LMFDB - Embedded newform 1502.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1502,2,Mod(1,1502)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1502, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1502.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1502 = 2 \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1502.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:	$11.9935303836$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} + \cdots + 4222 $ x^19 - 6x^18 - 23x^17 + 190x^16 + 128x^15 - 2394x^14 + 749x^13 + 15539x^12 - 11338x^11 - 56744x^10 + 50183x^9 + 120237x^8 - 102992x^7 - 148539x^6 + 99519x^5 + 104388x^4 - 41081x^3 - 35733x^2 + 5542x + 4222
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.01539$ of defining polynomial
Character	$\chi$	$=$	1502.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.01539 q^{3} +1.00000 q^{4} -2.64906 q^{5} +3.01539 q^{6} +2.56889 q^{7} -1.00000 q^{8} +6.09259 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.01539 q^{3} +1.00000 q^{4} -2.64906 q^{5} +3.01539 q^{6} +2.56889 q^{7} -1.00000 q^{8} +6.09259 q^{9} +2.64906 q^{10} -4.11236 q^{11} -3.01539 q^{12} -0.291418 q^{13} -2.56889 q^{14} +7.98796 q^{15} +1.00000 q^{16} +4.20214 q^{17} -6.09259 q^{18} -2.36721 q^{19} -2.64906 q^{20} -7.74620 q^{21} +4.11236 q^{22} -6.58360 q^{23} +3.01539 q^{24} +2.01752 q^{25} +0.291418 q^{26} -9.32537 q^{27} +2.56889 q^{28} +0.310105 q^{29} -7.98796 q^{30} +0.795398 q^{31} -1.00000 q^{32} +12.4004 q^{33} -4.20214 q^{34} -6.80514 q^{35} +6.09259 q^{36} -2.31520 q^{37} +2.36721 q^{38} +0.878740 q^{39} +2.64906 q^{40} -10.3816 q^{41} +7.74620 q^{42} -6.46147 q^{43} -4.11236 q^{44} -16.1396 q^{45} +6.58360 q^{46} +1.11419 q^{47} -3.01539 q^{48} -0.400824 q^{49} -2.01752 q^{50} -12.6711 q^{51} -0.291418 q^{52} +2.49862 q^{53} +9.32537 q^{54} +10.8939 q^{55} -2.56889 q^{56} +7.13808 q^{57} -0.310105 q^{58} -6.43815 q^{59} +7.98796 q^{60} +3.73626 q^{61} -0.795398 q^{62} +15.6512 q^{63} +1.00000 q^{64} +0.771984 q^{65} -12.4004 q^{66} -2.21498 q^{67} +4.20214 q^{68} +19.8521 q^{69} +6.80514 q^{70} +5.40982 q^{71} -6.09259 q^{72} +9.50064 q^{73} +2.31520 q^{74} -6.08363 q^{75} -2.36721 q^{76} -10.5642 q^{77} -0.878740 q^{78} +12.9489 q^{79} -2.64906 q^{80} +9.84189 q^{81} +10.3816 q^{82} +14.7199 q^{83} -7.74620 q^{84} -11.1317 q^{85} +6.46147 q^{86} -0.935089 q^{87} +4.11236 q^{88} -7.31750 q^{89} +16.1396 q^{90} -0.748620 q^{91} -6.58360 q^{92} -2.39844 q^{93} -1.11419 q^{94} +6.27089 q^{95} +3.01539 q^{96} +10.9490 q^{97} +0.400824 q^{98} -25.0549 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) $ 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9 - 2 * q^10 - 7 * q^11 + 6 * q^12 + 19 * q^13 - 13 * q^14 + 6 * q^15 + 19 * q^16 + 11 * q^17 - 25 * q^18 + 7 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 12 * q^23 - 6 * q^24 + 37 * q^25 - 19 * q^26 + 24 * q^27 + 13 * q^28 - 8 * q^29 - 6 * q^30 + 32 * q^31 - 19 * q^32 + 24 * q^33 - 11 * q^34 - 19 * q^35 + 25 * q^36 + 41 * q^37 - 7 * q^38 - 2 * q^39 - 2 * q^40 + 15 * q^41 - 7 * q^42 + 15 * q^43 - 7 * q^44 + 28 * q^45 - 12 * q^46 - 6 * q^47 + 6 * q^48 + 42 * q^49 - 37 * q^50 + 8 * q^51 + 19 * q^52 + 6 * q^53 - 24 * q^54 + 22 * q^55 - 13 * q^56 + 24 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^60 + 13 * q^61 - 32 * q^62 + 37 * q^63 + 19 * q^64 + 20 * q^65 - 24 * q^66 + 47 * q^67 + 11 * q^68 + 15 * q^69 + 19 * q^70 - 25 * q^72 + 64 * q^73 - 41 * q^74 - 3 * q^75 + 7 * q^76 - 4 * q^77 + 2 * q^78 + 34 * q^79 + 2 * q^80 + 27 * q^81 - 15 * q^82 + 4 * q^83 + 7 * q^84 + 21 * q^85 - 15 * q^86 + 7 * q^88 + 18 * q^89 - 28 * q^90 + 28 * q^91 + 12 * q^92 + 43 * q^93 + 6 * q^94 - 8 * q^95 - 6 * q^96 + 82 * q^97 - 42 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.01539	−1.74094	−0.870469	−	0.492224i	$-0.836184\pi$
$3$	−3.01539	−1.74094	−0.870469	+	0.492224i	$0.836184\pi$
$4$	1.00000	0.500000
$5$	−2.64906	−1.18470	−0.592348	−	0.805682i	$-0.701799\pi$
$5$	−2.64906	−1.18470	−0.592348	+	0.805682i	$0.701799\pi$
$6$	3.01539	1.23103
$7$	2.56889	0.970948	0.485474	−	0.874251i	$-0.338647\pi$
$7$	2.56889	0.970948	0.485474	+	0.874251i	$0.338647\pi$
$8$	−1.00000	−0.353553
$9$	6.09259	2.03086
$10$	2.64906	0.837707
$11$	−4.11236	−1.23992	−0.619962	−	0.784632i	$-0.712852\pi$
$11$	−4.11236	−1.23992	−0.619962	+	0.784632i	$0.712852\pi$
$12$	−3.01539	−0.870469
$13$	−0.291418	−0.0808248	−0.0404124	−	0.999183i	$-0.512867\pi$
$13$	−0.291418	−0.0808248	−0.0404124	+	0.999183i	$0.512867\pi$
$14$	−2.56889	−0.686564
$15$	7.98796	2.06248
$16$	1.00000	0.250000
$17$	4.20214	1.01917	0.509584	−	0.860421i	$-0.329799\pi$
$17$	4.20214	1.01917	0.509584	+	0.860421i	$0.329799\pi$
$18$	−6.09259	−1.43604
$19$	−2.36721	−0.543076	−0.271538	−	0.962428i	$-0.587532\pi$
$19$	−2.36721	−0.543076	−0.271538	+	0.962428i	$0.587532\pi$
$20$	−2.64906	−0.592348
$21$	−7.74620	−1.69036
$22$	4.11236	0.876758
$23$	−6.58360	−1.37277	−0.686387	−	0.727236i	$-0.740804\pi$
$23$	−6.58360	−1.37277	−0.686387	+	0.727236i	$0.740804\pi$
$24$	3.01539	0.615514
$25$	2.01752	0.403505
$26$	0.291418	0.0571518
$27$	−9.32537	−1.79467
$28$	2.56889	0.485474
$29$	0.310105	0.0575851	0.0287926	−	0.999585i	$-0.490834\pi$
$29$	0.310105	0.0575851	0.0287926	+	0.999585i	$0.490834\pi$
$30$	−7.98796	−1.45839
$31$	0.795398	0.142858	0.0714288	−	0.997446i	$-0.477244\pi$
$31$	0.795398	0.142858	0.0714288	+	0.997446i	$0.477244\pi$
$32$	−1.00000	−0.176777
$33$	12.4004	2.15863
$34$	−4.20214	−0.720660
$35$	−6.80514	−1.15028
$36$	6.09259	1.01543
$37$	−2.31520	−0.380616	−0.190308	−	0.981724i	$-0.560949\pi$
$37$	−2.31520	−0.380616	−0.190308	+	0.981724i	$0.560949\pi$
$38$	2.36721	0.384013
$39$	0.878740	0.140711
$40$	2.64906	0.418853
$41$	−10.3816	−1.62133	−0.810667	−	0.585508i	$-0.800895\pi$
$41$	−10.3816	−1.62133	−0.810667	+	0.585508i	$0.800895\pi$
$42$	7.74620	1.19526
$43$	−6.46147	−0.985365	−0.492682	−	0.870209i	$-0.663984\pi$
