LMFDB - Embedded newform 625.8.a.d.1.3

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:	$0$
Dimension:	$34$
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-18.1517 q^{2} -84.5499 q^{3} +201.484 q^{4} +1534.72 q^{6} +935.944 q^{7} -1333.85 q^{8} +4961.69 q^{9} +O(q^{10})$	\(q-18.1517 q^{2} -84.5499 q^{3} +201.484 q^{4} +1534.72 q^{6} +935.944 q^{7} -1333.85 q^{8} +4961.69 q^{9} +4135.69 q^{11} -17035.4 q^{12} -8419.28 q^{13} -16988.9 q^{14} -1578.28 q^{16} -18752.8 q^{17} -90063.0 q^{18} -20736.6 q^{19} -79133.9 q^{21} -75069.7 q^{22} -8185.47 q^{23} +112777. q^{24} +152824. q^{26} -234600. q^{27} +188577. q^{28} -91508.4 q^{29} +299897. q^{31} +199381. q^{32} -349672. q^{33} +340395. q^{34} +999698. q^{36} -163912. q^{37} +376403. q^{38} +711849. q^{39} +74148.7 q^{41} +1.43641e6 q^{42} -190254. q^{43} +833273. q^{44} +148580. q^{46} -1.33034e6 q^{47} +133443. q^{48} +52447.3 q^{49} +1.58555e6 q^{51} -1.69635e6 q^{52} +1.34358e6 q^{53} +4.25838e6 q^{54} -1.24841e6 q^{56} +1.75327e6 q^{57} +1.66103e6 q^{58} -1.19044e6 q^{59} +894528. q^{61} -5.44364e6 q^{62} +4.64386e6 q^{63} -3.41708e6 q^{64} +6.34713e6 q^{66} -1.45903e6 q^{67} -3.77838e6 q^{68} +692080. q^{69} +486817. q^{71} -6.61815e6 q^{72} +674373. q^{73} +2.97528e6 q^{74} -4.17808e6 q^{76} +3.87077e6 q^{77} -1.29213e7 q^{78} -3.28279e6 q^{79} +8.98416e6 q^{81} -1.34592e6 q^{82} +4.80072e6 q^{83} -1.59442e7 q^{84} +3.45343e6 q^{86} +7.73703e6 q^{87} -5.51638e6 q^{88} +4.83478e6 q^{89} -7.87997e6 q^{91} -1.64924e6 q^{92} -2.53563e7 q^{93} +2.41479e7 q^{94} -1.68577e7 q^{96} +6.67435e6 q^{97} -952006. q^{98} +2.05200e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 17 q^{2} + 14 q^{3} + 2177 q^{4} + 993 q^{6} + 867 q^{7} + 2700 q^{8} + 23708 q^{9}+O(q^{10}) $ 34 * q + 17 * q^2 + 14 * q^3 + 2177 * q^4 + 993 * q^6 + 867 * q^7 + 2700 * q^8 + 23708 * q^9	\( 34 q + 17 q^{2} + 14 q^{3} + 2177 q^{4} + 993 q^{6} + 867 q^{7} + 2700 q^{8} + 23708 q^{9} + 8718 q^{11} + 2377 q^{12} + 8769 q^{13} + 17674 q^{14} + 139109 q^{16} - 48258 q^{17} + 58104 q^{18} + 61955 q^{19} + 96943 q^{21} - 52556 q^{22} + 55584 q^{23} + 276990 q^{24} + 337623 q^{26} - 24370 q^{27} + 392941 q^{28} + 449815 q^{29} + 450438 q^{31} + 522987 q^{32} - 689282 q^{33} + 639889 q^{34} + 2024094 q^{36} - 373008 q^{37} + 1272835 q^{38} + 877321 q^{39} + 926418 q^{41} + 580554 q^{42} - 1719911 q^{43} + 1809244 q^{44} + 1186623 q^{46} + 640247 q^{47} + 3924149 q^{48} + 3292177 q^{49} + 1961038 q^{51} + 3900212 q^{52} + 1609214 q^{53} + 3802395 q^{54} + 5709425 q^{56} - 7299205 q^{57} - 4001010 q^{58} + 6056265 q^{59} + 2885628 q^{61} - 14351151 q^{62} + 8834194 q^{63} + 12281872 q^{64} + 8819186 q^{66} - 164318 q^{67} - 14549609 q^{68} + 5884976 q^{69} + 10286173 q^{71} + 41997930 q^{72} + 20595894 q^{73} + 7025034 q^{74} + 11251000 q^{76} + 14569234 q^{77} - 32702012 q^{78} + 11772835 q^{79} + 24408674 q^{81} - 38307271 q^{82} + 2455444 q^{83} + 17190739 q^{84} + 7636893 q^{86} - 39372965 q^{87} + 45397760 q^{88} + 17788560 q^{89} + 15865973 q^{91} + 37183957 q^{92} - 59352382 q^{93} + 29349634 q^{94} + 38027693 q^{96} + 3708847 q^{97} + 6000776 q^{98} + 35633116 q^{99}+O(q^{100}) \) 34 * q + 17 * q^2 + 14 * q^3 + 2177 * q^4 + 993 * q^6 + 867 * q^7 + 2700 * q^8 + 23708 * q^9 + 8718 * q^11 + 2377 * q^12 + 8769 * q^13 + 17674 * q^14 + 139109 * q^16 - 48258 * q^17 + 58104 * q^18 + 61955 * q^19 + 96943 * q^21 - 52556 * q^22 + 55584 * q^23 + 276990 * q^24 + 337623 * q^26 - 24370 * q^27 + 392941 * q^28 + 449815 * q^29 + 450438 * q^31 + 522987 * q^32 - 689282 * q^33 + 639889 * q^34 + 2024094 * q^36 - 373008 * q^37 + 1272835 * q^38 + 877321 * q^39 + 926418 * q^41 + 580554 * q^42 - 1719911 * q^43 + 1809244 * q^44 + 1186623 * q^46 + 640247 * q^47 + 3924149 * q^48 + 3292177 * q^49 + 1961038 * q^51 + 3900212 * q^52 + 1609214 * q^53 + 3802395 * q^54 + 5709425 * q^56 - 7299205 * q^57 - 4001010 * q^58 + 6056265 * q^59 + 2885628 * q^61 - 14351151 * q^62 + 8834194 * q^63 + 12281872 * q^64 + 8819186 * q^66 - 164318 * q^67 - 14549609 * q^68 + 5884976 * q^69 + 10286173 * q^71 + 41997930 * q^72 + 20595894 * q^73 + 7025034 * q^74 + 11251000 * q^76 + 14569234 * q^77 - 32702012 * q^78 + 11772835 * q^79 + 24408674 * q^81 - 38307271 * q^82 + 2455444 * q^83 + 17190739 * q^84 + 7636893 * q^86 - 39372965 * q^87 + 45397760 * q^88 + 17788560 * q^89 + 15865973 * q^91 + 37183957 * q^92 - 59352382 * q^93 + 29349634 * q^94 + 38027693 * q^96 + 3708847 * q^97 + 6000776 * q^98 + 35633116 * 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 $

$2$	−18.1517	−1.60440	−0.802199	−	0.597057i	$-0.796336\pi$
$2$	−18.1517	−1.60440	−0.802199	+	0.597057i	$0.796336\pi$
$3$	−84.5499	−1.80796	−0.903980	−	0.427575i	$-0.859368\pi$
$3$	−84.5499	−1.80796	−0.903980	+	0.427575i	$0.859368\pi$
$4$	201.484	1.57409
$5$	0	0
$5$	0	0
$6$	1534.72	2.90069
$7$	935.944	1.03135	0.515676	−	0.856784i	$-0.327541\pi$
$7$	935.944	1.03135	0.515676	+	0.856784i	$0.327541\pi$
$8$	−1333.85	−0.921069
$9$	4961.69	2.26872
$10$	0	0
$11$	4135.69	0.936857	0.468428	−	0.883502i	$-0.344820\pi$
$11$	4135.69	0.936857	0.468428	+	0.883502i	$0.344820\pi$
$12$	−17035.4	−2.84589
$13$	−8419.28	−1.06285	−0.531426	−	0.847104i	$-0.678344\pi$
$13$	−8419.28	−1.06285	−0.531426	+	0.847104i	$0.678344\pi$
$14$	−16988.9	−1.65470
$15$	0	0
$16$	−1578.28	−0.0963303
$17$	−18752.8	−0.925752	−0.462876	−	0.886423i	$-0.653182\pi$
$17$	−18752.8	−0.925752	−0.462876	+	0.886423i	$0.653182\pi$
$18$	−90063.0	−3.63993
$19$	−20736.6	−0.693584	−0.346792	−	0.937942i	$-0.612729\pi$
$19$	−20736.6	−0.693584	−0.346792	+	0.937942i	$0.612729\pi$
$20$	0	0
$21$	−79133.9	−1.86464
$22$	−75069.7	−1.50309
$23$	−8185.47	−0.140280	−0.0701400	−	0.997537i	$-0.522345\pi$
$23$	−8185.47	−0.140280	−0.0701400	+	0.997537i	$0.522345\pi$
$24$	112777.	1.66526
$25$	0	0
$26$	152824.	1.70524
$27$	−234600.	−2.29379
$28$	188577.	1.62344
$29$	−91508.4	−0.696736	−0.348368	−	0.937358i	$-0.613264\pi$
$29$	−91508.4	−0.696736	−0.348368	+	0.937358i	$0.613264\pi$
$30$	0	0
$31$	299897.	1.80804	0.904018	−	0.427495i	$-0.140604\pi$
