LMFDB - Embedded newform 1040.2.k.e.961.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(961,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1040.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.k (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$8.30444181021$
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:	$\Z[a_1, \ldots, a_{5}]$
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			961.6
Root			$1.47812 - 1.47812i$ of defining polynomial
Character	$\chi$	$=$	1040.961
Dual form			1040.2.k.e.961.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.369700 q^{3} +1.00000i q^{5} -0.956248i q^{7} -2.86332 q^{9} +O(q^{10})$	$q-0.369700 q^{3} +1.00000i q^{5} -0.956248i q^{7} -2.86332 q^{9} -2.58655i q^{11} +(-0.369700 + 3.58655i) q^{13} -0.369700i q^{15} +6.81957 q^{17} -5.49362i q^{19} +0.353525i q^{21} +6.80339 q^{23} -1.00000 q^{25} +2.16767 q^{27} +6.60272 q^{29} +5.40612i q^{31} +0.956248i q^{33} +0.956248 q^{35} -3.69565i q^{37} +(0.136678 - 1.32595i) q^{39} +8.08017i q^{41} +0.723226 q^{43} -2.86332i q^{45} -12.6829i q^{47} +6.08559 q^{49} -2.52120 q^{51} -8.08017 q^{53} +2.58655 q^{55} +2.03099i q^{57} +6.23302i q^{59} +14.2970 q^{61} +2.73804i q^{63} +(-3.58655 - 0.369700i) q^{65} +7.50980i q^{67} -2.51522 q^{69} +9.05259i q^{71} -8.12934i q^{73} +0.369700 q^{75} -2.47338 q^{77} +8.43369 q^{79} +7.78857 q^{81} -9.69565i q^{83} +6.81957i q^{85} -2.44103 q^{87} -2.08750i q^{89} +(3.42963 + 0.353525i) q^{91} -1.99864i q^{93} +5.49362 q^{95} -10.4337i q^{97} +7.40612i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 16 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 + 16 * q^9	$ 8 q - 4 q^{3} + 16 q^{9} - 4 q^{13} - 4 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 12 q^{35} + 40 q^{39} + 24 q^{43} - 32 q^{49} - 16 q^{51} - 4 q^{53} + 32 q^{61} - 8 q^{65} + 56 q^{69} + 4 q^{75} - 44 q^{77} + 24 q^{79} + 64 q^{81} - 76 q^{87} + 32 q^{91} + 4 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 + 16 * q^9 - 4 * q^13 - 4 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 + 16 * q^29 - 12 * q^35 + 40 * q^39 + 24 * q^43 - 32 * q^49 - 16 * q^51 - 4 * q^53 + 32 * q^61 - 8 * q^65 + 56 * q^69 + 4 * q^75 - 44 * q^77 + 24 * q^79 + 64 * q^81 - 76 * q^87 + 32 * q^91 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		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	$-0.872497\pi$
$11$		−	2.58655i		−	0.779873i	0.920842	−	0.389937i	$-0.127503\pi$
$12$		0			0
$13$	−0.369700	+	3.58655i	−0.102536	+	0.994729i
$13$	−0.369700	+	3.58655i	−0.102536	+	0.994729i
$14$		0			0
$15$		−	0.369700i		−	0.0954563i
$16$		0			0
$17$	6.81957			1.65399			0.826994	−	0.562210i	$-0.190049\pi$
$17$	6.81957			1.65399			0.826994	+	0.562210i	$0.190049\pi$
$18$		0			0
$19$		−	5.49362i		−	1.26032i	−0.776464	−	0.630162i	$-0.782989\pi$
$19$		−	5.49362i		−	1.26032i	0.776464	−	0.630162i	$-0.217011\pi$
$20$		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
$25$	−1.00000			−0.200000
$26$		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	$0.161356\pi$
$31$			5.40612i			0.970967i	−0.874246	+	0.485484i	$0.838644\pi$
$32$		0			0
$33$			0.956248i			0.166461i
$34$		0			0
$35$	0.956248			0.161635
$36$		0			0
$37$		−	3.69565i		−	0.607561i	−0.952742	−	0.303780i	$-0.901751\pi$
