LMFDB - Embedded newform 400.6.c.d.49.1

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.0000i q^{3} -24.0000i q^{7} -157.000 q^{9} +O(q^{10})$	$q-20.0000i q^{3} -24.0000i q^{7} -157.000 q^{9} -124.000 q^{11} +478.000i q^{13} +1198.00i q^{17} +3044.00 q^{19} -480.000 q^{21} -184.000i q^{23} -1720.00i q^{27} +3282.00 q^{29} +5728.00 q^{31} +2480.00i q^{33} -10326.0i q^{37} +9560.00 q^{39} -8886.00 q^{41} +9188.00i q^{43} +23664.0i q^{47} +16231.0 q^{49} +23960.0 q^{51} +11686.0i q^{53} -60880.0i q^{57} +16876.0 q^{59} -18482.0 q^{61} +3768.00i q^{63} -15532.0i q^{67} -3680.00 q^{69} +31960.0 q^{71} -4886.00i q^{73} +2976.00i q^{77} +44560.0 q^{79} -72551.0 q^{81} -67364.0i q^{83} -65640.0i q^{87} -71994.0 q^{89} +11472.0 q^{91} -114560. i q^{93} -48866.0i q^{97} +19468.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 314 q^{9}+O(q^{10}) $ 2 * q - 314 * q^9	$ 2 q - 314 q^{9} - 248 q^{11} + 6088 q^{19} - 960 q^{21} + 6564 q^{29} + 11456 q^{31} + 19120 q^{39} - 17772 q^{41} + 32462 q^{49} + 47920 q^{51} + 33752 q^{59} - 36964 q^{61} - 7360 q^{69} + 63920 q^{71} + 89120 q^{79} - 145102 q^{81} - 143988 q^{89} + 22944 q^{91} + 38936 q^{99}+O(q^{100}) $ 2 * q - 314 * q^9 - 248 * q^11 + 6088 * q^19 - 960 * q^21 + 6564 * q^29 + 11456 * q^31 + 19120 * q^39 - 17772 * q^41 + 32462 * q^49 + 47920 * q^51 + 33752 * q^59 - 36964 * q^61 - 7360 * q^69 + 63920 * q^71 + 89120 * q^79 - 145102 * q^81 - 143988 * q^89 + 22944 * q^91 + 38936 * 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
$3$	− 20.0000i	− 1.28300i	−0.767123	−	0.641500i	$-0.778312\pi$
$3$	− 20.0000i	− 1.28300i	0.767123	−	0.641500i	$-0.221688\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 24.0000i	− 0.185125i	−0.995707	−	0.0925627i	$-0.970494\pi$
$7$	− 24.0000i	− 0.185125i	0.995707	−	0.0925627i	$-0.0295059\pi$
$8$	0	0
$9$	−157.000	−0.646091
$10$	0	0
$11$	−124.000	−0.308987	−0.154493	−	0.987994i	$-0.549375\pi$
$11$	−124.000	−0.308987	−0.154493	+	0.987994i	$0.549375\pi$
$12$	0	0
$13$	478.000i	0.784458i	0.919868	+	0.392229i	$0.128296\pi$
$13$	478.000i	0.784458i	−0.919868	+	0.392229i	$0.871704\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1198.00i	1.00539i	0.864464	+	0.502695i	$0.167658\pi$
$17$	1198.00i	1.00539i	−0.864464	+	0.502695i	$0.832342\pi$
$18$	0	0
$19$	3044.00	1.93446	0.967232	−	0.253894i	$-0.0817115\pi$
$19$	3044.00	1.93446	0.967232	+	0.253894i	$0.0817115\pi$
$20$	0	0
$21$	−480.000	−0.237516
$22$	0	0
$23$	− 184.000i	− 0.0725268i	−0.999342	−	0.0362634i	$-0.988454\pi$
$23$	− 184.000i	− 0.0725268i	0.999342	−	0.0362634i	$-0.0115455\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1720.00i	− 0.454066i
$28$	0	0
$29$	3282.00	0.724676	0.362338	−	0.932047i	$-0.381979\pi$
$29$	3282.00	0.724676	0.362338	+	0.932047i	$0.381979\pi$
$30$	0	0
$31$	5728.00	1.07053	0.535265	−	0.844684i	$-0.320212\pi$
$31$	5728.00	1.07053	0.535265	+	0.844684i	$0.320212\pi$
$32$	0	0
$33$	2480.00i	0.396430i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10326.0i	− 1.24002i	−0.784595	−	0.620009i	$-0.787129\pi$
$37$	− 10326.0i	− 1.24002i	0.784595	−	0.620009i	$-0.212871\pi$
$38$	0	0
$39$	9560.00	1.00646
$40$	0	0
$41$	−8886.00	−0.825556	−0.412778	−	0.910832i	$-0.635442\pi$
$41$	−8886.00	−0.825556	−0.412778	+	0.910832i	$0.635442\pi$
$42$	0	0
$43$	9188.00i	0.757792i	0.925439	+	0.378896i	$0.123696\pi$
$43$	9188.00i	0.757792i	−0.925439	+	0.378896i	$0.876304\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	23664.0i	1.56258i	0.624165	+	0.781292i	$0.285439\pi$
$47$	23664.0i	1.56258i	−0.624165	+	0.781292i	$0.714561\pi$
$48$	0	0
$49$	16231.0	0.965729
$50$	0	0
$51$	23960.0	1.28992
$52$	0	0
$53$	11686.0i	0.571447i	0.958312	+	0.285724i	$0.0922339\pi$
$53$	11686.0i	0.571447i	−0.958312	+	0.285724i	$0.907766\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 60880.0i	− 2.48192i
$58$	0	0
$59$	16876.0	0.631160	0.315580	−	0.948899i	$-0.397801\pi$
$59$	16876.0	0.631160	0.315580	+	0.948899i	$0.397801\pi$
$60$	0	0
$61$	−18482.0	−0.635952	−0.317976	−	0.948099i	$-0.603003\pi$
$61$	−18482.0	−0.635952	−0.317976	+	0.948099i	$0.603003\pi$
$62$	0	0
$63$	3768.00i	0.119608i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 15532.0i	− 0.422708i	−0.977410	−	0.211354i	$-0.932213\pi$
$67$	− 15532.0i	− 0.422708i	0.977410	−	0.211354i	$-0.0677873\pi$
$68$	0	0
$69$	−3680.00	−0.0930519
$70$	0	0
$71$	31960.0	0.752421	0.376210	−	0.926534i	$-0.377227\pi$
$71$	31960.0	0.752421	0.376210	+	0.926534i	$0.377227\pi$
$72$	0	0
$73$	− 4886.00i	− 0.107312i	−0.998559	−	0.0536558i	$-0.982913\pi$
$73$	− 4886.00i	− 0.107312i	0.998559	−	0.0536558i	$-0.0170874\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2976.00i	0.0572013i
$78$	0	0
$79$	44560.0	0.803299	0.401650	−	0.915793i	$-0.368437\pi$
$79$	44560.0	0.803299	0.401650	+	0.915793i	$0.368437\pi$
$80$	0	0
$81$	−72551.0	−1.22866
$82$	0	0
$83$	− 67364.0i	− 1.07333i	−0.843796	−	0.536664i	$-0.819684\pi$
