LMFDB - Embedded newform 2352.3.f.g.97.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(97,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.f (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:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.35911766016.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 $ x^8 - 2x^7 - 7x^6 - 2x^5 + 78x^4 - 18x^3 - 153x^2 - 230*x + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.3
Root			$2.40015 - 0.808379i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.g.97.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +5.40561i q^{5} -3.00000 q^{9} -10.5344 q^{11} -12.0366i q^{13} +9.36280 q^{15} -24.0714i q^{17} +35.6891i q^{19} +41.3104 q^{23} -4.22066 q^{25} +5.19615i q^{27} -28.6732 q^{29} +5.19884i q^{31} +18.2461i q^{33} -3.81864 q^{37} -20.8481 q^{39} -9.20741i q^{41} -54.2960 q^{43} -16.2168i q^{45} +16.5526i q^{47} -41.6929 q^{51} +19.1664 q^{53} -56.9447i q^{55} +61.8154 q^{57} -12.5009i q^{59} -97.1310i q^{61} +65.0655 q^{65} -0.998156 q^{67} -71.5517i q^{69} +14.8401 q^{71} -59.0037i q^{73} +7.31040i q^{75} +101.512 q^{79} +9.00000 q^{81} -22.7269i q^{83} +130.121 q^{85} +49.6635i q^{87} +114.029i q^{89} +9.00465 q^{93} -192.922 q^{95} -153.333i q^{97} +31.6031 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9} - 44 q^{11} + 12 q^{15} + 96 q^{23} - 84 q^{25} + 68 q^{29} + 236 q^{37} - 36 q^{39} + 92 q^{43} + 72 q^{51} - 20 q^{53} + 84 q^{57} + 296 q^{65} + 44 q^{67} + 392 q^{71} + 328 q^{79} + 72 q^{81}+ \cdots + 132 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 - 44 * q^11 + 12 * q^15 + 96 * q^23 - 84 * q^25 + 68 * q^29 + 236 * q^37 - 36 * q^39 + 92 * q^43 + 72 * q^51 - 20 * q^53 + 84 * q^57 + 296 * q^65 + 44 * q^67 + 392 * q^71 + 328 * q^79 + 72 * q^81 + 200 * q^85 + 120 * q^93 + 132 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	5.40561i	1.08112i	0.841305	+	0.540561i	$0.181788\pi$
$5$	5.40561i	1.08112i	−0.841305	+	0.540561i	$0.818212\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−10.5344	−0.957670	−0.478835	−	0.877905i	$-0.658941\pi$
$11$	−10.5344	−0.957670	−0.478835	+	0.877905i	$0.658941\pi$
$12$	0	0
$13$	− 12.0366i	− 0.925896i	−0.886386	−	0.462948i	$-0.846792\pi$
$13$	− 12.0366i	− 0.925896i	0.886386	−	0.462948i	$-0.153208\pi$
$14$	0	0
$15$	9.36280	0.624187
$16$	0	0
$17$	− 24.0714i	− 1.41596i	−0.706230	−	0.707982i	$-0.749606\pi$
$17$	− 24.0714i	− 1.41596i	0.706230	−	0.707982i	$-0.250394\pi$
$18$	0	0
$19$	35.6891i	1.87837i	0.343406	+	0.939187i	$0.388419\pi$
$19$	35.6891i	1.87837i	−0.343406	+	0.939187i	$0.611581\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	41.3104	1.79611	0.898053	−	0.439888i	$-0.144982\pi$
$23$	41.3104	1.79611	0.898053	+	0.439888i	$0.144982\pi$
$24$	0	0
$25$	−4.22066	−0.168826
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−28.6732	−0.988732	−0.494366	−	0.869254i	$-0.664600\pi$
$29$	−28.6732	−0.988732	−0.494366	+	0.869254i	$0.664600\pi$
$30$	0	0
$31$	5.19884i	0.167704i	0.996478	+	0.0838522i	$0.0267224\pi$
$31$	5.19884i	0.167704i	−0.996478	+	0.0838522i	$0.973278\pi$
$32$	0	0
$33$	18.2461i	0.552911i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.81864	−0.103207	−0.0516033	−	0.998668i	$-0.516433\pi$
$37$	−3.81864	−0.103207	−0.0516033	+	0.998668i	$0.516433\pi$
$38$	0	0
$39$	−20.8481	−0.534566
$40$	0	0
$41$	− 9.20741i	− 0.224571i	−0.993676	−	0.112286i	$-0.964183\pi$
$41$	− 9.20741i	− 0.224571i	0.993676	−	0.112286i	$-0.0358171\pi$
$42$	0	0
$43$	−54.2960	−1.26270	−0.631349	−	0.775499i	$-0.717498\pi$
$43$	−54.2960	−1.26270	−0.631349	+	0.775499i	$0.717498\pi$
$44$	0	0
$45$	− 16.2168i	− 0.360374i
$46$	0	0
$47$	16.5526i	0.352184i	0.984374	+	0.176092i	$0.0563456\pi$
$47$	16.5526i	0.352184i	−0.984374	+	0.176092i	$0.943654\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−41.6929	−0.817507
$52$	0	0
$53$	19.1664	0.361631	0.180815	−	0.983517i	$-0.442126\pi$
$53$	19.1664	0.361631	0.180815	+	0.983517i	$0.442126\pi$
$54$	0	0
$55$	− 56.9447i	− 1.03536i
$56$	0	0
$57$	61.8154	1.08448
$58$	0	0
$59$	− 12.5009i	− 0.211879i	−0.994373	−	0.105939i	$-0.966215\pi$
$59$	− 12.5009i	− 0.211879i	0.994373	−	0.105939i	$-0.0337850\pi$
$60$	0	0
$61$	− 97.1310i	− 1.59231i	−0.605092	−	0.796156i	$-0.706864\pi$
$61$	− 97.1310i	− 1.59231i	0.605092	−	0.796156i	$-0.293136\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	65.0655	1.00101
$66$	0	0
$67$	−0.998156	−0.0148979	−0.00744893	−	0.999972i	$-0.502371\pi$
$67$	−0.998156	−0.0148979	−0.00744893	+	0.999972i	$0.502371\pi$
$68$	0	0
$69$	− 71.5517i	− 1.03698i
$70$	0	0
$71$	14.8401	0.209015	0.104507	−	0.994524i	$-0.466673\pi$
$71$	14.8401	0.209015	0.104507	+	0.994524i	$0.466673\pi$
$72$	0	0
$73$	− 59.0037i	− 0.808270i	−0.914699	−	0.404135i	$-0.867573\pi$
$73$	− 59.0037i	− 0.808270i	0.914699	−	0.404135i	$-0.132427\pi$
