LMFDB - Embedded newform 784.4.f.i.783.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.16928550682624.32
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 62x^{6} - 152x^{5} + 1187x^{4} - 1424x^{3} + 7038x^{2} + 1452x + 5287 $ x^8 - 4x^7 + 62x^6 - 152x^5 + 1187x^4 - 1424x^3 + 7038x^2 + 1452*x + 5287
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.7
Root			$-0.207107 - 4.54966i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.i.783.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.98491 q^{3} -9.37011i q^{5} +72.6985 q^{9} +O(q^{10})$	$q+9.98491 q^{3} -9.37011i q^{5} +72.6985 q^{9} +44.5415i q^{11} +71.8225i q^{13} -93.5597i q^{15} +52.6114i q^{17} +68.8909 q^{19} +155.783i q^{23} +37.2010 q^{25} +456.295 q^{27} -131.799 q^{29} +2.00707 q^{31} +444.743i q^{33} -277.186 q^{37} +717.141i q^{39} -272.652i q^{41} -445.640i q^{43} -681.193i q^{45} +135.775 q^{47} +525.320i q^{51} +362.583 q^{53} +417.359 q^{55} +687.869 q^{57} +488.257 q^{59} -113.696i q^{61} +672.985 q^{65} -463.322i q^{67} +1555.48i q^{69} -480.779i q^{71} +146.445i q^{73} +371.449 q^{75} -1104.36i q^{79} +2593.21 q^{81} -727.895 q^{83} +492.975 q^{85} -1316.00 q^{87} +51.3114i q^{89} +20.0404 q^{93} -645.515i q^{95} +556.128i q^{97} +3238.10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 344 q^{9}+O(q^{10}) $ 8 * q + 344 * q^9	$ 8 q + 344 q^{9} + 456 q^{25} - 896 q^{29} + 208 q^{53} + 672 q^{57} + 3008 q^{65} + 6728 q^{81} - 16 q^{85} + 6496 q^{93}+O(q^{100}) $ 8 * q + 344 * q^9 + 456 * q^25 - 896 * q^29 + 208 * q^53 + 672 * q^57 + 3008 * q^65 + 6728 * q^81 - 16 * q^85 + 6496 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	9.98491	1.92160	0.960799	−	0.277247i	$-0.0894220\pi$
$3$	9.98491	1.92160	0.960799	+	0.277247i	$0.0894220\pi$
$4$	0	0
$5$	− 9.37011i	− 0.838088i	−0.907966	−	0.419044i	$-0.862365\pi$
$5$	− 9.37011i	− 0.838088i	0.907966	−	0.419044i	$-0.137635\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	72.6985	2.69254
$10$	0	0
$11$	44.5415i	1.22089i	0.792059	+	0.610445i	$0.209009\pi$
$11$	44.5415i	1.22089i	−0.792059	+	0.610445i	$0.790991\pi$
$12$	0	0
$13$	71.8225i	1.53231i	0.642658	+	0.766153i	$0.277832\pi$
$13$	71.8225i	1.53231i	−0.642658	+	0.766153i	$0.722168\pi$
$14$	0	0
$15$	− 93.5597i	− 1.61047i
$16$	0	0
$17$	52.6114i	0.750596i	0.926904	+	0.375298i	$0.122460\pi$
$17$	52.6114i	0.750596i	−0.926904	+	0.375298i	$0.877540\pi$
$18$	0	0
$19$	68.8909	0.831823	0.415912	−	0.909405i	$-0.363463\pi$
$19$	68.8909	0.831823	0.415912	+	0.909405i	$0.363463\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	155.783i	1.41230i	0.708060	+	0.706152i	$0.249570\pi$
$23$	155.783i	1.41230i	−0.708060	+	0.706152i	$0.750430\pi$
$24$	0	0
$25$	37.2010	0.297608
$26$	0	0
$27$	456.295	3.25237
$28$	0	0
$29$	−131.799	−0.843947	−0.421973	−	0.906608i	$-0.638662\pi$
$29$	−131.799	−0.843947	−0.421973	+	0.906608i	$0.638662\pi$
$30$	0	0
$31$	2.00707	0.0116284	0.00581420	−	0.999983i	$-0.498149\pi$
$31$	2.00707	0.0116284	0.00581420	+	0.999983i	$0.498149\pi$
$32$	0	0
$33$	444.743i	2.34606i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−277.186	−1.23160	−0.615798	−	0.787904i	$-0.711166\pi$
$37$	−277.186	−1.23160	−0.615798	+	0.787904i	$0.711166\pi$
$38$	0	0
$39$	717.141i	2.94448i
$40$	0	0
$41$	− 272.652i	− 1.03856i	−0.854603	−	0.519282i	$-0.826199\pi$
$41$	− 272.652i	− 1.03856i	0.854603	−	0.519282i	$-0.173801\pi$
$42$	0	0
$43$	− 445.640i	− 1.58045i	−0.612814	−	0.790227i	$-0.709963\pi$
$43$	− 445.640i	− 1.58045i	0.612814	−	0.790227i	$-0.290037\pi$
$44$	0	0
$45$	− 681.193i	− 2.25658i
$46$	0	0
$47$	135.775	0.421378	0.210689	−	0.977553i	$-0.432429\pi$
$47$	135.775	0.421378	0.210689	+	0.977553i	$0.432429\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	525.320i	1.44234i
$52$	0	0
$53$	362.583	0.939709	0.469855	−	0.882744i	$-0.344306\pi$
$53$	362.583	0.939709	0.469855	+	0.882744i	$0.344306\pi$
$54$	0	0
$55$	417.359	1.02321
$56$	0	0
$57$	687.869	1.59843
$58$	0	0
$59$	488.257	1.07738	0.538692	−	0.842503i	$-0.318919\pi$
$59$	488.257	1.07738	0.538692	+	0.842503i	$0.318919\pi$
$60$	0	0
$61$	− 113.696i	− 0.238644i	−0.992856	−	0.119322i	$-0.961928\pi$
$61$	− 113.696i	− 0.238644i	0.992856	−	0.119322i	$-0.0380722\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	672.985	1.28421
$66$	0	0
$67$	− 463.322i	− 0.844833i	−0.906402	−	0.422416i	$-0.861182\pi$
$67$	− 463.322i	− 0.844833i	0.906402	−	0.422416i	$-0.138818\pi$
$68$	0	0
$69$	1555.48i	2.71388i
$70$	0	0
$71$	− 480.779i	− 0.803633i	−0.915720	−	0.401816i	$-0.868379\pi$
$71$	− 480.779i	− 0.803633i	0.915720	−	0.401816i	$-0.131621\pi$
$72$	0	0
$73$	146.445i	0.234796i	0.993085	+	0.117398i	$0.0374554\pi$
