LMFDB - Embedded newform 7225.2.a.bt.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$1$
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} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 $ x^15 - 21x^13 - 2x^12 + 171x^11 + 30x^10 - 678x^9 - 153x^8 + 1350x^7 + 301x^6 - 1284x^5 - 183x^4 + 555x^3 + 18x^2 - 81*x + 3
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			1.12
Root			$1.71131$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71131 q^{2} +0.109608 q^{3} +0.928572 q^{4} +0.187572 q^{6} +4.34210 q^{7} -1.83354 q^{8} -2.98799 q^{9} +O(q^{10})$	\(q+1.71131 q^{2} +0.109608 q^{3} +0.928572 q^{4} +0.187572 q^{6} +4.34210 q^{7} -1.83354 q^{8} -2.98799 q^{9} +5.20553 q^{11} +0.101779 q^{12} -4.88189 q^{13} +7.43066 q^{14} -4.99490 q^{16} -5.11336 q^{18} -1.64053 q^{19} +0.475927 q^{21} +8.90825 q^{22} -8.39280 q^{23} -0.200970 q^{24} -8.35442 q^{26} -0.656330 q^{27} +4.03195 q^{28} -0.485258 q^{29} -6.30076 q^{31} -4.88072 q^{32} +0.570566 q^{33} -2.77456 q^{36} -0.930097 q^{37} -2.80745 q^{38} -0.535093 q^{39} -5.44822 q^{41} +0.814458 q^{42} -7.53358 q^{43} +4.83371 q^{44} -14.3627 q^{46} -4.88962 q^{47} -0.547479 q^{48} +11.8538 q^{49} -4.53319 q^{52} +3.10894 q^{53} -1.12318 q^{54} -7.96142 q^{56} -0.179815 q^{57} -0.830425 q^{58} -7.94104 q^{59} +14.1498 q^{61} -10.7825 q^{62} -12.9741 q^{63} +1.63738 q^{64} +0.976414 q^{66} -11.5894 q^{67} -0.919916 q^{69} -0.201195 q^{71} +5.47860 q^{72} -5.37389 q^{73} -1.59168 q^{74} -1.52335 q^{76} +22.6029 q^{77} -0.915709 q^{78} +6.62508 q^{79} +8.89202 q^{81} -9.32358 q^{82} +10.5231 q^{83} +0.441933 q^{84} -12.8923 q^{86} -0.0531880 q^{87} -9.54455 q^{88} -13.0569 q^{89} -21.1977 q^{91} -7.79333 q^{92} -0.690612 q^{93} -8.36764 q^{94} -0.534965 q^{96} +1.17959 q^{97} +20.2855 q^{98} -15.5540 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 9 q^{3} + 12 q^{4} - 9 q^{6} - 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) $ 15 * q - 9 * q^3 + 12 * q^4 - 9 * q^6 - 12 * q^7 + 6 * q^8 + 12 * q^9	$ 15 q - 9 q^{3} + 12 q^{4} - 9 q^{6} - 12 q^{7} + 6 q^{8} + 12 q^{9} - 6 q^{11} - 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} - 12 q^{22} - 36 q^{23} - 18 q^{24} + 36 q^{26} - 36 q^{27} - 24 q^{28} + 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} - 12 q^{37} - 6 q^{38} - 9 q^{39} + 18 q^{41} + 36 q^{42} - 3 q^{43} + 12 q^{44} - 21 q^{46} - 3 q^{47} + 12 q^{48} + 15 q^{49} - 27 q^{52} - 21 q^{54} + 6 q^{56} - 39 q^{57} - 18 q^{58} - 12 q^{59} + 15 q^{61} - 54 q^{62} - 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} - 6 q^{71} + 66 q^{72} + 9 q^{73} + 36 q^{74} - 18 q^{76} + 30 q^{77} - 30 q^{78} + 9 q^{79} + 51 q^{81} + 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} - 30 q^{88} - 24 q^{89} - 27 q^{91} - 15 q^{92} - 42 q^{93} - 57 q^{94} - 42 q^{96} - 48 q^{97} - 12 q^{99}+O(q^{100}) $ 15 * q - 9 * q^3 + 12 * q^4 - 9 * q^6 - 12 * q^7 + 6 * q^8 + 12 * q^9 - 6 * q^11 - 24 * q^12 + 6 * q^16 + 12 * q^18 + 6 * q^19 + 30 * q^21 - 12 * q^22 - 36 * q^23 - 18 * q^24 + 36 * q^26 - 36 * q^27 - 24 * q^28 + 18 * q^29 + 12 * q^32 + 12 * q^33 - 9 * q^36 - 12 * q^37 - 6 * q^38 - 9 * q^39 + 18 * q^41 + 36 * q^42 - 3 * q^43 + 12 * q^44 - 21 * q^46 - 3 * q^47 + 12 * q^48 + 15 * q^49 - 27 * q^52 - 21 * q^54 + 6 * q^56 - 39 * q^57 - 18 * q^58 - 12 * q^59 + 15 * q^61 - 54 * q^62 - 60 * q^63 - 36 * q^64 + 18 * q^66 - 24 * q^67 + 42 * q^69 - 6 * q^71 + 66 * q^72 + 9 * q^73 + 36 * q^74 - 18 * q^76 + 30 * q^77 - 30 * q^78 + 9 * q^79 + 51 * q^81 + 36 * q^82 + 15 * q^83 + 9 * q^84 - 36 * q^86 - 51 * q^87 - 30 * q^88 - 24 * q^89 - 27 * q^91 - 15 * q^92 - 42 * q^93 - 57 * q^94 - 42 * q^96 - 48 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.71131	1.21008	0.605038	−	0.796196i	$-0.293158\pi$
$2$	1.71131	1.21008	0.605038	+	0.796196i	$0.293158\pi$
$3$	0.109608	0.0632820	0.0316410	−	0.999499i	$-0.489927\pi$
$3$	0.109608	0.0632820	0.0316410	+	0.999499i	$0.489927\pi$
$4$	0.928572	0.464286
$5$	0	0
$5$	0	0
$6$	0.187572	0.0765761
$7$	4.34210	1.64116	0.820579	−	0.571533i	$-0.193651\pi$
$7$	4.34210	1.64116	0.820579	+	0.571533i	$0.193651\pi$
$8$	−1.83354	−0.648255
$9$	−2.98799	−0.995995
$10$	0	0
$11$	5.20553	1.56953	0.784763	−	0.619796i	$-0.212785\pi$
$11$	5.20553	1.56953	0.784763	+	0.619796i	$0.212785\pi$
$12$	0.101779	0.0293810
$13$	−4.88189	−1.35399	−0.676997	−	0.735986i	$-0.736719\pi$
$13$	−4.88189	−1.35399	−0.676997	+	0.735986i	$0.736719\pi$
$14$	7.43066	1.98593
$15$	0	0
$16$	−4.99490	−1.24872
$17$	0	0
$17$	0	0
$18$	−5.11336	−1.20523
$19$	−1.64053	−0.376364	−0.188182	−	0.982134i	$-0.560259\pi$
$19$	−1.64053	−0.376364	−0.188182	+	0.982134i	$0.560259\pi$
$20$	0	0
$21$	0.475927	0.103856
$22$	8.90825	1.89925
$23$	−8.39280	−1.75002	−0.875010	−	0.484105i	$-0.839145\pi$
$23$	−8.39280	−1.75002	−0.875010	+	0.484105i	$0.839145\pi$
$24$	−0.200970	−0.0410229
$25$	0	0
$26$	−8.35442	−1.63844
$27$	−0.656330	−0.126311
$28$	4.03195	0.761967
$29$	−0.485258	−0.0901101	−0.0450551	−	0.998985i	$-0.514346\pi$
$29$	−0.485258	−0.0901101	−0.0450551	+	0.998985i	$0.514346\pi$
$30$	0	0
$31$	−6.30076	−1.13165	−0.565825	−	0.824525i	$-0.691442\pi$
$31$	−6.30076	−1.13165	−0.565825	+	0.824525i	$0.691442\pi$
$32$	−4.88072	−0.862798
$33$	0.570566	0.0993228
$34$	0	0
$35$	0	0
$36$	−2.77456	−0.462427
$37$	−0.930097	−0.152907	−0.0764535	−	0.997073i	$-0.524360\pi$
$37$	−0.930097	−0.152907	−0.0764535	+	0.997073i	$0.524360\pi$
$38$	−2.80745	−0.455429
$39$	−0.535093	−0.0856835
$40$	0	0
$41$	−5.44822	−0.850869	−0.425434	−	0.904989i	$-0.639879\pi$
