LMFDB - Embedded newform 175.8.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,8,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("175.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	175.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:	$54.6673794597$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 706x^{6} + 1459x^{5} + 140610x^{4} - 269712x^{3} - 8960496x^{2} + 18558112x + 98712192 $ x^8 - 2x^7 - 706x^6 + 1459x^5 + 140610x^4 - 269712x^3 - 8960496x^2 + 18558112*x + 98712192
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-11.2353$ of defining polynomial
Character	$\chi$	$=$	175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.23526 q^{2} +79.3182 q^{3} -42.7100 q^{4} +732.524 q^{6} -343.000 q^{7} -1576.55 q^{8} +4104.37 q^{9} +O(q^{10})$	\(q+9.23526 q^{2} +79.3182 q^{3} -42.7100 q^{4} +732.524 q^{6} -343.000 q^{7} -1576.55 q^{8} +4104.37 q^{9} +7560.19 q^{11} -3387.68 q^{12} -6570.26 q^{13} -3167.69 q^{14} -9092.98 q^{16} +38530.0 q^{17} +37904.9 q^{18} +27842.9 q^{19} -27206.1 q^{21} +69820.3 q^{22} -56508.5 q^{23} -125049. q^{24} -60678.1 q^{26} +152082. q^{27} +14649.5 q^{28} +213939. q^{29} +18914.9 q^{31} +117822. q^{32} +599660. q^{33} +355834. q^{34} -175297. q^{36} +158204. q^{37} +257136. q^{38} -521141. q^{39} +371840. q^{41} -251256. q^{42} -39351.7 q^{43} -322895. q^{44} -521871. q^{46} +452269. q^{47} -721239. q^{48} +117649. q^{49} +3.05613e6 q^{51} +280616. q^{52} -1.21612e6 q^{53} +1.40452e6 q^{54} +540757. q^{56} +2.20844e6 q^{57} +1.97578e6 q^{58} -2.62913e6 q^{59} +981864. q^{61} +174684. q^{62} -1.40780e6 q^{63} +2.25202e6 q^{64} +5.53802e6 q^{66} -3.20605e6 q^{67} -1.64561e6 q^{68} -4.48215e6 q^{69} -1.27584e6 q^{71} -6.47075e6 q^{72} +3.51670e6 q^{73} +1.46106e6 q^{74} -1.18917e6 q^{76} -2.59315e6 q^{77} -4.81287e6 q^{78} +2.45207e6 q^{79} +3.08662e6 q^{81} +3.43404e6 q^{82} -6.84085e6 q^{83} +1.16197e6 q^{84} -363423. q^{86} +1.69692e7 q^{87} -1.19190e7 q^{88} -7.49365e6 q^{89} +2.25360e6 q^{91} +2.41348e6 q^{92} +1.50029e6 q^{93} +4.17682e6 q^{94} +9.34546e6 q^{96} -1.88197e6 q^{97} +1.08652e6 q^{98} +3.10298e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 18 q^{2} - 54 q^{3} + 432 q^{4} - 53 q^{6} - 2744 q^{7} - 3843 q^{8} + 12334 q^{9}+O(q^{10}) $ 8 * q - 18 * q^2 - 54 * q^3 + 432 * q^4 - 53 * q^6 - 2744 * q^7 - 3843 * q^8 + 12334 * q^9	\( 8 q - 18 q^{2} - 54 q^{3} + 432 q^{4} - 53 q^{6} - 2744 q^{7} - 3843 q^{8} + 12334 q^{9} - 672 q^{11} - 37071 q^{12} + 9410 q^{13} + 6174 q^{14} + 39152 q^{16} + 34626 q^{17} - 1861 q^{18} - 1876 q^{19} + 18522 q^{21} - 57686 q^{22} - 2892 q^{23} + 115419 q^{24} - 126624 q^{26} + 119664 q^{27} - 148176 q^{28} + 375306 q^{29} + 401106 q^{31} - 782562 q^{32} + 57268 q^{33} + 115141 q^{34} + 1043493 q^{36} + 510970 q^{37} + 176355 q^{38} - 229864 q^{39} + 1725360 q^{41} + 18179 q^{42} - 2583780 q^{43} + 417810 q^{44} - 3420221 q^{46} + 830988 q^{47} - 6605405 q^{48} + 941192 q^{49} + 2465784 q^{51} + 6335102 q^{52} - 3081888 q^{53} - 504707 q^{54} + 1318149 q^{56} + 699682 q^{57} - 84077 q^{58} - 503982 q^{59} + 5355454 q^{61} + 10940010 q^{62} - 4230562 q^{63} + 12425931 q^{64} + 23945921 q^{66} + 2025904 q^{67} + 3202389 q^{68} + 4197918 q^{69} + 12457176 q^{71} + 6598620 q^{72} + 1059500 q^{73} + 10942191 q^{74} + 15089099 q^{76} + 230496 q^{77} - 18142970 q^{78} + 10473540 q^{79} + 17137732 q^{81} + 14606391 q^{82} - 18993798 q^{83} + 12715353 q^{84} + 37213797 q^{86} + 18782326 q^{87} - 46963547 q^{88} + 14400636 q^{89} - 3227630 q^{91} + 26180415 q^{92} - 7950312 q^{93} - 9402748 q^{94} + 91197911 q^{96} + 24116506 q^{97} - 2117682 q^{98} + 11403964 q^{99}+O(q^{100}) \) 8 * q - 18 * q^2 - 54 * q^3 + 432 * q^4 - 53 * q^6 - 2744 * q^7 - 3843 * q^8 + 12334 * q^9 - 672 * q^11 - 37071 * q^12 + 9410 * q^13 + 6174 * q^14 + 39152 * q^16 + 34626 * q^17 - 1861 * q^18 - 1876 * q^19 + 18522 * q^21 - 57686 * q^22 - 2892 * q^23 + 115419 * q^24 - 126624 * q^26 + 119664 * q^27 - 148176 * q^28 + 375306 * q^29 + 401106 * q^31 - 782562 * q^32 + 57268 * q^33 + 115141 * q^34 + 1043493 * q^36 + 510970 * q^37 + 176355 * q^38 - 229864 * q^39 + 1725360 * q^41 + 18179 * q^42 - 2583780 * q^43 + 417810 * q^44 - 3420221 * q^46 + 830988 * q^47 - 6605405 * q^48 + 941192 * q^49 + 2465784 * q^51 + 6335102 * q^52 - 3081888 * q^53 - 504707 * q^54 + 1318149 * q^56 + 699682 * q^57 - 84077 * q^58 - 503982 * q^59 + 5355454 * q^61 + 10940010 * q^62 - 4230562 * q^63 + 12425931 * q^64 + 23945921 * q^66 + 2025904 * q^67 + 3202389 * q^68 + 4197918 * q^69 + 12457176 * q^71 + 6598620 * q^72 + 1059500 * q^73 + 10942191 * q^74 + 15089099 * q^76 + 230496 * q^77 - 18142970 * q^78 + 10473540 * q^79 + 17137732 * q^81 + 14606391 * q^82 - 18993798 * q^83 + 12715353 * q^84 + 37213797 * q^86 + 18782326 * q^87 - 46963547 * q^88 + 14400636 * q^89 - 3227630 * q^91 + 26180415 * q^92 - 7950312 * q^93 - 9402748 * q^94 + 91197911 * q^96 + 24116506 * q^97 - 2117682 * q^98 + 11403964 * 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 $

$2$	9.23526	0.816289	0.408145	−	0.912917i	$-0.366176\pi$
$2$	9.23526	0.816289	0.408145	+	0.912917i	$0.366176\pi$
$3$	79.3182	1.69609	0.848044	−	0.529926i	$-0.177780\pi$
$3$	79.3182	1.69609	0.848044	+	0.529926i	$0.177780\pi$
$4$	−42.7100	−0.333672
$5$	0	0
$5$	0	0
$6$	732.524	1.38450
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−1576.55	−1.08866
$9$	4104.37	1.87671
$10$	0	0
$11$	7560.19	1.71261	0.856305	−	0.516471i	$-0.172755\pi$
$11$	7560.19	1.71261	0.856305	+	0.516471i	$0.172755\pi$
$12$	−3387.68	−0.565936
$13$	−6570.26	−0.829432	−0.414716	−	0.909951i	$-0.636119\pi$
$13$	−6570.26	−0.829432	−0.414716	+	0.909951i	$0.636119\pi$
$14$	−3167.69	−0.308528
$15$	0	0
$16$	−9092.98	−0.554992
$17$	38530.0	1.90207	0.951037	−	0.309077i	$-0.100020\pi$
$17$	38530.0	1.90207	0.951037	+	0.309077i	$0.100020\pi$
$18$	37904.9	1.53194
$19$	27842.9	0.931271	0.465636	−	0.884977i	$-0.345826\pi$
$19$	27842.9	0.931271	0.465636	+	0.884977i	$0.345826\pi$
$20$	0	0
$21$	−27206.1	−0.641061
$22$	69820.3	1.39799
$23$	−56508.5	−0.968426	−0.484213	−	0.874950i	$-0.660894\pi$
$23$	−56508.5	−0.968426	−0.484213	+	0.874950i	$0.660894\pi$
