LMFDB - Embedded newform 5225.2.a.y.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,5,4,17,0,-1,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 $ x^15 - 5x^14 - 11x^13 + 87x^12 - 4x^11 - 545x^10 + 431x^9 + 1480x^8 - 1763x^7 - 1609x^6 + 2516x^5 + 391x^4 - 1081x^3 - 45x^2 + 153x + 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.527123$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.527123 q^{2} +1.24468 q^{3} -1.72214 q^{4} -0.656097 q^{6} +3.22241 q^{7} +1.96203 q^{8} -1.45078 q^{9} +1.00000 q^{11} -2.14351 q^{12} -4.87506 q^{13} -1.69861 q^{14} +2.41005 q^{16} +7.11649 q^{17} +0.764741 q^{18} +1.00000 q^{19} +4.01085 q^{21} -0.527123 q^{22} +1.12671 q^{23} +2.44209 q^{24} +2.56976 q^{26} -5.53978 q^{27} -5.54944 q^{28} +8.47231 q^{29} -0.926610 q^{31} -5.19445 q^{32} +1.24468 q^{33} -3.75127 q^{34} +2.49845 q^{36} -10.7386 q^{37} -0.527123 q^{38} -6.06787 q^{39} +7.49805 q^{41} -2.11421 q^{42} -7.22055 q^{43} -1.72214 q^{44} -0.593913 q^{46} +12.4483 q^{47} +2.99973 q^{48} +3.38391 q^{49} +8.85772 q^{51} +8.39555 q^{52} +7.92848 q^{53} +2.92015 q^{54} +6.32245 q^{56} +1.24468 q^{57} -4.46595 q^{58} +1.91691 q^{59} -11.5308 q^{61} +0.488438 q^{62} -4.67502 q^{63} -2.08199 q^{64} -0.656097 q^{66} +10.2103 q^{67} -12.2556 q^{68} +1.40239 q^{69} +0.979715 q^{71} -2.84647 q^{72} -15.3116 q^{73} +5.66057 q^{74} -1.72214 q^{76} +3.22241 q^{77} +3.19852 q^{78} -3.05834 q^{79} -2.54288 q^{81} -3.95240 q^{82} -2.60802 q^{83} -6.90725 q^{84} +3.80612 q^{86} +10.5453 q^{87} +1.96203 q^{88} +5.38303 q^{89} -15.7094 q^{91} -1.94035 q^{92} -1.15333 q^{93} -6.56179 q^{94} -6.46540 q^{96} +15.6399 q^{97} -1.78374 q^{98} -1.45078 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) $ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * 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 $

$2$	−0.527123	−0.372732	−0.186366	−	0.982480i	$-0.559671\pi$
$2$	−0.527123	−0.372732	−0.186366	+	0.982480i	$0.559671\pi$
$3$	1.24468	0.718614	0.359307	−	0.933219i	$-0.383013\pi$
$3$	1.24468	0.718614	0.359307	+	0.933219i	$0.383013\pi$
$4$	−1.72214	−0.861071
$5$	0	0
$5$	0	0
$6$	−0.656097	−0.267850
$7$	3.22241	1.21796	0.608978	−	0.793187i	$-0.291580\pi$
$7$	3.22241	1.21796	0.608978	+	0.793187i	$0.291580\pi$
$8$	1.96203	0.693681
$9$	−1.45078	−0.483594
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.14351	−0.618777
$13$	−4.87506	−1.35210	−0.676050	−	0.736856i	$-0.736310\pi$
$13$	−4.87506	−1.35210	−0.676050	+	0.736856i	$0.736310\pi$
$14$	−1.69861	−0.453971
$15$	0	0
$16$	2.41005	0.602513
$17$	7.11649	1.72600	0.863002	−	0.505201i	$-0.168582\pi$
$17$	7.11649	1.72600	0.863002	+	0.505201i	$0.168582\pi$
$18$	0.764741	0.180251
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.01085	0.875240
$22$	−0.527123	−0.112383
$23$	1.12671	0.234935	0.117467	−	0.993077i	$-0.462522\pi$
$23$	1.12671	0.234935	0.117467	+	0.993077i	$0.462522\pi$
$24$	2.44209	0.498489
$25$	0	0
$26$	2.56976	0.503971
$27$	−5.53978	−1.06613
$28$	−5.54944	−1.04875
$29$	8.47231	1.57327	0.786634	−	0.617420i	$-0.211822\pi$
$29$	8.47231	1.57327	0.786634	+	0.617420i	$0.211822\pi$
$30$	0	0
$31$	−0.926610	−0.166424	−0.0832121	−	0.996532i	$-0.526518\pi$
$31$	−0.926610	−0.166424	−0.0832121	+	0.996532i	$0.526518\pi$
$32$	−5.19445	−0.918257
$33$	1.24468	0.216670
$34$	−3.75127	−0.643337
$35$	0	0
$36$	2.49845	0.416409
$37$	−10.7386	−1.76542	−0.882709	−	0.469920i	$-0.844283\pi$
$37$	−10.7386	−1.76542	−0.882709	+	0.469920i	$0.844283\pi$
$38$	−0.527123	−0.0855106
$39$	−6.06787	−0.971637
$40$	0	0
$41$	7.49805	1.17100	0.585499	−	0.810673i	$-0.300898\pi$
$41$	7.49805	1.17100	0.585499	+	0.810673i	$0.300898\pi$
$42$	−2.11421	−0.326230
$43$	−7.22055	−1.10112	−0.550562	−	0.834794i	$-0.685586\pi$
$43$	−7.22055	−1.10112	−0.550562	+	0.834794i	$0.685586\pi$
$44$	−1.72214	−0.259623
$45$	0	0
$46$	−0.593913	−0.0875678
$47$	12.4483	1.81577	0.907886	−	0.419217i	$-0.137695\pi$
$47$	12.4483	1.81577	0.907886	+	0.419217i	$0.137695\pi$
$48$	2.99973	0.432974
$49$	3.38391	0.483416
$50$	0	0
$51$	8.85772	1.24033
$52$	8.39555	1.16425
$53$	7.92848	1.08906	0.544530	−	0.838741i	$-0.316708\pi$
$53$	7.92848	1.08906	0.544530	+	0.838741i	$0.316708\pi$
$54$	2.92015	0.397381
$55$	0	0
$56$	6.32245	0.844873
$57$	1.24468	0.164861
$58$	−4.46595	−0.586408
$59$	1.91691	0.249561	0.124780	−	0.992184i	$-0.460177\pi$
$59$	1.91691	0.249561	0.124780	+	0.992184i	$0.460177\pi$
$60$	0	0
$61$	−11.5308	−1.47637	−0.738185	−	0.674599i	$-0.764317\pi$
$61$	−11.5308	−1.47637	−0.738185	+	0.674599i	$0.764317\pi$
$62$	0.488438	0.0620316
$63$	−4.67502	−0.588997
$64$	−2.08199	−0.260249
$65$	0	0
$66$	−0.656097	−0.0807600
$67$	10.2103	1.24739	0.623693	−	0.781669i	$-0.285631\pi$
$67$	10.2103	1.24739	0.623693	+	0.781669i	$0.285631\pi$
$68$	−12.2556	−1.48621
$69$	1.40239	0.168827
$70$	0	0
$71$	0.979715	0.116271	0.0581354	−	0.998309i	$-0.481484\pi$
$71$	0.979715	0.116271	0.0581354	+	0.998309i	$0.481484\pi$
$72$	−2.84647	−0.335460
$73$	−15.3116	−1.79209	−0.896044	−	0.443965i	$-0.853571\pi$
$73$	−15.3116	−1.79209	−0.896044	+	0.443965i	$0.853571\pi$
$74$	5.66057	0.658028
$75$	0	0
