LMFDB - Embedded newform 882.5.b.i.197.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,Mod(197,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("882.197");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	882.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:	$91.1723074400$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4494128644096.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 967x^{4} + 279841 $ x^8 + 967*x^4 + 279841
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.2
Root			$3.72624 + 3.01913i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.i.197.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -27.6182i q^{5} +22.6274i q^{8} +O(q^{10})$	$q-2.82843i q^{2} -8.00000 q^{4} -27.6182i q^{5} +22.6274i q^{8} -78.1160 q^{10} -68.7565i q^{11} -234.014 q^{13} +64.0000 q^{16} -232.091i q^{17} -358.259 q^{19} +220.945i q^{20} -194.473 q^{22} +673.885i q^{23} -137.764 q^{25} +661.891i q^{26} -791.599i q^{29} +705.898 q^{31} -181.019i q^{32} -656.452 q^{34} -1377.96 q^{37} +1013.31i q^{38} +624.928 q^{40} +56.5270i q^{41} +263.382 q^{43} +550.052i q^{44} +1906.04 q^{46} +2287.31i q^{47} +389.654i q^{50} +1872.11 q^{52} -2325.76i q^{53} -1898.93 q^{55} -2238.98 q^{58} +2058.36i q^{59} -5583.42 q^{61} -1996.58i q^{62} -512.000 q^{64} +6463.03i q^{65} -5836.14 q^{67} +1856.73i q^{68} +8126.58i q^{71} +2198.92 q^{73} +3897.47i q^{74} +2866.07 q^{76} +10566.5 q^{79} -1767.56i q^{80} +159.882 q^{82} -7943.31i q^{83} -6409.93 q^{85} -744.955i q^{86} +1555.78 q^{88} -6287.14i q^{89} -5391.08i q^{92} +6469.49 q^{94} +9894.45i q^{95} -852.642 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} + 512 q^{16} - 640 q^{22} - 1560 q^{25} - 1408 q^{37} - 10256 q^{43} - 320 q^{46} - 512 q^{58} - 4096 q^{64} - 9600 q^{67} + 17680 q^{79} - 32048 q^{85} + 5120 q^{88}+O(q^{100}) $ 8 * q - 64 * q^4 + 512 * q^16 - 640 * q^22 - 1560 * q^25 - 1408 * q^37 - 10256 * q^43 - 320 * q^46 - 512 * q^58 - 4096 * q^64 - 9600 * q^67 + 17680 * q^79 - 32048 * q^85 + 5120 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 27.6182i	− 1.10473i	−0.833603	−	0.552364i	$-0.813726\pi$
$5$	− 27.6182i	− 1.10473i	0.833603	−	0.552364i	$-0.186274\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−78.1160	−0.781160
$11$	− 68.7565i	− 0.568235i	−0.958789	−	0.284118i	$-0.908299\pi$
$11$	− 68.7565i	− 0.568235i	0.958789	−	0.284118i	$-0.0917006\pi$
$12$	0	0
$13$	−234.014	−1.38470	−0.692348	−	0.721563i	$-0.743424\pi$
$13$	−234.014	−1.38470	−0.692348	+	0.721563i	$0.743424\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 232.091i	− 0.803083i	−0.915841	−	0.401541i	$-0.868475\pi$
$17$	− 232.091i	− 0.803083i	0.915841	−	0.401541i	$-0.131525\pi$
$18$	0	0
$19$	−358.259	−0.992407	−0.496203	−	0.868206i	$-0.665273\pi$
$19$	−358.259	−0.992407	−0.496203	+	0.868206i	$0.665273\pi$
$20$	220.945i	0.552364i
$21$	0	0
$22$	−194.473	−0.401803
$23$	673.885i	1.27389i	0.770911	+	0.636943i	$0.219801\pi$
$23$	673.885i	1.27389i	−0.770911	+	0.636943i	$0.780199\pi$
$24$	0	0
$25$	−137.764	−0.220422
$26$	661.891i	0.979128i
$27$	0	0
$28$	0	0
$29$	− 791.599i	− 0.941260i	−0.882331	−	0.470630i	$-0.844027\pi$
$29$	− 791.599i	− 0.941260i	0.882331	−	0.470630i	$-0.155973\pi$
$30$	0	0
$31$	705.898	0.734546	0.367273	−	0.930113i	$-0.380292\pi$
$31$	705.898	0.734546	0.367273	+	0.930113i	$0.380292\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−656.452	−0.567865
$35$	0	0
$36$	0	0
$37$	−1377.96	−1.00655	−0.503274	−	0.864127i	$-0.667871\pi$
$37$	−1377.96	−1.00655	−0.503274	+	0.864127i	$0.667871\pi$
$38$	1013.31i	0.701737i
$39$	0	0
$40$	624.928	0.390580
$41$	56.5270i	0.0336270i	0.999859	+	0.0168135i	$0.00535215\pi$
$41$	56.5270i	0.0336270i	−0.999859	+	0.0168135i	$0.994648\pi$
$42$	0	0
$43$	263.382	0.142445	0.0712227	−	0.997460i	$-0.477310\pi$
$43$	263.382	0.142445	0.0712227	+	0.997460i	$0.477310\pi$
$44$	550.052i	0.284118i
$45$	0	0
$46$	1906.04	0.900773
$47$	2287.31i	1.03545i	0.855547	+	0.517725i	$0.173221\pi$
$47$	2287.31i	1.03545i	−0.855547	+	0.517725i	$0.826779\pi$
$48$	0	0
$49$	0	0
$50$	389.654i	0.155862i
$51$	0	0
$52$	1872.11	0.692348
$53$	− 2325.76i	− 0.827969i	−0.910284	−	0.413984i	$-0.864137\pi$
$53$	− 2325.76i	− 0.827969i	0.910284	−	0.413984i	$-0.135863\pi$
$54$	0	0
$55$	−1898.93	−0.627745
$56$	0	0
$57$	0	0
$58$	−2238.98	−0.665571
$59$	2058.36i	0.591314i	0.955294	+	0.295657i	$0.0955385\pi$
$59$	2058.36i	0.591314i	−0.955294	+	0.295657i	$0.904461\pi$
$60$	0	0
$61$	−5583.42	−1.50052	−0.750258	−	0.661146i	$-0.770071\pi$
$61$	−5583.42	−1.50052	−0.750258	+	0.661146i	$0.770071\pi$
$62$	− 1996.58i	− 0.519402i
$63$	0	0
$64$	−512.000	−0.125000
$65$	6463.03i	1.52971i
$66$	0	0
$67$	−5836.14	−1.30010	−0.650050	−	0.759892i	$-0.725252\pi$
$67$	−5836.14	−1.30010	−0.650050	+	0.759892i	$0.725252\pi$
$68$	1856.73i	0.401541i
$69$	0	0
$70$	0	0
$71$	8126.58i	1.61210i	0.591849	+	0.806049i	$0.298398\pi$
