LMFDB - Embedded newform 1075.6.a.l.1.41

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:	$0$
Dimension:	$52$
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.41
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)

$f(q)$	$=$	$q+7.10848 q^{2} +0.0987616 q^{3} +18.5304 q^{4} +0.702045 q^{6} +118.060 q^{7} -95.7481 q^{8} -242.990 q^{9} +O(q^{10})$	\(q+7.10848 q^{2} +0.0987616 q^{3} +18.5304 q^{4} +0.702045 q^{6} +118.060 q^{7} -95.7481 q^{8} -242.990 q^{9} -426.303 q^{11} +1.83009 q^{12} +825.742 q^{13} +839.225 q^{14} -1273.60 q^{16} +1739.57 q^{17} -1727.29 q^{18} +1596.09 q^{19} +11.6598 q^{21} -3030.37 q^{22} -2789.28 q^{23} -9.45624 q^{24} +5869.77 q^{26} -47.9972 q^{27} +2187.70 q^{28} +1971.56 q^{29} -3997.81 q^{31} -5989.39 q^{32} -42.1024 q^{33} +12365.7 q^{34} -4502.71 q^{36} -16139.7 q^{37} +11345.7 q^{38} +81.5516 q^{39} +20568.3 q^{41} +82.8832 q^{42} -1849.00 q^{43} -7899.58 q^{44} -19827.5 q^{46} -5425.07 q^{47} -125.782 q^{48} -2868.91 q^{49} +171.803 q^{51} +15301.4 q^{52} +27172.3 q^{53} -341.187 q^{54} -11304.0 q^{56} +157.632 q^{57} +14014.8 q^{58} +21246.4 q^{59} +22730.7 q^{61} -28418.4 q^{62} -28687.4 q^{63} -1820.35 q^{64} -299.284 q^{66} +438.272 q^{67} +32235.1 q^{68} -275.474 q^{69} -33484.9 q^{71} +23265.9 q^{72} +52666.0 q^{73} -114729. q^{74} +29576.2 q^{76} -50329.2 q^{77} +579.708 q^{78} +46616.7 q^{79} +59041.9 q^{81} +146209. q^{82} +10496.4 q^{83} +216.060 q^{84} -13143.6 q^{86} +194.715 q^{87} +40817.7 q^{88} +108581. q^{89} +97486.9 q^{91} -51686.5 q^{92} -394.830 q^{93} -38564.0 q^{94} -591.522 q^{96} +173286. q^{97} -20393.5 q^{98} +103588. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	7.10848	1.25661	0.628306	−	0.777966i	$-0.283748\pi$
$2$	7.10848	1.25661	0.628306	+	0.777966i	$0.283748\pi$
$3$	0.0987616	0.00633556	0.00316778	−	0.999995i	$-0.498992\pi$
$3$	0.0987616	0.00633556	0.00316778	+	0.999995i	$0.498992\pi$
$4$	18.5304	0.579076
$5$	0	0
$5$	0	0
$6$	0.702045	0.00796135
$7$	118.060	0.910661	0.455330	−	0.890323i	$-0.349521\pi$
$7$	118.060	0.910661	0.455330	+	0.890323i	$0.349521\pi$
$8$	−95.7481	−0.528939
$9$	−242.990	−0.999960
$10$	0	0
$11$	−426.303	−1.06228	−0.531138	−	0.847286i	$-0.678235\pi$
$11$	−426.303	−1.06228	−0.531138	+	0.847286i	$0.678235\pi$
$12$	1.83009	0.00366877
$13$	825.742	1.35515	0.677573	−	0.735456i	$-0.263032\pi$
$13$	825.742	1.35515	0.677573	+	0.735456i	$0.263032\pi$
$14$	839.225	1.14435
$15$	0	0
$16$	−1273.60	−1.24375
$17$	1739.57	1.45989	0.729946	−	0.683505i	$-0.239545\pi$
$17$	1739.57	1.45989	0.729946	+	0.683505i	$0.239545\pi$
$18$	−1727.29	−1.25656
$19$	1596.09	1.01431	0.507157	−	0.861854i	$-0.330696\pi$
$19$	1596.09	1.01431	0.507157	+	0.861854i	$0.330696\pi$
$20$	0	0
$21$	11.6598	0.00576955
$22$	−3030.37	−1.33487
$23$	−2789.28	−1.09944	−0.549721	−	0.835348i	$-0.685266\pi$
$23$	−2789.28	−1.09944	−0.549721	+	0.835348i	$0.685266\pi$
$24$	−9.45624	−0.00335112
$25$	0	0
$26$	5869.77	1.70289
$27$	−47.9972	−0.0126709
$28$	2187.70	0.527342
$29$	1971.56	0.435327	0.217664	−	0.976024i	$-0.430156\pi$
$29$	1971.56	0.435327	0.217664	+	0.976024i	$0.430156\pi$
$30$	0	0
$31$	−3997.81	−0.747168	−0.373584	−	0.927596i	$-0.621871\pi$
$31$	−3997.81	−0.747168	−0.373584	+	0.927596i	$0.621871\pi$
$32$	−5989.39	−1.03397
$33$	−42.1024	−0.00673011
$34$	12365.7	1.83452
$35$	0	0
$36$	−4502.71	−0.579053
$37$	−16139.7	−1.93817	−0.969083	−	0.246736i	$-0.920642\pi$
$37$	−16139.7	−1.93817	−0.969083	+	0.246736i	$0.920642\pi$
$38$	11345.7	1.27460
$39$	81.5516	0.00858561
$40$	0	0
$41$	20568.3	1.91090	0.955451	−	0.295149i	$-0.0953694\pi$
$41$	20568.3	1.91090	0.955451	+	0.295149i	$0.0953694\pi$
$42$	82.8832	0.00725009
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	−7899.58	−0.615138
$45$	0	0
$46$	−19827.5	−1.38157
$47$	−5425.07	−0.358229	−0.179115	−	0.983828i	$-0.557323\pi$
$47$	−5425.07	−0.358229	−0.179115	+	0.983828i	$0.557323\pi$
$48$	−125.782	−0.00787983
$49$	−2868.91	−0.170697
$50$	0	0
$51$	171.803	0.00924923
$52$	15301.4	0.784732
$53$	27172.3	1.32873	0.664365	−	0.747409i	$-0.268702\pi$
$53$	27172.3	1.32873	0.664365	+	0.747409i	$0.268702\pi$
$54$	−341.187	−0.0159224
$55$	0	0
$56$	−11304.0	−0.481684
$57$	157.632	0.00642625
$58$	14014.8	0.547038
$59$	21246.4	0.794614	0.397307	−	0.917686i	$-0.369945\pi$
$59$	21246.4	0.794614	0.397307	+	0.917686i	$0.369945\pi$
$60$	0	0
$61$	22730.7	0.782148	0.391074	−	0.920359i	$-0.372104\pi$
$61$	22730.7	0.782148	0.391074	+	0.920359i	$0.372104\pi$
$62$	−28418.4	−0.938901
$63$	−28687.4	−0.910624
$64$	−1820.35	−0.0555526
$65$	0	0
$66$	−299.284	−0.00845714
$67$	438.272	0.0119277	0.00596385	−	0.999982i	$-0.498102\pi$
$67$	438.272	0.0119277	0.00596385	+	0.999982i	$0.498102\pi$
$68$	32235.1	0.845388
$69$	−275.474	−0.00696558
$70$	0	0
$71$	−33484.9	−0.788320	−0.394160	−	0.919042i	$-0.628964\pi$
$71$	−33484.9	−0.788320	−0.394160	+	0.919042i	$0.628964\pi$
$72$	23265.9	0.528917
$73$	52666.0	1.15671	0.578353	−	0.815787i	$-0.303696\pi$
$73$	52666.0	1.15671	0.578353	+	0.815787i	$0.303696\pi$
$74$	−114729.	−2.43552
$75$	0	0
$76$	29576.2	0.587365
$77$	−50329.2	−0.967372
$78$	579.708	0.0107888
$79$	46616.7	0.840376	0.420188	−	0.907437i	$-0.361964\pi$
$79$	46616.7	0.840376	0.420188	+	0.907437i	$0.361964\pi$
$80$	0	0
$81$	59041.9	0.999880
