LMFDB - Embedded newform 175.8.a.h.1.4

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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 542x^{4} + 982x^{3} + 69485x^{2} - 187765x - 953200 $ x^6 - x^5 - 542x^4 + 982x^3 + 69485x^2 - 187765x - 953200
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.67453$ 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)

$f(q)$	$=$	$q+3.67453 q^{2} -52.1592 q^{3} -114.498 q^{4} -191.660 q^{6} -343.000 q^{7} -891.065 q^{8} +533.578 q^{9} +O(q^{10})$	\(q+3.67453 q^{2} -52.1592 q^{3} -114.498 q^{4} -191.660 q^{6} -343.000 q^{7} -891.065 q^{8} +533.578 q^{9} +2402.61 q^{11} +5972.11 q^{12} +8971.92 q^{13} -1260.36 q^{14} +11381.5 q^{16} +16827.7 q^{17} +1960.65 q^{18} +15357.7 q^{19} +17890.6 q^{21} +8828.45 q^{22} -81244.7 q^{23} +46477.2 q^{24} +32967.6 q^{26} +86241.1 q^{27} +39272.8 q^{28} -203872. q^{29} +88474.9 q^{31} +155878. q^{32} -125318. q^{33} +61833.9 q^{34} -61093.5 q^{36} +261256. q^{37} +56432.3 q^{38} -467968. q^{39} +179461. q^{41} +65739.5 q^{42} +520767. q^{43} -275093. q^{44} -298536. q^{46} -123436. q^{47} -593648. q^{48} +117649. q^{49} -877719. q^{51} -1.02727e6 q^{52} -936049. q^{53} +316895. q^{54} +305635. q^{56} -801044. q^{57} -749135. q^{58} -1.77448e6 q^{59} -2.84579e6 q^{61} +325104. q^{62} -183017. q^{63} -884051. q^{64} -460485. q^{66} +2.45723e6 q^{67} -1.92674e6 q^{68} +4.23765e6 q^{69} -2.59302e6 q^{71} -475453. q^{72} -1.98548e6 q^{73} +959993. q^{74} -1.75842e6 q^{76} -824094. q^{77} -1.71956e6 q^{78} -1.13717e6 q^{79} -5.66520e6 q^{81} +659433. q^{82} -2.32030e6 q^{83} -2.04843e6 q^{84} +1.91357e6 q^{86} +1.06338e7 q^{87} -2.14088e6 q^{88} -1.88285e6 q^{89} -3.07737e6 q^{91} +9.30234e6 q^{92} -4.61478e6 q^{93} -453569. q^{94} -8.13046e6 q^{96} +5.66884e6 q^{97} +432305. q^{98} +1.28198e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} - 28 q^{3} + 321 q^{4} - 917 q^{6} - 2058 q^{7} + 3297 q^{8} + 1800 q^{9}+O(q^{10}) $ 6 * q + 5 * q^2 - 28 * q^3 + 321 * q^4 - 917 * q^6 - 2058 * q^7 + 3297 * q^8 + 1800 * q^9	\( 6 q + 5 q^{2} - 28 q^{3} + 321 q^{4} - 917 q^{6} - 2058 q^{7} + 3297 q^{8} + 1800 q^{9} - 2636 q^{11} - 8043 q^{12} + 14196 q^{13} - 1715 q^{14} - 255 q^{16} - 11606 q^{17} + 28720 q^{18} - 45276 q^{19} + 9604 q^{21} + 1455 q^{22} + 28324 q^{23} - 156751 q^{24} - 10038 q^{26} + 51044 q^{27} - 110103 q^{28} - 285224 q^{29} - 158928 q^{31} + 68721 q^{32} - 163590 q^{33} - 259371 q^{34} - 360694 q^{36} + 348468 q^{37} - 676585 q^{38} - 40352 q^{39} - 819574 q^{41} + 314531 q^{42} - 299484 q^{43} + 838563 q^{44} - 1032870 q^{46} + 1308496 q^{47} - 1674519 q^{48} + 705894 q^{49} - 1148440 q^{51} + 1528170 q^{52} + 198568 q^{53} + 525273 q^{54} - 1130871 q^{56} + 4094722 q^{57} - 5881332 q^{58} - 3779552 q^{59} - 3079860 q^{61} + 1750056 q^{62} - 617400 q^{63} - 3754335 q^{64} - 9954427 q^{66} + 1673172 q^{67} - 11552737 q^{68} - 7859880 q^{69} - 9136988 q^{71} + 7139950 q^{72} - 4605594 q^{73} - 9029106 q^{74} - 23890251 q^{76} + 904148 q^{77} - 11618738 q^{78} - 2417100 q^{79} - 23407970 q^{81} + 6341937 q^{82} - 1365532 q^{83} + 2758749 q^{84} - 30428682 q^{86} + 14992936 q^{87} - 5525277 q^{88} - 13845594 q^{89} - 4869228 q^{91} + 41315508 q^{92} - 9319440 q^{93} - 18396630 q^{94} - 28657783 q^{96} - 2963436 q^{97} + 588245 q^{98} - 4220084 q^{99}+O(q^{100}) \) 6 * q + 5 * q^2 - 28 * q^3 + 321 * q^4 - 917 * q^6 - 2058 * q^7 + 3297 * q^8 + 1800 * q^9 - 2636 * q^11 - 8043 * q^12 + 14196 * q^13 - 1715 * q^14 - 255 * q^16 - 11606 * q^17 + 28720 * q^18 - 45276 * q^19 + 9604 * q^21 + 1455 * q^22 + 28324 * q^23 - 156751 * q^24 - 10038 * q^26 + 51044 * q^27 - 110103 * q^28 - 285224 * q^29 - 158928 * q^31 + 68721 * q^32 - 163590 * q^33 - 259371 * q^34 - 360694 * q^36 + 348468 * q^37 - 676585 * q^38 - 40352 * q^39 - 819574 * q^41 + 314531 * q^42 - 299484 * q^43 + 838563 * q^44 - 1032870 * q^46 + 1308496 * q^47 - 1674519 * q^48 + 705894 * q^49 - 1148440 * q^51 + 1528170 * q^52 + 198568 * q^53 + 525273 * q^54 - 1130871 * q^56 + 4094722 * q^57 - 5881332 * q^58 - 3779552 * q^59 - 3079860 * q^61 + 1750056 * q^62 - 617400 * q^63 - 3754335 * q^64 - 9954427 * q^66 + 1673172 * q^67 - 11552737 * q^68 - 7859880 * q^69 - 9136988 * q^71 + 7139950 * q^72 - 4605594 * q^73 - 9029106 * q^74 - 23890251 * q^76 + 904148 * q^77 - 11618738 * q^78 - 2417100 * q^79 - 23407970 * q^81 + 6341937 * q^82 - 1365532 * q^83 + 2758749 * q^84 - 30428682 * q^86 + 14992936 * q^87 - 5525277 * q^88 - 13845594 * q^89 - 4869228 * q^91 + 41315508 * q^92 - 9319440 * q^93 - 18396630 * q^94 - 28657783 * q^96 - 2963436 * q^97 + 588245 * q^98 - 4220084 * 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 $

$2$	3.67453	0.324786	0.162393	−	0.986726i	$-0.448079\pi$
$2$	3.67453	0.324786	0.162393	+	0.986726i	$0.448079\pi$
$3$	−52.1592	−1.11534	−0.557669	−	0.830064i	$-0.688304\pi$
$3$	−52.1592	−1.11534	−0.557669	+	0.830064i	$0.688304\pi$
$4$	−114.498	−0.894514
$5$	0	0
$5$	0	0
$6$	−191.660	−0.362245
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−891.065	−0.615311
$9$	533.578	0.243977
$10$	0	0
$11$	2402.61	0.544262	0.272131	−	0.962260i	$-0.412272\pi$
$11$	2402.61	0.544262	0.272131	+	0.962260i	$0.412272\pi$
$12$	5972.11	0.997685
$13$	8971.92	1.13262	0.566310	−	0.824193i	$-0.308371\pi$
$13$	8971.92	1.13262	0.566310	+	0.824193i	$0.308371\pi$
$14$	−1260.36	−0.122757
$15$	0	0
$16$	11381.5	0.694670
$17$	16827.7	0.830718	0.415359	−	0.909658i	$-0.363656\pi$
$17$	16827.7	0.830718	0.415359	+	0.909658i	$0.363656\pi$
$18$	1960.65	0.0792402
$19$	15357.7	0.513675	0.256838	−	0.966455i	$-0.417319\pi$
$19$	15357.7	0.513675	0.256838	+	0.966455i	$0.417319\pi$
$20$	0	0
$21$	17890.6	0.421558
$22$	8828.45	0.176769
$23$	−81244.7	−1.39235	−0.696174	−	0.717874i	$-0.745116\pi$
$23$	−81244.7	−1.39235	−0.696174	+	0.717874i	$0.745116\pi$
$24$	46477.2	0.686279
$25$	0	0
$26$	32967.6	0.367858
$27$	86241.1	0.843221
$28$	39272.8	0.338095
$29$	−203872.	−1.55226	−0.776132	−	0.630571i	$-0.782821\pi$
$29$	−203872.	−1.55226	−0.776132	+	0.630571i	$0.782821\pi$
$30$	0	0