$71$	8126.58i	1.61210i	−0.591849	+	0.806049i	$0.701602\pi$
$72$	0	0
$73$	2198.92	0.412633	0.206317	−	0.978485i	$-0.433852\pi$
$73$	2198.92	0.412633	0.206317	+	0.978485i	$0.433852\pi$
$74$	3897.47i	0.711736i
$75$	0	0
$76$	2866.07	0.496203
$77$	0	0
$78$	0	0
$79$	10566.5	1.69308	0.846540	−	0.532326i	$-0.178682\pi$
$79$	10566.5	1.69308	0.846540	+	0.532326i	$0.178682\pi$
$80$	− 1767.56i	− 0.276182i
$81$	0	0
$82$	159.882	0.0237779
$83$	− 7943.31i	− 1.15304i	−0.817082	−	0.576521i	$-0.804410\pi$
$83$	− 7943.31i	− 1.15304i	0.817082	−	0.576521i	$-0.195590\pi$
$84$	0	0
$85$	−6409.93	−0.887187
$86$	− 744.955i	− 0.100724i
$87$	0	0
$88$	1555.78	0.200902
$89$	− 6287.14i	− 0.793731i	−0.917877	−	0.396866i	$-0.870098\pi$
$89$	− 6287.14i	− 0.793731i	0.917877	−	0.396866i	$-0.129902\pi$
$90$	0	0
$91$	0	0
$92$	− 5391.08i	− 0.636943i
$93$	0	0
$94$	6469.49	0.732174
$95$	9894.45i	1.09634i
$96$	0	0
$97$	−852.642	−0.0906199	−0.0453099	−	0.998973i	$-0.514428\pi$
$97$	−852.642	−0.0906199	−0.0453099	+	0.998973i	$0.514428\pi$
$98$	0	0
$99$	0	0
$100$	1102.11	0.110211
$101$	− 7566.93i	− 0.741783i	−0.928676	−	0.370891i	$-0.879052\pi$
$101$	− 7566.93i	− 0.741783i	0.928676	−	0.370891i	$-0.120948\pi$
$102$	0	0
$103$	−9887.74	−0.932015	−0.466007	−	0.884781i	$-0.654308\pi$
$103$	−9887.74	−0.932015	−0.466007	+	0.884781i	$0.654308\pi$
$104$	− 5295.13i	− 0.489564i
$105$	0	0
$106$	−6578.25	−0.585462
$107$	802.041i	0.0700534i	0.999386	+	0.0350267i	$0.0111516\pi$
$107$	802.041i	0.0700534i	−0.999386	+	0.0350267i	$0.988848\pi$
$108$	0	0
$109$	16648.8	1.40129	0.700647	−	0.713509i	$-0.252895\pi$
$109$	16648.8	1.40129	0.700647	+	0.713509i	$0.252895\pi$
$110$	5370.98i	0.443883i
$111$	0	0
$112$	0	0
$113$	23326.5i	1.82681i	0.407053	+	0.913405i	$0.366556\pi$
$113$	23326.5i	1.82681i	−0.407053	+	0.913405i	$0.633444\pi$
$114$	0	0
$115$	18611.5	1.40730
$116$	6332.80i	0.470630i
$117$	0	0
$118$	5821.93	0.418122
$119$	0	0
$120$	0	0
$121$	9913.55	0.677109
$122$	15792.3i	1.06102i
$123$	0	0
$124$	−5647.19	−0.367273
$125$	− 13456.6i	− 0.861221i
$126$	0	0
$127$	2986.40	0.185157	0.0925786	−	0.995705i	$-0.470489\pi$
$127$	2986.40	0.185157	0.0925786	+	0.995705i	$0.470489\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	18280.2	1.08167
$131$	6240.58i	0.363649i	0.983331	+	0.181825i	$0.0582003\pi$
$131$	6240.58i	0.363649i	−0.983331	+	0.181825i	$0.941800\pi$
$132$	0	0
$133$	0	0
$134$	16507.1i	0.919309i
$135$	0	0
$136$	5251.62	0.283933
$137$	19860.2i	1.05814i	0.848579	+	0.529069i	$0.177459\pi$
$137$	19860.2i	1.05814i	−0.848579	+	0.529069i	$0.822541\pi$
$138$	0	0
$139$	7532.10	0.389840	0.194920	−	0.980819i	$-0.437555\pi$
$139$	7532.10	0.389840	0.194920	+	0.980819i	$0.437555\pi$
$140$	0	0
$141$	0	0
$142$	22985.5	1.13993
$143$	16090.0i	0.786834i
$144$	0	0
$145$	−21862.5	−1.03984
$146$	− 6219.49i	− 0.291776i
$147$	0	0
$148$	11023.7	0.503274
$149$	30207.3i	1.36063i	0.732920	+	0.680314i	$0.238157\pi$
$149$	30207.3i	1.36063i	−0.732920	+	0.680314i	$0.761843\pi$
$150$	0	0
$151$	−9611.96	−0.421559	−0.210779	−	0.977534i	$-0.567600\pi$
$151$	−9611.96	−0.421559	−0.210779	+	0.977534i	$0.567600\pi$
$152$	− 8106.47i	− 0.350869i
$153$	0	0
$154$	0	0
$155$	− 19495.6i	− 0.811472i
$156$	0	0
$157$	42834.5	1.73778	0.868890	−	0.495006i	$-0.164834\pi$
$157$	42834.5	1.73778	0.868890	+	0.495006i	$0.164834\pi$
$158$	− 29886.6i	− 1.19719i
$159$	0	0
$160$	−4999.42	−0.195290
$161$	0	0
$162$	0	0
$163$	26357.2	0.992030	0.496015	−	0.868314i	$-0.334796\pi$
$163$	26357.2	0.992030	0.496015	+	0.868314i	$0.334796\pi$
$164$	− 452.216i	− 0.0168135i
$165$	0	0
$166$	−22467.1	−0.815324
$167$	30744.5i	1.10239i	0.834378	+	0.551193i	$0.185827\pi$
$167$	30744.5i	1.10239i	−0.834378	+	0.551193i	$0.814173\pi$
$168$	0	0
$169$	26201.4	0.917385
$170$	18130.0i	0.627336i
$171$	0	0
$172$	−2107.05	−0.0712227
$173$	− 34960.7i	− 1.16812i	−0.811710	−	0.584060i	$-0.801463\pi$
$173$	− 34960.7i	− 1.16812i	0.811710	−	0.584060i	$-0.198537\pi$
$174$	0	0
$175$	0	0
$176$	− 4400.41i	− 0.142059i
$177$	0	0
$178$	−17782.7	−0.561253
$179$	29486.2i	0.920265i	0.887850	+	0.460132i	$0.152198\pi$
$179$	29486.2i	0.920265i	−0.887850	+	0.460132i	$0.847802\pi$
$180$	0	0
$181$	−55075.2	−1.68112	−0.840560	−	0.541719i	$-0.817774\pi$
$181$	−55075.2	−1.68112	−0.840560	+	0.541719i	$0.817774\pi$
$182$	0	0
$183$	0	0
$184$	−15248.3	−0.450387
$185$	38056.8i	1.11196i
$186$	0	0
$187$	−15957.8	−0.456340
$188$	− 18298.5i	− 0.517725i
$189$	0	0
$190$	27985.7	0.775228
$191$	68263.7i	1.87121i	0.353047	+	0.935606i	$0.385146\pi$
$191$	68263.7i	1.87121i	−0.353047	+	0.935606i	$0.614854\pi$
$192$	0	0
$193$	61260.9	1.64463	0.822316	−	0.569032i	$-0.192682\pi$
$193$	61260.9	1.64463	0.822316	+	0.569032i	$0.192682\pi$
$194$	2411.64i	0.0640779i
$195$	0	0
$196$	0	0
$197$	8850.17i	0.228044i	0.993478	+	0.114022i	$0.0363735\pi$
$197$	8850.17i	0.228044i	−0.993478	+	0.114022i	$0.963627\pi$
$198$	0	0
