LMFDB - Embedded newform 5225.2.a.y.1.7

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			1.7
Root			$-0.102671$ 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)

$f(q)$	$=$	\(q-0.102671 q^{2} -2.39685 q^{3} -1.98946 q^{4} +0.246087 q^{6} -2.47527 q^{7} +0.409602 q^{8} +2.74488 q^{9} +1.00000 q^{11} +4.76843 q^{12} +4.28964 q^{13} +0.254139 q^{14} +3.93686 q^{16} +6.76526 q^{17} -0.281820 q^{18} +1.00000 q^{19} +5.93286 q^{21} -0.102671 q^{22} +4.01461 q^{23} -0.981755 q^{24} -0.440422 q^{26} +0.611478 q^{27} +4.92445 q^{28} -5.47946 q^{29} +3.31201 q^{31} -1.22341 q^{32} -2.39685 q^{33} -0.694597 q^{34} -5.46083 q^{36} -6.41117 q^{37} -0.102671 q^{38} -10.2816 q^{39} -7.20691 q^{41} -0.609133 q^{42} +10.3647 q^{43} -1.98946 q^{44} -0.412185 q^{46} +3.38016 q^{47} -9.43606 q^{48} -0.873022 q^{49} -16.2153 q^{51} -8.53406 q^{52} -6.42147 q^{53} -0.0627811 q^{54} -1.01388 q^{56} -2.39685 q^{57} +0.562583 q^{58} -3.10281 q^{59} -0.0582058 q^{61} -0.340048 q^{62} -6.79433 q^{63} -7.74812 q^{64} +0.246087 q^{66} -1.24707 q^{67} -13.4592 q^{68} -9.62241 q^{69} +9.20338 q^{71} +1.12431 q^{72} +11.4157 q^{73} +0.658242 q^{74} -1.98946 q^{76} -2.47527 q^{77} +1.05562 q^{78} -11.4226 q^{79} -9.70027 q^{81} +0.739942 q^{82} -3.70335 q^{83} -11.8032 q^{84} -1.06416 q^{86} +13.1334 q^{87} +0.409602 q^{88} +8.44912 q^{89} -10.6180 q^{91} -7.98690 q^{92} -7.93840 q^{93} -0.347044 q^{94} +2.93232 q^{96} +4.30242 q^{97} +0.0896341 q^{98} +2.74488 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.102671	−0.0725995	−0.0362997	−	0.999341i	$-0.511557\pi$
$2$	−0.102671	−0.0725995	−0.0362997	+	0.999341i	$0.511557\pi$
$3$	−2.39685	−1.38382	−0.691911	−	0.721983i	$-0.743231\pi$
$3$	−2.39685	−1.38382	−0.691911	+	0.721983i	$0.743231\pi$
$4$	−1.98946	−0.994729
$5$	0	0
$5$	0	0
$6$	0.246087	0.100465
$7$	−2.47527	−0.935565	−0.467783	−	0.883844i	$-0.654947\pi$
$7$	−2.47527	−0.935565	−0.467783	+	0.883844i	$0.654947\pi$
$8$	0.409602	0.144816
$9$	2.74488	0.914961
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	4.76843	1.37653
$13$	4.28964	1.18973	0.594866	−	0.803825i	$-0.297205\pi$
$13$	4.28964	1.18973	0.594866	+	0.803825i	$0.297205\pi$
$14$	0.254139	0.0679216
$15$	0	0
$16$	3.93686	0.984216
$17$	6.76526	1.64082	0.820408	−	0.571778i	$-0.193746\pi$
$17$	6.76526	1.64082	0.820408	+	0.571778i	$0.193746\pi$
$18$	−0.281820	−0.0664257
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	5.93286	1.29466
$22$	−0.102671	−0.0218896
$23$	4.01461	0.837104	0.418552	−	0.908193i	$-0.362538\pi$
$23$	4.01461	0.837104	0.418552	+	0.908193i	$0.362538\pi$
$24$	−0.981755	−0.200400
$25$	0	0
$26$	−0.440422	−0.0863739
$27$	0.611478	0.117679
$28$	4.92445	0.930634
$29$	−5.47946	−1.01751	−0.508755	−	0.860911i	$-0.669894\pi$
$29$	−5.47946	−1.01751	−0.508755	+	0.860911i	$0.669894\pi$
$30$	0	0
$31$	3.31201	0.594855	0.297428	−	0.954744i	$-0.403871\pi$
$31$	3.31201	0.594855	0.297428	+	0.954744i	$0.403871\pi$
$32$	−1.22341	−0.216270
$33$	−2.39685	−0.417238
$34$	−0.694597	−0.119122
$35$	0	0
$36$	−5.46083	−0.910138
$37$	−6.41117	−1.05399	−0.526995	−	0.849868i	$-0.676681\pi$
$37$	−6.41117	−1.05399	−0.526995	+	0.849868i	$0.676681\pi$
$38$	−0.102671	−0.0166555
$39$	−10.2816	−1.64638
$40$	0	0
$41$	−7.20691	−1.12553	−0.562765	−	0.826617i	$-0.690263\pi$
$41$	−7.20691	−1.12553	−0.562765	+	0.826617i	$0.690263\pi$
$42$	−0.609133	−0.0939913
$43$	10.3647	1.58061	0.790304	−	0.612715i	$-0.209923\pi$
$43$	10.3647	1.58061	0.790304	+	0.612715i	$0.209923\pi$
$44$	−1.98946	−0.299922
$45$	0	0
$46$	−0.412185	−0.0607733
$47$	3.38016	0.493046	0.246523	−	0.969137i	$-0.420712\pi$
$47$	3.38016	0.493046	0.246523	+	0.969137i	$0.420712\pi$
$48$	−9.43606	−1.36198
$49$	−0.873022	−0.124717
$50$	0	0
$51$	−16.2153	−2.27060
$52$	−8.53406	−1.18346
$53$	−6.42147	−0.882056	−0.441028	−	0.897493i	$-0.645386\pi$
$53$	−6.42147	−0.882056	−0.441028	+	0.897493i	$0.645386\pi$
$54$	−0.0627811	−0.00854343
$55$	0	0
$56$	−1.01388	−0.135485
$57$	−2.39685	−0.317470
$58$	0.562583	0.0738707
$59$	−3.10281	−0.403951	−0.201976	−	0.979391i	$-0.564736\pi$
$59$	−3.10281	−0.403951	−0.201976	+	0.979391i	$0.564736\pi$
$60$	0	0
$61$	−0.0582058	−0.00745249	−0.00372624	−	0.999993i	$-0.501186\pi$
$61$	−0.0582058	−0.00745249	−0.00372624	+	0.999993i	$0.501186\pi$
$62$	−0.340048	−0.0431862
$63$	−6.79433	−0.856006
$64$	−7.74812	−0.968515
$65$	0	0
$66$	0.246087	0.0302912
$67$	−1.24707	−0.152354	−0.0761769	−	0.997094i	$-0.524271\pi$
$67$	−1.24707	−0.152354	−0.0761769	+	0.997094i	$0.524271\pi$
$68$	−13.4592	−1.63217
$69$	−9.62241	−1.15840
$70$	0	0
$71$	9.20338	1.09224	0.546120	−	0.837707i	$-0.316104\pi$
$71$	9.20338	1.09224	0.546120	+	0.837707i	$0.316104\pi$
$72$	1.12431	0.132501
$73$	11.4157	1.33611	0.668053	−	0.744113i	$-0.267128\pi$
$73$	11.4157	1.33611	0.668053	+	0.744113i	$0.267128\pi$
$74$	0.658242	0.0765191
$75$	0	0
$76$	−1.98946	−0.228207
$77$	−2.47527	−0.282084
$78$	1.05562	0.119526
$79$	−11.4226	−1.28515	−0.642573	−	0.766225i	$-0.722133\pi$
$79$	−11.4226	−1.28515	−0.642573	+	0.766225i	$0.722133\pi$
$80$	0	0
$81$	−9.70027	−1.07781
$82$	0.739942	0.0817129
