LMFDB - Embedded newform 847.6.a.p.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,6,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	847.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:	$135.845095382$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.04168 q^{2} -8.44784 q^{3} -15.6648 q^{4} -80.4152 q^{5} +34.1435 q^{6} +49.0000 q^{7} +192.646 q^{8} -171.634 q^{9} +O(q^{10})$	\(q-4.04168 q^{2} -8.44784 q^{3} -15.6648 q^{4} -80.4152 q^{5} +34.1435 q^{6} +49.0000 q^{7} +192.646 q^{8} -171.634 q^{9} +325.013 q^{10} +132.334 q^{12} -1041.00 q^{13} -198.043 q^{14} +679.335 q^{15} -277.341 q^{16} +1377.06 q^{17} +693.690 q^{18} -3073.28 q^{19} +1259.69 q^{20} -413.944 q^{21} -2802.75 q^{23} -1627.44 q^{24} +3341.60 q^{25} +4207.41 q^{26} +3502.76 q^{27} -767.575 q^{28} +1964.27 q^{29} -2745.66 q^{30} +3667.06 q^{31} -5043.75 q^{32} -5565.65 q^{34} -3940.34 q^{35} +2688.61 q^{36} -2896.98 q^{37} +12421.2 q^{38} +8794.24 q^{39} -15491.7 q^{40} -12022.8 q^{41} +1673.03 q^{42} +3038.75 q^{43} +13802.0 q^{45} +11327.8 q^{46} -77.6245 q^{47} +2342.94 q^{48} +2401.00 q^{49} -13505.7 q^{50} -11633.2 q^{51} +16307.1 q^{52} +10813.7 q^{53} -14157.1 q^{54} +9439.66 q^{56} +25962.6 q^{57} -7938.97 q^{58} +50600.7 q^{59} -10641.6 q^{60} +3498.86 q^{61} -14821.1 q^{62} -8410.06 q^{63} +29260.2 q^{64} +83712.5 q^{65} -2376.54 q^{67} -21571.4 q^{68} +23677.2 q^{69} +15925.6 q^{70} -8444.12 q^{71} -33064.6 q^{72} +28364.7 q^{73} +11708.7 q^{74} -28229.4 q^{75} +48142.2 q^{76} -35543.5 q^{78} +70294.9 q^{79} +22302.5 q^{80} +12116.2 q^{81} +48592.4 q^{82} +123238. q^{83} +6484.35 q^{84} -110737. q^{85} -12281.7 q^{86} -16593.9 q^{87} -58546.5 q^{89} -55783.2 q^{90} -51009.2 q^{91} +43904.5 q^{92} -30978.8 q^{93} +313.734 q^{94} +247138. q^{95} +42608.8 q^{96} -2415.92 q^{97} -9704.08 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 14 q^{2} - 28 q^{3} + 374 q^{4} - 258 q^{5} + 4 q^{6} + 1372 q^{7} - 339 q^{8} + 1660 q^{9}+O(q^{10}) $ 28 * q - 14 * q^2 - 28 * q^3 + 374 * q^4 - 258 * q^5 + 4 * q^6 + 1372 * q^7 - 339 * q^8 + 1660 * q^9	\( 28 q - 14 q^{2} - 28 q^{3} + 374 q^{4} - 258 q^{5} + 4 q^{6} + 1372 q^{7} - 339 q^{8} + 1660 q^{9} - 536 q^{10} - 996 q^{12} + 326 q^{13} - 686 q^{14} - 2561 q^{15} + 5642 q^{16} - 2159 q^{17} - 6642 q^{18} + 2253 q^{19} - 13417 q^{20} - 1372 q^{21} - 6620 q^{23} + 3009 q^{24} + 15576 q^{25} - 4960 q^{26} - 24718 q^{27} + 18326 q^{28} - 15747 q^{29} - 9484 q^{30} - 14455 q^{31} - 13922 q^{32} - 13463 q^{34} - 12642 q^{35} + 27148 q^{36} - 14447 q^{37} - 62211 q^{38} - 28537 q^{39} - 31765 q^{40} - 14604 q^{41} + 196 q^{42} - 6828 q^{43} - 2703 q^{45} - 110251 q^{46} - 51396 q^{47} - 24310 q^{48} + 67228 q^{49} - 17179 q^{50} - 23461 q^{51} + 71461 q^{52} - 134505 q^{53} + 135496 q^{54} - 16611 q^{56} - 45846 q^{57} + 18657 q^{58} - 22255 q^{59} - 176135 q^{60} - 61141 q^{61} + 186610 q^{62} + 81340 q^{63} + 83377 q^{64} - 8616 q^{65} - 34675 q^{67} - 106726 q^{68} - 178024 q^{69} - 26264 q^{70} - 81779 q^{71} - 191763 q^{72} + 91273 q^{73} - 24102 q^{74} + 95115 q^{75} + 578339 q^{76} - 230091 q^{78} - 244684 q^{79} - 296482 q^{80} - 55548 q^{81} - 129920 q^{82} + 89009 q^{83} - 48804 q^{84} + 95335 q^{85} - 32043 q^{86} + 63248 q^{87} - 450187 q^{89} + 408246 q^{90} + 15974 q^{91} - 290507 q^{92} - 411705 q^{93} + 482748 q^{94} - 164289 q^{95} + 149697 q^{96} - 266929 q^{97} - 33614 q^{98}+O(q^{100}) \) 28 * q - 14 * q^2 - 28 * q^3 + 374 * q^4 - 258 * q^5 + 4 * q^6 + 1372 * q^7 - 339 * q^8 + 1660 * q^9 - 536 * q^10 - 996 * q^12 + 326 * q^13 - 686 * q^14 - 2561 * q^15 + 5642 * q^16 - 2159 * q^17 - 6642 * q^18 + 2253 * q^19 - 13417 * q^20 - 1372 * q^21 - 6620 * q^23 + 3009 * q^24 + 15576 * q^25 - 4960 * q^26 - 24718 * q^27 + 18326 * q^28 - 15747 * q^29 - 9484 * q^30 - 14455 * q^31 - 13922 * q^32 - 13463 * q^34 - 12642 * q^35 + 27148 * q^36 - 14447 * q^37 - 62211 * q^38 - 28537 * q^39 - 31765 * q^40 - 14604 * q^41 + 196 * q^42 - 6828 * q^43 - 2703 * q^45 - 110251 * q^46 - 51396 * q^47 - 24310 * q^48 + 67228 * q^49 - 17179 * q^50 - 23461 * q^51 + 71461 * q^52 - 134505 * q^53 + 135496 * q^54 - 16611 * q^56 - 45846 * q^57 + 18657 * q^58 - 22255 * q^59 - 176135 * q^60 - 61141 * q^61 + 186610 * q^62 + 81340 * q^63 + 83377 * q^64 - 8616 * q^65 - 34675 * q^67 - 106726 * q^68 - 178024 * q^69 - 26264 * q^70 - 81779 * q^71 - 191763 * q^72 + 91273 * q^73 - 24102 * q^74 + 95115 * q^75 + 578339 * q^76 - 230091 * q^78 - 244684 * q^79 - 296482 * q^80 - 55548 * q^81 - 129920 * q^82 + 89009 * q^83 - 48804 * q^84 + 95335 * q^85 - 32043 * q^86 + 63248 * q^87 - 450187 * q^89 + 408246 * q^90 + 15974 * q^91 - 290507 * q^92 - 411705 * q^93 + 482748 * q^94 - 164289 * q^95 + 149697 * q^96 - 266929 * q^97 - 33614 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.04168	−0.714476	−0.357238	−	0.934013i	$-0.616281\pi$
$2$	−4.04168	−0.714476	−0.357238	+	0.934013i	$0.616281\pi$
$3$	−8.44784	−0.541929	−0.270965	−	0.962589i	$-0.587343\pi$
$3$	−8.44784	−0.541929	−0.270965	+	0.962589i	$0.587343\pi$
$4$	−15.6648	−0.489525
$5$	−80.4152	−1.43851	−0.719255	−	0.694746i	$-0.755517\pi$
$5$	−80.4152	−1.43851	−0.719255	+	0.694746i	$0.755517\pi$
$6$	34.1435	0.387195
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	192.646	1.06423
$9$	−171.634	−0.706312
$10$	325.013	1.02778
$11$	0	0
$11$	0	0
$12$	132.334	0.265288
$13$	−1041.00	−1.70842	−0.854209	−	0.519930i	$-0.825958\pi$
$13$	−1041.00	−1.70842	−0.854209	+	0.519930i	$0.825958\pi$
$14$	−198.043	−0.270046
$15$	679.335	0.779571
$16$	−277.341	−0.270841
$17$	1377.06	1.15566	0.577832	−	0.816156i	$-0.303899\pi$
$17$	1377.06	1.15566	0.577832	+	0.816156i	$0.303899\pi$
$18$	693.690	0.504643
$19$	−3073.28	−1.95307	−0.976534	−	0.215363i	$-0.930907\pi$
$19$	−3073.28	−1.95307	−0.976534	+	0.215363i	$0.930907\pi$
$20$	1259.69	0.704186
$21$	−413.944	−0.204830
$22$	0	0
$23$	−2802.75	−1.10475	−0.552377	−	0.833595i	$-0.686279\pi$
$23$	−2802.75	−1.10475	−0.552377	+	0.833595i	$0.686279\pi$
$24$	−1627.44	−0.576737
$25$	3341.60	1.06931
$26$	4207.41	1.22062
$27$	3502.76	0.924701
$28$	−767.575	−0.185023
$29$	1964.27	0.433717	0.216859	−	0.976203i	$-0.430419\pi$
$29$	1964.27	0.433717	0.216859	+	0.976203i	$0.430419\pi$
$30$	−2745.66	−0.556985
