LMFDB - Embedded newform 625.8.a.e.1.5

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:	$48$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-19.8925 q^{2} -60.7396 q^{3} +267.712 q^{4} +1208.26 q^{6} -431.550 q^{7} -2779.23 q^{8} +1502.29 q^{9} +O(q^{10})$	\(q-19.8925 q^{2} -60.7396 q^{3} +267.712 q^{4} +1208.26 q^{6} -431.550 q^{7} -2779.23 q^{8} +1502.29 q^{9} +6063.98 q^{11} -16260.7 q^{12} -12971.8 q^{13} +8584.62 q^{14} +21018.7 q^{16} +32334.7 q^{17} -29884.4 q^{18} -8796.05 q^{19} +26212.2 q^{21} -120628. q^{22} -33502.1 q^{23} +168809. q^{24} +258041. q^{26} +41588.7 q^{27} -115531. q^{28} +176202. q^{29} -217697. q^{31} -62373.7 q^{32} -368323. q^{33} -643218. q^{34} +402183. q^{36} +118169. q^{37} +174976. q^{38} +787900. q^{39} +804103. q^{41} -521426. q^{42} -428967. q^{43} +1.62340e6 q^{44} +666441. q^{46} -878263. q^{47} -1.27667e6 q^{48} -637307. q^{49} -1.96399e6 q^{51} -3.47270e6 q^{52} -439199. q^{53} -827304. q^{54} +1.19938e6 q^{56} +534268. q^{57} -3.50511e6 q^{58} -2.29164e6 q^{59} -1.50269e6 q^{61} +4.33054e6 q^{62} -648316. q^{63} -1.44963e6 q^{64} +7.32688e6 q^{66} +534078. q^{67} +8.65638e6 q^{68} +2.03490e6 q^{69} -228556. q^{71} -4.17522e6 q^{72} +1.42332e6 q^{73} -2.35068e6 q^{74} -2.35481e6 q^{76} -2.61691e6 q^{77} -1.56733e7 q^{78} -500505. q^{79} -5.81160e6 q^{81} -1.59956e7 q^{82} +4.31498e6 q^{83} +7.01732e6 q^{84} +8.53324e6 q^{86} -1.07025e7 q^{87} -1.68532e7 q^{88} +4.29958e6 q^{89} +5.59797e6 q^{91} -8.96892e6 q^{92} +1.32228e7 q^{93} +1.74709e7 q^{94} +3.78855e6 q^{96} -1.33481e7 q^{97} +1.26776e7 q^{98} +9.10988e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9}+O(q^{10}) $ 48 * q - 25 * q^2 - 95 * q^3 + 3419 * q^4 + 431 * q^6 - 4030 * q^7 - 3375 * q^8 + 36231 * q^9	\( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9} + 781 q^{11} - 3925 q^{12} - 4290 q^{13} - 20762 q^{14} + 270603 q^{16} - 75075 q^{17} - 89950 q^{18} + 17750 q^{19} - 48034 q^{21} - 331305 q^{22} - 343890 q^{23} - 271570 q^{24} - 304129 q^{26} - 474740 q^{27} - 1146535 q^{28} - 59330 q^{29} - 385989 q^{31} - 1887300 q^{32} - 879805 q^{33} + 286938 q^{34} + 3553198 q^{36} - 935610 q^{37} - 984745 q^{38} - 294888 q^{39} + 160466 q^{41} + 783725 q^{42} + 146400 q^{43} + 2261658 q^{44} - 2639009 q^{46} - 4446810 q^{47} - 3994240 q^{48} + 7532484 q^{49} - 2294894 q^{51} - 4582065 q^{52} - 3977030 q^{53} - 3979475 q^{54} - 743430 q^{56} - 2455430 q^{57} - 14413560 q^{58} - 1614425 q^{59} + 7720866 q^{61} - 20362850 q^{62} - 26297840 q^{63} + 21801809 q^{64} + 945327 q^{66} - 3017910 q^{67} - 17494265 q^{68} - 13519553 q^{69} - 9483549 q^{71} - 21929370 q^{72} + 388070 q^{73} + 16144878 q^{74} - 13507955 q^{76} - 25473115 q^{77} + 3108110 q^{78} - 10950620 q^{79} + 34443488 q^{81} + 354040 q^{82} - 47217920 q^{83} - 27843102 q^{84} + 20021766 q^{86} - 56120960 q^{87} - 54397660 q^{88} + 10850545 q^{89} - 8553794 q^{91} + 20734425 q^{92} + 11206260 q^{93} - 52545997 q^{94} - 28125034 q^{96} - 32784020 q^{97} - 14131170 q^{98} + 27513602 q^{99}+O(q^{100}) \) 48 * q - 25 * q^2 - 95 * q^3 + 3419 * q^4 + 431 * q^6 - 4030 * q^7 - 3375 * q^8 + 36231 * q^9 + 781 * q^11 - 3925 * q^12 - 4290 * q^13 - 20762 * q^14 + 270603 * q^16 - 75075 * q^17 - 89950 * q^18 + 17750 * q^19 - 48034 * q^21 - 331305 * q^22 - 343890 * q^23 - 271570 * q^24 - 304129 * q^26 - 474740 * q^27 - 1146535 * q^28 - 59330 * q^29 - 385989 * q^31 - 1887300 * q^32 - 879805 * q^33 + 286938 * q^34 + 3553198 * q^36 - 935610 * q^37 - 984745 * q^38 - 294888 * q^39 + 160466 * q^41 + 783725 * q^42 + 146400 * q^43 + 2261658 * q^44 - 2639009 * q^46 - 4446810 * q^47 - 3994240 * q^48 + 7532484 * q^49 - 2294894 * q^51 - 4582065 * q^52 - 3977030 * q^53 - 3979475 * q^54 - 743430 * q^56 - 2455430 * q^57 - 14413560 * q^58 - 1614425 * q^59 + 7720866 * q^61 - 20362850 * q^62 - 26297840 * q^63 + 21801809 * q^64 + 945327 * q^66 - 3017910 * q^67 - 17494265 * q^68 - 13519553 * q^69 - 9483549 * q^71 - 21929370 * q^72 + 388070 * q^73 + 16144878 * q^74 - 13507955 * q^76 - 25473115 * q^77 + 3108110 * q^78 - 10950620 * q^79 + 34443488 * q^81 + 354040 * q^82 - 47217920 * q^83 - 27843102 * q^84 + 20021766 * q^86 - 56120960 * q^87 - 54397660 * q^88 + 10850545 * q^89 - 8553794 * q^91 + 20734425 * q^92 + 11206260 * q^93 - 52545997 * q^94 - 28125034 * q^96 - 32784020 * q^97 - 14131170 * q^98 + 27513602 * 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 $

$2$	−19.8925	−1.75827	−0.879133	−	0.476576i	$-0.841878\pi$
$2$	−19.8925	−1.75827	−0.879133	+	0.476576i	$0.841878\pi$
$3$	−60.7396	−1.29881	−0.649407	−	0.760441i	$-0.724983\pi$
$3$	−60.7396	−1.29881	−0.649407	+	0.760441i	$0.724983\pi$
$4$	267.712	2.09150
$5$	0	0
$5$	0	0
$6$	1208.26	2.28366
$7$	−431.550	−0.475541	−0.237771	−	0.971321i	$-0.576417\pi$
$7$	−431.550	−0.475541	−0.237771	+	0.971321i	$0.576417\pi$
$8$	−2779.23	−1.91915
$9$	1502.29	0.686920
$10$	0	0
$11$	6063.98	1.37367	0.686836	−	0.726812i	$-0.258999\pi$
$11$	6063.98	1.37367	0.686836	+	0.726812i	$0.258999\pi$
$12$	−16260.7	−2.71647
$13$	−12971.8	−1.63756	−0.818781	−	0.574106i	$-0.805350\pi$
$13$	−12971.8	−1.63756	−0.818781	+	0.574106i	$0.805350\pi$
$14$	8584.62	0.836129
$15$	0	0
$16$	21018.7	1.28288
$17$	32334.7	1.59624	0.798118	−	0.602502i	$-0.205829\pi$
$17$	32334.7	1.59624	0.798118	+	0.602502i	$0.205829\pi$
$18$	−29884.4	−1.20779
$19$	−8796.05	−0.294205	−0.147103	−	0.989121i	$-0.546995\pi$
$19$	−8796.05	−0.294205	−0.147103	+	0.989121i	$0.546995\pi$
$20$	0	0
$21$	26212.2	0.617640
$22$	−120628.	−2.41528
$23$	−33502.1	−0.574149	−0.287074	−	0.957908i	$-0.592683\pi$
$23$	−33502.1	−0.574149	−0.287074	+	0.957908i	$0.592683\pi$
$24$	168809.	2.49262
$25$	0	0
$26$	258041.	2.87927
$27$	41588.7	0.406633
$28$	−115531.	−0.994596
$29$	176202.	1.34159	0.670793	−	0.741644i	$-0.265954\pi$
$29$	176202.	1.34159	0.670793	+	0.741644i	$0.265954\pi$
$30$	0	0
$31$	−217697.	−1.31246	−0.656231	−	0.754560i	$-0.727850\pi$
$31$	−217697.	−1.31246	−0.656231	+	0.754560i	$0.727850\pi$
$32$	−62373.7	−0.336493
