LMFDB - Embedded newform 4425.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4425,2,Mod(1,4425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4425 = 3 \cdot 5^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4425.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:	$35.3338028944$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 885)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	4425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} -1.81361 q^{6} +2.52444 q^{7} +1.28917 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} -1.81361 q^{6} +2.52444 q^{7} +1.28917 q^{8} +1.00000 q^{9} -5.39194 q^{11} +1.28917 q^{12} +5.91638 q^{13} -4.57834 q^{14} -4.91638 q^{16} -2.23527 q^{17} -1.81361 q^{18} -0.0836184 q^{19} +2.52444 q^{21} +9.77886 q^{22} -4.57834 q^{23} +1.28917 q^{24} -10.7300 q^{26} +1.00000 q^{27} +3.25443 q^{28} -2.89722 q^{29} +2.20555 q^{31} +6.33804 q^{32} -5.39194 q^{33} +4.05390 q^{34} +1.28917 q^{36} -3.71083 q^{37} +0.151651 q^{38} +5.91638 q^{39} -6.00000 q^{41} -4.57834 q^{42} +0.661956 q^{43} -6.95112 q^{44} +8.30330 q^{46} -4.91638 q^{48} -0.627213 q^{49} -2.23527 q^{51} +7.62721 q^{52} -10.6464 q^{53} -1.81361 q^{54} +3.25443 q^{56} -0.0836184 q^{57} +5.25443 q^{58} +1.00000 q^{59} -13.9844 q^{61} -4.00000 q^{62} +2.52444 q^{63} -1.66196 q^{64} +9.77886 q^{66} +10.7597 q^{67} -2.88164 q^{68} -4.57834 q^{69} -6.20555 q^{71} +1.28917 q^{72} -15.5436 q^{73} +6.72999 q^{74} -0.107798 q^{76} -13.6116 q^{77} -10.7300 q^{78} -10.2892 q^{79} +1.00000 q^{81} +10.8816 q^{82} +14.7839 q^{83} +3.25443 q^{84} -1.20053 q^{86} -2.89722 q^{87} -6.95112 q^{88} +10.1758 q^{89} +14.9355 q^{91} -5.90225 q^{92} +2.20555 q^{93} +6.33804 q^{96} -16.0383 q^{97} +1.13752 q^{98} -5.39194 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + 3 q^{4} + q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 + 3 * q^3 + 3 * q^4 + q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + q^{2} + 3 q^{3} + 3 q^{4} + q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} - 8 q^{11} + 3 q^{12} + 4 q^{13} - 12 q^{14} - q^{16} - 2 q^{17} + q^{18} - 14 q^{19} + 2 q^{21} - 2 q^{22} - 12 q^{23} + 3 q^{24} - 12 q^{26} + 3 q^{27} - 16 q^{28} - 16 q^{29} - 8 q^{31} + 7 q^{32} - 8 q^{33} + 16 q^{34} + 3 q^{36} - 12 q^{37} - 18 q^{38} + 4 q^{39} - 18 q^{41} - 12 q^{42} + 14 q^{43} - 32 q^{44} - 12 q^{46} - q^{48} + 11 q^{49} - 2 q^{51} + 10 q^{52} + 2 q^{53} + q^{54} - 16 q^{56} - 14 q^{57} - 10 q^{58} + 3 q^{59} + 4 q^{61} - 12 q^{62} + 2 q^{63} - 17 q^{64} - 2 q^{66} + 22 q^{67} + 30 q^{68} - 12 q^{69} - 4 q^{71} + 3 q^{72} - 20 q^{73} - 8 q^{76} + 18 q^{77} - 12 q^{78} - 30 q^{79} + 3 q^{81} - 6 q^{82} + 28 q^{83} - 16 q^{84} + 26 q^{86} - 16 q^{87} - 32 q^{88} + 6 q^{89} + 10 q^{91} - 40 q^{92} - 8 q^{93} + 7 q^{96} - 6 q^{97} + 21 q^{98} - 8 q^{99}+O(q^{100}) $ 3 * q + q^2 + 3 * q^3 + 3 * q^4 + q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 - 8 * q^11 + 3 * q^12 + 4 * q^13 - 12 * q^14 - q^16 - 2 * q^17 + q^18 - 14 * q^19 + 2 * q^21 - 2 * q^22 - 12 * q^23 + 3 * q^24 - 12 * q^26 + 3 * q^27 - 16 * q^28 - 16 * q^29 - 8 * q^31 + 7 * q^32 - 8 * q^33 + 16 * q^34 + 3 * q^36 - 12 * q^37 - 18 * q^38 + 4 * q^39 - 18 * q^41 - 12 * q^42 + 14 * q^43 - 32 * q^44 - 12 * q^46 - q^48 + 11 * q^49 - 2 * q^51 + 10 * q^52 + 2 * q^53 + q^54 - 16 * q^56 - 14 * q^57 - 10 * q^58 + 3 * q^59 + 4 * q^61 - 12 * q^62 + 2 * q^63 - 17 * q^64 - 2 * q^66 + 22 * q^67 + 30 * q^68 - 12 * q^69 - 4 * q^71 + 3 * q^72 - 20 * q^73 - 8 * q^76 + 18 * q^77 - 12 * q^78 - 30 * q^79 + 3 * q^81 - 6 * q^82 + 28 * q^83 - 16 * q^84 + 26 * q^86 - 16 * q^87 - 32 * q^88 + 6 * q^89 + 10 * q^91 - 40 * q^92 - 8 * q^93 + 7 * q^96 - 6 * q^97 + 21 * q^98 - 8 * 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 $

$2$	−1.81361	−1.28241	−0.641207	−	0.767368i	$-0.721566\pi$
$2$	−1.81361	−1.28241	−0.641207	+	0.767368i	$0.721566\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.28917	0.644584
$5$	0	0
$5$	0	0
$6$	−1.81361	−0.740402
$7$	2.52444	0.954148	0.477074	−	0.878863i	$-0.341697\pi$
$7$	2.52444	0.954148	0.477074	+	0.878863i	$0.341697\pi$
$8$	1.28917	0.455790
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.39194	−1.62573	−0.812866	−	0.582451i	$-0.802094\pi$
$11$	−5.39194	−1.62573	−0.812866	+	0.582451i	$0.802094\pi$
$12$	1.28917	0.372151
$13$	5.91638	1.64091	0.820455	−	0.571712i	$-0.193720\pi$
$13$	5.91638	1.64091	0.820455	+	0.571712i	$0.193720\pi$
$14$	−4.57834	−1.22361
$15$	0	0
$16$	−4.91638	−1.22910
$17$	−2.23527	−0.542132	−0.271066	−	0.962561i	$-0.587376\pi$
$17$	−2.23527	−0.542132	−0.271066	+	0.962561i	$0.587376\pi$
$18$	−1.81361	−0.427471
$19$	−0.0836184	−0.0191834	−0.00959169	−	0.999954i	$-0.503053\pi$
$19$	−0.0836184	−0.0191834	−0.00959169	+	0.999954i	$0.503053\pi$
$20$	0	0
$21$	2.52444	0.550878
$22$	9.77886	2.08486
$23$	−4.57834	−0.954649	−0.477325	−	0.878727i	$-0.658393\pi$
$23$	−4.57834	−0.954649	−0.477325	+	0.878727i	$0.658393\pi$
$24$	1.28917	0.263150
$25$	0	0
$26$	−10.7300	−2.10432
$27$	1.00000	0.192450
$28$	3.25443	0.615029
$29$	−2.89722	−0.538001	−0.269001	−	0.963140i	$-0.586693\pi$
$29$	−2.89722	−0.538001	−0.269001	+	0.963140i	$0.586693\pi$
$30$	0	0
$31$	2.20555	0.396128	0.198064	−	0.980189i	$-0.436535\pi$
$31$	2.20555	0.396128	0.198064	+	0.980189i	$0.436535\pi$
$32$	6.33804	1.12042
$33$	−5.39194	−0.938617
$34$	4.05390	0.695238
$35$	0	0
$36$	1.28917	0.214861
$37$	−3.71083	−0.610057	−0.305028	−	0.952343i	$-0.598666\pi$
$37$	−3.71083	−0.610057	−0.305028	+	0.952343i	$0.598666\pi$
$38$	0.151651	0.0246010
