LMFDB - Embedded newform 384.5.b.c.319.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(319,384)]

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([1, 1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.b (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:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.7
Root			$-1.09445 + 0.895644i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.5.b.c.319.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} +23.7490i q^{5} +9.38251i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} +23.7490i q^{5} +9.38251i q^{7} +27.0000 q^{9} +112.001 q^{11} +56.5020i q^{13} +123.404i q^{15} -79.9921 q^{17} +211.297 q^{19} +48.7530i q^{21} -217.355i q^{23} +60.9843 q^{25} +140.296 q^{27} +616.753i q^{29} +1111.81i q^{31} +581.976 q^{33} -222.825 q^{35} +802.980i q^{37} +293.593i q^{39} -2411.93 q^{41} -2130.38 q^{43} +641.223i q^{45} +3596.58i q^{47} +2312.97 q^{49} -415.651 q^{51} -833.725i q^{53} +2659.92i q^{55} +1097.93 q^{57} -1309.43 q^{59} -4785.91i q^{61} +253.328i q^{63} -1341.87 q^{65} +4025.23 q^{67} -1129.41i q^{69} +9487.13i q^{71} -266.031 q^{73} +316.883 q^{75} +1050.85i q^{77} +5756.11i q^{79} +729.000 q^{81} +7287.16 q^{83} -1899.73i q^{85} +3204.74i q^{87} -1414.08 q^{89} -530.130 q^{91} +5777.13i q^{93} +5018.09i q^{95} -3110.08 q^{97} +3024.04 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 216 q^{9}+O(q^{10}) $ 8 * q + 216 * q^9	$ 8 q + 216 q^{9} + 1392 q^{17} - 3576 q^{25} - 1440 q^{33} - 1008 q^{41} + 10376 q^{49} - 9504 q^{57} + 23808 q^{65} - 10256 q^{73} + 5832 q^{81} - 31632 q^{89} - 45200 q^{97}+O(q^{100}) $ 8 * q + 216 * q^9 + 1392 * q^17 - 3576 * q^25 - 1440 * q^33 - 1008 * q^41 + 10376 * q^49 - 9504 * q^57 + 23808 * q^65 - 10256 * q^73 + 5832 * q^81 - 31632 * q^89 - 45200 * q^97

Display 100 coefficients 10 coefficients

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
$3$	5.19615	0.577350
$3$	5.19615	0.577350
$4$	0	0
$5$	23.7490i	0.949961i	0.879996	+	0.474980i	$0.157545\pi$
$5$	23.7490i	0.949961i	−0.879996	+	0.474980i	$0.842455\pi$
$6$	0	0
$7$	9.38251i	0.191480i	0.995406	+	0.0957399i	$0.0305217\pi$
$7$	9.38251i	0.191480i	−0.995406	+	0.0957399i	$0.969478\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	112.001	0.925631	0.462816	−	0.886455i	$-0.346839\pi$
$11$	112.001	0.925631	0.462816	+	0.886455i	$0.346839\pi$
$12$	0	0
$13$	56.5020i	0.334331i	0.985929	+	0.167166i	$0.0534615\pi$
$13$	56.5020i	0.334331i	−0.985929	+	0.167166i	$0.946539\pi$
$14$	0	0
$15$	123.404i	0.548460i
$16$	0	0
$17$	−79.9921	−0.276789	−0.138395	−	0.990377i	$-0.544194\pi$
$17$	−79.9921	−0.276789	−0.138395	+	0.990377i	$0.544194\pi$
$18$	0	0
$19$	211.297	0.585309	0.292655	−	0.956218i	$-0.405461\pi$
$19$	211.297	0.585309	0.292655	+	0.956218i	$0.405461\pi$
$20$	0	0
$21$	48.7530i	0.110551i
$22$	0	0
$23$	− 217.355i	− 0.410880i	−0.978670	−	0.205440i	$-0.934138\pi$
$23$	− 217.355i	− 0.410880i	0.978670	−	0.205440i	$-0.0658625\pi$
$24$	0	0
$25$	60.9843	0.0975748
$26$	0	0
$27$	140.296	0.192450
$28$	0	0
$29$	616.753i	0.733357i	0.930348	+	0.366678i	$0.119505\pi$
$29$	616.753i	0.733357i	−0.930348	+	0.366678i	$0.880495\pi$
$30$	0	0
$31$	1111.81i	1.15693i	0.815707	+	0.578465i	$0.196348\pi$
$31$	1111.81i	1.15693i	−0.815707	+	0.578465i	$0.803652\pi$
$32$	0	0
$33$	581.976	0.534414
$34$	0	0
$35$	−222.825	−0.181898
$36$	0	0
$37$	802.980i	0.586545i	0.956029	+	0.293273i	$0.0947443\pi$
$37$	802.980i	0.586545i	−0.956029	+	0.293273i	$0.905256\pi$
$38$	0	0
$39$	293.593i	0.193026i
$40$	0	0
$41$	−2411.93	−1.43482	−0.717409	−	0.696652i	$-0.754672\pi$
$41$	−2411.93	−1.43482	−0.717409	+	0.696652i	$0.754672\pi$
$42$	0	0
$43$	−2130.38	−1.15218	−0.576090	−	0.817386i	$-0.695422\pi$
$43$	−2130.38	−1.15218	−0.576090	+	0.817386i	$0.695422\pi$
$44$	0	0
$45$	641.223i	0.316654i
$46$	0	0
$47$	3596.58i	1.62815i	0.580761	+	0.814074i	$0.302755\pi$
$47$	3596.58i	1.62815i	−0.580761	+	0.814074i	$0.697245\pi$
$48$	0	0
$49$	2312.97	0.963335
$50$	0	0
$51$	−415.651	−0.159804
$52$	0	0
$53$	− 833.725i	− 0.296805i	−0.988927	−	0.148403i	$-0.952587\pi$
$53$	− 833.725i	− 0.296805i	0.988927	−	0.148403i	$-0.0474131\pi$
$54$	0	0
$55$	2659.92i	0.879313i
$56$	0	0
$57$	1097.93	0.337928
$58$	0	0
$59$	−1309.43	−0.376165	−0.188083	−	0.982153i	$-0.560227\pi$
$59$	−1309.43	−0.376165	−0.188083	+	0.982153i	$0.560227\pi$
$60$	0	0
$61$	− 4785.91i	− 1.28619i	−0.765786	−	0.643095i	$-0.777650\pi$
$61$	− 4785.91i	− 1.28619i	0.765786	−	0.643095i	$-0.222350\pi$
$62$	0	0
$63$	253.328i	0.0638266i
$64$	0	0
$65$	−1341.87	−0.317601
$66$	0	0
$67$	4025.23	0.896688	0.448344	−	0.893861i	$-0.352014\pi$
$67$	4025.23	0.896688	0.448344	+	0.893861i	$0.352014\pi$
$68$	0	0
$69$	− 1129.41i	− 0.237221i
$70$	0	0
$71$	9487.13i	1.88199i	0.338416	+	0.940997i	$0.390109\pi$
