LMFDB - Embedded newform 400.6.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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, 0]))

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

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

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:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 100)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.61187$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+30.2237 q^{3} -141.342 q^{7} +670.475 q^{9} +O(q^{10})$	$q+30.2237 q^{3} -141.342 q^{7} +670.475 q^{9} -273.356 q^{11} -945.370 q^{13} -1212.32 q^{17} -135.931 q^{19} -4271.90 q^{21} -2730.88 q^{23} +12919.9 q^{27} -3077.70 q^{29} +1410.44 q^{31} -8261.85 q^{33} +3152.03 q^{37} -28572.6 q^{39} -13499.5 q^{41} +5433.70 q^{43} -3269.53 q^{47} +3170.70 q^{49} -36640.7 q^{51} -21421.3 q^{53} -4108.35 q^{57} +41403.1 q^{59} -24373.9 q^{61} -94766.6 q^{63} -32307.0 q^{67} -82537.3 q^{69} +38675.0 q^{71} -8083.79 q^{73} +38636.8 q^{77} +15979.3 q^{79} +227562. q^{81} +48494.1 q^{83} -93019.6 q^{87} +89187.5 q^{89} +133621. q^{91} +42628.7 q^{93} +56822.2 q^{97} -183279. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{3} - 40 q^{7} + 532 q^{9}+O(q^{10}) $ 2 * q + 20 * q^3 - 40 * q^7 + 532 * q^9	$ 2 q + 20 q^{3} - 40 q^{7} + 532 q^{9} + 60 q^{11} - 920 q^{13} - 2910 q^{17} - 2092 q^{19} - 5308 q^{21} + 120 q^{23} + 16820 q^{27} + 3552 q^{29} + 8888 q^{31} - 11670 q^{33} - 12140 q^{37} - 28832 q^{39} - 12438 q^{41} + 1160 q^{43} - 1200 q^{47} - 3366 q^{49} - 19284 q^{51} - 26340 q^{53} + 15890 q^{57} + 36696 q^{59} + 19204 q^{61} - 108800 q^{63} - 90460 q^{67} - 111684 q^{69} - 2736 q^{71} - 12770 q^{73} + 72420 q^{77} + 16184 q^{79} + 221338 q^{81} - 30300 q^{83} - 160800 q^{87} - 47322 q^{89} + 136192 q^{91} - 33820 q^{93} + 2980 q^{97} - 229440 q^{99}+O(q^{100}) $ 2 * q + 20 * q^3 - 40 * q^7 + 532 * q^9 + 60 * q^11 - 920 * q^13 - 2910 * q^17 - 2092 * q^19 - 5308 * q^21 + 120 * q^23 + 16820 * q^27 + 3552 * q^29 + 8888 * q^31 - 11670 * q^33 - 12140 * q^37 - 28832 * q^39 - 12438 * q^41 + 1160 * q^43 - 1200 * q^47 - 3366 * q^49 - 19284 * q^51 - 26340 * q^53 + 15890 * q^57 + 36696 * q^59 + 19204 * q^61 - 108800 * q^63 - 90460 * q^67 - 111684 * q^69 - 2736 * q^71 - 12770 * q^73 + 72420 * q^77 + 16184 * q^79 + 221338 * q^81 - 30300 * q^83 - 160800 * q^87 - 47322 * q^89 + 136192 * q^91 - 33820 * q^93 + 2980 * q^97 - 229440 * q^99

Display 100 coefficients 10 coefficients

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
$3$	30.2237	1.93885	0.969427	−	0.245379i	$-0.0789125\pi$
$3$	30.2237	1.93885	0.969427	+	0.245379i	$0.0789125\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−141.342	−1.09025	−0.545127	−	0.838354i	$-0.683519\pi$
$7$	−141.342	−1.09025	−0.545127	+	0.838354i	$0.683519\pi$
$8$	0	0
$9$	670.475	2.75916
$10$	0	0
$11$	−273.356	−0.681157	−0.340579	−	0.940216i	$-0.610623\pi$