$33$	−368323.	−1.78415
$34$	−643218.	−2.80661
$35$	0	0
$36$	402183.	1.43670
$37$	118169.	0.383528	0.191764	−	0.981441i	$-0.438579\pi$
$37$	118169.	0.383528	0.191764	+	0.981441i	$0.438579\pi$
$38$	174976.	0.517291
$39$	787900.	2.12689
$40$	0	0
$41$	804103.	1.82208	0.911041	−	0.412315i	$-0.135280\pi$
$41$	804103.	1.82208	0.911041	+	0.412315i	$0.135280\pi$
$42$	−521426.	−1.08598
$43$	−428967.	−0.822782	−0.411391	−	0.911459i	$-0.634957\pi$
$43$	−428967.	−0.822782	−0.411391	+	0.911459i	$0.634957\pi$
$44$	1.62340e6	2.87304
$45$	0	0
$46$	666441.	1.00951
$47$	−878263.	−1.23391	−0.616953	−	0.787000i	$-0.711633\pi$
$47$	−878263.	−1.23391	−0.616953	+	0.787000i	$0.711633\pi$
$48$	−1.27667e6	−1.66622
$49$	−637307.	−0.773860
$50$	0	0
$51$	−1.96399e6	−2.07321
$52$	−3.47270e6	−3.42496
$53$	−439199.	−0.405225	−0.202612	−	0.979259i	$-0.564943\pi$
$53$	−439199.	−0.405225	−0.202612	+	0.979259i	$0.564943\pi$
$54$	−827304.	−0.714969
$55$	0	0
$56$	1.19938e6	0.912636
$57$	534268.	0.382118
$58$	−3.50511e6	−2.35887
$59$	−2.29164e6	−1.45266	−0.726331	−	0.687345i	$-0.758776\pi$
$59$	−2.29164e6	−1.45266	−0.726331	+	0.687345i	$0.758776\pi$
$60$	0	0
$61$	−1.50269e6	−0.847649	−0.423824	−	0.905744i	$-0.639313\pi$
$61$	−1.50269e6	−0.847649	−0.423824	+	0.905744i	$0.639313\pi$
$62$	4.33054e6	2.30766
$63$	−648316.	−0.326659
$64$	−1.44963e6	−0.691235
$65$	0	0
$66$	7.32688e6	3.13700
$67$	534078.	0.216942	0.108471	−	0.994100i	$-0.465405\pi$
$67$	534078.	0.216942	0.108471	+	0.994100i	$0.465405\pi$
$68$	8.65638e6	3.33853
$69$	2.03490e6	0.745713
$70$	0	0
$71$	−228556.	−0.0757861	−0.0378930	−	0.999282i	$-0.512065\pi$
$71$	−228556.	−0.0757861	−0.0378930	+	0.999282i	$0.512065\pi$
$72$	−4.17522e6	−1.31830
$73$	1.42332e6	0.428225	0.214112	−	0.976809i	$-0.431314\pi$
$73$	1.42332e6	0.428225	0.214112	+	0.976809i	$0.431314\pi$
$74$	−2.35068e6	−0.674345
$75$	0	0
$76$	−2.35481e6	−0.615331
$77$	−2.61691e6	−0.653238
$78$	−1.56733e7	−3.73964
$79$	−500505.	−0.114212	−0.0571062	−	0.998368i	$-0.518187\pi$
$79$	−500505.	−0.114212	−0.0571062	+	0.998368i	$0.518187\pi$
$80$	0	0
$81$	−5.81160e6	−1.21506
$82$	−1.59956e7	−3.20371
$83$	4.31498e6	0.828335	0.414167	−	0.910201i	$-0.364073\pi$
$83$	4.31498e6	0.828335	0.414167	+	0.910201i	$0.364073\pi$
$84$	7.01732e6	1.29180
$85$	0	0
$86$	8.53324e6	1.44667
$87$	−1.07025e7	−1.74247
$88$	−1.68532e7	−2.63629
$89$	4.29958e6	0.646489	0.323244	−	0.946316i	$-0.395226\pi$
$89$	4.29958e6	0.646489	0.323244	+	0.946316i	$0.395226\pi$
$90$	0	0
$91$	5.59797e6	0.778728
$92$	−8.96892e6	−1.20083
$93$	1.32228e7	1.70464
$94$	1.74709e7	2.16954
$95$	0	0
$96$	3.78855e6	0.437043
$97$	−1.33481e7	−1.48498	−0.742488	−	0.669859i	$-0.766354\pi$
$97$	−1.33481e7	−1.48498	−0.742488	+	0.669859i	$0.766354\pi$
$98$	1.26776e7	1.36065
$99$	9.10988e6	0.943603
$100$	0	0
$101$	−1.54002e7	−1.48731	−0.743656	−	0.668563i	$-0.766910\pi$
$101$	−1.54002e7	−1.48731	−0.743656	+	0.668563i	$0.766910\pi$
$102$	3.90688e7	3.64526
$103$	−6.14075e6	−0.553721	−0.276861	−	0.960910i	$-0.589294\pi$
$103$	−6.14075e6	−0.553721	−0.276861	+	0.960910i	$0.589294\pi$
$104$	3.60515e7	3.14273
$105$	0	0
$106$	8.73678e6	0.712493
$107$	−2.78183e6	−0.219526	−0.109763	−	0.993958i	$-0.535009\pi$
$107$	−2.78183e6	−0.219526	−0.109763	+	0.993958i	$0.535009\pi$
$108$	1.11338e7	0.850473
$109$	2.00938e7	1.48617	0.743087	−	0.669195i	$-0.233361\pi$
$109$	2.00938e7	1.48617	0.743087	+	0.669195i	$0.233361\pi$
$110$	0	0
$111$	−7.17753e6	−0.498132
$112$	−9.07063e6	−0.610063
$113$	2.38486e7	1.55485	0.777424	−	0.628976i	$-0.216526\pi$
$113$	2.38486e7	1.55485	0.777424	+	0.628976i	$0.216526\pi$
$114$	−1.06279e7	−0.671865
$115$	0	0
$116$	4.71715e7	2.80593
$117$	−1.94874e7	−1.12487
$118$	4.55865e7	2.55417
$119$	−1.39540e7	−0.759076
$120$	0	0
$121$	1.72846e7	0.886975
$122$	2.98924e7	1.49039
$123$	−4.88408e7	−2.36655
$124$	−5.82802e7	−2.74502
$125$	0	0
$126$	1.28966e7	0.574354
$127$	1.57287e7	0.681365	0.340682	−	0.940178i	$-0.389342\pi$
$127$	1.57287e7	0.681365	0.340682	+	0.940178i	$0.389342\pi$
$128$	3.68205e7	1.55187
$129$	2.60553e7	1.06864
$130$	0	0
$131$	5.04307e7	1.95995	0.979975	−	0.199119i	$-0.0638081\pi$
$131$	5.04307e7	1.95995	0.979975	+	0.199119i	$0.0638081\pi$
$132$	−9.86047e7	−3.73155
$133$	3.79594e6	0.139907
$134$	−1.06242e7	−0.381441
$135$	0	0
$136$	−8.98654e7	−3.06342
$137$	3.23700e7	1.07553	0.537763	−	0.843096i	$-0.319270\pi$
$137$	3.23700e7	1.07553	0.537763	+	0.843096i	$0.319270\pi$
$138$	−4.04793e7	−1.31116
$139$	2.73302e7	0.863160	0.431580	−	0.902075i	$-0.357956\pi$
$139$	2.73302e7	0.863160	0.431580	+	0.902075i	$0.357956\pi$
$140$	0	0
$141$	5.33453e7	1.60261
$142$	4.54656e6	0.133252
$143$	−7.86605e7	−2.24947
$144$	3.15763e7	0.881237
$145$	0	0
$146$	−2.83134e7	−0.752933
$147$	3.87098e7	1.00510
$148$	3.16353e7	0.802150
$149$	−1.65243e7	−0.409233	−0.204616	−	0.978842i	$-0.565595\pi$
$149$	−1.65243e7	−0.409233	−0.204616	+	0.978842i	$0.565595\pi$
$150$	0	0
$151$	1.78095e6	0.0420951	0.0210476	−	0.999778i	$-0.493300\pi$
$151$	1.78095e6	0.0420951	0.0210476	+	0.999778i	$0.493300\pi$
$152$	2.44463e7	0.564625
$153$	4.85762e7	1.09649
$154$	5.20569e7	1.14857
$155$	0	0
$156$	2.10931e8	4.44840
$157$	3.26554e7	0.673452	0.336726	−	0.941603i	$-0.390680\pi$
$157$	3.26554e7	0.673452	0.336726	+	0.941603i	$0.390680\pi$
$158$	9.95630e6	0.200816
$159$	2.66768e7	0.526312
$160$	0	0
$161$	1.44578e7	0.273032
$162$	1.15607e8	2.13640
$163$	1.88789e7	0.341444	0.170722	−	0.985319i	$-0.445390\pi$
$163$	1.88789e7	0.341444	0.170722	+	0.985319i	$0.445390\pi$
$164$	2.15268e8	3.81089
$165$	0	0
$166$	−8.58359e7	−1.45643
$167$	4.84024e7	0.804190	0.402095	−	0.915598i	$-0.368282\pi$
$167$	4.84024e7	0.804190	0.402095	+	0.915598i	$0.368282\pi$
$168$	−7.28497e7	−1.18535
$169$	1.05518e8	1.68161
$170$	0	0
$171$	−1.32143e7	−0.202095
$172$	−1.14840e8	−1.72085
$173$	−1.64846e7	−0.242057	−0.121029	−	0.992649i	$-0.538619\pi$
$173$	−1.64846e7	−0.242057	−0.121029	+	0.992649i	$0.538619\pi$
$174$	2.12899e8	3.06373
$175$	0	0
$176$	1.27457e8	1.76226
$177$	1.39193e8	1.88674
$178$	−8.55294e7	−1.13670