$41$	−5.44822	−0.850869	−0.425434	+	0.904989i	$0.639879\pi$
$42$	0.814458	0.125674
$43$	−7.53358	−1.14886	−0.574430	−	0.818554i	$-0.694776\pi$
$43$	−7.53358	−1.14886	−0.574430	+	0.818554i	$0.694776\pi$
$44$	4.83371	0.728709
$45$	0	0
$46$	−14.3627	−2.11766
$47$	−4.88962	−0.713224	−0.356612	−	0.934253i	$-0.616068\pi$
$47$	−4.88962	−0.713224	−0.356612	+	0.934253i	$0.616068\pi$
$48$	−0.547479	−0.0790218
$49$	11.8538	1.69340
$50$	0	0
$51$	0	0
$52$	−4.53319	−0.628641
$53$	3.10894	0.427046	0.213523	−	0.976938i	$-0.431506\pi$
$53$	3.10894	0.427046	0.213523	+	0.976938i	$0.431506\pi$
$54$	−1.12318	−0.152846
$55$	0	0
$56$	−7.96142	−1.06389
$57$	−0.179815	−0.0238171
$58$	−0.830425	−0.109040
$59$	−7.94104	−1.03384	−0.516918	−	0.856035i	$-0.672921\pi$
$59$	−7.94104	−1.03384	−0.516918	+	0.856035i	$0.672921\pi$
$60$	0	0
$61$	14.1498	1.81169	0.905845	−	0.423608i	$-0.139237\pi$
$61$	14.1498	1.81169	0.905845	+	0.423608i	$0.139237\pi$
$62$	−10.7825	−1.36938
$63$	−12.9741	−1.63459
$64$	1.63738	0.204673
$65$	0	0
$66$	0.976414	0.120188
$67$	−11.5894	−1.41588	−0.707938	−	0.706275i	$-0.750374\pi$
$67$	−11.5894	−1.41588	−0.707938	+	0.706275i	$0.750374\pi$
$68$	0	0
$69$	−0.919916	−0.110745
$70$	0	0
$71$	−0.201195	−0.0238775	−0.0119387	−	0.999929i	$-0.503800\pi$
$71$	−0.201195	−0.0238775	−0.0119387	+	0.999929i	$0.503800\pi$
$72$	5.47860	0.645659
$73$	−5.37389	−0.628967	−0.314483	−	0.949263i	$-0.601831\pi$
$73$	−5.37389	−0.628967	−0.314483	+	0.949263i	$0.601831\pi$
$74$	−1.59168	−0.185029
$75$	0	0
$76$	−1.52335	−0.174740
$77$	22.6029	2.57584
$78$	−0.915709	−0.103684
$79$	6.62508	0.745380	0.372690	−	0.927956i	$-0.378436\pi$
$79$	6.62508	0.745380	0.372690	+	0.927956i	$0.378436\pi$
$80$	0	0
$81$	8.89202	0.988002
$82$	−9.32358	−1.02962
$83$	10.5231	1.15507	0.577533	−	0.816367i	$-0.304016\pi$
$83$	10.5231	1.15507	0.577533	+	0.816367i	$0.304016\pi$
$84$	0.441933	0.0482188
$85$	0	0
$86$	−12.8923	−1.39021
$87$	−0.0531880	−0.00570235
$88$	−9.54455	−1.01745
$89$	−13.0569	−1.38403	−0.692016	−	0.721882i	$-0.743277\pi$
$89$	−13.0569	−1.38403	−0.692016	+	0.721882i	$0.743277\pi$
$90$	0	0
$91$	−21.1977	−2.22212
$92$	−7.79333	−0.812510
$93$	−0.690612	−0.0716132
$94$	−8.36764	−0.863056
$95$	0	0
$96$	−0.534965	−0.0545996
$97$	1.17959	0.119769	0.0598846	−	0.998205i	$-0.480927\pi$
$97$	1.17959	0.119769	0.0598846	+	0.998205i	$0.480927\pi$
$98$	20.2855	2.04915
$99$	−15.5540	−1.56324
$100$	0	0
$101$	−3.46756	−0.345035	−0.172518	−	0.985006i	$-0.555190\pi$
$101$	−3.46756	−0.345035	−0.172518	+	0.985006i	$0.555190\pi$
$102$	0	0
$103$	12.5490	1.23649	0.618244	−	0.785986i	$-0.287844\pi$
$103$	12.5490	1.23649	0.618244	+	0.785986i	$0.287844\pi$
$104$	8.95116	0.877733
$105$	0	0
$106$	5.32035	0.516758
$107$	−4.36167	−0.421659	−0.210829	−	0.977523i	$-0.567616\pi$
$107$	−4.36167	−0.421659	−0.210829	+	0.977523i	$0.567616\pi$
$108$	−0.609450	−0.0586443
$109$	−4.02991	−0.385995	−0.192998	−	0.981199i	$-0.561821\pi$
$109$	−4.02991	−0.385995	−0.192998	+	0.981199i	$0.561821\pi$
$110$	0	0
$111$	−0.101946	−0.00967627
$112$	−21.6883	−2.04935
$113$	9.65991	0.908728	0.454364	−	0.890816i	$-0.349867\pi$
$113$	9.65991	0.908728	0.454364	+	0.890816i	$0.349867\pi$
$114$	−0.307718	−0.0288205
$115$	0	0
$116$	−0.450597	−0.0418369
$117$	14.5870	1.34857
$118$	−13.5896	−1.25102
$119$	0	0
$120$	0	0
$121$	16.0975	1.46341
$122$	24.2146	2.19229
$123$	−0.597167	−0.0538447
$124$	−5.85071	−0.525410
$125$	0	0
$126$	−22.2027	−1.97798
$127$	12.0444	1.06877	0.534385	−	0.845241i	$-0.320543\pi$
$127$	12.0444	1.06877	0.534385	+	0.845241i	$0.320543\pi$
$128$	12.5635	1.11047
$129$	−0.825739	−0.0727022
$130$	0	0
$131$	1.82113	0.159112	0.0795562	−	0.996830i	$-0.474650\pi$
$131$	1.82113	0.159112	0.0795562	+	0.996830i	$0.474650\pi$
$132$	0.529812	0.0461142
$133$	−7.12334	−0.617672
$134$	−19.8331	−1.71332
$135$	0	0
$136$	0	0
$137$	−14.8662	−1.27010	−0.635052	−	0.772470i	$-0.719021\pi$
$137$	−14.8662	−1.27010	−0.635052	+	0.772470i	$0.719021\pi$
$138$	−1.57426	−0.134010
$139$	−16.3516	−1.38693	−0.693463	−	0.720493i	$-0.743916\pi$
$139$	−16.3516	−1.38693	−0.693463	+	0.720493i	$0.743916\pi$
$140$	0	0
$141$	−0.535940	−0.0451343
$142$	−0.344307	−0.0288936
$143$	−25.4128	−2.12513
$144$	14.9247	1.24372
$145$	0	0
$146$	−9.19639	−0.761098
$147$	1.29927	0.107162
$148$	−0.863663	−0.0709926
$149$	−2.23082	−0.182756	−0.0913779	−	0.995816i	$-0.529127\pi$
$149$	−2.23082	−0.182756	−0.0913779	+	0.995816i	$0.529127\pi$
$150$	0	0
$151$	−4.11034	−0.334495	−0.167247	−	0.985915i	$-0.553488\pi$
$151$	−4.11034	−0.334495	−0.167247	+	0.985915i	$0.553488\pi$
$152$	3.00798	0.243980
$153$	0	0
$154$	38.6805	3.11696
$155$	0	0
$156$	−0.496873	−0.0397817
$157$	−2.58687	−0.206454	−0.103227	−	0.994658i	$-0.532917\pi$
$157$	−2.58687	−0.206454	−0.103227	+	0.994658i	$0.532917\pi$
$158$	11.3375	0.901967
$159$	0.340764	0.0270243
$160$	0	0
$161$	−36.4424	−2.87206
$162$	15.2170	1.19556
$163$	4.78278	0.374616	0.187308	−	0.982301i	$-0.440024\pi$
$163$	4.78278	0.374616	0.187308	+	0.982301i	$0.440024\pi$
$164$	−5.05907	−0.395047
$165$	0	0
$166$	18.0083	1.39772
$167$	−1.16937	−0.0904888	−0.0452444	−	0.998976i	$-0.514407\pi$
$167$	−1.16937	−0.0904888	−0.0452444	+	0.998976i	$0.514407\pi$
$168$	−0.872633	−0.0673251
$169$	10.8329	0.833299
$170$	0	0
$171$	4.90188	0.374856
$172$	−6.99548	−0.533400
$173$	−1.81101	−0.137688	−0.0688441	−	0.997627i	$-0.521931\pi$
$173$	−1.81101	−0.137688	−0.0688441	+	0.997627i	$0.521931\pi$
$174$	−0.0910210	−0.00690029
$175$	0	0
$176$	−26.0011	−1.95990
$177$	−0.870400	−0.0654232
$178$	−22.3444	−1.67479
