LMFDB - Embedded newform 4425.2.a.bg.1.5

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.22298624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 8x^{3} + 14x^{2} - 4x - 2 $ x^6 - 2x^5 - 7x^4 + 8x^3 + 14x^2 - 4*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 885)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.45188$ 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)

$f(q)$	$=$	$q+2.45188 q^{2} +1.00000 q^{3} +4.01172 q^{4} +2.45188 q^{6} -2.82843 q^{7} +4.93251 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.45188 q^{2} +1.00000 q^{3} +4.01172 q^{4} +2.45188 q^{6} -2.82843 q^{7} +4.93251 q^{8} +1.00000 q^{9} -1.15357 q^{11} +4.01172 q^{12} +5.14234 q^{13} -6.93497 q^{14} +4.07048 q^{16} +0.846426 q^{17} +2.45188 q^{18} +4.73705 q^{19} -2.82843 q^{21} -2.82843 q^{22} +5.46748 q^{23} +4.93251 q^{24} +12.6084 q^{26} +1.00000 q^{27} -11.3469 q^{28} +3.89062 q^{29} +1.84157 q^{31} +0.115318 q^{32} -1.15357 q^{33} +2.07534 q^{34} +4.01172 q^{36} +5.14234 q^{37} +11.6147 q^{38} +5.14234 q^{39} -2.38845 q^{41} -6.93497 q^{42} -7.32422 q^{43} -4.62782 q^{44} +13.4056 q^{46} +0.839664 q^{47} +4.07048 q^{48} +1.00000 q^{49} +0.846426 q^{51} +20.6296 q^{52} -1.17702 q^{53} +2.45188 q^{54} -13.9512 q^{56} +4.73705 q^{57} +9.53934 q^{58} -1.00000 q^{59} -7.88091 q^{61} +4.51531 q^{62} -2.82843 q^{63} -7.85821 q^{64} -2.82843 q^{66} +14.0461 q^{67} +3.39563 q^{68} +5.46748 q^{69} +1.15089 q^{71} +4.93251 q^{72} -2.83519 q^{73} +12.6084 q^{74} +19.0037 q^{76} +3.26280 q^{77} +12.6084 q^{78} -0.763332 q^{79} +1.00000 q^{81} -5.85620 q^{82} +5.78434 q^{83} -11.3469 q^{84} -17.9581 q^{86} +3.89062 q^{87} -5.69001 q^{88} -5.77364 q^{89} -14.5447 q^{91} +21.9340 q^{92} +1.84157 q^{93} +2.05876 q^{94} +0.115318 q^{96} +15.2985 q^{97} +2.45188 q^{98} -1.15357 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9	$ 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9} - 2 q^{11} + 8 q^{12} - 2 q^{13} + 12 q^{16} + 10 q^{17} + 4 q^{18} - 2 q^{19} + 12 q^{23} + 12 q^{24} - 8 q^{26} + 6 q^{27} + 16 q^{28} - 12 q^{29} + 8 q^{31} + 28 q^{32} - 2 q^{33} + 8 q^{34} + 8 q^{36} - 2 q^{37} + 24 q^{38} - 2 q^{39} - 4 q^{41} + 18 q^{43} + 4 q^{44} + 16 q^{47} + 12 q^{48} + 6 q^{49} + 10 q^{51} - 12 q^{52} + 30 q^{53} + 4 q^{54} - 2 q^{57} + 16 q^{58} - 6 q^{59} + 4 q^{61} + 16 q^{62} + 20 q^{64} + 30 q^{67} + 20 q^{68} + 12 q^{69} - 24 q^{71} + 12 q^{72} + 6 q^{73} - 8 q^{74} - 4 q^{76} + 16 q^{77} - 8 q^{78} - 2 q^{79} + 6 q^{81} + 20 q^{83} + 16 q^{84} - 12 q^{87} + 16 q^{88} + 14 q^{89} - 48 q^{91} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 28 q^{96} - 4 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9 - 2 * q^11 + 8 * q^12 - 2 * q^13 + 12 * q^16 + 10 * q^17 + 4 * q^18 - 2 * q^19 + 12 * q^23 + 12 * q^24 - 8 * q^26 + 6 * q^27 + 16 * q^28 - 12 * q^29 + 8 * q^31 + 28 * q^32 - 2 * q^33 + 8 * q^34 + 8 * q^36 - 2 * q^37 + 24 * q^38 - 2 * q^39 - 4 * q^41 + 18 * q^43 + 4 * q^44 + 16 * q^47 + 12 * q^48 + 6 * q^49 + 10 * q^51 - 12 * q^52 + 30 * q^53 + 4 * q^54 - 2 * q^57 + 16 * q^58 - 6 * q^59 + 4 * q^61 + 16 * q^62 + 20 * q^64 + 30 * q^67 + 20 * q^68 + 12 * q^69 - 24 * q^71 + 12 * q^72 + 6 * q^73 - 8 * q^74 - 4 * q^76 + 16 * q^77 - 8 * q^78 - 2 * q^79 + 6 * q^81 + 20 * q^83 + 16 * q^84 - 12 * q^87 + 16 * q^88 + 14 * q^89 - 48 * q^91 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 28 * q^96 - 4 * q^97 + 4 * q^98 - 2 * 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 $

$2$	2.45188	1.73374	0.866871	−	0.498532i	$-0.166128\pi$
$2$	2.45188	1.73374	0.866871	+	0.498532i	$0.166128\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.01172	2.00586
$5$	0	0
$5$	0	0
$6$	2.45188	1.00098
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	4.93251	1.74391
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.15357	−0.347816	−0.173908	−	0.984762i	$-0.555639\pi$
$11$	−1.15357	−0.347816	−0.173908	+	0.984762i	$0.555639\pi$
$12$	4.01172	1.15808
$13$	5.14234	1.42623	0.713114	−	0.701048i	$-0.247284\pi$
$13$	5.14234	1.42623	0.713114	+	0.701048i	$0.247284\pi$
$14$	−6.93497	−1.85345
$15$	0	0
$16$	4.07048	1.01762
$17$	0.846426	0.205288	0.102644	−	0.994718i	$-0.467270\pi$
$17$	0.846426	0.205288	0.102644	+	0.994718i	$0.467270\pi$
$18$	2.45188	0.577914
$19$	4.73705	1.08675	0.543377	−	0.839489i	$-0.317146\pi$
$19$	4.73705	1.08675	0.543377	+	0.839489i	$0.317146\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	−2.82843	−0.603023
$23$	5.46748	1.14005	0.570025	−	0.821628i	$-0.306934\pi$
$23$	5.46748	1.14005	0.570025	+	0.821628i	$0.306934\pi$
$24$	4.93251	1.00684
$25$	0	0
$26$	12.6084	2.47271
$27$	1.00000	0.192450
$28$	−11.3469	−2.14436
$29$	3.89062	0.722470	0.361235	−	0.932475i	$-0.382355\pi$
$29$	3.89062	0.722470	0.361235	+	0.932475i	$0.382355\pi$
$30$	0	0
$31$	1.84157	0.330756	0.165378	−	0.986230i	$-0.447116\pi$
$31$	1.84157	0.330756	0.165378	+	0.986230i	$0.447116\pi$
$32$	0.115318	0.0203856
$33$	−1.15357	−0.200811
$34$	2.07534	0.355917
$35$	0	0
$36$	4.01172	0.668621
$37$	5.14234	0.845395	0.422698	−	0.906271i	$-0.361083\pi$
$37$	5.14234	0.845395	0.422698	+	0.906271i	$0.361083\pi$
$38$	11.6147	1.88415
$39$	5.14234	0.823433
$40$	0	0
$41$	−2.38845	−0.373014	−0.186507	−	0.982454i	$-0.559717\pi$
$41$	−2.38845	−0.373014	−0.186507	+	0.982454i	$0.559717\pi$
$42$	−6.93497	−1.07009
$43$	−7.32422	−1.11693	−0.558466	−	0.829527i	$-0.688610\pi$
$43$	−7.32422	−1.11693	−0.558466	+	0.829527i	$0.688610\pi$
$44$	−4.62782	−0.697670
$45$	0	0
$46$	13.4056	1.97655