$199$	63447.4	1.60217	0.801083	−	0.598553i	$-0.204258\pi$
$199$	63447.4	1.60217	0.801083	+	0.598553i	$0.204258\pi$
$200$	− 3117.24i	− 0.0779309i
$201$	0	0
$202$	−21402.5	−0.524520
$203$	0	0
$204$	0	0
$205$	1561.17	0.0371487
$206$	27966.8i	0.659034i
$207$	0	0
$208$	−14976.9	−0.346174
$209$	24632.6i	0.563921i
$210$	0	0
$211$	−67144.1	−1.50814	−0.754072	−	0.656792i	$-0.771913\pi$
$211$	−67144.1	−1.50814	−0.754072	+	0.656792i	$0.771913\pi$
$212$	18606.1i	0.413984i
$213$	0	0
$214$	2268.52	0.0495352
$215$	− 7274.12i	− 0.157363i
$216$	0	0
$217$	0	0
$218$	− 47089.8i	− 0.990864i
$219$	0	0
$220$	15191.4	0.313873
$221$	54312.5i	1.11203i
$222$	0	0
$223$	−90939.4	−1.82870	−0.914350	−	0.404925i	$-0.867298\pi$
$223$	−90939.4	−1.82870	−0.914350	+	0.404925i	$0.867298\pi$
$224$	0	0
$225$	0	0
$226$	65977.4	1.29175
$227$	67084.1i	1.30187i	0.759133	+	0.650935i	$0.225623\pi$
$227$	67084.1i	1.30187i	−0.759133	+	0.650935i	$0.774377\pi$
$228$	0	0
$229$	53029.6	1.01122	0.505612	−	0.862761i	$-0.331267\pi$
$229$	53029.6	1.01122	0.505612	+	0.862761i	$0.331267\pi$
$230$	− 52641.2i	− 0.995108i
$231$	0	0
$232$	17911.9	0.332786
$233$	− 52122.9i	− 0.960101i	−0.877241	−	0.480050i	$-0.840618\pi$
$233$	− 52122.9i	− 0.960101i	0.877241	−	0.480050i	$-0.159382\pi$
$234$	0	0
$235$	63171.3	1.14389
$236$	− 16466.9i	− 0.295657i
$237$	0	0
$238$	0	0
$239$	− 27196.3i	− 0.476117i	−0.971251	−	0.238059i	$-0.923489\pi$
$239$	− 27196.3i	− 0.476117i	0.971251	−	0.238059i	$-0.0765111\pi$
$240$	0	0
$241$	37110.2	0.638938	0.319469	−	0.947597i	$-0.396495\pi$
$241$	37110.2	0.638938	0.319469	+	0.947597i	$0.396495\pi$
$242$	− 28039.7i	− 0.478788i
$243$	0	0
$244$	44667.3	0.750258
$245$	0	0
$246$	0	0
$247$	83837.5	1.37418
$248$	15972.7i	0.259701i
$249$	0	0
$250$	−38061.0	−0.608975
$251$	38179.3i	0.606011i	0.952989	+	0.303005i	$0.0979900\pi$
$251$	38179.3i	0.606011i	−0.952989	+	0.303005i	$0.902010\pi$
$252$	0	0
$253$	46334.0	0.723867
$254$	− 8446.82i	− 0.130926i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 63380.8i	− 0.959603i	−0.877377	−	0.479801i	$-0.840709\pi$
$257$	− 63380.8i	− 0.959603i	0.877377	−	0.479801i	$-0.159291\pi$
$258$	0	0
$259$	0	0
$260$	− 51704.3i	− 0.764856i
$261$	0	0
$262$	17651.0	0.257139
$263$	− 27453.2i	− 0.396901i	−0.980111	−	0.198450i	$-0.936409\pi$
$263$	− 27453.2i	− 0.396901i	0.980111	−	0.198450i	$-0.0635908\pi$
$264$	0	0
$265$	−64233.4	−0.914679
$266$	0	0
$267$	0	0
$268$	46689.2	0.650050
$269$	20657.3i	0.285475i	0.989761	+	0.142737i	$0.0455905\pi$
$269$	20657.3i	0.285475i	−0.989761	+	0.142737i	$0.954410\pi$
$270$	0	0
$271$	70407.7	0.958698	0.479349	−	0.877624i	$-0.340873\pi$
$271$	70407.7	0.958698	0.479349	+	0.877624i	$0.340873\pi$
$272$	− 14853.8i	− 0.200771i
$273$	0	0
$274$	56173.1	0.748216
$275$	9472.14i	0.125251i
$276$	0	0
$277$	−126579.	−1.64969	−0.824844	−	0.565361i	$-0.808737\pi$
$277$	−126579.	−1.64969	−0.824844	+	0.565361i	$0.808737\pi$
$278$	− 21304.0i	− 0.275659i
$279$	0	0
$280$	0	0
$281$	40285.6i	0.510196i	0.966915	+	0.255098i	$0.0821077\pi$
$281$	40285.6i	0.510196i	−0.966915	+	0.255098i	$0.917892\pi$
$282$	0	0
$283$	75484.6	0.942509	0.471254	−	0.881997i	$-0.343801\pi$
$283$	75484.6	0.942509	0.471254	+	0.881997i	$0.343801\pi$
$284$	− 65012.7i	− 0.806049i
$285$	0	0
$286$	45509.3	0.556375
$287$	0	0
$288$	0	0
$289$	29654.8	0.355058
$290$	61836.6i	0.735274i
$291$	0	0
$292$	−17591.4	−0.206317
$293$	− 71196.2i	− 0.829319i	−0.909977	−	0.414660i	$-0.863901\pi$
$293$	− 71196.2i	− 0.829319i	0.909977	−	0.414660i	$-0.136099\pi$
$294$	0	0
$295$	56848.2	0.653240
$296$	− 31179.8i	− 0.355868i
$297$	0	0
$298$	85439.2	0.962110
$299$	− 157698.i	− 1.76395i
$300$	0	0
$301$	0	0
$302$	27186.7i	0.298087i
$303$	0	0
$304$	−22928.6	−0.248102
$305$	154204.i	1.65766i
$306$	0	0
$307$	53894.7	0.571833	0.285917	−	0.958255i	$-0.407702\pi$
$307$	53894.7	0.571833	0.285917	+	0.958255i	$0.407702\pi$
$308$	0	0
$309$	0	0
$310$	−55142.0	−0.573798
$311$	− 122336.i	− 1.26484i	−0.774627	−	0.632419i	$-0.782062\pi$
$311$	− 122336.i	− 1.26484i	0.774627	−	0.632419i	$-0.217938\pi$
$312$	0	0
$313$	−134653.	−1.37445	−0.687225	−	0.726445i	$-0.741171\pi$
$313$	−134653.	−1.37445	−0.687225	+	0.726445i	$0.741171\pi$
$314$	− 121154.i	− 1.22880i
$315$	0	0
$316$	−84532.1	−0.846540
$317$	− 99383.3i	− 0.988997i	−0.869179	−	0.494498i	$-0.835352\pi$
$317$	− 99383.3i	− 0.988997i	0.869179	−	0.494498i	$-0.164648\pi$
$318$	0	0
$319$	−54427.6	−0.534857
$320$	14140.5i	0.138091i
$321$	0	0
$322$	0	0
$323$	83148.6i	0.796985i
$324$	0	0
$325$	32238.6	0.305217
$326$	− 74549.5i	− 0.701471i
$327$	0	0
$328$	−1279.06	−0.0118889
$329$	0	0
$330$	0	0
$331$	183598.	1.67576	0.837879	−	0.545856i	$-0.183795\pi$
$331$	183598.	1.67576	0.837879	+	0.545856i	$0.183795\pi$
$332$	63546.5i	0.576521i
$333$	0	0
$334$	86958.5	0.779505
$335$	161184.i	1.43625i
$336$	0	0
$337$	−33778.0	−0.297423	−0.148712	−	0.988881i	$-0.547513\pi$
$337$	−33778.0	−0.297423	−0.148712	+	0.988881i	$0.547513\pi$
