LMFDB - Embedded newform 2600.2.k.c.2001.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2600,2,Mod(2001,2600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2600.2001"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196 $ x^8 - 2x^7 + 2x^6 + 6x^5 + 36x^4 - 52x^3 + 50x^2 + 140*x + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2001.5
Root			$1.47812 - 1.47812i$ of defining polynomial
Character	$\chi$	$=$	2600.2001
Dual form			2600.2.k.c.2001.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.369700 q^{3} -0.956248i q^{7} -2.86332 q^{9} +2.58655i q^{11} +(0.369700 - 3.58655i) q^{13} -6.81957 q^{17} +5.49362i q^{19} +0.353525i q^{21} +6.80339 q^{23} +2.16767 q^{27} +6.60272 q^{29} -5.40612i q^{31} -0.956248i q^{33} +3.69565i q^{37} +(-0.136678 + 1.32595i) q^{39} +8.08017i q^{41} +0.723226 q^{43} -12.6829i q^{47} +6.08559 q^{49} +2.52120 q^{51} +8.08017 q^{53} -2.03099i q^{57} -6.23302i q^{59} +14.2970 q^{61} +2.73804i q^{63} +7.50980i q^{67} -2.51522 q^{69} -9.05259i q^{71} +8.12934i q^{73} +2.47338 q^{77} -8.43369 q^{79} +7.78857 q^{81} -9.69565i q^{83} -2.44103 q^{87} -2.08750i q^{89} +(-3.42963 - 0.353525i) q^{91} +1.99864i q^{93} +10.4337i q^{97} -7.40612i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 16 q^{9} + 4 q^{13} + 4 q^{17} + 12 q^{23} - 4 q^{27} + 16 q^{29} - 40 q^{39} + 24 q^{43} - 32 q^{49} + 16 q^{51} + 4 q^{53} + 32 q^{61} + 56 q^{69} + 44 q^{77} - 24 q^{79} + 64 q^{81} - 76 q^{87}+ \cdots - 32 q^{91}+O(q^{100}) $ 8 * q - 4 * q^3 + 16 * q^9 + 4 * q^13 + 4 * q^17 + 12 * q^23 - 4 * q^27 + 16 * q^29 - 40 * q^39 + 24 * q^43 - 32 * q^49 + 16 * q^51 + 4 * q^53 + 32 * q^61 + 56 * q^69 + 44 * q^77 - 24 * q^79 + 64 * q^81 - 76 * q^87 - 32 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$-1$	$1$	$1$

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			0
$2$		0			0
$3$	−0.369700			−0.213447			−0.106723	−	0.994289i	$-0.534036\pi$
$3$	−0.369700			−0.213447			−0.106723	+	0.994289i	$0.534036\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	0.956248i		−	0.361428i	−0.983536	−	0.180714i	$-0.942159\pi$
$7$		−	0.956248i		−	0.361428i	0.983536	−	0.180714i	$-0.0578408\pi$
$8$		0			0
$9$	−2.86332			−0.954441
$10$		0			0
$11$			2.58655i			0.779873i	0.920842	+	0.389937i	$0.127503\pi$
$11$			2.58655i			0.779873i	−0.920842	+	0.389937i	$0.872497\pi$
$12$		0			0
$13$	0.369700	−	3.58655i	0.102536	−	0.994729i
$13$	0.369700	−	3.58655i	0.102536	−	0.994729i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.81957			−1.65399			−0.826994	−	0.562210i	$-0.809951\pi$
$17$	−6.81957			−1.65399			−0.826994	+	0.562210i	$0.809951\pi$
$18$		0			0
$19$			5.49362i			1.26032i	0.776464	+	0.630162i	$0.217011\pi$
$19$			5.49362i			1.26032i	−0.776464	+	0.630162i	$0.782989\pi$
$20$		0			0
$21$			0.353525i			0.0771455i
$22$		0			0
$23$	6.80339			1.41861			0.709303	−	0.704904i	$-0.249010\pi$
$23$	6.80339			1.41861			0.709303	+	0.704904i	$0.249010\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.16767			0.417169
$28$		0			0
$29$	6.60272			1.22609			0.613047	−	0.790046i	$-0.289943\pi$
$29$	6.60272			1.22609			0.613047	+	0.790046i	$0.289943\pi$
$30$		0			0
$31$		−	5.40612i		−	0.970967i	−0.874246	−	0.485484i	$-0.838644\pi$
$31$		−	5.40612i		−	0.970967i	0.874246	−	0.485484i	$-0.161356\pi$
$32$		0			0
$33$		−	0.956248i		−	0.166461i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			3.69565i			0.607561i	0.952742	+	0.303780i	$0.0982489\pi$
$37$			3.69565i			0.607561i	−0.952742	+	0.303780i	$0.901751\pi$
$38$		0			0
$39$	−0.136678	+	1.32595i	−0.0218861	+	0.212322i
$40$		0			0
$41$			8.08017i			1.26191i	0.775819	+	0.630955i	$0.217337\pi$
$41$			8.08017i			1.26191i	−0.775819	+	0.630955i	$0.782663\pi$
$42$		0			0
$43$	0.723226			0.110291			0.0551454	−	0.998478i	$-0.482438\pi$
$43$	0.723226			0.110291			0.0551454	+	0.998478i	$0.482438\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	12.6829i		−	1.84999i	−0.379980	−	0.924995i	$-0.624069\pi$
$47$		−	12.6829i		−	1.84999i	0.379980	−	0.924995i	$-0.375931\pi$
$48$		0			0
$49$	6.08559			0.869370
$50$		0			0
$51$	2.52120			0.353038
$52$		0			0
$53$	8.08017			1.10990			0.554948	−	0.831885i	$-0.312738\pi$
$53$	8.08017			1.10990			0.554948	+	0.831885i	$0.312738\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	2.03099i		−	0.269012i
$58$		0			0
$59$		−	6.23302i		−	0.811470i	−0.913991	−	0.405735i	$-0.867016\pi$
$59$		−	6.23302i		−	0.811470i	0.913991	−	0.405735i	$-0.132984\pi$
$60$		0			0
$61$	14.2970			1.83055			0.915273	−	0.402835i	$-0.131975\pi$
$61$	14.2970			1.83055			0.915273	+	0.402835i	$0.131975\pi$
$62$		0			0
$63$			2.73804i			0.344961i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			7.50980i			0.917468i	0.888574	+	0.458734i	$0.151697\pi$
$67$			7.50980i			0.917468i	−0.888574	+	0.458734i	$0.848303\pi$
$68$		0			0
$69$	−2.51522			−0.302797
$70$		0			0
$71$		−	9.05259i		−	1.07434i	−0.843472	−	0.537172i	$-0.819492\pi$
$71$		−	9.05259i		−	1.07434i	0.843472	−	0.537172i	$-0.180508\pi$
$72$		0			0
$73$			8.12934i			0.951468i	0.879589	+	0.475734i	$0.157817\pi$
$73$			8.12934i			0.951468i	−0.879589	+	0.475734i	$0.842183\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.47338			0.281868
