LMFDB - Embedded newform 7225.2.a.bv.1.6

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:	$0$
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} - 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			1.6
Root			$0.718828$ 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)

$f(q)$	$=$	$q-0.718828 q^{2} +0.0658528 q^{3} -1.48329 q^{4} -0.0473368 q^{6} +2.64277 q^{7} +2.50388 q^{8} -2.99566 q^{9} +O(q^{10})$	\(q-0.718828 q^{2} +0.0658528 q^{3} -1.48329 q^{4} -0.0473368 q^{6} +2.64277 q^{7} +2.50388 q^{8} -2.99566 q^{9} -4.10961 q^{11} -0.0976785 q^{12} -5.95910 q^{13} -1.89969 q^{14} +1.16671 q^{16} +2.15337 q^{18} +7.10576 q^{19} +0.174034 q^{21} +2.95411 q^{22} +3.02089 q^{23} +0.164888 q^{24} +4.28357 q^{26} -0.394831 q^{27} -3.91998 q^{28} +7.45909 q^{29} +0.0419433 q^{31} -5.84643 q^{32} -0.270630 q^{33} +4.44343 q^{36} -10.8152 q^{37} -5.10782 q^{38} -0.392423 q^{39} -9.07749 q^{41} -0.125100 q^{42} -5.19901 q^{43} +6.09573 q^{44} -2.17150 q^{46} +3.02440 q^{47} +0.0768311 q^{48} -0.0157829 q^{49} +8.83905 q^{52} -4.00575 q^{53} +0.283816 q^{54} +6.61718 q^{56} +0.467934 q^{57} -5.36181 q^{58} -1.12288 q^{59} -0.502359 q^{61} -0.0301500 q^{62} -7.91684 q^{63} +1.86916 q^{64} +0.194536 q^{66} +6.95612 q^{67} +0.198934 q^{69} -7.61164 q^{71} -7.50079 q^{72} +2.83503 q^{73} +7.77428 q^{74} -10.5399 q^{76} -10.8608 q^{77} +0.282085 q^{78} -1.13392 q^{79} +8.96099 q^{81} +6.52516 q^{82} +11.1816 q^{83} -0.258142 q^{84} +3.73720 q^{86} +0.491202 q^{87} -10.2900 q^{88} +5.33403 q^{89} -15.7485 q^{91} -4.48085 q^{92} +0.00276208 q^{93} -2.17402 q^{94} -0.385004 q^{96} +12.4444 q^{97} +0.0113452 q^{98} +12.3110 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.718828	−0.508288	−0.254144	−	0.967166i	$-0.581794\pi$
$2$	−0.718828	−0.508288	−0.254144	+	0.967166i	$0.581794\pi$
$3$	0.0658528	0.0380201	0.0190101	−	0.999819i	$-0.493949\pi$
$3$	0.0658528	0.0380201	0.0190101	+	0.999819i	$0.493949\pi$
$4$	−1.48329	−0.741643
$5$	0	0
$5$	0	0
$6$	−0.0473368	−0.0193252
$7$	2.64277	0.998872	0.499436	−	0.866351i	$-0.333541\pi$
$7$	2.64277	0.998872	0.499436	+	0.866351i	$0.333541\pi$
$8$	2.50388	0.885257
$9$	−2.99566	−0.998554
$10$	0	0
$11$	−4.10961	−1.23910	−0.619548	−	0.784959i	$-0.712684\pi$
$11$	−4.10961	−1.23910	−0.619548	+	0.784959i	$0.712684\pi$
$12$	−0.0976785	−0.0281974
$13$	−5.95910	−1.65276	−0.826379	−	0.563115i	$-0.809603\pi$
$13$	−5.95910	−1.65276	−0.826379	+	0.563115i	$0.809603\pi$
$14$	−1.89969	−0.507715
$15$	0	0
$16$	1.16671	0.291678
$17$	0	0
$17$	0	0
$18$	2.15337	0.507553
$19$	7.10576	1.63017	0.815086	−	0.579340i	$-0.196690\pi$
$19$	7.10576	1.63017	0.815086	+	0.579340i	$0.196690\pi$
$20$	0	0
$21$	0.174034	0.0379772
$22$	2.95411	0.629817
$23$	3.02089	0.629899	0.314950	−	0.949108i	$-0.398012\pi$
$23$	3.02089	0.629899	0.314950	+	0.949108i	$0.398012\pi$
$24$	0.164888	0.0336576
$25$	0	0
$26$	4.28357	0.840077
$27$	−0.394831	−0.0759853
$28$	−3.91998	−0.740807
$29$	7.45909	1.38512	0.692559	−	0.721361i	$-0.256483\pi$
$29$	7.45909	1.38512	0.692559	+	0.721361i	$0.256483\pi$
$30$	0	0
$31$	0.0419433	0.00753324	0.00376662	−	0.999993i	$-0.498801\pi$
$31$	0.0419433	0.00753324	0.00376662	+	0.999993i	$0.498801\pi$
$32$	−5.84643	−1.03351
$33$	−0.270630	−0.0471105
$34$	0	0
$35$	0	0
$36$	4.44343	0.740571
$37$	−10.8152	−1.77801	−0.889005	−	0.457897i	$-0.848603\pi$
$37$	−10.8152	−1.77801	−0.889005	+	0.457897i	$0.848603\pi$
$38$	−5.10782	−0.828597
$39$	−0.392423	−0.0628380
$40$	0	0
$41$	−9.07749	−1.41767	−0.708833	−	0.705376i	$-0.750778\pi$
$41$	−9.07749	−1.41767	−0.708833	+	0.705376i	$0.750778\pi$
$42$	−0.125100	−0.0193034
$43$	−5.19901	−0.792842	−0.396421	−	0.918069i	$-0.629748\pi$
$43$	−5.19901	−0.792842	−0.396421	+	0.918069i	$0.629748\pi$
$44$	6.09573	0.918966
$45$	0	0
$46$	−2.17150	−0.320170
$47$	3.02440	0.441154	0.220577	−	0.975370i	$-0.429206\pi$
$47$	3.02440	0.441154	0.220577	+	0.975370i	$0.429206\pi$
$48$	0.0768311	0.0110896
$49$	−0.0157829	−0.00225471
$50$	0	0
$51$	0	0
$52$	8.83905	1.22576
$53$	−4.00575	−0.550231	−0.275116	−	0.961411i	$-0.588716\pi$
$53$	−4.00575	−0.550231	−0.275116	+	0.961411i	$0.588716\pi$
$54$	0.283816	0.0386224
$55$	0	0
$56$	6.61718	0.884258
$57$	0.467934	0.0619793
$58$	−5.36181	−0.704040
$59$	−1.12288	−0.146186	−0.0730930	−	0.997325i	$-0.523287\pi$
$59$	−1.12288	−0.146186	−0.0730930	+	0.997325i	$0.523287\pi$
$60$	0	0
$61$	−0.502359	−0.0643204	−0.0321602	−	0.999483i	$-0.510239\pi$
$61$	−0.502359	−0.0643204	−0.0321602	+	0.999483i	$0.510239\pi$
$62$	−0.0301500	−0.00382906
$63$	−7.91684	−0.997428
$64$	1.86916	0.233645
$65$	0	0
$66$	0.194536	0.0239457
$67$	6.95612	0.849825	0.424913	−	0.905234i	$-0.360305\pi$
$67$	6.95612	0.849825	0.424913	+	0.905234i	$0.360305\pi$
$68$	0	0
$69$	0.198934	0.0239489
$70$	0	0
$71$	−7.61164	−0.903336	−0.451668	−	0.892186i	$-0.649171\pi$
$71$	−7.61164	−0.903336	−0.451668	+	0.892186i	$0.649171\pi$
$72$	−7.50079	−0.883977
$73$	2.83503	0.331815	0.165907	−	0.986141i	$-0.446945\pi$
$73$	2.83503	0.331815	0.165907	+	0.986141i	$0.446945\pi$
$74$	7.77428	0.903742
$75$	0	0
$76$	−10.5399	−1.20901
$77$	−10.8608	−1.23770