$71$	9487.13i	1.88199i	−0.338416	+	0.940997i	$0.609891\pi$
$72$	0	0
$73$	−266.031	−0.0499215	−0.0249607	−	0.999688i	$-0.507946\pi$
$73$	−266.031	−0.0499215	−0.0249607	+	0.999688i	$0.507946\pi$
$74$	0	0
$75$	316.883	0.0563348
$76$	0	0
$77$	1050.85i	0.177240i
$78$	0	0
$79$	5756.11i	0.922306i	0.887321	+	0.461153i	$0.152564\pi$
$79$	5756.11i	0.922306i	−0.887321	+	0.461153i	$0.847436\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	7287.16	1.05780	0.528898	−	0.848685i	$-0.322605\pi$
$83$	7287.16	1.05780	0.528898	+	0.848685i	$0.322605\pi$
$84$	0	0
$85$	− 1899.73i	− 0.262939i
$86$	0	0
$87$	3204.74i	0.423404i
$88$	0	0
$89$	−1414.08	−0.178523	−0.0892614	−	0.996008i	$-0.528451\pi$
$89$	−1414.08	−0.178523	−0.0892614	+	0.996008i	$0.528451\pi$
$90$	0	0
$91$	−530.130	−0.0640177
$92$	0	0
$93$	5777.13i	0.667953i
$94$	0	0
$95$	5018.09i	0.556021i
$96$	0	0
$97$	−3110.08	−0.330543	−0.165271	−	0.986248i	$-0.552850\pi$
$97$	−3110.08	−0.330543	−0.165271	+	0.986248i	$0.552850\pi$
$98$	0	0
$99$	3024.04	0.308544
$100$	0	0
$101$	− 488.501i	− 0.0478876i	−0.999713	−	0.0239438i	$-0.992378\pi$
$101$	− 488.501i	− 0.0478876i	0.999713	−	0.0239438i	$-0.00762227\pi$
$102$	0	0
$103$	20097.0i	1.89434i	0.320739	+	0.947168i	$0.396069\pi$
$103$	20097.0i	1.89434i	−0.320739	+	0.947168i	$0.603931\pi$
$104$	0	0
$105$	−1157.83	−0.105019
$106$	0	0
$107$	−12383.0	−1.08158	−0.540790	−	0.841158i	$-0.681874\pi$
$107$	−12383.0	−1.08158	−0.540790	+	0.841158i	$0.681874\pi$
$108$	0	0
$109$	− 15542.1i	− 1.30814i	−0.756433	−	0.654072i	$-0.773059\pi$
$109$	− 15542.1i	− 1.30814i	0.756433	−	0.654072i	$-0.226941\pi$
$110$	0	0
$111$	4172.41i	0.338642i
$112$	0	0
$113$	11653.6	0.912651	0.456325	−	0.889813i	$-0.349165\pi$
$113$	11653.6	0.912651	0.456325	+	0.889813i	$0.349165\pi$
$114$	0	0
$115$	5161.98	0.390319
$116$	0	0
$117$	1525.55i	0.111444i
$118$	0	0
$119$	− 750.527i	− 0.0529996i
$120$	0	0
$121$	−2096.69	−0.143206
$122$	0	0
$123$	−12532.8	−0.828393
$124$	0	0
$125$	16291.5i	1.04265i
$126$	0	0
$127$	4724.04i	0.292891i	0.989219	+	0.146445i	$0.0467833\pi$
$127$	4724.04i	0.292891i	−0.989219	+	0.146445i	$0.953217\pi$
$128$	0	0
$129$	−11069.8	−0.665212
$130$	0	0
$131$	9632.86	0.561323	0.280661	−	0.959807i	$-0.409446\pi$
$131$	9632.86	0.561323	0.280661	+	0.959807i	$0.409446\pi$
$132$	0	0
$133$	1982.49i	0.112075i
$134$	0	0
$135$	3331.89i	0.182820i
$136$	0	0
$137$	16407.6	0.874185	0.437093	−	0.899417i	$-0.356008\pi$
$137$	16407.6	0.874185	0.437093	+	0.899417i	$0.356008\pi$
$138$	0	0
$139$	−25544.1	−1.32209	−0.661046	−	0.750346i	$-0.729887\pi$
$139$	−25544.1	−1.32209	−0.661046	+	0.750346i	$0.729887\pi$
$140$	0	0
$141$	18688.4i	0.940012i
$142$	0	0
$143$	6328.30i	0.309467i
$144$	0	0
$145$	−14647.3	−0.696660
$146$	0	0
$147$	12018.5	0.556182
$148$	0	0
$149$	18716.6i	0.843054i	0.906816	+	0.421527i	$0.138506\pi$
$149$	18716.6i	0.843054i	−0.906816	+	0.421527i	$0.861494\pi$
$150$	0	0
$151$	− 28860.0i	− 1.26574i	−0.774260	−	0.632868i	$-0.781878\pi$
$151$	− 28860.0i	− 1.26574i	0.774260	−	0.632868i	$-0.218122\pi$
$152$	0	0
$153$	−2159.79	−0.0922631
$154$	0	0
$155$	−26404.4	−1.09904
$156$	0	0
$157$	− 23963.1i	− 0.972172i	−0.873911	−	0.486086i	$-0.838424\pi$
$157$	− 23963.1i	− 0.972172i	0.873911	−	0.486086i	$-0.161576\pi$
$158$	0	0
$159$	− 4332.16i	− 0.171360i
$160$	0	0
$161$	2039.34	0.0786752
$162$	0	0
$163$	24424.8	0.919297	0.459648	−	0.888101i	$-0.347975\pi$
$163$	24424.8	0.919297	0.459648	+	0.888101i	$0.347975\pi$
$164$	0	0
$165$	13821.4i	0.507672i
$166$	0	0
$167$	− 3088.40i	− 0.110739i	−0.998466	−	0.0553696i	$-0.982366\pi$
$167$	− 3088.40i	− 0.110739i	0.998466	−	0.0553696i	$-0.0176337\pi$
$168$	0	0
$169$	25368.5	0.888223
$170$	0	0
$171$	5705.01	0.195103
$172$	0	0
$173$	18621.0i	0.622171i	0.950382	+	0.311085i	$0.100693\pi$
$173$	18621.0i	0.622171i	−0.950382	+	0.311085i	$0.899307\pi$
$174$	0	0
$175$	572.185i	0.0186836i
$176$	0	0
$177$	−6804.00	−0.217179
$178$	0	0
$179$	50739.3	1.58357	0.791787	−	0.610797i	$-0.209151\pi$
$179$	50739.3	1.58357	0.791787	+	0.610797i	$0.209151\pi$
$180$	0	0
$181$	− 58050.5i	− 1.77194i	−0.463742	−	0.885970i	$-0.653494\pi$
$181$	− 58050.5i	− 1.77194i	0.463742	−	0.885970i	$-0.346506\pi$
$182$	0	0
$183$	− 24868.3i	− 0.742582i
$184$	0	0
$185$	−19070.0	−0.557195
$186$	0	0
$187$	−8959.23	−0.256205
$188$	0	0
$189$	1316.33i	0.0368503i
$190$	0	0
$191$	− 21285.6i	− 0.583471i	−0.956499	−	0.291736i	$-0.905767\pi$
$191$	− 21285.6i	− 0.583471i	0.956499	−	0.291736i	$-0.0942327\pi$
$192$	0	0
$193$	−26074.3	−0.700001	−0.350001	−	0.936749i	$-0.613819\pi$
$193$	−26074.3	−0.700001	−0.350001	+	0.936749i	$0.613819\pi$
$194$	0	0
$195$	−6972.54	−0.183367
$196$	0	0
$197$	4731.26i	0.121911i	0.998140	+	0.0609557i	$0.0194149\pi$
$197$	4731.26i	0.121911i	−0.998140	+	0.0609557i	$0.980585\pi$
$198$	0	0
$199$	− 38561.7i	− 0.973756i	−0.873470	−	0.486878i	$-0.838136\pi$