$78$		0			0
$79$	−8.43369			−0.948865			−0.474432	−	0.880292i	$-0.657347\pi$
$79$	−8.43369			−0.948865			−0.474432	+	0.880292i	$0.657347\pi$
$80$		0			0
$81$	7.78857			0.865397
$82$		0			0
$83$		−	9.69565i		−	1.06424i	−0.846670	−	0.532118i	$-0.821396\pi$
$83$		−	9.69565i		−	1.06424i	0.846670	−	0.532118i	$-0.178604\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.44103			−0.261706
$88$		0			0
$89$		−	2.08750i		−	0.221275i	−0.993861	−	0.110638i	$-0.964711\pi$
$89$		−	2.08750i		−	0.221275i	0.993861	−	0.110638i	$-0.0352893\pi$
$90$		0			0
$91$	−3.42963	−	0.353525i	−0.359523	−	0.0370595i
$92$		0			0
$93$			1.99864i			0.207250i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			10.4337i			1.05938i	0.848191	+	0.529691i	$0.177692\pi$
$97$			10.4337i			1.05938i	−0.848191	+	0.529691i	$0.822308\pi$
$98$		0			0
$99$		−	7.40612i		−	0.744343i
$100$		0			0
$101$	−5.26060			−0.523449			−0.261725	−	0.965143i	$-0.584291\pi$
$101$	−5.26060			−0.523449			−0.261725	+	0.965143i	$0.584291\pi$
$102$		0			0
$103$	19.0143			1.87353			0.936765	−	0.349958i	$-0.113804\pi$
$103$	19.0143			1.87353			0.936765	+	0.349958i	$0.113804\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−17.9287			−1.73323			−0.866615	−	0.498977i	$-0.833709\pi$
$107$	−17.9287			−1.73323			−0.866615	+	0.498977i	$0.833709\pi$
$108$		0			0
$109$		−	11.9125i		−	1.14101i	−0.821294	−	0.570505i	$-0.806747\pi$
$109$		−	11.9125i		−	1.14101i	0.821294	−	0.570505i	$-0.193253\pi$
$110$		0			0
$111$		−	1.36628i		−	0.129682i
$112$		0			0
$113$	−0.907074			−0.0853303			−0.0426652	−	0.999089i	$-0.513585\pi$
$113$	−0.907074			−0.0853303			−0.0426652	+	0.999089i	$0.513585\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.05857	+	10.2694i	−0.0978650	+	0.949410i
$118$		0			0
$119$			6.52120i			0.597797i
$120$		0			0
$121$	4.30977			0.391798
$122$		0			0
$123$		−	2.98724i		−	0.269351i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.63572			0.766296			0.383148	−	0.923687i	$-0.374840\pi$
$127$	8.63572			0.766296			0.383148	+	0.923687i	$0.374840\pi$
$128$		0			0
$129$	−0.267377			−0.0235412
$130$		0			0
$131$	18.0728			1.57903			0.789515	−	0.613731i	$-0.210332\pi$
$131$	18.0728			1.57903			0.789515	+	0.613731i	$0.210332\pi$
$132$		0			0
$133$	5.25326			0.455516
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.89974i			0.589484i	0.955577	+	0.294742i	$0.0952338\pi$
$137$			6.89974i			0.589484i	−0.955577	+	0.294742i	$0.904766\pi$
$138$		0			0
$139$	22.0728			1.87219			0.936097	−	0.351743i	$-0.114411\pi$
$139$	22.0728			1.87219			0.936097	+	0.351743i	$0.114411\pi$
$140$		0			0
$141$			4.68887i			0.394874i
$142$		0			0
$143$	9.27677	+	0.956248i	0.775763	+	0.0799655i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.24985			−0.185564
$148$		0			0
$149$			11.2054i			0.917986i	0.888440	+	0.458993i	$0.151790\pi$
$149$			11.2054i			0.917986i	−0.888440	+	0.458993i	$0.848210\pi$
$150$		0			0
$151$		−	8.28084i		−	0.673885i	−0.941525	−	0.336943i	$-0.890607\pi$
$151$		−	8.28084i		−	0.673885i	0.941525	−	0.336943i	$-0.109393\pi$
$152$		0			0
$153$	19.5266			1.57863
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.899738			−0.0718069			−0.0359034	−	0.999355i	$-0.511431\pi$
$157$	−0.899738			−0.0718069			−0.0359034	+	0.999355i	$0.511431\pi$
$158$		0			0
$159$	−2.98724			−0.236904
$160$		0			0
$161$		−	6.50573i		−	0.512723i
$162$		0			0
$163$			14.5954i			1.14320i	0.820533	+	0.571599i	$0.193677\pi$
$163$			14.5954i			1.14320i	−0.820533	+	0.571599i	$0.806323\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.40270i		−	0.185926i	−0.995670	−	0.0929632i	$-0.970366\pi$
$167$		−	2.40270i		−	0.185926i	0.995670	−	0.0929632i	$-0.0296339\pi$
$168$		0			0
$169$	−12.7266	−	2.65190i	−0.978973	−	0.203992i
$170$		0			0
$171$		−	15.7300i		−	1.20290i
$172$		0			0
$173$	1.99266			0.151499			0.0757497	−	0.997127i	$-0.475865\pi$
$173$	1.99266			0.151499			0.0757497	+	0.997127i	$0.475865\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			2.30435i			0.173206i
$178$		0			0
$179$	−10.5535			−0.788809			−0.394405	−	0.918937i	$-0.629049\pi$
$179$	−10.5535			−0.788809			−0.394405	+	0.918937i	$0.629049\pi$
$180$		0			0
$181$	4.51522			0.335614			0.167807	−	0.985820i	$-0.446332\pi$
$181$	4.51522			0.335614			0.167807	+	0.985820i	$0.446332\pi$
$182$		0			0
$183$	−5.28561			−0.390724
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	17.6391i		−	1.28990i
$188$		0			0
$189$		−	2.07283i		−	0.150776i
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	13.2606i		−	0.954519i	−0.878762	−	0.477259i	$-0.841630\pi$
$193$		−	13.2606i		−	0.954519i	0.878762	−	0.477259i	$-0.158370\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.34810i		−	0.238542i	−0.992862	−	0.119271i	$-0.961944\pi$
$197$		−	3.34810i		−	0.238542i	0.992862	−	0.119271i	$-0.0380558\pi$
$198$		0			0
$199$	10.5535			0.748121			0.374061	−	0.927404i	$-0.377965\pi$
$199$	10.5535			0.748121			0.374061	+	0.927404i	$0.377965\pi$
$200$		0			0
$201$		−	2.77638i		−	0.195830i
$202$		0			0
$203$		−	6.31384i		−	0.443145i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−19.4803			−1.35397
$208$		0			0
$209$	−14.2095			−0.982892