$43$	−6.46147	−0.985365	−0.492682	+	0.870209i	$0.663984\pi$
$44$	−4.11236	−0.619962
$45$	−16.1396	−2.40596
$46$	6.58360	0.970698
$47$	1.11419	0.162522	0.0812608	−	0.996693i	$-0.474105\pi$
$47$	1.11419	0.162522	0.0812608	+	0.996693i	$0.474105\pi$
$48$	−3.01539	−0.435234
$49$	−0.400824	−0.0572606
$50$	−2.01752	−0.285321
$51$	−12.6711	−1.77431
$52$	−0.291418	−0.0404124
$53$	2.49862	0.343212	0.171606	−	0.985166i	$-0.445104\pi$
$53$	2.49862	0.343212	0.171606	+	0.985166i	$0.445104\pi$
$54$	9.32537	1.26902
$55$	10.8939	1.46893
$56$	−2.56889	−0.343282
$57$	7.13808	0.945462
$58$	−0.310105	−0.0407188
$59$	−6.43815	−0.838175	−0.419088	−	0.907946i	$-0.637650\pi$
$59$	−6.43815	−0.838175	−0.419088	+	0.907946i	$0.637650\pi$
$60$	7.98796	1.03124
$61$	3.73626	0.478380	0.239190	−	0.970973i	$-0.423118\pi$
$61$	3.73626	0.478380	0.239190	+	0.970973i	$0.423118\pi$
$62$	−0.795398	−0.101016
$63$	15.6512	1.97186
$64$	1.00000	0.125000
$65$	0.771984	0.0957529
$66$	−12.4004	−1.52638
$67$	−2.21498	−0.270603	−0.135302	−	0.990804i	$-0.543200\pi$
$67$	−2.21498	−0.270603	−0.135302	+	0.990804i	$0.543200\pi$
$68$	4.20214	0.509584
$69$	19.8521	2.38991
$70$	6.80514	0.813369
$71$	5.40982	0.642027	0.321014	−	0.947075i	$-0.395976\pi$
$71$	5.40982	0.642027	0.321014	+	0.947075i	$0.395976\pi$
$72$	−6.09259	−0.718019
$73$	9.50064	1.11197	0.555983	−	0.831194i	$-0.312342\pi$
$73$	9.50064	1.11197	0.555983	+	0.831194i	$0.312342\pi$
$74$	2.31520	0.269136
$75$	−6.08363	−0.702477
$76$	−2.36721	−0.271538
$77$	−10.5642	−1.20390
$78$	−0.878740	−0.0994977
$79$	12.9489	1.45687	0.728434	−	0.685116i	$-0.240249\pi$
$79$	12.9489	1.45687	0.728434	+	0.685116i	$0.240249\pi$
$80$	−2.64906	−0.296174
$81$	9.84189	1.09354
$82$	10.3816	1.14646
$83$	14.7199	1.61572	0.807861	−	0.589372i	$-0.200625\pi$
$83$	14.7199	1.61572	0.807861	+	0.589372i	$0.200625\pi$
$84$	−7.74620	−0.845180
$85$	−11.1317	−1.20740
$86$	6.46147	0.696758
$87$	−0.935089	−0.100252
$88$	4.11236	0.438379
$89$	−7.31750	−0.775653	−0.387827	−	0.921732i	$-0.626774\pi$
$89$	−7.31750	−0.775653	−0.387827	+	0.921732i	$0.626774\pi$
$90$	16.1396	1.70127
$91$	−0.748620	−0.0784767
$92$	−6.58360	−0.686387
$93$	−2.39844	−0.248706
$94$	−1.11419	−0.114920
$95$	6.27089	0.643380
$96$	3.01539	0.307757
$97$	10.9490	1.11170	0.555851	−	0.831282i	$-0.312392\pi$
$97$	10.9490	1.11170	0.555851	+	0.831282i	$0.312392\pi$
$98$	0.400824	0.0404893
$99$	−25.0549	−2.51812
$100$	2.01752	0.201752
$101$	−4.08518	−0.406490	−0.203245	−	0.979128i	$-0.565149\pi$
$101$	−4.08518	−0.406490	−0.203245	+	0.979128i	$0.565149\pi$
$102$	12.6711	1.25462
$103$	−0.758520	−0.0747392	−0.0373696	−	0.999302i	$-0.511898\pi$
$103$	−0.758520	−0.0747392	−0.0373696	+	0.999302i	$0.511898\pi$
$104$	0.291418	0.0285759
$105$	20.5202	2.00256
$106$	−2.49862	−0.242688
$107$	−8.77940	−0.848737	−0.424368	−	0.905490i	$-0.639504\pi$
$107$	−8.77940	−0.848737	−0.424368	+	0.905490i	$0.639504\pi$
$108$	−9.32537	−0.897334
$109$	17.2722	1.65438	0.827189	−	0.561924i	$-0.189939\pi$
$109$	17.2722	1.65438	0.827189	+	0.561924i	$0.189939\pi$
$110$	−10.8939	−1.03869
$111$	6.98123	0.662629
$112$	2.56889	0.242737
$113$	7.96353	0.749146	0.374573	−	0.927197i	$-0.377789\pi$
$113$	7.96353	0.749146	0.374573	+	0.927197i	$0.377789\pi$
$114$	−7.13808	−0.668542
$115$	17.4403	1.62632
$116$	0.310105	0.0287926
$117$	−1.77549	−0.164144
$118$	6.43815	0.592679
$119$	10.7948	0.989559
$120$	−7.98796	−0.729197
$121$	5.91152	0.537411
$122$	−3.73626	−0.338265
$123$	31.3046	2.82264
$124$	0.795398	0.0714288
$125$	7.90076	0.706665
$126$	−15.6512	−1.39432
$127$	0.412707	0.0366218	0.0183109	−	0.999832i	$-0.494171\pi$
$127$	0.412707	0.0366218	0.0183109	+	0.999832i	$0.494171\pi$
$128$	−1.00000	−0.0883883
$129$	19.4839	1.71546
$130$	−0.771984	−0.0677075
$131$	−0.0114496	−0.00100036	−0.000500179	−	1.00000i	$-0.500159\pi$
$131$	−0.0114496	−0.00100036	−0.000500179		1.00000i	$0.500159\pi$
$132$	12.4004	1.07931
$133$	−6.08110	−0.527298
$134$	2.21498	0.191345
$135$	24.7035	2.12614
$136$	−4.20214	−0.360330
$137$	9.38002	0.801389	0.400695	−	0.916212i	$-0.368769\pi$
$137$	9.38002	0.801389	0.400695	+	0.916212i	$0.368769\pi$
$138$	−19.8521	−1.68992
$139$	8.32702	0.706289	0.353144	−	0.935569i	$-0.385112\pi$
$139$	8.32702	0.706289	0.353144	+	0.935569i	$0.385112\pi$
$140$	−6.80514	−0.575139
$141$	−3.35973	−0.282940
$142$	−5.40982	−0.453982
$143$	1.19842	0.100217
$144$	6.09259	0.507716
$145$	−0.821488	−0.0682209
$146$	−9.50064	−0.786279
$147$	1.20864	0.0996871
$148$	−2.31520	−0.190308
$149$	−1.98547	−0.162656	−0.0813279	−	0.996687i	$-0.525916\pi$
$149$	−1.98547	−0.162656	−0.0813279	+	0.996687i	$0.525916\pi$
$150$	6.08363	0.496726
$151$	−17.8175	−1.44997	−0.724985	−	0.688764i	$-0.758153\pi$
$151$	−17.8175	−1.44997	−0.724985	+	0.688764i	$0.758153\pi$
$152$	2.36721	0.192006
$153$	25.6019	2.06979
$154$	10.5642	0.851287
$155$	−2.10706	−0.169243
$156$	0.878740	0.0703555
$157$	13.6815	1.09190	0.545952	−	0.837816i	$-0.316168\pi$
$157$	13.6815	1.09190	0.545952	+	0.837816i	$0.316168\pi$
$158$	−12.9489	−1.03016
$159$	−7.53432	−0.597511
$160$	2.64906	0.209427
$161$	−16.9125	−1.33289
$162$	−9.84189	−0.773252
$163$	11.3457	0.888660	0.444330	−	0.895863i	$-0.353442\pi$
$163$	11.3457	0.888660	0.444330	+	0.895863i	$0.353442\pi$
$164$	−10.3816	−0.810667
$165$	−32.8494	−2.55732
$166$	−14.7199	−1.14249
$167$	−9.24997	−0.715784	−0.357892	−	0.933763i	$-0.616504\pi$
$167$	−9.24997	−0.715784	−0.357892	+	0.933763i	$0.616504\pi$
$168$	7.74620	0.597632
$169$	−12.9151	−0.993467
$170$	11.1317	0.853764
$171$	−14.4225	−1.10291
$172$	−6.46147	−0.492682
$173$	14.1437	1.07532	0.537662	−	0.843161i	$-0.319308\pi$
$173$	14.1437	1.07532	0.537662	+	0.843161i	$0.319308\pi$
$174$	0.935089	0.0708889
$175$	5.18279	0.391782
$176$	−4.11236	−0.309981
$177$	19.4135	1.45921
$178$	7.31750	0.548470
$179$	8.57379	0.640835	0.320418	−	0.947276i	$-0.396177\pi$
