LMFDB - Embedded newform 1400.4.g.l.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,4,Mod(449,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1400.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$82.6026740080$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.7807489600.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 63x^{4} + 1041x^{2} + 1600 $ x^6 + 63x^4 + 1041x^2 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$5.68734i$ of defining polynomial
Character	$\chi$	$=$	1400.449
Dual form			1400.4.g.l.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.238730i q^{3} -7.00000i q^{7} +26.9430 q^{9} +O(q^{10})$	$q+0.238730i q^{3} -7.00000i q^{7} +26.9430 q^{9} +54.5770 q^{11} -52.2149i q^{13} -49.8939i q^{17} +161.039 q^{19} +1.67111 q^{21} -119.590i q^{23} +12.8778i q^{27} -49.5506 q^{29} -164.972 q^{31} +13.0292i q^{33} +216.787i q^{37} +12.4653 q^{39} -252.670 q^{41} -68.7387i q^{43} +84.4004i q^{47} -49.0000 q^{49} +11.9112 q^{51} -204.554i q^{53} +38.4448i q^{57} +351.538 q^{59} +552.739 q^{61} -188.601i q^{63} +178.159i q^{67} +28.5498 q^{69} -1037.44 q^{71} +415.770i q^{73} -382.039i q^{77} -931.328 q^{79} +724.387 q^{81} -223.567i q^{83} -11.8292i q^{87} +1483.80 q^{89} -365.504 q^{91} -39.3838i q^{93} -1709.21i q^{97} +1470.47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 50 q^{9}+O(q^{10}) $ 6 * q - 50 * q^9	$ 6 q - 50 q^{9} - 12 q^{11} + 304 q^{19} - 84 q^{21} + 936 q^{29} - 536 q^{31} + 1348 q^{39} - 148 q^{41} - 294 q^{49} - 732 q^{51} + 2176 q^{59} + 348 q^{61} + 3568 q^{69} - 2112 q^{71} - 724 q^{79} + 5158 q^{81} - 316 q^{89} + 112 q^{91} + 2544 q^{99}+O(q^{100}) $ 6 * q - 50 * q^9 - 12 * q^11 + 304 * q^19 - 84 * q^21 + 936 * q^29 - 536 * q^31 + 1348 * q^39 - 148 * q^41 - 294 * q^49 - 732 * q^51 + 2176 * q^59 + 348 * q^61 + 3568 * q^69 - 2112 * q^71 - 724 * q^79 + 5158 * q^81 - 316 * q^89 + 112 * q^91 + 2544 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.238730i	0.0459437i	0.999736	+	0.0229718i	$0.00731280\pi$
$3$	0.238730i	0.0459437i	−0.999736	+	0.0229718i	$0.992687\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	26.9430	0.997889
$10$	0	0
$11$	54.5770	1.49596	0.747981	−	0.663721i	$-0.231024\pi$
$11$	54.5770	1.49596	0.747981	+	0.663721i	$0.231024\pi$
$12$	0	0
$13$	− 52.2149i	− 1.11399i	−0.830517	−	0.556993i	$-0.811955\pi$
$13$	− 52.2149i	− 1.11399i	0.830517	−	0.556993i	$-0.188045\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 49.8939i	− 0.711827i	−0.934519	−	0.355913i	$-0.884170\pi$
$17$	− 49.8939i	− 0.711827i	0.934519	−	0.355913i	$-0.115830\pi$
$18$	0	0
$19$	161.039	1.94446	0.972231	−	0.234024i	$-0.0751895\pi$
$19$	161.039	1.94446	0.972231	+	0.234024i	$0.0751895\pi$
$20$	0	0
$21$	1.67111	0.0173651
$22$	0	0
$23$	− 119.590i	− 1.08419i	−0.840318	−	0.542093i	$-0.817632\pi$
$23$	− 119.590i	− 1.08419i	0.840318	−	0.542093i	$-0.182368\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	12.8778i	0.0917903i
$28$	0	0
$29$	−49.5506	−0.317286	−0.158643	−	0.987336i	$-0.550712\pi$
$29$	−49.5506	−0.317286	−0.158643	+	0.987336i	$0.550712\pi$
$30$	0	0
$31$	−164.972	−0.955803	−0.477901	−	0.878414i	$-0.658602\pi$
$31$	−164.972	−0.955803	−0.477901	+	0.878414i	$0.658602\pi$
$32$	0	0
$33$	13.0292i	0.0687299i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	216.787i	0.963230i	0.876383	+	0.481615i	$0.159950\pi$
$37$	216.787i	0.963230i	−0.876383	+	0.481615i	$0.840050\pi$
$38$	0	0
$39$	12.4653	0.0511806
$40$	0	0
$41$	−252.670	−0.962449	−0.481224	−	0.876597i	$-0.659808\pi$
$41$	−252.670	−0.962449	−0.481224	+	0.876597i	$0.659808\pi$
$42$	0	0
$43$	− 68.7387i	− 0.243780i	−0.992544	−	0.121890i	$-0.961104\pi$
$43$	− 68.7387i	− 0.243780i	0.992544	−	0.121890i	$-0.0388955\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	84.4004i	0.261937i	0.991387	+	0.130969i	$0.0418087\pi$
$47$	84.4004i	0.261937i	−0.991387	+	0.130969i	$0.958191\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	11.9112	0.0327039
$52$	0	0
$53$	− 204.554i	− 0.530146i	−0.964228	−	0.265073i	$-0.914604\pi$
$53$	− 204.554i	− 0.530146i	0.964228	−	0.265073i	$-0.0853960\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	38.4448i	0.0893357i
$58$	0	0
$59$	351.538	0.775702	0.387851	−	0.921722i	$-0.373218\pi$
$59$	351.538	0.775702	0.387851	+	0.921722i	$0.373218\pi$
$60$	0	0
$61$	552.739	1.16018	0.580090	−	0.814552i	$-0.303017\pi$
$61$	552.739	1.16018	0.580090	+	0.814552i	$0.303017\pi$
$62$	0	0
$63$	− 188.601i	− 0.377167i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	178.159i	0.324860i	0.986720	+	0.162430i	$0.0519331\pi$
$67$	178.159i	0.324860i	−0.986720	+	0.162430i	$0.948067\pi$
$68$	0	0
$69$	28.5498	0.0498115
$70$	0	0
$71$	−1037.44	−1.73411	−0.867056	−	0.498211i	$-0.833990\pi$
