LMFDB - Embedded newform 2475.2.a.x.1.1

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:	$0$
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 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} +1.58579 q^{8} +O(q^{10})$	$q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} +1.58579 q^{8} -1.00000 q^{11} +6.82843 q^{13} -0.828427 q^{14} +3.00000 q^{16} +1.17157 q^{17} +0.414214 q^{22} +2.82843 q^{23} -2.82843 q^{26} -3.65685 q^{28} -7.65685 q^{29} -4.41421 q^{32} -0.485281 q^{34} -3.65685 q^{37} -6.00000 q^{41} +6.00000 q^{43} +1.82843 q^{44} -1.17157 q^{46} -2.82843 q^{47} -3.00000 q^{49} -12.4853 q^{52} +0.343146 q^{53} +3.17157 q^{56} +3.17157 q^{58} +9.65685 q^{59} +13.3137 q^{61} -4.17157 q^{64} +4.48528 q^{67} -2.14214 q^{68} +11.3137 q^{71} +6.82843 q^{73} +1.51472 q^{74} -2.00000 q^{77} +4.00000 q^{79} +2.48528 q^{82} -6.00000 q^{83} -2.48528 q^{86} -1.58579 q^{88} -9.31371 q^{89} +13.6569 q^{91} -5.17157 q^{92} +1.17157 q^{94} +7.65685 q^{97} +1.24264 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8} - 2 q^{11} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} - 2 q^{22} + 4 q^{28} - 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{37} - 12 q^{41} + 12 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{49} - 8 q^{52} + 12 q^{53} + 12 q^{56} + 12 q^{58} + 8 q^{59} + 4 q^{61} - 14 q^{64} - 8 q^{67} + 24 q^{68} + 8 q^{73} + 20 q^{74} - 4 q^{77} + 8 q^{79} - 12 q^{82} - 12 q^{83} + 12 q^{86} - 6 q^{88} + 4 q^{89} + 16 q^{91} - 16 q^{92} + 8 q^{94} + 4 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 6 * q^8 - 2 * q^11 + 8 * q^13 + 4 * q^14 + 6 * q^16 + 8 * q^17 - 2 * q^22 + 4 * q^28 - 4 * q^29 - 6 * q^32 + 16 * q^34 + 4 * q^37 - 12 * q^41 + 12 * q^43 - 2 * q^44 - 8 * q^46 - 6 * q^49 - 8 * q^52 + 12 * q^53 + 12 * q^56 + 12 * q^58 + 8 * q^59 + 4 * q^61 - 14 * q^64 - 8 * q^67 + 24 * q^68 + 8 * q^73 + 20 * q^74 - 4 * q^77 + 8 * q^79 - 12 * q^82 - 12 * q^83 + 12 * q^86 - 6 * q^88 + 4 * q^89 + 16 * q^91 - 16 * q^92 + 8 * q^94 + 4 * q^97 - 6 * 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	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.58579	0.560660
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.82843	1.89386	0.946932	−	0.321433i	$-0.104164\pi$
$13$	6.82843	1.89386	0.946932	+	0.321433i	$0.104164\pi$
$14$	−0.828427	−0.221406
$15$	0	0
$16$	3.00000	0.750000
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0.414214	0.0883106
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	0	0
$26$	−2.82843	−0.554700
$27$	0	0
$28$	−3.65685	−0.691080
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−4.41421	−0.780330
$33$	0	0
$34$	−0.485281	−0.0832251
$35$	0	0
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	1.82843	0.275646
$45$	0	0
$46$	−1.17157	−0.172739
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	−12.4853	−1.73140
$53$	0.343146	0.0471347	0.0235673	−	0.999722i	$-0.492498\pi$
$53$	0.343146	0.0471347	0.0235673	+	0.999722i	$0.492498\pi$
$54$	0	0
$55$	0	0
$56$	3.17157	0.423819
$57$	0	0
$58$	3.17157	0.416448
$59$	9.65685	1.25722	0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685	1.25722	0.628608	+	0.777723i	$0.283625\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	0	0
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0	0
$66$	0	0
$67$	4.48528	0.547964	0.273982	−	0.961735i	$-0.411659\pi$
$67$	4.48528	0.547964	0.273982	+	0.961735i	$0.411659\pi$
$68$	−2.14214	−0.259772
$69$	0	0
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	6.82843	0.799207	0.399603	−	0.916688i	$-0.369148\pi$
$73$	6.82843	0.799207	0.399603	+	0.916688i	$0.369148\pi$
$74$	1.51472	0.176082
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	2.48528	0.274453
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	−2.48528	−0.267995
$87$	0	0
$88$	−1.58579	−0.169045
$89$	−9.31371	−0.987251	−0.493626	−	0.869675i	$-0.664329\pi$
$89$	−9.31371	−0.987251	−0.493626	+	0.869675i	$0.664329\pi$
$90$	0	0
$91$	13.6569	1.43163
$92$	−5.17157	−0.539174
$93$	0	0
$94$	1.17157	0.120839
$95$	0	0
$96$	0	0
$97$	7.65685	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$97$	7.65685	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$98$	1.24264	0.125526
$99$	0	0
$100$	0	0
$101$	13.3137	1.32476	0.662382	−	0.749166i	$-0.269546\pi$
$101$	13.3137	1.32476	0.662382	+	0.749166i	$0.269546\pi$
$102$	0	0
$103$	−1.17157	−0.115439	−0.0577193	−	0.998333i	$-0.518383\pi$