$39$	5.91638	0.947379
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−4.57834	−0.706453
$43$	0.661956	0.100947	0.0504736	−	0.998725i	$-0.483927\pi$
$43$	0.661956	0.100947	0.0504736	+	0.998725i	$0.483927\pi$
$44$	−6.95112	−1.04792
$45$	0	0
$46$	8.30330	1.22426
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−4.91638	−0.709619
$49$	−0.627213	−0.0896019
$50$	0	0
$51$	−2.23527	−0.313000
$52$	7.62721	1.05770
$53$	−10.6464	−1.46239	−0.731196	−	0.682168i	$-0.761037\pi$
$53$	−10.6464	−1.46239	−0.731196	+	0.682168i	$0.761037\pi$
$54$	−1.81361	−0.246801
$55$	0	0
$56$	3.25443	0.434891
$57$	−0.0836184	−0.0110755
$58$	5.25443	0.689940
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−13.9844	−1.79052	−0.895260	−	0.445543i	$-0.853011\pi$
$61$	−13.9844	−1.79052	−0.895260	+	0.445543i	$0.853011\pi$
$62$	−4.00000	−0.508001
$63$	2.52444	0.318049
$64$	−1.66196	−0.207744
$65$	0	0
$66$	9.77886	1.20369
$67$	10.7597	1.31451	0.657254	−	0.753669i	$-0.271718\pi$
$67$	10.7597	1.31451	0.657254	+	0.753669i	$0.271718\pi$
$68$	−2.88164	−0.349450
$69$	−4.57834	−0.551167
$70$	0	0
$71$	−6.20555	−0.736463	−0.368232	−	0.929734i	$-0.620037\pi$
$71$	−6.20555	−0.736463	−0.368232	+	0.929734i	$0.620037\pi$
$72$	1.28917	0.151930
$73$	−15.5436	−1.81924	−0.909620	−	0.415441i	$-0.863627\pi$
$73$	−15.5436	−1.81924	−0.909620	+	0.415441i	$0.863627\pi$
$74$	6.72999	0.782345
$75$	0	0
$76$	−0.107798	−0.0123653
$77$	−13.6116	−1.55119
$78$	−10.7300	−1.21493
$79$	−10.2892	−1.15762	−0.578811	−	0.815462i	$-0.696483\pi$
$79$	−10.2892	−1.15762	−0.578811	+	0.815462i	$0.696483\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.8816	1.20168
$83$	14.7839	1.62274	0.811371	−	0.584531i	$-0.198722\pi$
$83$	14.7839	1.62274	0.811371	+	0.584531i	$0.198722\pi$
$84$	3.25443	0.355087
$85$	0	0
$86$	−1.20053	−0.129456
$87$	−2.89722	−0.310615
$88$	−6.95112	−0.740992
$89$	10.1758	1.07864	0.539318	−	0.842102i	$-0.318682\pi$
$89$	10.1758	1.07864	0.539318	+	0.842102i	$0.318682\pi$
$90$	0	0
$91$	14.9355	1.56567
$92$	−5.90225	−0.615352
$93$	2.20555	0.228705
$94$	0	0
$95$	0	0
$96$	6.33804	0.646874
$97$	−16.0383	−1.62844	−0.814222	−	0.580554i	$-0.802836\pi$
$97$	−16.0383	−1.62844	−0.814222	+	0.580554i	$0.802836\pi$
$98$	1.13752	0.114907
$99$	−5.39194	−0.541911
$100$	0	0
$101$	0.205550	0.0204530	0.0102265	−	0.999948i	$-0.496745\pi$
$101$	0.205550	0.0204530	0.0102265	+	0.999948i	$0.496745\pi$
$102$	4.05390	0.401396
$103$	−4.47054	−0.440495	−0.220248	−	0.975444i	$-0.570687\pi$
$103$	−4.47054	−0.440495	−0.220248	+	0.975444i	$0.570687\pi$
$104$	7.62721	0.747910
$105$	0	0
$106$	19.3083	1.87539
$107$	2.60806	0.252130	0.126065	−	0.992022i	$-0.459765\pi$
$107$	2.60806	0.252130	0.126065	+	0.992022i	$0.459765\pi$
$108$	1.28917	0.124050
$109$	−6.57834	−0.630090	−0.315045	−	0.949077i	$-0.602020\pi$
$109$	−6.57834	−0.630090	−0.315045	+	0.949077i	$0.602020\pi$
$110$	0	0
$111$	−3.71083	−0.352217
$112$	−12.4111	−1.17274
$113$	−5.10278	−0.480029	−0.240014	−	0.970769i	$-0.577152\pi$
$113$	−5.10278	−0.480029	−0.240014	+	0.970769i	$0.577152\pi$
$114$	0.151651	0.0142034
$115$	0	0
$116$	−3.73501	−0.346787
$117$	5.91638	0.546970
$118$	−1.81361	−0.166956
$119$	−5.64280	−0.517275
$120$	0	0
$121$	18.0731	1.64301
$122$	25.3622	2.29619
$123$	−6.00000	−0.541002
$124$	2.84333	0.255338
$125$	0	0
$126$	−4.57834	−0.407871
$127$	17.4600	1.54932	0.774661	−	0.632377i	$-0.217920\pi$
$127$	17.4600	1.54932	0.774661	+	0.632377i	$0.217920\pi$
$128$	−9.66196	−0.854004
$129$	0.661956	0.0582819
$130$	0	0
$131$	−20.0680	−1.75335	−0.876676	−	0.481081i	$-0.840244\pi$
$131$	−20.0680	−1.75335	−0.876676	+	0.481081i	$0.840244\pi$
$132$	−6.95112	−0.605018
$133$	−0.211090	−0.0183038
$134$	−19.5139	−1.68574
$135$	0	0
$136$	−2.88164	−0.247099
$137$	7.45998	0.637349	0.318674	−	0.947864i	$-0.396762\pi$
$137$	7.45998	0.637349	0.318674	+	0.947864i	$0.396762\pi$
$138$	8.30330	0.706824
$139$	1.21611	0.103149	0.0515747	−	0.998669i	$-0.483576\pi$
$139$	1.21611	0.103149	0.0515747	+	0.998669i	$0.483576\pi$
$140$	0	0
$141$	0	0
$142$	11.2544	0.944450
$143$	−31.9008	−2.66768
$144$	−4.91638	−0.409698
$145$	0	0
$146$	28.1900	2.33302
$147$	−0.627213	−0.0517317
$148$	−4.78389	−0.393233
$149$	8.27358	0.677798	0.338899	−	0.940823i	$-0.389945\pi$
$149$	8.27358	0.677798	0.338899	+	0.940823i	$0.389945\pi$
$150$	0	0
$151$	17.7789	1.44682	0.723412	−	0.690417i	$-0.242573\pi$
$151$	17.7789	1.44682	0.723412	+	0.690417i	$0.242573\pi$
$152$	−0.107798	−0.00874359
$153$	−2.23527	−0.180711
$154$	24.6861	1.98927
$155$	0	0
$156$	7.62721	0.610666
$157$	6.88164	0.549215	0.274607	−	0.961556i	$-0.411452\pi$
$157$	6.88164	0.549215	0.274607	+	0.961556i	$0.411452\pi$
$158$	18.6605	1.48455
$159$	−10.6464	−0.844312
$160$	0	0
$161$	−11.5577	−0.910877
$162$	−1.81361	−0.142490
$163$	−21.9305	−1.71773	−0.858865	−	0.512202i	$-0.828830\pi$
$163$	−21.9305	−1.71773	−0.858865	+	0.512202i	$0.828830\pi$
$164$	−7.73501	−0.604003
$165$	0	0
$166$	−26.8122	−2.08103
$167$	14.4408	1.11746	0.558732	−	0.829348i	$-0.311288\pi$
$167$	14.4408	1.11746	0.558732	+	0.829348i	$0.311288\pi$
$168$	3.25443	0.251084
$169$	22.0036	1.69258
$170$	0	0
$171$	−0.0836184	−0.00639446
$172$	0.853372	0.0650690
$173$	−5.42166	−0.412201	−0.206101	−	0.978531i	$-0.566077\pi$
$173$	−5.42166	−0.412201	−0.206101	+	0.978531i	$0.566077\pi$
$174$	5.25443	0.398337
$175$	0	0
$176$	26.5089	1.99818
$177$	1.00000	0.0751646
$178$	−18.4550	−1.38326
$179$	−18.4408	−1.37833	−0.689166	−	0.724604i	$-0.742023\pi$
$179$	−18.4408	−1.37833	−0.689166	+	0.724604i	$0.742023\pi$
$180$	0	0