$31$	3667.06	0.685352	0.342676	−	0.939454i	$-0.388667\pi$
$31$	3667.06	0.685352	0.342676	+	0.939454i	$0.388667\pi$
$32$	−5043.75	−0.870720
$33$	0	0
$34$	−5565.65	−0.825693
$35$	−3940.34	−0.543706
$36$	2688.61	0.345757
$37$	−2896.98	−0.347890	−0.173945	−	0.984755i	$-0.555651\pi$
$37$	−2896.98	−0.347890	−0.173945	+	0.984755i	$0.555651\pi$
$38$	12421.2	1.39542
$39$	8794.24	0.925842
$40$	−15491.7	−1.53091
$41$	−12022.8	−1.11698	−0.558491	−	0.829510i	$-0.688620\pi$
$41$	−12022.8	−1.11698	−0.558491	+	0.829510i	$0.688620\pi$
$42$	1673.03	0.146346
$43$	3038.75	0.250625	0.125312	−	0.992117i	$-0.460007\pi$
$43$	3038.75	0.250625	0.125312	+	0.992117i	$0.460007\pi$
$44$	0	0
$45$	13802.0	1.01604
$46$	11327.8	0.789319
$47$	−77.6245	−0.00512571	−0.00256285	−	0.999997i	$-0.500816\pi$
$47$	−77.6245	−0.00512571	−0.00256285	+	0.999997i	$0.500816\pi$
$48$	2342.94	0.146777
$49$	2401.00	0.142857
$50$	−13505.7	−0.763998
$51$	−11633.2	−0.626288
$52$	16307.1	0.836312
$53$	10813.7	0.528792	0.264396	−	0.964414i	$-0.414827\pi$
$53$	10813.7	0.528792	0.264396	+	0.964414i	$0.414827\pi$
$54$	−14157.1	−0.660676
$55$	0	0
$56$	9439.66	0.402241
$57$	25962.6	1.05843
$58$	−7938.97	−0.309881
$59$	50600.7	1.89246	0.946229	−	0.323498i	$-0.104859\pi$
$59$	50600.7	1.89246	0.946229	+	0.323498i	$0.104859\pi$
$60$	−10641.6	−0.381619
$61$	3498.86	0.120393	0.0601966	−	0.998187i	$-0.480827\pi$
$61$	3498.86	0.120393	0.0601966	+	0.998187i	$0.480827\pi$
$62$	−14821.1	−0.489668
$63$	−8410.06	−0.266961
$64$	29260.2	0.892949
$65$	83712.5	2.45758
$66$	0	0
$67$	−2376.54	−0.0646781	−0.0323391	−	0.999477i	$-0.510296\pi$
$67$	−2376.54	−0.0646781	−0.0323391	+	0.999477i	$0.510296\pi$
$68$	−21571.4	−0.565726
$69$	23677.2	0.598699
$70$	15925.6	0.388465
$71$	−8444.12	−0.198796	−0.0993982	−	0.995048i	$-0.531692\pi$
$71$	−8444.12	−0.198796	−0.0993982	+	0.995048i	$0.531692\pi$
$72$	−33064.6	−0.751678
$73$	28364.7	0.622976	0.311488	−	0.950250i	$-0.399173\pi$
$73$	28364.7	0.622976	0.311488	+	0.950250i	$0.399173\pi$
$74$	11708.7	0.248559
$75$	−28229.4	−0.579493
$76$	48142.2	0.956075
$77$	0	0
$78$	−35543.5	−0.661491
$79$	70294.9	1.26723	0.633616	−	0.773648i	$-0.281570\pi$
$79$	70294.9	1.26723	0.633616	+	0.773648i	$0.281570\pi$
$80$	22302.5	0.389608
$81$	12116.2	0.205190
$82$	48592.4	0.798057
$83$	123238.	1.96358	0.981789	−	0.189974i	$-0.0608405\pi$
$83$	123238.	1.96358	0.981789	+	0.189974i	$0.0608405\pi$
$84$	6484.35	0.100269
$85$	−110737.	−1.66243
$86$	−12281.7	−0.179065
$87$	−16593.9	−0.235044
$88$	0	0
$89$	−58546.5	−0.783476	−0.391738	−	0.920077i	$-0.628126\pi$
$89$	−58546.5	−0.783476	−0.391738	+	0.920077i	$0.628126\pi$
$90$	−55783.2	−0.725934
$91$	−51009.2	−0.645721
$92$	43904.5	0.540804
$93$	−30978.8	−0.371413
$94$	313.734	0.00366219
$95$	247138.	2.80951
$96$	42608.8	0.471869
$97$	−2415.92	−0.0260707	−0.0130354	−	0.999915i	$-0.504149\pi$
$97$	−2415.92	−0.0260707	−0.0130354	+	0.999915i	$0.504149\pi$
$98$	−9704.08	−0.102068
$99$	0	0
$100$	−52345.5	−0.523455
$101$	42738.4	0.416883	0.208442	−	0.978035i	$-0.433161\pi$
$101$	42738.4	0.416883	0.208442	+	0.978035i	$0.433161\pi$
$102$	47017.8	0.447468
$103$	60291.9	0.559971	0.279986	−	0.960004i	$-0.409670\pi$
$103$	60291.9	0.559971	0.279986	+	0.960004i	$0.409670\pi$
$104$	−200545.	−1.81815
$105$	33287.4	0.294650
$106$	−43705.6	−0.377809
$107$	102675.	0.866973	0.433486	−	0.901160i	$-0.357283\pi$
$107$	102675.	0.866973	0.433486	+	0.901160i	$0.357283\pi$
$108$	−54870.0	−0.452664
$109$	−127112.	−1.02475	−0.512376	−	0.858761i	$-0.671235\pi$
$109$	−127112.	−1.02475	−0.512376	+	0.858761i	$0.671235\pi$
$110$	0	0
$111$	24473.3	0.188532
$112$	−13589.7	−0.102368
$113$	160528.	1.18264	0.591321	−	0.806436i	$-0.298607\pi$
$113$	160528.	1.18264	0.591321	+	0.806436i	$0.298607\pi$
$114$	−104932.	−0.756219
$115$	225384.	1.58920
$116$	−30769.9	−0.212315
$117$	178672.	1.20668
$118$	−204512.	−1.35212
$119$	67476.1	0.436800
$120$	130871.	0.829643
$121$	0	0
$122$	−14141.3	−0.0860181
$123$	101567.	0.605326
$124$	−57443.7	−0.335497
$125$	−17418.3	−0.0997083
$126$	33990.8	0.190737
$127$	−274071.	−1.50783	−0.753917	−	0.656969i	$-0.771838\pi$
$127$	−274071.	−1.50783	−0.753917	+	0.656969i	$0.771838\pi$
$128$	43139.6	0.232729
$129$	−25670.9	−0.135821
$130$	−338340.	−1.75588
$131$	−77645.7	−0.395311	−0.197656	−	0.980272i	$-0.563333\pi$
$131$	−77645.7	−0.395311	−0.197656	+	0.980272i	$0.563333\pi$
$132$	0	0
$133$	−150590.	−0.738190
$134$	9605.21	0.0462109
$135$	−281675.	−1.33019
$136$	265286.	1.22989
$137$	−305053.	−1.38859	−0.694295	−	0.719691i	$-0.744284\pi$
$137$	−305053.	−1.38859	−0.694295	+	0.719691i	$0.744284\pi$
$138$	−95695.9	−0.427756
$139$	−122876.	−0.539423	−0.269712	−	0.962941i	$-0.586928\pi$
$139$	−122876.	−0.539423	−0.269712	+	0.962941i	$0.586928\pi$
$140$	61724.7	0.266157
$141$	655.759	0.00277777
$142$	34128.5	0.142035
$143$	0	0
$144$	47601.2	0.191298
$145$	−157957.	−0.623907
$146$	−114641.	−0.445101
$147$	−20283.3	−0.0774185
$148$	45380.6	0.170301
$149$	−15689.1	−0.0578938	−0.0289469	−	0.999581i	$-0.509215\pi$
$149$	−15689.1	−0.0578938	−0.0289469	+	0.999581i	$0.509215\pi$
$150$	114094.	0.414033
$151$	−161012.	−0.574666	−0.287333	−	0.957831i	$-0.592769\pi$
$151$	−161012.	−0.574666	−0.287333	+	0.957831i	$0.592769\pi$
$152$	−592054.	−2.07851
$153$	−236351.	−0.816260
$154$	0	0
$155$	−294888.	−0.985887
$156$	−137760.	−0.453222
$157$	228805.	0.740825	0.370413	−	0.928867i	$-0.379216\pi$
$157$	228805.	0.740825	0.370413	+	0.928867i	$0.379216\pi$
$158$	−284110.	−0.905407
$159$	−91352.6	−0.286568
$160$	405594.	1.25254
$161$	−137335.	−0.417558
$162$	−48970.0	−0.146603
$163$	−605399.	−1.78473	−0.892365	−	0.451314i	$-0.850955\pi$
$163$	−605399.	−1.78473	−0.892365	+	0.451314i	$0.850955\pi$
$164$	188335.	0.546790
$165$	0	0
$166$	−498088.	−1.40293
$167$	89774.9	0.249094	0.124547	−	0.992214i	$-0.460252\pi$
$167$	89774.9	0.249094	0.124547	+	0.992214i	$0.460252\pi$
$168$	−79744.7	−0.217986
$169$	712396.	1.91869
$170$	447563.	1.18777
$171$	527478.	1.37948
$172$	−47601.4	−0.122687
$173$	628032.	1.59539	0.797694	−	0.603062i	$-0.206053\pi$
$173$	628032.	1.59539	0.797694	+	0.603062i	$0.206053\pi$
$174$	67067.2	0.167933
$175$	163739.	0.404163
$176$	0	0
$177$	−427467.	−1.02558
$178$	236627.	0.559775
$179$	306225.	0.714345	0.357173	−	0.934038i	$-0.383741\pi$
