LMFDB - Embedded newform 7406.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7406,2,Mod(1,7406)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7406, 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("7406.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7406 = 2 \cdot 7 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7406.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:	$59.1372077370$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.74912$ of defining polynomial
Character	$\chi$	$=$	7406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.334904 q^{3} +1.00000 q^{4} +2.22274 q^{5} +0.334904 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.88784 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.334904 q^{3} +1.00000 q^{4} +2.22274 q^{5} +0.334904 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.88784 q^{9} +2.22274 q^{10} -0.585786 q^{11} +0.334904 q^{12} -5.97186 q^{13} +1.00000 q^{14} +0.744406 q^{15} +1.00000 q^{16} -4.22274 q^{17} -2.88784 q^{18} -2.66981 q^{19} +2.22274 q^{20} +0.334904 q^{21} -0.585786 q^{22} +0.334904 q^{24} -0.0594122 q^{25} -5.97186 q^{26} -1.97186 q^{27} +1.00000 q^{28} -4.43882 q^{29} +0.744406 q^{30} +6.11058 q^{31} +1.00000 q^{32} -0.196182 q^{33} -4.22274 q^{34} +2.22274 q^{35} -2.88784 q^{36} +3.53304 q^{37} -2.66981 q^{38} -2.00000 q^{39} +2.22274 q^{40} +2.27744 q^{41} +0.334904 q^{42} +5.78510 q^{43} -0.585786 q^{44} -6.41893 q^{45} +3.04451 q^{47} +0.334904 q^{48} +1.00000 q^{49} -0.0594122 q^{50} -1.41421 q^{51} -5.97186 q^{52} -9.19932 q^{53} -1.97186 q^{54} -1.30205 q^{55} +1.00000 q^{56} -0.894129 q^{57} -4.43882 q^{58} -4.11058 q^{59} +0.744406 q^{60} -1.23570 q^{61} +6.11058 q^{62} -2.88784 q^{63} +1.00000 q^{64} -13.2739 q^{65} -0.196182 q^{66} -6.46696 q^{67} -4.22274 q^{68} +2.22274 q^{70} +1.21137 q^{71} -2.88784 q^{72} -12.6948 q^{73} +3.53304 q^{74} -0.0198974 q^{75} -2.66981 q^{76} -0.585786 q^{77} -2.00000 q^{78} -12.7672 q^{79} +2.22274 q^{80} +8.00313 q^{81} +2.27744 q^{82} +9.97186 q^{83} +0.334904 q^{84} -9.38607 q^{85} +5.78510 q^{86} -1.48658 q^{87} -0.585786 q^{88} -14.6588 q^{89} -6.41893 q^{90} -5.97186 q^{91} +2.04646 q^{93} +3.04451 q^{94} -5.93430 q^{95} +0.334904 q^{96} -2.85499 q^{97} +1.00000 q^{98} +1.69166 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8} - 4 q^{10} - 8 q^{11} - 4 q^{13} + 4 q^{14} + 4 q^{16} - 4 q^{17} - 8 q^{19} - 4 q^{20} - 8 q^{22} - 4 q^{26} + 12 q^{27} + 4 q^{28} - 4 q^{29} + 4 q^{32} - 4 q^{33} - 4 q^{34} - 4 q^{35} + 8 q^{37} - 8 q^{38} - 8 q^{39} - 4 q^{40} - 8 q^{43} - 8 q^{44} - 16 q^{45} + 4 q^{49} - 4 q^{52} + 12 q^{54} + 12 q^{55} + 4 q^{56} - 24 q^{57} - 4 q^{58} + 8 q^{59} - 12 q^{61} + 4 q^{64} - 16 q^{65} - 4 q^{66} - 32 q^{67} - 4 q^{68} - 4 q^{70} + 8 q^{71} + 4 q^{73} + 8 q^{74} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 8 q^{78} - 16 q^{79} - 4 q^{80} - 8 q^{81} + 20 q^{83} - 12 q^{85} - 8 q^{86} - 20 q^{87} - 8 q^{88} - 28 q^{89} - 16 q^{90} - 4 q^{91} - 12 q^{93} + 8 q^{95} + 16 q^{97} + 4 q^{98} - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 + 4 * q^8 - 4 * q^10 - 8 * q^11 - 4 * q^13 + 4 * q^14 + 4 * q^16 - 4 * q^17 - 8 * q^19 - 4 * q^20 - 8 * q^22 - 4 * q^26 + 12 * q^27 + 4 * q^28 - 4 * q^29 + 4 * q^32 - 4 * q^33 - 4 * q^34 - 4 * q^35 + 8 * q^37 - 8 * q^38 - 8 * q^39 - 4 * q^40 - 8 * q^43 - 8 * q^44 - 16 * q^45 + 4 * q^49 - 4 * q^52 + 12 * q^54 + 12 * q^55 + 4 * q^56 - 24 * q^57 - 4 * q^58 + 8 * q^59 - 12 * q^61 + 4 * q^64 - 16 * q^65 - 4 * q^66 - 32 * q^67 - 4 * q^68 - 4 * q^70 + 8 * q^71 + 4 * q^73 + 8 * q^74 + 4 * q^75 - 8 * q^76 - 8 * q^77 - 8 * q^78 - 16 * q^79 - 4 * q^80 - 8 * q^81 + 20 * q^83 - 12 * q^85 - 8 * q^86 - 20 * q^87 - 8 * q^88 - 28 * q^89 - 16 * q^90 - 4 * q^91 - 12 * q^93 + 8 * q^95 + 16 * q^97 + 4 * q^98 - 8 * 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
$3$	0.334904	0.193357	0.0966785	−	0.995316i	$-0.469178\pi$
$3$	0.334904	0.193357	0.0966785	+	0.995316i	$0.469178\pi$
$4$	1.00000	0.500000
$5$	2.22274	0.994041	0.497021	−	0.867739i	$-0.334427\pi$
$5$	2.22274	0.994041	0.497021	+	0.867739i	$0.334427\pi$
$6$	0.334904	0.136724
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.88784	−0.962613
$10$	2.22274	0.702893
$11$	−0.585786	−0.176621	−0.0883106	−	0.996093i	$-0.528147\pi$
$11$	−0.585786	−0.176621	−0.0883106	+	0.996093i	$0.528147\pi$
$12$	0.334904	0.0966785
$13$	−5.97186	−1.65630	−0.828148	−	0.560509i	$-0.810605\pi$
$13$	−5.97186	−1.65630	−0.828148	+	0.560509i	$0.810605\pi$
$14$	1.00000	0.267261
$15$	0.744406	0.192205
$16$	1.00000	0.250000
$17$	−4.22274	−1.02417	−0.512083	−	0.858936i	$-0.671126\pi$
$17$	−4.22274	−1.02417	−0.512083	+	0.858936i	$0.671126\pi$
$18$	−2.88784	−0.680670
$19$	−2.66981	−0.612496	−0.306248	−	0.951952i	$-0.599074\pi$
$19$	−2.66981	−0.612496	−0.306248	+	0.951952i	$0.599074\pi$
$20$	2.22274	0.497021
$21$	0.334904	0.0730820
$22$	−0.585786	−0.124890
$23$	0	0
$23$	0	0
$24$	0.334904	0.0683620
$25$	−0.0594122	−0.0118824
$26$	−5.97186	−1.17118
$27$	−1.97186	−0.379485
$28$	1.00000	0.188982
$29$	−4.43882	−0.824269	−0.412134	−	0.911123i	$-0.635217\pi$
$29$	−4.43882	−0.824269	−0.412134	+	0.911123i	$0.635217\pi$
$30$	0.744406	0.135909
$31$	6.11058	1.09749	0.548747	−	0.835989i	$-0.315105\pi$
$31$	6.11058	1.09749	0.548747	+	0.835989i	$0.315105\pi$
$32$	1.00000	0.176777
$33$	−0.196182	−0.0341509
$34$	−4.22274	−0.724195
$35$	2.22274	0.375712
$36$	−2.88784	−0.481307
$37$	3.53304	0.580828	0.290414	−	0.956901i	$-0.406207\pi$
$37$	3.53304	0.580828	0.290414	+	0.956901i	$0.406207\pi$
$38$	−2.66981	−0.433100
$39$	−2.00000	−0.320256
$40$	2.22274	0.351447
$41$	2.27744	0.355677	0.177838	−	0.984060i	$-0.443090\pi$
$41$	2.27744	0.355677	0.177838	+	0.984060i	$0.443090\pi$
$42$	0.334904	0.0516768
$43$	5.78510	0.882220	0.441110	−	0.897453i	$-0.354585\pi$
$43$	5.78510	0.882220	0.441110	+	0.897453i	$0.354585\pi$
$44$	−0.585786	−0.0883106
