LMFDB - Embedded newform 384.4.d.d.193.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.d (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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.4.d.d.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -2.82843i q^{5} +14.1421 q^{7} -9.00000 q^{9} +20.0000i q^{11} +39.5980i q^{13} +8.48528 q^{15} -34.0000 q^{17} -52.0000i q^{19} +42.4264i q^{21} +62.2254 q^{23} +117.000 q^{25} -27.0000i q^{27} +200.818i q^{29} +110.309 q^{31} -60.0000 q^{33} -40.0000i q^{35} +271.529i q^{37} -118.794 q^{39} +26.0000 q^{41} +252.000i q^{43} +25.4558i q^{45} -345.068 q^{47} -143.000 q^{49} -102.000i q^{51} +681.651i q^{53} +56.5685 q^{55} +156.000 q^{57} +364.000i q^{59} +735.391i q^{61} -127.279 q^{63} +112.000 q^{65} -628.000i q^{67} +186.676i q^{69} -333.754 q^{71} -338.000 q^{73} +351.000i q^{75} +282.843i q^{77} +789.131 q^{79} +81.0000 q^{81} -1036.00i q^{83} +96.1665i q^{85} -602.455 q^{87} -234.000 q^{89} +560.000i q^{91} +330.926i q^{93} -147.078 q^{95} -178.000 q^{97} -180.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9} - 136 q^{17} + 468 q^{25} - 240 q^{33} + 104 q^{41} - 572 q^{49} + 624 q^{57} + 448 q^{65} - 1352 q^{73} + 324 q^{81} - 936 q^{89} - 712 q^{97}+O(q^{100}) $ 4 * q - 36 * q^9 - 136 * q^17 + 468 * q^25 - 240 * q^33 + 104 * q^41 - 572 * q^49 + 624 * q^57 + 448 * q^65 - 1352 * q^73 + 324 * q^81 - 936 * q^89 - 712 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	− 2.82843i	− 0.252982i	−0.991968	−	0.126491i	$-0.959628\pi$
$5$	− 2.82843i	− 0.252982i	0.991968	−	0.126491i	$-0.0403715\pi$
$6$	0	0
$7$	14.1421	0.763604	0.381802	−	0.924244i	$-0.375304\pi$
$7$	14.1421	0.763604	0.381802	+	0.924244i	$0.375304\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	20.0000i	0.548202i	0.961701	+	0.274101i	$0.0883803\pi$
$11$	20.0000i	0.548202i	−0.961701	+	0.274101i	$0.911620\pi$
$12$	0	0
$13$	39.5980i	0.844808i	0.906408	+	0.422404i	$0.138814\pi$
$13$	39.5980i	0.844808i	−0.906408	+	0.422404i	$0.861186\pi$
$14$	0	0
$15$	8.48528	0.146059
$16$	0	0
$17$	−34.0000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−34.0000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 52.0000i	− 0.627875i	−0.949444	−	0.313937i	$-0.898352\pi$
$19$	− 52.0000i	− 0.627875i	0.949444	−	0.313937i	$-0.101648\pi$
$20$	0	0
$21$	42.4264i	0.440867i
$22$	0	0
$23$	62.2254	0.564126	0.282063	−	0.959396i	$-0.408981\pi$
$23$	62.2254	0.564126	0.282063	+	0.959396i	$0.408981\pi$
$24$	0	0
$25$	117.000	0.936000
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	200.818i	1.28590i	0.765909	+	0.642949i	$0.222289\pi$
$29$	200.818i	1.28590i	−0.765909	+	0.642949i	$0.777711\pi$
$30$	0	0
$31$	110.309	0.639097	0.319549	−	0.947570i	$-0.396469\pi$
$31$	110.309	0.639097	0.319549	+	0.947570i	$0.396469\pi$
$32$	0	0
$33$	−60.0000	−0.316505
$34$	0	0
$35$	− 40.0000i	− 0.193178i
$36$	0	0
$37$	271.529i	1.20646i	0.797567	+	0.603231i	$0.206120\pi$
$37$	271.529i	1.20646i	−0.797567	+	0.603231i	$0.793880\pi$
$38$	0	0
$39$	−118.794	−0.487750
$40$	0	0
$41$	26.0000	0.0990370	0.0495185	−	0.998773i	$-0.484231\pi$
$41$	26.0000	0.0990370	0.0495185	+	0.998773i	$0.484231\pi$
$42$	0	0
$43$	252.000i	0.893713i	0.894606	+	0.446856i	$0.147456\pi$
$43$	252.000i	0.893713i	−0.894606	+	0.446856i	$0.852544\pi$
$44$	0	0
$45$	25.4558i	0.0843274i
$46$	0	0
$47$	−345.068	−1.07092	−0.535461	−	0.844560i	$-0.679862\pi$
$47$	−345.068	−1.07092	−0.535461	+	0.844560i	$0.679862\pi$
$48$	0	0
$49$	−143.000	−0.416910
$50$	0	0
$51$	− 102.000i	− 0.280056i
$52$	0	0
$53$	681.651i	1.76664i	0.468770	+	0.883320i	$0.344697\pi$
$53$	681.651i	1.76664i	−0.468770	+	0.883320i	$0.655303\pi$
$54$	0	0
$55$	56.5685	0.138685
$56$	0	0
$57$	156.000	0.362504
$58$	0	0
$59$	364.000i	0.803199i	0.915815	+	0.401600i	$0.131546\pi$
$59$	364.000i	0.803199i	−0.915815	+	0.401600i	$0.868454\pi$
$60$	0	0
$61$	735.391i	1.54356i	0.635889	+	0.771780i	$0.280633\pi$
$61$	735.391i	1.54356i	−0.635889	+	0.771780i	$0.719367\pi$
$62$	0	0
$63$	−127.279	−0.254535
$64$	0	0
$65$	112.000	0.213721
$66$	0	0
$67$	− 628.000i	− 1.14511i	−0.819866	−	0.572555i	$-0.805952\pi$
$67$	− 628.000i	− 1.14511i	0.819866	−	0.572555i	$-0.194048\pi$
$68$	0	0
$69$	186.676i	0.325698i
$70$	0	0
$71$	−333.754	−0.557878	−0.278939	−	0.960309i	$-0.589983\pi$
$71$	−333.754	−0.557878	−0.278939	+	0.960309i	$0.589983\pi$
$72$	0	0
$73$	−338.000	−0.541917	−0.270958	−	0.962591i	$-0.587341\pi$
$73$	−338.000	−0.541917	−0.270958	+	0.962591i	$0.587341\pi$
$74$	0	0
$75$	351.000i	0.540400i
$76$	0	0
$77$	282.843i	0.418609i
$78$	0	0
$79$	789.131	1.12385	0.561925	−	0.827188i	$-0.310061\pi$
$79$	789.131	1.12385	0.561925	+	0.827188i	$0.310061\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1036.00i	− 1.37007i	−0.728510	−	0.685035i	$-0.759787\pi$
