LMFDB - Embedded newform 1344.4.c.a.673.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 722x^{3} + 11881x^{2} + 54936x + 127008 $ x^6 - 2x^5 + 2x^4 + 722x^3 + 11881x^2 + 54936*x + 127008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.1
Root			$-4.87742 + 4.87742i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.a.673.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -11.7548i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -11.7548i q^{5} -7.00000 q^{7} -9.00000 q^{9} +72.4604i q^{11} -50.7056i q^{13} -35.2645 q^{15} +60.4604 q^{17} +33.8235i q^{19} +21.0000i q^{21} -116.852 q^{23} -13.1765 q^{25} +27.0000i q^{27} -13.1959i q^{29} +250.117 q^{31} +217.381 q^{33} +82.2839i q^{35} +92.7056i q^{37} -152.117 q^{39} -69.1663 q^{41} +69.6471i q^{43} +105.794i q^{45} +346.039 q^{47} +49.0000 q^{49} -181.381i q^{51} +585.232i q^{53} +851.761 q^{55} +101.471 q^{57} +66.1946i q^{59} -492.057i q^{61} +63.0000 q^{63} -596.036 q^{65} +543.704i q^{67} +350.556i q^{69} +365.128 q^{71} +374.174 q^{73} +39.5294i q^{75} -507.223i q^{77} +670.176 q^{79} +81.0000 q^{81} -595.115i q^{83} -710.703i q^{85} -39.5876 q^{87} +1036.53 q^{89} +354.939i q^{91} -750.350i q^{93} +397.590 q^{95} -218.998 q^{97} -652.144i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 42 q^{7} - 54 q^{9}+O(q^{10}) $ 6 * q - 42 * q^7 - 54 * q^9	$ 6 q - 42 q^{7} - 54 q^{9} - 24 q^{15} - 16 q^{17} + 60 q^{23} - 138 q^{25} + 552 q^{31} + 168 q^{33} + 36 q^{39} - 272 q^{41} + 1576 q^{47} + 294 q^{49} + 1632 q^{55} + 432 q^{57} + 378 q^{63} - 664 q^{65} + 2548 q^{71} + 444 q^{73} + 3528 q^{79} + 486 q^{81} + 336 q^{87} + 16 q^{89} + 4776 q^{95} + 1548 q^{97}+O(q^{100}) $ 6 * q - 42 * q^7 - 54 * q^9 - 24 * q^15 - 16 * q^17 + 60 * q^23 - 138 * q^25 + 552 * q^31 + 168 * q^33 + 36 * q^39 - 272 * q^41 + 1576 * q^47 + 294 * q^49 + 1632 * q^55 + 432 * q^57 + 378 * q^63 - 664 * q^65 + 2548 * q^71 + 444 * q^73 + 3528 * q^79 + 486 * q^81 + 336 * q^87 + 16 * q^89 + 4776 * q^95 + 1548 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	− 11.7548i	− 1.05139i	−0.850674	−	0.525693i	$-0.823806\pi$
$5$	− 11.7548i	− 1.05139i	0.850674	−	0.525693i	$-0.176194\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	72.4604i	1.98615i	0.117487	+	0.993074i	$0.462516\pi$
$11$	72.4604i	1.98615i	−0.117487	+	0.993074i	$0.537484\pi$
$12$	0	0
$13$	− 50.7056i	− 1.08178i	−0.841092	−	0.540892i	$-0.818087\pi$
$13$	− 50.7056i	− 1.08178i	0.841092	−	0.540892i	$-0.181913\pi$
$14$	0	0
$15$	−35.2645	−0.607018
$16$	0	0
$17$	60.4604	0.862577	0.431288	−	0.902214i	$-0.358059\pi$
$17$	60.4604	0.862577	0.431288	+	0.902214i	$0.358059\pi$
$18$	0	0
$19$	33.8235i	0.408403i	0.978929	+	0.204201i	$0.0654597\pi$
$19$	33.8235i	0.408403i	−0.978929	+	0.204201i	$0.934540\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−116.852	−1.05936	−0.529682	−	0.848197i	$-0.677689\pi$
$23$	−116.852	−1.05936	−0.529682	+	0.848197i	$0.677689\pi$
$24$	0	0
$25$	−13.1765	−0.105412
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	− 13.1959i	− 0.0844969i	−0.999107	−	0.0422485i	$-0.986548\pi$
$29$	− 13.1959i	− 0.0844969i	0.999107	−	0.0422485i	$-0.0134521\pi$
$30$	0	0
$31$	250.117	1.44911	0.724553	−	0.689219i	$-0.242046\pi$
$31$	250.117	1.44911	0.724553	+	0.689219i	$0.242046\pi$
$32$	0	0
$33$	217.381	1.14670
$34$	0	0
$35$	82.2839i	0.397386i
$36$	0	0
$37$	92.7056i	0.411911i	0.978561	+	0.205955i	$0.0660302\pi$
$37$	92.7056i	0.411911i	−0.978561	+	0.205955i	$0.933970\pi$
$38$	0	0
$39$	−152.117	−0.624568
$40$	0	0
$41$	−69.1663	−0.263462	−0.131731	−	0.991285i	$-0.542054\pi$
$41$	−69.1663	−0.263462	−0.131731	+	0.991285i	$0.542054\pi$
$42$	0	0
$43$	69.6471i	0.247002i	0.992344	+	0.123501i	$0.0394122\pi$
$43$	69.6471i	0.247002i	−0.992344	+	0.123501i	$0.960588\pi$
$44$	0	0
$45$	105.794i	0.350462i
$46$	0	0
$47$	346.039	1.07394	0.536968	−	0.843603i	$-0.319570\pi$
$47$	346.039	1.07394	0.536968	+	0.843603i	$0.319570\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	− 181.381i	− 0.498009i
$52$	0	0
$53$	585.232i	1.51675i	0.651818	+	0.758376i	$0.274007\pi$
$53$	585.232i	1.51675i	−0.651818	+	0.758376i	$0.725993\pi$
$54$	0	0
$55$	851.761	2.08821
$56$	0	0
$57$	101.471	0.235791
$58$	0	0
$59$	66.1946i	0.146064i	0.997330	+	0.0730322i	$0.0232676\pi$
$59$	66.1946i	0.146064i	−0.997330	+	0.0730322i	$0.976732\pi$
$60$	0	0
$61$	− 492.057i	− 1.03281i	−0.856344	−	0.516405i	$-0.827270\pi$
$61$	− 492.057i	− 1.03281i	0.856344	−	0.516405i	$-0.172730\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−596.036	−1.13737
$66$	0	0
$67$	543.704i	0.991403i	0.868493	+	0.495701i	$0.165089\pi$
$67$	543.704i	0.991403i	−0.868493	+	0.495701i	$0.834911\pi$
$68$	0	0
$69$	350.556i	0.611624i
$70$	0	0
$71$	365.128	0.610319	0.305160	−	0.952301i	$-0.401290\pi$