$47$	0.839664	0.122478	0.0612388	−	0.998123i	$-0.480495\pi$
$47$	0.839664	0.122478	0.0612388	+	0.998123i	$0.480495\pi$
$48$	4.07048	0.587523
$49$	1.00000	0.142857
$50$	0	0
$51$	0.846426	0.118523
$52$	20.6296	2.86082
$53$	−1.17702	−0.161676	−0.0808382	−	0.996727i	$-0.525760\pi$
$53$	−1.17702	−0.161676	−0.0808382	+	0.996727i	$0.525760\pi$
$54$	2.45188	0.333659
$55$	0	0
$56$	−13.9512	−1.86431
$57$	4.73705	0.627437
$58$	9.53934	1.25258
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−7.88091	−1.00905	−0.504523	−	0.863398i	$-0.668332\pi$
$61$	−7.88091	−1.00905	−0.504523	+	0.863398i	$0.668332\pi$
$62$	4.51531	0.573445
$63$	−2.82843	−0.356348
$64$	−7.85821	−0.982277
$65$	0	0
$66$	−2.82843	−0.348155
$67$	14.0461	1.71600	0.858002	−	0.513646i	$-0.171706\pi$
$67$	14.0461	1.71600	0.858002	+	0.513646i	$0.171706\pi$
$68$	3.39563	0.411780
$69$	5.46748	0.658208
$70$	0	0
$71$	1.15089	0.136585	0.0682927	−	0.997665i	$-0.478245\pi$
$71$	1.15089	0.136585	0.0682927	+	0.997665i	$0.478245\pi$
$72$	4.93251	0.581302
$73$	−2.83519	−0.331834	−0.165917	−	0.986140i	$-0.553058\pi$
$73$	−2.83519	−0.331834	−0.165917	+	0.986140i	$0.553058\pi$
$74$	12.6084	1.46570
$75$	0	0
$76$	19.0037	2.17988
$77$	3.26280	0.371831
$78$	12.6084	1.42762
$79$	−0.763332	−0.0858816	−0.0429408	−	0.999078i	$-0.513673\pi$
$79$	−0.763332	−0.0858816	−0.0429408	+	0.999078i	$0.513673\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.85620	−0.646709
$83$	5.78434	0.634914	0.317457	−	0.948273i	$-0.397171\pi$
$83$	5.78434	0.634914	0.317457	+	0.948273i	$0.397171\pi$
$84$	−11.3469	−1.23804
$85$	0	0
$86$	−17.9581	−1.93647
$87$	3.89062	0.417118
$88$	−5.69001	−0.606558
$89$	−5.77364	−0.612004	−0.306002	−	0.952031i	$-0.598991\pi$
$89$	−5.77364	−0.612004	−0.306002	+	0.952031i	$0.598991\pi$
$90$	0	0
$91$	−14.5447	−1.52470
$92$	21.9340	2.28678
$93$	1.84157	0.190962
$94$	2.05876	0.212345
$95$	0	0
$96$	0.115318	0.0117696
$97$	15.2985	1.55332	0.776662	−	0.629918i	$-0.216912\pi$
$97$	15.2985	1.55332	0.776662	+	0.629918i	$0.216912\pi$
$98$	2.45188	0.247677
$99$	−1.15357	−0.115939
$100$	0	0
$101$	8.13596	0.809558	0.404779	−	0.914415i	$-0.367348\pi$
$101$	8.13596	0.809558	0.404779	+	0.914415i	$0.367348\pi$
$102$	2.07534	0.205489
$103$	−11.2731	−1.11077	−0.555384	−	0.831594i	$-0.687429\pi$
$103$	−11.2731	−1.11077	−0.555384	+	0.831594i	$0.687429\pi$
$104$	25.3646	2.48721
$105$	0	0
$106$	−2.88592	−0.280305
$107$	19.8861	1.92247	0.961233	−	0.275737i	$-0.0889218\pi$
$107$	19.8861	1.92247	0.961233	+	0.275737i	$0.0889218\pi$
$108$	4.01172	0.386028
$109$	−15.3703	−1.47221	−0.736105	−	0.676868i	$-0.763337\pi$
$109$	−15.3703	−1.47221	−0.736105	+	0.676868i	$0.763337\pi$
$110$	0	0
$111$	5.14234	0.488089
$112$	−11.5131	−1.08788
$113$	5.10464	0.480204	0.240102	−	0.970748i	$-0.422819\pi$
$113$	5.10464	0.480204	0.240102	+	0.970748i	$0.422819\pi$
$114$	11.6147	1.08781
$115$	0	0
$116$	15.6081	1.44918
$117$	5.14234	0.475409
$118$	−2.45188	−0.225714
$119$	−2.39405	−0.219463
$120$	0	0
$121$	−9.66927	−0.879024
$122$	−19.3231	−1.74943
$123$	−2.38845	−0.215359
$124$	7.38787	0.663450
$125$	0	0
$126$	−6.93497	−0.617816
$127$	11.0856	0.983692	0.491846	−	0.870682i	$-0.336322\pi$
$127$	11.0856	0.983692	0.491846	+	0.870682i	$0.336322\pi$
$128$	−19.4980	−1.72340
$129$	−7.32422	−0.644861
$130$	0	0
$131$	−14.2295	−1.24324	−0.621619	−	0.783320i	$-0.713525\pi$
$131$	−14.2295	−1.24324	−0.621619	+	0.783320i	$0.713525\pi$
$132$	−4.62782	−0.402800
$133$	−13.3984	−1.16179
$134$	34.4394	2.97511
$135$	0	0
$136$	4.17500	0.358004
$137$	−5.20315	−0.444535	−0.222268	−	0.974986i	$-0.571346\pi$
$137$	−5.20315	−0.444535	−0.222268	+	0.974986i	$0.571346\pi$
$138$	13.4056	1.14116
$139$	1.50618	0.127753	0.0638763	−	0.997958i	$-0.479654\pi$
$139$	1.50618	0.127753	0.0638763	+	0.997958i	$0.479654\pi$
$140$	0	0
$141$	0.839664	0.0707124
$142$	2.82184	0.236804
$143$	−5.93207	−0.496064
$144$	4.07048	0.339207
$145$	0	0
$146$	−6.95155	−0.575314
$147$	1.00000	0.0824786
$148$	20.6296	1.69575
$149$	8.49552	0.695980	0.347990	−	0.937498i	$-0.386864\pi$
$149$	8.49552	0.695980	0.347990	+	0.937498i	$0.386864\pi$
$150$	0	0
$151$	−18.2787	−1.48750	−0.743750	−	0.668458i	$-0.766955\pi$
$151$	−18.2787	−1.48750	−0.743750	+	0.668458i	$0.766955\pi$
$152$	23.3655	1.89519
$153$	0.846426	0.0684295
$154$	8.00000	0.644658
$155$	0	0
$156$	20.6296	1.65169
$157$	−6.69968	−0.534692	−0.267346	−	0.963601i	$-0.586147\pi$
$157$	−6.69968	−0.534692	−0.267346	+	0.963601i	$0.586147\pi$
$158$	−1.87160	−0.148896
$159$	−1.17702	−0.0933439
$160$	0	0
$161$	−15.4644	−1.21876
$162$	2.45188	0.192638
$163$	11.4815	0.899299	0.449650	−	0.893205i	$-0.351549\pi$
$163$	11.4815	0.899299	0.449650	+	0.893205i	$0.351549\pi$
$164$	−9.58181	−0.748214
$165$	0	0
$166$	14.1825	1.10078
$167$	−8.74952	−0.677058	−0.338529	−	0.940956i	$-0.609929\pi$
$167$	−8.74952	−0.677058	−0.338529	+	0.940956i	$0.609929\pi$
$168$	−13.9512	−1.07636
$169$	13.4436	1.03413
$170$	0	0
$171$	4.73705	0.362251
$172$	−29.3827	−2.24041
$173$	1.85841	0.141292	0.0706461	−	0.997501i	$-0.477494\pi$
$173$	1.85841	0.141292	0.0706461	+	0.997501i	$0.477494\pi$
$174$	9.53934	0.723176
$175$	0	0
$176$	−4.69560	−0.353944
$177$	−1.00000	−0.0751646
$178$	−14.1563	−1.06106
$179$	−12.7164	−0.950466	−0.475233	−	0.879860i	$-0.657636\pi$
$179$	−12.7164	−0.950466	−0.475233	+	0.879860i	$0.657636\pi$
$180$	0	0
$181$	−22.5928	−1.67931	−0.839655	−	0.543121i	$-0.817243\pi$
$181$	−22.5928	−1.67931	−0.839655	+	0.543121i	$0.817243\pi$
$182$	−35.6619	−2.64344
$183$	−7.88091	−0.582574