$71$	−1037.44	−1.73411	−0.867056	+	0.498211i	$0.833990\pi$
$72$	0	0
$73$	415.770i	0.666605i	0.942820	+	0.333303i	$0.108163\pi$
$73$	415.770i	0.666605i	−0.942820	+	0.333303i	$0.891837\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 382.039i	− 0.565420i
$78$	0	0
$79$	−931.328	−1.32636	−0.663181	−	0.748459i	$-0.730794\pi$
$79$	−931.328	−1.32636	−0.663181	+	0.748459i	$0.730794\pi$
$80$	0	0
$81$	724.387	0.993672
$82$	0	0
$83$	− 223.567i	− 0.295659i	−0.989013	−	0.147830i	$-0.952771\pi$
$83$	− 223.567i	− 0.295659i	0.989013	−	0.147830i	$-0.0472287\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 11.8292i	− 0.0145773i
$88$	0	0
$89$	1483.80	1.76722	0.883610	−	0.468224i	$-0.155106\pi$
$89$	1483.80	1.76722	0.883610	+	0.468224i	$0.155106\pi$
$90$	0	0
$91$	−365.504	−0.421047
$92$	0	0
$93$	− 39.3838i	− 0.0439131i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1709.21i	− 1.78911i	−0.446959	−	0.894554i	$-0.647493\pi$
$97$	− 1709.21i	− 1.78911i	0.446959	−	0.894554i	$-0.352507\pi$
$98$	0	0
$99$	1470.47	1.49280
$100$	0	0
$101$	−1147.25	−1.13025	−0.565125	−	0.825005i	$-0.691172\pi$
$101$	−1147.25	−1.13025	−0.565125	+	0.825005i	$0.691172\pi$
$102$	0	0
$103$	193.450i	0.185061i	0.995710	+	0.0925303i	$0.0294955\pi$
$103$	193.450i	0.185061i	−0.995710	+	0.0925303i	$0.970505\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 357.788i	− 0.323258i	−0.986852	−	0.161629i	$-0.948325\pi$
$107$	− 357.788i	− 0.323258i	0.986852	−	0.161629i	$-0.0516748\pi$
$108$	0	0
$109$	−538.325	−0.473048	−0.236524	−	0.971626i	$-0.576008\pi$
$109$	−538.325	−0.473048	−0.236524	+	0.971626i	$0.576008\pi$
$110$	0	0
$111$	−51.7535	−0.0442543
$112$	0	0
$113$	1872.84i	1.55913i	0.626322	+	0.779564i	$0.284559\pi$
$113$	1872.84i	1.55913i	−0.626322	+	0.779564i	$0.715441\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1406.83i	− 1.11163i
$118$	0	0
$119$	−349.258	−0.269045
$120$	0	0
$121$	1647.64	1.23790
$122$	0	0
$123$	− 60.3199i	− 0.0442184i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2335.02i	− 1.63149i	−0.578410	−	0.815746i	$-0.696327\pi$
$127$	− 2335.02i	− 1.63149i	0.578410	−	0.815746i	$-0.303673\pi$
$128$	0	0
$129$	16.4100	0.0112002
$130$	0	0
$131$	2034.75	1.35707	0.678536	−	0.734567i	$-0.262615\pi$
$131$	2034.75	1.35707	0.678536	+	0.734567i	$0.262615\pi$
$132$	0	0
$133$	− 1127.27i	− 0.734937i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3170.67i	− 1.97729i	−0.150264	−	0.988646i	$-0.548012\pi$
$137$	− 3170.67i	− 1.97729i	0.150264	−	0.988646i	$-0.451988\pi$
$138$	0	0
$139$	769.758	0.469713	0.234856	−	0.972030i	$-0.424538\pi$
$139$	769.758	0.469713	0.234856	+	0.972030i	$0.424538\pi$
$140$	0	0
$141$	−20.1489	−0.0120344
$142$	0	0
$143$	− 2849.73i	− 1.66648i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 11.6978i	− 0.00656338i
$148$	0	0
$149$	1178.84	0.648150	0.324075	−	0.946031i	$-0.394947\pi$
$149$	1178.84	0.648150	0.324075	+	0.946031i	$0.394947\pi$
$150$	0	0
$151$	−1183.88	−0.638029	−0.319015	−	0.947750i	$-0.603352\pi$
$151$	−1183.88	−0.638029	−0.319015	+	0.947750i	$0.603352\pi$
$152$	0	0
$153$	− 1344.29i	− 0.710324i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1437.53i	0.730747i	0.930861	+	0.365374i	$0.119059\pi$
$157$	1437.53i	0.730747i	−0.930861	+	0.365374i	$0.880941\pi$
$158$	0	0
$159$	48.8333	0.0243568
$160$	0	0
$161$	−837.132	−0.409784
$162$	0	0
$163$	− 2492.06i	− 1.19750i	−0.800935	−	0.598752i	$-0.795664\pi$
$163$	− 2492.06i	− 1.19750i	0.800935	−	0.598752i	$-0.204336\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1511.53i	0.700394i	0.936676	+	0.350197i	$0.113885\pi$
$167$	1511.53i	0.700394i	−0.936676	+	0.350197i	$0.886115\pi$
$168$	0	0
$169$	−529.396	−0.240963
$170$	0	0
$171$	4338.86	1.94036
$172$	0	0
$173$	1717.16i	0.754645i	0.926082	+	0.377322i	$0.123155\pi$
$173$	1717.16i	0.754645i	−0.926082	+	0.377322i	$0.876845\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	83.9228i	0.0356386i
$178$	0	0
$179$	−3003.12	−1.25399	−0.626993	−	0.779025i	$-0.715714\pi$
$179$	−3003.12	−1.25399	−0.626993	+	0.779025i	$0.715714\pi$
$180$	0	0
$181$	−200.183	−0.0822072	−0.0411036	−	0.999155i	$-0.513087\pi$
$181$	−200.183	−0.0822072	−0.0411036	+	0.999155i	$0.513087\pi$
$182$	0	0
$183$	131.955i	0.0533029i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2723.06i	− 1.06487i
$188$	0	0
$189$	90.1448	0.0346935
$190$	0	0
$191$	−1214.78	−0.460203	−0.230101	−	0.973167i	$-0.573906\pi$
$191$	−1214.78	−0.460203	−0.230101	+	0.973167i	$0.573906\pi$
$192$	0	0
$193$	2533.21i	0.944788i	0.881387	+	0.472394i	$0.156610\pi$
$193$	2533.21i	0.944788i	−0.881387	+	0.472394i	$0.843390\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2057.14i	0.743986i	0.928235	+	0.371993i	$0.121326\pi$
$197$	2057.14i	0.743986i	−0.928235	+	0.371993i	$0.878674\pi$
$198$	0	0