$71$	365.128	0.610319	0.305160	+	0.952301i	$0.401290\pi$
$72$	0	0
$73$	374.174	0.599915	0.299957	−	0.953953i	$-0.403028\pi$
$73$	374.174	0.599915	0.299957	+	0.953953i	$0.403028\pi$
$74$	0	0
$75$	39.5294i	0.0608595i
$76$	0	0
$77$	− 507.223i	− 0.750694i
$78$	0	0
$79$	670.176	0.954439	0.477220	−	0.878784i	$-0.341645\pi$
$79$	670.176	0.954439	0.477220	+	0.878784i	$0.341645\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 595.115i	− 0.787017i	−0.919321	−	0.393508i	$-0.871261\pi$
$83$	− 595.115i	− 0.787017i	0.919321	−	0.393508i	$-0.128739\pi$
$84$	0	0
$85$	− 710.703i	− 0.906901i
$86$	0	0
$87$	−39.5876	−0.0487843
$88$	0	0
$89$	1036.53	1.23452	0.617260	−	0.786759i	$-0.288243\pi$
$89$	1036.53	1.23452	0.617260	+	0.786759i	$0.288243\pi$
$90$	0	0
$91$	354.939i	0.408876i
$92$	0	0
$93$	− 750.350i	− 0.836642i
$94$	0	0
$95$	397.590	0.429389
$96$	0	0
$97$	−218.998	−0.229236	−0.114618	−	0.993410i	$-0.536564\pi$
$97$	−218.998	−0.229236	−0.114618	+	0.993410i	$0.536564\pi$
$98$	0	0
$99$	− 652.144i	− 0.662050i
$100$	0	0
$101$	446.361i	0.439748i	0.975528	+	0.219874i	$0.0705647\pi$
$101$	446.361i	0.439748i	−0.975528	+	0.219874i	$0.929435\pi$
$102$	0	0
$103$	−335.995	−0.321423	−0.160712	−	0.987001i	$-0.551379\pi$
$103$	−335.995	−0.321423	−0.160712	+	0.987001i	$0.551379\pi$
$104$	0	0
$105$	246.852	0.229431
$106$	0	0
$107$	− 1966.85i	− 1.77703i	−0.458849	−	0.888514i	$-0.651738\pi$
$107$	− 1966.85i	− 1.77703i	0.458849	−	0.888514i	$-0.348262\pi$
$108$	0	0
$109$	− 1161.76i	− 1.02089i	−0.859911	−	0.510444i	$-0.829481\pi$
$109$	− 1161.76i	− 1.02089i	0.859911	−	0.510444i	$-0.170519\pi$
$110$	0	0
$111$	278.117	0.237817
$112$	0	0
$113$	252.726	0.210394	0.105197	−	0.994451i	$-0.466453\pi$
$113$	252.726	0.210394	0.105197	+	0.994451i	$0.466453\pi$
$114$	0	0
$115$	1373.58i	1.11380i
$116$	0	0
$117$	456.350i	0.360595i
$118$	0	0
$119$	−423.223	−0.326023
$120$	0	0
$121$	−3919.51	−2.94479
$122$	0	0
$123$	207.499i	0.152110i
$124$	0	0
$125$	− 1314.47i	− 0.940557i
$126$	0	0
$127$	−2119.46	−1.48088	−0.740438	−	0.672124i	$-0.765382\pi$
$127$	−2119.46	−1.48088	−0.740438	+	0.672124i	$0.765382\pi$
$128$	0	0
$129$	208.941	0.142607
$130$	0	0
$131$	1199.47i	0.799985i	0.916518	+	0.399993i	$0.130987\pi$
$131$	1199.47i	0.799985i	−0.916518	+	0.399993i	$0.869013\pi$
$132$	0	0
$133$	− 236.765i	− 0.154362i
$134$	0	0
$135$	317.381	0.202339
$136$	0	0
$137$	−466.885	−0.291158	−0.145579	−	0.989347i	$-0.546504\pi$
$137$	−466.885	−0.291158	−0.145579	+	0.989347i	$0.546504\pi$
$138$	0	0
$139$	− 2256.57i	− 1.37698i	−0.725246	−	0.688490i	$-0.758274\pi$
$139$	− 2256.57i	− 1.37698i	0.725246	−	0.688490i	$-0.241726\pi$
$140$	0	0
$141$	− 1038.12i	− 0.620037i
$142$	0	0
$143$	3674.15	2.14858
$144$	0	0
$145$	−155.115	−0.0888389
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	− 1777.97i	− 0.977566i	−0.872405	−	0.488783i	$-0.837441\pi$
$149$	− 1777.97i	− 0.977566i	0.872405	−	0.488783i	$-0.162559\pi$
$150$	0	0
$151$	1853.75	0.999049	0.499525	−	0.866300i	$-0.333508\pi$
$151$	1853.75	0.999049	0.499525	+	0.866300i	$0.333508\pi$
$152$	0	0
$153$	−544.144	−0.287526
$154$	0	0
$155$	− 2940.08i	− 1.52357i
$156$	0	0
$157$	800.278i	0.406810i	0.979095	+	0.203405i	$0.0652008\pi$
$157$	800.278i	0.406810i	−0.979095	+	0.203405i	$0.934799\pi$
$158$	0	0
$159$	1755.70	0.875697
$160$	0	0
$161$	817.965	0.400402
$162$	0	0
$163$	− 923.582i	− 0.443807i	−0.975069	−	0.221903i	$-0.928773\pi$
$163$	− 923.582i	− 0.443807i	0.975069	−	0.221903i	$-0.0712270\pi$
$164$	0	0
$165$	− 2555.28i	− 1.20563i
$166$	0	0
$167$	2439.11	1.13020	0.565102	−	0.825021i	$-0.308837\pi$
$167$	2439.11	1.13020	0.565102	+	0.825021i	$0.308837\pi$
$168$	0	0
$169$	−374.054	−0.170257
$170$	0	0
$171$	− 304.412i	− 0.136134i
$172$	0	0
$173$	− 814.093i	− 0.357771i	−0.983870	−	0.178885i	$-0.942751\pi$
$173$	− 814.093i	− 0.357771i	0.983870	−	0.178885i	$-0.0572491\pi$
$174$	0	0
$175$	92.2353	0.0398419
$176$	0	0
$177$	198.584	0.0843304
$178$	0	0
$179$	1490.42i	0.622340i	0.950354	+	0.311170i	$0.100721\pi$
$179$	1490.42i	0.622340i	−0.950354	+	0.311170i	$0.899279\pi$
$180$	0	0
$181$	− 41.6575i	− 0.0171071i	−0.999963	−	0.00855353i	$-0.997277\pi$
$181$	− 41.6575i	− 0.0171071i	0.999963	−	0.00855353i	$-0.00272271\pi$
$182$	0	0
$183$	−1476.17	−0.596293
$184$	0	0
$185$	1089.74	0.433077
$186$	0	0
$187$	4380.99i	1.71321i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	2597.64	0.984076	0.492038	−	0.870574i	$-0.336252\pi$
$191$	2597.64	0.984076	0.492038	+	0.870574i	$0.336252\pi$
$192$	0	0
$193$	1468.75	0.547788	0.273894	−	0.961760i	$-0.411688\pi$
$193$	1468.75	0.547788	0.273894	+	0.961760i	$0.411688\pi$
$194$	0	0
$195$	1788.11i	0.656662i
$196$	0	0
$197$	− 1366.23i	− 0.494112i	−0.969001	−	0.247056i	$-0.920537\pi$
$197$	− 1366.23i	− 0.494112i	0.969001	−	0.247056i	$-0.0794632\pi$
