LMFDB - Embedded newform 625.8.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,8,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	625.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:	$195.240640928$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.8316 q^{2} +60.3488 q^{3} +155.304 q^{4} -1015.77 q^{6} +1500.12 q^{7} -459.578 q^{8} +1454.98 q^{9} +O(q^{10})$	\(q-16.8316 q^{2} +60.3488 q^{3} +155.304 q^{4} -1015.77 q^{6} +1500.12 q^{7} -459.578 q^{8} +1454.98 q^{9} +791.439 q^{11} +9372.44 q^{12} -10075.6 q^{13} -25249.5 q^{14} -12143.5 q^{16} +17159.0 q^{17} -24489.7 q^{18} -8753.75 q^{19} +90530.5 q^{21} -13321.2 q^{22} +68732.5 q^{23} -27735.0 q^{24} +169589. q^{26} -44176.4 q^{27} +232975. q^{28} -189289. q^{29} -202629. q^{31} +263221. q^{32} +47762.4 q^{33} -288814. q^{34} +225965. q^{36} +111920. q^{37} +147340. q^{38} -608052. q^{39} -493756. q^{41} -1.52378e6 q^{42} +641476. q^{43} +122914. q^{44} -1.15688e6 q^{46} -587091. q^{47} -732846. q^{48} +1.42682e6 q^{49} +1.03553e6 q^{51} -1.56479e6 q^{52} -414382. q^{53} +743562. q^{54} -689423. q^{56} -528279. q^{57} +3.18605e6 q^{58} -2.31846e6 q^{59} -1.90838e6 q^{61} +3.41058e6 q^{62} +2.18265e6 q^{63} -2.87608e6 q^{64} -803921. q^{66} -4.85246e6 q^{67} +2.66487e6 q^{68} +4.14793e6 q^{69} +5.10643e6 q^{71} -668678. q^{72} -3.79268e6 q^{73} -1.88380e6 q^{74} -1.35950e6 q^{76} +1.18725e6 q^{77} +1.02345e7 q^{78} +1.32033e6 q^{79} -5.84804e6 q^{81} +8.31072e6 q^{82} +550110. q^{83} +1.40598e7 q^{84} -1.07971e7 q^{86} -1.14234e7 q^{87} -363728. q^{88} -2.28476e6 q^{89} -1.51147e7 q^{91} +1.06745e7 q^{92} -1.22284e7 q^{93} +9.88172e6 q^{94} +1.58851e7 q^{96} +575335. q^{97} -2.40157e7 q^{98} +1.15153e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9}+O(q^{10}) $ 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9} - 781 q^{11} - 12608 q^{12} - 14686 q^{13} + 20762 q^{14} + 16117 q^{16} - 45648 q^{17} - 47171 q^{18} + 6185 q^{19} + 14149 q^{21} + 71124 q^{22} - 126921 q^{23} + 271570 q^{24} + 304129 q^{26} - 546520 q^{27} - 2019 q^{28} + 59330 q^{29} + 394804 q^{31} + 74397 q^{32} - 49067 q^{33} - 286938 q^{34} - 287278 q^{36} + 792122 q^{37} - 1338860 q^{38} + 635223 q^{39} - 160466 q^{41} - 3420191 q^{42} - 1527256 q^{43} - 1154853 q^{44} + 2653604 q^{46} - 1300863 q^{47} - 1885241 q^{48} + 1652981 q^{49} + 3408539 q^{51} - 1423303 q^{52} - 755656 q^{53} + 3117755 q^{54} - 2132625 q^{56} - 3026890 q^{57} + 5941470 q^{58} - 1548370 q^{59} - 6029951 q^{61} - 79936 q^{62} + 6962459 q^{63} - 5858224 q^{64} - 5380407 q^{66} - 7608838 q^{67} - 10737124 q^{68} + 13519553 q^{69} + 9483549 q^{71} - 6806340 q^{72} - 13548801 q^{73} - 15016023 q^{74} + 19635315 q^{76} + 2145019 q^{77} - 17222402 q^{78} + 10769160 q^{79} + 2757382 q^{81} - 12087571 q^{82} + 9632744 q^{83} + 19168542 q^{84} - 8511651 q^{86} - 298330 q^{87} + 21641425 q^{88} - 10850545 q^{89} - 6648131 q^{91} - 52978503 q^{92} - 51294822 q^{93} + 5863777 q^{94} + 2611654 q^{96} - 11579993 q^{97} - 7468074 q^{98} - 13552997 q^{99}+O(q^{100}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9 - 781 * q^11 - 12608 * q^12 - 14686 * q^13 + 20762 * q^14 + 16117 * q^16 - 45648 * q^17 - 47171 * q^18 + 6185 * q^19 + 14149 * q^21 + 71124 * q^22 - 126921 * q^23 + 271570 * q^24 + 304129 * q^26 - 546520 * q^27 - 2019 * q^28 + 59330 * q^29 + 394804 * q^31 + 74397 * q^32 - 49067 * q^33 - 286938 * q^34 - 287278 * q^36 + 792122 * q^37 - 1338860 * q^38 + 635223 * q^39 - 160466 * q^41 - 3420191 * q^42 - 1527256 * q^43 - 1154853 * q^44 + 2653604 * q^46 - 1300863 * q^47 - 1885241 * q^48 + 1652981 * q^49 + 3408539 * q^51 - 1423303 * q^52 - 755656 * q^53 + 3117755 * q^54 - 2132625 * q^56 - 3026890 * q^57 + 5941470 * q^58 - 1548370 * q^59 - 6029951 * q^61 - 79936 * q^62 + 6962459 * q^63 - 5858224 * q^64 - 5380407 * q^66 - 7608838 * q^67 - 10737124 * q^68 + 13519553 * q^69 + 9483549 * q^71 - 6806340 * q^72 - 13548801 * q^73 - 15016023 * q^74 + 19635315 * q^76 + 2145019 * q^77 - 17222402 * q^78 + 10769160 * q^79 + 2757382 * q^81 - 12087571 * q^82 + 9632744 * q^83 + 19168542 * q^84 - 8511651 * q^86 - 298330 * q^87 + 21641425 * q^88 - 10850545 * q^89 - 6648131 * q^91 - 52978503 * q^92 - 51294822 * q^93 + 5863777 * q^94 + 2611654 * q^96 - 11579993 * q^97 - 7468074 * q^98 - 13552997 * 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 $

$2$	−16.8316	−1.48772	−0.743861	−	0.668335i	$-0.767007\pi$
$2$	−16.8316	−1.48772	−0.743861	+	0.668335i	$0.767007\pi$
$3$	60.3488	1.29046	0.645230	−	0.763988i	$-0.276762\pi$
$3$	60.3488	1.29046	0.645230	+	0.763988i	$0.276762\pi$
$4$	155.304	1.21332
$5$	0	0
$5$	0	0
$6$	−1015.77	−1.91985
$7$	1500.12	1.65304	0.826520	−	0.562908i	$-0.190317\pi$
$7$	1500.12	1.65304	0.826520	+	0.562908i	$0.190317\pi$
$8$	−459.578	−0.317354
$9$	1454.98	0.665287
$10$	0	0
$11$	791.439	0.179285	0.0896423	−	0.995974i	$-0.471428\pi$
$11$	791.439	0.179285	0.0896423	+	0.995974i	$0.471428\pi$
$12$	9372.44	1.56574
$13$	−10075.6	−1.27195	−0.635975	−	0.771709i	$-0.719402\pi$
$13$	−10075.6	−1.27195	−0.635975	+	0.771709i	$0.719402\pi$
$14$	−25249.5	−2.45926
$15$	0	0
$16$	−12143.5	−0.741181
$17$	17159.0	0.847073	0.423537	−	0.905879i	$-0.360788\pi$
$17$	17159.0	0.847073	0.423537	+	0.905879i	$0.360788\pi$
$18$	−24489.7	−0.989761
$19$	−8753.75	−0.292790	−0.146395	−	0.989226i	$-0.546767\pi$
$19$	−8753.75	−0.292790	−0.146395	+	0.989226i	$0.546767\pi$
$20$	0	0
$21$	90530.5	2.13318
$22$	−13321.2	−0.266726
$23$	68732.5	1.17792	0.588959	−	0.808163i	$-0.299538\pi$
$23$	68732.5	1.17792	0.588959	+	0.808163i	$0.299538\pi$
$24$	−27735.0	−0.409533
$25$	0	0
$26$	169589.	1.89231
$27$	−44176.4	−0.431934
$28$	232975.	2.00566
$29$	−189289.	−1.44123	−0.720614	−	0.693336i	$-0.756140\pi$
$29$	−189289.	−1.44123	−0.720614	+	0.693336i	$0.756140\pi$
$30$	0	0
$31$	−202629.	−1.22162	−0.610809	−	0.791778i	$-0.709156\pi$
$31$	−202629.	−1.22162	−0.610809	+	0.791778i	$0.709156\pi$
$32$	263221.	1.42002
$33$	47762.4	0.231360
$34$	−288814.	−1.26021
$35$	0	0
