LMFDB - Embedded newform 1075.6.a.k.1.36

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:	$52$
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.36
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)

$f(q)$	$=$	$q+4.61181 q^{2} -26.3914 q^{3} -10.7312 q^{4} -121.712 q^{6} -179.304 q^{7} -197.068 q^{8} +453.507 q^{9} +O(q^{10})$	\(q+4.61181 q^{2} -26.3914 q^{3} -10.7312 q^{4} -121.712 q^{6} -179.304 q^{7} -197.068 q^{8} +453.507 q^{9} -605.380 q^{11} +283.212 q^{12} -90.2912 q^{13} -826.915 q^{14} -565.441 q^{16} -986.422 q^{17} +2091.49 q^{18} +424.242 q^{19} +4732.09 q^{21} -2791.90 q^{22} +4051.21 q^{23} +5200.91 q^{24} -416.406 q^{26} -5555.59 q^{27} +1924.15 q^{28} -2811.81 q^{29} +3674.76 q^{31} +3698.48 q^{32} +15976.8 q^{33} -4549.19 q^{34} -4866.69 q^{36} -9300.00 q^{37} +1956.52 q^{38} +2382.91 q^{39} +16617.4 q^{41} +21823.5 q^{42} +1849.00 q^{43} +6496.47 q^{44} +18683.4 q^{46} -5790.02 q^{47} +14922.8 q^{48} +15342.9 q^{49} +26033.1 q^{51} +968.936 q^{52} -19222.5 q^{53} -25621.3 q^{54} +35335.1 q^{56} -11196.3 q^{57} -12967.5 q^{58} +34858.4 q^{59} -55423.2 q^{61} +16947.3 q^{62} -81315.7 q^{63} +35150.8 q^{64} +73682.1 q^{66} -32520.3 q^{67} +10585.5 q^{68} -106917. q^{69} +66019.3 q^{71} -89371.9 q^{72} +47388.4 q^{73} -42889.8 q^{74} -4552.64 q^{76} +108547. q^{77} +10989.5 q^{78} +51948.6 q^{79} +36417.7 q^{81} +76636.5 q^{82} +32767.2 q^{83} -50781.1 q^{84} +8527.23 q^{86} +74207.7 q^{87} +119301. q^{88} +117907. q^{89} +16189.6 q^{91} -43474.5 q^{92} -96982.1 q^{93} -26702.5 q^{94} -97608.1 q^{96} +61287.5 q^{97} +70758.5 q^{98} -274544. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q - 20 q^{2} - 54 q^{3} + 826 q^{4} - 162 q^{6} - 196 q^{7} - 960 q^{8} + 4098 q^{9}+O(q^{10}) $ 52 * q - 20 * q^2 - 54 * q^3 + 826 * q^4 - 162 * q^6 - 196 * q^7 - 960 * q^8 + 4098 * q^9	\( 52 q - 20 q^{2} - 54 q^{3} + 826 q^{4} - 162 q^{6} - 196 q^{7} - 960 q^{8} + 4098 q^{9} - 664 q^{11} + 523 q^{12} - 2704 q^{13} + 150 q^{14} + 13474 q^{16} - 7266 q^{17} - 4860 q^{18} - 1970 q^{19} + 800 q^{21} - 14477 q^{22} - 9522 q^{23} + 314 q^{24} + 5514 q^{26} - 22926 q^{27} - 9408 q^{28} - 7188 q^{29} - 11556 q^{31} - 48390 q^{32} - 26136 q^{33} + 16774 q^{34} + 51872 q^{36} - 42558 q^{37} - 46208 q^{38} + 4682 q^{39} - 7746 q^{41} - 174265 q^{42} + 96148 q^{43} - 48600 q^{44} + 16182 q^{46} - 87136 q^{47} + 2912 q^{48} + 142286 q^{49} - 3710 q^{51} - 146868 q^{52} - 127034 q^{53} - 49563 q^{54} - 2849 q^{56} - 101594 q^{57} - 9480 q^{58} - 55924 q^{59} + 73702 q^{61} - 186016 q^{62} - 50120 q^{63} + 157750 q^{64} + 58211 q^{66} - 131996 q^{67} - 298560 q^{68} + 128436 q^{69} - 56284 q^{71} - 343775 q^{72} - 128620 q^{73} - 17721 q^{74} - 170410 q^{76} - 448438 q^{77} - 237616 q^{78} + 106204 q^{79} + 478568 q^{81} - 249596 q^{82} - 348616 q^{83} - 131855 q^{84} - 36980 q^{86} - 267478 q^{87} - 525216 q^{88} + 80410 q^{89} + 226376 q^{91} - 581456 q^{92} - 902902 q^{93} + 180980 q^{94} + 38543 q^{96} - 316148 q^{97} - 295095 q^{98} + 68428 q^{99}+O(q^{100}) \) 52 * q - 20 * q^2 - 54 * q^3 + 826 * q^4 - 162 * q^6 - 196 * q^7 - 960 * q^8 + 4098 * q^9 - 664 * q^11 + 523 * q^12 - 2704 * q^13 + 150 * q^14 + 13474 * q^16 - 7266 * q^17 - 4860 * q^18 - 1970 * q^19 + 800 * q^21 - 14477 * q^22 - 9522 * q^23 + 314 * q^24 + 5514 * q^26 - 22926 * q^27 - 9408 * q^28 - 7188 * q^29 - 11556 * q^31 - 48390 * q^32 - 26136 * q^33 + 16774 * q^34 + 51872 * q^36 - 42558 * q^37 - 46208 * q^38 + 4682 * q^39 - 7746 * q^41 - 174265 * q^42 + 96148 * q^43 - 48600 * q^44 + 16182 * q^46 - 87136 * q^47 + 2912 * q^48 + 142286 * q^49 - 3710 * q^51 - 146868 * q^52 - 127034 * q^53 - 49563 * q^54 - 2849 * q^56 - 101594 * q^57 - 9480 * q^58 - 55924 * q^59 + 73702 * q^61 - 186016 * q^62 - 50120 * q^63 + 157750 * q^64 + 58211 * q^66 - 131996 * q^67 - 298560 * q^68 + 128436 * q^69 - 56284 * q^71 - 343775 * q^72 - 128620 * q^73 - 17721 * q^74 - 170410 * q^76 - 448438 * q^77 - 237616 * q^78 + 106204 * q^79 + 478568 * q^81 - 249596 * q^82 - 348616 * q^83 - 131855 * q^84 - 36980 * q^86 - 267478 * q^87 - 525216 * q^88 + 80410 * q^89 + 226376 * q^91 - 581456 * q^92 - 902902 * q^93 + 180980 * q^94 + 38543 * q^96 - 316148 * q^97 - 295095 * q^98 + 68428 * 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 $

$2$	4.61181	0.815260	0.407630	−	0.913147i	$-0.366355\pi$
$2$	4.61181	0.815260	0.407630	+	0.913147i	$0.366355\pi$
$3$	−26.3914	−1.69301	−0.846505	−	0.532380i	$-0.821298\pi$
$3$	−26.3914	−1.69301	−0.846505	+	0.532380i	$0.821298\pi$
$4$	−10.7312	−0.335351
$5$	0	0
$5$	0	0
$6$	−121.712	−1.38024
$7$	−179.304	−1.38307	−0.691536	−	0.722342i	$-0.743066\pi$
$7$	−179.304	−1.38307	−0.691536	+	0.722342i	$0.743066\pi$
$8$	−197.068	−1.08866
$9$	453.507	1.86629
$10$	0	0
$11$	−605.380	−1.50850	−0.754252	−	0.656586i	$-0.772000\pi$
$11$	−605.380	−1.50850	−0.754252	+	0.656586i	$0.772000\pi$
$12$	283.212	0.567753
$13$	−90.2912	−0.148179	−0.0740896	−	0.997252i	$-0.523605\pi$
$13$	−90.2912	−0.148179	−0.0740896	+	0.997252i	$0.523605\pi$
$14$	−826.915	−1.12756
$15$	0	0
$16$	−565.441	−0.552189
$17$	−986.422	−0.827829	−0.413914	−	0.910316i	$-0.635839\pi$
$17$	−986.422	−0.827829	−0.413914	+	0.910316i	$0.635839\pi$
$18$	2091.49	1.52151
$19$	424.242	0.269606	0.134803	−	0.990872i	$-0.456960\pi$
$19$	424.242	0.269606	0.134803	+	0.990872i	$0.456960\pi$
$20$	0	0
$21$	4732.09	2.34156
$22$	−2791.90	−1.22982
$23$	4051.21	1.59685	0.798427	−	0.602091i	$-0.205666\pi$
$23$	4051.21	1.59685	0.798427	+	0.602091i	$0.205666\pi$
$24$	5200.91	1.84311
$25$	0	0
$26$	−416.406	−0.120805
$27$	−5555.59	−1.46663
$28$	1924.15	0.463814
$29$	−2811.81	−0.620856	−0.310428	−	0.950597i	$-0.600472\pi$
$29$	−2811.81	−0.620856	−0.310428	+	0.950597i	$0.600472\pi$
$30$	0	0
$31$	3674.76	0.686791	0.343395	−	0.939191i	$-0.388423\pi$
$31$	3674.76	0.686791	0.343395	+	0.939191i	$0.388423\pi$
$32$	3698.48	0.638481
$33$	15976.8	2.55391
