LMFDB - Embedded newform 7381.2.a.i.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7381,2,Mod(1,7381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7381, 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("7381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7381 = 11^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7381.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:	$58.9375817319$
Analytic rank:	$1$
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} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.436741$ of defining polynomial
Character	$\chi$	$=$	7381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.436741 q^{2} -0.0103706 q^{3} -1.80926 q^{4} +3.47884 q^{5} -0.00452927 q^{6} +1.93516 q^{7} -1.66366 q^{8} -2.99989 q^{9} +O(q^{10})$	\(q+0.436741 q^{2} -0.0103706 q^{3} -1.80926 q^{4} +3.47884 q^{5} -0.00452927 q^{6} +1.93516 q^{7} -1.66366 q^{8} -2.99989 q^{9} +1.51935 q^{10} +0.0187631 q^{12} -4.56403 q^{13} +0.845165 q^{14} -0.0360777 q^{15} +2.89193 q^{16} -2.96582 q^{17} -1.31017 q^{18} +4.35587 q^{19} -6.29411 q^{20} -0.0200689 q^{21} +6.48962 q^{23} +0.0172532 q^{24} +7.10230 q^{25} -1.99330 q^{26} +0.0622226 q^{27} -3.50121 q^{28} -8.12635 q^{29} -0.0157566 q^{30} -7.92562 q^{31} +4.59034 q^{32} -1.29529 q^{34} +6.73212 q^{35} +5.42758 q^{36} -1.05106 q^{37} +1.90239 q^{38} +0.0473318 q^{39} -5.78759 q^{40} -9.87724 q^{41} -0.00876488 q^{42} -0.495326 q^{43} -10.4361 q^{45} +2.83428 q^{46} +10.4314 q^{47} -0.0299911 q^{48} -3.25514 q^{49} +3.10186 q^{50} +0.0307574 q^{51} +8.25751 q^{52} +11.7850 q^{53} +0.0271751 q^{54} -3.21945 q^{56} -0.0451731 q^{57} -3.54911 q^{58} -1.04645 q^{59} +0.0652738 q^{60} +1.00000 q^{61} -3.46144 q^{62} -5.80529 q^{63} -3.77907 q^{64} -15.8775 q^{65} +6.10340 q^{67} +5.36593 q^{68} -0.0673013 q^{69} +2.94019 q^{70} +0.0603962 q^{71} +4.99079 q^{72} -2.52552 q^{73} -0.459042 q^{74} -0.0736552 q^{75} -7.88089 q^{76} +0.0206717 q^{78} -5.40838 q^{79} +10.0605 q^{80} +8.99903 q^{81} -4.31379 q^{82} -10.9265 q^{83} +0.0363097 q^{84} -10.3176 q^{85} -0.216329 q^{86} +0.0842753 q^{87} -18.5756 q^{89} -4.55788 q^{90} -8.83215 q^{91} -11.7414 q^{92} +0.0821935 q^{93} +4.55580 q^{94} +15.1534 q^{95} -0.0476046 q^{96} -17.6748 q^{97} -1.42165 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9	$ 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9} - 7 q^{10} - 4 q^{12} - 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} - 9 q^{17} - 10 q^{18} - 17 q^{19} - 6 q^{20} - 18 q^{21} - 10 q^{23} - 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} - 36 q^{28} - 27 q^{29} + 30 q^{30} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} - 24 q^{39} - 10 q^{40} - 19 q^{41} + 21 q^{42} - 20 q^{43} - 32 q^{45} - 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} - 36 q^{50} - 47 q^{51} + 28 q^{52} + 3 q^{53} + 33 q^{54} - 44 q^{56} - 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} + 19 q^{61} + 11 q^{62} + 32 q^{63} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} - 19 q^{71} - 34 q^{72} - 20 q^{73} + 22 q^{74} - 50 q^{75} + 25 q^{76} - 94 q^{78} - 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} - q^{83} + 28 q^{84} - 24 q^{85} - 27 q^{86} + 58 q^{87} - 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} - 64 q^{94} + 3 q^{95} + 26 q^{96} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9 - 7 * q^10 - 4 * q^12 - 8 * q^13 - 11 * q^14 - 5 * q^15 + 31 * q^16 - 9 * q^17 - 10 * q^18 - 17 * q^19 - 6 * q^20 - 18 * q^21 - 10 * q^23 - 26 * q^24 + 45 * q^25 + 5 * q^26 - 33 * q^27 - 36 * q^28 - 27 * q^29 + 30 * q^30 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 38 * q^36 + 20 * q^37 - 37 * q^38 - 24 * q^39 - 10 * q^40 - 19 * q^41 + 21 * q^42 - 20 * q^43 - 32 * q^45 - 41 * q^46 - 19 * q^47 + 5 * q^48 + 42 * q^49 - 36 * q^50 - 47 * q^51 + 28 * q^52 + 3 * q^53 + 33 * q^54 - 44 * q^56 - 11 * q^57 + 23 * q^58 - 28 * q^59 - 96 * q^60 + 19 * q^61 + 11 * q^62 + 32 * q^63 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 - 19 * q^71 - 34 * q^72 - 20 * q^73 + 22 * q^74 - 50 * q^75 + 25 * q^76 - 94 * q^78 - 69 * q^79 - 36 * q^80 + 47 * q^81 - 61 * q^82 - q^83 + 28 * q^84 - 24 * q^85 - 27 * q^86 + 58 * q^87 - 36 * q^90 + 24 * q^91 - 67 * q^92 - 14 * q^93 - 64 * q^94 + 3 * q^95 + 26 * q^96 + 21 * q^97 + 87 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.436741	0.308822	0.154411	−	0.988007i	$-0.450652\pi$
$2$	0.436741	0.308822	0.154411	+	0.988007i	$0.450652\pi$
$3$	−0.0103706	−0.00598748	−0.00299374	−	0.999996i	$-0.500953\pi$
$3$	−0.0103706	−0.00598748	−0.00299374	+	0.999996i	$0.500953\pi$
$4$	−1.80926	−0.904629
$5$	3.47884	1.55578	0.777891	−	0.628399i	$-0.216289\pi$
$5$	3.47884	1.55578	0.777891	+	0.628399i	$0.216289\pi$
$6$	−0.00452927	−0.00184907
$7$	1.93516	0.731424	0.365712	−	0.930728i	$-0.380826\pi$
$7$	1.93516	0.731424	0.365712	+	0.930728i	$0.380826\pi$
$8$	−1.66366	−0.588192
$9$	−2.99989	−0.999964
$10$	1.51935	0.480460
$11$	0	0
$11$	0	0
$12$	0.0187631	0.00541645
$13$	−4.56403	−1.26583	−0.632917	−	0.774220i	$-0.718143\pi$
$13$	−4.56403	−1.26583	−0.632917	+	0.774220i	$0.718143\pi$
$14$	0.845165	0.225880
$15$	−0.0360777	−0.00931522
$16$	2.89193	0.722982
$17$	−2.96582	−0.719316	−0.359658	−	0.933084i	$-0.617107\pi$
$17$	−2.96582	−0.719316	−0.359658	+	0.933084i	$0.617107\pi$
$18$	−1.31017	−0.308811
$19$	4.35587	0.999305	0.499653	−	0.866226i	$-0.333461\pi$
$19$	4.35587	0.999305	0.499653	+	0.866226i	$0.333461\pi$
$20$	−6.29411	−1.40741
$21$	−0.0200689	−0.00437938
$22$	0	0
$23$	6.48962	1.35318	0.676589	−	0.736360i	$-0.263457\pi$
$23$	6.48962	1.35318	0.676589	+	0.736360i	$0.263457\pi$
$24$	0.0172532	0.00352179
$25$	7.10230	1.42046
$26$	−1.99330	−0.390918
$27$	0.0622226	0.0119747
$28$	−3.50121	−0.661667
$29$	−8.12635	−1.50903	−0.754513	−	0.656286i	$-0.772127\pi$
$29$	−8.12635	−1.50903	−0.754513	+	0.656286i	$0.772127\pi$
$30$	−0.0157566	−0.00287675
$31$	−7.92562	−1.42348	−0.711741	−	0.702442i	$-0.752093\pi$
$31$	−7.92562	−1.42348	−0.711741	+	0.702442i	$0.752093\pi$
