LMFDB - Embedded newform 378.3.b.c.323.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,3,Mod(323,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("378.323");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	378.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:	$10.2997539928$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.3
Root			$-1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.3.b.c.323.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +2.15587i q^{5} -2.64575 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} +2.15587i q^{5} -2.64575 q^{7} +2.82843i q^{8} +3.04886 q^{10} -0.218617i q^{11} +7.82186 q^{13} +3.74166i q^{14} +4.00000 q^{16} +13.7186i q^{17} +28.4049 q^{19} -4.31174i q^{20} -0.309171 q^{22} +29.7010i q^{23} +20.3522 q^{25} -11.0618i q^{26} +5.29150 q^{28} +8.12846i q^{29} +13.3012 q^{31} -5.65685i q^{32} +19.4011 q^{34} -5.70390i q^{35} +45.8097 q^{37} -40.1705i q^{38} -6.09772 q^{40} +49.4028i q^{41} -30.4441 q^{43} +0.437234i q^{44} +42.0035 q^{46} -67.7431i q^{47} +7.00000 q^{49} -28.7824i q^{50} -15.6437 q^{52} +10.7431i q^{53} +0.471310 q^{55} -7.48331i q^{56} +11.4954 q^{58} +7.58815i q^{59} +20.9845 q^{61} -18.8108i q^{62} -8.00000 q^{64} +16.8629i q^{65} +45.4987 q^{67} -27.4373i q^{68} -8.06653 q^{70} +16.5274i q^{71} -67.5611 q^{73} -64.7847i q^{74} -56.8097 q^{76} +0.578406i q^{77} -44.0134 q^{79} +8.62348i q^{80} +69.8662 q^{82} -42.6135i q^{83} -29.5756 q^{85} +43.0544i q^{86} +0.618342 q^{88} +70.4999i q^{89} -20.6947 q^{91} -59.4019i q^{92} -95.8032 q^{94} +61.2372i q^{95} -88.6462 q^{97} -9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} + 16 q^{10} - 40 q^{13} + 32 q^{16} + 40 q^{19} - 16 q^{22} - 40 q^{31} - 16 q^{34} - 8 q^{37} - 32 q^{40} + 40 q^{43} - 64 q^{46} + 56 q^{49} + 80 q^{52} + 56 q^{55} + 112 q^{58} - 88 q^{61} - 64 q^{64} + 240 q^{67} - 112 q^{70} - 288 q^{73} - 80 q^{76} - 424 q^{79} + 144 q^{82} + 200 q^{85} + 32 q^{88} + 56 q^{91} + 288 q^{94} + 600 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^10 - 40 * q^13 + 32 * q^16 + 40 * q^19 - 16 * q^22 - 40 * q^31 - 16 * q^34 - 8 * q^37 - 32 * q^40 + 40 * q^43 - 64 * q^46 + 56 * q^49 + 80 * q^52 + 56 * q^55 + 112 * q^58 - 88 * q^61 - 64 * q^64 + 240 * q^67 - 112 * q^70 - 288 * q^73 - 80 * q^76 - 424 * q^79 + 144 * q^82 + 200 * q^85 + 32 * q^88 + 56 * q^91 + 288 * q^94 + 600 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	2.15587i	0.431174i	0.976485	+	0.215587i	$0.0691665\pi$
$5$	2.15587i	0.431174i	−0.976485	+	0.215587i	$0.930833\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	2.82843i	0.353553i
$9$	0	0
$10$	3.04886	0.304886
$11$	− 0.218617i	− 0.0198743i	−0.999951	−	0.00993714i	$-0.996837\pi$
$11$	− 0.218617i	− 0.0198743i	0.999951	−	0.00993714i	$-0.00316314\pi$
$12$	0	0
$13$	7.82186	0.601682	0.300841	−	0.953674i	$-0.402733\pi$
$13$	7.82186	0.601682	0.300841	+	0.953674i	$0.402733\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	13.7186i	0.806979i	0.914984	+	0.403489i	$0.132203\pi$
$17$	13.7186i	0.806979i	−0.914984	+	0.403489i	$0.867797\pi$
$18$	0	0
$19$	28.4049	1.49499	0.747496	−	0.664266i	$-0.231256\pi$
$19$	28.4049	1.49499	0.747496	+	0.664266i	$0.231256\pi$
$20$	− 4.31174i	− 0.215587i
$21$	0	0
$22$	−0.309171	−0.0140532
$23$	29.7010i	1.29135i	0.763614	+	0.645673i	$0.223423\pi$
$23$	29.7010i	1.29135i	−0.763614	+	0.645673i	$0.776577\pi$
$24$	0	0
$25$	20.3522	0.814089
$26$	− 11.0618i	− 0.425453i
$27$	0	0
$28$	5.29150	0.188982
$29$	8.12846i	0.280292i	0.990131	+	0.140146i	$0.0447572\pi$
$29$	8.12846i	0.280292i	−0.990131	+	0.140146i	$0.955243\pi$
$30$	0	0
$31$	13.3012	0.429072	0.214536	−	0.976716i	$-0.431176\pi$
$31$	13.3012	0.429072	0.214536	+	0.976716i	$0.431176\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	19.4011	0.570620
$35$	− 5.70390i	− 0.162969i
$36$	0	0
$37$	45.8097	1.23810	0.619050	−	0.785351i	$-0.287518\pi$
$37$	45.8097	1.23810	0.619050	+	0.785351i	$0.287518\pi$
$38$	− 40.1705i	− 1.05712i
$39$	0	0
$40$	−6.09772	−0.152443
$41$	49.4028i	1.20495i	0.798139	+	0.602474i	$0.205818\pi$
$41$	49.4028i	1.20495i	−0.798139	+	0.602474i	$0.794182\pi$
$42$	0	0
$43$	−30.4441	−0.708002	−0.354001	−	0.935245i	$-0.615179\pi$
$43$	−30.4441	−0.708002	−0.354001	+	0.935245i	$0.615179\pi$
$44$	0.437234i	0.00993714i
$45$	0	0
$46$	42.0035	0.913120
$47$	− 67.7431i	− 1.44134i	−0.693277	−	0.720671i	$-0.743834\pi$
$47$	− 67.7431i	− 1.44134i	0.693277	−	0.720671i	$-0.256166\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	− 28.7824i	− 0.575648i
$51$	0	0
$52$	−15.6437	−0.300841
$53$	10.7431i	0.202700i	0.994851	+	0.101350i	$0.0323162\pi$
$53$	10.7431i	0.202700i	−0.994851	+	0.101350i	$0.967684\pi$
$54$	0	0
$55$	0.471310	0.00856927
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	11.4954	0.198196
$59$	7.58815i	0.128613i	0.997930	+	0.0643063i	$0.0204835\pi$
$59$	7.58815i	0.128613i	−0.997930	+	0.0643063i	$0.979517\pi$
$60$	0	0
$61$	20.9845	0.344009	0.172004	−	0.985096i	$-0.444976\pi$
$61$	20.9845	0.344009	0.172004	+	0.985096i	$0.444976\pi$
$62$	− 18.8108i	− 0.303400i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	16.8629i	0.259430i
$66$	0	0
$67$	45.4987	0.679086	0.339543	−	0.940591i	$-0.389728\pi$
$67$	45.4987	0.679086	0.339543	+	0.940591i	$0.389728\pi$
