LMFDB - Embedded newform 400.6.c.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+26.0000i q^{3} -22.0000i q^{7} -433.000 q^{9} +O(q^{10})$	$q+26.0000i q^{3} -22.0000i q^{7} -433.000 q^{9} +768.000 q^{11} -46.0000i q^{13} -378.000i q^{17} +1100.00 q^{19} +572.000 q^{21} +1986.00i q^{23} -4940.00i q^{27} +5610.00 q^{29} +3988.00 q^{31} +19968.0i q^{33} +142.000i q^{37} +1196.00 q^{39} +1542.00 q^{41} +5026.00i q^{43} +24738.0i q^{47} +16323.0 q^{49} +9828.00 q^{51} -14166.0i q^{53} +28600.0i q^{57} +28380.0 q^{59} +5522.00 q^{61} +9526.00i q^{63} -24742.0i q^{67} -51636.0 q^{69} -42372.0 q^{71} -52126.0i q^{73} -16896.0i q^{77} -39640.0 q^{79} +23221.0 q^{81} +59826.0i q^{83} +145860. i q^{87} -57690.0 q^{89} -1012.00 q^{91} +103688. i q^{93} +144382. i q^{97} -332544. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 866 q^{9}+O(q^{10}) $ 2 * q - 866 * q^9	$ 2 q - 866 q^{9} + 1536 q^{11} + 2200 q^{19} + 1144 q^{21} + 11220 q^{29} + 7976 q^{31} + 2392 q^{39} + 3084 q^{41} + 32646 q^{49} + 19656 q^{51} + 56760 q^{59} + 11044 q^{61} - 103272 q^{69} - 84744 q^{71} - 79280 q^{79} + 46442 q^{81} - 115380 q^{89} - 2024 q^{91} - 665088 q^{99}+O(q^{100}) $ 2 * q - 866 * q^9 + 1536 * q^11 + 2200 * q^19 + 1144 * q^21 + 11220 * q^29 + 7976 * q^31 + 2392 * q^39 + 3084 * q^41 + 32646 * q^49 + 19656 * q^51 + 56760 * q^59 + 11044 * q^61 - 103272 * q^69 - 84744 * q^71 - 79280 * q^79 + 46442 * q^81 - 115380 * q^89 - 2024 * q^91 - 665088 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	26.0000i	1.66790i	0.551839	+	0.833950i	$0.313926\pi$
$3$	26.0000i	1.66790i	−0.551839	+	0.833950i	$0.686074\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 22.0000i	− 0.169698i	−0.996394	−	0.0848492i	$-0.972959\pi$
$7$	− 22.0000i	− 0.169698i	0.996394	−	0.0848492i	$-0.0270408\pi$
$8$	0	0
$9$	−433.000	−1.78189
$10$	0	0
$11$	768.000	1.91372	0.956862	−	0.290541i	$-0.0938354\pi$
$11$	768.000	1.91372	0.956862	+	0.290541i	$0.0938354\pi$
$12$	0	0
$13$	− 46.0000i	− 0.0754917i	−0.999287	−	0.0377459i	$-0.987982\pi$
$13$	− 46.0000i	− 0.0754917i	0.999287	−	0.0377459i	$-0.0120177\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 378.000i	− 0.317227i	−0.987341	−	0.158613i	$-0.949298\pi$
$17$	− 378.000i	− 0.317227i	0.987341	−	0.158613i	$-0.0507023\pi$
$18$	0	0
$19$	1100.00	0.699051	0.349525	−	0.936927i	$-0.386343\pi$
$19$	1100.00	0.699051	0.349525	+	0.936927i	$0.386343\pi$
$20$	0	0
$21$	572.000	0.283040
$22$	0	0
$23$	1986.00i	0.782816i	0.920217	+	0.391408i	$0.128012\pi$
$23$	1986.00i	0.782816i	−0.920217	+	0.391408i	$0.871988\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4940.00i	− 1.30412i
$28$	0	0
$29$	5610.00	1.23870	0.619352	−	0.785113i	$-0.287395\pi$
$29$	5610.00	1.23870	0.619352	+	0.785113i	$0.287395\pi$
$30$	0	0
$31$	3988.00	0.745334	0.372667	−	0.927965i	$-0.378443\pi$
$31$	3988.00	0.745334	0.372667	+	0.927965i	$0.378443\pi$
$32$	0	0
$33$	19968.0i	3.19190i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	142.000i	0.0170523i	0.999964	+	0.00852617i	$0.00271400\pi$
$37$	142.000i	0.0170523i	−0.999964	+	0.00852617i	$0.997286\pi$
$38$	0	0
$39$	1196.00	0.125913
$40$	0	0
$41$	1542.00	0.143260	0.0716300	−	0.997431i	$-0.477180\pi$
$41$	1542.00	0.143260	0.0716300	+	0.997431i	$0.477180\pi$
$42$	0	0
$43$	5026.00i	0.414526i	0.978285	+	0.207263i	$0.0664555\pi$
$43$	5026.00i	0.414526i	−0.978285	+	0.207263i	$0.933544\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	24738.0i	1.63350i	0.576990	+	0.816752i	$0.304227\pi$
$47$	24738.0i	1.63350i	−0.576990	+	0.816752i	$0.695773\pi$
$48$	0	0
$49$	16323.0	0.971202
$50$	0	0
$51$	9828.00	0.529102
$52$	0	0
$53$	− 14166.0i	− 0.692720i	−0.938102	−	0.346360i	$-0.887418\pi$
$53$	− 14166.0i	− 0.692720i	0.938102	−	0.346360i	$-0.112582\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	28600.0i	1.16595i
$58$	0	0
$59$	28380.0	1.06141	0.530704	−	0.847557i	$-0.321928\pi$
$59$	28380.0	1.06141	0.530704	+	0.847557i	$0.321928\pi$
$60$	0	0
$61$	5522.00	0.190008	0.0950040	−	0.995477i	$-0.469714\pi$
$61$	5522.00	0.190008	0.0950040	+	0.995477i	$0.469714\pi$
$62$	0	0
$63$	9526.00i	0.302384i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 24742.0i	− 0.673361i	−0.941619	−	0.336680i	$-0.890696\pi$
$67$	− 24742.0i	− 0.673361i	0.941619	−	0.336680i	$-0.109304\pi$
$68$	0	0
$69$	−51636.0	−1.30566
$70$	0	0
$71$	−42372.0	−0.997546	−0.498773	−	0.866733i	$-0.666216\pi$
$71$	−42372.0	−0.997546	−0.498773	+	0.866733i	$0.666216\pi$
$72$	0	0
$73$	− 52126.0i	− 1.14485i	−0.819958	−	0.572423i	$-0.806003\pi$
$73$	− 52126.0i	− 1.14485i	0.819958	−	0.572423i	$-0.193997\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 16896.0i	− 0.324756i
$78$	0	0
$79$	−39640.0	−0.714605	−0.357302	−	0.933989i	$-0.616303\pi$
$79$	−39640.0	−0.714605	−0.357302	+	0.933989i	$0.616303\pi$
$80$	0	0
$81$	23221.0	0.393250
$82$	0	0