$32$	4.59034	0.811465
$33$	0	0
$34$	−1.29529	−0.222141
$35$	6.73212	1.13794
$36$	5.42758	0.904596
$37$	−1.05106	−0.172794	−0.0863970	−	0.996261i	$-0.527535\pi$
$37$	−1.05106	−0.172794	−0.0863970	+	0.996261i	$0.527535\pi$
$38$	1.90239	0.308608
$39$	0.0473318	0.00757915
$40$	−5.78759	−0.915098
$41$	−9.87724	−1.54257	−0.771283	−	0.636493i	$-0.780385\pi$
$41$	−9.87724	−1.54257	−0.771283	+	0.636493i	$0.780385\pi$
$42$	−0.00876488	−0.00135245
$43$	−0.495326	−0.0755365	−0.0377683	−	0.999287i	$-0.512025\pi$
$43$	−0.495326	−0.0755365	−0.0377683	+	0.999287i	$0.512025\pi$
$44$	0	0
$45$	−10.4361	−1.55573
$46$	2.83428	0.417892
$47$	10.4314	1.52157	0.760785	−	0.649004i	$-0.224814\pi$
$47$	10.4314	1.52157	0.760785	+	0.649004i	$0.224814\pi$
$48$	−0.0299911	−0.00432884
$49$	−3.25514	−0.465020
$50$	3.10186	0.438669
$51$	0.0307574	0.00430689
$52$	8.25751	1.14511
$53$	11.7850	1.61879	0.809395	−	0.587265i	$-0.199795\pi$
$53$	11.7850	1.61879	0.809395	+	0.587265i	$0.199795\pi$
$54$	0.0271751	0.00369807
$55$	0	0
$56$	−3.21945	−0.430217
$57$	−0.0451731	−0.00598332
$58$	−3.54911	−0.466020
$59$	−1.04645	−0.136236	−0.0681179	−	0.997677i	$-0.521699\pi$
$59$	−1.04645	−0.136236	−0.0681179	+	0.997677i	$0.521699\pi$
$60$	0.0652738	0.00842681
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−3.46144	−0.439603
$63$	−5.80529	−0.731397
$64$	−3.77907	−0.472384
$65$	−15.8775	−1.96936
$66$	0	0
$67$	6.10340	0.745649	0.372825	−	0.927902i	$-0.378389\pi$
$67$	6.10340	0.745649	0.372825	+	0.927902i	$0.378389\pi$
$68$	5.36593	0.650714
$69$	−0.0673013	−0.00810213
$70$	2.94019	0.351420
$71$	0.0603962	0.00716771	0.00358386	−	0.999994i	$-0.498859\pi$
$71$	0.0603962	0.00716771	0.00358386	+	0.999994i	$0.498859\pi$
$72$	4.99079	0.588171
$73$	−2.52552	−0.295590	−0.147795	−	0.989018i	$-0.547218\pi$
$73$	−2.52552	−0.295590	−0.147795	+	0.989018i	$0.547218\pi$
$74$	−0.459042	−0.0533626
$75$	−0.0736552	−0.00850497
$76$	−7.88089	−0.904000
$77$	0	0
$78$	0.0206717	0.00234061
$79$	−5.40838	−0.608491	−0.304245	−	0.952594i	$-0.598404\pi$
$79$	−5.40838	−0.608491	−0.304245	+	0.952594i	$0.598404\pi$
$80$	10.0605	1.12480
$81$	8.99903	0.999892
$82$	−4.31379	−0.476378
$83$	−10.9265	−1.19934	−0.599672	−	0.800246i	$-0.704702\pi$
$83$	−10.9265	−1.19934	−0.599672	+	0.800246i	$0.704702\pi$
$84$	0.0363097	0.00396172
$85$	−10.3176	−1.11910
$86$	−0.216329	−0.0233274
$87$	0.0842753	0.00903526
$88$	0	0
$89$	−18.5756	−1.96901	−0.984503	−	0.175368i	$-0.943888\pi$
$89$	−18.5756	−1.96901	−0.984503	+	0.175368i	$0.943888\pi$
$90$	−4.55788	−0.480443
$91$	−8.83215	−0.925861
$92$	−11.7414	−1.22412
$93$	0.0821935	0.00852307
$94$	4.55580	0.469895
$95$	15.1534	1.55470
$96$	−0.0476046	−0.00485863
$97$	−17.6748	−1.79460	−0.897300	−	0.441421i	$-0.854474\pi$
$97$	−17.6748	−1.79460	−0.897300	+	0.441421i	$0.854474\pi$
$98$	−1.42165	−0.143608
$99$	0	0
$100$	−12.8499	−1.28499
$101$	5.63429	0.560633	0.280317	−	0.959908i	$-0.409561\pi$
$101$	5.63429	0.560633	0.280317	+	0.959908i	$0.409561\pi$
$102$	0.0134330	0.00133006
$103$	−16.7201	−1.64748	−0.823738	−	0.566971i	$-0.808115\pi$
$103$	−16.7201	−1.64748	−0.823738	+	0.566971i	$0.808115\pi$
$104$	7.59298	0.744553
$105$	−0.0698163	−0.00681337
$106$	5.14697	0.499918
$107$	10.2226	0.988261	0.494130	−	0.869388i	$-0.335487\pi$
$107$	10.2226	0.988261	0.494130	+	0.869388i	$0.335487\pi$
$108$	−0.112577	−0.0108327
$109$	−8.04155	−0.770241	−0.385120	−	0.922866i	$-0.625840\pi$
$109$	−8.04155	−0.770241	−0.385120	+	0.922866i	$0.625840\pi$
$110$	0	0
$111$	0.0109002	0.00103460
$112$	5.59636	0.528806
$113$	−5.03833	−0.473967	−0.236983	−	0.971514i	$-0.576159\pi$
$113$	−5.03833	−0.473967	−0.236983	+	0.971514i	$0.576159\pi$
$114$	−0.0197289	−0.00184778
$115$	22.5763	2.10525
$116$	14.7027	1.36511
$117$	13.6916	1.26579
$118$	−0.457026	−0.0420726
$119$	−5.73934	−0.526125
$120$	0.0600209	0.00547913
$121$	0	0
$122$	0.436741	0.0395406
$123$	0.102433	0.00923608
$124$	14.3395	1.28772
$125$	7.31354	0.654143
$126$	−2.53540	−0.225872
$127$	−12.0779	−1.07174	−0.535871	−	0.844300i	$-0.680017\pi$
$127$	−12.0779	−1.07174	−0.535871	+	0.844300i	$0.680017\pi$
$128$	−10.8311	−0.957347
$129$	0.00513684	0.000452273 0
$130$	−6.93435	−0.608183
$131$	19.0023	1.66024	0.830119	−	0.557587i	$-0.188273\pi$
$131$	19.0023	1.66024	0.830119	+	0.557587i	$0.188273\pi$
$132$	0	0
$133$	8.42933	0.730915
$134$	2.66560	0.230273
$135$	0.216462	0.0186301
$136$	4.93410	0.423096
$137$	−22.4442	−1.91754	−0.958771	−	0.284180i	$-0.908279\pi$
$137$	−22.4442	−1.91754	−0.958771	+	0.284180i	$0.908279\pi$
$138$	−0.0293932	−0.00250212
$139$	12.2835	1.04188	0.520938	−	0.853594i	$-0.325582\pi$
$139$	12.2835	1.04188	0.520938	+	0.853594i	$0.325582\pi$
$140$	−12.1801	−1.02941
$141$	−0.108180	−0.00911037
$142$	0.0263775	0.00221355
$143$	0	0
$144$	−8.67548	−0.722956
$145$	−28.2702	−2.34771
$146$	−1.10300	−0.0912847
$147$	0.0337578	0.00278429
$148$	1.90165	0.156314
$149$	−19.8821	−1.62880	−0.814401	−	0.580302i	$-0.802935\pi$
$149$	−19.8821	−1.62880	−0.814401	+	0.580302i	$0.802935\pi$
$150$	−0.0321682	−0.00262652
$151$	10.5885	0.861678	0.430839	−	0.902429i	$-0.358218\pi$
$151$	10.5885	0.861678	0.430839	+	0.902429i	$0.358218\pi$
$152$	−7.24668	−0.587783
$153$	8.89713	0.719290
$154$	0	0
$155$	−27.5719	−2.21463
$156$	−0.0856354	−0.00685632
$157$	−7.05272	−0.562868	−0.281434	−	0.959581i	$-0.590810\pi$
$157$	−7.05272	−0.562868	−0.281434	+	0.959581i	$0.590810\pi$
$158$	−2.36206	−0.187915
$159$	−0.122217	−0.00969247
$160$	15.9690	1.26246
$161$	12.5585	0.989747
$162$	3.93024	0.308789
$163$	5.13074	0.401870	0.200935	−	0.979605i	$-0.435602\pi$
$163$	5.13074	0.401870	0.200935	+	0.979605i	$0.435602\pi$
$164$	17.8705	1.39545
$165$	0	0
$166$	−4.77207	−0.370384
$167$	−22.6642	−1.75381	−0.876903	−	0.480668i	$-0.840394\pi$
$167$	−22.6642	−1.75381	−0.876903	+	0.480668i	$0.840394\pi$
$168$	0.0333877	0.00257592
$169$	7.83036	0.602336
$170$	−4.50611	−0.345603
$171$	−13.0671	−0.999269
$172$	0.896173	0.0683325
$173$	−12.3401	−0.938201	−0.469100	−	0.883145i	$-0.655422\pi$
$173$	−12.3401	−0.938201	−0.469100	+	0.883145i	$0.655422\pi$
