LMFDB - Embedded newform 3675.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
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 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.00000 q^{3} +1.41421 q^{6} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{2} +1.00000 q^{3} +1.41421 q^{6} -2.82843 q^{8} +1.00000 q^{9} +0.585786 q^{11} +4.41421 q^{13} -4.00000 q^{16} -2.24264 q^{17} +1.41421 q^{18} +4.65685 q^{19} +0.828427 q^{22} -2.24264 q^{23} -2.82843 q^{24} +6.24264 q^{26} +1.00000 q^{27} +8.24264 q^{29} -5.82843 q^{31} +0.585786 q^{33} -3.17157 q^{34} +8.41421 q^{37} +6.58579 q^{38} +4.41421 q^{39} -6.24264 q^{41} +7.58579 q^{43} -3.17157 q^{46} +13.3137 q^{47} -4.00000 q^{48} -2.24264 q^{51} -6.82843 q^{53} +1.41421 q^{54} +4.65685 q^{57} +11.6569 q^{58} -1.41421 q^{59} -4.48528 q^{61} -8.24264 q^{62} +8.00000 q^{64} +0.828427 q^{66} +13.7279 q^{67} -2.24264 q^{69} -0.585786 q^{71} -2.82843 q^{72} +12.0711 q^{73} +11.8995 q^{74} +6.24264 q^{78} +6.65685 q^{79} +1.00000 q^{81} -8.82843 q^{82} -2.58579 q^{83} +10.7279 q^{86} +8.24264 q^{87} -1.65685 q^{88} -12.2426 q^{89} -5.82843 q^{93} +18.8284 q^{94} +5.17157 q^{97} +0.585786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} + 4 q^{11} + 6 q^{13} - 8 q^{16} + 4 q^{17} - 2 q^{19} - 4 q^{22} + 4 q^{23} + 4 q^{26} + 2 q^{27} + 8 q^{29} - 6 q^{31} + 4 q^{33} - 12 q^{34} + 14 q^{37} + 16 q^{38} + 6 q^{39} - 4 q^{41} + 18 q^{43} - 12 q^{46} + 4 q^{47} - 8 q^{48} + 4 q^{51} - 8 q^{53} - 2 q^{57} + 12 q^{58} + 8 q^{61} - 8 q^{62} + 16 q^{64} - 4 q^{66} + 2 q^{67} + 4 q^{69} - 4 q^{71} + 10 q^{73} + 4 q^{74} + 4 q^{78} + 2 q^{79} + 2 q^{81} - 12 q^{82} - 8 q^{83} - 4 q^{86} + 8 q^{87} + 8 q^{88} - 16 q^{89} - 6 q^{93} + 32 q^{94} + 16 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 4 * q^11 + 6 * q^13 - 8 * q^16 + 4 * q^17 - 2 * q^19 - 4 * q^22 + 4 * q^23 + 4 * q^26 + 2 * q^27 + 8 * q^29 - 6 * q^31 + 4 * q^33 - 12 * q^34 + 14 * q^37 + 16 * q^38 + 6 * q^39 - 4 * q^41 + 18 * q^43 - 12 * q^46 + 4 * q^47 - 8 * q^48 + 4 * q^51 - 8 * q^53 - 2 * q^57 + 12 * q^58 + 8 * q^61 - 8 * q^62 + 16 * q^64 - 4 * q^66 + 2 * q^67 + 4 * q^69 - 4 * q^71 + 10 * q^73 + 4 * q^74 + 4 * q^78 + 2 * q^79 + 2 * q^81 - 12 * q^82 - 8 * q^83 - 4 * q^86 + 8 * q^87 + 8 * q^88 - 16 * q^89 - 6 * q^93 + 32 * q^94 + 16 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	1.41421	0.577350
$7$	0	0
$7$	0	0
$8$	−2.82843	−1.00000
$9$	1.00000	0.333333
$10$	0	0
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	0	0
$13$	4.41421	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$13$	4.41421	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−2.24264	−0.543920	−0.271960	−	0.962309i	$-0.587672\pi$
$17$	−2.24264	−0.543920	−0.271960	+	0.962309i	$0.587672\pi$
$18$	1.41421	0.333333
$19$	4.65685	1.06836	0.534178	−	0.845372i	$-0.320621\pi$
$19$	4.65685	1.06836	0.534178	+	0.845372i	$0.320621\pi$
$20$	0	0
$21$	0	0
$22$	0.828427	0.176621
$23$	−2.24264	−0.467623	−0.233811	−	0.972282i	$-0.575120\pi$
$23$	−2.24264	−0.467623	−0.233811	+	0.972282i	$0.575120\pi$
$24$	−2.82843	−0.577350
$25$	0	0
$26$	6.24264	1.22428
$27$	1.00000	0.192450
$28$	0	0
$29$	8.24264	1.53062	0.765310	−	0.643662i	$-0.222586\pi$
$29$	8.24264	1.53062	0.765310	+	0.643662i	$0.222586\pi$
$30$	0	0
$31$	−5.82843	−1.04682	−0.523408	−	0.852082i	$-0.675340\pi$
$31$	−5.82843	−1.04682	−0.523408	+	0.852082i	$0.675340\pi$
$32$	0	0
$33$	0.585786	0.101972
$34$	−3.17157	−0.543920
$35$	0	0
$36$	0	0
$37$	8.41421	1.38329	0.691644	−	0.722238i	$-0.256887\pi$
$37$	8.41421	1.38329	0.691644	+	0.722238i	$0.256887\pi$
$38$	6.58579	1.06836
$39$	4.41421	0.706840
$40$	0	0
$41$	−6.24264	−0.974937	−0.487468	−	0.873141i	$-0.662080\pi$
$41$	−6.24264	−0.974937	−0.487468	+	0.873141i	$0.662080\pi$
$42$	0	0
$43$	7.58579	1.15682	0.578411	−	0.815746i	$-0.303673\pi$
$43$	7.58579	1.15682	0.578411	+	0.815746i	$0.303673\pi$
$44$	0	0
$45$	0	0
$46$	−3.17157	−0.467623
$47$	13.3137	1.94200	0.971002	−	0.239071i	$-0.0768430\pi$
$47$	13.3137	1.94200	0.971002	+	0.239071i	$0.0768430\pi$
$48$	−4.00000	−0.577350
$49$	0	0
$50$	0	0
$51$	−2.24264	−0.314033
$52$	0	0
$53$	−6.82843	−0.937957	−0.468978	−	0.883210i	$-0.655378\pi$
$53$	−6.82843	−0.937957	−0.468978	+	0.883210i	$0.655378\pi$
$54$	1.41421	0.192450
$55$	0	0
$56$	0	0
$57$	4.65685	0.616815
$58$	11.6569	1.53062
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	0	0
$61$	−4.48528	−0.574281	−0.287141	−	0.957888i	$-0.592705\pi$
$61$	−4.48528	−0.574281	−0.287141	+	0.957888i	$0.592705\pi$
$62$	−8.24264	−1.04682
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0.828427	0.101972
$67$	13.7279	1.67713	0.838566	−	0.544800i	$-0.183394\pi$
$67$	13.7279	1.67713	0.838566	+	0.544800i	$0.183394\pi$
$68$	0	0
$69$	−2.24264	−0.269982
$70$	0	0
$71$	−0.585786	−0.0695201	−0.0347600	−	0.999396i	$-0.511067\pi$
$71$	−0.585786	−0.0695201	−0.0347600	+	0.999396i	$0.511067\pi$
$72$	−2.82843	−0.333333
$73$	12.0711	1.41281	0.706406	−	0.707807i	$-0.250315\pi$
