LMFDB - Embedded newform 3150.3.c.f.449.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (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:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.9671731157401600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 $ x^16 + 7x^12 + 753x^8 + 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.11
Root			$0.941471 - 2.08559i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.f.449.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} +2.82843 q^{8} +11.2717i q^{11} -22.9496i q^{13} -3.74166i q^{14} +4.00000 q^{16} -8.36199 q^{17} -3.57092 q^{19} +15.9406i q^{22} +3.92441 q^{23} -32.4556i q^{26} -5.29150i q^{28} +43.5101i q^{29} +46.3702 q^{31} +5.65685 q^{32} -11.8256 q^{34} -27.9648i q^{37} -5.05005 q^{38} -32.8306i q^{41} -65.5572i q^{43} +22.5434i q^{44} +5.54995 q^{46} +5.65248 q^{47} -7.00000 q^{49} -45.8992i q^{52} -24.7975 q^{53} -7.48331i q^{56} +61.5325i q^{58} -58.6564i q^{59} +28.2418 q^{61} +65.5774 q^{62} +8.00000 q^{64} +10.7504i q^{67} -16.7240 q^{68} -20.8535i q^{71} -113.896i q^{73} -39.5482i q^{74} -7.14185 q^{76} +29.8222 q^{77} -133.236 q^{79} -46.4295i q^{82} -30.9830 q^{83} -92.7119i q^{86} +31.8812i q^{88} -79.6261i q^{89} -60.7189 q^{91} +7.84882 q^{92} +7.99382 q^{94} -26.1819i q^{97} -9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

Display 100 coefficients 10 coefficients

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	2.82843	0.353553
$9$	0	0
$10$	0	0
$11$	11.2717i	1.02470i	0.858776	+	0.512351i	$0.171225\pi$
$11$	11.2717i	1.02470i	−0.858776	+	0.512351i	$0.828775\pi$
$12$	0	0
$13$	− 22.9496i	− 1.76535i	−0.469980	−	0.882677i	$-0.655739\pi$
$13$	− 22.9496i	− 1.76535i	0.469980	−	0.882677i	$-0.344261\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	−8.36199	−0.491882	−0.245941	−	0.969285i	$-0.579097\pi$
$17$	−8.36199	−0.491882	−0.245941	+	0.969285i	$0.579097\pi$
$18$	0	0
$19$	−3.57092	−0.187943	−0.0939717	−	0.995575i	$-0.529956\pi$
$19$	−3.57092	−0.187943	−0.0939717	+	0.995575i	$0.529956\pi$
$20$	0	0
$21$	0	0
$22$	15.9406i	0.724573i
$23$	3.92441	0.170626	0.0853132	−	0.996354i	$-0.472811\pi$
$23$	3.92441	0.170626	0.0853132	+	0.996354i	$0.472811\pi$
$24$	0	0
$25$	0	0
$26$	− 32.4556i	− 1.24829i
$27$	0	0
$28$	− 5.29150i	− 0.188982i
$29$	43.5101i	1.50035i	0.661241	+	0.750173i	$0.270030\pi$
$29$	43.5101i	1.50035i	−0.661241	+	0.750173i	$0.729970\pi$
$30$	0	0
$31$	46.3702	1.49581	0.747907	−	0.663804i	$-0.231059\pi$
$31$	46.3702	1.49581	0.747907	+	0.663804i	$0.231059\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	−11.8256	−0.347813
$35$	0	0
$36$	0	0
$37$	− 27.9648i	− 0.755805i	−0.925845	−	0.377902i	$-0.876645\pi$
$37$	− 27.9648i	− 0.755805i	0.925845	−	0.377902i	$-0.123355\pi$
$38$	−5.05005	−0.132896
$39$	0	0
$40$	0	0
$41$	− 32.8306i	− 0.800748i	−0.916352	−	0.400374i	$-0.868880\pi$
$41$	− 32.8306i	− 0.800748i	0.916352	−	0.400374i	$-0.131120\pi$
$42$	0	0
$43$	− 65.5572i	− 1.52459i	−0.647232	−	0.762293i	$-0.724073\pi$
$43$	− 65.5572i	− 1.52459i	0.647232	−	0.762293i	$-0.275927\pi$
$44$	22.5434i	0.512351i
$45$	0	0
$46$	5.54995	0.120651
$47$	5.65248	0.120266	0.0601328	−	0.998190i	$-0.480848\pi$
$47$	5.65248	0.120266	0.0601328	+	0.998190i	$0.480848\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 45.8992i	− 0.882677i
$53$	−24.7975	−0.467878	−0.233939	−	0.972251i	$-0.575162\pi$
$53$	−24.7975	−0.467878	−0.233939	+	0.972251i	$0.575162\pi$
$54$	0	0
$55$	0	0
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	61.5325i	1.06091i
$59$	− 58.6564i	− 0.994176i	−0.867700	−	0.497088i	$-0.834403\pi$
$59$	− 58.6564i	− 0.994176i	0.867700	−	0.497088i	$-0.165597\pi$
$60$	0	0
$61$	28.2418	0.462980	0.231490	−	0.972837i	$-0.425640\pi$
$61$	28.2418	0.462980	0.231490	+	0.972837i	$0.425640\pi$
$62$	65.5774	1.05770
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.7504i	0.160454i	0.996777	+	0.0802271i	$0.0255646\pi$
$67$	10.7504i	0.160454i	−0.996777	+	0.0802271i	$0.974435\pi$
$68$	−16.7240	−0.245941
$69$	0	0
$70$	0	0
$71$	− 20.8535i	− 0.293711i	−0.989158	−	0.146856i	$-0.953085\pi$
$71$	− 20.8535i	− 0.293711i	0.989158	−	0.146856i	$-0.0469153\pi$
$72$	0	0
$73$	− 113.896i	− 1.56022i	−0.625642	−	0.780110i	$-0.715163\pi$
$73$	− 113.896i	− 1.56022i	0.625642	−	0.780110i	$-0.284837\pi$
$74$	− 39.5482i	− 0.534435i
$75$	0	0
$76$	−7.14185	−0.0939717
$77$	29.8222	0.387301
$78$	0	0
$79$	−133.236	−1.68654	−0.843268	−	0.537494i	$-0.819371\pi$
$79$	−133.236	−1.68654	−0.843268	+	0.537494i	$0.819371\pi$
$80$	0	0
$81$	0	0
$82$	− 46.4295i	− 0.566214i
$83$	−30.9830	−0.373289	−0.186644	−	0.982428i	$-0.559761\pi$
$83$	−30.9830	−0.373289	−0.186644	+	0.982428i	$0.559761\pi$
$84$	0	0
$85$	0	0
$86$	− 92.7119i	− 1.07805i