$82$	146209.	2.40126
$83$	10496.4	0.167241	0.0836206	−	0.996498i	$-0.473352\pi$
$83$	10496.4	0.167241	0.0836206	+	0.996498i	$0.473352\pi$
$84$	216.060	0.00334100
$85$	0	0
$86$	−13143.6	−0.191632
$87$	194.715	0.00275804
$88$	40817.7	0.561878
$89$	108581.	1.45305	0.726525	−	0.687140i	$-0.241134\pi$
$89$	108581.	1.45305	0.726525	+	0.687140i	$0.241134\pi$
$90$	0	0
$91$	97486.9	1.23408
$92$	−51686.5	−0.636660
$93$	−394.830	−0.00473373
$94$	−38564.0	−0.450155
$95$	0	0
$96$	−591.522	−0.00655078
$97$	173286.	1.86997	0.934983	−	0.354693i	$-0.115415\pi$
$97$	173286.	1.86997	0.934983	+	0.354693i	$0.115415\pi$
$98$	−20393.5	−0.214500
$99$	103588.	1.06223
$100$	0	0
$101$	106866.	1.04241	0.521204	−	0.853432i	$-0.325483\pi$
$101$	106866.	1.04241	0.521204	+	0.853432i	$0.325483\pi$
$102$	1221.26	0.0116227
$103$	−52284.6	−0.485602	−0.242801	−	0.970076i	$-0.578066\pi$
$103$	−52284.6	−0.485602	−0.242801	+	0.970076i	$0.578066\pi$
$104$	−79063.3	−0.716789
$105$	0	0
$106$	193154.	1.66970
$107$	16861.1	0.142373	0.0711863	−	0.997463i	$-0.477322\pi$
$107$	16861.1	0.142373	0.0711863	+	0.997463i	$0.477322\pi$
$108$	−889.408	−0.00733739
$109$	54284.4	0.437632	0.218816	−	0.975766i	$-0.429781\pi$
$109$	54284.4	0.437632	0.218816	+	0.975766i	$0.429781\pi$
$110$	0	0
$111$	−1593.98	−0.0122794
$112$	−150360.	−1.13263
$113$	−11169.6	−0.0822890	−0.0411445	−	0.999153i	$-0.513100\pi$
$113$	−11169.6	−0.0822890	−0.0411445	+	0.999153i	$0.513100\pi$
$114$	1120.52	0.00807531
$115$	0	0
$116$	36533.9	0.252088
$117$	−200647.	−1.35509
$118$	151030.	0.998522
$119$	205374.	1.32947
$120$	0	0
$121$	20683.5	0.128428
$122$	161581.	0.982857
$123$	2031.36	0.0121066
$124$	−74081.2	−0.432667
$125$	0	0
$126$	−203923.	−1.14430
$127$	201912.	1.11084	0.555421	−	0.831569i	$-0.312557\pi$
$127$	201912.	1.11084	0.555421	+	0.831569i	$0.312557\pi$
$128$	178721.	0.964162
$129$	−182.610	−0.000966164 0
$130$	0	0
$131$	−369600.	−1.88172	−0.940858	−	0.338802i	$-0.889978\pi$
$131$	−369600.	−1.88172	−0.940858	+	0.338802i	$0.889978\pi$
$132$	−780.176	−0.00389724
$133$	188434.	0.923696
$134$	3115.45	0.0149885
$135$	0	0
$136$	−166561.	−0.772193
$137$	145830.	0.663813	0.331906	−	0.943312i	$-0.392308\pi$
$137$	145830.	0.663813	0.331906	+	0.943312i	$0.392308\pi$
$138$	−1958.20	−0.00875304
$139$	161638.	0.709587	0.354793	−	0.934945i	$-0.384551\pi$
$139$	161638.	0.709587	0.354793	+	0.934945i	$0.384551\pi$
$140$	0	0
$141$	−535.789	−0.00226958
$142$	−238026.	−0.990613
$143$	−352017.	−1.43954
$144$	309472.	1.24370
$145$	0	0
$146$	374375.	1.45353
$147$	−283.338	−0.00108146
$148$	−299075.	−1.12234
$149$	338981.	1.25086	0.625431	−	0.780279i	$-0.284923\pi$
$149$	338981.	1.25086	0.625431	+	0.780279i	$0.284923\pi$
$150$	0	0
$151$	−308534.	−1.10119	−0.550594	−	0.834773i	$-0.685599\pi$
$151$	−308534.	−1.10119	−0.550594	+	0.834773i	$0.685599\pi$
$152$	−152822.	−0.536510
$153$	−422700.	−1.45983
$154$	−357764.	−1.21561
$155$	0	0
$156$	1511.19	0.00497172
$157$	101442.	0.328449	0.164224	−	0.986423i	$-0.447488\pi$
$157$	101442.	0.328449	0.164224	+	0.986423i	$0.447488\pi$
$158$	331374.	1.05603
$159$	2683.58	0.00841825
$160$	0	0
$161$	−329301.	−1.00122
$162$	419698.	1.25646
$163$	−267784.	−0.789435	−0.394717	−	0.918803i	$-0.629158\pi$
$163$	−267784.	−0.789435	−0.394717	+	0.918803i	$0.629158\pi$
$164$	381139.	1.10656
$165$	0	0
$166$	74613.1	0.210158
$167$	−559006.	−1.55105	−0.775524	−	0.631318i	$-0.782514\pi$
$167$	−559006.	−1.55105	−0.775524	+	0.631318i	$0.782514\pi$
$168$	−1116.40	−0.00305174
$169$	310557.	0.836421
$170$	0	0
$171$	−387834.	−1.01427
$172$	−34262.8	−0.0883082
$173$	−40882.9	−0.103855	−0.0519273	−	0.998651i	$-0.516536\pi$
$173$	−40882.9	−0.103855	−0.0519273	+	0.998651i	$0.516536\pi$
$174$	1384.13	0.00346579
$175$	0	0
$176$	542939.	1.32120
$177$	2098.33	0.00503432
$178$	771849.	1.82592
$179$	153552.	0.358199	0.179099	−	0.983831i	$-0.442682\pi$
$179$	153552.	0.358199	0.179099	+	0.983831i	$0.442682\pi$
$180$	0	0
$181$	15461.1	0.0350786	0.0175393	−	0.999846i	$-0.494417\pi$
$181$	15461.1	0.0350786	0.0175393	+	0.999846i	$0.494417\pi$
$182$	692983.	1.55076
$183$	2244.92	0.00495534
$184$	267068.	0.581538
$185$	0	0
$186$	−2806.64	−0.00594846
$187$	−741586.	−1.55081
$188$	−100529.	−0.207442
$189$	−5666.53	−0.0115389
$190$	0	0
$191$	828388.	1.64305	0.821524	−	0.570174i	$-0.193124\pi$
$191$	828388.	1.64305	0.821524	+	0.570174i	$0.193124\pi$
$192$	−179.781	−0.000351957 0
$193$	87192.7	0.168495	0.0842475	−	0.996445i	$-0.473151\pi$
$193$	87192.7	0.168495	0.0842475	+	0.996445i	$0.473151\pi$
$194$	1.23180e6	2.34982
$195$	0	0
$196$	−53162.1	−0.0988466
$197$	76744.4	0.140890	0.0704451	−	0.997516i	$-0.477558\pi$
$197$	76744.4	0.140890	0.0704451	+	0.997516i	$0.477558\pi$
$198$	736350.	1.33481
$199$	634093.	1.13506	0.567532	−	0.823351i	$-0.307898\pi$
$199$	634093.	1.13506	0.567532	+	0.823351i	$0.307898\pi$
$200$	0	0
$201$	43.2845	7.55687e−5 0
$202$	759658.	1.30990
$203$	232762.	0.396435
$204$	3183.59	0.00535601
$205$	0	0
$206$	−371664.	−0.610214
$207$	677768.	1.09940
$208$	−1.05166e6	−1.68546
$209$	−680417.	−1.07748
$210$	0	0
$211$	656240.	1.01474	0.507372	−	0.861727i	$-0.330617\pi$
$211$	656240.	1.01474	0.507372	+	0.861727i	$0.330617\pi$
$212$	503514.	0.769435
$213$	−3307.02	−0.00499445
$214$	119857.	0.178907
$215$	0	0