$338$	− 74108.8i	− 0.648689i
$339$	0	0
$340$	51279.4	0.443594
$341$	− 48535.1i	− 0.417395i
$342$	0	0
$343$	0	0
$344$	5959.64i	0.0503620i
$345$	0	0
$346$	−98883.7	−0.825986
$347$	106731.i	0.886406i	0.896421	+	0.443203i	$0.146158\pi$
$347$	106731.i	0.886406i	−0.896421	+	0.443203i	$0.853842\pi$
$348$	0	0
$349$	−108939.	−0.894398	−0.447199	−	0.894434i	$-0.647578\pi$
$349$	−108939.	−0.894398	−0.447199	+	0.894434i	$0.647578\pi$
$350$	0	0
$351$	0	0
$352$	−12446.3	−0.100451
$353$	35761.6i	0.286991i	0.989651	+	0.143495i	$0.0458342\pi$
$353$	35761.6i	0.286991i	−0.989651	+	0.143495i	$0.954166\pi$
$354$	0	0
$355$	224441.	1.78093
$356$	50297.1i	0.396866i
$357$	0	0
$358$	83399.6	0.650725
$359$	163816.i	1.27106i	0.772075	+	0.635531i	$0.219219\pi$
$359$	163816.i	1.27106i	−0.772075	+	0.635531i	$0.780781\pi$
$360$	0	0
$361$	−1971.63	−0.0151290
$362$	155776.i	1.18873i
$363$	0	0
$364$	0	0
$365$	− 60730.2i	− 0.455847i
$366$	0	0
$367$	−82221.2	−0.610452	−0.305226	−	0.952280i	$-0.598732\pi$
$367$	−82221.2	−0.610452	−0.305226	+	0.952280i	$0.598732\pi$
$368$	43128.7i	0.318471i
$369$	0	0
$370$	107641.	0.786275
$371$	0	0
$372$	0	0
$373$	−274694.	−1.97438	−0.987192	−	0.159537i	$-0.949000\pi$
$373$	−274694.	−1.97438	−0.987192	+	0.159537i	$0.949000\pi$
$374$	45135.3i	0.322681i
$375$	0	0
$376$	−51755.9	−0.366087
$377$	185245.i	1.30336i
$378$	0	0
$379$	−136008.	−0.946858	−0.473429	−	0.880832i	$-0.656984\pi$
$379$	−136008.	−0.946858	−0.473429	+	0.880832i	$0.656984\pi$
$380$	− 79155.6i	− 0.548169i
$381$	0	0
$382$	193079.	1.32315
$383$	55694.5i	0.379678i	0.981815	+	0.189839i	$0.0607966\pi$
$383$	55694.5i	0.379678i	−0.981815	+	0.189839i	$0.939203\pi$
$384$	0	0
$385$	0	0
$386$	− 173272.i	− 1.16293i
$387$	0	0
$388$	6821.14	0.0453099
$389$	− 7102.94i	− 0.0469395i	−0.999725	−	0.0234698i	$-0.992529\pi$
$389$	− 7102.94i	− 0.0469395i	0.999725	−	0.0234698i	$-0.00747135\pi$
$390$	0	0
$391$	156403.	1.02304
$392$	0	0
$393$	0	0
$394$	25032.1	0.161252
$395$	− 291828.i	− 1.87039i
$396$	0	0
$397$	−131128.	−0.831984	−0.415992	−	0.909368i	$-0.636566\pi$
$397$	−131128.	−0.831984	−0.415992	+	0.909368i	$0.636566\pi$
$398$	− 179456.i	− 1.13290i
$399$	0	0
$400$	−8816.87	−0.0551055
$401$	160110.i	0.995701i	0.867263	+	0.497851i	$0.165877\pi$
$401$	160110.i	0.995701i	−0.867263	+	0.497851i	$0.834123\pi$
$402$	0	0
$403$	−165190.	−1.01712
$404$	60535.4i	0.370891i
$405$	0	0
$406$	0	0
$407$	94743.9i	0.571956i
$408$	0	0
$409$	−244283.	−1.46031	−0.730156	−	0.683280i	$-0.760553\pi$
$409$	−244283.	−1.46031	−0.730156	+	0.683280i	$0.760553\pi$
$410$	− 4415.66i	− 0.0262681i
$411$	0	0
$412$	79102.0	0.466007
$413$	0	0
$414$	0	0
$415$	−219380.	−1.27380
$416$	42361.0i	0.244782i
$417$	0	0
$418$	69671.6	0.398752
$419$	62200.1i	0.354293i	0.984184	+	0.177147i	$0.0566867\pi$
$419$	62200.1i	0.354293i	−0.984184	+	0.177147i	$0.943313\pi$
$420$	0	0
$421$	−19878.2	−0.112153	−0.0560767	−	0.998426i	$-0.517859\pi$
$421$	−19878.2	−0.112153	−0.0560767	+	0.998426i	$0.517859\pi$
$422$	189912.i	1.06642i
$423$	0	0
$424$	52626.0	0.292731
$425$	31973.7i	0.177017i
$426$	0	0
$427$	0	0
$428$	− 6416.33i	− 0.0350267i
$429$	0	0
$430$	−20574.3	−0.111273
$431$	− 251651.i	− 1.35470i	−0.735660	−	0.677351i	$-0.763128\pi$
$431$	− 251651.i	− 1.35470i	0.735660	−	0.677351i	$-0.236872\pi$
$432$	0	0
$433$	−240935.	−1.28506	−0.642531	−	0.766259i	$-0.722116\pi$
$433$	−240935.	−1.28506	−0.642531	+	0.766259i	$0.722116\pi$
$434$	0	0
$435$	0	0
$436$	−133190.	−0.700647
$437$	− 241425.i	− 1.26421i
$438$	0	0
$439$	−302222.	−1.56819	−0.784093	−	0.620643i	$-0.786872\pi$
$439$	−302222.	−1.56819	−0.784093	+	0.620643i	$0.786872\pi$
$440$	− 42967.9i	− 0.221941i
$441$	0	0
$442$	153619.	0.786321
$443$	− 229704.i	− 1.17047i	−0.810863	−	0.585237i	$-0.801002\pi$
$443$	− 229704.i	− 1.17047i	0.810863	−	0.585237i	$-0.198998\pi$
$444$	0	0
$445$	−173639.	−0.876856
$446$	257216.i	1.29309i
$447$	0	0
$448$	0	0
$449$	298491.i	1.48060i	0.672274	+	0.740302i	$0.265318\pi$
$449$	298491.i	1.48060i	−0.672274	+	0.740302i	$0.734682\pi$
$450$	0	0
$451$	3886.60	0.0191080
$452$	− 186612.i	− 0.913405i
$453$	0	0
$454$	189743.	0.920562
$455$	0	0
$456$	0	0
$457$	−154255.	−0.738597	−0.369299	−	0.929311i	$-0.620402\pi$
$457$	−154255.	−0.738597	−0.369299	+	0.929311i	$0.620402\pi$
$458$	− 149990.i	− 0.715044i
$459$	0	0
$460$	−148892.	−0.703648
$461$	127462.i	0.599760i	0.953977	+	0.299880i	$0.0969467\pi$
$461$	127462.i	0.599760i	−0.953977	+	0.299880i	$0.903053\pi$
$462$	0	0
$463$	15503.0	0.0723194	0.0361597	−	0.999346i	$-0.488488\pi$
$463$	15503.0	0.0723194	0.0361597	+	0.999346i	$0.488488\pi$
$464$	− 50662.4i	− 0.235315i
$465$	0	0
$466$	−147426.	−0.678894
$467$	278201.i	1.27563i	0.770190	+	0.637815i	$0.220162\pi$
$467$	278201.i	1.27563i	−0.770190	+	0.637815i	$0.779838\pi$
$468$	0	0
$469$	0	0
$470$	− 178675.i	− 0.808852i
$471$	0	0
$472$	−46575.4	−0.209061
$473$	− 18109.2i	− 0.0809425i
$474$	0	0
