LMFDB - Embedded newform 3024.2.k.i.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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 25 $ x^4 - 5*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.2
Root			$-1.93649 + 1.11803i$ of defining polynomial
Character	$\chi$	$=$	3024.1889
Dual form			3024.2.k.i.1889.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.87298 q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})$	$q-3.87298 q^{5} +(2.00000 + 1.73205i) q^{7} -2.23607i q^{11} +3.46410i q^{13} -5.19615i q^{19} -2.23607i q^{23} +10.0000 q^{25} -4.47214i q^{29} +1.73205i q^{31} +(-7.74597 - 6.70820i) q^{35} -1.00000 q^{37} -3.87298 q^{41} -2.00000 q^{43} +7.74597 q^{47} +(1.00000 + 6.92820i) q^{49} +8.94427i q^{53} +8.66025i q^{55} -7.74597 q^{59} +6.92820i q^{61} -13.4164i q^{65} +10.0000 q^{67} +11.1803i q^{71} +10.3923i q^{73} +(3.87298 - 4.47214i) q^{77} -2.00000 q^{79} +7.74597 q^{83} +11.6190 q^{89} +(-6.00000 + 6.92820i) q^{91} +20.1246i q^{95} -13.8564i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} + 40 q^{25} - 4 q^{37} - 8 q^{43} + 4 q^{49} + 40 q^{67} - 8 q^{79} - 24 q^{91}+O(q^{100}) $ 4 * q + 8 * q^7 + 40 * q^25 - 4 * q^37 - 8 * q^43 + 4 * q^49 + 40 * q^67 - 8 * q^79 - 24 * 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
$5$	−3.87298			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$5$	−3.87298			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.23607i		−	0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$11$		−	2.23607i		−	0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$12$		0			0
$13$			3.46410i			0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$			3.46410i			0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		−	5.19615i		−	1.19208i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$		−	5.19615i		−	1.19208i	0.802955	−	0.596040i	$-0.203260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.23607i		−	0.466252i	−0.972446	−	0.233126i	$-0.925104\pi$
$23$		−	2.23607i		−	0.466252i	0.972446	−	0.233126i	$-0.0748955\pi$
$24$		0			0
$25$	10.0000			2.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.47214i		−	0.830455i	−0.909718	−	0.415227i	$-0.863702\pi$
$29$		−	4.47214i		−	0.830455i	0.909718	−	0.415227i	$-0.136298\pi$
$30$		0			0
$31$			1.73205i			0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$			1.73205i			0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.74597	−	6.70820i	−1.30931	−	1.13389i
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.87298			−0.604858			−0.302429	−	0.953172i	$-0.597798\pi$
$41$	−3.87298			−0.604858			−0.302429	+	0.953172i	$0.597798\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.74597			1.12987			0.564933	−	0.825137i	$-0.308902\pi$
$47$	7.74597			1.12987			0.564933	+	0.825137i	$0.308902\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			8.94427i			1.22859i	0.789076	+	0.614295i	$0.210560\pi$
$53$			8.94427i			1.22859i	−0.789076	+	0.614295i	$0.789440\pi$
$54$		0			0
$55$			8.66025i			1.16775i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.74597			−1.00844			−0.504219	−	0.863576i	$-0.668220\pi$
$59$	−7.74597			−1.00844			−0.504219	+	0.863576i	$0.668220\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	13.4164i		−	1.66410i
$66$		0			0
$67$	10.0000			1.22169			0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000			1.22169			0.610847	+	0.791748i	$0.290829\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.1803i			1.32686i	0.748237	+	0.663431i	$0.230900\pi$
$71$			11.1803i			1.32686i	−0.748237	+	0.663431i	$0.769100\pi$
$72$		0			0
$73$			10.3923i			1.21633i	0.793812	+	0.608164i	$0.208094\pi$
$73$			10.3923i			1.21633i	−0.793812	+	0.608164i	$0.791906\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.87298	−	4.47214i	0.441367	−	0.509647i
$78$		0			0
$79$	−2.00000			−0.225018			−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000			−0.225018			−0.112509	+	0.993651i	$0.535889\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.74597			0.850230			0.425115	−	0.905139i	$-0.360234\pi$
$83$	7.74597			0.850230			0.425115	+	0.905139i	$0.360234\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.6190			1.23161			0.615803	−	0.787900i	$-0.288832\pi$
$89$	11.6190			1.23161			0.615803	+	0.787900i	$0.288832\pi$
$90$		0			0
$91$	−6.00000	+	6.92820i	−0.628971	+	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			20.1246i			2.06474i
$96$		0			0
$97$		−	13.8564i		−	1.40690i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$		−	13.8564i		−	1.40690i	0.710742	−	0.703452i	$-0.248359\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.4919			1.54150			0.770752	−	0.637135i	$-0.219880\pi$