$31$	299897.	1.80804	0.904018	+	0.427495i	$0.140604\pi$
$32$	199381.	1.07562
$33$	−349672.	−1.69380
$34$	340395.	1.48527
$35$	0	0
$36$	999698.	3.57117
$37$	−163912.	−0.531993	−0.265996	−	0.963974i	$-0.585701\pi$
$37$	−163912.	−0.531993	−0.265996	+	0.963974i	$0.585701\pi$
$38$	376403.	1.11278
$39$	711849.	1.92160
$40$	0	0
$41$	74148.7	0.168020	0.0840098	−	0.996465i	$-0.473227\pi$
$41$	74148.7	0.168020	0.0840098	+	0.996465i	$0.473227\pi$
$42$	1.43641e6	2.99162
$43$	−190254.	−0.364917	−0.182459	−	0.983214i	$-0.558406\pi$
$43$	−190254.	−0.364917	−0.182459	+	0.983214i	$0.558406\pi$
$44$	833273.	1.47470
$45$	0	0
$46$	148580.	0.225065
$47$	−1.33034e6	−1.86905	−0.934523	−	0.355902i	$-0.884174\pi$
$47$	−1.33034e6	−1.86905	−0.934523	+	0.355902i	$0.884174\pi$
$48$	133443.	0.174161
$49$	52447.3	0.0636849
$50$	0	0
$51$	1.58555e6	1.67372
$52$	−1.69635e6	−1.67303
$53$	1.34358e6	1.23965	0.619825	−	0.784740i	$-0.287204\pi$
$53$	1.34358e6	1.23965	0.619825	+	0.784740i	$0.287204\pi$
$54$	4.25838e6	3.68015
$55$	0	0
$56$	−1.24841e6	−0.949945
$57$	1.75327e6	1.25397
$58$	1.66103e6	1.11784
$59$	−1.19044e6	−0.754615	−0.377307	−	0.926088i	$-0.623150\pi$
$59$	−1.19044e6	−0.754615	−0.377307	+	0.926088i	$0.623150\pi$
$60$	0	0
$61$	894528.	0.504591	0.252296	−	0.967650i	$-0.418814\pi$
$61$	894528.	0.504591	0.252296	+	0.967650i	$0.418814\pi$
$62$	−5.44364e6	−2.90081
$63$	4.64386e6	2.33984
$64$	−3.41708e6	−1.62939
$65$	0	0
$66$	6.34713e6	2.71753
$67$	−1.45903e6	−0.592654	−0.296327	−	0.955087i	$-0.595762\pi$
$67$	−1.45903e6	−0.592654	−0.296327	+	0.955087i	$0.595762\pi$
$68$	−3.77838e6	−1.45722
$69$	692080.	0.253621
$70$	0	0
$71$	486817.	0.161421	0.0807107	−	0.996738i	$-0.474281\pi$
$71$	486817.	0.161421	0.0807107	+	0.996738i	$0.474281\pi$
$72$	−6.61815e6	−2.08965
$73$	674373.	0.202894	0.101447	−	0.994841i	$-0.467653\pi$
$73$	674373.	0.202894	0.101447	+	0.994841i	$0.467653\pi$
$74$	2.97528e6	0.853528
$75$	0	0
$76$	−4.17808e6	−1.09176
$77$	3.87077e6	0.966228
$78$	−1.29213e7	−3.08300
$79$	−3.28279e6	−0.749114	−0.374557	−	0.927204i	$-0.622205\pi$
$79$	−3.28279e6	−0.749114	−0.374557	+	0.927204i	$0.622205\pi$
$80$	0	0
$81$	8.98416e6	1.87836
$82$	−1.34592e6	−0.269570
$83$	4.80072e6	0.921581	0.460790	−	0.887509i	$-0.347566\pi$
$83$	4.80072e6	0.921581	0.460790	+	0.887509i	$0.347566\pi$
$84$	−1.59442e7	−2.93511
$85$	0	0
$86$	3.45343e6	0.585472
$87$	7.73703e6	1.25967
$88$	−5.51638e6	−0.862909
$89$	4.83478e6	0.726962	0.363481	−	0.931602i	$-0.381588\pi$
$89$	4.83478e6	0.726962	0.363481	+	0.931602i	$0.381588\pi$
$90$	0	0
$91$	−7.87997e6	−1.09617
$92$	−1.64924e6	−0.220814
$93$	−2.53563e7	−3.26885
$94$	2.41479e7	2.99869
$95$	0	0
$96$	−1.68577e7	−1.94468
$97$	6.67435e6	0.742520	0.371260	−	0.928529i	$-0.378926\pi$
$97$	6.67435e6	0.742520	0.371260	+	0.928529i	$0.378926\pi$
$98$	−952006.	−0.102176
$99$	2.05200e7	2.12546
$100$	0	0
$101$	9.78559e6	0.945066	0.472533	−	0.881313i	$-0.343340\pi$
$101$	9.78559e6	0.945066	0.472533	+	0.881313i	$0.343340\pi$
$102$	−2.87803e7	−2.68532
$103$	−994573.	−0.0896822	−0.0448411	−	0.998994i	$-0.514278\pi$
$103$	−994573.	−0.0896822	−0.0448411	+	0.998994i	$0.514278\pi$
$104$	1.12301e7	0.978961
$105$	0	0
$106$	−2.43883e7	−1.98889
$107$	−2.23938e7	−1.76720	−0.883600	−	0.468243i	$-0.844887\pi$
$107$	−2.23938e7	−1.76720	−0.883600	+	0.468243i	$0.844887\pi$
$108$	−4.72679e7	−3.61063
$109$	6.15081e6	0.454924	0.227462	−	0.973787i	$-0.426957\pi$
$109$	6.15081e6	0.454924	0.227462	+	0.973787i	$0.426957\pi$
$110$	0	0
$111$	1.38588e7	0.961821
$112$	−1.47718e6	−0.0993503
$113$	−6.30823e6	−0.411276	−0.205638	−	0.978628i	$-0.565927\pi$
$113$	−6.30823e6	−0.411276	−0.205638	+	0.978628i	$0.565927\pi$
$114$	−3.18249e7	−2.01187
$115$	0	0
$116$	−1.84374e7	−1.09672
$117$	−4.17738e7	−2.41131
$118$	2.16085e7	1.21070
$119$	−1.75516e7	−0.954775
$120$	0	0
$121$	−2.38327e6	−0.122300
$122$	−1.62372e7	−0.809565
$123$	−6.26927e6	−0.303773
$124$	6.04244e7	2.84601
$125$	0	0
$126$	−8.42938e7	−3.75404
$127$	2.69832e7	1.16891	0.584454	−	0.811427i	$-0.301309\pi$
$127$	2.69832e7	1.16891	0.584454	+	0.811427i	$0.301309\pi$
$128$	3.65050e7	1.53857
$129$	1.60860e7	0.659756
$130$	0	0
$131$	−1.36625e7	−0.530983	−0.265491	−	0.964113i	$-0.585534\pi$
$131$	−1.36625e7	−0.530983	−0.265491	+	0.964113i	$0.585534\pi$
$132$	−7.04531e7	−2.66619
$133$	−1.94082e7	−0.715329
$134$	2.64838e7	0.950852
$135$	0	0
$136$	2.50134e7	0.852681
$137$	2.00484e7	0.666128	0.333064	−	0.942904i	$-0.391917\pi$
$137$	2.00484e7	0.666128	0.333064	+	0.942904i	$0.391917\pi$
$138$	−1.25624e7	−0.406908
$139$	−2.11898e6	−0.0669230	−0.0334615	−	0.999440i	$-0.510653\pi$
$139$	−2.11898e6	−0.0669230	−0.0334615	+	0.999440i	$0.510653\pi$
$140$	0	0
$141$	1.12480e8	3.37916
$142$	−8.83654e6	−0.258984
$143$	−3.48195e7	−0.995741
$144$	−7.83091e6	−0.218546
$145$	0	0
$146$	−1.22410e7	−0.325523
$147$	−4.43441e6	−0.115140
$148$	−3.30256e7	−0.837404
$149$	−1.32162e7	−0.327306	−0.163653	−	0.986518i	$-0.552328\pi$
$149$	−1.32162e7	−0.327306	−0.163653	+	0.986518i	$0.552328\pi$
$150$	0	0
$151$	−5.97267e7	−1.41172	−0.705861	−	0.708351i	$-0.749440\pi$
$151$	−5.97267e7	−1.41172	−0.705861	+	0.708351i	$0.749440\pi$
$152$	2.76595e7	0.638839
$153$	−9.30455e7	−2.10027
$154$	−7.02610e7	−1.55021
$155$	0	0
$156$	1.43426e8	3.02476
$157$	3.94508e7	0.813593	0.406797	−	0.913519i	$-0.366646\pi$
$157$	3.94508e7	0.813593	0.406797	+	0.913519i	$0.366646\pi$
$158$	5.95881e7	1.20188
$159$	−1.13600e8	−2.24124
$160$	0	0
$161$	−7.66113e6	−0.144678
$162$	−1.63078e8	−3.01364
$163$	−4.32585e7	−0.782375	−0.391188	−	0.920311i	$-0.627936\pi$
$163$	−4.32585e7	−0.782375	−0.391188	+	0.920311i	$0.627936\pi$
$164$	1.49397e7	0.264478
$165$	0	0
$166$	−8.71412e7	−1.47858
$167$	2.15827e7	0.358590	0.179295	−	0.983795i	$-0.442618\pi$
$167$	2.15827e7	0.358590	0.179295	+	0.983795i	$0.442618\pi$
$168$	1.05553e8	1.71746
$169$	8.13575e6	0.129656
$170$	0	0
$171$	−1.02888e8	−1.57355
$172$	−3.83331e7	−0.574413
$173$	−2.72963e6	−0.0400814	−0.0200407	−	0.999799i	$-0.506380\pi$
$173$	−2.72963e6	−0.0400814	−0.0200407	+	0.999799i	$0.506380\pi$
$174$	−1.40440e8	−2.02101