$179$	−18.3075	−1.36837	−0.684183	−	0.729310i	$-0.739841\pi$
$179$	−18.3075	−1.36837	−0.684183	+	0.729310i	$0.739841\pi$
$180$	0	0
$181$	4.61305	0.342885	0.171443	−	0.985194i	$-0.445157\pi$
$181$	4.61305	0.342885	0.171443	+	0.985194i	$0.445157\pi$
$182$	−36.2757	−2.68893
$183$	1.55092	0.114648
$184$	15.3886	1.13446
$185$	0	0
$186$	−1.18185	−0.0866574
$187$	0	0
$188$	−4.54036	−0.331140
$189$	−2.84985	−0.207296
$190$	0	0
$191$	11.6370	0.842026	0.421013	−	0.907055i	$-0.361675\pi$
$191$	11.6370	0.842026	0.421013	+	0.907055i	$0.361675\pi$
$192$	0.179470	0.0129521
$193$	−7.61788	−0.548347	−0.274174	−	0.961680i	$-0.588404\pi$
$193$	−7.61788	−0.548347	−0.274174	+	0.961680i	$0.588404\pi$
$194$	2.01864	0.144930
$195$	0	0
$196$	11.0071	0.786223
$197$	5.91667	0.421545	0.210773	−	0.977535i	$-0.432402\pi$
$197$	5.91667	0.421545	0.210773	+	0.977535i	$0.432402\pi$
$198$	−26.6177	−1.89164
$199$	21.8358	1.54790	0.773949	−	0.633248i	$-0.218279\pi$
$199$	21.8358	1.54790	0.773949	+	0.633248i	$0.218279\pi$
$200$	0	0
$201$	−1.27029	−0.0895995
$202$	−5.93407	−0.417519
$203$	−2.10704	−0.147885
$204$	0	0
$205$	0	0
$206$	21.4752	1.49625
$207$	25.0776	1.74301
$208$	24.3846	1.69077
$209$	−8.53983	−0.590712
$210$	0	0
$211$	−2.27702	−0.156757	−0.0783784	−	0.996924i	$-0.524974\pi$
$211$	−2.27702	−0.156757	−0.0783784	+	0.996924i	$0.524974\pi$
$212$	2.88688	0.198272
$213$	−0.0220525	−0.00151101
$214$	−7.46416	−0.510240
$215$	0	0
$216$	1.20341	0.0818815
$217$	−27.3585	−1.85722
$218$	−6.89641	−0.467084
$219$	−0.589020	−0.0398023
$220$	0	0
$221$	0	0
$222$	−0.174461	−0.0117090
$223$	15.9071	1.06522	0.532608	−	0.846362i	$-0.321212\pi$
$223$	15.9071	1.06522	0.532608	+	0.846362i	$0.321212\pi$
$224$	−21.1926	−1.41599
$225$	0	0
$226$	16.5311	1.09963
$227$	−12.0149	−0.797456	−0.398728	−	0.917069i	$-0.630548\pi$
$227$	−12.0149	−0.797456	−0.398728	+	0.917069i	$0.630548\pi$
$228$	−0.166971	−0.0110579
$229$	−23.5914	−1.55897	−0.779483	−	0.626423i	$-0.784518\pi$
$229$	−23.5914	−1.55897	−0.779483	+	0.626423i	$0.784518\pi$
$230$	0	0
$231$	2.47745	0.163004
$232$	0.889741	0.0584143
$233$	−12.9704	−0.849717	−0.424859	−	0.905260i	$-0.639676\pi$
$233$	−12.9704	−0.849717	−0.424859	+	0.905260i	$0.639676\pi$
$234$	24.9629	1.63188
$235$	0	0
$236$	−7.37383	−0.479996
$237$	0.726160	0.0471691
$238$	0	0
$239$	−26.2648	−1.69893	−0.849465	−	0.527646i	$-0.823075\pi$
$239$	−26.2648	−1.69893	−0.849465	+	0.527646i	$0.823075\pi$
$240$	0	0
$241$	−22.5879	−1.45501	−0.727506	−	0.686101i	$-0.759321\pi$
$241$	−22.5879	−1.45501	−0.727506	+	0.686101i	$0.759321\pi$
$242$	27.5478	1.77084
$243$	2.94362	0.188833
$244$	13.1391	0.841143
$245$	0	0
$246$	−1.02194	−0.0651563
$247$	8.00890	0.509594
$248$	11.5527	0.733598
$249$	1.15342	0.0730949
$250$	0	0
$251$	28.9971	1.83028	0.915141	−	0.403133i	$-0.132079\pi$
$251$	28.9971	1.83028	0.915141	+	0.403133i	$0.132079\pi$
$252$	−12.0474	−0.758916
$253$	−43.6890	−2.74670
$254$	20.6117	1.29329
$255$	0	0
$256$	18.2253	1.13908
$257$	−21.6041	−1.34762	−0.673812	−	0.738902i	$-0.735344\pi$
$257$	−21.6041	−1.34762	−0.673812	+	0.738902i	$0.735344\pi$
$258$	−1.41309	−0.0879753
$259$	−4.03857	−0.250945
$260$	0	0
$261$	1.44994	0.0897493
$262$	3.11651	0.192538
$263$	10.2248	0.630489	0.315245	−	0.949010i	$-0.397913\pi$
$263$	10.2248	0.630489	0.315245	+	0.949010i	$0.397913\pi$
$264$	−1.04616	−0.0643865
$265$	0	0
$266$	−12.1902	−0.747431
$267$	−1.43114	−0.0875844
$268$	−10.7616	−0.657372
$269$	30.5487	1.86259	0.931294	−	0.364269i	$-0.118681\pi$
$269$	30.5487	1.86259	0.931294	+	0.364269i	$0.118681\pi$
$270$	0	0
$271$	−10.7654	−0.653953	−0.326976	−	0.945033i	$-0.606030\pi$
$271$	−10.7654	−0.653953	−0.326976	+	0.945033i	$0.606030\pi$
$272$	0	0
$273$	−2.32343	−0.140620
$274$	−25.4406	−1.53692
$275$	0	0
$276$	−0.854209	−0.0514173
$277$	22.9465	1.37872	0.689361	−	0.724418i	$-0.257891\pi$
$277$	22.9465	1.37872	0.689361	+	0.724418i	$0.257891\pi$
$278$	−27.9826	−1.67829
$279$	18.8266	1.12712
$280$	0	0
$281$	21.3918	1.27612	0.638062	−	0.769985i	$-0.279736\pi$
$281$	21.3918	1.27612	0.638062	+	0.769985i	$0.279736\pi$
$282$	−0.917158	−0.0546160
$283$	−15.5072	−0.921807	−0.460904	−	0.887450i	$-0.652475\pi$
$283$	−15.5072	−0.921807	−0.460904	+	0.887450i	$0.652475\pi$
$284$	−0.186824	−0.0110860
$285$	0	0
$286$	−43.4892	−2.57157
$287$	−23.6567	−1.39641
$288$	14.5835	0.859343
$289$	0	0
$290$	0	0
$291$	0.129292	0.00757924
$292$	−4.99005	−0.292021
$293$	−11.1577	−0.651842	−0.325921	−	0.945397i	$-0.605674\pi$
$293$	−11.1577	−0.651842	−0.325921	+	0.945397i	$0.605674\pi$
$294$	2.22345	0.129674
$295$	0	0
$296$	1.70537	0.0991227
$297$	−3.41654	−0.198248
$298$	−3.81762	−0.221149
$299$	40.9728	2.36952
$300$	0	0
$301$	−32.7115	−1.88546
$302$	−7.03406	−0.404764
$303$	−0.380072	−0.0218345
$304$	8.19428	0.469974
$305$	0	0
$306$	0	0
$307$	−32.7577	−1.86958	−0.934790	−	0.355201i	$-0.884412\pi$
$307$	−32.7577	−1.86958	−0.934790	+	0.355201i	$0.884412\pi$
$308$	20.9884	1.19593
$309$	1.37547	0.0782475
$310$	0	0
$311$	19.1267	1.08458	0.542289	−	0.840192i	$-0.317558\pi$
$311$	19.1267	1.08458	0.542289	+	0.840192i	$0.317558\pi$
$312$	0.981116	0.0555447
$313$	−9.62567	−0.544075	−0.272037	−	0.962287i	$-0.587697\pi$
$313$	−9.62567	−0.544075	−0.272037	+	0.962287i	$0.587697\pi$
$314$	−4.42692	−0.249826
$315$	0	0
$316$	6.15187	0.346069
$317$	7.48875	0.420610	0.210305	−	0.977636i	$-0.432554\pi$
$317$	7.48875	0.420610	0.210305	+	0.977636i	$0.432554\pi$
$318$	0.583152	0.0327015
$319$	−2.52602	−0.141430
$320$	0	0
$321$	−0.478073	−0.0266834
$322$	−62.3641	−3.47541
$323$	0	0
$324$	8.25688	0.458716
$325$	0	0