$83$	− 67364.0i	− 1.07333i	0.843796	−	0.536664i	$-0.180316\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 65640.0i	− 0.929759i
$88$	0	0
$89$	−71994.0	−0.963432	−0.481716	−	0.876327i	$-0.659986\pi$
$89$	−71994.0	−0.963432	−0.481716	+	0.876327i	$0.659986\pi$
$90$	0	0
$91$	11472.0	0.145223
$92$	0	0
$93$	− 114560.i	− 1.37349i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 48866.0i	− 0.527324i	−0.964615	−	0.263662i	$-0.915070\pi$
$97$	− 48866.0i	− 0.527324i	0.964615	−	0.263662i	$-0.0849303\pi$
$98$	0	0
$99$	19468.0	0.199633
$100$	0	0
$101$	51606.0	0.503381	0.251690	−	0.967808i	$-0.419014\pi$
$101$	51606.0	0.503381	0.251690	+	0.967808i	$0.419014\pi$
$102$	0	0
$103$	− 180424.i	− 1.67572i	−0.545886	−	0.837860i	$-0.683807\pi$
$103$	− 180424.i	− 1.67572i	0.545886	−	0.837860i	$-0.316193\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 65700.0i	− 0.554761i	−0.960760	−	0.277381i	$-0.910534\pi$
$107$	− 65700.0i	− 0.554761i	0.960760	−	0.277381i	$-0.0894663\pi$
$108$	0	0
$109$	112706.	0.908617	0.454308	−	0.890844i	$-0.349886\pi$
$109$	112706.	0.908617	0.454308	+	0.890844i	$0.349886\pi$
$110$	0	0
$111$	−206520.	−1.59094
$112$	0	0
$113$	− 23502.0i	− 0.173145i	−0.996246	−	0.0865723i	$-0.972409\pi$
$113$	− 23502.0i	− 0.173145i	0.996246	−	0.0865723i	$-0.0275913\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 75046.0i	− 0.506831i
$118$	0	0
$119$	28752.0	0.186123
$120$	0	0
$121$	−145675.	−0.904527
$122$	0	0
$123$	177720.i	1.05919i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 94592.0i	− 0.520409i	−0.965553	−	0.260205i	$-0.916210\pi$
$127$	− 94592.0i	− 0.520409i	0.965553	−	0.260205i	$-0.0837900\pi$
$128$	0	0
$129$	183760.	0.972247
$130$	0	0
$131$	−70292.0	−0.357872	−0.178936	−	0.983861i	$-0.557265\pi$
$131$	−70292.0	−0.357872	−0.178936	+	0.983861i	$0.557265\pi$
$132$	0	0
$133$	− 73056.0i	− 0.358119i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 277290.i	− 1.26221i	−0.775696	−	0.631107i	$-0.782601\pi$
$137$	− 277290.i	− 1.26221i	0.775696	−	0.631107i	$-0.217399\pi$
$138$	0	0
$139$	−130308.	−0.572050	−0.286025	−	0.958222i	$-0.592334\pi$
$139$	−130308.	−0.572050	−0.286025	+	0.958222i	$0.592334\pi$
$140$	0	0
$141$	473280.	2.00480
$142$	0	0
$143$	− 59272.0i	− 0.242387i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 324620.i	− 1.23903i
$148$	0	0
$149$	401530.	1.48167	0.740836	−	0.671685i	$-0.234429\pi$
$149$	401530.	1.48167	0.740836	+	0.671685i	$0.234429\pi$
$150$	0	0
$151$	75976.0	0.271165	0.135583	−	0.990766i	$-0.456709\pi$
$151$	75976.0	0.271165	0.135583	+	0.990766i	$0.456709\pi$
$152$	0	0
$153$	− 188086.i	− 0.649573i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	394322.i	1.27674i	0.769730	+	0.638369i	$0.220391\pi$
$157$	394322.i	1.27674i	−0.769730	+	0.638369i	$0.779609\pi$
$158$	0	0
$159$	233720.	0.733167
$160$	0	0
$161$	−4416.00	−0.0134265
$162$	0	0
$163$	11724.0i	0.0345626i	0.999851	+	0.0172813i	$0.00550109\pi$
$163$	11724.0i	0.0345626i	−0.999851	+	0.0172813i	$0.994499\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 551928.i	− 1.53141i	−0.643192	−	0.765705i	$-0.722390\pi$
$167$	− 551928.i	− 1.53141i	0.643192	−	0.765705i	$-0.277610\pi$
$168$	0	0
$169$	142809.	0.384626
$170$	0	0
$171$	−477908.	−1.24984
$172$	0	0
$173$	432894.i	1.09968i	0.835270	+	0.549840i	$0.185311\pi$
$173$	432894.i	1.09968i	−0.835270	+	0.549840i	$0.814689\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 337520.i	− 0.809779i
$178$	0	0
$179$	559620.	1.30545	0.652726	−	0.757594i	$-0.273625\pi$
$179$	559620.	1.30545	0.652726	+	0.757594i	$0.273625\pi$
$180$	0	0
$181$	604710.	1.37199	0.685995	−	0.727607i	$-0.259367\pi$
$181$	604710.	1.37199	0.685995	+	0.727607i	$0.259367\pi$
$182$	0	0
$183$	369640.i	0.815927i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 148552.i	− 0.310652i
$188$	0	0
$189$	−41280.0	−0.0840592
$190$	0	0
$191$	409152.	0.811524	0.405762	−	0.913979i	$-0.367006\pi$
$191$	409152.	0.811524	0.405762	+	0.913979i	$0.367006\pi$
$192$	0	0
$193$	540866.i	1.04519i	0.852580	+	0.522596i	$0.175037\pi$
$193$	540866.i	1.04519i	−0.852580	+	0.522596i	$0.824963\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	629898.i	1.15639i	0.815898	+	0.578195i	$0.196243\pi$
$197$	629898.i	1.15639i	−0.815898	+	0.578195i	$0.803757\pi$
$198$	0	0
$199$	283048.	0.506673	0.253336	−	0.967378i	$-0.418472\pi$
$199$	283048.	0.506673	0.253336	+	0.967378i	$0.418472\pi$
$200$	0	0
$201$	−310640.	−0.542335
$202$	0	0
$203$	− 78768.0i	− 0.134156i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	28888.0i	0.0468588i
$208$	0	0
$209$	−377456.	−0.597724
$210$	0	0
$211$	−142756.	−0.220744	−0.110372	−	0.993890i	$-0.535204\pi$
$211$	−142756.	−0.220744	−0.110372	+	0.993890i	$0.535204\pi$
$212$	0	0
$213$	− 639200.i	− 0.965357i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 137472.i	− 0.198182i
$218$	0	0
$219$	−97720.0	−0.137681
$220$	0	0
$221$	−572644.	−0.788686
$222$	0	0
$223$	− 889696.i	− 1.19806i	−0.800726	−	0.599031i	$-0.795553\pi$