$101$	15.4919			1.54150			0.770752	+	0.637135i	$0.219880\pi$
$102$		0			0
$103$			1.73205i			0.170664i	0.996353	+	0.0853320i	$0.0271951\pi$
$103$			1.73205i			0.170664i	−0.996353	+	0.0853320i	$0.972805\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	8.94427i		−	0.864675i	−0.901712	−	0.432338i	$-0.857689\pi$
$107$		−	8.94427i		−	0.864675i	0.901712	−	0.432338i	$-0.142311\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.94427i			0.841406i	0.907198	+	0.420703i	$0.138217\pi$
$113$			8.94427i			0.841406i	−0.907198	+	0.420703i	$0.861783\pi$
$114$		0			0
$115$			8.66025i			0.807573i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	6.00000			0.545455
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−19.3649			−1.73205
$126$		0			0
$127$	10.0000			0.887357			0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000			0.887357			0.443678	+	0.896186i	$0.353673\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.74597			−0.676768			−0.338384	−	0.941008i	$-0.609880\pi$
$131$	−7.74597			−0.676768			−0.338384	+	0.941008i	$0.609880\pi$
$132$		0			0
$133$	9.00000	−	10.3923i	0.780399	−	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			22.3607i			1.91040i	0.295958	+	0.955201i	$0.404361\pi$
$137$			22.3607i			1.91040i	−0.295958	+	0.955201i	$0.595639\pi$
$138$		0			0
$139$		−	3.46410i		−	0.293821i	−0.989150	−	0.146911i	$-0.953067\pi$
$139$		−	3.46410i		−	0.293821i	0.989150	−	0.146911i	$-0.0469330\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	7.74597			0.647750
$144$		0			0
$145$			17.3205i			1.43839i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.94427i			0.732743i	0.930469	+	0.366372i	$0.119400\pi$
$149$			8.94427i			0.732743i	−0.930469	+	0.366372i	$0.880600\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	6.70820i		−	0.538816i
$156$		0			0
$157$			3.46410i			0.276465i	0.990400	+	0.138233i	$0.0441422\pi$
$157$			3.46410i			0.276465i	−0.990400	+	0.138233i	$0.955858\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.87298	−	4.47214i	0.305234	−	0.352454i
$162$		0			0
$163$	10.0000			0.783260			0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000			0.783260			0.391630	+	0.920123i	$0.371911\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.4919			1.19880			0.599401	−	0.800449i	$-0.295406\pi$
$167$	15.4919			1.19880			0.599401	+	0.800449i	$0.295406\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.87298			0.294457			0.147229	−	0.989102i	$-0.452965\pi$
$173$	3.87298			0.294457			0.147229	+	0.989102i	$0.452965\pi$
$174$		0			0
$175$	20.0000	+	17.3205i	1.51186	+	1.30931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			17.8885i			1.33705i	0.743689	+	0.668526i	$0.233075\pi$
$179$			17.8885i			1.33705i	−0.743689	+	0.668526i	$0.766925\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.87298			0.284747
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	2.23607i		−	0.161796i	−0.996722	−	0.0808981i	$-0.974221\pi$
$191$		−	2.23607i		−	0.161796i	0.996722	−	0.0808981i	$-0.0257788\pi$
$192$		0			0
$193$	2.00000			0.143963			0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000			0.143963			0.0719816	+	0.997406i	$0.477068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.94427i			0.637253i	0.947880	+	0.318626i	$0.103222\pi$
$197$			8.94427i			0.637253i	−0.947880	+	0.318626i	$0.896778\pi$
$198$		0			0
$199$			25.9808i			1.84173i	0.389885	+	0.920864i	$0.372515\pi$
$199$			25.9808i			1.84173i	−0.389885	+	0.920864i	$0.627485\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.74597	−	8.94427i	0.543660	−	0.627765i
$204$		0			0
$205$	15.0000			1.04765
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−11.6190			−0.803700
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.74597			0.528271
$216$		0			0
$217$	−3.00000	+	3.46410i	−0.203653	+	0.235159i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	1.73205i		−	0.115987i	−0.998317	−	0.0579934i	$-0.981530\pi$
$223$		−	1.73205i		−	0.115987i	0.998317	−	0.0579934i	$-0.0184702\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.4919			−1.02824			−0.514118	−	0.857720i	$-0.671881\pi$
$227$	−15.4919			−1.02824			−0.514118	+	0.857720i	$0.671881\pi$
$228$		0			0
$229$			13.8564i			0.915657i	0.889041	+	0.457829i	$0.151373\pi$
$229$			13.8564i			0.915657i	−0.889041	+	0.457829i	$0.848627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	17.8885i		−	1.17192i	−0.810341	−	0.585959i	$-0.800718\pi$
$233$		−	17.8885i		−	1.17192i	0.810341	−	0.585959i	$-0.199282\pi$
$234$		0			0
$235$	−30.0000			−1.95698
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.94427i		−	0.578557i	−0.957245	−	0.289278i	$-0.906585\pi$
