092 q^{51} + 5408 q^{52} + 44108 q^{53} + 1008 q^{54} - 8320 q^{56} - 16576 q^{57} - 27248 q^{58} - 21888 q^{59} - 41272 q^{61} - 1280 q^{62} + 13310 q^{63} + 8192 q^{64} + 32352 q^{66} - 103230 q^{67} + 6464 q^{68} + 45156 q^{69} - 15480 q^{71} - 11392 q^{72} - 54088 q^{73} + 53104 q^{74} + 1792 q^{76} - 21960 q^{77} - 4056 q^{78} + 27208 q^{79} - 48562 q^{81} - 140320 q^{82} - 42690 q^{83} + 1600 q^{84} - 10136 q^{86} - 38724 q^{87} + 13056 q^{88} - 68132 q^{89} - 21970 q^{91} - 4192 q^{92} - 8320 q^{93} + 88152 q^{94} - 6144 q^{96} - 209456 q^{97} - 99496 q^{98} - 70356 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 - 6 * q^3 + 32 * q^4 - 24 * q^6 - 130 * q^7 + 128 * q^8 - 178 * q^9 + 204 * q^11 - 96 * q^12 + 338 * q^13 - 520 * q^14 + 512 * q^16 + 404 * q^17 - 712 * q^18 + 112 * q^19 + 100 * q^21 + 816 * q^22 - 262 * q^23 - 384 * q^24 + 1352 * q^26 + 252 * q^27 - 2080 * q^28 - 6812 * q^29 - 320 * q^31 + 2048 * q^32 + 8088 * q^33 + 1616 * q^34 - 2848 * q^36 + 13276 * q^37 + 448 * q^38 - 1014 * q^39 - 35080 * q^41 + 400 * q^42 - 2534 * q^43 + 3264 * q^44 - 1048 * q^46 + 22038 * q^47 - 1536 * q^48 - 24874 * q^49 - 51092 * q^51 + 5408 * q^52 + 44108 * q^53 + 1008 * q^54 - 8320 * q^56 - 16576 * q^57 - 27248 * q^58 - 21888 * q^59 - 41272 * q^61 - 1280 * q^62 + 13310 * q^63 + 8192 * q^64 + 32352 * q^66 - 103230 * q^67 + 6464 * q^68 + 45156 * q^69 - 15480 * q^71 - 11392 * q^72 - 54088 * q^73 + 53104 * q^74 + 1792 * q^76 - 21960 * q^77 - 4056 * q^78 + 27208 * q^79 - 48562 * q^81 - 140320 * q^82 - 42690 * q^83 + 1600 * q^84 - 10136 * q^86 - 38724 * q^87 + 13056 * q^88 - 68132 * q^89 - 21970 * q^91 - 4192 * q^92 - 8320 * q^93 + 88152 * q^94 - 6144 * q^96 - 209456 * q^97 - 99496 * q^98 - 70356 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−15.0416	−0.964919	−0.482459	−	0.875918i	$-0.660256\pi$
$3$	−15.0416	−0.964919	−0.482459	+	0.875918i	$0.660256\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−60.1664	−0.682301
$7$	−52.9584	−0.408498	−0.204249	−	0.978919i	$-0.565475\pi$
$7$	−52.9584	−0.408498	−0.204249	+	0.978919i	$0.565475\pi$
$8$	64.0000	0.353553
$9$	−16.7504	−0.0689318
$10$	0	0
$11$	−259.248	−0.646001	−0.323001	−	0.946399i	$-0.604692\pi$
$11$	−259.248	−0.646001	−0.323001	+	0.946399i	$0.604692\pi$
$12$	−240.666	−0.482459
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−211.834	−0.288852
$15$	0	0
$16$	256.000	0.250000
$17$	2273.15	1.90768	0.953842	−	0.300308i	$-0.0970894\pi$
$17$	2273.15	1.90768	0.953842	+	0.300308i	$0.0970894\pi$
$18$	−67.0017	−0.0487422
$19$	730.329	0.464125	0.232062	−	0.972701i	$-0.425453\pi$
$19$	730.329	0.464125	0.232062	+	0.972701i	$0.425453\pi$
$20$	0	0
$21$	796.579	0.394167
$22$	−1036.99	−0.456792
$23$	−1973.36	−0.777835	−0.388918	−	0.921273i	$-0.627151\pi$
$23$	−1973.36	−0.777835	−0.388918	+	0.921273i	$0.627151\pi$
$24$	−962.662	−0.341150
$25$	0	0
$26$	676.000	0.196116
$27$	3907.06	1.03143
$28$	−847.334	−0.204249
$29$	−949.515	−0.209656	−0.104828	−	0.994490i	$-0.533429\pi$
$29$	−949.515	−0.209656	−0.104828	+	0.994490i	$0.533429\pi$
$30$	0	0
$31$	225.331	0.0421131	0.0210565	−	0.999778i	$-0.493297\pi$
$31$	225.331	0.0421131	0.0210565	+	0.999778i	$0.493297\pi$
$32$	1024.00	0.176777
$33$	3899.50	0.623339
$34$	9092.62	1.34894
$35$	0	0
$36$	−268.007	−0.0344659
$37$	954.367	0.114607	0.0573035	−	0.998357i	$-0.481750\pi$
$37$	954.367	0.114607	0.0573035	+	0.998357i	$0.481750\pi$
$38$	2921.32	0.328186
$39$	−2542.03	−0.267620
$40$	0	0
$41$	−17371.4	−1.61390	−0.806948	−	0.590622i	$-0.798882\pi$
$41$	−17371.4	−1.61390	−0.806948	+	0.590622i	$0.798882\pi$
$42$	3186.32	0.278718
$43$	10088.2	0.832039	0.416019	−	0.909356i	$-0.363425\pi$
$43$	10088.2	0.832039	0.416019	+	0.909356i	$0.363425\pi$
$44$	−4147.97	−0.323001
$45$	0	0
$46$	−7893.46	−0.550013
$47$	6069.90	0.400809	0.200404	−	0.979713i	$-0.435774\pi$
$47$	6069.90	0.400809	0.200404	+	0.979713i	$0.435774\pi$
$48$	−3850.65	−0.241230
$49$	−14002.4	−0.833129
$50$	0	0
$51$	−34191.9	−1.84076
$52$	2704.00	0.138675
$53$	16177.7	0.791092	0.395546	−	0.918446i	$-0.370555\pi$
$53$	16177.7	0.791092	0.395546	+	0.918446i	$0.370555\pi$
$54$	15628.2	0.729333
$55$	0	0
$56$	−3389.34	−0.144426
$57$	−10985.3	−0.447843
$58$	−3798.06	−0.148249
$59$	326.933	0.0122272	0.00611362	−	0.999981i	$-0.498054\pi$
$59$	326.933	0.0122272	0.00611362	+	0.999981i	$0.498054\pi$
$60$	0	0
$61$	−46814.4	−1.61085	−0.805425	−	0.592698i	$-0.798063\pi$
$61$	−46814.4	−1.61085	−0.805425	+	0.592698i	$0.798063\pi$
$62$	901.324	0.0297784
$63$	887.076	0.0281585
$64$	4096.00	0.125000
$65$	0	0
$66$	15598.0	0.440767
$67$	−43583.3	−1.18613	−0.593066	−	0.805154i	$-0.702082\pi$
$67$	−43583.3	−1.18613	−0.593066	+	0.805154i	$0.702082\pi$
$68$	36370.5	0.953842
$69$	29682.5	0.750548
$70$	0	0
$71$	51745.5	1.21822	0.609111	−	0.793085i	$-0.291526\pi$
$71$	51745.5	1.21822	0.609111	+	0.793085i	$0.291526\pi$
$72$	−1072.03	−0.0243711
$73$	13295.3	0.292006	0.146003	−	0.989284i	$-0.453359\pi$
$73$	13295.3	0.292006	0.146003	+	0.989284i	$0.453359\pi$
$74$	3817.47	0.0810394
$75$	0	0
$76$	11685.3	0.232062
$77$	13729.4	0.263890
$78$	−10168.1	−0.189236
$79$	76268.5	1.37492	0.687460	−	0.726223i	$-0.258726\pi$
$79$	76268.5	1.37492	0.687460	+	0.726223i	$0.258726\pi$
$80$	0	0
$81$	−54698.1	−0.926317
$82$	−69485.7	−1.14120
$83$	−30147.4	−0.480347	−0.240173	−	0.970730i	$-0.577204\pi$
$83$	−30147.4	−0.480347	−0.240173	+	0.970730i	$0.577204\pi$
$84$	12745.3	0.197084
$85$	0	0
$86$	40352.9	0.588340
$87$	14282.2	0.202301
$88$	−16591.9	−0.228396
$89$	−66771.0	−0.893537	−0.446769	−	0.894650i	$-0.647425\pi$