$223$	− 889696.i	− 1.19806i	0.800726	−	0.599031i	$-0.204447\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.14316e6i	1.47245i	0.676736	+	0.736226i	$0.263394\pi$
$227$	1.14316e6i	1.47245i	−0.676736	+	0.736226i	$0.736606\pi$
$228$	0	0
$229$	695786.	0.876773	0.438386	−	0.898787i	$-0.355550\pi$
$229$	695786.	0.876773	0.438386	+	0.898787i	$0.355550\pi$
$230$	0	0
$231$	59520.0	0.0733893
$232$	0	0
$233$	− 347126.i	− 0.418887i	−0.977821	−	0.209444i	$-0.932835\pi$
$233$	− 347126.i	− 0.418887i	0.977821	−	0.209444i	$-0.0671653\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 891200.i	− 1.03063i
$238$	0	0
$239$	−1.64296e6	−1.86051	−0.930255	−	0.366912i	$-0.880415\pi$
$239$	−1.64296e6	−1.86051	−0.930255	+	0.366912i	$0.880415\pi$
$240$	0	0
$241$	−1.16744e6	−1.29477	−0.647383	−	0.762165i	$-0.724137\pi$
$241$	−1.16744e6	−1.29477	−0.647383	+	0.762165i	$0.724137\pi$
$242$	0	0
$243$	1.03306e6i	1.12230i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.45503e6i	1.51751i
$248$	0	0
$249$	−1.34728e6	−1.37708
$250$	0	0
$251$	790612.	0.792098	0.396049	−	0.918229i	$-0.370381\pi$
$251$	790612.	0.792098	0.396049	+	0.918229i	$0.370381\pi$
$252$	0	0
$253$	22816.0i	0.0224098i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	129790.i	0.122577i	0.998120	+	0.0612884i	$0.0195209\pi$
$257$	129790.i	0.122577i	−0.998120	+	0.0612884i	$0.980479\pi$
$258$	0	0
$259$	−247824.	−0.229559
$260$	0	0
$261$	−515274.	−0.468206
$262$	0	0
$263$	− 70888.0i	− 0.0631951i	−0.999501	−	0.0315975i	$-0.989941\pi$
$263$	− 70888.0i	− 0.0631951i	0.999501	−	0.0315975i	$-0.0100595\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.43988e6i	1.23608i
$268$	0	0
$269$	−1.79017e6	−1.50839	−0.754197	−	0.656649i	$-0.771973\pi$
$269$	−1.79017e6	−1.50839	−0.754197	+	0.656649i	$0.771973\pi$
$270$	0	0
$271$	1.77362e6	1.46702	0.733511	−	0.679678i	$-0.237880\pi$
$271$	1.77362e6	1.46702	0.733511	+	0.679678i	$0.237880\pi$
$272$	0	0
$273$	− 229440.i	− 0.186321i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	275450.i	0.215697i	0.994167	+	0.107848i	$0.0343961\pi$
$277$	275450.i	0.215697i	−0.994167	+	0.107848i	$0.965604\pi$
$278$	0	0
$279$	−899296.	−0.691659
$280$	0	0
$281$	594170.	0.448895	0.224448	−	0.974486i	$-0.427942\pi$
$281$	594170.	0.448895	0.224448	+	0.974486i	$0.427942\pi$
$282$	0	0
$283$	− 1.09243e6i	− 0.810824i	−0.914134	−	0.405412i	$-0.867128\pi$
$283$	− 1.09243e6i	− 0.810824i	0.914134	−	0.405412i	$-0.132872\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	213264.i	0.152831i
$288$	0	0
$289$	−15347.0	−0.0108088
$290$	0	0
$291$	−977320.	−0.676557
$292$	0	0
$293$	333654.i	0.227053i	0.993535	+	0.113527i	$0.0362147\pi$
$293$	333654.i	0.227053i	−0.993535	+	0.113527i	$0.963785\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	213280.i	0.140300i
$298$	0	0
$299$	87952.0	0.0568942
$300$	0	0
$301$	220512.	0.140287
$302$	0	0
$303$	− 1.03212e6i	− 0.645838i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.05997e6i	0.641872i	0.947101	+	0.320936i	$0.103997\pi$
$307$	1.05997e6i	0.641872i	−0.947101	+	0.320936i	$0.896003\pi$
$308$	0	0
$309$	−3.60848e6	−2.14995
$310$	0	0
$311$	1.33649e6	0.783545	0.391773	−	0.920062i	$-0.371862\pi$
$311$	1.33649e6	0.783545	0.391773	+	0.920062i	$0.371862\pi$
$312$	0	0
$313$	1.64419e6i	0.948615i	0.880359	+	0.474308i	$0.157302\pi$
$313$	1.64419e6i	0.948615i	−0.880359	+	0.474308i	$0.842698\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.72370e6i	0.963414i	0.876332	+	0.481707i	$0.159983\pi$
$317$	1.72370e6i	0.963414i	−0.876332	+	0.481707i	$0.840017\pi$
$318$	0	0
$319$	−406968.	−0.223915
$320$	0	0
$321$	−1.31400e6	−0.711759
$322$	0	0
$323$	3.64671e6i	1.94489i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.25412e6i	− 1.16576i
$328$	0	0
$329$	567936.	0.289274
$330$	0	0
$331$	−2.74963e6	−1.37944	−0.689722	−	0.724074i	$-0.742267\pi$
$331$	−2.74963e6	−1.37944	−0.689722	+	0.724074i	$0.742267\pi$
$332$	0	0
$333$	1.62118e6i	0.801164i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.41489e6i	1.63796i	0.573824	+	0.818978i	$0.305459\pi$
$337$	3.41489e6i	1.63796i	−0.573824	+	0.818978i	$0.694541\pi$
$338$	0	0
$339$	−470040.	−0.222145
$340$	0	0
$341$	−710272.	−0.330780
$342$	0	0
$343$	− 792912.i	− 0.363906i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	730764.i	0.325802i	0.986642	+	0.162901i	$0.0520851\pi$
$347$	730764.i	0.325802i	−0.986642	+	0.162901i	$0.947915\pi$
$348$	0	0
$349$	2.29749e6	1.00969	0.504847	−	0.863209i	$-0.331549\pi$
$349$	2.29749e6	1.00969	0.504847	+	0.863209i	$0.331549\pi$
$350$	0	0
$351$	822160.	0.356196
$352$	0	0
$353$	− 1.17072e6i	− 0.500052i	−0.968239	−	0.250026i	$-0.919561\pi$
$353$	− 1.17072e6i	− 0.500052i	0.968239	−	0.250026i	$-0.0804392\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 575040.i	− 0.238796i
$358$	0	0
$359$	3.88654e6	1.59157	0.795787	−	0.605577i	$-0.207058\pi$
$359$	3.88654e6	1.59157	0.795787	+	0.605577i	$0.207058\pi$
$360$	0	0
$361$	6.78984e6	2.74215