$37$		−	3.69565i		−	0.607561i	0.952742	−	0.303780i	$-0.0982489\pi$
$38$		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
$45$		−	2.86332i		−	0.426839i
$46$		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	$-0.687262\pi$
$53$	−8.08017			−1.10990			−0.554948	+	0.831885i	$0.687262\pi$
$54$		0			0
$55$	2.58655			0.348770
$56$		0			0
$57$			2.03099i			0.269012i
$58$		0			0
$59$			6.23302i			0.811470i	0.913991	+	0.405735i	$0.132984\pi$
$59$			6.23302i			0.811470i	−0.913991	+	0.405735i	$0.867016\pi$
$60$		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
$65$	−3.58655	−	0.369700i	−0.444856	−	0.0458557i
$66$		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	$0.180508\pi$
$71$			9.05259i			1.07434i	−0.843472	+	0.537172i	$0.819492\pi$
$72$		0			0
$73$		−	8.12934i		−	0.951468i	−0.879589	−	0.475734i	$-0.842183\pi$
$73$		−	8.12934i		−	0.951468i	0.879589	−	0.475734i	$-0.157817\pi$
$74$		0			0
$75$	0.369700			0.0426893
$76$		0			0
$77$	−2.47338			−0.281868
$78$		0			0
$79$	8.43369			0.948865			0.474432	−	0.880292i	$-0.342653\pi$
$79$	8.43369			0.948865			0.474432	+	0.880292i	$0.342653\pi$
$80$		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
$85$			6.81957i			0.739686i
$86$		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
$95$	5.49362			0.563634
$96$		0			0
$97$		−	10.4337i		−	1.05938i	−0.848191	−	0.529691i	$-0.822308\pi$
$97$		−	10.4337i		−	1.05938i	0.848191	−	0.529691i	$-0.177692\pi$
$98$		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
$105$	−0.353525			−0.0345005
$106$		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	$-0.486415\pi$
$113$	0.907074			0.0853303			0.0426652	+	0.999089i	$0.486415\pi$
$114$		0			0
$115$			6.80339i			0.634420i
$116$		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
$125$		−	1.00000i		−	0.0894427i
$126$		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	$-0.789668\pi$
$131$	−18.0728			−1.57903			−0.789515	+	0.613731i	$0.789668\pi$
$132$		0			0
$133$	−5.25326			−0.455516
$134$		0			0
$135$			2.16767i			0.186564i
$136$		0			0
$137$		−	6.89974i		−	0.589484i	−0.955577	−	0.294742i	$-0.904766\pi$
$137$		−	6.89974i		−	0.589484i	0.955577	−	0.294742i	$-0.0952338\pi$
$138$		0			0
$139$	−22.0728			−1.87219			−0.936097	−	0.351743i	$-0.885589\pi$
$139$	−22.0728			−1.87219			−0.936097	+	0.351743i	$0.885589\pi$
$140$		0			0
$141$			4.68887i			0.394874i
$142$		0			0
$143$	9.27677	+	0.956248i	0.775763	+	0.0799655i
$144$		0			0
$145$			6.60272i			0.548326i
$146$		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	$0.109393\pi$
$151$			8.28084i			0.673885i	−0.941525	+	0.336943i	$0.890607\pi$
$152$		0			0
$153$	−19.5266			−1.57863
$154$		0			0
$155$	−5.40612			−0.434230
$156$		0			0
$157$	0.899738			0.0718069			0.0359034	−	0.999355i	$-0.488569\pi$
$157$	0.899738			0.0718069			0.0359034	+	0.999355i	$0.488569\pi$
$158$		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
$165$	−0.956248			−0.0744438
$166$		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	$-0.524135\pi$
$173$	−1.99266			−0.151499			−0.0757497	+	0.997127i	$0.524135\pi$
$174$		0			0
$175$			0.956248i			0.0722855i
$176$		0			0
$177$		−	2.30435i		−	0.173206i
$178$		0			0
$179$	10.5535			0.788809			0.394405	−	0.918937i	$-0.370951\pi$
$179$	10.5535			0.788809			0.394405	+	0.918937i	$0.370951\pi$