$83$	− 1036.00i	− 1.37007i	0.728510	−	0.685035i	$-0.240213\pi$
$84$	0	0
$85$	96.1665i	0.122714i
$86$	0	0
$87$	−602.455	−0.742413
$88$	0	0
$89$	−234.000	−0.278696	−0.139348	−	0.990243i	$-0.544501\pi$
$89$	−234.000	−0.278696	−0.139348	+	0.990243i	$0.544501\pi$
$90$	0	0
$91$	560.000i	0.645098i
$92$	0	0
$93$	330.926i	0.368983i
$94$	0	0
$95$	−147.078	−0.158841
$96$	0	0
$97$	−178.000	−0.186321	−0.0931606	−	0.995651i	$-0.529697\pi$
$97$	−178.000	−0.186321	−0.0931606	+	0.995651i	$0.529697\pi$
$98$	0	0
$99$	− 180.000i	− 0.182734i
$100$	0	0
$101$	− 257.387i	− 0.253574i	−0.991930	−	0.126787i	$-0.959534\pi$
$101$	− 257.387i	− 0.253574i	0.991930	−	0.126787i	$-0.0404664\pi$
$102$	0	0
$103$	1886.56	1.80474	0.902371	−	0.430961i	$-0.141825\pi$
$103$	1886.56	1.80474	0.902371	+	0.430961i	$0.141825\pi$
$104$	0	0
$105$	120.000	0.111531
$106$	0	0
$107$	1404.00i	1.26850i	0.773127	+	0.634251i	$0.218692\pi$
$107$	1404.00i	1.26850i	−0.773127	+	0.634251i	$0.781308\pi$
$108$	0	0
$109$	− 39.5980i	− 0.0347963i	−0.999849	−	0.0173982i	$-0.994462\pi$
$109$	− 39.5980i	− 0.0347963i	0.999849	−	0.0173982i	$-0.00553829\pi$
$110$	0	0
$111$	−814.587	−0.696551
$112$	0	0
$113$	1378.00	1.14718	0.573590	−	0.819143i	$-0.305550\pi$
$113$	1378.00	1.14718	0.573590	+	0.819143i	$0.305550\pi$
$114$	0	0
$115$	− 176.000i	− 0.142714i
$116$	0	0
$117$	− 356.382i	− 0.281603i
$118$	0	0
$119$	−480.833	−0.370402
$120$	0	0
$121$	931.000	0.699474
$122$	0	0
$123$	78.0000i	0.0571791i
$124$	0	0
$125$	− 684.479i	− 0.489774i
$126$	0	0
$127$	1790.39	1.25096	0.625480	−	0.780241i	$-0.284903\pi$
$127$	1790.39	1.25096	0.625480	+	0.780241i	$0.284903\pi$
$128$	0	0
$129$	−756.000	−0.515985
$130$	0	0
$131$	− 1572.00i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 1572.00i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	− 735.391i	− 0.479447i
$134$	0	0
$135$	−76.3675	−0.0486864
$136$	0	0
$137$	−2854.00	−1.77981	−0.889904	−	0.456148i	$-0.849229\pi$
$137$	−2854.00	−1.77981	−0.889904	+	0.456148i	$0.849229\pi$
$138$	0	0
$139$	1964.00i	1.19845i	0.800581	+	0.599224i	$0.204524\pi$
$139$	1964.00i	1.19845i	−0.800581	+	0.599224i	$0.795476\pi$
$140$	0	0
$141$	− 1035.20i	− 0.618297i
$142$	0	0
$143$	−791.960	−0.463126
$144$	0	0
$145$	568.000	0.325309
$146$	0	0
$147$	− 429.000i	− 0.240703i
$148$	0	0
$149$	− 1507.55i	− 0.828882i	−0.910076	−	0.414441i	$-0.863977\pi$
$149$	− 1507.55i	− 0.828882i	0.910076	−	0.414441i	$-0.136023\pi$
$150$	0	0
$151$	2265.57	1.22099	0.610495	−	0.792020i	$-0.290971\pi$
$151$	2265.57	1.22099	0.610495	+	0.792020i	$0.290971\pi$
$152$	0	0
$153$	306.000	0.161690
$154$	0	0
$155$	− 312.000i	− 0.161680i
$156$	0	0
$157$	− 3529.88i	− 1.79436i	−0.441663	−	0.897181i	$-0.645611\pi$
$157$	− 3529.88i	− 1.79436i	0.441663	−	0.897181i	$-0.354389\pi$
$158$	0	0
$159$	−2044.95	−1.01997
$160$	0	0
$161$	880.000	0.430768
$162$	0	0
$163$	− 2932.00i	− 1.40891i	−0.709750	−	0.704454i	$-0.751192\pi$
$163$	− 2932.00i	− 1.40891i	0.709750	−	0.704454i	$-0.248808\pi$
$164$	0	0
$165$	169.706i	0.0800701i
$166$	0	0
$167$	−3676.96	−1.70378	−0.851890	−	0.523720i	$-0.824544\pi$
$167$	−3676.96	−1.70378	−0.851890	+	0.523720i	$0.824544\pi$
$168$	0	0
$169$	629.000	0.286299
$170$	0	0
$171$	468.000i	0.209292i
$172$	0	0
$173$	− 1445.33i	− 0.635180i	−0.948228	−	0.317590i	$-0.897126\pi$
$173$	− 1445.33i	− 0.635180i	0.948228	−	0.317590i	$-0.102874\pi$
$174$	0	0
$175$	1654.63	0.714733
$176$	0	0
$177$	−1092.00	−0.463727
$178$	0	0
$179$	1308.00i	0.546170i	0.961990	+	0.273085i	$0.0880441\pi$
$179$	1308.00i	0.546170i	−0.961990	+	0.273085i	$0.911956\pi$
$180$	0	0
$181$	− 1996.87i	− 0.820034i	−0.912078	−	0.410017i	$-0.865523\pi$
$181$	− 1996.87i	− 0.820034i	0.912078	−	0.410017i	$-0.134477\pi$
$182$	0	0
$183$	−2206.17	−0.891175
$184$	0	0
$185$	768.000	0.305213
$186$	0	0
$187$	− 680.000i	− 0.265917i
$188$	0	0
$189$	− 381.838i	− 0.146956i
$190$	0	0
$191$	−939.038	−0.355740	−0.177870	−	0.984054i	$-0.556921\pi$
$191$	−939.038	−0.355740	−0.177870	+	0.984054i	$0.556921\pi$
$192$	0	0
$193$	−2490.00	−0.928674	−0.464337	−	0.885659i	$-0.653707\pi$
$193$	−2490.00	−0.928674	−0.464337	+	0.885659i	$0.653707\pi$
$194$	0	0
$195$	336.000i	0.123392i
$196$	0	0
$197$	− 2723.78i	− 0.985081i	−0.870290	−	0.492540i	$-0.836068\pi$
$197$	− 2723.78i	− 0.985081i	0.870290	−	0.492540i	$-0.163932\pi$
$198$	0	0
$199$	2158.09	0.768758	0.384379	−	0.923175i	$-0.374416\pi$
$199$	2158.09	0.768758	0.384379	+	0.923175i	$0.374416\pi$
$200$	0	0
$201$	1884.00	0.661130
$202$	0	0
$203$	2840.00i	0.981916i
$204$	0	0
$205$	− 73.5391i	− 0.0250546i
$206$	0	0
$207$	−560.029	−0.188042
$208$	0	0
$209$	1040.00	0.344202
$210$	0	0
$211$	924.000i	0.301473i	0.988574	+	0.150736i	$0.0481645\pi$
$211$	924.000i	0.301473i	−0.988574	+	0.150736i	$0.951836\pi$
