LMFDB - Embedded newform 275.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,2,Mod(1,275)] 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("275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.19588605559$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.41421 q^{2} +2.82843 q^{3} +3.82843 q^{4} -6.82843 q^{6} +2.00000 q^{7} -4.41421 q^{8} +5.00000 q^{9} +1.00000 q^{11} +10.8284 q^{12} +1.17157 q^{13} -4.82843 q^{14} +3.00000 q^{16} -6.82843 q^{17} -12.0711 q^{18} +5.65685 q^{21} -2.41421 q^{22} +2.82843 q^{23} -12.4853 q^{24} -2.82843 q^{26} +5.65685 q^{27} +7.65685 q^{28} -3.65685 q^{29} +1.58579 q^{32} +2.82843 q^{33} +16.4853 q^{34} +19.1421 q^{36} +7.65685 q^{37} +3.31371 q^{39} +6.00000 q^{41} -13.6569 q^{42} +6.00000 q^{43} +3.82843 q^{44} -6.82843 q^{46} -2.82843 q^{47} +8.48528 q^{48} -3.00000 q^{49} -19.3137 q^{51} +4.48528 q^{52} -11.6569 q^{53} -13.6569 q^{54} -8.82843 q^{56} +8.82843 q^{58} +1.65685 q^{59} -9.31371 q^{61} +10.0000 q^{63} -9.82843 q^{64} -6.82843 q^{66} -12.4853 q^{67} -26.1421 q^{68} +8.00000 q^{69} +11.3137 q^{71} -22.0711 q^{72} +1.17157 q^{73} -18.4853 q^{74} +2.00000 q^{77} -8.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} -14.4853 q^{82} +6.00000 q^{83} +21.6569 q^{84} -14.4853 q^{86} -10.3431 q^{87} -4.41421 q^{88} -13.3137 q^{89} +2.34315 q^{91} +10.8284 q^{92} +6.82843 q^{94} +4.48528 q^{96} -3.65685 q^{97} +7.24264 q^{98} +5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 8 q^{6} + 4 q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{11} + 16 q^{12} + 8 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 10 q^{18} - 2 q^{22} - 8 q^{24} + 4 q^{28} + 4 q^{29} + 6 q^{32} + 16 q^{34}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 8 * q^6 + 4 * q^7 - 6 * q^8 + 10 * q^9 + 2 * q^11 + 16 * q^12 + 8 * q^13 - 4 * q^14 + 6 * q^16 - 8 * q^17 - 10 * q^18 - 2 * q^22 - 8 * q^24 + 4 * q^28 + 4 * q^29 + 6 * q^32 + 16 * q^34 + 10 * q^36 + 4 * q^37 - 16 * q^39 + 12 * q^41 - 16 * q^42 + 12 * q^43 + 2 * q^44 - 8 * q^46 - 6 * q^49 - 16 * q^51 - 8 * q^52 - 12 * q^53 - 16 * q^54 - 12 * q^56 + 12 * q^58 - 8 * q^59 + 4 * q^61 + 20 * q^63 - 14 * q^64 - 8 * q^66 - 8 * q^67 - 24 * q^68 + 16 * q^69 - 30 * q^72 + 8 * q^73 - 20 * q^74 + 4 * q^77 - 16 * q^78 + 8 * q^79 + 2 * q^81 - 12 * q^82 + 12 * q^83 + 32 * q^84 - 12 * q^86 - 32 * q^87 - 6 * q^88 - 4 * q^89 + 16 * q^91 + 16 * q^92 + 8 * q^94 - 8 * q^96 + 4 * q^97 + 6 * q^98 + 10 * 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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	2.82843	1.63299	0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843	1.63299	0.816497	+	0.577350i	$0.195913\pi$
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	−6.82843	−2.78769
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−4.41421	−1.56066
$9$	5.00000	1.66667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	10.8284	3.12590
$13$	1.17157	0.324936	0.162468	−	0.986714i	$-0.448055\pi$
$13$	1.17157	0.324936	0.162468	+	0.986714i	$0.448055\pi$
$14$	−4.82843	−1.29045
$15$	0	0
$16$	3.00000	0.750000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	−12.0711	−2.84518
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	5.65685	1.23443
$22$	−2.41421	−0.514712
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	−12.4853	−2.54855
$25$	0	0
$26$	−2.82843	−0.554700
$27$	5.65685	1.08866
$28$	7.65685	1.44701
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.58579	0.280330
$33$	2.82843	0.492366
$34$	16.4853	2.82720
$35$	0	0
$36$	19.1421	3.19036
$37$	7.65685	1.25878	0.629390	−	0.777090i	$-0.283305\pi$
$37$	7.65685	1.25878	0.629390	+	0.777090i	$0.283305\pi$
$38$	0	0
$39$	3.31371	0.530618
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−13.6569	−2.10730
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	3.82843	0.577157
$45$	0	0
$46$	−6.82843	−1.00680
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	8.48528	1.22474
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−19.3137	−2.70446
$52$	4.48528	0.621997
$53$	−11.6569	−1.60119	−0.800596	−	0.599204i	$-0.795484\pi$
$53$	−11.6569	−1.60119	−0.800596	+	0.599204i	$0.795484\pi$
$54$	−13.6569	−1.85846
$55$	0	0
$56$	−8.82843	−1.17975
$57$	0	0
$58$	8.82843	1.15923
$59$	1.65685	0.215704	0.107852	−	0.994167i	$-0.465603\pi$
$59$	1.65685	0.215704	0.107852	+	0.994167i	$0.465603\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	0	0
$63$	10.0000	1.25988
$64$	−9.82843	−1.22855
$65$	0	0
$66$	−6.82843	−0.840521
$67$	−12.4853	−1.52532	−0.762660	−	0.646800i	$-0.776107\pi$
$67$	−12.4853	−1.52532	−0.762660	+	0.646800i	$0.776107\pi$
$68$	−26.1421	−3.17020
$69$	8.00000	0.963087
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	−22.0711	−2.60110
$73$	1.17157	0.137122	0.0685611	−	0.997647i	$-0.478159\pi$
$73$	1.17157	0.137122	0.0685611	+	0.997647i	$0.478159\pi$
$74$	−18.4853	−2.14887
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	−8.00000	−0.905822
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−14.4853	−1.59963