$179$	306225.	0.714345	0.357173	+	0.934038i	$0.383741\pi$
$180$	−216205.	−0.497376
$181$	−44942.3	−0.101967	−0.0509834	−	0.998700i	$-0.516236\pi$
$181$	−44942.3	−0.101967	−0.0509834	+	0.998700i	$0.516236\pi$
$182$	206163.	0.461352
$183$	−29557.8	−0.0652447
$184$	−539939.	−1.17571
$185$	232961.	0.500443
$186$	125206.	0.265365
$187$	0	0
$188$	1215.97	0.00250916
$189$	171635.	0.349504
$190$	−998854.	−2.00733
$191$	195457.	0.387675	0.193838	−	0.981034i	$-0.437907\pi$
$191$	195457.	0.387675	0.193838	+	0.981034i	$0.437907\pi$
$192$	−247185.	−0.483915
$193$	−379935.	−0.734202	−0.367101	−	0.930181i	$-0.619650\pi$
$193$	−379935.	−0.734202	−0.367101	+	0.930181i	$0.619650\pi$
$194$	9764.38	0.0186269
$195$	−707191.	−1.33183
$196$	−37611.2	−0.0699321
$197$	747608.	1.37249	0.686243	−	0.727372i	$-0.259258\pi$
$197$	747608.	1.37249	0.686243	+	0.727372i	$0.259258\pi$
$198$	0	0
$199$	58740.4	0.105149	0.0525744	−	0.998617i	$-0.483257\pi$
$199$	58740.4	0.105149	0.0525744	+	0.998617i	$0.483257\pi$
$200$	643747.	1.13799
$201$	20076.6	0.0350510
$202$	−172735.	−0.297853
$203$	96249.4	0.163930
$204$	182232.	0.306583
$205$	966817.	1.60679
$206$	−243681.	−0.400086
$207$	481048.	0.780301
$208$	288713.	0.462710
$209$	0	0
$210$	−134537.	−0.210520
$211$	202629.	0.313325	0.156663	−	0.987652i	$-0.449926\pi$
$211$	202629.	0.313325	0.156663	+	0.987652i	$0.449926\pi$
$212$	−169395.	−0.258857
$213$	71334.6	0.107734
$214$	−414980.	−0.619431
$215$	−244362.	−0.360527
$216$	674793.	0.984094
$217$	179686.	0.259039
$218$	513745.	0.732161
$219$	−239621.	−0.337609
$220$	0	0
$221$	−1.43353e6	−1.97436
$222$	−98913.2	−0.134701
$223$	−701691.	−0.944895	−0.472448	−	0.881359i	$-0.656629\pi$
$223$	−701691.	−0.944895	−0.472448	+	0.881359i	$0.656629\pi$
$224$	−247144.	−0.329101
$225$	−573533.	−0.755269
$226$	−648802.	−0.844969
$227$	548781.	0.706862	0.353431	−	0.935461i	$-0.385015\pi$
$227$	548781.	0.706862	0.353431	+	0.935461i	$0.385015\pi$
$228$	−406698.	−0.518125
$229$	1.08509e6	1.36734	0.683672	−	0.729790i	$-0.260382\pi$
$229$	1.08509e6	1.36734	0.683672	+	0.729790i	$0.260382\pi$
$230$	−910931.	−1.13544
$231$	0	0
$232$	378409.	0.461575
$233$	−987539.	−1.19169	−0.595846	−	0.803098i	$-0.703183\pi$
$233$	−987539.	−1.19169	−0.595846	+	0.803098i	$0.703183\pi$
$234$	−722134.	−0.862141
$235$	6242.19	0.00737339
$236$	−792649.	−0.926405
$237$	−593841.	−0.686751
$238$	−272717.	−0.312083
$239$	−693190.	−0.784978	−0.392489	−	0.919757i	$-0.628386\pi$
$239$	−693190.	−0.784978	−0.392489	+	0.919757i	$0.628386\pi$
$240$	−188408.	−0.211140
$241$	−1.44741e6	−1.60527	−0.802637	−	0.596467i	$-0.796571\pi$
$241$	−1.44741e6	−1.60527	−0.802637	+	0.596467i	$0.796571\pi$
$242$	0	0
$243$	−953528.	−1.03590
$244$	−54808.9	−0.0589355
$245$	−193077.	−0.205502
$246$	−410501.	−0.432491
$247$	3.19929e6	3.33666
$248$	706445.	0.729372
$249$	−1.04109e6	−1.06412
$250$	70399.4	0.0712392
$251$	−970504.	−0.972329	−0.486164	−	0.873867i	$-0.661604\pi$
$251$	−970504.	−0.972329	−0.486164	+	0.873867i	$0.661604\pi$
$252$	131742.	0.130684
$253$	0	0
$254$	1.10771e6	1.07731
$255$	935487.	0.900922
$256$	−1.11068e6	−1.05923
$257$	1.29134e6	1.21957	0.609787	−	0.792565i	$-0.291255\pi$
$257$	1.29134e6	1.21957	0.609787	+	0.792565i	$0.291255\pi$
$258$	103754.	0.0970408
$259$	−141952.	−0.131490
$260$	−1.31134e6	−1.20304
$261$	−337136.	−0.306340
$262$	313819.	0.282440
$263$	−848635.	−0.756540	−0.378270	−	0.925695i	$-0.623481\pi$
$263$	−848635.	−0.756540	−0.378270	+	0.925695i	$0.623481\pi$
$264$	0	0
$265$	−869587.	−0.760674
$266$	608639.	0.527419
$267$	494592.	0.424589
$268$	37227.9	0.0316615
$269$	100164.	0.0843975	0.0421988	−	0.999109i	$-0.486564\pi$
$269$	100164.	0.0843975	0.0421988	+	0.999109i	$0.486564\pi$
$270$	1.13844e6	0.950390
$271$	119477.	0.0988234	0.0494117	−	0.998778i	$-0.484265\pi$
$271$	119477.	0.0988234	0.0494117	+	0.998778i	$0.484265\pi$
$272$	−381916.	−0.313001
$273$	430918.	0.349935
$274$	1.23293e6	0.992113
$275$	0	0
$276$	−370899.	−0.293078
$277$	1.66014e6	1.30001	0.650005	−	0.759930i	$-0.274767\pi$
$277$	1.66014e6	1.30001	0.650005	+	0.759930i	$0.274767\pi$
$278$	496625.	0.385405
$279$	−629392.	−0.484073
$280$	−759092.	−0.578628
$281$	−2.09121e6	−1.57991	−0.789955	−	0.613165i	$-0.789896\pi$
$281$	−2.09121e6	−1.57991	−0.789955	+	0.613165i	$0.789896\pi$
$282$	−2650.37	−0.00198465
$283$	−56449.2	−0.0418979	−0.0209489	−	0.999781i	$-0.506669\pi$
$283$	−56449.2	−0.0418979	−0.0209489	+	0.999781i	$0.506669\pi$
$284$	132275.	0.0973157
$285$	−2.08778e6	−1.52256
$286$	0	0
$287$	−589118.	−0.422180
$288$	865678.	0.615000
$289$	476445.	0.335558
$290$	638414.	0.445767
$291$	20409.3	0.0141285
$292$	−444328.	−0.304962
$293$	340996.	0.232049	0.116025	−	0.993246i	$-0.462985\pi$
$293$	340996.	0.232049	0.116025	+	0.993246i	$0.462985\pi$
$294$	81978.6	0.0553136
$295$	−4.06906e6	−2.72232
$296$	−558092.	−0.370234
$297$	0	0
$298$	63410.3	0.0413637
$299$	2.91768e6	1.88738
$300$	442207.	0.283676
$301$	148899.	0.0947273
$302$	650759.	0.410585
$303$	−361047.	−0.225921
$304$	852346.	0.528971
$305$	−281362.	−0.173187
$306$	955255.	0.583198
$307$	1.82261e6	1.10369	0.551847	−	0.833946i	$-0.313923\pi$
$307$	1.82261e6	1.10369	0.551847	+	0.833946i	$0.313923\pi$
$308$	0	0
$309$	−509337.	−0.303465
$310$	1.19184e6	0.704392
$311$	−2.78579e6	−1.63323	−0.816616	−	0.577181i	$-0.804153\pi$
$311$	−2.78579e6	−1.63323	−0.816616	+	0.577181i	$0.804153\pi$
$312$	1.69418e6	0.985308
$313$	−952318.	−0.549441	−0.274721	−	0.961524i	$-0.588585\pi$
$313$	−952318.	−0.549441	−0.274721	+	0.961524i	$0.588585\pi$
$314$	−924756.	−0.529301
$315$	676297.	0.384026
$316$	−1.10115e6	−0.620341
$317$	−1.64001e6	−0.916636	−0.458318	−	0.888788i	$-0.651548\pi$
$317$	−1.64001e6	−0.916636	−0.458318	+	0.888788i	$0.651548\pi$
$318$	369218.	0.204746
$319$	0	0
$320$	−2.35296e6	−1.28452
$321$	−867383.	−0.469838
$322$	555065.	0.298335
$323$	−4.23209e6	−2.25709
$324$	−189798.	−0.100445
$325$	−3.47862e6	−1.82683
$326$	2.44683e6	1.27515
$327$	1.07382e6	0.555344
$328$	−2.31615e6	−1.18873
$329$	−3803.60	−0.00193734
$330$	0	0
$331$	2.10918e6	1.05814	0.529072	−	0.848577i	$-0.322540\pi$
$331$	2.10918e6	1.05814	0.529072	+	0.848577i	$0.322540\pi$
$332$	−1.93049e6	−0.961220
$333$	497220.	0.245719
$334$	−362842.	−0.177972