$212$	0	0
$213$	− 1001.26i	− 0.322091i
$214$	0	0
$215$	712.764	0.226093
$216$	0	0
$217$	1560.00	0.488017
$218$	0	0
$219$	− 1014.00i	− 0.312876i
$220$	0	0
$221$	− 1346.33i	− 0.409792i
$222$	0	0
$223$	2276.88	0.683728	0.341864	−	0.939749i	$-0.388942\pi$
$223$	2276.88	0.683728	0.341864	+	0.939749i	$0.388942\pi$
$224$	0	0
$225$	−1053.00	−0.312000
$226$	0	0
$227$	− 156.000i	− 0.0456127i	−0.999740	−	0.0228064i	$-0.992740\pi$
$227$	− 156.000i	− 0.0456127i	0.999740	−	0.0228064i	$-0.00726012\pi$
$228$	0	0
$229$	639.225i	0.184459i	0.995738	+	0.0922296i	$0.0293994\pi$
$229$	639.225i	0.184459i	−0.995738	+	0.0922296i	$0.970601\pi$
$230$	0	0
$231$	−848.528	−0.241684
$232$	0	0
$233$	−2826.00	−0.794581	−0.397291	−	0.917693i	$-0.630049\pi$
$233$	−2826.00	−0.794581	−0.397291	+	0.917693i	$0.630049\pi$
$234$	0	0
$235$	976.000i	0.270924i
$236$	0	0
$237$	2367.39i	0.648855i
$238$	0	0
$239$	−2466.39	−0.667521	−0.333760	−	0.942658i	$-0.608318\pi$
$239$	−2466.39	−0.667521	−0.333760	+	0.942658i	$0.608318\pi$
$240$	0	0
$241$	−3354.00	−0.896474	−0.448237	−	0.893915i	$-0.647948\pi$
$241$	−3354.00	−0.896474	−0.448237	+	0.893915i	$0.647948\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	404.465i	0.105471i
$246$	0	0
$247$	2059.09	0.530433
$248$	0	0
$249$	3108.00	0.791010
$250$	0	0
$251$	− 6396.00i	− 1.60841i	−0.594349	−	0.804207i	$-0.702590\pi$
$251$	− 6396.00i	− 1.60841i	0.594349	−	0.804207i	$-0.297410\pi$
$252$	0	0
$253$	1244.51i	0.309255i
$254$	0	0
$255$	−288.500	−0.0708492
$256$	0	0
$257$	6882.00	1.67038	0.835189	−	0.549962i	$-0.185358\pi$
$257$	6882.00	1.67038	0.835189	+	0.549962i	$0.185358\pi$
$258$	0	0
$259$	3840.00i	0.921259i
$260$	0	0
$261$	− 1807.36i	− 0.428632i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	1928.00	0.446929
$266$	0	0
$267$	− 702.000i	− 0.160905i
$268$	0	0
$269$	1434.01i	0.325031i	0.986706	+	0.162515i	$0.0519607\pi$
$269$	1434.01i	0.325031i	−0.986706	+	0.162515i	$0.948039\pi$
$270$	0	0
$271$	5942.53	1.33204	0.666020	−	0.745934i	$-0.267997\pi$
$271$	5942.53	1.33204	0.666020	+	0.745934i	$0.267997\pi$
$272$	0	0
$273$	−1680.00	−0.372448
$274$	0	0
$275$	2340.00i	0.513117i
$276$	0	0
$277$	1103.09i	0.239271i	0.992818	+	0.119635i	$0.0381726\pi$
$277$	1103.09i	0.239271i	−0.992818	+	0.119635i	$0.961827\pi$
$278$	0	0
$279$	−992.778	−0.213032
$280$	0	0
$281$	−6266.00	−1.33024	−0.665121	−	0.746735i	$-0.731620\pi$
$281$	−6266.00	−1.33024	−0.665121	+	0.746735i	$0.731620\pi$
$282$	0	0
$283$	− 8596.00i	− 1.80558i	−0.430082	−	0.902790i	$-0.641515\pi$
$283$	− 8596.00i	− 1.80558i	0.430082	−	0.902790i	$-0.358485\pi$
$284$	0	0
$285$	− 441.235i	− 0.0917070i
$286$	0	0
$287$	367.696	0.0756250
$288$	0	0
$289$	−3757.00	−0.764706
$290$	0	0
$291$	− 534.000i	− 0.107573i
$292$	0	0
$293$	8397.60i	1.67438i	0.546913	+	0.837189i	$0.315803\pi$
$293$	8397.60i	1.67438i	−0.546913	+	0.837189i	$0.684197\pi$
$294$	0	0
$295$	1029.55	0.203195
$296$	0	0
$297$	540.000	0.105502
$298$	0	0
$299$	2464.00i	0.476578i
$300$	0	0
$301$	3563.82i	0.682442i
$302$	0	0
$303$	772.161	0.146401
$304$	0	0
$305$	2080.00	0.390493
$306$	0	0
$307$	4940.00i	0.918374i	0.888340	+	0.459187i	$0.151859\pi$
$307$	4940.00i	0.918374i	−0.888340	+	0.459187i	$0.848141\pi$
$308$	0	0
$309$	5659.68i	1.04197i
$310$	0	0
$311$	3382.80	0.616788	0.308394	−	0.951259i	$-0.400209\pi$
$311$	3382.80	0.616788	0.308394	+	0.951259i	$0.400209\pi$
$312$	0	0
$313$	3106.00	0.560899	0.280450	−	0.959869i	$-0.409516\pi$
$313$	3106.00	0.560899	0.280450	+	0.959869i	$0.409516\pi$
$314$	0	0
$315$	360.000i	0.0643927i
$316$	0	0
$317$	6728.83i	1.19220i	0.802909	+	0.596102i	$0.203285\pi$
$317$	6728.83i	1.19220i	−0.802909	+	0.596102i	$0.796715\pi$
$318$	0	0
$319$	−4016.37	−0.704932
$320$	0	0
$321$	−4212.00	−0.732370
$322$	0	0
$323$	1768.00i	0.304564i
$324$	0	0
$325$	4632.96i	0.790740i
$326$	0	0
$327$	118.794	0.0200897
$328$	0	0
$329$	−4880.00	−0.817760
$330$	0	0
$331$	2908.00i	0.482895i	0.970414	+	0.241447i	$0.0776221\pi$
$331$	2908.00i	0.482895i	−0.970414	+	0.241447i	$0.922378\pi$
$332$	0	0
$333$	− 2443.76i	− 0.402154i
$334$	0	0
$335$	−1776.25	−0.289693
$336$	0	0
$337$	4298.00	0.694739	0.347369	−	0.937728i	$-0.387075\pi$
$337$	4298.00	0.694739	0.347369	+	0.937728i	$0.387075\pi$
$338$	0	0
$339$	4134.00i	0.662325i
$340$	0	0
$341$	2206.17i	0.350355i
$342$	0	0
$343$	−6873.08	−1.08196
$344$	0	0
$345$	528.000	0.0823958
$346$	0	0
$347$	− 9996.00i	− 1.54644i	−0.634140	−	0.773218i	$-0.718646\pi$
$347$	− 9996.00i	− 1.54644i	0.634140	−	0.773218i	$-0.281354\pi$
$348$	0	0
$349$	3993.74i	0.612550i	0.951943	+	0.306275i	$0.0990827\pi$
$349$	3993.74i	0.612550i	−0.951943	+	0.306275i	$0.900917\pi$
$350$	0	0
$351$	1069.15	0.162583
$352$	0	0
$353$	6738.00	1.01594	0.507971	−	0.861374i	$-0.330396\pi$