$74$	0	0
$75$	7.31040i	0.0974720i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	101.512	1.28496	0.642478	−	0.766304i	$-0.277906\pi$
$79$	101.512	1.28496	0.642478	+	0.766304i	$0.277906\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 22.7269i	− 0.273818i	−0.990584	−	0.136909i	$-0.956283\pi$
$83$	− 22.7269i	− 0.273818i	0.990584	−	0.136909i	$-0.0437168\pi$
$84$	0	0
$85$	130.121	1.53083
$86$	0	0
$87$	49.6635i	0.570844i
$88$	0	0
$89$	114.029i	1.28122i	0.767865	+	0.640611i	$0.221319\pi$
$89$	114.029i	1.28122i	−0.767865	+	0.640611i	$0.778681\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.00465	0.0968242
$94$	0	0
$95$	−192.922	−2.03075
$96$	0	0
$97$	− 153.333i	− 1.58075i	−0.612621	−	0.790377i	$-0.709885\pi$
$97$	− 153.333i	− 1.58075i	0.612621	−	0.790377i	$-0.290115\pi$
$98$	0	0
$99$	31.6031	0.319223
$100$	0	0
$101$	− 17.6413i	− 0.174666i	−0.996179	−	0.0873329i	$-0.972166\pi$
$101$	− 17.6413i	− 0.174666i	0.996179	−	0.0873329i	$-0.0278344\pi$
$102$	0	0
$103$	− 138.230i	− 1.34204i	−0.741441	−	0.671018i	$-0.765857\pi$
$103$	− 138.230i	− 1.34204i	0.741441	−	0.671018i	$-0.234143\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	82.1486	0.767744	0.383872	−	0.923386i	$-0.374590\pi$
$107$	82.1486	0.767744	0.383872	+	0.923386i	$0.374590\pi$
$108$	0	0
$109$	90.3779	0.829155	0.414577	−	0.910014i	$-0.363929\pi$
$109$	90.3779	0.829155	0.414577	+	0.910014i	$0.363929\pi$
$110$	0	0
$111$	6.61408i	0.0595863i
$112$	0	0
$113$	206.994	1.83180	0.915901	−	0.401405i	$-0.131478\pi$
$113$	206.994	1.83180	0.915901	+	0.401405i	$0.131478\pi$
$114$	0	0
$115$	223.308i	1.94181i
$116$	0	0
$117$	36.1099i	0.308632i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0270	−0.0828680
$122$	0	0
$123$	−15.9477	−0.129656
$124$	0	0
$125$	112.325i	0.898601i
$126$	0	0
$127$	155.159	1.22172	0.610862	−	0.791737i	$-0.290823\pi$
$127$	155.159	1.22172	0.610862	+	0.791737i	$0.290823\pi$
$128$	0	0
$129$	94.0434i	0.729019i
$130$	0	0
$131$	71.6322i	0.546811i	0.961899	+	0.273405i	$0.0881500\pi$
$131$	71.6322i	0.546811i	−0.961899	+	0.273405i	$0.911850\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−28.0884	−0.208062
$136$	0	0
$137$	152.245	1.11128	0.555639	−	0.831424i	$-0.312474\pi$
$137$	152.245	1.11128	0.555639	+	0.831424i	$0.312474\pi$
$138$	0	0
$139$	− 119.318i	− 0.858401i	−0.903209	−	0.429200i	$-0.858795\pi$
$139$	− 119.318i	− 0.858401i	0.903209	−	0.429200i	$-0.141205\pi$
$140$	0	0
$141$	28.6700	0.203333
$142$	0	0
$143$	126.798i	0.886703i
$144$	0	0
$145$	− 154.996i	− 1.06894i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−54.1020	−0.363101	−0.181550	−	0.983382i	$-0.558112\pi$
$149$	−54.1020	−0.363101	−0.181550	+	0.983382i	$0.558112\pi$
$150$	0	0
$151$	−131.751	−0.872525	−0.436262	−	0.899819i	$-0.643698\pi$
$151$	−131.751	−0.872525	−0.436262	+	0.899819i	$0.643698\pi$
$152$	0	0
$153$	72.2142i	0.471988i
$154$	0	0
$155$	−28.1029	−0.181309
$156$	0	0
$157$	− 240.385i	− 1.53112i	−0.643367	−	0.765558i	$-0.722463\pi$
$157$	− 240.385i	− 1.53112i	0.643367	−	0.765558i	$-0.277537\pi$
$158$	0	0
$159$	− 33.1972i	− 0.208788i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	269.170	1.65135	0.825674	−	0.564147i	$-0.190795\pi$
$163$	269.170	1.65135	0.825674	+	0.564147i	$0.190795\pi$
$164$	0	0
$165$	−98.6312	−0.597765
$166$	0	0
$167$	123.116i	0.737219i	0.929584	+	0.368609i	$0.120166\pi$
$167$	123.116i	0.737219i	−0.929584	+	0.368609i	$0.879834\pi$
$168$	0	0
$169$	24.1192	0.142717
$170$	0	0
$171$	− 107.067i	− 0.626125i
$172$	0	0
$173$	− 225.630i	− 1.30422i	−0.758126	−	0.652109i	$-0.773885\pi$
$173$	− 225.630i	− 1.30422i	0.758126	−	0.652109i	$-0.226115\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.6521	−0.122328
$178$	0	0
$179$	267.249	1.49301	0.746505	−	0.665380i	$-0.231730\pi$
$179$	267.249	1.49301	0.746505	+	0.665380i	$0.231730\pi$
$180$	0	0
$181$	64.5441i	0.356597i	0.983976	+	0.178299i	$0.0570593\pi$
$181$	64.5441i	0.356597i	−0.983976	+	0.178299i	$0.942941\pi$
$182$	0	0
$183$	−168.236	−0.919321
$184$	0	0
$185$	− 20.6421i	− 0.111579i
$186$	0	0
$187$	253.577i	1.35603i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−148.742	−0.778754	−0.389377	−	0.921078i	$-0.627310\pi$
$191$	−148.742	−0.778754	−0.389377	+	0.921078i	$0.627310\pi$
$192$	0	0
$193$	29.8631	0.154731	0.0773655	−	0.997003i	$-0.475349\pi$
$193$	29.8631	0.154731	0.0773655	+	0.997003i	$0.475349\pi$
$194$	0	0
$195$	− 112.697i	− 0.577932i
$196$	0	0
$197$	191.905	0.974138	0.487069	−	0.873364i	$-0.338066\pi$
$197$	191.905	0.974138	0.487069	+	0.873364i	$0.338066\pi$
$198$	0	0
$199$	− 21.4733i	− 0.107906i	−0.998543	−	0.0539531i	$-0.982818\pi$
$199$	− 21.4733i	− 0.107906i	0.998543	−	0.0539531i	$-0.0171821\pi$
$200$	0	0
$201$	1.72886i	0.00860128i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	49.7717	0.242789
$206$	0	0
