LMFDB - Embedded newform 637.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.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:	$5.08647060876$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.65544$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.65544 q^{2} -2.39593 q^{3} +5.05137 q^{4} -3.65544 q^{5} +6.36226 q^{6} -8.10275 q^{8} +2.74049 q^{9} +O(q^{10})$	\(q-2.65544 q^{2} -2.39593 q^{3} +5.05137 q^{4} -3.65544 q^{5} +6.36226 q^{6} -8.10275 q^{8} +2.74049 q^{9} +9.70682 q^{10} +0.655442 q^{11} -12.1027 q^{12} +1.00000 q^{13} +8.75819 q^{15} +11.4136 q^{16} -2.39593 q^{17} -7.27721 q^{18} +2.70682 q^{19} -18.4650 q^{20} -1.74049 q^{22} +7.36226 q^{23} +19.4136 q^{24} +8.36226 q^{25} -2.65544 q^{26} +0.621770 q^{27} -0.208136 q^{29} -23.2569 q^{30} +1.13642 q^{31} -14.1027 q^{32} -1.57040 q^{33} +6.36226 q^{34} +13.8432 q^{36} -7.44731 q^{37} -7.18780 q^{38} -2.39593 q^{39} +29.6191 q^{40} +10.2055 q^{41} -3.10275 q^{43} +3.31088 q^{44} -10.0177 q^{45} -19.5501 q^{46} -4.60407 q^{47} -27.3463 q^{48} -22.2055 q^{50} +5.74049 q^{51} +5.05137 q^{52} +5.25951 q^{53} -1.65107 q^{54} -2.39593 q^{55} -6.48535 q^{57} +0.552694 q^{58} -8.25951 q^{59} +44.2409 q^{60} +1.89725 q^{61} -3.01770 q^{62} +14.6218 q^{64} -3.65544 q^{65} +4.17009 q^{66} -12.8946 q^{67} -12.1027 q^{68} -17.6395 q^{69} -6.75819 q^{71} -22.2055 q^{72} -12.5367 q^{73} +19.7759 q^{74} -20.0354 q^{75} +13.6731 q^{76} +6.36226 q^{78} -1.51902 q^{79} -41.7219 q^{80} -9.71119 q^{81} -27.1001 q^{82} -15.7582 q^{83} +8.75819 q^{85} +8.23917 q^{86} +0.498680 q^{87} -5.31088 q^{88} +14.8096 q^{89} +26.6014 q^{90} +37.1895 q^{92} -2.72279 q^{93} +12.2258 q^{94} -9.89461 q^{95} +33.7892 q^{96} +10.0177 q^{97} +1.79623 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} - 6 q^{8} + 11 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 - 4 * q^3 + 6 * q^4 - 5 * q^5 - 2 * q^6 - 6 * q^8 + 11 * q^9	$ 3 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} - 6 q^{8} + 11 q^{9} + 14 q^{10} - 4 q^{11} - 18 q^{12} + 3 q^{13} + 2 q^{15} + 4 q^{16} - 4 q^{17} + 8 q^{18} - 7 q^{19} - 16 q^{20} - 8 q^{22} + q^{23} + 28 q^{24} + 4 q^{25} - 2 q^{26} - 22 q^{27} - 7 q^{29} - 24 q^{30} + 3 q^{31} - 24 q^{32} + 10 q^{33} - 2 q^{34} + 26 q^{36} - 10 q^{37} - 12 q^{38} - 4 q^{39} + 22 q^{40} - 6 q^{41} + 9 q^{43} - 2 q^{44} - 3 q^{45} - 28 q^{46} - 17 q^{47} - 16 q^{48} - 30 q^{50} + 20 q^{51} + 6 q^{52} + 13 q^{53} - 28 q^{54} - 4 q^{55} + 4 q^{57} + 14 q^{58} - 22 q^{59} + 42 q^{60} + 24 q^{61} + 18 q^{62} + 20 q^{64} - 5 q^{65} + 30 q^{66} - 14 q^{67} - 18 q^{68} - 2 q^{69} + 4 q^{71} - 30 q^{72} - 5 q^{73} + 8 q^{74} - 6 q^{75} + 8 q^{76} - 2 q^{78} + q^{79} - 40 q^{80} + 15 q^{81} - 20 q^{82} - 23 q^{83} + 2 q^{85} + 6 q^{86} - 20 q^{87} - 4 q^{88} + 11 q^{89} + 40 q^{90} + 30 q^{92} - 38 q^{93} + 16 q^{94} - 5 q^{95} + 52 q^{96} + 3 q^{97} - 30 q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 - 4 * q^3 + 6 * q^4 - 5 * q^5 - 2 * q^6 - 6 * q^8 + 11 * q^9 + 14 * q^10 - 4 * q^11 - 18 * q^12 + 3 * q^13 + 2 * q^15 + 4 * q^16 - 4 * q^17 + 8 * q^18 - 7 * q^19 - 16 * q^20 - 8 * q^22 + q^23 + 28 * q^24 + 4 * q^25 - 2 * q^26 - 22 * q^27 - 7 * q^29 - 24 * q^30 + 3 * q^31 - 24 * q^32 + 10 * q^33 - 2 * q^34 + 26 * q^36 - 10 * q^37 - 12 * q^38 - 4 * q^39 + 22 * q^40 - 6 * q^41 + 9 * q^43 - 2 * q^44 - 3 * q^45 - 28 * q^46 - 17 * q^47 - 16 * q^48 - 30 * q^50 + 20 * q^51 + 6 * q^52 + 13 * q^53 - 28 * q^54 - 4 * q^55 + 4 * q^57 + 14 * q^58 - 22 * q^59 + 42 * q^60 + 24 * q^61 + 18 * q^62 + 20 * q^64 - 5 * q^65 + 30 * q^66 - 14 * q^67 - 18 * q^68 - 2 * q^69 + 4 * q^71 - 30 * q^72 - 5 * q^73 + 8 * q^74 - 6 * q^75 + 8 * q^76 - 2 * q^78 + q^79 - 40 * q^80 + 15 * q^81 - 20 * q^82 - 23 * q^83 + 2 * q^85 + 6 * q^86 - 20 * q^87 - 4 * q^88 + 11 * q^89 + 40 * q^90 + 30 * q^92 - 38 * q^93 + 16 * q^94 - 5 * q^95 + 52 * q^96 + 3 * q^97 - 30 * 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 $

$2$	−2.65544	−1.87768	−0.938841	−	0.344352i	$-0.888099\pi$
$2$	−2.65544	−1.87768	−0.938841	+	0.344352i	$0.888099\pi$
$3$	−2.39593	−1.38329	−0.691646	−	0.722237i	$-0.743114\pi$
$3$	−2.39593	−1.38329	−0.691646	+	0.722237i	$0.743114\pi$
$4$	5.05137	2.52569
$5$	−3.65544	−1.63476	−0.817382	−	0.576096i	$-0.804575\pi$
$5$	−3.65544	−1.63476	−0.817382	+	0.576096i	$0.804575\pi$
$6$	6.36226	2.59738
$7$	0	0
$7$	0	0
$8$	−8.10275	−2.86475
$9$	2.74049	0.913496
$10$	9.70682	3.06956
$11$	0.655442	0.197623	0.0988117	−	0.995106i	$-0.468496\pi$
$11$	0.655442	0.197623	0.0988117	+	0.995106i	$0.468496\pi$
$12$	−12.1027	−3.49376
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	8.75819	2.26136
$16$	11.4136	2.85341
$17$	−2.39593	−0.581099	−0.290549	−	0.956860i	$-0.593838\pi$
$17$	−2.39593	−0.581099	−0.290549	+	0.956860i	$0.593838\pi$
$18$	−7.27721	−1.71526
$19$	2.70682	0.620986	0.310493	−	0.950576i	$-0.399506\pi$
$19$	2.70682	0.620986	0.310493	+	0.950576i	$0.399506\pi$
$20$	−18.4650	−4.12890
$21$	0	0
$22$	−1.74049	−0.371074
$23$	7.36226	1.53514	0.767569	−	0.640967i	$-0.221466\pi$
$23$	7.36226	1.53514	0.767569	+	0.640967i	$0.221466\pi$
$24$	19.4136	3.96279
$25$	8.36226	1.67245
$26$	−2.65544	−0.520775
$27$	0.621770	0.119660
$28$	0	0
$29$	−0.208136	−0.0386499	−0.0193250	−	0.999813i	$-0.506152\pi$
$29$	−0.208136	−0.0386499	−0.0193250	+	0.999813i	$0.506152\pi$
$30$	−23.2569	−4.24610
$31$	1.13642	0.204107	0.102054	−	0.994779i	$-0.467459\pi$
$31$	1.13642	0.204107	0.102054	+	0.994779i	$0.467459\pi$
$32$	−14.1027	−2.49304
$33$	−1.57040	−0.273371
$34$	6.36226	1.09112
$35$	0	0
$36$	13.8432	2.30721
$37$	−7.44731	−1.22433	−0.612165	−	0.790730i	$-0.709701\pi$
$37$	−7.44731	−1.22433	−0.612165	+	0.790730i	$0.709701\pi$
$38$	−7.18780	−1.16601
$39$	−2.39593	−0.383656
$40$	29.6191	4.68320
$41$	10.2055	1.59383	0.796915	−	0.604091i	$-0.206464\pi$