$180$		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
$185$	3.69565			0.271709
$186$		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	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$			13.2606i			0.954519i	0.878762	+	0.477259i	$0.158370\pi$
$193$			13.2606i			0.954519i	−0.878762	+	0.477259i	$0.841630\pi$
$194$		0			0
$195$	1.32595	+	0.136678i	0.0949531	+	0.00978775i
$196$		0			0
$197$			3.34810i			0.238542i	0.992862	+	0.119271i	$0.0380558\pi$
$197$			3.34810i			0.238542i	−0.992862	+	0.119271i	$0.961944\pi$
$198$		0			0
$199$	−10.5535			−0.748121			−0.374061	−	0.927404i	$-0.622035\pi$
$199$	−10.5535			−0.748121			−0.374061	+	0.927404i	$0.622035\pi$
$200$		0			0
$201$		−	2.77638i		−	0.195830i
$202$		0			0
$203$		−	6.31384i		−	0.443145i
$204$		0			0
$205$	−8.08017			−0.564343
$206$		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	$-0.580404\pi$
$211$	−7.26060			−0.499840			−0.249920	+	0.968266i	$0.580404\pi$
$212$		0			0
$213$		−	3.34675i		−	0.229315i
$214$		0			0
$215$			0.723226i			0.0493236i
$216$		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
$225$	2.86332			0.190888
$226$		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	$-0.506460\pi$
$233$	−0.619546			−0.0405878			−0.0202939	+	0.999794i	$0.506460\pi$
$234$		0			0
$235$	12.6829			0.827341
$236$		0			0
$237$	−3.11794			−0.202532
$238$		0			0
$239$		−	24.9578i		−	1.61438i	−0.590290	−	0.807191i	$-0.700987\pi$
$239$		−	24.9578i		−	1.61438i	0.590290	−	0.807191i	$-0.299013\pi$
$240$		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
$245$			6.08559i			0.388794i
$246$		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	$-0.653015\pi$
$251$	−14.6519			−0.924820			−0.462410	+	0.886666i	$0.653015\pi$
$252$		0			0
$253$		−	17.5973i		−	1.10633i
$254$		0			0
$255$		−	2.52120i		−	0.157884i
$256$		0			0
$257$	−13.4533			−0.839193			−0.419596	−	0.907711i	$-0.637828\pi$
$257$	−13.4533			−0.839193			−0.419596	+	0.907711i	$0.637828\pi$
$258$		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
$265$		−	8.08017i		−	0.496361i
$266$		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	$0.0766033\pi$
$271$			7.84715i			0.476680i	−0.971182	+	0.238340i	$0.923397\pi$
$272$		0			0
$273$	−1.26794	−	0.130698i	−0.0767389	−	0.00791023i
$274$		0			0
$275$			2.58655i			0.155975i
$276$		0			0
$277$	3.03777			0.182522			0.0912610	−	0.995827i	$-0.470910\pi$
$277$	3.03777			0.182522			0.0912610	+	0.995827i	$0.470910\pi$
$278$		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
$285$	−2.03099			−0.120306
$286$		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	$0.214166\pi$
$293$			21.3348i			1.24639i	−0.782066	+	0.623196i	$0.785834\pi$
$294$		0			0
$295$	−6.23302			−0.362901
$296$		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
$305$			14.2970i			0.818645i
$306$		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	$-0.366343\pi$
$311$	14.3785			0.815332			0.407666	+	0.913131i	$0.366343\pi$
$312$		0			0
$313$	−3.86468			−0.218444			−0.109222	−	0.994017i	$-0.534836\pi$
$313$	−3.86468			−0.218444			−0.109222	+	0.994017i	$0.534836\pi$
$314$		0			0
$315$	−2.73804			−0.154271
$316$		0			0
$317$		−	7.87066i		−	0.442060i	−0.975267	−	0.221030i	$-0.929058\pi$
$317$		−	7.87066i		−	0.442060i	0.975267	−	0.221030i	$-0.0709419\pi$