$239$		−	8.94427i		−	0.578557i	0.957245	−	0.289278i	$-0.0934153\pi$
$240$		0			0
$241$			17.3205i			1.11571i	0.829938	+	0.557856i	$0.188376\pi$
$241$			17.3205i			1.11571i	−0.829938	+	0.557856i	$0.811624\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.87298	−	26.8328i	−0.247436	−	1.71429i
$246$		0			0
$247$	18.0000			1.14531
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−5.00000			−0.314347
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−27.1109			−1.69113			−0.845565	−	0.533872i	$-0.820736\pi$
$257$	−27.1109			−1.69113			−0.845565	+	0.533872i	$0.820736\pi$
$258$		0			0
$259$	−2.00000	−	1.73205i	−0.124274	−	0.107624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2.23607i		−	0.137882i	−0.997621	−	0.0689409i	$-0.978038\pi$
$263$		−	2.23607i		−	0.137882i	0.997621	−	0.0689409i	$-0.0219620\pi$
$264$		0			0
$265$		−	34.6410i		−	2.12798i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.6190			−0.708420			−0.354210	−	0.935166i	$-0.615250\pi$
$269$	−11.6190			−0.708420			−0.354210	+	0.935166i	$0.615250\pi$
$270$		0			0
$271$			10.3923i			0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$			10.3923i			0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	22.3607i		−	1.34840i
$276$		0			0
$277$	11.0000			0.660926			0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000			0.660926			0.330463	+	0.943819i	$0.392795\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			22.3607i			1.33393i	0.745091	+	0.666963i	$0.232406\pi$
$281$			22.3607i			1.33393i	−0.745091	+	0.666963i	$0.767594\pi$
$282$		0			0
$283$		−	24.2487i		−	1.44144i	−0.693228	−	0.720718i	$-0.743812\pi$
$283$		−	24.2487i		−	1.44144i	0.693228	−	0.720718i	$-0.256188\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.74597	−	6.70820i	−0.457230	−	0.395973i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	30.9839			1.81010			0.905048	−	0.425309i	$-0.139834\pi$
$293$	30.9839			1.81010			0.905048	+	0.425309i	$0.139834\pi$
$294$		0			0
$295$	30.0000			1.74667
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.74597			0.447961
$300$		0			0
$301$	−4.00000	−	3.46410i	−0.230556	−	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	26.8328i		−	1.53644i
$306$		0			0
$307$		−	5.19615i		−	0.296560i	−0.988945	−	0.148280i	$-0.952626\pi$
$307$		−	5.19615i		−	0.296560i	0.988945	−	0.148280i	$-0.0473737\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.74597			−0.439233			−0.219617	−	0.975586i	$-0.570481\pi$
$311$	−7.74597			−0.439233			−0.219617	+	0.975586i	$0.570481\pi$
$312$		0			0
$313$		−	3.46410i		−	0.195803i	−0.995196	−	0.0979013i	$-0.968787\pi$
$313$		−	3.46410i		−	0.195803i	0.995196	−	0.0979013i	$-0.0312129\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.47214i		−	0.251180i	−0.992082	−	0.125590i	$-0.959918\pi$
$317$		−	4.47214i		−	0.251180i	0.992082	−	0.125590i	$-0.0400824\pi$
$318$		0			0
$319$	−10.0000			−0.559893
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$			34.6410i			1.92154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	15.4919	+	13.4164i	0.854098	+	0.739671i
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−38.7298			−2.11604
$336$		0			0
$337$	35.0000			1.90657			0.953286	−	0.302070i	$-0.0976776\pi$
$337$	35.0000			1.90657			0.953286	+	0.302070i	$0.0976776\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.87298			0.209734
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			11.1803i			0.600192i	0.953909	+	0.300096i	$0.0970187\pi$
$347$			11.1803i			0.600192i	−0.953909	+	0.300096i	$0.902981\pi$
$348$		0			0
$349$		−	13.8564i		−	0.741716i	−0.928689	−	0.370858i	$-0.879064\pi$
$349$		−	13.8564i		−	0.741716i	0.928689	−	0.370858i	$-0.120936\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.3649			−1.03069			−0.515345	−	0.856983i	$-0.672336\pi$
$353$	−19.3649			−1.03069			−0.515345	+	0.856983i	$0.672336\pi$
$354$		0			0
$355$		−	43.3013i		−	2.29819i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.94427i		−	0.472061i	−0.971746	−	0.236030i	$-0.924154\pi$
$359$		−	8.94427i		−	0.472061i	0.971746	−	0.236030i	$-0.0758465\pi$
$360$		0			0
$361$	−8.00000			−0.421053
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	40.2492i		−	2.10674i
$366$		0			0
$367$		−	1.73205i		−	0.0904123i	−0.998978	−	0.0452062i	$-0.985606\pi$
$367$		−	1.73205i		−	0.0904123i	0.998978	−	0.0452062i	$-0.0143945\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−15.4919	+	17.8885i	−0.804301	+	0.928727i
$372$		0			0
$373$	17.0000			0.880227			0.440113	−	0.897942i	$-0.354938\pi$
$373$	17.0000			0.880227			0.440113	+	0.897942i	$0.354938\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.4919			0.797875