$181$	22.1361	1.64536	0.822680	−	0.568504i	$-0.192478\pi$
$181$	22.1361	1.64536	0.822680	+	0.568504i	$0.192478\pi$
$182$	−27.0872	−2.00784
$183$	−13.9844	−1.03376
$184$	−5.90225	−0.435120
$185$	0	0
$186$	−4.00000	−0.293294
$187$	12.0524	0.881362
$188$	0	0
$189$	2.52444	0.183626
$190$	0	0
$191$	−10.8519	−0.785217	−0.392609	−	0.919706i	$-0.628427\pi$
$191$	−10.8519	−0.785217	−0.392609	+	0.919706i	$0.628427\pi$
$192$	−1.66196	−0.119941
$193$	19.0489	1.37117	0.685584	−	0.727993i	$-0.259547\pi$
$193$	19.0489	1.37117	0.685584	+	0.727993i	$0.259547\pi$
$194$	29.0872	2.08834
$195$	0	0
$196$	−0.808583	−0.0577559
$197$	9.90080	0.705402	0.352701	−	0.935736i	$-0.385263\pi$
$197$	9.90080	0.705402	0.352701	+	0.935736i	$0.385263\pi$
$198$	9.77886	0.694954
$199$	6.01413	0.426331	0.213165	−	0.977016i	$-0.431623\pi$
$199$	6.01413	0.426331	0.213165	+	0.977016i	$0.431623\pi$
$200$	0	0
$201$	10.7597	0.758931
$202$	−0.372787	−0.0262292
$203$	−7.31386	−0.513333
$204$	−2.88164	−0.201755
$205$	0	0
$206$	8.10780	0.564897
$207$	−4.57834	−0.318216
$208$	−29.0872	−2.01683
$209$	0.450866	0.0311871
$210$	0	0
$211$	−10.6917	−0.736045	−0.368023	−	0.929817i	$-0.619965\pi$
$211$	−10.6917	−0.736045	−0.368023	+	0.929817i	$0.619965\pi$
$212$	−13.7250	−0.942634
$213$	−6.20555	−0.425197
$214$	−4.72999	−0.323335
$215$	0	0
$216$	1.28917	0.0877168
$217$	5.56777	0.377965
$218$	11.9305	0.808036
$219$	−15.5436	−1.05034
$220$	0	0
$221$	−13.2247	−0.889590
$222$	6.72999	0.451687
$223$	−24.5628	−1.64484	−0.822422	−	0.568878i	$-0.807378\pi$
$223$	−24.5628	−1.64484	−0.822422	+	0.568878i	$0.807378\pi$
$224$	16.0000	1.06904
$225$	0	0
$226$	9.25443	0.615595
$227$	6.82774	0.453173	0.226587	−	0.973991i	$-0.427243\pi$
$227$	6.82774	0.453173	0.226587	+	0.973991i	$0.427243\pi$
$228$	−0.107798	−0.00713912
$229$	−13.5139	−0.893022	−0.446511	−	0.894778i	$-0.647334\pi$
$229$	−13.5139	−0.893022	−0.446511	+	0.894778i	$0.647334\pi$
$230$	0	0
$231$	−13.6116	−0.895579
$232$	−3.73501	−0.245216
$233$	3.79445	0.248583	0.124291	−	0.992246i	$-0.460334\pi$
$233$	3.79445	0.248583	0.124291	+	0.992246i	$0.460334\pi$
$234$	−10.7300	−0.701441
$235$	0	0
$236$	1.28917	0.0839177
$237$	−10.2892	−0.668353
$238$	10.2338	0.663360
$239$	−3.68111	−0.238111	−0.119056	−	0.992888i	$-0.537987\pi$
$239$	−3.68111	−0.238111	−0.119056	+	0.992888i	$0.537987\pi$
$240$	0	0
$241$	0.396967	0.0255709	0.0127854	−	0.999918i	$-0.495930\pi$
$241$	0.396967	0.0255709	0.0127854	+	0.999918i	$0.495930\pi$
$242$	−32.7774	−2.10701
$243$	1.00000	0.0641500
$244$	−18.0283	−1.15414
$245$	0	0
$246$	10.8816	0.693788
$247$	−0.494719	−0.0314782
$248$	2.84333	0.180551
$249$	14.7839	0.936891
$250$	0	0
$251$	−24.2439	−1.53026	−0.765130	−	0.643876i	$-0.777325\pi$
$251$	−24.2439	−1.53026	−0.765130	+	0.643876i	$0.777325\pi$
$252$	3.25443	0.205010
$253$	24.6861	1.55200
$254$	−31.6655	−1.98687
$255$	0	0
$256$	20.8469	1.30293
$257$	−3.69525	−0.230503	−0.115252	−	0.993336i	$-0.536767\pi$
$257$	−3.69525	−0.230503	−0.115252	+	0.993336i	$0.536767\pi$
$258$	−1.20053	−0.0747415
$259$	−9.36776	−0.582085
$260$	0	0
$261$	−2.89722	−0.179334
$262$	36.3955	2.24852
$263$	22.9114	1.41277	0.706387	−	0.707825i	$-0.250324\pi$
$263$	22.9114	1.41277	0.706387	+	0.707825i	$0.250324\pi$
$264$	−6.95112	−0.427812
$265$	0	0
$266$	0.382833	0.0234730
$267$	10.1758	0.622751
$268$	13.8711	0.847311
$269$	5.70529	0.347858	0.173929	−	0.984758i	$-0.444354\pi$
$269$	5.70529	0.347858	0.173929	+	0.984758i	$0.444354\pi$
$270$	0	0
$271$	−14.3416	−0.871191	−0.435596	−	0.900143i	$-0.643462\pi$
$271$	−14.3416	−0.871191	−0.435596	+	0.900143i	$0.643462\pi$
$272$	10.9894	0.666333
$273$	14.9355	0.903940
$274$	−13.5295	−0.817345
$275$	0	0
$276$	−5.90225	−0.355274
$277$	−29.2388	−1.75679	−0.878396	−	0.477934i	$-0.841386\pi$
$277$	−29.2388	−1.75679	−0.878396	+	0.477934i	$0.841386\pi$
$278$	−2.20555	−0.132280
$279$	2.20555	0.132043
$280$	0	0
$281$	−16.3572	−0.975789	−0.487894	−	0.872903i	$-0.662235\pi$
$281$	−16.3572	−0.975789	−0.487894	+	0.872903i	$0.662235\pi$
$282$	0	0
$283$	2.26499	0.134640	0.0673198	−	0.997731i	$-0.478555\pi$
$283$	2.26499	0.134640	0.0673198	+	0.997731i	$0.478555\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	57.8555	3.42107
$287$	−15.1466	−0.894077
$288$	6.33804	0.373473
$289$	−12.0036	−0.706092
$290$	0	0
$291$	−16.0383	−0.940183
$292$	−20.0383	−1.17265
$293$	25.4983	1.48963	0.744813	−	0.667273i	$-0.232539\pi$
$293$	25.4983	1.48963	0.744813	+	0.667273i	$0.232539\pi$
$294$	1.13752	0.0663414
$295$	0	0
$296$	−4.78389	−0.278058
$297$	−5.39194	−0.312872
$298$	−15.0050	−0.869218
$299$	−27.0872	−1.56649
$300$	0	0
$301$	1.67107	0.0963186
$302$	−32.2439	−1.85543
$303$	0.205550	0.0118085
$304$	0.411100	0.0235782
$305$	0	0
$306$	4.05390	0.231746
$307$	−10.2494	−0.584964	−0.292482	−	0.956271i	$-0.594481\pi$
$307$	−10.2494	−0.584964	−0.292482	+	0.956271i	$0.594481\pi$
$308$	−17.5477	−0.999872
$309$	−4.47054	−0.254320
$310$	0	0
$311$	−6.84333	−0.388049	−0.194025	−	0.980997i	$-0.562154\pi$
$311$	−6.84333	−0.388049	−0.194025	+	0.980997i	$0.562154\pi$
$312$	7.62721	0.431806
$313$	13.4983	0.762968	0.381484	−	0.924375i	$-0.375413\pi$
$313$	13.4983	0.762968	0.381484	+	0.924375i	$0.375413\pi$
$314$	−12.4806	−0.704320
$315$	0	0
$316$	−13.2645	−0.746185
$317$	11.9108	0.668980	0.334490	−	0.942399i	$-0.391436\pi$
$317$	11.9108	0.668980	0.334490	+	0.942399i	$0.391436\pi$
$318$	19.3083	1.08276
$319$	15.6217	0.874646
$320$	0	0
$321$	2.60806	0.145568
$322$	20.9612	1.16812
$323$	0.186910	0.0103999
$324$	1.28917	0.0716205
$325$	0	0
$326$	39.7733	2.20284
$327$	−6.57834	−0.363783