$175$	0	0
$176$	−6.52725e6	−0.0902477
$177$	1.00652e8	1.36431
$178$	−8.77593e7	−1.16634
$179$	−1.23495e8	−1.60940	−0.804702	−	0.593679i	$-0.797675\pi$
$179$	−1.23495e8	−1.60940	−0.804702	+	0.593679i	$0.797675\pi$
$180$	0	0
$181$	1.11929e8	1.40303	0.701517	−	0.712653i	$-0.252506\pi$
$181$	1.11929e8	1.40303	0.701517	+	0.712653i	$0.252506\pi$
$182$	1.43035e8	1.75870
$183$	−7.56323e7	−0.912281
$184$	1.09182e7	0.129208
$185$	0	0
$186$	4.60260e8	5.24454
$187$	−7.75556e7	−0.867297
$188$	−2.68042e8	−2.94205
$189$	−2.19572e8	−2.36570
$190$	0	0
$191$	1.76590e8	1.83379	0.916896	−	0.399127i	$-0.130687\pi$
$191$	1.76590e8	1.83379	0.916896	+	0.399127i	$0.130687\pi$
$192$	2.88914e8	2.94588
$193$	−5.00303e7	−0.500937	−0.250468	−	0.968125i	$-0.580585\pi$
$193$	−5.00303e7	−0.500937	−0.250468	+	0.968125i	$0.580585\pi$
$194$	−1.21151e8	−1.19130
$195$	0	0
$196$	1.05673e7	0.100246
$197$	−5.93990e7	−0.553538	−0.276769	−	0.960936i	$-0.589264\pi$
$197$	−5.93990e7	−0.553538	−0.276769	+	0.960936i	$0.589264\pi$
$198$	−3.72472e8	−3.41009
$199$	−1.55951e8	−1.40282	−0.701411	−	0.712757i	$-0.747446\pi$
$199$	−1.55951e8	−1.40282	−0.701411	+	0.712757i	$0.747446\pi$
$200$	0	0
$201$	1.23361e8	1.07149
$202$	−1.77625e8	−1.51626
$203$	−8.56467e7	−0.718579
$204$	3.19462e8	2.63459
$205$	0	0
$206$	1.80532e7	0.143886
$207$	−4.06137e7	−0.318256
$208$	1.32879e7	0.102385
$209$	−8.57599e7	−0.649789
$210$	0	0
$211$	1.28341e8	0.940538	0.470269	−	0.882523i	$-0.344157\pi$
$211$	1.28341e8	0.940538	0.470269	+	0.882523i	$0.344157\pi$
$212$	2.70710e8	1.95132
$213$	−4.11603e7	−0.291843
$214$	4.06486e8	2.83529
$215$	0	0
$216$	3.12921e8	2.11274
$217$	2.80687e8	1.86472
$218$	−1.11647e8	−0.729879
$219$	−5.70182e7	−0.366825
$220$	0	0
$221$	1.57885e8	0.983938
$222$	−2.51560e8	−1.54314
$223$	3.23272e8	1.95210	0.976048	−	0.217554i	$-0.0698079\pi$
$223$	3.23272e8	1.95210	0.976048	+	0.217554i	$0.0698079\pi$
$224$	1.86609e8	1.10934
$225$	0	0
$226$	1.14505e8	0.659849
$227$	1.60193e8	0.908978	0.454489	−	0.890752i	$-0.349822\pi$
$227$	1.60193e8	0.908978	0.454489	+	0.890752i	$0.349822\pi$
$228$	3.53256e8	1.97387
$229$	−2.12550e8	−1.16960	−0.584798	−	0.811179i	$-0.698826\pi$
$229$	−2.12550e8	−1.16960	−0.584798	+	0.811179i	$0.698826\pi$
$230$	0	0
$231$	−3.27273e8	−1.74690
$232$	1.22058e8	0.641741
$233$	8.82526e7	0.457069	0.228534	−	0.973536i	$-0.426607\pi$
$233$	8.82526e7	0.457069	0.228534	+	0.973536i	$0.426607\pi$
$234$	7.58265e8	3.86871
$235$	0	0
$236$	−2.39854e8	−1.18783
$237$	2.77559e8	1.35437
$238$	3.18590e8	1.53184
$239$	1.08075e8	0.512075	0.256037	−	0.966667i	$-0.417583\pi$
$239$	1.08075e8	0.512075	0.256037	+	0.966667i	$0.417583\pi$
$240$	0	0
$241$	−2.87279e8	−1.32204	−0.661020	−	0.750368i	$-0.729876\pi$
$241$	−2.87279e8	−1.32204	−0.661020	+	0.750368i	$0.729876\pi$
$242$	4.32604e7	0.196217
$243$	−2.46541e8	−1.10222
$244$	1.80233e8	0.794272
$245$	0	0
$246$	1.13798e8	0.487372
$247$	1.74587e8	0.737178
$248$	−4.00018e8	−1.66532
$249$	−4.05901e8	−1.66618
$250$	0	0
$251$	−4.53219e7	−0.180905	−0.0904525	−	0.995901i	$-0.528831\pi$
$251$	−4.53219e7	−0.180905	−0.0904525	+	0.995901i	$0.528831\pi$
$252$	9.35661e8	3.68313
$253$	−3.38525e7	−0.131422
$254$	−4.89790e8	−1.87539
$255$	0	0
$256$	−2.25241e8	−0.839088
$257$	3.95610e8	1.45379	0.726895	−	0.686749i	$-0.240963\pi$
$257$	3.95610e8	1.45379	0.726895	+	0.686749i	$0.240963\pi$
$258$	−2.91987e8	−1.05851
$259$	−1.53413e8	−0.548671
$260$	0	0
$261$	−4.54036e8	−1.58070
$262$	2.47997e8	0.851907
$263$	2.05430e8	0.696335	0.348168	−	0.937432i	$-0.386804\pi$
$263$	2.05430e8	0.696335	0.348168	+	0.937432i	$0.386804\pi$
$264$	4.66410e8	1.56011
$265$	0	0
$266$	3.52292e8	1.14767
$267$	−4.08780e8	−1.31432
$268$	−2.93970e8	−0.932891
$269$	1.05681e7	0.0331028	0.0165514	−	0.999863i	$-0.494731\pi$
$269$	1.05681e7	0.0331028	0.0165514	+	0.999863i	$0.494731\pi$
$270$	0	0
$271$	−4.36881e8	−1.33343	−0.666716	−	0.745312i	$-0.732301\pi$
$271$	−4.36881e8	−1.33343	−0.666716	+	0.745312i	$0.732301\pi$
$272$	2.95971e7	0.0891779
$273$	6.66251e8	1.98184
$274$	−3.63912e8	−1.06873
$275$	0	0
$276$	1.39443e8	0.399222
$277$	−3.85316e8	−1.08928	−0.544638	−	0.838671i	$-0.683333\pi$
$277$	−3.85316e8	−1.08928	−0.544638	+	0.838671i	$0.683333\pi$
$278$	3.84631e7	0.107371
$279$	1.48800e9	4.10192
$280$	0	0
$281$	2.67532e8	0.719290	0.359645	−	0.933089i	$-0.382898\pi$
$281$	2.67532e8	0.719290	0.359645	+	0.933089i	$0.382898\pi$
$282$	−2.04170e9	−5.42152
$283$	−1.79888e8	−0.471790	−0.235895	−	0.971779i	$-0.575802\pi$
$283$	−1.79888e8	−0.471790	−0.235895	+	0.971779i	$0.575802\pi$
$284$	9.80855e7	0.254092
$285$	0	0
$286$	6.32032e8	1.59756
$287$	6.93990e7	0.173287
$288$	9.89267e8	2.44028
$289$	−5.86716e7	−0.142983
$290$	0	0
$291$	−5.64316e8	−1.34245
$292$	1.35875e8	0.319374
$293$	−1.53578e8	−0.356691	−0.178346	−	0.983968i	$-0.557075\pi$
$293$	−1.53578e8	−0.356691	−0.178346	+	0.983968i	$0.557075\pi$
$294$	8.04920e7	0.184730
$295$	0	0
$296$	2.18634e8	0.490002
$297$	−9.70230e8	−2.14895
$298$	2.39896e8	0.525129
$299$	6.89157e7	0.149097
$300$	0	0
$301$	−1.78067e8	−0.376358
$302$	1.08414e9	2.26496
$303$	−8.27371e8	−1.70864
$304$	3.27280e7	0.0668131
$305$	0	0
$306$	1.68893e9	3.36967
$307$	5.02185e7	0.0990557	0.0495278	−	0.998773i	$-0.484228\pi$
$307$	5.02185e7	0.0990557	0.0495278	+	0.998773i	$0.484228\pi$
$308$	7.79896e8	1.52093
$309$	8.40910e7	0.162142
$310$	0	0
$311$	−7.27910e8	−1.37220	−0.686098	−	0.727509i	$-0.740678\pi$
$311$	−7.27910e8	−1.37220	−0.686098	+	0.727509i	$0.740678\pi$
$312$	−9.49500e8	−1.76992
$313$	−8.13351e8	−1.49925	−0.749623	−	0.661865i	$-0.769765\pi$
$313$	−8.13351e8	−1.49925	−0.749623	+	0.661865i	$0.769765\pi$
$314$	−7.16099e8	−1.30533
$315$	0	0
$316$	−6.61428e8	−1.17917
$317$	−2.06277e8	−0.363700	−0.181850	−	0.983326i	$-0.558209\pi$
$317$	−2.06277e8	−0.363700	−0.181850	+	0.983326i	$0.558209\pi$
$318$	2.06203e9	3.59583
$319$	−3.78450e8	−0.652741
$320$	0	0
$321$	1.89340e9	3.19502
$322$	1.39062e8	0.232121
$323$	3.88868e8	0.642087
$324$	1.81016e9	2.95672
$325$	0	0
$326$	7.85215e8	1.25524
$327$	−5.20050e8	−0.822485
$328$	−9.89032e7	−0.154758