$378$		0			0
$379$	−26.0000			−1.33553			−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000			−1.33553			−0.667765	+	0.744372i	$0.732749\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−30.9839			−1.58320			−0.791601	−	0.611039i	$-0.790752\pi$
$383$	−30.9839			−1.58320			−0.791601	+	0.611039i	$0.790752\pi$
$384$		0			0
$385$	−15.0000	+	17.3205i	−0.764471	+	0.882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			35.7771i			1.81397i	0.421163	+	0.906985i	$0.361622\pi$
$389$			35.7771i			1.81397i	−0.421163	+	0.906985i	$0.638378\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.74597			0.389742
$396$		0			0
$397$		−	20.7846i		−	1.04315i	−0.853206	−	0.521575i	$-0.825345\pi$
$397$		−	20.7846i		−	1.04315i	0.853206	−	0.521575i	$-0.174655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	31.3050i		−	1.56329i	−0.623721	−	0.781647i	$-0.714380\pi$
$401$		−	31.3050i		−	1.56329i	0.623721	−	0.781647i	$-0.285620\pi$
$402$		0			0
$403$	−6.00000			−0.298881
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.23607i			0.110838i
$408$		0			0
$409$		−	17.3205i		−	0.856444i	−0.903674	−	0.428222i	$-0.859140\pi$
$409$		−	17.3205i		−	0.856444i	0.903674	−	0.428222i	$-0.140860\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15.4919	−	13.4164i	−0.762308	−	0.660178i
$414$		0			0
$415$	−30.0000			−1.47264
$416$		0			0
$417$		0			0
$418$		0			0
$419$	15.4919			0.756830			0.378415	−	0.925636i	$-0.376469\pi$
$419$	15.4919			0.756830			0.378415	+	0.925636i	$0.376469\pi$
$420$		0			0
$421$	11.0000			0.536107			0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000			0.536107			0.268054	+	0.963404i	$0.413620\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−12.0000	+	13.8564i	−0.580721	+	0.670559i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	2.23607i		−	0.107708i	−0.998549	−	0.0538538i	$-0.982850\pi$
$431$		−	2.23607i		−	0.107708i	0.998549	−	0.0538538i	$-0.0171505\pi$
$432$		0			0
$433$			10.3923i			0.499422i	0.968320	+	0.249711i	$0.0803357\pi$
$433$			10.3923i			0.499422i	−0.968320	+	0.249711i	$0.919664\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.6190			−0.555810
$438$		0			0
$439$		−	38.1051i		−	1.81866i	−0.416078	−	0.909329i	$-0.636596\pi$
$439$		−	38.1051i		−	1.81866i	0.416078	−	0.909329i	$-0.363404\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			24.5967i			1.16863i	0.811528	+	0.584313i	$0.198636\pi$
$443$			24.5967i			1.16863i	−0.811528	+	0.584313i	$0.801364\pi$
$444$		0			0
$445$	−45.0000			−2.13320
$446$		0			0
$447$		0			0
$448$		0			0
$449$			8.94427i			0.422106i	0.977475	+	0.211053i	$0.0676893\pi$
$449$			8.94427i			0.422106i	−0.977475	+	0.211053i	$0.932311\pi$
$450$		0			0
$451$			8.66025i			0.407795i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	23.2379	−	26.8328i	1.08941	−	1.25794i
$456$		0			0
$457$	−25.0000			−1.16945			−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000			−1.16945			−0.584725	+	0.811231i	$0.698798\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.87298			0.180383			0.0901914	−	0.995924i	$-0.471252\pi$
$461$	3.87298			0.180383			0.0901914	+	0.995924i	$0.471252\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	23.2379			1.07532			0.537661	−	0.843161i	$-0.319308\pi$
$467$	23.2379			1.07532			0.537661	+	0.843161i	$0.319308\pi$
$468$		0			0
$469$	20.0000	+	17.3205i	0.923514	+	0.799787i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.47214i			0.205629i
$474$		0			0
$475$		−	51.9615i		−	2.38416i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.9839			1.41569			0.707845	−	0.706368i	$-0.249668\pi$
$479$	30.9839			1.41569			0.707845	+	0.706368i	$0.249668\pi$
$480$		0			0
$481$		−	3.46410i		−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			53.6656i			2.43683i
$486$		0			0
$487$	10.0000			0.453143			0.226572	−	0.973995i	$-0.427248\pi$
$487$	10.0000			0.453143			0.226572	+	0.973995i	$0.427248\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			38.0132i			1.71551i	0.514059	+	0.857755i	$0.328141\pi$
$491$			38.0132i			1.71551i	−0.514059	+	0.857755i	$0.671859\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−19.3649	+	22.3607i	−0.868635	+	1.00301i
$498$		0			0
$499$	−38.0000			−1.70111			−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000			−1.70111			−0.850557	+	0.525883i	$0.823735\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−23.2379			−1.03613			−0.518063	−	0.855342i	$-0.673347\pi$
$503$	−23.2379			−1.03613			−0.518063	+	0.855342i	$0.673347\pi$
$504$		0			0
$505$	−60.0000			−2.66996
