LMFDB - Embedded newform 3024.2.k.k.1889.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1889,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3024.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 $ x^16 - 24x^14 + 230x^12 - 1052x^10 + 2139x^8 - 1244x^6 + 1134x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.2
Root			$2.62616 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3024.1889
Dual form			3024.2.k.k.1889.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.86833 q^{5} +(-2.43500 + 1.03478i) q^{7} +O(q^{10})$	$q-2.86833 q^{5} +(-2.43500 + 1.03478i) q^{7} +3.32538i q^{11} +0.821835i q^{13} -3.52027 q^{17} -5.25233i q^{19} +5.12921i q^{23} +3.22729 q^{25} +1.64271i q^{29} +2.55389i q^{31} +(6.98438 - 2.96809i) q^{35} -8.93617 q^{37} -3.49721 q^{41} +0.161123 q^{43} +5.34071 q^{47} +(4.85846 - 5.03938i) q^{49} +3.28542i q^{53} -9.53826i q^{55} -4.08295 q^{59} -8.43509i q^{61} -2.35729i q^{65} +11.8780 q^{67} +5.68267i q^{71} -7.86238i q^{73} +(-3.44103 - 8.09729i) q^{77} +15.1635 q^{79} +15.6686 q^{83} +10.0973 q^{85} -16.2082 q^{89} +(-0.850418 - 2.00117i) q^{91} +15.0654i q^{95} -12.8678i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{7}+O(q^{10}) $ 16 * q + 2 * q^7	$ 16 q + 2 q^{7} - 12 q^{25} - 8 q^{37} - 8 q^{43} + 2 q^{49} + 28 q^{67} + 44 q^{79} + 16 q^{85} - 18 q^{91}+O(q^{100}) $ 16 * q + 2 * q^7 - 12 * q^25 - 8 * q^37 - 8 * q^43 + 2 * q^49 + 28 * q^67 + 44 * q^79 + 16 * q^85 - 18 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−2.86833			−1.28275			−0.641377	−	0.767226i	$-0.721637\pi$
$5$	−2.86833			−1.28275			−0.641377	+	0.767226i	$0.721637\pi$
$6$		0			0
$7$	−2.43500	+	1.03478i	−0.920344	+	0.391110i
$7$	−2.43500	+	1.03478i	−0.920344	+	0.391110i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.32538i			1.00264i	0.865262	+	0.501319i	$0.167152\pi$
$11$			3.32538i			1.00264i	−0.865262	+	0.501319i	$0.832848\pi$
$12$		0			0
$13$			0.821835i			0.227936i	0.993484	+	0.113968i	$0.0363561\pi$
$13$			0.821835i			0.227936i	−0.993484	+	0.113968i	$0.963644\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.52027			−0.853792			−0.426896	−	0.904301i	$-0.640393\pi$
$17$	−3.52027			−0.853792			−0.426896	+	0.904301i	$0.640393\pi$
$18$		0			0
$19$		−	5.25233i		−	1.20497i	−0.798132	−	0.602483i	$-0.794178\pi$
$19$		−	5.25233i		−	1.20497i	0.798132	−	0.602483i	$-0.205822\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.12921i			1.06951i	0.845006	+	0.534757i	$0.179597\pi$
$23$			5.12921i			1.06951i	−0.845006	+	0.534757i	$0.820403\pi$
$24$		0			0
$25$	3.22729			0.645459
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.64271i			0.305044i	0.988300	+	0.152522i	$0.0487394\pi$
$29$			1.64271i			0.305044i	−0.988300	+	0.152522i	$0.951261\pi$
$30$		0			0
$31$			2.55389i			0.458691i	0.973345	+	0.229346i	$0.0736586\pi$
$31$			2.55389i			0.458691i	−0.973345	+	0.229346i	$0.926341\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.98438	−	2.96809i	1.18058	−	0.501698i
$36$		0			0
$37$	−8.93617			−1.46910			−0.734549	−	0.678556i	$-0.762606\pi$
$37$	−8.93617			−1.46910			−0.734549	+	0.678556i	$0.762606\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.49721			−0.546172			−0.273086	−	0.961990i	$-0.588044\pi$
$41$	−3.49721			−0.546172			−0.273086	+	0.961990i	$0.588044\pi$
$42$		0			0
$43$	0.161123			0.0245710			0.0122855	−	0.999925i	$-0.496089\pi$
$43$	0.161123			0.0245710			0.0122855	+	0.999925i	$0.496089\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.34071			0.779022			0.389511	−	0.921022i	$-0.372644\pi$
$47$	5.34071			0.779022			0.389511	+	0.921022i	$0.372644\pi$
$48$		0			0
$49$	4.85846	−	5.03938i	0.694066	−	0.719912i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.28542i			0.451287i	0.974210	+	0.225644i	$0.0724485\pi$
$53$			3.28542i			0.451287i	−0.974210	+	0.225644i	$0.927552\pi$
$54$		0			0
$55$		−	9.53826i		−	1.28614i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.08295			−0.531554			−0.265777	−	0.964034i	$-0.585629\pi$
$59$	−4.08295			−0.531554			−0.265777	+	0.964034i	$0.585629\pi$
$60$		0			0
$61$		−	8.43509i		−	1.08000i	−0.841664	−	0.540001i	$-0.818424\pi$
$61$		−	8.43509i		−	1.08000i	0.841664	−	0.540001i	$-0.181576\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.35729i		−	0.292386i
$66$		0			0
$67$	11.8780			1.45113			0.725567	−	0.688151i	$-0.241578\pi$
$67$	11.8780			1.45113			0.725567	+	0.688151i	$0.241578\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.68267i			0.674408i	0.941432	+	0.337204i	$0.109481\pi$
$71$			5.68267i			0.674408i	−0.941432	+	0.337204i	$0.890519\pi$
$72$		0			0
$73$		−	7.86238i		−	0.920223i	−0.887861	−	0.460111i	$-0.847810\pi$
$73$		−	7.86238i		−	0.920223i	0.887861	−	0.460111i	$-0.152190\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.44103	−	8.09729i	−0.392142	−	0.922772i
$78$		0			0
$79$	15.1635			1.70602			0.853012	−	0.521892i	$-0.174774\pi$
$79$	15.1635			1.70602			0.853012	+	0.521892i	$0.174774\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	15.6686			1.71985			0.859926	−	0.510419i	$-0.170510\pi$
