LMFDB - Embedded newform 415.2.a.d.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [415,2,Mod(1,415)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("415.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 415 = 5 \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	415.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.31379168388$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	7.7.179711353.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 16x^{3} - 9x^{2} - 8x + 4 $ x^7 - x^6 - 8x^5 + 6x^4 + 16x^3 - 9x^2 - 8*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.51525$ of defining polynomial
Character	$\chi$	$=$	415.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.24504 q^{2} -0.166440 q^{3} -0.449886 q^{4} -1.00000 q^{5} -0.207224 q^{6} -2.60638 q^{7} -3.05020 q^{8} -2.97230 q^{9} -1.24504 q^{10} -5.50271 q^{11} +0.0748791 q^{12} +5.27316 q^{13} -3.24504 q^{14} +0.166440 q^{15} -2.89783 q^{16} -2.31115 q^{17} -3.70062 q^{18} +5.82565 q^{19} +0.449886 q^{20} +0.433806 q^{21} -6.85108 q^{22} +2.07800 q^{23} +0.507675 q^{24} +1.00000 q^{25} +6.56527 q^{26} +0.994030 q^{27} +1.17257 q^{28} -8.09462 q^{29} +0.207224 q^{30} +8.07873 q^{31} +2.49249 q^{32} +0.915872 q^{33} -2.87746 q^{34} +2.60638 q^{35} +1.33719 q^{36} -10.9885 q^{37} +7.25314 q^{38} -0.877665 q^{39} +3.05020 q^{40} -4.75856 q^{41} +0.540104 q^{42} -1.47228 q^{43} +2.47559 q^{44} +2.97230 q^{45} +2.58718 q^{46} -9.21328 q^{47} +0.482315 q^{48} -0.206786 q^{49} +1.24504 q^{50} +0.384668 q^{51} -2.37232 q^{52} -2.54774 q^{53} +1.23760 q^{54} +5.50271 q^{55} +7.94997 q^{56} -0.969621 q^{57} -10.0781 q^{58} +12.9647 q^{59} -0.0748791 q^{60} -8.32727 q^{61} +10.0583 q^{62} +7.74694 q^{63} +8.89890 q^{64} -5.27316 q^{65} +1.14029 q^{66} -4.09786 q^{67} +1.03975 q^{68} -0.345862 q^{69} +3.24504 q^{70} +1.35468 q^{71} +9.06609 q^{72} +9.69125 q^{73} -13.6810 q^{74} -0.166440 q^{75} -2.62088 q^{76} +14.3422 q^{77} -1.09272 q^{78} -7.99410 q^{79} +2.89783 q^{80} +8.75145 q^{81} -5.92458 q^{82} -1.00000 q^{83} -0.195163 q^{84} +2.31115 q^{85} -1.83304 q^{86} +1.34727 q^{87} +16.7844 q^{88} -7.45495 q^{89} +3.70062 q^{90} -13.7439 q^{91} -0.934862 q^{92} -1.34462 q^{93} -11.4709 q^{94} -5.82565 q^{95} -0.414850 q^{96} -10.1954 q^{97} -0.257455 q^{98} +16.3557 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{2} - 5 q^{3} + 7 q^{4} - 7 q^{5} - 6 q^{7} - 12 q^{8} + 2 q^{9} + 3 q^{10} - 2 q^{11} - 11 q^{12} - 5 q^{13} - 11 q^{14} + 5 q^{15} + 7 q^{16} - 25 q^{17} - 4 q^{18} + 6 q^{19} - 7 q^{20} + 2 q^{21}+ \cdots + 38 q^{99}+O(q^{100}) $ 7 * q - 3 * q^2 - 5 * q^3 + 7 * q^4 - 7 * q^5 - 6 * q^7 - 12 * q^8 + 2 * q^9 + 3 * q^10 - 2 * q^11 - 11 * q^12 - 5 * q^13 - 11 * q^14 + 5 * q^15 + 7 * q^16 - 25 * q^17 - 4 * q^18 + 6 * q^19 - 7 * q^20 + 2 * q^21 - 2 * q^22 - 9 * q^23 - q^24 + 7 * q^25 - 5 * q^26 - 17 * q^27 - 3 * q^28 - 2 * q^29 + q^31 - 17 * q^32 - 17 * q^33 + q^34 + 6 * q^35 + 21 * q^36 - 13 * q^37 + q^38 + 12 * q^40 - 15 * q^41 + 10 * q^42 - 5 * q^43 - 12 * q^44 - 2 * q^45 + 27 * q^46 - 12 * q^47 + 4 * q^48 - 15 * q^49 - 3 * q^50 + 23 * q^51 - 20 * q^52 - 31 * q^53 + 5 * q^54 + 2 * q^55 + 26 * q^56 - 9 * q^57 + 13 * q^58 - 14 * q^59 + 11 * q^60 + 7 * q^61 - 14 * q^62 + 4 * q^63 + 10 * q^64 + 5 * q^65 + 46 * q^66 + 10 * q^67 + 4 * q^68 - 19 * q^69 + 11 * q^70 + 9 * q^71 + 21 * q^72 - 15 * q^73 - 19 * q^74 - 5 * q^75 + 61 * q^76 - 9 * q^77 + 67 * q^78 + 6 * q^79 - 7 * q^80 + 3 * q^81 + 12 * q^82 - 7 * q^83 + 17 * q^84 + 25 * q^85 + 42 * q^86 + 23 * q^87 + 20 * q^88 - 37 * q^89 + 4 * q^90 + 9 * q^91 - 13 * q^92 + 2 * q^93 + 22 * q^94 - 6 * q^95 + 59 * q^96 - 20 * q^97 + 44 * q^98 + 38 * q^99

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 $

$2$	1.24504	0.880373	0.440187	−	0.897906i	$-0.354912\pi$
$2$	1.24504	0.880373	0.440187	+	0.897906i	$0.354912\pi$
$3$	−0.166440	−0.0960942	−0.0480471	−	0.998845i	$-0.515300\pi$
$3$	−0.166440	−0.0960942	−0.0480471	+	0.998845i	$0.515300\pi$
$4$	−0.449886	−0.224943
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.207224	−0.0845988
$7$	−2.60638	−0.985119	−0.492559	−	0.870279i	$-0.663939\pi$
$7$	−2.60638	−0.985119	−0.492559	+	0.870279i	$0.663939\pi$
$8$	−3.05020	−1.07841
$9$	−2.97230	−0.990766
$10$	−1.24504	−0.393715
$11$	−5.50271	−1.65913	−0.829565	−	0.558409i	$-0.811412\pi$
$11$	−5.50271	−1.65913	−0.829565	+	0.558409i	$0.811412\pi$
$12$	0.0748791	0.0216157
$13$	5.27316	1.46251	0.731256	−	0.682103i	$-0.238935\pi$
$13$	5.27316	1.46251	0.731256	+	0.682103i	$0.238935\pi$
$14$	−3.24504	−0.867272
$15$	0.166440	0.0429746
$16$	−2.89783	−0.724458
$17$	−2.31115	−0.560536	−0.280268	−	0.959922i	$-0.590423\pi$
$17$	−2.31115	−0.560536	−0.280268	+	0.959922i	$0.590423\pi$
$18$	−3.70062	−0.872244
$19$	5.82565	1.33650	0.668248	−	0.743939i	$-0.267045\pi$
$19$	5.82565	1.33650	0.668248	+	0.743939i	$0.267045\pi$
$20$	0.449886	0.100598
$21$	0.433806	0.0946642
$22$	−6.85108	−1.46065
$23$	2.07800	0.433292	0.216646	−	0.976250i	$-0.430488\pi$
$23$	2.07800	0.433292	0.216646	+	0.976250i	$0.430488\pi$
$24$	0.507675	0.103629
$25$	1.00000	0.200000
$26$	6.56527	1.28756
$27$	0.994030	0.191301
$28$	1.17257	0.221596
$29$	−8.09462	−1.50313	−0.751566	−	0.659658i	$-0.770701\pi$
$29$	−8.09462	−1.50313	−0.751566	+	0.659658i	$0.770701\pi$
$30$	0.207224	0.0378337
$31$	8.07873	1.45098	0.725492	−	0.688231i	$-0.241612\pi$
$31$	8.07873	1.45098	0.725492	+	0.688231i	$0.241612\pi$
$32$	2.49249	0.440614
$33$	0.915872	0.159433
$34$	−2.87746	−0.493481
$35$	2.60638	0.440559
$36$	1.33719	0.222866
$37$	−10.9885	−1.80649	−0.903247	−	0.429121i	$-0.858823\pi$
$37$	−10.9885	−1.80649	−0.903247	+	0.429121i	$0.858823\pi$
$38$	7.25314	1.17661
$39$	−0.877665	−0.140539
$40$	3.05020	0.482278
$41$	−4.75856	−0.743162	−0.371581	−	0.928401i	$-0.621184\pi$
$41$	−4.75856	−0.743162	−0.371581	+	0.928401i	$0.621184\pi$
$42$	0.540104	0.0833399
$43$	−1.47228	−0.224521	−0.112260	−	0.993679i	$-0.535809\pi$
$43$	−1.47228	−0.224521	−0.112260	+	0.993679i	$0.535809\pi$