$216$	4595.64	0.00670211
$217$	−471981.	−0.680416
$218$	385879.	0.549933
$219$	5201.38	0.00732838
$220$	0	0
$221$	1.43644e6	1.97837
$222$	−11330.8	−0.0154304
$223$	782619.	1.05387	0.526937	−	0.849905i	$-0.323341\pi$
$223$	782619.	1.05387	0.526937	+	0.849905i	$0.323341\pi$
$224$	−707106.	−0.941596
$225$	0	0
$226$	−79398.9	−0.103405
$227$	142382.	0.183397	0.0916984	−	0.995787i	$-0.470770\pi$
$227$	142382.	0.183397	0.0916984	+	0.995787i	$0.470770\pi$
$228$	2920.99	0.00372129
$229$	1.09050e6	1.37416	0.687078	−	0.726584i	$-0.258893\pi$
$229$	1.09050e6	1.37416	0.687078	+	0.726584i	$0.258893\pi$
$230$	0	0
$231$	−4970.60	−0.00612885
$232$	−188774.	−0.230261
$233$	−608888.	−0.734763	−0.367382	−	0.930070i	$-0.619746\pi$
$233$	−608888.	−0.734763	−0.367382	+	0.930070i	$0.619746\pi$
$234$	−1.42630e6	−1.70283
$235$	0	0
$236$	393705.	0.460142
$237$	4603.94	0.00532425
$238$	1.45989e6	1.67062
$239$	−1.62349e6	−1.83847	−0.919233	−	0.393713i	$-0.871190\pi$
$239$	−1.62349e6	−1.83847	−0.919233	+	0.393713i	$0.871190\pi$
$240$	0	0
$241$	−966711.	−1.07215	−0.536073	−	0.844172i	$-0.680093\pi$
$241$	−966711.	−1.07215	−0.536073	+	0.844172i	$0.680093\pi$
$242$	147028.	0.161385
$243$	17494.4	0.0190057
$244$	421210.	0.452923
$245$	0	0
$246$	14439.9	0.0152134
$247$	1.31796e6	1.37454
$248$	382783.	0.395206
$249$	1036.64	0.00105957
$250$	0	0
$251$	−554836.	−0.555879	−0.277939	−	0.960599i	$-0.589651\pi$
$251$	−554836.	−0.555879	−0.277939	+	0.960599i	$0.589651\pi$
$252$	−531589.	−0.527320
$253$	1.18908e6	1.16791
$254$	1.43528e6	1.39590
$255$	0	0
$256$	1.32868e6	1.26713
$257$	631564.	0.596464	0.298232	−	0.954493i	$-0.403603\pi$
$257$	631564.	0.596464	0.298232	+	0.954493i	$0.403603\pi$
$258$	−1298.08	−0.00121409
$259$	−1.90545e6	−1.76501
$260$	0	0
$261$	−479071.	−0.435310
$262$	−2.62729e6	−2.36459
$263$	222799.	0.198620	0.0993101	−	0.995057i	$-0.468336\pi$
$263$	222799.	0.198620	0.0993101	+	0.995057i	$0.468336\pi$
$264$	4031.23	0.00355981
$265$	0	0
$266$	1.33948e6	1.16073
$267$	10723.7	0.00920589
$268$	8121.37	0.00690704
$269$	1.63282e6	1.37581	0.687905	−	0.725801i	$-0.258531\pi$
$269$	1.63282e6	1.37581	0.687905	+	0.725801i	$0.258531\pi$
$270$	0	0
$271$	511602.	0.423164	0.211582	−	0.977360i	$-0.432138\pi$
$271$	511602.	0.423164	0.211582	+	0.977360i	$0.432138\pi$
$272$	−2.21552e6	−1.81574
$273$	9627.96	0.00781858
$274$	1.03663e6	0.834156
$275$	0	0
$276$	−5104.65	−0.00403360
$277$	1.59765e6	1.25108	0.625538	−	0.780194i	$-0.284880\pi$
$277$	1.59765e6	1.25108	0.625538	+	0.780194i	$0.284880\pi$
$278$	1.14900e6	0.891676
$279$	971429.	0.747138
$280$	0	0
$281$	512674.	0.387325	0.193662	−	0.981068i	$-0.437963\pi$
$281$	512674.	0.387325	0.193662	+	0.981068i	$0.437963\pi$
$282$	−3808.64	−0.00285199
$283$	1.13933e6	0.845635	0.422817	−	0.906215i	$-0.361041\pi$
$283$	1.13933e6	0.845635	0.422817	+	0.906215i	$0.361041\pi$
$284$	−620489.	−0.456497
$285$	0	0
$286$	−2.50230e6	−1.80894
$287$	2.42829e6	1.74018
$288$	1.45536e6	1.03393
$289$	1.60626e6	1.13128
$290$	0	0
$291$	17114.0	0.0118473
$292$	975923.	0.669820
$293$	−901397.	−0.613405	−0.306702	−	0.951805i	$-0.599226\pi$
$293$	−901397.	−0.613405	−0.306702	+	0.951805i	$0.599226\pi$
$294$	−2014.10	−0.00135898
$295$	0	0
$296$	1.54535e6	1.02517
$297$	20461.4	0.0134599
$298$	2.40964e6	1.57185
$299$	−2.30323e6	−1.48990
$300$	0	0
$301$	−218292.	−0.138874
$302$	−2.19321e6	−1.38377
$303$	10554.3	0.00660424
$304$	−2.03277e6	−1.26155
$305$	0	0
$306$	−3.00475e6	−1.83445
$307$	−2.63809e6	−1.59751	−0.798755	−	0.601657i	$-0.794508\pi$
$307$	−2.63809e6	−1.59751	−0.798755	+	0.601657i	$0.794508\pi$
$308$	−932622.	−0.560182
$309$	−5163.71	−0.00307656
$310$	0	0
$311$	−1.94498e6	−1.14029	−0.570144	−	0.821545i	$-0.693113\pi$
$311$	−1.94498e6	−1.14029	−0.570144	+	0.821545i	$0.693113\pi$
$312$	−7808.42	−0.00454126
$313$	−143531.	−0.0828105	−0.0414053	−	0.999142i	$-0.513183\pi$
$313$	−143531.	−0.0828105	−0.0414053	+	0.999142i	$0.513183\pi$
$314$	721097.	0.412733
$315$	0	0
$316$	863827.	0.486642
$317$	686730.	0.383829	0.191915	−	0.981412i	$-0.438530\pi$
$317$	686730.	0.383829	0.191915	+	0.981412i	$0.438530\pi$
$318$	19076.2	0.0105785
$319$	−840484.	−0.462437
$320$	0	0
$321$	1665.23	0.000902010 0
$322$	−2.34083e6	−1.25814
$323$	2.77651e6	1.48079
$324$	1.09407e6	0.579006
$325$	0	0
$326$	−1.90354e6	−0.992014
$327$	5361.21	0.00277264
$328$	−1.96938e6	−1.01075
$329$	−640482.	−0.326225
$330$	0	0
$331$	−3.63006e6	−1.82114	−0.910571	−	0.413353i	$-0.864357\pi$
$331$	−3.63006e6	−1.82114	−0.910571	+	0.413353i	$0.864357\pi$
$332$	194502.	0.0968454
$333$	3.92179e6	1.93809
$334$	−3.97368e6	−1.94907
$335$	0	0
$336$	−14849.8	−0.00717586
$337$	580940.	0.278648	0.139324	−	0.990247i	$-0.455507\pi$
$337$	580940.	0.278648	0.139324	+	0.990247i	$0.455507\pi$
$338$	2.20759e6	1.05106
$339$	−1103.13	−0.000521347 0
$340$	0	0
$341$	1.70428e6	0.793698
$342$	−2.75691e6	−1.27455
$343$	−2.32293e6	−1.06611
$344$	177038.	0.0806624
$345$	0	0
$346$	−290615.	−0.130505
$347$	3.48086e6	1.55190	0.775949	−	0.630795i	$-0.217271\pi$
$347$	3.48086e6	1.55190	0.775949	+	0.630795i	$0.217271\pi$
$348$	3608.15	0.00159712
$349$	1.82280e6	0.801081	0.400540	−	0.916279i	$-0.368822\pi$
$349$	1.82280e6	0.801081	0.400540	+	0.916279i	$0.368822\pi$
$350$	0	0
$351$	−39633.3	−0.0171709
$352$	2.55330e6	1.09836