$179$	8.57379	0.640835	0.320418	+	0.947276i	$0.396177\pi$
$180$	−16.1396	−1.20298
$181$	17.0599	1.26805	0.634026	−	0.773311i	$-0.281401\pi$
$181$	17.0599	1.26805	0.634026	+	0.773311i	$0.281401\pi$
$182$	0.748620	0.0554914
$183$	−11.2663	−0.832829
$184$	6.58360	0.485349
$185$	6.13310	0.450914
$186$	2.39844	0.175862
$187$	−17.2807	−1.26369
$188$	1.11419	0.0812608
$189$	−23.9558	−1.74253
$190$	−6.27089	−0.454938
$191$	−23.5207	−1.70190	−0.850949	−	0.525248i	$-0.823973\pi$
$191$	−23.5207	−1.70190	−0.850949	+	0.525248i	$0.823973\pi$
$192$	−3.01539	−0.217617
$193$	21.6349	1.55731	0.778657	−	0.627450i	$-0.215901\pi$
$193$	21.6349	1.55731	0.778657	+	0.627450i	$0.215901\pi$
$194$	−10.9490	−0.786092
$195$	−2.32784	−0.166700
$196$	−0.400824	−0.0286303
$197$	−11.4549	−0.816126	−0.408063	−	0.912954i	$-0.633796\pi$
$197$	−11.4549	−0.816126	−0.408063	+	0.912954i	$0.633796\pi$
$198$	25.0549	1.78058
$199$	−20.8937	−1.48112	−0.740559	−	0.671991i	$-0.765439\pi$
$199$	−20.8937	−1.48112	−0.740559	+	0.671991i	$0.765439\pi$
$200$	−2.01752	−0.142661
$201$	6.67904	0.471103
$202$	4.08518	0.287432
$203$	0.796625	0.0559121
$204$	−12.6711	−0.887154
$205$	27.5015	1.92079
$206$	0.758520	0.0528486
$207$	−40.1112	−2.78792
$208$	−0.291418	−0.0202062
$209$	9.73484	0.673373
$210$	−20.5202	−1.41603
$211$	−24.8102	−1.70801	−0.854004	−	0.520267i	$-0.825832\pi$
$211$	−24.8102	−1.70801	−0.854004	+	0.520267i	$0.825832\pi$
$212$	2.49862	0.171606
$213$	−16.3127	−1.11773
$214$	8.77940	0.600148
$215$	17.1168	1.16736
$216$	9.32537	0.634511
$217$	2.04329	0.138707
$218$	−17.2722	−1.16982
$219$	−28.6482	−1.93586
$220$	10.8939	0.734466
$221$	−1.22458	−0.0823741
$222$	−6.98123	−0.468549
$223$	17.6328	1.18078	0.590391	−	0.807117i	$-0.298974\pi$
$223$	17.6328	1.18078	0.590391	+	0.807117i	$0.298974\pi$
$224$	−2.56889	−0.171641
$225$	12.2920	0.819463
$226$	−7.96353	−0.529726
$227$	11.4610	0.760692	0.380346	−	0.924844i	$-0.375805\pi$
$227$	11.4610	0.760692	0.380346	+	0.924844i	$0.375805\pi$
$228$	7.13808	0.472731
$229$	−21.5997	−1.42735	−0.713675	−	0.700477i	$-0.752971\pi$
$229$	−21.5997	−1.42735	−0.713675	+	0.700477i	$0.752971\pi$
$230$	−17.4403	−1.14998
$231$	31.8552	2.09592
$232$	−0.310105	−0.0203594
$233$	−12.9972	−0.851472	−0.425736	−	0.904847i	$-0.639985\pi$
$233$	−12.9972	−0.851472	−0.425736	+	0.904847i	$0.639985\pi$
$234$	1.77549	0.116067
$235$	−2.95156	−0.192539
$236$	−6.43815	−0.419088
$237$	−39.0461	−2.53632
$238$	−10.7948	−0.699724
$239$	13.9555	0.902706	0.451353	−	0.892345i	$-0.350941\pi$
$239$	13.9555	0.902706	0.451353	+	0.892345i	$0.350941\pi$
$240$	7.98796	0.515620
$241$	8.41273	0.541911	0.270956	−	0.962592i	$-0.412660\pi$
$241$	8.41273	0.541911	0.270956	+	0.962592i	$0.412660\pi$
$242$	−5.91152	−0.380007
$243$	−1.70103	−0.109121
$244$	3.73626	0.239190
$245$	1.06181	0.0678364
$246$	−31.3046	−1.99591
$247$	0.689849	0.0438940
$248$	−0.795398	−0.0505078
$249$	−44.3864	−2.81287
$250$	−7.90076	−0.499688
$251$	−19.8263	−1.25142	−0.625711	−	0.780055i	$-0.715191\pi$
$251$	−19.8263	−1.25142	−0.625711	+	0.780055i	$0.715191\pi$
$252$	15.6512	0.985931
$253$	27.0741	1.70214
$254$	−0.412707	−0.0258956
$255$	33.5665	2.10202
$256$	1.00000	0.0625000
$257$	27.8302	1.73600	0.868000	−	0.496565i	$-0.165406\pi$
$257$	27.8302	1.73600	0.868000	+	0.496565i	$0.165406\pi$
$258$	−19.4839	−1.21301
$259$	−5.94748	−0.369558
$260$	0.771984	0.0478764
$261$	1.88934	0.116948
$262$	0.0114496	0.000707360 0
$263$	8.56585	0.528193	0.264097	−	0.964496i	$-0.414926\pi$
$263$	8.56585	0.528193	0.264097	+	0.964496i	$0.414926\pi$
$264$	−12.4004	−0.763191
$265$	−6.61900	−0.406602
$266$	6.08110	0.372856
$267$	22.0651	1.35036
$268$	−2.21498	−0.135302
$269$	31.7172	1.93383	0.966916	−	0.255096i	$-0.0821072\pi$
$269$	31.7172	1.93383	0.966916	+	0.255096i	$0.0821072\pi$
$270$	−24.7035	−1.50341
$271$	−13.2998	−0.807904	−0.403952	−	0.914780i	$-0.632364\pi$
$271$	−13.2998	−0.807904	−0.403952	+	0.914780i	$0.632364\pi$
$272$	4.20214	0.254792
$273$	2.25738	0.136623
$274$	−9.38002	−0.566668
$275$	−8.29679	−0.500315
$276$	19.8521	1.19496
$277$	29.5673	1.77653	0.888265	−	0.459332i	$-0.151911\pi$
$277$	29.5673	1.77653	0.888265	+	0.459332i	$0.151911\pi$
$278$	−8.32702	−0.499422
$279$	4.84603	0.290124
$280$	6.80514	0.406685
$281$	24.9127	1.48617	0.743085	−	0.669197i	$-0.233362\pi$
$281$	24.9127	1.48617	0.743085	+	0.669197i	$0.233362\pi$
$282$	3.35973	0.200069
$283$	−5.03165	−0.299101	−0.149550	−	0.988754i	$-0.547783\pi$
$283$	−5.03165	−0.299101	−0.149550	+	0.988754i	$0.547783\pi$
$284$	5.40982	0.321014
$285$	−18.9092	−1.12008
$286$	−1.19842	−0.0708638
$287$	−26.6691	−1.57423
$288$	−6.09259	−0.359009
$289$	0.657951	0.0387030
$290$	0.821488	0.0482394
$291$	−33.0155	−1.93540
$292$	9.50064	0.555983
$293$	20.3536	1.18907	0.594534	−	0.804070i	$-0.297336\pi$
$293$	20.3536	1.18907	0.594534	+	0.804070i	$0.297336\pi$
$294$	−1.20864	−0.0704894
$295$	17.0550	0.992983
$296$	2.31520	0.134568
$297$	38.3493	2.22525
$298$	1.98547	0.115015
$299$	1.91858	0.110954
$300$	−6.08363	−0.351238
$301$	−16.5988	−0.956738
$302$	17.8175	1.02528
$303$	12.3184	0.707674
$304$	−2.36721	−0.135769
$305$	−9.89759	−0.566734
$306$	−25.6019	−1.46356
$307$	20.8145	1.18795	0.593974	−	0.804484i	$-0.297558\pi$
$307$	20.8145	1.18795	0.593974	+	0.804484i	$0.297558\pi$
$308$	−10.5642	−0.601951
$309$	2.28724	0.130116
$310$	2.10706	0.119673
$311$	19.2816	1.09336	0.546680	−	0.837341i	$-0.315891\pi$
$311$	19.2816	1.09336	0.546680	+	0.837341i	$0.315891\pi$
$312$	−0.878740	−0.0497488
$313$	5.73692	0.324270	0.162135	−	0.986769i	$-0.448162\pi$
$313$	5.73692	0.324270	0.162135	+	0.986769i	$0.448162\pi$
$314$	−13.6815	−0.772093
$315$	−41.4609	−2.33606
$316$	12.9489	0.728434
$317$	−21.2412	−1.19302	−0.596512	−	0.802604i	$-0.703447\pi$
$317$	−21.2412	−1.19302	−0.596512	+	0.802604i	$0.703447\pi$
$318$	7.53432	0.422504
$319$	−1.27527	−0.0714011
$320$	−2.64906	−0.148087
$321$	26.4733	1.47760
