LMFDB - Embedded newform 75.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,6,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, 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("75.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	75.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:	$12.0287864860$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.6119$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.6119 q^{2} +9.00000 q^{3} +80.6119 q^{4} +95.5069 q^{6} -105.790 q^{7} +515.863 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+10.6119 q^{2} +9.00000 q^{3} +80.6119 q^{4} +95.5069 q^{6} -105.790 q^{7} +515.863 q^{8} +81.0000 q^{9} +447.580 q^{11} +725.507 q^{12} -276.210 q^{13} -1122.63 q^{14} +2894.69 q^{16} -1826.95 q^{17} +859.562 q^{18} -1371.27 q^{19} -952.110 q^{21} +4749.66 q^{22} +1122.63 q^{23} +4642.77 q^{24} -2931.11 q^{26} +729.000 q^{27} -8527.93 q^{28} +1621.16 q^{29} -443.690 q^{31} +14210.5 q^{32} +4028.22 q^{33} -19387.4 q^{34} +6529.56 q^{36} -12585.3 q^{37} -14551.7 q^{38} -2485.89 q^{39} +1686.86 q^{41} -10103.7 q^{42} +8867.16 q^{43} +36080.3 q^{44} +11913.2 q^{46} -2777.83 q^{47} +26052.2 q^{48} -5615.48 q^{49} -16442.5 q^{51} -22265.8 q^{52} +30152.2 q^{53} +7736.06 q^{54} -54573.2 q^{56} -12341.4 q^{57} +17203.5 q^{58} -33133.6 q^{59} +25965.4 q^{61} -4708.38 q^{62} -8568.99 q^{63} +58170.0 q^{64} +42747.0 q^{66} +19395.2 q^{67} -147274. q^{68} +10103.7 q^{69} -52846.0 q^{71} +41784.9 q^{72} -35710.0 q^{73} -133554. q^{74} -110541. q^{76} -47349.5 q^{77} -26380.0 q^{78} +91820.6 q^{79} +6561.00 q^{81} +17900.7 q^{82} +20272.9 q^{83} -76751.4 q^{84} +94097.2 q^{86} +14590.4 q^{87} +230890. q^{88} +126629. q^{89} +29220.3 q^{91} +90497.3 q^{92} -3993.21 q^{93} -29478.0 q^{94} +127895. q^{96} +138578. q^{97} -59590.8 q^{98} +36254.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 18 q^{3} + 141 q^{4} + 9 q^{6} + 112 q^{7} + 243 q^{8} + 162 q^{9}+O(q^{10}) $ 2 * q + q^2 + 18 * q^3 + 141 * q^4 + 9 * q^6 + 112 * q^7 + 243 * q^8 + 162 * q^9	\( 2 q + q^{2} + 18 q^{3} + 141 q^{4} + 9 q^{6} + 112 q^{7} + 243 q^{8} + 162 q^{9} + 248 q^{11} + 1269 q^{12} - 876 q^{13} - 3216 q^{14} + 3585 q^{16} - 2036 q^{17} + 81 q^{18} + 1464 q^{19} + 1008 q^{21} + 6668 q^{22} + 3216 q^{23} + 2187 q^{24} + 2834 q^{26} + 1458 q^{27} + 4624 q^{28} + 1948 q^{29} + 2672 q^{31} + 16307 q^{32} + 2232 q^{33} - 17378 q^{34} + 11421 q^{36} - 8668 q^{37} - 41804 q^{38} - 7884 q^{39} - 7628 q^{41} - 28944 q^{42} + 16440 q^{43} + 24028 q^{44} - 8208 q^{46} + 19360 q^{47} + 32265 q^{48} + 25010 q^{49} - 18324 q^{51} - 58486 q^{52} + 14356 q^{53} + 729 q^{54} - 114000 q^{56} + 13176 q^{57} + 14062 q^{58} - 904 q^{59} + 20220 q^{61} - 34656 q^{62} + 9072 q^{63} + 15929 q^{64} + 60012 q^{66} + 12904 q^{67} - 159898 q^{68} + 28944 q^{69} - 40976 q^{71} + 19683 q^{72} - 59124 q^{73} - 171206 q^{74} + 60676 q^{76} - 90816 q^{77} + 25506 q^{78} + 107600 q^{79} + 13122 q^{81} + 107434 q^{82} + 122088 q^{83} + 41616 q^{84} + 21308 q^{86} + 17532 q^{87} + 285348 q^{88} + 103764 q^{89} - 101408 q^{91} + 216912 q^{92} + 24048 q^{93} - 242264 q^{94} + 146763 q^{96} + 24764 q^{97} - 353959 q^{98} + 20088 q^{99}+O(q^{100}) \) 2 * q + q^2 + 18 * q^3 + 141 * q^4 + 9 * q^6 + 112 * q^7 + 243 * q^8 + 162 * q^9 + 248 * q^11 + 1269 * q^12 - 876 * q^13 - 3216 * q^14 + 3585 * q^16 - 2036 * q^17 + 81 * q^18 + 1464 * q^19 + 1008 * q^21 + 6668 * q^22 + 3216 * q^23 + 2187 * q^24 + 2834 * q^26 + 1458 * q^27 + 4624 * q^28 + 1948 * q^29 + 2672 * q^31 + 16307 * q^32 + 2232 * q^33 - 17378 * q^34 + 11421 * q^36 - 8668 * q^37 - 41804 * q^38 - 7884 * q^39 - 7628 * q^41 - 28944 * q^42 + 16440 * q^43 + 24028 * q^44 - 8208 * q^46 + 19360 * q^47 + 32265 * q^48 + 25010 * q^49 - 18324 * q^51 - 58486 * q^52 + 14356 * q^53 + 729 * q^54 - 114000 * q^56 + 13176 * q^57 + 14062 * q^58 - 904 * q^59 + 20220 * q^61 - 34656 * q^62 + 9072 * q^63 + 15929 * q^64 + 60012 * q^66 + 12904 * q^67 - 159898 * q^68 + 28944 * q^69 - 40976 * q^71 + 19683 * q^72 - 59124 * q^73 - 171206 * q^74 + 60676 * q^76 - 90816 * q^77 + 25506 * q^78 + 107600 * q^79 + 13122 * q^81 + 107434 * q^82 + 122088 * q^83 + 41616 * q^84 + 21308 * q^86 + 17532 * q^87 + 285348 * q^88 + 103764 * q^89 - 101408 * q^91 + 216912 * q^92 + 24048 * q^93 - 242264 * q^94 + 146763 * q^96 + 24764 * q^97 - 353959 * q^98 + 20088 * 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 $

$2$	10.6119	1.87593	0.937966	−	0.346727i	$-0.112707\pi$
$2$	10.6119	1.87593	0.937966	+	0.346727i	$0.112707\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	80.6119	2.51912
$5$	0	0
$5$	0	0
$6$	95.5069	1.08307
$7$	−105.790	−0.816017	−0.408009	−	0.912978i	$-0.633777\pi$
$7$	−105.790	−0.816017	−0.408009	+	0.912978i	$0.633777\pi$
$8$	515.863	2.84977
$9$	81.0000	0.333333
$10$	0	0
$11$	447.580	1.11529	0.557646	−	0.830079i	$-0.311705\pi$
$11$	447.580	1.11529	0.557646	+	0.830079i	$0.311705\pi$
$12$	725.507	1.45442
$13$	−276.210	−0.453295	−0.226648	−	0.973977i	$-0.572777\pi$
$13$	−276.210	−0.453295	−0.226648	+	0.973977i	$0.572777\pi$
$14$	−1122.63	−1.53079
$15$	0	0
$16$	2894.69	2.82685
$17$	−1826.95	−1.53322	−0.766610	−	0.642113i	$-0.778058\pi$
$17$	−1826.95	−1.53322	−0.766610	+	0.642113i	$0.778058\pi$
$18$	859.562	0.625311
$19$	−1371.27	−0.871443	−0.435721	−	0.900082i	$-0.643507\pi$
$19$	−1371.27	−0.871443	−0.435721	+	0.900082i	$0.643507\pi$
$20$	0	0
$21$	−952.110	−0.471128
$22$	4749.66	2.09221
$23$	1122.63	0.442504	0.221252	−	0.975217i	$-0.428986\pi$
$23$	1122.63	0.442504	0.221252	+	0.975217i	$0.428986\pi$
$24$	4642.77	1.64531
$25$	0	0
$26$	−2931.11	−0.850351
$27$	729.000	0.192450
$28$	−8527.93	−2.05565
$29$	1621.16	0.357957	0.178979	−	0.983853i	$-0.442721\pi$
$29$	1621.16	0.357957	0.178979	+	0.983853i	$0.442721\pi$
$30$	0	0
$31$	−443.690	−0.0829231	−0.0414615	−	0.999140i	$-0.513201\pi$
$31$	−443.690	−0.0829231	−0.0414615	+	0.999140i	$0.513201\pi$
$32$	14210.5	2.45321
$33$	4028.22	0.643915