$199$	− 38561.7i	− 0.973756i	0.873470	−	0.486878i	$-0.161864\pi$
$200$	0	0
$201$	20915.7	0.517703
$202$	0	0
$203$	−5786.69	−0.140423
$204$	0	0
$205$	− 57280.9i	− 1.36302i
$206$	0	0
$207$	− 5868.59i	− 0.136960i
$208$	0	0
$209$	23665.5	0.541780
$210$	0	0
$211$	48656.4	1.09289	0.546444	−	0.837496i	$-0.315981\pi$
$211$	48656.4	1.09289	0.546444	+	0.837496i	$0.315981\pi$
$212$	0	0
$213$	49296.6i	1.08657i
$214$	0	0
$215$	− 50594.5i	− 1.09453i
$216$	0	0
$217$	−10431.6	−0.221529
$218$	0	0
$219$	−1382.34	−0.0288222
$220$	0	0
$221$	− 4519.71i	− 0.0925393i
$222$	0	0
$223$	− 73299.7i	− 1.47398i	−0.675902	−	0.736992i	$-0.736246\pi$
$223$	− 73299.7i	− 1.47398i	0.675902	−	0.736992i	$-0.263754\pi$
$224$	0	0
$225$	1646.57	0.0325249
$226$	0	0
$227$	−14996.0	−0.291020	−0.145510	−	0.989357i	$-0.546482\pi$
$227$	−14996.0	−0.291020	−0.145510	+	0.989357i	$0.546482\pi$
$228$	0	0
$229$	− 74694.9i	− 1.42436i	−0.701996	−	0.712181i	$-0.747708\pi$
$229$	− 74694.9i	− 1.42436i	0.701996	−	0.712181i	$-0.252292\pi$
$230$	0	0
$231$	5460.40i	0.102329i
$232$	0	0
$233$	−69476.6	−1.27975	−0.639877	−	0.768477i	$-0.721015\pi$
$233$	−69476.6	−1.27975	−0.639877	+	0.768477i	$0.721015\pi$
$234$	0	0
$235$	−85415.2	−1.54668
$236$	0	0
$237$	29909.6i	0.532494i
$238$	0	0
$239$	− 87478.2i	− 1.53145i	−0.643166	−	0.765727i	$-0.722380\pi$
$239$	− 87478.2i	− 1.53145i	0.643166	−	0.765727i	$-0.277620\pi$
$240$	0	0
$241$	74951.4	1.29046	0.645232	−	0.763987i	$-0.276761\pi$
$241$	74951.4	1.29046	0.645232	+	0.763987i	$0.276761\pi$
$242$	0	0
$243$	3788.00	0.0641500
$244$	0	0
$245$	54930.7i	0.915131i
$246$	0	0
$247$	11938.7i	0.195687i
$248$	0	0
$249$	37865.2	0.610719
$250$	0	0
$251$	−43677.8	−0.693287	−0.346643	−	0.937997i	$-0.612679\pi$
$251$	−43677.8	−0.693287	−0.346643	+	0.937997i	$0.612679\pi$
$252$	0	0
$253$	− 24344.1i	− 0.380323i
$254$	0	0
$255$	− 9871.31i	− 0.151808i
$256$	0	0
$257$	−1164.55	−0.0176316	−0.00881581	−	0.999961i	$-0.502806\pi$
$257$	−1164.55	−0.0176316	−0.00881581	+	0.999961i	$0.502806\pi$
$258$	0	0
$259$	−7533.97	−0.112312
$260$	0	0
$261$	16652.3i	0.244452i
$262$	0	0
$263$	− 35001.0i	− 0.506021i	−0.967464	−	0.253010i	$-0.918579\pi$
$263$	− 35001.0i	− 0.506021i	0.967464	−	0.253010i	$-0.0814207\pi$
$264$	0	0
$265$	19800.2	0.281953
$266$	0	0
$267$	−7347.77	−0.103070
$268$	0	0
$269$	41077.9i	0.567681i	0.958872	+	0.283840i	$0.0916085\pi$
$269$	41077.9i	0.567681i	−0.958872	+	0.283840i	$0.908391\pi$
$270$	0	0
$271$	− 46419.1i	− 0.632060i	−0.948749	−	0.316030i	$-0.897650\pi$
$271$	− 46419.1i	− 0.632060i	0.948749	−	0.316030i	$-0.102350\pi$
$272$	0	0
$273$	−2754.64	−0.0369606
$274$	0	0
$275$	6830.32	0.0903183
$276$	0	0
$277$	− 150297.i	− 1.95881i	−0.201907	−	0.979405i	$-0.564714\pi$
$277$	− 150297.i	− 1.95881i	0.201907	−	0.979405i	$-0.435286\pi$
$278$	0	0
$279$	30018.8i	0.385643i
$280$	0	0
$281$	−117793.	−1.49179	−0.745895	−	0.666063i	$-0.767978\pi$
$281$	−117793.	−1.49179	−0.745895	+	0.666063i	$0.767978\pi$
$282$	0	0
$283$	41243.6	0.514973	0.257486	−	0.966282i	$-0.417106\pi$
$283$	41243.6	0.514973	0.257486	+	0.966282i	$0.417106\pi$
$284$	0	0
$285$	26074.7i	0.321019i
$286$	0	0
$287$	− 22629.9i	− 0.274739i
$288$	0	0
$289$	−77122.3	−0.923388
$290$	0	0
$291$	−16160.4	−0.190839
$292$	0	0
$293$	71937.8i	0.837957i	0.907996	+	0.418979i	$0.137612\pi$
$293$	71937.8i	0.837957i	−0.907996	+	0.418979i	$0.862388\pi$
$294$	0	0
$295$	− 31097.7i	− 0.357342i
$296$	0	0
$297$	15713.4	0.178138
$298$	0	0
$299$	12281.0	0.137370
$300$	0	0
$301$	− 19988.3i	− 0.220619i
$302$	0	0
$303$	− 2538.33i	− 0.0276479i
$304$	0	0
$305$	113661.	1.22183
$306$	0	0
$307$	54572.7	0.579027	0.289514	−	0.957174i	$-0.406506\pi$
$307$	54572.7	0.579027	0.289514	+	0.957174i	$0.406506\pi$
$308$	0	0
$309$	104427.i	1.09370i
$310$	0	0
$311$	60945.8i	0.630119i	0.949072	+	0.315060i	$0.102025\pi$
$311$	60945.8i	0.630119i	−0.949072	+	0.315060i	$0.897975\pi$
$312$	0	0
$313$	102774.	1.04904	0.524521	−	0.851397i	$-0.324244\pi$
$313$	102774.	1.04904	0.524521	+	0.851397i	$0.324244\pi$
$314$	0	0
$315$	−6016.29	−0.0606328
$316$	0	0
$317$	39779.2i	0.395857i	0.980217	+	0.197928i	$0.0634213\pi$
$317$	39779.2i	0.395857i	−0.980217	+	0.197928i	$0.936579\pi$
$318$	0	0
$319$	69077.2i	0.678818i
$320$	0	0
$321$	−64344.0	−0.624450
$322$	0	0
$323$	−16902.1	−0.162007
$324$	0	0
$325$	3445.73i	0.0326223i
$326$	0	0
$327$	− 80758.9i	− 0.755257i
$328$	0	0
$329$	−33744.9	−0.311758
$330$	0	0
$331$	87804.6	0.801422	0.400711	−	0.916204i	$-0.368763\pi$
$331$	87804.6	0.801422	0.400711	+	0.916204i	$0.368763\pi$
$332$	0	0
$333$	21680.5i	0.195515i
$334$	0	0
$335$	95595.3i	0.851818i
$336$	0	0
$337$	−57972.0	−0.510456	−0.255228	−	0.966881i	$-0.582151\pi$
$337$	−57972.0	−0.510456	−0.255228	+	0.966881i	$0.582151\pi$
$338$	0	0
$339$	60554.1	0.526919
$340$	0	0
$341$	124524.i	1.07089i
$342$	0	0
$343$	44228.9i	0.375939i
