LMFDB - Embedded newform 4009.2.a.d.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37585 q^{2} +1.43559 q^{3} +3.64464 q^{4} +2.52541 q^{5} -3.41075 q^{6} -1.13884 q^{7} -3.90741 q^{8} -0.939073 q^{9} +O(q^{10})$	\(q-2.37585 q^{2} +1.43559 q^{3} +3.64464 q^{4} +2.52541 q^{5} -3.41075 q^{6} -1.13884 q^{7} -3.90741 q^{8} -0.939073 q^{9} -5.99997 q^{10} +2.48039 q^{11} +5.23222 q^{12} -4.58353 q^{13} +2.70570 q^{14} +3.62545 q^{15} +1.99412 q^{16} +3.27872 q^{17} +2.23109 q^{18} -1.00000 q^{19} +9.20419 q^{20} -1.63490 q^{21} -5.89302 q^{22} -4.03869 q^{23} -5.60945 q^{24} +1.37767 q^{25} +10.8897 q^{26} -5.65491 q^{27} -4.15065 q^{28} -6.76587 q^{29} -8.61352 q^{30} -0.791289 q^{31} +3.07710 q^{32} +3.56083 q^{33} -7.78974 q^{34} -2.87602 q^{35} -3.42258 q^{36} +2.80955 q^{37} +2.37585 q^{38} -6.58008 q^{39} -9.86779 q^{40} +9.81028 q^{41} +3.88428 q^{42} -1.18720 q^{43} +9.04012 q^{44} -2.37154 q^{45} +9.59529 q^{46} -4.37762 q^{47} +2.86274 q^{48} -5.70305 q^{49} -3.27314 q^{50} +4.70691 q^{51} -16.7053 q^{52} +6.47163 q^{53} +13.4352 q^{54} +6.26398 q^{55} +4.44990 q^{56} -1.43559 q^{57} +16.0747 q^{58} -6.53629 q^{59} +13.2135 q^{60} -12.8991 q^{61} +1.87998 q^{62} +1.06945 q^{63} -11.2990 q^{64} -11.5753 q^{65} -8.45997 q^{66} +9.80152 q^{67} +11.9498 q^{68} -5.79791 q^{69} +6.83298 q^{70} +11.7363 q^{71} +3.66934 q^{72} -4.09255 q^{73} -6.67505 q^{74} +1.97778 q^{75} -3.64464 q^{76} -2.82475 q^{77} +15.6332 q^{78} -0.441291 q^{79} +5.03596 q^{80} -5.30092 q^{81} -23.3077 q^{82} -4.83824 q^{83} -5.95864 q^{84} +8.28011 q^{85} +2.82061 q^{86} -9.71304 q^{87} -9.69189 q^{88} +1.75765 q^{89} +5.63441 q^{90} +5.21988 q^{91} -14.7196 q^{92} -1.13597 q^{93} +10.4006 q^{94} -2.52541 q^{95} +4.41746 q^{96} -1.64020 q^{97} +13.5496 q^{98} -2.32926 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * 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 $

$2$	−2.37585	−1.67998	−0.839988	−	0.542605i	$-0.817438\pi$
$2$	−2.37585	−1.67998	−0.839988	+	0.542605i	$0.817438\pi$
$3$	1.43559	0.828840	0.414420	−	0.910086i	$-0.363984\pi$
$3$	1.43559	0.828840	0.414420	+	0.910086i	$0.363984\pi$
$4$	3.64464	1.82232
$5$	2.52541	1.12940	0.564698	−	0.825298i	$-0.308993\pi$
$5$	2.52541	1.12940	0.564698	+	0.825298i	$0.308993\pi$
$6$	−3.41075	−1.39243
$7$	−1.13884	−0.430439	−0.215220	−	0.976566i	$-0.569047\pi$
$7$	−1.13884	−0.430439	−0.215220	+	0.976566i	$0.569047\pi$
$8$	−3.90741	−1.38148
$9$	−0.939073	−0.313024
$10$	−5.99997	−1.89736
$11$	2.48039	0.747865	0.373933	−	0.927456i	$-0.378009\pi$
$11$	2.48039	0.747865	0.373933	+	0.927456i	$0.378009\pi$
$12$	5.23222	1.51041
$13$	−4.58353	−1.27124	−0.635621	−	0.772002i	$-0.719256\pi$
$13$	−4.58353	−1.27124	−0.635621	+	0.772002i	$0.719256\pi$
$14$	2.70570	0.723128
$15$	3.62545	0.936088
$16$	1.99412	0.498530
$17$	3.27872	0.795207	0.397604	−	0.917557i	$-0.369842\pi$
$17$	3.27872	0.795207	0.397604	+	0.917557i	$0.369842\pi$
$18$	2.23109	0.525873
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	9.20419	2.05812
$21$	−1.63490	−0.356765
$22$	−5.89302	−1.25640
$23$	−4.03869	−0.842124	−0.421062	−	0.907032i	$-0.638343\pi$
$23$	−4.03869	−0.842124	−0.421062	+	0.907032i	$0.638343\pi$
$24$	−5.60945	−1.14502
$25$	1.37767	0.275535
$26$	10.8897	2.13566
$27$	−5.65491	−1.08829
$28$	−4.15065	−0.784398
$29$	−6.76587	−1.25639	−0.628195	−	0.778056i	$-0.716206\pi$
$29$	−6.76587	−1.25639	−0.628195	+	0.778056i	$0.716206\pi$
$30$	−8.61352	−1.57261
$31$	−0.791289	−0.142120	−0.0710599	−	0.997472i	$-0.522638\pi$
$31$	−0.791289	−0.142120	−0.0710599	+	0.997472i	$0.522638\pi$
$32$	3.07710	0.543959
$33$	3.56083	0.619860
$34$	−7.78974	−1.33593
$35$	−2.87602	−0.486136
$36$	−3.42258	−0.570430
$37$	2.80955	0.461887	0.230943	−	0.972967i	$-0.425819\pi$
$37$	2.80955	0.461887	0.230943	+	0.972967i	$0.425819\pi$
$38$	2.37585	0.385413
$39$	−6.58008	−1.05366
$40$	−9.86779	−1.56024
$41$	9.81028	1.53211	0.766054	−	0.642776i	$-0.222217\pi$
$41$	9.81028	1.53211	0.766054	+	0.642776i	$0.222217\pi$
$42$	3.88428	0.599357
$43$	−1.18720	−0.181047	−0.0905233	−	0.995894i	$-0.528854\pi$
$43$	−1.18720	−0.181047	−0.0905233	+	0.995894i	$0.528854\pi$
$44$	9.04012	1.36285
$45$	−2.37154	−0.353528
$46$	9.59529	1.41475
$47$	−4.37762	−0.638542	−0.319271	−	0.947663i	$-0.603438\pi$
$47$	−4.37762	−0.638542	−0.319271	+	0.947663i	$0.603438\pi$
$48$	2.86274	0.413202
$49$	−5.70305	−0.814722
$50$	−3.27314	−0.462891
$51$	4.70691	0.659100
$52$	−16.7053	−2.31661
$53$	6.47163	0.888947	0.444474	−	0.895792i	$-0.353391\pi$
$53$	6.47163	0.888947	0.444474	+	0.895792i	$0.353391\pi$
$54$	13.4352	1.82830
$55$	6.26398	0.844636
$56$	4.44990	0.594642
$57$	−1.43559	−0.190149
$58$	16.0747	2.11071
$59$	−6.53629	−0.850953	−0.425476	−	0.904970i	$-0.639893\pi$
$59$	−6.53629	−0.850953	−0.425476	+	0.904970i	$0.639893\pi$
$60$	13.2135	1.70585
$61$	−12.8991	−1.65156	−0.825778	−	0.563996i	$-0.809263\pi$
$61$	−12.8991	−1.65156	−0.825778	+	0.563996i	$0.809263\pi$
$62$	1.87998	0.238758
$63$	1.06945	0.134738
$64$	−11.2990	−1.41237
$65$	−11.5753	−1.43573
$66$	−8.45997	−1.04135
$67$	9.80152	1.19745	0.598723	−	0.800956i	$-0.295675\pi$
$67$	9.80152	1.19745	0.598723	+	0.800956i	$0.295675\pi$
$68$	11.9498	1.44912
$69$	−5.79791	−0.697986
$70$	6.83298	0.816697
$71$	11.7363	1.39284	0.696420	−	0.717635i	$-0.254775\pi$
$71$	11.7363	1.39284	0.696420	+	0.717635i	$0.254775\pi$
$72$	3.66934	0.432436
$73$	−4.09255	−0.478997	−0.239499	−	0.970897i	$-0.576983\pi$
$73$	−4.09255	−0.478997	−0.239499	+	0.970897i	$0.576983\pi$
