LMFDB - Embedded newform 3475.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3475,2,Mod(1,3475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3475 = 5^{2} \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3475.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:	$27.7480147024$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 11x^{5} + 8x^{4} + 35x^{3} - 10x^{2} - 32x - 8 $ x^7 - x^6 - 11x^5 + 8x^4 + 35x^3 - 10x^2 - 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 139)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.308806$ of defining polynomial
Character	$\chi$	$=$	3475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.308806 q^{2} +1.39811 q^{3} -1.90464 q^{4} +0.431745 q^{6} -4.16776 q^{7} -1.20578 q^{8} -1.04529 q^{9} +O(q^{10})$	\(q+0.308806 q^{2} +1.39811 q^{3} -1.90464 q^{4} +0.431745 q^{6} -4.16776 q^{7} -1.20578 q^{8} -1.04529 q^{9} +1.93164 q^{11} -2.66290 q^{12} -4.12937 q^{13} -1.28703 q^{14} +3.43693 q^{16} -1.03363 q^{17} -0.322790 q^{18} -7.34157 q^{19} -5.82699 q^{21} +0.596503 q^{22} +4.09697 q^{23} -1.68581 q^{24} -1.27517 q^{26} -5.65576 q^{27} +7.93807 q^{28} +4.49067 q^{29} -1.28386 q^{31} +3.47289 q^{32} +2.70065 q^{33} -0.319192 q^{34} +1.99089 q^{36} +7.77849 q^{37} -2.26712 q^{38} -5.77331 q^{39} +11.5603 q^{41} -1.79941 q^{42} +7.87867 q^{43} -3.67908 q^{44} +1.26517 q^{46} +6.84669 q^{47} +4.80521 q^{48} +10.3702 q^{49} -1.44513 q^{51} +7.86495 q^{52} -3.68473 q^{53} -1.74653 q^{54} +5.02538 q^{56} -10.2643 q^{57} +1.38675 q^{58} +1.24534 q^{59} +0.281096 q^{61} -0.396462 q^{62} +4.35649 q^{63} -5.80140 q^{64} +0.833977 q^{66} +8.51902 q^{67} +1.96870 q^{68} +5.72803 q^{69} -6.40133 q^{71} +1.26038 q^{72} +5.92608 q^{73} +2.40204 q^{74} +13.9830 q^{76} -8.05062 q^{77} -1.78283 q^{78} -8.65375 q^{79} -4.77152 q^{81} +3.56988 q^{82} -15.8366 q^{83} +11.0983 q^{84} +2.43298 q^{86} +6.27846 q^{87} -2.32913 q^{88} -3.23844 q^{89} +17.2102 q^{91} -7.80326 q^{92} -1.79497 q^{93} +2.11430 q^{94} +4.85549 q^{96} +7.27325 q^{97} +3.20237 q^{98} -2.01912 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 2 q^{3} + 9 q^{4} - 7 q^{6} + 5 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) $ 7 * q - q^2 + 2 * q^3 + 9 * q^4 - 7 * q^6 + 5 * q^7 - 6 * q^8 + 13 * q^9	$ 7 q - q^{2} + 2 q^{3} + 9 q^{4} - 7 q^{6} + 5 q^{7} - 6 q^{8} + 13 q^{9} + 2 q^{11} + 8 q^{12} - 6 q^{13} + 7 q^{14} + 5 q^{16} - 5 q^{17} + 10 q^{18} - 10 q^{19} - 5 q^{21} + 18 q^{22} + q^{23} - 21 q^{24} - 8 q^{26} + 11 q^{27} + 28 q^{28} + 30 q^{29} - 20 q^{31} + 12 q^{32} + 20 q^{33} - 17 q^{34} + 2 q^{36} - 6 q^{37} - 6 q^{38} + 11 q^{39} + 19 q^{41} - 6 q^{42} + 12 q^{43} + 25 q^{44} + 22 q^{46} + 3 q^{47} - 15 q^{48} - 8 q^{49} + 23 q^{51} + 8 q^{52} - 38 q^{53} - 7 q^{54} + 21 q^{56} + 19 q^{57} + 21 q^{58} - 14 q^{59} + 4 q^{61} + q^{62} + 18 q^{63} - 16 q^{64} + 18 q^{66} - 9 q^{67} + 25 q^{68} + 9 q^{69} + 24 q^{71} - 41 q^{72} + 5 q^{73} + 9 q^{74} + 3 q^{76} + 13 q^{77} - 20 q^{78} + 8 q^{79} + 39 q^{81} - 56 q^{82} + 9 q^{83} - q^{84} + 39 q^{86} + 25 q^{87} + 29 q^{88} + 10 q^{89} + 7 q^{91} - 29 q^{92} + 15 q^{93} - 36 q^{94} - 11 q^{96} + 5 q^{97} + 49 q^{98} - 31 q^{99}+O(q^{100}) $ 7 * q - q^2 + 2 * q^3 + 9 * q^4 - 7 * q^6 + 5 * q^7 - 6 * q^8 + 13 * q^9 + 2 * q^11 + 8 * q^12 - 6 * q^13 + 7 * q^14 + 5 * q^16 - 5 * q^17 + 10 * q^18 - 10 * q^19 - 5 * q^21 + 18 * q^22 + q^23 - 21 * q^24 - 8 * q^26 + 11 * q^27 + 28 * q^28 + 30 * q^29 - 20 * q^31 + 12 * q^32 + 20 * q^33 - 17 * q^34 + 2 * q^36 - 6 * q^37 - 6 * q^38 + 11 * q^39 + 19 * q^41 - 6 * q^42 + 12 * q^43 + 25 * q^44 + 22 * q^46 + 3 * q^47 - 15 * q^48 - 8 * q^49 + 23 * q^51 + 8 * q^52 - 38 * q^53 - 7 * q^54 + 21 * q^56 + 19 * q^57 + 21 * q^58 - 14 * q^59 + 4 * q^61 + q^62 + 18 * q^63 - 16 * q^64 + 18 * q^66 - 9 * q^67 + 25 * q^68 + 9 * q^69 + 24 * q^71 - 41 * q^72 + 5 * q^73 + 9 * q^74 + 3 * q^76 + 13 * q^77 - 20 * q^78 + 8 * q^79 + 39 * q^81 - 56 * q^82 + 9 * q^83 - q^84 + 39 * q^86 + 25 * q^87 + 29 * q^88 + 10 * q^89 + 7 * q^91 - 29 * q^92 + 15 * q^93 - 36 * q^94 - 11 * q^96 + 5 * q^97 + 49 * q^98 - 31 * 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 $

$2$	0.308806	0.218359	0.109179	−	0.994022i	$-0.465178\pi$
$2$	0.308806	0.218359	0.109179	+	0.994022i	$0.465178\pi$
$3$	1.39811	0.807200	0.403600	−	0.914936i	$-0.367759\pi$
$3$	1.39811	0.807200	0.403600	+	0.914936i	$0.367759\pi$
$4$	−1.90464	−0.952319
$5$	0	0
$5$	0	0
$6$	0.431745	0.176259
$7$	−4.16776	−1.57526	−0.787632	−	0.616146i	$-0.788693\pi$
$7$	−4.16776	−1.57526	−0.787632	+	0.616146i	$0.788693\pi$
$8$	−1.20578	−0.426306
$9$	−1.04529	−0.348428
$10$	0	0
$11$	1.93164	0.582412	0.291206	−	0.956660i	$-0.405943\pi$
$11$	1.93164	0.582412	0.291206	+	0.956660i	$0.405943\pi$
$12$	−2.66290	−0.768712
$13$	−4.12937	−1.14528	−0.572640	−	0.819807i	$-0.694081\pi$
$13$	−4.12937	−1.14528	−0.572640	+	0.819807i	$0.694081\pi$
$14$	−1.28703	−0.343972
$15$	0	0
$16$	3.43693	0.859232
$17$	−1.03363	−0.250693	−0.125347	−	0.992113i	$-0.540004\pi$
$17$	−1.03363	−0.250693	−0.125347	+	0.992113i	$0.540004\pi$
$18$	−0.322790	−0.0760824
$19$	−7.34157	−1.68427	−0.842135	−	0.539266i	$-0.818702\pi$
$19$	−7.34157	−1.68427	−0.842135	+	0.539266i	$0.818702\pi$
$20$	0	0
$21$	−5.82699	−1.27155
$22$	0.596503	0.127175
$23$	4.09697	0.854278	0.427139	−	0.904186i	$-0.359521\pi$
$23$	4.09697	0.854278	0.427139	+	0.904186i	$0.359521\pi$
$24$	−1.68581	−0.344114
$25$	0	0
$26$	−1.27517	−0.250082
$27$	−5.65576	−1.08845
$28$	7.93807	1.50015
$29$	4.49067	0.833897	0.416948	−	0.908930i	$-0.363100\pi$
$29$	4.49067	0.833897	0.416948	+	0.908930i	$0.363100\pi$
$30$	0	0
$31$	−1.28386	−0.230587	−0.115294	−	0.993331i	$-0.536781\pi$