$41$	10.2055	1.59383	0.796915	+	0.604091i	$0.206464\pi$
$42$	0	0
$43$	−3.10275	−0.473165	−0.236582	−	0.971611i	$-0.576027\pi$
$43$	−3.10275	−0.473165	−0.236582	+	0.971611i	$0.576027\pi$
$44$	3.31088	0.499135
$45$	−10.0177	−1.49335
$46$	−19.5501	−2.88250
$47$	−4.60407	−0.671572	−0.335786	−	0.941938i	$-0.609002\pi$
$47$	−4.60407	−0.671572	−0.335786	+	0.941938i	$0.609002\pi$
$48$	−27.3463	−3.94710
$49$	0	0
$50$	−22.2055	−3.14033
$51$	5.74049	0.803829
$52$	5.05137	0.700500
$53$	5.25951	0.722449	0.361225	−	0.932479i	$-0.382359\pi$
$53$	5.25951	0.722449	0.361225	+	0.932479i	$0.382359\pi$
$54$	−1.65107	−0.224683
$55$	−2.39593	−0.323067
$56$	0	0
$57$	−6.48535	−0.859005
$58$	0.552694	0.0725723
$59$	−8.25951	−1.07530	−0.537648	−	0.843169i	$-0.680687\pi$
$59$	−8.25951	−1.07530	−0.537648	+	0.843169i	$0.680687\pi$
$60$	44.2409	5.71148
$61$	1.89725	0.242918	0.121459	−	0.992596i	$-0.461243\pi$
$61$	1.89725	0.242918	0.121459	+	0.992596i	$0.461243\pi$
$62$	−3.01770	−0.383248
$63$	0	0
$64$	14.6218	1.82772
$65$	−3.65544	−0.453402
$66$	4.17009	0.513303
$67$	−12.8946	−1.57533	−0.787664	−	0.616105i	$-0.788710\pi$
$67$	−12.8946	−1.57533	−0.787664	+	0.616105i	$0.788710\pi$
$68$	−12.1027	−1.46767
$69$	−17.6395	−2.12354
$70$	0	0
$71$	−6.75819	−0.802050	−0.401025	−	0.916067i	$-0.631346\pi$
$71$	−6.75819	−0.802050	−0.401025	+	0.916067i	$0.631346\pi$
$72$	−22.2055	−2.61694
$73$	−12.5367	−1.46731	−0.733656	−	0.679521i	$-0.762188\pi$
$73$	−12.5367	−1.46731	−0.733656	+	0.679521i	$0.762188\pi$
$74$	19.7759	2.29890
$75$	−20.0354	−2.31349
$76$	13.6731	1.56842
$77$	0	0
$78$	6.36226	0.720384
$79$	−1.51902	−0.170903	−0.0854516	−	0.996342i	$-0.527233\pi$
$79$	−1.51902	−0.170903	−0.0854516	+	0.996342i	$0.527233\pi$
$80$	−41.7219	−4.66465
$81$	−9.71119	−1.07902
$82$	−27.1001	−2.99271
$83$	−15.7582	−1.72969	−0.864843	−	0.502042i	$-0.832582\pi$
$83$	−15.7582	−1.72969	−0.864843	+	0.502042i	$0.832582\pi$
$84$	0	0
$85$	8.75819	0.949959
$86$	8.23917	0.888453
$87$	0.498680	0.0534641
$88$	−5.31088	−0.566142
$89$	14.8096	1.56981	0.784905	−	0.619616i	$-0.212712\pi$
$89$	14.8096	1.56981	0.784905	+	0.619616i	$0.212712\pi$
$90$	26.6014	2.80404
$91$	0	0
$92$	37.1895	3.87728
$93$	−2.72279	−0.282340
$94$	12.2258	1.26100
$95$	−9.89461	−1.01517
$96$	33.7892	3.44860
$97$	10.0177	1.01714	0.508572	−	0.861020i	$-0.330174\pi$
$97$	10.0177	1.01714	0.508572	+	0.861020i	$0.330174\pi$
$98$	0	0
$99$	1.79623	0.180528
$100$	42.2409	4.22409
$101$	3.01770	0.300273	0.150136	−	0.988665i	$-0.452029\pi$
$101$	3.01770	0.300273	0.150136	+	0.988665i	$0.452029\pi$
$102$	−15.2435	−1.50934
$103$	5.03804	0.496413	0.248207	−	0.968707i	$-0.420159\pi$
$103$	5.03804	0.496413	0.248207	+	0.968707i	$0.420159\pi$
$104$	−8.10275	−0.794540
$105$	0	0
$106$	−13.9663	−1.35653
$107$	11.8432	1.14493	0.572465	−	0.819929i	$-0.305987\pi$
$107$	11.8432	1.14493	0.572465	+	0.819929i	$0.305987\pi$
$108$	3.14079	0.302223
$109$	3.55005	0.340034	0.170017	−	0.985441i	$-0.445618\pi$
$109$	3.55005	0.340034	0.170017	+	0.985441i	$0.445618\pi$
$110$	6.36226	0.606618
$111$	17.8432	1.69361
$112$	0	0
$113$	−9.46501	−0.890393	−0.445197	−	0.895433i	$-0.646866\pi$
$113$	−9.46501	−0.890393	−0.445197	+	0.895433i	$0.646866\pi$
$114$	17.2215	1.61294
$115$	−26.9123	−2.50959
$116$	−1.05137	−0.0976176
$117$	2.74049	0.253358
$118$	21.9327	2.01906
$119$	0	0
$120$	−70.9654	−6.47823
$121$	−10.5704	−0.960945
$122$	−5.03804	−0.456123
$123$	−24.4517	−2.20473
$124$	5.74049	0.515511
$125$	−12.2905	−1.09930
$126$	0	0
$127$	5.46765	0.485175	0.242588	−	0.970130i	$-0.422004\pi$
$127$	5.46765	0.485175	0.242588	+	0.970130i	$0.422004\pi$
$128$	−10.6218	−0.938841
$129$	7.43397	0.654525
$130$	9.70682	0.851344
$131$	−9.82991	−0.858843	−0.429421	−	0.903104i	$-0.641282\pi$
$131$	−9.82991	−0.858843	−0.429421	+	0.903104i	$0.641282\pi$
$132$	−7.93265	−0.690449
$133$	0	0
$134$	34.2409	2.95796
$135$	−2.27284	−0.195615
$136$	19.4136	1.66471
$137$	17.5501	1.49940	0.749701	−	0.661777i	$-0.230197\pi$
$137$	17.5501	1.49940	0.749701	+	0.661777i	$0.230197\pi$
$138$	46.8406	3.98734
$139$	4.91495	0.416881	0.208440	−	0.978035i	$-0.433161\pi$
$139$	4.91495	0.416881	0.208440	+	0.978035i	$0.433161\pi$
$140$	0	0
$141$	11.0310	0.928981
$142$	17.9460	1.50599
$143$	0.655442	0.0548108
$144$	31.2789	2.60658
$145$	0.760830	0.0631835
$146$	33.2905	2.75515
$147$	0	0
$148$	−37.6191	−3.09227
$149$	−10.3419	−0.847243	−0.423621	−	0.905839i	$-0.639241\pi$
$149$	−10.3419	−0.847243	−0.423621	+	0.905839i	$0.639241\pi$
$150$	53.2029	4.34400
$151$	5.07171	0.412730	0.206365	−	0.978475i	$-0.433837\pi$
$151$	5.07171	0.412730	0.206365	+	0.978475i	$0.433837\pi$
$152$	−21.9327	−1.77897
$153$	−6.56603	−0.530832
$154$	0	0
$155$	−4.15412	−0.333667
$156$	−12.1027	−0.968995
$157$	12.6014	1.00570	0.502852	−	0.864373i	$-0.332284\pi$
$157$	12.6014	1.00570	0.502852	+	0.864373i	$0.332284\pi$
$158$	4.03367	0.320902
$159$	−12.6014	−0.999358
$160$	51.5518	4.07553
$161$	0	0
$162$	25.7875	2.02606
$163$	3.20814	0.251281	0.125640	−	0.992076i	$-0.459901\pi$
$163$	3.20814	0.251281	0.125640	+	0.992076i	$0.459901\pi$
$164$	51.5518	4.02552
$165$	5.74049	0.446896
$166$	41.8450	3.24780
$167$	−8.12045	−0.628379	−0.314190	−	0.949360i	$-0.601733\pi$
$167$	−8.12045	−0.628379	−0.314190	+	0.949360i	$0.601733\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−23.2569	−1.78372
$171$	7.41800	0.567269
$172$	−15.6731	−1.19507
$173$	10.3286	0.785268	0.392634	−	0.919695i	$-0.371564\pi$
$173$	10.3286	0.785268	0.392634	+	0.919695i	$0.371564\pi$
$174$	−1.32422	−0.100389
$175$	0	0
$176$	7.48098	0.563900
$177$	19.7892	1.48745
$178$	−39.3259	−2.94760
$179$	−2.37823	−0.177757	−0.0888786	−	0.996042i	$-0.528328\pi$
$179$	−2.37823	−0.177757	−0.0888786	+	0.996042i	$0.528328\pi$
$180$	−50.6032	−3.77174
$181$	−21.8096	−1.62109	−0.810546	−	0.585675i	$-0.800830\pi$