$11$	−273.356	−0.681157	−0.340579	+	0.940216i	$0.610623\pi$
$12$	0	0
$13$	−945.370	−1.55147	−0.775735	−	0.631059i	$-0.782621\pi$
$13$	−945.370	−1.55147	−0.775735	+	0.631059i	$0.782621\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1212.32	−1.01740	−0.508702	−	0.860943i	$-0.669874\pi$
$17$	−1212.32	−1.01740	−0.508702	+	0.860943i	$0.669874\pi$
$18$	0	0
$19$	−135.931	−0.0863844	−0.0431922	−	0.999067i	$-0.513753\pi$
$19$	−135.931	−0.0863844	−0.0431922	+	0.999067i	$0.513753\pi$
$20$	0	0
$21$	−4271.90	−2.11384
$22$	0	0
$23$	−2730.88	−1.07642	−0.538211	−	0.842810i	$-0.680900\pi$
$23$	−2730.88	−1.07642	−0.538211	+	0.842810i	$0.680900\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	12919.9	3.41075
$28$	0	0
$29$	−3077.70	−0.679565	−0.339783	−	0.940504i	$-0.610354\pi$
$29$	−3077.70	−0.679565	−0.339783	+	0.940504i	$0.610354\pi$
$30$	0	0
$31$	1410.44	0.263603	0.131801	−	0.991276i	$-0.457924\pi$
$31$	1410.44	0.263603	0.131801	+	0.991276i	$0.457924\pi$
$32$	0	0
$33$	−8261.85	−1.32066
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3152.03	0.378517	0.189259	−	0.981927i	$-0.439392\pi$
$37$	3152.03	0.378517	0.189259	+	0.981927i	$0.439392\pi$
$38$	0	0
$39$	−28572.6	−3.00807
$40$	0	0
$41$	−13499.5	−1.25418	−0.627090	−	0.778947i	$-0.715754\pi$
$41$	−13499.5	−1.25418	−0.627090	+	0.778947i	$0.715754\pi$
$42$	0	0
$43$	5433.70	0.448151	0.224076	−	0.974572i	$-0.428064\pi$
$43$	5433.70	0.448151	0.224076	+	0.974572i	$0.428064\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3269.53	−0.215894	−0.107947	−	0.994157i	$-0.534428\pi$
$47$	−3269.53	−0.215894	−0.107947	+	0.994157i	$0.534428\pi$
$48$	0	0
$49$	3170.70	0.188654
$50$	0	0
$51$	−36640.7	−1.97260
$52$	0	0
$53$	−21421.3	−1.04750	−0.523752	−	0.851871i	$-0.675468\pi$
$53$	−21421.3	−1.04750	−0.523752	+	0.851871i	$0.675468\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4108.35	−0.167487
$58$	0	0
$59$	41403.1	1.54847	0.774235	−	0.632899i	$-0.218135\pi$
$59$	41403.1	1.54847	0.774235	+	0.632899i	$0.218135\pi$
$60$	0	0
$61$	−24373.9	−0.838688	−0.419344	−	0.907827i	$-0.637740\pi$
$61$	−24373.9	−0.838688	−0.419344	+	0.907827i	$0.637740\pi$
$62$	0	0
$63$	−94766.6	−3.00818
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−32307.0	−0.879245	−0.439623	−	0.898183i	$-0.644888\pi$
$67$	−32307.0	−0.879245	−0.439623	+	0.898183i	$0.644888\pi$
$68$	0	0
$69$	−82537.3	−2.08703
$70$	0	0
$71$	38675.0	0.910510	0.455255	−	0.890361i	$-0.349548\pi$
$71$	38675.0	0.910510	0.455255	+	0.890361i	$0.349548\pi$
$72$	0	0
$73$	−8083.79	−0.177545	−0.0887724	−	0.996052i	$-0.528294\pi$
$73$	−8083.79	−0.177545	−0.0887724	+	0.996052i	$0.528294\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	38636.8	0.742634
$78$	0	0
$79$	15979.3	0.288064	0.144032	−	0.989573i	$-0.453993\pi$
$79$	15979.3	0.288064	0.144032	+	0.989573i	$0.453993\pi$
$80$	0	0
$81$	227562.	3.85379
$82$	0	0
$83$	48494.1	0.772670	0.386335	−	0.922359i	$-0.373741\pi$