$31$	−1.28386	−0.230587	−0.115294	+	0.993331i	$0.536781\pi$
$32$	3.47289	0.613927
$33$	2.70065	0.470123
$34$	−0.319192	−0.0547410
$35$	0	0
$36$	1.99089	0.331815
$37$	7.77849	1.27878	0.639388	−	0.768884i	$-0.279188\pi$
$37$	7.77849	1.27878	0.639388	+	0.768884i	$0.279188\pi$
$38$	−2.26712	−0.367775
$39$	−5.77331	−0.924470
$40$	0	0
$41$	11.5603	1.80541	0.902706	−	0.430258i	$-0.141577\pi$
$41$	11.5603	1.80541	0.902706	+	0.430258i	$0.141577\pi$
$42$	−1.79941	−0.277655
$43$	7.87867	1.20149	0.600743	−	0.799442i	$-0.294871\pi$
$43$	7.87867	1.20149	0.600743	+	0.799442i	$0.294871\pi$
$44$	−3.67908	−0.554643
$45$	0	0
$46$	1.26517	0.186539
$47$	6.84669	0.998692	0.499346	−	0.866403i	$-0.333574\pi$
$47$	6.84669	0.998692	0.499346	+	0.866403i	$0.333574\pi$
$48$	4.80521	0.693572
$49$	10.3702	1.48145
$50$	0	0
$51$	−1.44513	−0.202359
$52$	7.86495	1.09067
$53$	−3.68473	−0.506136	−0.253068	−	0.967449i	$-0.581440\pi$
$53$	−3.68473	−0.506136	−0.253068	+	0.967449i	$0.581440\pi$
$54$	−1.74653	−0.237673
$55$	0	0
$56$	5.02538	0.671544
$57$	−10.2643	−1.35954
$58$	1.38675	0.182089
$59$	1.24534	0.162129	0.0810647	−	0.996709i	$-0.474168\pi$
$59$	1.24534	0.162129	0.0810647	+	0.996709i	$0.474168\pi$
$60$	0	0
$61$	0.281096	0.0359907	0.0179953	−	0.999838i	$-0.494272\pi$
$61$	0.281096	0.0359907	0.0179953	+	0.999838i	$0.494272\pi$
$62$	−0.396462	−0.0503507
$63$	4.35649	0.548867
$64$	−5.80140	−0.725176
$65$	0	0
$66$	0.833977	0.102655
$67$	8.51902	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$67$	8.51902	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$68$	1.96870	0.238740
$69$	5.72803	0.689573
$70$	0	0
$71$	−6.40133	−0.759698	−0.379849	−	0.925048i	$-0.624024\pi$
$71$	−6.40133	−0.759698	−0.379849	+	0.925048i	$0.624024\pi$
$72$	1.26038	0.148537
$73$	5.92608	0.693595	0.346798	−	0.937940i	$-0.387269\pi$
$73$	5.92608	0.693595	0.346798	+	0.937940i	$0.387269\pi$
$74$	2.40204	0.279232
$75$	0	0
$76$	13.9830	1.60396
$77$	−8.05062	−0.917453
$78$	−1.78283	−0.201866
$79$	−8.65375	−0.973623	−0.486811	−	0.873507i	$-0.661840\pi$
$79$	−8.65375	−0.973623	−0.486811	+	0.873507i	$0.661840\pi$
$80$	0	0
$81$	−4.77152	−0.530169
$82$	3.56988	0.394228
$83$	−15.8366	−1.73829	−0.869146	−	0.494555i	$-0.835331\pi$
$83$	−15.8366	−1.73829	−0.869146	+	0.494555i	$0.835331\pi$
$84$	11.0983	1.21092
$85$	0	0
$86$	2.43298	0.262355
$87$	6.27846	0.673121
$88$	−2.32913	−0.248286
$89$	−3.23844	−0.343274	−0.171637	−	0.985160i	$-0.554906\pi$
$89$	−3.23844	−0.343274	−0.171637	+	0.985160i	$0.554906\pi$
$90$	0	0
$91$	17.2102	1.80412
$92$	−7.80326	−0.813546
$93$	−1.79497	−0.186130
$94$	2.11430	0.218073
$95$	0	0
$96$	4.85549	0.495562
$97$	7.27325	0.738487	0.369243	−	0.929333i	$-0.379617\pi$
$97$	7.27325	0.738487	0.369243	+	0.929333i	$0.379617\pi$
$98$	3.20237	0.323489
$99$	−2.01912	−0.202929
$100$	0	0
$101$	−13.0456	−1.29809	−0.649044	−	0.760751i	$-0.724831\pi$
$101$	−13.0456	−1.29809	−0.649044	+	0.760751i	$0.724831\pi$
$102$	−0.446266	−0.0441869
$103$	11.9048	1.17302	0.586510	−	0.809942i	$-0.300502\pi$
$103$	11.9048	1.17302	0.586510	+	0.809942i	$0.300502\pi$
$104$	4.97909	0.488240
$105$	0	0
$106$	−1.13786	−0.110519
$107$	−16.1058	−1.55700	−0.778502	−	0.627642i	$-0.784020\pi$
$107$	−16.1058	−1.55700	−0.778502	+	0.627642i	$0.784020\pi$
$108$	10.7722	1.03655
$109$	13.1464	1.25919	0.629597	−	0.776922i	$-0.283220\pi$
$109$	13.1464	1.25919	0.629597	+	0.776922i	$0.283220\pi$
$110$	0	0
$111$	10.8752	1.03223
$112$	−14.3243	−1.35352
$113$	14.7595	1.38846	0.694228	−	0.719755i	$-0.255746\pi$
$113$	14.7595	1.38846	0.694228	+	0.719755i	$0.255746\pi$
$114$	−3.16968	−0.296868
$115$	0	0
$116$	−8.55311	−0.794136
$117$	4.31636	0.399048
$118$	0.384568	0.0354024
$119$	4.30793	0.394908
$120$	0	0
$121$	−7.26876	−0.660796
$122$	0.0868042	0.00785888
$123$	16.1626	1.45733
$124$	2.44528	0.219593
$125$	0	0
$126$	1.34531	0.119850
$127$	−11.0142	−0.977350	−0.488675	−	0.872466i	$-0.662520\pi$
$127$	−11.0142	−0.977350	−0.488675	+	0.872466i	$0.662520\pi$
$128$	−8.73730	−0.772275
$129$	11.0153	0.969840
$130$	0	0
$131$	−5.25402	−0.459046	−0.229523	−	0.973303i	$-0.573717\pi$
$131$	−5.25402	−0.459046	−0.229523	+	0.973303i	$0.573717\pi$
$132$	−5.14377	−0.447707
$133$	30.5979	2.65317
$134$	2.63072	0.227260
$135$	0	0
$136$	1.24633	0.106872
$137$	1.68115	0.143630	0.0718151	−	0.997418i	$-0.477121\pi$
$137$	1.68115	0.143630	0.0718151	+	0.997418i	$0.477121\pi$
$138$	1.76885	0.150574
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	0	0
$141$	9.57243	0.806144
$142$	−1.97677	−0.165887
$143$	−7.97646	−0.667025
$144$	−3.59257	−0.299381
$145$	0	0
$146$	1.83001	0.151453
$147$	14.4987	1.19583
$148$	−14.8152	−1.21780
$149$	13.9257	1.14084	0.570421	−	0.821353i	$-0.306780\pi$
$149$	13.9257	1.14084	0.570421	+	0.821353i	$0.306780\pi$
$150$	0	0
$151$	8.63888	0.703022	0.351511	−	0.936184i	$-0.385668\pi$
$151$	8.63888	0.703022	0.351511	+	0.936184i	$0.385668\pi$
$152$	8.85228	0.718015
$153$	1.08044	0.0873486
$154$	−2.48608	−0.200334
$155$	0	0
$156$	10.9961	0.880390
$157$	−10.4362	−0.832900	−0.416450	−	0.909159i	$-0.636726\pi$
$157$	−10.4362	−0.832900	−0.416450	+	0.909159i	$0.636726\pi$
$158$	−2.67233	−0.212599
$159$	−5.15166	−0.408553
$160$	0	0
$161$	−17.0752	−1.34571
$162$	−1.47347	−0.115767
$163$	3.73406	0.292474	0.146237	−	0.989250i	$-0.453284\pi$
$163$	3.73406	0.292474	0.146237	+	0.989250i	$0.453284\pi$
$164$	−22.0182	−1.71933
$165$	0	0
$166$	−4.89043	−0.379571
$167$	19.7503	1.52832	0.764161	−	0.645025i	$-0.223153\pi$
$167$	19.7503	1.52832	0.764161	+	0.645025i	$0.223153\pi$
$168$	7.02604	0.542070
$169$	4.05166	0.311666
$170$	0	0
$171$	7.67403	0.586848
$172$	−15.0060	−1.14420
$173$	7.51428	0.571300	0.285650	−	0.958334i	$-0.407790\pi$
$173$	7.51428	0.571300	0.285650	+	0.958334i	$0.407790\pi$