$329$	−1.24512e9	−1.92764
$330$	0	0
$331$	−5.18179e8	−0.785384	−0.392692	−	0.919670i	$-0.628456\pi$
$331$	−5.18179e8	−0.785384	−0.392692	+	0.919670i	$0.628456\pi$
$332$	9.67267e8	1.45065
$333$	−8.13282e8	−1.20694
$334$	−3.91762e8	−0.575320
$335$	0	0
$336$	1.24895e8	0.179621
$337$	−9.16585e7	−0.130457	−0.0652286	−	0.997870i	$-0.520778\pi$
$337$	−9.16585e7	−0.130457	−0.0652286	+	0.997870i	$0.520778\pi$
$338$	−1.47678e8	−0.208020
$339$	5.33360e8	0.743570
$340$	0	0
$341$	1.24028e9	1.69387
$342$	1.86760e9	2.52459
$343$	−7.21702e8	−0.965670
$344$	2.53770e8	0.336114
$345$	0	0
$346$	4.95474e7	0.0643064
$347$	7.60937e8	0.977677	0.488839	−	0.872374i	$-0.337421\pi$
$347$	7.60937e8	0.977677	0.488839	+	0.872374i	$0.337421\pi$
$348$	1.55888e9	1.98283
$349$	−2.72785e8	−0.343504	−0.171752	−	0.985140i	$-0.554943\pi$
$349$	−2.72785e8	−0.343504	−0.171752	+	0.985140i	$0.554943\pi$
$350$	0	0
$351$	1.97516e9	2.43796
$352$	8.24578e8	1.00770
$353$	1.62649e9	1.96807	0.984036	−	0.177969i	$-0.0569527\pi$
$353$	1.62649e9	1.96807	0.984036	+	0.177969i	$0.0569527\pi$
$354$	−1.82700e9	−2.18890
$355$	0	0
$356$	9.74128e8	1.14430
$357$	1.48398e9	1.72620
$358$	2.24165e9	2.58212
$359$	1.48230e8	0.169086	0.0845428	−	0.996420i	$-0.473057\pi$
$359$	1.48230e8	0.169086	0.0845428	+	0.996420i	$0.473057\pi$
$360$	0	0
$361$	−4.63867e8	−0.518941
$362$	−2.03170e9	−2.25102
$363$	2.01506e8	0.221113
$364$	−1.58768e9	−1.72548
$365$	0	0
$366$	1.37285e9	1.46366
$367$	1.20240e9	1.26975	0.634874	−	0.772616i	$-0.281052\pi$
$367$	1.20240e9	1.26975	0.634874	+	0.772616i	$0.281052\pi$
$368$	1.29189e7	0.0135132
$369$	3.67903e8	0.381189
$370$	0	0
$371$	1.25752e9	1.27851
$372$	−5.10888e9	−5.14547
$373$	−1.07027e9	−1.06785	−0.533927	−	0.845531i	$-0.679284\pi$
$373$	−1.07027e9	−1.06785	−0.533927	+	0.845531i	$0.679284\pi$
$374$	1.40777e9	1.39149
$375$	0	0
$376$	1.77447e9	1.72152
$377$	7.70435e8	0.740528
$378$	3.98560e9	3.79553
$379$	6.71477e8	0.633569	0.316784	−	0.948498i	$-0.397397\pi$
$379$	6.71477e8	0.633569	0.316784	+	0.948498i	$0.397397\pi$
$380$	0	0
$381$	−2.28143e9	−2.11334
$382$	−3.20541e9	−2.94213
$383$	5.70575e8	0.518940	0.259470	−	0.965751i	$-0.416452\pi$
$383$	5.70575e8	0.518940	0.259470	+	0.965751i	$0.416452\pi$
$384$	−3.08650e9	−2.78168
$385$	0	0
$386$	9.08135e8	0.803702
$387$	−9.43981e8	−0.827894
$388$	1.34477e9	1.16879
$389$	−2.09775e9	−1.80688	−0.903440	−	0.428714i	$-0.858967\pi$
$389$	−2.09775e9	−1.80688	−0.903440	+	0.428714i	$0.858967\pi$
$390$	0	0
$391$	1.53500e8	0.129865
$392$	−6.99568e7	−0.0586582
$393$	1.15516e9	0.959995
$394$	1.07819e9	0.888095
$395$	0	0
$396$	4.13444e9	3.34567
$397$	1.60552e8	0.128780	0.0643899	−	0.997925i	$-0.479490\pi$
$397$	1.60552e8	0.128780	0.0643899	+	0.997925i	$0.479490\pi$
$398$	2.83078e9	2.25068
$399$	1.64097e9	1.29329
$400$	0	0
$401$	5.56337e8	0.430856	0.215428	−	0.976520i	$-0.430885\pi$
$401$	5.56337e8	0.430856	0.215428	+	0.976520i	$0.430885\pi$
$402$	−2.23920e9	−1.71910
$403$	−2.52492e9	−1.92168
$404$	1.97164e9	1.48762
$405$	0	0
$406$	1.55463e9	1.15289
$407$	−6.77890e8	−0.498401
$408$	−2.11488e9	−1.54161
$409$	1.26589e9	0.914884	0.457442	−	0.889239i	$-0.348766\pi$
$409$	1.26589e9	0.914884	0.457442	+	0.889239i	$0.348766\pi$
$410$	0	0
$411$	−1.69509e9	−1.20433
$412$	−2.00390e8	−0.141168
$413$	−1.11418e9	−0.778273
$414$	7.37207e8	0.510609
$415$	0	0
$416$	−1.67865e9	−1.14323
$417$	1.79160e8	0.120994
$418$	1.55669e9	1.04252
$419$	−1.88046e9	−1.24887	−0.624433	−	0.781078i	$-0.714670\pi$
$419$	−1.88046e9	−1.24887	−0.624433	+	0.781078i	$0.714670\pi$
$420$	0	0
$421$	1.42794e8	0.0932662	0.0466331	−	0.998912i	$-0.485151\pi$
$421$	1.42794e8	0.0932662	0.0466331	+	0.998912i	$0.485151\pi$
$422$	−2.32960e9	−1.50900
$423$	−6.60073e9	−4.24034
$424$	−1.79214e9	−1.14180
$425$	0	0
$426$	7.47128e8	0.468233
$427$	8.37228e8	0.520411
$428$	−4.51199e9	−2.78173
$429$	2.94399e9	1.80026
$430$	0	0
$431$	1.27055e9	0.764398	0.382199	−	0.924080i	$-0.375167\pi$
$431$	1.27055e9	0.764398	0.382199	+	0.924080i	$0.375167\pi$
$432$	3.70263e8	0.220962
$433$	−1.29807e9	−0.768403	−0.384202	−	0.923249i	$-0.625523\pi$
$433$	−1.29807e9	−0.768403	−0.384202	+	0.923249i	$0.625523\pi$
$434$	−5.09494e9	−2.99175
$435$	0	0
$436$	1.23929e9	0.716092
$437$	1.69738e8	0.0972960
$438$	1.03498e9	0.588533
$439$	1.11226e9	0.627454	0.313727	−	0.949513i	$-0.398422\pi$
$439$	1.11226e9	0.627454	0.313727	+	0.949513i	$0.398422\pi$
$440$	0	0
$441$	2.60227e8	0.144483
$442$	−2.86588e9	−1.57863
$443$	2.97705e9	1.62695	0.813473	−	0.581602i	$-0.197574\pi$
$443$	2.97705e9	1.62695	0.813473	+	0.581602i	$0.197574\pi$
$444$	2.79231e9	1.51399
$445$	0	0
$446$	−5.86793e9	−3.13194
$447$	1.11743e9	0.591756
$448$	−3.19820e9	−1.68048
$449$	−2.92805e9	−1.52657	−0.763285	−	0.646062i	$-0.776415\pi$
$449$	−2.92805e9	−1.52657	−0.763285	+	0.646062i	$0.776415\pi$
$450$	0	0
$451$	3.06656e8	0.157410
$452$	−1.27100e9	−0.647385
$453$	5.04989e9	2.55234
$454$	−2.90777e9	−1.45836
$455$	0	0
$456$	−2.33860e9	−1.15499
$457$	−3.07172e9	−1.50548	−0.752739	−	0.658319i	$-0.771268\pi$
$457$	−3.07172e9	−1.50548	−0.752739	+	0.658319i	$0.771268\pi$
$458$	3.85813e9	1.87650
$459$	4.39940e9	2.12348
$460$	0	0
$461$	−2.93027e9	−1.39301	−0.696505	−	0.717552i	$-0.745263\pi$
$461$	−2.93027e9	−1.39301	−0.696505	+	0.717552i	$0.745263\pi$
$462$	5.94056e9	2.80272
$463$	−3.79849e9	−1.77860	−0.889298	−	0.457329i	$-0.848806\pi$
$463$	−3.79849e9	−1.77860	−0.889298	+	0.457329i	$0.848806\pi$
$464$	1.44425e8	0.0671167
$465$	0	0
$466$	−1.60193e9	−0.733320
$467$	1.13059e9	0.513683	0.256842	−	0.966454i	$-0.417318\pi$
$467$	1.13059e9	0.513683	0.256842	+	0.966454i	$0.417318\pi$
$468$	−8.41674e9	−3.79563
$469$	−1.36557e9	−0.611234
$470$	0	0
$471$	−3.33556e9	−1.47094
$472$	1.58787e9	0.695052
$473$	−7.86831e8	−0.341875
$474$	−5.03817e9	−2.17294
$475$	0	0
$476$	−3.53635e9	−1.50290
$477$	6.66644e9	2.81241
$478$	−1.96175e9	−0.821572
$479$	2.75865e9	1.14689	0.573446	−	0.819243i	$-0.305606\pi$
$479$	2.75865e9	1.14689	0.573446	+	0.819243i	$0.305606\pi$
$480$	0	0
$481$	1.38002e9	0.565430
$482$	5.21460e9	2.12108
$483$	6.47748e8	0.261572