$199$	2647.41	0.943063	0.471532	−	0.881849i	$-0.343701\pi$
$199$	2647.41	0.943063	0.471532	+	0.881849i	$0.343701\pi$
$200$	0	0
$201$	−42.5320	−0.0149252
$202$	0	0
$203$	346.854i	0.119923i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3222.12i	− 1.08190i
$208$	0	0
$209$	8788.99	2.90884
$210$	0	0
$211$	−5418.61	−1.76793	−0.883963	−	0.467557i	$-0.845134\pi$
$211$	−5418.61	−1.76793	−0.883963	+	0.467557i	$0.845134\pi$
$212$	0	0
$213$	− 247.669i	− 0.0796714i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1154.81i	0.361259i
$218$	0	0
$219$	−99.2568	−0.0306263
$220$	0	0
$221$	−2605.21	−0.792965
$222$	0	0
$223$	2473.81i	0.742864i	0.928460	+	0.371432i	$0.121133\pi$
$223$	2473.81i	0.742864i	−0.928460	+	0.371432i	$0.878867\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1296.70i	0.379141i	0.981867	+	0.189570i	$0.0607095\pi$
$227$	1296.70i	0.379141i	−0.981867	+	0.189570i	$0.939291\pi$
$228$	0	0
$229$	2480.71	0.715852	0.357926	−	0.933750i	$-0.383484\pi$
$229$	2480.71	0.715852	0.357926	+	0.933750i	$0.383484\pi$
$230$	0	0
$231$	91.2042	0.0259775
$232$	0	0
$233$	− 947.206i	− 0.266324i	−0.991094	−	0.133162i	$-0.957487\pi$
$233$	− 947.206i	− 0.266324i	0.991094	−	0.133162i	$-0.0425131\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 222.336i	− 0.0609379i
$238$	0	0
$239$	5214.56	1.41130	0.705652	−	0.708558i	$-0.250654\pi$
$239$	5214.56	1.41130	0.705652	+	0.708558i	$0.250654\pi$
$240$	0	0
$241$	−1461.62	−0.390670	−0.195335	−	0.980737i	$-0.562579\pi$
$241$	−1461.62	−0.390670	−0.195335	+	0.980737i	$0.562579\pi$
$242$	0	0
$243$	520.634i	0.137443i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8408.61i	− 2.16610i
$248$	0	0
$249$	53.3723	0.0135837
$250$	0	0
$251$	−2828.11	−0.711189	−0.355595	−	0.934640i	$-0.615722\pi$
$251$	−2828.11	−0.711189	−0.355595	+	0.934640i	$0.615722\pi$
$252$	0	0
$253$	− 6526.87i	− 1.62190i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2215.03i	− 0.537625i	−0.963193	−	0.268812i	$-0.913369\pi$
$257$	− 2215.03i	− 0.537625i	0.963193	−	0.268812i	$-0.0866312\pi$
$258$	0	0
$259$	1517.51	0.364067
$260$	0	0
$261$	−1335.04	−0.316617
$262$	0	0
$263$	1232.83i	0.289047i	0.989501	+	0.144523i	$0.0461649\pi$
$263$	1232.83i	0.289047i	−0.989501	+	0.144523i	$0.953835\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	354.228i	0.0811925i
$268$	0	0
$269$	8081.76	1.83180	0.915899	−	0.401408i	$-0.131479\pi$
$269$	8081.76	1.83180	0.915899	+	0.401408i	$0.131479\pi$
$270$	0	0
$271$	310.428	0.0695837	0.0347918	−	0.999395i	$-0.488923\pi$
$271$	310.428	0.0695837	0.0347918	+	0.999395i	$0.488923\pi$
$272$	0	0
$273$	− 87.2569i	− 0.0193444i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7015.75i	− 1.52179i	−0.648876	−	0.760894i	$-0.724760\pi$
$277$	− 7015.75i	− 1.52179i	0.648876	−	0.760894i	$-0.275240\pi$
$278$	0	0
$279$	−4444.85	−0.953785
$280$	0	0
$281$	6127.65	1.30087	0.650436	−	0.759561i	$-0.274586\pi$
$281$	6127.65	1.30087	0.650436	+	0.759561i	$0.274586\pi$
$282$	0	0
$283$	− 8228.73i	− 1.72844i	−0.503118	−	0.864218i	$-0.667814\pi$
$283$	− 8228.73i	− 1.72844i	0.503118	−	0.864218i	$-0.332186\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1768.69i	0.363772i
$288$	0	0
$289$	2423.60	0.493303
$290$	0	0
$291$	408.039	0.0821982
$292$	0	0
$293$	− 7726.68i	− 1.54061i	−0.637678	−	0.770303i	$-0.720105\pi$
$293$	− 7726.68i	− 1.54061i	0.637678	−	0.770303i	$-0.279895\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	702.833i	0.137315i
$298$	0	0
$299$	−6244.40	−1.20777
$300$	0	0
$301$	−481.171	−0.0921403
$302$	0	0
$303$	− 273.882i	− 0.0519278i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	162.694i	0.0302457i	0.999886	+	0.0151229i	$0.00481394\pi$
$307$	162.694i	0.0302457i	−0.999886	+	0.0151229i	$0.995186\pi$
$308$	0	0
$309$	−46.1825	−0.00850236
$310$	0	0
$311$	−2702.64	−0.492774	−0.246387	−	0.969172i	$-0.579243\pi$
$311$	−2702.64	−0.492774	−0.246387	+	0.969172i	$0.579243\pi$
$312$	0	0
$313$	744.313i	0.134412i	0.997739	+	0.0672062i	$0.0214085\pi$
$313$	744.313i	0.134412i	−0.997739	+	0.0672062i	$0.978591\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3764.61i	0.667008i	0.942749	+	0.333504i	$0.108231\pi$
$317$	3764.61i	0.667008i	−0.942749	+	0.333504i	$0.891769\pi$
$318$	0	0
$319$	−2704.32	−0.474648
$320$	0	0
$321$	85.4147	0.0148517
$322$	0	0
$323$	− 8034.84i	− 1.38412i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 128.515i	− 0.0217335i
$328$	0	0
$329$	590.803	0.0990030
$330$	0	0
$331$	3209.40	0.532944	0.266472	−	0.963843i	$-0.414142\pi$
$331$	3209.40	0.532944	0.266472	+	0.963843i	$0.414142\pi$
$332$	0	0
$333$	5840.89i	0.961197i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9304.77i	− 1.50404i	−0.659138	−	0.752022i	$-0.729079\pi$
$337$	− 9304.77i	− 1.50404i	0.659138	−	0.752022i	$-0.270921\pi$
$338$	0	0