$73$	12.0711	1.41281	0.706406	+	0.707807i	$0.250315\pi$
$74$	11.8995	1.38329
$75$	0	0
$76$	0	0
$77$	0	0
$78$	6.24264	0.706840
$79$	6.65685	0.748955	0.374477	−	0.927236i	$-0.377822\pi$
$79$	6.65685	0.748955	0.374477	+	0.927236i	$0.377822\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−8.82843	−0.974937
$83$	−2.58579	−0.283827	−0.141913	−	0.989879i	$-0.545325\pi$
$83$	−2.58579	−0.283827	−0.141913	+	0.989879i	$0.545325\pi$
$84$	0	0
$85$	0	0
$86$	10.7279	1.15682
$87$	8.24264	0.883704
$88$	−1.65685	−0.176621
$89$	−12.2426	−1.29772	−0.648859	−	0.760909i	$-0.724753\pi$
$89$	−12.2426	−1.29772	−0.648859	+	0.760909i	$0.724753\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.82843	−0.604380
$94$	18.8284	1.94200
$95$	0	0
$96$	0	0
$97$	5.17157	0.525094	0.262547	−	0.964919i	$-0.415438\pi$
$97$	5.17157	0.525094	0.262547	+	0.964919i	$0.415438\pi$
$98$	0	0
$99$	0.585786	0.0588738
$100$	0	0
$101$	2.24264	0.223151	0.111576	−	0.993756i	$-0.464410\pi$
$101$	2.24264	0.223151	0.111576	+	0.993756i	$0.464410\pi$
$102$	−3.17157	−0.314033
$103$	−7.24264	−0.713639	−0.356819	−	0.934173i	$-0.616139\pi$
$103$	−7.24264	−0.713639	−0.356819	+	0.934173i	$0.616139\pi$
$104$	−12.4853	−1.22428
$105$	0	0
$106$	−9.65685	−0.937957
$107$	−12.5858	−1.21671	−0.608357	−	0.793664i	$-0.708171\pi$
$107$	−12.5858	−1.21671	−0.608357	+	0.793664i	$0.708171\pi$
$108$	0	0
$109$	3.48528	0.333829	0.166915	−	0.985971i	$-0.446620\pi$
$109$	3.48528	0.333829	0.166915	+	0.985971i	$0.446620\pi$
$110$	0	0
$111$	8.41421	0.798642
$112$	0	0
$113$	15.6569	1.47287	0.736436	−	0.676507i	$-0.236507\pi$
$113$	15.6569	1.47287	0.736436	+	0.676507i	$0.236507\pi$
$114$	6.58579	0.616815
$115$	0	0
$116$	0	0
$117$	4.41421	0.408094
$118$	−2.00000	−0.184115
$119$	0	0
$120$	0	0
$121$	−10.6569	−0.968805
$122$	−6.34315	−0.574281
$123$	−6.24264	−0.562880
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.92893	0.348636	0.174318	−	0.984689i	$-0.444228\pi$
$127$	3.92893	0.348636	0.174318	+	0.984689i	$0.444228\pi$
$128$	11.3137	1.00000
$129$	7.58579	0.667891
$130$	0	0
$131$	6.48528	0.566622	0.283311	−	0.959028i	$-0.408567\pi$
$131$	6.48528	0.566622	0.283311	+	0.959028i	$0.408567\pi$
$132$	0	0
$133$	0	0
$134$	19.4142	1.67713
$135$	0	0
$136$	6.34315	0.543920
$137$	17.0711	1.45848	0.729240	−	0.684258i	$-0.239874\pi$
$137$	17.0711	1.45848	0.729240	+	0.684258i	$0.239874\pi$
$138$	−3.17157	−0.269982
$139$	−5.48528	−0.465255	−0.232628	−	0.972566i	$-0.574732\pi$
$139$	−5.48528	−0.465255	−0.232628	+	0.972566i	$0.574732\pi$
$140$	0	0
$141$	13.3137	1.12122
$142$	−0.828427	−0.0695201
$143$	2.58579	0.216234
$144$	−4.00000	−0.333333
$145$	0	0
$146$	17.0711	1.41281
$147$	0	0
$148$	0	0
$149$	6.34315	0.519651	0.259825	−	0.965656i	$-0.416335\pi$
$149$	6.34315	0.519651	0.259825	+	0.965656i	$0.416335\pi$
$150$	0	0
$151$	10.4853	0.853280	0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853	0.853280	0.426640	+	0.904422i	$0.359697\pi$
$152$	−13.1716	−1.06836
$153$	−2.24264	−0.181307
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.1421	−0.969048	−0.484524	−	0.874778i	$-0.661007\pi$
$157$	−12.1421	−0.969048	−0.484524	+	0.874778i	$0.661007\pi$
$158$	9.41421	0.748955
$159$	−6.82843	−0.541529
$160$	0	0
$161$	0	0
$162$	1.41421	0.111111
$163$	−3.31371	−0.259550	−0.129775	−	0.991543i	$-0.541425\pi$
$163$	−3.31371	−0.259550	−0.129775	+	0.991543i	$0.541425\pi$
$164$	0	0
$165$	0	0
$166$	−3.65685	−0.283827
$167$	−20.2426	−1.56642	−0.783211	−	0.621756i	$-0.786420\pi$
$167$	−20.2426	−1.56642	−0.783211	+	0.621756i	$0.786420\pi$
$168$	0	0
$169$	6.48528	0.498868
$170$	0	0
$171$	4.65685	0.356119
$172$	0	0
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	11.6569	0.883704
$175$	0	0
$176$	−2.34315	−0.176621
$177$	−1.41421	−0.106299
$178$	−17.3137	−1.29772
$179$	8.34315	0.623596	0.311798	−	0.950148i	$-0.399069\pi$
$179$	8.34315	0.623596	0.311798	+	0.950148i	$0.399069\pi$
$180$	0	0
$181$	−10.6569	−0.792118	−0.396059	−	0.918225i	$-0.629622\pi$
$181$	−10.6569	−0.792118	−0.396059	+	0.918225i	$0.629622\pi$
$182$	0	0
$183$	−4.48528	−0.331562
$184$	6.34315	0.467623
$185$	0	0
$186$	−8.24264	−0.604380
$187$	−1.31371	−0.0960679
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.9706	−1.08323	−0.541616	−	0.840626i	$-0.682187\pi$
$191$	−14.9706	−1.08323	−0.541616	+	0.840626i	$0.682187\pi$
$192$	8.00000	0.577350
$193$	15.2426	1.09719	0.548595	−	0.836088i	$-0.315163\pi$
$193$	15.2426	1.09719	0.548595	+	0.836088i	$0.315163\pi$
$194$	7.31371	0.525094
$195$	0	0
$196$	0	0
$197$	25.5563	1.82081	0.910407	−	0.413713i	$-0.135768\pi$
$197$	25.5563	1.82081	0.910407	+	0.413713i	$0.135768\pi$
$198$	0.828427	0.0588738
$199$	22.4853	1.59394	0.796970	−	0.604019i	$-0.206435\pi$
$199$	22.4853	1.59394	0.796970	+	0.604019i	$0.206435\pi$
$200$	0	0
$201$	13.7279	0.968293
$202$	3.17157	0.223151
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−10.2426	−0.713639
$207$	−2.24264	−0.155874
$208$	−17.6569	−1.22428
$209$	2.72792	0.188694
$210$	0	0
$211$	−28.1421	−1.93738	−0.968692	−	0.248265i	$-0.920140\pi$