$174$	1.93882	0.146982
$175$	0	0
$176$	6.63892	0.500427
$177$	1.74112	0.130871
$178$	−1.00005	−0.0749568
$179$	24.2625	1.81346	0.906732	−	0.421708i	$-0.138569\pi$
$179$	24.2625	1.81346	0.906732	+	0.421708i	$0.138569\pi$
$180$	0	0
$181$	−3.06020	−0.227463	−0.113732	−	0.993512i	$-0.536280\pi$
$181$	−3.06020	−0.227463	−0.113732	+	0.993512i	$0.536280\pi$
$182$	5.31461	0.393945
$183$	0.393004	0.0290517
$184$	−4.94003	−0.364184
$185$	0	0
$186$	−0.554298	−0.0406431
$187$	−1.99661	−0.146007
$188$	−13.0405	−0.951074
$189$	23.5718	1.71460
$190$	0	0
$191$	8.83544	0.639310	0.319655	−	0.947534i	$-0.396433\pi$
$191$	8.83544	0.639310	0.319655	+	0.947534i	$0.396433\pi$
$192$	−8.11101	−0.585362
$193$	−14.9309	−1.07475	−0.537374	−	0.843344i	$-0.680584\pi$
$193$	−14.9309	−1.07475	−0.537374	+	0.843344i	$0.680584\pi$
$194$	2.24602	0.161255
$195$	0	0
$196$	−19.7515	−1.41082
$197$	11.8981	0.847702	0.423851	−	0.905732i	$-0.360678\pi$
$197$	11.8981	0.847702	0.423851	+	0.905732i	$0.360678\pi$
$198$	−0.623515	−0.0443113
$199$	−4.40723	−0.312420	−0.156210	−	0.987724i	$-0.549928\pi$
$199$	−4.40723	−0.312420	−0.156210	+	0.987724i	$0.549928\pi$
$200$	0	0
$201$	11.9105	0.840104
$202$	−4.02857	−0.283449
$203$	−18.7160	−1.31361
$204$	2.75246	0.192711
$205$	0	0
$206$	3.67629	0.256139
$207$	−4.28251	−0.297655
$208$	−14.1923	−0.984061
$209$	−14.1813	−0.980940
$210$	0	0
$211$	21.4218	1.47474	0.737368	−	0.675492i	$-0.236069\pi$
$211$	21.4218	1.47474	0.737368	+	0.675492i	$0.236069\pi$
$212$	7.01807	0.482003
$213$	−8.94977	−0.613228
$214$	−4.97356	−0.339985
$215$	0	0
$216$	6.81957	0.464013
$217$	5.35079	0.363236
$218$	4.05968	0.274956
$219$	8.28532	0.559870
$220$	0	0
$221$	4.26825	0.287114
$222$	3.35833	0.225396
$223$	7.93581	0.531421	0.265711	−	0.964053i	$-0.414393\pi$
$223$	7.93581	0.531421	0.265711	+	0.964053i	$0.414393\pi$
$224$	−14.4742	−0.967096
$225$	0	0
$226$	4.55782	0.303181
$227$	−15.9354	−1.05767	−0.528836	−	0.848724i	$-0.677371\pi$
$227$	−15.9354	−1.05767	−0.528836	+	0.848724i	$0.677371\pi$
$228$	19.5498	1.29472
$229$	13.4712	0.890203	0.445102	−	0.895480i	$-0.353168\pi$
$229$	13.4712	0.890203	0.445102	+	0.895480i	$0.353168\pi$
$230$	0	0
$231$	−11.2557	−0.740568
$232$	−5.41474	−0.355495
$233$	−18.4315	−1.20749	−0.603745	−	0.797177i	$-0.706326\pi$
$233$	−18.4315	−1.20749	−0.603745	+	0.797177i	$0.706326\pi$
$234$	1.33292	0.0871356
$235$	0	0
$236$	−2.37192	−0.154399
$237$	−12.0989	−0.785908
$238$	1.33031	0.0862315
$239$	0.889441	0.0575331	0.0287666	−	0.999586i	$-0.490842\pi$
$239$	0.889441	0.0575331	0.0287666	+	0.999586i	$0.490842\pi$
$240$	0	0
$241$	−11.7554	−0.757233	−0.378616	−	0.925554i	$-0.623600\pi$
$241$	−11.7554	−0.757233	−0.378616	+	0.925554i	$0.623600\pi$
$242$	−2.24463	−0.144291
$243$	10.2962	0.660499
$244$	−0.535387	−0.0342746
$245$	0	0
$246$	4.99109	0.318220
$247$	30.3160	1.92896
$248$	1.54804	0.0983007
$249$	−22.1413	−1.40315
$250$	0	0
$251$	−14.4551	−0.912397	−0.456199	−	0.889878i	$-0.650789\pi$
$251$	−14.4551	−0.912397	−0.456199	+	0.889878i	$0.650789\pi$
$252$	−8.29755	−0.522696
$253$	7.91389	0.497542
$254$	−3.40124	−0.213413
$255$	0	0
$256$	8.90468	0.556543
$257$	−17.2989	−1.07907	−0.539537	−	0.841962i	$-0.681401\pi$
$257$	−17.2989	−1.07907	−0.539537	+	0.841962i	$0.681401\pi$
$258$	3.40158	0.211773
$259$	−32.4189	−2.01441
$260$	0	0
$261$	−4.69403	−0.290553
$262$	−1.62247	−0.100237
$263$	−4.00876	−0.247191	−0.123595	−	0.992333i	$-0.539442\pi$
$263$	−4.00876	−0.247191	−0.123595	+	0.992333i	$0.539442\pi$
$264$	−3.25638	−0.200416
$265$	0	0
$266$	9.44880	0.579343
$267$	−4.52770	−0.277091
$268$	−16.2257	−0.991140
$269$	28.0301	1.70902	0.854511	−	0.519433i	$-0.173857\pi$
$269$	28.0301	1.70902	0.854511	+	0.519433i	$0.173857\pi$
$270$	0	0
$271$	16.3058	0.990507	0.495253	−	0.868749i	$-0.335075\pi$
$271$	16.3058	0.990507	0.495253	+	0.868749i	$0.335075\pi$
$272$	−3.55252	−0.215403
$273$	24.0617	1.45628
$274$	0.519149	0.0313629
$275$	0	0
$276$	−10.9098	−0.656694
$277$	26.3241	1.58166	0.790831	−	0.612035i	$-0.209649\pi$
$277$	26.3241	1.58166	0.790831	+	0.612035i	$0.209649\pi$
$278$	0.308806	0.0185209
$279$	1.34199	0.0803431
$280$	0	0
$281$	−7.58167	−0.452285	−0.226142	−	0.974094i	$-0.572611\pi$
$281$	−7.58167	−0.452285	−0.226142	+	0.974094i	$0.572611\pi$
$282$	2.95602	0.176029
$283$	14.2221	0.845417	0.422709	−	0.906266i	$-0.361079\pi$
$283$	14.2221	0.845417	0.422709	+	0.906266i	$0.361079\pi$
$284$	12.1922	0.723475
$285$	0	0
$286$	−2.46318	−0.145651
$287$	−48.1804	−2.84400
$288$	−3.63016	−0.213910
$289$	−15.9316	−0.937153
$290$	0	0
$291$	10.1688	0.596107
$292$	−11.2870	−0.660524
$293$	4.68983	0.273983	0.136991	−	0.990572i	$-0.456257\pi$
$293$	4.68983	0.273983	0.136991	+	0.990572i	$0.456257\pi$
$294$	4.47727	0.261120
$295$	0	0
$296$	−9.37912	−0.545150
$297$	−10.9249	−0.633927
$298$	4.30035	0.249113
$299$	−16.9179	−0.978388
$300$	0	0
$301$	−32.8364	−1.89266
$302$	2.66774	0.153511
$303$	−18.2392	−1.04782
$304$	−25.2324	−1.44718
$305$	0	0
$306$	0.333647	0.0190733
$307$	−10.6916	−0.610199	−0.305100	−	0.952320i	$-0.598690\pi$
$307$	−10.6916	−0.610199	−0.305100	+	0.952320i	$0.598690\pi$
$308$	15.3335	0.873708
$309$	16.6443	0.946861
$310$	0	0
$311$	−21.0455	−1.19338	−0.596690	−	0.802472i	$-0.703518\pi$
$311$	−21.0455	−1.19338	−0.596690	+	0.802472i	$0.703518\pi$
$312$	6.96132	0.394107
$313$	14.2033	0.802819	0.401409	−	0.915899i	$-0.368520\pi$
$313$	14.2033	0.802819	0.401409	+	0.915899i	$0.368520\pi$
$314$	−3.22276	−0.181871
$315$	0	0
$316$	16.4823	0.927200
$317$	−2.85738	−0.160487	−0.0802433	−	0.996775i	$-0.525570\pi$
$317$	−2.85738	−0.160487	−0.0802433	+	0.996775i	$0.525570\pi$
$318$	−1.59086	−0.0892111
$319$	8.67437	0.485672
$320$	0	0
$321$	−22.5177	−1.25681