$322$	16.9125	0.942497
$323$	−9.94736	−0.553486
$324$	9.84189	0.546771
$325$	−0.587943	−0.0326132
$326$	−11.3457	−0.628378
$327$	−52.0825	−2.88017
$328$	10.3816	0.573228
$329$	2.86223	0.157800
$330$	32.8494	1.80830
$331$	−13.8752	−0.762649	−0.381325	−	0.924441i	$-0.624532\pi$
$331$	−13.8752	−0.762649	−0.381325	+	0.924441i	$0.624532\pi$
$332$	14.7199	0.807861
$333$	−14.1055	−0.772979
$334$	9.24997	0.506136
$335$	5.86762	0.320582
$336$	−7.74620	−0.422590
$337$	−12.3010	−0.670080	−0.335040	−	0.942204i	$-0.608750\pi$
$337$	−12.3010	−0.670080	−0.335040	+	0.942204i	$0.608750\pi$
$338$	12.9151	0.702487
$339$	−24.0132	−1.30422
$340$	−11.1317	−0.603702
$341$	−3.27096	−0.177133
$342$	14.4225	0.779878
$343$	−19.0119	−1.02654
$344$	6.46147	0.348379
$345$	−52.5895	−2.83132
$346$	−14.1437	−0.760369
$347$	−12.2895	−0.659738	−0.329869	−	0.944027i	$-0.607005\pi$
$347$	−12.2895	−0.659738	−0.329869	+	0.944027i	$0.607005\pi$
$348$	−0.935089	−0.0501260
$349$	8.46772	0.453267	0.226633	−	0.973980i	$-0.427228\pi$
$349$	8.46772	0.453267	0.226633	+	0.973980i	$0.427228\pi$
$350$	−5.18279	−0.277032
$351$	2.71758	0.145054
$352$	4.11236	0.219190
$353$	0.583103	0.0310354	0.0155177	−	0.999880i	$-0.495060\pi$
$353$	0.583103	0.0310354	0.0155177	+	0.999880i	$0.495060\pi$
$354$	−19.4135	−1.03182
$355$	−14.3309	−0.760607
$356$	−7.31750	−0.387827
$357$	−32.5506	−1.72276
$358$	−8.57379	−0.453139
$359$	14.5366	0.767210	0.383605	−	0.923497i	$-0.374682\pi$
$359$	14.5366	0.767210	0.383605	+	0.923497i	$0.374682\pi$
$360$	16.1396	0.850634
$361$	−13.3963	−0.705068
$362$	−17.0599	−0.896649
$363$	−17.8255	−0.935599
$364$	−0.748620	−0.0392383
$365$	−25.1678	−1.31734
$366$	11.2663	0.588899
$367$	14.6528	0.764872	0.382436	−	0.923982i	$-0.375085\pi$
$367$	14.6528	0.764872	0.382436	+	0.923982i	$0.375085\pi$
$368$	−6.58360	−0.343194
$369$	−63.2508	−3.29271
$370$	−6.13310	−0.318845
$371$	6.41867	0.333241
$372$	−2.39844	−0.124353
$373$	−13.9768	−0.723693	−0.361846	−	0.932238i	$-0.617853\pi$
$373$	−13.9768	−0.723693	−0.361846	+	0.932238i	$0.617853\pi$
$374$	17.2807	0.893564
$375$	−23.8239	−1.23026
$376$	−1.11419	−0.0574601
$377$	−0.0903703	−0.00465431
$378$	23.9558	1.23215
$379$	32.5779	1.67342	0.836708	−	0.547649i	$-0.184477\pi$
$379$	32.5779	1.67342	0.836708	+	0.547649i	$0.184477\pi$
$380$	6.27089	0.321690
$381$	−1.24447	−0.0637563
$382$	23.5207	1.20342
$383$	25.2463	1.29002	0.645012	−	0.764173i	$-0.276853\pi$
$383$	25.2463	1.29002	0.645012	+	0.764173i	$0.276853\pi$
$384$	3.01539	0.153879
$385$	27.9852	1.42626
$386$	−21.6349	−1.10119
$387$	−39.3671	−2.00114
$388$	10.9490	0.555851
$389$	−14.4208	−0.731162	−0.365581	−	0.930780i	$-0.619130\pi$
$389$	−14.4208	−0.731162	−0.365581	+	0.930780i	$0.619130\pi$
$390$	2.32784	0.117875
$391$	−27.6652	−1.39909
$392$	0.400824	0.0202447
$393$	0.0345251	0.00174156
$394$	11.4549	0.577088
$395$	−34.3025	−1.72595
$396$	−25.0549	−1.25906
$397$	12.6025	0.632504	0.316252	−	0.948675i	$-0.397576\pi$
$397$	12.6025	0.632504	0.316252	+	0.948675i	$0.397576\pi$
$398$	20.8937	1.04731
$399$	18.3369	0.917994
$400$	2.01752	0.100876
$401$	−6.20545	−0.309886	−0.154943	−	0.987923i	$-0.549519\pi$
$401$	−6.20545	−0.309886	−0.154943	+	0.987923i	$0.549519\pi$
$402$	−6.67904	−0.333120
$403$	−0.231793	−0.0115464
$404$	−4.08518	−0.203245
$405$	−26.0718	−1.29552
$406$	−0.796625	−0.0395358
$407$	9.52093	0.471935
$408$	12.6711	0.627312
$409$	−15.9932	−0.790815	−0.395408	−	0.918506i	$-0.629397\pi$
$409$	−15.9932	−0.790815	−0.395408	+	0.918506i	$0.629397\pi$
$410$	−27.5015	−1.35820
$411$	−28.2844	−1.39517
$412$	−0.758520	−0.0373696
$413$	−16.5389	−0.813824
$414$	40.1112	1.97136
$415$	−38.9940	−1.91414
$416$	0.291418	0.0142879
$417$	−25.1092	−1.22960
$418$	−9.73484	−0.476147
$419$	22.1648	1.08282	0.541411	−	0.840758i	$-0.317890\pi$
$419$	22.1648	1.08282	0.541411	+	0.840758i	$0.317890\pi$
$420$	20.5202	1.00128
$421$	−19.1498	−0.933306	−0.466653	−	0.884440i	$-0.654540\pi$
$421$	−19.1498	−0.933306	−0.466653	+	0.884440i	$0.654540\pi$
$422$	24.8102	1.20774
$423$	6.78832	0.330059
$424$	−2.49862	−0.121344
$425$	8.47791	0.411239
$426$	16.3127	0.790354
$427$	9.59804	0.464481
$428$	−8.77940	−0.424368
$429$	−3.61370	−0.174471
$430$	−17.1168	−0.825447
$431$	18.9414	0.912377	0.456189	−	0.889883i	$-0.349214\pi$
$431$	18.9414	0.912377	0.456189	+	0.889883i	$0.349214\pi$
$432$	−9.32537	−0.448667
$433$	−23.6693	−1.13747	−0.568736	−	0.822520i	$-0.692568\pi$
$433$	−23.6693	−1.13747	−0.568736	+	0.822520i	$0.692568\pi$
$434$	−2.04329	−0.0980809
$435$	2.47711	0.118768
$436$	17.2722	0.827189
$437$	15.5848	0.745521
$438$	28.6482	1.36886
$439$	15.4317	0.736514	0.368257	−	0.929724i	$-0.379955\pi$
$439$	15.4317	0.736514	0.368257	+	0.929724i	$0.379955\pi$
$440$	−10.8939	−0.519346
$441$	−2.44206	−0.116288
$442$	1.22458	0.0582473
$443$	7.99739	0.379967	0.189984	−	0.981787i	$-0.439157\pi$
$443$	7.99739	0.379967	0.189984	+	0.981787i	$0.439157\pi$
$444$	6.98123	0.331314
$445$	19.3845	0.918913
$446$	−17.6328	−0.834939
$447$	5.98696	0.283173
$448$	2.56889	0.121368
$449$	18.7212	0.883509	0.441754	−	0.897136i	$-0.354356\pi$
$449$	18.7212	0.883509	0.441754	+	0.897136i	$0.354356\pi$
$450$	−12.2920	−0.579448
$451$	42.6929	2.01033
$452$	7.96353	0.374573
$453$	53.7269	2.52431
$454$	−11.4610	−0.537891
$455$	1.98314	0.0929710
$456$	−7.13808	−0.334271
$457$	29.7423	1.39129	0.695644	−	0.718387i	$-0.255119\pi$
$457$	29.7423	1.39129	0.695644	+	0.718387i	$0.255119\pi$
$458$	21.5997	1.00929
$459$	−39.1865	−1.82907
$460$	17.4403	0.813160
$461$	15.4167	0.718027	0.359014	−	0.933332i	$-0.383113\pi$
$461$	15.4167	0.718027	0.359014	+	0.933332i	$0.383113\pi$
$462$	−31.8552	−1.48204
$463$	−8.15722	−0.379098	−0.189549	−	0.981871i	$-0.560703\pi$
$463$	−8.15722	−0.379098	−0.189549	+	0.981871i	$0.560703\pi$
$464$	0.310105	0.0143963
$465$	6.35360	0.294641
$466$	12.9972	0.602082
$467$	−25.4749	−1.17884	−0.589419	−	0.807828i	$-0.700643\pi$