$83$	48494.1	0.772670	0.386335	+	0.922359i	$0.373741\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−93019.6	−1.31758
$88$	0	0
$89$	89187.5	1.19352	0.596759	−	0.802420i	$-0.296455\pi$
$89$	89187.5	1.19352	0.596759	+	0.802420i	$0.296455\pi$
$90$	0	0
$91$	133621.	1.69150
$92$	0	0
$93$	42628.7	0.511087
$94$	0	0
$95$	0	0
$96$	0	0
$97$	56822.2	0.613181	0.306590	−	0.951842i	$-0.400812\pi$
$97$	56822.2	0.613181	0.306590	+	0.951842i	$0.400812\pi$
$98$	0	0
$99$	−183279.	−1.87942
$100$	0	0
$101$	67419.2	0.657627	0.328814	−	0.944395i	$-0.393351\pi$
$101$	67419.2	0.657627	0.328814	+	0.944395i	$0.393351\pi$
$102$	0	0
$103$	61779.3	0.573786	0.286893	−	0.957963i	$-0.407378\pi$
$103$	61779.3	0.573786	0.286893	+	0.957963i	$0.407378\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−79150.3	−0.668333	−0.334167	−	0.942514i	$-0.608455\pi$
$107$	−79150.3	−0.668333	−0.334167	+	0.942514i	$0.608455\pi$
$108$	0	0
$109$	−177041.	−1.42728	−0.713638	−	0.700515i	$-0.752954\pi$
$109$	−177041.	−1.42728	−0.713638	+	0.700515i	$0.752954\pi$
$110$	0	0
$111$	95266.1	0.733890
$112$	0	0
$113$	−197220.	−1.45297	−0.726483	−	0.687184i	$-0.758847\pi$
$113$	−197220.	−1.45297	−0.726483	+	0.687184i	$0.758847\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−633847.	−4.28075
$118$	0	0
$119$	171352.	1.10923
$120$	0	0
$121$	−86327.4	−0.536025
$122$	0	0
$123$	−408007.	−2.43167
$124$	0	0
$125$	0	0
$126$	0	0
$127$	75992.4	0.418081	0.209041	−	0.977907i	$-0.432966\pi$
$127$	75992.4	0.418081	0.209041	+	0.977907i	$0.432966\pi$
$128$	0	0
$129$	164227.	0.868900
$130$	0	0
$131$	297561.	1.51495	0.757474	−	0.652866i	$-0.226433\pi$
$131$	297561.	1.51495	0.757474	+	0.652866i	$0.226433\pi$
$132$	0	0
$133$	19212.9	0.0941810
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−57190.7	−0.260330	−0.130165	−	0.991492i	$-0.541551\pi$
$137$	−57190.7	−0.260330	−0.130165	+	0.991492i	$0.541551\pi$
$138$	0	0
$139$	−297610.	−1.30651	−0.653253	−	0.757140i	$-0.726596\pi$
$139$	−297610.	−1.30651	−0.653253	+	0.757140i	$0.726596\pi$
$140$	0	0
$141$	−98817.6	−0.418588
$142$	0	0
$143$	258423.	1.05679
$144$	0	0
$145$	0	0
$146$	0	0
$147$	95830.4	0.365772
$148$	0	0
$149$	−271940.	−1.00348	−0.501739	−	0.865019i	$-0.667306\pi$
$149$	−271940.	−1.00348	−0.501739	+	0.865019i	$0.667306\pi$
$150$	0	0
$151$	−144441.	−0.515523	−0.257762	−	0.966209i	$-0.582985\pi$
$151$	−144441.	−0.515523	−0.257762	+	0.966209i	$0.582985\pi$
$152$	0	0
$153$	−812827.	−2.80717
$154$	0	0
$155$	0	0
$156$	0	0
$157$	407289.	1.31872	0.659361	−	0.751826i	$-0.270827\pi$
$157$	407289.	1.31872	0.659361	+	0.751826i	$0.270827\pi$
$158$	0	0
$159$	−647432.	−2.03096
$160$	0	0
$161$	385989.	1.17357
$162$	0	0
$163$	−473883.	−1.39702	−0.698509	−	0.715601i	$-0.746153\pi$
$163$	−473883.	−1.39702	−0.698509	+	0.715601i	$0.746153\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	291047.	0.807555	0.403777	−	0.914857i	$-0.367697\pi$
$167$	291047.	0.807555	0.403777	+	0.914857i	$0.367697\pi$
$168$	0	0