$76$	−1.72214	−0.197543
$77$	3.22241	0.367227
$78$	3.19852	0.362161
$79$	−3.05834	−0.344091	−0.172045	−	0.985089i	$-0.555038\pi$
$79$	−3.05834	−0.344091	−0.172045	+	0.985089i	$0.555038\pi$
$80$	0	0
$81$	−2.54288	−0.282542
$82$	−3.95240	−0.436469
$83$	−2.60802	−0.286268	−0.143134	−	0.989703i	$-0.545718\pi$
$83$	−2.60802	−0.286268	−0.143134	+	0.989703i	$0.545718\pi$
$84$	−6.90725	−0.753643
$85$	0	0
$86$	3.80612	0.410424
$87$	10.5453	1.13057
$88$	1.96203	0.209153
$89$	5.38303	0.570600	0.285300	−	0.958438i	$-0.407907\pi$
$89$	5.38303	0.570600	0.285300	+	0.958438i	$0.407907\pi$
$90$	0	0
$91$	−15.7094	−1.64680
$92$	−1.94035	−0.202295
$93$	−1.15333	−0.119595
$94$	−6.56179	−0.676797
$95$	0	0
$96$	−6.46540	−0.659872
$97$	15.6399	1.58799	0.793995	−	0.607924i	$-0.207997\pi$
$97$	15.6399	1.58799	0.793995	+	0.607924i	$0.207997\pi$
$98$	−1.78374	−0.180185
$99$	−1.45078	−0.145809
$100$	0	0
$101$	14.5602	1.44879	0.724396	−	0.689384i	$-0.242119\pi$
$101$	14.5602	1.44879	0.724396	+	0.689384i	$0.242119\pi$
$102$	−4.66911	−0.462311
$103$	10.2341	1.00839	0.504197	−	0.863589i	$-0.331789\pi$
$103$	10.2341	1.00839	0.504197	+	0.863589i	$0.331789\pi$
$104$	−9.56501	−0.937926
$105$	0	0
$106$	−4.17928	−0.405928
$107$	5.30542	0.512895	0.256447	−	0.966558i	$-0.417448\pi$
$107$	5.30542	0.512895	0.256447	+	0.966558i	$0.417448\pi$
$108$	9.54028	0.918014
$109$	−6.54304	−0.626709	−0.313355	−	0.949636i	$-0.601453\pi$
$109$	−6.54304	−0.626709	−0.313355	+	0.949636i	$0.601453\pi$
$110$	0	0
$111$	−13.3661	−1.26865
$112$	7.76618	0.733835
$113$	−5.52729	−0.519963	−0.259982	−	0.965614i	$-0.583717\pi$
$113$	−5.52729	−0.519963	−0.259982	+	0.965614i	$0.583717\pi$
$114$	−0.656097	−0.0614491
$115$	0	0
$116$	−14.5905	−1.35470
$117$	7.07266	0.653868
$118$	−1.01045	−0.0930194
$119$	22.9322	2.10220
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.07816	0.550291
$123$	9.33264	0.841496
$124$	1.59575	0.143303
$125$	0	0
$126$	2.46431	0.219538
$127$	2.97437	0.263932	0.131966	−	0.991254i	$-0.457871\pi$
$127$	2.97437	0.263932	0.131966	+	0.991254i	$0.457871\pi$
$128$	11.4864	1.01526
$129$	−8.98724	−0.791283
$130$	0	0
$131$	11.1638	0.975388	0.487694	−	0.873015i	$-0.337838\pi$
$131$	11.1638	0.975388	0.487694	+	0.873015i	$0.337838\pi$
$132$	−2.14351	−0.186568
$133$	3.22241	0.279418
$134$	−5.38208	−0.464941
$135$	0	0
$136$	13.9627	1.19730
$137$	−13.3569	−1.14116	−0.570580	−	0.821242i	$-0.693282\pi$
$137$	−13.3569	−1.14116	−0.570580	+	0.821242i	$0.693282\pi$
$138$	−0.739229	−0.0629274
$139$	17.5311	1.48697	0.743485	−	0.668753i	$-0.233171\pi$
$139$	17.5311	1.48697	0.743485	+	0.668753i	$0.233171\pi$
$140$	0	0
$141$	15.4941	1.30484
$142$	−0.516430	−0.0433379
$143$	−4.87506	−0.407673
$144$	−3.49647	−0.291372
$145$	0	0
$146$	8.07110	0.667969
$147$	4.21187	0.347389
$148$	18.4934	1.52015
$149$	−9.45886	−0.774900	−0.387450	−	0.921891i	$-0.626644\pi$
$149$	−9.45886	−0.774900	−0.387450	+	0.921891i	$0.626644\pi$
$150$	0	0
$151$	−0.0675175	−0.00549449	−0.00274725	−	0.999996i	$-0.500874\pi$
$151$	−0.0675175	−0.00549449	−0.00274725	+	0.999996i	$0.500874\pi$
$152$	1.96203	0.159141
$153$	−10.3245	−0.834685
$154$	−1.69861	−0.136878
$155$	0	0
$156$	10.4497	0.836648
$157$	13.6996	1.09334	0.546672	−	0.837347i	$-0.315894\pi$
$157$	13.6996	1.09334	0.546672	+	0.837347i	$0.315894\pi$
$158$	1.61212	0.128254
$159$	9.86838	0.782613
$160$	0	0
$161$	3.63071	0.286140
$162$	1.34041	0.105313
$163$	9.50798	0.744722	0.372361	−	0.928088i	$-0.378548\pi$
$163$	9.50798	0.744722	0.372361	+	0.928088i	$0.378548\pi$
$164$	−12.9127	−1.00831
$165$	0	0
$166$	1.37475	0.106701
$167$	−3.51330	−0.271868	−0.135934	−	0.990718i	$-0.543403\pi$
$167$	−3.51330	−0.271868	−0.135934	+	0.990718i	$0.543403\pi$
$168$	7.86940	0.607137
$169$	10.7663	0.828174
$170$	0	0
$171$	−1.45078	−0.110944
$172$	12.4348	0.948145
$173$	−3.64800	−0.277352	−0.138676	−	0.990338i	$-0.544285\pi$
$173$	−3.64800	−0.277352	−0.138676	+	0.990338i	$0.544285\pi$
$174$	−5.55866	−0.421401
$175$	0	0
$176$	2.41005	0.181665
$177$	2.38594	0.179338
$178$	−2.83752	−0.212681
$179$	−18.2698	−1.36555	−0.682773	−	0.730630i	$-0.739226\pi$
$179$	−18.2698	−1.36555	−0.682773	+	0.730630i	$0.739226\pi$
$180$	0	0
$181$	4.87683	0.362492	0.181246	−	0.983438i	$-0.441987\pi$
$181$	4.87683	0.362492	0.181246	+	0.983438i	$0.441987\pi$
$182$	8.28081	0.613815
$183$	−14.3521	−1.06094
$184$	2.21063	0.162970
$185$	0	0
$186$	0.607946	0.0445768
$187$	7.11649	0.520409
$188$	−21.4377	−1.56351
$189$	−17.8514	−1.29850
$190$	0	0
$191$	−3.01617	−0.218242	−0.109121	−	0.994028i	$-0.534804\pi$
$191$	−3.01617	−0.218242	−0.109121	+	0.994028i	$0.534804\pi$
$192$	−2.59141	−0.187019
$193$	11.8494	0.852940	0.426470	−	0.904502i	$-0.359757\pi$
$193$	11.8494	0.852940	0.426470	+	0.904502i	$0.359757\pi$
$194$	−8.24415	−0.591895
$195$	0	0
$196$	−5.82758	−0.416256
$197$	22.9156	1.63267	0.816333	−	0.577581i	$-0.196003\pi$
$197$	22.9156	1.63267	0.816333	+	0.577581i	$0.196003\pi$
$198$	0.764741	0.0543478
$199$	−16.6587	−1.18090	−0.590452	−	0.807072i	$-0.701051\pi$
$199$	−16.6587	−1.18090	−0.590452	+	0.807072i	$0.701051\pi$
$200$	0	0
$201$	12.7085	0.896389
$202$	−7.67500	−0.540011
