LMFDB - Embedded newform 2475.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.82843 q^{4} -2.41421 q^{7} -1.58579 q^{8} +O(q^{10})$	$q+0.414214 q^{2} -1.82843 q^{4} -2.41421 q^{7} -1.58579 q^{8} +1.00000 q^{11} +2.82843 q^{13} -1.00000 q^{14} +3.00000 q^{16} +0.414214 q^{17} +3.58579 q^{19} +0.414214 q^{22} -1.00000 q^{23} +1.17157 q^{26} +4.41421 q^{28} -6.82843 q^{29} +8.48528 q^{31} +4.41421 q^{32} +0.171573 q^{34} -5.82843 q^{37} +1.48528 q^{38} -8.89949 q^{41} -0.343146 q^{43} -1.82843 q^{44} -0.414214 q^{46} -9.48528 q^{47} -1.17157 q^{49} -5.17157 q^{52} +3.65685 q^{53} +3.82843 q^{56} -2.82843 q^{58} -11.0000 q^{59} +3.17157 q^{61} +3.51472 q^{62} -4.17157 q^{64} -11.6569 q^{67} -0.757359 q^{68} -2.17157 q^{71} -3.17157 q^{73} -2.41421 q^{74} -6.55635 q^{76} -2.41421 q^{77} +4.75736 q^{79} -3.68629 q^{82} -12.4853 q^{83} -0.142136 q^{86} -1.58579 q^{88} +7.65685 q^{89} -6.82843 q^{91} +1.82843 q^{92} -3.92893 q^{94} +0.171573 q^{97} -0.485281 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8} + 2 q^{11} - 2 q^{14} + 6 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{22} - 2 q^{23} + 8 q^{26} + 6 q^{28} - 8 q^{29} + 6 q^{32} + 6 q^{34} - 6 q^{37} - 14 q^{38} + 2 q^{41} - 12 q^{43} + 2 q^{44} + 2 q^{46} - 2 q^{47} - 8 q^{49} - 16 q^{52} - 4 q^{53} + 2 q^{56} - 22 q^{59} + 12 q^{61} + 24 q^{62} - 14 q^{64} - 12 q^{67} - 10 q^{68} - 10 q^{71} - 12 q^{73} - 2 q^{74} + 18 q^{76} - 2 q^{77} + 18 q^{79} - 30 q^{82} - 8 q^{83} + 28 q^{86} - 6 q^{88} + 4 q^{89} - 8 q^{91} - 2 q^{92} - 22 q^{94} + 6 q^{97} + 16 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 6 * q^8 + 2 * q^11 - 2 * q^14 + 6 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^22 - 2 * q^23 + 8 * q^26 + 6 * q^28 - 8 * q^29 + 6 * q^32 + 6 * q^34 - 6 * q^37 - 14 * q^38 + 2 * q^41 - 12 * q^43 + 2 * q^44 + 2 * q^46 - 2 * q^47 - 8 * q^49 - 16 * q^52 - 4 * q^53 + 2 * q^56 - 22 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 - 12 * q^67 - 10 * q^68 - 10 * q^71 - 12 * q^73 - 2 * q^74 + 18 * q^76 - 2 * q^77 + 18 * q^79 - 30 * q^82 - 8 * q^83 + 28 * q^86 - 6 * q^88 + 4 * q^89 - 8 * q^91 - 2 * q^92 - 22 * q^94 + 6 * q^97 + 16 * q^98

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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.41421	−0.912487	−0.456243	−	0.889855i	$-0.650805\pi$
$7$	−2.41421	−0.912487	−0.456243	+	0.889855i	$0.650805\pi$
$8$	−1.58579	−0.560660
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	3.00000	0.750000
$17$	0.414214	0.100462	0.0502308	−	0.998738i	$-0.484004\pi$
$17$	0.414214	0.100462	0.0502308	+	0.998738i	$0.484004\pi$
$18$	0	0
$19$	3.58579	0.822636	0.411318	−	0.911492i	$-0.365069\pi$
$19$	3.58579	0.822636	0.411318	+	0.911492i	$0.365069\pi$
$20$	0	0
$21$	0	0
$22$	0.414214	0.0883106
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	0	0
$26$	1.17157	0.229764
$27$	0	0
$28$	4.41421	0.834208
$29$	−6.82843	−1.26801	−0.634004	−	0.773330i	$-0.718590\pi$
$29$	−6.82843	−1.26801	−0.634004	+	0.773330i	$0.718590\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	4.41421	0.780330
$33$	0	0
$34$	0.171573	0.0294245
$35$	0	0
$36$	0	0
$37$	−5.82843	−0.958188	−0.479094	−	0.877764i	$-0.659035\pi$
$37$	−5.82843	−0.958188	−0.479094	+	0.877764i	$0.659035\pi$
$38$	1.48528	0.240944
$39$	0	0
$40$	0	0
$41$	−8.89949	−1.38987	−0.694934	−	0.719074i	$-0.744566\pi$
$41$	−8.89949	−1.38987	−0.694934	+	0.719074i	$0.744566\pi$
$42$	0	0
$43$	−0.343146	−0.0523292	−0.0261646	−	0.999658i	$-0.508329\pi$
$43$	−0.343146	−0.0523292	−0.0261646	+	0.999658i	$0.508329\pi$
$44$	−1.82843	−0.275646
$45$	0	0
$46$	−0.414214	−0.0610725
$47$	−9.48528	−1.38357	−0.691785	−	0.722103i	$-0.743176\pi$
$47$	−9.48528	−1.38357	−0.691785	+	0.722103i	$0.743176\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	0	0
$52$	−5.17157	−0.717168
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	0	0
$56$	3.82843	0.511595
$57$	0	0
$58$	−2.82843	−0.371391
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	3.17157	0.406078	0.203039	−	0.979171i	$-0.434918\pi$
$61$	3.17157	0.406078	0.203039	+	0.979171i	$0.434918\pi$
$62$	3.51472	0.446370
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0	0
$66$	0	0
$67$	−11.6569	−1.42411	−0.712056	−	0.702123i	$-0.752236\pi$
$67$	−11.6569	−1.42411	−0.712056	+	0.702123i	$0.752236\pi$
$68$	−0.757359	−0.0918433
$69$	0	0
$70$	0	0
$71$	−2.17157	−0.257718	−0.128859	−	0.991663i	$-0.541132\pi$
$71$	−2.17157	−0.257718	−0.128859	+	0.991663i	$0.541132\pi$
$72$	0	0
$73$	−3.17157	−0.371205	−0.185602	−	0.982625i	$-0.559424\pi$
$73$	−3.17157	−0.371205	−0.185602	+	0.982625i	$0.559424\pi$
$74$	−2.41421	−0.280647
$75$	0	0
$76$	−6.55635	−0.752065