$169$	522431.	1.40706
$170$	0	0
$171$	−91138.5	−0.238348
$172$	0	0
$173$	133697.	0.339631	0.169815	−	0.985476i	$-0.445683\pi$
$173$	133697.	0.339631	0.169815	+	0.985476i	$0.445683\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.25136e6	3.00226
$178$	0	0
$179$	392572.	0.915770	0.457885	−	0.889012i	$-0.348607\pi$
$179$	392572.	0.915770	0.457885	+	0.889012i	$0.348607\pi$
$180$	0	0
$181$	−374040.	−0.848636	−0.424318	−	0.905513i	$-0.639486\pi$
$181$	−374040.	−0.848636	−0.424318	+	0.905513i	$0.639486\pi$
$182$	0	0
$183$	−736671.	−1.62609
$184$	0	0
$185$	0	0
$186$	0	0
$187$	331394.	0.693011
$188$	0	0
$189$	−1.82613e6	−3.71858
$190$	0	0
$191$	−993045.	−1.96963	−0.984817	−	0.173597i	$-0.944461\pi$
$191$	−993045.	−1.96963	−0.984817	+	0.173597i	$0.944461\pi$
$192$	0	0
$193$	−132234.	−0.255535	−0.127768	−	0.991804i	$-0.540781\pi$
$193$	−132234.	−0.255535	−0.127768	+	0.991804i	$0.540781\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.00791e6	−1.85035	−0.925177	−	0.379535i	$-0.876084\pi$
$197$	−1.00791e6	−1.85035	−0.925177	+	0.379535i	$0.876084\pi$
$198$	0	0
$199$	756589.	1.35434	0.677169	−	0.735827i	$-0.263206\pi$
$199$	756589.	1.35434	0.677169	+	0.735827i	$0.263206\pi$
$200$	0	0
$201$	−976439.	−1.70473
$202$	0	0
$203$	435010.	0.740899
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.83098e6	−2.97002
$208$	0	0
$209$	37157.7	0.0588414
$210$	0	0
$211$	−446900.	−0.691041	−0.345521	−	0.938411i	$-0.612298\pi$
$211$	−446900.	−0.691041	−0.345521	+	0.938411i	$0.612298\pi$
$212$	0	0
$213$	1.16890e6	1.76535
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−199355.	−0.287394
$218$	0	0
$219$	−244323.	−0.344234
$220$	0	0
$221$	1.14609e6	1.57847
$222$	0	0
$223$	−1.28822e6	−1.73471	−0.867357	−	0.497687i	$-0.834183\pi$
$223$	−1.28822e6	−1.73471	−0.867357	+	0.497687i	$0.834183\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−109761.	−0.141379	−0.0706894	−	0.997498i	$-0.522520\pi$
$227$	−109761.	−0.141379	−0.0706894	+	0.997498i	$0.522520\pi$
$228$	0	0
$229$	1.34217e6	1.69130	0.845648	−	0.533741i	$-0.179214\pi$
$229$	1.34217e6	1.69130	0.845648	+	0.533741i	$0.179214\pi$
$230$	0	0
$231$	1.16775e6	1.43986
$232$	0	0
$233$	−987929.	−1.19216	−0.596082	−	0.802924i	$-0.703277\pi$
$233$	−987929.	−1.19216	−0.596082	+	0.802924i	$0.703277\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	482953.	0.558514
$238$	0	0
$239$	78478.1	0.0888697	0.0444349	−	0.999012i	$-0.485851\pi$
$239$	78478.1	0.0888697	0.0444349	+	0.999012i	$0.485851\pi$
$240$	0	0
$241$	1.40078e6	1.55356	0.776780	−	0.629772i	$-0.216852\pi$
$241$	1.40078e6	1.55356	0.776780	+	0.629772i	$0.216852\pi$
$242$	0	0
$243$	3.73825e6	4.06118
$244$	0	0
$245$	0	0
$246$	0	0
$247$	128505.	0.134023
$248$	0	0
$249$	1.46567e6	1.49809
$250$	0	0
$251$	−181711.	−0.182053	−0.0910264	−	0.995848i	$-0.529015\pi$
$251$	−181711.	−0.182053	−0.0910264	+	0.995848i	$0.529015\pi$
$252$	0	0
$253$	746502.	0.733212
$254$	0	0
$255$	0	0
$256$	0	0