$44$	2.47559	0.373210
$45$	2.97230	0.443084
$46$	2.58718	0.381459
$47$	−9.21328	−1.34389	−0.671947	−	0.740599i	$-0.734542\pi$
$47$	−9.21328	−1.34389	−0.671947	+	0.740599i	$0.734542\pi$
$48$	0.482315	0.0696162
$49$	−0.206786	−0.0295408
$50$	1.24504	0.176075
$51$	0.384668	0.0538642
$52$	−2.37232	−0.328982
$53$	−2.54774	−0.349959	−0.174980	−	0.984572i	$-0.555986\pi$
$53$	−2.54774	−0.349959	−0.174980	+	0.984572i	$0.555986\pi$
$54$	1.23760	0.168416
$55$	5.50271	0.741986
$56$	7.94997	1.06236
$57$	−0.969621	−0.128429
$58$	−10.0781	−1.32332
$59$	12.9647	1.68787	0.843933	−	0.536449i	$-0.180234\pi$
$59$	12.9647	1.68787	0.843933	+	0.536449i	$0.180234\pi$
$60$	−0.0748791	−0.00966684
$61$	−8.32727	−1.06620	−0.533099	−	0.846053i	$-0.678973\pi$
$61$	−8.32727	−1.06620	−0.533099	+	0.846053i	$0.678973\pi$
$62$	10.0583	1.27741
$63$	7.74694	0.976022
$64$	8.89890	1.11236
$65$	−5.27316	−0.654055
$66$	1.14029	0.140360
$67$	−4.09786	−0.500633	−0.250316	−	0.968164i	$-0.580535\pi$
$67$	−4.09786	−0.500633	−0.250316	+	0.968164i	$0.580535\pi$
$68$	1.03975	0.126089
$69$	−0.345862	−0.0416369
$70$	3.24504	0.387856
$71$	1.35468	0.160771	0.0803856	−	0.996764i	$-0.474385\pi$
$71$	1.35468	0.160771	0.0803856	+	0.996764i	$0.474385\pi$
$72$	9.06609	1.06845
$73$	9.69125	1.13427	0.567137	−	0.823623i	$-0.308051\pi$
$73$	9.69125	1.13427	0.567137	+	0.823623i	$0.308051\pi$
$74$	−13.6810	−1.59039
$75$	−0.166440	−0.0192188
$76$	−2.62088	−0.300635
$77$	14.3422	1.63444
$78$	−1.09272	−0.123727
$79$	−7.99410	−0.899407	−0.449703	−	0.893178i	$-0.648470\pi$
$79$	−7.99410	−0.899407	−0.449703	+	0.893178i	$0.648470\pi$
$80$	2.89783	0.323987
$81$	8.75145	0.972383
$82$	−5.92458	−0.654260
$83$	−1.00000	−0.109764
$83$	−1.00000	−0.109764
$84$	−0.195163	−0.0212941
$85$	2.31115	0.250679
$86$	−1.83304	−0.197662
$87$	1.34727	0.144442
$88$	16.7844	1.78922
$89$	−7.45495	−0.790223	−0.395111	−	0.918633i	$-0.629294\pi$
$89$	−7.45495	−0.790223	−0.395111	+	0.918633i	$0.629294\pi$
$90$	3.70062	0.390079
$91$	−13.7439	−1.44075
$92$	−0.934862	−0.0974661
$93$	−1.34462	−0.139431
$94$	−11.4709	−1.18313
$95$	−5.82565	−0.597699
$96$	−0.414850	−0.0423404
$97$	−10.1954	−1.03519	−0.517595	−	0.855626i	$-0.673173\pi$
$97$	−10.1954	−1.03519	−0.517595	+	0.855626i	$0.673173\pi$
$98$	−0.257455	−0.0260069
$99$	16.3557	1.64381
$100$	−0.449886	−0.0449886
$101$	12.4843	1.24223	0.621117	−	0.783718i	$-0.286679\pi$
$101$	12.4843	1.24223	0.621117	+	0.783718i	$0.286679\pi$
$102$	0.478925	0.0474206
$103$	−1.08906	−0.107309	−0.0536544	−	0.998560i	$-0.517087\pi$
$103$	−1.08906	−0.107309	−0.0536544	+	0.998560i	$0.517087\pi$
$104$	−16.0842	−1.57718
$105$	−0.433806	−0.0423351
$106$	−3.17203	−0.308095
$107$	−6.96239	−0.673079	−0.336540	−	0.941669i	$-0.609257\pi$
$107$	−6.96239	−0.673079	−0.336540	+	0.941669i	$0.609257\pi$
$108$	−0.447200	−0.0430318
$109$	1.12751	0.107996	0.0539978	−	0.998541i	$-0.482804\pi$
$109$	1.12751	0.107996	0.0539978	+	0.998541i	$0.482804\pi$
$110$	6.85108	0.653225
$111$	1.82892	0.173594
$112$	7.55285	0.713677
$113$	−10.3145	−0.970305	−0.485152	−	0.874430i	$-0.661236\pi$
$113$	−10.3145	−0.970305	−0.485152	+	0.874430i	$0.661236\pi$
$114$	−1.20721	−0.113066
$115$	−2.07800	−0.193774
$116$	3.64165	0.338119
$117$	−15.6734	−1.44901
$118$	16.1416	1.48595
$119$	6.02373	0.552194
$120$	−0.507675	−0.0463442
$121$	19.2799	1.75272
$122$	−10.3677	−0.938652
$123$	0.792015	0.0714136
$124$	−3.63451	−0.326388
$125$	−1.00000	−0.0894427
$126$	9.64521	0.859264
$127$	1.29708	0.115097	0.0575487	−	0.998343i	$-0.481672\pi$
$127$	1.29708	0.115097	0.0575487	+	0.998343i	$0.481672\pi$
$128$	6.09447	0.538680
$129$	0.245046	0.0215751
$130$	−6.56527	−0.575813
$131$	−4.44343	−0.388225	−0.194112	−	0.980979i	$-0.562183\pi$
$131$	−4.44343	−0.388225	−0.194112	+	0.980979i	$0.562183\pi$
$132$	−0.412038	−0.0358633
$133$	−15.1839	−1.31661
$134$	−5.10198	−0.440744
$135$	−0.994030	−0.0855525
$136$	7.04945	0.604486
$137$	10.1106	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$137$	10.1106	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$138$	−0.430611	−0.0366560
$139$	16.7621	1.42174	0.710870	−	0.703324i	$-0.248302\pi$
$139$	16.7621	1.42174	0.710870	+	0.703324i	$0.248302\pi$
$140$	−1.17257	−0.0991006
$141$	1.53346	0.129141
$142$	1.68663	0.141539
$143$	−29.0167	−2.42650
$144$	8.61322	0.717768
$145$	8.09462	0.672221
$146$	12.0659	0.998585
$147$	0.0344174	0.00283870
$148$	4.94356	0.406358
$149$	−8.18290	−0.670369	−0.335185	−	0.942152i	$-0.608799\pi$
$149$	−8.18290	−0.670369	−0.335185	+	0.942152i	$0.608799\pi$
$150$	−0.207224	−0.0169198
$151$	−2.13120	−0.173434	−0.0867172	−	0.996233i	$-0.527638\pi$
$151$	−2.13120	−0.173434	−0.0867172	+	0.996233i	$0.527638\pi$
$152$	−17.7694	−1.44129
$153$	6.86942	0.555360
$154$	17.8565	1.43892
$155$	−8.07873	−0.648899
$156$	0.394849	0.0316132
$157$	−15.5472	−1.24080	−0.620399	−	0.784286i	$-0.713029\pi$
$157$	−15.5472	−1.24080	−0.620399	+	0.784286i	$0.713029\pi$
$158$	−9.95294	−0.791814
$159$	0.424046	0.0336290
$160$	−2.49249	−0.197049
$161$	−5.41605	−0.426845
$162$	10.8959	0.856060
$163$	−20.8580	−1.63373	−0.816863	−	0.576832i	$-0.804289\pi$
$163$	−20.8580	−1.63373	−0.816863	+	0.576832i	$0.804289\pi$
$164$	2.14081	0.167169
$165$	−0.915872	−0.0713006
$166$	−1.24504	−0.0966335
$167$	9.23380	0.714533	0.357267	−	0.934002i	$-0.383709\pi$
$167$	9.23380	0.714533	0.357267	+	0.934002i	$0.383709\pi$
$168$	−1.32319	−0.102087
$169$	14.8062	1.13894
$170$	2.87746	0.220691
$171$	−17.3156	−1.32415
$172$	0.662358	0.0505043
$173$	−13.0427	−0.991618	−0.495809	−	0.868432i	$-0.665128\pi$
$173$	−13.0427	−0.991618	−0.495809	+	0.868432i	$0.665128\pi$
$174$	1.67740	0.127163
$175$	−2.60638	−0.197024
$176$	15.9459	1.20197
$177$	−2.15785	−0.162194
$178$	−9.28167	−0.695691
$179$	−0.700134	−0.0523305	−0.0261652	−	0.999658i	$-0.508330\pi$
$179$	−0.700134	−0.0523305	−0.0261652	+	0.999658i	$0.508330\pi$
$180$	−1.33719	−0.0996686
$181$	19.7238	1.46606	0.733031	−	0.680196i	$-0.238105\pi$