$77$	−2.41421	−0.275125
$78$	0	0
$79$	4.75736	0.535245	0.267622	−	0.963524i	$-0.413762\pi$
$79$	4.75736	0.535245	0.267622	+	0.963524i	$0.413762\pi$
$80$	0	0
$81$	0	0
$82$	−3.68629	−0.407083
$83$	−12.4853	−1.37044	−0.685219	−	0.728337i	$-0.740293\pi$
$83$	−12.4853	−1.37044	−0.685219	+	0.728337i	$0.740293\pi$
$84$	0	0
$85$	0	0
$86$	−0.142136	−0.0153269
$87$	0	0
$88$	−1.58579	−0.169045
$89$	7.65685	0.811625	0.405812	−	0.913956i	$-0.366989\pi$
$89$	7.65685	0.811625	0.405812	+	0.913956i	$0.366989\pi$
$90$	0	0
$91$	−6.82843	−0.715814
$92$	1.82843	0.190627
$93$	0	0
$94$	−3.92893	−0.405238
$95$	0	0
$96$	0	0
$97$	0.171573	0.0174206	0.00871029	−	0.999962i	$-0.497227\pi$
$97$	0.171573	0.0174206	0.00871029	+	0.999962i	$0.497227\pi$
$98$	−0.485281	−0.0490208
$99$	0	0
$100$	0	0
$101$	4.89949	0.487518	0.243759	−	0.969836i	$-0.421619\pi$
$101$	4.89949	0.487518	0.243759	+	0.969836i	$0.421619\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	−4.48528	−0.439818
$105$	0	0
$106$	1.51472	0.147122
$107$	−17.3137	−1.67378	−0.836890	−	0.547372i	$-0.815628\pi$
$107$	−17.3137	−1.67378	−0.836890	+	0.547372i	$0.815628\pi$
$108$	0	0
$109$	−17.3137	−1.65835	−0.829176	−	0.558987i	$-0.811190\pi$
$109$	−17.3137	−1.65835	−0.829176	+	0.558987i	$0.811190\pi$
$110$	0	0
$111$	0	0
$112$	−7.24264	−0.684365
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	12.4853	1.15923
$117$	0	0
$118$	−4.55635	−0.419446
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.31371	0.118938
$123$	0	0
$124$	−15.5147	−1.39326
$125$	0	0
$126$	0	0
$127$	−1.24264	−0.110267	−0.0551333	−	0.998479i	$-0.517558\pi$
$127$	−1.24264	−0.110267	−0.0551333	+	0.998479i	$0.517558\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	0	0
$131$	−1.17157	−0.102361	−0.0511804	−	0.998689i	$-0.516298\pi$
$131$	−1.17157	−0.102361	−0.0511804	+	0.998689i	$0.516298\pi$
$132$	0	0
$133$	−8.65685	−0.750644
$134$	−4.82843	−0.417113
$135$	0	0
$136$	−0.656854	−0.0563248
$137$	−16.1421	−1.37912	−0.689558	−	0.724231i	$-0.742195\pi$
$137$	−16.1421	−1.37912	−0.689558	+	0.724231i	$0.742195\pi$
$138$	0	0
$139$	14.9706	1.26979	0.634893	−	0.772600i	$-0.281044\pi$
$139$	14.9706	1.26979	0.634893	+	0.772600i	$0.281044\pi$
$140$	0	0
$141$	0	0
$142$	−0.899495	−0.0754839
$143$	2.82843	0.236525
$144$	0	0
$145$	0	0
$146$	−1.31371	−0.108723
$147$	0	0
$148$	10.6569	0.875988
$149$	17.7279	1.45233	0.726164	−	0.687522i	$-0.241301\pi$
$149$	17.7279	1.45233	0.726164	+	0.687522i	$0.241301\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−5.68629	−0.461219
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	1.97056	0.156770
$159$	0	0
$160$	0	0
$161$	2.41421	0.190267
$162$	0	0
$163$	−23.7990	−1.86408	−0.932040	−	0.362354i	$-0.881973\pi$
$163$	−23.7990	−1.86408	−0.932040	+	0.362354i	$0.881973\pi$
$164$	16.2721	1.27064
$165$	0	0
$166$	−5.17157	−0.401392
$167$	17.7990	1.37733	0.688664	−	0.725081i	$-0.258198\pi$
$167$	17.7990	1.37733	0.688664	+	0.725081i	$0.258198\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0.627417	0.0478401
$173$	−18.5563	−1.41081	−0.705407	−	0.708803i	$-0.749236\pi$
$173$	−18.5563	−1.41081	−0.705407	+	0.708803i	$0.749236\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	3.17157	0.237719
$179$	22.7990	1.70408	0.852038	−	0.523480i	$-0.175366\pi$
$179$	22.7990	1.70408	0.852038	+	0.523480i	$0.175366\pi$
$180$	0	0
$181$	11.9706	0.889765	0.444882	−	0.895589i	$-0.353245\pi$
$181$	11.9706	0.889765	0.444882	+	0.895589i	$0.353245\pi$
$182$	−2.82843	−0.209657
$183$	0	0
$184$	1.58579	0.116906
$185$	0	0
$186$	0	0
$187$	0.414214	0.0302903
$188$	17.3431	1.26488
$189$	0	0
$190$	0	0
$191$	−11.8284	−0.855875	−0.427937	−	0.903808i	$-0.640760\pi$
$191$	−11.8284	−0.855875	−0.427937	+	0.903808i	$0.640760\pi$
$192$	0	0
$193$	−19.3137	−1.39023	−0.695116	−	0.718898i	$-0.744647\pi$
$193$	−19.3137	−1.39023	−0.695116	+	0.718898i	$0.744647\pi$
$194$	0.0710678	0.00510237
$195$	0	0
$196$	2.14214	0.153010
$197$	−13.2426	−0.943499	−0.471750	−	0.881733i	$-0.656377\pi$
$197$	−13.2426	−0.943499	−0.471750	+	0.881733i	$0.656377\pi$
$198$	0	0
$199$	5.17157	0.366603	0.183302	−	0.983057i	$-0.441322\pi$
$199$	5.17157	0.366603	0.183302	+	0.983057i	$0.441322\pi$
$200$	0	0
$201$	0	0
$202$	2.02944	0.142791
$203$	16.4853	1.15704
$204$	0	0
$205$	0	0
$206$	−0.970563	−0.0676223
$207$	0	0
$208$	8.48528	0.588348
$209$	3.58579	0.248034
$210$	0	0
$211$	9.31371	0.641182	0.320591	−	0.947218i	$-0.396118\pi$
$211$	9.31371	0.641182	0.320591	+	0.947218i	$0.396118\pi$
$212$	−6.68629	−0.459216
$213$	0	0
$214$	−7.17157	−0.490239
$215$	0	0
$216$	0	0
$217$	−20.4853	−1.39063
$218$	−7.17157	−0.485720
$219$	0	0
$220$	0	0
$221$	1.17157	0.0788085
$222$	0	0