$257$	267824.	0.252940	0.126470	−	0.991970i	$-0.459635\pi$
$257$	267824.	0.252940	0.126470	+	0.991970i	$0.459635\pi$
$258$	0	0
$259$	−445516.	−0.412680
$260$	0	0
$261$	−2.06352e6	−1.87503
$262$	0	0
$263$	1.11398e6	0.993090	0.496545	−	0.868011i	$-0.334602\pi$
$263$	1.11398e6	0.993090	0.496545	+	0.868011i	$0.334602\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.69558e6	2.31406
$268$	0	0
$269$	124320.	0.104751	0.0523757	−	0.998627i	$-0.483321\pi$
$269$	124320.	0.104751	0.0523757	+	0.998627i	$0.483321\pi$
$270$	0	0
$271$	1.81145e6	1.49832	0.749158	−	0.662391i	$-0.230458\pi$
$271$	1.81145e6	1.49832	0.749158	+	0.662391i	$0.230458\pi$
$272$	0	0
$273$	4.03853e6	3.27957
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−544092.	−0.426062	−0.213031	−	0.977045i	$-0.568334\pi$
$277$	−544092.	−0.426062	−0.213031	+	0.977045i	$0.568334\pi$
$278$	0	0
$279$	945663.	0.727321
$280$	0	0
$281$	1.18452e6	0.894902	0.447451	−	0.894308i	$-0.352332\pi$
$281$	1.18452e6	0.894902	0.447451	+	0.894308i	$0.352332\pi$
$282$	0	0
$283$	−999650.	−0.741963	−0.370981	−	0.928640i	$-0.620979\pi$
$283$	−999650.	−0.741963	−0.370981	+	0.928640i	$0.620979\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.90806e6	1.36737
$288$	0	0
$289$	49850.7	0.0351097
$290$	0	0
$291$	1.71738e6	1.18887
$292$	0	0
$293$	−2.56682e6	−1.74673	−0.873365	−	0.487065i	$-0.838067\pi$
$293$	−2.56682e6	−1.74673	−0.873365	+	0.487065i	$0.838067\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.53173e6	−2.32325
$298$	0	0
$299$	2.58169e6	1.67004
$300$	0	0
$301$	−768013.	−0.488598
$302$	0	0
$303$	2.03766e6	1.27504
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−925658.	−0.560537	−0.280269	−	0.959922i	$-0.590424\pi$
$307$	−925658.	−0.560537	−0.280269	+	0.959922i	$0.590424\pi$
$308$	0	0
$309$	1.86720e6	1.11249
$310$	0	0
$311$	−2.46013e6	−1.44231	−0.721153	−	0.692776i	$-0.756387\pi$
$311$	−2.46013e6	−1.44231	−0.721153	+	0.692776i	$0.756387\pi$
$312$	0	0
$313$	1.43344e6	0.827025	0.413513	−	0.910498i	$-0.364302\pi$
$313$	1.43344e6	0.827025	0.413513	+	0.910498i	$0.364302\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	400932.	0.224090	0.112045	−	0.993703i	$-0.464260\pi$
$317$	400932.	0.224090	0.112045	+	0.993703i	$0.464260\pi$
$318$	0	0
$319$	841308.	0.462891
$320$	0	0
$321$	−2.39222e6	−1.29580
$322$	0	0
$323$	164792.	0.0878878
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.35084e6	−2.76728
$328$	0	0
$329$	462124.	0.235380
$330$	0	0
$331$	337080.	0.169108	0.0845539	−	0.996419i	$-0.473053\pi$
$331$	337080.	0.169108	0.0845539	+	0.996419i	$0.473053\pi$
$332$	0	0
$333$	2.11336e6	1.04439
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.52668e6	1.21192	0.605962	−	0.795494i	$-0.292788\pi$
$337$	2.52668e6	1.21192	0.605962	+	0.795494i	$0.292788\pi$
$338$	0	0
$339$	−5.96074e6	−2.81709
$340$	0	0
$341$	−385552.	−0.179555
$342$	0	0
$343$	1.92739e6	0.884574
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.74945e6	1.22581	0.612904	−	0.790158i	$-0.290001\pi$