$89$	−66771.0	−0.893537	−0.446769	+	0.894650i	$0.647425\pi$
$90$	0	0
$91$	−8949.97	−0.113297
$92$	−31573.8	−0.388918
$93$	−3389.34	−0.0406357
$94$	24279.6	0.283415
$95$	0	0
$96$	−15402.6	−0.170575
$97$	−151425.	−1.63406	−0.817032	−	0.576592i	$-0.804382\pi$
$97$	−151425.	−1.63406	−0.817032	+	0.576592i	$0.804382\pi$
$98$	−56009.6	−0.589112
$99$	4342.51	0.0445300
$100$	0	0
$101$	−145198.	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$101$	−145198.	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$102$	−136767.	−1.30161
$103$	71524.1	0.664293	0.332146	−	0.943228i	$-0.392227\pi$
$103$	71524.1	0.664293	0.332146	+	0.943228i	$0.392227\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	64710.8	0.559387
$107$	−93356.6	−0.788289	−0.394145	−	0.919048i	$-0.628959\pi$
$107$	−93356.6	−0.788289	−0.394145	+	0.919048i	$0.628959\pi$
$108$	62513.0	0.515716
$109$	102373.	0.825315	0.412658	−	0.910886i	$-0.364601\pi$
$109$	102373.	0.825315	0.412658	+	0.910886i	$0.364601\pi$
$110$	0	0
$111$	−14355.2	−0.110586
$112$	−13557.4	−0.102124
$113$	−132979.	−0.979685	−0.489843	−	0.871811i	$-0.662946\pi$
$113$	−132979.	−0.979685	−0.489843	+	0.871811i	$0.662946\pi$
$114$	−43941.3	−0.316673
$115$	0	0
$116$	−15192.2	−0.104828
$117$	−2830.82	−0.0191182
$118$	1307.73	0.00864596
$119$	−120383.	−0.779285
$120$	0	0
$121$	−93841.6	−0.582682
$122$	−187258.	−1.13904
$123$	261294.	1.55728
$124$	3605.30	0.0210565
$125$	0	0
$126$	3548.30	0.0199111
$127$	−53858.8	−0.296311	−0.148155	−	0.988964i	$-0.547334\pi$
$127$	−53858.8	−0.296311	−0.148155	+	0.988964i	$0.547334\pi$
$128$	16384.0	0.0883883
$129$	−151743.	−0.802850
$130$	0	0
$131$	125815.	0.640552	0.320276	−	0.947324i	$-0.396224\pi$
$131$	125815.	0.640552	0.320276	+	0.947324i	$0.396224\pi$
$132$	62392.0	0.311669
$133$	−38677.1	−0.189594
$134$	−174333.	−0.838721
$135$	0	0
$136$	145482.	0.674468
$137$	55609.4	0.253132	0.126566	−	0.991958i	$-0.459604\pi$
$137$	55609.4	0.253132	0.126566	+	0.991958i	$0.459604\pi$
$138$	118730.	0.530717
$139$	−200026.	−0.878113	−0.439056	−	0.898460i	$-0.644687\pi$
$139$	−200026.	−0.878113	−0.439056	+	0.898460i	$0.644687\pi$
$140$	0	0
$141$	−91301.0	−0.386748
$142$	206982.	0.861413
$143$	−43812.9	−0.179169
$144$	−4288.11	−0.0172330
$145$	0	0
$146$	53181.4	0.206480
$147$	210619.	0.803902
$148$	15269.9	0.0573035
$149$	−126007.	−0.464974	−0.232487	−	0.972599i	$-0.574686\pi$
$149$	−126007.	−0.464974	−0.232487	+	0.972599i	$0.574686\pi$
$150$	0	0
$151$	13524.5	0.0482701	0.0241351	−	0.999709i	$-0.492317\pi$
$151$	13524.5	0.0482701	0.0241351	+	0.999709i	$0.492317\pi$
$152$	46741.1	0.164093
$153$	−38076.3	−0.131500
$154$	54917.4	0.186599
$155$	0	0
$156$	−40672.5	−0.133810
$157$	−422943.	−1.36941	−0.684704	−	0.728821i	$-0.740069\pi$
$157$	−422943.	−1.36941	−0.684704	+	0.728821i	$0.740069\pi$
$158$	305074.	0.972215
$159$	−243338.	−0.763340
$160$	0	0
$161$	104506.	0.317744
$162$	−218792.	−0.655005
$163$	−668710.	−1.97137	−0.985687	−	0.168585i	$-0.946080\pi$
$163$	−668710.	−1.97137	−0.985687	+	0.168585i	$0.946080\pi$
$164$	−277943.	−0.806948
$165$	0	0
$166$	−120590.	−0.339656
$167$	−328083.	−0.910316	−0.455158	−	0.890411i	$-0.650417\pi$
$167$	−328083.	−0.910316	−0.455158	+	0.890411i	$0.650417\pi$
$168$	50981.0	0.139359
$169$	28561.0	0.0769231
$170$	0	0
$171$	−12233.3	−0.0319930
$172$	161412.	0.416019
$173$	−62008.7	−0.157520	−0.0787602	−	0.996894i	$-0.525096\pi$
$173$	−62008.7	−0.157520	−0.0787602	+	0.996894i	$0.525096\pi$
$174$	57128.9	0.143048
$175$	0	0
$176$	−66367.4	−0.161500
$177$	−4917.59	−0.0117983
$178$	−267084.	−0.631826
$179$	110227.	0.257131	0.128566	−	0.991701i	$-0.458963\pi$
$179$	110227.	0.257131	0.128566	+	0.991701i	$0.458963\pi$
$180$	0	0
$181$	−371375.	−0.842589	−0.421295	−	0.906924i	$-0.638424\pi$
$181$	−371375.	−0.842589	−0.421295	+	0.906924i	$0.638424\pi$
$182$	−35799.9	−0.0801130
$183$	704164.	1.55434
$184$	−126295.	−0.275006
$185$	0	0
$186$	−13557.4	−0.0287338
$187$	−589310.	−1.23237
$188$	97118.5	0.200404
$189$	−206912.	−0.421338
$190$	0	0
$191$	−973641.	−1.93115	−0.965573	−	0.260131i	$-0.916234\pi$
$191$	−973641.	−1.93115	−0.965573	+	0.260131i	$0.916234\pi$
$192$	−61610.4	−0.120615
$193$	17937.3	0.0346628	0.0173314	−	0.999850i	$-0.494483\pi$
$193$	17937.3	0.0346628	0.0173314	+	0.999850i	$0.494483\pi$
$194$	−605701.	−1.15546
$195$	0	0
$196$	−224039.	−0.416565
$197$	−396284.	−0.727514	−0.363757	−	0.931494i	$-0.618506\pi$
$197$	−396284.	−0.727514	−0.363757	+	0.931494i	$0.618506\pi$
$198$	17370.1	0.0314875
$199$	−374690.	−0.670718	−0.335359	−	0.942090i	$-0.608858\pi$
$199$	−374690.	−0.670718	−0.335359	+	0.942090i	$0.608858\pi$
$200$	0	0
$201$	655562.	1.14452
$202$	−580794.	−1.00148
$203$	50284.8	0.0856439
$204$	−547070.	−0.920380
$205$	0	0
$206$	286096.	0.469726
$207$	33054.7	0.0536176
$208$	43264.0	0.0693375
$209$	−189336.	−0.299825
$210$	0	0
$211$	666091.	1.02998	0.514988	−	0.857197i	$-0.327796\pi$
$211$	666091.	1.02998	0.514988	+	0.857197i	$0.327796\pi$
$212$	258843.	0.395546
$213$	−778334.	−1.17549
$214$	−373426.	−0.557405
$215$	0	0
$216$	250052.	0.364666
$217$	−11933.2	−0.0172031
$218$	409493.	0.583586
$219$	−199983.	−0.281762
$220$	0	0
$221$	384163.	0.529097
$222$	−57420.8	−0.0781964
$223$	840415.	1.13170	0.565851	−	0.824508i	$-0.308548\pi$
$223$	840415.	1.13170	0.565851	+	0.824508i	$0.308548\pi$
$224$	−54229.4	−0.0722129
$225$	0	0
$226$	−531915.	−0.692742
$227$	880953.	1.13472	0.567359	−	0.823470i	$-0.307965\pi$
$227$	880953.	1.13472	0.567359	+	0.823470i	$0.307965\pi$
$228$	−175765.	−0.223921
$229$	−622076.	−0.783889	−0.391945	−	0.919989i	$-0.628198\pi$