$475$	49355.0	0.218748
$476$	0	0
$477$	0	0
$478$	−76922.8	−0.336666
$479$	37164.5i	0.161978i	0.996715	+	0.0809891i	$0.0258079\pi$
$479$	37164.5i	0.161978i	−0.996715	+	0.0809891i	$0.974192\pi$
$480$	0	0
$481$	322462.	1.39376
$482$	− 104963.i	− 0.451797i
$483$	0	0
$484$	−79308.4	−0.338554
$485$	23548.4i	0.100110i
$486$	0	0
$487$	−230932.	−0.973701	−0.486851	−	0.873485i	$-0.661854\pi$
$487$	−230932.	−0.973701	−0.486851	+	0.873485i	$0.661854\pi$
$488$	− 126338.i	− 0.530512i
$489$	0	0
$490$	0	0
$491$	85126.0i	0.353101i	0.984292	+	0.176551i	$0.0564939\pi$
$491$	85126.0i	0.353101i	−0.984292	+	0.176551i	$0.943506\pi$
$492$	0	0
$493$	−183723.	−0.755909
$494$	− 237128.i	− 0.971694i
$495$	0	0
$496$	45177.5	0.183636
$497$	0	0
$498$	0	0
$499$	−214349.	−0.860835	−0.430418	−	0.902630i	$-0.641634\pi$
$499$	−214349.	−0.860835	−0.430418	+	0.902630i	$0.641634\pi$
$500$	107653.i	0.430611i
$501$	0	0
$502$	107987.	0.428514
$503$	455331.i	1.79966i	0.436236	+	0.899832i	$0.356311\pi$
$503$	455331.i	1.79966i	−0.436236	+	0.899832i	$0.643689\pi$
$504$	0	0
$505$	−208985.	−0.819467
$506$	− 131052.i	− 0.511851i
$507$	0	0
$508$	−23891.2	−0.0925786
$509$	229262.i	0.884903i	0.896792	+	0.442452i	$0.145891\pi$
$509$	229262.i	0.884903i	−0.896792	+	0.442452i	$0.854109\pi$
$510$	0	0
$511$	0	0
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−179268.	−0.678542
$515$	273081.i	1.02962i
$516$	0	0
$517$	157267.	0.588379
$518$	0	0
$519$	0	0
$520$	−146242.	−0.540835
$521$	− 226306.i	− 0.833722i	−0.908970	−	0.416861i	$-0.863130\pi$
$521$	− 226306.i	− 0.833722i	0.908970	−	0.416861i	$-0.136870\pi$
$522$	0	0
$523$	243589.	0.890543	0.445271	−	0.895396i	$-0.353107\pi$
$523$	243589.	0.890543	0.445271	+	0.895396i	$0.353107\pi$
$524$	− 49924.6i	− 0.181825i
$525$	0	0
$526$	−77649.4	−0.280651
$527$	− 163833.i	− 0.589901i
$528$	0	0
$529$	−174281.	−0.622785
$530$	181679.i	0.646776i
$531$	0	0
$532$	0	0
$533$	− 13228.1i	− 0.0465632i
$534$	0	0
$535$	22150.9	0.0773899
$536$	− 132057.i	− 0.459654i
$537$	0	0
$538$	58427.5	0.201861
$539$	0	0
$540$	0	0
$541$	−215267.	−0.735502	−0.367751	−	0.929924i	$-0.619872\pi$
$541$	−215267.	−0.735502	−0.367751	+	0.929924i	$0.619872\pi$
$542$	− 199143.i	− 0.677902i
$543$	0	0
$544$	−42012.9	−0.141966
$545$	− 459808.i	− 1.54805i
$546$	0	0
$547$	−426624.	−1.42584	−0.712920	−	0.701246i	$-0.752628\pi$
$547$	−426624.	−1.42584	−0.712920	+	0.701246i	$0.752628\pi$
$548$	− 158881.i	− 0.529069i
$549$	0	0
$550$	26791.3	0.0885662
$551$	283597.i	0.934112i
$552$	0	0
$553$	0	0
$554$	358019.i	1.16651i
$555$	0	0
$556$	−60256.8	−0.194920
$557$	259435.i	0.836216i	0.908397	+	0.418108i	$0.137307\pi$
$557$	259435.i	0.836216i	−0.908397	+	0.418108i	$0.862693\pi$
$558$	0	0
$559$	−61634.9	−0.197244
$560$	0	0
$561$	0	0
$562$	113945.	0.360763
$563$	− 466206.i	− 1.47082i	−0.677620	−	0.735412i	$-0.736989\pi$
$563$	− 466206.i	− 1.47082i	0.677620	−	0.735412i	$-0.263011\pi$
$564$	0	0
$565$	644236.	2.01813
$566$	− 213503.i	− 0.666454i
$567$	0	0
$568$	−183884.	−0.569963
$569$	− 135184.i	− 0.417543i	−0.977964	−	0.208772i	$-0.933053\pi$
$569$	− 135184.i	− 0.417543i	0.977964	−	0.208772i	$-0.0669465\pi$
$570$	0	0
$571$	−194732.	−0.597262	−0.298631	−	0.954369i	$-0.596530\pi$
$571$	−194732.	−0.597262	−0.298631	+	0.954369i	$0.596530\pi$
$572$	− 128720.i	− 0.393417i
$573$	0	0
$574$	0	0
$575$	− 92836.9i	− 0.280792i
$576$	0	0
$577$	151193.	0.454130	0.227065	−	0.973880i	$-0.427087\pi$
$577$	151193.	0.454130	0.227065	+	0.973880i	$0.427087\pi$
$578$	− 83876.5i	− 0.251064i
$579$	0	0
$580$	174900.	0.519918
$581$	0	0
$582$	0	0
$583$	−159911.	−0.470481
$584$	49755.9i	0.145888i
$585$	0	0
$586$	−201373.	−0.586417
$587$	− 339023.i	− 0.983903i	−0.870623	−	0.491952i	$-0.836284\pi$
$587$	− 339023.i	− 0.983903i	0.870623	−	0.491952i	$-0.163716\pi$
$588$	0	0
$589$	−252894.	−0.728968
$590$	− 160791.i	− 0.461911i
$591$	0	0
$592$	−88189.7	−0.251637
$593$	− 342139.i	− 0.972957i	−0.873693	−	0.486478i	$-0.838281\pi$
$593$	− 342139.i	− 0.972957i	0.873693	−	0.486478i	$-0.161719\pi$
$594$	0	0
$595$	0	0
$596$	− 241659.i	− 0.680314i
$597$	0	0
$598$	−446039.	−1.24730
$599$	19818.5i	0.0552355i	0.999619	+	0.0276177i	$0.00879212\pi$
$599$	19818.5i	0.0552355i	−0.999619	+	0.0276177i	$0.991208\pi$
$600$	0	0
$601$	496947.	1.37582	0.687909	−	0.725797i	$-0.258529\pi$
$601$	496947.	1.37582	0.687909	+	0.725797i	$0.258529\pi$
$602$	0	0
$603$	0	0
$604$	76895.7	0.210779
$605$	− 273794.i	− 0.748020i
$606$	0	0
$607$	56342.4	0.152918	0.0764589	−	0.997073i	$-0.475639\pi$
$607$	56342.4	0.152918	0.0764589	+	0.997073i	$0.475639\pi$
$608$	64851.8i	0.175434i
$609$	0	0
$610$	436154.	1.17214
$611$	− 535262.i	− 1.43378i
$612$	0	0
$613$	−77210.9	−0.205474	−0.102737	−	0.994709i	$-0.532760\pi$
$613$	−77210.9	−0.205474	−0.102737	+	0.994709i	$0.532760\pi$
$614$	− 152437.i	− 0.404347i
$615$	0	0
$616$	0	0
$617$	− 24653.6i	− 0.0647606i	−0.999476	−	0.0323803i	$-0.989691\pi$