$203$	27.3012	1.91617
$204$	−15.2543	−1.06801
$205$	0	0
$206$	−5.39462	−0.375861
$207$	−1.63461	−0.113613
$208$	−11.7492	−0.814658
$209$	1.00000	0.0691714
$210$	0	0
$211$	−11.0707	−0.762138	−0.381069	−	0.924547i	$-0.624444\pi$
$211$	−11.0707	−0.762138	−0.381069	+	0.924547i	$0.624444\pi$
$212$	−13.6540	−0.937757
$213$	1.21943	0.0835538
$214$	−2.79661	−0.191172
$215$	0	0
$216$	−10.8692	−0.739555
$217$	−2.98592	−0.202697
$218$	3.44898	0.233595
$219$	−19.0580	−1.28782
$220$	0	0
$221$	−34.6934	−2.33373
$222$	7.04558	0.472868
$223$	−2.30152	−0.154121	−0.0770606	−	0.997026i	$-0.524553\pi$
$223$	−2.30152	−0.154121	−0.0770606	+	0.997026i	$0.524553\pi$
$224$	−16.7386	−1.11840
$225$	0	0
$226$	2.91356	0.193807
$227$	−15.8244	−1.05030	−0.525152	−	0.851008i	$-0.675992\pi$
$227$	−15.8244	−1.05030	−0.525152	+	0.851008i	$0.675992\pi$
$228$	−2.14351	−0.141957
$229$	21.0470	1.39082	0.695412	−	0.718612i	$-0.255222\pi$
$229$	21.0470	1.39082	0.695412	+	0.718612i	$0.255222\pi$
$230$	0	0
$231$	4.01085	0.263895
$232$	16.6229	1.09135
$233$	−1.45166	−0.0951017	−0.0475508	−	0.998869i	$-0.515142\pi$
$233$	−1.45166	−0.0951017	−0.0475508	+	0.998869i	$0.515142\pi$
$234$	−3.72816	−0.243718
$235$	0	0
$236$	−3.30120	−0.214890
$237$	−3.80665	−0.247268
$238$	−12.0881	−0.783556
$239$	3.11689	0.201615	0.100807	−	0.994906i	$-0.467857\pi$
$239$	3.11689	0.201615	0.100807	+	0.994906i	$0.467857\pi$
$240$	0	0
$241$	−25.3469	−1.63274	−0.816369	−	0.577531i	$-0.804016\pi$
$241$	−25.3469	−1.63274	−0.816369	+	0.577531i	$0.804016\pi$
$242$	−0.527123	−0.0338847
$243$	13.4543	0.863093
$244$	19.8577	1.27126
$245$	0	0
$246$	−4.91945	−0.313653
$247$	−4.87506	−0.310193
$248$	−1.81803	−0.115445
$249$	−3.24614	−0.205716
$250$	0	0
$251$	12.0511	0.760656	0.380328	−	0.924852i	$-0.375811\pi$
$251$	12.0511	0.760656	0.380328	+	0.924852i	$0.375811\pi$
$252$	8.05104	0.507168
$253$	1.12671	0.0708355
$254$	−1.56786	−0.0983761
$255$	0	0
$256$	−1.89074	−0.118171
$257$	17.1129	1.06747	0.533737	−	0.845651i	$-0.320787\pi$
$257$	17.1129	1.06747	0.533737	+	0.845651i	$0.320787\pi$
$258$	4.73738	0.294937
$259$	−34.6042	−2.15020
$260$	0	0
$261$	−12.2915	−0.760824
$262$	−5.88471	−0.363559
$263$	23.8861	1.47288	0.736440	−	0.676503i	$-0.236505\pi$
$263$	23.8861	1.47288	0.736440	+	0.676503i	$0.236505\pi$
$264$	2.44209	0.150300
$265$	0	0
$266$	−1.69861	−0.104148
$267$	6.70012	0.410041
$268$	−17.5836	−1.07409
$269$	−5.15565	−0.314346	−0.157173	−	0.987571i	$-0.550238\pi$
$269$	−5.15565	−0.314346	−0.157173	+	0.987571i	$0.550238\pi$
$270$	0	0
$271$	9.25556	0.562235	0.281118	−	0.959673i	$-0.409295\pi$
$271$	9.25556	0.562235	0.281118	+	0.959673i	$0.409295\pi$
$272$	17.1511	1.03994
$273$	−19.5532	−1.18341
$274$	7.04075	0.425347
$275$	0	0
$276$	−2.41511	−0.145372
$277$	−7.65078	−0.459691	−0.229845	−	0.973227i	$-0.573822\pi$
$277$	−7.65078	−0.459691	−0.229845	+	0.973227i	$0.573822\pi$
$278$	−9.24106	−0.554242
$279$	1.34431	0.0804818
$280$	0	0
$281$	16.4752	0.982829	0.491415	−	0.870926i	$-0.336480\pi$
$281$	16.4752	0.982829	0.491415	+	0.870926i	$0.336480\pi$
$282$	−8.16730	−0.486355
$283$	10.5470	0.626953	0.313476	−	0.949596i	$-0.398506\pi$
$283$	10.5470	0.626953	0.313476	+	0.949596i	$0.398506\pi$
$284$	−1.68721	−0.100117
$285$	0	0
$286$	2.56976	0.151953
$287$	24.1618	1.42622
$288$	7.53602	0.444064
$289$	33.6445	1.97909
$290$	0	0
$291$	19.4666	1.14115
$292$	26.3688	1.54311
$293$	27.1073	1.58362	0.791811	−	0.610766i	$-0.209138\pi$
$293$	27.1073	1.58362	0.791811	+	0.610766i	$0.209138\pi$
$294$	−2.22018	−0.129483
$295$	0	0
$296$	−21.0695	−1.22464
$297$	−5.53978	−0.321451
$298$	4.98598	0.288830
$299$	−5.49277	−0.317655
$300$	0	0
$301$	−23.2676	−1.34112
$302$	0.0355900	0.00204798
$303$	18.1227	1.04112
$304$	2.41005	0.138226
$305$	0	0
$306$	5.44227	0.311114
$307$	15.9437	0.909957	0.454978	−	0.890503i	$-0.349647\pi$
$307$	15.9437	0.909957	0.454978	+	0.890503i	$0.349647\pi$
$308$	−5.54944	−0.316209
$309$	12.7381	0.724645
$310$	0	0
$311$	23.0972	1.30972	0.654862	−	0.755749i	$-0.272727\pi$
$311$	23.0972	1.30972	0.654862	+	0.755749i	$0.272727\pi$
$312$	−11.9053	−0.674006
$313$	19.7892	1.11855	0.559275	−	0.828982i	$-0.311079\pi$
$313$	19.7892	1.11855	0.559275	+	0.828982i	$0.311079\pi$
$314$	−7.22135	−0.407525
$315$	0	0
$316$	5.26690	0.296286
$317$	−6.50111	−0.365139	−0.182569	−	0.983193i	$-0.558441\pi$
$317$	−6.50111	−0.365139	−0.182569	+	0.983193i	$0.558441\pi$
$318$	−5.20185	−0.291705
$319$	8.47231	0.474358
$320$	0	0
$321$	6.60353	0.368573
$322$	−1.91383	−0.106654
$323$	7.11649	0.395972
$324$	4.37920	0.243289
$325$	0	0
$326$	−5.01187	−0.277582
$327$	−8.14396	−0.450362
$328$	14.7114	0.812300
$329$	40.1135	2.21153
$330$	0	0
$331$	0.931117	0.0511788	0.0255894	−	0.999673i	$-0.491854\pi$
$331$	0.931117	0.0511788	0.0255894	+	0.999673i	$0.491854\pi$
$332$	4.49139	0.246497
$333$	15.5794	0.853746
$334$	1.85194	0.101334
$335$	0	0
$336$	9.66637	0.527344
$337$	13.9841	0.761761	0.380881	−	0.924624i	$-0.375621\pi$
$337$	13.9841	0.761761	0.380881	+	0.924624i	$0.375621\pi$
$338$	−5.67514	−0.308687
$339$	−6.87968	−0.373653
$340$	0	0
$341$	−0.926610	−0.0501788
