LMFDB - Embedded newform 3150.2.b.b.251.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(251,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3150.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.b (of order $2$, degree $1$, 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:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.5
Root			$-0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	3150.251
Dual form			3150.2.b.b.251.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-1.41421 - 2.23607i) q^{7} -1.00000i q^{8} +5.65685i q^{11} -4.47214i q^{13} +(2.23607 - 1.41421i) q^{14} +1.00000 q^{16} -3.16228 q^{17} -3.16228i q^{19} -5.65685 q^{22} +4.00000i q^{23} +4.47214 q^{26} +(1.41421 + 2.23607i) q^{28} -2.82843i q^{29} +6.32456i q^{31} +1.00000i q^{32} -3.16228i q^{34} +9.89949 q^{37} +3.16228 q^{38} -4.47214 q^{41} -1.41421 q^{43} -5.65685i q^{44} -4.00000 q^{46} +9.48683 q^{47} +(-3.00000 + 6.32456i) q^{49} +4.47214i q^{52} +4.00000i q^{53} +(-2.23607 + 1.41421i) q^{56} +2.82843 q^{58} +4.47214 q^{59} +9.48683i q^{61} -6.32456 q^{62} -1.00000 q^{64} +7.07107 q^{67} +3.16228 q^{68} +1.41421i q^{71} -13.4164i q^{73} +9.89949i q^{74} +3.16228i q^{76} +(12.6491 - 8.00000i) q^{77} -6.00000 q^{79} -4.47214i q^{82} +12.6491 q^{83} -1.41421i q^{86} +5.65685 q^{88} -4.47214 q^{89} +(-10.0000 + 6.32456i) q^{91} -4.00000i q^{92} +9.48683i q^{94} +(-6.32456 - 3.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 8 q^{16} - 32 q^{46} - 24 q^{49} - 8 q^{64} - 48 q^{79} - 80 q^{91}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 - 32 * q^46 - 24 * q^49 - 8 * q^64 - 48 * q^79 - 80 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.41421	−	2.23607i	−0.534522	−	0.845154i
$7$	−1.41421	−	2.23607i	−0.534522	−	0.845154i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		0			0
$11$			5.65685i			1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$			5.65685i			1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$		−	4.47214i		−	1.24035i	−0.784465	−	0.620174i	$-0.787062\pi$
$13$		−	4.47214i		−	1.24035i	0.784465	−	0.620174i	$-0.212938\pi$
$14$	2.23607	−	1.41421i	0.597614	−	0.377964i
$15$		0			0
$16$	1.00000			0.250000
$17$	−3.16228			−0.766965			−0.383482	−	0.923548i	$-0.625275\pi$
$17$	−3.16228			−0.766965			−0.383482	+	0.923548i	$0.625275\pi$
$18$		0			0
$19$		−	3.16228i		−	0.725476i	−0.931891	−	0.362738i	$-0.881842\pi$
$19$		−	3.16228i		−	0.725476i	0.931891	−	0.362738i	$-0.118158\pi$
$20$		0			0
$21$		0			0
$22$	−5.65685			−1.20605
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$		0			0
$26$	4.47214			0.877058
$27$		0			0
$28$	1.41421	+	2.23607i	0.267261	+	0.422577i
$29$		−	2.82843i		−	0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$		−	2.82843i		−	0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$		0			0
$31$			6.32456i			1.13592i	0.823055	+	0.567962i	$0.192268\pi$
$31$			6.32456i			1.13592i	−0.823055	+	0.567962i	$0.807732\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	3.16228i		−	0.542326i
$35$		0			0
$36$		0			0
$37$	9.89949			1.62747			0.813733	−	0.581238i	$-0.197432\pi$
$37$	9.89949			1.62747			0.813733	+	0.581238i	$0.197432\pi$
$38$	3.16228			0.512989
$39$		0			0
$40$		0			0
$41$	−4.47214			−0.698430			−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214			−0.698430			−0.349215	+	0.937043i	$0.613552\pi$
$42$		0			0
$43$	−1.41421			−0.215666			−0.107833	−	0.994169i	$-0.534391\pi$
$43$	−1.41421			−0.215666			−0.107833	+	0.994169i	$0.534391\pi$
$44$		−	5.65685i		−	0.852803i
$45$		0			0
$46$	−4.00000			−0.589768
$47$	9.48683			1.38380			0.691898	−	0.721995i	$-0.256775\pi$
$47$	9.48683			1.38380			0.691898	+	0.721995i	$0.256775\pi$
$48$		0			0
$49$	−3.00000	+	6.32456i	−0.428571	+	0.903508i
$50$		0			0
$51$		0			0
$52$			4.47214i			0.620174i
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$	−2.23607	+	1.41421i	−0.298807	+	0.188982i
$57$		0			0
$58$	2.82843			0.371391
$59$	4.47214			0.582223			0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214			0.582223			0.291111	+	0.956689i	$0.405975\pi$
$60$		0			0
$61$			9.48683i			1.21466i	0.794448	+	0.607332i	$0.207760\pi$
$61$			9.48683i			1.21466i	−0.794448	+	0.607332i	$0.792240\pi$
$62$	−6.32456			−0.803219
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	7.07107			0.863868			0.431934	−	0.901905i	$-0.357831\pi$
$67$	7.07107			0.863868			0.431934	+	0.901905i	$0.357831\pi$
$68$	3.16228			0.383482
$69$		0			0
$70$		0			0
$71$			1.41421i			0.167836i	0.996473	+	0.0839181i	$0.0267434\pi$
$71$			1.41421i			0.167836i	−0.996473	+	0.0839181i	$0.973257\pi$
$72$		0			0
$73$		−	13.4164i		−	1.57027i	−0.619324	−	0.785136i	$-0.712593\pi$
$73$		−	13.4164i		−	1.57027i	0.619324	−	0.785136i	$-0.287407\pi$
$74$			9.89949i			1.15079i
$75$		0			0
$76$			3.16228i			0.362738i
$77$	12.6491	−	8.00000i	1.44150	−	0.911685i
$78$		0			0
$79$	−6.00000			−0.675053			−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000			−0.675053			−0.337526	+	0.941316i	$0.609590\pi$
$80$		0			0
$81$		0			0
$82$		−	4.47214i		−	0.493865i
$83$	12.6491			1.38842			0.694210	−	0.719772i	$-0.255754\pi$