$211$	−28.1421	−1.93738	−0.968692	+	0.248265i	$0.920140\pi$
$212$	0	0
$213$	−0.585786	−0.0401374
$214$	−17.7990	−1.21671
$215$	0	0
$216$	−2.82843	−0.192450
$217$	0	0
$218$	4.92893	0.333829
$219$	12.0711	0.815687
$220$	0	0
$221$	−9.89949	−0.665912
$222$	11.8995	0.798642
$223$	−22.8284	−1.52870	−0.764352	−	0.644799i	$-0.776941\pi$
$223$	−22.8284	−1.52870	−0.764352	+	0.644799i	$0.776941\pi$
$224$	0	0
$225$	0	0
$226$	22.1421	1.47287
$227$	−8.92893	−0.592634	−0.296317	−	0.955090i	$-0.595758\pi$
$227$	−8.92893	−0.592634	−0.296317	+	0.955090i	$0.595758\pi$
$228$	0	0
$229$	8.31371	0.549385	0.274693	−	0.961532i	$-0.411424\pi$
$229$	8.31371	0.549385	0.274693	+	0.961532i	$0.411424\pi$
$230$	0	0
$231$	0	0
$232$	−23.3137	−1.53062
$233$	5.51472	0.361281	0.180641	−	0.983549i	$-0.442183\pi$
$233$	5.51472	0.361281	0.180641	+	0.983549i	$0.442183\pi$
$234$	6.24264	0.408094
$235$	0	0
$236$	0	0
$237$	6.65685	0.432409
$238$	0	0
$239$	−5.31371	−0.343715	−0.171858	−	0.985122i	$-0.554977\pi$
$239$	−5.31371	−0.343715	−0.171858	+	0.985122i	$0.554977\pi$
$240$	0	0
$241$	−21.6569	−1.39504	−0.697520	−	0.716565i	$-0.745713\pi$
$241$	−21.6569	−1.39504	−0.697520	+	0.716565i	$0.745713\pi$
$242$	−15.0711	−0.968805
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	−8.82843	−0.562880
$247$	20.5563	1.30797
$248$	16.4853	1.04682
$249$	−2.58579	−0.163868
$250$	0	0
$251$	−2.58579	−0.163213	−0.0816067	−	0.996665i	$-0.526005\pi$
$251$	−2.58579	−0.163213	−0.0816067	+	0.996665i	$0.526005\pi$
$252$	0	0
$253$	−1.31371	−0.0825921
$254$	5.55635	0.348636
$255$	0	0
$256$	0	0
$257$	10.1005	0.630052	0.315026	−	0.949083i	$-0.397987\pi$
$257$	10.1005	0.630052	0.315026	+	0.949083i	$0.397987\pi$
$258$	10.7279	0.667891
$259$	0	0
$260$	0	0
$261$	8.24264	0.510207
$262$	9.17157	0.566622
$263$	−6.82843	−0.421059	−0.210529	−	0.977588i	$-0.567519\pi$
$263$	−6.82843	−0.421059	−0.210529	+	0.977588i	$0.567519\pi$
$264$	−1.65685	−0.101972
$265$	0	0
$266$	0	0
$267$	−12.2426	−0.749237
$268$	0	0
$269$	−20.1421	−1.22809	−0.614044	−	0.789272i	$-0.710458\pi$
$269$	−20.1421	−1.22809	−0.614044	+	0.789272i	$0.710458\pi$
$270$	0	0
$271$	−8.14214	−0.494600	−0.247300	−	0.968939i	$-0.579543\pi$
$271$	−8.14214	−0.494600	−0.247300	+	0.968939i	$0.579543\pi$
$272$	8.97056	0.543920
$273$	0	0
$274$	24.1421	1.45848
$275$	0	0
$276$	0	0
$277$	−13.3848	−0.804213	−0.402107	−	0.915593i	$-0.631722\pi$
$277$	−13.3848	−0.804213	−0.402107	+	0.915593i	$0.631722\pi$
$278$	−7.75736	−0.465255
$279$	−5.82843	−0.348939
$280$	0	0
$281$	−1.65685	−0.0988396	−0.0494198	−	0.998778i	$-0.515737\pi$
$281$	−1.65685	−0.0988396	−0.0494198	+	0.998778i	$0.515737\pi$
$282$	18.8284	1.12122
$283$	4.75736	0.282796	0.141398	−	0.989953i	$-0.454840\pi$
$283$	4.75736	0.282796	0.141398	+	0.989953i	$0.454840\pi$
$284$	0	0
$285$	0	0
$286$	3.65685	0.216234
$287$	0	0
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	5.17157	0.303163
$292$	0	0
$293$	−7.31371	−0.427271	−0.213636	−	0.976913i	$-0.568531\pi$
$293$	−7.31371	−0.427271	−0.213636	+	0.976913i	$0.568531\pi$
$294$	0	0
$295$	0	0
$296$	−23.7990	−1.38329
$297$	0.585786	0.0339908
$298$	8.97056	0.519651
$299$	−9.89949	−0.572503
$300$	0	0
$301$	0	0
$302$	14.8284	0.853280
$303$	2.24264	0.128836
$304$	−18.6274	−1.06836
$305$	0	0
$306$	−3.17157	−0.181307
$307$	−4.41421	−0.251932	−0.125966	−	0.992035i	$-0.540203\pi$
$307$	−4.41421	−0.251932	−0.125966	+	0.992035i	$0.540203\pi$
$308$	0	0
$309$	−7.24264	−0.412019
$310$	0	0
$311$	−2.92893	−0.166085	−0.0830423	−	0.996546i	$-0.526464\pi$
$311$	−2.92893	−0.166085	−0.0830423	+	0.996546i	$0.526464\pi$
$312$	−12.4853	−0.706840
$313$	16.2132	0.916424	0.458212	−	0.888843i	$-0.348490\pi$
$313$	16.2132	0.916424	0.458212	+	0.888843i	$0.348490\pi$
$314$	−17.1716	−0.969048
$315$	0	0
$316$	0	0
$317$	−19.8995	−1.11767	−0.558833	−	0.829280i	$-0.688751\pi$
$317$	−19.8995	−1.11767	−0.558833	+	0.829280i	$0.688751\pi$
$318$	−9.65685	−0.541529
$319$	4.82843	0.270340
$320$	0	0
$321$	−12.5858	−0.702470
$322$	0	0
$323$	−10.4437	−0.581100
$324$	0	0
$325$	0	0
$326$	−4.68629	−0.259550
$327$	3.48528	0.192737
$328$	17.6569	0.974937
$329$	0	0
$330$	0	0
$331$	17.6274	0.968890	0.484445	−	0.874822i	$-0.339021\pi$
$331$	17.6274	0.968890	0.484445	+	0.874822i	$0.339021\pi$
$332$	0	0
$333$	8.41421	0.461096
$334$	−28.6274	−1.56642
$335$	0	0
$336$	0	0
$337$	−18.8995	−1.02952	−0.514761	−	0.857334i	$-0.672119\pi$
$337$	−18.8995	−1.02952	−0.514761	+	0.857334i	$0.672119\pi$
$338$	9.17157	0.498868
$339$	15.6569	0.850364
$340$	0	0
$341$	−3.41421	−0.184890
$342$	6.58579	0.356119
$343$	0	0
$344$	−21.4558	−1.15682
$345$	0	0
$346$	−4.00000	−0.215041
$347$	−8.97056	−0.481565	−0.240783	−	0.970579i	$-0.577404\pi$
$347$	−8.97056	−0.481565	−0.240783	+	0.970579i	$0.577404\pi$
$348$	0	0
$349$	28.6274	1.53239	0.766195	−	0.642608i	$-0.222148\pi$
$349$	28.6274	1.53239	0.766195	+	0.642608i	$0.222148\pi$
$350$	0	0
$351$	4.41421	0.235613
$352$	0	0
$353$	13.8995	0.739795	0.369898	−	0.929072i	$-0.379393\pi$