$78$	0.282085	0.0319398
$79$	−1.13392	−0.127576	−0.0637882	−	0.997963i	$-0.520318\pi$
$79$	−1.13392	−0.127576	−0.0637882	+	0.997963i	$0.520318\pi$
$80$	0	0
$81$	8.96099	0.995665
$82$	6.52516	0.720583
$83$	11.1816	1.22734	0.613669	−	0.789563i	$-0.289693\pi$
$83$	11.1816	1.22734	0.613669	+	0.789563i	$0.289693\pi$
$84$	−0.258142	−0.0281656
$85$	0	0
$86$	3.73720	0.402992
$87$	0.491202	0.0526624
$88$	−10.2900	−1.09692
$89$	5.33403	0.565406	0.282703	−	0.959208i	$-0.408769\pi$
$89$	5.33403	0.565406	0.282703	+	0.959208i	$0.408769\pi$
$90$	0	0
$91$	−15.7485	−1.65089
$92$	−4.48085	−0.467161
$93$	0.00276208	0.000286415 0
$94$	−2.17402	−0.224234
$95$	0	0
$96$	−0.385004	−0.0392943
$97$	12.4444	1.26354	0.631770	−	0.775156i	$-0.282329\pi$
$97$	12.4444	1.26354	0.631770	+	0.775156i	$0.282329\pi$
$98$	0.0113452	0.00114604
$99$	12.3110	1.23730
$100$	0	0
$101$	−14.2640	−1.41933	−0.709663	−	0.704542i	$-0.751153\pi$
$101$	−14.2640	−1.41933	−0.709663	+	0.704542i	$0.751153\pi$
$102$	0	0
$103$	6.57354	0.647710	0.323855	−	0.946107i	$-0.395021\pi$
$103$	6.57354	0.647710	0.323855	+	0.946107i	$0.395021\pi$
$104$	−14.9209	−1.46311
$105$	0	0
$106$	2.87944	0.279676
$107$	−16.1803	−1.56421	−0.782104	−	0.623148i	$-0.785854\pi$
$107$	−16.1803	−1.56421	−0.782104	+	0.623148i	$0.785854\pi$
$108$	0.585648	0.0563540
$109$	6.28111	0.601622	0.300811	−	0.953684i	$-0.402743\pi$
$109$	6.28111	0.601622	0.300811	+	0.953684i	$0.402743\pi$
$110$	0	0
$111$	−0.712212	−0.0676002
$112$	3.08334	0.291349
$113$	9.38888	0.883232	0.441616	−	0.897204i	$-0.354405\pi$
$113$	9.38888	0.883232	0.441616	+	0.897204i	$0.354405\pi$
$114$	−0.336364	−0.0315034
$115$	0	0
$116$	−11.0640	−1.02726
$117$	17.8515	1.65037
$118$	0.807155	0.0743047
$119$	0	0
$120$	0	0
$121$	5.88892	0.535356
$122$	0.361109	0.0326933
$123$	−0.597778	−0.0538998
$124$	−0.0622139	−0.00558697
$125$	0	0
$126$	5.69085	0.506981
$127$	−11.0154	−0.977459	−0.488729	−	0.872435i	$-0.662539\pi$
$127$	−11.0154	−0.977459	−0.488729	+	0.872435i	$0.662539\pi$
$128$	10.3493	0.914754
$129$	−0.342370	−0.0301440
$130$	0	0
$131$	−21.6986	−1.89582	−0.947909	−	0.318542i	$-0.896807\pi$
$131$	−21.6986	−1.89582	−0.947909	+	0.318542i	$0.896807\pi$
$132$	0.401421	0.0349392
$133$	18.7789	1.62833
$134$	−5.00025	−0.431956
$135$	0	0
$136$	0	0
$137$	9.79996	0.837267	0.418634	−	0.908155i	$-0.362509\pi$
$137$	9.79996	0.837267	0.418634	+	0.908155i	$0.362509\pi$
$138$	−0.142999	−0.0121729
$139$	−7.08063	−0.600571	−0.300286	−	0.953849i	$-0.597082\pi$
$139$	−7.08063	−0.600571	−0.300286	+	0.953849i	$0.597082\pi$
$140$	0	0
$141$	0.199165	0.0167727
$142$	5.47146	0.459155
$143$	24.4896	2.04792
$144$	−3.49507	−0.291256
$145$	0	0
$146$	−2.03790	−0.168658
$147$	−0.00103935	−8.57242e−5 0
$148$	16.0421	1.31865
$149$	17.7017	1.45018	0.725089	−	0.688656i	$-0.241799\pi$
$149$	17.7017	1.45018	0.725089	+	0.688656i	$0.241799\pi$
$150$	0	0
$151$	2.15268	0.175183	0.0875913	−	0.996156i	$-0.472083\pi$
$151$	2.15268	0.175183	0.0875913	+	0.996156i	$0.472083\pi$
$152$	17.7920	1.44312
$153$	0	0
$154$	7.80701	0.629107
$155$	0	0
$156$	0.582076	0.0466034
$157$	−10.0386	−0.801170	−0.400585	−	0.916260i	$-0.631193\pi$
$157$	−10.0386	−0.801170	−0.400585	+	0.916260i	$0.631193\pi$
$158$	0.815096	0.0648455
$159$	−0.263790	−0.0209199
$160$	0	0
$161$	7.98351	0.629189
$162$	−6.44141	−0.506085
$163$	−17.3990	−1.36279	−0.681396	−	0.731915i	$-0.738627\pi$
$163$	−17.3990	−1.36279	−0.681396	+	0.731915i	$0.738627\pi$
$164$	13.4645	1.05140
$165$	0	0
$166$	−8.03764	−0.623842
$167$	17.3077	1.33931	0.669657	−	0.742671i	$-0.266441\pi$
$167$	17.3077	1.33931	0.669657	+	0.742671i	$0.266441\pi$
$168$	0.435760	0.0336196
$169$	22.5109	1.73161
$170$	0	0
$171$	−21.2865	−1.62782
$172$	7.71163	0.588006
$173$	8.77362	0.667046	0.333523	−	0.942742i	$-0.391763\pi$
$173$	8.77362	0.667046	0.333523	+	0.942742i	$0.391763\pi$
$174$	−0.353090	−0.0267677
$175$	0	0
$176$	−4.79473	−0.361416
$177$	−0.0739446	−0.00555801
$178$	−3.83425	−0.287389
$179$	−17.9881	−1.34450	−0.672248	−	0.740326i	$-0.734671\pi$
$179$	−17.9881	−1.34450	−0.672248	+	0.740326i	$0.734671\pi$
$180$	0	0
$181$	23.3626	1.73653	0.868263	−	0.496104i	$-0.165236\pi$
$181$	23.3626	1.73653	0.868263	+	0.496104i	$0.165236\pi$
$182$	11.3205	0.839129
$183$	−0.0330817	−0.00244547
$184$	7.56396	0.557623
$185$	0	0
$186$	−0.00198546	−0.000145581 0
$187$	0	0
$188$	−4.48605	−0.327179
$189$	−1.04345	−0.0758996
$190$	0	0
$191$	1.38386	0.100133	0.0500664	−	0.998746i	$-0.484057\pi$
$191$	1.38386	0.100133	0.0500664	+	0.998746i	$0.484057\pi$
$192$	0.123089	0.00888320
$193$	9.17525	0.660449	0.330225	−	0.943902i	$-0.392876\pi$
$193$	9.17525	0.660449	0.330225	+	0.943902i	$0.392876\pi$
$194$	−8.94540	−0.642243
$195$	0	0
$196$	0.0234106	0.00167219
$197$	18.0845	1.28846	0.644232	−	0.764830i	$-0.277177\pi$
$197$	18.0845	1.28846	0.644232	+	0.764830i	$0.277177\pi$
$198$	−8.84951	−0.628907
$199$	1.81227	0.128468	0.0642342	−	0.997935i	$-0.479540\pi$
$199$	1.81227	0.128468	0.0642342	+	0.997935i	$0.479540\pi$
$200$	0	0
$201$	0.458080	0.0323105
$202$	10.2534	0.721426
$203$	19.7126	1.38356
$204$	0	0
$205$	0	0
$206$	−4.72524	−0.329223