$318$		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
$325$	0.369700	−	3.58655i	0.0205073	−	0.198946i
$326$		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	$-0.928854\pi$
$331$		−	8.06535i		−	0.443312i	0.975125	−	0.221656i	$-0.0711462\pi$
$332$		0			0
$333$			10.5818i			0.579881i
$334$		0			0
$335$	−7.50980			−0.410304
$336$		0			0
$337$	−12.6337			−0.688202			−0.344101	−	0.938933i	$-0.611816\pi$
$337$	−12.6337			−0.688202			−0.344101	+	0.938933i	$0.611816\pi$
$338$		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
$345$		−	2.51522i		−	0.135415i
$346$		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	$-0.871038\pi$
$353$		−	14.8109i		−	0.788303i	0.919045	−	0.394152i	$-0.128962\pi$
$354$		0			0
$355$	−9.05259			−0.480462
$356$		0			0
$357$			2.41089i			0.127598i
$358$		0			0
$359$		−	21.0526i		−	1.11111i	−0.831479	−	0.555557i	$-0.812505\pi$
$359$		−	21.0526i		−	1.11111i	0.831479	−	0.555557i	$-0.187495\pi$
$360$		0			0
$361$	−11.1799			−0.588414
$362$		0			0
$363$	−1.59333			−0.0836279
$364$		0			0
$365$	8.12934			0.425509
$366$		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	$-0.388359\pi$
$373$	13.2714			0.687169			0.343585	+	0.939122i	$0.388359\pi$
$374$		0			0
$375$			0.369700i			0.0190913i
$376$		0			0
$377$	−2.44103	+	23.6810i	−0.125719	+	1.21963i
$378$		0			0
$379$			4.77240i			0.245142i	0.992460	+	0.122571i	$0.0391139\pi$
$379$			4.77240i			0.245142i	−0.992460	+	0.122571i	$0.960886\pi$
$380$		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
$385$		−	2.47338i		−	0.126055i
$386$		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
$395$			8.43369i			0.424345i
$396$		0			0
$397$			3.26196i			0.163713i	0.996644	+	0.0818564i	$0.0260849\pi$
$397$			3.26196i			0.163713i	−0.996644	+	0.0818564i	$0.973915\pi$
$398$		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
$405$			7.78857i			0.387017i
$406$		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
$415$	9.69565			0.475941
$416$		0			0
$417$	8.16034			0.399613
$418$		0			0
$419$	−7.53396			−0.368058			−0.184029	−	0.982921i	$-0.558914\pi$
$419$	−7.53396			−0.368058			−0.184029	+	0.982921i	$0.558914\pi$
$420$		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
$425$	−6.81957			−0.330798
$426$		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	$0.137693\pi$
$431$			17.4061i			0.838423i	−0.907889	+	0.419212i	$0.862307\pi$
$432$		0			0
$433$	28.1712			1.35382			0.676910	−	0.736066i	$-0.263319\pi$
$433$	28.1712			1.35382			0.676910	+	0.736066i	$0.263319\pi$
$434$		0			0
$435$		−	2.44103i		−	0.117038i
$436$		0			0
$437$		−	37.3753i		−	1.78790i
$438$		0			0
$439$	24.8014			1.18371			0.591853	−	0.806046i	$-0.298397\pi$
$439$	24.8014			1.18371			0.591853	+	0.806046i	$0.298397\pi$
$440$		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
$445$	2.08750			0.0989572
$446$		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
$455$	−0.353525	+	3.42963i	−0.0165735	+	0.160783i
$456$		0			0
$457$		−	6.60870i		−	0.309142i	−0.987982	−	0.154571i	$-0.950600\pi$
$457$		−	6.60870i		−	0.309142i	0.987982	−	0.154571i	$-0.0493995\pi$
$458$		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
$465$	1.99864			0.0926849
$466$		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
$475$			5.49362i			0.252065i
$476$		0			0
$477$	23.1361			1.05933
$478$		0			0
