LMFDB - Embedded newform 882.5.c.f.685.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,Mod(685,882)]

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([0, 1]))

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

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

chi := DirichletCharacter("882.685");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	882.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$91.1723074400$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 294)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.5
Root			$0.662827 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.5.c.f.685.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{2} +8.00000 q^{4} -43.8369i q^{5} +22.6274 q^{8} +O(q^{10})$	$q+2.82843 q^{2} +8.00000 q^{4} -43.8369i q^{5} +22.6274 q^{8} -123.989i q^{10} -66.2583 q^{11} +273.359i q^{13} +64.0000 q^{16} -118.377i q^{17} +588.891i q^{19} -350.695i q^{20} -187.407 q^{22} -157.762 q^{23} -1296.67 q^{25} +773.175i q^{26} -1601.22 q^{29} -592.286i q^{31} +181.019 q^{32} -334.821i q^{34} +100.514 q^{37} +1665.64i q^{38} -991.915i q^{40} -1786.17i q^{41} +22.2640 q^{43} -530.067 q^{44} -446.220 q^{46} +2455.42i q^{47} -3667.54 q^{50} +2186.87i q^{52} +165.545 q^{53} +2904.56i q^{55} -4528.92 q^{58} +5848.25i q^{59} +7070.76i q^{61} -1675.24i q^{62} +512.000 q^{64} +11983.2 q^{65} -3600.31 q^{67} -947.016i q^{68} +4380.13 q^{71} -8657.35i q^{73} +284.297 q^{74} +4711.13i q^{76} +227.232 q^{79} -2805.56i q^{80} -5052.04i q^{82} +11546.1i q^{83} -5189.28 q^{85} +62.9721 q^{86} -1499.26 q^{88} +370.148i q^{89} -1262.10 q^{92} +6944.98i q^{94} +25815.1 q^{95} +12020.7i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 64 q^{4}+O(q^{10}) $ 8 * q + 64 * q^4	$ 8 q + 64 q^{4} + 512 q^{16} - 1312 q^{22} + 272 q^{23} - 2808 q^{25} - 400 q^{29} - 3328 q^{37} + 656 q^{43} - 2400 q^{46} - 800 q^{50} - 9264 q^{53} - 11488 q^{58} + 4096 q^{64} + 15696 q^{65} + 26816 q^{67} - 28192 q^{71} + 4512 q^{74} + 19728 q^{79} - 49632 q^{85} - 5888 q^{86} - 10496 q^{88} + 2176 q^{92} + 92752 q^{95}+O(q^{100}) $ 8 * q + 64 * q^4 + 512 * q^16 - 1312 * q^22 + 272 * q^23 - 2808 * q^25 - 400 * q^29 - 3328 * q^37 + 656 * q^43 - 2400 * q^46 - 800 * q^50 - 9264 * q^53 - 11488 * q^58 + 4096 * q^64 + 15696 * q^65 + 26816 * q^67 - 28192 * q^71 + 4512 * q^74 + 19728 * q^79 - 49632 * q^85 - 5888 * q^86 - 10496 * q^88 + 2176 * q^92 + 92752 * q^95

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 $

$2$	2.82843	0.707107
$2$	2.82843	0.707107
$3$	0	0
$3$	0	0
$4$	8.00000	0.500000
$5$	− 43.8369i	− 1.75347i	−0.480970	−	0.876737i	$-0.659715\pi$
$5$	− 43.8369i	− 1.75347i	0.480970	−	0.876737i	$-0.340285\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274	0.353553
$9$	0	0
$10$	− 123.989i	− 1.23989i
$11$	−66.2583	−0.547590	−0.273795	−	0.961788i	$-0.588279\pi$
$11$	−66.2583	−0.547590	−0.273795	+	0.961788i	$0.588279\pi$
$12$	0	0
$13$	273.359i	1.61751i	0.588147	+	0.808754i	$0.299858\pi$
$13$	273.359i	1.61751i	−0.588147	+	0.808754i	$0.700142\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 118.377i	− 0.409609i	−0.978803	−	0.204805i	$-0.934344\pi$
$17$	− 118.377i	− 0.409609i	0.978803	−	0.204805i	$-0.0656559\pi$
$18$	0	0
$19$	588.891i	1.63128i	0.578561	+	0.815639i	$0.303614\pi$
$19$	588.891i	1.63128i	−0.578561	+	0.815639i	$0.696386\pi$
$20$	− 350.695i	− 0.876737i
$21$	0	0
$22$	−187.407	−0.387204
$23$	−157.762	−0.298228	−0.149114	−	0.988820i	$-0.547642\pi$
$23$	−157.762	−0.298228	−0.149114	+	0.988820i	$0.547642\pi$
$24$	0	0
$25$	−1296.67	−2.07467
$26$	773.175i	1.14375i
$27$	0	0
$28$	0	0
$29$	−1601.22	−1.90394	−0.951972	−	0.306187i	$-0.900947\pi$
$29$	−1601.22	−1.90394	−0.951972	+	0.306187i	$0.900947\pi$
$30$	0	0
$31$	− 592.286i	− 0.616322i	−0.951334	−	0.308161i	$-0.900286\pi$
$31$	− 592.286i	− 0.616322i	0.951334	−	0.308161i	$-0.0997136\pi$
$32$	181.019	0.176777
$33$	0	0
$34$	− 334.821i	− 0.289637i
$35$	0	0
$36$	0	0
$37$	100.514	0.0734216	0.0367108	−	0.999326i	$-0.488312\pi$
$37$	100.514	0.0734216	0.0367108	+	0.999326i	$0.488312\pi$
$38$	1665.64i	1.15349i
$39$	0	0
$40$	− 991.915i	− 0.619947i
$41$	− 1786.17i	− 1.06256i	−0.847196	−	0.531281i	$-0.821711\pi$
$41$	− 1786.17i	− 1.06256i	0.847196	−	0.531281i	$-0.178289\pi$
$42$	0	0
$43$	22.2640	0.0120411	0.00602055	−	0.999982i	$-0.498084\pi$
$43$	22.2640	0.0120411	0.00602055	+	0.999982i	$0.498084\pi$
$44$	−530.067	−0.273795
$45$	0	0
$46$	−446.220	−0.210879
$47$	2455.42i	1.11155i	0.831332	+	0.555776i	$0.187579\pi$
$47$	2455.42i	1.11155i	−0.831332	+	0.555776i	$0.812421\pi$
$48$	0	0
$49$	0	0
$50$	−3667.54	−1.46701
$51$	0	0
$52$	2186.87i	0.808754i
$53$	165.545	0.0589338	0.0294669	−	0.999566i	$-0.490619\pi$
$53$	165.545	0.0589338	0.0294669	+	0.999566i	$0.490619\pi$
$54$	0	0
$55$	2904.56i	0.960184i
$56$	0	0
$57$	0	0
$58$	−4528.92	−1.34629
$59$	5848.25i	1.68005i	0.542549	+	0.840024i	$0.317459\pi$
$59$	5848.25i	1.68005i	−0.542549	+	0.840024i	$0.682541\pi$
$60$	0	0
$61$	7070.76i	1.90023i	0.311898	+	0.950116i	$0.399035\pi$