$328$	−7.73501	−0.427095
$329$	0	0
$330$	0	0
$331$	1.37636	0.0756515	0.0378257	−	0.999284i	$-0.487957\pi$
$331$	1.37636	0.0756515	0.0378257	+	0.999284i	$0.487957\pi$
$332$	19.0589	1.04599
$333$	−3.71083	−0.203352
$334$	−26.1900	−1.43305
$335$	0	0
$336$	−12.4111	−0.677081
$337$	8.45641	0.460650	0.230325	−	0.973114i	$-0.426021\pi$
$337$	8.45641	0.460650	0.230325	+	0.973114i	$0.426021\pi$
$338$	−39.9058	−2.17059
$339$	−5.10278	−0.277145
$340$	0	0
$341$	−11.8922	−0.643999
$342$	0.151651	0.00820034
$343$	−19.2544	−1.03964
$344$	0.853372	0.0460107
$345$	0	0
$346$	9.83276	0.528613
$347$	−15.6811	−0.841806	−0.420903	−	0.907106i	$-0.638287\pi$
$347$	−15.6811	−0.841806	−0.420903	+	0.907106i	$0.638287\pi$
$348$	−3.73501	−0.200218
$349$	32.8277	1.75723	0.878614	−	0.477532i	$-0.158469\pi$
$349$	32.8277	1.75723	0.878614	+	0.477532i	$0.158469\pi$
$350$	0	0
$351$	5.91638	0.315793
$352$	−34.1744	−1.82150
$353$	11.8867	0.632663	0.316332	−	0.948649i	$-0.397549\pi$
$353$	11.8867	0.632663	0.316332	+	0.948649i	$0.397549\pi$
$354$	−1.81361	−0.0963921
$355$	0	0
$356$	13.1184	0.695272
$357$	−5.64280	−0.298649
$358$	33.4444	1.76759
$359$	−0.945585	−0.0499060	−0.0249530	−	0.999689i	$-0.507944\pi$
$359$	−0.945585	−0.0499060	−0.0249530	+	0.999689i	$0.507944\pi$
$360$	0	0
$361$	−18.9930	−0.999632
$362$	−40.1461	−2.11003
$363$	18.0731	0.948589
$364$	19.2544	1.00921
$365$	0	0
$366$	25.3622	1.32570
$367$	−2.22973	−0.116391	−0.0581955	−	0.998305i	$-0.518535\pi$
$367$	−2.22973	−0.116391	−0.0581955	+	0.998305i	$0.518535\pi$
$368$	22.5089	1.17336
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−26.8761	−1.39534
$372$	2.84333	0.147420
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−21.8584	−1.13027
$375$	0	0
$376$	0	0
$377$	−17.1411	−0.882811
$378$	−4.57834	−0.235484
$379$	−10.5330	−0.541046	−0.270523	−	0.962714i	$-0.587197\pi$
$379$	−10.5330	−0.541046	−0.270523	+	0.962714i	$0.587197\pi$
$380$	0	0
$381$	17.4600	0.894502
$382$	19.6811	1.00697
$383$	−28.3119	−1.44667	−0.723335	−	0.690497i	$-0.757392\pi$
$383$	−28.3119	−1.44667	−0.723335	+	0.690497i	$0.757392\pi$
$384$	−9.66196	−0.493060
$385$	0	0
$386$	−34.5472	−1.75840
$387$	0.661956	0.0336491
$388$	−20.6761	−1.04967
$389$	−4.82774	−0.244776	−0.122388	−	0.992482i	$-0.539055\pi$
$389$	−4.82774	−0.244776	−0.122388	+	0.992482i	$0.539055\pi$
$390$	0	0
$391$	10.2338	0.517546
$392$	−0.808583	−0.0408396
$393$	−20.0680	−1.01230
$394$	−17.9561	−0.904618
$395$	0	0
$396$	−6.95112	−0.349307
$397$	−13.1708	−0.661024	−0.330512	−	0.943802i	$-0.607221\pi$
$397$	−13.1708	−0.661024	−0.330512	+	0.943802i	$0.607221\pi$
$398$	−10.9073	−0.546732
$399$	−0.211090	−0.0105677
$400$	0	0
$401$	−18.5189	−0.924790	−0.462395	−	0.886674i	$-0.653010\pi$
$401$	−18.5189	−0.924790	−0.462395	+	0.886674i	$0.653010\pi$
$402$	−19.5139	−0.973264
$403$	13.0489	0.650011
$404$	0.264989	0.0131837
$405$	0	0
$406$	13.2645	0.658305
$407$	20.0086	0.991789
$408$	−2.88164	−0.142662
$409$	32.1305	1.58875	0.794376	−	0.607426i	$-0.207798\pi$
$409$	32.1305	1.58875	0.794376	+	0.607426i	$0.207798\pi$
$410$	0	0
$411$	7.45998	0.367973
$412$	−5.76328	−0.283936
$413$	2.52444	0.124219
$414$	8.30330	0.408085
$415$	0	0
$416$	37.4983	1.83850
$417$	1.21611	0.0595533
$418$	−0.817693	−0.0399947
$419$	37.8313	1.84818	0.924090	−	0.382174i	$-0.124825\pi$
$419$	37.8313	1.84818	0.924090	+	0.382174i	$0.124825\pi$
$420$	0	0
$421$	6.51890	0.317712	0.158856	−	0.987302i	$-0.449219\pi$
$421$	6.51890	0.317712	0.158856	+	0.987302i	$0.449219\pi$
$422$	19.3905	0.943914
$423$	0	0
$424$	−13.7250	−0.666543
$425$	0	0
$426$	11.2544	0.545279
$427$	−35.3028	−1.70842
$428$	3.36222	0.162519
$429$	−31.9008	−1.54018
$430$	0	0
$431$	5.97028	0.287578	0.143789	−	0.989608i	$-0.454071\pi$
$431$	5.97028	0.287578	0.143789	+	0.989608i	$0.454071\pi$
$432$	−4.91638	−0.236540
$433$	−13.9406	−0.669941	−0.334970	−	0.942229i	$-0.608726\pi$
$433$	−13.9406	−0.669941	−0.334970	+	0.942229i	$0.608726\pi$
$434$	−10.0978	−0.484708
$435$	0	0
$436$	−8.48059	−0.406146
$437$	0.382833	0.0183134
$438$	28.1900	1.34697
$439$	−1.65139	−0.0788167	−0.0394083	−	0.999223i	$-0.512547\pi$
$439$	−1.65139	−0.0788167	−0.0394083	+	0.999223i	$0.512547\pi$
$440$	0	0
$441$	−0.627213	−0.0298673
$442$	23.9844	1.14082
$443$	12.4705	0.592493	0.296247	−	0.955111i	$-0.404265\pi$
$443$	12.4705	0.592493	0.296247	+	0.955111i	$0.404265\pi$
$444$	−4.78389	−0.227033
$445$	0	0
$446$	44.5472	2.10937
$447$	8.27358	0.391327
$448$	−4.19550	−0.198219
$449$	−22.8661	−1.07912	−0.539558	−	0.841949i	$-0.681409\pi$
$449$	−22.8661	−1.07912	−0.539558	+	0.841949i	$0.681409\pi$
$450$	0	0
$451$	32.3517	1.52338
$452$	−6.57834	−0.309419
$453$	17.7789	0.835324
$454$	−12.3828	−0.581155
$455$	0	0
$456$	−0.107798	−0.00504812
$457$	5.64135	0.263891	0.131946	−	0.991257i	$-0.457878\pi$
$457$	5.64135	0.263891	0.131946	+	0.991257i	$0.457878\pi$
$458$	24.5089	1.14522
$459$	−2.23527	−0.104333
$460$	0	0
$461$	−37.3905	−1.74145	−0.870724	−	0.491771i	$-0.836350\pi$
$461$	−37.3905	−1.74145	−0.870724	+	0.491771i	$0.836350\pi$
$462$	24.6861	1.14850
$463$	−1.73501	−0.0806328	−0.0403164	−	0.999187i	$-0.512837\pi$
$463$	−1.73501	−0.0806328	−0.0403164	+	0.999187i	$0.512837\pi$
$464$	14.2439	0.661255
$465$	0	0
$466$	−6.88164	−0.318786
$467$	−31.0433	−1.43651	−0.718257	−	0.695778i	$-0.755060\pi$
$467$	−31.0433	−1.43651	−0.718257	+	0.695778i	$0.755060\pi$
$468$	7.62721	0.352568
$469$	27.1622	1.25423
$470$	0	0
$471$	6.88164	0.317089
$472$	1.28917	0.0593388
$473$	−3.56923	−0.164113
$474$	18.6605	0.857105
$475$	0	0
$476$	−7.27452	−0.333427