$24$	−125049.	−1.84647
$25$	0	0
$26$	−60678.1	−0.677057
$27$	152082.	1.48698
$28$	14649.5	0.126116
$29$	213939.	1.62891	0.814453	−	0.580229i	$-0.197037\pi$
$29$	213939.	1.62891	0.814453	+	0.580229i	$0.197037\pi$
$30$	0	0
$31$	18914.9	0.114035	0.0570174	−	0.998373i	$-0.481841\pi$
$31$	18914.9	0.114035	0.0570174	+	0.998373i	$0.481841\pi$
$32$	117822.	0.635628
$33$	599660.	2.90474
$34$	355834.	1.55264
$35$	0	0
$36$	−175297.	−0.626205
$37$	158204.	0.513466	0.256733	−	0.966482i	$-0.417354\pi$
$37$	158204.	0.513466	0.256733	+	0.966482i	$0.417354\pi$
$38$	257136.	0.760187
$39$	−521141.	−1.40679
$40$	0	0
$41$	371840.	0.842582	0.421291	−	0.906926i	$-0.361577\pi$
$41$	371840.	0.842582	0.421291	+	0.906926i	$0.361577\pi$
$42$	−251256.	−0.523291
$43$	−39351.7	−0.0754785	−0.0377393	−	0.999288i	$-0.512016\pi$
$43$	−39351.7	−0.0754785	−0.0377393	+	0.999288i	$0.512016\pi$
$44$	−322895.	−0.571449
$45$	0	0
$46$	−521871.	−0.790516
$47$	452269.	0.635410	0.317705	−	0.948190i	$-0.397088\pi$
$47$	452269.	0.635410	0.317705	+	0.948190i	$0.397088\pi$
$48$	−721239.	−0.941314
$49$	117649.	0.142857
$50$	0	0
$51$	3.05613e6	3.22608
$52$	280616.	0.276758
$53$	−1.21612e6	−1.12204	−0.561021	−	0.827802i	$-0.689591\pi$
$53$	−1.21612e6	−1.12204	−0.561021	+	0.827802i	$0.689591\pi$
$54$	1.40452e6	1.21381
$55$	0	0
$56$	540757.	0.411476
$57$	2.20844e6	1.57952
$58$	1.97578e6	1.32966
$59$	−2.62913e6	−1.66660	−0.833299	−	0.552823i	$-0.813551\pi$
$59$	−2.62913e6	−1.66660	−0.833299	+	0.552823i	$0.813551\pi$
$60$	0	0
$61$	981864.	0.553856	0.276928	−	0.960891i	$-0.410684\pi$
$61$	981864.	0.553856	0.276928	+	0.960891i	$0.410684\pi$
$62$	174684.	0.0930854
$63$	−1.40780e6	−0.709330
$64$	2.25202e6	1.07385
$65$	0	0
$66$	5.53802e6	2.37110
$67$	−3.20605e6	−1.30229	−0.651147	−	0.758952i	$-0.725712\pi$
$67$	−3.20605e6	−1.30229	−0.651147	+	0.758952i	$0.725712\pi$
$68$	−1.64561e6	−0.634668
$69$	−4.48215e6	−1.64254
$70$	0	0
$71$	−1.27584e6	−0.423049	−0.211524	−	0.977373i	$-0.567843\pi$
$71$	−1.27584e6	−0.423049	−0.211524	+	0.977373i	$0.567843\pi$
$72$	−6.47075e6	−2.04311
$73$	3.51670e6	1.05805	0.529023	−	0.848607i	$-0.322558\pi$
$73$	3.51670e6	1.05805	0.529023	+	0.848607i	$0.322558\pi$
$74$	1.46106e6	0.419137
$75$	0	0
$76$	−1.18917e6	−0.310739
$77$	−2.59315e6	−0.647306
$78$	−4.81287e6	−1.14835
$79$	2.45207e6	0.559548	0.279774	−	0.960066i	$-0.409740\pi$
$79$	2.45207e6	0.559548	0.279774	+	0.960066i	$0.409740\pi$
$80$	0	0
$81$	3.08662e6	0.645336
$82$	3.43404e6	0.687791
$83$	−6.84085e6	−1.31322	−0.656609	−	0.754231i	$-0.728010\pi$
$83$	−6.84085e6	−1.31322	−0.656609	+	0.754231i	$0.728010\pi$
$84$	1.16197e6	0.213904
$85$	0	0
$86$	−363423.	−0.0616123
$87$	1.69692e7	2.76277
$88$	−1.19190e7	−1.86445
$89$	−7.49365e6	−1.12675	−0.563376	−	0.826200i	$-0.690498\pi$
$89$	−7.49365e6	−1.12675	−0.563376	+	0.826200i	$0.690498\pi$
$90$	0	0
$91$	2.25360e6	0.313496
$92$	2.41348e6	0.323136
$93$	1.50029e6	0.193413
$94$	4.17682e6	0.518678
$95$	0	0
$96$	9.34546e6	1.07808
$97$	−1.88197e6	−0.209369	−0.104685	−	0.994505i	$-0.533383\pi$
$97$	−1.88197e6	−0.209369	−0.104685	+	0.994505i	$0.533383\pi$
$98$	1.08652e6	0.116613
$99$	3.10298e7	3.21408
$100$	0	0
$101$	−1.58480e6	−0.153056	−0.0765281	−	0.997067i	$-0.524383\pi$
$101$	−1.58480e6	−0.153056	−0.0765281	+	0.997067i	$0.524383\pi$
$102$	2.82241e7	2.63342
$103$	−1.34911e7	−1.21651	−0.608257	−	0.793740i	$-0.708131\pi$
$103$	−1.34911e7	−1.21651	−0.608257	+	0.793740i	$0.708131\pi$
$104$	1.03584e7	0.902971
$105$	0	0
$106$	−1.12311e7	−0.915911
$107$	8.53266e6	0.673351	0.336675	−	0.941621i	$-0.390698\pi$
$107$	8.53266e6	0.673351	0.336675	+	0.941621i	$0.390698\pi$
$108$	−6.49542e6	−0.496163
$109$	4.40208e6	0.325586	0.162793	−	0.986660i	$-0.447950\pi$
$109$	4.40208e6	0.325586	0.162793	+	0.986660i	$0.447950\pi$
$110$	0	0
$111$	1.25485e7	0.870884
$112$	3.11889e6	0.209767
$113$	2.41828e7	1.57664	0.788319	−	0.615267i	$-0.210952\pi$
$113$	2.41828e7	1.57664	0.788319	+	0.615267i	$0.210952\pi$
$114$	2.03956e7	1.28934
$115$	0	0
$116$	−9.13731e6	−0.543520
$117$	−2.69668e7	−1.55661
$118$	−2.42807e7	−1.36043
$119$	−1.32158e7	−0.718917
$120$	0	0
$121$	3.76693e7	1.93303
$122$	9.06777e6	0.452107
$123$	2.94936e7	1.42909
$124$	−807853.	−0.0380502
$125$	0	0
$126$	−1.30014e7	−0.579019
$127$	1.10170e7	0.477255	0.238627	−	0.971111i	$-0.423303\pi$
$127$	1.10170e7	0.477255	0.238627	+	0.971111i	$0.423303\pi$
$128$	5.71675e6	0.240943
$129$	−3.12130e6	−0.128018
$130$	0	0
$131$	1.45605e7	0.565882	0.282941	−	0.959137i	$-0.408690\pi$
$131$	1.45605e7	0.565882	0.282941	+	0.959137i	$0.408690\pi$
$132$	−2.56115e7	−0.969228
$133$	−9.55010e6	−0.351987
$134$	−2.96087e7	−1.06305
$135$	0	0
$136$	−6.07445e7	−2.07072
$137$	8.01376e6	0.266265	0.133133	−	0.991098i	$-0.457496\pi$
$137$	8.01376e6	0.266265	0.133133	+	0.991098i	$0.457496\pi$
$138$	−4.13938e7	−1.34078
$139$	−2.56361e7	−0.809656	−0.404828	−	0.914393i	$-0.632669\pi$
$139$	−2.56361e7	−0.809656	−0.404828	+	0.914393i	$0.632669\pi$
$140$	0	0
$141$	3.58731e7	1.07771
$142$	−1.17827e7	−0.345330
$143$	−4.96724e7	−1.42049
$144$	−3.73210e7	−1.04156
$145$	0	0
$146$	3.24776e7	0.863672
$147$	9.33170e6	0.242298
$148$	−6.75690e6	−0.171329
$149$	4.78456e7	1.18492	0.592461	−	0.805599i	$-0.298156\pi$
$149$	4.78456e7	1.18492	0.592461	+	0.805599i	$0.298156\pi$
$150$	0	0
$151$	−7.88885e6	−0.186464	−0.0932319	−	0.995644i	$-0.529720\pi$
$151$	−7.88885e6	−0.186464	−0.0932319	+	0.995644i	$0.529720\pi$
$152$	−4.38957e7	−1.01384
$153$	1.58141e8	3.56965
$154$	−2.39484e7	−0.528389
$155$	0	0
$156$	2.22579e7	0.469406
$157$	−1.19330e7	−0.246094	−0.123047	−	0.992401i	$-0.539267\pi$
$157$	−1.19330e7	−0.246094	−0.123047	+	0.992401i	$0.539267\pi$
$158$	2.26455e7	0.456753
$159$	−9.64600e7	−1.90308
$160$	0	0
$161$	1.93824e7	0.366031
$162$	2.85058e7	0.526781
$163$	4.50349e6	0.0814503	0.0407251	−	0.999170i	$-0.487033\pi$
$163$	4.50349e6	0.0814503	0.0407251	+	0.999170i	$0.487033\pi$
$164$	−1.58813e7	−0.281146
$165$	0	0
$166$	−6.31770e7	−1.07197
$167$	−6.67726e7	−1.10941	−0.554703	−	0.832048i	$-0.687168\pi$
$167$	−6.67726e7	−1.10941	−0.554703	+	0.832048i	$0.687168\pi$
$168$	4.28918e7	0.697898