$344$	0	0
$345$	26822.4	0.225351
$346$	0	0
$347$	159188.	1.32206	0.661031	−	0.750359i	$-0.270119\pi$
$347$	159188.	1.32206	0.661031	+	0.750359i	$0.270119\pi$
$348$	0	0
$349$	− 17396.8i	− 0.142830i	−0.997447	−	0.0714149i	$-0.977249\pi$
$349$	− 17396.8i	− 0.142830i	0.997447	−	0.0714149i	$-0.0227514\pi$
$350$	0	0
$351$	7927.01i	0.0643421i
$352$	0	0
$353$	150532.	1.20803	0.604017	−	0.796972i	$-0.293566\pi$
$353$	150532.	1.20803	0.604017	+	0.796972i	$0.293566\pi$
$354$	0	0
$355$	−225310.	−1.78782
$356$	0	0
$357$	− 3899.85i	− 0.0305993i
$358$	0	0
$359$	24818.5i	0.192569i	0.995354	+	0.0962846i	$0.0306959\pi$
$359$	24818.5i	0.192569i	−0.995354	+	0.0962846i	$0.969304\pi$
$360$	0	0
$361$	−85674.8	−0.657413
$362$	0	0
$363$	−10894.7	−0.0826803
$364$	0	0
$365$	− 6317.99i	− 0.0474234i
$366$	0	0
$367$	89484.6i	0.664379i	0.943213	+	0.332190i	$0.107787\pi$
$367$	89484.6i	0.664379i	−0.943213	+	0.332190i	$0.892213\pi$
$368$	0	0
$369$	−65122.1	−0.478273
$370$	0	0
$371$	7822.44	0.0568322
$372$	0	0
$373$	− 168536.i	− 1.21137i	−0.795705	−	0.605684i	$-0.792900\pi$
$373$	− 168536.i	− 1.21137i	0.795705	−	0.605684i	$-0.207100\pi$
$374$	0	0
$375$	84652.9i	0.601976i
$376$	0	0
$377$	−34847.8	−0.245184
$378$	0	0
$379$	126865.	0.883208	0.441604	−	0.897210i	$-0.354410\pi$
$379$	126865.	0.883208	0.441604	+	0.897210i	$0.354410\pi$
$380$	0	0
$381$	24546.8i	0.169101i
$382$	0	0
$383$	257277.i	1.75389i	0.480587	+	0.876947i	$0.340424\pi$
$383$	257277.i	1.75389i	−0.480587	+	0.876947i	$0.659576\pi$
$384$	0	0
$385$	−24956.8	−0.168371
$386$	0	0
$387$	−57520.3	−0.384060
$388$	0	0
$389$	− 150562.i	− 0.994983i	−0.867469	−	0.497492i	$-0.834255\pi$
$389$	− 150562.i	− 0.994983i	0.867469	−	0.497492i	$-0.165745\pi$
$390$	0	0
$391$	17386.7i	0.113727i
$392$	0	0
$393$	50053.8	0.324080
$394$	0	0
$395$	−136702.	−0.876155
$396$	0	0
$397$	− 185402.i	− 1.17634i	−0.808736	−	0.588171i	$-0.799848\pi$
$397$	− 185402.i	− 1.17634i	0.808736	−	0.588171i	$-0.200152\pi$
$398$	0	0
$399$	10301.3i	0.0647064i
$400$	0	0
$401$	167681.	1.04279	0.521393	−	0.853317i	$-0.325413\pi$
$401$	167681.	1.04279	0.521393	+	0.853317i	$0.325413\pi$
$402$	0	0
$403$	−62819.4	−0.386798
$404$	0	0
$405$	17313.0i	0.105551i
$406$	0	0
$407$	89934.9i	0.542925i
$408$	0	0
$409$	60544.6	0.361934	0.180967	−	0.983489i	$-0.442077\pi$
$409$	60544.6	0.361934	0.180967	+	0.983489i	$0.442077\pi$
$410$	0	0
$411$	85256.3	0.504711
$412$	0	0
$413$	− 12285.7i	− 0.0720280i
$414$	0	0
$415$	173063.i	1.00486i
$416$	0	0
$417$	−132731.	−0.763310
$418$	0	0
$419$	−307306.	−1.75042	−0.875211	−	0.483741i	$-0.839277\pi$
$419$	−307306.	−1.75042	−0.875211	+	0.483741i	$0.839277\pi$
$420$	0	0
$421$	172189.i	0.971496i	0.874099	+	0.485748i	$0.161453\pi$
$421$	172189.i	0.971496i	−0.874099	+	0.485748i	$0.838547\pi$
$422$	0	0
$423$	97107.7i	0.542716i
$424$	0	0
$425$	−4878.26	−0.0270077
$426$	0	0
$427$	44903.9	0.246279
$428$	0	0
$429$	32882.8i	0.178671i
$430$	0	0
$431$	28153.8i	0.151559i	0.997125	+	0.0757797i	$0.0241446\pi$
$431$	28153.8i	0.151559i	−0.997125	+	0.0757797i	$0.975855\pi$
$432$	0	0
$433$	−322037.	−1.71763	−0.858815	−	0.512285i	$-0.828799\pi$
$433$	−322037.	−1.71763	−0.858815	+	0.512285i	$0.828799\pi$
$434$	0	0
$435$	−76109.5	−0.402217
$436$	0	0
$437$	− 45926.4i	− 0.240492i
$438$	0	0
$439$	292438.i	1.51742i	0.651430	+	0.758709i	$0.274170\pi$
$439$	292438.i	1.51742i	−0.651430	+	0.758709i	$0.725830\pi$
$440$	0	0
$441$	62450.1	0.321112
$442$	0	0
$443$	227681.	1.16016	0.580082	−	0.814558i	$-0.303020\pi$
$443$	227681.	1.16016	0.580082	+	0.814558i	$0.303020\pi$
$444$	0	0
$445$	− 33583.0i	− 0.169590i
$446$	0	0
$447$	97254.5i	0.486737i
$448$	0	0
$449$	159097.	0.789169	0.394585	−	0.918860i	$-0.370889\pi$
$449$	159097.	0.789169	0.394585	+	0.918860i	$0.370889\pi$
$450$	0	0
$451$	−270139.	−1.32811
$452$	0	0
$453$	− 149961.i	− 0.730773i
$454$	0	0
$455$	− 12590.1i	− 0.0608143i
$456$	0	0
$457$	340590.	1.63079	0.815397	−	0.578902i	$-0.196519\pi$
$457$	340590.	1.63079	0.815397	+	0.578902i	$0.196519\pi$
$458$	0	0
$459$	−11222.6	−0.0532681
$460$	0	0
$461$	178561.i	0.840204i	0.907477	+	0.420102i	$0.138006\pi$
$461$	178561.i	0.840204i	−0.907477	+	0.420102i	$0.861994\pi$
$462$	0	0
$463$	− 4179.11i	− 0.0194950i	−0.999952	−	0.00974748i	$-0.996897\pi$
$463$	− 4179.11i	− 0.0194950i	0.999952	−	0.00974748i	$-0.00310277\pi$
$464$	0	0
$465$	−137201.	−0.634529
$466$	0	0
$467$	160583.	0.736318	0.368159	−	0.929763i	$-0.379988\pi$
$467$	160583.	0.736318	0.368159	+	0.929763i	$0.379988\pi$
$468$	0	0
$469$	37766.8i	0.171698i
$470$	0	0
$471$	− 124516.i	− 0.561284i
$472$	0	0
$473$	−238606.	−1.06649
$474$	0	0
$475$	12885.8	0.0571114
$476$	0	0
$477$	− 22510.6i	− 0.0989350i
$478$	0	0
$479$	− 153260.i	− 0.667970i	−0.942578	−	0.333985i	$-0.891607\pi$
$479$	− 153260.i	− 0.667970i	0.942578	−	0.333985i	$-0.108393\pi$
$480$	0	0