$74$	−6.67505	−0.775959
$75$	1.97778	0.228374
$76$	−3.64464	−0.418069
$77$	−2.82475	−0.321911
$78$	15.6332	1.77012
$79$	−0.441291	−0.0496491	−0.0248246	−	0.999692i	$-0.507903\pi$
$79$	−0.441291	−0.0496491	−0.0248246	+	0.999692i	$0.507903\pi$
$80$	5.03596	0.563038
$81$	−5.30092	−0.588992
$82$	−23.3077	−2.57391
$83$	−4.83824	−0.531065	−0.265533	−	0.964102i	$-0.585548\pi$
$83$	−4.83824	−0.531065	−0.265533	+	0.964102i	$0.585548\pi$
$84$	−5.95864	−0.650141
$85$	8.28011	0.898104
$86$	2.82061	0.304154
$87$	−9.71304	−1.04135
$88$	−9.69189	−1.03316
$89$	1.75765	0.186310	0.0931551	−	0.995652i	$-0.470305\pi$
$89$	1.75765	0.186310	0.0931551	+	0.995652i	$0.470305\pi$
$90$	5.63441	0.593919
$91$	5.21988	0.547192
$92$	−14.7196	−1.53462
$93$	−1.13597	−0.117795
$94$	10.4006	1.07274
$95$	−2.52541	−0.259101
$96$	4.41746	0.450855
$97$	−1.64020	−0.166538	−0.0832688	−	0.996527i	$-0.526536\pi$
$97$	−1.64020	−0.166538	−0.0832688	+	0.996527i	$0.526536\pi$
$98$	13.5496	1.36871
$99$	−2.32926	−0.234100
$100$	5.02112	0.502112
$101$	−11.3772	−1.13208	−0.566038	−	0.824379i	$-0.691525\pi$
$101$	−11.3772	−1.13208	−0.566038	+	0.824379i	$0.691525\pi$
$102$	−11.1829	−1.10727
$103$	−9.48867	−0.934946	−0.467473	−	0.884007i	$-0.654835\pi$
$103$	−9.48867	−0.934946	−0.467473	+	0.884007i	$0.654835\pi$
$104$	17.9097	1.75619
$105$	−4.12880	−0.402929
$106$	−15.3756	−1.49341
$107$	−16.8674	−1.63064	−0.815319	−	0.579012i	$-0.803438\pi$
$107$	−16.8674	−1.63064	−0.815319	+	0.579012i	$0.803438\pi$
$108$	−20.6101	−1.98321
$109$	−3.22273	−0.308681	−0.154341	−	0.988018i	$-0.549325\pi$
$109$	−3.22273	−0.308681	−0.154341	+	0.988018i	$0.549325\pi$
$110$	−14.8823	−1.41897
$111$	4.03337	0.382830
$112$	−2.27097	−0.214587
$113$	−7.92656	−0.745668	−0.372834	−	0.927898i	$-0.621614\pi$
$113$	−7.92656	−0.745668	−0.372834	+	0.927898i	$0.621614\pi$
$114$	3.41075	0.319446
$115$	−10.1993	−0.951092
$116$	−24.6592	−2.28955
$117$	4.30426	0.397929
$118$	15.5292	1.42958
$119$	−3.73393	−0.342289
$120$	−14.1661	−1.29319
$121$	−4.84768	−0.440698
$122$	30.6462	2.77457
$123$	14.0836	1.26987
$124$	−2.88396	−0.258988
$125$	−9.14784	−0.818208
$126$	−2.54085	−0.226357
$127$	−9.03490	−0.801717	−0.400859	−	0.916140i	$-0.631288\pi$
$127$	−9.03490	−0.801717	−0.400859	+	0.916140i	$0.631288\pi$
$128$	20.6904	1.82879
$129$	−1.70434	−0.150059
$130$	27.5010	2.41200
$131$	−15.7822	−1.37890	−0.689451	−	0.724333i	$-0.742148\pi$
$131$	−15.7822	−1.37890	−0.689451	+	0.724333i	$0.742148\pi$
$132$	12.9779	1.12958
$133$	1.13884	0.0987496
$134$	−23.2869	−2.01168
$135$	−14.2809	−1.22911
$136$	−12.8113	−1.09856
$137$	12.2765	1.04885	0.524426	−	0.851456i	$-0.324280\pi$
$137$	12.2765	1.04885	0.524426	+	0.851456i	$0.324280\pi$
$138$	13.7749	1.17260
$139$	−9.43871	−0.800581	−0.400291	−	0.916388i	$-0.631091\pi$
$139$	−9.43871	−0.800581	−0.400291	+	0.916388i	$0.631091\pi$
$140$	−10.4821	−0.885896
$141$	−6.28449	−0.529249
$142$	−27.8836	−2.33994
$143$	−11.3689	−0.950717
$144$	−1.87262	−0.156052
$145$	−17.0866	−1.41896
$146$	9.72327	0.804704
$147$	−8.18726	−0.675274
$148$	10.2398	0.841706
$149$	−5.47204	−0.448287	−0.224144	−	0.974556i	$-0.571958\pi$
$149$	−5.47204	−0.448287	−0.224144	+	0.974556i	$0.571958\pi$
$150$	−4.69889	−0.383663
$151$	−3.25674	−0.265030	−0.132515	−	0.991181i	$-0.542305\pi$
$151$	−3.25674	−0.265030	−0.132515	+	0.991181i	$0.542305\pi$
$152$	3.90741	0.316933
$153$	−3.07896	−0.248919
$154$	6.71118	0.540802
$155$	−1.99833	−0.160509
$156$	−23.9820	−1.92010
$157$	12.8663	1.02685	0.513423	−	0.858136i	$-0.328377\pi$
$157$	12.8663	1.02685	0.513423	+	0.858136i	$0.328377\pi$
$158$	1.04844	0.0834093
$159$	9.29063	0.736795
$160$	7.77092	0.614345
$161$	4.59940	0.362483
$162$	12.5942	0.989492
$163$	8.64580	0.677191	0.338595	−	0.940932i	$-0.390048\pi$
$163$	8.64580	0.677191	0.338595	+	0.940932i	$0.390048\pi$
$164$	35.7550	2.79199
$165$	8.99253	0.700068
$166$	11.4949	0.892177
$167$	8.45198	0.654034	0.327017	−	0.945018i	$-0.393957\pi$
$167$	8.45198	0.654034	0.327017	+	0.945018i	$0.393957\pi$
$168$	6.38824	0.492863
$169$	8.00871	0.616055
$170$	−19.6723	−1.50879
$171$	0.939073	0.0718127
$172$	−4.32692	−0.329925
$173$	13.5067	1.02689	0.513446	−	0.858122i	$-0.328368\pi$
$173$	13.5067	1.02689	0.513446	+	0.858122i	$0.328368\pi$
$174$	23.0767	1.74944
$175$	−1.56894	−0.118601
$176$	4.94619	0.372833
$177$	−9.38346	−0.705304
$178$	−4.17590	−0.312997
$179$	17.7818	1.32907	0.664537	−	0.747256i	$-0.268629\pi$
$179$	17.7818	1.32907	0.664537	+	0.747256i	$0.268629\pi$
$180$	−8.64341	−0.644241
$181$	−1.94319	−0.144437	−0.0722183	−	0.997389i	$-0.523008\pi$
$181$	−1.94319	−0.144437	−0.0722183	+	0.997389i	$0.523008\pi$
$182$	−12.4016	−0.919270
$183$	−18.5178	−1.36887
$184$	15.7808	1.16338
$185$	7.09525	0.521653
$186$	2.69889	0.197892
$187$	8.13251	0.594708
$188$	−15.9549	−1.16363
$189$	6.44001	0.468442
$190$	5.99997	0.435284
$191$	−8.12944	−0.588226	−0.294113	−	0.955771i	$-0.595024\pi$
$191$	−8.12944	−0.588226	−0.294113	+	0.955771i	$0.595024\pi$
$192$	−16.2207	−1.17063
$193$	16.4678	1.18538	0.592691	−	0.805430i	$-0.298066\pi$
$193$	16.4678	1.18538	0.592691	+	0.805430i	$0.298066\pi$
$194$	3.89687	0.279779
$195$	−16.6174	−1.18999
$196$	−20.7856	−1.48468
$197$	−25.9281	−1.84730	−0.923651	−	0.383235i	$-0.874810\pi$
$197$	−25.9281	−1.84730	−0.923651	+	0.383235i	$0.874810\pi$
$198$	5.53397	0.393282
$199$	−8.10515	−0.574559	−0.287280	−	0.957847i	$-0.592751\pi$
$199$	−8.10515	−0.574559	−0.287280	+	0.957847i	$0.592751\pi$
$200$	−5.38313	−0.380645