$198$	0	0
$199$	2876.46	1.02466	0.512329	−	0.858789i	$-0.328783\pi$
$199$	2876.46	1.02466	0.512329	+	0.858789i	$0.328783\pi$
$200$	0	0
$201$	1631.11	0.572387
$202$	0	0
$203$	92.3711i	0.0319368i
$204$	0	0
$205$	813.039i	0.277001i
$206$	0	0
$207$	1051.67	0.353121
$208$	0	0
$209$	−2450.87	−0.811148
$210$	0	0
$211$	5569.49i	1.81716i	0.417716	+	0.908578i	$0.362831\pi$
$211$	5569.49i	1.81716i	−0.417716	+	0.908578i	$0.637169\pi$
$212$	0	0
$213$	− 1095.38i	− 0.352368i
$214$	0	0
$215$	818.691	0.259694
$216$	0	0
$217$	−1750.82	−0.547711
$218$	0	0
$219$	− 1122.52i	− 0.346361i
$220$	0	0
$221$	− 3065.68i	− 0.933122i
$222$	0	0
$223$	−5713.73	−1.71578	−0.857891	−	0.513831i	$-0.828226\pi$
$223$	−5713.73	−1.71578	−0.857891	+	0.513831i	$0.828226\pi$
$224$	0	0
$225$	118.588	0.0351373
$226$	0	0
$227$	1186.64i	0.346962i	0.984837	+	0.173481i	$0.0555015\pi$
$227$	1186.64i	0.346962i	−0.984837	+	0.173481i	$0.944498\pi$
$228$	0	0
$229$	− 5136.22i	− 1.48214i	−0.671426	−	0.741072i	$-0.734318\pi$
$229$	− 5136.22i	− 1.48214i	0.671426	−	0.741072i	$-0.265682\pi$
$230$	0	0
$231$	−1521.67	−0.433413
$232$	0	0
$233$	3084.93	0.867383	0.433692	−	0.901061i	$-0.357211\pi$
$233$	3084.93	0.867383	0.433692	+	0.901061i	$0.357211\pi$
$234$	0	0
$235$	− 4067.63i	− 1.12912i
$236$	0	0
$237$	− 2010.53i	− 0.551046i
$238$	0	0
$239$	5795.47	1.56853	0.784263	−	0.620429i	$-0.213041\pi$
$239$	5795.47	1.56853	0.784263	+	0.620429i	$0.213041\pi$
$240$	0	0
$241$	4918.74	1.31471	0.657353	−	0.753583i	$-0.271676\pi$
$241$	4918.74	1.31471	0.657353	+	0.753583i	$0.271676\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 575.988i	− 0.150198i
$246$	0	0
$247$	1715.04	0.441803
$248$	0	0
$249$	−1785.35	−0.454384
$250$	0	0
$251$	6190.85i	1.55682i	0.627753	+	0.778412i	$0.283975\pi$
$251$	6190.85i	1.55682i	−0.627753	+	0.778412i	$0.716025\pi$
$252$	0	0
$253$	− 8467.16i	− 2.10405i
$254$	0	0
$255$	−2132.11	−0.523599
$256$	0	0
$257$	1065.97	0.258730	0.129365	−	0.991597i	$-0.458706\pi$
$257$	1065.97	0.258730	0.129365	+	0.991597i	$0.458706\pi$
$258$	0	0
$259$	− 648.939i	− 0.155688i
$260$	0	0
$261$	118.763i	0.0281656i
$262$	0	0
$263$	−7817.25	−1.83282	−0.916411	−	0.400238i	$-0.868927\pi$
$263$	−7817.25	−1.83282	−0.916411	+	0.400238i	$0.868927\pi$
$264$	0	0
$265$	6879.32	1.59469
$266$	0	0
$267$	− 3109.60i	− 0.712750i
$268$	0	0
$269$	3801.54i	0.861651i	0.902435	+	0.430826i	$0.141778\pi$
$269$	3801.54i	0.861651i	−0.902435	+	0.430826i	$0.858222\pi$
$270$	0	0
$271$	6241.27	1.39900	0.699502	−	0.714630i	$-0.253405\pi$
$271$	6241.27	1.39900	0.699502	+	0.714630i	$0.253405\pi$
$272$	0	0
$273$	1064.82	0.236065
$274$	0	0
$275$	− 954.773i	− 0.209363i
$276$	0	0
$277$	− 5487.86i	− 1.19037i	−0.803587	−	0.595187i	$-0.797078\pi$
$277$	− 5487.86i	− 1.19037i	0.803587	−	0.595187i	$-0.202922\pi$
$278$	0	0
$279$	−2251.05	−0.483035
$280$	0	0
$281$	3378.67	0.717276	0.358638	−	0.933477i	$-0.383241\pi$
$281$	3378.67	0.717276	0.358638	+	0.933477i	$0.383241\pi$
$282$	0	0
$283$	− 8802.26i	− 1.84891i	−0.381297	−	0.924453i	$-0.624522\pi$
$283$	− 8802.26i	− 1.84891i	0.381297	−	0.924453i	$-0.375478\pi$
$284$	0	0
$285$	− 1192.77i	− 0.247908i
$286$	0	0
$287$	484.164	0.0995795
$288$	0	0
$289$	−1257.54	−0.255961
$290$	0	0
$291$	656.993i	0.132349i
$292$	0	0
$293$	− 5034.83i	− 1.00388i	−0.864901	−	0.501942i	$-0.832619\pi$
$293$	− 5034.83i	− 1.00388i	0.864901	−	0.501942i	$-0.167381\pi$
$294$	0	0
$295$	778.108	0.153570
$296$	0	0
$297$	−1956.43	−0.382235
$298$	0	0
$299$	5925.05i	1.14600i
$300$	0	0
$301$	− 487.529i	− 0.0933579i
$302$	0	0
$303$	1339.08	0.253889
$304$	0	0
$305$	−5784.05	−1.08588
$306$	0	0
$307$	− 8342.24i	− 1.55087i	−0.631428	−	0.775435i	$-0.717531\pi$
$307$	− 8342.24i	− 1.55087i	0.631428	−	0.775435i	$-0.282469\pi$
$308$	0	0
$309$	1007.99i	0.185574i
$310$	0	0
$311$	8071.66	1.47171	0.735855	−	0.677139i	$-0.236781\pi$
$311$	8071.66	1.47171	0.735855	+	0.677139i	$0.236781\pi$
$312$	0	0
$313$	−6596.89	−1.19130	−0.595652	−	0.803242i	$-0.703106\pi$
$313$	−6596.89	−1.19130	−0.595652	+	0.803242i	$0.703106\pi$
$314$	0	0
$315$	− 740.555i	− 0.132462i
$316$	0	0
$317$	10263.7i	1.81851i	0.416242	+	0.909254i	$0.363347\pi$
$317$	10263.7i	1.81851i	−0.416242	+	0.909254i	$0.636653\pi$
$318$	0	0
$319$	956.178	0.167823
$320$	0	0
$321$	−5900.54	−1.02597
$322$	0	0
$323$	2044.98i	0.352279i
$324$	0	0
$325$	668.120i	0.114033i
$326$	0	0
$327$	−3485.29	−0.589410
$328$	0	0
$329$	−2422.27	−0.405909
$330$	0	0
$331$	11311.7i	1.87839i	0.343383	+	0.939195i	$0.388427\pi$
$331$	11311.7i	1.87839i	−0.343383	+	0.939195i	$0.611573\pi$
$332$	0	0
$333$	− 834.350i	− 0.137304i
$334$	0	0
$335$	6391.15	1.04235
$336$	0	0
$337$	4700.58	0.759813	0.379907	−	0.925025i	$-0.375956\pi$
$337$	4700.58	0.759813	0.379907	+	0.925025i	$0.375956\pi$
$338$	0	0
$339$	− 758.179i	− 0.121471i