$83$	59826.0i	0.953223i	0.879114	+	0.476612i	$0.158135\pi$
$83$	59826.0i	0.953223i	−0.879114	+	0.476612i	$0.841865\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	145860.i	2.06604i
$88$	0	0
$89$	−57690.0	−0.772015	−0.386007	−	0.922496i	$-0.626146\pi$
$89$	−57690.0	−0.772015	−0.386007	+	0.922496i	$0.626146\pi$
$90$	0	0
$91$	−1012.00	−0.0128108
$92$	0	0
$93$	103688.i	1.24314i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	144382.i	1.55806i	0.626988	+	0.779029i	$0.284288\pi$
$97$	144382.i	1.55806i	−0.626988	+	0.779029i	$0.715712\pi$
$98$	0	0
$99$	−332544.	−3.41005
$100$	0	0
$101$	−141258.	−1.37787	−0.688937	−	0.724821i	$-0.741922\pi$
$101$	−141258.	−1.37787	−0.688937	+	0.724821i	$0.741922\pi$
$102$	0	0
$103$	− 139814.i	− 1.29855i	−0.760555	−	0.649273i	$-0.775073\pi$
$103$	− 139814.i	− 1.29855i	0.760555	−	0.649273i	$-0.224927\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	86418.0i	0.729701i	0.931066	+	0.364850i	$0.118880\pi$
$107$	86418.0i	0.729701i	−0.931066	+	0.364850i	$0.881120\pi$
$108$	0	0
$109$	−218450.	−1.76111	−0.880554	−	0.473947i	$-0.842829\pi$
$109$	−218450.	−1.76111	−0.880554	+	0.473947i	$0.842829\pi$
$110$	0	0
$111$	−3692.00	−0.0284416
$112$	0	0
$113$	− 28806.0i	− 0.212220i	−0.994354	−	0.106110i	$-0.966160\pi$
$113$	− 28806.0i	− 0.212220i	0.994354	−	0.106110i	$-0.0338396\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	19918.0i	0.134518i
$118$	0	0
$119$	−8316.00	−0.0538328
$120$	0	0
$121$	428773.	2.66234
$122$	0	0
$123$	40092.0i	0.238943i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 216502.i	− 1.19111i	−0.803314	−	0.595556i	$-0.796932\pi$
$127$	− 216502.i	− 1.19111i	0.803314	−	0.595556i	$-0.203068\pi$
$128$	0	0
$129$	−130676.	−0.691388
$130$	0	0
$131$	244608.	1.24535	0.622676	−	0.782479i	$-0.286045\pi$
$131$	244608.	1.24535	0.622676	+	0.782479i	$0.286045\pi$
$132$	0	0
$133$	− 24200.0i	− 0.118628i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	239502.i	1.09020i	0.838370	+	0.545102i	$0.183509\pi$
$137$	239502.i	1.09020i	−0.838370	+	0.545102i	$0.816491\pi$
$138$	0	0
$139$	30860.0	0.135475	0.0677375	−	0.997703i	$-0.478422\pi$
$139$	30860.0	0.135475	0.0677375	+	0.997703i	$0.478422\pi$
$140$	0	0
$141$	−643188.	−2.72452
$142$	0	0
$143$	− 35328.0i	− 0.144470i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	424398.i	1.61987i
$148$	0	0
$149$	100950.	0.372512	0.186256	−	0.982501i	$-0.440365\pi$
$149$	100950.	0.372512	0.186256	+	0.982501i	$0.440365\pi$
$150$	0	0
$151$	−12452.0	−0.0444423	−0.0222212	−	0.999753i	$-0.507074\pi$
$151$	−12452.0	−0.0444423	−0.0222212	+	0.999753i	$0.507074\pi$
$152$	0	0
$153$	163674.i	0.565264i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6022.00i	0.0194981i	0.999952	+	0.00974903i	$0.00310326\pi$
$157$	6022.00i	0.0194981i	−0.999952	+	0.00974903i	$0.996897\pi$
$158$	0	0
$159$	368316.	1.15539
$160$	0	0
$161$	43692.0	0.132843
$162$	0	0
$163$	500866.i	1.47656i	0.674492	+	0.738282i	$0.264363\pi$
$163$	500866.i	1.47656i	−0.674492	+	0.738282i	$0.735637\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	555258.i	1.54065i	0.637652	+	0.770324i	$0.279906\pi$
$167$	555258.i	1.54065i	−0.637652	+	0.770324i	$0.720094\pi$
$168$	0	0
$169$	369177.	0.994301
$170$	0	0
$171$	−476300.	−1.24563
$172$	0	0
$173$	417354.i	1.06020i	0.847934	+	0.530102i	$0.177846\pi$
$173$	417354.i	1.06020i	−0.847934	+	0.530102i	$0.822154\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	737880.i	1.77032i
$178$	0	0
$179$	−52380.0	−0.122189	−0.0610946	−	0.998132i	$-0.519459\pi$
$179$	−52380.0	−0.122189	−0.0610946	+	0.998132i	$0.519459\pi$
$180$	0	0
$181$	546662.	1.24029	0.620144	−	0.784488i	$-0.287074\pi$
$181$	546662.	1.24029	0.620144	+	0.784488i	$0.287074\pi$
$182$	0	0
$183$	143572.i	0.316914i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 290304.i	− 0.607084i
$188$	0	0
$189$	−108680.	−0.221307
$190$	0	0
$191$	452028.	0.896565	0.448283	−	0.893892i	$-0.352036\pi$
$191$	452028.	0.896565	0.448283	+	0.893892i	$0.352036\pi$
$192$	0	0
$193$	485594.i	0.938383i	0.883097	+	0.469191i	$0.155455\pi$
$193$	485594.i	0.938383i	−0.883097	+	0.469191i	$0.844545\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.01018e6i	− 1.85452i	−0.374414	−	0.927262i	$-0.622156\pi$
$197$	− 1.01018e6i	− 1.85452i	0.374414	−	0.927262i	$-0.377844\pi$
$198$	0	0
$199$	−807640.	−1.44572	−0.722862	−	0.690993i	$-0.757174\pi$
$199$	−807640.	−1.44572	−0.722862	+	0.690993i	$0.757174\pi$
$200$	0	0
$201$	643292.	1.12310
$202$	0	0
$203$	− 123420.i	− 0.210206i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 859938.i	− 1.39489i
$208$	0	0
$209$	844800.	1.33779
$210$	0	0
$211$	−149552.	−0.231252	−0.115626	−	0.993293i	$-0.536887\pi$
$211$	−149552.	−0.231252	−0.115626	+	0.993293i	$0.536887\pi$
$212$	0	0
$213$	− 1.10167e6i	− 1.66381i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 87736.0i	− 0.126482i