$201$	14.0710	0.992491
$202$	27.0305	1.90186
$203$	7.70522	0.540800
$204$	17.1550	1.20109
$205$	24.7749	1.73036
$206$	22.5436	1.57069
$207$	3.79262	0.263605
$208$	−9.14010	−0.633752
$209$	−2.48039	−0.171572
$210$	9.80938	0.676912
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	23.5868	1.61995
$213$	16.8485	1.15444
$214$	40.0744	2.73943
$215$	−2.99817	−0.204473
$216$	22.0960	1.50344
$217$	0.901148	0.0611739
$218$	7.65670	0.518577
$219$	−5.87524	−0.397012
$220$	22.8300	1.53920
$221$	−15.0281	−1.01090
$222$	−9.58266	−0.643146
$223$	6.32018	0.423231	0.211615	−	0.977353i	$-0.432128\pi$
$223$	6.32018	0.423231	0.211615	+	0.977353i	$0.432128\pi$
$224$	−3.50431	−0.234142
$225$	−1.29373	−0.0862490
$226$	18.8323	1.25270
$227$	14.5826	0.967878	0.483939	−	0.875102i	$-0.339206\pi$
$227$	14.5826	0.967878	0.483939	+	0.875102i	$0.339206\pi$
$228$	−5.23222	−0.346512
$229$	9.06210	0.598841	0.299420	−	0.954121i	$-0.403207\pi$
$229$	9.06210	0.598841	0.299420	+	0.954121i	$0.403207\pi$
$230$	24.2320	1.59781
$231$	−4.05520	−0.266812
$232$	26.4370	1.73568
$233$	−6.15922	−0.403504	−0.201752	−	0.979437i	$-0.564663\pi$
$233$	−6.15922	−0.403504	−0.201752	+	0.979437i	$0.564663\pi$
$234$	−10.2263	−0.668512
$235$	−11.0553	−0.721167
$236$	−23.8224	−1.55071
$237$	−0.633514	−0.0411512
$238$	8.87123	0.575037
$239$	−17.0406	−1.10226	−0.551132	−	0.834418i	$-0.685804\pi$
$239$	−17.0406	−1.10226	−0.551132	+	0.834418i	$0.685804\pi$
$240$	7.22959	0.466668
$241$	9.10958	0.586800	0.293400	−	0.955990i	$-0.405213\pi$
$241$	9.10958	0.586800	0.293400	+	0.955990i	$0.405213\pi$
$242$	11.5173	0.740362
$243$	9.35475	0.600107
$244$	−47.0124	−3.00966
$245$	−14.4025	−0.920143
$246$	−33.4604	−2.13336
$247$	4.58353	0.291643
$248$	3.09189	0.196335
$249$	−6.94574	−0.440168
$250$	21.7339	1.37457
$251$	−1.22010	−0.0770123	−0.0385061	−	0.999258i	$-0.512260\pi$
$251$	−1.22010	−0.0770123	−0.0385061	+	0.999258i	$0.512260\pi$
$252$	3.89776	0.245536
$253$	−10.0175	−0.629795
$254$	21.4655	1.34687
$255$	11.8869	0.744384
$256$	−26.5592	−1.65995
$257$	−27.4765	−1.71394	−0.856969	−	0.515368i	$-0.827655\pi$
$257$	−27.4765	−1.71394	−0.856969	+	0.515368i	$0.827655\pi$
$258$	4.04924	0.252095
$259$	−3.19961	−0.198814
$260$	−42.1877	−2.61637
$261$	6.35365	0.393281
$262$	37.4962	2.31652
$263$	−26.1880	−1.61482	−0.807409	−	0.589992i	$-0.799131\pi$
$263$	−26.1880	−1.61482	−0.807409	+	0.589992i	$0.799131\pi$
$264$	−13.9136	−0.856323
$265$	16.3435	1.00397
$266$	−2.70570	−0.165897
$267$	2.52327	0.154421
$268$	35.7230	2.18213
$269$	1.68408	0.102680	0.0513400	−	0.998681i	$-0.483651\pi$
$269$	1.68408	0.102680	0.0513400	+	0.998681i	$0.483651\pi$
$270$	33.9293	2.06487
$271$	7.80014	0.473825	0.236912	−	0.971531i	$-0.423865\pi$
$271$	7.80014	0.473825	0.236912	+	0.971531i	$0.423865\pi$
$272$	6.53817	0.396435
$273$	7.49363	0.453535
$274$	−29.1670	−1.76204
$275$	3.41716	0.206063
$276$	−21.1313	−1.27195
$277$	10.2282	0.614555	0.307278	−	0.951620i	$-0.400582\pi$
$277$	10.2282	0.614555	0.307278	+	0.951620i	$0.400582\pi$
$278$	22.4249	1.34496
$279$	0.743078	0.0444869
$280$	11.2378	0.671587
$281$	10.0706	0.600763	0.300382	−	0.953819i	$-0.402886\pi$
$281$	10.0706	0.600763	0.300382	+	0.953819i	$0.402886\pi$
$282$	14.9310	0.889126
$283$	14.2157	0.845034	0.422517	−	0.906355i	$-0.361147\pi$
$283$	14.2157	0.845034	0.422517	+	0.906355i	$0.361147\pi$
$284$	42.7745	2.53820
$285$	−3.62545	−0.214753
$286$	27.0108	1.59718
$287$	−11.1723	−0.659480
$288$	−2.88962	−0.170272
$289$	−6.24997	−0.367645
$290$	40.5950	2.38382
$291$	−2.35467	−0.138033
$292$	−14.9159	−0.872886
$293$	−12.1129	−0.707641	−0.353820	−	0.935313i	$-0.615118\pi$
$293$	−12.1129	−0.707641	−0.353820	+	0.935313i	$0.615118\pi$
$294$	19.4517	1.13444
$295$	−16.5068	−0.961062
$296$	−10.9781	−0.638086
$297$	−14.0264	−0.813892
$298$	13.0007	0.753112
$299$	18.5114	1.07054
$300$	7.20829	0.416171
$301$	1.35203	0.0779295
$302$	7.73752	0.445244
$303$	−16.3331	−0.938310
$304$	−1.99412	−0.114371
$305$	−32.5754	−1.86526
$306$	7.31513	0.418178
$307$	9.73043	0.555345	0.277673	−	0.960676i	$-0.410437\pi$
$307$	9.73043	0.555345	0.277673	+	0.960676i	$0.410437\pi$
$308$	−10.2952	−0.586624
$309$	−13.6219	−0.774921
$310$	4.74771	0.269652
$311$	11.1637	0.633034	0.316517	−	0.948587i	$-0.397487\pi$
$311$	11.1637	0.633034	0.316517	+	0.948587i	$0.397487\pi$
$312$	25.7111	1.45560
$313$	12.7130	0.718582	0.359291	−	0.933226i	$-0.383019\pi$
$313$	12.7130	0.718582	0.359291	+	0.933226i	$0.383019\pi$
$314$	−30.5684	−1.72508
$315$	2.70079	0.152172
$316$	−1.60835	−0.0904766
$317$	5.22892	0.293686	0.146843	−	0.989160i	$-0.453089\pi$
$317$	5.22892	0.293686	0.146843	+	0.989160i	$0.453089\pi$
$318$	−22.0731	−1.23780
$319$	−16.7820	−0.939611
$320$	−28.5344	−1.59512
$321$	−24.2148	−1.35154
$322$	−10.9275	−0.608964
$323$	−3.27872	−0.182433
$324$	−19.3200	−1.07333
$325$	−6.31460	−0.350271
$326$	−20.5411	−1.13766
$327$	−4.62653	−0.255848
$328$	−38.3328	−2.11657
$329$	4.98539	0.274854
$330$	−21.3649	−1.17610
$331$	−18.8572	−1.03648	−0.518242	−	0.855234i	$-0.673413\pi$
$331$	−18.8572	−1.03648	−0.518242	+	0.855234i	$0.673413\pi$
$332$	−17.6336	−0.967771
$333$	−2.63837	−0.144582
$334$	−20.0806	−1.09876
$335$	24.7528	1.35239
$336$	−3.26019	−0.177858
$337$	24.1191	1.31385	0.656926	−	0.753955i	$-0.271856\pi$
$337$	24.1191	1.31385	0.656926	+	0.753955i	$0.271856\pi$
$338$	−19.0275	−1.03496
$339$	−11.3793	−0.618040
$340$	30.1780	1.63663
$341$	−1.96270	−0.106286
$342$	−2.23109	−0.120644
$343$	14.4667	0.781128