$467$	−25.4749	−1.17884	−0.589419	+	0.807828i	$0.700643\pi$
$468$	−1.77549	−0.0820721
$469$	−5.69004	−0.262741
$470$	2.95156	0.136145
$471$	−41.2551	−1.90094
$472$	6.43815	0.296340
$473$	26.5719	1.22178
$474$	39.0461	1.79345
$475$	−4.77591	−0.219134
$476$	10.7948	0.494779
$477$	15.2231	0.697017
$478$	−13.9555	−0.638310
$479$	−10.3904	−0.474751	−0.237375	−	0.971418i	$-0.576287\pi$
$479$	−10.3904	−0.474751	−0.237375	+	0.971418i	$0.576287\pi$
$480$	−7.98796	−0.364599
$481$	0.674690	0.0307632
$482$	−8.41273	−0.383189
$483$	50.9978	2.32048
$484$	5.91152	0.268705
$485$	−29.0046	−1.31703
$486$	1.70103	0.0771602
$487$	3.66457	0.166058	0.0830288	−	0.996547i	$-0.473541\pi$
$487$	3.66457	0.166058	0.0830288	+	0.996547i	$0.473541\pi$
$488$	−3.73626	−0.169133
$489$	−34.2116	−1.54710
$490$	−1.06181	−0.0479676
$491$	39.0102	1.76050	0.880252	−	0.474506i	$-0.157373\pi$
$491$	39.0102	1.76050	0.880252	+	0.474506i	$0.157373\pi$
$492$	31.3046	1.41132
$493$	1.30310	0.0586889
$494$	−0.689849	−0.0310378
$495$	66.3721	2.98320
$496$	0.795398	0.0357144
$497$	13.8972	0.623375
$498$	44.3864	1.98900
$499$	−41.5493	−1.86000	−0.930001	−	0.367557i	$-0.880194\pi$
$499$	−41.5493	−1.86000	−0.930001	+	0.367557i	$0.880194\pi$
$500$	7.90076	0.353333
$501$	27.8923	1.24614
$502$	19.8263	0.884889
$503$	13.9543	0.622193	0.311096	−	0.950378i	$-0.399304\pi$
$503$	13.9543	0.622193	0.311096	+	0.950378i	$0.399304\pi$
$504$	−15.6512	−0.697159
$505$	10.8219	0.481567
$506$	−27.0741	−1.20359
$507$	38.9440	1.72956
$508$	0.412707	0.0183109
$509$	4.25069	0.188408	0.0942042	−	0.995553i	$-0.469969\pi$
$509$	4.25069	0.188408	0.0942042	+	0.995553i	$0.469969\pi$
$510$	−33.5665	−1.48635
$511$	24.4061	1.07966
$512$	−1.00000	−0.0441942
$513$	22.0752	0.974642
$514$	−27.8302	−1.22754
$515$	2.00937	0.0885433
$516$	19.4839	0.857729
$517$	−4.58196	−0.201514
$518$	5.94748	0.261317
$519$	−42.6487	−1.87207
$520$	−0.771984	−0.0338537
$521$	43.4018	1.90147	0.950734	−	0.310006i	$-0.100331\pi$
$521$	43.4018	1.90147	0.950734	+	0.310006i	$0.100331\pi$
$522$	−1.88934	−0.0826944
$523$	−26.5082	−1.15912	−0.579561	−	0.814929i	$-0.696776\pi$
$523$	−26.5082	−1.15912	−0.579561	+	0.814929i	$0.696776\pi$
$524$	−0.0114496	−0.000500179 0
$525$	−15.6281	−0.682068
$526$	−8.56585	−0.373489
$527$	3.34237	0.145596
$528$	12.4004	0.539657
$529$	20.3437	0.884510
$530$	6.61900	0.287511
$531$	−39.2250	−1.70222
$532$	−6.08110	−0.263649
$533$	3.02538	0.131044
$534$	−22.0651	−0.954851
$535$	23.2572	1.00550
$536$	2.21498	0.0956727
$537$	−25.8533	−1.11565
$538$	−31.7172	−1.36743
$539$	1.64833	0.0709988
$540$	24.7035	1.06307
$541$	−13.0859	−0.562608	−0.281304	−	0.959619i	$-0.590767\pi$
$541$	−13.0859	−0.562608	−0.281304	+	0.959619i	$0.590767\pi$
$542$	13.2998	0.571274
$543$	−51.4423	−2.20760
$544$	−4.20214	−0.180165
$545$	−45.7552	−1.95994
$546$	−2.25738	−0.0966070
$547$	−43.7444	−1.87038	−0.935188	−	0.354153i	$-0.884769\pi$
$547$	−43.7444	−1.87038	−0.935188	+	0.354153i	$0.884769\pi$
$548$	9.38002	0.400695
$549$	22.7635	0.971523
$550$	8.29679	0.353776
$551$	−0.734086	−0.0312731
$552$	−19.8521	−0.844962
$553$	33.2643	1.41454
$554$	−29.5673	−1.25620
$555$	−18.4937	−0.785014
$556$	8.32702	0.353144
$557$	−3.24907	−0.137667	−0.0688337	−	0.997628i	$-0.521928\pi$
$557$	−3.24907	−0.137667	−0.0688337	+	0.997628i	$0.521928\pi$
$558$	−4.84603	−0.205149
$559$	1.88299	0.0796419
$560$	−6.80514	−0.287569
$561$	52.1081	2.20001
$562$	−24.9127	−1.05088
$563$	−24.2249	−1.02096	−0.510479	−	0.859890i	$-0.670532\pi$
$563$	−24.2249	−1.02096	−0.510479	+	0.859890i	$0.670532\pi$
$564$	−3.35973	−0.141470
$565$	−21.0959	−0.887510
$566$	5.03165	0.211496
$567$	25.2827	1.06177
$568$	−5.40982	−0.226991
$569$	−19.8545	−0.832344	−0.416172	−	0.909286i	$-0.636629\pi$
$569$	−19.8545	−0.832344	−0.416172	+	0.909286i	$0.636629\pi$
$570$	18.9092	0.792019
$571$	37.0448	1.55028	0.775139	−	0.631791i	$-0.217680\pi$
$571$	37.0448	1.55028	0.775139	+	0.631791i	$0.217680\pi$
$572$	1.19842	0.0501083
$573$	70.9242	2.96290
$574$	26.6691	1.11315
$575$	−13.2826	−0.553921
$576$	6.09259	0.253858
$577$	18.0548	0.751632	0.375816	−	0.926694i	$-0.377363\pi$
$577$	18.0548	0.751632	0.375816	+	0.926694i	$0.377363\pi$
$578$	−0.657951	−0.0273672
$579$	−65.2377	−2.71119
$580$	−0.821488	−0.0341104
$581$	37.8138	1.56878
$582$	33.0155	1.36854
$583$	−10.2752	−0.425557
$584$	−9.50064	−0.393139
$585$	4.70338	0.194461
$586$	−20.3536	−0.840798
$587$	−4.99247	−0.206061	−0.103031	−	0.994678i	$-0.532854\pi$
$587$	−4.99247	−0.206061	−0.103031	+	0.994678i	$0.532854\pi$
$588$	1.20864	0.0498436
$589$	−1.88288	−0.0775826
$590$	−17.0550	−0.702145
$591$	34.5409	1.42082
$592$	−2.31520	−0.0951540
$593$	7.08518	0.290953	0.145477	−	0.989362i	$-0.453528\pi$
$593$	7.08518	0.290953	0.145477	+	0.989362i	$0.453528\pi$
$594$	−38.3493	−1.57349
$595$	−28.5961	−1.17233
$596$	−1.98547	−0.0813279
$597$	63.0028	2.57853
$598$	−1.91858	−0.0784565
$599$	−28.6962	−1.17249	−0.586247	−	0.810133i	$-0.699395\pi$
$599$	−28.6962	−1.17249	−0.586247	+	0.810133i	$0.699395\pi$
$600$	6.08363	0.248363
$601$	−10.1251	−0.413012	−0.206506	−	0.978445i	$-0.566209\pi$
$601$	−10.1251	−0.413012	−0.206506	+	0.978445i	$0.566209\pi$
$602$	16.5988	0.676516
$603$	−13.4950	−0.549558
$604$	−17.8175	−0.724985
$605$	−15.6600	−0.636668
$606$	−12.3184	−0.500401
$607$	−18.6132	−0.755485	−0.377743	−	0.925911i	$-0.623300\pi$
$607$	−18.6132	−0.755485	−0.377743	+	0.925911i	$0.623300\pi$
$608$	2.36721	0.0960032
$609$	−2.40214	−0.0973395
$610$	9.89759	0.400742
$611$	−0.324696	−0.0131358
$612$	25.6019	1.03490
$613$	16.3418	0.660038	0.330019	−	0.943974i	$-0.392945\pi$
$613$	16.3418	0.660038	0.330019	+	0.943974i	$0.392945\pi$
$614$	−20.8145	−0.840006
$615$	−82.9278	−3.34397
$616$	10.5642	0.425643
$617$	−6.26266	−0.252125	−0.126063	−	0.992022i	$-0.540234\pi$
$617$	−6.26266	−0.252125	−0.126063	+	0.992022i	$0.540234\pi$
$618$	−2.28724	−0.0920062