$353$	−1.21603e6	−0.519405	−0.259702	−	0.965689i	$-0.583624\pi$
$353$	−1.21603e6	−0.519405	−0.259702	+	0.965689i	$0.583624\pi$
$354$	14915.9	0.00632619
$355$	0	0
$356$	2.01206e6	0.841426
$357$	20283.0	0.00842291
$358$	1.09152e6	0.450117
$359$	−771900.	−0.316100	−0.158050	−	0.987431i	$-0.550521\pi$
$359$	−771900.	−0.316100	−0.158050	+	0.987431i	$0.550521\pi$
$360$	0	0
$361$	71395.1	0.0288337
$362$	109905.	0.0440803
$363$	2042.74	0.000813666 0
$364$	1.80647e6	0.714625
$365$	0	0
$366$	15958.0	0.00622695
$367$	3.24730e6	1.25851	0.629256	−	0.777198i	$-0.283360\pi$
$367$	3.24730e6	1.25851	0.629256	+	0.777198i	$0.283360\pi$
$368$	3.55242e6	1.36743
$369$	−4.99789e6	−1.91083
$370$	0	0
$371$	3.20795e6	1.21002
$372$	−7316.38	−0.00274119
$373$	−1.57980e6	−0.587937	−0.293969	−	0.955815i	$-0.594976\pi$
$373$	−1.57980e6	−0.587937	−0.293969	+	0.955815i	$0.594976\pi$
$374$	−5.27155e6	−1.94876
$375$	0	0
$376$	519441.	0.189481
$377$	1.62800e6	0.589932
$378$	−40280.4	−0.0144999
$379$	−4.33308e6	−1.54953	−0.774763	−	0.632252i	$-0.782131\pi$
$379$	−4.33308e6	−1.54953	−0.774763	+	0.632252i	$0.782131\pi$
$380$	0	0
$381$	19941.1	0.00703781
$382$	5.88858e6	2.06468
$383$	919600.	0.320333	0.160167	−	0.987090i	$-0.448797\pi$
$383$	919600.	0.320333	0.160167	+	0.987090i	$0.448797\pi$
$384$	17650.7	0.00610850
$385$	0	0
$386$	619807.	0.211733
$387$	449289.	0.152492
$388$	3.21106e6	1.08285
$389$	−3.13282e6	−1.04969	−0.524845	−	0.851198i	$-0.675877\pi$
$389$	−3.13282e6	−1.04969	−0.524845	+	0.851198i	$0.675877\pi$
$390$	0	0
$391$	−4.85216e6	−1.60507
$392$	274692.	0.0902883
$393$	−36502.3	−0.0119217
$394$	545535.	0.177044
$395$	0	0
$396$	1.91952e6	0.615113
$397$	−3.35221e6	−1.06747	−0.533735	−	0.845652i	$-0.679212\pi$
$397$	−3.35221e6	−1.06747	−0.533735	+	0.845652i	$0.679212\pi$
$398$	4.50744e6	1.42634
$399$	18610.0	0.00585213
$400$	0	0
$401$	5.38156e6	1.67127	0.835636	−	0.549283i	$-0.185099\pi$
$401$	5.38156e6	1.67127	0.835636	+	0.549283i	$0.185099\pi$
$402$	307.686	9.49606e−5 0
$403$	−3.30116e6	−1.01252
$404$	1.98028e6	0.603634
$405$	0	0
$406$	1.65459e6	0.498166
$407$	6.88040e6	2.05887
$408$	−16449.8	−0.00489228
$409$	1.25001e6	0.369493	0.184747	−	0.982786i	$-0.440854\pi$
$409$	1.25001e6	0.369493	0.184747	+	0.982786i	$0.440854\pi$
$410$	0	0
$411$	14402.4	0.00420563
$412$	−968856.	−0.281201
$413$	2.50835e6	0.723623
$414$	4.81789e6	1.38152
$415$	0	0
$416$	−4.94569e6	−1.40118
$417$	15963.6	0.00449563
$418$	−4.83673e6	−1.35398
$419$	6.20031e6	1.72535	0.862677	−	0.505755i	$-0.168786\pi$
$419$	6.20031e6	1.72535	0.862677	+	0.505755i	$0.168786\pi$
$420$	0	0
$421$	−302555.	−0.0831953	−0.0415977	−	0.999134i	$-0.513245\pi$
$421$	−302555.	−0.0831953	−0.0415977	+	0.999134i	$0.513245\pi$
$422$	4.66486e6	1.27514
$423$	1.31824e6	0.358215
$424$	−2.60170e6	−0.702816
$425$	0	0
$426$	−23507.9	−0.00627609
$427$	2.68358e6	0.712271
$428$	312443.	0.0824445
$429$	−34765.7	−0.00912028
$430$	0	0
$431$	−5.21483e6	−1.35222	−0.676109	−	0.736802i	$-0.736335\pi$
$431$	−5.21483e6	−1.35222	−0.676109	+	0.736802i	$0.736335\pi$
$432$	61129.1	0.0157594
$433$	−776857.	−0.199123	−0.0995614	−	0.995031i	$-0.531744\pi$
$433$	−776857.	−0.199123	−0.0995614	+	0.995031i	$0.531744\pi$
$434$	−3.35506e6	−0.855020
$435$	0	0
$436$	1.00591e6	0.253422
$437$	−4.45193e6	−1.11518
$438$	36973.9	0.00920894
$439$	1.24476e6	0.308264	0.154132	−	0.988050i	$-0.450742\pi$
$439$	1.24476e6	0.308264	0.154132	+	0.988050i	$0.450742\pi$
$440$	0	0
$441$	697116.	0.170690
$442$	1.02109e7	2.48604
$443$	−4.21055e6	−1.01937	−0.509683	−	0.860362i	$-0.670237\pi$
$443$	−4.21055e6	−1.01937	−0.509683	+	0.860362i	$0.670237\pi$
$444$	−29537.2	−0.00711068
$445$	0	0
$446$	5.56323e6	1.32431
$447$	33478.3	0.00792491
$448$	−214910.	−0.0505896
$449$	1.05348e6	0.246610	0.123305	−	0.992369i	$-0.460651\pi$
$449$	1.05348e6	0.246610	0.123305	+	0.992369i	$0.460651\pi$
$450$	0	0
$451$	−8.76833e6	−2.02990
$452$	−206978.	−0.0476516
$453$	−30471.4	−0.00697664
$454$	1.01212e6	0.230459
$455$	0	0
$456$	−15093.0	−0.00339909
$457$	−1.12128e6	−0.251145	−0.125573	−	0.992084i	$-0.540077\pi$
$457$	−1.12128e6	−0.251145	−0.125573	+	0.992084i	$0.540077\pi$
$458$	7.75178e6	1.72678
$459$	−83494.7	−0.0184981
$460$	0	0
$461$	3.01479e6	0.660702	0.330351	−	0.943858i	$-0.392833\pi$
$461$	3.01479e6	0.660702	0.330351	+	0.943858i	$0.392833\pi$
$462$	−35333.4	−0.00770159
$463$	4.23502e6	0.918128	0.459064	−	0.888403i	$-0.348185\pi$
$463$	4.23502e6	0.918128	0.459064	+	0.888403i	$0.348185\pi$
$464$	−2.51098e6	−0.541437
$465$	0	0
$466$	−4.32826e6	−0.923313
$467$	−1.20379e6	−0.255423	−0.127711	−	0.991811i	$-0.540763\pi$
$467$	−1.20379e6	−0.255423	−0.127711	+	0.991811i	$0.540763\pi$
$468$	−3.71808e6	−0.784701
$469$	51742.3	0.0108621
$470$	0	0
$471$	10018.6	0.00208091
$472$	−2.03431e6	−0.420302
$473$	788235.	0.161995
$474$	32727.0	0.00669053
$475$	0	0
$476$	3.80566e6	0.769862
$477$	−6.60260e6	−1.32868
$478$	−1.15406e7	−2.31024
$479$	−7.05576e6	−1.40509	−0.702546	−	0.711638i	$-0.747954\pi$
$479$	−7.05576e6	−1.40509	−0.702546	+	0.711638i	$0.747954\pi$
$480$	0	0
$481$	−1.33272e7	−2.62650
$482$	−6.87184e6	−1.34727
$483$	−32522.3	−0.00634328
$484$	383274.	0.0743698
$485$	0	0
$486$	124358.	0.0238828
$487$	−742868.	−0.141935	−0.0709675	−	0.997479i	$-0.522609\pi$