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	21.6569	2.36296
$85$	0	0
$86$	−14.4853	−1.56199
$87$	−10.3431	−1.10890
$88$	−4.41421	−0.470557
$89$	−13.3137	−1.41125	−0.705625	−	0.708585i	$-0.749334\pi$
$89$	−13.3137	−1.41125	−0.705625	+	0.708585i	$0.749334\pi$
$90$	0	0
$91$	2.34315	0.245628
$92$	10.8284	1.12894
$93$	0	0
$94$	6.82843	0.704298
$95$	0	0
$96$	4.48528	0.457777
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	7.24264	0.731617
$99$	5.00000	0.502519
$100$	0	0
$101$	9.31371	0.926749	0.463374	−	0.886163i	$-0.346639\pi$
$101$	9.31371	0.926749	0.463374	+	0.886163i	$0.346639\pi$
$102$	46.6274	4.61680
$103$	−6.82843	−0.672825	−0.336412	−	0.941715i	$-0.609214\pi$
$103$	−6.82843	−0.672825	−0.336412	+	0.941715i	$0.609214\pi$
$104$	−5.17157	−0.507114
$105$	0	0
$106$	28.1421	2.73341
$107$	−7.65685	−0.740216	−0.370108	−	0.928989i	$-0.620679\pi$
$107$	−7.65685	−0.740216	−0.370108	+	0.928989i	$0.620679\pi$
$108$	21.6569	2.08393
$109$	−7.65685	−0.733394	−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685	−0.733394	−0.366697	+	0.930341i	$0.619511\pi$
$110$	0	0
$111$	21.6569	2.05558
$112$	6.00000	0.566947
$113$	−19.6569	−1.84916	−0.924581	−	0.380986i	$-0.875584\pi$
$113$	−19.6569	−1.84916	−0.924581	+	0.380986i	$0.875584\pi$
$114$	0	0
$115$	0	0
$116$	−14.0000	−1.29987
$117$	5.85786	0.541560
$118$	−4.00000	−0.368230
$119$	−13.6569	−1.25192
$120$	0	0
$121$	1.00000	0.0909091
$122$	22.4853	2.03572
$123$	16.9706	1.53018
$124$	0	0
$125$	0	0
$126$	−24.1421	−2.15075
$127$	−4.34315	−0.385392	−0.192696	−	0.981259i	$-0.561723\pi$
$127$	−4.34315	−0.385392	−0.192696	+	0.981259i	$0.561723\pi$
$128$	20.5563	1.81694
$129$	16.9706	1.49417
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	10.8284	0.942494
$133$	0	0
$134$	30.1421	2.60388
$135$	0	0
$136$	30.1421	2.58467
$137$	10.9706	0.937278	0.468639	−	0.883390i	$-0.344744\pi$
$137$	10.9706	0.937278	0.468639	+	0.883390i	$0.344744\pi$
$138$	−19.3137	−1.64409
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−27.3137	−2.29212
$143$	1.17157	0.0979718
$144$	15.0000	1.25000
$145$	0	0
$146$	−2.82843	−0.234082
$147$	−8.48528	−0.699854
$148$	29.3137	2.40957
$149$	0.343146	0.0281116	0.0140558	−	0.999901i	$-0.495526\pi$
$149$	0.343146	0.0281116	0.0140558	+	0.999901i	$0.495526\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−34.1421	−2.76023
$154$	−4.82843	−0.389086
$155$	0	0
$156$	12.6863	1.01572
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−9.65685	−0.768258
$159$	−32.9706	−2.61474
$160$	0	0
$161$	5.65685	0.445823
$162$	−2.41421	−0.189679
$163$	−16.4853	−1.29123	−0.645613	−	0.763664i	$-0.723398\pi$
$163$	−16.4853	−1.29123	−0.645613	+	0.763664i	$0.723398\pi$
$164$	22.9706	1.79370
$165$	0	0
$166$	−14.4853	−1.12428
$167$	22.9706	1.77752	0.888758	−	0.458377i	$-0.151569\pi$
$167$	22.9706	1.77752	0.888758	+	0.458377i	$0.151569\pi$
$168$	−24.9706	−1.92652
$169$	−11.6274	−0.894417
$170$	0	0
$171$	0	0
$172$	22.9706	1.75149
$173$	22.1421	1.68344	0.841718	−	0.539918i	$-0.181545\pi$
$173$	22.1421	1.68344	0.841718	+	0.539918i	$0.181545\pi$
$174$	24.9706	1.89301
$175$	0	0
$176$	3.00000	0.226134
$177$	4.68629	0.352243
$178$	32.1421	2.40915
$179$	9.65685	0.721787	0.360894	−	0.932607i	$-0.382472\pi$
$179$	9.65685	0.721787	0.360894	+	0.932607i	$0.382472\pi$
$180$	0	0
$181$	21.3137	1.58424	0.792118	−	0.610368i	$-0.208979\pi$
$181$	21.3137	1.58424	0.792118	+	0.610368i	$0.208979\pi$
$182$	−5.65685	−0.419314
$183$	−26.3431	−1.94734
$184$	−12.4853	−0.920427
$185$	0	0
$186$	0	0
$187$	−6.82843	−0.499344
$188$	−10.8284	−0.789744
$189$	11.3137	0.822951
$190$	0	0
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	−27.7990	−2.00622
$193$	1.17157	0.0843317	0.0421658	−	0.999111i	$-0.486574\pi$
$193$	1.17157	0.0843317	0.0421658	+	0.999111i	$0.486574\pi$
$194$	8.82843	0.633844
$195$	0	0
$196$	−11.4853	−0.820377
$197$	10.8284	0.771493	0.385747	−	0.922605i	$-0.373944\pi$
$197$	10.8284	0.771493	0.385747	+	0.922605i	$0.373944\pi$
$198$	−12.0711	−0.857853
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	0	0
$201$	−35.3137	−2.49084
$202$	−22.4853	−1.58206
$203$	−7.31371	−0.513322
$204$	−73.9411	−5.17691
$205$	0	0
$206$	16.4853	1.14858
$207$	14.1421	0.982946
$208$	3.51472	0.243702
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−44.6274	−3.06502
$213$	32.0000	2.19260
$214$	18.4853	1.26363
$215$	0	0
$216$	−24.9706	−1.69903
$217$	0	0
$218$	18.4853	1.25198
$219$	3.31371	0.223920
$220$	0	0
$221$	−8.00000	−0.538138
$222$	−52.2843	−3.50909
$223$	10.8284	0.725125	0.362563	−	0.931959i	$-0.381902\pi$
$223$	10.8284	0.725125	0.362563	+	0.931959i	$0.381902\pi$
$224$	3.17157	0.211910
$225$	0	0
$226$	47.4558	3.15672
$227$	−25.3137	−1.68013	−0.840065	−	0.542486i	$-0.817483\pi$
$227$	−25.3137	−1.68013	−0.840065	+	0.542486i	$0.817483\pi$
$228$	0	0
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	16.1421	1.05978
$233$	6.14214	0.402385	0.201192	−	0.979552i	$-0.435518\pi$
$233$	6.14214	0.402385	0.201192	+	0.979552i	$0.435518\pi$