$83$	12.6491			1.38842			0.694210	+	0.719772i	$0.255754\pi$
$84$		0			0
$85$		0			0
$86$		−	1.41421i		−	0.152499i
$87$		0			0
$88$	5.65685			0.603023
$89$	−4.47214			−0.474045			−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214			−0.474045			−0.237023	+	0.971504i	$0.576172\pi$
$90$		0			0
$91$	−10.0000	+	6.32456i	−1.04828	+	0.662994i
$92$		−	4.00000i		−	0.417029i
$93$		0			0
$94$			9.48683i			0.978492i
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−6.32456	−	3.00000i	−0.638877	−	0.303046i
$99$		0			0
$100$		0			0
$101$	17.8885			1.77998			0.889988	−	0.455983i	$-0.150712\pi$
$101$	17.8885			1.77998			0.889988	+	0.455983i	$0.150712\pi$
$102$		0			0
$103$			8.94427i			0.881305i	0.897678	+	0.440653i	$0.145253\pi$
$103$			8.94427i			0.881305i	−0.897678	+	0.440653i	$0.854747\pi$
$104$	−4.47214			−0.438529
$105$		0			0
$106$	−4.00000			−0.388514
$107$			18.0000i			1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$			18.0000i			1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$	−1.41421	−	2.23607i	−0.133631	−	0.211289i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		0			0
$116$			2.82843i			0.262613i
$117$		0			0
$118$			4.47214i			0.411693i
$119$	4.47214	+	7.07107i	0.409960	+	0.648204i
$120$		0			0
$121$	−21.0000			−1.90909
$122$	−9.48683			−0.858898
$123$		0			0
$124$		−	6.32456i		−	0.567962i
$125$		0			0
$126$		0			0
$127$	−14.1421			−1.25491			−0.627456	−	0.778652i	$-0.715904\pi$
$127$	−14.1421			−1.25491			−0.627456	+	0.778652i	$0.715904\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		0			0
$131$	17.8885			1.56293			0.781465	−	0.623949i	$-0.214473\pi$
$131$	17.8885			1.56293			0.781465	+	0.623949i	$0.214473\pi$
$132$		0			0
$133$	−7.07107	+	4.47214i	−0.613139	+	0.387783i
$134$			7.07107i			0.610847i
$135$		0			0
$136$			3.16228i			0.271163i
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$			15.8114i			1.34110i	0.741862	+	0.670552i	$0.233943\pi$
$139$			15.8114i			1.34110i	−0.741862	+	0.670552i	$0.766057\pi$
$140$		0			0
$141$		0			0
$142$	−1.41421			−0.118678
$143$	25.2982			2.11554
$144$		0			0
$145$		0			0
$146$	13.4164			1.11035
$147$		0			0
$148$	−9.89949			−0.813733
$149$			16.9706i			1.39028i	0.718873	+	0.695141i	$0.244658\pi$
$149$			16.9706i			1.39028i	−0.718873	+	0.695141i	$0.755342\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$	−3.16228			−0.256495
$153$		0			0
$154$	8.00000	+	12.6491i	0.644658	+	1.01929i
$155$		0			0
$156$		0			0
$157$			4.47214i			0.356915i	0.983948	+	0.178458i	$0.0571108\pi$
$157$			4.47214i			0.356915i	−0.983948	+	0.178458i	$0.942889\pi$
$158$		−	6.00000i		−	0.477334i
$159$		0			0
$160$		0			0
$161$	8.94427	−	5.65685i	0.704907	−	0.445823i
$162$		0			0
$163$	21.2132			1.66155			0.830773	−	0.556611i	$-0.187899\pi$
$163$	21.2132			1.66155			0.830773	+	0.556611i	$0.187899\pi$
$164$	4.47214			0.349215
$165$		0			0
$166$			12.6491i			0.981761i
$167$	9.48683			0.734113			0.367057	−	0.930199i	$-0.380366\pi$
$167$	9.48683			0.734113			0.367057	+	0.930199i	$0.380366\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$		0			0
$172$	1.41421			0.107833
$173$	−25.2982			−1.92339			−0.961694	−	0.274125i	$-0.911612\pi$
$173$	−25.2982			−1.92339			−0.961694	+	0.274125i	$0.911612\pi$
$174$		0			0
$175$		0			0
$176$			5.65685i			0.426401i
$177$		0			0
$178$		−	4.47214i		−	0.335201i
$179$		−	2.82843i		−	0.211407i	−0.994398	−	0.105703i	$-0.966291\pi$
$179$		−	2.82843i		−	0.211407i	0.994398	−	0.105703i	$-0.0337094\pi$
$180$		0			0
$181$		−	3.16228i		−	0.235050i	−0.993070	−	0.117525i	$-0.962504\pi$
$181$		−	3.16228i		−	0.235050i	0.993070	−	0.117525i	$-0.0374961\pi$
$182$	−6.32456	−	10.0000i	−0.468807	−	0.741249i
$183$		0			0
$184$	4.00000			0.294884
$185$		0			0
$186$		0			0
$187$		−	17.8885i		−	1.30814i
$188$	−9.48683			−0.691898
$189$		0			0
$190$		0			0
$191$		−	1.41421i		−	0.102329i	−0.998690	−	0.0511645i	$-0.983707\pi$
$191$		−	1.41421i		−	0.102329i	0.998690	−	0.0511645i	$-0.0162933\pi$
$192$		0			0
$193$	8.48528			0.610784			0.305392	−	0.952227i	$-0.401213\pi$
$193$	8.48528			0.610784			0.305392	+	0.952227i	$0.401213\pi$
$194$		0			0
$195$		0			0
$196$	3.00000	−	6.32456i	0.214286	−	0.451754i
$197$			12.0000i			0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$			12.0000i			0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$			17.8885i			1.25863i
$203$	−6.32456	+	4.00000i	−0.443897	+	0.280745i
$204$		0			0
$205$		0			0
$206$	−8.94427			−0.623177
$207$		0			0
$208$		−	4.47214i		−	0.310087i
$209$	17.8885			1.23738
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		−	4.00000i		−	0.274721i
$213$		0			0
$214$	−18.0000			−1.23045
$215$		0			0
$216$		0			0
$217$	14.1421	−	8.94427i	0.960031	−	0.607177i
$218$			10.0000i			0.677285i
$219$		0			0
$220$		0			0
$221$			14.1421i			0.951303i
$222$		0			0
$223$		−	8.94427i		−	0.598953i	−0.954104	−	0.299476i	$-0.903188\pi$
$223$		−	8.94427i		−	0.598953i	0.954104	−	0.299476i	$-0.0968120\pi$
$224$	2.23607	−	1.41421i	0.149404	−	0.0944911i
$225$		0			0
$226$	−6.00000			−0.399114
$227$	−18.9737			−1.25933			−0.629663	−	0.776868i	$-0.716807\pi$