$362$	0	0
$363$	2.91350e6i	1.16051i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	933040.i	0.361606i	0.983519	+	0.180803i	$0.0578696\pi$
$367$	933040.i	0.361606i	−0.983519	+	0.180803i	$0.942130\pi$
$368$	0	0
$369$	1.39510e6	0.533384
$370$	0	0
$371$	280464.	0.105789
$372$	0	0
$373$	− 392218.i	− 0.145967i	−0.997333	−	0.0729836i	$-0.976748\pi$
$373$	− 392218.i	− 0.145967i	0.997333	−	0.0729836i	$-0.0232521\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.56880e6i	0.568477i
$378$	0	0
$379$	−4.72930e6	−1.69122	−0.845608	−	0.533805i	$-0.820762\pi$
$379$	−4.72930e6	−1.69122	−0.845608	+	0.533805i	$0.820762\pi$
$380$	0	0
$381$	−1.89184e6	−0.667686
$382$	0	0
$383$	− 1.89734e6i	− 0.660920i	−0.943820	−	0.330460i	$-0.892796\pi$
$383$	− 1.89734e6i	− 0.660920i	0.943820	−	0.330460i	$-0.107204\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.44252e6i	− 0.489602i
$388$	0	0
$389$	3.72295e6	1.24742	0.623711	−	0.781655i	$-0.285624\pi$
$389$	3.72295e6	1.24742	0.623711	+	0.781655i	$0.285624\pi$
$390$	0	0
$391$	220432.	0.0729177
$392$	0	0
$393$	1.40584e6i	0.459150i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3.33808e6i	− 1.06297i	−0.847068	−	0.531484i	$-0.821635\pi$
$397$	− 3.33808e6i	− 1.06297i	0.847068	−	0.531484i	$-0.178365\pi$
$398$	0	0
$399$	−1.46112e6	−0.459466
$400$	0	0
$401$	4.27490e6	1.32759	0.663796	−	0.747913i	$-0.268944\pi$
$401$	4.27490e6	1.32759	0.663796	+	0.747913i	$0.268944\pi$
$402$	0	0
$403$	2.73798e6i	0.839785i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.28042e6i	0.383149i
$408$	0	0
$409$	2.57319e6	0.760613	0.380306	−	0.924861i	$-0.375819\pi$
$409$	2.57319e6	0.760613	0.380306	+	0.924861i	$0.375819\pi$
$410$	0	0
$411$	−5.54580e6	−1.61942
$412$	0	0
$413$	− 405024.i	− 0.116844i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.60616e6i	0.733941i
$418$	0	0
$419$	5.26828e6	1.46600	0.732999	−	0.680230i	$-0.238120\pi$
$419$	5.26828e6	1.46600	0.732999	+	0.680230i	$0.238120\pi$
$420$	0	0
$421$	−973354.	−0.267649	−0.133824	−	0.991005i	$-0.542726\pi$
$421$	−973354.	−0.267649	−0.133824	+	0.991005i	$0.542726\pi$
$422$	0	0
$423$	− 3.71525e6i	− 1.00957i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	443568.i	0.117731i
$428$	0	0
$429$	−1.18544e6	−0.310983
$430$	0	0
$431$	−3.55736e6	−0.922433	−0.461216	−	0.887288i	$-0.652587\pi$
$431$	−3.55736e6	−0.922433	−0.461216	+	0.887288i	$0.652587\pi$
$432$	0	0
$433$	− 1.95496e6i	− 0.501092i	−0.968105	−	0.250546i	$-0.919390\pi$
$433$	− 1.95496e6i	− 0.501092i	0.968105	−	0.250546i	$-0.0806102\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 560096.i	− 0.140300i
$438$	0	0
$439$	−3.29681e6	−0.816455	−0.408228	−	0.912880i	$-0.633853\pi$
$439$	−3.29681e6	−0.816455	−0.408228	+	0.912880i	$0.633853\pi$
$440$	0	0
$441$	−2.54827e6	−0.623948
$442$	0	0
$443$	5.05820e6i	1.22458i	0.790634	+	0.612289i	$0.209751\pi$
$443$	5.05820e6i	1.22458i	−0.790634	+	0.612289i	$0.790249\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.03060e6i	− 1.90099i
$448$	0	0
$449$	−2.12730e6	−0.497981	−0.248990	−	0.968506i	$-0.580099\pi$
$449$	−2.12730e6	−0.497981	−0.248990	+	0.968506i	$0.580099\pi$
$450$	0	0
$451$	1.10186e6	0.255086
$452$	0	0
$453$	− 1.51952e6i	− 0.347905i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 289130.i	− 0.0647594i	−0.999476	−	0.0323797i	$-0.989691\pi$
$457$	− 289130.i	− 0.0647594i	0.999476	−	0.0323797i	$-0.0103086\pi$
$458$	0	0
$459$	2.06056e6	0.456513
$460$	0	0
$461$	2.66870e6	0.584854	0.292427	−	0.956288i	$-0.405537\pi$
$461$	2.66870e6	0.584854	0.292427	+	0.956288i	$0.405537\pi$
$462$	0	0
$463$	− 7.58619e6i	− 1.64464i	−0.569024	−	0.822321i	$-0.692679\pi$
$463$	− 7.58619e6i	− 1.64464i	0.569024	−	0.822321i	$-0.307321\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.41961e6i	− 0.301216i	−0.988594	−	0.150608i	$-0.951877\pi$
$467$	− 1.41961e6i	− 0.301216i	0.988594	−	0.150608i	$-0.0481231\pi$
$468$	0	0
$469$	−372768.	−0.0782540
$470$	0	0
$471$	7.88644e6	1.63806
$472$	0	0
$473$	− 1.13931e6i	− 0.234148i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.83470e6i	− 0.369207i
$478$	0	0
$479$	−1.88406e6	−0.375195	−0.187597	−	0.982246i	$-0.560070\pi$
$479$	−1.88406e6	−0.375195	−0.187597	+	0.982246i	$0.560070\pi$
$480$	0	0
$481$	4.93583e6	0.972741
$482$	0	0
$483$	88320.0i	0.0172263i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.01388e6i	− 1.14903i	−0.818493	−	0.574516i	$-0.805190\pi$
$487$	− 6.01388e6i	− 1.14903i	0.818493	−	0.574516i	$-0.194810\pi$
$488$	0	0
$489$	234480.	0.0443439
$490$	0	0
$491$	−4.29232e6	−0.803504	−0.401752	−	0.915749i	$-0.631599\pi$
$491$	−4.29232e6	−0.803504	−0.401752	+	0.915749i	$0.631599\pi$
$492$	0	0
$493$	3.93184e6i	0.728581i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 767040.i	− 0.139292i
$498$	0	0
$499$	1.34509e6	0.241825	0.120912	−	0.992663i	$-0.461418\pi$
$499$	1.34509e6	0.241825	0.120912	+	0.992663i	$0.461418\pi$
$500$	0	0
$501$	−1.10386e7	−1.96480
$502$	0	0