$234$	−14.1421	−0.924500
$235$	0	0
$236$	6.34315	0.412904
$237$	11.3137	0.734904
$238$	32.9706	2.13716
$239$	−23.3137	−1.50804	−0.754019	−	0.656852i	$-0.771887\pi$
$239$	−23.3137	−1.50804	−0.754019	+	0.656852i	$0.771887\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−2.41421	−0.155192
$243$	−14.1421	−0.907218
$244$	−35.6569	−2.28270
$245$	0	0
$246$	−40.9706	−2.61219
$247$	0	0
$248$	0	0
$249$	16.9706	1.07547
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	38.2843	2.41168
$253$	2.82843	0.177822
$254$	10.4853	0.657905
$255$	0	0
$256$	−29.9706	−1.87316
$257$	9.31371	0.580973	0.290487	−	0.956879i	$-0.406183\pi$
$257$	9.31371	0.580973	0.290487	+	0.956879i	$0.406183\pi$
$258$	−40.9706	−2.55072
$259$	15.3137	0.951548
$260$	0	0
$261$	−18.2843	−1.13177
$262$	27.3137	1.68745
$263$	10.9706	0.676474	0.338237	−	0.941061i	$-0.390169\pi$
$263$	10.9706	0.676474	0.338237	+	0.941061i	$0.390169\pi$
$264$	−12.4853	−0.768416
$265$	0	0
$266$	0	0
$267$	−37.6569	−2.30456
$268$	−47.7990	−2.91979
$269$	17.3137	1.05564	0.527818	−	0.849358i	$-0.323010\pi$
$269$	17.3137	1.05564	0.527818	+	0.849358i	$0.323010\pi$
$270$	0	0
$271$	7.31371	0.444276	0.222138	−	0.975015i	$-0.428696\pi$
$271$	7.31371	0.444276	0.222138	+	0.975015i	$0.428696\pi$
$272$	−20.4853	−1.24210
$273$	6.62742	0.401110
$274$	−26.4853	−1.60003
$275$	0	0
$276$	30.6274	1.84355
$277$	−6.82843	−0.410280	−0.205140	−	0.978733i	$-0.565765\pi$
$277$	−6.82843	−0.410280	−0.205140	+	0.978733i	$0.565765\pi$
$278$	9.65685	0.579180
$279$	0	0
$280$	0	0
$281$	17.3137	1.03285	0.516425	−	0.856333i	$-0.327263\pi$
$281$	17.3137	1.03285	0.516425	+	0.856333i	$0.327263\pi$
$282$	19.3137	1.15011
$283$	−32.6274	−1.93950	−0.969749	−	0.244103i	$-0.921507\pi$
$283$	−32.6274	−1.93950	−0.969749	+	0.244103i	$0.921507\pi$
$284$	43.3137	2.57020
$285$	0	0
$286$	−2.82843	−0.167248
$287$	12.0000	0.708338
$288$	7.92893	0.467217
$289$	29.6274	1.74279
$290$	0	0
$291$	−10.3431	−0.606326
$292$	4.48528	0.262481
$293$	9.17157	0.535809	0.267905	−	0.963445i	$-0.413669\pi$
$293$	9.17157	0.535809	0.267905	+	0.963445i	$0.413669\pi$
$294$	20.4853	1.19473
$295$	0	0
$296$	−33.7990	−1.96453
$297$	5.65685	0.328244
$298$	−0.828427	−0.0479895
$299$	3.31371	0.191637
$300$	0	0
$301$	12.0000	0.691669
$302$	28.9706	1.66707
$303$	26.3431	1.51337
$304$	0	0
$305$	0	0
$306$	82.4264	4.71200
$307$	16.3431	0.932753	0.466376	−	0.884586i	$-0.345559\pi$
$307$	16.3431	0.932753	0.466376	+	0.884586i	$0.345559\pi$
$308$	7.65685	0.436290
$309$	−19.3137	−1.09872
$310$	0	0
$311$	4.68629	0.265735	0.132868	−	0.991134i	$-0.457581\pi$
$311$	4.68629	0.265735	0.132868	+	0.991134i	$0.457581\pi$
$312$	−14.6274	−0.828114
$313$	1.31371	0.0742552	0.0371276	−	0.999311i	$-0.488179\pi$
$313$	1.31371	0.0742552	0.0371276	+	0.999311i	$0.488179\pi$
$314$	−33.7990	−1.90739
$315$	0	0
$316$	15.3137	0.861463
$317$	1.31371	0.0737852	0.0368926	−	0.999319i	$-0.488254\pi$
$317$	1.31371	0.0737852	0.0368926	+	0.999319i	$0.488254\pi$
$318$	79.5980	4.46363
$319$	−3.65685	−0.204745
$320$	0	0
$321$	−21.6569	−1.20877
$322$	−13.6569	−0.761067
$323$	0	0
$324$	3.82843	0.212690
$325$	0	0
$326$	39.7990	2.20426
$327$	−21.6569	−1.19763
$328$	−26.4853	−1.46241
$329$	−5.65685	−0.311872
$330$	0	0
$331$	−7.31371	−0.401998	−0.200999	−	0.979591i	$-0.564419\pi$
$331$	−7.31371	−0.401998	−0.200999	+	0.979591i	$0.564419\pi$
$332$	22.9706	1.26067
$333$	38.2843	2.09797
$334$	−55.4558	−3.03441
$335$	0	0
$336$	16.9706	0.925820
$337$	20.4853	1.11590	0.557952	−	0.829873i	$-0.311587\pi$
$337$	20.4853	1.11590	0.557952	+	0.829873i	$0.311587\pi$
$338$	28.0711	1.52686
$339$	−55.5980	−3.01967
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−26.4853	−1.42799
$345$	0	0
$346$	−53.4558	−2.87380
$347$	−10.9706	−0.588931	−0.294465	−	0.955662i	$-0.595142\pi$
$347$	−10.9706	−0.588931	−0.294465	+	0.955662i	$0.595142\pi$
$348$	−39.5980	−2.12267
$349$	26.9706	1.44370	0.721851	−	0.692049i	$-0.243292\pi$
$349$	26.9706	1.44370	0.721851	+	0.692049i	$0.243292\pi$
$350$	0	0
$351$	6.62742	0.353745
$352$	1.58579	0.0845227
$353$	−21.3137	−1.13441	−0.567207	−	0.823575i	$-0.691976\pi$
$353$	−21.3137	−1.13441	−0.567207	+	0.823575i	$0.691976\pi$
$354$	−11.3137	−0.601317
$355$	0	0
$356$	−50.9706	−2.70143
$357$	−38.6274	−2.04438
$358$	−23.3137	−1.23217
$359$	0.686292	0.0362211	0.0181105	−	0.999836i	$-0.494235\pi$
$359$	0.686292	0.0362211	0.0181105	+	0.999836i	$0.494235\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−51.4558	−2.70446
$363$	2.82843	0.148454
$364$	8.97056	0.470185
$365$	0	0
$366$	63.5980	3.32432
$367$	8.48528	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$368$	8.48528	0.442326
$369$	30.0000	1.56174
$370$	0	0
$371$	−23.3137	−1.21039
$372$	0	0
$373$	−35.7990	−1.85360	−0.926801	−	0.375554i	$-0.877453\pi$
$373$	−35.7990	−1.85360	−0.926801	+	0.375554i	$0.877453\pi$
$374$	16.4853	0.852434
$375$	0	0
$376$	12.4853	0.643879
$377$	−4.28427	−0.220651
$378$	−27.3137	−1.40487
$379$	33.6569	1.72884	0.864418	−	0.502773i	$-0.167687\pi$