$83$	15.6686			1.71985			0.859926	+	0.510419i	$0.170510\pi$
$84$		0			0
$85$	10.0973			1.09521
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−16.2082			−1.71807			−0.859033	−	0.511921i	$-0.828934\pi$
$89$	−16.2082			−1.71807			−0.859033	+	0.511921i	$0.828934\pi$
$90$		0			0
$91$	−0.850418	−	2.00117i	−0.0891481	−	0.209780i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			15.0654i			1.54568i
$96$		0			0
$97$		−	12.8678i		−	1.30653i	−0.757129	−	0.653266i	$-0.773398\pi$
$97$		−	12.8678i		−	1.30653i	0.757129	−	0.653266i	$-0.226602\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.40793			0.339102			0.169551	−	0.985521i	$-0.445768\pi$
$101$	3.40793			0.339102			0.169551	+	0.985521i	$0.445768\pi$
$102$		0			0
$103$		−	0.484326i		−	0.0477220i	−0.999715	−	0.0238610i	$-0.992404\pi$
$103$		−	0.484326i		−	0.0477220i	0.999715	−	0.0238610i	$-0.00759592\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17.3827i		−	1.68045i	−0.542238	−	0.840225i	$-0.682423\pi$
$107$		−	17.3827i		−	1.68045i	0.542238	−	0.840225i	$-0.317577\pi$
$108$		0			0
$109$	1.13804			0.109005			0.0545023	−	0.998514i	$-0.482643\pi$
$109$	1.13804			0.109005			0.0545023	+	0.998514i	$0.482643\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	16.1946i		−	1.52346i	−0.647895	−	0.761729i	$-0.724351\pi$
$113$		−	16.1946i		−	1.52346i	0.647895	−	0.761729i	$-0.275649\pi$
$114$		0			0
$115$		−	14.7122i		−	1.37192i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.57187	−	3.64271i	0.785782	−	0.333927i
$120$		0			0
$121$	−0.0581269			−0.00528426
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.08470			0.454790
$126$		0			0
$127$	−16.0335			−1.42274			−0.711370	−	0.702818i	$-0.751925\pi$
$127$	−16.0335			−1.42274			−0.711370	+	0.702818i	$0.751925\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.27801			0.635883			0.317941	−	0.948110i	$-0.397008\pi$
$131$	7.27801			0.635883			0.317941	+	0.948110i	$0.397008\pi$
$132$		0			0
$133$	5.43500	+	12.7894i	0.471274	+	1.10898i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			14.2296i			1.21572i	0.794044	+	0.607860i	$0.207972\pi$
$137$			14.2296i			1.21572i	−0.794044	+	0.607860i	$0.792028\pi$
$138$		0			0
$139$		−	21.7288i		−	1.84301i	−0.388361	−	0.921507i	$-0.626958\pi$
$139$		−	21.7288i		−	1.84301i	0.388361	−	0.921507i	$-0.373042\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.73291			−0.228537
$144$		0			0
$145$		−	4.71183i		−	0.391296i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	19.8723i		−	1.62801i	−0.580861	−	0.814003i	$-0.697284\pi$
$149$		−	19.8723i		−	1.62801i	0.580861	−	0.814003i	$-0.302716\pi$
$150$		0			0
$151$	0.870002			0.0707998			0.0353999	−	0.999373i	$-0.488730\pi$
$151$	0.870002			0.0707998			0.0353999	+	0.999373i	$0.488730\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	7.32538i		−	0.588388i
$156$		0			0
$157$		−	3.47414i		−	0.277266i	−0.990344	−	0.138633i	$-0.955729\pi$
$157$		−	3.47414i		−	0.277266i	0.990344	−	0.138633i	$-0.0442709\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.30760	−	12.4896i	−0.418298	−	0.984321i
$162$		0			0
$163$	2.13804			0.167464			0.0837321	−	0.996488i	$-0.473316\pi$
$163$	2.13804			0.167464			0.0837321	+	0.996488i	$0.473316\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−19.7515			−1.52842			−0.764210	−	0.644967i	$-0.776871\pi$
$167$	−19.7515			−1.52842			−0.764210	+	0.644967i	$0.776871\pi$
$168$		0			0
$169$	12.3246			0.948045
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.5084			−1.10305			−0.551525	−	0.834158i	$-0.685954\pi$
$173$	−14.5084			−1.10305			−0.551525	+	0.834158i	$0.685954\pi$
$174$		0			0
$175$	−7.85846	+	3.33954i	−0.594044	+	0.252445i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	6.81188i		−	0.509143i	−0.967054	−	0.254572i	$-0.918066\pi$
$179$		−	6.81188i		−	0.509143i	0.967054	−	0.254572i	$-0.0819345\pi$
$180$		0			0
$181$		−	23.6415i		−	1.75726i	−0.477502	−	0.878631i	$-0.658458\pi$
$181$		−	23.6415i		−	1.75726i	0.477502	−	0.878631i	$-0.341542\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	25.6319			1.88449
$186$		0			0
$187$		−	11.7062i		−	0.856045i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	7.06538i		−	0.511233i	−0.966778	−	0.255617i	$-0.917722\pi$
$191$		−	7.06538i		−	0.511233i	0.966778	−	0.255617i	$-0.0822784\pi$
$192$		0			0
$193$	−5.71692			−0.411513			−0.205757	−	0.978603i	$-0.565965\pi$
$193$	−5.71692			−0.411513			−0.205757	+	0.978603i	$0.565965\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			24.7815i			1.76561i	0.469740	+	0.882805i	$0.344348\pi$
$197$			24.7815i			1.76561i	−0.469740	+	0.882805i	$0.655652\pi$
$198$		0			0
$199$			5.15001i			0.365075i	0.983199	+	0.182537i	$0.0584311\pi$
$199$			5.15001i			0.365075i	−0.983199	+	0.182537i	$0.941569\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.69984	−	4.00000i	−0.119306	−	0.280745i
$204$		0			0
$205$	10.0311			0.700604
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.4660			1.20815
