LMFDB - Embedded newform 637.2.a.i.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:	$0$
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	$-0.256886\pi$
$3$	2.39593	1.38329	0.691646	+	0.722237i	$0.256886\pi$
$4$	5.05137	2.52569
$5$	3.65544	1.63476	0.817382	−	0.576096i	$-0.195425\pi$
$5$	3.65544	1.63476	0.817382	+	0.576096i	$0.195425\pi$
$6$	−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	$-0.406162\pi$
$17$	2.39593	0.581099	0.290549	+	0.956860i	$0.406162\pi$
$18$	−7.27721	−1.71526
$19$	−2.70682	−0.620986	−0.310493	−	0.950576i	$-0.600494\pi$
$19$	−2.70682	−0.620986	−0.310493	+	0.950576i	$0.600494\pi$
$20$	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	$-0.532541\pi$
$31$	−1.13642	−0.204107	−0.102054	+	0.994779i	$0.532541\pi$
$32$	−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	$-0.793536\pi$
$41$	−10.2055	−1.59383	−0.796915	+	0.604091i	$0.793536\pi$
$42$	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	$-0.390998\pi$
$47$	4.60407	0.671572	0.335786	+	0.941938i	$0.390998\pi$
$48$	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	$-0.319313\pi$
$59$	8.25951	1.07530	0.537648	+	0.843169i	$0.319313\pi$
$60$	44.2409	5.71148
$61$	−1.89725	−0.242918	−0.121459	−	0.992596i	$-0.538757\pi$
$61$	−1.89725	−0.242918	−0.121459	+	0.992596i	$0.538757\pi$
$62$	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	$-0.237812\pi$
$73$	12.5367	1.46731	0.733656	+	0.679521i	$0.237812\pi$
$74$	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	$-0.167418\pi$
$83$	15.7582	1.72969	0.864843	+	0.502042i	$0.167418\pi$
$84$	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	$-0.787288\pi$
$89$	−14.8096	−1.56981	−0.784905	+	0.619616i	$0.787288\pi$
$90$	−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	$-0.669826\pi$
$97$	−10.0177	−1.01714	−0.508572	+	0.861020i	$0.669826\pi$
$98$	0	0
$99$	1.79623	0.180528
$100$	42.2409	4.22409
$101$	−3.01770	−0.300273	−0.150136	−	0.988665i	$-0.547971\pi$
$101$	−3.01770	−0.300273	−0.150136	+	0.988665i	$0.547971\pi$
$102$	−15.2435	−1.50934
$103$	−5.03804	−0.496413	−0.248207	−	0.968707i	$-0.579841\pi$
$103$	−5.03804	−0.496413	−0.248207	+	0.968707i	$0.579841\pi$
$104$	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	$-0.358718\pi$
$131$	9.82991	0.858843	0.429421	+	0.903104i	$0.358718\pi$
$132$	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	$-0.566839\pi$
$139$	−4.91495	−0.416881	−0.208440	+	0.978035i	$0.566839\pi$
$140$	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	$-0.667716\pi$
$157$	−12.6014	−1.00570	−0.502852	+	0.864373i	$0.667716\pi$
$158$	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	$-0.398267\pi$
$167$	8.12045	0.628379	0.314190	+	0.949360i	$0.398267\pi$
$168$	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	$-0.628436\pi$
$173$	−10.3286	−0.785268	−0.392634	+	0.919695i	$0.628436\pi$
$174$	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	$-0.199170\pi$
$181$	21.8096	1.62109	0.810546	+	0.585675i	$0.199170\pi$
$182$	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	$-0.240538\pi$
$199$	20.5341	1.45562	0.727811	+	0.685777i	$0.240538\pi$
$200$	−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	$-0.591275\pi$
$223$	−8.44731	−0.565673	−0.282837	+	0.959168i	$0.591275\pi$
$224$	0	0
$225$	22.9167	1.52778
$226$	25.1338	1.67187
$227$	6.68912	0.443972	0.221986	−	0.975050i	$-0.428746\pi$
$227$	6.68912	0.443972	0.221986	+	0.975050i	$0.428746\pi$
$228$	−32.7599	−2.16958
$229$	8.63510	0.570624	0.285312	−	0.958435i	$-0.407903\pi$
$229$	8.63510	0.570624	0.285312	+	0.958435i	$0.407903\pi$
$230$	−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	$-0.352695\pi$
$241$	13.8609	0.892862	0.446431	+	0.894818i	$0.352695\pi$
$242$	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	$-0.345931\pi$
$251$	14.7449	0.930687	0.465344	+	0.885130i	$0.345931\pi$
$252$	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	$-0.235112\pi$
$257$	23.7068	1.47879	0.739395	+	0.673272i	$0.235112\pi$
$258$	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	$-0.609938\pi$
$269$	−11.1054	−0.677107	−0.338554	+	0.940947i	$0.609938\pi$
$270$	6.03540	0.367303
$271$	−12.7245	−0.772959	−0.386480	−	0.922298i	$-0.626309\pi$