$87$	0	0
$88$	31.8812i	0.362287i
$89$	− 79.6261i	− 0.894675i	−0.894365	−	0.447337i	$-0.852372\pi$
$89$	− 79.6261i	− 0.894675i	0.894365	−	0.447337i	$-0.147628\pi$
$90$	0	0
$91$	−60.7189	−0.667241
$92$	7.84882	0.0853132
$93$	0	0
$94$	7.99382	0.0850406
$95$	0	0
$96$	0	0
$97$	− 26.1819i	− 0.269917i	−0.990851	−	0.134958i	$-0.956910\pi$
$97$	− 26.1819i	− 0.269917i	0.990851	−	0.134958i	$-0.0430901\pi$
$98$	−9.89949	−0.101015
$99$	0	0
$100$	0	0
$101$	− 170.729i	− 1.69038i	−0.534465	−	0.845191i	$-0.679487\pi$
$101$	− 170.729i	− 1.69038i	0.534465	−	0.845191i	$-0.320513\pi$
$102$	0	0
$103$	159.260i	1.54622i	0.634274	+	0.773108i	$0.281299\pi$
$103$	159.260i	1.54622i	−0.634274	+	0.773108i	$0.718701\pi$
$104$	− 64.9113i	− 0.624147i
$105$	0	0
$106$	−35.0690	−0.330840
$107$	62.7625	0.586565	0.293283	−	0.956026i	$-0.405252\pi$
$107$	62.7625	0.586565	0.293283	+	0.956026i	$0.405252\pi$
$108$	0	0
$109$	−44.5696	−0.408895	−0.204448	−	0.978878i	$-0.565540\pi$
$109$	−44.5696	−0.408895	−0.204448	+	0.978878i	$0.565540\pi$
$110$	0	0
$111$	0	0
$112$	− 10.5830i	− 0.0944911i
$113$	128.003	1.13277	0.566385	−	0.824141i	$-0.308341\pi$
$113$	128.003	1.13277	0.566385	+	0.824141i	$0.308341\pi$
$114$	0	0
$115$	0	0
$116$	87.0201i	0.750173i
$117$	0	0
$118$	− 82.9526i	− 0.702988i
$119$	22.1237i	0.185914i
$120$	0	0
$121$	−6.05158	−0.0500130
$122$	39.9399	0.327376
$123$	0	0
$124$	92.7405	0.747907
$125$	0	0
$126$	0	0
$127$	− 99.0952i	− 0.780277i	−0.920756	−	0.390139i	$-0.872427\pi$
$127$	− 99.0952i	− 0.780277i	0.920756	−	0.390139i	$-0.127573\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	0	0
$131$	139.901i	1.06795i	0.845500	+	0.533975i	$0.179302\pi$
$131$	139.901i	1.06795i	−0.845500	+	0.533975i	$0.820698\pi$
$132$	0	0
$133$	9.44777i	0.0710359i
$134$	15.2034i	0.113458i
$135$	0	0
$136$	−23.6513	−0.173906
$137$	−200.351	−1.46242	−0.731208	−	0.682154i	$-0.761043\pi$
$137$	−200.351	−1.46242	−0.731208	+	0.682154i	$0.761043\pi$
$138$	0	0
$139$	89.8593	0.646470	0.323235	−	0.946319i	$-0.395230\pi$
$139$	89.8593	0.646470	0.323235	+	0.946319i	$0.395230\pi$
$140$	0	0
$141$	0	0
$142$	− 29.4913i	− 0.207685i
$143$	258.681	1.80896
$144$	0	0
$145$	0	0
$146$	− 161.073i	− 1.10324i
$147$	0	0
$148$	− 55.9296i	− 0.377902i
$149$	154.434i	1.03647i	0.855239	+	0.518234i	$0.173411\pi$
$149$	154.434i	1.03647i	−0.855239	+	0.518234i	$0.826589\pi$
$150$	0	0
$151$	293.829	1.94588	0.972942	−	0.231048i	$-0.0742153\pi$
$151$	293.829	1.94588	0.972942	+	0.231048i	$0.0742153\pi$
$152$	−10.1001	−0.0664480
$153$	0	0
$154$	42.1749	0.273863
$155$	0	0
$156$	0	0
$157$	− 275.050i	− 1.75191i	−0.482395	−	0.875954i	$-0.660233\pi$
$157$	− 275.050i	− 1.75191i	0.482395	−	0.875954i	$-0.339767\pi$
$158$	−188.425	−1.19256
$159$	0	0
$160$	0	0
$161$	− 10.3830i	− 0.0644907i
$162$	0	0
$163$	− 121.606i	− 0.746048i	−0.927822	−	0.373024i	$-0.878321\pi$
$163$	− 121.606i	− 0.746048i	0.927822	−	0.373024i	$-0.121679\pi$
$164$	− 65.6613i	− 0.400374i
$165$	0	0
$166$	−43.8165	−0.263955
$167$	155.389	0.930475	0.465238	−	0.885186i	$-0.345969\pi$
$167$	155.389	0.930475	0.465238	+	0.885186i	$0.345969\pi$
$168$	0	0
$169$	−357.684	−2.11648
$170$	0	0
$171$	0	0
$172$	− 131.114i	− 0.762293i
$173$	191.015	1.10413	0.552067	−	0.833800i	$-0.313839\pi$
$173$	191.015	1.10413	0.552067	+	0.833800i	$0.313839\pi$
$174$	0	0
$175$	0	0
$176$	45.0869i	0.256175i
$177$	0	0
$178$	− 112.608i	− 0.632631i
$179$	− 321.028i	− 1.79345i	−0.442586	−	0.896726i	$-0.645939\pi$
$179$	− 321.028i	− 1.79345i	0.442586	−	0.896726i	$-0.354061\pi$
$180$	0	0
$181$	−21.3100	−0.117735	−0.0588675	−	0.998266i	$-0.518749\pi$
$181$	−21.3100	−0.117735	−0.0588675	+	0.998266i	$0.518749\pi$
$182$	−85.8696	−0.471811
$183$	0	0
$184$	11.0999	0.0603256
$185$	0	0
$186$	0	0
$187$	− 94.2540i	− 0.504032i
$188$	11.3050	0.0601328
$189$	0	0
$190$	0	0
$191$	− 44.6968i	− 0.234015i	−0.993131	−	0.117007i	$-0.962670\pi$
$191$	− 44.6968i	− 0.234015i	0.993131	−	0.117007i	$-0.0373301\pi$
$192$	0	0
$193$	237.337i	1.22973i	0.788634	+	0.614863i	$0.210789\pi$
$193$	237.337i	1.22973i	−0.788634	+	0.614863i	$0.789211\pi$
$194$	− 37.0268i	− 0.190860i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	−199.475	−1.01256	−0.506282	−	0.862368i	$-0.668980\pi$
$197$	−199.475	−1.01256	−0.506282	+	0.862368i	$0.668980\pi$
$198$	0	0
$199$	−75.1828	−0.377803	−0.188902	−	0.981996i	$-0.560493\pi$
$199$	−75.1828	−0.377803	−0.188902	+	0.981996i	$0.560493\pi$
$200$	0	0
$201$	0	0
$202$	− 241.447i	− 1.19528i
$203$	115.117	0.567078
$204$	0	0
$205$	0	0
$206$	225.228i	1.09334i
$207$	0	0
$208$	− 91.7984i	− 0.441339i
$209$	− 40.2504i	− 0.192586i
$210$	0	0
$211$	177.079	0.839237	0.419619	−	0.907701i	$-0.362164\pi$
$211$	177.079	0.839237	0.419619	+	0.907701i	$0.362164\pi$