$207$	−123.931	−0.598702
$208$	0	0
$209$	− 375.962i	− 1.79886i
$210$	0	0
$211$	179.565	0.851020	0.425510	−	0.904954i	$-0.360095\pi$
$211$	179.565	0.851020	0.425510	+	0.904954i	$0.360095\pi$
$212$	0	0
$213$	− 25.7037i	− 0.120675i
$214$	0	0
$215$	− 293.503i	− 1.36513i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−102.197	−0.466655
$220$	0	0
$221$	−289.739	−1.31104
$222$	0	0
$223$	− 296.009i	− 1.32739i	−0.748001	−	0.663697i	$-0.768986\pi$
$223$	− 296.009i	− 1.32739i	0.748001	−	0.663697i	$-0.231014\pi$
$224$	0	0
$225$	12.6620	0.0562755
$226$	0	0
$227$	− 176.367i	− 0.776946i	−0.921460	−	0.388473i	$-0.873003\pi$
$227$	− 176.367i	− 0.776946i	0.921460	−	0.388473i	$-0.126997\pi$
$228$	0	0
$229$	48.3660i	0.211205i	0.994408	+	0.105603i	$0.0336772\pi$
$229$	48.3660i	0.211205i	−0.994408	+	0.105603i	$0.966323\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	458.429	1.96751	0.983753	−	0.179527i	$-0.0574568\pi$
$233$	458.429	1.96751	0.983753	+	0.179527i	$0.0574568\pi$
$234$	0	0
$235$	−89.4772	−0.380754
$236$	0	0
$237$	− 175.823i	− 0.741870i
$238$	0	0
$239$	−124.419	−0.520581	−0.260291	−	0.965530i	$-0.583818\pi$
$239$	−124.419	−0.520581	−0.260291	+	0.965530i	$0.583818\pi$
$240$	0	0
$241$	− 219.761i	− 0.911869i	−0.890013	−	0.455935i	$-0.849305\pi$
$241$	− 219.761i	− 0.911869i	0.890013	−	0.455935i	$-0.150695\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	429.577	1.73918
$248$	0	0
$249$	−39.3642	−0.158089
$250$	0	0
$251$	− 69.5582i	− 0.277124i	−0.990354	−	0.138562i	$-0.955752\pi$
$251$	− 69.5582i	− 0.277124i	0.990354	−	0.138562i	$-0.0442481\pi$
$252$	0	0
$253$	−435.179	−1.72008
$254$	0	0
$255$	− 225.376i	− 0.883826i
$256$	0	0
$257$	101.797i	0.396098i	0.980192	+	0.198049i	$0.0634605\pi$
$257$	101.797i	0.396098i	−0.980192	+	0.198049i	$0.936540\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	86.0197	0.329577
$262$	0	0
$263$	211.575	0.804466	0.402233	−	0.915537i	$-0.368234\pi$
$263$	211.575	0.804466	0.402233	+	0.915537i	$0.368234\pi$
$264$	0	0
$265$	103.606i	0.390967i
$266$	0	0
$267$	197.504	0.739714
$268$	0	0
$269$	311.835i	1.15924i	0.814887	+	0.579620i	$0.196799\pi$
$269$	311.835i	1.15924i	−0.814887	+	0.579620i	$0.803201\pi$
$270$	0	0
$271$	256.574i	0.946767i	0.880856	+	0.473383i	$0.156967\pi$
$271$	256.574i	0.946767i	−0.880856	+	0.473383i	$0.843033\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	44.4620	0.161680
$276$	0	0
$277$	−279.093	−1.00756	−0.503778	−	0.863833i	$-0.668057\pi$
$277$	−279.093	−1.00756	−0.503778	+	0.863833i	$0.668057\pi$
$278$	0	0
$279$	− 15.5965i	− 0.0559015i
$280$	0	0
$281$	−84.1671	−0.299527	−0.149763	−	0.988722i	$-0.547851\pi$
$281$	−84.1671	−0.299527	−0.149763	+	0.988722i	$0.547851\pi$
$282$	0	0
$283$	− 465.427i	− 1.64462i	−0.569042	−	0.822309i	$-0.692686\pi$
$283$	− 465.427i	− 1.64462i	0.569042	−	0.822309i	$-0.307314\pi$
$284$	0	0
$285$	334.150i	1.17246i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−290.432	−1.00495
$290$	0	0
$291$	−265.581	−0.912648
$292$	0	0
$293$	− 289.553i	− 0.988237i	−0.869395	−	0.494119i	$-0.835491\pi$
$293$	− 289.553i	− 0.988237i	0.869395	−	0.494119i	$-0.164509\pi$
$294$	0	0
$295$	67.5748	0.229067
$296$	0	0
$297$	− 54.7382i	− 0.184304i
$298$	0	0
$299$	− 497.239i	− 1.66301i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−30.5556	−0.100843
$304$	0	0
$305$	525.053	1.72148
$306$	0	0
$307$	339.471i	1.10577i	0.833258	+	0.552884i	$0.186473\pi$
$307$	339.471i	1.10577i	−0.833258	+	0.552884i	$0.813527\pi$
$308$	0	0
$309$	−239.421	−0.774825
$310$	0	0
$311$	− 98.7284i	− 0.317455i	−0.987322	−	0.158727i	$-0.949261\pi$
$311$	− 98.7284i	− 0.317455i	0.987322	−	0.158727i	$-0.0507391\pi$
$312$	0	0
$313$	− 214.621i	− 0.685689i	−0.939392	−	0.342845i	$-0.888610\pi$
$313$	− 214.621i	− 0.685689i	0.939392	−	0.342845i	$-0.111390\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	193.607	0.610746	0.305373	−	0.952233i	$-0.401219\pi$
$317$	193.607	0.610746	0.305373	+	0.952233i	$0.401219\pi$
$318$	0	0
$319$	302.054	0.946879
$320$	0	0
$321$	− 142.286i	− 0.443257i
$322$	0	0
$323$	859.086	2.65971
$324$	0	0
$325$	50.8026i	0.156316i
$326$	0	0
$327$	− 156.539i	− 0.478713i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−488.170	−1.47483	−0.737417	−	0.675438i	$-0.763955\pi$
$331$	−488.170	−1.47483	−0.737417	+	0.675438i	$0.763955\pi$
$332$	0	0
$333$	11.4559	0.0344022
$334$	0	0
$335$	− 5.39565i	− 0.0161064i
$336$	0	0
$337$	535.060	1.58771	0.793857	−	0.608104i	$-0.208070\pi$
$337$	535.060	1.58771	0.793857	+	0.608104i	$0.208070\pi$
$338$	0	0
$339$	− 358.523i	− 1.05759i
$340$	0	0
$341$	− 54.7665i	− 0.160606i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	386.781	1.12110
$346$	0	0
$347$	−335.384	−0.966524	−0.483262	−	0.875476i	$-0.660548\pi$
$347$	−335.384	−0.966524	−0.483262	+	0.875476i	$0.660548\pi$
$348$	0	0