$210$		0			0
$211$	7.26060			0.499840			0.249920	−	0.968266i	$-0.419596\pi$
$211$	7.26060			0.499840			0.249920	+	0.968266i	$0.419596\pi$
$212$		0			0
$213$			3.34675i			0.229315i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−5.16959			−0.350934
$218$		0			0
$219$		−	3.00542i		−	0.203088i
$220$		0			0
$221$	−2.52120	+	24.4587i	−0.169594	+	1.64527i
$222$		0			0
$223$		−	0.347545i		−	0.0232734i	−0.999932	−	0.0116367i	$-0.996296\pi$
$223$		−	0.347545i		−	0.0232734i	0.999932	−	0.0116367i	$-0.00370415\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	22.8984i		−	1.51982i	−0.650029	−	0.759909i	$-0.725243\pi$
$227$		−	22.8984i		−	1.51982i	0.650029	−	0.759909i	$-0.274757\pi$
$228$		0			0
$229$		−	0.00733606i		−	0.000484780i	−1.00000		0.000242390i	$-0.999923\pi$
$229$		−	0.00733606i		−	0.000484780i	1.00000		0.000242390i	$-7.71552e-5\pi$
$230$		0			0
$231$	−0.914410			−0.0601637
$232$		0			0
$233$	0.619546			0.0405878			0.0202939	−	0.999794i	$-0.493540\pi$
$233$	0.619546			0.0405878			0.0202939	+	0.999794i	$0.493540\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.11794			0.202532
$238$		0			0
$239$			24.9578i			1.61438i	0.590290	+	0.807191i	$0.299013\pi$
$239$			24.9578i			1.61438i	−0.590290	+	0.807191i	$0.700987\pi$
$240$		0			0
$241$			23.8068i			1.53353i	0.641927	+	0.766766i	$0.278135\pi$
$241$			23.8068i			1.53353i	−0.641927	+	0.766766i	$0.721865\pi$
$242$		0			0
$243$	−9.38246			−0.601885
$244$		0			0
$245$		0			0
$246$		0			0
$247$	19.7031	+	2.03099i	1.25368	+	0.129229i
$248$		0			0
$249$			3.58449i			0.227158i
$250$		0			0
$251$	14.6519			0.924820			0.462410	−	0.886666i	$-0.346985\pi$
$251$	14.6519			0.924820			0.462410	+	0.886666i	$0.346985\pi$
$252$		0			0
$253$			17.5973i			1.10633i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.4533			0.839193			0.419596	−	0.907711i	$-0.362172\pi$
$257$	13.4533			0.839193			0.419596	+	0.907711i	$0.362172\pi$
$258$		0			0
$259$	3.53396			0.219589
$260$		0			0
$261$	−18.9057			−1.17023
$262$		0			0
$263$	−14.0162			−0.864274			−0.432137	−	0.901808i	$-0.642240\pi$
$263$	−14.0162			−0.864274			−0.432137	+	0.901808i	$0.642240\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			0.771751i			0.0472304i
$268$		0			0
$269$	10.5644			0.644122			0.322061	−	0.946719i	$-0.395624\pi$
$269$	10.5644			0.644122			0.322061	+	0.946719i	$0.395624\pi$
$270$		0			0
$271$		−	7.84715i		−	0.476680i	−0.971182	−	0.238340i	$-0.923397\pi$
$271$		−	7.84715i		−	0.476680i	0.971182	−	0.238340i	$-0.0766033\pi$
$272$		0			0
$273$	1.26794	+	0.130698i	0.0767389	+	0.00791023i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.03777			−0.182522			−0.0912610	−	0.995827i	$-0.529090\pi$
$277$	−3.03777			−0.182522			−0.0912610	+	0.995827i	$0.529090\pi$
$278$		0			0
$279$			15.4795i			0.926730i
$280$		0			0
$281$			16.9503i			1.01117i	0.862777	+	0.505584i	$0.168723\pi$
$281$			16.9503i			1.01117i	−0.862777	+	0.505584i	$0.831277\pi$
$282$		0			0
$283$	0.191184			0.0113647			0.00568236	−	0.999984i	$-0.498191\pi$
$283$	0.191184			0.0113647			0.00568236	+	0.999984i	$0.498191\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.72664			0.456089
$288$		0			0
$289$	29.5065			1.73568
$290$		0			0
$291$		−	3.85734i		−	0.226121i
$292$		0			0
$293$		−	21.3348i		−	1.24639i	−0.782066	−	0.623196i	$-0.785834\pi$
$293$		−	21.3348i		−	1.24639i	0.782066	−	0.623196i	$-0.214166\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			5.60679i			0.325339i
$298$		0			0
$299$	2.51522	−	24.4007i	0.145459	−	1.41113i
$300$		0			0
$301$		−	0.691583i		−	0.0398622i
$302$		0			0
$303$	1.94485			0.111728
$304$		0			0
$305$		0			0
$306$		0			0
$307$			7.94349i			0.453359i	0.973969	+	0.226679i	$0.0727870\pi$
$307$			7.94349i			0.453359i	−0.973969	+	0.226679i	$0.927213\pi$
$308$		0			0
$309$	−7.02958			−0.399899
$310$		0			0
$311$	−14.3785			−0.815332			−0.407666	−	0.913131i	$-0.633657\pi$
$311$	−14.3785			−0.815332			−0.407666	+	0.913131i	$0.633657\pi$
$312$		0			0
$313$	3.86468			0.218444			0.109222	−	0.994017i	$-0.465164\pi$
$313$	3.86468			0.218444			0.109222	+	0.994017i	$0.465164\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			7.87066i			0.442060i	0.975267	+	0.221030i	$0.0709419\pi$
$317$			7.87066i			0.442060i	−0.975267	+	0.221030i	$0.929058\pi$
$318$		0			0
$319$			17.0783i			0.956199i
$320$		0			0
$321$	6.62824			0.369952
$322$		0			0
$323$		−	37.4641i		−	2.08456i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			4.40406i			0.243545i
$328$		0			0
$329$	−12.1280			−0.668638
$330$		0			0
$331$			8.06535i			0.443312i	0.975125	+	0.221656i	$0.0711462\pi$
$331$			8.06535i			0.443312i	−0.975125	+	0.221656i	$0.928854\pi$
$332$		0			0
$333$		−	10.5818i		−	0.579881i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12.6337			0.688202			0.344101	−	0.938933i	$-0.388184\pi$
$337$	12.6337			0.688202			0.344101	+	0.938933i	$0.388184\pi$
$338$		0			0
$339$	0.335346			0.0182135
$340$		0			0
$341$	13.9832			0.757231
$342$		0			0
$343$		−	12.5131i		−	0.675642i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.3569			−1.03914			−0.519568	−	0.854429i	$-0.673907\pi$
$347$	−19.3569			−1.03914			−0.519568	+	0.854429i	$0.673907\pi$