$83$	−3.70335	−0.406496	−0.203248	−	0.979127i	$-0.565150\pi$
$83$	−3.70335	−0.406496	−0.203248	+	0.979127i	$0.565150\pi$
$84$	−11.8032	−1.28783
$85$	0	0
$86$	−1.06416	−0.114751
$87$	13.1334	1.40805
$88$	0.409602	0.0436638
$89$	8.44912	0.895605	0.447803	−	0.894132i	$-0.352207\pi$
$89$	8.44912	0.895605	0.447803	+	0.894132i	$0.352207\pi$
$90$	0	0
$91$	−10.6180	−1.11307
$92$	−7.98690	−0.832692
$93$	−7.93840	−0.823173
$94$	−0.347044	−0.0357949
$95$	0	0
$96$	2.93232	0.299279
$97$	4.30242	0.436844	0.218422	−	0.975854i	$-0.429909\pi$
$97$	4.30242	0.436844	0.218422	+	0.975854i	$0.429909\pi$
$98$	0.0896341	0.00905441
$99$	2.74488	0.275871
$100$	0	0
$101$	14.5786	1.45062	0.725311	−	0.688421i	$-0.241696\pi$
$101$	14.5786	1.45062	0.725311	+	0.688421i	$0.241696\pi$
$102$	1.66484	0.164844
$103$	8.38445	0.826145	0.413072	−	0.910698i	$-0.364456\pi$
$103$	8.38445	0.826145	0.413072	+	0.910698i	$0.364456\pi$
$104$	1.75705	0.172292
$105$	0	0
$106$	0.659299	0.0640368
$107$	2.44565	0.236430	0.118215	−	0.992988i	$-0.462283\pi$
$107$	2.44565	0.236430	0.118215	+	0.992988i	$0.462283\pi$
$108$	−1.21651	−0.117059
$109$	−14.8048	−1.41804	−0.709021	−	0.705187i	$-0.750863\pi$
$109$	−14.8048	−1.41804	−0.709021	+	0.705187i	$0.750863\pi$
$110$	0	0
$111$	15.3666	1.45853
$112$	−9.74481	−0.920798
$113$	−5.81699	−0.547216	−0.273608	−	0.961841i	$-0.588217\pi$
$113$	−5.81699	−0.547216	−0.273608	+	0.961841i	$0.588217\pi$
$114$	0.246087	0.0230482
$115$	0	0
$116$	10.9012	1.01215
$117$	11.7745	1.08856
$118$	0.318569	0.0293266
$119$	−16.7459	−1.53509
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.00597605	0.000541047 0
$123$	17.2739	1.55753
$124$	−6.58911	−0.591720
$125$	0	0
$126$	0.697582	0.0621456
$127$	9.69084	0.859923	0.429962	−	0.902847i	$-0.358527\pi$
$127$	9.69084	0.859923	0.429962	+	0.902847i	$0.358527\pi$
$128$	3.24232	0.286583
$129$	−24.8427	−2.18728
$130$	0	0
$131$	7.65630	0.668934	0.334467	−	0.942407i	$-0.391444\pi$
$131$	7.65630	0.668934	0.334467	+	0.942407i	$0.391444\pi$
$132$	4.76843	0.415039
$133$	−2.47527	−0.214633
$134$	0.128038	0.0110608
$135$	0	0
$136$	2.77107	0.237617
$137$	−5.33128	−0.455482	−0.227741	−	0.973722i	$-0.573134\pi$
$137$	−5.33128	−0.455482	−0.227741	+	0.973722i	$0.573134\pi$
$138$	0.987944	0.0840994
$139$	7.77150	0.659170	0.329585	−	0.944126i	$-0.393091\pi$
$139$	7.77150	0.659170	0.329585	+	0.944126i	$0.393091\pi$
$140$	0	0
$141$	−8.10172	−0.682288
$142$	−0.944922	−0.0792961
$143$	4.28964	0.358717
$144$	10.8062	0.900519
$145$	0	0
$146$	−1.17206	−0.0970007
$147$	2.09250	0.172587
$148$	12.7548	1.04843
$149$	−7.14887	−0.585658	−0.292829	−	0.956165i	$-0.594597\pi$
$149$	−7.14887	−0.585658	−0.292829	+	0.956165i	$0.594597\pi$
$150$	0	0
$151$	4.14593	0.337391	0.168696	−	0.985668i	$-0.446044\pi$
$151$	4.14593	0.337391	0.168696	+	0.985668i	$0.446044\pi$
$152$	0.409602	0.0332231
$153$	18.5698	1.50128
$154$	0.254139	0.0204791
$155$	0	0
$156$	20.4548	1.63770
$157$	13.6699	1.09097	0.545487	−	0.838119i	$-0.316345\pi$
$157$	13.6699	1.09097	0.545487	+	0.838119i	$0.316345\pi$
$158$	1.17277	0.0933009
$159$	15.3913	1.22061
$160$	0	0
$161$	−9.93726	−0.783166
$162$	0.995938	0.0782483
$163$	1.85837	0.145558	0.0727792	−	0.997348i	$-0.476813\pi$
$163$	1.85837	0.145558	0.0727792	+	0.997348i	$0.476813\pi$
$164$	14.3379	1.11960
$165$	0	0
$166$	0.380227	0.0295114
$167$	−11.2750	−0.872486	−0.436243	−	0.899829i	$-0.643691\pi$
$167$	−11.2750	−0.872486	−0.436243	+	0.899829i	$0.643691\pi$
$168$	2.43011	0.187487
$169$	5.40099	0.415460
$170$	0	0
$171$	2.74488	0.209906
$172$	−20.6202	−1.57228
$173$	13.0474	0.991973	0.495986	−	0.868330i	$-0.334807\pi$
$173$	13.0474	0.991973	0.495986	+	0.868330i	$0.334807\pi$
$174$	−1.34843	−0.102224
$175$	0	0
$176$	3.93686	0.296752
$177$	7.43696	0.558996
$178$	−0.867481	−0.0650205
$179$	−10.1413	−0.757996	−0.378998	−	0.925398i	$-0.623731\pi$
$179$	−10.1413	−0.757996	−0.378998	+	0.925398i	$0.623731\pi$
$180$	0	0
$181$	−1.19695	−0.0889688	−0.0444844	−	0.999010i	$-0.514164\pi$
$181$	−1.19695	−0.0889688	−0.0444844	+	0.999010i	$0.514164\pi$
$182$	1.09016	0.0808084
$183$	0.139510	0.0103129
$184$	1.64439	0.121226
$185$	0	0
$186$	0.815044	0.0597619
$187$	6.76526	0.494725
$188$	−6.72468	−0.490448
$189$	−1.51357	−0.110096
$190$	0	0
$191$	12.4942	0.904046	0.452023	−	0.892006i	$-0.350702\pi$
$191$	12.4942	0.904046	0.452023	+	0.892006i	$0.350702\pi$
$192$	18.5711	1.34025
$193$	−11.8564	−0.853441	−0.426721	−	0.904383i	$-0.640331\pi$
$193$	−11.8564	−0.853441	−0.426721	+	0.904383i	$0.640331\pi$
$194$	−0.441734	−0.0317147
$195$	0	0
$196$	1.73684	0.124060
$197$	−26.1005	−1.85959	−0.929793	−	0.368083i	$-0.880014\pi$
$197$	−26.1005	−1.85959	−0.929793	+	0.368083i	$0.880014\pi$
$198$	−0.281820	−0.0200281
$199$	12.0041	0.850945	0.425472	−	0.904971i	$-0.360108\pi$
$199$	12.0041	0.850945	0.425472	+	0.904971i	$0.360108\pi$
$200$	0	0
$201$	2.98904	0.210830
$202$	−1.49680	−0.105314
$203$	13.5632	0.951948
$204$	32.2597	2.25863
$205$	0	0
$206$	−0.860841	−0.0599777
$207$	11.0196	0.765918
$208$	16.8877	1.17095
$209$	1.00000	0.0691714
$210$	0	0