$619$	35.7176	1.43561	0.717807	−	0.696242i	$-0.245146\pi$
$619$	35.7176	1.43561	0.717807	+	0.696242i	$0.245146\pi$
$620$	−2.10706	−0.0846214
$621$	61.3945	2.46368
$622$	−19.2816	−0.773123
$623$	−18.7978	−0.753119
$624$	0.878740	0.0351777
$625$	−31.0172	−1.24069
$626$	−5.73692	−0.229293
$627$	−29.3544	−1.17230
$628$	13.6815	0.545952
$629$	−9.72877	−0.387912
$630$	41.4609	1.65184
$631$	7.73784	0.308039	0.154019	−	0.988068i	$-0.450778\pi$
$631$	7.73784	0.308039	0.154019	+	0.988068i	$0.450778\pi$
$632$	−12.9489	−0.515080
$633$	74.8126	2.97353
$634$	21.2412	0.843595
$635$	−1.09329	−0.0433858
$636$	−7.53432	−0.298755
$637$	0.116807	0.00462808
$638$	1.27527	0.0504882
$639$	32.9598	1.30387
$640$	2.64906	0.104713
$641$	2.25646	0.0891249	0.0445625	−	0.999007i	$-0.485811\pi$
$641$	2.25646	0.0891249	0.0445625	+	0.999007i	$0.485811\pi$
$642$	−26.4733	−1.04482
$643$	19.8556	0.783028	0.391514	−	0.920172i	$-0.371952\pi$
$643$	19.8556	0.783028	0.391514	+	0.920172i	$0.371952\pi$
$644$	−16.9125	−0.666446
$645$	−51.6139	−2.03230
$646$	9.94736	0.391373
$647$	−12.1109	−0.476128	−0.238064	−	0.971250i	$-0.576513\pi$
$647$	−12.1109	−0.476128	−0.238064	+	0.971250i	$0.576513\pi$
$648$	−9.84189	−0.386626
$649$	26.4760	1.03927
$650$	0.587943	0.0230610
$651$	−6.16131	−0.241481
$652$	11.3457	0.444330
$653$	−34.7356	−1.35931	−0.679655	−	0.733531i	$-0.737871\pi$
$653$	−34.7356	−1.35931	−0.679655	+	0.733531i	$0.737871\pi$
$654$	52.0825	2.03659
$655$	0.0303307	0.00118512
$656$	−10.3816	−0.405333
$657$	57.8835	2.25825
$658$	−2.86223	−0.111581
$659$	−3.74767	−0.145989	−0.0729943	−	0.997332i	$-0.523255\pi$
$659$	−3.74767	−0.145989	−0.0729943	+	0.997332i	$0.523255\pi$
$660$	−32.8494	−1.27866
$661$	26.9669	1.04889	0.524445	−	0.851444i	$-0.324273\pi$
$661$	26.9669	1.04889	0.524445	+	0.851444i	$0.324273\pi$
$662$	13.8752	0.539274
$663$	3.69258	0.143408
$664$	−14.7199	−0.571244
$665$	16.1092	0.624688
$666$	14.1055	0.546579
$667$	−2.04161	−0.0790514
$668$	−9.24997	−0.357892
$669$	−53.1699	−2.05567
$670$	−5.86762	−0.226686
$671$	−15.3649	−0.593154
$672$	7.74620	0.298816
$673$	30.8670	1.18983	0.594917	−	0.803787i	$-0.297185\pi$
$673$	30.8670	1.18983	0.594917	+	0.803787i	$0.297185\pi$
$674$	12.3010	0.473818
$675$	−18.8142	−0.724158
$676$	−12.9151	−0.496734
$677$	19.2400	0.739454	0.369727	−	0.929140i	$-0.379451\pi$
$677$	19.2400	0.739454	0.369727	+	0.929140i	$0.379451\pi$
$678$	24.0132	0.922220
$679$	28.1267	1.07940
$680$	11.1317	0.426882
$681$	−34.5594	−1.32432
$682$	3.27096	0.125252
$683$	50.3847	1.92792	0.963959	−	0.266051i	$-0.0857190\pi$
$683$	50.3847	1.92792	0.963959	+	0.266051i	$0.0857190\pi$
$684$	−14.4225	−0.551457
$685$	−24.8482	−0.949403
$686$	19.0119	0.725877
$687$	65.1316	2.48493
$688$	−6.46147	−0.246341
$689$	−0.728143	−0.0277401
$690$	52.5895	2.00205
$691$	−34.0546	−1.29550	−0.647749	−	0.761853i	$-0.724290\pi$
$691$	−34.0546	−1.29550	−0.647749	+	0.761853i	$0.724290\pi$
$692$	14.1437	0.537662
$693$	−64.3633	−2.44496
$694$	12.2895	0.466505
$695$	−22.0588	−0.836738
$696$	0.935089	0.0354445
$697$	−43.6249	−1.65241
$698$	−8.46772	−0.320508
$699$	39.1915	1.48236
$700$	5.18279	0.195891
$701$	32.6822	1.23439	0.617194	−	0.786811i	$-0.288269\pi$
$701$	32.6822	1.23439	0.617194	+	0.786811i	$0.288269\pi$
$702$	−2.71758	−0.102569
$703$	5.48057	0.206703
$704$	−4.11236	−0.154990
$705$	8.90012	0.335198
$706$	−0.583103	−0.0219454
$707$	−10.4944	−0.394681
$708$	19.4135	0.729605
$709$	0.347730	0.0130593	0.00652964	−	0.999979i	$-0.497922\pi$
$709$	0.347730	0.0130593	0.00652964	+	0.999979i	$0.497922\pi$
$710$	14.3309	0.537831
$711$	78.8925	2.95870
$712$	7.31750	0.274235
$713$	−5.23658	−0.196111
$714$	32.5506	1.21818
$715$	−3.17468	−0.118726
$716$	8.57379	0.320418
$717$	−42.0813	−1.57156
$718$	−14.5366	−0.542500
$719$	10.3554	0.386191	0.193095	−	0.981180i	$-0.438147\pi$
$719$	10.3554	0.386191	0.193095	+	0.981180i	$0.438147\pi$
$720$	−16.1396	−0.601489
$721$	−1.94855	−0.0725679
$722$	13.3963	0.498559
$723$	−25.3677	−0.943434
$724$	17.0599	0.634026
$725$	0.625645	0.0232359
$726$	17.8255	0.661568
$727$	5.05276	0.187396	0.0936982	−	0.995601i	$-0.470131\pi$
$727$	5.05276	0.187396	0.0936982	+	0.995601i	$0.470131\pi$
$728$	0.748620	0.0277457
$729$	−24.3964	−0.903570
$730$	25.1678	0.931501
$731$	−27.1520	−1.00425
$732$	−11.2663	−0.416414
$733$	36.6679	1.35436	0.677180	−	0.735818i	$-0.263202\pi$
$733$	36.6679	1.35436	0.677180	+	0.735818i	$0.263202\pi$
$734$	−14.6528	−0.540846
$735$	−3.20177	−0.118099
$736$	6.58360	0.242675
$737$	9.10881	0.335527
$738$	63.2508	2.32829
$739$	9.56164	0.351731	0.175865	−	0.984414i	$-0.443728\pi$
$739$	9.56164	0.351731	0.175865	+	0.984414i	$0.443728\pi$
$740$	6.13310	0.225457
$741$	−2.08016	−0.0764168
$742$	−6.41867	−0.235637
$743$	−23.8821	−0.876149	−0.438074	−	0.898939i	$-0.644339\pi$
$743$	−23.8821	−0.876149	−0.438074	+	0.898939i	$0.644339\pi$
$744$	2.39844	0.0879309
$745$	5.25962	0.192698
$746$	13.9768	0.511728
$747$	89.6825	3.28131
$748$	−17.2807	−0.631845
$749$	−22.5533	−0.824079
$750$	23.8239	0.869925
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	1.11419	0.0406304
$753$	59.7839	2.17865
$754$	0.0903703	0.00329109
$755$	47.1997	1.71777
$756$	−23.9558	−0.871265
$757$	−52.2738	−1.89992	−0.949961	−	0.312368i	$-0.898878\pi$
$757$	−52.2738	−1.89992	−0.949961	+	0.312368i	$0.898878\pi$
$758$	−32.5779	−1.18328
$759$	−81.6391	−2.96331
$760$	−6.27089	−0.227469
$761$	21.3873	0.775290	0.387645	−	0.921809i	$-0.373289\pi$
$761$	21.3873	0.775290	0.387645	+	0.921809i	$0.373289\pi$
$762$	1.24447	0.0450825
$763$	44.3704	1.60631
$764$	−23.5207	−0.850949
$765$	−67.8210	−2.45207
$766$	−25.2463	−0.912184
$767$	1.87619	0.0677454
$768$	−3.01539	−0.108809
$769$	−12.5102	−0.451131	−0.225565	−	0.974228i	$-0.572423\pi$
$769$	−12.5102	−0.451131	−0.225565	+	0.974228i	$0.572423\pi$
$770$	−27.9852	−1.00852
$771$	−83.9189	−3.02227
$772$	21.6349	0.778657
$773$	−4.91284	−0.176702	−0.0883512	−	0.996089i	$-0.528160\pi$