$340$	0	0
$341$	18123.6i	2.87814i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	4120.74	0.643052
$346$	0	0
$347$	− 7390.19i	− 1.14330i	−0.820496	−	0.571652i	$-0.806303\pi$
$347$	− 7390.19i	− 1.14330i	0.820496	−	0.571652i	$-0.193697\pi$
$348$	0	0
$349$	− 10093.8i	− 1.54816i	−0.633088	−	0.774080i	$-0.718213\pi$
$349$	− 10093.8i	− 1.54816i	0.633088	−	0.774080i	$-0.281787\pi$
$350$	0	0
$351$	1369.05	0.208189
$352$	0	0
$353$	−10557.2	−1.59180	−0.795898	−	0.605431i	$-0.793001\pi$
$353$	−10557.2	−1.59180	−0.795898	+	0.605431i	$0.793001\pi$
$354$	0	0
$355$	− 4292.02i	− 0.641681i
$356$	0	0
$357$	1269.67i	0.188230i
$358$	0	0
$359$	8857.59	1.30219	0.651094	−	0.758997i	$-0.274310\pi$
$359$	8857.59	1.30219	0.651094	+	0.758997i	$0.274310\pi$
$360$	0	0
$361$	5714.97	0.833207
$362$	0	0
$363$	11758.5i	1.70017i
$364$	0	0
$365$	− 4398.36i	− 0.630742i
$366$	0	0
$367$	10701.6	1.52212	0.761061	−	0.648680i	$-0.224679\pi$
$367$	10701.6	1.52212	0.761061	+	0.648680i	$0.224679\pi$
$368$	0	0
$369$	622.497	0.0878208
$370$	0	0
$371$	− 4096.62i	− 0.573278i
$372$	0	0
$373$	2451.52i	0.340308i	0.985417	+	0.170154i	$0.0544265\pi$
$373$	2451.52i	0.340308i	−0.985417	+	0.170154i	$0.945573\pi$
$374$	0	0
$375$	−3943.41	−0.543031
$376$	0	0
$377$	−669.104	−0.0914074
$378$	0	0
$379$	− 170.113i	− 0.0230557i	−0.999934	−	0.0115279i	$-0.996330\pi$
$379$	− 170.113i	− 0.0230557i	0.999934	−	0.0115279i	$-0.00366952\pi$
$380$	0	0
$381$	6358.37i	0.854985i
$382$	0	0
$383$	4339.37	0.578933	0.289467	−	0.957188i	$-0.406522\pi$
$383$	4339.37	0.578933	0.289467	+	0.957188i	$0.406522\pi$
$384$	0	0
$385$	−5962.33	−0.789269
$386$	0	0
$387$	− 626.824i	− 0.0823339i
$388$	0	0
$389$	14598.8i	1.90280i	0.307954	+	0.951401i	$0.400356\pi$
$389$	14598.8i	1.90280i	−0.307954	+	0.951401i	$0.599644\pi$
$390$	0	0
$391$	−7064.93	−0.913782
$392$	0	0
$393$	3598.41	0.461872
$394$	0	0
$395$	− 7877.82i	− 1.00348i
$396$	0	0
$397$	1185.22i	0.149835i	0.997190	+	0.0749175i	$0.0238693\pi$
$397$	1185.22i	0.149835i	−0.997190	+	0.0749175i	$0.976131\pi$
$398$	0	0
$399$	−710.294	−0.0891208
$400$	0	0
$401$	3672.99	0.457407	0.228704	−	0.973496i	$-0.426551\pi$
$401$	3672.99	0.457407	0.228704	+	0.973496i	$0.426551\pi$
$402$	0	0
$403$	− 12682.3i	− 1.56762i
$404$	0	0
$405$	− 952.143i	− 0.116821i
$406$	0	0
$407$	−6717.48	−0.818116
$408$	0	0
$409$	−8842.03	−1.06897	−0.534487	−	0.845177i	$-0.679495\pi$
$409$	−8842.03	−1.06897	−0.534487	+	0.845177i	$0.679495\pi$
$410$	0	0
$411$	1400.65i	0.168100i
$412$	0	0
$413$	− 463.362i	− 0.0552072i
$414$	0	0
$415$	−6995.49	−0.827458
$416$	0	0
$417$	−6769.72	−0.795000
$418$	0	0
$419$	10691.7i	1.24660i	0.781984	+	0.623298i	$0.214208\pi$
$419$	10691.7i	1.24660i	−0.781984	+	0.623298i	$0.785792\pi$
$420$	0	0
$421$	15909.2i	1.84173i	0.389881	+	0.920865i	$0.372516\pi$
$421$	15909.2i	1.84173i	−0.389881	+	0.920865i	$0.627484\pi$
$422$	0	0
$423$	−3114.35	−0.357978
$424$	0	0
$425$	−796.655	−0.0909257
$426$	0	0
$427$	3444.40i	0.390366i
$428$	0	0
$429$	− 11022.4i	− 1.24049i
$430$	0	0
$431$	6689.77	0.747644	0.373822	−	0.927500i	$-0.378047\pi$
$431$	6689.77	0.747644	0.373822	+	0.927500i	$0.378047\pi$
$432$	0	0
$433$	13265.8	1.47232	0.736160	−	0.676808i	$-0.236637\pi$
$433$	13265.8	1.47232	0.736160	+	0.676808i	$0.236637\pi$
$434$	0	0
$435$	465.346i	0.0512911i
$436$	0	0
$437$	− 3952.35i	− 0.432647i
$438$	0	0
$439$	−6722.13	−0.730820	−0.365410	−	0.930847i	$-0.619071\pi$
$439$	−6722.13	−0.730820	−0.365410	+	0.930847i	$0.619071\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	8937.62i	0.958554i	0.877664	+	0.479277i	$0.159101\pi$
$443$	8937.62i	0.958554i	−0.877664	+	0.479277i	$0.840899\pi$
$444$	0	0
$445$	− 12184.3i	− 1.29796i
$446$	0	0
$447$	−5333.92	−0.564398
$448$	0	0
$449$	12718.8	1.33683	0.668414	−	0.743790i	$-0.266974\pi$
$449$	12718.8	1.33683	0.668414	+	0.743790i	$0.266974\pi$
$450$	0	0
$451$	− 5011.82i	− 0.523276i
$452$	0	0
$453$	− 5561.26i	− 0.576801i
$454$	0	0
$455$	4172.25	0.429886
$456$	0	0
$457$	−6941.85	−0.710560	−0.355280	−	0.934760i	$-0.615615\pi$
$457$	−6941.85	−0.710560	−0.355280	+	0.934760i	$0.615615\pi$
$458$	0	0
$459$	1632.43i	0.166003i
$460$	0	0
$461$	10090.3i	1.01942i	0.860347	+	0.509709i	$0.170247\pi$
$461$	10090.3i	1.01942i	−0.860347	+	0.509709i	$0.829753\pi$
$462$	0	0
$463$	−7973.79	−0.800375	−0.400187	−	0.916433i	$-0.631055\pi$
$463$	−7973.79	−0.800375	−0.400187	+	0.916433i	$0.631055\pi$
$464$	0	0
$465$	−8820.25	−0.879633
$466$	0	0
$467$	− 9435.71i	− 0.934973i	−0.884000	−	0.467486i	$-0.845160\pi$
$467$	− 9435.71i	− 0.934973i	0.884000	−	0.467486i	$-0.154840\pi$
$468$	0	0
$469$	− 3805.93i	− 0.374715i
$470$	0	0
$471$	2400.83	0.234872
$472$	0	0
$473$	−5046.65	−0.490582
$474$	0	0
$475$	− 445.675i	− 0.0430504i
$476$	0	0
$477$	− 5267.09i	− 0.505584i
$478$	0	0