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.4919			−0.686668			−0.343334	−	0.939213i	$-0.611556\pi$
$509$	−15.4919			−0.686668			−0.343334	+	0.939213i	$0.611556\pi$
$510$		0			0
$511$	−18.0000	+	20.7846i	−0.796273	+	0.919457i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	6.70820i		−	0.295599i
$516$		0			0
$517$		−	17.3205i		−	0.761755i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	34.8569			1.52711			0.763553	−	0.645745i	$-0.223453\pi$
$521$	34.8569			1.52711			0.763553	+	0.645745i	$0.223453\pi$
$522$		0			0
$523$		−	15.5885i		−	0.681636i	−0.940129	−	0.340818i	$-0.889296\pi$
$523$		−	15.5885i		−	0.681636i	0.940129	−	0.340818i	$-0.110704\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	18.0000			0.782609
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	13.4164i		−	0.581129i
$534$		0			0
$535$			34.6410i			1.49766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	15.4919	−	2.23607i	0.667285	−	0.0963143i
$540$		0			0
$541$	−1.00000			−0.0429934			−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000			−0.0429934			−0.0214967	+	0.999769i	$0.506843\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	27.1109			1.16130
$546$		0			0
$547$	16.0000			0.684111			0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000			0.684111			0.342055	+	0.939680i	$0.388877\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−23.2379			−0.989968
$552$		0			0
$553$	−4.00000	−	3.46410i	−0.170097	−	0.147309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			8.94427i			0.378981i	0.981883	+	0.189490i	$0.0606836\pi$
$557$			8.94427i			0.378981i	−0.981883	+	0.189490i	$0.939316\pi$
$558$		0			0
$559$		−	6.92820i		−	0.293032i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	38.7298			1.63227			0.816134	−	0.577863i	$-0.196113\pi$
$563$	38.7298			1.63227			0.816134	+	0.577863i	$0.196113\pi$
$564$		0			0
$565$		−	34.6410i		−	1.45736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	17.8885i		−	0.749927i	−0.927040	−	0.374963i	$-0.877655\pi$
$569$		−	17.8885i		−	0.749927i	0.927040	−	0.374963i	$-0.122345\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	22.3607i		−	0.932505i
$576$		0			0
$577$		−	20.7846i		−	0.865275i	−0.901568	−	0.432637i	$-0.857583\pi$
$577$		−	20.7846i		−	0.865275i	0.901568	−	0.432637i	$-0.142417\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.4919	+	13.4164i	0.642714	+	0.556606i
$582$		0			0
$583$	20.0000			0.828315
$584$		0			0
$585$		0			0
$586$		0			0
$587$	30.9839			1.27884			0.639421	−	0.768857i	$-0.279174\pi$
$587$	30.9839			1.27884			0.639421	+	0.768857i	$0.279174\pi$
$588$		0			0
$589$	9.00000			0.370839
$590$		0			0
$591$		0			0
$592$		0			0
$593$	34.8569			1.43140			0.715700	−	0.698408i	$-0.246108\pi$
$593$	34.8569			1.43140			0.715700	+	0.698408i	$0.246108\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	29.0689i		−	1.18772i	−0.804568	−	0.593861i	$-0.797603\pi$
$599$		−	29.0689i		−	1.18772i	0.804568	−	0.593861i	$-0.202397\pi$
$600$		0			0
$601$		−	24.2487i		−	0.989126i	−0.869142	−	0.494563i	$-0.835328\pi$
$601$		−	24.2487i		−	0.989126i	0.869142	−	0.494563i	$-0.164672\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−23.2379			−0.944755
$606$		0			0
$607$			38.1051i			1.54664i	0.634017	+	0.773320i	$0.281405\pi$
$607$			38.1051i			1.54664i	−0.634017	+	0.773320i	$0.718595\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			26.8328i			1.08554i
$612$		0			0
$613$	17.0000			0.686624			0.343312	−	0.939222i	$-0.388451\pi$
$613$	17.0000			0.686624			0.343312	+	0.939222i	$0.388451\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			22.3607i			0.900207i	0.892976	+	0.450104i	$0.148613\pi$
$617$			22.3607i			0.900207i	−0.892976	+	0.450104i	$0.851387\pi$
$618$		0			0
$619$		−	22.5167i		−	0.905021i	−0.891759	−	0.452510i	$-0.850529\pi$
$619$		−	22.5167i		−	0.905021i	0.891759	−	0.452510i	$-0.149471\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	23.2379	+	20.1246i	0.931007	+	0.806276i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−2.00000			−0.0796187			−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000			−0.0796187			−0.0398094	+	0.999207i	$0.512675\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−38.7298			−1.53695
$636$		0			0
$637$	−24.0000	+	3.46410i	−0.950915	+	0.137253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	17.8885i		−	0.706555i	−0.935519	−	0.353278i	$-0.885067\pi$
$641$		−	17.8885i		−	0.706555i	0.935519	−	0.353278i	$-0.114933\pi$
$642$		0			0
$643$		−	39.8372i		−	1.57102i	−0.618846	−	0.785512i	$-0.712400\pi$
$643$		−	39.8372i		−	1.57102i	0.618846	−	0.785512i	$-0.287600\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−23.2379			−0.913576			−0.456788	−	0.889576i	$-0.651000\pi$