$223$	−26.8284	−1.79656	−0.898282	−	0.439419i	$-0.855184\pi$
$223$	−26.8284	−1.79656	−0.898282	+	0.439419i	$0.855184\pi$
$224$	−10.6569	−0.712041
$225$	0	0
$226$	4.14214	0.275531
$227$	1.51472	0.100535	0.0502677	−	0.998736i	$-0.483993\pi$
$227$	1.51472	0.100535	0.0502677	+	0.998736i	$0.483993\pi$
$228$	0	0
$229$	−19.4853	−1.28762	−0.643812	−	0.765184i	$-0.722648\pi$
$229$	−19.4853	−1.28762	−0.643812	+	0.765184i	$0.722648\pi$
$230$	0	0
$231$	0	0
$232$	10.8284	0.710921
$233$	14.5563	0.953618	0.476809	−	0.879007i	$-0.341793\pi$
$233$	14.5563	0.953618	0.476809	+	0.879007i	$0.341793\pi$
$234$	0	0
$235$	0	0
$236$	20.1127	1.30923
$237$	0	0
$238$	−0.414214	−0.0268495
$239$	−12.3431	−0.798412	−0.399206	−	0.916861i	$-0.630714\pi$
$239$	−12.3431	−0.798412	−0.399206	+	0.916861i	$0.630714\pi$
$240$	0	0
$241$	−14.1421	−0.910975	−0.455488	−	0.890242i	$-0.650535\pi$
$241$	−14.1421	−0.910975	−0.455488	+	0.890242i	$0.650535\pi$
$242$	0.414214	0.0266267
$243$	0	0
$244$	−5.79899	−0.371242
$245$	0	0
$246$	0	0
$247$	10.1421	0.645329
$248$	−13.4558	−0.854447
$249$	0	0
$250$	0	0
$251$	24.9706	1.57613	0.788064	−	0.615593i	$-0.211084\pi$
$251$	24.9706	1.57613	0.788064	+	0.615593i	$0.211084\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	−0.514719	−0.0322963
$255$	0	0
$256$	3.97056	0.248160
$257$	13.3137	0.830486	0.415243	−	0.909710i	$-0.363696\pi$
$257$	13.3137	0.830486	0.415243	+	0.909710i	$0.363696\pi$
$258$	0	0
$259$	14.0711	0.874334
$260$	0	0
$261$	0	0
$262$	−0.485281	−0.0299808
$263$	10.9706	0.676474	0.338237	−	0.941061i	$-0.390169\pi$
$263$	10.9706	0.676474	0.338237	+	0.941061i	$0.390169\pi$
$264$	0	0
$265$	0	0
$266$	−3.58579	−0.219859
$267$	0	0
$268$	21.3137	1.30194
$269$	−23.7990	−1.45105	−0.725525	−	0.688196i	$-0.758403\pi$
$269$	−23.7990	−1.45105	−0.725525	+	0.688196i	$0.758403\pi$
$270$	0	0
$271$	−10.8995	−0.662097	−0.331049	−	0.943614i	$-0.607402\pi$
$271$	−10.8995	−0.662097	−0.331049	+	0.943614i	$0.607402\pi$
$272$	1.24264	0.0753462
$273$	0	0
$274$	−6.68629	−0.403934
$275$	0	0
$276$	0	0
$277$	−0.828427	−0.0497754	−0.0248877	−	0.999690i	$-0.507923\pi$
$277$	−0.828427	−0.0497754	−0.0248877	+	0.999690i	$0.507923\pi$
$278$	6.20101	0.371912
$279$	0	0
$280$	0	0
$281$	−17.9289	−1.06955	−0.534775	−	0.844994i	$-0.679604\pi$
$281$	−17.9289	−1.06955	−0.534775	+	0.844994i	$0.679604\pi$
$282$	0	0
$283$	18.8995	1.12346	0.561729	−	0.827321i	$-0.310136\pi$
$283$	18.8995	1.12346	0.561729	+	0.827321i	$0.310136\pi$
$284$	3.97056	0.235610
$285$	0	0
$286$	1.17157	0.0692766
$287$	21.4853	1.26824
$288$	0	0
$289$	−16.8284	−0.989907
$290$	0	0
$291$	0	0
$292$	5.79899	0.339360
$293$	−20.4142	−1.19261	−0.596306	−	0.802758i	$-0.703365\pi$
$293$	−20.4142	−1.19261	−0.596306	+	0.802758i	$0.703365\pi$
$294$	0	0
$295$	0	0
$296$	9.24264	0.537218
$297$	0	0
$298$	7.34315	0.425377
$299$	−2.82843	−0.163572
$300$	0	0
$301$	0.828427	0.0477497
$302$	5.79899	0.333694
$303$	0	0
$304$	10.7574	0.616977
$305$	0	0
$306$	0	0
$307$	29.3137	1.67302	0.836511	−	0.547950i	$-0.184592\pi$
$307$	29.3137	1.67302	0.836511	+	0.547950i	$0.184592\pi$
$308$	4.41421	0.251523
$309$	0	0
$310$	0	0
$311$	−2.34315	−0.132868	−0.0664338	−	0.997791i	$-0.521162\pi$
$311$	−2.34315	−0.132868	−0.0664338	+	0.997791i	$0.521162\pi$
$312$	0	0
$313$	1.14214	0.0645573	0.0322787	−	0.999479i	$-0.489724\pi$
$313$	1.14214	0.0645573	0.0322787	+	0.999479i	$0.489724\pi$
$314$	2.48528	0.140253
$315$	0	0
$316$	−8.69848	−0.489328
$317$	−25.1716	−1.41378	−0.706888	−	0.707325i	$-0.749902\pi$
$317$	−25.1716	−1.41378	−0.706888	+	0.707325i	$0.749902\pi$
$318$	0	0
$319$	−6.82843	−0.382319
$320$	0	0
$321$	0	0
$322$	1.00000	0.0557278
$323$	1.48528	0.0826433
$324$	0	0
$325$	0	0
$326$	−9.85786	−0.545977
$327$	0	0
$328$	14.1127	0.779243
$329$	22.8995	1.26249
$330$	0	0
$331$	−3.85786	−0.212047	−0.106024	−	0.994364i	$-0.533812\pi$
$331$	−3.85786	−0.212047	−0.106024	+	0.994364i	$0.533812\pi$
$332$	22.8284	1.25287
$333$	0	0
$334$	7.37258	0.403410
$335$	0	0
$336$	0	0
$337$	−24.1421	−1.31511	−0.657553	−	0.753408i	$-0.728408\pi$
$337$	−24.1421	−1.31511	−0.657553	+	0.753408i	$0.728408\pi$
$338$	−2.07107	−0.112651
$339$	0	0
$340$	0	0
$341$	8.48528	0.459504
$342$	0	0
$343$	19.7279	1.06521
$344$	0.544156	0.0293389
$345$	0	0
$346$	−7.68629	−0.413218
$347$	26.8284	1.44023	0.720113	−	0.693857i	$-0.244090\pi$
$347$	26.8284	1.44023	0.720113	+	0.693857i	$0.244090\pi$
$348$	0	0
$349$	14.4853	0.775379	0.387690	−	0.921790i	$-0.373273\pi$
$349$	14.4853	0.775379	0.387690	+	0.921790i	$0.373273\pi$
$350$	0	0
$351$	0	0
$352$	4.41421	0.235278
$353$	−12.4853	−0.664524	−0.332262	−	0.943187i	$-0.607812\pi$
$353$	−12.4853	−0.664524	−0.332262	+	0.943187i	$0.607812\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	9.44365	0.499112