$211$	17.1841	1.18300	0.591500	−	0.806305i	$-0.298536\pi$
$211$	17.1841	1.18300	0.591500	+	0.806305i	$0.298536\pi$
$212$	12.7752	0.877407
$213$	−22.0591	−1.51147
$214$	−0.251098	−0.0171647
$215$	0	0
$216$	0.250463	0.0170418
$217$	−8.19814	−0.556526
$218$	1.52003	0.102949
$219$	−27.3617	−1.84893
$220$	0	0
$221$	29.0205	1.95213
$222$	−1.57771	−0.105889
$223$	−18.1146	−1.21305	−0.606523	−	0.795066i	$-0.707436\pi$
$223$	−18.1146	−1.21305	−0.606523	+	0.795066i	$0.707436\pi$
$224$	3.02827	0.202335
$225$	0	0
$226$	0.597237	0.0397276
$227$	18.6952	1.24084	0.620422	−	0.784268i	$-0.286961\pi$
$227$	18.6952	1.24084	0.620422	+	0.784268i	$0.286961\pi$
$228$	4.76843	0.315797
$229$	−27.1453	−1.79381	−0.896906	−	0.442220i	$-0.854191\pi$
$229$	−27.1453	−1.79381	−0.896906	+	0.442220i	$0.854191\pi$
$230$	0	0
$231$	5.93286	0.390353
$232$	−2.24440	−0.147352
$233$	−5.95158	−0.389901	−0.194950	−	0.980813i	$-0.562455\pi$
$233$	−5.95158	−0.389901	−0.194950	+	0.980813i	$0.562455\pi$
$234$	−1.20891	−0.0790287
$235$	0	0
$236$	6.17291	0.401822
$237$	27.3783	1.77841
$238$	1.71932	0.111447
$239$	6.12898	0.396451	0.198225	−	0.980156i	$-0.436482\pi$
$239$	6.12898	0.396451	0.198225	+	0.980156i	$0.436482\pi$
$240$	0	0
$241$	−27.8382	−1.79322	−0.896609	−	0.442823i	$-0.853977\pi$
$241$	−27.8382	−1.79322	−0.896609	+	0.442823i	$0.853977\pi$
$242$	−0.102671	−0.00659995
$243$	21.4156	1.37381
$244$	0.115798	0.00741321
$245$	0	0
$246$	−1.77353	−0.113076
$247$	4.28964	0.272943
$248$	1.35661	0.0861447
$249$	8.87637	0.562517
$250$	0	0
$251$	−16.9502	−1.06989	−0.534943	−	0.844888i	$-0.679667\pi$
$251$	−16.9502	−1.06989	−0.534943	+	0.844888i	$0.679667\pi$
$252$	13.5170	0.851494
$253$	4.01461	0.252396
$254$	−0.994970	−0.0624300
$255$	0	0
$256$	15.1633	0.947709
$257$	23.6602	1.47588	0.737942	−	0.674864i	$-0.235798\pi$
$257$	23.6602	1.47588	0.737942	+	0.674864i	$0.235798\pi$
$258$	2.55063	0.158795
$259$	15.8694	0.986077
$260$	0	0
$261$	−15.0405	−0.930983
$262$	−0.786082	−0.0485643
$263$	15.7211	0.969406	0.484703	−	0.874679i	$-0.338928\pi$
$263$	15.7211	0.969406	0.484703	+	0.874679i	$0.338928\pi$
$264$	−0.981755	−0.0604228
$265$	0	0
$266$	0.254139	0.0155823
$267$	−20.2513	−1.23936
$268$	2.48099	0.151551
$269$	18.5449	1.13070	0.565351	−	0.824850i	$-0.308741\pi$
$269$	18.5449	1.13070	0.565351	+	0.824850i	$0.308741\pi$
$270$	0	0
$271$	−8.57424	−0.520848	−0.260424	−	0.965494i	$-0.583862\pi$
$271$	−8.57424	−0.520848	−0.260424	+	0.965494i	$0.583862\pi$
$272$	26.6339	1.61492
$273$	25.4498	1.54029
$274$	0.547369	0.0330678
$275$	0	0
$276$	19.1434	1.15230
$277$	−4.01070	−0.240980	−0.120490	−	0.992715i	$-0.538447\pi$
$277$	−4.01070	−0.240980	−0.120490	+	0.992715i	$0.538447\pi$
$278$	−0.797909	−0.0478554
$279$	9.09109	0.544269
$280$	0	0
$281$	−23.6412	−1.41031	−0.705157	−	0.709051i	$-0.749124\pi$
$281$	−23.6412	−1.41031	−0.705157	+	0.709051i	$0.749124\pi$
$282$	0.831813	0.0495337
$283$	−14.3662	−0.853979	−0.426990	−	0.904256i	$-0.640426\pi$
$283$	−14.3662	−0.853979	−0.426990	+	0.904256i	$0.640426\pi$
$284$	−18.3098	−1.08648
$285$	0	0
$286$	−0.440422	−0.0260427
$287$	17.8391	1.05301
$288$	−3.35811	−0.197878
$289$	28.7688	1.69228
$290$	0	0
$291$	−10.3122	−0.604514
$292$	−22.7111	−1.32906
$293$	9.21535	0.538366	0.269183	−	0.963089i	$-0.413246\pi$
$293$	9.21535	0.538366	0.269183	+	0.963089i	$0.413246\pi$
$294$	−0.214839	−0.0125297
$295$	0	0
$296$	−2.62603	−0.152635
$297$	0.611478	0.0354815
$298$	0.733983	0.0425185
$299$	17.2212	0.995929
$300$	0	0
$301$	−25.6556	−1.47876
$302$	−0.425668	−0.0244944
$303$	−34.9426	−2.00740
$304$	3.93686	0.225795
$305$	0	0
$306$	−1.90659	−0.108992
$307$	−23.7059	−1.35297	−0.676483	−	0.736458i	$-0.736497\pi$
$307$	−23.7059	−1.35297	−0.676483	+	0.736458i	$0.736497\pi$
$308$	4.92445	0.280597
$309$	−20.0963	−1.14324
$310$	0	0
$311$	13.5580	0.768806	0.384403	−	0.923165i	$-0.374407\pi$
$311$	13.5580	0.768806	0.384403	+	0.923165i	$0.374407\pi$
$312$	−4.21137	−0.238422
$313$	−7.33205	−0.414432	−0.207216	−	0.978295i	$-0.566440\pi$
$313$	−7.33205	−0.414432	−0.207216	+	0.978295i	$0.566440\pi$
$314$	−1.40350	−0.0792041
$315$	0	0
$316$	22.7248	1.27837
$317$	13.8194	0.776177	0.388088	−	0.921622i	$-0.373136\pi$
$317$	13.8194	0.776177	0.388088	+	0.921622i	$0.373136\pi$
$318$	−1.58024	−0.0886155
$319$	−5.47946	−0.306791
$320$	0	0
$321$	−5.86185	−0.327177
$322$	1.02027	0.0568574
$323$	6.76526	0.376429
$324$	19.2983	1.07213
$325$	0	0
$326$	−0.190801	−0.0105675
$327$	35.4848	1.96232
$328$	−2.95197	−0.162995
$329$	−8.36681	−0.461277
$330$	0	0
$331$	−20.7661	−1.14141	−0.570705	−	0.821155i	$-0.693330\pi$
$331$	−20.7661	−1.14141	−0.570705	+	0.821155i	$0.693330\pi$
$332$	7.36766	0.404353
$333$	−17.5979	−0.964360
$334$	1.15762	0.0633421
$335$	0	0
$336$	23.3568	1.27422
$337$	26.5298	1.44517	0.722586	−	0.691281i	$-0.242953\pi$
$337$	26.5298	1.44517	0.722586	+	0.691281i	$0.242953\pi$
$338$	−0.554525	−0.0301622
$339$	13.9424	0.757249
$340$	0	0
$341$	3.31201	0.179356
$342$	−0.281820	−0.0152391
$343$	19.4879	1.05225
$344$	4.24542	0.228898
$345$	0	0
$346$	−1.33959	−0.0720167
$347$	11.9044	0.639062	0.319531	−	0.947576i	$-0.396475\pi$