$477$	−10.6464	−0.487464
$478$	6.67609	0.305357
$479$	10.4494	0.477446	0.238723	−	0.971088i	$-0.423271\pi$
$479$	10.4494	0.477446	0.238723	+	0.971088i	$0.423271\pi$
$480$	0	0
$481$	−21.9547	−1.00105
$482$	−0.719942	−0.0327924
$483$	−11.5577	−0.525895
$484$	23.2992	1.05906
$485$	0	0
$486$	−1.81361	−0.0822669
$487$	37.2333	1.68720	0.843601	−	0.536971i	$-0.180431\pi$
$487$	37.2333	1.68720	0.843601	+	0.536971i	$0.180431\pi$
$488$	−18.0283	−0.816101
$489$	−21.9305	−0.991732
$490$	0	0
$491$	−8.78943	−0.396661	−0.198331	−	0.980135i	$-0.563552\pi$
$491$	−8.78943	−0.396661	−0.198331	+	0.980135i	$0.563552\pi$
$492$	−7.73501	−0.348721
$493$	6.47608	0.291668
$494$	0.897225	0.0403681
$495$	0	0
$496$	−10.8433	−0.486880
$497$	−15.6655	−0.702695
$498$	−26.8122	−1.20148
$499$	−16.2439	−0.727175	−0.363588	−	0.931560i	$-0.618448\pi$
$499$	−16.2439	−0.727175	−0.363588	+	0.931560i	$0.618448\pi$
$500$	0	0
$501$	14.4408	0.645168
$502$	43.9688	1.96242
$503$	−7.05442	−0.314541	−0.157270	−	0.987556i	$-0.550269\pi$
$503$	−7.05442	−0.314541	−0.157270	+	0.987556i	$0.550269\pi$
$504$	3.25443	0.144964
$505$	0	0
$506$	−44.7709	−1.99031
$507$	22.0036	0.977213
$508$	22.5089	0.998669
$509$	−5.07860	−0.225105	−0.112552	−	0.993646i	$-0.535903\pi$
$509$	−5.07860	−0.225105	−0.112552	+	0.993646i	$0.535903\pi$
$510$	0	0
$511$	−39.2388	−1.73582
$512$	−18.4842	−0.816892
$513$	−0.0836184	−0.00369184
$514$	6.70172	0.295600
$515$	0	0
$516$	0.853372	0.0375676
$517$	0	0
$518$	16.9894	0.746473
$519$	−5.42166	−0.237985
$520$	0	0
$521$	3.88666	0.170278	0.0851389	−	0.996369i	$-0.472867\pi$
$521$	3.88666	0.170278	0.0851389	+	0.996369i	$0.472867\pi$
$522$	5.25443	0.229980
$523$	10.0383	0.438945	0.219472	−	0.975619i	$-0.429566\pi$
$523$	10.0383	0.438945	0.219472	+	0.975619i	$0.429566\pi$
$524$	−25.8711	−1.13018
$525$	0	0
$526$	−41.5522	−1.81176
$527$	−4.93000	−0.214754
$528$	26.5089	1.15365
$529$	−2.03883	−0.0886448
$530$	0	0
$531$	1.00000	0.0433963
$532$	−0.272130	−0.0117983
$533$	−35.4983	−1.53760
$534$	−18.4550	−0.798624
$535$	0	0
$536$	13.8711	0.599139
$537$	−18.4408	−0.795780
$538$	−10.3472	−0.446098
$539$	3.38190	0.145669
$540$	0	0
$541$	−8.03831	−0.345594	−0.172797	−	0.984957i	$-0.555280\pi$
$541$	−8.03831	−0.345594	−0.172797	+	0.984957i	$0.555280\pi$
$542$	26.0100	1.11723
$543$	22.1361	0.949949
$544$	−14.1672	−0.607415
$545$	0	0
$546$	−27.0872	−1.15922
$547$	−33.0661	−1.41380	−0.706901	−	0.707312i	$-0.749908\pi$
$547$	−33.0661	−1.41380	−0.706901	+	0.707312i	$0.749908\pi$
$548$	9.61717	0.410825
$549$	−13.9844	−0.596840
$550$	0	0
$551$	0.242261	0.0103207
$552$	−5.90225	−0.251216
$553$	−25.9744	−1.10454
$554$	53.0278	2.25293
$555$	0	0
$556$	1.56777	0.0664884
$557$	7.58745	0.321490	0.160745	−	0.986996i	$-0.448610\pi$
$557$	7.58745	0.321490	0.160745	+	0.986996i	$0.448610\pi$
$558$	−4.00000	−0.169334
$559$	3.91638	0.165645
$560$	0	0
$561$	12.0524	0.508855
$562$	29.6655	1.25136
$563$	−28.9583	−1.22045	−0.610223	−	0.792230i	$-0.708920\pi$
$563$	−28.9583	−1.22045	−0.610223	+	0.792230i	$0.708920\pi$
$564$	0	0
$565$	0	0
$566$	−4.10780	−0.172664
$567$	2.52444	0.106016
$568$	−8.00000	−0.335673
$569$	25.5577	1.07144	0.535718	−	0.844397i	$-0.320041\pi$
$569$	25.5577	1.07144	0.535718	+	0.844397i	$0.320041\pi$
$570$	0	0
$571$	−15.6655	−0.655582	−0.327791	−	0.944750i	$-0.606304\pi$
$571$	−15.6655	−0.655582	−0.327791	+	0.944750i	$0.606304\pi$
$572$	−41.1255	−1.71954
$573$	−10.8519	−0.453345
$574$	27.4700	1.14658
$575$	0	0
$576$	−1.66196	−0.0692481
$577$	−8.84333	−0.368152	−0.184076	−	0.982912i	$-0.558929\pi$
$577$	−8.84333	−0.368152	−0.184076	+	0.982912i	$0.558929\pi$
$578$	21.7698	0.905502
$579$	19.0489	0.791644
$580$	0	0
$581$	37.3210	1.54834
$582$	29.0872	1.20570
$583$	57.4046	2.37746
$584$	−20.0383	−0.829191
$585$	0	0
$586$	−46.2439	−1.91032
$587$	−28.9894	−1.19652	−0.598261	−	0.801301i	$-0.704141\pi$
$587$	−28.9894	−1.19652	−0.598261	+	0.801301i	$0.704141\pi$
$588$	−0.808583	−0.0333454
$589$	−0.184425	−0.00759909
$590$	0	0
$591$	9.90080	0.407264
$592$	18.2439	0.749818
$593$	8.85192	0.363505	0.181752	−	0.983344i	$-0.441823\pi$
$593$	8.85192	0.363505	0.181752	+	0.983344i	$0.441823\pi$
$594$	9.77886	0.401232
$595$	0	0
$596$	10.6660	0.436898
$597$	6.01413	0.246142
$598$	49.1255	2.00889
$599$	−25.5522	−1.04403	−0.522017	−	0.852935i	$-0.674820\pi$
$599$	−25.5522	−1.04403	−0.522017	+	0.852935i	$0.674820\pi$
$600$	0	0
$601$	5.21057	0.212544	0.106272	−	0.994337i	$-0.466109\pi$
$601$	5.21057	0.212544	0.106272	+	0.994337i	$0.466109\pi$
$602$	−3.03066	−0.123520
$603$	10.7597	0.438169
$604$	22.9200	0.932600
$605$	0	0
$606$	−0.372787	−0.0151434
$607$	−47.6655	−1.93468	−0.967342	−	0.253475i	$-0.918427\pi$
$607$	−47.6655	−1.93468	−0.967342	+	0.253475i	$0.918427\pi$
$608$	−0.529977	−0.0214934
$609$	−7.31386	−0.296373
$610$	0	0
$611$	0	0
$612$	−2.88164	−0.116483
$613$	−11.7350	−0.473973	−0.236986	−	0.971513i	$-0.576160\pi$
$613$	−11.7350	−0.473973	−0.236986	+	0.971513i	$0.576160\pi$
$614$	18.5884	0.750166
$615$	0	0
$616$	−17.5477	−0.707016
$617$	−11.5592	−0.465355	−0.232678	−	0.972554i	$-0.574749\pi$
$617$	−11.5592	−0.465355	−0.232678	+	0.972554i	$0.574749\pi$
$618$	8.10780	0.326143
$619$	−34.3658	−1.38128	−0.690639	−	0.723200i	$-0.742671\pi$
$619$	−34.3658	−1.38128	−0.690639	+	0.723200i	$0.742671\pi$
$620$	0	0
$621$	−4.57834	−0.183722
$622$	12.4111	0.497640
$623$	25.6883	1.02918
$624$	−29.0872	−1.16442
$625$	0	0
$626$	−24.4806	−0.978441
$627$	0.450866	0.0180059
$628$	8.87159	0.354015
$629$	8.29471	0.330732
$630$	0	0
$631$	−34.3064	−1.36571	−0.682857	−	0.730552i	$-0.739263\pi$