$61$	7070.76i	1.90023i	−0.311898	+	0.950116i	$0.600965\pi$
$62$	− 1675.24i	− 0.435806i
$63$	0	0
$64$	512.000	0.125000
$65$	11983.2	2.83626
$66$	0	0
$67$	−3600.31	−0.802030	−0.401015	−	0.916071i	$-0.631342\pi$
$67$	−3600.31	−0.802030	−0.401015	+	0.916071i	$0.631342\pi$
$68$	− 947.016i	− 0.204805i
$69$	0	0
$70$	0	0
$71$	4380.13	0.868901	0.434451	−	0.900696i	$-0.356943\pi$
$71$	4380.13	0.868901	0.434451	+	0.900696i	$0.356943\pi$
$72$	0	0
$73$	− 8657.35i	− 1.62457i	−0.583258	−	0.812287i	$-0.698222\pi$
$73$	− 8657.35i	− 1.62457i	0.583258	−	0.812287i	$-0.301778\pi$
$74$	284.297	0.0519169
$75$	0	0
$76$	4711.13i	0.815639i
$77$	0	0
$78$	0	0
$79$	227.232	0.0364095	0.0182048	−	0.999834i	$-0.494205\pi$
$79$	227.232	0.0364095	0.0182048	+	0.999834i	$0.494205\pi$
$80$	− 2805.56i	− 0.438369i
$81$	0	0
$82$	− 5052.04i	− 0.751345i
$83$	11546.1i	1.67602i	0.545656	+	0.838009i	$0.316280\pi$
$83$	11546.1i	1.67602i	−0.545656	+	0.838009i	$0.683720\pi$
$84$	0	0
$85$	−5189.28	−0.718239
$86$	62.9721	0.00851434
$87$	0	0
$88$	−1499.26	−0.193602
$89$	370.148i	0.0467299i	0.999727	+	0.0233650i	$0.00743798\pi$
$89$	370.148i	0.0467299i	−0.999727	+	0.0233650i	$0.992562\pi$
$90$	0	0
$91$	0	0
$92$	−1262.10	−0.149114
$93$	0	0
$94$	6944.98i	0.785986i
$95$	25815.1	2.86040
$96$	0	0
$97$	12020.7i	1.27757i	0.769385	+	0.638785i	$0.220563\pi$
$97$	12020.7i	1.27757i	−0.769385	+	0.638785i	$0.779437\pi$
$98$	0	0
$99$	0	0
$100$	−10373.4	−1.03734
$101$	240.586i	0.0235846i	0.999930	+	0.0117923i	$0.00375369\pi$
$101$	240.586i	0.0235846i	−0.999930	+	0.0117923i	$0.996246\pi$
$102$	0	0
$103$	− 7014.48i	− 0.661182i	−0.943774	−	0.330591i	$-0.892752\pi$
$103$	− 7014.48i	− 0.661182i	0.943774	−	0.330591i	$-0.107248\pi$
$104$	6185.40i	0.571875i
$105$	0	0
$106$	468.232	0.0416725
$107$	−11922.0	−1.04131	−0.520655	−	0.853767i	$-0.674312\pi$
$107$	−11922.0	−1.04131	−0.520655	+	0.853767i	$0.674312\pi$
$108$	0	0
$109$	2685.95	0.226071	0.113036	−	0.993591i	$-0.463943\pi$
$109$	2685.95	0.226071	0.113036	+	0.993591i	$0.463943\pi$
$110$	8215.33i	0.678953i
$111$	0	0
$112$	0	0
$113$	9689.06	0.758795	0.379398	−	0.925234i	$-0.376131\pi$
$113$	9689.06	0.758795	0.379398	+	0.925234i	$0.376131\pi$
$114$	0	0
$115$	6915.81i	0.522935i
$116$	−12809.7	−0.951972
$117$	0	0
$118$	16541.3i	1.18797i
$119$	0	0
$120$	0	0
$121$	−10250.8	−0.700146
$122$	19999.1i	1.34367i
$123$	0	0
$124$	− 4738.29i	− 0.308161i
$125$	29443.9i	1.88441i
$126$	0	0
$127$	−5749.81	−0.356489	−0.178245	−	0.983986i	$-0.557042\pi$
$127$	−5749.81	−0.356489	−0.178245	+	0.983986i	$0.557042\pi$
$128$	1448.15	0.0883883
$129$	0	0
$130$	33893.6	2.00554
$131$	− 20762.8i	− 1.20988i	−0.796270	−	0.604941i	$-0.793197\pi$
$131$	− 20762.8i	− 1.20988i	0.796270	−	0.604941i	$-0.206803\pi$
$132$	0	0
$133$	0	0
$134$	−10183.2	−0.567121
$135$	0	0
$136$	− 2678.57i	− 0.144819i
$137$	−4795.67	−0.255510	−0.127755	−	0.991806i	$-0.540777\pi$
$137$	−4795.67	−0.255510	−0.127755	+	0.991806i	$0.540777\pi$
$138$	0	0
$139$	3154.68i	0.163277i	0.996662	+	0.0816386i	$0.0260153\pi$
$139$	3154.68i	0.163277i	−0.996662	+	0.0816386i	$0.973985\pi$
$140$	0	0
$141$	0	0
$142$	12388.9	0.614406
$143$	− 18112.3i	− 0.885730i
$144$	0	0
$145$	70192.3i	3.33851i
$146$	− 24486.7i	− 1.14875i
$147$	0	0
$148$	804.114	0.0367108
$149$	393.903	0.0177426	0.00887128	−	0.999961i	$-0.497176\pi$
$149$	393.903	0.0177426	0.00887128	+	0.999961i	$0.497176\pi$
$150$	0	0
$151$	43146.6	1.89231	0.946157	−	0.323709i	$-0.104930\pi$
$151$	43146.6	1.89231	0.946157	+	0.323709i	$0.104930\pi$
$152$	13325.1i	0.576744i
$153$	0	0
$154$	0	0
$155$	−25963.9	−1.08071
$156$	0	0
$157$	− 10573.1i	− 0.428945i	−0.976730	−	0.214472i	$-0.931197\pi$
$157$	− 10573.1i	− 0.428945i	0.976730	−	0.214472i	$-0.0688032\pi$
$158$	642.709	0.0257454
$159$	0	0
$160$	− 7935.32i	− 0.309973i
$161$	0	0
$162$	0	0
$163$	27672.6	1.04154	0.520768	−	0.853698i	$-0.325646\pi$
$163$	27672.6	1.04154	0.520768	+	0.853698i	$0.325646\pi$
$164$	− 14289.3i	− 0.531281i
$165$	0	0
$166$	32657.3i	1.18512i
$167$	30859.2i	1.10650i	0.833015	+	0.553251i	$0.186613\pi$
$167$	30859.2i	1.10650i	−0.833015	+	0.553251i	$0.813387\pi$
$168$	0	0
$169$	−46164.0	−1.61633
$170$	−14677.5	−0.507872
$171$	0	0
$172$	178.112	0.00602055
$173$	22164.5i	0.740569i	0.928918	+	0.370284i	$0.120740\pi$
$173$	22164.5i	0.740569i	−0.928918	+	0.370284i	$0.879260\pi$
$174$	0	0
$175$	0	0
$176$	−4240.53	−0.136897
$177$	0	0
$178$	1046.94i	0.0330431i
$179$	−26136.9	−0.815734	−0.407867	−	0.913041i	$-0.633727\pi$
$179$	−26136.9	−0.815734	−0.407867	+	0.913041i	$0.633727\pi$
$180$	0	0
$181$	− 35842.9i	− 1.09407i	−0.837109	−	0.547036i	$-0.815756\pi$
$181$	− 35842.9i	− 1.09407i	0.837109	−	0.547036i	$-0.184244\pi$
$182$	0	0
$183$	0	0
$184$	−3569.76	−0.105439
$185$	− 4406.23i	− 0.128743i
$186$	0	0
$187$	7843.47i	0.224298i
$188$	19643.4i	0.555776i
$189$	0	0
$190$	73016.2	2.02261
$191$	−66172.3	−1.81388	−0.906942	−	0.421256i	$-0.861589\pi$
$191$	−66172.3	−1.81388	−0.906942	+	0.421256i	$0.861589\pi$
$192$	0	0
$193$	14157.4	0.380076	0.190038	−	0.981777i	$-0.439139\pi$