$218$	0	0
$219$	1.35528e6	1.90949
$220$	0	0
$221$	−17388.0	−0.0239480
$222$	0	0
$223$	443506.i	0.597224i	0.954375	+	0.298612i	$0.0965237\pi$
$223$	443506.i	0.597224i	−0.954375	+	0.298612i	$0.903476\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	420018.i	0.541007i	0.962719	+	0.270504i	$0.0871902\pi$
$227$	420018.i	0.541007i	−0.962719	+	0.270504i	$0.912810\pi$
$228$	0	0
$229$	−1.05875e6	−1.33415	−0.667075	−	0.744990i	$-0.732454\pi$
$229$	−1.05875e6	−1.33415	−0.667075	+	0.744990i	$0.732454\pi$
$230$	0	0
$231$	439296.	0.541661
$232$	0	0
$233$	− 1.27345e6i	− 1.53671i	−0.640026	−	0.768353i	$-0.721077\pi$
$233$	− 1.27345e6i	− 1.53671i	0.640026	−	0.768353i	$-0.278923\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.03064e6i	− 1.19189i
$238$	0	0
$239$	−370680.	−0.419763	−0.209882	−	0.977727i	$-0.567308\pi$
$239$	−370680.	−0.419763	−0.209882	+	0.977727i	$0.567308\pi$
$240$	0	0
$241$	−561298.	−0.622517	−0.311258	−	0.950325i	$-0.600750\pi$
$241$	−561298.	−0.622517	−0.311258	+	0.950325i	$0.600750\pi$
$242$	0	0
$243$	− 596674.i	− 0.648219i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 50600.0i	− 0.0527726i
$248$	0	0
$249$	−1.55548e6	−1.58988
$250$	0	0
$251$	−577152.	−0.578237	−0.289119	−	0.957293i	$-0.593362\pi$
$251$	−577152.	−0.578237	−0.289119	+	0.957293i	$0.593362\pi$
$252$	0	0
$253$	1.52525e6i	1.49809i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	651462.i	0.615257i	0.951507	+	0.307628i	$0.0995353\pi$
$257$	651462.i	0.615257i	−0.951507	+	0.307628i	$0.900465\pi$
$258$	0	0
$259$	3124.00	0.00289375
$260$	0	0
$261$	−2.42913e6	−2.20724
$262$	0	0
$263$	− 917574.i	− 0.817997i	−0.912535	−	0.408999i	$-0.865878\pi$
$263$	− 917574.i	− 0.817997i	0.912535	−	0.408999i	$-0.134122\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.49994e6i	− 1.28764i
$268$	0	0
$269$	735390.	0.619637	0.309818	−	0.950796i	$-0.399732\pi$
$269$	735390.	0.619637	0.309818	+	0.950796i	$0.399732\pi$
$270$	0	0
$271$	1.12131e6	0.927474	0.463737	−	0.885973i	$-0.346508\pi$
$271$	1.12131e6	0.927474	0.463737	+	0.885973i	$0.346508\pi$
$272$	0	0
$273$	− 26312.0i	− 0.0213672i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.66034e6i	1.30016i	0.759864	+	0.650082i	$0.225265\pi$
$277$	1.66034e6i	1.30016i	−0.759864	+	0.650082i	$0.774735\pi$
$278$	0	0
$279$	−1.72680e6	−1.32811
$280$	0	0
$281$	1.45210e6	1.09706	0.548531	−	0.836130i	$-0.315187\pi$
$281$	1.45210e6	1.09706	0.548531	+	0.836130i	$0.315187\pi$
$282$	0	0
$283$	− 309014.i	− 0.229357i	−0.993403	−	0.114679i	$-0.963416\pi$
$283$	− 309014.i	− 0.229357i	0.993403	−	0.114679i	$-0.0365838\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 33924.0i	− 0.0243110i
$288$	0	0
$289$	1.27697e6	0.899367
$290$	0	0
$291$	−3.75393e6	−2.59869
$292$	0	0
$293$	− 1.59301e6i	− 1.08405i	−0.840363	−	0.542024i	$-0.817658\pi$
$293$	− 1.59301e6i	− 1.08405i	0.840363	−	0.542024i	$-0.182342\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.79392e6i	− 2.49573i
$298$	0	0
$299$	91356.0	0.0590961
$300$	0	0
$301$	110572.	0.0703443
$302$	0	0
$303$	− 3.67271e6i	− 2.29816i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.24726e6i	0.755284i	0.925952	+	0.377642i	$0.123265\pi$
$307$	1.24726e6i	0.755284i	−0.925952	+	0.377642i	$0.876735\pi$
$308$	0	0
$309$	3.63516e6	2.16585
$310$	0	0
$311$	665988.	0.390450	0.195225	−	0.980758i	$-0.437456\pi$
$311$	665988.	0.390450	0.195225	+	0.980758i	$0.437456\pi$
$312$	0	0
$313$	− 591286.i	− 0.341143i	−0.985345	−	0.170572i	$-0.945439\pi$
$313$	− 591286.i	− 0.341143i	0.985345	−	0.170572i	$-0.0545614\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	516342.i	0.288595i	0.989534	+	0.144298i	$0.0460923\pi$
$317$	516342.i	0.288595i	−0.989534	+	0.144298i	$0.953908\pi$
$318$	0	0
$319$	4.30848e6	2.37054
$320$	0	0
$321$	−2.24687e6	−1.21707
$322$	0	0
$323$	− 415800.i	− 0.221757i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.67970e6i	− 2.93735i
$328$	0	0
$329$	544236.	0.277203
$330$	0	0
$331$	3.29577e6	1.65343	0.826717	−	0.562619i	$-0.190206\pi$
$331$	3.29577e6	1.65343	0.826717	+	0.562619i	$0.190206\pi$
$332$	0	0
$333$	− 61486.0i	− 0.0303854i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.91098e6i	− 0.916602i	−0.888797	−	0.458301i	$-0.848458\pi$
$337$	− 1.91098e6i	− 0.916602i	0.888797	−	0.458301i	$-0.151542\pi$
$338$	0	0
$339$	748956.	0.353962
$340$	0	0
$341$	3.06278e6	1.42636
$342$	0	0
$343$	− 728860.i	− 0.334510i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.42006e6i	1.07895i	0.842001	+	0.539476i	$0.181378\pi$
$347$	2.42006e6i	1.07895i	−0.842001	+	0.539476i	$0.818622\pi$
$348$	0	0
$349$	−2.50727e6	−1.10189	−0.550944	−	0.834542i	$-0.685732\pi$
$349$	−2.50727e6	−1.10189	−0.550944	+	0.834542i	$0.685732\pi$
$350$	0	0
$351$	−227240.	−0.0984503
$352$	0	0
$353$	− 413166.i	− 0.176477i	−0.996099	−	0.0882384i	$-0.971876\pi$