$34$	−19387.4	−2.87622
$35$	0	0
$36$	6529.56	0.839707
$37$	−12585.3	−1.51133	−0.755664	−	0.654959i	$-0.772686\pi$
$37$	−12585.3	−1.51133	−0.755664	+	0.654959i	$0.772686\pi$
$38$	−14551.7	−1.63477
$39$	−2485.89	−0.261710
$40$	0	0
$41$	1686.86	0.156718	0.0783591	−	0.996925i	$-0.475032\pi$
$41$	1686.86	0.156718	0.0783591	+	0.996925i	$0.475032\pi$
$42$	−10103.7	−0.883804
$43$	8867.16	0.731330	0.365665	−	0.930747i	$-0.380842\pi$
$43$	8867.16	0.731330	0.365665	+	0.930747i	$0.380842\pi$
$44$	36080.3	2.80956
$45$	0	0
$46$	11913.2	0.830107
$47$	−2777.83	−0.183426	−0.0917130	−	0.995785i	$-0.529234\pi$
$47$	−2777.83	−0.183426	−0.0917130	+	0.995785i	$0.529234\pi$
$48$	26052.2	1.63208
$49$	−5615.48	−0.334115
$50$	0	0
$51$	−16442.5	−0.885205
$52$	−22265.8	−1.14191
$53$	30152.2	1.47445	0.737223	−	0.675649i	$-0.236137\pi$
$53$	30152.2	1.47445	0.737223	+	0.675649i	$0.236137\pi$
$54$	7736.06	0.361023
$55$	0	0
$56$	−54573.2	−2.32546
$57$	−12341.4	−0.503128
$58$	17203.5	0.671503
$59$	−33133.6	−1.23919	−0.619596	−	0.784921i	$-0.712703\pi$
$59$	−33133.6	−1.23919	−0.619596	+	0.784921i	$0.712703\pi$
$60$	0	0
$61$	25965.4	0.893451	0.446726	−	0.894671i	$-0.352590\pi$
$61$	25965.4	0.893451	0.446726	+	0.894671i	$0.352590\pi$
$62$	−4708.38	−0.155558
$63$	−8568.99	−0.272006
$64$	58170.0	1.77521
$65$	0	0
$66$	42747.0	1.20794
$67$	19395.2	0.527846	0.263923	−	0.964544i	$-0.414984\pi$
$67$	19395.2	0.527846	0.263923	+	0.964544i	$0.414984\pi$
$68$	−147274.	−3.86237
$69$	10103.7	0.255480
$70$	0	0
$71$	−52846.0	−1.24413	−0.622066	−	0.782965i	$-0.713706\pi$
$71$	−52846.0	−1.24413	−0.622066	+	0.782965i	$0.713706\pi$
$72$	41784.9	0.949923
$73$	−35710.0	−0.784301	−0.392151	−	0.919901i	$-0.628269\pi$
$73$	−35710.0	−0.784301	−0.392151	+	0.919901i	$0.628269\pi$
$74$	−133554.	−2.83515
$75$	0	0
$76$	−110541.	−2.19527
$77$	−47349.5	−0.910099
$78$	−26380.0	−0.490950
$79$	91820.6	1.65528	0.827642	−	0.561256i	$-0.189682\pi$
$79$	91820.6	1.65528	0.827642	+	0.561256i	$0.189682\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	17900.7	0.293993
$83$	20272.9	0.323014	0.161507	−	0.986872i	$-0.448365\pi$
$83$	20272.9	0.323014	0.161507	+	0.986872i	$0.448365\pi$
$84$	−76751.4	−1.18683
$85$	0	0
$86$	94097.2	1.37193
$87$	14590.4	0.206667
$88$	230890.	3.17833
$89$	126629.	1.69456	0.847282	−	0.531143i	$-0.178237\pi$
$89$	126629.	1.69456	0.847282	+	0.531143i	$0.178237\pi$
$90$	0	0
$91$	29220.3	0.369897
$92$	90497.3	1.11472
$93$	−3993.21	−0.0478756
$94$	−29478.0	−0.344095
$95$	0	0
$96$	127895.	1.41636
$97$	138578.	1.49543	0.747714	−	0.664021i	$-0.231151\pi$
$97$	138578.	1.49543	0.747714	+	0.664021i	$0.231151\pi$
$98$	−59590.8	−0.626778
$99$	36254.0	0.371764
$100$	0	0
$101$	−24568.9	−0.239653	−0.119826	−	0.992795i	$-0.538234\pi$
$101$	−24568.9	−0.239653	−0.119826	+	0.992795i	$0.538234\pi$
$102$	−174486.	−1.66058
$103$	134516.	1.24934	0.624672	−	0.780887i	$-0.285233\pi$
$103$	134516.	1.24934	0.624672	+	0.780887i	$0.285233\pi$
$104$	−142487.	−1.29179
$105$	0	0
$106$	319971.	2.76596
$107$	69262.6	0.584843	0.292422	−	0.956289i	$-0.405539\pi$
$107$	69262.6	0.584843	0.292422	+	0.956289i	$0.405539\pi$
$108$	58766.1	0.484805
$109$	−68878.7	−0.555289	−0.277644	−	0.960684i	$-0.589554\pi$
$109$	−68878.7	−0.555289	−0.277644	+	0.960684i	$0.589554\pi$
$110$	0	0
$111$	−113268.	−0.872566
$112$	−306230.	−2.30676
$113$	−14831.2	−0.109265	−0.0546325	−	0.998507i	$-0.517399\pi$
$113$	−14831.2	−0.109265	−0.0546325	+	0.998507i	$0.517399\pi$
$114$	−130966.	−0.943834
$115$	0	0
$116$	130685.	0.901737
$117$	−22373.0	−0.151098
$118$	−351609.	−2.32464
$119$	193273.	1.25113
$120$	0	0
$121$	39276.8	0.243878
$122$	275542.	1.67605
$123$	15181.7	0.0904813
$124$	−35766.7	−0.208893
$125$	0	0
$126$	−90933.0	−0.510264
$127$	247133.	1.35963	0.679817	−	0.733382i	$-0.262059\pi$
$127$	247133.	1.35963	0.679817	+	0.733382i	$0.262059\pi$
$128$	162556.	0.876956
$129$	79804.4	0.422234
$130$	0	0
$131$	−48024.5	−0.244503	−0.122252	−	0.992499i	$-0.539011\pi$
$131$	−48024.5	−0.244503	−0.122252	+	0.992499i	$0.539011\pi$
$132$	324722.	1.62210
$133$	145067.	0.711113
$134$	205819.	0.990203
$135$	0	0
$136$	−942456.	−4.36932
$137$	−250261.	−1.13918	−0.569590	−	0.821929i	$-0.692898\pi$
$137$	−250261.	−1.13918	−0.569590	+	0.821929i	$0.692898\pi$
$138$	107219.	0.479263
$139$	19048.7	0.0836234	0.0418117	−	0.999126i	$-0.486687\pi$
$139$	19048.7	0.0836234	0.0418117	+	0.999126i	$0.486687\pi$
$140$	0	0
$141$	−25000.5	−0.105901
$142$	−560795.	−2.33391
$143$	−123626.	−0.505557
$144$	234470.	0.942283
$145$	0	0
$146$	−378950.	−1.47130
$147$	−50539.3	−0.192902
$148$	−1.01452e6	−3.80722
$149$	−180224.	−0.665039	−0.332519	−	0.943096i	$-0.607899\pi$
$149$	−180224.	−0.665039	−0.332519	+	0.943096i	$0.607899\pi$
$150$	0	0
$151$	553375.	1.97504	0.987522	−	0.157479i	$-0.0503365\pi$
$151$	553375.	1.97504	0.987522	+	0.157479i	$0.0503365\pi$
$152$	−707388.	−2.48341
$153$	−147983.	−0.511073
$154$	−502467.	−1.70728
$155$	0	0
$156$	−200392.	−0.659279
$157$	544773.	1.76387	0.881935	−	0.471372i	$-0.156241\pi$
$157$	544773.	1.76387	0.881935	+	0.471372i	$0.156241\pi$
$158$	974389.	3.10520
$159$	271370.	0.851272
$160$	0	0
$161$	−118763.	−0.361091
$162$	69624.5	0.208437
$163$	−401608.	−1.18395	−0.591975	−	0.805957i	$-0.701651\pi$
$163$	−401608.	−1.18395	−0.591975	+	0.805957i	$0.701651\pi$
$164$	135981.	0.394792
$165$	0	0
$166$	215134.	0.605952
$167$	−411892.	−1.14286	−0.571428	−	0.820652i	$-0.693611\pi$
$167$	−411892.	−1.14286	−0.571428	+	0.820652i	$0.693611\pi$
$168$	−491158.	−1.34261
$169$	−295001.	−0.794524
$170$	0	0
$171$	−111073.	−0.290481
$172$	714798.	1.84231
$173$	391028.	0.993329	0.496665	−	0.867943i	$-0.334558\pi$
$173$	391028.	0.993329	0.496665	+	0.867943i	$0.334558\pi$
$174$	154832.	0.387692
$175$	0	0
$176$	1.29561e6	3.15277
$177$	−298202.	−0.715447
$178$	1.34377e6	3.17889
$179$	−562896.	−1.31309	−0.656546	−	0.754286i	$-0.727983\pi$
$179$	−562896.	−1.31309	−0.656546	+	0.754286i	$0.727983\pi$
$180$	0	0
$181$	−20889.1	−0.0473939	−0.0236970	−	0.999719i	$-0.507544\pi$
