LMFDB - Embedded newform 6825.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6825,2,Mod(1,6825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6825.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:	$54.4978993795$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.52616$ of defining polynomial
Character	$\chi$	$=$	6825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.670843 q^{2} -1.00000 q^{3} -1.54997 q^{4} -0.670843 q^{6} -1.00000 q^{7} -2.38147 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.670843 q^{2} -1.00000 q^{3} -1.54997 q^{4} -0.670843 q^{6} -1.00000 q^{7} -2.38147 q^{8} +1.00000 q^{9} -3.05232 q^{11} +1.54997 q^{12} -1.00000 q^{13} -0.670843 q^{14} +1.50235 q^{16} -1.34169 q^{17} +0.670843 q^{18} +7.60228 q^{19} +1.00000 q^{21} -2.04762 q^{22} -1.84403 q^{23} +2.38147 q^{24} -0.670843 q^{26} -1.00000 q^{27} +1.54997 q^{28} -5.60228 q^{29} +10.2857 q^{31} +5.77078 q^{32} +3.05232 q^{33} -0.900061 q^{34} -1.54997 q^{36} +9.20457 q^{37} +5.09994 q^{38} +1.00000 q^{39} -8.04762 q^{41} +0.670843 q^{42} -1.49765 q^{43} +4.73100 q^{44} -1.23706 q^{46} +3.89166 q^{47} -1.50235 q^{48} +1.00000 q^{49} +1.34169 q^{51} +1.54997 q^{52} -0.502345 q^{53} -0.670843 q^{54} +2.38147 q^{56} -7.60228 q^{57} -3.75825 q^{58} -4.28937 q^{59} -0.683372 q^{61} +6.90006 q^{62} -1.00000 q^{63} +0.866598 q^{64} +2.04762 q^{66} +7.68806 q^{67} +2.07957 q^{68} +1.84403 q^{69} -14.2569 q^{71} -2.38147 q^{72} +12.0189 q^{73} +6.17482 q^{74} -11.7833 q^{76} +3.05232 q^{77} +0.670843 q^{78} +4.91891 q^{79} +1.00000 q^{81} -5.39869 q^{82} +1.20828 q^{83} -1.54997 q^{84} -1.00469 q^{86} +5.60228 q^{87} +7.26900 q^{88} +13.7545 q^{89} +1.00000 q^{91} +2.85819 q^{92} -10.2857 q^{93} +2.61069 q^{94} -5.77078 q^{96} +7.18572 q^{97} +0.670843 q^{98} -3.05232 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9	$ 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24} + q^{26} - 4 q^{27} - 7 q^{28} + q^{29} + 3 q^{31} - 7 q^{32} + 2 q^{33} - 30 q^{34} + 7 q^{36} - 10 q^{37} - 6 q^{38} + 4 q^{39} - 16 q^{41} - q^{42} - 3 q^{43} - 12 q^{44} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 4 q^{49} - 2 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 3 q^{56} - 7 q^{57} + 4 q^{58} - 20 q^{59} + 12 q^{61} + 54 q^{62} - 4 q^{63} + 5 q^{64} - 8 q^{66} + 22 q^{67} + 10 q^{68} + 3 q^{69} - 3 q^{72} + 13 q^{73} - 6 q^{74} - 6 q^{76} + 2 q^{77} - q^{78} + 11 q^{79} + 4 q^{81} - 10 q^{82} - q^{83} + 7 q^{84} - 10 q^{86} - q^{87} + 60 q^{88} - 5 q^{89} + 4 q^{91} - 34 q^{92} - 3 q^{93} + 34 q^{94} + 7 q^{96} + 17 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) $ 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 - 7 * q^12 - 4 * q^13 + q^14 + 9 * q^16 + 2 * q^17 - q^18 + 7 * q^19 + 4 * q^21 + 8 * q^22 - 3 * q^23 + 3 * q^24 + q^26 - 4 * q^27 - 7 * q^28 + q^29 + 3 * q^31 - 7 * q^32 + 2 * q^33 - 30 * q^34 + 7 * q^36 - 10 * q^37 - 6 * q^38 + 4 * q^39 - 16 * q^41 - q^42 - 3 * q^43 - 12 * q^44 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 4 * q^49 - 2 * q^51 - 7 * q^52 - 5 * q^53 + q^54 + 3 * q^56 - 7 * q^57 + 4 * q^58 - 20 * q^59 + 12 * q^61 + 54 * q^62 - 4 * q^63 + 5 * q^64 - 8 * q^66 + 22 * q^67 + 10 * q^68 + 3 * q^69 - 3 * q^72 + 13 * q^73 - 6 * q^74 - 6 * q^76 + 2 * q^77 - q^78 + 11 * q^79 + 4 * q^81 - 10 * q^82 - q^83 + 7 * q^84 - 10 * q^86 - q^87 + 60 * q^88 - 5 * q^89 + 4 * q^91 - 34 * q^92 - 3 * q^93 + 34 * q^94 + 7 * q^96 + 17 * q^97 - q^98 - 2 * 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 $

$2$	0.670843	0.474358	0.237179	−	0.971466i	$-0.423777\pi$
$2$	0.670843	0.474358	0.237179	+	0.971466i	$0.423777\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.54997	−0.774985
$5$	0	0
$5$	0	0
$6$	−0.670843	−0.273870
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.38147	−0.841978
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.05232	−0.920308	−0.460154	−	0.887839i	$-0.652206\pi$
$11$	−3.05232	−0.920308	−0.460154	+	0.887839i	$0.652206\pi$
$12$	1.54997	0.447438
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.670843	−0.179290
$15$	0	0
$16$	1.50235	0.375586
$17$	−1.34169	−0.325407	−0.162703	−	0.986675i	$-0.552021\pi$
$17$	−1.34169	−0.325407	−0.162703	+	0.986675i	$0.552021\pi$
$18$	0.670843	0.158119
$19$	7.60228	1.74408	0.872042	−	0.489431i	$-0.162796\pi$
$19$	7.60228	1.74408	0.872042	+	0.489431i	$0.162796\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−2.04762	−0.436555
$23$	−1.84403	−0.384507	−0.192254	−	0.981345i	$-0.561580\pi$
$23$	−1.84403	−0.384507	−0.192254	+	0.981345i	$0.561580\pi$
$24$	2.38147	0.486116
$25$	0	0
$26$	−0.670843	−0.131563
$27$	−1.00000	−0.192450
$28$	1.54997	0.292917
$29$	−5.60228	−1.04032	−0.520159	−	0.854069i	$-0.674127\pi$
$29$	−5.60228	−1.04032	−0.520159	+	0.854069i	$0.674127\pi$
$30$	0	0
$31$	10.2857	1.84736	0.923679	−	0.383167i	$-0.125166\pi$
$31$	10.2857	1.84736	0.923679	+	0.383167i	$0.125166\pi$
$32$	5.77078	1.02014
$33$	3.05232	0.531340
$34$	−0.900061	−0.154359
$35$	0	0
$36$	−1.54997	−0.258328
$37$	9.20457	1.51322	0.756611	−	0.653865i	$-0.226854\pi$
$37$	9.20457	1.51322	0.756611	+	0.653865i	$0.226854\pi$
$38$	5.09994	0.827319
$39$	1.00000	0.160128
$40$	0	0
$41$	−8.04762	−1.25683	−0.628414	−	0.777879i	$-0.716296\pi$
$41$	−8.04762	−1.25683	−0.628414	+	0.777879i	$0.716296\pi$
$42$	0.670843	0.103513
$43$	−1.49765	−0.228390	−0.114195	−	0.993458i	$-0.536429\pi$
$43$	−1.49765	−0.228390	−0.114195	+	0.993458i	$0.536429\pi$
$44$	4.73100	0.713224
$45$	0	0
$46$	−1.23706	−0.182394
$47$	3.89166	0.567656	0.283828	−	0.958875i	$-0.408395\pi$