$212$	−49.5951	−0.233939
$213$	0	0
$214$	88.7595	0.414764
$215$	0	0
$216$	0	0
$217$	− 122.684i	− 0.565364i
$218$	−63.0309	−0.289133
$219$	0	0
$220$	0	0
$221$	191.904i	0.868346i
$222$	0	0
$223$	− 257.329i	− 1.15394i	−0.816765	−	0.576971i	$-0.804235\pi$
$223$	− 257.329i	− 1.15394i	0.816765	−	0.576971i	$-0.195765\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	181.024	0.800990
$227$	−111.108	−0.489464	−0.244732	−	0.969591i	$-0.578700\pi$
$227$	−111.108	−0.489464	−0.244732	+	0.969591i	$0.578700\pi$
$228$	0	0
$229$	−100.914	−0.440671	−0.220336	−	0.975424i	$-0.570715\pi$
$229$	−100.914	−0.440671	−0.220336	+	0.975424i	$0.570715\pi$
$230$	0	0
$231$	0	0
$232$	123.065i	0.530453i
$233$	−33.7780	−0.144970	−0.0724850	−	0.997370i	$-0.523093\pi$
$233$	−33.7780	−0.144970	−0.0724850	+	0.997370i	$0.523093\pi$
$234$	0	0
$235$	0	0
$236$	− 117.313i	− 0.497088i
$237$	0	0
$238$	31.2877i	0.131461i
$239$	− 440.484i	− 1.84303i	−0.388342	−	0.921515i	$-0.626952\pi$
$239$	− 440.484i	− 1.84303i	0.388342	−	0.921515i	$-0.373048\pi$
$240$	0	0
$241$	−204.074	−0.846780	−0.423390	−	0.905947i	$-0.639160\pi$
$241$	−204.074	−0.846780	−0.423390	+	0.905947i	$0.639160\pi$
$242$	−8.55822	−0.0353645
$243$	0	0
$244$	56.4835	0.231490
$245$	0	0
$246$	0	0
$247$	81.9513i	0.331787i
$248$	131.155	0.528850
$249$	0	0
$250$	0	0
$251$	5.18183i	0.0206447i	0.999947	+	0.0103224i	$0.00328577\pi$
$251$	5.18183i	0.0206447i	−0.999947	+	0.0103224i	$0.996714\pi$
$252$	0	0
$253$	44.2348i	0.174841i
$254$	− 140.142i	− 0.551739i
$255$	0	0
$256$	16.0000	0.0625000
$257$	236.312	0.919501	0.459750	−	0.888048i	$-0.347939\pi$
$257$	236.312	0.919501	0.459750	+	0.888048i	$0.347939\pi$
$258$	0	0
$259$	−73.9878	−0.285667
$260$	0	0
$261$	0	0
$262$	197.851i	0.755155i
$263$	17.5835	0.0668572	0.0334286	−	0.999441i	$-0.489357\pi$
$263$	17.5835	0.0668572	0.0334286	+	0.999441i	$0.489357\pi$
$264$	0	0
$265$	0	0
$266$	13.3612i	0.0502300i
$267$	0	0
$268$	21.5009i	0.0802271i
$269$	− 19.9314i	− 0.0740944i	−0.999314	−	0.0370472i	$-0.988205\pi$
$269$	− 19.9314i	− 0.0740944i	0.999314	−	0.0370472i	$-0.0117952\pi$
$270$	0	0
$271$	−87.0972	−0.321392	−0.160696	−	0.987004i	$-0.551374\pi$
$271$	−87.0972	−0.321392	−0.160696	+	0.987004i	$0.551374\pi$
$272$	−33.4480	−0.122970
$273$	0	0
$274$	−283.339	−1.03408
$275$	0	0
$276$	0	0
$277$	− 384.515i	− 1.38814i	−0.719907	−	0.694071i	$-0.755815\pi$
$277$	− 384.515i	− 1.38814i	0.719907	−	0.694071i	$-0.244185\pi$
$278$	127.080	0.457123
$279$	0	0
$280$	0	0
$281$	− 246.478i	− 0.877147i	−0.898695	−	0.438573i	$-0.855484\pi$
$281$	− 246.478i	− 0.877147i	0.898695	−	0.438573i	$-0.144516\pi$
$282$	0	0
$283$	318.082i	1.12396i	0.827150	+	0.561982i	$0.189961\pi$
$283$	318.082i	1.12396i	−0.827150	+	0.561982i	$0.810039\pi$
$284$	− 41.7070i	− 0.146856i
$285$	0	0
$286$	365.831	1.27913
$287$	−86.8617	−0.302654
$288$	0	0
$289$	−219.077	−0.758052
$290$	0	0
$291$	0	0
$292$	− 227.792i	− 0.780110i
$293$	−397.217	−1.35569	−0.677845	−	0.735205i	$-0.737086\pi$
$293$	−397.217	−1.35569	−0.677845	+	0.735205i	$0.737086\pi$
$294$	0	0
$295$	0	0
$296$	− 79.0963i	− 0.267217i
$297$	0	0
$298$	218.402i	0.732894i
$299$	− 90.0636i	− 0.301216i
$300$	0	0
$301$	−173.448	−0.576239
$302$	415.536	1.37595
$303$	0	0
$304$	−14.2837	−0.0469858
$305$	0	0
$306$	0	0
$307$	− 450.564i	− 1.46764i	−0.679346	−	0.733818i	$-0.737737\pi$
$307$	− 450.564i	− 1.46764i	0.679346	−	0.733818i	$-0.262263\pi$
$308$	59.6443	0.193650
$309$	0	0
$310$	0	0
$311$	484.602i	1.55821i	0.626897	+	0.779103i	$0.284325\pi$
$311$	484.602i	1.55821i	−0.626897	+	0.779103i	$0.715675\pi$
$312$	0	0
$313$	364.372i	1.16413i	0.813143	+	0.582064i	$0.197755\pi$
$313$	364.372i	1.16413i	−0.813143	+	0.582064i	$0.802245\pi$
$314$	− 388.979i	− 1.23879i
$315$	0	0
$316$	−266.473	−0.843268
$317$	510.623	1.61080	0.805399	−	0.592733i	$-0.201951\pi$
$317$	510.623	1.61080	0.805399	+	0.592733i	$0.201951\pi$
$318$	0	0
$319$	−490.433	−1.53741
$320$	0	0
$321$	0	0
$322$	− 14.6838i	− 0.0456018i
$323$	29.8600	0.0924459
$324$	0	0
$325$	0	0
$326$	− 171.977i	− 0.527536i
$327$	0	0
$328$	− 92.8591i	− 0.283107i
$329$	− 14.9551i	− 0.0454561i
$330$	0	0
$331$	199.957	0.604101	0.302050	−	0.953292i	$-0.402329\pi$
$331$	199.957	0.604101	0.302050	+	0.953292i	$0.402329\pi$
$332$	−61.9660	−0.186644
$333$	0	0
$334$	219.754	0.657945
$335$	0	0
$336$	0	0
$337$	− 197.218i	− 0.585217i	−0.956232	−	0.292609i	$-0.905477\pi$
$337$	− 197.218i	− 0.585217i	0.956232	−	0.292609i	$-0.0945233\pi$
$338$	−505.842	−1.49657
$339$	0	0
$340$	0	0
$341$	522.672i	1.53276i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	− 185.424i	− 0.539023i
$345$	0	0
$346$	270.136	0.780741
$347$	265.136	0.764080	0.382040	−	0.924146i	$-0.375222\pi$