$181$	−20889.1	−0.0473939	−0.0236970	+	0.999719i	$0.507544\pi$
$182$	310082.	0.693901
$183$	233689.	0.515834
$184$	579123.	1.26103
$185$	0	0
$186$	−42375.4	−0.0898115
$187$	−817706.	−1.70999
$188$	−223926.	−0.462072
$189$	−77120.9	−0.157043
$190$	0	0
$191$	−464789.	−0.921875	−0.460938	−	0.887433i	$-0.652487\pi$
$191$	−464789.	−0.921875	−0.460938	+	0.887433i	$0.652487\pi$
$192$	523530.	1.02492
$193$	306696.	0.592673	0.296336	−	0.955084i	$-0.404235\pi$
$193$	306696.	0.592673	0.296336	+	0.955084i	$0.404235\pi$
$194$	1.47057e6	2.80532
$195$	0	0
$196$	−452674.	−0.841677
$197$	119348.	0.219104	0.109552	−	0.993981i	$-0.465058\pi$
$197$	119348.	0.219104	0.109552	+	0.993981i	$0.465058\pi$
$198$	384723.	0.697405
$199$	224688.	0.402204	0.201102	−	0.979570i	$-0.435548\pi$
$199$	224688.	0.402204	0.201102	+	0.979570i	$0.435548\pi$
$200$	0	0
$201$	174557.	0.304752
$202$	−260722.	−0.449572
$203$	−171502.	−0.292099
$204$	−1.32546e6	−2.22994
$205$	0	0
$206$	1.42747e6	2.34369
$207$	90933.0	0.147501
$208$	−799544.	−1.28140
$209$	−613753.	−0.971914
$210$	0	0
$211$	−362491.	−0.560520	−0.280260	−	0.959924i	$-0.590421\pi$
$211$	−362491.	−0.560520	−0.280260	+	0.959924i	$0.590421\pi$
$212$	2.43062e6	3.71431
$213$	−475614.	−0.718300
$214$	735006.	1.09713
$215$	0	0
$216$	376064.	0.548438
$217$	46937.9	0.0676667
$218$	−730932.	−1.04168
$219$	−321390.	−0.452817
$220$	0	0
$221$	504622.	0.695001
$222$	−1.20198e6	−1.63687
$223$	−748336.	−1.00771	−0.503854	−	0.863789i	$-0.668085\pi$
$223$	−748336.	−1.00771	−0.503854	+	0.863789i	$0.668085\pi$
$224$	−1.50333e6	−2.00186
$225$	0	0
$226$	−157387.	−0.204974
$227$	−1.22456e6	−1.57731	−0.788654	−	0.614837i	$-0.789222\pi$
$227$	−1.22456e6	−1.57731	−0.788654	+	0.614837i	$0.789222\pi$
$228$	−994866.	−1.26744
$229$	−1.22358e6	−1.54186	−0.770929	−	0.636921i	$-0.780208\pi$
$229$	−1.22358e6	−1.54186	−0.770929	+	0.636921i	$0.780208\pi$
$230$	0	0
$231$	−426145.	−0.525446
$232$	836297.	1.02009
$233$	−548513.	−0.661907	−0.330954	−	0.943647i	$-0.607370\pi$
$233$	−548513.	−0.661907	−0.330954	+	0.943647i	$0.607370\pi$
$234$	−237420.	−0.283450
$235$	0	0
$236$	−2.67096e6	−3.12167
$237$	826386.	0.955679
$238$	2.05099e6	2.34704
$239$	−858038.	−0.971654	−0.485827	−	0.874055i	$-0.661482\pi$
$239$	−858038.	−0.971654	−0.485827	+	0.874055i	$0.661482\pi$
$240$	0	0
$241$	1.31732e6	1.46100	0.730499	−	0.682914i	$-0.239288\pi$
$241$	1.31732e6	1.46100	0.730499	+	0.682914i	$0.239288\pi$
$242$	416801.	0.457499
$243$	59049.0	0.0641500
$244$	2.09312e6	2.25071
$245$	0	0
$246$	161107.	0.169737
$247$	378758.	0.395021
$248$	−228883.	−0.236311
$249$	182456.	0.186492
$250$	0	0
$251$	−905122.	−0.906824	−0.453412	−	0.891301i	$-0.649793\pi$
$251$	−905122.	−0.906824	−0.453412	+	0.891301i	$0.649793\pi$
$252$	−690762.	−0.685216
$253$	502467.	0.493521
$254$	2.62255e6	2.55058
$255$	0	0
$256$	−136416.	−0.130097
$257$	−296690.	−0.280202	−0.140101	−	0.990137i	$-0.544743\pi$
$257$	−296690.	−0.280202	−0.140101	+	0.990137i	$0.544743\pi$
$258$	846875.	0.792082
$259$	1.33140e6	1.23327
$260$	0	0
$261$	131314.	0.119319
$262$	−509630.	−0.458671
$263$	295333.	0.263283	0.131641	−	0.991297i	$-0.457975\pi$
$263$	295333.	0.263283	0.131641	+	0.991297i	$0.457975\pi$
$264$	2.07801e6	1.83501
$265$	0	0
$266$	1.53943e6	1.33400
$267$	1.13966e6	0.978357
$268$	1.56348e6	1.32971
$269$	262660.	0.221316	0.110658	−	0.993859i	$-0.464704\pi$
$269$	262660.	0.221316	0.110658	+	0.993859i	$0.464704\pi$
$270$	0	0
$271$	−689179.	−0.570044	−0.285022	−	0.958521i	$-0.592001\pi$
$271$	−689179.	−0.570044	−0.285022	+	0.958521i	$0.592001\pi$
$272$	−5.28846e6	−4.33418
$273$	262982.	0.213560
$274$	−2.65574e6	−2.13702
$275$	0	0
$276$	814476.	0.643584
$277$	1.14961e6	0.900225	0.450112	−	0.892972i	$-0.351384\pi$
$277$	1.14961e6	0.900225	0.450112	+	0.892972i	$0.351384\pi$
$278$	202142.	0.156872
$279$	−35938.9	−0.0276410
$280$	0	0
$281$	2.41477e6	1.82436	0.912179	−	0.409792i	$-0.134399\pi$
$281$	2.41477e6	1.82436	0.912179	+	0.409792i	$0.134399\pi$
$282$	−265302.	−0.198663
$283$	−922080.	−0.684388	−0.342194	−	0.939629i	$-0.611170\pi$
$283$	−922080.	−0.684388	−0.342194	+	0.939629i	$0.611170\pi$
$284$	−4.26001e6	−3.13412
$285$	0	0
$286$	−1.31190e6	−0.948390
$287$	−178453.	−0.127885
$288$	1.15105e6	0.817737
$289$	1.91789e6	1.35076
$290$	0	0
$291$	1.24720e6	0.863386
$292$	−2.87865e6	−1.97575
$293$	347703.	0.236613	0.118307	−	0.992977i	$-0.462253\pi$
$293$	347703.	0.236613	0.118307	+	0.992977i	$0.462253\pi$
$294$	−536317.	−0.361870
$295$	0	0
$296$	−6.49229e6	−4.30694
$297$	326286.	0.214638
$298$	−1.91251e6	−1.24757
$299$	−310082.	−0.200585
$300$	0	0
$301$	−938057.	−0.596778
$302$	5.87234e6	3.70505
$303$	−221120.	−0.138364
$304$	−3.96941e6	−2.46344
$305$	0	0
$306$	−1.57038e6	−0.958739
$307$	−1.66131e6	−1.00602	−0.503008	−	0.864282i	$-0.667774\pi$
$307$	−1.66131e6	−1.00602	−0.503008	+	0.864282i	$0.667774\pi$
$308$	−3.81693e6	−2.29265
$309$	1.21065e6	0.721309
$310$	0	0
$311$	335376.	0.196622	0.0983108	−	0.995156i	$-0.468656\pi$
$311$	335376.	0.196622	0.0983108	+	0.995156i	$0.468656\pi$
$312$	−1.28238e6	−0.745813
$313$	−2.99160e6	−1.72601	−0.863004	−	0.505196i	$-0.831420\pi$
$313$	−2.99160e6	−1.72601	−0.863004	+	0.505196i	$0.831420\pi$
$314$	5.78106e6	3.30890
$315$	0	0
$316$	7.40183e6	4.16986
$317$	1.35871e6	0.759412	0.379706	−	0.925107i	$-0.376025\pi$
$317$	1.35871e6	0.759412	0.379706	+	0.925107i	$0.376025\pi$
$318$	2.87974e6	1.59693
$319$	725599.	0.399227
$320$	0	0
$321$	623364.	0.337659
$322$	−1.26030e6	−0.677382
$323$	2.50524e6	1.33611
$324$	528895.	0.279902
$325$	0	0
$326$	−4.26181e6	−2.22101
$327$	−619908.	−0.320596
$328$	870189.	0.446610
$329$	293866.	0.149679
$330$	0	0
$331$	−3.46060e6	−1.73613	−0.868063	−	0.496453i	$-0.834635\pi$
$331$	−3.46060e6	−1.73613	−0.868063	+	0.496453i	$0.834635\pi$
$332$	1.63424e6	0.813711
$333$	−1.01941e6	−0.503776
$334$	−4.37094e6	−2.14392
$335$	0	0
$336$	−2.75607e6	−1.33181
$337$	2.15998e6	1.03603	0.518017	−	0.855370i	$-0.326670\pi$