$647$	−23.2379			−0.913576			−0.456788	+	0.889576i	$0.651000\pi$
$648$		0			0
$649$			17.3205i			0.679889i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			49.1935i			1.92509i	0.271122	+	0.962545i	$0.412605\pi$
$653$			49.1935i			1.92509i	−0.271122	+	0.962545i	$0.587395\pi$
$654$		0			0
$655$	30.0000			1.17220
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.23607i		−	0.0871048i	−0.999051	−	0.0435524i	$-0.986132\pi$
$659$		−	2.23607i		−	0.0871048i	0.999051	−	0.0435524i	$-0.0138676\pi$
$660$		0			0
$661$			45.0333i			1.75159i	0.482680	+	0.875797i	$0.339663\pi$
$661$			45.0333i			1.75159i	−0.482680	+	0.875797i	$0.660337\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−34.8569	+	40.2492i	−1.35169	+	1.56080i
$666$		0			0
$667$	−10.0000			−0.387202
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.4919			0.598059
$672$		0			0
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	27.1109			1.04196			0.520978	−	0.853570i	$-0.325567\pi$
$677$	27.1109			1.04196			0.520978	+	0.853570i	$0.325567\pi$
$678$		0			0
$679$	24.0000	−	27.7128i	0.921035	−	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	29.0689i		−	1.11229i	−0.831085	−	0.556145i	$-0.812280\pi$
$683$		−	29.0689i		−	1.11229i	0.831085	−	0.556145i	$-0.187720\pi$
$684$		0			0
$685$		−	86.6025i		−	3.30891i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30.9839			−1.18039
$690$		0			0
$691$		−	17.3205i		−	0.658903i	−0.944172	−	0.329452i	$-0.893136\pi$
$691$		−	17.3205i		−	0.658903i	0.944172	−	0.329452i	$-0.106864\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			13.4164i			0.508913i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	31.3050i		−	1.18237i	−0.806535	−	0.591186i	$-0.798660\pi$
$701$		−	31.3050i		−	1.18237i	0.806535	−	0.591186i	$-0.201340\pi$
$702$		0			0
$703$			5.19615i			0.195977i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30.9839	+	26.8328i	1.16527	+	1.00915i
$708$		0			0
$709$	−43.0000			−1.61490			−0.807449	−	0.589937i	$-0.799153\pi$
$709$	−43.0000			−1.61490			−0.807449	+	0.589937i	$0.799153\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.87298			0.145044
$714$		0			0
$715$	−30.0000			−1.12194
$716$		0			0
$717$		0			0
$718$		0			0
$719$	23.2379			0.866627			0.433314	−	0.901243i	$-0.357344\pi$
$719$	23.2379			0.866627			0.433314	+	0.901243i	$0.357344\pi$
$720$		0			0
$721$	−3.00000	+	3.46410i	−0.111726	+	0.129010i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	44.7214i		−	1.66091i
$726$		0			0
$727$			3.46410i			0.128476i	0.997935	+	0.0642382i	$0.0204617\pi$
$727$			3.46410i			0.128476i	−0.997935	+	0.0642382i	$0.979538\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	38.1051i		−	1.40744i	−0.710475	−	0.703722i	$-0.751520\pi$
$733$		−	38.1051i		−	1.40744i	0.710475	−	0.703722i	$-0.248480\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	22.3607i		−	0.823666i
$738$		0			0
$739$	−44.0000			−1.61857			−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000			−1.61857			−0.809283	+	0.587419i	$0.800144\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			11.1803i			0.410167i	0.978744	+	0.205083i	$0.0657466\pi$
$743$			11.1803i			0.410167i	−0.978744	+	0.205083i	$0.934253\pi$
$744$		0			0
$745$		−	34.6410i		−	1.26915i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.4919	−	17.8885i	0.566063	−	0.653633i
$750$		0			0
$751$	46.0000			1.67856			0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000			1.67856			0.839282	+	0.543696i	$0.182976\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−61.9677			−2.25524
$756$		0			0
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.4919			−0.561582			−0.280791	−	0.959769i	$-0.590597\pi$
$761$	−15.4919			−0.561582			−0.280791	+	0.959769i	$0.590597\pi$
$762$		0			0
$763$	−14.0000	−	12.1244i	−0.506834	−	0.438931i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	26.8328i		−	0.968877i
$768$		0			0
$769$		−	17.3205i		−	0.624593i	−0.949985	−	0.312297i	$-0.898902\pi$
$769$		−	17.3205i		−	0.624593i	0.949985	−	0.312297i	$-0.101098\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.6190			0.417905			0.208952	−	0.977926i	$-0.432995\pi$
$773$	11.6190			0.417905			0.208952	+	0.977926i	$0.432995\pi$
$774$		0			0
$775$			17.3205i			0.622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			20.1246i			0.721039i
$780$		0			0
$781$	25.0000			0.894570
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	13.4164i		−	0.478852i
$786$		0			0
$787$			38.1051i			1.35830i	0.733999	+	0.679150i	$0.237652\pi$