$174$	0.0368064	0.00279029
$175$	13.7441	1.03896
$176$	0	0
$177$	0.0108523	0.000815709 0
$178$	−8.11270	−0.608073
$179$	4.14843	0.310069	0.155034	−	0.987909i	$-0.450451\pi$
$179$	4.14843	0.310069	0.155034	+	0.987909i	$0.450451\pi$
$180$	18.8817	1.40736
$181$	15.7435	1.17021	0.585104	−	0.810958i	$-0.301054\pi$
$181$	15.7435	1.17021	0.585104	+	0.810958i	$0.301054\pi$
$182$	−3.85736	−0.285926
$183$	−0.0103706	−0.000766618 0
$184$	−10.7965	−0.795928
$185$	−3.65648	−0.268830
$186$	0.0358972	0.00263211
$187$	0	0
$188$	−18.8730	−1.37646
$189$	0.120411	0.00875861
$190$	6.61808	0.480126
$191$	−14.6638	−1.06104	−0.530518	−	0.847674i	$-0.678003\pi$
$191$	−14.6638	−1.06104	−0.530518	+	0.847674i	$0.678003\pi$
$192$	0.0391913	0.00282839
$193$	19.6530	1.41466	0.707328	−	0.706885i	$-0.249900\pi$
$193$	19.6530	1.41466	0.707328	+	0.706885i	$0.249900\pi$
$194$	−7.71928	−0.554212
$195$	0.164660	0.0117915
$196$	5.88938	0.420670
$197$	4.17761	0.297643	0.148821	−	0.988864i	$-0.452452\pi$
$197$	4.17761	0.297643	0.148821	+	0.988864i	$0.452452\pi$
$198$	0	0
$199$	−14.5828	−1.03375	−0.516875	−	0.856061i	$-0.672905\pi$
$199$	−14.5828	−1.03375	−0.516875	+	0.856061i	$0.672905\pi$
$200$	−11.8158	−0.835502
$201$	−0.0632961	−0.00446456
$202$	2.46072	0.173136
$203$	−15.7258	−1.10374
$204$	−0.0556480	−0.00389614
$205$	−34.3613	−2.39990
$206$	−7.30233	−0.508777
$207$	−19.4682	−1.35313
$208$	−13.1988	−0.915175
$209$	0	0
$210$	−0.0304916	−0.00210412
$211$	11.4294	0.786830	0.393415	−	0.919361i	$-0.371294\pi$
$211$	11.4294	0.786830	0.393415	+	0.919361i	$0.371294\pi$
$212$	−21.3220	−1.46440
$213$	−0.000626346 0	−4.29165e−5 0
$214$	4.46464	0.305197
$215$	−1.72316	−0.117518
$216$	−0.103517	−0.00704344
$217$	−15.3374	−1.04117
$218$	−3.51207	−0.237868
$219$	0.0261912	0.00176984
$220$	0	0
$221$	13.5361	0.910535
$222$	0.00476055	0.000319507 0
$223$	3.82865	0.256385	0.128193	−	0.991749i	$-0.459082\pi$
$223$	3.82865	0.256385	0.128193	+	0.991749i	$0.459082\pi$
$224$	8.88306	0.593524
$225$	−21.3061	−1.42041
$226$	−2.20044	−0.146371
$227$	10.1929	0.676527	0.338264	−	0.941051i	$-0.390160\pi$
$227$	10.1929	0.676527	0.338264	+	0.941051i	$0.390160\pi$
$228$	0.0817297	0.00541268
$229$	4.69673	0.310369	0.155184	−	0.987886i	$-0.450403\pi$
$229$	4.69673	0.310369	0.155184	+	0.987886i	$0.450403\pi$
$230$	9.85999	0.650149
$231$	0	0
$232$	13.5195	0.887596
$233$	3.20209	0.209776	0.104888	−	0.994484i	$-0.466552\pi$
$233$	3.20209	0.209776	0.104888	+	0.994484i	$0.466552\pi$
$234$	5.97968	0.390904
$235$	36.2890	2.36723
$236$	1.89329	0.123243
$237$	0.0560883	0.00364333
$238$	−2.50660	−0.162479
$239$	15.0328	0.972388	0.486194	−	0.873851i	$-0.338385\pi$
$239$	15.0328	0.972388	0.486194	+	0.873851i	$0.338385\pi$
$240$	−0.104334	−0.00673474
$241$	−10.7597	−0.693092	−0.346546	−	0.938033i	$-0.612645\pi$
$241$	−10.7597	−0.693092	−0.346546	+	0.938033i	$0.612645\pi$
$242$	0	0
$243$	−0.279993	−0.0179616
$244$	−1.80926	−0.115826
$245$	−11.3241	−0.723469
$246$	0.0447367	0.00285231
$247$	−19.8803	−1.26495
$248$	13.1855	0.837281
$249$	0.113315	0.00718105
$250$	3.19412	0.202014
$251$	25.8084	1.62901	0.814504	−	0.580157i	$-0.197009\pi$
$251$	25.8084	1.62901	0.814504	+	0.580157i	$0.197009\pi$
$252$	10.5033	0.661643
$253$	0	0
$254$	−5.27492	−0.330978
$255$	0.107000	0.00670059
$256$	2.82774	0.176734
$257$	−23.8547	−1.48801	−0.744007	−	0.668172i	$-0.767077\pi$
$257$	−23.8547	−1.48801	−0.744007	+	0.668172i	$0.767077\pi$
$258$	0.00224347	0.000139672 0
$259$	−2.03398	−0.126386
$260$	28.7265	1.78154
$261$	24.3782	1.50897
$262$	8.29907	0.512718
$263$	15.0382	0.927294	0.463647	−	0.886020i	$-0.346541\pi$
$263$	15.0382	0.927294	0.463647	+	0.886020i	$0.346541\pi$
$264$	0	0
$265$	40.9980	2.51848
$266$	3.68143	0.225723
$267$	0.192640	0.0117894
$268$	−11.0426	−0.674536
$269$	5.09501	0.310648	0.155324	−	0.987864i	$-0.450358\pi$
$269$	5.09501	0.310648	0.155324	+	0.987864i	$0.450358\pi$
$270$	0.0945378	0.00575339
$271$	13.6962	0.831986	0.415993	−	0.909368i	$-0.363434\pi$
$271$	13.6962	0.831986	0.415993	+	0.909368i	$0.363434\pi$
$272$	−8.57693	−0.520053
$273$	0.0915949	0.00554357
$274$	−9.80231	−0.592179
$275$	0	0
$276$	0.121765	0.00732942
$277$	−12.0575	−0.724465	−0.362233	−	0.932088i	$-0.617985\pi$
$277$	−12.0575	−0.724465	−0.362233	+	0.932088i	$0.617985\pi$
$278$	5.36472	0.321754
$279$	23.7760	1.42343
$280$	−11.1999	−0.669324
$281$	−28.6925	−1.71165	−0.855827	−	0.517263i	$-0.826951\pi$
$281$	−28.6925	−1.71165	−0.855827	+	0.517263i	$0.826951\pi$
$282$	−0.0472464	−0.00281349
$283$	4.02652	0.239352	0.119676	−	0.992813i	$-0.461814\pi$
$283$	4.02652	0.239352	0.119676	+	0.992813i	$0.461814\pi$
$284$	−0.109272	−0.00648412
$285$	−0.157150	−0.00930874
$286$	0	0
$287$	−19.1141	−1.12827
$288$	−13.7705	−0.811436
$289$	−8.20393	−0.482584
$290$	−12.3468	−0.725026
$291$	0.183298	0.0107451
$292$	4.56932	0.267399
$293$	−1.96841	−0.114996	−0.0574979	−	0.998346i	$-0.518312\pi$
$293$	−1.96841	−0.114996	−0.0574979	+	0.998346i	$0.518312\pi$
$294$	0.0147434	0.000859852 0
$295$	−3.64042	−0.211953
$296$	1.74861	0.101636
$297$	0	0
$298$	−8.68331	−0.503010
$299$	−29.6188	−1.71290
$300$	0.133261	0.00769384
$301$	−0.958538	−0.0552492
$302$	4.62442	0.266105
$303$	−0.0584311	−0.00335678
$304$	12.5969	0.722480
$305$	3.47884	0.199198
$306$	3.88574	0.222133
$307$	−23.5477	−1.34394	−0.671969	−	0.740580i	$-0.734551\pi$
$307$	−23.5477	−1.34394	−0.671969	+	0.740580i	$0.734551\pi$
$308$	0	0
$309$	0.173397	0.00986423
$310$	−12.0418	−0.683927
$311$	21.8661	1.23991	0.619956	−	0.784636i	$-0.287150\pi$
$311$	21.8661	1.23991	0.619956	+	0.784636i	$0.287150\pi$
$312$	−0.0787439	−0.00445800
$313$	−8.85138	−0.500310	−0.250155	−	0.968206i	$-0.580482\pi$
$313$	−8.85138	−0.500310	−0.250155	+	0.968206i	$0.580482\pi$
$314$	−3.08021	−0.173826
$315$	−20.1956	−1.13790
$316$	9.78516	0.550458
$317$	0.553452	0.0310849	0.0155425	−	0.999879i	$-0.495052\pi$
$317$	0.553452	0.0310849	0.0155425	+	0.999879i	$0.495052\pi$
$318$	−0.0533773	−0.00299325
$319$	0	0
$320$	−13.1468	−0.734927
$321$	−0.106015	−0.00591719
$322$	5.48480	0.305656