$487$	−742868.	−0.141935	−0.0709675	+	0.997479i	$0.522609\pi$
$488$	−2.17642e6	−0.413708
$489$	−26446.8	−0.00500151
$490$	0	0
$491$	−5.83817e6	−1.09288	−0.546440	−	0.837498i	$-0.684017\pi$
$491$	−5.83817e6	−1.09288	−0.546440	+	0.837498i	$0.684017\pi$
$492$	37641.9	0.00701066
$493$	3.42968e6	0.635531
$494$	9.36866e6	1.72727
$495$	0	0
$496$	5.09160e6	0.929288
$497$	−3.95321e6	−0.717892
$498$	7368.91	0.00133147
$499$	−5.22798e6	−0.939902	−0.469951	−	0.882693i	$-0.655728\pi$
$499$	−5.22798e6	−0.939902	−0.469951	+	0.882693i	$0.655728\pi$
$500$	0	0
$501$	−55208.3	−0.00982675
$502$	−3.94403e6	−0.698524
$503$	−6.26373e6	−1.10386	−0.551929	−	0.833891i	$-0.686108\pi$
$503$	−6.26373e6	−1.10386	−0.551929	+	0.833891i	$0.686108\pi$
$504$	2.74676e6	0.481664
$505$	0	0
$506$	8.45254e6	1.46761
$507$	30671.1	0.00529919
$508$	3.74151e6	0.643262
$509$	−5.54880e6	−0.949302	−0.474651	−	0.880174i	$-0.657426\pi$
$509$	−5.54880e6	−0.949302	−0.474651	+	0.880174i	$0.657426\pi$
$510$	0	0
$511$	6.21773e6	1.05337
$512$	3.72585e6	0.628131
$513$	−76607.7	−0.0128522
$514$	4.48945e6	0.749524
$515$	0	0
$516$	−3383.85	−0.000559482 0
$517$	2.31273e6	0.380538
$518$	−1.35448e7	−2.21794
$519$	−4037.66	−0.000657978 0
$520$	0	0
$521$	−5.64304e6	−0.910791	−0.455395	−	0.890289i	$-0.650502\pi$
$521$	−5.64304e6	−0.910791	−0.455395	+	0.890289i	$0.650502\pi$
$522$	−3.40546e6	−0.547016
$523$	−6.39179e6	−1.02181	−0.510903	−	0.859638i	$-0.670689\pi$
$523$	−6.39179e6	−1.02181	−0.510903	+	0.859638i	$0.670689\pi$
$524$	−6.84885e6	−1.08966
$525$	0	0
$526$	1.58376e6	0.249589
$527$	−6.95449e6	−1.09078
$528$	53621.5	0.00837055
$529$	1.34373e6	0.208773
$530$	0	0
$531$	−5.16268e6	−0.794582
$532$	3.49175e6	0.534890
$533$	1.69841e7	2.58955
$534$	76229.0	0.0115682
$535$	0	0
$536$	−41963.7	−0.00630902
$537$	15165.1	0.00226939
$538$	1.16069e7	1.72886
$539$	1.22302e6	0.181327
$540$	0	0
$541$	−8.15939e6	−1.19857	−0.599287	−	0.800534i	$-0.704549\pi$
$541$	−8.15939e6	−1.19857	−0.599287	+	0.800534i	$0.704549\pi$
$542$	3.63671e6	0.531753
$543$	1526.96	0.000222243 0
$544$	−1.04190e7	−1.50948
$545$	0	0
$546$	68440.1	0.00982492
$547$	−8.65857e6	−1.23731	−0.618654	−	0.785664i	$-0.712322\pi$
$547$	−8.65857e6	−1.23731	−0.618654	+	0.785664i	$0.712322\pi$
$548$	2.70229e6	0.384398
$549$	−5.52335e6	−0.782116
$550$	0	0
$551$	3.14679e6	0.441559
$552$	26376.1	0.00368437
$553$	5.50355e6	0.765298
$554$	1.13569e7	1.57212
$555$	0	0
$556$	2.99521e6	0.410904
$557$	−2.47692e6	−0.338278	−0.169139	−	0.985592i	$-0.554099\pi$
$557$	−2.47692e6	−0.338278	−0.169139	+	0.985592i	$0.554099\pi$
$558$	6.90538e6	0.938863
$559$	−1.52680e6	−0.206658
$560$	0	0
$561$	−73240.3	−0.00982523
$562$	3.64433e6	0.486717
$563$	3.69026e6	0.490666	0.245333	−	0.969439i	$-0.421103\pi$
$563$	3.69026e6	0.490666	0.245333	+	0.969439i	$0.421103\pi$
$564$	−9928.40	−0.00131426
$565$	0	0
$566$	8.09889e6	1.06264
$567$	6.97047e6	0.910551
$568$	3.20611e6	0.416973
$569$	2.54720e6	0.329823	0.164912	−	0.986308i	$-0.447266\pi$
$569$	2.54720e6	0.329823	0.164912	+	0.986308i	$0.447266\pi$
$570$	0	0
$571$	4.10189e6	0.526494	0.263247	−	0.964728i	$-0.415207\pi$
$571$	4.10189e6	0.526494	0.263247	+	0.964728i	$0.415207\pi$
$572$	−6.52302e6	−0.833602
$573$	81812.9	0.0104096
$574$	1.72614e7	2.18674
$575$	0	0
$576$	442327.	0.0555504
$577$	−5.90027e6	−0.737790	−0.368895	−	0.929471i	$-0.620264\pi$
$577$	−5.90027e6	−0.737790	−0.368895	+	0.929471i	$0.620264\pi$
$578$	1.14181e7	1.42159
$579$	8611.29	0.00106751
$580$	0	0
$581$	1.23920e6	0.152300
$582$	121654.	0.0148874
$583$	−1.15836e7	−1.41148
$584$	−5.04267e6	−0.611827
$585$	0	0
$586$	−6.40756e6	−0.770812
$587$	1.20263e7	1.44058	0.720288	−	0.693675i	$-0.244010\pi$
$587$	1.20263e7	1.44058	0.720288	+	0.693675i	$0.244010\pi$
$588$	−5250.37	−0.000626248 0
$589$	−6.38086e6	−0.757863
$590$	0	0
$591$	7579.40	0.000892619 0
$592$	2.05555e7	2.41059
$593$	1.50115e7	1.75302	0.876511	−	0.481382i	$-0.159865\pi$
$593$	1.50115e7	1.75302	0.876511	+	0.481382i	$0.159865\pi$
$594$	145449.	0.0169139
$595$	0	0
$596$	6.28146e6	0.724344
$597$	62624.1	0.00719127
$598$	−1.63724e7	−1.87223
$599$	1.51427e7	1.72439	0.862197	−	0.506573i	$-0.169088\pi$
$599$	1.51427e7	1.72439	0.862197	+	0.506573i	$0.169088\pi$
$600$	0	0
$601$	−775936.	−0.0876274	−0.0438137	−	0.999040i	$-0.513951\pi$
$601$	−775936.	−0.0876274	−0.0438137	+	0.999040i	$0.513951\pi$
$602$	−1.55173e6	−0.174511
$603$	−106496.	−0.0119272
$604$	−5.71728e6	−0.637671
$605$	0	0
$606$	75025.0	0.00829898
$607$	−1.94645e6	−0.214423	−0.107212	−	0.994236i	$-0.534192\pi$
$607$	−1.94645e6	−0.214423	−0.107212	+	0.994236i	$0.534192\pi$
$608$	−9.55959e6	−1.04877
$609$	22988.0	0.00251164
$610$	0	0
$611$	−4.47971e6	−0.485453
$612$	−7.83280e6	−0.845354
$613$	−1.06888e6	−0.114889	−0.0574443	−	0.998349i	$-0.518295\pi$
$613$	−1.06888e6	−0.114889	−0.0574443	+	0.998349i	$0.518295\pi$
$614$	−1.87528e7	−2.00745
$615$	0	0
$616$	4.81893e6	0.511681
$617$	−7.57280e6	−0.800836	−0.400418	−	0.916333i	$-0.631135\pi$
$617$	−7.57280e6	−0.800836	−0.400418	+	0.916333i	$0.631135\pi$
$618$	−36706.1	−0.00386605
$619$	1.54036e7	1.61583	0.807915	−	0.589299i	$-0.200596\pi$
$619$	1.54036e7	1.61583	0.807915	+	0.589299i	$0.200596\pi$
$620$	0	0
$621$	133878.	0.0139309
$622$	−1.38259e7	−1.43290
$623$	1.28191e7	1.32324
$624$	−103864.	−0.0106783
$625$	0	0