$335$	191110.	0.0930402
$336$	114804.	0.0554764
$337$	1.37557e6	0.659794	0.329897	−	0.944017i	$-0.392986\pi$
$337$	1.37557e6	0.659794	0.329897	+	0.944017i	$0.392986\pi$
$338$	−2.87928e6	−1.37086
$339$	−1.35611e6	−0.640909
$340$	1.73467e6	0.813803
$341$	0	0
$342$	−2.13190e6	−0.985602
$343$	117649.	0.0539949
$344$	585403.	0.266722
$345$	−1.90401e6	−0.861234
$346$	−2.53831e6	−1.13987
$347$	−1.34456e6	−0.599456	−0.299728	−	0.954025i	$-0.596896\pi$
$347$	−1.34456e6	−0.599456	−0.299728	+	0.954025i	$0.596896\pi$
$348$	259939.	0.115060
$349$	−2.30128e6	−1.01136	−0.505681	−	0.862721i	$-0.668759\pi$
$349$	−2.30128e6	−1.01136	−0.505681	+	0.862721i	$0.668759\pi$
$350$	−661780.	−0.288764
$351$	−3.64639e6	−1.57978
$352$	0	0
$353$	−975475.	−0.416658	−0.208329	−	0.978059i	$-0.566802\pi$
$353$	−975475.	−0.416658	−0.208329	+	0.978059i	$0.566802\pi$
$354$	1.72769e6	0.732751
$355$	679036.	0.285971
$356$	917119.	0.383531
$357$	−570027.	−0.236715
$358$	−1.23767e6	−0.510382
$359$	−2.55771e6	−1.04740	−0.523702	−	0.851901i	$-0.675450\pi$
$359$	−2.55771e6	−1.04740	−0.523702	+	0.851901i	$0.675450\pi$
$360$	2.65890e6	1.08130
$361$	6.96892e6	2.81448
$362$	181643.	0.0728528
$363$	0	0
$364$	799048.	0.316096
$365$	−2.28096e6	−0.896158
$366$	119463.	0.0466157
$367$	1.88863e6	0.731952	0.365976	−	0.930624i	$-0.380735\pi$
$367$	1.88863e6	0.731952	0.365976	+	0.930624i	$0.380735\pi$
$368$	777319.	0.299213
$369$	2.06352e6	0.788939
$370$	−941556.	−0.357554
$371$	529872.	0.199865
$372$	485276.	0.181816
$373$	4.41752e6	1.64402	0.822008	−	0.569476i	$-0.192854\pi$
$373$	4.41752e6	1.64402	0.822008	+	0.569476i	$0.192854\pi$
$374$	0	0
$375$	147147.	0.0540349
$376$	−14954.0	−0.00545493
$377$	−2.04482e6	−0.740970
$378$	−693696.	−0.249712
$379$	−4.90055e6	−1.75246	−0.876228	−	0.481898i	$-0.839948\pi$
$379$	−4.90055e6	−1.75246	−0.876228	+	0.481898i	$0.839948\pi$
$380$	−3.87136e6	−1.37532
$381$	2.31531e6	0.817140
$382$	−789976.	−0.276984
$383$	−1.31663e6	−0.458634	−0.229317	−	0.973352i	$-0.573649\pi$
$383$	−1.31663e6	−0.458634	−0.229317	+	0.973352i	$0.573649\pi$
$384$	−364437.	−0.126123
$385$	0	0
$386$	1.53558e6	0.524570
$387$	−521553.	−0.177019
$388$	37844.8	0.0127623
$389$	−436160.	−0.146141	−0.0730704	−	0.997327i	$-0.523280\pi$
$389$	−436160.	−0.146141	−0.0730704	+	0.997327i	$0.523280\pi$
$390$	2.85824e6	0.951563
$391$	−3.85957e6	−1.27672
$392$	462543.	0.152033
$393$	655939.	0.214231
$394$	−3.02159e6	−0.980608
$395$	−5.65278e6	−1.82293
$396$	0	0
$397$	3.78433e6	1.20507	0.602536	−	0.798091i	$-0.294157\pi$
$397$	3.78433e6	1.20507	0.602536	+	0.798091i	$0.294157\pi$
$398$	−237410.	−0.0751262
$399$	1.27216e6	0.400047
$400$	−926765.	−0.289614
$401$	−2.63394e6	−0.817985	−0.408992	−	0.912538i	$-0.634120\pi$
$401$	−2.63394e6	−0.817985	−0.408992	+	0.912538i	$0.634120\pi$
$402$	−81143.3	−0.0250431
$403$	−3.81743e6	−1.17087
$404$	−669488.	−0.204075
$405$	−974330.	−0.295168
$406$	−389010.	−0.117124
$407$	0	0
$408$	−2.24109e6	−0.666514
$409$	2.73854e6	0.809490	0.404745	−	0.914430i	$-0.367360\pi$
$409$	2.73854e6	0.809490	0.404745	+	0.914430i	$0.367360\pi$
$410$	−3.90757e6	−1.14801
$411$	2.57704e6	0.752517
$412$	−944460.	−0.274120
$413$	2.47943e6	0.715282
$414$	−1.94424e6	−0.557506
$415$	−9.91018e6	−2.82463
$416$	5.25056e6	1.48755
$417$	1.03804e6	0.292329
$418$	0	0
$419$	3.42263e6	0.952413	0.476206	−	0.879333i	$-0.342011\pi$
$419$	3.42263e6	0.952413	0.476206	+	0.879333i	$0.342011\pi$
$420$	−521440.	−0.144239
$421$	−786541.	−0.216280	−0.108140	−	0.994136i	$-0.534489\pi$
$421$	−786541.	−0.216280	−0.108140	+	0.994136i	$0.534489\pi$
$422$	−818962.	−0.223863
$423$	13323.0	0.00362035
$424$	2.08322e6	0.562756
$425$	4.60160e6	1.23577
$426$	−288312.	−0.0769731
$427$	171444.	0.0455044
$428$	−1.60838e6	−0.424405
$429$	0	0
$430$	987633.	0.257587
$431$	−4.67165e6	−1.21137	−0.605686	−	0.795704i	$-0.707101\pi$
$431$	−4.67165e6	−1.21137	−0.605686	+	0.795704i	$0.707101\pi$
$432$	−971461.	−0.250447
$433$	4.45249e6	1.14126	0.570628	−	0.821209i	$-0.306700\pi$
$433$	4.45249e6	1.14126	0.570628	+	0.821209i	$0.306700\pi$
$434$	−726234.	−0.185077
$435$	1.33440e6	0.338114
$436$	1.99118e6	0.501642
$437$	8.61363e6	2.15766
$438$	968472.	0.241214
$439$	4.52816e6	1.12140	0.560700	−	0.828019i	$-0.310532\pi$
$439$	4.52816e6	1.12140	0.560700	+	0.828019i	$0.310532\pi$
$440$	0	0
$441$	−412093.	−0.100902
$442$	5.79387e6	1.41063
$443$	−4.66865e6	−1.13027	−0.565134	−	0.824999i	$-0.691176\pi$
$443$	−4.66865e6	−1.13027	−0.565134	+	0.824999i	$0.691176\pi$
$444$	−383368.	−0.0922909
$445$	4.70803e6	1.12704
$446$	2.83601e6	0.675105
$447$	132539.	0.0313743
$448$	1.43375e6	0.337503
$449$	149336.	0.0349583	0.0174791	−	0.999847i	$-0.494436\pi$
$449$	149336.	0.0349583	0.0174791	+	0.999847i	$0.494436\pi$
$450$	2.31804e6	0.539622
$451$	0	0
$452$	−2.51463e6	−0.578933
$453$	1.36020e6	0.311428
$454$	−2.21800e6	−0.505036
$455$	4.10191e6	0.928877
$456$	5.00158e6	1.12641
$457$	8.37642e6	1.87615	0.938076	−	0.346429i	$-0.112606\pi$
$457$	8.37642e6	1.87615	0.938076	+	0.346429i	$0.112606\pi$
$458$	−4.38560e6	−0.976934
$459$	4.82352e6	1.06864
$460$	−3.53059e6	−0.777952
$461$	1.36430e6	0.298990	0.149495	−	0.988762i	$-0.452235\pi$
$461$	1.36430e6	0.298990	0.149495	+	0.988762i	$0.452235\pi$
$462$	0	0
$463$	−1.86639e6	−0.404621	−0.202311	−	0.979321i	$-0.564845\pi$
$463$	−1.86639e6	−0.404621	−0.202311	+	0.979321i	$0.564845\pi$
$464$	−544774.	−0.117468
$465$	2.49116e6	0.534281
$466$	3.99132e6	0.851436
$467$	−4.46227e6	−0.946813	−0.473406	−	0.880844i	$-0.656976\pi$
$467$	−4.46227e6	−0.946813	−0.473406	+	0.880844i	$0.656976\pi$
$468$	−2.79885e6	−0.590698
$469$	−116450.	−0.0244460
$470$	−25229.0	−0.00526811
$471$	−1.93291e6	−0.401475
$472$	9.74802e6	2.01401
$473$	0	0
$474$	2.40012e6	0.490666
$475$	−1.02697e7	−2.08844
$476$	−1.05700e6	−0.213824
$477$	−1.85600e6	−0.373493
$478$	2.80166e6	0.560848
$479$	6.46066e6	1.28658	0.643292	−	0.765621i	$-0.277568\pi$
$479$	6.46066e6	1.28658	0.643292	+	0.765621i	$0.277568\pi$
$480$	−3.42639e6	−0.678788
$481$	3.01577e6	0.594341
$482$	5.84998e6	1.14693
$483$	1.16018e6	0.226287
$484$	0	0
$485$	194277.	0.0375030
$486$	3.85386e6	0.740125
$487$	5.85803e6	1.11925	0.559627	−	0.828744i	$-0.310944\pi$
$487$	5.85803e6	1.11925	0.559627	+	0.828744i	$0.310944\pi$