$323$	−12.9187	−0.718816
$324$	−16.2816	−0.904532
$325$	−32.4151	−1.79807
$326$	2.24080	0.124106
$327$	0.0833959	0.00461180
$328$	16.4323	0.907324
$329$	20.1864	1.11291
$330$	0	0
$331$	−2.04463	−0.112383	−0.0561915	−	0.998420i	$-0.517896\pi$
$331$	−2.04463	−0.112383	−0.0561915	+	0.998420i	$0.517896\pi$
$332$	19.7689	1.08496
$333$	3.15308	0.172788
$334$	−9.89836	−0.541614
$335$	21.2327	1.16007
$336$	−0.0580377	−0.00316622
$337$	4.08881	0.222732	0.111366	−	0.993779i	$-0.464477\pi$
$337$	4.08881	0.222732	0.111366	+	0.993779i	$0.464477\pi$
$338$	3.41984	0.186015
$339$	0.0522506	0.00283787
$340$	18.6672	1.01237
$341$	0	0
$342$	−5.70695	−0.308597
$343$	−19.8454	−1.07155
$344$	0.824053	0.0444300
$345$	−0.234130	−0.0126052
$346$	−5.38942	−0.289737
$347$	3.85730	0.207071	0.103535	−	0.994626i	$-0.466985\pi$
$347$	3.85730	0.207071	0.103535	+	0.994626i	$0.466985\pi$
$348$	−0.152476	−0.00817355
$349$	−22.3728	−1.19759	−0.598794	−	0.800903i	$-0.704353\pi$
$349$	−22.3728	−1.19759	−0.598794	+	0.800903i	$0.704353\pi$
$350$	6.00261	0.320853
$351$	−0.283986	−0.0151580
$352$	0	0
$353$	−2.36325	−0.125783	−0.0628917	−	0.998020i	$-0.520032\pi$
$353$	−2.36325	−0.125783	−0.0628917	+	0.998020i	$0.520032\pi$
$354$	0.00473964	0.000251909 0
$355$	0.210109	0.0111514
$356$	33.6080	1.78122
$357$	0.0595205	0.00315016
$358$	1.81179	0.0957560
$359$	−5.06403	−0.267269	−0.133635	−	0.991031i	$-0.542665\pi$
$359$	−5.06403	−0.267269	−0.133635	+	0.991031i	$0.542665\pi$
$360$	17.3621	0.915065
$361$	−0.0263928	−0.00138910
$362$	6.87584	0.361386
$363$	0	0
$364$	15.9796	0.837560
$365$	−8.78587	−0.459873
$366$	−0.00452927	−0.000236749 0
$367$	−3.23910	−0.169080	−0.0845399	−	0.996420i	$-0.526942\pi$
$367$	−3.23910	−0.169080	−0.0845399	+	0.996420i	$0.526942\pi$
$368$	18.7675	0.978324
$369$	29.6307	1.54251
$370$	−1.59693	−0.0830206
$371$	22.8059	1.18402
$372$	−0.148709	−0.00771022
$373$	3.70872	0.192030	0.0960150	−	0.995380i	$-0.469390\pi$
$373$	3.70872	0.192030	0.0960150	+	0.995380i	$0.469390\pi$
$374$	0	0
$375$	−0.0758460	−0.00391667
$376$	−17.3542	−0.894975
$377$	37.0889	1.91018
$378$	0.0525884	0.00270485
$379$	−21.3298	−1.09564	−0.547820	−	0.836596i	$-0.684542\pi$
$379$	−21.3298	−1.09564	−0.547820	+	0.836596i	$0.684542\pi$
$380$	−27.4163	−1.40643
$381$	0.125256	0.00641704
$382$	−6.40428	−0.327671
$383$	4.42601	0.226158	0.113079	−	0.993586i	$-0.463929\pi$
$383$	4.42601	0.226158	0.113079	+	0.993586i	$0.463929\pi$
$384$	0.112326	0.00573210
$385$	0	0
$386$	8.58328	0.436877
$387$	1.48593	0.0755338
$388$	31.9782	1.62345
$389$	18.0980	0.917606	0.458803	−	0.888538i	$-0.348278\pi$
$389$	18.0980	0.917606	0.458803	+	0.888538i	$0.348278\pi$
$390$	0.0719135	0.00364148
$391$	−19.2470	−0.973363
$392$	5.41543	0.273521
$393$	−0.197065	−0.00994064
$394$	1.82453	0.0919187
$395$	−18.8149	−0.946679
$396$	0	0
$397$	25.6066	1.28516	0.642579	−	0.766219i	$-0.277864\pi$
$397$	25.6066	1.28516	0.642579	+	0.766219i	$0.277864\pi$
$398$	−6.36892	−0.319245
$399$	−0.0874173	−0.00437634
$400$	20.5393	1.02697
$401$	−24.9495	−1.24592	−0.622958	−	0.782255i	$-0.714069\pi$
$401$	−24.9495	−1.24592	−0.622958	+	0.782255i	$0.714069\pi$
$402$	−0.0276440	−0.00137876
$403$	36.1727	1.80189
$404$	−10.1939	−0.507165
$405$	31.3062	1.55562
$406$	−6.86811	−0.340858
$407$	0	0
$408$	−0.0511697	−0.00253328
$409$	−26.3498	−1.30292	−0.651458	−	0.758685i	$-0.725842\pi$
$409$	−26.3498	−1.30292	−0.651458	+	0.758685i	$0.725842\pi$
$410$	−15.0070	−0.741141
$411$	0.232761	0.0114812
$412$	30.2509	1.49035
$413$	−2.02505	−0.0996460
$414$	−8.50253	−0.417877
$415$	−38.0117	−1.86592
$416$	−20.9504	−1.02718
$417$	−0.127388	−0.00623821
$418$	0	0
$419$	21.0273	1.02725	0.513626	−	0.858014i	$-0.328302\pi$
$419$	21.0273	1.02725	0.513626	+	0.858014i	$0.328302\pi$
$420$	0.126316	0.00616357
$421$	−24.6599	−1.20185	−0.600924	−	0.799306i	$-0.705201\pi$
$421$	−24.6599	−1.20185	−0.600924	+	0.799306i	$0.705201\pi$
$422$	4.99167	0.242991
$423$	−31.2930	−1.52152
$424$	−19.6061	−0.952159
$425$	−21.0641	−1.02176
$426$	−0.000273551 0	−1.32536e−5 0
$427$	1.93516	0.0936492
$428$	−18.4954	−0.894009
$429$	0	0
$430$	−0.752573	−0.0362923
$431$	−34.0658	−1.64089	−0.820447	−	0.571723i	$-0.806275\pi$
$431$	−34.0658	−1.64089	−0.820447	+	0.571723i	$0.806275\pi$
$432$	0.179943	0.00865753
$433$	−26.2258	−1.26033	−0.630165	−	0.776461i	$-0.717013\pi$
$433$	−26.2258	−1.26033	−0.630165	+	0.776461i	$0.717013\pi$
$434$	−6.69845	−0.321536
$435$	0.293180	0.0140569
$436$	14.5492	0.696782
$437$	28.2679	1.35224
$438$	0.0114388	0.000546565 0
$439$	31.2884	1.49331	0.746657	−	0.665209i	$-0.231657\pi$
$439$	31.2884	1.49331	0.746657	+	0.665209i	$0.231657\pi$
$440$	0	0
$441$	9.76506	0.465003
$442$	5.91175	0.281193
$443$	−4.24854	−0.201854	−0.100927	−	0.994894i	$-0.532181\pi$
$443$	−4.24854	−0.201854	−0.100927	+	0.994894i	$0.532181\pi$
$444$	−0.0197213	−0.000935929 0
$445$	−64.6213	−3.06334
$446$	1.67213	0.0791775
$447$	0.206189	0.00975242
$448$	−7.31313	−0.345513
$449$	18.4688	0.871594	0.435797	−	0.900045i	$-0.356467\pi$
$449$	18.4688	0.871594	0.435797	+	0.900045i	$0.356467\pi$
$450$	−9.30525	−0.438654
$451$	0	0
$452$	9.11564	0.428764
$453$	−0.109809	−0.00515928
$454$	4.45166	0.208927
$455$	−30.7256	−1.44044
$456$	0.0751525	0.00351934
$457$	−28.5285	−1.33451	−0.667254	−	0.744830i	$-0.732531\pi$
$457$	−28.5285	−1.33451	−0.667254	+	0.744830i	$0.732531\pi$
$458$	2.05125	0.0958488
$459$	−0.184541	−0.00861363
$460$	−40.8464	−1.90447
$461$	32.7053	1.52324	0.761620	−	0.648024i	$-0.224404\pi$
$461$	32.7053	1.52324	0.761620	+	0.648024i	$0.224404\pi$
$462$	0	0
$463$	−17.7971	−0.827102	−0.413551	−	0.910481i	$-0.635712\pi$
$463$	−17.7971	−0.827102	−0.413551	+	0.910481i	$0.635712\pi$
$464$	−23.5008	−1.09100
$465$	0.285938	0.0132600
$466$	1.39848	0.0647834
$467$	10.2527	0.474439	0.237220	−	0.971456i	$-0.423764\pi$
$467$	10.2527	0.474439	0.237220	+	0.971456i	$0.423764\pi$
$468$	−24.7716	−1.14507
$469$	11.8111	0.545386
$470$	15.8489	0.731054
$471$	0.0731411	0.00337016
$472$	1.74093	0.0801327
$473$	0	0
$474$	0.0244960	0.00112514