$34$	−4549.19	−0.674896
$35$	0	0
$36$	−4866.69	−0.625861
$37$	−9300.00	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$37$	−9300.00	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$38$	1956.52	0.219799
$39$	2382.91	0.250869
$40$	0	0
$41$	16617.4	1.54385	0.771924	−	0.635715i	$-0.219294\pi$
$41$	16617.4	1.54385	0.771924	+	0.635715i	$0.219294\pi$
$42$	21823.5	1.90898
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	6496.47	0.505878
$45$	0	0
$46$	18683.4	1.30185
$47$	−5790.02	−0.382328	−0.191164	−	0.981558i	$-0.561226\pi$
$47$	−5790.02	−0.382328	−0.191164	+	0.981558i	$0.561226\pi$
$48$	14922.8	0.934862
$49$	15342.9	0.912887
$50$	0	0
$51$	26033.1	1.40152
$52$	968.936	0.0496920
$53$	−19222.5	−0.939984	−0.469992	−	0.882671i	$-0.655743\pi$
$53$	−19222.5	−0.939984	−0.469992	+	0.882671i	$0.655743\pi$
$54$	−25621.3	−1.19569
$55$	0	0
$56$	35335.1	1.50569
$57$	−11196.3	−0.456446
$58$	−12967.5	−0.506159
$59$	34858.4	1.30370	0.651849	−	0.758348i	$-0.273993\pi$
$59$	34858.4	1.30370	0.651849	+	0.758348i	$0.273993\pi$
$60$	0	0
$61$	−55423.2	−1.90707	−0.953537	−	0.301277i	$-0.902587\pi$
$61$	−55423.2	−1.90707	−0.953537	+	0.301277i	$0.902587\pi$
$62$	16947.3	0.559913
$63$	−81315.7	−2.58121
$64$	35150.8	1.07272
$65$	0	0
$66$	73682.1	2.08210
$67$	−32520.3	−0.885050	−0.442525	−	0.896756i	$-0.645917\pi$
$67$	−32520.3	−0.885050	−0.442525	+	0.896756i	$0.645917\pi$
$68$	10585.5	0.277613
$69$	−106917.	−2.70349
$70$	0	0
$71$	66019.3	1.55427	0.777133	−	0.629337i	$-0.216673\pi$
$71$	66019.3	1.55427	0.777133	+	0.629337i	$0.216673\pi$
$72$	−89371.9	−2.03175
$73$	47388.4	1.04079	0.520397	−	0.853924i	$-0.325784\pi$
$73$	47388.4	1.04079	0.520397	+	0.853924i	$0.325784\pi$
$74$	−42889.8	−0.910489
$75$	0	0
$76$	−4552.64	−0.0904126
$77$	108547.	2.08637
$78$	10989.5	0.204523
$79$	51948.6	0.936497	0.468248	−	0.883597i	$-0.344885\pi$
$79$	51948.6	0.936497	0.468248	+	0.883597i	$0.344885\pi$
$80$	0	0
$81$	36417.7	0.616737
$82$	76636.5	1.25864
$83$	32767.2	0.522088	0.261044	−	0.965327i	$-0.415933\pi$
$83$	32767.2	0.522088	0.261044	+	0.965327i	$0.415933\pi$
$84$	−50781.1	−0.785243
$85$	0	0
$86$	8527.23	0.124326
$87$	74207.7	1.05112
$88$	119301.	1.64224
$89$	117907.	1.57785	0.788926	−	0.614489i	$-0.210638\pi$
$89$	117907.	1.57785	0.788926	+	0.614489i	$0.210638\pi$
$90$	0	0
$91$	16189.6	0.204942
$92$	−43474.5	−0.535507
$93$	−96982.1	−1.16274
$94$	−26702.5	−0.311696
$95$	0	0
$96$	−97608.1	−1.08095
$97$	61287.5	0.661367	0.330684	−	0.943742i	$-0.392721\pi$
$97$	61287.5	0.661367	0.330684	+	0.943742i	$0.392721\pi$
$98$	70758.5	0.744240
$99$	−274544.	−2.81530
$100$	0	0
$101$	6816.89	0.0664940	0.0332470	−	0.999447i	$-0.489415\pi$
$101$	6816.89	0.0664940	0.0332470	+	0.999447i	$0.489415\pi$
$102$	120060.	1.14261
$103$	75455.0	0.700801	0.350401	−	0.936600i	$-0.386045\pi$
$103$	75455.0	0.700801	0.350401	+	0.936600i	$0.386045\pi$
$104$	17793.5	0.161316
$105$	0	0
$106$	−88650.6	−0.766332
$107$	−14834.4	−0.125259	−0.0626295	−	0.998037i	$-0.519949\pi$
$107$	−14834.4	−0.125259	−0.0626295	+	0.998037i	$0.519949\pi$
$108$	59618.3	0.491836
$109$	36666.4	0.295598	0.147799	−	0.989017i	$-0.452781\pi$
$109$	36666.4	0.295598	0.147799	+	0.989017i	$0.452781\pi$
$110$	0	0
$111$	245440.	1.89077
$112$	101386.	0.763717
$113$	−89722.0	−0.661002	−0.330501	−	0.943806i	$-0.607218\pi$
$113$	−89722.0	−0.661002	−0.330501	+	0.943806i	$0.607218\pi$
$114$	−51635.4	−0.372122
$115$	0	0
$116$	30174.2	0.208205
$117$	−40947.7	−0.276545
$118$	160760.	1.06285
$119$	176869.	1.14495
$120$	0	0
$121$	205434.	1.27558
$122$	−255601.	−1.55476
$123$	−438558.	−2.61375
$124$	−39434.7	−0.230316
$125$	0	0
$126$	−375012.	−2.10435
$127$	−227853.	−1.25356	−0.626781	−	0.779196i	$-0.715628\pi$
$127$	−227853.	−1.25356	−0.626781	+	0.779196i	$0.715628\pi$
$128$	43757.4	0.236062
$129$	−48797.7	−0.258182
$130$	0	0
$131$	51474.7	0.262069	0.131034	−	0.991378i	$-0.458170\pi$
$131$	51474.7	0.262069	0.131034	+	0.991378i	$0.458170\pi$
$132$	−171451.	−0.856457
$133$	−76068.2	−0.372884
$134$	−149977.	−0.721546
$135$	0	0
$136$	194392.	0.901222
$137$	77000.1	0.350501	0.175251	−	0.984524i	$-0.443926\pi$
$137$	77000.1	0.350501	0.175251	+	0.984524i	$0.443926\pi$
$138$	−493082.	−2.20405
$139$	−308544.	−1.35450	−0.677252	−	0.735751i	$-0.736829\pi$
$139$	−308544.	−1.35450	−0.677252	+	0.735751i	$0.736829\pi$
$140$	0	0
$141$	152807.	0.647285
$142$	304468.	1.26713
$143$	54660.5	0.223529
$144$	−256432.	−1.03054
$145$	0	0
$146$	218546.	0.848518
$147$	−404921.	−1.54553
$148$	99800.4	0.374523
$149$	95203.0	0.351306	0.175653	−	0.984452i	$-0.443796\pi$
$149$	95203.0	0.351306	0.175653	+	0.984452i	$0.443796\pi$
$150$	0	0
$151$	−329766.	−1.17697	−0.588483	−	0.808510i	$-0.700275\pi$
$151$	−329766.	−1.17697	−0.588483	+	0.808510i	$0.700275\pi$
$152$	−83604.6	−0.293509
$153$	−447350.	−1.54496
$154$	500598.	1.70093
$155$	0	0
$156$	−25571.6	−0.0841291
$157$	−48024.4	−0.155494	−0.0777469	−	0.996973i	$-0.524773\pi$
$157$	−48024.4	−0.155494	−0.0777469	+	0.996973i	$0.524773\pi$
$158$	239577.	0.763489
$159$	507310.	1.59140
$160$	0	0
$161$	−726398.	−2.20856
$162$	167951.	0.502801
$163$	−388807.	−1.14621	−0.573107	−	0.819481i	$-0.694262\pi$
$163$	−388807.	−1.14621	−0.573107	+	0.819481i	$0.694262\pi$
$164$	−178326.	−0.517731
$165$	0	0
$166$	151116.	0.425638
$167$	−185373.	−0.514347	−0.257173	−	0.966365i	$-0.582791\pi$
$167$	−185373.	−0.514347	−0.257173	+	0.966365i	$0.582791\pi$
$168$	−932544.	−2.54915
$169$	−363140.	−0.978043
$170$	0	0
$171$	192397.	0.503162
$172$	−19842.0	−0.0511405
$173$	−722256.	−1.83475	−0.917373	−	0.398028i	$-0.869695\pi$
$173$	−722256.	−1.83475	−0.917373	+	0.398028i	$0.869695\pi$
$174$	342232.	0.856933
$175$	0	0
$176$	342307.	0.832979
$177$	−919963.	−2.20718
$178$	543766.	1.28636
$179$	519960.	1.21293	0.606467	−	0.795108i	$-0.292586\pi$
$179$	519960.	1.21293	0.606467	+	0.795108i	$0.292586\pi$
$180$	0	0
$181$	−541702.	−1.22903	−0.614517	−	0.788903i	$-0.710649\pi$