$179$	−1.12052e7	−0.146027	−0.0730134	−	0.997331i	$-0.523262\pi$
$179$	−1.12052e7	−0.146027	−0.0730134	+	0.997331i	$0.523262\pi$
$180$	0	0
$181$	−1.02550e8	−1.28547	−0.642734	−	0.766089i	$-0.722200\pi$
$181$	−1.02550e8	−1.28547	−0.642734	+	0.766089i	$0.722200\pi$
$182$	−1.11358e8	−1.36921
$183$	9.12729e7	1.10094
$184$	9.31100e7	1.10188
$185$	0	0
$186$	−2.63035e8	−2.99722
$187$	1.96077e8	2.19270
$188$	−2.35122e8	−2.58072
$189$	−1.79476e7	−0.193371
$190$	0	0
$191$	3.43926e7	0.357148	0.178574	−	0.983926i	$-0.442852\pi$
$191$	3.43926e7	0.357148	0.178574	+	0.983926i	$0.442852\pi$
$192$	8.80496e7	0.897787
$193$	−1.19687e8	−1.19839	−0.599194	−	0.800604i	$-0.704512\pi$
$193$	−1.19687e8	−1.19839	−0.599194	+	0.800604i	$0.704512\pi$
$194$	2.65528e8	2.61098
$195$	0	0
$196$	−1.70615e8	−1.61853
$197$	4.34924e7	0.405304	0.202652	−	0.979251i	$-0.435044\pi$
$197$	4.34924e7	0.405304	0.202652	+	0.979251i	$0.435044\pi$
$198$	−1.81218e8	−1.65911
$199$	6.69853e7	0.602551	0.301276	−	0.953537i	$-0.402588\pi$
$199$	6.69853e7	0.602551	0.301276	+	0.953537i	$0.402588\pi$
$200$	0	0
$201$	−3.24397e7	−0.281767
$202$	3.06349e8	2.61509
$203$	−7.60402e7	−0.637980
$204$	−5.25785e8	−4.33613
$205$	0	0
$206$	1.22155e8	0.973590
$207$	−5.03300e7	−0.394395
$208$	−2.72650e8	−2.10080
$209$	−5.33391e7	−0.404141
$210$	0	0
$211$	3.55501e7	0.260527	0.130263	−	0.991479i	$-0.458418\pi$
$211$	3.55501e7	0.260527	0.130263	+	0.991479i	$0.458418\pi$
$212$	−1.17579e8	−0.847528
$213$	1.38824e7	0.0984321
$214$	5.53375e7	0.385986
$215$	0	0
$216$	−1.15585e8	−0.780390
$217$	9.39472e7	0.624130
$218$	−3.99716e8	−2.61309
$219$	−8.64517e7	−0.556185
$220$	0	0
$221$	−4.19438e8	−2.61393
$222$	1.42779e8	0.875849
$223$	−1.42013e8	−0.857555	−0.428777	−	0.903410i	$-0.641056\pi$
$223$	−1.42013e8	−0.857555	−0.428777	+	0.903410i	$0.641056\pi$
$224$	2.69174e7	0.160017
$225$	0	0
$226$	−4.74409e8	−2.73384
$227$	−2.48569e8	−1.41044	−0.705222	−	0.708987i	$-0.749153\pi$
$227$	−2.48569e8	−1.41044	−0.705222	+	0.708987i	$0.749153\pi$
$228$	1.43030e8	0.799201
$229$	2.80671e8	1.54445	0.772224	−	0.635351i	$-0.219144\pi$
$229$	2.80671e8	1.54445	0.772224	+	0.635351i	$0.219144\pi$
$230$	0	0
$231$	1.58950e8	0.848435
$232$	−4.89707e8	−2.57471
$233$	4.97426e7	0.257622	0.128811	−	0.991669i	$-0.458884\pi$
$233$	4.97426e7	0.257622	0.128811	+	0.991669i	$0.458884\pi$
$234$	3.87654e8	1.97783
$235$	0	0
$236$	−6.13501e8	−3.03825
$237$	3.04004e7	0.148341
$238$	2.77581e8	1.33466
$239$	−1.07514e8	−0.509418	−0.254709	−	0.967018i	$-0.581980\pi$
$239$	−1.07514e8	−0.509418	−0.254709	+	0.967018i	$0.581980\pi$
$240$	0	0
$241$	1.91654e8	0.881979	0.440990	−	0.897512i	$-0.354628\pi$
$241$	1.91654e8	0.881979	0.440990	+	0.897512i	$0.354628\pi$
$242$	−3.43835e8	−1.55954
$243$	2.62039e8	1.17151
$244$	−4.02290e8	−1.77286
$245$	0	0
$246$	9.71567e8	4.16102
$247$	1.14100e8	0.481779
$248$	6.05030e8	2.51881
$249$	−2.62090e8	−1.07585
$250$	0	0
$251$	−3.24792e8	−1.29642	−0.648212	−	0.761460i	$-0.724483\pi$
$251$	−3.24792e8	−1.29642	−0.648212	+	0.761460i	$0.724483\pi$
$252$	−1.73562e8	−0.683208
$253$	−2.03156e8	−0.788692
$254$	−3.12883e8	−1.19802
$255$	0	0
$256$	−5.46901e8	−2.03737
$257$	1.98268e8	0.728595	0.364297	−	0.931283i	$-0.381309\pi$
$257$	1.98268e8	0.728595	0.364297	+	0.931283i	$0.381309\pi$
$258$	−5.18305e8	−1.87896
$259$	−5.09958e7	−0.182384
$260$	0	0
$261$	2.64708e8	0.921563
$262$	−1.00319e9	−3.44612
$263$	5.53609e8	1.87654	0.938270	−	0.345904i	$-0.112428\pi$
$263$	5.53609e8	1.87654	0.938270	+	0.345904i	$0.112428\pi$
$264$	1.02366e9	3.42405
$265$	0	0
$266$	−7.55108e7	−0.245993
$267$	−2.61154e8	−0.839669
$268$	1.42979e8	0.453734
$269$	−2.12806e8	−0.666578	−0.333289	−	0.942825i	$-0.608159\pi$
$269$	−2.12806e8	−0.666578	−0.333289	+	0.942825i	$0.608159\pi$
$270$	0	0
$271$	−2.20255e8	−0.672255	−0.336127	−	0.941817i	$-0.609117\pi$
$271$	−2.20255e8	−0.672255	−0.336127	+	0.941817i	$0.609117\pi$
$272$	6.79633e8	2.04778
$273$	−3.40018e8	−1.01142
$274$	−6.43921e8	−1.89106
$275$	0	0
$276$	5.44769e8	1.55966
$277$	5.41582e8	1.53104	0.765518	−	0.643415i	$-0.222483\pi$
$277$	5.41582e8	1.53104	0.765518	+	0.643415i	$0.222483\pi$
$278$	−5.43667e8	−1.51767
$279$	−3.27045e8	−0.901556
$280$	0	0
$281$	−2.23053e8	−0.599703	−0.299851	−	0.953986i	$-0.596937\pi$
$281$	−2.23053e8	−0.599703	−0.299851	+	0.953986i	$0.596937\pi$
$282$	−1.06117e9	−2.81782
$283$	−3.79933e8	−0.996447	−0.498223	−	0.867049i	$-0.666014\pi$
$283$	−3.79933e8	−0.996447	−0.498223	+	0.867049i	$0.666014\pi$
$284$	−6.11874e7	−0.158507
$285$	0	0
$286$	1.56476e9	3.95517
$287$	−3.47011e8	−0.866475
$288$	−9.37037e7	−0.231144
$289$	6.35191e8	1.54797
$290$	0	0
$291$	8.10760e8	1.92871
$292$	3.81040e8	0.895633
$293$	4.16751e8	0.967920	0.483960	−	0.875090i	$-0.339198\pi$
$293$	4.16751e8	0.967920	0.483960	+	0.875090i	$0.339198\pi$
$294$	−7.70035e8	−1.76724
$295$	0	0
$296$	−3.28419e8	−0.736049
$297$	2.52193e8	0.558580
$298$	3.28709e8	0.719540
$299$	4.34582e8	0.940204
$300$	0	0
$301$	1.85121e8	0.391267
$302$	−3.54275e7	−0.0740144
$303$	9.35402e8	1.93174
$304$	−1.84882e8	−0.377430
$305$	0	0
$306$	−9.66302e8	−1.92792
$307$	4.56174e8	0.899800	0.449900	−	0.893079i	$-0.351460\pi$
$307$	4.56174e8	0.899800	0.449900	+	0.893079i	$0.351460\pi$
$308$	−7.00579e8	−1.36625
$309$	3.72986e8	0.719181
$310$	0	0
$311$	3.11164e8	0.586581	0.293291	−	0.956023i	$-0.405250\pi$
$311$	3.11164e8	0.586581	0.293291	+	0.956023i	$0.405250\pi$
$312$	−2.18976e9	−4.08183
$313$	8.18451e7	0.150865	0.0754323	−	0.997151i	$-0.475966\pi$
$313$	8.18451e7	0.150865	0.0754323	+	0.997151i	$0.475966\pi$
$314$	−6.49598e8	−1.18411
$315$	0	0
$316$	−1.33991e8	−0.238875
$317$	5.62098e8	0.991070	0.495535	−	0.868588i	$-0.334972\pi$
$317$	5.62098e8	0.991070	0.495535	+	0.868588i	$0.334972\pi$
$318$	−5.30668e8	−0.925397
$319$	1.06849e9	1.84290
$320$	0	0
$321$	1.68967e8	0.285124
$322$	−2.87603e8	−0.480062
$323$	−2.84417e8	−0.469621
$324$	−1.55584e9	−2.54130
$325$	0	0
$326$	−3.75548e8	−0.600349
$327$	−1.22049e9	−1.93026
$328$	−2.23479e9	−3.49685
$329$	3.79014e8	0.586773
$330$	0	0