$36$	225965.	0.807203
$37$	111920.	0.363248	0.181624	−	0.983368i	$-0.441865\pi$
$37$	111920.	0.363248	0.181624	+	0.983368i	$0.441865\pi$
$38$	147340.	0.435591
$39$	−608052.	−1.64140
$40$	0	0
$41$	−493756.	−1.11884	−0.559421	−	0.828884i	$-0.688976\pi$
$41$	−493756.	−1.11884	−0.559421	+	0.828884i	$0.688976\pi$
$42$	−1.52378e6	−3.17358
$43$	641476.	1.23038	0.615192	−	0.788377i	$-0.289079\pi$
$43$	641476.	1.23038	0.615192	+	0.788377i	$0.289079\pi$
$44$	122914.	0.217529
$45$	0	0
$46$	−1.15688e6	−1.75241
$47$	−587091.	−0.824828	−0.412414	−	0.910997i	$-0.635314\pi$
$47$	−587091.	−0.824828	−0.412414	+	0.910997i	$0.635314\pi$
$48$	−732846.	−0.956464
$49$	1.42682e6	1.73254
$50$	0	0
$51$	1.03553e6	1.09311
$52$	−1.56479e6	−1.54328
$53$	−414382.	−0.382328	−0.191164	−	0.981558i	$-0.561226\pi$
$53$	−414382.	−0.382328	−0.191164	+	0.981558i	$0.561226\pi$
$54$	743562.	0.642598
$55$	0	0
$56$	−689423.	−0.524599
$57$	−528279.	−0.377834
$58$	3.18605e6	2.14415
$59$	−2.31846e6	−1.46966	−0.734830	−	0.678251i	$-0.762738\pi$
$59$	−2.31846e6	−1.46966	−0.734830	+	0.678251i	$0.762738\pi$
$60$	0	0
$61$	−1.90838e6	−1.07649	−0.538247	−	0.842787i	$-0.680913\pi$
$61$	−1.90838e6	−1.07649	−0.538247	+	0.842787i	$0.680913\pi$
$62$	3.41058e6	1.81743
$63$	2.18265e6	1.09974
$64$	−2.87608e6	−1.37142
$65$	0	0
$66$	−803921.	−0.344199
$67$	−4.85246e6	−1.97106	−0.985531	−	0.169492i	$-0.945787\pi$
$67$	−4.85246e6	−1.97106	−0.985531	+	0.169492i	$0.945787\pi$
$68$	2.66487e6	1.02777
$69$	4.14793e6	1.52006
$70$	0	0
$71$	5.10643e6	1.69322	0.846609	−	0.532215i	$-0.178640\pi$
$71$	5.10643e6	1.69322	0.846609	+	0.532215i	$0.178640\pi$
$72$	−668678.	−0.211132
$73$	−3.79268e6	−1.14108	−0.570541	−	0.821269i	$-0.693266\pi$
$73$	−3.79268e6	−1.14108	−0.570541	+	0.821269i	$0.693266\pi$
$74$	−1.88380e6	−0.540412
$75$	0	0
$76$	−1.35950e6	−0.355247
$77$	1.18725e6	0.296365
$78$	1.02345e7	2.44195
$79$	1.32033e6	0.301293	0.150647	−	0.988588i	$-0.451864\pi$
$79$	1.32033e6	0.301293	0.150647	+	0.988588i	$0.451864\pi$
$80$	0	0
$81$	−5.84804e6	−1.22268
$82$	8.31072e6	1.66453
$83$	550110.	0.105603	0.0528015	−	0.998605i	$-0.483185\pi$
$83$	550110.	0.105603	0.0528015	+	0.998605i	$0.483185\pi$
$84$	1.40598e7	2.58822
$85$	0	0
$86$	−1.07971e7	−1.83047
$87$	−1.14234e7	−1.85985
$88$	−363728.	−0.0568968
$89$	−2.28476e6	−0.343539	−0.171770	−	0.985137i	$-0.554948\pi$
$89$	−2.28476e6	−0.343539	−0.171770	+	0.985137i	$0.554948\pi$
$90$	0	0
$91$	−1.51147e7	−2.10258
$92$	1.06745e7	1.42919
$93$	−1.22284e7	−1.57645
$94$	9.88172e6	1.22711
$95$	0	0
$96$	1.58851e7	1.83249
$97$	575335.	0.0640058	0.0320029	−	0.999488i	$-0.489811\pi$
$97$	575335.	0.0640058	0.0320029	+	0.999488i	$0.489811\pi$
$98$	−2.40157e7	−2.57753
$99$	1.15153e6	0.119276
$100$	0	0
$101$	−1.31891e7	−1.27377	−0.636884	−	0.770959i	$-0.719777\pi$
$101$	−1.31891e7	−1.27377	−0.636884	+	0.770959i	$0.719777\pi$
$102$	−1.74296e7	−1.62625
$103$	−1.56945e7	−1.41520	−0.707601	−	0.706612i	$-0.750223\pi$
$103$	−1.56945e7	−1.41520	−0.707601	+	0.706612i	$0.750223\pi$
$104$	4.63054e6	0.403659
$105$	0	0
$106$	6.97474e6	0.568797
$107$	−2.17080e7	−1.71308	−0.856540	−	0.516081i	$-0.827390\pi$
$107$	−2.17080e7	−1.71308	−0.856540	+	0.516081i	$0.827390\pi$
$108$	−6.86079e6	−0.524072
$109$	1.53492e7	1.13526	0.567628	−	0.823285i	$-0.307861\pi$
$109$	1.53492e7	1.13526	0.567628	+	0.823285i	$0.307861\pi$
$110$	0	0
$111$	6.75426e6	0.468757
$112$	−1.82167e7	−1.22520
$113$	−7.46311e6	−0.486570	−0.243285	−	0.969955i	$-0.578225\pi$
$113$	−7.46311e6	−0.486570	−0.243285	+	0.969955i	$0.578225\pi$
$114$	8.89180e6	0.562112
$115$	0	0
$116$	−2.93974e7	−1.74867
$117$	−1.46599e7	−0.846212
$118$	3.90234e7	2.18645
$119$	2.57406e7	1.40025
$120$	0	0
$121$	−1.88608e7	−0.967857
$122$	3.21213e7	1.60152
$123$	−2.97976e7	−1.44382
$124$	−3.14692e7	−1.48221
$125$	0	0
$126$	−3.67376e7	−1.63611
$127$	−2.06230e7	−0.893386	−0.446693	−	0.894687i	$-0.647398\pi$
$127$	−2.06230e7	−0.893386	−0.446693	+	0.894687i	$0.647398\pi$
$128$	1.47168e7	0.620268
$129$	3.87123e7	1.58776
$130$	0	0
$131$	1.74603e7	0.678581	0.339290	−	0.940682i	$-0.389813\pi$
$131$	1.74603e7	0.678581	0.339290	+	0.940682i	$0.389813\pi$
$132$	7.41772e6	0.280712
$133$	−1.31317e7	−0.483994
$134$	8.16750e7	2.93239
$135$	0	0
$136$	−7.88591e6	−0.268822
$137$	3.40793e7	1.13232	0.566159	−	0.824296i	$-0.308429\pi$
$137$	3.40793e7	1.13232	0.566159	+	0.824296i	$0.308429\pi$
$138$	−6.98165e7	−2.26142
$139$	2.71101e7	0.856208	0.428104	−	0.903730i	$-0.359182\pi$
$139$	2.71101e7	0.856208	0.428104	+	0.903730i	$0.359182\pi$
$140$	0	0
$141$	−3.54303e7	−1.06441
$142$	−8.59496e7	−2.51904
$143$	−7.97425e6	−0.228041
$144$	−1.76686e7	−0.493098
$145$	0	0
$146$	6.38371e7	1.69761
$147$	8.61069e7	2.23577
$148$	1.73817e7	0.440734
$149$	1.54608e7	0.382895	0.191447	−	0.981503i	$-0.438682\pi$
$149$	1.54608e7	0.382895	0.191447	+	0.981503i	$0.438682\pi$
$150$	0	0
$151$	−1.14876e7	−0.271525	−0.135763	−	0.990741i	$-0.543348\pi$
$151$	−1.14876e7	−0.271525	−0.135763	+	0.990741i	$0.543348\pi$
$152$	4.02304e6	0.0929183
$153$	2.49660e7	0.563547
$154$	−1.99835e7	−0.440908
$155$	0	0
$156$	−9.44332e7	−1.99154
$157$	7.57408e7	1.56200	0.781000	−	0.624531i	$-0.214710\pi$
$157$	7.57408e7	1.56200	0.781000	+	0.624531i	$0.214710\pi$
$158$	−2.22234e7	−0.448240
$159$	−2.50075e7	−0.493378
$160$	0	0
$161$	1.03107e8	1.94714
$162$	9.84322e7	1.81901
$163$	5.76189e7	1.04210	0.521048	−	0.853527i	$-0.325541\pi$
$163$	5.76189e7	1.04210	0.521048	+	0.853527i	$0.325541\pi$
$164$	−7.66824e7	−1.35751
$165$	0	0
$166$	−9.25926e6	−0.157108
$167$	−1.47260e7	−0.244667	−0.122334	−	0.992489i	$-0.539038\pi$
$167$	−1.47260e7	−0.244667	−0.122334	+	0.992489i	$0.539038\pi$
$168$	−4.16059e7	−0.676974
$169$	3.87697e7	0.617859
$170$	0	0
$171$	−1.27366e7	−0.194790
$172$	9.96240e7	1.49284
$173$	9.49842e7	1.39473	0.697365	−	0.716716i	$-0.254356\pi$
$173$	9.49842e7	1.39473	0.697365	+	0.716716i	$0.254356\pi$
$174$	1.92274e8	2.76694
$175$	0	0
$176$	−9.61084e6	−0.132882
$177$	−1.39916e8	−1.89654
$178$	3.84563e7	0.511091