$184$	26.9684	1.98814
$185$	0	0
$186$	4.51531	0.331079
$187$	−0.976415	−0.0714025
$188$	3.36850	0.245673
$189$	−2.82843	−0.205738
$190$	0	0
$191$	12.1354	0.878089	0.439045	−	0.898465i	$-0.355317\pi$
$191$	12.1354	0.878089	0.439045	+	0.898465i	$0.355317\pi$
$192$	−7.85821	−0.567118
$193$	−17.4162	−1.25365	−0.626824	−	0.779161i	$-0.715646\pi$
$193$	−17.4162	−1.25365	−0.626824	+	0.779161i	$0.715646\pi$
$194$	37.5100	2.69306
$195$	0	0
$196$	4.01172	0.286552
$197$	−14.0857	−1.00356	−0.501782	−	0.864994i	$-0.667322\pi$
$197$	−14.0857	−1.00356	−0.501782	+	0.864994i	$0.667322\pi$
$198$	−2.82843	−0.201008
$199$	20.4148	1.44717	0.723584	−	0.690236i	$-0.242493\pi$
$199$	20.4148	1.44717	0.723584	+	0.690236i	$0.242493\pi$
$200$	0	0
$201$	14.0461	0.990736
$202$	19.9484	1.40356
$203$	−11.0043	−0.772353
$204$	3.39563	0.237741
$205$	0	0
$206$	−27.6402	−1.92578
$207$	5.46748	0.380016
$208$	20.9318	1.45136
$209$	−5.46453	−0.377990
$210$	0	0
$211$	5.09900	0.351030	0.175515	−	0.984477i	$-0.443841\pi$
$211$	5.09900	0.351030	0.175515	+	0.984477i	$0.443841\pi$
$212$	−4.72189	−0.324300
$213$	1.15089	0.0788576
$214$	48.7585	3.33306
$215$	0	0
$216$	4.93251	0.335615
$217$	−5.20875	−0.353593
$218$	−37.6862	−2.55243
$219$	−2.83519	−0.191584
$220$	0	0
$221$	4.35261	0.292788
$222$	12.6084	0.846221
$223$	−21.9112	−1.46728	−0.733642	−	0.679536i	$-0.762181\pi$
$223$	−21.9112	−1.46728	−0.733642	+	0.679536i	$0.762181\pi$
$224$	−0.326169	−0.0217931
$225$	0	0
$226$	12.5160	0.832549
$227$	−12.7772	−0.848051	−0.424026	−	0.905650i	$-0.639383\pi$
$227$	−12.7772	−0.848051	−0.424026	+	0.905650i	$0.639383\pi$
$228$	19.0037	1.25855
$229$	−10.3387	−0.683202	−0.341601	−	0.939845i	$-0.610969\pi$
$229$	−10.3387	−0.683202	−0.341601	+	0.939845i	$0.610969\pi$
$230$	0	0
$231$	3.26280	0.214676
$232$	19.1905	1.25992
$233$	−15.9131	−1.04250	−0.521252	−	0.853403i	$-0.674535\pi$
$233$	−15.9131	−1.04250	−0.521252	+	0.853403i	$0.674535\pi$
$234$	12.6084	0.824237
$235$	0	0
$236$	−4.01172	−0.261141
$237$	−0.763332	−0.0495837
$238$	−5.86994	−0.380492
$239$	−30.3206	−1.96128	−0.980638	−	0.195828i	$-0.937261\pi$
$239$	−30.3206	−1.96128	−0.980638	+	0.195828i	$0.937261\pi$
$240$	0	0
$241$	17.2836	1.11333	0.556666	−	0.830736i	$-0.312080\pi$
$241$	17.2836	1.11333	0.556666	+	0.830736i	$0.312080\pi$
$242$	−23.7079	−1.52400
$243$	1.00000	0.0641500
$244$	−31.6160	−2.02401
$245$	0	0
$246$	−5.85620	−0.373378
$247$	24.3595	1.54996
$248$	9.08356	0.576807
$249$	5.78434	0.366568
$250$	0	0
$251$	−28.7494	−1.81464	−0.907322	−	0.420436i	$-0.861877\pi$
$251$	−28.7494	−1.81464	−0.907322	+	0.420436i	$0.861877\pi$
$252$	−11.3469	−0.714786
$253$	−6.30715	−0.396527
$254$	27.1807	1.70547
$255$	0	0
$256$	−32.0905	−2.00565
$257$	−0.882635	−0.0550572	−0.0275286	−	0.999621i	$-0.508764\pi$
$257$	−0.882635	−0.0550572	−0.0275286	+	0.999621i	$0.508764\pi$
$258$	−17.9581	−1.11802
$259$	−14.5447	−0.903765
$260$	0	0
$261$	3.89062	0.240823
$262$	−34.8891	−2.15545
$263$	19.3569	1.19360	0.596800	−	0.802390i	$-0.296439\pi$
$263$	19.3569	1.19360	0.596800	+	0.802390i	$0.296439\pi$
$264$	−5.69001	−0.350196
$265$	0	0
$266$	−32.8513	−2.01424
$267$	−5.77364	−0.353341
$268$	56.3491	3.44207
$269$	11.9793	0.730393	0.365196	−	0.930931i	$-0.381002\pi$
$269$	11.9793	0.730393	0.365196	+	0.930931i	$0.381002\pi$
$270$	0	0
$271$	−11.0729	−0.672630	−0.336315	−	0.941750i	$-0.609181\pi$
$271$	−11.0729	−0.672630	−0.336315	+	0.941750i	$0.609181\pi$
$272$	3.44536	0.208906
$273$	−14.5447	−0.880287
$274$	−12.7575	−0.770710
$275$	0	0
$276$	21.9340	1.32027
$277$	−13.0716	−0.785397	−0.392698	−	0.919667i	$-0.628458\pi$
$277$	−13.0716	−0.785397	−0.392698	+	0.919667i	$0.628458\pi$
$278$	3.69298	0.221490
$279$	1.84157	0.110252
$280$	0	0
$281$	−4.24729	−0.253372	−0.126686	−	0.991943i	$-0.540434\pi$
$281$	−4.24729	−0.253372	−0.126686	+	0.991943i	$0.540434\pi$
$282$	2.05876	0.122597
$283$	27.8649	1.65640	0.828198	−	0.560436i	$-0.189367\pi$
$283$	27.8649	1.65640	0.828198	+	0.560436i	$0.189367\pi$
$284$	4.61705	0.273972
$285$	0	0
$286$	−14.5447	−0.860048
$287$	6.75556	0.398768
$288$	0.115318	0.00679519
$289$	−16.2836	−0.957857
$290$	0	0
$291$	15.2985	0.896812
$292$	−11.3740	−0.665613
$293$	26.1118	1.52547	0.762735	−	0.646711i	$-0.223856\pi$
$293$	26.1118	1.52547	0.762735	+	0.646711i	$0.223856\pi$
$294$	2.45188	0.142997
$295$	0	0
$296$	25.3646	1.47429
$297$	−1.15357	−0.0669372
$298$	20.8300	1.20665
$299$	28.1156	1.62597
$300$	0	0
$301$	20.7160	1.19405
$302$	−44.8172	−2.57894
$303$	8.13596	0.467399
$304$	19.2821	1.10590
$305$	0	0
$306$	2.07534	0.118639
$307$	31.9632	1.82424	0.912118	−	0.409928i	$-0.134446\pi$
$307$	31.9632	1.82424	0.912118	+	0.409928i	$0.134446\pi$
$308$	13.0895	0.745841
$309$	−11.2731	−0.641302
$310$	0	0
$311$	−11.2294	−0.636758	−0.318379	−	0.947963i	$-0.603139\pi$
$311$	−11.2294	−0.636758	−0.318379	+	0.947963i	$0.603139\pi$
$312$	25.3646	1.43599
$313$	−22.4522	−1.26907	−0.634536	−	0.772893i	$-0.718809\pi$
$313$	−22.4522	−1.26907	−0.634536	+	0.772893i	$0.718809\pi$
$314$	−16.4268	−0.927019
$315$	0	0
$316$	−3.06228	−0.172267
$317$	−19.9579	−1.12095	−0.560475	−	0.828171i	$-0.689381\pi$
$317$	−19.9579	−1.12095	−0.560475	+	0.828171i	$0.689381\pi$
$318$	−2.88592	−0.161834
$319$	−4.48812	−0.251286
$320$	0	0
$321$	19.8861	1.10994
$322$	−37.9168	−2.11302
$323$	4.00956	0.223098
$324$	4.01172	0.222874
$325$	0	0
$326$	28.1512	1.55915
$327$	−15.3703	−0.849980
$328$	−11.7811	−0.650500
$329$	−2.37493	−0.130934
$330$	0	0
$331$	−31.5489	−1.73408	−0.867042	−	0.498234i	$-0.833982\pi$