$379$	33.6569	1.72884	0.864418	+	0.502773i	$0.167687\pi$
$380$	0	0
$381$	−12.2843	−0.629342
$382$	−8.00000	−0.409316
$383$	5.85786	0.299323	0.149661	−	0.988737i	$-0.452182\pi$
$383$	5.85786	0.299323	0.149661	+	0.988737i	$0.452182\pi$
$384$	58.1421	2.96705
$385$	0	0
$386$	−2.82843	−0.143963
$387$	30.0000	1.52499
$388$	−14.0000	−0.710742
$389$	20.6274	1.04585	0.522926	−	0.852378i	$-0.324840\pi$
$389$	20.6274	1.04585	0.522926	+	0.852378i	$0.324840\pi$
$390$	0	0
$391$	−19.3137	−0.976736
$392$	13.2426	0.668854
$393$	−32.0000	−1.61419
$394$	−26.1421	−1.31702
$395$	0	0
$396$	19.1421	0.961929
$397$	9.31371	0.467442	0.233721	−	0.972304i	$-0.424910\pi$
$397$	9.31371	0.467442	0.233721	+	0.972304i	$0.424910\pi$
$398$	−24.9706	−1.25166
$399$	0	0
$400$	0	0
$401$	−5.31371	−0.265354	−0.132677	−	0.991159i	$-0.542357\pi$
$401$	−5.31371	−0.265354	−0.132677	+	0.991159i	$0.542357\pi$
$402$	85.2548	4.25212
$403$	0	0
$404$	35.6569	1.77399
$405$	0	0
$406$	17.6569	0.876295
$407$	7.65685	0.379536
$408$	85.2548	4.22074
$409$	1.02944	0.0509024	0.0254512	−	0.999676i	$-0.491898\pi$
$409$	1.02944	0.0509024	0.0254512	+	0.999676i	$0.491898\pi$
$410$	0	0
$411$	31.0294	1.53057
$412$	−26.1421	−1.28793
$413$	3.31371	0.163057
$414$	−34.1421	−1.67799
$415$	0	0
$416$	1.85786	0.0910893
$417$	−11.3137	−0.554035
$418$	0	0
$419$	−25.6569	−1.25342	−0.626710	−	0.779253i	$-0.715599\pi$
$419$	−25.6569	−1.25342	−0.626710	+	0.779253i	$0.715599\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	38.6274	1.88035
$423$	−14.1421	−0.687614
$424$	51.4558	2.49892
$425$	0	0
$426$	−77.2548	−3.74301
$427$	−18.6274	−0.901444
$428$	−29.3137	−1.41693
$429$	3.31371	0.159987
$430$	0	0
$431$	−11.3137	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$431$	−11.3137	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$432$	16.9706	0.816497
$433$	7.65685	0.367965	0.183982	−	0.982930i	$-0.441101\pi$
$433$	7.65685	0.367965	0.183982	+	0.982930i	$0.441101\pi$
$434$	0	0
$435$	0	0
$436$	−29.3137	−1.40387
$437$	0	0
$438$	−8.00000	−0.382255
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	19.3137	0.918659
$443$	26.8284	1.27466	0.637329	−	0.770592i	$-0.280039\pi$
$443$	26.8284	1.27466	0.637329	+	0.770592i	$0.280039\pi$
$444$	82.9117	3.93481
$445$	0	0
$446$	−26.1421	−1.23787
$447$	0.970563	0.0459060
$448$	−19.6569	−0.928699
$449$	28.6274	1.35101	0.675506	−	0.737355i	$-0.263925\pi$
$449$	28.6274	1.35101	0.675506	+	0.737355i	$0.263925\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−75.2548	−3.53969
$453$	−33.9411	−1.59469
$454$	61.1127	2.86816
$455$	0	0
$456$	0	0
$457$	−0.485281	−0.0227005	−0.0113503	−	0.999936i	$-0.503613\pi$
$457$	−0.485281	−0.0227005	−0.0113503	+	0.999936i	$0.503613\pi$
$458$	−3.17157	−0.148198
$459$	−38.6274	−1.80297
$460$	0	0
$461$	12.6274	0.588117	0.294059	−	0.955787i	$-0.404994\pi$
$461$	12.6274	0.588117	0.294059	+	0.955787i	$0.404994\pi$
$462$	−13.6569	−0.635374
$463$	6.14214	0.285449	0.142725	−	0.989762i	$-0.454414\pi$
$463$	6.14214	0.285449	0.142725	+	0.989762i	$0.454414\pi$
$464$	−10.9706	−0.509296
$465$	0	0
$466$	−14.8284	−0.686914
$467$	14.8284	0.686178	0.343089	−	0.939303i	$-0.388527\pi$
$467$	14.8284	0.686178	0.343089	+	0.939303i	$0.388527\pi$
$468$	22.4264	1.03666
$469$	−24.9706	−1.15303
$470$	0	0
$471$	39.5980	1.82458
$472$	−7.31371	−0.336641
$473$	6.00000	0.275880
$474$	−27.3137	−1.25456
$475$	0	0
$476$	−52.2843	−2.39645
$477$	−58.2843	−2.66865
$478$	56.2843	2.57438
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	8.97056	0.409022
$482$	−14.4853	−0.659786
$483$	16.0000	0.728025
$484$	3.82843	0.174019
$485$	0	0
$486$	34.1421	1.54872
$487$	24.4853	1.10953	0.554767	−	0.832006i	$-0.312807\pi$
$487$	24.4853	1.10953	0.554767	+	0.832006i	$0.312807\pi$
$488$	41.1127	1.86108
$489$	−46.6274	−2.10856
$490$	0	0
$491$	−0.686292	−0.0309719	−0.0154860	−	0.999880i	$-0.504930\pi$
$491$	−0.686292	−0.0309719	−0.0154860	+	0.999880i	$0.504930\pi$
$492$	64.9706	2.92910
$493$	24.9706	1.12462
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.6274	1.01498
$498$	−40.9706	−1.83593
$499$	9.65685	0.432300	0.216150	−	0.976360i	$-0.430650\pi$
$499$	9.65685	0.432300	0.216150	+	0.976360i	$0.430650\pi$
$500$	0	0
$501$	64.9706	2.90267
$502$	−28.9706	−1.29302
$503$	−16.6274	−0.741380	−0.370690	−	0.928757i	$-0.620879\pi$
$503$	−16.6274	−0.741380	−0.370690	+	0.928757i	$0.620879\pi$
$504$	−44.1421	−1.96625
$505$	0	0
$506$	−6.82843	−0.303561
$507$	−32.8873	−1.46058
$508$	−16.6274	−0.737722
$509$	−13.3137	−0.590120	−0.295060	−	0.955479i	$-0.595340\pi$
$509$	−13.3137	−0.590120	−0.295060	+	0.955479i	$0.595340\pi$
$510$	0	0
$511$	2.34315	0.103655
$512$	31.2426	1.38074
$513$	0	0
$514$	−22.4853	−0.991783
$515$	0	0
$516$	64.9706	2.86017
$517$	−2.82843	−0.124394
$518$	−36.9706	−1.62439
$519$	62.6274	2.74904
$520$	0	0
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	44.1421	1.93205
$523$	41.5980	1.81895	0.909476	−	0.415756i	$-0.136483\pi$
$523$	41.5980	1.81895	0.909476	+	0.415756i	$0.136483\pi$