$179$	1.01538e8	1.32326	0.661628	−	0.749832i	$-0.269866\pi$
$179$	1.01538e8	1.32326	0.661628	+	0.749832i	$0.269866\pi$
$180$	0	0
$181$	−1.16142e7	−0.145584	−0.0727920	−	0.997347i	$-0.523191\pi$
$181$	−1.16142e7	−0.145584	−0.0727920	+	0.997347i	$0.523191\pi$
$182$	2.54405e8	3.12806
$183$	−1.15169e8	−1.38917
$184$	−3.15880e7	−0.373818
$185$	0	0
$186$	2.05825e8	2.34532
$187$	1.35803e7	0.151867
$188$	−9.11779e7	−1.00078
$189$	−6.62700e7	−0.714004
$190$	0	0
$191$	−1.16949e8	−1.21445	−0.607227	−	0.794528i	$-0.707718\pi$
$191$	−1.16949e8	−1.21445	−0.607227	+	0.794528i	$0.707718\pi$
$192$	−1.73568e8	−1.76976
$193$	1.43969e8	1.44151	0.720754	−	0.693191i	$-0.243796\pi$
$193$	1.43969e8	1.44151	0.720754	+	0.693191i	$0.243796\pi$
$194$	−9.68383e6	−0.0952228
$195$	0	0
$196$	2.21591e8	2.10212
$197$	−1.32547e8	−1.23520	−0.617599	−	0.786493i	$-0.711895\pi$
$197$	−1.32547e8	−1.23520	−0.617599	+	0.786493i	$0.711895\pi$
$198$	−1.93821e7	−0.177449
$199$	−1.52122e8	−1.36838	−0.684190	−	0.729304i	$-0.739844\pi$
$199$	−1.52122e8	−1.36838	−0.684190	+	0.729304i	$0.739844\pi$
$200$	0	0
$201$	−2.92841e8	−2.54358
$202$	2.21994e8	1.89501
$203$	−2.83957e8	−2.38241
$204$	1.60822e8	1.32629
$205$	0	0
$206$	2.64165e8	2.10543
$207$	1.00005e8	0.783653
$208$	1.22353e8	0.942745
$209$	−6.92806e6	−0.0524928
$210$	0	0
$211$	1.08526e8	0.795323	0.397662	−	0.917532i	$-0.369822\pi$
$211$	1.08526e8	0.795323	0.397662	+	0.917532i	$0.369822\pi$
$212$	−6.43554e7	−0.463884
$213$	3.08167e8	2.18503
$214$	3.65382e8	2.54859
$215$	0	0
$216$	2.03025e7	0.137076
$217$	−3.03968e8	−2.01938
$218$	−2.58353e8	−1.68894
$219$	−2.28884e8	−1.47252
$220$	0	0
$221$	−1.72888e8	−1.07744
$222$	−1.13685e8	−0.697380
$223$	1.81367e8	1.09519	0.547596	−	0.836743i	$-0.315543\pi$
$223$	1.81367e8	1.09519	0.547596	+	0.836743i	$0.315543\pi$
$224$	3.94864e8	2.34736
$225$	0	0
$226$	1.25616e8	0.723881
$227$	−2.34031e8	−1.32795	−0.663976	−	0.747754i	$-0.731132\pi$
$227$	−2.34031e8	−1.32795	−0.663976	+	0.747754i	$0.731132\pi$
$228$	−8.20440e7	−0.458432
$229$	2.12882e8	1.17143	0.585713	−	0.810519i	$-0.300815\pi$
$229$	2.12882e8	1.17143	0.585713	+	0.810519i	$0.300815\pi$
$230$	0	0
$231$	7.16494e7	0.382447
$232$	8.69932e7	0.457380
$233$	1.05451e8	0.546139	0.273070	−	0.961994i	$-0.411961\pi$
$233$	1.05451e8	0.546139	0.273070	+	0.961994i	$0.411961\pi$
$234$	2.46750e8	1.25893
$235$	0	0
$236$	−3.60067e8	−1.78316
$237$	7.96807e7	0.388807
$238$	−4.33257e8	−2.08317
$239$	1.58071e8	0.748960	0.374480	−	0.927235i	$-0.377821\pi$
$239$	1.58071e8	0.748960	0.374480	+	0.927235i	$0.377821\pi$
$240$	0	0
$241$	4.69849e7	0.216221	0.108111	−	0.994139i	$-0.465520\pi$
$241$	4.69849e7	0.216221	0.108111	+	0.994139i	$0.465520\pi$
$242$	3.17458e8	1.43990
$243$	−2.56309e8	−1.14589
$244$	−2.96381e8	−1.30613
$245$	0	0
$246$	5.01542e8	2.14800
$247$	8.81996e7	0.372415
$248$	9.31239e7	0.387686
$249$	3.31985e7	0.136276
$250$	0	0
$251$	7.37911e7	0.294541	0.147270	−	0.989096i	$-0.452951\pi$
$251$	7.37911e7	0.294541	0.147270	+	0.989096i	$0.452951\pi$
$252$	3.38975e8	1.33434
$253$	5.43976e7	0.211183
$254$	3.47119e8	1.32911
$255$	0	0
$256$	1.20429e8	0.448635
$257$	−3.95895e8	−1.45483	−0.727417	−	0.686195i	$-0.759279\pi$
$257$	−3.95895e8	−1.45483	−0.727417	+	0.686195i	$0.759279\pi$
$258$	−6.51592e8	−2.36215
$259$	1.67894e8	0.600463
$260$	0	0
$261$	−2.75412e8	−0.958830
$262$	−2.93885e8	−1.00954
$263$	−2.86426e8	−0.970883	−0.485441	−	0.874269i	$-0.661341\pi$
$263$	−2.86426e8	−0.970883	−0.485441	+	0.874269i	$0.661341\pi$
$264$	−2.19506e7	−0.0734230
$265$	0	0
$266$	2.21028e8	0.720048
$267$	−1.37883e8	−0.443323
$268$	−7.53609e8	−2.39152
$269$	−1.83989e8	−0.576313	−0.288157	−	0.957583i	$-0.593042\pi$
$269$	−1.83989e8	−0.576313	−0.288157	+	0.957583i	$0.593042\pi$
$270$	0	0
$271$	8.70531e7	0.265700	0.132850	−	0.991136i	$-0.457587\pi$
$271$	8.70531e7	0.265700	0.132850	+	0.991136i	$0.457587\pi$
$272$	−2.08370e8	−0.627834
$273$	−9.12152e8	−2.71330
$274$	−5.73610e8	−1.68457
$275$	0	0
$276$	6.44192e8	1.84431
$277$	2.50900e8	0.709287	0.354643	−	0.935002i	$-0.384602\pi$
$277$	2.50900e8	0.709287	0.354643	+	0.935002i	$0.384602\pi$
$278$	−4.56307e8	−1.27380
$279$	−2.94822e8	−0.812727
$280$	0	0
$281$	−1.22909e8	−0.330456	−0.165228	−	0.986255i	$-0.552836\pi$
$281$	−1.22909e8	−0.330456	−0.165228	+	0.986255i	$0.552836\pi$
$282$	5.96350e8	1.58354
$283$	2.94692e8	0.772888	0.386444	−	0.922313i	$-0.373703\pi$
$283$	2.94692e8	0.772888	0.386444	+	0.922313i	$0.373703\pi$
$284$	7.93051e8	2.05441
$285$	0	0
$286$	1.34220e8	0.339262
$287$	−7.40693e8	−1.84949
$288$	3.82982e8	0.944724
$289$	−1.15907e8	−0.282467
$290$	0	0
$291$	3.47208e7	0.0825969
$292$	−5.89021e8	−1.38449
$293$	−2.72584e8	−0.633086	−0.316543	−	0.948578i	$-0.602522\pi$
$293$	−2.72584e8	−0.633086	−0.316543	+	0.948578i	$0.602522\pi$
$294$	−1.44932e9	−3.32620
$295$	0	0
$296$	−5.14362e7	−0.115278
$297$	−3.49630e7	−0.0774392
$298$	−2.60231e8	−0.569641
$299$	−6.92523e8	−1.49825
$300$	0	0
$301$	9.62291e8	2.03387
$302$	1.93355e8	0.403954
$303$	−7.95947e8	−1.64375
$304$	1.06301e8	0.217011
$305$	0	0
$306$	−4.20220e8	−0.838400
$307$	−4.22505e8	−0.833388	−0.416694	−	0.909047i	$-0.636811\pi$
$307$	−4.22505e8	−0.833388	−0.416694	+	0.909047i	$0.636811\pi$
$308$	1.84386e8	0.359584
$309$	−9.47147e8	−1.82626
$310$	0	0
$311$	5.30851e8	1.00072	0.500359	−	0.865818i	$-0.333201\pi$
$311$	5.30851e8	1.00072	0.500359	+	0.865818i	$0.333201\pi$
$312$	2.79448e8	0.520906
$313$	2.02268e8	0.372839	0.186420	−	0.982470i	$-0.440312\pi$
$313$	2.02268e8	0.372839	0.186420	+	0.982470i	$0.440312\pi$
$314$	−1.27484e9	−2.32382
$315$	0	0
$316$	2.05054e8	0.365564
$317$	−5.49415e8	−0.968709	−0.484355	−	0.874872i	$-0.660946\pi$
$317$	−5.49415e8	−0.968709	−0.484355	+	0.874872i	$0.660946\pi$
$318$	4.20917e8	0.734010
$319$	−1.49811e8	−0.258390
$320$	0	0
$321$	−1.31006e9	−2.21066
$322$	−1.73546e9	−2.89681
$323$	−1.50206e8	−0.248015
$324$	−9.08227e8	−1.48350
$325$	0	0
$326$	−9.69820e8	−1.55035
$327$	9.26308e8	1.46500
$328$	2.26919e8	0.355069
$329$	−8.80708e8	−1.36347
$330$	0	0