$45$	−6.41893	−0.956877
$46$	0	0
$47$	3.04451	0.444087	0.222043	−	0.975037i	$-0.428727\pi$
$47$	3.04451	0.444087	0.222043	+	0.975037i	$0.428727\pi$
$48$	0.334904	0.0483392
$49$	1.00000	0.142857
$50$	−0.0594122	−0.00840215
$51$	−1.41421	−0.198030
$52$	−5.97186	−0.828148
$53$	−9.19932	−1.26362	−0.631812	−	0.775122i	$-0.717688\pi$
$53$	−9.19932	−1.26362	−0.631812	+	0.775122i	$0.717688\pi$
$54$	−1.97186	−0.268336
$55$	−1.30205	−0.175569
$56$	1.00000	0.133631
$57$	−0.894129	−0.118430
$58$	−4.43882	−0.582846
$59$	−4.11058	−0.535152	−0.267576	−	0.963537i	$-0.586223\pi$
$59$	−4.11058	−0.535152	−0.267576	+	0.963537i	$0.586223\pi$
$60$	0.744406	0.0961023
$61$	−1.23570	−0.158215	−0.0791074	−	0.996866i	$-0.525207\pi$
$61$	−1.23570	−0.158215	−0.0791074	+	0.996866i	$0.525207\pi$
$62$	6.11058	0.776045
$63$	−2.88784	−0.363834
$64$	1.00000	0.125000
$65$	−13.2739	−1.64643
$66$	−0.196182	−0.0241484
$67$	−6.46696	−0.790065	−0.395033	−	0.918667i	$-0.629267\pi$
$67$	−6.46696	−0.790065	−0.395033	+	0.918667i	$0.629267\pi$
$68$	−4.22274	−0.512083
$69$	0	0
$70$	2.22274	0.265669
$71$	1.21137	0.143763	0.0718814	−	0.997413i	$-0.477100\pi$
$71$	1.21137	0.143763	0.0718814	+	0.997413i	$0.477100\pi$
$72$	−2.88784	−0.340335
$73$	−12.6948	−1.48581	−0.742906	−	0.669395i	$-0.766553\pi$
$73$	−12.6948	−1.48581	−0.742906	+	0.669395i	$0.766553\pi$
$74$	3.53304	0.410707
$75$	−0.0198974	−0.00229755
$76$	−2.66981	−0.306248
$77$	−0.585786	−0.0667566
$78$	−2.00000	−0.226455
$79$	−12.7672	−1.43642	−0.718209	−	0.695828i	$-0.755038\pi$
$79$	−12.7672	−1.43642	−0.718209	+	0.695828i	$0.755038\pi$
$80$	2.22274	0.248510
$81$	8.00313	0.889237
$82$	2.27744	0.251502
$83$	9.97186	1.09455	0.547277	−	0.836952i	$-0.315664\pi$
$83$	9.97186	1.09455	0.547277	+	0.836952i	$0.315664\pi$
$84$	0.334904	0.0365410
$85$	−9.38607	−1.01806
$86$	5.78510	0.623824
$87$	−1.48658	−0.159378
$88$	−0.585786	−0.0624450
$89$	−14.6588	−1.55383	−0.776915	−	0.629605i	$-0.783217\pi$
$89$	−14.6588	−1.55383	−0.776915	+	0.629605i	$0.783217\pi$
$90$	−6.41893	−0.676614
$91$	−5.97186	−0.626021
$92$	0	0
$93$	2.04646	0.212208
$94$	3.04451	0.314017
$95$	−5.93430	−0.608846
$96$	0.334904	0.0341810
$97$	−2.85499	−0.289880	−0.144940	−	0.989440i	$-0.546299\pi$
$97$	−2.85499	−0.289880	−0.144940	+	0.989440i	$0.546299\pi$
$98$	1.00000	0.101015
$99$	1.69166	0.170018
$100$	−0.0594122	−0.00594122
$101$	7.18323	0.714758	0.357379	−	0.933959i	$-0.383670\pi$
$101$	7.18323	0.714758	0.357379	+	0.933959i	$0.383670\pi$
$102$	−1.41421	−0.140028
$103$	−0.277444	−0.0273373	−0.0136687	−	0.999907i	$-0.504351\pi$
$103$	−0.277444	−0.0273373	−0.0136687	+	0.999907i	$0.504351\pi$
$104$	−5.97186	−0.585589
$105$	0.744406	0.0726465
$106$	−9.19932	−0.893517
$107$	−4.98705	−0.482116	−0.241058	−	0.970511i	$-0.577494\pi$
$107$	−4.98705	−0.482116	−0.241058	+	0.970511i	$0.577494\pi$
$108$	−1.97186	−0.189742
$109$	−9.29539	−0.890337	−0.445168	−	0.895447i	$-0.646856\pi$
$109$	−9.29539	−0.890337	−0.445168	+	0.895447i	$0.646856\pi$
$110$	−1.30205	−0.124146
$111$	1.18323	0.112307
$112$	1.00000	0.0944911
$113$	−15.0313	−1.41402	−0.707012	−	0.707202i	$-0.749957\pi$
$113$	−15.0313	−1.41402	−0.707012	+	0.707202i	$0.749957\pi$
$114$	−0.894129	−0.0837429
$115$	0	0
$116$	−4.43882	−0.412134
$117$	17.2458	1.59437
$118$	−4.11058	−0.378410
$119$	−4.22274	−0.387098
$120$	0.744406	0.0679546
$121$	−10.6569	−0.968805
$122$	−1.23570	−0.111875
$123$	0.762725	0.0687726
$124$	6.11058	0.548747
$125$	−11.2458	−1.00585
$126$	−2.88784	−0.257269
$127$	−18.0720	−1.60363	−0.801814	−	0.597574i	$-0.796132\pi$
$127$	−18.0720	−1.60363	−0.801814	+	0.597574i	$0.796132\pi$
$128$	1.00000	0.0883883
$129$	1.93745	0.170583
$130$	−13.2739	−1.16420
$131$	0.703033	0.0614243	0.0307121	−	0.999528i	$-0.490222\pi$
$131$	0.703033	0.0614243	0.0307121	+	0.999528i	$0.490222\pi$
$132$	−0.196182	−0.0170755
$133$	−2.66981	−0.231502
$134$	−6.46696	−0.558660
$135$	−4.38294	−0.377223
$136$	−4.22274	−0.362097
$137$	8.52951	0.728725	0.364363	−	0.931257i	$-0.381287\pi$
$137$	8.52951	0.728725	0.364363	+	0.931257i	$0.381287\pi$
$138$	0	0
$139$	4.87646	0.413616	0.206808	−	0.978382i	$-0.433692\pi$
$139$	4.87646	0.413616	0.206808	+	0.978382i	$0.433692\pi$
$140$	2.22274	0.187856
$141$	1.01962	0.0858673
$142$	1.21137	0.101656
$143$	3.49824	0.292537
$144$	−2.88784	−0.240653
$145$	−9.86636	−0.819357
$146$	−12.6948	−1.05063
$147$	0.334904	0.0276224
$148$	3.53304	0.290414
$149$	−19.9128	−1.63132	−0.815661	−	0.578530i	$-0.803627\pi$
$149$	−19.9128	−1.63132	−0.815661	+	0.578530i	$0.803627\pi$
$150$	−0.0198974	−0.00162461
$151$	19.9531	1.62376	0.811882	−	0.583822i	$-0.198443\pi$
$151$	19.9531	1.62376	0.811882	+	0.583822i	$0.198443\pi$
$152$	−2.66981	−0.216550
$153$	12.1946	0.985875
$154$	−0.585786	−0.0472040
$155$	13.5823	1.09095
$156$	−2.00000	−0.160128
$157$	5.46852	0.436435	0.218218	−	0.975900i	$-0.429976\pi$
$157$	5.46852	0.436435	0.218218	+	0.975900i	$0.429976\pi$
$158$	−12.7672	−1.01570
$159$	−3.08089	−0.244330
$160$	2.22274	0.175723
$161$	0	0
$162$	8.00313	0.628786
$163$	−1.01295	−0.0793407	−0.0396703	−	0.999213i	$-0.512631\pi$
$163$	−1.01295	−0.0793407	−0.0396703	+	0.999213i	$0.512631\pi$
$164$	2.27744	0.177838
$165$	−0.436063	−0.0339474
$166$	9.97186	0.773967
$167$	−16.9225	−1.30950	−0.654752	−	0.755844i	$-0.727227\pi$
$167$	−16.9225	−1.30950	−0.654752	+	0.755844i	$0.727227\pi$
$168$	0.334904	0.0258384
$169$	22.6631	1.74332
$170$	−9.38607	−0.719879
$171$	7.70998	0.589597
$172$	5.78510	0.441110
$173$	25.0192	1.90218	0.951088	−	0.308920i	$-0.0999675\pi$
$173$	25.0192	1.90218	0.951088	+	0.308920i	$0.0999675\pi$
$174$	−1.48658	−0.112697
$175$	−0.0594122	−0.00449114
$176$	−0.585786	−0.0441553
$177$	−1.37665	−0.103475
$178$	−14.6588	−1.09872
$179$	24.7292	1.84835	0.924174	−	0.381973i	$-0.124755\pi$
$179$	24.7292	1.84835	0.924174	+	0.381973i	$0.124755\pi$
$180$	−6.41893	−0.478438
$181$	−16.3001	−1.21158	−0.605788	−	0.795626i	$-0.707142\pi$