$181$	−541702.	−1.22903	−0.614517	+	0.788903i	$0.710649\pi$
$182$	74663.2	0.167081
$183$	1.46270e6	3.22870
$184$	−798365.	−1.73843
$185$	0	0
$186$	−447263.	−0.947939
$187$	597160.	1.24878
$188$	62134.1	0.128214
$189$	996139.	2.02846
$190$	0	0
$191$	150569.	0.298644	0.149322	−	0.988789i	$-0.452291\pi$
$191$	150569.	0.298644	0.149322	+	0.988789i	$0.452291\pi$
$192$	−927679.	−1.81612
$193$	741423.	1.43276	0.716379	−	0.697711i	$-0.245798\pi$
$193$	741423.	1.43276	0.716379	+	0.697711i	$0.245798\pi$
$194$	282646.	0.539187
$195$	0	0
$196$	−164648.	−0.306138
$197$	−349332.	−0.641318	−0.320659	−	0.947195i	$-0.603904\pi$
$197$	−349332.	−0.641318	−0.320659	+	0.947195i	$0.603904\pi$
$198$	−1.26615e6	−2.29520
$199$	748460.	1.33979	0.669893	−	0.742457i	$-0.266340\pi$
$199$	748460.	1.33979	0.669893	+	0.742457i	$0.266340\pi$
$200$	0	0
$201$	858257.	1.49840
$202$	31438.2	0.0542099
$203$	504169.	0.858689
$204$	−279367.	−0.470002
$205$	0	0
$206$	347984.	0.571335
$207$	1.83725e6	2.98019
$208$	51054.4	0.0818229
$209$	−256827.	−0.406701
$210$	0	0
$211$	−72893.7	−0.112715	−0.0563577	−	0.998411i	$-0.517949\pi$
$211$	−72893.7	−0.112715	−0.0563577	+	0.998411i	$0.517949\pi$
$212$	206281.	0.315225
$213$	−1.74234e6	−2.63139
$214$	−68413.2	−0.102119
$215$	0	0
$216$	1.09483e6	1.59666
$217$	−658899.	−0.949881
$218$	169098.	0.240990
$219$	−1.25065e6	−1.76208
$220$	0	0
$221$	89065.2	0.122667
$222$	1.13192e6	1.54147
$223$	−214743.	−0.289172	−0.144586	−	0.989492i	$-0.546185\pi$
$223$	−214743.	−0.289172	−0.144586	+	0.989492i	$0.546185\pi$
$224$	−663151.	−0.883065
$225$	0	0
$226$	−413781.	−0.538889
$227$	−692510.	−0.891993	−0.445997	−	0.895035i	$-0.647151\pi$
$227$	−692510.	−0.891993	−0.445997	+	0.895035i	$0.647151\pi$
$228$	120151.	0.153070
$229$	370800.	0.467251	0.233626	−	0.972327i	$-0.424941\pi$
$229$	370800.	0.467251	0.233626	+	0.972327i	$0.424941\pi$
$230$	0	0
$231$	−2.86471e6	−3.53224
$232$	554119.	0.675900
$233$	−20287.2	−0.0244811	−0.0122406	−	0.999925i	$-0.503896\pi$
$233$	−20287.2	−0.0244811	−0.0122406	+	0.999925i	$0.503896\pi$
$234$	−188843.	−0.225456
$235$	0	0
$236$	−374073.	−0.437197
$237$	−1.37100e6	−1.58550
$238$	815687.	0.933429
$239$	−1.36377e6	−1.54435	−0.772174	−	0.635412i	$-0.780830\pi$
$239$	−1.36377e6	−1.54435	−0.772174	+	0.635412i	$0.780830\pi$
$240$	0	0
$241$	1.54617e6	1.71481	0.857403	−	0.514646i	$-0.172076\pi$
$241$	1.54617e6	1.71481	0.857403	+	0.514646i	$0.172076\pi$
$242$	947421.	1.03993
$243$	388894.	0.422489
$244$	594760.	0.639539
$245$	0	0
$246$	−2.02255e6	−2.13089
$247$	−38305.3	−0.0399500
$248$	−724178.	−0.747681
$249$	−864773.	−0.883901
$250$	0	0
$251$	−4262.58	−0.00427060	−0.00213530	−	0.999998i	$-0.500680\pi$
$251$	−4262.58	−0.00427060	−0.00213530	+	0.999998i	$0.500680\pi$
$252$	872617.	0.865610
$253$	−2.45252e6	−2.40886
$254$	−1.05081e6	−1.02198
$255$	0	0
$256$	−923024.	−0.880265
$257$	1.46404e6	1.38267	0.691335	−	0.722534i	$-0.257023\pi$
$257$	1.46404e6	1.38267	0.691335	+	0.722534i	$0.257023\pi$
$258$	−225046.	−0.210485
$259$	1.66753e6	1.54463
$260$	0	0
$261$	−1.27518e6	−1.15870
$262$	237391.	0.213654
$263$	1.37607e6	1.22673	0.613367	−	0.789798i	$-0.289815\pi$
$263$	1.37607e6	1.22673	0.613367	+	0.789798i	$0.289815\pi$
$264$	−3.14853e6	−2.78034
$265$	0	0
$266$	−350812.	−0.303998
$267$	−3.11174e6	−2.67132
$268$	348983.	0.296802
$269$	−898408.	−0.756995	−0.378498	−	0.925602i	$-0.623559\pi$
$269$	−898408.	−0.756995	−0.378498	+	0.925602i	$0.623559\pi$
$270$	0	0
$271$	−723701.	−0.598599	−0.299300	−	0.954159i	$-0.596753\pi$
$271$	−723701.	−0.598599	−0.299300	+	0.954159i	$0.596753\pi$
$272$	557764.	0.457118
$273$	−427266.	−0.346970
$274$	355109.	0.285750
$275$	0	0
$276$	1.14735e6	0.906618
$277$	952147.	0.745598	0.372799	−	0.927912i	$-0.378398\pi$
$277$	952147.	0.745598	0.372799	+	0.927912i	$0.378398\pi$
$278$	−1.42295e6	−1.10427
$279$	1.66653e6	1.28175
$280$	0	0
$281$	1.58468e6	1.19723	0.598614	−	0.801038i	$-0.295718\pi$
$281$	1.58468e6	1.19723	0.598614	+	0.801038i	$0.295718\pi$
$282$	704716.	0.527705
$283$	2.42072e6	1.79671	0.898357	−	0.439266i	$-0.144761\pi$
$283$	2.42072e6	1.79671	0.898357	+	0.439266i	$0.144761\pi$
$284$	−708468.	−0.521224
$285$	0	0
$286$	252084.	0.182234
$287$	−2.97957e6	−2.13525
$288$	1.67729e6	1.19159
$289$	−446829.	−0.314700
$290$	0	0
$291$	−1.61747e6	−1.11970
$292$	−508536.	−0.349031
$293$	−2.49568e6	−1.69832	−0.849161	−	0.528135i	$-0.822892\pi$
$293$	−2.49568e6	−1.69832	−0.849161	+	0.528135i	$0.822892\pi$
$294$	−1.86742e6	−1.26001
$295$	0	0
$296$	1.83273e6	1.21582
$297$	3.36324e6	2.21242
$298$	439058.	0.286405
$299$	−365789.	−0.236621
$300$	0	0
$301$	−331533.	−0.210916
$302$	−1.52082e6	−0.959533
$303$	−179907.	−0.112575
$304$	−239884.	−0.148873
$305$	0	0
$306$	−2.06309e6	−1.25955
$307$	−2.02105e6	−1.22386	−0.611930	−	0.790912i	$-0.709607\pi$
$307$	−2.02105e6	−1.22386	−0.611930	+	0.790912i	$0.709607\pi$
$308$	−1.16484e6	−0.699665
$309$	−1.99136e6	−1.18646
$310$	0	0
$311$	1.40428e6	0.823289	0.411645	−	0.911344i	$-0.364954\pi$
$311$	1.40428e6	0.823289	0.411645	+	0.911344i	$0.364954\pi$
$312$	−469597.	−0.273111
$313$	1.21370e6	0.700248	0.350124	−	0.936703i	$-0.386140\pi$
$313$	1.21370e6	0.700248	0.350124	+	0.936703i	$0.386140\pi$
$314$	−221479.	−0.126768
$315$	0	0
$316$	−557473.	−0.314055
$317$	−1.97487e6	−1.10380	−0.551899	−	0.833911i	$-0.686096\pi$
$317$	−1.97487e6	−1.10380	−0.551899	+	0.833911i	$0.686096\pi$
$318$	2.33962e6	1.29741
$319$	1.70221e6	0.936564
$320$	0	0
$321$	391500.	0.212065
$322$	−3.35001e6	−1.80055
$323$	−418481.	−0.223187
$324$	−390807.	−0.206823
$325$	0	0
$326$	−1.79311e6	−0.934462
$327$	−967679.	−0.500451
$328$	−3.27477e6	−1.68072
$329$	1.03817e6	0.528786
$330$	0	0
$331$	3.00297e6	1.50654	0.753270	−	0.657712i	$-0.228475\pi$
$331$	3.00297e6	1.50654	0.753270	+	0.657712i	$0.228475\pi$
$332$	−351632.	−0.175083
$333$	−4.21762e6	−2.08428
$334$	−854906.	−0.419327
$335$	0	0