$47$	3.89166	0.567656	0.283828	+	0.958875i	$0.408395\pi$
$48$	−1.50235	−0.216845
$49$	1.00000	0.142857
$50$	0	0
$51$	1.34169	0.187874
$52$	1.54997	0.214942
$53$	−0.502345	−0.0690025	−0.0345012	−	0.999405i	$-0.510984\pi$
$53$	−0.502345	−0.0690025	−0.0345012	+	0.999405i	$0.510984\pi$
$54$	−0.670843	−0.0912902
$55$	0	0
$56$	2.38147	0.318238
$57$	−7.60228	−1.00695
$58$	−3.75825	−0.493483
$59$	−4.28937	−0.558429	−0.279214	−	0.960229i	$-0.590074\pi$
$59$	−4.28937	−0.558429	−0.279214	+	0.960229i	$0.590074\pi$
$60$	0	0
$61$	−0.683372	−0.0874968	−0.0437484	−	0.999043i	$-0.513930\pi$
$61$	−0.683372	−0.0874968	−0.0437484	+	0.999043i	$0.513930\pi$
$62$	6.90006	0.876309
$63$	−1.00000	−0.125988
$64$	0.866598	0.108325
$65$	0	0
$66$	2.04762	0.252045
$67$	7.68806	0.939246	0.469623	−	0.882867i	$-0.344390\pi$
$67$	7.68806	0.939246	0.469623	+	0.882867i	$0.344390\pi$
$68$	2.07957	0.252185
$69$	1.84403	0.221995
$70$	0	0
$71$	−14.2569	−1.69198	−0.845990	−	0.533198i	$-0.820990\pi$
$71$	−14.2569	−1.69198	−0.845990	+	0.533198i	$0.820990\pi$
$72$	−2.38147	−0.280659
$73$	12.0189	1.40670	0.703350	−	0.710844i	$-0.251687\pi$
$73$	12.0189	1.40670	0.703350	+	0.710844i	$0.251687\pi$
$74$	6.17482	0.717808
$75$	0	0
$76$	−11.7833	−1.35164
$77$	3.05232	0.347844
$78$	0.670843	0.0759580
$79$	4.91891	0.553421	0.276710	−	0.960953i	$-0.410756\pi$
$79$	4.91891	0.553421	0.276710	+	0.960953i	$0.410756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.39869	−0.596186
$83$	1.20828	0.132626	0.0663132	−	0.997799i	$-0.478876\pi$
$83$	1.20828	0.132626	0.0663132	+	0.997799i	$0.478876\pi$
$84$	−1.54997	−0.169116
$85$	0	0
$86$	−1.00469	−0.108339
$87$	5.60228	0.600628
$88$	7.26900	0.774878
$89$	13.7545	1.45798	0.728989	−	0.684525i	$-0.239990\pi$
$89$	13.7545	1.45798	0.728989	+	0.684525i	$0.239990\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	2.85819	0.297987
$93$	−10.2857	−1.06657
$94$	2.61069	0.269272
$95$	0	0
$96$	−5.77078	−0.588978
$97$	7.18572	0.729599	0.364800	−	0.931086i	$-0.381138\pi$
$97$	7.18572	0.729599	0.364800	+	0.931086i	$0.381138\pi$
$98$	0.670843	0.0677654
$99$	−3.05232	−0.306769
$100$	0	0
$101$	−18.4416	−1.83501	−0.917505	−	0.397724i	$-0.869800\pi$
$101$	−18.4416	−1.83501	−0.917505	+	0.397724i	$0.869800\pi$
$102$	0.900061	0.0891193
$103$	−1.89537	−0.186756	−0.0933782	−	0.995631i	$-0.529767\pi$
$103$	−1.89537	−0.186756	−0.0933782	+	0.995631i	$0.529767\pi$
$104$	2.38147	0.233523
$105$	0	0
$106$	−0.336995	−0.0327318
$107$	−18.5463	−1.79293	−0.896467	−	0.443110i	$-0.853875\pi$
$107$	−18.5463	−1.79293	−0.896467	+	0.443110i	$0.853875\pi$
$108$	1.54997	0.149146
$109$	−1.42126	−0.136132	−0.0680659	−	0.997681i	$-0.521683\pi$
$109$	−1.42126	−0.136132	−0.0680659	+	0.997681i	$0.521683\pi$
$110$	0	0
$111$	−9.20457	−0.873659
$112$	−1.50235	−0.141958
$113$	−5.60228	−0.527019	−0.263509	−	0.964657i	$-0.584880\pi$
$113$	−5.60228	−0.527019	−0.263509	+	0.964657i	$0.584880\pi$
$114$	−5.09994	−0.477653
$115$	0	0
$116$	8.68337	0.806231
$117$	−1.00000	−0.0924500
$118$	−2.87749	−0.264895
$119$	1.34169	0.122992
$120$	0	0
$121$	−1.68337	−0.153034
$122$	−0.458435	−0.0415048
$123$	8.04762	0.725630
$124$	−15.9425	−1.43167
$125$	0	0
$126$	−0.670843	−0.0597634
$127$	−14.1046	−1.25158	−0.625792	−	0.779990i	$-0.715224\pi$
$127$	−14.1046	−1.25158	−0.625792	+	0.779990i	$0.715224\pi$
$128$	−10.9602	−0.968755
$129$	1.49765	0.131861
$130$	0	0
$131$	−8.78800	−0.767811	−0.383906	−	0.923372i	$-0.625421\pi$
$131$	−8.78800	−0.767811	−0.383906	+	0.923372i	$0.625421\pi$
$132$	−4.73100	−0.411780
$133$	−7.60228	−0.659202
$134$	5.15748	0.445539
$135$	0	0
$136$	3.19519	0.273985
$137$	1.97743	0.168944	0.0844718	−	0.996426i	$-0.473080\pi$
$137$	1.97743	0.168944	0.0844718	+	0.996426i	$0.473080\pi$
$138$	1.23706	0.105305
$139$	13.0999	1.11112	0.555561	−	0.831476i	$-0.312503\pi$
$139$	13.0999	1.11112	0.555561	+	0.831476i	$0.312503\pi$
$140$	0	0
$141$	−3.89166	−0.327737
$142$	−9.56413	−0.802604
$143$	3.05232	0.255247
$144$	1.50235	0.125195
$145$	0	0
$146$	8.06276	0.667279
$147$	−1.00000	−0.0824786
$148$	−14.2668	−1.17272
$149$	14.5939	1.19558	0.597789	−	0.801654i	$-0.296046\pi$
$149$	14.5939	1.19558	0.597789	+	0.801654i	$0.296046\pi$
$150$	0	0
$151$	−5.41188	−0.440412	−0.220206	−	0.975453i	$-0.570673\pi$
$151$	−5.41188	−0.440412	−0.220206	+	0.975453i	$0.570673\pi$
$152$	−18.1046	−1.46848
$153$	−1.34169	−0.108469
$154$	2.04762	0.165002
$155$	0	0
$156$	−1.54997	−0.124097
$157$	−23.4044	−1.86788	−0.933939	−	0.357432i	$-0.883652\pi$
$157$	−23.4044	−1.86788	−0.933939	+	0.357432i	$0.883652\pi$
$158$	3.29982	0.262519
$159$	0.502345	0.0398386
$160$	0	0
$161$	1.84403	0.145330
$162$	0.670843	0.0527064
$163$	10.1046	0.791456	0.395728	−	0.918368i	$-0.370492\pi$
$163$	10.1046	0.791456	0.395728	+	0.918368i	$0.370492\pi$
$164$	12.4736	0.974022
$165$	0	0
$166$	0.810569	0.0629123
$167$	3.09623	0.239593	0.119797	−	0.992798i	$-0.461776\pi$
$167$	3.09623	0.239593	0.119797	+	0.992798i	$0.461776\pi$
$168$	−2.38147	−0.183735
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.60228	0.581361
$172$	2.32132	0.176999
$173$	−2.75356	−0.209349	−0.104675	−	0.994507i	$-0.533380\pi$
$173$	−2.75356	−0.209349	−0.104675	+	0.994507i	$0.533380\pi$
$174$	3.75825	0.284912
$175$	0	0
$176$	−4.58563	−0.345655
$177$	4.28937	0.322409
$178$	9.22714	0.691603
$179$	−1.57723	−0.117887	−0.0589437	−	0.998261i	$-0.518773\pi$
$179$	−1.57723	−0.117887	−0.0589437	+	0.998261i	$0.518773\pi$
$180$	0	0
$181$	9.51651	0.707356	0.353678	−	0.935367i	$-0.384931\pi$
$181$	9.51651	0.707356	0.353678	+	0.935367i	$0.384931\pi$
$182$	0.670843	0.0497262
$183$	0.683372	0.0505163