$626$	−1.02029e6	−0.104061
$627$	−67199.1	−0.00682645
$628$	1.87976e6	0.190197
$629$	−2.80762e7	−2.82951
$630$	0	0
$631$	1.62272e7	1.62245	0.811225	−	0.584734i	$-0.198801\pi$
$631$	1.62272e7	1.62245	0.811225	+	0.584734i	$0.198801\pi$
$632$	−4.46346e6	−0.444508
$633$	64811.3	0.00642897
$634$	4.88160e6	0.482325
$635$	0	0
$636$	49727.9	0.00487480
$637$	−2.36898e6	−0.231319
$638$	−5.97456e6	−0.581105
$639$	8.13650e6	0.788289
$640$	0	0
$641$	−9.49485e6	−0.912732	−0.456366	−	0.889792i	$-0.650849\pi$
$641$	−9.49485e6	−0.912732	−0.456366	+	0.889792i	$0.650849\pi$
$642$	11837.2	0.00113348
$643$	−1.29480e7	−1.23503	−0.617514	−	0.786560i	$-0.711860\pi$
$643$	−1.29480e7	−1.23503	−0.617514	+	0.786560i	$0.711860\pi$
$644$	−6.10210e6	−0.579782
$645$	0	0
$646$	1.97368e7	1.86078
$647$	−1.41462e7	−1.32855	−0.664276	−	0.747488i	$-0.731260\pi$
$647$	−1.41462e7	−1.32855	−0.664276	+	0.747488i	$0.731260\pi$
$648$	−5.65315e6	−0.528875
$649$	−9.05742e6	−0.844098
$650$	0	0
$651$	−46613.6	−0.00431082
$652$	−4.96216e6	−0.457143
$653$	−2.05138e7	−1.88262	−0.941310	−	0.337543i	$-0.890404\pi$
$653$	−2.05138e7	−1.88262	−0.941310	+	0.337543i	$0.890404\pi$
$654$	38110.0	0.00348414
$655$	0	0
$656$	−2.61957e7	−2.37668
$657$	−1.27973e7	−1.15666
$658$	−4.55285e6	−0.409939
$659$	9.17922e6	0.823365	0.411682	−	0.911327i	$-0.364941\pi$
$659$	9.17922e6	0.823365	0.411682	+	0.911327i	$0.364941\pi$
$660$	0	0
$661$	1.71241e7	1.52442	0.762210	−	0.647330i	$-0.224114\pi$
$661$	1.71241e7	1.52442	0.762210	+	0.647330i	$0.224114\pi$
$662$	−2.58042e7	−2.28847
$663$	141865.	0.0125341
$664$	−1.00501e6	−0.0884604
$665$	0	0
$666$	2.78779e7	2.43543
$667$	−5.49924e6	−0.478617
$668$	−1.03586e7	−0.898174
$669$	77292.7	0.00667688
$670$	0	0
$671$	−9.69019e6	−0.830856
$672$	−69834.9	−0.00596554
$673$	−1.25597e7	−1.06891	−0.534457	−	0.845195i	$-0.679484\pi$
$673$	−1.25597e7	−1.06891	−0.534457	+	0.845195i	$0.679484\pi$
$674$	4.12960e6	0.350153
$675$	0	0
$676$	5.75476e6	0.484351
$677$	2.11693e7	1.77515	0.887573	−	0.460666i	$-0.152390\pi$
$677$	2.11693e7	1.77515	0.887573	+	0.460666i	$0.152390\pi$
$678$	−7841.56	−0.000655131 0
$679$	2.04581e7	1.70290
$680$	0	0
$681$	14061.9	0.00116192
$682$	1.21148e7	0.997371
$683$	1.71086e7	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$683$	1.71086e7	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$684$	−7.18672e6	−0.587341
$685$	0	0
$686$	−1.65125e7	−1.33968
$687$	107699.	0.00870605
$688$	2.35488e6	0.189670
$689$	2.24373e7	1.80062
$690$	0	0
$691$	2.52987e6	0.201559	0.100780	−	0.994909i	$-0.467866\pi$
$691$	2.52987e6	0.201559	0.100780	+	0.994909i	$0.467866\pi$
$692$	−757577.	−0.0601397
$693$	1.22295e7	0.967333
$694$	2.47436e7	1.95014
$695$	0	0
$696$	−18643.6	−0.00145884
$697$	3.57801e7	2.78971
$698$	1.29574e7	1.00665
$699$	−60134.7	−0.00465514
$700$	0	0
$701$	1.55389e7	1.19433	0.597166	−	0.802118i	$-0.296293\pi$
$701$	1.55389e7	1.19433	0.597166	+	0.802118i	$0.296293\pi$
$702$	−281732.	−0.0215771
$703$	−2.57604e7	−1.96591
$704$	776021.	0.0590122
$705$	0	0
$706$	−8.64408e6	−0.652690
$707$	1.26166e7	0.949280
$708$	38883.0	0.00291525
$709$	−1.62447e7	−1.21366	−0.606828	−	0.794833i	$-0.707558\pi$
$709$	−1.62447e7	−1.21366	−0.606828	+	0.794833i	$0.707558\pi$
$710$	0	0
$711$	−1.13274e7	−0.840342
$712$	−1.03965e7	−0.768575
$713$	1.11510e7	0.821468
$714$	144181.	0.0105843
$715$	0	0
$716$	2.84539e6	0.207424
$717$	−160339.	−0.0116477
$718$	−5.48703e6	−0.397216
$719$	3.37931e6	0.243784	0.121892	−	0.992543i	$-0.461104\pi$
$719$	3.37931e6	0.243784	0.121892	+	0.992543i	$0.461104\pi$
$720$	0	0
$721$	−6.17270e6	−0.442219
$722$	507510.	0.0362328
$723$	−95473.9	−0.00679265
$724$	286500.	0.0203132
$725$	0	0
$726$	14520.7	0.00102246
$727$	9.54555e6	0.669831	0.334915	−	0.942248i	$-0.391292\pi$
$727$	9.54555e6	0.669831	0.334915	+	0.942248i	$0.391292\pi$
$728$	−9.33419e6	−0.652752
$729$	−1.43455e7	−0.999759
$730$	0	0
$731$	−3.21647e6	−0.222631
$732$	41599.4	0.00286952
$733$	3.41149e6	0.234522	0.117261	−	0.993101i	$-0.462589\pi$
$733$	3.41149e6	0.234522	0.117261	+	0.993101i	$0.462589\pi$
$734$	2.30834e7	1.58146
$735$	0	0
$736$	1.67061e7	1.13679
$737$	−186837.	−0.0126705
$738$	−3.55274e7	−2.40117
$739$	1.81197e7	1.22051	0.610253	−	0.792207i	$-0.291068\pi$
$739$	1.81197e7	1.22051	0.610253	+	0.792207i	$0.291068\pi$
$740$	0	0
$741$	130164.	0.00870851
$742$	2.28037e7	1.52053
$743$	1.89052e7	1.25634	0.628172	−	0.778074i	$-0.283803\pi$
$743$	1.89052e7	1.25634	0.628172	+	0.778074i	$0.283803\pi$
$744$	37804.3	0.00250385
$745$	0	0
$746$	−1.12300e7	−0.738810
$747$	−2.55051e6	−0.167235
$748$	−1.37419e7	−0.898035
$749$	1.99062e6	0.129653
$750$	0	0
$751$	1.28977e7	0.834474	0.417237	−	0.908798i	$-0.362998\pi$
$751$	1.28977e7	0.834474	0.417237	+	0.908798i	$0.362998\pi$
$752$	6.90936e6	0.445546
$753$	−54796.5	−0.00352180
$754$	1.15726e7	0.741316
$755$	0	0
$756$	−105003.	−0.00668188
$757$	−2.27614e7	−1.44364	−0.721822	−	0.692079i	$-0.756695\pi$
$757$	−2.27614e7	−1.44364	−0.721822	+	0.692079i	$0.756695\pi$
$758$	−3.08016e7	−1.94715
$759$	117435.	0.00739936
$760$	0	0
$761$	−2.59285e7	−1.62299	−0.811496	−	0.584358i	$-0.801346\pi$
$761$	−2.59285e7	−1.62299	−0.811496	+	0.584358i	$0.801346\pi$
$762$	141751.	0.00884380
$763$	6.40880e6	0.398534
$764$	1.53504e7	0.951450
$765$	0	0