$631$	−34.3064	−1.36571	−0.682857	+	0.730552i	$0.739263\pi$
$632$	−13.2645	−0.527632
$633$	−10.6917	−0.424956
$634$	−21.6016	−0.857908
$635$	0	0
$636$	−13.7250	−0.544230
$637$	−3.71083	−0.147028
$638$	−28.3316	−1.12166
$639$	−6.20555	−0.245488
$640$	0	0
$641$	−30.8222	−1.21740	−0.608702	−	0.793399i	$-0.708309\pi$
$641$	−30.8222	−1.21740	−0.608702	+	0.793399i	$0.708309\pi$
$642$	−4.72999	−0.186678
$643$	−32.6222	−1.28649	−0.643247	−	0.765659i	$-0.722413\pi$
$643$	−32.6222	−1.28649	−0.643247	+	0.765659i	$0.722413\pi$
$644$	−14.8999	−0.587137
$645$	0	0
$646$	−0.338981	−0.0133370
$647$	−45.3013	−1.78098	−0.890490	−	0.455004i	$-0.849638\pi$
$647$	−45.3013	−1.78098	−0.890490	+	0.455004i	$0.849638\pi$
$648$	1.28917	0.0506433
$649$	−5.39194	−0.211652
$650$	0	0
$651$	5.56777	0.218218
$652$	−28.2721	−1.10722
$653$	32.7441	1.28138	0.640688	−	0.767801i	$-0.278649\pi$
$653$	32.7441	1.28138	0.640688	+	0.767801i	$0.278649\pi$
$654$	11.9305	0.466520
$655$	0	0
$656$	29.4983	1.15171
$657$	−15.5436	−0.606413
$658$	0	0
$659$	−24.1758	−0.941757	−0.470878	−	0.882198i	$-0.656063\pi$
$659$	−24.1758	−0.941757	−0.470878	+	0.882198i	$0.656063\pi$
$660$	0	0
$661$	−20.9441	−0.814632	−0.407316	−	0.913287i	$-0.633535\pi$
$661$	−20.9441	−0.814632	−0.407316	+	0.913287i	$0.633535\pi$
$662$	−2.49617	−0.0970164
$663$	−13.2247	−0.513605
$664$	19.0589	0.739630
$665$	0	0
$666$	6.72999	0.260782
$667$	13.2645	0.513602
$668$	18.6167	0.720300
$669$	−24.5628	−0.949651
$670$	0	0
$671$	75.4032	2.91091
$672$	16.0000	0.617213
$673$	37.1638	1.43256	0.716280	−	0.697813i	$-0.245843\pi$
$673$	37.1638	1.43256	0.716280	+	0.697813i	$0.245843\pi$
$674$	−15.3366	−0.590743
$675$	0	0
$676$	28.3663	1.09101
$677$	9.96883	0.383133	0.191567	−	0.981480i	$-0.438643\pi$
$677$	9.96883	0.383133	0.191567	+	0.981480i	$0.438643\pi$
$678$	9.25443	0.355414
$679$	−40.4877	−1.55378
$680$	0	0
$681$	6.82774	0.261640
$682$	21.5678	0.825873
$683$	19.1511	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$683$	19.1511	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$684$	−0.107798	−0.00412177
$685$	0	0
$686$	34.9200	1.33325
$687$	−13.5139	−0.515586
$688$	−3.25443	−0.124074
$689$	−62.9880	−2.39965
$690$	0	0
$691$	−2.18996	−0.0833102	−0.0416551	−	0.999132i	$-0.513263\pi$
$691$	−2.18996	−0.0833102	−0.0416551	+	0.999132i	$0.513263\pi$
$692$	−6.98944	−0.265699
$693$	−13.6116	−0.517063
$694$	28.4394	1.07954
$695$	0	0
$696$	−3.73501	−0.141575
$697$	13.4116	0.508001
$698$	−59.5366	−2.25349
$699$	3.79445	0.143519
$700$	0	0
$701$	−14.7058	−0.555431	−0.277715	−	0.960663i	$-0.589577\pi$
$701$	−14.7058	−0.555431	−0.277715	+	0.960663i	$0.589577\pi$
$702$	−10.7300	−0.404977
$703$	0.310294	0.0117030
$704$	8.96117	0.337737
$705$	0	0
$706$	−21.5577	−0.811336
$707$	0.518898	0.0195152
$708$	1.28917	0.0484499
$709$	6.32748	0.237634	0.118817	−	0.992916i	$-0.462090\pi$
$709$	6.32748	0.237634	0.118817	+	0.992916i	$0.462090\pi$
$710$	0	0
$711$	−10.2892	−0.385874
$712$	13.1184	0.491631
$713$	−10.0978	−0.378164
$714$	10.2338	0.382991
$715$	0	0
$716$	−23.7733	−0.888451
$717$	−3.68111	−0.137474
$718$	1.71492	0.0640002
$719$	25.9985	0.969582	0.484791	−	0.874630i	$-0.338896\pi$
$719$	25.9985	0.969582	0.484791	+	0.874630i	$0.338896\pi$
$720$	0	0
$721$	−11.2856	−0.420298
$722$	34.4458	1.28194
$723$	0.396967	0.0147634
$724$	28.5371	1.06057
$725$	0	0
$726$	−32.7774	−1.21648
$727$	48.0172	1.78086	0.890429	−	0.455121i	$-0.150404\pi$
$727$	48.0172	1.78086	0.890429	+	0.455121i	$0.150404\pi$
$728$	19.2544	0.713617
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.47965	−0.0547268
$732$	−18.0283	−0.666344
$733$	−22.8222	−0.842957	−0.421479	−	0.906838i	$-0.638489\pi$
$733$	−22.8222	−0.842957	−0.421479	+	0.906838i	$0.638489\pi$
$734$	4.04385	0.149261
$735$	0	0
$736$	−29.0177	−1.06961
$737$	−58.0157	−2.13704
$738$	10.8816	0.400559
$739$	48.9583	1.80096	0.900479	−	0.434899i	$-0.143216\pi$
$739$	48.9583	1.80096	0.900479	+	0.434899i	$0.143216\pi$
$740$	0	0
$741$	−0.494719	−0.0181739
$742$	48.7427	1.78940
$743$	−51.8414	−1.90187	−0.950937	−	0.309383i	$-0.899877\pi$
$743$	−51.8414	−1.90187	−0.950937	+	0.309383i	$0.899877\pi$
$744$	2.84333	0.104241
$745$	0	0
$746$	−18.1361	−0.664009
$747$	14.7839	0.540914
$748$	15.5376	0.568112
$749$	6.58388	0.240570
$750$	0	0
$751$	4.94108	0.180302	0.0901512	−	0.995928i	$-0.471265\pi$
$751$	4.94108	0.180302	0.0901512	+	0.995928i	$0.471265\pi$
$752$	0	0
$753$	−24.2439	−0.883495
$754$	31.0872	1.13213
$755$	0	0
$756$	3.25443	0.118362
$757$	15.6428	0.568547	0.284274	−	0.958743i	$-0.408248\pi$
$757$	15.6428	0.568547	0.284274	+	0.958743i	$0.408248\pi$
$758$	19.1028	0.693844
$759$	24.6861	0.896050
$760$	0	0
$761$	−26.1517	−0.947997	−0.473998	−	0.880526i	$-0.657190\pi$
$761$	−26.1517	−0.947997	−0.473998	+	0.880526i	$0.657190\pi$
$762$	−31.6655	−1.14712
$763$	−16.6066	−0.601199
$764$	−13.9900	−0.506139
$765$	0	0
$766$	51.3466	1.85523
$767$	5.91638	0.213628
$768$	20.8469	0.752248
$769$	36.1900	1.30504	0.652522	−	0.757770i	$-0.273711\pi$
$769$	36.1900	1.30504	0.652522	+	0.757770i	$0.273711\pi$
$770$	0	0
$771$	−3.69525	−0.133081
$772$	24.5572	0.883833
$773$	6.42669	0.231152	0.115576	−	0.993299i	$-0.463129\pi$
$773$	6.42669	0.231152	0.115576	+	0.993299i	$0.463129\pi$
$774$	−1.20053	−0.0431520
$775$	0	0
$776$	−20.6761	−0.742228
$777$	−9.36776	−0.336067
$778$	8.75562	0.313904
$779$	0.501711	0.0179757
$780$	0	0
$781$	33.4600	1.19729
$782$	−18.5601	−0.663708
$783$	−2.89722	−0.103538
$784$	3.08362	0.110129
$785$	0	0
$786$	36.3955	1.29819