$229$	−622076.	−0.783889	−0.391945	+	0.919989i	$0.628198\pi$
$230$	0	0
$231$	−206511.	−0.254633
$232$	−60768.9	−0.0741245
$233$	1.09326e6	1.31927	0.659633	−	0.751588i	$-0.270712\pi$
$233$	1.09326e6	1.31927	0.659633	+	0.751588i	$0.270712\pi$
$234$	−11323.3	−0.0135186
$235$	0	0
$236$	5230.92	0.00611362
$237$	−1.14720e6	−1.32669
$238$	−481531.	−0.551038
$239$	153202.	0.173488	0.0867442	−	0.996231i	$-0.472354\pi$
$239$	153202.	0.173488	0.0867442	+	0.996231i	$0.472354\pi$
$240$	0	0
$241$	1.24027e6	1.37554	0.687771	−	0.725927i	$-0.258589\pi$
$241$	1.24027e6	1.37554	0.687771	+	0.725927i	$0.258589\pi$
$242$	−375366.	−0.412019
$243$	−126670.	−0.137612
$244$	−749031.	−0.805425
$245$	0	0
$246$	1.04518e6	1.10116
$247$	123426.	0.128725
$248$	14421.2	0.0148892
$249$	453465.	0.463495
$250$	0	0
$251$	−1.52653e6	−1.52940	−0.764702	−	0.644384i	$-0.777114\pi$
$251$	−1.52653e6	−1.52940	−0.764702	+	0.644384i	$0.777114\pi$
$252$	14193.2	0.0140793
$253$	511590.	0.502483
$254$	−215435.	−0.209523
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.27524e6	−1.20436	−0.602182	−	0.798359i	$-0.705702\pi$
$257$	−1.27524e6	−1.20436	−0.602182	+	0.798359i	$0.705702\pi$
$258$	−606972.	−0.567701
$259$	−50541.8	−0.0468167
$260$	0	0
$261$	15904.8	0.0144520
$262$	503260.	0.452938
$263$	−942046.	−0.839813	−0.419907	−	0.907567i	$-0.637937\pi$
$263$	−942046.	−0.839813	−0.419907	+	0.907567i	$0.637937\pi$
$264$	249568.	0.220384
$265$	0	0
$266$	−154708.	−0.134063
$267$	1.00434e6	0.862191
$268$	−697332.	−0.593066
$269$	806087.	0.679205	0.339603	−	0.940569i	$-0.389707\pi$
$269$	806087.	0.679205	0.339603	+	0.940569i	$0.389707\pi$
$270$	0	0
$271$	600088.	0.496354	0.248177	−	0.968715i	$-0.420169\pi$
$271$	600088.	0.496354	0.248177	+	0.968715i	$0.420169\pi$
$272$	581927.	0.476921
$273$	134622.	0.109322
$274$	222438.	0.178991
$275$	0	0
$276$	474921.	0.375274
$277$	−1.62719e6	−1.27421	−0.637103	−	0.770779i	$-0.719867\pi$
$277$	−1.62719e6	−1.27421	−0.637103	+	0.770779i	$0.719867\pi$
$278$	−800105.	−0.620919
$279$	−3774.39	−0.00290293
$280$	0	0
$281$	−228378.	−0.172539	−0.0862697	−	0.996272i	$-0.527495\pi$
$281$	−228378.	−0.172539	−0.0862697	+	0.996272i	$0.527495\pi$
$282$	−365204.	−0.273472
$283$	1.27404e6	0.945619	0.472809	−	0.881165i	$-0.343240\pi$
$283$	1.27404e6	0.945619	0.472809	+	0.881165i	$0.343240\pi$
$284$	827928.	0.609111
$285$	0	0
$286$	−175252.	−0.126691
$287$	919963.	0.659273
$288$	−17152.4	−0.0121855
$289$	3.74737e6	2.63926
$290$	0	0
$291$	2.27768e6	1.57674
$292$	212725.	0.146003
$293$	2.84127e6	1.93350	0.966749	−	0.255725i	$-0.0823143\pi$
$293$	2.84127e6	1.93350	0.966749	+	0.255725i	$0.0823143\pi$
$294$	842474.	0.568445
$295$	0	0
$296$	61079.5	0.0405197
$297$	−1.01290e6	−0.666307
$298$	−504028.	−0.328786
$299$	−333499.	−0.215733
$300$	0	0
$301$	−534256.	−0.339886
$302$	54097.9	0.0341321
$303$	2.18402e6	1.36662
$304$	186964.	0.116031
$305$	0	0
$306$	−152305.	−0.0929847
$307$	452866.	0.274235	0.137118	−	0.990555i	$-0.456216\pi$
$307$	452866.	0.274235	0.137118	+	0.990555i	$0.456216\pi$
$308$	219670.	0.131945
$309$	−1.07584e6	−0.640988
$310$	0	0
$311$	−1.43388e6	−0.840645	−0.420323	−	0.907375i	$-0.638083\pi$
$311$	−1.43388e6	−0.840645	−0.420323	+	0.907375i	$0.638083\pi$
$312$	−162690.	−0.0946181
$313$	165281.	0.0953588	0.0476794	−	0.998863i	$-0.484817\pi$
$313$	165281.	0.0953588	0.0476794	+	0.998863i	$0.484817\pi$
$314$	−1.69177e6	−0.968318
$315$	0	0
$316$	1.22030e6	0.687460
$317$	1.53580e6	0.858395	0.429198	−	0.903211i	$-0.358796\pi$
$317$	1.53580e6	0.858395	0.429198	+	0.903211i	$0.358796\pi$
$318$	−973354.	−0.539763
$319$	246160.	0.135438
$320$	0	0
$321$	1.40423e6	0.760635
$322$	418025.	0.224679
$323$	1.66015e6	0.885404
$324$	−875169.	−0.463158
$325$	0	0
$326$	−2.67484e6	−1.39397
$327$	−1.53986e6	−0.796362
$328$	−1.11177e6	−0.570599
$329$	−321452.	−0.163730
$330$	0	0
$331$	−1.57759e6	−0.791449	−0.395725	−	0.918369i	$-0.629507\pi$
$331$	−1.57759e6	−0.791449	−0.395725	+	0.918369i	$0.629507\pi$
$332$	−482358.	−0.240173
$333$	−15986.1	−0.00790007
$334$	−1.31233e6	−0.643691
$335$	0	0
$336$	203924.	0.0985418
$337$	303917.	0.145774	0.0728869	−	0.997340i	$-0.476779\pi$
$337$	303917.	0.145774	0.0728869	+	0.997340i	$0.476779\pi$
$338$	114244.	0.0543928
$339$	2.00021e6	0.945316
$340$	0	0
$341$	−58416.6	−0.0272051
$342$	−48933.3	−0.0226224
$343$	1.63162e6	0.748829
$344$	645646.	0.294170
$345$	0	0
$346$	−248035.	−0.111384
$347$	3.29844e6	1.47057	0.735283	−	0.677760i	$-0.237049\pi$
$347$	3.29844e6	1.47057	0.735283	+	0.677760i	$0.237049\pi$
$348$	228515.	0.101150
$349$	−356002.	−0.156455	−0.0782273	−	0.996936i	$-0.524926\pi$
$349$	−356002.	−0.156455	−0.0782273	+	0.996936i	$0.524926\pi$
$350$	0	0
$351$	660293.	0.286068
$352$	−265470.	−0.114198
$353$	−3.72901e6	−1.59279	−0.796393	−	0.604780i	$-0.793261\pi$
$353$	−3.72901e6	−1.59279	−0.796393	+	0.604780i	$0.793261\pi$
$354$	−19670.3	−0.00834265
$355$	0	0
$356$	−1.06834e6	−0.446769
$357$	1.81075e6	0.751947
$358$	440907.	0.181819
$359$	−2.27503e6	−0.931647	−0.465824	−	0.884878i	$-0.654242\pi$
$359$	−2.27503e6	−0.931647	−0.465824	+	0.884878i	$0.654242\pi$
$360$	0	0
$361$	−1.94272e6	−0.784588
$362$	−1.48550e6	−0.595801
$363$	1.41153e6	0.562241
$364$	−143200.	−0.0566485
$365$	0	0
$366$	2.81665e6	1.09908
$367$	−1.21410e6	−0.470534	−0.235267	−	0.971931i	$-0.575596\pi$
$367$	−1.21410e6	−0.470534	−0.235267	+	0.971931i	$0.575596\pi$
$368$	−505181.	−0.194459
$369$	290979.	0.111249
$370$	0	0
$371$	−856745.	−0.323160
$372$	−54229.4	−0.0203178
$373$	1.56733e6	0.583294	0.291647	−	0.956526i	$-0.405797\pi$
$373$	1.56733e6	0.583294	0.291647	+	0.956526i	$0.405797\pi$