$347$	265.136	0.764080	0.382040	+	0.924146i	$0.375222\pi$
$348$	0	0
$349$	−77.8495	−0.223065	−0.111532	−	0.993761i	$-0.535576\pi$
$349$	−77.8495	−0.223065	−0.111532	+	0.993761i	$0.535576\pi$
$350$	0	0
$351$	0	0
$352$	63.7625i	0.181143i
$353$	272.107	0.770842	0.385421	−	0.922741i	$-0.374056\pi$
$353$	272.107	0.770842	0.385421	+	0.922741i	$0.374056\pi$
$354$	0	0
$355$	0	0
$356$	− 159.252i	− 0.447337i
$357$	0	0
$358$	− 454.002i	− 1.26816i
$359$	695.157i	1.93637i	0.250236	+	0.968185i	$0.419492\pi$
$359$	695.157i	1.93637i	−0.250236	+	0.968185i	$0.580508\pi$
$360$	0	0
$361$	−348.249	−0.964677
$362$	−30.1369	−0.0832512
$363$	0	0
$364$	−121.438	−0.333621
$365$	0	0
$366$	0	0
$367$	− 161.491i	− 0.440031i	−0.975496	−	0.220015i	$-0.929389\pi$
$367$	− 161.491i	− 0.440031i	0.975496	−	0.220015i	$-0.0706108\pi$
$368$	15.6976	0.0426566
$369$	0	0
$370$	0	0
$371$	65.6081i	0.176841i
$372$	0	0
$373$	− 459.127i	− 1.23090i	−0.788175	−	0.615451i	$-0.788974\pi$
$373$	− 459.127i	− 1.23090i	0.788175	−	0.615451i	$-0.211026\pi$
$374$	− 133.295i	− 0.356404i
$375$	0	0
$376$	15.9876	0.0425203
$377$	998.539	2.64864
$378$	0	0
$379$	132.150	0.348680	0.174340	−	0.984685i	$-0.444221\pi$
$379$	132.150	0.348680	0.174340	+	0.984685i	$0.444221\pi$
$380$	0	0
$381$	0	0
$382$	− 63.2109i	− 0.165474i
$383$	178.047	0.464876	0.232438	−	0.972611i	$-0.425330\pi$
$383$	178.047	0.464876	0.232438	+	0.972611i	$0.425330\pi$
$384$	0	0
$385$	0	0
$386$	335.646i	0.869548i
$387$	0	0
$388$	− 52.3639i	− 0.134958i
$389$	432.024i	1.11060i	0.831650	+	0.555301i	$0.187397\pi$
$389$	432.024i	1.11060i	−0.831650	+	0.555301i	$0.812603\pi$
$390$	0	0
$391$	−32.8159	−0.0839281
$392$	−19.7990	−0.0505076
$393$	0	0
$394$	−282.100	−0.715990
$395$	0	0
$396$	0	0
$397$	394.193i	0.992929i	0.868057	+	0.496464i	$0.165369\pi$
$397$	394.193i	0.992929i	−0.868057	+	0.496464i	$0.834631\pi$
$398$	−106.325	−0.267147
$399$	0	0
$400$	0	0
$401$	− 711.381i	− 1.77402i	−0.461752	−	0.887009i	$-0.652779\pi$
$401$	− 711.381i	− 1.77402i	0.461752	−	0.887009i	$-0.347221\pi$
$402$	0	0
$403$	− 1064.18i	− 2.64064i
$404$	− 341.457i	− 0.845191i
$405$	0	0
$406$	162.800	0.400985
$407$	315.211	0.774474
$408$	0	0
$409$	532.984	1.30314	0.651570	−	0.758588i	$-0.274111\pi$
$409$	532.984	1.30314	0.651570	+	0.758588i	$0.274111\pi$
$410$	0	0
$411$	0	0
$412$	318.521i	0.773108i
$413$	−155.190	−0.375763
$414$	0	0
$415$	0	0
$416$	− 129.823i	− 0.312073i
$417$	0	0
$418$	− 56.9227i	− 0.136179i
$419$	537.395i	1.28257i	0.767305	+	0.641283i	$0.221597\pi$
$419$	537.395i	1.28257i	−0.767305	+	0.641283i	$0.778403\pi$
$420$	0	0
$421$	41.5394	0.0986685	0.0493342	−	0.998782i	$-0.484290\pi$
$421$	41.5394	0.0986685	0.0493342	+	0.998782i	$0.484290\pi$
$422$	250.428	0.593430
$423$	0	0
$424$	−70.1380	−0.165420
$425$	0	0
$426$	0	0
$427$	− 74.7206i	− 0.174990i
$428$	125.525	0.293283
$429$	0	0
$430$	0	0
$431$	19.8004i	0.0459406i	0.999736	+	0.0229703i	$0.00731232\pi$
$431$	19.8004i	0.0459406i	−0.999736	+	0.0229703i	$0.992688\pi$
$432$	0	0
$433$	709.349i	1.63822i	0.573637	+	0.819110i	$0.305532\pi$
$433$	709.349i	1.63822i	−0.573637	+	0.819110i	$0.694468\pi$
$434$	− 173.502i	− 0.399773i
$435$	0	0
$436$	−89.1391	−0.204448
$437$	−14.0138	−0.0320681
$438$	0	0
$439$	−148.273	−0.337751	−0.168875	−	0.985637i	$-0.554014\pi$
$439$	−148.273	−0.337751	−0.168875	+	0.985637i	$0.554014\pi$
$440$	0	0
$441$	0	0
$442$	271.394i	0.614013i
$443$	119.132	0.268921	0.134461	−	0.990919i	$-0.457070\pi$
$443$	119.132	0.268921	0.134461	+	0.990919i	$0.457070\pi$
$444$	0	0
$445$	0	0
$446$	− 363.918i	− 0.815960i
$447$	0	0
$448$	− 21.1660i	− 0.0472456i
$449$	299.789i	0.667682i	0.942629	+	0.333841i	$0.108345\pi$
$449$	299.789i	0.667682i	−0.942629	+	0.333841i	$0.891655\pi$
$450$	0	0
$451$	370.058	0.820527
$452$	256.006	0.566385
$453$	0	0
$454$	−157.131	−0.346103
$455$	0	0
$456$	0	0
$457$	210.083i	0.459701i	0.973226	+	0.229850i	$0.0738236\pi$
$457$	210.083i	0.459701i	−0.973226	+	0.229850i	$0.926176\pi$
$458$	−142.713	−0.311602
$459$	0	0
$460$	0	0
$461$	635.944i	1.37949i	0.724053	+	0.689744i	$0.242277\pi$
$461$	635.944i	1.37949i	−0.724053	+	0.689744i	$0.757723\pi$
$462$	0	0
$463$	630.339i	1.36142i	0.732551	+	0.680712i	$0.238329\pi$
$463$	630.339i	1.36142i	−0.732551	+	0.680712i	$0.761671\pi$
$464$	174.040i	0.375087i
$465$	0	0
$466$	−47.7693	−0.102509
$467$	−169.232	−0.362381	−0.181190	−	0.983448i	$-0.557995\pi$
$467$	−169.232	−0.362381	−0.181190	+	0.983448i	$0.557995\pi$
$468$	0	0
$469$	28.4430	0.0606460
$470$	0	0
$471$	0	0
$472$	− 165.905i	− 0.351494i
$473$	738.942	1.56225
$474$	0	0
$475$	0	0
$476$	44.2475i	0.0929569i
$477$	0	0
$478$	− 622.939i	− 1.30322i
$479$	572.103i	1.19437i	0.802103	+	0.597185i	$0.203714\pi$
$479$	572.103i	1.19437i	−0.802103	+	0.597185i	$0.796286\pi$