$331$	−5.34213e8	−0.809686	−0.404843	−	0.914386i	$-0.632674\pi$
$331$	−5.34213e8	−0.809686	−0.404843	+	0.914386i	$0.632674\pi$
$332$	8.54345e7	0.128130
$333$	1.62842e8	0.241664
$334$	2.47862e8	0.363997
$335$	0	0
$336$	−1.09936e9	−1.58107
$337$	7.73832e8	1.10139	0.550696	−	0.834706i	$-0.314362\pi$
$337$	7.73832e8	1.10139	0.550696	+	0.834706i	$0.314362\pi$
$338$	−6.52558e8	−0.919202
$339$	−4.50390e8	−0.627899
$340$	0	0
$341$	−1.60369e8	−0.219017
$342$	2.14377e8	0.289793
$343$	9.04988e8	1.21091
$344$	−2.94808e8	−0.390468
$345$	0	0
$346$	−1.59874e9	−2.07497
$347$	−3.20019e8	−0.411172	−0.205586	−	0.978639i	$-0.565910\pi$
$347$	−3.20019e8	−0.411172	−0.205586	+	0.978639i	$0.565910\pi$
$348$	−1.77410e9	−2.25658
$349$	1.02380e9	1.28922	0.644610	−	0.764511i	$-0.277020\pi$
$349$	1.02380e9	1.28922	0.644610	+	0.764511i	$0.277020\pi$
$350$	0	0
$351$	4.45105e8	0.549399
$352$	2.08324e8	0.254589
$353$	4.07525e8	0.493109	0.246554	−	0.969129i	$-0.420702\pi$
$353$	4.07525e8	0.493109	0.246554	+	0.969129i	$0.420702\pi$
$354$	2.35502e9	2.82152
$355$	0	0
$356$	−3.54834e8	−0.416821
$357$	1.55341e9	1.80696
$358$	−1.70906e9	−1.96864
$359$	−4.50330e7	−0.0513689	−0.0256844	−	0.999670i	$-0.508177\pi$
$359$	−4.50330e7	−0.0513689	−0.0256844	+	0.999670i	$0.508177\pi$
$360$	0	0
$361$	−8.17244e8	−0.914274
$362$	1.95486e8	0.216588
$363$	−1.13823e9	−1.24898
$364$	−2.34737e9	−2.55110
$365$	0	0
$366$	1.93848e9	2.06670
$367$	−6.52337e8	−0.688876	−0.344438	−	0.938809i	$-0.611930\pi$
$367$	−6.52337e8	−0.688876	−0.344438	+	0.938809i	$0.611930\pi$
$368$	−8.34654e8	−0.873050
$369$	−7.18406e8	−0.744351
$370$	0	0
$371$	−6.21623e8	−0.632002
$372$	−1.89913e9	−1.91273
$373$	5.53493e8	0.552244	0.276122	−	0.961123i	$-0.410951\pi$
$373$	5.53493e8	0.552244	0.276122	+	0.961123i	$0.410951\pi$
$374$	−2.28579e8	−0.225936
$375$	0	0
$376$	2.69814e8	0.261763
$377$	1.90721e9	1.83317
$378$	1.11543e9	1.06224
$379$	1.10394e9	1.04161	0.520807	−	0.853674i	$-0.325631\pi$
$379$	1.10394e9	1.04161	0.520807	+	0.853674i	$0.325631\pi$
$380$	0	0
$381$	−1.24457e9	−1.15288
$382$	1.96845e9	1.80677
$383$	4.68767e7	0.0426345	0.0213173	−	0.999773i	$-0.493214\pi$
$383$	4.68767e7	0.0426345	0.0213173	+	0.999773i	$0.493214\pi$
$384$	8.88144e8	0.800431
$385$	0	0
$386$	−2.42323e9	−2.14456
$387$	9.33335e8	0.818558
$388$	8.93520e7	0.0776593
$389$	−2.81292e8	−0.242289	−0.121145	−	0.992635i	$-0.538657\pi$
$389$	−2.81292e8	−0.242289	−0.121145	+	0.992635i	$0.538657\pi$
$390$	0	0
$391$	1.17938e9	0.997783
$392$	−6.55735e8	−0.549829
$393$	1.05371e9	0.875681
$394$	2.23098e9	1.83763
$395$	0	0
$396$	1.78838e8	0.144719
$397$	−8.71454e8	−0.699000	−0.349500	−	0.936936i	$-0.613649\pi$
$397$	−8.71454e8	−0.699000	−0.349500	+	0.936936i	$0.613649\pi$
$398$	2.56047e9	2.03577
$399$	−7.92482e8	−0.624575
$400$	0	0
$401$	6.29376e8	0.487421	0.243711	−	0.969848i	$-0.421635\pi$
$401$	6.29376e8	0.487421	0.243711	+	0.969848i	$0.421635\pi$
$402$	4.92899e9	3.78414
$403$	2.04161e9	1.55384
$404$	−2.04833e9	−1.54548
$405$	0	0
$406$	4.77946e9	3.54436
$407$	8.85781e7	0.0651248
$408$	−4.75905e8	−0.346905
$409$	−7.83733e8	−0.566417	−0.283209	−	0.959058i	$-0.591399\pi$
$409$	−7.83733e8	−0.566417	−0.283209	+	0.959058i	$0.591399\pi$
$410$	0	0
$411$	2.05664e9	1.46121
$412$	−2.43743e9	−1.71709
$413$	−3.47797e9	−2.42941
$414$	−1.68324e9	−1.16586
$415$	0	0
$416$	−2.65212e9	−1.80620
$417$	1.63606e9	1.10490
$418$	1.16611e8	0.0780947
$419$	−1.46069e9	−0.970083	−0.485042	−	0.874491i	$-0.661196\pi$
$419$	−1.46069e9	−0.970083	−0.485042	+	0.874491i	$0.661196\pi$
$420$	0	0
$421$	−7.25668e8	−0.473970	−0.236985	−	0.971513i	$-0.576159\pi$
$421$	−7.25668e8	−0.473970	−0.236985	+	0.971513i	$0.576159\pi$
$422$	−1.82667e9	−1.18322
$423$	−8.54207e8	−0.548747
$424$	1.90441e8	0.121333
$425$	0	0
$426$	−5.18696e9	−3.25072
$427$	−2.86281e9	−1.77949
$428$	−3.37136e9	−2.07851
$429$	−4.81236e8	−0.294278
$430$	0	0
$431$	−2.00485e9	−1.20618	−0.603091	−	0.797673i	$-0.706064\pi$
$431$	−2.00485e9	−1.20618	−0.603091	+	0.797673i	$0.706064\pi$
$432$	5.36457e8	0.320141
$433$	−1.96068e9	−1.16064	−0.580321	−	0.814388i	$-0.697073\pi$
$433$	−1.96068e9	−1.16064	−0.580321	+	0.814388i	$0.697073\pi$
$434$	5.11628e9	3.00428
$435$	0	0
$436$	2.38380e9	1.37742
$437$	−6.01668e8	−0.344883
$438$	3.85250e9	2.19070
$439$	−3.36187e8	−0.189651	−0.0948256	−	0.995494i	$-0.530229\pi$
$439$	−3.36187e8	−0.189651	−0.0948256	+	0.995494i	$0.530229\pi$
$440$	0	0
$441$	2.07600e9	1.15263
$442$	2.90999e9	1.60292
$443$	1.89614e9	1.03623	0.518116	−	0.855310i	$-0.326634\pi$
$443$	1.89614e9	1.03623	0.518116	+	0.855310i	$0.326634\pi$
$444$	1.04897e9	0.568750
$445$	0	0
$446$	−3.05270e9	−1.62934
$447$	9.33040e8	0.494111
$448$	−4.31447e9	−2.26701
$449$	−1.66970e9	−0.870514	−0.435257	−	0.900306i	$-0.643343\pi$
$449$	−1.66970e9	−0.870514	−0.435257	+	0.900306i	$0.643343\pi$
$450$	0	0
$451$	−3.90778e8	−0.200591
$452$	−1.15905e9	−0.590363
$453$	−6.93264e8	−0.350393
$454$	3.93912e9	1.97562
$455$	0	0
$456$	2.42786e8	0.119907
$457$	−2.02487e9	−0.992407	−0.496204	−	0.868206i	$-0.665273\pi$
$457$	−2.02487e9	−0.992407	−0.496204	+	0.868206i	$0.665273\pi$
$458$	−3.58315e9	−1.74276
$459$	−7.58024e8	−0.365880
$460$	0	0
$461$	−7.51321e8	−0.357168	−0.178584	−	0.983925i	$-0.557152\pi$
$461$	−7.51321e8	−0.357168	−0.178584	+	0.983925i	$0.557152\pi$
$462$	−1.20598e9	−0.568974
$463$	−3.91961e9	−1.83531	−0.917655	−	0.397377i	$-0.869920\pi$
$463$	−3.91961e9	−1.83531	−0.917655	+	0.397377i	$0.869920\pi$
$464$	2.29863e9	1.06821
$465$	0	0
$466$	−1.77491e9	−0.812503
$467$	−1.82834e9	−0.830707	−0.415353	−	0.909660i	$-0.636342\pi$
$467$	−1.82834e9	−0.830707	−0.415353	+	0.909660i	$0.636342\pi$
$468$	−2.27674e9	−1.02672
$469$	−7.27928e9	−3.25824
$470$	0	0
$471$	4.57087e9	2.01570
$472$	1.06551e9	0.466403
$473$	5.07689e8	0.220589
$474$	−1.34116e9	−0.578436
$475$	0	0
$476$	3.99763e9	1.69894
$477$	−6.02919e8	−0.254357
$478$	−2.66059e9	−1.11424
$479$	−1.00031e9	−0.415871	−0.207935	−	0.978143i	$-0.566674\pi$
$479$	−1.00031e9	−0.415871	−0.207935	+	0.978143i	$0.566674\pi$