$349$	− 105.238i	− 0.301542i	−0.988569	−	0.150771i	$-0.951824\pi$
$349$	− 105.238i	− 0.301542i	0.988569	−	0.150771i	$-0.0481756\pi$
$350$	0	0
$351$	62.5442	0.178189
$352$	0	0
$353$	328.060i	0.929349i	0.885482	+	0.464674i	$0.153829\pi$
$353$	328.060i	0.929349i	−0.885482	+	0.464674i	$0.846171\pi$
$354$	0	0
$355$	80.2196i	0.225971i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−534.835	−1.48979	−0.744895	−	0.667182i	$-0.767500\pi$
$359$	−534.835	−1.48979	−0.744895	+	0.667182i	$0.767500\pi$
$360$	0	0
$361$	−912.713	−2.52829
$362$	0	0
$363$	17.3673i	0.0478439i
$364$	0	0
$365$	318.951	0.873839
$366$	0	0
$367$	− 83.1428i	− 0.226547i	−0.993564	−	0.113274i	$-0.963866\pi$
$367$	− 83.1428i	− 0.226547i	0.993564	−	0.113274i	$-0.0361336\pi$
$368$	0	0
$369$	27.6222i	0.0748570i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−540.669	−1.44951	−0.724757	−	0.689004i	$-0.758048\pi$
$373$	−540.669	−1.44951	−0.724757	+	0.689004i	$0.758048\pi$
$374$	0	0
$375$	194.553	0.518807
$376$	0	0
$377$	345.129i	0.915462i
$378$	0	0
$379$	−74.8470	−0.197486	−0.0987428	−	0.995113i	$-0.531482\pi$
$379$	−74.8470	−0.197486	−0.0987428	+	0.995113i	$0.531482\pi$
$380$	0	0
$381$	− 268.743i	− 0.705363i
$382$	0	0
$383$	117.164i	0.305910i	0.988233	+	0.152955i	$0.0488790\pi$
$383$	117.164i	0.305910i	−0.988233	+	0.152955i	$0.951121\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	162.888	0.420899
$388$	0	0
$389$	−529.607	−1.36146	−0.680729	−	0.732535i	$-0.738337\pi$
$389$	−529.607	−1.36146	−0.680729	+	0.732535i	$0.738337\pi$
$390$	0	0
$391$	− 994.399i	− 2.54322i
$392$	0	0
$393$	124.071	0.315701
$394$	0	0
$395$	548.732i	1.38920i
$396$	0	0
$397$	− 348.048i	− 0.876696i	−0.898805	−	0.438348i	$-0.855564\pi$
$397$	− 348.048i	− 0.876696i	0.898805	−	0.438348i	$-0.144436\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−453.288	−1.13039	−0.565197	−	0.824956i	$-0.691200\pi$
$401$	−453.288	−1.13039	−0.565197	+	0.824956i	$0.691200\pi$
$402$	0	0
$403$	62.5766	0.155277
$404$	0	0
$405$	48.6505i	0.120125i
$406$	0	0
$407$	40.2270	0.0988378
$408$	0	0
$409$	− 555.437i	− 1.35804i	−0.734122	−	0.679018i	$-0.762406\pi$
$409$	− 555.437i	− 1.35804i	0.734122	−	0.679018i	$-0.237594\pi$
$410$	0	0
$411$	− 263.696i	− 0.641597i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	122.853	0.296031
$416$	0	0
$417$	−206.664	−0.495598
$418$	0	0
$419$	54.3155i	0.129631i	0.997897	+	0.0648156i	$0.0206459\pi$
$419$	54.3155i	0.129631i	−0.997897	+	0.0648156i	$0.979354\pi$
$420$	0	0
$421$	−578.890	−1.37504	−0.687518	−	0.726168i	$-0.741300\pi$
$421$	−578.890	−1.37504	−0.687518	+	0.726168i	$0.741300\pi$
$422$	0	0
$423$	− 49.6579i	− 0.117395i
$424$	0	0
$425$	101.597i	0.239052i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	219.621	0.511938
$430$	0	0
$431$	533.398	1.23758	0.618791	−	0.785556i	$-0.287623\pi$
$431$	533.398	1.23758	0.618791	+	0.785556i	$0.287623\pi$
$432$	0	0
$433$	110.744i	0.255760i	0.991790	+	0.127880i	$0.0408173\pi$
$433$	110.744i	0.255760i	−0.991790	+	0.127880i	$0.959183\pi$
$434$	0	0
$435$	−268.462	−0.617153
$436$	0	0
$437$	1474.33i	3.37376i
$438$	0	0
$439$	622.569i	1.41815i	0.705132	+	0.709076i	$0.250888\pi$
$439$	622.569i	1.41815i	−0.705132	+	0.709076i	$0.749112\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	422.345	0.953375	0.476688	−	0.879073i	$-0.341837\pi$
$443$	422.345	0.953375	0.476688	+	0.879073i	$0.341837\pi$
$444$	0	0
$445$	−616.396	−1.38516
$446$	0	0
$447$	93.7075i	0.209636i
$448$	0	0
$449$	−365.850	−0.814810	−0.407405	−	0.913248i	$-0.633566\pi$
$449$	−365.850	−0.814810	−0.407405	+	0.913248i	$0.633566\pi$
$450$	0	0
$451$	96.9943i	0.215065i
$452$	0	0
$453$	228.200i	0.503752i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.6553	−0.0714557	−0.0357279	−	0.999362i	$-0.511375\pi$
$457$	−32.6553	−0.0714557	−0.0357279	+	0.999362i	$0.511375\pi$
$458$	0	0
$459$	125.079	0.272502
$460$	0	0
$461$	770.259i	1.67084i	0.549609	+	0.835422i	$0.314777\pi$
$461$	770.259i	1.67084i	−0.549609	+	0.835422i	$0.685223\pi$
$462$	0	0
$463$	308.030	0.665292	0.332646	−	0.943052i	$-0.392059\pi$
$463$	308.030	0.665292	0.332646	+	0.943052i	$0.392059\pi$
$464$	0	0
$465$	48.6757i	0.104679i
$466$	0	0
$467$	− 93.5893i	− 0.200405i	−0.994967	−	0.100203i	$-0.968051\pi$
$467$	− 93.5893i	− 0.200405i	0.994967	−	0.100203i	$-0.0319491\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−416.359	−0.883990
$472$	0	0
$473$	571.974	1.20925
$474$	0	0
$475$	− 150.632i	− 0.317119i
$476$	0	0
$477$	−57.4993	−0.120544
$478$	0	0
$479$	76.8169i	0.160369i	0.996780	+	0.0801846i	$0.0255510\pi$
$479$	76.8169i	0.160369i	−0.996780	+	0.0801846i	$0.974449\pi$
$480$	0	0
$481$	45.9636i	0.0955585i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	828.859	1.70899
$486$	0	0
$487$	−609.974	−1.25251	−0.626256	−	0.779617i	$-0.715414\pi$