$359$	−32.4853	−1.71451	−0.857254	−	0.514894i	$-0.827831\pi$
$359$	−32.4853	−1.71451	−0.857254	+	0.514894i	$0.827831\pi$
$360$	0	0
$361$	−6.14214	−0.323270
$362$	4.95837	0.260606
$363$	0	0
$364$	12.4853	0.654407
$365$	0	0
$366$	0	0
$367$	−21.3137	−1.11257	−0.556283	−	0.830993i	$-0.687773\pi$
$367$	−21.3137	−1.11257	−0.556283	+	0.830993i	$0.687773\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	0	0
$371$	−8.82843	−0.458349
$372$	0	0
$373$	−12.3431	−0.639104	−0.319552	−	0.947569i	$-0.603532\pi$
$373$	−12.3431	−0.639104	−0.319552	+	0.947569i	$0.603532\pi$
$374$	0.171573	0.00887182
$375$	0	0
$376$	15.0416	0.775713
$377$	−19.3137	−0.994707
$378$	0	0
$379$	14.8284	0.761685	0.380843	−	0.924640i	$-0.375634\pi$
$379$	14.8284	0.761685	0.380843	+	0.924640i	$0.375634\pi$
$380$	0	0
$381$	0	0
$382$	−4.89949	−0.250680
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	0	0
$388$	−0.313708	−0.0159261
$389$	6.34315	0.321610	0.160805	−	0.986986i	$-0.448591\pi$
$389$	6.34315	0.321610	0.160805	+	0.986986i	$0.448591\pi$
$390$	0	0
$391$	−0.414214	−0.0209477
$392$	1.85786	0.0938363
$393$	0	0
$394$	−5.48528	−0.276344
$395$	0	0
$396$	0	0
$397$	31.9411	1.60308	0.801540	−	0.597942i	$-0.204015\pi$
$397$	31.9411	1.60308	0.801540	+	0.597942i	$0.204015\pi$
$398$	2.14214	0.107376
$399$	0	0
$400$	0	0
$401$	7.79899	0.389463	0.194731	−	0.980857i	$-0.437616\pi$
$401$	7.79899	0.389463	0.194731	+	0.980857i	$0.437616\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	−8.95837	−0.445696
$405$	0	0
$406$	6.82843	0.338889
$407$	−5.82843	−0.288904
$408$	0	0
$409$	24.1421	1.19375	0.596876	−	0.802334i	$-0.296408\pi$
$409$	24.1421	1.19375	0.596876	+	0.802334i	$0.296408\pi$
$410$	0	0
$411$	0	0
$412$	4.28427	0.211071
$413$	26.5563	1.30675
$414$	0	0
$415$	0	0
$416$	12.4853	0.612141
$417$	0	0
$418$	1.48528	0.0726475
$419$	8.51472	0.415971	0.207986	−	0.978132i	$-0.433309\pi$
$419$	8.51472	0.415971	0.207986	+	0.978132i	$0.433309\pi$
$420$	0	0
$421$	−27.0000	−1.31590	−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000	−1.31590	−0.657950	+	0.753062i	$0.728576\pi$
$422$	3.85786	0.187798
$423$	0	0
$424$	−5.79899	−0.281624
$425$	0	0
$426$	0	0
$427$	−7.65685	−0.370541
$428$	31.6569	1.53019
$429$	0	0
$430$	0	0
$431$	−6.82843	−0.328914	−0.164457	−	0.986384i	$-0.552587\pi$
$431$	−6.82843	−0.328914	−0.164457	+	0.986384i	$0.552587\pi$
$432$	0	0
$433$	9.31371	0.447588	0.223794	−	0.974636i	$-0.428156\pi$
$433$	9.31371	0.447588	0.223794	+	0.974636i	$0.428156\pi$
$434$	−8.48528	−0.407307
$435$	0	0
$436$	31.6569	1.51609
$437$	−3.58579	−0.171531
$438$	0	0
$439$	27.7279	1.32338	0.661691	−	0.749777i	$-0.269839\pi$
$439$	27.7279	1.32338	0.661691	+	0.749777i	$0.269839\pi$
$440$	0	0
$441$	0	0
$442$	0.485281	0.0230825
$443$	−25.9706	−1.23390	−0.616949	−	0.787003i	$-0.711632\pi$
$443$	−25.9706	−1.23390	−0.616949	+	0.787003i	$0.711632\pi$
$444$	0	0
$445$	0	0
$446$	−11.1127	−0.526202
$447$	0	0
$448$	10.0711	0.475813
$449$	−10.4853	−0.494831	−0.247416	−	0.968909i	$-0.579581\pi$
$449$	−10.4853	−0.494831	−0.247416	+	0.968909i	$0.579581\pi$
$450$	0	0
$451$	−8.89949	−0.419061
$452$	−18.2843	−0.860020
$453$	0	0
$454$	0.627417	0.0294461
$455$	0	0
$456$	0	0
$457$	32.1421	1.50355	0.751773	−	0.659422i	$-0.229199\pi$
$457$	32.1421	1.50355	0.751773	+	0.659422i	$0.229199\pi$
$458$	−8.07107	−0.377136
$459$	0	0
$460$	0	0
$461$	−40.7696	−1.89883	−0.949414	−	0.314028i	$-0.898321\pi$
$461$	−40.7696	−1.89883	−0.949414	+	0.314028i	$0.898321\pi$
$462$	0	0
$463$	1.02944	0.0478420	0.0239210	−	0.999714i	$-0.492385\pi$
$463$	1.02944	0.0478420	0.0239210	+	0.999714i	$0.492385\pi$
$464$	−20.4853	−0.951005
$465$	0	0
$466$	6.02944	0.279308
$467$	34.6274	1.60237	0.801183	−	0.598420i	$-0.204204\pi$
$467$	34.6274	1.60237	0.801183	+	0.598420i	$0.204204\pi$
$468$	0	0
$469$	28.1421	1.29948
$470$	0	0
$471$	0	0
$472$	17.4437	0.802909
$473$	−0.343146	−0.0157779
$474$	0	0
$475$	0	0
$476$	1.82843	0.0838058
$477$	0	0
$478$	−5.11270	−0.233849
$479$	7.51472	0.343356	0.171678	−	0.985153i	$-0.445081\pi$
$479$	7.51472	0.343356	0.171678	+	0.985153i	$0.445081\pi$
$480$	0	0
$481$	−16.4853	−0.751664
$482$	−5.85786	−0.266818
$483$	0	0
$484$	−1.82843	−0.0831103
$485$	0	0
$486$	0	0
$487$	10.4853	0.475133	0.237567	−	0.971371i	$-0.423650\pi$
$487$	10.4853	0.475133	0.237567	+	0.971371i	$0.423650\pi$
$488$	−5.02944	−0.227672
$489$	0	0
$490$	0	0
$491$	−4.14214	−0.186932	−0.0934660	−	0.995622i	$-0.529795\pi$
$491$	−4.14214	−0.186932	−0.0934660	+	0.995622i	$0.529795\pi$
$492$	0	0
$493$	−2.82843	−0.127386
$494$	4.20101	0.189012
$495$	0	0
$496$	25.4558	1.14300
$497$	5.24264	0.235165
$498$	0	0