$488$	674042.	0.128126
$489$	5.11432e6	0.967198
$490$	780356.	0.146826
$491$	1.79499e6	0.336014	0.168007	−	0.985786i	$-0.446267\pi$
$491$	1.79499e6	0.336014	0.168007	+	0.985786i	$0.446267\pi$
$492$	−1.59102e6	−0.296322
$493$	2.70493e6	0.501231
$494$	−1.29305e7	−2.38396
$495$	0	0
$496$	−1.01703e6	−0.185622
$497$	−413762.	−0.0751380
$498$	4.20777e6	0.760288
$499$	4.04836e6	0.727826	0.363913	−	0.931433i	$-0.381441\pi$
$499$	4.04836e6	0.727826	0.363913	+	0.931433i	$0.381441\pi$
$500$	272854.	0.0488097
$501$	−758405.	−0.134992
$502$	3.92247e6	0.694705
$503$	−1.16210e6	−0.204796	−0.102398	−	0.994743i	$-0.532652\pi$
$503$	−1.16210e6	−0.204796	−0.102398	+	0.994743i	$0.532652\pi$
$504$	−1.62017e6	−0.284108
$505$	−3.43682e6	−0.599691
$506$	0	0
$507$	−6.01821e6	−1.03979
$508$	4.29326e6	0.738122
$509$	5.96545e6	1.02058	0.510292	−	0.860001i	$-0.329537\pi$
$509$	5.96545e6	1.02058	0.510292	+	0.860001i	$0.329537\pi$
$510$	−3.78094e6	−0.643687
$511$	1.38987e6	0.235463
$512$	3.10856e6	0.524064
$513$	−1.07650e7	−1.80600
$514$	−5.21920e6	−0.871357
$515$	−4.84839e6	−0.805525
$516$	402129.	0.0664877
$517$	0	0
$518$	573726.	0.0939463
$519$	−5.30552e6	−0.864588
$520$	1.61269e7	2.61542
$521$	−4.44220e6	−0.716975	−0.358487	−	0.933535i	$-0.616707\pi$
$521$	−4.44220e6	−0.716975	−0.358487	+	0.933535i	$0.616707\pi$
$522$	1.36260e6	0.218872
$523$	2.96980e6	0.474758	0.237379	−	0.971417i	$-0.423712\pi$
$523$	2.96980e6	0.474758	0.237379	+	0.971417i	$0.423712\pi$
$524$	1.21630e6	0.193515
$525$	−1.38324e6	−0.219028
$526$	3.42992e6	0.540529
$527$	5.04977e6	0.792037
$528$	0	0
$529$	1.41909e6	0.220480
$530$	3.51460e6	0.543483
$531$	−8.68479e6	−1.33667
$532$	2.35897e6	0.361362
$533$	1.25158e7	1.90827
$534$	−1.99898e6	−0.303358
$535$	−8.25664e6	−1.24715
$536$	−457830.	−0.0688323
$537$	−2.58694e6	−0.387125
$538$	−404830.	−0.0603000
$539$	0	0
$540$	4.41238e6	0.651162
$541$	−1.13523e7	−1.66759	−0.833795	−	0.552074i	$-0.813837\pi$
$541$	−1.13523e7	−1.66759	−0.833795	+	0.552074i	$0.813837\pi$
$542$	−482887.	−0.0706069
$543$	379666.	0.0552588
$544$	−6.94556e6	−1.00626
$545$	1.02217e7	1.47412
$546$	−1.74163e6	−0.250020
$547$	3.90310e6	0.557752	0.278876	−	0.960327i	$-0.410038\pi$
$547$	3.90310e6	0.557752	0.278876	+	0.960327i	$0.410038\pi$
$548$	4.77859e6	0.679749
$549$	−600523.	−0.0850353
$550$	0	0
$551$	−6.03675e6	−0.847080
$552$	4.56132e6	0.637152
$553$	3.44445e6	0.478969
$554$	−6.70978e6	−0.928825
$555$	−1.96802e6	−0.271205
$556$	1.92482e6	0.264061
$557$	2.03455e6	0.277863	0.138932	−	0.990302i	$-0.455633\pi$
$557$	2.03455e6	0.277863	0.138932	+	0.990302i	$0.455633\pi$
$558$	2.54380e6	0.345858
$559$	−3.16335e6	−0.428172
$560$	1.09282e6	0.147258
$561$	0	0
$562$	8.45202e6	1.12881
$563$	−1.02551e6	−0.136354	−0.0681769	−	0.997673i	$-0.521718\pi$
$563$	−1.02551e6	−0.136354	−0.0681769	+	0.997673i	$0.521718\pi$
$564$	−10272.3	−0.00135979
$565$	−1.29089e7	−1.70124
$566$	228150.	0.0299350
$567$	593696.	0.0775544
$568$	−1.62673e6	−0.211565
$569$	1.20762e7	1.56369	0.781844	−	0.623474i	$-0.214279\pi$
$569$	1.20762e7	1.56369	0.781844	+	0.623474i	$0.214279\pi$
$570$	8.43816e6	1.08783
$571$	−581743.	−0.0746691	−0.0373345	−	0.999303i	$-0.511887\pi$
$571$	−581743.	−0.0746691	−0.0373345	+	0.999303i	$0.511887\pi$
$572$	0	0
$573$	−1.65119e6	−0.210093
$574$	2.38103e6	0.301637
$575$	−9.36570e6	−1.18133
$576$	−5.02204e6	−0.630701
$577$	−1.02212e7	−1.27810	−0.639048	−	0.769167i	$-0.720672\pi$
$577$	−1.02212e7	−1.27810	−0.639048	+	0.769167i	$0.720672\pi$
$578$	−1.92564e6	−0.239748
$579$	3.20963e6	0.397886
$580$	2.47437e6	0.305418
$581$	6.03864e6	0.742163
$582$	−82488.0	−0.0100945
$583$	0	0
$584$	5.46435e6	0.662990
$585$	−1.43679e7	−1.73582
$586$	−1.37820e6	−0.165794
$587$	3.11236e6	0.372816	0.186408	−	0.982472i	$-0.440315\pi$
$587$	3.11236e6	0.372816	0.186408	+	0.982472i	$0.440315\pi$
$588$	317733.	0.0378983
$589$	−1.12699e7	−1.33854
$590$	1.64459e7	1.94503
$591$	−6.31567e6	−0.743791
$592$	803453.	0.0942228
$593$	4.71911e6	0.551091	0.275545	−	0.961288i	$-0.411142\pi$
$593$	4.71911e6	0.551091	0.275545	+	0.961288i	$0.411142\pi$
$594$	0	0
$595$	−5.42610e6	−0.628341
$596$	245766.	0.0283404
$597$	−496230.	−0.0569832
$598$	−1.17923e7	−1.34849
$599$	−9.88888e6	−1.12611	−0.563054	−	0.826420i	$-0.690374\pi$
$599$	−9.88888e6	−1.12611	−0.563054	+	0.826420i	$0.690374\pi$
$600$	−5.43827e6	−0.616713
$601$	4.09646e6	0.462618	0.231309	−	0.972880i	$-0.425699\pi$
$601$	4.09646e6	0.462618	0.231309	+	0.972880i	$0.425699\pi$
$602$	−601802.	−0.0676803
$603$	407894.	0.0456830
$604$	2.52221e6	0.281313
$605$	0	0
$606$	1.45924e6	0.161415
$607$	−2.05567e6	−0.226455	−0.113227	−	0.993569i	$-0.536119\pi$
$607$	−2.05567e6	−0.226455	−0.113227	+	0.993569i	$0.536119\pi$
$608$	1.55008e7	1.70057
$609$	−813100.	−0.0888384
$610$	1.13718e6	0.123738
$611$	80807.4	0.00875685
$612$	3.70238e6	0.399579
$613$	4.46863e6	0.480311	0.240156	−	0.970734i	$-0.422802\pi$
$613$	4.46863e6	0.480311	0.240156	+	0.970734i	$0.422802\pi$
$614$	−7.36643e6	−0.788562
$615$	−8.16752e6	−0.870768
$616$	0	0
$617$	1.22886e7	1.29954	0.649771	−	0.760130i	$-0.274865\pi$
$617$	1.22886e7	1.29954	0.649771	+	0.760130i	$0.274865\pi$
$618$	2.05858e6	0.216818
$619$	−2.79648e6	−0.293349	−0.146675	−	0.989185i	$-0.546857\pi$
$619$	−2.79648e6	−0.293349	−0.146675	+	0.989185i	$0.546857\pi$
$620$	4.61935e6	0.482616
$621$	−9.81738e6	−1.02157
$622$	1.12593e7	1.16690
$623$	−2.86878e6	−0.296126
$624$	−2.43901e6	−0.250756
$625$	−9.04182e6	−0.925882
$626$	3.84897e6	0.392562
$627$	0	0
$628$	−3.58418e6	−0.362652
$629$	−3.98933e6	−0.402043
$630$	−2.73338e6	−0.274377
$631$	1.36725e6	0.136702	0.0683511	−	0.997661i	$-0.478226\pi$
$631$	1.36725e6	0.136702	0.0683511	+	0.997661i	$0.478226\pi$
$632$	1.35420e7	1.34863
$633$	−1.71178e6	−0.169800
$634$	6.62838e6	0.654914
$635$	2.20395e7	2.16904
$636$	1.43102e6	0.140282
$637$	−2.49945e6	−0.244060
$638$	0	0
$639$	1.44930e6	0.140412
$640$	−3.46908e6	−0.334784
$641$	−4.03788e6	−0.388158	−0.194079	−	0.980986i	$-0.562172\pi$
$641$	−4.03788e6	−0.388158	−0.194079	+	0.980986i	$0.562172\pi$
$642$	3.50569e6	0.335688
$643$	8.54991e6	0.815519	0.407759	−	0.913089i	$-0.366310\pi$
$643$	8.54991e6	0.815519	0.407759	+	0.913089i	$0.366310\pi$
$644$	2.15132e6	0.204405
$645$	2.06433e6	0.195380
$646$	1.71048e7	1.61264