$326$	8.18480	0.453314
$327$	−0.441709	−0.0244266
$328$	9.98954	0.551580
$329$	−21.2312	−1.17051
$330$	0	0
$331$	12.6115	0.693189	0.346594	−	0.938015i	$-0.387338\pi$
$331$	12.6115	0.693189	0.346594	+	0.938015i	$0.387338\pi$
$332$	9.77151	0.536281
$333$	2.77912	0.152295
$334$	−2.00116	−0.109498
$335$	0	0
$336$	−2.37721	−0.129687
$337$	−31.7383	−1.72889	−0.864447	−	0.502724i	$-0.832331\pi$
$337$	−31.7383	−1.72889	−0.864447	+	0.502724i	$0.832331\pi$
$338$	18.5384	1.00836
$339$	1.05880	0.0575062
$340$	0	0
$341$	−32.7988	−1.77615
$342$	8.38863	0.453605
$343$	21.0757	1.13798
$344$	13.8131	0.744754
$345$	0	0
$346$	−3.09919	−0.166613
$347$	3.79210	0.203571	0.101785	−	0.994806i	$-0.467544\pi$
$347$	3.79210	0.203571	0.101785	+	0.994806i	$0.467544\pi$
$348$	−0.0493889	−0.00264752
$349$	16.8636	0.902689	0.451344	−	0.892350i	$-0.350945\pi$
$349$	16.8636	0.902689	0.451344	+	0.892350i	$0.350945\pi$
$350$	0	0
$351$	3.20413	0.171024
$352$	−25.4067	−1.35418
$353$	24.7095	1.31516	0.657578	−	0.753387i	$-0.271581\pi$
$353$	24.7095	1.31516	0.657578	+	0.753387i	$0.271581\pi$
$354$	−1.48952	−0.0791672
$355$	0	0
$356$	−12.1243	−0.642587
$357$	0	0
$358$	−31.3298	−1.65583
$359$	16.5864	0.875395	0.437697	−	0.899122i	$-0.355794\pi$
$359$	16.5864	0.875395	0.437697	+	0.899122i	$0.355794\pi$
$360$	0	0
$361$	−16.3087	−0.858350
$362$	7.89435	0.414918
$363$	1.76441	0.0926075
$364$	−19.6836	−1.03170
$365$	0	0
$366$	2.65411	0.138732
$367$	−23.5105	−1.22724	−0.613619	−	0.789602i	$-0.710287\pi$
$367$	−23.5105	−1.22724	−0.613619	+	0.789602i	$0.710287\pi$
$368$	41.9212	2.18529
$369$	16.2792	0.847461
$370$	0	0
$371$	13.4993	0.700850
$372$	−0.641284	−0.0332490
$373$	−6.07680	−0.314645	−0.157322	−	0.987547i	$-0.550286\pi$
$373$	−6.07680	−0.314645	−0.157322	+	0.987547i	$0.550286\pi$
$374$	0	0
$375$	0	0
$376$	8.96532	0.462351
$377$	2.36898	0.122009
$378$	−4.87696	−0.250844
$379$	0.531157	0.0272837	0.0136419	−	0.999907i	$-0.495658\pi$
$379$	0.531157	0.0272837	0.0136419	+	0.999907i	$0.495658\pi$
$380$	0	0
$381$	1.32016	0.0676340
$382$	19.9145	1.01892
$383$	−4.35039	−0.222294	−0.111147	−	0.993804i	$-0.535453\pi$
$383$	−4.35039	−0.222294	−0.111147	+	0.993804i	$0.535453\pi$
$384$	1.37706	0.0702727
$385$	0	0
$386$	−13.0365	−0.663542
$387$	22.5102	1.14426
$388$	1.09533	0.0556072
$389$	6.47149	0.328118	0.164059	−	0.986451i	$-0.447541\pi$
$389$	6.47149	0.328118	0.164059	+	0.986451i	$0.447541\pi$
$390$	0	0
$391$	0	0
$392$	−21.7344	−1.09776
$393$	0.199609	0.0100690
$394$	10.1252	0.510102
$395$	0	0
$396$	−14.4431	−0.725791
$397$	−23.8954	−1.19928	−0.599639	−	0.800271i	$-0.704689\pi$
$397$	−23.8954	−1.19928	−0.599639	+	0.800271i	$0.704689\pi$
$398$	37.3677	1.87307
$399$	−0.780773	−0.0390876
$400$	0	0
$401$	−18.5433	−0.926007	−0.463003	−	0.886357i	$-0.653228\pi$
$401$	−18.5433	−0.926007	−0.463003	+	0.886357i	$0.653228\pi$
$402$	−2.17386	−0.108422
$403$	30.7597	1.53225
$404$	−3.21988	−0.160195
$405$	0	0
$406$	−3.60579	−0.178952
$407$	−4.84164	−0.239991
$408$	0	0
$409$	10.7079	0.529469	0.264735	−	0.964321i	$-0.414716\pi$
$409$	10.7079	0.529469	0.264735	+	0.964321i	$0.414716\pi$
$410$	0	0
$411$	−1.62945	−0.0803747
$412$	11.6526	0.574085
$413$	−34.4808	−1.69669
$414$	42.9154	2.10918
$415$	0	0
$416$	23.8272	1.16822
$417$	−1.79226	−0.0877675
$418$	−14.6143	−0.714807
$419$	10.3145	0.503895	0.251947	−	0.967741i	$-0.418929\pi$
$419$	10.3145	0.503895	0.251947	+	0.967741i	$0.418929\pi$
$420$	0	0
$421$	−28.0974	−1.36939	−0.684693	−	0.728832i	$-0.740064\pi$
$421$	−28.0974	−1.36939	−0.684693	+	0.728832i	$0.740064\pi$
$422$	−3.89669	−0.189688
$423$	14.6101	0.710368
$424$	−5.70037	−0.276835
$425$	0	0
$426$	−0.0377387	−0.00182844
$427$	61.4396	2.97327
$428$	−4.05013	−0.195770
$429$	−2.78544	−0.134482
$430$	0	0
$431$	−17.0458	−0.821066	−0.410533	−	0.911846i	$-0.634657\pi$
$431$	−17.0458	−0.821066	−0.410533	+	0.911846i	$0.634657\pi$
$432$	3.27830	0.157727
$433$	0.719105	0.0345580	0.0172790	−	0.999851i	$-0.494500\pi$
$433$	0.719105	0.0345580	0.0172790	+	0.999851i	$0.494500\pi$
$434$	−46.8188	−2.24738
$435$	0	0
$436$	−3.74206	−0.179212
$437$	13.7686	0.658644
$438$	−1.00799	−0.0481639
$439$	−11.8427	−0.565222	−0.282611	−	0.959235i	$-0.591201\pi$
$439$	−11.8427	−0.565222	−0.282611	+	0.959235i	$0.591201\pi$
$440$	0	0
$441$	−35.4190	−1.68662
$442$	0	0
$443$	2.37057	0.112629	0.0563147	−	0.998413i	$-0.482065\pi$
$443$	2.37057	0.112629	0.0563147	+	0.998413i	$0.482065\pi$
$444$	−0.0946641	−0.00449256
$445$	0	0
$446$	27.2219	1.28899
$447$	−0.244515	−0.0115652
$448$	7.10967	0.335900
$449$	14.1762	0.669016	0.334508	−	0.942393i	$-0.391430\pi$
$449$	14.1762	0.669016	0.334508	+	0.942393i	$0.391430\pi$
$450$	0	0
$451$	−28.3608	−1.33546
$452$	8.96993	0.421910
$453$	−0.450525	−0.0211675
$454$	−20.5612	−0.964984
$455$	0	0
$456$	0.329698	0.0154395
$457$	24.3340	1.13830	0.569149	−	0.822235i	$-0.307273\pi$
$457$	24.3340	1.13830	0.569149	+	0.822235i	$0.307273\pi$
$458$	−40.3722	−1.88647
$459$	0	0
$460$	0	0
$461$	−36.8238	−1.71506	−0.857528	−	0.514437i	$-0.828001\pi$
$461$	−36.8238	−1.71506	−0.857528	+	0.514437i	$0.828001\pi$
$462$	4.23968	0.197248
$463$	25.0303	1.16326	0.581628	−	0.813455i	$-0.302416\pi$
$463$	25.0303	1.16326	0.581628	+	0.813455i	$0.302416\pi$
$464$	2.42381	0.112523
$465$	0	0
$466$	−22.1963	−1.02822
$467$	−14.2892	−0.661224	−0.330612	−	0.943767i	$-0.607255\pi$
$467$	−14.2892	−0.661224	−0.330612	+	0.943767i	$0.607255\pi$
$468$	13.5451	0.626123
$469$	−50.3225	−2.32368
$470$	0	0
$471$	−0.283541	−0.0130649
$472$	14.5602	0.670189
$473$	−39.2163	−1.80317
$474$	1.24268	0.0570783