$480$	0	0
$481$	−641.781	−1.33426
$482$	−288.604	−0.598764
$483$	0	0
$484$	−12.1032	−0.0250065
$485$	0	0
$486$	0	0
$487$	− 468.392i	− 0.961791i	−0.876778	−	0.480896i	$-0.840312\pi$
$487$	− 468.392i	− 0.961791i	0.876778	−	0.480896i	$-0.159688\pi$
$488$	79.8797	0.163688
$489$	0	0
$490$	0	0
$491$	− 294.928i	− 0.600668i	−0.953834	−	0.300334i	$-0.902902\pi$
$491$	− 294.928i	− 0.600668i	0.953834	−	0.300334i	$-0.0970981\pi$
$492$	0	0
$493$	− 363.831i	− 0.737993i
$494$	115.897i	0.234609i
$495$	0	0
$496$	185.481	0.373953
$497$	−55.1732	−0.111012
$498$	0	0
$499$	−610.716	−1.22388	−0.611940	−	0.790904i	$-0.709611\pi$
$499$	−610.716	−1.22388	−0.611940	+	0.790904i	$0.709611\pi$
$500$	0	0
$501$	0	0
$502$	7.32821i	0.0145980i
$503$	58.6435	0.116587	0.0582937	−	0.998299i	$-0.481434\pi$
$503$	58.6435	0.116587	0.0582937	+	0.998299i	$0.481434\pi$
$504$	0	0
$505$	0	0
$506$	62.5575i	0.123631i
$507$	0	0
$508$	− 198.190i	− 0.390139i
$509$	− 225.419i	− 0.442866i	−0.975176	−	0.221433i	$-0.928927\pi$
$509$	− 225.419i	− 0.442866i	0.975176	−	0.221433i	$-0.0710734\pi$
$510$	0	0
$511$	−301.341	−0.589708
$512$	22.6274	0.0441942
$513$	0	0
$514$	334.195	0.650185
$515$	0	0
$516$	0	0
$517$	63.7132i	0.123236i
$518$	−104.635	−0.201997
$519$	0	0
$520$	0	0
$521$	− 6.65155i	− 0.0127669i	−0.999980	−	0.00638344i	$-0.997968\pi$
$521$	− 6.65155i	− 0.0127669i	0.999980	−	0.00638344i	$-0.00203193\pi$
$522$	0	0
$523$	− 516.493i	− 0.987558i	−0.869587	−	0.493779i	$-0.835615\pi$
$523$	− 516.493i	− 0.987558i	0.869587	−	0.493779i	$-0.164385\pi$
$524$	279.803i	0.533975i
$525$	0	0
$526$	24.8668	0.0472752
$527$	−387.747	−0.735764
$528$	0	0
$529$	−513.599	−0.970887
$530$	0	0
$531$	0	0
$532$	18.8955i	0.0355179i
$533$	−753.450	−1.41360
$534$	0	0
$535$	0	0
$536$	30.4068i	0.0567292i
$537$	0	0
$538$	− 28.1872i	− 0.0523926i
$539$	− 78.9020i	− 0.146386i
$540$	0	0
$541$	−70.6311	−0.130556	−0.0652782	−	0.997867i	$-0.520793\pi$
$541$	−70.6311	−0.130556	−0.0652782	+	0.997867i	$0.520793\pi$
$542$	−123.174	−0.227259
$543$	0	0
$544$	−47.3026	−0.0869532
$545$	0	0
$546$	0	0
$547$	707.126i	1.29273i	0.763026	+	0.646367i	$0.223713\pi$
$547$	707.126i	1.29273i	−0.763026	+	0.646367i	$0.776287\pi$
$548$	−400.702	−0.731208
$549$	0	0
$550$	0	0
$551$	− 155.371i	− 0.281980i
$552$	0	0
$553$	352.510i	0.637451i
$554$	− 543.787i	− 0.981565i
$555$	0	0
$556$	179.719	0.323235
$557$	−420.729	−0.755348	−0.377674	−	0.925939i	$-0.623276\pi$
$557$	−420.729	−0.755348	−0.377674	+	0.925939i	$0.623276\pi$
$558$	0	0
$559$	−1504.51	−2.69143
$560$	0	0
$561$	0	0
$562$	− 348.573i	− 0.620236i
$563$	425.266	0.755357	0.377679	−	0.925937i	$-0.376722\pi$
$563$	425.266	0.755357	0.377679	+	0.925937i	$0.376722\pi$
$564$	0	0
$565$	0	0
$566$	449.835i	0.794762i
$567$	0	0
$568$	− 58.9826i	− 0.103843i
$569$	565.501i	0.993851i	0.867793	+	0.496925i	$0.165538\pi$
$569$	565.501i	0.993851i	−0.867793	+	0.496925i	$0.834462\pi$
$570$	0	0
$571$	−655.541	−1.14806	−0.574029	−	0.818835i	$-0.694620\pi$
$571$	−655.541	−1.14806	−0.574029	+	0.818835i	$0.694620\pi$
$572$	517.363	0.904481
$573$	0	0
$574$	−122.841	−0.214009
$575$	0	0
$576$	0	0
$577$	− 866.848i	− 1.50234i	−0.660111	−	0.751168i	$-0.729491\pi$
$577$	− 866.848i	− 1.50234i	0.660111	−	0.751168i	$-0.270509\pi$
$578$	−309.822	−0.536024
$579$	0	0
$580$	0	0
$581$	81.9733i	0.141090i
$582$	0	0
$583$	− 279.511i	− 0.479435i
$584$	− 322.147i	− 0.551621i
$585$	0	0
$586$	−561.750	−0.958617
$587$	72.2212	0.123034	0.0615172	−	0.998106i	$-0.480406\pi$
$587$	72.2212	0.123034	0.0615172	+	0.998106i	$0.480406\pi$
$588$	0	0
$589$	−165.585	−0.281128
$590$	0	0
$591$	0	0
$592$	− 111.859i	− 0.188951i
$593$	445.912	0.751959	0.375979	−	0.926628i	$-0.377306\pi$
$593$	445.912	0.751959	0.375979	+	0.926628i	$0.377306\pi$
$594$	0	0
$595$	0	0
$596$	308.868i	0.518234i
$597$	0	0
$598$	− 127.369i	− 0.212992i
$599$	− 624.774i	− 1.04303i	−0.853243	−	0.521514i	$-0.825367\pi$
$599$	− 624.774i	− 1.04303i	0.853243	−	0.521514i	$-0.174633\pi$
$600$	0	0
$601$	366.209	0.609333	0.304667	−	0.952459i	$-0.401455\pi$
$601$	366.209	0.609333	0.304667	+	0.952459i	$0.401455\pi$
$602$	−245.293	−0.407463
$603$	0	0
$604$	587.657	0.972942
$605$	0	0
$606$	0	0
$607$	550.680i	0.907215i	0.891201	+	0.453608i	$0.149863\pi$
$607$	550.680i	0.907215i	−0.891201	+	0.453608i	$0.850137\pi$
$608$	−20.2002	−0.0332240
$609$	0	0
$610$	0	0
$611$	− 129.722i	− 0.212311i
$612$	0	0
$613$	380.620i	0.620914i	0.950587	+	0.310457i	$0.100482\pi$
$613$	380.620i	0.620914i	−0.950587	+	0.310457i	$0.899518\pi$
$614$	− 637.194i	− 1.03778i
$615$	0	0
$616$	84.3498	0.136931
$617$	1097.16	1.77821	0.889106	−	0.457702i	$-0.151327\pi$
$617$	1097.16	1.77821	0.889106	+	0.457702i	$0.151327\pi$
$618$	0	0