$339$	−447.102	−0.0716321
$340$	0	0
$341$	−9003.68	−1.42984
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3875.13i	− 0.599504i	−0.954017	−	0.299752i	$-0.903096\pi$
$347$	− 3875.13i	− 0.599504i	0.954017	−	0.299752i	$-0.0969040\pi$
$348$	0	0
$349$	8398.52	1.28814	0.644072	−	0.764964i	$-0.277244\pi$
$349$	8398.52	1.28814	0.644072	+	0.764964i	$0.277244\pi$
$350$	0	0
$351$	672.415	0.102253
$352$	0	0
$353$	11946.0i	1.80120i	0.434652	+	0.900599i	$0.356871\pi$
$353$	11946.0i	1.80120i	−0.434652	+	0.900599i	$0.643129\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 83.3783i	− 0.0123609i
$358$	0	0
$359$	−5956.20	−0.875644	−0.437822	−	0.899062i	$-0.644250\pi$
$359$	−5956.20	−0.875644	−0.437822	+	0.899062i	$0.644250\pi$
$360$	0	0
$361$	19074.4	2.78093
$362$	0	0
$363$	393.343i	0.0568736i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4354.19i	0.619311i	0.950849	+	0.309655i	$0.100214\pi$
$367$	4354.19i	0.619311i	−0.950849	+	0.309655i	$0.899786\pi$
$368$	0	0
$369$	−6807.69	−0.960417
$370$	0	0
$371$	−1431.88	−0.200376
$372$	0	0
$373$	12330.8i	1.71171i	0.517220	+	0.855853i	$0.326967\pi$
$373$	12330.8i	1.71171i	−0.517220	+	0.855853i	$0.673033\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2587.28i	0.353452i
$378$	0	0
$379$	−11286.2	−1.52964	−0.764819	−	0.644246i	$-0.777172\pi$
$379$	−11286.2	−1.52964	−0.764819	+	0.644246i	$0.777172\pi$
$380$	0	0
$381$	557.440	0.0749567
$382$	0	0
$383$	− 6412.87i	− 0.855567i	−0.903881	−	0.427784i	$-0.859295\pi$
$383$	− 6412.87i	− 0.855567i	0.903881	−	0.427784i	$-0.140705\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1852.03i	− 0.243266i
$388$	0	0
$389$	9838.47	1.28234	0.641170	−	0.767399i	$-0.278449\pi$
$389$	9838.47	1.28234	0.641170	+	0.767399i	$0.278449\pi$
$390$	0	0
$391$	−5966.83	−0.771753
$392$	0	0
$393$	485.755i	0.0623489i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9755.57i	1.23329i	0.787240	+	0.616647i	$0.211510\pi$
$397$	9755.57i	1.23329i	−0.787240	+	0.616647i	$0.788490\pi$
$398$	0	0
$399$	269.113	0.0337657
$400$	0	0
$401$	−948.092	−0.118069	−0.0590343	−	0.998256i	$-0.518802\pi$
$401$	−948.092	−0.118069	−0.0590343	+	0.998256i	$0.518802\pi$
$402$	0	0
$403$	8614.01i	1.06475i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11831.6i	1.44095i
$408$	0	0
$409$	−5121.47	−0.619169	−0.309585	−	0.950872i	$-0.600190\pi$
$409$	−5121.47	−0.619169	−0.309585	+	0.950872i	$0.600190\pi$
$410$	0	0
$411$	756.936	0.0908440
$412$	0	0
$413$	− 2460.77i	− 0.293188i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	183.765i	0.0215803i
$418$	0	0
$419$	3024.21	0.352607	0.176303	−	0.984336i	$-0.443586\pi$
$419$	3024.21	0.352607	0.176303	+	0.984336i	$0.443586\pi$
$420$	0	0
$421$	2583.60	0.299090	0.149545	−	0.988755i	$-0.452219\pi$
$421$	2583.60	0.299090	0.149545	+	0.988755i	$0.452219\pi$
$422$	0	0
$423$	2274.00i	0.261385i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3869.17i	− 0.438507i
$428$	0	0
$429$	680.317	0.0765641
$430$	0	0
$431$	16302.5	1.82196	0.910981	−	0.412448i	$-0.135326\pi$
$431$	16302.5	1.82196	0.910981	+	0.412448i	$0.135326\pi$
$432$	0	0
$433$	14137.9i	1.56911i	0.620062	+	0.784553i	$0.287107\pi$
$433$	14137.9i	1.56911i	−0.620062	+	0.784553i	$0.712893\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 19258.6i	− 2.10816i
$438$	0	0
$439$	−9389.80	−1.02084	−0.510422	−	0.859924i	$-0.670511\pi$
$439$	−9389.80	−1.02084	−0.510422	+	0.859924i	$0.670511\pi$
$440$	0	0
$441$	−1320.21	−0.142556
$442$	0	0
$443$	− 2999.50i	− 0.321694i	−0.986979	−	0.160847i	$-0.948577\pi$
$443$	− 2999.50i	− 0.321694i	0.986979	−	0.160847i	$-0.0514225\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	281.425i	0.0297784i
$448$	0	0
$449$	745.619	0.0783695	0.0391848	−	0.999232i	$-0.487524\pi$
$449$	745.619	0.0783695	0.0391848	+	0.999232i	$0.487524\pi$
$450$	0	0
$451$	−13790.0	−1.43979
$452$	0	0
$453$	− 282.627i	− 0.0293134i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2858.68i	0.292611i	0.989239	+	0.146305i	$0.0467382\pi$
$457$	2858.68i	0.292611i	−0.989239	+	0.146305i	$0.953262\pi$
$458$	0	0
$459$	642.525	0.0653388
$460$	0	0
$461$	2094.48	0.211605	0.105802	−	0.994387i	$-0.466259\pi$
$461$	2094.48	0.211605	0.105802	+	0.994387i	$0.466259\pi$
$462$	0	0
$463$	− 16672.5i	− 1.67351i	−0.547577	−	0.836755i	$-0.684450\pi$
$463$	− 16672.5i	− 1.67351i	0.547577	−	0.836755i	$-0.315550\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8898.67i	− 0.881759i	−0.897566	−	0.440879i	$-0.854667\pi$
$467$	− 8898.67i	− 0.881759i	0.897566	−	0.440879i	$-0.145333\pi$
$468$	0	0
$469$	1247.11	0.122785
$470$	0	0
$471$	−343.182	−0.0335732
$472$	0	0
$473$	− 3751.55i	− 0.364686i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5511.31i	− 0.529027i
$478$	0	0
$479$	−4033.27	−0.384728	−0.192364	−	0.981324i	$-0.561615\pi$