$617$	− 24653.6i	− 0.0647606i	0.999476	−	0.0323803i	$-0.0103088\pi$
$618$	0	0
$619$	−339421.	−0.885843	−0.442922	−	0.896560i	$-0.646058\pi$
$619$	−339421.	−0.885843	−0.442922	+	0.896560i	$0.646058\pi$
$620$	155965.i	0.405736i
$621$	0	0
$622$	−346020.	−0.894375
$623$	0	0
$624$	0	0
$625$	−457748.	−1.17184
$626$	380858.i	0.971883i
$627$	0	0
$628$	−342676.	−0.868890
$629$	319813.i	0.808341i
$630$	0	0
$631$	282702.	0.710019	0.355009	−	0.934863i	$-0.384478\pi$
$631$	282702.	0.710019	0.355009	+	0.934863i	$0.384478\pi$
$632$	239093.i	0.598594i
$633$	0	0
$634$	−281098.	−0.699326
$635$	− 82478.9i	− 0.204548i
$636$	0	0
$637$	0	0
$638$	153944.i	0.378201i
$639$	0	0
$640$	39995.4	0.0976450
$641$	− 414696.i	− 1.00928i	−0.863329	−	0.504642i	$-0.831625\pi$
$641$	− 414696.i	− 1.00928i	0.863329	−	0.504642i	$-0.168375\pi$
$642$	0	0
$643$	−267343.	−0.646616	−0.323308	−	0.946294i	$-0.604795\pi$
$643$	−267343.	−0.646616	−0.323308	+	0.946294i	$0.604795\pi$
$644$	0	0
$645$	0	0
$646$	235180.	0.563553
$647$	180110.i	0.430257i	0.976586	+	0.215129i	$0.0690171\pi$
$647$	180110.i	0.430257i	−0.976586	+	0.215129i	$0.930983\pi$
$648$	0	0
$649$	141526.	0.336005
$650$	− 91184.5i	− 0.215821i
$651$	0	0
$652$	−210858.	−0.496015
$653$	10510.4i	0.0246486i	0.999924	+	0.0123243i	$0.00392304\pi$
$653$	10510.4i	0.0246486i	−0.999924	+	0.0123243i	$0.996077\pi$
$654$	0	0
$655$	172353.	0.401733
$656$	3617.73i	0.00840675i
$657$	0	0
$658$	0	0
$659$	− 463404.i	− 1.06706i	−0.845781	−	0.533531i	$-0.820865\pi$
$659$	− 463404.i	− 1.06706i	0.845781	−	0.533531i	$-0.179135\pi$
$660$	0	0
$661$	−820578.	−1.87809	−0.939047	−	0.343790i	$-0.888289\pi$
$661$	−820578.	−1.87809	−0.939047	+	0.343790i	$0.888289\pi$
$662$	− 519293.i	− 1.18494i
$663$	0	0
$664$	179737.	0.407662
$665$	0	0
$666$	0	0
$667$	533447.	1.19906
$668$	− 245956.i	− 0.551193i
$669$	0	0
$670$	455896.	1.01559
$671$	383896.i	0.852646i
$672$	0	0
$673$	140419.	0.310025	0.155012	−	0.987913i	$-0.450458\pi$
$673$	140419.	0.310025	0.155012	+	0.987913i	$0.450458\pi$
$674$	95538.7i	0.210310i
$675$	0	0
$676$	−209611.	−0.458692
$677$	− 258346.i	− 0.563670i	−0.959463	−	0.281835i	$-0.909057\pi$
$677$	− 258346.i	− 0.563670i	0.959463	−	0.281835i	$-0.0909431\pi$
$678$	0	0
$679$	0	0
$680$	− 145040.i	− 0.313668i
$681$	0	0
$682$	−137278.	−0.295143
$683$	55733.0i	0.119473i	0.998214	+	0.0597367i	$0.0190261\pi$
$683$	55733.0i	0.119473i	−0.998214	+	0.0597367i	$0.980974\pi$
$684$	0	0
$685$	548502.	1.16895
$686$	0	0
$687$	0	0
$688$	16856.4	0.0356113
$689$	544261.i	1.14649i
$690$	0	0
$691$	−240461.	−0.503603	−0.251802	−	0.967779i	$-0.581023\pi$
$691$	−240461.	−0.503603	−0.251802	+	0.967779i	$0.581023\pi$
$692$	279685.i	0.584060i
$693$	0	0
$694$	301881.	0.626783
$695$	− 208023.i	− 0.430667i
$696$	0	0
$697$	13119.4	0.0270053
$698$	308125.i	0.632435i
$699$	0	0
$700$	0	0
$701$	561901.i	1.14347i	0.820439	+	0.571734i	$0.193729\pi$
$701$	561901.i	1.14347i	−0.820439	+	0.571734i	$0.806271\pi$
$702$	0	0
$703$	493668.	0.998904
$704$	35203.3i	0.0710294i
$705$	0	0
$706$	101149.	0.202933
$707$	0	0
$708$	0	0
$709$	99133.1	0.197209	0.0986044	−	0.995127i	$-0.468562\pi$
$709$	99133.1	0.197209	0.0986044	+	0.995127i	$0.468562\pi$
$710$	− 634816.i	− 1.25931i
$711$	0	0
$712$	142262.	0.280626
$713$	475695.i	0.935727i
$714$	0	0
$715$	444375.	0.869236
$716$	− 235890.i	− 0.460132i
$717$	0	0
$718$	463341.	0.898777
$719$	203270.i	0.393202i	0.980484	+	0.196601i	$0.0629903\pi$
$719$	203270.i	0.393202i	−0.980484	+	0.196601i	$0.937010\pi$
$720$	0	0
$721$	0	0
$722$	5576.60i	0.0106978i
$723$	0	0
$724$	440601.	0.840560
$725$	109054.i	0.207474i
$726$	0	0
$727$	−767390.	−1.45194	−0.725968	−	0.687729i	$-0.758608\pi$
$727$	−767390.	−1.45194	−0.725968	+	0.687729i	$0.758608\pi$
$728$	0	0
$729$	0	0
$730$	−171771.	−0.322333
$731$	− 61128.4i	− 0.114395i
$732$	0	0
$733$	−212536.	−0.395572	−0.197786	−	0.980245i	$-0.563375\pi$
$733$	−212536.	−0.395572	−0.197786	+	0.980245i	$0.563375\pi$
$734$	232557.i	0.431655i
$735$	0	0
$736$	121986.	0.225193
$737$	401273.i	0.738762i
$738$	0	0
$739$	824475.	1.50969	0.754847	−	0.655901i	$-0.227711\pi$
$739$	824475.	1.50969	0.754847	+	0.655901i	$0.227711\pi$
$740$	− 304455.i	− 0.555980i
$741$	0	0
$742$	0	0
$743$	− 359935.i	− 0.651999i	−0.945370	−	0.326000i	$-0.894299\pi$
$743$	− 359935.i	− 0.651999i	0.945370	−	0.326000i	$-0.105701\pi$
$744$	0	0
$745$	834271.	1.50312
$746$	776952.i	1.39610i
$747$	0	0
$748$	127662.	0.228170
$749$	0	0
$750$	0	0
$751$	−863199.	−1.53049	−0.765246	−	0.643738i	$-0.777383\pi$
$751$	−863199.	−1.53049	−0.765246	+	0.643738i	$0.777383\pi$
$752$	146388.i	0.258862i
$753$	0	0
$754$	523952.	0.921614
$755$	265465.i	0.465707i
$756$	0	0
$757$	196829.	0.343477	0.171738	−	0.985143i	$-0.445062\pi$
$757$	196829.	0.343477	0.171738	+	0.985143i	$0.445062\pi$
$758$	384688.i	0.669530i
$759$	0	0
$760$	−223886.	−0.387614
$761$	− 137737.i	− 0.237839i	−0.992904	−	0.118919i	$-0.962057\pi$