$227$	−18.9737			−1.25933			−0.629663	+	0.776868i	$0.716807\pi$
$228$		0			0
$229$			15.8114i			1.04485i	0.852686	+	0.522423i	$0.174972\pi$
$229$			15.8114i			1.04485i	−0.852686	+	0.522423i	$0.825028\pi$
$230$		0			0
$231$		0			0
$232$	−2.82843			−0.185695
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$		0			0
$236$	−4.47214			−0.291111
$237$		0			0
$238$	−7.07107	+	4.47214i	−0.458349	+	0.289886i
$239$		−	4.24264i		−	0.274434i	−0.990541	−	0.137217i	$-0.956184\pi$
$239$		−	4.24264i		−	0.274434i	0.990541	−	0.137217i	$-0.0438157\pi$
$240$		0			0
$241$			25.2982i			1.62960i	0.579741	+	0.814801i	$0.303154\pi$
$241$			25.2982i			1.62960i	−0.579741	+	0.814801i	$0.696846\pi$
$242$		−	21.0000i		−	1.34993i
$243$		0			0
$244$		−	9.48683i		−	0.607332i
$245$		0			0
$246$		0			0
$247$	−14.1421			−0.899843
$248$	6.32456			0.401610
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−22.6274			−1.42257
$254$		−	14.1421i		−	0.887357i
$255$		0			0
$256$	1.00000			0.0625000
$257$	15.8114			0.986287			0.493144	−	0.869948i	$-0.335848\pi$
$257$	15.8114			0.986287			0.493144	+	0.869948i	$0.335848\pi$
$258$		0			0
$259$	−14.0000	−	22.1359i	−0.869918	−	1.37546i
$260$		0			0
$261$		0			0
$262$			17.8885i			1.10516i
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$		0			0
$266$	−4.47214	−	7.07107i	−0.274204	−	0.433555i
$267$		0			0
$268$	−7.07107			−0.431934
$269$	−17.8885			−1.09068			−0.545342	−	0.838214i	$-0.683600\pi$
$269$	−17.8885			−1.09068			−0.545342	+	0.838214i	$0.683600\pi$
$270$		0			0
$271$			6.32456i			0.384189i	0.981376	+	0.192095i	$0.0615281\pi$
$271$			6.32456i			0.384189i	−0.981376	+	0.192095i	$0.938472\pi$
$272$	−3.16228			−0.191741
$273$		0			0
$274$	−2.00000			−0.120824
$275$		0			0
$276$		0			0
$277$	21.2132			1.27458			0.637289	−	0.770625i	$-0.280056\pi$
$277$	21.2132			1.27458			0.637289	+	0.770625i	$0.280056\pi$
$278$	−15.8114			−0.948304
$279$		0			0
$280$		0			0
$281$			12.7279i			0.759284i	0.925133	+	0.379642i	$0.123953\pi$
$281$			12.7279i			0.759284i	−0.925133	+	0.379642i	$0.876047\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		−	1.41421i		−	0.0839181i
$285$		0			0
$286$			25.2982i			1.49592i
$287$	6.32456	+	10.0000i	0.373327	+	0.590281i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		0			0
$292$			13.4164i			0.785136i
$293$	−6.32456			−0.369484			−0.184742	−	0.982787i	$-0.559145\pi$
$293$	−6.32456			−0.369484			−0.184742	+	0.982787i	$0.559145\pi$
$294$		0			0
$295$		0			0
$296$		−	9.89949i		−	0.575396i
$297$		0			0
$298$	−16.9706			−0.983078
$299$	17.8885			1.03452
$300$		0			0
$301$	2.00000	+	3.16228i	0.115278	+	0.182271i
$302$			10.0000i			0.575435i
$303$		0			0
$304$		−	3.16228i		−	0.181369i
$305$		0			0
$306$		0			0
$307$		−	26.8328i		−	1.53143i	−0.643180	−	0.765715i	$-0.722385\pi$
$307$		−	26.8328i		−	1.53143i	0.643180	−	0.765715i	$-0.277615\pi$
$308$	−12.6491	+	8.00000i	−0.720750	+	0.455842i
$309$		0			0
$310$		0			0
$311$	−8.94427			−0.507183			−0.253592	−	0.967311i	$-0.581612\pi$
$311$	−8.94427			−0.507183			−0.253592	+	0.967311i	$0.581612\pi$
$312$		0			0
$313$			13.4164i			0.758340i	0.925327	+	0.379170i	$0.123790\pi$
$313$			13.4164i			0.758340i	−0.925327	+	0.379170i	$0.876210\pi$
$314$	−4.47214			−0.252377
$315$		0			0
$316$	6.00000			0.337526
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$	16.0000			0.895828
$320$		0			0
$321$		0			0
$322$	5.65685	+	8.94427i	0.315244	+	0.498445i
$323$			10.0000i			0.556415i
$324$		0			0
$325$		0			0
$326$			21.2132i			1.17489i
$327$		0			0
$328$			4.47214i			0.246932i
$329$	−13.4164	−	21.2132i	−0.739671	−	1.16952i
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$	−12.6491			−0.694210
$333$		0			0
$334$			9.48683i			0.519096i
$335$		0			0
$336$		0			0
$337$	−25.4558			−1.38667			−0.693334	−	0.720616i	$-0.743859\pi$
$337$	−25.4558			−1.38667			−0.693334	+	0.720616i	$0.743859\pi$
$338$		−	7.00000i		−	0.380750i
$339$		0			0
$340$		0			0
$341$	−35.7771			−1.93744
$342$		0			0
$343$	18.3848	−	2.23607i	0.992685	−	0.120736i
$344$			1.41421i			0.0762493i
$345$		0			0
$346$		−	25.2982i		−	1.36004i
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$			15.8114i			0.846364i	0.906045	+	0.423182i	$0.139087\pi$
$349$			15.8114i			0.846364i	−0.906045	+	0.423182i	$0.860913\pi$
$350$		0			0
$351$		0			0
$352$	−5.65685			−0.301511
$353$	3.16228			0.168311			0.0841555	−	0.996453i	$-0.473181\pi$
$353$	3.16228			0.168311			0.0841555	+	0.996453i	$0.473181\pi$
$354$		0			0
$355$		0			0
$356$	4.47214			0.237023
$357$		0			0
$358$	2.82843			0.149487
$359$		−	18.3848i		−	0.970311i	−0.874428	−	0.485156i	$-0.838763\pi$
$359$		−	18.3848i		−	0.970311i	0.874428	−	0.485156i	$-0.161237\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$	3.16228			0.166206
$363$		0			0
$364$	10.0000	−	6.32456i	0.524142	−	0.331497i
$365$		0			0
$366$		0			0
$367$			13.4164i			0.700331i	0.936688	+	0.350165i	$0.113875\pi$
$367$			13.4164i			0.700331i	−0.936688	+	0.350165i	$0.886125\pi$
$368$			4.00000i			0.208514i
$369$		0			0
$370$		0			0
$371$	8.94427	−	5.65685i	0.464363	−	0.293689i
$372$		0			0