$353$	6738.00	1.01594	0.507971	+	0.861374i	$0.330396\pi$
$354$	0	0
$355$	944.000i	0.141133i
$356$	0	0
$357$	− 1442.50i	− 0.213852i
$358$	0	0
$359$	2132.63	0.313527	0.156763	−	0.987636i	$-0.449894\pi$
$359$	2132.63	0.313527	0.156763	+	0.987636i	$0.449894\pi$
$360$	0	0
$361$	4155.00	0.605773
$362$	0	0
$363$	2793.00i	0.403842i
$364$	0	0
$365$	956.008i	0.137095i
$366$	0	0
$367$	−7628.27	−1.08499	−0.542496	−	0.840058i	$-0.682521\pi$
$367$	−7628.27	−1.08499	−0.542496	+	0.840058i	$0.682521\pi$
$368$	0	0
$369$	−234.000	−0.0330123
$370$	0	0
$371$	9640.00i	1.34901i
$372$	0	0
$373$	8383.46i	1.16375i	0.813278	+	0.581875i	$0.197681\pi$
$373$	8383.46i	1.16375i	−0.813278	+	0.581875i	$0.802319\pi$
$374$	0	0
$375$	2053.44	0.282771
$376$	0	0
$377$	−7952.00	−1.08634
$378$	0	0
$379$	− 12788.0i	− 1.73318i	−0.499020	−	0.866590i	$-0.666307\pi$
$379$	− 12788.0i	− 1.73318i	0.499020	−	0.866590i	$-0.333693\pi$
$380$	0	0
$381$	5371.18i	0.722242i
$382$	0	0
$383$	−2319.31	−0.309429	−0.154714	−	0.987959i	$-0.549446\pi$
$383$	−2319.31	−0.309429	−0.154714	+	0.987959i	$0.549446\pi$
$384$	0	0
$385$	800.000	0.105901
$386$	0	0
$387$	− 2268.00i	− 0.297904i
$388$	0	0
$389$	− 2684.18i	− 0.349854i	−0.984581	−	0.174927i	$-0.944031\pi$
$389$	− 2684.18i	− 0.349854i	0.984581	−	0.174927i	$-0.0559689\pi$
$390$	0	0
$391$	−2115.66	−0.273641
$392$	0	0
$393$	4716.00	0.605320
$394$	0	0
$395$	− 2232.00i	− 0.284314i
$396$	0	0
$397$	2206.17i	0.278903i	0.990229	+	0.139452i	$0.0445340\pi$
$397$	2206.17i	0.278903i	−0.990229	+	0.139452i	$0.955466\pi$
$398$	0	0
$399$	2206.17	0.276809
$400$	0	0
$401$	3582.00	0.446076	0.223038	−	0.974810i	$-0.428403\pi$
$401$	3582.00	0.446076	0.223038	+	0.974810i	$0.428403\pi$
$402$	0	0
$403$	4368.00i	0.539915i
$404$	0	0
$405$	− 229.103i	− 0.0281091i
$406$	0	0
$407$	−5430.58	−0.661385
$408$	0	0
$409$	−5126.00	−0.619717	−0.309859	−	0.950783i	$-0.600282\pi$
$409$	−5126.00	−0.619717	−0.309859	+	0.950783i	$0.600282\pi$
$410$	0	0
$411$	− 8562.00i	− 1.02757i
$412$	0	0
$413$	5147.74i	0.613326i
$414$	0	0
$415$	−2930.25	−0.346603
$416$	0	0
$417$	−5892.00	−0.691924
$418$	0	0
$419$	− 2924.00i	− 0.340923i	−0.985364	−	0.170462i	$-0.945474\pi$
$419$	− 2924.00i	− 0.340923i	0.985364	−	0.170462i	$-0.0545259\pi$
$420$	0	0
$421$	− 7314.31i	− 0.846741i	−0.905957	−	0.423370i	$-0.860847\pi$
$421$	− 7314.31i	− 0.846741i	0.905957	−	0.423370i	$-0.139153\pi$
$422$	0	0
$423$	3105.61	0.356974
$424$	0	0
$425$	−3978.00	−0.454027
$426$	0	0
$427$	10400.0i	1.17867i
$428$	0	0
$429$	− 2375.88i	− 0.267386i
$430$	0	0
$431$	15844.8	1.77081	0.885405	−	0.464819i	$-0.153881\pi$
$431$	15844.8	1.77081	0.885405	+	0.464819i	$0.153881\pi$
$432$	0	0
$433$	−6274.00	−0.696326	−0.348163	−	0.937434i	$-0.613194\pi$
$433$	−6274.00	−0.696326	−0.348163	+	0.937434i	$0.613194\pi$
$434$	0	0
$435$	1704.00i	0.187817i
$436$	0	0
$437$	− 3235.72i	− 0.354200i
$438$	0	0
$439$	−4596.19	−0.499691	−0.249846	−	0.968286i	$-0.580380\pi$
$439$	−4596.19	−0.499691	−0.249846	+	0.968286i	$0.580380\pi$
$440$	0	0
$441$	1287.00	0.138970
$442$	0	0
$443$	− 5084.00i	− 0.545255i	−0.962120	−	0.272628i	$-0.912107\pi$
$443$	− 5084.00i	− 0.545255i	0.962120	−	0.272628i	$-0.0878927\pi$
$444$	0	0
$445$	661.852i	0.0705051i
$446$	0	0
$447$	4522.65	0.478555
$448$	0	0
$449$	14190.0	1.49146	0.745732	−	0.666246i	$-0.232100\pi$
$449$	14190.0	1.49146	0.745732	+	0.666246i	$0.232100\pi$
$450$	0	0
$451$	520.000i	0.0542923i
$452$	0	0
$453$	6796.71i	0.704939i
$454$	0	0
$455$	1583.92	0.163198
$456$	0	0
$457$	6474.00	0.662672	0.331336	−	0.943513i	$-0.392501\pi$
$457$	6474.00	0.662672	0.331336	+	0.943513i	$0.392501\pi$
$458$	0	0
$459$	918.000i	0.0933520i
$460$	0	0
$461$	− 6321.53i	− 0.638662i	−0.947643	−	0.319331i	$-0.896542\pi$
$461$	− 6321.53i	− 0.638662i	0.947643	−	0.319331i	$-0.103458\pi$
$462$	0	0
$463$	−11435.3	−1.14783	−0.573915	−	0.818915i	$-0.694576\pi$
$463$	−11435.3	−1.14783	−0.573915	+	0.818915i	$0.694576\pi$
$464$	0	0
$465$	936.000	0.0933462
$466$	0	0
$467$	3796.00i	0.376141i	0.982156	+	0.188071i	$0.0602234\pi$
$467$	3796.00i	0.376141i	−0.982156	+	0.188071i	$0.939777\pi$
$468$	0	0
$469$	− 8881.26i	− 0.874411i
$470$	0	0
$471$	10589.6	1.03598
$472$	0	0
$473$	−5040.00	−0.489935
$474$	0	0
$475$	− 6084.00i	− 0.587691i
$476$	0	0
$477$	− 6134.86i	− 0.588880i
$478$	0	0
$479$	10493.5	1.00096	0.500479	−	0.865749i	$-0.333157\pi$
$479$	10493.5	1.00096	0.500479	+	0.865749i	$0.333157\pi$
$480$	0	0
$481$	−10752.0	−1.01923
$482$	0	0
$483$	2640.00i	0.248704i
$484$	0	0
$485$	503.460i	0.0471360i
$486$	0	0
$487$	−15406.4	−1.43354	−0.716769	−	0.697311i	$-0.754380\pi$
$487$	−15406.4	−1.43354	−0.716769	+	0.697311i	$0.754380\pi$
$488$	0	0
$489$	8796.00	0.813433
$490$	0	0
$491$	15452.0i	1.42024i	0.704079	+	0.710121i	$0.251360\pi$