$480$	0	0
$481$	−1.12767e9	−0.462033
$482$	−7.90833e8	−0.321677
$483$	6.22239e9	2.51271
$484$	−2.92916e9	−1.17432
$485$	0	0
$486$	4.31410e9	1.70476
$487$	4.38298e9	1.71956	0.859782	−	0.510662i	$-0.170600\pi$
$487$	4.38298e9	1.71956	0.859782	+	0.510662i	$0.170600\pi$
$488$	8.77052e8	0.341630
$489$	3.47723e9	1.34478
$490$	0	0
$491$	−2.86278e9	−1.09145	−0.545724	−	0.837965i	$-0.683745\pi$
$491$	−2.86278e9	−1.09145	−0.545724	+	0.837965i	$0.683745\pi$
$492$	−4.62770e9	−1.75181
$493$	−3.24801e9	−1.22083
$494$	−1.48454e9	−0.554050
$495$	0	0
$496$	2.46063e9	0.905440
$497$	7.66026e9	2.79896
$498$	−5.58785e8	−0.202741
$499$	1.34104e9	0.483159	0.241580	−	0.970381i	$-0.422334\pi$
$499$	1.34104e9	0.483159	0.241580	+	0.970381i	$0.422334\pi$
$500$	0	0
$501$	−8.88695e8	−0.315733
$502$	−1.24203e9	−0.438195
$503$	−8.69989e7	−0.0304808	−0.0152404	−	0.999884i	$-0.504851\pi$
$503$	−8.69989e7	−0.0304808	−0.0152404	+	0.999884i	$0.504851\pi$
$504$	−1.00310e9	−0.349009
$505$	0	0
$506$	−9.15602e8	−0.314181
$507$	2.33971e9	0.797322
$508$	−3.20285e9	−1.08396
$509$	2.38461e9	0.801502	0.400751	−	0.916187i	$-0.368749\pi$
$509$	2.38461e9	0.801502	0.400751	+	0.916187i	$0.368749\pi$
$510$	0	0
$511$	−5.68949e9	−1.88625
$512$	−3.91078e9	−1.28771
$513$	3.86710e8	0.126466
$514$	6.66356e9	2.16439
$515$	0	0
$516$	6.01219e9	1.92646
$517$	−4.64647e8	−0.147879
$518$	−2.82593e9	−0.893322
$519$	5.73219e9	1.79984
$520$	0	0
$521$	−3.58767e9	−1.11142	−0.555712	−	0.831375i	$-0.687554\pi$
$521$	−3.58767e9	−1.11142	−0.555712	+	0.831375i	$0.687554\pi$
$522$	4.63564e9	1.42647
$523$	4.17147e9	1.27507	0.637534	−	0.770422i	$-0.279955\pi$
$523$	4.17147e9	1.27507	0.637534	+	0.770422i	$0.279955\pi$
$524$	2.71166e9	0.823333
$525$	0	0
$526$	4.82102e9	1.44440
$527$	−3.47691e9	−1.03480
$528$	−5.80003e8	−0.171479
$529$	1.31934e9	0.387490
$530$	0	0
$531$	−3.37331e9	−0.977745
$532$	−2.03941e9	−0.587237
$533$	4.97490e9	1.42311
$534$	2.32080e9	0.659542
$535$	0	0
$536$	2.23009e9	0.625526
$537$	6.12772e9	1.70761
$538$	3.09684e9	0.857394
$539$	1.12924e9	0.310618
$540$	0	0
$541$	−4.56176e9	−1.23863	−0.619315	−	0.785143i	$-0.712590\pi$
$541$	−4.56176e9	−1.23863	−0.619315	+	0.785143i	$0.712590\pi$
$542$	−1.46525e9	−0.395288
$543$	−7.00902e8	−0.187870
$544$	4.51662e9	1.20287
$545$	0	0
$546$	1.53530e10	4.03664
$547$	3.56179e9	0.930491	0.465246	−	0.885182i	$-0.345966\pi$
$547$	3.56179e9	0.930491	0.465246	+	0.885182i	$0.345966\pi$
$548$	5.29266e9	1.37386
$549$	−2.77667e9	−0.716177
$550$	0	0
$551$	1.65699e9	0.421978
$552$	−1.90630e9	−0.482397
$553$	1.98066e9	0.498049
$554$	−4.22306e9	−1.05522
$555$	0	0
$556$	4.21032e9	1.03885
$557$	−3.08425e9	−0.756235	−0.378118	−	0.925758i	$-0.623429\pi$
$557$	−3.08425e9	−0.756235	−0.378118	+	0.925758i	$0.623429\pi$
$558$	4.96233e9	1.20911
$559$	−6.46327e9	−1.56499
$560$	0	0
$561$	8.19556e8	0.195979
$562$	2.06877e9	0.491626
$563$	1.14730e9	0.270954	0.135477	−	0.990780i	$-0.456743\pi$
$563$	1.14730e9	0.270954	0.135477	+	0.990780i	$0.456743\pi$
$564$	−5.50248e9	−1.29146
$565$	0	0
$566$	−4.96016e9	−1.14984
$567$	−8.77277e9	−2.02114
$568$	−2.34680e9	−0.537351
$569$	−2.62808e9	−0.598062	−0.299031	−	0.954243i	$-0.596663\pi$
$569$	−2.62808e9	−0.598062	−0.299031	+	0.954243i	$0.596663\pi$
$570$	0	0
$571$	3.76164e9	0.845572	0.422786	−	0.906229i	$-0.361052\pi$
$571$	3.76164e9	0.845572	0.422786	+	0.906229i	$0.361052\pi$
$572$	−1.23844e9	−0.276686
$573$	−7.05776e9	−1.56720
$574$	1.24671e10	2.75152
$575$	0	0
$576$	−4.18464e9	−0.912388
$577$	−5.08828e9	−1.10270	−0.551348	−	0.834275i	$-0.685886\pi$
$577$	−5.08828e9	−1.10270	−0.551348	+	0.834275i	$0.685886\pi$
$578$	1.95091e9	0.420232
$579$	8.68834e9	1.86021
$580$	0	0
$581$	8.25231e8	0.174566
$582$	−5.84408e8	−0.122881
$583$	−3.27958e8	−0.0685455
$584$	1.74304e9	0.362127
$585$	0	0
$586$	4.58803e9	0.941856
$587$	−6.77298e9	−1.38212	−0.691061	−	0.722796i	$-0.742857\pi$
$587$	−6.77298e9	−1.38212	−0.691061	+	0.722796i	$0.742857\pi$
$588$	1.33728e10	2.71270
$589$	1.77376e9	0.357678
$590$	0	0
$591$	−7.99903e9	−1.59397
$592$	−1.35911e9	−0.269232
$593$	−5.77140e9	−1.13655	−0.568277	−	0.822838i	$-0.692390\pi$
$593$	−5.77140e9	−1.13655	−0.568277	+	0.822838i	$0.692390\pi$
$594$	5.88484e8	0.115208
$595$	0	0
$596$	2.40113e9	0.464572
$597$	−9.18039e9	−1.76584
$598$	1.16563e10	2.22898
$599$	6.42032e9	1.22057	0.610285	−	0.792182i	$-0.291055\pi$
$599$	6.42032e9	1.22057	0.610285	+	0.792182i	$0.291055\pi$
$600$	0	0
$601$	4.52468e8	0.0850212	0.0425106	−	0.999096i	$-0.486464\pi$
$601$	4.52468e8	0.0850212	0.0425106	+	0.999096i	$0.486464\pi$
$602$	−1.61969e10	−3.02584
$603$	−7.06025e9	−1.31132
$604$	−1.78408e9	−0.329446
$605$	0	0
$606$	1.33971e10	2.44544
$607$	9.37841e9	1.70204	0.851018	−	0.525137i	$-0.175986\pi$
$607$	9.37841e9	1.70204	0.851018	+	0.525137i	$0.175986\pi$
$608$	−2.30417e9	−0.415770
$609$	−1.71365e10	−3.07440
$610$	0	0
$611$	5.91531e9	1.04914
$612$	3.87734e9	0.683760
$613$	1.49180e9	0.261577	0.130788	−	0.991410i	$-0.458249\pi$
$613$	1.49180e9	0.261577	0.130788	+	0.991410i	$0.458249\pi$
$614$	7.11146e9	1.23985
$615$	0	0
$616$	−5.45637e8	−0.0940526
$617$	6.17627e9	1.05859	0.529296	−	0.848438i	$-0.322456\pi$
$617$	6.17627e9	1.05859	0.529296	+	0.848438i	$0.322456\pi$
$618$	1.59421e10	2.71697
$619$	4.13146e9	0.700142	0.350071	−	0.936723i	$-0.386157\pi$
$619$	4.13146e9	0.700142	0.350071	+	0.936723i	$0.386157\pi$
$620$	0	0
$621$	−3.03636e9	−0.508783
$622$	−8.93510e9	−1.48879
$623$	−3.42742e9	−0.567884
$624$	7.38388e9	1.21657
$625$	0	0
$626$	−3.40450e9	−0.554681
$627$	−4.18101e8	−0.0677399
$628$	1.17629e10	1.89520
$629$	1.92044e9	0.307697
$630$	0	0
$631$	−7.38957e9	−1.17089	−0.585445	−	0.810712i	$-0.699080\pi$
$631$	−7.38957e9	−1.17089	−0.585445	+	0.810712i	$0.699080\pi$
$632$	−6.06797e8	−0.0956167
$633$	6.54940e9	1.02633
$634$	9.24757e9	1.44117
$635$	0	0
$636$	−3.88377e9	−0.598624
$637$	−1.43761e10	−2.20370
$638$	2.52156e9	0.384413
$639$	7.42976e9	1.12648
$640$	0	0
$641$	−6.00844e8	−0.0901070	−0.0450535	−	0.998985i	$-0.514346\pi$