$373$	−21.2132			−1.09838			−0.549189	−	0.835698i	$-0.685063\pi$
$373$	−21.2132			−1.09838			−0.549189	+	0.835698i	$0.685063\pi$
$374$	17.8885			0.924995
$375$		0			0
$376$		−	9.48683i		−	0.489246i
$377$	−12.6491			−0.651462
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	1.41421			0.0723575
$383$	9.48683			0.484755			0.242377	−	0.970182i	$-0.422073\pi$
$383$	9.48683			0.484755			0.242377	+	0.970182i	$0.422073\pi$
$384$		0			0
$385$		0			0
$386$			8.48528i			0.431889i
$387$		0			0
$388$		0			0
$389$			16.9706i			0.860442i	0.902724	+	0.430221i	$0.141564\pi$
$389$			16.9706i			0.860442i	−0.902724	+	0.430221i	$0.858436\pi$
$390$		0			0
$391$		−	12.6491i		−	0.639693i
$392$	6.32456	+	3.00000i	0.319438	+	0.151523i
$393$		0			0
$394$	−12.0000			−0.604551
$395$		0			0
$396$		0			0
$397$		−	4.47214i		−	0.224450i	−0.993683	−	0.112225i	$-0.964202\pi$
$397$		−	4.47214i		−	0.224450i	0.993683	−	0.112225i	$-0.0357978\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.41421i		−	0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$		−	1.41421i		−	0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$		0			0
$403$	28.2843			1.40894
$404$	−17.8885			−0.889988
$405$		0			0
$406$	−4.00000	−	6.32456i	−0.198517	−	0.313882i
$407$			56.0000i			2.77582i
$408$		0			0
$409$		−	37.9473i		−	1.87637i	−0.346128	−	0.938187i	$-0.612504\pi$
$409$		−	37.9473i		−	1.87637i	0.346128	−	0.938187i	$-0.387496\pi$
$410$		0			0
$411$		0			0
$412$		−	8.94427i		−	0.440653i
$413$	−6.32456	−	10.0000i	−0.311211	−	0.492068i
$414$		0			0
$415$		0			0
$416$	4.47214			0.219265
$417$		0			0
$418$			17.8885i			0.874957i
$419$	13.4164			0.655434			0.327717	−	0.944776i	$-0.393721\pi$
$419$	13.4164			0.655434			0.327717	+	0.944776i	$0.393721\pi$
$420$		0			0
$421$	38.0000			1.85201			0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000			1.85201			0.926003	+	0.377515i	$0.123221\pi$
$422$		−	20.0000i		−	0.973585i
$423$		0			0
$424$	4.00000			0.194257
$425$		0			0
$426$		0			0
$427$	21.2132	−	13.4164i	1.02658	−	0.649265i
$428$		−	18.0000i		−	0.870063i
$429$		0			0
$430$		0			0
$431$			15.5563i			0.749323i	0.927162	+	0.374661i	$0.122241\pi$
$431$			15.5563i			0.749323i	−0.927162	+	0.374661i	$0.877759\pi$
$432$		0			0
$433$			26.8328i			1.28950i	0.764392	+	0.644751i	$0.223039\pi$
$433$			26.8328i			1.28950i	−0.764392	+	0.644751i	$0.776961\pi$
$434$	8.94427	+	14.1421i	0.429339	+	0.678844i
$435$		0			0
$436$	−10.0000			−0.478913
$437$	12.6491			0.605089
$438$		0			0
$439$		−	6.32456i		−	0.301855i	−0.988545	−	0.150927i	$-0.951774\pi$
$439$		−	6.32456i		−	0.301855i	0.988545	−	0.150927i	$-0.0482259\pi$
$440$		0			0
$441$		0			0
$442$	−14.1421			−0.672673
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$		0			0
$446$	8.94427			0.423524
$447$		0			0
$448$	1.41421	+	2.23607i	0.0668153	+	0.105644i
$449$		−	9.89949i		−	0.467186i	−0.972334	−	0.233593i	$-0.924952\pi$
$449$		−	9.89949i		−	0.467186i	0.972334	−	0.233593i	$-0.0750483\pi$
$450$		0			0
$451$		−	25.2982i		−	1.19125i
$452$		−	6.00000i		−	0.282216i
$453$		0			0
$454$		−	18.9737i		−	0.890478i
$455$		0			0
$456$		0			0
$457$	−16.9706			−0.793849			−0.396925	−	0.917851i	$-0.629923\pi$
$457$	−16.9706			−0.793849			−0.396925	+	0.917851i	$0.629923\pi$
$458$	−15.8114			−0.738818
$459$		0			0
$460$		0			0
$461$	8.94427			0.416576			0.208288	−	0.978068i	$-0.433211\pi$
$461$	8.94427			0.416576			0.208288	+	0.978068i	$0.433211\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		−	2.82843i		−	0.131306i
$465$		0			0
$466$	14.0000			0.648537
$467$	12.6491			0.585331			0.292666	−	0.956215i	$-0.405458\pi$
$467$	12.6491			0.585331			0.292666	+	0.956215i	$0.405458\pi$
$468$		0			0
$469$	−10.0000	−	15.8114i	−0.461757	−	0.730102i
$470$		0			0
$471$		0			0
$472$		−	4.47214i		−	0.205847i
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$		0			0
$476$	−4.47214	−	7.07107i	−0.204980	−	0.324102i
$477$		0			0
$478$	4.24264			0.194054
$479$	−17.8885			−0.817348			−0.408674	−	0.912680i	$-0.634009\pi$
$479$	−17.8885			−0.817348			−0.408674	+	0.912680i	$0.634009\pi$
$480$		0			0
$481$		−	44.2719i		−	2.01862i
$482$	−25.2982			−1.15230
$483$		0			0
$484$	21.0000			0.954545
$485$		0			0
$486$		0			0
$487$	25.4558			1.15351			0.576757	−	0.816916i	$-0.304318\pi$
$487$	25.4558			1.15351			0.576757	+	0.816916i	$0.304318\pi$
$488$	9.48683			0.429449
$489$		0			0
$490$		0			0
$491$		−	19.7990i		−	0.893516i	−0.894655	−	0.446758i	$-0.852579\pi$
$491$		−	19.7990i		−	0.893516i	0.894655	−	0.446758i	$-0.147421\pi$
$492$		0			0
$493$			8.94427i			0.402830i
$494$		−	14.1421i		−	0.636285i
$495$		0			0
$496$			6.32456i			0.283981i
$497$	3.16228	−	2.00000i	0.141848	−	0.0897123i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9.48683			0.422997			0.211498	−	0.977378i	$-0.432166\pi$
$503$	9.48683			0.422997			0.211498	+	0.977378i	$0.432166\pi$
$504$		0			0
$505$		0			0
$506$		−	22.6274i		−	1.00591i
$507$		0			0
$508$	14.1421			0.627456
$509$	13.4164			0.594672			0.297336	−	0.954773i	$-0.403902\pi$
$509$	13.4164			0.594672			0.297336	+	0.954773i	$0.403902\pi$
$510$		0			0