$322$	−5.27292	−0.293848
$323$	7.58849	0.422235
$324$	9.08803	0.504890
$325$	0	0
$326$	1.15310	0.0638642
$327$	18.3801	1.01642
$328$	−13.9391	−0.769658
$329$	−28.5353	−1.57320
$330$	0	0
$331$	12.1172	0.666019	0.333010	−	0.942923i	$-0.391936\pi$
$331$	12.1172	0.666019	0.333010	+	0.942923i	$0.391936\pi$
$332$	30.1630	1.65541
$333$	−8.13074	−0.445562
$334$	6.09900	0.333722
$335$	0	0
$336$	−20.0269	−1.09256
$337$	12.4929	0.680532	0.340266	−	0.940329i	$-0.389483\pi$
$337$	12.4929	0.680532	0.340266	+	0.940329i	$0.389483\pi$
$338$	1.25117	0.0680549
$339$	20.6354	1.12076
$340$	0	0
$341$	−2.47995	−0.134297
$342$	2.36979	0.128143
$343$	−14.0461	−0.758418
$344$	−9.49991	−0.512201
$345$	0	0
$346$	2.32045	0.124748
$347$	11.1030	0.596039	0.298020	−	0.954560i	$-0.403674\pi$
$347$	11.1030	0.596039	0.298020	+	0.954560i	$0.403674\pi$
$348$	−11.9582	−0.641026
$349$	16.3372	0.874509	0.437254	−	0.899338i	$-0.355951\pi$
$349$	16.3372	0.874509	0.437254	+	0.899338i	$0.355951\pi$
$350$	0	0
$351$	23.3547	1.24658
$352$	6.70839	0.357558
$353$	26.2175	1.39542	0.697709	−	0.716382i	$-0.254203\pi$
$353$	26.2175	1.39542	0.697709	+	0.716382i	$0.254203\pi$
$354$	0.537669	0.0285768
$355$	0	0
$356$	6.16806	0.326906
$357$	6.02297	0.318769
$358$	7.49240	0.395986
$359$	18.6480	0.984204	0.492102	−	0.870537i	$-0.336229\pi$
$359$	18.6480	0.984204	0.492102	+	0.870537i	$0.336229\pi$
$360$	0	0
$361$	34.8986	1.83677
$362$	−0.945009	−0.0496686
$363$	−10.1625	−0.533394
$364$	−32.7792	−1.71810
$365$	0	0
$366$	0.121362	0.00634369
$367$	3.92562	0.204916	0.102458	−	0.994737i	$-0.467329\pi$
$367$	3.92562	0.204916	0.102458	+	0.994737i	$0.467329\pi$
$368$	14.0810	0.734023
$369$	−12.0838	−0.629057
$370$	0	0
$371$	15.3570	0.797297
$372$	3.41877	0.177255
$373$	−14.3279	−0.741870	−0.370935	−	0.928659i	$-0.620963\pi$
$373$	−14.3279	−0.741870	−0.370935	+	0.928659i	$0.620963\pi$
$374$	−0.616565	−0.0318818
$375$	0	0
$376$	−8.25557	−0.425749
$377$	−18.5436	−0.955045
$378$	7.27912	0.374397
$379$	5.65835	0.290650	0.145325	−	0.989384i	$-0.453577\pi$
$379$	5.65835	0.290650	0.145325	+	0.989384i	$0.453577\pi$
$380$	0	0
$381$	−15.3990	−0.788917
$382$	2.72844	0.139599
$383$	−4.23323	−0.216308	−0.108154	−	0.994134i	$-0.534494\pi$
$383$	−4.23323	−0.216308	−0.108154	+	0.994134i	$0.534494\pi$
$384$	−12.2157	−0.623380
$385$	0	0
$386$	−4.61074	−0.234681
$387$	−8.23546	−0.418632
$388$	−13.8529	−0.703275
$389$	−5.21409	−0.264365	−0.132182	−	0.991225i	$-0.542198\pi$
$389$	−5.21409	−0.264365	−0.132182	+	0.991225i	$0.542198\pi$
$390$	0	0
$391$	−4.23477	−0.214162
$392$	−12.5041	−0.631553
$393$	−7.34570	−0.370542
$394$	3.67419	0.185103
$395$	0	0
$396$	3.84569	0.193253
$397$	−9.63263	−0.483448	−0.241724	−	0.970345i	$-0.577713\pi$
$397$	−9.63263	−0.483448	−0.241724	+	0.970345i	$0.577713\pi$
$398$	−1.36098	−0.0682196
$399$	42.7792	2.14164
$400$	0	0
$401$	24.8722	1.24206	0.621030	−	0.783787i	$-0.286714\pi$
$401$	24.8722	1.24206	0.621030	+	0.783787i	$0.286714\pi$
$402$	3.67804	0.183444
$403$	5.30151	0.264087
$404$	24.8472	1.23619
$405$	0	0
$406$	−5.77962	−0.286837
$407$	15.0253	0.744775
$408$	1.74251	0.0862670
$409$	−28.4449	−1.40651	−0.703255	−	0.710938i	$-0.748271\pi$
$409$	−28.4449	−1.40651	−0.703255	+	0.710938i	$0.748271\pi$
$410$	0	0
$411$	2.35043	0.115938
$412$	−22.6744	−1.11709
$413$	−5.19027	−0.255397
$414$	−1.32246	−0.0649955
$415$	0	0
$416$	−14.3408	−0.703118
$417$	1.39811	0.0684658
$418$	−4.37926	−0.214197
$419$	−4.59075	−0.224273	−0.112136	−	0.993693i	$-0.535769\pi$
$419$	−4.59075	−0.224273	−0.112136	+	0.993693i	$0.535769\pi$
$420$	0	0
$421$	20.9586	1.02146	0.510729	−	0.859742i	$-0.329375\pi$
$421$	20.9586	1.02146	0.510729	+	0.859742i	$0.329375\pi$
$422$	6.61517	0.322021
$423$	−7.15674	−0.347973
$424$	4.44295	0.215769
$425$	0	0
$426$	−2.76374	−0.133904
$427$	−1.17154	−0.0566948
$428$	30.6757	1.48276
$429$	−11.1520	−0.538423
$430$	0	0
$431$	14.1727	0.682676	0.341338	−	0.939941i	$-0.389120\pi$
$431$	14.1727	0.682676	0.341338	+	0.939941i	$0.389120\pi$
$432$	−19.4384	−0.935232
$433$	−7.32714	−0.352120	−0.176060	−	0.984379i	$-0.556335\pi$
$433$	−7.32714	−0.352120	−0.176060	+	0.984379i	$0.556335\pi$
$434$	1.65236	0.0793157
$435$	0	0
$436$	−25.0391	−1.19916
$437$	−30.0782	−1.43884
$438$	2.55856	0.122253
$439$	3.07088	0.146565	0.0732825	−	0.997311i	$-0.476653\pi$
$439$	3.07088	0.146565	0.0732825	+	0.997311i	$0.476653\pi$
$440$	0	0
$441$	−10.8398	−0.516181
$442$	1.31806	0.0626938
$443$	−14.9982	−0.712586	−0.356293	−	0.934374i	$-0.615960\pi$
$443$	−14.9982	−0.712586	−0.356293	+	0.934374i	$0.615960\pi$
$444$	−20.7133	−0.983011
$445$	0	0
$446$	2.45063	0.116041
$447$	19.4697	0.920887
$448$	24.1788	1.14234
$449$	−22.4064	−1.05742	−0.528711	−	0.848802i	$-0.677324\pi$
$449$	−22.4064	−1.05742	−0.528711	+	0.848802i	$0.677324\pi$
$450$	0	0
$451$	22.3303	1.05149
$452$	−28.1115	−1.32225
$453$	12.0781	0.567479
$454$	−4.92096	−0.230952
$455$	0	0
$456$	12.3765	0.579581
$457$	12.9317	0.604919	0.302460	−	0.953162i	$-0.402192\pi$
$457$	12.9317	0.604919	0.302460	+	0.953162i	$0.402192\pi$
$458$	4.15999	0.194384
$459$	5.84598	0.272867
$460$	0	0
$461$	−39.7115	−1.84955	−0.924774	−	0.380517i	$-0.875746\pi$
$461$	−39.7115	−1.84955	−0.924774	+	0.380517i	$0.875746\pi$
$462$	−3.47581	−0.161709
$463$	−21.4987	−0.999130	−0.499565	−	0.866276i	$-0.666507\pi$
$463$	−21.4987	−0.999130	−0.499565	+	0.866276i	$0.666507\pi$
$464$	15.4341	0.716510
$465$	0	0
$466$	−5.69177	−0.263666
$467$	19.2688	0.891655	0.445827	−	0.895119i	$-0.352910\pi$
$467$	19.2688	0.891655	0.445827	+	0.895119i	$0.352910\pi$
$468$	−8.22112	−0.380021
$469$	−35.5052	−1.63948
$470$	0	0
$471$	−14.5910	−0.672317
$472$	−1.50160	−0.0691168
$473$	15.2188	0.699761
$474$	−3.73621	−0.171610
$475$	0	0