$347$	11.9044	0.639062	0.319531	+	0.947576i	$0.396475\pi$
$348$	−26.1284	−1.40063
$349$	1.01766	0.0544741	0.0272370	−	0.999629i	$-0.491329\pi$
$349$	1.01766	0.0544741	0.0272370	+	0.999629i	$0.491329\pi$
$350$	0	0
$351$	2.62302	0.140006
$352$	−1.22341	−0.0652078
$353$	18.3943	0.979029	0.489514	−	0.871995i	$-0.337174\pi$
$353$	18.3943	0.979029	0.489514	+	0.871995i	$0.337174\pi$
$354$	−0.763561	−0.0405828
$355$	0	0
$356$	−16.8092	−0.890885
$357$	40.1373	2.12429
$358$	1.04122	0.0550301
$359$	9.94068	0.524649	0.262325	−	0.964980i	$-0.415511\pi$
$359$	9.94068	0.524649	0.262325	+	0.964980i	$0.415511\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.122892	0.00645909
$363$	−2.39685	−0.125802
$364$	21.1241	1.10720
$365$	0	0
$366$	−0.0143237	−0.000748712 0
$367$	−2.77988	−0.145108	−0.0725542	−	0.997364i	$-0.523115\pi$
$367$	−2.77988	−0.145108	−0.0725542	+	0.997364i	$0.523115\pi$
$368$	15.8050	0.823891
$369$	−19.7821	−1.02982
$370$	0	0
$371$	15.8949	0.825221
$372$	15.7931	0.818835
$373$	20.7775	1.07582	0.537908	−	0.843004i	$-0.319215\pi$
$373$	20.7775	1.07582	0.537908	+	0.843004i	$0.319215\pi$
$374$	−0.694597	−0.0359168
$375$	0	0
$376$	1.38452	0.0714011
$377$	−23.5049	−1.21056
$378$	0.155400	0.00799294
$379$	20.9040	1.07377	0.536883	−	0.843657i	$-0.319602\pi$
$379$	20.9040	1.07377	0.536883	+	0.843657i	$0.319602\pi$
$380$	0	0
$381$	−23.2275	−1.18998
$382$	−1.28279	−0.0656333
$383$	−24.0145	−1.22708	−0.613542	−	0.789662i	$-0.710256\pi$
$383$	−24.0145	−1.22708	−0.613542	+	0.789662i	$0.710256\pi$
$384$	−7.77135	−0.396580
$385$	0	0
$386$	1.21731	0.0619594
$387$	28.4500	1.44619
$388$	−8.55948	−0.434542
$389$	−33.8949	−1.71854	−0.859270	−	0.511522i	$-0.829082\pi$
$389$	−33.8949	−1.71854	−0.859270	+	0.511522i	$0.829082\pi$
$390$	0	0
$391$	27.1599	1.37353
$392$	−0.357592	−0.0180611
$393$	−18.3510	−0.925686
$394$	2.67977	0.135005
$395$	0	0
$396$	−5.46083	−0.274417
$397$	30.2669	1.51905	0.759527	−	0.650475i	$-0.225430\pi$
$397$	30.2669	1.51905	0.759527	+	0.650475i	$0.225430\pi$
$398$	−1.23247	−0.0617781
$399$	5.93286	0.297014
$400$	0	0
$401$	−28.1503	−1.40576	−0.702878	−	0.711310i	$-0.748102\pi$
$401$	−28.1503	−1.40576	−0.702878	+	0.711310i	$0.748102\pi$
$402$	−0.306888	−0.0153062
$403$	14.2073	0.707718
$404$	−29.0035	−1.44298
$405$	0	0
$406$	−1.39255	−0.0691109
$407$	−6.41117	−0.317790
$408$	−6.64183	−0.328819
$409$	−5.79745	−0.286666	−0.143333	−	0.989675i	$-0.545782\pi$
$409$	−5.79745	−0.286666	−0.143333	+	0.989675i	$0.545782\pi$
$410$	0	0
$411$	12.7783	0.630306
$412$	−16.6805	−0.821790
$413$	7.68030	0.377923
$414$	−1.13140	−0.0556052
$415$	0	0
$416$	−5.24797	−0.257303
$417$	−18.6271	−0.912173
$418$	−0.102671	−0.00502181
$419$	12.2699	0.599425	0.299712	−	0.954030i	$-0.403109\pi$
$419$	12.2699	0.599425	0.299712	+	0.954030i	$0.403109\pi$
$420$	0	0
$421$	−36.8040	−1.79372	−0.896860	−	0.442315i	$-0.854157\pi$
$421$	−36.8040	−1.79372	−0.896860	+	0.442315i	$0.854157\pi$
$422$	−1.76431	−0.0858852
$423$	9.27813	0.451118
$424$	−2.63025	−0.127736
$425$	0	0
$426$	2.26483	0.109732
$427$	0.144075	0.00697229
$428$	−4.86552	−0.235184
$429$	−10.2816	−0.496401
$430$	0	0
$431$	28.2190	1.35926	0.679630	−	0.733556i	$-0.262141\pi$
$431$	28.2190	1.35926	0.679630	+	0.733556i	$0.262141\pi$
$432$	2.40730	0.115821
$433$	−7.43295	−0.357205	−0.178602	−	0.983921i	$-0.557158\pi$
$433$	−7.43295	−0.357205	−0.178602	+	0.983921i	$0.557158\pi$
$434$	0.841713	0.0404035
$435$	0	0
$436$	29.4535	1.41057
$437$	4.01461	0.192045
$438$	2.80926	0.134232
$439$	4.04293	0.192959	0.0964793	−	0.995335i	$-0.469242\pi$
$439$	4.04293	0.192959	0.0964793	+	0.995335i	$0.469242\pi$
$440$	0	0
$441$	−2.39634	−0.114112
$442$	−2.97957	−0.141724
$443$	−33.0706	−1.57123	−0.785615	−	0.618715i	$-0.787653\pi$
$443$	−33.0706	−1.57123	−0.785615	+	0.618715i	$0.787653\pi$
$444$	−30.5712	−1.45085
$445$	0	0
$446$	1.85985	0.0880665
$447$	17.1348	0.810446
$448$	19.1787	0.906109
$449$	37.4339	1.76661	0.883306	−	0.468796i	$-0.155312\pi$
$449$	37.4339	1.76661	0.883306	+	0.468796i	$0.155312\pi$
$450$	0	0
$451$	−7.20691	−0.339360
$452$	11.5727	0.544332
$453$	−9.93718	−0.466889
$454$	−1.91946	−0.0900847
$455$	0	0
$456$	−0.981755	−0.0459749
$457$	−25.3943	−1.18789	−0.593947	−	0.804504i	$-0.702431\pi$
$457$	−25.3943	−1.18789	−0.593947	+	0.804504i	$0.702431\pi$
$458$	2.78704	0.130230
$459$	4.13681	0.193090
$460$	0	0
$461$	−12.2994	−0.572839	−0.286420	−	0.958104i	$-0.592465\pi$
$461$	−12.2994	−0.572839	−0.286420	+	0.958104i	$0.592465\pi$
$462$	−0.609133	−0.0283394
$463$	24.7142	1.14857	0.574283	−	0.818657i	$-0.305281\pi$
$463$	24.7142	1.14857	0.574283	+	0.818657i	$0.305281\pi$
$464$	−21.5719	−1.00145
$465$	0	0
$466$	0.611055	0.0283066
$467$	23.6362	1.09375	0.546877	−	0.837213i	$-0.315817\pi$
$467$	23.6362	1.09375	0.546877	+	0.837213i	$0.315817\pi$
$468$	−23.4250	−1.08282
$469$	3.08684	0.142537
$470$	0	0
$471$	−32.7646	−1.50971
$472$	−1.27092	−0.0584987
$473$	10.3647	0.476571
$474$	−2.81096	−0.129112
$475$	0	0
$476$	33.3152	1.52700
$477$	−17.6262	−0.807047
$478$	−0.629270	−0.0287821
$479$	28.9089	1.32088	0.660440	−	0.750879i	$-0.270370\pi$