$181$	19.7238	1.46606	0.733031	+	0.680196i	$0.238105\pi$
$182$	−17.1116	−1.26840
$183$	1.38599	0.102455
$184$	−6.33830	−0.467266
$185$	10.9885	0.807888
$186$	−1.67411	−0.122751
$187$	12.7176	0.930002
$188$	4.14492	0.302300
$189$	−2.59082	−0.188454
$190$	−7.25314	−0.526198
$191$	23.1169	1.67268	0.836339	−	0.548213i	$-0.184692\pi$
$191$	23.1169	1.67268	0.836339	+	0.548213i	$0.184692\pi$
$192$	−1.48113	−0.106892
$193$	5.49746	0.395716	0.197858	−	0.980231i	$-0.436602\pi$
$193$	5.49746	0.395716	0.197858	+	0.980231i	$0.436602\pi$
$194$	−12.6937	−0.911353
$195$	0.877665	0.0628509
$196$	0.0930299	0.00664499
$197$	−12.7033	−0.905072	−0.452536	−	0.891746i	$-0.649481\pi$
$197$	−12.7033	−0.905072	−0.452536	+	0.891746i	$0.649481\pi$
$198$	20.3634	1.44717
$199$	16.0755	1.13956	0.569781	−	0.821796i	$-0.307028\pi$
$199$	16.0755	1.13956	0.569781	+	0.821796i	$0.307028\pi$
$200$	−3.05020	−0.215681
$201$	0.682048	0.0481079
$202$	15.5434	1.09363
$203$	21.0976	1.48076
$204$	−0.173057	−0.0121164
$205$	4.75856	0.332352
$206$	−1.35592	−0.0944717
$207$	−6.17643	−0.429291
$208$	−15.2807	−1.05953
$209$	−32.0569	−2.21742
$210$	−0.540104	−0.0372707
$211$	3.67598	0.253065	0.126532	−	0.991962i	$-0.459615\pi$
$211$	3.67598	0.253065	0.126532	+	0.991962i	$0.459615\pi$
$212$	1.14619	0.0787208
$213$	−0.225473	−0.0154492
$214$	−8.66842	−0.592561
$215$	1.47228	0.100409
$216$	−3.03198	−0.206300
$217$	−21.0562	−1.42939
$218$	1.40379	0.0950764
$219$	−1.61301	−0.108997
$220$	−2.47559	−0.166905
$221$	−12.1871	−0.819790
$222$	2.27707	0.152827
$223$	−20.3938	−1.36567	−0.682834	−	0.730573i	$-0.739253\pi$
$223$	−20.3938	−1.36567	−0.682834	+	0.730573i	$0.739253\pi$
$224$	−6.49637	−0.434057
$225$	−2.97230	−0.198153
$226$	−12.8419	−0.854230
$227$	5.88815	0.390810	0.195405	−	0.980723i	$-0.437398\pi$
$227$	5.88815	0.390810	0.195405	+	0.980723i	$0.437398\pi$
$228$	0.436219	0.0288893
$229$	−7.64169	−0.504977	−0.252489	−	0.967600i	$-0.581249\pi$
$229$	−7.64169	−0.504977	−0.252489	+	0.967600i	$0.581249\pi$
$230$	−2.58718	−0.170594
$231$	−2.38711	−0.157060
$232$	24.6902	1.62099
$233$	−12.3326	−0.807933	−0.403966	−	0.914774i	$-0.632369\pi$
$233$	−12.3326	−0.807933	−0.403966	+	0.914774i	$0.632369\pi$
$234$	−19.5139	−1.27567
$235$	9.21328	0.601008
$236$	−5.83266	−0.379674
$237$	1.33054	0.0864278
$238$	7.49976	0.486137
$239$	−13.8285	−0.894494	−0.447247	−	0.894410i	$-0.647596\pi$
$239$	−13.8285	−0.894494	−0.447247	+	0.894410i	$0.647596\pi$
$240$	−0.482315	−0.0311333
$241$	−23.3707	−1.50544	−0.752720	−	0.658341i	$-0.771259\pi$
$241$	−23.3707	−1.50544	−0.752720	+	0.658341i	$0.771259\pi$
$242$	24.0041	1.54304
$243$	−4.43868	−0.284741
$244$	3.74632	0.239834
$245$	0.206786	0.0132110
$246$	0.986087	0.0628706
$247$	30.7196	1.95464
$248$	−24.6417	−1.56475
$249$	0.166440	0.0105477
$250$	−1.24504	−0.0787430
$251$	7.61182	0.480454	0.240227	−	0.970717i	$-0.422778\pi$
$251$	7.61182	0.480454	0.240227	+	0.970717i	$0.422778\pi$
$252$	−3.48524	−0.219549
$253$	−11.4346	−0.718889
$254$	1.61491	0.101329
$255$	−0.384668	−0.0240888
$256$	−10.2100	−0.638123
$257$	−7.04602	−0.439519	−0.219759	−	0.975554i	$-0.570527\pi$
$257$	−7.04602	−0.439519	−0.219759	+	0.975554i	$0.570527\pi$
$258$	0.305091	0.0189942
$259$	28.6401	1.77961
$260$	2.37232	0.147125
$261$	24.0596	1.48925
$262$	−5.53223	−0.341783
$263$	26.8343	1.65467	0.827337	−	0.561705i	$-0.189854\pi$
$263$	26.8343	1.65467	0.827337	+	0.561705i	$0.189854\pi$
$264$	−2.79359	−0.171934
$265$	2.54774	0.156506
$266$	−18.9044	−1.15911
$267$	1.24080	0.0759358
$268$	1.84357	0.112614
$269$	19.0639	1.16235	0.581173	−	0.813780i	$-0.302594\pi$
$269$	19.0639	1.16235	0.581173	+	0.813780i	$0.302594\pi$
$270$	−1.23760	−0.0753181
$271$	16.0663	0.975958	0.487979	−	0.872855i	$-0.337734\pi$
$271$	16.0663	0.975958	0.487979	+	0.872855i	$0.337734\pi$
$272$	6.69732	0.406084
$273$	2.28753	0.138448
$274$	12.5880	0.760469
$275$	−5.50271	−0.331826
$276$	0.155598	0.00936593
$277$	8.87248	0.533096	0.266548	−	0.963822i	$-0.414117\pi$
$277$	8.87248	0.533096	0.266548	+	0.963822i	$0.414117\pi$
$278$	20.8694	1.25166
$279$	−24.0124	−1.43758
$280$	−7.94997	−0.475101
$281$	−30.5416	−1.82196	−0.910980	−	0.412450i	$-0.864673\pi$
$281$	−30.5416	−1.82196	−0.910980	+	0.412450i	$0.864673\pi$
$282$	1.90921	0.113692
$283$	−16.4272	−0.976498	−0.488249	−	0.872704i	$-0.662364\pi$
$283$	−16.4272	−0.976498	−0.488249	+	0.872704i	$0.662364\pi$
$284$	−0.609452	−0.0361643
$285$	0.969621	0.0574354
$286$	−36.1268	−2.13622
$287$	12.4026	0.732103
$288$	−7.40842	−0.436545
$289$	−11.6586	−0.685800
$290$	10.0781	0.591806
$291$	1.69693	0.0994757
$292$	−4.35996	−0.255147
$293$	−5.26648	−0.307671	−0.153836	−	0.988096i	$-0.549163\pi$
$293$	−5.26648	−0.307671	−0.153836	+	0.988096i	$0.549163\pi$
$294$	0.0428509	0.00249912
$295$	−12.9647	−0.754837
$296$	33.5170	1.94814
$297$	−5.46986	−0.317394
$298$	−10.1880	−0.590175
$299$	10.9576	0.633695
$300$	0.0748791	0.00432314
$301$	3.83732	0.221179
$302$	−2.65342	−0.152687
$303$	−2.07789	−0.119372
$304$	−16.8817	−0.968234
$305$	8.32727	0.476818
$306$	8.55267	0.488924
$307$	−15.1037	−0.862011	−0.431006	−	0.902349i	$-0.641841\pi$
$307$	−15.1037	−0.862011	−0.431006	+	0.902349i	$0.641841\pi$
$308$	−6.45234	−0.367656
$309$	0.181264	0.0103118
$310$	−10.0583	−0.571274
$311$	−5.33336	−0.302427	−0.151214	−	0.988501i	$-0.548318\pi$
$311$	−5.33336	−0.302427	−0.151214	+	0.988501i	$0.548318\pi$
$312$	2.67705	0.151558
$313$	−4.59120	−0.259510	−0.129755	−	0.991546i	$-0.541419\pi$
$313$	−4.59120	−0.259510	−0.129755	+	0.991546i	$0.541419\pi$
$314$	−19.3568	−1.09237
$315$	−7.74694	−0.436490
$316$	3.59643	0.202315
$317$	27.9269	1.56853	0.784266	−	0.620425i	$-0.213040\pi$
$317$	27.9269	1.56853	0.784266	+	0.620425i	$0.213040\pi$
$318$	0.527953	0.0296061
$319$	44.5424	2.49389
$320$	−8.89890	−0.497464
$321$	1.15882	0.0646790
$322$	−6.74318	−0.375783
$323$	−13.4639	−0.749153
$324$	−3.93715	−0.218731
$325$	5.27316	0.292502
$326$	−25.9690	−1.43829
$327$	−0.187662	−0.0103778