$479$	444.695	0.0424189	0.0212094	−	0.999775i	$-0.493248\pi$
$479$	444.695	0.0424189	0.0212094	+	0.999775i	$0.493248\pi$
$480$	0	0
$481$	4700.69	0.445599
$482$	0	0
$483$	− 2453.90i	− 0.231172i
$484$	0	0
$485$	2574.29i	0.241015i
$486$	0	0
$487$	−12965.2	−1.20638	−0.603191	−	0.797597i	$-0.706104\pi$
$487$	−12965.2	−1.20638	−0.603191	+	0.797597i	$0.706104\pi$
$488$	0	0
$489$	−2770.74	−0.256232
$490$	0	0
$491$	− 17323.6i	− 1.59227i	−0.605120	−	0.796134i	$-0.706875\pi$
$491$	− 17323.6i	− 1.59227i	0.605120	−	0.796134i	$-0.293125\pi$
$492$	0	0
$493$	− 797.828i	− 0.0728851i
$494$	0	0
$495$	−7665.85	−0.696069
$496$	0	0
$497$	−2555.89	−0.230679
$498$	0	0
$499$	3053.26i	0.273913i	0.990577	+	0.136957i	$0.0437321\pi$
$499$	3053.26i	0.273913i	−0.990577	+	0.136957i	$0.956268\pi$
$500$	0	0
$501$	− 7317.33i	− 0.652523i
$502$	0	0
$503$	−4076.65	−0.361369	−0.180685	−	0.983541i	$-0.557831\pi$
$503$	−4076.65	−0.361369	−0.180685	+	0.983541i	$0.557831\pi$
$504$	0	0
$505$	5246.90	0.462345
$506$	0	0
$507$	1122.16i	0.0982979i
$508$	0	0
$509$	− 458.202i	− 0.0399007i	−0.999801	−	0.0199503i	$-0.993649\pi$
$509$	− 458.202i	− 0.0399007i	0.999801	−	0.0199503i	$-0.00635081\pi$
$510$	0	0
$511$	−2619.22	−0.226747
$512$	0	0
$513$	−913.235	−0.0785971
$514$	0	0
$515$	3949.57i	0.337940i
$516$	0	0
$517$	25074.1i	2.13299i
$518$	0	0
$519$	−2442.28	−0.206559
$520$	0	0
$521$	6244.54	0.525103	0.262551	−	0.964918i	$-0.415436\pi$
$521$	6244.54	0.525103	0.262551	+	0.964918i	$0.415436\pi$
$522$	0	0
$523$	22863.4i	1.91156i	0.294086	+	0.955779i	$0.404985\pi$
$523$	22863.4i	1.91156i	−0.294086	+	0.955779i	$0.595015\pi$
$524$	0	0
$525$	− 276.706i	− 0.0230027i
$526$	0	0
$527$	15122.2	1.24997
$528$	0	0
$529$	1487.42	0.122251
$530$	0	0
$531$	− 595.751i	− 0.0486882i
$532$	0	0
$533$	3507.12i	0.285010i
$534$	0	0
$535$	−23120.0	−1.86834
$536$	0	0
$537$	4471.25	0.359308
$538$	0	0
$539$	3550.56i	0.283736i
$540$	0	0
$541$	20780.9i	1.65146i	0.564066	+	0.825730i	$0.309236\pi$
$541$	20780.9i	1.65146i	−0.564066	+	0.825730i	$0.690764\pi$
$542$	0	0
$543$	−124.973	−0.00987677
$544$	0	0
$545$	−13656.4	−1.07335
$546$	0	0
$547$	9848.10i	0.769788i	0.922961	+	0.384894i	$0.125762\pi$
$547$	9848.10i	0.769788i	−0.922961	+	0.384894i	$0.874238\pi$
$548$	0	0
$549$	4428.51i	0.344270i
$550$	0	0
$551$	446.331	0.0345088
$552$	0	0
$553$	−4691.23	−0.360744
$554$	0	0
$555$	− 3269.22i	− 0.250037i
$556$	0	0
$557$	20834.5i	1.58489i	0.609942	+	0.792446i	$0.291193\pi$
$557$	20834.5i	1.58489i	−0.609942	+	0.792446i	$0.708807\pi$
$558$	0	0
$559$	3531.49	0.267203
$560$	0	0
$561$	13143.0	0.989120
$562$	0	0
$563$	2168.31i	0.162315i	0.996701	+	0.0811575i	$0.0258617\pi$
$563$	2168.31i	0.162315i	−0.996701	+	0.0811575i	$0.974138\pi$
$564$	0	0
$565$	− 2970.76i	− 0.221205i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−3909.14	−0.288013	−0.144007	−	0.989577i	$-0.545999\pi$
$569$	−3909.14	−0.288013	−0.144007	+	0.989577i	$0.545999\pi$
$570$	0	0
$571$	− 6744.18i	− 0.494282i	−0.968979	−	0.247141i	$-0.920509\pi$
$571$	− 6744.18i	− 0.494282i	0.968979	−	0.247141i	$-0.0794912\pi$
$572$	0	0
$573$	− 7792.92i	− 0.568157i
$574$	0	0
$575$	1539.70	0.111669
$576$	0	0
$577$	16895.9	1.21904	0.609521	−	0.792770i	$-0.291362\pi$
$577$	16895.9	1.21904	0.609521	+	0.792770i	$0.291362\pi$
$578$	0	0
$579$	− 4406.25i	− 0.316265i
$580$	0	0
$581$	4165.81i	0.297464i
$582$	0	0
$583$	−42406.2	−3.01249
$584$	0	0
$585$	5364.33	0.379124
$586$	0	0
$587$	14853.0i	1.04438i	0.852831	+	0.522188i	$0.174884\pi$
$587$	14853.0i	1.04438i	−0.852831	+	0.522188i	$0.825116\pi$
$588$	0	0
$589$	8459.83i	0.591819i
$590$	0	0
$591$	−4098.70	−0.285276
$592$	0	0
$593$	13967.4	0.967239	0.483620	−	0.875278i	$-0.339322\pi$
$593$	13967.4	0.967239	0.483620	+	0.875278i	$0.339322\pi$
$594$	0	0
$595$	4974.92i	0.342776i
$596$	0	0
$597$	− 8629.39i	− 0.591587i
$598$	0	0
$599$	1474.31	0.100565	0.0502826	−	0.998735i	$-0.483988\pi$
$599$	1474.31	0.100565	0.0502826	+	0.998735i	$0.483988\pi$
$600$	0	0
$601$	3720.31	0.252503	0.126252	−	0.991998i	$-0.459705\pi$
$601$	3720.31	0.252503	0.126252	+	0.991998i	$0.459705\pi$
$602$	0	0
$603$	− 4893.33i	− 0.330468i
$604$	0	0
$605$	46073.3i	3.09611i
$606$	0	0
$607$	10969.4	0.733501	0.366750	−	0.930319i	$-0.380470\pi$
$607$	10969.4	0.733501	0.366750	+	0.930319i	$0.380470\pi$
$608$	0	0
$609$	277.113	0.0184387
$610$	0	0
$611$	− 17546.1i	− 1.16177i
$612$	0	0
$613$	− 768.228i	− 0.0506173i	−0.999680	−	0.0253087i	$-0.991943\pi$
$613$	− 768.228i	− 0.0506173i	0.999680	−	0.0253087i	$-0.00805686\pi$
$614$	0	0
$615$	2439.12	0.159926
$616$	0	0
$617$	9912.22	0.646760	0.323380	−	0.946269i	$-0.395181\pi$
$617$	9912.22	0.646760	0.323380	+	0.946269i	$0.395181\pi$
$618$	0	0
$619$	− 10821.1i	− 0.702647i	−0.936254	−	0.351323i	$-0.885732\pi$
$619$	− 10821.1i	− 0.702647i	0.936254	−	0.351323i	$-0.114268\pi$