$484$	−4.80190e8	−0.192511
$485$	0	0
$486$	4.47513e9	1.76839
$487$	−1.69705e8	−0.0665798	−0.0332899	−	0.999446i	$-0.510598\pi$
$487$	−1.69705e8	−0.0665798	−0.0332899	+	0.999446i	$0.510598\pi$
$488$	−1.19317e9	−0.464763
$489$	3.65750e9	1.41450
$490$	0	0
$491$	2.66606e9	1.01645	0.508223	−	0.861225i	$-0.330303\pi$
$491$	2.66606e9	1.01645	0.508223	+	0.861225i	$0.330303\pi$
$492$	−1.26315e9	−0.478166
$493$	1.71604e9	0.645004
$494$	−3.16905e9	−1.18273
$495$	0	0
$496$	−4.73321e8	−0.174169
$497$	4.55633e8	0.166482
$498$	7.36778e9	2.67322
$499$	4.56518e9	1.64477	0.822387	−	0.568928i	$-0.192642\pi$
$499$	4.56518e9	1.64477	0.822387	+	0.568928i	$0.192642\pi$
$500$	0	0
$501$	−1.82481e9	−0.648316
$502$	8.22670e8	0.290243
$503$	−2.09748e9	−0.734867	−0.367434	−	0.930050i	$-0.619763\pi$
$503$	−2.09748e9	−0.734867	−0.367434	+	0.930050i	$0.619763\pi$
$504$	−6.19421e9	−2.15516
$505$	0	0
$506$	6.14480e8	0.210854
$507$	−6.87877e8	−0.234414
$508$	5.43667e9	1.83997
$509$	2.38113e9	0.800335	0.400167	−	0.916442i	$-0.368952\pi$
$509$	2.38113e9	0.800335	0.400167	+	0.916442i	$0.368952\pi$
$510$	0	0
$511$	6.31175e8	0.209255
$512$	−5.84142e8	−0.192342
$513$	4.86479e9	1.59094
$514$	−7.18099e9	−2.33246
$515$	0	0
$516$	3.24106e9	1.03851
$517$	−5.50187e9	−1.75103
$518$	2.78470e9	0.880286
$519$	2.30790e8	0.0724655
$520$	0	0
$521$	6.20183e9	1.92127	0.960634	−	0.277818i	$-0.0896113\pi$
$521$	6.20183e9	1.92127	0.960634	+	0.277818i	$0.0896113\pi$
$522$	8.24152e9	2.53607
$523$	6.08679e9	1.86051	0.930256	−	0.366911i	$-0.119585\pi$
$523$	6.08679e9	1.86051	0.930256	+	0.366911i	$0.119585\pi$
$524$	−2.75277e9	−0.835814
$525$	0	0
$526$	−3.72890e9	−1.11720
$527$	−5.62391e9	−1.67379
$528$	5.51878e8	0.163164
$529$	−3.33782e9	−0.980321
$530$	0	0
$531$	−5.90659e9	−1.71201
$532$	−3.91044e9	−1.12599
$533$	−6.24279e8	−0.178580
$534$	7.42004e9	2.10869
$535$	0	0
$536$	1.94612e9	0.545875
$537$	1.04415e10	2.90974
$538$	−1.91829e8	−0.0531101
$539$	2.16905e8	0.0596636
$540$	0	0
$541$	−5.88106e9	−1.59685	−0.798427	−	0.602091i	$-0.794334\pi$
$541$	−5.88106e9	−1.59685	−0.798427	+	0.602091i	$0.794334\pi$
$542$	7.93013e9	2.13936
$543$	−9.46360e9	−2.53663
$544$	−3.73895e9	−0.995758
$545$	0	0
$546$	−1.20936e10	−3.17966
$547$	−2.66732e9	−0.696819	−0.348409	−	0.937342i	$-0.613278\pi$
$547$	−2.66732e9	−0.696819	−0.348409	+	0.937342i	$0.613278\pi$
$548$	4.03942e9	1.04855
$549$	4.43837e9	1.14478
$550$	0	0
$551$	1.89757e9	0.483245
$552$	−9.23131e8	−0.233602
$553$	−3.07250e9	−0.772600
$554$	6.99413e9	1.74763
$555$	0	0
$556$	−4.26940e8	−0.105343
$557$	−1.03490e8	−0.0253750	−0.0126875	−	0.999920i	$-0.504039\pi$
$557$	−1.03490e8	−0.0253750	−0.0126875	+	0.999920i	$0.504039\pi$
$558$	−2.70097e10	−6.58111
$559$	1.60180e9	0.387853
$560$	0	0
$561$	6.55732e9	1.56804
$562$	−4.85616e9	−1.15403
$563$	−2.09575e9	−0.494948	−0.247474	−	0.968894i	$-0.579601\pi$
$563$	−2.09575e9	−0.494948	−0.247474	+	0.968894i	$0.579601\pi$
$564$	2.26629e10	5.31910
$565$	0	0
$566$	3.26526e9	0.756939
$567$	8.40867e9	1.93725
$568$	−6.49340e8	−0.148680
$569$	−1.59475e9	−0.362911	−0.181455	−	0.983399i	$-0.558081\pi$
$569$	−1.59475e9	−0.362911	−0.181455	+	0.983399i	$0.558081\pi$
$570$	0	0
$571$	4.47556e9	1.00605	0.503027	−	0.864271i	$-0.332220\pi$
$571$	4.47556e9	1.00605	0.503027	+	0.864271i	$0.332220\pi$
$572$	−7.01556e9	−1.56739
$573$	−1.49307e10	−3.31542
$574$	−1.25971e9	−0.278022
$575$	0	0
$576$	−1.69545e10	−3.69663
$577$	1.18565e9	0.256946	0.128473	−	0.991713i	$-0.458992\pi$
$577$	1.18565e9	0.256946	0.128473	+	0.991713i	$0.458992\pi$
$578$	1.06499e9	0.229402
$579$	4.23006e9	0.905674
$580$	0	0
$581$	4.49321e9	0.950473
$582$	1.02433e10	2.15382
$583$	5.55663e9	1.16137
$584$	−8.99512e8	−0.186880
$585$	0	0
$586$	2.78770e9	0.572275
$587$	−5.39076e9	−1.10006	−0.550030	−	0.835145i	$-0.685384\pi$
$587$	−5.39076e9	−1.10006	−0.550030	+	0.835145i	$0.685384\pi$
$588$	−8.93461e8	−0.181240
$589$	−6.21884e9	−1.25402
$590$	0	0
$591$	5.02218e9	1.00077
$592$	2.58699e8	0.0512470
$593$	4.53623e8	0.0893314	0.0446657	−	0.999002i	$-0.485778\pi$
$593$	4.53623e8	0.0893314	0.0446657	+	0.999002i	$0.485778\pi$
$594$	1.76113e10	3.44778
$595$	0	0
$596$	−2.66284e9	−0.515209
$597$	1.31857e10	2.53625
$598$	−1.25094e9	−0.239211
$599$	6.15429e8	0.117000	0.0584998	−	0.998287i	$-0.481368\pi$
$599$	6.15429e8	0.117000	0.0584998	+	0.998287i	$0.481368\pi$
$600$	0	0
$601$	2.63776e9	0.495650	0.247825	−	0.968805i	$-0.420284\pi$
$601$	2.63776e9	0.495650	0.247825	+	0.968805i	$0.420284\pi$
$602$	3.23222e9	0.603827
$603$	−7.23923e9	−1.34457
$604$	−1.20339e10	−2.22218
$605$	0	0
$606$	1.50182e10	2.74134
$607$	9.20548e9	1.67065	0.835326	−	0.549755i	$-0.185279\pi$
$607$	9.20548e9	1.67065	0.835326	+	0.549755i	$0.185279\pi$
$608$	−4.13448e9	−0.746033
$609$	7.24142e9	1.29916
$610$	0	0
$611$	1.12005e10	1.98652
$612$	−1.87471e10	−3.30601
$613$	−1.60716e9	−0.281804	−0.140902	−	0.990024i	$-0.545000\pi$
$613$	−1.60716e9	−0.281804	−0.140902	+	0.990024i	$0.545000\pi$
$614$	−9.11550e8	−0.158925
$615$	0	0
$616$	−5.16302e9	−0.889962
$617$	1.06412e10	1.82386	0.911928	−	0.410350i	$-0.134594\pi$
$617$	1.06412e10	1.82386	0.911928	+	0.410350i	$0.134594\pi$
$618$	−1.52639e9	−0.260140
$619$	1.17265e9	0.198724	0.0993619	−	0.995051i	$-0.468320\pi$
$619$	1.17265e9	0.198724	0.0993619	+	0.995051i	$0.468320\pi$
$620$	0	0
$621$	1.92031e9	0.321773
$622$	1.32128e10	2.20155
$623$	4.52508e9	0.749753
$624$	−1.12349e9	−0.185108
$625$	0	0
$626$	1.47637e10	2.40539
$627$	7.25099e9	1.17479
$628$	7.94869e9	1.28067
$629$	3.07381e9	0.492493
$630$	0	0
$631$	−2.63743e8	−0.0417906	−0.0208953	−	0.999782i	$-0.506652\pi$
$631$	−2.63743e8	−0.0417906	−0.0208953	+	0.999782i	$0.506652\pi$
$632$	4.37875e9	0.689986
$633$	−1.08512e10	−1.70046
$634$	3.74427e9	0.583519
$635$	0	0
$636$	−2.28885e10	−3.52791
$637$	−4.41568e8	−0.0676877
$638$	6.86950e9	1.04726
$639$	2.41543e9	0.366220
$640$	0	0
$641$	−1.18253e9	−0.177340	−0.0886701	−	0.996061i	$-0.528262\pi$
$641$	−1.18253e9	−0.177340	−0.0886701	+	0.996061i	$0.528262\pi$
$642$	−3.43684e10	−5.12609
$643$	−1.07875e10	−1.60022	−0.800112	−	0.599851i	$-0.795227\pi$