$328$	14.5145	0.801431
$329$	24.0133	1.32390
$330$	−1.14029	−0.0627711
$331$	3.27519	0.180021	0.0900103	−	0.995941i	$-0.471310\pi$
$331$	3.27519	0.180021	0.0900103	+	0.995941i	$0.471310\pi$
$332$	0.449886	0.0246907
$333$	32.6610	1.78981
$334$	11.4964	0.629056
$335$	4.09786	0.223890
$336$	−1.25710	−0.0685802
$337$	−17.9030	−0.975238	−0.487619	−	0.873057i	$-0.662134\pi$
$337$	−17.9030	−0.975238	−0.487619	+	0.873057i	$0.662134\pi$
$338$	18.4343	1.00269
$339$	1.71674	0.0932407
$340$	−1.03975	−0.0563885
$341$	−44.4550	−2.40737
$342$	−21.5585	−1.16575
$343$	18.7836	1.01422
$344$	4.49074	0.242125
$345$	0.345862	0.0186206
$346$	−16.2386	−0.872994
$347$	−13.1318	−0.704950	−0.352475	−	0.935821i	$-0.614660\pi$
$347$	−13.1318	−0.704950	−0.352475	+	0.935821i	$0.614660\pi$
$348$	−0.606117	−0.0324913
$349$	16.3519	0.875296	0.437648	−	0.899146i	$-0.355812\pi$
$349$	16.3519	0.875296	0.437648	+	0.899146i	$0.355812\pi$
$350$	−3.24504	−0.173454
$351$	5.24168	0.279780
$352$	−13.7155	−0.731036
$353$	−15.8664	−0.844484	−0.422242	−	0.906483i	$-0.638757\pi$
$353$	−15.8664	−0.844484	−0.422242	+	0.906483i	$0.638757\pi$
$354$	−2.68660	−0.142791
$355$	−1.35468	−0.0718990
$356$	3.35388	0.177755
$357$	−1.00259	−0.0530627
$358$	−0.871692	−0.0460703
$359$	−29.0900	−1.53531	−0.767657	−	0.640861i	$-0.778578\pi$
$359$	−29.0900	−1.53531	−0.767657	+	0.640861i	$0.778578\pi$
$360$	−9.06609	−0.477825
$361$	14.9382	0.786220
$362$	24.5569	1.29068
$363$	−3.20894	−0.168426
$364$	6.18317	0.324086
$365$	−9.69125	−0.507263
$366$	1.72561	0.0901990
$367$	−11.2739	−0.588494	−0.294247	−	0.955729i	$-0.595069\pi$
$367$	−11.2739	−0.588494	−0.294247	+	0.955729i	$0.595069\pi$
$368$	−6.02168	−0.313902
$369$	14.1439	0.736300
$370$	13.6810	0.711243
$371$	6.64038	0.344751
$372$	0.604928	0.0313640
$373$	35.1170	1.81829	0.909143	−	0.416483i	$-0.136737\pi$
$373$	35.1170	1.81829	0.909143	+	0.416483i	$0.136737\pi$
$374$	15.8339	0.818749
$375$	0.166440	0.00859493
$376$	28.1023	1.44927
$377$	−42.6842	−2.19835
$378$	−3.22566	−0.165910
$379$	22.0532	1.13280	0.566400	−	0.824131i	$-0.308336\pi$
$379$	22.0532	1.13280	0.566400	+	0.824131i	$0.308336\pi$
$380$	2.62088	0.134448
$381$	−0.215887	−0.0110602
$382$	28.7813	1.47258
$383$	17.5000	0.894208	0.447104	−	0.894482i	$-0.352455\pi$
$383$	17.5000	0.894208	0.447104	+	0.894482i	$0.352455\pi$
$384$	−1.01436	−0.0517641
$385$	−14.3422	−0.730944
$386$	6.84453	0.348377
$387$	4.37605	0.222447
$388$	4.58678	0.232859
$389$	−8.92745	−0.452640	−0.226320	−	0.974053i	$-0.572669\pi$
$389$	−8.92745	−0.452640	−0.226320	+	0.974053i	$0.572669\pi$
$390$	1.09272	0.0553323
$391$	−4.80256	−0.242876
$392$	0.630736	0.0318570
$393$	0.739565	0.0373061
$394$	−15.8161	−0.796801
$395$	7.99410	0.402227
$396$	−7.35820	−0.369764
$397$	12.1910	0.611851	0.305925	−	0.952056i	$-0.401034\pi$
$397$	12.1910	0.611851	0.305925	+	0.952056i	$0.401034\pi$
$398$	20.0146	1.00324
$399$	2.52720	0.126518
$400$	−2.89783	−0.144892
$401$	−24.0817	−1.20258	−0.601292	−	0.799030i	$-0.705347\pi$
$401$	−24.0817	−1.20258	−0.601292	+	0.799030i	$0.705347\pi$
$402$	0.849174	0.0423529
$403$	42.6005	2.12208
$404$	−5.61651	−0.279432
$405$	−8.75145	−0.434863
$406$	26.2673	1.30363
$407$	60.4664	2.99721
$408$	−1.17331	−0.0580876
$409$	19.9182	0.984892	0.492446	−	0.870343i	$-0.336103\pi$
$409$	19.9182	0.984892	0.492446	+	0.870343i	$0.336103\pi$
$410$	5.92458	0.292594
$411$	−1.68280	−0.0830065
$412$	0.489955	0.0241383
$413$	−33.7910	−1.66275
$414$	−7.68987	−0.377937
$415$	1.00000	0.0490881
$416$	13.1433	0.644403
$417$	−2.78988	−0.136621
$418$	−39.9120	−1.95216
$419$	23.7145	1.15853	0.579265	−	0.815140i	$-0.303340\pi$
$419$	23.7145	1.15853	0.579265	+	0.815140i	$0.303340\pi$
$420$	0.195163	0.00952299
$421$	28.9755	1.41218	0.706090	−	0.708123i	$-0.250458\pi$
$421$	28.9755	1.41218	0.706090	+	0.708123i	$0.250458\pi$
$422$	4.57672	0.222791
$423$	27.3846	1.33149
$424$	7.77111	0.377398
$425$	−2.31115	−0.112107
$426$	−0.280722	−0.0136010
$427$	21.7040	1.05033
$428$	3.13228	0.151404
$429$	4.82954	0.233172
$430$	1.83304	0.0883971
$431$	−19.2694	−0.928173	−0.464086	−	0.885790i	$-0.653617\pi$
$431$	−19.2694	−0.928173	−0.464086	+	0.885790i	$0.653617\pi$
$432$	−2.88053	−0.138590
$433$	2.96194	0.142342	0.0711710	−	0.997464i	$-0.477326\pi$
$433$	2.96194	0.142342	0.0711710	+	0.997464i	$0.477326\pi$
$434$	−26.2158	−1.25840
$435$	−1.34727	−0.0645966
$436$	−0.507250	−0.0242929
$437$	12.1057	0.579093
$438$	−2.00826	−0.0959582
$439$	−22.7285	−1.08477	−0.542385	−	0.840130i	$-0.682479\pi$
$439$	−22.7285	−1.08477	−0.542385	+	0.840130i	$0.682479\pi$
$440$	−16.7844	−0.800163
$441$	0.614628	0.0292680
$442$	−15.1733	−0.721721
$443$	−1.07815	−0.0512246	−0.0256123	−	0.999672i	$-0.508154\pi$
$443$	−1.07815	−0.0512246	−0.0256123	+	0.999672i	$0.508154\pi$
$444$	−0.822806	−0.0390487
$445$	7.45495	0.353398
$446$	−25.3910	−1.20230
$447$	1.36196	0.0644186
$448$	−23.1939	−1.09581
$449$	−18.4216	−0.869371	−0.434685	−	0.900582i	$-0.643140\pi$
$449$	−18.4216	−0.869371	−0.434685	+	0.900582i	$0.643140\pi$
$450$	−3.70062	−0.174449
$451$	26.1850	1.23300
$452$	4.64034	0.218263
$453$	0.354717	0.0166660
$454$	7.33096	0.344059
$455$	13.7439	0.644322
$456$	2.95753	0.138499
$457$	41.0275	1.91919	0.959594	−	0.281390i	$-0.0907954\pi$
$457$	41.0275	1.91919	0.959594	+	0.281390i	$0.0907954\pi$
$458$	−9.51418	−0.444568
$459$	−2.29735	−0.107231
$460$	0.934862	0.0435882
$461$	28.2754	1.31692	0.658459	−	0.752617i	$-0.271209\pi$
$461$	28.2754	1.31692	0.658459	+	0.752617i	$0.271209\pi$
$462$	−2.97204	−0.138272
$463$	−30.4928	−1.41712	−0.708560	−	0.705650i	$-0.750655\pi$
$463$	−30.4928	−1.41712	−0.708560	+	0.705650i	$0.750655\pi$
$464$	23.4568	1.08896
$465$	1.34462	0.0623555
$466$	−15.3545	−0.711282
$467$	21.0201	0.972694	0.486347	−	0.873766i	$-0.338329\pi$
$467$	21.0201	0.972694	0.486347	+	0.873766i	$0.338329\pi$
$468$	7.05124	0.325944
$469$	10.6806	0.493183
$470$	11.4709	0.529111
$471$	2.58767	0.119234
$472$	−39.5450	−1.82021
$473$	8.10154	0.372509