$184$	4.39151	0.323746
$185$	0	0
$186$	−6.90006	−0.505937
$187$	4.09525	0.299474
$188$	−6.03195	−0.439925
$189$	1.00000	0.0727393
$190$	0	0
$191$	−22.8508	−1.65342	−0.826712	−	0.562626i	$-0.809791\pi$
$191$	−22.8508	−1.65342	−0.826712	+	0.562626i	$0.809791\pi$
$192$	−0.866598	−0.0625413
$193$	−19.5760	−1.40911	−0.704556	−	0.709649i	$-0.748854\pi$
$193$	−19.5760	−1.40911	−0.704556	+	0.709649i	$0.748854\pi$
$194$	4.82049	0.346091
$195$	0	0
$196$	−1.54997	−0.110712
$197$	21.2271	1.51237	0.756185	−	0.654357i	$-0.227061\pi$
$197$	21.2271	1.51237	0.756185	+	0.654357i	$0.227061\pi$
$198$	−2.04762	−0.145518
$199$	−6.68337	−0.473772	−0.236886	−	0.971537i	$-0.576127\pi$
$199$	−6.68337	−0.473772	−0.236886	+	0.971537i	$0.576127\pi$
$200$	0	0
$201$	−7.68806	−0.542274
$202$	−12.3714	−0.870451
$203$	5.60228	0.393203
$204$	−2.07957	−0.145599
$205$	0	0
$206$	−1.27150	−0.0885893
$207$	−1.84403	−0.128169
$208$	−1.50235	−0.104169
$209$	−23.2046	−1.60509
$210$	0	0
$211$	−4.60698	−0.317157	−0.158579	−	0.987346i	$-0.550691\pi$
$211$	−4.60698	−0.317157	−0.158579	+	0.987346i	$0.550691\pi$
$212$	0.778620	0.0534759
$213$	14.2569	0.976866
$214$	−12.4416	−0.850492
$215$	0	0
$216$	2.38147	0.162039
$217$	−10.2857	−0.698236
$218$	−0.953441	−0.0645752
$219$	−12.0189	−0.812159
$220$	0	0
$221$	1.34169	0.0902516
$222$	−6.17482	−0.414427
$223$	−8.60698	−0.576366	−0.288183	−	0.957575i	$-0.593051\pi$
$223$	−8.60698	−0.576366	−0.288183	+	0.957575i	$0.593051\pi$
$224$	−5.77078	−0.385577
$225$	0	0
$226$	−3.75825	−0.249995
$227$	14.4892	0.961685	0.480843	−	0.876807i	$-0.340331\pi$
$227$	14.4892	0.961685	0.480843	+	0.876807i	$0.340331\pi$
$228$	11.7833	0.780369
$229$	−20.1999	−1.33485	−0.667423	−	0.744679i	$-0.732603\pi$
$229$	−20.1999	−1.33485	−0.667423	+	0.744679i	$0.732603\pi$
$230$	0	0
$231$	−3.05232	−0.200828
$232$	13.3417	0.875925
$233$	−7.28097	−0.476992	−0.238496	−	0.971143i	$-0.576654\pi$
$233$	−7.28097	−0.476992	−0.238496	+	0.971143i	$0.576654\pi$
$234$	−0.670843	−0.0438544
$235$	0	0
$236$	6.64839	0.432774
$237$	−4.91891	−0.319518
$238$	0.900061	0.0583423
$239$	27.6688	1.78974	0.894872	−	0.446323i	$-0.147267\pi$
$239$	27.6688	1.78974	0.894872	+	0.446323i	$0.147267\pi$
$240$	0	0
$241$	−22.2187	−1.43123	−0.715617	−	0.698493i	$-0.753854\pi$
$241$	−22.2187	−1.43123	−0.715617	+	0.698493i	$0.753854\pi$
$242$	−1.12928	−0.0725928
$243$	−1.00000	−0.0641500
$244$	1.05921	0.0678087
$245$	0	0
$246$	5.39869	0.344208
$247$	−7.60228	−0.483722
$248$	−24.4950	−1.55543
$249$	−1.20828	−0.0765719
$250$	0	0
$251$	17.3092	1.09255	0.546274	−	0.837607i	$-0.316046\pi$
$251$	17.3092	1.09255	0.546274	+	0.837607i	$0.316046\pi$
$252$	1.54997	0.0976389
$253$	5.62857	0.353865
$254$	−9.46199	−0.593698
$255$	0	0
$256$	−9.08578	−0.567861
$257$	5.55837	0.346722	0.173361	−	0.984858i	$-0.444537\pi$
$257$	5.55837	0.346722	0.173361	+	0.984858i	$0.444537\pi$
$258$	1.00469	0.0625493
$259$	−9.20457	−0.571944
$260$	0	0
$261$	−5.60228	−0.346773
$262$	−5.89537	−0.364217
$263$	−25.0486	−1.54456	−0.772281	−	0.635281i	$-0.780884\pi$
$263$	−25.0486	−1.54456	−0.772281	+	0.635281i	$0.780884\pi$
$264$	−7.26900	−0.447376
$265$	0	0
$266$	−5.09994	−0.312697
$267$	−13.7545	−0.841764
$268$	−11.9163	−0.727902
$269$	−4.55368	−0.277643	−0.138821	−	0.990317i	$-0.544331\pi$
$269$	−4.55368	−0.277643	−0.138821	+	0.990317i	$0.544331\pi$
$270$	0	0
$271$	8.88325	0.539619	0.269810	−	0.962914i	$-0.413039\pi$
$271$	8.88325	0.539619	0.269810	+	0.962914i	$0.413039\pi$
$272$	−2.01568	−0.122218
$273$	−1.00000	−0.0605228
$274$	1.32655	0.0801397
$275$	0	0
$276$	−2.85819	−0.172043
$277$	5.38560	0.323589	0.161795	−	0.986824i	$-0.448272\pi$
$277$	5.38560	0.323589	0.161795	+	0.986824i	$0.448272\pi$
$278$	8.78800	0.527069
$279$	10.2857	0.615786
$280$	0	0
$281$	−12.9727	−0.773889	−0.386944	−	0.922103i	$-0.626469\pi$
$281$	−12.9727	−0.773889	−0.386944	+	0.922103i	$0.626469\pi$
$282$	−2.61069	−0.155464
$283$	1.52589	0.0907047	0.0453523	−	0.998971i	$-0.485559\pi$
$283$	1.52589	0.0907047	0.0453523	+	0.998971i	$0.485559\pi$
$284$	22.0977	1.31126
$285$	0	0
$286$	2.04762	0.121079
$287$	8.04762	0.475036
$288$	5.77078	0.340047
$289$	−15.1999	−0.894111
$290$	0	0
$291$	−7.18572	−0.421234
$292$	−18.6289	−1.09017
$293$	−18.8545	−1.10149	−0.550745	−	0.834673i	$-0.685656\pi$
$293$	−18.8545	−1.10149	−0.550745	+	0.834673i	$0.685656\pi$
$294$	−0.670843	−0.0391244
$295$	0	0
$296$	−21.9204	−1.27410
$297$	3.05232	0.177113
$298$	9.79020	0.567131
$299$	1.84403	0.106643
$300$	0	0
$301$	1.49765	0.0863234
$302$	−3.63052	−0.208913
$303$	18.4416	1.05944
$304$	11.4213	0.655054
$305$	0	0
$306$	−0.900061	−0.0514530
$307$	−15.3856	−0.878102	−0.439051	−	0.898462i	$-0.644685\pi$
$307$	−15.3856	−0.878102	−0.439051	+	0.898462i	$0.644685\pi$
$308$	−4.73100	−0.269574
$309$	1.89537	0.107824
$310$	0	0
$311$	−17.0999	−0.969649	−0.484824	−	0.874612i	$-0.661116\pi$
$311$	−17.0999	−0.969649	−0.484824	+	0.874612i	$0.661116\pi$
$312$	−2.38147	−0.134824
$313$	−4.89263	−0.276548	−0.138274	−	0.990394i	$-0.544155\pi$
$313$	−4.89263	−0.276548	−0.138274	+	0.990394i	$0.544155\pi$
$314$	−15.7007	−0.886042
$315$	0	0
$316$	−7.62417	−0.428893
$317$	−1.44382	−0.0810933	−0.0405466	−	0.999178i	$-0.512910\pi$
$317$	−1.44382	−0.0810933	−0.0405466	+	0.999178i	$0.512910\pi$
$318$	0.336995	0.0188977
$319$	17.0999	0.957413
$320$	0	0
$321$	18.5463	1.03515
$322$	1.23706	0.0689384
$323$	−10.1999	−0.567536
$324$	−1.54997	−0.0861094
$325$	0	0
$326$	6.77862	0.375433
$327$	1.42126	0.0785958
$328$	19.1652	1.05822
$329$	−3.89166	−0.214554
$330$	0	0
$331$	−14.9953	−0.824217	−0.412108	−	0.911135i	$-0.635207\pi$