$479$	−4033.27	−0.384728	−0.192364	+	0.981324i	$0.561615\pi$
$480$	0	0
$481$	11319.5	1.07302
$482$	0	0
$483$	− 199.849i	− 0.0188270i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20567.3i	1.91374i	0.290513	+	0.956871i	$0.406174\pi$
$487$	20567.3i	1.91374i	−0.290513	+	0.956871i	$0.593826\pi$
$488$	0	0
$489$	594.930	0.0550177
$490$	0	0
$491$	−9445.63	−0.868178	−0.434089	−	0.900870i	$-0.642930\pi$
$491$	−9445.63	−0.868178	−0.434089	+	0.900870i	$0.642930\pi$
$492$	0	0
$493$	2472.27i	0.225853i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7262.11i	0.655432i
$498$	0	0
$499$	6043.96	0.542214	0.271107	−	0.962549i	$-0.412610\pi$
$499$	6043.96	0.542214	0.271107	+	0.962549i	$0.412610\pi$
$500$	0	0
$501$	−360.848	−0.0321787
$502$	0	0
$503$	− 420.284i	− 0.0372555i	−0.999826	−	0.0186278i	$-0.994070\pi$
$503$	− 420.284i	− 0.0372555i	0.999826	−	0.0186278i	$-0.00592975\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 126.383i	− 0.0110707i
$508$	0	0
$509$	−9619.80	−0.837702	−0.418851	−	0.908055i	$-0.637567\pi$
$509$	−9619.80	−0.837702	−0.418851	+	0.908055i	$0.637567\pi$
$510$	0	0
$511$	2910.39	0.251953
$512$	0	0
$513$	2073.83i	0.178483i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4606.32i	0.391848i
$518$	0	0
$519$	−409.939	−0.0346711
$520$	0	0
$521$	−10113.4	−0.850436	−0.425218	−	0.905091i	$-0.639803\pi$
$521$	−10113.4	−0.850436	−0.425218	+	0.905091i	$0.639803\pi$
$522$	0	0
$523$	7459.91i	0.623707i	0.950130	+	0.311854i	$0.100950\pi$
$523$	7459.91i	0.623707i	−0.950130	+	0.311854i	$0.899050\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8231.11i	0.680366i
$528$	0	0
$529$	−2134.84	−0.175461
$530$	0	0
$531$	9471.50	0.774064
$532$	0	0
$533$	13193.1i	1.07215i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 716.934i	− 0.0576127i
$538$	0	0
$539$	−2674.27	−0.213709
$540$	0	0
$541$	−7127.40	−0.566415	−0.283208	−	0.959059i	$-0.591399\pi$
$541$	−7127.40	−0.566415	−0.283208	+	0.959059i	$0.591399\pi$
$542$	0	0
$543$	− 47.7898i	− 0.00377690i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24741.3i	1.93393i	0.254902	+	0.966967i	$0.417957\pi$
$547$	24741.3i	1.93393i	−0.254902	+	0.966967i	$0.582043\pi$
$548$	0	0
$549$	14892.4	1.15773
$550$	0	0
$551$	−7979.55	−0.616951
$552$	0	0
$553$	6519.30i	0.501318i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7053.82i	0.536589i	0.963337	+	0.268295i	$0.0864601\pi$
$557$	7053.82i	0.536589i	−0.963337	+	0.268295i	$0.913540\pi$
$558$	0	0
$559$	−3589.18	−0.271568
$560$	0	0
$561$	650.077	0.0489238
$562$	0	0
$563$	− 12659.8i	− 0.947689i	−0.880609	−	0.473844i	$-0.842866\pi$
$563$	− 12659.8i	− 0.947689i	0.880609	−	0.473844i	$-0.157134\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 5070.71i	− 0.375573i
$568$	0	0
$569$	−21504.2	−1.58436	−0.792182	−	0.610285i	$-0.791055\pi$
$569$	−21504.2	−1.58436	−0.792182	+	0.610285i	$0.791055\pi$
$570$	0	0
$571$	−13590.0	−0.996016	−0.498008	−	0.867172i	$-0.665935\pi$
$571$	−13590.0	−0.996016	−0.498008	+	0.867172i	$0.665935\pi$
$572$	0	0
$573$	− 290.006i	− 0.0211434i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13310.1i	0.960323i	0.877180	+	0.480161i	$0.159422\pi$
$577$	13310.1i	0.960323i	−0.877180	+	0.480161i	$0.840578\pi$
$578$	0	0
$579$	−604.753	−0.0434070
$580$	0	0
$581$	−1564.97	−0.111749
$582$	0	0
$583$	− 11164.0i	− 0.793077i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4873.26i	0.342659i	0.985214	+	0.171329i	$0.0548063\pi$
$587$	4873.26i	0.342659i	−0.985214	+	0.171329i	$0.945194\pi$
$588$	0	0
$589$	−26566.9	−1.85852
$590$	0	0
$591$	−491.102	−0.0341815
$592$	0	0
$593$	− 26496.0i	− 1.83484i	−0.397921	−	0.917420i	$-0.630268\pi$
$593$	− 26496.0i	− 1.83484i	0.397921	−	0.917420i	$-0.369732\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	632.016i	0.0433278i
$598$	0	0
$599$	15300.2	1.04365	0.521826	−	0.853052i	$-0.325251\pi$
$599$	15300.2	1.04365	0.521826	+	0.853052i	$0.325251\pi$
$600$	0	0
$601$	793.950	0.0538867	0.0269434	−	0.999637i	$-0.491423\pi$
$601$	793.950	0.0538867	0.0269434	+	0.999637i	$0.491423\pi$
$602$	0	0
$603$	4800.14i	0.324174i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3938.86i	0.263383i	0.991291	+	0.131691i	$0.0420408\pi$
$607$	3938.86i	0.263383i	−0.991291	+	0.131691i	$0.957959\pi$
$608$	0	0
$609$	−82.8045	−0.00550970
$610$	0	0
$611$	4406.96	0.291794
$612$	0	0
$613$	24436.3i	1.61007i	0.593229	+	0.805034i	$0.297853\pi$
$613$	24436.3i	1.61007i	−0.593229	+	0.805034i	$0.702147\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8997.67i	0.587087i	0.955946	+	0.293543i	$0.0948345\pi$
$617$	8997.67i	0.587087i	−0.955946	+	0.293543i	$0.905165\pi$
$618$	0	0
$619$	−9643.79	−0.626198	−0.313099	−	0.949720i	$-0.601367\pi$
$619$	−9643.79	−0.626198	−0.313099	+	0.949720i	$0.601367\pi$
$620$	0	0
$621$	1540.06	0.0995179