$337$	2.15998e6	1.03603	0.518017	+	0.855370i	$0.326670\pi$
$338$	−3.13051e6	−1.49047
$339$	−133481.	−0.0630842
$340$	0	0
$341$	−198587.	−0.0924835
$342$	−1.17869e6	−0.544923
$343$	2.37207e6	1.08866
$344$	4.57424e6	2.08412
$345$	0	0
$346$	4.14955e6	1.86342
$347$	1.17748e6	0.524966	0.262483	−	0.964937i	$-0.415459\pi$
$347$	1.17748e6	0.524966	0.262483	+	0.964937i	$0.415459\pi$
$348$	1.17616e6	0.520618
$349$	−437128.	−0.192108	−0.0960539	−	0.995376i	$-0.530622\pi$
$349$	−437128.	−0.192108	−0.0960539	+	0.995376i	$0.530622\pi$
$350$	0	0
$351$	−201357.	−0.0872367
$352$	6.36034e6	2.73605
$353$	−511587.	−0.218516	−0.109258	−	0.994013i	$-0.534847\pi$
$353$	−511587.	−0.218516	−0.109258	+	0.994013i	$0.534847\pi$
$354$	−3.16448e6	−1.34213
$355$	0	0
$356$	1.02078e7	4.26881
$357$	1.73946e6	0.722343
$358$	−5.97338e6	−2.46327
$359$	−396627.	−0.162422	−0.0812112	−	0.996697i	$-0.525879\pi$
$359$	−396627.	−0.162422	−0.0812112	+	0.996697i	$0.525879\pi$
$360$	0	0
$361$	−595718.	−0.240587
$362$	−221672.	−0.0889078
$363$	353492.	0.140803
$364$	2.35550e6	0.931815
$365$	0	0
$366$	2.47988e6	0.967670
$367$	−3.07015e6	−1.18986	−0.594928	−	0.803779i	$-0.702819\pi$
$367$	−3.07015e6	−1.18986	−0.594928	+	0.803779i	$0.702819\pi$
$368$	3.24967e6	1.25089
$369$	136636.	0.0522394
$370$	0	0
$371$	−3.18980e6	−1.20317
$372$	−321900.	−0.120605
$373$	2.96347e6	1.10288	0.551440	−	0.834215i	$-0.314079\pi$
$373$	2.96347e6	1.10288	0.551440	+	0.834215i	$0.314079\pi$
$374$	−8.67740e6	−3.20782
$375$	0	0
$376$	−1.43298e6	−0.522721
$377$	−447781.	−0.162260
$378$	−818397.	−0.294601
$379$	48151.7	0.0172192	0.00860961	−	0.999963i	$-0.497259\pi$
$379$	48151.7	0.0172192	0.00860961	+	0.999963i	$0.497259\pi$
$380$	0	0
$381$	2.22420e6	0.784985
$382$	−4.93228e6	−1.72938
$383$	−750637.	−0.261477	−0.130738	−	0.991417i	$-0.541735\pi$
$383$	−750637.	−0.261477	−0.130738	+	0.991417i	$0.541735\pi$
$384$	1.46300e6	0.506311
$385$	0	0
$386$	3.25462e6	1.11181
$387$	718240.	0.243777
$388$	1.11710e7	3.76716
$389$	−3.42194e6	−1.14656	−0.573281	−	0.819359i	$-0.694330\pi$
$389$	−3.42194e6	−1.14656	−0.573281	+	0.819359i	$0.694330\pi$
$390$	0	0
$391$	−2.05099e6	−0.678456
$392$	−2.89682e6	−0.952152
$393$	−432220.	−0.141164
$394$	1.26651e6	0.411024
$395$	0	0
$396$	2.92250e6	0.936519
$397$	−972175.	−0.309577	−0.154788	−	0.987948i	$-0.549470\pi$
$397$	−972175.	−0.309577	−0.154788	+	0.987948i	$0.549470\pi$
$398$	2.38436e6	0.754508
$399$	1.30560e6	0.410561
$400$	0	0
$401$	5.73421e6	1.78079	0.890394	−	0.455190i	$-0.150429\pi$
$401$	5.73421e6	1.78079	0.890394	+	0.455190i	$0.150429\pi$
$402$	1.85237e6	0.571694
$403$	122552.	0.0375886
$404$	−1.98055e6	−0.603714
$405$	0	0
$406$	−1.81996e6	−0.547958
$407$	−5.63292e6	−1.68557
$408$	−8.48210e6	−2.52263
$409$	3.62492e6	1.07149	0.535747	−	0.844379i	$-0.320030\pi$
$409$	3.62492e6	1.07149	0.535747	+	0.844379i	$0.320030\pi$
$410$	0	0
$411$	−2.25235e6	−0.657706
$412$	1.08436e7	3.14725
$413$	3.50520e6	1.01120
$414$	964970.	0.276702
$415$	0	0
$416$	−3.92509e6	−1.11203
$417$	171438.	0.0482800
$418$	−6.51307e6	−1.82324
$419$	4.93470e6	1.37317	0.686587	−	0.727048i	$-0.259108\pi$
$419$	4.93470e6	1.37317	0.686587	+	0.727048i	$0.259108\pi$
$420$	0	0
$421$	−1.23702e6	−0.340150	−0.170075	−	0.985431i	$-0.554401\pi$
$421$	−1.23702e6	−0.340150	−0.170075	+	0.985431i	$0.554401\pi$
$422$	−3.84671e6	−1.05150
$423$	−225004.	−0.0611420
$424$	1.55544e7	4.20183
$425$	0	0
$426$	−5.04716e6	−1.34748
$427$	−2.74688e6	−0.729072
$428$	5.58339e6	1.47329
$429$	−1.11263e6	−0.291883
$430$	0	0
$431$	−2.06478e6	−0.535402	−0.267701	−	0.963502i	$-0.586264\pi$
$431$	−2.06478e6	−0.535402	−0.267701	+	0.963502i	$0.586264\pi$
$432$	2.11023e6	0.544028
$433$	2.00096e6	0.512882	0.256441	−	0.966560i	$-0.417450\pi$
$433$	2.00096e6	0.512882	0.256441	+	0.966560i	$0.417450\pi$
$434$	498100.	0.126938
$435$	0	0
$436$	−5.55244e6	−1.39884
$437$	−1.53943e6	−0.385617
$438$	−3.41055e6	−0.849453
$439$	−1.52639e6	−0.378010	−0.189005	−	0.981976i	$-0.560526\pi$
$439$	−1.52639e6	−0.378010	−0.189005	+	0.981976i	$0.560526\pi$
$440$	0	0
$441$	−454854.	−0.111372
$442$	5.35498e6	1.30377
$443$	−5.34148e6	−1.29316	−0.646580	−	0.762846i	$-0.723801\pi$
$443$	−5.34148e6	−1.29316	−0.646580	+	0.762846i	$0.723801\pi$
$444$	−9.13071e6	−2.19810
$445$	0	0
$446$	−7.94125e6	−1.89039
$447$	−1.62202e6	−0.383960
$448$	−6.15380e6	−1.44860
$449$	−5.63920e6	−1.32008	−0.660042	−	0.751229i	$-0.729461\pi$
$449$	−5.63920e6	−1.32008	−0.660042	+	0.751229i	$0.729461\pi$
$450$	0	0
$451$	755005.	0.174787
$452$	−1.19557e6	−0.275252
$453$	4.98037e6	1.14029
$454$	−1.29949e7	−2.95892
$455$	0	0
$456$	−6.36649e6	−1.43380
$457$	1.58458e6	0.354915	0.177457	−	0.984128i	$-0.443213\pi$
$457$	1.58458e6	0.354915	0.177457	+	0.984128i	$0.443213\pi$
$458$	−1.29845e7	−2.89242
$459$	−1.33185e6	−0.295068
$460$	0	0
$461$	280734.	0.0615238	0.0307619	−	0.999527i	$-0.490207\pi$
$461$	280734.	0.0615238	0.0307619	+	0.999527i	$0.490207\pi$
$462$	−4.52220e6	−0.985700
$463$	1.58025e6	0.342590	0.171295	−	0.985220i	$-0.445205\pi$
$463$	1.58025e6	0.342590	0.171295	+	0.985220i	$0.445205\pi$
$464$	4.69276e6	1.01189
$465$	0	0
$466$	−5.82075e6	−1.24169
$467$	432682.	0.0918071	0.0459036	−	0.998946i	$-0.485383\pi$
$467$	432682.	0.0918071	0.0459036	+	0.998946i	$0.485383\pi$
$468$	−1.80353e6	−0.380635
$469$	−2.05182e6	−0.430732
$470$	0	0
$471$	4.90296e6	1.01837
$472$	−1.70924e7	−3.53141
$473$	3.96876e6	0.815647
$474$	8.76950e6	1.79279
$475$	0	0
$476$	1.55801e7	3.15176
$477$	2.44233e6	0.491482
$478$	−9.10539e6	−1.82276
$479$	4.30302e6	0.856908	0.428454	−	0.903564i	$-0.359058\pi$
$479$	4.30302e6	0.856908	0.428454	+	0.903564i	$0.359058\pi$
$480$	0	0
$481$	3.47618e6	0.685078
$482$	1.39793e7	2.74073
$483$	−1.06887e6	−0.208476
$484$	3.16618e6	0.614359
$485$	0	0
$486$	626621.	0.120341
$487$	2.16178e6	0.413036	0.206518	−	0.978443i	$-0.433787\pi$
$487$	2.16178e6	0.413036	0.206518	+	0.978443i	$0.433787\pi$
$488$	1.33946e7	2.54613
$489$	−3.61447e6	−0.683554
$490$	0	0