$73$	146.445i	0.234796i	−0.993085	+	0.117398i	$0.962545\pi$
$74$	0	0
$75$	371.449	0.571883
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 1104.36i	− 1.57279i	−0.617725	−	0.786394i	$-0.711946\pi$
$79$	− 1104.36i	− 1.57279i	0.617725	−	0.786394i	$-0.288054\pi$
$80$	0	0
$81$	2593.21	3.55722
$82$	0	0
$83$	−727.895	−0.962613	−0.481306	−	0.876552i	$-0.659838\pi$
$83$	−727.895	−0.962613	−0.481306	+	0.876552i	$0.659838\pi$
$84$	0	0
$85$	492.975	0.629066
$86$	0	0
$87$	−1316.00	−1.62173
$88$	0	0
$89$	51.3114i	0.0611124i	0.999533	+	0.0305562i	$0.00972785\pi$
$89$	51.3114i	0.0611124i	−0.999533	+	0.0305562i	$0.990272\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	20.0404	0.0223451
$94$	0	0
$95$	− 645.515i	− 0.697141i
$96$	0	0
$97$	556.128i	0.582126i	0.956704	+	0.291063i	$0.0940090\pi$
$97$	556.128i	0.582126i	−0.956704	+	0.291063i	$0.905991\pi$
$98$	0	0
$99$	3238.10i	3.28729i
$100$	0	0
$101$	− 1286.22i	− 1.26716i	−0.773677	−	0.633580i	$-0.781585\pi$
$101$	− 1286.22i	− 1.26716i	0.773677	−	0.633580i	$-0.218415\pi$
$102$	0	0
$103$	−1785.24	−1.70782	−0.853908	−	0.520423i	$-0.825774\pi$
$103$	−1785.24	−1.70782	−0.853908	+	0.520423i	$0.825774\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	566.735i	0.512041i	0.966671	+	0.256020i	$0.0824114\pi$
$107$	566.735i	0.512041i	−0.966671	+	0.256020i	$0.917589\pi$
$108$	0	0
$109$	189.206	0.166263	0.0831314	−	0.996539i	$-0.473508\pi$
$109$	189.206	0.166263	0.0831314	+	0.996539i	$0.473508\pi$
$110$	0	0
$111$	−2767.68	−2.36663
$112$	0	0
$113$	−396.935	−0.330447	−0.165223	−	0.986256i	$-0.552835\pi$
$113$	−396.935	−0.330447	−0.165223	+	0.986256i	$0.552835\pi$
$114$	0	0
$115$	1459.70	1.18364
$116$	0	0
$117$	5221.39i	4.12579i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−652.949	−0.490571
$122$	0	0
$123$	− 2722.41i	− 1.99570i
$124$	0	0
$125$	− 1519.84i	− 1.08751i
$126$	0	0
$127$	− 163.386i	− 0.114159i	−0.998370	−	0.0570795i	$-0.981821\pi$
$127$	− 163.386i	− 0.114159i	0.998370	−	0.0570795i	$-0.0181789\pi$
$128$	0	0
$129$	− 4449.68i	− 3.03700i
$130$	0	0
$131$	−726.791	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$131$	−726.791	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 4275.54i	− 2.72578i
$136$	0	0
$137$	133.497	0.0832515	0.0416258	−	0.999133i	$-0.486746\pi$
$137$	133.497	0.0832515	0.0416258	+	0.999133i	$0.486746\pi$
$138$	0	0
$139$	−1431.86	−0.873730	−0.436865	−	0.899527i	$-0.643911\pi$
$139$	−1431.86	−0.873730	−0.436865	+	0.899527i	$0.643911\pi$
$140$	0	0
$141$	1355.70	0.809719
$142$	0	0
$143$	−3199.09	−1.87078
$144$	0	0
$145$	1234.97i	0.707302i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	485.839	0.267124	0.133562	−	0.991040i	$-0.457358\pi$
$149$	485.839	0.267124	0.133562	+	0.991040i	$0.457358\pi$
$150$	0	0
$151$	− 386.319i	− 0.208200i	−0.994567	−	0.104100i	$-0.966804\pi$
$151$	− 386.319i	− 0.208200i	0.994567	−	0.104100i	$-0.0331962\pi$
$152$	0	0
$153$	3824.77i	2.02101i
$154$	0	0
$155$	− 18.8065i	− 0.00974562i
$156$	0	0
$157$	2533.11i	1.28767i	0.765165	+	0.643835i	$0.222658\pi$
$157$	2533.11i	1.28767i	−0.765165	+	0.643835i	$0.777342\pi$
$158$	0	0
$159$	3620.36	1.80574
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2993.24i	1.43833i	0.694837	+	0.719167i	$0.255476\pi$
$163$	2993.24i	1.43833i	−0.694837	+	0.719167i	$0.744524\pi$
$164$	0	0
$165$	4167.30	1.96620
$166$	0	0
$167$	−375.413	−0.173954	−0.0869769	−	0.996210i	$-0.527721\pi$
$167$	−375.413	−0.173954	−0.0869769	+	0.996210i	$0.527721\pi$
$168$	0	0
$169$	−2961.47	−1.34796
$170$	0	0
$171$	5008.26	2.23972
$172$	0	0
$173$	− 453.129i	− 0.199137i	−0.995031	−	0.0995687i	$-0.968254\pi$
$173$	− 453.129i	− 0.199137i	0.995031	−	0.0995687i	$-0.0317463\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4875.21	2.07030
$178$	0	0
$179$	− 218.456i	− 0.0912188i	−0.998959	−	0.0456094i	$-0.985477\pi$
$179$	− 218.456i	− 0.0912188i	0.998959	−	0.0456094i	$-0.0145230\pi$
$180$	0	0
$181$	1634.88i	0.671378i	0.941973	+	0.335689i	$0.108969\pi$
$181$	1634.88i	0.671378i	−0.941973	+	0.335689i	$0.891031\pi$
$182$	0	0
$183$	− 1135.25i	− 0.458579i
$184$	0	0
$185$	2597.26i	1.03219i
$186$	0	0
$187$	−2343.39	−0.916395
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1887.30i	0.714976i	0.933918	+	0.357488i	$0.116367\pi$
$191$	1887.30i	0.714976i	−0.933918	+	0.357488i	$0.883633\pi$
$192$	0	0
$193$	3167.95	1.18152	0.590762	−	0.806846i	$-0.298827\pi$
$193$	3167.95	1.18152	0.590762	+	0.806846i	$0.298827\pi$
$194$	0	0
$195$	6719.70	2.46773
$196$	0	0
$197$	4767.83	1.72433	0.862167	−	0.506625i	$-0.169107\pi$
$197$	4767.83	1.72433	0.862167	+	0.506625i	$0.169107\pi$
$198$	0	0
$199$	−3977.71	−1.41695	−0.708473	−	0.705738i	$-0.750616\pi$