$271$	−12.7245	−0.772959	−0.386480	+	0.922298i	$0.626309\pi$
$272$	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	$-0.620944\pi$
$283$	−12.4783	−0.741760	−0.370880	+	0.928681i	$0.620944\pi$
$284$	−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	$-0.629371\pi$
$293$	−13.5341	−0.790670	−0.395335	+	0.918537i	$0.629371\pi$
$294$	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	$-0.201751\pi$
$307$	28.2365	1.61154	0.805772	+	0.592226i	$0.201751\pi$
$308$	0	0
$309$	−12.0708	−0.686684
$310$	11.0310	0.626521
$311$	−24.7042	−1.40085	−0.700423	−	0.713728i	$-0.747005\pi$
$311$	−24.7042	−1.40085	−0.700423	+	0.713728i	$0.747005\pi$
$312$	19.4136	1.09908
$313$	−20.8122	−1.17638	−0.588188	−	0.808724i	$-0.700158\pi$
$313$	−20.8122	−1.17638	−0.588188	+	0.808724i	$0.700158\pi$
$314$	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	$-0.515672\pi$
$349$	−1.83887	−0.0984324	−0.0492162	+	0.998788i	$0.515672\pi$
$350$	0	0
$351$	0.621770	0.0331876
$352$	−9.24354	−0.492682
$353$	−23.2569	−1.23784	−0.618919	−	0.785455i	$-0.712429\pi$
$353$	−23.2569	−1.23784	−0.618919	+	0.785455i	$0.712429\pi$
$354$	−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	$-0.509310\pi$
$367$	−1.12045	−0.0584870	−0.0292435	+	0.999572i	$0.509310\pi$
$368$	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	$-0.164400\pi$
$383$	34.0354	1.73913	0.869564	+	0.493820i	$0.164400\pi$
$384$	−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	$-0.539881\pi$
$397$	−4.97966	−0.249922	−0.124961	+	0.992162i	$0.539881\pi$
$398$	−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	$-0.335571\pi$
$409$	19.9770	0.987800	0.493900	+	0.869519i	$0.335571\pi$
$410$	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	$-0.340500\pi$
$419$	19.6661	0.960754	0.480377	+	0.877062i	$0.340500\pi$
$420$	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	$-0.569087\pi$
$433$	−8.96196	−0.430684	−0.215342	+	0.976539i	$0.569087\pi$
$434$	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	$-0.560623\pi$
$439$	−7.93265	−0.378605	−0.189302	+	0.981919i	$0.560623\pi$
$440$	−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	$-0.483144\pi$
$461$	2.27284	0.105857	0.0529284	+	0.998598i	$0.483144\pi$
$462$	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	$-0.224433\pi$
$467$	32.9150	1.52312	0.761561	+	0.648093i	$0.224433\pi$
$468$	−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	$-0.568262\pi$
$479$	−9.31525	−0.425625	−0.212812	+	0.977093i	$0.568262\pi$
$480$	−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	$-0.481195\pi$
$503$	2.64843	0.118088	0.0590439	+	0.998255i	$0.481195\pi$
$504$	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	$-0.599986\pi$
$509$	−13.9416	−0.617951	−0.308976	+	0.951070i	$0.599986\pi$
$510$	−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	$-0.603782\pi$
$521$	−14.6218	−0.640591	−0.320296	+	0.947318i	$0.603782\pi$
$522$	1.51465	0.0662945
$523$	16.5190	0.722326	0.361163	−	0.932503i	$-0.382380\pi$
$523$	16.5190	0.722326	0.361163	+	0.932503i	$0.382380\pi$
$524$	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	$-0.266587\pi$
$563$	31.7626	1.33863	0.669316	+	0.742978i	$0.266587\pi$
$564$	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	$-0.376766\pi$
$577$	18.1382	0.755101	0.377551	+	0.925989i	$0.376766\pi$
$578$	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	$-0.369554\pi$
$587$	19.3065	0.796865	0.398433	+	0.917198i	$0.369554\pi$
$588$	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	$-0.446143\pi$
$593$	8.20113	0.336780	0.168390	+	0.985720i	$0.446143\pi$
$594$	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	$-0.558511\pi$
$601$	−8.96196	−0.365566	−0.182783	+	0.983153i	$0.558511\pi$
$602$	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	$-0.156145\pi$
$607$	43.4641	1.76415	0.882077	+	0.471106i	$0.156145\pi$
$608$	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	$-0.417687\pi$
$619$	12.7245	0.511442	0.255721	+	0.966751i	$0.417687\pi$
$620$	−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	$-0.504715\pi$
$643$	−0.751182	−0.0296237	−0.0148119	+	0.999890i	$0.504715\pi$
$644$	0	0