$68$	− 27.4373i	− 0.403489i
$69$	0	0
$70$	−8.06653	−0.115236
$71$	16.5274i	0.232780i	0.993204	+	0.116390i	$0.0371323\pi$
$71$	16.5274i	0.232780i	−0.993204	+	0.116390i	$0.962868\pi$
$72$	0	0
$73$	−67.5611	−0.925495	−0.462748	−	0.886490i	$-0.653136\pi$
$73$	−67.5611	−0.925495	−0.462748	+	0.886490i	$0.653136\pi$
$74$	− 64.7847i	− 0.875470i
$75$	0	0
$76$	−56.8097	−0.747496
$77$	0.578406i	0.00751177i
$78$	0	0
$79$	−44.0134	−0.557132	−0.278566	−	0.960417i	$-0.589859\pi$
$79$	−44.0134	−0.557132	−0.278566	+	0.960417i	$0.589859\pi$
$80$	8.62348i	0.107794i
$81$	0	0
$82$	69.8662	0.852026
$83$	− 42.6135i	− 0.513416i	−0.966489	−	0.256708i	$-0.917362\pi$
$83$	− 42.6135i	− 0.513416i	0.966489	−	0.256708i	$-0.0826379\pi$
$84$	0	0
$85$	−29.5756	−0.347948
$86$	43.0544i	0.500633i
$87$	0	0
$88$	0.618342	0.00702662
$89$	70.4999i	0.792133i	0.918222	+	0.396067i	$0.129625\pi$
$89$	70.4999i	0.792133i	−0.918222	+	0.396067i	$0.870375\pi$
$90$	0	0
$91$	−20.6947	−0.227414
$92$	− 59.4019i	− 0.645673i
$93$	0	0
$94$	−95.8032	−1.01918
$95$	61.2372i	0.644602i
$96$	0	0
$97$	−88.6462	−0.913878	−0.456939	−	0.889498i	$-0.651054\pi$
$97$	−88.6462	−0.913878	−0.456939	+	0.889498i	$0.651054\pi$
$98$	− 9.89949i	− 0.101015i
$99$	0	0
$100$	−40.7044	−0.407044
$101$	− 114.464i	− 1.13331i	−0.823955	−	0.566656i	$-0.808237\pi$
$101$	− 114.464i	− 1.13331i	0.823955	−	0.566656i	$-0.191763\pi$
$102$	0	0
$103$	52.2420	0.507204	0.253602	−	0.967309i	$-0.418385\pi$
$103$	52.2420	0.507204	0.253602	+	0.967309i	$0.418385\pi$
$104$	22.1236i	0.212727i
$105$	0	0
$106$	15.1931	0.143331
$107$	145.416i	1.35903i	0.733661	+	0.679516i	$0.237810\pi$
$107$	145.416i	1.35903i	−0.733661	+	0.679516i	$0.762190\pi$
$108$	0	0
$109$	−4.11113	−0.0377168	−0.0188584	−	0.999822i	$-0.506003\pi$
$109$	−4.11113	−0.0377168	−0.0188584	+	0.999822i	$0.506003\pi$
$110$	− 0.666533i	− 0.00605939i
$111$	0	0
$112$	−10.5830	−0.0944911
$113$	196.330i	1.73744i	0.495307	+	0.868718i	$0.335055\pi$
$113$	196.330i	1.73744i	−0.495307	+	0.868718i	$0.664945\pi$
$114$	0	0
$115$	−64.0315	−0.556795
$116$	− 16.2569i	− 0.140146i
$117$	0	0
$118$	10.7313	0.0909429
$119$	− 36.2961i	− 0.305009i
$120$	0	0
$121$	120.952	0.999605
$122$	− 29.6766i	− 0.243251i
$123$	0	0
$124$	−26.6025	−0.214536
$125$	97.7735i	0.782188i
$126$	0	0
$127$	−197.694	−1.55664	−0.778322	−	0.627866i	$-0.783929\pi$
$127$	−197.694	−1.55664	−0.778322	+	0.627866i	$0.783929\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	23.8478	0.183444
$131$	− 154.363i	− 1.17834i	−0.808009	−	0.589170i	$-0.799455\pi$
$131$	− 154.363i	− 1.17834i	0.808009	−	0.589170i	$-0.200545\pi$
$132$	0	0
$133$	−75.1522	−0.565054
$134$	− 64.3449i	− 0.480186i
$135$	0	0
$136$	−38.8022	−0.285310
$137$	− 55.4411i	− 0.404679i	−0.979315	−	0.202340i	$-0.935145\pi$
$137$	− 55.4411i	− 0.404679i	0.979315	−	0.202340i	$-0.0648545\pi$
$138$	0	0
$139$	−212.311	−1.52742	−0.763708	−	0.645562i	$-0.776623\pi$
$139$	−212.311	−1.52742	−0.763708	+	0.645562i	$0.776623\pi$
$140$	11.4078i	0.0814843i
$141$	0	0
$142$	23.3733	0.164600
$143$	− 1.70999i	− 0.0119580i
$144$	0	0
$145$	−17.5239	−0.120855
$146$	95.5459i	0.654424i
$147$	0	0
$148$	−91.6195	−0.619050
$149$	162.733i	1.09217i	0.837730	+	0.546084i	$0.183882\pi$
$149$	162.733i	1.09217i	−0.837730	+	0.546084i	$0.816118\pi$
$150$	0	0
$151$	264.701	1.75299	0.876495	−	0.481411i	$-0.159876\pi$
$151$	264.701	1.75299	0.876495	+	0.481411i	$0.159876\pi$
$152$	80.3411i	0.528560i
$153$	0	0
$154$	0.817990	0.00531162
$155$	28.6758i	0.185005i
$156$	0	0
$157$	−54.8371	−0.349281	−0.174640	−	0.984632i	$-0.555876\pi$
$157$	−54.8371	−0.349281	−0.174640	+	0.984632i	$0.555876\pi$
$158$	62.2444i	0.393952i
$159$	0	0
$160$	12.1954	0.0762215
$161$	− 78.5814i	− 0.488083i
$162$	0	0
$163$	−20.0871	−0.123234	−0.0616168	−	0.998100i	$-0.519626\pi$
$163$	−20.0871	−0.123234	−0.0616168	+	0.998100i	$0.519626\pi$
$164$	− 98.8057i	− 0.602474i
$165$	0	0
$166$	−60.2646	−0.363040
$167$	− 85.5452i	− 0.512247i	−0.966644	−	0.256123i	$-0.917555\pi$
$167$	− 85.5452i	− 0.512247i	0.966644	−	0.256123i	$-0.0824453\pi$
$168$	0	0
$169$	−107.818	−0.637979
$170$	41.8262i	0.246037i
$171$	0	0
$172$	60.8882	0.354001
$173$	18.7079i	0.108138i	0.998537	+	0.0540690i	$0.0172191\pi$
$173$	18.7079i	0.108138i	−0.998537	+	0.0540690i	$0.982781\pi$
$174$	0	0
$175$	−53.8469	−0.307697
$176$	− 0.874468i	− 0.00496857i
$177$	0	0
$178$	99.7019	0.560123
$179$	− 253.608i	− 1.41680i	−0.705810	−	0.708401i	$-0.749417\pi$
$179$	− 253.608i	− 1.41680i	0.705810	−	0.708401i	$-0.250583\pi$
$180$	0	0
$181$	−288.252	−1.59256	−0.796278	−	0.604931i	$-0.793200\pi$
$181$	−288.252	−1.59256	−0.796278	+	0.604931i	$0.793200\pi$
$182$	29.2667i	0.160806i
$183$	0	0
$184$	−84.0070	−0.456560
$185$	98.7599i	0.533837i
$186$	0	0
$187$	2.99913	0.0160381
$188$	135.486i	0.720671i
$189$	0	0
$190$	86.6025	0.455803
$191$	− 320.955i	− 1.68039i	−0.542284	−	0.840195i	$-0.682440\pi$
$191$	− 320.955i	− 1.68039i	0.542284	−	0.840195i	$-0.317560\pi$
$192$	0	0
$193$	353.509	1.83165	0.915827	−	0.401573i	$-0.131536\pi$
$193$	353.509	1.83165	0.915827	+	0.401573i	$0.131536\pi$
$194$	125.365i	0.646209i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	105.750i	0.536803i	0.963307	+	0.268401i	$0.0864953\pi$
$197$	105.750i	0.536803i	−0.963307	+	0.268401i	$0.913505\pi$