$31$	88474.9	0.533401	0.266701	−	0.963779i	$-0.414067\pi$
$31$	88474.9	0.533401	0.266701	+	0.963779i	$0.414067\pi$
$32$	155878.	0.840930
$33$	−125318.	−0.607036
$34$	61833.9	0.269805
$35$	0	0
$36$	−61093.5	−0.218241
$37$	261256.	0.847931	0.423965	−	0.905678i	$-0.360638\pi$
$37$	261256.	0.847931	0.423965	+	0.905678i	$0.360638\pi$
$38$	56432.3	0.166834
$39$	−467968.	−1.26325
$40$	0	0
$41$	179461.	0.406655	0.203327	−	0.979111i	$-0.434824\pi$
$41$	179461.	0.406655	0.203327	+	0.979111i	$0.434824\pi$
$42$	65739.5	0.136916
$43$	520767.	0.998858	0.499429	−	0.866355i	$-0.333543\pi$
$43$	520767.	0.998858	0.499429	+	0.866355i	$0.333543\pi$
$44$	−275093.	−0.486851
$45$	0	0
$46$	−298536.	−0.452214
$47$	−123436.	−0.173420	−0.0867100	−	0.996234i	$-0.527635\pi$
$47$	−123436.	−0.173420	−0.0867100	+	0.996234i	$0.527635\pi$
$48$	−593648.	−0.774792
$49$	117649.	0.142857
$50$	0	0
$51$	−877719.	−0.926531
$52$	−1.02727e6	−1.01314
$53$	−936049.	−0.863641	−0.431820	−	0.901960i	$-0.642129\pi$
$53$	−936049.	−0.863641	−0.431820	+	0.901960i	$0.642129\pi$
$54$	316895.	0.273866
$55$	0	0
$56$	305635.	0.232566
$57$	−801044.	−0.572921
$58$	−749135.	−0.504153
$59$	−1.77448e6	−1.12484	−0.562419	−	0.826852i	$-0.690129\pi$
$59$	−1.77448e6	−1.12484	−0.562419	+	0.826852i	$0.690129\pi$
$60$	0	0
$61$	−2.84579e6	−1.60527	−0.802634	−	0.596471i	$-0.796569\pi$
$61$	−2.84579e6	−1.60527	−0.802634	+	0.596471i	$0.796569\pi$
$62$	325104.	0.173241
$63$	−183017.	−0.0922147
$64$	−884051.	−0.421548
$65$	0	0
$66$	−460485.	−0.197157
$67$	2.45723e6	0.998124	0.499062	−	0.866566i	$-0.333678\pi$
$67$	2.45723e6	0.998124	0.499062	+	0.866566i	$0.333678\pi$
$68$	−1.92674e6	−0.743089
$69$	4.23765e6	1.55294
$70$	0	0
$71$	−2.59302e6	−0.859809	−0.429905	−	0.902874i	$-0.641453\pi$
$71$	−2.59302e6	−0.859809	−0.429905	+	0.902874i	$0.641453\pi$
$72$	−475453.	−0.150122
$73$	−1.98548e6	−0.597360	−0.298680	−	0.954353i	$-0.596546\pi$
$73$	−1.98548e6	−0.597360	−0.298680	+	0.954353i	$0.596546\pi$
$74$	959993.	0.275396
$75$	0	0
$76$	−1.75842e6	−0.459490
$77$	−824094.	−0.205712
$78$	−1.71956e6	−0.410286
$79$	−1.13717e6	−0.259495	−0.129748	−	0.991547i	$-0.541417\pi$
$79$	−1.13717e6	−0.259495	−0.129748	+	0.991547i	$0.541417\pi$
$80$	0	0
$81$	−5.66520e6	−1.18445
$82$	659433.	0.132076
$83$	−2.32030e6	−0.445420	−0.222710	−	0.974885i	$-0.571490\pi$
$83$	−2.32030e6	−0.445420	−0.222710	+	0.974885i	$0.571490\pi$
$84$	−2.04843e6	−0.377090
$85$	0	0
$86$	1.91357e6	0.324415
$87$	1.06338e7	1.73130
$88$	−2.14088e6	−0.334891
$89$	−1.88285e6	−0.283108	−0.141554	−	0.989931i	$-0.545210\pi$
$89$	−1.88285e6	−0.283108	−0.141554	+	0.989931i	$0.545210\pi$
$90$	0	0
$91$	−3.07737e6	−0.428090
$92$	9.30234e6	1.24547
$93$	−4.61478e6	−0.594922
$94$	−453569.	−0.0563243
$95$	0	0
$96$	−8.13046e6	−0.937920
$97$	5.66884e6	0.630656	0.315328	−	0.948983i	$-0.397885\pi$
$97$	5.66884e6	0.630656	0.315328	+	0.948983i	$0.397885\pi$
$98$	432305.	0.0463979
$99$	1.28198e6	0.132788
$100$	0	0
$101$	−4.87704e6	−0.471012	−0.235506	−	0.971873i	$-0.575675\pi$
$101$	−4.87704e6	−0.471012	−0.235506	+	0.971873i	$0.575675\pi$
$102$	−3.22520e6	−0.300924
$103$	1.86759e7	1.68404	0.842019	−	0.539447i	$-0.181367\pi$
$103$	1.86759e7	1.68404	0.842019	+	0.539447i	$0.181367\pi$
$104$	−7.99457e6	−0.696913
$105$	0	0
$106$	−3.43954e6	−0.280498
$107$	8.17763e6	0.645333	0.322667	−	0.946513i	$-0.395421\pi$
$107$	8.17763e6	0.645333	0.322667	+	0.946513i	$0.395421\pi$
$108$	−9.87442e6	−0.754273
$109$	−1.20799e7	−0.893452	−0.446726	−	0.894671i	$-0.647410\pi$
$109$	−1.20799e7	−0.893452	−0.446726	+	0.894671i	$0.647410\pi$
$110$	0	0
$111$	−1.36269e7	−0.945729
$112$	−3.90385e6	−0.262561
$113$	1.08439e7	0.706988	0.353494	−	0.935437i	$-0.384994\pi$
$113$	1.08439e7	0.706988	0.353494	+	0.935437i	$0.384994\pi$
$114$	−2.94346e6	−0.186076
$115$	0	0
$116$	2.33430e7	1.38852
$117$	4.78722e6	0.276333
$118$	−6.52039e6	−0.365331
$119$	−5.77190e6	−0.313982
$120$	0	0
$121$	−1.37146e7	−0.703778
$122$	−1.04569e7	−0.521368
$123$	−9.36052e6	−0.453557
$124$	−1.01302e7	−0.477135
$125$	0	0
$126$	−672502.	−0.0299500
$127$	2.76641e7	1.19841	0.599203	−	0.800597i	$-0.295484\pi$
$127$	2.76641e7	1.19841	0.599203	+	0.800597i	$0.295484\pi$
$128$	−2.32008e7	−0.977843
$129$	−2.71628e7	−1.11406
$130$	0	0
$131$	−1.60892e7	−0.625295	−0.312647	−	0.949869i	$-0.601216\pi$
$131$	−1.60892e7	−0.625295	−0.312647	+	0.949869i	$0.601216\pi$
$132$	1.43486e7	0.543003
$133$	−5.26769e6	−0.194151
$134$	9.02917e6	0.324176
$135$	0	0
$136$	−1.49946e7	−0.511150
$137$	−1.69816e7	−0.564231	−0.282116	−	0.959380i	$-0.591036\pi$
$137$	−1.69816e7	−0.564231	−0.282116	+	0.959380i	$0.591036\pi$
$138$	1.55714e7	0.504371
$139$	3.31021e7	1.04545	0.522725	−	0.852501i	$-0.324915\pi$
$139$	3.31021e7	1.04545	0.522725	+	0.852501i	$0.324915\pi$
$140$	0	0
$141$	6.43831e6	0.193422
$142$	−9.52814e6	−0.279254
$143$	2.15560e7	0.616442
$144$	6.07290e6	0.169484
$145$	0	0
$146$	−7.29571e6	−0.194014
$147$	−6.13647e6	−0.159334
$148$	−2.99133e7	−0.758486
$149$	6.37347e7	1.57843	0.789213	−	0.614119i	$-0.210489\pi$
$149$	6.37347e7	1.57843	0.789213	+	0.614119i	$0.210489\pi$
$150$	0	0
$151$	−9.83946e6	−0.232569	−0.116285	−	0.993216i	$-0.537098\pi$
$151$	−9.83946e6	−0.232569	−0.116285	+	0.993216i	$0.537098\pi$
$152$	−1.36847e7	−0.316070
$153$	8.97889e6	0.202676
$154$	−3.02816e6	−0.0668122
$155$	0	0
$156$	5.35813e7	1.13000
$157$	−7.00482e7	−1.44460	−0.722301	−	0.691578i	$-0.756916\pi$
$157$	−7.00482e7	−1.44460	−0.722301	+	0.691578i	$0.756916\pi$
$158$	−4.17856e6	−0.0842804
$159$	4.88235e7	0.963251
$160$	0	0
$161$	2.78669e7	0.526258
$162$	−2.08169e7	−0.384693
$163$	−9.64289e7	−1.74402	−0.872009	−	0.489491i	$-0.837183\pi$
$163$	−9.64289e7	−1.74402	−0.872009	+	0.489491i	$0.837183\pi$
$164$	−2.05479e7	−0.363758
$165$	0	0
$166$	−8.52599e6	−0.144666
$167$	−4.58188e7	−0.761266	−0.380633	−	0.924726i	$-0.624294\pi$
$167$	−4.58188e7	−0.761266	−0.380633	+	0.924726i	$0.624294\pi$
$168$	−1.59417e7	−0.259389
$169$	1.77469e7	0.282826
$170$	0	0
$171$	8.19453e6	0.125325
$172$	−5.96267e7	−0.893493
$173$	8.69029e7	1.27606	0.638032	−	0.770009i	$-0.279749\pi$
$173$	8.69029e7	1.27606	0.638032	+	0.770009i	$0.279749\pi$