$210$		0			0
$211$	12.0392			0.828810			0.414405	−	0.910092i	$-0.363990\pi$
$211$	12.0392			0.828810			0.414405	+	0.910092i	$0.363990\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.462153			−0.0315186
$216$		0			0
$217$	−2.64271	−	6.21871i	−0.179399	−	0.422154i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.89308i		−	0.194610i
$222$		0			0
$223$			22.7676i			1.52463i	0.647206	+	0.762315i	$0.275937\pi$
$223$			22.7676i			1.52463i	−0.647206	+	0.762315i	$0.724063\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−25.3297			−1.68119			−0.840596	−	0.541663i	$-0.817795\pi$
$227$	−25.3297			−1.68119			−0.840596	+	0.541663i	$0.817795\pi$
$228$		0			0
$229$			2.49545i			0.164904i	0.996595	+	0.0824520i	$0.0262751\pi$
$229$			2.49545i			0.164904i	−0.996595	+	0.0824520i	$0.973725\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	−15.3189			−0.999294
$236$		0			0
$237$		0			0
$238$		0			0
$239$			8.66813i			0.560695i	0.959899	+	0.280347i	$0.0904497\pi$
$239$			8.66813i			0.560695i	−0.959899	+	0.280347i	$0.909550\pi$
$240$		0			0
$241$			30.2907i			1.95119i	0.219571	+	0.975597i	$0.429534\pi$
$241$			30.2907i			1.95119i	−0.219571	+	0.975597i	$0.570466\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.9356	+	14.4546i	−0.890316	+	0.923470i
$246$		0			0
$247$	4.31654			0.274655
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.78279			0.365006			0.182503	−	0.983205i	$-0.441580\pi$
$251$	5.78279			0.365006			0.182503	+	0.983205i	$0.441580\pi$
$252$		0			0
$253$	−17.0565			−1.07234
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.3454			1.08197			0.540987	−	0.841031i	$-0.318051\pi$
$257$	17.3454			1.08197			0.540987	+	0.841031i	$0.318051\pi$
$258$		0			0
$259$	21.7596	−	9.24697i	1.35208	−	0.574579i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			8.80696i			0.543061i	0.962430	+	0.271530i	$0.0875297\pi$
$263$			8.80696i			0.543061i	−0.962430	+	0.271530i	$0.912470\pi$
$264$		0			0
$265$		−	9.42365i		−	0.578890i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	28.4310			1.73347			0.866733	−	0.498772i	$-0.166215\pi$
$269$	28.4310			1.73347			0.866733	+	0.498772i	$0.166215\pi$
$270$		0			0
$271$			2.47554i			0.150378i	0.997169	+	0.0751892i	$0.0239561\pi$
$271$			2.47554i			0.150378i	−0.997169	+	0.0751892i	$0.976044\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			10.7320i			0.647162i
$276$		0			0
$277$	−5.93383			−0.356529			−0.178265	−	0.983983i	$-0.557048\pi$
$277$	−5.93383			−0.356529			−0.178265	+	0.983983i	$0.557048\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.96496i			0.594459i	0.954806	+	0.297230i	$0.0960627\pi$
$281$			9.96496i			0.594459i	−0.954806	+	0.297230i	$0.903937\pi$
$282$		0			0
$283$			13.6796i			0.813170i	0.913613	+	0.406585i	$0.133281\pi$
$283$			13.6796i			0.813170i	−0.913613	+	0.406585i	$0.866719\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.51570	−	3.61884i	0.502666	−	0.213613i
$288$		0			0
$289$	−4.60767			−0.271039
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.47062			0.0859148			0.0429574	−	0.999077i	$-0.486322\pi$
$293$	1.47062			0.0859148			0.0429574	+	0.999077i	$0.486322\pi$
$294$		0			0
$295$	11.7112			0.681854
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.21536			−0.243781
$300$		0			0
$301$	−0.392335	+	0.166727i	−0.0226138	+	0.00960997i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			24.1946i			1.38538i
$306$		0			0
$307$		−	31.6829i		−	1.80824i	−0.427277	−	0.904121i	$-0.640527\pi$
$307$		−	31.6829i		−	1.80824i	0.427277	−	0.904121i	$-0.359473\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−19.1971			−1.08857			−0.544284	−	0.838901i	$-0.683199\pi$
$311$	−19.1971			−1.08857			−0.544284	+	0.838901i	$0.683199\pi$
$312$		0			0
$313$		−	13.4406i		−	0.759705i	−0.925047	−	0.379852i	$-0.875975\pi$
$313$		−	13.4406i		−	0.759705i	0.925047	−	0.379852i	$-0.124025\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	29.5151i		−	1.65773i	−0.559448	−	0.828865i	$-0.688987\pi$
$317$		−	29.5151i		−	1.65773i	0.559448	−	0.828865i	$-0.311013\pi$
$318$		0			0
$319$	−5.46263			−0.305848
$320$		0			0
$321$		0			0
$322$		0			0
$323$			18.4896i			1.02879i
$324$		0			0
$325$			2.65230i			0.147123i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−13.0046	+	5.52646i	−0.716968	+	0.304683i
$330$		0			0
$331$	−0.547756			−0.0301074			−0.0150537	−	0.999887i	$-0.504792\pi$
$331$	−0.547756			−0.0301074			−0.0150537	+	0.999887i	$0.504792\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−34.0701			−1.86145
$336$		0			0
$337$	−17.1946			−0.936649			−0.468324	−	0.883557i	$-0.655142\pi$
$337$	−17.1946			−0.936649			−0.468324	+	0.883557i	$0.655142\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.49263			−0.459902
$342$		0			0
$343$	−6.61571	+	17.2983i	−0.357215	+	0.934022i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			22.6443i			1.21561i	0.794087	+	0.607804i	$0.207949\pi$
$347$			22.6443i			1.21561i	−0.794087	+	0.607804i	$0.792051\pi$