$198$	0	0
$199$	−143.140	−0.719298	−0.359649	−	0.933088i	$-0.617103\pi$
$199$	−143.140	−0.719298	−0.359649	+	0.933088i	$0.617103\pi$
$200$	57.5648i	0.287824i
$201$	0	0
$202$	−161.877	−0.801372
$203$	− 21.5059i	− 0.105940i
$204$	0	0
$205$	−106.506	−0.519542
$206$	− 73.8814i	− 0.358648i
$207$	0	0
$208$	31.2874	0.150420
$209$	− 6.20979i	− 0.0297119i
$210$	0	0
$211$	338.905	1.60619	0.803094	−	0.595853i	$-0.203186\pi$
$211$	338.905	1.60619	0.803094	+	0.595853i	$0.203186\pi$
$212$	− 21.4862i	− 0.101350i
$213$	0	0
$214$	205.650	0.960981
$215$	− 65.6335i	− 0.305272i
$216$	0	0
$217$	−35.1918	−0.162174
$218$	5.81402i	0.0266698i
$219$	0	0
$220$	−0.942620	−0.00428464
$221$	107.305i	0.485544i
$222$	0	0
$223$	275.449	1.23520	0.617598	−	0.786494i	$-0.288106\pi$
$223$	275.449	1.23520	0.617598	+	0.786494i	$0.288106\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	277.653	1.22855
$227$	− 49.9671i	− 0.220119i	−0.993925	−	0.110060i	$-0.964896\pi$
$227$	− 49.9671i	− 0.220119i	0.993925	−	0.110060i	$-0.0351042\pi$
$228$	0	0
$229$	341.137	1.48968	0.744840	−	0.667243i	$-0.232526\pi$
$229$	341.137	1.48968	0.744840	+	0.667243i	$0.232526\pi$
$230$	90.5542i	0.393714i
$231$	0	0
$232$	−22.9908	−0.0990981
$233$	− 145.863i	− 0.626019i	−0.949750	−	0.313010i	$-0.898663\pi$
$233$	− 145.863i	− 0.626019i	0.949750	−	0.313010i	$-0.101337\pi$
$234$	0	0
$235$	146.045	0.621470
$236$	− 15.1763i	− 0.0643063i
$237$	0	0
$238$	−51.3304	−0.215674
$239$	− 338.858i	− 1.41782i	−0.705300	−	0.708909i	$-0.749188\pi$
$239$	− 338.858i	− 1.41782i	0.705300	−	0.708909i	$-0.250812\pi$
$240$	0	0
$241$	−63.6954	−0.264296	−0.132148	−	0.991230i	$-0.542187\pi$
$241$	−63.6954	−0.264296	−0.132148	+	0.991230i	$0.542187\pi$
$242$	− 171.052i	− 0.706827i
$243$	0	0
$244$	−41.9691	−0.172004
$245$	15.0911i	0.0615963i
$246$	0	0
$247$	222.179	0.899510
$248$	37.6216i	0.151700i
$249$	0	0
$250$	138.273	0.553091
$251$	− 319.643i	− 1.27348i	−0.771079	−	0.636739i	$-0.780283\pi$
$251$	− 319.643i	− 1.27348i	0.771079	−	0.636739i	$-0.219717\pi$
$252$	0	0
$253$	6.49314	0.0256646
$254$	279.581i	1.10071i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 459.293i	− 1.78713i	−0.448930	−	0.893567i	$-0.648195\pi$
$257$	− 459.293i	− 1.78713i	0.448930	−	0.893567i	$-0.351805\pi$
$258$	0	0
$259$	−121.201	−0.467958
$260$	− 33.7258i	− 0.129715i
$261$	0	0
$262$	−218.302	−0.833212
$263$	− 33.4521i	− 0.127194i	−0.997976	−	0.0635972i	$-0.979743\pi$
$263$	− 33.4521i	− 0.127194i	0.997976	−	0.0635972i	$-0.0202573\pi$
$264$	0	0
$265$	−23.1608	−0.0873991
$266$	106.281i	0.399554i
$267$	0	0
$268$	−90.9975	−0.339543
$269$	156.125i	0.580389i	0.956968	+	0.290194i	$0.0937200\pi$
$269$	156.125i	0.580389i	−0.956968	+	0.290194i	$0.906280\pi$
$270$	0	0
$271$	392.487	1.44829	0.724145	−	0.689648i	$-0.242235\pi$
$271$	392.487	1.44829	0.724145	+	0.689648i	$0.242235\pi$
$272$	54.8745i	0.201745i
$273$	0	0
$274$	−78.4055	−0.286152
$275$	− 4.44934i	− 0.0161794i
$276$	0	0
$277$	412.741	1.49004	0.745019	−	0.667043i	$-0.232440\pi$
$277$	412.741	1.49004	0.745019	+	0.667043i	$0.232440\pi$
$278$	300.253i	1.08005i
$279$	0	0
$280$	16.1331	0.0576181
$281$	497.582i	1.77076i	0.464872	+	0.885378i	$0.346100\pi$
$281$	497.582i	1.77076i	−0.464872	+	0.885378i	$0.653900\pi$
$282$	0	0
$283$	−119.223	−0.421281	−0.210640	−	0.977564i	$-0.567555\pi$
$283$	−119.223	−0.421281	−0.210640	+	0.977564i	$0.567555\pi$
$284$	− 33.0548i	− 0.116390i
$285$	0	0
$286$	−2.41829	−0.00845557
$287$	− 130.708i	− 0.455427i
$288$	0	0
$289$	100.799	0.348785
$290$	24.7826i	0.0854571i
$291$	0	0
$292$	135.122	0.462748
$293$	− 143.232i	− 0.488846i	−0.969669	−	0.244423i	$-0.921401\pi$
$293$	− 143.232i	− 0.488846i	0.969669	−	0.244423i	$-0.0785986\pi$
$294$	0	0
$295$	−16.3591	−0.0554545
$296$	129.569i	0.437735i
$297$	0	0
$298$	230.139	0.772279
$299$	232.317i	0.776979i
$300$	0	0
$301$	80.5475	0.267600
$302$	− 374.344i	− 1.23955i
$303$	0	0
$304$	113.619	0.373748
$305$	45.2399i	0.148328i
$306$	0	0
$307$	−424.312	−1.38212	−0.691061	−	0.722796i	$-0.742857\pi$
$307$	−424.312	−1.38212	−0.691061	+	0.722796i	$0.742857\pi$
$308$	− 1.15681i	− 0.00375588i
$309$	0	0
$310$	40.5537	0.130818
$311$	− 235.641i	− 0.757687i	−0.925461	−	0.378843i	$-0.876322\pi$
$311$	− 235.641i	− 0.757687i	0.925461	−	0.378843i	$-0.123678\pi$
$312$	0	0
$313$	−329.722	−1.05342	−0.526712	−	0.850044i	$-0.676575\pi$
$313$	−329.722	−1.05342	−0.526712	+	0.850044i	$0.676575\pi$
$314$	77.5513i	0.246979i
$315$	0	0
$316$	88.0269	0.278566
$317$	− 40.0026i	− 0.126191i	−0.998007	−	0.0630956i	$-0.979903\pi$
$317$	− 40.0026i	− 0.126191i	0.998007	−	0.0630956i	$-0.0200973\pi$
$318$	0	0
$319$	1.77702	0.00557060
$320$	− 17.2470i	− 0.0538968i
$321$	0	0
$322$	−111.131	−0.345127
$323$	389.676i	1.20643i
$324$	0	0
$325$	159.192	0.489822
$326$	28.4074i	0.0871393i
$327$	0	0
$328$	−139.732	−0.426013
$329$	179.231i	0.544776i
$330$	0	0
$331$	96.4638	0.291431	0.145716	−	0.989327i	$-0.453452\pi$
$331$	96.4638	0.291431	0.145716	+	0.989327i	$0.453452\pi$
$332$	85.2271i	0.256708i
$333$	0	0
$334$	−120.979	−0.362213
$335$	98.0894i	0.292804i
$336$	0	0
$337$	−91.0689	−0.270234	−0.135117	−	0.990830i	$-0.543141\pi$
$337$	−91.0689	−0.270234	−0.135117	+	0.990830i	$0.543141\pi$
$338$	152.478i	0.451119i
$339$	0	0