$344$	4.63888	0.250112
$345$	−14.6421	−0.788303
$346$	−32.0898	−1.72516
$347$	4.13597	0.222031	0.111015	−	0.993819i	$-0.464590\pi$
$347$	4.13597	0.222031	0.111015	+	0.993819i	$0.464590\pi$
$348$	−35.4005	−1.89767
$349$	−7.95756	−0.425959	−0.212979	−	0.977057i	$-0.568317\pi$
$349$	−7.95756	−0.425959	−0.212979	+	0.977057i	$0.568317\pi$
$350$	3.72756	0.199247
$351$	25.9194	1.38348
$352$	7.63240	0.406808
$353$	−20.2183	−1.07611	−0.538057	−	0.842909i	$-0.680841\pi$
$353$	−20.2183	−1.07611	−0.538057	+	0.842909i	$0.680841\pi$
$354$	22.2936	1.18489
$355$	29.6388	1.57307
$356$	6.40599	0.339517
$357$	−5.36040	−0.283702
$358$	−42.2468	−2.23281
$359$	−12.9616	−0.684086	−0.342043	−	0.939684i	$-0.611119\pi$
$359$	−12.9616	−0.684086	−0.342043	+	0.939684i	$0.611119\pi$
$360$	9.26657	0.488391
$361$	1.00000	0.0526316
$362$	4.61673	0.242650
$363$	−6.95929	−0.365268
$364$	19.0246	0.997159
$365$	−10.3354	−0.540977
$366$	43.9954	2.29968
$367$	27.3452	1.42741	0.713705	−	0.700447i	$-0.247016\pi$
$367$	27.3452	1.42741	0.713705	+	0.700447i	$0.247016\pi$
$368$	−8.05362	−0.419824
$369$	−9.21257	−0.479587
$370$	−16.8572	−0.876365
$371$	−7.37013	−0.382638
$372$	−4.14020	−0.214659
$373$	17.0738	0.884046	0.442023	−	0.897004i	$-0.354261\pi$
$373$	17.0738	0.884046	0.442023	+	0.897004i	$0.354261\pi$
$374$	−19.3216	−0.999095
$375$	−13.1326	−0.678164
$376$	17.1052	0.882132
$377$	31.0116	1.59718
$378$	−15.3005	−0.786971
$379$	0.263502	0.0135352	0.00676760	−	0.999977i	$-0.497846\pi$
$379$	0.263502	0.0135352	0.00676760	+	0.999977i	$0.497846\pi$
$380$	−9.20419	−0.472165
$381$	−12.9704	−0.664495
$382$	19.3143	0.988205
$383$	−12.9510	−0.661763	−0.330881	−	0.943672i	$-0.607346\pi$
$383$	−12.9510	−0.661763	−0.330881	+	0.943672i	$0.607346\pi$
$384$	29.7029	1.51577
$385$	−7.13365	−0.363564
$386$	−39.1251	−1.99141
$387$	1.11487	0.0566720
$388$	−5.97796	−0.303485
$389$	28.5310	1.44658	0.723289	−	0.690545i	$-0.242629\pi$
$389$	28.5310	1.44658	0.723289	+	0.690545i	$0.242629\pi$
$390$	39.4803	1.99916
$391$	−13.2417	−0.669663
$392$	22.2842	1.12552
$393$	−22.6569	−1.14289
$394$	61.6012	3.10342
$395$	−1.11444	−0.0560735
$396$	−8.48933	−0.426605
$397$	5.73172	0.287667	0.143833	−	0.989602i	$-0.454057\pi$
$397$	5.73172	0.287667	0.143833	+	0.989602i	$0.454057\pi$
$398$	19.2566	0.965246
$399$	1.63490	0.0818476
$400$	2.74724	0.137362
$401$	−14.0156	−0.699906	−0.349953	−	0.936767i	$-0.613802\pi$
$401$	−14.0156	−0.699906	−0.349953	+	0.936767i	$0.613802\pi$
$402$	−33.4305	−1.66736
$403$	3.62689	0.180668
$404$	−41.4659	−2.06301
$405$	−13.3870	−0.665205
$406$	−18.3064	−0.908531
$407$	6.96877	0.345429
$408$	−18.3918	−0.910532
$409$	−31.5277	−1.55895	−0.779473	−	0.626436i	$-0.784513\pi$
$409$	−31.5277	−1.55895	−0.779473	+	0.626436i	$0.784513\pi$
$410$	−58.8614	−2.90696
$411$	17.6240	0.869330
$412$	−34.5828	−1.70377
$413$	7.44376	0.366284
$414$	−9.01068	−0.442851
$415$	−12.2185	−0.599783
$416$	−14.1040	−0.691504
$417$	−13.5501	−0.663554
$418$	5.89302	0.288237
$419$	28.1075	1.37314	0.686571	−	0.727063i	$-0.259115\pi$
$419$	28.1075	1.37314	0.686571	+	0.727063i	$0.259115\pi$
$420$	−15.0480	−0.734266
$421$	−27.9136	−1.36042	−0.680212	−	0.733015i	$-0.738112\pi$
$421$	−27.9136	−1.36042	−0.680212	+	0.733015i	$0.738112\pi$
$422$	2.37585	0.115654
$423$	4.11091	0.199879
$424$	−25.2873	−1.22806
$425$	4.51701	0.219107
$426$	−40.0294	−1.93943
$427$	14.6899	0.710894
$428$	−61.4758	−2.97154
$429$	−16.3211	−0.787992
$430$	7.12318	0.343510
$431$	15.1471	0.729610	0.364805	−	0.931084i	$-0.381136\pi$
$431$	15.1471	0.729610	0.364805	+	0.931084i	$0.381136\pi$
$432$	−11.2766	−0.542544
$433$	33.8713	1.62775	0.813876	−	0.581039i	$-0.197354\pi$
$433$	33.8713	1.62775	0.813876	+	0.581039i	$0.197354\pi$
$434$	−2.14099	−0.102771
$435$	−24.5294	−1.17609
$436$	−11.7457	−0.562516
$437$	4.03869	0.193197
$438$	13.9587	0.666971
$439$	−32.5478	−1.55342	−0.776711	−	0.629858i	$-0.783113\pi$
$439$	−32.5478	−1.55342	−0.776711	+	0.629858i	$0.783113\pi$
$440$	−24.4760	−1.16685
$441$	5.35558	0.255028
$442$	35.7045	1.69829
$443$	−41.9668	−1.99390	−0.996952	−	0.0780194i	$-0.975140\pi$
$443$	−41.9668	−1.99390	−0.996952	+	0.0780194i	$0.975140\pi$
$444$	14.7002	0.697639
$445$	4.43877	0.210418
$446$	−15.0158	−0.711017
$447$	−7.85563	−0.371559
$448$	12.8676	0.607939
$449$	−15.7600	−0.743761	−0.371880	−	0.928281i	$-0.621287\pi$
$449$	−15.7600	−0.743761	−0.371880	+	0.928281i	$0.621287\pi$
$450$	3.07371	0.144896
$451$	24.3333	1.14581
$452$	−28.8895	−1.35885
$453$	−4.67536	−0.219668
$454$	−34.6459	−1.62601
$455$	13.1823	0.617997
$456$	5.60945	0.262687
$457$	9.88190	0.462256	0.231128	−	0.972923i	$-0.425758\pi$
$457$	9.88190	0.462256	0.231128	+	0.972923i	$0.425758\pi$
$458$	−21.5302	−1.00604
$459$	−18.5409	−0.865414
$460$	−37.1728	−1.73319
$461$	−25.0509	−1.16674	−0.583368	−	0.812208i	$-0.698265\pi$
$461$	−25.0509	−1.16674	−0.583368	+	0.812208i	$0.698265\pi$
$462$	9.63452	0.448238
$463$	−22.1604	−1.02988	−0.514941	−	0.857226i	$-0.672186\pi$
$463$	−22.1604	−1.02988	−0.514941	+	0.857226i	$0.672186\pi$
$464$	−13.4920	−0.626349
$465$	−2.86878	−0.133037
$466$	14.6333	0.677877
$467$	1.72160	0.0796662	0.0398331	−	0.999206i	$-0.487317\pi$
$467$	1.72160	0.0796662	0.0398331	+	0.999206i	$0.487317\pi$
$468$	15.6875	0.725155
$469$	−11.1623	−0.515428
$470$	26.2656	1.21154
$471$	18.4708	0.851091
$472$	25.5400	1.17557
$473$	−2.94472	−0.135398
$474$	1.50513	0.0691330
$475$	−1.37767	−0.0632120
$476$	−13.6088	−0.623759
$477$	−6.07733	−0.278262