$331$	−31.5489	−1.73408	−0.867042	+	0.498234i	$0.833982\pi$
$332$	23.2052	1.27355
$333$	5.14234	0.281798
$334$	−21.4528	−1.17384
$335$	0	0
$336$	−11.5131	−0.628089
$337$	−0.284861	−0.0155174	−0.00775868	−	0.999970i	$-0.502470\pi$
$337$	−0.284861	−0.0155174	−0.00775868	+	0.999970i	$0.502470\pi$
$338$	32.9622	1.79291
$339$	5.10464	0.277246
$340$	0	0
$341$	−2.12439	−0.115042
$342$	11.6147	0.628050
$343$	16.9706	0.916324
$344$	−36.1268	−1.94782
$345$	0	0
$346$	4.55660	0.244964
$347$	8.20087	0.440246	0.220123	−	0.975472i	$-0.429354\pi$
$347$	8.20087	0.440246	0.220123	+	0.975472i	$0.429354\pi$
$348$	15.6081	0.836682
$349$	−24.6204	−1.31790	−0.658949	−	0.752188i	$-0.728999\pi$
$349$	−24.6204	−1.31790	−0.658949	+	0.752188i	$0.728999\pi$
$350$	0	0
$351$	5.14234	0.274478
$352$	−0.133028	−0.00709042
$353$	−27.8962	−1.48477	−0.742384	−	0.669975i	$-0.766305\pi$
$353$	−27.8962	−1.48477	−0.742384	+	0.669975i	$0.766305\pi$
$354$	−2.45188	−0.130316
$355$	0	0
$356$	−23.1622	−1.22760
$357$	−2.39405	−0.126707
$358$	−31.1790	−1.64786
$359$	22.8256	1.20469	0.602344	−	0.798236i	$-0.294233\pi$
$359$	22.8256	1.20469	0.602344	+	0.798236i	$0.294233\pi$
$360$	0	0
$361$	3.43961	0.181032
$362$	−55.3949	−2.91149
$363$	−9.66927	−0.507505
$364$	−58.3494	−3.05834
$365$	0	0
$366$	−19.3231	−1.01003
$367$	−28.0833	−1.46594	−0.732968	−	0.680263i	$-0.761865\pi$
$367$	−28.0833	−1.46594	−0.732968	+	0.680263i	$0.761865\pi$
$368$	22.2553	1.16014
$369$	−2.38845	−0.124338
$370$	0	0
$371$	3.32912	0.172839
$372$	7.38787	0.383043
$373$	9.36934	0.485126	0.242563	−	0.970136i	$-0.422012\pi$
$373$	9.36934	0.485126	0.242563	+	0.970136i	$0.422012\pi$
$374$	−2.39405	−0.123794
$375$	0	0
$376$	4.14165	0.213589
$377$	20.0069	1.03041
$378$	−6.93497	−0.356696
$379$	−15.1689	−0.779173	−0.389587	−	0.920990i	$-0.627382\pi$
$379$	−15.1689	−0.779173	−0.389587	+	0.920990i	$0.627382\pi$
$380$	0	0
$381$	11.0856	0.567935
$382$	29.7547	1.52238
$383$	−23.0688	−1.17876	−0.589381	−	0.807855i	$-0.700628\pi$
$383$	−23.0688	−1.17876	−0.589381	+	0.807855i	$0.700628\pi$
$384$	−19.4980	−0.995006
$385$	0	0
$386$	−42.7026	−2.17350
$387$	−7.32422	−0.372311
$388$	61.3732	3.11575
$389$	−0.390242	−0.0197861	−0.00989303	−	0.999951i	$-0.503149\pi$
$389$	−0.390242	−0.0197861	−0.00989303	+	0.999951i	$0.503149\pi$
$390$	0	0
$391$	4.62782	0.234039
$392$	4.93251	0.249129
$393$	−14.2295	−0.717783
$394$	−34.5365	−1.73992
$395$	0	0
$396$	−4.62782	−0.232557
$397$	−1.92978	−0.0968529	−0.0484264	−	0.998827i	$-0.515421\pi$
$397$	−1.92978	−0.0968529	−0.0484264	+	0.998827i	$0.515421\pi$
$398$	50.0547	2.50902
$399$	−13.3984	−0.670759
$400$	0	0
$401$	−15.1367	−0.755891	−0.377946	−	0.925828i	$-0.623369\pi$
$401$	−15.1367	−0.755891	−0.377946	+	0.925828i	$0.623369\pi$
$402$	34.4394	1.71768
$403$	9.46997	0.471733
$404$	32.6392	1.62386
$405$	0	0
$406$	−26.9813	−1.33906
$407$	−5.93207	−0.294042
$408$	4.17500	0.206693
$409$	19.5286	0.965626	0.482813	−	0.875723i	$-0.339615\pi$
$409$	19.5286	0.965626	0.482813	+	0.875723i	$0.339615\pi$
$410$	0	0
$411$	−5.20315	−0.256653
$412$	−45.2244	−2.22805
$413$	2.82843	0.139178
$414$	13.4056	0.658851
$415$	0	0
$416$	0.593005	0.0290745
$417$	1.50618	0.0737580
$418$	−13.3984	−0.655337
$419$	17.1629	0.838462	0.419231	−	0.907880i	$-0.362300\pi$
$419$	17.1629	0.838462	0.419231	+	0.907880i	$0.362300\pi$
$420$	0	0
$421$	3.70638	0.180638	0.0903189	−	0.995913i	$-0.471211\pi$
$421$	3.70638	0.180638	0.0903189	+	0.995913i	$0.471211\pi$
$422$	12.5021	0.608595
$423$	0.839664	0.0408259
$424$	−5.80567	−0.281948
$425$	0	0
$426$	2.82184	0.136719
$427$	22.2906	1.07872
$428$	79.7777	3.85620
$429$	−5.93207	−0.286403
$430$	0	0
$431$	25.0108	1.20473	0.602363	−	0.798222i	$-0.294226\pi$
$431$	25.0108	1.20473	0.602363	+	0.798222i	$0.294226\pi$
$432$	4.07048	0.195841
$433$	−27.6153	−1.32711	−0.663553	−	0.748130i	$-0.730952\pi$
$433$	−27.6153	−1.32711	−0.663553	+	0.748130i	$0.730952\pi$
$434$	−12.7712	−0.613039
$435$	0	0
$436$	−61.6615	−2.95305
$437$	25.8997	1.23895
$438$	−6.95155	−0.332158
$439$	4.44647	0.212219	0.106109	−	0.994354i	$-0.466161\pi$
$439$	4.44647	0.212219	0.106109	+	0.994354i	$0.466161\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	10.6721	0.507619
$443$	6.89574	0.327626	0.163813	−	0.986491i	$-0.447621\pi$
$443$	6.89574	0.327626	0.163813	+	0.986491i	$0.447621\pi$
$444$	20.6296	0.979039
$445$	0	0
$446$	−53.7237	−2.54389
$447$	8.49552	0.401824
$448$	22.2264	1.05010
$449$	21.7249	1.02526	0.512630	−	0.858610i	$-0.328671\pi$
$449$	21.7249	1.02526	0.512630	+	0.858610i	$0.328671\pi$
$450$	0	0
$451$	2.75526	0.129740
$452$	20.4784	0.963222
$453$	−18.2787	−0.858809
$454$	−31.3281	−1.47030
$455$	0	0
$456$	23.3655	1.09419
$457$	−31.0382	−1.45190	−0.725952	−	0.687746i	$-0.758600\pi$
$457$	−31.0382	−1.45190	−0.725952	+	0.687746i	$0.758600\pi$
$458$	−25.3493	−1.18450
$459$	0.846426	0.0395078
$460$	0	0
$461$	−10.8506	−0.505364	−0.252682	−	0.967549i	$-0.581313\pi$
$461$	−10.8506	−0.505364	−0.252682	+	0.967549i	$0.581313\pi$
$462$	8.00000	0.372194
$463$	26.8718	1.24884	0.624420	−	0.781089i	$-0.285335\pi$
$463$	26.8718	1.24884	0.624420	+	0.781089i	$0.285335\pi$
$464$	15.8367	0.735200
$465$	0	0
$466$	−39.0171	−1.80743
$467$	16.9959	0.786478	0.393239	−	0.919436i	$-0.371354\pi$
$467$	16.9959	0.786478	0.393239	+	0.919436i	$0.371354\pi$
$468$	20.6296	0.953605
$469$	−39.7284	−1.83449
$470$	0	0
$471$	−6.69968	−0.308705
$472$	−4.93251	−0.227037
$473$	8.44902	0.388487
$474$	−1.87160	−0.0859654
$475$	0	0
$476$	−9.60428	−0.440212
$477$	−1.17702	−0.0538921