$481$	−45370.0	−0.196100
$482$	0	0
$483$	10596.7	0.0454231
$484$	0	0
$485$	− 73861.3i	− 0.314003i
$486$	0	0
$487$	113347.i	0.477917i	0.971030	+	0.238958i	$0.0768059\pi$
$487$	113347.i	0.477917i	−0.971030	+	0.238958i	$0.923194\pi$
$488$	0	0
$489$	126915.	0.530756
$490$	0	0
$491$	248336.	1.03009	0.515046	−	0.857162i	$-0.327775\pi$
$491$	248336.	1.03009	0.515046	+	0.857162i	$0.327775\pi$
$492$	0	0
$493$	− 49335.4i	− 0.202985i
$494$	0	0
$495$	71817.9i	0.293104i
$496$	0	0
$497$	−89013.1	−0.360364
$498$	0	0
$499$	378612.	1.52052	0.760262	−	0.649617i	$-0.225071\pi$
$499$	378612.	1.52052	0.760262	+	0.649617i	$0.225071\pi$
$500$	0	0
$501$	− 16047.8i	− 0.0639353i
$502$	0	0
$503$	255639.i	1.01040i	0.863004	+	0.505198i	$0.168580\pi$
$503$	255639.i	1.01040i	−0.863004	+	0.505198i	$0.831420\pi$
$504$	0	0
$505$	11601.4	0.0454913
$506$	0	0
$507$	131819.	0.512816
$508$	0	0
$509$	− 204106.i	− 0.787809i	−0.919151	−	0.393904i	$-0.871124\pi$
$509$	− 204106.i	− 0.787809i	0.919151	−	0.393904i	$-0.128876\pi$
$510$	0	0
$511$	− 2496.04i	− 0.00955895i
$512$	0	0
$513$	29644.1	0.112643
$514$	0	0
$515$	−477284.	−1.79954
$516$	0	0
$517$	402822.i	1.50707i
$518$	0	0
$519$	96757.3i	0.359211i
$520$	0	0
$521$	−295131.	−1.08727	−0.543637	−	0.839320i	$-0.682953\pi$
$521$	−295131.	−1.08727	−0.543637	+	0.839320i	$0.682953\pi$
$522$	0	0
$523$	212950.	0.778527	0.389263	−	0.921127i	$-0.372730\pi$
$523$	212950.	0.778527	0.389263	+	0.921127i	$0.372730\pi$
$524$	0	0
$525$	2973.16i	0.0107870i
$526$	0	0
$527$	− 88936.0i	− 0.320226i
$528$	0	0
$529$	232598.	0.831178
$530$	0	0
$531$	−35354.6	−0.125388
$532$	0	0
$533$	− 136279.i	− 0.479704i
$534$	0	0
$535$	− 294084.i	− 1.02746i
$536$	0	0
$537$	263649.	0.914277
$538$	0	0
$539$	259056.	0.891694
$540$	0	0
$541$	387756.i	1.32484i	0.749132	+	0.662421i	$0.230471\pi$
$541$	387756.i	1.32484i	−0.749132	+	0.662421i	$0.769529\pi$
$542$	0	0
$543$	− 301639.i	− 1.02303i
$544$	0	0
$545$	369108.	1.24268
$546$	0	0
$547$	−241227.	−0.806215	−0.403107	−	0.915153i	$-0.632070\pi$
$547$	−241227.	−0.806215	−0.403107	+	0.915153i	$0.632070\pi$
$548$	0	0
$549$	− 129220.i	− 0.428730i
$550$	0	0
$551$	130318.i	0.429240i
$552$	0	0
$553$	−54006.8	−0.176603
$554$	0	0
$555$	−99090.6	−0.321697
$556$	0	0
$557$	86454.9i	0.278663i	0.990246	+	0.139331i	$0.0444954\pi$
$557$	86454.9i	0.278663i	−0.990246	+	0.139331i	$0.955505\pi$
$558$	0	0
$559$	− 120371.i	− 0.385210i
$560$	0	0
$561$	−46553.5	−0.147920
$562$	0	0
$563$	99585.2	0.314179	0.157090	−	0.987584i	$-0.449789\pi$
$563$	99585.2	0.314179	0.157090	+	0.987584i	$0.449789\pi$
$564$	0	0
$565$	276762.i	0.866982i
$566$	0	0
$567$	6839.85i	0.0212755i
$568$	0	0
$569$	−524085.	−1.61874	−0.809371	−	0.587298i	$-0.800192\pi$
$569$	−524085.	−1.61874	−0.809371	+	0.587298i	$0.800192\pi$
$570$	0	0
$571$	32749.2	0.100445	0.0502225	−	0.998738i	$-0.484007\pi$
$571$	32749.2	0.100445	0.0502225	+	0.998738i	$0.484007\pi$
$572$	0	0
$573$	− 110603.i	− 0.336867i
$574$	0	0
$575$	− 13255.3i	− 0.0400915i
$576$	0	0
$577$	−140276.	−0.421338	−0.210669	−	0.977557i	$-0.567564\pi$
$577$	−140276.	−0.421338	−0.210669	+	0.977557i	$0.567564\pi$
$578$	0	0
$579$	−135486.	−0.404146
$580$	0	0
$581$	68371.8i	0.202547i
$582$	0	0
$583$	− 93378.4i	− 0.274732i
$584$	0	0
$585$	−36230.4	−0.105867
$586$	0	0
$587$	−303094.	−0.879634	−0.439817	−	0.898087i	$-0.644957\pi$
$587$	−303094.	−0.879634	−0.439817	+	0.898087i	$0.644957\pi$
$588$	0	0
$589$	234921.i	0.677161i
$590$	0	0
$591$	24584.4i	0.0703856i
$592$	0	0
$593$	117438.	0.333963	0.166981	−	0.985960i	$-0.446598\pi$
$593$	117438.	0.333963	0.166981	+	0.985960i	$0.446598\pi$
$594$	0	0
$595$	17824.3	0.0503475
$596$	0	0
$597$	− 200373.i	− 0.562198i
$598$	0	0
$599$	− 537187.i	− 1.49717i	−0.663037	−	0.748587i	$-0.730733\pi$
$599$	− 537187.i	− 1.49717i	0.663037	−	0.748587i	$-0.269267\pi$
$600$	0	0
$601$	382705.	1.05953	0.529767	−	0.848143i	$-0.322279\pi$
$601$	382705.	1.05953	0.529767	+	0.848143i	$0.322279\pi$
$602$	0	0
$603$	108681.	0.298896
$604$	0	0
$605$	− 49794.2i	− 0.136040i
$606$	0	0
$607$	− 578934.i	− 1.57127i	−0.618688	−	0.785637i	$-0.712336\pi$
$607$	− 578934.i	− 1.57127i	0.618688	−	0.785637i	$-0.287664\pi$
$608$	0	0
$609$	−30068.5	−0.0810732
$610$	0	0
$611$	−203214.	−0.544341
$612$	0	0
$613$	287542.i	0.765209i	0.923912	+	0.382605i	$0.124973\pi$
$613$	287542.i	0.765209i	−0.923912	+	0.382605i	$0.875027\pi$
$614$	0	0
$615$	− 297641.i	− 0.786940i
$616$	0	0
$617$	−400806.	−1.05284	−0.526422	−	0.850223i	$-0.676467\pi$
$617$	−400806.	−1.05284	−0.526422	+	0.850223i	$0.676467\pi$
$618$	0	0
$619$	−233469.	−0.609323	−0.304661	−	0.952461i	$-0.598543\pi$
$619$	−233469.	−0.609323	−0.304661	+	0.952461i	$0.598543\pi$
$620$	0	0
$621$	− 30494.1i	− 0.0790738i
$622$	0	0
$623$	− 13267.6i	− 0.0341835i
$624$	0	0
$625$	−348791.	−0.892904
$626$	0	0
$627$	122970.	0.312797
$628$	0	0