$353$	− 413166.i	− 0.176477i	0.996099	−	0.0882384i	$-0.0281237\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 216216.i	− 0.0897878i
$358$	0	0
$359$	1.73772e6	0.711613	0.355806	−	0.934560i	$-0.384206\pi$
$359$	1.73772e6	0.711613	0.355806	+	0.934560i	$0.384206\pi$
$360$	0	0
$361$	−1.26610e6	−0.511328
$362$	0	0
$363$	1.11481e7i	4.44052i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.16098e6i	0.449944i	0.974365	+	0.224972i	$0.0722291\pi$
$367$	1.16098e6i	0.449944i	−0.974365	+	0.224972i	$0.927771\pi$
$368$	0	0
$369$	−667686.	−0.255274
$370$	0	0
$371$	−311652.	−0.117553
$372$	0	0
$373$	343754.i	0.127931i	0.997952	+	0.0639655i	$0.0203748\pi$
$373$	343754.i	0.127931i	−0.997952	+	0.0639655i	$0.979625\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 258060.i	− 0.0935120i
$378$	0	0
$379$	573140.	0.204957	0.102478	−	0.994735i	$-0.467323\pi$
$379$	573140.	0.204957	0.102478	+	0.994735i	$0.467323\pi$
$380$	0	0
$381$	5.62905e6	1.98666
$382$	0	0
$383$	2.88055e6i	1.00341i	0.865039	+	0.501704i	$0.167293\pi$
$383$	2.88055e6i	1.00341i	−0.865039	+	0.501704i	$0.832707\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.17626e6i	− 0.738640i
$388$	0	0
$389$	3.08559e6	1.03387	0.516933	−	0.856026i	$-0.327074\pi$
$389$	3.08559e6	1.03387	0.516933	+	0.856026i	$0.327074\pi$
$390$	0	0
$391$	750708.	0.248330
$392$	0	0
$393$	6.35981e6i	2.07712i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 885458.i	− 0.281963i	−0.990012	−	0.140981i	$-0.954974\pi$
$397$	− 885458.i	− 0.281963i	0.990012	−	0.140981i	$-0.0450258\pi$
$398$	0	0
$399$	629200.	0.197859
$400$	0	0
$401$	−3.75344e6	−1.16565	−0.582825	−	0.812598i	$-0.698053\pi$
$401$	−3.75344e6	−1.16565	−0.582825	+	0.812598i	$0.698053\pi$
$402$	0	0
$403$	− 183448.i	− 0.0562666i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	109056.i	0.0326335i
$408$	0	0
$409$	1.94653e6	0.575377	0.287689	−	0.957724i	$-0.407113\pi$
$409$	1.94653e6	0.575377	0.287689	+	0.957724i	$0.407113\pi$
$410$	0	0
$411$	−6.22705e6	−1.81835
$412$	0	0
$413$	− 624360.i	− 0.180119i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	802360.i	0.225959i
$418$	0	0
$419$	−2.99166e6	−0.832486	−0.416243	−	0.909253i	$-0.636654\pi$
$419$	−2.99166e6	−0.832486	−0.416243	+	0.909253i	$0.636654\pi$
$420$	0	0
$421$	3.96660e6	1.09072	0.545360	−	0.838202i	$-0.316393\pi$
$421$	3.96660e6	1.09072	0.545360	+	0.838202i	$0.316393\pi$
$422$	0	0
$423$	− 1.07116e7i	− 2.91073i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 121484.i	− 0.0322440i
$428$	0	0
$429$	918528.	0.240962
$430$	0	0
$431$	5.17115e6	1.34089	0.670446	−	0.741958i	$-0.266103\pi$
$431$	5.17115e6	1.34089	0.670446	+	0.741958i	$0.266103\pi$
$432$	0	0
$433$	− 4.53485e6i	− 1.16237i	−0.813773	−	0.581183i	$-0.802590\pi$
$433$	− 4.53485e6i	− 1.16237i	0.813773	−	0.581183i	$-0.197410\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.18460e6i	0.547228i
$438$	0	0
$439$	−1.08220e6	−0.268007	−0.134004	−	0.990981i	$-0.542783\pi$
$439$	−1.08220e6	−0.268007	−0.134004	+	0.990981i	$0.542783\pi$
$440$	0	0
$441$	−7.06786e6	−1.73058
$442$	0	0
$443$	1.08079e6i	0.261656i	0.991405	+	0.130828i	$0.0417635\pi$
$443$	1.08079e6i	0.261656i	−0.991405	+	0.130828i	$0.958236\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.62470e6i	0.621314i
$448$	0	0
$449$	−2.61783e6	−0.612810	−0.306405	−	0.951901i	$-0.599126\pi$
$449$	−2.61783e6	−0.612810	−0.306405	+	0.951901i	$0.599126\pi$
$450$	0	0
$451$	1.18426e6	0.274160
$452$	0	0
$453$	− 323752.i	− 0.0741254i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.59046e6i	− 0.356231i	−0.984010	−	0.178115i	$-0.943000\pi$
$457$	− 1.59046e6i	− 0.356231i	0.984010	−	0.178115i	$-0.0570001\pi$
$458$	0	0
$459$	−1.86732e6	−0.413701
$460$	0	0
$461$	4.25470e6	0.932431	0.466216	−	0.884671i	$-0.345617\pi$
$461$	4.25470e6	0.932431	0.466216	+	0.884671i	$0.345617\pi$
$462$	0	0
$463$	− 3.26605e6i	− 0.708061i	−0.935234	−	0.354031i	$-0.884811\pi$
$463$	− 3.26605e6i	− 0.708061i	0.935234	−	0.354031i	$-0.115189\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 601542.i	− 0.127636i	−0.997962	−	0.0638181i	$-0.979672\pi$
$467$	− 601542.i	− 0.127636i	0.997962	−	0.0638181i	$-0.0203277\pi$
$468$	0	0
$469$	−544324.	−0.114268
$470$	0	0
$471$	−156572.	−0.0325208
$472$	0	0
$473$	3.85997e6i	0.793288i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.13388e6i	1.23435i
$478$	0	0
$479$	−4.57932e6	−0.911931	−0.455966	−	0.889997i	$-0.650706\pi$
$479$	−4.57932e6	−0.911931	−0.455966	+	0.889997i	$0.650706\pi$
$480$	0	0
$481$	6532.00	0.00128731
$482$	0	0
$483$	1.13599e6i	0.221568i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.05226e6i	1.34743i	0.738992	+	0.673714i	$0.235302\pi$
$487$	7.05226e6i	1.34743i	−0.738992	+	0.673714i	$0.764698\pi$
$488$	0	0
$489$	−1.30225e7	−2.46276
$490$	0	0
$491$	2.62349e6	0.491106	0.245553	−	0.969383i	$-0.421030\pi$
$491$	2.62349e6	0.491106	0.245553	+	0.969383i	$0.421030\pi$