$348$		0			0
$349$		−	25.0378i		−	1.34024i	−0.742252	−	0.670121i	$-0.766242\pi$
$349$		−	25.0378i		−	1.34024i	0.742252	−	0.670121i	$-0.233758\pi$
$350$		0			0
$351$	0.801390	−	7.77446i	0.0427750	−	0.414970i
$352$		0			0
$353$			14.8109i			0.788303i	0.919045	+	0.394152i	$0.128962\pi$
$353$			14.8109i			0.788303i	−0.919045	+	0.394152i	$0.871038\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	2.41089i		−	0.127598i
$358$		0			0
$359$			21.0526i			1.11111i	0.831479	+	0.555557i	$0.187495\pi$
$359$			21.0526i			1.11111i	−0.831479	+	0.555557i	$0.812505\pi$
$360$		0			0
$361$	−11.1799			−0.588414
$362$		0			0
$363$	−1.59333			−0.0836279
$364$		0			0
$365$		0			0
$366$		0			0
$367$	9.24442			0.482555			0.241277	−	0.970456i	$-0.422434\pi$
$367$	9.24442			0.482555			0.241277	+	0.970456i	$0.422434\pi$
$368$		0			0
$369$		−	23.1361i		−	1.20442i
$370$		0			0
$371$		−	7.72664i		−	0.401147i
$372$		0			0
$373$	−13.2714			−0.687169			−0.343585	−	0.939122i	$-0.611641\pi$
$373$	−13.2714			−0.687169			−0.343585	+	0.939122i	$0.611641\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.44103	−	23.6810i	0.125719	−	1.21963i
$378$		0			0
$379$		−	4.77240i		−	0.245142i	−0.992460	−	0.122571i	$-0.960886\pi$
$379$		−	4.77240i		−	0.245142i	0.992460	−	0.122571i	$-0.0391139\pi$
$380$		0			0
$381$	−3.19263			−0.163563
$382$		0			0
$383$			34.4527i			1.76045i	0.474555	+	0.880226i	$0.342609\pi$
$383$			34.4527i			1.76045i	−0.474555	+	0.880226i	$0.657391\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.07083			−0.105266
$388$		0			0
$389$	−14.8674			−0.753806			−0.376903	−	0.926253i	$-0.623011\pi$
$389$	−14.8674			−0.753806			−0.376903	+	0.926253i	$0.623011\pi$
$390$		0			0
$391$	−46.3962			−2.34636
$392$		0			0
$393$	−6.68153			−0.337039
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	3.26196i		−	0.163713i	−0.996644	−	0.0818564i	$-0.973915\pi$
$397$		−	3.26196i		−	0.163713i	0.996644	−	0.0818564i	$-0.0260849\pi$
$398$		0			0
$399$	−1.94213			−0.0972283
$400$		0			0
$401$		−	4.16034i		−	0.207757i	−0.994590	−	0.103879i	$-0.966875\pi$
$401$		−	4.16034i		−	0.207757i	0.994590	−	0.103879i	$-0.0331254\pi$
$402$		0			0
$403$	−19.3893	−	1.99864i	−0.965849	−	0.0995595i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.55897			−0.473821
$408$		0			0
$409$		−	7.90516i		−	0.390885i	−0.980715	−	0.195442i	$-0.937386\pi$
$409$		−	7.90516i		−	0.390885i	0.980715	−	0.195442i	$-0.0626143\pi$
$410$		0			0
$411$		−	2.55084i		−	0.125823i
$412$		0			0
$413$	−5.96031			−0.293288
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−8.16034			−0.399613
$418$		0			0
$419$	7.53396			0.368058			0.184029	−	0.982921i	$-0.441086\pi$
$419$	7.53396			0.368058			0.184029	+	0.982921i	$0.441086\pi$
$420$		0			0
$421$			2.60870i			0.127140i	0.997977	+	0.0635702i	$0.0202487\pi$
$421$			2.60870i			0.127140i	−0.997977	+	0.0635702i	$0.979751\pi$
$422$		0			0
$423$			36.3152i			1.76571i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	13.6715i		−	0.661610i
$428$		0			0
$429$	−3.42963	−	0.353525i	−0.165584	−	0.0170684i
$430$		0			0
$431$		−	17.4061i		−	0.838423i	−0.907889	−	0.419212i	$-0.862307\pi$
$431$		−	17.4061i		−	0.838423i	0.907889	−	0.419212i	$-0.137693\pi$
$432$		0			0
$433$	−28.1712			−1.35382			−0.676910	−	0.736066i	$-0.736681\pi$
$433$	−28.1712			−1.35382			−0.676910	+	0.736066i	$0.736681\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			37.3753i			1.78790i
$438$		0			0
$439$	−24.8014			−1.18371			−0.591853	−	0.806046i	$-0.701603\pi$
$439$	−24.8014			−1.18371			−0.591853	+	0.806046i	$0.701603\pi$
$440$		0			0
$441$	−17.4250			−0.829762
$442$		0			0
$443$	22.4749			1.06781			0.533907	−	0.845543i	$-0.320723\pi$
$443$	22.4749			1.06781			0.533907	+	0.845543i	$0.320723\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	4.14266i		−	0.195941i
$448$		0			0
$449$			26.3280i			1.24250i	0.783614	+	0.621248i	$0.213374\pi$
$449$			26.3280i			1.24250i	−0.783614	+	0.621248i	$0.786626\pi$
$450$		0			0
$451$	−20.8997			−0.984130
$452$		0			0
$453$			3.06143i			0.143839i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			6.60870i			0.309142i	0.987982	+	0.154571i	$0.0493995\pi$
$457$			6.60870i			0.309142i	−0.987982	+	0.154571i	$0.950600\pi$
$458$		0			0
$459$	−14.7826			−0.689992
$460$		0			0
$461$		−	22.3389i		−	1.04042i	−0.854037	−	0.520212i	$-0.825853\pi$
$461$		−	22.3389i		−	1.04042i	0.854037	−	0.520212i	$-0.174147\pi$
$462$		0			0
$463$		−	9.72529i		−	0.451972i	−0.974131	−	0.225986i	$-0.927440\pi$
$463$		−	9.72529i		−	0.451972i	0.974131	−	0.225986i	$-0.0725604\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.3246			0.524040			0.262020	−	0.965062i	$-0.415611\pi$
$467$	11.3246			0.524040			0.262020	+	0.965062i	$0.415611\pi$
$468$		0			0
$469$	7.18123			0.331598
$470$		0			0
$471$	0.332633			0.0153269
$472$		0			0
$473$			1.87066i			0.0860129i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−23.1361			−1.05933
$478$		0			0
$479$		−	36.7469i		−	1.67901i	−0.543354	−	0.839504i	$-0.682846\pi$
$479$		−	36.7469i		−	1.67901i	0.543354	−	0.839504i	$-0.317154\pi$
$480$		0			0
$481$	13.2546	+	1.36628i	0.604359	+	0.0622971i
$482$		0			0