$331$	−14.9953	−0.824217	−0.412108	+	0.911135i	$0.635207\pi$
$332$	−1.87280	−0.102783
$333$	9.20457	0.504407
$334$	2.07708	0.113653
$335$	0	0
$336$	1.50235	0.0819597
$337$	14.9115	0.812280	0.406140	−	0.913811i	$-0.366874\pi$
$337$	14.9115	0.812280	0.406140	+	0.913811i	$0.366874\pi$
$338$	0.670843	0.0364890
$339$	5.60228	0.304274
$340$	0	0
$341$	−31.3951	−1.70014
$342$	5.09994	0.275773
$343$	−1.00000	−0.0539949
$344$	3.56662	0.192299
$345$	0	0
$346$	−1.84721	−0.0993065
$347$	−7.07488	−0.379800	−0.189900	−	0.981803i	$-0.560816\pi$
$347$	−7.07488	−0.379800	−0.189900	+	0.981803i	$0.560816\pi$
$348$	−8.68337	−0.465478
$349$	21.1188	1.13046	0.565232	−	0.824932i	$-0.308787\pi$
$349$	21.1188	1.13046	0.565232	+	0.824932i	$0.308787\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−17.6142	−0.938843
$353$	17.3643	0.924206	0.462103	−	0.886826i	$-0.347095\pi$
$353$	17.3643	0.924206	0.462103	+	0.886826i	$0.347095\pi$
$354$	2.87749	0.152937
$355$	0	0
$356$	−21.3191	−1.12991
$357$	−1.34169	−0.0710096
$358$	−1.05807	−0.0559208
$359$	18.3521	0.968589	0.484294	−	0.874905i	$-0.339076\pi$
$359$	18.3521	0.968589	0.484294	+	0.874905i	$0.339076\pi$
$360$	0	0
$361$	38.7947	2.04183
$362$	6.38408	0.335540
$363$	1.68337	0.0883541
$364$	−1.54997	−0.0812405
$365$	0	0
$366$	0.458435	0.0239628
$367$	−8.40719	−0.438852	−0.219426	−	0.975629i	$-0.570418\pi$
$367$	−8.40719	−0.438852	−0.219426	+	0.975629i	$0.570418\pi$
$368$	−2.77037	−0.144416
$369$	−8.04762	−0.418943
$370$	0	0
$371$	0.502345	0.0260805
$372$	15.9425	0.826578
$373$	−27.0925	−1.40280	−0.701399	−	0.712769i	$-0.747441\pi$
$373$	−27.0925	−1.40280	−0.701399	+	0.712769i	$0.747441\pi$
$374$	2.74727	0.142058
$375$	0	0
$376$	−9.26787	−0.477954
$377$	5.60228	0.288532
$378$	0.670843	0.0345044
$379$	−6.41657	−0.329597	−0.164798	−	0.986327i	$-0.552697\pi$
$379$	−6.41657	−0.329597	−0.164798	+	0.986327i	$0.552697\pi$
$380$	0	0
$381$	14.1046	0.722602
$382$	−15.3293	−0.784314
$383$	−32.5939	−1.66547	−0.832735	−	0.553672i	$-0.813226\pi$
$383$	−32.5939	−1.66547	−0.832735	+	0.553672i	$0.813226\pi$
$384$	10.9602	0.559311
$385$	0	0
$386$	−13.1324	−0.668423
$387$	−1.49765	−0.0761301
$388$	−11.1376	−0.565428
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	2.47411	0.125121
$392$	−2.38147	−0.120283
$393$	8.78800	0.443296
$394$	14.2401	0.717405
$395$	0	0
$396$	4.73100	0.237741
$397$	−15.7520	−0.790573	−0.395286	−	0.918558i	$-0.629355\pi$
$397$	−15.7520	−0.790573	−0.395286	+	0.918558i	$0.629355\pi$
$398$	−4.48349	−0.224737
$399$	7.60228	0.380590
$400$	0	0
$401$	26.1867	1.30770	0.653851	−	0.756624i	$-0.273152\pi$
$401$	26.1867	1.30770	0.653851	+	0.756624i	$0.273152\pi$
$402$	−5.15748	−0.257232
$403$	−10.2857	−0.512365
$404$	28.5840	1.42211
$405$	0	0
$406$	3.75825	0.186519
$407$	−28.0952	−1.39263
$408$	−3.19519	−0.158185
$409$	−17.3856	−0.859662	−0.429831	−	0.902909i	$-0.641427\pi$
$409$	−17.3856	−0.859662	−0.429831	+	0.902909i	$0.641427\pi$
$410$	0	0
$411$	−1.97743	−0.0975396
$412$	2.93777	0.144733
$413$	4.28937	0.211066
$414$	−1.23706	−0.0607980
$415$	0	0
$416$	−5.77078	−0.282936
$417$	−13.0999	−0.641507
$418$	−15.5666	−0.761388
$419$	−12.4761	−0.609496	−0.304748	−	0.952433i	$-0.598572\pi$
$419$	−12.4761	−0.609496	−0.304748	+	0.952433i	$0.598572\pi$
$420$	0	0
$421$	9.57405	0.466611	0.233305	−	0.972404i	$-0.425046\pi$
$421$	9.57405	0.466611	0.233305	+	0.972404i	$0.425046\pi$
$422$	−3.09056	−0.150446
$423$	3.89166	0.189219
$424$	1.19632	0.0580985
$425$	0	0
$426$	9.56413	0.463384
$427$	0.683372	0.0330707
$428$	28.7461	1.38950
$429$	−3.05232	−0.147367
$430$	0	0
$431$	5.20208	0.250575	0.125288	−	0.992120i	$-0.460015\pi$
$431$	5.20208	0.250575	0.125288	+	0.992120i	$0.460015\pi$
$432$	−1.50235	−0.0722816
$433$	17.2547	0.829207	0.414604	−	0.910002i	$-0.363920\pi$
$433$	17.2547	0.829207	0.414604	+	0.910002i	$0.363920\pi$
$434$	−6.90006	−0.331214
$435$	0	0
$436$	2.20291	0.105500
$437$	−14.0189	−0.670613
$438$	−8.06276	−0.385254
$439$	−22.1925	−1.05919	−0.529594	−	0.848251i	$-0.677656\pi$
$439$	−22.1925	−1.05919	−0.529594	+	0.848251i	$0.677656\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0.900061	0.0428115
$443$	37.9319	1.80220	0.901098	−	0.433615i	$-0.142762\pi$
$443$	37.9319	1.80220	0.901098	+	0.433615i	$0.142762\pi$
$444$	14.2668	0.677073
$445$	0	0
$446$	−5.77393	−0.273403
$447$	−14.5939	−0.690267
$448$	−0.866598	−0.0409429
$449$	−33.3150	−1.57223	−0.786115	−	0.618080i	$-0.787911\pi$
$449$	−33.3150	−1.57223	−0.786115	+	0.618080i	$0.787911\pi$
$450$	0	0
$451$	24.5639	1.15667
$452$	8.68337	0.408431
$453$	5.41188	0.254272
$454$	9.72001	0.456183
$455$	0	0
$456$	18.1046	0.847827
$457$	7.79269	0.364527	0.182263	−	0.983250i	$-0.441658\pi$
$457$	7.79269	0.364527	0.182263	+	0.983250i	$0.441658\pi$
$458$	−13.5509	−0.633194
$459$	1.34169	0.0626245
$460$	0	0
$461$	30.4568	1.41851	0.709256	−	0.704951i	$-0.249031\pi$
$461$	30.4568	1.41851	0.709256	+	0.704951i	$0.249031\pi$
$462$	−2.04762	−0.0952641
$463$	−39.4639	−1.83405	−0.917023	−	0.398835i	$-0.869415\pi$
$463$	−39.4639	−1.83405	−0.917023	+	0.398835i	$0.869415\pi$
$464$	−8.41657	−0.390729
$465$	0	0
$466$	−4.88438	−0.226265
$467$	9.93307	0.459648	0.229824	−	0.973232i	$-0.426185\pi$
$467$	9.93307	0.459648	0.229824	+	0.973232i	$0.426185\pi$
$468$	1.54997	0.0716474
$469$	−7.68806	−0.355002
$470$	0	0
$471$	23.4044	1.07842
$472$	10.2150	0.470184
$473$	4.57131	0.210189
$474$	−3.29982	−0.151566
$475$	0	0
$476$	−2.07957	−0.0953171
$477$	−0.502345	−0.0230008
$478$	18.5614	0.848978
$479$	−0.941479	−0.0430173	−0.0215086	−	0.999769i	$-0.506847\pi$