$331$	−7.88503e8	−1.19510	−0.597552	−	0.801830i	$-0.703860\pi$
$331$	−7.88503e8	−1.19510	−0.597552	+	0.801830i	$0.703860\pi$
$332$	1.15517e9	1.73246
$333$	1.77525e8	0.263453
$334$	−9.62845e8	−1.41398
$335$	0	0
$336$	5.50946e8	0.792358
$337$	−4.10327e8	−0.584016	−0.292008	−	0.956416i	$-0.594323\pi$
$337$	−4.10327e8	−0.584016	−0.292008	+	0.956416i	$0.594323\pi$
$338$	−2.09903e9	−2.95672
$339$	−1.44855e9	−2.01946
$340$	0	0
$341$	−1.32011e9	−1.80289
$342$	2.62865e8	0.355338
$343$	6.30430e8	0.843544
$344$	1.19220e9	1.57904
$345$	0	0
$346$	3.27921e8	0.425601
$347$	1.41272e9	1.81512	0.907558	−	0.419926i	$-0.137944\pi$
$347$	1.41272e9	1.81512	0.907558	+	0.419926i	$0.137944\pi$
$348$	−2.86518e9	−3.64439
$349$	1.88980e6	0.00237972	0.00118986	−	0.999999i	$-0.499621\pi$
$349$	1.88980e6	0.00237972	0.00118986	+	0.999999i	$0.499621\pi$
$350$	0	0
$351$	−5.39479e8	−0.665886
$352$	−3.78233e8	−0.462232
$353$	−1.10083e9	−1.33202	−0.666010	−	0.745943i	$-0.731999\pi$
$353$	−1.10083e9	−1.33202	−0.666010	+	0.745943i	$0.731999\pi$
$354$	−2.76891e9	−3.31739
$355$	0	0
$356$	1.15105e9	1.35213
$357$	8.47561e8	0.985899
$358$	2.22899e8	0.256754
$359$	−4.29616e8	−0.490061	−0.245030	−	0.969515i	$-0.578798\pi$
$359$	−4.29616e8	−0.490061	−0.245030	+	0.969515i	$0.578798\pi$
$360$	0	0
$361$	−8.16501e8	−0.913443
$362$	2.03998e9	2.26020
$363$	−1.04986e9	−1.15202
$364$	1.49865e9	1.62871
$365$	0	0
$366$	−1.81565e9	−1.93574
$367$	−3.05602e8	−0.322720	−0.161360	−	0.986896i	$-0.551588\pi$
$367$	−3.05602e8	−0.322720	−0.161360	+	0.986896i	$0.551588\pi$
$368$	−7.04171e8	−0.736564
$369$	1.20800e9	1.25163
$370$	0	0
$371$	1.89536e8	0.192701
$372$	3.53991e9	3.56527
$373$	1.50973e9	1.50633	0.753164	−	0.657833i	$-0.228527\pi$
$373$	1.50973e9	1.50633	0.753164	+	0.657833i	$0.228527\pi$
$374$	−3.90046e9	−3.85536
$375$	0	0
$376$	2.44089e9	2.36805
$377$	−2.28566e9	−2.19693
$378$	3.57023e8	0.339997
$379$	−1.05720e9	−0.997515	−0.498758	−	0.866741i	$-0.666210\pi$
$379$	−1.05720e9	−0.997515	−0.498758	+	0.866741i	$0.666210\pi$
$380$	0	0
$381$	−9.55354e8	−0.884967
$382$	−6.84155e8	−0.627961
$383$	4.36490e7	0.0396989	0.0198494	−	0.999803i	$-0.493681\pi$
$383$	4.36490e7	0.0396989	0.0198494	+	0.999803i	$0.493681\pi$
$384$	−2.23646e9	−2.01559
$385$	0	0
$386$	2.38088e9	2.10709
$387$	−6.44435e8	−0.565185
$388$	−3.57346e9	−3.10583
$389$	−2.91290e7	−0.0250900	−0.0125450	−	0.999921i	$-0.503993\pi$
$389$	−2.91290e7	−0.0250900	−0.0125450	+	0.999921i	$0.503993\pi$
$390$	0	0
$391$	−1.08328e9	−0.916477
$392$	1.77122e9	1.48516
$393$	−3.06314e9	−2.54561
$394$	−8.65173e8	−0.712633
$395$	0	0
$396$	2.43883e9	1.97355
$397$	−1.94643e9	−1.56125	−0.780624	−	0.625000i	$-0.785099\pi$
$397$	−1.94643e9	−1.56125	−0.780624	+	0.625000i	$0.785099\pi$
$398$	−1.33251e9	−1.05945
$399$	−2.30564e8	−0.181713
$400$	0	0
$401$	1.99990e9	1.54883	0.774415	−	0.632677i	$-0.218044\pi$
$401$	1.99990e9	1.54883	0.774415	+	0.632677i	$0.218044\pi$
$402$	6.45306e8	0.495421
$403$	2.82392e9	2.14924
$404$	−4.12283e9	−3.11072
$405$	0	0
$406$	1.51263e9	1.12174
$407$	7.16574e8	0.526842
$408$	5.45839e9	3.97882
$409$	1.86943e9	1.35107	0.675534	−	0.737329i	$-0.263913\pi$
$409$	1.86943e9	1.35107	0.675534	+	0.737329i	$0.263913\pi$
$410$	0	0
$411$	−1.96614e9	−1.39691
$412$	−1.64395e9	−1.15811
$413$	9.88959e8	0.690801
$414$	1.00119e9	0.693451
$415$	0	0
$416$	8.09098e8	0.551029
$417$	−1.66003e9	−1.12109
$418$	1.06105e9	0.710588
$419$	9.42661e8	0.626046	0.313023	−	0.949746i	$-0.398658\pi$
$419$	9.42661e8	0.626046	0.313023	+	0.949746i	$0.398658\pi$
$420$	0	0
$421$	−2.34089e9	−1.52895	−0.764477	−	0.644652i	$-0.777002\pi$
$421$	−2.34089e9	−1.52895	−0.764477	+	0.644652i	$0.777002\pi$
$422$	−7.07181e8	−0.458075
$423$	−1.31941e9	−0.847595
$424$	1.22064e9	0.777688
$425$	0	0
$426$	−2.76156e8	−0.173070
$427$	6.48488e8	0.403092
$428$	−7.44729e8	−0.459140
$429$	4.77781e9	2.92165
$430$	0	0
$431$	2.00195e9	1.20443	0.602217	−	0.798333i	$-0.294284\pi$
$431$	2.00195e9	1.20443	0.602217	+	0.798333i	$0.294284\pi$
$432$	8.74141e8	0.521661
$433$	−1.21120e9	−0.716984	−0.358492	−	0.933533i	$-0.616709\pi$
$433$	−1.21120e9	−0.716984	−0.358492	+	0.933533i	$0.616709\pi$
$434$	−1.86885e9	−1.09739
$435$	0	0
$436$	5.37936e9	3.10834
$437$	2.94686e8	0.168918
$438$	1.71974e9	0.977921
$439$	4.69070e8	0.264614	0.132307	−	0.991209i	$-0.457762\pi$
$439$	4.69070e8	0.264614	0.132307	+	0.991209i	$0.457762\pi$
$440$	0	0
$441$	−9.57423e8	−0.531580
$442$	8.34367e9	4.59599
$443$	−1.31675e8	−0.0719600	−0.0359800	−	0.999353i	$-0.511455\pi$
$443$	−1.31675e8	−0.0719600	−0.0359800	+	0.999353i	$0.511455\pi$
$444$	−1.92151e9	−1.04184
$445$	0	0
$446$	2.82500e9	1.50781
$447$	1.00368e9	0.531518
$448$	6.25586e8	0.328711
$449$	−2.42762e9	−1.26567	−0.632833	−	0.774289i	$-0.718108\pi$
$449$	−2.42762e9	−1.26567	−0.632833	+	0.774289i	$0.718108\pi$
$450$	0	0
$451$	4.87606e9	2.50294
$452$	6.38456e9	3.25197
$453$	−1.08174e8	−0.0546737
$454$	4.94466e9	2.47994
$455$	0	0
$456$	−1.48485e9	−0.733343
$457$	−2.57432e8	−0.126170	−0.0630850	−	0.998008i	$-0.520094\pi$
$457$	−2.57432e8	−0.126170	−0.0630850	+	0.998008i	$0.520094\pi$
$458$	−5.58325e9	−2.71555
$459$	1.34476e9	0.649082
$460$	0	0
$461$	−2.41056e9	−1.14595	−0.572975	−	0.819573i	$-0.694211\pi$
$461$	−2.41056e9	−1.14595	−0.572975	+	0.819573i	$0.694211\pi$
$462$	−3.16192e9	−1.49178
$463$	1.07040e7	0.00501202	0.00250601	−	0.999997i	$-0.499202\pi$
$463$	1.07040e7	0.00501202	0.00250601	+	0.999997i	$0.499202\pi$
$464$	3.70355e9	1.72110
$465$	0	0
$466$	−9.89506e8	−0.452968
$467$	−9.37901e7	−0.0426136	−0.0213068	−	0.999773i	$-0.506783\pi$
$467$	−9.37901e7	−0.0426136	−0.0213068	+	0.999773i	$0.506783\pi$
$468$	−5.21702e9	−2.35268
$469$	−2.30481e8	−0.103165
$470$	0	0
$471$	−1.98348e9	−0.874689
$472$	6.36900e9	2.78788
$473$	−2.60125e9	−1.13023
$474$	−6.04741e8	−0.260823
$475$	0	0
$476$	−3.73567e9	−1.58761
$477$	−6.59806e8	−0.278357
$478$	2.13873e9	0.895693
$479$	−1.45809e9	−0.606191	−0.303096	−	0.952960i	$-0.598020\pi$
$479$	−1.45809e9	−0.606191	−0.303096	+	0.952960i	$0.598020\pi$
$480$	0	0
$481$	−1.53286e9	−0.628051