$478$	40.4858	1.85178
$479$	−2.15984	−0.0986856	−0.0493428	−	0.998782i	$-0.515713\pi$
$479$	−2.15984	−0.0986856	−0.0493428	+	0.998782i	$0.515713\pi$
$480$	11.1559	0.509194
$481$	−12.8776	−0.587170
$482$	−21.6430	−0.985810
$483$	6.60287	0.300441
$484$	−17.6680	−0.803093
$485$	−4.14218	−0.188087
$486$	−22.2254	−1.00817
$487$	9.97500	0.452010	0.226005	−	0.974126i	$-0.427433\pi$
$487$	9.97500	0.452010	0.226005	+	0.974126i	$0.427433\pi$
$488$	50.4019	2.28159
$489$	12.4118	0.561283
$490$	34.2182	1.54582
$491$	8.31159	0.375097	0.187548	−	0.982255i	$-0.439946\pi$
$491$	8.31159	0.375097	0.187548	+	0.982255i	$0.439946\pi$
$492$	51.3296	2.31412
$493$	−22.1834	−0.999091
$494$	−10.8897	−0.489953
$495$	−5.88234	−0.264391
$496$	−1.57793	−0.0708509
$497$	−13.3657	−0.599533
$498$	16.5020	0.739472
$499$	25.3529	1.13495	0.567476	−	0.823390i	$-0.307920\pi$
$499$	25.3529	1.13495	0.567476	+	0.823390i	$0.307920\pi$
$500$	−33.3406	−1.49104
$501$	12.1336	0.542090
$502$	2.89878	0.129379
$503$	−21.8199	−0.972903	−0.486451	−	0.873708i	$-0.661709\pi$
$503$	−21.8199	−0.972903	−0.486451	+	0.873708i	$0.661709\pi$
$504$	−4.17878	−0.186137
$505$	−28.7321	−1.27856
$506$	23.8000	1.05804
$507$	11.4972	0.510611
$508$	−32.9289	−1.46099
$509$	−22.5365	−0.998912	−0.499456	−	0.866339i	$-0.666467\pi$
$509$	−22.5365	−0.998912	−0.499456	+	0.866339i	$0.666467\pi$
$510$	−28.2413	−1.25055
$511$	4.66074	0.206179
$512$	21.7198	0.959888
$513$	5.65491	0.249670
$514$	65.2799	2.87937
$515$	−23.9627	−1.05592
$516$	−6.21170	−0.273455
$517$	−10.8582	−0.477543
$518$	7.60178	0.334003
$519$	19.3901	0.851130
$520$	45.2293	1.98344
$521$	22.0123	0.964377	0.482189	−	0.876067i	$-0.339842\pi$
$521$	22.0123	0.964377	0.482189	+	0.876067i	$0.339842\pi$
$522$	−15.0953	−0.660702
$523$	−39.7805	−1.73948	−0.869739	−	0.493511i	$-0.835713\pi$
$523$	−39.7805	−1.73948	−0.869739	+	0.493511i	$0.835713\pi$
$524$	−57.5206	−2.51280
$525$	−2.25236	−0.0983012
$526$	62.2185	2.71286
$527$	−2.59442	−0.113015
$528$	7.10072	0.309019
$529$	−6.68901	−0.290827
$530$	−38.8296	−1.68665
$531$	6.13805	0.266369
$532$	4.15065	0.179953
$533$	−44.9657	−1.94768
$534$	−5.99489	−0.259424
$535$	−42.5971	−1.84163
$536$	−38.2986	−1.65425
$537$	25.5274	1.10159
$538$	−4.00110	−0.172500
$539$	−14.1458	−0.609302
$540$	−52.0488	−2.23983
$541$	32.8912	1.41410	0.707051	−	0.707163i	$-0.250025\pi$
$541$	32.8912	1.41410	0.707051	+	0.707163i	$0.250025\pi$
$542$	−18.5319	−0.796014
$543$	−2.78964	−0.119715
$544$	10.0890	0.432561
$545$	−8.13870	−0.348623
$546$	−17.8037	−0.761928
$547$	−21.9743	−0.939551	−0.469776	−	0.882786i	$-0.655665\pi$
$547$	−21.9743	−0.939551	−0.469776	+	0.882786i	$0.655665\pi$
$548$	44.7434	1.91134
$549$	12.1132	0.516977
$550$	−8.11865	−0.346180
$551$	6.76587	0.288236
$552$	22.6548	0.964253
$553$	0.502558	0.0213709
$554$	−24.3007	−1.03244
$555$	10.1859	0.432367
$556$	−34.4007	−1.45891
$557$	−21.1039	−0.894201	−0.447101	−	0.894484i	$-0.647543\pi$
$557$	−21.1039	−0.894201	−0.447101	+	0.894484i	$0.647543\pi$
$558$	−1.76544	−0.0747370
$559$	5.44157	0.230154
$560$	−5.73513	−0.242354
$561$	11.6750	0.492918
$562$	−23.9262	−1.00927
$563$	45.6019	1.92189	0.960945	−	0.276738i	$-0.0892535\pi$
$563$	45.6019	1.92189	0.960945	+	0.276738i	$0.0892535\pi$
$564$	−22.9047	−0.964461
$565$	−20.0178	−0.842154
$566$	−33.7742	−1.41964
$567$	6.03688	0.253525
$568$	−45.8584	−1.92418
$569$	−35.9478	−1.50701	−0.753504	−	0.657443i	$-0.771638\pi$
$569$	−35.9478	−1.50701	−0.753504	+	0.657443i	$0.771638\pi$
$570$	8.61352	0.360781
$571$	29.1319	1.21913	0.609565	−	0.792736i	$-0.291344\pi$
$571$	29.1319	1.21913	0.609565	+	0.792736i	$0.291344\pi$
$572$	−41.4356	−1.73251
$573$	−11.6706	−0.487545
$574$	26.5437	1.10791
$575$	−5.56399	−0.232034
$576$	10.6105	0.442106
$577$	35.2791	1.46869	0.734343	−	0.678778i	$-0.237490\pi$
$577$	35.2791	1.46869	0.734343	+	0.678778i	$0.237490\pi$
$578$	14.8490	0.617635
$579$	23.6411	0.982492
$580$	−62.2744	−2.58580
$581$	5.50995	0.228591
$582$	5.59432	0.231892
$583$	16.0522	0.664813
$584$	15.9913	0.661724
$585$	10.8700	0.449420
$586$	28.7783	1.18882
$587$	26.7137	1.10259	0.551296	−	0.834309i	$-0.314133\pi$
$587$	26.7137	1.10259	0.551296	+	0.834309i	$0.314133\pi$
$588$	−29.8396	−1.23057
$589$	0.791289	0.0326045
$590$	39.2176	1.61456
$591$	−37.2222	−1.53112
$592$	5.60258	0.230264
$593$	−7.93842	−0.325992	−0.162996	−	0.986627i	$-0.552116\pi$
$593$	−7.93842	−0.325992	−0.162996	+	0.986627i	$0.552116\pi$
$594$	33.3245	1.36732
$595$	−9.42968	−0.386579
$596$	−19.9436	−0.816923
$597$	−11.6357	−0.476218
$598$	−43.9803	−1.79849
$599$	−4.44037	−0.181428	−0.0907142	−	0.995877i	$-0.528915\pi$
$599$	−4.44037	−0.181428	−0.0907142	+	0.995877i	$0.528915\pi$
$600$	−7.72799	−0.315494
$601$	−8.37157	−0.341484	−0.170742	−	0.985316i	$-0.554616\pi$
$601$	−8.37157	−0.341484	−0.170742	+	0.985316i	$0.554616\pi$
$602$	−3.21221	−0.130920
$603$	−9.20434	−0.374830
$604$	−11.8697	−0.482970
$605$	−12.2424	−0.497722
$606$	38.8048	1.57634
$607$	−46.3908	−1.88294	−0.941472	−	0.337092i	$-0.890557\pi$
$607$	−46.3908	−1.88294	−0.941472	+	0.337092i	$0.890557\pi$
$608$	−3.07710	−0.124793
$609$	11.0616	0.448237
$610$	77.3940	3.13359
$611$	20.0650	0.811741
$612$	−11.2217	−0.453610
$613$	−6.46283	−0.261031	−0.130516	−	0.991446i	$-0.541663\pi$
$613$	−6.46283	−0.261031	−0.130516	+	0.991446i	$0.541663\pi$
$614$	−23.1180	−0.932966
$615$	35.5667	1.43419
$616$	11.0375	0.444712
$617$	6.79713	0.273642	0.136821	−	0.990596i	$-0.456311\pi$
$617$	6.79713	0.273642	0.136821	+	0.990596i	$0.456311\pi$