$340$	59.1512	0.173974
$341$	− 2.90788i	− 0.00852750i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	− 86.1088i	− 0.250316i
$345$	0	0
$346$	26.4569	0.0764650
$347$	457.429i	1.31824i	0.752039	+	0.659119i	$0.229071\pi$
$347$	457.429i	1.31824i	−0.752039	+	0.659119i	$0.770929\pi$
$348$	0	0
$349$	−54.8066	−0.157039	−0.0785194	−	0.996913i	$-0.525019\pi$
$349$	−54.8066	−0.157039	−0.0785194	+	0.996913i	$0.525019\pi$
$350$	76.1510i	0.217574i
$351$	0	0
$352$	−1.23668	−0.00351331
$353$	603.210i	1.70881i	0.519607	+	0.854405i	$0.326078\pi$
$353$	603.210i	1.70881i	−0.519607	+	0.854405i	$0.673922\pi$
$354$	0	0
$355$	−35.6309	−0.100369
$356$	− 141.000i	− 0.396067i
$357$	0	0
$358$	−358.655	−1.00183
$359$	− 703.760i	− 1.96033i	−0.198171	−	0.980167i	$-0.563500\pi$
$359$	− 703.760i	− 1.96033i	0.198171	−	0.980167i	$-0.436500\pi$
$360$	0	0
$361$	445.836	1.23500
$362$	407.651i	1.12611i
$363$	0	0
$364$	41.3894	0.113707
$365$	− 145.653i	− 0.399050i
$366$	0	0
$367$	−367.070	−1.00019	−0.500095	−	0.865971i	$-0.666702\pi$
$367$	−367.070	−1.00019	−0.500095	+	0.865971i	$0.666702\pi$
$368$	118.804i	0.322837i
$369$	0	0
$370$	139.668	0.377480
$371$	− 28.4236i	− 0.0766135i
$372$	0	0
$373$	140.081	0.375553	0.187776	−	0.982212i	$-0.439872\pi$
$373$	140.081	0.375553	0.187776	+	0.982212i	$0.439872\pi$
$374$	− 4.24141i	− 0.0113407i
$375$	0	0
$376$	191.606	0.509592
$377$	63.5797i	0.168646i
$378$	0	0
$379$	−244.188	−0.644295	−0.322147	−	0.946690i	$-0.604405\pi$
$379$	−244.188	−0.644295	−0.322147	+	0.946690i	$0.604405\pi$
$380$	− 122.474i	− 0.322301i
$381$	0	0
$382$	−453.898	−1.18822
$383$	− 634.185i	− 1.65584i	−0.560849	−	0.827918i	$-0.689525\pi$
$383$	− 634.185i	− 1.65584i	0.560849	−	0.827918i	$-0.310475\pi$
$384$	0	0
$385$	−1.24697	−0.00323888
$386$	− 499.938i	− 1.29517i
$387$	0	0
$388$	177.292	0.456939
$389$	459.064i	1.18011i	0.807362	+	0.590056i	$0.200895\pi$
$389$	459.064i	1.18011i	−0.807362	+	0.590056i	$0.799105\pi$
$390$	0	0
$391$	−407.457	−1.04209
$392$	19.7990i	0.0505076i
$393$	0	0
$394$	149.553	0.379577
$395$	− 94.8873i	− 0.240221i
$396$	0	0
$397$	−145.132	−0.365573	−0.182786	−	0.983153i	$-0.558512\pi$
$397$	−145.132	−0.365573	−0.182786	+	0.983153i	$0.558512\pi$
$398$	202.431i	0.508620i
$399$	0	0
$400$	81.4089	0.203522
$401$	− 81.4307i	− 0.203069i	−0.994832	−	0.101534i	$-0.967625\pi$
$401$	− 81.4307i	− 0.203069i	0.994832	−	0.101534i	$-0.0323752\pi$
$402$	0	0
$403$	104.040	0.258165
$404$	228.929i	0.566656i
$405$	0	0
$406$	−30.4139	−0.0749111
$407$	− 10.0148i	− 0.0246064i
$408$	0	0
$409$	−470.737	−1.15095	−0.575473	−	0.817821i	$-0.695182\pi$
$409$	−470.737	−1.15095	−0.575473	+	0.817821i	$0.695182\pi$
$410$	150.622i	0.367372i
$411$	0	0
$412$	−104.484	−0.253602
$413$	− 20.0763i	− 0.0486110i
$414$	0	0
$415$	91.8693	0.221372
$416$	− 44.2471i	− 0.106363i
$417$	0	0
$418$	−8.78197	−0.0210095
$419$	− 700.211i	− 1.67115i	−0.549377	−	0.835574i	$-0.685135\pi$
$419$	− 700.211i	− 1.67115i	0.549377	−	0.835574i	$-0.314865\pi$
$420$	0	0
$421$	−597.965	−1.42035	−0.710173	−	0.704027i	$-0.751383\pi$
$421$	−597.965	−1.42035	−0.710173	+	0.704027i	$0.751383\pi$
$422$	− 479.285i	− 1.13575i
$423$	0	0
$424$	−30.3861	−0.0716654
$425$	279.205i	0.656952i
$426$	0	0
$427$	−55.5199	−0.130023
$428$	− 290.833i	− 0.679516i
$429$	0	0
$430$	−92.8198	−0.215860
$431$	− 72.6764i	− 0.168623i	−0.996439	−	0.0843114i	$-0.973131\pi$
$431$	− 72.6764i	− 0.168623i	0.996439	−	0.0843114i	$-0.0268691\pi$
$432$	0	0
$433$	129.145	0.298256	0.149128	−	0.988818i	$-0.452353\pi$
$433$	129.145	0.298256	0.149128	+	0.988818i	$0.452353\pi$
$434$	49.7687i	0.114674i
$435$	0	0
$436$	8.22226	0.0188584
$437$	843.652i	1.93055i
$438$	0	0
$439$	298.510	0.679978	0.339989	−	0.940429i	$-0.389577\pi$
$439$	298.510	0.679978	0.339989	+	0.940429i	$0.389577\pi$
$440$	1.33307i	0.00302970i
$441$	0	0
$442$	151.753	0.343332
$443$	171.292i	0.386663i	0.981133	+	0.193332i	$0.0619293\pi$
$443$	171.292i	0.386663i	−0.981133	+	0.193332i	$0.938071\pi$
$444$	0	0
$445$	−151.989	−0.341547
$446$	− 389.543i	− 0.873416i
$447$	0	0
$448$	21.1660	0.0472456
$449$	134.626i	0.299835i	0.988698	+	0.149918i	$0.0479009\pi$
$449$	134.626i	0.299835i	−0.988698	+	0.149918i	$0.952099\pi$
$450$	0	0
$451$	10.8003	0.0239475
$452$	− 392.660i	− 0.868718i
$453$	0	0
$454$	−70.6641	−0.155648
$455$	− 44.6151i	− 0.0980552i
$456$	0	0
$457$	−771.061	−1.68722	−0.843612	−	0.536953i	$-0.819575\pi$
$457$	−771.061	−1.68722	−0.843612	+	0.536953i	$0.819575\pi$
$458$	− 482.440i	− 1.05336i
$459$	0	0
$460$	128.063	0.278398
$461$	− 321.704i	− 0.697840i	−0.937152	−	0.348920i	$-0.886548\pi$
$461$	− 321.704i	− 0.697840i	0.937152	−	0.348920i	$-0.113452\pi$
$462$	0	0
$463$	−317.176	−0.685046	−0.342523	−	0.939509i	$-0.611281\pi$
$463$	−317.176	−0.685046	−0.342523	+	0.939509i	$0.611281\pi$
$464$	32.5138i	0.0700729i
$465$	0	0
$466$	−206.281	−0.442663
$467$	740.066i	1.58472i	0.610051	+	0.792362i	$0.291149\pi$
$467$	740.066i	1.58472i	−0.610051	+	0.792362i	$0.708851\pi$
$468$	0	0
$469$	−120.378	−0.256670
$470$	− 206.539i	− 0.439446i
$471$	0	0
$472$	−21.4625	−0.0454714
$473$	6.65559i	0.0140710i
$474$	0	0
$475$	578.102	1.21706
$476$	72.5922i	0.152505i
$477$	0	0
$478$	−479.218	−1.00255
$479$	46.9277i	0.0979702i	0.998800	+	0.0489851i	$0.0155987\pi$