$483$			2.40517i			0.109439i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	11.2068i		−	0.507829i	−0.967227	−	0.253914i	$-0.918282\pi$
$487$		−	11.2068i		−	0.507829i	0.967227	−	0.253914i	$-0.0817181\pi$
$488$		0			0
$489$		−	5.39592i		−	0.244012i
$490$		0			0
$491$	−37.0277			−1.67104			−0.835519	−	0.549462i	$-0.814833\pi$
$491$	−37.0277			−1.67104			−0.835519	+	0.549462i	$0.814833\pi$
$492$		0			0
$493$	−45.0277			−2.02795
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.65652			−0.388298
$498$		0			0
$499$		−	20.4809i		−	0.916849i	−0.888733	−	0.458425i	$-0.848414\pi$
$499$		−	20.4809i		−	0.916849i	0.888733	−	0.458425i	$-0.151586\pi$
$500$		0			0
$501$			0.888279i			0.0396854i
$502$		0			0
$503$	−36.0385			−1.60688			−0.803438	−	0.595388i	$-0.796998\pi$
$503$	−36.0385			−1.60688			−0.803438	+	0.595388i	$0.796998\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.70505	+	0.980407i	0.208958	+	0.0435414i
$508$		0			0
$509$			32.3280i			1.43291i	0.697631	+	0.716457i	$0.254237\pi$
$509$			32.3280i			1.43291i	−0.697631	+	0.716457i	$0.745763\pi$
$510$		0			0
$511$	7.77367			0.343887
$512$		0			0
$513$			11.9084i			0.525768i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	32.8049			1.44276
$518$		0			0
$519$	−0.736689			−0.0323371
$520$		0			0
$521$	−7.47282			−0.327390			−0.163695	−	0.986511i	$-0.552341\pi$
$521$	−7.47282			−0.327390			−0.163695	+	0.986511i	$0.552341\pi$
$522$		0			0
$523$	−13.2371			−0.578817			−0.289409	−	0.957206i	$-0.593459\pi$
$523$	−13.2371			−0.578817			−0.289409	+	0.957206i	$0.593459\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			36.8674i			1.60597i
$528$		0			0
$529$	23.2862			1.01244
$530$		0			0
$531$			17.8471i			0.774500i
$532$		0			0
$533$	28.9799	+	2.98724i	1.25526	+	0.129392i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.90165			0.168369
$538$		0			0
$539$			15.7407i			0.677999i
$540$		0			0
$541$		−	20.4264i		−	0.878198i	−0.898439	−	0.439099i	$-0.855298\pi$
$541$		−	20.4264i		−	0.878198i	0.898439	−	0.439099i	$-0.144702\pi$
$542$		0			0
$543$	−1.66928			−0.0716356
$544$		0			0
$545$		0			0
$546$		0			0
$547$	15.3751			0.657393			0.328696	−	0.944436i	$-0.393391\pi$
$547$	15.3751			0.657393			0.328696	+	0.944436i	$0.393391\pi$
$548$		0			0
$549$	−40.9370			−1.74715
$550$		0			0
$551$			36.2729i			1.54528i
$552$		0			0
$553$			8.06470i			0.342946i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	17.3321i		−	0.734384i	−0.930145	−	0.367192i	$-0.880319\pi$
$557$		−	17.3321i		−	0.734384i	0.930145	−	0.367192i	$-0.119681\pi$
$558$		0			0
$559$	0.267377	−	2.59388i	0.0113088	−	0.109710i
$560$		0			0
$561$			6.52120i			0.275325i
$562$		0			0
$563$	−2.39471			−0.100925			−0.0504626	−	0.998726i	$-0.516070\pi$
$563$	−2.39471			−0.100925			−0.0504626	+	0.998726i	$0.516070\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	7.44781i		−	0.312779i
$568$		0			0
$569$	15.0688			0.631716			0.315858	−	0.948807i	$-0.397708\pi$
$569$	15.0688			0.631716			0.315858	+	0.948807i	$0.397708\pi$
$570$		0			0
$571$	26.8446			1.12341			0.561705	−	0.827337i	$-0.310145\pi$
$571$	26.8446			1.12341			0.561705	+	0.827337i	$0.310145\pi$
$572$		0			0
$573$	−2.95760			−0.123556
$574$		0			0
$575$		0			0
$576$		0			0
$577$			42.7989i			1.78174i	0.454256	+	0.890871i	$0.349905\pi$
$577$			42.7989i			1.78174i	−0.454256	+	0.890871i	$0.650095\pi$
$578$		0			0
$579$			4.90245i			0.203739i
$580$		0			0
$581$	−9.27144			−0.384644
$582$		0			0
$583$			20.8997i			0.865579i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			0.697563i			0.0287915i	0.999896	+	0.0143958i	$0.00458247\pi$
$587$			0.697563i			0.0287915i	−0.999896	+	0.0143958i	$0.995418\pi$
$588$		0			0
$589$	29.6992			1.22373
$590$		0			0
$591$			1.23780i			0.0509161i
$592$		0			0
$593$			34.4837i			1.41608i	0.706174	+	0.708038i	$0.250420\pi$
$593$			34.4837i			1.41608i	−0.706174	+	0.708038i	$0.749580\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.90165			−0.159684
$598$		0			0
$599$	21.0601			0.860491			0.430246	−	0.902712i	$-0.358427\pi$
$599$	21.0601			0.860491			0.430246	+	0.902712i	$0.358427\pi$
$600$		0			0
$601$	−23.2027			−0.946459			−0.473230	−	0.880939i	$-0.656912\pi$
$601$	−23.2027			−0.946459			−0.473230	+	0.880939i	$0.656912\pi$
$602$		0			0
$603$		−	21.5030i		−	0.875668i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	46.0493			1.86908			0.934542	−	0.355852i	$-0.115809\pi$
$607$	46.0493			1.86908			0.934542	+	0.355852i	$0.115809\pi$
$608$		0			0
$609$			2.33423i			0.0945877i
$610$		0			0
$611$	−45.4878	−	4.68887i	−1.84024	−	0.189691i
$612$		0			0
$613$			4.18585i			0.169065i	0.996421	+	0.0845325i	$0.0269397\pi$
$613$			4.18585i			0.169065i	−0.996421	+	0.0845325i	$0.973060\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			13.4856i			0.542911i	0.962451	+	0.271456i	$0.0875050\pi$
$617$			13.4856i			0.542911i	−0.962451	+	0.271456i	$0.912495\pi$
$618$		0			0
$619$		−	12.2439i		−	0.492123i	−0.969254	−	0.246061i	$-0.920864\pi$
$619$		−	12.2439i		−	0.492123i	0.969254	−	0.246061i	$-0.0791364\pi$
$620$		0			0
$621$	14.7475			0.591798
$622$		0			0
$623$	−1.99617			−0.0799749
$624$		0			0
$625$		0			0
$626$		0			0
$627$	5.25326			0.209795
$628$		0			0