$503$	− 202008.i	− 0.0355999i	−0.999842	−	0.0177999i	$-0.994334\pi$
$503$	− 202008.i	− 0.0355999i	0.999842	−	0.0177999i	$-0.00566620\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.85618e6i	− 0.493476i
$508$	0	0
$509$	−9.78344e6	−1.67377	−0.836887	−	0.547375i	$-0.815627\pi$
$509$	−9.78344e6	−1.67377	−0.836887	+	0.547375i	$0.815627\pi$
$510$	0	0
$511$	−117264.	−0.0198661
$512$	0	0
$513$	− 5.23568e6i	− 0.878374i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.93434e6i	− 0.482818i
$518$	0	0
$519$	8.65788e6	1.41089
$520$	0	0
$521$	−1.04830e7	−1.69197	−0.845985	−	0.533207i	$-0.820987\pi$
$521$	−1.04830e7	−1.69197	−0.845985	+	0.533207i	$0.820987\pi$
$522$	0	0
$523$	− 6.21017e6i	− 0.992772i	−0.868102	−	0.496386i	$-0.834660\pi$
$523$	− 6.21017e6i	− 0.992772i	0.868102	−	0.496386i	$-0.165340\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.86214e6i	1.07630i
$528$	0	0
$529$	6.40249e6	0.994740
$530$	0	0
$531$	−2.64953e6	−0.407787
$532$	0	0
$533$	− 4.24751e6i	− 0.647614i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.11924e7i	− 1.67489i
$538$	0	0
$539$	−2.01264e6	−0.298397
$540$	0	0
$541$	5.08088e6	0.746355	0.373178	−	0.927760i	$-0.378268\pi$
$541$	5.08088e6	0.746355	0.373178	+	0.927760i	$0.378268\pi$
$542$	0	0
$543$	− 1.20942e7i	− 1.76026i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.34687e6i	0.478267i	0.970987	+	0.239133i	$0.0768633\pi$
$547$	3.34687e6i	0.478267i	−0.970987	+	0.239133i	$0.923137\pi$
$548$	0	0
$549$	2.90167e6	0.410883
$550$	0	0
$551$	9.99041e6	1.40186
$552$	0	0
$553$	− 1.06944e6i	− 0.148711i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.00377e6i	− 0.956520i	−0.878218	−	0.478260i	$-0.841268\pi$
$557$	− 7.00377e6i	− 0.956520i	0.878218	−	0.478260i	$-0.158732\pi$
$558$	0	0
$559$	−4.39186e6	−0.594456
$560$	0	0
$561$	−2.97104e6	−0.398567
$562$	0	0
$563$	1.29819e7i	1.72610i	0.505116	+	0.863052i	$0.331450\pi$
$563$	1.29819e7i	1.72610i	−0.505116	+	0.863052i	$0.668550\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.74122e6i	0.227456i
$568$	0	0
$569$	−1.89942e6	−0.245946	−0.122973	−	0.992410i	$-0.539243\pi$
$569$	−1.89942e6	−0.245946	−0.122973	+	0.992410i	$0.539243\pi$
$570$	0	0
$571$	1.66300e6	0.213452	0.106726	−	0.994288i	$-0.465963\pi$
$571$	1.66300e6	0.213452	0.106726	+	0.994288i	$0.465963\pi$
$572$	0	0
$573$	− 8.18304e6i	− 1.04119i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8.77344e6i	− 1.09706i	−0.836131	−	0.548530i	$-0.815188\pi$
$577$	− 8.77344e6i	− 1.09706i	0.836131	−	0.548530i	$-0.184812\pi$
$578$	0	0
$579$	1.08173e7	1.34098
$580$	0	0
$581$	−1.61674e6	−0.198700
$582$	0	0
$583$	− 1.44906e6i	− 0.176570i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.18393e6i	0.620961i	0.950580	+	0.310480i	$0.100490\pi$
$587$	5.18393e6i	0.620961i	−0.950580	+	0.310480i	$0.899510\pi$
$588$	0	0
$589$	1.74360e7	2.07090
$590$	0	0
$591$	1.25980e7	1.48365
$592$	0	0
$593$	8.49858e6i	0.992452i	0.868193	+	0.496226i	$0.165281\pi$
$593$	8.49858e6i	0.992452i	−0.868193	+	0.496226i	$0.834719\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5.66096e6i	− 0.650061i
$598$	0	0
$599$	1.12471e7	1.28078	0.640388	−	0.768051i	$-0.278773\pi$
$599$	1.12471e7	1.28078	0.640388	+	0.768051i	$0.278773\pi$
$600$	0	0
$601$	−3.46439e6	−0.391238	−0.195619	−	0.980680i	$-0.562672\pi$
$601$	−3.46439e6	−0.391238	−0.195619	+	0.980680i	$0.562672\pi$
$602$	0	0
$603$	2.43852e6i	0.273108i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 999712.i	− 0.110129i	−0.998483	−	0.0550647i	$-0.982463\pi$
$607$	− 999712.i	− 0.110129i	0.998483	−	0.0550647i	$-0.0175365\pi$
$608$	0	0
$609$	−1.57536e6	−0.172122
$610$	0	0
$611$	−1.13114e7	−1.22578
$612$	0	0
$613$	9.81340e6i	1.05480i	0.849619	+	0.527398i	$0.176832\pi$
$613$	9.81340e6i	1.05480i	−0.849619	+	0.527398i	$0.823168\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.34745e6i	0.565501i	0.959193	+	0.282751i	$0.0912469\pi$
$617$	5.34745e6i	0.565501i	−0.959193	+	0.282751i	$0.908753\pi$
$618$	0	0
$619$	−6.82768e6	−0.716221	−0.358110	−	0.933679i	$-0.616579\pi$
$619$	−6.82768e6	−0.716221	−0.358110	+	0.933679i	$0.616579\pi$
$620$	0	0
$621$	−316480.	−0.0329319
$622$	0	0
$623$	1.72786e6i	0.178356i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.54912e6i	0.766880i
$628$	0	0
$629$	1.23705e7	1.24670
$630$	0	0
$631$	3.60970e6	0.360909	0.180455	−	0.983583i	$-0.442243\pi$
$631$	3.60970e6	0.360909	0.180455	+	0.983583i	$0.442243\pi$
$632$	0	0
$633$	2.85512e6i	0.283214i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.75842e6i	0.757573i
$638$	0	0
$639$	−5.01772e6	−0.486132
$640$	0	0
$641$	−1.33853e7	−1.28672	−0.643361	−	0.765563i	$-0.722460\pi$
$641$	−1.33853e7	−1.28672	−0.643361	+	0.765563i	$0.722460\pi$
$642$	0	0
$643$	9.91115e6i	0.945358i	0.881235	+	0.472679i	$0.156713\pi$
$643$	9.91115e6i	0.945358i	−0.881235	+	0.472679i	$0.843287\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.78359e7i	− 1.67508i	−0.546378	−	0.837539i	$-0.683994\pi$