$647$	7.96505e6	0.748045	0.374023	−	0.927420i	$-0.377978\pi$
$647$	7.96505e6	0.748045	0.374023	+	0.927420i	$0.377978\pi$
$648$	2.33415e6	0.218369
$649$	0	0
$650$	1.40595e7	1.30523
$651$	−1.51796e6	−0.140381
$652$	9.48344e6	0.873669
$653$	5.20984e6	0.478125	0.239062	−	0.971004i	$-0.423160\pi$
$653$	5.20984e6	0.478125	0.239062	+	0.971004i	$0.423160\pi$
$654$	−4.34004e6	−0.396779
$655$	6.24390e6	0.568660
$656$	3.33442e6	0.302525
$657$	−4.86835e6	−0.440016
$658$	15372.9	0.00138418
$659$	−8.18627e6	−0.734298	−0.367149	−	0.930162i	$-0.619666\pi$
$659$	−8.18627e6	−0.734298	−0.367149	+	0.930162i	$0.619666\pi$
$660$	0	0
$661$	1.60839e7	1.43182	0.715910	−	0.698193i	$-0.246012\pi$
$661$	1.60839e7	1.43182	0.715910	+	0.698193i	$0.246012\pi$
$662$	−8.52466e6	−0.756017
$663$	1.21102e7	1.06996
$664$	2.37412e7	2.08970
$665$	1.21098e7	1.06189
$666$	−2.00961e6	−0.175560
$667$	−5.50537e6	−0.479151
$668$	−1.40631e6	−0.121938
$669$	5.92778e6	0.512067
$670$	−772405.	−0.0664749
$671$	0	0
$672$	2.08783e6	0.178350
$673$	−7.38053e6	−0.628131	−0.314065	−	0.949401i	$-0.601691\pi$
$673$	−7.38053e6	−0.628131	−0.314065	+	0.949401i	$0.601691\pi$
$674$	−5.55963e6	−0.471407
$675$	1.17048e7	0.988795
$676$	−1.11595e7	−0.939246
$677$	−1.31438e7	−1.10217	−0.551087	−	0.834448i	$-0.685787\pi$
$677$	−1.31438e7	−1.10217	−0.551087	+	0.834448i	$0.685787\pi$
$678$	5.48098e6	0.457914
$679$	−118380.	−0.00985380
$680$	−2.13330e7	−1.76921
$681$	−4.63602e6	−0.383069
$682$	0	0
$683$	−1.02336e7	−0.839416	−0.419708	−	0.907659i	$-0.637868\pi$
$683$	−1.02336e7	−0.839416	−0.419708	+	0.907659i	$0.637868\pi$
$684$	−8.26283e6	−0.675288
$685$	2.45309e7	1.99750
$686$	−475500.	−0.0385781
$687$	−9.16668e6	−0.741004
$688$	−842771.	−0.0678795
$689$	−1.12571e7	−0.903398
$690$	7.69541e6	0.615331
$691$	−6.17195e6	−0.491731	−0.245865	−	0.969304i	$-0.579072\pi$
$691$	−6.17195e6	−0.491731	−0.245865	+	0.969304i	$0.579072\pi$
$692$	−9.83799e6	−0.780982
$693$	0	0
$694$	5.43429e6	0.428296
$695$	9.88109e6	0.775966
$696$	−3.19674e6	−0.250141
$697$	−1.65562e7	−1.29086
$698$	9.30106e6	0.722593
$699$	8.34257e6	0.645814
$700$	−2.56493e6	−0.197847
$701$	2.21105e7	1.69943	0.849714	−	0.527244i	$-0.176775\pi$
$701$	2.21105e7	1.69943	0.849714	+	0.527244i	$0.176775\pi$
$702$	1.47376e7	1.12871
$703$	8.90322e6	0.679452
$704$	0	0
$705$	−52733.0	−0.00399586
$706$	3.94256e6	0.297692
$707$	2.09418e6	0.157567
$708$	6.69617e6	0.502046
$709$	−3.92613e6	−0.293325	−0.146663	−	0.989187i	$-0.546853\pi$
$709$	−3.92613e6	−0.293325	−0.146663	+	0.989187i	$0.546853\pi$
$710$	−2.74445e6	−0.204319
$711$	−1.20650e7	−0.895062
$712$	−1.12788e7	−0.833798
$713$	−1.02779e7	−0.757146
$714$	2.30387e6	0.169127
$715$	0	0
$716$	−4.79695e6	−0.349690
$717$	5.85596e6	0.425403
$718$	1.03374e7	0.748345
$719$	−1.17116e7	−0.844877	−0.422438	−	0.906392i	$-0.638826\pi$
$719$	−1.17116e7	−0.844877	−0.422438	+	0.906392i	$0.638826\pi$
$720$	−3.82786e6	−0.275185
$721$	2.95430e6	0.211649
$722$	−2.81662e7	−2.01087
$723$	1.22275e7	0.869946
$724$	704012.	0.0499153
$725$	6.56382e6	0.463780
$726$	0	0
$727$	−1.45893e7	−1.02376	−0.511880	−	0.859057i	$-0.671051\pi$
$727$	−1.45893e7	−1.02376	−0.511880	+	0.859057i	$0.671051\pi$
$728$	−9.82672e6	−0.687195
$729$	5.11101e6	0.356195
$730$	9.21891e6	0.640283
$731$	4.18455e6	0.289638
$732$	463017.	0.0319389
$733$	−1.07191e7	−0.736880	−0.368440	−	0.929652i	$-0.620108\pi$
$733$	−1.07191e7	−0.736880	−0.368440	+	0.929652i	$0.620108\pi$
$734$	−7.63326e6	−0.522962
$735$	1.63108e6	0.111367
$736$	1.41364e7	0.961931
$737$	0	0
$738$	−8.34011e6	−0.563677
$739$	1.37202e7	0.924167	0.462083	−	0.886836i	$-0.347102\pi$
$739$	1.37202e7	0.924167	0.462083	+	0.886836i	$0.347102\pi$
$740$	−3.64929e6	−0.244979
$741$	−2.70271e7	−1.80823
$742$	−2.14158e6	−0.142798
$743$	−3.80126e6	−0.252613	−0.126307	−	0.991991i	$-0.540312\pi$
$743$	−3.80126e6	−0.252613	−0.126307	+	0.991991i	$0.540312\pi$
$744$	−5.96794e6	−0.395268
$745$	1.26164e6	0.0832808
$746$	−1.78542e7	−1.17461
$747$	−2.11518e7	−1.38690
$748$	0	0
$749$	5.03108e6	0.327685
$750$	−594723.	−0.0386066
$751$	−2.44326e7	−1.58078	−0.790388	−	0.612607i	$-0.790121\pi$
$751$	−2.44326e7	−1.58078	−0.790388	+	0.612607i	$0.790121\pi$
$752$	21528.5	0.00138825
$753$	8.19867e6	0.526934
$754$	8.26450e6	0.529405
$755$	1.29478e7	0.826663
$756$	−2.68863e6	−0.171091
$757$	2.82220e7	1.78998	0.894990	−	0.446086i	$-0.147182\pi$
$757$	2.82220e7	1.78998	0.894990	+	0.446086i	$0.147182\pi$
$758$	1.98065e7	1.25209
$759$	0	0
$760$	4.76102e7	2.98996
$761$	−1.80560e7	−1.13021	−0.565106	−	0.825018i	$-0.691165\pi$
$761$	−1.80560e7	−1.13021	−0.565106	+	0.825018i	$0.691165\pi$
$762$	−9.35775e6	−0.583827
$763$	−6.22847e6	−0.387320
$764$	−3.06179e6	−0.189776
$765$	1.90062e7	1.17420
$766$	5.32140e6	0.327683
$767$	−5.26755e7	−3.23311
$768$	9.38287e6	0.574027
$769$	−7.90576e6	−0.482090	−0.241045	−	0.970514i	$-0.577490\pi$
$769$	−7.90576e6	−0.482090	−0.241045	+	0.970514i	$0.577490\pi$
$770$	0	0
$771$	−1.09091e7	−0.660924
$772$	5.95160e6	0.359410
$773$	−4.26660e6	−0.256822	−0.128411	−	0.991721i	$-0.540988\pi$
$773$	−4.26660e6	−0.256822	−0.128411	+	0.991721i	$0.540988\pi$
$774$	2.10795e6	0.126476
$775$	1.22539e7	0.732857
$776$	−465417.	−0.0277452
$777$	1.19919e6	0.0712583
$778$	1.76282e6	0.104414
$779$	3.69494e7	2.18154
$780$	1.10780e7	0.651965
$781$	0	0
$782$	1.55992e7	0.912188
$783$	6.88038e6	0.401059
$784$	−665896.	−0.0386916
$785$	−1.83994e7	−1.06568
$786$	−2.65110e6	−0.153063
$787$	−1.34007e7	−0.771244	−0.385622	−	0.922657i	$-0.626013\pi$
$787$	−1.34007e7	−0.771244	−0.385622	+	0.922657i	$0.626013\pi$
$788$	−1.17111e7	−0.671866
$789$	7.16914e6	0.409991
$790$	2.28468e7	1.30244
$791$	7.86585e6	0.446997
$792$	0	0
$793$	−3.64233e6	−0.205682
$794$	−1.52951e7	−0.860995
$795$	7.34614e6	0.412231
$796$	−920156.	−0.0514729
$797$	−2.66880e7	−1.48823	−0.744114	−	0.668052i	$-0.767128\pi$
$797$	−2.66880e7	−1.48823	−0.744114	+	0.668052i	$0.767128\pi$
$798$	−5.14169e6	−0.285824
$799$	−106894.	−0.00592360
$800$	−1.68542e7	−0.931072
$801$	1.00486e7	0.553379
$802$	1.06456e7	0.584430
$803$	0	0
$804$	−314496.	−0.0171583
$805$	1.10438e7	0.600661
$806$	1.54288e7	0.836557
$807$	−846168.	−0.0457375
$808$	8.23338e6	0.443659
$809$	2.58769e7	1.39009	0.695043	−	0.718968i	$-0.255385\pi$