$479$	46.9277i	0.0979702i	−0.998800	+	0.0489851i	$0.984401\pi$
$480$	0	0
$481$	358.317	0.744943
$482$	90.0789i	0.186886i
$483$	0	0
$484$	−241.904	−0.499803
$485$	− 191.110i	− 0.394041i
$486$	0	0
$487$	−605.503	−1.24333	−0.621666	−	0.783282i	$-0.713544\pi$
$487$	−605.503	−1.24333	−0.621666	+	0.783282i	$0.713544\pi$
$488$	59.3532i	0.121625i
$489$	0	0
$490$	21.3420	0.0435552
$491$	471.667i	0.960625i	0.877097	+	0.480312i	$0.159477\pi$
$491$	471.667i	0.960625i	−0.877097	+	0.480312i	$0.840523\pi$
$492$	0	0
$493$	−111.511	−0.226189
$494$	− 314.208i	− 0.636050i
$495$	0	0
$496$	53.2050	0.107268
$497$	− 43.7274i	− 0.0879826i
$498$	0	0
$499$	−414.969	−0.831600	−0.415800	−	0.909456i	$-0.636498\pi$
$499$	−414.969	−0.831600	−0.415800	+	0.909456i	$0.636498\pi$
$500$	− 195.547i	− 0.391094i
$501$	0	0
$502$	−452.044	−0.900485
$503$	− 237.703i	− 0.472571i	−0.971684	−	0.236286i	$-0.924070\pi$
$503$	− 237.703i	− 0.472571i	0.971684	−	0.236286i	$-0.0759301\pi$
$504$	0	0
$505$	246.771	0.488655
$506$	− 9.18268i	− 0.0181476i
$507$	0	0
$508$	395.387	0.778322
$509$	− 835.313i	− 1.64109i	−0.571584	−	0.820544i	$-0.693671\pi$
$509$	− 835.313i	− 1.64109i	0.571584	−	0.820544i	$-0.306329\pi$
$510$	0	0
$511$	178.750	0.349804
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−649.539	−1.26369
$515$	112.627i	0.218693i
$516$	0	0
$517$	−14.8098	−0.0286456
$518$	171.404i	0.330896i
$519$	0	0
$520$	−47.6956	−0.0917222
$521$	− 85.4808i	− 0.164071i	−0.996629	−	0.0820353i	$-0.973858\pi$
$521$	− 85.4808i	− 0.164071i	0.996629	−	0.0820353i	$-0.0261420\pi$
$522$	0	0
$523$	−129.321	−0.247268	−0.123634	−	0.992328i	$-0.539455\pi$
$523$	−129.321	−0.247268	−0.123634	+	0.992328i	$0.539455\pi$
$524$	308.725i	0.589170i
$525$	0	0
$526$	−47.3084	−0.0899400
$527$	182.475i	0.346252i
$528$	0	0
$529$	−353.148	−0.667576
$530$	32.7543i	0.0618005i
$531$	0	0
$532$	150.304	0.282527
$533$	386.422i	0.724995i
$534$	0	0
$535$	−313.499	−0.585980
$536$	128.690i	0.240093i
$537$	0	0
$538$	220.794	0.410397
$539$	− 1.53032i	− 0.00283918i
$540$	0	0
$541$	750.291	1.38686	0.693430	−	0.720524i	$-0.256099\pi$
$541$	750.291	1.38686	0.693430	+	0.720524i	$0.256099\pi$
$542$	− 555.060i	− 1.02410i
$543$	0	0
$544$	77.6043	0.142655
$545$	− 8.86307i	− 0.0162625i
$546$	0	0
$547$	333.136	0.609023	0.304512	−	0.952509i	$-0.401507\pi$
$547$	333.136	0.609023	0.304512	+	0.952509i	$0.401507\pi$
$548$	110.882i	0.202340i
$549$	0	0
$550$	−6.29232	−0.0114406
$551$	230.888i	0.419034i
$552$	0	0
$553$	116.449	0.210576
$554$	− 583.703i	− 1.05362i
$555$	0	0
$556$	424.622	0.763708
$557$	− 452.721i	− 0.812785i	−0.913699	−	0.406393i	$-0.866787\pi$
$557$	− 452.721i	− 0.812785i	0.913699	−	0.406393i	$-0.133213\pi$
$558$	0	0
$559$	−238.129	−0.425992
$560$	− 22.8156i	− 0.0407421i
$561$	0	0
$562$	703.688	1.25211
$563$	811.021i	1.44053i	0.693697	+	0.720267i	$0.255981\pi$
$563$	811.021i	1.44053i	−0.693697	+	0.720267i	$0.744019\pi$
$564$	0	0
$565$	−423.263	−0.749137
$566$	168.606i	0.297891i
$567$	0	0
$568$	−46.7465	−0.0823002
$569$	− 668.538i	− 1.17493i	−0.809248	−	0.587467i	$-0.800125\pi$
$569$	− 668.538i	− 1.17493i	0.809248	−	0.587467i	$-0.199875\pi$
$570$	0	0
$571$	149.728	0.262220	0.131110	−	0.991368i	$-0.458146\pi$
$571$	149.728	0.262220	0.131110	+	0.991368i	$0.458146\pi$
$572$	3.41998i	0.00597899i
$573$	0	0
$574$	−184.848	−0.322036
$575$	604.481i	1.05127i
$576$	0	0
$577$	497.623	0.862431	0.431215	−	0.902249i	$-0.358085\pi$
$577$	497.623	0.862431	0.431215	+	0.902249i	$0.358085\pi$
$578$	− 142.551i	− 0.246629i
$579$	0	0
$580$	35.0478	0.0604273
$581$	112.745i	0.194053i
$582$	0	0
$583$	2.34863	0.00402852
$584$	− 191.092i	− 0.327212i
$585$	0	0
$586$	−202.561	−0.345666
$587$	− 89.4505i	− 0.152386i	−0.997093	−	0.0761929i	$-0.975724\pi$
$587$	− 89.4505i	− 0.152386i	0.997093	−	0.0761929i	$-0.0242765\pi$
$588$	0	0
$589$	377.820	0.641460
$590$	23.1352i	0.0392122i
$591$	0	0
$592$	183.239	0.309525
$593$	945.591i	1.59459i	0.603591	+	0.797294i	$0.293736\pi$
$593$	945.591i	1.59459i	−0.603591	+	0.797294i	$0.706264\pi$
$594$	0	0
$595$	78.2497	0.131512
$596$	− 325.466i	− 0.546084i
$597$	0	0
$598$	328.546	0.549407
$599$	904.850i	1.51060i	0.655378	+	0.755301i	$0.272509\pi$
$599$	904.850i	1.51060i	−0.655378	+	0.755301i	$0.727491\pi$
$600$	0	0
$601$	724.737	1.20589	0.602943	−	0.797784i	$-0.293995\pi$
$601$	724.737	1.20589	0.602943	+	0.797784i	$0.293995\pi$
$602$	− 113.911i	− 0.189221i
$603$	0	0
$604$	−529.403	−0.876495
$605$	260.757i	0.431004i
$606$	0	0
$607$	−573.921	−0.945504	−0.472752	−	0.881195i	$-0.656739\pi$
$607$	−573.921	−0.945504	−0.472752	+	0.881195i	$0.656739\pi$
$608$	− 160.682i	− 0.264280i
$609$	0	0
$610$	63.9789	0.104884
$611$	− 529.877i	− 0.867230i
$612$	0	0
$613$	−768.686	−1.25397	−0.626987	−	0.779030i	$-0.715712\pi$
$613$	−768.686	−1.25397	−0.626987	+	0.779030i	$0.715712\pi$
$614$	600.067i	0.977308i
$615$	0	0
$616$	−1.63598	−0.00265581
$617$	− 94.4180i	− 0.153028i	−0.997069	−	0.0765138i	$-0.975621\pi$
$617$	− 94.4180i	− 0.153028i	0.997069	−	0.0765138i	$-0.0243789\pi$
$618$	0	0
$619$	709.579	1.14633	0.573166	−	0.819439i	$-0.305715\pi$
$619$	709.579	1.14633	0.573166	+	0.819439i	$0.305715\pi$
$620$	− 57.3515i	− 0.0925025i
$621$	0	0
$622$	−333.246	−0.535765
$623$	− 186.525i	− 0.299398i
$624$	0	0
$625$	298.018	0.476829