$766$	6.53695e6	0.402535
$767$	1.75441e7	1.07682
$768$	131223.	0.00802798
$769$	3.35573e6	0.204631	0.102315	−	0.994752i	$-0.467375\pi$
$769$	3.35573e6	0.204631	0.102315	+	0.994752i	$0.467375\pi$
$770$	0	0
$771$	62374.2	0.00377893
$772$	1.61572e6	0.0975714
$773$	3.66397e6	0.220548	0.110274	−	0.993901i	$-0.464827\pi$
$773$	3.66397e6	0.220548	0.110274	+	0.993901i	$0.464827\pi$
$774$	3.19376e6	0.191624
$775$	0	0
$776$	−1.65918e7	−0.989097
$777$	−188185.	−0.0111823
$778$	−2.22696e7	−1.31906
$779$	3.28288e7	1.93826
$780$	0	0
$781$	1.42747e7	0.837413
$782$	−3.44914e7	−2.01695
$783$	−94629.5	−0.00551597
$784$	3.65383e6	0.212304
$785$	0	0
$786$	−259476.	−0.0149810
$787$	−1.16621e7	−0.671182	−0.335591	−	0.942008i	$-0.608936\pi$
$787$	−1.16621e7	−0.671182	−0.335591	+	0.942008i	$0.608936\pi$
$788$	1.42211e6	0.0815861
$789$	22004.0	0.00125837
$790$	0	0
$791$	−1.31868e6	−0.0749373
$792$	−9.91831e6	−0.561856
$793$	1.87697e7	1.05992
$794$	−2.38291e7	−1.34140
$795$	0	0
$796$	1.17500e7	0.657288
$797$	2.94091e7	1.63997	0.819985	−	0.572385i	$-0.193982\pi$
$797$	2.94091e7	1.63997	0.819985	+	0.572385i	$0.193982\pi$
$798$	132289.	0.00735387
$799$	−9.43732e6	−0.522976
$800$	0	0
$801$	−2.63842e7	−1.45299
$802$	3.82547e7	2.10014
$803$	−2.24517e7	−1.22874
$804$	802.079	4.37600e−5 0
$805$	0	0
$806$	−2.34662e7	−1.27235
$807$	161260.	0.00871653
$808$	−1.02323e7	−0.551370
$809$	7.70945e6	0.414145	0.207072	−	0.978326i	$-0.433606\pi$
$809$	7.70945e6	0.414145	0.207072	+	0.978326i	$0.433606\pi$
$810$	0	0
$811$	−8.12802e6	−0.433943	−0.216971	−	0.976178i	$-0.569618\pi$
$811$	−8.12802e6	−0.433943	−0.216971	+	0.976178i	$0.569618\pi$
$812$	4.31318e6	0.229566
$813$	50526.6	0.00268098
$814$	4.89092e7	2.58720
$815$	0	0
$816$	−218808.	−0.0115037
$817$	−2.95117e6	−0.154681
$818$	8.88569e6	0.464310
$819$	−2.36884e7	−1.23403
$820$	0	0
$821$	2.49664e7	1.29270	0.646351	−	0.763040i	$-0.276294\pi$
$821$	2.49664e7	1.29270	0.646351	+	0.763040i	$0.276294\pi$
$822$	102379.	0.00528484
$823$	−5.06452e6	−0.260639	−0.130319	−	0.991472i	$-0.541600\pi$
$823$	−5.06452e6	−0.260639	−0.130319	+	0.991472i	$0.541600\pi$
$824$	5.00615e6	0.256854
$825$	0	0
$826$	1.78305e7	0.909314
$827$	1.44859e7	0.736516	0.368258	−	0.929724i	$-0.379954\pi$
$827$	1.44859e7	0.736516	0.368258	+	0.929724i	$0.379954\pi$
$828$	1.25593e7	0.636635
$829$	−8.61625e6	−0.435444	−0.217722	−	0.976011i	$-0.569863\pi$
$829$	−8.61625e6	−0.435444	−0.217722	+	0.976011i	$0.569863\pi$
$830$	0	0
$831$	157787.	0.00792626
$832$	−1.50314e6	−0.0752819
$833$	−4.99067e6	−0.249199
$834$	113477.	0.00564927
$835$	0	0
$836$	−1.26084e7	−0.623943
$837$	191884.	0.00946727
$838$	4.40748e7	2.16810
$839$	−1.30605e7	−0.640552	−0.320276	−	0.947324i	$-0.603776\pi$
$839$	−1.30605e7	−0.640552	−0.320276	+	0.947324i	$0.603776\pi$
$840$	0	0
$841$	−1.66241e7	−0.810490
$842$	−2.15070e6	−0.104544
$843$	50632.5	0.00245392
$844$	1.21604e7	0.587614
$845$	0	0
$846$	9.37067e6	0.450137
$847$	2.44189e6	0.116955
$848$	−3.46065e7	−1.65260
$849$	112522.	0.00535757
$850$	0	0
$851$	4.50181e7	2.13090
$852$	−61280.5	−0.00289217
$853$	−1.82534e7	−0.858954	−0.429477	−	0.903078i	$-0.641302\pi$
$853$	−1.82534e7	−0.858954	−0.429477	+	0.903078i	$0.641302\pi$
$854$	1.90762e7	0.895049
$855$	0	0
$856$	−1.61442e6	−0.0753064
$857$	−1.92196e7	−0.893908	−0.446954	−	0.894557i	$-0.647491\pi$
$857$	−1.92196e7	−0.893908	−0.446954	+	0.894557i	$0.647491\pi$
$858$	−247131.	−0.0114607
$859$	−1.42483e7	−0.658839	−0.329419	−	0.944184i	$-0.606853\pi$
$859$	−1.42483e7	−0.658839	−0.329419	+	0.944184i	$0.606853\pi$
$860$	0	0
$861$	239821.	0.0110250
$862$	−3.70695e7	−1.69921
$863$	3.46110e7	1.58193	0.790965	−	0.611861i	$-0.209579\pi$
$863$	3.46110e7	1.58193	0.790965	+	0.611861i	$0.209579\pi$
$864$	287474.	0.0131013
$865$	0	0
$866$	−5.52227e6	−0.250220
$867$	158637.	0.00716732
$868$	−8.74600e6	−0.394013
$869$	−1.98729e7	−0.892711
$870$	0	0
$871$	361900.	0.0161638
$872$	−5.19763e6	−0.231480
$873$	−4.21067e7	−1.86989
$874$	−3.16465e7	−1.40135
$875$	0	0
$876$	96383.7	0.00424369
$877$	−3.01990e7	−1.32585	−0.662924	−	0.748687i	$-0.730685\pi$
$877$	−3.01990e7	−1.32585	−0.662924	+	0.748687i	$0.730685\pi$
$878$	8.84833e6	0.387369
$879$	−89023.4	−0.00388626
$880$	0	0
$881$	−2.91457e6	−0.126513	−0.0632564	−	0.997997i	$-0.520149\pi$
$881$	−2.91457e6	−0.126513	−0.0632564	+	0.997997i	$0.520149\pi$
$882$	4.95543e6	0.214492
$883$	−3.53875e7	−1.52738	−0.763692	−	0.645581i	$-0.776615\pi$
$883$	−3.53875e7	−1.52738	−0.763692	+	0.645581i	$0.776615\pi$
$884$	2.66178e7	1.14562
$885$	0	0
$886$	−2.99306e7	−1.28095
$887$	1.56425e7	0.667571	0.333785	−	0.942649i	$-0.391674\pi$
$887$	1.56425e7	0.667571	0.333785	+	0.942649i	$0.391674\pi$
$888$	152621.	0.00649503
$889$	2.38376e7	1.01160
$890$	0	0
$891$	−2.51698e7	−1.06215
$892$	1.45023e7	0.610273
$893$	−8.65889e6	−0.363357
$894$	237980.	0.00995855
$895$	0	0
$896$	2.10997e7	0.878024
$897$	−227470.	−0.00943938
$898$	7.48865e6	0.309894
$899$	−7.88194e6	−0.325263
$900$	0	0
$901$	4.72682e7	1.93980
$902$	−6.23295e7	−2.55080
$903$	−21558.9	−0.000879848 0
$904$	1.06947e6	0.0435258
$905$	0	0
$906$	−216605.	−0.00876694
$907$	1.98853e7	0.802627	0.401314	−	0.915941i	$-0.368554\pi$
$907$	1.98853e7	0.802627	0.401314	+	0.915941i	$0.368554\pi$
$908$	2.63841e6	0.106201
$909$	−2.59675e7	−1.04237