$482$	−3.81248e9	−1.55075
$483$	−8.78163e8	−0.354617
$484$	4.62731e9	1.85511
$485$	0	0
$486$	−5.21262e9	−2.05982
$487$	−2.63514e9	−1.03384	−0.516918	−	0.856035i	$-0.672921\pi$
$487$	−2.63514e9	−1.03384	−0.516918	+	0.856035i	$0.672921\pi$
$488$	4.17633e9	1.62677
$489$	−1.14669e9	−0.443472
$490$	0	0
$491$	5.77488e8	0.220170	0.110085	−	0.993922i	$-0.464888\pi$
$491$	5.77488e8	0.220170	0.110085	+	0.993922i	$0.464888\pi$
$492$	−1.30753e10	−4.94964
$493$	5.69744e9	2.14149
$494$	−2.26974e9	−0.847096
$495$	0	0
$496$	−4.57571e9	−1.68373
$497$	9.86336e7	0.0360394
$498$	5.21364e9	1.89164
$499$	1.74925e8	0.0630233	0.0315116	−	0.999503i	$-0.489968\pi$
$499$	1.74925e8	0.0630233	0.0315116	+	0.999503i	$0.489968\pi$
$500$	0	0
$501$	−2.93994e9	−1.04449
$502$	6.46093e9	2.27946
$503$	3.64156e9	1.27585	0.637926	−	0.770098i	$-0.279793\pi$
$503$	3.64156e9	1.27585	0.637926	+	0.770098i	$0.279793\pi$
$504$	1.80182e9	0.626908
$505$	0	0
$506$	4.04128e9	1.38673
$507$	−6.40914e9	−2.18410
$508$	4.21076e9	1.42508
$509$	−1.51887e9	−0.510516	−0.255258	−	0.966873i	$-0.582160\pi$
$509$	−1.51887e9	−0.510516	−0.255258	+	0.966873i	$0.582160\pi$
$510$	0	0
$511$	−6.14233e8	−0.203639
$512$	6.16621e9	2.03036
$513$	−3.65816e8	−0.119633
$514$	−3.94404e9	−1.28106
$515$	0	0
$516$	6.97532e9	2.23507
$517$	−5.32576e9	−1.69498
$518$	1.01444e9	0.320679
$519$	1.00127e9	0.314388
$520$	0	0
$521$	−2.16090e9	−0.669427	−0.334714	−	0.942320i	$-0.608640\pi$
$521$	−2.16090e9	−0.669427	−0.334714	+	0.942320i	$0.608640\pi$
$522$	−5.26571e9	−1.62035
$523$	1.50945e9	0.461384	0.230692	−	0.973027i	$-0.425901\pi$
$523$	1.50945e9	0.461384	0.230692	+	0.973027i	$0.425901\pi$
$524$	1.35009e10	4.09924
$525$	0	0
$526$	−1.10127e10	−3.29946
$527$	−7.03916e9	−2.09500
$528$	−7.74168e9	−2.28885
$529$	−2.28244e9	−0.670353
$530$	0	0
$531$	−3.44272e9	−0.997863
$532$	1.01622e9	0.292615
$533$	−1.04306e10	−2.98377
$534$	5.19502e9	1.47636
$535$	0	0
$536$	−1.48433e9	−0.416344
$537$	6.80597e8	0.189662
$538$	4.23325e9	1.17202
$539$	−3.86462e9	−1.06303
$540$	0	0
$541$	3.70352e9	1.00560	0.502799	−	0.864403i	$-0.332303\pi$
$541$	3.70352e9	1.00560	0.502799	+	0.864403i	$0.332303\pi$
$542$	4.38143e9	1.18200
$543$	6.22886e9	1.66959
$544$	−2.01683e9	−0.537123
$545$	0	0
$546$	6.76382e9	1.77835
$547$	2.12472e9	0.555067	0.277534	−	0.960716i	$-0.410483\pi$
$547$	2.12472e9	0.555067	0.277534	+	0.960716i	$0.410483\pi$
$548$	8.66585e9	2.24946
$549$	−2.25749e9	−0.582267
$550$	0	0
$551$	−1.54989e9	−0.394702
$552$	−5.65546e9	−1.43114
$553$	2.15993e8	0.0543127
$554$	−1.07734e10	−2.69197
$555$	0	0
$556$	7.31664e9	1.80530
$557$	5.35048e9	1.31190	0.655949	−	0.754806i	$-0.272269\pi$
$557$	5.35048e9	1.31190	0.655949	+	0.754806i	$0.272269\pi$
$558$	6.50575e9	1.58518
$559$	5.56447e9	1.34736
$560$	0	0
$561$	−1.19096e10	−2.84792
$562$	4.43709e9	1.05444
$563$	−5.22504e9	−1.23399	−0.616993	−	0.786968i	$-0.711649\pi$
$563$	−5.22504e9	−1.23399	−0.616993	+	0.786968i	$0.711649\pi$
$564$	1.42812e10	3.35187
$565$	0	0
$566$	7.55782e9	1.75202
$567$	2.50800e9	0.577812
$568$	6.35211e8	0.145445
$569$	6.85964e9	1.56102	0.780510	−	0.625144i	$-0.214960\pi$
$569$	6.85964e9	1.56102	0.780510	+	0.625144i	$0.214960\pi$
$570$	0	0
$571$	−4.95971e9	−1.11488	−0.557442	−	0.830216i	$-0.688217\pi$
$571$	−4.95971e9	−1.11488	−0.557442	+	0.830216i	$0.688217\pi$
$572$	−2.10584e10	−4.70478
$573$	−2.08899e9	−0.463869
$574$	6.90292e9	1.52350
$575$	0	0
$576$	−2.17776e9	−0.474823
$577$	6.59694e9	1.42964	0.714821	−	0.699308i	$-0.246508\pi$
$577$	6.59694e9	1.42964	0.714821	+	0.699308i	$0.246508\pi$
$578$	−1.26355e10	−2.72174
$579$	7.26976e9	1.55649
$580$	0	0
$581$	−1.86213e9	−0.393907
$582$	−1.61281e10	−3.39119
$583$	−2.66329e9	−0.556646
$584$	−3.95573e9	−0.821829
$585$	0	0
$586$	−8.29022e9	−1.70186
$587$	2.88898e9	0.589537	0.294768	−	0.955569i	$-0.404757\pi$
$587$	2.88898e9	0.589537	0.294768	+	0.955569i	$0.404757\pi$
$588$	1.03631e10	2.10217
$589$	1.91487e9	0.386133
$590$	0	0
$591$	−2.64171e9	−0.526415
$592$	2.48376e9	0.492021
$593$	−4.28692e9	−0.844217	−0.422108	−	0.906545i	$-0.638710\pi$
$593$	−4.28692e9	−0.844217	−0.422108	+	0.906545i	$0.638710\pi$
$594$	−5.01675e9	−0.982133
$595$	0	0
$596$	−4.42375e9	−0.855911
$597$	−4.06866e9	−0.782602
$598$	−8.64492e9	−1.65313
$599$	4.38110e9	0.832893	0.416447	−	0.909160i	$-0.363275\pi$
$599$	4.38110e9	0.832893	0.416447	+	0.909160i	$0.363275\pi$
$600$	0	0
$601$	8.13846e7	0.0152926	0.00764630	−	0.999971i	$-0.497566\pi$
$601$	8.13846e7	0.0152926	0.00764630	+	0.999971i	$0.497566\pi$
$602$	−3.68252e9	−0.687951
$603$	8.02342e8	0.149022
$604$	4.76781e8	0.0880420
$605$	0	0
$606$	−1.86075e10	−3.39652
$607$	−1.28101e9	−0.232483	−0.116242	−	0.993221i	$-0.537085\pi$
$607$	−1.28101e9	−0.232483	−0.116242	+	0.993221i	$0.537085\pi$
$608$	5.48642e8	0.0989981
$609$	4.61865e9	0.828618
$610$	0	0
$611$	1.13926e10	2.02060
$612$	1.30044e10	2.29330
$613$	5.19321e9	0.910593	0.455297	−	0.890340i	$-0.349533\pi$
$613$	5.19321e9	0.910593	0.455297	+	0.890340i	$0.349533\pi$
$614$	−9.07445e9	−1.58209
$615$	0	0
$616$	7.27300e9	1.25366
$617$	−4.11690e8	−0.0705622	−0.0352811	−	0.999377i	$-0.511233\pi$
$617$	−4.11690e8	−0.0705622	−0.0352811	+	0.999377i	$0.511233\pi$
$618$	−7.41964e9	−1.26451
$619$	9.86123e9	1.67114	0.835572	−	0.549381i	$-0.185136\pi$
$619$	9.86123e9	1.67114	0.835572	+	0.549381i	$0.185136\pi$
$620$	0	0
$621$	−1.39331e9	−0.233468
$622$	−6.18984e9	−1.03137
$623$	−1.85548e9	−0.307432
$624$	1.65606e10	2.72855
$625$	0	0
$626$	−1.62810e9	−0.265260
$627$	3.23979e9	0.524905
$628$	8.74226e9	1.40853
$629$	3.82095e9	0.612201
$630$	0	0
$631$	5.03166e9	0.797275	0.398638	−	0.917109i	$-0.369483\pi$
$631$	5.03166e9	0.797275	0.398638	+	0.917109i	$0.369483\pi$
$632$	1.39102e9	0.219191
$633$	−2.15930e9	−0.338376
$634$	−1.11815e10	−1.74257
$635$	0	0
$636$	7.14170e9	1.10078
$637$	8.26701e9	1.26724
$638$	−2.12549e10	−3.24031
$639$	−3.43359e8	−0.0520590
$640$	0	0
$641$	−4.03624e9	−0.605304	−0.302652	−	0.953101i	$-0.597872\pi$
$641$	−4.03624e9	−0.605304	−0.302652	+	0.953101i	$0.597872\pi$