$618$	32.3634	1.30185
$619$	13.0655	0.525145	0.262572	−	0.964912i	$-0.415429\pi$
$619$	13.0655	0.525145	0.262572	+	0.964912i	$0.415429\pi$
$620$	−7.28318	−0.292499
$621$	22.8384	0.916473
$622$	−26.5231	−1.06348
$623$	−2.00167	−0.0801952
$624$	−13.1215	−0.525279
$625$	−29.9904	−1.19962
$626$	−30.2041	−1.20720
$627$	−3.56083	−0.142206
$628$	46.8932	1.87124
$629$	9.21173	0.367296
$630$	−6.41667	−0.255646
$631$	8.05858	0.320807	0.160404	−	0.987052i	$-0.448720\pi$
$631$	8.05858	0.320807	0.160404	+	0.987052i	$0.448720\pi$
$632$	1.72430	0.0685892
$633$	−1.43559	−0.0570597
$634$	−12.4231	−0.493385
$635$	−22.8168	−0.905456
$636$	33.8610	1.34268
$637$	26.1401	1.03571
$638$	39.8714	1.57852
$639$	−11.0212	−0.435992
$640$	52.2515	2.06542
$641$	−26.4728	−1.04561	−0.522807	−	0.852451i	$-0.675115\pi$
$641$	−26.4728	−1.04561	−0.522807	+	0.852451i	$0.675115\pi$
$642$	57.5306	2.27055
$643$	−38.1211	−1.50335	−0.751675	−	0.659534i	$-0.770754\pi$
$643$	−38.1211	−1.50335	−0.751675	+	0.659534i	$0.770754\pi$
$644$	16.7632	0.660561
$645$	−4.30414	−0.169476
$646$	7.78974	0.306483
$647$	10.6570	0.418968	0.209484	−	0.977812i	$-0.432822\pi$
$647$	10.6570	0.418968	0.209484	+	0.977812i	$0.432822\pi$
$648$	20.7129	0.813679
$649$	−16.2125	−0.636398
$650$	15.0025	0.588447
$651$	1.29368	0.0507034
$652$	31.5108	1.23406
$653$	25.2146	0.986722	0.493361	−	0.869825i	$-0.335768\pi$
$653$	25.2146	0.986722	0.493361	+	0.869825i	$0.335768\pi$
$654$	10.9919	0.429818
$655$	−39.8566	−1.55733
$656$	19.5629	0.763802
$657$	3.84320	0.149938
$658$	−11.8445	−0.461748
$659$	−11.9460	−0.465350	−0.232675	−	0.972555i	$-0.574748\pi$
$659$	−11.9460	−0.465350	−0.232675	+	0.972555i	$0.574748\pi$
$660$	32.7745	1.27575
$661$	−17.5333	−0.681968	−0.340984	−	0.940069i	$-0.610760\pi$
$661$	−17.5333	−0.681968	−0.340984	+	0.940069i	$0.610760\pi$
$662$	44.8017	1.74127
$663$	−21.5743	−0.837875
$664$	18.9050	0.733655
$665$	2.87602	0.111527
$666$	6.26836	0.242894
$667$	27.3252	1.05804
$668$	30.8044	1.19186
$669$	9.07321	0.350791
$670$	−58.8089	−2.27198
$671$	−31.9947	−1.23514
$672$	−5.03076	−0.194066
$673$	7.73041	0.297985	0.148993	−	0.988838i	$-0.452397\pi$
$673$	7.73041	0.297985	0.148993	+	0.988838i	$0.452397\pi$
$674$	−57.3033	−2.20724
$675$	−7.79061	−0.299861
$676$	29.1889	1.12265
$677$	−9.94825	−0.382342	−0.191171	−	0.981557i	$-0.561229\pi$
$677$	−9.94825	−0.382342	−0.191171	+	0.981557i	$0.561229\pi$
$678$	27.0355	1.03829
$679$	1.86792	0.0716843
$680$	−32.3538	−1.24071
$681$	20.9346	0.802216
$682$	4.66308	0.178559
$683$	3.12007	0.119386	0.0596930	−	0.998217i	$-0.480988\pi$
$683$	3.12007	0.119386	0.0596930	+	0.998217i	$0.480988\pi$
$684$	3.42258	0.130866
$685$	31.0031	1.18457
$686$	−34.3706	−1.31228
$687$	13.0095	0.496343
$688$	−2.36742	−0.0902571
$689$	−29.6629	−1.13007
$690$	34.7873	1.32433
$691$	−11.8140	−0.449426	−0.224713	−	0.974425i	$-0.572145\pi$
$691$	−11.8140	−0.449426	−0.224713	+	0.974425i	$0.572145\pi$
$692$	49.2269	1.87133
$693$	2.65265	0.100766
$694$	−9.82643	−0.373006
$695$	−23.8366	−0.904173
$696$	37.9528	1.43860
$697$	32.1652	1.21834
$698$	18.9059	0.715600
$699$	−8.84213	−0.334440
$700$	−5.71823	−0.216129
$701$	−43.4059	−1.63942	−0.819709	−	0.572780i	$-0.805865\pi$
$701$	−43.4059	−1.63942	−0.819709	+	0.572780i	$0.805865\pi$
$702$	−61.5805	−2.32421
$703$	−2.80955	−0.105964
$704$	−28.0258	−1.05626
$705$	−15.8709	−0.597732
$706$	48.0356	1.80784
$707$	12.9568	0.487290
$708$	−34.1993	−1.28529
$709$	−12.8071	−0.480982	−0.240491	−	0.970651i	$-0.577308\pi$
$709$	−12.8071	−0.480982	−0.240491	+	0.970651i	$0.577308\pi$
$710$	−70.4173	−2.64271
$711$	0.414404	0.0155414
$712$	−6.86784	−0.257383
$713$	3.19577	0.119682
$714$	12.7355	0.476613
$715$	−28.7111	−1.07374
$716$	64.8082	2.42200
$717$	−24.4634	−0.913601
$718$	30.7947	1.14925
$719$	23.7402	0.885361	0.442681	−	0.896679i	$-0.354028\pi$
$719$	23.7402	0.885361	0.442681	+	0.896679i	$0.354028\pi$
$720$	−4.72913	−0.176244
$721$	10.8060	0.402438
$722$	−2.37585	−0.0884198
$723$	13.0777	0.486363
$724$	−7.08224	−0.263210
$725$	−9.32116	−0.346179
$726$	16.5342	0.613642
$727$	−25.4903	−0.945383	−0.472692	−	0.881228i	$-0.656718\pi$
$727$	−25.4903	−0.945383	−0.472692	+	0.881228i	$0.656718\pi$
$728$	−20.3962	−0.755934
$729$	29.3324	1.08638
$730$	24.5552	0.908829
$731$	−3.89251	−0.143970
$732$	−67.4907	−2.49453
$733$	28.4500	1.05083	0.525413	−	0.850848i	$-0.323911\pi$
$733$	28.4500	1.05083	0.525413	+	0.850848i	$0.323911\pi$
$734$	−64.9680	−2.39801
$735$	−20.6762	−0.762652
$736$	−12.4274	−0.458081
$737$	24.3116	0.895528
$738$	21.8876	0.805695
$739$	23.5866	0.867648	0.433824	−	0.900998i	$-0.357164\pi$
$739$	23.5866	0.867648	0.433824	+	0.900998i	$0.357164\pi$
$740$	25.8596	0.950619
$741$	6.58008	0.241725
$742$	17.5103	0.642823
$743$	−7.74097	−0.283989	−0.141994	−	0.989867i	$-0.545351\pi$
$743$	−7.74097	−0.283989	−0.141994	+	0.989867i	$0.545351\pi$
$744$	4.43870	0.162730
$745$	−13.8191	−0.506294
$746$	−40.5646	−1.48518
$747$	4.54345	0.166236
$748$	29.6401	1.08375
$749$	19.2092	0.701891
$750$	31.2010	1.13930
$751$	5.68396	0.207411	0.103705	−	0.994608i	$-0.466930\pi$
$751$	5.68396	0.207411	0.103705	+	0.994608i	$0.466930\pi$
$752$	−8.72951	−0.318332
$753$	−1.75157	−0.0638309
$754$	−73.6786	−2.68322
$755$	−8.22460	−0.299324
$756$	23.4715	0.853650
$757$	16.3405	0.593907	0.296954	−	0.954892i	$-0.404029\pi$
$757$	16.3405	0.593907	0.296954	+	0.954892i	$0.404029\pi$
$758$	−0.626040	−0.0227388
$759$	−14.3811	−0.522000
$760$	9.86779	0.357942