$622$	0	0
$623$	− 10386.6i	− 0.667946i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2098.20i	0.133643i
$628$	0	0
$629$	10816.3	0.685653
$630$	0	0
$631$	−12535.4	−0.790849	−0.395425	−	0.918498i	$-0.629403\pi$
$631$	−12535.4	−0.790849	−0.395425	+	0.918498i	$0.629403\pi$
$632$	0	0
$633$	− 1293.59i	− 0.0812250i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2558.53i	0.159141i
$638$	0	0
$639$	−27951.9	−1.73045
$640$	0	0
$641$	4442.01	0.273711	0.136856	−	0.990591i	$-0.456300\pi$
$641$	4442.01	0.273711	0.136856	+	0.990591i	$0.456300\pi$
$642$	0	0
$643$	19188.2i	1.17684i	0.808556	+	0.588419i	$0.200250\pi$
$643$	19188.2i	1.17684i	−0.808556	+	0.588419i	$0.799750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3797.43i	− 0.230746i	−0.993322	−	0.115373i	$-0.963194\pi$
$647$	− 3797.43i	− 0.230746i	0.993322	−	0.115373i	$-0.0368063\pi$
$648$	0	0
$649$	19185.9	1.16042
$650$	0	0
$651$	−275.687	−0.0165976
$652$	0	0
$653$	− 28736.4i	− 1.72212i	−0.508504	−	0.861060i	$-0.669801\pi$
$653$	− 28736.4i	− 1.72212i	0.508504	−	0.861060i	$-0.330199\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11202.1i	0.665198i
$658$	0	0
$659$	−3673.08	−0.217121	−0.108561	−	0.994090i	$-0.534624\pi$
$659$	−3673.08	−0.217121	−0.108561	+	0.994090i	$0.534624\pi$
$660$	0	0
$661$	24299.2	1.42985	0.714925	−	0.699201i	$-0.246461\pi$
$661$	24299.2	1.42985	0.714925	+	0.699201i	$0.246461\pi$
$662$	0	0
$663$	− 621.942i	− 0.0364317i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5925.77i	0.343998i
$668$	0	0
$669$	−590.574	−0.0341299
$670$	0	0
$671$	30166.8	1.73558
$672$	0	0
$673$	20354.4i	1.16583i	0.812533	+	0.582915i	$0.198088\pi$
$673$	20354.4i	1.16583i	−0.812533	+	0.582915i	$0.801912\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23460.0i	1.33182i	0.746034	+	0.665908i	$0.231956\pi$
$677$	23460.0i	1.33182i	−0.746034	+	0.665908i	$0.768044\pi$
$678$	0	0
$679$	−11964.4	−0.676219
$680$	0	0
$681$	−309.561	−0.0174191
$682$	0	0
$683$	− 15399.9i	− 0.862751i	−0.902172	−	0.431376i	$-0.858028\pi$
$683$	− 15399.9i	− 0.862751i	0.902172	−	0.431376i	$-0.141972\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	592.221i	0.0328889i
$688$	0	0
$689$	−10680.8	−0.590574
$690$	0	0
$691$	−8545.82	−0.470475	−0.235238	−	0.971938i	$-0.575587\pi$
$691$	−8545.82	−0.470475	−0.235238	+	0.971938i	$0.575587\pi$
$692$	0	0
$693$	− 10293.3i	− 0.564227i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12606.7i	0.685097i
$698$	0	0
$699$	226.127	0.0122359
$700$	0	0
$701$	19864.4	1.07028	0.535141	−	0.844763i	$-0.320258\pi$
$701$	19864.4	1.07028	0.535141	+	0.844763i	$0.320258\pi$
$702$	0	0
$703$	34911.0i	1.87296i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8030.72i	0.427194i
$708$	0	0
$709$	4831.78	0.255940	0.127970	−	0.991778i	$-0.459154\pi$
$709$	4831.78	0.255940	0.127970	+	0.991778i	$0.459154\pi$
$710$	0	0
$711$	−25092.8	−1.32356
$712$	0	0
$713$	19729.1i	1.03627i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1244.87i	0.0648405i
$718$	0	0
$719$	−11434.3	−0.593082	−0.296541	−	0.955020i	$-0.595833\pi$
$719$	−11434.3	−0.593082	−0.296541	+	0.955020i	$0.595833\pi$
$720$	0	0
$721$	1354.15	0.0699463
$722$	0	0
$723$	− 348.934i	− 0.0179488i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 11809.6i	− 0.602467i	−0.953550	−	0.301234i	$-0.902602\pi$
$727$	− 11809.6i	− 0.602467i	0.953550	−	0.301234i	$-0.0973984\pi$
$728$	0	0
$729$	19434.2	0.987357
$730$	0	0
$731$	−3429.64	−0.173529
$732$	0	0
$733$	21093.2i	1.06289i	0.847094	+	0.531443i	$0.178350\pi$
$733$	21093.2i	1.06289i	−0.847094	+	0.531443i	$0.821650\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9723.39i	0.485978i
$738$	0	0
$739$	−12100.7	−0.602343	−0.301172	−	0.953570i	$-0.597378\pi$
$739$	−12100.7	−0.602343	−0.301172	+	0.953570i	$0.597378\pi$
$740$	0	0
$741$	2007.39	0.0995186
$742$	0	0
$743$	13422.4i	0.662745i	0.943500	+	0.331373i	$0.107512\pi$
$743$	13422.4i	0.662745i	−0.943500	+	0.331373i	$0.892488\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6023.58i	− 0.295035i
$748$	0	0
$749$	−2504.51	−0.122180
$750$	0	0
$751$	−14422.7	−0.700787	−0.350393	−	0.936603i	$-0.613952\pi$
$751$	−14422.7	−0.700787	−0.350393	+	0.936603i	$0.613952\pi$
$752$	0	0
$753$	− 675.154i	− 0.0326746i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3910.02i	− 0.187730i	−0.995585	−	0.0938652i	$-0.970078\pi$
$757$	− 3910.02i	− 0.187730i	0.995585	−	0.0938652i	$-0.0299223\pi$
$758$	0	0
$759$	1558.16	0.0745161
$760$	0	0
$761$	3806.66	0.181329	0.0906646	−	0.995881i	$-0.471101\pi$
$761$	3806.66	0.181329	0.0906646	+	0.995881i	$0.471101\pi$
$762$	0	0
$763$	3768.28i	0.178795i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 18355.5i	− 0.864120i
$768$	0	0
$769$	34062.2	1.59729	0.798643	−	0.601805i	$-0.205552\pi$