$641$	−6.00844e8	−0.0901070	−0.0450535	+	0.998985i	$0.514346\pi$
$642$	2.20504e10	3.28885
$643$	1.13759e10	1.68751	0.843754	−	0.536730i	$-0.180341\pi$
$643$	1.13759e10	1.68751	0.843754	+	0.536730i	$0.180341\pi$
$644$	1.60130e10	2.36250
$645$	0	0
$646$	2.52821e9	0.368977
$647$	1.13081e10	1.64144	0.820720	−	0.571331i	$-0.193573\pi$
$647$	1.13081e10	1.64144	0.820720	+	0.571331i	$0.193573\pi$
$648$	2.68763e9	0.388023
$649$	−1.83492e9	−0.263488
$650$	0	0
$651$	−1.83441e10	−2.60593
$652$	8.94846e9	1.26439
$653$	9.13881e8	0.128438	0.0642190	−	0.997936i	$-0.479544\pi$
$653$	9.13881e8	0.128438	0.0642190	+	0.997936i	$0.479544\pi$
$654$	−1.55913e10	−2.17951
$655$	0	0
$656$	5.99592e9	0.829264
$657$	−5.51829e9	−0.759146
$658$	1.48238e10	2.02847
$659$	2.98685e9	0.406550	0.203275	−	0.979122i	$-0.434841\pi$
$659$	2.98685e9	0.406550	0.203275	+	0.979122i	$0.434841\pi$
$660$	0	0
$661$	−7.93450e8	−0.106860	−0.0534299	−	0.998572i	$-0.517015\pi$
$661$	−7.93450e8	−0.106860	−0.0534299	+	0.998572i	$0.517015\pi$
$662$	8.99169e9	1.20459
$663$	−1.04336e10	−1.39039
$664$	−2.52819e8	−0.0335136
$665$	0	0
$666$	−2.74090e9	−0.359529
$667$	−1.30103e10	−1.69765
$668$	−2.28701e9	−0.296859
$669$	1.09453e10	1.41330
$670$	0	0
$671$	−1.51037e9	−0.192999
$672$	2.38296e10	3.02917
$673$	1.26562e10	1.60048	0.800239	−	0.599682i	$-0.204706\pi$
$673$	1.26562e10	1.60048	0.800239	+	0.599682i	$0.204706\pi$
$674$	−1.30249e10	−1.63857
$675$	0	0
$676$	6.02111e9	0.749658
$677$	8.91501e9	1.10424	0.552118	−	0.833766i	$-0.313820\pi$
$677$	8.91501e9	1.10424	0.552118	+	0.833766i	$0.313820\pi$
$678$	7.58081e9	0.934139
$679$	8.63071e8	0.105804
$680$	0	0
$681$	−1.41235e10	−1.71367
$682$	2.69927e9	0.325837
$683$	−6.76424e9	−0.812357	−0.406178	−	0.913794i	$-0.633139\pi$
$683$	−6.76424e9	−0.812357	−0.406178	+	0.913794i	$0.633139\pi$
$684$	−1.97804e9	−0.236341
$685$	0	0
$686$	−1.52324e10	−1.80150
$687$	1.28472e10	1.51168
$688$	−7.78976e9	−0.911936
$689$	4.17516e9	0.486302
$690$	0	0
$691$	8.47398e9	0.977044	0.488522	−	0.872551i	$-0.337536\pi$
$691$	8.47398e9	0.977044	0.488522	+	0.872551i	$0.337536\pi$
$692$	1.47515e10	1.69225
$693$	1.72743e9	0.197167
$694$	5.38646e9	0.611709
$695$	0	0
$696$	5.24994e9	0.590231
$697$	−8.47236e9	−0.947741
$698$	−1.72323e10	−1.91800
$699$	6.36382e9	0.704771
$700$	0	0
$701$	−4.05891e9	−0.445037	−0.222519	−	0.974928i	$-0.571428\pi$
$701$	−4.05891e9	−0.445037	−0.222519	+	0.974928i	$0.571428\pi$
$702$	−7.49185e9	−0.817353
$703$	−9.79723e8	−0.106355
$704$	−2.27624e9	−0.245875
$705$	0	0
$706$	−6.85932e9	−0.733609
$707$	−1.97853e10	−2.10559
$708$	−2.17296e10	−2.30110
$709$	−6.66015e9	−0.701815	−0.350907	−	0.936410i	$-0.614127\pi$
$709$	−6.66015e9	−0.701815	−0.350907	+	0.936410i	$0.614127\pi$
$710$	0	0
$711$	1.92106e9	0.200446
$712$	1.05003e9	0.109024
$713$	−1.39272e10	−1.43897
$714$	−2.61465e10	−2.68825
$715$	0	0
$716$	1.57693e10	1.60553
$717$	9.53938e9	0.966503
$718$	7.57979e8	0.0764226
$719$	−5.98027e9	−0.600025	−0.300013	−	0.953935i	$-0.596991\pi$
$719$	−5.98027e9	−0.600025	−0.300013	+	0.953935i	$0.596991\pi$
$720$	0	0
$721$	−2.35437e10	−2.33938
$722$	1.37556e10	1.36018
$723$	2.83548e9	0.279025
$724$	−1.80373e9	−0.176639
$725$	0	0
$726$	1.91582e10	1.85814
$727$	−7.07253e9	−0.682660	−0.341330	−	0.939944i	$-0.610877\pi$
$727$	−7.07253e9	−0.682660	−0.341330	+	0.939944i	$0.610877\pi$
$728$	6.94637e9	0.667265
$729$	−2.67826e9	−0.256039
$730$	0	0
$731$	1.10071e10	1.04222
$732$	−1.78862e10	−1.68550
$733$	4.94718e9	0.463975	0.231987	−	0.972719i	$-0.425477\pi$
$733$	4.94718e9	0.463975	0.231987	+	0.972719i	$0.425477\pi$
$734$	1.09799e10	1.02486
$735$	0	0
$736$	1.80919e10	1.67267
$737$	−3.84043e9	−0.353381
$738$	1.20920e10	1.10739
$739$	2.10162e10	1.91557	0.957786	−	0.287482i	$-0.0928180\pi$
$739$	2.10162e10	1.91557	0.957786	+	0.287482i	$0.0928180\pi$
$740$	0	0
$741$	5.32274e9	0.480586
$742$	1.04629e10	0.940244
$743$	6.83100e9	0.610975	0.305488	−	0.952196i	$-0.401181\pi$
$743$	6.83100e9	0.610975	0.305488	+	0.952196i	$0.401181\pi$
$744$	5.61992e9	0.500293
$745$	0	0
$746$	−9.31620e9	−0.821586
$747$	8.00400e8	0.0702563
$748$	2.10908e9	0.184263
$749$	−3.25647e10	−2.83179
$750$	0	0
$751$	−1.71247e10	−1.47531	−0.737655	−	0.675178i	$-0.764067\pi$
$751$	−1.71247e10	−1.47531	−0.737655	+	0.675178i	$0.764067\pi$
$752$	7.12935e9	0.611346
$753$	4.45320e9	0.380093
$754$	−3.21014e10	−2.72725
$755$	0	0
$756$	−1.02920e10	−0.866312
$757$	−6.52334e9	−0.546556	−0.273278	−	0.961935i	$-0.588108\pi$
$757$	−6.52334e9	−0.546556	−0.273278	+	0.961935i	$0.588108\pi$
$758$	−1.85811e10	−1.54963
$759$	3.28283e9	0.272523
$760$	0	0
$761$	5.07409e9	0.417361	0.208681	−	0.977984i	$-0.433083\pi$
$761$	5.07409e9	0.417361	0.208681	+	0.977984i	$0.433083\pi$
$762$	2.09482e10	1.71516
$763$	2.30257e10	1.87662
$764$	−1.81628e10	−1.47352
$765$	0	0
$766$	−7.89013e8	−0.0634283
$767$	2.33599e10	1.86934
$768$	7.26778e9	0.578945
$769$	−2.25880e9	−0.179116	−0.0895581	−	0.995982i	$-0.528545\pi$
$769$	−2.25880e9	−0.179116	−0.0895581	+	0.995982i	$0.528545\pi$
$770$	0	0
$771$	−2.38918e10	−1.87741
$772$	2.23590e10	1.74900
$773$	−1.32954e10	−1.03532	−0.517660	−	0.855586i	$-0.673197\pi$
$773$	−1.32954e10	−1.03532	−0.517660	+	0.855586i	$0.673197\pi$
$774$	−1.57096e10	−1.21779
$775$	0	0
$776$	−2.64411e8	−0.0203125
$777$	1.01322e10	0.774873
$778$	4.73462e9	0.360459
$779$	4.32222e9	0.327586
$780$	0	0
$781$	4.04143e9	0.303568
$782$	−1.98509e10	−1.48442
$783$	8.36212e9	0.622516
$784$	−1.73266e10	−1.28412
$785$	0	0
$786$	−1.77356e10	−1.30277
$787$	−2.07324e10	−1.51614	−0.758068	−	0.652175i	$-0.773857\pi$
$787$	−2.07324e10	−1.51614	−0.758068	+	0.652175i	$0.773857\pi$
$788$	−2.05851e10	−1.49869
$789$	−1.72855e10	−1.25289
$790$	0	0
$791$	−1.11956e10	−0.804319
$792$	−5.29218e8	−0.0378527
$793$	1.92282e10	1.36925
$794$	1.46680e10	1.03992
$795$	0	0
$796$	−2.36252e10	−1.66028
$797$	−2.12621e10	−1.48765	−0.743826	−	0.668374i	$-0.766991\pi$
$797$	−2.12621e10	−1.48765	−0.743826	+	0.668374i	$0.766991\pi$
$798$	1.33388e10	0.929193
$799$	−1.00739e10	−0.698689
$800$	0	0