$353$	13.8995	0.739795	0.369898	+	0.929072i	$0.379393\pi$
$354$	−2.00000	−0.106299
$355$	0	0
$356$	0	0
$357$	0	0
$358$	11.7990	0.623596
$359$	−29.4142	−1.55242	−0.776211	−	0.630473i	$-0.782861\pi$
$359$	−29.4142	−1.55242	−0.776211	+	0.630473i	$0.782861\pi$
$360$	0	0
$361$	2.68629	0.141384
$362$	−15.0711	−0.792118
$363$	−10.6569	−0.559340
$364$	0	0
$365$	0	0
$366$	−6.34315	−0.331562
$367$	22.4142	1.17001	0.585006	−	0.811029i	$-0.301092\pi$
$367$	22.4142	1.17001	0.585006	+	0.811029i	$0.301092\pi$
$368$	8.97056	0.467623
$369$	−6.24264	−0.324979
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.7574	−0.556995	−0.278497	−	0.960437i	$-0.589836\pi$
$373$	−10.7574	−0.556995	−0.278497	+	0.960437i	$0.589836\pi$
$374$	−1.85786	−0.0960679
$375$	0	0
$376$	−37.6569	−1.94200
$377$	36.3848	1.87391
$378$	0	0
$379$	−24.7990	−1.27384	−0.636919	−	0.770931i	$-0.719792\pi$
$379$	−24.7990	−1.27384	−0.636919	+	0.770931i	$0.719792\pi$
$380$	0	0
$381$	3.92893	0.201285
$382$	−21.1716	−1.08323
$383$	4.48528	0.229187	0.114594	−	0.993412i	$-0.463443\pi$
$383$	4.48528	0.229187	0.114594	+	0.993412i	$0.463443\pi$
$384$	11.3137	0.577350
$385$	0	0
$386$	21.5563	1.09719
$387$	7.58579	0.385607
$388$	0	0
$389$	−36.8701	−1.86939	−0.934693	−	0.355456i	$-0.884326\pi$
$389$	−36.8701	−1.86939	−0.934693	+	0.355456i	$0.884326\pi$
$390$	0	0
$391$	5.02944	0.254350
$392$	0	0
$393$	6.48528	0.327139
$394$	36.1421	1.82081
$395$	0	0
$396$	0	0
$397$	−18.2132	−0.914094	−0.457047	−	0.889442i	$-0.651093\pi$
$397$	−18.2132	−0.914094	−0.457047	+	0.889442i	$0.651093\pi$
$398$	31.7990	1.59394
$399$	0	0
$400$	0	0
$401$	−16.4853	−0.823236	−0.411618	−	0.911357i	$-0.635036\pi$
$401$	−16.4853	−0.823236	−0.411618	+	0.911357i	$0.635036\pi$
$402$	19.4142	0.968293
$403$	−25.7279	−1.28160
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.92893	0.244318
$408$	6.34315	0.314033
$409$	−3.68629	−0.182275	−0.0911377	−	0.995838i	$-0.529050\pi$
$409$	−3.68629	−0.182275	−0.0911377	+	0.995838i	$0.529050\pi$
$410$	0	0
$411$	17.0711	0.842054
$412$	0	0
$413$	0	0
$414$	−3.17157	−0.155874
$415$	0	0
$416$	0	0
$417$	−5.48528	−0.268615
$418$	3.85786	0.188694
$419$	−8.82843	−0.431297	−0.215648	−	0.976471i	$-0.569187\pi$
$419$	−8.82843	−0.431297	−0.215648	+	0.976471i	$0.569187\pi$
$420$	0	0
$421$	10.5147	0.512456	0.256228	−	0.966616i	$-0.417520\pi$
$421$	10.5147	0.512456	0.256228	+	0.966616i	$0.417520\pi$
$422$	−39.7990	−1.93738
$423$	13.3137	0.647335
$424$	19.3137	0.937957
$425$	0	0
$426$	−0.828427	−0.0401374
$427$	0	0
$428$	0	0
$429$	2.58579	0.124843
$430$	0	0
$431$	31.1716	1.50148	0.750741	−	0.660597i	$-0.229697\pi$
$431$	31.1716	1.50148	0.750741	+	0.660597i	$0.229697\pi$
$432$	−4.00000	−0.192450
$433$	−6.55635	−0.315078	−0.157539	−	0.987513i	$-0.550356\pi$
$433$	−6.55635	−0.315078	−0.157539	+	0.987513i	$0.550356\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.4437	−0.499588
$438$	17.0711	0.815687
$439$	11.6569	0.556351	0.278176	−	0.960530i	$-0.410270\pi$
$439$	11.6569	0.556351	0.278176	+	0.960530i	$0.410270\pi$
$440$	0	0
$441$	0	0
$442$	−14.0000	−0.665912
$443$	16.4853	0.783239	0.391620	−	0.920127i	$-0.371915\pi$
$443$	16.4853	0.783239	0.391620	+	0.920127i	$0.371915\pi$
$444$	0	0
$445$	0	0
$446$	−32.2843	−1.52870
$447$	6.34315	0.300020
$448$	0	0
$449$	26.9706	1.27282	0.636410	−	0.771351i	$-0.280419\pi$
$449$	26.9706	1.27282	0.636410	+	0.771351i	$0.280419\pi$
$450$	0	0
$451$	−3.65685	−0.172195
$452$	0	0
$453$	10.4853	0.492641
$454$	−12.6274	−0.592634
$455$	0	0
$456$	−13.1716	−0.616815
$457$	1.10051	0.0514795	0.0257397	−	0.999669i	$-0.491806\pi$
$457$	1.10051	0.0514795	0.0257397	+	0.999669i	$0.491806\pi$
$458$	11.7574	0.549385
$459$	−2.24264	−0.104678
$460$	0	0
$461$	−12.9289	−0.602160	−0.301080	−	0.953599i	$-0.597347\pi$
$461$	−12.9289	−0.602160	−0.301080	+	0.953599i	$0.597347\pi$
$462$	0	0
$463$	7.58579	0.352541	0.176271	−	0.984342i	$-0.443597\pi$
$463$	7.58579	0.352541	0.176271	+	0.984342i	$0.443597\pi$
$464$	−32.9706	−1.53062
$465$	0	0
$466$	7.79899	0.361281
$467$	−29.3137	−1.35648	−0.678238	−	0.734842i	$-0.737256\pi$
$467$	−29.3137	−1.35648	−0.678238	+	0.734842i	$0.737256\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−12.1421	−0.559480
$472$	4.00000	0.184115
$473$	4.44365	0.204319
$474$	9.41421	0.432409
$475$	0	0
$476$	0	0
$477$	−6.82843	−0.312652
$478$	−7.51472	−0.343715
$479$	10.6274	0.485579	0.242790	−	0.970079i	$-0.421938\pi$
$479$	10.6274	0.485579	0.242790	+	0.970079i	$0.421938\pi$
$480$	0	0
$481$	37.1421	1.69354
$482$	−30.6274	−1.39504
$483$	0	0
$484$	0	0
$485$	0	0
$486$	1.41421	0.0641500
$487$	18.8995	0.856418	0.428209	−	0.903680i	$-0.359145\pi$
$487$	18.8995	0.856418	0.428209	+	0.903680i	$0.359145\pi$
$488$	12.6863	0.574281
$489$	−3.31371	−0.149851
$490$	0	0
$491$	18.1421	0.818743	0.409372	−	0.912368i	$-0.365748\pi$
$491$	18.1421	0.818743	0.409372	+	0.912368i	$0.365748\pi$
$492$	0	0
$493$	−18.4853	−0.832535
$494$	29.0711	1.30797
$495$	0	0
$496$	23.3137	1.04682
$497$	0	0
$498$	−3.65685	−0.163868