$193$	14157.4	0.380076	0.190038	+	0.981777i	$0.439139\pi$
$194$	33999.6i	0.903379i
$195$	0	0
$196$	0	0
$197$	−45172.5	−1.16397	−0.581985	−	0.813200i	$-0.697724\pi$
$197$	−45172.5	−1.16397	−0.581985	+	0.813200i	$0.697724\pi$
$198$	0	0
$199$	1134.69i	0.0286531i	0.999897	+	0.0143265i	$0.00456044\pi$
$199$	1134.69i	0.0286531i	−0.999897	+	0.0143265i	$0.995440\pi$
$200$	−29340.3	−0.733507
$201$	0	0
$202$	680.480i	0.0166768i
$203$	0	0
$204$	0	0
$205$	−78299.9	−1.86317
$206$	− 19840.0i	− 0.467527i
$207$	0	0
$208$	17495.0i	0.404377i
$209$	− 39019.0i	− 0.893271i
$210$	0	0
$211$	−73423.2	−1.64918	−0.824590	−	0.565730i	$-0.808594\pi$
$211$	−73423.2	−1.64918	−0.824590	+	0.565730i	$0.808594\pi$
$212$	1324.36	0.0294669
$213$	0	0
$214$	−33720.4	−0.736317
$215$	− 975.984i	− 0.0211138i
$216$	0	0
$217$	0	0
$218$	7597.02	0.159856
$219$	0	0
$220$	23236.5i	0.480092i
$221$	32359.4	0.662546
$222$	0	0
$223$	55677.6i	1.11962i	0.828621	+	0.559810i	$0.189126\pi$
$223$	55677.6i	1.11962i	−0.828621	+	0.559810i	$0.810874\pi$
$224$	0	0
$225$	0	0
$226$	27404.8	0.536549
$227$	− 58625.5i	− 1.13772i	−0.822435	−	0.568859i	$-0.807385\pi$
$227$	− 58625.5i	− 1.13772i	0.822435	−	0.568859i	$-0.192615\pi$
$228$	0	0
$229$	− 7310.93i	− 0.139412i	−0.997568	−	0.0697062i	$-0.977794\pi$
$229$	− 7310.93i	− 0.139412i	0.997568	−	0.0697062i	$-0.0222062\pi$
$230$	19560.9i	0.369771i
$231$	0	0
$232$	−36231.4	−0.673146
$233$	−33351.5	−0.614332	−0.307166	−	0.951656i	$-0.599381\pi$
$233$	−33351.5	−0.614332	−0.307166	+	0.951656i	$0.599381\pi$
$234$	0	0
$235$	107638.	1.94908
$236$	46786.0i	0.840024i
$237$	0	0
$238$	0	0
$239$	55488.1	0.971413	0.485706	−	0.874122i	$-0.338562\pi$
$239$	55488.1	0.971413	0.485706	+	0.874122i	$0.338562\pi$
$240$	0	0
$241$	33459.2i	0.576078i	0.957619	+	0.288039i	$0.0930034\pi$
$241$	33459.2i	0.576078i	−0.957619	+	0.288039i	$0.906997\pi$
$242$	−28993.7	−0.495078
$243$	0	0
$244$	56566.1i	0.950116i
$245$	0	0
$246$	0	0
$247$	−160979.	−2.63860
$248$	− 13401.9i	− 0.217903i
$249$	0	0
$250$	83279.9i	1.33248i
$251$	− 79312.8i	− 1.25891i	−0.777036	−	0.629457i	$-0.783277\pi$
$251$	− 79312.8i	− 1.25891i	0.777036	−	0.629457i	$-0.216723\pi$
$252$	0	0
$253$	10453.1	0.163306
$254$	−16262.9	−0.252076
$255$	0	0
$256$	4096.00	0.0625000
$257$	27095.4i	0.410232i	0.978738	+	0.205116i	$0.0657572\pi$
$257$	27095.4i	0.410232i	−0.978738	+	0.205116i	$0.934243\pi$
$258$	0	0
$259$	0	0
$260$	95865.5	1.41813
$261$	0	0
$262$	− 58726.0i	− 0.855515i
$263$	54875.8	0.793359	0.396679	−	0.917957i	$-0.370162\pi$
$263$	54875.8	0.793359	0.396679	+	0.917957i	$0.370162\pi$
$264$	0	0
$265$	− 7256.98i	− 0.103339i
$266$	0	0
$267$	0	0
$268$	−28802.5	−0.401015
$269$	− 11532.1i	− 0.159368i	−0.996820	−	0.0796842i	$-0.974609\pi$
$269$	− 11532.1i	− 0.159368i	0.996820	−	0.0796842i	$-0.0253912\pi$
$270$	0	0
$271$	98459.2i	1.34066i	0.742065	+	0.670328i	$0.233847\pi$
$271$	98459.2i	1.34066i	−0.742065	+	0.670328i	$0.766153\pi$
$272$	− 7576.13i	− 0.102402i
$273$	0	0
$274$	−13564.2	−0.180673
$275$	85915.2	1.13607
$276$	0	0
$277$	−64987.4	−0.846973	−0.423487	−	0.905902i	$-0.639194\pi$
$277$	−64987.4	−0.846973	−0.423487	+	0.905902i	$0.639194\pi$
$278$	8922.78i	0.115454i
$279$	0	0
$280$	0	0
$281$	−39118.4	−0.495414	−0.247707	−	0.968835i	$-0.579677\pi$
$281$	−39118.4	−0.495414	−0.247707	+	0.968835i	$0.579677\pi$
$282$	0	0
$283$	58187.8i	0.726539i	0.931684	+	0.363269i	$0.118340\pi$
$283$	58187.8i	0.726539i	−0.931684	+	0.363269i	$0.881660\pi$
$284$	35041.1	0.434451
$285$	0	0
$286$	− 51229.3i	− 0.626306i
$287$	0	0
$288$	0	0
$289$	69507.9	0.832220
$290$	198534.i	2.36069i
$291$	0	0
$292$	− 69258.8i	− 0.812287i
$293$	38775.9i	0.451676i	0.974165	+	0.225838i	$0.0725120\pi$
$293$	38775.9i	0.451676i	−0.974165	+	0.225838i	$0.927488\pi$
$294$	0	0
$295$	256369.	2.94592
$296$	2274.38	0.0259585
$297$	0	0
$298$	1114.13	0.0125459
$299$	− 43125.8i	− 0.482386i
$300$	0	0
$301$	0	0
$302$	122037.	1.33807
$303$	0	0
$304$	37689.0i	0.407819i
$305$	309960.	3.33201
$306$	0	0
$307$	135801.i	1.44088i	0.693518	+	0.720439i	$0.256060\pi$
$307$	135801.i	1.44088i	−0.693518	+	0.720439i	$0.743940\pi$
$308$	0	0
$309$	0	0
$310$	−73437.1	−0.764174
$311$	− 88411.9i	− 0.914092i	−0.889443	−	0.457046i	$-0.848907\pi$
$311$	− 88411.9i	− 0.914092i	0.889443	−	0.457046i	$-0.151093\pi$
$312$	0	0
$313$	− 146877.i	− 1.49922i	−0.661878	−	0.749612i	$-0.730240\pi$
$313$	− 146877.i	− 1.49922i	0.661878	−	0.749612i	$-0.269760\pi$
$314$	− 29905.1i	− 0.303310i
$315$	0	0
$316$	1817.85	0.0182048
$317$	−20416.2	−0.203168	−0.101584	−	0.994827i	$-0.532391\pi$
$317$	−20416.2	−0.203168	−0.101584	+	0.994827i	$0.532391\pi$
$318$	0	0
$319$	106094.	1.04258
$320$	− 22444.5i	− 0.219184i
$321$	0	0
$322$	0	0
$323$	69711.2	0.668186
$324$	0	0
$325$	− 354456.i	− 3.35580i
$326$	78269.8	0.736477
$327$	0	0
$328$	− 40416.3i	− 0.375672i
$329$	0	0
$330$	0	0
$331$	80110.8	0.731198	0.365599	−	0.930772i	$-0.380864\pi$
$331$	80110.8	0.731198	0.365599	+	0.930772i	$0.380864\pi$
$332$	92368.7i	0.838009i
$333$	0	0
$334$	87283.1i	0.782415i
$335$	157826.i	1.40634i
$336$	0	0
