LMFDB - Embedded newform 768.5.b.g.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(127,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$79.3881316484$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.592240896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 $ x^8 - 7x^6 + 40x^4 - 63*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$-1.12824 - 0.651388i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.5.b.g.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615 q^{3} -34.8444i q^{5} +43.0318i q^{7} +27.0000 q^{9} +O(q^{10})$	$q-5.19615 q^{3} -34.8444i q^{5} +43.0318i q^{7} +27.0000 q^{9} +65.2790 q^{11} -99.0665i q^{13} +181.057i q^{15} -207.689 q^{17} +569.960 q^{19} -223.600i q^{21} -371.198i q^{23} -589.133 q^{25} -140.296 q^{27} +423.911i q^{29} +1174.18i q^{31} -339.199 q^{33} +1499.42 q^{35} +1448.13i q^{37} +514.764i q^{39} +265.822 q^{41} +699.900 q^{43} -940.799i q^{45} +1250.00i q^{47} +549.266 q^{49} +1079.18 q^{51} -787.645i q^{53} -2274.61i q^{55} -2961.60 q^{57} -3012.53 q^{59} +4519.33i q^{61} +1161.86i q^{63} -3451.91 q^{65} +5215.70 q^{67} +1928.80i q^{69} -4693.16i q^{71} +5404.92 q^{73} +3061.22 q^{75} +2809.07i q^{77} -7453.44i q^{79} +729.000 q^{81} -8950.92 q^{83} +7236.79i q^{85} -2202.71i q^{87} +616.496 q^{89} +4263.01 q^{91} -6101.20i q^{93} -19859.9i q^{95} +13723.1 q^{97} +1762.53 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 216 q^{9}+O(q^{10}) $ 8 * q + 216 * q^9	$ 8 q + 216 q^{9} - 1200 q^{17} - 1944 q^{25} + 1440 q^{33} - 1104 q^{41} - 1144 q^{49} - 11232 q^{57} - 36384 q^{65} - 17680 q^{73} + 5832 q^{81} + 50160 q^{89} + 46096 q^{97}+O(q^{100}) $ 8 * q + 216 * q^9 - 1200 * q^17 - 1944 * q^25 + 1440 * q^33 - 1104 * q^41 - 1144 * q^49 - 11232 * q^57 - 36384 * q^65 - 17680 * q^73 + 5832 * q^81 + 50160 * q^89 + 46096 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 34.8444i	− 1.39378i	−0.717180	−	0.696888i	$-0.754567\pi$
$5$	− 34.8444i	− 1.39378i	0.717180	−	0.696888i	$-0.245433\pi$
$6$	0	0
$7$	43.0318i	0.878200i	0.898438	+	0.439100i	$0.144703\pi$
$7$	43.0318i	0.878200i	−0.898438	+	0.439100i	$0.855297\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	65.2790	0.539495	0.269748	−	0.962931i	$-0.413060\pi$
$11$	65.2790	0.539495	0.269748	+	0.962931i	$0.413060\pi$
$12$	0	0
$13$	− 99.0665i	− 0.586192i	−0.956083	−	0.293096i	$-0.905314\pi$
$13$	− 99.0665i	− 0.586192i	0.956083	−	0.293096i	$-0.0946856\pi$
$14$	0	0
$15$	181.057i	0.804697i
$16$	0	0
$17$	−207.689	−0.718646	−0.359323	−	0.933213i	$-0.616992\pi$
$17$	−207.689	−0.718646	−0.359323	+	0.933213i	$0.616992\pi$
$18$	0	0
$19$	569.960	1.57884	0.789418	−	0.613856i	$-0.210382\pi$
$19$	569.960	1.57884	0.789418	+	0.613856i	$0.210382\pi$
$20$	0	0
$21$	− 223.600i	− 0.507029i
$22$	0	0
$23$	− 371.198i	− 0.701697i	−0.936432	−	0.350849i	$-0.885893\pi$
$23$	− 371.198i	− 0.701697i	0.936432	−	0.350849i	$-0.114107\pi$
$24$	0	0
$25$	−589.133	−0.942613
$26$	0	0
$27$	−140.296	−0.192450
$28$	0	0
$29$	423.911i	0.504056i	0.967720	+	0.252028i	$0.0810975\pi$
$29$	423.911i	0.504056i	−0.967720	+	0.252028i	$0.918903\pi$
$30$	0	0
$31$	1174.18i	1.22183i	0.791697	+	0.610914i	$0.209198\pi$
$31$	1174.18i	1.22183i	−0.791697	+	0.610914i	$0.790802\pi$
$32$	0	0
$33$	−339.199	−0.311478
$34$	0	0
$35$	1499.42	1.22401
$36$	0	0
$37$	1448.13i	1.05780i	0.848683	+	0.528902i	$0.177396\pi$
$37$	1448.13i	1.05780i	−0.848683	+	0.528902i	$0.822604\pi$
$38$	0	0
$39$	514.764i	0.338438i
$40$	0	0
$41$	265.822	0.158133	0.0790666	−	0.996869i	$-0.474806\pi$
$41$	265.822	0.158133	0.0790666	+	0.996869i	$0.474806\pi$
$42$	0	0
$43$	699.900	0.378529	0.189265	−	0.981926i	$-0.439390\pi$
$43$	699.900	0.378529	0.189265	+	0.981926i	$0.439390\pi$
$44$	0	0
$45$	− 940.799i	− 0.464592i
$46$	0	0
$47$	1250.00i	0.565868i	0.959139	+	0.282934i	$0.0913077\pi$
$47$	1250.00i	0.565868i	−0.959139	+	0.282934i	$0.908692\pi$
$48$	0	0
$49$	549.266	0.228765
$50$	0	0
$51$	1079.18	0.414911
$52$	0	0
$53$	− 787.645i	− 0.280401i	−0.990123	−	0.140200i	$-0.955225\pi$
$53$	− 787.645i	− 0.280401i	0.990123	−	0.140200i	$-0.0447746\pi$
$54$	0	0
$55$	− 2274.61i	− 0.751936i
$56$	0	0
$57$	−2961.60	−0.911541
$58$	0	0
$59$	−3012.53	−0.865422	−0.432711	−	0.901533i	$-0.642443\pi$
$59$	−3012.53	−0.865422	−0.432711	+	0.901533i	$0.642443\pi$
$60$	0	0
$61$	4519.33i	1.21455i	0.794493	+	0.607273i	$0.207737\pi$
$61$	4519.33i	1.21455i	−0.794493	+	0.607273i	$0.792263\pi$
$62$	0	0
$63$	1161.86i	0.292733i
$64$	0	0
$65$	−3451.91	−0.817021
$66$	0	0
$67$	5215.70	1.16189	0.580943	−	0.813945i	$-0.302684\pi$
$67$	5215.70	1.16189	0.580943	+	0.813945i	$0.302684\pi$
$68$	0	0
$69$	1928.80i	0.405125i
$70$	0	0
$71$	− 4693.16i	− 0.930998i	−0.885048	−	0.465499i	$-0.845875\pi$
$71$	− 4693.16i	− 0.930998i	0.885048	−	0.465499i	$-0.154125\pi$
$72$	0	0
$73$	5404.92	1.01425	0.507124	−	0.861873i	$-0.330709\pi$