$181$	−16.3001	−1.21158	−0.605788	+	0.795626i	$0.707142\pi$
$182$	−5.97186	−0.442664
$183$	−0.413840	−0.0305919
$184$	0	0
$185$	7.85304	0.577367
$186$	2.04646	0.150054
$187$	2.47363	0.180889
$188$	3.04451	0.222043
$189$	−1.97186	−0.143432
$190$	−5.93430	−0.430519
$191$	21.3857	1.54741	0.773707	−	0.633544i	$-0.218400\pi$
$191$	21.3857	1.54741	0.773707	+	0.633544i	$0.218400\pi$
$192$	0.334904	0.0241696
$193$	−12.6041	−0.907263	−0.453632	−	0.891189i	$-0.649872\pi$
$193$	−12.6041	−0.907263	−0.453632	+	0.891189i	$0.649872\pi$
$194$	−2.85499	−0.204976
$195$	−4.44549	−0.318348
$196$	1.00000	0.0714286
$197$	12.8910	0.918444	0.459222	−	0.888322i	$-0.348128\pi$
$197$	12.8910	0.918444	0.459222	+	0.888322i	$0.348128\pi$
$198$	1.69166	0.120221
$199$	−16.8025	−1.19110	−0.595550	−	0.803319i	$-0.703066\pi$
$199$	−16.8025	−1.19110	−0.595550	+	0.803319i	$0.703066\pi$
$200$	−0.0594122	−0.00420108
$201$	−2.16581	−0.152765
$202$	7.18323	0.505410
$203$	−4.43882	−0.311544
$204$	−1.41421	−0.0990148
$205$	5.06217	0.353557
$206$	−0.277444	−0.0193304
$207$	0	0
$208$	−5.97186	−0.414074
$209$	1.56394	0.108180
$210$	0.744406	0.0513689
$211$	−6.70960	−0.461908	−0.230954	−	0.972965i	$-0.574185\pi$
$211$	−6.70960	−0.461908	−0.230954	+	0.972965i	$0.574185\pi$
$212$	−9.19932	−0.631812
$213$	0.405692	0.0277975
$214$	−4.98705	−0.340907
$215$	12.8588	0.876963
$216$	−1.97186	−0.134168
$217$	6.11058	0.414813
$218$	−9.29539	−0.629563
$219$	−4.25154	−0.287292
$220$	−1.30205	−0.0877844
$221$	25.2176	1.69632
$222$	1.18323	0.0794131
$223$	23.4658	1.57138	0.785692	−	0.618618i	$-0.212307\pi$
$223$	23.4658	1.57138	0.785692	+	0.618618i	$0.212307\pi$
$224$	1.00000	0.0668153
$225$	0.171573	0.0114382
$226$	−15.0313	−0.999865
$227$	−14.5018	−0.962516	−0.481258	−	0.876579i	$-0.659820\pi$
$227$	−14.5018	−0.962516	−0.481258	+	0.876579i	$0.659820\pi$
$228$	−0.894129	−0.0592152
$229$	−18.7683	−1.24025	−0.620123	−	0.784504i	$-0.712917\pi$
$229$	−18.7683	−1.24025	−0.620123	+	0.784504i	$0.712917\pi$
$230$	0	0
$231$	−0.196182	−0.0129078
$232$	−4.43882	−0.291423
$233$	−11.7823	−0.771887	−0.385943	−	0.922522i	$-0.626124\pi$
$233$	−11.7823	−0.771887	−0.385943	+	0.922522i	$0.626124\pi$
$234$	17.2458	1.12739
$235$	6.76716	0.441441
$236$	−4.11058	−0.267576
$237$	−4.27577	−0.277741
$238$	−4.22274	−0.273720
$239$	−13.4420	−0.869488	−0.434744	−	0.900554i	$-0.643161\pi$
$239$	−13.4420	−0.869488	−0.434744	+	0.900554i	$0.643161\pi$
$240$	0.744406	0.0480512
$241$	14.9153	0.960779	0.480390	−	0.877055i	$-0.340495\pi$
$241$	14.9153	0.960779	0.480390	+	0.877055i	$0.340495\pi$
$242$	−10.6569	−0.685049
$243$	8.59586	0.551425
$244$	−1.23570	−0.0791074
$245$	2.22274	0.142006
$246$	0.762725	0.0486296
$247$	15.9437	1.01447
$248$	6.11058	0.388022
$249$	3.33962	0.211640
$250$	−11.2458	−0.711245
$251$	−22.1814	−1.40008	−0.700038	−	0.714106i	$-0.746834\pi$
$251$	−22.1814	−1.40008	−0.700038	+	0.714106i	$0.746834\pi$
$252$	−2.88784	−0.181917
$253$	0	0
$254$	−18.0720	−1.13394
$255$	−3.14343	−0.196849
$256$	1.00000	0.0625000
$257$	0.356473	0.0222362	0.0111181	−	0.999938i	$-0.496461\pi$
$257$	0.356473	0.0222362	0.0111181	+	0.999938i	$0.496461\pi$
$258$	1.93745	0.120621
$259$	3.53304	0.219532
$260$	−13.2739	−0.823213
$261$	12.8186	0.793452
$262$	0.703033	0.0434335
$263$	20.3052	1.25207	0.626036	−	0.779794i	$-0.284676\pi$
$263$	20.3052	1.25207	0.626036	+	0.779794i	$0.284676\pi$
$264$	−0.196182	−0.0120742
$265$	−20.4477	−1.25609
$266$	−2.66981	−0.163696
$267$	−4.90929	−0.300444
$268$	−6.46696	−0.395033
$269$	3.12235	0.190373	0.0951866	−	0.995459i	$-0.469655\pi$
$269$	3.12235	0.190373	0.0951866	+	0.995459i	$0.469655\pi$
$270$	−4.38294	−0.266737
$271$	25.5016	1.54911	0.774557	−	0.632504i	$-0.217973\pi$
$271$	25.5016	1.54911	0.774557	+	0.632504i	$0.217973\pi$
$272$	−4.22274	−0.256041
$273$	−2.00000	−0.121046
$274$	8.52951	0.515286
$275$	0.0348029	0.00209869
$276$	0	0
$277$	16.1449	0.970053	0.485026	−	0.874500i	$-0.338810\pi$
$277$	16.1449	0.970053	0.485026	+	0.874500i	$0.338810\pi$
$278$	4.87646	0.292471
$279$	−17.6464	−1.05646
$280$	2.22274	0.132834
$281$	21.7338	1.29653	0.648265	−	0.761414i	$-0.275495\pi$
$281$	21.7338	1.29653	0.648265	+	0.761414i	$0.275495\pi$
$282$	1.01962	0.0607173
$283$	0.956675	0.0568685	0.0284342	−	0.999596i	$-0.490948\pi$
$283$	0.956675	0.0568685	0.0284342	+	0.999596i	$0.490948\pi$
$284$	1.21137	0.0718814
$285$	−1.98742	−0.117725
$286$	3.49824	0.206855
$287$	2.27744	0.134433
$288$	−2.88784	−0.170168
$289$	0.831561	0.0489154
$290$	−9.86636	−0.579373
$291$	−0.956147	−0.0560503
$292$	−12.6948	−0.742906
$293$	−19.0176	−1.11102	−0.555511	−	0.831509i	$-0.687477\pi$
$293$	−19.0176	−1.11102	−0.555511	+	0.831509i	$0.687477\pi$
$294$	0.334904	0.0195320
$295$	−9.13677	−0.531963
$296$	3.53304	0.205354
$297$	1.15509	0.0670251
$298$	−19.9128	−1.15352
$299$	0	0
$300$	−0.0198974	−0.00114878
$301$	5.78510	0.333448
$302$	19.9531	1.14817
$303$	2.40569	0.138203
$304$	−2.66981	−0.153124
$305$	−2.74664	−0.157272
$306$	12.1946	0.697119
$307$	25.9566	1.48142	0.740710	−	0.671825i	$-0.234489\pi$
$307$	25.9566	1.48142	0.740710	+	0.671825i	$0.234489\pi$
$308$	−0.585786	−0.0333783
$309$	−0.0929169	−0.00528586
$310$	13.5823	0.771420
$311$	−18.2036	−1.03223	−0.516117	−	0.856518i	$-0.672623\pi$
$311$	−18.2036	−1.03223	−0.516117	+	0.856518i	$0.672623\pi$
$312$	−2.00000	−0.113228
$313$	6.26198	0.353948	0.176974	−	0.984216i	$-0.443369\pi$
$313$	6.26198	0.353948	0.176974	+	0.984216i	$0.443369\pi$
$314$	5.46852	0.308606
$315$	−6.41893	−0.361665
$316$	−12.7672	−0.718209
$317$	−0.564707	−0.0317171	−0.0158585	−	0.999874i	$-0.505048\pi$
$317$	−0.564707	−0.0317171	−0.0158585	+	0.999874i	$0.505048\pi$
$318$	−3.08089	−0.172768
$319$	2.60020	0.145583
$320$	2.22274	0.124255
$321$	−1.67018	−0.0932205
$322$	0	0
$323$	11.2739	0.627297
$324$	8.00313	0.444619
$325$	0.354801	0.0196808
$326$	−1.01295	−0.0561023
$327$	−3.11306	−0.172153
$328$	2.27744	0.125751
$329$	3.04451	0.167849