$511$	−30.0000	+	18.9737i	−1.32712	+	0.839346i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			15.8114i			0.697410i
$515$		0			0
$516$		0			0
$517$			53.6656i			2.36021i
$518$	22.1359	−	14.0000i	0.972598	−	0.615125i
$519$		0			0
$520$		0			0
$521$	−13.4164			−0.587784			−0.293892	−	0.955839i	$-0.594951\pi$
$521$	−13.4164			−0.587784			−0.293892	+	0.955839i	$0.594951\pi$
$522$		0			0
$523$		−	17.8885i		−	0.782211i	−0.920346	−	0.391106i	$-0.872093\pi$
$523$		−	17.8885i		−	0.782211i	0.920346	−	0.391106i	$-0.127907\pi$
$524$	−17.8885			−0.781465
$525$		0			0
$526$	16.0000			0.697633
$527$		−	20.0000i		−	0.871214i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$	7.07107	−	4.47214i	0.306570	−	0.193892i
$533$			20.0000i			0.866296i
$534$		0			0
$535$		0			0
$536$		−	7.07107i		−	0.305424i
$537$		0			0
$538$		−	17.8885i		−	0.771230i
$539$	−35.7771	−	16.9706i	−1.54103	−	0.730974i
$540$		0			0
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$	−6.32456			−0.271663
$543$		0			0
$544$		−	3.16228i		−	0.135582i
$545$		0			0
$546$		0			0
$547$	−21.2132			−0.907011			−0.453506	−	0.891253i	$-0.649827\pi$
$547$	−21.2132			−0.907011			−0.453506	+	0.891253i	$0.649827\pi$
$548$		−	2.00000i		−	0.0854358i
$549$		0			0
$550$		0			0
$551$	−8.94427			−0.381039
$552$		0			0
$553$	8.48528	+	13.4164i	0.360831	+	0.570524i
$554$			21.2132i			0.901263i
$555$		0			0
$556$		−	15.8114i		−	0.670552i
$557$		−	38.0000i		−	1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$		−	38.0000i		−	1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$		0			0
$559$			6.32456i			0.267500i
$560$		0			0
$561$		0			0
$562$	−12.7279			−0.536895
$563$	−18.9737			−0.799645			−0.399822	−	0.916593i	$-0.630928\pi$
$563$	−18.9737			−0.799645			−0.399822	+	0.916593i	$0.630928\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.41421			0.0593391
$569$			24.0416i			1.00788i	0.863739	+	0.503939i	$0.168116\pi$
$569$			24.0416i			1.00788i	−0.863739	+	0.503939i	$0.831884\pi$
$570$		0			0
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$	−25.2982			−1.05777
$573$		0			0
$574$	−10.0000	+	6.32456i	−0.417392	+	0.263982i
$575$		0			0
$576$		0			0
$577$		−	22.3607i		−	0.930887i	−0.885078	−	0.465444i	$-0.845895\pi$
$577$		−	22.3607i		−	0.930887i	0.885078	−	0.465444i	$-0.154105\pi$
$578$		−	7.00000i		−	0.291162i
$579$		0			0
$580$		0			0
$581$	−17.8885	−	28.2843i	−0.742142	−	1.17343i
$582$		0			0
$583$	−22.6274			−0.937132
$584$	−13.4164			−0.555175
$585$		0			0
$586$		−	6.32456i		−	0.261265i
$587$	31.6228			1.30521			0.652606	−	0.757698i	$-0.273676\pi$
$587$	31.6228			1.30521			0.652606	+	0.757698i	$0.273676\pi$
$588$		0			0
$589$	20.0000			0.824086
$590$		0			0
$591$		0			0
$592$	9.89949			0.406867
$593$	9.48683			0.389578			0.194789	−	0.980845i	$-0.437598\pi$
$593$	9.48683			0.389578			0.194789	+	0.980845i	$0.437598\pi$
$594$		0			0
$595$		0			0
$596$		−	16.9706i		−	0.695141i
$597$		0			0
$598$			17.8885i			0.731517i
$599$			32.5269i			1.32901i	0.747282	+	0.664507i	$0.231358\pi$
$599$			32.5269i			1.32901i	−0.747282	+	0.664507i	$0.768642\pi$
$600$		0			0
$601$		−	25.2982i		−	1.03194i	−0.856608	−	0.515968i	$-0.827432\pi$
$601$		−	25.2982i		−	1.03194i	0.856608	−	0.515968i	$-0.172568\pi$
$602$	−3.16228	+	2.00000i	−0.128885	+	0.0815139i
$603$		0			0
$604$	−10.0000			−0.406894
$605$		0			0
$606$		0			0
$607$			8.94427i			0.363037i	0.983388	+	0.181518i	$0.0581012\pi$
$607$			8.94427i			0.363037i	−0.983388	+	0.181518i	$0.941899\pi$
$608$	3.16228			0.128247
$609$		0			0
$610$		0			0
$611$		−	42.4264i		−	1.71639i
$612$		0			0
$613$	−35.3553			−1.42799			−0.713994	−	0.700151i	$-0.753116\pi$
$613$	−35.3553			−1.42799			−0.713994	+	0.700151i	$0.753116\pi$
$614$	26.8328			1.08288
$615$		0			0
$616$	−8.00000	−	12.6491i	−0.322329	−	0.509647i
$617$			38.0000i			1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$			38.0000i			1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$			28.4605i			1.14392i	0.820280	+	0.571962i	$0.193818\pi$
$619$			28.4605i			1.14392i	−0.820280	+	0.571962i	$0.806182\pi$
$620$		0			0
$621$		0			0
$622$		−	8.94427i		−	0.358633i
$623$	6.32456	+	10.0000i	0.253388	+	0.400642i
$624$		0			0
$625$		0			0
$626$	−13.4164			−0.536228
$627$		0			0
$628$		−	4.47214i		−	0.178458i
$629$	−31.3050			−1.24821
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$			6.00000i			0.238667i
$633$		0			0
$634$	18.0000			0.714871
$635$		0			0
$636$		0			0
$637$	28.2843	+	13.4164i	1.12066	+	0.531577i
$638$			16.0000i			0.633446i
$639$		0			0
$640$		0			0
$641$			1.41421i			0.0558581i	0.999610	+	0.0279290i	$0.00889125\pi$
$641$			1.41421i			0.0558581i	−0.999610	+	0.0279290i	$0.991109\pi$
$642$		0			0
$643$		−	44.7214i		−	1.76364i	−0.471588	−	0.881819i	$-0.656319\pi$
$643$		−	44.7214i		−	1.76364i	0.471588	−	0.881819i	$-0.343681\pi$
$644$	−8.94427	+	5.65685i	−0.352454	+	0.222911i
$645$		0			0
$646$	−10.0000			−0.393445
$647$	34.7851			1.36754			0.683771	−	0.729697i	$-0.260339\pi$
$647$	34.7851			1.36754			0.683771	+	0.729697i	$0.260339\pi$
$648$		0			0
$649$			25.2982i			0.993042i
$650$		0			0
$651$		0			0
$652$	−21.2132			−0.830773