$491$	−4.58618e6	−0.858514	−0.429257	−	0.903182i	$-0.641225\pi$
$491$	−4.58618e6	−0.858514	−0.429257	+	0.903182i	$0.641225\pi$
$492$	1.22383e6	0.227933
$493$	−2.96178e6	−0.548827
$494$	4.01934e6	0.741032
$495$	0	0
$496$	−1.28435e6	−0.234411
$497$	5.59058e6	1.01523
$498$	1.93620e6	0.349847
$499$	3.89509e6	0.700271	0.350136	−	0.936699i	$-0.386135\pi$
$499$	3.89509e6	0.700271	0.350136	+	0.936699i	$0.386135\pi$
$500$	0	0
$501$	−3.70703e6	−0.659829
$502$	−9.60504e6	−1.70114
$503$	2.16926e6	0.382290	0.191145	−	0.981562i	$-0.438780\pi$
$503$	2.16926e6	0.382290	0.191145	+	0.981562i	$0.438780\pi$
$504$	−4.42043e6	−0.775153
$505$	0	0
$506$	5.33211e6	0.925813
$507$	−2.65501e6	−0.458718
$508$	1.99219e7	3.42508
$509$	4.42112e6	0.756376	0.378188	−	0.925729i	$-0.376547\pi$
$509$	4.42112e6	0.756376	0.378188	+	0.925729i	$0.376547\pi$
$510$	0	0
$511$	3.77776e6	0.640004
$512$	−6.64942e6	−1.12101
$513$	−999656.	−0.167709
$514$	−3.14844e6	−0.525639
$515$	0	0
$516$	6.43319e6	1.06366
$517$	−1.24330e6	−0.204574
$518$	1.41286e7	2.31353
$519$	3.51926e6	0.573499
$520$	0	0
$521$	−2.88426e6	−0.465522	−0.232761	−	0.972534i	$-0.574776\pi$
$521$	−2.88426e6	−0.465522	−0.232761	+	0.972534i	$0.574776\pi$
$522$	1.39349e6	0.223834
$523$	−5.08193e6	−0.812408	−0.406204	−	0.913782i	$-0.633148\pi$
$523$	−5.08193e6	−0.812408	−0.406204	+	0.913782i	$0.633148\pi$
$524$	−3.87134e6	−0.615933
$525$	0	0
$526$	3.13404e6	0.493901
$527$	810599.	0.127139
$528$	1.16605e7	1.82025
$529$	−5.17604e6	−0.804190
$530$	0	0
$531$	−2.68382e6	−0.413064
$532$	1.16941e7	1.79138
$533$	−465928.	−0.0710396
$534$	1.20939e7	1.83533
$535$	0	0
$536$	1.00053e7	1.50424
$537$	−5.06606e6	−0.758114
$538$	2.78731e6	0.415174
$539$	−2.51338e6	−0.372637
$540$	0	0
$541$	1.19634e7	1.75737	0.878685	−	0.477402i	$-0.158421\pi$
$541$	1.19634e7	1.75737	0.878685	+	0.477402i	$0.158421\pi$
$542$	−7.31348e6	−1.06936
$543$	−188002.	−0.0273629
$544$	−2.59619e7	−3.76131
$545$	0	0
$546$	2.79073e6	0.400624
$547$	9.41845e6	1.34589	0.672947	−	0.739690i	$-0.265028\pi$
$547$	9.41845e6	1.34589	0.672947	+	0.739690i	$0.265028\pi$
$548$	−2.01740e7	−2.86973
$549$	2.10320e6	0.297817
$550$	0	0
$551$	−2.22305e6	−0.311939
$552$	5.21211e6	0.728058
$553$	−9.71371e6	−1.35074
$554$	1.21995e7	1.68876
$555$	0	0
$556$	1.53555e6	0.210658
$557$	6.13295e6	0.837589	0.418795	−	0.908081i	$-0.362453\pi$
$557$	6.13295e6	0.837589	0.418795	+	0.908081i	$0.362453\pi$
$558$	−381379.	−0.0518527
$559$	−2.44920e6	−0.331508
$560$	0	0
$561$	−7.35936e6	−0.987263
$562$	2.56252e7	3.42237
$563$	−1.25000e7	−1.66204	−0.831019	−	0.556245i	$-0.812242\pi$
$563$	−1.25000e7	−1.66204	−0.831019	+	0.556245i	$0.812242\pi$
$564$	−2.01533e6	−0.266778
$565$	0	0
$566$	−9.78499e6	−1.28387
$567$	−694088.	−0.0906686
$568$	−2.72613e7	−3.54549
$569$	−3.54884e6	−0.459521	−0.229761	−	0.973247i	$-0.573794\pi$
$569$	−3.54884e6	−0.459521	−0.229761	+	0.973247i	$0.573794\pi$
$570$	0	0
$571$	6.09787e6	0.782687	0.391344	−	0.920245i	$-0.372010\pi$
$571$	6.09787e6	0.782687	0.391344	+	0.920245i	$0.372010\pi$
$572$	−9.96573e6	−1.27356
$573$	−4.18310e6	−0.532245
$574$	−1.89372e6	−0.239903
$575$	0	0
$576$	4.71177e6	0.591735
$577$	408501.	0.0510803	0.0255401	−	0.999674i	$-0.491869\pi$
$577$	408501.	0.0510803	0.0255401	+	0.999674i	$0.491869\pi$
$578$	2.03524e7	2.53394
$579$	2.76027e6	0.342180
$580$	0	0
$581$	−2.14467e6	−0.263585
$582$	1.32352e7	1.61965
$583$	1.34955e7	1.64444
$584$	−1.84215e7	−2.23508
$585$	0	0
$586$	3.68978e6	0.443870
$587$	5.07435e6	0.607835	0.303917	−	0.952698i	$-0.401705\pi$
$587$	5.07435e6	0.607835	0.303917	+	0.952698i	$0.401705\pi$
$588$	−4.07407e6	−0.485943
$589$	608419.	0.0722627
$590$	0	0
$591$	1.07413e6	0.126500
$592$	−3.64306e7	−4.27230
$593$	6.10529e6	0.712968	0.356484	−	0.934301i	$-0.383975\pi$
$593$	6.10529e6	0.712968	0.356484	+	0.934301i	$0.383975\pi$
$594$	3.46250e6	0.402647
$595$	0	0
$596$	−1.45282e7	−1.67531
$597$	2.02219e6	0.232213
$598$	−3.29055e6	−0.376284
$599$	993739.	0.113163	0.0565816	−	0.998398i	$-0.481980\pi$
$599$	993739.	0.113163	0.0565816	+	0.998398i	$0.481980\pi$
$600$	0	0
$601$	−1.45037e6	−0.163792	−0.0818959	−	0.996641i	$-0.526098\pi$
$601$	−1.45037e6	−0.163792	−0.0818959	+	0.996641i	$0.526098\pi$
$602$	−9.95454e6	−1.11952
$603$	1.57101e6	0.175949
$604$	4.46086e7	4.97538
$605$	0	0
$606$	−2.34650e6	−0.259561
$607$	1.27149e6	0.140068	0.0700341	−	0.997545i	$-0.477689\pi$
$607$	1.27149e6	0.140068	0.0700341	+	0.997545i	$0.477689\pi$
$608$	−1.94864e7	−2.13783
$609$	−1.54352e6	−0.168644
$610$	0	0
$611$	767264.	0.0831461
$612$	−1.19292e7	−1.28746
$613$	−3.99513e6	−0.429417	−0.214709	−	0.976678i	$-0.568880\pi$
$613$	−3.99513e6	−0.429417	−0.214709	+	0.976678i	$0.568880\pi$
$614$	−1.76296e7	−1.88722
$615$	0	0
$616$	−2.44258e7	−2.59357
$617$	−5.17424e6	−0.547184	−0.273592	−	0.961846i	$-0.588212\pi$
$617$	−5.17424e6	−0.547184	−0.273592	+	0.961846i	$0.588212\pi$
$618$	1.28472e7	1.35313
$619$	−4.91770e6	−0.515864	−0.257932	−	0.966163i	$-0.583041\pi$
$619$	−4.91770e6	−0.515864	−0.257932	+	0.966163i	$0.583041\pi$
$620$	0	0
$621$	818397.	0.0851599
$622$	3.55897e6	0.368849
$623$	−1.33961e7	−1.38279
$624$	−7.19589e6	−0.739815
$625$	0	0
$626$	−3.17465e7	−3.23788
$627$	−5.52378e6	−0.561135
$628$	4.39152e7	4.44340
$629$	2.29927e7	2.31720
$630$	0	0
$631$	−1.71486e7	−1.71457	−0.857286	−	0.514841i	$-0.827851\pi$
$631$	−1.71486e7	−1.71457	−0.857286	+	0.514841i	$0.827851\pi$
$632$	4.73669e7	4.71718
$633$	−3.26242e6	−0.323616
$634$	1.44184e7	1.42460
$635$	0	0
$636$	2.18756e7	2.14446
$637$	1.55105e6	0.151453
$638$	7.69996e6	0.748923
$639$	−4.28053e6	−0.414710
$640$	0	0
$641$	1.87181e7	1.79936	0.899679	−	0.436552i	$-0.143801\pi$
$641$	1.87181e7	1.79936	0.899679	+	0.436552i	$0.143801\pi$
$642$	6.61506e6	0.633426
$643$	−4.37631e6	−0.417427	−0.208714	−	0.977977i	$-0.566928\pi$
$643$	−4.37631e6	−0.417427	−0.208714	+	0.977977i	$0.566928\pi$
$644$	−9.57371e6	−0.909632
$645$	0	0
$646$	2.65853e7	2.50646
$647$	787451.	0.0739542	0.0369771	−	0.999316i	$-0.488227\pi$