$330$	−0.436063	−0.0240045
$331$	−29.2480	−1.60762	−0.803808	−	0.594889i	$-0.797196\pi$
$331$	−29.2480	−1.60762	−0.803808	+	0.594889i	$0.797196\pi$
$332$	9.97186	0.547277
$333$	−10.2028	−0.559113
$334$	−16.9225	−0.925959
$335$	−14.3744	−0.785357
$336$	0.334904	0.0182705
$337$	2.27301	0.123819	0.0619094	−	0.998082i	$-0.480281\pi$
$337$	2.27301	0.123819	0.0619094	+	0.998082i	$0.480281\pi$
$338$	22.6631	1.23271
$339$	−5.03403	−0.273411
$340$	−9.38607	−0.509031
$341$	−3.57950	−0.193841
$342$	7.70998	0.416908
$343$	1.00000	0.0539949
$344$	5.78510	0.311912
$345$	0	0
$346$	25.0192	1.34504
$347$	−19.0192	−1.02101	−0.510503	−	0.859876i	$-0.670541\pi$
$347$	−19.0192	−1.02101	−0.510503	+	0.859876i	$0.670541\pi$
$348$	−1.48658	−0.0796890
$349$	3.97353	0.212698	0.106349	−	0.994329i	$-0.466084\pi$
$349$	3.97353	0.212698	0.106349	+	0.994329i	$0.466084\pi$
$350$	−0.0594122	−0.00317572
$351$	11.7757	0.628539
$352$	−0.585786	−0.0312225
$353$	14.0774	0.749262	0.374631	−	0.927174i	$-0.377769\pi$
$353$	14.0774	0.749262	0.374631	+	0.927174i	$0.377769\pi$
$354$	−1.37665	−0.0731681
$355$	2.69256	0.142906
$356$	−14.6588	−0.776915
$357$	−1.41421	−0.0748481
$358$	24.7292	1.30698
$359$	15.7073	0.829002	0.414501	−	0.910049i	$-0.363956\pi$
$359$	15.7073	0.829002	0.414501	+	0.910049i	$0.363956\pi$
$360$	−6.41893	−0.338307
$361$	−11.8721	−0.624849
$362$	−16.3001	−0.856714
$363$	−3.56902	−0.187325
$364$	−5.97186	−0.313011
$365$	−28.2173	−1.47696
$366$	−0.413840	−0.0216318
$367$	20.5689	1.07369	0.536844	−	0.843681i	$-0.319616\pi$
$367$	20.5689	1.07369	0.536844	+	0.843681i	$0.319616\pi$
$368$	0	0
$369$	−6.57689	−0.342379
$370$	7.85304	0.408260
$371$	−9.19932	−0.477605
$372$	2.04646	0.106104
$373$	−10.8727	−0.562965	−0.281482	−	0.959566i	$-0.590826\pi$
$373$	−10.8727	−0.562965	−0.281482	+	0.959566i	$0.590826\pi$
$374$	2.47363	0.127908
$375$	−3.76625	−0.194489
$376$	3.04451	0.157008
$377$	26.5080	1.36523
$378$	−1.97186	−0.101422
$379$	27.4697	1.41102	0.705512	−	0.708698i	$-0.250717\pi$
$379$	27.4697	1.41102	0.705512	+	0.708698i	$0.250717\pi$
$380$	−5.93430	−0.304423
$381$	−6.05237	−0.310073
$382$	21.3857	1.09419
$383$	34.4634	1.76100	0.880499	−	0.474048i	$-0.157208\pi$
$383$	34.4634	1.76100	0.880499	+	0.474048i	$0.157208\pi$
$384$	0.334904	0.0170905
$385$	−1.30205	−0.0663588
$386$	−12.6041	−0.641532
$387$	−16.7064	−0.849236
$388$	−2.85499	−0.144940
$389$	−6.52324	−0.330741	−0.165371	−	0.986231i	$-0.552882\pi$
$389$	−6.52324	−0.330741	−0.165371	+	0.986231i	$0.552882\pi$
$390$	−4.44549	−0.225106
$391$	0	0
$392$	1.00000	0.0505076
$393$	0.235448	0.0118768
$394$	12.8910	0.649438
$395$	−28.3781	−1.42786
$396$	1.69166	0.0850090
$397$	−3.07146	−0.154152	−0.0770762	−	0.997025i	$-0.524558\pi$
$397$	−3.07146	−0.154152	−0.0770762	+	0.997025i	$0.524558\pi$
$398$	−16.8025	−0.842234
$399$	−0.894129	−0.0447625
$400$	−0.0594122	−0.00297061
$401$	29.5823	1.47727	0.738634	−	0.674107i	$-0.235471\pi$
$401$	29.5823	1.47727	0.738634	+	0.674107i	$0.235471\pi$
$402$	−2.16581	−0.108021
$403$	−36.4915	−1.81777
$404$	7.18323	0.357379
$405$	17.7889	0.883938
$406$	−4.43882	−0.220295
$407$	−2.06961	−0.102587
$408$	−1.41421	−0.0700140
$409$	15.3950	0.761232	0.380616	−	0.924733i	$-0.375712\pi$
$409$	15.3950	0.761232	0.380616	+	0.924733i	$0.375712\pi$
$410$	5.06217	0.250003
$411$	2.85657	0.140904
$412$	−0.277444	−0.0136687
$413$	−4.11058	−0.202269
$414$	0	0
$415$	22.1649	1.08803
$416$	−5.97186	−0.292795
$417$	1.63315	0.0799756
$418$	1.56394	0.0764947
$419$	−25.2708	−1.23456	−0.617279	−	0.786745i	$-0.711765\pi$
$419$	−25.2708	−1.23456	−0.617279	+	0.786745i	$0.711765\pi$
$420$	0.744406	0.0363233
$421$	29.8660	1.45558	0.727789	−	0.685801i	$-0.240548\pi$
$421$	29.8660	1.45558	0.727789	+	0.685801i	$0.240548\pi$
$422$	−6.70960	−0.326618
$423$	−8.79205	−0.427484
$424$	−9.19932	−0.446758
$425$	0.250882	0.0121696
$426$	0.405692	0.0196558
$427$	−1.23570	−0.0597996
$428$	−4.98705	−0.241058
$429$	1.17157	0.0565641
$430$	12.8588	0.620106
$431$	14.8551	0.715544	0.357772	−	0.933809i	$-0.383536\pi$
$431$	14.8551	0.715544	0.357772	+	0.933809i	$0.383536\pi$
$432$	−1.97186	−0.0948712
$433$	17.0722	0.820440	0.410220	−	0.911987i	$-0.365452\pi$
$433$	17.0722	0.820440	0.410220	+	0.911987i	$0.365452\pi$
$434$	6.11058	0.293317
$435$	−3.30428	−0.158428
$436$	−9.29539	−0.445168
$437$	0	0
$438$	−4.25154	−0.203146
$439$	37.9769	1.81254	0.906270	−	0.422698i	$-0.138917\pi$
$439$	37.9769	1.81254	0.906270	+	0.422698i	$0.138917\pi$
$440$	−1.30205	−0.0620729
$441$	−2.88784	−0.137516
$442$	25.2176	1.19948
$443$	−23.0723	−1.09620	−0.548100	−	0.836413i	$-0.684649\pi$
$443$	−23.0723	−1.09620	−0.548100	+	0.836413i	$0.684649\pi$
$444$	1.18323	0.0561535
$445$	−32.5828	−1.54457
$446$	23.4658	1.11114
$447$	−6.66888	−0.315427
$448$	1.00000	0.0472456
$449$	−29.6261	−1.39814	−0.699071	−	0.715053i	$-0.746403\pi$
$449$	−29.6261	−1.39814	−0.699071	+	0.715053i	$0.746403\pi$
$450$	0.171573	0.00808802
$451$	−1.33410	−0.0628201
$452$	−15.0313	−0.707012
$453$	6.68239	0.313966
$454$	−14.5018	−0.680602
$455$	−13.2739	−0.622291
$456$	−0.894129	−0.0418714
$457$	−19.5027	−0.912296	−0.456148	−	0.889904i	$-0.650771\pi$
$457$	−19.5027	−0.912296	−0.456148	+	0.889904i	$0.650771\pi$
$458$	−18.7683	−0.876987
$459$	8.32666	0.388655
$460$	0	0
$461$	−4.81138	−0.224088	−0.112044	−	0.993703i	$-0.535740\pi$
$461$	−4.81138	−0.224088	−0.112044	+	0.993703i	$0.535740\pi$
$462$	−0.196182	−0.00912722
$463$	32.1251	1.49298	0.746490	−	0.665397i	$-0.231738\pi$
$463$	32.1251	1.49298	0.746490	+	0.665397i	$0.231738\pi$
$464$	−4.43882	−0.206067
$465$	4.54875	0.210943
$466$	−11.7823	−0.545806
$467$	−9.49824	−0.439526	−0.219763	−	0.975553i	$-0.570528\pi$
$467$	−9.49824	−0.439526	−0.219763	+	0.975553i	$0.570528\pi$
$468$	17.2458	0.797186
$469$	−6.46696	−0.298617
$470$	6.76716	0.312146
$471$	1.83143	0.0843878
$472$	−4.11058	−0.189205
$473$	−3.38883	−0.155819
$474$	−4.27577	−0.196393
$475$	0.158619	0.00727795