$181$	−21.8096	−1.62109	−0.810546	+	0.585675i	$0.800830\pi$
$182$	0	0
$183$	−4.54569	−0.336027
$184$	−59.6545	−4.39779
$185$	27.2232	2.00149
$186$	7.23021	0.530144
$187$	−1.57040	−0.114839
$188$	−23.2569	−1.69618
$189$	0	0
$190$	26.2746	1.90616
$191$	−25.0868	−1.81522	−0.907608	−	0.419819i	$-0.862093\pi$
$191$	−25.0868	−1.81522	−0.907608	+	0.419819i	$0.862093\pi$
$192$	−35.0328	−2.52827
$193$	11.3109	0.814175	0.407088	−	0.913389i	$-0.366544\pi$
$193$	11.3109	0.814175	0.407088	+	0.913389i	$0.366544\pi$
$194$	−26.6014	−1.90987
$195$	8.75819	0.627187
$196$	0	0
$197$	−16.7919	−1.19637	−0.598185	−	0.801358i	$-0.704111\pi$
$197$	−16.7919	−1.19637	−0.598185	+	0.801358i	$0.704111\pi$
$198$	−4.76979	−0.338974
$199$	−20.5341	−1.45562	−0.727811	−	0.685777i	$-0.759462\pi$
$199$	−20.5341	−1.45562	−0.727811	+	0.685777i	$0.759462\pi$
$200$	−67.7573	−4.79116
$201$	30.8946	2.17914
$202$	−8.01333	−0.563816
$203$	0	0
$204$	28.9974	2.03022
$205$	−37.3056	−2.60554
$206$	−13.3782	−0.932105
$207$	20.1762	1.40234
$208$	11.4136	0.791393
$209$	1.77416	0.122721
$210$	0	0
$211$	−15.7785	−1.08624	−0.543119	−	0.839655i	$-0.682757\pi$
$211$	−15.7785	−1.08624	−0.543119	+	0.839655i	$0.682757\pi$
$212$	26.5678	1.82468
$213$	16.1922	1.10947
$214$	−31.4490	−2.14981
$215$	11.3419	0.773512
$216$	−5.03804	−0.342795
$217$	0	0
$218$	−9.42697	−0.638475
$219$	30.0371	2.02972
$220$	−12.1027	−0.815967
$221$	−2.39593	−0.161168
$222$	−47.3817	−3.18005
$223$	8.44731	0.565673	0.282837	−	0.959168i	$-0.408725\pi$
$223$	8.44731	0.565673	0.282837	+	0.959168i	$0.408725\pi$
$224$	0	0
$225$	22.9167	1.52778
$226$	25.1338	1.67187
$227$	−6.68912	−0.443972	−0.221986	−	0.975050i	$-0.571254\pi$
$227$	−6.68912	−0.443972	−0.221986	+	0.975050i	$0.571254\pi$
$228$	−32.7599	−2.16958
$229$	−8.63510	−0.570624	−0.285312	−	0.958435i	$-0.592097\pi$
$229$	−8.63510	−0.570624	−0.285312	+	0.958435i	$0.592097\pi$
$230$	71.4641	4.71220
$231$	0	0
$232$	1.68648	0.110723
$233$	−4.16745	−0.273019	−0.136510	−	0.990639i	$-0.543588\pi$
$233$	−4.16745	−0.273019	−0.136510	+	0.990639i	$0.543588\pi$
$234$	−7.27721	−0.475726
$235$	16.8299	1.09786
$236$	−41.7219	−2.71586
$237$	3.63947	0.236409
$238$	0	0
$239$	−1.79450	−0.116077	−0.0580384	−	0.998314i	$-0.518485\pi$
$239$	−1.79450	−0.116077	−0.0580384	+	0.998314i	$0.518485\pi$
$240$	99.9628	6.45257
$241$	−13.8609	−0.892862	−0.446431	−	0.894818i	$-0.647305\pi$
$241$	−13.8609	−0.892862	−0.446431	+	0.894818i	$0.647305\pi$
$242$	28.0691	1.80435
$243$	21.4020	1.37294
$244$	9.58373	0.613535
$245$	0	0
$246$	64.9300	4.13979
$247$	2.70682	0.172231
$248$	−9.20814	−0.584717
$249$	37.7556	2.39266
$250$	32.6368	2.06413
$251$	−14.7449	−0.930687	−0.465344	−	0.885130i	$-0.654069\pi$
$251$	−14.7449	−0.930687	−0.465344	+	0.885130i	$0.654069\pi$
$252$	0	0
$253$	4.82554	0.303379
$254$	−14.5190	−0.911004
$255$	−20.9840	−1.31407
$256$	−1.03804	−0.0648776
$257$	−23.7068	−1.47879	−0.739395	−	0.673272i	$-0.764888\pi$
$257$	−23.7068	−1.47879	−0.739395	+	0.673272i	$0.764888\pi$
$258$	−19.7405	−1.22899
$259$	0	0
$260$	−18.4650	−1.14515
$261$	−0.570395	−0.0353066
$262$	26.1027	1.61263
$263$	11.3756	0.701449	0.350725	−	0.936479i	$-0.385935\pi$
$263$	11.3756	0.701449	0.350725	+	0.936479i	$0.385935\pi$
$264$	12.7245	0.783140
$265$	−19.2258	−1.18103
$266$	0	0
$267$	−35.4827	−2.17151
$268$	−65.1355	−3.97878
$269$	11.1054	0.677107	0.338554	−	0.940947i	$-0.390062\pi$
$269$	11.1054	0.677107	0.338554	+	0.940947i	$0.390062\pi$
$270$	6.03540	0.367303
$271$	12.7245	0.772959	0.386480	−	0.922298i	$-0.373691\pi$
$271$	12.7245	0.772959	0.386480	+	0.922298i	$0.373691\pi$
$272$	−27.3463	−1.65811
$273$	0	0
$274$	−46.6032	−2.81540
$275$	5.48098	0.330515
$276$	−89.1036	−5.36340
$277$	−3.00000	−0.180253	−0.0901263	−	0.995930i	$-0.528727\pi$
$277$	−3.00000	−0.180253	−0.0901263	+	0.995930i	$0.528727\pi$
$278$	−13.0514	−0.782769
$279$	3.11435	0.186451
$280$	0	0
$281$	3.44731	0.205649	0.102825	−	0.994700i	$-0.467212\pi$
$281$	3.44731	0.205649	0.102825	+	0.994700i	$0.467212\pi$
$282$	−29.2923	−1.74433
$283$	12.4783	0.741760	0.370880	−	0.928681i	$-0.379056\pi$
$283$	12.4783	0.741760	0.370880	+	0.928681i	$0.379056\pi$
$284$	−34.1382	−2.02573
$285$	23.7068	1.40427
$286$	−1.74049	−0.102917
$287$	0	0
$288$	−38.6484	−2.27738
$289$	−11.2595	−0.662324
$290$	−2.02034	−0.118638
$291$	−24.0017	−1.40701
$292$	−63.3277	−3.70597
$293$	13.5341	0.790670	0.395335	−	0.918537i	$-0.370629\pi$
$293$	13.5341	0.790670	0.395335	+	0.918537i	$0.370629\pi$
$294$	0	0
$295$	30.1922	1.75786
$296$	60.3436	3.50740
$297$	0.407534	0.0236475
$298$	27.4624	1.59085
$299$	7.36226	0.425770
$300$	−101.206	−5.84315
$301$	0	0
$302$	−13.4676	−0.774976
$303$	−7.23021	−0.415365
$304$	30.8946	1.77193
$305$	−6.93529	−0.397114
$306$	17.4357	0.996733
$307$	−28.2365	−1.61154	−0.805772	−	0.592226i	$-0.798249\pi$
$307$	−28.2365	−1.61154	−0.805772	+	0.592226i	$0.798249\pi$
$308$	0	0
$309$	−12.0708	−0.686684
$310$	11.0310	0.626521
$311$	24.7042	1.40085	0.700423	−	0.713728i	$-0.252995\pi$
$311$	24.7042	1.40085	0.700423	+	0.713728i	$0.252995\pi$
$312$	19.4136	1.09908
$313$	20.8122	1.17638	0.588188	−	0.808724i	$-0.299842\pi$
$313$	20.8122	1.17638	0.588188	+	0.808724i	$0.299842\pi$
$314$	−33.4624	−1.88839
$315$	0	0
$316$	−7.67314	−0.431648
$317$	26.1382	1.46806	0.734032	−	0.679114i	$-0.237636\pi$
$317$	26.1382	1.46806	0.734032	+	0.679114i	$0.237636\pi$
$318$	33.4624	1.87648
$319$	−0.136421	−0.00763813
$320$	−53.4490	−2.98789
$321$	−28.3756	−1.58377
$322$	0	0
$323$	−6.48535	−0.360854
$324$	−49.0548	−2.72527
$325$	8.36226	0.463855
$326$	−8.51902	−0.471825
$327$	−8.50569	−0.470366
$328$	−82.6926	−4.56593
$329$	0	0
$330$	−15.2435	−0.839129
$331$	−24.1382	−1.32675	−0.663376	−	0.748286i	$-0.730877\pi$
$331$	−24.1382	−1.32675	−0.663376	+	0.748286i	$0.730877\pi$