$374$	−2.35724e6	−0.871415
$375$	0	0
$376$	388474.	0.141707
$377$	−160468.	−0.0581480
$378$	−827647.	−0.297931
$379$	4.40321e6	1.57461	0.787303	−	0.616567i	$-0.211477\pi$
$379$	4.40321e6	1.57461	0.787303	+	0.616567i	$0.211477\pi$
$380$	0	0
$381$	810122.	0.285916
$382$	−3.89456e6	−1.36553
$383$	3.45596e6	1.20385	0.601924	−	0.798554i	$-0.294401\pi$
$383$	3.45596e6	1.20385	0.601924	+	0.798554i	$0.294401\pi$
$384$	−246441.	−0.0852876
$385$	0	0
$386$	71749.3	0.0245103
$387$	−168982.	−0.0573539
$388$	−2.42280e6	−0.817032
$389$	556620.	0.186503	0.0932513	−	0.995643i	$-0.470274\pi$
$389$	556620.	0.186503	0.0932513	+	0.995643i	$0.470274\pi$
$390$	0	0
$391$	−4.48576e6	−1.48386
$392$	−896154.	−0.294556
$393$	−1.89246e6	−0.618080
$394$	−1.58514e6	−0.514430
$395$	0	0
$396$	69480.2	0.0222650
$397$	225504.	0.0718089	0.0359045	−	0.999355i	$-0.488569\pi$
$397$	225504.	0.0718089	0.0359045	+	0.999355i	$0.488569\pi$
$398$	−1.49876e6	−0.474269
$399$	581765.	0.182943
$400$	0	0
$401$	898293.	0.278970	0.139485	−	0.990224i	$-0.455455\pi$
$401$	898293.	0.278970	0.139485	+	0.990224i	$0.455455\pi$
$402$	2.62225e6	0.809298
$403$	38080.9	0.0116801
$404$	−2.32317e6	−0.708155
$405$	0	0
$406$	201139.	0.0605594
$407$	−247418.	−0.0740363
$408$	−2.18828e6	−0.650807
$409$	−1.77186e6	−0.523746	−0.261873	−	0.965102i	$-0.584340\pi$
$409$	−1.77186e6	−0.523746	−0.261873	+	0.965102i	$0.584340\pi$
$410$	0	0
$411$	−836454.	−0.244252
$412$	1.14439e6	0.332146
$413$	−17313.8	−0.00499480
$414$	132219.	0.0379134
$415$	0	0
$416$	173056.	0.0490290
$417$	3.00871e6	0.847307
$418$	−757345.	−0.212008
$419$	617010.	0.171695	0.0858474	−	0.996308i	$-0.472640\pi$
$419$	617010.	0.171695	0.0858474	+	0.996308i	$0.472640\pi$
$420$	0	0
$421$	−6.27988e6	−1.72682	−0.863409	−	0.504505i	$-0.831675\pi$
$421$	−6.27988e6	−1.72682	−0.863409	+	0.504505i	$0.831675\pi$
$422$	2.66436e6	0.728303
$423$	−101674.	−0.0276285
$424$	1.03537e6	0.279693
$425$	0	0
$426$	−3.11334e6	−0.831194
$427$	2.47922e6	0.658029
$428$	−1.49371e6	−0.394145
$429$	659016.	0.172883
$430$	0	0
$431$	2.03052e6	0.526519	0.263259	−	0.964725i	$-0.415202\pi$
$431$	2.03052e6	0.526519	0.263259	+	0.964725i	$0.415202\pi$
$432$	1.00021e6	0.257858
$433$	1.29410e6	0.331702	0.165851	−	0.986151i	$-0.446963\pi$
$433$	1.29410e6	0.331702	0.165851	+	0.986151i	$0.446963\pi$
$434$	−47732.7	−0.0121644
$435$	0	0
$436$	1.63797e6	0.412658
$437$	−1.44121e6	−0.361013
$438$	−799933.	−0.199236
$439$	−2.53819e6	−0.628584	−0.314292	−	0.949326i	$-0.601767\pi$
$439$	−2.53819e6	−0.628584	−0.314292	+	0.949326i	$0.601767\pi$
$440$	0	0
$441$	234546.	0.0574291
$442$	1.53665e6	0.374128
$443$	−6.27678e6	−1.51959	−0.759797	−	0.650160i	$-0.774702\pi$
$443$	−6.27678e6	−1.51959	−0.759797	+	0.650160i	$0.774702\pi$
$444$	−229683.	−0.0552932
$445$	0	0
$446$	3.36166e6	0.800234
$447$	1.89535e6	0.448662
$448$	−216918.	−0.0510622
$449$	−1.91186e6	−0.447549	−0.223775	−	0.974641i	$-0.571838\pi$
$449$	−1.91186e6	−0.447549	−0.223775	+	0.974641i	$0.571838\pi$
$450$	0	0
$451$	4.50350e6	1.04258
$452$	−2.12766e6	−0.489843
$453$	−203430.	−0.0465767
$454$	3.52381e6	0.802367
$455$	0	0
$456$	−703060.	−0.158336
$457$	−7.83658e6	−1.75524	−0.877619	−	0.479360i	$-0.840869\pi$
$457$	−7.83658e6	−1.75524	−0.877619	+	0.479360i	$0.840869\pi$
$458$	−2.48830e6	−0.554294
$459$	8.88135e6	1.96765
$460$	0	0
$461$	−3.95695e6	−0.867178	−0.433589	−	0.901111i	$-0.642753\pi$
$461$	−3.95695e6	−0.867178	−0.433589	+	0.901111i	$0.642753\pi$
$462$	−826045.	−0.180052
$463$	−2.70615e6	−0.586677	−0.293338	−	0.956009i	$-0.594766\pi$
$463$	−2.70615e6	−0.586677	−0.293338	+	0.956009i	$0.594766\pi$
$464$	−243076.	−0.0524139
$465$	0	0
$466$	4.37303e6	0.932862
$467$	5.13059e6	1.08862	0.544308	−	0.838885i	$-0.316792\pi$
$467$	5.13059e6	1.08862	0.544308	+	0.838885i	$0.316792\pi$
$468$	−45293.2	−0.00955912
$469$	2.30810e6	0.484532
$470$	0	0
$471$	6.36174e6	1.32137
$472$	20923.7	0.00432298
$473$	−2.61535e6	−0.537498
$474$	−4.58880e6	−0.938108
$475$	0	0
$476$	−1.92612e6	−0.389643
$477$	−270984.	−0.0545314
$478$	612809.	0.122675
$479$	2.22034e6	0.442162	0.221081	−	0.975255i	$-0.429042\pi$
$479$	2.22034e6	0.442162	0.221081	+	0.975255i	$0.429042\pi$
$480$	0	0
$481$	161288.	0.0317863
$482$	4.96108e6	0.972656
$483$	−1.57194e6	−0.306597
$484$	−1.50146e6	−0.291341
$485$	0	0
$486$	−506678.	−0.0973065
$487$	5.01311e6	0.957822	0.478911	−	0.877863i	$-0.341032\pi$
$487$	5.01311e6	0.957822	0.478911	+	0.877863i	$0.341032\pi$
$488$	−2.99612e6	−0.569522
$489$	1.00585e7	1.90222
$490$	0	0
$491$	−3.79096e6	−0.709652	−0.354826	−	0.934932i	$-0.615460\pi$
$491$	−3.79096e6	−0.709652	−0.354826	+	0.934932i	$0.615460\pi$
$492$	4.18070e6	0.778639
$493$	−2.15839e6	−0.399957
$494$	493703.	0.0910224
$495$	0	0
$496$	57684.7	0.0105283
$497$	−2.74036e6	−0.497641
$498$	1.81386e6	0.327741
$499$	7.59448e6	1.36536	0.682679	−	0.730718i	$-0.260815\pi$
$499$	7.59448e6	1.36536	0.682679	+	0.730718i	$0.260815\pi$
$500$	0	0
$501$	4.93489e6	0.878381
$502$	−6.10614e6	−1.08145
$503$	−2.64250e6	−0.465688	−0.232844	−	0.972514i	$-0.574803\pi$
$503$	−2.64250e6	−0.465688	−0.232844	+	0.972514i	$0.574803\pi$
$504$	56772.9	0.00995553
$505$	0	0
$506$	2.04636e6	0.355309
$507$	−429603.	−0.0742245
$508$	−861741.	−0.148155
$509$	9.84097e6	1.68362	0.841809	−	0.539776i	$-0.181491\pi$
$509$	9.84097e6	1.68362	0.841809	+	0.539776i	$0.181491\pi$
$510$	0	0
$511$	−704100.	−0.119284
$512$	262144.	0.0441942
$513$	2.85344e6	0.478713
$514$	−5.10095e6	−0.851614
$515$	0	0
$516$	−2.42789e6	−0.401425
$517$	−1.57361e6	−0.258923
$518$	−202167.	−0.0331044
$519$	932709.	0.151994
$520$	0	0