$207$	−9.04958	−0.628989
$208$	−6.95255	−0.482072
$209$	−29.2019	−2.01994
$210$	0	0
$211$	−5.29806	−0.364733	−0.182367	−	0.983231i	$-0.558376\pi$
$211$	−5.29806	−0.364733	−0.182367	+	0.983231i	$0.558376\pi$
$212$	5.94167	0.408075
$213$	−0.501248	−0.0343449
$214$	11.6309	0.795069
$215$	0	0
$216$	−0.988611	−0.0672665
$217$	0.110846	0.00752474
$218$	−4.51504	−0.305797
$219$	0.186694	0.0126156
$220$	0	0
$221$	0	0
$222$	0.511958	0.0343604
$223$	−6.70575	−0.449050	−0.224525	−	0.974468i	$-0.572083\pi$
$223$	−6.70575	−0.449050	−0.224525	+	0.974468i	$0.572083\pi$
$224$	−15.4508	−1.03235
$225$	0	0
$226$	−6.74899	−0.448936
$227$	20.9624	1.39133	0.695663	−	0.718368i	$-0.255111\pi$
$227$	20.9624	1.39133	0.695663	+	0.718368i	$0.255111\pi$
$228$	−0.694080	−0.0459666
$229$	0.655994	0.0433493	0.0216747	−	0.999765i	$-0.493100\pi$
$229$	0.655994	0.0433493	0.0216747	+	0.999765i	$0.493100\pi$
$230$	0	0
$231$	−0.715211	−0.0470574
$232$	18.6767	1.22619
$233$	22.8976	1.50007	0.750037	−	0.661396i	$-0.230036\pi$
$233$	22.8976	1.50007	0.750037	+	0.661396i	$0.230036\pi$
$234$	−12.8321	−0.838863
$235$	0	0
$236$	1.66555	0.108418
$237$	−0.0746720	−0.00485047
$238$	0	0
$239$	27.6861	1.79087	0.895433	−	0.445196i	$-0.146866\pi$
$239$	27.6861	1.79087	0.895433	+	0.445196i	$0.146866\pi$
$240$	0	0
$241$	19.2424	1.23951	0.619756	−	0.784795i	$-0.287232\pi$
$241$	19.2424	1.23951	0.619756	+	0.784795i	$0.287232\pi$
$242$	−4.23312	−0.272115
$243$	1.77460	0.113841
$244$	0.745142	0.0477028
$245$	0	0
$246$	0.429700	0.0273967
$247$	−42.3439	−2.69428
$248$	0.105021	0.00666885
$249$	0.736339	0.0466636
$250$	0	0
$251$	5.00419	0.315862	0.157931	−	0.987450i	$-0.449518\pi$
$251$	5.00419	0.315862	0.157931	+	0.987450i	$0.449518\pi$
$252$	11.7429	0.739736
$253$	−12.4147	−0.780505
$254$	7.91818	0.496831
$255$	0	0
$256$	−11.1777	−0.698603
$257$	−29.7359	−1.85488	−0.927438	−	0.373976i	$-0.877994\pi$
$257$	−29.7359	−1.85488	−0.927438	+	0.373976i	$0.877994\pi$
$258$	0.246105	0.0153218
$259$	−28.5821	−1.77600
$260$	0	0
$261$	−22.3449	−1.38312
$262$	15.5976	0.963622
$263$	8.84249	0.545251	0.272626	−	0.962120i	$-0.412108\pi$
$263$	8.84249	0.545251	0.272626	+	0.962120i	$0.412108\pi$
$264$	−0.677625	−0.0417049
$265$	0	0
$266$	−13.4988	−0.827663
$267$	0.351261	0.0214968
$268$	−10.3179	−0.630267
$269$	−12.9759	−0.791158	−0.395579	−	0.918432i	$-0.629456\pi$
$269$	−12.9759	−0.791158	−0.395579	+	0.918432i	$0.629456\pi$
$270$	0	0
$271$	−11.4090	−0.693046	−0.346523	−	0.938042i	$-0.612638\pi$
$271$	−11.4090	−0.693046	−0.346523	+	0.938042i	$0.612638\pi$
$272$	0	0
$273$	−1.03708	−0.0627672
$274$	−7.04449	−0.425573
$275$	0	0
$276$	−0.295076	−0.0177615
$277$	0.0544841	0.00327363	0.00163682	−	0.999999i	$-0.499479\pi$
$277$	0.0544841	0.00327363	0.00163682	+	0.999999i	$0.499479\pi$
$278$	5.08976	0.305263
$279$	−0.125648	−0.00752235
$280$	0	0
$281$	5.47612	0.326678	0.163339	−	0.986570i	$-0.447774\pi$
$281$	5.47612	0.326678	0.163339	+	0.986570i	$0.447774\pi$
$282$	−0.143166	−0.00852539
$283$	−1.74768	−0.103889	−0.0519443	−	0.998650i	$-0.516542\pi$
$283$	−1.74768	−0.103889	−0.0519443	+	0.998650i	$0.516542\pi$
$284$	11.2902	0.669953
$285$	0	0
$286$	−17.6038	−1.04094
$287$	−23.9897	−1.41607
$288$	17.5139	1.03202
$289$	0	0
$290$	0	0
$291$	0.819500	0.0480400
$292$	−4.20516	−0.246088
$293$	0.868085	0.0507141	0.0253570	−	0.999678i	$-0.491928\pi$
$293$	0.868085	0.0507141	0.0253570	+	0.999678i	$0.491928\pi$
$294$	0.000747114 0	4.35726e−5 0
$295$	0	0
$296$	−27.0800	−1.57400
$297$	1.62260	0.0941530
$298$	−12.7245	−0.737108
$299$	−18.0018	−1.04107
$300$	0	0
$301$	−13.7398	−0.791948
$302$	−1.54741	−0.0890432
$303$	−0.939327	−0.0539629
$304$	8.29036	0.475485
$305$	0	0
$306$	0	0
$307$	28.4506	1.62376	0.811882	−	0.583822i	$-0.198443\pi$
$307$	28.4506	1.62376	0.811882	+	0.583822i	$0.198443\pi$
$308$	16.1096	0.917930
$309$	0.432886	0.0246260
$310$	0	0
$311$	−5.93811	−0.336719	−0.168360	−	0.985726i	$-0.553847\pi$
$311$	−5.93811	−0.336719	−0.168360	+	0.985726i	$0.553847\pi$
$312$	−0.982583	−0.0556278
$313$	−5.54022	−0.313152	−0.156576	−	0.987666i	$-0.550046\pi$
$313$	−5.54022	−0.313152	−0.156576	+	0.987666i	$0.550046\pi$
$314$	7.21605	0.407225
$315$	0	0
$316$	1.68193	0.0946161
$317$	−0.618323	−0.0347285	−0.0173642	−	0.999849i	$-0.505527\pi$
$317$	−0.618323	−0.0347285	−0.0173642	+	0.999849i	$0.505527\pi$
$318$	0.189619	0.0106333
$319$	−30.6540	−1.71629
$320$	0	0
$321$	−1.06552	−0.0594714
$322$	−5.73877	−0.319809
$323$	0	0
$324$	−13.2917	−0.738428
$325$	0	0
$326$	12.5069	0.692691
$327$	0.413629	0.0228737
$328$	−22.7290	−1.25500
$329$	7.99279	0.440657
$330$	0	0
$331$	−31.4469	−1.72848	−0.864238	−	0.503083i	$-0.832199\pi$
$331$	−31.4469	−1.72848	−0.864238	+	0.503083i	$0.832199\pi$
$332$	−16.5855	−0.910247
$333$	32.3987	1.77544
$334$	−12.4413	−0.680757
$335$	0	0
$336$	0.203047	0.0110771
$337$	6.72488	0.366327	0.183164	−	0.983082i	$-0.441366\pi$
$337$	6.72488	0.366327	0.183164	+	0.983082i	$0.441366\pi$
$338$	−16.1815	−0.880155
$339$	0.618284	0.0335806
$340$	0	0
$341$	−0.172371	−0.00933440
$342$	15.3013	0.827399
$343$	−18.5411	−1.00112
$344$	−13.0177	−0.701869