$524$	−43.3137	−1.89217
$525$	0	0
$526$	−26.4853	−1.15481
$527$	0	0
$528$	8.48528	0.369274
$529$	−15.0000	−0.652174
$530$	0	0
$531$	8.28427	0.359507
$532$	0	0
$533$	7.02944	0.304479
$534$	90.9117	3.93413
$535$	0	0
$536$	55.1127	2.38051
$537$	27.3137	1.17867
$538$	−41.7990	−1.80208
$539$	−3.00000	−0.129219
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	−17.6569	−0.758427
$543$	60.2843	2.58705
$544$	−10.8284	−0.464265
$545$	0	0
$546$	−16.0000	−0.684737
$547$	34.0000	1.45374	0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000	1.45374	0.726868	+	0.686778i	$0.240975\pi$
$548$	42.0000	1.79415
$549$	−46.5685	−1.98750
$550$	0	0
$551$	0	0
$552$	−35.3137	−1.50305
$553$	8.00000	0.340195
$554$	16.4853	0.700392
$555$	0	0
$556$	−15.3137	−0.649446
$557$	−9.85786	−0.417691	−0.208846	−	0.977949i	$-0.566971\pi$
$557$	−9.85786	−0.417691	−0.208846	+	0.977949i	$0.566971\pi$
$558$	0	0
$559$	7.02944	0.297314
$560$	0	0
$561$	−19.3137	−0.815425
$562$	−41.7990	−1.76318
$563$	−0.343146	−0.0144619	−0.00723093	−	0.999974i	$-0.502302\pi$
$563$	−0.343146	−0.0144619	−0.00723093	+	0.999974i	$0.502302\pi$
$564$	−30.6274	−1.28965
$565$	0	0
$566$	78.7696	3.31093
$567$	2.00000	0.0839921
$568$	−49.9411	−2.09548
$569$	31.6569	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$569$	31.6569	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$570$	0	0
$571$	−21.9411	−0.918208	−0.459104	−	0.888383i	$-0.651829\pi$
$571$	−21.9411	−0.918208	−0.459104	+	0.888383i	$0.651829\pi$
$572$	4.48528	0.187539
$573$	9.37258	0.391545
$574$	−28.9706	−1.20921
$575$	0	0
$576$	−49.1421	−2.04759
$577$	26.9706	1.12280	0.561400	−	0.827545i	$-0.310263\pi$
$577$	26.9706	1.12280	0.561400	+	0.827545i	$0.310263\pi$
$578$	−71.5269	−2.97513
$579$	3.31371	0.137713
$580$	0	0
$581$	12.0000	0.497844
$582$	24.9706	1.03506
$583$	−11.6569	−0.482778
$584$	−5.17157	−0.214001
$585$	0	0
$586$	−22.1421	−0.914683
$587$	2.14214	0.0884154	0.0442077	−	0.999022i	$-0.485924\pi$
$587$	2.14214	0.0884154	0.0442077	+	0.999022i	$0.485924\pi$
$588$	−32.4853	−1.33967
$589$	0	0
$590$	0	0
$591$	30.6274	1.25984
$592$	22.9706	0.944084
$593$	−3.51472	−0.144332	−0.0721661	−	0.997393i	$-0.522991\pi$
$593$	−3.51472	−0.144332	−0.0721661	+	0.997393i	$0.522991\pi$
$594$	−13.6569	−0.560348
$595$	0	0
$596$	1.31371	0.0538116
$597$	29.2548	1.19732
$598$	−8.00000	−0.327144
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	23.9411	0.976579	0.488289	−	0.872682i	$-0.337621\pi$
$601$	23.9411	0.976579	0.488289	+	0.872682i	$0.337621\pi$
$602$	−28.9706	−1.18075
$603$	−62.4264	−2.54220
$604$	−45.9411	−1.86932
$605$	0	0
$606$	−63.5980	−2.58349
$607$	−38.2843	−1.55391	−0.776955	−	0.629556i	$-0.783237\pi$
$607$	−38.2843	−1.55391	−0.776955	+	0.629556i	$0.783237\pi$
$608$	0	0
$609$	−20.6863	−0.838251
$610$	0	0
$611$	−3.31371	−0.134058
$612$	−130.711	−5.28367
$613$	25.4558	1.02815	0.514076	−	0.857745i	$-0.328135\pi$
$613$	25.4558	1.02815	0.514076	+	0.857745i	$0.328135\pi$
$614$	−39.4558	−1.59231
$615$	0	0
$616$	−8.82843	−0.355707
$617$	−0.343146	−0.0138145	−0.00690726	−	0.999976i	$-0.502199\pi$
$617$	−0.343146	−0.0138145	−0.00690726	+	0.999976i	$0.502199\pi$
$618$	46.6274	1.87563
$619$	−14.3431	−0.576500	−0.288250	−	0.957555i	$-0.593073\pi$
$619$	−14.3431	−0.576500	−0.288250	+	0.957555i	$0.593073\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	−11.3137	−0.453638
$623$	−26.6274	−1.06680
$624$	9.94113	0.397964
$625$	0	0
$626$	−3.17157	−0.126762
$627$	0	0
$628$	53.5980	2.13879
$629$	−52.2843	−2.08471
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−17.6569	−0.702352
$633$	−45.2548	−1.79872
$634$	−3.17157	−0.125959
$635$	0	0
$636$	−126.225	−5.00516
$637$	−3.51472	−0.139258
$638$	8.82843	0.349521
$639$	56.5685	2.23782
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	52.2843	2.06350
$643$	1.45584	0.0574129	0.0287064	−	0.999588i	$-0.490861\pi$
$643$	1.45584	0.0574129	0.0287064	+	0.999588i	$0.490861\pi$
$644$	21.6569	0.853400
$645$	0	0
$646$	0	0
$647$	−27.1127	−1.06591	−0.532955	−	0.846144i	$-0.678919\pi$
$647$	−27.1127	−1.06591	−0.532955	+	0.846144i	$0.678919\pi$
$648$	−4.41421	−0.173407
$649$	1.65685	0.0650372
$650$	0	0
$651$	0	0
$652$	−63.1127	−2.47168
$653$	−11.6569	−0.456168	−0.228084	−	0.973641i	$-0.573246\pi$
$653$	−11.6569	−0.456168	−0.228084	+	0.973641i	$0.573246\pi$
$654$	52.2843	2.04448
$655$	0	0
$656$	18.0000	0.702782
$657$	5.85786	0.228537
$658$	13.6569	0.532400
$659$	45.9411	1.78961	0.894806	−	0.446455i	$-0.147314\pi$
$659$	45.9411	1.78961	0.894806	+	0.446455i	$0.147314\pi$
$660$	0	0
$661$	44.6274	1.73581	0.867903	−	0.496734i	$-0.165468\pi$
$661$	44.6274	1.73581	0.867903	+	0.496734i	$0.165468\pi$
$662$	17.6569	0.686253
$663$	−22.6274	−0.878776
$664$	−26.4853	−1.02783
$665$	0	0
$666$	−92.4264	−3.58145
$667$	−10.3431	−0.400488
$668$	87.9411	3.40254
$669$	30.6274	1.18412
$670$	0	0
$671$	−9.31371	−0.359552
$672$	8.97056	0.346047
$673$	12.4853	0.481272	0.240636	−	0.970615i	$-0.422644\pi$
$673$	12.4853	0.481272	0.240636	+	0.970615i	$0.422644\pi$