$521$	994770.	0.160557	0.0802783	−	0.996772i	$-0.474419\pi$
$521$	994770.	0.160557	0.0802783	+	0.996772i	$0.474419\pi$
$522$	63619.1	0.0102191
$523$	−5.82741e6	−0.931582	−0.465791	−	0.884895i	$-0.654230\pi$
$523$	−5.82741e6	−0.931582	−0.465791	+	0.884895i	$0.654230\pi$
$524$	2.01304e6	0.320276
$525$	0	0
$526$	−3.76818e6	−0.593838
$527$	512212.	0.0803384
$528$	998272.	0.155835
$529$	−2.54218e6	−0.394972
$530$	0	0
$531$	−5476.26	−0.000842845 0
$532$	−618833.	−0.0947970
$533$	−2.93577e6	−0.447614
$534$	4.01737e6	0.609661
$535$	0	0
$536$	−2.78933e6	−0.419361
$537$	−1.65799e6	−0.248111
$538$	3.22435e6	0.480271
$539$	3.63009e6	0.538203
$540$	0	0
$541$	−4.57878e6	−0.672599	−0.336300	−	0.941755i	$-0.609175\pi$
$541$	−4.57878e6	−0.672599	−0.336300	+	0.941755i	$0.609175\pi$
$542$	2.40035e6	0.350975
$543$	5.58607e6	0.813030
$544$	2.32771e6	0.337234
$545$	0	0
$546$	538487.	0.0773026
$547$	7.15998e6	1.02316	0.511580	−	0.859236i	$-0.329060\pi$
$547$	7.15998e6	1.02316	0.511580	+	0.859236i	$0.329060\pi$
$548$	889751.	0.126566
$549$	784162.	0.111039
$550$	0	0
$551$	−693458.	−0.0973064
$552$	1.89968e6	0.265359
$553$	−4.03906e6	−0.561652
$554$	−6.50877e6	−0.901000
$555$	0	0
$556$	−3.20042e6	−0.439056
$557$	2.24322e6	0.306361	0.153180	−	0.988198i	$-0.451048\pi$
$557$	2.24322e6	0.306361	0.153180	+	0.988198i	$0.451048\pi$
$558$	−15097.6	−0.00205268
$559$	1.70491e6	0.230766
$560$	0	0
$561$	8.86417e6	1.18913
$562$	−913512.	−0.122004
$563$	−9.93603e6	−1.32112	−0.660560	−	0.750774i	$-0.729681\pi$
$563$	−9.93603e6	−1.32112	−0.660560	+	0.750774i	$0.729681\pi$
$564$	−1.46082e6	−0.193374
$565$	0	0
$566$	5.09615e6	0.668653
$567$	2.89672e6	0.378398
$568$	3.31171e6	0.430707
$569$	−1.17630e7	−1.52313	−0.761567	−	0.648086i	$-0.775570\pi$
$569$	−1.17630e7	−1.52313	−0.761567	+	0.648086i	$0.775570\pi$
$570$	0	0
$571$	−9.88308e6	−1.26853	−0.634267	−	0.773114i	$-0.718698\pi$
$571$	−9.88308e6	−1.26853	−0.634267	+	0.773114i	$0.718698\pi$
$572$	−701006.	−0.0895843
$573$	1.46451e7	1.86340
$574$	3.67985e6	0.466177
$575$	0	0
$576$	−68609.8	−0.00861648
$577$	6.86460e6	0.858372	0.429186	−	0.903216i	$-0.358800\pi$
$577$	6.86460e6	0.858372	0.429186	+	0.903216i	$0.358800\pi$
$578$	1.49895e7	1.86624
$579$	−269806.	−0.0334468
$580$	0	0
$581$	1.59656e6	0.196221
$582$	9.11071e6	1.11492
$583$	−4.19403e6	−0.511047
$584$	850902.	0.103240
$585$	0	0
$586$	1.13651e7	1.36719
$587$	3.09692e6	0.370966	0.185483	−	0.982647i	$-0.440615\pi$
$587$	3.09692e6	0.370966	0.185483	+	0.982647i	$0.440615\pi$
$588$	3.36990e6	0.401951
$589$	164566.	0.0195457
$590$	0	0
$591$	5.96075e6	0.701992
$592$	244318.	0.0286518
$593$	5.60192e6	0.654184	0.327092	−	0.944992i	$-0.393931\pi$
$593$	5.60192e6	0.654184	0.327092	+	0.944992i	$0.393931\pi$
$594$	−4.05159e6	−0.471150
$595$	0	0
$596$	−2.01611e6	−0.232487
$597$	5.63594e6	0.647188
$598$	−1.33399e6	−0.152546
$599$	1.17534e7	1.33843	0.669214	−	0.743070i	$-0.266631\pi$
$599$	1.17534e7	1.33843	0.669214	+	0.743070i	$0.266631\pi$
$600$	0	0
$601$	−8.93997e6	−1.00960	−0.504800	−	0.863236i	$-0.668434\pi$
$601$	−8.93997e6	−1.00960	−0.504800	+	0.863236i	$0.668434\pi$
$602$	−2.13702e6	−0.240336
$603$	730038.	0.0817622
$604$	216392.	0.0241351
$605$	0	0
$606$	8.73606e6	0.966349
$607$	1.73574e7	1.91211	0.956053	−	0.293194i	$-0.0947183\pi$
$607$	1.73574e7	1.91211	0.956053	+	0.293194i	$0.0947183\pi$
$608$	747857.	0.0820464
$609$	−756363.	−0.0826394
$610$	0	0
$611$	1.02581e6	0.111164
$612$	−609221.	−0.0657501
$613$	1.63664e7	1.75915	0.879575	−	0.475760i	$-0.157827\pi$
$613$	1.63664e7	1.75915	0.879575	+	0.475760i	$0.157827\pi$
$614$	1.81146e6	0.193914
$615$	0	0
$616$	878679.	0.0932993
$617$	6.81573e6	0.720775	0.360387	−	0.932803i	$-0.382645\pi$
$617$	6.81573e6	0.720775	0.360387	+	0.932803i	$0.382645\pi$
$618$	−4.30335e6	−0.453247
$619$	−3.32032e6	−0.348300	−0.174150	−	0.984719i	$-0.555718\pi$
$619$	−3.32032e6	−0.348300	−0.174150	+	0.984719i	$0.555718\pi$
$620$	0	0
$621$	−7.71005e6	−0.802284
$622$	−5.73553e6	−0.594426
$623$	3.53608e6	0.365008
$624$	−650760.	−0.0669051
$625$	0	0
$626$	661122.	0.0674289
$627$	2.84792e6	0.289307
$628$	−6.76709e6	−0.684704
$629$	2.16942e6	0.218634
$630$	0	0
$631$	−1.25729e7	−1.25708	−0.628541	−	0.777777i	$-0.716347\pi$
$631$	−1.25729e7	−1.25708	−0.628541	+	0.777777i	$0.716347\pi$
$632$	4.88118e6	0.486107
$633$	−1.00191e7	−0.993844
$634$	6.14321e6	0.606977
$635$	0	0
$636$	−3.89341e6	−0.381670
$637$	−2.36641e6	−0.231069
$638$	984639.	0.0957690
$639$	−866759.	−0.0839743
$640$	0	0
$641$	5.56432e6	0.534893	0.267446	−	0.963573i	$-0.413820\pi$
$641$	5.56432e6	0.534893	0.267446	+	0.963573i	$0.413820\pi$
$642$	5.61693e6	0.537850
$643$	1.05294e7	1.00433	0.502165	−	0.864772i	$-0.332537\pi$
$643$	1.05294e7	1.00433	0.502165	+	0.864772i	$0.332537\pi$
$644$	1.67210e6	0.158872
$645$	0	0
$646$	6.64060e6	0.626075
$647$	−1.29138e7	−1.21281	−0.606406	−	0.795155i	$-0.707389\pi$
$647$	−1.29138e7	−1.21281	−0.606406	+	0.795155i	$0.707389\pi$
$648$	−3.50068e6	−0.327502
$649$	−84756.6	−0.00789881
$650$	0	0
$651$	179494.	0.0165996
$652$	−1.06994e7	−0.985687
$653$	1.36537e7	1.25305	0.626523	−	0.779403i	$-0.284477\pi$
$653$	1.36537e7	1.25305	0.626523	+	0.779403i	$0.284477\pi$
$654$	−6.15942e6	−0.563113
$655$	0	0
$656$	−4.44708e6	−0.403474
$657$	−222703.	−0.0201285
$658$	−1.28581e6	−0.115774
$659$	−612348.	−0.0549268	−0.0274634	−	0.999623i	$-0.508743\pi$
$659$	−612348.	−0.0549268	−0.0274634	+	0.999623i	$0.508743\pi$
$660$	0	0
$661$	1.10774e7	0.986129	0.493064	−	0.869993i	$-0.335877\pi$
$661$	1.10774e7	0.986129	0.493064	+	0.869993i	$0.335877\pi$
$662$	−6.31034e6	−0.559639