$491$	15452.0i	1.42024i	−0.704079	+	0.710121i	$0.748640\pi$
$492$	0	0
$493$	− 6827.82i	− 0.623752i
$494$	0	0
$495$	−509.117	−0.0462285
$496$	0	0
$497$	−4720.00	−0.425998
$498$	0	0
$499$	− 52.0000i	− 0.00466501i	−0.999997	−	0.00233250i	$-0.999258\pi$
$499$	− 52.0000i	− 0.00466501i	0.999997	−	0.00233250i	$-0.000742460\pi$
$500$	0	0
$501$	− 11030.9i	− 0.983678i
$502$	0	0
$503$	12428.1	1.10167	0.550837	−	0.834613i	$-0.314309\pi$
$503$	12428.1	1.10167	0.550837	+	0.834613i	$0.314309\pi$
$504$	0	0
$505$	−728.000	−0.0641497
$506$	0	0
$507$	1887.00i	0.165295i
$508$	0	0
$509$	− 16362.5i	− 1.42486i	−0.701744	−	0.712429i	$-0.747595\pi$
$509$	− 16362.5i	− 1.42486i	0.701744	−	0.712429i	$-0.252405\pi$
$510$	0	0
$511$	−4780.04	−0.413809
$512$	0	0
$513$	−1404.00	−0.120835
$514$	0	0
$515$	− 5336.00i	− 0.456567i
$516$	0	0
$517$	− 6901.36i	− 0.587082i
$518$	0	0
$519$	4335.98	0.366721
$520$	0	0
$521$	714.000	0.0600401	0.0300201	−	0.999549i	$-0.490443\pi$
$521$	714.000	0.0600401	0.0300201	+	0.999549i	$0.490443\pi$
$522$	0	0
$523$	5980.00i	0.499975i	0.968249	+	0.249988i	$0.0804266\pi$
$523$	5980.00i	0.499975i	−0.968249	+	0.249988i	$0.919573\pi$
$524$	0	0
$525$	4963.89i	0.412651i
$526$	0	0
$527$	−3750.49	−0.310008
$528$	0	0
$529$	−8295.00	−0.681762
$530$	0	0
$531$	− 3276.00i	− 0.267733i
$532$	0	0
$533$	1029.55i	0.0836673i
$534$	0	0
$535$	3971.11	0.320909
$536$	0	0
$537$	−3924.00	−0.315332
$538$	0	0
$539$	− 2860.00i	− 0.228551i
$540$	0	0
$541$	− 13729.2i	− 1.09106i	−0.838091	−	0.545530i	$-0.816328\pi$
$541$	− 13729.2i	− 1.09106i	0.838091	−	0.545530i	$-0.183672\pi$
$542$	0	0
$543$	5990.61	0.473447
$544$	0	0
$545$	−112.000	−0.00880285
$546$	0	0
$547$	− 18500.0i	− 1.44607i	−0.690809	−	0.723037i	$-0.742745\pi$
$547$	− 18500.0i	− 1.44607i	0.690809	−	0.723037i	$-0.257255\pi$
$548$	0	0
$549$	− 6618.52i	− 0.514520i
$550$	0	0
$551$	10442.6	0.807382
$552$	0	0
$553$	11160.0	0.858176
$554$	0	0
$555$	2304.00i	0.176215i
$556$	0	0
$557$	− 8765.30i	− 0.666782i	−0.942789	−	0.333391i	$-0.891807\pi$
$557$	− 8765.30i	− 0.666782i	0.942789	−	0.333391i	$-0.108193\pi$
$558$	0	0
$559$	−9978.69	−0.755015
$560$	0	0
$561$	2040.00	0.153527
$562$	0	0
$563$	− 268.000i	− 0.0200619i	−0.999950	−	0.0100310i	$-0.996807\pi$
$563$	− 268.000i	− 0.0200619i	0.999950	−	0.0100310i	$-0.00319301\pi$
$564$	0	0
$565$	− 3897.57i	− 0.290216i
$566$	0	0
$567$	1145.51	0.0848448
$568$	0	0
$569$	13866.0	1.02160	0.510802	−	0.859698i	$-0.329348\pi$
$569$	13866.0	1.02160	0.510802	+	0.859698i	$0.329348\pi$
$570$	0	0
$571$	− 5140.00i	− 0.376712i	−0.982101	−	0.188356i	$-0.939684\pi$
$571$	− 5140.00i	− 0.376712i	0.982101	−	0.188356i	$-0.0603158\pi$
$572$	0	0
$573$	− 2817.11i	− 0.205387i
$574$	0	0
$575$	7280.37	0.528022
$576$	0	0
$577$	9386.00	0.677200	0.338600	−	0.940930i	$-0.390047\pi$
$577$	9386.00	0.677200	0.338600	+	0.940930i	$0.390047\pi$
$578$	0	0
$579$	− 7470.00i	− 0.536170i
$580$	0	0
$581$	− 14651.3i	− 1.04619i
$582$	0	0
$583$	−13633.0	−0.968477
$584$	0	0
$585$	−1008.00	−0.0712405
$586$	0	0
$587$	8844.00i	0.621859i	0.950433	+	0.310929i	$0.100640\pi$
$587$	8844.00i	0.621859i	−0.950433	+	0.310929i	$0.899360\pi$
$588$	0	0
$589$	− 5736.05i	− 0.401273i
$590$	0	0
$591$	8171.33	0.568737
$592$	0	0
$593$	−9406.00	−0.651363	−0.325681	−	0.945480i	$-0.605594\pi$
$593$	−9406.00	−0.651363	−0.325681	+	0.945480i	$0.605594\pi$
$594$	0	0
$595$	1360.00i	0.0937051i
$596$	0	0
$597$	6474.27i	0.443843i
$598$	0	0
$599$	−23459.0	−1.60018	−0.800090	−	0.599880i	$-0.795215\pi$
$599$	−23459.0	−1.60018	−0.800090	+	0.599880i	$0.795215\pi$
$600$	0	0
$601$	1262.00	0.0856540	0.0428270	−	0.999083i	$-0.486364\pi$
$601$	1262.00	0.0856540	0.0428270	+	0.999083i	$0.486364\pi$
$602$	0	0
$603$	5652.00i	0.381704i
$604$	0	0
$605$	− 2633.27i	− 0.176955i
$606$	0	0
$607$	−16288.9	−1.08920	−0.544602	−	0.838695i	$-0.683319\pi$
$607$	−16288.9	−1.08920	−0.544602	+	0.838695i	$0.683319\pi$
$608$	0	0
$609$	−8520.00	−0.566909
$610$	0	0
$611$	− 13664.0i	− 0.904724i
$612$	0	0
$613$	7138.95i	0.470374i	0.971950	+	0.235187i	$0.0755703\pi$
$613$	7138.95i	0.470374i	−0.971950	+	0.235187i	$0.924430\pi$
$614$	0	0
$615$	220.617	0.0144653
$616$	0	0
$617$	−16874.0	−1.10101	−0.550504	−	0.834833i	$-0.685564\pi$
$617$	−16874.0	−1.10101	−0.550504	+	0.834833i	$0.685564\pi$
$618$	0	0
$619$	20748.0i	1.34723i	0.739085	+	0.673613i	$0.235258\pi$
$619$	20748.0i	1.34723i	−0.739085	+	0.673613i	$0.764742\pi$
$620$	0	0
$621$	− 1680.09i	− 0.108566i
$622$	0	0
$623$	−3309.26	−0.212813
$624$	0	0
$625$	12689.0	0.812096
$626$	0	0
$627$	3120.00i	0.198725i
$628$	0	0
$629$	− 9231.99i	− 0.585220i
$630$	0	0
$631$	−14840.8	−0.936294	−0.468147	−	0.883651i	$-0.655078\pi$
$631$	−14840.8	−0.936294	−0.468147	+	0.883651i	$0.655078\pi$
$632$	0	0
$633$	−2772.00	−0.174055