$342$	0.764741	0.0413525
$343$	−11.6525	−0.629176
$344$	−14.1669	−0.763829
$345$	0	0
$346$	1.92294	0.103378
$347$	−30.7347	−1.64992	−0.824961	−	0.565189i	$-0.808803\pi$
$347$	−30.7347	−1.64992	−0.824961	+	0.565189i	$0.808803\pi$
$348$	−18.1605	−0.973502
$349$	−12.9948	−0.695595	−0.347798	−	0.937570i	$-0.613070\pi$
$349$	−12.9948	−0.695595	−0.347798	+	0.937570i	$0.613070\pi$
$350$	0	0
$351$	27.0068	1.44152
$352$	−5.19445	−0.276865
$353$	18.3337	0.975803	0.487901	−	0.872899i	$-0.337763\pi$
$353$	18.3337	0.975803	0.487901	+	0.872899i	$0.337763\pi$
$354$	−1.25768	−0.0668450
$355$	0	0
$356$	−9.27033	−0.491327
$357$	28.5432	1.51067
$358$	9.63041	0.508983
$359$	−7.55341	−0.398654	−0.199327	−	0.979933i	$-0.563876\pi$
$359$	−7.55341	−0.398654	−0.199327	+	0.979933i	$0.563876\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−2.57069	−0.135113
$363$	1.24468	0.0653285
$364$	27.0539	1.41801
$365$	0	0
$366$	7.56533	0.395446
$367$	6.80846	0.355399	0.177699	−	0.984085i	$-0.443135\pi$
$367$	6.80846	0.355399	0.177699	+	0.984085i	$0.443135\pi$
$368$	2.71543	0.141551
$369$	−10.8780	−0.566289
$370$	0	0
$371$	25.5488	1.32643
$372$	1.98620	0.102979
$373$	−0.104052	−0.00538763	−0.00269381	−	0.999996i	$-0.500857\pi$
$373$	−0.104052	−0.00538763	−0.00269381	+	0.999996i	$0.500857\pi$
$374$	−3.75127	−0.193973
$375$	0	0
$376$	24.4239	1.25957
$377$	−41.3031	−2.12722
$378$	9.40990	0.483993
$379$	−29.4765	−1.51411	−0.757053	−	0.653353i	$-0.773362\pi$
$379$	−29.4765	−1.51411	−0.757053	+	0.653353i	$0.773362\pi$
$380$	0	0
$381$	3.70212	0.189665
$382$	1.58989	0.0813460
$383$	−15.0758	−0.770337	−0.385169	−	0.922846i	$-0.625857\pi$
$383$	−15.0758	−0.770337	−0.385169	+	0.922846i	$0.625857\pi$
$384$	14.2968	0.729580
$385$	0	0
$386$	−6.24610	−0.317918
$387$	10.4755	0.532497
$388$	−26.9341	−1.36737
$389$	−8.16895	−0.414182	−0.207091	−	0.978322i	$-0.566400\pi$
$389$	−8.16895	−0.414182	−0.207091	+	0.978322i	$0.566400\pi$
$390$	0	0
$391$	8.01821	0.405498
$392$	6.63933	0.335337
$393$	13.8953	0.700927
$394$	−12.0793	−0.608548
$395$	0	0
$396$	2.49845	0.125552
$397$	14.6856	0.737050	0.368525	−	0.929618i	$-0.379863\pi$
$397$	14.6856	0.737050	0.368525	+	0.929618i	$0.379863\pi$
$398$	8.78119	0.440161
$399$	4.01085	0.200794
$400$	0	0
$401$	1.14341	0.0570994	0.0285497	−	0.999592i	$-0.490911\pi$
$401$	1.14341	0.0570994	0.0285497	+	0.999592i	$0.490911\pi$
$402$	−6.69895	−0.334113
$403$	4.51729	0.225022
$404$	−25.0747	−1.24751
$405$	0	0
$406$	−14.3911	−0.714219
$407$	−10.7386	−0.532294
$408$	17.3791	0.860393
$409$	34.6619	1.71392	0.856961	−	0.515381i	$-0.172350\pi$
$409$	34.6619	1.71392	0.856961	+	0.515381i	$0.172350\pi$
$410$	0	0
$411$	−16.6250	−0.820053
$412$	−17.6245	−0.868298
$413$	6.17708	0.303954
$414$	0.861640	0.0423473
$415$	0	0
$416$	25.3233	1.24158
$417$	21.8206	1.06856
$418$	−0.527123	−0.0257824
$419$	−19.8862	−0.971505	−0.485752	−	0.874096i	$-0.661454\pi$
$419$	−19.8862	−0.971505	−0.485752	+	0.874096i	$0.661454\pi$
$420$	0	0
$421$	32.9388	1.60534	0.802669	−	0.596424i	$-0.203412\pi$
$421$	32.9388	1.60534	0.802669	+	0.596424i	$0.203412\pi$
$422$	5.83562	0.284073
$423$	−18.0598	−0.878097
$424$	15.5559	0.755460
$425$	0	0
$426$	−0.642788	−0.0311432
$427$	−37.1570	−1.79815
$428$	−9.13669	−0.441639
$429$	−6.06787	−0.292960
$430$	0	0
$431$	−21.4153	−1.03154	−0.515770	−	0.856727i	$-0.672494\pi$
$431$	−21.4153	−1.03154	−0.515770	+	0.856727i	$0.672494\pi$
$432$	−13.3512	−0.642358
$433$	−31.7437	−1.52551	−0.762753	−	0.646690i	$-0.776153\pi$
$433$	−31.7437	−1.52551	−0.762753	+	0.646690i	$0.776153\pi$
$434$	1.57395	0.0755518
$435$	0	0
$436$	11.2680	0.539641
$437$	1.12671	0.0538977
$438$	10.0459	0.480012
$439$	9.40014	0.448644	0.224322	−	0.974515i	$-0.427983\pi$
$439$	9.40014	0.448644	0.224322	+	0.974515i	$0.427983\pi$
$440$	0	0
$441$	−4.90932	−0.233777
$442$	18.2877	0.869856
$443$	21.5223	1.02256	0.511278	−	0.859415i	$-0.329172\pi$
$443$	21.5223	1.02256	0.511278	+	0.859415i	$0.329172\pi$
$444$	23.0183	1.09240
$445$	0	0
$446$	1.21318	0.0574459
$447$	−11.7732	−0.556854
$448$	−6.70904	−0.316972
$449$	30.7497	1.45117	0.725583	−	0.688134i	$-0.241570\pi$
$449$	30.7497	1.45117	0.725583	+	0.688134i	$0.241570\pi$
$450$	0	0
$451$	7.49805	0.353069
$452$	9.51877	0.447725
$453$	−0.0840373	−0.00394842
$454$	8.34142	0.391482
$455$	0	0
$456$	2.44209	0.114361
$457$	−0.507061	−0.0237193	−0.0118596	−	0.999930i	$-0.503775\pi$
$457$	−0.507061	−0.0237193	−0.0118596	+	0.999930i	$0.503775\pi$
$458$	−11.0943	−0.518405
$459$	−39.4238	−1.84015
$460$	0	0
$461$	5.60128	0.260878	0.130439	−	0.991456i	$-0.458361\pi$
$461$	5.60128	0.260878	0.130439	+	0.991456i	$0.458361\pi$
$462$	−2.11421	−0.0983621
$463$	22.5309	1.04710	0.523550	−	0.851995i	$-0.324607\pi$
$463$	22.5309	1.04710	0.523550	+	0.851995i	$0.324607\pi$
$464$	20.4187	0.947915
$465$	0	0
$466$	0.765205	0.0354475
$467$	−15.1056	−0.699003	−0.349502	−	0.936936i	$-0.613649\pi$
$467$	−15.1056	−0.699003	−0.349502	+	0.936936i	$0.613649\pi$
$468$	−12.1801	−0.563026
$469$	32.9018	1.51926
$470$	0	0
$471$	17.0515	0.785692
$472$	3.76103	0.173116
$473$	−7.22055	−0.332001
$474$	2.00657	0.0921648
$475$	0	0