$73$	5404.92	1.01425	0.507124	+	0.861873i	$0.330709\pi$
$74$	0	0
$75$	3061.22	0.544218
$76$	0	0
$77$	2809.07i	0.473785i
$78$	0	0
$79$	− 7453.44i	− 1.19427i	−0.802141	−	0.597135i	$-0.796306\pi$
$79$	− 7453.44i	− 1.19427i	0.802141	−	0.597135i	$-0.203694\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	−8950.92	−1.29931	−0.649653	−	0.760231i	$-0.725086\pi$
$83$	−8950.92	−1.29931	−0.649653	+	0.760231i	$0.725086\pi$
$84$	0	0
$85$	7236.79i	1.00163i
$86$	0	0
$87$	− 2202.71i	− 0.291017i
$88$	0	0
$89$	616.496	0.0778305	0.0389153	−	0.999243i	$-0.487610\pi$
$89$	616.496	0.0778305	0.0389153	+	0.999243i	$0.487610\pi$
$90$	0	0
$91$	4263.01	0.514794
$92$	0	0
$93$	− 6101.20i	− 0.705422i
$94$	0	0
$95$	− 19859.9i	− 2.20054i
$96$	0	0
$97$	13723.1	1.45850	0.729252	−	0.684246i	$-0.239868\pi$
$97$	13723.1	1.45850	0.729252	+	0.684246i	$0.239868\pi$
$98$	0	0
$99$	1762.53	0.179832
$100$	0	0
$101$	− 14758.8i	− 1.44680i	−0.690428	−	0.723401i	$-0.742578\pi$
$101$	− 14758.8i	− 1.44680i	0.690428	−	0.723401i	$-0.257422\pi$
$102$	0	0
$103$	− 1859.14i	− 0.175242i	−0.996154	−	0.0876210i	$-0.972074\pi$
$103$	− 1859.14i	− 0.175242i	0.996154	−	0.0876210i	$-0.0279264\pi$
$104$	0	0
$105$	−7791.20	−0.706685
$106$	0	0
$107$	10664.4	0.931466	0.465733	−	0.884925i	$-0.345791\pi$
$107$	10664.4	0.931466	0.465733	+	0.884925i	$0.345791\pi$
$108$	0	0
$109$	− 4479.34i	− 0.377017i	−0.982072	−	0.188508i	$-0.939635\pi$
$109$	− 4479.34i	− 0.377017i	0.982072	−	0.188508i	$-0.0603653\pi$
$110$	0	0
$111$	− 7524.72i	− 0.610723i
$112$	0	0
$113$	−21225.5	−1.66227	−0.831135	−	0.556070i	$-0.812309\pi$
$113$	−21225.5	−1.66227	−0.831135	+	0.556070i	$0.812309\pi$
$114$	0	0
$115$	−12934.2	−0.978009
$116$	0	0
$117$	− 2674.79i	− 0.195397i
$118$	0	0
$119$	− 8937.22i	− 0.631115i
$120$	0	0
$121$	−10379.7	−0.708945
$122$	0	0
$123$	−1381.25	−0.0912982
$124$	0	0
$125$	− 1249.77i	− 0.0799851i
$126$	0	0
$127$	− 4676.78i	− 0.289961i	−0.989435	−	0.144980i	$-0.953688\pi$
$127$	− 4676.78i	− 0.289961i	0.989435	−	0.144980i	$-0.0463119\pi$
$128$	0	0
$129$	−3636.79	−0.218544
$130$	0	0
$131$	29922.6	1.74364	0.871819	−	0.489828i	$-0.162940\pi$
$131$	29922.6	1.74364	0.871819	+	0.489828i	$0.162940\pi$
$132$	0	0
$133$	24526.4i	1.38653i
$134$	0	0
$135$	4888.54i	0.268232i
$136$	0	0
$137$	19273.3	1.02687	0.513434	−	0.858129i	$-0.328373\pi$
$137$	19273.3	1.02687	0.513434	+	0.858129i	$0.328373\pi$
$138$	0	0
$139$	24086.9	1.24667	0.623335	−	0.781955i	$-0.285778\pi$
$139$	24086.9	1.24667	0.623335	+	0.781955i	$0.285778\pi$
$140$	0	0
$141$	− 6495.20i	− 0.326704i
$142$	0	0
$143$	− 6466.95i	− 0.316248i
$144$	0	0
$145$	14770.9	0.702541
$146$	0	0
$147$	−2854.07	−0.132078
$148$	0	0
$149$	− 43993.2i	− 1.98159i	−0.135378	−	0.990794i	$-0.543225\pi$
$149$	− 43993.2i	− 1.98159i	0.135378	−	0.990794i	$-0.456775\pi$
$150$	0	0
$151$	1388.00i	0.0608746i	0.999537	+	0.0304373i	$0.00969000\pi$
$151$	1388.00i	0.0608746i	−0.999537	+	0.0304373i	$0.990310\pi$
$152$	0	0
$153$	−5607.60	−0.239549
$154$	0	0
$155$	40913.5	1.70295
$156$	0	0
$157$	7464.93i	0.302849i	0.988469	+	0.151425i	$0.0483861\pi$
$157$	7464.93i	0.302849i	−0.988469	+	0.151425i	$0.951614\pi$
$158$	0	0
$159$	4092.72i	0.161889i
$160$	0	0
$161$	15973.3	0.616230
$162$	0	0
$163$	47291.0	1.77993	0.889965	−	0.456029i	$-0.150729\pi$
$163$	47291.0	1.77993	0.889965	+	0.456029i	$0.150729\pi$
$164$	0	0
$165$	11819.2i	0.434130i
$166$	0	0
$167$	90.1917i	0.00323395i	0.999999	+	0.00161698i	$0.000514700\pi$
$167$	90.1917i	0.00323395i	−0.999999	+	0.00161698i	$0.999485\pi$
$168$	0	0
$169$	18746.8	0.656379
$170$	0	0
$171$	15388.9	0.526279
$172$	0	0
$173$	− 1212.10i	− 0.0404993i	−0.999795	−	0.0202497i	$-0.993554\pi$
$173$	− 1212.10i	− 0.0404993i	0.999795	−	0.0202497i	$-0.00644611\pi$
$174$	0	0
$175$	− 25351.4i	− 0.827802i
$176$	0	0
$177$	15653.6	0.499652
$178$	0	0
$179$	20697.3	0.645964	0.322982	−	0.946405i	$-0.395315\pi$
$179$	20697.3	0.645964	0.322982	+	0.946405i	$0.395315\pi$
$180$	0	0
$181$	− 18722.8i	− 0.571496i	−0.958305	−	0.285748i	$-0.907758\pi$
$181$	− 18722.8i	− 0.571496i	0.958305	−	0.285748i	$-0.0922421\pi$
$182$	0	0
$183$	− 23483.1i	− 0.701219i
$184$	0	0
$185$	50459.3	1.47434
$186$	0	0
$187$	−13557.7	−0.387706
$188$	0	0
$189$	− 6037.19i	− 0.169010i
$190$	0	0
$191$	37058.2i	1.01582i	0.861410	+	0.507910i	$0.169582\pi$
$191$	37058.2i	1.01582i	−0.861410	+	0.507910i	$0.830418\pi$
$192$	0	0
$193$	−23019.8	−0.617999	−0.308999	−	0.951062i	$-0.599994\pi$
$193$	−23019.8	−0.617999	−0.308999	+	0.951062i	$0.599994\pi$
$194$	0	0
$195$	17936.7	0.471707
$196$	0	0
$197$	42137.6i	1.08577i	0.839807	+	0.542885i	$0.182668\pi$
$197$	42137.6i	1.08577i	−0.839807	+	0.542885i	$0.817332\pi$
$198$	0	0
$199$	− 68550.6i	− 1.73103i	−0.500881	−	0.865516i	$-0.666991\pi$
$199$	− 68550.6i	− 1.73103i	0.500881	−	0.865516i	$-0.333009\pi$
$200$	0	0
$201$	−27101.6	−0.670815
$202$	0	0
$203$	−18241.6	−0.442662
$204$	0	0