$647$	787451.	0.0739542	0.0369771	+	0.999316i	$0.488227\pi$
$648$	3.38458e6	0.316641
$649$	−1.48299e7	−1.38206
$650$	0	0
$651$	422442.	0.0390674
$652$	−3.23744e7	−2.98251
$653$	−5.20974e6	−0.478116	−0.239058	−	0.971005i	$-0.576839\pi$
$653$	−5.20974e6	−0.478116	−0.239058	+	0.971005i	$0.576839\pi$
$654$	−6.57839e6	−0.601416
$655$	0	0
$656$	4.88294e6	0.443019
$657$	−2.89251e6	−0.261434
$658$	3.11847e6	0.280787
$659$	−5.95464e6	−0.534124	−0.267062	−	0.963679i	$-0.586053\pi$
$659$	−5.95464e6	−0.534124	−0.267062	+	0.963679i	$0.586053\pi$
$660$	0	0
$661$	1.16361e7	1.03587	0.517934	−	0.855421i	$-0.326701\pi$
$661$	1.16361e7	1.03587	0.517934	+	0.855421i	$0.326701\pi$
$662$	−3.67235e7	−3.25686
$663$	4.54160e6	0.401259
$664$	1.04581e7	0.920515
$665$	0	0
$666$	−1.08178e7	−0.945050
$667$	1.81996e6	0.158397
$668$	−3.32034e7	−2.87899
$669$	−6.73502e6	−0.581800
$670$	0	0
$671$	1.16216e7	0.996460
$672$	−1.35300e7	−1.15578
$673$	888669.	0.0756315	0.0378157	−	0.999285i	$-0.487960\pi$
$673$	888669.	0.0756315	0.0378157	+	0.999285i	$0.487960\pi$
$674$	2.29214e7	1.94353
$675$	0	0
$676$	−2.37806e7	−2.00150
$677$	−1.10703e7	−0.928299	−0.464149	−	0.885757i	$-0.653640\pi$
$677$	−1.10703e7	−0.928299	−0.464149	+	0.885757i	$0.653640\pi$
$678$	−1.41648e6	−0.118342
$679$	−1.46602e7	−1.22030
$680$	0	0
$681$	−1.10211e7	−0.910660
$682$	−2.10738e6	−0.173493
$683$	6.70735e6	0.550173	0.275087	−	0.961419i	$-0.411293\pi$
$683$	6.70735e6	0.550173	0.275087	+	0.961419i	$0.411293\pi$
$684$	−8.95379e6	−0.731757
$685$	0	0
$686$	2.51721e7	2.04226
$687$	−1.10122e7	−0.890192
$688$	2.56677e7	2.06736
$689$	−8.32833e6	−0.668359
$690$	0	0
$691$	1.18223e7	0.941901	0.470950	−	0.882160i	$-0.343911\pi$
$691$	1.18223e7	0.941901	0.470950	+	0.882160i	$0.343911\pi$
$692$	3.15215e7	2.50232
$693$	−3.83531e6	−0.303366
$694$	1.24953e7	0.984800
$695$	0	0
$696$	7.52667e6	0.588952
$697$	−3.08181e6	−0.240283
$698$	−4.63875e6	−0.360381
$699$	−4.93662e6	−0.382152
$700$	0	0
$701$	−1.39654e7	−1.07339	−0.536695	−	0.843777i	$-0.680327\pi$
$701$	−1.39654e7	−1.07339	−0.536695	+	0.843777i	$0.680327\pi$
$702$	−2.13678e6	−0.163650
$703$	1.72578e7	1.31704
$704$	2.60357e7	1.97987
$705$	0	0
$706$	−5.42889e6	−0.409920
$707$	2.59915e6	0.195561
$708$	−2.40386e7	−1.80230
$709$	9.11846e6	0.681249	0.340625	−	0.940199i	$-0.389362\pi$
$709$	9.11846e6	0.681249	0.340625	+	0.940199i	$0.389362\pi$
$710$	0	0
$711$	7.43747e6	0.551761
$712$	6.53232e7	4.82912
$713$	−498100.	−0.0366938
$714$	1.84589e7	1.35507
$715$	0	0
$716$	−4.53761e7	−3.30784
$717$	−7.72234e6	−0.560985
$718$	−4.20896e6	−0.304694
$719$	4.48606e6	0.323625	0.161813	−	0.986822i	$-0.448266\pi$
$719$	4.48606e6	0.323625	0.161813	+	0.986822i	$0.448266\pi$
$720$	0	0
$721$	−1.42305e7	−1.01949
$722$	−6.32168e6	−0.451325
$723$	1.18559e7	0.843507
$724$	−1.68391e6	−0.119391
$725$	0	0
$726$	3.75121e6	0.264137
$727$	2.53427e6	0.177835	0.0889175	−	0.996039i	$-0.471659\pi$
$727$	2.53427e6	0.177835	0.0889175	+	0.996039i	$0.471659\pi$
$728$	1.50737e7	1.05412
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.61999e7	−1.12129
$732$	1.88381e7	1.29945
$733$	6.77813e6	0.465961	0.232981	−	0.972481i	$-0.425152\pi$
$733$	6.77813e6	0.465961	0.232981	+	0.972481i	$0.425152\pi$
$734$	−3.25800e7	−2.23209
$735$	0	0
$736$	1.59531e7	1.08556
$737$	8.68090e6	0.588703
$738$	1.44996e6	0.0979975
$739$	−7.73044e6	−0.520707	−0.260353	−	0.965513i	$-0.583839\pi$
$739$	−7.73044e6	−0.520707	−0.260353	+	0.965513i	$0.583839\pi$
$740$	0	0
$741$	3.40883e6	0.228065
$742$	−3.38497e7	−2.25707
$743$	2.83687e6	0.188525	0.0942623	−	0.995547i	$-0.469951\pi$
$743$	2.83687e6	0.188525	0.0942623	+	0.995547i	$0.469951\pi$
$744$	−2.05995e6	−0.136434
$745$	0	0
$746$	3.14479e7	2.06893
$747$	1.64211e6	0.107671
$748$	−6.59168e7	−4.30767
$749$	−7.32729e6	−0.477242
$750$	0	0
$751$	3.97981e6	0.257491	0.128746	−	0.991678i	$-0.458905\pi$
$751$	3.97981e6	0.257491	0.128746	+	0.991678i	$0.458905\pi$
$752$	−8.04097e6	−0.518518
$753$	−8.14610e6	−0.523555
$754$	−4.75179e6	−0.304389
$755$	0	0
$756$	−6.21686e6	−0.395609
$757$	−1.72106e7	−1.09158	−0.545789	−	0.837922i	$-0.683770\pi$
$757$	−1.72106e7	−1.09158	−0.545789	+	0.837922i	$0.683770\pi$
$758$	510979.	0.0323021
$759$	4.52220e6	0.284935
$760$	0	0
$761$	−9.81793e6	−0.614552	−0.307276	−	0.951621i	$-0.599417\pi$
$761$	−9.81793e6	−0.614552	−0.307276	+	0.951621i	$0.599417\pi$
$762$	2.36029e7	1.47258
$763$	7.28668e6	0.453125
$764$	−3.74675e7	−2.32232
$765$	0	0
$766$	−7.96567e6	−0.490513
$767$	9.15183e6	0.561719
$768$	−1.22775e6	−0.0751114
$769$	8.65401e6	0.527718	0.263859	−	0.964561i	$-0.415005\pi$
$769$	8.65401e6	0.527718	0.263859	+	0.964561i	$0.415005\pi$
$770$	0	0
$771$	−2.67021e6	−0.161775
$772$	2.47234e7	1.49301
$773$	2.44011e7	1.46879	0.734396	−	0.678722i	$-0.237466\pi$
$773$	2.44011e7	1.46879	0.734396	+	0.678722i	$0.237466\pi$
$774$	7.62187e6	0.457308
$775$	0	0
$776$	7.14874e7	4.26162
$777$	1.19826e7	0.712029
$778$	−3.63132e7	−2.15087
$779$	−2.31314e6	−0.136571
$780$	0	0
$781$	−2.36528e7	−1.38757
$782$	−2.17648e7	−1.27274
$783$	1.18183e6	0.0688889
$784$	−1.62551e7	−0.944494
$785$	0	0
$786$	−4.58667e6	−0.264814
$787$	−2.10471e7	−1.21131	−0.605655	−	0.795728i	$-0.707089\pi$
$787$	−2.10471e7	−1.21131	−0.605655	+	0.795728i	$0.707089\pi$
$788$	9.62087e6	0.551949
$789$	2.65800e6	0.152006
$790$	0	0
$791$	1.56900e6	0.0891621
$792$	1.87021e7	1.05944
$793$	−7.17191e6	−0.404997
$794$	−1.03166e7	−0.580745
$795$	0	0
$796$	1.81125e7	1.01320
$797$	−2.60267e7	−1.45135	−0.725676	−	0.688036i	$-0.758473\pi$
$797$	−2.60267e7	−1.45135	−0.725676	+	0.688036i	$0.758473\pi$
$798$	1.38549e7	0.770185
$799$	5.07495e6	0.281232
$800$	0	0
$801$	1.02569e7	0.564855
$802$	6.08507e7	3.34064
$803$	−1.59831e7	−0.874726
$804$	1.40714e7	0.767707
$805$	0	0
$806$	1.30050e6	0.0705137
$807$	2.36394e6	0.127777
$808$	−1.26742e7	−0.682955
$809$	3.30770e7	1.77687	0.888434	−	0.459004i	$-0.151794\pi$
$809$	3.30770e7	1.77687	0.888434	+	0.459004i	$0.151794\pi$
$810$	0	0
$811$	2.88531e7	1.54042	0.770212	−	0.637788i	$-0.220151\pi$