$487$	−609.974	−1.25251	−0.626256	+	0.779617i	$0.715414\pi$
$488$	0	0
$489$	− 466.216i	− 0.953407i
$490$	0	0
$491$	−132.287	−0.269424	−0.134712	−	0.990885i	$-0.543011\pi$
$491$	−132.287	−0.269424	−0.134712	+	0.990885i	$0.543011\pi$
$492$	0	0
$493$	690.204i	1.40001i
$494$	0	0
$495$	170.834i	0.345120i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	393.614	0.788806	0.394403	−	0.918938i	$-0.370951\pi$
$499$	393.614	0.788806	0.394403	+	0.918938i	$0.370951\pi$
$500$	0	0
$501$	213.242	0.425634
$502$	0	0
$503$	− 79.0334i	− 0.157124i	−0.996909	−	0.0785620i	$-0.974967\pi$
$503$	− 79.0334i	− 0.157124i	0.996909	−	0.0785620i	$-0.0250329\pi$
$504$	0	0
$505$	95.3618	0.188835
$506$	0	0
$507$	− 41.7757i	− 0.0823978i
$508$	0	0
$509$	321.769i	0.632158i	0.948733	+	0.316079i	$0.102367\pi$
$509$	321.769i	0.632158i	−0.948733	+	0.316079i	$0.897633\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−185.446	−0.361493
$514$	0	0
$515$	747.216	1.45091
$516$	0	0
$517$	− 174.372i	− 0.337276i
$518$	0	0
$519$	−390.802	−0.752990
$520$	0	0
$521$	− 500.533i	− 0.960716i	−0.877073	−	0.480358i	$-0.840507\pi$
$521$	− 500.533i	− 0.960716i	0.877073	−	0.480358i	$-0.159493\pi$
$522$	0	0
$523$	− 133.011i	− 0.254323i	−0.991882	−	0.127161i	$-0.959413\pi$
$523$	− 133.011i	− 0.254323i	0.991882	−	0.127161i	$-0.0405866\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	125.143	0.237463
$528$	0	0
$529$	1177.55	2.22599
$530$	0	0
$531$	37.5026i	0.0706263i
$532$	0	0
$533$	−110.826	−0.207929
$534$	0	0
$535$	444.064i	0.830026i
$536$	0	0
$537$	− 462.889i	− 0.861990i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	899.673	1.66298	0.831491	−	0.555539i	$-0.187488\pi$
$541$	899.673	1.66298	0.831491	+	0.555539i	$0.187488\pi$
$542$	0	0
$543$	111.794	0.205881
$544$	0	0
$545$	488.548i	0.896418i
$546$	0	0
$547$	59.3373	0.108478	0.0542388	−	0.998528i	$-0.482727\pi$
$547$	59.3373	0.108478	0.0542388	+	0.998528i	$0.482727\pi$
$548$	0	0
$549$	291.393i	0.530770i
$550$	0	0
$551$	− 1023.32i	− 1.85721i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−35.7532	−0.0644201
$556$	0	0
$557$	696.966	1.25129	0.625643	−	0.780110i	$-0.284837\pi$
$557$	696.966	1.25129	0.625643	+	0.780110i	$0.284837\pi$
$558$	0	0
$559$	653.541i	1.16913i
$560$	0	0
$561$	439.208	0.782902
$562$	0	0
$563$	− 85.7945i	− 0.152388i	−0.997093	−	0.0761940i	$-0.975723\pi$
$563$	− 85.7945i	− 0.152388i	0.997093	−	0.0761940i	$-0.0242768\pi$
$564$	0	0
$565$	1118.93i	1.98040i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−520.704	−0.915121	−0.457561	−	0.889178i	$-0.651277\pi$
$569$	−520.704	−0.915121	−0.457561	+	0.889178i	$0.651277\pi$
$570$	0	0
$571$	180.744	0.316539	0.158270	−	0.987396i	$-0.449408\pi$
$571$	180.744	0.316539	0.158270	+	0.987396i	$0.449408\pi$
$572$	0	0
$573$	257.629i	0.449614i
$574$	0	0
$575$	−174.357	−0.303230
$576$	0	0
$577$	− 518.343i	− 0.898341i	−0.893446	−	0.449170i	$-0.851720\pi$
$577$	− 518.343i	− 0.898341i	0.893446	−	0.449170i	$-0.148280\pi$
$578$	0	0
$579$	− 51.7244i	− 0.0893340i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−201.906	−0.346323
$584$	0	0
$585$	−195.196	−0.333669
$586$	0	0
$587$	1072.05i	1.82632i	0.407602	+	0.913160i	$0.366365\pi$
$587$	1072.05i	1.82632i	−0.407602	+	0.913160i	$0.633635\pi$
$588$	0	0
$589$	−185.542	−0.315012
$590$	0	0
$591$	− 332.389i	− 0.562419i
$592$	0	0
$593$	− 266.145i	− 0.448812i	−0.974496	−	0.224406i	$-0.927956\pi$
$593$	− 266.145i	− 0.448812i	0.974496	−	0.224406i	$-0.0720441\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−37.1929	−0.0622997
$598$	0	0
$599$	906.845	1.51393	0.756966	−	0.653454i	$-0.226681\pi$
$599$	906.845	1.51393	0.756966	+	0.653454i	$0.226681\pi$
$600$	0	0
$601$	− 472.642i	− 0.786426i	−0.919448	−	0.393213i	$-0.871364\pi$
$601$	− 472.642i	− 0.786426i	0.919448	−	0.393213i	$-0.128636\pi$
$602$	0	0
$603$	2.99447	0.00496595
$604$	0	0
$605$	− 54.2022i	− 0.0895905i
$606$	0	0
$607$	− 22.6125i	− 0.0372529i	−0.999827	−	0.0186264i	$-0.994071\pi$
$607$	− 22.6125i	− 0.0372529i	0.999827	−	0.0186264i	$-0.00592933\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	199.238	0.326085
$612$	0	0
$613$	−4.78301	−0.00780263	−0.00390132	−	0.999992i	$-0.501242\pi$
$613$	−4.78301	−0.00780263	−0.00390132	+	0.999992i	$0.501242\pi$
$614$	0	0
$615$	− 86.2071i	− 0.140174i
$616$	0	0
$617$	−537.983	−0.871934	−0.435967	−	0.899963i	$-0.643594\pi$
$617$	−537.983	−0.871934	−0.435967	+	0.899963i	$0.643594\pi$
$618$	0	0
$619$	419.575i	0.677827i	0.940817	+	0.338914i	$0.110059\pi$
$619$	419.575i	0.677827i	−0.940817	+	0.338914i	$0.889941\pi$
$620$	0	0
$621$	214.655i	0.345661i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−712.703	−1.14032
$626$	0	0
$627$	−651.186	−1.03857
$628$	0	0
$629$	91.9200i	0.146137i
$630$	0	0
$631$	−346.433	−0.549022	−0.274511	−	0.961584i	$-0.588516\pi$
$631$	−346.433	−0.549022	−0.274511	+	0.961584i	$0.588516\pi$