$205$	− 9262.40i	− 0.220402i
$206$	0	0
$207$	− 10022.3i	− 0.233899i
$208$	0	0
$209$	37206.4	0.851775
$210$	0	0
$211$	37678.5	0.846308	0.423154	−	0.906058i	$-0.360923\pi$
$211$	37678.5	0.846308	0.423154	+	0.906058i	$0.360923\pi$
$212$	0	0
$213$	24386.4i	0.537512i
$214$	0	0
$215$	− 24387.6i	− 0.527585i
$216$	0	0
$217$	−50526.9	−1.07301
$218$	0	0
$219$	−28084.8	−0.585576
$220$	0	0
$221$	20575.0i	0.421265i
$222$	0	0
$223$	9238.47i	0.185776i	0.995677	+	0.0928882i	$0.0296099\pi$
$223$	9238.47i	0.185776i	−0.995677	+	0.0928882i	$0.970390\pi$
$224$	0	0
$225$	−15906.6	−0.314204
$226$	0	0
$227$	48884.8	0.948685	0.474342	−	0.880340i	$-0.342686\pi$
$227$	48884.8	0.948685	0.474342	+	0.880340i	$0.342686\pi$
$228$	0	0
$229$	− 17654.5i	− 0.336654i	−0.985731	−	0.168327i	$-0.946164\pi$
$229$	− 17654.5i	− 0.336654i	0.985731	−	0.168327i	$-0.0538365\pi$
$230$	0	0
$231$	− 14596.4i	− 0.273540i
$232$	0	0
$233$	−52029.9	−0.958387	−0.479193	−	0.877709i	$-0.659071\pi$
$233$	−52029.9	−0.958387	−0.479193	+	0.877709i	$0.659071\pi$
$234$	0	0
$235$	43555.6	0.788693
$236$	0	0
$237$	38729.2i	0.689512i
$238$	0	0
$239$	− 53899.3i	− 0.943598i	−0.881706	−	0.471799i	$-0.843605\pi$
$239$	− 53899.3i	− 0.943598i	0.881706	−	0.471799i	$-0.156395\pi$
$240$	0	0
$241$	−56311.0	−0.969526	−0.484763	−	0.874646i	$-0.661094\pi$
$241$	−56311.0	−0.969526	−0.484763	+	0.874646i	$0.661094\pi$
$242$	0	0
$243$	−3788.00	−0.0641500
$244$	0	0
$245$	− 19138.8i	− 0.318848i
$246$	0	0
$247$	− 56463.9i	− 0.925501i
$248$	0	0
$249$	46510.3	0.750155
$250$	0	0
$251$	11951.6	0.189705	0.0948525	−	0.995491i	$-0.469762\pi$
$251$	11951.6	0.189705	0.0948525	+	0.995491i	$0.469762\pi$
$252$	0	0
$253$	− 24231.4i	− 0.378562i
$254$	0	0
$255$	− 37603.5i	− 0.578293i
$256$	0	0
$257$	−15483.1	−0.234418	−0.117209	−	0.993107i	$-0.537395\pi$
$257$	−15483.1	−0.234418	−0.117209	+	0.993107i	$0.537395\pi$
$258$	0	0
$259$	−62315.7	−0.928963
$260$	0	0
$261$	11445.6i	0.168019i
$262$	0	0
$263$	− 16337.3i	− 0.236194i	−0.993002	−	0.118097i	$-0.962321\pi$
$263$	− 16337.3i	− 0.236194i	0.993002	−	0.118097i	$-0.0376793\pi$
$264$	0	0
$265$	−27445.0	−0.390816
$266$	0	0
$267$	−3203.41	−0.0449355
$268$	0	0
$269$	− 82062.5i	− 1.13407i	−0.823694	−	0.567035i	$-0.808090\pi$
$269$	− 82062.5i	− 1.13407i	0.823694	−	0.567035i	$-0.191910\pi$
$270$	0	0
$271$	− 75123.6i	− 1.02291i	−0.859310	−	0.511456i	$-0.829106\pi$
$271$	− 75123.6i	− 1.02291i	0.859310	−	0.511456i	$-0.170894\pi$
$272$	0	0
$273$	−22151.2	−0.297216
$274$	0	0
$275$	−38458.0	−0.508535
$276$	0	0
$277$	− 9651.88i	− 0.125792i	−0.998020	−	0.0628959i	$-0.979966\pi$
$277$	− 9651.88i	− 0.125792i	0.998020	−	0.0628959i	$-0.0200336\pi$
$278$	0	0
$279$	31702.8i	0.407276i
$280$	0	0
$281$	90483.6	1.14593	0.572964	−	0.819581i	$-0.305794\pi$
$281$	90483.6	1.14593	0.572964	+	0.819581i	$0.305794\pi$
$282$	0	0
$283$	−44901.3	−0.560643	−0.280321	−	0.959906i	$-0.590441\pi$
$283$	−44901.3	−0.560643	−0.280321	+	0.959906i	$0.590441\pi$
$284$	0	0
$285$	103195.i	1.27048i
$286$	0	0
$287$	11438.8i	0.138872i
$288$	0	0
$289$	−40386.4	−0.483547
$290$	0	0
$291$	−71307.1	−0.842067
$292$	0	0
$293$	− 30326.7i	− 0.353257i	−0.984278	−	0.176628i	$-0.943481\pi$
$293$	− 30326.7i	− 0.353257i	0.984278	−	0.176628i	$-0.0565191\pi$
$294$	0	0
$295$	104970.i	1.20620i
$296$	0	0
$297$	−9158.38	−0.103826
$298$	0	0
$299$	−36773.3	−0.411329
$300$	0	0
$301$	30118.0i	0.332424i
$302$	0	0
$303$	76689.2i	0.835312i
$304$	0	0
$305$	157473.	1.69281
$306$	0	0
$307$	35547.6	0.377167	0.188583	−	0.982057i	$-0.439610\pi$
$307$	35547.6	0.377167	0.188583	+	0.982057i	$0.439610\pi$
$308$	0	0
$309$	9660.38i	0.101176i
$310$	0	0
$311$	− 72202.2i	− 0.746500i	−0.927731	−	0.373250i	$-0.878243\pi$
$311$	− 72202.2i	− 0.746500i	0.927731	−	0.373250i	$-0.121757\pi$
$312$	0	0
$313$	−162750.	−1.66124	−0.830618	−	0.556842i	$-0.812013\pi$
$313$	−162750.	−1.66124	−0.830618	+	0.556842i	$0.812013\pi$
$314$	0	0
$315$	40484.3	0.408005
$316$	0	0
$317$	− 64556.9i	− 0.642428i	−0.947007	−	0.321214i	$-0.895909\pi$
$317$	− 64556.9i	− 0.642428i	0.947007	−	0.321214i	$-0.104091\pi$
$318$	0	0
$319$	27672.5i	0.271936i
$320$	0	0
$321$	−55413.6	−0.537782
$322$	0	0
$323$	−118374.	−1.13462
$324$	0	0
$325$	58363.3i	0.552552i
$326$	0	0
$327$	23275.3i	0.217671i
$328$	0	0
$329$	−53789.8	−0.496945
$330$	0	0
$331$	−11544.1	−0.105367	−0.0526833	−	0.998611i	$-0.516777\pi$
$331$	−11544.1	−0.105367	−0.0526833	+	0.998611i	$0.516777\pi$
$332$	0	0
$333$	39099.6i	0.352601i
$334$	0	0
$335$	− 181738.i	− 1.61941i
$336$	0	0
$337$	−209011.	−1.84039	−0.920195	−	0.391461i	$-0.871970\pi$
$337$	−209011.	−1.84039	−0.920195	+	0.391461i	$0.871970\pi$
$338$	0	0
$339$	110291.	0.959712
$340$	0	0
$341$	76649.0i	0.659170i
$342$	0	0
$343$	126955.i	1.07910i
$344$	0	0
$345$	67207.9	0.564654
$346$	0	0
$347$	161304.	1.33964	0.669819	−	0.742525i	$-0.266372\pi$
$347$	161304.	1.33964	0.669819	+	0.742525i	$0.266372\pi$