$801$	−3.32429e9	−0.228552
$802$	−1.05934e10	−0.725147
$803$	−3.00168e9	−0.204578
$804$	−4.54794e10	−3.08616
$805$	0	0
$806$	−3.43637e10	−2.31168
$807$	−1.11035e10	−0.743709
$808$	6.06143e9	0.404236
$809$	−3.17723e9	−0.210974	−0.105487	−	0.994421i	$-0.533640\pi$
$809$	−3.17723e9	−0.210974	−0.105487	+	0.994421i	$0.533640\pi$
$810$	0	0
$811$	7.18484e9	0.472981	0.236491	−	0.971634i	$-0.424003\pi$
$811$	7.18484e9	0.472981	0.236491	+	0.971634i	$0.424003\pi$
$812$	−4.40997e10	−2.89061
$813$	5.25355e9	0.342875
$814$	−1.49092e9	−0.0968875
$815$	0	0
$816$	−1.25749e10	−0.810195
$817$	−5.61532e9	−0.360244
$818$	1.31915e10	0.842671
$819$	−2.19916e10	−1.39882
$820$	0	0
$821$	−1.69583e10	−1.06950	−0.534750	−	0.845010i	$-0.679594\pi$
$821$	−1.69583e10	−1.06950	−0.534750	+	0.845010i	$0.679594\pi$
$822$	−3.46167e10	−2.17387
$823$	−1.77974e8	−0.0111290	−0.00556451	−	0.999985i	$-0.501771\pi$
$823$	−1.77974e8	−0.0111290	−0.00556451	+	0.999985i	$0.501771\pi$
$824$	7.21287e9	0.449121
$825$	0	0
$826$	5.85399e10	3.61428
$827$	−2.13225e10	−1.31090	−0.655448	−	0.755240i	$-0.727520\pi$
$827$	−2.13225e10	−1.31090	−0.655448	+	0.755240i	$0.727520\pi$
$828$	1.55312e10	0.950819
$829$	9.40282e9	0.573214	0.286607	−	0.958048i	$-0.407473\pi$
$829$	9.40282e9	0.573214	0.286607	+	0.958048i	$0.407473\pi$
$830$	0	0
$831$	1.51415e10	0.915306
$832$	2.89783e10	1.74438
$833$	2.44828e10	1.46759
$834$	−2.75376e10	−1.64379
$835$	0	0
$836$	−1.07596e9	−0.0636904
$837$	8.95142e9	0.527659
$838$	2.45858e10	1.44321
$839$	2.00156e10	1.17004	0.585020	−	0.811019i	$-0.301086\pi$
$839$	2.00156e10	1.17004	0.585020	+	0.811019i	$0.301086\pi$
$840$	0	0
$841$	1.85805e10	1.07714
$842$	1.22142e10	0.705135
$843$	−7.41744e9	−0.426440
$844$	1.68545e10	0.964978
$845$	0	0
$846$	1.43777e10	0.816383
$847$	−2.82935e10	−1.59991
$848$	5.03205e9	0.283374
$849$	1.77843e10	0.997381
$850$	0	0
$851$	7.69257e9	0.427876
$852$	4.78597e10	2.65113
$853$	−2.60147e10	−1.43515	−0.717575	−	0.696481i	$-0.754748\pi$
$853$	−2.60147e10	−1.43515	−0.717575	+	0.696481i	$0.754748\pi$
$854$	4.81858e10	2.64738
$855$	0	0
$856$	9.97655e9	0.543654
$857$	1.75408e10	0.951954	0.475977	−	0.879458i	$-0.342095\pi$
$857$	1.75408e10	0.951954	0.475977	+	0.879458i	$0.342095\pi$
$858$	8.10000e9	0.437804
$859$	−9.51931e9	−0.512424	−0.256212	−	0.966621i	$-0.582475\pi$
$859$	−9.51931e9	−0.512424	−0.256212	+	0.966621i	$0.582475\pi$
$860$	0	0
$861$	−4.47000e10	−2.38669
$862$	3.37450e10	1.79446
$863$	−3.40004e10	−1.80072	−0.900359	−	0.435147i	$-0.856696\pi$
$863$	−3.40004e10	−1.80072	−0.900359	+	0.435147i	$0.856696\pi$
$864$	−1.16282e10	−0.613357
$865$	0	0
$866$	3.30014e10	1.72671
$867$	−6.99486e9	−0.364512
$868$	−4.72076e10	−2.45015
$869$	1.04496e9	0.0540172
$870$	0	0
$871$	4.88916e10	2.50710
$872$	−7.05417e9	−0.360278
$873$	8.37101e8	0.0425822
$874$	1.01271e10	0.513090
$875$	0	0
$876$	−3.55467e10	−1.78663
$877$	2.15407e10	1.07836	0.539178	−	0.842192i	$-0.318735\pi$
$877$	2.15407e10	1.07836	0.539178	+	0.842192i	$0.318735\pi$
$878$	5.65859e9	0.282148
$879$	−1.64501e10	−0.816973
$880$	0	0
$881$	1.80427e10	0.888965	0.444483	−	0.895787i	$-0.353388\pi$
$881$	1.80427e10	0.888965	0.444483	+	0.895787i	$0.353388\pi$
$882$	−3.49425e10	−1.71480
$883$	2.97926e10	1.45628	0.728142	−	0.685427i	$-0.240384\pi$
$883$	2.97926e10	1.45628	0.728142	+	0.685427i	$0.240384\pi$
$884$	−2.68502e10	−1.30727
$885$	0	0
$886$	−3.19151e10	−1.54162
$887$	−8.11714e8	−0.0390544	−0.0195272	−	0.999809i	$-0.506216\pi$
$887$	−8.11714e8	−0.0390544	−0.0195272	+	0.999809i	$0.506216\pi$
$888$	−3.10411e9	−0.148762
$889$	−3.09370e10	−1.47680
$890$	0	0
$891$	−4.62837e9	−0.219208
$892$	2.81670e10	1.32881
$893$	5.13925e9	0.241502
$894$	−1.57046e10	−0.735099
$895$	0	0
$896$	2.20770e10	1.02533
$897$	−4.17930e10	−1.93344
$898$	2.81038e10	1.29508
$899$	3.83555e10	1.76063
$900$	0	0
$901$	−7.11039e9	−0.323859
$902$	6.57743e9	0.298424
$903$	5.80731e10	2.62463
$904$	3.42988e9	0.154415
$905$	0	0
$906$	1.16688e10	0.521287
$907$	4.05135e9	0.180291	0.0901455	−	0.995929i	$-0.471267\pi$
$907$	4.05135e9	0.180291	0.0901455	+	0.995929i	$0.471267\pi$
$908$	−3.63460e10	−1.61123
$909$	−1.91899e10	−0.847421
$910$	0	0
$911$	−1.17458e10	−0.514716	−0.257358	−	0.966316i	$-0.582852\pi$
$911$	−1.17458e10	−0.514716	−0.257358	+	0.966316i	$0.582852\pi$
$912$	6.41516e9	0.280043
$913$	4.35379e8	0.0189330
$914$	3.40819e10	1.47643
$915$	0	0
$916$	3.30615e10	1.42131
$917$	2.61925e10	1.12172
$918$	1.27588e10	0.544327
$919$	−1.42786e10	−0.606848	−0.303424	−	0.952856i	$-0.598130\pi$
$919$	−1.42786e10	−0.606848	−0.303424	+	0.952856i	$0.598130\pi$
$920$	0	0
$921$	−2.54977e10	−1.07545
$922$	1.26460e10	0.531366
$923$	−5.14505e10	−2.15369
$924$	1.11275e10	0.464028
$925$	0	0
$926$	6.59735e10	2.73043
$927$	−2.28353e10	−0.941515
$928$	−4.98249e10	−2.04658
$929$	−1.04119e10	−0.426064	−0.213032	−	0.977045i	$-0.568334\pi$
$929$	−1.04119e10	−0.426064	−0.213032	+	0.977045i	$0.568334\pi$
$930$	0	0
$931$	−1.24900e10	−0.507270
$932$	1.63769e10	0.662639
$933$	3.20363e10	1.29139
$934$	3.07740e10	1.23586
$935$	0	0
$936$	6.73735e9	0.268549
$937$	−2.11954e10	−0.841690	−0.420845	−	0.907133i	$-0.638266\pi$
$937$	−2.11954e10	−0.841690	−0.420845	+	0.907133i	$0.638266\pi$
$938$	1.22522e11	4.84736
$939$	1.22066e10	0.481134
$940$	0	0
$941$	3.91163e10	1.53036	0.765181	−	0.643816i	$-0.222650\pi$
$941$	3.91163e10	1.53036	0.765181	+	0.643816i	$0.222650\pi$
$942$	−7.69353e10	−2.99880
$943$	−3.39371e10	−1.31790
$944$	2.81542e10	1.08928
$945$	0	0
$946$	−8.54524e9	−0.328175
$947$	6.61675e9	0.253174	0.126587	−	0.991955i	$-0.459598\pi$
$947$	6.61675e9	0.253174	0.126587	+	0.991955i	$0.459598\pi$
$948$	1.23748e10	0.471745
$949$	3.82137e10	1.45140
$950$	0	0
$951$	−3.31566e10	−1.25008
$952$	−1.18298e10	−0.444374
$953$	3.97784e9	0.148875	0.0744376	−	0.997226i	$-0.476284\pi$
$953$	3.97784e9	0.148875	0.0744376	+	0.997226i	$0.476284\pi$
$954$	1.01481e10	0.378413
$955$	0	0
$956$	2.45491e10	0.908725
$957$	−9.04091e9	−0.333442
$958$	1.68368e10	0.618700
$959$	5.11230e10	1.87176