$475$	30.9367	1.41947
$476$	10.3840	0.475948
$477$	−35.3536	−1.61873
$478$	6.56541	0.300295
$479$	37.5407	1.71528	0.857639	−	0.514252i	$-0.171930\pi$
$479$	37.5407	1.71528	0.857639	+	0.514252i	$0.171930\pi$
$480$	−0.165609	−0.00755897
$481$	4.79709	0.218728
$482$	−4.69919	−0.214042
$483$	−0.130239	−0.00592609
$484$	0	0
$485$	−61.4876	−2.79201
$486$	−0.122284	−0.00554693
$487$	−23.1356	−1.04837	−0.524186	−	0.851604i	$-0.675630\pi$
$487$	−23.1356	−1.04837	−0.524186	+	0.851604i	$0.675630\pi$
$488$	−1.66366	−0.0753102
$489$	−0.0532089	−0.00240619
$490$	−4.94569	−0.223423
$491$	3.37400	0.152267	0.0761333	−	0.997098i	$-0.475743\pi$
$491$	3.37400	0.152267	0.0761333	+	0.997098i	$0.475743\pi$
$492$	−0.185328	−0.00835522
$493$	24.1013	1.08547
$494$	−8.68254	−0.390646
$495$	0	0
$496$	−22.9203	−1.02915
$497$	0.116877	0.00524263
$498$	0.0494893	0.00221767
$499$	−3.50167	−0.156756	−0.0783781	−	0.996924i	$-0.524974\pi$
$499$	−3.50167	−0.156756	−0.0783781	+	0.996924i	$0.524974\pi$
$500$	−13.2321	−0.591757
$501$	0.235041	0.0105009
$502$	11.2716	0.503074
$503$	−33.4010	−1.48928	−0.744639	−	0.667468i	$-0.767378\pi$
$503$	−33.4010	−1.48928	−0.744639	+	0.667468i	$0.767378\pi$
$504$	9.65801	0.430202
$505$	19.6008	0.872223
$506$	0	0
$507$	−0.0812057	−0.00360647
$508$	21.8521	0.969530
$509$	4.81578	0.213456	0.106728	−	0.994288i	$-0.465963\pi$
$509$	4.81578	0.213456	0.106728	+	0.994288i	$0.465963\pi$
$510$	0.0467311	0.00206929
$511$	−4.88730	−0.216201
$512$	22.8973	1.01193
$513$	0.271034	0.0119664
$514$	−10.4183	−0.459532
$515$	−58.1663	−2.56311
$516$	−0.00929387	−0.000409140 0
$517$	0	0
$518$	−0.888323	−0.0390307
$519$	0.127975	0.00561746
$520$	26.4147	1.15836
$521$	−14.9388	−0.654481	−0.327241	−	0.944941i	$-0.606119\pi$
$521$	−14.9388	−0.654481	−0.327241	+	0.944941i	$0.606119\pi$
$522$	10.6469	0.466004
$523$	11.8191	0.516812	0.258406	−	0.966036i	$-0.416803\pi$
$523$	11.8191	0.516812	0.258406	+	0.966036i	$0.416803\pi$
$524$	−34.3800	−1.50190
$525$	−0.142535	−0.00622074
$526$	6.56778	0.286369
$527$	23.5059	1.02393
$528$	0	0
$529$	19.1151	0.831093
$530$	17.9055	0.777764
$531$	3.13923	0.136231
$532$	−15.2508	−0.661207
$533$	45.0800	1.95263
$534$	0.0841337	0.00364082
$535$	35.5629	1.53752
$536$	−10.1540	−0.438585
$537$	−0.0430218	−0.00185653
$538$	2.22520	0.0959351
$539$	0	0
$540$	−0.391636	−0.0168533
$541$	−23.8977	−1.02744	−0.513721	−	0.857957i	$-0.671733\pi$
$541$	−23.8977	−1.02744	−0.513721	+	0.857957i	$0.671733\pi$
$542$	5.98169	0.256936
$543$	−0.163270	−0.00700660
$544$	−13.6141	−0.583700
$545$	−27.9752	−1.19833
$546$	0.0400032	0.00171198
$547$	−3.31882	−0.141902	−0.0709512	−	0.997480i	$-0.522603\pi$
$547$	−3.31882	−0.141902	−0.0709512	+	0.997480i	$0.522603\pi$
$548$	40.6074	1.73466
$549$	−2.99989	−0.128032
$550$	0	0
$551$	−35.3973	−1.50798
$552$	0.111966	0.00476561
$553$	−10.4661	−0.445065
$554$	−5.26600	−0.223731
$555$	0.0379200	0.00160961
$556$	−22.2241	−0.942511
$557$	−5.44841	−0.230856	−0.115428	−	0.993316i	$-0.536824\pi$
$557$	−5.44841	−0.230856	−0.115428	+	0.993316i	$0.536824\pi$
$558$	10.3839	0.439587
$559$	2.26068	0.0956167
$560$	19.4688	0.822708
$561$	0	0
$562$	−12.5312	−0.528597
$563$	2.97850	0.125529	0.0627643	−	0.998028i	$-0.480008\pi$
$563$	2.97850	0.125529	0.0627643	+	0.998028i	$0.480008\pi$
$564$	0.195725	0.00824151
$565$	−17.5275	−0.737389
$566$	1.75854	0.0739171
$567$	17.4146	0.731345
$568$	−0.100479	−0.00421599
$569$	15.1867	0.636658	0.318329	−	0.947980i	$-0.396878\pi$
$569$	15.1867	0.636658	0.318329	+	0.947980i	$0.396878\pi$
$570$	−0.0686336	−0.00287475
$571$	0.197248	0.00825458	0.00412729	−	0.999991i	$-0.498686\pi$
$571$	0.197248	0.00825458	0.00412729	+	0.999991i	$0.498686\pi$
$572$	0	0
$573$	0.152073	0.00635293
$574$	−8.34790	−0.348434
$575$	46.0912	1.92214
$576$	11.3368	0.472367
$577$	−23.8728	−0.993839	−0.496919	−	0.867797i	$-0.665536\pi$
$577$	−23.8728	−0.993839	−0.496919	+	0.867797i	$0.665536\pi$
$578$	−3.58299	−0.149033
$579$	−0.203814	−0.00847023
$580$	51.1481	2.12381
$581$	−21.1447	−0.877229
$582$	0.0800537	0.00331833
$583$	0	0
$584$	4.20160	0.173863
$585$	47.6308	1.96929
$586$	−0.859685	−0.0355132
$587$	−25.9108	−1.06945	−0.534727	−	0.845025i	$-0.679585\pi$
$587$	−25.9108	−1.06945	−0.534727	+	0.845025i	$0.679585\pi$
$588$	−0.0610765	−0.00251875
$589$	−34.5230	−1.42249
$590$	−1.58992	−0.0654558
$591$	−0.0433244	−0.00178213
$592$	−3.03960	−0.124927
$593$	−0.815514	−0.0334891	−0.0167446	−	0.999860i	$-0.505330\pi$
$593$	−0.815514	−0.0334891	−0.0167446	+	0.999860i	$0.505330\pi$
$594$	0	0
$595$	−19.9662	−0.818536
$596$	35.9718	1.47346
$597$	0.151233	0.00618956
$598$	−12.9357	−0.528981
$599$	6.38692	0.260963	0.130481	−	0.991451i	$-0.458348\pi$
$599$	6.38692	0.260963	0.130481	+	0.991451i	$0.458348\pi$
$600$	0.122537	0.00500255
$601$	29.4015	1.19931	0.599656	−	0.800258i	$-0.295304\pi$
$601$	29.4015	1.19931	0.599656	+	0.800258i	$0.295304\pi$
$602$	−0.418632	−0.0170622
$603$	−18.3096	−0.745623
$604$	−19.1573	−0.779499
$605$	0	0
$606$	−0.0255192	−0.00103665
$607$	−9.41610	−0.382188	−0.191094	−	0.981572i	$-0.561203\pi$
$607$	−9.41610	−0.382188	−0.191094	+	0.981572i	$0.561203\pi$
$608$	19.9949	0.810901
$609$	0.163087	0.00660860
$610$	1.51935	0.0615166
$611$	−47.6090	−1.92606
$612$	−16.0972	−0.650691
$613$	3.24986	0.131261	0.0656304	−	0.997844i	$-0.479094\pi$
$613$	3.24986	0.131261	0.0656304	+	0.997844i	$0.479094\pi$
$614$	−10.2842	−0.415038
$615$	0.356348	0.0143693
$616$	0	0
$617$	−8.37130	−0.337016	−0.168508	−	0.985700i	$-0.553895\pi$
$617$	−8.37130	−0.337016	−0.168508	+	0.985700i	$0.553895\pi$
$618$	0.0757296	0.00304629
$619$	−8.62770	−0.346777	−0.173388	−	0.984854i	$-0.555472\pi$
$619$	−8.62770	−0.346777	−0.173388	+	0.984854i	$0.555472\pi$
$620$	49.8847	2.00342
$621$	0.403801	0.0162040
$622$	9.54981	0.382913
$623$	−35.9468	−1.44018
$624$	0.136880	0.00547959
$625$	−10.0689	−0.402755
$626$	−3.86576	−0.154507
$627$	0	0
$628$	12.7602	0.509187
$629$	3.11726	0.124293
$630$	−8.82025	−0.351407
$631$	28.1700	1.12143	0.560716	−	0.828008i	$-0.310526\pi$
$631$	28.1700	1.12143	0.560716	+	0.828008i	$0.310526\pi$