$479$	−0.941479	−0.0430173	−0.0215086	+	0.999769i	$0.506847\pi$
$480$	0	0
$481$	−9.20457	−0.419692
$482$	−14.9053	−0.678917
$483$	−1.84403	−0.0839063
$484$	2.60918	0.118599
$485$	0	0
$486$	−0.670843	−0.0304301
$487$	39.3499	1.78312	0.891558	−	0.452907i	$-0.149613\pi$
$487$	39.3499	1.78312	0.891558	+	0.452907i	$0.149613\pi$
$488$	1.62743	0.0736703
$489$	−10.1046	−0.456947
$490$	0	0
$491$	5.45374	0.246124	0.123062	−	0.992399i	$-0.460729\pi$
$491$	5.45374	0.246124	0.123062	+	0.992399i	$0.460729\pi$
$492$	−12.4736	−0.562352
$493$	7.51651	0.338526
$494$	−5.09994	−0.229457
$495$	0	0
$496$	15.4526	0.693843
$497$	14.2569	0.639509
$498$	−0.810569	−0.0363225
$499$	−31.3574	−1.40375	−0.701874	−	0.712301i	$-0.747653\pi$
$499$	−31.3574	−1.40375	−0.701874	+	0.712301i	$0.747653\pi$
$500$	0	0
$501$	−3.09623	−0.138329
$502$	11.6118	0.518258
$503$	−43.9926	−1.96153	−0.980766	−	0.195188i	$-0.937468\pi$
$503$	−43.9926	−1.96153	−0.980766	+	0.195188i	$0.937468\pi$
$504$	2.38147	0.106079
$505$	0	0
$506$	3.77588	0.167858
$507$	−1.00000	−0.0444116
$508$	21.8617	0.969958
$509$	−41.8498	−1.85496	−0.927480	−	0.373874i	$-0.878029\pi$
$509$	−41.8498	−1.85496	−0.927480	+	0.373874i	$0.878029\pi$
$510$	0	0
$511$	−12.0189	−0.531683
$512$	15.8253	0.699386
$513$	−7.60228	−0.335649
$514$	3.72880	0.164470
$515$	0	0
$516$	−2.32132	−0.102190
$517$	−11.8786	−0.522418
$518$	−6.17482	−0.271306
$519$	2.75356	0.120868
$520$	0	0
$521$	8.32957	0.364925	0.182462	−	0.983213i	$-0.441593\pi$
$521$	8.32957	0.364925	0.182462	+	0.983213i	$0.441593\pi$
$522$	−3.75825	−0.164494
$523$	−25.3092	−1.10669	−0.553347	−	0.832951i	$-0.686650\pi$
$523$	−25.3092	−1.10669	−0.553347	+	0.832951i	$0.686650\pi$
$524$	13.6211	0.595042
$525$	0	0
$526$	−16.8037	−0.732675
$527$	−13.8001	−0.601143
$528$	4.58563	0.199564
$529$	−19.5995	−0.852154
$530$	0	0
$531$	−4.28937	−0.186143
$532$	11.7833	0.510871
$533$	8.04762	0.348581
$534$	−9.22714	−0.399297
$535$	0	0
$536$	−18.3089	−0.790824
$537$	1.57723	0.0680624
$538$	−3.05481	−0.131702
$539$	−3.05232	−0.131473
$540$	0	0
$541$	27.1501	1.16727	0.583636	−	0.812015i	$-0.301630\pi$
$541$	27.1501	1.16727	0.583636	+	0.812015i	$0.301630\pi$
$542$	5.95927	0.255972
$543$	−9.51651	−0.408392
$544$	−7.74258	−0.331960
$545$	0	0
$546$	−0.670843	−0.0287094
$547$	−27.5949	−1.17987	−0.589935	−	0.807450i	$-0.700847\pi$
$547$	−27.5949	−1.17987	−0.589935	+	0.807450i	$0.700847\pi$
$548$	−3.06496	−0.130929
$549$	−0.683372	−0.0291656
$550$	0	0
$551$	−42.5902	−1.81440
$552$	−4.39151	−0.186915
$553$	−4.91891	−0.209173
$554$	3.61289	0.153497
$555$	0	0
$556$	−20.3045	−0.861103
$557$	−7.19881	−0.305024	−0.152512	−	0.988302i	$-0.548736\pi$
$557$	−7.19881	−0.305024	−0.152512	+	0.988302i	$0.548736\pi$
$558$	6.90006	0.292103
$559$	1.49765	0.0633440
$560$	0	0
$561$	−4.09525	−0.172902
$562$	−8.70267	−0.367100
$563$	20.2594	0.853831	0.426915	−	0.904292i	$-0.359600\pi$
$563$	20.2594	0.853831	0.426915	+	0.904292i	$0.359600\pi$
$564$	6.03195	0.253991
$565$	0	0
$566$	1.02363	0.0430265
$567$	−1.00000	−0.0419961
$568$	33.9524	1.42461
$569$	7.28097	0.305234	0.152617	−	0.988285i	$-0.451230\pi$
$569$	7.28097	0.305234	0.152617	+	0.988285i	$0.451230\pi$
$570$	0	0
$571$	−22.1141	−0.925446	−0.462723	−	0.886503i	$-0.653128\pi$
$571$	−22.1141	−0.925446	−0.462723	+	0.886503i	$0.653128\pi$
$572$	−4.73100	−0.197813
$573$	22.8508	0.954604
$574$	5.39869	0.225337
$575$	0	0
$576$	0.866598	0.0361082
$577$	−20.3139	−0.845678	−0.422839	−	0.906205i	$-0.638966\pi$
$577$	−20.3139	−0.845678	−0.422839	+	0.906205i	$0.638966\pi$
$578$	−10.1967	−0.424128
$579$	19.5760	0.813551
$580$	0	0
$581$	−1.20828	−0.0501281
$582$	−4.82049	−0.199816
$583$	1.53332	0.0635035
$584$	−28.6226	−1.18441
$585$	0	0
$586$	−12.6484	−0.522500
$587$	−35.8842	−1.48110	−0.740550	−	0.672001i	$-0.765435\pi$
$587$	−35.8842	−1.48110	−0.740550	+	0.672001i	$0.765435\pi$
$588$	1.54997	0.0639197
$589$	78.1945	3.22195
$590$	0	0
$591$	−21.2271	−0.873168
$592$	13.8284	0.568346
$593$	−20.2664	−0.832239	−0.416120	−	0.909310i	$-0.636610\pi$
$593$	−20.2664	−0.832239	−0.416120	+	0.909310i	$0.636610\pi$
$594$	2.04762	0.0840150
$595$	0	0
$596$	−22.6201	−0.926554
$597$	6.68337	0.273532
$598$	1.23706	0.0505870
$599$	14.9365	0.610291	0.305145	−	0.952306i	$-0.401295\pi$
$599$	14.9365	0.610291	0.305145	+	0.952306i	$0.401295\pi$
$600$	0	0
$601$	4.89263	0.199575	0.0997873	−	0.995009i	$-0.468184\pi$
$601$	4.89263	0.199575	0.0997873	+	0.995009i	$0.468184\pi$
$602$	1.00469	0.0409481
$603$	7.68806	0.313082
$604$	8.38824	0.341313
$605$	0	0
$606$	12.3714	0.502555
$607$	−20.6164	−0.836796	−0.418398	−	0.908264i	$-0.637408\pi$
$607$	−20.6164	−0.836796	−0.418398	+	0.908264i	$0.637408\pi$
$608$	43.8711	1.77921
$609$	−5.60228	−0.227016
$610$	0	0
$611$	−3.89166	−0.157440
$612$	2.07957	0.0840617
$613$	9.73818	0.393321	0.196661	−	0.980472i	$-0.436990\pi$
$613$	9.73818	0.393321	0.196661	+	0.980472i	$0.436990\pi$
$614$	−10.3213	−0.416535
$615$	0	0
$616$	−7.26900	−0.292877
$617$	0.763482	0.0307366	0.0153683	−	0.999882i	$-0.495108\pi$
$617$	0.763482	0.0307366	0.0153683	+	0.999882i	$0.495108\pi$
$618$	1.27150	0.0511470
$619$	−41.0925	−1.65165	−0.825824	−	0.563928i	$-0.809289\pi$
$619$	−41.0925	−1.65165	−0.825824	+	0.563928i	$0.809289\pi$
$620$	0	0
$621$	1.84403	0.0739984
$622$	−11.4714	−0.459960
$623$	−13.7545	−0.551064
$624$	1.50235	0.0601420
$625$	0	0
$626$	−3.28219	−0.131183
$627$	23.2046	0.926701
$628$	36.2762	1.44758
$629$	−12.3496	−0.492412
$630$	0	0
$631$	−14.4667	−0.575910	−0.287955	−	0.957644i	$-0.592975\pi$
$631$	−14.4667	−0.575910	−0.287955	+	0.957644i	$0.592975\pi$