$345$	0	0
$346$	−6.30672	−0.339052
$347$	20.3365	1.09172	0.545861	−	0.837876i	$-0.316203\pi$
$347$	20.3365	1.09172	0.545861	+	0.837876i	$0.316203\pi$
$348$	−0.728593	−0.0390567
$349$	−23.5876	−1.26262	−0.631309	−	0.775532i	$-0.717482\pi$
$349$	−23.5876	−1.26262	−0.631309	+	0.775532i	$0.717482\pi$
$350$	0	0
$351$	2.35284	0.125585
$352$	24.0266	1.28062
$353$	21.4356	1.14090	0.570451	−	0.821332i	$-0.306768\pi$
$353$	21.4356	1.14090	0.570451	+	0.821332i	$0.306768\pi$
$354$	0.0531534	0.00282507
$355$	0	0
$356$	−7.91189	−0.419329
$357$	0	0
$358$	12.9304	0.683391
$359$	25.6751	1.35508	0.677540	−	0.735486i	$-0.263046\pi$
$359$	25.6751	1.35508	0.677540	+	0.735486i	$0.263046\pi$
$360$	0	0
$361$	31.4918	1.65746
$362$	−16.7937	−0.882656
$363$	0.387802	0.0203543
$364$	23.3596	1.22437
$365$	0	0
$366$	0.0237801	0.00124300
$367$	16.8597	0.880068	0.440034	−	0.897981i	$-0.354966\pi$
$367$	16.8597	0.880068	0.440034	+	0.897981i	$0.354966\pi$
$368$	3.52451	0.183728
$369$	27.1931	1.41562
$370$	0	0
$371$	−10.5863	−0.549611
$372$	−0.00409696	−0.000212417 0
$373$	−29.1725	−1.51049	−0.755246	−	0.655441i	$-0.772483\pi$
$373$	−29.1725	−1.51049	−0.755246	+	0.655441i	$0.772483\pi$
$374$	0	0
$375$	0	0
$376$	7.57275	0.390535
$377$	−44.4495	−2.28927
$378$	0.750059	0.0385789
$379$	25.9783	1.33442	0.667208	−	0.744871i	$-0.267489\pi$
$379$	25.9783	1.33442	0.667208	+	0.744871i	$0.267489\pi$
$380$	0	0
$381$	−0.725395	−0.0371631
$382$	−0.994760	−0.0508963
$383$	−11.3541	−0.580165	−0.290083	−	0.957002i	$-0.593683\pi$
$383$	−11.3541	−0.580165	−0.290083	+	0.957002i	$0.593683\pi$
$384$	0.681528	0.0347791
$385$	0	0
$386$	−6.59543	−0.335698
$387$	15.5745	0.791696
$388$	−18.4586	−0.937096
$389$	9.75640	0.494669	0.247335	−	0.968930i	$-0.420445\pi$
$389$	9.75640	0.494669	0.247335	+	0.968930i	$0.420445\pi$
$390$	0	0
$391$	0	0
$392$	−0.0395186	−0.00199599
$393$	−1.42892	−0.0720792
$394$	−12.9996	−0.654911
$395$	0	0
$396$	−18.2608	−0.917638
$397$	26.9841	1.35429	0.677145	−	0.735849i	$-0.263217\pi$
$397$	26.9841	1.35429	0.677145	+	0.735849i	$0.263217\pi$
$398$	−1.30271	−0.0652990
$399$	1.23664	0.0619094
$400$	0	0
$401$	14.8964	0.743890	0.371945	−	0.928255i	$-0.378691\pi$
$401$	14.8964	0.743890	0.371945	+	0.928255i	$0.378691\pi$
$402$	−0.329281	−0.0164230
$403$	−0.249944	−0.0124506
$404$	21.1577	1.05263
$405$	0	0
$406$	−14.1700	−0.703245
$407$	44.4463	2.20312
$408$	0	0
$409$	−7.04947	−0.348574	−0.174287	−	0.984695i	$-0.555762\pi$
$409$	−7.04947	−0.348574	−0.174287	+	0.984695i	$0.555762\pi$
$410$	0	0
$411$	0.645355	0.0318330
$412$	−9.75044	−0.480370
$413$	−2.96750	−0.146021
$414$	6.50509	0.319708
$415$	0	0
$416$	34.8395	1.70815
$417$	−0.466279	−0.0228338
$418$	20.9911	1.02671
$419$	16.1587	0.789402	0.394701	−	0.918810i	$-0.370848\pi$
$419$	16.1587	0.789402	0.394701	+	0.918810i	$0.370848\pi$
$420$	0	0
$421$	13.9891	0.681789	0.340895	−	0.940102i	$-0.389270\pi$
$421$	13.9891	0.681789	0.340895	+	0.940102i	$0.389270\pi$
$422$	3.80839	0.185390
$423$	−9.06009	−0.440517
$424$	−10.0299	−0.487096
$425$	0	0
$426$	0.360311	0.0174571
$427$	−1.32762	−0.0642479
$428$	24.0000	1.16008
$429$	1.61271	0.0778623
$430$	0	0
$431$	27.0906	1.30491	0.652455	−	0.757828i	$-0.273739\pi$
$431$	27.0906	1.30491	0.652455	+	0.757828i	$0.273739\pi$
$432$	−0.460654	−0.0221632
$433$	−0.367293	−0.0176510	−0.00882549	−	0.999961i	$-0.502809\pi$
$433$	−0.367293	−0.0176510	−0.00882549	+	0.999961i	$0.502809\pi$
$434$	−0.0796795	−0.00382474
$435$	0	0
$436$	−9.31669	−0.446188
$437$	21.4657	1.02684
$438$	−0.134201	−0.00641238
$439$	38.9369	1.85836	0.929178	−	0.369633i	$-0.120517\pi$
$439$	38.9369	1.85836	0.929178	+	0.369633i	$0.120517\pi$
$440$	0	0
$441$	0.0472804	0.00225145
$442$	0	0
$443$	11.2375	0.533912	0.266956	−	0.963709i	$-0.413982\pi$
$443$	11.2375	0.533912	0.266956	+	0.963709i	$0.413982\pi$
$444$	1.05641	0.0501352
$445$	0	0
$446$	4.82028	0.228247
$447$	1.16570	0.0551359
$448$	4.93975	0.233381
$449$	7.45829	0.351978	0.175989	−	0.984392i	$-0.443688\pi$
$449$	7.45829	0.351978	0.175989	+	0.984392i	$0.443688\pi$
$450$	0	0
$451$	37.3050	1.75662
$452$	−13.9264	−0.655043
$453$	0.141760	0.00666046
$454$	−15.0684	−0.707195
$455$	0	0
$456$	1.17165	0.0548676
$457$	−11.3791	−0.532292	−0.266146	−	0.963933i	$-0.585750\pi$
$457$	−11.3791	−0.532292	−0.266146	+	0.963933i	$0.585750\pi$
$458$	−0.471547	−0.0220339
$459$	0	0
$460$	0	0
$461$	3.91639	0.182404	0.0912022	−	0.995832i	$-0.470929\pi$
$461$	3.91639	0.182404	0.0912022	+	0.995832i	$0.470929\pi$
$462$	0.514114	0.0239187
$463$	26.4826	1.23075	0.615374	−	0.788235i	$-0.289005\pi$
$463$	26.4826	1.23075	0.615374	+	0.788235i	$0.289005\pi$
$464$	8.70260	0.404008
$465$	0	0
$466$	−16.4595	−0.762469
$467$	10.2294	0.473359	0.236680	−	0.971588i	$-0.423941\pi$
$467$	10.2294	0.473359	0.236680	+	0.971588i	$0.423941\pi$
$468$	−26.4788	−1.22398
$469$	18.3834	0.848866
$470$	0	0
$471$	−0.661071	−0.0304606
$472$	−2.81155	−0.129412
$473$	21.3659	0.982407
$474$	0.0536763	0.00246544
$475$	0	0
$476$	0	0
$477$	11.9999	0.549436
$478$	−19.9016	−0.910276
$479$	−6.00274	−0.274272	−0.137136	−	0.990552i	$-0.543790\pi$