$348$		0			0
$349$		−	24.2288i		−	1.29694i	−0.761241	−	0.648469i	$-0.775410\pi$
$349$		−	24.2288i		−	1.29694i	0.761241	−	0.648469i	$-0.224590\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.0497			0.854240			0.427120	−	0.904195i	$-0.359528\pi$
$353$	16.0497			0.854240			0.427120	+	0.904195i	$0.359528\pi$
$354$		0			0
$355$		−	16.2997i		−	0.865100i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	26.8119i		−	1.41508i	−0.706675	−	0.707538i	$-0.749806\pi$
$359$		−	26.8119i		−	1.41508i	0.706675	−	0.707538i	$-0.250194\pi$
$360$		0			0
$361$	−8.58692			−0.451943
$362$		0			0
$363$		0			0
$364$		0			0
$365$			22.5519i			1.18042i
$366$		0			0
$367$			22.9643i			1.19872i	0.800478	+	0.599362i	$0.204579\pi$
$367$			22.9643i			1.19872i	−0.800478	+	0.599362i	$0.795421\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.39969	−	8.00000i	−0.176503	−	0.415339i
$372$		0			0
$373$	17.6043			0.911516			0.455758	−	0.890104i	$-0.349368\pi$
$373$	17.6043			0.911516			0.455758	+	0.890104i	$0.349368\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.35004			−0.0695304
$378$		0			0
$379$	14.7250			0.756371			0.378185	−	0.925730i	$-0.376548\pi$
$379$	14.7250			0.756371			0.378185	+	0.925730i	$0.376548\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	36.8030			1.88055			0.940273	−	0.340421i	$-0.110570\pi$
$383$	36.8030			1.88055			0.940273	+	0.340421i	$0.110570\pi$
$384$		0			0
$385$	9.87000	+	23.2257i	0.503022	+	1.18369i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			4.32225i			0.219147i	0.993979	+	0.109573i	$0.0349484\pi$
$389$			4.32225i			0.219147i	−0.993979	+	0.109573i	$0.965052\pi$
$390$		0			0
$391$		−	18.0562i		−	0.913143i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−43.4938			−2.18841
$396$		0			0
$397$		−	3.84549i		−	0.192999i	−0.995333	−	0.0964997i	$-0.969235\pi$
$397$		−	3.84549i		−	0.192999i	0.995333	−	0.0964997i	$-0.0307647\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.33655i			0.166619i	0.996524	+	0.0833097i	$0.0265491\pi$
$401$			3.33655i			0.166619i	−0.996524	+	0.0833097i	$0.973451\pi$
$402$		0			0
$403$	−2.09887			−0.104552
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	29.7161i		−	1.47297i
$408$		0			0
$409$			5.99402i			0.296385i	0.988959	+	0.148193i	$0.0473456\pi$
$409$			5.99402i			0.296385i	−0.988959	+	0.148193i	$0.952654\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.94198	−	4.22495i	0.489213	−	0.207896i
$414$		0			0
$415$	−44.9426			−2.20615
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−31.1326			−1.52093			−0.760463	−	0.649382i	$-0.775028\pi$
$419$	−31.1326			−1.52093			−0.760463	+	0.649382i	$0.775028\pi$
$420$		0			0
$421$	2.93617			0.143100			0.0715501	−	0.997437i	$-0.477205\pi$
$421$	2.93617			0.143100			0.0715501	+	0.997437i	$0.477205\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11.3610			−0.551087
$426$		0			0
$427$	8.72846	+	20.5395i	0.422400	+	0.993974i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	23.0654i		−	1.11102i	−0.831510	−	0.555510i	$-0.812523\pi$
$431$		−	23.0654i		−	1.11102i	0.831510	−	0.555510i	$-0.187477\pi$
$432$		0			0
$433$			27.8831i			1.33998i	0.742371	+	0.669989i	$0.233701\pi$
$433$			27.8831i			1.33998i	−0.742371	+	0.669989i	$0.766299\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	26.9403			1.28873
$438$		0			0
$439$			25.7312i			1.22808i	0.789274	+	0.614041i	$0.210457\pi$
$439$			25.7312i			1.22808i	−0.789274	+	0.614041i	$0.789543\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			6.73845i			0.320154i	0.987105	+	0.160077i	$0.0511742\pi$
$443$			6.73845i			0.320154i	−0.987105	+	0.160077i	$0.948826\pi$
$444$		0			0
$445$	46.4904			2.20386
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	38.7654i		−	1.82945i	−0.404073	−	0.914727i	$-0.632406\pi$
$449$		−	38.7654i		−	1.82945i	0.404073	−	0.914727i	$-0.367594\pi$
$450$		0			0
$451$		−	11.6295i		−	0.547613i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.43928	+	5.74000i	0.114355	+	0.269096i
$456$		0			0
$457$	31.7423			1.48484			0.742422	−	0.669932i	$-0.233677\pi$
$457$	31.7423			1.48484			0.742422	+	0.669932i	$0.233677\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.14740			−0.146589			−0.0732945	−	0.997310i	$-0.523351\pi$
$461$	−3.14740			−0.146589			−0.0732945	+	0.997310i	$0.523351\pi$
$462$		0			0
$463$	−5.46833			−0.254135			−0.127067	−	0.991894i	$-0.540556\pi$
$463$	−5.46833			−0.254135			−0.127067	+	0.991894i	$0.540556\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.5471			0.950809			0.475404	−	0.879767i	$-0.342302\pi$
$467$	20.5471			0.950809			0.475404	+	0.879767i	$0.342302\pi$
$468$		0			0
$469$	−28.9231	+	12.2912i	−1.33554	+	0.567553i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.535795i			0.0246359i
$474$		0			0
$475$		−	16.9508i		−	0.777756i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.0645			−0.596933			−0.298466	−	0.954420i	$-0.596475\pi$
$479$	−13.0645			−0.596933			−0.298466	+	0.954420i	$0.596475\pi$
$480$		0			0