$476$	−39.4926	−1.81014
$477$	−11.5025	−0.526663
$478$	−1.64298	−0.0751482
$479$	−1.97446	−0.0902153	−0.0451076	−	0.998982i	$-0.514363\pi$
$479$	−1.97446	−0.0902153	−0.0451076	+	0.998982i	$0.514363\pi$
$480$	0	0
$481$	52.3515	2.38702
$482$	13.3609	0.608574
$483$	4.51906	0.205624
$484$	−1.72214	−0.0782792
$485$	0	0
$486$	−7.09206	−0.321702
$487$	−6.23748	−0.282647	−0.141324	−	0.989963i	$-0.545136\pi$
$487$	−6.23748	−0.282647	−0.141324	+	0.989963i	$0.545136\pi$
$488$	−22.6238	−1.02413
$489$	11.8343	0.535168
$490$	0	0
$491$	29.4073	1.32713	0.663566	−	0.748117i	$-0.269042\pi$
$491$	29.4073	1.32713	0.663566	+	0.748117i	$0.269042\pi$
$492$	−16.0721	−0.724587
$493$	60.2931	2.71547
$494$	2.56976	0.115619
$495$	0	0
$496$	−2.23318	−0.100273
$497$	3.15704	0.141613
$498$	1.71112	0.0766770
$499$	−8.33612	−0.373176	−0.186588	−	0.982438i	$-0.559743\pi$
$499$	−8.33612	−0.373176	−0.186588	+	0.982438i	$0.559743\pi$
$500$	0	0
$501$	−4.37292	−0.195368
$502$	−6.35239	−0.283521
$503$	36.9242	1.64637	0.823184	−	0.567774i	$-0.192195\pi$
$503$	36.9242	1.64637	0.823184	+	0.567774i	$0.192195\pi$
$504$	−9.17250	−0.408576
$505$	0	0
$506$	−0.593913	−0.0264027
$507$	13.4005	0.595137
$508$	−5.12228	−0.227265
$509$	−14.0280	−0.621782	−0.310891	−	0.950446i	$-0.600627\pi$
$509$	−14.0280	−0.621782	−0.310891	+	0.950446i	$0.600627\pi$
$510$	0	0
$511$	−49.3402	−2.18268
$512$	−21.9761	−0.971214
$513$	−5.53978	−0.244587
$514$	−9.02061	−0.397882
$515$	0	0
$516$	15.4773	0.681350
$517$	12.4483	0.547476
$518$	18.2407	0.801449
$519$	−4.54057	−0.199309
$520$	0	0
$521$	6.56794	0.287747	0.143873	−	0.989596i	$-0.454044\pi$
$521$	6.56794	0.287747	0.143873	+	0.989596i	$0.454044\pi$
$522$	6.47912	0.283583
$523$	6.72747	0.294172	0.147086	−	0.989124i	$-0.453011\pi$
$523$	6.72747	0.294172	0.147086	+	0.989124i	$0.453011\pi$
$524$	−19.2257	−0.839878
$525$	0	0
$526$	−12.5909	−0.548990
$527$	−6.59422	−0.287249
$528$	2.99973	0.130547
$529$	−21.7305	−0.944806
$530$	0	0
$531$	−2.78103	−0.120686
$532$	−5.54944	−0.240599
$533$	−36.5535	−1.58331
$534$	−3.53179	−0.152835
$535$	0	0
$536$	20.0329	0.865288
$537$	−22.7399	−0.981300
$538$	2.71766	0.117167
$539$	3.38391	0.145755
$540$	0	0
$541$	−5.50043	−0.236482	−0.118241	−	0.992985i	$-0.537726\pi$
$541$	−5.50043	−0.236482	−0.118241	+	0.992985i	$0.537726\pi$
$542$	−4.87882	−0.209563
$543$	6.07008	0.260492
$544$	−36.9662	−1.58491
$545$	0	0
$546$	10.3069	0.441096
$547$	30.6112	1.30884	0.654420	−	0.756131i	$-0.272913\pi$
$547$	30.6112	1.30884	0.654420	+	0.756131i	$0.272913\pi$
$548$	23.0025	0.982619
$549$	16.7287	0.713964
$550$	0	0
$551$	8.47231	0.360932
$552$	2.75152	0.117112
$553$	−9.85523	−0.419087
$554$	4.03290	0.171342
$555$	0	0
$556$	−30.1911	−1.28039
$557$	−14.5219	−0.615313	−0.307656	−	0.951498i	$-0.599545\pi$
$557$	−14.5219	−0.615313	−0.307656	+	0.951498i	$0.599545\pi$
$558$	−0.708617	−0.0299982
$559$	35.2007	1.48883
$560$	0	0
$561$	8.85772	0.373973
$562$	−8.68447	−0.366332
$563$	18.6917	0.787762	0.393881	−	0.919161i	$-0.371132\pi$
$563$	18.6917	0.787762	0.393881	+	0.919161i	$0.371132\pi$
$564$	−26.6830	−1.12356
$565$	0	0
$566$	−5.55955	−0.233685
$567$	−8.19419	−0.344124
$568$	1.92223	0.0806549
$569$	4.12594	0.172968	0.0864842	−	0.996253i	$-0.472437\pi$
$569$	4.12594	0.172968	0.0864842	+	0.996253i	$0.472437\pi$
$570$	0	0
$571$	−20.6081	−0.862422	−0.431211	−	0.902251i	$-0.641913\pi$
$571$	−20.6081	−0.862422	−0.431211	+	0.902251i	$0.641913\pi$
$572$	8.39555	0.351036
$573$	−3.75415	−0.156832
$574$	−12.7362	−0.531600
$575$	0	0
$576$	3.02052	0.125855
$577$	18.0882	0.753020	0.376510	−	0.926413i	$-0.377124\pi$
$577$	18.0882	0.753020	0.376510	+	0.926413i	$0.377124\pi$
$578$	−17.7348	−0.737669
$579$	14.7487	0.612934
$580$	0	0
$581$	−8.40412	−0.348662
$582$	−10.2613	−0.425344
$583$	7.92848	0.328364
$584$	−30.0418	−1.24314
$585$	0	0
$586$	−14.2889	−0.590267
$587$	−31.2026	−1.28787	−0.643934	−	0.765081i	$-0.722699\pi$
$587$	−31.2026	−1.28787	−0.643934	+	0.765081i	$0.722699\pi$
$588$	−7.25344	−0.299127
$589$	−0.926610	−0.0381803
$590$	0	0
$591$	28.5224	1.17326
$592$	−25.8806	−1.06369
$593$	31.0847	1.27649	0.638247	−	0.769831i	$-0.279660\pi$
$593$	31.0847	1.27649	0.638247	+	0.769831i	$0.279660\pi$
$594$	2.92015	0.119815
$595$	0	0
$596$	16.2895	0.667244
$597$	−20.7347	−0.848614
$598$	2.89537	0.118400
$599$	−3.46676	−0.141648	−0.0708240	−	0.997489i	$-0.522563\pi$
$599$	−3.46676	−0.141648	−0.0708240	+	0.997489i	$0.522563\pi$
$600$	0	0
$601$	−22.6643	−0.924498	−0.462249	−	0.886750i	$-0.652957\pi$
$601$	−22.6643	−0.924498	−0.462249	+	0.886750i	$0.652957\pi$
$602$	12.2649	0.499879
$603$	−14.8129	−0.603229
$604$	0.116275	0.00473115
$605$	0	0
$606$	−9.55289	−0.388060
$607$	−45.2747	−1.83764	−0.918821	−	0.394675i	$-0.870857\pi$
$607$	−45.2747	−1.83764	−0.918821	+	0.394675i	$0.870857\pi$
$608$	−5.19445	−0.210663
$609$	33.9812	1.37699
$610$	0	0
$611$	−60.6863	−2.45511
$612$	17.7802	0.718723
$613$	32.5470	1.31456	0.657280	−	0.753646i	$-0.271707\pi$
$613$	32.5470	1.31456	0.657280	+	0.753646i	$0.271707\pi$
$614$	−8.40431	−0.339170
$615$	0	0
$616$	6.32245	0.254739
$617$	0.114512	0.00461007	0.00230503	−	0.999997i	$-0.499266\pi$