$479$	28.9089	1.32088	0.660440	+	0.750879i	$0.270370\pi$
$480$	0	0
$481$	−27.5016	−1.25396
$482$	2.85818	0.130187
$483$	23.8181	1.08376
$484$	−1.98946	−0.0904299
$485$	0	0
$486$	−2.19877	−0.0997382
$487$	26.6045	1.20556	0.602782	−	0.797906i	$-0.294059\pi$
$487$	26.6045	1.20556	0.602782	+	0.797906i	$0.294059\pi$
$488$	−0.0238412	−0.00107924
$489$	−4.45422	−0.201427
$490$	0	0
$491$	1.92046	0.0866693	0.0433347	−	0.999061i	$-0.486202\pi$
$491$	1.92046	0.0866693	0.0433347	+	0.999061i	$0.486202\pi$
$492$	−34.3657	−1.54932
$493$	−37.0700	−1.66955
$494$	−0.440422	−0.0198155
$495$	0	0
$496$	13.0389	0.585466
$497$	−22.7809	−1.02186
$498$	−0.911347	−0.0408384
$499$	4.95869	0.221981	0.110991	−	0.993821i	$-0.464598\pi$
$499$	4.95869	0.221981	0.110991	+	0.993821i	$0.464598\pi$
$500$	0	0
$501$	27.0245	1.20737
$502$	1.74030	0.0776732
$503$	−6.81042	−0.303661	−0.151831	−	0.988407i	$-0.548517\pi$
$503$	−6.81042	−0.303661	−0.151831	+	0.988407i	$0.548517\pi$
$504$	−2.78298	−0.123964
$505$	0	0
$506$	−0.412185	−0.0183238
$507$	−12.9453	−0.574923
$508$	−19.2795	−0.855391
$509$	10.4847	0.464727	0.232363	−	0.972629i	$-0.425354\pi$
$509$	10.4847	0.464727	0.232363	+	0.972629i	$0.425354\pi$
$510$	0	0
$511$	−28.2570	−1.25002
$512$	−8.04148	−0.355387
$513$	0.611478	0.0269974
$514$	−2.42922	−0.107148
$515$	0	0
$516$	49.4235	2.17575
$517$	3.38016	0.148659
$518$	−1.62933	−0.0715886
$519$	−31.2726	−1.37271
$520$	0	0
$521$	29.7694	1.30422	0.652110	−	0.758124i	$-0.273884\pi$
$521$	29.7694	1.30422	0.652110	+	0.758124i	$0.273884\pi$
$522$	1.54422	0.0675888
$523$	34.3546	1.50222	0.751111	−	0.660176i	$-0.229518\pi$
$523$	34.3546	1.50222	0.751111	+	0.660176i	$0.229518\pi$
$524$	−15.2319	−0.665409
$525$	0	0
$526$	−1.61411	−0.0703783
$527$	22.4066	0.976048
$528$	−9.43606	−0.410652
$529$	−6.88290	−0.299257
$530$	0	0
$531$	−8.51684	−0.369599
$532$	4.92445	0.213502
$533$	−30.9150	−1.33908
$534$	2.07922	0.0899767
$535$	0	0
$536$	−0.510802	−0.0220633
$537$	24.3071	1.04893
$538$	−1.90403	−0.0820884
$539$	−0.873022	−0.0376037
$540$	0	0
$541$	35.9823	1.54700	0.773499	−	0.633797i	$-0.218505\pi$
$541$	35.9823	1.54700	0.773499	+	0.633797i	$0.218505\pi$
$542$	0.880328	0.0378133
$543$	2.86891	0.123117
$544$	−8.27667	−0.354859
$545$	0	0
$546$	−2.61296	−0.111824
$547$	−5.44079	−0.232631	−0.116316	−	0.993212i	$-0.537108\pi$
$547$	−5.44079	−0.232631	−0.116316	+	0.993212i	$0.537108\pi$
$548$	10.6064	0.453082
$549$	−0.159768	−0.00681873
$550$	0	0
$551$	−5.47946	−0.233433
$552$	−3.94136	−0.167756
$553$	28.2741	1.20234
$554$	0.411783	0.0174950
$555$	0	0
$556$	−15.4611	−0.655696
$557$	−15.9004	−0.673722	−0.336861	−	0.941554i	$-0.609365\pi$
$557$	−15.9004	−0.673722	−0.336861	+	0.941554i	$0.609365\pi$
$558$	−0.933393	−0.0395137
$559$	44.4609	1.88050
$560$	0	0
$561$	−16.2153	−0.684611
$562$	2.42727	0.102388
$563$	−29.3287	−1.23606	−0.618029	−	0.786155i	$-0.712069\pi$
$563$	−29.3287	−1.23606	−0.618029	+	0.786155i	$0.712069\pi$
$564$	16.1180	0.678692
$565$	0	0
$566$	1.47499	0.0619985
$567$	24.0108	1.00836
$568$	3.76973	0.158174
$569$	6.18252	0.259185	0.129592	−	0.991567i	$-0.458633\pi$
$569$	6.18252	0.259185	0.129592	+	0.991567i	$0.458633\pi$
$570$	0	0
$571$	16.1288	0.674971	0.337485	−	0.941331i	$-0.390424\pi$
$571$	16.1288	0.674971	0.337485	+	0.941331i	$0.390424\pi$
$572$	−8.53406	−0.356827
$573$	−29.9466	−1.25104
$574$	−1.83156	−0.0764478
$575$	0	0
$576$	−21.2677	−0.886153
$577$	−13.1699	−0.548270	−0.274135	−	0.961691i	$-0.588392\pi$
$577$	−13.1699	−0.548270	−0.274135	+	0.961691i	$0.588392\pi$
$578$	−2.95372	−0.122859
$579$	28.4180	1.18101
$580$	0	0
$581$	9.16680	0.380303
$582$	1.05877	0.0438874
$583$	−6.42147	−0.265950
$584$	4.67590	0.193490
$585$	0	0
$586$	−0.946151	−0.0390851
$587$	−14.4749	−0.597444	−0.298722	−	0.954340i	$-0.596560\pi$
$587$	−14.4749	−0.597444	−0.298722	+	0.954340i	$0.596560\pi$
$588$	−4.16294	−0.171677
$589$	3.31201	0.136469
$590$	0	0
$591$	62.5590	2.57333
$592$	−25.2399	−1.03735
$593$	13.5088	0.554739	0.277370	−	0.960763i	$-0.410537\pi$
$593$	13.5088	0.554739	0.277370	+	0.960763i	$0.410537\pi$
$594$	−0.0627811	−0.00257594
$595$	0	0
$596$	14.2224	0.582572
$597$	−28.7719	−1.17756
$598$	−1.76812	−0.0723039
$599$	18.4470	0.753723	0.376861	−	0.926270i	$-0.377003\pi$
$599$	18.4470	0.753723	0.376861	+	0.926270i	$0.377003\pi$
$600$	0	0
$601$	37.1149	1.51395	0.756974	−	0.653445i	$-0.226677\pi$
$601$	37.1149	1.51395	0.756974	+	0.653445i	$0.226677\pi$
$602$	2.63409	0.107357
$603$	−3.42306	−0.139398
$604$	−8.24816	−0.335613
$605$	0	0
$606$	3.58760	0.145736
$607$	20.2696	0.822717	0.411358	−	0.911474i	$-0.365055\pi$
$607$	20.2696	0.822717	0.411358	+	0.911474i	$0.365055\pi$
$608$	−1.22341	−0.0496157
$609$	−32.5089	−1.31733
$610$	0	0
$611$	14.4996	0.586593
$612$	−36.9439	−1.49337
$613$	38.4564	1.55324	0.776621	−	0.629969i	$-0.216932\pi$
$613$	38.4564	1.55324	0.776621	+	0.629969i	$0.216932\pi$
$614$	2.43391	0.0982246
$615$	0	0
$616$	−1.01388	−0.0408503
$617$	9.13912	0.367927	0.183964	−	0.982933i	$-0.441107\pi$
$617$	9.13912	0.367927	0.183964	+	0.982933i	$0.441107\pi$