$348$	0	0
$349$	88041.4i	0.722830i	0.932405	+	0.361415i	$0.117706\pi$
$349$	88041.4i	0.722830i	−0.932405	+	0.361415i	$0.882294\pi$
$350$	0	0
$351$	13898.6i	0.112813i
$352$	0	0
$353$	186762.	1.49878	0.749391	−	0.662128i	$-0.230347\pi$
$353$	186762.	1.49878	0.749391	+	0.662128i	$0.230347\pi$
$354$	0	0
$355$	−163530.	−1.29760
$356$	0	0
$357$	46439.2i	0.364374i
$358$	0	0
$359$	226517.i	1.75756i	0.477224	+	0.878782i	$0.341643\pi$
$359$	226517.i	1.75756i	−0.477224	+	0.878782i	$0.658357\pi$
$360$	0	0
$361$	194533.	1.49272
$362$	0	0
$363$	53934.3	0.409309
$364$	0	0
$365$	− 188331.i	− 1.41363i
$366$	0	0
$367$	106652.i	0.791839i	0.918285	+	0.395920i	$0.129574\pi$
$367$	106652.i	0.791839i	−0.918285	+	0.395920i	$0.870426\pi$
$368$	0	0
$369$	7177.19	0.0527110
$370$	0	0
$371$	33893.8	0.246248
$372$	0	0
$373$	− 129184.i	− 0.928518i	−0.885699	−	0.464259i	$-0.846321\pi$
$373$	− 129184.i	− 0.928518i	0.885699	−	0.464259i	$-0.153679\pi$
$374$	0	0
$375$	6493.98i	0.0461794i
$376$	0	0
$377$	41995.3	0.295473
$378$	0	0
$379$	156156.	1.08713	0.543563	−	0.839368i	$-0.317075\pi$
$379$	156156.	1.08713	0.543563	+	0.839368i	$0.317075\pi$
$380$	0	0
$381$	24301.3i	0.167409i
$382$	0	0
$383$	261403.i	1.78202i	0.453980	+	0.891012i	$0.350004\pi$
$383$	261403.i	1.78202i	−0.453980	+	0.891012i	$0.649996\pi$
$384$	0	0
$385$	97880.4	0.660350
$386$	0	0
$387$	18897.3	0.126176
$388$	0	0
$389$	1046.60i	0.00691644i	0.999994	+	0.00345822i	$0.00110079\pi$
$389$	1046.60i	0.00691644i	−0.999994	+	0.00345822i	$0.998899\pi$
$390$	0	0
$391$	77093.6i	0.504272i
$392$	0	0
$393$	−155482.	−1.00669
$394$	0	0
$395$	−259711.	−1.66454
$396$	0	0
$397$	181472.i	1.15140i	0.817659	+	0.575702i	$0.195271\pi$
$397$	181472.i	1.15140i	−0.817659	+	0.575702i	$0.804729\pi$
$398$	0	0
$399$	− 127443.i	− 0.800515i
$400$	0	0
$401$	144404.	0.898029	0.449015	−	0.893524i	$-0.351775\pi$
$401$	144404.	0.898029	0.449015	+	0.893524i	$0.351775\pi$
$402$	0	0
$403$	116321.	0.716226
$404$	0	0
$405$	− 25401.6i	− 0.154864i
$406$	0	0
$407$	94532.6i	0.570680i
$408$	0	0
$409$	−1867.96	−0.0111666	−0.00558330	−	0.999984i	$-0.501777\pi$
$409$	−1867.96	−0.0111666	−0.00558330	+	0.999984i	$0.501777\pi$
$410$	0	0
$411$	−100147.	−0.592863
$412$	0	0
$413$	− 129635.i	− 0.760013i
$414$	0	0
$415$	311889.i	1.81094i
$416$	0	0
$417$	−125159.	−0.719765
$418$	0	0
$419$	−112803.	−0.642531	−0.321266	−	0.946989i	$-0.604108\pi$
$419$	−112803.	−0.642531	−0.321266	+	0.946989i	$0.604108\pi$
$420$	0	0
$421$	77595.0i	0.437794i	0.975748	+	0.218897i	$0.0702458\pi$
$421$	77595.0i	0.437794i	−0.975748	+	0.218897i	$0.929754\pi$
$422$	0	0
$423$	33750.0i	0.188623i
$424$	0	0
$425$	122356.	0.677405
$426$	0	0
$427$	−194475.	−1.06661
$428$	0	0
$429$	33603.3i	0.182586i
$430$	0	0
$431$	156726.i	0.843699i	0.906666	+	0.421850i	$0.138619\pi$
$431$	156726.i	0.843699i	−0.906666	+	0.421850i	$0.861381\pi$
$432$	0	0
$433$	−41267.2	−0.220105	−0.110052	−	0.993926i	$-0.535102\pi$
$433$	−41267.2	−0.220105	−0.110052	+	0.993926i	$0.535102\pi$
$434$	0	0
$435$	−76752.0	−0.405612
$436$	0	0
$437$	− 211568.i	− 1.10786i
$438$	0	0
$439$	− 154804.i	− 0.803252i	−0.915804	−	0.401626i	$-0.868445\pi$
$439$	− 154804.i	− 0.803252i	0.915804	−	0.401626i	$-0.131555\pi$
$440$	0	0
$441$	14830.2	0.0762551
$442$	0	0
$443$	−251131.	−1.27966	−0.639828	−	0.768518i	$-0.720994\pi$
$443$	−251131.	−1.27966	−0.639828	+	0.768518i	$0.720994\pi$
$444$	0	0
$445$	− 21481.4i	− 0.108478i
$446$	0	0
$447$	228596.i	1.14407i
$448$	0	0
$449$	−31888.6	−0.158177	−0.0790884	−	0.996868i	$-0.525201\pi$
$449$	−31888.6	−0.158177	−0.0790884	+	0.996868i	$0.525201\pi$
$450$	0	0
$451$	17352.6	0.0853121
$452$	0	0
$453$	− 7212.27i	− 0.0351460i
$454$	0	0
$455$	− 148542.i	− 0.717507i
$456$	0	0
$457$	166922.	0.799247	0.399624	−	0.916679i	$-0.369141\pi$
$457$	166922.	0.799247	0.399624	+	0.916679i	$0.369141\pi$
$458$	0	0
$459$	29137.9	0.138304
$460$	0	0
$461$	− 130544.i	− 0.614266i	−0.951667	−	0.307133i	$-0.900630\pi$
$461$	− 130544.i	− 0.614266i	0.951667	−	0.307133i	$-0.0993696\pi$
$462$	0	0
$463$	− 273167.i	− 1.27428i	−0.770746	−	0.637142i	$-0.780116\pi$
$463$	− 273167.i	− 1.27428i	0.770746	−	0.637142i	$-0.219884\pi$
$464$	0	0
$465$	−212593.	−0.983201
$466$	0	0
$467$	−56070.3	−0.257098	−0.128549	−	0.991703i	$-0.541032\pi$
$467$	−56070.3	−0.257098	−0.128549	+	0.991703i	$0.541032\pi$
$468$	0	0
$469$	224441.i	1.02037i
$470$	0	0
$471$	− 38788.9i	− 0.174850i
$472$	0	0
$473$	45688.8	0.204215
$474$	0	0
$475$	−335782.	−1.48823
$476$	0	0
$477$	− 21266.4i	− 0.0934668i
$478$	0	0
$479$	160400.i	0.699091i	0.936919	+	0.349546i	$0.113664\pi$
$479$	160400.i	0.699091i	−0.936919	+	0.349546i	$0.886336\pi$
$480$	0	0
$481$	143461.	0.620076
$482$	0	0
$483$	−82999.7	−0.355781
$484$	0	0
$485$	− 478172.i	− 2.03283i
$486$	0	0
$487$	314133.i	1.32451i	0.749278	+	0.662256i	$0.230401\pi$