$761$	− 137737.i	− 0.237839i	0.992904	−	0.118919i	$-0.0379430\pi$
$762$	0	0
$763$	0	0
$764$	− 546109.i	− 0.935606i
$765$	0	0
$766$	157528.	0.268473
$767$	− 481685.i	− 0.818790i
$768$	0	0
$769$	−118263.	−0.199985	−0.0999926	−	0.994988i	$-0.531882\pi$
$769$	−118263.	−0.199985	−0.0999926	+	0.994988i	$0.531882\pi$
$770$	0	0
$771$	0	0
$772$	−490087.	−0.822316
$773$	1.10986e6i	1.85742i	0.370807	+	0.928710i	$0.379081\pi$
$773$	1.10986e6i	1.85742i	−0.370807	+	0.928710i	$0.620919\pi$
$774$	0	0
$775$	−97247.1	−0.161910
$776$	− 19293.1i	− 0.0320390i
$777$	0	0
$778$	−20090.1	−0.0331913
$779$	− 20251.3i	− 0.0333717i
$780$	0	0
$781$	558755.	0.916051
$782$	− 442374.i	− 0.723395i
$783$	0	0
$784$	0	0
$785$	− 1.18301e6i	− 1.91977i
$786$	0	0
$787$	−821921.	−1.32703	−0.663514	−	0.748163i	$-0.730936\pi$
$787$	−821921.	−1.32703	−0.663514	+	0.748163i	$0.730936\pi$
$788$	− 70801.4i	− 0.114022i
$789$	0	0
$790$	−825413.	−1.32257
$791$	0	0
$792$	0	0
$793$	1.30660e6	2.07776
$794$	370887.i	0.588302i
$795$	0	0
$796$	−507579.	−0.801083
$797$	300705.i	0.473396i	0.971583	+	0.236698i	$0.0760652\pi$
$797$	300705.i	0.473396i	−0.971583	+	0.236698i	$0.923935\pi$
$798$	0	0
$799$	530864.	0.831552
$800$	24937.9i	0.0389654i
$801$	0	0
$802$	452859.	0.704067
$803$	− 151190.i	− 0.234473i
$804$	0	0
$805$	0	0
$806$	467228.i	0.719214i
$807$	0	0
$808$	171220.	0.262260
$809$	494584.i	0.755688i	0.925869	+	0.377844i	$0.123334\pi$
$809$	494584.i	0.755688i	−0.925869	+	0.377844i	$0.876666\pi$
$810$	0	0
$811$	−744159.	−1.13142	−0.565710	−	0.824604i	$-0.691398\pi$
$811$	−744159.	−1.13142	−0.565710	+	0.824604i	$0.691398\pi$
$812$	0	0
$813$	0	0
$814$	267976.	0.404434
$815$	− 727939.i	− 1.09592i
$816$	0	0
$817$	−94358.7	−0.141364
$818$	690936.i	1.03260i
$819$	0	0
$820$	−12489.4	−0.0185743
$821$	814616.i	1.20856i	0.796773	+	0.604278i	$0.206538\pi$
$821$	814616.i	1.20856i	−0.796773	+	0.604278i	$0.793462\pi$
$822$	0	0
$823$	−253130.	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$823$	−253130.	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$824$	− 223734.i	− 0.329517i
$825$	0	0
$826$	0	0
$827$	462238.i	0.675856i	0.941172	+	0.337928i	$0.109726\pi$
$827$	462238.i	0.675856i	−0.941172	+	0.337928i	$0.890274\pi$
$828$	0	0
$829$	−592960.	−0.862812	−0.431406	−	0.902158i	$-0.641982\pi$
$829$	−592960.	−0.862812	−0.431406	+	0.902158i	$0.641982\pi$
$830$	620499.i	0.900710i
$831$	0	0
$832$	119815.	0.173087
$833$	0	0
$834$	0	0
$835$	849106.	1.21784
$836$	− 197061.i	− 0.281960i
$837$	0	0
$838$	175928.	0.250523
$839$	− 1.16715e6i	− 1.65807i	−0.559195	−	0.829036i	$-0.688890\pi$
$839$	− 1.16715e6i	− 1.65807i	0.559195	−	0.829036i	$-0.311110\pi$
$840$	0	0
$841$	80651.3	0.114030
$842$	56224.0i	0.0793044i
$843$	0	0
$844$	537152.	0.754072
$845$	− 723636.i	− 1.01346i
$846$	0	0
$847$	0	0
$848$	− 148849.i	− 0.206992i
$849$	0	0
$850$	90435.2	0.125170
$851$	− 928590.i	− 1.28223i
$852$	0	0
$853$	562911.	0.773645	0.386822	−	0.922154i	$-0.373573\pi$
$853$	562911.	0.773645	0.386822	+	0.922154i	$0.373573\pi$
$854$	0	0
$855$	0	0
$856$	−18148.1	−0.0247676
$857$	1.11462e6i	1.51762i	0.651310	+	0.758812i	$0.274220\pi$
$857$	1.11462e6i	1.51762i	−0.651310	+	0.758812i	$0.725780\pi$
$858$	0	0
$859$	−769113.	−1.04233	−0.521163	−	0.853457i	$-0.674502\pi$
$859$	−769113.	−1.04233	−0.521163	+	0.853457i	$0.674502\pi$
$860$	58192.9i	0.0786816i
$861$	0	0
$862$	−711776.	−0.957919
$863$	− 290180.i	− 0.389624i	−0.980841	−	0.194812i	$-0.937590\pi$
$863$	− 290180.i	− 0.389624i	0.980841	−	0.194812i	$-0.0624097\pi$
$864$	0	0
$865$	−965550.	−1.29045
$866$	681468.i	0.908677i
$867$	0	0
$868$	0	0
$869$	− 726516.i	− 0.962068i
$870$	0	0
$871$	1.36574e6	1.80024
$872$	376719.i	0.495432i
$873$	0	0
$874$	−682854.	−0.893933
$875$	0	0
$876$	0	0
$877$	691767.	0.899417	0.449708	−	0.893175i	$-0.351528\pi$
$877$	691767.	0.899417	0.449708	+	0.893175i	$0.351528\pi$
$878$	854814.i	1.10888i
$879$	0	0
$880$	−121531.	−0.156936
$881$	− 812553.i	− 1.04689i	−0.852060	−	0.523443i	$-0.824647\pi$
$881$	− 812553.i	− 1.04689i	0.852060	−	0.523443i	$-0.175353\pi$
$882$	0	0
$883$	70306.0	0.0901719	0.0450859	−	0.998983i	$-0.485644\pi$
$883$	70306.0	0.0901719	0.0450859	+	0.998983i	$0.485644\pi$
$884$	− 434500.i	− 0.556013i
$885$	0	0
$886$	−649702.	−0.827650
$887$	− 699113.i	− 0.888587i	−0.895881	−	0.444293i	$-0.853455\pi$
$887$	− 699113.i	− 0.888587i	0.895881	−	0.444293i	$-0.146545\pi$
$888$	0	0
$889$	0	0
$890$	491126.i	0.620031i
$891$	0	0
$892$	727515.	0.914350
$893$	− 819449.i	− 1.02759i
$894$	0	0
$895$	814355.	1.01664
$896$	0	0
$897$	0	0
$898$	844261.	1.04695
$899$	− 558789.i	− 0.691398i
$900$	0	0
$901$	−539789.	−0.664927
$902$	− 10993.0i	− 0.0135114i
$903$	0	0
$904$	−527819.	−0.645875
$905$	1.52108e6i	1.85718i
$906$	0	0
$907$	1.41499e6	1.72004	0.860022	−	0.510257i	$-0.170450\pi$
$907$	1.41499e6	1.72004	0.860022	+	0.510257i	$0.170450\pi$
$908$	− 536673.i	− 0.650935i