$632$	−11.7143	−0.465968
$633$	4.60698	0.183111
$634$	−0.968580	−0.0384672
$635$	0	0
$636$	−0.778620	−0.0308743
$637$	−1.00000	−0.0396214
$638$	11.4714	0.454156
$639$	−14.2569	−0.563994
$640$	0	0
$641$	35.7069	1.41034	0.705169	−	0.709039i	$-0.250871\pi$
$641$	35.7069	1.41034	0.705169	+	0.709039i	$0.250871\pi$
$642$	12.4416	0.491032
$643$	26.4091	1.04147	0.520737	−	0.853717i	$-0.325657\pi$
$643$	26.4091	1.04147	0.520737	+	0.853717i	$0.325657\pi$
$644$	−2.85819	−0.112629
$645$	0	0
$646$	−6.84252	−0.269215
$647$	−37.3543	−1.46855	−0.734275	−	0.678852i	$-0.762478\pi$
$647$	−37.3543	−1.46855	−0.734275	+	0.678852i	$0.762478\pi$
$648$	−2.38147	−0.0935531
$649$	13.0925	0.513926
$650$	0	0
$651$	10.2857	0.403127
$652$	−15.6619	−0.613366
$653$	0.987881	0.0386588	0.0193294	−	0.999813i	$-0.493847\pi$
$653$	0.987881	0.0386588	0.0193294	+	0.999813i	$0.493847\pi$
$654$	0.953441	0.0372825
$655$	0	0
$656$	−12.0903	−0.472047
$657$	12.0189	0.468900
$658$	−2.61069	−0.101775
$659$	−1.24653	−0.0485578	−0.0242789	−	0.999705i	$-0.507729\pi$
$659$	−1.24653	−0.0485578	−0.0242789	+	0.999705i	$0.507729\pi$
$660$	0	0
$661$	−33.8994	−1.31853	−0.659266	−	0.751910i	$-0.729133\pi$
$661$	−33.8994	−1.31853	−0.659266	+	0.751910i	$0.729133\pi$
$662$	−10.0595	−0.390973
$663$	−1.34169	−0.0521068
$664$	−2.87749	−0.111668
$665$	0	0
$666$	6.17482	0.239269
$667$	10.3308	0.400010
$668$	−4.79906	−0.185681
$669$	8.60698	0.332765
$670$	0	0
$671$	2.08587	0.0805240
$672$	5.77078	0.222613
$673$	−42.5808	−1.64137	−0.820684	−	0.571382i	$-0.806408\pi$
$673$	−42.5808	−1.64137	−0.820684	+	0.571382i	$0.806408\pi$
$674$	10.0033	0.385311
$675$	0	0
$676$	−1.54997	−0.0596142
$677$	33.8629	1.30146	0.650728	−	0.759311i	$-0.274464\pi$
$677$	33.8629	1.30146	0.650728	+	0.759311i	$0.274464\pi$
$678$	3.75825	0.144335
$679$	−7.18572	−0.275763
$680$	0	0
$681$	−14.4892	−0.555229
$682$	−21.0612	−0.806473
$683$	34.5163	1.32073	0.660364	−	0.750946i	$-0.270402\pi$
$683$	34.5163	1.32073	0.660364	+	0.750946i	$0.270402\pi$
$684$	−11.7833	−0.450546
$685$	0	0
$686$	−0.670843	−0.0256129
$687$	20.1999	0.770673
$688$	−2.24999	−0.0857802
$689$	0.502345	0.0191378
$690$	0	0
$691$	−7.90679	−0.300789	−0.150394	−	0.988626i	$-0.548054\pi$
$691$	−7.90679	−0.300789	−0.150394	+	0.988626i	$0.548054\pi$
$692$	4.26794	0.162243
$693$	3.05232	0.115948
$694$	−4.74613	−0.180161
$695$	0	0
$696$	−13.3417	−0.505715
$697$	10.7974	0.408980
$698$	14.1674	0.536244
$699$	7.28097	0.275391
$700$	0	0
$701$	−5.19510	−0.196216	−0.0981081	−	0.995176i	$-0.531279\pi$
$701$	−5.19510	−0.196216	−0.0981081	+	0.995176i	$0.531279\pi$
$702$	0.670843	0.0253193
$703$	69.9758	2.63919
$704$	−2.64513	−0.0996921
$705$	0	0
$706$	11.6487	0.438404
$707$	18.4416	0.693569
$708$	−6.64839	−0.249862
$709$	33.1971	1.24674	0.623372	−	0.781925i	$-0.285762\pi$
$709$	33.1971	1.24674	0.623372	+	0.781925i	$0.285762\pi$
$710$	0	0
$711$	4.91891	0.184474
$712$	−32.7561	−1.22759
$713$	−18.9671	−0.710323
$714$	−0.900061	−0.0336839
$715$	0	0
$716$	2.44465	0.0913610
$717$	−27.6688	−1.03331
$718$	12.3114	0.459457
$719$	30.7261	1.14589	0.572944	−	0.819594i	$-0.305801\pi$
$719$	30.7261	1.14589	0.572944	+	0.819594i	$0.305801\pi$
$720$	0	0
$721$	1.89537	0.0705873
$722$	26.0252	0.968557
$723$	22.2187	0.826324
$724$	−14.7503	−0.548190
$725$	0	0
$726$	1.12928	0.0419114
$727$	−1.83783	−0.0681612	−0.0340806	−	0.999419i	$-0.510850\pi$
$727$	−1.83783	−0.0681612	−0.0340806	+	0.999419i	$0.510850\pi$
$728$	−2.38147	−0.0882632
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.00938	0.0743197
$732$	−1.05921	−0.0391494
$733$	19.1857	0.708641	0.354320	−	0.935124i	$-0.384712\pi$
$733$	19.1857	0.708641	0.354320	+	0.935124i	$0.384712\pi$
$734$	−5.63990	−0.208173
$735$	0	0
$736$	−10.6415	−0.392251
$737$	−23.4664	−0.864396
$738$	−5.39869	−0.198729
$739$	−21.4882	−0.790456	−0.395228	−	0.918583i	$-0.629334\pi$
$739$	−21.4882	−0.790456	−0.395228	+	0.918583i	$0.629334\pi$
$740$	0	0
$741$	7.60228	0.279277
$742$	0.336995	0.0123715
$743$	−19.9430	−0.731637	−0.365819	−	0.930686i	$-0.619211\pi$
$743$	−19.9430	−0.731637	−0.365819	+	0.930686i	$0.619211\pi$
$744$	24.4950	0.898030
$745$	0	0
$746$	−18.1748	−0.665427
$747$	1.20828	0.0442088
$748$	−6.34751	−0.232088
$749$	18.5463	0.677665
$750$	0	0
$751$	38.4498	1.40305	0.701526	−	0.712644i	$-0.252502\pi$
$751$	38.4498	1.40305	0.701526	+	0.712644i	$0.252502\pi$
$752$	5.84661	0.213204
$753$	−17.3092	−0.630782
$754$	3.75825	0.136868
$755$	0	0
$756$	−1.54997	−0.0563719
$757$	16.2931	0.592182	0.296091	−	0.955160i	$-0.404317\pi$
$757$	16.2931	0.592182	0.296091	+	0.955160i	$0.404317\pi$
$758$	−4.30451	−0.156347
$759$	−5.62857	−0.204304
$760$	0	0
$761$	35.3380	1.28100	0.640500	−	0.767958i	$-0.278727\pi$
$761$	35.3380	1.28100	0.640500	+	0.767958i	$0.278727\pi$
$762$	9.46199	0.342772
$763$	1.42126	0.0514530
$764$	35.4180	1.28138
$765$	0	0
$766$	−21.8654	−0.790028
$767$	4.28937	0.154880
$768$	9.08578	0.327855
$769$	−11.5428	−0.416244	−0.208122	−	0.978103i	$-0.566735\pi$
$769$	−11.5428	−0.416244	−0.208122	+	0.978103i	$0.566735\pi$
$770$	0	0
$771$	−5.55837	−0.200180
$772$	30.3422	1.09204
$773$	5.89786	0.212131	0.106066	−	0.994359i	$-0.466175\pi$
$773$	5.89786	0.212131	0.106066	+	0.994359i	$0.466175\pi$
$774$	−1.00469	−0.0361129
$775$	0	0
$776$	−17.1126	−0.614306
$777$	9.20457	0.330212
$778$	4.02506	0.144305
$779$	−61.1803	−2.19201
$780$	0	0
$781$	43.5165	1.55714
$782$	1.65974	0.0593522
$783$	5.60228	0.200209
$784$	1.50235	0.0536552
$785$	0	0
$786$	5.89537	0.210281
$787$	−6.33079	−0.225668	−0.112834	−	0.993614i	$-0.535993\pi$
$787$	−6.33079	−0.225668	−0.112834	+	0.993614i	$0.535993\pi$