$643$	−1.07875e10	−1.60022	−0.800112	+	0.599851i	$0.795227\pi$
$644$	−1.54359e9	−0.227736
$645$	0	0
$646$	−7.05861e9	−1.03016
$647$	2.04299e8	0.0296552	0.0148276	−	0.999890i	$-0.495280\pi$
$647$	2.04299e8	0.0296552	0.0148276	+	0.999890i	$0.495280\pi$
$648$	−1.19835e10	−1.73010
$649$	−4.92328e9	−0.706966
$650$	0	0
$651$	−2.37321e10	−3.37134
$652$	−8.71588e9	−1.23153
$653$	−6.82646e9	−0.959399	−0.479699	−	0.877433i	$-0.659254\pi$
$653$	−6.82646e9	−0.959399	−0.479699	+	0.877433i	$0.659254\pi$
$654$	9.43978e9	1.31959
$655$	0	0
$656$	−1.17027e8	−0.0161854
$657$	3.34603e9	0.460310
$658$	2.26011e10	3.09271
$659$	−1.75782e9	−0.239262	−0.119631	−	0.992818i	$-0.538171\pi$
$659$	−1.75782e9	−0.239262	−0.119631	+	0.992818i	$0.538171\pi$
$660$	0	0
$661$	−4.30192e9	−0.579372	−0.289686	−	0.957122i	$-0.593551\pi$
$661$	−4.30192e9	−0.579372	−0.289686	+	0.957122i	$0.593551\pi$
$662$	9.40582e9	1.26007
$663$	−1.33492e10	−1.77892
$664$	−6.40344e9	−0.848839
$665$	0	0
$666$	1.47624e10	1.93641
$667$	7.49039e8	0.0977381
$668$	4.34856e9	0.564452
$669$	−2.73326e10	−3.52931
$670$	0	0
$671$	3.69949e9	0.472730
$672$	−1.57778e10	−2.00565
$673$	−4.86334e9	−0.615010	−0.307505	−	0.951546i	$-0.599494\pi$
$673$	−4.86334e9	−0.615010	−0.307505	+	0.951546i	$0.599494\pi$
$674$	1.66376e9	0.209305
$675$	0	0
$676$	1.63922e9	0.204091
$677$	−1.30016e10	−1.61042	−0.805208	−	0.592993i	$-0.797946\pi$
$677$	−1.30016e10	−1.61042	−0.805208	+	0.592993i	$0.797946\pi$
$678$	−9.68139e9	−1.19298
$679$	6.24682e9	0.765799
$680$	0	0
$681$	−1.35443e10	−1.64340
$682$	−2.25132e10	−2.71764
$683$	1.40721e9	0.169000	0.0845002	−	0.996423i	$-0.473071\pi$
$683$	1.40721e9	0.169000	0.0845002	+	0.996423i	$0.473071\pi$
$684$	−2.07303e10	−2.47690
$685$	0	0
$686$	1.31001e10	1.54932
$687$	1.79710e10	2.11458
$688$	3.00273e8	0.0351526
$689$	−1.13120e10	−1.31756
$690$	0	0
$691$	−3.41978e9	−0.394299	−0.197149	−	0.980373i	$-0.563168\pi$
$691$	−3.41978e9	−0.394299	−0.197149	+	0.980373i	$0.563168\pi$
$692$	−5.49976e8	−0.0630917
$693$	1.92055e10	2.19210
$694$	−1.38123e10	−1.56858
$695$	0	0
$696$	−1.03200e10	−1.16024
$697$	−1.39050e9	−0.155545
$698$	4.95151e9	0.551117
$699$	−7.46175e9	−0.826362
$700$	0	0
$701$	1.39609e10	1.53074	0.765368	−	0.643593i	$-0.222557\pi$
$701$	1.39609e10	1.53074	0.765368	+	0.643593i	$0.222557\pi$
$702$	−3.58525e10	−3.91146
$703$	3.39898e9	0.368982
$704$	−1.41320e10	−1.52651
$705$	0	0
$706$	−2.95236e10	−3.15757
$707$	9.15876e9	0.974695
$708$	2.02796e10	2.14755
$709$	−1.38420e10	−1.45861	−0.729303	−	0.684191i	$-0.760155\pi$
$709$	−1.38420e10	−1.45861	−0.729303	+	0.684191i	$0.760155\pi$
$710$	0	0
$711$	−1.62882e10	−1.69953
$712$	−6.44887e9	−0.669582
$713$	−2.45480e9	−0.253631
$714$	−2.69368e10	−2.76950
$715$	0	0
$716$	−2.48823e10	−2.53335
$717$	−9.13775e9	−0.925811
$718$	−2.69063e9	−0.271281
$719$	1.40240e10	1.40709	0.703545	−	0.710651i	$-0.251600\pi$
$719$	1.40240e10	1.40709	0.703545	+	0.710651i	$0.251600\pi$
$720$	0	0
$721$	−9.30864e8	−0.0924938
$722$	8.41996e9	0.832588
$723$	2.42895e10	2.39020
$724$	2.25519e10	2.20850
$725$	0	0
$726$	−3.65767e9	−0.354753
$727$	−4.48749e9	−0.433145	−0.216573	−	0.976267i	$-0.569488\pi$
$727$	−4.48749e9	−0.433145	−0.216573	+	0.976267i	$0.569488\pi$
$728$	1.05107e10	1.00965
$729$	1.19664e9	0.114397
$730$	0	0
$731$	3.56779e9	0.337823
$732$	−1.52387e10	−1.43601
$733$	8.95166e9	0.839536	0.419768	−	0.907631i	$-0.362111\pi$
$733$	8.95166e9	0.839536	0.419768	+	0.907631i	$0.362111\pi$
$734$	−2.18256e10	−2.03718
$735$	0	0
$736$	−1.63203e9	−0.150888
$737$	−6.03407e9	−0.555232
$738$	−6.67805e9	−0.611579
$739$	−6.90000e9	−0.628917	−0.314459	−	0.949271i	$-0.601823\pi$
$739$	−6.90000e9	−0.628917	−0.314459	+	0.949271i	$0.601823\pi$
$740$	0	0
$741$	−1.47613e10	−1.33279
$742$	−2.28261e10	−2.05124
$743$	1.44134e10	1.28915	0.644576	−	0.764540i	$-0.277034\pi$
$743$	1.44134e10	1.28915	0.644576	+	0.764540i	$0.277034\pi$
$744$	3.38215e10	3.01084
$745$	0	0
$746$	1.94272e10	1.71326
$747$	2.38197e10	2.09081
$748$	−1.56262e10	−1.36520
$749$	−2.09594e10	−1.82260
$750$	0	0
$751$	−3.95843e9	−0.341023	−0.170511	−	0.985356i	$-0.554542\pi$
$751$	−3.95843e9	−0.341023	−0.170511	+	0.985356i	$0.554542\pi$
$752$	2.09964e9	0.180046
$753$	3.83197e9	0.327069
$754$	−1.39847e10	−1.18810
$755$	0	0
$756$	−4.42401e10	−3.72383
$757$	−6.97998e9	−0.584815	−0.292407	−	0.956294i	$-0.594456\pi$
$757$	−6.97998e9	−0.584815	−0.292407	+	0.956294i	$0.594456\pi$
$758$	−1.21884e10	−1.01650
$759$	2.86223e9	0.237606
$760$	0	0
$761$	−9.92015e9	−0.815966	−0.407983	−	0.912990i	$-0.633768\pi$
$761$	−9.92015e9	−0.815966	−0.407983	+	0.912990i	$0.633768\pi$
$762$	4.14117e10	3.39063
$763$	5.75681e9	0.469187
$764$	3.55800e10	2.88655
$765$	0	0
$766$	−1.03569e10	−0.832586
$767$	1.00226e10	0.802045
$768$	1.90441e10	1.51704
$769$	1.69604e10	1.34491	0.672454	−	0.740139i	$-0.265240\pi$
$769$	1.69604e10	1.34491	0.672454	+	0.740139i	$0.265240\pi$
$770$	0	0
$771$	−3.34488e10	−2.62839
$772$	−1.00803e10	−0.788520
$773$	7.75745e9	0.604075	0.302037	−	0.953296i	$-0.402333\pi$
$773$	7.75745e9	0.604075	0.302037	+	0.953296i	$0.402333\pi$
$774$	1.71348e10	1.32827
$775$	0	0
$776$	−8.90259e9	−0.683912
$777$	1.29710e10	0.991975
$778$	3.80776e10	2.89895
$779$	−1.53759e9	−0.116536
$780$	0	0
$781$	2.01332e9	0.151229
$782$	−2.78629e9	−0.208354
$783$	2.14678e10	1.59817
$784$	−8.27762e7	−0.00613478
$785$	0	0
$786$	−2.09681e10	−1.54021
$787$	2.23080e10	1.63136	0.815679	−	0.578504i	$-0.196363\pi$
$787$	2.23080e10	1.63136	0.815679	+	0.578504i	$0.196363\pi$
$788$	−1.19679e10	−0.871319
$789$	−1.73691e10	−1.25895
$790$	0	0
$791$	−5.90415e9	−0.424170
$792$	−2.73706e10	−1.95770
$793$	−7.53128e9	−0.536306
$794$	−2.91428e9	−0.206614
$795$	0	0
$796$	−3.14216e10	−2.20817
$797$	−1.54662e10	−1.08213	−0.541065	−	0.840981i	$-0.681979\pi$
$797$	−1.54662e10	−1.08213	−0.541065	+	0.840981i	$0.681979\pi$
$798$	−2.97863e10	−2.07494
$799$	2.49476e10	1.73027
$800$	0	0
$801$	2.39887e10	1.64927
$802$	−1.00984e10	−0.691264
$803$	2.78899e9	0.190083
$804$	2.48551e10	1.68663
$805$	0	0
$806$	4.58316e10	3.08313
$807$	−8.93534e8	−0.0598486