$632$	0	0
$633$	− 311.016i	− 0.491337i
$634$	0	0
$635$	838.730i	1.32083i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−44.5202	−0.0696716
$640$	0	0
$641$	826.947	1.29009	0.645044	−	0.764145i	$-0.276839\pi$
$641$	826.947	1.29009	0.645044	+	0.764145i	$0.276839\pi$
$642$	0	0
$643$	1041.58i	1.61988i	0.586514	+	0.809939i	$0.300500\pi$
$643$	1041.58i	1.61988i	−0.586514	+	0.809939i	$0.699500\pi$
$644$	0	0
$645$	−508.362	−0.788159
$646$	0	0
$647$	− 536.371i	− 0.829013i	−0.910046	−	0.414507i	$-0.863954\pi$
$647$	− 536.371i	− 0.829013i	0.910046	−	0.414507i	$-0.136046\pi$
$648$	0	0
$649$	131.689i	0.202910i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	796.821	1.22025	0.610123	−	0.792307i	$-0.291120\pi$
$653$	796.821	1.22025	0.610123	+	0.792307i	$0.291120\pi$
$654$	0	0
$655$	−387.216	−0.591169
$656$	0	0
$657$	177.011i	0.269423i
$658$	0	0
$659$	−668.820	−1.01490	−0.507451	−	0.861681i	$-0.669412\pi$
$659$	−668.820	−1.01490	−0.507451	+	0.861681i	$0.669412\pi$
$660$	0	0
$661$	262.603i	0.397282i	0.980072	+	0.198641i	$0.0636527\pi$
$661$	262.603i	0.397282i	−0.980072	+	0.198641i	$0.936347\pi$
$662$	0	0
$663$	501.842i	0.756926i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1184.50	−1.77587
$668$	0	0
$669$	−512.702	−0.766371
$670$	0	0
$671$	1023.21i	1.52491i
$672$	0	0
$673$	−234.458	−0.348378	−0.174189	−	0.984712i	$-0.555730\pi$
$673$	−234.458	−0.348378	−0.174189	+	0.984712i	$0.555730\pi$
$674$	0	0
$675$	− 21.9312i	− 0.0324907i
$676$	0	0
$677$	− 523.889i	− 0.773839i	−0.922114	−	0.386919i	$-0.873539\pi$
$677$	− 523.889i	− 0.773839i	0.922114	−	0.386919i	$-0.126461\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−305.476	−0.448570
$682$	0	0
$683$	−225.386	−0.329994	−0.164997	−	0.986294i	$-0.552761\pi$
$683$	−225.386	−0.329994	−0.164997	+	0.986294i	$0.552761\pi$
$684$	0	0
$685$	822.978i	1.20143i
$686$	0	0
$687$	83.7724	0.121939
$688$	0	0
$689$	− 230.700i	− 0.334833i
$690$	0	0
$691$	558.965i	0.808922i	0.914555	+	0.404461i	$0.132541\pi$
$691$	558.965i	0.808922i	−0.914555	+	0.404461i	$0.867459\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	644.985	0.928037
$696$	0	0
$697$	−221.635	−0.317984
$698$	0	0
$699$	− 794.022i	− 1.13594i
$700$	0	0
$701$	−270.005	−0.385171	−0.192585	−	0.981280i	$-0.561687\pi$
$701$	−270.005	−0.385171	−0.192585	+	0.981280i	$0.561687\pi$
$702$	0	0
$703$	− 136.284i	− 0.193861i
$704$	0	0
$705$	154.979i	0.219828i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1253.29	−1.76769	−0.883846	−	0.467778i	$-0.845055\pi$
$709$	−1253.29	−1.76769	−0.883846	+	0.467778i	$0.845055\pi$
$710$	0	0
$711$	−304.535	−0.428319
$712$	0	0
$713$	214.766i	0.301215i
$714$	0	0
$715$	−685.424	−0.958634
$716$	0	0
$717$	215.500i	0.300558i
$718$	0	0
$719$	593.147i	0.824961i	0.910966	+	0.412481i	$0.135338\pi$
$719$	593.147i	0.824961i	−0.910966	+	0.412481i	$0.864662\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−380.636	−0.526468
$724$	0	0
$725$	121.020	0.166924
$726$	0	0
$727$	− 752.400i	− 1.03494i	−0.855702	−	0.517469i	$-0.826874\pi$
$727$	− 752.400i	− 1.03494i	0.855702	−	0.517469i	$-0.173126\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	1306.98i	1.78793i
$732$	0	0
$733$	735.085i	1.00284i	0.865203	+	0.501422i	$0.167190\pi$
$733$	735.085i	1.00284i	−0.865203	+	0.501422i	$0.832810\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.5149	0.0142672
$738$	0	0
$739$	68.7526	0.0930346	0.0465173	−	0.998917i	$-0.485188\pi$
$739$	68.7526	0.0930346	0.0465173	+	0.998917i	$0.485188\pi$
$740$	0	0
$741$	− 744.049i	− 1.00412i
$742$	0	0
$743$	−1023.37	−1.37735	−0.688674	−	0.725071i	$-0.741807\pi$
$743$	−1023.37	−1.37735	−0.688674	+	0.725071i	$0.741807\pi$
$744$	0	0
$745$	− 292.455i	− 0.392557i
$746$	0	0
$747$	68.1807i	0.0912727i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	73.6350	0.0980492	0.0490246	−	0.998798i	$-0.484389\pi$
$751$	73.6350	0.0980492	0.0490246	+	0.998798i	$0.484389\pi$
$752$	0	0
$753$	−120.478	−0.159998
$754$	0	0
$755$	− 712.196i	− 0.943307i
$756$	0	0
$757$	14.6651	0.0193727	0.00968634	−	0.999953i	$-0.496917\pi$
$757$	14.6651	0.0193727	0.00968634	+	0.999953i	$0.496917\pi$
$758$	0	0
$759$	753.753i	0.993086i
$760$	0	0
$761$	− 799.548i	− 1.05065i	−0.850901	−	0.525327i	$-0.823943\pi$
$761$	− 799.548i	− 1.05065i	0.850901	−	0.525327i	$-0.176057\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−390.362	−0.510277
$766$	0	0
$767$	−150.468	−0.196178
$768$	0	0
$769$	665.988i	0.866044i	0.901383	+	0.433022i	$0.142553\pi$
$769$	665.988i	0.866044i	−0.901383	+	0.433022i	$0.857447\pi$
$770$	0	0
$771$	176.318	0.228687
$772$	0	0
$773$	− 54.6730i	− 0.0707283i	−0.999374	−	0.0353642i	$-0.988741\pi$
$773$	− 54.6730i	− 0.0707283i	0.999374	−	0.0353642i	$-0.0112591\pi$
$774$	0	0
$775$	− 21.9425i	− 0.0283130i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	328.604	0.421828