$617$	0.114512	0.00461007	0.00230503	+	0.999997i	$0.499266\pi$
$618$	−6.71455	−0.270099
$619$	−21.7981	−0.876139	−0.438069	−	0.898941i	$-0.644338\pi$
$619$	−21.7981	−0.876139	−0.438069	+	0.898941i	$0.644338\pi$
$620$	0	0
$621$	−6.24171	−0.250471
$622$	−12.1751	−0.488176
$623$	17.3463	0.694965
$624$	−14.6239	−0.585425
$625$	0	0
$626$	−10.4313	−0.416920
$627$	1.24468	0.0497075
$628$	−23.5926	−0.941447
$629$	−76.4213	−3.04712
$630$	0	0
$631$	−7.64209	−0.304227	−0.152113	−	0.988363i	$-0.548608\pi$
$631$	−7.64209	−0.304227	−0.152113	+	0.988363i	$0.548608\pi$
$632$	−6.00055	−0.238689
$633$	−13.7794	−0.547683
$634$	3.42688	0.136099
$635$	0	0
$636$	−16.9947	−0.673885
$637$	−16.4968	−0.653627
$638$	−4.46595	−0.176809
$639$	−1.42135	−0.0562279
$640$	0	0
$641$	−15.6286	−0.617291	−0.308646	−	0.951177i	$-0.599876\pi$
$641$	−15.6286	−0.617291	−0.308646	+	0.951177i	$0.599876\pi$
$642$	−3.48087	−0.137379
$643$	−21.8689	−0.862425	−0.431213	−	0.902250i	$-0.641914\pi$
$643$	−21.8689	−0.862425	−0.431213	+	0.902250i	$0.641914\pi$
$644$	−6.25260	−0.246387
$645$	0	0
$646$	−3.75127	−0.147592
$647$	44.3165	1.74226	0.871131	−	0.491051i	$-0.163387\pi$
$647$	44.3165	1.74226	0.871131	+	0.491051i	$0.163387\pi$
$648$	−4.98919	−0.195994
$649$	1.91691	0.0752454
$650$	0	0
$651$	−3.71650	−0.145661
$652$	−16.3741	−0.641259
$653$	−23.2868	−0.911284	−0.455642	−	0.890163i	$-0.650590\pi$
$653$	−23.2868	−0.911284	−0.455642	+	0.890163i	$0.650590\pi$
$654$	4.29287	0.167864
$655$	0	0
$656$	18.0707	0.705543
$657$	22.2138	0.866644
$658$	−21.1448	−0.824309
$659$	−33.4706	−1.30383	−0.651914	−	0.758293i	$-0.726034\pi$
$659$	−33.4706	−1.30383	−0.651914	+	0.758293i	$0.726034\pi$
$660$	0	0
$661$	4.30725	0.167533	0.0837663	−	0.996485i	$-0.473305\pi$
$661$	4.30725	0.167533	0.0837663	+	0.996485i	$0.473305\pi$
$662$	−0.490813	−0.0190760
$663$	−43.1820	−1.67705
$664$	−5.11701	−0.198579
$665$	0	0
$666$	−8.21226	−0.318219
$667$	9.54581	0.369615
$668$	6.05041	0.234097
$669$	−2.86464	−0.110754
$670$	0	0
$671$	−11.5308	−0.445142
$672$	−20.8342	−0.803695
$673$	−43.9621	−1.69461	−0.847307	−	0.531103i	$-0.821778\pi$
$673$	−43.9621	−1.69461	−0.847307	+	0.531103i	$0.821778\pi$
$674$	−7.37133	−0.283933
$675$	0	0
$676$	−18.5410	−0.713116
$677$	19.9483	0.766676	0.383338	−	0.923608i	$-0.374774\pi$
$677$	19.9483	0.766676	0.383338	+	0.923608i	$0.374774\pi$
$678$	3.62644	0.139272
$679$	50.3981	1.93410
$680$	0	0
$681$	−19.6963	−0.754763
$682$	0.488438	0.0187032
$683$	−41.1720	−1.57540	−0.787701	−	0.616058i	$-0.788729\pi$
$683$	−41.1720	−1.57540	−0.787701	+	0.616058i	$0.788729\pi$
$684$	2.49845	0.0955308
$685$	0	0
$686$	6.14230	0.234514
$687$	26.1967	0.999465
$688$	−17.4019	−0.663442
$689$	−38.6518	−1.47252
$690$	0	0
$691$	−7.48838	−0.284872	−0.142436	−	0.989804i	$-0.545493\pi$
$691$	−7.48838	−0.284872	−0.142436	+	0.989804i	$0.545493\pi$
$692$	6.28237	0.238820
$693$	−4.67502	−0.177589
$694$	16.2009	0.614979
$695$	0	0
$696$	20.6901	0.784256
$697$	53.3598	2.02115
$698$	6.84985	0.259271
$699$	−1.80685	−0.0683414
$700$	0	0
$701$	17.9421	0.677666	0.338833	−	0.940847i	$-0.389968\pi$
$701$	17.9421	0.677666	0.338833	+	0.940847i	$0.389968\pi$
$702$	−14.2359	−0.537299
$703$	−10.7386	−0.405015
$704$	−2.08199	−0.0784681
$705$	0	0
$706$	−9.66410	−0.363713
$707$	46.9188	1.76456
$708$	−4.10892	−0.154423
$709$	12.2738	0.460954	0.230477	−	0.973078i	$-0.425971\pi$
$709$	12.2738	0.460954	0.230477	+	0.973078i	$0.425971\pi$
$710$	0	0
$711$	4.43699	0.166400
$712$	10.5616	0.395814
$713$	−1.04402	−0.0390988
$714$	−15.0458	−0.563074
$715$	0	0
$716$	31.4631	1.17583
$717$	3.87951	0.144883
$718$	3.98158	0.148591
$719$	34.2879	1.27872	0.639361	−	0.768907i	$-0.279199\pi$
$719$	34.2879	1.27872	0.639361	+	0.768907i	$0.279199\pi$
$720$	0	0
$721$	32.9784	1.22818
$722$	−0.527123	−0.0196175
$723$	−31.5487	−1.17331
$724$	−8.39860	−0.312131
$725$	0	0
$726$	−0.656097	−0.0243500
$727$	15.7447	0.583939	0.291970	−	0.956428i	$-0.405689\pi$
$727$	15.7447	0.583939	0.291970	+	0.956428i	$0.405689\pi$
$728$	−30.8223	−1.14235
$729$	24.3748	0.902772
$730$	0	0
$731$	−51.3850	−1.90054
$732$	24.7164	0.913544
$733$	51.1191	1.88813	0.944063	−	0.329765i	$-0.106970\pi$
$733$	51.1191	1.88813	0.944063	+	0.329765i	$0.106970\pi$
$734$	−3.58890	−0.132469
$735$	0	0
$736$	−5.85262	−0.215731
$737$	10.2103	0.376101
$738$	5.73407	0.211074
$739$	−50.3516	−1.85221	−0.926107	−	0.377260i	$-0.876866\pi$
$739$	−50.3516	−1.85221	−0.926107	+	0.377260i	$0.876866\pi$
$740$	0	0
$741$	−6.06787	−0.222909
$742$	−13.4673	−0.494402
$743$	2.11671	0.0776544	0.0388272	−	0.999246i	$-0.487638\pi$
$743$	2.11671	0.0776544	0.0388272	+	0.999246i	$0.487638\pi$
$744$	−2.26286	−0.0829606
$745$	0	0
$746$	0.0548484	0.00200814
$747$	3.78368	0.138438
$748$	−12.2556	−0.448109
$749$	17.0962	0.624683
$750$	0	0
$751$	−7.61971	−0.278047	−0.139024	−	0.990289i	$-0.544396\pi$
$751$	−7.61971	−0.278047	−0.139024	+	0.990289i	$0.544396\pi$
$752$	30.0011	1.09403
$753$	14.9997	0.546618
$754$	21.7718	0.792882
$755$	0	0
$756$	30.7427	1.11810
$757$	−45.4268	−1.65107	−0.825533	−	0.564354i	$-0.809125\pi$