$773$	−4.91284	−0.176702	−0.0883512	+	0.996089i	$0.528160\pi$
$774$	39.3671	1.41502
$775$	1.60473	0.0576438
$776$	−10.9490	−0.393046
$777$	17.9340	0.643378
$778$	14.4208	0.517010
$779$	24.5755	0.880507
$780$	−2.32784	−0.0833499
$781$	−22.2471	−0.796065
$782$	27.6652	0.989304
$783$	−2.89185	−0.103346
$784$	−0.400824	−0.0143151
$785$	−36.2432	−1.29357
$786$	−0.0345251	−0.00123147
$787$	36.8930	1.31509	0.657547	−	0.753414i	$-0.271594\pi$
$787$	36.8930	1.31509	0.657547	+	0.753414i	$0.271594\pi$
$788$	−11.4549	−0.408063
$789$	−25.8294	−0.919551
$790$	34.3025	1.22043
$791$	20.4574	0.727381
$792$	25.0549	0.890288
$793$	−1.08881	−0.0386649
$794$	−12.6025	−0.447248
$795$	19.9589	0.707869
$796$	−20.8937	−0.740559
$797$	50.1958	1.77803	0.889013	−	0.457881i	$-0.151392\pi$
$797$	50.1958	1.77803	0.889013	+	0.457881i	$0.151392\pi$
$798$	−18.3369	−0.649120
$799$	4.68199	0.165637
$800$	−2.01752	−0.0713303
$801$	−44.5825	−1.57525
$802$	6.20545	0.219122
$803$	−39.0701	−1.37875
$804$	6.67904	0.235552
$805$	44.8023	1.57907
$806$	0.231793	0.00816457
$807$	−95.6398	−3.36668
$808$	4.08518	0.143716
$809$	−18.2168	−0.640470	−0.320235	−	0.947338i	$-0.603762\pi$
$809$	−18.2168	−0.640470	−0.320235	+	0.947338i	$0.603762\pi$
$810$	26.0718	0.916068
$811$	55.6365	1.95366	0.976831	−	0.214013i	$-0.0686536\pi$
$811$	55.6365	1.95366	0.976831	+	0.214013i	$0.0686536\pi$
$812$	0.796625	0.0279561
$813$	40.1041	1.40651
$814$	−9.52093	−0.333708
$815$	−30.0553	−1.05279
$816$	−12.6711	−0.443577
$817$	15.2957	0.535128
$818$	15.9932	0.559191
$819$	−4.56103	−0.159375
$820$	27.5015	0.960394
$821$	22.8214	0.796472	0.398236	−	0.917283i	$-0.369623\pi$
$821$	22.8214	0.796472	0.398236	+	0.917283i	$0.369623\pi$
$822$	28.2844	0.986533
$823$	43.1810	1.50519	0.752597	−	0.658482i	$-0.228801\pi$
$823$	43.1810	1.50519	0.752597	+	0.658482i	$0.228801\pi$
$824$	0.758520	0.0264243
$825$	25.0181	0.871018
$826$	16.5389	0.575461
$827$	−33.2153	−1.15501	−0.577504	−	0.816388i	$-0.695973\pi$
$827$	−33.2153	−1.15501	−0.577504	+	0.816388i	$0.695973\pi$
$828$	−40.1112	−1.39396
$829$	19.6391	0.682094	0.341047	−	0.940046i	$-0.389218\pi$
$829$	19.6391	0.682094	0.341047	+	0.940046i	$0.389218\pi$
$830$	38.9940	1.35350
$831$	−89.1571	−3.09283
$832$	−0.291418	−0.0101031
$833$	−1.68432	−0.0583581
$834$	25.1092	0.869462
$835$	24.5037	0.847987
$836$	9.73484	0.336686
$837$	−7.41738	−0.256382
$838$	−22.1648	−0.765670
$839$	−17.1061	−0.590567	−0.295284	−	0.955410i	$-0.595414\pi$
$839$	−17.1061	−0.590567	−0.295284	+	0.955410i	$0.595414\pi$
$840$	−20.5202	−0.708013
$841$	−28.9038	−0.996684
$842$	19.1498	0.659947
$843$	−75.1217	−2.58733
$844$	−24.8102	−0.854004
$845$	34.2128	1.17696
$846$	−6.78832	−0.233387
$847$	15.1860	0.521798
$848$	2.49862	0.0858030
$849$	15.1724	0.520716
$850$	−8.47791	−0.290790
$851$	15.2423	0.522500
$852$	−16.3127	−0.558865
$853$	−38.8947	−1.33173	−0.665864	−	0.746073i	$-0.731937\pi$
$853$	−38.8947	−1.33173	−0.665864	+	0.746073i	$0.731937\pi$
$854$	−9.59804	−0.328438
$855$	38.2060	1.30662
$856$	8.77940	0.300074
$857$	18.6963	0.638653	0.319326	−	0.947645i	$-0.396543\pi$
$857$	18.6963	0.638653	0.319326	+	0.947645i	$0.396543\pi$
$858$	3.61370	0.123370
$859$	38.9915	1.33037	0.665187	−	0.746677i	$-0.268352\pi$
$859$	38.9915	1.33037	0.665187	+	0.746677i	$0.268352\pi$
$860$	17.1168	0.583679
$861$	80.4179	2.74064
$862$	−18.9414	−0.645148
$863$	15.5622	0.529742	0.264871	−	0.964284i	$-0.414671\pi$
$863$	15.5622	0.529742	0.264871	+	0.964284i	$0.414671\pi$
$864$	9.32537	0.317256
$865$	−37.4675	−1.27393
$866$	23.6693	0.804315
$867$	−1.98398	−0.0673795
$868$	2.04329	0.0693536
$869$	−53.2507	−1.80640
$870$	−2.47711	−0.0839818
$871$	0.645486	0.0218715
$872$	−17.2722	−0.584911
$873$	66.7078	2.25772
$874$	−15.5848	−0.527163
$875$	20.2962	0.686135
$876$	−28.6482	−0.967931
$877$	−7.82643	−0.264280	−0.132140	−	0.991231i	$-0.542185\pi$
$877$	−7.82643	−0.264280	−0.132140	+	0.991231i	$0.542185\pi$
$878$	−15.4317	−0.520794
$879$	−61.3740	−2.07009
$880$	10.8939	0.367233
$881$	−17.3009	−0.582883	−0.291441	−	0.956589i	$-0.594135\pi$
$881$	−17.3009	−0.582883	−0.291441	+	0.956589i	$0.594135\pi$
$882$	2.44206	0.0822283
$883$	37.5198	1.26264	0.631321	−	0.775521i	$-0.282513\pi$
$883$	37.5198	1.26264	0.631321	+	0.775521i	$0.282513\pi$
$884$	−1.22458	−0.0411870
$885$	−51.4276	−1.72872
$886$	−7.99739	−0.268677
$887$	10.1431	0.340572	0.170286	−	0.985395i	$-0.445531\pi$
$887$	10.1431	0.340572	0.170286	+	0.985395i	$0.445531\pi$
$888$	−6.98123	−0.234275
$889$	1.06020	0.0355579
$890$	−19.3845	−0.649770
$891$	−40.4734	−1.35591
$892$	17.6328	0.590391
$893$	−2.63753	−0.0882616
$894$	−5.98696	−0.200234
$895$	−22.7125	−0.759195
$896$	−2.56889	−0.0858205
$897$	−5.78527	−0.193164
$898$	−18.7212	−0.624735
$899$	0.246657	0.00822647
$900$	12.2920	0.409732
$901$	10.4995	0.349791
$902$	−42.6929	−1.42152
$903$	50.0518	1.66562
$904$	−7.96353	−0.264863
$905$	−45.1927	−1.50226
$906$	−53.7269	−1.78496
$907$	16.4946	0.547693	0.273846	−	0.961773i	$-0.411704\pi$
$907$	16.4946	0.547693	0.273846	+	0.961773i	$0.411704\pi$
$908$	11.4610	0.380346
$909$	−24.8893	−0.825526
$910$	−1.98314	−0.0657404
$911$	26.6923	0.884355	0.442177	−	0.896928i	$-0.354206\pi$
$911$	26.6923	0.884355	0.442177	+	0.896928i	$0.354206\pi$
$912$	7.13808	0.236365
$913$	−60.5337	−2.00337
$914$	−29.7423	−0.983789
$915$	29.8451	0.986649
$916$	−21.5997	−0.713675
$917$	−0.0294128	−0.000971295 0
$918$	39.1865	1.29335
$919$	7.43967	0.245412	0.122706	−	0.992443i	$-0.460843\pi$
$919$	7.43967	0.245412	0.122706	+	0.992443i	$0.460843\pi$
$920$	−17.4403	−0.574991
$921$	−62.7640	−2.06814
$922$	−15.4167	−0.507722
$923$	−1.57652	−0.0518917
$924$	31.8552	1.04796
$925$	−4.67097	−0.153580
$926$	8.15722	0.268063
$927$	−4.62135	−0.151785
$928$	−0.310105	−0.0101797
$929$	−20.8428	−0.683829	−0.341915	−	0.939731i	$-0.611075\pi$
$929$	−20.8428	−0.683829	−0.341915	+	0.939731i	$0.611075\pi$