$103$	−1.17157	−0.115439	−0.0577193	+	0.998333i	$0.518383\pi$
$104$	10.8284	1.06181
$105$	0	0
$106$	−0.142136	−0.0138054
$107$	−3.65685	−0.353521	−0.176761	−	0.984254i	$-0.556562\pi$
$107$	−3.65685	−0.353521	−0.176761	+	0.984254i	$0.556562\pi$
$108$	0	0
$109$	3.65685	0.350263	0.175132	−	0.984545i	$-0.443965\pi$
$109$	3.65685	0.350263	0.175132	+	0.984545i	$0.443965\pi$
$110$	0	0
$111$	0	0
$112$	6.00000	0.566947
$113$	8.34315	0.784857	0.392429	−	0.919782i	$-0.371635\pi$
$113$	8.34315	0.784857	0.392429	+	0.919782i	$0.371635\pi$
$114$	0	0
$115$	0	0
$116$	14.0000	1.29987
$117$	0	0
$118$	−4.00000	−0.368230
$119$	2.34315	0.214796
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.51472	−0.499279
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.6569	−1.38932	−0.694661	−	0.719338i	$-0.744445\pi$
$127$	−15.6569	−1.38932	−0.694661	+	0.719338i	$0.744445\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	−1.85786	−0.160495
$135$	0	0
$136$	1.85786	0.159311
$137$	22.9706	1.96251	0.981254	−	0.192720i	$-0.0617309\pi$
$137$	22.9706	1.96251	0.981254	+	0.192720i	$0.0617309\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	−4.68629	−0.393265
$143$	−6.82843	−0.571022
$144$	0	0
$145$	0	0
$146$	−2.82843	−0.234082
$147$	0	0
$148$	6.68629	0.549610
$149$	−11.6569	−0.954967	−0.477483	−	0.878641i	$-0.658451\pi$
$149$	−11.6569	−0.954967	−0.477483	+	0.878641i	$0.658451\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	0	0
$154$	0.828427	0.0667566
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−1.65685	−0.131812
$159$	0	0
$160$	0	0
$161$	5.65685	0.445823
$162$	0	0
$163$	0.485281	0.0380102	0.0190051	−	0.999819i	$-0.493950\pi$
$163$	0.485281	0.0380102	0.0190051	+	0.999819i	$0.493950\pi$
$164$	10.9706	0.856657
$165$	0	0
$166$	2.48528	0.192895
$167$	10.9706	0.848928	0.424464	−	0.905445i	$-0.360463\pi$
$167$	10.9706	0.848928	0.424464	+	0.905445i	$0.360463\pi$
$168$	0	0
$169$	33.6274	2.58672
$170$	0	0
$171$	0	0
$172$	−10.9706	−0.836498
$173$	6.14214	0.466978	0.233489	−	0.972359i	$-0.424986\pi$
$173$	6.14214	0.466978	0.233489	+	0.972359i	$0.424986\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	3.85786	0.289159
$179$	1.65685	0.123839	0.0619196	−	0.998081i	$-0.480278\pi$
$179$	1.65685	0.123839	0.0619196	+	0.998081i	$0.480278\pi$
$180$	0	0
$181$	−1.31371	−0.0976472	−0.0488236	−	0.998807i	$-0.515547\pi$
$181$	−1.31371	−0.0976472	−0.0488236	+	0.998807i	$0.515547\pi$
$182$	−5.65685	−0.419314
$183$	0	0
$184$	4.48528	0.330659
$185$	0	0
$186$	0	0
$187$	−1.17157	−0.0856739
$188$	5.17157	0.377176
$189$	0	0
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	6.82843	0.491521	0.245760	−	0.969331i	$-0.420962\pi$
$193$	6.82843	0.491521	0.245760	+	0.969331i	$0.420962\pi$
$194$	−3.17157	−0.227706
$195$	0	0
$196$	5.48528	0.391806
$197$	−5.17157	−0.368459	−0.184230	−	0.982883i	$-0.558979\pi$
$197$	−5.17157	−0.368459	−0.184230	+	0.982883i	$0.558979\pi$
$198$	0	0
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	0	0
$201$	0	0
$202$	−5.51472	−0.388014
$203$	−15.3137	−1.07481
$204$	0	0
$205$	0	0
$206$	0.485281	0.0338112
$207$	0	0
$208$	20.4853	1.42040
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−0.627417	−0.0430912
$213$	0	0
$214$	1.51472	0.103544
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−1.51472	−0.102590
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	5.17157	0.346314	0.173157	−	0.984894i	$-0.444603\pi$
$223$	5.17157	0.346314	0.173157	+	0.984894i	$0.444603\pi$
$224$	−8.82843	−0.589874
$225$	0	0
$226$	−3.45584	−0.229879
$227$	2.68629	0.178295	0.0891477	−	0.996018i	$-0.471586\pi$
$227$	2.68629	0.178295	0.0891477	+	0.996018i	$0.471586\pi$
$228$	0	0
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	0	0
$231$	0	0
$232$	−12.1421	−0.797170
$233$	22.1421	1.45058	0.725290	−	0.688444i	$-0.241706\pi$
$233$	22.1421	1.45058	0.725290	+	0.688444i	$0.241706\pi$
$234$	0	0
$235$	0	0
$236$	−17.6569	−1.14936
$237$	0	0
$238$	−0.970563	−0.0629122
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−0.414214	−0.0266267
$243$	0	0
$244$	−24.3431	−1.55841
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	6.48528	0.406923
$255$	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
$259$	−7.31371	−0.454452
$260$	0	0
$261$	0	0
$262$	4.68629	0.289520
$263$	22.9706	1.41643	0.708213	−	0.705999i	$-0.249502\pi$
$263$	22.9706	1.41643	0.708213	+	0.705999i	$0.249502\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−8.20101	−0.500956