$479$	−6.00274	−0.274272	−0.137136	+	0.990552i	$0.543790\pi$
$480$	0	0
$481$	64.4490	2.93862
$482$	−13.8320	−0.630029
$483$	0.525737	0.0239218
$484$	−8.73496	−0.397043
$485$	0	0
$486$	−1.27563	−0.0578638
$487$	11.2506	0.509815	0.254907	−	0.966965i	$-0.417955\pi$
$487$	11.2506	0.509815	0.254907	+	0.966965i	$0.417955\pi$
$488$	−1.25785	−0.0569401
$489$	−1.14577	−0.0518135
$490$	0	0
$491$	−13.5286	−0.610538	−0.305269	−	0.952266i	$-0.598746\pi$
$491$	−13.5286	−0.610538	−0.305269	+	0.952266i	$0.598746\pi$
$492$	0.886676	0.0399744
$493$	0	0
$494$	30.4380	1.36947
$495$	0	0
$496$	0.0489357	0.00219728
$497$	−20.1158	−0.902317
$498$	−0.529301	−0.0237185
$499$	9.43647	0.422434	0.211217	−	0.977439i	$-0.432257\pi$
$499$	9.43647	0.422434	0.211217	+	0.977439i	$0.432257\pi$
$500$	0	0
$501$	1.13976	0.0509209
$502$	−3.59716	−0.160549
$503$	−19.0021	−0.847262	−0.423631	−	0.905835i	$-0.639245\pi$
$503$	−19.0021	−0.847262	−0.423631	+	0.905835i	$0.639245\pi$
$504$	−19.8228	−0.882980
$505$	0	0
$506$	8.92403	0.396722
$507$	1.48241	0.0658359
$508$	16.3390	0.724926
$509$	2.83056	0.125462	0.0627312	−	0.998030i	$-0.480019\pi$
$509$	2.83056	0.125462	0.0627312	+	0.998030i	$0.480019\pi$
$510$	0	0
$511$	7.49231	0.331440
$512$	−12.6637	−0.559662
$513$	−2.80557	−0.123869
$514$	21.3750	0.942812
$515$	0	0
$516$	0.507832	0.0223561
$517$	−12.4291	−0.546632
$518$	20.5456	0.902722
$519$	0.577767	0.0253612
$520$	0	0
$521$	−27.5026	−1.20491	−0.602456	−	0.798152i	$-0.705811\pi$
$521$	−27.5026	−1.20491	−0.602456	+	0.798152i	$0.705811\pi$
$522$	16.0622	0.703022
$523$	22.3229	0.976111	0.488055	−	0.872813i	$-0.337706\pi$
$523$	22.3229	0.976111	0.488055	+	0.872813i	$0.337706\pi$
$524$	32.1853	1.40602
$525$	0	0
$526$	−6.35623	−0.277145
$527$	0	0
$528$	−0.315746	−0.0137411
$529$	−13.8742	−0.603227
$530$	0	0
$531$	3.36376	0.145975
$532$	−27.8544	−1.20764
$533$	54.0937	2.34306
$534$	−0.252496	−0.0109266
$535$	0	0
$536$	17.4173	0.752313
$537$	−1.18457	−0.0511179
$538$	9.32748	0.402136
$539$	0.0648618	0.00279379
$540$	0	0
$541$	−7.16862	−0.308203	−0.154101	−	0.988055i	$-0.549248\pi$
$541$	−7.16862	−0.308203	−0.154101	+	0.988055i	$0.549248\pi$
$542$	8.20109	0.352267
$543$	1.53849	0.0660230
$544$	0	0
$545$	0	0
$546$	0.745485	0.0319038
$547$	37.3872	1.59856	0.799280	−	0.600958i	$-0.205214\pi$
$547$	37.3872	1.59856	0.799280	+	0.600958i	$0.205214\pi$
$548$	−14.5361	−0.620953
$549$	1.50490	0.0642274
$550$	0	0
$551$	53.0025	2.25798
$552$	0.498108	0.0212009
$553$	−2.99669	−0.127432
$554$	−0.0391647	−0.00166395
$555$	0	0
$556$	10.5026	0.445410
$557$	−12.1210	−0.513582	−0.256791	−	0.966467i	$-0.582665\pi$
$557$	−12.1210	−0.513582	−0.256791	+	0.966467i	$0.582665\pi$
$558$	0.0903193	0.00382352
$559$	30.9815	1.31038
$560$	0	0
$561$	0	0
$562$	−3.93639	−0.166046
$563$	−34.4147	−1.45041	−0.725204	−	0.688534i	$-0.758255\pi$
$563$	−34.4147	−1.45041	−0.725204	+	0.688534i	$0.758255\pi$
$564$	−0.295419	−0.0124394
$565$	0	0
$566$	1.25628	0.0528053
$567$	23.6818	0.994542
$568$	−19.0587	−0.799684
$569$	−15.7694	−0.661086	−0.330543	−	0.943791i	$-0.607232\pi$
$569$	−15.7694	−0.661086	−0.330543	+	0.943791i	$0.607232\pi$
$570$	0	0
$571$	−29.2810	−1.22537	−0.612686	−	0.790327i	$-0.709911\pi$
$571$	−29.2810	−1.22537	−0.612686	+	0.790327i	$0.709911\pi$
$572$	−36.3251	−1.51883
$573$	0.0911313	0.00380706
$574$	17.2445	0.719770
$575$	0	0
$576$	−5.59937	−0.233307
$577$	21.7531	0.905593	0.452796	−	0.891614i	$-0.350426\pi$
$577$	21.7531	0.905593	0.452796	+	0.891614i	$0.350426\pi$
$578$	0	0
$579$	0.604216	0.0251104
$580$	0	0
$581$	29.5503	1.22595
$582$	−0.589080	−0.0244181
$583$	16.4621	0.681789
$584$	7.09858	0.293741
$585$	0	0
$586$	−0.624004	−0.0257774
$587$	36.0678	1.48868	0.744338	−	0.667803i	$-0.232765\pi$
$587$	36.0678	1.48868	0.744338	+	0.667803i	$0.232765\pi$
$588$	0.00154165	6.35767e−5 0
$589$	0.298039	0.0122805
$590$	0	0
$591$	1.19091	0.0489876
$592$	−12.6182	−0.518606
$593$	16.4875	0.677061	0.338531	−	0.940955i	$-0.390070\pi$
$593$	16.4875	0.677061	0.338531	+	0.940955i	$0.390070\pi$
$594$	−1.16637	−0.0478569
$595$	0	0
$596$	−26.2566	−1.07551
$597$	0.119343	0.00488439
$598$	12.9402	0.529164
$599$	−21.8659	−0.893418	−0.446709	−	0.894679i	$-0.647404\pi$
$599$	−21.8659	−0.893418	−0.446709	+	0.894679i	$0.647404\pi$
$600$	0	0
$601$	−4.77356	−0.194718	−0.0973588	−	0.995249i	$-0.531039\pi$
$601$	−4.77356	−0.194718	−0.0973588	+	0.995249i	$0.531039\pi$
$602$	9.87654	0.402538
$603$	−20.8382	−0.848597
$604$	−3.19304	−0.129923
$605$	0	0
$606$	0.675215	0.0274287
$607$	15.3175	0.621719	0.310860	−	0.950456i	$-0.399383\pi$
$607$	15.3175	0.621719	0.310860	+	0.950456i	$0.399383\pi$
$608$	−41.5433	−1.68480
$609$	1.29813	0.0526030
$610$	0	0
$611$	−18.0227	−0.729121
$612$	0	0
$613$	−25.4721	−1.02881	−0.514405	−	0.857548i	$-0.671987\pi$
$613$	−25.4721	−1.02881	−0.514405	+	0.857548i	$0.671987\pi$
$614$	−20.4511	−0.825340
$615$	0	0
$616$	−27.1941	−1.09568
$617$	18.9989	0.764867	0.382433	−	0.923983i	$-0.375086\pi$
$617$	18.9989	0.764867	0.382433	+	0.923983i	$0.375086\pi$
$618$	−0.311171	−0.0125171
$619$	−7.65925	−0.307851	−0.153926	−	0.988082i	$-0.549192\pi$