$788$	−32.9014	−1.17206
$789$	25.0486	0.891754
$790$	0	0
$791$	5.60228	0.199194
$792$	7.26900	0.258293
$793$	0.683372	0.0242672
$794$	−10.5672	−0.375014
$795$	0	0
$796$	10.3590	0.367166
$797$	22.0721	0.781835	0.390918	−	0.920426i	$-0.372158\pi$
$797$	22.0721	0.781835	0.390918	+	0.920426i	$0.372158\pi$
$798$	5.09994	0.180536
$799$	−5.22138	−0.184719
$800$	0	0
$801$	13.7545	0.485993
$802$	17.5672	0.620318
$803$	−36.6853	−1.29460
$804$	11.9163	0.420254
$805$	0	0
$806$	−6.90006	−0.243044
$807$	4.55368	0.160297
$808$	43.9182	1.54504
$809$	36.2019	1.27279	0.636396	−	0.771363i	$-0.280424\pi$
$809$	36.2019	1.27279	0.636396	+	0.771363i	$0.280424\pi$
$810$	0	0
$811$	−52.4950	−1.84335	−0.921674	−	0.387964i	$-0.873178\pi$
$811$	−52.4950	−1.84335	−0.921674	+	0.387964i	$0.873178\pi$
$812$	−8.68337	−0.304727
$813$	−8.88325	−0.311549
$814$	−18.8475	−0.660605
$815$	0	0
$816$	2.01568	0.0705628
$817$	−11.3856	−0.398332
$818$	−11.6630	−0.407787
$819$	1.00000	0.0349428
$820$	0	0
$821$	−0.972743	−0.0339490	−0.0169745	−	0.999856i	$-0.505403\pi$
$821$	−0.972743	−0.0339490	−0.0169745	+	0.999856i	$0.505403\pi$
$822$	−1.32655	−0.0462687
$823$	29.5572	1.03030	0.515150	−	0.857100i	$-0.327736\pi$
$823$	29.5572	1.03030	0.515150	+	0.857100i	$0.327736\pi$
$824$	4.51377	0.157245
$825$	0	0
$826$	2.87749	0.100121
$827$	−15.3191	−0.532698	−0.266349	−	0.963877i	$-0.585817\pi$
$827$	−15.3191	−0.532698	−0.266349	+	0.963877i	$0.585817\pi$
$828$	2.85819	0.0993291
$829$	27.0236	0.938570	0.469285	−	0.883047i	$-0.344512\pi$
$829$	27.0236	0.938570	0.469285	+	0.883047i	$0.344512\pi$
$830$	0	0
$831$	−5.38560	−0.186824
$832$	−0.866598	−0.0300439
$833$	−1.34169	−0.0464867
$834$	−8.78800	−0.304304
$835$	0	0
$836$	35.9664	1.24392
$837$	−10.2857	−0.355524
$838$	−8.36948	−0.289119
$839$	46.2200	1.59569	0.797846	−	0.602862i	$-0.205973\pi$
$839$	46.2200	1.59569	0.797846	+	0.602862i	$0.205973\pi$
$840$	0	0
$841$	2.38560	0.0822619
$842$	6.42268	0.221340
$843$	12.9727	0.446805
$844$	7.14067	0.245792
$845$	0	0
$846$	2.61069	0.0897574
$847$	1.68337	0.0578413
$848$	−0.754696	−0.0259164
$849$	−1.52589	−0.0523684
$850$	0	0
$851$	−16.9735	−0.581845
$852$	−22.0977	−0.757056
$853$	15.1094	0.517336	0.258668	−	0.965966i	$-0.416716\pi$
$853$	15.1094	0.517336	0.258668	+	0.965966i	$0.416716\pi$
$854$	0.458435	0.0156873
$855$	0	0
$856$	44.1674	1.50961
$857$	0.191631	0.00654599	0.00327299	−	0.999995i	$-0.498958\pi$
$857$	0.191631	0.00654599	0.00327299	+	0.999995i	$0.498958\pi$
$858$	−2.04762	−0.0699047
$859$	7.06419	0.241027	0.120514	−	0.992712i	$-0.461546\pi$
$859$	7.06419	0.241027	0.120514	+	0.992712i	$0.461546\pi$
$860$	0	0
$861$	−8.04762	−0.274262
$862$	3.48978	0.118862
$863$	−41.6142	−1.41657	−0.708283	−	0.705929i	$-0.750530\pi$
$863$	−41.6142	−1.41657	−0.708283	+	0.705929i	$0.750530\pi$
$864$	−5.77078	−0.196326
$865$	0	0
$866$	11.5752	0.393341
$867$	15.1999	0.516215
$868$	15.9425	0.541122
$869$	−15.0141	−0.509318
$870$	0	0
$871$	−7.68806	−0.260500
$872$	3.38469	0.114620
$873$	7.18572	0.243200
$874$	−9.40445	−0.318110
$875$	0	0
$876$	18.6289	0.629411
$877$	52.5471	1.77439	0.887194	−	0.461396i	$-0.152651\pi$
$877$	52.5471	1.77439	0.887194	+	0.461396i	$0.152651\pi$
$878$	−14.8876	−0.502434
$879$	18.8545	0.635946
$880$	0	0
$881$	0.934500	0.0314841	0.0157421	−	0.999876i	$-0.494989\pi$
$881$	0.934500	0.0314841	0.0157421	+	0.999876i	$0.494989\pi$
$882$	0.670843	0.0225885
$883$	42.6448	1.43511	0.717555	−	0.696501i	$-0.245261\pi$
$883$	42.6448	1.43511	0.717555	+	0.696501i	$0.245261\pi$
$884$	−2.07957	−0.0699436
$885$	0	0
$886$	25.4463	0.854886
$887$	−24.8833	−0.835498	−0.417749	−	0.908563i	$-0.637181\pi$
$887$	−24.8833	−0.835498	−0.417749	+	0.908563i	$0.637181\pi$
$888$	21.9204	0.735601
$889$	14.1046	0.473054
$890$	0	0
$891$	−3.05232	−0.102256
$892$	13.3406	0.446675
$893$	29.5855	0.990040
$894$	−9.79020	−0.327433
$895$	0	0
$896$	10.9602	0.366155
$897$	−1.84403	−0.0615704
$898$	−22.3491	−0.745799
$899$	−57.6232	−1.92184
$900$	0	0
$901$	0.673990	0.0224539
$902$	16.4785	0.548674
$903$	−1.49765	−0.0498388
$904$	13.3417	0.443738
$905$	0	0
$906$	3.63052	0.120616
$907$	−42.2207	−1.40191	−0.700957	−	0.713203i	$-0.747244\pi$
$907$	−42.2207	−1.40191	−0.700957	+	0.713203i	$0.747244\pi$
$908$	−22.4579	−0.745292
$909$	−18.4416	−0.611670
$910$	0	0
$911$	14.1108	0.467513	0.233756	−	0.972295i	$-0.424898\pi$
$911$	14.1108	0.467513	0.233756	+	0.972295i	$0.424898\pi$
$912$	−11.4213	−0.378196
$913$	−3.68806	−0.122057
$914$	5.22767	0.172916
$915$	0	0
$916$	31.3092	1.03449
$917$	8.78800	0.290205
$918$	0.900061	0.0297064
$919$	−18.1547	−0.598870	−0.299435	−	0.954117i	$-0.596798\pi$
$919$	−18.1547	−0.598870	−0.299435	+	0.954117i	$0.596798\pi$
$920$	0	0
$921$	15.3856	0.506973
$922$	20.4317	0.672882
$923$	14.2569	0.469271
$924$	4.73100	0.155638
$925$	0	0
$926$	−26.4741	−0.869993
$927$	−1.89537	−0.0622521
$928$	−32.3296	−1.06127
$929$	−8.24741	−0.270589	−0.135294	−	0.990805i	$-0.543198\pi$
$929$	−8.24741	−0.270589	−0.135294	+	0.990805i	$0.543198\pi$
$930$	0	0
$931$	7.60228	0.249155
$932$	11.2853	0.369662
$933$	17.0999	0.559827
$934$	6.66353	0.218037
$935$	0	0
$936$	2.38147	0.0778409
$937$	21.6693	0.707905	0.353953	−	0.935263i	$-0.384837\pi$
$937$	21.6693	0.707905	0.353953	+	0.935263i	$0.384837\pi$
$938$	−5.15748	−0.168398
$939$	4.89263	0.159665
$940$	0	0
$941$	42.5238	1.38624	0.693118	−	0.720824i	$-0.256237\pi$
$941$	42.5238	1.38624	0.693118	+	0.720824i	$0.256237\pi$
$942$	15.7007	0.511557
$943$	14.8401	0.483259
$944$	−6.44412	−0.209738
$945$	0	0
$946$	3.06663	0.0997049