$629$		−	25.2027i		−	1.00490i
$630$		0			0
$631$		−	38.9469i		−	1.55045i	−0.631684	−	0.775226i	$-0.717636\pi$
$631$		−	38.9469i		−	1.55045i	0.631684	−	0.775226i	$-0.282364\pi$
$632$		0			0
$633$	−2.68425			−0.106689
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.24985	−	21.8263i	0.0891421	−	0.864788i
$638$		0			0
$639$			25.9205i			1.02540i
$640$		0			0
$641$	−25.2929			−0.999011			−0.499506	−	0.866311i	$-0.666485\pi$
$641$	−25.2929			−0.999011			−0.499506	+	0.866311i	$0.666485\pi$
$642$		0			0
$643$		−	2.70569i		−	0.106702i	−0.998576	−	0.0533511i	$-0.983010\pi$
$643$		−	2.70569i		−	0.106702i	0.998576	−	0.0533511i	$-0.0169902\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.2849			−0.600912			−0.300456	−	0.953796i	$-0.597139\pi$
$647$	−15.2849			−0.600912			−0.300456	+	0.953796i	$0.597139\pi$
$648$		0			0
$649$	16.1220			0.632844
$650$		0			0
$651$	1.91120			0.0749058
$652$		0			0
$653$	−37.8655			−1.48179			−0.740895	−	0.671621i	$-0.765599\pi$
$653$	−37.8655			−1.48179			−0.740895	+	0.671621i	$0.765599\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	23.2769i		−	0.908119i
$658$		0			0
$659$	−36.2910			−1.41370			−0.706849	−	0.707364i	$-0.749884\pi$
$659$	−36.2910			−1.41370			−0.706849	+	0.707364i	$0.749884\pi$
$660$		0			0
$661$			28.6815i			1.11558i	0.829981	+	0.557791i	$0.188351\pi$
$661$			28.6815i			1.11558i	−0.829981	+	0.557791i	$0.811649\pi$
$662$		0			0
$663$	0.932088	−	9.04240i	0.0361993	−	0.351178i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	44.9209			1.73935
$668$		0			0
$669$			0.128488i			0.00496762i
$670$		0			0
$671$			36.9799i			1.42759i
$672$		0			0
$673$	−18.3712			−0.708158			−0.354079	−	0.935216i	$-0.615206\pi$
$673$	−18.3712			−0.708158			−0.354079	+	0.935216i	$0.615206\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.36437			0.0908701			0.0454350	−	0.998967i	$-0.485533\pi$
$677$	2.36437			0.0908701			0.0454350	+	0.998967i	$0.485533\pi$
$678$		0			0
$679$	9.97720			0.382890
$680$		0			0
$681$			8.46554i			0.324400i
$682$		0			0
$683$		−	42.1767i		−	1.61385i	−0.590657	−	0.806923i	$-0.701131\pi$
$683$		−	42.1767i		−	1.61385i	0.590657	−	0.806923i	$-0.298869\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			0.00271214i			0.000103475i
$688$		0			0
$689$	2.98724	−	28.9799i	0.113805	−	1.10405i
$690$		0			0
$691$		−	14.9542i		−	0.568886i	−0.958693	−	0.284443i	$-0.908191\pi$
$691$		−	14.9542i		−	0.568886i	0.958693	−	0.284443i	$-0.0918087\pi$
$692$		0			0
$693$	−7.08208			−0.269026
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	55.1033i		−	2.08719i
$698$		0			0
$699$	−0.229046			−0.00866333
$700$		0			0
$701$	−22.7394			−0.858855			−0.429428	−	0.903101i	$-0.641285\pi$
$701$	−22.7394			−0.858855			−0.429428	+	0.903101i	$0.641285\pi$
$702$		0			0
$703$	−20.3025			−0.765723
$704$		0			0
$705$		0			0
$706$		0			0
$707$			5.03044i			0.189189i
$708$		0			0
$709$			17.5481i			0.659034i	0.944150	+	0.329517i	$0.106886\pi$
$709$			17.5481i			0.659034i	−0.944150	+	0.329517i	$0.893114\pi$
$710$		0			0
$711$	24.1484			0.905635
$712$		0			0
$713$		−	36.7799i		−	1.37742i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	9.22689i		−	0.344585i
$718$		0			0
$719$	20.2587			0.755521			0.377761	−	0.925903i	$-0.376694\pi$
$719$	20.2587			0.755521			0.377761	+	0.925903i	$0.376694\pi$
$720$		0			0
$721$		−	18.1823i		−	0.677146i
$722$		0			0
$723$		−	8.80139i		−	0.327327i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−19.7502			−0.732493			−0.366246	−	0.930518i	$-0.619357\pi$
$727$	−19.7502			−0.732493			−0.366246	+	0.930518i	$0.619357\pi$
$728$		0			0
$729$	−19.8970			−0.736927
$730$		0			0
$731$	−4.93209			−0.182420
$732$		0			0
$733$		−	38.1160i		−	1.40785i	−0.710276	−	0.703924i	$-0.751430\pi$
$733$		−	38.1160i		−	1.40785i	0.710276	−	0.703924i	$-0.248570\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.4244			−0.715509
$738$		0			0
$739$		−	36.9974i		−	1.36097i	−0.732761	−	0.680486i	$-0.761769\pi$
$739$		−	36.9974i		−	1.36097i	0.732761	−	0.680486i	$-0.238231\pi$
$740$		0			0
$741$	−7.28426	−	0.750860i	−0.267594	−	0.0275835i
$742$		0			0
$743$			40.3625i			1.48076i	0.672190	+	0.740378i	$0.265354\pi$
$743$			40.3625i			1.48076i	−0.672190	+	0.740378i	$0.734646\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			27.7618i			1.01575i
$748$		0			0
$749$			17.1443i			0.626437i
$750$		0			0
$751$	2.89161			0.105516			0.0527581	−	0.998607i	$-0.483199\pi$
$751$	2.89161			0.105516			0.0527581	+	0.998607i	$0.483199\pi$
$752$		0			0
$753$	−5.41681			−0.197400
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−3.63180			−0.132000			−0.0660001	−	0.997820i	$-0.521024\pi$
$757$	−3.63180			−0.132000			−0.0660001	+	0.997820i	$0.521024\pi$
$758$		0			0
$759$		−	6.50573i		−	0.236143i
$760$		0			0
$761$		−	31.2016i		−	1.13106i	−0.824729	−	0.565529i	$-0.808672\pi$
$761$		−	31.2016i		−	1.13106i	0.824729	−	0.565529i	$-0.191328\pi$
$762$		0			0
$763$	−11.3913			−0.412393
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−22.3550	−	2.30435i	−0.807193	−	0.0832053i
$768$		0			0
$769$			53.0897i			1.91446i	0.289321	+	0.957232i	$0.406570\pi$
$769$			53.0897i			1.91446i	−0.289321	+	0.957232i	$0.593430\pi$