$332$	−79.6005	−4.36865
$333$	−20.4093	−1.11842
$334$	21.5634	1.17990
$335$	47.1355	2.57529
$336$	0	0
$337$	−30.5297	−1.66306	−0.831530	−	0.555480i	$-0.812534\pi$
$337$	−30.5297	−1.66306	−0.831530	+	0.555480i	$0.812534\pi$
$338$	−2.65544	−0.144437
$339$	22.6775	1.23167
$340$	44.2409	2.39930
$341$	0.744859	0.0403364
$342$	−19.6981	−1.06515
$343$	0	0
$344$	25.1408	1.35550
$345$	64.4801	3.47149
$346$	−27.4270	−1.47448
$347$	−24.9974	−1.34193	−0.670964	−	0.741490i	$-0.734120\pi$
$347$	−24.9974	−1.34193	−0.670964	+	0.741490i	$0.734120\pi$
$348$	2.51902	0.135034
$349$	1.83887	0.0984324	0.0492162	−	0.998788i	$-0.484328\pi$
$349$	1.83887	0.0984324	0.0492162	+	0.998788i	$0.484328\pi$
$350$	0	0
$351$	0.621770	0.0331876
$352$	−9.24354	−0.492682
$353$	23.2569	1.23784	0.618919	−	0.785455i	$-0.287571\pi$
$353$	23.2569	1.23784	0.618919	+	0.785455i	$0.287571\pi$
$354$	−52.5491	−2.79296
$355$	24.7042	1.31116
$356$	74.8087	3.96485
$357$	0	0
$358$	6.31525	0.333772
$359$	−21.4473	−1.13195	−0.565973	−	0.824424i	$-0.691499\pi$
$359$	−21.4473	−1.13195	−0.565973	+	0.824424i	$0.691499\pi$
$360$	81.1709	4.27808
$361$	−11.6731	−0.614376
$362$	57.9140	3.04389
$363$	25.3259	1.32927
$364$	0	0
$365$	45.8273	2.39871
$366$	12.0708	0.630951
$367$	1.12045	0.0584870	0.0292435	−	0.999572i	$-0.490690\pi$
$367$	1.12045	0.0584870	0.0292435	+	0.999572i	$0.490690\pi$
$368$	84.0301	4.38037
$369$	27.9681	1.45596
$370$	−72.2896	−3.75816
$371$	0	0
$372$	−13.7538	−0.713102
$373$	15.6058	0.808038	0.404019	−	0.914751i	$-0.367613\pi$
$373$	15.6058	0.808038	0.404019	+	0.914751i	$0.367613\pi$
$374$	4.17009	0.215630
$375$	29.4473	1.52065
$376$	37.3056	1.92389
$377$	−0.208136	−0.0107196
$378$	0	0
$379$	12.7849	0.656714	0.328357	−	0.944554i	$-0.393505\pi$
$379$	12.7849	0.656714	0.328357	+	0.944554i	$0.393505\pi$
$380$	−49.9814	−2.56399
$381$	−13.1001	−0.671139
$382$	66.6165	3.40840
$383$	−34.0354	−1.73913	−0.869564	−	0.493820i	$-0.835600\pi$
$383$	−34.0354	−1.73913	−0.869564	+	0.493820i	$0.835600\pi$
$384$	25.4490	1.29869
$385$	0	0
$386$	−30.0354	−1.52876
$387$	−8.50305	−0.432234
$388$	50.6032	2.56899
$389$	−24.6705	−1.25084	−0.625422	−	0.780287i	$-0.715073\pi$
$389$	−24.6705	−1.25084	−0.625422	+	0.780287i	$0.715073\pi$
$390$	−23.2569	−1.17766
$391$	−17.6395	−0.892066
$392$	0	0
$393$	23.5518	1.18803
$394$	44.5898	2.24640
$395$	5.55269	0.279386
$396$	9.07344	0.455958
$397$	4.97966	0.249922	0.124961	−	0.992162i	$-0.460119\pi$
$397$	4.97966	0.249922	0.124961	+	0.992162i	$0.460119\pi$
$398$	54.5271	2.73320
$399$	0	0
$400$	95.4438	4.77219
$401$	0.689115	0.0344128	0.0172064	−	0.999852i	$-0.494523\pi$
$401$	0.689115	0.0344128	0.0172064	+	0.999852i	$0.494523\pi$
$402$	−82.0389	−4.09173
$403$	1.13642	0.0566092
$404$	15.2435	0.758394
$405$	35.4987	1.76394
$406$	0	0
$407$	−4.88128	−0.241956
$408$	−46.5137	−2.30277
$409$	−19.9770	−0.987800	−0.493900	−	0.869519i	$-0.664429\pi$
$409$	−19.9770	−0.987800	−0.493900	+	0.869519i	$0.664429\pi$
$410$	99.0629	4.89237
$411$	−42.0487	−2.07411
$412$	25.4490	1.25378
$413$	0	0
$414$	−53.5767	−2.63315
$415$	57.6032	2.82763
$416$	−14.1027	−0.691444
$417$	−11.7759	−0.576668
$418$	−4.71119	−0.230432
$419$	−19.6661	−0.960754	−0.480377	−	0.877062i	$-0.659500\pi$
$419$	−19.6661	−0.960754	−0.480377	+	0.877062i	$0.659500\pi$
$420$	0	0
$421$	14.9283	0.727560	0.363780	−	0.931485i	$-0.381486\pi$
$421$	14.9283	0.727560	0.363780	+	0.931485i	$0.381486\pi$
$422$	41.8990	2.03961
$423$	−12.6174	−0.613479
$424$	−42.6165	−2.06964
$425$	−20.0354	−0.971860
$426$	−42.9974	−2.08323
$427$	0	0
$428$	59.8246	2.89173
$429$	−1.57040	−0.0758194
$430$	−30.1178	−1.45241
$431$	−32.6838	−1.57433	−0.787163	−	0.616746i	$-0.788451\pi$
$431$	−32.6838	−1.57433	−0.787163	+	0.616746i	$0.788451\pi$
$432$	7.09665	0.341438
$433$	8.96196	0.430684	0.215342	−	0.976539i	$-0.430913\pi$
$433$	8.96196	0.430684	0.215342	+	0.976539i	$0.430913\pi$
$434$	0	0
$435$	−1.82290	−0.0874012
$436$	17.9327	0.858818
$437$	19.9283	0.953299
$438$	−79.7619	−3.81117
$439$	7.93265	0.378605	0.189302	−	0.981919i	$-0.439377\pi$
$439$	7.93265	0.378605	0.189302	+	0.981919i	$0.439377\pi$
$440$	19.4136	0.925509
$441$	0	0
$442$	6.36226	0.302622
$443$	−8.91058	−0.423355	−0.211677	−	0.977340i	$-0.567893\pi$
$443$	−8.91058	−0.423355	−0.211677	+	0.977340i	$0.567893\pi$
$444$	90.1329	4.27752
$445$	−54.1355	−2.56627
$446$	−22.4313	−1.06215
$447$	24.7785	1.17198
$448$	0	0
$449$	−8.45168	−0.398859	−0.199430	−	0.979912i	$-0.563909\pi$
$449$	−8.45168	−0.398859	−0.199430	+	0.979912i	$0.563909\pi$
$450$	−60.8539	−2.86868
$451$	6.68912	0.314978
$452$	−47.8113	−2.24885
$453$	−12.1515	−0.570926
$454$	17.7626	0.833638
$455$	0	0
$456$	52.5491	2.46084
$457$	23.1692	1.08381	0.541904	−	0.840440i	$-0.317703\pi$
$457$	23.1692	1.08381	0.541904	+	0.840440i	$0.317703\pi$
$458$	22.9300	1.07145
$459$	−1.48972	−0.0695341
$460$	−135.944	−6.33843
$461$	−2.27284	−0.105857	−0.0529284	−	0.998598i	$-0.516856\pi$
$461$	−2.27284	−0.105857	−0.0529284	+	0.998598i	$0.516856\pi$
$462$	0	0
$463$	−4.10976	−0.190997	−0.0954983	−	0.995430i	$-0.530444\pi$
$463$	−4.10976	−0.190997	−0.0954983	+	0.995430i	$0.530444\pi$
$464$	−2.37559	−0.110284
$465$	9.95299	0.461559
$466$	11.0664	0.512643
$467$	−32.9150	−1.52312	−0.761561	−	0.648093i	$-0.775567\pi$
$467$	−32.9150	−1.52312	−0.761561	+	0.648093i	$0.775567\pi$
$468$	13.8432	0.639904
$469$	0	0
$470$	−44.6908	−2.06143
$471$	−30.1922	−1.39118
$472$	66.9247	3.08046
$473$	−2.03367	−0.0935084
$474$	−9.66441	−0.443901
$475$	22.6351	1.03857
$476$	0	0
$477$	14.4136	0.659955
$478$	4.76520	0.217955
$479$	9.31525	0.425625	0.212812	−	0.977093i	$-0.431738\pi$
$479$	9.31525	0.425625	0.212812	+	0.977093i	$0.431738\pi$
$480$	−123.515	−5.63764
$481$	−7.44731	−0.339568
$482$	36.8069	1.67651
$483$	0	0
$484$	−53.3950	−2.42705
$485$	−36.6191	−1.66279