$492$	0	0
$493$	− 2.12058e6i	− 0.392950i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	932184.i	0.169282i
$498$	0	0
$499$	−3.61234e6	−0.649437	−0.324719	−	0.945811i	$-0.605270\pi$
$499$	−3.61234e6	−0.649437	−0.324719	+	0.945811i	$0.605270\pi$
$500$	0	0
$501$	−1.44367e7	−2.56965
$502$	0	0
$503$	− 9.15629e6i	− 1.61361i	−0.590815	−	0.806807i	$-0.701194\pi$
$503$	− 9.15629e6i	− 1.61361i	0.590815	−	0.806807i	$-0.298806\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.59860e6i	1.65840i
$508$	0	0
$509$	−7.26159e6	−1.24233	−0.621165	−	0.783679i	$-0.713340\pi$
$509$	−7.26159e6	−1.24233	−0.621165	+	0.783679i	$0.713340\pi$
$510$	0	0
$511$	−1.14677e6	−0.194279
$512$	0	0
$513$	− 5.43400e6i	− 0.911646i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.89988e7i	3.12608i
$518$	0	0
$519$	−1.08512e7	−1.76831
$520$	0	0
$521$	5.81020e6	0.937771	0.468886	−	0.883259i	$-0.344656\pi$
$521$	5.81020e6	0.937771	0.468886	+	0.883259i	$0.344656\pi$
$522$	0	0
$523$	8.17067e6i	1.30618i	0.757280	+	0.653090i	$0.226528\pi$
$523$	8.17067e6i	1.30618i	−0.757280	+	0.653090i	$0.773472\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.50746e6i	− 0.236440i
$528$	0	0
$529$	2.49215e6	0.387199
$530$	0	0
$531$	−1.22885e7	−1.89132
$532$	0	0
$533$	− 70932.0i	− 0.0108149i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.36188e6i	− 0.203800i
$538$	0	0
$539$	1.25361e7	1.85861
$540$	0	0
$541$	−817378.	−0.120069	−0.0600343	−	0.998196i	$-0.519121\pi$
$541$	−817378.	−0.120069	−0.0600343	+	0.998196i	$0.519121\pi$
$542$	0	0
$543$	1.42132e7i	2.06868i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.50750e6i	− 0.501221i	−0.968088	−	0.250611i	$-0.919369\pi$
$547$	− 3.50750e6i	− 0.501221i	0.968088	−	0.250611i	$-0.0806314\pi$
$548$	0	0
$549$	−2.39103e6	−0.338574
$550$	0	0
$551$	6.17100e6	0.865918
$552$	0	0
$553$	872080.i	0.121267i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.61490e6i	− 1.31313i	−0.754271	−	0.656563i	$-0.772009\pi$
$557$	− 9.61490e6i	− 1.31313i	0.754271	−	0.656563i	$-0.227991\pi$
$558$	0	0
$559$	231196.	0.0312933
$560$	0	0
$561$	7.54790e6	1.01256
$562$	0	0
$563$	− 2.01941e6i	− 0.268506i	−0.990947	−	0.134253i	$-0.957136\pi$
$563$	− 2.01941e6i	− 0.268506i	0.990947	−	0.134253i	$-0.0428635\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 510862.i	− 0.0667338i
$568$	0	0
$569$	−1.37859e6	−0.178507	−0.0892533	−	0.996009i	$-0.528448\pi$
$569$	−1.37859e6	−0.178507	−0.0892533	+	0.996009i	$0.528448\pi$
$570$	0	0
$571$	−8.54295e6	−1.09652	−0.548261	−	0.836307i	$-0.684710\pi$
$571$	−8.54295e6	−1.09652	−0.548261	+	0.836307i	$0.684710\pi$
$572$	0	0
$573$	1.17527e7i	1.49538i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.31458e6i	0.289423i	0.989474	+	0.144711i	$0.0462254\pi$
$577$	2.31458e6i	0.289423i	−0.989474	+	0.144711i	$0.953775\pi$
$578$	0	0
$579$	−1.26254e7	−1.56513
$580$	0	0
$581$	1.31617e6	0.161760
$582$	0	0
$583$	− 1.08795e7i	− 1.32568i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	928338.i	0.111202i	0.998453	+	0.0556008i	$0.0177074\pi$
$587$	928338.i	0.111202i	−0.998453	+	0.0556008i	$0.982293\pi$
$588$	0	0
$589$	4.38680e6	0.521026
$590$	0	0
$591$	2.62646e7	3.09316
$592$	0	0
$593$	− 909486.i	− 0.106209i	−0.998589	−	0.0531043i	$-0.983088\pi$
$593$	− 909486.i	− 0.106209i	0.998589	−	0.0531043i	$-0.0169116\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.09986e7i	− 2.41132i
$598$	0	0
$599$	−8.51136e6	−0.969241	−0.484621	−	0.874724i	$-0.661042\pi$
$599$	−8.51136e6	−0.969241	−0.484621	+	0.874724i	$0.661042\pi$
$600$	0	0
$601$	6.12498e6	0.691701	0.345851	−	0.938290i	$-0.387590\pi$
$601$	6.12498e6	0.691701	0.345851	+	0.938290i	$0.387590\pi$
$602$	0	0
$603$	1.07133e7i	1.19986i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.51646e6i	− 0.497538i	−0.968563	−	0.248769i	$-0.919974\pi$
$607$	− 4.51646e6i	− 0.497538i	0.968563	−	0.248769i	$-0.0800261\pi$
$608$	0	0
$609$	3.20892e6	0.350603
$610$	0	0
$611$	1.13795e6	0.123316
$612$	0	0
$613$	9.63979e6i	1.03614i	0.855340	+	0.518068i	$0.173349\pi$
$613$	9.63979e6i	1.03614i	−0.855340	+	0.518068i	$0.826651\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.92650e6i	1.04974i	0.851181	+	0.524872i	$0.175887\pi$
$617$	9.92650e6i	1.04974i	−0.851181	+	0.524872i	$0.824113\pi$
$618$	0	0
$619$	7.63322e6	0.800721	0.400360	−	0.916358i	$-0.368885\pi$
$619$	7.63322e6	0.800721	0.400360	+	0.916358i	$0.368885\pi$
$620$	0	0
$621$	9.81084e6	1.02089
$622$	0	0
$623$	1.26918e6i	0.131010i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.19648e7i	2.23130i
$628$	0	0
$629$	53676.0	0.00540946
$630$	0	0
$631$	−1.80314e7	−1.80284	−0.901418	−	0.432949i	$-0.857473\pi$
$631$	−1.80314e7	−1.80284	−0.901418	+	0.432949i	$0.857473\pi$
$632$	0	0
$633$	− 3.88835e6i	− 0.385706i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 750858.i	− 0.0733178i
$638$	0	0
$639$	1.83471e7	1.77752