$634$	0	0
$635$	− 5064.00i	− 0.316470i
$636$	0	0
$637$	− 5662.51i	− 0.352209i
$638$	0	0
$639$	3003.79	0.185959
$640$	0	0
$641$	17758.0	1.09423	0.547113	−	0.837059i	$-0.315727\pi$
$641$	17758.0	1.09423	0.547113	+	0.837059i	$0.315727\pi$
$642$	0	0
$643$	1148.00i	0.0704086i	0.999380	+	0.0352043i	$0.0112082\pi$
$643$	1148.00i	0.0704086i	−0.999380	+	0.0352043i	$0.988792\pi$
$644$	0	0
$645$	2138.29i	0.130535i
$646$	0	0
$647$	26988.9	1.63994	0.819970	−	0.572406i	$-0.193990\pi$
$647$	26988.9	1.63994	0.819970	+	0.572406i	$0.193990\pi$
$648$	0	0
$649$	−7280.00	−0.440316
$650$	0	0
$651$	4680.00i	0.281757i
$652$	0	0
$653$	21069.0i	1.26262i	0.775530	+	0.631311i	$0.217483\pi$
$653$	21069.0i	1.26262i	−0.775530	+	0.631311i	$0.782517\pi$
$654$	0	0
$655$	−4446.29	−0.265238
$656$	0	0
$657$	3042.00	0.180639
$658$	0	0
$659$	− 18356.0i	− 1.08505i	−0.840040	−	0.542525i	$-0.817468\pi$
$659$	− 18356.0i	− 1.08505i	0.840040	−	0.542525i	$-0.182532\pi$
$660$	0	0
$661$	− 15250.9i	− 0.897414i	−0.893679	−	0.448707i	$-0.851885\pi$
$661$	− 15250.9i	− 0.897414i	0.893679	−	0.448707i	$-0.148115\pi$
$662$	0	0
$663$	4038.99	0.236594
$664$	0	0
$665$	−2080.00	−0.121292
$666$	0	0
$667$	12496.0i	0.725408i
$668$	0	0
$669$	6830.65i	0.394751i
$670$	0	0
$671$	−14707.8	−0.846184
$672$	0	0
$673$	−12082.0	−0.692016	−0.346008	−	0.938232i	$-0.612463\pi$
$673$	−12082.0	−0.692016	−0.346008	+	0.938232i	$0.612463\pi$
$674$	0	0
$675$	− 3159.00i	− 0.180133i
$676$	0	0
$677$	− 12742.1i	− 0.723364i	−0.932302	−	0.361682i	$-0.882203\pi$
$677$	− 12742.1i	− 0.723364i	0.932302	−	0.361682i	$-0.117797\pi$
$678$	0	0
$679$	−2517.30	−0.142276
$680$	0	0
$681$	468.000	0.0263345
$682$	0	0
$683$	33508.0i	1.87723i	0.344967	+	0.938615i	$0.387890\pi$
$683$	33508.0i	1.87723i	−0.344967	+	0.938615i	$0.612110\pi$
$684$	0	0
$685$	8072.33i	0.450260i
$686$	0	0
$687$	−1917.67	−0.106498
$688$	0	0
$689$	−26992.0	−1.49247
$690$	0	0
$691$	364.000i	0.0200394i	0.999950	+	0.0100197i	$0.00318942\pi$
$691$	364.000i	0.0200394i	−0.999950	+	0.0100197i	$0.996811\pi$
$692$	0	0
$693$	− 2545.58i	− 0.139536i
$694$	0	0
$695$	5555.03	0.303186
$696$	0	0
$697$	−884.000	−0.0480400
$698$	0	0
$699$	− 8478.00i	− 0.458752i
$700$	0	0
$701$	− 3849.49i	− 0.207408i	−0.994608	−	0.103704i	$-0.966930\pi$
$701$	− 3849.49i	− 0.207408i	0.994608	−	0.103704i	$-0.0330695\pi$
$702$	0	0
$703$	14119.5	0.757507
$704$	0	0
$705$	−2928.00	−0.156418
$706$	0	0
$707$	− 3640.00i	− 0.193630i
$708$	0	0
$709$	23606.1i	1.25041i	0.780459	+	0.625207i	$0.214986\pi$
$709$	23606.1i	1.25041i	−0.780459	+	0.625207i	$0.785014\pi$
$710$	0	0
$711$	−7102.18	−0.374617
$712$	0	0
$713$	6864.00	0.360531
$714$	0	0
$715$	2240.00i	0.117163i
$716$	0	0
$717$	− 7399.17i	− 0.385393i
$718$	0	0
$719$	15799.6	0.819507	0.409753	−	0.912196i	$-0.365615\pi$
$719$	15799.6	0.819507	0.409753	+	0.912196i	$0.365615\pi$
$720$	0	0
$721$	26680.0	1.37811
$722$	0	0
$723$	− 10062.0i	− 0.517579i
$724$	0	0
$725$	23495.7i	1.20360i
$726$	0	0
$727$	4607.51	0.235052	0.117526	−	0.993070i	$-0.462504\pi$
$727$	4607.51	0.235052	0.117526	+	0.993070i	$0.462504\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 8568.00i	− 0.433514i
$732$	0	0
$733$	26219.5i	1.32120i	0.750738	+	0.660600i	$0.229698\pi$
$733$	26219.5i	1.32120i	−0.750738	+	0.660600i	$0.770302\pi$
$734$	0	0
$735$	−1213.40	−0.0608935
$736$	0	0
$737$	12560.0	0.627752
$738$	0	0
$739$	− 27924.0i	− 1.38999i	−0.719016	−	0.694994i	$-0.755407\pi$
$739$	− 27924.0i	− 1.38999i	0.719016	−	0.694994i	$-0.244593\pi$
$740$	0	0
$741$	6177.28i	0.306246i
$742$	0	0
$743$	−8937.83	−0.441315	−0.220658	−	0.975351i	$-0.570820\pi$
$743$	−8937.83	−0.441315	−0.220658	+	0.975351i	$0.570820\pi$
$744$	0	0
$745$	−4264.00	−0.209692
$746$	0	0
$747$	9324.00i	0.456690i
$748$	0	0
$749$	19855.6i	0.968633i
$750$	0	0
$751$	14082.7	0.684270	0.342135	−	0.939651i	$-0.388850\pi$
$751$	14082.7	0.684270	0.342135	+	0.939651i	$0.388850\pi$
$752$	0	0
$753$	19188.0	0.928618
$754$	0	0
$755$	− 6408.00i	− 0.308889i
$756$	0	0
$757$	14871.9i	0.714039i	0.934097	+	0.357019i	$0.116207\pi$
$757$	14871.9i	0.714039i	−0.934097	+	0.357019i	$0.883793\pi$
$758$	0	0
$759$	−3733.52	−0.178549
$760$	0	0
$761$	15834.0	0.754247	0.377124	−	0.926163i	$-0.376913\pi$
$761$	15834.0	0.754247	0.377124	+	0.926163i	$0.376913\pi$
$762$	0	0
$763$	− 560.000i	− 0.0265706i
$764$	0	0
$765$	− 865.499i	− 0.0409048i
$766$	0	0
$767$	−14413.7	−0.678549
$768$	0	0
$769$	−16666.0	−0.781523	−0.390762	−	0.920492i	$-0.627788\pi$
$769$	−16666.0	−0.781523	−0.390762	+	0.920492i	$0.627788\pi$
$770$	0	0
$771$	20646.0i	0.964394i
$772$	0	0
$773$	30957.1i	1.44043i	0.693752	+	0.720214i	$0.255956\pi$
$773$	30957.1i	1.44043i	−0.693752	+	0.720214i	$0.744044\pi$
$774$	0	0
$775$	12906.1	0.598195
$776$	0	0
$777$	−11520.0	−0.531889