$336$	−2.67572e6	−1.29298
$337$	649238.	0.311407	0.155704	−	0.987804i	$-0.450235\pi$
$337$	649238.	0.311407	0.155704	+	0.987804i	$0.450235\pi$
$338$	−1.67473e6	−0.797359
$339$	2.36789e6	1.11908
$340$	0	0
$341$	−2.22462e6	−1.03603
$342$	887297.	0.410208
$343$	262520.	0.120483
$344$	−364379.	−0.166019
$345$	0	0
$346$	−3.33091e6	−1.49580
$347$	3.76978e6	1.68071	0.840353	−	0.542039i	$-0.182348\pi$
$347$	3.76978e6	1.68071	0.840353	+	0.542039i	$0.182348\pi$
$348$	−796340.	−0.352493
$349$	1.50224e6	0.660200	0.330100	−	0.943946i	$-0.392918\pi$
$349$	1.50224e6	0.660200	0.330100	+	0.943946i	$0.392918\pi$
$350$	0	0
$351$	501621.	0.217324
$352$	−2.23898e6	−0.963150
$353$	−3.44264e6	−1.47046	−0.735232	−	0.677815i	$-0.762927\pi$
$353$	−3.44264e6	−1.47046	−0.735232	+	0.677815i	$0.762927\pi$
$354$	−4.24269e6	−1.79942
$355$	0	0
$356$	−1.26529e6	−0.529134
$357$	−4.66783e6	−1.93841
$358$	2.39796e6	0.988858
$359$	1.47389e6	0.603572	0.301786	−	0.953376i	$-0.402417\pi$
$359$	1.47389e6	0.603572	0.301786	+	0.953376i	$0.402417\pi$
$360$	0	0
$361$	−2.29612e6	−0.927313
$362$	−2.49823e6	−1.00198
$363$	−5.42169e6	−2.15957
$364$	−173734.	−0.0687276
$365$	0	0
$366$	6.74568e6	2.63223
$367$	−3.50663e6	−1.35901	−0.679507	−	0.733669i	$-0.737806\pi$
$367$	−3.50663e6	−1.35901	−0.679507	+	0.733669i	$0.737806\pi$
$368$	−2.29072e6	−0.881765
$369$	7.53614e6	2.88126
$370$	0	0
$371$	3.44667e6	1.30007
$372$	1.04074e6	0.389927
$373$	−3.94314e6	−1.46747	−0.733737	−	0.679434i	$-0.762225\pi$
$373$	−3.94314e6	−1.46747	−0.733737	+	0.679434i	$0.762225\pi$
$374$	2.75399e6	1.01808
$375$	0	0
$376$	1.14103e6	0.416224
$377$	253882.	0.0919980
$378$	4.59400e6	1.65372
$379$	3.45846e6	1.23676	0.618379	−	0.785880i	$-0.287790\pi$
$379$	3.45846e6	1.23676	0.618379	+	0.785880i	$0.287790\pi$
$380$	0	0
$381$	6.01337e6	2.12229
$382$	694398.	0.243472
$383$	−1.50075e6	−0.522771	−0.261385	−	0.965235i	$-0.584179\pi$
$383$	−1.50075e6	−0.522771	−0.261385	+	0.965235i	$0.584179\pi$
$384$	−1.15482e6	−0.399656
$385$	0	0
$386$	3.41930e6	1.16807
$387$	838535.	0.284606
$388$	−657691.	−0.221790
$389$	−1.76087e6	−0.590001	−0.295000	−	0.955497i	$-0.595320\pi$
$389$	−1.76087e6	−0.590001	−0.295000	+	0.955497i	$0.595320\pi$
$390$	0	0
$391$	−3.99620e6	−1.32192
$392$	−3.02360e6	−0.993822
$393$	−1.35849e6	−0.443685
$394$	−1.61105e6	−0.522841
$395$	0	0
$396$	2.94620e6	0.944113
$397$	1.08233e6	0.344656	0.172328	−	0.985040i	$-0.444871\pi$
$397$	1.08233e6	0.344656	0.172328	+	0.985040i	$0.444871\pi$
$398$	3.45175e6	1.09227
$399$	2.00755e6	0.631297
$400$	0	0
$401$	3.64479e6	1.13191	0.565955	−	0.824436i	$-0.308508\pi$
$401$	3.64479e6	1.13191	0.565955	+	0.824436i	$0.308508\pi$
$402$	3.95812e6	1.22158
$403$	−331798.	−0.101768
$404$	−73153.6	−0.0222988
$405$	0	0
$406$	2.32513e6	0.700055
$407$	5.63003e6	1.68471
$408$	−5.13029e6	−1.52578
$409$	2.50856e6	0.741509	0.370754	−	0.928731i	$-0.379099\pi$
$409$	2.50856e6	0.741509	0.370754	+	0.928731i	$0.379099\pi$
$410$	0	0
$411$	−2.03214e6	−0.593402
$412$	−809724.	−0.235014
$413$	−6.25025e6	−1.80311
$414$	8.47306e6	2.42963
$415$	0	0
$416$	−333940.	−0.0946096
$417$	8.14292e6	2.29319
$418$	−1.18444e6	−0.331567
$419$	5.00932e6	1.39394	0.696970	−	0.717101i	$-0.254531\pi$
$419$	5.00932e6	1.39394	0.696970	+	0.717101i	$0.254531\pi$
$420$	0	0
$421$	21774.2	0.00598738	0.00299369	−	0.999996i	$-0.499047\pi$
$421$	21774.2	0.00598738	0.00299369	+	0.999996i	$0.499047\pi$
$422$	−336172.	−0.0918924
$423$	−2.62582e6	−0.713532
$424$	3.78815e6	1.02332
$425$	0	0
$426$	−8.03536e6	−2.14527
$427$	9.93760e6	2.63762
$428$	159191.	0.0420057
$429$	−1.44257e6	−0.378437
$430$	0	0
$431$	3.84729e6	0.997612	0.498806	−	0.866714i	$-0.333772\pi$
$431$	3.84729e6	0.997612	0.498806	+	0.866714i	$0.333772\pi$
$432$	3.14136e6	0.809857
$433$	1.65555e6	0.424348	0.212174	−	0.977232i	$-0.431946\pi$
$433$	1.65555e6	0.424348	0.212174	+	0.977232i	$0.431946\pi$
$434$	−3.03871e6	−0.774400
$435$	0	0
$436$	−393476.	−0.0991292
$437$	1.71869e6	0.430521
$438$	−5.76775e6	−1.43655
$439$	1.85570e6	0.459564	0.229782	−	0.973242i	$-0.426199\pi$
$439$	1.85570e6	0.459564	0.229782	+	0.973242i	$0.426199\pi$
$440$	0	0
$441$	6.95812e6	1.70371
$442$	410752.	0.100005
$443$	5.98612e6	1.44923	0.724613	−	0.689156i	$-0.242019\pi$
$443$	5.98612e6	1.44923	0.724613	+	0.689156i	$0.242019\pi$
$444$	−2.63388e6	−0.634071
$445$	0	0
$446$	−990351.	−0.235750
$447$	−2.51254e6	−0.594764
$448$	−6.30267e6	−1.48364
$449$	−3.86739e6	−0.905321	−0.452660	−	0.891683i	$-0.649525\pi$
$449$	−3.86739e6	−0.905321	−0.452660	+	0.891683i	$0.649525\pi$
$450$	0	0
$451$	−1.00599e7	−2.32890
$452$	962827.	0.221668
$453$	8.70300e6	1.99262
$454$	−3.19372e6	−0.727206
$455$	0	0
$456$	2.20644e6	0.496914
$457$	733464.	0.164281	0.0821407	−	0.996621i	$-0.473824\pi$
$457$	733464.	0.164281	0.0821407	+	0.996621i	$0.473824\pi$
$458$	1.71006e6	0.380931
$459$	5.48016e6	1.21412
$460$	0	0
$461$	4.86415e6	1.06599	0.532997	−	0.846117i	$-0.321066\pi$
$461$	4.86415e6	1.06599	0.532997	+	0.846117i	$0.321066\pi$
$462$	−1.32115e7	−2.87970
$463$	2.33879e6	0.507035	0.253517	−	0.967331i	$-0.418412\pi$
$463$	2.33879e6	0.507035	0.253517	+	0.967331i	$0.418412\pi$
$464$	1.58991e6	0.342830
$465$	0	0
$466$	−93560.5	−0.0199585
$467$	−548499.	−0.116381	−0.0581907	−	0.998305i	$-0.518533\pi$
$467$	−548499.	−0.116381	−0.0581907	+	0.998305i	$0.518533\pi$
$468$	439420.	0.0927395
$469$	5.83102e6	1.22409
$470$	0	0
$471$	1.26743e6	0.263253
$472$	−6.86948e6	−1.41928
$473$	−1.11935e6	−0.230045
$474$	−6.32278e6	−1.29259
$475$	0	0
$476$	−1.89803e6	−0.383959
$477$	−8.71756e6	−1.75428
$478$	−6.28942e6	−1.25904
$479$	−2.33509e6	−0.465012	−0.232506	−	0.972595i	$-0.574693\pi$
$479$	−2.33509e6	−0.465012	−0.232506	+	0.972595i	$0.574693\pi$
$480$	0	0
$481$	839708.	0.165488
$482$	7.13064e6	1.39801
$483$	1.91707e7	3.73912
$484$	−2.20456e6	−0.427768
$485$	0	0
$486$	1.79350e6	0.344439
$487$	4.65123e6	0.888680	0.444340	−	0.895858i	$-0.353438\pi$