$619$	−7.65925	−0.307851	−0.153926	+	0.988082i	$0.549192\pi$
$620$	0	0
$621$	−1.19274	−0.0478631
$622$	4.26848	0.171151
$623$	14.0966	0.564768
$624$	−0.457845	−0.0183285
$625$	0	0
$626$	3.98247	0.159171
$627$	−1.92303	−0.0767983
$628$	14.8902	0.594182
$629$	0	0
$630$	0	0
$631$	−11.8884	−0.473269	−0.236635	−	0.971599i	$-0.576044\pi$
$631$	−11.8884	−0.473269	−0.236635	+	0.971599i	$0.576044\pi$
$632$	−2.83921	−0.112938
$633$	−0.348892	−0.0138672
$634$	0.444468	0.0176521
$635$	0	0
$636$	0.391275	0.0155151
$637$	0.0940521	0.00372648
$638$	22.0349	0.872372
$639$	22.8019	0.902030
$640$	0	0
$641$	26.6113	1.05108	0.525541	−	0.850768i	$-0.323863\pi$
$641$	26.6113	1.05108	0.525541	+	0.850768i	$0.323863\pi$
$642$	0.765924	0.0302286
$643$	41.1931	1.62450	0.812248	−	0.583312i	$-0.198243\pi$
$643$	41.1931	1.62450	0.812248	+	0.583312i	$0.198243\pi$
$644$	−11.8418	−0.466634
$645$	0	0
$646$	0	0
$647$	23.2759	0.915070	0.457535	−	0.889192i	$-0.348732\pi$
$647$	23.2759	0.915070	0.457535	+	0.889192i	$0.348732\pi$
$648$	22.4373	0.881419
$649$	4.61459	0.181138
$650$	0	0
$651$	0.00729954	0.000286092 0
$652$	25.8076	1.01070
$653$	39.2072	1.53430	0.767149	−	0.641469i	$-0.221675\pi$
$653$	39.2072	1.53430	0.767149	+	0.641469i	$0.221675\pi$
$654$	−0.297328	−0.0116264
$655$	0	0
$656$	−10.5908	−0.413502
$657$	−8.49278	−0.331335
$658$	−5.74544	−0.223981
$659$	1.25054	0.0487140	0.0243570	−	0.999703i	$-0.492246\pi$
$659$	1.25054	0.0487140	0.0243570	+	0.999703i	$0.492246\pi$
$660$	0	0
$661$	0.848429	0.0330001	0.0165000	−	0.999864i	$-0.494748\pi$
$661$	0.848429	0.0330001	0.0165000	+	0.999864i	$0.494748\pi$
$662$	22.6049	0.878564
$663$	0	0
$664$	27.9974	1.08651
$665$	0	0
$666$	−23.2891	−0.902435
$667$	22.5331	0.872486
$668$	−25.6723	−0.993293
$669$	−0.441592	−0.0170729
$670$	0	0
$671$	2.06450	0.0796991
$672$	−1.01748	−0.0392500
$673$	24.0883	0.928535	0.464268	−	0.885695i	$-0.346318\pi$
$673$	24.0883	0.928535	0.464268	+	0.885695i	$0.346318\pi$
$674$	−4.83403	−0.186200
$675$	0	0
$676$	−33.3901	−1.28423
$677$	44.8742	1.72466	0.862328	−	0.506350i	$-0.169005\pi$
$677$	44.8742	1.72466	0.862328	+	0.506350i	$0.169005\pi$
$678$	−0.444440	−0.0170686
$679$	32.8877	1.26211
$680$	0	0
$681$	1.38044	0.0528984
$682$	0.123905	0.00474456
$683$	−19.2516	−0.736644	−0.368322	−	0.929698i	$-0.620068\pi$
$683$	−19.2516	−0.736644	−0.368322	+	0.929698i	$0.620068\pi$
$684$	31.5739	1.20726
$685$	0	0
$686$	13.3278	0.508860
$687$	0.0431990	0.00164815
$688$	−6.06574	−0.231254
$689$	23.8706	0.909399
$690$	0	0
$691$	−4.38326	−0.166747	−0.0833735	−	0.996518i	$-0.526569\pi$
$691$	−4.38326	−0.166747	−0.0833735	+	0.996518i	$0.526569\pi$
$692$	−13.0138	−0.494710
$693$	32.5352	1.23591
$694$	−14.6185	−0.554910
$695$	0	0
$696$	1.22991	0.0466197
$697$	0	0
$698$	16.9555	0.641773
$699$	1.50787	0.0570330
$700$	0	0
$701$	30.0554	1.13518	0.567589	−	0.823312i	$-0.307876\pi$
$701$	30.0554	1.13518	0.567589	+	0.823312i	$0.307876\pi$
$702$	−1.69129	−0.0638335
$703$	−76.8503	−2.89846
$704$	−7.68152	−0.289508
$705$	0	0
$706$	−15.4085	−0.579907
$707$	−37.6965	−1.41772
$708$	0.109681	0.00412206
$709$	37.8930	1.42310	0.711551	−	0.702635i	$-0.247993\pi$
$709$	37.8930	1.42310	0.711551	+	0.702635i	$0.247993\pi$
$710$	0	0
$711$	3.39685	0.127392
$712$	13.3558	0.500529
$713$	0.126706	0.00474518
$714$	0	0
$715$	0	0
$716$	26.6815	0.997136
$717$	1.82321	0.0680890
$718$	−18.4560	−0.688771
$719$	−27.6613	−1.03159	−0.515796	−	0.856712i	$-0.672504\pi$
$719$	−27.6613	−1.03159	−0.515796	+	0.856712i	$0.672504\pi$
$720$	0	0
$721$	17.3723	0.646980
$722$	−22.6372	−0.842468
$723$	1.26717	0.0471264
$724$	−34.6534	−1.28788
$725$	0	0
$726$	−0.278763	−0.0103459
$727$	9.90050	0.367189	0.183595	−	0.983002i	$-0.441227\pi$
$727$	9.90050	0.367189	0.183595	+	0.983002i	$0.441227\pi$
$728$	−39.4325	−1.46146
$729$	−26.7661	−0.991337
$730$	0	0
$731$	0	0
$732$	0.0490697	0.00181367
$733$	41.6098	1.53689	0.768446	−	0.639914i	$-0.221030\pi$
$733$	41.6098	1.53689	0.768446	+	0.639914i	$0.221030\pi$
$734$	−12.1192	−0.447328
$735$	0	0
$736$	−17.6614	−0.651009
$737$	−28.5870	−1.05301
$738$	−19.5472	−0.719541
$739$	14.4169	0.530334	0.265167	−	0.964202i	$-0.414573\pi$
$739$	14.4169	0.530334	0.265167	+	0.964202i	$0.414573\pi$
$740$	0	0
$741$	−2.78847	−0.102437
$742$	7.60969	0.279361
$743$	6.29072	0.230784	0.115392	−	0.993320i	$-0.463188\pi$
$743$	6.29072	0.230784	0.115392	+	0.993320i	$0.463188\pi$
$744$	0.00691594	0.000253550 0
$745$	0	0
$746$	20.9700	0.767765
$747$	−33.4963	−1.22556
$748$	0	0
$749$	−42.7608	−1.56244
$750$	0	0
$751$	−29.0142	−1.05874	−0.529371	−	0.848390i	$-0.677572\pi$
$751$	−29.0142	−1.05874	−0.529371	+	0.848390i	$0.677572\pi$
$752$	3.52860	0.128675
$753$	0.329540	0.0120091
$754$	31.9515	1.16361
$755$	0	0
$756$	1.54773	0.0562904
$757$	1.07574	0.0390984	0.0195492	−	0.999809i	$-0.493777\pi$
$757$	1.07574	0.0390984	0.0195492	+	0.999809i	$0.493777\pi$
$758$	−18.6739	−0.678268
$759$	−0.817542	−0.0296749
$760$	0	0
$761$	−35.3361	−1.28093	−0.640466	−	0.767987i	$-0.721259\pi$
$761$	−35.3361	−1.28093	−0.640466	+	0.767987i	$0.721259\pi$
$762$	0.521434	0.0188896
$763$	16.5995	0.600943