$499$	−30.7990	−1.37875	−0.689376	−	0.724404i	$-0.742115\pi$
$499$	−30.7990	−1.37875	−0.689376	+	0.724404i	$0.742115\pi$
$500$	0	0
$501$	−20.2426	−0.904374
$502$	−3.65685	−0.163213
$503$	1.51472	0.0675380	0.0337690	−	0.999430i	$-0.489249\pi$
$503$	1.51472	0.0675380	0.0337690	+	0.999430i	$0.489249\pi$
$504$	0	0
$505$	0	0
$506$	−1.85786	−0.0825921
$507$	6.48528	0.288021
$508$	0	0
$509$	−25.7990	−1.14352	−0.571760	−	0.820421i	$-0.693739\pi$
$509$	−25.7990	−1.14352	−0.571760	+	0.820421i	$0.693739\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−1.00000
$513$	4.65685	0.205605
$514$	14.2843	0.630052
$515$	0	0
$516$	0	0
$517$	7.79899	0.342999
$518$	0	0
$519$	−2.82843	−0.124154
$520$	0	0
$521$	23.1127	1.01259	0.506293	−	0.862362i	$-0.331015\pi$
$521$	23.1127	1.01259	0.506293	+	0.862362i	$0.331015\pi$
$522$	11.6569	0.510207
$523$	10.4142	0.455382	0.227691	−	0.973733i	$-0.426882\pi$
$523$	10.4142	0.455382	0.227691	+	0.973733i	$0.426882\pi$
$524$	0	0
$525$	0	0
$526$	−9.65685	−0.421059
$527$	13.0711	0.569385
$528$	−2.34315	−0.101972
$529$	−17.9706	−0.781329
$530$	0	0
$531$	−1.41421	−0.0613716
$532$	0	0
$533$	−27.5563	−1.19360
$534$	−17.3137	−0.749237
$535$	0	0
$536$	−38.8284	−1.67713
$537$	8.34315	0.360033
$538$	−28.4853	−1.22809
$539$	0	0
$540$	0	0
$541$	9.68629	0.416446	0.208223	−	0.978081i	$-0.433232\pi$
$541$	9.68629	0.416446	0.208223	+	0.978081i	$0.433232\pi$
$542$	−11.5147	−0.494600
$543$	−10.6569	−0.457329
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.79899	−0.333461	−0.166730	−	0.986003i	$-0.553321\pi$
$547$	−7.79899	−0.333461	−0.166730	+	0.986003i	$0.553321\pi$
$548$	0	0
$549$	−4.48528	−0.191427
$550$	0	0
$551$	38.3848	1.63525
$552$	6.34315	0.269982
$553$	0	0
$554$	−18.9289	−0.804213
$555$	0	0
$556$	0	0
$557$	19.6569	0.832888	0.416444	−	0.909161i	$-0.363276\pi$
$557$	19.6569	0.832888	0.416444	+	0.909161i	$0.363276\pi$
$558$	−8.24264	−0.348939
$559$	33.4853	1.41628
$560$	0	0
$561$	−1.31371	−0.0554648
$562$	−2.34315	−0.0988396
$563$	−42.2843	−1.78207	−0.891035	−	0.453935i	$-0.850020\pi$
$563$	−42.2843	−1.78207	−0.891035	+	0.453935i	$0.850020\pi$
$564$	0	0
$565$	0	0
$566$	6.72792	0.282796
$567$	0	0
$568$	1.65685	0.0695201
$569$	16.8701	0.707230	0.353615	−	0.935391i	$-0.384952\pi$
$569$	16.8701	0.707230	0.353615	+	0.935391i	$0.384952\pi$
$570$	0	0
$571$	−5.97056	−0.249860	−0.124930	−	0.992166i	$-0.539871\pi$
$571$	−5.97056	−0.249860	−0.124930	+	0.992166i	$0.539871\pi$
$572$	0	0
$573$	−14.9706	−0.625404
$574$	0	0
$575$	0	0
$576$	8.00000	0.333333
$577$	24.5563	1.02229	0.511147	−	0.859493i	$-0.329221\pi$
$577$	24.5563	1.02229	0.511147	+	0.859493i	$0.329221\pi$
$578$	−16.9289	−0.704151
$579$	15.2426	0.633463
$580$	0	0
$581$	0	0
$582$	7.31371	0.303163
$583$	−4.00000	−0.165663
$584$	−34.1421	−1.41281
$585$	0	0
$586$	−10.3431	−0.427271
$587$	−21.7574	−0.898022	−0.449011	−	0.893526i	$-0.648224\pi$
$587$	−21.7574	−0.898022	−0.449011	+	0.893526i	$0.648224\pi$
$588$	0	0
$589$	−27.1421	−1.11837
$590$	0	0
$591$	25.5563	1.05125
$592$	−33.6569	−1.38329
$593$	−10.5858	−0.434706	−0.217353	−	0.976093i	$-0.569742\pi$
$593$	−10.5858	−0.434706	−0.217353	+	0.976093i	$0.569742\pi$
$594$	0.828427	0.0339908
$595$	0	0
$596$	0	0
$597$	22.4853	0.920261
$598$	−14.0000	−0.572503
$599$	−13.8579	−0.566217	−0.283108	−	0.959088i	$-0.591366\pi$
$599$	−13.8579	−0.566217	−0.283108	+	0.959088i	$0.591366\pi$
$600$	0	0
$601$	−29.8284	−1.21673	−0.608363	−	0.793659i	$-0.708174\pi$
$601$	−29.8284	−1.21673	−0.608363	+	0.793659i	$0.708174\pi$
$602$	0	0
$603$	13.7279	0.559044
$604$	0	0
$605$	0	0
$606$	3.17157	0.128836
$607$	−7.24264	−0.293970	−0.146985	−	0.989139i	$-0.546957\pi$
$607$	−7.24264	−0.293970	−0.146985	+	0.989139i	$0.546957\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	58.7696	2.37756
$612$	0	0
$613$	18.1421	0.732754	0.366377	−	0.930467i	$-0.380598\pi$
$613$	18.1421	0.732754	0.366377	+	0.930467i	$0.380598\pi$
$614$	−6.24264	−0.251932
$615$	0	0
$616$	0	0
$617$	3.17157	0.127683	0.0638414	−	0.997960i	$-0.479665\pi$
$617$	3.17157	0.127683	0.0638414	+	0.997960i	$0.479665\pi$
$618$	−10.2426	−0.412019
$619$	6.02944	0.242344	0.121172	−	0.992632i	$-0.461335\pi$
$619$	6.02944	0.242344	0.121172	+	0.992632i	$0.461335\pi$
$620$	0	0
$621$	−2.24264	−0.0899941
$622$	−4.14214	−0.166085
$623$	0	0
$624$	−17.6569	−0.706840
$625$	0	0
$626$	22.9289	0.916424
$627$	2.72792	0.108943
$628$	0	0
$629$	−18.8701	−0.752398
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−18.8284	−0.748955
$633$	−28.1421	−1.11855
$634$	−28.1421	−1.11767
$635$	0	0
$636$	0	0
$637$	0	0
$638$	6.82843	0.270340
$639$	−0.585786	−0.0231734
$640$	0	0
$641$	29.2132	1.15385	0.576926	−	0.816796i	$-0.304252\pi$
$641$	29.2132	1.15385	0.576926	+	0.816796i	$0.304252\pi$
$642$	−17.7990	−0.702470
$643$	6.55635	0.258557	0.129279	−	0.991608i	$-0.458734\pi$
$643$	6.55635	0.258557	0.129279	+	0.991608i	$0.458734\pi$
$644$	0	0
$645$	0	0
$646$	−14.7696	−0.581100
$647$	−46.3848	−1.82357	−0.911787	−	0.410664i	$-0.865297\pi$
$647$	−46.3848	−1.82357	−0.911787	+	0.410664i	$0.865297\pi$