$487$	4.65123e6	0.888680	0.444340	+	0.895858i	$0.353438\pi$
$488$	1.09222e7	2.07615
$489$	1.02612e7	1.94055
$490$	0	0
$491$	2.46001e6	0.460504	0.230252	−	0.973131i	$-0.426045\pi$
$491$	2.46001e6	0.460504	0.230252	+	0.973131i	$0.426045\pi$
$492$	4.70627e6	0.876524
$493$	2.77363e6	0.513963
$494$	−176657.	−0.0325696
$495$	0	0
$496$	−2.07786e6	−0.379238
$497$	−1.18375e7	−2.14966
$498$	−3.98817e6	−0.720609
$499$	−8.70699e6	−1.56537	−0.782684	−	0.622419i	$-0.786150\pi$
$499$	−8.70699e6	−1.56537	−0.782684	+	0.622419i	$0.786150\pi$
$500$	0	0
$501$	4.89227e6	0.870795
$502$	−19658.2	−0.00348165
$503$	1.03253e6	0.181964	0.0909818	−	0.995853i	$-0.470999\pi$
$503$	1.03253e6	0.181964	0.0909818	+	0.995853i	$0.470999\pi$
$504$	1.60247e7	2.81005
$505$	0	0
$506$	−1.13106e7	−1.96385
$507$	9.58380e6	1.65584
$508$	2.44514e6	0.420383
$509$	5.85397e6	1.00151	0.500756	−	0.865588i	$-0.333055\pi$
$509$	5.85397e6	1.00151	0.500756	+	0.865588i	$0.333055\pi$
$510$	0	0
$511$	−8.49693e6	−1.43949
$512$	−5.65705e6	−0.953707
$513$	−2.35691e6	−0.395412
$514$	6.75185e6	1.12724
$515$	0	0
$516$	523660.	0.0865815
$517$	3.50516e6	0.576742
$518$	7.69031e6	1.25927
$519$	1.90614e7	3.10625
$520$	0	0
$521$	1.05488e7	1.70259	0.851294	−	0.524688i	$-0.175818\pi$
$521$	1.05488e7	1.70259	0.851294	+	0.524688i	$0.175818\pi$
$522$	−5.88087e6	−0.944638
$523$	−7.29899e6	−1.16683	−0.583416	−	0.812173i	$-0.698284\pi$
$523$	−7.29899e6	−1.16683	−0.583416	+	0.812173i	$0.698284\pi$
$524$	−552386.	−0.0878850
$525$	0	0
$526$	6.34616e6	1.00011
$527$	−3.62486e6	−0.568545
$528$	−9.03397e6	−1.41024
$529$	9.97597e6	1.54994
$530$	0	0
$531$	1.58085e7	2.43307
$532$	816305.	0.125047
$533$	−1.50041e6	−0.228766
$534$	−1.43508e7	−2.17782
$535$	0	0
$536$	6.40872e6	0.963517
$537$	−1.37225e7	−2.05351
$538$	−4.14329e6	−0.617148
$539$	−9.28828e6	−1.37709
$540$	0	0
$541$	−1.01864e7	−1.49634	−0.748169	−	0.663508i	$-0.769067\pi$
$541$	−1.01864e7	−1.49634	−0.748169	+	0.663508i	$0.769067\pi$
$542$	−3.33757e6	−0.488014
$543$	1.42963e7	2.08077
$544$	−3.64826e6	−0.528553
$545$	0	0
$546$	−1.97047e6	−0.282871
$547$	3.74431e6	0.535061	0.267531	−	0.963549i	$-0.413792\pi$
$547$	3.74431e6	0.535061	0.267531	+	0.963549i	$0.413792\pi$
$548$	−826305.	−0.117541
$549$	−2.51349e7	−3.55914
$550$	0	0
$551$	−1.19289e6	−0.167387
$552$	2.10700e7	2.94318
$553$	−9.31459e6	−1.29524
$554$	4.39112e6	0.607856
$555$	0	0
$556$	3.31106e6	0.454234
$557$	−4.12090e6	−0.562800	−0.281400	−	0.959591i	$-0.590799\pi$
$557$	−4.12090e6	−0.562800	−0.281400	+	0.959591i	$0.590799\pi$
$558$	7.68572e6	1.04496
$559$	−166948.	−0.0225971
$560$	0	0
$561$	−1.57599e7	−2.11420
$562$	7.30825e6	0.976052
$563$	7.40197e6	0.984184	0.492092	−	0.870543i	$-0.336232\pi$
$563$	7.40197e6	0.984184	0.492092	+	0.870543i	$0.336232\pi$
$564$	−1.63981e6	−0.217068
$565$	0	0
$566$	1.11639e7	1.46479
$567$	−6.52983e6	−0.852991
$568$	−1.30103e7	−1.69206
$569$	−1.09109e7	−1.41279	−0.706397	−	0.707816i	$-0.749680\pi$
$569$	−1.09109e7	−1.41279	−0.706397	+	0.707816i	$0.749680\pi$
$570$	0	0
$571$	8.01743e6	1.02907	0.514535	−	0.857469i	$-0.327965\pi$
$571$	8.01743e6	1.02907	0.514535	+	0.857469i	$0.327965\pi$
$572$	−586574.	−0.0749606
$573$	−3.97374e6	−0.505607
$574$	−1.37412e7	−1.74079
$575$	0	0
$576$	1.59411e7	2.00200
$577$	−1.26033e7	−1.57596	−0.787980	−	0.615701i	$-0.788873\pi$
$577$	−1.26033e7	−1.57596	−0.787980	+	0.615701i	$0.788873\pi$
$578$	−2.06069e6	−0.256562
$579$	−1.95672e7	−2.42567
$580$	0	0
$581$	−5.87528e6	−0.722085
$582$	−7.45944e6	−0.912849
$583$	1.16369e7	1.41797
$584$	−9.33875e6	−1.13307
$585$	0	0
$586$	−1.15096e7	−1.38457
$587$	313191.	0.0375157	0.0187579	−	0.999824i	$-0.494029\pi$
$587$	313191.	0.0375157	0.0187579	+	0.999824i	$0.494029\pi$
$588$	4.34530e6	0.518294
$589$	1.55899e6	0.185163
$590$	0	0
$591$	9.21938e6	1.08576
$592$	5.25861e6	0.616689
$593$	−2.33856e6	−0.273094	−0.136547	−	0.990634i	$-0.543600\pi$
$593$	−2.33856e6	−0.273094	−0.136547	+	0.990634i	$0.543600\pi$
$594$	1.55106e7	1.80370
$595$	0	0
$596$	−1.02165e6	−0.117811
$597$	−1.97529e7	−2.26827
$598$	−1.68695e6	−0.192907
$599$	−1.25687e7	−1.43128	−0.715640	−	0.698470i	$-0.753865\pi$
$599$	−1.25687e7	−1.43128	−0.715640	+	0.698470i	$0.753865\pi$
$600$	0	0
$601$	5.38110e6	0.607694	0.303847	−	0.952721i	$-0.401729\pi$
$601$	5.38110e6	0.607694	0.303847	+	0.952721i	$0.401729\pi$
$602$	−1.52897e6	−0.171952
$603$	−1.47482e7	−1.65176
$604$	3.53880e6	0.394697
$605$	0	0
$606$	−829698.	−0.0917780
$607$	750429.	0.0826681	0.0413341	−	0.999145i	$-0.486839\pi$
$607$	750429.	0.0826681	0.0413341	+	0.999145i	$0.486839\pi$
$608$	1.56905e6	0.172138
$609$	−1.33057e7	−1.45377
$610$	0	0
$611$	522788.	0.0566530
$612$	4.80061e6	0.518105
$613$	−9.02978e6	−0.970568	−0.485284	−	0.874357i	$-0.661284\pi$
$613$	−9.02978e6	−0.970568	−0.485284	+	0.874357i	$0.661284\pi$
$614$	−9.32071e6	−0.997765
$615$	0	0
$616$	−2.13912e7	−2.27134
$617$	−3.85785e6	−0.407974	−0.203987	−	0.978974i	$-0.565390\pi$
$617$	−3.85785e6	−0.407974	−0.203987	+	0.978974i	$0.565390\pi$
$618$	−9.18379e6	−0.967277
$619$	−746837.	−0.0783428	−0.0391714	−	0.999233i	$-0.512472\pi$
$619$	−746837.	−0.0783428	−0.0391714	+	0.999233i	$0.512472\pi$
$620$	0	0
$621$	−2.25069e7	−2.34200
$622$	6.47627e6	0.671195
$623$	−2.11413e7	−2.18228
$624$	−1.34740e6	−0.138527
$625$	0	0
$626$	5.59737e6	0.570884
$627$	6.77804e6	0.688550
$628$	515361.	0.0521450
$629$	9.17372e6	0.924526
$630$	0	0
$631$	5.41826e6	0.541734	0.270867	−	0.962617i	$-0.412690\pi$
$631$	5.41826e6	0.541734	0.270867	+	0.962617i	$0.412690\pi$
$632$	−1.02374e7	−1.01953
$633$	1.92377e6	0.190829
$634$	−9.10770e6	−0.899882
$635$	0	0
$636$	−5.44406e6	−0.533679
$637$	−1.38533e6	−0.135271
$638$	7.85028e6	0.763543
$639$	2.99403e7	2.90070
$640$	0	0
$641$	−4.70828e6	−0.452603	−0.226301	−	0.974057i	$-0.572663\pi$
$641$	−4.70828e6	−0.452603	−0.226301	+	0.974057i	$0.572663\pi$
$642$	1.80552e6	0.172888
$643$	6.77228e6	0.645963	0.322981	−	0.946405i	$-0.395315\pi$