$761$	15.3257	0.555557	0.277778	−	0.960645i	$-0.410402\pi$
$761$	15.3257	0.555557	0.277778	+	0.960645i	$0.410402\pi$
$762$	30.8157	1.11634
$763$	3.67016	0.132869
$764$	−29.6289	−1.07194
$765$	−7.77562	−0.281128
$766$	30.7695	1.11175
$767$	29.9593	1.08177
$768$	−38.1282	−1.37583
$769$	36.4857	1.31571	0.657855	−	0.753145i	$-0.271464\pi$
$769$	36.4857	1.31571	0.657855	+	0.753145i	$0.271464\pi$
$770$	16.9484	0.610779
$771$	−39.4451	−1.42058
$772$	60.0194	2.16014
$773$	−0.0399960	−0.00143855	−0.000719277	−	1.00000i	$-0.500229\pi$
$773$	−0.0399960	−0.00143855	−0.000719277		1.00000i	$0.500229\pi$
$774$	−2.64875	−0.0952075
$775$	−1.09014	−0.0391589
$776$	6.40895	0.230068
$777$	−4.59334	−0.164785
$778$	−67.7852	−2.43022
$779$	−9.81028	−0.351490
$780$	−60.5643	−2.16855
$781$	29.1105	1.04166
$782$	31.4603	1.12502
$783$	38.2604	1.36731
$784$	−11.3726	−0.406163
$785$	32.4927	1.15972
$786$	53.8293	1.92003
$787$	−13.1168	−0.467564	−0.233782	−	0.972289i	$-0.575110\pi$
$787$	−13.1168	−0.467564	−0.233782	+	0.972289i	$0.575110\pi$
$788$	−94.4986	−3.36637
$789$	−37.5952	−1.33843
$790$	2.64773	0.0942021
$791$	9.02705	0.320965
$792$	9.10139	0.323404
$793$	59.1232	2.09953
$794$	−13.6177	−0.483273
$795$	23.4626	0.832133
$796$	−29.5404	−1.04703
$797$	24.6833	0.874328	0.437164	−	0.899382i	$-0.355983\pi$
$797$	24.6833	0.874328	0.437164	+	0.899382i	$0.355983\pi$
$798$	−3.88428	−0.137502
$799$	−14.3530	−0.507773
$800$	4.23924	0.149880
$801$	−1.65056	−0.0583196
$802$	33.2989	1.17582
$803$	−10.1511	−0.358225
$804$	51.2837	1.80864
$805$	11.6153	0.409387
$806$	−8.61694	−0.303519
$807$	2.41765	0.0851052
$808$	44.4555	1.56394
$809$	19.7808	0.695456	0.347728	−	0.937595i	$-0.386953\pi$
$809$	19.7808	0.695456	0.347728	+	0.937595i	$0.386953\pi$
$810$	31.8054	1.11753
$811$	−0.362366	−0.0127244	−0.00636220	−	0.999980i	$-0.502025\pi$
$811$	−0.362366	−0.0127244	−0.00636220	+	0.999980i	$0.502025\pi$
$812$	28.0827	0.985511
$813$	11.1978	0.392725
$814$	−16.5567	−0.580313
$815$	21.8341	0.764817
$816$	9.38615	0.328581
$817$	1.18720	0.0415349
$818$	74.9050	2.61899
$819$	−4.90185	−0.171284
$820$	90.2958	3.15326
$821$	18.0022	0.628281	0.314140	−	0.949377i	$-0.398284\pi$
$821$	18.0022	0.628281	0.314140	+	0.949377i	$0.398284\pi$
$822$	−41.8720	−1.46045
$823$	−24.4761	−0.853185	−0.426592	−	0.904444i	$-0.640286\pi$
$823$	−24.4761	−0.853185	−0.426592	+	0.904444i	$0.640286\pi$
$824$	37.0761	1.29161
$825$	4.90565	0.170793
$826$	−17.6852	−0.615348
$827$	17.5790	0.611283	0.305641	−	0.952147i	$-0.401129\pi$
$827$	17.5790	0.611283	0.305641	+	0.952147i	$0.401129\pi$
$828$	13.8227	0.480373
$829$	21.5120	0.747143	0.373572	−	0.927601i	$-0.378133\pi$
$829$	21.5120	0.747143	0.373572	+	0.927601i	$0.378133\pi$
$830$	29.0293	1.00762
$831$	14.6836	0.509368
$832$	51.7890	1.79546
$833$	−18.6987	−0.647873
$834$	32.1931	1.11475
$835$	21.3447	0.738663
$836$	−9.04012	−0.312659
$837$	4.47466	0.154667
$838$	−66.7791	−2.30685
$839$	−3.06407	−0.105784	−0.0528918	−	0.998600i	$-0.516844\pi$
$839$	−3.06407	−0.105784	−0.0528918	+	0.998600i	$0.516844\pi$
$840$	16.1329	0.556638
$841$	16.7770	0.578518
$842$	66.3184	2.28548
$843$	14.4573	0.497937
$844$	−3.64464	−0.125454
$845$	20.2252	0.695770
$846$	−9.76688	−0.335792
$847$	5.52071	0.189694
$848$	12.9052	0.443167
$849$	20.4079	0.700398
$850$	−10.7317	−0.368095
$851$	−11.3469	−0.388966
$852$	61.4067	2.10376
$853$	−35.2973	−1.20856	−0.604279	−	0.796773i	$-0.706539\pi$
$853$	−35.2973	−1.20856	−0.604279	+	0.796773i	$0.706539\pi$
$854$	−34.9009	−1.19429
$855$	2.37154	0.0811049
$856$	65.9080	2.25269
$857$	6.64294	0.226918	0.113459	−	0.993543i	$-0.463807\pi$
$857$	6.64294	0.226918	0.113459	+	0.993543i	$0.463807\pi$
$858$	38.7765	1.32381
$859$	−29.8823	−1.01957	−0.509786	−	0.860301i	$-0.670275\pi$
$859$	−29.8823	−1.01957	−0.509786	+	0.860301i	$0.670275\pi$
$860$	−10.9272	−0.372616
$861$	−16.0389	−0.546603
$862$	−35.9872	−1.22573
$863$	17.8254	0.606784	0.303392	−	0.952866i	$-0.401881\pi$
$863$	17.8254	0.606784	0.303392	+	0.952866i	$0.401881\pi$
$864$	−17.4007	−0.591984
$865$	34.1098	1.15977
$866$	−80.4730	−2.73458
$867$	−8.97241	−0.304719
$868$	3.28436	0.111478
$869$	−1.09457	−0.0371308
$870$	58.2780	1.97581
$871$	−44.9255	−1.52224
$872$	12.5925	0.426437
$873$	1.54027	0.0521303
$874$	−9.59529	−0.324566
$875$	10.4179	0.352189
$876$	−21.4131	−0.723483
$877$	25.5704	0.863450	0.431725	−	0.902005i	$-0.357905\pi$
$877$	25.5704	0.863450	0.431725	+	0.902005i	$0.357905\pi$
$878$	77.3285	2.60971
$879$	−17.3891	−0.586521
$880$	12.4911	0.421076
$881$	8.43465	0.284170	0.142085	−	0.989854i	$-0.454619\pi$
$881$	8.43465	0.284170	0.142085	+	0.989854i	$0.454619\pi$
$882$	−12.7240	−0.428440
$883$	7.36444	0.247833	0.123917	−	0.992293i	$-0.460454\pi$
$883$	7.36444	0.247833	0.123917	+	0.992293i	$0.460454\pi$
$884$	−54.7721	−1.84218
$885$	−23.6970	−0.796567
$886$	99.7067	3.34971
$887$	−41.5244	−1.39425	−0.697126	−	0.716948i	$-0.745538\pi$
$887$	−41.5244	−1.39425	−0.697126	+	0.716948i	$0.745538\pi$
$888$	−15.7600	−0.528872
$889$	10.2893	0.345091
$890$	−10.5458	−0.353497
$891$	−13.1483	−0.440486
$892$	23.0348	0.771262
$893$	4.37762	0.146492
$894$	18.6638	0.624209
$895$	44.9062	1.50105
$896$	−23.5629	−0.787182
$897$	26.5749	0.887309
$898$	37.4433	1.24950
$899$	5.35376	0.178558
$900$	−4.71520	−0.157173
$901$	21.2187	0.706897
$902$	−57.8122	−1.92493
$903$	1.94096	0.0645911
$904$	30.9723	1.03012
$905$	−4.90735	−0.163126
$906$	11.1079	0.369036
$907$	59.4041	1.97248	0.986240	−	0.165320i	$-0.0528658\pi$