$653$			14.0000i			0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$			14.0000i			0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$		0			0
$655$		0			0
$656$	−4.47214			−0.174608
$657$		0			0
$658$	21.2132	−	13.4164i	0.826977	−	0.523026i
$659$			16.9706i			0.661079i	0.943792	+	0.330540i	$0.107231\pi$
$659$			16.9706i			0.661079i	−0.943792	+	0.330540i	$0.892769\pi$
$660$		0			0
$661$			3.16228i			0.122998i	0.998107	+	0.0614992i	$0.0195882\pi$
$661$			3.16228i			0.122998i	−0.998107	+	0.0614992i	$0.980412\pi$
$662$			20.0000i			0.777322i
$663$		0			0
$664$		−	12.6491i		−	0.490881i
$665$		0			0
$666$		0			0
$667$	11.3137			0.438069
$668$	−9.48683			−0.367057
$669$		0			0
$670$		0			0
$671$	−53.6656			−2.07174
$672$		0			0
$673$	−19.7990			−0.763195			−0.381597	−	0.924329i	$-0.624626\pi$
$673$	−19.7990			−0.763195			−0.381597	+	0.924329i	$0.624626\pi$
$674$		−	25.4558i		−	0.980522i
$675$		0			0
$676$	7.00000			0.269231
$677$	−37.9473			−1.45843			−0.729217	−	0.684282i	$-0.760116\pi$
$677$	−37.9473			−1.45843			−0.729217	+	0.684282i	$0.760116\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		−	35.7771i		−	1.36998i
$683$			14.0000i			0.535695i	0.963461	+	0.267848i	$0.0863124\pi$
$683$			14.0000i			0.535695i	−0.963461	+	0.267848i	$0.913688\pi$
$684$		0			0
$685$		0			0
$686$	2.23607	+	18.3848i	0.0853735	+	0.701934i
$687$		0			0
$688$	−1.41421			−0.0539164
$689$	17.8885			0.681499
$690$		0			0
$691$		−	28.4605i		−	1.08269i	−0.840801	−	0.541344i	$-0.817916\pi$
$691$		−	28.4605i		−	1.08269i	0.840801	−	0.541344i	$-0.182084\pi$
$692$	25.2982			0.961694
$693$		0			0
$694$	12.0000			0.455514
$695$		0			0
$696$		0			0
$697$	14.1421			0.535672
$698$	−15.8114			−0.598470
$699$		0			0
$700$		0			0
$701$		−	19.7990i		−	0.747798i	−0.927470	−	0.373899i	$-0.878021\pi$
$701$		−	19.7990i		−	0.747798i	0.927470	−	0.373899i	$-0.121979\pi$
$702$		0			0
$703$		−	31.3050i		−	1.18069i
$704$		−	5.65685i		−	0.213201i
$705$		0			0
$706$			3.16228i			0.119014i
$707$	−25.2982	−	40.0000i	−0.951438	−	1.50435i
$708$		0			0
$709$	30.0000			1.12667			0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000			1.12667			0.563337	+	0.826227i	$0.309517\pi$
$710$		0			0
$711$		0			0
$712$			4.47214i			0.167600i
$713$	−25.2982			−0.947426
$714$		0			0
$715$		0			0
$716$			2.82843i			0.105703i
$717$		0			0
$718$	18.3848			0.686114
$719$	−44.7214			−1.66783			−0.833913	−	0.551896i	$-0.813904\pi$
$719$	−44.7214			−1.66783			−0.833913	+	0.551896i	$0.813904\pi$
$720$		0			0
$721$	20.0000	−	12.6491i	0.744839	−	0.471077i
$722$			9.00000i			0.334945i
$723$		0			0
$724$			3.16228i			0.117525i
$725$		0			0
$726$		0			0
$727$		−	26.8328i		−	0.995174i	−0.867414	−	0.497587i	$-0.834220\pi$
$727$		−	26.8328i		−	0.995174i	0.867414	−	0.497587i	$-0.165780\pi$
$728$	6.32456	+	10.0000i	0.234404	+	0.370625i
$729$		0			0
$730$		0			0
$731$	4.47214			0.165408
$732$		0			0
$733$		−	40.2492i		−	1.48664i	−0.668937	−	0.743319i	$-0.733250\pi$
$733$		−	40.2492i		−	1.48664i	0.668937	−	0.743319i	$-0.266750\pi$
$734$	−13.4164			−0.495209
$735$		0			0
$736$	−4.00000			−0.147442
$737$			40.0000i			1.47342i
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$	5.65685	+	8.94427i	0.207670	+	0.328355i
$743$			36.0000i			1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$			36.0000i			1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$		0			0
$745$		0			0
$746$		−	21.2132i		−	0.776671i
$747$		0			0
$748$			17.8885i			0.654070i
$749$	40.2492	−	25.4558i	1.47067	−	0.930136i
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$	9.48683			0.345949
$753$		0			0
$754$		−	12.6491i		−	0.460653i
$755$		0			0
$756$		0			0
$757$	−32.5269			−1.18221			−0.591105	−	0.806594i	$-0.701308\pi$
$757$	−32.5269			−1.18221			−0.591105	+	0.806594i	$0.701308\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31.3050			1.13480			0.567402	−	0.823441i	$-0.307949\pi$
$761$	31.3050			1.13480			0.567402	+	0.823441i	$0.307949\pi$
$762$		0			0
$763$	−14.1421	−	22.3607i	−0.511980	−	0.809511i
$764$			1.41421i			0.0511645i
$765$		0			0
$766$			9.48683i			0.342773i
$767$		−	20.0000i		−	0.722158i
$768$		0			0
$769$		−	31.6228i		−	1.14035i	−0.821524	−	0.570173i	$-0.806876\pi$
$769$		−	31.6228i		−	1.14035i	0.821524	−	0.570173i	$-0.193124\pi$
$770$		0			0
$771$		0			0
$772$	−8.48528			−0.305392
$773$	−31.6228			−1.13739			−0.568696	−	0.822548i	$-0.692552\pi$
$773$	−31.6228			−1.13739			−0.568696	+	0.822548i	$0.692552\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−16.9706			−0.608424
$779$			14.1421i			0.506695i
$780$		0			0
$781$	−8.00000			−0.286263
$782$	12.6491			0.452331
$783$		0			0
$784$	−3.00000	+	6.32456i	−0.107143	+	0.225877i
$785$		0			0
$786$		0			0
$787$		−	26.8328i		−	0.956487i	−0.878227	−	0.478243i	$-0.841274\pi$
$787$		−	26.8328i		−	0.956487i	0.878227	−	0.478243i	$-0.158726\pi$
$788$		−	12.0000i		−	0.427482i
$789$		0			0
$790$		0			0
$791$	13.4164	−	8.48528i	0.477033	−	0.301702i
$792$		0			0
$793$	42.4264			1.50661
$794$	4.47214			0.158710
$795$		0			0
$796$		0			0
$797$	−25.2982			−0.896109			−0.448054	−	0.894006i	$-0.647883\pi$