$778$	0	0
$779$	− 1352.00i	− 0.0621828i
$780$	0	0
$781$	− 6675.09i	− 0.305830i
$782$	0	0
$783$	5422.09	0.247471
$784$	0	0
$785$	−9984.00	−0.453942
$786$	0	0
$787$	− 20228.0i	− 0.916201i	−0.888900	−	0.458101i	$-0.848530\pi$
$787$	− 20228.0i	− 0.916201i	0.888900	−	0.458101i	$-0.151470\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19487.9	0.875991
$792$	0	0
$793$	−29120.0	−1.30401
$794$	0	0
$795$	5784.00i	0.258034i
$796$	0	0
$797$	9008.54i	0.400375i	0.979758	+	0.200187i	$0.0641551\pi$
$797$	9008.54i	0.400375i	−0.979758	+	0.200187i	$0.935845\pi$
$798$	0	0
$799$	11732.3	0.519474
$800$	0	0
$801$	2106.00	0.0928987
$802$	0	0
$803$	− 6760.00i	− 0.297080i
$804$	0	0
$805$	− 2489.02i	− 0.108977i
$806$	0	0
$807$	−4302.04	−0.187657
$808$	0	0
$809$	9242.00	0.401646	0.200823	−	0.979628i	$-0.435638\pi$
$809$	9242.00	0.401646	0.200823	+	0.979628i	$0.435638\pi$
$810$	0	0
$811$	10972.0i	0.475067i	0.971379	+	0.237533i	$0.0763389\pi$
$811$	10972.0i	0.475067i	−0.971379	+	0.237533i	$0.923661\pi$
$812$	0	0
$813$	17827.6i	0.769053i
$814$	0	0
$815$	−8292.95	−0.356429
$816$	0	0
$817$	13104.0	0.561139
$818$	0	0
$819$	− 5040.00i	− 0.215033i
$820$	0	0
$821$	9336.64i	0.396895i	0.980112	+	0.198448i	$0.0635900\pi$
$821$	9336.64i	0.396895i	−0.980112	+	0.198448i	$0.936410\pi$
$822$	0	0
$823$	3566.65	0.151064	0.0755319	−	0.997143i	$-0.475935\pi$
$823$	3566.65	0.151064	0.0755319	+	0.997143i	$0.475935\pi$
$824$	0	0
$825$	−7020.00	−0.296249
$826$	0	0
$827$	18876.0i	0.793691i	0.917885	+	0.396846i	$0.129895\pi$
$827$	18876.0i	0.793691i	−0.917885	+	0.396846i	$0.870105\pi$
$828$	0	0
$829$	6974.90i	0.292218i	0.989269	+	0.146109i	$0.0466749\pi$
$829$	6974.90i	0.292218i	−0.989269	+	0.146109i	$0.953325\pi$
$830$	0	0
$831$	−3309.26	−0.138143
$832$	0	0
$833$	4862.00	0.202231
$834$	0	0
$835$	10400.0i	0.431026i
$836$	0	0
$837$	− 2978.33i	− 0.122994i
$838$	0	0
$839$	−30077.5	−1.23765	−0.618826	−	0.785528i	$-0.712392\pi$
$839$	−30077.5	−1.23765	−0.618826	+	0.785528i	$0.712392\pi$
$840$	0	0
$841$	−15939.0	−0.653532
$842$	0	0
$843$	− 18798.0i	− 0.768016i
$844$	0	0
$845$	− 1779.08i	− 0.0724287i
$846$	0	0
$847$	13166.3	0.534121
$848$	0	0
$849$	25788.0	1.04245
$850$	0	0
$851$	16896.0i	0.680596i
$852$	0	0
$853$	41159.3i	1.65213i	0.563575	+	0.826065i	$0.309426\pi$
$853$	41159.3i	1.65213i	−0.563575	+	0.826065i	$0.690574\pi$
$854$	0	0
$855$	1323.70	0.0529470
$856$	0	0
$857$	25194.0	1.00421	0.502107	−	0.864806i	$-0.332559\pi$
$857$	25194.0	1.00421	0.502107	+	0.864806i	$0.332559\pi$
$858$	0	0
$859$	9308.00i	0.369715i	0.982765	+	0.184857i	$0.0591823\pi$
$859$	9308.00i	0.369715i	−0.982765	+	0.184857i	$0.940818\pi$
$860$	0	0
$861$	1103.09i	0.0436621i
$862$	0	0
$863$	26802.2	1.05719	0.528596	−	0.848874i	$-0.322719\pi$
$863$	26802.2	1.05719	0.528596	+	0.848874i	$0.322719\pi$
$864$	0	0
$865$	−4088.00	−0.160689
$866$	0	0
$867$	− 11271.0i	− 0.441503i
$868$	0	0
$869$	15782.6i	0.616098i
$870$	0	0
$871$	24867.5	0.967399
$872$	0	0
$873$	1602.00	0.0621071
$874$	0	0
$875$	− 9680.00i	− 0.373993i
$876$	0	0
$877$	1436.84i	0.0553235i	0.999617	+	0.0276617i	$0.00880613\pi$
$877$	1436.84i	0.0553235i	−0.999617	+	0.0276617i	$0.991194\pi$
$878$	0	0
$879$	−25192.8	−0.966703
$880$	0	0
$881$	−42830.0	−1.63789	−0.818944	−	0.573873i	$-0.805440\pi$
$881$	−42830.0	−1.63789	−0.818944	+	0.573873i	$0.805440\pi$
$882$	0	0
$883$	23964.0i	0.913310i	0.889644	+	0.456655i	$0.150953\pi$
$883$	23964.0i	0.913310i	−0.889644	+	0.456655i	$0.849047\pi$
$884$	0	0
$885$	3088.64i	0.117315i
$886$	0	0
$887$	−28239.0	−1.06897	−0.534483	−	0.845179i	$-0.679494\pi$
$887$	−28239.0	−1.06897	−0.534483	+	0.845179i	$0.679494\pi$
$888$	0	0
$889$	25320.0	0.955237
$890$	0	0
$891$	1620.00i	0.0609114i
$892$	0	0
$893$	17943.5i	0.672405i
$894$	0	0
$895$	3699.58	0.138171
$896$	0	0
$897$	−7392.00	−0.275152
$898$	0	0
$899$	22152.0i	0.821814i
$900$	0	0
$901$	− 23176.1i	− 0.856947i
$902$	0	0
$903$	−10691.5	−0.394008
$904$	0	0
$905$	−5648.00	−0.207454
$906$	0	0
$907$	− 31972.0i	− 1.17047i	−0.810865	−	0.585233i	$-0.801003\pi$
$907$	− 31972.0i	− 1.17047i	0.810865	−	0.585233i	$-0.198997\pi$
$908$	0	0
$909$	2316.48i	0.0845246i
$910$	0	0
$911$	−26858.7	−0.976806	−0.488403	−	0.872618i	$-0.662420\pi$
$911$	−26858.7	−0.976806	−0.488403	+	0.872618i	$0.662420\pi$
$912$	0	0
$913$	20720.0	0.751075
$914$	0	0
$915$	6240.00i	0.225451i
$916$	0	0
$917$	− 22231.4i	− 0.800596i
$918$	0	0
$919$	40336.2	1.44784	0.723922	−	0.689882i	$-0.242338\pi$
$919$	40336.2	1.44784	0.723922	+	0.689882i	$0.242338\pi$
$920$	0	0
$921$	−14820.0	−0.530223
$922$	0	0
$923$	− 13216.0i	− 0.471300i
$924$	0	0
$925$	31768.9i	1.12925i
$926$	0	0
$927$	−16979.0	−0.601580
$928$	0	0
$929$	−13650.0	−0.482069	−0.241034	−	0.970517i	$-0.577487\pi$
$929$	−13650.0	−0.482069	−0.241034	+	0.970517i	$0.577487\pi$