$648$	−2.82843	−0.111111
$649$	−0.828427	−0.0325186
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.2426	−0.635624	−0.317812	−	0.948154i	$-0.602948\pi$
$653$	−16.2426	−0.635624	−0.317812	+	0.948154i	$0.602948\pi$
$654$	4.92893	0.192737
$655$	0	0
$656$	24.9706	0.974937
$657$	12.0711	0.470937
$658$	0	0
$659$	−4.68629	−0.182552	−0.0912760	−	0.995826i	$-0.529095\pi$
$659$	−4.68629	−0.182552	−0.0912760	+	0.995826i	$0.529095\pi$
$660$	0	0
$661$	9.68629	0.376753	0.188377	−	0.982097i	$-0.439677\pi$
$661$	9.68629	0.376753	0.188377	+	0.982097i	$0.439677\pi$
$662$	24.9289	0.968890
$663$	−9.89949	−0.384465
$664$	7.31371	0.283827
$665$	0	0
$666$	11.8995	0.461096
$667$	−18.4853	−0.715753
$668$	0	0
$669$	−22.8284	−0.882598
$670$	0	0
$671$	−2.62742	−0.101430
$672$	0	0
$673$	−27.7279	−1.06883	−0.534416	−	0.845221i	$-0.679469\pi$
$673$	−27.7279	−1.06883	−0.534416	+	0.845221i	$0.679469\pi$
$674$	−26.7279	−1.02952
$675$	0	0
$676$	0	0
$677$	15.8995	0.611067	0.305534	−	0.952181i	$-0.401165\pi$
$677$	15.8995	0.611067	0.305534	+	0.952181i	$0.401165\pi$
$678$	22.1421	0.850364
$679$	0	0
$680$	0	0
$681$	−8.92893	−0.342157
$682$	−4.82843	−0.184890
$683$	−23.4142	−0.895920	−0.447960	−	0.894054i	$-0.647849\pi$
$683$	−23.4142	−0.895920	−0.447960	+	0.894054i	$0.647849\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.31371	0.317188
$688$	−30.3431	−1.15682
$689$	−30.1421	−1.14832
$690$	0	0
$691$	7.68629	0.292400	0.146200	−	0.989255i	$-0.453296\pi$
$691$	7.68629	0.292400	0.146200	+	0.989255i	$0.453296\pi$
$692$	0	0
$693$	0	0
$694$	−12.6863	−0.481565
$695$	0	0
$696$	−23.3137	−0.883704
$697$	14.0000	0.530288
$698$	40.4853	1.53239
$699$	5.51472	0.208586
$700$	0	0
$701$	−1.21320	−0.0458221	−0.0229110	−	0.999738i	$-0.507293\pi$
$701$	−1.21320	−0.0458221	−0.0229110	+	0.999738i	$0.507293\pi$
$702$	6.24264	0.235613
$703$	39.1838	1.47784
$704$	4.68629	0.176621
$705$	0	0
$706$	19.6569	0.739795
$707$	0	0
$708$	0	0
$709$	51.1127	1.91958	0.959789	−	0.280723i	$-0.0905742\pi$
$709$	51.1127	1.91958	0.959789	+	0.280723i	$0.0905742\pi$
$710$	0	0
$711$	6.65685	0.249652
$712$	34.6274	1.29772
$713$	13.0711	0.489515
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.31371	−0.198444
$718$	−41.5980	−1.55242
$719$	34.4853	1.28608	0.643042	−	0.765831i	$-0.277672\pi$
$719$	34.4853	1.28608	0.643042	+	0.765831i	$0.277672\pi$
$720$	0	0
$721$	0	0
$722$	3.79899	0.141384
$723$	−21.6569	−0.805427
$724$	0	0
$725$	0	0
$726$	−15.0711	−0.559340
$727$	−13.2426	−0.491142	−0.245571	−	0.969379i	$-0.578976\pi$
$727$	−13.2426	−0.491142	−0.245571	+	0.969379i	$0.578976\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.0122	−0.629219
$732$	0	0
$733$	19.2426	0.710743	0.355372	−	0.934725i	$-0.384354\pi$
$733$	19.2426	0.710743	0.355372	+	0.934725i	$0.384354\pi$
$734$	31.6985	1.17001
$735$	0	0
$736$	0	0
$737$	8.04163	0.296217
$738$	−8.82843	−0.324979
$739$	−6.85786	−0.252271	−0.126135	−	0.992013i	$-0.540257\pi$
$739$	−6.85786	−0.252271	−0.126135	+	0.992013i	$0.540257\pi$
$740$	0	0
$741$	20.5563	0.755156
$742$	0	0
$743$	−20.7279	−0.760434	−0.380217	−	0.924897i	$-0.624151\pi$
$743$	−20.7279	−0.760434	−0.380217	+	0.924897i	$0.624151\pi$
$744$	16.4853	0.604380
$745$	0	0
$746$	−15.2132	−0.556995
$747$	−2.58579	−0.0946090
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.6274	−0.935158	−0.467579	−	0.883951i	$-0.654874\pi$
$751$	−25.6274	−0.935158	−0.467579	+	0.883951i	$0.654874\pi$
$752$	−53.2548	−1.94200
$753$	−2.58579	−0.0942313
$754$	51.4558	1.87391
$755$	0	0
$756$	0	0
$757$	37.5980	1.36652	0.683261	−	0.730174i	$-0.260561\pi$
$757$	37.5980	1.36652	0.683261	+	0.730174i	$0.260561\pi$
$758$	−35.0711	−1.27384
$759$	−1.31371	−0.0476846
$760$	0	0
$761$	−35.0711	−1.27133	−0.635663	−	0.771967i	$-0.719273\pi$
$761$	−35.0711	−1.27133	−0.635663	+	0.771967i	$0.719273\pi$
$762$	5.55635	0.201285
$763$	0	0
$764$	0	0
$765$	0	0
$766$	6.34315	0.229187
$767$	−6.24264	−0.225409
$768$	0	0
$769$	17.9706	0.648035	0.324018	−	0.946051i	$-0.394966\pi$
$769$	17.9706	0.648035	0.324018	+	0.946051i	$0.394966\pi$
$770$	0	0
$771$	10.1005	0.363761
$772$	0	0
$773$	−39.0711	−1.40529	−0.702644	−	0.711541i	$-0.747997\pi$
$773$	−39.0711	−1.40529	−0.702644	+	0.711541i	$0.747997\pi$
$774$	10.7279	0.385607
$775$	0	0
$776$	−14.6274	−0.525094
$777$	0	0
$778$	−52.1421	−1.86939
$779$	−29.0711	−1.04158
$780$	0	0
$781$	−0.343146	−0.0122787
$782$	7.11270	0.254350
$783$	8.24264	0.294568
$784$	0	0
$785$	0	0
$786$	9.17157	0.327139
$787$	−18.8284	−0.671161	−0.335580	−	0.942012i	$-0.608932\pi$
$787$	−18.8284	−0.671161	−0.335580	+	0.942012i	$0.608932\pi$
$788$	0	0
$789$	−6.82843	−0.243098
$790$	0	0
$791$	0	0
$792$	−1.65685	−0.0588738
$793$	−19.7990	−0.703083
$794$	−25.7574	−0.914094
$795$	0	0
$796$	0	0
$797$	−41.5563	−1.47200	−0.736001	−	0.676981i	$-0.763288\pi$
$797$	−41.5563	−1.47200	−0.736001	+	0.676981i	$0.763288\pi$
$798$	0	0
$799$	−29.8579	−1.05630
$800$	0	0
$801$	−12.2426	−0.432572
$802$	−23.3137	−0.823236
$803$	7.07107	0.249533