$620$	0	0
$621$	− 3155.01i	− 0.203875i
$622$	0	0
$623$	−7255.73	−0.466605
$624$	0	0
$625$	−17098.4	−1.09430
$626$	0	0
$627$	7352.60i	0.468317i
$628$	0	0
$629$	5605.02i	0.355305i
$630$	0	0
$631$	−21838.3	−1.37776	−0.688881	−	0.724874i	$-0.741898\pi$
$631$	−21838.3	−1.37776	−0.688881	+	0.724874i	$0.741898\pi$
$632$	0	0
$633$	16708.5	1.04914
$634$	0	0
$635$	24913.9i	1.55697i
$636$	0	0
$637$	− 2484.57i	− 0.154541i
$638$	0	0
$639$	−3286.15	−0.203440
$640$	0	0
$641$	3288.45	0.202630	0.101315	−	0.994854i	$-0.467695\pi$
$641$	3288.45	0.202630	0.101315	+	0.994854i	$0.467695\pi$
$642$	0	0
$643$	6563.13i	0.402527i	0.979537	+	0.201263i	$0.0645047\pi$
$643$	6563.13i	0.402527i	−0.979537	+	0.201263i	$0.935495\pi$
$644$	0	0
$645$	− 2456.07i	− 0.149934i
$646$	0	0
$647$	−13561.5	−0.824043	−0.412022	−	0.911174i	$-0.635177\pi$
$647$	−13561.5	−0.824043	−0.412022	+	0.911174i	$0.635177\pi$
$648$	0	0
$649$	−4796.49	−0.290106
$650$	0	0
$651$	5252.45i	0.316221i
$652$	0	0
$653$	− 9533.77i	− 0.571340i	−0.958328	−	0.285670i	$-0.907784\pi$
$653$	− 9533.77i	− 0.571340i	0.958328	−	0.285670i	$-0.0922162\pi$
$654$	0	0
$655$	14099.6	0.841093
$656$	0	0
$657$	−3367.57	−0.199972
$658$	0	0
$659$	− 253.185i	− 0.0149661i	−0.999972	−	0.00748307i	$-0.997618\pi$
$659$	− 253.185i	− 0.0149661i	0.999972	−	0.00748307i	$-0.00238196\pi$
$660$	0	0
$661$	− 29105.5i	− 1.71266i	−0.516425	−	0.856332i	$-0.672738\pi$
$661$	− 29105.5i	− 1.71266i	0.516425	−	0.856332i	$-0.327262\pi$
$662$	0	0
$663$	−9197.04	−0.538738
$664$	0	0
$665$	−2783.13	−0.162294
$666$	0	0
$667$	1541.97i	0.0895130i
$668$	0	0
$669$	17141.2i	0.990608i
$670$	0	0
$671$	35654.6	2.05132
$672$	0	0
$673$	−3546.16	−0.203112	−0.101556	−	0.994830i	$-0.532382\pi$
$673$	−3546.16	−0.203112	−0.101556	+	0.994830i	$0.532382\pi$
$674$	0	0
$675$	− 355.765i	− 0.0202865i
$676$	0	0
$677$	3670.94i	0.208399i	0.994556	+	0.104199i	$0.0332280\pi$
$677$	3670.94i	0.208399i	−0.994556	+	0.104199i	$0.966772\pi$
$678$	0	0
$679$	1532.98	0.0866429
$680$	0	0
$681$	3559.93	0.200319
$682$	0	0
$683$	− 12118.2i	− 0.678904i	−0.940623	−	0.339452i	$-0.889758\pi$
$683$	− 12118.2i	− 0.678904i	0.940623	−	0.339452i	$-0.110242\pi$
$684$	0	0
$685$	5488.16i	0.306119i
$686$	0	0
$687$	−15408.6	−0.855716
$688$	0	0
$689$	29674.5	1.64080
$690$	0	0
$691$	27270.4i	1.50132i	0.660686	+	0.750662i	$0.270265\pi$
$691$	27270.4i	1.50132i	−0.660686	+	0.750662i	$0.729735\pi$
$692$	0	0
$693$	4565.01i	0.250231i
$694$	0	0
$695$	−26525.7	−1.44774
$696$	0	0
$697$	−4181.82	−0.227257
$698$	0	0
$699$	− 9254.78i	− 0.500784i
$700$	0	0
$701$	28435.3i	1.53208i	0.642796	+	0.766038i	$0.277774\pi$
$701$	28435.3i	1.53208i	−0.642796	+	0.766038i	$0.722226\pi$
$702$	0	0
$703$	−3135.63	−0.168225
$704$	0	0
$705$	−12202.9	−0.651898
$706$	0	0
$707$	− 3124.53i	− 0.166209i
$708$	0	0
$709$	− 24828.2i	− 1.31515i	−0.753388	−	0.657577i	$-0.771582\pi$
$709$	− 24828.2i	− 1.31515i	0.753388	−	0.657577i	$-0.228418\pi$
$710$	0	0
$711$	−6031.59	−0.318146
$712$	0	0
$713$	−29226.7	−1.53513
$714$	0	0
$715$	− 43189.0i	− 2.25899i
$716$	0	0
$717$	− 17386.4i	− 0.905589i
$718$	0	0
$719$	−20755.0	−1.07654	−0.538269	−	0.842773i	$-0.680921\pi$
$719$	−20755.0	−1.07654	−0.538269	+	0.842773i	$0.680921\pi$
$720$	0	0
$721$	2351.97	0.121487
$722$	0	0
$723$	− 14756.2i	− 0.759046i
$724$	0	0
$725$	173.875i	0.00890697i
$726$	0	0
$727$	−32260.1	−1.64575	−0.822875	−	0.568223i	$-0.807631\pi$
$727$	−32260.1	−1.64575	−0.822875	+	0.568223i	$0.807631\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	4210.89i	0.213058i
$732$	0	0
$733$	− 26453.5i	− 1.33299i	−0.745508	−	0.666497i	$-0.767793\pi$
$733$	− 26453.5i	− 1.33299i	0.745508	−	0.666497i	$-0.232207\pi$
$734$	0	0
$735$	−1727.96	−0.0867168
$736$	0	0
$737$	−39397.0	−1.96907
$738$	0	0
$739$	3725.00i	0.185421i	0.995693	+	0.0927107i	$0.0295532\pi$
$739$	3725.00i	0.185421i	−0.995693	+	0.0927107i	$0.970447\pi$
$740$	0	0
$741$	− 5145.12i	− 0.255075i
$742$	0	0
$743$	25781.9	1.27301	0.636505	−	0.771273i	$-0.280379\pi$
$743$	25781.9	1.27301	0.636505	+	0.771273i	$0.280379\pi$
$744$	0	0
$745$	−20899.8	−1.02780
$746$	0	0
$747$	5356.04i	0.262339i
$748$	0	0
$749$	13767.9i	0.671654i
$750$	0	0
$751$	−8267.60	−0.401716	−0.200858	−	0.979620i	$-0.564373\pi$
$751$	−8267.60	−0.401716	−0.200858	+	0.979620i	$0.564373\pi$
$752$	0	0
$753$	18572.5	0.898833
$754$	0	0
$755$	− 21790.6i	− 1.05039i
$756$	0	0
$757$	− 20675.7i	− 0.992694i	−0.868124	−	0.496347i	$-0.834674\pi$
$757$	− 20675.7i	− 0.992694i	0.868124	−	0.496347i	$-0.165326\pi$
$758$	0	0
$759$	−25401.5	−1.21478
$760$	0	0
$761$	−8607.01	−0.409992	−0.204996	−	0.978763i	$-0.565718\pi$
$761$	−8607.01	−0.409992	−0.204996	+	0.978763i	$0.565718\pi$
$762$	0	0
$763$	8132.34i	0.385859i
$764$	0	0
$765$	6396.33i	0.302300i
$766$	0	0
$767$	3356.43	0.158010