$474$	1.65657	0.0760887
$475$	5.82565	0.267299
$476$	−2.70999	−0.124212
$477$	7.57264	0.346728
$478$	−17.2170	−0.787489
$479$	14.2217	0.649805	0.324903	−	0.945747i	$-0.394668\pi$
$479$	14.2217	0.649805	0.324903	+	0.945747i	$0.394668\pi$
$480$	0.414850	0.0189352
$481$	−57.9440	−2.64202
$482$	−29.0974	−1.32535
$483$	0.901448	0.0410173
$484$	−8.67374	−0.394261
$485$	10.1954	0.462951
$486$	−5.52632	−0.250679
$487$	20.0391	0.908058	0.454029	−	0.890987i	$-0.349986\pi$
$487$	20.0391	0.908058	0.454029	+	0.890987i	$0.349986\pi$
$488$	25.3998	1.14979
$489$	3.47161	0.156992
$490$	0.257455	0.0116307
$491$	−17.1588	−0.774368	−0.387184	−	0.922003i	$-0.626552\pi$
$491$	−17.1588	−0.774368	−0.387184	+	0.922003i	$0.626552\pi$
$492$	−0.356316	−0.0160640
$493$	18.7079	0.842560
$494$	38.2470	1.72081
$495$	−16.3557	−0.735134
$496$	−23.4108	−1.05118
$497$	−3.53081	−0.158379
$498$	0.207224	0.00928592
$499$	−21.5451	−0.964493	−0.482247	−	0.876036i	$-0.660179\pi$
$499$	−21.5451	−0.964493	−0.482247	+	0.876036i	$0.660179\pi$
$500$	0.449886	0.0201195
$501$	−1.53687	−0.0686625
$502$	9.47698	0.422978
$503$	29.7181	1.32506	0.662531	−	0.749034i	$-0.269482\pi$
$503$	29.7181	1.32506	0.662531	+	0.749034i	$0.269482\pi$
$504$	−23.6297	−1.05255
$505$	−12.4843	−0.555544
$506$	−14.2365	−0.632891
$507$	−2.46435	−0.109446
$508$	−0.583539	−0.0258904
$509$	27.6673	1.22633	0.613165	−	0.789955i	$-0.289896\pi$
$509$	27.6673	1.22633	0.613165	+	0.789955i	$0.289896\pi$
$510$	−0.478925	−0.0212072
$511$	−25.2591	−1.11740
$512$	−24.9007	−1.10047
$513$	5.79087	0.255673
$514$	−8.77255	−0.386940
$515$	1.08906	0.0479899
$516$	−0.110243	−0.00485317
$517$	50.6980	2.22970
$518$	35.6580	1.56672
$519$	2.17083	0.0952888
$520$	16.0842	0.705338
$521$	4.10603	0.179888	0.0899442	−	0.995947i	$-0.471331\pi$
$521$	4.10603	0.179888	0.0899442	+	0.995947i	$0.471331\pi$
$522$	29.9551	1.31110
$523$	−33.1895	−1.45128	−0.725639	−	0.688076i	$-0.758456\pi$
$523$	−33.1895	−1.45128	−0.725639	+	0.688076i	$0.758456\pi$
$524$	1.99904	0.0873284
$525$	0.433806	0.0189328
$526$	33.4097	1.45673
$527$	−18.6711	−0.813328
$528$	−2.65404	−0.115502
$529$	−18.6819	−0.812258
$530$	3.17203	0.137784
$531$	−38.5351	−1.67228
$532$	6.83100	0.296161
$533$	−25.0926	−1.08688
$534$	1.54484	0.0668519
$535$	6.96239	0.301010
$536$	12.4993	0.539886
$537$	0.116530	0.00502866
$538$	23.7352	1.02330
$539$	1.13788	0.0490121
$540$	0.447200	0.0192444
$541$	−0.492759	−0.0211854	−0.0105927	−	0.999944i	$-0.503372\pi$
$541$	−0.492759	−0.0211854	−0.0105927	+	0.999944i	$0.503372\pi$
$542$	20.0031	0.859207
$543$	−3.28284	−0.140880
$544$	−5.76051	−0.246980
$545$	−1.12751	−0.0482971
$546$	2.84806	0.121886
$547$	7.24431	0.309744	0.154872	−	0.987935i	$-0.450503\pi$
$547$	7.24431	0.309744	0.154872	+	0.987935i	$0.450503\pi$
$548$	−4.54860	−0.194306
$549$	24.7511	1.05635
$550$	−6.85108	−0.292131
$551$	−47.1564	−2.00893
$552$	1.05495	0.0449015
$553$	20.8357	0.886023
$554$	11.0466	0.469323
$555$	−1.82892	−0.0776334
$556$	−7.54101	−0.319810
$557$	−7.27145	−0.308101	−0.154051	−	0.988063i	$-0.549232\pi$
$557$	−7.27145	−0.308101	−0.154051	+	0.988063i	$0.549232\pi$
$558$	−29.8963	−1.26561
$559$	−7.76357	−0.328364
$560$	−7.55285	−0.319166
$561$	−2.11672	−0.0893678
$562$	−38.0254	−1.60401
$563$	−36.9237	−1.55615	−0.778074	−	0.628173i	$-0.783803\pi$
$563$	−36.9237	−1.55615	−0.778074	+	0.628173i	$0.783803\pi$
$564$	−0.689882	−0.0290493
$565$	10.3145	0.433934
$566$	−20.4525	−0.859682
$567$	−22.8096	−0.957913
$568$	−4.13204	−0.173377
$569$	−15.2488	−0.639264	−0.319632	−	0.947542i	$-0.603559\pi$
$569$	−15.2488	−0.639264	−0.319632	+	0.947542i	$0.603559\pi$
$570$	1.20721	0.0505646
$571$	−6.07235	−0.254120	−0.127060	−	0.991895i	$-0.540554\pi$
$571$	−6.07235	−0.254120	−0.127060	+	0.991895i	$0.540554\pi$
$572$	13.0542	0.545824
$573$	−3.84757	−0.160735
$574$	15.4417	0.644524
$575$	2.07800	0.0866585
$576$	−26.4502	−1.10209
$577$	−27.8497	−1.15940	−0.579699	−	0.814831i	$-0.696830\pi$
$577$	−27.8497	−1.15940	−0.579699	+	0.814831i	$0.696830\pi$
$578$	−14.5154	−0.603760
$579$	−0.914997	−0.0380260
$580$	−3.64165	−0.151211
$581$	2.60638	0.108131
$582$	2.11274	0.0875758
$583$	14.0195	0.580628
$584$	−29.5602	−1.22321
$585$	15.6734	0.648015
$586$	−6.55696	−0.270866
$587$	−8.50024	−0.350842	−0.175421	−	0.984493i	$-0.556129\pi$
$587$	−8.50024	−0.350842	−0.175421	+	0.984493i	$0.556129\pi$
$588$	−0.0154839	−0.000638546 0
$589$	47.0639	1.93923
$590$	−16.1416	−0.664538
$591$	2.11434	0.0869722
$592$	31.8427	1.30873
$593$	−17.0379	−0.699665	−0.349832	−	0.936812i	$-0.613761\pi$
$593$	−17.0379	−0.699665	−0.349832	+	0.936812i	$0.613761\pi$
$594$	−6.81017	−0.279425
$595$	−6.02373	−0.246949
$596$	3.68137	0.150795
$597$	−2.67561	−0.109505
$598$	13.6426	0.557888
$599$	−28.8532	−1.17891	−0.589455	−	0.807801i	$-0.700657\pi$
$599$	−28.8532	−1.17891	−0.589455	+	0.807801i	$0.700657\pi$
$600$	0.507675	0.0207257
$601$	−32.0017	−1.30538	−0.652689	−	0.757626i	$-0.726359\pi$
$601$	−32.0017	−1.30538	−0.652689	+	0.757626i	$0.726359\pi$
$602$	4.77760	0.194720
$603$	12.1801	0.496010
$604$	0.958796	0.0390129
$605$	−19.2799	−0.783838
$606$	−2.58705	−0.105092
$607$	−18.7927	−0.762773	−0.381386	−	0.924416i	$-0.624553\pi$
$607$	−18.7927	−0.762773	−0.381386	+	0.924416i	$0.624553\pi$
$608$	14.5204	0.588878
$609$	−3.51149	−0.142293
$610$	10.3677	0.419778
$611$	−48.5831	−1.96546
$612$	−3.09046	−0.124924
$613$	−32.8074	−1.32508	−0.662539	−	0.749028i	$-0.730521\pi$
$613$	−32.8074	−1.32508	−0.662539	+	0.749028i	$0.730521\pi$
$614$	−18.8046	−0.758892
$615$	−0.792015	−0.0319371
$616$	−43.7464	−1.76259
$617$	−18.5402	−0.746402	−0.373201	−	0.927750i	$-0.621740\pi$
$617$	−18.5402	−0.746402	−0.373201	+	0.927750i	$0.621740\pi$
$618$	0.225680	0.00907819
$619$	−15.5977	−0.626926	−0.313463	−	0.949600i	$-0.601489\pi$
$619$	−15.5977	−0.626926	−0.313463	+	0.949600i	$0.601489\pi$
$620$	3.63451	0.145965
$621$	2.06559	0.0828893
$622$	−6.64023	−0.266249
$623$	19.4304	0.778463
$624$	2.54333	0.101814