$169$	−1.95802e7	−0.312042
$170$	0	0
$171$	1.14277e8	1.74773
$172$	1.68071e6	0.0251850
$173$	−6.29382e7	−0.924173	−0.462086	−	0.886835i	$-0.652899\pi$
$173$	−6.29382e7	−0.924173	−0.462086	+	0.886835i	$0.652899\pi$
$174$	1.56715e8	2.25522
$175$	0	0
$176$	−6.87447e7	−0.950484
$177$	−2.08538e8	−2.82669
$178$	−6.92058e7	−0.919756
$179$	−1.10967e8	−1.44613	−0.723066	−	0.690779i	$-0.757268\pi$
$179$	−1.10967e8	−1.44613	−0.723066	+	0.690779i	$0.757268\pi$
$180$	0	0
$181$	−2.24619e7	−0.281560	−0.140780	−	0.990041i	$-0.544961\pi$
$181$	−2.24619e7	−0.281560	−0.140780	+	0.990041i	$0.544961\pi$
$182$	2.08126e7	0.255903
$183$	7.78796e7	0.939388
$184$	8.90886e7	1.05429
$185$	0	0
$186$	1.38556e7	0.157881
$187$	2.91294e8	3.25751
$188$	−1.93164e7	−0.212018
$189$	−5.21642e7	−0.562026
$190$	0	0
$191$	−9.39656e7	−0.975780	−0.487890	−	0.872905i	$-0.662233\pi$
$191$	−9.39656e7	−0.975780	−0.487890	+	0.872905i	$0.662233\pi$
$192$	1.78626e8	1.82134
$193$	4.11603e7	0.412124	0.206062	−	0.978539i	$-0.433935\pi$
$193$	4.11603e7	0.412124	0.206062	+	0.978539i	$0.433935\pi$
$194$	−1.73805e7	−0.170906
$195$	0	0
$196$	−5.02478e6	−0.0476674
$197$	−1.89027e8	−1.76154	−0.880770	−	0.473544i	$-0.842974\pi$
$197$	−1.89027e8	−1.76154	−0.880770	+	0.473544i	$0.842974\pi$
$198$	2.86568e8	2.62362
$199$	−1.55243e8	−1.39645	−0.698225	−	0.715878i	$-0.746027\pi$
$199$	−1.55243e8	−1.39645	−0.698225	+	0.715878i	$0.746027\pi$
$200$	0	0
$201$	−2.54298e8	−2.20880
$202$	−1.46361e7	−0.124938
$203$	−7.33809e7	−0.615669
$204$	−1.30527e8	−1.07645
$205$	0	0
$206$	−1.24594e8	−0.993028
$207$	−2.31932e8	−1.81746
$208$	5.97433e7	0.460328
$209$	2.10497e8	1.59490
$210$	0	0
$211$	−1.66141e8	−1.21755	−0.608777	−	0.793341i	$-0.708340\pi$
$211$	−1.66141e8	−1.21755	−0.608777	+	0.793341i	$0.708340\pi$
$212$	5.19402e7	0.374394
$213$	−1.01197e8	−0.717528
$214$	7.88013e7	0.549649
$215$	0	0
$216$	−2.39765e8	−1.61882
$217$	−6.48780e6	−0.0431011
$218$	4.06544e7	0.265772
$219$	2.78938e8	1.79454
$220$	0	0
$221$	−2.53152e8	−1.57764
$222$	1.15888e8	0.710893
$223$	1.64063e8	0.990704	0.495352	−	0.868692i	$-0.335039\pi$
$223$	1.64063e8	0.990704	0.495352	+	0.868692i	$0.335039\pi$
$224$	−4.04131e7	−0.240245
$225$	0	0
$226$	2.23334e8	1.28699
$227$	−5.02061e7	−0.284882	−0.142441	−	0.989803i	$-0.545495\pi$
$227$	−5.02061e7	−0.284882	−0.142441	+	0.989803i	$0.545495\pi$
$228$	−9.43226e7	−0.527040
$229$	2.77148e8	1.52506	0.762531	−	0.646952i	$-0.223957\pi$
$229$	2.77148e8	1.52506	0.762531	+	0.646952i	$0.223957\pi$
$230$	0	0
$231$	−2.05684e8	−1.09789
$232$	−3.37285e8	−1.77333
$233$	2.11829e7	0.109708	0.0548542	−	0.998494i	$-0.482531\pi$
$233$	2.11829e7	0.109708	0.0548542	+	0.998494i	$0.482531\pi$
$234$	−2.49045e8	−1.27064
$235$	0	0
$236$	1.12290e8	0.556096
$237$	1.94493e8	0.949042
$238$	−1.22051e8	−0.586844
$239$	3.93164e8	1.86287	0.931433	−	0.363913i	$-0.118560\pi$
$239$	3.93164e8	1.86287	0.931433	+	0.363913i	$0.118560\pi$
$240$	0	0
$241$	3.44922e8	1.58731	0.793654	−	0.608369i	$-0.208176\pi$
$241$	3.44922e8	1.58731	0.793654	+	0.608369i	$0.208176\pi$
$242$	3.47886e8	1.57791
$243$	−8.77786e7	−0.392434
$244$	−4.19354e7	−0.184806
$245$	0	0
$246$	2.72381e8	1.16655
$247$	−1.82935e8	−0.772426
$248$	−2.98203e7	−0.124145
$249$	−5.42604e8	−2.22733
$250$	0	0
$251$	−1.84645e7	−0.0737021	−0.0368511	−	0.999321i	$-0.511733\pi$
$251$	−1.84645e7	−0.0737021	−0.0368511	+	0.999321i	$0.511733\pi$
$252$	6.01270e7	0.236683
$253$	−4.27215e8	−1.65854
$254$	1.01745e8	0.389578
$255$	0	0
$256$	−2.35463e8	−0.877169
$257$	−1.09956e8	−0.404067	−0.202033	−	0.979379i	$-0.564755\pi$
$257$	−1.09956e8	−0.404067	−0.202033	+	0.979379i	$0.564755\pi$
$258$	−2.88260e7	−0.104500
$259$	−5.42640e7	−0.194072
$260$	0	0
$261$	8.78083e8	3.05699
$262$	1.34470e8	0.461924
$263$	−4.84965e8	−1.64386	−0.821931	−	0.569587i	$-0.807103\pi$
$263$	−4.84965e8	−1.64386	−0.821931	+	0.569587i	$0.807103\pi$
$264$	−9.45395e8	−3.16228
$265$	0	0
$266$	−8.81977e7	−0.287324
$267$	−5.94383e8	−1.91107
$268$	1.36930e8	0.434538
$269$	−1.59008e8	−0.498065	−0.249033	−	0.968495i	$-0.580113\pi$
$269$	−1.59008e8	−0.498065	−0.249033	+	0.968495i	$0.580113\pi$
$270$	0	0
$271$	−7.76177e7	−0.236902	−0.118451	−	0.992960i	$-0.537793\pi$
$271$	−7.76177e7	−0.236902	−0.118451	+	0.992960i	$0.537793\pi$
$272$	−3.50352e8	−1.05564
$273$	1.78751e8	0.531716
$274$	7.40092e7	0.217350
$275$	0	0
$276$	1.91433e8	0.548067
$277$	−2.83761e8	−0.802183	−0.401091	−	0.916038i	$-0.631369\pi$
$277$	−2.83761e8	−0.802183	−0.401091	+	0.916038i	$0.631369\pi$
$278$	−2.36756e8	−0.660914
$279$	7.76336e7	0.214010
$280$	0	0
$281$	1.17071e8	0.314758	0.157379	−	0.987538i	$-0.449696\pi$
$281$	1.17071e8	0.314758	0.157379	+	0.987538i	$0.449696\pi$
$282$	3.31297e8	0.879724
$283$	3.12939e7	0.0820743	0.0410372	−	0.999158i	$-0.486934\pi$
$283$	3.12939e7	0.0820743	0.0410372	+	0.999158i	$0.486934\pi$
$284$	5.44909e7	0.141159
$285$	0	0
$286$	−4.58738e8	−1.15953
$287$	−1.27541e8	−0.318466
$288$	4.83587e8	1.19289
$289$	1.07422e9	2.61789
$290$	0	0
$291$	−1.49275e8	−0.355108
$292$	−1.50198e8	−0.353040
$293$	1.20452e8	0.279754	0.139877	−	0.990169i	$-0.455329\pi$
$293$	1.20452e8	0.279754	0.139877	+	0.990169i	$0.455329\pi$
$294$	8.61807e7	0.197785
$295$	0	0
$296$	−2.49417e8	−0.558991
$297$	1.14977e9	2.54662
$298$	4.41867e8	0.967240
$299$	3.71276e8	0.803244
$300$	0	0
$301$	1.34976e7	0.0285282
$302$	−7.28556e7	−0.152208
$303$	−1.25704e8	−0.259597
$304$	−2.53175e8	−0.516848
$305$	0	0
$306$	1.46048e9	2.91386
$307$	5.66299e8	1.11702	0.558511	−	0.829497i	$-0.311373\pi$
$307$	5.66299e8	1.11702	0.558511	+	0.829497i	$0.311373\pi$
$308$	1.10753e8	0.215987
$309$	−1.07009e9	−2.06331
$310$	0	0
$311$	7.71752e8	1.45484	0.727422	−	0.686190i	$-0.240718\pi$
$311$	7.71752e8	1.45484	0.727422	+	0.686190i	$0.240718\pi$
$312$	8.21605e8	1.53152
$313$	−4.57295e7	−0.0842930	−0.0421465	−	0.999111i	$-0.513420\pi$
$313$	−4.57295e7	−0.0842930	−0.0421465	+	0.999111i	$0.513420\pi$
$314$	−1.10204e8	−0.200884
$315$	0	0
$316$	−1.04728e8	−0.186705
$317$	−1.11428e9	−1.96466	−0.982328	−	0.187166i	$-0.940070\pi$
$317$	−1.11428e9	−1.96466	−0.982328	+	0.187166i	$0.940070\pi$
$318$	−8.90833e8	−1.55347
$319$	1.61742e9	2.78968