$479$			36.7469i			1.67901i	0.543354	+	0.839504i	$0.317154\pi$
$479$			36.7469i			1.67901i	−0.543354	+	0.839504i	$0.682846\pi$
$480$		0			0
$481$	13.2546	+	1.36628i	0.604359	+	0.0622971i
$482$		0			0
$483$			2.40517i			0.109439i
$484$		0			0
$485$	10.4337			0.473770
$486$		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	$-0.185167\pi$
$491$	37.0277			1.67104			0.835519	+	0.549462i	$0.185167\pi$
$492$		0			0
$493$	45.0277			2.02795
$494$		0			0
$495$	−7.40612			−0.332880
$496$		0			0
$497$	8.65652			0.388298
$498$		0			0
$499$			20.4809i			0.916849i	0.888733	+	0.458425i	$0.151586\pi$
$499$			20.4809i			0.916849i	−0.888733	+	0.458425i	$0.848414\pi$
$500$		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
$505$		−	5.26060i		−	0.234094i
$506$		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
$515$			19.0143i			0.837868i
$516$		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
$525$		−	0.353525i		−	0.0154291i
$526$		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
$535$		−	17.9287i		−	0.775124i
$536$		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
$545$	11.9125			0.510275
$546$		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
$555$	−1.36628			−0.0579955
$556$		0			0
$557$			17.3321i			0.734384i	0.930145	+	0.367192i	$0.119681\pi$
$557$			17.3321i			0.734384i	−0.930145	+	0.367192i	$0.880319\pi$
$558$		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
$565$			0.907074i			0.0381609i
$566$		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	$-0.689855\pi$
$571$	−26.8446			−1.12341			−0.561705	+	0.827337i	$0.689855\pi$
$572$		0			0
$573$	2.95760			0.123556
$574$		0			0
$575$	−6.80339			−0.283721
$576$		0			0
$577$		−	42.7989i		−	1.78174i	−0.454256	−	0.890871i	$-0.650095\pi$
$577$		−	42.7989i		−	1.78174i	0.454256	−	0.890871i	$-0.349905\pi$
$578$		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
$585$	10.2694	+	1.05857i	0.424589	+	0.0437665i
$586$		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	$-0.749580\pi$
$593$		−	34.4837i		−	1.41608i	0.706174	−	0.708038i	$-0.250420\pi$
$594$		0			0
$595$	6.52120			0.267343
$596$		0			0
$597$	3.90165			0.159684
$598$		0			0
$599$	−21.0601			−0.860491			−0.430246	−	0.902712i	$-0.641573\pi$
$599$	−21.0601			−0.860491			−0.430246	+	0.902712i	$0.641573\pi$
$600$		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
$605$			4.30977i			0.175217i
$606$		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	$-0.973060\pi$
$613$		−	4.18585i		−	0.169065i	0.996421	−	0.0845325i	$-0.0269397\pi$
$614$		0			0
$615$	2.98724			0.120457
$616$		0			0
$617$		−	13.4856i		−	0.542911i	−0.962451	−	0.271456i	$-0.912495\pi$
$617$		−	13.4856i		−	0.542911i	0.962451	−	0.271456i	$-0.0875050\pi$
$618$		0			0
$619$			12.2439i			0.492123i	0.969254	+	0.246061i	$0.0791364\pi$
$619$			12.2439i			0.492123i	−0.969254	+	0.246061i	$0.920864\pi$
$620$		0			0
$621$	14.7475			0.591798
$622$		0			0
$623$	−1.99617			−0.0799749
$624$		0			0
$625$	1.00000			0.0400000
$626$		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	$0.282364\pi$
$631$			38.9469i			1.55045i	−0.631684	+	0.775226i	$0.717636\pi$
$632$		0			0
$633$	2.68425			0.106689
$634$		0			0
$635$			8.63572i			0.342698i
$636$		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
$645$		−	0.267377i		−	0.0105280i