$476$	−4.22274	−0.193549
$477$	26.5661	1.21638
$478$	−13.4420	−0.614821
$479$	−18.4947	−0.845045	−0.422522	−	0.906353i	$-0.638855\pi$
$479$	−18.4947	−0.845045	−0.422522	+	0.906353i	$0.638855\pi$
$480$	0.744406	0.0339773
$481$	−21.0988	−0.962023
$482$	14.9153	0.679373
$483$	0	0
$484$	−10.6569	−0.484402
$485$	−6.34591	−0.288153
$486$	8.59586	0.389916
$487$	−27.5613	−1.24892	−0.624462	−	0.781055i	$-0.714682\pi$
$487$	−27.5613	−1.24892	−0.624462	+	0.781055i	$0.714682\pi$
$488$	−1.23570	−0.0559374
$489$	−0.339242	−0.0153411
$490$	2.22274	0.100413
$491$	8.56041	0.386326	0.193163	−	0.981167i	$-0.438125\pi$
$491$	8.56041	0.386326	0.193163	+	0.981167i	$0.438125\pi$
$492$	0.762725	0.0343863
$493$	18.7440	0.844188
$494$	15.9437	0.717342
$495$	3.76012	0.169005
$496$	6.11058	0.274373
$497$	1.21137	0.0543373
$498$	3.33962	0.149652
$499$	−24.4986	−1.09671	−0.548354	−	0.836246i	$-0.684745\pi$
$499$	−24.4986	−1.09671	−0.548354	+	0.836246i	$0.684745\pi$
$500$	−11.2458	−0.502926
$501$	−5.66742	−0.253202
$502$	−22.1814	−0.990003
$503$	−8.56041	−0.381690	−0.190845	−	0.981620i	$-0.561123\pi$
$503$	−8.56041	−0.381690	−0.190845	+	0.981620i	$0.561123\pi$
$504$	−2.88784	−0.128635
$505$	15.9665	0.710499
$506$	0	0
$507$	7.58997	0.337082
$508$	−18.0720	−0.801814
$509$	20.3570	0.902310	0.451155	−	0.892446i	$-0.351012\pi$
$509$	20.3570	0.902310	0.451155	+	0.892446i	$0.351012\pi$
$510$	−3.14343	−0.139194
$511$	−12.6948	−0.561584
$512$	1.00000	0.0441942
$513$	5.26449	0.232433
$514$	0.356473	0.0157234
$515$	−0.616686	−0.0271744
$516$	1.93745	0.0852917
$517$	−1.78343	−0.0784352
$518$	3.53304	0.155233
$519$	8.37904	0.367799
$520$	−13.2739	−0.582100
$521$	−15.0463	−0.659192	−0.329596	−	0.944122i	$-0.606913\pi$
$521$	−15.0463	−0.659192	−0.329596	+	0.944122i	$0.606913\pi$
$522$	12.8186	0.561055
$523$	42.0181	1.83732	0.918661	−	0.395048i	$-0.129272\pi$
$523$	42.0181	1.83732	0.918661	+	0.395048i	$0.129272\pi$
$524$	0.703033	0.0307121
$525$	−0.0198974	−0.000868393 0
$526$	20.3052	0.885348
$527$	−25.8034	−1.12401
$528$	−0.196182	−0.00853773
$529$	0	0
$530$	−20.4477	−0.888192
$531$	11.8707	0.515145
$532$	−2.66981	−0.115751
$533$	−13.6006	−0.589106
$534$	−4.90929	−0.212446
$535$	−11.0849	−0.479243
$536$	−6.46696	−0.279330
$537$	8.28191	0.357391
$538$	3.12235	0.134614
$539$	−0.585786	−0.0252316
$540$	−4.38294	−0.188612
$541$	40.9696	1.76142	0.880710	−	0.473655i	$-0.157066\pi$
$541$	40.9696	1.76142	0.880710	+	0.473655i	$0.157066\pi$
$542$	25.5016	1.09539
$543$	−5.45897	−0.234267
$544$	−4.22274	−0.181049
$545$	−20.6613	−0.885031
$546$	−2.00000	−0.0855921
$547$	18.0631	0.772323	0.386161	−	0.922431i	$-0.373801\pi$
$547$	18.0631	0.772323	0.386161	+	0.922431i	$0.373801\pi$
$548$	8.52951	0.364363
$549$	3.56849	0.152300
$550$	0.0348029	0.00148400
$551$	11.8508	0.504861
$552$	0	0
$553$	−12.7672	−0.542915
$554$	16.1449	0.685931
$555$	2.63001	0.111638
$556$	4.87646	0.206808
$557$	−30.4207	−1.28897	−0.644483	−	0.764619i	$-0.722927\pi$
$557$	−30.4207	−1.28897	−0.644483	+	0.764619i	$0.722927\pi$
$558$	−17.6464	−0.747031
$559$	−34.5478	−1.46122
$560$	2.22274	0.0939280
$561$	0.828427	0.0349762
$562$	21.7338	0.916786
$563$	−14.7096	−0.619936	−0.309968	−	0.950747i	$-0.600318\pi$
$563$	−14.7096	−0.619936	−0.309968	+	0.950747i	$0.600318\pi$
$564$	1.01962	0.0429336
$565$	−33.4107	−1.40560
$566$	0.956675	0.0402121
$567$	8.00313	0.336100
$568$	1.21137	0.0508279
$569$	−36.6514	−1.53651	−0.768254	−	0.640145i	$-0.778874\pi$
$569$	−36.6514	−1.53651	−0.768254	+	0.640145i	$0.778874\pi$
$570$	−1.98742	−0.0832439
$571$	−20.3096	−0.849931	−0.424966	−	0.905210i	$-0.639714\pi$
$571$	−20.3096	−0.849931	−0.424966	+	0.905210i	$0.639714\pi$
$572$	3.49824	0.146269
$573$	7.16215	0.299203
$574$	2.27744	0.0950587
$575$	0	0
$576$	−2.88784	−0.120327
$577$	29.6842	1.23577	0.617885	−	0.786269i	$-0.287990\pi$
$577$	29.6842	1.23577	0.617885	+	0.786269i	$0.287990\pi$
$578$	0.831561	0.0345884
$579$	−4.22117	−0.175426
$580$	−9.86636	−0.409678
$581$	9.97186	0.413702
$582$	−0.956147	−0.0396336
$583$	5.38883	0.223183
$584$	−12.6948	−0.525314
$585$	38.3329	1.58487
$586$	−19.0176	−0.785612
$587$	7.01098	0.289374	0.144687	−	0.989477i	$-0.453782\pi$
$587$	7.01098	0.289374	0.144687	+	0.989477i	$0.453782\pi$
$588$	0.334904	0.0138112
$589$	−16.3141	−0.672210
$590$	−9.13677	−0.376155
$591$	4.31724	0.177587
$592$	3.53304	0.145207
$593$	2.69962	0.110860	0.0554300	−	0.998463i	$-0.482347\pi$
$593$	2.69962	0.110860	0.0554300	+	0.998463i	$0.482347\pi$
$594$	1.15509	0.0473939
$595$	−9.38607	−0.384792
$596$	−19.9128	−0.815661
$597$	−5.62723	−0.230307
$598$	0	0
$599$	36.6230	1.49637	0.748187	−	0.663488i	$-0.230925\pi$
$599$	36.6230	1.49637	0.748187	+	0.663488i	$0.230925\pi$
$600$	−0.0198974	−0.000812307 0
$601$	−16.0603	−0.655114	−0.327557	−	0.944831i	$-0.606225\pi$
$601$	−16.0603	−0.655114	−0.327557	+	0.944831i	$0.606225\pi$
$602$	5.78510	0.235783
$603$	18.6755	0.760527
$604$	19.9531	0.811882
$605$	−23.6875	−0.963032
$606$	2.40569	0.0977246
$607$	−11.3319	−0.459949	−0.229975	−	0.973197i	$-0.573864\pi$
$607$	−11.3319	−0.459949	−0.229975	+	0.973197i	$0.573864\pi$
$608$	−2.66981	−0.108275
$609$	−1.48658	−0.0602392
$610$	−2.74664	−0.111208
$611$	−18.1814	−0.735540
$612$	12.1946	0.492938
$613$	21.4579	0.866677	0.433338	−	0.901231i	$-0.357335\pi$
$613$	21.4579	0.866677	0.433338	+	0.901231i	$0.357335\pi$
$614$	25.9566	1.04752
$615$	1.69534	0.0683628
$616$	−0.585786	−0.0236020
$617$	36.0599	1.45172	0.725859	−	0.687844i	$-0.241443\pi$
$617$	36.0599	1.45172	0.725859	+	0.687844i	$0.241443\pi$
$618$	−0.0929169	−0.00373767
$619$	48.8393	1.96302	0.981508	−	0.191419i	$-0.0613089\pi$
$619$	48.8393	1.96302	0.981508	+	0.191419i	$0.0613089\pi$
$620$	13.5823	0.545477
$621$	0	0
$622$	−18.2036	−0.729899
$623$	−14.6588	−0.587293
$624$	−2.00000	−0.0800641
$625$	−24.6994	−0.987976
$626$	6.26198	0.250279
$627$	0.523769	0.0209173
$628$	5.46852	0.218218
$629$	−14.9191	−0.594864