$487$	314133.i	1.32451i	−0.749278	+	0.662256i	$0.769599\pi$
$488$	0	0
$489$	−245731.	−1.02764
$490$	0	0
$491$	−57822.1	−0.239845	−0.119923	−	0.992783i	$-0.538265\pi$
$491$	−57822.1	−0.239845	−0.119923	+	0.992783i	$0.538265\pi$
$492$	0	0
$493$	− 88041.5i	− 0.362238i
$494$	0	0
$495$	− 61414.4i	− 0.250645i
$496$	0	0
$497$	201955.	0.817602
$498$	0	0
$499$	−276054.	−1.10865	−0.554324	−	0.832301i	$-0.687023\pi$
$499$	−276054.	−1.10865	−0.554324	+	0.832301i	$0.687023\pi$
$500$	0	0
$501$	− 468.650i	− 0.00186712i
$502$	0	0
$503$	− 124228.i	− 0.491002i	−0.969396	−	0.245501i	$-0.921048\pi$
$503$	− 124228.i	− 0.491002i	0.969396	−	0.245501i	$-0.0789524\pi$
$504$	0	0
$505$	−514263.	−2.01652
$506$	0	0
$507$	−97411.4	−0.378961
$508$	0	0
$509$	− 335116.i	− 1.29348i	−0.762711	−	0.646739i	$-0.776132\pi$
$509$	− 335116.i	− 1.29348i	0.762711	−	0.646739i	$-0.223868\pi$
$510$	0	0
$511$	232584.i	0.890712i
$512$	0	0
$513$	−79963.2	−0.303847
$514$	0	0
$515$	−64780.7	−0.244248
$516$	0	0
$517$	81598.8i	0.305283i
$518$	0	0
$519$	6298.28i	0.0233823i
$520$	0	0
$521$	320361.	1.18022	0.590111	−	0.807322i	$-0.299084\pi$
$521$	320361.	1.18022	0.590111	+	0.807322i	$0.299084\pi$
$522$	0	0
$523$	−202708.	−0.741085	−0.370542	−	0.928816i	$-0.620828\pi$
$523$	−202708.	−0.741085	−0.370542	+	0.928816i	$0.620828\pi$
$524$	0	0
$525$	131730.i	0.477932i
$526$	0	0
$527$	− 243863.i	− 0.878062i
$528$	0	0
$529$	142053.	0.507621
$530$	0	0
$531$	−81338.4	−0.288474
$532$	0	0
$533$	− 26334.0i	− 0.0926964i
$534$	0	0
$535$	− 371593.i	− 1.29826i
$536$	0	0
$537$	−107546.	−0.372947
$538$	0	0
$539$	35855.5	0.123418
$540$	0	0
$541$	195927.i	0.669421i	0.942321	+	0.334710i	$0.108639\pi$
$541$	195927.i	0.669421i	−0.942321	+	0.334710i	$0.891361\pi$
$542$	0	0
$543$	97286.5i	0.329954i
$544$	0	0
$545$	−156080.	−0.525477
$546$	0	0
$547$	−71049.9	−0.237459	−0.118730	−	0.992927i	$-0.537882\pi$
$547$	−71049.9	−0.237459	−0.118730	+	0.992927i	$0.537882\pi$
$548$	0	0
$549$	122022.i	0.404849i
$550$	0	0
$551$	241612.i	0.795821i
$552$	0	0
$553$	320735.	1.04881
$554$	0	0
$555$	−262194.	−0.851211
$556$	0	0
$557$	152777.i	0.492432i	0.969215	+	0.246216i	$0.0791873\pi$
$557$	152777.i	0.492432i	−0.969215	+	0.246216i	$0.920813\pi$
$558$	0	0
$559$	− 69336.6i	− 0.221891i
$560$	0	0
$561$	70447.9	0.223842
$562$	0	0
$563$	128620.	0.405782	0.202891	−	0.979201i	$-0.434966\pi$
$563$	128620.	0.405782	0.202891	+	0.979201i	$0.434966\pi$
$564$	0	0
$565$	739591.i	2.31683i
$566$	0	0
$567$	31370.2i	0.0975777i
$568$	0	0
$569$	353147.	1.09076	0.545382	−	0.838188i	$-0.316385\pi$
$569$	353147.	1.09076	0.545382	+	0.838188i	$0.316385\pi$
$570$	0	0
$571$	−9607.19	−0.0294662	−0.0147331	−	0.999891i	$-0.504690\pi$
$571$	−9607.19	−0.0294662	−0.0147331	+	0.999891i	$0.504690\pi$
$572$	0	0
$573$	− 192560.i	− 0.586484i
$574$	0	0
$575$	218685.i	0.661429i
$576$	0	0
$577$	−35488.3	−0.106594	−0.0532971	−	0.998579i	$-0.516973\pi$
$577$	−35488.3	−0.106594	−0.0532971	+	0.998579i	$0.516973\pi$
$578$	0	0
$579$	119615.	0.356802
$580$	0	0
$581$	− 385174.i	− 1.14105i
$582$	0	0
$583$	− 51416.6i	− 0.151275i
$584$	0	0
$585$	−93201.6	−0.272340
$586$	0	0
$587$	396523.	1.15078	0.575390	−	0.817879i	$-0.304850\pi$
$587$	396523.	1.15078	0.575390	+	0.817879i	$0.304850\pi$
$588$	0	0
$589$	669233.i	1.92907i
$590$	0	0
$591$	− 218954.i	− 0.626869i
$592$	0	0
$593$	−199308.	−0.566780	−0.283390	−	0.959005i	$-0.591459\pi$
$593$	−199308.	−0.566780	−0.283390	+	0.959005i	$0.591459\pi$
$594$	0	0
$595$	−311412.	−0.879633
$596$	0	0
$597$	356199.i	0.999412i
$598$	0	0
$599$	596802.i	1.66332i	0.555284	+	0.831661i	$0.312610\pi$
$599$	596802.i	1.66332i	−0.555284	+	0.831661i	$0.687390\pi$
$600$	0	0
$601$	−171774.	−0.475563	−0.237782	−	0.971319i	$-0.576420\pi$
$601$	−171774.	−0.475563	−0.237782	+	0.971319i	$0.576420\pi$
$602$	0	0
$603$	140824.	0.387295
$604$	0	0
$605$	361673.i	0.988110i
$606$	0	0
$607$	204299.i	0.554485i	0.960800	+	0.277243i	$0.0894205\pi$
$607$	204299.i	0.554485i	−0.960800	+	0.277243i	$0.910579\pi$
$608$	0	0
$609$	94786.3	0.255571
$610$	0	0
$611$	123833.	0.331707
$612$	0	0
$613$	− 72753.9i	− 0.193613i	−0.995303	−	0.0968066i	$-0.969137\pi$
$613$	− 72753.9i	− 0.193613i	0.995303	−	0.0968066i	$-0.0308628\pi$
$614$	0	0
$615$	48128.9i	0.127249i
$616$	0	0
$617$	−387966.	−1.01911	−0.509557	−	0.860437i	$-0.670191\pi$
$617$	−387966.	−1.01911	−0.509557	+	0.860437i	$0.670191\pi$
$618$	0	0
$619$	676575.	1.76577	0.882886	−	0.469587i	$-0.155597\pi$
$619$	676575.	1.76577	0.882886	+	0.469587i	$0.155597\pi$
$620$	0	0
$621$	52077.6i	0.135042i
$622$	0	0
$623$	26528.9i	0.0683507i
$624$	0	0
$625$	−411755.	−1.05409
$626$	0	0
$627$	−193330.	−0.491772
$628$	0	0
$629$	− 300761.i	− 0.760187i
$630$	0	0
$631$	383055.i	0.962059i	0.876704	+	0.481030i	$0.159737\pi$
$631$	383055.i	0.962059i	−0.876704	+	0.481030i	$0.840263\pi$
$632$	0	0
$633$	−195783.	−0.488616
$634$	0	0
$635$	−162960.	−0.404141