$486$	−56.8319	−2.57795
$487$	20.2409	0.917203	0.458601	−	0.888642i	$-0.348351\pi$
$487$	20.2409	0.917203	0.458601	+	0.888642i	$0.348351\pi$
$488$	−15.3730	−0.695901
$489$	−7.68648	−0.347594
$490$	0	0
$491$	−4.36226	−0.196866	−0.0984330	−	0.995144i	$-0.531383\pi$
$491$	−4.36226	−0.196866	−0.0984330	+	0.995144i	$0.531383\pi$
$492$	−123.515	−5.56847
$493$	0.498680	0.0224594
$494$	−7.18780	−0.323394
$495$	−6.56603	−0.295121
$496$	12.9707	0.582401
$497$	0	0
$498$	−100.258	−4.49265
$499$	9.69348	0.433940	0.216970	−	0.976178i	$-0.430383\pi$
$499$	9.69348	0.433940	0.216970	+	0.976178i	$0.430383\pi$
$500$	−62.0841	−2.77649
$501$	19.4560	0.869232
$502$	39.1541	1.74753
$503$	−2.64843	−0.118088	−0.0590439	−	0.998255i	$-0.518805\pi$
$503$	−2.64843	−0.118088	−0.0590439	+	0.998255i	$0.518805\pi$
$504$	0	0
$505$	−11.0310	−0.490875
$506$	−12.8139	−0.569649
$507$	−2.39593	−0.106407
$508$	27.6191	1.22540
$509$	13.9416	0.617951	0.308976	−	0.951070i	$-0.400014\pi$
$509$	13.9416	0.617951	0.308976	+	0.951070i	$0.400014\pi$
$510$	55.7219	2.46741
$511$	0	0
$512$	24.0000	1.06066
$513$	1.68302	0.0743070
$514$	62.9521	2.77670
$515$	−18.4163	−0.811518
$516$	37.5518	1.65313
$517$	−3.01770	−0.132718
$518$	0	0
$519$	−24.7466	−1.08625
$520$	29.6191	1.29888
$521$	14.6218	0.640591	0.320296	−	0.947318i	$-0.396218\pi$
$521$	14.6218	0.640591	0.320296	+	0.947318i	$0.396218\pi$
$522$	1.51465	0.0662945
$523$	−16.5190	−0.722326	−0.361163	−	0.932503i	$-0.617620\pi$
$523$	−16.5190	−0.722326	−0.361163	+	0.932503i	$0.617620\pi$
$524$	−49.6545	−2.16917
$525$	0	0
$526$	−30.2072	−1.31710
$527$	−2.72279	−0.118607
$528$	−17.9239	−0.780038
$529$	31.2029	1.35665
$530$	51.0531	2.21761
$531$	−22.6351	−0.982280
$532$	0	0
$533$	10.2055	0.442049
$534$	94.2223	4.07740
$535$	−43.2923	−1.87169
$536$	104.482	4.51293
$537$	5.69808	0.245890
$538$	−29.4897	−1.27139
$539$	0	0
$540$	−11.4810	−0.494063
$541$	−43.1018	−1.85309	−0.926546	−	0.376181i	$-0.877237\pi$
$541$	−43.1018	−1.85309	−0.926546	+	0.376181i	$0.877237\pi$
$542$	−33.7892	−1.45137
$543$	52.2542	2.24244
$544$	33.7892	1.44870
$545$	−12.9770	−0.555874
$546$	0	0
$547$	−13.5057	−0.577462	−0.288731	−	0.957410i	$-0.593233\pi$
$547$	−13.5057	−0.577462	−0.288731	+	0.957410i	$0.593233\pi$
$548$	88.6519	3.78702
$549$	5.19940	0.221905
$550$	−14.5544	−0.620603
$551$	−0.563387	−0.0240011
$552$	142.928	6.08343
$553$	0	0
$554$	7.96633	0.338457
$555$	−65.2249	−2.76864
$556$	24.8273	1.05291
$557$	1.35157	0.0572677	0.0286338	−	0.999590i	$-0.490884\pi$
$557$	1.35157	0.0572677	0.0286338	+	0.999590i	$0.490884\pi$
$558$	−8.26998	−0.350096
$559$	−3.10275	−0.131232
$560$	0	0
$561$	3.76256	0.158855
$562$	−9.15412	−0.386143
$563$	−31.7626	−1.33863	−0.669316	−	0.742978i	$-0.733413\pi$
$563$	−31.7626	−1.33863	−0.669316	+	0.742978i	$0.733413\pi$
$564$	55.7219	2.34631
$565$	34.5988	1.45558
$566$	−33.1355	−1.39279
$567$	0	0
$568$	54.7599	2.29768
$569$	30.3730	1.27330	0.636650	−	0.771153i	$-0.280320\pi$
$569$	30.3730	1.27330	0.636650	+	0.771153i	$0.280320\pi$
$570$	−62.9521	−2.63677
$571$	−0.432244	−0.0180888	−0.00904442	−	0.999959i	$-0.502879\pi$
$571$	−0.432244	−0.0180888	−0.00904442	+	0.999959i	$0.502879\pi$
$572$	3.31088	0.138435
$573$	60.1062	2.51097
$574$	0	0
$575$	61.5651	2.56744
$576$	40.0708	1.66962
$577$	−18.1382	−0.755101	−0.377551	−	0.925989i	$-0.623234\pi$
$577$	−18.1382	−0.755101	−0.377551	+	0.925989i	$0.623234\pi$
$578$	29.8990	1.24363
$579$	−27.1001	−1.12624
$580$	3.84324	0.159582
$581$	0	0
$582$	63.7352	2.64191
$583$	3.44731	0.142773
$584$	101.582	4.20349
$585$	−10.0177	−0.414181
$586$	−35.9390	−1.48463
$587$	−19.3065	−0.796865	−0.398433	−	0.917198i	$-0.630446\pi$
$587$	−19.3065	−0.796865	−0.398433	+	0.917198i	$0.630446\pi$
$588$	0	0
$589$	3.07608	0.126748
$590$	−80.1736	−3.30069
$591$	40.2322	1.65493
$592$	−85.0008	−3.49351
$593$	−8.20113	−0.336780	−0.168390	−	0.985720i	$-0.553857\pi$
$593$	−8.20113	−0.336780	−0.168390	+	0.985720i	$0.553857\pi$
$594$	−1.08218	−0.0444025
$595$	0	0
$596$	−52.2409	−2.13987
$597$	49.1983	2.01355
$598$	−19.5501	−0.799461
$599$	23.4783	0.959299	0.479649	−	0.877460i	$-0.340764\pi$
$599$	23.4783	0.959299	0.479649	+	0.877460i	$0.340764\pi$
$600$	162.342	6.62758
$601$	8.96196	0.365566	0.182783	−	0.983153i	$-0.441489\pi$
$601$	8.96196	0.365566	0.182783	+	0.983153i	$0.441489\pi$
$602$	0	0
$603$	−35.3375	−1.43906
$604$	25.6191	1.04243
$605$	38.6395	1.57092
$606$	19.1994	0.779922
$607$	−43.4641	−1.76415	−0.882077	−	0.471106i	$-0.843855\pi$
$607$	−43.4641	−1.76415	−0.882077	+	0.471106i	$0.843855\pi$
$608$	−38.1736	−1.54814
$609$	0	0
$610$	18.4163	0.745653
$611$	−4.60407	−0.186261
$612$	−33.1675	−1.34071
$613$	−21.4490	−0.866318	−0.433159	−	0.901317i	$-0.642601\pi$
$613$	−21.4490	−0.866318	−0.433159	+	0.901317i	$0.642601\pi$
$614$	74.9805	3.02597
$615$	89.3817	3.60422
$616$	0	0
$617$	−12.1294	−0.488312	−0.244156	−	0.969736i	$-0.578511\pi$
$617$	−12.1294	−0.488312	−0.244156	+	0.969736i	$0.578511\pi$
$618$	32.0533	1.28937
$619$	−12.7245	−0.511442	−0.255721	−	0.966751i	$-0.582313\pi$
$619$	−12.7245	−0.511442	−0.255721	+	0.966751i	$0.582313\pi$
$620$	−20.9840	−0.842739
$621$	4.57763	0.183694
$622$	−65.6005	−2.63034
$623$	0	0
$624$	−27.3463	−1.09473
$625$	3.11608	0.124643
$626$	−55.2656	−2.20886
$627$	−4.25077	−0.169759
$628$	63.6545	2.54009
$629$	17.8432	0.711456
$630$	0	0
$631$	−11.7538	−0.467912	−0.233956	−	0.972247i	$-0.575167\pi$
$631$	−11.7538	−0.467912	−0.233956	+	0.972247i	$0.575167\pi$
$632$	12.3082	0.489596
$633$	37.8043	1.50259
$634$	−69.4084	−2.75656
$635$	−19.9867	−0.793147
$636$	−63.6545	−2.52407
$637$	0	0
$638$	0.362259	0.0143420
$639$	−18.5208	−0.732670
$640$	38.8273	1.53478
$641$	4.68648	0.185105	0.0925523	−	0.995708i	$-0.470497\pi$
$641$	4.68648	0.185105	0.0925523	+	0.995708i	$0.470497\pi$
$642$	75.3497	2.97382