$269$	5.31371	0.323983	0.161991	−	0.986792i	$-0.448208\pi$
$269$	5.31371	0.323983	0.161991	+	0.986792i	$0.448208\pi$
$270$	0	0
$271$	−15.3137	−0.930242	−0.465121	−	0.885247i	$-0.653989\pi$
$271$	−15.3137	−0.930242	−0.465121	+	0.885247i	$0.653989\pi$
$272$	3.51472	0.213111
$273$	0	0
$274$	−9.51472	−0.574805
$275$	0	0
$276$	0	0
$277$	−1.17157	−0.0703930	−0.0351965	−	0.999380i	$-0.511206\pi$
$277$	−1.17157	−0.0703930	−0.0351965	+	0.999380i	$0.511206\pi$
$278$	1.65685	0.0993715
$279$	0	0
$280$	0	0
$281$	5.31371	0.316989	0.158495	−	0.987360i	$-0.449336\pi$
$281$	5.31371	0.316989	0.158495	+	0.987360i	$0.449336\pi$
$282$	0	0
$283$	12.6274	0.750622	0.375311	−	0.926899i	$-0.377536\pi$
$283$	12.6274	0.750622	0.375311	+	0.926899i	$0.377536\pi$
$284$	−20.6863	−1.22751
$285$	0	0
$286$	2.82843	0.167248
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	0	0
$292$	−12.4853	−0.730646
$293$	−14.8284	−0.866286	−0.433143	−	0.901325i	$-0.642595\pi$
$293$	−14.8284	−0.866286	−0.433143	+	0.901325i	$0.642595\pi$
$294$	0	0
$295$	0	0
$296$	−5.79899	−0.337059
$297$	0	0
$298$	4.82843	0.279703
$299$	19.3137	1.11694
$300$	0	0
$301$	12.0000	0.691669
$302$	4.97056	0.286024
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	27.6569	1.57846	0.789230	−	0.614098i	$-0.210480\pi$
$307$	27.6569	1.57846	0.789230	+	0.614098i	$0.210480\pi$
$308$	3.65685	0.208369
$309$	0	0
$310$	0	0
$311$	−27.3137	−1.54882	−0.774409	−	0.632685i	$-0.781953\pi$
$311$	−27.3137	−1.54882	−0.774409	+	0.632685i	$0.781953\pi$
$312$	0	0
$313$	−21.3137	−1.20472	−0.602361	−	0.798224i	$-0.705773\pi$
$313$	−21.3137	−1.20472	−0.602361	+	0.798224i	$0.705773\pi$
$314$	−5.79899	−0.327256
$315$	0	0
$316$	−7.31371	−0.411428
$317$	21.3137	1.19710	0.598549	−	0.801087i	$-0.295744\pi$
$317$	21.3137	1.19710	0.598549	+	0.801087i	$0.295744\pi$
$318$	0	0
$319$	7.65685	0.428702
$320$	0	0
$321$	0	0
$322$	−2.34315	−0.130578
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−0.201010	−0.0111329
$327$	0	0
$328$	−9.51472	−0.525362
$329$	−5.65685	−0.311872
$330$	0	0
$331$	15.3137	0.841718	0.420859	−	0.907126i	$-0.361729\pi$
$331$	15.3137	0.841718	0.420859	+	0.907126i	$0.361729\pi$
$332$	10.9706	0.602088
$333$	0	0
$334$	−4.54416	−0.248645
$335$	0	0
$336$	0	0
$337$	3.51472	0.191459	0.0957295	−	0.995407i	$-0.469482\pi$
$337$	3.51472	0.191459	0.0957295	+	0.995407i	$0.469482\pi$
$338$	−13.9289	−0.757634
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	9.51472	0.512999
$345$	0	0
$346$	−2.54416	−0.136775
$347$	−22.9706	−1.23312	−0.616562	−	0.787306i	$-0.711475\pi$
$347$	−22.9706	−1.23312	−0.616562	+	0.787306i	$0.711475\pi$
$348$	0	0
$349$	−6.97056	−0.373126	−0.186563	−	0.982443i	$-0.559735\pi$
$349$	−6.97056	−0.373126	−0.186563	+	0.982443i	$0.559735\pi$
$350$	0	0
$351$	0	0
$352$	4.41421	0.235278
$353$	−1.31371	−0.0699216	−0.0349608	−	0.999389i	$-0.511131\pi$
$353$	−1.31371	−0.0699216	−0.0349608	+	0.999389i	$0.511131\pi$
$354$	0	0
$355$	0	0
$356$	17.0294	0.902558
$357$	0	0
$358$	−0.686292	−0.0362716
$359$	−23.3137	−1.23045	−0.615225	−	0.788351i	$-0.710935\pi$
$359$	−23.3137	−1.23045	−0.615225	+	0.788351i	$0.710935\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0.544156	0.0286002
$363$	0	0
$364$	−24.9706	−1.30881
$365$	0	0
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	8.48528	0.442326
$369$	0	0
$370$	0	0
$371$	0.686292	0.0356305
$372$	0	0
$373$	3.79899	0.196704	0.0983521	−	0.995152i	$-0.468643\pi$
$373$	3.79899	0.196704	0.0983521	+	0.995152i	$0.468643\pi$
$374$	0.485281	0.0250933
$375$	0	0
$376$	−4.48528	−0.231311
$377$	−52.2843	−2.69278
$378$	0	0
$379$	22.3431	1.14769	0.573845	−	0.818964i	$-0.305451\pi$
$379$	22.3431	1.14769	0.573845	+	0.818964i	$0.305451\pi$
$380$	0	0
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−34.1421	−1.74458	−0.872291	−	0.488987i	$-0.837366\pi$
$383$	−34.1421	−1.74458	−0.872291	+	0.488987i	$0.837366\pi$
$384$	0	0
$385$	0	0
$386$	−2.82843	−0.143963
$387$	0	0
$388$	−14.0000	−0.710742
$389$	24.6274	1.24866	0.624330	−	0.781161i	$-0.285372\pi$
$389$	24.6274	1.24866	0.624330	+	0.781161i	$0.285372\pi$
$390$	0	0
$391$	3.31371	0.167581
$392$	−4.75736	−0.240283
$393$	0	0
$394$	2.14214	0.107919
$395$	0	0
$396$	0	0
$397$	−13.3137	−0.668196	−0.334098	−	0.942538i	$-0.608432\pi$
$397$	−13.3137	−0.668196	−0.334098	+	0.942538i	$0.608432\pi$
$398$	−8.97056	−0.449654
$399$	0	0