$478$	−74.3425	−3.40035
$479$	19.0107	0.868620	0.434310	−	0.900763i	$-0.356992\pi$
$479$	19.0107	0.868620	0.434310	+	0.900763i	$0.356992\pi$
$480$	0	0
$481$	26.4436	1.20573
$482$	42.3773	1.93023
$483$	−15.4644	−0.703654
$484$	−38.7904	−1.76320
$485$	0	0
$486$	2.45188	0.111220
$487$	−9.05629	−0.410380	−0.205190	−	0.978722i	$-0.565781\pi$
$487$	−9.05629	−0.410380	−0.205190	+	0.978722i	$0.565781\pi$
$488$	−38.8726	−1.75968
$489$	11.4815	0.519211
$490$	0	0
$491$	5.05015	0.227910	0.113955	−	0.993486i	$-0.463648\pi$
$491$	5.05015	0.227910	0.113955	+	0.993486i	$0.463648\pi$
$492$	−9.58181	−0.431981
$493$	3.29312	0.148315
$494$	59.7266	2.68723
$495$	0	0
$496$	7.49607	0.336584
$497$	−3.25521	−0.146016
$498$	14.1825	0.635534
$499$	38.6109	1.72846	0.864230	−	0.503097i	$-0.167806\pi$
$499$	38.6109	1.72846	0.864230	+	0.503097i	$0.167806\pi$
$500$	0	0
$501$	−8.74952	−0.390900
$502$	−70.4901	−3.14613
$503$	−20.0785	−0.895257	−0.447629	−	0.894220i	$-0.647731\pi$
$503$	−20.0785	−0.895257	−0.447629	+	0.894220i	$0.647731\pi$
$504$	−13.9512	−0.621438
$505$	0	0
$506$	−15.4644	−0.687476
$507$	13.4436	0.597053
$508$	44.4725	1.97315
$509$	26.1692	1.15993	0.579964	−	0.814642i	$-0.303067\pi$
$509$	26.1692	1.15993	0.579964	+	0.814642i	$0.303067\pi$
$510$	0	0
$511$	8.01913	0.354745
$512$	−39.6860	−1.75389
$513$	4.73705	0.209146
$514$	−2.16412	−0.0954550
$515$	0	0
$516$	−29.3827	−1.29350
$517$	−0.968614	−0.0425996
$518$	−35.6619	−1.56690
$519$	1.85841	0.0815751
$520$	0	0
$521$	10.7803	0.472293	0.236146	−	0.971717i	$-0.424116\pi$
$521$	10.7803	0.472293	0.236146	+	0.971717i	$0.424116\pi$
$522$	9.53934	0.417526
$523$	−38.6586	−1.69042	−0.845211	−	0.534433i	$-0.820525\pi$
$523$	−38.6586	−1.69042	−0.845211	+	0.534433i	$0.820525\pi$
$524$	−57.0848	−2.49376
$525$	0	0
$526$	47.4609	2.06940
$527$	1.55875	0.0679003
$528$	−4.69560	−0.204350
$529$	6.89338	0.299712
$530$	0	0
$531$	−1.00000	−0.0433963
$532$	−53.7506	−2.33039
$533$	−12.2822	−0.532002
$534$	−14.1563	−0.612602
$535$	0	0
$536$	69.2825	2.99255
$537$	−12.7164	−0.548752
$538$	29.3719	1.26631
$539$	−1.15357	−0.0496880
$540$	0	0
$541$	−8.03718	−0.345545	−0.172773	−	0.984962i	$-0.555273\pi$
$541$	−8.03718	−0.345545	−0.172773	+	0.984962i	$0.555273\pi$
$542$	−27.1494	−1.16617
$543$	−22.5928	−0.969550
$544$	0.0976083	0.00418492
$545$	0	0
$546$	−35.6619	−1.52619
$547$	27.4916	1.17546	0.587729	−	0.809058i	$-0.300022\pi$
$547$	27.4916	1.17546	0.587729	+	0.809058i	$0.300022\pi$
$548$	−20.8736	−0.891677
$549$	−7.88091	−0.336349
$550$	0	0
$551$	18.4301	0.785147
$552$	26.9684	1.14785
$553$	2.15903	0.0918112
$554$	−32.0500	−1.36168
$555$	0	0
$556$	6.04239	0.256254
$557$	−36.9495	−1.56560	−0.782801	−	0.622272i	$-0.786210\pi$
$557$	−36.9495	−1.56560	−0.782801	+	0.622272i	$0.786210\pi$
$558$	4.51531	0.191148
$559$	−37.6636	−1.59300
$560$	0	0
$561$	−0.976415	−0.0412243
$562$	−10.4139	−0.439282
$563$	42.7700	1.80254	0.901271	−	0.433256i	$-0.142636\pi$
$563$	42.7700	1.80254	0.901271	+	0.433256i	$0.142636\pi$
$564$	3.36850	0.141839
$565$	0	0
$566$	68.3214	2.87176
$567$	−2.82843	−0.118783
$568$	5.67677	0.238192
$569$	−24.2182	−1.01528	−0.507640	−	0.861569i	$-0.669482\pi$
$569$	−24.2182	−1.01528	−0.507640	+	0.861569i	$0.669482\pi$
$570$	0	0
$571$	27.4583	1.14910	0.574548	−	0.818471i	$-0.305178\pi$
$571$	27.4583	1.14910	0.574548	+	0.818471i	$0.305178\pi$
$572$	−23.7978	−0.995037
$573$	12.1354	0.506965
$574$	16.5638	0.691361
$575$	0	0
$576$	−7.85821	−0.327426
$577$	−0.132594	−0.00551996	−0.00275998	−	0.999996i	$-0.500879\pi$
$577$	−0.132594	−0.00551996	−0.00275998	+	0.999996i	$0.500879\pi$
$578$	−39.9254	−1.66068
$579$	−17.4162	−0.723794
$580$	0	0
$581$	−16.3606	−0.678752
$582$	37.5100	1.55484
$583$	1.35778	0.0562336
$584$	−13.9846	−0.578687
$585$	0	0
$586$	64.0231	2.64477
$587$	10.4098	0.429660	0.214830	−	0.976651i	$-0.431080\pi$
$587$	10.4098	0.429660	0.214830	+	0.976651i	$0.431080\pi$
$588$	4.01172	0.165441
$589$	8.72360	0.359450
$590$	0	0
$591$	−14.0857	−0.579408
$592$	20.9318	0.860291
$593$	−11.9970	−0.492658	−0.246329	−	0.969186i	$-0.579224\pi$
$593$	−11.9970	−0.492658	−0.246329	+	0.969186i	$0.579224\pi$
$594$	−2.82843	−0.116052
$595$	0	0
$596$	34.0817	1.39604
$597$	20.4148	0.835523
$598$	68.9362	2.81901
$599$	−35.6309	−1.45584	−0.727920	−	0.685662i	$-0.759513\pi$
$599$	−35.6309	−1.45584	−0.727920	+	0.685662i	$0.759513\pi$
$600$	0	0
$601$	15.9562	0.650866	0.325433	−	0.945565i	$-0.394490\pi$
$601$	15.9562	0.650866	0.325433	+	0.945565i	$0.394490\pi$
$602$	50.7932	2.07018
$603$	14.0461	0.572001
$604$	−73.3291	−2.98372
$605$	0	0
$606$	19.9484	0.810349
$607$	20.1088	0.816190	0.408095	−	0.912939i	$-0.366193\pi$
$607$	20.1088	0.816190	0.408095	+	0.912939i	$0.366193\pi$
$608$	0.546268	0.0221541
$609$	−11.0043	−0.445918
$610$	0	0
$611$	4.31783	0.174681
$612$	3.39563	0.137260
$613$	−10.0653	−0.406533	−0.203267	−	0.979123i	$-0.565156\pi$
$613$	−10.0653	−0.406533	−0.203267	+	0.979123i	$0.565156\pi$
$614$	78.3700	3.16275
$615$	0	0
$616$	16.0938	0.648437
$617$	38.4113	1.54638	0.773191	−	0.634173i	$-0.218659\pi$
$617$	38.4113	1.54638	0.773191	+	0.634173i	$0.218659\pi$
$618$	−27.6402	−1.11185
$619$	33.6132	1.35103	0.675515	−	0.737346i	$-0.263921\pi$
$619$	33.6132	1.35103	0.675515	+	0.737346i	$0.263921\pi$
$620$	0	0
$621$	5.46748	0.219403
$622$	−27.5331	−1.10397
$623$	16.3303	0.654260
$624$	20.9318	0.837942
$625$	0	0
$626$	−55.0501	−2.20024
$627$	−5.46453	−0.218232
$628$	−26.8773	−1.07252
$629$	4.35261	0.173550
$630$	0	0
$631$	44.9913	1.79108	0.895539	−	0.444984i	$-0.146791\pi$