$320$	0	0
$321$	6.76795e8	1.14206
$322$	1.79002e8	0.298787
$323$	1.07278e9	1.77135
$324$	−1.31830e8	−0.215330
$325$	0	0
$326$	4.15909e7	0.0664870
$327$	3.49165e8	0.552222
$328$	−5.86224e8	−0.917287
$329$	−1.55128e8	−0.240162
$330$	0	0
$331$	−9.22898e7	−0.139880	−0.0699400	−	0.997551i	$-0.522281\pi$
$331$	−9.22898e7	−0.139880	−0.0699400	+	0.997551i	$0.522281\pi$
$332$	2.92172e8	0.438183
$333$	6.49328e8	0.963629
$334$	−6.16662e8	−0.905596
$335$	0	0
$336$	2.47385e8	0.355783
$337$	1.28496e9	1.82888	0.914441	−	0.404720i	$-0.132631\pi$
$337$	1.28496e9	1.82888	0.914441	+	0.404720i	$0.132631\pi$
$338$	−1.80828e8	−0.254717
$339$	1.91814e9	2.67412
$340$	0	0
$341$	1.43000e8	0.195297
$342$	1.05538e9	1.42665
$343$	−4.03536e7	−0.0539949
$344$	6.20399e7	0.0821706
$345$	0	0
$346$	−5.81251e8	−0.754392
$347$	4.16453e8	0.535073	0.267537	−	0.963548i	$-0.413790\pi$
$347$	4.16453e8	0.535073	0.267537	+	0.963548i	$0.413790\pi$
$348$	−7.24755e8	−0.921857
$349$	−2.99825e7	−0.0377554	−0.0188777	−	0.999822i	$-0.506009\pi$
$349$	−2.99825e7	−0.0377554	−0.0188777	+	0.999822i	$0.506009\pi$
$350$	0	0
$351$	−9.99220e8	−1.23335
$352$	8.90760e8	1.08858
$353$	−1.57977e9	−1.91153	−0.955765	−	0.294130i	$-0.904970\pi$
$353$	−1.57977e9	−1.91153	−0.955765	+	0.294130i	$0.904970\pi$
$354$	−1.92590e9	−2.30740
$355$	0	0
$356$	3.20054e8	0.375965
$357$	−1.04825e9	−1.21935
$358$	−1.02481e9	−1.18046
$359$	−1.33098e9	−1.51824	−0.759119	−	0.650952i	$-0.774370\pi$
$359$	−1.33098e9	−1.51824	−0.759119	+	0.650952i	$0.774370\pi$
$360$	0	0
$361$	−1.18647e8	−0.132734
$362$	−2.07442e8	−0.229835
$363$	2.98786e9	3.27859
$364$	−9.62512e7	−0.104605
$365$	0	0
$366$	7.19238e8	0.766812
$367$	1.00006e9	1.05607	0.528035	−	0.849223i	$-0.322929\pi$
$367$	1.00006e9	1.05607	0.528035	+	0.849223i	$0.322929\pi$
$368$	5.13831e8	0.537469
$369$	1.52617e9	1.58128
$370$	0	0
$371$	4.17127e8	0.424092
$372$	−6.40774e7	−0.0645364
$373$	−1.31150e9	−1.30854	−0.654270	−	0.756261i	$-0.727024\pi$
$373$	−1.31150e9	−1.30854	−0.654270	+	0.756261i	$0.727024\pi$
$374$	2.69018e9	2.65907
$375$	0	0
$376$	−7.13024e8	−0.691746
$377$	−1.40563e9	−1.35107
$378$	−4.81750e8	−0.458776
$379$	7.55529e8	0.712876	0.356438	−	0.934319i	$-0.383991\pi$
$379$	7.55529e8	0.712876	0.356438	+	0.934319i	$0.383991\pi$
$380$	0	0
$381$	8.73848e8	0.809466
$382$	−8.67797e8	−0.796519
$383$	5.92335e8	0.538731	0.269365	−	0.963038i	$-0.413186\pi$
$383$	5.92335e8	0.538731	0.269365	+	0.963038i	$0.413186\pi$
$384$	4.53442e8	0.408660
$385$	0	0
$386$	3.80126e8	0.336412
$387$	−1.61514e8	−0.141651
$388$	8.03791e7	0.0698605
$389$	−1.13635e8	−0.0978784	−0.0489392	−	0.998802i	$-0.515584\pi$
$389$	−1.13635e8	−0.0978784	−0.0489392	+	0.998802i	$0.515584\pi$
$390$	0	0
$391$	−2.17727e9	−1.84202
$392$	−1.85480e8	−0.155523
$393$	1.15491e9	0.959786
$394$	−1.74572e9	−1.43793
$395$	0	0
$396$	−1.32528e9	−1.07245
$397$	−1.63458e9	−1.31111	−0.655557	−	0.755145i	$-0.727566\pi$
$397$	−1.63458e9	−1.31111	−0.655557	+	0.755145i	$0.727566\pi$
$398$	−1.43371e9	−1.13991
$399$	−7.57496e8	−0.597001
$400$	0	0
$401$	7.10604e8	0.550329	0.275165	−	0.961397i	$-0.411268\pi$
$401$	7.10604e8	0.550329	0.275165	+	0.961397i	$0.411268\pi$
$402$	−2.34851e9	−1.80302
$403$	−1.24276e8	−0.0945841
$404$	6.76869e7	0.0510705
$405$	0	0
$406$	−6.77692e8	−0.502564
$407$	1.19605e9	0.879367
$408$	−4.81814e9	−3.51212
$409$	−9.79887e8	−0.708181	−0.354090	−	0.935211i	$-0.615210\pi$
$409$	−9.79887e8	−0.708181	−0.354090	+	0.935211i	$0.615210\pi$
$410$	0	0
$411$	6.35637e8	0.451609
$412$	5.76204e8	0.405916
$413$	9.01793e8	0.629915
$414$	−2.14195e9	−1.48357
$415$	0	0
$416$	−7.74124e8	−0.527210
$417$	−2.03341e9	−1.37325
$418$	1.94400e9	1.30190
$419$	5.71474e8	0.379531	0.189766	−	0.981829i	$-0.439227\pi$
$419$	5.71474e8	0.379531	0.189766	+	0.981829i	$0.439227\pi$
$420$	0	0
$421$	−2.47927e8	−0.161933	−0.0809667	−	0.996717i	$-0.525801\pi$
$421$	−2.47927e8	−0.161933	−0.0809667	+	0.996717i	$0.525801\pi$
$422$	−1.53436e9	−0.993876
$423$	1.85628e9	1.19248
$424$	1.91727e9	1.22152
$425$	0	0
$426$	−9.34580e8	−0.585710
$427$	−3.36779e8	−0.209338
$428$	−3.64430e8	−0.224678
$429$	−3.93993e9	−2.40928
$430$	0	0
$431$	−2.65433e8	−0.159693	−0.0798463	−	0.996807i	$-0.525443\pi$
$431$	−2.65433e8	−0.159693	−0.0798463	+	0.996807i	$0.525443\pi$
$432$	−1.38288e9	−0.825262
$433$	−1.66732e9	−0.986989	−0.493494	−	0.869749i	$-0.664281\pi$
$433$	−1.66732e9	−0.986989	−0.493494	+	0.869749i	$0.664281\pi$
$434$	−5.99165e7	−0.0351830
$435$	0	0
$436$	−1.88013e8	−0.108639
$437$	−1.57336e9	−0.901868
$438$	2.57606e9	1.46486
$439$	−1.21657e9	−0.686294	−0.343147	−	0.939282i	$-0.611493\pi$
$439$	−1.21657e9	−0.686294	−0.343147	+	0.939282i	$0.611493\pi$
$440$	0	0
$441$	4.82875e8	0.268102
$442$	−2.33792e9	−1.28781
$443$	−1.24380e9	−0.679734	−0.339867	−	0.940474i	$-0.610382\pi$
$443$	−1.24380e9	−0.679734	−0.339867	+	0.940474i	$0.610382\pi$
$444$	−5.35944e8	−0.290589
$445$	0	0
$446$	1.51517e9	0.808701
$447$	3.79502e9	2.00973
$448$	−7.72444e8	−0.405876
$449$	1.95796e8	0.102080	0.0510402	−	0.998697i	$-0.483746\pi$
$449$	1.95796e8	0.102080	0.0510402	+	0.998697i	$0.483746\pi$
$450$	0	0
$451$	2.81118e9	1.44301
$452$	−1.03285e9	−0.526079
$453$	−6.25729e8	−0.316259
$454$	−4.63666e8	−0.232547
$455$	0	0
$456$	−3.48172e9	−1.71956
$457$	3.69582e9	1.81136	0.905679	−	0.423965i	$-0.139362\pi$
$457$	3.69582e9	1.81136	0.905679	+	0.423965i	$0.139362\pi$
$458$	2.55953e9	1.24489
$459$	5.85972e9	2.82835
$460$	0	0
$461$	−3.05324e9	−1.45147	−0.725733	−	0.687976i	$-0.758499\pi$
$461$	−3.05324e9	−1.45147	−0.725733	+	0.687976i	$0.758499\pi$
$462$	−1.89954e9	−0.896193
$463$	−9.99765e7	−0.0468128	−0.0234064	−	0.999726i	$-0.507451\pi$
$463$	−9.99765e7	−0.0468128	−0.0234064	+	0.999726i	$0.507451\pi$
$464$	−1.94534e9	−0.904030
$465$	0	0
$466$	1.95630e8	0.0895539
$467$	4.57973e8	0.208080	0.104040	−	0.994573i	$-0.466823\pi$
$467$	4.57973e8	0.208080	0.104040	+	0.994573i	$0.466823\pi$
$468$	1.15175e9	0.519395
$469$	1.09968e9	0.492221
$470$	0	0
$471$	−9.46504e8	−0.417397
$472$	4.14496e9	1.81436
$473$	−2.97506e8	−0.129265
$474$	1.79620e9	0.774693
$475$	0	0
$476$	5.64445e8	0.239882
$477$	−4.99139e9	−2.10575
$478$	3.63098e9	1.52064