$337$	−160979.	−1.41746	−0.708730	−	0.705480i	$-0.750732\pi$
$337$	−160979.	−1.41746	−0.708730	+	0.705480i	$0.750732\pi$
$338$	−130572.	−1.14292
$339$	0	0
$340$	−41514.2	−0.359120
$341$	39243.9i	0.337492i
$342$	0	0
$343$	0	0
$344$	503.777	0.00425717
$345$	0	0
$346$	62690.6i	0.523661i
$347$	−103883.	−0.862749	−0.431375	−	0.902173i	$-0.641971\pi$
$347$	−103883.	−0.862749	−0.431375	+	0.902173i	$0.641971\pi$
$348$	0	0
$349$	42377.0i	0.347920i	0.984753	+	0.173960i	$0.0556563\pi$
$349$	42377.0i	0.347920i	−0.984753	+	0.173960i	$0.944344\pi$
$350$	0	0
$351$	0	0
$352$	−11994.0	−0.0968011
$353$	− 38676.6i	− 0.310384i	−0.987884	−	0.155192i	$-0.950400\pi$
$353$	− 38676.6i	− 0.310384i	0.987884	−	0.155192i	$-0.0495996\pi$
$354$	0	0
$355$	− 192011.i	− 1.52360i
$356$	2961.18i	0.0233650i
$357$	0	0
$358$	−73926.4	−0.576811
$359$	52162.6	0.404735	0.202367	−	0.979310i	$-0.435136\pi$
$359$	52162.6	0.404735	0.202367	+	0.979310i	$0.435136\pi$
$360$	0	0
$361$	−216472.	−1.66107
$362$	− 101379.i	− 0.773626i
$363$	0	0
$364$	0	0
$365$	−379511.	−2.84865
$366$	0	0
$367$	− 91221.7i	− 0.677276i	−0.940917	−	0.338638i	$-0.890034\pi$
$367$	− 91221.7i	− 0.677276i	0.940917	−	0.338638i	$-0.109966\pi$
$368$	−10096.8	−0.0745569
$369$	0	0
$370$	− 12462.7i	− 0.0910350i
$371$	0	0
$372$	0	0
$373$	−147511.	−1.06025	−0.530125	−	0.847920i	$-0.677855\pi$
$373$	−147511.	−1.06025	−0.530125	+	0.847920i	$0.677855\pi$
$374$	22184.7i	0.158602i
$375$	0	0
$376$	55559.8i	0.392993i
$377$	− 437706.i	− 3.07964i
$378$	0	0
$379$	165402.	1.15150	0.575749	−	0.817627i	$-0.304711\pi$
$379$	165402.	1.15150	0.575749	+	0.817627i	$0.304711\pi$
$380$	206521.	1.43020
$381$	0	0
$382$	−187163.	−1.28261
$383$	− 160959.i	− 1.09728i	−0.836058	−	0.548642i	$-0.815145\pi$
$383$	− 160959.i	− 1.09728i	0.836058	−	0.548642i	$-0.184855\pi$
$384$	0	0
$385$	0	0
$386$	40043.3	0.268754
$387$	0	0
$388$	96165.3i	0.638785i
$389$	−222028.	−1.46726	−0.733632	−	0.679547i	$-0.762176\pi$
$389$	−222028.	−1.46726	−0.733632	+	0.679547i	$0.762176\pi$
$390$	0	0
$391$	18675.5i	0.122157i
$392$	0	0
$393$	0	0
$394$	−127767.	−0.823051
$395$	− 9961.13i	− 0.0638431i
$396$	0	0
$397$	190906.i	1.21127i	0.795744	+	0.605633i	$0.207080\pi$
$397$	190906.i	1.21127i	−0.795744	+	0.605633i	$0.792920\pi$
$398$	3209.39i	0.0202608i
$399$	0	0
$400$	−82986.8	−0.518668
$401$	97080.3	0.603730	0.301865	−	0.953351i	$-0.402391\pi$
$401$	97080.3	0.603730	0.301865	+	0.953351i	$0.402391\pi$
$402$	0	0
$403$	161907.	0.996906
$404$	1924.69i	0.0117923i
$405$	0	0
$406$	0	0
$407$	−6659.90	−0.0402049
$408$	0	0
$409$	38787.2i	0.231868i	0.993257	+	0.115934i	$0.0369862\pi$
$409$	38787.2i	0.231868i	−0.993257	+	0.115934i	$0.963014\pi$
$410$	−221466.	−1.31746
$411$	0	0
$412$	− 56115.9i	− 0.330591i
$413$	0	0
$414$	0	0
$415$	506144.	2.93885
$416$	49483.2i	0.285938i
$417$	0	0
$418$	− 110362.i	− 0.631638i
$419$	19366.1i	0.110310i	0.998478	+	0.0551548i	$0.0175652\pi$
$419$	19366.1i	0.110310i	−0.998478	+	0.0551548i	$0.982435\pi$
$420$	0	0
$421$	−179607.	−1.01335	−0.506674	−	0.862138i	$-0.669125\pi$
$421$	−179607.	−1.01335	−0.506674	+	0.862138i	$0.669125\pi$
$422$	−207672.	−1.16615
$423$	0	0
$424$	3745.86	0.0208363
$425$	153496.i	0.849804i
$426$	0	0
$427$	0	0
$428$	−95375.7	−0.520655
$429$	0	0
$430$	− 2760.50i	− 0.0149297i
$431$	−23412.8	−0.126037	−0.0630187	−	0.998012i	$-0.520073\pi$
$431$	−23412.8	−0.126037	−0.0630187	+	0.998012i	$0.520073\pi$
$432$	0	0
$433$	179691.i	0.958407i	0.877704	+	0.479204i	$0.159074\pi$
$433$	179691.i	0.958407i	−0.877704	+	0.479204i	$0.840926\pi$
$434$	0	0
$435$	0	0
$436$	21487.6	0.113036
$437$	− 92904.9i	− 0.486492i
$438$	0	0
$439$	− 342493.i	− 1.77715i	−0.458736	−	0.888573i	$-0.651698\pi$
$439$	− 342493.i	− 1.77715i	0.458736	−	0.888573i	$-0.348302\pi$
$440$	65722.6i	0.339476i
$441$	0	0
$442$	91526.2	0.468491
$443$	53356.7	0.271883	0.135941	−	0.990717i	$-0.456594\pi$
$443$	53356.7	0.271883	0.135941	+	0.990717i	$0.456594\pi$
$444$	0	0
$445$	16226.1	0.0819397
$446$	157480.i	0.791692i
$447$	0	0
$448$	0	0
$449$	−88798.6	−0.440467	−0.220234	−	0.975447i	$-0.570682\pi$
$449$	−88798.6	−0.440467	−0.220234	+	0.975447i	$0.570682\pi$
$450$	0	0
$451$	118348.i	0.581848i
$452$	77512.4	0.379398
$453$	0	0
$454$	− 165818.i	− 0.804488i
$455$	0	0
$456$	0	0
$457$	296922.	1.42171	0.710854	−	0.703339i	$-0.248308\pi$
$457$	296922.	1.42171	0.710854	+	0.703339i	$0.248308\pi$
$458$	− 20678.4i	− 0.0985795i
$459$	0	0
$460$	55326.5i	0.261467i
$461$	193726.i	0.911562i	0.890092	+	0.455781i	$0.150640\pi$
$461$	193726.i	0.911562i	−0.890092	+	0.455781i	$0.849360\pi$
$462$	0	0
$463$	−25440.7	−0.118677	−0.0593385	−	0.998238i	$-0.518899\pi$
$463$	−25440.7	−0.118677	−0.0593385	+	0.998238i	$0.518899\pi$
$464$	−102478.	−0.475986
$465$	0	0
$466$	−94332.2	−0.434398
$467$	32439.9i	0.148746i	0.997230	+	0.0743732i	$0.0236956\pi$
$467$	32439.9i	0.148746i	−0.997230	+	0.0743732i	$0.976304\pi$
$468$	0	0
$469$	0	0
$470$	304446.	1.37821
$471$	0	0
$472$	132331.i	0.593987i
$473$	−1475.18	−0.00659358
$474$	0	0
$475$	− 763597.i	− 3.38436i
$476$	0	0
$477$	0	0