$475$	0	0
$476$	0	0
$477$	−9.28947	−0.425336
$478$	−44.9472	−2.05584
$479$	32.6409	1.49140	0.745701	−	0.666281i	$-0.232115\pi$
$479$	32.6409	1.49140	0.745701	+	0.666281i	$0.232115\pi$
$480$	0	0
$481$	4.54063	0.207035
$482$	−38.6548	−1.76068
$483$	−3.99436	−0.181750
$484$	14.9477	0.679441
$485$	0	0
$486$	5.03744	0.228503
$487$	−21.4885	−0.973739	−0.486869	−	0.873475i	$-0.661861\pi$
$487$	−21.4885	−0.973739	−0.486869	+	0.873475i	$0.661861\pi$
$488$	−25.9442	−1.17444
$489$	0.524229	0.0237065
$490$	0	0
$491$	15.9151	0.718240	0.359120	−	0.933291i	$-0.383077\pi$
$491$	15.9151	0.718240	0.359120	+	0.933291i	$0.383077\pi$
$492$	−0.554513	−0.0249994
$493$	0	0
$494$	13.7057	0.616648
$495$	0	0
$496$	31.4717	1.41312
$497$	−0.873608	−0.0391867
$498$	1.97385	0.0884505
$499$	11.1698	0.500031	0.250015	−	0.968242i	$-0.419564\pi$
$499$	11.1698	0.500031	0.250015	+	0.968242i	$0.419564\pi$
$500$	0	0
$501$	−0.128172	−0.00572631
$502$	49.6230	2.21478
$503$	−24.0544	−1.07253	−0.536266	−	0.844049i	$-0.680166\pi$
$503$	−24.0544	−1.07253	−0.536266	+	0.844049i	$0.680166\pi$
$504$	23.7886	1.05963
$505$	0	0
$506$	−74.7652	−3.32372
$507$	1.18737	0.0527329
$508$	11.1841	0.496215
$509$	8.62627	0.382353	0.191176	−	0.981556i	$-0.438770\pi$
$509$	8.62627	0.382353	0.191176	+	0.981556i	$0.438770\pi$
$510$	0	0
$511$	−23.3340	−1.03223
$512$	6.06200	0.267905
$513$	1.07673	0.0475387
$514$	−36.9712	−1.63073
$515$	0	0
$516$	−0.766758	−0.0337547
$517$	−25.4530	−1.11942
$518$	−6.91124	−0.303662
$519$	−0.198500	−0.00871319
$520$	0	0
$521$	−3.15348	−0.138157	−0.0690783	−	0.997611i	$-0.522006\pi$
$521$	−3.15348	−0.138157	−0.0690783	+	0.997611i	$0.522006\pi$
$522$	2.48130	0.108604
$523$	−14.8175	−0.647923	−0.323962	−	0.946070i	$-0.605015\pi$
$523$	−14.8175	−0.647923	−0.323962	+	0.946070i	$0.605015\pi$
$524$	1.69105	0.0738737
$525$	0	0
$526$	17.4978	0.762940
$527$	0	0
$528$	−2.84992	−0.124027
$529$	47.4391	2.06257
$530$	0	0
$531$	23.7277	1.02970
$532$	−6.61454	−0.286777
$533$	26.5976	1.15207
$534$	−2.44912	−0.105984
$535$	0	0
$536$	21.2497	0.917848
$537$	−2.00664	−0.0865930
$538$	52.2782	2.25387
$539$	61.7053	2.65784
$540$	0	0
$541$	25.4124	1.09256	0.546282	−	0.837602i	$-0.316043\pi$
$541$	25.4124	1.09256	0.546282	+	0.837602i	$0.316043\pi$
$542$	−18.4229	−0.791333
$543$	0.505626	0.0216985
$544$	0	0
$545$	0	0
$546$	−3.97610	−0.170161
$547$	13.1706	0.563133	0.281566	−	0.959542i	$-0.409146\pi$
$547$	13.1706	0.563133	0.281566	+	0.959542i	$0.409146\pi$
$548$	−13.8043	−0.589692
$549$	−42.2793	−1.80444
$550$	0	0
$551$	0.796080	0.0339142
$552$	1.68670	0.0717909
$553$	28.7667	1.22329
$554$	39.2685	1.66836
$555$	0	0
$556$	−15.1837	−0.643930
$557$	27.4786	1.16431	0.582154	−	0.813079i	$-0.302210\pi$
$557$	27.4786	1.16431	0.582154	+	0.813079i	$0.302210\pi$
$558$	32.2181	1.36390
$559$	36.7781	1.55555
$560$	0	0
$561$	0	0
$562$	36.6079	1.54421
$563$	−4.57643	−0.192874	−0.0964368	−	0.995339i	$-0.530745\pi$
$563$	−4.57643	−0.192874	−0.0964368	+	0.995339i	$0.530745\pi$
$564$	−0.497659	−0.0209552
$565$	0	0
$566$	−26.5376	−1.11546
$567$	38.6100	1.62147
$568$	0.368900	0.0154787
$569$	26.9526	1.12991	0.564956	−	0.825121i	$-0.308893\pi$
$569$	26.9526	1.12991	0.564956	+	0.825121i	$0.308893\pi$
$570$	0	0
$571$	−30.6171	−1.28128	−0.640642	−	0.767840i	$-0.721332\pi$
$571$	−30.6171	−1.28128	−0.640642	+	0.767840i	$0.721332\pi$
$572$	−23.5976	−0.986667
$573$	1.27551	0.0532851
$574$	−40.4839	−1.68976
$575$	0	0
$576$	−4.89247	−0.203853
$577$	28.7634	1.19744	0.598718	−	0.800960i	$-0.295677\pi$
$577$	28.7634	1.19744	0.598718	+	0.800960i	$0.295677\pi$
$578$	0	0
$579$	−0.834979	−0.0347005
$580$	0	0
$581$	45.6925	1.89565
$582$	0.221259	0.00917146
$583$	16.1837	0.670259
$584$	9.85326	0.407731
$585$	0	0
$586$	−19.0943	−0.788779
$587$	−11.2016	−0.462339	−0.231170	−	0.972913i	$-0.574255\pi$
$587$	−11.2016	−0.462339	−0.231170	+	0.972913i	$0.574255\pi$
$588$	1.20647	0.0497538
$589$	10.3366	0.425912
$590$	0	0
$591$	0.648512	0.0266762
$592$	4.64574	0.190939
$593$	−5.70078	−0.234103	−0.117051	−	0.993126i	$-0.537344\pi$
$593$	−5.70078	−0.234103	−0.117051	+	0.993126i	$0.537344\pi$
$594$	−5.84675	−0.239895
$595$	0	0
$596$	−2.07148	−0.0848510
$597$	2.39337	0.0979541
$598$	70.1170	2.86730
$599$	8.28563	0.338542	0.169271	−	0.985570i	$-0.445859\pi$
$599$	8.28563	0.338542	0.169271	+	0.985570i	$0.445859\pi$
$600$	0	0
$601$	42.9633	1.75251	0.876254	−	0.481849i	$-0.160035\pi$
$601$	42.9633	1.75251	0.876254	+	0.481849i	$0.160035\pi$
$602$	−55.9795	−2.28155
$603$	34.6291	1.41021
$604$	−3.81675	−0.155301
$605$	0	0
$606$	−0.650419	−0.0264215
$607$	−18.1339	−0.736032	−0.368016	−	0.929820i	$-0.619963\pi$
$607$	−18.1339	−0.736032	−0.368016	+	0.929820i	$0.619963\pi$
$608$	8.00697	0.324726
$609$	−0.230948	−0.00935847
$610$	0	0
$611$	23.8706	0.965701
$612$	0	0
$613$	−17.0246	−0.687616	−0.343808	−	0.939040i	$-0.611717\pi$
$613$	−17.0246	−0.687616	−0.343808	+	0.939040i	$0.611717\pi$
$614$	−56.0585	−2.26234
$615$	0	0
$616$	−41.4434	−1.66980
$617$	7.40951	0.298296	0.149148	−	0.988815i	$-0.452347\pi$
$617$	7.40951	0.298296	0.149148	+	0.988815i	$0.452347\pi$
$618$	2.35384	0.0946855
$619$	22.0128	0.884769	0.442385	−	0.896825i	$-0.354133\pi$
$619$	22.0128	0.884769	0.442385	+	0.896825i	$0.354133\pi$
$620$	0	0
$621$	5.50844	0.221046
$622$	32.7317	1.31242
$623$	−56.6945	−2.27142
$624$	2.67274	0.106995
$625$	0	0
$626$	−16.4725	−0.658372
$627$	−0.936031	−0.0373815
$628$	−2.40209	−0.0958540
$629$	0	0
$630$	0	0
$631$	−32.5360	−1.29524	−0.647619	−	0.761965i	$-0.724235\pi$
$631$	−32.5360	−1.29524	−0.647619	+	0.761965i	$0.724235\pi$