$674$	−49.4558	−1.90497
$675$	0	0
$676$	−44.5147	−1.71210
$677$	−22.8284	−0.877368	−0.438684	−	0.898641i	$-0.644555\pi$
$677$	−22.8284	−0.877368	−0.438684	+	0.898641i	$0.644555\pi$
$678$	134.225	5.15490
$679$	−7.31371	−0.280674
$680$	0	0
$681$	−71.5980	−2.74364
$682$	0	0
$683$	7.79899	0.298420	0.149210	−	0.988806i	$-0.452327\pi$
$683$	7.79899	0.298420	0.149210	+	0.988806i	$0.452327\pi$
$684$	0	0
$685$	0	0
$686$	48.2843	1.84350
$687$	3.71573	0.141764
$688$	18.0000	0.686244
$689$	−13.6569	−0.520285
$690$	0	0
$691$	−39.3137	−1.49556	−0.747782	−	0.663944i	$-0.768881\pi$
$691$	−39.3137	−1.49556	−0.747782	+	0.663944i	$0.768881\pi$
$692$	84.7696	3.22245
$693$	10.0000	0.379869
$694$	26.4853	1.00537
$695$	0	0
$696$	45.6569	1.73062
$697$	−40.9706	−1.55187
$698$	−65.1127	−2.46455
$699$	17.3726	0.657091
$700$	0	0
$701$	−12.6274	−0.476931	−0.238465	−	0.971151i	$-0.576644\pi$
$701$	−12.6274	−0.476931	−0.238465	+	0.971151i	$0.576644\pi$
$702$	−16.0000	−0.603881
$703$	0	0
$704$	−9.82843	−0.370423
$705$	0	0
$706$	51.4558	1.93657
$707$	18.6274	0.700556
$708$	17.9411	0.674269
$709$	24.6274	0.924902	0.462451	−	0.886645i	$-0.346970\pi$
$709$	24.6274	0.924902	0.462451	+	0.886645i	$0.346970\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	58.7696	2.20248
$713$	0	0
$714$	93.2548	3.48997
$715$	0	0
$716$	36.9706	1.38165
$717$	−65.9411	−2.46262
$718$	−1.65685	−0.0618333
$719$	18.3431	0.684084	0.342042	−	0.939685i	$-0.388882\pi$
$719$	18.3431	0.684084	0.342042	+	0.939685i	$0.388882\pi$
$720$	0	0
$721$	−13.6569	−0.508608
$722$	45.8701	1.70711
$723$	16.9706	0.631142
$724$	81.5980	3.03257
$725$	0	0
$726$	−6.82843	−0.253427
$727$	19.5147	0.723761	0.361880	−	0.932225i	$-0.382135\pi$
$727$	19.5147	0.723761	0.361880	+	0.932225i	$0.382135\pi$
$728$	−10.3431	−0.383342
$729$	−43.0000	−1.59259
$730$	0	0
$731$	−40.9706	−1.51535
$732$	−100.853	−3.72763
$733$	17.4558	0.644746	0.322373	−	0.946613i	$-0.395519\pi$
$733$	17.4558	0.644746	0.322373	+	0.946613i	$0.395519\pi$
$734$	−20.4853	−0.756126
$735$	0	0
$736$	4.48528	0.165330
$737$	−12.4853	−0.459901
$738$	−72.4264	−2.66605
$739$	29.9411	1.10140	0.550701	−	0.834703i	$-0.314360\pi$
$739$	29.9411	1.10140	0.550701	+	0.834703i	$0.314360\pi$
$740$	0	0
$741$	0	0
$742$	56.2843	2.06626
$743$	49.5980	1.81957	0.909787	−	0.415076i	$-0.136245\pi$
$743$	49.5980	1.81957	0.909787	+	0.415076i	$0.136245\pi$
$744$	0	0
$745$	0	0
$746$	86.4264	3.16430
$747$	30.0000	1.09764
$748$	−26.1421	−0.955851
$749$	−15.3137	−0.559551
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−8.48528	−0.309426
$753$	33.9411	1.23688
$754$	10.3431	0.376675
$755$	0	0
$756$	43.3137	1.57530
$757$	−13.3137	−0.483895	−0.241947	−	0.970289i	$-0.577786\pi$
$757$	−13.3137	−0.483895	−0.241947	+	0.970289i	$0.577786\pi$
$758$	−81.2548	−2.95131
$759$	8.00000	0.290382
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	29.6569	1.07435
$763$	−15.3137	−0.554393
$764$	12.6863	0.458974
$765$	0	0
$766$	−14.1421	−0.510976
$767$	1.94113	0.0700900
$768$	−84.7696	−3.05886
$769$	−18.9706	−0.684096	−0.342048	−	0.939682i	$-0.611121\pi$
$769$	−18.9706	−0.684096	−0.342048	+	0.939682i	$0.611121\pi$
$770$	0	0
$771$	26.3431	0.948725
$772$	4.48528	0.161429
$773$	−26.2843	−0.945380	−0.472690	−	0.881229i	$-0.656717\pi$
$773$	−26.2843	−0.945380	−0.472690	+	0.881229i	$0.656717\pi$
$774$	−72.4264	−2.60331
$775$	0	0
$776$	16.1421	0.579469
$777$	43.3137	1.55387
$778$	−49.7990	−1.78538
$779$	0	0
$780$	0	0
$781$	11.3137	0.404836
$782$	46.6274	1.66739
$783$	−20.6863	−0.739268
$784$	−9.00000	−0.321429
$785$	0	0
$786$	77.2548	2.75559
$787$	14.9706	0.533643	0.266821	−	0.963746i	$-0.414027\pi$
$787$	14.9706	0.533643	0.266821	+	0.963746i	$0.414027\pi$
$788$	41.4558	1.47680
$789$	31.0294	1.10468
$790$	0	0
$791$	−39.3137	−1.39783
$792$	−22.0711	−0.784261
$793$	−10.9117	−0.387485
$794$	−22.4853	−0.797973
$795$	0	0
$796$	39.5980	1.40351
$797$	−32.6274	−1.15572	−0.577861	−	0.816135i	$-0.696113\pi$
$797$	−32.6274	−1.15572	−0.577861	+	0.816135i	$0.696113\pi$
$798$	0	0
$799$	19.3137	0.683270
$800$	0	0
$801$	−66.5685	−2.35208
$802$	12.8284	0.452988
$803$	1.17157	0.0413439
$804$	−135.196	−4.76799
$805$	0	0
$806$	0	0
$807$	48.9706	1.72385
$808$	−41.1127	−1.44634
$809$	−10.9706	−0.385704	−0.192852	−	0.981228i	$-0.561774\pi$
$809$	−10.9706	−0.385704	−0.192852	+	0.981228i	$0.561774\pi$
$810$	0	0
$811$	53.9411	1.89413	0.947065	−	0.321043i	$-0.104033\pi$
$811$	53.9411	1.89413	0.947065	+	0.321043i	$0.104033\pi$
$812$	−28.0000	−0.982607
$813$	20.6863	0.725500
$814$	−18.4853	−0.647909
$815$	0	0
$816$	−57.9411	−2.02835
$817$	0	0
$818$	−2.48528	−0.0868958
$819$	11.7157	0.409381
$820$	0	0
$821$	−41.3137	−1.44186	−0.720929	−	0.693009i	$-0.756285\pi$
$821$	−41.3137	−1.44186	−0.720929	+	0.693009i	$0.756285\pi$
$822$	−74.9117	−2.61285
$823$	−19.5147	−0.680240	−0.340120	−	0.940382i	$-0.610468\pi$
$823$	−19.5147	−0.680240	−0.340120	+	0.940382i	$0.610468\pi$
$824$	30.1421	1.05005
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−22.2843	−0.774900	−0.387450	−	0.921891i	$-0.626644\pi$