$629$	− 64232.1i	− 0.162349i
$630$	0	0
$631$	− 499538.i	− 1.25461i	−0.778773	−	0.627306i	$-0.784157\pi$
$631$	− 499538.i	− 1.25461i	0.778773	−	0.627306i	$-0.215843\pi$
$632$	0	0
$633$	252826.	0.630979
$634$	0	0
$635$	−112191.	−0.278235
$636$	0	0
$637$	130687.i	0.322073i
$638$	0	0
$639$	256152.i	0.627331i
$640$	0	0
$641$	405774.	0.987571	0.493786	−	0.869584i	$-0.335613\pi$
$641$	405774.	0.987571	0.493786	+	0.869584i	$0.335613\pi$
$642$	0	0
$643$	−550571.	−1.33165	−0.665827	−	0.746106i	$-0.731921\pi$
$643$	−550571.	−1.33165	−0.665827	+	0.746106i	$0.731921\pi$
$644$	0	0
$645$	− 262897.i	− 0.631925i
$646$	0	0
$647$	− 428220.i	− 1.02296i	−0.859295	−	0.511480i	$-0.829097\pi$
$647$	− 428220.i	− 1.02296i	0.859295	−	0.511480i	$-0.170903\pi$
$648$	0	0
$649$	−146658.	−0.348190
$650$	0	0
$651$	−54204.0	−0.127900
$652$	0	0
$653$	− 802147.i	− 1.88117i	−0.339561	−	0.940584i	$-0.610279\pi$
$653$	− 802147.i	− 1.88117i	0.339561	−	0.940584i	$-0.389721\pi$
$654$	0	0
$655$	228771.i	0.533234i
$656$	0	0
$657$	−7182.85	−0.0166405
$658$	0	0
$659$	662958.	1.52656	0.763282	−	0.646065i	$-0.223586\pi$
$659$	662958.	1.52656	0.763282	+	0.646065i	$0.223586\pi$
$660$	0	0
$661$	231396.i	0.529607i	0.964302	+	0.264804i	$0.0853071\pi$
$661$	231396.i	0.529607i	−0.964302	+	0.264804i	$0.914693\pi$
$662$	0	0
$663$	− 23485.1i	− 0.0534276i
$664$	0	0
$665$	−47082.2	−0.106467
$666$	0	0
$667$	134055.	0.301321
$668$	0	0
$669$	− 380877.i	− 0.851005i
$670$	0	0
$671$	− 536029.i	− 1.19054i
$672$	0	0
$673$	311867.	0.688557	0.344278	−	0.938868i	$-0.388124\pi$
$673$	311867.	0.688557	0.344278	+	0.938868i	$0.388124\pi$
$674$	0	0
$675$	8555.85	0.0187783
$676$	0	0
$677$	− 655173.i	− 1.42948i	−0.699389	−	0.714741i	$-0.746545\pi$
$677$	− 655173.i	− 1.42948i	0.699389	−	0.714741i	$-0.253455\pi$
$678$	0	0
$679$	− 29180.3i	− 0.0632923i
$680$	0	0
$681$	−77921.4	−0.168021
$682$	0	0
$683$	−358002.	−0.767439	−0.383719	−	0.923450i	$-0.625357\pi$
$683$	−358002.	−0.767439	−0.383719	+	0.923450i	$0.625357\pi$
$684$	0	0
$685$	389664.i	0.830442i
$686$	0	0
$687$	− 388126.i	− 0.822355i
$688$	0	0
$689$	47107.1	0.0992312
$690$	0	0
$691$	826006.	1.72992	0.864962	−	0.501838i	$-0.167343\pi$
$691$	826006.	1.72992	0.864962	+	0.501838i	$0.167343\pi$
$692$	0	0
$693$	28373.1i	0.0590799i
$694$	0	0
$695$	− 606648.i	− 1.25593i
$696$	0	0
$697$	192935.	0.397142
$698$	0	0
$699$	−361011.	−0.738866
$700$	0	0
$701$	229057.i	0.466130i	0.972461	+	0.233065i	$0.0748756\pi$
$701$	229057.i	0.466130i	−0.972461	+	0.233065i	$0.925124\pi$
$702$	0	0
$703$	169667.i	0.343310i
$704$	0	0
$705$	−443831.	−0.892974
$706$	0	0
$707$	4583.37	0.00916950
$708$	0	0
$709$	6312.88i	0.0125584i	0.999980	+	0.00627921i	$0.00199875\pi$
$709$	6312.88i	0.0125584i	−0.999980	+	0.00627921i	$0.998001\pi$
$710$	0	0
$711$	155415.i	0.307435i
$712$	0	0
$713$	241658.	0.475359
$714$	0	0
$715$	−150291.	−0.293982
$716$	0	0
$717$	− 454550.i	− 0.884185i
$718$	0	0
$719$	− 179483.i	− 0.347189i	−0.984817	−	0.173594i	$-0.944462\pi$
$719$	− 179483.i	− 0.347189i	0.984817	−	0.173594i	$-0.0555382\pi$
$720$	0	0
$721$	−188560.	−0.362727
$722$	0	0
$723$	389459.	0.745049
$724$	0	0
$725$	37612.2i	0.0715571i
$726$	0	0
$727$	− 310931.i	− 0.588295i	−0.955760	−	0.294147i	$-0.904964\pi$
$727$	− 310931.i	− 0.588295i	0.955760	−	0.294147i	$-0.0950357\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	170414.	0.318911
$732$	0	0
$733$	602271.i	1.12094i	0.828173	+	0.560472i	$0.189380\pi$
$733$	602271.i	1.12094i	−0.828173	+	0.560472i	$0.810620\pi$
$734$	0	0
$735$	285428.i	0.528351i
$736$	0	0
$737$	450832.	0.830002
$738$	0	0
$739$	−64180.3	−0.117520	−0.0587602	−	0.998272i	$-0.518715\pi$
$739$	−64180.3	−0.117520	−0.0587602	+	0.998272i	$0.518715\pi$
$740$	0	0
$741$	62035.2i	0.112980i
$742$	0	0
$743$	525793.i	0.952440i	0.879326	+	0.476220i	$0.157993\pi$
$743$	525793.i	0.952440i	−0.879326	+	0.476220i	$0.842007\pi$
$744$	0	0
$745$	−444502.	−0.800868
$746$	0	0
$747$	196753.	0.352599
$748$	0	0
$749$	− 116184.i	− 0.207101i
$750$	0	0
$751$	− 405381.i	− 0.718759i	−0.933191	−	0.359380i	$-0.882988\pi$
$751$	− 405381.i	− 0.718759i	0.933191	−	0.359380i	$-0.117012\pi$
$752$	0	0
$753$	−226956.	−0.400269
$754$	0	0
$755$	685398.	1.20240
$756$	0	0
$757$	5071.58i	0.00885016i	0.999990	+	0.00442508i	$0.00140855\pi$
$757$	5071.58i	0.00885016i	−0.999990	+	0.00442508i	$0.998591\pi$
$758$	0	0
$759$	− 126496.i	− 0.219580i
$760$	0	0
$761$	−29104.4	−0.0502561	−0.0251280	−	0.999684i	$-0.507999\pi$
$761$	−29104.4	−0.0502561	−0.0251280	+	0.999684i	$0.507999\pi$
$762$	0	0
$763$	145823.	0.250483
$764$	0	0
$765$	− 51292.8i	− 0.0876463i
$766$	0	0
$767$	− 73985.4i	− 0.125764i
$768$	0	0
$769$	−235095.	−0.397548	−0.198774	−	0.980045i	$-0.563696\pi$
$769$	−235095.	−0.397548	−0.198774	+	0.980045i	$0.563696\pi$
$770$	0	0
$771$	−6051.19	−0.0101796
$772$	0	0