$632$	−12.1474	−0.483196
$633$	−0.249579	−0.00991989
$634$	12.8156	0.508971
$635$	0	0
$636$	0.316424	0.0125470
$637$	−57.8690	−2.29285
$638$	−4.32280	−0.171141
$639$	0.601168	0.0237818
$640$	0	0
$641$	3.52036	0.139046	0.0695229	−	0.997580i	$-0.477852\pi$
$641$	3.52036	0.139046	0.0695229	+	0.997580i	$0.477852\pi$
$642$	−0.818130	−0.0322890
$643$	−21.1594	−0.834445	−0.417223	−	0.908804i	$-0.636996\pi$
$643$	−21.1594	−0.834445	−0.417223	+	0.908804i	$0.636996\pi$
$644$	−33.8394	−1.33346
$645$	0	0
$646$	0	0
$647$	9.06885	0.356533	0.178267	−	0.983982i	$-0.442951\pi$
$647$	9.06885	0.356533	0.178267	+	0.983982i	$0.442951\pi$
$648$	−16.3039	−0.640477
$649$	−41.3373	−1.62263
$650$	0	0
$651$	−2.99871	−0.117529
$652$	4.44116	0.173929
$653$	−15.2290	−0.595958	−0.297979	−	0.954572i	$-0.596313\pi$
$653$	−15.2290	−0.595958	−0.297979	+	0.954572i	$0.596313\pi$
$654$	−0.755900	−0.0295580
$655$	0	0
$656$	27.2133	1.06250
$657$	16.0571	0.626448
$658$	−36.3331	−1.41641
$659$	−4.19934	−0.163583	−0.0817916	−	0.996649i	$-0.526064\pi$
$659$	−4.19934	−0.163583	−0.0817916	+	0.996649i	$0.526064\pi$
$660$	0	0
$661$	46.7564	1.81861	0.909306	−	0.416127i	$-0.136613\pi$
$661$	46.7564	1.81861	0.909306	+	0.416127i	$0.136613\pi$
$662$	21.5821	0.838812
$663$	0	0
$664$	−19.2946	−0.748777
$665$	0	0
$666$	4.75592	0.184288
$667$	4.07267	0.157695
$668$	−1.08585	−0.0420127
$669$	1.74354	0.0674090
$670$	0	0
$671$	73.6569	2.84349
$672$	−2.32287	−0.0896066
$673$	10.3268	0.398068	0.199034	−	0.979993i	$-0.436220\pi$
$673$	10.3268	0.398068	0.199034	+	0.979993i	$0.436220\pi$
$674$	−54.3139	−2.09209
$675$	0	0
$676$	10.0591	0.386889
$677$	−32.2638	−1.24000	−0.619999	−	0.784602i	$-0.712867\pi$
$677$	−32.2638	−1.24000	−0.619999	+	0.784602i	$0.712867\pi$
$678$	1.81193	0.0695869
$679$	5.12189	0.196560
$680$	0	0
$681$	−1.31692	−0.0504647
$682$	−56.1288	−2.14928
$683$	−20.7965	−0.795755	−0.397877	−	0.917439i	$-0.630253\pi$
$683$	−20.7965	−0.795755	−0.397877	+	0.917439i	$0.630253\pi$
$684$	4.55175	0.174041
$685$	0	0
$686$	36.0670	1.37704
$687$	−2.58580	−0.0986546
$688$	37.6295	1.43461
$689$	−15.1775	−0.578217
$690$	0	0
$691$	19.7441	0.751102	0.375551	−	0.926802i	$-0.377453\pi$
$691$	19.7441	0.751102	0.375551	+	0.926802i	$0.377453\pi$
$692$	−1.68165	−0.0639267
$693$	−67.5372	−2.56552
$694$	6.48946	0.246336
$695$	0	0
$696$	0.0975224	0.00369658
$697$	0	0
$698$	28.8588	1.09232
$699$	−1.42165	−0.0537718
$700$	0	0
$701$	−9.51246	−0.359281	−0.179640	−	0.983732i	$-0.557493\pi$
$701$	−9.51246	−0.359281	−0.179640	+	0.983732i	$0.557493\pi$
$702$	5.48325	0.206952
$703$	1.52585	0.0575486
$704$	8.52343	0.321239
$705$	0	0
$706$	42.2856	1.59144
$707$	−15.0565	−0.566258
$708$	−0.808229	−0.0303751
$709$	−26.5052	−0.995425	−0.497713	−	0.867342i	$-0.665827\pi$
$709$	−26.5052	−0.995425	−0.497713	+	0.867342i	$0.665827\pi$
$710$	0	0
$711$	−19.7956	−0.742395
$712$	23.9404	0.897205
$713$	52.8811	1.98041
$714$	0	0
$715$	0	0
$716$	−16.9998	−0.635314
$717$	−2.87883	−0.107512
$718$	28.3844	1.05929
$719$	24.3830	0.909331	0.454666	−	0.890662i	$-0.349759\pi$
$719$	24.3830	0.909331	0.454666	+	0.890662i	$0.349759\pi$
$720$	0	0
$721$	54.4889	2.02927
$722$	−27.9091	−1.03867
$723$	−2.47580	−0.0920761
$724$	4.28355	0.159197
$725$	0	0
$726$	3.01945	0.112062
$727$	−0.0836850	−0.00310371	−0.00155185	−	0.999999i	$-0.500494\pi$
$727$	−0.0836850	−0.00310371	−0.00155185	+	0.999999i	$0.500494\pi$
$728$	38.8668	1.44050
$729$	−26.3534	−0.976052
$730$	0	0
$731$	0	0
$732$	1.44014	0.0532293
$733$	−3.01445	−0.111341	−0.0556706	−	0.998449i	$-0.517730\pi$
$733$	−3.01445	−0.111341	−0.0556706	+	0.998449i	$0.517730\pi$
$734$	−40.2337	−1.48505
$735$	0	0
$736$	40.9629	1.50991
$737$	−60.3292	−2.22225
$738$	27.8587	1.02549
$739$	21.4747	0.789958	0.394979	−	0.918690i	$-0.370752\pi$
$739$	21.4747	0.789958	0.394979	+	0.918690i	$0.370752\pi$
$740$	0	0
$741$	0.877837	0.0322481
$742$	23.1015	0.848082
$743$	−1.29230	−0.0474101	−0.0237050	−	0.999719i	$-0.507546\pi$
$743$	−1.29230	−0.0474101	−0.0237050	+	0.999719i	$0.507546\pi$
$744$	1.26627	0.0464236
$745$	0	0
$746$	−10.3993	−0.380744
$747$	−31.4430	−1.15044
$748$	0	0
$749$	−18.9388	−0.692009
$750$	0	0
$751$	29.3569	1.07125	0.535623	−	0.844457i	$-0.320077\pi$
$751$	29.3569	1.07125	0.535623	+	0.844457i	$0.320077\pi$
$752$	24.4231	0.890620
$753$	3.17831	0.115824
$754$	4.05405	0.147640
$755$	0	0
$756$	−2.64629	−0.0962446
$757$	−1.20743	−0.0438848	−0.0219424	−	0.999759i	$-0.506985\pi$
$757$	−1.20743	−0.0438848	−0.0219424	+	0.999759i	$0.506985\pi$
$758$	0.908973	0.0330154
$759$	−4.78865	−0.173817
$760$	0	0
$761$	−7.48319	−0.271265	−0.135633	−	0.990759i	$-0.543307\pi$
$761$	−7.48319	−0.271265	−0.135633	+	0.990759i	$0.543307\pi$
$762$	2.25920	0.0818423
$763$	−17.4983	−0.633479
$764$	10.8058	0.390941
$765$	0	0
$766$	−7.44485	−0.268993
$767$	38.7673	1.39981
$768$	1.99763	0.0720832
$769$	−43.3632	−1.56372	−0.781859	−	0.623456i	$-0.785728\pi$
$769$	−43.3632	−1.56372	−0.781859	+	0.623456i	$0.785728\pi$
$770$	0	0
$771$	−2.36797	−0.0852805
$772$	−7.07376	−0.254590
$773$	−35.7762	−1.28678	−0.643390	−	0.765539i	$-0.722472\pi$
$773$	−35.7762	−1.28678	−0.643390	+	0.765539i	$0.722472\pi$
$774$	38.5219	1.38464
$775$	0	0
$776$	−2.16283	−0.0776409
$777$	−0.442659	−0.0158803
$778$	11.0747	0.397047
$779$	8.93797	0.320236
$780$	0	0
$781$	−1.04733	−0.0374763
$782$	0	0
$783$	0.318489	0.0113819
$784$	−59.2086	−2.11459
$785$	0	0
$786$	0.341593	0.0121842
$787$	−29.8797	−1.06510	−0.532549	−	0.846399i	$-0.678766\pi$
$787$	−29.8797	−1.06510	−0.532549	+	0.846399i	$0.678766\pi$