$632$	8.99770	0.357909
$633$	−0.118530	−0.00471113
$634$	0.241715	0.00959972
$635$	−42.0171	−1.66740
$636$	0.221123	0.00876809
$637$	14.8565	0.588637
$638$	0	0
$639$	−0.181182	−0.00716746
$640$	−37.6798	−1.48942
$641$	−4.34993	−0.171812	−0.0859059	−	0.996303i	$-0.527378\pi$
$641$	−4.34993	−0.171812	−0.0859059	+	0.996303i	$0.527378\pi$
$642$	−0.0463011	−0.00182736
$643$	33.0246	1.30236	0.651181	−	0.758923i	$-0.274274\pi$
$643$	33.0246	1.30236	0.651181	+	0.758923i	$0.274274\pi$
$644$	−22.7215	−0.895354
$645$	0.0178702	0.000703639 0
$646$	−5.64213	−0.221986
$647$	−21.7241	−0.854061	−0.427031	−	0.904237i	$-0.640440\pi$
$647$	−21.7241	−0.854061	−0.427031	+	0.904237i	$0.640440\pi$
$648$	−14.9713	−0.588128
$649$	0	0
$650$	−14.1570	−0.555283
$651$	0.159058	0.00623398
$652$	−9.28283	−0.363543
$653$	−17.0226	−0.666146	−0.333073	−	0.942901i	$-0.608085\pi$
$653$	−17.0226	−0.666146	−0.333073	+	0.942901i	$0.608085\pi$
$654$	0.0364224	0.00142423
$655$	66.1058	2.58297
$656$	−28.5643	−1.11525
$657$	7.57629	0.295579
$658$	8.81622	0.343692
$659$	−45.0710	−1.75572	−0.877858	−	0.478921i	$-0.841028\pi$
$659$	−45.0710	−1.75572	−0.877858	+	0.478921i	$0.841028\pi$
$660$	0	0
$661$	17.6243	0.685504	0.342752	−	0.939426i	$-0.388641\pi$
$661$	17.6243	0.685504	0.342752	+	0.939426i	$0.388641\pi$
$662$	−0.892972	−0.0347063
$663$	−0.140377	−0.00545181
$664$	18.1780	0.705444
$665$	29.3242	1.13715
$666$	1.37708	0.0533607
$667$	−52.7369	−2.04198
$668$	41.0053	1.58654
$669$	−0.0397055	−0.00153510
$670$	9.27320	0.358255
$671$	0	0
$672$	−0.0921228	−0.00355372
$673$	9.44548	0.364096	0.182048	−	0.983290i	$-0.441727\pi$
$673$	9.44548	0.364096	0.182048	+	0.983290i	$0.441727\pi$
$674$	1.78575	0.0687845
$675$	0.441923	0.0170096
$676$	−14.1671	−0.544890
$677$	42.3833	1.62892	0.814461	−	0.580218i	$-0.197033\pi$
$677$	42.3833	1.62892	0.814461	+	0.580218i	$0.197033\pi$
$678$	0.0228200	0.000876396 0
$679$	−34.2036	−1.31261
$680$	17.1649	0.658245
$681$	−0.105707	−0.00405069
$682$	0	0
$683$	−7.58574	−0.290260	−0.145130	−	0.989413i	$-0.546360\pi$
$683$	−7.58574	−0.290260	−0.145130	+	0.989413i	$0.546360\pi$
$684$	23.6418	0.903968
$685$	−78.0798	−2.98328
$686$	−8.66728	−0.330918
$687$	−0.0487080	−0.00185833
$688$	−1.43245	−0.0546116
$689$	−53.7869	−2.04912
$690$	−0.102254	−0.00389275
$691$	6.46933	0.246105	0.123052	−	0.992400i	$-0.460732\pi$
$691$	6.46933	0.246105	0.123052	+	0.992400i	$0.460732\pi$
$692$	22.3264	0.848723
$693$	0	0
$694$	1.68464	0.0639480
$695$	42.7324	1.62093
$696$	−0.140205	−0.00531446
$697$	29.2941	1.10959
$698$	−9.77111	−0.369842
$699$	−0.0332076	−0.00125603
$700$	−24.8666	−0.939871
$701$	−13.5910	−0.513325	−0.256663	−	0.966501i	$-0.582623\pi$
$701$	−13.5910	−0.513325	−0.256663	+	0.966501i	$0.582623\pi$
$702$	−0.124028	−0.00468114
$703$	−4.57830	−0.172674
$704$	0	0
$705$	−0.376339	−0.0141738
$706$	−1.03213	−0.0388447
$707$	10.9033	0.410060
$708$	−0.0196346	−0.000737914 0
$709$	9.54923	0.358629	0.179314	−	0.983792i	$-0.442612\pi$
$709$	9.54923	0.358629	0.179314	+	0.983792i	$0.442612\pi$
$710$	0.0917629	0.00344380
$711$	16.2246	0.608469
$712$	30.9034	1.15815
$713$	−51.4342	−1.92623
$714$	0.0259950	0.000972840 0
$715$	0	0
$716$	−7.50559	−0.280497
$717$	−0.155899	−0.00582215
$718$	−2.21167	−0.0825387
$719$	−6.51267	−0.242881	−0.121441	−	0.992599i	$-0.538751\pi$
$719$	−6.51267	−0.242881	−0.121441	+	0.992599i	$0.538751\pi$
$720$	−30.1806	−1.12476
$721$	−32.3561	−1.20500
$722$	−0.0115268	−0.000428984 0
$723$	0.111585	0.00414987
$724$	−28.4841	−1.05860
$725$	−57.7157	−2.14351
$726$	0	0
$727$	3.58509	0.132964	0.0664818	−	0.997788i	$-0.478823\pi$
$727$	3.58509	0.132964	0.0664818	+	0.997788i	$0.478823\pi$
$728$	14.6937	0.544584
$729$	−26.9942	−0.999785
$730$	−3.83714	−0.142019
$731$	1.46905	0.0543347
$732$	0.0187631	0.000693505 0
$733$	17.0361	0.629243	0.314622	−	0.949217i	$-0.398122\pi$
$733$	17.0361	0.629243	0.314622	+	0.949217i	$0.398122\pi$
$734$	−1.41465	−0.0522156
$735$	0.117438	0.00433176
$736$	29.7895	1.09806
$737$	0	0
$738$	12.9409	0.476361
$739$	−41.5324	−1.52779	−0.763897	−	0.645339i	$-0.776716\pi$
$739$	−41.5324	−1.52779	−0.763897	+	0.645339i	$0.776716\pi$
$740$	6.61552	0.243191
$741$	0.206171	0.00757389
$742$	9.96024	0.365652
$743$	−18.1772	−0.666855	−0.333428	−	0.942776i	$-0.608205\pi$
$743$	−18.1772	−0.666855	−0.333428	+	0.942776i	$0.608205\pi$
$744$	−0.136742	−0.00501320
$745$	−69.1665	−2.53406
$746$	1.61975	0.0593031
$747$	32.7785	1.19930
$748$	0	0
$749$	19.7825	0.722837
$750$	−0.0331250	−0.00120955
$751$	45.6729	1.66663	0.833314	−	0.552799i	$-0.186440\pi$
$751$	45.6729	1.66663	0.833314	+	0.552799i	$0.186440\pi$
$752$	30.1668	1.10007
$753$	−0.267649	−0.00975366
$754$	16.1982	0.589905
$755$	36.8356	1.34058
$756$	−0.217855	−0.00792329
$757$	−22.2550	−0.808873	−0.404436	−	0.914566i	$-0.632532\pi$
$757$	−22.2550	−0.808873	−0.404436	+	0.914566i	$0.632532\pi$
$758$	−9.31560	−0.338358
$759$	0	0
$760$	−25.2100	−0.914462
$761$	14.1456	0.512779	0.256390	−	0.966573i	$-0.417467\pi$
$761$	14.1456	0.512779	0.256390	+	0.966573i	$0.417467\pi$
$762$	0.0547042	0.00198172
$763$	−15.5617	−0.563372
$764$	26.5306	0.959844
$765$	30.9517	1.11906
$766$	1.93302	0.0698427
$767$	4.77601	0.172452
$768$	−0.0293254	−0.00105819
$769$	13.7367	0.495357	0.247678	−	0.968842i	$-0.420332\pi$
$769$	13.7367	0.495357	0.247678	+	0.968842i	$0.420332\pi$
$770$	0	0
$771$	0.247388	0.00890945
$772$	−35.5574	−1.27974
$773$	−36.9060	−1.32742	−0.663709	−	0.747991i	$-0.731019\pi$
$773$	−36.9060	−1.32742	−0.663709	+	0.747991i	$0.731019\pi$
$774$	0.648964	0.0233265
$775$	−56.2901	−2.02200
$776$	29.4047	1.05557
$777$	0.0210937	0.000756731 0
$778$	7.90414	0.283377
$779$	−43.0240	−1.54149
$780$	−0.297912	−0.0106669
$781$	0	0
$782$	−8.40595	−0.300596
$783$	−0.505643	−0.0180702
$784$	−9.41362	−0.336201
$785$	−24.5353	−0.875701
$786$	−0.0860665	−0.00306989
$787$	50.6675	1.80610	0.903051	−	0.429534i	$-0.141322\pi$
$787$	50.6675	1.80610	0.903051	+	0.429534i	$0.141322\pi$
$788$	−7.55838	−0.269256
$789$	−0.155955	−0.00555215
$790$	−8.21722	−0.292356