$930$	−6.35360	−0.208343
$931$	0.948836	0.0310969
$932$	−12.9972	−0.425736
$933$	−58.1417	−1.90347
$934$	25.4749	0.833564
$935$	45.7776	1.49709
$936$	1.77549	0.0580337
$937$	46.5624	1.52113	0.760563	−	0.649264i	$-0.224923\pi$
$937$	46.5624	1.52113	0.760563	+	0.649264i	$0.224923\pi$
$938$	5.69004	0.185786
$939$	−17.2991	−0.564533
$940$	−2.95156	−0.0962694
$941$	−36.4811	−1.18925	−0.594624	−	0.804004i	$-0.702699\pi$
$941$	−36.4811	−1.18925	−0.594624	+	0.804004i	$0.702699\pi$
$942$	41.2551	1.34417
$943$	68.3482	2.22572
$944$	−6.43815	−0.209544
$945$	63.4604	2.06437
$946$	−26.5719	−0.863927
$947$	−47.6784	−1.54934	−0.774670	−	0.632366i	$-0.782084\pi$
$947$	−47.6784	−1.54934	−0.774670	+	0.632366i	$0.782084\pi$
$948$	−39.0461	−1.26816
$949$	−2.76866	−0.0898744
$950$	4.77591	0.154951
$951$	64.0505	2.07698
$952$	−10.7948	−0.349862
$953$	−57.6394	−1.86712	−0.933562	−	0.358416i	$-0.883317\pi$
$953$	−57.6394	−1.86712	−0.933562	+	0.358416i	$0.883317\pi$
$954$	−15.2231	−0.492865
$955$	62.3078	2.01623
$956$	13.9555	0.451353
$957$	3.84543	0.124305
$958$	10.3904	0.335700
$959$	24.0962	0.778107
$960$	7.98796	0.257810
$961$	−30.3673	−0.979592
$962$	−0.674690	−0.0217529
$963$	−53.4893	−1.72367
$964$	8.41273	0.270956
$965$	−57.3122	−1.84494
$966$	−50.9978	−1.64083
$967$	−16.4773	−0.529874	−0.264937	−	0.964266i	$-0.585351\pi$
$967$	−16.4773	−0.529874	−0.264937	+	0.964266i	$0.585351\pi$
$968$	−5.91152	−0.190003
$969$	29.9952	0.963584
$970$	29.0046	0.931280
$971$	−56.3875	−1.80956	−0.904781	−	0.425877i	$-0.859966\pi$
$971$	−56.3875	−1.80956	−0.904781	+	0.425877i	$0.859966\pi$
$972$	−1.70103	−0.0545605
$973$	21.3912	0.685770
$974$	−3.66457	−0.117420
$975$	1.77288	0.0567776
$976$	3.73626	0.119595
$977$	−33.8742	−1.08373	−0.541865	−	0.840465i	$-0.682282\pi$
$977$	−33.8742	−1.08373	−0.541865	+	0.840465i	$0.682282\pi$
$978$	34.2116	1.09397
$979$	30.0922	0.961751
$980$	1.06181	0.0339182
$981$	105.233	3.35982
$982$	−39.0102	−1.24486
$983$	−7.90688	−0.252190	−0.126095	−	0.992018i	$-0.540244\pi$
$983$	−7.90688	−0.252190	−0.126095	+	0.992018i	$0.540244\pi$
$984$	−31.3046	−0.997954
$985$	30.3447	0.966861
$986$	−1.30310	−0.0414993
$987$	−8.63076	−0.274720
$988$	0.689849	0.0219470
$989$	42.5397	1.35268
$990$	−66.3721	−2.10944
$991$	−15.4126	−0.489596	−0.244798	−	0.969574i	$-0.578722\pi$
$991$	−15.4126	−0.489596	−0.244798	+	0.969574i	$0.578722\pi$
$992$	−0.795398	−0.0252539
$993$	41.8391	1.32772
$994$	−13.8972	−0.440793
$995$	55.3488	1.75467
$996$	−44.3864	−1.40644
$997$	−40.2492	−1.27471	−0.637353	−	0.770572i	$-0.719971\pi$
$997$	−40.2492	−1.27471	−0.637353	+	0.770572i	$0.719971\pi$
$998$	41.5493	1.31522
$999$	21.5901	0.683080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1502.2.a.h.1.1	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1502.2.a.h.1.1	✓	19	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:	\( N \)	\(=\)	\( 1502 = 2 \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1502.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.01539 q^{3} +1.00000 q^{4} -2.64906 q^{5} +3.01539 q^{6} +2.56889 q^{7} -1.00000 q^{8} +6.09259 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.01539 q^{3} +1.00000 q^{4} -2.64906 q^{5} +3.01539 q^{6} +2.56889 q^{7} -1.00000 q^{8} +6.09259 q^{9} +2.64906 q^{10} -4.11236 q^{11} -3.01539 q^{12} -0.291418 q^{13} -2.56889 q^{14} +7.98796 q^{15} +1.00000 q^{16} +4.20214 q^{17} -6.09259 q^{18} -2.36721 q^{19} -2.64906 q^{20} -7.74620 q^{21} +4.11236 q^{22} -6.58360 q^{23} +3.01539 q^{24} +2.01752 q^{25} +0.291418 q^{26} -9.32537 q^{27} +2.56889 q^{28} +0.310105 q^{29} -7.98796 q^{30} +0.795398 q^{31} -1.00000 q^{32} +12.4004 q^{33} -4.20214 q^{34} -6.80514 q^{35} +6.09259 q^{36} -2.31520 q^{37} +2.36721 q^{38} +0.878740 q^{39} +2.64906 q^{40} -10.3816 q^{41} +7.74620 q^{42} -6.46147 q^{43} -4.11236 q^{44} -16.1396 q^{45} +6.58360 q^{46} +1.11419 q^{47} -3.01539 q^{48} -0.400824 q^{49} -2.01752 q^{50} -12.6711 q^{51} -0.291418 q^{52} +2.49862 q^{53} +9.32537 q^{54} +10.8939 q^{55} -2.56889 q^{56} +7.13808 q^{57} -0.310105 q^{58} -6.43815 q^{59} +7.98796 q^{60} +3.73626 q^{61} -0.795398 q^{62} +15.6512 q^{63} +1.00000 q^{64} +0.771984 q^{65} -12.4004 q^{66} -2.21498 q^{67} +4.20214 q^{68} +19.8521 q^{69} +6.80514 q^{70} +5.40982 q^{71} -6.09259 q^{72} +9.50064 q^{73} +2.31520 q^{74} -6.08363 q^{75} -2.36721 q^{76} -10.5642 q^{77} -0.878740 q^{78} +12.9489 q^{79} -2.64906 q^{80} +9.84189 q^{81} +10.3816 q^{82} +14.7199 q^{83} -7.74620 q^{84} -11.1317 q^{85} +6.46147 q^{86} -0.935089 q^{87} +4.11236 q^{88} -7.31750 q^{89} +16.1396 q^{90} -0.748620 q^{91} -6.58360 q^{92} -2.39844 q^{93} -1.11419 q^{94} +6.27089 q^{95} +3.01539 q^{96} +10.9490 q^{97} +0.400824 q^{98} -25.0549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9 - 2 * q^10 - 7 * q^11 + 6 * q^12 + 19 * q^13 - 13 * q^14 + 6 * q^15 + 19 * q^16 + 11 * q^17 - 25 * q^18 + 7 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 12 * q^23 - 6 * q^24 + 37 * q^25 - 19 * q^26 + 24 * q^27 + 13 * q^28 - 8 * q^29 - 6 * q^30 + 32 * q^31 - 19 * q^32 + 24 * q^33 - 11 * q^34 - 19 * q^35 + 25 * q^36 + 41 * q^37 - 7 * q^38 - 2 * q^39 - 2 * q^40 + 15 * q^41 - 7 * q^42 + 15 * q^43 - 7 * q^44 + 28 * q^45 - 12 * q^46 - 6 * q^47 + 6 * q^48 + 42 * q^49 - 37 * q^50 + 8 * q^51 + 19 * q^52 + 6 * q^53 - 24 * q^54 + 22 * q^55 - 13 * q^56 + 24 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^60 + 13 * q^61 - 32 * q^62 + 37 * q^63 + 19 * q^64 + 20 * q^65 - 24 * q^66 + 47 * q^67 + 11 * q^68 + 15 * q^69 + 19 * q^70 - 25 * q^72 + 64 * q^73 - 41 * q^74 - 3 * q^75 + 7 * q^76 - 4 * q^77 + 2 * q^78 + 34 * q^79 + 2 * q^80 + 27 * q^81 - 15 * q^82 + 4 * q^83 + 7 * q^84 + 21 * q^85 - 15 * q^86 + 7 * q^88 + 18 * q^89 - 28 * q^90 + 28 * q^91 + 12 * q^92 + 43 * q^93 + 6 * q^94 - 8 * q^95 - 6 * q^96 + 82 * q^97 - 42 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more