$643$	6.77228e6	0.645963	0.322981	+	0.946405i	$0.395315\pi$
$644$	7.79514e6	0.740644
$645$	0	0
$646$	−1.92996e6	−0.181956
$647$	1.75890e7	1.65188	0.825942	−	0.563756i	$-0.190644\pi$
$647$	1.75890e7	1.65188	0.825942	+	0.563756i	$0.190644\pi$
$648$	−7.17677e6	−0.671416
$649$	−2.11026e7	−1.96663
$650$	0	0
$651$	1.73893e7	1.60816
$652$	4.17238e6	0.384384
$653$	6.28891e6	0.577155	0.288577	−	0.957457i	$-0.406818\pi$
$653$	6.28891e6	0.577155	0.288577	+	0.957457i	$0.406818\pi$
$654$	−4.46275e6	−0.407998
$655$	0	0
$656$	−9.39619e6	−0.852496
$657$	2.14910e7	1.94242
$658$	4.78786e6	0.431098
$659$	2.05618e7	1.84437	0.922184	−	0.386751i	$-0.126403\pi$
$659$	2.05618e7	1.84437	0.922184	+	0.386751i	$0.126403\pi$
$660$	0	0
$661$	−4.61410e6	−0.410756	−0.205378	−	0.978683i	$-0.565842\pi$
$661$	−4.61410e6	−0.410756	−0.205378	+	0.978683i	$0.565842\pi$
$662$	1.38491e7	1.22822
$663$	−2.35056e6	−0.207676
$664$	−6.45737e6	−0.568376
$665$	0	0
$666$	−1.94509e7	−1.69923
$667$	−1.13912e7	−0.991417
$668$	1.98928e6	0.172487
$669$	5.66736e6	0.489571
$670$	0	0
$671$	3.35521e7	2.87683
$672$	1.75015e7	1.49504
$673$	1.13028e6	0.0961938	0.0480969	−	0.998843i	$-0.484684\pi$
$673$	1.13028e6	0.0961938	0.0480969	+	0.998843i	$0.484684\pi$
$674$	2.99416e6	0.253878
$675$	0	0
$676$	3.89694e6	0.327988
$677$	1.60026e7	1.34189	0.670946	−	0.741506i	$-0.265888\pi$
$677$	1.60026e7	1.34189	0.670946	+	0.741506i	$0.265888\pi$
$678$	1.09203e7	0.912344
$679$	−1.09891e7	−0.914719
$680$	0	0
$681$	1.82763e7	1.51015
$682$	−1.02595e7	−0.844631
$683$	−1.55196e7	−1.27300	−0.636500	−	0.771276i	$-0.719619\pi$
$683$	−1.55196e7	−1.27300	−0.636500	+	0.771276i	$0.719619\pi$
$684$	−2.06465e6	−0.168736
$685$	0	0
$686$	1.21069e6	0.0982253
$687$	−9.78593e6	−0.791062
$688$	−1.04550e6	−0.0842080
$689$	1.73562e6	0.139286
$690$	0	0
$691$	−3.57603e6	−0.284909	−0.142454	−	0.989801i	$-0.545499\pi$
$691$	−3.57603e6	−0.284909	−0.142454	+	0.989801i	$0.545499\pi$
$692$	7.75070e6	0.615284
$693$	4.92269e7	3.89376
$694$	1.73855e7	1.37021
$695$	0	0
$696$	−1.46240e7	−1.14431
$697$	−1.63918e7	−1.27804
$698$	6.92804e6	0.538234
$699$	535407.	0.0414468
$700$	0	0
$701$	−1.21733e7	−0.935650	−0.467825	−	0.883821i	$-0.654962\pi$
$701$	−1.21733e7	−0.935650	−0.467825	+	0.883821i	$0.654962\pi$
$702$	2.31338e6	0.177176
$703$	−3.94545e6	−0.301098
$704$	−2.12796e7	−1.61820
$705$	0	0
$706$	−1.58768e7	−1.19881
$707$	−1.22229e6	−0.0919660
$708$	9.87233e6	0.740179
$709$	−1.43439e7	−1.07165	−0.535824	−	0.844330i	$-0.679999\pi$
$709$	−1.43439e7	−1.07165	−0.535824	+	0.844330i	$0.679999\pi$
$710$	0	0
$711$	2.35591e7	1.74777
$712$	−2.32358e7	−1.71774
$713$	1.48872e7	1.09671
$714$	−2.15271e7	−1.58031
$715$	0	0
$716$	−5.57981e6	−0.406759
$717$	3.59917e7	2.61460
$718$	6.79730e6	0.492068
$719$	−1.29622e7	−0.935098	−0.467549	−	0.883967i	$-0.654863\pi$
$719$	−1.29622e7	−0.935098	−0.467549	+	0.883967i	$0.654863\pi$
$720$	0	0
$721$	−1.35294e7	−0.969258
$722$	−1.05893e7	−0.756001
$723$	−4.08057e7	−2.90319
$724$	5.81313e6	0.412158
$725$	0	0
$726$	−2.50038e7	−1.76061
$727$	−1.40156e7	−0.983503	−0.491752	−	0.870735i	$-0.663643\pi$
$727$	−1.40156e7	−0.983503	−0.491752	+	0.870735i	$0.663643\pi$
$728$	−3.19045e6	−0.223112
$729$	−1.91130e7	−1.33202
$730$	0	0
$731$	−1.82389e6	−0.126243
$732$	−1.56966e7	−1.08275
$733$	−5.32267e6	−0.365906	−0.182953	−	0.983122i	$-0.558566\pi$
$733$	−5.32267e6	−0.365906	−0.182953	+	0.983122i	$0.558566\pi$
$734$	−1.61719e7	−1.10795
$735$	0	0
$736$	1.49833e7	1.01956
$737$	1.96871e7	1.33510
$738$	3.47552e7	2.34898
$739$	2.74758e7	1.85071	0.925356	−	0.379099i	$-0.123766\pi$
$739$	2.74758e7	1.85071	0.925356	+	0.379099i	$0.123766\pi$
$740$	0	0
$741$	1.01093e6	0.0676358
$742$	1.58954e7	1.05989
$743$	2.51397e7	1.67066	0.835330	−	0.549749i	$-0.185277\pi$
$743$	2.51397e7	1.67066	0.835330	+	0.549749i	$0.185277\pi$
$744$	1.91121e7	1.26583
$745$	0	0
$746$	−1.81850e7	−1.19637
$747$	1.48602e7	0.974366
$748$	−6.40826e6	−0.418780
$749$	2.65986e6	0.173242
$750$	0	0
$751$	−4.38512e6	−0.283715	−0.141857	−	0.989887i	$-0.545307\pi$
$751$	−4.38512e6	−0.283715	−0.141857	+	0.989887i	$0.545307\pi$
$752$	3.27392e6	0.211117
$753$	112496.	0.00723016
$754$	1.17085e6	0.0750023
$755$	0	0
$756$	−1.06898e7	−0.680245
$757$	−3.05247e7	−1.93603	−0.968014	−	0.250897i	$-0.919275\pi$
$757$	−3.05247e7	−1.93603	−0.968014	+	0.250897i	$0.919275\pi$
$758$	1.59497e7	1.00828
$759$	6.47255e7	4.07823
$760$	0	0
$761$	1.48688e7	0.930710	0.465355	−	0.885124i	$-0.345927\pi$
$761$	1.48688e7	0.930710	0.465355	+	0.885124i	$0.345927\pi$
$762$	2.77325e7	1.73022
$763$	−6.57443e6	−0.408834
$764$	−1.61580e6	−0.100150
$765$	0	0
$766$	−6.92117e6	−0.426194
$767$	−3.14741e6	−0.193181
$768$	2.43599e7	1.49030
$769$	−3.35916e6	−0.204840	−0.102420	−	0.994741i	$-0.532659\pi$
$769$	−3.35916e6	−0.204840	−0.102420	+	0.994741i	$0.532659\pi$
$770$	0	0
$771$	−3.86380e7	−2.34088
$772$	−7.95638e6	−0.480477
$773$	1.86568e7	1.12302	0.561512	−	0.827469i	$-0.310220\pi$
$773$	1.86568e7	1.12302	0.561512	+	0.827469i	$0.310220\pi$
$774$	3.86716e6	0.232028
$775$	0	0
$776$	−1.20778e7	−0.720003
$777$	−4.40084e7	−2.61507
$778$	−8.12078e6	−0.481004
$779$	7.04982e6	0.416231
$780$	0	0
$781$	−3.99668e7	−2.34461
$782$	−1.84297e7	−1.07771
$783$	1.56213e7	0.910567
$784$	−8.67551e6	−0.504086
$785$	0	0
$786$	−6.26509e6	−0.361719
$787$	3.25746e7	1.87475	0.937373	−	0.348328i	$-0.113251\pi$
$787$	3.25746e7	1.87475	0.937373	+	0.348328i	$0.113251\pi$
$788$	3.74877e6	0.215067
$789$	−3.63164e7	−2.07687
$790$	0	0
$791$	1.60875e7	0.914213
$792$	5.41039e7	3.06490
$793$	5.00423e6	0.282589
$794$	4.99152e6	0.280984
$795$	0	0
$796$	−8.03189e6	−0.449299
$797$	−4.95515e6	−0.276319	−0.138160	−	0.990410i	$-0.544119\pi$
$797$	−4.95515e6	−0.276319	−0.138160	+	0.990410i	$0.544119\pi$
$798$	9.25843e6	0.514671
$799$	5.71140e6	0.316502
$800$	0	0
$801$	5.34719e7	2.94472
$802$	1.68091e7	0.922801
$803$	−2.86880e7	−1.57004
$804$	−9.21016e6	−0.502489