$640$	0	0
$641$	9.30190e6	0.894184	0.447092	−	0.894488i	$-0.352460\pi$
$641$	9.30190e6	0.894184	0.447092	+	0.894488i	$0.352460\pi$
$642$	0	0
$643$	1.38332e7i	1.31946i	0.751503	+	0.659730i	$0.229329\pi$
$643$	1.38332e7i	1.31946i	−0.751503	+	0.659730i	$0.770671\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.48997e7i	− 1.39932i	−0.714478	−	0.699658i	$-0.753336\pi$
$647$	− 1.48997e7i	− 1.39932i	0.714478	−	0.699658i	$-0.246664\pi$
$648$	0	0
$649$	2.17958e7	2.03124
$650$	0	0
$651$	2.28114e6	0.210959
$652$	0	0
$653$	− 1.93306e7i	− 1.77403i	−0.461738	−	0.887016i	$-0.652774\pi$
$653$	− 1.93306e7i	− 1.77403i	0.461738	−	0.887016i	$-0.347226\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.25706e7i	2.03999i
$658$	0	0
$659$	−4.06110e6	−0.364276	−0.182138	−	0.983273i	$-0.558302\pi$
$659$	−4.06110e6	−0.364276	−0.182138	+	0.983273i	$0.558302\pi$
$660$	0	0
$661$	−1.35152e7	−1.20315	−0.601575	−	0.798816i	$-0.705460\pi$
$661$	−1.35152e7	−1.20315	−0.601575	+	0.798816i	$0.705460\pi$
$662$	0	0
$663$	− 452088.i	− 0.0399429i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.11415e7i	0.969678i
$668$	0	0
$669$	−1.15312e7	−0.996111
$670$	0	0
$671$	4.24090e6	0.363623
$672$	0	0
$673$	1.43520e7i	1.22144i	0.791845	+	0.610722i	$0.209121\pi$
$673$	1.43520e7i	1.22144i	−0.791845	+	0.610722i	$0.790879\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.89530e6i	− 0.158930i	−0.996838	−	0.0794650i	$-0.974679\pi$
$677$	− 1.89530e6i	− 0.158930i	0.996838	−	0.0794650i	$-0.0253212\pi$
$678$	0	0
$679$	3.17640e6	0.264400
$680$	0	0
$681$	−1.09205e7	−0.902347
$682$	0	0
$683$	− 2.91641e6i	− 0.239220i	−0.992821	−	0.119610i	$-0.961836\pi$
$683$	− 2.91641e6i	− 0.239220i	0.992821	−	0.119610i	$-0.0381644\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.75275e7i	− 2.22523i
$688$	0	0
$689$	−651636.	−0.0522946
$690$	0	0
$691$	−1.44278e7	−1.14949	−0.574743	−	0.818334i	$-0.694898\pi$
$691$	−1.44278e7	−1.14949	−0.574743	+	0.818334i	$0.694898\pi$
$692$	0	0
$693$	7.31597e6i	0.578680i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 582876.i	− 0.0454458i
$698$	0	0
$699$	3.31096e7	2.56307
$700$	0	0
$701$	−1.58679e7	−1.21962	−0.609811	−	0.792547i	$-0.708754\pi$
$701$	−1.58679e7	−1.21962	−0.609811	+	0.792547i	$0.708754\pi$
$702$	0	0
$703$	156200.i	0.0119205i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.10768e6i	0.233823i
$708$	0	0
$709$	301810.	0.0225485	0.0112743	−	0.999936i	$-0.496411\pi$
$709$	301810.	0.0225485	0.0112743	+	0.999936i	$0.496411\pi$
$710$	0	0
$711$	1.71641e7	1.27335
$712$	0	0
$713$	7.92017e6i	0.583459i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 9.63768e6i	− 0.700123i
$718$	0	0
$719$	2.12677e7	1.53426	0.767130	−	0.641492i	$-0.221684\pi$
$719$	2.12677e7	1.53426	0.767130	+	0.641492i	$0.221684\pi$
$720$	0	0
$721$	−3.07591e6	−0.220361
$722$	0	0
$723$	− 1.45937e7i	− 1.03830i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.55009e7i	1.08773i	0.839174	+	0.543863i	$0.183039\pi$
$727$	1.55009e7i	1.08773i	−0.839174	+	0.543863i	$0.816961\pi$
$728$	0	0
$729$	2.11562e7	1.47441
$730$	0	0
$731$	1.89983e6	0.131499
$732$	0	0
$733$	− 1.21850e7i	− 0.837653i	−0.908066	−	0.418827i	$-0.862441\pi$
$733$	− 1.21850e7i	− 0.837653i	0.908066	−	0.418827i	$-0.137559\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.90019e7i	− 1.28863i
$738$	0	0
$739$	−2.90282e7	−1.95528	−0.977641	−	0.210282i	$-0.932562\pi$
$739$	−2.90282e7	−1.95528	−0.977641	+	0.210282i	$0.932562\pi$
$740$	0	0
$741$	1.31560e6	0.0880194
$742$	0	0
$743$	− 1.61145e7i	− 1.07089i	−0.844570	−	0.535445i	$-0.820144\pi$
$743$	− 1.61145e7i	− 1.07089i	0.844570	−	0.535445i	$-0.179856\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.59047e7i	− 1.69854i
$748$	0	0
$749$	1.90120e6	0.123829
$750$	0	0
$751$	2.92431e6	0.189201	0.0946005	−	0.995515i	$-0.469843\pi$
$751$	2.92431e6	0.189201	0.0946005	+	0.995515i	$0.469843\pi$
$752$	0	0
$753$	− 1.50060e7i	− 0.964442i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.60325e7i	− 1.65111i	−0.564319	−	0.825557i	$-0.690861\pi$
$757$	− 2.60325e7i	− 1.65111i	0.564319	−	0.825557i	$-0.309139\pi$
$758$	0	0
$759$	−3.96564e7	−2.49867
$760$	0	0
$761$	1.63263e7	1.02194	0.510970	−	0.859598i	$-0.329286\pi$
$761$	1.63263e7	1.02194	0.510970	+	0.859598i	$0.329286\pi$
$762$	0	0
$763$	4.80590e6i	0.298857i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.30548e6i	− 0.0801275i
$768$	0	0
$769$	−2.58132e7	−1.57408	−0.787040	−	0.616902i	$-0.788388\pi$
$769$	−2.58132e7	−1.57408	−0.787040	+	0.616902i	$0.788388\pi$
$770$	0	0
$771$	−1.69380e7	−1.02619
$772$	0	0
$773$	− 1.90592e7i	− 1.14725i	−0.819119	−	0.573624i	$-0.805537\pi$
$773$	− 1.90592e7i	− 1.14725i	0.819119	−	0.573624i	$-0.194463\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	81224.0i	0.00482649i
$778$	0	0
$779$	1.69620e6	0.100146
$780$	0	0
$781$	−3.25417e7	−1.90903
$782$	0	0