$773$	89776.8i	0.150247i	0.997174	+	0.0751234i	$0.0239351\pi$
$773$	89776.8i	0.150247i	−0.997174	+	0.0751234i	$0.976065\pi$
$774$	0	0
$775$	67802.8i	0.112887i
$776$	0	0
$777$	−39147.7	−0.0648431
$778$	0	0
$779$	−509632.	−0.839812
$780$	0	0
$781$	1.06257e6i	1.74203i
$782$	0	0
$783$	86528.0i	0.141135i
$784$	0	0
$785$	569099.	0.923525
$786$	0	0
$787$	406715.	0.656660	0.328330	−	0.944563i	$-0.393514\pi$
$787$	406715.	0.656660	0.328330	+	0.944563i	$0.393514\pi$
$788$	0	0
$789$	− 181870.i	− 0.292151i
$790$	0	0
$791$	109340.i	0.174754i
$792$	0	0
$793$	270414.	0.430013
$794$	0	0
$795$	102885.	0.162786
$796$	0	0
$797$	− 1.02634e6i	− 1.61576i	−0.589350	−	0.807878i	$-0.700616\pi$
$797$	− 1.02634e6i	− 1.61576i	0.589350	−	0.807878i	$-0.299384\pi$
$798$	0	0
$799$	− 287698.i	− 0.450654i
$800$	0	0
$801$	−38180.1	−0.0595076
$802$	0	0
$803$	−29795.9	−0.0462089
$804$	0	0
$805$	48432.3i	0.0747383i
$806$	0	0
$807$	213447.i	0.327751i
$808$	0	0
$809$	245306.	0.374810	0.187405	−	0.982283i	$-0.439992\pi$
$809$	245306.	0.374810	0.187405	+	0.982283i	$0.439992\pi$
$810$	0	0
$811$	−1.26760e6	−1.92726	−0.963628	−	0.267246i	$-0.913886\pi$
$811$	−1.26760e6	−1.92726	−0.963628	+	0.267246i	$0.913886\pi$
$812$	0	0
$813$	− 241201.i	− 0.364920i
$814$	0	0
$815$	580065.i	0.873296i
$816$	0	0
$817$	−450142.	−0.674382
$818$	0	0
$819$	−14313.5	−0.0213392
$820$	0	0
$821$	− 163484.i	− 0.242542i	−0.992619	−	0.121271i	$-0.961303\pi$
$821$	− 163484.i	− 0.242542i	0.992619	−	0.121271i	$-0.0386971\pi$
$822$	0	0
$823$	913000.i	1.34794i	0.738758	+	0.673971i	$0.235413\pi$
$823$	913000.i	1.34794i	−0.738758	+	0.673971i	$0.764587\pi$
$824$	0	0
$825$	35491.4	0.0521453
$826$	0	0
$827$	−378235.	−0.553032	−0.276516	−	0.961009i	$-0.589180\pi$
$827$	−378235.	−0.553032	−0.276516	+	0.961009i	$0.589180\pi$
$828$	0	0
$829$	− 70195.3i	− 0.102141i	−0.998695	−	0.0510704i	$-0.983737\pi$
$829$	− 70195.3i	− 0.102141i	0.998695	−	0.0510704i	$-0.0162633\pi$
$830$	0	0
$831$	− 780969.i	− 1.13092i
$832$	0	0
$833$	−185019.	−0.266641
$834$	0	0
$835$	73346.5	0.105198
$836$	0	0
$837$	155982.i	0.222651i
$838$	0	0
$839$	410513.i	0.583180i	0.956543	+	0.291590i	$0.0941843\pi$
$839$	410513.i	0.583180i	−0.956543	+	0.291590i	$0.905816\pi$
$840$	0	0
$841$	326897.	0.462188
$842$	0	0
$843$	−612072.	−0.861286
$844$	0	0
$845$	602478.i	0.843777i
$846$	0	0
$847$	− 19672.2i	− 0.0274211i
$848$	0	0
$849$	214308.	0.297320
$850$	0	0
$851$	174532.	0.240999
$852$	0	0
$853$	577725.i	0.794005i	0.917817	+	0.397003i	$0.129950\pi$
$853$	577725.i	0.794005i	−0.917817	+	0.397003i	$0.870050\pi$
$854$	0	0
$855$	135488.i	0.185340i
$856$	0	0
$857$	−388294.	−0.528687	−0.264344	−	0.964429i	$-0.585155\pi$
$857$	−388294.	−0.528687	−0.264344	+	0.964429i	$0.585155\pi$
$858$	0	0
$859$	−124443.	−0.168650	−0.0843248	−	0.996438i	$-0.526873\pi$
$859$	−124443.	−0.168650	−0.0843248	+	0.996438i	$0.526873\pi$
$860$	0	0
$861$	− 117589.i	− 0.158620i
$862$	0	0
$863$	652166.i	0.875662i	0.899057	+	0.437831i	$0.144253\pi$
$863$	652166.i	0.875662i	−0.899057	+	0.437831i	$0.855747\pi$
$864$	0	0
$865$	−442229.	−0.591038
$866$	0	0
$867$	−400739.	−0.533118
$868$	0	0
$869$	644693.i	0.853716i
$870$	0	0
$871$	227434.i	0.299791i
$872$	0	0
$873$	−83972.1	−0.110181
$874$	0	0
$875$	−152855.	−0.199647
$876$	0	0
$877$	75235.9i	0.0978196i	0.998803	+	0.0489098i	$0.0155747\pi$
$877$	75235.9i	0.0978196i	−0.998803	+	0.0489098i	$0.984425\pi$
$878$	0	0
$879$	373800.i	0.483795i
$880$	0	0
$881$	−916209.	−1.18044	−0.590218	−	0.807244i	$-0.700958\pi$
$881$	−916209.	−1.18044	−0.590218	+	0.807244i	$0.700958\pi$
$882$	0	0
$883$	1.14981e6	1.47471	0.737354	−	0.675506i	$-0.236075\pi$
$883$	1.14981e6	1.47471	0.737354	+	0.675506i	$0.236075\pi$
$884$	0	0
$885$	− 161588.i	− 0.206311i
$886$	0	0
$887$	1.01630e6i	1.29174i	0.763448	+	0.645870i	$0.223505\pi$
$887$	1.01630e6i	1.29174i	−0.763448	+	0.645870i	$0.776495\pi$
$888$	0	0
$889$	−44323.3	−0.0560827
$890$	0	0
$891$	81649.0	0.102848
$892$	0	0
$893$	759945.i	0.952970i
$894$	0	0
$895$	1.20501e6i	1.50433i
$896$	0	0
$897$	63814.0	0.0793105
$898$	0	0
$899$	−685711.	−0.848442
$900$	0	0
$901$	66691.5i	0.0821525i
$902$	0	0
$903$	− 103862.i	− 0.127375i
$904$	0	0
$905$	1.37864e6	1.68327
$906$	0	0
$907$	−1.00415e6	−1.22063	−0.610315	−	0.792159i	$-0.708957\pi$
$907$	−1.00415e6	−1.22063	−0.610315	+	0.792159i	$0.708957\pi$
$908$	0	0
$909$	− 13189.5i	− 0.0159625i
$910$	0	0
$911$	231087.i	0.278444i	0.990261	+	0.139222i	$0.0444602\pi$
$911$	231087.i	0.278444i	−0.990261	+	0.139222i	$0.955540\pi$
$912$	0	0
$913$	816172.	0.979129
$914$	0	0
$915$	590598.	0.705424
$916$	0	0
$917$	90380.4i	0.107482i
$918$	0	0
$919$	545658.i	0.646084i	0.946385	+	0.323042i	$0.104706\pi$
$919$	545658.i	0.646084i	−0.946385	+	0.323042i	$0.895294\pi$
$920$	0	0
$921$	283568.	0.334302
$922$	0	0
$923$	−536041.	−0.629209
$924$	0	0
$925$	48969.2i	0.0572320i