$619$	223.332	0.360795	0.180398	−	0.983594i	$-0.442262\pi$
$619$	223.332	0.360795	0.180398	+	0.983594i	$0.442262\pi$
$620$	0	0
$621$	0	0
$622$	685.330i	1.10182i
$623$	−210.671	−0.338155
$624$	0	0
$625$	0	0
$626$	515.300i	0.823163i
$627$	0	0
$628$	− 550.099i	− 0.875954i
$629$	233.841i	0.371767i
$630$	0	0
$631$	−724.190	−1.14769	−0.573843	−	0.818965i	$-0.694548\pi$
$631$	−724.190	−1.14769	−0.573843	+	0.818965i	$0.694548\pi$
$632$	−376.849	−0.596280
$633$	0	0
$634$	722.130	1.13901
$635$	0	0
$636$	0	0
$637$	160.647i	0.252193i
$638$	−693.577	−1.08711
$639$	0	0
$640$	0	0
$641$	459.829i	0.717362i	0.933460	+	0.358681i	$0.116773\pi$
$641$	459.829i	0.717362i	−0.933460	+	0.358681i	$0.883227\pi$
$642$	0	0
$643$	816.842i	1.27036i	0.772364	+	0.635180i	$0.219074\pi$
$643$	816.842i	1.27036i	−0.772364	+	0.635180i	$0.780926\pi$
$644$	− 20.7660i	− 0.0322454i
$645$	0	0
$646$	42.2284	0.0653691
$647$	−726.332	−1.12261	−0.561307	−	0.827607i	$-0.689701\pi$
$647$	−726.332	−1.12261	−0.561307	+	0.827607i	$0.689701\pi$
$648$	0	0
$649$	661.158	1.01873
$650$	0	0
$651$	0	0
$652$	− 243.212i	− 0.373024i
$653$	763.701	1.16953	0.584763	−	0.811204i	$-0.301187\pi$
$653$	763.701	1.16953	0.584763	+	0.811204i	$0.301187\pi$
$654$	0	0
$655$	0	0
$656$	− 131.323i	− 0.200187i
$657$	0	0
$658$	− 21.1497i	− 0.0321423i
$659$	343.226i	0.520828i	0.965497	+	0.260414i	$0.0838591\pi$
$659$	343.226i	0.520828i	−0.965497	+	0.260414i	$0.916141\pi$
$660$	0	0
$661$	−541.334	−0.818962	−0.409481	−	0.912319i	$-0.634290\pi$
$661$	−541.334	−0.818962	−0.409481	+	0.912319i	$0.634290\pi$
$662$	282.782	0.427164
$663$	0	0
$664$	−87.6331	−0.131978
$665$	0	0
$666$	0	0
$667$	170.751i	0.255999i
$668$	310.779	0.465238
$669$	0	0
$670$	0	0
$671$	318.333i	0.474416i
$672$	0	0
$673$	984.277i	1.46252i	0.682098	+	0.731261i	$0.261068\pi$
$673$	984.277i	1.46252i	−0.682098	+	0.731261i	$0.738932\pi$
$674$	− 278.909i	− 0.413811i
$675$	0	0
$676$	−715.369	−1.05824
$677$	1156.51	1.70829	0.854145	−	0.520034i	$-0.174081\pi$
$677$	1156.51	1.70829	0.854145	+	0.520034i	$0.174081\pi$
$678$	0	0
$679$	−69.2709	−0.102019
$680$	0	0
$681$	0	0
$682$	739.170i	1.08383i
$683$	110.138	0.161256	0.0806281	−	0.996744i	$-0.474307\pi$
$683$	110.138	0.161256	0.0806281	+	0.996744i	$0.474307\pi$
$684$	0	0
$685$	0	0
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	− 262.229i	− 0.381147i
$689$	569.094i	0.825971i
$690$	0	0
$691$	858.618	1.24257	0.621286	−	0.783584i	$-0.286610\pi$
$691$	858.618	1.24257	0.621286	+	0.783584i	$0.286610\pi$
$692$	382.031	0.552067
$693$	0	0
$694$	374.958	0.540286
$695$	0	0
$696$	0	0
$697$	274.530i	0.393873i
$698$	−110.096	−0.157730
$699$	0	0
$700$	0	0
$701$	− 54.6278i	− 0.0779285i	−0.999241	−	0.0389642i	$-0.987594\pi$
$701$	− 54.6278i	− 0.0779285i	0.999241	−	0.0389642i	$-0.0124058\pi$
$702$	0	0
$703$	99.8601i	0.142048i
$704$	90.1737i	0.128088i
$705$	0	0
$706$	384.818	0.545067
$707$	−451.705	−0.638904
$708$	0	0
$709$	−98.3377	−0.138699	−0.0693496	−	0.997592i	$-0.522092\pi$
$709$	−98.3377	−0.138699	−0.0693496	+	0.997592i	$0.522092\pi$
$710$	0	0
$711$	0	0
$712$	− 225.216i	− 0.316315i
$713$	181.976	0.255225
$714$	0	0
$715$	0	0
$716$	− 642.056i	− 0.896726i
$717$	0	0
$718$	983.100i	1.36922i
$719$	− 1302.52i	− 1.81157i	−0.423735	−	0.905786i	$-0.639281\pi$
$719$	− 1302.52i	− 1.81157i	0.423735	−	0.905786i	$-0.360719\pi$
$720$	0	0
$721$	421.363	0.584415
$722$	−492.498	−0.682130
$723$	0	0
$724$	−42.6201	−0.0588675
$725$	0	0
$726$	0	0
$727$	− 29.4141i	− 0.0404596i	−0.999795	−	0.0202298i	$-0.993560\pi$
$727$	− 29.4141i	− 0.0404596i	0.999795	−	0.0202298i	$-0.00643978\pi$
$728$	−171.739	−0.235905
$729$	0	0
$730$	0	0
$731$	548.189i	0.749916i
$732$	0	0
$733$	13.8012i	0.0188284i	0.999956	+	0.00941421i	$0.00299668\pi$
$733$	13.8012i	0.0188284i	−0.999956	+	0.00941421i	$0.997003\pi$
$734$	− 228.383i	− 0.311149i
$735$	0	0
$736$	22.1998	0.0301628
$737$	−121.176	−0.164418
$738$	0	0
$739$	−25.7420	−0.0348335	−0.0174168	−	0.999848i	$-0.505544\pi$
$739$	−25.7420	−0.0348335	−0.0174168	+	0.999848i	$0.505544\pi$
$740$	0	0
$741$	0	0
$742$	92.7839i	0.125046i
$743$	567.032	0.763165	0.381583	−	0.924335i	$-0.375379\pi$
$743$	567.032	0.763165	0.381583	+	0.924335i	$0.375379\pi$
$744$	0	0
$745$	0	0
$746$	− 649.303i	− 0.870380i
$747$	0	0
$748$	− 188.508i	− 0.252016i
$749$	− 166.054i	− 0.221701i
$750$	0	0
$751$	−328.985	−0.438062	−0.219031	−	0.975718i	$-0.570290\pi$
$751$	−328.985	−0.438062	−0.219031	+	0.975718i	$0.570290\pi$
$752$	22.6099	0.0300664
$753$	0	0
$754$	1412.15	1.87287
$755$	0	0
$756$	0	0
$757$	323.996i	0.428000i	0.976834	+	0.214000i	$0.0686493\pi$
$757$	323.996i	0.428000i	−0.976834	+	0.214000i	$0.931351\pi$
$758$	186.888	0.246554
$759$	0	0
$760$	0	0