$663$	−5.77843e6	−0.510535
$664$	−1.92943e6	−0.169828
$665$	0	0
$666$	−63944.3	−0.00558619
$667$	1.87374e6	0.163078
$668$	−5.24932e6	−0.455158
$669$	−1.26412e7	−1.09200
$670$	0	0
$671$	1.21365e7	1.04061
$672$	815697.	0.0696796
$673$	−1.39729e6	−0.118918	−0.0594590	−	0.998231i	$-0.518938\pi$
$673$	−1.39729e6	−0.118918	−0.0594590	+	0.998231i	$0.518938\pi$
$674$	1.21567e6	0.103078
$675$	0	0
$676$	456976.	0.0384615
$677$	−1.49879e7	−1.25681	−0.628406	−	0.777886i	$-0.716292\pi$
$677$	−1.49879e7	−1.25681	−0.628406	+	0.777886i	$0.716292\pi$
$678$	8.00086e6	0.668440
$679$	8.01924e6	0.667512
$680$	0	0
$681$	−1.32509e7	−1.09491
$682$	−233666.	−0.0192369
$683$	5.44672e6	0.446769	0.223384	−	0.974730i	$-0.428289\pi$
$683$	5.44672e6	0.446769	0.223384	+	0.974730i	$0.428289\pi$
$684$	−195733.	−0.0159965
$685$	0	0
$686$	6.52647e6	0.529502
$687$	9.35702e6	0.756390
$688$	2.58259e6	0.208010
$689$	2.73403e6	0.219410
$690$	0	0
$691$	1.20363e7	0.958951	0.479475	−	0.877555i	$-0.340827\pi$
$691$	1.20363e7	0.958951	0.479475	+	0.877555i	$0.340827\pi$
$692$	−992138.	−0.0787602
$693$	−229973.	−0.0181904
$694$	1.31937e7	1.03985
$695$	0	0
$696$	914062.	0.0715241
$697$	−3.94879e7	−3.07881
$698$	−1.42401e6	−0.110630
$699$	−1.64443e7	−1.27299
$700$	0	0
$701$	793102.	0.0609585	0.0304792	−	0.999535i	$-0.490297\pi$
$701$	793102.	0.0609585	0.0304792	+	0.999535i	$0.490297\pi$
$702$	2.64117e6	0.202281
$703$	697002.	0.0531920
$704$	−1.06188e6	−0.0807502
$705$	0	0
$706$	−1.49161e7	−1.12627
$707$	7.68947e6	0.578560
$708$	−78681.4	−0.00589914
$709$	−1.94248e7	−1.45125	−0.725623	−	0.688092i	$-0.758448\pi$
$709$	−1.94248e7	−1.45125	−0.725623	+	0.688092i	$0.758448\pi$
$710$	0	0
$711$	−1.27753e6	−0.0947757
$712$	−4.27334e6	−0.315913
$713$	−444660.	−0.0327570
$714$	7.24299e6	0.531707
$715$	0	0
$716$	1.76363e6	0.128566
$717$	−2.30441e6	−0.167402
$718$	−9.10013e6	−0.658774
$719$	−5.30618e6	−0.382789	−0.191395	−	0.981513i	$-0.561301\pi$
$719$	−5.30618e6	−0.382789	−0.191395	+	0.981513i	$0.561301\pi$
$720$	0	0
$721$	−3.78780e6	−0.271362
$722$	−7.77087e6	−0.554788
$723$	−1.86557e7	−1.32729
$724$	−5.94200e6	−0.421295
$725$	0	0
$726$	5.64611e6	0.397564
$727$	2.30593e7	1.61812	0.809059	−	0.587728i	$-0.199977\pi$
$727$	2.30593e7	1.61812	0.809059	+	0.587728i	$0.199977\pi$
$728$	−572798.	−0.0400565
$729$	1.51969e7	1.05910
$730$	0	0
$731$	2.29321e7	1.58727
$732$	1.12666e7	0.777170
$733$	−1.84481e7	−1.26821	−0.634104	−	0.773248i	$-0.718631\pi$
$733$	−1.84481e7	−1.26821	−0.634104	+	0.773248i	$0.718631\pi$
$734$	−4.85641e6	−0.332717
$735$	0	0
$736$	−2.02072e6	−0.137503
$737$	1.12989e7	0.766242
$738$	1.16392e6	0.0786648
$739$	−6.14417e6	−0.413859	−0.206929	−	0.978356i	$-0.566347\pi$
$739$	−6.14417e6	−0.413859	−0.206929	+	0.978356i	$0.566347\pi$
$740$	0	0
$741$	−1.85652e6	−0.124209
$742$	−3.42698e6	−0.228508
$743$	1.18784e7	0.789383	0.394691	−	0.918814i	$-0.370852\pi$
$743$	1.18784e7	0.789383	0.394691	+	0.918814i	$0.370852\pi$
$744$	−216918.	−0.0143669
$745$	0	0
$746$	6.26931e6	0.412451
$747$	504982.	0.0331112
$748$	−9.42897e6	−0.616183
$749$	4.94402e6	0.322014
$750$	0	0
$751$	−3.38234e6	−0.218835	−0.109418	−	0.993996i	$-0.534899\pi$
$751$	−3.38234e6	−0.218835	−0.109418	+	0.993996i	$0.534899\pi$
$752$	1.55390e6	0.100202
$753$	2.29615e7	1.47575
$754$	−641872.	−0.0411169
$755$	0	0
$756$	−3.31059e6	−0.210669
$757$	2.42823e6	0.154010	0.0770052	−	0.997031i	$-0.475464\pi$
$757$	2.42823e6	0.154010	0.0770052	+	0.997031i	$0.475464\pi$
$758$	1.76129e7	1.11341
$759$	−7.69513e6	−0.484855
$760$	0	0
$761$	−2.18931e7	−1.37040	−0.685199	−	0.728356i	$-0.740285\pi$
$761$	−2.18931e7	−1.37040	−0.685199	+	0.728356i	$0.740285\pi$
$762$	3.24049e6	0.202173
$763$	−5.42152e6	−0.337139
$764$	−1.55782e7	−0.965573
$765$	0	0
$766$	1.38238e7	0.851249
$767$	55251.6	0.00339122
$768$	−985766.	−0.0603074
$769$	−1.96376e7	−1.19749	−0.598747	−	0.800939i	$-0.704334\pi$
$769$	−1.96376e7	−1.19749	−0.598747	+	0.800939i	$0.704334\pi$
$770$	0	0
$771$	1.91816e7	1.16211
$772$	286997.	0.0173314
$773$	−1.21015e7	−0.728433	−0.364216	−	0.931314i	$-0.618663\pi$
$773$	−1.21015e7	−0.728433	−0.364216	+	0.931314i	$0.618663\pi$
$774$	−675928.	−0.0405554
$775$	0	0
$776$	−9.69122e6	−0.577729
$777$	760229.	0.0451743
$778$	2.22648e6	0.131877
$779$	−1.26869e7	−0.749049
$780$	0	0
$781$	−1.34149e7	−0.786973
$782$	−1.79430e7	−1.04925
$783$	−3.70981e6	−0.216246
$784$	−3.58462e6	−0.208282
$785$	0	0
$786$	−7.56983e6	−0.437049
$787$	−2.30778e7	−1.32818	−0.664092	−	0.747651i	$-0.731182\pi$
$787$	−2.30778e7	−1.32818	−0.664092	+	0.747651i	$0.731182\pi$
$788$	−6.34055e6	−0.363757
$789$	1.41699e7	0.810351
$790$	0	0
$791$	7.04235e6	0.400199
$792$	277921.	0.0157438
$793$	−7.91164e6	−0.446769
$794$	902017.	0.0507766
$795$	0	0
$796$	−5.99505e6	−0.335359
$797$	−8.88351e6	−0.495380	−0.247690	−	0.968839i	$-0.579672\pi$
$797$	−8.88351e6	−0.495380	−0.247690	+	0.968839i	$0.579672\pi$
$798$	2.32706e6	0.129360
$799$	1.37978e7	0.764617
$800$	0	0
$801$	1.11844e6	0.0615932
$802$	3.59317e6	0.197261
$803$	−3.44679e6	−0.188637
$804$	1.04890e7	0.572260
$805$	0	0
$806$	152324.	0.00825905
$807$	−1.21248e7	−0.655378
$808$	−9.29270e6	−0.500741
$809$	−1.27959e7	−0.687382	−0.343691	−	0.939083i	$-0.611677\pi$
$809$	−1.27959e7	−0.687382	−0.343691	+	0.939083i	$0.611677\pi$
$810$	0	0
$811$	3.55180e6	0.189626	0.0948128	−	0.995495i	$-0.469775\pi$
$811$	3.55180e6	0.189626	0.0948128	+	0.995495i	$0.469775\pi$
$812$	804557.	0.0428220
$813$	−9.02627e6	−0.478941
$814$	−989671.	−0.0523516
$815$	0	0
$816$	−8.75312e6	−0.460190
$817$	7.36773e6	0.386170
$818$	−7.08744e6	−0.370345
$819$	149916.	0.00780976