$645$	−27.1745	−1.06999
$646$	17.2215	0.677570
$647$	40.8476	1.60589	0.802943	−	0.596056i	$-0.203267\pi$
$647$	40.8476	1.60589	0.802943	+	0.596056i	$0.203267\pi$
$648$	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	$-0.500663\pi$
$661$	−0.107118	−0.00416640	−0.00208320	+	0.999998i	$0.500663\pi$
$662$	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	$-0.630756\pi$
$677$	−20.7803	−0.798650	−0.399325	+	0.916809i	$0.630756\pi$
$678$	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	$-0.187892\pi$
$691$	43.6775	1.66157	0.830785	+	0.556593i	$0.187892\pi$
$692$	−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	$-0.571710\pi$
$719$	−11.9797	−0.446766	−0.223383	+	0.974731i	$0.571710\pi$
$720$	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	$-0.352039\pi$
$727$	24.1736	0.896547	0.448274	+	0.893896i	$0.352039\pi$
$728$	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	$-0.735040\pi$
$733$	−36.4473	−1.34621	−0.673106	+	0.739546i	$0.735040\pi$
$734$	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	$-0.413981\pi$
$761$	14.7289	0.533922	0.266961	+	0.963707i	$0.413981\pi$
$762$	−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	$-0.726132\pi$
$769$	−36.1692	−1.30429	−0.652147	+	0.758092i	$0.726132\pi$
$770$	0	0
$771$	56.7999	2.04560
$772$	57.1355	2.05635
$773$	38.9567	1.40117	0.700587	−	0.713567i	$-0.252921\pi$
$773$	38.9567	1.40117	0.700587	+	0.713567i	$0.252921\pi$
$774$	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	$-0.326405\pi$
$787$	29.1045	1.03746	0.518731	+	0.854937i	$0.326405\pi$
$788$	−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	$-0.602636\pi$
$797$	−17.8920	−0.633766	−0.316883	+	0.948465i	$0.602636\pi$
$798$	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	$-0.687518\pi$
$811$	−31.6458	−1.11123	−0.555617	+	0.831438i	$0.687518\pi$
$812$	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	$-0.213454\pi$
$829$	45.1152	1.56691	0.783457	+	0.621446i	$0.213454\pi$
$830$	−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	$-0.676408\pi$
$839$	−30.4871	−1.05253	−0.526265	+	0.850320i	$0.676408\pi$
$840$	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	$-0.692921\pi$
$853$	−33.2746	−1.13930	−0.569650	+	0.821888i	$0.692921\pi$
$854$	0	0
$855$	−27.1161	−0.927350
$856$	−95.9628	−3.27994
$857$	45.6139	1.55814	0.779070	−	0.626937i	$-0.215692\pi$
$857$	45.6139	1.55814	0.779070	+	0.626937i	$0.215692\pi$
$858$	4.17009	0.142365
$859$	20.9353	0.714303	0.357151	−	0.934046i	$-0.383748\pi$
$859$	20.9353	0.714303	0.357151	+	0.934046i	$0.383748\pi$
$860$	−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	$-0.325771\pi$
$881$	30.8946	1.04087	0.520433	+	0.853903i	$0.325771\pi$
$882$	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	$-0.830372\pi$
$887$	−51.3056	−1.72267	−0.861337	+	0.508034i	$0.830372\pi$
$888$	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	$-0.393992\pi$
$929$	19.9283	0.653826	0.326913	+	0.945054i	$0.393992\pi$
$930$	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	$-0.508553\pi$
$937$	−1.64475	−0.0537316	−0.0268658	+	0.999639i	$0.508553\pi$
$938$	0	0
$939$	−49.8646	−1.62727
$940$	85.0142	2.77286
$941$	−17.7849	−0.579770	−0.289885	−	0.957062i	$-0.593617\pi$
$941$	−17.7849	−0.579770	−0.289885	+	0.957062i	$0.593617\pi$
$942$	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	$-0.279342\pi$
$971$	39.8246	1.27803	0.639017	+	0.769193i	$0.279342\pi$
$972$	−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	$-0.740689\pi$
$983$	−43.0238	−1.37225	−0.686123	+	0.727485i	$0.740689\pi$
$984$	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	$-0.565847\pi$
$997$	−12.9707	−0.410786	−0.205393	+	0.978680i	$0.565847\pi$
$998$	−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.i.1.1	yes	3
3.2	odd	2		5733.2.a.bd.1.3		3
7.2	even	3		637.2.e.k.508.3		6
7.3	odd	6		637.2.e.l.79.3		6
7.4	even	3		637.2.e.k.79.3		6
7.5	odd	6		637.2.e.l.508.3		6
7.6	odd	2		637.2.a.h.1.1	✓	3
13.12	even	2		8281.2.a.bk.1.3		3
21.20	even	2		5733.2.a.be.1.3		3
91.90	odd	2		8281.2.a.bh.1.3		3

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

Modular forms

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