$827$	−22.2843	−0.774900	−0.387450	+	0.921891i	$0.626644\pi$
$828$	54.1421	1.88157
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−19.3137	−0.669985
$832$	−11.5147	−0.399201
$833$	20.4853	0.709773
$834$	27.3137	0.945796
$835$	0	0
$836$	0	0
$837$	0	0
$838$	61.9411	2.13972
$839$	26.3431	0.909466	0.454733	−	0.890628i	$-0.349735\pi$
$839$	26.3431	0.909466	0.454733	+	0.890628i	$0.349735\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	14.4853	0.499196
$843$	48.9706	1.68664
$844$	−61.2548	−2.10848
$845$	0	0
$846$	34.1421	1.17383
$847$	2.00000	0.0687208
$848$	−34.9706	−1.20089
$849$	−92.2843	−3.16719
$850$	0	0
$851$	21.6569	0.742387
$852$	122.510	4.19711
$853$	15.5147	0.531214	0.265607	−	0.964081i	$-0.414428\pi$
$853$	15.5147	0.531214	0.265607	+	0.964081i	$0.414428\pi$
$854$	44.9706	1.53886
$855$	0	0
$856$	33.7990	1.15523
$857$	−24.7696	−0.846112	−0.423056	−	0.906104i	$-0.639043\pi$
$857$	−24.7696	−0.846112	−0.423056	+	0.906104i	$0.639043\pi$
$858$	−8.00000	−0.273115
$859$	24.2843	0.828569	0.414284	−	0.910148i	$-0.364032\pi$
$859$	24.2843	0.828569	0.414284	+	0.910148i	$0.364032\pi$
$860$	0	0
$861$	33.9411	1.15671
$862$	27.3137	0.930309
$863$	9.17157	0.312204	0.156102	−	0.987741i	$-0.450107\pi$
$863$	9.17157	0.312204	0.156102	+	0.987741i	$0.450107\pi$
$864$	8.97056	0.305185
$865$	0	0
$866$	−18.4853	−0.628155
$867$	83.7990	2.84596
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−14.6274	−0.495631
$872$	33.7990	1.14458
$873$	−18.2843	−0.618829
$874$	0	0
$875$	0	0
$876$	12.6863	0.428630
$877$	49.4558	1.67001	0.835003	−	0.550246i	$-0.185466\pi$
$877$	49.4558	1.67001	0.835003	+	0.550246i	$0.185466\pi$
$878$	38.6274	1.30361
$879$	25.9411	0.874972
$880$	0	0
$881$	−7.37258	−0.248389	−0.124194	−	0.992258i	$-0.539635\pi$
$881$	−7.37258	−0.248389	−0.124194	+	0.992258i	$0.539635\pi$
$882$	36.2132	1.21936
$883$	−37.1716	−1.25092	−0.625462	−	0.780255i	$-0.715089\pi$
$883$	−37.1716	−1.25092	−0.625462	+	0.780255i	$0.715089\pi$
$884$	−30.6274	−1.03011
$885$	0	0
$886$	−64.7696	−2.17598
$887$	−38.2843	−1.28546	−0.642730	−	0.766093i	$-0.722198\pi$
$887$	−38.2843	−1.28546	−0.642730	+	0.766093i	$0.722198\pi$
$888$	−95.5980	−3.20806
$889$	−8.68629	−0.291329
$890$	0	0
$891$	1.00000	0.0335013
$892$	41.4558	1.38804
$893$	0	0
$894$	−2.34315	−0.0783665
$895$	0	0
$896$	41.1127	1.37348
$897$	9.37258	0.312941
$898$	−69.1127	−2.30632
$899$	0	0
$900$	0	0
$901$	79.5980	2.65179
$902$	−14.4853	−0.482307
$903$	33.9411	1.12949
$904$	86.7696	2.88591
$905$	0	0
$906$	81.9411	2.72231
$907$	27.5147	0.913611	0.456806	−	0.889567i	$-0.348993\pi$
$907$	27.5147	0.913611	0.456806	+	0.889567i	$0.348993\pi$
$908$	−96.9117	−3.21613
$909$	46.5685	1.54458
$910$	0	0
$911$	−9.94113	−0.329364	−0.164682	−	0.986347i	$-0.552660\pi$
$911$	−9.94113	−0.329364	−0.164682	+	0.986347i	$0.552660\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	1.17157	0.0387522
$915$	0	0
$916$	5.02944	0.166177
$917$	−22.6274	−0.747223
$918$	93.2548	3.07787
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	46.2254	1.52318
$922$	−30.4853	−1.00398
$923$	13.2548	0.436288
$924$	21.6569	0.712458
$925$	0	0
$926$	−14.8284	−0.487292
$927$	−34.1421	−1.12137
$928$	−5.79899	−0.190361
$929$	5.31371	0.174337	0.0871686	−	0.996194i	$-0.472218\pi$
$929$	5.31371	0.174337	0.0871686	+	0.996194i	$0.472218\pi$
$930$	0	0
$931$	0	0
$932$	23.5147	0.770250
$933$	13.2548	0.433944
$934$	−35.7990	−1.17138
$935$	0	0
$936$	−25.8579	−0.845191
$937$	1.45584	0.0475604	0.0237802	−	0.999717i	$-0.492430\pi$
$937$	1.45584	0.0475604	0.0237802	+	0.999717i	$0.492430\pi$
$938$	60.2843	1.96835
$939$	3.71573	0.121258
$940$	0	0
$941$	−6.68629	−0.217967	−0.108983	−	0.994044i	$-0.534760\pi$
$941$	−6.68629	−0.217967	−0.108983	+	0.994044i	$0.534760\pi$
$942$	−95.5980	−3.11475
$943$	16.9706	0.552638
$944$	4.97056	0.161778
$945$	0	0
$946$	−14.4853	−0.470957
$947$	−41.1716	−1.33790	−0.668948	−	0.743309i	$-0.733255\pi$
$947$	−41.1716	−1.33790	−0.668948	+	0.743309i	$0.733255\pi$
$948$	43.3137	1.40676
$949$	1.37258	0.0445559
$950$	0	0
$951$	3.71573	0.120491
$952$	60.2843	1.95382
$953$	−53.1716	−1.72240	−0.861198	−	0.508269i	$-0.830285\pi$
$953$	−53.1716	−1.72240	−0.861198	+	0.508269i	$0.830285\pi$
$954$	140.711	4.55568
$955$	0	0
$956$	−89.2548	−2.88671
$957$	−10.3431	−0.334346
$958$	86.9117	2.80799
$959$	21.9411	0.708516
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−21.6569	−0.698245
$963$	−38.2843	−1.23369
$964$	22.9706	0.739832
$965$	0	0
$966$	−38.6274	−1.24282
$967$	14.9706	0.481421	0.240710	−	0.970597i	$-0.422620\pi$
$967$	14.9706	0.481421	0.240710	+	0.970597i	$0.422620\pi$
$968$	−4.41421	−0.141878
$969$	0	0
$970$	0	0
$971$	−8.68629	−0.278756	−0.139378	−	0.990239i	$-0.544510\pi$
$971$	−8.68629	−0.278756	−0.139378	+	0.990239i	$0.544510\pi$
$972$	−54.1421	−1.73661
$973$	−8.00000	−0.256468
$974$	−59.1127	−1.89409
$975$	0	0
$976$	−27.9411	−0.894374
$977$	−32.3431	−1.03475	−0.517374	−	0.855759i	$-0.673091\pi$