$787$			38.1051i			1.35830i	−0.733999	+	0.679150i	$0.762348\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15.4919	+	17.8885i	−0.550830	+	0.636043i
$792$		0			0
$793$	−24.0000			−0.852265
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.87298			−0.137188			−0.0685941	−	0.997645i	$-0.521851\pi$
$797$	−3.87298			−0.137188			−0.0685941	+	0.997645i	$0.521851\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.2379			0.820048
$804$		0			0
$805$	−15.0000	+	17.3205i	−0.528681	+	0.610468i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	31.3050i		−	1.10062i	−0.834959	−	0.550312i	$-0.814509\pi$
$809$		−	31.3050i		−	1.10062i	0.834959	−	0.550312i	$-0.185491\pi$
$810$		0			0
$811$		−	36.3731i		−	1.27723i	−0.769526	−	0.638616i	$-0.779507\pi$
$811$		−	36.3731i		−	1.27723i	0.769526	−	0.638616i	$-0.220493\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−38.7298			−1.35665
$816$		0			0
$817$			10.3923i			0.363581i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			22.3607i			0.780393i	0.920732	+	0.390197i	$0.127593\pi$
$821$			22.3607i			0.780393i	−0.920732	+	0.390197i	$0.872407\pi$
$822$		0			0
$823$	−20.0000			−0.697156			−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000			−0.697156			−0.348578	+	0.937280i	$0.613335\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			11.1803i			0.388779i	0.980924	+	0.194389i	$0.0622725\pi$
$827$			11.1803i			0.388779i	−0.980924	+	0.194389i	$0.937728\pi$
$828$		0			0
$829$			31.1769i			1.08282i	0.840759	+	0.541409i	$0.182109\pi$
$829$			31.1769i			1.08282i	−0.840759	+	0.541409i	$0.817891\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−60.0000			−2.07639
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−38.7298			−1.33710			−0.668551	−	0.743666i	$-0.733085\pi$
$839$	−38.7298			−1.33710			−0.668551	+	0.743666i	$0.733085\pi$
$840$		0			0
$841$	9.00000			0.310345
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.87298			−0.133235
$846$		0			0
$847$	12.0000	+	10.3923i	0.412325	+	0.357084i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			2.23607i			0.0766514i
$852$		0			0
$853$		−	13.8564i		−	0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$		−	13.8564i		−	0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	27.1109			0.926090			0.463045	−	0.886335i	$-0.346757\pi$
$857$	27.1109			0.926090			0.463045	+	0.886335i	$0.346757\pi$
$858$		0			0
$859$			22.5167i			0.768259i	0.923279	+	0.384129i	$0.125498\pi$
$859$			22.5167i			0.768259i	−0.923279	+	0.384129i	$0.874502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	8.94427i		−	0.304467i	−0.988345	−	0.152233i	$-0.951353\pi$
$863$		−	8.94427i		−	0.304467i	0.988345	−	0.152233i	$-0.0486465\pi$
$864$		0			0
$865$	−15.0000			−0.510015
$866$		0			0
$867$		0			0
$868$		0			0
$869$			4.47214i			0.151707i
$870$		0			0
$871$			34.6410i			1.17377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−38.7298	−	33.5410i	−1.30931	−	1.13389i
$876$		0			0
$877$	−46.0000			−1.55331			−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000			−1.55331			−0.776655	+	0.629926i	$0.783085\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−34.8569			−1.17436			−0.587179	−	0.809457i	$-0.699761\pi$
$881$	−34.8569			−1.17436			−0.587179	+	0.809457i	$0.699761\pi$
$882$		0			0
$883$	−8.00000			−0.269221			−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000			−0.269221			−0.134611	+	0.990899i	$0.542978\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.74597			−0.260084			−0.130042	−	0.991508i	$-0.541511\pi$
$887$	−7.74597			−0.260084			−0.130042	+	0.991508i	$0.541511\pi$
$888$		0			0
$889$	20.0000	+	17.3205i	0.670778	+	0.580911i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	40.2492i		−	1.34689i
$894$		0			0
$895$		−	69.2820i		−	2.31584i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7.74597			0.258342
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	16.0000			0.531271			0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000			0.531271			0.265636	+	0.964073i	$0.414418\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	35.7771i		−	1.18535i	−0.805443	−	0.592674i	$-0.798072\pi$
$911$		−	35.7771i		−	1.18535i	0.805443	−	0.592674i	$-0.201928\pi$
$912$		0			0
$913$		−	17.3205i		−	0.573225i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.4919	−	13.4164i	−0.511589	−	0.443049i
$918$		0			0
$919$	28.0000			0.923635			0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000			0.923635			0.461817	+	0.886975i	$0.347198\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−38.7298			−1.27481
$924$		0			0
$925$	−10.0000			−0.328798
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.4919			0.508274			0.254137	−	0.967168i	$-0.418209\pi$