$646$		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	$-0.234401\pi$
$653$	37.8655			1.48179			0.740895	+	0.671621i	$0.234401\pi$
$654$		0			0
$655$		−	18.0728i		−	0.706164i
$656$		0			0
$657$			23.2769i			0.908119i
$658$		0			0
$659$	36.2910			1.41370			0.706849	−	0.707364i	$-0.250116\pi$
$659$	36.2910			1.41370			0.706849	+	0.707364i	$0.250116\pi$
$660$		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
$665$		−	5.25326i		−	0.203713i
$666$		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	$-0.384794\pi$
$673$	18.3712			0.708158			0.354079	+	0.935216i	$0.384794\pi$
$674$		0			0
$675$	−2.16767			−0.0834338
$676$		0			0
$677$	−2.36437			−0.0908701			−0.0454350	−	0.998967i	$-0.514467\pi$
$677$	−2.36437			−0.0908701			−0.0454350	+	0.998967i	$0.514467\pi$
$678$		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
$685$	6.89974			0.263625
$686$		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	$0.0918087\pi$
$691$			14.9542i			0.568886i	−0.958693	+	0.284443i	$0.908191\pi$
$692$		0			0
$693$	7.08208			0.269026
$694$		0			0
$695$		−	22.0728i		−	0.837270i
$696$		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
$705$	−4.68887			−0.176593
$706$		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
$715$	−0.956248	+	9.27677i	−0.0357616	+	0.346932i
$716$		0			0
$717$			9.22689i			0.344585i
$718$		0			0
$719$	−20.2587			−0.755521			−0.377761	−	0.925903i	$-0.623306\pi$
$719$	−20.2587			−0.755521			−0.377761	+	0.925903i	$0.623306\pi$
$720$		0			0
$721$		−	18.1823i		−	0.677146i
$722$		0			0
$723$		−	8.80139i		−	0.327327i
$724$		0			0
$725$	−6.60272			−0.245219
$726$		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	$0.248570\pi$
$733$			38.1160i			1.40785i	−0.710276	+	0.703924i	$0.751430\pi$
$734$		0			0
$735$		−	2.24985i		−	0.0829868i
$736$		0			0
$737$	19.4244			0.715509
$738$		0			0
$739$			36.9974i			1.36097i	0.732761	+	0.680486i	$0.238231\pi$
$739$			36.9974i			1.36097i	−0.732761	+	0.680486i	$0.761769\pi$
$740$		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
$745$	−11.2054			−0.410536
$746$		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	$-0.516801\pi$
$751$	−2.89161			−0.105516			−0.0527581	+	0.998607i	$0.516801\pi$
$752$		0			0
$753$	5.41681			0.197400
$754$		0			0
$755$	−8.28084			−0.301371
$756$		0			0
$757$	3.63180			0.132000			0.0660001	−	0.997820i	$-0.478976\pi$
$757$	3.63180			0.132000			0.0660001	+	0.997820i	$0.478976\pi$
$758$		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
$765$		−	19.5266i		−	0.705986i
$766$		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	$-0.644896\pi$
$773$		−	49.9435i		−	1.79634i	0.439646	−	0.898171i	$-0.355104\pi$
$774$		0			0
$775$		−	5.40612i		−	0.194193i
$776$		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
$785$			0.899738i			0.0321130i
$786$		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
$795$			2.98724i			0.105947i
$796$		0			0
$797$	−8.22633			−0.291392			−0.145696	−	0.989329i	$-0.546542\pi$
$797$	−8.22633			−0.291392			−0.145696	+	0.989329i	$0.546542\pi$
$798$		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
$805$	6.50573			0.229297
$806$		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	$-0.793067\pi$
$811$		−	34.4735i		−	1.21053i	0.796024	−	0.605265i	$-0.206933\pi$
$812$		0			0