$787$	51.2388	1.82647	0.913234	−	0.407436i	$-0.133577\pi$
$787$	51.2388	1.82647	0.913234	+	0.407436i	$0.133577\pi$
$788$	12.7638	0.454691
$789$	22.9114	0.815666
$790$	0	0
$791$	−12.8816	−0.458018
$792$	−6.95112	−0.246997
$793$	−82.7371	−2.93808
$794$	23.8867	0.847706
$795$	0	0
$796$	7.75323	0.274806
$797$	53.6344	1.89983	0.949913	−	0.312514i	$-0.101171\pi$
$797$	53.6344	1.89983	0.949913	+	0.312514i	$0.101171\pi$
$798$	0.382833	0.0135522
$799$	0	0
$800$	0	0
$801$	10.1758	0.359545
$802$	33.5860	1.18596
$803$	83.8102	2.95760
$804$	13.8711	0.489195
$805$	0	0
$806$	−23.6655	−0.833583
$807$	5.70529	0.200836
$808$	0.264989	0.00932227
$809$	44.0071	1.54721	0.773604	−	0.633669i	$-0.218452\pi$
$809$	44.0071	1.54721	0.773604	+	0.633669i	$0.218452\pi$
$810$	0	0
$811$	6.67609	0.234429	0.117215	−	0.993107i	$-0.462603\pi$
$811$	6.67609	0.234429	0.117215	+	0.993107i	$0.462603\pi$
$812$	−9.42880	−0.330886
$813$	−14.3416	−0.502982
$814$	−36.2877	−1.27188
$815$	0	0
$816$	10.9894	0.384707
$817$	−0.0553517	−0.00193651
$818$	−58.2721	−2.03744
$819$	14.9355	0.521890
$820$	0	0
$821$	34.8222	1.21530	0.607652	−	0.794204i	$-0.292112\pi$
$821$	34.8222	1.21530	0.607652	+	0.794204i	$0.292112\pi$
$822$	−13.5295	−0.471894
$823$	−10.0625	−0.350756	−0.175378	−	0.984501i	$-0.556115\pi$
$823$	−10.0625	−0.350756	−0.175378	+	0.984501i	$0.556115\pi$
$824$	−5.76328	−0.200773
$825$	0	0
$826$	−4.57834	−0.159301
$827$	−45.1255	−1.56917	−0.784584	−	0.620023i	$-0.787123\pi$
$827$	−45.1255	−1.56917	−0.784584	+	0.620023i	$0.787123\pi$
$828$	−5.90225	−0.205117
$829$	54.1119	1.87938	0.939692	−	0.342023i	$-0.111112\pi$
$829$	54.1119	1.87938	0.939692	+	0.342023i	$0.111112\pi$
$830$	0	0
$831$	−29.2388	−1.01428
$832$	−9.83276	−0.340890
$833$	1.40199	0.0485761
$834$	−2.20555	−0.0763720
$835$	0	0
$836$	0.581242	0.0201027
$837$	2.20555	0.0762350
$838$	−68.6111	−2.37013
$839$	−5.14808	−0.177731	−0.0888657	−	0.996044i	$-0.528324\pi$
$839$	−5.14808	−0.177731	−0.0888657	+	0.996044i	$0.528324\pi$
$840$	0	0
$841$	−20.6061	−0.710555
$842$	−11.8227	−0.407438
$843$	−16.3572	−0.563372
$844$	−13.7834	−0.474443
$845$	0	0
$846$	0	0
$847$	45.6243	1.56767
$848$	52.3416	1.79742
$849$	2.26499	0.0777342
$850$	0	0
$851$	16.9894	0.582390
$852$	−8.00000	−0.274075
$853$	30.9739	1.06052	0.530262	−	0.847834i	$-0.322093\pi$
$853$	30.9739	1.06052	0.530262	+	0.847834i	$0.322093\pi$
$854$	64.0254	2.19090
$855$	0	0
$856$	3.36222	0.114918
$857$	45.2278	1.54495	0.772475	−	0.635045i	$-0.219018\pi$
$857$	45.2278	1.54495	0.772475	+	0.635045i	$0.219018\pi$
$858$	57.8555	1.97515
$859$	5.38335	0.183678	0.0918388	−	0.995774i	$-0.470726\pi$
$859$	5.38335	0.183678	0.0918388	+	0.995774i	$0.470726\pi$
$860$	0	0
$861$	−15.1466	−0.516196
$862$	−10.8277	−0.368794
$863$	32.9583	1.12191	0.560956	−	0.827845i	$-0.310434\pi$
$863$	32.9583	1.12191	0.560956	+	0.827845i	$0.310434\pi$
$864$	6.33804	0.215625
$865$	0	0
$866$	25.2827	0.859141
$867$	−12.0036	−0.407663
$868$	7.17780	0.243630
$869$	55.4786	1.88198
$870$	0	0
$871$	63.6585	2.15699
$872$	−8.48059	−0.287189
$873$	−16.0383	−0.542815
$874$	−0.694309	−0.0234854
$875$	0	0
$876$	−20.0383	−0.677032
$877$	−27.9461	−0.943673	−0.471836	−	0.881686i	$-0.656409\pi$
$877$	−27.9461	−0.943673	−0.471836	+	0.881686i	$0.656409\pi$
$878$	2.99498	0.101076
$879$	25.4983	0.860036
$880$	0	0
$881$	−14.5300	−0.489527	−0.244764	−	0.969583i	$-0.578710\pi$
$881$	−14.5300	−0.489527	−0.244764	+	0.969583i	$0.578710\pi$
$882$	1.13752	0.0383022
$883$	−18.7995	−0.632653	−0.316326	−	0.948650i	$-0.602449\pi$
$883$	−18.7995	−0.632653	−0.316326	+	0.948650i	$0.602449\pi$
$884$	−17.0489	−0.573416
$885$	0	0
$886$	−22.6167	−0.759821
$887$	14.1305	0.474457	0.237228	−	0.971454i	$-0.423761\pi$
$887$	14.1305	0.474457	0.237228	+	0.971454i	$0.423761\pi$
$888$	−4.78389	−0.160537
$889$	44.0766	1.47828
$890$	0	0
$891$	−5.39194	−0.180637
$892$	−31.6655	−1.06024
$893$	0	0
$894$	−15.0050	−0.501843
$895$	0	0
$896$	−24.3910	−0.814846
$897$	−27.0872	−0.904415
$898$	41.4700	1.38387
$899$	−6.38997	−0.213118
$900$	0	0
$901$	23.7975	0.792810
$902$	−58.6732	−1.95360
$903$	1.67107	0.0556096
$904$	−6.57834	−0.218792
$905$	0	0
$906$	−32.2439	−1.07123
$907$	23.4217	0.777704	0.388852	−	0.921300i	$-0.372872\pi$
$907$	23.4217	0.777704	0.388852	+	0.921300i	$0.372872\pi$
$908$	8.80211	0.292108
$909$	0.205550	0.00681767
$910$	0	0
$911$	53.0661	1.75816	0.879079	−	0.476677i	$-0.158159\pi$
$911$	53.0661	1.75816	0.879079	+	0.476677i	$0.158159\pi$
$912$	0.411100	0.0136129
$913$	−79.7139	−2.63814
$914$	−10.2312	−0.338417
$915$	0	0
$916$	−17.4217	−0.575628
$917$	−50.6605	−1.67296
$918$	4.05390	0.133799
$919$	−48.0455	−1.58487	−0.792437	−	0.609954i	$-0.791188\pi$
$919$	−48.0455	−1.58487	−0.792437	+	0.609954i	$0.791188\pi$
$920$	0	0
$921$	−10.2494	−0.337729
$922$	67.8116	2.23326
$923$	−36.7144	−1.20847
$924$	−17.5477	−0.577276
$925$	0	0
$926$	3.14663	0.103405
$927$	−4.47054	−0.146832
$928$	−18.3627	−0.602786
$929$	33.5577	1.10099	0.550497	−	0.834837i	$-0.314438\pi$
$929$	33.5577	1.10099	0.550497	+	0.834837i	$0.314438\pi$
$930$	0	0
$931$	0.0524466	0.00171887
$932$	4.89169	0.160232
$933$	−6.84333	−0.224040
$934$	56.3004	1.84221
$935$	0	0
$936$	7.62721	0.249303
$937$	19.3522	0.632208	0.316104	−	0.948725i	$-0.397625\pi$
$937$	19.3522	0.632208	0.316104	+	0.948725i	$0.397625\pi$
$938$	−49.2616	−1.60845
$939$	13.4983	0.440500
$940$	0	0
$941$	53.2913	1.73725	0.868623	−	0.495473i	$-0.165005\pi$
$941$	53.2913	1.73725	0.868623	+	0.495473i	$0.165005\pi$
$942$	−12.4806	−0.406639
$943$	27.4700	0.894547
$944$	−4.91638	−0.160015
$945$	0	0
$946$	6.47317	0.210461