$642$	−3.36118e9	−0.501324
$643$	7.33702e8	0.108838	0.0544191	−	0.998518i	$-0.482669\pi$
$643$	7.33702e8	0.108838	0.0544191	+	0.998518i	$0.482669\pi$
$644$	3.87054e9	0.571046
$645$	0	0
$646$	5.65778e9	0.825718
$647$	−1.10518e10	−1.60424	−0.802119	−	0.597165i	$-0.796294\pi$
$647$	−1.10518e10	−1.60424	−0.802119	+	0.597165i	$0.796294\pi$
$648$	1.61518e10	2.33189
$649$	−1.38965e10	−1.99548
$650$	0	0
$651$	−5.70631e9	−0.810629
$652$	5.05410e9	0.714130
$653$	6.10895e9	0.858559	0.429280	−	0.903172i	$-0.358767\pi$
$653$	6.10895e9	0.858559	0.429280	+	0.903172i	$0.358767\pi$
$654$	2.42786e10	3.39392
$655$	0	0
$656$	1.69012e10	2.33751
$657$	2.13824e9	0.294156
$658$	−7.53955e9	−1.03170
$659$	−1.02963e10	−1.40146	−0.700730	−	0.713427i	$-0.747142\pi$
$659$	−1.02963e10	−1.40146	−0.700730	+	0.713427i	$0.747142\pi$
$660$	0	0
$661$	−1.56934e9	−0.211354	−0.105677	−	0.994400i	$-0.533701\pi$
$661$	−1.56934e9	−0.211354	−0.105677	+	0.994400i	$0.533701\pi$
$662$	1.56853e10	2.10131
$663$	2.54765e10	3.39502
$664$	−1.19923e10	−1.58970
$665$	0	0
$666$	−3.53141e9	−0.463221
$667$	−5.90315e9	−0.770271
$668$	1.29579e10	1.68197
$669$	8.62582e9	1.11380
$670$	0	0
$671$	−9.11230e9	−1.16439
$672$	−1.63495e9	−0.207832
$673$	−1.25997e10	−1.59334	−0.796671	−	0.604413i	$-0.793408\pi$
$673$	−1.25997e10	−1.59334	−0.796671	+	0.604413i	$0.793408\pi$
$674$	8.16243e9	1.02686
$675$	0	0
$676$	2.82486e10	3.51709
$677$	−4.86676e9	−0.602809	−0.301405	−	0.953496i	$-0.597455\pi$
$677$	−4.86676e9	−0.602809	−0.301405	+	0.953496i	$0.597455\pi$
$678$	2.88154e10	3.55075
$679$	5.76039e9	0.706167
$680$	0	0
$681$	1.50979e10	1.83191
$682$	2.62603e10	3.16996
$683$	−1.12495e9	−0.135102	−0.0675511	−	0.997716i	$-0.521519\pi$
$683$	−1.12495e9	−0.135102	−0.0675511	+	0.997716i	$0.521519\pi$
$684$	−3.53762e9	−0.422683
$685$	0	0
$686$	−1.25408e10	−1.48318
$687$	−1.70478e10	−2.00595
$688$	−9.01634e9	−1.05553
$689$	5.69719e9	0.663580
$690$	0	0
$691$	−8.16337e9	−0.941232	−0.470616	−	0.882338i	$-0.655968\pi$
$691$	−8.16337e9	−0.941232	−0.470616	+	0.882338i	$0.655968\pi$
$692$	−4.41314e9	−0.506264
$693$	−3.93137e9	−0.448722
$694$	−2.81027e10	−3.19146
$695$	0	0
$696$	2.97446e10	3.34407
$697$	2.60004e10	2.90847
$698$	−3.75928e7	−0.00418419
$699$	−3.02135e9	−0.334603
$700$	0	0
$701$	−7.21526e9	−0.791114	−0.395557	−	0.918441i	$-0.629448\pi$
$701$	−7.21526e9	−0.791114	−0.395557	+	0.918441i	$0.629448\pi$
$702$	1.07316e10	1.17081
$703$	−1.03942e9	−0.112836
$704$	−8.79049e9	−0.949531
$705$	0	0
$706$	2.18984e10	2.34205
$707$	6.64597e9	0.707278
$708$	3.72638e10	3.94612
$709$	7.47050e8	0.0787205	0.0393603	−	0.999225i	$-0.487468\pi$
$709$	7.47050e8	0.0787205	0.0393603	+	0.999225i	$0.487468\pi$
$710$	0	0
$711$	−7.51905e8	−0.0784548
$712$	−1.19495e10	−1.24071
$713$	7.29331e9	0.753548
$714$	−1.68601e10	−1.73347
$715$	0	0
$716$	−2.99976e9	−0.305415
$717$	6.53038e9	0.661640
$718$	8.54615e9	0.861658
$719$	8.04093e9	0.806780	0.403390	−	0.915028i	$-0.367832\pi$
$719$	8.04093e9	0.806780	0.403390	+	0.915028i	$0.367832\pi$
$720$	0	0
$721$	2.65004e9	0.263317
$722$	1.62423e10	1.60608
$723$	−1.16410e10	−1.14553
$724$	−2.74540e10	−2.68856
$725$	0	0
$726$	2.08844e10	2.02555
$727$	1.94846e9	0.188070	0.0940352	−	0.995569i	$-0.470023\pi$
$727$	1.94846e9	0.188070	0.0940352	+	0.995569i	$0.470023\pi$
$728$	−1.55581e10	−1.49450
$729$	−3.20619e9	−0.306509
$730$	0	0
$731$	−1.38705e10	−1.31335
$732$	2.44349e10	2.30262
$733$	1.62129e10	1.52053	0.760267	−	0.649611i	$-0.225068\pi$
$733$	1.62129e10	1.52053	0.760267	+	0.649611i	$0.225068\pi$
$734$	6.07920e9	0.567428
$735$	0	0
$736$	2.08965e9	0.193197
$737$	3.23864e9	0.298007
$738$	−2.40301e10	−2.20069
$739$	−2.14807e10	−1.95791	−0.978954	−	0.204081i	$-0.934579\pi$
$739$	−2.14807e10	−1.95791	−0.978954	+	0.204081i	$0.934579\pi$
$740$	0	0
$741$	−6.93041e9	−0.625742
$742$	−3.77036e9	−0.338820
$743$	−1.74000e10	−1.55628	−0.778142	−	0.628089i	$-0.783837\pi$
$743$	−1.74000e10	−1.55628	−0.778142	+	0.628089i	$0.783837\pi$
$744$	−3.67493e10	−3.27147
$745$	0	0
$746$	−3.00324e10	−2.64853
$747$	6.48238e9	0.569000
$748$	5.24921e10	4.58605
$749$	1.20050e9	0.104394
$750$	0	0
$751$	−1.49074e10	−1.28429	−0.642143	−	0.766585i	$-0.721955\pi$
$751$	−1.49074e10	−1.28429	−0.642143	+	0.766585i	$0.721955\pi$
$752$	−1.84600e10	−1.58295
$753$	1.97277e10	1.68382
$754$	4.54675e10	3.86279
$755$	0	0
$756$	−4.80480e9	−0.404435
$757$	−1.76732e10	−1.48074	−0.740372	−	0.672197i	$-0.765351\pi$
$757$	−1.76732e10	−1.48074	−0.740372	+	0.672197i	$0.765351\pi$
$758$	2.10304e10	1.75390
$759$	1.23396e10	1.02437
$760$	0	0
$761$	9.01678e9	0.741660	0.370830	−	0.928701i	$-0.379073\pi$
$761$	9.01678e9	0.741660	0.370830	+	0.928701i	$0.379073\pi$
$762$	1.90044e10	1.55601
$763$	−8.67149e9	−0.706737
$764$	9.20732e9	0.746976
$765$	0	0
$766$	−8.68288e8	−0.0698012
$767$	2.97267e10	2.37882
$768$	3.32185e10	2.64616
$769$	−1.63961e10	−1.30017	−0.650084	−	0.759862i	$-0.725266\pi$
$769$	−1.63961e10	−1.30017	−0.650084	+	0.759862i	$0.725266\pi$
$770$	0	0
$771$	−1.20427e10	−0.946310
$772$	−3.20418e10	−2.50643
$773$	−1.70561e10	−1.32816	−0.664081	−	0.747660i	$-0.731177\pi$
$773$	−1.70561e10	−1.32816	−0.664081	+	0.747660i	$0.731177\pi$
$774$	1.28194e10	0.993747
$775$	0	0
$776$	3.70975e10	2.84990
$777$	3.09746e9	0.236882
$778$	5.79448e8	0.0441150
$779$	−7.07293e9	−0.536066
$780$	0	0
$781$	−1.38596e9	−0.104105
$782$	2.15491e10	1.61141
$783$	7.32803e9	0.545533
$784$	−1.33954e10	−0.992771
$785$	0	0
$786$	6.09335e10	4.47587
$787$	−4.74568e9	−0.347046	−0.173523	−	0.984830i	$-0.555515\pi$
$787$	−4.74568e9	−0.347046	−0.173523	+	0.984830i	$0.555515\pi$
$788$	1.16434e10	0.847695
$789$	−3.36259e10	−2.43728
$790$	0	0
$791$	−1.02919e10	−0.739395
$792$	−2.53185e10	−1.81092
$793$	1.94926e10	1.38808
$794$	3.87194e10	2.74509
$795$	0	0
$796$	1.79328e10	1.26024
$797$	−1.85509e10	−1.29796	−0.648981	−	0.760805i	$-0.724804\pi$
$797$	−1.85509e10	−1.29796	−0.648981	+	0.760805i	$0.724804\pi$
$798$	4.58649e9	0.319500
$799$	−2.83983e10	−1.96960
$800$	0	0
$801$	6.45923e9	0.444086
$802$	−3.97831e10	−2.72326