$647$	− 1.78359e7i	− 1.67508i	0.546378	−	0.837539i	$-0.316006\pi$
$648$	0	0
$649$	−2.09262e6	−0.195020
$650$	0	0
$651$	−2.74944e6	−0.254268
$652$	0	0
$653$	− 4.32323e6i	− 0.396758i	−0.980125	−	0.198379i	$-0.936432\pi$
$653$	− 4.32323e6i	− 0.396758i	0.980125	−	0.198379i	$-0.0635677\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	767102.i	0.0693330i
$658$	0	0
$659$	1.97858e7	1.77476	0.887382	−	0.461035i	$-0.152522\pi$
$659$	1.97858e7	1.77476	0.887382	+	0.461035i	$0.152522\pi$
$660$	0	0
$661$	1.57772e7	1.40451	0.702255	−	0.711925i	$-0.252176\pi$
$661$	1.57772e7	1.40451	0.702255	+	0.711925i	$0.252176\pi$
$662$	0	0
$663$	1.14529e7i	1.01188i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 603888.i	− 0.0525584i
$668$	0	0
$669$	−1.77939e7	−1.53711
$670$	0	0
$671$	2.29177e6	0.196501
$672$	0	0
$673$	6.78762e6i	0.577670i	0.957379	+	0.288835i	$0.0932679\pi$
$673$	6.78762e6i	0.577670i	−0.957379	+	0.288835i	$0.906732\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.49942e7i	1.25734i	0.777673	+	0.628669i	$0.216400\pi$
$677$	1.49942e7i	1.25734i	−0.777673	+	0.628669i	$0.783600\pi$
$678$	0	0
$679$	−1.17278e6	−0.0976211
$680$	0	0
$681$	2.28631e7	1.88916
$682$	0	0
$683$	− 1.15580e7i	− 0.948053i	−0.880511	−	0.474026i	$-0.842800\pi$
$683$	− 1.15580e7i	− 0.948053i	0.880511	−	0.474026i	$-0.157200\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.39157e7i	− 1.12490i
$688$	0	0
$689$	−5.58591e6	−0.448276
$690$	0	0
$691$	220156.	0.0175402	0.00877012	−	0.999962i	$-0.497208\pi$
$691$	220156.	0.0175402	0.00877012	+	0.999962i	$0.497208\pi$
$692$	0	0
$693$	− 467232.i	− 0.0369572i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.06454e7i	− 0.830006i
$698$	0	0
$699$	−6.94252e6	−0.537433
$700$	0	0
$701$	4.78933e6	0.368111	0.184056	−	0.982916i	$-0.441077\pi$
$701$	4.78933e6	0.368111	0.184056	+	0.982916i	$0.441077\pi$
$702$	0	0
$703$	− 3.14323e7i	− 2.39877i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.23854e6i	− 0.0931886i
$708$	0	0
$709$	−4.26892e6	−0.318935	−0.159468	−	0.987203i	$-0.550978\pi$
$709$	−4.26892e6	−0.318935	−0.159468	+	0.987203i	$0.550978\pi$
$710$	0	0
$711$	−6.99592e6	−0.519004
$712$	0	0
$713$	− 1.05395e6i	− 0.0776421i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.28592e7i	2.38704i
$718$	0	0
$719$	−1.61960e7	−1.16838	−0.584190	−	0.811617i	$-0.698588\pi$
$719$	−1.61960e7	−1.16838	−0.584190	+	0.811617i	$0.698588\pi$
$720$	0	0
$721$	−4.33018e6	−0.310218
$722$	0	0
$723$	2.33488e7i	1.66119i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.53426e6i	0.458522i	0.973365	+	0.229261i	$0.0736310\pi$
$727$	6.53426e6i	0.458522i	−0.973365	+	0.229261i	$0.926369\pi$
$728$	0	0
$729$	3.03131e6	0.211257
$730$	0	0
$731$	−1.10072e7	−0.761876
$732$	0	0
$733$	1.31617e7i	0.904800i	0.891815	+	0.452400i	$0.149432\pi$
$733$	1.31617e7i	0.904800i	−0.891815	+	0.452400i	$0.850568\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.92597e6i	0.130611i
$738$	0	0
$739$	−1.42348e7	−0.958825	−0.479412	−	0.877590i	$-0.659150\pi$
$739$	−1.42348e7	−0.958825	−0.479412	+	0.877590i	$0.659150\pi$
$740$	0	0
$741$	2.91006e7	1.94696
$742$	0	0
$743$	2.15835e7i	1.43434i	0.696901	+	0.717168i	$0.254562\pi$
$743$	2.15835e7i	1.43434i	−0.696901	+	0.717168i	$0.745438\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.05761e7i	0.693467i
$748$	0	0
$749$	−1.57680e6	−0.102700
$750$	0	0
$751$	−1.86594e7	−1.20725	−0.603625	−	0.797268i	$-0.706278\pi$
$751$	−1.86594e7	−1.20725	−0.603625	+	0.797268i	$0.706278\pi$
$752$	0	0
$753$	− 1.58122e7i	− 1.01626i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.56681e6i	0.162800i	0.996682	+	0.0813999i	$0.0259391\pi$
$757$	2.56681e6i	0.162800i	−0.996682	+	0.0813999i	$0.974061\pi$
$758$	0	0
$759$	456320.	0.0287518
$760$	0	0
$761$	−2.59586e7	−1.62487	−0.812436	−	0.583051i	$-0.801859\pi$
$761$	−2.59586e7	−1.62487	−0.812436	+	0.583051i	$0.801859\pi$
$762$	0	0
$763$	− 2.70494e6i	− 0.168208i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.06673e6i	0.495118i
$768$	0	0
$769$	−5.53267e6	−0.337380	−0.168690	−	0.985669i	$-0.553954\pi$
$769$	−5.53267e6	−0.337380	−0.168690	+	0.985669i	$0.553954\pi$
$770$	0	0
$771$	2.59580e6	0.157266
$772$	0	0
$773$	8.32940e6i	0.501378i	0.968068	+	0.250689i	$0.0806571\pi$
$773$	8.32940e6i	0.501378i	−0.968068	+	0.250689i	$0.919343\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.95648e6i	0.294524i
$778$	0	0
$779$	−2.70490e7	−1.59701
$780$	0	0
$781$	−3.96304e6	−0.232488
$782$	0	0
$783$	− 5.64504e6i	− 0.329051i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.36523e7i	− 0.785719i	−0.919598	−	0.392860i	$-0.871486\pi$
$787$	− 1.36523e7i	− 0.785719i	0.919598	−	0.392860i	$-0.128514\pi$
$788$	0	0
$789$	−1.41776e6	−0.0810793
$790$	0	0
$791$	−564048.	−0.0320535
$792$	0	0
$793$	− 8.83440e6i	− 0.498877i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.54626e6i	0.476574i	0.971195	+	0.238287i	$0.0765859\pi$
$797$	8.54626e6i	0.476574i	−0.971195	+	0.238287i	$0.923414\pi$
$798$	0	0