$797$	−25.2982			−0.896109			−0.448054	+	0.894006i	$0.647883\pi$
$798$		0			0
$799$	−30.0000			−1.06132
$800$		0			0
$801$		0			0
$802$	1.41421			0.0499376
$803$	75.8947			2.67826
$804$		0			0
$805$		0			0
$806$			28.2843i			0.996271i
$807$		0			0
$808$		−	17.8885i		−	0.629317i
$809$		−	18.3848i		−	0.646374i	−0.946335	−	0.323187i	$-0.895246\pi$
$809$		−	18.3848i		−	0.646374i	0.946335	−	0.323187i	$-0.104754\pi$
$810$		0			0
$811$		−	22.1359i		−	0.777298i	−0.921386	−	0.388649i	$-0.872942\pi$
$811$		−	22.1359i		−	0.777298i	0.921386	−	0.388649i	$-0.127058\pi$
$812$	6.32456	−	4.00000i	0.221948	−	0.140372i
$813$		0			0
$814$	−56.0000			−1.96280
$815$		0			0
$816$		0			0
$817$			4.47214i			0.156460i
$818$	37.9473			1.32680
$819$		0			0
$820$		0			0
$821$			19.7990i			0.690990i	0.938421	+	0.345495i	$0.112289\pi$
$821$			19.7990i			0.690990i	−0.938421	+	0.345495i	$0.887711\pi$
$822$		0			0
$823$	28.2843			0.985928			0.492964	−	0.870050i	$-0.335913\pi$
$823$	28.2843			0.985928			0.492964	+	0.870050i	$0.335913\pi$
$824$	8.94427			0.311588
$825$		0			0
$826$	10.0000	−	6.32456i	0.347945	−	0.220059i
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		0			0
$829$		−	41.1096i		−	1.42780i	−0.700250	−	0.713898i	$-0.746928\pi$
$829$		−	41.1096i		−	1.42780i	0.700250	−	0.713898i	$-0.253072\pi$
$830$		0			0
$831$		0			0
$832$			4.47214i			0.155043i
$833$	9.48683	−	20.0000i	0.328699	−	0.692959i
$834$		0			0
$835$		0			0
$836$	−17.8885			−0.618688
$837$		0			0
$838$			13.4164i			0.463462i
$839$	17.8885			0.617581			0.308791	−	0.951130i	$-0.400076\pi$
$839$	17.8885			0.617581			0.308791	+	0.951130i	$0.400076\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$			38.0000i			1.30957i
$843$		0			0
$844$	20.0000			0.688428
$845$		0			0
$846$		0			0
$847$	29.6985	+	46.9574i	1.02045	+	1.61348i
$848$			4.00000i			0.137361i
$849$		0			0
$850$		0			0
$851$			39.5980i			1.35740i
$852$		0			0
$853$			49.1935i			1.68435i	0.539202	+	0.842177i	$0.318726\pi$
$853$			49.1935i			1.68435i	−0.539202	+	0.842177i	$0.681274\pi$
$854$	13.4164	+	21.2132i	0.459100	+	0.725901i
$855$		0			0
$856$	18.0000			0.615227
$857$	41.1096			1.40428			0.702139	−	0.712040i	$-0.252229\pi$
$857$	41.1096			1.40428			0.702139	+	0.712040i	$0.252229\pi$
$858$		0			0
$859$		−	34.7851i		−	1.18685i	−0.804889	−	0.593425i	$-0.797775\pi$
$859$		−	34.7851i		−	1.18685i	0.804889	−	0.593425i	$-0.202225\pi$
$860$		0			0
$861$		0			0
$862$	−15.5563			−0.529851
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$		0			0
$866$	−26.8328			−0.911816
$867$		0			0
$868$	−14.1421	+	8.94427i	−0.480015	+	0.303588i
$869$		−	33.9411i		−	1.15137i
$870$		0			0
$871$		−	31.6228i		−	1.07150i
$872$		−	10.0000i		−	0.338643i
$873$		0			0
$874$			12.6491i			0.427863i
$875$		0			0
$876$		0			0
$877$	−21.2132			−0.716319			−0.358159	−	0.933660i	$-0.616596\pi$
$877$	−21.2132			−0.716319			−0.358159	+	0.933660i	$0.616596\pi$
$878$	6.32456			0.213443
$879$		0			0
$880$		0			0
$881$	−4.47214			−0.150670			−0.0753350	−	0.997158i	$-0.524003\pi$
$881$	−4.47214			−0.150670			−0.0753350	+	0.997158i	$0.524003\pi$
$882$		0			0
$883$	−12.7279			−0.428329			−0.214164	−	0.976798i	$-0.568703\pi$
$883$	−12.7279			−0.428329			−0.214164	+	0.976798i	$0.568703\pi$
$884$		−	14.1421i		−	0.475651i
$885$		0			0
$886$	4.00000			0.134383
$887$	−9.48683			−0.318537			−0.159268	−	0.987235i	$-0.550914\pi$
$887$	−9.48683			−0.318537			−0.159268	+	0.987235i	$0.550914\pi$
$888$		0			0
$889$	20.0000	+	31.6228i	0.670778	+	1.06059i
$890$		0			0
$891$		0			0
$892$			8.94427i			0.299476i
$893$		−	30.0000i		−	1.00391i
$894$		0			0
$895$		0			0
$896$	−2.23607	+	1.41421i	−0.0747018	+	0.0472456i
$897$		0			0
$898$	9.89949			0.330350
$899$	17.8885			0.596616
$900$		0			0
$901$		−	12.6491i		−	0.421403i
$902$	25.2982			0.842339
$903$		0			0
$904$	6.00000			0.199557
$905$		0			0
$906$		0			0
$907$	46.6690			1.54962			0.774810	−	0.632194i	$-0.217845\pi$
$907$	46.6690			1.54962			0.774810	+	0.632194i	$0.217845\pi$
$908$	18.9737			0.629663
$909$		0			0
$910$		0			0
$911$			43.8406i			1.45250i	0.687428	+	0.726252i	$0.258740\pi$
$911$			43.8406i			1.45250i	−0.687428	+	0.726252i	$0.741260\pi$
$912$		0			0
$913$			71.5542i			2.36810i
$914$		−	16.9706i		−	0.561336i
$915$		0			0
$916$		−	15.8114i		−	0.522423i
$917$	−25.2982	−	40.0000i	−0.835421	−	1.32092i
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		0			0
$922$			8.94427i			0.294564i
$923$	6.32456			0.208175
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	2.82843			0.0928477
$929$	22.3607			0.733630			0.366815	−	0.930294i	$-0.380448\pi$
$929$	22.3607			0.733630			0.366815	+	0.930294i	$0.380448\pi$
$930$		0			0
$931$	20.0000	+	9.48683i	0.655474	+	0.310918i
$932$			14.0000i			0.458585i
$933$		0			0
$934$			12.6491i			0.413892i
$935$		0			0
$936$		0			0
$937$		−	53.6656i		−	1.75318i	−0.481238	−	0.876590i	$-0.659813\pi$
$937$		−	53.6656i		−	1.75318i	0.481238	−	0.876590i	$-0.340187\pi$
$938$	15.8114	−	10.0000i	0.516260	−	0.326512i
$939$		0			0
$940$		0			0
$941$	26.8328			0.874725			0.437362	−	0.899285i	$-0.355913\pi$