$499$	40.8284	1.82773	0.913866	−	0.406017i	$-0.133082\pi$
$499$	40.8284	1.82773	0.913866	+	0.406017i	$0.133082\pi$
$500$	0	0
$501$	0	0
$502$	10.3431	0.461637
$503$	−22.2843	−0.993607	−0.496803	−	0.867863i	$-0.665493\pi$
$503$	−22.2843	−0.993607	−0.496803	+	0.867863i	$0.665493\pi$
$504$	0	0
$505$	0	0
$506$	−0.414214	−0.0184140
$507$	0	0
$508$	2.27208	0.100807
$509$	40.6274	1.80078	0.900389	−	0.435085i	$-0.143282\pi$
$509$	40.6274	1.80078	0.900389	+	0.435085i	$0.143282\pi$
$510$	0	0
$511$	7.65685	0.338719
$512$	22.7574	1.00574
$513$	0	0
$514$	5.51472	0.243244
$515$	0	0
$516$	0	0
$517$	−9.48528	−0.417162
$518$	5.82843	0.256086
$519$	0	0
$520$	0	0
$521$	7.85786	0.344259	0.172130	−	0.985074i	$-0.444935\pi$
$521$	7.85786	0.344259	0.172130	+	0.985074i	$0.444935\pi$
$522$	0	0
$523$	−0.213203	−0.00932274	−0.00466137	−	0.999989i	$-0.501484\pi$
$523$	−0.213203	−0.00932274	−0.00466137	+	0.999989i	$0.501484\pi$
$524$	2.14214	0.0935796
$525$	0	0
$526$	4.54416	0.198135
$527$	3.51472	0.153104
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	15.8284	0.686249
$533$	−25.1716	−1.09030
$534$	0	0
$535$	0	0
$536$	18.4853	0.798443
$537$	0	0
$538$	−9.85786	−0.425003
$539$	−1.17157	−0.0504632
$540$	0	0
$541$	−11.3137	−0.486414	−0.243207	−	0.969974i	$-0.578199\pi$
$541$	−11.3137	−0.486414	−0.243207	+	0.969974i	$0.578199\pi$
$542$	−4.51472	−0.193924
$543$	0	0
$544$	1.82843	0.0783932
$545$	0	0
$546$	0	0
$547$	−17.8701	−0.764068	−0.382034	−	0.924148i	$-0.624776\pi$
$547$	−17.8701	−0.764068	−0.382034	+	0.924148i	$0.624776\pi$
$548$	29.5147	1.26081
$549$	0	0
$550$	0	0
$551$	−24.4853	−1.04311
$552$	0	0
$553$	−11.4853	−0.488404
$554$	−0.343146	−0.0145789
$555$	0	0
$556$	−27.3726	−1.16086
$557$	10.8284	0.458815	0.229408	−	0.973330i	$-0.426321\pi$
$557$	10.8284	0.458815	0.229408	+	0.973330i	$0.426321\pi$
$558$	0	0
$559$	−0.970563	−0.0410504
$560$	0	0
$561$	0	0
$562$	−7.42641	−0.313264
$563$	7.31371	0.308236	0.154118	−	0.988052i	$-0.450746\pi$
$563$	7.31371	0.308236	0.154118	+	0.988052i	$0.450746\pi$
$564$	0	0
$565$	0	0
$566$	7.82843	0.329053
$567$	0	0
$568$	3.44365	0.144492
$569$	−6.75736	−0.283283	−0.141642	−	0.989918i	$-0.545238\pi$
$569$	−6.75736	−0.283283	−0.141642	+	0.989918i	$0.545238\pi$
$570$	0	0
$571$	−42.9706	−1.79826	−0.899131	−	0.437680i	$-0.855800\pi$
$571$	−42.9706	−1.79826	−0.899131	+	0.437680i	$0.855800\pi$
$572$	−5.17157	−0.216234
$573$	0	0
$574$	8.89949	0.371458
$575$	0	0
$576$	0	0
$577$	−9.97056	−0.415080	−0.207540	−	0.978227i	$-0.566546\pi$
$577$	−9.97056	−0.415080	−0.207540	+	0.978227i	$0.566546\pi$
$578$	−6.97056	−0.289937
$579$	0	0
$580$	0	0
$581$	30.1421	1.25051
$582$	0	0
$583$	3.65685	0.151451
$584$	5.02944	0.208120
$585$	0	0
$586$	−8.45584	−0.349308
$587$	25.3431	1.04602	0.523012	−	0.852325i	$-0.324808\pi$
$587$	25.3431	1.04602	0.523012	+	0.852325i	$0.324808\pi$
$588$	0	0
$589$	30.4264	1.25370
$590$	0	0
$591$	0	0
$592$	−17.4853	−0.718641
$593$	35.7990	1.47009	0.735044	−	0.678019i	$-0.237161\pi$
$593$	35.7990	1.47009	0.735044	+	0.678019i	$0.237161\pi$
$594$	0	0
$595$	0	0
$596$	−32.4142	−1.32774
$597$	0	0
$598$	−1.17157	−0.0479092
$599$	−13.6863	−0.559207	−0.279603	−	0.960116i	$-0.590203\pi$
$599$	−13.6863	−0.559207	−0.279603	+	0.960116i	$0.590203\pi$
$600$	0	0
$601$	−9.17157	−0.374116	−0.187058	−	0.982349i	$-0.559895\pi$
$601$	−9.17157	−0.374116	−0.187058	+	0.982349i	$0.559895\pi$
$602$	0.343146	0.0139856
$603$	0	0
$604$	−25.5980	−1.04157
$605$	0	0
$606$	0	0
$607$	30.9706	1.25706	0.628528	−	0.777787i	$-0.283658\pi$
$607$	30.9706	1.25706	0.628528	+	0.777787i	$0.283658\pi$
$608$	15.8284	0.641927
$609$	0	0
$610$	0	0
$611$	−26.8284	−1.08536
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	12.1421	0.490017
$615$	0	0
$616$	3.82843	0.154252
$617$	38.1421	1.53554	0.767772	−	0.640723i	$-0.221365\pi$
$617$	38.1421	1.53554	0.767772	+	0.640723i	$0.221365\pi$
$618$	0	0
$619$	−6.62742	−0.266378	−0.133189	−	0.991091i	$-0.542522\pi$
$619$	−6.62742	−0.266378	−0.133189	+	0.991091i	$0.542522\pi$
$620$	0	0
$621$	0	0
$622$	−0.970563	−0.0389160
$623$	−18.4853	−0.740597
$624$	0	0
$625$	0	0
$626$	0.473088	0.0189084
$627$	0	0
$628$	−10.9706	−0.437773
$629$	−2.41421	−0.0962610
$630$	0	0
$631$	−18.6274	−0.741546	−0.370773	−	0.928724i	$-0.620907\pi$
$631$	−18.6274	−0.741546	−0.370773	+	0.928724i	$0.620907\pi$
$632$	−7.54416	−0.300090
$633$	0	0
$634$	−10.4264	−0.414086
$635$	0	0
$636$	0	0
$637$	−3.31371	−0.131294
$638$	−2.82843	−0.111979
$639$	0	0
$640$	0	0
$641$	25.5147	1.00777	0.503885	−	0.863771i	$-0.331903\pi$
$641$	25.5147	1.00777	0.503885	+	0.863771i	$0.331903\pi$
$642$	0	0
$643$	0.970563	0.0382753	0.0191376	−	0.999817i	$-0.493908\pi$