$636$	0	0
$637$	− 54413.8i	− 0.134100i
$638$	0	0
$639$	− 126715.i	− 0.310333i
$640$	0	0
$641$	218038.	0.530660	0.265330	−	0.964158i	$-0.414519\pi$
$641$	218038.	0.530660	0.265330	+	0.964158i	$0.414519\pi$
$642$	0	0
$643$	329758.	0.797579	0.398789	−	0.917043i	$-0.369430\pi$
$643$	329758.	0.797579	0.398789	+	0.917043i	$0.369430\pi$
$644$	0	0
$645$	126722.i	0.304601i
$646$	0	0
$647$	166751.i	0.398346i	0.979964	+	0.199173i	$0.0638256\pi$
$647$	166751.i	0.398346i	−0.979964	+	0.199173i	$0.936174\pi$
$648$	0	0
$649$	−196655.	−0.466891
$650$	0	0
$651$	262545.	0.619502
$652$	0	0
$653$	− 684629.i	− 1.60557i	−0.596269	−	0.802785i	$-0.703351\pi$
$653$	− 684629.i	− 1.60557i	0.596269	−	0.802785i	$-0.296649\pi$
$654$	0	0
$655$	− 1.04263e6i	− 2.43024i
$656$	0	0
$657$	145933.	0.338082
$658$	0	0
$659$	−580631.	−1.33699	−0.668497	−	0.743714i	$-0.733062\pi$
$659$	−580631.	−1.33699	−0.668497	+	0.743714i	$0.733062\pi$
$660$	0	0
$661$	− 324617.i	− 0.742964i	−0.928440	−	0.371482i	$-0.878850\pi$
$661$	− 324617.i	− 0.742964i	0.928440	−	0.371482i	$-0.121150\pi$
$662$	0	0
$663$	− 106911.i	− 0.243217i
$664$	0	0
$665$	854607.	1.93252
$666$	0	0
$667$	157355.	0.353695
$668$	0	0
$669$	− 48004.5i	− 0.107258i
$670$	0	0
$671$	295017.i	0.655243i
$672$	0	0
$673$	46209.4	0.102024	0.0510118	−	0.998698i	$-0.483755\pi$
$673$	46209.4	0.102024	0.0510118	+	0.998698i	$0.483755\pi$
$674$	0	0
$675$	82653.1	0.181406
$676$	0	0
$677$	98225.8i	0.214313i	0.994242	+	0.107156i	$0.0341746\pi$
$677$	98225.8i	0.214313i	−0.994242	+	0.107156i	$0.965825\pi$
$678$	0	0
$679$	590528.i	1.28086i
$680$	0	0
$681$	−254013.	−0.547723
$682$	0	0
$683$	454709.	0.974747	0.487373	−	0.873194i	$-0.337955\pi$
$683$	454709.	0.974747	0.487373	+	0.873194i	$0.337955\pi$
$684$	0	0
$685$	− 671567.i	− 1.43123i
$686$	0	0
$687$	91735.4i	0.194367i
$688$	0	0
$689$	−78029.2	−0.164369
$690$	0	0
$691$	190359.	0.398673	0.199337	−	0.979931i	$-0.436121\pi$
$691$	190359.	0.398673	0.199337	+	0.979931i	$0.436121\pi$
$692$	0	0
$693$	75844.9i	0.157928i
$694$	0	0
$695$	− 839294.i	− 1.73758i
$696$	0	0
$697$	−55208.2	−0.113642
$698$	0	0
$699$	270355.	0.553325
$700$	0	0
$701$	504089.i	1.02582i	0.858442	+	0.512910i	$0.171433\pi$
$701$	504089.i	1.02582i	−0.858442	+	0.512910i	$0.828567\pi$
$702$	0	0
$703$	825378.i	1.67010i
$704$	0	0
$705$	−226321.	−0.455352
$706$	0	0
$707$	635099.	1.27058
$708$	0	0
$709$	160106.i	0.318504i	0.987238	+	0.159252i	$0.0509082\pi$
$709$	160106.i	0.318504i	−0.987238	+	0.159252i	$0.949092\pi$
$710$	0	0
$711$	− 201243.i	− 0.398090i
$712$	0	0
$713$	435852.	0.857353
$714$	0	0
$715$	−225337.	−0.440779
$716$	0	0
$717$	280069.i	0.544786i
$718$	0	0
$719$	− 918988.i	− 1.77767i	−0.458224	−	0.888836i	$-0.651514\pi$
$719$	− 918988.i	− 1.77767i	0.458224	−	0.888836i	$-0.348486\pi$
$720$	0	0
$721$	80002.2	0.153897
$722$	0	0
$723$	292601.	0.559756
$724$	0	0
$725$	− 249740.i	− 0.475129i
$726$	0	0
$727$	− 339605.i	− 0.642548i	−0.946986	−	0.321274i	$-0.895889\pi$
$727$	− 339605.i	− 0.642548i	0.946986	−	0.321274i	$-0.104111\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	−145361.	−0.272029
$732$	0	0
$733$	− 310186.i	− 0.577317i	−0.957432	−	0.288659i	$-0.906791\pi$
$733$	− 310186.i	− 0.577317i	0.957432	−	0.288659i	$-0.0932092\pi$
$734$	0	0
$735$	99448.4i	0.184087i
$736$	0	0
$737$	340476.	0.626832
$738$	0	0
$739$	561855.	1.02881	0.514405	−	0.857547i	$-0.328013\pi$
$739$	561855.	1.02881	0.514405	+	0.857547i	$0.328013\pi$
$740$	0	0
$741$	293395.i	0.534338i
$742$	0	0
$743$	89658.6i	0.162411i	0.996697	+	0.0812053i	$0.0258769\pi$
$743$	89658.6i	0.162411i	−0.996697	+	0.0812053i	$0.974123\pi$
$744$	0	0
$745$	−1.53292e6	−2.76189
$746$	0	0
$747$	−241675.	−0.433102
$748$	0	0
$749$	458906.i	0.818013i
$750$	0	0
$751$	58229.8i	0.103244i	0.998667	+	0.0516221i	$0.0164391\pi$
$751$	58229.8i	0.103244i	−0.998667	+	0.0516221i	$0.983561\pi$
$752$	0	0
$753$	−62102.4	−0.109526
$754$	0	0
$755$	48364.1	0.0848456
$756$	0	0
$757$	77925.2i	0.135983i	0.997686	+	0.0679917i	$0.0216592\pi$
$757$	77925.2i	0.135983i	−0.997686	+	0.0679917i	$0.978341\pi$
$758$	0	0
$759$	125910.i	0.218563i
$760$	0	0
$761$	−241107.	−0.416332	−0.208166	−	0.978094i	$-0.566749\pi$
$761$	−241107.	−0.416332	−0.208166	+	0.978094i	$0.566749\pi$
$762$	0	0
$763$	192754.	0.331096
$764$	0	0
$765$	195393.i	0.333877i
$766$	0	0
$767$	298441.i	0.507303i
$768$	0	0
$769$	−805186.	−1.36158	−0.680791	−	0.732478i	$-0.738364\pi$
$769$	−805186.	−1.36158	−0.680791	+	0.732478i	$0.738364\pi$
$770$	0	0
$771$	80452.5	0.135342
$772$	0	0
$773$	− 431024.i	− 0.721344i	−0.932693	−	0.360672i	$-0.882547\pi$
$773$	− 431024.i	− 0.721344i	0.932693	−	0.360672i	$-0.117453\pi$
$774$	0	0
$775$	− 691746.i	− 1.15171i
$776$	0	0
$777$	323802.	0.536337
$778$	0	0
$779$	151508.	0.249666
$780$	0	0
$781$	− 306365.i	− 0.502269i
$782$	0	0
$783$	− 59473.0i	− 0.0970056i
$784$	0	0
$785$	260111.	0.422104
$786$	0	0