$643$	0.751182	0.0296237	0.0148119	−	0.999890i	$-0.495285\pi$
$643$	0.751182	0.0296237	0.0148119	+	0.999890i	$0.495285\pi$
$644$	0	0
$645$	−27.1745	−1.06999
$646$	17.2215	0.677570
$647$	−40.8476	−1.60589	−0.802943	−	0.596056i	$-0.796733\pi$
$647$	−40.8476	−1.60589	−0.802943	+	0.596056i	$0.796733\pi$
$648$	78.6873	3.09113
$649$	−5.41363	−0.212504
$650$	−22.2055	−0.870971
$651$	0	0
$652$	16.2055	0.634656
$653$	−46.4783	−1.81884	−0.909419	−	0.415881i	$-0.863473\pi$
$653$	−46.4783	−1.81884	−0.909419	+	0.415881i	$0.863473\pi$
$654$	22.5864	0.883197
$655$	35.9327	1.40400
$656$	116.482	4.54785
$657$	−34.3568	−1.34038
$658$	0	0
$659$	30.3596	1.18264	0.591321	−	0.806436i	$-0.298606\pi$
$659$	30.3596	1.18264	0.591321	+	0.806436i	$0.298606\pi$
$660$	28.9974	1.12872
$661$	0.107118	0.00416640	0.00208320	−	0.999998i	$-0.499337\pi$
$661$	0.107118	0.00416640	0.00208320	+	0.999998i	$0.499337\pi$
$662$	64.0975	2.49122
$663$	5.74049	0.222942
$664$	127.685	4.95513
$665$	0	0
$666$	54.1956	2.10004
$667$	−1.53235	−0.0593330
$668$	−41.0194	−1.58709
$669$	−20.2392	−0.782492
$670$	−125.166	−4.83557
$671$	1.24354	0.0480063
$672$	0	0
$673$	38.5385	1.48555	0.742774	−	0.669542i	$-0.233510\pi$
$673$	38.5385	1.48555	0.742774	+	0.669542i	$0.233510\pi$
$674$	81.0699	3.12270
$675$	5.19940	0.200125
$676$	5.05137	0.194284
$677$	20.7803	0.798650	0.399325	−	0.916809i	$-0.369244\pi$
$677$	20.7803	0.798650	0.399325	+	0.916809i	$0.369244\pi$
$678$	−60.2188	−2.31269
$679$	0	0
$680$	−70.9654	−2.72140
$681$	16.0267	0.614143
$682$	−1.97793	−0.0757388
$683$	27.5837	1.05546	0.527731	−	0.849412i	$-0.323043\pi$
$683$	27.5837	1.05546	0.527731	+	0.849412i	$0.323043\pi$
$684$	37.4711	1.43274
$685$	−64.1532	−2.45117
$686$	0	0
$687$	20.6891	0.789339
$688$	−35.4136	−1.35013
$689$	5.25951	0.200371
$690$	−171.223	−6.51835
$691$	−43.6775	−1.66157	−0.830785	−	0.556593i	$-0.812108\pi$
$691$	−43.6775	−1.66157	−0.830785	+	0.556593i	$0.812108\pi$
$692$	52.1736	1.98334
$693$	0	0
$694$	66.3791	2.51971
$695$	−17.9663	−0.681502
$696$	−4.04068	−0.153162
$697$	−24.4517	−0.926173
$698$	−4.88301	−0.184825
$699$	9.98494	0.377665
$700$	0	0
$701$	−20.8973	−0.789278	−0.394639	−	0.918836i	$-0.629130\pi$
$701$	−20.8973	−0.789278	−0.394639	+	0.918836i	$0.629130\pi$
$702$	−1.65107	−0.0623157
$703$	−20.1585	−0.760292
$704$	9.58373	0.361200
$705$	−40.3233	−1.51866
$706$	−61.7573	−2.32427
$707$	0	0
$708$	99.9628	3.75683
$709$	9.93966	0.373292	0.186646	−	0.982427i	$-0.440238\pi$
$709$	9.93966	0.373292	0.186646	+	0.982427i	$0.440238\pi$
$710$	−65.6005	−2.46194
$711$	−4.16286	−0.156119
$712$	−119.998	−4.49712
$713$	8.36663	0.313333
$714$	0	0
$715$	−2.39593	−0.0896028
$716$	−12.0133	−0.448959
$717$	4.29951	0.160568
$718$	56.9521	2.12543
$719$	11.9797	0.446766	0.223383	−	0.974731i	$-0.428290\pi$
$719$	11.9797	0.446766	0.223383	+	0.974731i	$0.428290\pi$
$720$	−114.338	−4.26114
$721$	0	0
$722$	30.9974	1.15360
$723$	33.2099	1.23509
$724$	−110.168	−4.09437
$725$	−1.74049	−0.0646402
$726$	−67.2516	−2.49594
$727$	−24.1736	−0.896547	−0.448274	−	0.893896i	$-0.647961\pi$
$727$	−24.1736	−0.896547	−0.448274	+	0.893896i	$0.647961\pi$
$728$	0	0
$729$	−22.1443	−0.820157
$730$	−121.692	−4.50401
$731$	7.43397	0.274955
$732$	−22.9620	−0.848698
$733$	36.4473	1.34621	0.673106	−	0.739546i	$-0.264960\pi$
$733$	36.4473	1.34621	0.673106	+	0.739546i	$0.264960\pi$
$734$	−2.97529	−0.109820
$735$	0	0
$736$	−103.828	−3.82715
$737$	−8.45168	−0.311321
$738$	−74.2676	−2.73383
$739$	43.2772	1.59198	0.795989	−	0.605311i	$-0.206951\pi$
$739$	43.2772	1.59198	0.795989	+	0.605311i	$0.206951\pi$
$740$	137.515	5.05514
$741$	−6.48535	−0.238245
$742$	0	0
$743$	22.6572	0.831211	0.415606	−	0.909545i	$-0.363570\pi$
$743$	22.6572	0.831211	0.415606	+	0.909545i	$0.363570\pi$
$744$	22.0621	0.808835
$745$	37.8043	1.38504
$746$	−41.4403	−1.51724
$747$	−43.1852	−1.58006
$748$	−7.93265	−0.290047
$749$	0	0
$750$	−78.1956	−2.85530
$751$	29.8679	1.08990	0.544948	−	0.838470i	$-0.316549\pi$
$751$	29.8679	1.08990	0.544948	+	0.838470i	$0.316549\pi$
$752$	−52.5491	−1.91627
$753$	35.3277	1.28741
$754$	0.552694	0.0201279
$755$	−18.5394	−0.674716
$756$	0	0
$757$	2.55706	0.0929380	0.0464690	−	0.998920i	$-0.485203\pi$
$757$	2.55706	0.0929380	0.0464690	+	0.998920i	$0.485203\pi$
$758$	−33.9494	−1.23310
$759$	−11.5617	−0.419662
$760$	80.1736	2.90820
$761$	−14.7289	−0.533922	−0.266961	−	0.963707i	$-0.586019\pi$
$761$	−14.7289	−0.533922	−0.266961	+	0.963707i	$0.586019\pi$
$762$	34.7866	1.26019
$763$	0	0
$764$	−126.723	−4.58467
$765$	24.0017	0.867784
$766$	90.3791	3.26553
$767$	−8.25951	−0.298234
$768$	2.48708	0.0897447
$769$	36.1692	1.30429	0.652147	−	0.758092i	$-0.273868\pi$
$769$	36.1692	1.30429	0.652147	+	0.758092i	$0.273868\pi$
$770$	0	0
$771$	56.7999	2.04560
$772$	57.1355	2.05635
$773$	−38.9567	−1.40117	−0.700587	−	0.713567i	$-0.747079\pi$
$773$	−38.9567	−1.40117	−0.700587	+	0.713567i	$0.747079\pi$
$774$	22.5794	0.811598
$775$	9.50305	0.341360
$776$	−81.1709	−2.91387
$777$	0	0
$778$	65.5111	2.34869
$779$	27.6244	0.989747
$780$	44.2409	1.58408
$781$	−4.42960	−0.158504
$782$	46.8406	1.67502
$783$	−0.129413	−0.00462484
$784$	0	0
$785$	−46.0638	−1.64409
$786$	−62.5404	−2.23074
$787$	−29.1045	−1.03746	−0.518731	−	0.854937i	$-0.673595\pi$
$787$	−29.1045	−1.03746	−0.518731	+	0.854937i	$0.673595\pi$
$788$	−84.8220	−3.02166
$789$	−27.2551	−0.970309
$790$	−14.7449	−0.524599
$791$	0	0
$792$	−14.5544	−0.517169
$793$	1.89725	0.0673734
$794$	−13.2232	−0.469274
$795$	46.0638	1.63371
$796$	−103.725	−3.67645
$797$	17.8920	0.633766	0.316883	−	0.948465i	$-0.397364\pi$
$797$	17.8920	0.633766	0.316883	+	0.948465i	$0.397364\pi$
$798$	0	0
$799$	11.0310	0.390250
$800$	−117.931	−4.16948
$801$	40.5855	1.43402
$802$	−1.82991	−0.0646162
$803$	−8.21710	−0.289975
$804$	156.060	5.50382
$805$	0	0