$481$		−	7.34406i		−	0.334860i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			36.9092i			1.67596i
$486$		0			0
$487$	−1.74571			−0.0791055			−0.0395527	−	0.999217i	$-0.512593\pi$
$487$	−1.74571			−0.0791055			−0.0395527	+	0.999217i	$0.512593\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	4.51350i		−	0.203691i	−0.994800	−	0.101846i	$-0.967525\pi$
$491$		−	4.51350i		−	0.203691i	0.994800	−	0.101846i	$-0.0324748\pi$
$492$		0			0
$493$		−	5.78279i		−	0.260444i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.88031	−	13.8373i	−0.263768	−	0.620688i
$498$		0			0
$499$	−14.7320			−0.659493			−0.329747	−	0.944069i	$-0.606963\pi$
$499$	−14.7320			−0.659493			−0.329747	+	0.944069i	$0.606963\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.64812			0.385600			0.192800	−	0.981238i	$-0.438243\pi$
$503$	8.64812			0.385600			0.192800	+	0.981238i	$0.438243\pi$
$504$		0			0
$505$	−9.77505			−0.434984
$506$		0			0
$507$		0			0
$508$		0			0
$509$	29.8666			1.32381			0.661906	−	0.749587i	$-0.269748\pi$
$509$	29.8666			1.32381			0.661906	+	0.749587i	$0.269748\pi$
$510$		0			0
$511$	8.13584	+	19.1449i	0.359908	+	0.846921i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			1.38920i			0.0612157i
$516$		0			0
$517$			17.7599i			0.781078i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.5756			0.550946			0.275473	−	0.961309i	$-0.411166\pi$
$521$	12.5756			0.550946			0.275473	+	0.961309i	$0.411166\pi$
$522$		0			0
$523$		−	24.9477i		−	1.09089i	−0.838148	−	0.545443i	$-0.816361\pi$
$523$		−	24.9477i		−	1.09089i	0.838148	−	0.545443i	$-0.183639\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8.99038i		−	0.391627i
$528$		0			0
$529$	−3.30878			−0.143860
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2.87413i		−	0.124492i
$534$		0			0
$535$			49.8593i			2.15560i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.7578	+	16.1562i	0.721811	+	0.695897i
$540$		0			0
$541$	−12.5438			−0.539302			−0.269651	−	0.962958i	$-0.586908\pi$
$541$	−12.5438			−0.539302			−0.269651	+	0.962958i	$0.586908\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.26427			−0.139826
$546$		0			0
$547$	8.87000			0.379254			0.189627	−	0.981856i	$-0.439272\pi$
$547$	8.87000			0.379254			0.189627	+	0.981856i	$0.439272\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.62805			0.367567
$552$		0			0
$553$	−36.9231	+	15.6908i	−1.57013	+	0.667243i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	32.7815i		−	1.38900i	−0.719494	−	0.694499i	$-0.755626\pi$
$557$		−	32.7815i		−	1.38900i	0.719494	−	0.694499i	$-0.244374\pi$
$558$		0			0
$559$			0.132416i			0.00560062i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.97060			−0.167341			−0.0836704	−	0.996493i	$-0.526664\pi$
$563$	−3.97060			−0.167341			−0.0836704	+	0.996493i	$0.526664\pi$
$564$		0			0
$565$			46.4514i			1.95422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			20.0669i			0.841250i	0.907235	+	0.420625i	$0.138189\pi$
$569$			20.0669i			0.841250i	−0.907235	+	0.420625i	$0.861811\pi$
$570$		0			0
$571$	22.9323			0.959685			0.479842	−	0.877355i	$-0.340694\pi$
$571$	22.9323			0.959685			0.479842	+	0.877355i	$0.340694\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			16.5535i			0.690327i
$576$		0			0
$577$			27.5760i			1.14801i	0.818853	+	0.574003i	$0.194610\pi$
$577$			27.5760i			1.14801i	−0.818853	+	0.574003i	$0.805390\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−38.1530	+	16.2135i	−1.58286	+	0.672651i
$582$		0			0
$583$	−10.9253			−0.452478
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.6684			0.605428			0.302714	−	0.953081i	$-0.402107\pi$
$587$	14.6684			0.605428			0.302714	+	0.953081i	$0.402107\pi$
$588$		0			0
$589$	13.4138			0.552708
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−27.7560			−1.13980			−0.569900	−	0.821714i	$-0.693018\pi$
$593$	−27.7560			−1.13980			−0.569900	+	0.821714i	$0.693018\pi$
$594$		0			0
$595$	−24.5869	+	10.4485i	−1.00797	+	0.428346i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	31.2600i		−	1.27725i	−0.769519	−	0.638624i	$-0.779504\pi$
$599$		−	31.2600i		−	1.27725i	0.769519	−	0.638624i	$-0.220496\pi$
$600$		0			0
$601$		−	26.8665i		−	1.09591i	−0.836508	−	0.547955i	$-0.815407\pi$
$601$		−	26.8665i		−	1.09591i	0.836508	−	0.547955i	$-0.184593\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.166727			0.00677841
$606$		0			0
$607$			22.6331i			0.918648i	0.888269	+	0.459324i	$0.151908\pi$
$607$			22.6331i			0.918648i	−0.888269	+	0.459324i	$0.848092\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			4.38918i			0.177567i
$612$		0			0
$613$	−42.8518			−1.73077			−0.865384	−	0.501109i	$-0.832926\pi$
$613$	−42.8518			−1.73077			−0.865384	+	0.501109i	$0.832926\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	10.9442i		−	0.440597i	−0.975432	−	0.220299i	$-0.929297\pi$
$617$		−	10.9442i		−	0.440597i	0.975432	−	0.220299i	$-0.0707032\pi$
$618$		0			0
$619$			14.4430i			0.580515i	0.956949	+	0.290257i	$0.0937409\pi$