$174$	3.90743e7	0.562300
$175$	0	0
$176$	2.73452e7	0.378083
$177$	9.25556e7	1.25457
$178$	−6.91860e6	−0.0919493
$179$	−1.48385e8	−1.93377	−0.966884	−	0.255216i	$-0.917853\pi$
$179$	−1.48385e8	−1.93377	−0.966884	+	0.255216i	$0.917853\pi$
$180$	0	0
$181$	1.45089e8	1.81869	0.909345	−	0.416042i	$-0.136583\pi$
$181$	1.45089e8	1.81869	0.909345	+	0.416042i	$0.136583\pi$
$182$	−1.13079e7	−0.139037
$183$	1.48434e8	1.79042
$184$	7.23943e7	0.856726
$185$	0	0
$186$	−1.69571e7	−0.193222
$187$	4.04304e7	0.452129
$188$	1.41331e7	0.155127
$189$	−2.95807e7	−0.318707
$190$	0	0
$191$	1.70854e8	1.77422	0.887112	−	0.461555i	$-0.152708\pi$
$191$	1.70854e8	1.77422	0.887112	+	0.461555i	$0.152708\pi$
$192$	4.61114e7	0.470169
$193$	6.73834e6	0.0674687	0.0337343	−	0.999431i	$-0.489260\pi$
$193$	6.73834e6	0.0674687	0.0337343	+	0.999431i	$0.489260\pi$
$194$	2.08303e7	0.204828
$195$	0	0
$196$	−1.34706e7	−0.127788
$197$	−1.11483e8	−1.03891	−0.519454	−	0.854498i	$-0.673865\pi$
$197$	−1.11483e8	−1.03891	−0.519454	+	0.854498i	$0.673865\pi$
$198$	4.71066e6	0.0431275
$199$	−1.57863e8	−1.42002	−0.710012	−	0.704189i	$-0.751311\pi$
$199$	−1.57863e8	−1.42002	−0.710012	+	0.704189i	$0.751311\pi$
$200$	0	0
$201$	−1.28167e8	−1.11324
$202$	−1.79208e7	−0.152978
$203$	6.99282e7	0.586701
$204$	1.00497e8	0.828795
$205$	0	0
$206$	6.86253e7	0.546951
$207$	−4.33504e7	−0.339701
$208$	1.02114e8	0.786797
$209$	3.68985e7	0.279574
$210$	0	0
$211$	9.19808e7	0.674076	0.337038	−	0.941491i	$-0.390575\pi$
$211$	9.19808e7	0.674076	0.337038	+	0.941491i	$0.390575\pi$
$212$	1.07176e8	0.772539
$213$	1.35250e8	0.958977
$214$	3.00489e7	0.209595
$215$	0	0
$216$	−7.68465e7	−0.518843
$217$	−3.03469e7	−0.201607
$218$	−4.43880e7	−0.290180
$219$	1.03561e8	0.666257
$220$	0	0
$221$	1.50977e8	0.940887
$222$	−5.00724e7	−0.307159
$223$	6.32103e7	0.381699	0.190849	−	0.981619i	$-0.438876\pi$
$223$	6.32103e7	0.381699	0.190849	+	0.981619i	$0.438876\pi$
$224$	−5.34661e7	−0.317842
$225$	0	0
$226$	3.98463e7	0.229619
$227$	−1.72910e8	−0.981138	−0.490569	−	0.871402i	$-0.663211\pi$
$227$	−1.72910e8	−0.981138	−0.490569	+	0.871402i	$0.663211\pi$
$228$	9.17179e7	0.512486
$229$	−3.40481e8	−1.87357	−0.936784	−	0.349909i	$-0.886212\pi$
$229$	−3.40481e8	−1.87357	−0.936784	+	0.349909i	$0.886212\pi$
$230$	0	0
$231$	4.29841e7	0.229438
$232$	1.81664e8	0.955125
$233$	−9.00565e7	−0.466412	−0.233206	−	0.972427i	$-0.574922\pi$
$233$	−9.00565e7	−0.466412	−0.233206	+	0.972427i	$0.574922\pi$
$234$	1.75908e7	0.0897490
$235$	0	0
$236$	2.03175e8	1.00618
$237$	5.93137e7	0.289425
$238$	−2.12090e7	−0.101977
$239$	2.50065e8	1.18484	0.592421	−	0.805628i	$-0.298172\pi$
$239$	2.50065e8	1.18484	0.592421	+	0.805628i	$0.298172\pi$
$240$	0	0
$241$	−1.76848e8	−0.813842	−0.406921	−	0.913463i	$-0.633397\pi$
$241$	−1.76848e8	−0.813842	−0.406921	+	0.913463i	$0.633397\pi$
$242$	−5.03949e7	−0.228577
$243$	1.06883e8	0.477843
$244$	3.25836e8	1.43594
$245$	0	0
$246$	−3.43955e7	−0.147309
$247$	1.37788e8	0.581798
$248$	−7.88369e7	−0.328208
$249$	1.21025e8	0.496794
$250$	0	0
$251$	−3.48482e7	−0.139098	−0.0695492	−	0.997579i	$-0.522156\pi$
$251$	−3.48482e7	−0.139098	−0.0695492	+	0.997579i	$0.522156\pi$
$252$	2.09551e7	0.0824873
$253$	−1.95199e8	−0.757802
$254$	1.01653e8	0.389225
$255$	0	0
$256$	2.79064e7	0.103959
$257$	−2.80632e8	−1.03127	−0.515633	−	0.856810i	$-0.672443\pi$
$257$	−2.80632e8	−1.03127	−0.515633	+	0.856810i	$0.672443\pi$
$258$	−9.98103e7	−0.361832
$259$	−8.96109e7	−0.320488
$260$	0	0
$261$	−1.08782e8	−0.378717
$262$	−5.91202e7	−0.203087
$263$	1.37112e7	0.0464763	0.0232381	−	0.999730i	$-0.492602\pi$
$263$	1.37112e7	0.0464763	0.0232381	+	0.999730i	$0.492602\pi$
$264$	1.11666e8	0.373516
$265$	0	0
$266$	−1.93563e7	−0.0630574
$267$	9.82081e7	0.315761
$268$	−2.81348e8	−0.892836
$269$	−1.60114e8	−0.501530	−0.250765	−	0.968048i	$-0.580682\pi$
$269$	−1.60114e8	−0.501530	−0.250765	+	0.968048i	$0.580682\pi$
$270$	0	0
$271$	−1.68123e8	−0.513140	−0.256570	−	0.966526i	$-0.582592\pi$
$271$	−1.68123e8	−0.513140	−0.256570	+	0.966526i	$0.582592\pi$
$272$	1.91524e8	0.577075
$273$	1.60513e8	0.477464
$274$	−6.23995e7	−0.183254
$275$	0	0
$276$	−4.85202e8	−1.38912
$277$	−4.17131e8	−1.17922	−0.589608	−	0.807690i	$-0.700718\pi$
$277$	−4.17131e8	−1.17922	−0.589608	+	0.807690i	$0.700718\pi$
$278$	1.21635e8	0.339547
$279$	4.72082e7	0.130138
$280$	0	0
$281$	4.23209e7	0.113784	0.0568921	−	0.998380i	$-0.481881\pi$
$281$	4.23209e7	0.113784	0.0568921	+	0.998380i	$0.481881\pi$
$282$	2.36578e7	0.0628206
$283$	−4.33200e8	−1.13615	−0.568075	−	0.822976i	$-0.692312\pi$
$283$	−4.33200e8	−1.13615	−0.568075	+	0.822976i	$0.692312\pi$
$284$	2.96895e8	0.769112
$285$	0	0
$286$	7.92082e7	0.200211
$287$	−6.15550e7	−0.153701
$288$	8.31730e7	0.205168
$289$	−1.27167e8	−0.309908
$290$	0	0
$291$	−2.95682e8	−0.703395
$292$	2.27333e8	0.534347
$293$	−7.50146e8	−1.74224	−0.871122	−	0.491067i	$-0.836607\pi$
$293$	−7.50146e8	−1.74224	−0.871122	+	0.491067i	$0.836607\pi$
$294$	−2.25486e7	−0.0517493
$295$	0	0
$296$	−2.32796e8	−0.521741
$297$	2.07204e8	0.458933
$298$	2.34195e8	0.512650
$299$	−7.28921e8	−1.57700
$300$	0	0
$301$	−1.78623e8	−0.377533
$302$	−3.61554e7	−0.0755351
$303$	2.54382e8	0.525337
$304$	1.74793e8	0.356835
$305$	0	0
$306$	3.29932e7	0.0658263
$307$	−5.40678e8	−1.06648	−0.533242	−	0.845963i	$-0.679027\pi$
$307$	−5.40678e8	−1.06648	−0.533242	+	0.845963i	$0.679027\pi$
$308$	9.43570e7	0.184012
$309$	−9.74121e8	−1.87827
$310$	0	0
$311$	−4.63814e8	−0.874344	−0.437172	−	0.899378i	$-0.644020\pi$
$311$	−4.63814e8	−0.874344	−0.437172	+	0.899378i	$0.644020\pi$
$312$	4.16990e8	0.777293
$313$	−7.31063e8	−1.34757	−0.673783	−	0.738930i	$-0.735332\pi$
$313$	−7.31063e8	−1.34757	−0.673783	+	0.738930i	$0.735332\pi$
$314$	−2.57394e8	−0.469186
$315$	0	0
$316$	1.30203e8	0.232122
$317$	7.29343e8	1.28595	0.642976	−	0.765886i	$-0.277700\pi$
$317$	7.29343e8	1.28595	0.642976	+	0.765886i	$0.277700\pi$
$318$	1.79403e8	0.312850
$319$	−4.89825e8	−0.844839
$320$	0	0
$321$	−4.26538e8	−0.719764
$322$	1.02398e8	0.170921
$323$	2.58435e8	0.426719
$324$	6.48653e8	1.05951