$643$	0.970563	0.0382753	0.0191376	+	0.999817i	$0.493908\pi$
$644$	−4.41421	−0.173944
$645$	0	0
$646$	0.615224	0.0242057
$647$	−28.6569	−1.12662	−0.563309	−	0.826247i	$-0.690472\pi$
$647$	−28.6569	−1.12662	−0.563309	+	0.826247i	$0.690472\pi$
$648$	0	0
$649$	−11.0000	−0.431788
$650$	0	0
$651$	0	0
$652$	43.5147	1.70417
$653$	23.5147	0.920202	0.460101	−	0.887867i	$-0.347813\pi$
$653$	23.5147	0.920202	0.460101	+	0.887867i	$0.347813\pi$
$654$	0	0
$655$	0	0
$656$	−26.6985	−1.04240
$657$	0	0
$658$	9.48528	0.369775
$659$	47.1127	1.83525	0.917625	−	0.397447i	$-0.130104\pi$
$659$	47.1127	1.83525	0.917625	+	0.397447i	$0.130104\pi$
$660$	0	0
$661$	−23.3431	−0.907943	−0.453972	−	0.891016i	$-0.649993\pi$
$661$	−23.3431	−0.907943	−0.453972	+	0.891016i	$0.649993\pi$
$662$	−1.59798	−0.0621072
$663$	0	0
$664$	19.7990	0.768350
$665$	0	0
$666$	0	0
$667$	6.82843	0.264398
$668$	−32.5442	−1.25917
$669$	0	0
$670$	0	0
$671$	3.17157	0.122437
$672$	0	0
$673$	0.343146	0.0132273	0.00661365	−	0.999978i	$-0.497895\pi$
$673$	0.343146	0.0132273	0.00661365	+	0.999978i	$0.497895\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	9.14214	0.351621
$677$	−23.5147	−0.903744	−0.451872	−	0.892083i	$-0.649244\pi$
$677$	−23.5147	−0.903744	−0.451872	+	0.892083i	$0.649244\pi$
$678$	0	0
$679$	−0.414214	−0.0158961
$680$	0	0
$681$	0	0
$682$	3.51472	0.134586
$683$	−12.5147	−0.478862	−0.239431	−	0.970913i	$-0.576961\pi$
$683$	−12.5147	−0.478862	−0.239431	+	0.970913i	$0.576961\pi$
$684$	0	0
$685$	0	0
$686$	8.17157	0.311992
$687$	0	0
$688$	−1.02944	−0.0392469
$689$	10.3431	0.394042
$690$	0	0
$691$	−35.4558	−1.34880	−0.674402	−	0.738364i	$-0.735598\pi$
$691$	−35.4558	−1.34880	−0.674402	+	0.738364i	$0.735598\pi$
$692$	33.9289	1.28978
$693$	0	0
$694$	11.1127	0.421832
$695$	0	0
$696$	0	0
$697$	−3.68629	−0.139628
$698$	6.00000	0.227103
$699$	0	0
$700$	0	0
$701$	41.3848	1.56308	0.781541	−	0.623854i	$-0.214434\pi$
$701$	41.3848	1.56308	0.781541	+	0.623854i	$0.214434\pi$
$702$	0	0
$703$	−20.8995	−0.788239
$704$	−4.17157	−0.157222
$705$	0	0
$706$	−5.17157	−0.194635
$707$	−11.8284	−0.444854
$708$	0	0
$709$	−29.1421	−1.09446	−0.547228	−	0.836984i	$-0.684317\pi$
$709$	−29.1421	−1.09446	−0.547228	+	0.836984i	$0.684317\pi$
$710$	0	0
$711$	0	0
$712$	−12.1421	−0.455046
$713$	−8.48528	−0.317776
$714$	0	0
$715$	0	0
$716$	−41.6863	−1.55789
$717$	0	0
$718$	−13.4558	−0.502168
$719$	−9.65685	−0.360140	−0.180070	−	0.983654i	$-0.557632\pi$
$719$	−9.65685	−0.360140	−0.180070	+	0.983654i	$0.557632\pi$
$720$	0	0
$721$	5.65685	0.210672
$722$	−2.54416	−0.0946837
$723$	0	0
$724$	−21.8873	−0.813435
$725$	0	0
$726$	0	0
$727$	16.9706	0.629403	0.314702	−	0.949191i	$-0.398096\pi$
$727$	16.9706	0.629403	0.314702	+	0.949191i	$0.398096\pi$
$728$	10.8284	0.401328
$729$	0	0
$730$	0	0
$731$	−0.142136	−0.00525708
$732$	0	0
$733$	32.1421	1.18720	0.593598	−	0.804761i	$-0.297707\pi$
$733$	32.1421	1.18720	0.593598	+	0.804761i	$0.297707\pi$
$734$	−8.82843	−0.325863
$735$	0	0
$736$	−4.41421	−0.162710
$737$	−11.6569	−0.429386
$738$	0	0
$739$	20.4142	0.750949	0.375474	−	0.926833i	$-0.377480\pi$
$739$	20.4142	0.750949	0.375474	+	0.926833i	$0.377480\pi$
$740$	0	0
$741$	0	0
$742$	−3.65685	−0.134247
$743$	31.1127	1.14141	0.570707	−	0.821154i	$-0.306669\pi$
$743$	31.1127	1.14141	0.570707	+	0.821154i	$0.306669\pi$
$744$	0	0
$745$	0	0
$746$	−5.11270	−0.187189
$747$	0	0
$748$	−0.757359	−0.0276918
$749$	41.7990	1.52730
$750$	0	0
$751$	31.5980	1.15303	0.576513	−	0.817088i	$-0.304413\pi$
$751$	31.5980	1.15303	0.576513	+	0.817088i	$0.304413\pi$
$752$	−28.4558	−1.03768
$753$	0	0
$754$	−8.00000	−0.291343
$755$	0	0
$756$	0	0
$757$	−10.6863	−0.388400	−0.194200	−	0.980962i	$-0.562211\pi$
$757$	−10.6863	−0.388400	−0.194200	+	0.980962i	$0.562211\pi$
$758$	6.14214	0.223092
$759$	0	0
$760$	0	0
$761$	5.17157	0.187469	0.0937347	−	0.995597i	$-0.470119\pi$
$761$	5.17157	0.187469	0.0937347	+	0.995597i	$0.470119\pi$
$762$	0	0
$763$	41.7990	1.51323
$764$	21.6274	0.782452
$765$	0	0
$766$	−8.28427	−0.299323
$767$	−31.1127	−1.12341
$768$	0	0
$769$	−7.65685	−0.276113	−0.138057	−	0.990424i	$-0.544086\pi$
$769$	−7.65685	−0.276113	−0.138057	+	0.990424i	$0.544086\pi$
$770$	0	0
$771$	0	0
$772$	35.3137	1.27097
$773$	0.828427	0.0297965	0.0148982	−	0.999889i	$-0.495258\pi$
$773$	0.828427	0.0297965	0.0148982	+	0.999889i	$0.495258\pi$
$774$	0	0
$775$	0	0
$776$	−0.272078	−0.00976703
$777$	0	0
$778$	2.62742	0.0941975
$779$	−31.9117	−1.14335
$780$	0	0
$781$	−2.17157	−0.0777050
$782$	−0.171573	−0.00613543
$783$	0	0
$784$	−3.51472	−0.125526
$785$	0	0
$786$	0	0
$787$	−7.92893	−0.282636	−0.141318	−	0.989964i	$-0.545134\pi$
$787$	−7.92893	−0.282636	−0.141318	+	0.989964i	$0.545134\pi$
$788$	24.2132	0.862560