$631$	44.9913	1.79108	0.895539	+	0.444984i	$0.146791\pi$
$632$	−3.76514	−0.149769
$633$	5.09900	0.202667
$634$	−48.9345	−1.94344
$635$	0	0
$636$	−4.72189	−0.187235
$637$	5.14234	0.203747
$638$	−11.0043	−0.435666
$639$	1.15089	0.0455285
$640$	0	0
$641$	21.6468	0.854999	0.427499	−	0.904016i	$-0.359395\pi$
$641$	21.6468	0.854999	0.427499	+	0.904016i	$0.359395\pi$
$642$	48.7585	1.92434
$643$	−21.3467	−0.841830	−0.420915	−	0.907100i	$-0.638291\pi$
$643$	−21.3467	−0.841830	−0.420915	+	0.907100i	$0.638291\pi$
$644$	−62.0388	−2.44467
$645$	0	0
$646$	9.83096	0.386794
$647$	40.3255	1.58536	0.792680	−	0.609638i	$-0.208685\pi$
$647$	40.3255	1.58536	0.792680	+	0.609638i	$0.208685\pi$
$648$	4.93251	0.193767
$649$	1.15357	0.0452817
$650$	0	0
$651$	−5.20875	−0.204147
$652$	46.0605	1.80387
$653$	−2.67085	−0.104518	−0.0522591	−	0.998634i	$-0.516642\pi$
$653$	−2.67085	−0.104518	−0.0522591	+	0.998634i	$0.516642\pi$
$654$	−37.6862	−1.47365
$655$	0	0
$656$	−9.72215	−0.379586
$657$	−2.83519	−0.110611
$658$	−5.82304	−0.227006
$659$	−24.6368	−0.959715	−0.479858	−	0.877346i	$-0.659312\pi$
$659$	−24.6368	−0.959715	−0.479858	+	0.877346i	$0.659312\pi$
$660$	0	0
$661$	44.0120	1.71187	0.855933	−	0.517086i	$-0.172983\pi$
$661$	44.0120	1.71187	0.855933	+	0.517086i	$0.172983\pi$
$662$	−77.3542	−3.00646
$663$	4.35261	0.169041
$664$	28.5313	1.10723
$665$	0	0
$666$	12.6084	0.488566
$667$	21.2719	0.823652
$668$	−35.1007	−1.35809
$669$	−21.9112	−0.847137
$670$	0	0
$671$	9.09121	0.350962
$672$	−0.326169	−0.0125823
$673$	24.9740	0.962677	0.481339	−	0.876535i	$-0.340151\pi$
$673$	24.9740	0.962677	0.481339	+	0.876535i	$0.340151\pi$
$674$	−0.698445	−0.0269031
$675$	0	0
$676$	53.9321	2.07431
$677$	22.1287	0.850476	0.425238	−	0.905082i	$-0.360190\pi$
$677$	22.1287	0.850476	0.425238	+	0.905082i	$0.360190\pi$
$678$	12.5160	0.480673
$679$	−43.2706	−1.66057
$680$	0	0
$681$	−12.7772	−0.489623
$682$	−5.20875	−0.199453
$683$	−22.2609	−0.851789	−0.425895	−	0.904773i	$-0.640041\pi$
$683$	−22.2609	−0.851789	−0.425895	+	0.904773i	$0.640041\pi$
$684$	19.0037	0.726625
$685$	0	0
$686$	41.6098	1.58867
$687$	−10.3387	−0.394447
$688$	−29.8131	−1.13661
$689$	−6.05264	−0.230587
$690$	0	0
$691$	−36.2412	−1.37868	−0.689339	−	0.724439i	$-0.742099\pi$
$691$	−36.2412	−1.37868	−0.689339	+	0.724439i	$0.742099\pi$
$692$	7.45542	0.283413
$693$	3.26280	0.123944
$694$	20.1076	0.763273
$695$	0	0
$696$	19.1905	0.727415
$697$	−2.02165	−0.0765754
$698$	−60.3662	−2.28489
$699$	−15.9131	−0.601890
$700$	0	0
$701$	−18.0978	−0.683545	−0.341773	−	0.939783i	$-0.611027\pi$
$701$	−18.0978	−0.683545	−0.341773	+	0.939783i	$0.611027\pi$
$702$	12.6084	0.475873
$703$	24.3595	0.918736
$704$	9.06503	0.341651
$705$	0	0
$706$	−68.3983	−2.57420
$707$	−23.0120	−0.865454
$708$	−4.01172	−0.150770
$709$	30.9960	1.16408	0.582039	−	0.813161i	$-0.302255\pi$
$709$	30.9960	1.16408	0.582039	+	0.813161i	$0.302255\pi$
$710$	0	0
$711$	−0.763332	−0.0286272
$712$	−28.4785	−1.06728
$713$	10.0688	0.377078
$714$	−5.86994	−0.219677
$715$	0	0
$716$	−51.0145	−1.90650
$717$	−30.3206	−1.13234
$718$	55.9656	2.08862
$719$	30.7535	1.14691	0.573457	−	0.819236i	$-0.305602\pi$
$719$	30.7535	1.14691	0.573457	+	0.819236i	$0.305602\pi$
$720$	0	0
$721$	31.8850	1.18746
$722$	8.43352	0.313863
$723$	17.2836	0.642783
$724$	−90.6361	−3.36846
$725$	0	0
$726$	−23.7079	−0.879883
$727$	17.3402	0.643113	0.321556	−	0.946890i	$-0.395794\pi$
$727$	17.3402	0.643113	0.321556	+	0.946890i	$0.395794\pi$
$728$	−71.7420	−2.65894
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.19941	−0.229293
$732$	−31.6160	−1.16856
$733$	−8.82695	−0.326031	−0.163015	−	0.986624i	$-0.552122\pi$
$733$	−8.82695	−0.326031	−0.163015	+	0.986624i	$0.552122\pi$
$734$	−68.8569	−2.54155
$735$	0	0
$736$	0.630501	0.0232406
$737$	−16.2032	−0.596853
$738$	−5.85620	−0.215570
$739$	16.8386	0.619419	0.309710	−	0.950831i	$-0.399768\pi$
$739$	16.8386	0.619419	0.309710	+	0.950831i	$0.399768\pi$
$740$	0	0
$741$	24.3595	0.894868
$742$	8.16261	0.299659
$743$	27.5526	1.01081	0.505404	−	0.862883i	$-0.331344\pi$
$743$	27.5526	1.01081	0.505404	+	0.862883i	$0.331344\pi$
$744$	9.08356	0.333019
$745$	0	0
$746$	22.9725	0.841084
$747$	5.78434	0.211638
$748$	−3.91711	−0.143224
$749$	−56.2465	−2.05520
$750$	0	0
$751$	8.33181	0.304032	0.152016	−	0.988378i	$-0.451423\pi$
$751$	8.33181	0.304032	0.152016	+	0.988378i	$0.451423\pi$
$752$	3.41783	0.124636
$753$	−28.7494	−1.04769
$754$	49.0545	1.78646
$755$	0	0
$756$	−11.3469	−0.412682
$757$	−25.3705	−0.922107	−0.461054	−	0.887372i	$-0.652528\pi$
$757$	−25.3705	−0.922107	−0.461054	+	0.887372i	$0.652528\pi$
$758$	−37.1923	−1.35089
$759$	−6.30715	−0.228935
$760$	0	0
$761$	40.1113	1.45403	0.727017	−	0.686619i	$-0.240906\pi$
$761$	40.1113	1.45403	0.727017	+	0.686619i	$0.240906\pi$
$762$	27.1807	0.984652
$763$	43.4738	1.57386
$764$	48.6840	1.76133
$765$	0	0
$766$	−56.5620	−2.04367
$767$	−5.14234	−0.185679
$768$	−32.0905	−1.15797
$769$	36.0410	1.29967	0.649836	−	0.760074i	$-0.274837\pi$
$769$	36.0410	1.29967	0.649836	+	0.760074i	$0.274837\pi$
$770$	0	0
$771$	−0.882635	−0.0317873
$772$	−69.8691	−2.51465
$773$	34.3013	1.23373	0.616866	−	0.787069i	$-0.288402\pi$
$773$	34.3013	1.23373	0.616866	+	0.787069i	$0.288402\pi$
$774$	−17.9581	−0.645491
$775$	0	0
$776$	75.4598	2.70885
$777$	−14.5447	−0.521789
$778$	−0.956827	−0.0343039
$779$	−11.3142	−0.405374
$780$	0	0
$781$	−1.32764	−0.0475066
$782$	11.3469	0.405763
$783$	3.89062	0.139039
$784$	4.07048	0.145374
$785$	0	0