$791$	−9.75001	−0.346670
$792$	0	0
$793$	−4.56403	−0.162073
$794$	11.1834	0.396885
$795$	−0.425174	−0.0150794
$796$	26.3841	0.935160
$797$	47.1764	1.67107	0.835536	−	0.549435i	$-0.185157\pi$
$797$	47.1764	1.67107	0.835536	+	0.549435i	$0.185157\pi$
$798$	−0.0381787	−0.00135151
$799$	−30.9375	−1.09449
$800$	32.6019	1.15265
$801$	55.7247	1.96894
$802$	−10.8964	−0.384767
$803$	0	0
$804$	0.114519	0.00403877
$805$	43.6889	1.53983
$806$	15.7981	0.556464
$807$	−0.0528384	−0.00186000
$808$	−9.37353	−0.329760
$809$	9.36222	0.329158	0.164579	−	0.986364i	$-0.447373\pi$
$809$	9.36222	0.329158	0.164579	+	0.986364i	$0.447373\pi$
$810$	13.6727	0.480408
$811$	48.1058	1.68922	0.844611	−	0.535380i	$-0.179832\pi$
$811$	48.1058	1.68922	0.844611	+	0.535380i	$0.179832\pi$
$812$	28.4521	0.998472
$813$	−0.142038	−0.00498150
$814$	0	0
$815$	17.8490	0.625223
$816$	0.0889481	0.00311381
$817$	−2.15758	−0.0754841
$818$	−11.5080	−0.402369
$819$	26.4955	0.925828
$820$	62.1684	2.17102
$821$	−12.1493	−0.424014	−0.212007	−	0.977268i	$-0.568000\pi$
$821$	−12.1493	−0.424014	−0.212007	+	0.977268i	$0.568000\pi$
$822$	0.101656	0.00354566
$823$	−52.0806	−1.81541	−0.907707	−	0.419604i	$-0.862169\pi$
$823$	−52.0806	−1.81541	−0.907707	+	0.419604i	$0.862169\pi$
$824$	27.8164	0.969032
$825$	0	0
$826$	−0.884420	−0.0307729
$827$	17.3050	0.601753	0.300876	−	0.953663i	$-0.402721\pi$
$827$	17.3050	0.601753	0.300876	+	0.953663i	$0.402721\pi$
$828$	35.2229	1.22408
$829$	−32.5453	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$829$	−32.5453	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$830$	−16.6012	−0.576237
$831$	0.125044	0.00433772
$832$	17.2478	0.597960
$833$	9.65414	0.334496
$834$	−0.0556355	−0.00192650
$835$	−78.8449	−2.72854
$836$	0	0
$837$	−0.493152	−0.0170458
$838$	9.18348	0.317238
$839$	−10.1679	−0.351035	−0.175517	−	0.984476i	$-0.556160\pi$
$839$	−10.1679	−0.351035	−0.175517	+	0.984476i	$0.556160\pi$
$840$	0.116150	0.00400757
$841$	37.0375	1.27716
$842$	−10.7700	−0.371157
$843$	0.297559	0.0102485
$844$	−20.6787	−0.711789
$845$	27.2405	0.937103
$846$	−13.6669	−0.469878
$847$	0	0
$848$	34.0813	1.17036
$849$	−0.0417575	−0.00143311
$850$	−9.19955	−0.315542
$851$	−6.82101	−0.233821
$852$	0.00113322	3.88235e−5 0
$853$	10.7159	0.366907	0.183453	−	0.983028i	$-0.441272\pi$
$853$	10.7159	0.366907	0.183453	+	0.983028i	$0.441272\pi$
$854$	0.845165	0.0289210
$855$	−45.4584	−1.55465
$856$	−17.0070	−0.581287
$857$	10.1298	0.346027	0.173014	−	0.984919i	$-0.444650\pi$
$857$	10.1298	0.346027	0.173014	+	0.984919i	$0.444650\pi$
$858$	0	0
$859$	−8.76656	−0.299111	−0.149556	−	0.988753i	$-0.547784\pi$
$859$	−8.76656	−0.299111	−0.149556	+	0.988753i	$0.547784\pi$
$860$	3.11764	0.106311
$861$	0.198225	0.00675549
$862$	−14.8779	−0.506744
$863$	0.329472	0.0112153	0.00560767	−	0.999984i	$-0.498215\pi$
$863$	0.329472	0.0112153	0.00560767	+	0.999984i	$0.498215\pi$
$864$	0.285623	0.00971708
$865$	−42.9292	−1.45964
$866$	−11.4539	−0.389218
$867$	0.0850799	0.00288946
$868$	27.7493	0.941871
$869$	0	0
$870$	0.128043	0.00434108
$871$	−27.8561	−0.943868
$872$	13.3784	0.453049
$873$	53.0224	1.79454
$874$	12.3458	0.417601
$875$	14.1529	0.478456
$876$	−0.0473866	−0.00160105
$877$	−19.8266	−0.669497	−0.334748	−	0.942308i	$-0.608651\pi$
$877$	−19.8266	−0.669497	−0.334748	+	0.942308i	$0.608651\pi$
$878$	13.6649	0.461169
$879$	0.0204136	0.000688535 0
$880$	0	0
$881$	43.9146	1.47952	0.739761	−	0.672870i	$-0.234939\pi$
$881$	43.9146	1.47952	0.739761	+	0.672870i	$0.234939\pi$
$882$	4.26480	0.143603
$883$	−1.12961	−0.0380143	−0.0190071	−	0.999819i	$-0.506051\pi$
$883$	−1.12961	−0.0380143	−0.0190071	+	0.999819i	$0.506051\pi$
$884$	−24.4902	−0.823696
$885$	0.0377534	0.00126907
$886$	−1.85551	−0.0623370
$887$	30.5280	1.02503	0.512515	−	0.858678i	$-0.328714\pi$
$887$	30.5280	1.02503	0.512515	+	0.858678i	$0.328714\pi$
$888$	−0.0181342	−0.000608543 0
$889$	−23.3728	−0.783898
$890$	−28.2228	−0.946029
$891$	0	0
$892$	−6.92702	−0.231934
$893$	45.4377	1.52051
$894$	0.0900513	0.00301176
$895$	14.4317	0.482399
$896$	−20.9601	−0.700226
$897$	0.307165	0.0102560
$898$	8.06605	0.269168
$899$	64.4063	2.14807
$900$	38.5483	1.28494
$901$	−34.9520	−1.16442
$902$	0	0
$903$	0.00994063	0.000330803 0
$904$	8.38206	0.278783
$905$	54.7692	1.82059
$906$	−0.0479581	−0.00159330
$907$	−6.59359	−0.218937	−0.109468	−	0.993990i	$-0.534915\pi$
$907$	−6.59359	−0.218937	−0.109468	+	0.993990i	$0.534915\pi$
$908$	−18.4416	−0.612006
$909$	−16.9023	−0.560613
$910$	−13.4191	−0.444839
$911$	−25.2887	−0.837850	−0.418925	−	0.908021i	$-0.637593\pi$
$911$	−25.2887	−0.837850	−0.418925	+	0.908021i	$0.637593\pi$
$912$	−0.130637	−0.00432583
$913$	0	0
$914$	−12.4596	−0.412126
$915$	−0.0360777	−0.00119269
$916$	−8.49760	−0.280769
$917$	36.7725	1.21434
$918$	−0.0805964	−0.00266008
$919$	39.4731	1.30210	0.651049	−	0.759036i	$-0.274329\pi$
$919$	39.4731	1.30210	0.651049	+	0.759036i	$0.274329\pi$
$920$	−37.5592	−1.23829
$921$	0.244204	0.00804680
$922$	14.2837	0.470410
$923$	−0.275650	−0.00907313
$924$	0	0
$925$	−7.46497	−0.245447
$926$	−7.77272	−0.255428
$927$	50.1584	1.64742
$928$	−37.3027	−1.22452
$929$	51.4929	1.68943	0.844713	−	0.535219i	$-0.179771\pi$
$929$	51.4929	1.68943	0.844713	+	0.535219i	$0.179771\pi$
$930$	0.124881	0.00409500
$931$	−14.1790	−0.464696
$932$	−5.79340	−0.189769
$933$	−0.226765	−0.00742395
$934$	4.47778	0.146517
$935$	0	0
$936$	−22.7781	−0.744526
$937$	−5.68848	−0.185835	−0.0929173	−	0.995674i	$-0.529619\pi$
$937$	−5.68848	−0.185835	−0.0929173	+	0.995674i	$0.529619\pi$
$938$	5.15838	0.168427
$939$	0.0917943	0.00299559
$940$	−65.6561	−2.14147
$941$	9.84589	0.320967	0.160483	−	0.987039i	$-0.448695\pi$
$941$	9.84589	0.320967	0.160483	+	0.987039i	$0.448695\pi$
$942$	0.0319437	0.00104078
$943$	−64.0995	−2.08737
$944$	−3.02625	−0.0984960
$945$	0.418890	0.0136265
$946$	0	0
$947$	−9.33207	−0.303251	−0.151626	−	0.988438i	$-0.548451\pi$
$947$	−9.33207	−0.303251	−0.151626	+	0.988438i	$0.548451\pi$
$948$	−0.101478	−0.00329586
$949$	11.5265	0.374167
$950$	13.5113	0.438365