$806$	−3.01770	−0.106294
$807$	−26.6078	−0.936637
$808$	−24.4517	−0.860207
$809$	20.4543	0.719135	0.359568	−	0.933119i	$-0.382924\pi$
$809$	20.4543	0.719135	0.359568	+	0.933119i	$0.382924\pi$
$810$	−94.2647	−3.31212
$811$	31.6458	1.11123	0.555617	−	0.831438i	$-0.312482\pi$
$811$	31.6458	1.11123	0.555617	+	0.831438i	$0.312482\pi$
$812$	0	0
$813$	−30.4871	−1.06923
$814$	12.9620	0.454316
$815$	−11.7272	−0.410784
$816$	65.5198	2.29365
$817$	−8.39857	−0.293829
$818$	53.0478	1.85477
$819$	0	0
$820$	−188.445	−6.58077
$821$	18.0761	0.630860	0.315430	−	0.948949i	$-0.397851\pi$
$821$	18.0761	0.630860	0.315430	+	0.948949i	$0.397851\pi$
$822$	111.658	3.89452
$823$	−23.0514	−0.803520	−0.401760	−	0.915745i	$-0.631601\pi$
$823$	−23.0514	−0.803520	−0.401760	+	0.915745i	$0.631601\pi$
$824$	−40.8220	−1.42210
$825$	−13.1321	−0.457199
$826$	0	0
$827$	−26.3756	−0.917169	−0.458585	−	0.888651i	$-0.651643\pi$
$827$	−26.3756	−0.917169	−0.458585	+	0.888651i	$0.651643\pi$
$828$	101.918	3.54188
$829$	−45.1152	−1.56691	−0.783457	−	0.621446i	$-0.786546\pi$
$829$	−45.1152	−1.56691	−0.783457	+	0.621446i	$0.786546\pi$
$830$	−152.962	−5.30938
$831$	7.18780	0.249342
$832$	14.6218	0.506919
$833$	0	0
$834$	31.2702	1.08280
$835$	29.6838	1.02725
$836$	8.96196	0.309956
$837$	0.706592	0.0244234
$838$	52.2223	1.80399
$839$	30.4871	1.05253	0.526265	−	0.850320i	$-0.323592\pi$
$839$	30.4871	1.05253	0.526265	+	0.850320i	$0.323592\pi$
$840$	0	0
$841$	−28.9567	−0.998506
$842$	−39.6412	−1.36613
$843$	−8.25951	−0.284473
$844$	−79.7033	−2.74350
$845$	−3.65544	−0.125751
$846$	33.5048	1.15192
$847$	0	0
$848$	60.0301	2.06144
$849$	−29.8973	−1.02607
$850$	53.2029	1.82484
$851$	−54.8290	−1.87951
$852$	81.7927	2.80217
$853$	33.2746	1.13930	0.569650	−	0.821888i	$-0.307079\pi$
$853$	33.2746	1.13930	0.569650	+	0.821888i	$0.307079\pi$
$854$	0	0
$855$	−27.1161	−0.927350
$856$	−95.9628	−3.27994
$857$	−45.6139	−1.55814	−0.779070	−	0.626937i	$-0.784308\pi$
$857$	−45.6139	−1.55814	−0.779070	+	0.626937i	$0.784308\pi$
$858$	4.17009	0.142365
$859$	−20.9353	−0.714303	−0.357151	−	0.934046i	$-0.616252\pi$
$859$	−20.9353	−0.714303	−0.357151	+	0.934046i	$0.616252\pi$
$860$	57.2923	1.95365
$861$	0	0
$862$	86.7900	2.95608
$863$	2.96196	0.100826	0.0504131	−	0.998728i	$-0.483946\pi$
$863$	2.96196	0.100826	0.0504131	+	0.998728i	$0.483946\pi$
$864$	−8.76866	−0.298316
$865$	−37.7556	−1.28373
$866$	−23.7980	−0.808688
$867$	26.9770	0.916188
$868$	0	0
$869$	−0.995631	−0.0337745
$870$	4.84060	0.164112
$871$	−12.8946	−0.436917
$872$	−28.7652	−0.974113
$873$	27.4534	0.929157
$874$	−52.9184	−1.78999
$875$	0	0
$876$	151.729	5.12644
$877$	−36.9370	−1.24727	−0.623637	−	0.781714i	$-0.714346\pi$
$877$	−36.9370	−1.24727	−0.623637	+	0.781714i	$0.714346\pi$
$878$	−21.0647	−0.710899
$879$	−32.4267	−1.09373
$880$	−27.3463	−0.921843
$881$	−30.8946	−1.04087	−0.520433	−	0.853903i	$-0.674229\pi$
$881$	−30.8946	−1.04087	−0.520433	+	0.853903i	$0.674229\pi$
$882$	0	0
$883$	−9.64648	−0.324630	−0.162315	−	0.986739i	$-0.551896\pi$
$883$	−9.64648	−0.324630	−0.162315	+	0.986739i	$0.551896\pi$
$884$	−12.1027	−0.407059
$885$	−72.3384	−2.43163
$886$	23.6615	0.794925
$887$	51.3056	1.72267	0.861337	−	0.508034i	$-0.169628\pi$
$887$	51.3056	1.72267	0.861337	+	0.508034i	$0.169628\pi$
$888$	−144.579	−4.85176
$889$	0	0
$890$	143.754	4.81864
$891$	−6.36512	−0.213240
$892$	42.6705	1.42871
$893$	−12.4624	−0.417037
$894$	−65.7980	−2.20061
$895$	8.69348	0.290591
$896$	0	0
$897$	−17.6395	−0.588965
$898$	22.4429	0.748931
$899$	−0.236531	−0.00788873
$900$	115.761	3.85869
$901$	−12.6014	−0.419814
$902$	−17.7626	−0.591429
$903$	0	0
$904$	76.6926	2.55076
$905$	79.7236	2.65010
$906$	32.2676	1.07202
$907$	−18.5385	−0.615559	−0.307780	−	0.951458i	$-0.599586\pi$
$907$	−18.5385	−0.615559	−0.307780	+	0.951458i	$0.599586\pi$
$908$	−33.7892	−1.12133
$909$	8.26998	0.274298
$910$	0	0
$911$	20.7272	0.686721	0.343361	−	0.939204i	$-0.388435\pi$
$911$	20.7272	0.686721	0.343361	+	0.939204i	$0.388435\pi$
$912$	−74.0214	−2.45109
$913$	−10.3286	−0.341826
$914$	−61.5244	−2.03505
$915$	16.6165	0.549324
$916$	−43.6191	−1.44122
$917$	0	0
$918$	3.95586	0.130563
$919$	−5.13205	−0.169291	−0.0846454	−	0.996411i	$-0.526976\pi$
$919$	−5.13205	−0.169291	−0.0846454	+	0.996411i	$0.526976\pi$
$920$	218.064	7.18935
$921$	67.6528	2.22924
$922$	6.03540	0.198765
$923$	−6.75819	−0.222449
$924$	0	0
$925$	−62.2763	−2.04763
$926$	10.9132	0.358631
$927$	13.8067	0.453472
$928$	2.93529	0.0963557
$929$	−19.9283	−0.653826	−0.326913	−	0.945054i	$-0.606008\pi$
$929$	−19.9283	−0.653826	−0.326913	+	0.945054i	$0.606008\pi$
$930$	−26.4296	−0.866661
$931$	0	0
$932$	−21.0514	−0.689561
$933$	−59.1895	−1.93778
$934$	87.4038	2.85994
$935$	5.74049	0.187734
$936$	−22.2055	−0.725809
$937$	1.64475	0.0537316	0.0268658	−	0.999639i	$-0.491447\pi$
$937$	1.64475	0.0537316	0.0268658	+	0.999639i	$0.491447\pi$
$938$	0	0
$939$	−49.8646	−1.62727
$940$	85.0142	2.77286
$941$	17.7849	0.579770	0.289885	−	0.957062i	$-0.406383\pi$
$941$	17.7849	0.579770	0.289885	+	0.957062i	$0.406383\pi$
$942$	80.1736	2.61220
$943$	75.1355	2.44675
$944$	−94.2710	−3.06826
$945$	0	0
$946$	5.40030	0.175579
$947$	10.6484	0.346028	0.173014	−	0.984919i	$-0.444649\pi$
$947$	10.6484	0.346028	0.173014	+	0.984919i	$0.444649\pi$
$948$	18.3843	0.597095
$949$	−12.5367	−0.406959
$950$	−60.1062	−1.95010
$951$	−62.6252	−2.03076
$952$	0	0
$953$	−41.5544	−1.34608	−0.673040	−	0.739606i	$-0.735012\pi$
$953$	−41.5544	−1.34608	−0.673040	+	0.739606i	$0.735012\pi$
$954$	−38.2746	−1.23919
$955$	91.7033	2.96745
$956$	−9.06471	−0.293174
$957$	0.326856	0.0105658
$958$	−24.7361	−0.799188
$959$	0	0
$960$	128.060	4.13313
$961$	−29.7085	−0.958340
$962$	19.7759	0.637600
$963$	32.4563	1.04589
$964$	−70.0168	−2.25509
$965$	−41.3463	−1.33098
$966$	0	0
$967$	−5.51465	−0.177339	−0.0886696	−	0.996061i	$-0.528262\pi$