$618$	2.06331	0.0829983
$619$	35.3756	1.42187	0.710933	−	0.703260i	$-0.248273\pi$
$619$	35.3756	1.42187	0.710933	+	0.703260i	$0.248273\pi$
$620$	0	0
$621$	2.45485	0.0985096
$622$	−1.39202	−0.0558149
$623$	−20.9139	−0.837897
$624$	−40.4773	−1.62039
$625$	0	0
$626$	0.752790	0.0300876
$627$	−2.39685	−0.0957209
$628$	−27.1956	−1.08522
$629$	−43.3732	−1.72940
$630$	0	0
$631$	−3.69774	−0.147205	−0.0736024	−	0.997288i	$-0.523450\pi$
$631$	−3.69774	−0.147205	−0.0736024	+	0.997288i	$0.523450\pi$
$632$	−4.67873	−0.186110
$633$	−41.1876	−1.63706
$634$	−1.41886	−0.0563500
$635$	0	0
$636$	−30.6203	−1.21417
$637$	−3.74495	−0.148380
$638$	0.562583	0.0222729
$639$	25.2622	0.999357
$640$	0	0
$641$	39.7510	1.57007	0.785035	−	0.619451i	$-0.212645\pi$
$641$	39.7510	1.57007	0.785035	+	0.619451i	$0.212645\pi$
$642$	0.601843	0.0237529
$643$	−40.3082	−1.58960	−0.794800	−	0.606872i	$-0.792424\pi$
$643$	−40.3082	−1.58960	−0.794800	+	0.606872i	$0.792424\pi$
$644$	19.7698	0.779038
$645$	0	0
$646$	−0.694597	−0.0273286
$647$	−31.8685	−1.25288	−0.626440	−	0.779469i	$-0.715489\pi$
$647$	−31.8685	−1.25288	−0.626440	+	0.779469i	$0.715489\pi$
$648$	−3.97325	−0.156084
$649$	−3.10281	−0.121796
$650$	0	0
$651$	19.6497	0.770132
$652$	−3.69714	−0.144791
$653$	35.7351	1.39842	0.699211	−	0.714916i	$-0.253535\pi$
$653$	35.7351	1.39842	0.699211	+	0.714916i	$0.253535\pi$
$654$	−3.64327	−0.142463
$655$	0	0
$656$	−28.3726	−1.10777
$657$	31.3348	1.22249
$658$	0.859030	0.0334885
$659$	−8.31768	−0.324011	−0.162005	−	0.986790i	$-0.551796\pi$
$659$	−8.31768	−0.324011	−0.162005	+	0.986790i	$0.551796\pi$
$660$	0	0
$661$	11.0879	0.431268	0.215634	−	0.976474i	$-0.430818\pi$
$661$	11.0879	0.431268	0.215634	+	0.976474i	$0.430818\pi$
$662$	2.13208	0.0828658
$663$	−69.5578	−2.70140
$664$	−1.51690	−0.0588672
$665$	0	0
$666$	1.80680	0.0700120
$667$	−21.9979	−0.851762
$668$	22.4312	0.867888
$669$	43.4180	1.67864
$670$	0	0
$671$	−0.0582058	−0.00224701
$672$	−7.25830	−0.279995
$673$	−29.4826	−1.13647	−0.568236	−	0.822866i	$-0.692374\pi$
$673$	−29.4826	−1.13647	−0.568236	+	0.822866i	$0.692374\pi$
$674$	−2.72385	−0.104919
$675$	0	0
$676$	−10.7450	−0.413271
$677$	−6.60442	−0.253829	−0.126914	−	0.991914i	$-0.540507\pi$
$677$	−6.60442	−0.253829	−0.126914	+	0.991914i	$0.540507\pi$
$678$	−1.43149	−0.0549759
$679$	−10.6497	−0.408696
$680$	0	0
$681$	−44.8096	−1.71711
$682$	−0.340048	−0.0130211
$683$	16.8710	0.645550	0.322775	−	0.946476i	$-0.395384\pi$
$683$	16.8710	0.645550	0.322775	+	0.946476i	$0.395384\pi$
$684$	−5.46083	−0.208800
$685$	0	0
$686$	−2.00084	−0.0763926
$687$	65.0632	2.48232
$688$	40.8045	1.55566
$689$	−27.5458	−1.04941
$690$	0	0
$691$	−10.2317	−0.389231	−0.194615	−	0.980880i	$-0.562346\pi$
$691$	−10.2317	−0.389231	−0.194615	+	0.980880i	$0.562346\pi$
$692$	−25.9572	−0.986745
$693$	−6.79433	−0.258095
$694$	−1.22224	−0.0463955
$695$	0	0
$696$	5.37949	0.203909
$697$	−48.7566	−1.84679
$698$	−0.104484	−0.00395479
$699$	14.2650	0.539553
$700$	0	0
$701$	−47.1809	−1.78200	−0.890998	−	0.454006i	$-0.849994\pi$
$701$	−47.1809	−1.78200	−0.890998	+	0.454006i	$0.849994\pi$
$702$	−0.269308	−0.0101644
$703$	−6.41117	−0.241802
$704$	−7.74812	−0.292018
$705$	0	0
$706$	−1.88856	−0.0710770
$707$	−36.0860	−1.35715
$708$	−14.7955	−0.556050
$709$	39.0837	1.46782	0.733910	−	0.679247i	$-0.237693\pi$
$709$	39.0837	1.46782	0.733910	+	0.679247i	$0.237693\pi$
$710$	0	0
$711$	−31.3538	−1.17586
$712$	3.46078	0.129698
$713$	13.2964	0.497956
$714$	−4.12094	−0.154222
$715$	0	0
$716$	20.1757	0.754001
$717$	−14.6902	−0.548617
$718$	−1.02062	−0.0380893
$719$	28.7476	1.07210	0.536052	−	0.844185i	$-0.319915\pi$
$719$	28.7476	1.07210	0.536052	+	0.844185i	$0.319915\pi$
$720$	0	0
$721$	−20.7538	−0.772912
$722$	−0.102671	−0.00382102
$723$	66.7240	2.48149
$724$	2.38129	0.0884999
$725$	0	0
$726$	0.246087	0.00913315
$727$	48.2185	1.78833	0.894163	−	0.447741i	$-0.147771\pi$
$727$	48.2185	1.78833	0.894163	+	0.447741i	$0.147771\pi$
$728$	−4.34917	−0.161191
$729$	−22.2292	−0.823305
$730$	0	0
$731$	70.1201	2.59349
$732$	−0.277550	−0.0102586
$733$	−20.7545	−0.766584	−0.383292	−	0.923627i	$-0.625210\pi$
$733$	−20.7545	−0.766584	−0.383292	+	0.923627i	$0.625210\pi$
$734$	0.285413	0.0105348
$735$	0	0
$736$	−4.91150	−0.181040
$737$	−1.24707	−0.0459364
$738$	2.03105	0.0747641
$739$	−34.2862	−1.26124	−0.630619	−	0.776092i	$-0.717199\pi$
$739$	−34.2862	−1.26124	−0.630619	+	0.776092i	$0.717199\pi$
$740$	0	0
$741$	−10.2816	−0.377704
$742$	−1.63195	−0.0599106
$743$	−27.6918	−1.01591	−0.507956	−	0.861383i	$-0.669599\pi$
$743$	−27.6918	−1.01591	−0.507956	+	0.861383i	$0.669599\pi$
$744$	−3.25159	−0.119209
$745$	0	0
$746$	−2.13325	−0.0781037
$747$	−10.1653	−0.371927
$748$	−13.4592	−0.492117
$749$	−6.05365	−0.221196
$750$	0	0
$751$	5.00411	0.182602	0.0913012	−	0.995823i	$-0.470897\pi$
$751$	5.00411	0.182602	0.0913012	+	0.995823i	$0.470897\pi$
$752$	13.3072	0.485264
$753$	40.6270	1.48053
$754$	2.41328	0.0878863
$755$	0	0
$756$	3.01119	0.109516
$757$	−44.0885	−1.60242	−0.801212	−	0.598380i	$-0.795811\pi$
$757$	−44.0885	−1.60242	−0.801212	+	0.598380i	$0.795811\pi$