$479$	2.97851e9	1.23830	0.619149	−	0.785274i	$-0.287478\pi$
$479$	2.97851e9	1.23830	0.619149	+	0.785274i	$0.287478\pi$
$480$	0	0
$481$	−1.03944e9	−0.425886
$482$	3.18545e9	1.29570
$483$	1.53738e9	0.620820
$484$	−1.60886e9	−0.644998
$485$	0	0
$486$	−8.10658e8	−0.320340
$487$	3.51262e9	1.37810	0.689048	−	0.724716i	$-0.258029\pi$
$487$	3.51262e9	1.37810	0.689048	+	0.724716i	$0.258029\pi$
$488$	−1.54796e9	−0.602962
$489$	3.57209e8	0.138147
$490$	0	0
$491$	3.13708e9	1.19603	0.598013	−	0.801487i	$-0.295957\pi$
$491$	3.13708e9	1.19603	0.598013	+	0.801487i	$0.295957\pi$
$492$	−1.25967e9	−0.476848
$493$	8.24305e9	3.09830
$494$	−1.68945e9	−0.630524
$495$	0	0
$496$	−1.71993e8	−0.0632884
$497$	4.37612e8	0.159897
$498$	−5.01109e9	−1.81815
$499$	−2.35673e9	−0.849097	−0.424549	−	0.905405i	$-0.639567\pi$
$499$	−2.35673e9	−0.849097	−0.424549	+	0.905405i	$0.639567\pi$
$500$	0	0
$501$	−5.29628e9	−1.88165
$502$	−1.70525e8	−0.0601623
$503$	4.78527e9	1.67656	0.838279	−	0.545242i	$-0.183562\pi$
$503$	4.78527e9	1.67656	0.838279	+	0.545242i	$0.183562\pi$
$504$	2.21947e9	0.772221
$505$	0	0
$506$	−3.94545e9	−1.35385
$507$	−1.55306e9	−0.529251
$508$	−4.70536e8	−0.159246
$509$	−2.10557e9	−0.707713	−0.353856	−	0.935300i	$-0.615130\pi$
$509$	−2.10557e9	−0.707713	−0.353856	+	0.935300i	$0.615130\pi$
$510$	0	0
$511$	−1.20623e9	−0.399904
$512$	−2.90631e9	−0.956967
$513$	4.23440e9	1.38478
$514$	−1.01547e9	−0.329836
$515$	0	0
$516$	1.33311e8	0.0427160
$517$	3.41924e9	1.08821
$518$	−5.01143e8	−0.158419
$519$	−4.99214e9	−1.56748
$520$	0	0
$521$	2.25442e9	0.698397	0.349198	−	0.937049i	$-0.386454\pi$
$521$	2.25442e9	0.698397	0.349198	+	0.937049i	$0.386454\pi$
$522$	8.10933e9	2.49539
$523$	3.32175e9	1.01534	0.507670	−	0.861552i	$-0.330507\pi$
$523$	3.32175e9	1.01534	0.507670	+	0.861552i	$0.330507\pi$
$524$	−6.21877e8	−0.188819
$525$	0	0
$526$	−4.47878e9	−1.34187
$527$	7.28789e8	0.216903
$528$	−5.45270e9	−1.61210
$529$	−2.11611e8	−0.0621502
$530$	0	0
$531$	−1.07909e10	−3.12772
$532$	4.07884e8	0.117448
$533$	−2.44308e9	−0.698865
$534$	−5.48928e9	−1.55999
$535$	0	0
$536$	5.05451e9	1.41776
$537$	−8.80169e9	−2.45277
$538$	−1.46848e9	−0.406565
$539$	8.89449e8	0.244659
$540$	0	0
$541$	1.22668e9	0.333074	0.166537	−	0.986035i	$-0.446741\pi$
$541$	1.22668e9	0.333074	0.166537	+	0.986035i	$0.446741\pi$
$542$	−7.16820e8	−0.193380
$543$	−1.78164e9	−0.477551
$544$	4.53970e9	1.20901
$545$	0	0
$546$	1.65082e9	0.434035
$547$	−2.52060e9	−0.658489	−0.329244	−	0.944245i	$-0.606794\pi$
$547$	−2.52060e9	−0.658489	−0.329244	+	0.944245i	$0.606794\pi$
$548$	−3.42268e8	−0.0888452
$549$	4.02993e9	1.03943
$550$	0	0
$551$	5.95666e9	1.51695
$552$	7.06634e9	1.78817
$553$	−8.41059e8	−0.211489
$554$	−2.62061e9	−0.654813
$555$	0	0
$556$	1.09492e9	0.270159
$557$	−6.27626e9	−1.53889	−0.769445	−	0.638713i	$-0.779467\pi$
$557$	−6.27626e9	−1.53889	−0.769445	+	0.638713i	$0.779467\pi$
$558$	7.16967e8	0.174694
$559$	2.58551e8	0.0626043
$560$	0	0
$561$	2.31049e10	5.52502
$562$	1.08118e9	0.256933
$563$	−1.45170e9	−0.342844	−0.171422	−	0.985198i	$-0.554836\pi$
$563$	−1.45170e9	−0.342844	−0.171422	+	0.985198i	$0.554836\pi$
$564$	−1.53214e9	−0.359601
$565$	0	0
$566$	2.89007e8	0.0669964
$567$	−1.05871e9	−0.243914
$568$	2.01142e9	0.460557
$569$	5.99543e9	1.36436	0.682178	−	0.731186i	$-0.261033\pi$
$569$	5.99543e9	1.36436	0.682178	+	0.731186i	$0.261033\pi$
$570$	0	0
$571$	1.09750e9	0.246705	0.123353	−	0.992363i	$-0.460635\pi$
$571$	1.09750e9	0.246705	0.123353	+	0.992363i	$0.460635\pi$
$572$	2.12151e9	0.473978
$573$	−7.45318e9	−1.65501
$574$	−1.17787e9	−0.259960
$575$	0	0
$576$	9.24313e9	2.01530
$577$	3.33317e9	0.722341	0.361170	−	0.932500i	$-0.382377\pi$
$577$	3.33317e9	0.722341	0.361170	+	0.932500i	$0.382377\pi$
$578$	9.92070e9	2.13695
$579$	3.26476e9	0.698998
$580$	0	0
$581$	2.34641e9	0.496350
$582$	−1.37859e9	−0.289871
$583$	−9.19406e9	−1.92162
$584$	−5.54425e9	−1.15185
$585$	0	0
$586$	1.11240e9	0.228360
$587$	5.29160e9	1.07983	0.539913	−	0.841721i	$-0.318457\pi$
$587$	5.29160e9	1.07983	0.539913	+	0.841721i	$0.318457\pi$
$588$	−3.98557e8	−0.0808480
$589$	5.26644e8	0.106197
$590$	0	0
$591$	−1.49933e10	−2.98773
$592$	−1.43855e9	−0.284970
$593$	−2.56026e9	−0.504189	−0.252094	−	0.967703i	$-0.581119\pi$
$593$	−2.56026e9	−0.504189	−0.252094	+	0.967703i	$0.581119\pi$
$594$	1.06184e10	2.07878
$595$	0	0
$596$	−2.04348e9	−0.395375
$597$	−1.23136e10	−2.36850
$598$	3.42883e9	0.655680
$599$	3.35967e9	0.638708	0.319354	−	0.947636i	$-0.396534\pi$
$599$	3.35967e9	0.638708	0.319354	+	0.947636i	$0.396534\pi$
$600$	0	0
$601$	4.80116e9	0.902164	0.451082	−	0.892482i	$-0.351038\pi$
$601$	4.80116e9	0.902164	0.451082	+	0.892482i	$0.351038\pi$
$602$	1.24654e8	0.0232873
$603$	−1.31588e10	−2.44403
$604$	3.36933e8	0.0622177
$605$	0	0
$606$	−1.16091e9	−0.211906
$607$	−3.29532e9	−0.598049	−0.299024	−	0.954245i	$-0.596661\pi$
$607$	−3.29532e9	−0.598049	−0.299024	+	0.954245i	$0.596661\pi$
$608$	3.28051e9	0.591942
$609$	−5.82044e9	−1.04423
$610$	0	0
$611$	−2.97152e9	−0.527029
$612$	−6.75421e9	−1.19109
$613$	−7.72517e9	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$613$	−7.72517e9	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$614$	5.22992e9	0.911813
$615$	0	0
$616$	4.08823e9	0.704697
$617$	−3.53848e9	−0.606483	−0.303242	−	0.952914i	$-0.598069\pi$
$617$	−3.53848e9	−0.606483	−0.303242	+	0.952914i	$0.598069\pi$
$618$	−9.88255e9	−1.68426
$619$	−3.48297e9	−0.590246	−0.295123	−	0.955459i	$-0.595361\pi$
$619$	−3.48297e9	−0.590246	−0.295123	+	0.955459i	$0.595361\pi$
$620$	0	0
$621$	−8.59394e9	−1.44003
$622$	7.12733e9	1.18757
$623$	2.57032e9	0.425872
$624$	4.73873e9	0.780757
$625$	0	0
$626$	−4.22324e8	−0.0688075
$627$	1.66963e10	2.70510
$628$	5.09658e8	0.0821146
$629$	6.09560e9	0.976651
$630$	0	0
$631$	−4.08998e9	−0.648065	−0.324033	−	0.946046i	$-0.605039\pi$
$631$	−4.08998e9	−0.648065	−0.324033	+	0.946046i	$0.605039\pi$
$632$	−3.86581e9	−0.609159
$633$	−1.31780e10	−2.06508
$634$	−1.02907e10	−1.60373
$635$	0	0
$636$	4.11980e9	0.635004
$637$	−7.72985e8	−0.118490
$638$	1.49373e10	2.27719
$639$	−5.23650e9	−0.793941
$640$	0	0
$641$	2.42218e9	0.363249	0.181624	−	0.983368i	$-0.441865\pi$