$400$	0	0
$401$	−17.3137	−0.864605	−0.432303	−	0.901729i	$-0.642299\pi$
$401$	−17.3137	−0.864605	−0.432303	+	0.901729i	$0.642299\pi$
$402$	0	0
$403$	0	0
$404$	−24.3431	−1.21112
$405$	0	0
$406$	6.34315	0.314805
$407$	3.65685	0.181264
$408$	0	0
$409$	34.9706	1.72918	0.864592	−	0.502475i	$-0.167577\pi$
$409$	34.9706	1.72918	0.864592	+	0.502475i	$0.167577\pi$
$410$	0	0
$411$	0	0
$412$	2.14214	0.105535
$413$	19.3137	0.950365
$414$	0	0
$415$	0	0
$416$	−30.1421	−1.47784
$417$	0	0
$418$	0	0
$419$	14.3431	0.700709	0.350354	−	0.936617i	$-0.386061\pi$
$419$	14.3431	0.700709	0.350354	+	0.936617i	$0.386061\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	6.62742	0.322618
$423$	0	0
$424$	0.544156	0.0264265
$425$	0	0
$426$	0	0
$427$	26.6274	1.28859
$428$	6.68629	0.323194
$429$	0	0
$430$	0	0
$431$	−11.3137	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$431$	−11.3137	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$432$	0	0
$433$	−3.65685	−0.175737	−0.0878686	−	0.996132i	$-0.528006\pi$
$433$	−3.65685	−0.175737	−0.0878686	+	0.996132i	$0.528006\pi$
$434$	0	0
$435$	0	0
$436$	−6.68629	−0.320215
$437$	0	0
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	0	0
$442$	−3.31371	−0.157617
$443$	−21.1716	−1.00589	−0.502946	−	0.864318i	$-0.667751\pi$
$443$	−21.1716	−1.00589	−0.502946	+	0.864318i	$0.667751\pi$
$444$	0	0
$445$	0	0
$446$	−2.14214	−0.101433
$447$	0	0
$448$	−8.34315	−0.394177
$449$	16.6274	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$449$	16.6274	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−15.2548	−0.717527
$453$	0	0
$454$	−1.11270	−0.0522215
$455$	0	0
$456$	0	0
$457$	16.4853	0.771149	0.385574	−	0.922677i	$-0.374003\pi$
$457$	16.4853	0.771149	0.385574	+	0.922677i	$0.374003\pi$
$458$	8.82843	0.412525
$459$	0	0
$460$	0	0
$461$	32.6274	1.51961	0.759805	−	0.650151i	$-0.225294\pi$
$461$	32.6274	1.51961	0.759805	+	0.650151i	$0.225294\pi$
$462$	0	0
$463$	−22.1421	−1.02903	−0.514516	−	0.857481i	$-0.672028\pi$
$463$	−22.1421	−1.02903	−0.514516	+	0.857481i	$0.672028\pi$
$464$	−22.9706	−1.06638
$465$	0	0
$466$	−9.17157	−0.424865
$467$	−9.17157	−0.424410	−0.212205	−	0.977225i	$-0.568064\pi$
$467$	−9.17157	−0.424410	−0.212205	+	0.977225i	$0.568064\pi$
$468$	0	0
$469$	8.97056	0.414222
$470$	0	0
$471$	0	0
$472$	15.3137	0.704871
$473$	−6.00000	−0.275880
$474$	0	0
$475$	0	0
$476$	−4.28427	−0.196369
$477$	0	0
$478$	−0.284271	−0.0130023
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−24.9706	−1.13856
$482$	−2.48528	−0.113201
$483$	0	0
$484$	−1.82843	−0.0831103
$485$	0	0
$486$	0	0
$487$	7.51472	0.340524	0.170262	−	0.985399i	$-0.445539\pi$
$487$	7.51472	0.340524	0.170262	+	0.985399i	$0.445539\pi$
$488$	21.1127	0.955727
$489$	0	0
$490$	0	0
$491$	23.3137	1.05213	0.526066	−	0.850443i	$-0.323666\pi$
$491$	23.3137	1.05213	0.526066	+	0.850443i	$0.323666\pi$
$492$	0	0
$493$	−8.97056	−0.404014
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.6274	1.01498
$498$	0	0
$499$	−1.65685	−0.0741710	−0.0370855	−	0.999312i	$-0.511807\pi$
$499$	−1.65685	−0.0741710	−0.0370855	+	0.999312i	$0.511807\pi$
$500$	0	0
$501$	0	0
$502$	4.97056	0.221847
$503$	−28.6274	−1.27643	−0.638217	−	0.769857i	$-0.720328\pi$
$503$	−28.6274	−1.27643	−0.638217	+	0.769857i	$0.720328\pi$
$504$	0	0
$505$	0	0
$506$	1.17157	0.0520828
$507$	0	0
$508$	28.6274	1.27014
$509$	−9.31371	−0.412823	−0.206411	−	0.978465i	$-0.566179\pi$
$509$	−9.31371	−0.412823	−0.206411	+	0.978465i	$0.566179\pi$
$510$	0	0
$511$	13.6569	0.604144
$512$	−22.7574	−1.00574
$513$	0	0
$514$	−5.51472	−0.243244
$515$	0	0
$516$	0	0
$517$	2.82843	0.124394
$518$	3.02944	0.133106
$519$	0	0
$520$	0	0
$521$	−2.68629	−0.117689	−0.0588443	−	0.998267i	$-0.518742\pi$
$521$	−2.68629	−0.117689	−0.0588443	+	0.998267i	$0.518742\pi$
$522$	0	0
$523$	−37.5980	−1.64404	−0.822022	−	0.569455i	$-0.807154\pi$
$523$	−37.5980	−1.64404	−0.822022	+	0.569455i	$0.807154\pi$
$524$	20.6863	0.903685
$525$	0	0
$526$	−9.51472	−0.414861
$527$	0	0
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−40.9706	−1.77463
$534$	0	0
$535$	0	0
$536$	7.11270	0.307222
$537$	0	0
$538$	−2.20101	−0.0948923
$539$	3.00000	0.129219
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	6.34315	0.272461
$543$	0	0
$544$	−5.17157	−0.221729
$545$	0	0
$546$	0	0
$547$	34.0000	1.45374	0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000	1.45374	0.726868	+	0.686778i	$0.240975\pi$
$548$	−42.0000	−1.79415