$199$	−3977.71	−1.41695	−0.708473	+	0.705738i	$0.750616\pi$
$200$	0	0
$201$	− 4626.23i	− 1.62343i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2554.78	−0.870409
$206$	0	0
$207$	11325.2i	3.80268i
$208$	0	0
$209$	3068.51i	1.01556i
$210$	0	0
$211$	− 1882.40i	− 0.614169i	−0.951682	−	0.307084i	$-0.900647\pi$
$211$	− 1882.40i	− 0.614169i	0.951682	−	0.307084i	$-0.0993534\pi$
$212$	0	0
$213$	− 4800.53i	− 1.54426i
$214$	0	0
$215$	−4175.70	−1.32456
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1462.24i	0.451184i
$220$	0	0
$221$	−3778.68	−1.15014
$222$	0	0
$223$	4664.81	1.40080	0.700401	−	0.713750i	$-0.253005\pi$
$223$	4664.81	1.40080	0.700401	+	0.713750i	$0.253005\pi$
$224$	0	0
$225$	2704.46	0.801321
$226$	0	0
$227$	1008.58	0.294897	0.147448	−	0.989070i	$-0.452894\pi$
$227$	1008.58	0.294897	0.147448	+	0.989070i	$0.452894\pi$
$228$	0	0
$229$	152.488i	0.0440030i	0.999758	+	0.0220015i	$0.00700386\pi$
$229$	152.488i	0.0440030i	−0.999758	+	0.0220015i	$0.992996\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2329.92	−0.655099	−0.327550	−	0.944834i	$-0.606223\pi$
$233$	−2329.92	−0.655099	−0.327550	+	0.944834i	$0.606223\pi$
$234$	0	0
$235$	− 1272.22i	− 0.353152i
$236$	0	0
$237$	− 11026.9i	− 3.02227i
$238$	0	0
$239$	− 6739.43i	− 1.82401i	−0.410184	−	0.912003i	$-0.634535\pi$
$239$	− 6739.43i	− 1.82401i	0.410184	−	0.912003i	$-0.365465\pi$
$240$	0	0
$241$	− 473.861i	− 0.126656i	−0.997993	−	0.0633279i	$-0.979829\pi$
$241$	− 473.861i	− 0.126656i	0.997993	−	0.0633279i	$-0.0201714\pi$
$242$	0	0
$243$	13573.0	3.58316
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4947.91i	1.27461i
$248$	0	0
$249$	−7267.97	−1.84975
$250$	0	0
$251$	4796.42	1.20616	0.603082	−	0.797679i	$-0.293939\pi$
$251$	4796.42	1.20616	0.603082	+	0.797679i	$0.293939\pi$
$252$	0	0
$253$	−6938.81	−1.72427
$254$	0	0
$255$	4922.31	1.20881
$256$	0	0
$257$	3956.97i	0.960425i	0.877152	+	0.480212i	$0.159440\pi$
$257$	3956.97i	0.960425i	−0.877152	+	0.480212i	$0.840560\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−9581.59	−2.27236
$262$	0	0
$263$	− 1449.04i	− 0.339740i	−0.985466	−	0.169870i	$-0.945665\pi$
$263$	− 1449.04i	− 0.339740i	0.985466	−	0.169870i	$-0.0543348\pi$
$264$	0	0
$265$	− 3397.44i	− 0.787559i
$266$	0	0
$267$	512.340i	0.117433i
$268$	0	0
$269$	− 3513.91i	− 0.796457i	−0.917286	−	0.398228i	$-0.869625\pi$
$269$	− 3513.91i	− 0.796457i	0.917286	−	0.398228i	$-0.130375\pi$
$270$	0	0
$271$	−375.413	−0.0841501	−0.0420751	−	0.999114i	$-0.513397\pi$
$271$	−375.413	−0.0841501	−0.0420751	+	0.999114i	$0.513397\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1656.99i	0.363347i
$276$	0	0
$277$	1183.53	0.256720	0.128360	−	0.991728i	$-0.459029\pi$
$277$	1183.53	0.256720	0.128360	+	0.991728i	$0.459029\pi$
$278$	0	0
$279$	145.911	0.0313099
$280$	0	0
$281$	5391.53	1.14460	0.572298	−	0.820046i	$-0.306052\pi$
$281$	5391.53	1.14460	0.572298	+	0.820046i	$0.306052\pi$
$282$	0	0
$283$	−4112.53	−0.863832	−0.431916	−	0.901914i	$-0.642162\pi$
$283$	−4112.53	−0.863832	−0.431916	+	0.901914i	$0.642162\pi$
$284$	0	0
$285$	− 6445.41i	− 1.33963i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2145.04	0.436605
$290$	0	0
$291$	5552.89i	1.11861i
$292$	0	0
$293$	− 321.668i	− 0.0641367i	−0.999486	−	0.0320683i	$-0.989791\pi$
$293$	− 321.668i	− 0.0641367i	0.999486	−	0.0320683i	$-0.0102094\pi$
$294$	0	0
$295$	− 4575.02i	− 0.902943i
$296$	0	0
$297$	20324.1i	3.97079i
$298$	0	0
$299$	−11188.7	−2.16408
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 12842.7i	− 2.43497i
$304$	0	0
$305$	−1065.35	−0.200005
$306$	0	0
$307$	−9844.77	−1.83020	−0.915099	−	0.403229i	$-0.867888\pi$
$307$	−9844.77	−1.83020	−0.915099	+	0.403229i	$0.867888\pi$
$308$	0	0
$309$	−17825.5	−3.28174
$310$	0	0
$311$	3862.10	0.704179	0.352090	−	0.935966i	$-0.385471\pi$
$311$	3862.10	0.704179	0.352090	+	0.935966i	$0.385471\pi$
$312$	0	0
$313$	− 7799.27i	− 1.40844i	−0.709983	−	0.704218i	$-0.751298\pi$
$313$	− 7799.27i	− 1.40844i	0.709983	−	0.704218i	$-0.248702\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8229.84	1.45815	0.729075	−	0.684434i	$-0.239950\pi$
$317$	8229.84	1.45815	0.729075	+	0.684434i	$0.239950\pi$
$318$	0	0
$319$	− 5870.53i	− 1.03037i
$320$	0	0
$321$	5658.80i	0.983936i
$322$	0	0
$323$	3624.44i	0.624364i
$324$	0	0
$325$	2671.87i	0.456027i
$326$	0	0
$327$	1889.21	0.319490
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5778.57i	0.959574i	0.877385	+	0.479787i	$0.159286\pi$
$331$	5778.57i	0.959574i	−0.877385	+	0.479787i	$0.840714\pi$
$332$	0	0
$333$	−20151.0	−3.31612
$334$	0	0
$335$	−4341.38	−0.708044
$336$	0	0
$337$	8187.84	1.32350	0.661751	−	0.749724i	$-0.269814\pi$