$625$	1.00000	0.0400000
$626$	−5.71621	−0.228466
$627$	5.33555	0.213081
$628$	6.99445	0.279109
$629$	25.3960	1.01260
$630$	−9.64521	−0.384274
$631$	26.4361	1.05240	0.526202	−	0.850359i	$-0.323615\pi$
$631$	26.4361	1.05240	0.526202	+	0.850359i	$0.323615\pi$
$632$	24.3836	0.969927
$633$	−0.611830	−0.0243181
$634$	34.7700	1.38089
$635$	−1.29708	−0.0514732
$636$	−0.190772	−0.00756462
$637$	−1.09041	−0.0432038
$638$	55.4568	2.19556
$639$	−4.02652	−0.159287
$640$	−6.09447	−0.240905
$641$	24.2326	0.957129	0.478565	−	0.878052i	$-0.341157\pi$
$641$	24.2326	0.957129	0.478565	+	0.878052i	$0.341157\pi$
$642$	1.44277	0.0569417
$643$	8.86847	0.349738	0.174869	−	0.984592i	$-0.444050\pi$
$643$	8.86847	0.349738	0.174869	+	0.984592i	$0.444050\pi$
$644$	2.43660	0.0960157
$645$	−0.245046	−0.00964869
$646$	−16.7631	−0.659535
$647$	3.91936	0.154086	0.0770429	−	0.997028i	$-0.475452\pi$
$647$	3.91936	0.154086	0.0770429	+	0.997028i	$0.475452\pi$
$648$	−26.6936	−1.04862
$649$	−71.3413	−2.80039
$650$	6.56527	0.257511
$651$	3.50460	0.137356
$652$	9.38373	0.367495
$653$	23.4455	0.917494	0.458747	−	0.888567i	$-0.348298\pi$
$653$	23.4455	0.917494	0.458747	+	0.888567i	$0.348298\pi$
$654$	−0.233646	−0.00913630
$655$	4.44343	0.173619
$656$	13.7895	0.538389
$657$	−28.8053	−1.12380
$658$	29.8974	1.16552
$659$	17.7838	0.692758	0.346379	−	0.938095i	$-0.387411\pi$
$659$	17.7838	0.692758	0.346379	+	0.938095i	$0.387411\pi$
$660$	0.412038	0.0160386
$661$	−12.9871	−0.505138	−0.252569	−	0.967579i	$-0.581276\pi$
$661$	−12.9871	−0.505138	−0.252569	+	0.967579i	$0.581276\pi$
$662$	4.07772	0.158485
$663$	2.02841	0.0787771
$664$	3.05020	0.118371
$665$	15.1839	0.588804
$666$	40.6641	1.57570
$667$	−16.8206	−0.651296
$668$	−4.15416	−0.160729
$669$	3.39434	0.131233
$670$	5.10198	0.197107
$671$	45.8226	1.76896
$672$	1.08126	0.0417104
$673$	−20.0636	−0.773394	−0.386697	−	0.922207i	$-0.626384\pi$
$673$	−20.0636	−0.773394	−0.386697	+	0.922207i	$0.626384\pi$
$674$	−22.2899	−0.858573
$675$	0.994030	0.0382602
$676$	−6.66111	−0.256197
$677$	−20.0807	−0.771764	−0.385882	−	0.922548i	$-0.626103\pi$
$677$	−20.0807	−0.771764	−0.385882	+	0.922548i	$0.626103\pi$
$678$	2.13741	0.0820866
$679$	26.5732	1.01978
$680$	−7.04945	−0.270334
$681$	−0.980024	−0.0375546
$682$	−55.3480	−2.11939
$683$	32.8832	1.25824	0.629121	−	0.777307i	$-0.283415\pi$
$683$	32.8832	1.25824	0.629121	+	0.777307i	$0.283415\pi$
$684$	7.79003	0.297859
$685$	−10.1106	−0.386305
$686$	23.3863	0.892892
$687$	1.27188	0.0485254
$688$	4.26642	0.162656
$689$	−13.4346	−0.511819
$690$	0.430611	0.0163931
$691$	5.40542	0.205632	0.102816	−	0.994700i	$-0.467215\pi$
$691$	5.40542	0.205632	0.102816	+	0.994700i	$0.467215\pi$
$692$	5.86773	0.223058
$693$	−42.6292	−1.61935
$694$	−16.3495	−0.620619
$695$	−16.7621	−0.635821
$696$	−4.10943	−0.155768
$697$	10.9977	0.416569
$698$	20.3587	0.770587
$699$	2.05263	0.0776377
$700$	1.17257	0.0443191
$701$	−48.6086	−1.83592	−0.917960	−	0.396672i	$-0.870165\pi$
$701$	−48.6086	−1.83592	−0.917960	+	0.396672i	$0.870165\pi$
$702$	6.52608	0.246311
$703$	−64.0150	−2.41437
$704$	−48.9681	−1.84555
$705$	−1.53346	−0.0577534
$706$	−19.7543	−0.743461
$707$	−32.5388	−1.22375
$708$	0.970788	0.0364844
$709$	−10.4181	−0.391260	−0.195630	−	0.980678i	$-0.562675\pi$
$709$	−10.4181	−0.391260	−0.195630	+	0.980678i	$0.562675\pi$
$710$	−1.68663	−0.0632980
$711$	23.7609	0.891102
$712$	22.7390	0.852182
$713$	16.7876	0.628700
$714$	−1.24826	−0.0467150
$715$	29.0167	1.08516
$716$	0.314980	0.0117714
$717$	2.30162	0.0859557
$718$	−36.2182	−1.35165
$719$	−25.8556	−0.964252	−0.482126	−	0.876102i	$-0.660135\pi$
$719$	−25.8556	−0.964252	−0.482126	+	0.876102i	$0.660135\pi$
$720$	−8.61322	−0.320996
$721$	2.83852	0.105712
$722$	18.5986	0.692167
$723$	3.88982	0.144664
$724$	−8.87348	−0.329780
$725$	−8.09462	−0.300627
$726$	−3.99525	−0.148278
$727$	−36.0203	−1.33592	−0.667960	−	0.744197i	$-0.732832\pi$
$727$	−36.0203	−1.33592	−0.667960	+	0.744197i	$0.732832\pi$
$728$	41.9215	1.55371
$729$	−25.5156	−0.945021
$730$	−12.0659	−0.446581
$731$	3.40266	0.125852
$732$	−0.623538	−0.0230466
$733$	47.2287	1.74443	0.872216	−	0.489121i	$-0.162682\pi$
$733$	47.2287	1.74443	0.872216	+	0.489121i	$0.162682\pi$
$734$	−14.0364	−0.518094
$735$	−0.0344174	−0.00126951
$736$	5.17938	0.190915
$737$	22.5493	0.830616
$738$	17.6096	0.648218
$739$	15.3655	0.565228	0.282614	−	0.959234i	$-0.408798\pi$
$739$	15.3655	0.565228	0.282614	+	0.959234i	$0.408798\pi$
$740$	−4.94356	−0.181729
$741$	−5.11297	−0.187830
$742$	8.26751	0.303510
$743$	2.88804	0.105952	0.0529760	−	0.998596i	$-0.483129\pi$
$743$	2.88804	0.105952	0.0529760	+	0.998596i	$0.483129\pi$
$744$	4.10137	0.150363
$745$	8.18290	0.299798
$746$	43.7219	1.60077
$747$	2.97230	0.108751
$748$	−5.72146	−0.209197
$749$	18.1466	0.663063
$750$	0.207224	0.00756674
$751$	−35.4009	−1.29180	−0.645899	−	0.763423i	$-0.723517\pi$
$751$	−35.4009	−1.29180	−0.645899	+	0.763423i	$0.723517\pi$
$752$	26.6985	0.973595
$753$	−1.26691	−0.0461688
$754$	−53.1434	−1.93537
$755$	2.13120	0.0775622
$756$	1.16557	0.0423915
$757$	−32.0281	−1.16408	−0.582041	−	0.813159i	$-0.697746\pi$
$757$	−32.0281	−1.16408	−0.582041	+	0.813159i	$0.697746\pi$
$758$	27.4571	0.997286
$759$	1.90318	0.0690811
$760$	17.7694	0.644563
$761$	37.2950	1.35194	0.675972	−	0.736927i	$-0.263724\pi$
$761$	37.2950	1.35194	0.675972	+	0.736927i	$0.263724\pi$
$762$	−0.268786	−0.00973711
$763$	−2.93871	−0.106389
$764$	−10.3999	−0.376257
$765$	−6.86942	−0.248364
$766$	21.7881	0.787237
$767$	68.3652	2.46852
$768$	1.69935	0.0613199
$769$	41.1722	1.48471	0.742354	−	0.670008i	$-0.233709\pi$
$769$	41.1722	1.48471	0.742354	+	0.670008i	$0.233709\pi$
$770$	−17.8565	−0.643504
$771$	1.17274	0.0422352
$772$	−2.47323	−0.0890134
$773$	−7.60205	−0.273427	−0.136713	−	0.990611i	$-0.543654\pi$
$773$	−7.60205	−0.273427	−0.136713	+	0.990611i	$0.543654\pi$
$774$	5.44834	0.195837
$775$	8.07873	0.290197
$776$	31.0981	1.11636
$777$	−4.76686	−0.171010
$778$	−11.1150	−0.398492