$804$	0	0
$805$	0	0
$806$	−36.3848	−1.28160
$807$	−20.1421	−0.709037
$808$	−6.34315	−0.223151
$809$	40.6274	1.42838	0.714192	−	0.699950i	$-0.246794\pi$
$809$	40.6274	1.42838	0.714192	+	0.699950i	$0.246794\pi$
$810$	0	0
$811$	10.9706	0.385229	0.192614	−	0.981275i	$-0.438303\pi$
$811$	10.9706	0.385229	0.192614	+	0.981275i	$0.438303\pi$
$812$	0	0
$813$	−8.14214	−0.285557
$814$	6.97056	0.244318
$815$	0	0
$816$	8.97056	0.314033
$817$	35.3259	1.23590
$818$	−5.21320	−0.182275
$819$	0	0
$820$	0	0
$821$	−17.5563	−0.612721	−0.306360	−	0.951916i	$-0.599111\pi$
$821$	−17.5563	−0.612721	−0.306360	+	0.951916i	$0.599111\pi$
$822$	24.1421	0.842054
$823$	34.2843	1.19507	0.597537	−	0.801841i	$-0.296146\pi$
$823$	34.2843	1.19507	0.597537	+	0.801841i	$0.296146\pi$
$824$	20.4853	0.713639
$825$	0	0
$826$	0	0
$827$	−7.17157	−0.249380	−0.124690	−	0.992196i	$-0.539794\pi$
$827$	−7.17157	−0.249380	−0.124690	+	0.992196i	$0.539794\pi$
$828$	0	0
$829$	−7.34315	−0.255038	−0.127519	−	0.991836i	$-0.540701\pi$
$829$	−7.34315	−0.255038	−0.127519	+	0.991836i	$0.540701\pi$
$830$	0	0
$831$	−13.3848	−0.464313
$832$	35.3137	1.22428
$833$	0	0
$834$	−7.75736	−0.268615
$835$	0	0
$836$	0	0
$837$	−5.82843	−0.201460
$838$	−12.4853	−0.431297
$839$	−32.7279	−1.12989	−0.564947	−	0.825127i	$-0.691103\pi$
$839$	−32.7279	−1.12989	−0.564947	+	0.825127i	$0.691103\pi$
$840$	0	0
$841$	38.9411	1.34280
$842$	14.8701	0.512456
$843$	−1.65685	−0.0570651
$844$	0	0
$845$	0	0
$846$	18.8284	0.647335
$847$	0	0
$848$	27.3137	0.937957
$849$	4.75736	0.163272
$850$	0	0
$851$	−18.8701	−0.646857
$852$	0	0
$853$	44.0122	1.50695	0.753474	−	0.657477i	$-0.228376\pi$
$853$	44.0122	1.50695	0.753474	+	0.657477i	$0.228376\pi$
$854$	0	0
$855$	0	0
$856$	35.5980	1.21671
$857$	−2.24264	−0.0766071	−0.0383036	−	0.999266i	$-0.512195\pi$
$857$	−2.24264	−0.0766071	−0.0383036	+	0.999266i	$0.512195\pi$
$858$	3.65685	0.124843
$859$	4.34315	0.148186	0.0740931	−	0.997251i	$-0.476394\pi$
$859$	4.34315	0.148186	0.0740931	+	0.997251i	$0.476394\pi$
$860$	0	0
$861$	0	0
$862$	44.0833	1.50148
$863$	−29.7990	−1.01437	−0.507185	−	0.861837i	$-0.669314\pi$
$863$	−29.7990	−1.01437	−0.507185	+	0.861837i	$0.669314\pi$
$864$	0	0
$865$	0	0
$866$	−9.27208	−0.315078
$867$	−11.9706	−0.406542
$868$	0	0
$869$	3.89949	0.132281
$870$	0	0
$871$	60.5980	2.05328
$872$	−9.85786	−0.333829
$873$	5.17157	0.175031
$874$	−14.7696	−0.499588
$875$	0	0
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	16.4853	0.556351
$879$	−7.31371	−0.246685
$880$	0	0
$881$	−10.3431	−0.348469	−0.174235	−	0.984704i	$-0.555745\pi$
$881$	−10.3431	−0.348469	−0.174235	+	0.984704i	$0.555745\pi$
$882$	0	0
$883$	6.07107	0.204308	0.102154	−	0.994769i	$-0.467427\pi$
$883$	6.07107	0.204308	0.102154	+	0.994769i	$0.467427\pi$
$884$	0	0
$885$	0	0
$886$	23.3137	0.783239
$887$	37.2132	1.24950	0.624749	−	0.780826i	$-0.285201\pi$
$887$	37.2132	1.24950	0.624749	+	0.780826i	$0.285201\pi$
$888$	−23.7990	−0.798642
$889$	0	0
$890$	0	0
$891$	0.585786	0.0196246
$892$	0	0
$893$	62.0000	2.07475
$894$	8.97056	0.300020
$895$	0	0
$896$	0	0
$897$	−9.89949	−0.330535
$898$	38.1421	1.27282
$899$	−48.0416	−1.60228
$900$	0	0
$901$	15.3137	0.510174
$902$	−5.17157	−0.172195
$903$	0	0
$904$	−44.2843	−1.47287
$905$	0	0
$906$	14.8284	0.492641
$907$	45.3848	1.50698	0.753488	−	0.657461i	$-0.228370\pi$
$907$	45.3848	1.50698	0.753488	+	0.657461i	$0.228370\pi$
$908$	0	0
$909$	2.24264	0.0743837
$910$	0	0
$911$	−4.24264	−0.140565	−0.0702825	−	0.997527i	$-0.522390\pi$
$911$	−4.24264	−0.140565	−0.0702825	+	0.997527i	$0.522390\pi$
$912$	−18.6274	−0.616815
$913$	−1.51472	−0.0501299
$914$	1.55635	0.0514795
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−3.17157	−0.104678
$919$	−35.3431	−1.16586	−0.582931	−	0.812521i	$-0.698094\pi$
$919$	−35.3431	−1.16586	−0.582931	+	0.812521i	$0.698094\pi$
$920$	0	0
$921$	−4.41421	−0.145453
$922$	−18.2843	−0.602160
$923$	−2.58579	−0.0851122
$924$	0	0
$925$	0	0
$926$	10.7279	0.352541
$927$	−7.24264	−0.237880
$928$	0	0
$929$	31.0122	1.01748	0.508739	−	0.860921i	$-0.330112\pi$
$929$	31.0122	1.01748	0.508739	+	0.860921i	$0.330112\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.92893	−0.0958889
$934$	−41.4558	−1.35648
$935$	0	0
$936$	−12.4853	−0.408094
$937$	22.0711	0.721030	0.360515	−	0.932753i	$-0.382601\pi$
$937$	22.0711	0.721030	0.360515	+	0.932753i	$0.382601\pi$
$938$	0	0
$939$	16.2132	0.529098
$940$	0	0
$941$	−36.7279	−1.19730	−0.598648	−	0.801012i	$-0.704295\pi$
$941$	−36.7279	−1.19730	−0.598648	+	0.801012i	$0.704295\pi$
$942$	−17.1716	−0.559480
$943$	14.0000	0.455903
$944$	5.65685	0.184115
$945$	0	0
$946$	6.28427	0.204319
$947$	19.0711	0.619726	0.309863	−	0.950781i	$-0.399717\pi$
$947$	19.0711	0.619726	0.309863	+	0.950781i	$0.399717\pi$
$948$	0	0
$949$	53.2843	1.72968
$950$	0	0
$951$	−19.8995	−0.645285
$952$	0	0
$953$	−31.5147	−1.02086	−0.510431	−	0.859919i	$-0.670514\pi$
$953$	−31.5147	−1.02086	−0.510431	+	0.859919i	$0.670514\pi$
$954$	−9.65685	−0.312652
$955$	0	0
$956$	0	0
$957$	4.82843	0.156081