$337$	8187.84	1.32350	0.661751	+	0.749724i	$0.269814\pi$
$338$	0	0
$339$	−3963.36	−0.634986
$340$	0	0
$341$	89.3979i	0.0141970i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	14575.0	2.27447
$346$	0	0
$347$	− 7716.85i	− 1.19384i	−0.802301	−	0.596919i	$-0.796391\pi$
$347$	− 7716.85i	− 1.19384i	0.802301	−	0.596919i	$-0.203609\pi$
$348$	0	0
$349$	3609.62i	0.553634i	0.960923	+	0.276817i	$0.0892796\pi$
$349$	3609.62i	0.553634i	−0.960923	+	0.276817i	$0.910720\pi$
$350$	0	0
$351$	32772.3i	4.98363i
$352$	0	0
$353$	6932.46i	1.04526i	0.852559	+	0.522631i	$0.175049\pi$
$353$	6932.46i	1.04526i	−0.852559	+	0.522631i	$0.824951\pi$
$354$	0	0
$355$	−4504.95	−0.673515
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6166.91i	0.906622i	0.891352	+	0.453311i	$0.149757\pi$
$359$	6166.91i	0.906622i	−0.891352	+	0.453311i	$0.850243\pi$
$360$	0	0
$361$	−2113.05	−0.308070
$362$	0	0
$363$	−6519.64	−0.942679
$364$	0	0
$365$	1372.21	0.196780
$366$	0	0
$367$	−4113.88	−0.585131	−0.292565	−	0.956245i	$-0.594509\pi$
$367$	−4113.88	−0.585131	−0.292565	+	0.956245i	$0.594509\pi$
$368$	0	0
$369$	− 19821.4i	− 2.79637i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7657.56	−1.06298	−0.531492	−	0.847063i	$-0.678369\pi$
$373$	−7657.56	−1.06298	−0.531492	+	0.847063i	$0.678369\pi$
$374$	0	0
$375$	− 15175.5i	− 2.08976i
$376$	0	0
$377$	− 9466.13i	− 1.29318i
$378$	0	0
$379$	− 9044.59i	− 1.22583i	−0.790149	−	0.612915i	$-0.789997\pi$
$379$	− 9044.59i	− 1.22583i	0.790149	−	0.612915i	$-0.210003\pi$
$380$	0	0
$381$	− 1631.40i	− 0.219368i
$382$	0	0
$383$	2346.71	0.313084	0.156542	−	0.987671i	$-0.449965\pi$
$383$	2346.71	0.313084	0.156542	+	0.987671i	$0.449965\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 32397.4i	− 4.25543i
$388$	0	0
$389$	−1944.72	−0.253474	−0.126737	−	0.991936i	$-0.540450\pi$
$389$	−1944.72	−0.253474	−0.126737	+	0.991936i	$0.540450\pi$
$390$	0	0
$391$	−8195.96	−1.06007
$392$	0	0
$393$	−7256.94	−0.931461
$394$	0	0
$395$	−10348.0	−1.31814
$396$	0	0
$397$	10331.6i	1.30612i	0.757306	+	0.653060i	$0.226515\pi$
$397$	10331.6i	1.30612i	−0.757306	+	0.653060i	$0.773485\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4176.29	−0.520085	−0.260042	−	0.965597i	$-0.583737\pi$
$401$	−4176.29	−0.520085	−0.260042	+	0.965597i	$0.583737\pi$
$402$	0	0
$403$	144.153i	0.0178183i
$404$	0	0
$405$	− 24298.7i	− 2.98126i
$406$	0	0
$407$	− 12346.3i	− 1.50364i
$408$	0	0
$409$	− 6370.70i	− 0.770198i	−0.922875	−	0.385099i	$-0.874167\pi$
$409$	− 6370.70i	− 0.770198i	0.922875	−	0.385099i	$-0.125833\pi$
$410$	0	0
$411$	1332.96	0.159976
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6820.46i	0.806755i
$416$	0	0
$417$	−14297.0	−1.67896
$418$	0	0
$419$	−12152.0	−1.41686	−0.708432	−	0.705779i	$-0.750597\pi$
$419$	−12152.0	−1.41686	−0.708432	+	0.705779i	$0.750597\pi$
$420$	0	0
$421$	−3412.11	−0.395003	−0.197501	−	0.980303i	$-0.563283\pi$
$421$	−3412.11	−0.395003	−0.197501	+	0.980303i	$0.563283\pi$
$422$	0	0
$423$	9870.61	1.13458
$424$	0	0
$425$	1957.20i	0.223384i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−31942.6	−3.59488
$430$	0	0
$431$	− 12838.3i	− 1.43480i	−0.696662	−	0.717400i	$-0.745332\pi$
$431$	− 12838.3i	− 1.43480i	0.696662	−	0.717400i	$-0.254668\pi$
$432$	0	0
$433$	7230.12i	0.802442i	0.915981	+	0.401221i	$0.131414\pi$
$433$	7230.12i	0.802442i	−0.915981	+	0.401221i	$0.868586\pi$
$434$	0	0
$435$	12331.1i	1.35915i
$436$	0	0
$437$	10732.0i	1.17479i
$438$	0	0
$439$	−10429.8	−1.13391	−0.566954	−	0.823749i	$-0.691878\pi$
$439$	−10429.8	−1.13391	−0.566954	+	0.823749i	$0.691878\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2284.40i	0.245000i	0.992469	+	0.122500i	$0.0390911\pi$
$443$	2284.40i	0.245000i	−0.992469	+	0.122500i	$0.960909\pi$
$444$	0	0
$445$	480.794	0.0512176
$446$	0	0
$447$	4851.06	0.513305
$448$	0	0
$449$	−6250.59	−0.656979	−0.328490	−	0.944508i	$-0.606540\pi$
$449$	−6250.59	−0.656979	−0.328490	+	0.944508i	$0.606540\pi$
$450$	0	0
$451$	12144.4	1.26797
$452$	0	0
$453$	− 3857.36i	− 0.400077i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4529.34	−0.463618	−0.231809	−	0.972761i	$-0.574464\pi$
$457$	−4529.34	−0.463618	−0.231809	+	0.972761i	$0.574464\pi$
$458$	0	0
$459$	24006.3i	2.44122i
$460$	0	0
$461$	− 17621.4i	− 1.78028i	−0.455687	−	0.890140i	$-0.650607\pi$
$461$	− 17621.4i	− 1.78028i	0.455687	−	0.890140i	$-0.349393\pi$
$462$	0	0
$463$	1856.39i	0.186337i	0.995650	+	0.0931683i	$0.0296995\pi$
$463$	1856.39i	0.186337i	−0.995650	+	0.0931683i	$0.970301\pi$
$464$	0	0
$465$	− 187.781i	− 0.0187272i
$466$	0	0
$467$	−1862.77	−0.184579	−0.0922897	−	0.995732i	$-0.529419\pi$
$467$	−1862.77	−0.184579	−0.0922897	+	0.995732i	$0.529419\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	25292.9i	2.47438i