$947$	6.79792	0.220903	0.110451	−	0.993882i	$-0.464770\pi$
$947$	6.79792	0.220903	0.110451	+	0.993882i	$0.464770\pi$
$948$	7.62417	0.247621
$949$	−12.0189	−0.390148
$950$	0	0
$951$	1.44382	0.0468192
$952$	−3.19519	−0.103557
$953$	−57.3782	−1.85866	−0.929331	−	0.369249i	$-0.879615\pi$
$953$	−57.3782	−1.85866	−0.929331	+	0.369249i	$0.879615\pi$
$954$	−0.336995	−0.0109106
$955$	0	0
$956$	−42.8857	−1.38702
$957$	−17.0999	−0.552763
$958$	−0.631585	−0.0204056
$959$	−1.97743	−0.0638547
$960$	0	0
$961$	74.7947	2.41273
$962$	−6.17482	−0.199084
$963$	−18.5463	−0.597645
$964$	34.4384	1.10918
$965$	0	0
$966$	−1.23706	−0.0398016
$967$	−7.52393	−0.241953	−0.120977	−	0.992655i	$-0.538603\pi$
$967$	−7.52393	−0.241953	−0.120977	+	0.992655i	$0.538603\pi$
$968$	4.00890	0.128851
$969$	10.1999	0.327667
$970$	0	0
$971$	−18.5807	−0.596283	−0.298141	−	0.954522i	$-0.596367\pi$
$971$	−18.5807	−0.596283	−0.298141	+	0.954522i	$0.596367\pi$
$972$	1.54997	0.0497153
$973$	−13.0999	−0.419965
$974$	26.3976	0.845835
$975$	0	0
$976$	−1.02666	−0.0328626
$977$	38.5939	1.23473	0.617364	−	0.786678i	$-0.288201\pi$
$977$	38.5939	1.23473	0.617364	+	0.786678i	$0.288201\pi$
$978$	−6.77862	−0.216756
$979$	−41.9832	−1.34179
$980$	0	0
$981$	−1.42126	−0.0453773
$982$	3.65861	0.116751
$983$	0.591837	0.0188767	0.00943834	−	0.999955i	$-0.496996\pi$
$983$	0.591837	0.0188767	0.00943834	+	0.999955i	$0.496996\pi$
$984$	−19.1652	−0.610964
$985$	0	0
$986$	5.04240	0.160583
$987$	3.89166	0.123873
$988$	11.7833	0.374877
$989$	2.76172	0.0878177
$990$	0	0
$991$	9.41686	0.299136	0.149568	−	0.988751i	$-0.452212\pi$
$991$	9.41686	0.299136	0.149568	+	0.988751i	$0.452212\pi$
$992$	59.3563	1.88456
$993$	14.9953	0.475862
$994$	9.56413	0.303356
$995$	0	0
$996$	1.87280	0.0593420
$997$	−22.1716	−0.702180	−0.351090	−	0.936342i	$-0.614189\pi$
$997$	−22.1716	−0.702180	−0.351090	+	0.936342i	$0.614189\pi$
$998$	−21.0359	−0.665879
$999$	−9.20457	−0.291220

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bg.1.3		4
5.4	even	2		273.2.a.e.1.2	✓	4
15.14	odd	2		819.2.a.k.1.3		4
20.19	odd	2		4368.2.a.br.1.4		4
35.34	odd	2		1911.2.a.s.1.2		4
65.64	even	2		3549.2.a.w.1.3		4
105.104	even	2		5733.2.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.2	✓	4	5.4	even	2
819.2.a.k.1.3		4	15.14	odd	2
1911.2.a.s.1.2		4	35.34	odd	2
3549.2.a.w.1.3		4	65.64	even	2
4368.2.a.br.1.4		4	20.19	odd	2
5733.2.a.bf.1.3		4	105.104	even	2
6825.2.a.bg.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6825.a (trivial)

\(f(q)\)	\(=\)	\(q+0.670843 q^{2} -1.00000 q^{3} -1.54997 q^{4} -0.670843 q^{6} -1.00000 q^{7} -2.38147 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.670843 q^{2} -1.00000 q^{3} -1.54997 q^{4} -0.670843 q^{6} -1.00000 q^{7} -2.38147 q^{8} +1.00000 q^{9} -3.05232 q^{11} +1.54997 q^{12} -1.00000 q^{13} -0.670843 q^{14} +1.50235 q^{16} -1.34169 q^{17} +0.670843 q^{18} +7.60228 q^{19} +1.00000 q^{21} -2.04762 q^{22} -1.84403 q^{23} +2.38147 q^{24} -0.670843 q^{26} -1.00000 q^{27} +1.54997 q^{28} -5.60228 q^{29} +10.2857 q^{31} +5.77078 q^{32} +3.05232 q^{33} -0.900061 q^{34} -1.54997 q^{36} +9.20457 q^{37} +5.09994 q^{38} +1.00000 q^{39} -8.04762 q^{41} +0.670843 q^{42} -1.49765 q^{43} +4.73100 q^{44} -1.23706 q^{46} +3.89166 q^{47} -1.50235 q^{48} +1.00000 q^{49} +1.34169 q^{51} +1.54997 q^{52} -0.502345 q^{53} -0.670843 q^{54} +2.38147 q^{56} -7.60228 q^{57} -3.75825 q^{58} -4.28937 q^{59} -0.683372 q^{61} +6.90006 q^{62} -1.00000 q^{63} +0.866598 q^{64} +2.04762 q^{66} +7.68806 q^{67} +2.07957 q^{68} +1.84403 q^{69} -14.2569 q^{71} -2.38147 q^{72} +12.0189 q^{73} +6.17482 q^{74} -11.7833 q^{76} +3.05232 q^{77} +0.670843 q^{78} +4.91891 q^{79} +1.00000 q^{81} -5.39869 q^{82} +1.20828 q^{83} -1.54997 q^{84} -1.00469 q^{86} +5.60228 q^{87} +7.26900 q^{88} +13.7545 q^{89} +1.00000 q^{91} +2.85819 q^{92} -10.2857 q^{93} +2.61069 q^{94} -5.77078 q^{96} +7.18572 q^{97} +0.670843 q^{98} -3.05232 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9	\( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24} + q^{26} - 4 q^{27} - 7 q^{28} + q^{29} + 3 q^{31} - 7 q^{32} + 2 q^{33} - 30 q^{34} + 7 q^{36} - 10 q^{37} - 6 q^{38} + 4 q^{39} - 16 q^{41} - q^{42} - 3 q^{43} - 12 q^{44} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 4 q^{49} - 2 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 3 q^{56} - 7 q^{57} + 4 q^{58} - 20 q^{59} + 12 q^{61} + 54 q^{62} - 4 q^{63} + 5 q^{64} - 8 q^{66} + 22 q^{67} + 10 q^{68} + 3 q^{69} - 3 q^{72} + 13 q^{73} - 6 q^{74} - 6 q^{76} + 2 q^{77} - q^{78} + 11 q^{79} + 4 q^{81} - 10 q^{82} - q^{83} + 7 q^{84} - 10 q^{86} - q^{87} + 60 q^{88} - 5 q^{89} + 4 q^{91} - 34 q^{92} - 3 q^{93} + 34 q^{94} + 7 q^{96} + 17 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 - 7 * q^12 - 4 * q^13 + q^14 + 9 * q^16 + 2 * q^17 - q^18 + 7 * q^19 + 4 * q^21 + 8 * q^22 - 3 * q^23 + 3 * q^24 + q^26 - 4 * q^27 - 7 * q^28 + q^29 + 3 * q^31 - 7 * q^32 + 2 * q^33 - 30 * q^34 + 7 * q^36 - 10 * q^37 - 6 * q^38 + 4 * q^39 - 16 * q^41 - q^42 - 3 * q^43 - 12 * q^44 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 4 * q^49 - 2 * q^51 - 7 * q^52 - 5 * q^53 + q^54 + 3 * q^56 - 7 * q^57 + 4 * q^58 - 20 * q^59 + 12 * q^61 + 54 * q^62 - 4 * q^63 + 5 * q^64 - 8 * q^66 + 22 * q^67 + 10 * q^68 + 3 * q^69 - 3 * q^72 + 13 * q^73 - 6 * q^74 - 6 * q^76 + 2 * q^77 - q^78 + 11 * q^79 + 4 * q^81 - 10 * q^82 - q^83 + 7 * q^84 - 10 * q^86 - q^87 + 60 * q^88 - 5 * q^89 + 4 * q^91 - 34 * q^92 - 3 * q^93 + 34 * q^94 + 7 * q^96 + 17 * q^97 - q^98 - 2 * q^99

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