$787$	−347948.	−0.561778	−0.280889	−	0.959740i	$-0.590629\pi$
$787$	−347948.	−0.561778	−0.280889	+	0.959740i	$0.590629\pi$
$788$	0	0
$789$	84891.0i	0.136367i
$790$	0	0
$791$	− 913373.i	− 1.45981i
$792$	0	0
$793$	447714.	0.711958
$794$	0	0
$795$	142609.	0.225637
$796$	0	0
$797$	− 460443.i	− 0.724868i	−0.932009	−	0.362434i	$-0.881946\pi$
$797$	− 460443.i	− 0.724868i	0.932009	−	0.362434i	$-0.118054\pi$
$798$	0	0
$799$	− 259611.i	− 0.406659i
$800$	0	0
$801$	16645.4	0.0259435
$802$	0	0
$803$	352828.	0.547182
$804$	0	0
$805$	− 556580.i	− 0.858887i
$806$	0	0
$807$	426409.i	0.654756i
$808$	0	0
$809$	621711.	0.949930	0.474965	−	0.880005i	$-0.342461\pi$
$809$	621711.	0.949930	0.474965	+	0.880005i	$0.342461\pi$
$810$	0	0
$811$	−25991.9	−0.0395181	−0.0197591	−	0.999805i	$-0.506290\pi$
$811$	−25991.9	−0.0395181	−0.0197591	+	0.999805i	$0.506290\pi$
$812$	0	0
$813$	390354.i	0.590578i
$814$	0	0
$815$	− 1.64783e6i	− 2.48082i
$816$	0	0
$817$	398915.	0.597635
$818$	0	0
$819$	115101.	0.171598
$820$	0	0
$821$	917945.i	1.36185i	0.732352	+	0.680927i	$0.238423\pi$
$821$	917945.i	1.36185i	−0.732352	+	0.680927i	$0.761577\pi$
$822$	0	0
$823$	− 1.22951e6i	− 1.81523i	−0.419807	−	0.907613i	$-0.637902\pi$
$823$	− 1.22951e6i	− 1.81523i	0.419807	−	0.907613i	$-0.362098\pi$
$824$	0	0
$825$	199834.	0.293603
$826$	0	0
$827$	−254347.	−0.371891	−0.185946	−	0.982560i	$-0.559535\pi$
$827$	−254347.	−0.371891	−0.185946	+	0.982560i	$0.559535\pi$
$828$	0	0
$829$	− 456708.i	− 0.664552i	−0.943182	−	0.332276i	$-0.892183\pi$
$829$	− 456708.i	− 0.664552i	0.943182	−	0.332276i	$-0.107817\pi$
$830$	0	0
$831$	50152.6i	0.0726259i
$832$	0	0
$833$	−114076.	−0.164401
$834$	0	0
$835$	3142.68	0.00450740
$836$	0	0
$837$	− 164732.i	− 0.235141i
$838$	0	0
$839$	− 397834.i	− 0.565168i	−0.959243	−	0.282584i	$-0.908808\pi$
$839$	− 397834.i	− 0.565168i	0.959243	−	0.282584i	$-0.0911916\pi$
$840$	0	0
$841$	527581.	0.745928
$842$	0	0
$843$	−470167.	−0.661602
$844$	0	0
$845$	− 653222.i	− 0.914845i
$846$	0	0
$847$	− 446655.i	− 0.622595i
$848$	0	0
$849$	233314.	0.323687
$850$	0	0
$851$	537544.	0.742258
$852$	0	0
$853$	− 683508.i	− 0.939389i	−0.882829	−	0.469694i	$-0.844364\pi$
$853$	− 683508.i	− 0.939389i	0.882829	−	0.469694i	$-0.155636\pi$
$854$	0	0
$855$	− 536218.i	− 0.733515i
$856$	0	0
$857$	116155.	0.158152	0.0790760	−	0.996869i	$-0.474803\pi$
$857$	116155.	0.158152	0.0790760	+	0.996869i	$0.474803\pi$
$858$	0	0
$859$	169632.	0.229890	0.114945	−	0.993372i	$-0.463331\pi$
$859$	169632.	0.229890	0.114945	+	0.993372i	$0.463331\pi$
$860$	0	0
$861$	− 59437.7i	− 0.0801780i
$862$	0	0
$863$	651371.i	0.874595i	0.899317	+	0.437298i	$0.144064\pi$
$863$	651371.i	0.874595i	−0.899317	+	0.437298i	$0.855936\pi$
$864$	0	0
$865$	−42235.1	−0.0564470
$866$	0	0
$867$	209854.	0.279176
$868$	0	0
$869$	− 486552.i	− 0.644303i
$870$	0	0
$871$	− 516701.i	− 0.681088i
$872$	0	0
$873$	370523.	0.486168
$874$	0	0
$875$	53779.7	0.0702429
$876$	0	0
$877$	1.36009e6i	1.76836i	0.467151	+	0.884178i	$0.345280\pi$
$877$	1.36009e6i	1.76836i	−0.467151	+	0.884178i	$0.654720\pi$
$878$	0	0
$879$	157582.i	0.203953i
$880$	0	0
$881$	976763.	1.25845	0.629227	−	0.777222i	$-0.283372\pi$
$881$	976763.	1.25845	0.629227	+	0.777222i	$0.283372\pi$
$882$	0	0
$883$	1.29155e6	1.65650	0.828248	−	0.560362i	$-0.189338\pi$
$883$	1.29155e6	1.65650	0.828248	+	0.560362i	$0.189338\pi$
$884$	0	0
$885$	− 545440.i	− 0.696403i
$886$	0	0
$887$	− 1.40632e6i	− 1.78747i	−0.448599	−	0.893733i	$-0.648077\pi$
$887$	− 1.40632e6i	− 1.78747i	0.448599	−	0.893733i	$-0.351923\pi$
$888$	0	0
$889$	201250.	0.254644
$890$	0	0
$891$	47588.4	0.0599439
$892$	0	0
$893$	712451.i	0.893412i
$894$	0	0
$895$	− 721186.i	− 0.900329i
$896$	0	0
$897$	191079.	0.237481
$898$	0	0
$899$	−497746.	−0.615869
$900$	0	0
$901$	163585.i	0.201509i
$902$	0	0
$903$	− 156497.i	− 0.191925i
$904$	0	0
$905$	−652385.	−0.796538
$906$	0	0
$907$	−373379.	−0.453874	−0.226937	−	0.973909i	$-0.572871\pi$
$907$	−373379.	−0.453874	−0.226937	+	0.973909i	$0.572871\pi$
$908$	0	0
$909$	− 398489.i	− 0.482268i
$910$	0	0
$911$	− 62632.1i	− 0.0754675i	−0.999288	−	0.0377338i	$-0.987986\pi$
$911$	− 62632.1i	− 0.0754675i	0.999288	−	0.0377338i	$-0.0120139\pi$
$912$	0	0
$913$	−584307.	−0.700970
$914$	0	0
$915$	−818256.	−0.977343
$916$	0	0
$917$	1.28762e6i	1.53126i
$918$	0	0
$919$	829993.i	0.982751i	0.870948	+	0.491375i	$0.163506\pi$
$919$	829993.i	0.982751i	−0.870948	+	0.491375i	$0.836494\pi$
$920$	0	0
$921$	−184711.	−0.217757
$922$	0	0
$923$	−464935.	−0.545744
$924$	0	0
$925$	− 853143.i	− 0.997099i
$926$	0	0
$927$	− 50196.8i	− 0.0584140i
$928$	0	0
$929$	1.22106e6	1.41483	0.707416	−	0.706798i	$-0.249861\pi$
$929$	1.22106e6	1.41483	0.707416	+	0.706798i	$0.249861\pi$
$930$	0	0
$931$	313059.	0.361183
$932$	0	0
$933$	375174.i	0.430992i
$934$	0	0
$935$	472410.i	0.540376i
$936$	0	0
$937$	238403.	0.271539	0.135770	−	0.990740i	$-0.456649\pi$