$967$	−5.51465	−0.177339	−0.0886696	+	0.996061i	$0.528262\pi$
$968$	85.6493	2.75287
$969$	15.5385	0.499167
$970$	97.2400	3.12219
$971$	−39.8246	−1.27803	−0.639017	−	0.769193i	$-0.720658\pi$
$971$	−39.8246	−1.27803	−0.639017	+	0.769193i	$0.720658\pi$
$972$	108.110	3.46762
$973$	0	0
$974$	−53.7485	−1.72221
$975$	−20.0354	−0.641646
$976$	21.6545	0.693145
$977$	1.93092	0.0617757	0.0308879	−	0.999523i	$-0.490167\pi$
$977$	1.93092	0.0617757	0.0308879	+	0.999523i	$0.490167\pi$
$978$	20.4110	0.652672
$979$	9.70682	0.310231
$980$	0	0
$981$	9.72889	0.310619
$982$	11.5837	0.369652
$983$	43.0238	1.37225	0.686123	−	0.727485i	$-0.259311\pi$
$983$	43.0238	1.37225	0.686123	+	0.727485i	$0.259311\pi$
$984$	198.126	6.31602
$985$	61.3817	1.95578
$986$	−1.32422	−0.0421717
$987$	0	0
$988$	13.6731	0.435001
$989$	−22.8432	−0.726373
$990$	17.4357	0.554143
$991$	−17.6058	−0.559267	−0.279633	−	0.960107i	$-0.590213\pi$
$991$	−17.6058	−0.559267	−0.279633	+	0.960107i	$0.590213\pi$
$992$	−16.0267	−0.508847
$993$	57.8334	1.83529
$994$	0	0
$995$	75.0612	2.37960
$996$	190.717	6.04311
$997$	12.9707	0.410786	0.205393	−	0.978680i	$-0.434153\pi$
$997$	12.9707	0.410786	0.205393	+	0.978680i	$0.434153\pi$
$998$	−25.7405	−0.814801
$999$	−4.63051	−0.146503

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.h.1.1	✓	3
3.2	odd	2		5733.2.a.be.1.3		3
7.2	even	3		637.2.e.l.508.3		6
7.3	odd	6		637.2.e.k.79.3		6
7.4	even	3		637.2.e.l.79.3		6
7.5	odd	6		637.2.e.k.508.3		6
7.6	odd	2		637.2.a.i.1.1	yes	3
13.12	even	2		8281.2.a.bh.1.3		3
21.20	even	2		5733.2.a.bd.1.3		3
91.90	odd	2		8281.2.a.bk.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.h.1.1	✓	3	1.1	even	1	trivial
637.2.a.i.1.1	yes	3	7.6	odd	2
637.2.e.k.79.3		6	7.3	odd	6
637.2.e.k.508.3		6	7.5	odd	6
637.2.e.l.79.3		6	7.4	even	3
637.2.e.l.508.3		6	7.2	even	3
5733.2.a.bd.1.3		3	21.20	even	2
5733.2.a.be.1.3		3	3.2	odd	2
8281.2.a.bh.1.3		3	13.12	even	2
8281.2.a.bk.1.3		3	91.90	odd	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:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q-2.65544 q^{2} -2.39593 q^{3} +5.05137 q^{4} -3.65544 q^{5} +6.36226 q^{6} -8.10275 q^{8} +2.74049 q^{9} +O(q^{10})\)	\(q-2.65544 q^{2} -2.39593 q^{3} +5.05137 q^{4} -3.65544 q^{5} +6.36226 q^{6} -8.10275 q^{8} +2.74049 q^{9} +9.70682 q^{10} +0.655442 q^{11} -12.1027 q^{12} +1.00000 q^{13} +8.75819 q^{15} +11.4136 q^{16} -2.39593 q^{17} -7.27721 q^{18} +2.70682 q^{19} -18.4650 q^{20} -1.74049 q^{22} +7.36226 q^{23} +19.4136 q^{24} +8.36226 q^{25} -2.65544 q^{26} +0.621770 q^{27} -0.208136 q^{29} -23.2569 q^{30} +1.13642 q^{31} -14.1027 q^{32} -1.57040 q^{33} +6.36226 q^{34} +13.8432 q^{36} -7.44731 q^{37} -7.18780 q^{38} -2.39593 q^{39} +29.6191 q^{40} +10.2055 q^{41} -3.10275 q^{43} +3.31088 q^{44} -10.0177 q^{45} -19.5501 q^{46} -4.60407 q^{47} -27.3463 q^{48} -22.2055 q^{50} +5.74049 q^{51} +5.05137 q^{52} +5.25951 q^{53} -1.65107 q^{54} -2.39593 q^{55} -6.48535 q^{57} +0.552694 q^{58} -8.25951 q^{59} +44.2409 q^{60} +1.89725 q^{61} -3.01770 q^{62} +14.6218 q^{64} -3.65544 q^{65} +4.17009 q^{66} -12.8946 q^{67} -12.1027 q^{68} -17.6395 q^{69} -6.75819 q^{71} -22.2055 q^{72} -12.5367 q^{73} +19.7759 q^{74} -20.0354 q^{75} +13.6731 q^{76} +6.36226 q^{78} -1.51902 q^{79} -41.7219 q^{80} -9.71119 q^{81} -27.1001 q^{82} -15.7582 q^{83} +8.75819 q^{85} +8.23917 q^{86} +0.498680 q^{87} -5.31088 q^{88} +14.8096 q^{89} +26.6014 q^{90} +37.1895 q^{92} -2.72279 q^{93} +12.2258 q^{94} -9.89461 q^{95} +33.7892 q^{96} +10.0177 q^{97} +1.79623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} - 6 q^{8} + 11 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 - 4 * q^3 + 6 * q^4 - 5 * q^5 - 2 * q^6 - 6 * q^8 + 11 * q^9	\( 3 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} - 6 q^{8} + 11 q^{9} + 14 q^{10} - 4 q^{11} - 18 q^{12} + 3 q^{13} + 2 q^{15} + 4 q^{16} - 4 q^{17} + 8 q^{18} - 7 q^{19} - 16 q^{20} - 8 q^{22} + q^{23} + 28 q^{24} + 4 q^{25} - 2 q^{26} - 22 q^{27} - 7 q^{29} - 24 q^{30} + 3 q^{31} - 24 q^{32} + 10 q^{33} - 2 q^{34} + 26 q^{36} - 10 q^{37} - 12 q^{38} - 4 q^{39} + 22 q^{40} - 6 q^{41} + 9 q^{43} - 2 q^{44} - 3 q^{45} - 28 q^{46} - 17 q^{47} - 16 q^{48} - 30 q^{50} + 20 q^{51} + 6 q^{52} + 13 q^{53} - 28 q^{54} - 4 q^{55} + 4 q^{57} + 14 q^{58} - 22 q^{59} + 42 q^{60} + 24 q^{61} + 18 q^{62} + 20 q^{64} - 5 q^{65} + 30 q^{66} - 14 q^{67} - 18 q^{68} - 2 q^{69} + 4 q^{71} - 30 q^{72} - 5 q^{73} + 8 q^{74} - 6 q^{75} + 8 q^{76} - 2 q^{78} + q^{79} - 40 q^{80} + 15 q^{81} - 20 q^{82} - 23 q^{83} + 2 q^{85} + 6 q^{86} - 20 q^{87} - 4 q^{88} + 11 q^{89} + 40 q^{90} + 30 q^{92} - 38 q^{93} + 16 q^{94} - 5 q^{95} + 52 q^{96} + 3 q^{97} - 30 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 - 4 * q^3 + 6 * q^4 - 5 * q^5 - 2 * q^6 - 6 * q^8 + 11 * q^9 + 14 * q^10 - 4 * q^11 - 18 * q^12 + 3 * q^13 + 2 * q^15 + 4 * q^16 - 4 * q^17 + 8 * q^18 - 7 * q^19 - 16 * q^20 - 8 * q^22 + q^23 + 28 * q^24 + 4 * q^25 - 2 * q^26 - 22 * q^27 - 7 * q^29 - 24 * q^30 + 3 * q^31 - 24 * q^32 + 10 * q^33 - 2 * q^34 + 26 * q^36 - 10 * q^37 - 12 * q^38 - 4 * q^39 + 22 * q^40 - 6 * q^41 + 9 * q^43 - 2 * q^44 - 3 * q^45 - 28 * q^46 - 17 * q^47 - 16 * q^48 - 30 * q^50 + 20 * q^51 + 6 * q^52 + 13 * q^53 - 28 * q^54 - 4 * q^55 + 4 * q^57 + 14 * q^58 - 22 * q^59 + 42 * q^60 + 24 * q^61 + 18 * q^62 + 20 * q^64 - 5 * q^65 + 30 * q^66 - 14 * q^67 - 18 * q^68 - 2 * q^69 + 4 * q^71 - 30 * q^72 - 5 * q^73 + 8 * q^74 - 6 * q^75 + 8 * q^76 - 2 * q^78 + q^79 - 40 * q^80 + 15 * q^81 - 20 * q^82 - 23 * q^83 + 2 * q^85 + 6 * q^86 - 20 * q^87 - 4 * q^88 + 11 * q^89 + 40 * q^90 + 30 * q^92 - 38 * q^93 + 16 * q^94 - 5 * q^95 + 52 * q^96 + 3 * q^97 - 30 * q^99

Introduction

L-functions

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