$780$	0	0
$781$	−156.331	−0.200167
$782$	0	0
$783$	− 148.990i	− 0.190281i
$784$	0	0
$785$	1299.43	1.65532
$786$	0	0
$787$	− 439.045i	− 0.557872i	−0.960310	−	0.278936i	$-0.910018\pi$
$787$	− 439.045i	− 0.557872i	0.960310	−	0.278936i	$-0.0899818\pi$
$788$	0	0
$789$	− 366.458i	− 0.464459i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1169.13	−1.47431
$794$	0	0
$795$	179.452	0.225725
$796$	0	0
$797$	479.187i	0.601239i	0.953744	+	0.300619i	$0.0971934\pi$
$797$	479.187i	0.601239i	−0.953744	+	0.300619i	$0.902807\pi$
$798$	0	0
$799$	398.445	0.498680
$800$	0	0
$801$	− 342.086i	− 0.427074i
$802$	0	0
$803$	621.567i	0.774056i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	540.115	0.669287
$808$	0	0
$809$	740.232	0.914996	0.457498	−	0.889211i	$-0.348746\pi$
$809$	740.232	0.914996	0.457498	+	0.889211i	$0.348746\pi$
$810$	0	0
$811$	− 157.469i	− 0.194166i	−0.995276	−	0.0970831i	$-0.969049\pi$
$811$	− 157.469i	− 0.194166i	0.995276	−	0.0970831i	$-0.0309513\pi$
$812$	0	0
$813$	444.399	0.546616
$814$	0	0
$815$	1455.03i	1.78531i
$816$	0	0
$817$	− 1937.78i	− 2.37182i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−833.399	−1.01510	−0.507551	−	0.861622i	$-0.669449\pi$
$821$	−833.399	−1.01510	−0.507551	+	0.861622i	$0.669449\pi$
$822$	0	0
$823$	−580.360	−0.705176	−0.352588	−	0.935779i	$-0.614698\pi$
$823$	−580.360	−0.705176	−0.352588	+	0.935779i	$0.614698\pi$
$824$	0	0
$825$	− 77.0105i	− 0.0933460i
$826$	0	0
$827$	−310.093	−0.374962	−0.187481	−	0.982268i	$-0.560032\pi$
$827$	−310.093	−0.374962	−0.187481	+	0.982268i	$0.560032\pi$
$828$	0	0
$829$	− 589.980i	− 0.711677i	−0.934548	−	0.355838i	$-0.884195\pi$
$829$	− 589.980i	− 0.711677i	0.934548	−	0.355838i	$-0.115805\pi$
$830$	0	0
$831$	483.404i	0.581713i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−665.515	−0.797024
$836$	0	0
$837$	−27.0140	−0.0322747
$838$	0	0
$839$	1658.50i	1.97676i	0.152018	+	0.988378i	$0.451423\pi$
$839$	1658.50i	1.97676i	−0.152018	+	0.988378i	$0.548577\pi$
$840$	0	0
$841$	−18.8466	−0.0224098
$842$	0	0
$843$	145.782i	0.172932i
$844$	0	0
$845$	130.379i	0.154295i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−806.143	−0.949520
$850$	0	0
$851$	−157.750	−0.185370
$852$	0	0
$853$	− 921.244i	− 1.08001i	−0.841663	−	0.540003i	$-0.818423\pi$
$853$	− 921.244i	− 1.08001i	0.841663	−	0.540003i	$-0.181577\pi$
$854$	0	0
$855$	578.765	0.676918
$856$	0	0
$857$	− 1278.69i	− 1.49206i	−0.665914	−	0.746028i	$-0.731958\pi$
$857$	− 1278.69i	− 1.49206i	0.665914	−	0.746028i	$-0.268042\pi$
$858$	0	0
$859$	− 924.393i	− 1.07613i	−0.842904	−	0.538064i	$-0.819156\pi$
$859$	− 924.393i	− 1.07613i	0.842904	−	0.538064i	$-0.180844\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1222.69	−1.41679	−0.708394	−	0.705817i	$-0.750580\pi$
$863$	−1222.69	−1.41679	−0.708394	+	0.705817i	$0.750580\pi$
$864$	0	0
$865$	1219.67	1.41002
$866$	0	0
$867$	503.043i	0.580211i
$868$	0	0
$869$	−1069.36	−1.23056
$870$	0	0
$871$	12.0145i	0.0137939i
$872$	0	0
$873$	459.999i	0.526918i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1130.37	1.28891	0.644453	−	0.764644i	$-0.277085\pi$
$877$	1130.37	1.28891	0.644453	+	0.764644i	$0.277085\pi$
$878$	0	0
$879$	−501.521	−0.570559
$880$	0	0
$881$	− 37.1187i	− 0.0421325i	−0.999778	−	0.0210663i	$-0.993294\pi$
$881$	− 37.1187i	− 0.0421325i	0.999778	−	0.0210663i	$-0.00670609\pi$
$882$	0	0
$883$	−1517.07	−1.71809	−0.859045	−	0.511901i	$-0.828942\pi$
$883$	−1517.07	−1.71809	−0.859045	+	0.511901i	$0.828942\pi$
$884$	0	0
$885$	− 117.043i	− 0.132252i
$886$	0	0
$887$	− 1332.36i	− 1.50209i	−0.660248	−	0.751047i	$-0.729549\pi$
$887$	− 1332.36i	− 1.50209i	0.660248	−	0.751047i	$-0.270451\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−94.8093	−0.106408
$892$	0	0
$893$	−590.749	−0.661533
$894$	0	0
$895$	1444.64i	1.61413i
$896$	0	0
$897$	−861.243	−0.960137
$898$	0	0
$899$	− 149.067i	− 0.165815i
$900$	0	0
$901$	− 461.363i	− 0.512056i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−348.900	−0.385525
$906$	0	0
$907$	−65.4828	−0.0721971	−0.0360986	−	0.999348i	$-0.511493\pi$
$907$	−65.4828	−0.0721971	−0.0360986	+	0.999348i	$0.511493\pi$
$908$	0	0
$909$	52.9238i	0.0582220i
$910$	0	0
$911$	242.277	0.265946	0.132973	−	0.991120i	$-0.457548\pi$
$911$	242.277	0.265946	0.132973	+	0.991120i	$0.457548\pi$
$912$	0	0
$913$	239.414i	0.262227i
$914$	0	0
$915$	− 909.418i	− 0.993899i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	803.171	0.873962	0.436981	−	0.899471i	$-0.356048\pi$
$919$	803.171	0.873962	0.436981	+	0.899471i	$0.356048\pi$
$920$	0	0
$921$	587.980	0.638415
$922$	0	0
$923$	− 178.624i	− 0.193526i
$924$	0	0
$925$	16.1172	0.0174240
$926$	0	0
$927$	414.689i	0.447345i
$928$	0	0
$929$	629.140i	0.677223i	0.940926	+	0.338611i	$0.109957\pi$