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	0.485281	0.0206176
$555$	0	0
$556$	7.31371	0.310170
$557$	38.1421	1.61613	0.808067	−	0.589090i	$-0.200514\pi$
$557$	38.1421	1.61613	0.808067	+	0.589090i	$0.200514\pi$
$558$	0	0
$559$	40.9706	1.73287
$560$	0	0
$561$	0	0
$562$	−2.20101	−0.0928440
$563$	11.6569	0.491278	0.245639	−	0.969361i	$-0.421002\pi$
$563$	11.6569	0.491278	0.245639	+	0.969361i	$0.421002\pi$
$564$	0	0
$565$	0	0
$566$	−5.23045	−0.219852
$567$	0	0
$568$	17.9411	0.752793
$569$	−20.3431	−0.852829	−0.426415	−	0.904528i	$-0.640224\pi$
$569$	−20.3431	−0.852829	−0.426415	+	0.904528i	$0.640224\pi$
$570$	0	0
$571$	45.9411	1.92258	0.961288	−	0.275545i	$-0.0888584\pi$
$571$	45.9411	1.92258	0.961288	+	0.275545i	$0.0888584\pi$
$572$	12.4853	0.522036
$573$	0	0
$574$	4.97056	0.207467
$575$	0	0
$576$	0	0
$577$	−6.97056	−0.290188	−0.145094	−	0.989418i	$-0.546349\pi$
$577$	−6.97056	−0.290188	−0.145094	+	0.989418i	$0.546349\pi$
$578$	6.47309	0.269245
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−0.343146	−0.0142116
$584$	10.8284	0.448084
$585$	0	0
$586$	6.14214	0.253729
$587$	26.1421	1.07900	0.539501	−	0.841985i	$-0.318613\pi$
$587$	26.1421	1.07900	0.539501	+	0.841985i	$0.318613\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−10.9706	−0.450887
$593$	20.4853	0.841230	0.420615	−	0.907239i	$-0.361814\pi$
$593$	20.4853	0.841230	0.420615	+	0.907239i	$0.361814\pi$
$594$	0	0
$595$	0	0
$596$	21.3137	0.873044
$597$	0	0
$598$	−8.00000	−0.327144
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	−43.9411	−1.79240	−0.896198	−	0.443654i	$-0.853682\pi$
$601$	−43.9411	−1.79240	−0.896198	+	0.443654i	$0.853682\pi$
$602$	−4.97056	−0.202585
$603$	0	0
$604$	21.9411	0.892772
$605$	0	0
$606$	0	0
$607$	18.2843	0.742136	0.371068	−	0.928606i	$-0.378992\pi$
$607$	18.2843	0.742136	0.371068	+	0.928606i	$0.378992\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−19.3137	−0.781349
$612$	0	0
$613$	−25.4558	−1.02815	−0.514076	−	0.857745i	$-0.671865\pi$
$613$	−25.4558	−1.02815	−0.514076	+	0.857745i	$0.671865\pi$
$614$	−11.4558	−0.462320
$615$	0	0
$616$	−3.17157	−0.127786
$617$	11.6569	0.469287	0.234644	−	0.972081i	$-0.424608\pi$
$617$	11.6569	0.469287	0.234644	+	0.972081i	$0.424608\pi$
$618$	0	0
$619$	−25.6569	−1.03124	−0.515618	−	0.856819i	$-0.672438\pi$
$619$	−25.6569	−1.03124	−0.515618	+	0.856819i	$0.672438\pi$
$620$	0	0
$621$	0	0
$622$	11.3137	0.453638
$623$	−18.6274	−0.746292
$624$	0	0
$625$	0	0
$626$	8.82843	0.352855
$627$	0	0
$628$	−25.5980	−1.02147
$629$	−4.28427	−0.170825
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	6.34315	0.252317
$633$	0	0
$634$	−8.82843	−0.350622
$635$	0	0
$636$	0	0
$637$	−20.4853	−0.811656
$638$	−3.17157	−0.125564
$639$	0	0
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−49.4558	−1.95035	−0.975174	−	0.221440i	$-0.928924\pi$
$643$	−49.4558	−1.95035	−0.975174	+	0.221440i	$0.928924\pi$
$644$	−10.3431	−0.407577
$645$	0	0
$646$	0	0
$647$	−35.1127	−1.38042	−0.690211	−	0.723608i	$-0.742482\pi$
$647$	−35.1127	−1.38042	−0.690211	+	0.723608i	$0.742482\pi$
$648$	0	0
$649$	−9.65685	−0.379065
$650$	0	0
$651$	0	0
$652$	−0.887302	−0.0347494
$653$	0.343146	0.0134283	0.00671417	−	0.999977i	$-0.497863\pi$
$653$	0.343146	0.0134283	0.00671417	+	0.999977i	$0.497863\pi$
$654$	0	0
$655$	0	0
$656$	−18.0000	−0.702782
$657$	0	0
$658$	2.34315	0.0913453
$659$	21.9411	0.854705	0.427352	−	0.904085i	$-0.359446\pi$
$659$	21.9411	0.854705	0.427352	+	0.904085i	$0.359446\pi$
$660$	0	0
$661$	−0.627417	−0.0244037	−0.0122018	−	0.999926i	$-0.503884\pi$
$661$	−0.627417	−0.0244037	−0.0122018	+	0.999926i	$0.503884\pi$
$662$	−6.34315	−0.246533
$663$	0	0
$664$	−9.51472	−0.369243
$665$	0	0
$666$	0	0
$667$	−21.6569	−0.838557
$668$	−20.0589	−0.776101
$669$	0	0
$670$	0	0
$671$	−13.3137	−0.513970
$672$	0	0
$673$	−4.48528	−0.172895	−0.0864474	−	0.996256i	$-0.527551\pi$
$673$	−4.48528	−0.172895	−0.0864474	+	0.996256i	$0.527551\pi$
$674$	−1.45584	−0.0560770
$675$	0	0
$676$	−61.4853	−2.36482
$677$	17.1716	0.659957	0.329979	−	0.943988i	$-0.392958\pi$
$677$	17.1716	0.659957	0.329979	+	0.943988i	$0.392958\pi$
$678$	0	0
$679$	15.3137	0.587686
$680$	0	0
$681$	0	0
$682$	0	0
$683$	31.7990	1.21675	0.608377	−	0.793648i	$-0.291821\pi$
$683$	31.7990	1.21675	0.608377	+	0.793648i	$0.291821\pi$
$684$	0	0
$685$	0	0
$686$	8.28427	0.316295
$687$	0	0
$688$	18.0000	0.686244
$689$	2.34315	0.0892667
$690$	0	0