$619$			14.4430i			0.580515i	−0.956949	+	0.290257i	$0.906259\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	39.4670	−	16.7719i	1.58121	−	0.671953i
$624$		0			0
$625$	−30.7210			−1.22884
$626$		0			0
$627$		0			0
$628$		0			0
$629$	31.4578			1.25430
$630$		0			0
$631$	−14.3326			−0.570573			−0.285286	−	0.958442i	$-0.592089\pi$
$631$	−14.3326			−0.570573			−0.285286	+	0.958442i	$0.592089\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	45.9892			1.82503
$636$		0			0
$637$	4.14154	+	3.99285i	0.164094	+	0.158203i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	31.8023i		−	1.25611i	−0.778168	−	0.628057i	$-0.783851\pi$
$641$		−	31.8023i		−	1.25611i	0.778168	−	0.628057i	$-0.216149\pi$
$642$		0			0
$643$		−	0.345751i		−	0.0136351i	−0.999977	−	0.00681755i	$-0.997830\pi$
$643$		−	0.345751i		−	0.0136351i	0.999977	−	0.00681755i	$-0.00217011\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.85274			0.348037			0.174019	−	0.984742i	$-0.444325\pi$
$647$	8.85274			0.348037			0.174019	+	0.984742i	$0.444325\pi$
$648$		0			0
$649$		−	13.5773i		−	0.532957i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	40.9603i		−	1.60290i	−0.598062	−	0.801450i	$-0.704062\pi$
$653$		−	40.9603i		−	1.60290i	0.598062	−	0.801450i	$-0.295938\pi$
$654$		0			0
$655$	−20.8757			−0.815681
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	5.38920i		−	0.209934i	−0.994476	−	0.104967i	$-0.966526\pi$
$659$		−	5.38920i		−	0.209934i	0.994476	−	0.104967i	$-0.0334736\pi$
$660$		0			0
$661$			25.9703i			1.01013i	0.863083	+	0.505063i	$0.168531\pi$
$661$			25.9703i			1.01013i	−0.863083	+	0.505063i	$0.831469\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15.5894	−	36.6842i	−0.604529	−	1.42255i
$666$		0			0
$667$	−8.42580			−0.326248
$668$		0			0
$669$		0			0
$670$		0			0
$671$	28.0499			1.08285
$672$		0			0
$673$	−28.0553			−1.08145			−0.540725	−	0.841199i	$-0.681850\pi$
$673$	−28.0553			−1.08145			−0.540725	+	0.841199i	$0.681850\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−31.5977			−1.21440			−0.607200	−	0.794549i	$-0.707707\pi$
$677$	−31.5977			−1.21440			−0.607200	+	0.794549i	$0.707707\pi$
$678$		0			0
$679$	13.3154	+	31.3332i	0.510998	+	1.20246i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			8.55679i			0.327417i	0.986509	+	0.163708i	$0.0523456\pi$
$683$			8.55679i			0.327417i	−0.986509	+	0.163708i	$0.947654\pi$
$684$		0			0
$685$		−	40.8152i		−	1.55947i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.70007			−0.102865
$690$		0			0
$691$			19.5748i			0.744660i	0.928100	+	0.372330i	$0.121441\pi$
$691$			19.5748i			0.744660i	−0.928100	+	0.372330i	$0.878559\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			62.3254i			2.36414i
$696$		0			0
$697$	12.3111			0.466317
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	31.9810i		−	1.20791i	−0.797019	−	0.603954i	$-0.793591\pi$
$701$		−	31.9810i		−	1.20791i	0.797019	−	0.603954i	$-0.206409\pi$
$702$		0			0
$703$			46.9357i			1.77021i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.29831	+	3.52646i	−0.312090	+	0.132626i
$708$		0			0
$709$	−2.07029			−0.0777515			−0.0388757	−	0.999244i	$-0.512378\pi$
$709$	−2.07029			−0.0777515			−0.0388757	+	0.999244i	$0.512378\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.0994			−0.490577
$714$		0			0
$715$	7.83888			0.293157
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33.4828			−1.24870			−0.624350	−	0.781145i	$-0.714636\pi$
$719$	−33.4828			−1.24870			−0.624350	+	0.781145i	$0.714636\pi$
$720$		0			0
$721$	0.501171	+	1.17933i	0.0186646	+	0.0439207i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			5.30150i			0.196893i
$726$		0			0
$727$		−	12.7754i		−	0.473814i	−0.971532	−	0.236907i	$-0.923866\pi$
$727$		−	12.7754i		−	0.473814i	0.971532	−	0.236907i	$-0.0761336\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.567197			−0.0209785
$732$		0			0
$733$			30.2462i			1.11717i	0.829448	+	0.558584i	$0.188655\pi$
$733$			30.2462i			1.11717i	−0.829448	+	0.558584i	$0.811345\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			39.4990i			1.45496i
$738$		0			0
$739$	9.10692			0.335003			0.167502	−	0.985872i	$-0.446430\pi$
$739$	9.10692			0.335003			0.167502	+	0.985872i	$0.446430\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	36.9551i		−	1.35575i	−0.735177	−	0.677876i	$-0.762901\pi$
$743$		−	36.9551i		−	1.35575i	0.735177	−	0.677876i	$-0.237099\pi$
$744$		0			0
$745$			57.0004i			2.08833i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	17.9873	+	42.3269i	0.657241	+	1.54659i
$750$		0			0
$751$	−6.89879			−0.251740			−0.125870	−	0.992047i	$-0.540172\pi$
$751$	−6.89879			−0.251740			−0.125870	+	0.992047i	$0.540172\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.49545			−0.0908187
$756$		0			0
$757$	−12.8861			−0.468353			−0.234176	−	0.972194i	$-0.575239\pi$
$757$	−12.8861			−0.468353			−0.234176	+	0.972194i	$0.575239\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−32.8078			−1.18928			−0.594641	−	0.803991i	$-0.702706\pi$