$472$	0	0
$473$	19849.5	1.92956
$474$	0	0
$475$	2562.81	0.247557
$476$	0	0
$477$	26359.2	2.53020
$478$	0	0
$479$	9006.29	0.859098	0.429549	−	0.903044i	$-0.358673\pi$
$479$	9006.29	0.859098	0.429549	+	0.903044i	$0.358673\pi$
$480$	0	0
$481$	− 19908.2i	− 1.88718i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5210.98	0.487873
$486$	0	0
$487$	2416.92i	0.224889i	0.993658	+	0.112445i	$0.0358681\pi$
$487$	2416.92i	0.224889i	−0.993658	+	0.112445i	$0.964132\pi$
$488$	0	0
$489$	29887.2i	2.76390i
$490$	0	0
$491$	− 17100.0i	− 1.57171i	−0.618410	−	0.785855i	$-0.712223\pi$
$491$	− 17100.0i	− 1.57171i	0.618410	−	0.785855i	$-0.287777\pi$
$492$	0	0
$493$	− 6934.13i	− 0.633464i
$494$	0	0
$495$	30341.4	2.75504
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 16112.4i	− 1.44547i	−0.691123	−	0.722737i	$-0.742884\pi$
$499$	− 16112.4i	− 1.44547i	0.691123	−	0.722737i	$-0.257116\pi$
$500$	0	0
$501$	−3748.46	−0.334269
$502$	0	0
$503$	15531.4	1.37676	0.688381	−	0.725350i	$-0.258322\pi$
$503$	15531.4	1.37676	0.688381	+	0.725350i	$0.258322\pi$
$504$	0	0
$505$	−12052.0	−1.06199
$506$	0	0
$507$	−29570.0	−2.59024
$508$	0	0
$509$	6159.53i	0.536378i	0.963366	+	0.268189i	$0.0864252\pi$
$509$	6159.53i	0.536378i	−0.963366	+	0.268189i	$0.913575\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	31434.6	2.70540
$514$	0	0
$515$	16727.9i	1.43130i
$516$	0	0
$517$	6047.61i	0.514456i
$518$	0	0
$519$	− 4524.45i	− 0.382662i
$520$	0	0
$521$	9860.29i	0.829150i	0.910015	+	0.414575i	$0.136070\pi$
$521$	9860.29i	0.829150i	−0.910015	+	0.414575i	$0.863930\pi$
$522$	0	0
$523$	−15816.1	−1.32235	−0.661174	−	0.750233i	$-0.729941\pi$
$523$	−15816.1	−1.32235	−0.661174	+	0.750233i	$0.729941\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	105.595i	0.00872823i
$528$	0	0
$529$	−12101.3	−0.994602
$530$	0	0
$531$	35495.6	2.90090
$532$	0	0
$533$	19582.6	1.59140
$534$	0	0
$535$	5310.37	0.429135
$536$	0	0
$537$	− 2181.26i	− 0.175286i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21732.0	1.72704	0.863522	−	0.504312i	$-0.168254\pi$
$541$	21732.0	1.72704	0.863522	+	0.504312i	$0.168254\pi$
$542$	0	0
$543$	16324.1i	1.29012i
$544$	0	0
$545$	− 1772.88i	− 0.139343i
$546$	0	0
$547$	− 14469.3i	− 1.13101i	−0.824746	−	0.565503i	$-0.808682\pi$
$547$	− 14469.3i	− 1.13101i	0.824746	−	0.565503i	$-0.191318\pi$
$548$	0	0
$549$	− 8265.54i	− 0.642559i
$550$	0	0
$551$	−9079.75	−0.702015
$552$	0	0
$553$	0	0
$554$	0	0
$555$	25933.4i	1.98345i
$556$	0	0
$557$	11976.7	0.911074	0.455537	−	0.890217i	$-0.349447\pi$
$557$	11976.7	0.911074	0.455537	+	0.890217i	$0.349447\pi$
$558$	0	0
$559$	32007.0	2.42174
$560$	0	0
$561$	−23398.6	−1.76094
$562$	0	0
$563$	−11099.0	−0.830844	−0.415422	−	0.909629i	$-0.636366\pi$
$563$	−11099.0	−0.830844	−0.415422	+	0.909629i	$0.636366\pi$
$564$	0	0
$565$	3719.32i	0.276944i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15503.4	1.14224	0.571121	−	0.820866i	$-0.306509\pi$
$569$	15503.4	1.14224	0.571121	+	0.820866i	$0.306509\pi$
$570$	0	0
$571$	− 17956.0i	− 1.31600i	−0.753017	−	0.658001i	$-0.771402\pi$
$571$	− 17956.0i	− 1.31600i	0.753017	−	0.658001i	$-0.228598\pi$
$572$	0	0
$573$	18844.5i	1.37390i
$574$	0	0
$575$	5795.28i	0.420313i
$576$	0	0
$577$	− 12632.1i	− 0.911403i	−0.890133	−	0.455701i	$-0.849389\pi$
$577$	− 12632.1i	− 0.911403i	0.890133	−	0.455701i	$-0.150611\pi$
$578$	0	0
$579$	31631.7	2.27041
$580$	0	0
$581$	0	0
$582$	0	0
$583$	16150.0i	1.14728i
$584$	0	0
$585$	48925.0	3.45778
$586$	0	0
$587$	−12331.1	−0.867049	−0.433524	−	0.901142i	$-0.642730\pi$
$587$	−12331.1	−0.867049	−0.433524	+	0.901142i	$0.642730\pi$
$588$	0	0
$589$	138.269	0.00967277
$590$	0	0
$591$	47606.3	3.31347
$592$	0	0
$593$	13057.5i	0.904226i	0.891961	+	0.452113i	$0.149330\pi$
$593$	13057.5i	0.904226i	−0.891961	+	0.452113i	$0.850670\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−39717.1	−2.72280
$598$	0	0
$599$	12969.9i	0.884702i	0.896842	+	0.442351i	$0.145855\pi$
$599$	12969.9i	0.884702i	−0.896842	+	0.442351i	$0.854145\pi$
$600$	0	0
$601$	− 602.499i	− 0.0408926i	−0.999791	−	0.0204463i	$-0.993491\pi$
$601$	− 602.499i	− 0.0408926i	0.999791	−	0.0204463i	$-0.00650871\pi$
$602$	0	0
$603$	− 33682.8i	− 2.27474i
$604$	0	0
$605$	6118.21i	0.411141i
$606$	0	0
$607$	−80.6862	−0.00539531	−0.00269765	−	0.999996i	$-0.500859\pi$
$607$	−80.6862	−0.00539531	−0.00269765	+	0.999996i	$0.500859\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9751.67i	0.645680i
$612$	0	0
$613$	−26381.5	−1.73824	−0.869120	−	0.494602i	$-0.835314\pi$
$613$	−26381.5	−1.73824	−0.869120	+	0.494602i	$0.835314\pi$
$614$	0	0
$615$	−25509.3	−1.67258
$616$	0	0