$786$	−34.8891	−1.24445
$787$	7.30274	0.260314	0.130157	−	0.991493i	$-0.458452\pi$
$787$	7.30274	0.260314	0.130157	+	0.991493i	$0.458452\pi$
$788$	−56.5080	−2.01301
$789$	19.3569	0.689125
$790$	0	0
$791$	−14.4381	−0.513359
$792$	−5.69001	−0.202186
$793$	−40.5263	−1.43913
$794$	−4.73159	−0.167918
$795$	0	0
$796$	81.8986	2.90282
$797$	37.2470	1.31936	0.659679	−	0.751548i	$-0.270692\pi$
$797$	37.2470	1.31936	0.659679	+	0.751548i	$0.270692\pi$
$798$	−32.8513	−1.16292
$799$	0.710713	0.0251432
$800$	0	0
$801$	−5.77364	−0.204001
$802$	−37.1134	−1.31052
$803$	3.27060	0.115417
$804$	56.3491	1.98728
$805$	0	0
$806$	23.2193	0.817863
$807$	11.9793	0.421692
$808$	40.1307	1.41179
$809$	−5.41407	−0.190349	−0.0951743	−	0.995461i	$-0.530341\pi$
$809$	−5.41407	−0.190349	−0.0951743	+	0.995461i	$0.530341\pi$
$810$	0	0
$811$	−13.2232	−0.464329	−0.232164	−	0.972677i	$-0.574581\pi$
$811$	−13.2232	−0.464329	−0.232164	+	0.972677i	$0.574581\pi$
$812$	−44.1464	−1.54923
$813$	−11.0729	−0.388343
$814$	−14.5447	−0.509792
$815$	0	0
$816$	3.44536	0.120612
$817$	−34.6951	−1.21383
$818$	47.8818	1.67415
$819$	−14.5447	−0.508234
$820$	0	0
$821$	13.2853	0.463662	0.231831	−	0.972756i	$-0.425528\pi$
$821$	13.2853	0.463662	0.231831	+	0.972756i	$0.425528\pi$
$822$	−12.7575	−0.444970
$823$	−21.5918	−0.752642	−0.376321	−	0.926489i	$-0.622811\pi$
$823$	−21.5918	−0.752642	−0.376321	+	0.926489i	$0.622811\pi$
$824$	−55.6044	−1.93707
$825$	0	0
$826$	6.93497	0.241298
$827$	16.8282	0.585175	0.292587	−	0.956239i	$-0.405484\pi$
$827$	16.8282	0.585175	0.292587	+	0.956239i	$0.405484\pi$
$828$	21.9340	0.762260
$829$	11.8798	0.412603	0.206301	−	0.978488i	$-0.433857\pi$
$829$	11.8798	0.412603	0.206301	+	0.978488i	$0.433857\pi$
$830$	0	0
$831$	−13.0716	−0.453449
$832$	−40.4096	−1.40095
$833$	0.846426	0.0293269
$834$	3.69298	0.127877
$835$	0	0
$836$	−21.9222	−0.758195
$837$	1.84157	0.0636540
$838$	42.0814	1.45368
$839$	28.5711	0.986383	0.493191	−	0.869921i	$-0.335830\pi$
$839$	28.5711	0.986383	0.493191	+	0.869921i	$0.335830\pi$
$840$	0	0
$841$	−13.8631	−0.478037
$842$	9.08760	0.313179
$843$	−4.24729	−0.146284
$844$	20.4558	0.704117
$845$	0	0
$846$	2.05876	0.0707815
$847$	27.3488	0.939716
$848$	−4.79104	−0.164525
$849$	27.8649	0.956320
$850$	0	0
$851$	28.1156	0.963792
$852$	4.61705	0.158178
$853$	8.92759	0.305675	0.152837	−	0.988251i	$-0.451159\pi$
$853$	8.92759	0.305675	0.152837	+	0.988251i	$0.451159\pi$
$854$	54.6539	1.87022
$855$	0	0
$856$	98.0886	3.35260
$857$	−15.6867	−0.535849	−0.267925	−	0.963440i	$-0.586338\pi$
$857$	−15.6867	−0.535849	−0.267925	+	0.963440i	$0.586338\pi$
$858$	−14.5447	−0.496549
$859$	19.6147	0.669245	0.334622	−	0.942352i	$-0.391391\pi$
$859$	19.6147	0.669245	0.334622	+	0.942352i	$0.391391\pi$
$860$	0	0
$861$	6.75556	0.230229
$862$	61.3234	2.08868
$863$	−28.6813	−0.976324	−0.488162	−	0.872753i	$-0.662332\pi$
$863$	−28.6813	−0.976324	−0.488162	+	0.872753i	$0.662332\pi$
$864$	0.115318	0.00392321
$865$	0	0
$866$	−67.7094	−2.30086
$867$	−16.2836	−0.553019
$868$	−20.8960	−0.709258
$869$	0.880560	0.0298710
$870$	0	0
$871$	72.2298	2.44741
$872$	−75.8142	−2.56739
$873$	15.2985	0.517774
$874$	63.5031	2.14802
$875$	0	0
$876$	−11.3740	−0.384292
$877$	5.41951	0.183004	0.0915019	−	0.995805i	$-0.470833\pi$
$877$	5.41951	0.183004	0.0915019	+	0.995805i	$0.470833\pi$
$878$	10.9022	0.367932
$879$	26.1118	0.880730
$880$	0	0
$881$	−47.3058	−1.59377	−0.796887	−	0.604129i	$-0.793521\pi$
$881$	−47.3058	−1.59377	−0.796887	+	0.604129i	$0.793521\pi$
$882$	2.45188	0.0825592
$883$	−10.1519	−0.341640	−0.170820	−	0.985302i	$-0.554642\pi$
$883$	−10.1519	−0.341640	−0.170820	+	0.985302i	$0.554642\pi$
$884$	17.4615	0.587292
$885$	0	0
$886$	16.9075	0.568020
$887$	20.4138	0.685428	0.342714	−	0.939440i	$-0.388654\pi$
$887$	20.4138	0.685428	0.342714	+	0.939440i	$0.388654\pi$
$888$	25.3646	0.851181
$889$	−31.3549	−1.05161
$890$	0	0
$891$	−1.15357	−0.0386462
$892$	−87.9018	−2.94317
$893$	3.97753	0.133103
$894$	20.8300	0.696660
$895$	0	0
$896$	55.1488	1.84239
$897$	28.1156	0.938754
$898$	53.2668	1.77754
$899$	7.16485	0.238961
$900$	0	0
$901$	−0.996262	−0.0331903
$902$	6.75556	0.224936
$903$	20.7160	0.689386
$904$	25.1787	0.837430
$905$	0	0
$906$	−44.8172	−1.48895
$907$	20.9540	0.695765	0.347883	−	0.937538i	$-0.386901\pi$
$907$	20.9540	0.695765	0.347883	+	0.937538i	$0.386901\pi$
$908$	−51.2585	−1.70107
$909$	8.13596	0.269853
$910$	0	0
$911$	−54.1112	−1.79278	−0.896391	−	0.443264i	$-0.853821\pi$
$911$	−54.1112	−1.79278	−0.896391	+	0.443264i	$0.853821\pi$
$912$	19.2821	0.638493
$913$	−6.67267	−0.220833
$914$	−76.1019	−2.51723
$915$	0	0
$916$	−41.4761	−1.37041
$917$	40.2471	1.32908
$918$	2.07534	0.0684963
$919$	−3.44976	−0.113797	−0.0568986	−	0.998380i	$-0.518121\pi$
$919$	−3.44976	−0.113797	−0.0568986	+	0.998380i	$0.518121\pi$
$920$	0	0
$921$	31.9632	1.05322
$922$	−26.6044	−0.876170
$923$	5.91826	0.194802
$924$	13.0895	0.430611
$925$	0	0
$926$	65.8866	2.16517
$927$	−11.2731	−0.370256
$928$	0.448659	0.0147280
$929$	−23.1134	−0.758325	−0.379163	−	0.925330i	$-0.623788\pi$
$929$	−23.1134	−0.758325	−0.379163	+	0.925330i	$0.623788\pi$
$930$	0	0
$931$	4.73705	0.155250
$932$	−63.8390	−2.09112
$933$	−11.2294	−0.367633
$934$	41.6720	1.36355
$935$	0	0
$936$	25.3646	0.829069
$937$	−54.0397	−1.76540	−0.882700	−	0.469937i	$-0.844277\pi$
$937$	−54.0397	−1.76540	−0.882700	+	0.469937i	$0.844277\pi$
$938$	−97.4093	−3.18053
$939$	−22.4522	−0.732699
$940$	0	0
$941$	−7.88481	−0.257037	−0.128519	−	0.991707i	$-0.541022\pi$