$630$	−6.41893	−0.255736
$631$	−32.7109	−1.30220	−0.651100	−	0.758992i	$-0.725692\pi$
$631$	−32.7109	−1.30220	−0.651100	+	0.758992i	$0.725692\pi$
$632$	−12.7672	−0.507850
$633$	−2.24707	−0.0893131
$634$	−0.564707	−0.0224274
$635$	−40.1694	−1.59407
$636$	−3.08089	−0.122165
$637$	−5.97186	−0.236614
$638$	2.60020	0.102943
$639$	−3.49824	−0.138388
$640$	2.22274	0.0878616
$641$	−4.01776	−0.158692	−0.0793460	−	0.996847i	$-0.525283\pi$
$641$	−4.01776	−0.158692	−0.0793460	+	0.996847i	$0.525283\pi$
$642$	−1.67018	−0.0659168
$643$	−25.9272	−1.02247	−0.511235	−	0.859441i	$-0.670812\pi$
$643$	−25.9272	−1.02247	−0.511235	+	0.859441i	$0.670812\pi$
$644$	0	0
$645$	4.30646	0.169567
$646$	11.2739	0.443566
$647$	−32.2582	−1.26820	−0.634101	−	0.773250i	$-0.718630\pi$
$647$	−32.2582	−1.26820	−0.634101	+	0.773250i	$0.718630\pi$
$648$	8.00313	0.314393
$649$	2.40792	0.0945193
$650$	0.354801	0.0139165
$651$	2.04646	0.0802070
$652$	−1.01295	−0.0396703
$653$	18.0125	0.704885	0.352442	−	0.935833i	$-0.385351\pi$
$653$	18.0125	0.704885	0.352442	+	0.935833i	$0.385351\pi$
$654$	−3.11306	−0.121730
$655$	1.56266	0.0610582
$656$	2.27744	0.0889192
$657$	36.6605	1.43026
$658$	3.04451	0.118687
$659$	−1.24745	−0.0485936	−0.0242968	−	0.999705i	$-0.507735\pi$
$659$	−1.24745	−0.0485936	−0.0242968	+	0.999705i	$0.507735\pi$
$660$	−0.436063	−0.0169737
$661$	24.1494	0.939304	0.469652	−	0.882852i	$-0.344379\pi$
$661$	24.1494	0.939304	0.469652	+	0.882852i	$0.344379\pi$
$662$	−29.2480	−1.13676
$663$	8.44549	0.327996
$664$	9.97186	0.386983
$665$	−5.93430	−0.230122
$666$	−10.2028	−0.395352
$667$	0	0
$668$	−16.9225	−0.654752
$669$	7.85878	0.303838
$670$	−14.3744	−0.555331
$671$	0.723855	0.0279441
$672$	0.334904	0.0129192
$673$	−10.5367	−0.406160	−0.203080	−	0.979162i	$-0.565095\pi$
$673$	−10.5367	−0.406160	−0.203080	+	0.979162i	$0.565095\pi$
$674$	2.27301	0.0875531
$675$	0.117153	0.00450921
$676$	22.6631	0.871659
$677$	−13.7443	−0.528236	−0.264118	−	0.964490i	$-0.585081\pi$
$677$	−13.7443	−0.528236	−0.264118	+	0.964490i	$0.585081\pi$
$678$	−5.03403	−0.193331
$679$	−2.85499	−0.109564
$680$	−9.38607	−0.359940
$681$	−4.85670	−0.186109
$682$	−3.57950	−0.137066
$683$	42.3916	1.62207	0.811034	−	0.584999i	$-0.198905\pi$
$683$	42.3916	1.62207	0.811034	+	0.584999i	$0.198905\pi$
$684$	7.70998	0.294798
$685$	18.9589	0.724383
$686$	1.00000	0.0381802
$687$	−6.28559	−0.239810
$688$	5.78510	0.220555
$689$	54.9370	2.09293
$690$	0	0
$691$	−34.6936	−1.31981	−0.659904	−	0.751350i	$-0.729403\pi$
$691$	−34.6936	−1.31981	−0.659904	+	0.751350i	$0.729403\pi$
$692$	25.0192	0.951088
$693$	1.69166	0.0642607
$694$	−19.0192	−0.721960
$695$	10.8391	0.411152
$696$	−1.48658	−0.0563486
$697$	−9.61706	−0.364272
$698$	3.97353	0.150400
$699$	−3.94595	−0.149250
$700$	−0.0594122	−0.00224557
$701$	14.8467	0.560754	0.280377	−	0.959890i	$-0.409541\pi$
$701$	14.8467	0.560754	0.280377	+	0.959890i	$0.409541\pi$
$702$	11.7757	0.444444
$703$	−9.43253	−0.355755
$704$	−0.585786	−0.0220777
$705$	2.26635	0.0853556
$706$	14.0774	0.529808
$707$	7.18323	0.270153
$708$	−1.37665	−0.0517377
$709$	−31.5095	−1.18336	−0.591682	−	0.806171i	$-0.701536\pi$
$709$	−31.5095	−1.18336	−0.591682	+	0.806171i	$0.701536\pi$
$710$	2.69256	0.101050
$711$	36.8695	1.38271
$712$	−14.6588	−0.549362
$713$	0	0
$714$	−1.41421	−0.0529256
$715$	7.77568	0.290794
$716$	24.7292	0.924174
$717$	−4.50176	−0.168121
$718$	15.7073	0.586193
$719$	−8.08231	−0.301419	−0.150710	−	0.988578i	$-0.548156\pi$
$719$	−8.08231	−0.301419	−0.150710	+	0.988578i	$0.548156\pi$
$720$	−6.41893	−0.239219
$721$	−0.277444	−0.0103325
$722$	−11.8721	−0.441835
$723$	4.99519	0.185773
$724$	−16.3001	−0.605788
$725$	0.263720	0.00979432
$726$	−3.56902	−0.132459
$727$	13.4912	0.500360	0.250180	−	0.968199i	$-0.419510\pi$
$727$	13.4912	0.500360	0.250180	+	0.968199i	$0.419510\pi$
$728$	−5.97186	−0.221332
$729$	−21.1306	−0.782615
$730$	−28.2173	−1.04437
$731$	−24.4290	−0.903539
$732$	−0.413840	−0.0152960
$733$	−18.7611	−0.692959	−0.346479	−	0.938058i	$-0.612623\pi$
$733$	−18.7611	−0.692959	−0.346479	+	0.938058i	$0.612623\pi$
$734$	20.5689	0.759212
$735$	0.744406	0.0274578
$736$	0	0
$737$	3.78826	0.139542
$738$	−6.57689	−0.242099
$739$	6.28923	0.231353	0.115677	−	0.993287i	$-0.463096\pi$
$739$	6.28923	0.231353	0.115677	+	0.993287i	$0.463096\pi$
$740$	7.85304	0.288683
$741$	5.33962	0.196156
$742$	−9.19932	−0.337718
$743$	−29.4901	−1.08189	−0.540943	−	0.841059i	$-0.681933\pi$
$743$	−29.4901	−1.08189	−0.540943	+	0.841059i	$0.681933\pi$
$744$	2.04646	0.0750268
$745$	−44.2611	−1.62160
$746$	−10.8727	−0.398076
$747$	−28.7971	−1.05363
$748$	2.47363	0.0904447
$749$	−4.98705	−0.182223
$750$	−3.76625	−0.137524
$751$	−15.0585	−0.549492	−0.274746	−	0.961517i	$-0.588594\pi$
$751$	−15.0585	−0.549492	−0.274746	+	0.961517i	$0.588594\pi$
$752$	3.04451	0.111022
$753$	−7.42863	−0.270714
$754$	26.5080	0.965366
$755$	44.3507	1.61409
$756$	−1.97186	−0.0717159
$757$	23.5066	0.854361	0.427180	−	0.904166i	$-0.359507\pi$
$757$	23.5066	0.854361	0.427180	+	0.904166i	$0.359507\pi$
$758$	27.4697	0.997744
$759$	0	0
$760$	−5.93430	−0.215260
$761$	32.4736	1.17717	0.588584	−	0.808436i	$-0.299686\pi$
$761$	32.4736	1.17717	0.588584	+	0.808436i	$0.299686\pi$
$762$	−6.05237	−0.219254
$763$	−9.29539	−0.336516
$764$	21.3857	0.773707
$765$	27.1055	0.980001
$766$	34.4634	1.24521
$767$	24.5478	0.886371
$768$	0.334904	0.0120848
$769$	−5.08706	−0.183444	−0.0917221	−	0.995785i	$-0.529237\pi$
$769$	−5.08706	−0.183444	−0.0917221	+	0.995785i	$0.529237\pi$
$770$	−1.30205	−0.0469227
$771$	0.119384	0.00429952
$772$	−12.6041	−0.453632
$773$	8.22813	0.295945	0.147973	−	0.988991i	$-0.452725\pi$
$773$	8.22813	0.295945	0.147973	+	0.988991i	$0.452725\pi$
$774$	−16.7064	−0.600501
$775$	−0.363043	−0.0130409
$776$	−2.85499	−0.102488
$777$	1.18323	0.0424481
$778$	−6.52324	−0.233869
$779$	−6.08034	−0.217851
$780$	−4.44549	−0.159174
$781$	−0.709603	−0.0253916
$782$	0	0