$478$	156944.	0.686893
$479$	− 330216.i	− 1.43922i	−0.694378	−	0.719611i	$-0.744320\pi$
$479$	− 330216.i	− 1.43922i	0.694378	−	0.719611i	$-0.255680\pi$
$480$	0	0
$481$	27476.4i	0.118760i
$482$	94636.9i	0.407349i
$483$	0	0
$484$	−82006.7	−0.350073
$485$	526948.	2.24019
$486$	0	0
$487$	39786.3	0.167755	0.0838775	−	0.996476i	$-0.473270\pi$
$487$	39786.3	0.167755	0.0838775	+	0.996476i	$0.473270\pi$
$488$	159993.i	0.671833i
$489$	0	0
$490$	0	0
$491$	−398313.	−1.65219	−0.826097	−	0.563528i	$-0.809444\pi$
$491$	−398313.	−1.65219	−0.826097	+	0.563528i	$0.809444\pi$
$492$	0	0
$493$	189547.i	0.779873i
$494$	−455316.	−1.86577
$495$	0	0
$496$	− 37906.3i	− 0.154081i
$497$	0	0
$498$	0	0
$499$	82093.0	0.329689	0.164845	−	0.986320i	$-0.447288\pi$
$499$	82093.0	0.329689	0.164845	+	0.986320i	$0.447288\pi$
$500$	235551.i	0.942204i
$501$	0	0
$502$	− 224331.i	− 0.890186i
$503$	427740.i	1.69061i	0.534283	+	0.845306i	$0.320582\pi$
$503$	427740.i	1.69061i	−0.534283	+	0.845306i	$0.679418\pi$
$504$	0	0
$505$	10546.5	0.0413549
$506$	29565.8	0.115475
$507$	0	0
$508$	−45998.5	−0.178245
$509$	− 297217.i	− 1.14720i	−0.819136	−	0.573599i	$-0.805547\pi$
$509$	− 297217.i	− 1.14720i	0.819136	−	0.573599i	$-0.194453\pi$
$510$	0	0
$511$	0	0
$512$	11585.2	0.0441942
$513$	0	0
$514$	76637.4i	0.290078i
$515$	−307493.	−1.15937
$516$	0	0
$517$	− 162692.i	− 0.608675i
$518$	0	0
$519$	0	0
$520$	271149.	1.00277
$521$	− 30563.0i	− 0.112595i	−0.998414	−	0.0562977i	$-0.982070\pi$
$521$	− 30563.0i	− 0.112595i	0.998414	−	0.0562977i	$-0.0179296\pi$
$522$	0	0
$523$	− 44565.9i	− 0.162929i	−0.996676	−	0.0814646i	$-0.974040\pi$
$523$	− 44565.9i	− 0.162929i	0.996676	−	0.0814646i	$-0.0259598\pi$
$524$	− 166102.i	− 0.604941i
$525$	0	0
$526$	155212.	0.560990
$527$	−70113.1	−0.252451
$528$	0	0
$529$	−254952.	−0.911060
$530$	− 20525.8i	− 0.0730717i
$531$	0	0
$532$	0	0
$533$	488264.	1.71870
$534$	0	0
$535$	522621.i	1.82591i
$536$	−81465.8	−0.283560
$537$	0	0
$538$	− 32617.6i	− 0.112690i
$539$	0	0
$540$	0	0
$541$	−435375.	−1.48754	−0.743770	−	0.668436i	$-0.766964\pi$
$541$	−435375.	−1.48754	−0.743770	+	0.668436i	$0.766964\pi$
$542$	278485.i	0.947987i
$543$	0	0
$544$	− 21428.5i	− 0.0724094i
$545$	− 117744.i	− 0.396410i
$546$	0	0
$547$	−178673.	−0.597152	−0.298576	−	0.954386i	$-0.596512\pi$
$547$	−178673.	−0.597152	−0.298576	+	0.954386i	$0.596512\pi$
$548$	−38365.4	−0.127755
$549$	0	0
$550$	243005.	0.803322
$551$	− 942942.i	− 3.10586i
$552$	0	0
$553$	0	0
$554$	−183812.	−0.598900
$555$	0	0
$556$	25237.4i	0.0816386i
$557$	216125.	0.696618	0.348309	−	0.937380i	$-0.386756\pi$
$557$	216125.	0.696618	0.348309	+	0.937380i	$0.386756\pi$
$558$	0	0
$559$	6086.06i	0.0194766i
$560$	0	0
$561$	0	0
$562$	−110644.	−0.350311
$563$	33117.9i	0.104483i	0.998634	+	0.0522415i	$0.0166366\pi$
$563$	33117.9i	0.104483i	−0.998634	+	0.0522415i	$0.983363\pi$
$564$	0	0
$565$	− 424738.i	− 1.33053i
$566$	164580.i	0.513741i
$567$	0	0
$568$	99111.1	0.307203
$569$	−332191.	−1.02604	−0.513020	−	0.858377i	$-0.671473\pi$
$569$	−332191.	−1.02604	−0.513020	+	0.858377i	$0.671473\pi$
$570$	0	0
$571$	21249.0	0.0651728	0.0325864	−	0.999469i	$-0.489626\pi$
$571$	21249.0	0.0651728	0.0325864	+	0.999469i	$0.489626\pi$
$572$	− 144898.i	− 0.442865i
$573$	0	0
$574$	0	0
$575$	204566.	0.618725
$576$	0	0
$577$	− 262834.i	− 0.789459i	−0.918797	−	0.394730i	$-0.870838\pi$
$577$	− 262834.i	− 0.789459i	0.918797	−	0.394730i	$-0.129162\pi$
$578$	196598.	0.588469
$579$	0	0
$580$	561538.i	1.66926i
$581$	0	0
$582$	0	0
$583$	−10968.7	−0.0322716
$584$	− 195894.i	− 0.574374i
$585$	0	0
$586$	109675.i	0.319383i
$587$	241430.i	0.700671i	0.936624	+	0.350336i	$0.113932\pi$
$587$	241430.i	0.700671i	−0.936624	+	0.350336i	$0.886068\pi$
$588$	0	0
$589$	348792.	1.00539
$590$	725120.	2.08308
$591$	0	0
$592$	6432.91	0.0183554
$593$	− 402268.i	− 1.14395i	−0.820272	−	0.571973i	$-0.806178\pi$
$593$	− 402268.i	− 1.14395i	0.820272	−	0.571973i	$-0.193822\pi$
$594$	0	0
$595$	0	0
$596$	3151.22	0.00887128
$597$	0	0
$598$	− 121978.i	− 0.341098i
$599$	−103978.	−0.289793	−0.144897	−	0.989447i	$-0.546285\pi$
$599$	−103978.	−0.289793	−0.144897	+	0.989447i	$0.546285\pi$
$600$	0	0
$601$	234544.i	0.649345i	0.945826	+	0.324673i	$0.105254\pi$
$601$	234544.i	0.649345i	−0.945826	+	0.324673i	$0.894746\pi$
$602$	0	0
$603$	0	0
$604$	345173.	0.946157
$605$	449364.i	1.22769i
$606$	0	0
$607$	− 420933.i	− 1.14245i	−0.820795	−	0.571223i	$-0.806469\pi$
$607$	− 420933.i	− 1.14245i	0.820795	−	0.571223i	$-0.193531\pi$
$608$	106601.i	0.288372i
$609$	0	0
$610$	876699.	2.35608
$611$	−671210.	−1.79794
$612$	0	0
$613$	−120471.	−0.320599	−0.160300	−	0.987068i	$-0.551246\pi$
$613$	−120471.	−0.320599	−0.160300	+	0.987068i	$0.551246\pi$
$614$	384104.i	1.01886i
$615$	0	0
$616$	0	0
$617$	362513.	0.952254	0.476127	−	0.879377i	$-0.342040\pi$
$617$	362513.	0.952254	0.476127	+	0.879377i	$0.342040\pi$
$618$	0	0
$619$	− 240538.i	− 0.627773i	−0.949460	−	0.313887i	$-0.898369\pi$
$619$	− 240538.i	− 0.627773i	0.949460	−	0.313887i	$-0.101631\pi$
$620$	−207712.	−0.540353
$621$	0	0