$770$		0			0
$771$	−4.97369			−0.179123
$772$		0			0
$773$			49.9435i			1.79634i	0.439646	+	0.898171i	$0.355104\pi$
$773$			49.9435i			1.79634i	−0.439646	+	0.898171i	$0.644896\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.30651			−0.0468706
$778$		0			0
$779$	−44.3894			−1.59041
$780$		0			0
$781$	23.4150			0.837853
$782$		0			0
$783$	14.3125			0.511489
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	28.1065i		−	1.00189i	−0.865479	−	0.500945i	$-0.832986\pi$
$787$		−	28.1065i		−	1.00189i	0.865479	−	0.500945i	$-0.167014\pi$
$788$		0			0
$789$	5.18179			0.184476
$790$		0			0
$791$			0.867387i			0.0308407i
$792$		0			0
$793$	5.28561	−	51.2769i	0.187698	−	1.82090i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.22633			0.291392			0.145696	−	0.989329i	$-0.453458\pi$
$797$	8.22633			0.291392			0.145696	+	0.989329i	$0.453458\pi$
$798$		0			0
$799$			86.4919i			3.05986i
$800$		0			0
$801$			5.97720i			0.211194i
$802$		0			0
$803$	−21.0269			−0.742024
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−3.90566			−0.137486
$808$		0			0
$809$	−32.4978			−1.14256			−0.571281	−	0.820754i	$-0.693554\pi$
$809$	−32.4978			−1.14256			−0.571281	+	0.820754i	$0.693554\pi$
$810$		0			0
$811$			34.4735i			1.21053i	0.796024	+	0.605265i	$0.206933\pi$
$811$			34.4735i			1.21053i	−0.796024	+	0.605265i	$0.793067\pi$
$812$		0			0
$813$			2.90109i			0.101746i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			3.97313i			0.139002i
$818$		0			0
$819$	9.82013	+	1.01226i	0.343143	+	0.0353711i
$820$		0			0
$821$			2.76904i			0.0966401i	0.998832	+	0.0483201i	$0.0153867\pi$
$821$			2.76904i			0.0966401i	−0.998832	+	0.0483201i	$0.984613\pi$
$822$		0			0
$823$	27.4263			0.956020			0.478010	−	0.878355i	$-0.341358\pi$
$823$	27.4263			0.956020			0.478010	+	0.878355i	$0.341358\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			4.39186i			0.152720i	0.997080	+	0.0763599i	$0.0243298\pi$
$827$			4.39186i			0.152720i	−0.997080	+	0.0763599i	$0.975670\pi$
$828$		0			0
$829$	−35.2144			−1.22305			−0.611524	−	0.791226i	$-0.709443\pi$
$829$	−35.2144			−1.22305			−0.611524	+	0.791226i	$0.709443\pi$
$830$		0			0
$831$	1.12307			0.0389587
$832$		0			0
$833$	−41.5011			−1.43793
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	11.7187i		−	0.405057i
$838$		0			0
$839$		−	30.3086i		−	1.04637i	−0.852220	−	0.523184i	$-0.824744\pi$
$839$		−	30.3086i		−	1.04637i	0.852220	−	0.523184i	$-0.175256\pi$
$840$		0			0
$841$	14.5959			0.503308
$842$		0			0
$843$		−	6.26652i		−	0.215830i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	4.12121i		−	0.141606i
$848$		0			0
$849$	−0.0706808			−0.00242576
$850$		0			0
$851$			25.1430i			0.861889i
$852$		0			0
$853$			0.369052i			0.0126361i	0.999980	+	0.00631805i	$0.00201111\pi$
$853$			0.369052i			0.0126361i	−0.999980	+	0.00631805i	$0.997989\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.2163			0.861372			0.430686	−	0.902502i	$-0.358272\pi$
$857$	25.2163			0.861372			0.430686	+	0.902502i	$0.358272\pi$
$858$		0			0
$859$	15.6500			0.533971			0.266985	−	0.963701i	$-0.413972\pi$
$859$	15.6500			0.533971			0.266985	+	0.963701i	$0.413972\pi$
$860$		0			0
$861$	−2.85654			−0.0973507
$862$		0			0
$863$		−	5.22960i		−	0.178018i	−0.996031	−	0.0890089i	$-0.971630\pi$
$863$		−	5.22960i		−	0.178018i	0.996031	−	0.0890089i	$-0.0283700\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−10.9086			−0.370475
$868$		0			0
$869$		−	21.8141i		−	0.739994i
$870$		0			0
$871$	26.9342	+	2.77638i	0.912632	+	0.0940739i
$872$		0			0
$873$		−	29.8750i		−	1.01112i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			12.5859i			0.424996i	0.977162	+	0.212498i	$0.0681598\pi$
$877$			12.5859i			0.424996i	−0.977162	+	0.212498i	$0.931840\pi$
$878$		0			0
$879$			7.88748i			0.266038i
$880$		0			0
$881$	28.5449			0.961701			0.480850	−	0.876803i	$-0.340328\pi$
$881$	28.5449			0.961701			0.480850	+	0.876803i	$0.340328\pi$
$882$		0			0
$883$	13.7537			0.462848			0.231424	−	0.972853i	$-0.425662\pi$
$883$	13.7537			0.462848			0.231424	+	0.972853i	$0.425662\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.0289			−0.370315			−0.185158	−	0.982709i	$-0.559280\pi$
$887$	−11.0289			−0.370315			−0.185158	+	0.982709i	$0.559280\pi$
$888$		0			0
$889$		−	8.25789i		−	0.276961i
$890$		0			0
$891$			20.1455i			0.674900i
$892$		0			0
$893$	69.6750			2.33158
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.929877	+	9.02095i	−0.0310477	+	0.301201i
$898$		0			0
$899$		−	35.6951i		−	1.19050i
$900$		0			0
$901$	−55.1033			−1.83576
$902$		0			0
$903$			0.255679i			0.00850845i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	11.3893			0.378175			0.189088	−	0.981960i	$-0.439447\pi$
$907$	11.3893			0.378175			0.189088	+	0.981960i	$0.439447\pi$
$908$		0			0
$909$	15.0628			0.499601
$910$		0			0
$911$	25.9785			0.860706			0.430353	−	0.902661i	$-0.358389\pi$
$911$	25.9785			0.860706			0.430353	+	0.902661i	$0.358389\pi$
$912$		0			0
$913$	25.0783			0.829969
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	17.2821i		−	0.570705i
$918$		0			0
$919$	18.6492			0.615180			0.307590	−	0.951519i	$-0.400478\pi$