$811$	2.88531e7	1.54042	0.770212	+	0.637788i	$0.220151\pi$
$812$	−1.38251e7	−0.735833
$813$	−6.20261e6	−0.329115
$814$	−5.97759e7	−3.16202
$815$	0	0
$816$	−4.75962e7	−2.50234
$817$	−1.21593e7	−0.637312
$818$	3.84672e7	2.01005
$819$	2.36684e6	0.123299
$820$	0	0
$821$	1.82331e7	0.944069	0.472034	−	0.881580i	$-0.343520\pi$
$821$	1.82331e7	0.944069	0.472034	+	0.881580i	$0.343520\pi$
$822$	−2.39017e7	−1.23381
$823$	−1.08726e7	−0.559541	−0.279771	−	0.960067i	$-0.590258\pi$
$823$	−1.08726e7	−0.559541	−0.279771	+	0.960067i	$0.590258\pi$
$824$	6.93920e7	3.56034
$825$	0	0
$826$	3.71967e7	1.89695
$827$	3.22275e7	1.63856	0.819282	−	0.573391i	$-0.194372\pi$
$827$	3.22275e7	1.63856	0.819282	+	0.573391i	$0.194372\pi$
$828$	7.33028e6	0.371574
$829$	2.82837e7	1.42939	0.714693	−	0.699439i	$-0.246567\pi$
$829$	2.82837e7	1.42939	0.714693	+	0.699439i	$0.246567\pi$
$830$	0	0
$831$	1.03465e7	0.519745
$832$	−1.60671e7	−0.804692
$833$	1.02592e7	0.512272
$834$	1.81928e6	0.0905700
$835$	0	0
$836$	−4.94758e7	−2.44837
$837$	−323450.	−0.0159585
$838$	5.23664e7	2.57598
$839$	−1.23138e7	−0.603933	−0.301967	−	0.953319i	$-0.597643\pi$
$839$	−1.23138e7	−0.603933	−0.301967	+	0.953319i	$0.597643\pi$
$840$	0	0
$841$	−1.78830e7	−0.871867
$842$	−1.31271e7	−0.638098
$843$	2.17329e7	1.05329
$844$	−2.92211e7	−1.41202
$845$	0	0
$846$	−2.38772e6	−0.114698
$847$	−4.15510e6	−0.199009
$848$	8.72813e7	4.16804
$849$	−8.29872e6	−0.395132
$850$	0	0
$851$	−1.41286e7	−0.668769
$852$	−3.83401e7	−1.80948
$853$	−3.80421e6	−0.179016	−0.0895080	−	0.995986i	$-0.528529\pi$
$853$	−3.80421e6	−0.179016	−0.0895080	+	0.995986i	$0.528529\pi$
$854$	−2.91496e7	−1.36769
$855$	0	0
$856$	3.57300e7	1.66667
$857$	2.86121e7	1.33075	0.665376	−	0.746508i	$-0.268271\pi$
$857$	2.86121e7	1.33075	0.665376	+	0.746508i	$0.268271\pi$
$858$	−1.18071e7	−0.547553
$859$	2.86987e7	1.32702	0.663512	−	0.748165i	$-0.269065\pi$
$859$	2.86987e7	1.32702	0.663512	+	0.748165i	$0.269065\pi$
$860$	0	0
$861$	−1.60608e6	−0.0738343
$862$	−2.19112e7	−1.00438
$863$	−2.51054e7	−1.14747	−0.573734	−	0.819042i	$-0.694506\pi$
$863$	−2.51054e7	−1.14747	−0.573734	+	0.819042i	$0.694506\pi$
$864$	1.03595e7	0.472121
$865$	0	0
$866$	2.12339e7	0.962132
$867$	1.72610e7	0.779863
$868$	3.78376e6	0.170461
$869$	4.10971e7	1.84613
$870$	0	0
$871$	−5.35715e6	−0.239270
$872$	−3.55320e7	−1.58244
$873$	1.12248e7	0.498476
$874$	−1.63362e7	−0.723391
$875$	0	0
$876$	−2.59079e7	−1.14070
$877$	3.22931e7	1.41779	0.708893	−	0.705316i	$-0.249195\pi$
$877$	3.22931e7	1.41779	0.708893	+	0.705316i	$0.249195\pi$
$878$	−1.61978e7	−0.709121
$879$	3.12932e6	0.136609
$880$	0	0
$881$	7.21912e6	0.313361	0.156680	−	0.987649i	$-0.449921\pi$
$881$	7.21912e6	0.313361	0.156680	+	0.987649i	$0.449921\pi$
$882$	−4.82685e6	−0.208926
$883$	−1.54973e7	−0.668890	−0.334445	−	0.942415i	$-0.608549\pi$
$883$	−1.54973e7	−0.668890	−0.334445	+	0.942415i	$0.608549\pi$
$884$	4.06785e7	1.75079
$885$	0	0
$886$	−5.66831e7	−2.42588
$887$	4.01737e7	1.71448	0.857242	−	0.514914i	$-0.172176\pi$
$887$	4.01737e7	1.71448	0.857242	+	0.514914i	$0.172176\pi$
$888$	−5.84306e7	−2.48661
$889$	−2.61442e7	−1.10948
$890$	0	0
$891$	2.93657e6	0.123921
$892$	−6.03248e7	−2.53854
$893$	3.80915e6	0.159845
$894$	−1.72126e7	−0.720283
$895$	0	0
$896$	−1.71968e7	−0.715611
$897$	−2.79073e6	−0.115808
$898$	−5.98425e7	−2.47639
$899$	−719292.	−0.0296829
$900$	0	0
$901$	−5.50865e7	−2.26065
$902$	8.01201e6	0.327888
$903$	−8.44251e6	−0.344550
$904$	−7.65088e6	−0.311380
$905$	0	0
$906$	5.28511e7	2.13911
$907$	−3.33663e7	−1.34676	−0.673380	−	0.739297i	$-0.735158\pi$
$907$	−3.33663e7	−1.34676	−0.673380	+	0.739297i	$0.735158\pi$
$908$	−9.87144e7	−3.97343
$909$	−1.99008e6	−0.0798843
$910$	0	0
$911$	2.88206e7	1.15055	0.575277	−	0.817958i	$-0.304894\pi$
$911$	2.88206e7	1.15055	0.575277	+	0.817958i	$0.304894\pi$
$912$	−3.57247e7	−1.42227
$913$	9.07375e6	0.360255
$914$	1.68154e7	0.665796
$915$	0	0
$916$	−9.86352e7	−3.88413
$917$	5.08051e6	0.199519
$918$	−1.41334e7	−0.553528
$919$	9.79583e6	0.382607	0.191303	−	0.981531i	$-0.438729\pi$
$919$	9.79583e6	0.382607	0.191303	+	0.981531i	$0.438729\pi$
$920$	0	0
$921$	−1.49518e7	−0.580824
$922$	2.97912e6	0.115414
$923$	1.45966e7	0.563959
$924$	−3.43524e7	−1.32366
$925$	0	0
$926$	1.67695e7	0.642675
$927$	1.08958e7	0.416448
$928$	2.30375e7	0.878144
$929$	−4.71065e7	−1.79078	−0.895389	−	0.445284i	$-0.853103\pi$
$929$	−4.71065e7	−1.79078	−0.895389	+	0.445284i	$0.853103\pi$
$930$	0	0
$931$	7.70034e6	0.291163
$932$	−4.42166e7	−1.66742
$933$	3.01838e6	0.113520
$934$	4.59157e6	0.172224
$935$	0	0
$936$	−1.15414e7	−0.430595
$937$	−548115.	−0.0203950	−0.0101975	−	0.999948i	$-0.503246\pi$
$937$	−548115.	−0.0203950	−0.0101975	+	0.999948i	$0.503246\pi$
$938$	−2.17736e7	−0.808023
$939$	−2.69244e7	−0.996512
$940$	0	0
$941$	1.27644e7	0.469922	0.234961	−	0.972005i	$-0.424504\pi$
$941$	1.27644e7	0.469922	0.234961	+	0.972005i	$0.424504\pi$
$942$	5.20296e7	1.91039
$943$	1.89372e6	0.0693484
$944$	−9.59116e7	−3.50301
$945$	0	0
$946$	4.21160e7	1.53010
$947$	−1.06700e7	−0.386624	−0.193312	−	0.981137i	$-0.561923\pi$
$947$	−1.06700e7	−0.386624	−0.193312	+	0.981137i	$0.561923\pi$
$948$	6.66165e7	2.40747
$949$	9.86346e6	0.355520
$950$	0	0
$951$	1.22284e7	0.438446
$952$	9.97024e7	3.56544
$953$	−6.61145e6	−0.235811	−0.117906	−	0.993025i	$-0.537618\pi$
$953$	−6.61145e6	−0.235811	−0.117906	+	0.993025i	$0.537618\pi$
$954$	2.59177e7	0.921987
$955$	0	0
$956$	−6.91680e7	−2.44771
$957$	6.53039e6	0.230494
$958$	4.56631e7	1.60750
$959$	2.64752e7	0.929591
$960$	0	0
$961$	−2.84323e7	−0.993124
$962$	3.68888e7	1.28516
$963$	5.61027e6	0.194948
$964$	1.06192e8	3.68043
$965$	0	0
$966$	−1.13427e7	−0.391087
$967$	3.19820e7	1.09987	0.549933	−	0.835209i	$-0.314653\pi$
$967$	3.19820e7	1.09987	0.549933	+	0.835209i	$0.314653\pi$
$968$	2.02615e7	0.694996
$969$	2.25472e7	0.771405
$970$	0	0
$971$	−1.08607e6	−0.0369666	−0.0184833	−	0.999829i	$-0.505884\pi$
$971$	−1.08607e6	−0.0369666	−0.0184833	+	0.999829i	$0.505884\pi$