$617$	21916.5	1.43003	0.715013	−	0.699111i	$-0.246421\pi$
$617$	21916.5	1.43003	0.715013	+	0.699111i	$0.246421\pi$
$618$	0	0
$619$	−6523.46	−0.423586	−0.211793	−	0.977315i	$-0.567930\pi$
$619$	−6523.46	−0.423586	−0.211793	+	0.977315i	$0.567930\pi$
$620$	0	0
$621$	71083.0i	4.59334i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−9590.96	−0.613821
$626$	0	0
$627$	30638.8i	1.95151i
$628$	0	0
$629$	− 14583.1i	− 0.924432i
$630$	0	0
$631$	− 254.337i	− 0.0160460i	−0.999968	−	0.00802298i	$-0.997446\pi$
$631$	− 254.337i	− 0.0160460i	0.999968	−	0.00802298i	$-0.00255382\pi$
$632$	0	0
$633$	− 18795.6i	− 1.18019i
$634$	0	0
$635$	−1530.95	−0.0956754
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 34951.9i	− 2.16381i
$640$	0	0
$641$	−9171.59	−0.565142	−0.282571	−	0.959246i	$-0.591187\pi$
$641$	−9171.59	−0.565142	−0.282571	+	0.959246i	$0.591187\pi$
$642$	0	0
$643$	−566.427	−0.0347398	−0.0173699	−	0.999849i	$-0.505529\pi$
$643$	−566.427	−0.0347398	−0.0173699	+	0.999849i	$0.505529\pi$
$644$	0	0
$645$	−41694.0	−2.54527
$646$	0	0
$647$	−22582.8	−1.37221	−0.686105	−	0.727503i	$-0.740681\pi$
$647$	−22582.8	−1.37221	−0.686105	+	0.727503i	$0.740681\pi$
$648$	0	0
$649$	21747.7i	1.31537i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21180.9	−1.26933	−0.634666	−	0.772787i	$-0.718862\pi$
$653$	−21180.9	−1.26933	−0.634666	+	0.772787i	$0.718862\pi$
$654$	0	0
$655$	6810.11i	0.406249i
$656$	0	0
$657$	10646.4i	0.632198i
$658$	0	0
$659$	26581.0i	1.57125i	0.618706	+	0.785623i	$0.287657\pi$
$659$	26581.0i	1.57125i	−0.618706	+	0.785623i	$0.712343\pi$
$660$	0	0
$661$	− 33622.0i	− 1.97843i	−0.146468	−	0.989215i	$-0.546791\pi$
$661$	− 33622.0i	− 1.97843i	0.146468	−	0.989215i	$-0.453209\pi$
$662$	0	0
$663$	−37729.8	−2.21011
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 20532.0i	− 1.19191i
$668$	0	0
$669$	46577.7	2.69178
$670$	0	0
$671$	5064.21	0.291358
$672$	0	0
$673$	15410.3	0.882650	0.441325	−	0.897347i	$-0.354509\pi$
$673$	15410.3	0.882650	0.441325	+	0.897347i	$0.354509\pi$
$674$	0	0
$675$	16974.6	0.967933
$676$	0	0
$677$	− 235.019i	− 0.0133420i	−0.999978	−	0.00667098i	$-0.997877\pi$
$677$	− 235.019i	− 0.0133420i	0.999978	−	0.00667098i	$-0.00212345\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10070.6	0.566673
$682$	0	0
$683$	− 28591.2i	− 1.60177i	−0.598816	−	0.800886i	$-0.704362\pi$
$683$	− 28591.2i	− 1.60177i	0.598816	−	0.800886i	$-0.295638\pi$
$684$	0	0
$685$	− 1250.89i	− 0.0697721i
$686$	0	0
$687$	1522.58i	0.0845561i
$688$	0	0
$689$	26041.6i	1.43992i
$690$	0	0
$691$	25202.6	1.38748	0.693742	−	0.720224i	$-0.255961\pi$
$691$	25202.6	1.38748	0.693742	+	0.720224i	$0.255961\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13416.7i	0.732263i
$696$	0	0
$697$	14344.6	0.779543
$698$	0	0
$699$	−23264.0	−1.25884
$700$	0	0
$701$	15570.3	0.838919	0.419460	−	0.907774i	$-0.362220\pi$
$701$	15570.3	0.838919	0.419460	+	0.907774i	$0.362220\pi$
$702$	0	0
$703$	−19095.6	−1.02447
$704$	0	0
$705$	− 12703.0i	− 0.678616i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20099.0	1.06465	0.532323	−	0.846541i	$-0.321319\pi$
$709$	20099.0	1.06465	0.532323	+	0.846541i	$0.321319\pi$
$710$	0	0
$711$	− 80285.3i	− 4.23479i
$712$	0	0
$713$	312.667i	0.0164228i
$714$	0	0
$715$	29975.8i	1.56788i
$716$	0	0
$717$	− 67292.6i	− 3.50500i
$718$	0	0
$719$	17029.4	0.883298	0.441649	−	0.897188i	$-0.354394\pi$
$719$	17029.4	0.883298	0.441649	+	0.897188i	$0.354394\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 4731.46i	− 0.243382i
$724$	0	0
$725$	−4903.06	−0.251165
$726$	0	0
$727$	−14129.5	−0.720817	−0.360409	−	0.932795i	$-0.617363\pi$
$727$	−14129.5	−0.720817	−0.360409	+	0.932795i	$0.617363\pi$
$728$	0	0
$729$	65508.6	3.32818
$730$	0	0
$731$	23445.8	1.18628
$732$	0	0
$733$	− 32735.0i	− 1.64951i	−0.565487	−	0.824757i	$-0.691312\pi$
$733$	− 32735.0i	− 1.64951i	0.565487	−	0.824757i	$-0.308688\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20637.1	1.03145
$738$	0	0
$739$	− 13549.9i	− 0.674478i	−0.941419	−	0.337239i	$-0.890507\pi$
$739$	− 13549.9i	− 0.674478i	0.941419	−	0.337239i	$-0.109493\pi$
$740$	0	0
$741$	49404.5i	2.44928i
$742$	0	0
$743$	12007.9i	0.592905i	0.955048	+	0.296452i	$0.0958036\pi$
$743$	12007.9i	0.592905i	−0.955048	+	0.296452i	$0.904196\pi$
$744$	0	0
$745$	− 4552.37i	− 0.223874i
$746$	0	0
$747$	−52916.9	−2.59187
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 1831.36i	− 0.0889841i	−0.999010	−	0.0444921i	$-0.985833\pi$
$751$	− 1831.36i	− 0.0889841i	0.999010	−	0.0444921i	$-0.0141669\pi$
$752$	0	0
$753$	47891.8	2.31776
$754$	0	0
$755$	−3619.85	−0.174490
$756$	0	0
$757$	−40198.4	−1.93003	−0.965016	−	0.262190i	$-0.915555\pi$