$820$	0	0
$821$	−4.71136e6	−0.243943	−0.121971	−	0.992534i	$-0.538922\pi$
$821$	−4.71136e6	−0.243943	−0.121971	+	0.992534i	$0.538922\pi$
$822$	−3.34582e6	−0.172712
$823$	4.15156e6	0.213654	0.106827	−	0.994278i	$-0.465931\pi$
$823$	4.15156e6	0.213654	0.106827	+	0.994278i	$0.465931\pi$
$824$	4.57754e6	0.234863
$825$	0	0
$826$	−69255.3	−0.00353186
$827$	−2.57396e7	−1.30869	−0.654346	−	0.756195i	$-0.727056\pi$
$827$	−2.57396e7	−1.30869	−0.654346	+	0.756195i	$0.727056\pi$
$828$	528875.	0.0268088
$829$	−1.95125e7	−0.986114	−0.493057	−	0.869997i	$-0.664121\pi$
$829$	−1.95125e7	−0.986114	−0.493057	+	0.869997i	$0.664121\pi$
$830$	0	0
$831$	2.44756e7	1.22951
$832$	692224.	0.0346688
$833$	−3.18296e7	−1.58935
$834$	1.20349e7	0.599137
$835$	0	0
$836$	−3.02938e6	−0.149913
$837$	880382.	0.0434368
$838$	2.46804e6	0.121407
$839$	−1.56844e7	−0.769243	−0.384621	−	0.923074i	$-0.625668\pi$
$839$	−1.56844e7	−0.769243	−0.384621	+	0.923074i	$0.625668\pi$
$840$	0	0
$841$	−1.96096e7	−0.956044
$842$	−2.51195e7	−1.22104
$843$	3.43517e6	0.166487
$844$	1.06575e7	0.514988
$845$	0	0
$846$	−406694.	−0.0195363
$847$	4.96970e6	0.238024
$848$	4.14149e6	0.197773
$849$	−1.91635e7	−0.912445
$850$	0	0
$851$	−1.88331e6	−0.0891454
$852$	−1.24534e7	−0.587743
$853$	2.35674e7	1.10902	0.554510	−	0.832177i	$-0.312906\pi$
$853$	2.35674e7	1.10902	0.554510	+	0.832177i	$0.312906\pi$
$854$	9.91687e6	0.465297
$855$	0	0
$856$	−5.97482e6	−0.278702
$857$	−4.23900e6	−0.197157	−0.0985783	−	0.995129i	$-0.531429\pi$
$857$	−4.23900e6	−0.197157	−0.0985783	+	0.995129i	$0.531429\pi$
$858$	2.63606e6	0.122247
$859$	1.30196e7	0.602026	0.301013	−	0.953620i	$-0.402675\pi$
$859$	1.30196e7	0.602026	0.301013	+	0.953620i	$0.402675\pi$
$860$	0	0
$861$	−1.38377e7	−0.636145
$862$	8.12208e6	0.372305
$863$	−3.41837e7	−1.56240	−0.781200	−	0.624280i	$-0.785392\pi$
$863$	−3.41837e7	−1.56240	−0.781200	+	0.624280i	$0.785392\pi$
$864$	4.00083e6	0.182333
$865$	0	0
$866$	5.17640e6	0.234549
$867$	−5.63665e7	−2.54667
$868$	−190931.	−0.00860155
$869$	−1.97724e7	−0.888200
$870$	0	0
$871$	−7.36557e6	−0.328974
$872$	6.55188e6	0.291793
$873$	2.53644e6	0.112639
$874$	−5.76482e6	−0.255274
$875$	0	0
$876$	−3.19973e6	−0.140881
$877$	−3.07617e7	−1.35055	−0.675275	−	0.737566i	$-0.735975\pi$
$877$	−3.07617e7	−1.35055	−0.675275	+	0.737566i	$0.735975\pi$
$878$	−1.01528e7	−0.444476
$879$	−4.27373e7	−1.86567
$880$	0	0
$881$	8.83910e6	0.383679	0.191840	−	0.981426i	$-0.438555\pi$
$881$	8.83910e6	0.383679	0.191840	+	0.981426i	$0.438555\pi$
$882$	938186.	0.0406085
$883$	2.09708e7	0.905136	0.452568	−	0.891730i	$-0.350508\pi$
$883$	2.09708e7	0.905136	0.452568	+	0.891730i	$0.350508\pi$
$884$	6.14661e6	0.264548
$885$	0	0
$886$	−2.51071e7	−1.07452
$887$	−3.98177e7	−1.69929	−0.849644	−	0.527357i	$-0.823183\pi$
$887$	−3.98177e7	−1.69929	−0.849644	+	0.527357i	$0.823183\pi$
$888$	−918733.	−0.0390982
$889$	2.85228e6	0.121042
$890$	0	0
$891$	1.41804e7	0.598402
$892$	1.34466e7	0.565851
$893$	4.43303e6	0.186025
$894$	7.58138e6	0.317252
$895$	0	0
$896$	−867671.	−0.0361065
$897$	5.01635e6	0.208165
$898$	−7.64745e6	−0.316465
$899$	−213955.	−0.00882924
$900$	0	0
$901$	3.67744e7	1.50915
$902$	1.80140e7	0.737215
$903$	8.03607e6	0.327962
$904$	−8.51065e6	−0.346371
$905$	0	0
$906$	−813719.	−0.0329347
$907$	−3.42787e6	−0.138358	−0.0691792	−	0.997604i	$-0.522038\pi$
$907$	−3.42787e6	−0.138358	−0.0691792	+	0.997604i	$0.522038\pi$
$908$	1.40953e7	0.567359
$909$	2.43214e6	0.0976288
$910$	0	0
$911$	−3.60531e7	−1.43929	−0.719643	−	0.694344i	$-0.755695\pi$
$911$	−3.60531e7	−1.43929	−0.719643	+	0.694344i	$0.755695\pi$
$912$	−2.81224e6	−0.111961
$913$	7.81565e6	0.310305
$914$	−3.13463e7	−1.24114
$915$	0	0
$916$	−9.95322e6	−0.391945
$917$	−6.66296e6	−0.261664
$918$	3.55254e7	1.39134
$919$	−2.96521e7	−1.15816	−0.579078	−	0.815272i	$-0.696587\pi$
$919$	−2.96521e7	−1.15816	−0.579078	+	0.815272i	$0.696587\pi$
$920$	0	0
$921$	−6.81182e6	−0.264615
$922$	−1.58278e7	−0.613187
$923$	8.74499e6	0.337874
$924$	−3.30418e6	−0.127316
$925$	0	0
$926$	−1.08246e7	−0.414843
$927$	−1.19806e6	−0.0457909
$928$	−972303.	−0.0370622
$929$	−1.01663e7	−0.386478	−0.193239	−	0.981152i	$-0.561899\pi$
$929$	−1.01663e7	−0.386478	−0.193239	+	0.981152i	$0.561899\pi$
$930$	0	0
$931$	−1.02264e7	−0.386676
$932$	1.74921e7	0.659633
$933$	2.15679e7	0.811154
$934$	2.05223e7	0.769768
$935$	0	0
$936$	−181173.	−0.00675932
$937$	3.22787e7	1.20107	0.600534	−	0.799599i	$-0.294955\pi$
$937$	3.22787e7	1.20107	0.600534	+	0.799599i	$0.294955\pi$
$938$	9.23240e6	0.342616
$939$	−2.48608e6	−0.0920135
$940$	0	0
$941$	2.53184e7	0.932100	0.466050	−	0.884758i	$-0.345677\pi$
$941$	2.53184e7	0.932100	0.466050	+	0.884758i	$0.345677\pi$
$942$	2.54470e7	0.934348
$943$	3.42801e7	1.25535
$944$	83694.7	0.00305681
$945$	0	0
$946$	−1.04614e7	−0.380069
$947$	−4.70863e7	−1.70616	−0.853080	−	0.521780i	$-0.825268\pi$
$947$	−4.70863e7	−1.70616	−0.853080	+	0.521780i	$0.825268\pi$
$948$	−1.83552e7	−0.663343
$949$	2.24691e6	0.0809880
$950$	0	0
$951$	−2.31009e7	−0.828282
$952$	−7.70449e6	−0.275519
$953$	−6.95299e6	−0.247993	−0.123997	−	0.992283i	$-0.539571\pi$
$953$	−6.95299e6	−0.247993	−0.123997	+	0.992283i	$0.539571\pi$
$954$	−1.08393e6	−0.0385595
$955$	0	0
$956$	2.45124e6	0.0867442
$957$	−3.70263e6	−0.130687
$958$	8.88137e6	0.312656
$959$	−2.94499e6	−0.103404
$960$	0	0
$961$	−2.85784e7	−0.998226
$962$	645152.	0.0224763
$963$	1.56376e6	0.0543382
$964$	1.98443e7	0.687771
$965$	0	0
$966$	−6.28776e6	−0.216797
$967$	4.00072e7	1.37585	0.687927	−	0.725780i	$-0.258521\pi$
$967$	4.00072e7	1.37585	0.687927	+	0.725780i	$0.258521\pi$