$947$	−45.0575	−1.46417	−0.732086	−	0.681213i	$-0.761453\pi$
$947$	−45.0575	−1.46417	−0.732086	+	0.681213i	$0.761453\pi$
$948$	−13.2645	−0.430810
$949$	−91.9618	−2.98521
$950$	0	0
$951$	11.9108	0.386236
$952$	−7.27452	−0.235769
$953$	17.5209	0.567557	0.283778	−	0.958890i	$-0.408412\pi$
$953$	17.5209	0.567557	0.283778	+	0.958890i	$0.408412\pi$
$954$	19.3083	0.625130
$955$	0	0
$956$	−4.74557	−0.153483
$957$	15.6217	0.504977
$958$	−18.9511	−0.612283
$959$	18.8322	0.608125
$960$	0	0
$961$	−26.1355	−0.843082
$962$	39.8172	1.28376
$963$	2.60806	0.0840434
$964$	0.511757	0.0164826
$965$	0	0
$966$	20.9612	0.674415
$967$	−37.1013	−1.19310	−0.596549	−	0.802577i	$-0.703462\pi$
$967$	−37.1013	−1.19310	−0.596549	+	0.802577i	$0.703462\pi$
$968$	23.2992	0.748865
$969$	0.186910	0.00600441
$970$	0	0
$971$	−5.24438	−0.168300	−0.0841501	−	0.996453i	$-0.526818\pi$
$971$	−5.24438	−0.168300	−0.0841501	+	0.996453i	$0.526818\pi$
$972$	1.28917	0.0413501
$973$	3.07000	0.0984197
$974$	−67.5266	−2.16369
$975$	0	0
$976$	68.7527	2.20072
$977$	3.50834	0.112242	0.0561208	−	0.998424i	$-0.482127\pi$
$977$	3.50834	0.112242	0.0561208	+	0.998424i	$0.482127\pi$
$978$	39.7733	1.27181
$979$	−54.8675	−1.75357
$980$	0	0
$981$	−6.57834	−0.210030
$982$	15.9406	0.508684
$983$	−35.9094	−1.14533	−0.572666	−	0.819789i	$-0.694091\pi$
$983$	−35.9094	−1.14533	−0.572666	+	0.819789i	$0.694091\pi$
$984$	−7.73501	−0.246583
$985$	0	0
$986$	−11.7451	−0.374039
$987$	0	0
$988$	−0.637776	−0.0202903
$989$	−3.03066	−0.0963692
$990$	0	0
$991$	12.1672	0.386505	0.193253	−	0.981149i	$-0.438096\pi$
$991$	12.1672	0.386505	0.193253	+	0.981149i	$0.438096\pi$
$992$	13.9789	0.443830
$993$	1.37636	0.0436774
$994$	28.4111	0.901145
$995$	0	0
$996$	19.0589	0.603905
$997$	−55.7321	−1.76505	−0.882527	−	0.470262i	$-0.844159\pi$
$997$	−55.7321	−1.76505	−0.882527	+	0.470262i	$0.844159\pi$
$998$	29.4600	0.932539
$999$	−3.71083	−0.117406

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4425.2.a.x.1.1		3
5.4	even	2		885.2.a.g.1.3	✓	3
15.14	odd	2		2655.2.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
885.2.a.g.1.3	✓	3	5.4	even	2
2655.2.a.p.1.1		3	15.14	odd	2
4425.2.a.x.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4425 = 3 \cdot 5^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4425.a (trivial)

\(f(q)\)	\(=\)	\(q-1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} -1.81361 q^{6} +2.52444 q^{7} +1.28917 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} -1.81361 q^{6} +2.52444 q^{7} +1.28917 q^{8} +1.00000 q^{9} -5.39194 q^{11} +1.28917 q^{12} +5.91638 q^{13} -4.57834 q^{14} -4.91638 q^{16} -2.23527 q^{17} -1.81361 q^{18} -0.0836184 q^{19} +2.52444 q^{21} +9.77886 q^{22} -4.57834 q^{23} +1.28917 q^{24} -10.7300 q^{26} +1.00000 q^{27} +3.25443 q^{28} -2.89722 q^{29} +2.20555 q^{31} +6.33804 q^{32} -5.39194 q^{33} +4.05390 q^{34} +1.28917 q^{36} -3.71083 q^{37} +0.151651 q^{38} +5.91638 q^{39} -6.00000 q^{41} -4.57834 q^{42} +0.661956 q^{43} -6.95112 q^{44} +8.30330 q^{46} -4.91638 q^{48} -0.627213 q^{49} -2.23527 q^{51} +7.62721 q^{52} -10.6464 q^{53} -1.81361 q^{54} +3.25443 q^{56} -0.0836184 q^{57} +5.25443 q^{58} +1.00000 q^{59} -13.9844 q^{61} -4.00000 q^{62} +2.52444 q^{63} -1.66196 q^{64} +9.77886 q^{66} +10.7597 q^{67} -2.88164 q^{68} -4.57834 q^{69} -6.20555 q^{71} +1.28917 q^{72} -15.5436 q^{73} +6.72999 q^{74} -0.107798 q^{76} -13.6116 q^{77} -10.7300 q^{78} -10.2892 q^{79} +1.00000 q^{81} +10.8816 q^{82} +14.7839 q^{83} +3.25443 q^{84} -1.20053 q^{86} -2.89722 q^{87} -6.95112 q^{88} +10.1758 q^{89} +14.9355 q^{91} -5.90225 q^{92} +2.20555 q^{93} +6.33804 q^{96} -16.0383 q^{97} +1.13752 q^{98} -5.39194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 + q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} - 8 q^{11} + 3 q^{12} + 4 q^{13} - 12 q^{14} - q^{16} - 2 q^{17} + q^{18} - 14 q^{19} + 2 q^{21} - 2 q^{22} - 12 q^{23} + 3 q^{24} - 12 q^{26} + 3 q^{27} - 16 q^{28} - 16 q^{29} - 8 q^{31} + 7 q^{32} - 8 q^{33} + 16 q^{34} + 3 q^{36} - 12 q^{37} - 18 q^{38} + 4 q^{39} - 18 q^{41} - 12 q^{42} + 14 q^{43} - 32 q^{44} - 12 q^{46} - q^{48} + 11 q^{49} - 2 q^{51} + 10 q^{52} + 2 q^{53} + q^{54} - 16 q^{56} - 14 q^{57} - 10 q^{58} + 3 q^{59} + 4 q^{61} - 12 q^{62} + 2 q^{63} - 17 q^{64} - 2 q^{66} + 22 q^{67} + 30 q^{68} - 12 q^{69} - 4 q^{71} + 3 q^{72} - 20 q^{73} - 8 q^{76} + 18 q^{77} - 12 q^{78} - 30 q^{79} + 3 q^{81} - 6 q^{82} + 28 q^{83} - 16 q^{84} + 26 q^{86} - 16 q^{87} - 32 q^{88} + 6 q^{89} + 10 q^{91} - 40 q^{92} - 8 q^{93} + 7 q^{96} - 6 q^{97} + 21 q^{98} - 8 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 + q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 - 8 * q^11 + 3 * q^12 + 4 * q^13 - 12 * q^14 - q^16 - 2 * q^17 + q^18 - 14 * q^19 + 2 * q^21 - 2 * q^22 - 12 * q^23 + 3 * q^24 - 12 * q^26 + 3 * q^27 - 16 * q^28 - 16 * q^29 - 8 * q^31 + 7 * q^32 - 8 * q^33 + 16 * q^34 + 3 * q^36 - 12 * q^37 - 18 * q^38 + 4 * q^39 - 18 * q^41 - 12 * q^42 + 14 * q^43 - 32 * q^44 - 12 * q^46 - q^48 + 11 * q^49 - 2 * q^51 + 10 * q^52 + 2 * q^53 + q^54 - 16 * q^56 - 14 * q^57 - 10 * q^58 + 3 * q^59 + 4 * q^61 - 12 * q^62 + 2 * q^63 - 17 * q^64 - 2 * q^66 + 22 * q^67 + 30 * q^68 - 12 * q^69 - 4 * q^71 + 3 * q^72 - 20 * q^73 - 8 * q^76 + 18 * q^77 - 12 * q^78 - 30 * q^79 + 3 * q^81 - 6 * q^82 + 28 * q^83 - 16 * q^84 + 26 * q^86 - 16 * q^87 - 32 * q^88 + 6 * q^89 + 10 * q^91 - 40 * q^92 - 8 * q^93 + 7 * q^96 - 6 * q^97 + 21 * q^98 - 8 * q^99

Introduction

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