$757$	−40198.4	−1.93003	−0.965016	+	0.262190i	$0.915555\pi$
$758$	0	0
$759$	−69283.4	−3.31335
$760$	0	0
$761$	24225.4i	1.15397i	0.816756	+	0.576984i	$0.195770\pi$
$761$	24225.4i	1.15397i	−0.816756	+	0.576984i	$0.804230\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	35838.5	1.69378
$766$	0	0
$767$	35067.9i	1.65088i
$768$	0	0
$769$	5497.35i	0.257789i	0.991658	+	0.128894i	$0.0411428\pi$
$769$	5497.35i	0.257789i	−0.991658	+	0.128894i	$0.958857\pi$
$770$	0	0
$771$	39510.0i	1.84555i
$772$	0	0
$773$	41568.2i	1.93416i	0.254475	+	0.967079i	$0.418097\pi$
$773$	41568.2i	1.93416i	−0.254475	+	0.967079i	$0.581903\pi$
$774$	0	0
$775$	74.6650	0.00346070
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 18783.3i	− 0.863903i
$780$	0	0
$781$	21414.6	0.981147
$782$	0	0
$783$	−60139.3	−2.74483
$784$	0	0
$785$	23735.5	1.07918
$786$	0	0
$787$	−9169.82	−0.415335	−0.207668	−	0.978199i	$-0.566587\pi$
$787$	−9169.82	−0.415335	−0.207668	+	0.978199i	$0.566587\pi$
$788$	0	0
$789$	− 14468.5i	− 0.652844i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8165.95	0.365676
$794$	0	0
$795$	− 33923.2i	− 1.51337i
$796$	0	0
$797$	14252.4i	0.633435i	0.948520	+	0.316717i	$0.102581\pi$
$797$	14252.4i	0.633435i	−0.948520	+	0.316717i	$0.897419\pi$
$798$	0	0
$799$	7143.29i	0.316285i
$800$	0	0
$801$	3730.26i	0.164547i
$802$	0	0
$803$	−6522.91	−0.286660
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 35086.1i	− 1.53047i
$808$	0	0
$809$	−3169.13	−0.137727	−0.0688633	−	0.997626i	$-0.521937\pi$
$809$	−3169.13	−0.137727	−0.0688633	+	0.997626i	$0.521937\pi$
$810$	0	0
$811$	−18422.1	−0.797640	−0.398820	−	0.917029i	$-0.630580\pi$
$811$	−18422.1	−0.797640	−0.398820	+	0.917029i	$0.630580\pi$
$812$	0	0
$813$	−3748.46	−0.161703
$814$	0	0
$815$	28047.0	1.20545
$816$	0	0
$817$	− 30700.6i	− 1.31466i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38927.2	−1.65477	−0.827386	−	0.561633i	$-0.810173\pi$
$821$	−38927.2	−1.65477	−0.827386	+	0.561633i	$0.810173\pi$
$822$	0	0
$823$	39291.9i	1.66419i	0.554631	+	0.832097i	$0.312860\pi$
$823$	39291.9i	1.66419i	−0.554631	+	0.832097i	$0.687140\pi$
$824$	0	0
$825$	16544.9i	0.698206i
$826$	0	0
$827$	− 7699.59i	− 0.323750i	−0.986811	−	0.161875i	$-0.948246\pi$
$827$	− 7699.59i	− 0.323750i	0.986811	−	0.161875i	$-0.0517541\pi$
$828$	0	0
$829$	4971.23i	0.208273i	0.994563	+	0.104136i	$0.0332078\pi$
$829$	4971.23i	0.208273i	−0.994563	+	0.104136i	$0.966792\pi$
$830$	0	0
$831$	11817.4	0.493312
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3517.66i	0.145789i
$836$	0	0
$837$	915.816	0.0378199
$838$	0	0
$839$	326.749	0.0134453	0.00672266	−	0.999977i	$-0.497860\pi$
$839$	326.749	0.0134453	0.00672266	+	0.999977i	$0.497860\pi$
$840$	0	0
$841$	−7018.03	−0.287754
$842$	0	0
$843$	53833.9	2.19945
$844$	0	0
$845$	27749.3i	1.12971i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−41063.2	−1.65994
$850$	0	0
$851$	− 43180.8i	− 1.73939i
$852$	0	0
$853$	− 7306.34i	− 0.293276i	−0.989190	−	0.146638i	$-0.953155\pi$
$853$	− 7306.34i	− 0.293276i	0.989190	−	0.146638i	$-0.0468452\pi$
$854$	0	0
$855$	− 46928.0i	− 1.87708i
$856$	0	0
$857$	3886.28i	0.154904i	0.996996	+	0.0774521i	$0.0246785\pi$
$857$	3886.28i	0.154904i	−0.996996	+	0.0774521i	$0.975322\pi$
$858$	0	0
$859$	−8254.71	−0.327878	−0.163939	−	0.986470i	$-0.552420\pi$
$859$	−8254.71	−0.327878	−0.163939	+	0.986470i	$0.552420\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 42051.3i	− 1.65868i	−0.558742	−	0.829342i	$-0.688716\pi$
$863$	− 42051.3i	− 1.65868i	0.558742	−	0.829342i	$-0.311284\pi$
$864$	0	0
$865$	−4245.87	−0.166895
$866$	0	0
$867$	21418.0	0.838979
$868$	0	0
$869$	49189.9	1.92020
$870$	0	0
$871$	33277.0	1.29454
$872$	0	0
$873$	40429.7i	1.56740i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33485.8	−1.28932	−0.644661	−	0.764469i	$-0.723001\pi$
$877$	−33485.8	−1.28932	−0.644661	+	0.764469i	$0.723001\pi$
$878$	0	0
$879$	− 3211.83i	− 0.123245i
$880$	0	0
$881$	− 31744.2i	− 1.21395i	−0.794721	−	0.606975i	$-0.792383\pi$
$881$	− 31744.2i	− 1.21395i	0.794721	−	0.606975i	$-0.207617\pi$
$882$	0	0
$883$	30695.6i	1.16986i	0.811083	+	0.584931i	$0.198879\pi$
$883$	30695.6i	1.16986i	−0.811083	+	0.584931i	$0.801121\pi$
$884$	0	0
$885$	− 45681.2i	− 1.73509i
$886$	0	0
$887$	20840.2	0.788888	0.394444	−	0.918920i	$-0.370937\pi$
$887$	20840.2	0.788888	0.394444	+	0.918920i	$0.370937\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	115506.i	4.34297i
$892$	0	0
$893$	9353.63	0.350512
$894$	0	0
$895$	−2046.96	−0.0764494
$896$	0	0
$897$	−111718.	−4.15849
$898$	0	0
$899$	−264.530	−0.00981374
$900$	0	0
$901$	19076.0i	0.705342i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15319.0	0.562674
$906$	0	0