$768$	0	0
$769$	−35697.9	−1.67399	−0.836995	−	0.547210i	$-0.815690\pi$
$769$	−35697.9	−1.67399	−0.836995	+	0.547210i	$0.815690\pi$
$770$	0	0
$771$	− 3197.92i	− 0.149378i
$772$	0	0
$773$	− 13279.0i	− 0.617870i	−0.951083	−	0.308935i	$-0.900027\pi$
$773$	− 13279.0i	− 0.617870i	0.951083	−	0.308935i	$-0.0999726\pi$
$774$	0	0
$775$	−3295.66	−0.152753
$776$	0	0
$777$	−1946.82	−0.0898863
$778$	0	0
$779$	− 2339.45i	− 0.107599i
$780$	0	0
$781$	26457.3i	1.21218i
$782$	0	0
$783$	356.288	0.0162614
$784$	0	0
$785$	9407.15	0.427714
$786$	0	0
$787$	14156.7i	0.641209i	0.947213	+	0.320604i	$0.103886\pi$
$787$	14156.7i	0.641209i	−0.947213	+	0.320604i	$0.896114\pi$
$788$	0	0
$789$	23451.7i	1.05818i
$790$	0	0
$791$	−1769.08	−0.0795213
$792$	0	0
$793$	−24950.0	−1.11728
$794$	0	0
$795$	− 20637.9i	− 0.920695i
$796$	0	0
$797$	32003.6i	1.42236i	0.703008	+	0.711182i	$0.251840\pi$
$797$	32003.6i	1.42236i	−0.703008	+	0.711182i	$0.748160\pi$
$798$	0	0
$799$	20921.6	0.926351
$800$	0	0
$801$	−9328.80	−0.411507
$802$	0	0
$803$	27112.8i	1.19152i
$804$	0	0
$805$	− 9615.06i	− 0.420977i
$806$	0	0
$807$	11404.6	0.497475
$808$	0	0
$809$	23023.1	1.00056	0.500278	−	0.865865i	$-0.333231\pi$
$809$	23023.1	1.00056	0.500278	+	0.865865i	$0.333231\pi$
$810$	0	0
$811$	16578.4i	0.717812i	0.933374	+	0.358906i	$0.116850\pi$
$811$	16578.4i	0.717812i	−0.933374	+	0.358906i	$0.883150\pi$
$812$	0	0
$813$	− 18723.8i	− 0.807716i
$814$	0	0
$815$	−10856.6	−0.466612
$816$	0	0
$817$	−2355.71	−0.100876
$818$	0	0
$819$	− 3194.45i	− 0.136292i
$820$	0	0
$821$	− 5711.11i	− 0.242776i	−0.992605	−	0.121388i	$-0.961265\pi$
$821$	− 5711.11i	− 0.242776i	0.992605	−	0.121388i	$-0.0387345\pi$
$822$	0	0
$823$	−6254.66	−0.264913	−0.132457	−	0.991189i	$-0.542287\pi$
$823$	−6254.66	−0.264913	−0.132457	+	0.991189i	$0.542287\pi$
$824$	0	0
$825$	−2864.32	−0.120876
$826$	0	0
$827$	− 30670.6i	− 1.28963i	−0.764339	−	0.644814i	$-0.776935\pi$
$827$	− 30670.6i	− 1.28963i	0.764339	−	0.644814i	$-0.223065\pi$
$828$	0	0
$829$	− 34681.9i	− 1.45302i	−0.687156	−	0.726510i	$-0.741141\pi$
$829$	− 34681.9i	− 1.45302i	0.687156	−	0.726510i	$-0.258859\pi$
$830$	0	0
$831$	−16463.6	−0.687263
$832$	0	0
$833$	2962.56	0.123225
$834$	0	0
$835$	− 28671.4i	− 1.18828i
$836$	0	0
$837$	6753.15i	0.278881i
$838$	0	0
$839$	−11537.3	−0.474744	−0.237372	−	0.971419i	$-0.576286\pi$
$839$	−11537.3	−0.474744	−0.237372	+	0.971419i	$0.576286\pi$
$840$	0	0
$841$	24214.9	0.992860
$842$	0	0
$843$	− 10136.0i	− 0.414119i
$844$	0	0
$845$	4396.95i	0.179006i
$846$	0	0
$847$	27436.6	1.11302
$848$	0	0
$849$	−26406.8	−1.06747
$850$	0	0
$851$	− 10832.8i	− 0.436363i
$852$	0	0
$853$	13110.0i	0.526233i	0.964764	+	0.263116i	$0.0847503\pi$
$853$	13110.0i	0.526233i	−0.964764	+	0.263116i	$0.915250\pi$
$854$	0	0
$855$	−3578.31	−0.143130
$856$	0	0
$857$	39493.4	1.57418	0.787088	−	0.616841i	$-0.211588\pi$
$857$	39493.4	1.57418	0.787088	+	0.616841i	$0.211588\pi$
$858$	0	0
$859$	27189.9i	1.07999i	0.841669	+	0.539994i	$0.181573\pi$
$859$	27189.9i	1.07999i	−0.841669	+	0.539994i	$0.818427\pi$
$860$	0	0
$861$	− 1452.49i	− 0.0574922i
$862$	0	0
$863$	44314.3	1.74794	0.873972	−	0.485976i	$-0.161536\pi$
$863$	44314.3	1.74794	0.873972	+	0.485976i	$0.161536\pi$
$864$	0	0
$865$	−9569.53	−0.376155
$866$	0	0
$867$	3772.61i	0.147779i
$868$	0	0
$869$	48561.2i	1.89566i
$870$	0	0
$871$	27568.8	1.07248
$872$	0	0
$873$	1970.98	0.0764119
$874$	0	0
$875$	9201.28i	0.355497i
$876$	0	0
$877$	43050.0i	1.65758i	0.559563	+	0.828788i	$0.310969\pi$
$877$	43050.0i	1.65758i	−0.559563	+	0.828788i	$0.689031\pi$
$878$	0	0
$879$	−15104.5	−0.579593
$880$	0	0
$881$	21065.2	0.805569	0.402784	−	0.915295i	$-0.368042\pi$
$881$	21065.2	0.805569	0.402784	+	0.915295i	$0.368042\pi$
$882$	0	0
$883$	− 38004.2i	− 1.44841i	−0.689587	−	0.724203i	$-0.742208\pi$
$883$	− 38004.2i	− 1.44841i	0.689587	−	0.724203i	$-0.257792\pi$
$884$	0	0
$885$	− 2334.32i	− 0.0886637i
$886$	0	0
$887$	−8568.20	−0.324343	−0.162171	−	0.986763i	$-0.551850\pi$
$887$	−8568.20	−0.324343	−0.162171	+	0.986763i	$0.551850\pi$
$888$	0	0
$889$	14836.2	0.559719
$890$	0	0
$891$	5869.29i	0.220683i
$892$	0	0
$893$	11704.3i	0.438598i
$894$	0	0
$895$	17519.6	0.654319
$896$	0	0
$897$	17775.2	0.661645
$898$	0	0
$899$	− 3300.51i	− 0.122445i
$900$	0	0
$901$	35383.4i	1.30831i
$902$	0	0
$903$	−1462.59	−0.0539002
$904$	0	0
$905$	−489.678	−0.0179861
$906$	0	0
$907$	26976.9i	0.987600i	0.869575	+	0.493800i	$0.164393\pi$
$907$	26976.9i	0.987600i	−0.869575	+	0.493800i	$0.835607\pi$
$908$	0	0
$909$	− 4017.25i	− 0.146583i
$910$	0	0
$911$	48524.9	1.76477	0.882384	−	0.470531i	$-0.155937\pi$
$911$	48524.9	1.76477	0.882384	+	0.470531i	$0.155937\pi$
$912$	0	0
$913$	43122.3	1.56313
$914$	0	0
$915$	17352.2i	0.626934i
$916$	0	0
$917$	− 8396.28i	− 0.302366i
$918$	0	0