$803$	8.63096e9	0.588240
$804$	−8.68450e9	−0.589316
$805$	0	0
$806$	−5.61748e10	−3.77893
$807$	1.29258e10	0.865762
$808$	4.28007e10	2.85438
$809$	−2.94976e9	−0.195870	−0.0979349	−	0.995193i	$-0.531224\pi$
$809$	−2.94976e9	−0.195870	−0.0979349	+	0.995193i	$0.531224\pi$
$810$	0	0
$811$	−8.21711e9	−0.540936	−0.270468	−	0.962729i	$-0.587178\pi$
$811$	−8.21711e9	−0.540936	−0.270468	+	0.962729i	$0.587178\pi$
$812$	−2.03569e10	−1.33434
$813$	1.33782e10	0.873135
$814$	−1.42545e10	−0.926329
$815$	0	0
$816$	−4.12806e10	−2.65969
$817$	3.77322e9	0.242067
$818$	−3.71876e10	−2.37554
$819$	8.40980e9	0.534924
$820$	0	0
$821$	1.92868e10	1.21635	0.608177	−	0.793801i	$-0.291901\pi$
$821$	1.92868e10	1.21635	0.608177	+	0.793801i	$0.291901\pi$
$822$	3.91115e10	2.45614
$823$	−2.23579e10	−1.39808	−0.699038	−	0.715084i	$-0.746388\pi$
$823$	−2.23579e10	−1.39808	−0.699038	+	0.715084i	$0.746388\pi$
$824$	1.70666e10	1.06268
$825$	0	0
$826$	−1.96729e10	−1.21461
$827$	2.16073e9	0.132841	0.0664204	−	0.997792i	$-0.478842\pi$
$827$	2.16073e9	0.132841	0.0664204	+	0.997792i	$0.478842\pi$
$828$	−1.34740e10	−0.824877
$829$	−8.57416e9	−0.522698	−0.261349	−	0.965244i	$-0.584167\pi$
$829$	−8.57416e9	−0.522698	−0.261349	+	0.965244i	$0.584167\pi$
$830$	0	0
$831$	−3.28955e10	−1.98853
$832$	1.88042e10	1.13194
$833$	−2.06071e10	−1.23526
$834$	3.30221e10	1.97117
$835$	0	0
$836$	−1.42795e10	−0.845263
$837$	−9.05374e9	−0.533690
$838$	−1.87519e10	−1.10076
$839$	1.40941e10	0.823894	0.411947	−	0.911208i	$-0.364849\pi$
$839$	1.40941e10	0.823894	0.411947	+	0.911208i	$0.364849\pi$
$840$	0	0
$841$	1.37974e10	0.799855
$842$	4.65663e10	2.68831
$843$	1.35481e10	0.778903
$844$	9.51720e9	0.544892
$845$	0	0
$846$	2.62464e10	1.49030
$847$	−7.45919e9	−0.421793
$848$	−9.23140e9	−0.519855
$849$	2.30769e10	1.29420
$850$	0	0
$851$	−3.95891e9	−0.220202
$852$	3.71650e9	0.205871
$853$	−1.77782e10	−0.980767	−0.490384	−	0.871507i	$-0.663143\pi$
$853$	−1.77782e10	−0.980767	−0.490384	+	0.871507i	$0.663143\pi$
$854$	−1.29001e10	−0.708743
$855$	0	0
$856$	7.73134e9	0.421305
$857$	7.15883e9	0.388516	0.194258	−	0.980950i	$-0.437770\pi$
$857$	7.15883e9	0.388516	0.194258	+	0.980950i	$0.437770\pi$
$858$	−9.50426e10	−5.13704
$859$	−1.19425e10	−0.642863	−0.321432	−	0.946933i	$-0.604164\pi$
$859$	−1.19425e10	−0.642863	−0.321432	+	0.946933i	$0.604164\pi$
$860$	0	0
$861$	2.10773e10	1.12539
$862$	−3.98238e10	−2.11772
$863$	2.78861e9	0.147690	0.0738449	−	0.997270i	$-0.476473\pi$
$863$	2.78861e9	0.147690	0.0738449	+	0.997270i	$0.476473\pi$
$864$	−2.59404e9	−0.136829
$865$	0	0
$866$	2.40939e10	1.26065
$867$	−3.85812e10	−2.01052
$868$	2.51508e10	1.30537
$869$	−3.03505e9	−0.156890
$870$	0	0
$871$	−6.92794e9	−0.355255
$872$	−5.58453e10	−2.85219
$873$	−2.00528e10	−1.02006
$874$	−5.86205e9	−0.297002
$875$	0	0
$876$	−2.31442e10	−1.16326
$877$	−4.55128e9	−0.227843	−0.113921	−	0.993490i	$-0.536341\pi$
$877$	−4.55128e9	−0.227843	−0.113921	+	0.993490i	$0.536341\pi$
$878$	−9.33099e9	−0.465261
$879$	−2.53133e10	−1.25715
$880$	0	0
$881$	7.11173e9	0.350396	0.175198	−	0.984533i	$-0.443943\pi$
$881$	7.11173e9	0.350396	0.175198	+	0.984533i	$0.443943\pi$
$882$	1.90456e10	0.934660
$883$	2.16277e9	0.105718	0.0528589	−	0.998602i	$-0.483167\pi$
$883$	2.16277e9	0.105718	0.0528589	+	0.998602i	$0.483167\pi$
$884$	−1.12289e11	−5.46705
$885$	0	0
$886$	2.61935e9	0.126525
$887$	3.66524e10	1.76348	0.881738	−	0.471740i	$-0.156374\pi$
$887$	3.66524e10	1.76348	0.881738	+	0.471740i	$0.156374\pi$
$888$	1.99480e10	0.955992
$889$	−6.78772e9	−0.324017
$890$	0	0
$891$	−3.52414e10	−1.66910
$892$	−3.80187e10	−1.79358
$893$	7.72524e9	0.363021
$894$	−1.99657e10	−0.934550
$895$	0	0
$896$	−1.58899e10	−0.737978
$897$	−2.63963e10	−1.22115
$898$	4.82915e10	2.22538
$899$	−3.83587e10	−1.76078
$900$	0	0
$901$	−1.42013e10	−0.646834
$902$	−9.69971e10	−4.40084
$903$	−1.12442e10	−0.508183
$904$	−6.62807e10	−2.98399
$905$	0	0
$906$	2.15185e9	0.0961310
$907$	−2.34244e10	−1.04242	−0.521209	−	0.853429i	$-0.674519\pi$
$907$	−2.34244e10	−1.04242	−0.521209	+	0.853429i	$0.674519\pi$
$908$	−6.65449e10	−2.94995
$909$	−2.31357e10	−1.02166
$910$	0	0
$911$	−1.22849e10	−0.538341	−0.269170	−	0.963093i	$-0.586749\pi$
$911$	−1.22849e10	−0.538341	−0.269170	+	0.963093i	$0.586749\pi$
$912$	1.12296e10	0.490212
$913$	2.61660e10	1.13786
$914$	5.12097e9	0.221841
$915$	0	0
$916$	7.51391e10	3.23022
$917$	−2.17634e10	−0.932037
$918$	−2.67506e10	−1.14126
$919$	−1.83306e10	−0.779061	−0.389531	−	0.921014i	$-0.627363\pi$
$919$	−1.83306e10	−0.779061	−0.389531	+	0.921014i	$0.627363\pi$
$920$	0	0
$921$	−2.77078e10	−1.16867
$922$	4.79522e10	2.01488
$923$	2.96478e9	0.124104
$924$	4.25529e10	1.77450
$925$	0	0
$926$	−2.12929e8	−0.00881246
$927$	−9.22521e9	−0.380362
$928$	−1.09904e10	−0.451435
$929$	3.57153e10	1.46150	0.730750	−	0.682645i	$-0.239171\pi$
$929$	3.57153e10	1.46150	0.730750	+	0.682645i	$0.239171\pi$
$930$	0	0
$931$	5.60579e9	0.227674
$932$	1.33167e10	0.538817
$933$	−1.89000e10	−0.761861
$934$	1.86572e9	0.0749260
$935$	0	0
$936$	5.41600e10	2.15881
$937$	−1.24895e10	−0.495972	−0.247986	−	0.968764i	$-0.579769\pi$
$937$	−1.24895e10	−0.495972	−0.247986	+	0.968764i	$0.579769\pi$
$938$	4.58486e9	0.181391
$939$	−4.97123e9	−0.195945
$940$	0	0
$941$	6.63793e9	0.259698	0.129849	−	0.991534i	$-0.458551\pi$
$941$	6.63793e9	0.259698	0.129849	+	0.991534i	$0.458551\pi$
$942$	3.94563e10	1.53794
$943$	−2.69391e10	−1.04615
$944$	−4.81674e10	−1.86359
$945$	0	0
$946$	5.17454e10	1.98725
$947$	−2.77880e10	−1.06324	−0.531621	−	0.846982i	$-0.678417\pi$
$947$	−2.77880e10	−1.06324	−0.531621	+	0.846982i	$0.678417\pi$
$948$	8.13857e9	0.310255
$949$	−1.84630e10	−0.701244
$950$	0	0
$951$	−3.41416e10	−1.28722
$952$	3.87815e10	1.45678
$953$	−1.34702e10	−0.504137	−0.252069	−	0.967709i	$-0.581111\pi$
$953$	−1.34702e10	−0.504137	−0.252069	+	0.967709i	$0.581111\pi$
$954$	1.31252e10	0.489426
$955$	0	0
$956$	−2.87829e10	−1.06545
$957$	−6.48994e10	−2.39359
$958$	2.90051e10	1.06585
$959$	−1.39693e10	−0.511457
$960$	0	0
$961$	1.98794e10	0.722555
$962$	3.04925e10	1.10428