$476$	−8.20506	−0.376078
$477$	3.85159	0.176352
$478$	0.274665	0.0125629
$479$	−23.7465	−1.08500	−0.542502	−	0.840055i	$-0.682523\pi$
$479$	−23.7465	−1.08500	−0.542502	+	0.840055i	$0.682523\pi$
$480$	0	0
$481$	−32.1202	−1.46456
$482$	−3.63014	−0.165348
$483$	−23.8730	−1.08626
$484$	13.8444	0.629289
$485$	0	0
$486$	3.17951	0.144226
$487$	27.3504	1.23937	0.619683	−	0.784852i	$-0.287261\pi$
$487$	27.3504	1.23937	0.619683	+	0.784852i	$0.287261\pi$
$488$	−0.338939	−0.0153430
$489$	5.22062	0.236085
$490$	0	0
$491$	−29.0333	−1.31025	−0.655126	−	0.755519i	$-0.727385\pi$
$491$	−29.0333	−1.31025	−0.655126	+	0.755519i	$0.727385\pi$
$492$	−30.7838	−1.38784
$493$	−4.64171	−0.209052
$494$	9.36176	0.421206
$495$	0	0
$496$	−4.41252	−0.198128
$497$	26.6792	1.19672
$498$	−6.83737	−0.306390
$499$	26.6880	1.19472	0.597359	−	0.801974i	$-0.296217\pi$
$499$	26.6880	1.19472	0.597359	+	0.801974i	$0.296217\pi$
$500$	0	0
$501$	27.6131	1.23366
$502$	−4.46382	−0.199230
$503$	−7.41588	−0.330658	−0.165329	−	0.986238i	$-0.552869\pi$
$503$	−7.41588	−0.330658	−0.165329	+	0.986238i	$0.552869\pi$
$504$	−5.25295	−0.233985
$505$	0	0
$506$	2.44386	0.108643
$507$	5.66466	0.251577
$508$	20.9780	0.930749
$509$	36.1455	1.60212	0.801060	−	0.598583i	$-0.204270\pi$
$509$	36.1455	1.60212	0.801060	+	0.598583i	$0.204270\pi$
$510$	0	0
$511$	−24.6985	−1.09260
$512$	20.2244	0.893801
$513$	41.5221	1.83325
$514$	−5.34199	−0.235625
$515$	0	0
$516$	−20.9801	−0.923597
$517$	13.2254	0.581651
$518$	−10.0111	−0.439864
$519$	10.5058	0.461153
$520$	0	0
$521$	33.9636	1.48797	0.743986	−	0.668196i	$-0.232933\pi$
$521$	33.9636	1.48797	0.743986	+	0.668196i	$0.232933\pi$
$522$	−1.44954	−0.0634448
$523$	11.7578	0.514134	0.257067	−	0.966394i	$-0.417244\pi$
$523$	11.7578	0.514134	0.257067	+	0.966394i	$0.417244\pi$
$524$	10.0070	0.437158
$525$	0	0
$526$	−1.23793	−0.0539762
$527$	1.32704	0.0578066
$528$	9.28194	0.403945
$529$	−6.21480	−0.270209
$530$	0	0
$531$	−1.30174	−0.0564905
$532$	−58.2779	−2.52667
$533$	−47.7366	−2.06770
$534$	−1.39818	−0.0605051
$535$	0	0
$536$	−10.2720	−0.443684
$537$	33.9217	1.46383
$538$	8.65585	0.373180
$539$	20.0315	0.862818
$540$	0	0
$541$	0.804094	0.0345707	0.0172854	−	0.999851i	$-0.494498\pi$
$541$	0.804094	0.0345707	0.0172854	+	0.999851i	$0.494498\pi$
$542$	5.03533	0.216286
$543$	−4.27851	−0.183608
$544$	−3.58970	−0.153907
$545$	0	0
$546$	7.43041	0.317992
$547$	4.34094	0.185605	0.0928027	−	0.995685i	$-0.470417\pi$
$547$	4.34094	0.185605	0.0928027	+	0.995685i	$0.470417\pi$
$548$	−3.20198	−0.136782
$549$	−0.293826	−0.0125402
$550$	0	0
$551$	−32.9686	−1.40451
$552$	−6.90671	−0.293969
$553$	36.0667	1.53371
$554$	8.12904	0.345370
$555$	0	0
$556$	−1.90464	−0.0807747
$557$	29.2321	1.23861	0.619303	−	0.785152i	$-0.287415\pi$
$557$	29.2321	1.23861	0.619303	+	0.785152i	$0.287415\pi$
$558$	0.414416	0.0175436
$559$	−32.5339	−1.37604
$560$	0	0
$561$	−2.79148	−0.117857
$562$	−2.34126	−0.0987603
$563$	−27.9325	−1.17721	−0.588606	−	0.808420i	$-0.700323\pi$
$563$	−27.9325	−1.17721	−0.588606	+	0.808420i	$0.700323\pi$
$564$	−18.2320	−0.767707
$565$	0	0
$566$	4.39187	0.184604
$567$	19.8865	0.835156
$568$	7.71857	0.323864
$569$	21.5023	0.901424	0.450712	−	0.892669i	$-0.351170\pi$
$569$	21.5023	0.901424	0.450712	+	0.892669i	$0.351170\pi$
$570$	0	0
$571$	19.6384	0.821843	0.410922	−	0.911671i	$-0.365207\pi$
$571$	19.6384	0.821843	0.410922	+	0.911671i	$0.365207\pi$
$572$	15.1923	0.635221
$573$	12.3529	0.516051
$574$	−14.8784	−0.621012
$575$	0	0
$576$	6.06412	0.252672
$577$	−14.7632	−0.614599	−0.307299	−	0.951613i	$-0.599425\pi$
$577$	−14.7632	−0.614599	−0.307299	+	0.951613i	$0.599425\pi$
$578$	−4.91977	−0.204636
$579$	−20.8750	−0.867536
$580$	0	0
$581$	66.0031	2.73827
$582$	3.14019	0.130165
$583$	−7.11757	−0.294780
$584$	−7.14552	−0.295684
$585$	0	0
$586$	1.44825	0.0598265
$587$	6.27328	0.258926	0.129463	−	0.991584i	$-0.458675\pi$
$587$	6.27328	0.258926	0.129463	+	0.991584i	$0.458675\pi$
$588$	−27.6147	−1.13881
$589$	9.42551	0.388371
$590$	0	0
$591$	16.6348	0.684265
$592$	26.7341	1.09877
$593$	−40.8082	−1.67579	−0.837895	−	0.545831i	$-0.816214\pi$
$593$	−40.8082	−1.67579	−0.837895	+	0.545831i	$0.816214\pi$
$594$	−3.37368	−0.138424
$595$	0	0
$596$	−26.5235	−1.08645
$597$	−6.16179	−0.252185
$598$	−5.22435	−0.213639
$599$	−10.3330	−0.422193	−0.211096	−	0.977465i	$-0.567703\pi$
$599$	−10.3330	−0.422193	−0.211096	+	0.977465i	$0.567703\pi$
$600$	0	0
$601$	15.7883	0.644019	0.322009	−	0.946737i	$-0.395642\pi$
$601$	15.7883	0.644019	0.322009	+	0.946737i	$0.395642\pi$
$602$	−10.1401	−0.413278
$603$	−8.90480	−0.362632
$604$	−16.4539	−0.669502
$605$	0	0
$606$	−5.63238	−0.228800
$607$	−12.8843	−0.522958	−0.261479	−	0.965209i	$-0.584210\pi$
$607$	−12.8843	−0.522958	−0.261479	+	0.965209i	$0.584210\pi$
$608$	−25.4965	−1.03402
$609$	−26.1671	−1.06034
$610$	0	0
$611$	−28.2725	−1.14378
$612$	−2.05785	−0.0831837
$613$	−6.99917	−0.282694	−0.141347	−	0.989960i	$-0.545143\pi$
$613$	−6.99917	−0.282694	−0.141347	+	0.989960i	$0.545143\pi$
$614$	−3.30162	−0.133242
$615$	0	0
$616$	9.70723	0.391116
$617$	8.16208	0.328593	0.164297	−	0.986411i	$-0.447465\pi$
$617$	8.16208	0.328593	0.164297	+	0.986411i	$0.447465\pi$
$618$	5.13986	0.206755
$619$	−12.6031	−0.506559	−0.253280	−	0.967393i	$-0.581509\pi$
$619$	−12.6031	−0.506559	−0.253280	+	0.967393i	$0.581509\pi$
$620$	0	0
$621$	−23.1715	−0.929840
$622$	−6.49897	−0.260585
$623$	13.4970	0.540747
$624$	−19.8425	−0.794334
$625$	0	0
$626$	4.38607	0.175303
$627$	−19.8270	−0.791815
$628$	19.8772	0.793187
$629$	−8.04011	−0.320580
$630$	0	0
$631$	45.4769	1.81041	0.905203	−	0.424980i	$-0.139719\pi$
$631$	45.4769	1.81041	0.905203	+	0.424980i	$0.139719\pi$
$632$	10.4345	0.415061
$633$	29.9500	1.19041