$761$	−32.8078			−1.18928			−0.594641	+	0.803991i	$0.702706\pi$
$762$		0			0
$763$	−2.77113	+	1.17762i	−0.100322	+	0.0426328i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	3.35551i		−	0.121160i
$768$		0			0
$769$		−	18.9096i		−	0.681899i	−0.940082	−	0.340949i	$-0.889251\pi$
$769$		−	18.9096i		−	0.681899i	0.940082	−	0.340949i	$-0.110749\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15.6195			−0.561793			−0.280896	−	0.959738i	$-0.590632\pi$
$773$	−15.6195			−0.561793			−0.280896	+	0.959738i	$0.590632\pi$
$774$		0			0
$775$			8.24214i			0.296066i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			18.3685i			0.658118i
$780$		0			0
$781$	−18.8970			−0.676188
$782$		0			0
$783$		0			0
$784$		0			0
$785$			9.96496i			0.355665i
$786$		0			0
$787$		−	14.4471i		−	0.514983i	−0.966281	−	0.257492i	$-0.917104\pi$
$787$		−	14.4471i		−	0.514983i	0.966281	−	0.257492i	$-0.0828960\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	16.7578	+	39.4338i	0.595840	+	1.40211i
$792$		0			0
$793$	6.93225			0.246172
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−46.1246			−1.63382			−0.816909	−	0.576767i	$-0.804314\pi$
$797$	−46.1246			−1.63382			−0.816909	+	0.576767i	$0.804314\pi$
$798$		0			0
$799$	−18.8008			−0.665123
$800$		0			0
$801$		0			0
$802$		0			0
$803$	26.1454			0.922651
$804$		0			0
$805$	15.2239	+	35.8243i	0.536573	+	1.26264i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			4.17879i			0.146918i	0.997298	+	0.0734591i	$0.0234039\pi$
$809$			4.17879i			0.146918i	−0.997298	+	0.0734591i	$0.976596\pi$
$810$		0			0
$811$		−	14.7767i		−	0.518879i	−0.965759	−	0.259439i	$-0.916462\pi$
$811$		−	14.7767i		−	0.518879i	0.965759	−	0.259439i	$-0.0835378\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.13260			−0.214815
$816$		0			0
$817$		−	0.846270i		−	0.0296073i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	46.5519i		−	1.62467i	−0.583190	−	0.812336i	$-0.698196\pi$
$821$		−	46.5519i		−	1.62467i	0.583190	−	0.812336i	$-0.301804\pi$
$822$		0			0
$823$	−3.20833			−0.111835			−0.0559177	−	0.998435i	$-0.517808\pi$
$823$	−3.20833			−0.111835			−0.0559177	+	0.998435i	$0.517808\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			17.7927i			0.618711i	0.950946	+	0.309356i	$0.100113\pi$
$827$			17.7927i			0.618711i	−0.950946	+	0.309356i	$0.899887\pi$
$828$		0			0
$829$		−	6.66451i		−	0.231468i	−0.993280	−	0.115734i	$-0.963078\pi$
$829$		−	6.66451i		−	0.231468i	0.993280	−	0.115734i	$-0.0369220\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−17.1031	+	17.7400i	−0.592588	+	0.614655i
$834$		0			0
$835$	56.6539			1.96059
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.14564			0.0740758			0.0370379	−	0.999314i	$-0.488208\pi$
$839$	2.14564			0.0740758			0.0370379	+	0.999314i	$0.488208\pi$
$840$		0			0
$841$	26.3015			0.906948
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−35.3509			−1.21611
$846$		0			0
$847$	0.141539	−	0.0601485i	0.00486334	−	0.00206673i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	45.8355i		−	1.57122i
$852$		0			0
$853$			18.8954i			0.646966i	0.946234	+	0.323483i	$0.104854\pi$
$853$			18.8954i			0.646966i	−0.946234	+	0.323483i	$0.895146\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	35.2140			1.20289			0.601443	−	0.798915i	$-0.294593\pi$
$857$	35.2140			1.20289			0.601443	+	0.798915i	$0.294593\pi$
$858$		0			0
$859$		−	23.6538i		−	0.807058i	−0.914967	−	0.403529i	$-0.867783\pi$
$859$		−	23.6538i		−	0.807058i	0.914967	−	0.403529i	$-0.132217\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	0.993535i		−	0.0338203i	−0.999857	−	0.0169102i	$-0.994617\pi$
$863$		−	0.993535i		−	0.0338203i	0.999857	−	0.0169102i	$-0.00538293\pi$
$864$		0			0
$865$	41.6147			1.41494
$866$		0			0
$867$		0			0
$868$		0			0
$869$			50.4242i			1.71052i
$870$		0			0
$871$			9.76179i			0.330766i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.3813	+	5.26155i	−0.418563	+	0.177873i
$876$		0			0
$877$	−14.9401			−0.504491			−0.252246	−	0.967663i	$-0.581169\pi$
$877$	−14.9401			−0.504491			−0.252246	+	0.967663i	$0.581169\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	17.6044			0.593106			0.296553	−	0.955016i	$-0.404163\pi$
$881$	17.6044			0.593106			0.296553	+	0.955016i	$0.404163\pi$
$882$		0			0
$883$	−42.0439			−1.41489			−0.707444	−	0.706769i	$-0.750152\pi$
$883$	−42.0439			−1.41489			−0.707444	+	0.706769i	$0.750152\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.3186			0.480770			0.240385	−	0.970678i	$-0.422726\pi$
$887$	14.3186			0.480770			0.240385	+	0.970678i	$0.422726\pi$
$888$		0			0
$889$	39.0415	−	16.5911i	1.30941	−	0.556448i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	28.0511i		−	0.938695i
$894$		0			0
$895$			19.5387i			0.653106i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.19529			−0.139921
$900$		0			0
$901$		−	11.5656i		−	0.385305i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			67.8116i			2.25413i
$906$		0			0
$907$	−9.43616			−0.313322			−0.156661	−	0.987652i	$-0.550073\pi$