$641$	2.42218e9	0.363249	0.181624	+	0.983368i	$0.441865\pi$
$642$	6.25038e9	0.932253
$643$	6.08579e9	0.902773	0.451387	−	0.892328i	$-0.350930\pi$
$643$	6.08579e9	0.902773	0.451387	+	0.892328i	$0.350930\pi$
$644$	−8.27823e8	−0.122134
$645$	0	0
$646$	9.90745e9	1.44593
$647$	−5.52131e9	−0.801451	−0.400726	−	0.916198i	$-0.631242\pi$
$647$	−5.52131e9	−0.801451	−0.400726	+	0.916198i	$0.631242\pi$
$648$	−4.86622e9	−0.702553
$649$	−1.98768e10	−2.85423
$650$	0	0
$651$	−5.14600e8	−0.0731032
$652$	−1.92344e8	−0.0271776
$653$	7.06205e9	0.992510	0.496255	−	0.868177i	$-0.334708\pi$
$653$	7.06205e9	0.992510	0.496255	+	0.868177i	$0.334708\pi$
$654$	3.22463e9	0.450773
$655$	0	0
$656$	−3.38113e9	−0.467626
$657$	1.44338e10	1.98565
$658$	−1.43265e9	−0.196042
$659$	3.89902e9	0.530709	0.265354	−	0.964151i	$-0.414511\pi$
$659$	3.89902e9	0.530709	0.265354	+	0.964151i	$0.414511\pi$
$660$	0	0
$661$	4.85556e9	0.653934	0.326967	−	0.945036i	$-0.393973\pi$
$661$	4.85556e9	0.653934	0.326967	+	0.945036i	$0.393973\pi$
$662$	−8.52320e8	−0.114183
$663$	−2.00795e10	−2.67582
$664$	1.07850e10	1.42965
$665$	0	0
$666$	5.99672e9	0.786600
$667$	−1.20894e10	−1.57748
$668$	2.85185e9	0.370177
$669$	1.30132e10	1.68032
$670$	0	0
$671$	7.42308e9	0.948539
$672$	−3.20549e9	−0.407476
$673$	−4.47378e9	−0.565747	−0.282873	−	0.959157i	$-0.591288\pi$
$673$	−4.47378e9	−0.565747	−0.282873	+	0.959157i	$0.591288\pi$
$674$	1.18670e10	1.49290
$675$	0	0
$676$	8.36269e8	0.104120
$677$	2.69889e9	0.334291	0.167146	−	0.985932i	$-0.446545\pi$
$677$	2.69889e9	0.334291	0.167146	+	0.985932i	$0.446545\pi$
$678$	1.77145e10	2.18285
$679$	6.45517e8	0.0791341
$680$	0	0
$681$	−3.98225e9	−0.483185
$682$	1.32064e9	0.159419
$683$	1.01893e10	1.22370	0.611848	−	0.790975i	$-0.290426\pi$
$683$	1.01893e10	1.22370	0.611848	+	0.790975i	$0.290426\pi$
$684$	−4.88078e9	−0.583167
$685$	0	0
$686$	−3.72676e8	−0.0440755
$687$	2.19829e10	2.58664
$688$	3.57824e8	0.0418900
$689$	7.99019e9	0.930658
$690$	0	0
$691$	−1.48923e10	−1.71707	−0.858535	−	0.512755i	$-0.828625\pi$
$691$	−1.48923e10	−1.71707	−0.858535	+	0.512755i	$0.828625\pi$
$692$	2.68809e9	0.308370
$693$	−1.06432e10	−1.21481
$694$	3.84605e9	0.436775
$695$	0	0
$696$	−2.67528e10	−3.00772
$697$	1.43270e10	1.60265
$698$	−2.76896e8	−0.0308193
$699$	1.68019e9	0.186075
$700$	0	0
$701$	1.27757e10	1.40079	0.700395	−	0.713755i	$-0.253007\pi$
$701$	1.27757e10	1.40079	0.700395	+	0.713755i	$0.253007\pi$
$702$	−9.22805e9	−1.00677
$703$	4.40486e9	0.478177
$704$	1.70257e10	1.83908
$705$	0	0
$706$	−1.45895e10	−1.56036
$707$	5.43588e8	0.0578498
$708$	8.90665e9	0.943188
$709$	−7.19766e9	−0.758454	−0.379227	−	0.925304i	$-0.623810\pi$
$709$	−7.19766e9	−0.758454	−0.379227	+	0.925304i	$0.623810\pi$
$710$	0	0
$711$	1.00642e10	1.05011
$712$	1.18141e10	1.22665
$713$	−1.06885e9	−0.110434
$714$	−9.68087e9	−0.995339
$715$	0	0
$716$	4.73939e9	0.482533
$717$	3.11851e10	3.15958
$718$	−1.22919e10	−1.23932
$719$	−8.05856e9	−0.808549	−0.404275	−	0.914638i	$-0.632476\pi$
$719$	−8.05856e9	−0.808549	−0.404275	+	0.914638i	$0.632476\pi$
$720$	0	0
$721$	4.62745e9	0.459799
$722$	−1.09574e9	−0.108349
$723$	2.73586e10	2.69221
$724$	9.59347e8	0.0939487
$725$	0	0
$726$	2.75937e10	2.67628
$727$	−1.83829e10	−1.77437	−0.887184	−	0.461416i	$-0.847341\pi$
$727$	−1.83829e10	−1.77437	−0.887184	+	0.461416i	$0.847341\pi$
$728$	−3.55291e9	−0.341291
$729$	−1.37129e10	−1.31094
$730$	0	0
$731$	−1.51622e9	−0.143566
$732$	−3.32624e9	−0.313447
$733$	1.09787e10	1.02964	0.514822	−	0.857297i	$-0.327858\pi$
$733$	1.09787e10	1.02964	0.514822	+	0.857297i	$0.327858\pi$
$734$	9.23577e9	0.862059
$735$	0	0
$736$	−6.65797e9	−0.615559
$737$	−2.42384e10	−2.23032
$738$	1.40946e10	1.29079
$739$	−7.57794e9	−0.690710	−0.345355	−	0.938472i	$-0.612242\pi$
$739$	−7.57794e9	−0.690710	−0.345355	+	0.938472i	$0.612242\pi$
$740$	0	0
$741$	−1.45101e10	−1.31010
$742$	3.85228e9	0.346182
$743$	−1.79243e10	−1.60318	−0.801590	−	0.597875i	$-0.796012\pi$
$743$	−1.79243e10	−1.60318	−0.801590	+	0.597875i	$0.796012\pi$
$744$	−2.36529e9	−0.210561
$745$	0	0
$746$	−1.21120e10	−1.06815
$747$	−2.80774e10	−2.46453
$748$	−1.24412e10	−1.08694
$749$	−2.92670e9	−0.254503
$750$	0	0
$751$	2.31633e9	0.199554	0.0997771	−	0.995010i	$-0.468187\pi$
$751$	2.31633e9	0.199554	0.0997771	+	0.995010i	$0.468187\pi$
$752$	−4.11247e9	−0.352647
$753$	−1.46457e9	−0.125005
$754$	−1.29814e10	−1.10286
$755$	0	0
$756$	2.22793e9	0.187532
$757$	−3.38357e8	−0.0283491	−0.0141746	−	0.999900i	$-0.504512\pi$
$757$	−3.38357e8	−0.0283491	−0.0141746	+	0.999900i	$0.504512\pi$
$758$	6.97751e9	0.581913
$759$	−3.38859e10	−2.81302
$760$	0	0
$761$	1.74775e10	1.43758	0.718792	−	0.695225i	$-0.244695\pi$
$761$	1.74775e10	1.43758	0.718792	+	0.695225i	$0.244695\pi$
$762$	8.07021e9	0.660758
$763$	−1.50992e9	−0.123060
$764$	4.01327e9	0.325590
$765$	0	0
$766$	5.47037e9	0.439760
$767$	1.72741e10	1.38233
$768$	−1.86765e10	−1.48776
$769$	1.68650e10	1.33735	0.668673	−	0.743557i	$-0.266863\pi$
$769$	1.68650e10	1.33735	0.668673	+	0.743557i	$0.266863\pi$
$770$	0	0
$771$	−8.72151e9	−0.685333
$772$	−1.75795e9	−0.137514
$773$	1.28837e10	1.00326	0.501630	−	0.865082i	$-0.332734\pi$
$773$	1.28837e10	1.00326	0.501630	+	0.865082i	$0.332734\pi$
$774$	−1.49162e9	−0.115629
$775$	0	0
$776$	2.96703e9	0.227932
$777$	−4.30412e9	−0.329163
$778$	−1.04944e9	−0.0798971
$779$	1.03531e10	0.784673
$780$	0	0
$781$	−9.64556e9	−0.724518
$782$	−2.01077e10	−1.50362
$783$	3.25363e10	2.42215
$784$	−1.06978e9	−0.0792845
$785$	0	0
$786$	1.06659e10	0.783463
$787$	−2.67614e9	−0.195703	−0.0978514	−	0.995201i	$-0.531197\pi$
$787$	−2.67614e9	−0.195703	−0.0978514	+	0.995201i	$0.531197\pi$
$788$	8.07334e9	0.587776
$789$	−3.84665e10	−2.78813
$790$	0	0
$791$	−8.29470e9	−0.595913
$792$	−4.89201e10	−3.49904
$793$	−6.45110e9	−0.459386
$794$	−1.50958e10	−1.07025
$795$	0	0
$796$	6.63041e9	0.465956
$797$	−1.44598e10	−1.01171	−0.505857	−	0.862617i	$-0.668824\pi$
$797$	−1.44598e10	−1.01171	−0.505857	+	0.862617i	$0.668824\pi$
$798$	−6.99568e9	−0.487326
$799$	1.74259e10	1.20860
$800$	0	0
$801$	−3.07567e10	−2.11459
$802$	6.56262e9	0.449228
$803$	2.65869e10	1.81202
$804$	1.08611e10	0.737015