$779$	−27.7217	−0.993233
$780$	−0.394849	−0.0141379
$781$	−7.45443	−0.266740
$782$	−5.97936	−0.213821
$783$	−8.04629	−0.287551
$784$	0.599230	0.0214011
$785$	15.5472	0.554902
$786$	0.920785	0.0328433
$787$	32.1274	1.14522	0.572610	−	0.819828i	$-0.305931\pi$
$787$	32.1274	1.14522	0.572610	+	0.819828i	$0.305931\pi$
$788$	5.71503	0.203590
$789$	−4.46631	−0.159005
$790$	9.95294	0.354110
$791$	26.8835	0.955866
$792$	−49.8881	−1.77270
$793$	−43.9110	−1.55933
$794$	15.1783	0.538657
$795$	−0.424046	−0.0150394
$796$	−7.23215	−0.256337
$797$	−10.4340	−0.369590	−0.184795	−	0.982777i	$-0.559162\pi$
$797$	−10.4340	−0.369590	−0.184795	+	0.982777i	$0.559162\pi$
$798$	3.14646	0.111383
$799$	21.2933	0.753301
$800$	2.49249	0.0881228
$801$	22.1583	0.782926
$802$	−29.9826	−1.05872
$803$	−53.3282	−1.88191
$804$	−0.306844	−0.0108215
$805$	5.41605	0.190891
$806$	53.0391	1.86822
$807$	−3.17300	−0.111695
$808$	−38.0796	−1.33963
$809$	−6.02723	−0.211906	−0.105953	−	0.994371i	$-0.533789\pi$
$809$	−6.02723	−0.211906	−0.105953	+	0.994371i	$0.533789\pi$
$810$	−10.8959	−0.382842
$811$	41.5723	1.45980	0.729900	−	0.683554i	$-0.239567\pi$
$811$	41.5723	1.45980	0.729900	+	0.683554i	$0.239567\pi$
$812$	−9.49153	−0.333088
$813$	−2.67408	−0.0937839
$814$	75.2829	2.63866
$815$	20.8580	0.730625
$816$	−1.11470	−0.0390224
$817$	−8.57698	−0.300071
$818$	24.7989	0.867072
$819$	40.8508	1.42744
$820$	−2.14081	−0.0747603
$821$	29.8181	1.04066	0.520330	−	0.853965i	$-0.325809\pi$
$821$	29.8181	1.04066	0.520330	+	0.853965i	$0.325809\pi$
$822$	−2.09515	−0.0730767
$823$	−17.6053	−0.613682	−0.306841	−	0.951761i	$-0.599272\pi$
$823$	−17.6053	−0.613682	−0.306841	+	0.951761i	$0.599272\pi$
$824$	3.32186	0.115723
$825$	0.915872	0.0318866
$826$	−42.0711	−1.46384
$827$	4.09065	0.142246	0.0711230	−	0.997468i	$-0.477342\pi$
$827$	4.09065	0.142246	0.0711230	+	0.997468i	$0.477342\pi$
$828$	2.77869	0.0965661
$829$	−11.4055	−0.396130	−0.198065	−	0.980189i	$-0.563466\pi$
$829$	−11.4055	−0.396130	−0.198065	+	0.980189i	$0.563466\pi$
$830$	1.24504	0.0432158
$831$	−1.47674	−0.0512274
$832$	46.9253	1.62684
$833$	0.477912	0.0165587
$834$	−3.47350	−0.120277
$835$	−9.23380	−0.319549
$836$	14.4219	0.498793
$837$	8.03050	0.277575
$838$	29.5254	1.01994
$839$	34.1865	1.18025	0.590124	−	0.807313i	$-0.299079\pi$
$839$	34.1865	1.18025	0.590124	+	0.807313i	$0.299079\pi$
$840$	1.32319	0.0456545
$841$	36.5228	1.25941
$842$	36.0755	1.24324
$843$	5.08335	0.175080
$844$	−1.65377	−0.0569251
$845$	−14.8062	−0.509350
$846$	34.0948	1.17220
$847$	−50.2507	−1.72663
$848$	7.38292	0.253531
$849$	2.73415	0.0938358
$850$	−2.87746	−0.0986961
$851$	−22.8340	−0.782740
$852$	0.101437	0.00347518
$853$	19.2888	0.660435	0.330217	−	0.943905i	$-0.392878\pi$
$853$	19.2888	0.660435	0.330217	+	0.943905i	$0.392878\pi$
$854$	27.0223	0.924684
$855$	17.3156	0.592180
$856$	21.2366	0.725854
$857$	−37.7814	−1.29059	−0.645293	−	0.763935i	$-0.723265\pi$
$857$	−37.7814	−1.29059	−0.645293	+	0.763935i	$0.723265\pi$
$858$	6.01295	0.205279
$859$	0.577130	0.0196914	0.00984570	−	0.999952i	$-0.496866\pi$
$859$	0.577130	0.0196914	0.00984570	+	0.999952i	$0.496866\pi$
$860$	−0.662358	−0.0225862
$861$	−2.06429	−0.0703509
$862$	−23.9911	−0.817139
$863$	21.0115	0.715241	0.357621	−	0.933867i	$-0.383588\pi$
$863$	21.0115	0.715241	0.357621	+	0.933867i	$0.383588\pi$
$864$	2.47761	0.0842899
$865$	13.0427	0.443465
$866$	3.68773	0.125314
$867$	1.94046	0.0659014
$868$	9.47291	0.321531
$869$	43.9893	1.49223
$870$	−1.67740	−0.0568691
$871$	−21.6087	−0.732182
$872$	−3.43912	−0.116463
$873$	30.3039	1.02563
$874$	15.0720	0.509818
$875$	2.60638	0.0881117
$876$	0.725671	0.0245182
$877$	−36.7544	−1.24111	−0.620554	−	0.784163i	$-0.713092\pi$
$877$	−36.7544	−1.24111	−0.620554	+	0.784163i	$0.713092\pi$
$878$	−28.2977	−0.955003
$879$	0.876554	0.0295654
$880$	−15.9459	−0.537537
$881$	6.85472	0.230941	0.115471	−	0.993311i	$-0.463162\pi$
$881$	6.85472	0.230941	0.115471	+	0.993311i	$0.463162\pi$
$882$	0.765234	0.0257668
$883$	−22.2929	−0.750214	−0.375107	−	0.926981i	$-0.622394\pi$
$883$	−22.2929	−0.750214	−0.375107	+	0.926981i	$0.622394\pi$
$884$	5.48279	0.184406
$885$	2.15785	0.0725354
$886$	−1.34234	−0.0450967
$887$	44.7244	1.50170	0.750849	−	0.660473i	$-0.229644\pi$
$887$	44.7244	1.50170	0.750849	+	0.660473i	$0.229644\pi$
$888$	−5.57857	−0.187205
$889$	−3.38069	−0.113385
$890$	9.28167	0.311122
$891$	−48.1567	−1.61331
$892$	9.17488	0.307198
$893$	−53.6733	−1.79611
$894$	1.69569	0.0567124
$895$	0.700134	0.0234029
$896$	−15.8845	−0.530664
$897$	−1.82379	−0.0608944
$898$	−22.9356	−0.765371
$899$	−65.3942	−2.18102
$900$	1.33719	0.0445732
$901$	5.88821	0.196165
$902$	32.6012	1.08550
$903$	−0.638684	−0.0212541
$904$	31.4612	1.04638
$905$	−19.7238	−0.655643
$906$	0.441635	0.0146723
$907$	25.3486	0.841686	0.420843	−	0.907134i	$-0.361734\pi$
$907$	25.3486	0.841686	0.420843	+	0.907134i	$0.361734\pi$
$908$	−2.64900	−0.0879100
$909$	−37.1071	−1.23076
$910$	17.1116	0.567244
$911$	−9.88777	−0.327596	−0.163798	−	0.986494i	$-0.552375\pi$
$911$	−9.88777	−0.327596	−0.163798	+	0.986494i	$0.552375\pi$
$912$	2.80980	0.0930417
$913$	5.50271	0.182113
$914$	51.0807	1.68960
$915$	−1.38599	−0.0458195
$916$	3.43789	0.113591
$917$	11.5813	0.382447
$918$	−2.86028	−0.0944034
$919$	−44.6432	−1.47264	−0.736321	−	0.676632i	$-0.763439\pi$
$919$	−44.6432	−1.47264	−0.736321	+	0.676632i	$0.763439\pi$
$920$	6.33830	0.208968
$921$	2.51385	0.0828343
$922$	35.2039	1.15938
$923$	7.14345	0.235130
$924$	1.07393	0.0353296
$925$	−10.9885	−0.361299
$926$	−37.9646	−1.24759
$927$	3.23702	0.106318
$928$	−20.1757	−0.662301
$929$	−26.7919	−0.879013	−0.439507	−	0.898239i	$-0.644847\pi$
$929$	−26.7919	−0.879013	−0.439507	+	0.898239i	$0.644847\pi$
$930$	1.67411	0.0548961
$931$	−1.20466	−0.0394811
$932$	5.54825	0.181739
$933$	0.887685	0.0290615
$934$	26.1708	0.856334
$935$	−12.7176	−0.415910
$936$	47.8069	1.56262
$937$	11.5270	0.376571	0.188285	−	0.982114i	$-0.439707\pi$