$919$	−19675.1	−0.706227	−0.353114	−	0.935580i	$-0.614877\pi$
$919$	−19675.1	−0.706227	−0.353114	+	0.935580i	$0.614877\pi$
$920$	0	0
$921$	−25026.7	−0.895395
$922$	0	0
$923$	− 18514.0i	− 0.660234i
$924$	0	0
$925$	− 1221.53i	− 0.0434203i
$926$	0	0
$927$	3023.96	0.107141
$928$	0	0
$929$	−50122.0	−1.77013	−0.885065	−	0.465468i	$-0.845886\pi$
$929$	−50122.0	−1.77013	−0.885065	+	0.465468i	$0.845886\pi$
$930$	0	0
$931$	1657.35i	0.0583432i
$932$	0	0
$933$	− 24215.0i	− 0.849692i
$934$	0	0
$935$	51497.8	1.80124
$936$	0	0
$937$	−36136.2	−1.25989	−0.629945	−	0.776640i	$-0.716923\pi$
$937$	−36136.2	−1.25989	−0.629945	+	0.776640i	$0.716923\pi$
$938$	0	0
$939$	19790.7i	0.687800i
$940$	0	0
$941$	− 14214.9i	− 0.492445i	−0.969213	−	0.246223i	$-0.920811\pi$
$941$	− 14214.9i	− 0.492445i	0.969213	−	0.246223i	$-0.0791894\pi$
$942$	0	0
$943$	8082.23	0.279103
$944$	0	0
$945$	−2221.67	−0.0764771
$946$	0	0
$947$	40485.7i	1.38924i	0.719377	+	0.694619i	$0.244427\pi$
$947$	40485.7i	1.38924i	−0.719377	+	0.694619i	$0.755573\pi$
$948$	0	0
$949$	− 18972.7i	− 0.648978i
$950$	0	0
$951$	30791.1	1.04992
$952$	0	0
$953$	39737.8	1.35072	0.675359	−	0.737489i	$-0.263989\pi$
$953$	39737.8	1.35072	0.675359	+	0.737489i	$0.263989\pi$
$954$	0	0
$955$	− 30534.9i	− 1.03464i
$956$	0	0
$957$	− 2868.53i	− 0.0968929i
$958$	0	0
$959$	3268.19	0.110047
$960$	0	0
$961$	32767.4	1.09991
$962$	0	0
$963$	17701.6i	0.592343i
$964$	0	0
$965$	− 17265.0i	− 0.575936i
$966$	0	0
$967$	−1577.22	−0.0524508	−0.0262254	−	0.999656i	$-0.508349\pi$
$967$	−1577.22	−0.0524508	−0.0262254	+	0.999656i	$0.508349\pi$
$968$	0	0
$969$	6134.95	0.203388
$970$	0	0
$971$	3863.04i	0.127673i	0.997960	+	0.0638367i	$0.0203337\pi$
$971$	3863.04i	0.127673i	−0.997960	+	0.0638367i	$0.979666\pi$
$972$	0	0
$973$	15796.0i	0.520449i
$974$	0	0
$975$	2004.36	0.0658369
$976$	0	0
$977$	13916.9	0.455721	0.227861	−	0.973694i	$-0.426827\pi$
$977$	13916.9	0.455721	0.227861	+	0.973694i	$0.426827\pi$
$978$	0	0
$979$	75107.6i	2.45194i
$980$	0	0
$981$	10455.9i	0.340296i
$982$	0	0
$983$	−4570.14	−0.148286	−0.0741428	−	0.997248i	$-0.523622\pi$
$983$	−4570.14	−0.148286	−0.0741428	+	0.997248i	$0.523622\pi$
$984$	0	0
$985$	−16059.9	−0.519503
$986$	0	0
$987$	7266.81i	0.234352i
$988$	0	0
$989$	− 8138.41i	− 0.261665i
$990$	0	0
$991$	19916.6	0.638418	0.319209	−	0.947684i	$-0.396583\pi$
$991$	19916.6	0.638418	0.319209	+	0.947684i	$0.396583\pi$
$992$	0	0
$993$	33935.1	1.08449
$994$	0	0
$995$	− 33812.4i	− 1.07731i
$996$	0	0
$997$	− 8951.10i	− 0.284337i	−0.989842	−	0.142169i	$-0.954592\pi$
$997$	− 8951.10i	− 0.284337i	0.989842	−	0.142169i	$-0.0454075\pi$
$998$	0	0
$999$	−2503.05	−0.0792723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.a.673.1	✓	6
4.3	odd	2		1344.4.c.d.673.4	yes	6
8.3	odd	2		1344.4.c.d.673.3	yes	6
8.5	even	2	inner	1344.4.c.a.673.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.a.673.1	✓	6	1.1	even	1	trivial
1344.4.c.a.673.6	yes	6	8.5	even	2	inner
1344.4.c.d.673.3	yes	6	8.3	odd	2
1344.4.c.d.673.4	yes	6	4.3	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -11.7548i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -11.7548i q^{5} -7.00000 q^{7} -9.00000 q^{9} +72.4604i q^{11} -50.7056i q^{13} -35.2645 q^{15} +60.4604 q^{17} +33.8235i q^{19} +21.0000i q^{21} -116.852 q^{23} -13.1765 q^{25} +27.0000i q^{27} -13.1959i q^{29} +250.117 q^{31} +217.381 q^{33} +82.2839i q^{35} +92.7056i q^{37} -152.117 q^{39} -69.1663 q^{41} +69.6471i q^{43} +105.794i q^{45} +346.039 q^{47} +49.0000 q^{49} -181.381i q^{51} +585.232i q^{53} +851.761 q^{55} +101.471 q^{57} +66.1946i q^{59} -492.057i q^{61} +63.0000 q^{63} -596.036 q^{65} +543.704i q^{67} +350.556i q^{69} +365.128 q^{71} +374.174 q^{73} +39.5294i q^{75} -507.223i q^{77} +670.176 q^{79} +81.0000 q^{81} -595.115i q^{83} -710.703i q^{85} -39.5876 q^{87} +1036.53 q^{89} +354.939i q^{91} -750.350i q^{93} +397.590 q^{95} -218.998 q^{97} -652.144i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 42 q^{7} - 54 q^{9}+O(q^{10}) \) 6 * q - 42 * q^7 - 54 * q^9	\( 6 q - 42 q^{7} - 54 q^{9} - 24 q^{15} - 16 q^{17} + 60 q^{23} - 138 q^{25} + 552 q^{31} + 168 q^{33} + 36 q^{39} - 272 q^{41} + 1576 q^{47} + 294 q^{49} + 1632 q^{55} + 432 q^{57} + 378 q^{63} - 664 q^{65} + 2548 q^{71} + 444 q^{73} + 3528 q^{79} + 486 q^{81} + 336 q^{87} + 16 q^{89} + 4776 q^{95} + 1548 q^{97}+O(q^{100}) \) 6 * q - 42 * q^7 - 54 * q^9 - 24 * q^15 - 16 * q^17 + 60 * q^23 - 138 * q^25 + 552 * q^31 + 168 * q^33 + 36 * q^39 - 272 * q^41 + 1576 * q^47 + 294 * q^49 + 1632 * q^55 + 432 * q^57 + 378 * q^63 - 664 * q^65 + 2548 * q^71 + 444 * q^73 + 3528 * q^79 + 486 * q^81 + 336 * q^87 + 16 * q^89 + 4776 * q^95 + 1548 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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