$941$	−7.88481	−0.257037	−0.128519	+	0.991707i	$0.541022\pi$
$942$	−16.4268	−0.535215
$943$	−13.0588	−0.425254
$944$	−4.07048	−0.132483
$945$	0	0
$946$	20.7160	0.673536
$947$	−10.9029	−0.354298	−0.177149	−	0.984184i	$-0.556687\pi$
$947$	−10.9029	−0.354298	−0.177149	+	0.984184i	$0.556687\pi$
$948$	−3.06228	−0.0994581
$949$	−14.5795	−0.473271
$950$	0	0
$951$	−19.9579	−0.647181
$952$	−11.8087	−0.382722
$953$	32.4724	1.05188	0.525942	−	0.850521i	$-0.323713\pi$
$953$	32.4724	1.05188	0.525942	+	0.850521i	$0.323713\pi$
$954$	−2.88592	−0.0934350
$955$	0	0
$956$	−121.638	−3.93405
$957$	−4.48812	−0.145080
$958$	46.6119	1.50596
$959$	14.7167	0.475228
$960$	0	0
$961$	−27.6086	−0.890601
$962$	64.8367	2.09042
$963$	19.8861	0.640822
$964$	69.3369	2.23319
$965$	0	0
$966$	−37.9168	−1.21995
$967$	11.1843	0.359662	0.179831	−	0.983697i	$-0.442445\pi$
$967$	11.1843	0.359662	0.179831	+	0.983697i	$0.442445\pi$
$968$	−47.6937	−1.53293
$969$	4.00956	0.128806
$970$	0	0
$971$	−19.7138	−0.632645	−0.316322	−	0.948652i	$-0.602448\pi$
$971$	−19.7138	−0.632645	−0.316322	+	0.948652i	$0.602448\pi$
$972$	4.01172	0.128676
$973$	−4.26013	−0.136573
$974$	−22.2050	−0.711493
$975$	0	0
$976$	−32.0791	−1.02683
$977$	14.0373	0.449092	0.224546	−	0.974463i	$-0.427910\pi$
$977$	14.0373	0.449092	0.224546	+	0.974463i	$0.427910\pi$
$978$	28.1512	0.900177
$979$	6.66032	0.212865
$980$	0	0
$981$	−15.3703	−0.490736
$982$	12.3824	0.395137
$983$	35.9296	1.14598	0.572988	−	0.819564i	$-0.305784\pi$
$983$	35.9296	1.14598	0.572988	+	0.819564i	$0.305784\pi$
$984$	−11.7811	−0.375567
$985$	0	0
$986$	8.07435	0.257140
$987$	−2.37493	−0.0755948
$988$	97.7236	3.10900
$989$	−40.0450	−1.27336
$990$	0	0
$991$	−30.6333	−0.973098	−0.486549	−	0.873653i	$-0.661745\pi$
$991$	−30.6333	−0.973098	−0.486549	+	0.873653i	$0.661745\pi$
$992$	0.212367	0.00674265
$993$	−31.5489	−1.00117
$994$	−7.98138	−0.253154
$995$	0	0
$996$	23.2052	0.735285
$997$	−36.2124	−1.14686	−0.573429	−	0.819255i	$-0.694387\pi$
$997$	−36.2124	−1.14686	−0.573429	+	0.819255i	$0.694387\pi$
$998$	94.6693	2.99670
$999$	5.14234	0.162696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4425.2.a.bg.1.5		6
5.4	even	2		885.2.a.j.1.2	✓	6
15.14	odd	2		2655.2.a.v.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
885.2.a.j.1.2	✓	6	5.4	even	2
2655.2.a.v.1.5		6	15.14	odd	2
4425.2.a.bg.1.5		6	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)

\(f(q)\)	\(=\)	\(q+2.45188 q^{2} +1.00000 q^{3} +4.01172 q^{4} +2.45188 q^{6} -2.82843 q^{7} +4.93251 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.45188 q^{2} +1.00000 q^{3} +4.01172 q^{4} +2.45188 q^{6} -2.82843 q^{7} +4.93251 q^{8} +1.00000 q^{9} -1.15357 q^{11} +4.01172 q^{12} +5.14234 q^{13} -6.93497 q^{14} +4.07048 q^{16} +0.846426 q^{17} +2.45188 q^{18} +4.73705 q^{19} -2.82843 q^{21} -2.82843 q^{22} +5.46748 q^{23} +4.93251 q^{24} +12.6084 q^{26} +1.00000 q^{27} -11.3469 q^{28} +3.89062 q^{29} +1.84157 q^{31} +0.115318 q^{32} -1.15357 q^{33} +2.07534 q^{34} +4.01172 q^{36} +5.14234 q^{37} +11.6147 q^{38} +5.14234 q^{39} -2.38845 q^{41} -6.93497 q^{42} -7.32422 q^{43} -4.62782 q^{44} +13.4056 q^{46} +0.839664 q^{47} +4.07048 q^{48} +1.00000 q^{49} +0.846426 q^{51} +20.6296 q^{52} -1.17702 q^{53} +2.45188 q^{54} -13.9512 q^{56} +4.73705 q^{57} +9.53934 q^{58} -1.00000 q^{59} -7.88091 q^{61} +4.51531 q^{62} -2.82843 q^{63} -7.85821 q^{64} -2.82843 q^{66} +14.0461 q^{67} +3.39563 q^{68} +5.46748 q^{69} +1.15089 q^{71} +4.93251 q^{72} -2.83519 q^{73} +12.6084 q^{74} +19.0037 q^{76} +3.26280 q^{77} +12.6084 q^{78} -0.763332 q^{79} +1.00000 q^{81} -5.85620 q^{82} +5.78434 q^{83} -11.3469 q^{84} -17.9581 q^{86} +3.89062 q^{87} -5.69001 q^{88} -5.77364 q^{89} -14.5447 q^{91} +21.9340 q^{92} +1.84157 q^{93} +2.05876 q^{94} +0.115318 q^{96} +15.2985 q^{97} +2.45188 q^{98} -1.15357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9	\( 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9} - 2 q^{11} + 8 q^{12} - 2 q^{13} + 12 q^{16} + 10 q^{17} + 4 q^{18} - 2 q^{19} + 12 q^{23} + 12 q^{24} - 8 q^{26} + 6 q^{27} + 16 q^{28} - 12 q^{29} + 8 q^{31} + 28 q^{32} - 2 q^{33} + 8 q^{34} + 8 q^{36} - 2 q^{37} + 24 q^{38} - 2 q^{39} - 4 q^{41} + 18 q^{43} + 4 q^{44} + 16 q^{47} + 12 q^{48} + 6 q^{49} + 10 q^{51} - 12 q^{52} + 30 q^{53} + 4 q^{54} - 2 q^{57} + 16 q^{58} - 6 q^{59} + 4 q^{61} + 16 q^{62} + 20 q^{64} + 30 q^{67} + 20 q^{68} + 12 q^{69} - 24 q^{71} + 12 q^{72} + 6 q^{73} - 8 q^{74} - 4 q^{76} + 16 q^{77} - 8 q^{78} - 2 q^{79} + 6 q^{81} + 20 q^{83} + 16 q^{84} - 12 q^{87} + 16 q^{88} + 14 q^{89} - 48 q^{91} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 28 q^{96} - 4 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9 - 2 * q^11 + 8 * q^12 - 2 * q^13 + 12 * q^16 + 10 * q^17 + 4 * q^18 - 2 * q^19 + 12 * q^23 + 12 * q^24 - 8 * q^26 + 6 * q^27 + 16 * q^28 - 12 * q^29 + 8 * q^31 + 28 * q^32 - 2 * q^33 + 8 * q^34 + 8 * q^36 - 2 * q^37 + 24 * q^38 - 2 * q^39 - 4 * q^41 + 18 * q^43 + 4 * q^44 + 16 * q^47 + 12 * q^48 + 6 * q^49 + 10 * q^51 - 12 * q^52 + 30 * q^53 + 4 * q^54 - 2 * q^57 + 16 * q^58 - 6 * q^59 + 4 * q^61 + 16 * q^62 + 20 * q^64 + 30 * q^67 + 20 * q^68 + 12 * q^69 - 24 * q^71 + 12 * q^72 + 6 * q^73 - 8 * q^74 - 4 * q^76 + 16 * q^77 - 8 * q^78 - 2 * q^79 + 6 * q^81 + 20 * q^83 + 16 * q^84 - 12 * q^87 + 16 * q^88 + 14 * q^89 - 48 * q^91 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 28 * q^96 - 4 * q^97 + 4 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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