$757$	−45.4268	−1.65107	−0.825533	+	0.564354i	$0.809125\pi$
$758$	15.5377	0.564356
$759$	1.40239	0.0509034
$760$	0	0
$761$	3.78459	0.137191	0.0685957	−	0.997645i	$-0.478148\pi$
$761$	3.78459	0.137191	0.0685957	+	0.997645i	$0.478148\pi$
$762$	−1.95147	−0.0706944
$763$	−21.0843	−0.763304
$764$	5.19427	0.187922
$765$	0	0
$766$	7.94680	0.287130
$767$	−9.34508	−0.337431
$768$	−2.35335	−0.0849193
$769$	12.0500	0.434534	0.217267	−	0.976112i	$-0.430286\pi$
$769$	12.0500	0.434534	0.217267	+	0.976112i	$0.430286\pi$
$770$	0	0
$771$	21.3000	0.767101
$772$	−20.4064	−0.734441
$773$	5.54555	0.199460	0.0997298	−	0.995015i	$-0.468202\pi$
$773$	5.54555	0.199460	0.0997298	+	0.995015i	$0.468202\pi$
$774$	−5.52185	−0.198479
$775$	0	0
$776$	30.6859	1.10156
$777$	−43.0710	−1.54516
$778$	4.30604	0.154379
$779$	7.49805	0.268646
$780$	0	0
$781$	0.979715	0.0350570
$782$	−4.22658	−0.151142
$783$	−46.9347	−1.67731
$784$	8.15541	0.291265
$785$	0	0
$786$	−7.32456	−0.261258
$787$	−46.5114	−1.65795	−0.828976	−	0.559284i	$-0.811076\pi$
$787$	−46.5114	−1.65795	−0.828976	+	0.559284i	$0.811076\pi$
$788$	−39.4639	−1.40584
$789$	29.7304	1.05843
$790$	0	0
$791$	−17.8112	−0.633292
$792$	−2.84647	−0.101145
$793$	56.2135	1.99620
$794$	−7.74113	−0.274722
$795$	0	0
$796$	28.6887	1.01684
$797$	3.60512	0.127700	0.0638500	−	0.997960i	$-0.479662\pi$
$797$	3.60512	0.127700	0.0638500	+	0.997960i	$0.479662\pi$
$798$	−2.11421	−0.0748423
$799$	88.5883	3.13403
$800$	0	0
$801$	−7.80960	−0.275939
$802$	−0.602720	−0.0212828
$803$	−15.3116	−0.540335
$804$	−21.8858	−0.771854
$805$	0	0
$806$	−2.38117	−0.0838730
$807$	−6.41712	−0.225893
$808$	28.5674	1.00500
$809$	32.1749	1.13121	0.565605	−	0.824676i	$-0.308643\pi$
$809$	32.1749	1.13121	0.565605	+	0.824676i	$0.308643\pi$
$810$	0	0
$811$	−43.8016	−1.53808	−0.769041	−	0.639200i	$-0.779266\pi$
$811$	−43.8016	−1.53808	−0.769041	+	0.639200i	$0.779266\pi$
$812$	−47.0166	−1.64996
$813$	11.5202	0.404030
$814$	5.66057	0.198403
$815$	0	0
$816$	21.3476	0.747315
$817$	−7.22055	−0.252615
$818$	−18.2711	−0.638834
$819$	22.7910	0.796382
$820$	0	0
$821$	−37.6149	−1.31277	−0.656385	−	0.754426i	$-0.727915\pi$
$821$	−37.6149	−1.31277	−0.656385	+	0.754426i	$0.727915\pi$
$822$	8.76345	0.305660
$823$	8.52852	0.297285	0.148643	−	0.988891i	$-0.452510\pi$
$823$	8.52852	0.297285	0.148643	+	0.988891i	$0.452510\pi$
$824$	20.0795	0.699503
$825$	0	0
$826$	−3.25608	−0.113293
$827$	43.5444	1.51419	0.757093	−	0.653307i	$-0.226619\pi$
$827$	43.5444	1.51419	0.757093	+	0.653307i	$0.226619\pi$
$828$	2.81503	0.0978289
$829$	−1.16979	−0.0406286	−0.0203143	−	0.999794i	$-0.506467\pi$
$829$	−1.16979	−0.0406286	−0.0203143	+	0.999794i	$0.506467\pi$
$830$	0	0
$831$	−9.52273	−0.330340
$832$	10.1499	0.351883
$833$	24.0816	0.834378
$834$	−11.5021	−0.398286
$835$	0	0
$836$	−1.72214	−0.0595615
$837$	5.13322	0.177430
$838$	10.4825	0.362111
$839$	−45.7360	−1.57898	−0.789491	−	0.613762i	$-0.789656\pi$
$839$	−45.7360	−1.57898	−0.789491	+	0.613762i	$0.789656\pi$
$840$	0	0
$841$	42.7800	1.47517
$842$	−17.3628	−0.598362
$843$	20.5063	0.706275
$844$	19.0653	0.656255
$845$	0	0
$846$	9.51973	0.327295
$847$	3.22241	0.110723
$848$	19.1081	0.656173
$849$	13.1276	0.450537
$850$	0	0
$851$	−12.0993	−0.414758
$852$	−2.10003	−0.0719457
$853$	−56.7867	−1.94434	−0.972170	−	0.234277i	$-0.924728\pi$
$853$	−56.7867	−1.94434	−0.972170	+	0.234277i	$0.924728\pi$
$854$	19.5863	0.670230
$855$	0	0
$856$	10.4094	0.355785
$857$	−24.0415	−0.821241	−0.410620	−	0.911806i	$-0.634688\pi$
$857$	−24.0415	−0.821241	−0.410620	+	0.911806i	$0.634688\pi$
$858$	3.19852	0.109196
$859$	−21.4533	−0.731978	−0.365989	−	0.930619i	$-0.619269\pi$
$859$	−21.4533	−0.731978	−0.365989	+	0.930619i	$0.619269\pi$
$860$	0	0
$861$	30.0736	1.02490
$862$	11.2885	0.384488
$863$	−25.3546	−0.863082	−0.431541	−	0.902093i	$-0.642030\pi$
$863$	−25.3546	−0.863082	−0.431541	+	0.902093i	$0.642030\pi$
$864$	28.7761	0.978983
$865$	0	0
$866$	16.7329	0.568606
$867$	41.8764	1.42220
$868$	5.14217	0.174537
$869$	−3.05834	−0.103747
$870$	0	0
$871$	−49.7759	−1.68659
$872$	−12.8376	−0.434736
$873$	−22.6901	−0.767944
$874$	−0.593913	−0.0200894
$875$	0	0
$876$	32.8205	1.10890
$877$	−39.1208	−1.32102	−0.660508	−	0.750819i	$-0.729659\pi$
$877$	−39.1208	−1.32102	−0.660508	+	0.750819i	$0.729659\pi$
$878$	−4.95503	−0.167224
$879$	33.7397	1.13801
$880$	0	0
$881$	−18.4388	−0.621219	−0.310610	−	0.950538i	$-0.600533\pi$
$881$	−18.4388	−0.621219	−0.310610	+	0.950538i	$0.600533\pi$
$882$	2.58782	0.0871364
$883$	−4.68809	−0.157767	−0.0788834	−	0.996884i	$-0.525135\pi$
$883$	−4.68809	−0.157767	−0.0788834	+	0.996884i	$0.525135\pi$
$884$	59.7469	2.00950
$885$	0	0
$886$	−11.3449	−0.381140
$887$	−29.0079	−0.973990	−0.486995	−	0.873405i	$-0.661907\pi$
$887$	−29.0079	−0.973990	−0.486995	+	0.873405i	$0.661907\pi$
$888$	−26.2246	−0.880041
$889$	9.58463	0.321458
$890$	0	0
$891$	−2.54288	−0.0851896
$892$	3.96354	0.132709
$893$	12.4483	0.416567
$894$	6.20593	0.207557
$895$	0	0
$896$	37.0137	1.23654
$897$	−6.83672	−0.228271
$898$	−16.2089	−0.540897
$899$	−7.85053	−0.261830
$900$	0	0
$901$	56.4229	1.87972
$902$	−3.95240	−0.131600
$903$	−28.9606	−0.963747