$909$	0	0
$910$	0	0
$911$	1.30486e6i	1.57226i	0.618059	+	0.786132i	$0.287919\pi$
$911$	1.30486e6i	1.57226i	−0.618059	+	0.786132i	$0.712081\pi$
$912$	0	0
$913$	−546154.	−0.655199
$914$	436300.i	0.522267i
$915$	0	0
$916$	−424237.	−0.505612
$917$	0	0
$918$	0	0
$919$	1.14283e6	1.35317	0.676583	−	0.736366i	$-0.263460\pi$
$919$	1.14283e6	1.35317	0.676583	+	0.736366i	$0.263460\pi$
$920$	421130.i	0.497554i
$921$	0	0
$922$	360516.	0.424095
$923$	− 1.90173e6i	− 2.23227i
$924$	0	0
$925$	189833.	0.221865
$926$	− 43849.2i	− 0.0511375i
$927$	0	0
$928$	−143295.	−0.166393
$929$	836194.i	0.968893i	0.874821	+	0.484446i	$0.160979\pi$
$929$	836194.i	0.968893i	−0.874821	+	0.484446i	$0.839021\pi$
$930$	0	0
$931$	0	0
$932$	416983.i	0.480050i
$933$	0	0
$934$	786871.	0.902007
$935$	440724.i	0.504131i
$936$	0	0
$937$	−417434.	−0.475454	−0.237727	−	0.971332i	$-0.576402\pi$
$937$	−417434.	−0.475454	−0.237727	+	0.971332i	$0.576402\pi$
$938$	0	0
$939$	0	0
$940$	−505370.	−0.571945
$941$	1.01674e6i	1.14823i	0.818774	+	0.574115i	$0.194654\pi$
$941$	1.01674e6i	1.14823i	−0.818774	+	0.574115i	$0.805346\pi$
$942$	0	0
$943$	−38092.7	−0.0428369
$944$	131735.i	0.147828i
$945$	0	0
$946$	−51220.5	−0.0572350
$947$	− 550680.i	− 0.614044i	−0.951702	−	0.307022i	$-0.900668\pi$
$947$	− 550680.i	− 0.614044i	0.951702	−	0.307022i	$-0.0993325\pi$
$948$	0	0
$949$	−514578.	−0.571372
$950$	− 139597.i	− 0.154678i
$951$	0	0
$952$	0	0
$953$	301150.i	0.331587i	0.986160	+	0.165794i	$0.0530185\pi$
$953$	301150.i	0.331587i	−0.986160	+	0.165794i	$0.946981\pi$
$954$	0	0
$955$	1.88532e6	2.06718
$956$	217570.i	0.238059i
$957$	0	0
$958$	105117.	0.114536
$959$	0	0
$960$	0	0
$961$	−425229.	−0.460443
$962$	− 912061.i	− 0.985539i
$963$	0	0
$964$	−296881.	−0.319469
$965$	− 1.69191e6i	− 1.81687i
$966$	0	0
$967$	848232.	0.907114	0.453557	−	0.891227i	$-0.350155\pi$
$967$	848232.	0.907114	0.453557	+	0.891227i	$0.350155\pi$
$968$	224318.i	0.239394i
$969$	0	0
$970$	66605.0	0.0707886
$971$	− 1.07290e6i	− 1.13794i	−0.822358	−	0.568970i	$-0.807342\pi$
$971$	− 1.07290e6i	− 1.13794i	0.822358	−	0.568970i	$-0.192658\pi$
$972$	0	0
$973$	0	0
$974$	653174.i	0.688511i
$975$	0	0
$976$	−357339.	−0.375129
$977$	1.56714e6i	1.64179i	0.571077	+	0.820896i	$0.306526\pi$
$977$	1.56714e6i	1.64179i	−0.571077	+	0.820896i	$0.693474\pi$
$978$	0	0
$979$	−432282.	−0.451026
$980$	0	0
$981$	0	0
$982$	240773.	0.249680
$983$	1.57893e6i	1.63401i	0.576628	+	0.817007i	$0.304368\pi$
$983$	1.57893e6i	1.63401i	−0.576628	+	0.817007i	$0.695632\pi$
$984$	0	0
$985$	244426.	0.251927
$986$	519647.i	0.534509i
$987$	0	0
$988$	−670700.	−0.687091
$989$	177489.i	0.181459i
$990$	0	0
$991$	1.28966e6	1.31319	0.656594	−	0.754244i	$-0.271997\pi$
$991$	1.28966e6	1.31319	0.656594	+	0.754244i	$0.271997\pi$
$992$	− 127781.i	− 0.129851i
$993$	0	0
$994$	0	0
$995$	− 1.75230e6i	− 1.76996i
$996$	0	0
$997$	1.71724e6	1.72759	0.863793	−	0.503847i	$-0.168082\pi$
$997$	1.71724e6	1.72759	0.863793	+	0.503847i	$0.168082\pi$
$998$	606270.i	0.608702i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.i.197.2	✓	8
3.2	odd	2	inner	882.5.b.i.197.7	yes	8
7.6	odd	2	inner	882.5.b.i.197.3	yes	8
21.20	even	2	inner	882.5.b.i.197.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.5.b.i.197.2	✓	8	1.1	even	1	trivial
882.5.b.i.197.3	yes	8	7.6	odd	2	inner
882.5.b.i.197.6	yes	8	21.20	even	2	inner
882.5.b.i.197.7	yes	8	3.2	odd	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -27.6182i q^{5} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -27.6182i q^{5} +22.6274i q^{8} -78.1160 q^{10} -68.7565i q^{11} -234.014 q^{13} +64.0000 q^{16} -232.091i q^{17} -358.259 q^{19} +220.945i q^{20} -194.473 q^{22} +673.885i q^{23} -137.764 q^{25} +661.891i q^{26} -791.599i q^{29} +705.898 q^{31} -181.019i q^{32} -656.452 q^{34} -1377.96 q^{37} +1013.31i q^{38} +624.928 q^{40} +56.5270i q^{41} +263.382 q^{43} +550.052i q^{44} +1906.04 q^{46} +2287.31i q^{47} +389.654i q^{50} +1872.11 q^{52} -2325.76i q^{53} -1898.93 q^{55} -2238.98 q^{58} +2058.36i q^{59} -5583.42 q^{61} -1996.58i q^{62} -512.000 q^{64} +6463.03i q^{65} -5836.14 q^{67} +1856.73i q^{68} +8126.58i q^{71} +2198.92 q^{73} +3897.47i q^{74} +2866.07 q^{76} +10566.5 q^{79} -1767.56i q^{80} +159.882 q^{82} -7943.31i q^{83} -6409.93 q^{85} -744.955i q^{86} +1555.78 q^{88} -6287.14i q^{89} -5391.08i q^{92} +6469.49 q^{94} +9894.45i q^{95} -852.642 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} + 512 q^{16} - 640 q^{22} - 1560 q^{25} - 1408 q^{37} - 10256 q^{43} - 320 q^{46} - 512 q^{58} - 4096 q^{64} - 9600 q^{67} + 17680 q^{79} - 32048 q^{85} + 5120 q^{88}+O(q^{100}) \) 8 * q - 64 * q^4 + 512 * q^16 - 640 * q^22 - 1560 * q^25 - 1408 * q^37 - 10256 * q^43 - 320 * q^46 - 512 * q^58 - 4096 * q^64 - 9600 * q^67 + 17680 * q^79 - 32048 * q^85 + 5120 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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