$929$	15.4919			0.508274			0.254137	+	0.967168i	$0.418209\pi$
$930$		0			0
$931$	36.0000	−	5.19615i	1.17985	−	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			20.7846i			0.679004i	0.940605	+	0.339502i	$0.110258\pi$
$937$			20.7846i			0.679004i	−0.940605	+	0.339502i	$0.889742\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.87298			−0.126256			−0.0631278	−	0.998005i	$-0.520108\pi$
$941$	−3.87298			−0.126256			−0.0631278	+	0.998005i	$0.520108\pi$
$942$		0			0
$943$			8.66025i			0.282017i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	15.6525i		−	0.508637i	−0.967121	−	0.254319i	$-0.918149\pi$
$947$		−	15.6525i		−	0.508637i	0.967121	−	0.254319i	$-0.0818512\pi$
$948$		0			0
$949$	−36.0000			−1.16861
$950$		0			0
$951$		0			0
$952$		0			0
$953$			22.3607i			0.724333i	0.932113	+	0.362167i	$0.117963\pi$
$953$			22.3607i			0.724333i	−0.932113	+	0.362167i	$0.882037\pi$
$954$		0			0
$955$			8.66025i			0.280239i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−38.7298	+	44.7214i	−1.25065	+	1.44413i
$960$		0			0
$961$	28.0000			0.903226
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.74597			−0.249351
$966$		0			0
$967$	−56.0000			−1.80084			−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000			−1.80084			−0.900419	+	0.435023i	$0.856740\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	23.2379			0.745740			0.372870	−	0.927884i	$-0.378374\pi$
$971$	23.2379			0.745740			0.372870	+	0.927884i	$0.378374\pi$
$972$		0			0
$973$	6.00000	−	6.92820i	0.192351	−	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	17.8885i		−	0.572305i	−0.958184	−	0.286153i	$-0.907624\pi$
$977$		−	17.8885i		−	0.572305i	0.958184	−	0.286153i	$-0.0923764\pi$
$978$		0			0
$979$		−	25.9808i		−	0.830349i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	54.2218			1.72941			0.864703	−	0.502284i	$-0.167507\pi$
$983$	54.2218			1.72941			0.864703	+	0.502284i	$0.167507\pi$
$984$		0			0
$985$		−	34.6410i		−	1.10375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			4.47214i			0.142206i
$990$		0			0
$991$	−26.0000			−0.825917			−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000			−0.825917			−0.412959	+	0.910750i	$0.635505\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	100.623i		−	3.18997i
$996$		0			0
$997$			27.7128i			0.877674i	0.898567	+	0.438837i	$0.144609\pi$
$997$			27.7128i			0.877674i	−0.898567	+	0.438837i	$0.855391\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.k.i.1889.2		4
3.2	odd	2	inner	3024.2.k.i.1889.4		4
4.3	odd	2		189.2.c.b.188.3	yes	4
7.6	odd	2	inner	3024.2.k.i.1889.3		4
12.11	even	2		189.2.c.b.188.2	yes	4
21.20	even	2	inner	3024.2.k.i.1889.1		4
28.27	even	2		189.2.c.b.188.4	yes	4
36.7	odd	6		567.2.o.c.377.2		4
36.11	even	6		567.2.o.c.377.1		4
36.23	even	6		567.2.o.d.188.2		4
36.31	odd	6		567.2.o.d.188.1		4
84.83	odd	2		189.2.c.b.188.1	✓	4
252.83	odd	6		567.2.o.d.377.1		4
252.139	even	6		567.2.o.c.188.1		4
252.167	odd	6		567.2.o.c.188.2		4
252.223	even	6		567.2.o.d.377.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.c.b.188.1	✓	4	84.83	odd	2
189.2.c.b.188.2	yes	4	12.11	even	2
189.2.c.b.188.3	yes	4	4.3	odd	2
189.2.c.b.188.4	yes	4	28.27	even	2
567.2.o.c.188.1		4	252.139	even	6
567.2.o.c.188.2		4	252.167	odd	6
567.2.o.c.377.1		4	36.11	even	6
567.2.o.c.377.2		4	36.7	odd	6
567.2.o.d.188.1		4	36.31	odd	6
567.2.o.d.188.2		4	36.23	even	6
567.2.o.d.377.1		4	252.83	odd	6
567.2.o.d.377.2		4	252.223	even	6
3024.2.k.i.1889.1		4	21.20	even	2	inner
3024.2.k.i.1889.2		4	1.1	even	1	trivial
3024.2.k.i.1889.3		4	7.6	odd	2	inner
3024.2.k.i.1889.4		4	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)

\(f(q)\)	\(=\)	\(q-3.87298 q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q-3.87298 q^{5} +(2.00000 + 1.73205i) q^{7} -2.23607i q^{11} +3.46410i q^{13} -5.19615i q^{19} -2.23607i q^{23} +10.0000 q^{25} -4.47214i q^{29} +1.73205i q^{31} +(-7.74597 - 6.70820i) q^{35} -1.00000 q^{37} -3.87298 q^{41} -2.00000 q^{43} +7.74597 q^{47} +(1.00000 + 6.92820i) q^{49} +8.94427i q^{53} +8.66025i q^{55} -7.74597 q^{59} +6.92820i q^{61} -13.4164i q^{65} +10.0000 q^{67} +11.1803i q^{71} +10.3923i q^{73} +(3.87298 - 4.47214i) q^{77} -2.00000 q^{79} +7.74597 q^{83} +11.6190 q^{89} +(-6.00000 + 6.92820i) q^{91} +20.1246i q^{95} -13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} + 40 q^{25} - 4 q^{37} - 8 q^{43} + 4 q^{49} + 40 q^{67} - 8 q^{79} - 24 q^{91}+O(q^{100}) \) 4 * q + 8 * q^7 + 40 * q^25 - 4 * q^37 - 8 * q^43 + 4 * q^49 + 40 * q^67 - 8 * q^79 - 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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