$783$	− 2.77134e7i	− 1.61542i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.73411e7i	− 0.998021i	−0.866596	−	0.499011i	$-0.833697\pi$
$787$	− 1.73411e7i	− 0.998021i	0.866596	−	0.499011i	$-0.166303\pi$
$788$	0	0
$789$	2.38569e7	1.36434
$790$	0	0
$791$	−633732.	−0.0360134
$792$	0	0
$793$	− 254012.i	− 0.0143440i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.58169e7i	1.43965i	0.694153	+	0.719827i	$0.255779\pi$
$797$	2.58169e7i	1.43965i	−0.694153	+	0.719827i	$0.744221\pi$
$798$	0	0
$799$	9.35096e6	0.518190
$800$	0	0
$801$	2.49798e7	1.37565
$802$	0	0
$803$	− 4.00328e7i	− 2.19092i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.91201e7i	1.03349i
$808$	0	0
$809$	−8.88489e6	−0.477288	−0.238644	−	0.971107i	$-0.576703\pi$
$809$	−8.88489e6	−0.477288	−0.238644	+	0.971107i	$0.576703\pi$
$810$	0	0
$811$	2.46396e7	1.31547	0.657735	−	0.753249i	$-0.271515\pi$
$811$	2.46396e7	1.31547	0.657735	+	0.753249i	$0.271515\pi$
$812$	0	0
$813$	2.91540e7i	1.54693i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.52860e6i	0.289774i
$818$	0	0
$819$	438196.	0.0228275
$820$	0	0
$821$	1.13768e7	0.589062	0.294531	−	0.955642i	$-0.404837\pi$
$821$	1.13768e7	0.589062	0.294531	+	0.955642i	$0.404837\pi$
$822$	0	0
$823$	1.30783e7i	0.673057i	0.941673	+	0.336529i	$0.109253\pi$
$823$	1.30783e7i	0.673057i	−0.941673	+	0.336529i	$0.890747\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.57188e7i	− 1.81607i	−0.418891	−	0.908037i	$-0.637581\pi$
$827$	− 3.57188e7i	− 1.81607i	0.418891	−	0.908037i	$-0.362419\pi$
$828$	0	0
$829$	−1.61880e7	−0.818103	−0.409052	−	0.912511i	$-0.634140\pi$
$829$	−1.61880e7	−0.818103	−0.409052	+	0.912511i	$0.634140\pi$
$830$	0	0
$831$	−4.31689e7	−2.16854
$832$	0	0
$833$	− 6.17009e6i	− 0.308091i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.97007e7i	− 0.972005i
$838$	0	0
$839$	−2.55497e7	−1.25309	−0.626543	−	0.779387i	$-0.715531\pi$
$839$	−2.55497e7	−1.25309	−0.626543	+	0.779387i	$0.715531\pi$
$840$	0	0
$841$	1.09610e7	0.534390
$842$	0	0
$843$	3.77547e7i	1.82979i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 9.43301e6i	− 0.451795i
$848$	0	0
$849$	8.03436e6	0.382545
$850$	0	0
$851$	−282012.	−0.0133488
$852$	0	0
$853$	− 2.22953e7i	− 1.04916i	−0.851362	−	0.524579i	$-0.824223\pi$
$853$	− 2.22953e7i	− 1.04916i	0.851362	−	0.524579i	$-0.175777\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.96872e7i	− 0.915656i	−0.889041	−	0.457828i	$-0.848628\pi$
$857$	− 1.96872e7i	− 0.915656i	0.889041	−	0.457828i	$-0.151372\pi$
$858$	0	0
$859$	6.77582e6	0.313313	0.156657	−	0.987653i	$-0.449928\pi$
$859$	6.77582e6	0.313313	0.156657	+	0.987653i	$0.449928\pi$
$860$	0	0
$861$	882024.	0.0405483
$862$	0	0
$863$	2.63804e7i	1.20574i	0.797839	+	0.602871i	$0.205977\pi$
$863$	2.63804e7i	1.20574i	−0.797839	+	0.602871i	$0.794023\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.32013e7i	1.50006i
$868$	0	0
$869$	−3.04435e7	−1.36756
$870$	0	0
$871$	−1.13813e6	−0.0508332
$872$	0	0
$873$	− 6.25174e7i	− 2.77629i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.95161e7i	− 1.29587i	−0.761697	−	0.647934i	$-0.775633\pi$
$877$	− 2.95161e7i	− 1.29587i	0.761697	−	0.647934i	$-0.224367\pi$
$878$	0	0
$879$	4.14182e7	1.80808
$880$	0	0
$881$	−1.48565e7	−0.644877	−0.322438	−	0.946590i	$-0.604502\pi$
$881$	−1.48565e7	−0.644877	−0.322438	+	0.946590i	$0.604502\pi$
$882$	0	0
$883$	1.45340e7i	0.627313i	0.949537	+	0.313656i	$0.101554\pi$
$883$	1.45340e7i	0.627313i	−0.949537	+	0.313656i	$0.898446\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.72028e7i	− 0.734160i	−0.930189	−	0.367080i	$-0.880358\pi$
$887$	− 1.72028e7i	− 0.734160i	0.930189	−	0.367080i	$-0.119642\pi$
$888$	0	0
$889$	−4.76304e6	−0.202130
$890$	0	0
$891$	1.78337e7	0.752572
$892$	0	0
$893$	2.72118e7i	1.14190i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.37526e6i	0.0985665i
$898$	0	0
$899$	2.23727e7	0.923249
$900$	0	0
$901$	−5.35475e6	−0.219749
$902$	0	0
$903$	2.87487e6i	0.117327i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.44434e7i	− 1.39023i	−0.718897	−	0.695116i	$-0.755353\pi$
$907$	− 3.44434e7i	− 1.39023i	0.718897	−	0.695116i	$-0.244647\pi$
$908$	0	0
$909$	6.11647e7	2.45522
$910$	0	0
$911$	983748.	0.0392724	0.0196362	−	0.999807i	$-0.493749\pi$
$911$	983748.	0.0392724	0.0196362	+	0.999807i	$0.493749\pi$
$912$	0	0
$913$	4.59464e7i	1.82421i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.38138e6i	− 0.211334i
$918$	0	0
$919$	3.08857e7	1.20634	0.603168	−	0.797614i	$-0.293905\pi$
$919$	3.08857e7	1.20634	0.603168	+	0.797614i	$0.293905\pi$
$920$	0	0
$921$	−3.24287e7	−1.25974
$922$	0	0
$923$	1.94911e6i	0.0753065i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.05395e7i	2.31387i
$928$	0	0
$929$	3.20874e7	1.21982	0.609909	−	0.792472i	$-0.291206\pi$
$929$	3.20874e7	1.21982	0.609909	+	0.792472i	$0.291206\pi$
$930$	0	0
$931$	1.79553e7	0.678920