$937$	238403.	0.271539	0.135770	+	0.990740i	$0.456649\pi$
$938$	0	0
$939$	845672.	0.959115
$940$	0	0
$941$	− 1.17958e6i	− 1.33214i	−0.745889	−	0.666070i	$-0.767975\pi$
$941$	− 1.17958e6i	− 1.33214i	0.745889	−	0.666070i	$-0.232025\pi$
$942$	0	0
$943$	− 98672.5i	− 0.110962i
$944$	0	0
$945$	−210362.	−0.235562
$946$	0	0
$947$	886731.	0.988762	0.494381	−	0.869245i	$-0.335395\pi$
$947$	886731.	0.988762	0.494381	+	0.869245i	$0.335395\pi$
$948$	0	0
$949$	− 535447.i	− 0.594544i
$950$	0	0
$951$	335448.i	0.370906i
$952$	0	0
$953$	−1.72385e6	−1.89807	−0.949037	−	0.315166i	$-0.897940\pi$
$953$	−1.72385e6	−1.89807	−0.949037	+	0.315166i	$0.897940\pi$
$954$	0	0
$955$	1.29127e6	1.41583
$956$	0	0
$957$	− 143790.i	− 0.157002i
$958$	0	0
$959$	829364.i	0.901796i
$960$	0	0
$961$	−455168.	−0.492862
$962$	0	0
$963$	287938.	0.310489
$964$	0	0
$965$	802113.i	0.861352i
$966$	0	0
$967$	652513.i	0.697808i	0.937158	+	0.348904i	$0.113446\pi$
$967$	652513.i	0.697808i	−0.937158	+	0.348904i	$0.886554\pi$
$968$	0	0
$969$	615091.	0.655076
$970$	0	0
$971$	399968.	0.424216	0.212108	−	0.977246i	$-0.431967\pi$
$971$	399968.	0.424216	0.212108	+	0.977246i	$0.431967\pi$
$972$	0	0
$973$	1.03650e6i	1.09482i
$974$	0	0
$975$	− 303265.i	− 0.319016i
$976$	0	0
$977$	−1.43718e6	−1.50564	−0.752820	−	0.658227i	$-0.771307\pi$
$977$	−1.43718e6	−1.50564	−0.752820	+	0.658227i	$0.771307\pi$
$978$	0	0
$979$	40244.2	0.0419892
$980$	0	0
$981$	− 120942.i	− 0.125672i
$982$	0	0
$983$	1.64352e6i	1.70086i	0.526092	+	0.850428i	$0.323657\pi$
$983$	1.64352e6i	1.70086i	−0.526092	+	0.850428i	$0.676343\pi$
$984$	0	0
$985$	1.46826e6	1.51332
$986$	0	0
$987$	279500.	0.286911
$988$	0	0
$989$	− 259801.i	− 0.265613i
$990$	0	0
$991$	713551.i	0.726570i	0.931678	+	0.363285i	$0.118345\pi$
$991$	713551.i	0.726570i	−0.931678	+	0.363285i	$0.881655\pi$
$992$	0	0
$993$	59984.8	0.0608334
$994$	0	0
$995$	−2.38861e6	−2.41267
$996$	0	0
$997$	1.13252e6i	1.13935i	0.821871	+	0.569673i	$0.192930\pi$
$997$	1.13252e6i	1.13935i	−0.821871	+	0.569673i	$0.807070\pi$
$998$	0	0
$999$	− 203167.i	− 0.203574i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.b.g.127.1		8
4.3	odd	2	inner	768.5.b.g.127.5		8
8.3	odd	2	inner	768.5.b.g.127.4		8
8.5	even	2	inner	768.5.b.g.127.8		8
16.3	odd	4		192.5.g.d.127.3		4
16.5	even	4		12.5.d.a.7.2	yes	4
16.11	odd	4		12.5.d.a.7.1	✓	4
16.13	even	4		192.5.g.d.127.1		4
48.5	odd	4		36.5.d.b.19.3		4
48.11	even	4		36.5.d.b.19.4		4
48.29	odd	4		576.5.g.m.127.3		4
48.35	even	4		576.5.g.m.127.4		4
80.27	even	4		300.5.f.a.199.7		8
80.37	odd	4		300.5.f.a.199.1		8
80.43	even	4		300.5.f.a.199.2		8
80.53	odd	4		300.5.f.a.199.8		8
80.59	odd	4		300.5.c.a.151.4		4
80.69	even	4		300.5.c.a.151.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.5.d.a.7.1	✓	4	16.11	odd	4
12.5.d.a.7.2	yes	4	16.5	even	4
36.5.d.b.19.3		4	48.5	odd	4
36.5.d.b.19.4		4	48.11	even	4
192.5.g.d.127.1		4	16.13	even	4
192.5.g.d.127.3		4	16.3	odd	4
300.5.c.a.151.3		4	80.69	even	4
300.5.c.a.151.4		4	80.59	odd	4
300.5.f.a.199.1		8	80.37	odd	4
300.5.f.a.199.2		8	80.43	even	4
300.5.f.a.199.7		8	80.27	even	4
300.5.f.a.199.8		8	80.53	odd	4
576.5.g.m.127.3		4	48.29	odd	4
576.5.g.m.127.4		4	48.35	even	4
768.5.b.g.127.1		8	1.1	even	1	trivial
768.5.b.g.127.4		8	8.3	odd	2	inner
768.5.b.g.127.5		8	4.3	odd	2	inner
768.5.b.g.127.8		8	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.19615 q^{3} -34.8444i q^{5} +43.0318i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q-5.19615 q^{3} -34.8444i q^{5} +43.0318i q^{7} +27.0000 q^{9} +65.2790 q^{11} -99.0665i q^{13} +181.057i q^{15} -207.689 q^{17} +569.960 q^{19} -223.600i q^{21} -371.198i q^{23} -589.133 q^{25} -140.296 q^{27} +423.911i q^{29} +1174.18i q^{31} -339.199 q^{33} +1499.42 q^{35} +1448.13i q^{37} +514.764i q^{39} +265.822 q^{41} +699.900 q^{43} -940.799i q^{45} +1250.00i q^{47} +549.266 q^{49} +1079.18 q^{51} -787.645i q^{53} -2274.61i q^{55} -2961.60 q^{57} -3012.53 q^{59} +4519.33i q^{61} +1161.86i q^{63} -3451.91 q^{65} +5215.70 q^{67} +1928.80i q^{69} -4693.16i q^{71} +5404.92 q^{73} +3061.22 q^{75} +2809.07i q^{77} -7453.44i q^{79} +729.000 q^{81} -8950.92 q^{83} +7236.79i q^{85} -2202.71i q^{87} +616.496 q^{89} +4263.01 q^{91} -6101.20i q^{93} -19859.9i q^{95} +13723.1 q^{97} +1762.53 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 216 q^{9}+O(q^{10}) \) 8 * q + 216 * q^9	\( 8 q + 216 q^{9} - 1200 q^{17} - 1944 q^{25} + 1440 q^{33} - 1104 q^{41} - 1144 q^{49} - 11232 q^{57} - 36384 q^{65} - 17680 q^{73} + 5832 q^{81} + 50160 q^{89} + 46096 q^{97}+O(q^{100}) \) 8 * q + 216 * q^9 - 1200 * q^17 - 1944 * q^25 + 1440 * q^33 - 1104 * q^41 - 1144 * q^49 - 11232 * q^57 - 36384 * q^65 - 17680 * q^73 + 5832 * q^81 + 50160 * q^89 + 46096 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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