$788$	5.49405	0.195718
$789$	1.12072	0.0398986
$790$	0	0
$791$	41.9443	1.49137
$792$	28.5190	1.01338
$793$	−69.0776	−2.45302
$794$	−40.8924	−1.45122
$795$	0	0
$796$	20.2761	0.718667
$797$	−5.40524	−0.191463	−0.0957317	−	0.995407i	$-0.530519\pi$
$797$	−5.40524	−0.191463	−0.0957317	+	0.995407i	$0.530519\pi$
$798$	−1.33614	−0.0472990
$799$	0	0
$800$	0	0
$801$	39.0139	1.37849
$802$	−31.7332	−1.12054
$803$	−27.9740	−0.987179
$804$	−1.17956	−0.0415998
$805$	0	0
$806$	52.6392	1.85414
$807$	3.34837	0.117868
$808$	6.35792	0.223671
$809$	−30.2974	−1.06520	−0.532600	−	0.846367i	$-0.678785\pi$
$809$	−30.2974	−1.06520	−0.532600	+	0.846367i	$0.678785\pi$
$810$	0	0
$811$	44.5944	1.56592	0.782961	−	0.622071i	$-0.213708\pi$
$811$	44.5944	1.56592	0.782961	+	0.622071i	$0.213708\pi$
$812$	−1.95654	−0.0686610
$813$	−1.17997	−0.0413835
$814$	−8.28554	−0.290408
$815$	0	0
$816$	0	0
$817$	12.3591	0.432389
$818$	18.3244	0.640698
$819$	63.3383	2.21322
$820$	0	0
$821$	13.5220	0.471921	0.235961	−	0.971763i	$-0.424176\pi$
$821$	13.5220	0.471921	0.235961	+	0.971763i	$0.424176\pi$
$822$	−2.78849	−0.0972596
$823$	−12.8097	−0.446518	−0.223259	−	0.974759i	$-0.571670\pi$
$823$	−12.8097	−0.446518	−0.223259	+	0.974759i	$0.571670\pi$
$824$	−23.0091	−0.801560
$825$	0	0
$826$	−59.0072	−2.05312
$827$	32.6667	1.13593	0.567965	−	0.823052i	$-0.307731\pi$
$827$	32.6667	1.13593	0.567965	+	0.823052i	$0.307731\pi$
$828$	23.2863	0.809257
$829$	−21.7442	−0.755207	−0.377603	−	0.925967i	$-0.623252\pi$
$829$	−21.7442	−0.755207	−0.377603	+	0.925967i	$0.623252\pi$
$830$	0	0
$831$	2.51511	0.0872484
$832$	−7.99352	−0.277126
$833$	0	0
$834$	−3.06711	−0.106205
$835$	0	0
$836$	−7.92985	−0.274259
$837$	4.13538	0.142940
$838$	17.6512	0.609751
$839$	−26.6031	−0.918441	−0.459221	−	0.888322i	$-0.651871\pi$
$839$	−26.6031	−0.918441	−0.459221	+	0.888322i	$0.651871\pi$
$840$	0	0
$841$	−28.7645	−0.991880
$842$	−48.0834	−1.65706
$843$	2.34470	0.0807558
$844$	−2.11438	−0.0727800
$845$	0	0
$846$	25.0024	0.859600
$847$	69.8969	2.40169
$848$	−15.5288	−0.533263
$849$	−1.69971	−0.0583339
$850$	0	0
$851$	7.80612	0.267590
$852$	−0.0204774	−0.000701543 0
$853$	14.5178	0.497079	0.248539	−	0.968622i	$-0.420049\pi$
$853$	14.5178	0.497079	0.248539	+	0.968622i	$0.420049\pi$
$854$	105.142	3.59789
$855$	0	0
$856$	7.99731	0.273342
$857$	54.4262	1.85916	0.929582	−	0.368615i	$-0.120168\pi$
$857$	54.4262	1.85916	0.929582	+	0.368615i	$0.120168\pi$
$858$	−4.76675	−0.162734
$859$	32.4771	1.10811	0.554053	−	0.832482i	$-0.313081\pi$
$859$	32.4771	1.10811	0.554053	+	0.832482i	$0.313081\pi$
$860$	0	0
$861$	−2.59296	−0.0883677
$862$	−29.1706	−0.993553
$863$	−50.6141	−1.72292	−0.861462	−	0.507823i	$-0.830451\pi$
$863$	−50.6141	−1.72292	−0.861462	+	0.507823i	$0.830451\pi$
$864$	3.20336	0.108981
$865$	0	0
$866$	1.23061	0.0418178
$867$	0	0
$868$	−25.4044	−0.862281
$869$	34.4870	1.16989
$870$	0	0
$871$	56.5784	1.91709
$872$	7.38900	0.250223
$873$	−3.52460	−0.119290
$874$	23.5624	0.797010
$875$	0	0
$876$	−0.546948	−0.0184797
$877$	−8.66586	−0.292625	−0.146313	−	0.989238i	$-0.546741\pi$
$877$	−8.66586	−0.292625	−0.146313	+	0.989238i	$0.546741\pi$
$878$	−20.2665	−0.683963
$879$	−1.22297	−0.0412499
$880$	0	0
$881$	3.98342	0.134205	0.0671024	−	0.997746i	$-0.478625\pi$
$881$	3.98342	0.134205	0.0671024	+	0.997746i	$0.478625\pi$
$882$	−60.6128	−2.04094
$883$	16.9182	0.569341	0.284671	−	0.958625i	$-0.408116\pi$
$883$	16.9182	0.569341	0.284671	+	0.958625i	$0.408116\pi$
$884$	0	0
$885$	0	0
$886$	4.05678	0.136290
$887$	−31.6739	−1.06350	−0.531752	−	0.846900i	$-0.678466\pi$
$887$	−31.6739	−1.06350	−0.531752	+	0.846900i	$0.678466\pi$
$888$	0.186922	0.00627269
$889$	52.2981	1.75402
$890$	0	0
$891$	46.2876	1.55069
$892$	14.7709	0.494565
$893$	8.02157	0.268432
$894$	−0.418440	−0.0139947
$895$	0	0
$896$	54.5520	1.82245
$897$	4.49093	0.149948
$898$	24.2598	0.809561
$899$	3.05749	0.101973
$900$	0	0
$901$	0	0
$902$	−48.5341	−1.61601
$903$	−3.58544	−0.119316
$904$	−17.7118	−0.589087
$905$	0	0
$906$	−0.770987	−0.0256143
$907$	−30.6815	−1.01876	−0.509381	−	0.860541i	$-0.670126\pi$
$907$	−30.6815	−1.01876	−0.509381	+	0.860541i	$0.670126\pi$
$908$	−11.1567	−0.370248
$909$	10.3610	0.343654
$910$	0	0
$911$	−47.2836	−1.56658	−0.783288	−	0.621659i	$-0.786459\pi$
$911$	−47.2836	−1.56658	−0.783288	+	0.621659i	$0.786459\pi$
$912$	0.898157	0.0297409
$913$	54.7785	1.81290
$914$	41.6430	1.37743
$915$	0	0
$916$	−21.9064	−0.723807
$917$	7.90750	0.261129
$918$	0	0
$919$	44.6124	1.47163	0.735813	−	0.677184i	$-0.236800\pi$
$919$	44.6124	1.47163	0.735813	+	0.677184i	$0.236800\pi$
$920$	0	0
$921$	−3.59050	−0.118311
$922$	−63.0169	−2.07535
$923$	0.982213	0.0323299
$924$	2.30049	0.0756807
$925$	0	0
$926$	42.8345	1.40763
$927$	−37.4962	−1.23154
$928$	2.36841	0.0777468
$929$	8.94983	0.293634	0.146817	−	0.989164i	$-0.453097\pi$
$929$	8.94983	0.293634	0.146817	+	0.989164i	$0.453097\pi$
$930$	0	0
$931$	−19.4465	−0.637334
$932$	−12.0439	−0.394512
$933$	2.09644	0.0686343
$934$	−24.4532	−0.800132
$935$	0	0
$936$	−26.7459	−0.874218
$937$	4.68402	0.153020	0.0765102	−	0.997069i	$-0.475622\pi$
$937$	4.68402	0.153020	0.0765102	+	0.997069i	$0.475622\pi$
$938$	−86.1172	−2.81183
$939$	−1.05505	−0.0344302
$940$	0	0
$941$	47.5249	1.54927	0.774634	−	0.632409i	$-0.217934\pi$
$941$	47.5249	1.54927	0.774634	+	0.632409i	$0.217934\pi$
$942$	−0.485225	−0.0158095
$943$	45.7258	1.48904
$944$	39.6647	1.29098
$945$	0	0
$946$	−67.1111	−2.18197
$947$	−19.1409	−0.621997	−0.310998	−	0.950410i	$-0.600663\pi$
$947$	−19.1409	−0.621997	−0.310998	+	0.950410i	$0.600663\pi$