$813$		−	2.90109i		−	0.101746i
$814$		0			0
$815$	−14.5954			−0.511254
$816$		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
$825$		−	0.956248i		−	0.0332923i
$826$		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
$835$	2.40270			0.0831488
$836$		0			0
$837$			11.7187i			0.405057i
$838$		0			0
$839$			30.3086i			1.04637i	0.852220	+	0.523184i	$0.175256\pi$
$839$			30.3086i			1.04637i	−0.852220	+	0.523184i	$0.824744\pi$
$840$		0			0
$841$	14.5959			0.503308
$842$		0			0
$843$		−	6.26652i		−	0.215830i
$844$		0			0
$845$	2.65190	−	12.7266i	0.0912280	−	0.437810i
$846$		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	$-0.997989\pi$
$853$		−	0.369052i		−	0.0126361i	0.999980	−	0.00631805i	$-0.00201111\pi$
$854$		0			0
$855$	−15.7300			−0.537955
$856$		0			0
$857$	−25.2163			−0.861372			−0.430686	−	0.902502i	$-0.641728\pi$
$857$	−25.2163			−0.861372			−0.430686	+	0.902502i	$0.641728\pi$
$858$		0			0
$859$	−15.6500			−0.533971			−0.266985	−	0.963701i	$-0.586028\pi$
$859$	−15.6500			−0.533971			−0.266985	+	0.963701i	$0.586028\pi$
$860$		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
$865$		−	1.99266i		−	0.0677526i
$866$		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
$875$	−0.956248			−0.0323271
$876$		0			0
$877$		−	12.5859i		−	0.424996i	−0.977162	−	0.212498i	$-0.931840\pi$
$877$		−	12.5859i		−	0.424996i	0.977162	−	0.212498i	$-0.0681598\pi$
$878$		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
$885$	2.30435			0.0774599
$886$		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
$895$			10.5535i			0.352766i
$896$		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
$905$			4.51522i			0.150091i
$906$		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	$-0.641611\pi$
$911$	−25.9785			−0.860706			−0.430353	+	0.902661i	$0.641611\pi$
$912$		0			0
$913$	−25.0783			−0.829969
$914$		0			0
$915$		−	5.28561i		−	0.174737i
$916$		0			0
$917$			17.2821i			0.570705i
$918$		0			0
$919$	−18.6492			−0.615180			−0.307590	−	0.951519i	$-0.599522\pi$
$919$	−18.6492			−0.615180			−0.307590	+	0.951519i	$0.599522\pi$
$920$		0			0
$921$		−	2.93671i		−	0.0967680i
$922$		0			0
$923$	−32.4675	−	3.34675i	−1.06868	−	0.110160i
$924$		0			0
$925$			3.69565i			0.121512i
$926$		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
$935$	17.6391			0.576862
$936$		0			0
$937$	−50.5794			−1.65236			−0.826178	−	0.563409i	$-0.809489\pi$
$937$	−50.5794			−1.65236			−0.826178	+	0.563409i	$0.809489\pi$
$938$		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
$945$	2.07283			0.0674292
$946$		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	$-0.494668\pi$
$953$	1.03426			0.0335031			0.0167516	+	0.999860i	$0.494668\pi$
$954$		0			0
$955$		−	8.00000i		−	0.258874i
$956$		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
$965$	−13.2606			−0.426874
$966$		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	$-0.344368\pi$
$971$	29.2714			0.939365			0.469683	+	0.882835i	$0.344368\pi$
$972$		0			0
$973$			21.1071i			0.676662i
$974$		0			0
$975$	−0.136678	+	1.32595i	−0.00437721	+	0.0424643i
$976$		0			0
$977$			9.41145i			0.301099i	0.988602	+	0.150549i	$0.0481043\pi$
$977$			9.41145i			0.301099i	−0.988602	+	0.150549i	$0.951896\pi$
$978$		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
$985$	−3.34810			−0.106679