$919$	18.6492			0.615180			0.307590	+	0.951519i	$0.400478\pi$
$920$		0			0
$921$		−	2.93671i		−	0.0967680i
$922$		0			0
$923$	−32.4675	−	3.34675i	−1.06868	−	0.110160i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−54.4439			−1.78817
$928$		0			0
$929$			40.6769i			1.33457i	0.744804	+	0.667283i	$0.232543\pi$
$929$			40.6769i			1.33457i	−0.744804	+	0.667283i	$0.767457\pi$
$930$		0			0
$931$			33.4319i			1.09569i
$932$		0			0
$933$	5.31575			0.174030
$934$		0			0
$935$		0			0
$936$		0			0
$937$	50.5794			1.65236			0.826178	−	0.563409i	$-0.190511\pi$
$937$	50.5794			1.65236			0.826178	+	0.563409i	$0.190511\pi$
$938$		0			0
$939$	−1.42877			−0.0466263
$940$		0			0
$941$		−	51.9133i		−	1.69232i	−0.532925	−	0.846162i	$-0.678907\pi$
$941$		−	51.9133i		−	1.69232i	0.532925	−	0.846162i	$-0.321093\pi$
$942$		0			0
$943$			54.9726i			1.79015i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			5.68098i			0.184607i	0.995731	+	0.0923035i	$0.0294230\pi$
$947$			5.68098i			0.184607i	−0.995731	+	0.0923035i	$0.970577\pi$
$948$		0			0
$949$	29.1563	+	3.00542i	0.946453	+	0.0975601i
$950$		0			0
$951$		−	2.90979i		−	0.0943563i
$952$		0			0
$953$	−1.03426			−0.0335031			−0.0167516	−	0.999860i	$-0.505332\pi$
$953$	−1.03426			−0.0335031			−0.0167516	+	0.999860i	$0.505332\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	6.31384i		−	0.204097i
$958$		0			0
$959$	6.59786			0.213056
$960$		0			0
$961$	1.77390			0.0572227
$962$		0			0
$963$	51.3355			1.65426
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	14.2277i		−	0.457532i	−0.973481	−	0.228766i	$-0.926531\pi$
$967$		−	14.2277i		−	0.457532i	0.973481	−	0.228766i	$-0.0734690\pi$
$968$		0			0
$969$			13.8505i			0.444942i
$970$		0			0
$971$	−29.2714			−0.939365			−0.469683	−	0.882835i	$-0.655632\pi$
$971$	−29.2714			−0.939365			−0.469683	+	0.882835i	$0.655632\pi$
$972$		0			0
$973$		−	21.1071i		−	0.676662i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	9.41145i		−	0.301099i	−0.988602	−	0.150549i	$-0.951896\pi$
$977$		−	9.41145i		−	0.301099i	0.988602	−	0.150549i	$-0.0481043\pi$
$978$		0			0
$979$	5.39943			0.172566
$980$		0			0
$981$			34.1093i			1.08903i
$982$		0			0
$983$			19.1274i			0.610070i	0.952341	+	0.305035i	$0.0986682\pi$
$983$			19.1274i			0.610070i	−0.952341	+	0.305035i	$0.901332\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	4.48372			0.142718
$988$		0			0
$989$	4.92039			0.156459
$990$		0			0
$991$	−13.5231			−0.429576			−0.214788	−	0.976661i	$-0.568906\pi$
$991$	−13.5231			−0.429576			−0.214788	+	0.976661i	$0.568906\pi$
$992$		0			0
$993$		−	2.98176i		−	0.0946234i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	38.1530			1.20832			0.604159	−	0.796864i	$-0.293509\pi$
$997$	38.1530			1.20832			0.604159	+	0.796864i	$0.293509\pi$
$998$		0			0
$999$			8.01096i			0.253455i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.k.c.2001.5		8
5.2	odd	4		2600.2.f.e.649.5		8
5.3	odd	4		2600.2.f.f.649.4		8
5.4	even	2		520.2.k.b.441.4	yes	8
13.12	even	2	inner	2600.2.k.c.2001.6		8
15.14	odd	2		4680.2.g.k.2521.3		8
20.19	odd	2		1040.2.k.e.961.6		8
65.12	odd	4		2600.2.f.f.649.5		8
65.34	odd	4		6760.2.a.bc.1.2		4
65.38	odd	4		2600.2.f.e.649.4		8
65.44	odd	4		6760.2.a.bd.1.2		4
65.64	even	2		520.2.k.b.441.3	✓	8
195.194	odd	2		4680.2.g.k.2521.6		8
260.259	odd	2		1040.2.k.e.961.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.b.441.3	✓	8	65.64	even	2
520.2.k.b.441.4	yes	8	5.4	even	2
1040.2.k.e.961.5		8	260.259	odd	2
1040.2.k.e.961.6		8	20.19	odd	2
2600.2.f.e.649.4		8	65.38	odd	4
2600.2.f.e.649.5		8	5.2	odd	4
2600.2.f.f.649.4		8	5.3	odd	4
2600.2.f.f.649.5		8	65.12	odd	4
2600.2.k.c.2001.5		8	1.1	even	1	trivial
2600.2.k.c.2001.6		8	13.12	even	2	inner
4680.2.g.k.2521.3		8	15.14	odd	2
4680.2.g.k.2521.6		8	195.194	odd	2
6760.2.a.bc.1.2		4	65.34	odd	4
6760.2.a.bd.1.2		4	65.44	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.369700 q^{3} -0.956248i q^{7} -2.86332 q^{9} +2.58655i q^{11} +(0.369700 - 3.58655i) q^{13} -6.81957 q^{17} +5.49362i q^{19} +0.353525i q^{21} +6.80339 q^{23} +2.16767 q^{27} +6.60272 q^{29} -5.40612i q^{31} -0.956248i q^{33} +3.69565i q^{37} +(-0.136678 + 1.32595i) q^{39} +8.08017i q^{41} +0.723226 q^{43} -12.6829i q^{47} +6.08559 q^{49} +2.52120 q^{51} +8.08017 q^{53} -2.03099i q^{57} -6.23302i q^{59} +14.2970 q^{61} +2.73804i q^{63} +7.50980i q^{67} -2.51522 q^{69} -9.05259i q^{71} +8.12934i q^{73} +2.47338 q^{77} -8.43369 q^{79} +7.78857 q^{81} -9.69565i q^{83} -2.44103 q^{87} -2.08750i q^{89} +(-3.42963 - 0.353525i) q^{91} +1.99864i q^{93} +10.4337i q^{97} -7.40612i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 16 q^{9} + 4 q^{13} + 4 q^{17} + 12 q^{23} - 4 q^{27} + 16 q^{29} - 40 q^{39} + 24 q^{43} - 32 q^{49} + 16 q^{51} + 4 q^{53} + 32 q^{61} + 56 q^{69} + 44 q^{77} - 24 q^{79} + 64 q^{81} - 76 q^{87}+ \cdots - 32 q^{91}+O(q^{100}) \) 8 * q - 4 * q^3 + 16 * q^9 + 4 * q^13 + 4 * q^17 + 12 * q^23 - 4 * q^27 + 16 * q^29 - 40 * q^39 + 24 * q^43 - 32 * q^49 + 16 * q^51 + 4 * q^53 + 32 * q^61 + 56 * q^69 + 44 * q^77 - 24 * q^79 + 64 * q^81 - 76 * q^87 - 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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