$325$	0	0
$326$	−3.54331e8	−0.566432
$327$	6.30079e8	0.996501
$328$	−1.59911e8	−0.250219
$329$	4.23385e7	0.0655466
$330$	0	0
$331$	−9.87428e7	−0.149661	−0.0748303	−	0.997196i	$-0.523842\pi$
$331$	−9.87428e7	−0.149661	−0.0748303	+	0.997196i	$0.523842\pi$
$332$	2.65669e8	0.398435
$333$	1.39400e8	0.206876
$334$	−1.68363e8	−0.247248
$335$	0	0
$336$	2.03621e8	0.292844
$337$	−9.26763e7	−0.131906	−0.0659529	−	0.997823i	$-0.521009\pi$
$337$	−9.26763e7	−0.131906	−0.0659529	+	0.997823i	$0.521009\pi$
$338$	6.52115e7	0.0918578
$339$	−5.65610e8	−0.788530
$340$	0	0
$341$	2.12570e8	0.290310
$342$	3.01110e7	0.0407037
$343$	−4.03536e7	−0.0539949
$344$	−4.64037e8	−0.614608
$345$	0	0
$346$	3.19327e8	0.414447
$347$	−1.42087e8	−0.182558	−0.0912792	−	0.995825i	$-0.529096\pi$
$347$	−1.42087e8	−0.182558	−0.0912792	+	0.995825i	$0.529096\pi$
$348$	−1.21755e9	−1.54867
$349$	7.82143e8	0.984912	0.492456	−	0.870337i	$-0.336099\pi$
$349$	7.82143e8	0.984912	0.492456	+	0.870337i	$0.336099\pi$
$350$	0	0
$351$	7.73749e8	0.955048
$352$	3.74513e8	0.457686
$353$	5.74706e8	0.695399	0.347700	−	0.937606i	$-0.386963\pi$
$353$	5.74706e8	0.695399	0.347700	+	0.937606i	$0.386963\pi$
$354$	3.40098e8	0.407468
$355$	0	0
$356$	2.15583e8	0.253244
$357$	3.01058e8	0.350196
$358$	−5.45245e8	−0.628060
$359$	−1.42968e9	−1.63082	−0.815412	−	0.578881i	$-0.803489\pi$
$359$	−1.42968e9	−1.63082	−0.815412	+	0.578881i	$0.803489\pi$
$360$	0	0
$361$	−6.58013e8	−0.736138
$362$	5.33133e8	0.590684
$363$	7.15345e8	0.784950
$364$	3.52352e8	0.382932
$365$	0	0
$366$	5.45424e8	0.581501
$367$	8.75438e8	0.924473	0.462236	−	0.886757i	$-0.347047\pi$
$367$	8.75438e8	0.924473	0.462236	+	0.886757i	$0.347047\pi$
$368$	−9.24684e8	−0.967222
$369$	9.57562e7	0.0992144
$370$	0	0
$371$	3.21065e8	0.326425
$372$	5.28382e8	0.532167
$373$	7.33357e8	0.731703	0.365851	−	0.930673i	$-0.380778\pi$
$373$	7.33357e8	0.731703	0.365851	+	0.930673i	$0.380778\pi$
$374$	1.48563e8	0.146845
$375$	0	0
$376$	1.09989e8	0.106707
$377$	−1.82913e9	−1.75812
$378$	−1.08695e8	−0.103512
$379$	8.06937e8	0.761381	0.380691	−	0.924702i	$-0.375686\pi$
$379$	8.06937e8	0.761381	0.380691	+	0.924702i	$0.375686\pi$
$380$	0	0
$381$	−1.44294e9	−1.33663
$382$	6.27808e8	0.576242
$383$	−5.38367e7	−0.0489647	−0.0244823	−	0.999700i	$-0.507794\pi$
$383$	−5.38367e7	−0.0489647	−0.0244823	+	0.999700i	$0.507794\pi$
$384$	1.21014e9	1.09062
$385$	0	0
$386$	2.47602e7	0.0219129
$387$	2.77870e8	0.243698
$388$	−6.49070e8	−0.564131
$389$	1.01850e9	0.877278	0.438639	−	0.898663i	$-0.355461\pi$
$389$	1.01850e9	0.877278	0.438639	+	0.898663i	$0.355461\pi$
$390$	0	0
$391$	−1.36716e9	−1.15665
$392$	−1.04833e8	−0.0879016
$393$	8.39199e8	0.697414
$394$	−4.09648e8	−0.337423
$395$	0	0
$396$	−1.46784e8	−0.118780
$397$	−1.92918e9	−1.54742	−0.773708	−	0.633543i	$-0.781600\pi$
$397$	−1.92918e9	−1.54742	−0.773708	+	0.633543i	$0.781600\pi$
$398$	−5.80074e8	−0.461203
$399$	2.74758e8	0.216544
$400$	0	0
$401$	−2.34586e9	−1.81676	−0.908380	−	0.418146i	$-0.862680\pi$
$401$	−2.34586e9	−1.81676	−0.908380	+	0.418146i	$0.862680\pi$
$402$	−4.70954e8	−0.361566
$403$	7.93790e8	0.604140
$404$	5.58411e8	0.421327
$405$	0	0
$406$	2.56953e8	0.190552
$407$	6.27696e8	0.461497
$408$	7.82105e8	0.570104
$409$	1.95618e9	1.41376	0.706882	−	0.707332i	$-0.250101\pi$
$409$	1.95618e9	1.41376	0.706882	+	0.707332i	$0.250101\pi$
$410$	0	0
$411$	8.85747e8	0.629308
$412$	−2.13835e9	−1.50640
$413$	6.08648e8	0.425149
$414$	−1.59292e8	−0.110330
$415$	0	0
$416$	1.39852e9	0.952453
$417$	−1.72658e9	−1.16603
$418$	1.35585e8	0.0908016
$419$	−2.72762e9	−1.81149	−0.905744	−	0.423825i	$-0.860687\pi$
$419$	−2.72762e9	−1.81149	−0.905744	+	0.423825i	$0.860687\pi$
$420$	0	0
$421$	1.49203e9	0.974519	0.487260	−	0.873257i	$-0.337997\pi$
$421$	1.49203e9	0.974519	0.487260	+	0.873257i	$0.337997\pi$
$422$	3.37986e8	0.218930
$423$	−6.58627e7	−0.0423105
$424$	8.34081e8	0.531407
$425$	0	0
$426$	4.96980e8	0.311462
$427$	9.76105e8	0.606735
$428$	−9.36321e8	−0.577260
$429$	−1.12434e9	−0.687541
$430$	0	0
$431$	2.00007e9	1.20330	0.601650	−	0.798760i	$-0.294510\pi$
$431$	2.00007e9	1.20330	0.601650	+	0.798760i	$0.294510\pi$
$432$	9.81551e8	0.585760
$433$	6.56921e8	0.388871	0.194435	−	0.980915i	$-0.437713\pi$
$433$	6.56921e8	0.388871	0.194435	+	0.980915i	$0.437713\pi$
$434$	−1.11511e8	−0.0654790
$435$	0	0
$436$	1.38312e9	0.799206
$437$	−1.24773e9	−0.715214
$438$	3.80538e8	0.216391
$439$	1.07249e9	0.605015	0.302508	−	0.953147i	$-0.402176\pi$
$439$	1.07249e9	0.605015	0.302508	+	0.953147i	$0.402176\pi$
$440$	0	0
$441$	6.27749e7	0.0348539
$442$	5.54769e8	0.305587
$443$	1.74877e9	0.955696	0.477848	−	0.878443i	$-0.341417\pi$
$443$	1.74877e9	0.955696	0.477848	+	0.878443i	$0.341417\pi$
$444$	1.56025e9	0.845968
$445$	0	0
$446$	2.32268e8	0.123970
$447$	−3.32435e9	−1.76048
$448$	3.03230e8	0.159330
$449$	−1.66388e9	−0.867481	−0.433741	−	0.901038i	$-0.642807\pi$
$449$	−1.66388e9	−0.867481	−0.433741	+	0.901038i	$0.642807\pi$
$450$	0	0
$451$	4.31174e8	0.221327
$452$	−1.24161e9	−0.632411
$453$	5.13218e8	0.259393
$454$	−6.35364e8	−0.318659
$455$	0	0
$456$	7.13783e8	0.352524
$457$	−2.49377e9	−1.22222	−0.611110	−	0.791546i	$-0.709277\pi$
$457$	−2.49377e9	−1.22222	−0.611110	+	0.791546i	$0.709277\pi$
$458$	−1.25111e9	−0.608508
$459$	1.45124e9	0.700478
$460$	0	0
$461$	−3.96619e9	−1.88548	−0.942738	−	0.333535i	$-0.891758\pi$
$461$	−3.96619e9	−1.88548	−0.942738	+	0.333535i	$0.891758\pi$
$462$	1.57946e8	0.0745182
$463$	3.38794e9	1.58636	0.793181	−	0.608986i	$-0.208424\pi$
$463$	3.38794e9	1.58636	0.793181	+	0.608986i	$0.208424\pi$
$464$	−2.32037e9	−1.07831
$465$	0	0
$466$	−3.30915e8	−0.151484
$467$	3.18287e9	1.44614	0.723069	−	0.690776i	$-0.242731\pi$
$467$	3.18287e9	1.44614	0.723069	+	0.690776i	$0.242731\pi$
$468$	−5.48126e8	−0.247184
$469$	−8.42831e8	−0.377255
$470$	0	0
$471$	3.65366e9	1.61122
$472$	1.58118e9	0.692125
$473$	1.25120e9	0.543641
$474$	2.17950e8	0.0940010
$475$	0	0
$476$	6.60870e8	0.280861
$477$	−4.99455e8	−0.210709
$478$	9.18872e8	0.384820
$479$	1.47183e9	0.611906	0.305953	−	0.952047i	$-0.401025\pi$
$479$	1.47183e9	0.611906	0.305953	+	0.952047i	$0.401025\pi$