$805$	0	0
$806$	−1.14772e9	−0.0772080
$807$	−1.26122e10	−0.844762
$808$	2.49853e9	0.166626
$809$	−1.91699e10	−1.27292	−0.636459	−	0.771310i	$-0.719602\pi$
$809$	−1.91699e10	−1.27292	−0.636459	+	0.771310i	$0.719602\pi$
$810$	0	0
$811$	7.49695e9	0.493528	0.246764	−	0.969076i	$-0.420633\pi$
$811$	7.49695e9	0.493528	0.246764	+	0.969076i	$0.420633\pi$
$812$	3.13410e9	0.205431
$813$	−6.15649e9	−0.401806
$814$	1.10459e10	0.717818
$815$	0	0
$816$	−2.77893e10	−1.79045
$817$	−1.09566e9	−0.0702910
$818$	−9.04951e9	−0.578081
$819$	9.24961e9	0.588342
$820$	0	0
$821$	−8.43321e8	−0.0531853	−0.0265927	−	0.999646i	$-0.508466\pi$
$821$	−8.43321e8	−0.0531853	−0.0265927	+	0.999646i	$0.508466\pi$
$822$	5.87027e9	0.368644
$823$	−2.39654e9	−0.149860	−0.0749299	−	0.997189i	$-0.523873\pi$
$823$	−2.39654e9	−0.149860	−0.0749299	+	0.997189i	$0.523873\pi$
$824$	2.12694e10	1.32437
$825$	0	0
$826$	8.32829e9	0.514193
$827$	−6.20597e9	−0.381540	−0.190770	−	0.981635i	$-0.561099\pi$
$827$	−6.20597e9	−0.381540	−0.190770	+	0.981635i	$0.561099\pi$
$828$	9.90580e9	0.606434
$829$	−1.30007e10	−0.792550	−0.396275	−	0.918132i	$-0.629697\pi$
$829$	−1.30007e10	−0.792550	−0.396275	+	0.918132i	$0.629697\pi$
$830$	0	0
$831$	−2.25074e10	−1.36057
$832$	−1.47964e10	−0.890684
$833$	4.53301e9	0.271725
$834$	−1.87791e10	−1.12097
$835$	0	0
$836$	−8.99033e9	−0.532174
$837$	2.87661e9	0.169567
$838$	5.27771e9	0.309807
$839$	2.66826e9	0.155977	0.0779885	−	0.996954i	$-0.475150\pi$
$839$	2.66826e9	0.155977	0.0779885	+	0.996954i	$0.475150\pi$
$840$	0	0
$841$	2.85199e10	1.65334
$842$	−2.28967e9	−0.132185
$843$	9.28584e9	0.533857
$844$	7.09588e9	0.406263
$845$	0	0
$846$	1.71432e10	0.973410
$847$	−1.29206e10	−0.730617
$848$	1.10581e10	0.622724
$849$	2.48217e9	0.139205
$850$	0	0
$851$	−8.93989e9	−0.497254
$852$	4.32212e9	0.239419
$853$	−1.74908e9	−0.0964913	−0.0482457	−	0.998836i	$-0.515363\pi$
$853$	−1.74908e9	−0.0964913	−0.0482457	+	0.998836i	$0.515363\pi$
$854$	−3.11024e9	−0.170880
$855$	0	0
$856$	−1.34522e10	−0.733051
$857$	1.66810e10	0.905291	0.452645	−	0.891691i	$-0.350480\pi$
$857$	1.66810e10	0.905291	0.452645	+	0.891691i	$0.350480\pi$
$858$	−3.63862e10	−1.96667
$859$	−1.88459e10	−1.01447	−0.507237	−	0.861807i	$-0.669333\pi$
$859$	−1.88459e10	−1.01447	−0.507237	+	0.861807i	$0.669333\pi$
$860$	0	0
$861$	−1.01163e10	−0.540146
$862$	−2.45134e9	−0.130355
$863$	−2.40431e10	−1.27336	−0.636681	−	0.771127i	$-0.719693\pi$
$863$	−2.40431e10	−1.27336	−0.636681	+	0.771127i	$0.719693\pi$
$864$	1.79187e10	0.945166
$865$	0	0
$866$	−1.53982e10	−0.805668
$867$	8.52052e10	4.44016
$868$	2.77094e8	0.0143816
$869$	1.85381e10	0.958287
$870$	0	0
$871$	2.10646e10	1.08016
$872$	−6.94011e9	−0.354453
$873$	−7.72432e9	−0.392926
$874$	−1.45304e10	−0.736185
$875$	0	0
$876$	−1.19134e10	−0.598787
$877$	1.03189e10	0.516576	0.258288	−	0.966068i	$-0.416842\pi$
$877$	1.03189e10	0.516576	0.258288	+	0.966068i	$0.416842\pi$
$878$	−1.12353e10	−0.560215
$879$	9.55400e9	0.474487
$880$	0	0
$881$	−3.18216e10	−1.56786	−0.783928	−	0.620852i	$-0.786787\pi$
$881$	−3.18216e10	−1.56786	−0.783928	+	0.620852i	$0.786787\pi$
$882$	4.45948e9	0.218849
$883$	−2.46997e10	−1.20734	−0.603669	−	0.797235i	$-0.706295\pi$
$883$	−2.46997e10	−1.20734	−0.603669	+	0.797235i	$0.706295\pi$
$884$	1.08121e10	0.526414
$885$	0	0
$886$	−1.14869e10	−0.554859
$887$	4.91222e9	0.236344	0.118172	−	0.992993i	$-0.462297\pi$
$887$	4.91222e9	0.236344	0.118172	+	0.992993i	$0.462297\pi$
$888$	−1.97833e10	−0.948098
$889$	−3.77883e9	−0.180385
$890$	0	0
$891$	2.33355e10	1.10521
$892$	−7.00713e9	−0.330570
$893$	1.25924e10	0.591739
$894$	3.50480e10	1.64052
$895$	0	0
$896$	−1.96084e9	−0.0910678
$897$	2.94489e10	1.36237
$898$	1.80823e9	0.0833272
$899$	4.04662e9	0.185752
$900$	0	0
$901$	−4.68569e10	−2.13421
$902$	2.59620e10	1.17792
$903$	1.07061e9	0.0483863
$904$	−3.81254e10	−1.71643
$905$	0	0
$906$	−5.77877e9	−0.258159
$907$	−1.04163e10	−0.463543	−0.231771	−	0.972770i	$-0.574452\pi$
$907$	−1.04163e10	−0.463543	−0.231771	+	0.972770i	$0.574452\pi$
$908$	2.14430e9	0.0950572
$909$	−6.50462e9	−0.287242
$910$	0	0
$911$	2.28934e10	1.00322	0.501610	−	0.865094i	$-0.332741\pi$
$911$	2.28934e10	1.00322	0.501610	+	0.865094i	$0.332741\pi$
$912$	−2.00813e10	−0.876619
$913$	−5.17181e10	−2.24903
$914$	3.41319e10	1.47859
$915$	0	0
$916$	−1.18370e10	−0.508870
$917$	−4.99424e9	−0.213883
$918$	5.41161e10	2.30875
$919$	4.97399e9	0.211398	0.105699	−	0.994398i	$-0.466292\pi$
$919$	4.97399e9	0.211398	0.105699	+	0.994398i	$0.466292\pi$
$920$	0	0
$921$	4.49178e10	1.89457
$922$	−2.81974e10	−1.18482
$923$	8.38258e9	0.350890
$924$	8.78474e9	0.366334
$925$	0	0
$926$	−9.23309e8	−0.0382128
$927$	−5.53725e10	−2.28305
$928$	2.52068e10	1.03538
$929$	3.93240e10	1.60917	0.804587	−	0.593835i	$-0.202387\pi$
$929$	3.93240e10	1.60917	0.804587	+	0.593835i	$0.202387\pi$
$930$	0	0
$931$	3.27568e9	0.133039
$932$	−9.04722e8	−0.0366066
$933$	6.12140e10	2.46754
$934$	4.22950e9	0.169854
$935$	0	0
$936$	4.25145e10	1.69462
$937$	2.39127e9	0.0949599	0.0474800	−	0.998872i	$-0.484881\pi$
$937$	2.39127e9	0.0949599	0.0474800	+	0.998872i	$0.484881\pi$
$938$	1.01558e10	0.401794
$939$	−3.62718e9	−0.142968
$940$	0	0
$941$	2.09769e10	0.820686	0.410343	−	0.911931i	$-0.365409\pi$
$941$	2.09769e10	0.820686	0.410343	+	0.911931i	$0.365409\pi$
$942$	−8.74121e9	−0.340717
$943$	−2.10121e10	−0.815979
$944$	2.39067e10	0.924948
$945$	0	0
$946$	−2.74755e9	−0.105518
$947$	−1.66975e10	−0.638891	−0.319446	−	0.947605i	$-0.603497\pi$
$947$	−1.66975e10	−0.638891	−0.319446	+	0.947605i	$0.603497\pi$
$948$	−8.30680e9	−0.316668
$949$	−2.31056e10	−0.877578
$950$	0	0
$951$	−8.83826e10	−3.33223
$952$	2.08354e10	0.782657
$953$	−3.33403e10	−1.24780	−0.623899	−	0.781505i	$-0.714452\pi$
$953$	−3.33403e10	−1.24780	−0.623899	+	0.781505i	$0.714452\pi$
$954$	−4.60967e10	−1.71890
$955$	0	0
$956$	−1.67920e10	−0.621585
$957$	1.28291e11	4.73154
$958$	2.75073e10	1.01081
$959$	−2.74872e9	−0.100639
$960$	0	0
$961$	−2.71548e10	−0.986996
$962$	−9.59953e9	−0.347646
$963$	3.50212e10	1.26369
$964$	−1.47316e10	−0.529640
$965$	0	0
$966$	1.41981e10	0.506769
$967$	−3.78414e10	−1.34578	−0.672892	−	0.739741i	$-0.734948\pi$