$758$	−2.14624	−0.0779548
$759$	−9.62241	−0.349271
$760$	0	0
$761$	−0.838856	−0.0304085	−0.0152043	−	0.999884i	$-0.504840\pi$
$761$	−0.838856	−0.0304085	−0.0152043	+	0.999884i	$0.504840\pi$
$762$	2.38479	0.0863919
$763$	36.6459	1.32667
$764$	−24.8566	−0.899281
$765$	0	0
$766$	2.46560	0.0890856
$767$	−13.3099	−0.480593
$768$	−36.3442	−1.31146
$769$	−20.6026	−0.742950	−0.371475	−	0.928443i	$-0.621148\pi$
$769$	−20.6026	−0.742950	−0.371475	+	0.928443i	$0.621148\pi$
$770$	0	0
$771$	−56.7100	−2.04236
$772$	23.5878	0.848943
$773$	36.2748	1.30471	0.652357	−	0.757912i	$-0.273780\pi$
$773$	36.2748	1.30471	0.652357	+	0.757912i	$0.273780\pi$
$774$	−2.92099	−0.104993
$775$	0	0
$776$	1.76228	0.0632622
$777$	−38.0366	−1.36455
$778$	3.48003	0.124765
$779$	−7.20691	−0.258214
$780$	0	0
$781$	9.20338	0.329323
$782$	−2.78854	−0.0997179
$783$	−3.35057	−0.119740
$784$	−3.43697	−0.122749
$785$	0	0
$786$	1.88412	0.0672043
$787$	−22.2218	−0.792120	−0.396060	−	0.918225i	$-0.629623\pi$
$787$	−22.2218	−0.792120	−0.396060	+	0.918225i	$0.629623\pi$
$788$	51.9259	1.84978
$789$	−37.6811	−1.34148
$790$	0	0
$791$	14.3986	0.511956
$792$	1.12431	0.0399506
$793$	−0.249682	−0.00886646
$794$	−3.10754	−0.110283
$795$	0	0
$796$	−23.8816	−0.846460
$797$	−26.3406	−0.933032	−0.466516	−	0.884513i	$-0.654491\pi$
$797$	−26.3406	−0.933032	−0.466516	+	0.884513i	$0.654491\pi$
$798$	−0.609133	−0.0215631
$799$	22.8676	0.808999
$800$	0	0
$801$	23.1919	0.819444
$802$	2.89022	0.102057
$803$	11.4157	0.402851
$804$	−5.94656	−0.209719
$805$	0	0
$806$	−1.45868	−0.0513799
$807$	−44.4493	−1.56469
$808$	5.97142	0.210074
$809$	−23.6915	−0.832948	−0.416474	−	0.909148i	$-0.636734\pi$
$809$	−23.6915	−0.832948	−0.416474	+	0.909148i	$0.636734\pi$
$810$	0	0
$811$	−5.87587	−0.206330	−0.103165	−	0.994664i	$-0.532897\pi$
$811$	−5.87587	−0.206330	−0.103165	+	0.994664i	$0.532897\pi$
$812$	−26.9834	−0.946930
$813$	20.5512	0.720761
$814$	0.658242	0.0230714
$815$	0	0
$816$	−63.8374	−2.23476
$817$	10.3647	0.362616
$818$	0.595231	0.0208118
$819$	−29.1452	−1.01842
$820$	0	0
$821$	−2.87670	−0.100397	−0.0501987	−	0.998739i	$-0.515985\pi$
$821$	−2.87670	−0.100397	−0.0501987	+	0.998739i	$0.515985\pi$
$822$	−1.31196	−0.0457599
$823$	51.0614	1.77989	0.889945	−	0.456068i	$-0.150743\pi$
$823$	51.0614	1.77989	0.889945	+	0.456068i	$0.150743\pi$
$824$	3.43429	0.119639
$825$	0	0
$826$	−0.788545	−0.0274370
$827$	43.5840	1.51556	0.757782	−	0.652508i	$-0.226283\pi$
$827$	43.5840	1.51556	0.757782	+	0.652508i	$0.226283\pi$
$828$	−21.9231	−0.761881
$829$	11.9212	0.414040	0.207020	−	0.978337i	$-0.433623\pi$
$829$	11.9212	0.414040	0.207020	+	0.978337i	$0.433623\pi$
$830$	0	0
$831$	9.61304	0.333473
$832$	−33.2366	−1.15227
$833$	−5.90622	−0.204638
$834$	1.91247	0.0662233
$835$	0	0
$836$	−1.98946	−0.0688069
$837$	2.02522	0.0700020
$838$	−1.25977	−0.0435179
$839$	6.51682	0.224986	0.112493	−	0.993653i	$-0.464116\pi$
$839$	6.51682	0.224986	0.112493	+	0.993653i	$0.464116\pi$
$840$	0	0
$841$	1.02452	0.0353281
$842$	3.77871	0.130223
$843$	56.6643	1.95162
$844$	−34.1870	−1.17677
$845$	0	0
$846$	−0.952596	−0.0327509
$847$	−2.47527	−0.0850514
$848$	−25.2804	−0.868134
$849$	34.4335	1.18175
$850$	0	0
$851$	−25.7384	−0.882299
$852$	43.8857	1.50350
$853$	35.7807	1.22511	0.612554	−	0.790428i	$-0.290142\pi$
$853$	35.7807	1.22511	0.612554	+	0.790428i	$0.290142\pi$
$854$	−0.0147924	−0.000506184 0
$855$	0	0
$856$	1.00174	0.0342389
$857$	−37.9264	−1.29554	−0.647771	−	0.761835i	$-0.724299\pi$
$857$	−37.9264	−1.29554	−0.647771	+	0.761835i	$0.724299\pi$
$858$	1.05562	0.0360384
$859$	49.7834	1.69859	0.849293	−	0.527921i	$-0.177028\pi$
$859$	49.7834	1.69859	0.849293	+	0.527921i	$0.177028\pi$
$860$	0	0
$861$	−42.7576	−1.45717
$862$	−2.89727	−0.0986815
$863$	−39.7764	−1.35400	−0.677002	−	0.735981i	$-0.736721\pi$
$863$	−39.7764	−1.35400	−0.677002	+	0.735981i	$0.736721\pi$
$864$	−0.748086	−0.0254504
$865$	0	0
$866$	0.763149	0.0259329
$867$	−68.9543	−2.34181
$868$	16.3099	0.553593
$869$	−11.4226	−0.387486
$870$	0	0
$871$	−5.34947	−0.181260
$872$	−6.06408	−0.205356
$873$	11.8096	0.399695
$874$	−0.412185	−0.0139424
$875$	0	0
$876$	54.4350	1.83919
$877$	14.7361	0.497603	0.248802	−	0.968554i	$-0.419963\pi$
$877$	14.7361	0.497603	0.248802	+	0.968554i	$0.419963\pi$
$878$	−0.415092	−0.0140087
$879$	−22.0878	−0.745003
$880$	0	0
$881$	36.1210	1.21695	0.608473	−	0.793575i	$-0.291782\pi$
$881$	36.1210	1.21695	0.608473	+	0.793575i	$0.291782\pi$
$882$	0.246035	0.00828444
$883$	−54.5279	−1.83501	−0.917505	−	0.397723i	$-0.869800\pi$
$883$	−54.5279	−1.83501	−0.917505	+	0.397723i	$0.869800\pi$
$884$	−57.7351	−1.94184
$885$	0	0
$886$	3.39539	0.114070
$887$	−30.6419	−1.02885	−0.514427	−	0.857534i	$-0.671995\pi$
$887$	−30.6419	−1.02885	−0.514427	+	0.857534i	$0.671995\pi$
$888$	6.29420	0.211219
$889$	−23.9875	−0.804515
$890$	0	0
$891$	−9.70027	−0.324971
$892$	36.0383	1.20665
$893$	3.38016	0.113113
$894$	−1.75925	−0.0588380
$895$	0	0
$896$	−8.02563	−0.268118
$897$	−41.2767	−1.37819
$898$	−3.84338	−0.128255
$899$	−18.1481	−0.605272
$900$	0	0
$901$	−43.4429	−1.44729
$902$	0.739942	0.0246374