$789$	0	0
$790$	0	0
$791$	−24.1421	−0.858396
$792$	0	0
$793$	8.97056	0.318554
$794$	13.2304	0.469531
$795$	0	0
$796$	−9.45584	−0.335154
$797$	−40.9706	−1.45125	−0.725626	−	0.688089i	$-0.758450\pi$
$797$	−40.9706	−1.45125	−0.725626	+	0.688089i	$0.758450\pi$
$798$	0	0
$799$	−3.92893	−0.138996
$800$	0	0
$801$	0	0
$802$	3.23045	0.114071
$803$	−3.17157	−0.111922
$804$	0	0
$805$	0	0
$806$	9.94113	0.350161
$807$	0	0
$808$	−7.76955	−0.273332
$809$	−7.72792	−0.271699	−0.135850	−	0.990729i	$-0.543376\pi$
$809$	−7.72792	−0.271699	−0.135850	+	0.990729i	$0.543376\pi$
$810$	0	0
$811$	10.2132	0.358634	0.179317	−	0.983791i	$-0.442611\pi$
$811$	10.2132	0.358634	0.179317	+	0.983791i	$0.442611\pi$
$812$	−30.1421	−1.05778
$813$	0	0
$814$	−2.41421	−0.0846181
$815$	0	0
$816$	0	0
$817$	−1.23045	−0.0430479
$818$	10.0000	0.349642
$819$	0	0
$820$	0	0
$821$	−15.7990	−0.551389	−0.275694	−	0.961245i	$-0.588908\pi$
$821$	−15.7990	−0.551389	−0.275694	+	0.961245i	$0.588908\pi$
$822$	0	0
$823$	−14.9706	−0.521841	−0.260921	−	0.965360i	$-0.584026\pi$
$823$	−14.9706	−0.521841	−0.260921	+	0.965360i	$0.584026\pi$
$824$	3.71573	0.129444
$825$	0	0
$826$	11.0000	0.382739
$827$	12.6863	0.441146	0.220573	−	0.975371i	$-0.429207\pi$
$827$	12.6863	0.441146	0.220573	+	0.975371i	$0.429207\pi$
$828$	0	0
$829$	−47.9411	−1.66506	−0.832532	−	0.553977i	$-0.813110\pi$
$829$	−47.9411	−1.66506	−0.832532	+	0.553977i	$0.813110\pi$
$830$	0	0
$831$	0	0
$832$	−11.7990	−0.409056
$833$	−0.485281	−0.0168140
$834$	0	0
$835$	0	0
$836$	−6.55635	−0.226756
$837$	0	0
$838$	3.52691	0.121835
$839$	47.3137	1.63345	0.816725	−	0.577027i	$-0.195787\pi$
$839$	47.3137	1.63345	0.816725	+	0.577027i	$0.195787\pi$
$840$	0	0
$841$	17.6274	0.607842
$842$	−11.1838	−0.385418
$843$	0	0
$844$	−17.0294	−0.586177
$845$	0	0
$846$	0	0
$847$	−2.41421	−0.0829534
$848$	10.9706	0.376731
$849$	0	0
$850$	0	0
$851$	5.82843	0.199796
$852$	0	0
$853$	−19.1716	−0.656422	−0.328211	−	0.944604i	$-0.606446\pi$
$853$	−19.1716	−0.656422	−0.328211	+	0.944604i	$0.606446\pi$
$854$	−3.17157	−0.108529
$855$	0	0
$856$	27.4558	0.938421
$857$	36.6985	1.25360	0.626798	−	0.779182i	$-0.284365\pi$
$857$	36.6985	1.25360	0.626798	+	0.779182i	$0.284365\pi$
$858$	0	0
$859$	−20.4853	−0.698949	−0.349474	−	0.936946i	$-0.613640\pi$
$859$	−20.4853	−0.698949	−0.349474	+	0.936946i	$0.613640\pi$
$860$	0	0
$861$	0	0
$862$	−2.82843	−0.0963366
$863$	28.6863	0.976493	0.488246	−	0.872706i	$-0.337637\pi$
$863$	28.6863	0.976493	0.488246	+	0.872706i	$0.337637\pi$
$864$	0	0
$865$	0	0
$866$	3.85786	0.131096
$867$	0	0
$868$	37.4558	1.27133
$869$	4.75736	0.161382
$870$	0	0
$871$	−32.9706	−1.11716
$872$	27.4558	0.929772
$873$	0	0
$874$	−1.48528	−0.0502404
$875$	0	0
$876$	0	0
$877$	15.1127	0.510320	0.255160	−	0.966899i	$-0.417872\pi$
$877$	15.1127	0.510320	0.255160	+	0.966899i	$0.417872\pi$
$878$	11.4853	0.387609
$879$	0	0
$880$	0	0
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	38.6274	1.29992	0.649958	−	0.759970i	$-0.274786\pi$
$883$	38.6274	1.29992	0.649958	+	0.759970i	$0.274786\pi$
$884$	−2.14214	−0.0720478
$885$	0	0
$886$	−10.7574	−0.361401
$887$	22.1421	0.743460	0.371730	−	0.928341i	$-0.378765\pi$
$887$	22.1421	0.743460	0.371730	+	0.928341i	$0.378765\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	0	0
$891$	0	0
$892$	49.0538	1.64244
$893$	−34.0122	−1.13817
$894$	0	0
$895$	0	0
$896$	25.4853	0.851403
$897$	0	0
$898$	−4.34315	−0.144933
$899$	−57.9411	−1.93244
$900$	0	0
$901$	1.51472	0.0504626
$902$	−3.68629	−0.122740
$903$	0	0
$904$	−15.8579	−0.527425
$905$	0	0
$906$	0	0
$907$	−2.48528	−0.0825224	−0.0412612	−	0.999148i	$-0.513138\pi$
$907$	−2.48528	−0.0825224	−0.0412612	+	0.999148i	$0.513138\pi$
$908$	−2.76955	−0.0919108
$909$	0	0
$910$	0	0
$911$	−28.5147	−0.944735	−0.472367	−	0.881402i	$-0.656600\pi$
$911$	−28.5147	−0.944735	−0.472367	+	0.881402i	$0.656600\pi$
$912$	0	0
$913$	−12.4853	−0.413203
$914$	13.3137	0.440378
$915$	0	0
$916$	35.6274	1.17716
$917$	2.82843	0.0934029
$918$	0	0
$919$	1.78680	0.0589410	0.0294705	−	0.999566i	$-0.490618\pi$
$919$	1.78680	0.0589410	0.0294705	+	0.999566i	$0.490618\pi$
$920$	0	0
$921$	0	0
$922$	−16.8873	−0.556154
$923$	−6.14214	−0.202171
$924$	0	0
$925$	0	0
$926$	0.426407	0.0140126
$927$	0	0
$928$	−30.1421	−0.989464
$929$	13.7990	0.452730	0.226365	−	0.974043i	$-0.427316\pi$
$929$	13.7990	0.452730	0.226365	+	0.974043i	$0.427316\pi$
$930$	0	0
$931$	−4.20101	−0.137683
$932$	−26.6152	−0.871811
$933$	0	0
$934$	14.3431	0.469322
$935$	0	0
$936$	0	0
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	11.6569	0.380610
$939$	0	0
$940$	0	0
$941$	22.8995	0.746502	0.373251	−	0.927730i	$-0.378243\pi$