$626$	466.297i	0.744883i
$627$	0	0
$628$	109.674	0.174640
$629$	628.447i	0.999121i
$630$	0	0
$631$	471.531	0.747276	0.373638	−	0.927575i	$-0.378110\pi$
$631$	471.531	0.747276	0.373638	+	0.927575i	$0.378110\pi$
$632$	− 124.489i	− 0.196976i
$633$	0	0
$634$	−56.5722	−0.0892307
$635$	− 426.202i	− 0.671185i
$636$	0	0
$637$	54.7530	0.0859545
$638$	− 2.51309i	− 0.00393901i
$639$	0	0
$640$	−24.3909	−0.0381108
$641$	− 1066.95i	− 1.66452i	−0.554389	−	0.832258i	$-0.687048\pi$
$641$	− 1066.95i	− 1.66452i	0.554389	−	0.832258i	$-0.312952\pi$
$642$	0	0
$643$	203.958	0.317198	0.158599	−	0.987343i	$-0.449302\pi$
$643$	203.958	0.317198	0.158599	+	0.987343i	$0.449302\pi$
$644$	157.163i	0.244042i
$645$	0	0
$646$	551.085	0.853073
$647$	− 347.326i	− 0.536825i	−0.963304	−	0.268413i	$-0.913501\pi$
$647$	− 347.326i	− 0.536825i	0.963304	−	0.268413i	$-0.0864991\pi$
$648$	0	0
$649$	1.65890	0.00255608
$650$	− 225.132i	− 0.346357i
$651$	0	0
$652$	40.1742	0.0616168
$653$	666.957i	1.02137i	0.859767	+	0.510687i	$0.170609\pi$
$653$	666.957i	1.02137i	−0.859767	+	0.510687i	$0.829391\pi$
$654$	0	0
$655$	332.786	0.508070
$656$	197.611i	0.301237i
$657$	0	0
$658$	253.472	0.385215
$659$	430.849i	0.653792i	0.945060	+	0.326896i	$0.106003\pi$
$659$	430.849i	0.653792i	−0.945060	+	0.326896i	$0.893997\pi$
$660$	0	0
$661$	895.403	1.35462	0.677309	−	0.735699i	$-0.263146\pi$
$661$	895.403	1.35462	0.677309	+	0.735699i	$0.263146\pi$
$662$	− 136.420i	− 0.206073i
$663$	0	0
$664$	120.529	0.181520
$665$	− 162.018i	− 0.243637i
$666$	0	0
$667$	−241.423	−0.361954
$668$	171.090i	0.256123i
$669$	0	0
$670$	138.719	0.207044
$671$	− 4.58758i	− 0.00683692i
$672$	0	0
$673$	252.370	0.374993	0.187496	−	0.982265i	$-0.439963\pi$
$673$	252.370	0.374993	0.187496	+	0.982265i	$0.439963\pi$
$674$	128.791i	0.191084i
$675$	0	0
$676$	215.637	0.318990
$677$	− 338.121i	− 0.499441i	−0.968318	−	0.249720i	$-0.919661\pi$
$677$	− 338.121i	− 0.499441i	0.968318	−	0.249720i	$-0.0803387\pi$
$678$	0	0
$679$	234.536	0.345414
$680$	− 83.6525i	− 0.123018i
$681$	0	0
$682$	−4.11236	−0.00602985
$683$	891.516i	1.30529i	0.757662	+	0.652647i	$0.226341\pi$
$683$	891.516i	1.30529i	−0.757662	+	0.652647i	$0.773659\pi$
$684$	0	0
$685$	119.524	0.174487
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	−121.776	−0.177000
$689$	84.0312i	0.121961i
$690$	0	0
$691$	768.533	1.11220	0.556102	−	0.831114i	$-0.312296\pi$
$691$	768.533	1.11220	0.556102	+	0.831114i	$0.312296\pi$
$692$	− 37.4157i	− 0.0540690i
$693$	0	0
$694$	646.902	0.932135
$695$	− 457.715i	− 0.658582i
$696$	0	0
$697$	−677.740	−0.972367
$698$	77.5082i	0.111043i
$699$	0	0
$700$	107.694	0.153848
$701$	1079.86i	1.54046i	0.637767	+	0.770229i	$0.279858\pi$
$701$	1079.86i	1.54046i	−0.637767	+	0.770229i	$0.720142\pi$
$702$	0	0
$703$	1301.22	1.85095
$704$	1.74894i	0.00248428i
$705$	0	0
$706$	853.068	1.20831
$707$	302.844i	0.428351i
$708$	0	0
$709$	132.628	0.187063	0.0935316	−	0.995616i	$-0.470184\pi$
$709$	132.628	0.187063	0.0935316	+	0.995616i	$0.470184\pi$
$710$	50.3897i	0.0709714i
$711$	0	0
$712$	−199.404	−0.280061
$713$	395.060i	0.554081i
$714$	0	0
$715$	3.68652	0.00515598
$716$	507.215i	0.708401i
$717$	0	0
$718$	−995.267	−1.38617
$719$	− 488.887i	− 0.679954i	−0.940434	−	0.339977i	$-0.889581\pi$
$719$	− 488.887i	− 0.679954i	0.940434	−	0.339977i	$-0.110419\pi$
$720$	0	0
$721$	−138.219	−0.191705
$722$	− 630.508i	− 0.873280i
$723$	0	0
$724$	576.505	0.796278
$725$	165.432i	0.228182i
$726$	0	0
$727$	508.741	0.699782	0.349891	−	0.936790i	$-0.386219\pi$
$727$	508.741	0.699782	0.349891	+	0.936790i	$0.386219\pi$
$728$	− 58.5335i	− 0.0804031i
$729$	0	0
$730$	−205.985	−0.282171
$731$	− 417.651i	− 0.571342i
$732$	0	0
$733$	−496.372	−0.677179	−0.338589	−	0.940934i	$-0.609950\pi$
$733$	−496.372	−0.677179	−0.338589	+	0.940934i	$0.609950\pi$
$734$	519.115i	0.707241i
$735$	0	0
$736$	168.014	0.228280
$737$	− 9.94680i	− 0.0134963i
$738$	0	0
$739$	−735.132	−0.994766	−0.497383	−	0.867531i	$-0.665706\pi$
$739$	−735.132	−0.994766	−0.497383	+	0.867531i	$0.665706\pi$
$740$	− 197.520i	− 0.266919i
$741$	0	0
$742$	−40.1971	−0.0541739
$743$	− 1033.98i	− 1.39163i	−0.718219	−	0.695817i	$-0.755042\pi$
$743$	− 1033.98i	− 1.39163i	0.718219	−	0.695817i	$-0.244958\pi$
$744$	0	0
$745$	−350.831	−0.470915
$746$	− 198.105i	− 0.265556i
$747$	0	0
$748$	−5.99825	−0.00801906
$749$	− 384.736i	− 0.513666i
$750$	0	0
$751$	−538.317	−0.716800	−0.358400	−	0.933568i	$-0.616678\pi$
$751$	−538.317	−0.716800	−0.358400	+	0.933568i	$0.616678\pi$
$752$	− 270.972i	− 0.360336i
$753$	0	0
$754$	89.9153	0.119251
$755$	570.662i	0.755844i
$756$	0	0
$757$	−192.989	−0.254939	−0.127470	−	0.991842i	$-0.540686\pi$
$757$	−192.989	−0.254939	−0.127470	+	0.991842i	$0.540686\pi$
$758$	345.334i	0.455585i
$759$	0	0
$760$	−173.205	−0.227901
$761$	− 985.723i	− 1.29530i	−0.761938	−	0.647650i	$-0.775752\pi$
$761$	− 985.723i	− 1.29530i	0.761938	−	0.647650i	$-0.224248\pi$
$762$	0	0
$763$	10.8770	0.0142556
$764$	641.909i	0.840195i
$765$	0	0
$766$	−896.874	−1.17085
$767$	59.3534i	0.0773839i
$768$	0	0
$769$	−43.8838	−0.0570660	−0.0285330	−	0.999593i	$-0.509084\pi$
$769$	−43.8838	−0.0570660	−0.0285330	+	0.999593i	$0.509084\pi$
$770$	1.76348i	0.00229023i
$771$	0	0
$772$	−707.018	−0.915827
$773$	− 1117.15i	− 1.44521i	−0.691260	−	0.722606i	$-0.742944\pi$