$761$	725.410i	0.953232i	0.879112	+	0.476616i	$0.158137\pi$
$761$	725.410i	0.953232i	−0.879112	+	0.476616i	$0.841863\pi$
$762$	0	0
$763$	117.920i	0.154548i
$764$	− 89.3937i	− 0.117007i
$765$	0	0
$766$	251.797	0.328717
$767$	−1346.14	−1.75507
$768$	0	0
$769$	942.014	1.22499	0.612493	−	0.790476i	$-0.290167\pi$
$769$	942.014	1.22499	0.612493	+	0.790476i	$0.290167\pi$
$770$	0	0
$771$	0	0
$772$	474.675i	0.614863i
$773$	−707.864	−0.915737	−0.457868	−	0.889020i	$-0.651387\pi$
$773$	−707.864	−0.915737	−0.457868	+	0.889020i	$0.651387\pi$
$774$	0	0
$775$	0	0
$776$	− 74.0537i	− 0.0954300i
$777$	0	0
$778$	610.974i	0.785314i
$779$	117.236i	0.150495i
$780$	0	0
$781$	235.055	0.300966
$782$	−46.4086	−0.0593461
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 141.139i	− 0.179338i	−0.995972	−	0.0896688i	$-0.971419\pi$
$787$	− 141.139i	− 0.179338i	0.995972	−	0.0896688i	$-0.0285808\pi$
$788$	−398.950	−0.506282
$789$	0	0
$790$	0	0
$791$	− 338.664i	− 0.428147i
$792$	0	0
$793$	− 648.137i	− 0.817323i
$794$	557.473i	0.702107i
$795$	0	0
$796$	−150.366	−0.188902
$797$	−1183.67	−1.48515	−0.742575	−	0.669763i	$-0.766396\pi$
$797$	−1183.67	−1.48515	−0.742575	+	0.669763i	$0.766396\pi$
$798$	0	0
$799$	−47.2660	−0.0591565
$800$	0	0
$801$	0	0
$802$	− 1006.05i	− 1.25442i
$803$	1283.80	1.59876
$804$	0	0
$805$	0	0
$806$	− 1504.98i	− 1.86722i
$807$	0	0
$808$	− 482.893i	− 0.597640i
$809$	1300.44i	1.60747i	0.594988	+	0.803735i	$0.297157\pi$
$809$	1300.44i	1.60747i	−0.594988	+	0.803735i	$0.702843\pi$
$810$	0	0
$811$	180.853	0.223000	0.111500	−	0.993764i	$-0.464435\pi$
$811$	180.853	0.223000	0.111500	+	0.993764i	$0.464435\pi$
$812$	230.234	0.283539
$813$	0	0
$814$	445.776	0.547636
$815$	0	0
$816$	0	0
$817$	234.100i	0.286536i
$818$	753.754	0.921459
$819$	0	0
$820$	0	0
$821$	1312.33i	1.59845i	0.601029	+	0.799227i	$0.294757\pi$
$821$	1312.33i	1.59845i	−0.601029	+	0.799227i	$0.705243\pi$
$822$	0	0
$823$	120.713i	0.146675i	0.997307	+	0.0733373i	$0.0233650\pi$
$823$	120.713i	0.146675i	−0.997307	+	0.0733373i	$0.976635\pi$
$824$	450.456i	0.546670i
$825$	0	0
$826$	−219.472	−0.265705
$827$	−573.888	−0.693939	−0.346970	−	0.937876i	$-0.612789\pi$
$827$	−573.888	−0.693939	−0.346970	+	0.937876i	$0.612789\pi$
$828$	0	0
$829$	1607.19	1.93871	0.969357	−	0.245657i	$-0.0790038\pi$
$829$	1607.19	1.93871	0.969357	+	0.245657i	$0.0790038\pi$
$830$	0	0
$831$	0	0
$832$	− 183.597i	− 0.220669i
$833$	58.5339	0.0702688
$834$	0	0
$835$	0	0
$836$	− 80.5009i	− 0.0962929i
$837$	0	0
$838$	759.991i	0.906911i
$839$	115.880i	0.138116i	0.997613	+	0.0690582i	$0.0219994\pi$
$839$	115.880i	0.138116i	−0.997613	+	0.0690582i	$0.978001\pi$
$840$	0	0
$841$	−1052.12	−1.25104
$842$	58.7456	0.0697691
$843$	0	0
$844$	354.158	0.419619
$845$	0	0
$846$	0	0
$847$	16.0110i	0.0189031i
$848$	−99.1902	−0.116970
$849$	0	0
$850$	0	0
$851$	− 109.745i	− 0.128960i
$852$	0	0
$853$	− 93.1233i	− 0.109171i	−0.998509	−	0.0545857i	$-0.982616\pi$
$853$	− 93.1233i	− 0.109171i	0.998509	−	0.0545857i	$-0.0173838\pi$
$854$	− 105.671i	− 0.123736i
$855$	0	0
$856$	177.519	0.207382
$857$	347.355	0.405315	0.202658	−	0.979250i	$-0.435042\pi$
$857$	347.355	0.405315	0.202658	+	0.979250i	$0.435042\pi$
$858$	0	0
$859$	705.651	0.821479	0.410740	−	0.911753i	$-0.365271\pi$
$859$	705.651	0.821479	0.410740	+	0.911753i	$0.365271\pi$
$860$	0	0
$861$	0	0
$862$	28.0020i	0.0324849i
$863$	−1504.37	−1.74318	−0.871591	−	0.490235i	$-0.836911\pi$
$863$	−1504.37	−1.74318	−0.871591	+	0.490235i	$0.836911\pi$
$864$	0	0
$865$	0	0
$866$	1003.17i	1.15840i
$867$	0	0
$868$	− 245.368i	− 0.282682i
$869$	− 1501.80i	− 1.72820i
$870$	0	0
$871$	246.718	0.283259
$872$	−126.062	−0.144566
$873$	0	0
$874$	−19.8185	−0.0226756
$875$	0	0
$876$	0	0
$877$	552.772i	0.630298i	0.949042	+	0.315149i	$0.102055\pi$
$877$	552.772i	0.630298i	−0.949042	+	0.315149i	$0.897945\pi$
$878$	−209.689	−0.238826
$879$	0	0
$880$	0	0
$881$	− 131.749i	− 0.149545i	−0.997201	−	0.0747726i	$-0.976177\pi$
$881$	− 131.749i	− 0.149545i	0.997201	−	0.0747726i	$-0.0238231\pi$
$882$	0	0
$883$	870.659i	0.986024i	0.870022	+	0.493012i	$0.164104\pi$
$883$	870.659i	0.986024i	−0.870022	+	0.493012i	$0.835896\pi$
$884$	383.809i	0.434173i
$885$	0	0
$886$	168.478	0.190156
$887$	−1488.49	−1.67812	−0.839060	−	0.544039i	$-0.816894\pi$
$887$	−1488.49	−1.67812	−0.839060	+	0.544039i	$0.816894\pi$
$888$	0	0
$889$	−262.181	−0.294917
$890$	0	0
$891$	0	0
$892$	− 514.658i	− 0.576971i
$893$	−20.1846	−0.0226031
$894$	0	0
$895$	0	0
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	423.966i	0.472122i
$899$	2017.57i	2.24424i
$900$	0	0
$901$	207.357	0.230141
$902$	523.341	0.580200
$903$	0	0
$904$	362.047	0.400495
$905$	0	0
$906$	0	0
$907$	− 217.860i	− 0.240198i	−0.992762	−	0.120099i	$-0.961679\pi$