$799$	−2.83495e7	−1.57101
$800$	0	0
$801$	1.13031e7	0.622465
$802$	0	0
$803$	605864.i	0.0331578i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.58035e7i	1.93527i
$808$	0	0
$809$	−7.58484e6	−0.407451	−0.203725	−	0.979028i	$-0.565305\pi$
$809$	−7.58484e6	−0.407451	−0.203725	+	0.979028i	$0.565305\pi$
$810$	0	0
$811$	−6.18473e6	−0.330194	−0.165097	−	0.986277i	$-0.552794\pi$
$811$	−6.18473e6	−0.330194	−0.165097	+	0.986277i	$0.552794\pi$
$812$	0	0
$813$	− 3.54723e7i	− 1.88219i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.79683e7i	1.46592i
$818$	0	0
$819$	−1.80110e6	−0.0938273
$820$	0	0
$821$	−2.78102e6	−0.143995	−0.0719973	−	0.997405i	$-0.522937\pi$
$821$	−2.78102e6	−0.143995	−0.0719973	+	0.997405i	$0.522937\pi$
$822$	0	0
$823$	− 1.63895e7i	− 0.843461i	−0.906721	−	0.421731i	$-0.861423\pi$
$823$	− 1.63895e7i	− 0.843461i	0.906721	−	0.421731i	$-0.138577\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.29511e7i	1.16692i	0.812142	+	0.583459i	$0.198301\pi$
$827$	2.29511e7i	1.16692i	−0.812142	+	0.583459i	$0.801699\pi$
$828$	0	0
$829$	3.50136e6	0.176950	0.0884750	−	0.996078i	$-0.471801\pi$
$829$	3.50136e6	0.176950	0.0884750	+	0.996078i	$0.471801\pi$
$830$	0	0
$831$	5.50900e6	0.276739
$832$	0	0
$833$	1.94447e7i	0.970934i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 9.85216e6i	− 0.486091i
$838$	0	0
$839$	5.29668e6	0.259776	0.129888	−	0.991529i	$-0.458538\pi$
$839$	5.29668e6	0.259776	0.129888	+	0.991529i	$0.458538\pi$
$840$	0	0
$841$	−9.73962e6	−0.474845
$842$	0	0
$843$	− 1.18834e7i	− 0.575933i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.49620e6i	0.167451i
$848$	0	0
$849$	−2.18486e7	−1.04029
$850$	0	0
$851$	−1.89998e6	−0.0899344
$852$	0	0
$853$	− 2.02948e7i	− 0.955021i	−0.878626	−	0.477511i	$-0.841539\pi$
$853$	− 2.02948e7i	− 0.955021i	0.878626	−	0.477511i	$-0.158461\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.82785e6i	0.224544i	0.993678	+	0.112272i	$0.0358128\pi$
$857$	4.82785e6i	0.224544i	−0.993678	+	0.112272i	$0.964187\pi$
$858$	0	0
$859$	−1.30210e7	−0.602092	−0.301046	−	0.953610i	$-0.597336\pi$
$859$	−1.30210e7	−0.602092	−0.301046	+	0.953610i	$0.597336\pi$
$860$	0	0
$861$	4.26528e6	0.196083
$862$	0	0
$863$	3.92387e7i	1.79344i	0.442596	+	0.896721i	$0.354058\pi$
$863$	3.92387e7i	1.79344i	−0.442596	+	0.896721i	$0.645942\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	306940.i	0.0138677i
$868$	0	0
$869$	−5.52544e6	−0.248209
$870$	0	0
$871$	7.42430e6	0.331596
$872$	0	0
$873$	7.67196e6i	0.340699i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.34622e7i	− 0.591041i	−0.955336	−	0.295520i	$-0.904507\pi$
$877$	− 1.34622e7i	− 0.591041i	0.955336	−	0.295520i	$-0.0954930\pi$
$878$	0	0
$879$	6.67308e6	0.291309
$880$	0	0
$881$	−917710.	−0.0398351	−0.0199175	−	0.999802i	$-0.506340\pi$
$881$	−917710.	−0.0398351	−0.0199175	+	0.999802i	$0.506340\pi$
$882$	0	0
$883$	− 2.45488e7i	− 1.05957i	−0.848133	−	0.529784i	$-0.822273\pi$
$883$	− 2.45488e7i	− 1.05957i	0.848133	−	0.529784i	$-0.177727\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.61463e7i	− 0.689070i	−0.938773	−	0.344535i	$-0.888037\pi$
$887$	− 1.61463e7i	− 0.689070i	0.938773	−	0.344535i	$-0.111963\pi$
$888$	0	0
$889$	−2.27021e6	−0.0963410
$890$	0	0
$891$	8.99632e6	0.379639
$892$	0	0
$893$	7.20332e7i	3.02276i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.75904e6i	− 0.0729953i
$898$	0	0
$899$	1.87993e7	0.775787
$900$	0	0
$901$	−1.39998e7	−0.574527
$902$	0	0
$903$	− 4.41024e6i	− 0.179988i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 2.03361e7i	− 0.820824i	−0.911900	−	0.410412i	$-0.865385\pi$
$907$	− 2.03361e7i	− 0.820824i	0.911900	−	0.410412i	$-0.134615\pi$
$908$	0	0
$909$	−8.10214e6	−0.325230
$910$	0	0
$911$	−1.07726e7	−0.430054	−0.215027	−	0.976608i	$-0.568984\pi$
$911$	−1.07726e7	−0.430054	−0.215027	+	0.976608i	$0.568984\pi$
$912$	0	0
$913$	8.35314e6i	0.331644i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.68701e6i	0.0662512i
$918$	0	0
$919$	4.18566e7	1.63484	0.817419	−	0.576043i	$-0.195404\pi$
$919$	4.18566e7	1.63484	0.817419	+	0.576043i	$0.195404\pi$
$920$	0	0
$921$	2.11994e7	0.823522
$922$	0	0
$923$	1.52769e7i	0.590242i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.83266e7i	1.08267i
$928$	0	0
$929$	−2.99845e7	−1.13988	−0.569939	−	0.821687i	$-0.693033\pi$
$929$	−2.99845e7	−1.13988	−0.569939	+	0.821687i	$0.693033\pi$
$930$	0	0
$931$	4.94072e7	1.86817
$932$	0	0
$933$	− 2.67298e7i	− 1.00529i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.42402e7i	− 0.529867i	−0.964267	−	0.264934i	$-0.914650\pi$
$937$	− 1.42402e7i	− 0.529867i	0.964267	−	0.264934i	$-0.0853501\pi$
$938$	0	0
$939$	3.28837e7	1.21707
$940$	0	0
$941$	−4.14546e7	−1.52615	−0.763077	−	0.646307i	$-0.776313\pi$
$941$	−4.14546e7	−1.52615	−0.763077	+	0.646307i	$0.776313\pi$
$942$	0	0
$943$	1.63502e6i	0.0598749i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.54079e7i	− 0.558300i	−0.960248	−	0.279150i	$-0.909947\pi$