$958$	15.0294	0.485579
$959$	0	0
$960$	0	0
$961$	2.97056	0.0958246
$962$	52.5269	1.69354
$963$	−12.5858	−0.405571
$964$	0	0
$965$	0	0
$966$	0	0
$967$	13.3848	0.430425	0.215213	−	0.976567i	$-0.430956\pi$
$967$	13.3848	0.430425	0.215213	+	0.976567i	$0.430956\pi$
$968$	30.1421	0.968805
$969$	−10.4437	−0.335498
$970$	0	0
$971$	25.6569	0.823368	0.411684	−	0.911327i	$-0.364941\pi$
$971$	25.6569	0.823368	0.411684	+	0.911327i	$0.364941\pi$
$972$	0	0
$973$	0	0
$974$	26.7279	0.856418
$975$	0	0
$976$	17.9411	0.574281
$977$	45.6985	1.46202	0.731012	−	0.682365i	$-0.239048\pi$
$977$	45.6985	1.46202	0.731012	+	0.682365i	$0.239048\pi$
$978$	−4.68629	−0.149851
$979$	−7.17157	−0.229204
$980$	0	0
$981$	3.48528	0.111276
$982$	25.6569	0.818743
$983$	57.1543	1.82294	0.911470	−	0.411367i	$-0.134948\pi$
$983$	57.1543	1.82294	0.911470	+	0.411367i	$0.134948\pi$
$984$	17.6569	0.562880
$985$	0	0
$986$	−26.1421	−0.832535
$987$	0	0
$988$	0	0
$989$	−17.0122	−0.540956
$990$	0	0
$991$	−19.3431	−0.614455	−0.307228	−	0.951636i	$-0.599401\pi$
$991$	−19.3431	−0.614455	−0.307228	+	0.951636i	$0.599401\pi$
$992$	0	0
$993$	17.6274	0.559389
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9.24264	−0.292717	−0.146359	−	0.989232i	$-0.546755\pi$
$997$	−9.24264	−0.292717	−0.146359	+	0.989232i	$0.546755\pi$
$998$	−43.5563	−1.37875
$999$	8.41421	0.266214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.z.1.2		2
5.4	even	2		735.2.a.i.1.1		2
7.2	even	3		525.2.i.g.151.1		4
7.4	even	3		525.2.i.g.226.1		4
7.6	odd	2		3675.2.a.x.1.2		2
15.14	odd	2		2205.2.a.u.1.2		2
35.2	odd	12		525.2.r.g.424.4		8
35.4	even	6		105.2.i.c.16.2	✓	4
35.9	even	6		105.2.i.c.46.2	yes	4
35.18	odd	12		525.2.r.g.499.4		8
35.19	odd	6		735.2.i.j.361.2		4
35.23	odd	12		525.2.r.g.424.1		8
35.24	odd	6		735.2.i.j.226.2		4
35.32	odd	12		525.2.r.g.499.1		8
35.34	odd	2		735.2.a.j.1.1		2
105.44	odd	6		315.2.j.d.46.1		4
105.74	odd	6		315.2.j.d.226.1		4
105.104	even	2		2205.2.a.s.1.2		2
140.39	odd	6		1680.2.bg.p.961.2		4
140.79	odd	6		1680.2.bg.p.1201.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.i.c.16.2	✓	4	35.4	even	6
105.2.i.c.46.2	yes	4	35.9	even	6
315.2.j.d.46.1		4	105.44	odd	6
315.2.j.d.226.1		4	105.74	odd	6
525.2.i.g.151.1		4	7.2	even	3
525.2.i.g.226.1		4	7.4	even	3
525.2.r.g.424.1		8	35.23	odd	12
525.2.r.g.424.4		8	35.2	odd	12
525.2.r.g.499.1		8	35.32	odd	12
525.2.r.g.499.4		8	35.18	odd	12
735.2.a.i.1.1		2	5.4	even	2
735.2.a.j.1.1		2	35.34	odd	2
735.2.i.j.226.2		4	35.24	odd	6
735.2.i.j.361.2		4	35.19	odd	6
1680.2.bg.p.961.2		4	140.39	odd	6
1680.2.bg.p.1201.2		4	140.79	odd	6
2205.2.a.s.1.2		2	105.104	even	2
2205.2.a.u.1.2		2	15.14	odd	2
3675.2.a.x.1.2		2	7.6	odd	2
3675.2.a.z.1.2		2	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:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.00000 q^{3} +1.41421 q^{6} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.00000 q^{3} +1.41421 q^{6} -2.82843 q^{8} +1.00000 q^{9} +0.585786 q^{11} +4.41421 q^{13} -4.00000 q^{16} -2.24264 q^{17} +1.41421 q^{18} +4.65685 q^{19} +0.828427 q^{22} -2.24264 q^{23} -2.82843 q^{24} +6.24264 q^{26} +1.00000 q^{27} +8.24264 q^{29} -5.82843 q^{31} +0.585786 q^{33} -3.17157 q^{34} +8.41421 q^{37} +6.58579 q^{38} +4.41421 q^{39} -6.24264 q^{41} +7.58579 q^{43} -3.17157 q^{46} +13.3137 q^{47} -4.00000 q^{48} -2.24264 q^{51} -6.82843 q^{53} +1.41421 q^{54} +4.65685 q^{57} +11.6569 q^{58} -1.41421 q^{59} -4.48528 q^{61} -8.24264 q^{62} +8.00000 q^{64} +0.828427 q^{66} +13.7279 q^{67} -2.24264 q^{69} -0.585786 q^{71} -2.82843 q^{72} +12.0711 q^{73} +11.8995 q^{74} +6.24264 q^{78} +6.65685 q^{79} +1.00000 q^{81} -8.82843 q^{82} -2.58579 q^{83} +10.7279 q^{86} +8.24264 q^{87} -1.65685 q^{88} -12.2426 q^{89} -5.82843 q^{93} +18.8284 q^{94} +5.17157 q^{97} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} + 4 q^{11} + 6 q^{13} - 8 q^{16} + 4 q^{17} - 2 q^{19} - 4 q^{22} + 4 q^{23} + 4 q^{26} + 2 q^{27} + 8 q^{29} - 6 q^{31} + 4 q^{33} - 12 q^{34} + 14 q^{37} + 16 q^{38} + 6 q^{39} - 4 q^{41} + 18 q^{43} - 12 q^{46} + 4 q^{47} - 8 q^{48} + 4 q^{51} - 8 q^{53} - 2 q^{57} + 12 q^{58} + 8 q^{61} - 8 q^{62} + 16 q^{64} - 4 q^{66} + 2 q^{67} + 4 q^{69} - 4 q^{71} + 10 q^{73} + 4 q^{74} + 4 q^{78} + 2 q^{79} + 2 q^{81} - 12 q^{82} - 8 q^{83} - 4 q^{86} + 8 q^{87} + 8 q^{88} - 16 q^{89} - 6 q^{93} + 32 q^{94} + 16 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 4 * q^11 + 6 * q^13 - 8 * q^16 + 4 * q^17 - 2 * q^19 - 4 * q^22 + 4 * q^23 + 4 * q^26 + 2 * q^27 + 8 * q^29 - 6 * q^31 + 4 * q^33 - 12 * q^34 + 14 * q^37 + 16 * q^38 + 6 * q^39 - 4 * q^41 + 18 * q^43 - 12 * q^46 + 4 * q^47 - 8 * q^48 + 4 * q^51 - 8 * q^53 - 2 * q^57 + 12 * q^58 + 8 * q^61 - 8 * q^62 + 16 * q^64 - 4 * q^66 + 2 * q^67 + 4 * q^69 - 4 * q^71 + 10 * q^73 + 4 * q^74 + 4 * q^78 + 2 * q^79 + 2 * q^81 - 12 * q^82 - 8 * q^83 - 4 * q^86 + 8 * q^87 + 8 * q^88 - 16 * q^89 - 6 * q^93 + 32 * q^94 + 16 * q^97 + 4 * q^99

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