$941$	22.8995	0.746502	0.373251	+	0.927730i	$0.378243\pi$
$942$	0	0
$943$	8.89949	0.289807
$944$	−33.0000	−1.07406
$945$	0	0
$946$	−0.142136	−0.00462123
$947$	36.7990	1.19581	0.597903	−	0.801568i	$-0.296001\pi$
$947$	36.7990	1.19581	0.597903	+	0.801568i	$0.296001\pi$
$948$	0	0
$949$	−8.97056	−0.291197
$950$	0	0
$951$	0	0
$952$	1.58579	0.0513956
$953$	29.0416	0.940751	0.470375	−	0.882466i	$-0.344119\pi$
$953$	29.0416	0.940751	0.470375	+	0.882466i	$0.344119\pi$
$954$	0	0
$955$	0	0
$956$	22.5685	0.729919
$957$	0	0
$958$	3.11270	0.100567
$959$	38.9706	1.25843
$960$	0	0
$961$	41.0000	1.32258
$962$	−6.82843	−0.220157
$963$	0	0
$964$	25.8579	0.832826
$965$	0	0
$966$	0	0
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	−1.58579	−0.0509691
$969$	0	0
$970$	0	0
$971$	50.3137	1.61464	0.807322	−	0.590111i	$-0.200916\pi$
$971$	50.3137	1.61464	0.807322	+	0.590111i	$0.200916\pi$
$972$	0	0
$973$	−36.1421	−1.15866
$974$	4.34315	0.139163
$975$	0	0
$976$	9.51472	0.304559
$977$	−60.5685	−1.93776	−0.968880	−	0.247532i	$-0.920380\pi$
$977$	−60.5685	−1.93776	−0.968880	+	0.247532i	$0.920380\pi$
$978$	0	0
$979$	7.65685	0.244714
$980$	0	0
$981$	0	0
$982$	−1.71573	−0.0547511
$983$	−51.2843	−1.63571	−0.817857	−	0.575421i	$-0.804838\pi$
$983$	−51.2843	−1.63571	−0.817857	+	0.575421i	$0.804838\pi$
$984$	0	0
$985$	0	0
$986$	−1.17157	−0.0373105
$987$	0	0
$988$	−18.5442	−0.589968
$989$	0.343146	0.0109114
$990$	0	0
$991$	−42.2843	−1.34320	−0.671602	−	0.740912i	$-0.734394\pi$
$991$	−42.2843	−1.34320	−0.671602	+	0.740912i	$0.734394\pi$
$992$	37.4558	1.18922
$993$	0	0
$994$	2.17157	0.0688781
$995$	0	0
$996$	0	0
$997$	−47.2548	−1.49658	−0.748288	−	0.663374i	$-0.769124\pi$
$997$	−47.2548	−1.49658	−0.748288	+	0.663374i	$0.769124\pi$
$998$	16.9117	0.535330
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.l.1.2		2
3.2	odd	2		825.2.a.f.1.1	yes	2
5.2	odd	4		2475.2.c.o.199.3		4
5.3	odd	4		2475.2.c.o.199.2		4
5.4	even	2		2475.2.a.w.1.1		2
15.2	even	4		825.2.c.d.199.2		4
15.8	even	4		825.2.c.d.199.3		4
15.14	odd	2		825.2.a.d.1.2	✓	2
33.32	even	2		9075.2.a.w.1.2		2
165.164	even	2		9075.2.a.ca.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.d.1.2	✓	2	15.14	odd	2
825.2.a.f.1.1	yes	2	3.2	odd	2
825.2.c.d.199.2		4	15.2	even	4
825.2.c.d.199.3		4	15.8	even	4
2475.2.a.l.1.2		2	1.1	even	1	trivial
2475.2.a.w.1.1		2	5.4	even	2
2475.2.c.o.199.2		4	5.3	odd	4
2475.2.c.o.199.3		4	5.2	odd	4
9075.2.a.w.1.2		2	33.32	even	2
9075.2.a.ca.1.1		2	165.164	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.82843 q^{4} -2.41421 q^{7} -1.58579 q^{8} +O(q^{10})\)	\(q+0.414214 q^{2} -1.82843 q^{4} -2.41421 q^{7} -1.58579 q^{8} +1.00000 q^{11} +2.82843 q^{13} -1.00000 q^{14} +3.00000 q^{16} +0.414214 q^{17} +3.58579 q^{19} +0.414214 q^{22} -1.00000 q^{23} +1.17157 q^{26} +4.41421 q^{28} -6.82843 q^{29} +8.48528 q^{31} +4.41421 q^{32} +0.171573 q^{34} -5.82843 q^{37} +1.48528 q^{38} -8.89949 q^{41} -0.343146 q^{43} -1.82843 q^{44} -0.414214 q^{46} -9.48528 q^{47} -1.17157 q^{49} -5.17157 q^{52} +3.65685 q^{53} +3.82843 q^{56} -2.82843 q^{58} -11.0000 q^{59} +3.17157 q^{61} +3.51472 q^{62} -4.17157 q^{64} -11.6569 q^{67} -0.757359 q^{68} -2.17157 q^{71} -3.17157 q^{73} -2.41421 q^{74} -6.55635 q^{76} -2.41421 q^{77} +4.75736 q^{79} -3.68629 q^{82} -12.4853 q^{83} -0.142136 q^{86} -1.58579 q^{88} +7.65685 q^{89} -6.82843 q^{91} +1.82843 q^{92} -3.92893 q^{94} +0.171573 q^{97} -0.485281 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 6 q^{8} + 2 q^{11} - 2 q^{14} + 6 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{22} - 2 q^{23} + 8 q^{26} + 6 q^{28} - 8 q^{29} + 6 q^{32} + 6 q^{34} - 6 q^{37} - 14 q^{38} + 2 q^{41} - 12 q^{43} + 2 q^{44} + 2 q^{46} - 2 q^{47} - 8 q^{49} - 16 q^{52} - 4 q^{53} + 2 q^{56} - 22 q^{59} + 12 q^{61} + 24 q^{62} - 14 q^{64} - 12 q^{67} - 10 q^{68} - 10 q^{71} - 12 q^{73} - 2 q^{74} + 18 q^{76} - 2 q^{77} + 18 q^{79} - 30 q^{82} - 8 q^{83} + 28 q^{86} - 6 q^{88} + 4 q^{89} - 8 q^{91} - 2 q^{92} - 22 q^{94} + 6 q^{97} + 16 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 6 * q^8 + 2 * q^11 - 2 * q^14 + 6 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^22 - 2 * q^23 + 8 * q^26 + 6 * q^28 - 8 * q^29 + 6 * q^32 + 6 * q^34 - 6 * q^37 - 14 * q^38 + 2 * q^41 - 12 * q^43 + 2 * q^44 + 2 * q^46 - 2 * q^47 - 8 * q^49 - 16 * q^52 - 4 * q^53 + 2 * q^56 - 22 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 - 12 * q^67 - 10 * q^68 - 10 * q^71 - 12 * q^73 - 2 * q^74 + 18 * q^76 - 2 * q^77 + 18 * q^79 - 30 * q^82 - 8 * q^83 + 28 * q^86 - 6 * q^88 + 4 * q^89 - 8 * q^91 - 2 * q^92 - 22 * q^94 + 6 * q^97 + 16 * q^98

Introduction

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