$941$	26.8328			0.874725			0.437362	+	0.899285i	$0.355913\pi$
$942$		0			0
$943$		−	17.8885i		−	0.582531i
$944$	4.47214			0.145556
$945$		0			0
$946$	8.00000			0.260102
$947$			42.0000i			1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$			42.0000i			1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$		0			0
$949$	−60.0000			−1.94768
$950$		0			0
$951$		0			0
$952$	7.07107	−	4.47214i	0.229175	−	0.144943i
$953$		−	54.0000i		−	1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$		−	54.0000i		−	1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$		0			0
$955$		0			0
$956$			4.24264i			0.137217i
$957$		0			0
$958$		−	17.8885i		−	0.577953i
$959$	4.47214	−	2.82843i	0.144413	−	0.0913347i
$960$		0			0
$961$	−9.00000			−0.290323
$962$	44.2719			1.42738
$963$		0			0
$964$		−	25.2982i		−	0.814801i
$965$		0			0
$966$		0			0
$967$	−42.4264			−1.36434			−0.682171	−	0.731193i	$-0.738964\pi$
$967$	−42.4264			−1.36434			−0.682171	+	0.731193i	$0.738964\pi$
$968$			21.0000i			0.674966i
$969$		0			0
$970$		0			0
$971$	−35.7771			−1.14814			−0.574071	−	0.818806i	$-0.694637\pi$
$971$	−35.7771			−1.14814			−0.574071	+	0.818806i	$0.694637\pi$
$972$		0			0
$973$	35.3553	−	22.3607i	1.13344	−	0.716850i
$974$			25.4558i			0.815658i
$975$		0			0
$976$			9.48683i			0.303666i
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$		−	25.2982i		−	0.808535i
$980$		0			0
$981$		0			0
$982$	19.7990			0.631811
$983$	9.48683			0.302583			0.151291	−	0.988489i	$-0.451657\pi$
$983$	9.48683			0.302583			0.151291	+	0.988489i	$0.451657\pi$
$984$		0			0
$985$		0			0
$986$	−8.94427			−0.284844
$987$		0			0
$988$	14.1421			0.449921
$989$		−	5.65685i		−	0.179878i
$990$		0			0
$991$	40.0000			1.27064			0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000			1.27064			0.635321	+	0.772248i	$0.280868\pi$
$992$	−6.32456			−0.200805
$993$		0			0
$994$	2.00000	+	3.16228i	0.0634361	+	0.100301i
$995$		0			0
$996$		0			0
$997$			13.4164i			0.424902i	0.977172	+	0.212451i	$0.0681446\pi$
$997$			13.4164i			0.424902i	−0.977172	+	0.212451i	$0.931855\pi$
$998$			20.0000i			0.633089i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.b.b.251.5		8
3.2	odd	2	inner	3150.2.b.b.251.1		8
5.2	odd	4		630.2.d.b.629.1	✓	4
5.3	odd	4		630.2.d.c.629.2	yes	4
5.4	even	2	inner	3150.2.b.b.251.4		8
7.6	odd	2	inner	3150.2.b.b.251.6		8
15.2	even	4		630.2.d.c.629.3	yes	4
15.8	even	4		630.2.d.b.629.4	yes	4
15.14	odd	2	inner	3150.2.b.b.251.8		8
20.3	even	4		5040.2.k.c.1889.1		4
20.7	even	4		5040.2.k.b.1889.2		4
21.20	even	2	inner	3150.2.b.b.251.2		8
35.13	even	4		630.2.d.c.629.4	yes	4
35.27	even	4		630.2.d.b.629.3	yes	4
35.34	odd	2	inner	3150.2.b.b.251.3		8
60.23	odd	4		5040.2.k.b.1889.3		4
60.47	odd	4		5040.2.k.c.1889.4		4
105.62	odd	4		630.2.d.c.629.1	yes	4
105.83	odd	4		630.2.d.b.629.2	yes	4
105.104	even	2	inner	3150.2.b.b.251.7		8
140.27	odd	4		5040.2.k.b.1889.4		4
140.83	odd	4		5040.2.k.c.1889.3		4
420.83	even	4		5040.2.k.b.1889.1		4
420.167	even	4		5040.2.k.c.1889.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.d.b.629.1	✓	4	5.2	odd	4
630.2.d.b.629.2	yes	4	105.83	odd	4
630.2.d.b.629.3	yes	4	35.27	even	4
630.2.d.b.629.4	yes	4	15.8	even	4
630.2.d.c.629.1	yes	4	105.62	odd	4
630.2.d.c.629.2	yes	4	5.3	odd	4
630.2.d.c.629.3	yes	4	15.2	even	4
630.2.d.c.629.4	yes	4	35.13	even	4
3150.2.b.b.251.1		8	3.2	odd	2	inner
3150.2.b.b.251.2		8	21.20	even	2	inner
3150.2.b.b.251.3		8	35.34	odd	2	inner
3150.2.b.b.251.4		8	5.4	even	2	inner
3150.2.b.b.251.5		8	1.1	even	1	trivial
3150.2.b.b.251.6		8	7.6	odd	2	inner
3150.2.b.b.251.7		8	105.104	even	2	inner
3150.2.b.b.251.8		8	15.14	odd	2	inner
5040.2.k.b.1889.1		4	420.83	even	4
5040.2.k.b.1889.2		4	20.7	even	4
5040.2.k.b.1889.3		4	60.23	odd	4
5040.2.k.b.1889.4		4	140.27	odd	4
5040.2.k.c.1889.1		4	20.3	even	4
5040.2.k.c.1889.2		4	420.167	even	4
5040.2.k.c.1889.3		4	140.83	odd	4
5040.2.k.c.1889.4		4	60.47	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-1.41421 - 2.23607i) q^{7} -1.00000i q^{8} +5.65685i q^{11} -4.47214i q^{13} +(2.23607 - 1.41421i) q^{14} +1.00000 q^{16} -3.16228 q^{17} -3.16228i q^{19} -5.65685 q^{22} +4.00000i q^{23} +4.47214 q^{26} +(1.41421 + 2.23607i) q^{28} -2.82843i q^{29} +6.32456i q^{31} +1.00000i q^{32} -3.16228i q^{34} +9.89949 q^{37} +3.16228 q^{38} -4.47214 q^{41} -1.41421 q^{43} -5.65685i q^{44} -4.00000 q^{46} +9.48683 q^{47} +(-3.00000 + 6.32456i) q^{49} +4.47214i q^{52} +4.00000i q^{53} +(-2.23607 + 1.41421i) q^{56} +2.82843 q^{58} +4.47214 q^{59} +9.48683i q^{61} -6.32456 q^{62} -1.00000 q^{64} +7.07107 q^{67} +3.16228 q^{68} +1.41421i q^{71} -13.4164i q^{73} +9.89949i q^{74} +3.16228i q^{76} +(12.6491 - 8.00000i) q^{77} -6.00000 q^{79} -4.47214i q^{82} +12.6491 q^{83} -1.41421i q^{86} +5.65685 q^{88} -4.47214 q^{89} +(-10.0000 + 6.32456i) q^{91} -4.00000i q^{92} +9.48683i q^{94} +(-6.32456 - 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 8 q^{16} - 32 q^{46} - 24 q^{49} - 8 q^{64} - 48 q^{79} - 80 q^{91}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 - 32 * q^46 - 24 * q^49 - 8 * q^64 - 48 * q^79 - 80 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more