$773$	− 1117.15i	− 1.44521i	0.691260	−	0.722606i	$-0.257056\pi$
$774$	0	0
$775$	270.710	0.349303
$776$	− 250.729i	− 0.323105i
$777$	0	0
$778$	649.214	0.834466
$779$	1403.28i	1.80139i
$780$	0	0
$781$	3.61317	0.00462634
$782$	576.231i	0.736868i
$783$	0	0
$784$	28.0000	0.0357143
$785$	− 118.222i	− 0.150601i
$786$	0	0
$787$	−244.114	−0.310182	−0.155091	−	0.987900i	$-0.549567\pi$
$787$	−244.114	−0.310182	−0.155091	+	0.987900i	$0.549567\pi$
$788$	− 211.500i	− 0.268401i
$789$	0	0
$790$	−134.191	−0.169862
$791$	− 519.441i	− 0.656689i
$792$	0	0
$793$	164.138	0.206984
$794$	205.248i	0.258499i
$795$	0	0
$796$	286.280	0.359649
$797$	− 403.476i	− 0.506243i	−0.967434	−	0.253122i	$-0.918543\pi$
$797$	− 403.476i	− 0.506243i	0.967434	−	0.253122i	$-0.0814573\pi$
$798$	0	0
$799$	929.343	1.16313
$800$	− 115.130i	− 0.143912i
$801$	0	0
$802$	−115.160	−0.143591
$803$	14.7700i	0.0183935i
$804$	0	0
$805$	169.411	0.210449
$806$	− 147.135i	− 0.182550i
$807$	0	0
$808$	323.754	0.400686
$809$	− 1362.80i	− 1.68455i	−0.539046	−	0.842276i	$-0.681215\pi$
$809$	− 1362.80i	− 1.68455i	0.539046	−	0.842276i	$-0.318785\pi$
$810$	0	0
$811$	717.313	0.884479	0.442240	−	0.896897i	$-0.354184\pi$
$811$	717.313	0.884479	0.442240	+	0.896897i	$0.354184\pi$
$812$	43.0118i	0.0529702i
$813$	0	0
$814$	−14.1630	−0.0173993
$815$	− 43.3052i	− 0.0531352i
$816$	0	0
$817$	−864.760	−1.05846
$818$	665.723i	0.813842i
$819$	0	0
$820$	213.012	0.259771
$821$	− 514.408i	− 0.626562i	−0.949660	−	0.313281i	$-0.898572\pi$
$821$	− 514.408i	− 0.626562i	0.949660	−	0.313281i	$-0.101428\pi$
$822$	0	0
$823$	−1256.13	−1.52628	−0.763140	−	0.646233i	$-0.776344\pi$
$823$	−1256.13	−1.52628	−0.763140	+	0.646233i	$0.776344\pi$
$824$	147.763i	0.179324i
$825$	0	0
$826$	−28.3922	−0.0343732
$827$	297.901i	0.360219i	0.983647	+	0.180109i	$0.0576452\pi$
$827$	297.901i	0.360219i	−0.983647	+	0.180109i	$0.942355\pi$
$828$	0	0
$829$	752.404	0.907605	0.453802	−	0.891102i	$-0.350067\pi$
$829$	752.404	0.907605	0.453802	+	0.891102i	$0.350067\pi$
$830$	− 129.923i	− 0.156533i
$831$	0	0
$832$	−62.5749	−0.0752102
$833$	96.0305i	0.115283i
$834$	0	0
$835$	184.424	0.220868
$836$	12.4196i	0.0148560i
$837$	0	0
$838$	−990.248	−1.18168
$839$	− 1624.56i	− 1.93630i	−0.250365	−	0.968152i	$-0.580551\pi$
$839$	− 1624.56i	− 1.93630i	0.250365	−	0.968152i	$-0.419449\pi$
$840$	0	0
$841$	774.928	0.921437
$842$	845.651i	1.00434i
$843$	0	0
$844$	−677.811	−0.803094
$845$	− 232.443i	− 0.275080i
$846$	0	0
$847$	−320.009	−0.377815
$848$	42.9725i	0.0506751i
$849$	0	0
$850$	394.855	0.464535
$851$	1360.59i	1.59882i
$852$	0	0
$853$	−935.086	−1.09623	−0.548116	−	0.836402i	$-0.684655\pi$
$853$	−935.086	−1.09623	−0.548116	+	0.836402i	$0.684655\pi$
$854$	78.5169i	0.0919402i
$855$	0	0
$856$	−411.300	−0.480490
$857$	− 731.048i	− 0.853032i	−0.904480	−	0.426516i	$-0.859741\pi$
$857$	− 731.048i	− 0.853032i	0.904480	−	0.426516i	$-0.140259\pi$
$858$	0	0
$859$	−1275.71	−1.48512	−0.742558	−	0.669782i	$-0.766388\pi$
$859$	−1275.71	−1.48512	−0.742558	+	0.669782i	$0.766388\pi$
$860$	131.267i	0.152636i
$861$	0	0
$862$	−102.780	−0.119234
$863$	495.527i	0.574192i	0.957902	+	0.287096i	$0.0926898\pi$
$863$	495.527i	0.574192i	−0.957902	+	0.287096i	$0.907310\pi$
$864$	0	0
$865$	−40.3317	−0.0466263
$866$	− 182.638i	− 0.210899i
$867$	0	0
$868$	70.3836	0.0810871
$869$	9.62209i	0.0110726i
$870$	0	0
$871$	355.885	0.408593
$872$	− 11.6280i	− 0.0133349i
$873$	0	0
$874$	1193.10	1.36511
$875$	− 258.684i	− 0.295639i
$876$	0	0
$877$	−728.985	−0.831226	−0.415613	−	0.909542i	$-0.636433\pi$
$877$	−728.985	−0.831226	−0.415613	+	0.909542i	$0.636433\pi$
$878$	− 422.157i	− 0.480817i
$879$	0	0
$880$	1.88524	0.00214232
$881$	− 499.962i	− 0.567494i	−0.958899	−	0.283747i	$-0.908422\pi$
$881$	− 499.962i	− 0.567494i	0.958899	−	0.283747i	$-0.0915776\pi$
$882$	0	0
$883$	−366.063	−0.414568	−0.207284	−	0.978281i	$-0.566462\pi$
$883$	−366.063	−0.414568	−0.207284	+	0.978281i	$0.566462\pi$
$884$	− 214.611i	− 0.242772i
$885$	0	0
$886$	242.243	0.273412
$887$	− 687.811i	− 0.775435i	−0.921778	−	0.387717i	$-0.873264\pi$
$887$	− 687.811i	− 0.775435i	0.921778	−	0.387717i	$-0.126736\pi$
$888$	0	0
$889$	523.048	0.588356
$890$	214.944i	0.241511i
$891$	0	0
$892$	−550.898	−0.617598
$893$	− 1924.23i	− 2.15480i
$894$	0	0
$895$	546.745	0.610889
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	190.390	0.212016
$899$	108.119i	0.120265i
$900$	0	0
$901$	−147.381	−0.163575
$902$	− 15.2739i	− 0.0169334i
$903$	0	0
$904$	−555.306	−0.614276
$905$	− 621.435i	− 0.686669i
$906$	0	0
$907$	−620.187	−0.683778	−0.341889	−	0.939740i	$-0.611067\pi$
$907$	−620.187	−0.683778	−0.341889	+	0.939740i	$0.611067\pi$
$908$	99.9341i	0.110060i
$909$	0	0
$910$	−63.0953	−0.0693355
$911$	− 551.736i	− 0.605638i	−0.953048	−	0.302819i	$-0.902072\pi$
$911$	− 551.736i	− 0.605638i	0.953048	−	0.302819i	$-0.0979278\pi$
$912$	0	0
$913$	−9.31604	−0.0102038
$914$	1090.45i	1.19305i
$915$	0	0
$916$	−682.274	−0.744840
$917$	408.405i	0.445371i
$918$	0	0
$919$	−609.699	−0.663437	−0.331719	−	0.943378i	$-0.607628\pi$
$919$	−609.699	−0.663437	−0.331719	+	0.943378i	$0.607628\pi$
$920$	− 181.108i	− 0.196857i
$921$	0	0
$922$	−454.959	−0.493448
$923$	129.275i	0.140060i
$924$	0	0
$925$	932.330	1.00792