$947$	− 1.54079e7i	− 0.558300i	0.960248	−	0.279150i	$-0.0900527\pi$
$948$	0	0
$949$	2.33551e6	0.0841813
$950$	0	0
$951$	3.44740e7	1.23606
$952$	0	0
$953$	− 2.06328e7i	− 0.735912i	−0.929843	−	0.367956i	$-0.880058\pi$
$953$	− 2.06328e7i	− 0.735912i	0.929843	−	0.367956i	$-0.119942\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.13936e6i	0.287283i
$958$	0	0
$959$	−6.65496e6	−0.233668
$960$	0	0
$961$	4.18083e6	0.146034
$962$	0	0
$963$	1.03149e7i	0.358426i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.18724e7i	0.408294i	0.978940	+	0.204147i	$0.0654421\pi$
$967$	1.18724e7i	0.408294i	−0.978940	+	0.204147i	$0.934558\pi$
$968$	0	0
$969$	7.29342e7	2.49530
$970$	0	0
$971$	−1.53222e6	−0.0521523	−0.0260761	−	0.999660i	$-0.508301\pi$
$971$	−1.53222e6	−0.0521523	−0.0260761	+	0.999660i	$0.508301\pi$
$972$	0	0
$973$	3.12739e6i	0.105901i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.74321e7i	− 0.584269i	−0.956377	−	0.292135i	$-0.905635\pi$
$977$	− 1.74321e7i	− 0.584269i	0.956377	−	0.292135i	$-0.0943655\pi$
$978$	0	0
$979$	8.92726e6	0.297688
$980$	0	0
$981$	−1.76948e7	−0.587049
$982$	0	0
$983$	− 2.23270e6i	− 0.0736963i	−0.999321	−	0.0368482i	$-0.988268\pi$
$983$	− 2.23270e6i	− 0.0736963i	0.999321	−	0.0368482i	$-0.0117318\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.13587e7i	− 0.371139i
$988$	0	0
$989$	1.69059e6	0.0549602
$990$	0	0
$991$	−2.22501e7	−0.719693	−0.359847	−	0.933011i	$-0.617171\pi$
$991$	−2.22501e7	−0.719693	−0.359847	+	0.933011i	$0.617171\pi$
$992$	0	0
$993$	5.49926e7i	1.76983i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 5.32662e7i	− 1.69712i	−0.529095	−	0.848562i	$-0.677469\pi$
$997$	− 5.32662e7i	− 1.69712i	0.529095	−	0.848562i	$-0.322531\pi$
$998$	0	0
$999$	−1.77607e7	−0.563050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.d.49.1		2
4.3	odd	2		200.6.c.a.49.2		2
5.2	odd	4		16.6.a.a.1.1		1
5.3	odd	4		400.6.a.l.1.1		1
5.4	even	2	inner	400.6.c.d.49.2		2
15.2	even	4		144.6.a.k.1.1		1
20.3	even	4		200.6.a.a.1.1		1
20.7	even	4		8.6.a.a.1.1	✓	1
20.19	odd	2		200.6.c.a.49.1		2
35.27	even	4		784.6.a.l.1.1		1
40.27	even	4		64.6.a.a.1.1		1
40.37	odd	4		64.6.a.g.1.1		1
60.47	odd	4		72.6.a.f.1.1		1
80.27	even	4		256.6.b.f.129.1		2
80.37	odd	4		256.6.b.d.129.2		2
80.67	even	4		256.6.b.f.129.2		2
80.77	odd	4		256.6.b.d.129.1		2
120.77	even	4		576.6.a.h.1.1		1
120.107	odd	4		576.6.a.g.1.1		1
140.27	odd	4		392.6.a.b.1.1		1
140.47	odd	12		392.6.i.e.361.1		2
140.67	even	12		392.6.i.b.177.1		2
140.87	odd	12		392.6.i.e.177.1		2
140.107	even	12		392.6.i.b.361.1		2
220.87	odd	4		968.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.6.a.a.1.1	✓	1	20.7	even	4
16.6.a.a.1.1		1	5.2	odd	4
64.6.a.a.1.1		1	40.27	even	4
64.6.a.g.1.1		1	40.37	odd	4
72.6.a.f.1.1		1	60.47	odd	4
144.6.a.k.1.1		1	15.2	even	4
200.6.a.a.1.1		1	20.3	even	4
200.6.c.a.49.1		2	20.19	odd	2
200.6.c.a.49.2		2	4.3	odd	2
256.6.b.d.129.1		2	80.77	odd	4
256.6.b.d.129.2		2	80.37	odd	4
256.6.b.f.129.1		2	80.27	even	4
256.6.b.f.129.2		2	80.67	even	4
392.6.a.b.1.1		1	140.27	odd	4
392.6.i.b.177.1		2	140.67	even	12
392.6.i.b.361.1		2	140.107	even	12
392.6.i.e.177.1		2	140.87	odd	12
392.6.i.e.361.1		2	140.47	odd	12
400.6.a.l.1.1		1	5.3	odd	4
400.6.c.d.49.1		2	1.1	even	1	trivial
400.6.c.d.49.2		2	5.4	even	2	inner
576.6.a.g.1.1		1	120.107	odd	4
576.6.a.h.1.1		1	120.77	even	4
784.6.a.l.1.1		1	35.27	even	4
968.6.a.a.1.1		1	220.87	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-20.0000i q^{3} -24.0000i q^{7} -157.000 q^{9} +O(q^{10})\)	\(q-20.0000i q^{3} -24.0000i q^{7} -157.000 q^{9} -124.000 q^{11} +478.000i q^{13} +1198.00i q^{17} +3044.00 q^{19} -480.000 q^{21} -184.000i q^{23} -1720.00i q^{27} +3282.00 q^{29} +5728.00 q^{31} +2480.00i q^{33} -10326.0i q^{37} +9560.00 q^{39} -8886.00 q^{41} +9188.00i q^{43} +23664.0i q^{47} +16231.0 q^{49} +23960.0 q^{51} +11686.0i q^{53} -60880.0i q^{57} +16876.0 q^{59} -18482.0 q^{61} +3768.00i q^{63} -15532.0i q^{67} -3680.00 q^{69} +31960.0 q^{71} -4886.00i q^{73} +2976.00i q^{77} +44560.0 q^{79} -72551.0 q^{81} -67364.0i q^{83} -65640.0i q^{87} -71994.0 q^{89} +11472.0 q^{91} -114560. i q^{93} -48866.0i q^{97} +19468.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 314 q^{9}+O(q^{10}) \) 2 * q - 314 * q^9	\( 2 q - 314 q^{9} - 248 q^{11} + 6088 q^{19} - 960 q^{21} + 6564 q^{29} + 11456 q^{31} + 19120 q^{39} - 17772 q^{41} + 32462 q^{49} + 47920 q^{51} + 33752 q^{59} - 36964 q^{61} - 7360 q^{69} + 63920 q^{71} + 89120 q^{79} - 145102 q^{81} - 143988 q^{89} + 22944 q^{91} + 38936 q^{99}+O(q^{100}) \) 2 * q - 314 * q^9 - 248 * q^11 + 6088 * q^19 - 960 * q^21 + 6564 * q^29 + 11456 * q^31 + 19120 * q^39 - 17772 * q^41 + 32462 * q^49 + 47920 * q^51 + 33752 * q^59 - 36964 * q^61 - 7360 * q^69 + 63920 * q^71 + 89120 * q^79 - 145102 * q^81 - 143988 * q^89 + 22944 * q^91 + 38936 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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