$809$	2.58769e7	1.39009	0.695043	+	0.718968i	$0.255385\pi$
$810$	3.93794e6	0.210890
$811$	3.13209e7	1.67217	0.836087	−	0.548597i	$-0.184838\pi$
$811$	3.13209e7	1.67217	0.836087	+	0.548597i	$0.184838\pi$
$812$	−1.50773e6	−0.0802477
$813$	−1.00932e6	−0.0535553
$814$	0	0
$815$	4.86833e7	2.56735
$816$	3.22637e6	0.169625
$817$	−9.33892e6	−0.489487
$818$	−1.10683e7	−0.578361
$819$	8.75491e6	0.456081
$820$	−1.51450e7	−0.786564
$821$	−6.51072e6	−0.337110	−0.168555	−	0.985692i	$-0.553910\pi$
$821$	−6.51072e6	−0.337110	−0.168555	+	0.985692i	$0.553910\pi$
$822$	−1.04156e7	−0.537655
$823$	2.60561e7	1.34094	0.670469	−	0.741937i	$-0.266093\pi$
$823$	2.60561e7	1.34094	0.670469	+	0.741937i	$0.266093\pi$
$824$	1.16150e7	0.595938
$825$	0	0
$826$	−1.00211e7	−0.511051
$827$	2.27554e6	0.115696	0.0578482	−	0.998325i	$-0.481576\pi$
$827$	2.27554e6	0.115696	0.0578482	+	0.998325i	$0.481576\pi$
$828$	−7.53551e6	−0.381977
$829$	2.19768e7	1.11065	0.555326	−	0.831632i	$-0.312593\pi$
$829$	2.19768e7	1.11065	0.555326	+	0.831632i	$0.312593\pi$
$830$	4.00538e7	2.01813
$831$	−1.40246e7	−0.704513
$832$	−3.04599e7	−1.52553
$833$	3.30633e6	0.165095
$834$	−4.19541e6	−0.208862
$835$	−7.21927e6	−0.358325
$836$	0	0
$837$	1.28448e7	0.633746
$838$	−1.38332e7	−0.680476
$839$	2.28690e7	1.12161	0.560806	−	0.827947i	$-0.310491\pi$
$839$	2.28690e7	1.12161	0.560806	+	0.827947i	$0.310491\pi$
$840$	6.41269e6	0.313575
$841$	−1.66528e7	−0.811889
$842$	3.17895e6	0.154527
$843$	1.76662e7	0.856199
$844$	−3.17414e6	−0.153380
$845$	−5.72875e7	−2.76006
$846$	−53847.3	−0.00258665
$847$	0	0
$848$	−2.99909e6	−0.143219
$849$	476874.	0.0227057
$850$	−1.85982e7	−0.882925
$851$	8.11953e6	0.384332
$852$	−1.11744e6	−0.0527383
$853$	−6.23342e6	−0.293328	−0.146664	−	0.989186i	$-0.546854\pi$
$853$	−6.23342e6	−0.293328	−0.146664	+	0.989186i	$0.546854\pi$
$854$	−692923.	−0.0325118
$855$	−4.24173e7	−1.98439
$856$	1.97799e7	0.922658
$857$	−3.03302e7	−1.41066	−0.705332	−	0.708877i	$-0.749202\pi$
$857$	−3.03302e7	−1.41066	−0.705332	+	0.708877i	$0.749202\pi$
$858$	0	0
$859$	−2.36567e7	−1.09388	−0.546942	−	0.837171i	$-0.684208\pi$
$859$	−2.36567e7	−1.09388	−0.546942	+	0.837171i	$0.684208\pi$
$860$	3.82788e6	0.176487
$861$	4.97678e6	0.228792
$862$	1.88813e7	0.865495
$863$	−2.59863e7	−1.18773	−0.593865	−	0.804565i	$-0.702399\pi$
$863$	−2.59863e7	−1.18773	−0.593865	+	0.804565i	$0.702399\pi$
$864$	−1.76671e7	−0.805155
$865$	−5.05033e7	−2.29498
$866$	−1.79956e7	−0.815400
$867$	−4.02493e6	−0.181849
$868$	−2.81474e6	−0.126806
$869$	0	0
$870$	−5.39322e6	−0.241574
$871$	2.47398e6	0.110497
$872$	−2.44875e7	−1.09057
$873$	414654.	0.0184141
$874$	−3.48136e7	−1.54159
$875$	−853498.	−0.0376862
$876$	3.75361e6	0.165268
$877$	2.17841e7	0.956404	0.478202	−	0.878250i	$-0.341289\pi$
$877$	2.17841e7	0.956404	0.478202	+	0.878250i	$0.341289\pi$
$878$	−1.83014e7	−0.801213
$879$	−2.88068e6	−0.125754
$880$	0	0
$881$	−7.15165e6	−0.310432	−0.155216	−	0.987881i	$-0.549607\pi$
$881$	−7.15165e6	−0.310432	−0.155216	+	0.987881i	$0.549607\pi$
$882$	1.66555e6	0.0720919
$883$	−3.65362e6	−0.157696	−0.0788482	−	0.996887i	$-0.525124\pi$
$883$	−3.65362e6	−0.157696	−0.0788482	+	0.996887i	$0.525124\pi$
$884$	2.24559e7	0.966496
$885$	3.43748e7	1.47531
$886$	1.88692e7	0.807549
$887$	3.54202e7	1.51162	0.755808	−	0.654794i	$-0.227244\pi$
$887$	3.54202e7	1.51162	0.755808	+	0.654794i	$0.227244\pi$
$888$	4.71468e6	0.200641
$889$	−1.34295e7	−0.569908
$890$	−1.90284e7	−0.805242
$891$	0	0
$892$	1.09918e7	0.462549
$893$	238561.	0.0100109
$894$	−535681.	−0.0224162
$895$	−2.46252e7	−1.02759
$896$	2.11384e6	0.0879634
$897$	−2.46481e7	−1.02283
$898$	−603571.	−0.0249768
$899$	7.20311e6	0.297249
$900$	8.98427e6	0.369723
$901$	1.48912e7	0.611106
$902$	0	0
$903$	−1.25787e6	−0.0513355
$904$	3.09250e7	1.25860
$905$	3.61404e6	0.146680
$906$	−5.49751e6	−0.222508
$907$	4.42454e7	1.78587	0.892936	−	0.450184i	$-0.148642\pi$
$907$	4.42454e7	1.78587	0.892936	+	0.450184i	$0.148642\pi$
$908$	−8.59654e6	−0.346026
$909$	−7.33535e6	−0.294450
$910$	−1.65786e7	−0.663660
$911$	8.12527e6	0.324371	0.162185	−	0.986760i	$-0.448146\pi$
$911$	8.12527e6	0.324371	0.162185	+	0.986760i	$0.448146\pi$
$912$	−7.20049e6	−0.286665
$913$	0	0
$914$	−3.38549e7	−1.34047
$915$	2.37690e6	0.0938552
$916$	−1.69977e7	−0.669348
$917$	−3.80464e6	−0.149414
$918$	−1.94952e7	−0.763520
$919$	3.52689e7	1.37754	0.688768	−	0.724982i	$-0.258152\pi$
$919$	3.52689e7	1.37754	0.688768	+	0.724982i	$0.258152\pi$
$920$	4.34193e7	1.69127
$921$	−1.53971e7	−0.598124
$922$	−5.51406e6	−0.213621
$923$	8.79036e6	0.339627
$924$	0	0
$925$	−9.68057e6	−0.372003
$926$	7.54334e6	0.289092
$927$	−1.03481e7	−0.395515
$928$	−9.90730e6	−0.377646
$929$	−2.83744e7	−1.07867	−0.539333	−	0.842093i	$-0.681324\pi$
$929$	−2.83744e7	−1.07867	−0.539333	+	0.842093i	$0.681324\pi$
$930$	−1.00685e7	−0.381731
$931$	−7.37893e6	−0.279010
$932$	1.54696e7	0.583363
$933$	2.35339e7	0.885097
$934$	1.80351e7	0.676475
$935$	0	0
$936$	3.44204e7	1.28418
$937$	−1.02334e7	−0.380778	−0.190389	−	0.981709i	$-0.560975\pi$
$937$	−1.02334e7	−0.380778	−0.190389	+	0.981709i	$0.560975\pi$
$938$	470655.	0.0174661
$939$	8.04504e6	0.297758
$940$	−97782.5	−0.00360946
$941$	−3.95728e6	−0.145688	−0.0728439	−	0.997343i	$-0.523207\pi$
$941$	−3.95728e6	−0.145688	−0.0728439	+	0.997343i	$0.523207\pi$
$942$	7.81220e6	0.286844
$943$	3.36970e7	1.23399
$944$	−1.40337e7	−0.512555
$945$	−1.38021e7	−0.502766
$946$	0	0
$947$	−2.83470e7	−1.02715	−0.513573	−	0.858046i	$-0.671678\pi$
$947$	−2.83470e7	−1.02715	−0.513573	+	0.858046i	$0.671678\pi$
$948$	9.30239e6	0.336181
$949$	−2.95278e7	−1.06430
$950$	4.15068e7	1.49214
$951$	1.38545e7	0.496752
$952$	1.29990e7	0.464855
$953$	−781041.	−0.0278575	−0.0139287	−	0.999903i	$-0.504434\pi$
$953$	−781041.	−0.0278575	−0.0139287	+	0.999903i	$0.504434\pi$
$954$	7.50137e6	0.266851
$955$	−1.57177e7	−0.557675
$956$	1.08587e7	0.384266
$957$	0	0
$958$	−2.61120e7	−0.919233
$959$	−1.49476e7	−0.524837
$960$	1.98775e7	0.696118
$961$	−1.51818e7	−0.530292
$962$	−1.21888e7	−0.424642
$963$	−1.76225e7	−0.612354
$964$	2.26734e7	0.785822
$965$	3.05525e7	1.05616
$966$	−4.68910e6	−0.161676
$967$	−8.71090e6	−0.299569	−0.149785	−	0.988719i	$-0.547858\pi$
$967$	−8.71090e6	−0.299569	−0.149785	+	0.988719i	$0.547858\pi$