$622$	− 250067.i	− 0.646361i
$623$	0	0
$624$	0	0
$625$	480308.	1.22959
$626$	− 415432.i	− 1.06011i
$627$	0	0
$628$	− 84584.4i	− 0.214472i
$629$	− 11898.6i	− 0.0300742i
$630$	0	0
$631$	−263114.	−0.660822	−0.330411	−	0.943837i	$-0.607187\pi$
$631$	−263114.	−0.660822	−0.330411	+	0.943837i	$0.607187\pi$
$632$	5141.67	0.0128727
$633$	0	0
$634$	−57745.6	−0.143662
$635$	252054.i	0.625095i
$636$	0	0
$637$	0	0
$638$	300079.	0.737215
$639$	0	0
$640$	− 63482.5i	− 0.154987i
$641$	288511.	0.702176	0.351088	−	0.936343i	$-0.385812\pi$
$641$	288511.	0.702176	0.351088	+	0.936343i	$0.385812\pi$
$642$	0	0
$643$	− 566593.i	− 1.37041i	−0.728352	−	0.685203i	$-0.759713\pi$
$643$	− 566593.i	− 1.37041i	0.728352	−	0.685203i	$-0.240287\pi$
$644$	0	0
$645$	0	0
$646$	197173.	0.472479
$647$	264364.i	0.631531i	0.948837	+	0.315765i	$0.102261\pi$
$647$	264364.i	0.631531i	−0.948837	+	0.315765i	$0.897739\pi$
$648$	0	0
$649$	− 387495.i	− 0.919977i
$650$	− 1.00255e6i	− 2.37291i
$651$	0	0
$652$	221380.	0.520768
$653$	369486.	0.866507	0.433253	−	0.901272i	$-0.357366\pi$
$653$	369486.	0.866507	0.433253	+	0.901272i	$0.357366\pi$
$654$	0	0
$655$	−910175.	−2.12150
$656$	− 114315.i	− 0.265640i
$657$	0	0
$658$	0	0
$659$	−90347.1	−0.208038	−0.104019	−	0.994575i	$-0.533170\pi$
$659$	−90347.1	−0.208038	−0.104019	+	0.994575i	$0.533170\pi$
$660$	0	0
$661$	− 449282.i	− 1.02829i	−0.857703	−	0.514146i	$-0.828109\pi$
$661$	− 449282.i	− 1.02829i	0.857703	−	0.514146i	$-0.171891\pi$
$662$	226588.	0.517035
$663$	0	0
$664$	261258.i	0.592562i
$665$	0	0
$666$	0	0
$667$	252612.	0.567809
$668$	246874.i	0.553251i
$669$	0	0
$670$	446400.i	0.994432i
$671$	− 468497.i	− 1.04055i
$672$	0	0
$673$	−177367.	−0.391599	−0.195800	−	0.980644i	$-0.562730\pi$
$673$	−177367.	−0.391599	−0.195800	+	0.980644i	$0.562730\pi$
$674$	−455319.	−1.00230
$675$	0	0
$676$	−369312.	−0.808165
$677$	537289.i	1.17228i	0.810211	+	0.586138i	$0.199353\pi$
$677$	537289.i	1.17228i	−0.810211	+	0.586138i	$0.800647\pi$
$678$	0	0
$679$	0	0
$680$	−117420.	−0.253936
$681$	0	0
$682$	110998.i	0.238643i
$683$	126817.	0.271854	0.135927	−	0.990719i	$-0.456599\pi$
$683$	126817.	0.271854	0.135927	+	0.990719i	$0.456599\pi$
$684$	0	0
$685$	210227.i	0.448031i
$686$	0	0
$687$	0	0
$688$	1424.90	0.00301028
$689$	45253.2i	0.0953259i
$690$	0	0
$691$	429361.i	0.899220i	0.893225	+	0.449610i	$0.148437\pi$
$691$	429361.i	0.899220i	−0.893225	+	0.449610i	$0.851563\pi$
$692$	177316.i	0.370284i
$693$	0	0
$694$	−293825.	−0.610056
$695$	138291.	0.286302
$696$	0	0
$697$	−211441.	−0.435235
$698$	119860.i	0.246016i
$699$	0	0
$700$	0	0
$701$	−61476.0	−0.125104	−0.0625518	−	0.998042i	$-0.519924\pi$
$701$	−61476.0	−0.125104	−0.0625518	+	0.998042i	$0.519924\pi$
$702$	0	0
$703$	59191.9i	0.119771i
$704$	−33924.3	−0.0684487
$705$	0	0
$706$	− 109394.i	− 0.219475i
$707$	0	0
$708$	0	0
$709$	−170870.	−0.339917	−0.169958	−	0.985451i	$-0.554363\pi$
$709$	−170870.	−0.339917	−0.169958	+	0.985451i	$0.554363\pi$
$710$	− 543090.i	− 1.07735i
$711$	0	0
$712$	8375.49i	0.0165215i
$713$	93440.5i	0.183804i
$714$	0	0
$715$	−793986.	−1.55311
$716$	−209095.	−0.407867
$717$	0	0
$718$	147538.	0.286191
$719$	− 67384.1i	− 0.130347i	−0.997874	−	0.0651733i	$-0.979240\pi$
$719$	− 67384.1i	− 0.130347i	0.997874	−	0.0651733i	$-0.0207600\pi$
$720$	0	0
$721$	0	0
$722$	−612275.	−1.17455
$723$	0	0
$724$	− 286743.i	− 0.547036i
$725$	2.07625e6	3.95006
$726$	0	0
$727$	− 280290.i	− 0.530322i	−0.964204	−	0.265161i	$-0.914575\pi$
$727$	− 280290.i	− 0.530322i	0.964204	−	0.265161i	$-0.0854251\pi$
$728$	0	0
$729$	0	0
$730$	−1.07342e6	−2.01430
$731$	− 2635.55i	− 0.00493215i
$732$	0	0
$733$	363812.i	0.677126i	0.940944	+	0.338563i	$0.109941\pi$
$733$	363812.i	0.677126i	−0.940944	+	0.338563i	$0.890059\pi$
$734$	− 258014.i	− 0.478907i
$735$	0	0
$736$	−28558.1	−0.0527197
$737$	238551.	0.439183
$738$	0	0
$739$	1.02672e6	1.88003	0.940015	−	0.341132i	$-0.110810\pi$
$739$	1.02672e6	1.88003	0.940015	+	0.341132i	$0.110810\pi$
$740$	− 35249.8i	− 0.0643715i
$741$	0	0
$742$	0	0
$743$	−7359.82	−0.0133318	−0.00666591	−	0.999978i	$-0.502122\pi$
$743$	−7359.82	−0.0133318	−0.00666591	+	0.999978i	$0.502122\pi$
$744$	0	0
$745$	− 17267.5i	− 0.0311111i
$746$	−417225.	−0.749710
$747$	0	0
$748$	62747.7i	0.112149i
$749$	0	0
$750$	0	0
$751$	−701991.	−1.24466	−0.622331	−	0.782754i	$-0.713814\pi$
$751$	−701991.	−1.24466	−0.622331	+	0.782754i	$0.713814\pi$
$752$	157147.i	0.277888i
$753$	0	0
$754$	− 1.23802e6i	− 2.17764i
$755$	− 1.89141e6i	− 3.31812i
$756$	0	0
$757$	135437.	0.236344	0.118172	−	0.992993i	$-0.462297\pi$
$757$	135437.	0.236344	0.118172	+	0.992993i	$0.462297\pi$
$758$	467828.	0.814232
$759$	0	0
$760$	584130.	1.01131
$761$	76737.6i	0.132507i	0.997803	+	0.0662535i	$0.0211046\pi$
$761$	76737.6i	0.132507i	−0.997803	+	0.0662535i	$0.978895\pi$
$762$	0	0
$763$	0	0
$764$	−529378.	−0.906942
$765$	0	0
$766$	− 455262.i	− 0.775896i
$767$	−1.59867e6	−2.71749
$768$	0	0
$769$	356188.i	0.602318i	0.953574	+	0.301159i	$0.0973735\pi$
$769$	356188.i	0.602318i	−0.953574	+	0.301159i	$0.902626\pi$
$770$	0	0