$926$	0	0
$927$	542619.i	0.631445i
$928$	0	0
$929$	−243550.	−0.282200	−0.141100	−	0.989995i	$-0.545064\pi$
$929$	−243550.	−0.282200	−0.141100	+	0.989995i	$0.545064\pi$
$930$	0	0
$931$	488722.	0.563849
$932$	0	0
$933$	316684.i	0.363800i
$934$	0	0
$935$	− 212773.i	− 0.243385i
$936$	0	0
$937$	479076.	0.545664	0.272832	−	0.962062i	$-0.412040\pi$
$937$	479076.	0.545664	0.272832	+	0.962062i	$0.412040\pi$
$938$	0	0
$939$	534027.	0.605665
$940$	0	0
$941$	− 1.25655e6i	− 1.41906i	−0.704675	−	0.709530i	$-0.748907\pi$
$941$	− 1.25655e6i	− 1.41906i	0.704675	−	0.709530i	$-0.251093\pi$
$942$	0	0
$943$	524246.i	0.589538i
$944$	0	0
$945$	−31261.5	−0.0350063
$946$	0	0
$947$	1.22304e6	1.36377	0.681886	−	0.731459i	$-0.261160\pi$
$947$	1.22304e6	1.36377	0.681886	+	0.731459i	$0.261160\pi$
$948$	0	0
$949$	− 15031.3i	− 0.0166903i
$950$	0	0
$951$	206699.i	0.228548i
$952$	0	0
$953$	−1.12329e6	−1.23682	−0.618409	−	0.785856i	$-0.712223\pi$
$953$	−1.12329e6	−1.23682	−0.618409	+	0.785856i	$0.712223\pi$
$954$	0	0
$955$	505512.	0.554275
$956$	0	0
$957$	358936.i	0.391916i
$958$	0	0
$959$	153944.i	0.167389i
$960$	0	0
$961$	−312598.	−0.338485
$962$	0	0
$963$	−334341.	−0.360526
$964$	0	0
$965$	− 619240.i	− 0.664974i
$966$	0	0
$967$	310108.i	0.331635i	0.986156	+	0.165818i	$0.0530263\pi$
$967$	310108.i	0.331635i	−0.986156	+	0.165818i	$0.946974\pi$
$968$	0	0
$969$	−87825.7	−0.0935350
$970$	0	0
$971$	459519.	0.487376	0.243688	−	0.969854i	$-0.421643\pi$
$971$	459519.	0.487376	0.243688	+	0.969854i	$0.421643\pi$
$972$	0	0
$973$	− 239668.i	− 0.253154i
$974$	0	0
$975$	17904.5i	0.0188345i
$976$	0	0
$977$	−645770.	−0.676532	−0.338266	−	0.941050i	$-0.609840\pi$
$977$	−645770.	−0.676532	−0.338266	+	0.941050i	$0.609840\pi$
$978$	0	0
$979$	−158379.	−0.165246
$980$	0	0
$981$	− 419635.i	− 0.436048i
$982$	0	0
$983$	828941.i	0.857860i	0.903338	+	0.428930i	$0.141109\pi$
$983$	828941.i	0.857860i	−0.903338	+	0.428930i	$0.858891\pi$
$984$	0	0
$985$	−112363.	−0.115811
$986$	0	0
$987$	−175344.	−0.179993
$988$	0	0
$989$	463050.i	0.473407i
$990$	0	0
$991$	460245.i	0.468642i	0.972159	+	0.234321i	$0.0752867\pi$
$991$	460245.i	0.468642i	−0.972159	+	0.234321i	$0.924713\pi$
$992$	0	0
$993$	456246.	0.462701
$994$	0	0
$995$	915803.	0.925030
$996$	0	0
$997$	− 1.82730e6i	− 1.83831i	−0.393894	−	0.919156i	$-0.628872\pi$
$997$	− 1.82730e6i	− 1.83831i	0.393894	−	0.919156i	$-0.371128\pi$
$998$	0	0
$999$	112655.i	0.112881i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.b.c.319.7	yes	8
3.2	odd	2		1152.5.b.k.703.4		8
4.3	odd	2	inner	384.5.b.c.319.3	yes	8
8.3	odd	2	inner	384.5.b.c.319.6	yes	8
8.5	even	2	inner	384.5.b.c.319.2	✓	8
12.11	even	2		1152.5.b.k.703.3		8
16.3	odd	4		768.5.g.c.511.2		4
16.5	even	4		768.5.g.g.511.1		4
16.11	odd	4		768.5.g.g.511.3		4
16.13	even	4		768.5.g.c.511.4		4
24.5	odd	2		1152.5.b.k.703.6		8
24.11	even	2		1152.5.b.k.703.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.c.319.2	✓	8	8.5	even	2	inner
384.5.b.c.319.3	yes	8	4.3	odd	2	inner
384.5.b.c.319.6	yes	8	8.3	odd	2	inner
384.5.b.c.319.7	yes	8	1.1	even	1	trivial
768.5.g.c.511.2		4	16.3	odd	4
768.5.g.c.511.4		4	16.13	even	4
768.5.g.g.511.1		4	16.5	even	4
768.5.g.g.511.3		4	16.11	odd	4
1152.5.b.k.703.3		8	12.11	even	2
1152.5.b.k.703.4		8	3.2	odd	2
1152.5.b.k.703.5		8	24.11	even	2
1152.5.b.k.703.6		8	24.5	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} +23.7490i q^{5} +9.38251i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} +23.7490i q^{5} +9.38251i q^{7} +27.0000 q^{9} +112.001 q^{11} +56.5020i q^{13} +123.404i q^{15} -79.9921 q^{17} +211.297 q^{19} +48.7530i q^{21} -217.355i q^{23} +60.9843 q^{25} +140.296 q^{27} +616.753i q^{29} +1111.81i q^{31} +581.976 q^{33} -222.825 q^{35} +802.980i q^{37} +293.593i q^{39} -2411.93 q^{41} -2130.38 q^{43} +641.223i q^{45} +3596.58i q^{47} +2312.97 q^{49} -415.651 q^{51} -833.725i q^{53} +2659.92i q^{55} +1097.93 q^{57} -1309.43 q^{59} -4785.91i q^{61} +253.328i q^{63} -1341.87 q^{65} +4025.23 q^{67} -1129.41i q^{69} +9487.13i q^{71} -266.031 q^{73} +316.883 q^{75} +1050.85i q^{77} +5756.11i q^{79} +729.000 q^{81} +7287.16 q^{83} -1899.73i q^{85} +3204.74i q^{87} -1414.08 q^{89} -530.130 q^{91} +5777.13i q^{93} +5018.09i q^{95} -3110.08 q^{97} +3024.04 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 216 q^{9}+O(q^{10}) \) 8 * q + 216 * q^9	\( 8 q + 216 q^{9} + 1392 q^{17} - 3576 q^{25} - 1440 q^{33} - 1008 q^{41} + 10376 q^{49} - 9504 q^{57} + 23808 q^{65} - 10256 q^{73} + 5832 q^{81} - 31632 q^{89} - 45200 q^{97}+O(q^{100}) \) 8 * q + 216 * q^9 + 1392 * q^17 - 3576 * q^25 - 1440 * q^33 - 1008 * q^41 + 10376 * q^49 - 9504 * q^57 + 23808 * q^65 - 10256 * q^73 + 5832 * q^81 - 31632 * q^89 - 45200 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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