$480$	0	0
$481$	2.34397e9	0.960383
$482$	−6.49832e8	−0.264324
$483$	−1.45352e9	−0.586955
$484$	1.57030e9	0.629540
$485$	0	0
$486$	3.92743e8	0.155197
$487$	−6.97309e8	−0.273574	−0.136787	−	0.990601i	$-0.543678\pi$
$487$	−6.97309e8	−0.273574	−0.136787	+	0.990601i	$0.543678\pi$
$488$	2.53578e9	0.987739
$489$	5.02965e9	1.94517
$490$	0	0
$491$	7.85732e8	0.299564	0.149782	−	0.988719i	$-0.452143\pi$
$491$	7.85732e8	0.299564	0.149782	+	0.988719i	$0.452143\pi$
$492$	1.07176e9	0.405713
$493$	−3.43070e9	−1.28949
$494$	5.06306e8	0.188960
$495$	0	0
$496$	1.00697e9	0.370538
$497$	8.89407e8	0.324977
$498$	4.44709e8	0.161351
$499$	−5.18713e9	−1.86885	−0.934427	−	0.356155i	$-0.884088\pi$
$499$	−5.18713e9	−1.86885	−0.934427	+	0.356155i	$0.884088\pi$
$500$	0	0
$501$	2.38987e9	0.849068
$502$	−1.28051e8	−0.0451771
$503$	1.10364e9	0.386667	0.193334	−	0.981133i	$-0.438070\pi$
$503$	1.10364e9	0.386667	0.193334	+	0.981133i	$0.438070\pi$
$504$	1.63080e8	0.0567407
$505$	0	0
$506$	−7.17265e8	−0.246123
$507$	−9.25664e8	−0.315446
$508$	−3.16748e9	−1.07199
$509$	−1.13819e9	−0.382563	−0.191282	−	0.981535i	$-0.561264\pi$
$509$	−1.13819e9	−0.382563	−0.191282	+	0.981535i	$0.561264\pi$
$510$	0	0
$511$	6.81020e8	0.225781
$512$	3.07225e9	1.01161
$513$	1.32446e9	0.433141
$514$	−1.03119e9	−0.334940
$515$	0	0
$516$	3.11008e9	0.996546
$517$	−2.96568e8	−0.0943860
$518$	−3.29278e8	−0.104090
$519$	−4.53278e9	−1.42324
$520$	0	0
$521$	5.48972e8	0.170066	0.0850331	−	0.996378i	$-0.472900\pi$
$521$	5.48972e8	0.170066	0.0850331	+	0.996378i	$0.472900\pi$
$522$	−3.99722e8	−0.123002
$523$	5.67676e8	0.173518	0.0867590	−	0.996229i	$-0.472349\pi$
$523$	5.67676e8	0.173518	0.0867590	+	0.996229i	$0.472349\pi$
$524$	1.84218e9	0.559335
$525$	0	0
$526$	5.03823e7	0.0150948
$527$	1.48883e9	0.443106
$528$	−1.42630e9	−0.421690
$529$	3.19587e9	0.938630
$530$	0	0
$531$	−9.46825e8	−0.274435
$532$	6.03139e8	0.173671
$533$	1.61011e9	0.460585
$534$	3.60868e8	0.102554
$535$	0	0
$536$	−2.18955e9	−0.614156
$537$	7.73964e9	2.15680
$538$	−5.88344e8	−0.162890
$539$	2.82664e8	0.0777518
$540$	0	0
$541$	3.48367e9	0.945904	0.472952	−	0.881088i	$-0.343188\pi$
$541$	3.48367e9	0.945904	0.472952	+	0.881088i	$0.343188\pi$
$542$	−6.17774e8	−0.166660
$543$	−7.56771e9	−2.02845
$544$	2.62307e9	0.698575
$545$	0	0
$546$	5.89810e8	0.155074
$547$	3.85021e8	0.100584	0.0502920	−	0.998735i	$-0.483985\pi$
$547$	3.85021e8	0.100584	0.0502920	+	0.998735i	$0.483985\pi$
$548$	1.94436e9	0.504713
$549$	−1.51845e9	−0.391649
$550$	0	0
$551$	−3.13101e9	−0.797359
$552$	−3.77603e9	−0.955539
$553$	3.90048e8	0.0980800
$554$	−1.53276e9	−0.382992
$555$	0	0
$556$	−3.79012e9	−0.935170
$557$	−6.31036e9	−1.54725	−0.773626	−	0.633643i	$-0.781559\pi$
$557$	−6.31036e9	−1.54725	−0.773626	+	0.633643i	$0.781559\pi$
$558$	1.73468e8	0.0422668
$559$	4.67228e9	1.13133
$560$	0	0
$561$	−2.10881e9	−0.504276
$562$	1.55509e8	0.0369555
$563$	5.61957e9	1.32716	0.663581	−	0.748105i	$-0.269036\pi$
$563$	5.61957e9	1.32716	0.663581	+	0.748105i	$0.269036\pi$
$564$	−7.37173e8	−0.173018
$565$	0	0
$566$	−1.59181e9	−0.369005
$567$	1.94316e9	0.447681
$568$	2.31055e9	0.529050
$569$	−6.15281e9	−1.40017	−0.700085	−	0.714059i	$-0.746855\pi$
$569$	−6.15281e9	−1.40017	−0.700085	+	0.714059i	$0.746855\pi$
$570$	0	0
$571$	−7.39916e9	−1.66324	−0.831622	−	0.555342i	$-0.812587\pi$
$571$	−7.39916e9	−1.66324	−0.831622	+	0.555342i	$0.812587\pi$
$572$	−2.46812e9	−0.551416
$573$	−8.91160e9	−1.97886
$574$	−2.26186e8	−0.0499199
$575$	0	0
$576$	−4.71710e8	−0.102848
$577$	−5.93377e9	−1.28592	−0.642962	−	0.765898i	$-0.722295\pi$
$577$	−5.93377e9	−1.28592	−0.642962	+	0.765898i	$0.722295\pi$
$578$	−4.67279e8	−0.100653
$579$	−3.51466e8	−0.0752503
$580$	0	0
$581$	7.95861e8	0.168353
$582$	−1.08649e9	−0.228452
$583$	−2.24896e9	−0.470047
$584$	1.76919e9	0.367562
$585$	0	0
$586$	−2.75643e9	−0.565855
$587$	−8.51343e9	−1.73729	−0.868643	−	0.495439i	$-0.835007\pi$
$587$	−8.51343e9	−1.73729	−0.868643	+	0.495439i	$0.835007\pi$
$588$	7.02613e8	0.142526
$589$	1.35877e9	0.273995
$590$	0	0
$591$	5.81487e9	1.15873
$592$	2.97348e9	0.589032
$593$	3.13459e9	0.617290	0.308645	−	0.951177i	$-0.400125\pi$
$593$	3.13459e9	0.617290	0.308645	+	0.951177i	$0.400125\pi$
$594$	7.61375e8	0.149055
$595$	0	0
$596$	−7.29749e9	−1.41192
$597$	8.23403e9	1.58381
$598$	−2.67844e9	−0.512186
$599$	−4.23565e9	−0.805241	−0.402620	−	0.915367i	$-0.631901\pi$
$599$	−4.23565e9	−0.805241	−0.402620	+	0.915367i	$0.631901\pi$
$600$	0	0
$601$	−1.04662e10	−1.96666	−0.983328	−	0.181843i	$-0.941794\pi$
$601$	−1.04662e10	−1.96666	−0.983328	+	0.181843i	$0.941794\pi$
$602$	−6.56355e8	−0.122617
$603$	1.31112e9	0.243519
$604$	1.12660e9	0.208036
$605$	0	0
$606$	9.34735e8	0.170622
$607$	7.13282e9	1.29450	0.647248	−	0.762279i	$-0.275920\pi$
$607$	7.13282e9	1.29450	0.647248	+	0.762279i	$0.275920\pi$
$608$	2.39393e9	0.431965
$609$	−3.64740e9	−0.654369
$610$	0	0
$611$	−1.10746e9	−0.196419
$612$	−1.02806e9	−0.181297
$613$	−4.63327e9	−0.812412	−0.406206	−	0.913782i	$-0.633148\pi$
$613$	−4.63327e9	−0.812412	−0.406206	+	0.913782i	$0.633148\pi$
$614$	−1.98674e9	−0.346379
$615$	0	0
$616$	7.34322e8	0.126577
$617$	5.02424e9	0.861137	0.430568	−	0.902558i	$-0.358313\pi$
$617$	5.02424e9	0.861137	0.430568	+	0.902558i	$0.358313\pi$
$618$	−3.57944e9	−0.610035
$619$	5.70730e9	0.967194	0.483597	−	0.875291i	$-0.339330\pi$
$619$	5.70730e9	0.967194	0.483597	+	0.875291i	$0.339330\pi$
$620$	0	0
$621$	−7.00663e9	−1.17406
$622$	−1.70430e9	−0.283974
$623$	6.45819e8	0.107005
$624$	−5.32617e9	−0.877544
$625$	0	0
$626$	−2.68631e9	−0.437670
$627$	−1.92460e9	−0.311819
$628$	8.02037e9	1.29222
$629$	4.39634e9	0.704392
$630$	0	0
$631$	9.74258e9	1.54373	0.771865	−	0.635787i	$-0.219324\pi$
$631$	9.74258e9	1.54373	0.771865	+	0.635787i	$0.219324\pi$
$632$	1.01329e9	0.159670
$633$	−4.79764e9	−0.751822
$634$	2.67999e9	0.417659
$635$	0	0
$636$	−5.59019e9	−0.861641
$637$	1.05554e9	0.161803
$638$	−1.79988e9	−0.274391
$639$	−1.38358e9	−0.209774
$640$	0	0
$641$	−6.95897e9	−1.04362	−0.521809	−	0.853062i	$-0.674743\pi$
$641$	−6.95897e9	−1.04362	−0.521809	+	0.853062i	$0.674743\pi$