$691$	−16.6863	−0.634776	−0.317388	−	0.948296i	$-0.602806\pi$
$691$	−16.6863	−0.634776	−0.317388	+	0.948296i	$0.602806\pi$
$692$	−11.2304	−0.426918
$693$	0	0
$694$	9.51472	0.361174
$695$	0	0
$696$	0	0
$697$	−7.02944	−0.266259
$698$	2.88730	0.109286
$699$	0	0
$700$	0	0
$701$	−32.6274	−1.23232	−0.616160	−	0.787621i	$-0.711313\pi$
$701$	−32.6274	−1.23232	−0.616160	+	0.787621i	$0.711313\pi$
$702$	0	0
$703$	0	0
$704$	4.17157	0.157222
$705$	0	0
$706$	0.544156	0.0204796
$707$	26.6274	1.00143
$708$	0	0
$709$	−20.6274	−0.774679	−0.387339	−	0.921937i	$-0.626606\pi$
$709$	−20.6274	−0.774679	−0.387339	+	0.921937i	$0.626606\pi$
$710$	0	0
$711$	0	0
$712$	−14.7696	−0.553512
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3.02944	−0.113215
$717$	0	0
$718$	9.65685	0.360391
$719$	−29.6569	−1.10601	−0.553007	−	0.833177i	$-0.686520\pi$
$719$	−29.6569	−1.10601	−0.553007	+	0.833177i	$0.686520\pi$
$720$	0	0
$721$	−2.34315	−0.0872633
$722$	7.87006	0.292893
$723$	0	0
$724$	2.40202	0.0892704
$725$	0	0
$726$	0	0
$727$	36.4853	1.35316	0.676582	−	0.736367i	$-0.263460\pi$
$727$	36.4853	1.35316	0.676582	+	0.736367i	$0.263460\pi$
$728$	21.6569	0.802656
$729$	0	0
$730$	0	0
$731$	7.02944	0.259993
$732$	0	0
$733$	−33.4558	−1.23572	−0.617860	−	0.786288i	$-0.712000\pi$
$733$	−33.4558	−1.23572	−0.617860	+	0.786288i	$0.712000\pi$
$734$	3.51472	0.129731
$735$	0	0
$736$	−12.4853	−0.460214
$737$	−4.48528	−0.165217
$738$	0	0
$739$	−37.9411	−1.39569	−0.697843	−	0.716250i	$-0.745857\pi$
$739$	−37.9411	−1.39569	−0.697843	+	0.716250i	$0.745857\pi$
$740$	0	0
$741$	0	0
$742$	−0.284271	−0.0104359
$743$	29.5980	1.08584	0.542922	−	0.839783i	$-0.317318\pi$
$743$	29.5980	1.08584	0.542922	+	0.839783i	$0.317318\pi$
$744$	0	0
$745$	0	0
$746$	−1.57359	−0.0576133
$747$	0	0
$748$	2.14214	0.0783242
$749$	−7.31371	−0.267237
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−8.48528	−0.309426
$753$	0	0
$754$	21.6569	0.788696
$755$	0	0
$756$	0	0
$757$	9.31371	0.338512	0.169256	−	0.985572i	$-0.445863\pi$
$757$	9.31371	0.338512	0.169256	+	0.985572i	$0.445863\pi$
$758$	−9.25483	−0.336151
$759$	0	0
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	7.31371	0.264774
$764$	−35.3137	−1.27761
$765$	0	0
$766$	14.1421	0.510976
$767$	65.9411	2.38100
$768$	0	0
$769$	14.9706	0.539852	0.269926	−	0.962881i	$-0.413001\pi$
$769$	14.9706	0.539852	0.269926	+	0.962881i	$0.413001\pi$
$770$	0	0
$771$	0	0
$772$	−12.4853	−0.449355
$773$	−30.2843	−1.08925	−0.544625	−	0.838680i	$-0.683328\pi$
$773$	−30.2843	−1.08925	−0.544625	+	0.838680i	$0.683328\pi$
$774$	0	0
$775$	0	0
$776$	12.1421	0.435877
$777$	0	0
$778$	−10.2010	−0.365724
$779$	0	0
$780$	0	0
$781$	−11.3137	−0.404836
$782$	−1.37258	−0.0490835
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	−18.9706	−0.676228	−0.338114	−	0.941105i	$-0.609789\pi$
$787$	−18.9706	−0.676228	−0.338114	+	0.941105i	$0.609789\pi$
$788$	9.45584	0.336850
$789$	0	0
$790$	0	0
$791$	16.6863	0.593296
$792$	0	0
$793$	90.9117	3.22837
$794$	5.51472	0.195710
$795$	0	0
$796$	−39.5980	−1.40351
$797$	−12.6274	−0.447286	−0.223643	−	0.974671i	$-0.571795\pi$
$797$	−12.6274	−0.447286	−0.223643	+	0.974671i	$0.571795\pi$
$798$	0	0
$799$	−3.31371	−0.117231
$800$	0	0
$801$	0	0
$802$	7.17157	0.253237
$803$	−6.82843	−0.240970
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	21.1127	0.742742
$809$	−22.9706	−0.807602	−0.403801	−	0.914847i	$-0.632311\pi$
$809$	−22.9706	−0.807602	−0.403801	+	0.914847i	$0.632311\pi$
$810$	0	0
$811$	−13.9411	−0.489539	−0.244770	−	0.969581i	$-0.578712\pi$
$811$	−13.9411	−0.489539	−0.244770	+	0.969581i	$0.578712\pi$
$812$	28.0000	0.982607
$813$	0	0
$814$	−1.51472	−0.0530909
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−14.4853	−0.506466
$819$	0	0
$820$	0	0
$821$	18.6863	0.652156	0.326078	−	0.945343i	$-0.394273\pi$
$821$	18.6863	0.652156	0.326078	+	0.945343i	$0.394273\pi$
$822$	0	0
$823$	−36.4853	−1.27180	−0.635898	−	0.771773i	$-0.719370\pi$
$823$	−36.4853	−1.27180	−0.635898	+	0.771773i	$0.719370\pi$
$824$	−1.85786	−0.0647218
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−34.2843	−1.19218	−0.596090	−	0.802917i	$-0.703280\pi$
$827$	−34.2843	−1.19218	−0.596090	+	0.802917i	$0.703280\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	0	0
$832$	−28.4853	−0.987549
$833$	−3.51472	−0.121778
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−5.94113	−0.205233
$839$	−37.6569	−1.30006	−0.650029	−	0.759909i	$-0.725243\pi$