$783$	8.75274	0.312797
$784$	1.00000	0.0357143
$785$	12.1551	0.433835
$786$	0.235448	0.00839817
$787$	−40.5495	−1.44543	−0.722717	−	0.691144i	$-0.757107\pi$
$787$	−40.5495	−1.44543	−0.722717	+	0.691144i	$0.757107\pi$
$788$	12.8910	0.459222
$789$	6.80029	0.242097
$790$	−28.3781	−1.00965
$791$	−15.0313	−0.534451
$792$	1.69166	0.0601104
$793$	7.37941	0.262051
$794$	−3.07146	−0.109002
$795$	−6.84802	−0.242874
$796$	−16.8025	−0.595550
$797$	17.6209	0.624163	0.312082	−	0.950055i	$-0.398974\pi$
$797$	17.6209	0.624163	0.312082	+	0.950055i	$0.398974\pi$
$798$	−0.894129	−0.0316518
$799$	−12.8562	−0.454819
$800$	−0.0594122	−0.00210054
$801$	42.3323	1.49574
$802$	29.5823	1.04459
$803$	7.43644	0.262426
$804$	−2.16581	−0.0763823
$805$	0	0
$806$	−36.4915	−1.28536
$807$	1.04569	0.0368100
$808$	7.18323	0.252705
$809$	−10.2002	−0.358621	−0.179310	−	0.983793i	$-0.557387\pi$
$809$	−10.2002	−0.358621	−0.179310	+	0.983793i	$0.557387\pi$
$810$	17.7889	0.625039
$811$	15.4750	0.543401	0.271700	−	0.962382i	$-0.412414\pi$
$811$	15.4750	0.543401	0.271700	+	0.962382i	$0.412414\pi$
$812$	−4.43882	−0.155772
$813$	8.54060	0.299532
$814$	−2.06961	−0.0725396
$815$	−2.25154	−0.0788679
$816$	−1.41421	−0.0495074
$817$	−15.4451	−0.540356
$818$	15.3950	0.538272
$819$	17.2458	0.602616
$820$	5.06217	0.176779
$821$	−48.7812	−1.70247	−0.851237	−	0.524781i	$-0.824147\pi$
$821$	−48.7812	−1.70247	−0.851237	+	0.524781i	$0.824147\pi$
$822$	2.85657	0.0996342
$823$	−29.2828	−1.02073	−0.510367	−	0.859957i	$-0.670490\pi$
$823$	−29.2828	−1.02073	−0.510367	+	0.859957i	$0.670490\pi$
$824$	−0.277444	−0.00966520
$825$	0.0116556	0.000405796 0
$826$	−4.11058	−0.143025
$827$	−8.27800	−0.287854	−0.143927	−	0.989588i	$-0.545973\pi$
$827$	−8.27800	−0.287854	−0.143927	+	0.989588i	$0.545973\pi$
$828$	0	0
$829$	11.3467	0.394086	0.197043	−	0.980395i	$-0.436866\pi$
$829$	11.3467	0.394086	0.197043	+	0.980395i	$0.436866\pi$
$830$	22.1649	0.769354
$831$	5.40699	0.187566
$832$	−5.97186	−0.207037
$833$	−4.22274	−0.146309
$834$	1.63315	0.0565513
$835$	−37.6144	−1.30170
$836$	1.56394	0.0540899
$837$	−12.0492	−0.416482
$838$	−25.2708	−0.872964
$839$	0.486953	0.0168115	0.00840575	−	0.999965i	$-0.497324\pi$
$839$	0.486953	0.0168115	0.00840575	+	0.999965i	$0.497324\pi$
$840$	0.744406	0.0256844
$841$	−9.29685	−0.320581
$842$	29.8660	1.02925
$843$	7.27874	0.250693
$844$	−6.70960	−0.230954
$845$	50.3743	1.73293
$846$	−8.79205	−0.302277
$847$	−10.6569	−0.366174
$848$	−9.19932	−0.315906
$849$	0.320394	0.0109959
$850$	0.250882	0.00860520
$851$	0	0
$852$	0.405692	0.0138988
$853$	28.9487	0.991184	0.495592	−	0.868555i	$-0.334951\pi$
$853$	28.9487	0.991184	0.495592	+	0.868555i	$0.334951\pi$
$854$	−1.23570	−0.0422847
$855$	17.1373	0.586083
$856$	−4.98705	−0.170454
$857$	−26.0390	−0.889475	−0.444738	−	0.895661i	$-0.646703\pi$
$857$	−26.0390	−0.889475	−0.444738	+	0.895661i	$0.646703\pi$
$858$	1.17157	0.0399968
$859$	−9.69783	−0.330886	−0.165443	−	0.986219i	$-0.552905\pi$
$859$	−9.69783	−0.330886	−0.165443	+	0.986219i	$0.552905\pi$
$860$	12.8588	0.438481
$861$	0.762725	0.0259936
$862$	14.8551	0.505966
$863$	54.9237	1.86962	0.934812	−	0.355142i	$-0.115567\pi$
$863$	54.9237	1.86962	0.934812	+	0.355142i	$0.115567\pi$
$864$	−1.97186	−0.0670841
$865$	55.6113	1.89084
$866$	17.0722	0.580139
$867$	0.278493	0.00945812
$868$	6.11058	0.207407
$869$	7.47883	0.253702
$870$	−3.30428	−0.112026
$871$	38.6198	1.30858
$872$	−9.29539	−0.314782
$873$	8.24475	0.279042
$874$	0	0
$875$	−11.2458	−0.380177
$876$	−4.25154	−0.143646
$877$	−26.7390	−0.902912	−0.451456	−	0.892293i	$-0.649095\pi$
$877$	−26.7390	−0.902912	−0.451456	+	0.892293i	$0.649095\pi$
$878$	37.9769	1.28166
$879$	−6.36908	−0.214824
$880$	−1.30205	−0.0438922
$881$	−11.2027	−0.377430	−0.188715	−	0.982032i	$-0.560432\pi$
$881$	−11.2027	−0.377430	−0.188715	+	0.982032i	$0.560432\pi$
$882$	−2.88784	−0.0972386
$883$	−27.5498	−0.927125	−0.463563	−	0.886064i	$-0.653429\pi$
$883$	−27.5498	−0.927125	−0.463563	+	0.886064i	$0.653429\pi$
$884$	25.2176	0.848161
$885$	−3.05994	−0.102859
$886$	−23.0723	−0.775131
$887$	−15.4126	−0.517506	−0.258753	−	0.965944i	$-0.583312\pi$
$887$	−15.4126	−0.517506	−0.258753	+	0.965944i	$0.583312\pi$
$888$	1.18323	0.0397065
$889$	−18.0720	−0.606114
$890$	−32.5828	−1.09218
$891$	−4.68813	−0.157058
$892$	23.4658	0.785692
$893$	−8.12825	−0.272001
$894$	−6.66888	−0.223041
$895$	54.9667	1.83733
$896$	1.00000	0.0334077
$897$	0	0
$898$	−29.6261	−0.988635
$899$	−27.1238	−0.904629
$900$	0.171573	0.00571910
$901$	38.8464	1.29416
$902$	−1.33410	−0.0444205
$903$	1.93745	0.0644744
$904$	−15.0313	−0.499933
$905$	−36.2309	−1.20436
$906$	6.68239	0.222007
$907$	−32.0759	−1.06506	−0.532531	−	0.846411i	$-0.678759\pi$
$907$	−32.0759	−1.06506	−0.532531	+	0.846411i	$0.678759\pi$
$908$	−14.5018	−0.481258
$909$	−20.7440	−0.688035
$910$	−13.2739	−0.440026
$911$	−19.9979	−0.662561	−0.331281	−	0.943532i	$-0.607481\pi$
$911$	−19.9979	−0.662561	−0.331281	+	0.943532i	$0.607481\pi$
$912$	−0.894129	−0.0296076
$913$	−5.84138	−0.193321
$914$	−19.5027	−0.645091
$915$	−0.919860	−0.0304096
$916$	−18.7683	−0.620123
$917$	0.703033	0.0232162
$918$	8.32666	0.274821
$919$	−38.3070	−1.26363	−0.631816	−	0.775119i	$-0.717690\pi$
$919$	−38.3070	−1.26363	−0.631816	+	0.775119i	$0.717690\pi$
$920$	0	0
$921$	8.69296	0.286443
$922$	−4.81138	−0.158454
$923$	−7.23412	−0.238114
$924$	−0.196182	−0.00645392
$925$	−0.209906	−0.00690165
$926$	32.1251	1.05570
$927$	0.801212	0.0263153
$928$	−4.43882	−0.145712
$929$	37.6763	1.23612	0.618059	−	0.786132i	$-0.287919\pi$
$929$	37.6763	1.23612	0.618059	+	0.786132i	$0.287919\pi$
$930$	4.54875	0.149159
$931$	−2.66981	−0.0874994
$932$	−11.7823	−0.385943
$933$	−6.09647	−0.199589
$934$	−9.49824	−0.310792
$935$	5.49824	0.179812
$936$	17.2458	0.563696
$937$	18.4267	0.601973	0.300987	−	0.953628i	$-0.402684\pi$
$937$	18.4267	0.601973	0.300987	+	0.953628i	$0.402684\pi$
$938$	−6.46696	−0.211154
$939$	2.09716	0.0684383