$642$	−1.56733e9	−0.233769
$643$	−4.05867e9	−0.602067	−0.301034	−	0.953613i	$-0.597332\pi$
$643$	−4.05867e9	−0.602067	−0.301034	+	0.953613i	$0.597332\pi$
$644$	−3.19070e9	−0.470745
$645$	0	0
$646$	9.49626e8	0.138592
$647$	1.06287e9	0.154282	0.0771411	−	0.997020i	$-0.475421\pi$
$647$	1.06287e9	0.154282	0.0771411	+	0.997020i	$0.475421\pi$
$648$	5.04806e9	0.728806
$649$	−4.26339e9	−0.612207
$650$	0	0
$651$	1.58287e9	0.224859
$652$	1.10409e10	1.56005
$653$	−4.15262e9	−0.583615	−0.291807	−	0.956477i	$-0.594257\pi$
$653$	−4.15262e9	−0.583615	−0.291807	+	0.956477i	$0.594257\pi$
$654$	2.31524e9	0.323649
$655$	0	0
$656$	2.04253e9	0.282491
$657$	−1.05941e9	−0.145742
$658$	1.55574e8	0.0212886
$659$	1.98297e9	0.269909	0.134954	−	0.990852i	$-0.456911\pi$
$659$	1.98297e9	0.269909	0.134954	+	0.990852i	$0.456911\pi$
$660$	0	0
$661$	−2.36180e9	−0.318081	−0.159041	−	0.987272i	$-0.550840\pi$
$661$	−2.36180e9	−0.318081	−0.159041	+	0.987272i	$0.550840\pi$
$662$	−3.62833e8	−0.0486076
$663$	−7.87483e9	−1.04941
$664$	2.06754e9	0.274072
$665$	0	0
$666$	5.12231e8	0.0671902
$667$	1.65635e10	2.16129
$668$	5.24616e9	0.680963
$669$	−3.29700e9	−0.425723
$670$	0	0
$671$	−6.83731e9	−0.873688
$672$	2.78875e9	0.354501
$673$	−5.23236e9	−0.661676	−0.330838	−	0.943688i	$-0.607331\pi$
$673$	−5.23236e9	−0.661676	−0.330838	+	0.943688i	$0.607331\pi$
$674$	−3.40542e8	−0.0428411
$675$	0	0
$676$	−2.03198e9	−0.252992
$677$	−5.54225e9	−0.686476	−0.343238	−	0.939248i	$-0.611524\pi$
$677$	−5.54225e9	−0.686476	−0.343238	+	0.939248i	$0.611524\pi$
$678$	−2.07835e9	−0.256103
$679$	−1.94441e9	−0.238366
$680$	0	0
$681$	9.01885e9	1.09430
$682$	7.81096e8	0.0942886
$683$	2.76485e9	0.332047	0.166024	−	0.986122i	$-0.446907\pi$
$683$	2.76485e9	0.332047	0.166024	+	0.986122i	$0.446907\pi$
$684$	−9.38255e8	−0.112105
$685$	0	0
$686$	−1.48280e8	−0.0175368
$687$	1.77592e10	2.08966
$688$	5.92710e9	0.693877
$689$	−8.39816e9	−0.978176
$690$	0	0
$691$	1.20041e10	1.38406	0.692030	−	0.721868i	$-0.256716\pi$
$691$	1.20041e10	1.38406	0.692030	+	0.721868i	$0.256716\pi$
$692$	−9.95019e9	−1.14146
$693$	−4.39718e8	−0.0501890
$694$	−5.22104e8	−0.0592923
$695$	0	0
$696$	−9.47542e9	−1.06529
$697$	3.01991e9	0.337815
$698$	2.87401e9	0.319885
$699$	4.69727e9	0.520206
$700$	0	0
$701$	1.26875e10	1.39111	0.695557	−	0.718471i	$-0.255158\pi$
$701$	1.26875e10	1.39111	0.695557	+	0.718471i	$0.255158\pi$
$702$	2.84316e9	0.310186
$703$	4.01229e9	0.435561
$704$	−2.12403e9	−0.229433
$705$	0	0
$706$	2.11177e9	0.225856
$707$	1.67282e9	0.178026
$708$	−1.05974e10	−1.12223
$709$	1.72658e10	1.81939	0.909694	−	0.415280i	$-0.136316\pi$
$709$	1.72658e10	1.81939	0.909694	+	0.415280i	$0.136316\pi$
$710$	0	0
$711$	−6.06767e8	−0.0633109
$712$	1.67775e9	0.174199
$713$	−7.18811e9	−0.742680
$714$	1.10624e9	0.113738
$715$	0	0
$716$	1.69898e10	1.72978
$717$	−1.30432e10	−1.32150
$718$	−5.25339e9	−0.529668
$719$	−1.23080e10	−1.23491	−0.617455	−	0.786606i	$-0.711836\pi$
$719$	−1.23080e10	−1.23491	−0.617455	+	0.786606i	$0.711836\pi$
$720$	0	0
$721$	−6.40585e9	−0.636507
$722$	−2.41789e9	−0.239087
$723$	9.22423e9	0.907708
$724$	−1.66124e10	−1.62684
$725$	0	0
$726$	2.62855e9	0.254940
$727$	−1.18757e10	−1.14628	−0.573138	−	0.819459i	$-0.694274\pi$
$727$	−1.18757e10	−1.14628	−0.573138	+	0.819459i	$0.694274\pi$
$728$	2.74214e9	0.263408
$729$	6.81488e9	0.651496
$730$	0	0
$731$	8.76331e9	0.829769
$732$	−1.69953e10	−1.60155
$733$	3.88675e9	0.364521	0.182261	−	0.983250i	$-0.441659\pi$
$733$	3.88675e9	0.364521	0.182261	+	0.983250i	$0.441659\pi$
$734$	3.21682e9	0.300255
$735$	0	0
$736$	−1.26643e10	−1.17087
$737$	5.90376e9	0.543241
$738$	3.51859e8	0.0322234
$739$	1.21007e10	1.10295	0.551474	−	0.834192i	$-0.314065\pi$
$739$	1.21007e10	1.10295	0.551474	+	0.834192i	$0.314065\pi$
$740$	0	0
$741$	−7.18691e9	−0.648901
$742$	1.17976e9	0.106018
$743$	1.18364e10	1.05867	0.529333	−	0.848414i	$-0.322442\pi$
$743$	1.18364e10	1.05867	0.529333	+	0.848414i	$0.322442\pi$
$744$	4.11207e9	0.366062
$745$	0	0
$746$	2.69474e9	0.237646
$747$	−1.23806e9	−0.108672
$748$	−4.62919e9	−0.404436
$749$	−2.80493e9	−0.243913
$750$	0	0
$751$	8.53011e8	0.0734877	0.0367439	−	0.999325i	$-0.488301\pi$
$751$	8.53011e8	0.0734877	0.0367439	+	0.999325i	$0.488301\pi$
$752$	−1.40488e9	−0.120470
$753$	1.81765e9	0.155142
$754$	−6.72118e9	−0.571013
$755$	0	0
$756$	3.38693e9	0.285088
$757$	8.40844e9	0.704498	0.352249	−	0.935906i	$-0.385417\pi$
$757$	8.40844e9	0.704498	0.352249	+	0.935906i	$0.385417\pi$
$758$	2.96511e9	0.247286
$759$	1.01814e10	0.845205
$760$	0	0
$761$	−4.78648e9	−0.393704	−0.196852	−	0.980433i	$-0.563072\pi$
$761$	−4.78648e9	−0.393704	−0.196852	+	0.980433i	$0.563072\pi$
$762$	−5.30211e9	−0.434117
$763$	4.14341e9	0.337693
$764$	−1.95624e10	−1.58707
$765$	0	0
$766$	−1.97825e8	−0.0159030
$767$	−1.59205e10	−1.27401
$768$	−1.45557e9	−0.115950
$769$	1.23304e10	0.977764	0.488882	−	0.872350i	$-0.337405\pi$
$769$	1.23304e10	0.977764	0.488882	+	0.872350i	$0.337405\pi$
$770$	0	0
$771$	1.46375e10	1.15021
$772$	−7.71525e8	−0.0603517
$773$	1.65741e10	1.29063	0.645314	−	0.763917i	$-0.276726\pi$
$773$	1.65741e10	1.29063	0.645314	+	0.763917i	$0.276726\pi$
$774$	1.02104e9	0.0791497
$775$	0	0
$776$	−5.05130e9	−0.388050
$777$	4.67403e9	0.357452
$778$	3.74251e9	0.284927
$779$	2.75610e9	0.208888
$780$	0	0
$781$	−6.23001e9	−0.467962
$782$	−5.02367e9	−0.375662
$783$	−1.75822e10	−1.30890
$784$	1.33902e9	0.0992386
$785$	0	0
$786$	3.08366e9	0.226510
$787$	−1.49363e10	−1.09227	−0.546137	−	0.837696i	$-0.683902\pi$
$787$	−1.49363e10	−1.09227	−0.546137	+	0.837696i	$0.683902\pi$
$788$	1.27646e10	0.929319
$789$	−7.15166e8	−0.0518367
$790$	0	0
$791$	−3.71947e9	−0.267216
$792$	−1.14233e9	−0.0817056
$793$	−2.55322e10	−1.81816
$794$	−7.08884e9	−0.502578
$795$	0	0
$796$	1.80750e10	1.27023
$797$	−1.31070e10	−0.917064	−0.458532	−	0.888678i	$-0.651625\pi$
$797$	−1.31070e10	−0.917064	−0.458532	+	0.888678i	$0.651625\pi$
$798$	1.00961e9	0.0703303
$799$	−2.07714e9	−0.144063
$800$	0	0
$801$	−1.00465e9	−0.0690718
$802$	−8.61995e9	−0.590057
$803$	−4.77033e9	−0.325120