$805$	0	0
$806$	−1.53019e6	−0.0829675
$807$	2.37103e7	1.28160
$808$	−1.34339e6	−0.0723893
$809$	−1.45654e6	−0.0782440	−0.0391220	−	0.999234i	$-0.512456\pi$
$809$	−1.45654e6	−0.0782440	−0.0391220	+	0.999234i	$0.512456\pi$
$810$	0	0
$811$	2.25081e6	0.120168	0.0600838	−	0.998193i	$-0.480863\pi$
$811$	2.25081e6	0.120168	0.0600838	+	0.998193i	$0.480863\pi$
$812$	−5.41035e6	−0.287962
$813$	1.90995e7	1.01343
$814$	2.59646e7	1.37348
$815$	0	0
$816$	−1.47202e7	−0.773905
$817$	784423.	0.0411145
$818$	1.15690e7	0.604523
$819$	7.34209e6	0.382481
$820$	0	0
$821$	−2.74521e7	−1.42141	−0.710703	−	0.703492i	$-0.751623\pi$
$821$	−2.74521e7	−1.42141	−0.710703	+	0.703492i	$0.751623\pi$
$822$	−9.37184e6	−0.483777
$823$	1.39139e7	0.716057	0.358029	−	0.933711i	$-0.383449\pi$
$823$	1.39139e7	0.716057	0.358029	+	0.933711i	$0.383449\pi$
$824$	−1.48698e7	−0.762933
$825$	0	0
$826$	−2.88249e7	−1.47000
$827$	1.91938e7	0.975884	0.487942	−	0.872876i	$-0.337748\pi$
$827$	1.91938e7	0.975884	0.487942	+	0.872876i	$0.337748\pi$
$828$	−1.97160e7	−0.999408
$829$	−3.51096e7	−1.77435	−0.887176	−	0.461432i	$-0.847336\pi$
$829$	−3.51096e7	−1.77435	−0.887176	+	0.461432i	$0.847336\pi$
$830$	0	0
$831$	−2.51285e7	−1.26231
$832$	−3.17381e6	−0.158954
$833$	−1.51346e7	−0.755714
$834$	3.75536e7	1.86955
$835$	0	0
$836$	2.75607e6	0.136388
$837$	−2.04155e7	−1.00727
$838$	2.31020e7	1.13642
$839$	−2.15179e7	−1.05535	−0.527673	−	0.849447i	$-0.676935\pi$
$839$	−2.15179e7	−1.05535	−0.527673	+	0.849447i	$0.676935\pi$
$840$	0	0
$841$	−1.26049e7	−0.614537
$842$	100418.	0.00488127
$843$	−4.18220e7	−2.02692
$844$	782239.	0.0377992
$845$	0	0
$846$	−1.21098e7	−0.581715
$847$	−3.68351e7	−1.76422
$848$	1.08692e7	0.519049
$849$	−6.38863e7	−3.04186
$850$	0	0
$851$	−3.76763e7	−1.78338
$852$	1.86975e7	0.882438
$853$	−6.72797e6	−0.316600	−0.158300	−	0.987391i	$-0.550601\pi$
$853$	−6.72797e6	−0.316600	−0.158300	+	0.987391i	$0.550601\pi$
$854$	4.58303e7	2.15035
$855$	0	0
$856$	2.92338e6	0.136364
$857$	−3.45525e7	−1.60704	−0.803520	−	0.595277i	$-0.797042\pi$
$857$	−3.45525e7	−1.60704	−0.803520	+	0.595277i	$0.797042\pi$
$858$	−6.65285e6	−0.308524
$859$	−6.75537e6	−0.312368	−0.156184	−	0.987728i	$-0.549919\pi$
$859$	−6.75537e6	−0.312368	−0.156184	+	0.987728i	$0.549919\pi$
$860$	0	0
$861$	7.86352e7	3.61501
$862$	1.77430e7	0.813314
$863$	−4.51856e6	−0.206525	−0.103263	−	0.994654i	$-0.532928\pi$
$863$	−4.51856e6	−0.206525	−0.103263	+	0.994654i	$0.532928\pi$
$864$	−2.05472e7	−0.936416
$865$	0	0
$866$	7.63508e6	0.345954
$867$	1.17925e7	0.532790
$868$	7.07079e6	0.318543
$869$	−3.14487e7	−1.41271
$870$	0	0
$871$	2.93630e6	0.131146
$872$	−7.22578e6	−0.321806
$873$	2.77944e7	1.23430
$874$	7.92628e6	0.350987
$875$	0	0
$876$	1.34210e7	0.590914
$877$	1.39115e7	0.610766	0.305383	−	0.952230i	$-0.401216\pi$
$877$	1.39115e7	0.610766	0.305383	+	0.952230i	$0.401216\pi$
$878$	8.55812e6	0.374664
$879$	6.58646e7	2.87528
$880$	0	0
$881$	−7.67479e6	−0.333140	−0.166570	−	0.986030i	$-0.553269\pi$
$881$	−7.67479e6	−0.333140	−0.166570	+	0.986030i	$0.553269\pi$
$882$	3.20895e7	1.38897
$883$	2.54331e7	1.09774	0.548868	−	0.835909i	$-0.315059\pi$
$883$	2.54331e7	1.09774	0.548868	+	0.835909i	$0.315059\pi$
$884$	−955779.	−0.0411365
$885$	0	0
$886$	2.76068e7	1.18150
$887$	−1.68309e7	−0.718287	−0.359144	−	0.933282i	$-0.616931\pi$
$887$	−1.68309e7	−0.718287	−0.359144	+	0.933282i	$0.616931\pi$
$888$	−4.83685e7	−2.05840
$889$	4.08549e7	1.73376
$890$	0	0
$891$	−2.20465e7	−0.930349
$892$	2.30445e6	0.0969740
$893$	−2.45637e6	−0.103078
$894$	−1.15874e7	−0.484888
$895$	0	0
$896$	−7.84588e6	−0.326491
$897$	9.65369e6	0.400601
$898$	−1.78357e7	−0.738072
$899$	−1.03327e7	−0.426399
$900$	0	0
$901$	1.89615e7	0.778146
$902$	−4.63942e7	−1.89866
$903$	8.74963e6	0.357084
$904$	1.76814e7	0.719605
$905$	0	0
$906$	4.01366e7	1.62450
$907$	1.06470e7	0.429743	0.214871	−	0.976642i	$-0.431067\pi$
$907$	1.06470e7	0.429743	0.214871	+	0.976642i	$0.431067\pi$
$908$	7.43148e6	0.299131
$909$	3.09151e6	0.124097
$910$	0	0
$911$	−1.72035e7	−0.686786	−0.343393	−	0.939192i	$-0.611576\pi$
$911$	−1.72035e7	−0.686786	−0.343393	+	0.939192i	$0.611576\pi$
$912$	6.33088e6	0.252044
$913$	−1.98366e7	−0.787572
$914$	3.38259e6	0.133932
$915$	0	0
$916$	−3.97914e6	−0.156693
$917$	−9.22961e6	−0.362460
$918$	2.52734e7	0.989823
$919$	2.23878e7	0.874426	0.437213	−	0.899358i	$-0.355966\pi$
$919$	2.23878e7	0.874426	0.437213	+	0.899358i	$0.355966\pi$
$920$	0	0
$921$	5.33385e7	2.07201
$922$	2.24325e7	0.869062
$923$	−5.96096e6	−0.230310
$924$	3.07419e7	1.18454
$925$	0	0
$926$	1.07860e7	0.413365
$927$	3.42194e7	1.30789
$928$	−1.03994e7	−0.396405
$929$	−2.78067e7	−1.05708	−0.528542	−	0.848907i	$-0.677261\pi$
$929$	−2.78067e7	−1.05708	−0.528542	+	0.848907i	$0.677261\pi$
$930$	0	0
$931$	6.50910e6	0.246120
$932$	217706.	0.00820977
$933$	−3.70609e7	−1.39384
$934$	−2.52957e6	−0.0948811
$935$	0	0
$936$	8.06950e6	0.301063
$937$	−2.31532e7	−0.861514	−0.430757	−	0.902468i	$-0.641753\pi$
$937$	−2.31532e7	−0.861514	−0.430757	+	0.902468i	$0.641753\pi$
$938$	2.68915e7	0.997949
$939$	−3.20314e7	−1.18553
$940$	0	0
$941$	−1.78613e7	−0.657565	−0.328782	−	0.944406i	$-0.606638\pi$
$941$	−1.78613e7	−0.657565	−0.328782	+	0.944406i	$0.606638\pi$
$942$	5.84516e6	0.214619
$943$	6.73208e7	2.46530
$944$	−1.97104e7	−0.719888
$945$	0	0
$946$	−5.16221e6	−0.187546
$947$	−2.09157e7	−0.757874	−0.378937	−	0.925422i	$-0.623710\pi$
$947$	−2.09157e7	−0.757874	−0.378937	+	0.925422i	$0.623710\pi$
$948$	1.47125e7	0.531699
$949$	−4.27876e6	−0.154224
$950$	0	0
$951$	5.21195e7	1.86874
$952$	−3.48553e7	−1.24646
$953$	2.11409e7	0.754034	0.377017	−	0.926206i	$-0.376950\pi$
$953$	2.11409e7	0.754034	0.377017	+	0.926206i	$0.376950\pi$
$954$	−4.02037e7	−1.43019
$955$	0	0
$956$	1.46349e7	0.517898
$957$	−4.49238e7	−1.58561
$958$	−1.07690e7	−0.379106
$959$	−1.38064e7	−0.484768
$960$	0	0
$961$	−1.51253e7	−0.528318
$962$	3.87257e6	0.134916