$903$	61.4925	2.04634
$904$	−2.38265	−0.0792458
$905$	0	0
$906$	1.02026	0.0338959
$907$	52.2779	1.73586	0.867929	−	0.496688i	$-0.165451\pi$
$907$	52.2779	1.73586	0.867929	+	0.496688i	$0.165451\pi$
$908$	−37.1934	−1.23430
$909$	40.0165	1.32726
$910$	0	0
$911$	15.3410	0.508272	0.254136	−	0.967169i	$-0.418209\pi$
$911$	15.3410	0.508272	0.254136	+	0.967169i	$0.418209\pi$
$912$	−9.43606	−0.312459
$913$	−3.70335	−0.122563
$914$	2.60726	0.0862405
$915$	0	0
$916$	54.0045	1.78436
$917$	−18.9514	−0.625832
$918$	−0.424731	−0.0140182
$919$	32.0789	1.05818	0.529092	−	0.848564i	$-0.322532\pi$
$919$	32.0789	1.05818	0.529092	+	0.848564i	$0.322532\pi$
$920$	0	0
$921$	56.8194	1.87226
$922$	1.26279	0.0415878
$923$	39.4792	1.29947
$924$	−11.8032	−0.388296
$925$	0	0
$926$	−2.53743	−0.0833853
$927$	23.0143	0.755890
$928$	6.70361	0.220057
$929$	−6.23142	−0.204446	−0.102223	−	0.994761i	$-0.532596\pi$
$929$	−6.23142	−0.204446	−0.102223	+	0.994761i	$0.532596\pi$
$930$	0	0
$931$	−0.873022	−0.0286121
$932$	11.8404	0.387846
$933$	−32.4966	−1.06389
$934$	−2.42676	−0.0794059
$935$	0	0
$936$	4.82288	0.157641
$937$	−35.2669	−1.15212	−0.576059	−	0.817408i	$-0.695410\pi$
$937$	−35.2669	−1.15212	−0.576059	+	0.817408i	$0.695410\pi$
$938$	−0.316929	−0.0103481
$939$	17.5738	0.573500
$940$	0	0
$941$	−22.3104	−0.727299	−0.363649	−	0.931536i	$-0.618469\pi$
$941$	−22.3104	−0.727299	−0.363649	+	0.931536i	$0.618469\pi$
$942$	3.36398	0.109604
$943$	−28.9329	−0.942186
$944$	−12.2153	−0.397575
$945$	0	0
$946$	−1.06416	−0.0345988
$947$	−41.3286	−1.34300	−0.671499	−	0.741005i	$-0.734349\pi$
$947$	−41.3286	−1.34300	−0.671499	+	0.741005i	$0.734349\pi$
$948$	−54.4680	−1.76904
$949$	48.9692	1.58961
$950$	0	0
$951$	−33.1231	−1.07409
$952$	−6.85915	−0.222306
$953$	−19.8043	−0.641524	−0.320762	−	0.947160i	$-0.603939\pi$
$953$	−19.8043	−0.641524	−0.320762	+	0.947160i	$0.603939\pi$
$954$	1.80970	0.0585912
$955$	0	0
$956$	−12.1934	−0.394361
$957$	13.1334	0.424544
$958$	−2.96811	−0.0958951
$959$	13.1964	0.426134
$960$	0	0
$961$	−20.0306	−0.646147
$962$	2.82362	0.0910372
$963$	6.71302	0.216324
$964$	55.3830	1.78377
$965$	0	0
$966$	−2.44543	−0.0786805
$967$	−15.4967	−0.498341	−0.249170	−	0.968460i	$-0.580158\pi$
$967$	−15.4967	−0.498341	−0.249170	+	0.968460i	$0.580158\pi$
$968$	0.409602	0.0131651
$969$	−16.2153	−0.520911
$970$	0	0
$971$	−3.10819	−0.0997467	−0.0498733	−	0.998756i	$-0.515882\pi$
$971$	−3.10819	−0.0997467	−0.0498733	+	0.998756i	$0.515882\pi$
$972$	−42.6055	−1.36657
$973$	−19.2366	−0.616696
$974$	−2.73151	−0.0875233
$975$	0	0
$976$	−0.229148	−0.00733485
$977$	52.0716	1.66592	0.832960	−	0.553334i	$-0.186645\pi$
$977$	52.0716	1.66592	0.832960	+	0.553334i	$0.186645\pi$
$978$	0.457320	0.0146235
$979$	8.44912	0.270035
$980$	0	0
$981$	−40.6374	−1.29745
$982$	−0.197176	−0.00629215
$983$	−16.7488	−0.534204	−0.267102	−	0.963668i	$-0.586066\pi$
$983$	−16.7488	−0.534204	−0.267102	+	0.963668i	$0.586066\pi$
$984$	7.07542	0.225556
$985$	0	0
$986$	3.80602	0.121208
$987$	20.0540	0.638325
$988$	−8.53406	−0.271504
$989$	41.6104	1.32313
$990$	0	0
$991$	−22.5486	−0.716279	−0.358140	−	0.933668i	$-0.616589\pi$
$991$	−22.5486	−0.716279	−0.358140	+	0.933668i	$0.616589\pi$
$992$	−4.05194	−0.128649
$993$	49.7733	1.57951
$994$	2.33894	0.0741867
$995$	0	0
$996$	−17.6592	−0.559552
$997$	59.0193	1.86916	0.934580	−	0.355752i	$-0.115775\pi$
$997$	59.0193	1.86916	0.934580	+	0.355752i	$0.115775\pi$
$998$	−0.509114	−0.0161157
$999$	−3.92029	−0.124032

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.y.1.7	yes	15
5.4	even	2		5225.2.a.r.1.9	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.r.1.9	✓	15	5.4	even	2
5225.2.a.y.1.7	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)

\(f(q)\)	\(=\)	\(q-0.102671 q^{2} -2.39685 q^{3} -1.98946 q^{4} +0.246087 q^{6} -2.47527 q^{7} +0.409602 q^{8} +2.74488 q^{9} +1.00000 q^{11} +4.76843 q^{12} +4.28964 q^{13} +0.254139 q^{14} +3.93686 q^{16} +6.76526 q^{17} -0.281820 q^{18} +1.00000 q^{19} +5.93286 q^{21} -0.102671 q^{22} +4.01461 q^{23} -0.981755 q^{24} -0.440422 q^{26} +0.611478 q^{27} +4.92445 q^{28} -5.47946 q^{29} +3.31201 q^{31} -1.22341 q^{32} -2.39685 q^{33} -0.694597 q^{34} -5.46083 q^{36} -6.41117 q^{37} -0.102671 q^{38} -10.2816 q^{39} -7.20691 q^{41} -0.609133 q^{42} +10.3647 q^{43} -1.98946 q^{44} -0.412185 q^{46} +3.38016 q^{47} -9.43606 q^{48} -0.873022 q^{49} -16.2153 q^{51} -8.53406 q^{52} -6.42147 q^{53} -0.0627811 q^{54} -1.01388 q^{56} -2.39685 q^{57} +0.562583 q^{58} -3.10281 q^{59} -0.0582058 q^{61} -0.340048 q^{62} -6.79433 q^{63} -7.74812 q^{64} +0.246087 q^{66} -1.24707 q^{67} -13.4592 q^{68} -9.62241 q^{69} +9.20338 q^{71} +1.12431 q^{72} +11.4157 q^{73} +0.658242 q^{74} -1.98946 q^{76} -2.47527 q^{77} +1.05562 q^{78} -11.4226 q^{79} -9.70027 q^{81} +0.739942 q^{82} -3.70335 q^{83} -11.8032 q^{84} -1.06416 q^{86} +13.1334 q^{87} +0.409602 q^{88} +8.44912 q^{89} -10.6180 q^{91} -7.98690 q^{92} -7.93840 q^{93} -0.347044 q^{94} +2.93232 q^{96} +4.30242 q^{97} +0.0896341 q^{98} +2.74488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more