$804$	1.46749e10	0.995813
$805$	0	0
$806$	2.91680e9	0.196216
$807$	8.35142e9	0.559375
$808$	4.34576e9	0.289819
$809$	5.11703e8	0.0339780	0.0169890	−	0.999856i	$-0.494592\pi$
$809$	5.11703e8	0.0339780	0.0169890	+	0.999856i	$0.494592\pi$
$810$	0	0
$811$	6.98145e9	0.459592	0.229796	−	0.973239i	$-0.426194\pi$
$811$	6.98145e9	0.459592	0.229796	+	0.973239i	$0.426194\pi$
$812$	−8.00663e9	−0.524812
$813$	8.76917e9	0.572324
$814$	2.30649e9	0.149888
$815$	0	0
$816$	−9.98974e9	−0.643633
$817$	7.99778e9	0.513088
$818$	7.18803e9	0.459170
$819$	−1.64202e9	−0.104444
$820$	0	0
$821$	1.37608e10	0.867848	0.433924	−	0.900950i	$-0.357129\pi$
$821$	1.37608e10	0.867848	0.433924	+	0.900950i	$0.357129\pi$
$822$	3.25470e9	0.204390
$823$	1.36525e10	0.853716	0.426858	−	0.904319i	$-0.359620\pi$
$823$	1.36525e10	0.853716	0.426858	+	0.904319i	$0.359620\pi$
$824$	−1.66415e10	−1.03621
$825$	0	0
$826$	2.23650e9	0.138082
$827$	−2.97475e10	−1.82887	−0.914433	−	0.404737i	$-0.867363\pi$
$827$	−2.97475e10	−1.82887	−0.914433	+	0.404737i	$0.867363\pi$
$828$	4.96352e9	0.303867
$829$	2.10898e10	1.28567	0.642837	−	0.766003i	$-0.277757\pi$
$829$	2.10898e10	1.28567	0.642837	+	0.766003i	$0.277757\pi$
$830$	0	0
$831$	2.17572e10	1.31522
$832$	−7.93164e9	−0.477454
$833$	1.97976e9	0.118674
$834$	−6.34436e9	−0.378710
$835$	0	0
$836$	−4.22480e9	−0.250083
$837$	7.63017e9	0.449775
$838$	−1.00227e10	−0.588345
$839$	2.12030e10	1.23945	0.619727	−	0.784817i	$-0.287243\pi$
$839$	2.12030e10	1.23945	0.619727	+	0.784817i	$0.287243\pi$
$840$	0	0
$841$	2.43141e10	1.40952
$842$	5.48251e9	0.316510
$843$	−2.20742e9	−0.126908
$844$	−1.05316e10	−0.602970
$845$	0	0
$846$	−2.42014e8	−0.0137418
$847$	4.70412e9	0.266003
$848$	−1.06536e10	−0.599945
$849$	2.25954e10	1.26719
$850$	0	0
$851$	−2.12257e10	−1.18061
$852$	−1.54858e10	−0.857819
$853$	−3.82463e9	−0.210993	−0.105496	−	0.994420i	$-0.533643\pi$
$853$	−3.82463e9	−0.210993	−0.105496	+	0.994420i	$0.533643\pi$
$854$	3.58672e9	0.197059
$855$	0	0
$856$	−7.28680e9	−0.397081
$857$	−1.02342e10	−0.555417	−0.277708	−	0.960665i	$-0.589575\pi$
$857$	−1.02342e10	−0.555417	−0.277708	+	0.960665i	$0.589575\pi$
$858$	−4.13143e9	−0.223303
$859$	1.61175e10	0.867603	0.433801	−	0.901009i	$-0.357172\pi$
$859$	1.61175e10	0.867603	0.433801	+	0.901009i	$0.357172\pi$
$860$	0	0
$861$	3.21066e9	0.171428
$862$	7.34930e9	0.390815
$863$	3.16724e10	1.67742	0.838712	−	0.544575i	$-0.183309\pi$
$863$	3.16724e10	1.67742	0.838712	+	0.544575i	$0.183309\pi$
$864$	1.34431e10	0.709089
$865$	0	0
$866$	2.41387e9	0.126300
$867$	6.63293e9	0.345651
$868$	3.47465e9	0.180340
$869$	−2.73217e9	−0.141234
$870$	0	0
$871$	2.20461e10	1.13049
$872$	1.07640e10	0.549751
$873$	3.02477e9	0.153866
$874$	−4.58482e9	−0.232291
$875$	0	0
$876$	−1.18575e10	−0.595977
$877$	3.49078e10	1.74753	0.873764	−	0.486350i	$-0.161672\pi$
$877$	3.49078e10	1.74753	0.873764	+	0.486350i	$0.161672\pi$
$878$	3.94089e9	0.196500
$879$	3.91270e10	1.94319
$880$	0	0
$881$	−5.20797e9	−0.256598	−0.128299	−	0.991736i	$-0.540952\pi$
$881$	−5.20797e9	−0.256598	−0.128299	+	0.991736i	$0.540952\pi$
$882$	2.30668e8	0.0113200
$883$	1.95840e9	0.0957278	0.0478639	−	0.998854i	$-0.484759\pi$
$883$	1.95840e9	0.0957278	0.0478639	+	0.998854i	$0.484759\pi$
$884$	−1.72865e10	−0.841637
$885$	0	0
$886$	6.42591e9	0.310396
$887$	1.80819e10	0.869985	0.434992	−	0.900434i	$-0.356751\pi$
$887$	1.80819e10	0.869985	0.434992	+	0.900434i	$0.356751\pi$
$888$	1.21425e10	0.581917
$889$	−9.48879e9	−0.452955
$890$	0	0
$891$	−1.36112e10	−0.644653
$892$	−7.23744e9	−0.341435
$893$	−1.89569e9	−0.0890815
$894$	−1.22154e10	−0.571778
$895$	0	0
$896$	7.95789e9	0.369590
$897$	3.80199e10	1.75889
$898$	−6.11398e9	−0.281745
$899$	−1.80376e10	−0.827979
$900$	0	0
$901$	−1.57516e10	−0.717442
$902$	1.58436e9	0.0718838
$903$	9.31683e9	0.421076
$904$	−9.66264e9	−0.435017
$905$	0	0
$906$	1.88583e9	0.0842471
$907$	1.32060e10	0.587688	0.293844	−	0.955853i	$-0.405065\pi$
$907$	1.32060e10	0.587688	0.293844	+	0.955853i	$0.405065\pi$
$908$	1.97978e10	0.877642
$909$	−2.60228e9	−0.114916
$910$	0	0
$911$	−6.83495e9	−0.299517	−0.149758	−	0.988723i	$-0.547850\pi$
$911$	−6.83495e9	−0.299517	−0.149758	+	0.988723i	$0.547850\pi$
$912$	−9.11707e9	−0.397991
$913$	−5.57476e9	−0.242426
$914$	−9.16343e9	−0.396959
$915$	0	0
$916$	3.89844e10	1.67593
$917$	5.51859e9	0.236339
$918$	5.33262e9	0.227505
$919$	−3.96347e10	−1.68450	−0.842251	−	0.539086i	$-0.818770\pi$
$919$	−3.96347e10	−1.68450	−0.842251	+	0.539086i	$0.818770\pi$
$920$	0	0
$921$	2.82013e10	1.18949
$922$	−1.45739e10	−0.612375
$923$	−2.32644e10	−0.973836
$924$	−4.92158e9	−0.205236
$925$	0	0
$926$	1.24491e10	0.515227
$927$	9.96506e9	0.410867
$928$	−3.17792e10	−1.30534
$929$	−2.44116e10	−0.998944	−0.499472	−	0.866330i	$-0.666473\pi$
$929$	−2.44116e10	−0.998944	−0.499472	+	0.866330i	$0.666473\pi$
$930$	0	0
$931$	1.80682e9	0.0733821
$932$	1.03113e10	0.417212
$933$	2.41922e10	0.975189
$934$	1.16955e10	0.469685
$935$	0	0
$936$	−4.26573e9	−0.170031
$937$	5.38252e8	0.0213745	0.0106873	−	0.999943i	$-0.496598\pi$
$937$	5.38252e8	0.0213745	0.0106873	+	0.999943i	$0.496598\pi$
$938$	−3.09701e9	−0.122527
$939$	3.81316e10	1.50299
$940$	0	0
$941$	−2.81189e10	−1.10010	−0.550052	−	0.835130i	$-0.685392\pi$
$941$	−2.81189e10	−1.10010	−0.550052	+	0.835130i	$0.685392\pi$
$942$	1.34255e10	0.523301
$943$	−1.45802e10	−0.566204
$944$	−2.01963e10	−0.781392
$945$	0	0
$946$	4.59756e9	0.176567
$947$	−4.48215e10	−1.71499	−0.857495	−	0.514492i	$-0.827980\pi$
$947$	−4.48215e10	−1.71499	−0.857495	+	0.514492i	$0.827980\pi$
$948$	−6.79129e9	−0.258895
$949$	−1.78136e10	−0.676581
$950$	0	0
$951$	−3.80419e10	−1.43427
$952$	5.14314e9	0.193196
$953$	−2.41425e9	−0.0903561	−0.0451780	−	0.998979i	$-0.514386\pi$
$953$	−2.41425e9	−0.0903561	−0.0451780	+	0.998979i	$0.514386\pi$
$954$	−1.83526e9	−0.0684351
$955$	0	0
$956$	−2.86319e10	−1.05986
$957$	2.55489e10	0.942280
$958$	5.40830e9	0.198738
$959$	5.82470e9	0.213259
$960$	0	0
$961$	−1.96848e10	−0.715483
$962$	8.61299e9	0.311918
$963$	4.36340e9	0.157447
$964$	2.02487e10	0.727993
$965$	0	0
$966$	−5.34098e9	−0.190634