$907$	59.4041	1.97248	0.986240	+	0.165320i	$0.0528658\pi$
$908$	53.1482	1.76378
$909$	10.6840	0.354367
$910$	−31.3191	−1.03822
$911$	−21.9104	−0.725925	−0.362963	−	0.931804i	$-0.618235\pi$
$911$	−21.9104	−0.725925	−0.362963	+	0.931804i	$0.618235\pi$
$912$	−2.86274	−0.0947949
$913$	−12.0007	−0.397165
$914$	−23.4779	−0.776578
$915$	−46.7649	−1.54600
$916$	33.0281	1.09128
$917$	17.9734	0.593533
$918$	44.0502	1.45387
$919$	−13.6440	−0.450073	−0.225037	−	0.974350i	$-0.572250\pi$
$919$	−13.6440	−0.450073	−0.225037	+	0.974350i	$0.572250\pi$
$920$	39.8529	1.31391
$921$	13.9689	0.460292
$922$	59.5170	1.96009
$923$	−53.7935	−1.77063
$924$	−14.7797	−0.486217
$925$	3.87064	0.127266
$926$	52.6497	1.73018
$927$	8.91055	0.292661
$928$	−20.8193	−0.683426
$929$	−15.1140	−0.495874	−0.247937	−	0.968776i	$-0.579753\pi$
$929$	−15.1140	−0.495874	−0.247937	+	0.968776i	$0.579753\pi$
$930$	6.81578	0.223498
$931$	5.70305	0.186910
$932$	−22.4481	−0.735313
$933$	16.0265	0.524684
$934$	−4.09026	−0.133837
$935$	20.5379	0.671660
$936$	−16.8185	−0.549731
$937$	45.2576	1.47850	0.739250	−	0.673431i	$-0.235180\pi$
$937$	45.2576	1.47850	0.739250	+	0.673431i	$0.235180\pi$
$938$	26.5199	0.865907
$939$	18.2507	0.595589
$940$	−40.2925	−1.31420
$941$	47.1601	1.53738	0.768688	−	0.639624i	$-0.220910\pi$
$941$	47.1601	1.53738	0.768688	+	0.639624i	$0.220910\pi$
$942$	−43.8839	−1.42981
$943$	−39.6207	−1.29023
$944$	−13.0341	−0.424225
$945$	16.2636	0.529056
$946$	6.99620	0.227466
$947$	−45.4877	−1.47815	−0.739076	−	0.673623i	$-0.764737\pi$
$947$	−45.4877	−1.47815	−0.739076	+	0.673623i	$0.764737\pi$
$948$	−2.30893	−0.0749906
$949$	18.7583	0.608921
$950$	3.27314	0.106195
$951$	7.50661	0.243418
$952$	14.5900	0.472864
$953$	12.5349	0.406044	0.203022	−	0.979174i	$-0.434924\pi$
$953$	12.5349	0.406044	0.203022	+	0.979174i	$0.434924\pi$
$954$	14.4388	0.467474
$955$	−20.5301	−0.664340
$956$	−62.1069	−2.00868
$957$	−24.0921	−0.778787
$958$	5.13144	0.165789
$959$	−13.9809	−0.451467
$960$	−40.9638	−1.32210
$961$	−30.3739	−0.979802
$962$	30.5953	0.986431
$963$	15.8398	0.510429
$964$	33.2012	1.06934
$965$	41.5880	1.33876
$966$	−15.6874	−0.504733
$967$	10.3814	0.333843	0.166921	−	0.985970i	$-0.446617\pi$
$967$	10.3814	0.333843	0.166921	+	0.985970i	$0.446617\pi$
$968$	18.9419	0.608814
$969$	−4.70691	−0.151208
$970$	9.84118	0.315981
$971$	15.0361	0.482532	0.241266	−	0.970459i	$-0.422437\pi$
$971$	15.0361	0.482532	0.241266	+	0.970459i	$0.422437\pi$
$972$	34.0947	1.09359
$973$	10.7491	0.344602
$974$	−23.6990	−0.759366
$975$	−9.06519	−0.290319
$976$	−25.7223	−0.823350
$977$	27.8976	0.892524	0.446262	−	0.894902i	$-0.352755\pi$
$977$	27.8976	0.892524	0.446262	+	0.894902i	$0.352755\pi$
$978$	−29.4886	−0.942942
$979$	4.35964	0.139335
$980$	−52.4920	−1.67680
$981$	3.02638	0.0966248
$982$	−19.7471	−0.630154
$983$	4.13430	0.131864	0.0659319	−	0.997824i	$-0.478998\pi$
$983$	4.13430	0.131864	0.0659319	+	0.997824i	$0.478998\pi$
$984$	−55.0303	−1.75430
$985$	−65.4790	−2.08633
$986$	52.7044	1.67845
$987$	7.15700	0.227810
$988$	16.7053	0.531466
$989$	4.79473	0.152464
$990$	13.9755	0.444171
$991$	−29.0377	−0.922413	−0.461207	−	0.887293i	$-0.652583\pi$
$991$	−29.0377	−0.922413	−0.461207	+	0.887293i	$0.652583\pi$
$992$	−2.43487	−0.0773074
$993$	−27.0712	−0.859079
$994$	31.7548	1.00720
$995$	−20.4688	−0.648905
$996$	−25.3147	−0.802127
$997$	−27.1771	−0.860708	−0.430354	−	0.902660i	$-0.641611\pi$
$997$	−27.1771	−0.860708	−0.430354	+	0.902660i	$0.641611\pi$
$998$	−60.2345	−1.90669
$999$	−15.8877	−0.502665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.9	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.9	✓	75	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.37585 q^{2} +1.43559 q^{3} +3.64464 q^{4} +2.52541 q^{5} -3.41075 q^{6} -1.13884 q^{7} -3.90741 q^{8} -0.939073 q^{9} +O(q^{10})\)	\(q-2.37585 q^{2} +1.43559 q^{3} +3.64464 q^{4} +2.52541 q^{5} -3.41075 q^{6} -1.13884 q^{7} -3.90741 q^{8} -0.939073 q^{9} -5.99997 q^{10} +2.48039 q^{11} +5.23222 q^{12} -4.58353 q^{13} +2.70570 q^{14} +3.62545 q^{15} +1.99412 q^{16} +3.27872 q^{17} +2.23109 q^{18} -1.00000 q^{19} +9.20419 q^{20} -1.63490 q^{21} -5.89302 q^{22} -4.03869 q^{23} -5.60945 q^{24} +1.37767 q^{25} +10.8897 q^{26} -5.65491 q^{27} -4.15065 q^{28} -6.76587 q^{29} -8.61352 q^{30} -0.791289 q^{31} +3.07710 q^{32} +3.56083 q^{33} -7.78974 q^{34} -2.87602 q^{35} -3.42258 q^{36} +2.80955 q^{37} +2.37585 q^{38} -6.58008 q^{39} -9.86779 q^{40} +9.81028 q^{41} +3.88428 q^{42} -1.18720 q^{43} +9.04012 q^{44} -2.37154 q^{45} +9.59529 q^{46} -4.37762 q^{47} +2.86274 q^{48} -5.70305 q^{49} -3.27314 q^{50} +4.70691 q^{51} -16.7053 q^{52} +6.47163 q^{53} +13.4352 q^{54} +6.26398 q^{55} +4.44990 q^{56} -1.43559 q^{57} +16.0747 q^{58} -6.53629 q^{59} +13.2135 q^{60} -12.8991 q^{61} +1.87998 q^{62} +1.06945 q^{63} -11.2990 q^{64} -11.5753 q^{65} -8.45997 q^{66} +9.80152 q^{67} +11.9498 q^{68} -5.79791 q^{69} +6.83298 q^{70} +11.7363 q^{71} +3.66934 q^{72} -4.09255 q^{73} -6.67505 q^{74} +1.97778 q^{75} -3.64464 q^{76} -2.82475 q^{77} +15.6332 q^{78} -0.441291 q^{79} +5.03596 q^{80} -5.30092 q^{81} -23.3077 q^{82} -4.83824 q^{83} -5.95864 q^{84} +8.28011 q^{85} +2.82061 q^{86} -9.71304 q^{87} -9.69189 q^{88} +1.75765 q^{89} +5.63441 q^{90} +5.21988 q^{91} -14.7196 q^{92} -1.13597 q^{93} +10.4006 q^{94} -2.52541 q^{95} +4.41746 q^{96} -1.64020 q^{97} +13.5496 q^{98} -2.32926 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more