$907$	22901.1i	0.838388i	0.907897	+	0.419194i	$0.137687\pi$
$907$	22901.1i	0.838388i	−0.907897	+	0.419194i	$0.862313\pi$
$908$	0	0
$909$	− 93505.9i	− 3.41188i
$910$	0	0
$911$	4526.12i	0.164607i	0.996607	+	0.0823035i	$0.0262277\pi$
$911$	4526.12i	0.164607i	−0.996607	+	0.0823035i	$0.973772\pi$
$912$	0	0
$913$	− 32421.6i	− 1.17524i
$914$	0	0
$915$	−10637.4	−0.384329
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 8790.70i	− 0.315537i	−0.987476	−	0.157769i	$-0.949570\pi$
$919$	− 8790.70i	− 0.315537i	0.987476	−	0.157769i	$-0.0504300\pi$
$920$	0	0
$921$	−98299.2	−3.51690
$922$	0	0
$923$	34530.7	1.23141
$924$	0	0
$925$	−10311.6	−0.366533
$926$	0	0
$927$	−129784.	−4.59836
$928$	0	0
$929$	51590.8i	1.82200i	0.412405	+	0.911001i	$0.364689\pi$
$929$	51590.8i	1.82200i	−0.412405	+	0.911001i	$0.635311\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	38562.8	1.35315
$934$	0	0
$935$	21957.9i	0.768020i
$936$	0	0
$937$	25622.6i	0.893335i	0.894700	+	0.446668i	$0.147389\pi$
$937$	25622.6i	0.893335i	−0.894700	+	0.446668i	$0.852611\pi$
$938$	0	0
$939$	− 77875.0i	− 2.70645i
$940$	0	0
$941$	48959.9i	1.69612i	0.529902	+	0.848059i	$0.322229\pi$
$941$	48959.9i	1.69612i	−0.529902	+	0.848059i	$0.677771\pi$
$942$	0	0
$943$	42474.6	1.46677
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 29422.8i	− 1.00962i	−0.863230	−	0.504811i	$-0.831562\pi$
$947$	− 29422.8i	− 1.00962i	0.863230	−	0.504811i	$-0.168438\pi$
$948$	0	0
$949$	−10518.1	−0.359780
$950$	0	0
$951$	82174.2	2.80198
$952$	0	0
$953$	34930.5	1.18731	0.593657	−	0.804718i	$-0.297683\pi$
$953$	34930.5	1.18731	0.593657	+	0.804718i	$0.297683\pi$
$954$	0	0
$955$	17684.2	0.599213
$956$	0	0
$957$	− 58616.7i	− 1.97995i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29787.0	−0.999865
$962$	0	0
$963$	41200.8i	1.37869i
$964$	0	0
$965$	− 29684.0i	− 0.990221i
$966$	0	0
$967$	− 49631.3i	− 1.65050i	−0.564767	−	0.825250i	$-0.691034\pi$
$967$	− 49631.3i	− 1.65050i	0.564767	−	0.825250i	$-0.308966\pi$
$968$	0	0
$969$	36189.8i	1.19978i
$970$	0	0
$971$	56603.7	1.87075	0.935375	−	0.353658i	$-0.115062\pi$
$971$	56603.7	1.87075	0.935375	+	0.353658i	$0.115062\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	26678.4i	0.876300i
$976$	0	0
$977$	−11175.4	−0.365950	−0.182975	−	0.983118i	$-0.558573\pi$
$977$	−11175.4	−0.365950	−0.182975	+	0.983118i	$0.558573\pi$
$978$	0	0
$979$	−2285.49	−0.0746115
$980$	0	0
$981$	13755.0	0.447669
$982$	0	0
$983$	9444.83	0.306453	0.153226	−	0.988191i	$-0.451034\pi$
$983$	9444.83	0.306453	0.153226	+	0.988191i	$0.451034\pi$
$984$	0	0
$985$	− 44675.1i	− 1.44514i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	69423.2	2.23208
$990$	0	0
$991$	30310.1i	0.971575i	0.874077	+	0.485788i	$0.161467\pi$
$991$	30310.1i	0.971575i	−0.874077	+	0.485788i	$0.838533\pi$
$992$	0	0
$993$	57698.5i	1.84391i
$994$	0	0
$995$	37271.6i	1.18753i
$996$	0	0
$997$	− 19773.8i	− 0.628126i	−0.949402	−	0.314063i	$-0.898310\pi$
$997$	− 19773.8i	− 0.628126i	0.949402	−	0.314063i	$-0.101690\pi$
$998$	0	0
$999$	−126479.	−4.00561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.i.783.7	yes	8
4.3	odd	2	inner	784.4.f.i.783.1	✓	8
7.6	odd	2	inner	784.4.f.i.783.2	yes	8
28.27	even	2	inner	784.4.f.i.783.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.i.783.1	✓	8	4.3	odd	2	inner
784.4.f.i.783.2	yes	8	7.6	odd	2	inner
784.4.f.i.783.7	yes	8	1.1	even	1	trivial
784.4.f.i.783.8	yes	8	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.98491 q^{3} -9.37011i q^{5} +72.6985 q^{9} +O(q^{10})\)	\(q+9.98491 q^{3} -9.37011i q^{5} +72.6985 q^{9} +44.5415i q^{11} +71.8225i q^{13} -93.5597i q^{15} +52.6114i q^{17} +68.8909 q^{19} +155.783i q^{23} +37.2010 q^{25} +456.295 q^{27} -131.799 q^{29} +2.00707 q^{31} +444.743i q^{33} -277.186 q^{37} +717.141i q^{39} -272.652i q^{41} -445.640i q^{43} -681.193i q^{45} +135.775 q^{47} +525.320i q^{51} +362.583 q^{53} +417.359 q^{55} +687.869 q^{57} +488.257 q^{59} -113.696i q^{61} +672.985 q^{65} -463.322i q^{67} +1555.48i q^{69} -480.779i q^{71} +146.445i q^{73} +371.449 q^{75} -1104.36i q^{79} +2593.21 q^{81} -727.895 q^{83} +492.975 q^{85} -1316.00 q^{87} +51.3114i q^{89} +20.0404 q^{93} -645.515i q^{95} +556.128i q^{97} +3238.10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 344 q^{9}+O(q^{10}) \) 8 * q + 344 * q^9	\( 8 q + 344 q^{9} + 456 q^{25} - 896 q^{29} + 208 q^{53} + 672 q^{57} + 3008 q^{65} + 6728 q^{81} - 16 q^{85} + 6496 q^{93}+O(q^{100}) \) 8 * q + 344 * q^9 + 456 * q^25 - 896 * q^29 + 208 * q^53 + 672 * q^57 + 3008 * q^65 + 6728 * q^81 - 16 * q^85 + 6496 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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