$929$	629.140i	0.677223i	−0.940926	+	0.338611i	$0.890043\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−171.003	−0.183283
$934$	0	0
$935$	−1370.74	−1.46603
$936$	0	0
$937$	217.501i	0.232124i	0.993242	+	0.116062i	$0.0370272\pi$
$937$	217.501i	0.232124i	−0.993242	+	0.116062i	$0.962973\pi$
$938$	0	0
$939$	−371.734	−0.395883
$940$	0	0
$941$	1788.80i	1.90095i	0.310794	+	0.950477i	$0.399405\pi$
$941$	1788.80i	1.90095i	−0.310794	+	0.950477i	$0.600595\pi$
$942$	0	0
$943$	− 380.362i	− 0.403353i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1085.22	−1.14595	−0.572976	−	0.819572i	$-0.694211\pi$
$947$	−1085.22	−1.14595	−0.572976	+	0.819572i	$0.694211\pi$
$948$	0	0
$949$	−710.207	−0.748374
$950$	0	0
$951$	− 335.336i	− 0.352615i
$952$	0	0
$953$	1638.91	1.71973	0.859867	−	0.510518i	$-0.170546\pi$
$953$	1638.91	1.71973	0.859867	+	0.510518i	$0.170546\pi$
$954$	0	0
$955$	− 804.042i	− 0.841929i
$956$	0	0
$957$	− 523.173i	− 0.546681i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	933.972	0.971875
$962$	0	0
$963$	−246.446	−0.255915
$964$	0	0
$965$	161.428i	0.167283i
$966$	0	0
$967$	−486.560	−0.503164	−0.251582	−	0.967836i	$-0.580951\pi$
$967$	−486.560	−0.503164	−0.251582	+	0.967836i	$0.580951\pi$
$968$	0	0
$969$	− 1487.98i	− 1.53558i
$970$	0	0
$971$	349.516i	0.359955i	0.983671	+	0.179978i	$0.0576025\pi$
$971$	349.516i	0.359955i	−0.983671	+	0.179978i	$0.942397\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	87.9927	0.0902489
$976$	0	0
$977$	−240.292	−0.245949	−0.122974	−	0.992410i	$-0.539243\pi$
$977$	−240.292	−0.245949	−0.122974	+	0.992410i	$0.539243\pi$
$978$	0	0
$979$	− 1201.22i	− 1.22699i
$980$	0	0
$981$	−271.134	−0.276385
$982$	0	0
$983$	601.403i	0.611804i	0.952063	+	0.305902i	$0.0989580\pi$
$983$	601.403i	0.611804i	−0.952063	+	0.305902i	$0.901042\pi$
$984$	0	0
$985$	1037.36i	1.05316i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2242.99	−2.26794
$990$	0	0
$991$	−451.763	−0.455866	−0.227933	−	0.973677i	$-0.573197\pi$
$991$	−451.763	−0.455866	−0.227933	+	0.973677i	$0.573197\pi$
$992$	0	0
$993$	845.535i	0.851496i
$994$	0	0
$995$	116.077	0.116660
$996$	0	0
$997$	− 511.234i	− 0.512772i	−0.966574	−	0.256386i	$-0.917468\pi$
$997$	− 511.234i	− 0.512772i	0.966574	−	0.256386i	$-0.0825319\pi$
$998$	0	0
$999$	− 19.8422i	− 0.0198621i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.g.97.3		8
4.3	odd	2		1176.3.f.c.97.7		8
7.4	even	3		336.3.bh.g.145.2		8
7.5	odd	6		336.3.bh.g.241.2		8
7.6	odd	2	inner	2352.3.f.g.97.6		8
12.11	even	2		3528.3.f.b.2449.3		8
21.5	even	6		1008.3.cg.p.577.3		8
21.11	odd	6		1008.3.cg.p.145.3		8
28.3	even	6		1176.3.z.c.313.3		8
28.11	odd	6		168.3.z.b.145.2	yes	8
28.19	even	6		168.3.z.b.73.2	✓	8
28.23	odd	6		1176.3.z.c.913.3		8
28.27	even	2		1176.3.f.c.97.2		8
84.11	even	6		504.3.by.c.145.3		8
84.47	odd	6		504.3.by.c.73.3		8
84.83	odd	2		3528.3.f.b.2449.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.z.b.73.2	✓	8	28.19	even	6
168.3.z.b.145.2	yes	8	28.11	odd	6
336.3.bh.g.145.2		8	7.4	even	3
336.3.bh.g.241.2		8	7.5	odd	6
504.3.by.c.73.3		8	84.47	odd	6
504.3.by.c.145.3		8	84.11	even	6
1008.3.cg.p.145.3		8	21.11	odd	6
1008.3.cg.p.577.3		8	21.5	even	6
1176.3.f.c.97.2		8	28.27	even	2
1176.3.f.c.97.7		8	4.3	odd	2
1176.3.z.c.313.3		8	28.3	even	6
1176.3.z.c.913.3		8	28.23	odd	6
2352.3.f.g.97.3		8	1.1	even	1	trivial
2352.3.f.g.97.6		8	7.6	odd	2	inner
3528.3.f.b.2449.3		8	12.11	even	2
3528.3.f.b.2449.6		8	84.83	odd	2

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +5.40561i q^{5} -3.00000 q^{9} -10.5344 q^{11} -12.0366i q^{13} +9.36280 q^{15} -24.0714i q^{17} +35.6891i q^{19} +41.3104 q^{23} -4.22066 q^{25} +5.19615i q^{27} -28.6732 q^{29} +5.19884i q^{31} +18.2461i q^{33} -3.81864 q^{37} -20.8481 q^{39} -9.20741i q^{41} -54.2960 q^{43} -16.2168i q^{45} +16.5526i q^{47} -41.6929 q^{51} +19.1664 q^{53} -56.9447i q^{55} +61.8154 q^{57} -12.5009i q^{59} -97.1310i q^{61} +65.0655 q^{65} -0.998156 q^{67} -71.5517i q^{69} +14.8401 q^{71} -59.0037i q^{73} +7.31040i q^{75} +101.512 q^{79} +9.00000 q^{81} -22.7269i q^{83} +130.121 q^{85} +49.6635i q^{87} +114.029i q^{89} +9.00465 q^{93} -192.922 q^{95} -153.333i q^{97} +31.6031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} - 44 q^{11} + 12 q^{15} + 96 q^{23} - 84 q^{25} + 68 q^{29} + 236 q^{37} - 36 q^{39} + 92 q^{43} + 72 q^{51} - 20 q^{53} + 84 q^{57} + 296 q^{65} + 44 q^{67} + 392 q^{71} + 328 q^{79} + 72 q^{81}+ \cdots + 132 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 - 44 * q^11 + 12 * q^15 + 96 * q^23 - 84 * q^25 + 68 * q^29 + 236 * q^37 - 36 * q^39 + 92 * q^43 + 72 * q^51 - 20 * q^53 + 84 * q^57 + 296 * q^65 + 44 * q^67 + 392 * q^71 + 328 * q^79 + 72 * q^81 + 200 * q^85 + 120 * q^93 + 132 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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