$764$	−2.05267	−0.0742628
$765$	0	0
$766$	8.16161	0.294891
$767$	6.69134	0.241610
$768$	−0.736080	−0.0265610
$769$	−33.3317	−1.20197	−0.600986	−	0.799259i	$-0.705225\pi$
$769$	−33.3317	−1.20197	−0.600986	+	0.799259i	$0.705225\pi$
$770$	0	0
$771$	−1.95819	−0.0705226
$772$	−13.6095	−0.489818
$773$	−30.1799	−1.08550	−0.542749	−	0.839895i	$-0.682616\pi$
$773$	−30.1799	−1.08550	−0.542749	+	0.839895i	$0.682616\pi$
$774$	−11.1954	−0.402410
$775$	0	0
$776$	31.1594	1.11856
$777$	−1.88221	−0.0675239
$778$	−7.01317	−0.251434
$779$	−64.5024	−2.31104
$780$	0	0
$781$	31.2809	1.11932
$782$	0	0
$783$	−2.94508	−0.105249
$784$	−0.0184141	−0.000657647 0
$785$	0	0
$786$	1.02714	0.0366370
$787$	25.6469	0.914213	0.457106	−	0.889412i	$-0.348886\pi$
$787$	25.6469	0.914213	0.457106	+	0.889412i	$0.348886\pi$
$788$	−26.8244	−0.955580
$789$	0.582302	0.0207305
$790$	0	0
$791$	24.8126	0.882236
$792$	30.8254	1.09533
$793$	2.99361	0.106306
$794$	−19.3969	−0.688370
$795$	0	0
$796$	−2.68812	−0.0952778
$797$	39.0090	1.38177	0.690886	−	0.722964i	$-0.257221\pi$
$797$	39.0090	1.38177	0.690886	+	0.722964i	$0.257221\pi$
$798$	−0.888932	−0.0314678
$799$	0	0
$800$	0	0
$801$	−15.9790	−0.564589
$802$	−10.7079	−0.378111
$803$	−11.6509	−0.411150
$804$	−0.679463	−0.0239628
$805$	0	0
$806$	0.179667	0.00632850
$807$	−0.854503	−0.0300799
$808$	−35.7155	−1.25647
$809$	−15.3181	−0.538554	−0.269277	−	0.963063i	$-0.586785\pi$
$809$	−15.3181	−0.538554	−0.269277	+	0.963063i	$0.586785\pi$
$810$	0	0
$811$	−31.8325	−1.11779	−0.558895	−	0.829238i	$-0.688775\pi$
$811$	−31.8325	−1.11779	−0.558895	+	0.829238i	$0.688775\pi$
$812$	−29.2395	−1.02611
$813$	−0.751313	−0.0263497
$814$	−31.9493	−1.11982
$815$	0	0
$816$	0	0
$817$	−36.9429	−1.29247
$818$	5.06736	0.177176
$819$	47.1773	1.64851
$820$	0	0
$821$	5.86502	0.204691	0.102345	−	0.994749i	$-0.467365\pi$
$821$	5.86502	0.204691	0.102345	+	0.994749i	$0.467365\pi$
$822$	−0.463899	−0.0161803
$823$	2.04279	0.0712073	0.0356036	−	0.999366i	$-0.488665\pi$
$823$	2.04279	0.0712073	0.0356036	+	0.999366i	$0.488665\pi$
$824$	16.4594	0.573390
$825$	0	0
$826$	2.13312	0.0742208
$827$	47.1696	1.64025	0.820123	−	0.572187i	$-0.193905\pi$
$827$	47.1696	1.64025	0.820123	+	0.572187i	$0.193905\pi$
$828$	13.4231	0.466485
$829$	−54.0888	−1.87858	−0.939291	−	0.343122i	$-0.888515\pi$
$829$	−54.0888	−1.87858	−0.939291	+	0.343122i	$0.888515\pi$
$830$	0	0
$831$	0.00358793	0.000124464 0
$832$	−11.1385	−0.386158
$833$	0	0
$834$	0.335175	0.0116061
$835$	0	0
$836$	43.3148	1.49807
$837$	−0.0165605	−0.000572415 0
$838$	−11.6153	−0.401244
$839$	−33.6990	−1.16342	−0.581709	−	0.813397i	$-0.697616\pi$
$839$	−33.6990	−1.16342	−0.581709	+	0.813397i	$0.697616\pi$
$840$	0	0
$841$	26.6381	0.918554
$842$	−10.0558	−0.346545
$843$	0.360618	0.0124203
$844$	7.85854	0.270502
$845$	0	0
$846$	6.51265	0.223909
$847$	15.5630	0.534753
$848$	−4.67355	−0.160490
$849$	−0.115089	−0.00394985
$850$	0	0
$851$	−32.6716	−1.11997
$852$	0.743494	0.0254717
$853$	−45.6318	−1.56240	−0.781201	−	0.624280i	$-0.785392\pi$
$853$	−45.6318	−1.56240	−0.781201	+	0.624280i	$0.785392\pi$
$854$	0.954328	0.0326564
$855$	0	0
$856$	−40.5136	−1.38473
$857$	40.8825	1.39652	0.698260	−	0.715844i	$-0.253958\pi$
$857$	40.8825	1.39652	0.698260	+	0.715844i	$0.253958\pi$
$858$	−1.15926	−0.0395765
$859$	24.5116	0.836325	0.418162	−	0.908372i	$-0.362674\pi$
$859$	24.5116	0.836325	0.418162	+	0.908372i	$0.362674\pi$
$860$	0	0
$861$	−1.57979	−0.0538390
$862$	−19.4735	−0.663270
$863$	−5.80274	−0.197527	−0.0987637	−	0.995111i	$-0.531489\pi$
$863$	−5.80274	−0.197527	−0.0987637	+	0.995111i	$0.531489\pi$
$864$	2.30835	0.0785318
$865$	0	0
$866$	0.264021	0.00897178
$867$	0	0
$868$	−0.164417	−0.00558067
$869$	4.65999	0.158079
$870$	0	0
$871$	−41.4522	−1.40455
$872$	15.7272	0.532589
$873$	−37.2793	−1.26171
$874$	−15.4302	−0.521933
$875$	0	0
$876$	−0.276921	−0.00935630
$877$	27.0474	0.913325	0.456663	−	0.889640i	$-0.349045\pi$
$877$	27.0474	0.913325	0.456663	+	0.889640i	$0.349045\pi$
$878$	−27.9889	−0.944580
$879$	0.0571658	0.00192816
$880$	0	0
$881$	17.4441	0.587706	0.293853	−	0.955851i	$-0.405062\pi$
$881$	17.4441	0.587706	0.293853	+	0.955851i	$0.405062\pi$
$882$	−0.0339865	−0.00114438
$883$	13.1699	0.443201	0.221601	−	0.975138i	$-0.428872\pi$
$883$	13.1699	0.443201	0.221601	+	0.975138i	$0.428872\pi$
$884$	0	0
$885$	0	0
$886$	−8.07786	−0.271381
$887$	15.6234	0.524582	0.262291	−	0.964989i	$-0.415522\pi$
$887$	15.6234	0.524582	0.262291	+	0.964989i	$0.415522\pi$
$888$	−1.78330	−0.0598435
$889$	−29.1111	−0.976356
$890$	0	0
$891$	−36.8262	−1.23372
$892$	9.94654	0.333035
$893$	21.4907	0.719157
$894$	−0.837941	−0.0280249
$895$	0	0
$896$	27.3507	0.913722
$897$	−1.18547	−0.0395817
$898$	−5.36123	−0.178906
$899$	0.312859	0.0104344
$900$	0	0
$901$	0	0
$902$	−26.8159	−0.892871
$903$	−0.904803	−0.0301100
$904$	23.5087	0.781887
$905$	0	0
$906$	−0.101901	−0.00338543
$907$	−2.77839	−0.0922550	−0.0461275	−	0.998936i	$-0.514688\pi$
$907$	−2.77839	−0.0922550	−0.0461275	+	0.998936i	$0.514688\pi$
$908$	−31.0933	−1.03187
$909$	42.7303	1.41727
$910$	0	0
$911$	−18.4228	−0.610374	−0.305187	−	0.952292i	$-0.598719\pi$