$963$	−4.17912e9	−0.150797
$964$	5.13081e10	1.84466
$965$	0	0
$966$	1.74689e10	0.623512
$967$	3.87456e9	0.137794	0.0688969	−	0.997624i	$-0.478052\pi$
$967$	3.87456e9	0.137794	0.0688969	+	0.997624i	$0.478052\pi$
$968$	−4.80380e10	−1.70224
$969$	1.72754e10	0.609950
$970$	0	0
$971$	2.05915e10	0.721806	0.360903	−	0.932603i	$-0.382469\pi$
$971$	2.05915e10	0.721806	0.360903	+	0.932603i	$0.382469\pi$
$972$	7.01512e10	2.45021
$973$	−1.17944e10	−0.410468
$974$	5.24195e10	1.81776
$975$	0	0
$976$	−3.15847e10	−1.08743
$977$	1.16885e10	0.400984	0.200492	−	0.979695i	$-0.435746\pi$
$977$	1.16885e10	0.400984	0.200492	+	0.979695i	$0.435746\pi$
$978$	2.28106e10	0.779742
$979$	2.60725e10	0.888063
$980$	0	0
$981$	3.01868e10	1.02088
$982$	−1.14877e10	−0.387117
$983$	1.57945e9	0.0530356	0.0265178	−	0.999648i	$-0.491558\pi$
$983$	1.57945e9	0.0530356	0.0265178	+	0.999648i	$0.491558\pi$
$984$	1.35740e11	4.54177
$985$	0	0
$986$	−1.13336e11	−3.76531
$987$	−2.30212e10	−0.762110
$988$	3.05461e10	1.00764
$989$	1.43713e10	0.472399
$990$	0	0
$991$	−4.08499e10	−1.33332	−0.666658	−	0.745363i	$-0.732276\pi$
$991$	−4.08499e10	−1.33332	−0.666658	+	0.745363i	$0.732276\pi$
$992$	1.35786e10	0.441635
$993$	4.78933e10	1.55222
$994$	−1.96207e9	−0.0633669
$995$	0	0
$996$	−7.01648e10	−2.25015
$997$	7.71156e9	0.246439	0.123219	−	0.992379i	$-0.460678\pi$
$997$	7.71156e9	0.246439	0.123219	+	0.992379i	$0.460678\pi$
$998$	−3.47971e9	−0.110812
$999$	4.91449e9	0.155955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.8.a.e.1.5	✓	48
5.4	even	2		625.8.a.f.1.44	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.8.a.e.1.5	✓	48	1.1	even	1	trivial
625.8.a.f.1.44	yes	48	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)

\(f(q)\)	\(=\)	\(q-19.8925 q^{2} -60.7396 q^{3} +267.712 q^{4} +1208.26 q^{6} -431.550 q^{7} -2779.23 q^{8} +1502.29 q^{9} +O(q^{10})\)	\(q-19.8925 q^{2} -60.7396 q^{3} +267.712 q^{4} +1208.26 q^{6} -431.550 q^{7} -2779.23 q^{8} +1502.29 q^{9} +6063.98 q^{11} -16260.7 q^{12} -12971.8 q^{13} +8584.62 q^{14} +21018.7 q^{16} +32334.7 q^{17} -29884.4 q^{18} -8796.05 q^{19} +26212.2 q^{21} -120628. q^{22} -33502.1 q^{23} +168809. q^{24} +258041. q^{26} +41588.7 q^{27} -115531. q^{28} +176202. q^{29} -217697. q^{31} -62373.7 q^{32} -368323. q^{33} -643218. q^{34} +402183. q^{36} +118169. q^{37} +174976. q^{38} +787900. q^{39} +804103. q^{41} -521426. q^{42} -428967. q^{43} +1.62340e6 q^{44} +666441. q^{46} -878263. q^{47} -1.27667e6 q^{48} -637307. q^{49} -1.96399e6 q^{51} -3.47270e6 q^{52} -439199. q^{53} -827304. q^{54} +1.19938e6 q^{56} +534268. q^{57} -3.50511e6 q^{58} -2.29164e6 q^{59} -1.50269e6 q^{61} +4.33054e6 q^{62} -648316. q^{63} -1.44963e6 q^{64} +7.32688e6 q^{66} +534078. q^{67} +8.65638e6 q^{68} +2.03490e6 q^{69} -228556. q^{71} -4.17522e6 q^{72} +1.42332e6 q^{73} -2.35068e6 q^{74} -2.35481e6 q^{76} -2.61691e6 q^{77} -1.56733e7 q^{78} -500505. q^{79} -5.81160e6 q^{81} -1.59956e7 q^{82} +4.31498e6 q^{83} +7.01732e6 q^{84} +8.53324e6 q^{86} -1.07025e7 q^{87} -1.68532e7 q^{88} +4.29958e6 q^{89} +5.59797e6 q^{91} -8.96892e6 q^{92} +1.32228e7 q^{93} +1.74709e7 q^{94} +3.78855e6 q^{96} -1.33481e7 q^{97} +1.26776e7 q^{98} +9.10988e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9}+O(q^{10}) \) 48 * q - 25 * q^2 - 95 * q^3 + 3419 * q^4 + 431 * q^6 - 4030 * q^7 - 3375 * q^8 + 36231 * q^9	\( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9} + 781 q^{11} - 3925 q^{12} - 4290 q^{13} - 20762 q^{14} + 270603 q^{16} - 75075 q^{17} - 89950 q^{18} + 17750 q^{19} - 48034 q^{21} - 331305 q^{22} - 343890 q^{23} - 271570 q^{24} - 304129 q^{26} - 474740 q^{27} - 1146535 q^{28} - 59330 q^{29} - 385989 q^{31} - 1887300 q^{32} - 879805 q^{33} + 286938 q^{34} + 3553198 q^{36} - 935610 q^{37} - 984745 q^{38} - 294888 q^{39} + 160466 q^{41} + 783725 q^{42} + 146400 q^{43} + 2261658 q^{44} - 2639009 q^{46} - 4446810 q^{47} - 3994240 q^{48} + 7532484 q^{49} - 2294894 q^{51} - 4582065 q^{52} - 3977030 q^{53} - 3979475 q^{54} - 743430 q^{56} - 2455430 q^{57} - 14413560 q^{58} - 1614425 q^{59} + 7720866 q^{61} - 20362850 q^{62} - 26297840 q^{63} + 21801809 q^{64} + 945327 q^{66} - 3017910 q^{67} - 17494265 q^{68} - 13519553 q^{69} - 9483549 q^{71} - 21929370 q^{72} + 388070 q^{73} + 16144878 q^{74} - 13507955 q^{76} - 25473115 q^{77} + 3108110 q^{78} - 10950620 q^{79} + 34443488 q^{81} + 354040 q^{82} - 47217920 q^{83} - 27843102 q^{84} + 20021766 q^{86} - 56120960 q^{87} - 54397660 q^{88} + 10850545 q^{89} - 8553794 q^{91} + 20734425 q^{92} + 11206260 q^{93} - 52545997 q^{94} - 28125034 q^{96} - 32784020 q^{97} - 14131170 q^{98} + 27513602 q^{99}+O(q^{100}) \) 48 * q - 25 * q^2 - 95 * q^3 + 3419 * q^4 + 431 * q^6 - 4030 * q^7 - 3375 * q^8 + 36231 * q^9 + 781 * q^11 - 3925 * q^12 - 4290 * q^13 - 20762 * q^14 + 270603 * q^16 - 75075 * q^17 - 89950 * q^18 + 17750 * q^19 - 48034 * q^21 - 331305 * q^22 - 343890 * q^23 - 271570 * q^24 - 304129 * q^26 - 474740 * q^27 - 1146535 * q^28 - 59330 * q^29 - 385989 * q^31 - 1887300 * q^32 - 879805 * q^33 + 286938 * q^34 + 3553198 * q^36 - 935610 * q^37 - 984745 * q^38 - 294888 * q^39 + 160466 * q^41 + 783725 * q^42 + 146400 * q^43 + 2261658 * q^44 - 2639009 * q^46 - 4446810 * q^47 - 3994240 * q^48 + 7532484 * q^49 - 2294894 * q^51 - 4582065 * q^52 - 3977030 * q^53 - 3979475 * q^54 - 743430 * q^56 - 2455430 * q^57 - 14413560 * q^58 - 1614425 * q^59 + 7720866 * q^61 - 20362850 * q^62 - 26297840 * q^63 + 21801809 * q^64 + 945327 * q^66 - 3017910 * q^67 - 17494265 * q^68 - 13519553 * q^69 - 9483549 * q^71 - 21929370 * q^72 + 388070 * q^73 + 16144878 * q^74 - 13507955 * q^76 - 25473115 * q^77 + 3108110 * q^78 - 10950620 * q^79 + 34443488 * q^81 + 354040 * q^82 - 47217920 * q^83 - 27843102 * q^84 + 20021766 * q^86 - 56120960 * q^87 - 54397660 * q^88 + 10850545 * q^89 - 8553794 * q^91 + 20734425 * q^92 + 11206260 * q^93 - 52545997 * q^94 - 28125034 * q^96 - 32784020 * q^97 - 14131170 * q^98 + 27513602 * q^99

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