$904$	−10.8447	−0.360689
$905$	0	0
$906$	0.0442980	0.00147170
$907$	−31.2319	−1.03704	−0.518519	−	0.855066i	$-0.673516\pi$
$907$	−31.2319	−1.03704	−0.518519	+	0.855066i	$0.673516\pi$
$908$	27.2519	0.904386
$909$	−21.1237	−0.700628
$910$	0	0
$911$	−12.0740	−0.400031	−0.200015	−	0.979793i	$-0.564099\pi$
$911$	−12.0740	−0.400031	−0.200015	+	0.979793i	$0.564099\pi$
$912$	2.99973	0.0993311
$913$	−2.60802	−0.0863130
$914$	0.267283	0.00884095
$915$	0	0
$916$	−36.2459	−1.19760
$917$	35.9744	1.18798
$918$	20.7812	0.685882
$919$	−28.9331	−0.954416	−0.477208	−	0.878790i	$-0.658351\pi$
$919$	−28.9331	−0.954416	−0.477208	+	0.878790i	$0.658351\pi$
$920$	0	0
$921$	19.8448	0.653907
$922$	−2.95257	−0.0972376
$923$	−4.77618	−0.157210
$924$	−6.90725	−0.227232
$925$	0	0
$926$	−11.8766	−0.390288
$927$	−14.8474	−0.487653
$928$	−44.0090	−1.44466
$929$	−45.6501	−1.49773	−0.748866	−	0.662722i	$-0.769401\pi$
$929$	−45.6501	−1.49773	−0.748866	+	0.662722i	$0.769401\pi$
$930$	0	0
$931$	3.38391	0.110903
$932$	2.49997	0.0818893
$933$	28.7485	0.941185
$934$	7.96250	0.260541
$935$	0	0
$936$	13.8767	0.453576
$937$	−16.6566	−0.544146	−0.272073	−	0.962277i	$-0.587709\pi$
$937$	−16.6566	−0.544146	−0.272073	+	0.962277i	$0.587709\pi$
$938$	−17.3433	−0.566278
$939$	24.6311	0.803806
$940$	0	0
$941$	17.6743	0.576165	0.288082	−	0.957606i	$-0.406982\pi$
$941$	17.6743	0.576165	0.288082	+	0.957606i	$0.406982\pi$
$942$	−8.98824	−0.292853
$943$	8.44811	0.275108
$944$	4.61986	0.150364
$945$	0	0
$946$	3.80612	0.123748
$947$	18.3019	0.594731	0.297366	−	0.954764i	$-0.403892\pi$
$947$	18.3019	0.594731	0.297366	+	0.954764i	$0.403892\pi$
$948$	6.55558	0.212915
$949$	74.6451	2.42308
$950$	0	0
$951$	−8.09177	−0.262394
$952$	44.9937	1.45825
$953$	11.9242	0.386261	0.193131	−	0.981173i	$-0.438136\pi$
$953$	11.9242	0.386261	0.193131	+	0.981173i	$0.438136\pi$
$954$	6.06323	0.196304
$955$	0	0
$956$	−5.36772	−0.173604
$957$	10.5453	0.340880
$958$	1.04078	0.0336261
$959$	−43.0415	−1.38988
$960$	0	0
$961$	−30.1414	−0.972303
$962$	−27.5957	−0.889720
$963$	−7.69702	−0.248033
$964$	43.6510	1.40590
$965$	0	0
$966$	−2.38210	−0.0766428
$967$	−9.97275	−0.320702	−0.160351	−	0.987060i	$-0.551263\pi$
$967$	−9.97275	−0.320702	−0.160351	+	0.987060i	$0.551263\pi$
$968$	1.96203	0.0630619
$969$	8.85772	0.284551
$970$	0	0
$971$	−20.4714	−0.656960	−0.328480	−	0.944511i	$-0.606536\pi$
$971$	−20.4714	−0.656960	−0.328480	+	0.944511i	$0.606536\pi$
$972$	−23.1702	−0.743184
$973$	56.4924	1.81106
$974$	3.28792	0.105352
$975$	0	0
$976$	−27.7899	−0.889533
$977$	−15.9786	−0.511202	−0.255601	−	0.966782i	$-0.582273\pi$
$977$	−15.9786	−0.511202	−0.255601	+	0.966782i	$0.582273\pi$
$978$	−6.23816	−0.199474
$979$	5.38303	0.172042
$980$	0	0
$981$	9.49253	0.303073
$982$	−15.5013	−0.494665
$983$	10.3802	0.331077	0.165538	−	0.986203i	$-0.447064\pi$
$983$	10.3802	0.331077	0.165538	+	0.986203i	$0.447064\pi$
$984$	18.3109	0.583730
$985$	0	0
$986$	−31.7819	−1.01214
$987$	49.9283	1.58924
$988$	8.39555	0.267098
$989$	−8.13545	−0.258692
$990$	0	0
$991$	47.3252	1.50334	0.751668	−	0.659542i	$-0.229250\pi$
$991$	47.3252	1.50334	0.751668	+	0.659542i	$0.229250\pi$
$992$	4.81323	0.152820
$993$	1.15894	0.0367778
$994$	−1.66415	−0.0527836
$995$	0	0
$996$	5.59032	0.177136
$997$	1.92478	0.0609584	0.0304792	−	0.999535i	$-0.490297\pi$
$997$	1.92478	0.0609584	0.0304792	+	0.999535i	$0.490297\pi$
$998$	4.39416	0.139095
$999$	59.4896	1.88217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.y.1.5	yes	15
5.4	even	2		5225.2.a.r.1.11	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.r.1.11	✓	15	5.4	even	2
5225.2.a.y.1.5	yes	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.527123 q^{2} +1.24468 q^{3} -1.72214 q^{4} -0.656097 q^{6} +3.22241 q^{7} +1.96203 q^{8} -1.45078 q^{9} +1.00000 q^{11} -2.14351 q^{12} -4.87506 q^{13} -1.69861 q^{14} +2.41005 q^{16} +7.11649 q^{17} +0.764741 q^{18} +1.00000 q^{19} +4.01085 q^{21} -0.527123 q^{22} +1.12671 q^{23} +2.44209 q^{24} +2.56976 q^{26} -5.53978 q^{27} -5.54944 q^{28} +8.47231 q^{29} -0.926610 q^{31} -5.19445 q^{32} +1.24468 q^{33} -3.75127 q^{34} +2.49845 q^{36} -10.7386 q^{37} -0.527123 q^{38} -6.06787 q^{39} +7.49805 q^{41} -2.11421 q^{42} -7.22055 q^{43} -1.72214 q^{44} -0.593913 q^{46} +12.4483 q^{47} +2.99973 q^{48} +3.38391 q^{49} +8.85772 q^{51} +8.39555 q^{52} +7.92848 q^{53} +2.92015 q^{54} +6.32245 q^{56} +1.24468 q^{57} -4.46595 q^{58} +1.91691 q^{59} -11.5308 q^{61} +0.488438 q^{62} -4.67502 q^{63} -2.08199 q^{64} -0.656097 q^{66} +10.2103 q^{67} -12.2556 q^{68} +1.40239 q^{69} +0.979715 q^{71} -2.84647 q^{72} -15.3116 q^{73} +5.66057 q^{74} -1.72214 q^{76} +3.22241 q^{77} +3.19852 q^{78} -3.05834 q^{79} -2.54288 q^{81} -3.95240 q^{82} -2.60802 q^{83} -6.90725 q^{84} +3.80612 q^{86} +10.5453 q^{87} +1.96203 q^{88} +5.38303 q^{89} -15.7094 q^{91} -1.94035 q^{92} -1.15333 q^{93} -6.56179 q^{94} -6.46540 q^{96} +15.6399 q^{97} -1.78374 q^{98} -1.45078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) \) 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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