$951$	−0.00573964	−0.000186120 0
$952$	9.54830	0.309462
$953$	2.64276	0.0856074	0.0428037	−	0.999084i	$-0.486371\pi$
$953$	2.64276	0.0856074	0.0428037	+	0.999084i	$0.486371\pi$
$954$	−15.4404	−0.499900
$955$	−51.0130	−1.65074
$956$	−27.1981	−0.879650
$957$	0	0
$958$	16.3955	0.529716
$959$	−43.4333	−1.40254
$960$	0.136340	0.00440036
$961$	31.8154	1.02630
$962$	2.09508	0.0675482
$963$	−30.6668	−0.988225
$964$	19.4670	0.626991
$965$	68.3697	2.20090
$966$	−0.0568807	−0.00183011
$967$	38.0052	1.22217	0.611083	−	0.791567i	$-0.290734\pi$
$967$	38.0052	1.22217	0.611083	+	0.791567i	$0.290734\pi$
$968$	0	0
$969$	0.133975	0.00430390
$970$	−26.8541	−0.862234
$971$	−0.160161	−0.00513981	−0.00256991	−	0.999997i	$-0.500818\pi$
$971$	−0.160161	−0.00513981	−0.00256991	+	0.999997i	$0.500818\pi$
$972$	0.506580	0.0162486
$973$	23.7707	0.762053
$974$	−10.1042	−0.323761
$975$	0.336165	0.0107659
$976$	2.89193	0.0925684
$977$	20.4963	0.655735	0.327867	−	0.944724i	$-0.393670\pi$
$977$	20.4963	0.655735	0.327867	+	0.944724i	$0.393670\pi$
$978$	−0.0232385	−0.000743085 0
$979$	0	0
$980$	20.4882	0.654471
$981$	24.1238	0.770213
$982$	1.47356	0.0470233
$983$	10.2845	0.328024	0.164012	−	0.986458i	$-0.447556\pi$
$983$	10.2845	0.328024	0.164012	+	0.986458i	$0.447556\pi$
$984$	−0.170414	−0.00543258
$985$	14.5332	0.463067
$986$	10.5260	0.335216
$987$	−0.209346	−0.00666354
$988$	35.9686	1.14431
$989$	−3.21448	−0.102214
$990$	0	0
$991$	9.85979	0.313207	0.156603	−	0.987662i	$-0.449946\pi$
$991$	9.85979	0.313207	0.156603	+	0.987662i	$0.449946\pi$
$992$	−36.3812	−1.15511
$993$	0.0212041	0.000672890 0
$994$	0.0510448	0.00161904
$995$	−50.7313	−1.60829
$996$	−0.205016	−0.00649618
$997$	26.1240	0.827356	0.413678	−	0.910423i	$-0.364244\pi$
$997$	26.1240	0.827356	0.413678	+	0.910423i	$0.364244\pi$
$998$	−1.52932	−0.0484098
$999$	−0.0654000	−0.00206916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7381.2.a.i.1.12		19
11.10	odd	2		671.2.a.c.1.8	✓	19
33.32	even	2		6039.2.a.k.1.12		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.8	✓	19	11.10	odd	2
6039.2.a.k.1.12		19	33.32	even	2
7381.2.a.i.1.12		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:	\( N \)	\(=\)	\( 7381 = 11^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7381.a (trivial)

\(f(q)\)	\(=\)	\(q+0.436741 q^{2} -0.0103706 q^{3} -1.80926 q^{4} +3.47884 q^{5} -0.00452927 q^{6} +1.93516 q^{7} -1.66366 q^{8} -2.99989 q^{9} +O(q^{10})\)	\(q+0.436741 q^{2} -0.0103706 q^{3} -1.80926 q^{4} +3.47884 q^{5} -0.00452927 q^{6} +1.93516 q^{7} -1.66366 q^{8} -2.99989 q^{9} +1.51935 q^{10} +0.0187631 q^{12} -4.56403 q^{13} +0.845165 q^{14} -0.0360777 q^{15} +2.89193 q^{16} -2.96582 q^{17} -1.31017 q^{18} +4.35587 q^{19} -6.29411 q^{20} -0.0200689 q^{21} +6.48962 q^{23} +0.0172532 q^{24} +7.10230 q^{25} -1.99330 q^{26} +0.0622226 q^{27} -3.50121 q^{28} -8.12635 q^{29} -0.0157566 q^{30} -7.92562 q^{31} +4.59034 q^{32} -1.29529 q^{34} +6.73212 q^{35} +5.42758 q^{36} -1.05106 q^{37} +1.90239 q^{38} +0.0473318 q^{39} -5.78759 q^{40} -9.87724 q^{41} -0.00876488 q^{42} -0.495326 q^{43} -10.4361 q^{45} +2.83428 q^{46} +10.4314 q^{47} -0.0299911 q^{48} -3.25514 q^{49} +3.10186 q^{50} +0.0307574 q^{51} +8.25751 q^{52} +11.7850 q^{53} +0.0271751 q^{54} -3.21945 q^{56} -0.0451731 q^{57} -3.54911 q^{58} -1.04645 q^{59} +0.0652738 q^{60} +1.00000 q^{61} -3.46144 q^{62} -5.80529 q^{63} -3.77907 q^{64} -15.8775 q^{65} +6.10340 q^{67} +5.36593 q^{68} -0.0673013 q^{69} +2.94019 q^{70} +0.0603962 q^{71} +4.99079 q^{72} -2.52552 q^{73} -0.459042 q^{74} -0.0736552 q^{75} -7.88089 q^{76} +0.0206717 q^{78} -5.40838 q^{79} +10.0605 q^{80} +8.99903 q^{81} -4.31379 q^{82} -10.9265 q^{83} +0.0363097 q^{84} -10.3176 q^{85} -0.216329 q^{86} +0.0842753 q^{87} -18.5756 q^{89} -4.55788 q^{90} -8.83215 q^{91} -11.7414 q^{92} +0.0821935 q^{93} +4.55580 q^{94} +15.1534 q^{95} -0.0476046 q^{96} -17.6748 q^{97} -1.42165 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9	\( 19 q - 5 q^{2} + 23 q^{4} - q^{6} - 9 q^{7} - 9 q^{8} + 29 q^{9} - 7 q^{10} - 4 q^{12} - 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} - 9 q^{17} - 10 q^{18} - 17 q^{19} - 6 q^{20} - 18 q^{21} - 10 q^{23} - 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} - 36 q^{28} - 27 q^{29} + 30 q^{30} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} - 24 q^{39} - 10 q^{40} - 19 q^{41} + 21 q^{42} - 20 q^{43} - 32 q^{45} - 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} - 36 q^{50} - 47 q^{51} + 28 q^{52} + 3 q^{53} + 33 q^{54} - 44 q^{56} - 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} + 19 q^{61} + 11 q^{62} + 32 q^{63} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} - 19 q^{71} - 34 q^{72} - 20 q^{73} + 22 q^{74} - 50 q^{75} + 25 q^{76} - 94 q^{78} - 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} - q^{83} + 28 q^{84} - 24 q^{85} - 27 q^{86} + 58 q^{87} - 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} - 64 q^{94} + 3 q^{95} + 26 q^{96} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 - q^6 - 9 * q^7 - 9 * q^8 + 29 * q^9 - 7 * q^10 - 4 * q^12 - 8 * q^13 - 11 * q^14 - 5 * q^15 + 31 * q^16 - 9 * q^17 - 10 * q^18 - 17 * q^19 - 6 * q^20 - 18 * q^21 - 10 * q^23 - 26 * q^24 + 45 * q^25 + 5 * q^26 - 33 * q^27 - 36 * q^28 - 27 * q^29 + 30 * q^30 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 38 * q^36 + 20 * q^37 - 37 * q^38 - 24 * q^39 - 10 * q^40 - 19 * q^41 + 21 * q^42 - 20 * q^43 - 32 * q^45 - 41 * q^46 - 19 * q^47 + 5 * q^48 + 42 * q^49 - 36 * q^50 - 47 * q^51 + 28 * q^52 + 3 * q^53 + 33 * q^54 - 44 * q^56 - 11 * q^57 + 23 * q^58 - 28 * q^59 - 96 * q^60 + 19 * q^61 + 11 * q^62 + 32 * q^63 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 - 19 * q^71 - 34 * q^72 - 20 * q^73 + 22 * q^74 - 50 * q^75 + 25 * q^76 - 94 * q^78 - 69 * q^79 - 36 * q^80 + 47 * q^81 - 61 * q^82 - q^83 + 28 * q^84 - 24 * q^85 - 27 * q^86 + 58 * q^87 - 36 * q^90 + 24 * q^91 - 67 * q^92 - 14 * q^93 - 64 * q^94 + 3 * q^95 + 26 * q^96 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more