$963$	−6.72749e6	−0.233769
$964$	−1.65923e7	−0.575062
$965$	0	0
$966$	8.84115e7	3.04836
$967$	3.92113e7	1.34848	0.674242	−	0.738511i	$-0.264471\pi$
$967$	3.92113e7	1.34848	0.674242	+	0.738511i	$0.264471\pi$
$968$	−4.04845e7	−1.38867
$969$	1.10443e7	0.377859
$970$	0	0
$971$	2.02850e7	0.690443	0.345221	−	0.938521i	$-0.387804\pi$
$971$	2.02850e7	0.690443	0.345221	+	0.938521i	$0.387804\pi$
$972$	−4.17331e6	−0.141682
$973$	5.53232e7	1.87338
$974$	2.14506e7	0.724506
$975$	0	0
$976$	3.13386e7	1.05306
$977$	3.90519e6	0.130890	0.0654449	−	0.997856i	$-0.479153\pi$
$977$	3.90519e6	0.130890	0.0654449	+	0.997856i	$0.479153\pi$
$978$	4.73226e7	1.58205
$979$	−7.13788e7	−2.38019
$980$	0	0
$981$	1.66285e7	0.551671
$982$	1.13451e7	0.375430
$983$	2.70058e7	0.891400	0.445700	−	0.895182i	$-0.352955\pi$
$983$	2.70058e7	0.891400	0.445700	+	0.895182i	$0.352955\pi$
$984$	8.64259e7	2.84548
$985$	0	0
$986$	1.27915e7	0.419013
$987$	−2.73989e7	−0.895241
$988$	411063.	0.0133973
$989$	7.49069e6	0.243518
$990$	0	0
$991$	−6.52133e6	−0.210937	−0.105468	−	0.994423i	$-0.533634\pi$
$991$	−6.52133e6	−0.210937	−0.105468	+	0.994423i	$0.533634\pi$
$992$	1.35910e7	0.438503
$993$	−7.92526e7	−2.55059
$994$	−5.45924e7	−1.75253
$995$	0	0
$996$	9.28007e6	0.296417
$997$	2.75439e7	0.877582	0.438791	−	0.898589i	$-0.355407\pi$
$997$	2.75439e7	0.877582	0.438791	+	0.898589i	$0.355407\pi$
$998$	−4.01549e7	−1.27618
$999$	5.16670e7	1.63795

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.k.1.36		52
5.2	odd	4		215.6.b.a.44.71	yes	104
5.3	odd	4		215.6.b.a.44.34	✓	104
5.4	even	2		1075.6.a.l.1.17		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.6.b.a.44.34	✓	104	5.3	odd	4
215.6.b.a.44.71	yes	104	5.2	odd	4
1075.6.a.k.1.36		52	1.1	even	1	trivial
1075.6.a.l.1.17		52	5.4	even	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q+4.61181 q^{2} -26.3914 q^{3} -10.7312 q^{4} -121.712 q^{6} -179.304 q^{7} -197.068 q^{8} +453.507 q^{9} +O(q^{10})\)	\(q+4.61181 q^{2} -26.3914 q^{3} -10.7312 q^{4} -121.712 q^{6} -179.304 q^{7} -197.068 q^{8} +453.507 q^{9} -605.380 q^{11} +283.212 q^{12} -90.2912 q^{13} -826.915 q^{14} -565.441 q^{16} -986.422 q^{17} +2091.49 q^{18} +424.242 q^{19} +4732.09 q^{21} -2791.90 q^{22} +4051.21 q^{23} +5200.91 q^{24} -416.406 q^{26} -5555.59 q^{27} +1924.15 q^{28} -2811.81 q^{29} +3674.76 q^{31} +3698.48 q^{32} +15976.8 q^{33} -4549.19 q^{34} -4866.69 q^{36} -9300.00 q^{37} +1956.52 q^{38} +2382.91 q^{39} +16617.4 q^{41} +21823.5 q^{42} +1849.00 q^{43} +6496.47 q^{44} +18683.4 q^{46} -5790.02 q^{47} +14922.8 q^{48} +15342.9 q^{49} +26033.1 q^{51} +968.936 q^{52} -19222.5 q^{53} -25621.3 q^{54} +35335.1 q^{56} -11196.3 q^{57} -12967.5 q^{58} +34858.4 q^{59} -55423.2 q^{61} +16947.3 q^{62} -81315.7 q^{63} +35150.8 q^{64} +73682.1 q^{66} -32520.3 q^{67} +10585.5 q^{68} -106917. q^{69} +66019.3 q^{71} -89371.9 q^{72} +47388.4 q^{73} -42889.8 q^{74} -4552.64 q^{76} +108547. q^{77} +10989.5 q^{78} +51948.6 q^{79} +36417.7 q^{81} +76636.5 q^{82} +32767.2 q^{83} -50781.1 q^{84} +8527.23 q^{86} +74207.7 q^{87} +119301. q^{88} +117907. q^{89} +16189.6 q^{91} -43474.5 q^{92} -96982.1 q^{93} -26702.5 q^{94} -97608.1 q^{96} +61287.5 q^{97} +70758.5 q^{98} -274544. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 20 q^{2} - 54 q^{3} + 826 q^{4} - 162 q^{6} - 196 q^{7} - 960 q^{8} + 4098 q^{9}+O(q^{10}) \) 52 * q - 20 * q^2 - 54 * q^3 + 826 * q^4 - 162 * q^6 - 196 * q^7 - 960 * q^8 + 4098 * q^9	\( 52 q - 20 q^{2} - 54 q^{3} + 826 q^{4} - 162 q^{6} - 196 q^{7} - 960 q^{8} + 4098 q^{9} - 664 q^{11} + 523 q^{12} - 2704 q^{13} + 150 q^{14} + 13474 q^{16} - 7266 q^{17} - 4860 q^{18} - 1970 q^{19} + 800 q^{21} - 14477 q^{22} - 9522 q^{23} + 314 q^{24} + 5514 q^{26} - 22926 q^{27} - 9408 q^{28} - 7188 q^{29} - 11556 q^{31} - 48390 q^{32} - 26136 q^{33} + 16774 q^{34} + 51872 q^{36} - 42558 q^{37} - 46208 q^{38} + 4682 q^{39} - 7746 q^{41} - 174265 q^{42} + 96148 q^{43} - 48600 q^{44} + 16182 q^{46} - 87136 q^{47} + 2912 q^{48} + 142286 q^{49} - 3710 q^{51} - 146868 q^{52} - 127034 q^{53} - 49563 q^{54} - 2849 q^{56} - 101594 q^{57} - 9480 q^{58} - 55924 q^{59} + 73702 q^{61} - 186016 q^{62} - 50120 q^{63} + 157750 q^{64} + 58211 q^{66} - 131996 q^{67} - 298560 q^{68} + 128436 q^{69} - 56284 q^{71} - 343775 q^{72} - 128620 q^{73} - 17721 q^{74} - 170410 q^{76} - 448438 q^{77} - 237616 q^{78} + 106204 q^{79} + 478568 q^{81} - 249596 q^{82} - 348616 q^{83} - 131855 q^{84} - 36980 q^{86} - 267478 q^{87} - 525216 q^{88} + 80410 q^{89} + 226376 q^{91} - 581456 q^{92} - 902902 q^{93} + 180980 q^{94} + 38543 q^{96} - 316148 q^{97} - 295095 q^{98} + 68428 q^{99}+O(q^{100}) \) 52 * q - 20 * q^2 - 54 * q^3 + 826 * q^4 - 162 * q^6 - 196 * q^7 - 960 * q^8 + 4098 * q^9 - 664 * q^11 + 523 * q^12 - 2704 * q^13 + 150 * q^14 + 13474 * q^16 - 7266 * q^17 - 4860 * q^18 - 1970 * q^19 + 800 * q^21 - 14477 * q^22 - 9522 * q^23 + 314 * q^24 + 5514 * q^26 - 22926 * q^27 - 9408 * q^28 - 7188 * q^29 - 11556 * q^31 - 48390 * q^32 - 26136 * q^33 + 16774 * q^34 + 51872 * q^36 - 42558 * q^37 - 46208 * q^38 + 4682 * q^39 - 7746 * q^41 - 174265 * q^42 + 96148 * q^43 - 48600 * q^44 + 16182 * q^46 - 87136 * q^47 + 2912 * q^48 + 142286 * q^49 - 3710 * q^51 - 146868 * q^52 - 127034 * q^53 - 49563 * q^54 - 2849 * q^56 - 101594 * q^57 - 9480 * q^58 - 55924 * q^59 + 73702 * q^61 - 186016 * q^62 - 50120 * q^63 + 157750 * q^64 + 58211 * q^66 - 131996 * q^67 - 298560 * q^68 + 128436 * q^69 - 56284 * q^71 - 343775 * q^72 - 128620 * q^73 - 17721 * q^74 - 170410 * q^76 - 448438 * q^77 - 237616 * q^78 + 106204 * q^79 + 478568 * q^81 - 249596 * q^82 - 348616 * q^83 - 131855 * q^84 - 36980 * q^86 - 267478 * q^87 - 525216 * q^88 + 80410 * q^89 + 226376 * q^91 - 581456 * q^92 - 902902 * q^93 + 180980 * q^94 + 38543 * q^96 - 316148 * q^97 - 295095 * q^98 + 68428 * q^99

Introduction

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