$771$	0	0
$772$	113260.	0.190038
$773$	− 224878.i	− 0.376347i	−0.982136	−	0.188174i	$-0.939743\pi$
$773$	− 224878.i	− 0.376347i	0.982136	−	0.188174i	$-0.0602568\pi$
$774$	0	0
$775$	767999.i	1.27867i
$776$	271997.i	0.451689i
$777$	0	0
$778$	−627990.	−1.03751
$779$	1.05186e6	1.73333
$780$	0	0
$781$	−290220.	−0.475801
$782$	52822.2i	0.0863779i
$783$	0	0
$784$	0	0
$785$	−463489.	−0.752143
$786$	0	0
$787$	− 634743.i	− 1.02482i	−0.858740	−	0.512411i	$-0.828753\pi$
$787$	− 634743.i	− 1.02482i	0.858740	−	0.512411i	$-0.171247\pi$
$788$	−361380.	−0.581985
$789$	0	0
$790$	− 28174.3i	− 0.0451439i
$791$	0	0
$792$	0	0
$793$	−1.93285e6	−3.07364
$794$	539965.i	0.856494i
$795$	0	0
$796$	9077.52i	0.0143265i
$797$	391335.i	0.616072i	0.951375	+	0.308036i	$0.0996717\pi$
$797$	391335.i	0.616072i	−0.951375	+	0.308036i	$0.900328\pi$
$798$	0	0
$799$	290665.	0.455302
$800$	−234722.	−0.366754
$801$	0	0
$802$	274585.	0.426901
$803$	573622.i	0.889600i
$804$	0	0
$805$	0	0
$806$	457941.	0.704919
$807$	0	0
$808$	5443.84i	0.00833840i
$809$	−97284.4	−0.148644	−0.0743218	−	0.997234i	$-0.523679\pi$
$809$	−97284.4	−0.148644	−0.0743218	+	0.997234i	$0.523679\pi$
$810$	0	0
$811$	16226.2i	0.0246703i	0.999924	+	0.0123351i	$0.00392650\pi$
$811$	16226.2i	0.0246703i	−0.999924	+	0.0123351i	$0.996074\pi$
$812$	0	0
$813$	0	0
$814$	−18837.1	−0.0284292
$815$	− 1.21308e6i	− 1.82631i
$816$	0	0
$817$	13111.1i	0.0196424i
$818$	109707.i	0.163956i
$819$	0	0
$820$	−626399.	−0.931587
$821$	−1.24509e6	−1.84721	−0.923604	−	0.383349i	$-0.874771\pi$
$821$	−1.24509e6	−1.84721	−0.923604	+	0.383349i	$0.874771\pi$
$822$	0	0
$823$	−1.15336e6	−1.70280	−0.851400	−	0.524516i	$-0.824246\pi$
$823$	−1.15336e6	−1.70280	−0.851400	+	0.524516i	$0.824246\pi$
$824$	− 158720.i	− 0.233763i
$825$	0	0
$826$	0	0
$827$	609792.	0.891601	0.445800	−	0.895132i	$-0.352919\pi$
$827$	609792.	0.891601	0.445800	+	0.895132i	$0.352919\pi$
$828$	0	0
$829$	768105.i	1.11766i	0.829281	+	0.558832i	$0.188750\pi$
$829$	768105.i	1.11766i	−0.829281	+	0.558832i	$0.811250\pi$
$830$	1.43159e6	2.07808
$831$	0	0
$832$	139960.i	0.202188i
$833$	0	0
$834$	0	0
$835$	1.35277e6	1.94022
$836$	− 312152.i	− 0.446635i
$837$	0	0
$838$	54775.5i	0.0780006i
$839$	120318.i	0.170925i	0.996341	+	0.0854626i	$0.0272368\pi$
$839$	120318.i	0.170925i	−0.996341	+	0.0854626i	$0.972763\pi$
$840$	0	0
$841$	1.85661e6	2.62500
$842$	−508005.	−0.716546
$843$	0	0
$844$	−587385.	−0.824590
$845$	2.02369e6i	2.83419i
$846$	0	0
$847$	0	0
$848$	10594.9	0.0147335
$849$	0	0
$850$	434152.i	0.600902i
$851$	−15857.4	−0.0218964
$852$	0	0
$853$	− 25780.6i	− 0.0354320i	−0.999843	−	0.0177160i	$-0.994361\pi$
$853$	− 25780.6i	− 0.0354320i	0.999843	−	0.0177160i	$-0.00563947\pi$
$854$	0	0
$855$	0	0
$856$	−269763.	−0.368159
$857$	293315.i	0.399367i	0.979860	+	0.199684i	$0.0639915\pi$
$857$	293315.i	0.399367i	−0.979860	+	0.199684i	$0.936009\pi$
$858$	0	0
$859$	1.16184e6i	1.57457i	0.616590	+	0.787285i	$0.288514\pi$
$859$	1.16184e6i	1.57457i	−0.616590	+	0.787285i	$0.711486\pi$
$860$	− 7807.87i	− 0.0105569i
$861$	0	0
$862$	−66221.5	−0.0891219
$863$	367715.	0.493731	0.246865	−	0.969050i	$-0.420599\pi$
$863$	367715.	0.493731	0.246865	+	0.969050i	$0.420599\pi$
$864$	0	0
$865$	971621.	1.29857
$866$	508242.i	0.677696i
$867$	0	0
$868$	0	0
$869$	−15056.0	−0.0199375
$870$	0	0
$871$	− 984177.i	− 1.29729i
$872$	60776.1	0.0799282
$873$	0	0
$874$	− 262775.i	− 0.344002i
$875$	0	0
$876$	0	0
$877$	784503.	1.01999	0.509994	−	0.860178i	$-0.329648\pi$
$877$	784503.	1.01999	0.509994	+	0.860178i	$0.329648\pi$
$878$	− 968717.i	− 1.25663i
$879$	0	0
$880$	185892.i	0.240046i
$881$	1.26070e6i	1.62428i	0.583464	+	0.812139i	$0.301697\pi$
$881$	1.26070e6i	1.62428i	−0.583464	+	0.812139i	$0.698303\pi$
$882$	0	0
$883$	−1.01678e6	−1.30408	−0.652041	−	0.758183i	$-0.726087\pi$
$883$	−1.01678e6	−1.30408	−0.652041	+	0.758183i	$0.726087\pi$
$884$	258875.	0.331273
$885$	0	0
$886$	150916.	0.192250
$887$	− 1.18543e6i	− 1.50671i	−0.657615	−	0.753354i	$-0.728435\pi$
$887$	− 1.18543e6i	− 1.50671i	0.657615	−	0.753354i	$-0.271565\pi$
$888$	0	0
$889$	0	0
$890$	45894.4	0.0579402
$891$	0	0
$892$	445421.i	0.559810i
$893$	−1.44597e6	−1.81325
$894$	0	0
$895$	1.14576e6i	1.43037i
$896$	0	0
$897$	0	0
$898$	−251160.	−0.311457
$899$	948378.i	1.17344i
$900$	0	0
$901$	− 19596.8i	− 0.0241398i
$902$	334740.i	0.411429i
$903$	0	0
$904$	219238.	0.268275
$905$	−1.57124e6	−1.91843
$906$	0	0
$907$	708666.	0.861444	0.430722	−	0.902485i	$-0.358259\pi$
$907$	708666.	0.861444	0.430722	+	0.902485i	$0.358259\pi$
$908$	− 469004.i	− 0.568859i
$909$	0	0
$910$	0	0
$911$	1.15754e6	1.39475	0.697377	−	0.716705i	$-0.254350\pi$
$911$	1.15754e6	1.39475	0.697377	+	0.716705i	$0.254350\pi$
$912$	0	0
$913$	− 765025.i	− 0.917770i
$914$	839823.	1.00530
$915$	0	0
$916$	− 58487.4i	− 0.0697062i
$917$	0	0
$918$	0	0
$919$	−878487.	−1.04017	−0.520085	−	0.854115i	$-0.674100\pi$
$919$	−878487.	−1.04017	−0.520085	+	0.854115i	$0.674100\pi$
$920$	156487.i	0.184885i
$921$	0	0
$922$	547940.i	0.644572i
$923$	1.19735e6i	1.40545i
$924$	0	0