$968$	−6.00586e6	−0.206009
$969$	−2.49713e7	−0.854343
$970$	0	0
$971$	5.12102e7	1.74305	0.871523	−	0.490355i	$-0.163133\pi$
$971$	5.12102e7	1.74305	0.871523	+	0.490355i	$0.163133\pi$
$972$	−2.02671e6	−0.0688061
$973$	1.05931e7	0.358707
$974$	2.00524e7	0.677283
$975$	0	0
$976$	−1.19845e7	−0.402713
$977$	−7.74195e6	−0.259486	−0.129743	−	0.991548i	$-0.541415\pi$
$977$	−7.74195e6	−0.259486	−0.129743	+	0.991548i	$0.541415\pi$
$978$	4.02339e7	1.34507
$979$	1.73102e7	0.577226
$980$	0	0
$981$	−1.71479e6	−0.0568905
$982$	−1.51638e7	−0.501800
$983$	−2.28293e7	−0.753544	−0.376772	−	0.926306i	$-0.622966\pi$
$983$	−2.28293e7	−0.753544	−0.376772	+	0.926306i	$0.622966\pi$
$984$	1.67228e7	0.550581
$985$	0	0
$986$	−8.63357e6	−0.282812
$987$	4.83516e6	0.157986
$988$	1.97481e6	0.0643625
$989$	−1.99077e7	−0.647189
$990$	0	0
$991$	3.12601e7	1.01113	0.505564	−	0.862789i	$-0.331284\pi$
$991$	3.12601e7	1.01113	0.505564	+	0.862789i	$0.331284\pi$
$992$	230739.	0.00744461
$993$	2.37294e7	0.763684
$994$	−1.09614e7	−0.351885
$995$	0	0
$996$	7.25544e6	0.231748
$997$	−1.09102e7	−0.347611	−0.173805	−	0.984780i	$-0.555606\pi$
$997$	−1.09102e7	−0.347611	−0.173805	+	0.984780i	$0.555606\pi$
$998$	3.03779e7	0.965454
$999$	3.72877e6	0.118209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.a.f.1.1		2
5.2	odd	4		650.6.b.f.599.4		4
5.3	odd	4		650.6.b.f.599.1		4
5.4	even	2		130.6.a.c.1.2	✓	2
20.19	odd	2		1040.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.c.1.2	✓	2	5.4	even	2
650.6.a.f.1.1		2	1.1	even	1	trivial
650.6.b.f.599.1		4	5.3	odd	4
650.6.b.f.599.4		4	5.2	odd	4
1040.6.a.c.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -15.0416 q^{3} +16.0000 q^{4} -60.1664 q^{6} -52.9584 q^{7} +64.0000 q^{8} -16.7504 q^{9} +O(q^{10})\)	\(q+4.00000 q^{2} -15.0416 q^{3} +16.0000 q^{4} -60.1664 q^{6} -52.9584 q^{7} +64.0000 q^{8} -16.7504 q^{9} -259.248 q^{11} -240.666 q^{12} +169.000 q^{13} -211.834 q^{14} +256.000 q^{16} +2273.15 q^{17} -67.0017 q^{18} +730.329 q^{19} +796.579 q^{21} -1036.99 q^{22} -1973.36 q^{23} -962.662 q^{24} +676.000 q^{26} +3907.06 q^{27} -847.334 q^{28} -949.515 q^{29} +225.331 q^{31} +1024.00 q^{32} +3899.50 q^{33} +9092.62 q^{34} -268.007 q^{36} +954.367 q^{37} +2921.32 q^{38} -2542.03 q^{39} -17371.4 q^{41} +3186.32 q^{42} +10088.2 q^{43} -4147.97 q^{44} -7893.46 q^{46} +6069.90 q^{47} -3850.65 q^{48} -14002.4 q^{49} -34191.9 q^{51} +2704.00 q^{52} +16177.7 q^{53} +15628.2 q^{54} -3389.34 q^{56} -10985.3 q^{57} -3798.06 q^{58} +326.933 q^{59} -46814.4 q^{61} +901.324 q^{62} +887.076 q^{63} +4096.00 q^{64} +15598.0 q^{66} -43583.3 q^{67} +36370.5 q^{68} +29682.5 q^{69} +51745.5 q^{71} -1072.03 q^{72} +13295.3 q^{73} +3817.47 q^{74} +11685.3 q^{76} +13729.4 q^{77} -10168.1 q^{78} +76268.5 q^{79} -54698.1 q^{81} -69485.7 q^{82} -30147.4 q^{83} +12745.3 q^{84} +40352.9 q^{86} +14282.2 q^{87} -16591.9 q^{88} -66771.0 q^{89} -8949.97 q^{91} -31573.8 q^{92} -3389.34 q^{93} +24279.6 q^{94} -15402.6 q^{96} -151425. q^{97} -56009.6 q^{98} +4342.51 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} - 6 q^{3} + 32 q^{4} - 24 q^{6} - 130 q^{7} + 128 q^{8} - 178 q^{9}+O(q^{10}) \) 2 * q + 8 * q^2 - 6 * q^3 + 32 * q^4 - 24 * q^6 - 130 * q^7 + 128 * q^8 - 178 * q^9	\( 2 q + 8 q^{2} - 6 q^{3} + 32 q^{4} - 24 q^{6} - 130 q^{7} + 128 q^{8} - 178 q^{9} + 204 q^{11} - 96 q^{12} + 338 q^{13} - 520 q^{14} + 512 q^{16} + 404 q^{17} - 712 q^{18} + 112 q^{19} + 100 q^{21} + 816 q^{22} - 262 q^{23} - 384 q^{24} + 1352 q^{26} + 252 q^{27} - 2080 q^{28} - 6812 q^{29} - 320 q^{31} + 2048 q^{32} + 8088 q^{33} + 1616 q^{34} - 2848 q^{36} + 13276 q^{37} + 448 q^{38} - 1014 q^{39} - 35080 q^{41} + 400 q^{42} - 2534 q^{43} + 3264 q^{44} - 1048 q^{46} + 22038 q^{47} - 1536 q^{48} - 24874 q^{49} - 51092 q^{51} + 5408 q^{52} + 44108 q^{53} + 1008 q^{54} - 8320 q^{56} - 16576 q^{57} - 27248 q^{58} - 21888 q^{59} - 41272 q^{61} - 1280 q^{62} + 13310 q^{63} + 8192 q^{64} + 32352 q^{66} - 103230 q^{67} + 6464 q^{68} + 45156 q^{69} - 15480 q^{71} - 11392 q^{72} - 54088 q^{73} + 53104 q^{74} + 1792 q^{76} - 21960 q^{77} - 4056 q^{78} + 27208 q^{79} - 48562 q^{81} - 140320 q^{82} - 42690 q^{83} + 1600 q^{84} - 10136 q^{86} - 38724 q^{87} + 13056 q^{88} - 68132 q^{89} - 21970 q^{91} - 4192 q^{92} - 8320 q^{93} + 88152 q^{94} - 6144 q^{96} - 209456 q^{97} - 99496 q^{98} - 70356 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 - 6 * q^3 + 32 * q^4 - 24 * q^6 - 130 * q^7 + 128 * q^8 - 178 * q^9 + 204 * q^11 - 96 * q^12 + 338 * q^13 - 520 * q^14 + 512 * q^16 + 404 * q^17 - 712 * q^18 + 112 * q^19 + 100 * q^21 + 816 * q^22 - 262 * q^23 - 384 * q^24 + 1352 * q^26 + 252 * q^27 - 2080 * q^28 - 6812 * q^29 - 320 * q^31 + 2048 * q^32 + 8088 * q^33 + 1616 * q^34 - 2848 * q^36 + 13276 * q^37 + 448 * q^38 - 1014 * q^39 - 35080 * q^41 + 400 * q^42 - 2534 * q^43 + 3264 * q^44 - 1048 * q^46 + 22038 * q^47 - 1536 * q^48 - 24874 * q^49 - 51092 * q^51 + 5408 * q^52 + 44108 * q^53 + 1008 * q^54 - 8320 * q^56 - 16576 * q^57 - 27248 * q^58 - 21888 * q^59 - 41272 * q^61 - 1280 * q^62 + 13310 * q^63 + 8192 * q^64 + 32352 * q^66 - 103230 * q^67 + 6464 * q^68 + 45156 * q^69 - 15480 * q^71 - 11392 * q^72 - 54088 * q^73 + 53104 * q^74 + 1792 * q^76 - 21960 * q^77 - 4056 * q^78 + 27208 * q^79 - 48562 * q^81 - 140320 * q^82 - 42690 * q^83 + 1600 * q^84 - 10136 * q^86 - 38724 * q^87 + 13056 * q^88 - 68132 * q^89 - 21970 * q^91 - 4192 * q^92 - 8320 * q^93 + 88152 * q^94 - 6144 * q^96 - 209456 * q^97 - 99496 * q^98 - 70356 * q^99

Introduction

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