$968$	0	0
$969$	3.57521e7	1.22318
$970$	−785205.	−0.0267950
$971$	−3.01293e6	−0.102551	−0.0512756	−	0.998685i	$-0.516329\pi$
$971$	−3.01293e6	−0.102551	−0.0512756	+	0.998685i	$0.516329\pi$
$972$	1.49368e7	0.507098
$973$	−6.02092e6	−0.203883
$974$	−2.36763e7	−0.799680
$975$	2.93869e7	0.990015
$976$	−970379.	−0.0326074
$977$	2.26587e7	0.759450	0.379725	−	0.925099i	$-0.376019\pi$
$977$	2.26587e7	0.759450	0.379725	+	0.925099i	$0.376019\pi$
$978$	−2.06705e7	−0.691039
$979$	0	0
$980$	3.02451e6	0.100598
$981$	2.18167e7	0.723795
$982$	−7.25476e6	−0.240074
$983$	−3.94025e7	−1.30059	−0.650295	−	0.759682i	$-0.725355\pi$
$983$	−3.94025e7	−1.30059	−0.650295	+	0.759682i	$0.725355\pi$
$984$	1.95665e7	0.644205
$985$	−6.01190e7	−1.97434
$986$	−1.09325e7	−0.358118
$987$	32132.2	0.00104990
$988$	−5.01162e7	−1.63338
$989$	−8.51687e6	−0.276879
$990$	0	0
$991$	−3.90280e7	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$991$	−3.90280e7	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$992$	−1.84957e7	−0.596750
$993$	−1.78181e7	−0.573439
$994$	1.67229e6	0.0536843
$995$	−4.72362e6	−0.151258
$996$	1.63085e7	0.520913
$997$	3.68820e7	1.17511	0.587553	−	0.809186i	$-0.300092\pi$
$997$	3.68820e7	1.17511	0.587553	+	0.809186i	$0.300092\pi$
$998$	−1.63622e7	−0.520014
$999$	−1.01474e7	−0.321694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.6.a.p.1.10		28
11.2	odd	10		77.6.f.a.15.5	✓	56
11.6	odd	10		77.6.f.a.36.5	yes	56
11.10	odd	2		847.6.a.q.1.19		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.6.f.a.15.5	✓	56	11.2	odd	10
77.6.f.a.36.5	yes	56	11.6	odd	10
847.6.a.p.1.10		28	1.1	even	1	trivial
847.6.a.q.1.19		28	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-4.04168 q^{2} -8.44784 q^{3} -15.6648 q^{4} -80.4152 q^{5} +34.1435 q^{6} +49.0000 q^{7} +192.646 q^{8} -171.634 q^{9} +O(q^{10})\)	\(q-4.04168 q^{2} -8.44784 q^{3} -15.6648 q^{4} -80.4152 q^{5} +34.1435 q^{6} +49.0000 q^{7} +192.646 q^{8} -171.634 q^{9} +325.013 q^{10} +132.334 q^{12} -1041.00 q^{13} -198.043 q^{14} +679.335 q^{15} -277.341 q^{16} +1377.06 q^{17} +693.690 q^{18} -3073.28 q^{19} +1259.69 q^{20} -413.944 q^{21} -2802.75 q^{23} -1627.44 q^{24} +3341.60 q^{25} +4207.41 q^{26} +3502.76 q^{27} -767.575 q^{28} +1964.27 q^{29} -2745.66 q^{30} +3667.06 q^{31} -5043.75 q^{32} -5565.65 q^{34} -3940.34 q^{35} +2688.61 q^{36} -2896.98 q^{37} +12421.2 q^{38} +8794.24 q^{39} -15491.7 q^{40} -12022.8 q^{41} +1673.03 q^{42} +3038.75 q^{43} +13802.0 q^{45} +11327.8 q^{46} -77.6245 q^{47} +2342.94 q^{48} +2401.00 q^{49} -13505.7 q^{50} -11633.2 q^{51} +16307.1 q^{52} +10813.7 q^{53} -14157.1 q^{54} +9439.66 q^{56} +25962.6 q^{57} -7938.97 q^{58} +50600.7 q^{59} -10641.6 q^{60} +3498.86 q^{61} -14821.1 q^{62} -8410.06 q^{63} +29260.2 q^{64} +83712.5 q^{65} -2376.54 q^{67} -21571.4 q^{68} +23677.2 q^{69} +15925.6 q^{70} -8444.12 q^{71} -33064.6 q^{72} +28364.7 q^{73} +11708.7 q^{74} -28229.4 q^{75} +48142.2 q^{76} -35543.5 q^{78} +70294.9 q^{79} +22302.5 q^{80} +12116.2 q^{81} +48592.4 q^{82} +123238. q^{83} +6484.35 q^{84} -110737. q^{85} -12281.7 q^{86} -16593.9 q^{87} -58546.5 q^{89} -55783.2 q^{90} -51009.2 q^{91} +43904.5 q^{92} -30978.8 q^{93} +313.734 q^{94} +247138. q^{95} +42608.8 q^{96} -2415.92 q^{97} -9704.08 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 14 q^{2} - 28 q^{3} + 374 q^{4} - 258 q^{5} + 4 q^{6} + 1372 q^{7} - 339 q^{8} + 1660 q^{9}+O(q^{10}) \) 28 * q - 14 * q^2 - 28 * q^3 + 374 * q^4 - 258 * q^5 + 4 * q^6 + 1372 * q^7 - 339 * q^8 + 1660 * q^9	\( 28 q - 14 q^{2} - 28 q^{3} + 374 q^{4} - 258 q^{5} + 4 q^{6} + 1372 q^{7} - 339 q^{8} + 1660 q^{9} - 536 q^{10} - 996 q^{12} + 326 q^{13} - 686 q^{14} - 2561 q^{15} + 5642 q^{16} - 2159 q^{17} - 6642 q^{18} + 2253 q^{19} - 13417 q^{20} - 1372 q^{21} - 6620 q^{23} + 3009 q^{24} + 15576 q^{25} - 4960 q^{26} - 24718 q^{27} + 18326 q^{28} - 15747 q^{29} - 9484 q^{30} - 14455 q^{31} - 13922 q^{32} - 13463 q^{34} - 12642 q^{35} + 27148 q^{36} - 14447 q^{37} - 62211 q^{38} - 28537 q^{39} - 31765 q^{40} - 14604 q^{41} + 196 q^{42} - 6828 q^{43} - 2703 q^{45} - 110251 q^{46} - 51396 q^{47} - 24310 q^{48} + 67228 q^{49} - 17179 q^{50} - 23461 q^{51} + 71461 q^{52} - 134505 q^{53} + 135496 q^{54} - 16611 q^{56} - 45846 q^{57} + 18657 q^{58} - 22255 q^{59} - 176135 q^{60} - 61141 q^{61} + 186610 q^{62} + 81340 q^{63} + 83377 q^{64} - 8616 q^{65} - 34675 q^{67} - 106726 q^{68} - 178024 q^{69} - 26264 q^{70} - 81779 q^{71} - 191763 q^{72} + 91273 q^{73} - 24102 q^{74} + 95115 q^{75} + 578339 q^{76} - 230091 q^{78} - 244684 q^{79} - 296482 q^{80} - 55548 q^{81} - 129920 q^{82} + 89009 q^{83} - 48804 q^{84} + 95335 q^{85} - 32043 q^{86} + 63248 q^{87} - 450187 q^{89} + 408246 q^{90} + 15974 q^{91} - 290507 q^{92} - 411705 q^{93} + 482748 q^{94} - 164289 q^{95} + 149697 q^{96} - 266929 q^{97} - 33614 q^{98}+O(q^{100}) \) 28 * q - 14 * q^2 - 28 * q^3 + 374 * q^4 - 258 * q^5 + 4 * q^6 + 1372 * q^7 - 339 * q^8 + 1660 * q^9 - 536 * q^10 - 996 * q^12 + 326 * q^13 - 686 * q^14 - 2561 * q^15 + 5642 * q^16 - 2159 * q^17 - 6642 * q^18 + 2253 * q^19 - 13417 * q^20 - 1372 * q^21 - 6620 * q^23 + 3009 * q^24 + 15576 * q^25 - 4960 * q^26 - 24718 * q^27 + 18326 * q^28 - 15747 * q^29 - 9484 * q^30 - 14455 * q^31 - 13922 * q^32 - 13463 * q^34 - 12642 * q^35 + 27148 * q^36 - 14447 * q^37 - 62211 * q^38 - 28537 * q^39 - 31765 * q^40 - 14604 * q^41 + 196 * q^42 - 6828 * q^43 - 2703 * q^45 - 110251 * q^46 - 51396 * q^47 - 24310 * q^48 + 67228 * q^49 - 17179 * q^50 - 23461 * q^51 + 71461 * q^52 - 134505 * q^53 + 135496 * q^54 - 16611 * q^56 - 45846 * q^57 + 18657 * q^58 - 22255 * q^59 - 176135 * q^60 - 61141 * q^61 + 186610 * q^62 + 81340 * q^63 + 83377 * q^64 - 8616 * q^65 - 34675 * q^67 - 106726 * q^68 - 178024 * q^69 - 26264 * q^70 - 81779 * q^71 - 191763 * q^72 + 91273 * q^73 - 24102 * q^74 + 95115 * q^75 + 578339 * q^76 - 230091 * q^78 - 244684 * q^79 - 296482 * q^80 - 55548 * q^81 - 129920 * q^82 + 89009 * q^83 - 48804 * q^84 + 95335 * q^85 - 32043 * q^86 + 63248 * q^87 - 450187 * q^89 + 408246 * q^90 + 15974 * q^91 - 290507 * q^92 - 411705 * q^93 + 482748 * q^94 - 164289 * q^95 + 149697 * q^96 - 266929 * q^97 - 33614 * q^98

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