$932$	0	0
$933$	1.73157e7i	0.651232i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.52520e7i	− 0.567515i	−0.958896	−	0.283757i	$-0.908419\pi$
$937$	− 1.52520e7i	− 0.567515i	0.958896	−	0.283757i	$-0.0915810\pi$
$938$	0	0
$939$	1.53734e7	0.568993
$940$	0	0
$941$	3.48166e6	0.128178	0.0640889	−	0.997944i	$-0.479586\pi$
$941$	3.48166e6	0.128178	0.0640889	+	0.997944i	$0.479586\pi$
$942$	0	0
$943$	3.06241e6i	0.112146i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.54010e7i	− 0.920398i	−0.887816	−	0.460199i	$-0.847778\pi$
$947$	− 2.54010e7i	− 0.920398i	0.887816	−	0.460199i	$-0.152222\pi$
$948$	0	0
$949$	−2.39780e6	−0.0864265
$950$	0	0
$951$	−1.34249e7	−0.481348
$952$	0	0
$953$	− 4.97352e7i	− 1.77391i	−0.461856	−	0.886955i	$-0.652816\pi$
$953$	− 4.97352e7i	− 1.77391i	0.461856	−	0.886955i	$-0.347184\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.12020e8i	3.95383i
$958$	0	0
$959$	5.26904e6	0.185006
$960$	0	0
$961$	−1.27250e7	−0.444477
$962$	0	0
$963$	− 3.74190e7i	− 1.30025i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.05173e7i	1.04949i	0.851258	+	0.524747i	$0.175840\pi$
$967$	3.05173e7i	1.04949i	−0.851258	+	0.524747i	$0.824160\pi$
$968$	0	0
$969$	1.08108e7	0.369869
$970$	0	0
$971$	−3.19854e7	−1.08869	−0.544344	−	0.838862i	$-0.683221\pi$
$971$	−3.19854e7	−1.08869	−0.544344	+	0.838862i	$0.683221\pi$
$972$	0	0
$973$	− 678920.i	− 0.0229899i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.90786e6i	− 0.0974623i	−0.998812	−	0.0487312i	$-0.984482\pi$
$977$	− 2.90786e6i	− 0.0974623i	0.998812	−	0.0487312i	$-0.0155178\pi$
$978$	0	0
$979$	−4.43059e7	−1.47742
$980$	0	0
$981$	9.45888e7	3.13810
$982$	0	0
$983$	− 3.49621e7i	− 1.15402i	−0.816737	−	0.577010i	$-0.804219\pi$
$983$	− 3.49621e7i	− 1.15402i	0.816737	−	0.577010i	$-0.195781\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.41501e7i	0.462347i
$988$	0	0
$989$	−9.98164e6	−0.324497
$990$	0	0
$991$	−3.00465e6	−0.0971874	−0.0485937	−	0.998819i	$-0.515474\pi$
$991$	−3.00465e6	−0.0971874	−0.0485937	+	0.998819i	$0.515474\pi$
$992$	0	0
$993$	8.56900e7i	2.75776i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 3.20789e7i	− 1.02207i	−0.859560	−	0.511035i	$-0.829262\pi$
$997$	− 3.20789e7i	− 1.02207i	0.859560	−	0.511035i	$-0.170738\pi$
$998$	0	0
$999$	701480.	0.0222383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.a.49.2		2
4.3	odd	2		50.6.b.d.49.2		2
5.2	odd	4		80.6.a.h.1.1		1
5.3	odd	4		400.6.a.a.1.1		1
5.4	even	2	inner	400.6.c.a.49.1		2
12.11	even	2		450.6.c.o.199.1		2
15.2	even	4		720.6.a.r.1.1		1
20.3	even	4		50.6.a.g.1.1		1
20.7	even	4		10.6.a.a.1.1	✓	1
20.19	odd	2		50.6.b.d.49.1		2
40.27	even	4		320.6.a.p.1.1		1
40.37	odd	4		320.6.a.a.1.1		1
60.23	odd	4		450.6.a.h.1.1		1
60.47	odd	4		90.6.a.f.1.1		1
60.59	even	2		450.6.c.o.199.2		2
140.27	odd	4		490.6.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.a.1.1	✓	1	20.7	even	4
50.6.a.g.1.1		1	20.3	even	4
50.6.b.d.49.1		2	20.19	odd	2
50.6.b.d.49.2		2	4.3	odd	2
80.6.a.h.1.1		1	5.2	odd	4
90.6.a.f.1.1		1	60.47	odd	4
320.6.a.a.1.1		1	40.37	odd	4
320.6.a.p.1.1		1	40.27	even	4
400.6.a.a.1.1		1	5.3	odd	4
400.6.c.a.49.1		2	5.4	even	2	inner
400.6.c.a.49.2		2	1.1	even	1	trivial
450.6.a.h.1.1		1	60.23	odd	4
450.6.c.o.199.1		2	12.11	even	2
450.6.c.o.199.2		2	60.59	even	2
490.6.a.j.1.1		1	140.27	odd	4
720.6.a.r.1.1		1	15.2	even	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+26.0000i q^{3} -22.0000i q^{7} -433.000 q^{9} +O(q^{10})\)	\(q+26.0000i q^{3} -22.0000i q^{7} -433.000 q^{9} +768.000 q^{11} -46.0000i q^{13} -378.000i q^{17} +1100.00 q^{19} +572.000 q^{21} +1986.00i q^{23} -4940.00i q^{27} +5610.00 q^{29} +3988.00 q^{31} +19968.0i q^{33} +142.000i q^{37} +1196.00 q^{39} +1542.00 q^{41} +5026.00i q^{43} +24738.0i q^{47} +16323.0 q^{49} +9828.00 q^{51} -14166.0i q^{53} +28600.0i q^{57} +28380.0 q^{59} +5522.00 q^{61} +9526.00i q^{63} -24742.0i q^{67} -51636.0 q^{69} -42372.0 q^{71} -52126.0i q^{73} -16896.0i q^{77} -39640.0 q^{79} +23221.0 q^{81} +59826.0i q^{83} +145860. i q^{87} -57690.0 q^{89} -1012.00 q^{91} +103688. i q^{93} +144382. i q^{97} -332544. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 866 q^{9}+O(q^{10}) \) 2 * q - 866 * q^9	\( 2 q - 866 q^{9} + 1536 q^{11} + 2200 q^{19} + 1144 q^{21} + 11220 q^{29} + 7976 q^{31} + 2392 q^{39} + 3084 q^{41} + 32646 q^{49} + 19656 q^{51} + 56760 q^{59} + 11044 q^{61} - 103272 q^{69} - 84744 q^{71} - 79280 q^{79} + 46442 q^{81} - 115380 q^{89} - 2024 q^{91} - 665088 q^{99}+O(q^{100}) \) 2 * q - 866 * q^9 + 1536 * q^11 + 2200 * q^19 + 1144 * q^21 + 11220 * q^29 + 7976 * q^31 + 2392 * q^39 + 3084 * q^41 + 32646 * q^49 + 19656 * q^51 + 56760 * q^59 + 11044 * q^61 - 103272 * q^69 - 84744 * q^71 - 79280 * q^79 + 46442 * q^81 - 115380 * q^89 - 2024 * q^91 - 665088 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more