$634$	−0.882377	−0.0350437
$635$	0	0
$636$	9.81204	0.389073
$637$	−42.8223	−1.69668
$638$	2.67870	0.106051
$639$	6.69122	0.264700
$640$	0	0
$641$	12.1865	0.481339	0.240669	−	0.970607i	$-0.422633\pi$
$641$	12.1865	0.481339	0.240669	+	0.970607i	$0.422633\pi$
$642$	−6.95358	−0.274436
$643$	−29.7340	−1.17259	−0.586296	−	0.810097i	$-0.699415\pi$
$643$	−29.7340	−1.17259	−0.586296	+	0.810097i	$0.699415\pi$
$644$	32.5221	1.28155
$645$	0	0
$646$	2.34337	0.0921987
$647$	32.8228	1.29040	0.645199	−	0.764015i	$-0.276774\pi$
$647$	32.8228	1.29040	0.645199	+	0.764015i	$0.276774\pi$
$648$	5.75339	0.226014
$649$	2.40555	0.0944262
$650$	0	0
$651$	7.48100	0.293204
$652$	−7.11203	−0.278529
$653$	−10.3258	−0.404078	−0.202039	−	0.979377i	$-0.564757\pi$
$653$	−10.3258	−0.404078	−0.202039	+	0.979377i	$0.564757\pi$
$654$	5.67588	0.221944
$655$	0	0
$656$	39.7318	1.55127
$657$	−6.19445	−0.241668
$658$	−8.81188	−0.343523
$659$	25.8076	1.00532	0.502661	−	0.864484i	$-0.332354\pi$
$659$	25.8076	1.00532	0.502661	+	0.864484i	$0.332354\pi$
$660$	0	0
$661$	−13.3955	−0.521025	−0.260513	−	0.965470i	$-0.583892\pi$
$661$	−13.3955	−0.521025	−0.260513	+	0.965470i	$0.583892\pi$
$662$	3.74185	0.145431
$663$	5.96749	0.231758
$664$	19.0954	0.741044
$665$	0	0
$666$	−2.51082	−0.0972924
$667$	18.3982	0.712380
$668$	−37.6171	−1.45545
$669$	11.0952	0.428963
$670$	0	0
$671$	0.542978	0.0209614
$672$	−20.2365	−0.780640
$673$	6.82079	0.262922	0.131461	−	0.991321i	$-0.458033\pi$
$673$	6.82079	0.262922	0.131461	+	0.991321i	$0.458033\pi$
$674$	3.85788	0.148600
$675$	0	0
$676$	−7.71694	−0.296805
$677$	−10.5869	−0.406888	−0.203444	−	0.979087i	$-0.565213\pi$
$677$	−10.5869	−0.406888	−0.203444	+	0.979087i	$0.565213\pi$
$678$	6.37233	0.244728
$679$	−30.3131	−1.16331
$680$	0	0
$681$	−22.2795	−0.853753
$682$	−0.765823	−0.0293249
$683$	−43.3107	−1.65724	−0.828620	−	0.559812i	$-0.810873\pi$
$683$	−43.3107	−1.65724	−0.828620	+	0.559812i	$0.810873\pi$
$684$	−14.6163	−0.558867
$685$	0	0
$686$	−4.33752	−0.165607
$687$	18.8343	0.718572
$688$	27.0784	1.03236
$689$	15.2156	0.579667
$690$	0	0
$691$	−27.5238	−1.04706	−0.523528	−	0.852009i	$-0.675384\pi$
$691$	−27.5238	−1.04706	−0.523528	+	0.852009i	$0.675384\pi$
$692$	−14.3120	−0.544060
$693$	8.41519	0.319667
$694$	3.42867	0.130150
$695$	0	0
$696$	−7.57041	−0.286956
$697$	−11.9491	−0.452604
$698$	5.04501	0.190957
$699$	−25.7694	−0.974686
$700$	0	0
$701$	51.4490	1.94320	0.971601	−	0.236625i	$-0.0760413\pi$
$701$	51.4490	1.94320	0.971601	+	0.236625i	$0.0760413\pi$
$702$	7.21207	0.272202
$703$	−57.1063	−2.15381
$704$	−11.2062	−0.422351
$705$	0	0
$706$	8.09612	0.304702
$707$	54.3710	2.04483
$708$	−3.31621	−0.124631
$709$	16.8255	0.631894	0.315947	−	0.948777i	$-0.397678\pi$
$709$	16.8255	0.631894	0.315947	+	0.948777i	$0.397678\pi$
$710$	0	0
$711$	9.04563	0.339238
$712$	3.90483	0.146340
$713$	−5.25992	−0.196986
$714$	1.85993	0.0696061
$715$	0	0
$716$	−46.2113	−1.72700
$717$	1.24354	0.0464407
$718$	5.75862	0.214910
$719$	26.2933	0.980574	0.490287	−	0.871561i	$-0.336892\pi$
$719$	26.2933	0.980574	0.490287	+	0.871561i	$0.336892\pi$
$720$	0	0
$721$	−49.6165	−1.84781
$722$	10.7769	0.401074
$723$	−16.4354	−0.611238
$724$	5.82858	0.216618
$725$	0	0
$726$	−3.13825	−0.116471
$727$	−23.5378	−0.872967	−0.436484	−	0.899712i	$-0.643776\pi$
$727$	−23.5378	−0.872967	−0.436484	+	0.899712i	$0.643776\pi$
$728$	−20.7516	−0.769106
$729$	28.7097	1.06332
$730$	0	0
$731$	−8.14366	−0.301204
$732$	−0.748531	−0.0276665
$733$	10.0422	0.370919	0.185459	−	0.982652i	$-0.440623\pi$
$733$	10.0422	0.370919	0.185459	+	0.982652i	$0.440623\pi$
$734$	1.21225	0.0447451
$735$	0	0
$736$	14.2284	0.524464
$737$	16.4557	0.606154
$738$	−3.73154	−0.137360
$739$	8.15975	0.300161	0.150081	−	0.988674i	$-0.452047\pi$
$739$	8.15975	0.300161	0.150081	+	0.988674i	$0.452047\pi$
$740$	0	0
$741$	42.3851	1.55706
$742$	4.74234	0.174097
$743$	−7.78206	−0.285496	−0.142748	−	0.989759i	$-0.545594\pi$
$743$	−7.78206	−0.285496	−0.142748	+	0.989759i	$0.545594\pi$
$744$	2.16433	0.0793483
$745$	0	0
$746$	−4.42454	−0.161994
$747$	16.5538	0.605670
$748$	3.80282	0.139045
$749$	67.1249	2.45269
$750$	0	0
$751$	−8.30887	−0.303195	−0.151598	−	0.988442i	$-0.548442\pi$
$751$	−8.30887	−0.303195	−0.151598	+	0.988442i	$0.548442\pi$
$752$	23.5316	0.858108
$753$	−20.2098	−0.736487
$754$	−5.72638	−0.208542
$755$	0	0
$756$	−44.8958	−1.63284
$757$	38.8376	1.41158	0.705788	−	0.708424i	$-0.250593\pi$
$757$	38.8376	1.41158	0.705788	+	0.708424i	$0.250593\pi$
$758$	1.74733	0.0634659
$759$	11.0645	0.401616
$760$	0	0
$761$	−25.3906	−0.920408	−0.460204	−	0.887813i	$-0.652224\pi$
$761$	−25.3906	−0.920408	−0.460204	+	0.887813i	$0.652224\pi$
$762$	−4.75531	−0.172267
$763$	−54.7909	−1.98356
$764$	−16.8283	−0.608827
$765$	0	0
$766$	−1.30725	−0.0472327
$767$	−5.14246	−0.185684
$768$	12.4497	0.449241
$769$	39.8098	1.43558	0.717788	−	0.696261i	$-0.245155\pi$
$769$	39.8098	1.43558	0.717788	+	0.696261i	$0.245155\pi$
$770$	0	0
$771$	−24.1857	−0.871028
$772$	28.4379	1.02350
$773$	−9.95735	−0.358141	−0.179070	−	0.983836i	$-0.557309\pi$
$773$	−9.95735	−0.358141	−0.179070	+	0.983836i	$0.557309\pi$
$774$	−2.54316	−0.0914120
$775$	0	0
$776$	−8.76991	−0.314821
$777$	−45.3252	−1.62603
$778$	−1.61014	−0.0577263
$779$	−84.8706	−3.04080
$780$	0	0
$781$	−12.3651	−0.442458
$782$	−1.30772	−0.0467641
$783$	−25.3981	−0.907656
$784$	35.6416	1.27291
$785$	0	0
$786$	−2.26839	−0.0809110
$787$	−30.2211	−1.07727	−0.538633	−	0.842541i	$-0.681059\pi$
$787$	−30.2211	−1.07727	−0.538633	+	0.842541i	$0.681059\pi$
$788$	−22.6615	−0.807283
$789$	−5.60469	−0.199532
$790$	0	0
$791$	−61.5139	−2.18718
$792$	2.43460	0.0865098
$793$	−1.16075	−0.0412194
$794$	−2.97461	−0.105565
$795$	0	0