$808$	−1.30525e10	−0.870471
$809$	1.27223e10	0.844784	0.422392	−	0.906413i	$-0.361191\pi$
$809$	1.27223e10	0.844784	0.422392	+	0.906413i	$0.361191\pi$
$810$	0	0
$811$	−6.41618e8	−0.0422380	−0.0211190	−	0.999777i	$-0.506723\pi$
$811$	−6.41618e8	−0.0422380	−0.0211190	+	0.999777i	$0.506723\pi$
$812$	−1.72564e10	−1.13111
$813$	3.69383e10	2.41079
$814$	1.23048e10	0.799633
$815$	0	0
$816$	−2.50243e9	−0.161230
$817$	3.94522e9	0.253101
$818$	−2.29781e10	−1.46784
$819$	−3.90979e10	−2.48691
$820$	0	0
$821$	−1.72086e10	−1.08528	−0.542642	−	0.839964i	$-0.682576\pi$
$821$	−1.72086e10	−1.08528	−0.542642	+	0.839964i	$0.682576\pi$
$822$	3.07687e10	1.93223
$823$	1.38632e10	0.866887	0.433444	−	0.901181i	$-0.357298\pi$
$823$	1.38632e10	0.866887	0.433444	+	0.901181i	$0.357298\pi$
$824$	1.32661e9	0.0826034
$825$	0	0
$826$	2.02243e10	1.24866
$827$	1.11146e10	0.683319	0.341659	−	0.939824i	$-0.389011\pi$
$827$	1.11146e10	0.683319	0.341659	+	0.939824i	$0.389011\pi$
$828$	−8.18300e9	−0.500964
$829$	1.47485e10	0.899097	0.449548	−	0.893256i	$-0.351585\pi$
$829$	1.47485e10	0.899097	0.449548	+	0.893256i	$0.351585\pi$
$830$	0	0
$831$	3.25784e10	1.96937
$832$	2.87694e10	1.73180
$833$	−9.83532e8	−0.0589564
$834$	−3.25205e9	−0.194123
$835$	0	0
$836$	−1.72792e10	−1.02283
$837$	−7.03558e10	−4.14726
$838$	3.41336e10	2.00368
$839$	2.64994e9	0.154906	0.0774532	−	0.996996i	$-0.475321\pi$
$839$	2.64994e9	0.154906	0.0774532	+	0.996996i	$0.475321\pi$
$840$	0	0
$841$	−8.87609e9	−0.514560
$842$	−2.59196e9	−0.149636
$843$	−2.26198e10	−1.30045
$844$	2.58586e10	1.48049
$845$	0	0
$846$	1.19814e11	6.80319
$847$	−2.23061e9	−0.126134
$848$	−2.12054e9	−0.119416
$849$	1.52095e10	0.852978
$850$	0	0
$851$	1.34170e9	0.0746280
$852$	−8.29312e9	−0.459388
$853$	−1.51679e10	−0.836763	−0.418382	−	0.908271i	$-0.637403\pi$
$853$	−1.51679e10	−0.836763	−0.418382	+	0.908271i	$0.637403\pi$
$854$	−1.51971e10	−0.834945
$855$	0	0
$856$	2.98700e10	1.62771
$857$	2.49214e10	1.35251	0.676253	−	0.736670i	$-0.263603\pi$
$857$	2.49214e10	1.35251	0.676253	+	0.736670i	$0.263603\pi$
$858$	−5.34383e10	−2.88833
$859$	2.31196e9	0.124453	0.0622264	−	0.998062i	$-0.480180\pi$
$859$	2.31196e9	0.124453	0.0622264	+	0.998062i	$0.480180\pi$
$860$	0	0
$861$	−5.86768e9	−0.313296
$862$	−2.30625e10	−1.22640
$863$	6.67063e9	0.353288	0.176644	−	0.984275i	$-0.443476\pi$
$863$	6.67063e9	0.353288	0.176644	+	0.984275i	$0.443476\pi$
$864$	−4.67747e10	−2.46725
$865$	0	0
$866$	2.35621e10	1.23282
$867$	4.96068e9	0.258508
$868$	5.65538e10	2.93524
$869$	−1.35766e10	−0.701813
$870$	0	0
$871$	1.22839e10	0.629904
$872$	−8.20425e9	−0.419017
$873$	3.31161e10	1.68457
$874$	−3.08104e9	−0.156101
$875$	0	0
$876$	−1.14882e10	−0.577415
$877$	1.09077e10	0.546052	0.273026	−	0.962007i	$-0.411975\pi$
$877$	1.09077e10	0.546052	0.273026	+	0.962007i	$0.411975\pi$
$878$	−2.01895e10	−1.00669
$879$	1.29850e10	0.644884
$880$	0	0
$881$	−1.34129e10	−0.660856	−0.330428	−	0.943831i	$-0.607193\pi$
$881$	−1.34129e10	−0.660856	−0.330428	+	0.943831i	$0.607193\pi$
$882$	−4.72356e9	−0.231808
$883$	1.01152e10	0.494440	0.247220	−	0.968959i	$-0.420483\pi$
$883$	1.01152e10	0.494440	0.247220	+	0.968959i	$0.420483\pi$
$884$	3.18112e10	1.54881
$885$	0	0
$886$	−5.40385e10	−2.61027
$887$	−1.29302e10	−0.622118	−0.311059	−	0.950391i	$-0.600684\pi$
$887$	−1.29302e10	−0.622118	−0.311059	+	0.950391i	$0.600684\pi$
$888$	−1.84855e10	−0.885903
$889$	2.52547e10	1.20555
$890$	0	0
$891$	3.71557e10	1.75976
$892$	6.51340e10	3.07278
$893$	2.75867e10	1.29634
$894$	−2.02832e10	−0.949412
$895$	0	0
$896$	3.41666e10	1.58681
$897$	−5.82682e9	−0.269562
$898$	5.31491e10	2.44922
$899$	−2.74431e10	−1.25972
$900$	0	0
$901$	−2.51959e10	−1.14761
$902$	−5.56632e9	−0.252549
$903$	1.50556e10	0.680440
$904$	8.41423e9	0.378813
$905$	0	0
$906$	−9.16639e10	−4.09496
$907$	−3.07993e9	−0.137062	−0.0685308	−	0.997649i	$-0.521831\pi$
$907$	−3.07993e9	−0.137062	−0.0685308	+	0.997649i	$0.521831\pi$
$908$	3.22763e10	1.43081
$909$	4.85530e10	2.14409
$910$	0	0
$911$	1.13702e10	0.498258	0.249129	−	0.968470i	$-0.419856\pi$
$911$	1.13702e10	0.498258	0.249129	+	0.968470i	$0.419856\pi$
$912$	−2.76715e9	−0.120795
$913$	1.98543e10	0.863389
$914$	5.57568e10	2.41538
$915$	0	0
$916$	−4.28252e10	−1.84105
$917$	−1.27873e10	−0.547629
$918$	−7.98564e10	−3.40691
$919$	2.65703e10	1.12925	0.564627	−	0.825346i	$-0.309020\pi$
$919$	2.65703e10	1.12925	0.564627	+	0.825346i	$0.309020\pi$
$920$	0	0
$921$	−4.24597e9	−0.179089
$922$	5.31893e10	2.23494
$923$	−4.09864e9	−0.171567
$924$	−6.59401e10	−2.74978
$925$	0	0
$926$	6.89489e10	2.85357
$927$	−4.93476e9	−0.203464
$928$	−1.82451e10	−0.749423
$929$	9.10717e9	0.372673	0.186337	−	0.982486i	$-0.440338\pi$
$929$	9.10717e9	0.372673	0.186337	+	0.982486i	$0.440338\pi$
$930$	0	0
$931$	−1.08758e9	−0.0441708
$932$	1.77814e10	0.719467
$933$	6.15447e10	2.48088
$934$	−2.05221e10	−0.824152
$935$	0	0
$936$	5.57200e10	2.22099
$937$	4.66624e10	1.85301	0.926506	−	0.376280i	$-0.122797\pi$
$937$	4.66624e10	1.85301	0.926506	+	0.376280i	$0.122797\pi$
$938$	2.47873e10	0.980663
$939$	6.87688e10	2.71058
$940$	0	0
$941$	4.40281e10	1.72253	0.861264	−	0.508158i	$-0.169673\pi$
$941$	4.40281e10	1.72253	0.861264	+	0.508158i	$0.169673\pi$
$942$	6.05461e10	2.35998
$943$	−6.06942e8	−0.0235698
$944$	1.87884e9	0.0726923
$945$	0	0
$946$	1.42823e10	0.548504
$947$	−2.70062e9	−0.103333	−0.0516664	−	0.998664i	$-0.516453\pi$
$947$	−2.70062e9	−0.103333	−0.0516664	+	0.998664i	$0.516453\pi$
$948$	5.59237e10	2.13190
$949$	−5.67773e9	−0.215647
$950$	0	0
$951$	1.74407e10	0.657555
$952$	2.34111e10	0.879413
$953$	4.02541e10	1.50655	0.753277	−	0.657703i	$-0.228472\pi$
$953$	4.02541e10	1.50655	0.753277	+	0.657703i	$0.228472\pi$
$954$	−1.21007e11	−4.51223
$955$	0	0
$956$	2.17754e10	0.806052
$957$	3.19979e10	1.18013
$958$	−5.00742e10	−1.84007
$959$	1.87642e10	0.687012
$960$	0	0
$961$	6.24259e10	2.26899
$962$	−2.50498e10	−0.907174
$963$	−1.11111e11	−4.00928
$964$	−5.78821e10	−2.08101
$965$	0	0
$966$	−1.17577e10	−0.419665
$967$	−3.55726e10	−1.26509	−0.632547	−	0.774522i	$-0.717990\pi$
$967$	−3.55726e10	−1.26509	−0.632547	+	0.774522i	$0.717990\pi$