$967$	−4.34362e10	−1.54475	−0.772377	−	0.635164i	$-0.780932\pi$
$967$	−4.34362e10	−1.54475	−0.772377	+	0.635164i	$0.780932\pi$
$968$	1.22206e10	0.433042
$969$	−1.34797e10	−0.475936
$970$	0	0
$971$	−4.21172e10	−1.47636	−0.738180	−	0.674604i	$-0.764314\pi$
$971$	−4.21172e10	−1.47636	−0.738180	+	0.674604i	$0.764314\pi$
$972$	−1.22378e10	−0.427438
$973$	−1.13540e10	−0.395143
$974$	−2.56228e9	−0.0888527
$975$	0	0
$976$	−3.23892e10	−1.11513
$977$	−3.57143e10	−1.22521	−0.612606	−	0.790389i	$-0.709879\pi$
$977$	−3.57143e10	−1.22521	−0.612606	+	0.790389i	$0.709879\pi$
$978$	1.84816e10	0.631762
$979$	−4.52376e9	−0.154085
$980$	0	0
$981$	−6.44558e9	−0.217982
$982$	2.88719e9	0.0972940
$983$	−2.72118e10	−0.913735	−0.456868	−	0.889535i	$-0.651029\pi$
$983$	−2.72118e10	−0.913735	−0.456868	+	0.889535i	$0.651029\pi$
$984$	8.34083e9	0.279079
$985$	0	0
$986$	−1.26062e10	−0.418809
$987$	−2.20834e9	−0.0731065
$988$	−1.57764e10	−0.520427
$989$	−4.23095e10	−1.39076
$990$	0	0
$991$	3.43790e10	1.12211	0.561055	−	0.827778i	$-0.310396\pi$
$991$	3.43790e10	1.12211	0.561055	+	0.827778i	$0.310396\pi$
$992$	1.37913e10	0.448553
$993$	5.15034e9	0.166922
$994$	3.26815e9	0.105548
$995$	0	0
$996$	−1.38571e10	−0.444389
$997$	4.40554e10	1.40788	0.703940	−	0.710259i	$-0.251422\pi$
$997$	4.40554e10	1.40788	0.703940	+	0.710259i	$0.251422\pi$
$998$	−1.90603e10	−0.606977
$999$	2.25310e10	0.714993

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.8.a.h.1.4	yes	6
5.2	odd	4		175.8.b.g.99.7		12
5.3	odd	4		175.8.b.g.99.6		12
5.4	even	2		175.8.a.g.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.8.a.g.1.3	✓	6	5.4	even	2
175.8.a.h.1.4	yes	6	1.1	even	1	trivial
175.8.b.g.99.6		12	5.3	odd	4
175.8.b.g.99.7		12	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)

\(f(q)\)	\(=\)	\(q+3.67453 q^{2} -52.1592 q^{3} -114.498 q^{4} -191.660 q^{6} -343.000 q^{7} -891.065 q^{8} +533.578 q^{9} +O(q^{10})\)	\(q+3.67453 q^{2} -52.1592 q^{3} -114.498 q^{4} -191.660 q^{6} -343.000 q^{7} -891.065 q^{8} +533.578 q^{9} +2402.61 q^{11} +5972.11 q^{12} +8971.92 q^{13} -1260.36 q^{14} +11381.5 q^{16} +16827.7 q^{17} +1960.65 q^{18} +15357.7 q^{19} +17890.6 q^{21} +8828.45 q^{22} -81244.7 q^{23} +46477.2 q^{24} +32967.6 q^{26} +86241.1 q^{27} +39272.8 q^{28} -203872. q^{29} +88474.9 q^{31} +155878. q^{32} -125318. q^{33} +61833.9 q^{34} -61093.5 q^{36} +261256. q^{37} +56432.3 q^{38} -467968. q^{39} +179461. q^{41} +65739.5 q^{42} +520767. q^{43} -275093. q^{44} -298536. q^{46} -123436. q^{47} -593648. q^{48} +117649. q^{49} -877719. q^{51} -1.02727e6 q^{52} -936049. q^{53} +316895. q^{54} +305635. q^{56} -801044. q^{57} -749135. q^{58} -1.77448e6 q^{59} -2.84579e6 q^{61} +325104. q^{62} -183017. q^{63} -884051. q^{64} -460485. q^{66} +2.45723e6 q^{67} -1.92674e6 q^{68} +4.23765e6 q^{69} -2.59302e6 q^{71} -475453. q^{72} -1.98548e6 q^{73} +959993. q^{74} -1.75842e6 q^{76} -824094. q^{77} -1.71956e6 q^{78} -1.13717e6 q^{79} -5.66520e6 q^{81} +659433. q^{82} -2.32030e6 q^{83} -2.04843e6 q^{84} +1.91357e6 q^{86} +1.06338e7 q^{87} -2.14088e6 q^{88} -1.88285e6 q^{89} -3.07737e6 q^{91} +9.30234e6 q^{92} -4.61478e6 q^{93} -453569. q^{94} -8.13046e6 q^{96} +5.66884e6 q^{97} +432305. q^{98} +1.28198e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} - 28 q^{3} + 321 q^{4} - 917 q^{6} - 2058 q^{7} + 3297 q^{8} + 1800 q^{9}+O(q^{10}) \) 6 * q + 5 * q^2 - 28 * q^3 + 321 * q^4 - 917 * q^6 - 2058 * q^7 + 3297 * q^8 + 1800 * q^9	\( 6 q + 5 q^{2} - 28 q^{3} + 321 q^{4} - 917 q^{6} - 2058 q^{7} + 3297 q^{8} + 1800 q^{9} - 2636 q^{11} - 8043 q^{12} + 14196 q^{13} - 1715 q^{14} - 255 q^{16} - 11606 q^{17} + 28720 q^{18} - 45276 q^{19} + 9604 q^{21} + 1455 q^{22} + 28324 q^{23} - 156751 q^{24} - 10038 q^{26} + 51044 q^{27} - 110103 q^{28} - 285224 q^{29} - 158928 q^{31} + 68721 q^{32} - 163590 q^{33} - 259371 q^{34} - 360694 q^{36} + 348468 q^{37} - 676585 q^{38} - 40352 q^{39} - 819574 q^{41} + 314531 q^{42} - 299484 q^{43} + 838563 q^{44} - 1032870 q^{46} + 1308496 q^{47} - 1674519 q^{48} + 705894 q^{49} - 1148440 q^{51} + 1528170 q^{52} + 198568 q^{53} + 525273 q^{54} - 1130871 q^{56} + 4094722 q^{57} - 5881332 q^{58} - 3779552 q^{59} - 3079860 q^{61} + 1750056 q^{62} - 617400 q^{63} - 3754335 q^{64} - 9954427 q^{66} + 1673172 q^{67} - 11552737 q^{68} - 7859880 q^{69} - 9136988 q^{71} + 7139950 q^{72} - 4605594 q^{73} - 9029106 q^{74} - 23890251 q^{76} + 904148 q^{77} - 11618738 q^{78} - 2417100 q^{79} - 23407970 q^{81} + 6341937 q^{82} - 1365532 q^{83} + 2758749 q^{84} - 30428682 q^{86} + 14992936 q^{87} - 5525277 q^{88} - 13845594 q^{89} - 4869228 q^{91} + 41315508 q^{92} - 9319440 q^{93} - 18396630 q^{94} - 28657783 q^{96} - 2963436 q^{97} + 588245 q^{98} - 4220084 q^{99}+O(q^{100}) \) 6 * q + 5 * q^2 - 28 * q^3 + 321 * q^4 - 917 * q^6 - 2058 * q^7 + 3297 * q^8 + 1800 * q^9 - 2636 * q^11 - 8043 * q^12 + 14196 * q^13 - 1715 * q^14 - 255 * q^16 - 11606 * q^17 + 28720 * q^18 - 45276 * q^19 + 9604 * q^21 + 1455 * q^22 + 28324 * q^23 - 156751 * q^24 - 10038 * q^26 + 51044 * q^27 - 110103 * q^28 - 285224 * q^29 - 158928 * q^31 + 68721 * q^32 - 163590 * q^33 - 259371 * q^34 - 360694 * q^36 + 348468 * q^37 - 676585 * q^38 - 40352 * q^39 - 819574 * q^41 + 314531 * q^42 - 299484 * q^43 + 838563 * q^44 - 1032870 * q^46 + 1308496 * q^47 - 1674519 * q^48 + 705894 * q^49 - 1148440 * q^51 + 1528170 * q^52 + 198568 * q^53 + 525273 * q^54 - 1130871 * q^56 + 4094722 * q^57 - 5881332 * q^58 - 3779552 * q^59 - 3079860 * q^61 + 1750056 * q^62 - 617400 * q^63 - 3754335 * q^64 - 9954427 * q^66 + 1673172 * q^67 - 11552737 * q^68 - 7859880 * q^69 - 9136988 * q^71 + 7139950 * q^72 - 4605594 * q^73 - 9029106 * q^74 - 23890251 * q^76 + 904148 * q^77 - 11618738 * q^78 - 2417100 * q^79 - 23407970 * q^81 + 6341937 * q^82 - 1365532 * q^83 + 2758749 * q^84 - 30428682 * q^86 + 14992936 * q^87 - 5525277 * q^88 - 13845594 * q^89 - 4869228 * q^91 + 41315508 * q^92 - 9319440 * q^93 - 18396630 * q^94 - 28657783 * q^96 - 2963436 * q^97 + 588245 * q^98 - 4220084 * q^99

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