$907$	−9.43616			−0.313322			−0.156661	+	0.987652i	$0.550073\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			36.1803i			1.19871i	0.800484	+	0.599354i	$0.204576\pi$
$911$			36.1803i			1.19871i	−0.800484	+	0.599354i	$0.795424\pi$
$912$		0			0
$913$			52.1040i			1.72439i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−17.7220	+	7.53114i	−0.585231	+	0.248700i
$918$		0			0
$919$	−13.4292			−0.442987			−0.221494	−	0.975162i	$-0.571093\pi$
$919$	−13.4292			−0.442987			−0.221494	+	0.975162i	$0.571093\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.67021			−0.153722
$924$		0			0
$925$	−28.8396			−0.948242
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.4485			−0.342803			−0.171402	−	0.985201i	$-0.554830\pi$
$929$	−10.4485			−0.342803			−0.171402	+	0.985201i	$0.554830\pi$
$930$		0			0
$931$	−26.4685	−	25.5182i	−0.867469	−	0.836326i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			33.5773i			1.09810i
$936$		0			0
$937$		−	46.0300i		−	1.50373i	−0.659315	−	0.751867i	$-0.729153\pi$
$937$		−	46.0300i		−	1.50373i	0.659315	−	0.751867i	$-0.270847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−16.4836			−0.537349			−0.268674	−	0.963231i	$-0.586586\pi$
$941$	−16.4836			−0.537349			−0.268674	+	0.963231i	$0.586586\pi$
$942$		0			0
$943$		−	17.9379i		−	0.584138i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			49.0527i			1.59400i	0.603981	+	0.796999i	$0.293580\pi$
$947$			49.0527i			1.59400i	−0.603981	+	0.796999i	$0.706420\pi$
$948$		0			0
$949$	6.46158			0.209752
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	42.9442i		−	1.39110i	−0.718478	−	0.695550i	$-0.755161\pi$
$953$		−	42.9442i		−	1.39110i	0.718478	−	0.695550i	$-0.244839\pi$
$954$		0			0
$955$			20.2658i			0.655786i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.7245	−	34.6492i	−0.475480	−	1.11888i
$960$		0			0
$961$	24.4777			0.789602
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.3980			0.527870
$966$		0			0
$967$	4.03917			0.129891			0.0649454	−	0.997889i	$-0.479313\pi$
$967$	4.03917			0.129891			0.0649454	+	0.997889i	$0.479313\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.7312			−0.601114			−0.300557	−	0.953764i	$-0.597173\pi$
$971$	−18.7312			−0.601114			−0.300557	+	0.953764i	$0.597173\pi$
$972$		0			0
$973$	22.4846	+	52.9097i	0.720822	+	1.69621i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	35.1899i		−	1.12583i	−0.826516	−	0.562913i	$-0.809681\pi$
$977$		−	35.1899i		−	1.12583i	0.826516	−	0.562913i	$-0.190319\pi$
$978$		0			0
$979$		−	53.8984i		−	1.72260i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.0312			0.351841			0.175921	−	0.984404i	$-0.443710\pi$
$983$	11.0312			0.351841			0.175921	+	0.984404i	$0.443710\pi$
$984$		0			0
$985$		−	71.0815i		−	2.26484i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			0.826433i			0.0262791i
$990$		0			0
$991$	−4.31654			−0.137120			−0.0685598	−	0.997647i	$-0.521840\pi$
$991$	−4.31654			−0.137120			−0.0685598	+	0.997647i	$0.521840\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	14.7719i		−	0.468301i
$996$		0			0
$997$		−	10.5691i		−	0.334726i	−0.985895	−	0.167363i	$-0.946475\pi$
$997$		−	10.5691i		−	0.334726i	0.985895	−	0.167363i	$-0.0535251\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.k.k.1889.2		16
3.2	odd	2	inner	3024.2.k.k.1889.16		16
4.3	odd	2		1512.2.k.a.377.1	✓	16
7.6	odd	2	inner	3024.2.k.k.1889.15		16
12.11	even	2		1512.2.k.a.377.15	yes	16
21.20	even	2	inner	3024.2.k.k.1889.1		16
28.27	even	2		1512.2.k.a.377.16	yes	16
84.83	odd	2		1512.2.k.a.377.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.k.a.377.1	✓	16	4.3	odd	2
1512.2.k.a.377.2	yes	16	84.83	odd	2
1512.2.k.a.377.15	yes	16	12.11	even	2
1512.2.k.a.377.16	yes	16	28.27	even	2
3024.2.k.k.1889.1		16	21.20	even	2	inner
3024.2.k.k.1889.2		16	1.1	even	1	trivial
3024.2.k.k.1889.15		16	7.6	odd	2	inner
3024.2.k.k.1889.16		16	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.86833 q^{5} +(-2.43500 + 1.03478i) q^{7} +O(q^{10})\)	\(q-2.86833 q^{5} +(-2.43500 + 1.03478i) q^{7} +3.32538i q^{11} +0.821835i q^{13} -3.52027 q^{17} -5.25233i q^{19} +5.12921i q^{23} +3.22729 q^{25} +1.64271i q^{29} +2.55389i q^{31} +(6.98438 - 2.96809i) q^{35} -8.93617 q^{37} -3.49721 q^{41} +0.161123 q^{43} +5.34071 q^{47} +(4.85846 - 5.03938i) q^{49} +3.28542i q^{53} -9.53826i q^{55} -4.08295 q^{59} -8.43509i q^{61} -2.35729i q^{65} +11.8780 q^{67} +5.68267i q^{71} -7.86238i q^{73} +(-3.44103 - 8.09729i) q^{77} +15.1635 q^{79} +15.6686 q^{83} +10.0973 q^{85} -16.2082 q^{89} +(-0.850418 - 2.00117i) q^{91} +15.0654i q^{95} -12.8678i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{7}+O(q^{10}) \) 16 * q + 2 * q^7	\( 16 q + 2 q^{7} - 12 q^{25} - 8 q^{37} - 8 q^{43} + 2 q^{49} + 28 q^{67} + 44 q^{79} + 16 q^{85} - 18 q^{91}+O(q^{100}) \) 16 * q + 2 * q^7 - 12 * q^25 - 8 * q^37 - 8 * q^43 + 2 * q^49 + 28 * q^67 + 44 * q^79 + 16 * q^85 - 18 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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