$910$	0	0
$911$	−8.53179e6	−0.340600	−0.170300	−	0.985392i	$-0.554474\pi$
$911$	−8.53179e6	−0.340600	−0.170300	+	0.985392i	$0.554474\pi$
$912$	−200760.	−0.00799263
$913$	−4.47463e6	−0.177656
$914$	−7.97062e6	−0.315592
$915$	0	0
$916$	2.02074e7	0.795741
$917$	−4.36349e7	−1.71360
$918$	−593520.	−0.0232449
$919$	−4.78113e7	−1.86742	−0.933710	−	0.358031i	$-0.883448\pi$
$919$	−4.78113e7	−1.86742	−0.933710	+	0.358031i	$0.883448\pi$
$920$	0	0
$921$	−260542.	−0.0101211
$922$	2.14306e7	0.830246
$923$	−2.76499e7	−1.06829
$924$	−92107.3	−0.00354907
$925$	0	0
$926$	3.01046e7	1.15373
$927$	1.27046e7	0.485583
$928$	−1.18085e7	−0.450115
$929$	−3.45719e7	−1.31427	−0.657135	−	0.753773i	$-0.728232\pi$
$929$	−3.45719e7	−1.31427	−0.657135	+	0.753773i	$0.728232\pi$
$930$	0	0
$931$	−4.57902e6	−0.173141
$932$	−1.12829e7	−0.425484
$933$	−192090.	−0.00722437
$934$	−8.55714e6	−0.320968
$935$	0	0
$936$	1.92116e7	0.716760
$937$	1.64475e7	0.612001	0.306000	−	0.952031i	$-0.401009\pi$
$937$	1.64475e7	0.612001	0.306000	+	0.952031i	$0.401009\pi$
$938$	367809.	0.0136494
$939$	−14175.4	−0.000524651 0
$940$	0	0
$941$	−7.61355e6	−0.280293	−0.140147	−	0.990131i	$-0.544757\pi$
$941$	−7.61355e6	−0.280293	−0.140147	+	0.990131i	$0.544757\pi$
$942$	71216.7	0.00261490
$943$	−5.73707e7	−2.10093
$944$	−2.70594e7	−0.988298
$945$	0	0
$946$	5.60315e6	0.203566
$947$	2.75580e7	0.998556	0.499278	−	0.866442i	$-0.333599\pi$
$947$	2.75580e7	0.998556	0.499278	+	0.866442i	$0.333599\pi$
$948$	85313.0	0.00308315
$949$	4.34885e7	1.56751
$950$	0	0
$951$	67822.6	0.00243177
$952$	−1.96641e7	−0.703206
$953$	1.64542e7	0.586875	0.293437	−	0.955978i	$-0.405201\pi$
$953$	1.64542e7	0.586875	0.293437	+	0.955978i	$0.405201\pi$
$954$	−4.69344e7	−1.66963
$955$	0	0
$956$	−3.00840e7	−1.06461
$957$	−83007.6	−0.00292980
$958$	−5.01557e7	−1.76566
$959$	1.72167e7	0.604508
$960$	0	0
$961$	−1.26466e7	−0.441740
$962$	−9.47362e7	−3.30049
$963$	−4.09708e6	−0.142367
$964$	−1.79136e7	−0.620854
$965$	0	0
$966$	−231184.	−0.00797105
$967$	−1.15446e7	−0.397021	−0.198511	−	0.980099i	$-0.563610\pi$
$967$	−1.15446e7	−0.397021	−0.198511	+	0.980099i	$0.563610\pi$
$968$	−1.98041e6	−0.0679307
$969$	274213.	0.00938163
$970$	0	0
$971$	−1.66291e7	−0.566004	−0.283002	−	0.959119i	$-0.591330\pi$
$971$	−1.66291e7	−0.566004	−0.283002	+	0.959119i	$0.591330\pi$
$972$	324178.	0.0110057
$973$	1.90829e7	0.646193
$974$	−5.28066e6	−0.178357
$975$	0	0
$976$	−2.89498e7	−0.972794
$977$	3.56928e7	1.19631	0.598156	−	0.801380i	$-0.295900\pi$
$977$	3.56928e7	1.19631	0.598156	+	0.801380i	$0.295900\pi$
$978$	−187997.	−0.00628497
$979$	−4.62886e7	−1.54354
$980$	0	0
$981$	−1.31906e7	−0.437614
$982$	−4.15005e7	−1.37333
$983$	−2.26067e6	−0.0746197	−0.0373099	−	0.999304i	$-0.511879\pi$
$983$	−2.26067e6	−0.0746197	−0.0373099	+	0.999304i	$0.511879\pi$
$984$	−194499.	−0.00640367
$985$	0	0
$986$	2.43798e7	0.798616
$987$	−63255.1	−0.00206682
$988$	2.44223e7	0.795965
$989$	5.15738e6	0.167663
$990$	0	0
$991$	3.91331e7	1.26579	0.632893	−	0.774240i	$-0.281867\pi$
$991$	3.91331e7	1.26579	0.632893	+	0.774240i	$0.281867\pi$
$992$	2.39445e7	0.772549
$993$	−358510.	−0.0115380
$994$	−2.81013e7	−0.902113
$995$	0	0
$996$	19209.3	0.000613570 0
$997$	−1.77519e7	−0.565598	−0.282799	−	0.959179i	$-0.591263\pi$
$997$	−1.77519e7	−0.565598	−0.282799	+	0.959179i	$0.591263\pi$
$998$	−3.71630e7	−1.18109
$999$	774660.	0.0245582

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.l.1.41		52
5.2	odd	4		215.6.b.a.44.83	yes	104
5.3	odd	4		215.6.b.a.44.22	✓	104
5.4	even	2		1075.6.a.k.1.12		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.6.b.a.44.22	✓	104	5.3	odd	4
215.6.b.a.44.83	yes	104	5.2	odd	4
1075.6.a.k.1.12		52	5.4	even	2
1075.6.a.l.1.41		52	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+7.10848 q^{2} +0.0987616 q^{3} +18.5304 q^{4} +0.702045 q^{6} +118.060 q^{7} -95.7481 q^{8} -242.990 q^{9} +O(q^{10})\)	\(q+7.10848 q^{2} +0.0987616 q^{3} +18.5304 q^{4} +0.702045 q^{6} +118.060 q^{7} -95.7481 q^{8} -242.990 q^{9} -426.303 q^{11} +1.83009 q^{12} +825.742 q^{13} +839.225 q^{14} -1273.60 q^{16} +1739.57 q^{17} -1727.29 q^{18} +1596.09 q^{19} +11.6598 q^{21} -3030.37 q^{22} -2789.28 q^{23} -9.45624 q^{24} +5869.77 q^{26} -47.9972 q^{27} +2187.70 q^{28} +1971.56 q^{29} -3997.81 q^{31} -5989.39 q^{32} -42.1024 q^{33} +12365.7 q^{34} -4502.71 q^{36} -16139.7 q^{37} +11345.7 q^{38} +81.5516 q^{39} +20568.3 q^{41} +82.8832 q^{42} -1849.00 q^{43} -7899.58 q^{44} -19827.5 q^{46} -5425.07 q^{47} -125.782 q^{48} -2868.91 q^{49} +171.803 q^{51} +15301.4 q^{52} +27172.3 q^{53} -341.187 q^{54} -11304.0 q^{56} +157.632 q^{57} +14014.8 q^{58} +21246.4 q^{59} +22730.7 q^{61} -28418.4 q^{62} -28687.4 q^{63} -1820.35 q^{64} -299.284 q^{66} +438.272 q^{67} +32235.1 q^{68} -275.474 q^{69} -33484.9 q^{71} +23265.9 q^{72} +52666.0 q^{73} -114729. q^{74} +29576.2 q^{76} -50329.2 q^{77} +579.708 q^{78} +46616.7 q^{79} +59041.9 q^{81} +146209. q^{82} +10496.4 q^{83} +216.060 q^{84} -13143.6 q^{86} +194.715 q^{87} +40817.7 q^{88} +108581. q^{89} +97486.9 q^{91} -51686.5 q^{92} -394.830 q^{93} -38564.0 q^{94} -591.522 q^{96} +173286. q^{97} -20393.5 q^{98} +103588. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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