$948$	0.674292	0.0219000
$949$	26.2348	0.851617
$950$	0	0
$951$	0.820825	0.0266171
$952$	0	0
$953$	41.5915	1.34728	0.673640	−	0.739060i	$-0.264730\pi$
$953$	41.5915	1.34728	0.673640	+	0.739060i	$0.264730\pi$
$954$	−15.8971	−0.514689
$955$	0	0
$956$	−24.3888	−0.788789
$957$	−0.276872	−0.00894999
$958$	55.8586	1.80471
$959$	−64.5504	−2.08444
$960$	0	0
$961$	8.69961	0.280633
$962$	7.77042	0.250528
$963$	13.0326	0.419970
$964$	−20.9745	−0.675542
$965$	0	0
$966$	−6.83558	−0.219931
$967$	46.3577	1.49076	0.745381	−	0.666638i	$-0.232267\pi$
$967$	46.3577	1.49076	0.745381	+	0.666638i	$0.232267\pi$
$968$	−29.5154	−0.948662
$969$	0	0
$970$	0	0
$971$	−18.5657	−0.595802	−0.297901	−	0.954597i	$-0.596287\pi$
$971$	−18.5657	−0.595802	−0.297901	+	0.954597i	$0.596287\pi$
$972$	2.73337	0.0876728
$973$	−71.0003	−2.27616
$974$	−36.7735	−1.17830
$975$	0	0
$976$	−70.6766	−2.26230
$977$	46.3204	1.48192	0.740961	−	0.671548i	$-0.234371\pi$
$977$	46.3204	1.48192	0.740961	+	0.671548i	$0.234371\pi$
$978$	0.897118	0.0286867
$979$	−67.9682	−2.17227
$980$	0	0
$981$	12.0413	0.384449
$982$	27.2357	0.869125
$983$	−36.3105	−1.15813	−0.579063	−	0.815283i	$-0.696581\pi$
$983$	−36.3105	−1.15813	−0.579063	+	0.815283i	$0.696581\pi$
$984$	1.09493	0.0349051
$985$	0	0
$986$	0	0
$987$	−2.32710	−0.0740725
$988$	7.43684	0.236597
$989$	63.2279	2.01053
$990$	0	0
$991$	8.15237	0.258968	0.129484	−	0.991581i	$-0.458668\pi$
$991$	8.15237	0.258968	0.129484	+	0.991581i	$0.458668\pi$
$992$	30.7523	0.976386
$993$	1.38231	0.0438664
$994$	−1.49501	−0.0474189
$995$	0	0
$996$	1.07103	0.0339370
$997$	−21.6186	−0.684668	−0.342334	−	0.939578i	$-0.611217\pi$
$997$	−21.6186	−0.684668	−0.342334	+	0.939578i	$0.611217\pi$
$998$	19.1150	0.605075
$999$	0.610450	0.0193138

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bt.1.12	✓	15
5.4	even	2		7225.2.a.bv.1.4	yes	15
17.16	even	2		7225.2.a.bw.1.12	yes	15
85.84	even	2		7225.2.a.bu.1.4	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.12	✓	15	1.1	even	1	trivial
7225.2.a.bu.1.4	yes	15	85.84	even	2
7225.2.a.bv.1.4	yes	15	5.4	even	2
7225.2.a.bw.1.12	yes	15	17.16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71131 q^{2} +0.109608 q^{3} +0.928572 q^{4} +0.187572 q^{6} +4.34210 q^{7} -1.83354 q^{8} -2.98799 q^{9} +O(q^{10})\)	\(q+1.71131 q^{2} +0.109608 q^{3} +0.928572 q^{4} +0.187572 q^{6} +4.34210 q^{7} -1.83354 q^{8} -2.98799 q^{9} +5.20553 q^{11} +0.101779 q^{12} -4.88189 q^{13} +7.43066 q^{14} -4.99490 q^{16} -5.11336 q^{18} -1.64053 q^{19} +0.475927 q^{21} +8.90825 q^{22} -8.39280 q^{23} -0.200970 q^{24} -8.35442 q^{26} -0.656330 q^{27} +4.03195 q^{28} -0.485258 q^{29} -6.30076 q^{31} -4.88072 q^{32} +0.570566 q^{33} -2.77456 q^{36} -0.930097 q^{37} -2.80745 q^{38} -0.535093 q^{39} -5.44822 q^{41} +0.814458 q^{42} -7.53358 q^{43} +4.83371 q^{44} -14.3627 q^{46} -4.88962 q^{47} -0.547479 q^{48} +11.8538 q^{49} -4.53319 q^{52} +3.10894 q^{53} -1.12318 q^{54} -7.96142 q^{56} -0.179815 q^{57} -0.830425 q^{58} -7.94104 q^{59} +14.1498 q^{61} -10.7825 q^{62} -12.9741 q^{63} +1.63738 q^{64} +0.976414 q^{66} -11.5894 q^{67} -0.919916 q^{69} -0.201195 q^{71} +5.47860 q^{72} -5.37389 q^{73} -1.59168 q^{74} -1.52335 q^{76} +22.6029 q^{77} -0.915709 q^{78} +6.62508 q^{79} +8.89202 q^{81} -9.32358 q^{82} +10.5231 q^{83} +0.441933 q^{84} -12.8923 q^{86} -0.0531880 q^{87} -9.54455 q^{88} -13.0569 q^{89} -21.1977 q^{91} -7.79333 q^{92} -0.690612 q^{93} -8.36764 q^{94} -0.534965 q^{96} +1.17959 q^{97} +20.2855 q^{98} -15.5540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 9 q^{3} + 12 q^{4} - 9 q^{6} - 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 15 * q - 9 * q^3 + 12 * q^4 - 9 * q^6 - 12 * q^7 + 6 * q^8 + 12 * q^9	\( 15 q - 9 q^{3} + 12 q^{4} - 9 q^{6} - 12 q^{7} + 6 q^{8} + 12 q^{9} - 6 q^{11} - 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} - 12 q^{22} - 36 q^{23} - 18 q^{24} + 36 q^{26} - 36 q^{27} - 24 q^{28} + 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} - 12 q^{37} - 6 q^{38} - 9 q^{39} + 18 q^{41} + 36 q^{42} - 3 q^{43} + 12 q^{44} - 21 q^{46} - 3 q^{47} + 12 q^{48} + 15 q^{49} - 27 q^{52} - 21 q^{54} + 6 q^{56} - 39 q^{57} - 18 q^{58} - 12 q^{59} + 15 q^{61} - 54 q^{62} - 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} - 6 q^{71} + 66 q^{72} + 9 q^{73} + 36 q^{74} - 18 q^{76} + 30 q^{77} - 30 q^{78} + 9 q^{79} + 51 q^{81} + 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} - 30 q^{88} - 24 q^{89} - 27 q^{91} - 15 q^{92} - 42 q^{93} - 57 q^{94} - 42 q^{96} - 48 q^{97} - 12 q^{99}+O(q^{100}) \) 15 * q - 9 * q^3 + 12 * q^4 - 9 * q^6 - 12 * q^7 + 6 * q^8 + 12 * q^9 - 6 * q^11 - 24 * q^12 + 6 * q^16 + 12 * q^18 + 6 * q^19 + 30 * q^21 - 12 * q^22 - 36 * q^23 - 18 * q^24 + 36 * q^26 - 36 * q^27 - 24 * q^28 + 18 * q^29 + 12 * q^32 + 12 * q^33 - 9 * q^36 - 12 * q^37 - 6 * q^38 - 9 * q^39 + 18 * q^41 + 36 * q^42 - 3 * q^43 + 12 * q^44 - 21 * q^46 - 3 * q^47 + 12 * q^48 + 15 * q^49 - 27 * q^52 - 21 * q^54 + 6 * q^56 - 39 * q^57 - 18 * q^58 - 12 * q^59 + 15 * q^61 - 54 * q^62 - 60 * q^63 - 36 * q^64 + 18 * q^66 - 24 * q^67 + 42 * q^69 - 6 * q^71 + 66 * q^72 + 9 * q^73 + 36 * q^74 - 18 * q^76 + 30 * q^77 - 30 * q^78 + 9 * q^79 + 51 * q^81 + 36 * q^82 + 15 * q^83 + 9 * q^84 - 36 * q^86 - 51 * q^87 - 30 * q^88 - 24 * q^89 - 27 * q^91 - 15 * q^92 - 42 * q^93 - 57 * q^94 - 42 * q^96 - 48 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

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Database

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