$930$	0	0
$931$	7436.00i	0.261767i
$932$	0	0
$933$	10148.4i	0.356102i
$934$	0	0
$935$	−1923.33	−0.0672723
$936$	0	0
$937$	−7098.00	−0.247472	−0.123736	−	0.992315i	$-0.539488\pi$
$937$	−7098.00	−0.247472	−0.123736	+	0.992315i	$0.539488\pi$
$938$	0	0
$939$	9318.00i	0.323835i
$940$	0	0
$941$	− 41326.1i	− 1.43166i	−0.698274	−	0.715831i	$-0.746048\pi$
$941$	− 41326.1i	− 1.43166i	0.698274	−	0.715831i	$-0.253952\pi$
$942$	0	0
$943$	1617.86	0.0558693
$944$	0	0
$945$	−1080.00	−0.0371771
$946$	0	0
$947$	9900.00i	0.339711i	0.985469	+	0.169856i	$0.0543302\pi$
$947$	9900.00i	0.339711i	−0.985469	+	0.169856i	$0.945670\pi$
$948$	0	0
$949$	− 13384.1i	− 0.457815i
$950$	0	0
$951$	−20186.5	−0.688319
$952$	0	0
$953$	46938.0	1.59546	0.797729	−	0.603016i	$-0.206035\pi$
$953$	46938.0	1.59546	0.797729	+	0.603016i	$0.206035\pi$
$954$	0	0
$955$	2656.00i	0.0899960i
$956$	0	0
$957$	− 12049.1i	− 0.406993i
$958$	0	0
$959$	−40361.7	−1.35907
$960$	0	0
$961$	−17623.0	−0.591554
$962$	0	0
$963$	− 12636.0i	− 0.422834i
$964$	0	0
$965$	7042.78i	0.234938i
$966$	0	0
$967$	6989.04	0.232422	0.116211	−	0.993225i	$-0.462925\pi$
$967$	6989.04	0.232422	0.116211	+	0.993225i	$0.462925\pi$
$968$	0	0
$969$	−5304.00	−0.175840
$970$	0	0
$971$	− 53052.0i	− 1.75337i	−0.481067	−	0.876684i	$-0.659751\pi$
$971$	− 53052.0i	− 1.75337i	0.481067	−	0.876684i	$-0.340249\pi$
$972$	0	0
$973$	27775.2i	0.915139i
$974$	0	0
$975$	−13898.9	−0.456534
$976$	0	0
$977$	−41890.0	−1.37173	−0.685865	−	0.727729i	$-0.740576\pi$
$977$	−41890.0	−1.37173	−0.685865	+	0.727729i	$0.740576\pi$
$978$	0	0
$979$	− 4680.00i	− 0.152782i
$980$	0	0
$981$	356.382i	0.0115988i
$982$	0	0
$983$	−10861.2	−0.352408	−0.176204	−	0.984354i	$-0.556382\pi$
$983$	−10861.2	−0.352408	−0.176204	+	0.984354i	$0.556382\pi$
$984$	0	0
$985$	−7704.00	−0.249208
$986$	0	0
$987$	− 14640.0i	− 0.472134i
$988$	0	0
$989$	15680.8i	0.504166i
$990$	0	0
$991$	−330.926	−0.0106077	−0.00530384	−	0.999986i	$-0.501688\pi$
$991$	−330.926	−0.0106077	−0.00530384	+	0.999986i	$0.501688\pi$
$992$	0	0
$993$	−8724.00	−0.278799
$994$	0	0
$995$	− 6104.00i	− 0.194482i
$996$	0	0
$997$	− 39948.7i	− 1.26900i	−0.772925	−	0.634498i	$-0.781207\pi$
$997$	− 39948.7i	− 1.26900i	0.772925	−	0.634498i	$-0.218793\pi$
$998$	0	0
$999$	7331.28	0.232184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.d.d.193.3	yes	4
3.2	odd	2		1152.4.d.n.577.4		4
4.3	odd	2	inner	384.4.d.d.193.1	✓	4
8.3	odd	2	inner	384.4.d.d.193.4	yes	4
8.5	even	2	inner	384.4.d.d.193.2	yes	4
12.11	even	2		1152.4.d.n.577.3		4
16.3	odd	4		768.4.a.m.1.1		2
16.5	even	4		768.4.a.m.1.2		2
16.11	odd	4		768.4.a.h.1.2		2
16.13	even	4		768.4.a.h.1.1		2
24.5	odd	2		1152.4.d.n.577.2		4
24.11	even	2		1152.4.d.n.577.1		4
48.5	odd	4		2304.4.a.bh.1.1		2
48.11	even	4		2304.4.a.bb.1.1		2
48.29	odd	4		2304.4.a.bb.1.2		2
48.35	even	4		2304.4.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.d.193.1	✓	4	4.3	odd	2	inner
384.4.d.d.193.2	yes	4	8.5	even	2	inner
384.4.d.d.193.3	yes	4	1.1	even	1	trivial
384.4.d.d.193.4	yes	4	8.3	odd	2	inner
768.4.a.h.1.1		2	16.13	even	4
768.4.a.h.1.2		2	16.11	odd	4
768.4.a.m.1.1		2	16.3	odd	4
768.4.a.m.1.2		2	16.5	even	4
1152.4.d.n.577.1		4	24.11	even	2
1152.4.d.n.577.2		4	24.5	odd	2
1152.4.d.n.577.3		4	12.11	even	2
1152.4.d.n.577.4		4	3.2	odd	2
2304.4.a.bb.1.1		2	48.11	even	4
2304.4.a.bb.1.2		2	48.29	odd	4
2304.4.a.bh.1.1		2	48.5	odd	4
2304.4.a.bh.1.2		2	48.35	even	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -2.82843i q^{5} +14.1421 q^{7} -9.00000 q^{9} +20.0000i q^{11} +39.5980i q^{13} +8.48528 q^{15} -34.0000 q^{17} -52.0000i q^{19} +42.4264i q^{21} +62.2254 q^{23} +117.000 q^{25} -27.0000i q^{27} +200.818i q^{29} +110.309 q^{31} -60.0000 q^{33} -40.0000i q^{35} +271.529i q^{37} -118.794 q^{39} +26.0000 q^{41} +252.000i q^{43} +25.4558i q^{45} -345.068 q^{47} -143.000 q^{49} -102.000i q^{51} +681.651i q^{53} +56.5685 q^{55} +156.000 q^{57} +364.000i q^{59} +735.391i q^{61} -127.279 q^{63} +112.000 q^{65} -628.000i q^{67} +186.676i q^{69} -333.754 q^{71} -338.000 q^{73} +351.000i q^{75} +282.843i q^{77} +789.131 q^{79} +81.0000 q^{81} -1036.00i q^{83} +96.1665i q^{85} -602.455 q^{87} -234.000 q^{89} +560.000i q^{91} +330.926i q^{93} -147.078 q^{95} -178.000 q^{97} -180.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9} - 136 q^{17} + 468 q^{25} - 240 q^{33} + 104 q^{41} - 572 q^{49} + 624 q^{57} + 448 q^{65} - 1352 q^{73} + 324 q^{81} - 936 q^{89} - 712 q^{97}+O(q^{100}) \) 4 * q - 36 * q^9 - 136 * q^17 + 468 * q^25 - 240 * q^33 + 104 * q^41 - 572 * q^49 + 624 * q^57 + 448 * q^65 - 1352 * q^73 + 324 * q^81 - 936 * q^89 - 712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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