$769$	34062.2	1.59729	0.798643	+	0.601805i	$0.205552\pi$
$770$	0	0
$771$	528.794	0.0247005
$772$	0	0
$773$	18721.5i	0.871105i	0.900163	+	0.435552i	$0.143447\pi$
$773$	18721.5i	0.871105i	−0.900163	+	0.435552i	$0.856553\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	362.275i	0.0167266i
$778$	0	0
$779$	−40689.6	−1.87144
$780$	0	0
$781$	−56620.5	−2.59416
$782$	0	0
$783$	− 638.104i	− 0.0291238i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24041.7i	1.08894i	0.838781	+	0.544468i	$0.183269\pi$
$787$	24041.7i	1.08894i	−0.838781	+	0.544468i	$0.816731\pi$
$788$	0	0
$789$	−294.313	−0.0132799
$790$	0	0
$791$	13109.8	0.589295
$792$	0	0
$793$	− 28861.2i	− 1.29242i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 15102.9i	− 0.671231i	−0.941999	−	0.335615i	$-0.891056\pi$
$797$	− 15102.9i	− 0.671231i	0.941999	−	0.335615i	$-0.108944\pi$
$798$	0	0
$799$	4211.07	0.186454
$800$	0	0
$801$	39978.1	1.76349
$802$	0	0
$803$	22691.4i	0.997215i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1929.36i	0.0841595i
$808$	0	0
$809$	−323.993	−0.0140803	−0.00704016	−	0.999975i	$-0.502241\pi$
$809$	−323.993	−0.0140803	−0.00704016	+	0.999975i	$0.502241\pi$
$810$	0	0
$811$	35110.7	1.52023	0.760113	−	0.649791i	$-0.225144\pi$
$811$	35110.7	1.52023	0.760113	+	0.649791i	$0.225144\pi$
$812$	0	0
$813$	74.1086i	0.00319693i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11069.6i	− 0.474021i
$818$	0	0
$819$	−9847.79	−0.420158
$820$	0	0
$821$	−32175.3	−1.36775	−0.683877	−	0.729598i	$-0.739707\pi$
$821$	−32175.3	−1.36775	−0.683877	+	0.729598i	$0.739707\pi$
$822$	0	0
$823$	− 27363.1i	− 1.15895i	−0.814989	−	0.579476i	$-0.803257\pi$
$823$	− 27363.1i	− 1.15895i	0.814989	−	0.579476i	$-0.196743\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9970.57i	− 0.419239i	−0.977783	−	0.209620i	$-0.932777\pi$
$827$	− 9970.57i	− 0.419239i	0.977783	−	0.209620i	$-0.0672226\pi$
$828$	0	0
$829$	−31637.0	−1.32545	−0.662725	−	0.748863i	$-0.730600\pi$
$829$	−31637.0	−1.32545	−0.662725	+	0.748863i	$0.730600\pi$
$830$	0	0
$831$	1674.87	0.0699165
$832$	0	0
$833$	2444.80i	0.101690i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2124.48i	− 0.0877334i
$838$	0	0
$839$	13868.0	0.570653	0.285326	−	0.958430i	$-0.407898\pi$
$839$	13868.0	0.570653	0.285326	+	0.958430i	$0.407898\pi$
$840$	0	0
$841$	−21933.7	−0.899329
$842$	0	0
$843$	1462.86i	0.0597668i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 11533.5i	− 0.467882i
$848$	0	0
$849$	1964.45	0.0794107
$850$	0	0
$851$	25925.6	1.04432
$852$	0	0
$853$	30781.4i	1.23556i	0.786350	+	0.617781i	$0.211968\pi$
$853$	30781.4i	1.23556i	−0.786350	+	0.617781i	$0.788032\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16576.7i	0.660735i	0.943852	+	0.330367i	$0.107173\pi$
$857$	16576.7i	0.660735i	−0.943852	+	0.330367i	$0.892827\pi$
$858$	0	0
$859$	−6912.50	−0.274565	−0.137283	−	0.990532i	$-0.543837\pi$
$859$	−6912.50	−0.274565	−0.137283	+	0.990532i	$0.543837\pi$
$860$	0	0
$861$	−422.240	−0.0167130
$862$	0	0
$863$	10218.5i	0.403060i	0.979482	+	0.201530i	$0.0645913\pi$
$863$	10218.5i	0.403060i	−0.979482	+	0.201530i	$0.935409\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	578.586i	0.0226641i
$868$	0	0
$869$	−50829.0	−1.98419
$870$	0	0
$871$	9302.56	0.361889
$872$	0	0
$873$	− 46051.1i	− 1.78533i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 29190.4i	− 1.12393i	−0.827160	−	0.561967i	$-0.810045\pi$
$877$	− 29190.4i	− 1.12393i	0.827160	−	0.561967i	$-0.189955\pi$
$878$	0	0
$879$	1844.59	0.0707811
$880$	0	0
$881$	6304.66	0.241100	0.120550	−	0.992707i	$-0.461534\pi$
$881$	6304.66	0.241100	0.120550	+	0.992707i	$0.461534\pi$
$882$	0	0
$883$	− 50266.9i	− 1.91576i	−0.287169	−	0.957880i	$-0.592714\pi$
$883$	− 50266.9i	− 1.91576i	0.287169	−	0.957880i	$-0.407286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26704.1i	1.01086i	0.862867	+	0.505431i	$0.168666\pi$
$887$	26704.1i	1.01086i	−0.862867	+	0.505431i	$0.831334\pi$
$888$	0	0
$889$	−16345.1	−0.616646
$890$	0	0
$891$	39534.8	1.48649
$892$	0	0
$893$	13591.7i	0.509327i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1490.73i	− 0.0554893i
$898$	0	0
$899$	8174.46	0.303263
$900$	0	0
$901$	−10206.0	−0.377372
$902$	0	0
$903$	− 114.870i	− 0.00423326i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1833.79i	− 0.0671334i	−0.999436	−	0.0335667i	$-0.989313\pi$
$907$	− 1833.79i	− 0.0671334i	0.999436	−	0.0335667i	$-0.0106866\pi$
$908$	0	0
$909$	−30910.3	−1.12786
$910$	0	0
$911$	30890.1	1.12342	0.561710	−	0.827334i	$-0.310144\pi$
$911$	30890.1	1.12342	0.561710	+	0.827334i	$0.310144\pi$
$912$	0	0
$913$	− 12201.6i	− 0.442295i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14243.2i	− 0.512925i
$918$	0	0
$919$	29295.0	1.05152	0.525762	−	0.850631i	$-0.323780\pi$