$940$	6.76716	0.220720
$941$	16.9479	0.552486	0.276243	−	0.961088i	$-0.410911\pi$
$941$	16.9479	0.552486	0.276243	+	0.961088i	$0.410911\pi$
$942$	1.83143	0.0596712
$943$	0	0
$944$	−4.11058	−0.133788
$945$	−4.38294	−0.142577
$946$	−3.38883	−0.110181
$947$	−43.8754	−1.42576	−0.712880	−	0.701286i	$-0.752610\pi$
$947$	−43.8754	−1.42576	−0.712880	+	0.701286i	$0.752610\pi$
$948$	−4.27577	−0.138871
$949$	75.8115	2.46095
$950$	0.158619	0.00514628
$951$	−0.189123	−0.00613272
$952$	−4.22274	−0.136860
$953$	42.2262	1.36784	0.683920	−	0.729557i	$-0.260274\pi$
$953$	42.2262	1.36784	0.683920	+	0.729557i	$0.260274\pi$
$954$	26.5661	0.860111
$955$	47.5349	1.53819
$956$	−13.4420	−0.434744
$957$	0.870818	0.0281496
$958$	−18.4947	−0.597537
$959$	8.52951	0.275432
$960$	0.744406	0.0240256
$961$	6.33922	0.204491
$962$	−21.0988	−0.680253
$963$	14.4018	0.464091
$964$	14.9153	0.480390
$965$	−28.0157	−0.901857
$966$	0	0
$967$	−43.7453	−1.40675	−0.703377	−	0.710817i	$-0.748325\pi$
$967$	−43.7453	−1.40675	−0.703377	+	0.710817i	$0.748325\pi$
$968$	−10.6569	−0.342524
$969$	3.77568	0.121292
$970$	−6.34591	−0.203755
$971$	34.5734	1.10951	0.554756	−	0.832013i	$-0.312812\pi$
$971$	34.5734	1.10951	0.554756	+	0.832013i	$0.312812\pi$
$972$	8.59586	0.275712
$973$	4.87646	0.156332
$974$	−27.5613	−0.883122
$975$	0.118824	0.00380543
$976$	−1.23570	−0.0395537
$977$	−0.995382	−0.0318451	−0.0159225	−	0.999873i	$-0.505069\pi$
$977$	−0.995382	−0.0318451	−0.0159225	+	0.999873i	$0.505069\pi$
$978$	−0.339242	−0.0108478
$979$	8.58693	0.274439
$980$	2.22274	0.0710029
$981$	26.8436	0.857050
$982$	8.56041	0.273174
$983$	−2.64557	−0.0843807	−0.0421903	−	0.999110i	$-0.513434\pi$
$983$	−2.64557	−0.0843807	−0.0421903	+	0.999110i	$0.513434\pi$
$984$	0.762725	0.0243148
$985$	28.6533	0.912971
$986$	18.7440	0.596931
$987$	1.01962	0.0324548
$988$	15.9437	0.507237
$989$	0	0
$990$	3.76012	0.119504
$991$	−42.9667	−1.36488	−0.682440	−	0.730941i	$-0.739081\pi$
$991$	−42.9667	−1.36488	−0.682440	+	0.730941i	$0.739081\pi$
$992$	6.11058	0.194011
$993$	−9.79527	−0.310844
$994$	1.21137	0.0384222
$995$	−37.3477	−1.18400
$996$	3.33962	0.105820
$997$	3.05511	0.0967564	0.0483782	−	0.998829i	$-0.484595\pi$
$997$	3.05511	0.0967564	0.0483782	+	0.998829i	$0.484595\pi$
$998$	−24.4986	−0.775490
$999$	−6.96666	−0.220415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7406.2.a.bb.1.3	✓	4
23.22	odd	2		7406.2.a.bc.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7406.2.a.bb.1.3	✓	4	1.1	even	1	trivial
7406.2.a.bc.1.3	yes	4	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7406.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.334904 q^{3} +1.00000 q^{4} +2.22274 q^{5} +0.334904 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.88784 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.334904 q^{3} +1.00000 q^{4} +2.22274 q^{5} +0.334904 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.88784 q^{9} +2.22274 q^{10} -0.585786 q^{11} +0.334904 q^{12} -5.97186 q^{13} +1.00000 q^{14} +0.744406 q^{15} +1.00000 q^{16} -4.22274 q^{17} -2.88784 q^{18} -2.66981 q^{19} +2.22274 q^{20} +0.334904 q^{21} -0.585786 q^{22} +0.334904 q^{24} -0.0594122 q^{25} -5.97186 q^{26} -1.97186 q^{27} +1.00000 q^{28} -4.43882 q^{29} +0.744406 q^{30} +6.11058 q^{31} +1.00000 q^{32} -0.196182 q^{33} -4.22274 q^{34} +2.22274 q^{35} -2.88784 q^{36} +3.53304 q^{37} -2.66981 q^{38} -2.00000 q^{39} +2.22274 q^{40} +2.27744 q^{41} +0.334904 q^{42} +5.78510 q^{43} -0.585786 q^{44} -6.41893 q^{45} +3.04451 q^{47} +0.334904 q^{48} +1.00000 q^{49} -0.0594122 q^{50} -1.41421 q^{51} -5.97186 q^{52} -9.19932 q^{53} -1.97186 q^{54} -1.30205 q^{55} +1.00000 q^{56} -0.894129 q^{57} -4.43882 q^{58} -4.11058 q^{59} +0.744406 q^{60} -1.23570 q^{61} +6.11058 q^{62} -2.88784 q^{63} +1.00000 q^{64} -13.2739 q^{65} -0.196182 q^{66} -6.46696 q^{67} -4.22274 q^{68} +2.22274 q^{70} +1.21137 q^{71} -2.88784 q^{72} -12.6948 q^{73} +3.53304 q^{74} -0.0198974 q^{75} -2.66981 q^{76} -0.585786 q^{77} -2.00000 q^{78} -12.7672 q^{79} +2.22274 q^{80} +8.00313 q^{81} +2.27744 q^{82} +9.97186 q^{83} +0.334904 q^{84} -9.38607 q^{85} +5.78510 q^{86} -1.48658 q^{87} -0.585786 q^{88} -14.6588 q^{89} -6.41893 q^{90} -5.97186 q^{91} +2.04646 q^{93} +3.04451 q^{94} -5.93430 q^{95} +0.334904 q^{96} -2.85499 q^{97} +1.00000 q^{98} +1.69166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8} - 4 q^{10} - 8 q^{11} - 4 q^{13} + 4 q^{14} + 4 q^{16} - 4 q^{17} - 8 q^{19} - 4 q^{20} - 8 q^{22} - 4 q^{26} + 12 q^{27} + 4 q^{28} - 4 q^{29} + 4 q^{32} - 4 q^{33} - 4 q^{34} - 4 q^{35} + 8 q^{37} - 8 q^{38} - 8 q^{39} - 4 q^{40} - 8 q^{43} - 8 q^{44} - 16 q^{45} + 4 q^{49} - 4 q^{52} + 12 q^{54} + 12 q^{55} + 4 q^{56} - 24 q^{57} - 4 q^{58} + 8 q^{59} - 12 q^{61} + 4 q^{64} - 16 q^{65} - 4 q^{66} - 32 q^{67} - 4 q^{68} - 4 q^{70} + 8 q^{71} + 4 q^{73} + 8 q^{74} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 8 q^{78} - 16 q^{79} - 4 q^{80} - 8 q^{81} + 20 q^{83} - 12 q^{85} - 8 q^{86} - 20 q^{87} - 8 q^{88} - 28 q^{89} - 16 q^{90} - 4 q^{91} - 12 q^{93} + 8 q^{95} + 16 q^{97} + 4 q^{98} - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 + 4 * q^8 - 4 * q^10 - 8 * q^11 - 4 * q^13 + 4 * q^14 + 4 * q^16 - 4 * q^17 - 8 * q^19 - 4 * q^20 - 8 * q^22 - 4 * q^26 + 12 * q^27 + 4 * q^28 - 4 * q^29 + 4 * q^32 - 4 * q^33 - 4 * q^34 - 4 * q^35 + 8 * q^37 - 8 * q^38 - 8 * q^39 - 4 * q^40 - 8 * q^43 - 8 * q^44 - 16 * q^45 + 4 * q^49 - 4 * q^52 + 12 * q^54 + 12 * q^55 + 4 * q^56 - 24 * q^57 - 4 * q^58 + 8 * q^59 - 12 * q^61 + 4 * q^64 - 16 * q^65 - 4 * q^66 - 32 * q^67 - 4 * q^68 - 4 * q^70 + 8 * q^71 + 4 * q^73 + 8 * q^74 + 4 * q^75 - 8 * q^76 - 8 * q^77 - 8 * q^78 - 16 * q^79 - 4 * q^80 - 8 * q^81 + 20 * q^83 - 12 * q^85 - 8 * q^86 - 20 * q^87 - 8 * q^88 - 28 * q^89 - 16 * q^90 - 4 * q^91 - 12 * q^93 + 8 * q^95 + 16 * q^97 + 4 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more