$977$	−32.3431	−1.03475	−0.517374	+	0.855759i	$0.673091\pi$
$978$	112.569	3.59955
$979$	−13.3137	−0.425508
$980$	0	0
$981$	−38.2843	−1.22232
$982$	1.65685	0.0528723
$983$	21.8579	0.697158	0.348579	−	0.937279i	$-0.386664\pi$
$983$	21.8579	0.697158	0.348579	+	0.937279i	$0.386664\pi$
$984$	−74.9117	−2.38810
$985$	0	0
$986$	−60.2843	−1.91984
$987$	−16.0000	−0.509286
$988$	0	0
$989$	16.9706	0.539633
$990$	0	0
$991$	−57.9411	−1.84056	−0.920280	−	0.391260i	$-0.872039\pi$
$991$	−57.9411	−1.84056	−0.920280	+	0.391260i	$0.872039\pi$
$992$	0	0
$993$	−20.6863	−0.656460
$994$	−54.6274	−1.73268
$995$	0	0
$996$	64.9706	2.05867
$997$	41.4558	1.31292	0.656460	−	0.754361i	$-0.272053\pi$
$997$	41.4558	1.31292	0.656460	+	0.754361i	$0.272053\pi$
$998$	−23.3137	−0.737983
$999$	43.3137	1.37039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.a.c.1.1		2
3.2	odd	2		2475.2.a.x.1.2		2
4.3	odd	2		4400.2.a.bn.1.1		2
5.2	odd	4		275.2.b.d.199.1		4
5.3	odd	4		275.2.b.d.199.4		4
5.4	even	2		55.2.a.b.1.2	✓	2
11.10	odd	2		3025.2.a.o.1.2		2
15.2	even	4		2475.2.c.l.199.4		4
15.8	even	4		2475.2.c.l.199.1		4
15.14	odd	2		495.2.a.b.1.1		2
20.3	even	4		4400.2.b.q.4049.2		4
20.7	even	4		4400.2.b.q.4049.3		4
20.19	odd	2		880.2.a.m.1.2		2
35.34	odd	2		2695.2.a.f.1.2		2
40.19	odd	2		3520.2.a.bo.1.1		2
40.29	even	2		3520.2.a.bn.1.2		2
55.4	even	10		605.2.g.f.511.2		8
55.9	even	10		605.2.g.f.81.1		8
55.14	even	10		605.2.g.f.251.2		8
55.19	odd	10		605.2.g.l.251.1		8
55.24	odd	10		605.2.g.l.81.2		8
55.29	odd	10		605.2.g.l.511.1		8
55.39	odd	10		605.2.g.l.366.2		8
55.49	even	10		605.2.g.f.366.1		8
55.54	odd	2		605.2.a.d.1.1		2
60.59	even	2		7920.2.a.ch.1.2		2
65.64	even	2		9295.2.a.g.1.1		2
165.164	even	2		5445.2.a.y.1.2		2
220.219	even	2		9680.2.a.bn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.2	✓	2	5.4	even	2
275.2.a.c.1.1		2	1.1	even	1	trivial
275.2.b.d.199.1		4	5.2	odd	4
275.2.b.d.199.4		4	5.3	odd	4
495.2.a.b.1.1		2	15.14	odd	2
605.2.a.d.1.1		2	55.54	odd	2
605.2.g.f.81.1		8	55.9	even	10
605.2.g.f.251.2		8	55.14	even	10
605.2.g.f.366.1		8	55.49	even	10
605.2.g.f.511.2		8	55.4	even	10
605.2.g.l.81.2		8	55.24	odd	10
605.2.g.l.251.1		8	55.19	odd	10
605.2.g.l.366.2		8	55.39	odd	10
605.2.g.l.511.1		8	55.29	odd	10
880.2.a.m.1.2		2	20.19	odd	2
2475.2.a.x.1.2		2	3.2	odd	2
2475.2.c.l.199.1		4	15.8	even	4
2475.2.c.l.199.4		4	15.2	even	4
2695.2.a.f.1.2		2	35.34	odd	2
3025.2.a.o.1.2		2	11.10	odd	2
3520.2.a.bn.1.2		2	40.29	even	2
3520.2.a.bo.1.1		2	40.19	odd	2
4400.2.a.bn.1.1		2	4.3	odd	2
4400.2.b.q.4049.2		4	20.3	even	4
4400.2.b.q.4049.3		4	20.7	even	4
5445.2.a.y.1.2		2	165.164	even	2
7920.2.a.ch.1.2		2	60.59	even	2
9295.2.a.g.1.1		2	65.64	even	2
9680.2.a.bn.1.2		2	220.219	even	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +2.82843 q^{3} +3.82843 q^{4} -6.82843 q^{6} +2.00000 q^{7} -4.41421 q^{8} +5.00000 q^{9} +1.00000 q^{11} +10.8284 q^{12} +1.17157 q^{13} -4.82843 q^{14} +3.00000 q^{16} -6.82843 q^{17} -12.0711 q^{18} +5.65685 q^{21} -2.41421 q^{22} +2.82843 q^{23} -12.4853 q^{24} -2.82843 q^{26} +5.65685 q^{27} +7.65685 q^{28} -3.65685 q^{29} +1.58579 q^{32} +2.82843 q^{33} +16.4853 q^{34} +19.1421 q^{36} +7.65685 q^{37} +3.31371 q^{39} +6.00000 q^{41} -13.6569 q^{42} +6.00000 q^{43} +3.82843 q^{44} -6.82843 q^{46} -2.82843 q^{47} +8.48528 q^{48} -3.00000 q^{49} -19.3137 q^{51} +4.48528 q^{52} -11.6569 q^{53} -13.6569 q^{54} -8.82843 q^{56} +8.82843 q^{58} +1.65685 q^{59} -9.31371 q^{61} +10.0000 q^{63} -9.82843 q^{64} -6.82843 q^{66} -12.4853 q^{67} -26.1421 q^{68} +8.00000 q^{69} +11.3137 q^{71} -22.0711 q^{72} +1.17157 q^{73} -18.4853 q^{74} +2.00000 q^{77} -8.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} -14.4853 q^{82} +6.00000 q^{83} +21.6569 q^{84} -14.4853 q^{86} -10.3431 q^{87} -4.41421 q^{88} -13.3137 q^{89} +2.34315 q^{91} +10.8284 q^{92} +6.82843 q^{94} +4.48528 q^{96} -3.65685 q^{97} +7.24264 q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 8 q^{6} + 4 q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{11} + 16 q^{12} + 8 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 10 q^{18} - 2 q^{22} - 8 q^{24} + 4 q^{28} + 4 q^{29} + 6 q^{32} + 16 q^{34}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 8 * q^6 + 4 * q^7 - 6 * q^8 + 10 * q^9 + 2 * q^11 + 16 * q^12 + 8 * q^13 - 4 * q^14 + 6 * q^16 - 8 * q^17 - 10 * q^18 - 2 * q^22 - 8 * q^24 + 4 * q^28 + 4 * q^29 + 6 * q^32 + 16 * q^34 + 10 * q^36 + 4 * q^37 - 16 * q^39 + 12 * q^41 - 16 * q^42 + 12 * q^43 + 2 * q^44 - 8 * q^46 - 6 * q^49 - 16 * q^51 - 8 * q^52 - 12 * q^53 - 16 * q^54 - 12 * q^56 + 12 * q^58 - 8 * q^59 + 4 * q^61 + 20 * q^63 - 14 * q^64 - 8 * q^66 - 8 * q^67 - 24 * q^68 + 16 * q^69 - 30 * q^72 + 8 * q^73 - 20 * q^74 + 4 * q^77 - 16 * q^78 + 8 * q^79 + 2 * q^81 - 12 * q^82 + 12 * q^83 + 32 * q^84 - 12 * q^86 - 32 * q^87 - 6 * q^88 - 4 * q^89 + 16 * q^91 + 16 * q^92 + 8 * q^94 - 8 * q^96 + 4 * q^97 + 6 * q^98 + 10 * q^99

Introduction

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