$796$	8.39417	0.297524
$797$	10.6318	0.376597	0.188298	−	0.982112i	$-0.439703\pi$
$797$	10.6318	0.376597	0.188298	+	0.982112i	$0.439703\pi$
$798$	13.2105	0.467645
$799$	−7.07697	−0.250365
$800$	0	0
$801$	3.38509	0.119606
$802$	7.68069	0.271215
$803$	11.4471	0.403959
$804$	−22.6853	−0.800048
$805$	0	0
$806$	1.63714	0.0576657
$807$	39.1891	1.37952
$808$	15.7301	0.553383
$809$	−15.9047	−0.559178	−0.279589	−	0.960120i	$-0.590198\pi$
$809$	−15.9047	−0.559178	−0.279589	+	0.960120i	$0.590198\pi$
$810$	0	0
$811$	53.3099	1.87196	0.935981	−	0.352049i	$-0.114515\pi$
$811$	53.3099	1.87196	0.935981	+	0.352049i	$0.114515\pi$
$812$	35.6473	1.25097
$813$	22.7973	0.799537
$814$	4.63989	0.162628
$815$	0	0
$816$	−4.96682	−0.173874
$817$	−57.8418	−2.02363
$818$	−8.78395	−0.307124
$819$	−17.9895	−0.628606
$820$	0	0
$821$	−16.7614	−0.584975	−0.292488	−	0.956269i	$-0.594483\pi$
$821$	−16.7614	−0.584975	−0.292488	+	0.956269i	$0.594483\pi$
$822$	0.725827	0.0253161
$823$	−38.0271	−1.32554	−0.662770	−	0.748823i	$-0.730619\pi$
$823$	−38.0271	−1.32554	−0.662770	+	0.748823i	$0.730619\pi$
$824$	−14.3546	−0.500065
$825$	0	0
$826$	−1.60279	−0.0557681
$827$	−40.6301	−1.41285	−0.706424	−	0.707789i	$-0.749693\pi$
$827$	−40.6301	−1.41285	−0.706424	+	0.707789i	$0.749693\pi$
$828$	8.15663	0.283462
$829$	−48.8098	−1.69523	−0.847617	−	0.530608i	$-0.821964\pi$
$829$	−48.8098	−1.69523	−0.847617	+	0.530608i	$0.821964\pi$
$830$	0	0
$831$	36.8040	1.27672
$832$	23.9561	0.830529
$833$	−10.7190	−0.371390
$834$	0.431745	0.0149501
$835$	0	0
$836$	27.0102	0.934168
$837$	7.26117	0.250983
$838$	−1.41765	−0.0489719
$839$	−13.9510	−0.481641	−0.240821	−	0.970570i	$-0.577417\pi$
$839$	−13.9510	−0.481641	−0.240821	+	0.970570i	$0.577417\pi$
$840$	0	0
$841$	−8.83388	−0.304617
$842$	6.47213	0.223044
$843$	−10.6000	−0.365084
$844$	−40.8007	−1.40442
$845$	0	0
$846$	−2.21004	−0.0759829
$847$	30.2944	1.04093
$848$	−12.6641	−0.434888
$849$	19.8841	0.682421
$850$	0	0
$851$	31.8683	1.09243
$852$	17.0461	0.583989
$853$	48.3304	1.65480	0.827401	−	0.561612i	$-0.189819\pi$
$853$	48.3304	1.65480	0.827401	+	0.561612i	$0.189819\pi$
$854$	−0.361779	−0.0123798
$855$	0	0
$856$	19.4199	0.663760
$857$	47.8068	1.63305	0.816525	−	0.577310i	$-0.195897\pi$
$857$	47.8068	1.63305	0.816525	+	0.577310i	$0.195897\pi$
$858$	−3.44380	−0.117569
$859$	57.7110	1.96907	0.984537	−	0.175179i	$-0.0560503\pi$
$859$	57.7110	1.96907	0.984537	+	0.175179i	$0.0560503\pi$
$860$	0	0
$861$	−67.3616	−2.29568
$862$	4.37662	0.149068
$863$	−22.5913	−0.769018	−0.384509	−	0.923121i	$-0.625629\pi$
$863$	−22.5913	−0.769018	−0.384509	+	0.923121i	$0.625629\pi$
$864$	−19.6418	−0.668229
$865$	0	0
$866$	−2.26266	−0.0768884
$867$	−22.2741	−0.756470
$868$	−10.1913	−0.345916
$869$	−16.7160	−0.567050
$870$	0	0
$871$	−35.1781	−1.19197
$872$	−15.8516	−0.536802
$873$	−7.60262	−0.257310
$874$	−9.28833	−0.314182
$875$	0	0
$876$	−15.7805	−0.533175
$877$	−8.20763	−0.277152	−0.138576	−	0.990352i	$-0.544252\pi$
$877$	−8.20763	−0.277152	−0.138576	+	0.990352i	$0.544252\pi$
$878$	0.948305	0.0320037
$879$	6.55690	0.221159
$880$	0	0
$881$	54.9341	1.85078	0.925388	−	0.379020i	$-0.123739\pi$
$881$	54.9341	1.85078	0.925388	+	0.379020i	$0.123739\pi$
$882$	−3.34739	−0.112713
$883$	42.9747	1.44621	0.723107	−	0.690736i	$-0.242714\pi$
$883$	42.9747	1.44621	0.723107	+	0.690736i	$0.242714\pi$
$884$	−8.12948	−0.273424
$885$	0	0
$886$	−4.63153	−0.155599
$887$	16.0125	0.537647	0.268824	−	0.963189i	$-0.413365\pi$
$887$	16.0125	0.537647	0.268824	+	0.963189i	$0.413365\pi$
$888$	−13.1130	−0.440045
$889$	45.9044	1.53958
$890$	0	0
$891$	−9.21688	−0.308777
$892$	−15.1149	−0.506083
$893$	−50.2654	−1.68207
$894$	6.01237	0.201084
$895$	0	0
$896$	36.4149	1.21654
$897$	−23.6531	−0.789754
$898$	−6.91921	−0.230897
$899$	−5.76537	−0.192286
$900$	0	0
$901$	3.80866	0.126885
$902$	6.89574	0.229603
$903$	−45.9089	−1.52775
$904$	−17.7966	−0.591907
$905$	0	0
$906$	3.72979	0.123914
$907$	−17.4683	−0.580026	−0.290013	−	0.957023i	$-0.593660\pi$
$907$	−17.4683	−0.580026	−0.290013	+	0.957023i	$0.593660\pi$
$908$	30.3513	1.00724
$909$	13.6364	0.452291
$910$	0	0
$911$	16.6656	0.552157	0.276078	−	0.961135i	$-0.410965\pi$
$911$	16.6656	0.552157	0.276078	+	0.961135i	$0.410965\pi$
$912$	−35.2777	−1.16816
$913$	−30.5907	−1.01240
$914$	3.99338	0.132089
$915$	0	0
$916$	−25.6578	−0.847758
$917$	21.8975	0.723118
$918$	1.80527	0.0595829
$919$	18.5901	0.613230	0.306615	−	0.951834i	$-0.400804\pi$
$919$	18.5901	0.613230	0.306615	+	0.951834i	$0.400804\pi$
$920$	0	0
$921$	−14.9480	−0.492553
$922$	−12.2631	−0.403865
$923$	26.4334	0.870067
$924$	21.4380	0.705257
$925$	0	0
$926$	−6.63893	−0.218169
$927$	−12.4440	−0.408713
$928$	15.5956	0.511951
$929$	6.08296	0.199576	0.0997878	−	0.995009i	$-0.468184\pi$
$929$	6.08296	0.199576	0.0997878	+	0.995009i	$0.468184\pi$
$930$	0	0
$931$	−76.1334	−2.49517
$932$	35.1054	1.14992
$933$	−29.4239	−0.963296
$934$	5.95033	0.194701
$935$	0	0
$936$	−5.20457	−0.170117
$937$	50.4851	1.64928	0.824639	−	0.565660i	$-0.191378\pi$
$937$	50.4851	1.64928	0.824639	+	0.565660i	$0.191378\pi$
$938$	−10.9642	−0.357994
$939$	19.8578	0.648035
$940$	0	0
$941$	−26.6303	−0.868123	−0.434061	−	0.900883i	$-0.642920\pi$
$941$	−26.6303	−0.868123	−0.434061	+	0.900883i	$0.642920\pi$
$942$	−4.50578	−0.146806
$943$	47.3622	1.54232
$944$	4.28014	0.139307
$945$	0	0
$946$	4.69965	0.152799
$947$	41.7464	1.35658	0.678289	−	0.734796i	$-0.262722\pi$
$947$	41.7464	1.35658	0.678289	+	0.734796i	$0.262722\pi$
$948$	23.0440	0.748436
$949$	−24.4710	−0.794361
$950$	0	0
$951$	−3.99494	−0.129545
$952$	−5.19440	−0.168351
$953$	45.7416	1.48172	0.740858	−	0.671662i	$-0.234419\pi$
$953$	45.7416	1.48172	0.740858	+	0.671662i	$0.234419\pi$
$954$	1.18939	0.0385080