$960$	0	0
$961$	1.35459e10	0.492352
$962$	1.89805e10	0.687377
$963$	−3.15848e10	−1.13969
$964$	7.29696e9	0.262345
$965$	0	0
$966$	−1.04733e11	−3.73822
$967$	−8.23008e9	−0.292692	−0.146346	−	0.989233i	$-0.546751\pi$
$967$	−8.23008e9	−0.292692	−0.146346	+	0.989233i	$0.546751\pi$
$968$	8.66801e9	0.307154
$969$	−9.06474e9	−0.320053
$970$	0	0
$971$	2.22059e10	0.778396	0.389198	−	0.921154i	$-0.372752\pi$
$971$	2.22059e10	0.778396	0.389198	+	0.921154i	$0.372752\pi$
$972$	−3.98059e10	−1.39032
$973$	4.06684e10	1.41534
$974$	−7.37728e10	−2.55823
$975$	0	0
$976$	2.31745e10	0.797876
$977$	1.94112e9	0.0665919	0.0332960	−	0.999446i	$-0.489400\pi$
$977$	1.94112e9	0.0665919	0.0332960	+	0.999446i	$0.489400\pi$
$978$	−5.85275e10	−2.00066
$979$	−1.80825e9	−0.0615913
$980$	0	0
$981$	2.23328e10	0.755270
$982$	4.81853e10	1.62377
$983$	−1.97512e9	−0.0663219	−0.0331609	−	0.999450i	$-0.510557\pi$
$983$	−1.97512e9	−0.0663219	−0.0331609	+	0.999450i	$0.510557\pi$
$984$	1.36943e10	0.458203
$985$	0	0
$986$	5.46694e10	1.81625
$987$	−5.31497e10	−1.75951
$988$	1.36978e10	0.451857
$989$	4.40902e10	1.44929
$990$	0	0
$991$	3.96358e10	1.29369	0.646845	−	0.762622i	$-0.276088\pi$
$991$	3.96358e10	1.29369	0.646845	+	0.762622i	$0.276088\pi$
$992$	−5.33362e10	−1.73473
$993$	−3.22392e10	−1.04487
$994$	−1.28935e11	−4.16407
$995$	0	0
$996$	5.15587e9	0.165346
$997$	1.14988e9	0.0367467	0.0183733	−	0.999831i	$-0.494151\pi$
$997$	1.14988e9	0.0367467	0.0183733	+	0.999831i	$0.494151\pi$
$998$	−2.25719e10	−0.718807
$999$	−4.94424e9	−0.156899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.8.a.a.1.4	✓	22
5.4	even	2		625.8.a.b.1.19	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.8.a.a.1.4	✓	22	1.1	even	1	trivial
625.8.a.b.1.19	yes	22	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q-16.8316 q^{2} +60.3488 q^{3} +155.304 q^{4} -1015.77 q^{6} +1500.12 q^{7} -459.578 q^{8} +1454.98 q^{9} +O(q^{10})\)	\(q-16.8316 q^{2} +60.3488 q^{3} +155.304 q^{4} -1015.77 q^{6} +1500.12 q^{7} -459.578 q^{8} +1454.98 q^{9} +791.439 q^{11} +9372.44 q^{12} -10075.6 q^{13} -25249.5 q^{14} -12143.5 q^{16} +17159.0 q^{17} -24489.7 q^{18} -8753.75 q^{19} +90530.5 q^{21} -13321.2 q^{22} +68732.5 q^{23} -27735.0 q^{24} +169589. q^{26} -44176.4 q^{27} +232975. q^{28} -189289. q^{29} -202629. q^{31} +263221. q^{32} +47762.4 q^{33} -288814. q^{34} +225965. q^{36} +111920. q^{37} +147340. q^{38} -608052. q^{39} -493756. q^{41} -1.52378e6 q^{42} +641476. q^{43} +122914. q^{44} -1.15688e6 q^{46} -587091. q^{47} -732846. q^{48} +1.42682e6 q^{49} +1.03553e6 q^{51} -1.56479e6 q^{52} -414382. q^{53} +743562. q^{54} -689423. q^{56} -528279. q^{57} +3.18605e6 q^{58} -2.31846e6 q^{59} -1.90838e6 q^{61} +3.41058e6 q^{62} +2.18265e6 q^{63} -2.87608e6 q^{64} -803921. q^{66} -4.85246e6 q^{67} +2.66487e6 q^{68} +4.14793e6 q^{69} +5.10643e6 q^{71} -668678. q^{72} -3.79268e6 q^{73} -1.88380e6 q^{74} -1.35950e6 q^{76} +1.18725e6 q^{77} +1.02345e7 q^{78} +1.32033e6 q^{79} -5.84804e6 q^{81} +8.31072e6 q^{82} +550110. q^{83} +1.40598e7 q^{84} -1.07971e7 q^{86} -1.14234e7 q^{87} -363728. q^{88} -2.28476e6 q^{89} -1.51147e7 q^{91} +1.06745e7 q^{92} -1.22284e7 q^{93} +9.88172e6 q^{94} +1.58851e7 q^{96} +575335. q^{97} -2.40157e7 q^{98} +1.15153e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9}+O(q^{10}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9	\( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9} - 781 q^{11} - 12608 q^{12} - 14686 q^{13} + 20762 q^{14} + 16117 q^{16} - 45648 q^{17} - 47171 q^{18} + 6185 q^{19} + 14149 q^{21} + 71124 q^{22} - 126921 q^{23} + 271570 q^{24} + 304129 q^{26} - 546520 q^{27} - 2019 q^{28} + 59330 q^{29} + 394804 q^{31} + 74397 q^{32} - 49067 q^{33} - 286938 q^{34} - 287278 q^{36} + 792122 q^{37} - 1338860 q^{38} + 635223 q^{39} - 160466 q^{41} - 3420191 q^{42} - 1527256 q^{43} - 1154853 q^{44} + 2653604 q^{46} - 1300863 q^{47} - 1885241 q^{48} + 1652981 q^{49} + 3408539 q^{51} - 1423303 q^{52} - 755656 q^{53} + 3117755 q^{54} - 2132625 q^{56} - 3026890 q^{57} + 5941470 q^{58} - 1548370 q^{59} - 6029951 q^{61} - 79936 q^{62} + 6962459 q^{63} - 5858224 q^{64} - 5380407 q^{66} - 7608838 q^{67} - 10737124 q^{68} + 13519553 q^{69} + 9483549 q^{71} - 6806340 q^{72} - 13548801 q^{73} - 15016023 q^{74} + 19635315 q^{76} + 2145019 q^{77} - 17222402 q^{78} + 10769160 q^{79} + 2757382 q^{81} - 12087571 q^{82} + 9632744 q^{83} + 19168542 q^{84} - 8511651 q^{86} - 298330 q^{87} + 21641425 q^{88} - 10850545 q^{89} - 6648131 q^{91} - 52978503 q^{92} - 51294822 q^{93} + 5863777 q^{94} + 2611654 q^{96} - 11579993 q^{97} - 7468074 q^{98} - 13552997 q^{99}+O(q^{100}) \) 22 * q - 23 * q^2 - 121 * q^3 + 1061 * q^4 - 431 * q^6 - 843 * q^7 - 4980 * q^8 + 14799 * q^9 - 781 * q^11 - 12608 * q^12 - 14686 * q^13 + 20762 * q^14 + 16117 * q^16 - 45648 * q^17 - 47171 * q^18 + 6185 * q^19 + 14149 * q^21 + 71124 * q^22 - 126921 * q^23 + 271570 * q^24 + 304129 * q^26 - 546520 * q^27 - 2019 * q^28 + 59330 * q^29 + 394804 * q^31 + 74397 * q^32 - 49067 * q^33 - 286938 * q^34 - 287278 * q^36 + 792122 * q^37 - 1338860 * q^38 + 635223 * q^39 - 160466 * q^41 - 3420191 * q^42 - 1527256 * q^43 - 1154853 * q^44 + 2653604 * q^46 - 1300863 * q^47 - 1885241 * q^48 + 1652981 * q^49 + 3408539 * q^51 - 1423303 * q^52 - 755656 * q^53 + 3117755 * q^54 - 2132625 * q^56 - 3026890 * q^57 + 5941470 * q^58 - 1548370 * q^59 - 6029951 * q^61 - 79936 * q^62 + 6962459 * q^63 - 5858224 * q^64 - 5380407 * q^66 - 7608838 * q^67 - 10737124 * q^68 + 13519553 * q^69 + 9483549 * q^71 - 6806340 * q^72 - 13548801 * q^73 - 15016023 * q^74 + 19635315 * q^76 + 2145019 * q^77 - 17222402 * q^78 + 10769160 * q^79 + 2757382 * q^81 - 12087571 * q^82 + 9632744 * q^83 + 19168542 * q^84 - 8511651 * q^86 - 298330 * q^87 + 21641425 * q^88 - 10850545 * q^89 - 6648131 * q^91 - 52978503 * q^92 - 51294822 * q^93 + 5863777 * q^94 + 2611654 * q^96 - 11579993 * q^97 - 7468074 * q^98 - 13552997 * q^99

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