$907$	− 217.860i	− 0.240198i	0.992762	−	0.120099i	$-0.0383213\pi$
$908$	−222.216	−0.244732
$909$	0	0
$910$	0	0
$911$	− 752.667i	− 0.826198i	−0.910686	−	0.413099i	$-0.864446\pi$
$911$	− 752.667i	− 0.826198i	0.910686	−	0.413099i	$-0.135554\pi$
$912$	0	0
$913$	− 349.231i	− 0.382510i
$914$	297.102i	0.325057i
$915$	0	0
$916$	−201.827	−0.220336
$917$	370.144	0.403647
$918$	0	0
$919$	1765.94	1.92159	0.960793	−	0.277266i	$-0.0894282\pi$
$919$	1765.94	1.92159	0.960793	+	0.277266i	$0.0894282\pi$
$920$	0	0
$921$	0	0
$922$	899.361i	0.975446i
$923$	−478.580	−0.518505
$924$	0	0
$925$	0	0
$926$	891.434i	0.962672i
$927$	0	0
$928$	246.130i	0.265226i
$929$	1051.94i	1.13234i	0.824289	+	0.566169i	$0.191575\pi$
$929$	1051.94i	1.13234i	−0.824289	+	0.566169i	$0.808425\pi$
$930$	0	0
$931$	24.9965	0.0268490
$932$	−67.5560	−0.0724850
$933$	0	0
$934$	−239.330	−0.256242
$935$	0	0
$936$	0	0
$937$	934.930i	0.997791i	0.866662	+	0.498896i	$0.166261\pi$
$937$	934.930i	0.997791i	−0.866662	+	0.498896i	$0.833739\pi$
$938$	40.2245	0.0428832
$939$	0	0
$940$	0	0
$941$	962.012i	1.02233i	0.859483	+	0.511165i	$0.170786\pi$
$941$	962.012i	1.02233i	−0.859483	+	0.511165i	$0.829214\pi$
$942$	0	0
$943$	− 128.841i	− 0.136629i
$944$	− 234.625i	− 0.248544i
$945$	0	0
$946$	1045.02	1.10467
$947$	−1811.65	−1.91304	−0.956521	−	0.291665i	$-0.905791\pi$
$947$	−1811.65	−1.91304	−0.956521	+	0.291665i	$0.905791\pi$
$948$	0	0
$949$	−2613.87	−2.75434
$950$	0	0
$951$	0	0
$952$	62.5754i	0.0657305i
$953$	−1140.52	−1.19677	−0.598385	−	0.801209i	$-0.704191\pi$
$953$	−1140.52	−1.19677	−0.598385	+	0.801209i	$0.704191\pi$
$954$	0	0
$955$	0	0
$956$	− 880.969i	− 0.921515i
$957$	0	0
$958$	809.076i	0.844547i
$959$	530.079i	0.552742i
$960$	0	0
$961$	1189.20	1.23746
$962$	−907.615	−0.943467
$963$	0	0
$964$	−408.148	−0.423390
$965$	0	0
$966$	0	0
$967$	− 535.438i	− 0.553710i	−0.960912	−	0.276855i	$-0.910708\pi$
$967$	− 535.438i	− 0.553710i	0.960912	−	0.276855i	$-0.0892922\pi$
$968$	−17.1164	−0.0176823
$969$	0	0
$970$	0	0
$971$	− 182.412i	− 0.187860i	−0.995579	−	0.0939301i	$-0.970057\pi$
$971$	− 182.412i	− 0.187860i	0.995579	−	0.0939301i	$-0.0299430\pi$
$972$	0	0
$973$	− 237.745i	− 0.244343i
$974$	− 662.407i	− 0.680089i
$975$	0	0
$976$	112.967	0.115745
$977$	−1146.82	−1.17381	−0.586906	−	0.809655i	$-0.699654\pi$
$977$	−1146.82	−1.17381	−0.586906	+	0.809655i	$0.699654\pi$
$978$	0	0
$979$	897.522	0.916775
$980$	0	0
$981$	0	0
$982$	− 417.091i	− 0.424736i
$983$	198.963	0.202403	0.101202	−	0.994866i	$-0.467731\pi$
$983$	198.963	0.202403	0.101202	+	0.994866i	$0.467731\pi$
$984$	0	0
$985$	0	0
$986$	− 514.534i	− 0.521840i
$987$	0	0
$988$	163.903i	0.165893i
$989$	− 257.273i	− 0.260135i
$990$	0	0
$991$	1917.76	1.93517	0.967587	−	0.252539i	$-0.0812658\pi$
$991$	1917.76	1.93517	0.967587	+	0.252539i	$0.0812658\pi$
$992$	262.310	0.264425
$993$	0	0
$994$	−78.0267	−0.0784977
$995$	0	0
$996$	0	0
$997$	1709.94i	1.71509i	0.514410	+	0.857544i	$0.328011\pi$
$997$	1709.94i	1.71509i	−0.514410	+	0.857544i	$0.671989\pi$
$998$	−863.683	−0.865414
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.f.449.11		16
3.2	odd	2	inner	3150.3.c.f.449.2		16
5.2	odd	4		3150.3.e.f.701.8		8
5.3	odd	4		630.3.e.b.71.1	✓	8
5.4	even	2	inner	3150.3.c.f.449.7		16
15.2	even	4		3150.3.e.f.701.3		8
15.8	even	4		630.3.e.b.71.7	yes	8
15.14	odd	2	inner	3150.3.c.f.449.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.3.e.b.71.1	✓	8	5.3	odd	4
630.3.e.b.71.7	yes	8	15.8	even	4
3150.3.c.f.449.2		16	3.2	odd	2	inner
3150.3.c.f.449.7		16	5.4	even	2	inner
3150.3.c.f.449.11		16	1.1	even	1	trivial
3150.3.c.f.449.14		16	15.14	odd	2	inner
3150.3.e.f.701.3		8	15.2	even	4
3150.3.e.f.701.8		8	5.2	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} +2.82843 q^{8} +11.2717i q^{11} -22.9496i q^{13} -3.74166i q^{14} +4.00000 q^{16} -8.36199 q^{17} -3.57092 q^{19} +15.9406i q^{22} +3.92441 q^{23} -32.4556i q^{26} -5.29150i q^{28} +43.5101i q^{29} +46.3702 q^{31} +5.65685 q^{32} -11.8256 q^{34} -27.9648i q^{37} -5.05005 q^{38} -32.8306i q^{41} -65.5572i q^{43} +22.5434i q^{44} +5.54995 q^{46} +5.65248 q^{47} -7.00000 q^{49} -45.8992i q^{52} -24.7975 q^{53} -7.48331i q^{56} +61.5325i q^{58} -58.6564i q^{59} +28.2418 q^{61} +65.5774 q^{62} +8.00000 q^{64} +10.7504i q^{67} -16.7240 q^{68} -20.8535i q^{71} -113.896i q^{73} -39.5482i q^{74} -7.14185 q^{76} +29.8222 q^{77} -133.236 q^{79} -46.4295i q^{82} -30.9830 q^{83} -92.7119i q^{86} +31.8812i q^{88} -79.6261i q^{89} -60.7189 q^{91} +7.84882 q^{92} +7.99382 q^{94} -26.1819i q^{97} -9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

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Database

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