$839$	−37.6569	−1.30006	−0.650029	+	0.759909i	$0.725243\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	2.48528	0.0856485
$843$	0	0
$844$	29.2548	1.00699
$845$	0	0
$846$	0	0
$847$	2.00000	0.0687208
$848$	1.02944	0.0353510
$849$	0	0
$850$	0	0
$851$	−10.3431	−0.354558
$852$	0	0
$853$	32.4853	1.11227	0.556137	−	0.831090i	$-0.312283\pi$
$853$	32.4853	1.11227	0.556137	+	0.831090i	$0.312283\pi$
$854$	−11.0294	−0.377420
$855$	0	0
$856$	−5.79899	−0.198205
$857$	−48.7696	−1.66594	−0.832968	−	0.553321i	$-0.813360\pi$
$857$	−48.7696	−1.66594	−0.832968	+	0.553321i	$0.813360\pi$
$858$	0	0
$859$	−32.2843	−1.10153	−0.550763	−	0.834662i	$-0.685663\pi$
$859$	−32.2843	−1.10153	−0.550763	+	0.834662i	$0.685663\pi$
$860$	0	0
$861$	0	0
$862$	4.68629	0.159616
$863$	−14.8284	−0.504766	−0.252383	−	0.967627i	$-0.581214\pi$
$863$	−14.8284	−0.504766	−0.252383	+	0.967627i	$0.581214\pi$
$864$	0	0
$865$	0	0
$866$	1.51472	0.0514722
$867$	0	0
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	30.6274	1.03777
$872$	5.79899	0.196379
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.45584	−0.0491604	−0.0245802	−	0.999698i	$-0.507825\pi$
$877$	−1.45584	−0.0491604	−0.0245802	+	0.999698i	$0.507825\pi$
$878$	6.62742	0.223664
$879$	0	0
$880$	0	0
$881$	52.6274	1.77306	0.886531	−	0.462668i	$-0.153108\pi$
$881$	52.6274	1.77306	0.886531	+	0.462668i	$0.153108\pi$
$882$	0	0
$883$	−42.8284	−1.44129	−0.720646	−	0.693304i	$-0.756155\pi$
$883$	−42.8284	−1.44129	−0.720646	+	0.693304i	$0.756155\pi$
$884$	−14.6274	−0.491973
$885$	0	0
$886$	8.76955	0.294619
$887$	−18.2843	−0.613926	−0.306963	−	0.951721i	$-0.599313\pi$
$887$	−18.2843	−0.613926	−0.306963	+	0.951721i	$0.599313\pi$
$888$	0	0
$889$	−31.3137	−1.05023
$890$	0	0
$891$	0	0
$892$	−9.45584	−0.316605
$893$	0	0
$894$	0	0
$895$	0	0
$896$	21.1127	0.705326
$897$	0	0
$898$	−6.88730	−0.229832
$899$	0	0
$900$	0	0
$901$	0.402020	0.0133932
$902$	−2.48528	−0.0827508
$903$	0	0
$904$	13.2304	0.440038
$905$	0	0
$906$	0	0
$907$	44.4853	1.47711	0.738555	−	0.674193i	$-0.235509\pi$
$907$	44.4853	1.47711	0.738555	+	0.674193i	$0.235509\pi$
$908$	−4.91169	−0.163000
$909$	0	0
$910$	0	0
$911$	−57.9411	−1.91968	−0.959838	−	0.280556i	$-0.909481\pi$
$911$	−57.9411	−1.91968	−0.959838	+	0.280556i	$0.909481\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	−6.82843	−0.225864
$915$	0	0
$916$	38.9706	1.28762
$917$	−22.6274	−0.747223
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	−13.5147	−0.445084
$923$	77.2548	2.54287
$924$	0	0
$925$	0	0
$926$	9.17157	0.301397
$927$	0	0
$928$	33.7990	1.10951
$929$	17.3137	0.568044	0.284022	−	0.958818i	$-0.408331\pi$
$929$	17.3137	0.568044	0.284022	+	0.958818i	$0.408331\pi$
$930$	0	0
$931$	0	0
$932$	−40.4853	−1.32614
$933$	0	0
$934$	3.79899	0.124307
$935$	0	0
$936$	0	0
$937$	−49.4558	−1.61565	−0.807826	−	0.589421i	$-0.799356\pi$
$937$	−49.4558	−1.61565	−0.807826	+	0.589421i	$0.799356\pi$
$938$	−3.71573	−0.121323
$939$	0	0
$940$	0	0
$941$	29.3137	0.955600	0.477800	−	0.878469i	$-0.341434\pi$
$941$	29.3137	0.955600	0.477800	+	0.878469i	$0.341434\pi$
$942$	0	0
$943$	−16.9706	−0.552638
$944$	28.9706	0.942912
$945$	0	0
$946$	2.48528	0.0808035
$947$	46.8284	1.52172	0.760860	−	0.648916i	$-0.224778\pi$
$947$	46.8284	1.52172	0.760860	+	0.648916i	$0.224778\pi$
$948$	0	0
$949$	46.6274	1.51359
$950$	0	0
$951$	0	0
$952$	3.71573	0.120427
$953$	58.8284	1.90564	0.952820	−	0.303536i	$-0.0981674\pi$
$953$	58.8284	1.90564	0.952820	+	0.303536i	$0.0981674\pi$
$954$	0	0
$955$	0	0
$956$	−1.25483	−0.0405842
$957$	0	0
$958$	−14.9117	−0.481775
$959$	45.9411	1.48352
$960$	0	0
$961$	−31.0000	−1.00000
$962$	10.3431	0.333476
$963$	0	0
$964$	−10.9706	−0.353338
$965$	0	0
$966$	0	0
$967$	−18.9706	−0.610052	−0.305026	−	0.952344i	$-0.598665\pi$
$967$	−18.9706	−0.610052	−0.305026	+	0.952344i	$0.598665\pi$
$968$	1.58579	0.0509691
$969$	0	0
$970$	0	0
$971$	31.3137	1.00490	0.502452	−	0.864605i	$-0.332431\pi$
$971$	31.3137	1.00490	0.502452	+	0.864605i	$0.332431\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	−3.11270	−0.0997373
$975$	0	0
$976$	39.9411	1.27848
$977$	43.6569	1.39671	0.698353	−	0.715753i	$-0.253916\pi$
$977$	43.6569	1.39671	0.698353	+	0.715753i	$0.253916\pi$
$978$	0	0
$979$	9.31371	0.297667
$980$	0	0
$981$	0	0
$982$	−9.65685	−0.308163
$983$	−50.1421	−1.59929	−0.799643	−	0.600476i	$-0.794978\pi$
$983$	−50.1421	−1.59929	−0.799643	+	0.600476i	$0.794978\pi$