$968$	3.17893e9	0.112646
$969$	−3.28788e10	−1.16087
$970$	0	0
$971$	−2.66420e10	−0.933897	−0.466949	−	0.884284i	$-0.654647\pi$
$971$	−2.66420e10	−0.933897	−0.466949	+	0.884284i	$0.654647\pi$
$972$	−4.96739e10	−1.73499
$973$	−1.98325e9	−0.0690211
$974$	3.08042e9	0.106820
$975$	0	0
$976$	−1.41181e9	−0.0486074
$977$	1.55706e10	0.534162	0.267081	−	0.963674i	$-0.413941\pi$
$977$	1.55706e10	0.534162	0.267081	+	0.963674i	$0.413941\pi$
$978$	−6.63898e10	−2.26942
$979$	1.99951e10	0.681059
$980$	0	0
$981$	3.05184e10	1.03210
$982$	−4.83934e10	−1.63078
$983$	−4.15955e10	−1.39672	−0.698359	−	0.715748i	$-0.746086\pi$
$983$	−4.15955e10	−1.39672	−0.698359	+	0.715748i	$0.746086\pi$
$984$	8.36226e9	0.279796
$985$	0	0
$986$	−3.11490e10	−1.03484
$987$	1.05275e11	3.48510
$988$	3.51764e10	1.16038
$989$	1.55732e9	0.0511906
$990$	0	0
$991$	1.77382e10	0.578965	0.289483	−	0.957183i	$-0.406517\pi$
$991$	1.77382e10	0.578965	0.289483	+	0.957183i	$0.406517\pi$
$992$	5.97939e10	1.94476
$993$	4.38120e10	1.41994
$994$	−8.27050e9	−0.267103
$995$	0	0
$996$	−8.17823e10	−2.62272
$997$	−3.78343e10	−1.20907	−0.604537	−	0.796577i	$-0.706642\pi$
$997$	−3.78343e10	−1.20907	−0.604537	+	0.796577i	$0.706642\pi$
$998$	−8.28658e10	−2.63887
$999$	3.84538e10	1.22028

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.8.a.d.1.3		34
5.4	even	2		625.8.a.c.1.32		34
25.11	even	5		25.8.d.a.21.2	yes	68
25.16	even	5		25.8.d.a.6.2	✓	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.d.a.6.2	✓	68	25.16	even	5
25.8.d.a.21.2	yes	68	25.11	even	5
625.8.a.c.1.32		34	5.4	even	2
625.8.a.d.1.3		34	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-18.1517 q^{2} -84.5499 q^{3} +201.484 q^{4} +1534.72 q^{6} +935.944 q^{7} -1333.85 q^{8} +4961.69 q^{9} +O(q^{10})\)	\(q-18.1517 q^{2} -84.5499 q^{3} +201.484 q^{4} +1534.72 q^{6} +935.944 q^{7} -1333.85 q^{8} +4961.69 q^{9} +4135.69 q^{11} -17035.4 q^{12} -8419.28 q^{13} -16988.9 q^{14} -1578.28 q^{16} -18752.8 q^{17} -90063.0 q^{18} -20736.6 q^{19} -79133.9 q^{21} -75069.7 q^{22} -8185.47 q^{23} +112777. q^{24} +152824. q^{26} -234600. q^{27} +188577. q^{28} -91508.4 q^{29} +299897. q^{31} +199381. q^{32} -349672. q^{33} +340395. q^{34} +999698. q^{36} -163912. q^{37} +376403. q^{38} +711849. q^{39} +74148.7 q^{41} +1.43641e6 q^{42} -190254. q^{43} +833273. q^{44} +148580. q^{46} -1.33034e6 q^{47} +133443. q^{48} +52447.3 q^{49} +1.58555e6 q^{51} -1.69635e6 q^{52} +1.34358e6 q^{53} +4.25838e6 q^{54} -1.24841e6 q^{56} +1.75327e6 q^{57} +1.66103e6 q^{58} -1.19044e6 q^{59} +894528. q^{61} -5.44364e6 q^{62} +4.64386e6 q^{63} -3.41708e6 q^{64} +6.34713e6 q^{66} -1.45903e6 q^{67} -3.77838e6 q^{68} +692080. q^{69} +486817. q^{71} -6.61815e6 q^{72} +674373. q^{73} +2.97528e6 q^{74} -4.17808e6 q^{76} +3.87077e6 q^{77} -1.29213e7 q^{78} -3.28279e6 q^{79} +8.98416e6 q^{81} -1.34592e6 q^{82} +4.80072e6 q^{83} -1.59442e7 q^{84} +3.45343e6 q^{86} +7.73703e6 q^{87} -5.51638e6 q^{88} +4.83478e6 q^{89} -7.87997e6 q^{91} -1.64924e6 q^{92} -2.53563e7 q^{93} +2.41479e7 q^{94} -1.68577e7 q^{96} +6.67435e6 q^{97} -952006. q^{98} +2.05200e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 17 q^{2} + 14 q^{3} + 2177 q^{4} + 993 q^{6} + 867 q^{7} + 2700 q^{8} + 23708 q^{9}+O(q^{10}) \) 34 * q + 17 * q^2 + 14 * q^3 + 2177 * q^4 + 993 * q^6 + 867 * q^7 + 2700 * q^8 + 23708 * q^9	\( 34 q + 17 q^{2} + 14 q^{3} + 2177 q^{4} + 993 q^{6} + 867 q^{7} + 2700 q^{8} + 23708 q^{9} + 8718 q^{11} + 2377 q^{12} + 8769 q^{13} + 17674 q^{14} + 139109 q^{16} - 48258 q^{17} + 58104 q^{18} + 61955 q^{19} + 96943 q^{21} - 52556 q^{22} + 55584 q^{23} + 276990 q^{24} + 337623 q^{26} - 24370 q^{27} + 392941 q^{28} + 449815 q^{29} + 450438 q^{31} + 522987 q^{32} - 689282 q^{33} + 639889 q^{34} + 2024094 q^{36} - 373008 q^{37} + 1272835 q^{38} + 877321 q^{39} + 926418 q^{41} + 580554 q^{42} - 1719911 q^{43} + 1809244 q^{44} + 1186623 q^{46} + 640247 q^{47} + 3924149 q^{48} + 3292177 q^{49} + 1961038 q^{51} + 3900212 q^{52} + 1609214 q^{53} + 3802395 q^{54} + 5709425 q^{56} - 7299205 q^{57} - 4001010 q^{58} + 6056265 q^{59} + 2885628 q^{61} - 14351151 q^{62} + 8834194 q^{63} + 12281872 q^{64} + 8819186 q^{66} - 164318 q^{67} - 14549609 q^{68} + 5884976 q^{69} + 10286173 q^{71} + 41997930 q^{72} + 20595894 q^{73} + 7025034 q^{74} + 11251000 q^{76} + 14569234 q^{77} - 32702012 q^{78} + 11772835 q^{79} + 24408674 q^{81} - 38307271 q^{82} + 2455444 q^{83} + 17190739 q^{84} + 7636893 q^{86} - 39372965 q^{87} + 45397760 q^{88} + 17788560 q^{89} + 15865973 q^{91} + 37183957 q^{92} - 59352382 q^{93} + 29349634 q^{94} + 38027693 q^{96} + 3708847 q^{97} + 6000776 q^{98} + 35633116 q^{99}+O(q^{100}) \) 34 * q + 17 * q^2 + 14 * q^3 + 2177 * q^4 + 993 * q^6 + 867 * q^7 + 2700 * q^8 + 23708 * q^9 + 8718 * q^11 + 2377 * q^12 + 8769 * q^13 + 17674 * q^14 + 139109 * q^16 - 48258 * q^17 + 58104 * q^18 + 61955 * q^19 + 96943 * q^21 - 52556 * q^22 + 55584 * q^23 + 276990 * q^24 + 337623 * q^26 - 24370 * q^27 + 392941 * q^28 + 449815 * q^29 + 450438 * q^31 + 522987 * q^32 - 689282 * q^33 + 639889 * q^34 + 2024094 * q^36 - 373008 * q^37 + 1272835 * q^38 + 877321 * q^39 + 926418 * q^41 + 580554 * q^42 - 1719911 * q^43 + 1809244 * q^44 + 1186623 * q^46 + 640247 * q^47 + 3924149 * q^48 + 3292177 * q^49 + 1961038 * q^51 + 3900212 * q^52 + 1609214 * q^53 + 3802395 * q^54 + 5709425 * q^56 - 7299205 * q^57 - 4001010 * q^58 + 6056265 * q^59 + 2885628 * q^61 - 14351151 * q^62 + 8834194 * q^63 + 12281872 * q^64 + 8819186 * q^66 - 164318 * q^67 - 14549609 * q^68 + 5884976 * q^69 + 10286173 * q^71 + 41997930 * q^72 + 20595894 * q^73 + 7025034 * q^74 + 11251000 * q^76 + 14569234 * q^77 - 32702012 * q^78 + 11772835 * q^79 + 24408674 * q^81 - 38307271 * q^82 + 2455444 * q^83 + 17190739 * q^84 + 7636893 * q^86 - 39372965 * q^87 + 45397760 * q^88 + 17788560 * q^89 + 15865973 * q^91 + 37183957 * q^92 - 59352382 * q^93 + 29349634 * q^94 + 38027693 * q^96 + 3708847 * q^97 + 6000776 * q^98 + 35633116 * q^99

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