$925$	−130334.	−0.152326
$926$	−71957.1	−0.0839173
$927$	0	0
$928$	−289851.	−0.336573
$929$	327831.i	0.379856i	0.981798	+	0.189928i	$0.0608254\pi$
$929$	327831.i	0.379856i	−0.981798	+	0.189928i	$0.939175\pi$
$930$	0	0
$931$	0	0
$932$	−266812.	−0.307166
$933$	0	0
$934$	91754.0i	0.105180i
$935$	343833.	0.393300
$936$	0	0
$937$	918855.i	1.04657i	0.852158	+	0.523285i	$0.175294\pi$
$937$	918855.i	1.04657i	−0.852158	+	0.523285i	$0.824706\pi$
$938$	0	0
$939$	0	0
$940$	861103.	0.974539
$941$	536002.i	0.605323i	0.953098	+	0.302661i	$0.0978751\pi$
$941$	536002.i	0.605323i	−0.953098	+	0.302661i	$0.902125\pi$
$942$	0	0
$943$	281790.i	0.316885i
$944$	374288.i	0.420012i
$945$	0	0
$946$	−4172.43	−0.00466237
$947$	−480281.	−0.535545	−0.267772	−	0.963482i	$-0.586288\pi$
$947$	−480281.	−0.535545	−0.267772	+	0.963482i	$0.586288\pi$
$948$	0	0
$949$	2.36656e6	2.62776
$950$	− 2.15978e6i	− 2.39311i
$951$	0	0
$952$	0	0
$953$	307100.	0.338138	0.169069	−	0.985604i	$-0.445924\pi$
$953$	307100.	0.338138	0.169069	+	0.985604i	$0.445924\pi$
$954$	0	0
$955$	2.90078e6i	3.18060i
$956$	443905.	0.485706
$957$	0	0
$958$	− 933993.i	− 1.01768i
$959$	0	0
$960$	0	0
$961$	572718.	0.620147
$962$	77715.1i	0.0839760i
$963$	0	0
$964$	267674.i	0.288039i
$965$	− 620618.i	− 0.666453i
$966$	0	0
$967$	1.21890e6	1.30351	0.651755	−	0.758430i	$-0.274033\pi$
$967$	1.21890e6	1.30351	0.651755	+	0.758430i	$0.274033\pi$
$968$	−231950.	−0.247539
$969$	0	0
$970$	1.49043e6	1.58405
$971$	− 566221.i	− 0.600547i	−0.953853	−	0.300274i	$-0.902922\pi$
$971$	− 566221.i	− 0.600547i	0.953853	−	0.300274i	$-0.0970780\pi$
$972$	0	0
$973$	0	0
$974$	112533.	0.118621
$975$	0	0
$976$	452529.i	0.475058i
$977$	−409749.	−0.429268	−0.214634	−	0.976695i	$-0.568856\pi$
$977$	−409749.	−0.429268	−0.214634	+	0.976695i	$0.568856\pi$
$978$	0	0
$979$	− 24525.4i	− 0.0255888i
$980$	0	0
$981$	0	0
$982$	−1.12660e6	−1.16828
$983$	− 243710.i	− 0.252212i	−0.992017	−	0.126106i	$-0.959752\pi$
$983$	− 243710.i	− 0.252212i	0.992017	−	0.126106i	$-0.0402480\pi$
$984$	0	0
$985$	1.98022e6i	2.04099i
$986$	536121.i	0.551453i
$987$	0	0
$988$	−1.28783e6	−1.31930
$989$	−3512.42	−0.00359099
$990$	0	0
$991$	736653.	0.750094	0.375047	−	0.927006i	$-0.377627\pi$
$991$	736653.	0.750094	0.375047	+	0.927006i	$0.377627\pi$
$992$	− 107215.i	− 0.108951i
$993$	0	0
$994$	0	0
$995$	49741.3	0.0502424
$996$	0	0
$997$	1.80304e6i	1.81391i	0.421225	+	0.906956i	$0.361600\pi$
$997$	1.80304e6i	1.81391i	−0.421225	+	0.906956i	$0.638400\pi$
$998$	232194.	0.233126
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.c.f.685.5		8
3.2	odd	2		294.5.c.b.97.4	yes	8
7.6	odd	2	inner	882.5.c.f.685.8		8
21.2	odd	6		294.5.g.e.31.4		8
21.5	even	6		294.5.g.g.31.3		8
21.11	odd	6		294.5.g.g.19.3		8
21.17	even	6		294.5.g.e.19.4		8
21.20	even	2		294.5.c.b.97.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.5.c.b.97.1	✓	8	21.20	even	2
294.5.c.b.97.4	yes	8	3.2	odd	2
294.5.g.e.19.4		8	21.17	even	6
294.5.g.e.31.4		8	21.2	odd	6
294.5.g.g.19.3		8	21.11	odd	6
294.5.g.g.31.3		8	21.5	even	6
882.5.c.f.685.5		8	1.1	even	1	trivial
882.5.c.f.685.8		8	7.6	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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} +8.00000 q^{4} -43.8369i q^{5} +22.6274 q^{8} +O(q^{10})\)	\(q+2.82843 q^{2} +8.00000 q^{4} -43.8369i q^{5} +22.6274 q^{8} -123.989i q^{10} -66.2583 q^{11} +273.359i q^{13} +64.0000 q^{16} -118.377i q^{17} +588.891i q^{19} -350.695i q^{20} -187.407 q^{22} -157.762 q^{23} -1296.67 q^{25} +773.175i q^{26} -1601.22 q^{29} -592.286i q^{31} +181.019 q^{32} -334.821i q^{34} +100.514 q^{37} +1665.64i q^{38} -991.915i q^{40} -1786.17i q^{41} +22.2640 q^{43} -530.067 q^{44} -446.220 q^{46} +2455.42i q^{47} -3667.54 q^{50} +2186.87i q^{52} +165.545 q^{53} +2904.56i q^{55} -4528.92 q^{58} +5848.25i q^{59} +7070.76i q^{61} -1675.24i q^{62} +512.000 q^{64} +11983.2 q^{65} -3600.31 q^{67} -947.016i q^{68} +4380.13 q^{71} -8657.35i q^{73} +284.297 q^{74} +4711.13i q^{76} +227.232 q^{79} -2805.56i q^{80} -5052.04i q^{82} +11546.1i q^{83} -5189.28 q^{85} +62.9721 q^{86} -1499.26 q^{88} +370.148i q^{89} -1262.10 q^{92} +6944.98i q^{94} +25815.1 q^{95} +12020.7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 64 q^{4}+O(q^{10}) \) 8 * q + 64 * q^4	\( 8 q + 64 q^{4} + 512 q^{16} - 1312 q^{22} + 272 q^{23} - 2808 q^{25} - 400 q^{29} - 3328 q^{37} + 656 q^{43} - 2400 q^{46} - 800 q^{50} - 9264 q^{53} - 11488 q^{58} + 4096 q^{64} + 15696 q^{65} + 26816 q^{67} - 28192 q^{71} + 4512 q^{74} + 19728 q^{79} - 49632 q^{85} - 5888 q^{86} - 10496 q^{88} + 2176 q^{92} + 92752 q^{95}+O(q^{100}) \) 8 * q + 64 * q^4 + 512 * q^16 - 1312 * q^22 + 272 * q^23 - 2808 * q^25 - 400 * q^29 - 3328 * q^37 + 656 * q^43 - 2400 * q^46 - 800 * q^50 - 9264 * q^53 - 11488 * q^58 + 4096 * q^64 + 15696 * q^65 + 26816 * q^67 - 28192 * q^71 + 4512 * q^74 + 19728 * q^79 - 49632 * q^85 - 5888 * q^86 - 10496 * q^88 + 2176 * q^92 + 92752 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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