$984$	0	0
$985$	0	0
$986$	3.71573	0.118333
$987$	0	0
$988$	0	0
$989$	16.9706	0.539633
$990$	0	0
$991$	9.94113	0.315790	0.157895	−	0.987456i	$-0.449529\pi$
$991$	9.94113	0.315790	0.157895	+	0.987456i	$0.449529\pi$
$992$	0	0
$993$	0	0
$994$	−9.37258	−0.297280
$995$	0	0
$996$	0	0
$997$	−9.45584	−0.299470	−0.149735	−	0.988726i	$-0.547842\pi$
$997$	−9.45584	−0.299470	−0.149735	+	0.988726i	$0.547842\pi$
$998$	0.686292	0.0217242
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.x.1.1		2
3.2	odd	2		275.2.a.c.1.2		2
5.2	odd	4		2475.2.c.l.199.2		4
5.3	odd	4		2475.2.c.l.199.3		4
5.4	even	2		495.2.a.b.1.2		2
12.11	even	2		4400.2.a.bn.1.2		2
15.2	even	4		275.2.b.d.199.3		4
15.8	even	4		275.2.b.d.199.2		4
15.14	odd	2		55.2.a.b.1.1	✓	2
20.19	odd	2		7920.2.a.ch.1.1		2
33.32	even	2		3025.2.a.o.1.1		2
55.54	odd	2		5445.2.a.y.1.1		2
60.23	odd	4		4400.2.b.q.4049.4		4
60.47	odd	4		4400.2.b.q.4049.1		4
60.59	even	2		880.2.a.m.1.1		2
105.104	even	2		2695.2.a.f.1.1		2
120.29	odd	2		3520.2.a.bn.1.1		2
120.59	even	2		3520.2.a.bo.1.2		2
165.14	odd	10		605.2.g.f.251.1		8
165.29	even	10		605.2.g.l.511.2		8
165.59	odd	10		605.2.g.f.511.1		8
165.74	even	10		605.2.g.l.251.2		8
165.104	odd	10		605.2.g.f.366.2		8
165.119	odd	10		605.2.g.f.81.2		8
165.134	even	10		605.2.g.l.81.1		8
165.149	even	10		605.2.g.l.366.1		8
165.164	even	2		605.2.a.d.1.2		2
195.194	odd	2		9295.2.a.g.1.2		2
660.659	odd	2		9680.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.1	✓	2	15.14	odd	2
275.2.a.c.1.2		2	3.2	odd	2
275.2.b.d.199.2		4	15.8	even	4
275.2.b.d.199.3		4	15.2	even	4
495.2.a.b.1.2		2	5.4	even	2
605.2.a.d.1.2		2	165.164	even	2
605.2.g.f.81.2		8	165.119	odd	10
605.2.g.f.251.1		8	165.14	odd	10
605.2.g.f.366.2		8	165.104	odd	10
605.2.g.f.511.1		8	165.59	odd	10
605.2.g.l.81.1		8	165.134	even	10
605.2.g.l.251.2		8	165.74	even	10
605.2.g.l.366.1		8	165.149	even	10
605.2.g.l.511.2		8	165.29	even	10
880.2.a.m.1.1		2	60.59	even	2
2475.2.a.x.1.1		2	1.1	even	1	trivial
2475.2.c.l.199.2		4	5.2	odd	4
2475.2.c.l.199.3		4	5.3	odd	4
2695.2.a.f.1.1		2	105.104	even	2
3025.2.a.o.1.1		2	33.32	even	2
3520.2.a.bn.1.1		2	120.29	odd	2
3520.2.a.bo.1.2		2	120.59	even	2
4400.2.a.bn.1.2		2	12.11	even	2
4400.2.b.q.4049.1		4	60.47	odd	4
4400.2.b.q.4049.4		4	60.23	odd	4
5445.2.a.y.1.1		2	55.54	odd	2
7920.2.a.ch.1.1		2	20.19	odd	2
9295.2.a.g.1.2		2	195.194	odd	2
9680.2.a.bn.1.1		2	660.659	odd	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)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} +1.58579 q^{8} +O(q^{10})\)	\(q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} +1.58579 q^{8} -1.00000 q^{11} +6.82843 q^{13} -0.828427 q^{14} +3.00000 q^{16} +1.17157 q^{17} +0.414214 q^{22} +2.82843 q^{23} -2.82843 q^{26} -3.65685 q^{28} -7.65685 q^{29} -4.41421 q^{32} -0.485281 q^{34} -3.65685 q^{37} -6.00000 q^{41} +6.00000 q^{43} +1.82843 q^{44} -1.17157 q^{46} -2.82843 q^{47} -3.00000 q^{49} -12.4853 q^{52} +0.343146 q^{53} +3.17157 q^{56} +3.17157 q^{58} +9.65685 q^{59} +13.3137 q^{61} -4.17157 q^{64} +4.48528 q^{67} -2.14214 q^{68} +11.3137 q^{71} +6.82843 q^{73} +1.51472 q^{74} -2.00000 q^{77} +4.00000 q^{79} +2.48528 q^{82} -6.00000 q^{83} -2.48528 q^{86} -1.58579 q^{88} -9.31371 q^{89} +13.6569 q^{91} -5.17157 q^{92} +1.17157 q^{94} +7.65685 q^{97} +1.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8} - 2 q^{11} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} - 2 q^{22} + 4 q^{28} - 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{37} - 12 q^{41} + 12 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{49} - 8 q^{52} + 12 q^{53} + 12 q^{56} + 12 q^{58} + 8 q^{59} + 4 q^{61} - 14 q^{64} - 8 q^{67} + 24 q^{68} + 8 q^{73} + 20 q^{74} - 4 q^{77} + 8 q^{79} - 12 q^{82} - 12 q^{83} + 12 q^{86} - 6 q^{88} + 4 q^{89} + 16 q^{91} - 16 q^{92} + 8 q^{94} + 4 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 6 * q^8 - 2 * q^11 + 8 * q^13 + 4 * q^14 + 6 * q^16 + 8 * q^17 - 2 * q^22 + 4 * q^28 - 4 * q^29 - 6 * q^32 + 16 * q^34 + 4 * q^37 - 12 * q^41 + 12 * q^43 - 2 * q^44 - 8 * q^46 - 6 * q^49 - 8 * q^52 + 12 * q^53 + 12 * q^56 + 12 * q^58 + 8 * q^59 + 4 * q^61 - 14 * q^64 - 8 * q^67 + 24 * q^68 + 8 * q^73 + 20 * q^74 - 4 * q^77 + 8 * q^79 - 12 * q^82 - 12 * q^83 + 12 * q^86 - 6 * q^88 + 4 * q^89 + 16 * q^91 - 16 * q^92 + 8 * q^94 + 4 * q^97 - 6 * q^98

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