$986$		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	$-0.431094\pi$
$991$	13.5231			0.429576			0.214788	+	0.976661i	$0.431094\pi$
$992$		0			0
$993$			2.98176i			0.0946234i
$994$		0			0
$995$		−	10.5535i		−	0.334570i
$996$		0			0
$997$	−38.1530			−1.20832			−0.604159	−	0.796864i	$-0.706491\pi$
$997$	−38.1530			−1.20832			−0.604159	+	0.796864i	$0.706491\pi$
$998$		0			0
$999$		−	8.01096i		−	0.253455i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.k.e.961.6		8
4.3	odd	2		520.2.k.b.441.4	yes	8
12.11	even	2		4680.2.g.k.2521.3		8
13.12	even	2	inner	1040.2.k.e.961.5		8
20.3	even	4		2600.2.f.e.649.5		8
20.7	even	4		2600.2.f.f.649.4		8
20.19	odd	2		2600.2.k.c.2001.5		8
52.31	even	4		6760.2.a.bd.1.2		4
52.47	even	4		6760.2.a.bc.1.2		4
52.51	odd	2		520.2.k.b.441.3	✓	8
156.155	even	2		4680.2.g.k.2521.6		8
260.103	even	4		2600.2.f.f.649.5		8
260.207	even	4		2600.2.f.e.649.4		8
260.259	odd	2		2600.2.k.c.2001.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.b.441.3	✓	8	52.51	odd	2
520.2.k.b.441.4	yes	8	4.3	odd	2
1040.2.k.e.961.5		8	13.12	even	2	inner
1040.2.k.e.961.6		8	1.1	even	1	trivial
2600.2.f.e.649.4		8	260.207	even	4
2600.2.f.e.649.5		8	20.3	even	4
2600.2.f.f.649.4		8	20.7	even	4
2600.2.f.f.649.5		8	260.103	even	4
2600.2.k.c.2001.5		8	20.19	odd	2
2600.2.k.c.2001.6		8	260.259	odd	2
4680.2.g.k.2521.3		8	12.11	even	2
4680.2.g.k.2521.6		8	156.155	even	2
6760.2.a.bc.1.2		4	52.47	even	4
6760.2.a.bd.1.2		4	52.31	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.369700 q^{3} +1.00000i q^{5} -0.956248i q^{7} -2.86332 q^{9} +O(q^{10})\)	\(q-0.369700 q^{3} +1.00000i q^{5} -0.956248i q^{7} -2.86332 q^{9} -2.58655i q^{11} +(-0.369700 + 3.58655i) q^{13} -0.369700i q^{15} +6.81957 q^{17} -5.49362i q^{19} +0.353525i q^{21} +6.80339 q^{23} -1.00000 q^{25} +2.16767 q^{27} +6.60272 q^{29} +5.40612i q^{31} +0.956248i q^{33} +0.956248 q^{35} -3.69565i q^{37} +(0.136678 - 1.32595i) q^{39} +8.08017i q^{41} +0.723226 q^{43} -2.86332i q^{45} -12.6829i q^{47} +6.08559 q^{49} -2.52120 q^{51} -8.08017 q^{53} +2.58655 q^{55} +2.03099i q^{57} +6.23302i q^{59} +14.2970 q^{61} +2.73804i q^{63} +(-3.58655 - 0.369700i) q^{65} +7.50980i q^{67} -2.51522 q^{69} +9.05259i q^{71} -8.12934i q^{73} +0.369700 q^{75} -2.47338 q^{77} +8.43369 q^{79} +7.78857 q^{81} -9.69565i q^{83} +6.81957i q^{85} -2.44103 q^{87} -2.08750i q^{89} +(3.42963 + 0.353525i) q^{91} -1.99864i q^{93} +5.49362 q^{95} -10.4337i q^{97} +7.40612i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 16 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 + 16 * q^9	\( 8 q - 4 q^{3} + 16 q^{9} - 4 q^{13} - 4 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 12 q^{35} + 40 q^{39} + 24 q^{43} - 32 q^{49} - 16 q^{51} - 4 q^{53} + 32 q^{61} - 8 q^{65} + 56 q^{69} + 4 q^{75} - 44 q^{77} + 24 q^{79} + 64 q^{81} - 76 q^{87} + 32 q^{91} + 4 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 + 16 * q^9 - 4 * q^13 - 4 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 + 16 * q^29 - 12 * q^35 + 40 * q^39 + 24 * q^43 - 32 * q^49 - 16 * q^51 - 4 * q^53 + 32 * q^61 - 8 * q^65 + 56 * q^69 + 4 * q^75 - 44 * q^77 + 24 * q^79 + 64 * q^81 - 76 * q^87 + 32 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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