$937$	11.5270	0.376571	0.188285	+	0.982114i	$0.439707\pi$
$938$	13.2977	0.434185
$939$	0.764159	0.0249374
$940$	−4.14492	−0.135193
$941$	−7.24371	−0.236138	−0.118069	−	0.993005i	$-0.537670\pi$
$941$	−7.24371	−0.236138	−0.118069	+	0.993005i	$0.537670\pi$
$942$	3.22174	0.104970
$943$	−9.88827	−0.322006
$944$	−37.5696	−1.22279
$945$	2.59082	0.0842793
$946$	10.0867	0.327947
$947$	−2.92315	−0.0949898	−0.0474949	−	0.998871i	$-0.515124\pi$
$947$	−2.92315	−0.0949898	−0.0474949	+	0.998871i	$0.515124\pi$
$948$	−0.598591	−0.0194413
$949$	51.1035	1.65889
$950$	7.25314	0.235323
$951$	−4.64816	−0.150727
$952$	−18.3736	−0.595490
$953$	−41.8295	−1.35499	−0.677494	−	0.735528i	$-0.736934\pi$
$953$	−41.8295	−1.35499	−0.677494	+	0.735528i	$0.736934\pi$
$954$	9.42821	0.305250
$955$	−23.1169	−0.748044
$956$	6.22127	0.201210
$957$	−7.41363	−0.239649
$958$	17.7065	0.572071
$959$	−26.3519	−0.850949
$960$	1.48113	0.0478034
$961$	34.2659	1.10535
$962$	−72.1423	−2.32596
$963$	20.6943	0.666864
$964$	10.5142	0.338638
$965$	−5.49746	−0.176969
$966$	1.12233	0.0361105
$967$	39.4218	1.26772	0.633859	−	0.773448i	$-0.281470\pi$
$967$	39.4218	1.26772	0.633859	+	0.773448i	$0.281470\pi$
$968$	−58.8074	−1.89014
$969$	2.24094	0.0719893
$970$	12.6937	0.407570
$971$	21.9651	0.704892	0.352446	−	0.935832i	$-0.385350\pi$
$971$	21.9651	0.704892	0.352446	+	0.935832i	$0.385350\pi$
$972$	1.99690	0.0640506
$973$	−43.6883	−1.40058
$974$	24.9494	0.799430
$975$	−0.877665	−0.0281078
$976$	24.1310	0.772415
$977$	34.3780	1.09985	0.549924	−	0.835214i	$-0.314657\pi$
$977$	34.3780	1.09985	0.549924	+	0.835214i	$0.314657\pi$
$978$	4.32228	0.138211
$979$	41.0224	1.31108
$980$	−0.0930299	−0.00297173
$981$	−3.35129	−0.106998
$982$	−21.3634	−0.681733
$983$	37.3380	1.19090	0.595448	−	0.803394i	$-0.296975\pi$
$983$	37.3380	1.19090	0.595448	+	0.803394i	$0.296975\pi$
$984$	−2.41580	−0.0770129
$985$	12.7033	0.404761
$986$	23.2920	0.741767
$987$	−3.99678	−0.127219
$988$	−13.8203	−0.439683
$989$	−3.05939	−0.0972831
$990$	−20.3634	−0.647193
$991$	−23.8047	−0.756180	−0.378090	−	0.925769i	$-0.623419\pi$
$991$	−23.8047	−0.756180	−0.378090	+	0.925769i	$0.623419\pi$
$992$	20.1361	0.639323
$993$	−0.545122	−0.0172989
$994$	−4.39599	−0.139432
$995$	−16.0755	−0.509628
$996$	−0.0748791	−0.00237263
$997$	20.3102	0.643232	0.321616	−	0.946870i	$-0.395774\pi$
$997$	20.3102	0.643232	0.321616	+	0.946870i	$0.395774\pi$
$998$	−26.8245	−0.849114
$999$	−10.9229	−0.345584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	415.2.a.d.1.6	✓	7
3.2	odd	2		3735.2.a.o.1.2		7
4.3	odd	2		6640.2.a.be.1.4		7
5.4	even	2		2075.2.a.h.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.2.a.d.1.6	✓	7	1.1	even	1	trivial
2075.2.a.h.1.2		7	5.4	even	2
3735.2.a.o.1.2		7	3.2	odd	2
6640.2.a.be.1.4		7	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 415 = 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	415.a (trivial)

\(f(q)\)	\(=\)	\(q+1.24504 q^{2} -0.166440 q^{3} -0.449886 q^{4} -1.00000 q^{5} -0.207224 q^{6} -2.60638 q^{7} -3.05020 q^{8} -2.97230 q^{9} -1.24504 q^{10} -5.50271 q^{11} +0.0748791 q^{12} +5.27316 q^{13} -3.24504 q^{14} +0.166440 q^{15} -2.89783 q^{16} -2.31115 q^{17} -3.70062 q^{18} +5.82565 q^{19} +0.449886 q^{20} +0.433806 q^{21} -6.85108 q^{22} +2.07800 q^{23} +0.507675 q^{24} +1.00000 q^{25} +6.56527 q^{26} +0.994030 q^{27} +1.17257 q^{28} -8.09462 q^{29} +0.207224 q^{30} +8.07873 q^{31} +2.49249 q^{32} +0.915872 q^{33} -2.87746 q^{34} +2.60638 q^{35} +1.33719 q^{36} -10.9885 q^{37} +7.25314 q^{38} -0.877665 q^{39} +3.05020 q^{40} -4.75856 q^{41} +0.540104 q^{42} -1.47228 q^{43} +2.47559 q^{44} +2.97230 q^{45} +2.58718 q^{46} -9.21328 q^{47} +0.482315 q^{48} -0.206786 q^{49} +1.24504 q^{50} +0.384668 q^{51} -2.37232 q^{52} -2.54774 q^{53} +1.23760 q^{54} +5.50271 q^{55} +7.94997 q^{56} -0.969621 q^{57} -10.0781 q^{58} +12.9647 q^{59} -0.0748791 q^{60} -8.32727 q^{61} +10.0583 q^{62} +7.74694 q^{63} +8.89890 q^{64} -5.27316 q^{65} +1.14029 q^{66} -4.09786 q^{67} +1.03975 q^{68} -0.345862 q^{69} +3.24504 q^{70} +1.35468 q^{71} +9.06609 q^{72} +9.69125 q^{73} -13.6810 q^{74} -0.166440 q^{75} -2.62088 q^{76} +14.3422 q^{77} -1.09272 q^{78} -7.99410 q^{79} +2.89783 q^{80} +8.75145 q^{81} -5.92458 q^{82} -1.00000 q^{83} -0.195163 q^{84} +2.31115 q^{85} -1.83304 q^{86} +1.34727 q^{87} +16.7844 q^{88} -7.45495 q^{89} +3.70062 q^{90} -13.7439 q^{91} -0.934862 q^{92} -1.34462 q^{93} -11.4709 q^{94} -5.82565 q^{95} -0.414850 q^{96} -10.1954 q^{97} -0.257455 q^{98} +16.3557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{2} - 5 q^{3} + 7 q^{4} - 7 q^{5} - 6 q^{7} - 12 q^{8} + 2 q^{9} + 3 q^{10} - 2 q^{11} - 11 q^{12} - 5 q^{13} - 11 q^{14} + 5 q^{15} + 7 q^{16} - 25 q^{17} - 4 q^{18} + 6 q^{19} - 7 q^{20} + 2 q^{21}+ \cdots + 38 q^{99}+O(q^{100}) \) 7 * q - 3 * q^2 - 5 * q^3 + 7 * q^4 - 7 * q^5 - 6 * q^7 - 12 * q^8 + 2 * q^9 + 3 * q^10 - 2 * q^11 - 11 * q^12 - 5 * q^13 - 11 * q^14 + 5 * q^15 + 7 * q^16 - 25 * q^17 - 4 * q^18 + 6 * q^19 - 7 * q^20 + 2 * q^21 - 2 * q^22 - 9 * q^23 - q^24 + 7 * q^25 - 5 * q^26 - 17 * q^27 - 3 * q^28 - 2 * q^29 + q^31 - 17 * q^32 - 17 * q^33 + q^34 + 6 * q^35 + 21 * q^36 - 13 * q^37 + q^38 + 12 * q^40 - 15 * q^41 + 10 * q^42 - 5 * q^43 - 12 * q^44 - 2 * q^45 + 27 * q^46 - 12 * q^47 + 4 * q^48 - 15 * q^49 - 3 * q^50 + 23 * q^51 - 20 * q^52 - 31 * q^53 + 5 * q^54 + 2 * q^55 + 26 * q^56 - 9 * q^57 + 13 * q^58 - 14 * q^59 + 11 * q^60 + 7 * q^61 - 14 * q^62 + 4 * q^63 + 10 * q^64 + 5 * q^65 + 46 * q^66 + 10 * q^67 + 4 * q^68 - 19 * q^69 + 11 * q^70 + 9 * q^71 + 21 * q^72 - 15 * q^73 - 19 * q^74 - 5 * q^75 + 61 * q^76 - 9 * q^77 + 67 * q^78 + 6 * q^79 - 7 * q^80 + 3 * q^81 + 12 * q^82 - 7 * q^83 + 17 * q^84 + 25 * q^85 + 42 * q^86 + 23 * q^87 + 20 * q^88 - 37 * q^89 + 4 * q^90 + 9 * q^91 - 13 * q^92 + 2 * q^93 + 22 * q^94 - 6 * q^95 + 59 * q^96 - 20 * q^97 + 44 * q^98 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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