$967$	−3.78414e10	−1.34578	−0.672892	+	0.739741i	$0.734948\pi$
$968$	−5.93876e10	−2.10442
$969$	8.50913e10	3.00436
$970$	0	0
$971$	−2.26402e9	−0.0793622	−0.0396811	−	0.999212i	$-0.512634\pi$
$971$	−2.26402e9	−0.0793622	−0.0396811	+	0.999212i	$0.512634\pi$
$972$	3.74902e9	0.130944
$973$	8.79319e9	0.306021
$974$	3.24399e10	1.12493
$975$	0	0
$976$	−8.92807e9	−0.307385
$977$	−2.20843e10	−0.757622	−0.378811	−	0.925474i	$-0.623667\pi$
$977$	−2.20843e10	−0.757622	−0.378811	+	0.925474i	$0.623667\pi$
$978$	3.29891e9	0.112768
$979$	−5.66534e10	−1.92969
$980$	0	0
$981$	1.80678e10	0.611031
$982$	2.89717e10	0.976303
$983$	6.71759e9	0.225567	0.112784	−	0.993620i	$-0.464023\pi$
$983$	6.71759e9	0.225567	0.112784	+	0.993620i	$0.464023\pi$
$984$	−4.64982e10	−1.55580
$985$	0	0
$986$	7.61267e10	2.52911
$987$	−1.23045e10	−0.407336
$988$	7.81314e9	0.257737
$989$	2.22370e9	0.0730954
$990$	0	0
$991$	5.45207e10	1.77952	0.889761	−	0.456427i	$-0.150871\pi$
$991$	5.45207e10	1.77952	0.889761	+	0.456427i	$0.150871\pi$
$992$	2.22860e9	0.0724837
$993$	−7.32026e9	−0.237249
$994$	4.04146e9	0.130523
$995$	0	0
$996$	2.31746e10	0.743197
$997$	1.86308e10	0.595385	0.297692	−	0.954662i	$-0.403783\pi$
$997$	1.86308e10	0.595385	0.297692	+	0.954662i	$0.403783\pi$
$998$	−2.17650e10	−0.693109
$999$	2.40600e10	0.763514

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.8.a.i.1.6	✓	8
5.2	odd	4		175.8.b.h.99.11		16
5.3	odd	4		175.8.b.h.99.6		16
5.4	even	2		175.8.a.j.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.8.a.i.1.6	✓	8	1.1	even	1	trivial
175.8.a.j.1.3	yes	8	5.4	even	2
175.8.b.h.99.6		16	5.3	odd	4
175.8.b.h.99.11		16	5.2	odd	4

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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

\(f(q)\)	\(=\)	\(q+9.23526 q^{2} +79.3182 q^{3} -42.7100 q^{4} +732.524 q^{6} -343.000 q^{7} -1576.55 q^{8} +4104.37 q^{9} +O(q^{10})\)	\(q+9.23526 q^{2} +79.3182 q^{3} -42.7100 q^{4} +732.524 q^{6} -343.000 q^{7} -1576.55 q^{8} +4104.37 q^{9} +7560.19 q^{11} -3387.68 q^{12} -6570.26 q^{13} -3167.69 q^{14} -9092.98 q^{16} +38530.0 q^{17} +37904.9 q^{18} +27842.9 q^{19} -27206.1 q^{21} +69820.3 q^{22} -56508.5 q^{23} -125049. q^{24} -60678.1 q^{26} +152082. q^{27} +14649.5 q^{28} +213939. q^{29} +18914.9 q^{31} +117822. q^{32} +599660. q^{33} +355834. q^{34} -175297. q^{36} +158204. q^{37} +257136. q^{38} -521141. q^{39} +371840. q^{41} -251256. q^{42} -39351.7 q^{43} -322895. q^{44} -521871. q^{46} +452269. q^{47} -721239. q^{48} +117649. q^{49} +3.05613e6 q^{51} +280616. q^{52} -1.21612e6 q^{53} +1.40452e6 q^{54} +540757. q^{56} +2.20844e6 q^{57} +1.97578e6 q^{58} -2.62913e6 q^{59} +981864. q^{61} +174684. q^{62} -1.40780e6 q^{63} +2.25202e6 q^{64} +5.53802e6 q^{66} -3.20605e6 q^{67} -1.64561e6 q^{68} -4.48215e6 q^{69} -1.27584e6 q^{71} -6.47075e6 q^{72} +3.51670e6 q^{73} +1.46106e6 q^{74} -1.18917e6 q^{76} -2.59315e6 q^{77} -4.81287e6 q^{78} +2.45207e6 q^{79} +3.08662e6 q^{81} +3.43404e6 q^{82} -6.84085e6 q^{83} +1.16197e6 q^{84} -363423. q^{86} +1.69692e7 q^{87} -1.19190e7 q^{88} -7.49365e6 q^{89} +2.25360e6 q^{91} +2.41348e6 q^{92} +1.50029e6 q^{93} +4.17682e6 q^{94} +9.34546e6 q^{96} -1.88197e6 q^{97} +1.08652e6 q^{98} +3.10298e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 18 q^{2} - 54 q^{3} + 432 q^{4} - 53 q^{6} - 2744 q^{7} - 3843 q^{8} + 12334 q^{9}+O(q^{10}) \) 8 * q - 18 * q^2 - 54 * q^3 + 432 * q^4 - 53 * q^6 - 2744 * q^7 - 3843 * q^8 + 12334 * q^9	\( 8 q - 18 q^{2} - 54 q^{3} + 432 q^{4} - 53 q^{6} - 2744 q^{7} - 3843 q^{8} + 12334 q^{9} - 672 q^{11} - 37071 q^{12} + 9410 q^{13} + 6174 q^{14} + 39152 q^{16} + 34626 q^{17} - 1861 q^{18} - 1876 q^{19} + 18522 q^{21} - 57686 q^{22} - 2892 q^{23} + 115419 q^{24} - 126624 q^{26} + 119664 q^{27} - 148176 q^{28} + 375306 q^{29} + 401106 q^{31} - 782562 q^{32} + 57268 q^{33} + 115141 q^{34} + 1043493 q^{36} + 510970 q^{37} + 176355 q^{38} - 229864 q^{39} + 1725360 q^{41} + 18179 q^{42} - 2583780 q^{43} + 417810 q^{44} - 3420221 q^{46} + 830988 q^{47} - 6605405 q^{48} + 941192 q^{49} + 2465784 q^{51} + 6335102 q^{52} - 3081888 q^{53} - 504707 q^{54} + 1318149 q^{56} + 699682 q^{57} - 84077 q^{58} - 503982 q^{59} + 5355454 q^{61} + 10940010 q^{62} - 4230562 q^{63} + 12425931 q^{64} + 23945921 q^{66} + 2025904 q^{67} + 3202389 q^{68} + 4197918 q^{69} + 12457176 q^{71} + 6598620 q^{72} + 1059500 q^{73} + 10942191 q^{74} + 15089099 q^{76} + 230496 q^{77} - 18142970 q^{78} + 10473540 q^{79} + 17137732 q^{81} + 14606391 q^{82} - 18993798 q^{83} + 12715353 q^{84} + 37213797 q^{86} + 18782326 q^{87} - 46963547 q^{88} + 14400636 q^{89} - 3227630 q^{91} + 26180415 q^{92} - 7950312 q^{93} - 9402748 q^{94} + 91197911 q^{96} + 24116506 q^{97} - 2117682 q^{98} + 11403964 q^{99}+O(q^{100}) \) 8 * q - 18 * q^2 - 54 * q^3 + 432 * q^4 - 53 * q^6 - 2744 * q^7 - 3843 * q^8 + 12334 * q^9 - 672 * q^11 - 37071 * q^12 + 9410 * q^13 + 6174 * q^14 + 39152 * q^16 + 34626 * q^17 - 1861 * q^18 - 1876 * q^19 + 18522 * q^21 - 57686 * q^22 - 2892 * q^23 + 115419 * q^24 - 126624 * q^26 + 119664 * q^27 - 148176 * q^28 + 375306 * q^29 + 401106 * q^31 - 782562 * q^32 + 57268 * q^33 + 115141 * q^34 + 1043493 * q^36 + 510970 * q^37 + 176355 * q^38 - 229864 * q^39 + 1725360 * q^41 + 18179 * q^42 - 2583780 * q^43 + 417810 * q^44 - 3420221 * q^46 + 830988 * q^47 - 6605405 * q^48 + 941192 * q^49 + 2465784 * q^51 + 6335102 * q^52 - 3081888 * q^53 - 504707 * q^54 + 1318149 * q^56 + 699682 * q^57 - 84077 * q^58 - 503982 * q^59 + 5355454 * q^61 + 10940010 * q^62 - 4230562 * q^63 + 12425931 * q^64 + 23945921 * q^66 + 2025904 * q^67 + 3202389 * q^68 + 4197918 * q^69 + 12457176 * q^71 + 6598620 * q^72 + 1059500 * q^73 + 10942191 * q^74 + 15089099 * q^76 + 230496 * q^77 - 18142970 * q^78 + 10473540 * q^79 + 17137732 * q^81 + 14606391 * q^82 - 18993798 * q^83 + 12715353 * q^84 + 37213797 * q^86 + 18782326 * q^87 - 46963547 * q^88 + 14400636 * q^89 - 3227630 * q^91 + 26180415 * q^92 - 7950312 * q^93 - 9402748 * q^94 + 91197911 * q^96 + 24116506 * q^97 - 2117682 * q^98 + 11403964 * q^99

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