$926$	448.555i	0.484401i
$927$	0	0
$928$	45.9815	0.0495491
$929$	− 287.578i	− 0.309556i	−0.987949	−	0.154778i	$-0.950534\pi$
$929$	− 287.578i	− 0.309556i	0.987949	−	0.154778i	$-0.0494663\pi$
$930$	0	0
$931$	198.834	0.213570
$932$	291.725i	0.313010i
$933$	0	0
$934$	1046.61	1.12057
$935$	6.46573i	0.00691522i
$936$	0	0
$937$	604.089	0.644706	0.322353	−	0.946620i	$-0.395526\pi$
$937$	604.089	0.644706	0.322353	+	0.946620i	$0.395526\pi$
$938$	170.241i	0.181493i
$939$	0	0
$940$	−292.091	−0.310735
$941$	996.226i	1.05869i	0.848407	+	0.529344i	$0.177562\pi$
$941$	996.226i	1.05869i	−0.848407	+	0.529344i	$0.822438\pi$
$942$	0	0
$943$	−1467.31	−1.55600
$944$	30.3526i	0.0321532i
$945$	0	0
$946$	9.41243	0.00994971
$947$	1551.01i	1.63781i	0.573926	+	0.818907i	$0.305420\pi$
$947$	1551.01i	1.63781i	−0.573926	+	0.818907i	$0.694580\pi$
$948$	0	0
$949$	−528.454	−0.556853
$950$	− 817.560i	− 0.860589i
$951$	0	0
$952$	102.661	0.107837
$953$	− 289.179i	− 0.303441i	−0.988423	−	0.151720i	$-0.951519\pi$
$953$	− 289.179i	− 0.303441i	0.988423	−	0.151720i	$-0.0484813\pi$
$954$	0	0
$955$	691.937	0.724541
$956$	677.717i	0.708909i
$957$	0	0
$958$	66.3658	0.0692754
$959$	146.683i	0.152954i
$960$	0	0
$961$	−784.077	−0.815897
$962$	− 506.737i	− 0.526754i
$963$	0	0
$964$	127.391	0.132148
$965$	762.120i	0.789762i
$966$	0	0
$967$	−1149.84	−1.18908	−0.594541	−	0.804065i	$-0.702666\pi$
$967$	−1149.84	−1.18908	−0.594541	+	0.804065i	$0.702666\pi$
$968$	342.105i	0.353414i
$969$	0	0
$970$	−270.270	−0.278629
$971$	1197.89i	1.23366i	0.787096	+	0.616831i	$0.211584\pi$
$971$	1197.89i	1.23366i	−0.787096	+	0.616831i	$0.788416\pi$
$972$	0	0
$973$	561.722	0.577309
$974$	856.310i	0.879168i
$975$	0	0
$976$	83.9381	0.0860022
$977$	629.816i	0.644643i	0.946630	+	0.322321i	$0.104463\pi$
$977$	629.816i	0.644643i	−0.946630	+	0.322321i	$0.895537\pi$
$978$	0	0
$979$	15.4125	0.0157431
$980$	− 30.1822i	− 0.0307982i
$981$	0	0
$982$	667.038	0.679264
$983$	− 738.820i	− 0.751597i	−0.926701	−	0.375799i	$-0.877368\pi$
$983$	− 738.820i	− 0.751597i	0.926701	−	0.375799i	$-0.122632\pi$
$984$	0	0
$985$	−227.984	−0.231455
$986$	157.701i	0.159940i
$987$	0	0
$988$	−444.358	−0.449755
$989$	− 904.219i	− 0.914276i
$990$	0	0
$991$	−548.234	−0.553213	−0.276606	−	0.960983i	$-0.589210\pi$
$991$	−548.234	−0.553213	−0.276606	+	0.960983i	$0.589210\pi$
$992$	− 75.2432i	− 0.0758500i
$993$	0	0
$994$	−61.8398	−0.0622131
$995$	− 308.592i	− 0.310143i
$996$	0	0
$997$	−1667.30	−1.67232	−0.836161	−	0.548484i	$-0.815205\pi$
$997$	−1667.30	−1.67232	−0.836161	+	0.548484i	$0.815205\pi$
$998$	586.854i	0.588030i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.b.c.323.3	✓	8
3.2	odd	2	inner	378.3.b.c.323.6	yes	8
4.3	odd	2		3024.3.d.h.1457.6		8
9.2	odd	6		1134.3.q.f.701.7		16
9.4	even	3		1134.3.q.f.1079.7		16
9.5	odd	6		1134.3.q.f.1079.2		16
9.7	even	3		1134.3.q.f.701.2		16
12.11	even	2		3024.3.d.h.1457.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.3.b.c.323.3	✓	8	1.1	even	1	trivial
378.3.b.c.323.6	yes	8	3.2	odd	2	inner
1134.3.q.f.701.2		16	9.7	even	3
1134.3.q.f.701.7		16	9.2	odd	6
1134.3.q.f.1079.2		16	9.5	odd	6
1134.3.q.f.1079.7		16	9.4	even	3
3024.3.d.h.1457.3		8	12.11	even	2
3024.3.d.h.1457.6		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.15587i q^{5} -2.64575 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.15587i q^{5} -2.64575 q^{7} +2.82843i q^{8} +3.04886 q^{10} -0.218617i q^{11} +7.82186 q^{13} +3.74166i q^{14} +4.00000 q^{16} +13.7186i q^{17} +28.4049 q^{19} -4.31174i q^{20} -0.309171 q^{22} +29.7010i q^{23} +20.3522 q^{25} -11.0618i q^{26} +5.29150 q^{28} +8.12846i q^{29} +13.3012 q^{31} -5.65685i q^{32} +19.4011 q^{34} -5.70390i q^{35} +45.8097 q^{37} -40.1705i q^{38} -6.09772 q^{40} +49.4028i q^{41} -30.4441 q^{43} +0.437234i q^{44} +42.0035 q^{46} -67.7431i q^{47} +7.00000 q^{49} -28.7824i q^{50} -15.6437 q^{52} +10.7431i q^{53} +0.471310 q^{55} -7.48331i q^{56} +11.4954 q^{58} +7.58815i q^{59} +20.9845 q^{61} -18.8108i q^{62} -8.00000 q^{64} +16.8629i q^{65} +45.4987 q^{67} -27.4373i q^{68} -8.06653 q^{70} +16.5274i q^{71} -67.5611 q^{73} -64.7847i q^{74} -56.8097 q^{76} +0.578406i q^{77} -44.0134 q^{79} +8.62348i q^{80} +69.8662 q^{82} -42.6135i q^{83} -29.5756 q^{85} +43.0544i q^{86} +0.618342 q^{88} +70.4999i q^{89} -20.6947 q^{91} -59.4019i q^{92} -95.8032 q^{94} +61.2372i q^{95} -88.6462 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} + 16 q^{10} - 40 q^{13} + 32 q^{16} + 40 q^{19} - 16 q^{22} - 40 q^{31} - 16 q^{34} - 8 q^{37} - 32 q^{40} + 40 q^{43} - 64 q^{46} + 56 q^{49} + 80 q^{52} + 56 q^{55} + 112 q^{58} - 88 q^{61} - 64 q^{64} + 240 q^{67} - 112 q^{70} - 288 q^{73} - 80 q^{76} - 424 q^{79} + 144 q^{82} + 200 q^{85} + 32 q^{88} + 56 q^{91} + 288 q^{94} + 600 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^10 - 40 * q^13 + 32 * q^16 + 40 * q^19 - 16 * q^22 - 40 * q^31 - 16 * q^34 - 8 * q^37 - 32 * q^40 + 40 * q^43 - 64 * q^46 + 56 * q^49 + 80 * q^52 + 56 * q^55 + 112 * q^58 - 88 * q^61 - 64 * q^64 + 240 * q^67 - 112 * q^70 - 288 * q^73 - 80 * q^76 - 424 * q^79 + 144 * q^82 + 200 * q^85 + 32 * q^88 + 56 * q^91 + 288 * q^94 + 600 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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