$911$	−18.4228	−0.610374	−0.305187	+	0.952292i	$0.598719\pi$
$912$	0.545943	0.0180780
$913$	−45.9520	−1.52079
$914$	8.17962	0.270558
$915$	0	0
$916$	−0.973027	−0.0321497
$917$	−57.3444	−1.89368
$918$	0	0
$919$	−8.94370	−0.295025	−0.147513	−	0.989060i	$-0.547127\pi$
$919$	−8.94370	−0.295025	−0.147513	+	0.989060i	$0.547127\pi$
$920$	0	0
$921$	1.87355	0.0617357
$922$	−2.81521	−0.0927140
$923$	45.3585	1.49299
$924$	1.06086	0.0348998
$925$	0	0
$926$	−19.0364	−0.625575
$927$	−19.6921	−0.646774
$928$	−43.6091	−1.43154
$929$	14.5865	0.478566	0.239283	−	0.970950i	$-0.423088\pi$
$929$	14.5865	0.478566	0.239283	+	0.970950i	$0.423088\pi$
$930$	0	0
$931$	−0.112150	−0.00367556
$932$	−33.9637	−1.11252
$933$	−0.391041	−0.0128021
$934$	−7.35317	−0.240603
$935$	0	0
$936$	44.6980	1.46100
$937$	−16.9711	−0.554421	−0.277210	−	0.960809i	$-0.589410\pi$
$937$	−16.9711	−0.554421	−0.277210	+	0.960809i	$0.589410\pi$
$938$	−13.2145	−0.431469
$939$	−0.364839	−0.0119061
$940$	0	0
$941$	36.9549	1.20470	0.602348	−	0.798233i	$-0.294232\pi$
$941$	36.9549	1.20470	0.602348	+	0.798233i	$0.294232\pi$
$942$	0.475197	0.0154827
$943$	−27.4221	−0.892987
$944$	−1.31007	−0.0426392
$945$	0	0
$946$	−15.3584	−0.499346
$947$	−33.8102	−1.09868	−0.549342	−	0.835598i	$-0.685121\pi$
$947$	−33.8102	−1.09868	−0.549342	+	0.835598i	$0.685121\pi$
$948$	0.110760	0.00359732
$949$	−16.8942	−0.548409
$950$	0	0
$951$	−0.0407183	−0.00132038
$952$	0	0
$953$	−4.93585	−0.159888	−0.0799439	−	0.996799i	$-0.525474\pi$
$953$	−4.93585	−0.159888	−0.0799439	+	0.996799i	$0.525474\pi$
$954$	−8.62584	−0.279272
$955$	0	0
$956$	−41.0664	−1.32818
$957$	−2.01865	−0.0652537
$958$	4.31494	0.139409
$959$	25.8990	0.836323
$960$	0	0
$961$	−30.9982	−0.999943
$962$	−46.3277	−1.49367
$963$	48.4707	1.56195
$964$	−28.5420	−0.919276
$965$	0	0
$966$	−0.377914	−0.0121592
$967$	−35.0539	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$967$	−35.0539	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$968$	14.7452	0.473928
$969$	0	0
$970$	0	0
$971$	−43.4475	−1.39430	−0.697148	−	0.716928i	$-0.745548\pi$
$971$	−43.4475	−1.39430	−0.697148	+	0.716928i	$0.745548\pi$
$972$	−2.63224	−0.0844291
$973$	−18.7125	−0.599894
$974$	−8.08727	−0.259133
$975$	0	0
$976$	−0.586107	−0.0187608
$977$	58.6896	1.87765	0.938823	−	0.344399i	$-0.111917\pi$
$977$	58.6896	1.87765	0.938823	+	0.344399i	$0.111917\pi$
$978$	0.823612	0.0263362
$979$	−21.9208	−0.700592
$980$	0	0
$981$	−18.8161	−0.600752
$982$	9.72474	0.310329
$983$	−3.70994	−0.118329	−0.0591643	−	0.998248i	$-0.518844\pi$
$983$	−3.70994	−0.118329	−0.0591643	+	0.998248i	$0.518844\pi$
$984$	−1.49677	−0.0477152
$985$	0	0
$986$	0	0
$987$	0.526347	0.0167538
$988$	62.8081	1.99819
$989$	−15.7057	−0.499411
$990$	0	0
$991$	−47.6867	−1.51482	−0.757410	−	0.652940i	$-0.773535\pi$
$991$	−47.6867	−1.51482	−0.757410	+	0.652940i	$0.773535\pi$
$992$	−0.245219	−0.00778570
$993$	−2.07086	−0.0657169
$994$	14.4598	0.458637
$995$	0	0
$996$	−1.09220	−0.0346077
$997$	−58.6957	−1.85891	−0.929456	−	0.368933i	$-0.879723\pi$
$997$	−58.6957	−1.85891	−0.929456	+	0.368933i	$0.879723\pi$
$998$	−6.78320	−0.214718
$999$	4.27018	0.135103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bv.1.6	yes	15
5.4	even	2		7225.2.a.bt.1.10	✓	15
17.16	even	2		7225.2.a.bu.1.6	yes	15
85.84	even	2		7225.2.a.bw.1.10	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.10	✓	15	5.4	even	2
7225.2.a.bu.1.6	yes	15	17.16	even	2
7225.2.a.bv.1.6	yes	15	1.1	even	1	trivial
7225.2.a.bw.1.10	yes	15	85.84	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)

\(f(q)\)	\(=\)	\(q-0.718828 q^{2} +0.0658528 q^{3} -1.48329 q^{4} -0.0473368 q^{6} +2.64277 q^{7} +2.50388 q^{8} -2.99566 q^{9} +O(q^{10})\)	\(q-0.718828 q^{2} +0.0658528 q^{3} -1.48329 q^{4} -0.0473368 q^{6} +2.64277 q^{7} +2.50388 q^{8} -2.99566 q^{9} -4.10961 q^{11} -0.0976785 q^{12} -5.95910 q^{13} -1.89969 q^{14} +1.16671 q^{16} +2.15337 q^{18} +7.10576 q^{19} +0.174034 q^{21} +2.95411 q^{22} +3.02089 q^{23} +0.164888 q^{24} +4.28357 q^{26} -0.394831 q^{27} -3.91998 q^{28} +7.45909 q^{29} +0.0419433 q^{31} -5.84643 q^{32} -0.270630 q^{33} +4.44343 q^{36} -10.8152 q^{37} -5.10782 q^{38} -0.392423 q^{39} -9.07749 q^{41} -0.125100 q^{42} -5.19901 q^{43} +6.09573 q^{44} -2.17150 q^{46} +3.02440 q^{47} +0.0768311 q^{48} -0.0157829 q^{49} +8.83905 q^{52} -4.00575 q^{53} +0.283816 q^{54} +6.61718 q^{56} +0.467934 q^{57} -5.36181 q^{58} -1.12288 q^{59} -0.502359 q^{61} -0.0301500 q^{62} -7.91684 q^{63} +1.86916 q^{64} +0.194536 q^{66} +6.95612 q^{67} +0.198934 q^{69} -7.61164 q^{71} -7.50079 q^{72} +2.83503 q^{73} +7.77428 q^{74} -10.5399 q^{76} -10.8608 q^{77} +0.282085 q^{78} -1.13392 q^{79} +8.96099 q^{81} +6.52516 q^{82} +11.1816 q^{83} -0.258142 q^{84} +3.73720 q^{86} +0.491202 q^{87} -10.2900 q^{88} +5.33403 q^{89} -15.7485 q^{91} -4.48085 q^{92} +0.00276208 q^{93} -2.17402 q^{94} -0.385004 q^{96} +12.4444 q^{97} +0.0113452 q^{98} +12.3110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) 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

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