$919$	29295.0	1.05152	0.525762	+	0.850631i	$0.323780\pi$
$920$	0	0
$921$	−38.8400	−0.00138960
$922$	0	0
$923$	54170.0i	1.93177i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5212.14i	0.184670i
$928$	0	0
$929$	44472.0	1.57059	0.785295	−	0.619121i	$-0.212511\pi$
$929$	44472.0	1.57059	0.785295	+	0.619121i	$0.212511\pi$
$930$	0	0
$931$	−7890.89	−0.277780
$932$	0	0
$933$	− 645.202i	− 0.0226398i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28853.7i	− 1.00599i	−0.864290	−	0.502993i	$-0.832232\pi$
$937$	− 28853.7i	− 1.00599i	0.864290	−	0.502993i	$-0.167768\pi$
$938$	0	0
$939$	−177.690	−0.00617540
$940$	0	0
$941$	−10118.1	−0.350520	−0.175260	−	0.984522i	$-0.556077\pi$
$941$	−10118.1	−0.350520	−0.175260	+	0.984522i	$0.556077\pi$
$942$	0	0
$943$	30216.9i	1.04347i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17461.3i	0.599173i	0.954069	+	0.299586i	$0.0968486\pi$
$947$	17461.3i	0.599173i	−0.954069	+	0.299586i	$0.903151\pi$
$948$	0	0
$949$	21709.4	0.742588
$950$	0	0
$951$	−898.726	−0.0306448
$952$	0	0
$953$	− 18405.8i	− 0.625628i	−0.949814	−	0.312814i	$-0.898728\pi$
$953$	− 18405.8i	− 0.625628i	0.949814	−	0.312814i	$-0.101272\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 645.603i	− 0.0218071i
$958$	0	0
$959$	−22194.7	−0.747346
$960$	0	0
$961$	−2575.18	−0.0864416
$962$	0	0
$963$	− 9639.87i	− 0.322576i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 35686.8i	− 1.18678i	−0.804917	−	0.593388i	$-0.797790\pi$
$967$	− 35686.8i	− 1.18678i	0.804917	−	0.593388i	$-0.202210\pi$
$968$	0	0
$969$	1918.16	0.0635915
$970$	0	0
$971$	−405.185	−0.0133914	−0.00669568	−	0.999978i	$-0.502131\pi$
$971$	−405.185	−0.0133914	−0.00669568	+	0.999978i	$0.502131\pi$
$972$	0	0
$973$	− 5388.31i	− 0.177535i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56356.3i	1.84544i	0.385467	+	0.922722i	$0.374041\pi$
$977$	56356.3i	1.84544i	−0.385467	+	0.922722i	$0.625959\pi$
$978$	0	0
$979$	80981.3	2.64369
$980$	0	0
$981$	−14504.1	−0.472049
$982$	0	0
$983$	29881.0i	0.969538i	0.874642	+	0.484769i	$0.161096\pi$
$983$	29881.0i	0.969538i	−0.874642	+	0.484769i	$0.838904\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	141.042i	0.00454856i
$988$	0	0
$989$	−8220.48	−0.264303
$990$	0	0
$991$	7701.21	0.246859	0.123429	−	0.992353i	$-0.460611\pi$
$991$	7701.21	0.246859	0.123429	+	0.992353i	$0.460611\pi$
$992$	0	0
$993$	766.180i	0.0244854i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 15453.1i	− 0.490879i	−0.969412	−	0.245439i	$-0.921068\pi$
$997$	− 15453.1i	− 0.490879i	0.969412	−	0.245439i	$-0.0789322\pi$
$998$	0	0
$999$	−2791.74	−0.0884152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.4.g.l.449.4		6
5.2	odd	4		280.4.a.f.1.2	✓	3
5.3	odd	4		1400.4.a.m.1.2		3
5.4	even	2	inner	1400.4.g.l.449.3		6
20.7	even	4		560.4.a.w.1.2		3
35.27	even	4		1960.4.a.r.1.2		3
40.27	even	4		2240.4.a.br.1.2		3
40.37	odd	4		2240.4.a.bz.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.f.1.2	✓	3	5.2	odd	4
560.4.a.w.1.2		3	20.7	even	4
1400.4.a.m.1.2		3	5.3	odd	4
1400.4.g.l.449.3		6	5.4	even	2	inner
1400.4.g.l.449.4		6	1.1	even	1	trivial
1960.4.a.r.1.2		3	35.27	even	4
2240.4.a.br.1.2		3	40.27	even	4
2240.4.a.bz.1.2		3	40.37	odd	4

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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1400.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.238730i q^{3} -7.00000i q^{7} +26.9430 q^{9} +O(q^{10})\)	\(q+0.238730i q^{3} -7.00000i q^{7} +26.9430 q^{9} +54.5770 q^{11} -52.2149i q^{13} -49.8939i q^{17} +161.039 q^{19} +1.67111 q^{21} -119.590i q^{23} +12.8778i q^{27} -49.5506 q^{29} -164.972 q^{31} +13.0292i q^{33} +216.787i q^{37} +12.4653 q^{39} -252.670 q^{41} -68.7387i q^{43} +84.4004i q^{47} -49.0000 q^{49} +11.9112 q^{51} -204.554i q^{53} +38.4448i q^{57} +351.538 q^{59} +552.739 q^{61} -188.601i q^{63} +178.159i q^{67} +28.5498 q^{69} -1037.44 q^{71} +415.770i q^{73} -382.039i q^{77} -931.328 q^{79} +724.387 q^{81} -223.567i q^{83} -11.8292i q^{87} +1483.80 q^{89} -365.504 q^{91} -39.3838i q^{93} -1709.21i q^{97} +1470.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 50 q^{9}+O(q^{10}) \) 6 * q - 50 * q^9	\( 6 q - 50 q^{9} - 12 q^{11} + 304 q^{19} - 84 q^{21} + 936 q^{29} - 536 q^{31} + 1348 q^{39} - 148 q^{41} - 294 q^{49} - 732 q^{51} + 2176 q^{59} + 348 q^{61} + 3568 q^{69} - 2112 q^{71} - 724 q^{79} + 5158 q^{81} - 316 q^{89} + 112 q^{91} + 2544 q^{99}+O(q^{100}) \) 6 * q - 50 * q^9 - 12 * q^11 + 304 * q^19 - 84 * q^21 + 936 * q^29 - 536 * q^31 + 1348 * q^39 - 148 * q^41 - 294 * q^49 - 732 * q^51 + 2176 * q^59 + 348 * q^61 + 3568 * q^69 - 2112 * q^71 - 724 * q^79 + 5158 * q^81 - 316 * q^89 + 112 * q^91 + 2544 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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