$955$	0	0
$956$	−1.69406	−0.0547899
$957$	12.1277	0.392034
$958$	−7.33305	−0.236920
$959$	−7.00662	−0.226255
$960$	0	0
$961$	−29.3517	−0.946830
$962$	−9.91892	−0.319799
$963$	16.8351	0.542504
$964$	22.3898	0.721128
$965$	0	0
$966$	−7.37213	−0.237194
$967$	−29.9836	−0.964207	−0.482104	−	0.876114i	$-0.660127\pi$
$967$	−29.9836	−0.964207	−0.482104	+	0.876114i	$0.660127\pi$
$968$	8.76449	0.281701
$969$	10.6096	0.340828
$970$	0	0
$971$	−32.2248	−1.03414	−0.517071	−	0.855942i	$-0.672978\pi$
$971$	−32.2248	−1.03414	−0.517071	+	0.855942i	$0.672978\pi$
$972$	−19.6105	−0.629006
$973$	−4.16776	−0.133612
$974$	8.44597	0.270626
$975$	0	0
$976$	0.966108	0.0309244
$977$	1.45628	0.0465907	0.0232953	−	0.999729i	$-0.492584\pi$
$977$	1.45628	0.0465907	0.0232953	+	0.999729i	$0.492584\pi$
$978$	1.61216	0.0515512
$979$	−6.25551	−0.199927
$980$	0	0
$981$	−13.7417	−0.438739
$982$	−8.96564	−0.286105
$983$	35.1255	1.12033	0.560164	−	0.828382i	$-0.310738\pi$
$983$	35.1255	1.12033	0.560164	+	0.828382i	$0.310738\pi$
$984$	−19.4884	−0.621268
$985$	0	0
$986$	−1.43339	−0.0456483
$987$	−39.8956	−1.26989
$988$	−57.7410	−1.83699
$989$	32.2787	1.02640
$990$	0	0
$991$	−9.58149	−0.304366	−0.152183	−	0.988352i	$-0.548630\pi$
$991$	−9.58149	−0.304366	−0.152183	+	0.988352i	$0.548630\pi$
$992$	−4.45869	−0.141564
$993$	16.9411	0.537611
$994$	8.23869	0.261315
$995$	0	0
$996$	42.1712	1.33625
$997$	16.5332	0.523613	0.261807	−	0.965120i	$-0.415682\pi$
$997$	16.5332	0.523613	0.261807	+	0.965120i	$0.415682\pi$
$998$	8.24140	0.260877
$999$	−43.9933	−1.39189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3475.2.a.e.1.4		7
5.4	even	2		139.2.a.c.1.4	✓	7
15.14	odd	2		1251.2.a.k.1.4		7
20.19	odd	2		2224.2.a.o.1.5		7
35.34	odd	2		6811.2.a.p.1.4		7
40.19	odd	2		8896.2.a.bd.1.3		7
40.29	even	2		8896.2.a.be.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
139.2.a.c.1.4	✓	7	5.4	even	2
1251.2.a.k.1.4		7	15.14	odd	2
2224.2.a.o.1.5		7	20.19	odd	2
3475.2.a.e.1.4		7	1.1	even	1	trivial
6811.2.a.p.1.4		7	35.34	odd	2
8896.2.a.bd.1.3		7	40.19	odd	2
8896.2.a.be.1.5		7	40.29	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3475 = 5^{2} \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.308806 q^{2} +1.39811 q^{3} -1.90464 q^{4} +0.431745 q^{6} -4.16776 q^{7} -1.20578 q^{8} -1.04529 q^{9} +O(q^{10})\)	\(q+0.308806 q^{2} +1.39811 q^{3} -1.90464 q^{4} +0.431745 q^{6} -4.16776 q^{7} -1.20578 q^{8} -1.04529 q^{9} +1.93164 q^{11} -2.66290 q^{12} -4.12937 q^{13} -1.28703 q^{14} +3.43693 q^{16} -1.03363 q^{17} -0.322790 q^{18} -7.34157 q^{19} -5.82699 q^{21} +0.596503 q^{22} +4.09697 q^{23} -1.68581 q^{24} -1.27517 q^{26} -5.65576 q^{27} +7.93807 q^{28} +4.49067 q^{29} -1.28386 q^{31} +3.47289 q^{32} +2.70065 q^{33} -0.319192 q^{34} +1.99089 q^{36} +7.77849 q^{37} -2.26712 q^{38} -5.77331 q^{39} +11.5603 q^{41} -1.79941 q^{42} +7.87867 q^{43} -3.67908 q^{44} +1.26517 q^{46} +6.84669 q^{47} +4.80521 q^{48} +10.3702 q^{49} -1.44513 q^{51} +7.86495 q^{52} -3.68473 q^{53} -1.74653 q^{54} +5.02538 q^{56} -10.2643 q^{57} +1.38675 q^{58} +1.24534 q^{59} +0.281096 q^{61} -0.396462 q^{62} +4.35649 q^{63} -5.80140 q^{64} +0.833977 q^{66} +8.51902 q^{67} +1.96870 q^{68} +5.72803 q^{69} -6.40133 q^{71} +1.26038 q^{72} +5.92608 q^{73} +2.40204 q^{74} +13.9830 q^{76} -8.05062 q^{77} -1.78283 q^{78} -8.65375 q^{79} -4.77152 q^{81} +3.56988 q^{82} -15.8366 q^{83} +11.0983 q^{84} +2.43298 q^{86} +6.27846 q^{87} -2.32913 q^{88} -3.23844 q^{89} +17.2102 q^{91} -7.80326 q^{92} -1.79497 q^{93} +2.11430 q^{94} +4.85549 q^{96} +7.27325 q^{97} +3.20237 q^{98} -2.01912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 2 q^{3} + 9 q^{4} - 7 q^{6} + 5 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \) 7 * q - q^2 + 2 * q^3 + 9 * q^4 - 7 * q^6 + 5 * q^7 - 6 * q^8 + 13 * q^9	\( 7 q - q^{2} + 2 q^{3} + 9 q^{4} - 7 q^{6} + 5 q^{7} - 6 q^{8} + 13 q^{9} + 2 q^{11} + 8 q^{12} - 6 q^{13} + 7 q^{14} + 5 q^{16} - 5 q^{17} + 10 q^{18} - 10 q^{19} - 5 q^{21} + 18 q^{22} + q^{23} - 21 q^{24} - 8 q^{26} + 11 q^{27} + 28 q^{28} + 30 q^{29} - 20 q^{31} + 12 q^{32} + 20 q^{33} - 17 q^{34} + 2 q^{36} - 6 q^{37} - 6 q^{38} + 11 q^{39} + 19 q^{41} - 6 q^{42} + 12 q^{43} + 25 q^{44} + 22 q^{46} + 3 q^{47} - 15 q^{48} - 8 q^{49} + 23 q^{51} + 8 q^{52} - 38 q^{53} - 7 q^{54} + 21 q^{56} + 19 q^{57} + 21 q^{58} - 14 q^{59} + 4 q^{61} + q^{62} + 18 q^{63} - 16 q^{64} + 18 q^{66} - 9 q^{67} + 25 q^{68} + 9 q^{69} + 24 q^{71} - 41 q^{72} + 5 q^{73} + 9 q^{74} + 3 q^{76} + 13 q^{77} - 20 q^{78} + 8 q^{79} + 39 q^{81} - 56 q^{82} + 9 q^{83} - q^{84} + 39 q^{86} + 25 q^{87} + 29 q^{88} + 10 q^{89} + 7 q^{91} - 29 q^{92} + 15 q^{93} - 36 q^{94} - 11 q^{96} + 5 q^{97} + 49 q^{98} - 31 q^{99}+O(q^{100}) \) 7 * q - q^2 + 2 * q^3 + 9 * q^4 - 7 * q^6 + 5 * q^7 - 6 * q^8 + 13 * q^9 + 2 * q^11 + 8 * q^12 - 6 * q^13 + 7 * q^14 + 5 * q^16 - 5 * q^17 + 10 * q^18 - 10 * q^19 - 5 * q^21 + 18 * q^22 + q^23 - 21 * q^24 - 8 * q^26 + 11 * q^27 + 28 * q^28 + 30 * q^29 - 20 * q^31 + 12 * q^32 + 20 * q^33 - 17 * q^34 + 2 * q^36 - 6 * q^37 - 6 * q^38 + 11 * q^39 + 19 * q^41 - 6 * q^42 + 12 * q^43 + 25 * q^44 + 22 * q^46 + 3 * q^47 - 15 * q^48 - 8 * q^49 + 23 * q^51 + 8 * q^52 - 38 * q^53 - 7 * q^54 + 21 * q^56 + 19 * q^57 + 21 * q^58 - 14 * q^59 + 4 * q^61 + q^62 + 18 * q^63 - 16 * q^64 + 18 * q^66 - 9 * q^67 + 25 * q^68 + 9 * q^69 + 24 * q^71 - 41 * q^72 + 5 * q^73 + 9 * q^74 + 3 * q^76 + 13 * q^77 - 20 * q^78 + 8 * q^79 + 39 * q^81 - 56 * q^82 + 9 * q^83 - q^84 + 39 * q^86 + 25 * q^87 + 29 * q^88 + 10 * q^89 + 7 * q^91 - 29 * q^92 + 15 * q^93 - 36 * q^94 - 11 * q^96 + 5 * q^97 + 49 * q^98 - 31 * q^99

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