$972$	4.76005e6	0.161602
$973$	−2.01516e6	−0.0682382
$974$	2.29405e7	0.774828
$975$	0	0
$976$	7.51619e7	2.52565
$977$	−3.40058e7	−1.13977	−0.569884	−	0.821725i	$-0.693012\pi$
$977$	−3.40058e7	−1.13977	−0.569884	+	0.821725i	$0.693012\pi$
$978$	−3.83563e7	−1.28230
$979$	5.66766e7	1.88994
$980$	0	0
$981$	−5.57918e6	−0.185096
$982$	−4.86680e7	−1.61051
$983$	−2.75262e7	−0.908579	−0.454290	−	0.890854i	$-0.650107\pi$
$983$	−2.75262e7	−0.908579	−0.454290	+	0.890854i	$0.650107\pi$
$984$	7.83170e6	0.257851
$985$	0	0
$986$	−3.14300e7	−1.02956
$987$	2.64480e6	0.0864171
$988$	3.05324e7	0.995105
$989$	9.95454e6	0.323616
$990$	0	0
$991$	6.12961e7	1.98266	0.991331	−	0.131389i	$-0.0419438\pi$
$991$	6.12961e7	1.98266	0.991331	+	0.131389i	$0.0419438\pi$
$992$	−6.30506e6	−0.203428
$993$	−3.11454e7	−1.00235
$994$	5.93265e7	1.90451
$995$	0	0
$996$	1.47081e7	0.469796
$997$	1.40079e7	0.446307	0.223153	−	0.974783i	$-0.428365\pi$
$997$	1.40079e7	0.446307	0.223153	+	0.974783i	$0.428365\pi$
$998$	4.13342e7	1.31366
$999$	−9.17468e6	−0.290855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.a.h.1.2		2
3.2	odd	2		225.6.a.m.1.1		2
5.2	odd	4		75.6.b.e.49.4		4
5.3	odd	4		75.6.b.e.49.1		4
5.4	even	2		15.6.a.c.1.1	✓	2
15.2	even	4		225.6.b.g.199.1		4
15.8	even	4		225.6.b.g.199.4		4
15.14	odd	2		45.6.a.e.1.2		2
20.19	odd	2		240.6.a.q.1.1		2
35.34	odd	2		735.6.a.g.1.1		2
40.19	odd	2		960.6.a.bf.1.1		2
40.29	even	2		960.6.a.bj.1.2		2
60.59	even	2		720.6.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.6.a.c.1.1	✓	2	5.4	even	2
45.6.a.e.1.2		2	15.14	odd	2
75.6.a.h.1.2		2	1.1	even	1	trivial
75.6.b.e.49.1		4	5.3	odd	4
75.6.b.e.49.4		4	5.2	odd	4
225.6.a.m.1.1		2	3.2	odd	2
225.6.b.g.199.1		4	15.2	even	4
225.6.b.g.199.4		4	15.8	even	4
240.6.a.q.1.1		2	20.19	odd	2
720.6.a.bd.1.1		2	60.59	even	2
735.6.a.g.1.1		2	35.34	odd	2
960.6.a.bf.1.1		2	40.19	odd	2
960.6.a.bj.1.2		2	40.29	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+10.6119 q^{2} +9.00000 q^{3} +80.6119 q^{4} +95.5069 q^{6} -105.790 q^{7} +515.863 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+10.6119 q^{2} +9.00000 q^{3} +80.6119 q^{4} +95.5069 q^{6} -105.790 q^{7} +515.863 q^{8} +81.0000 q^{9} +447.580 q^{11} +725.507 q^{12} -276.210 q^{13} -1122.63 q^{14} +2894.69 q^{16} -1826.95 q^{17} +859.562 q^{18} -1371.27 q^{19} -952.110 q^{21} +4749.66 q^{22} +1122.63 q^{23} +4642.77 q^{24} -2931.11 q^{26} +729.000 q^{27} -8527.93 q^{28} +1621.16 q^{29} -443.690 q^{31} +14210.5 q^{32} +4028.22 q^{33} -19387.4 q^{34} +6529.56 q^{36} -12585.3 q^{37} -14551.7 q^{38} -2485.89 q^{39} +1686.86 q^{41} -10103.7 q^{42} +8867.16 q^{43} +36080.3 q^{44} +11913.2 q^{46} -2777.83 q^{47} +26052.2 q^{48} -5615.48 q^{49} -16442.5 q^{51} -22265.8 q^{52} +30152.2 q^{53} +7736.06 q^{54} -54573.2 q^{56} -12341.4 q^{57} +17203.5 q^{58} -33133.6 q^{59} +25965.4 q^{61} -4708.38 q^{62} -8568.99 q^{63} +58170.0 q^{64} +42747.0 q^{66} +19395.2 q^{67} -147274. q^{68} +10103.7 q^{69} -52846.0 q^{71} +41784.9 q^{72} -35710.0 q^{73} -133554. q^{74} -110541. q^{76} -47349.5 q^{77} -26380.0 q^{78} +91820.6 q^{79} +6561.00 q^{81} +17900.7 q^{82} +20272.9 q^{83} -76751.4 q^{84} +94097.2 q^{86} +14590.4 q^{87} +230890. q^{88} +126629. q^{89} +29220.3 q^{91} +90497.3 q^{92} -3993.21 q^{93} -29478.0 q^{94} +127895. q^{96} +138578. q^{97} -59590.8 q^{98} +36254.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 18 q^{3} + 141 q^{4} + 9 q^{6} + 112 q^{7} + 243 q^{8} + 162 q^{9}+O(q^{10}) \) 2 * q + q^2 + 18 * q^3 + 141 * q^4 + 9 * q^6 + 112 * q^7 + 243 * q^8 + 162 * q^9	\( 2 q + q^{2} + 18 q^{3} + 141 q^{4} + 9 q^{6} + 112 q^{7} + 243 q^{8} + 162 q^{9} + 248 q^{11} + 1269 q^{12} - 876 q^{13} - 3216 q^{14} + 3585 q^{16} - 2036 q^{17} + 81 q^{18} + 1464 q^{19} + 1008 q^{21} + 6668 q^{22} + 3216 q^{23} + 2187 q^{24} + 2834 q^{26} + 1458 q^{27} + 4624 q^{28} + 1948 q^{29} + 2672 q^{31} + 16307 q^{32} + 2232 q^{33} - 17378 q^{34} + 11421 q^{36} - 8668 q^{37} - 41804 q^{38} - 7884 q^{39} - 7628 q^{41} - 28944 q^{42} + 16440 q^{43} + 24028 q^{44} - 8208 q^{46} + 19360 q^{47} + 32265 q^{48} + 25010 q^{49} - 18324 q^{51} - 58486 q^{52} + 14356 q^{53} + 729 q^{54} - 114000 q^{56} + 13176 q^{57} + 14062 q^{58} - 904 q^{59} + 20220 q^{61} - 34656 q^{62} + 9072 q^{63} + 15929 q^{64} + 60012 q^{66} + 12904 q^{67} - 159898 q^{68} + 28944 q^{69} - 40976 q^{71} + 19683 q^{72} - 59124 q^{73} - 171206 q^{74} + 60676 q^{76} - 90816 q^{77} + 25506 q^{78} + 107600 q^{79} + 13122 q^{81} + 107434 q^{82} + 122088 q^{83} + 41616 q^{84} + 21308 q^{86} + 17532 q^{87} + 285348 q^{88} + 103764 q^{89} - 101408 q^{91} + 216912 q^{92} + 24048 q^{93} - 242264 q^{94} + 146763 q^{96} + 24764 q^{97} - 353959 q^{98} + 20088 q^{99}+O(q^{100}) \) 2 * q + q^2 + 18 * q^3 + 141 * q^4 + 9 * q^6 + 112 * q^7 + 243 * q^8 + 162 * q^9 + 248 * q^11 + 1269 * q^12 - 876 * q^13 - 3216 * q^14 + 3585 * q^16 - 2036 * q^17 + 81 * q^18 + 1464 * q^19 + 1008 * q^21 + 6668 * q^22 + 3216 * q^23 + 2187 * q^24 + 2834 * q^26 + 1458 * q^27 + 4624 * q^28 + 1948 * q^29 + 2672 * q^31 + 16307 * q^32 + 2232 * q^33 - 17378 * q^34 + 11421 * q^36 - 8668 * q^37 - 41804 * q^38 - 7884 * q^39 - 7628 * q^41 - 28944 * q^42 + 16440 * q^43 + 24028 * q^44 - 8208 * q^46 + 19360 * q^47 + 32265 * q^48 + 25010 * q^49 - 18324 * q^51 - 58486 * q^52 + 14356 * q^53 + 729 * q^54 - 114000 * q^56 + 13176 * q^57 + 14062 * q^58 - 904 * q^59 + 20220 * q^61 - 34656 * q^62 + 9072 * q^63 + 15929 * q^64 + 60012 * q^66 + 12904 * q^67 - 159898 * q^68 + 28944 * q^69 - 40976 * q^71 + 19683 * q^72 - 59124 * q^73 - 171206 * q^74 + 60676 * q^76 - 90816 * q^77 + 25506 * q^78 + 107600 * q^79 + 13122 * q^81 + 107434 * q^82 + 122088 * q^83 + 41616 * q^84 + 21308 * q^86 + 17532 * q^87 + 285348 * q^88 + 103764 * q^89 - 101408 * q^91 + 216912 * q^92 + 24048 * q^93 - 242264 * q^94 + 146763 * q^96 + 24764 * q^97 - 353959 * q^98 + 20088 * q^99

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