$347$	2.74945e6	1.22581	0.612904	+	0.790158i	$0.290001\pi$
$348$	0	0
$349$	2.70732e6	1.18981	0.594903	−	0.803797i	$-0.297190\pi$
$349$	2.70732e6	1.18981	0.594903	+	0.803797i	$0.297190\pi$
$350$	0	0
$351$	−1.22141e7	−5.29167
$352$	0	0
$353$	−3.20305e6	−1.36813	−0.684064	−	0.729422i	$-0.739789\pi$
$353$	−3.20305e6	−1.36813	−0.684064	+	0.729422i	$0.739789\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.17889e6	2.15063
$358$	0	0
$359$	4.39968e6	1.80171	0.900855	−	0.434121i	$-0.142941\pi$
$359$	4.39968e6	1.80171	0.900855	+	0.434121i	$0.142941\pi$
$360$	0	0
$361$	−2.45762e6	−0.992538
$362$	0	0
$363$	−2.60914e6	−1.03927
$364$	0	0
$365$	0	0
$366$	0	0
$367$	884314.	0.342721	0.171361	−	0.985208i	$-0.445184\pi$
$367$	884314.	0.342721	0.171361	+	0.985208i	$0.445184\pi$
$368$	0	0
$369$	−9.05111e6	−3.46048
$370$	0	0
$371$	3.02774e6	1.14205
$372$	0	0
$373$	−1.02175e6	−0.380251	−0.190126	−	0.981760i	$-0.560890\pi$
$373$	−1.02175e6	−0.380251	−0.190126	+	0.981760i	$0.560890\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.90956e6	1.05433
$378$	0	0
$379$	−623582.	−0.222995	−0.111498	−	0.993765i	$-0.535565\pi$
$379$	−623582.	−0.222995	−0.111498	+	0.993765i	$0.535565\pi$
$380$	0	0
$381$	2.29677e6	0.810599
$382$	0	0
$383$	−2.60949e6	−0.908988	−0.454494	−	0.890750i	$-0.650180\pi$
$383$	−2.60949e6	−0.908988	−0.454494	+	0.890750i	$0.650180\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.64316e6	1.23652
$388$	0	0
$389$	−1.48187e6	−0.496520	−0.248260	−	0.968693i	$-0.579859\pi$
$389$	−1.48187e6	−0.496520	−0.248260	+	0.968693i	$0.579859\pi$
$390$	0	0
$391$	3.31068e6	1.09516
$392$	0	0
$393$	8.99340e6	2.93726
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.62187e6	1.79021	0.895106	−	0.445853i	$-0.147100\pi$
$397$	5.62187e6	1.79021	0.895106	+	0.445853i	$0.147100\pi$
$398$	0	0
$399$	580685.	0.182603
$400$	0	0
$401$	−3.64060e6	−1.13061	−0.565304	−	0.824883i	$-0.691241\pi$
$401$	−3.64060e6	−1.13061	−0.565304	+	0.824883i	$0.691241\pi$
$402$	0	0
$403$	−1.33339e6	−0.408972
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−861627.	−0.257830
$408$	0	0
$409$	−3.12488e6	−0.923687	−0.461844	−	0.886961i	$-0.652812\pi$
$409$	−3.12488e6	−0.923687	−0.461844	+	0.886961i	$0.652812\pi$
$410$	0	0
$411$	−1.72852e6	−0.504742
$412$	0	0
$413$	−5.85201e6	−1.68822
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.99490e6	−2.53312
$418$	0	0
$419$	3.94722e6	1.09839	0.549195	−	0.835694i	$-0.314935\pi$
$419$	3.94722e6	1.09839	0.549195	+	0.835694i	$0.314935\pi$
$420$	0	0
$421$	3.97485e6	1.09299	0.546494	−	0.837463i	$-0.315962\pi$
$421$	3.97485e6	1.09299	0.546494	+	0.837463i	$0.315962\pi$
$422$	0	0
$423$	−2.19214e6	−0.595686
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.44507e6	0.914383
$428$	0	0
$429$	7.81050e6	2.04897
$430$	0	0
$431$	−1.02193e6	−0.264989	−0.132495	−	0.991184i	$-0.542299\pi$
$431$	−1.02193e6	−0.264989	−0.132495	+	0.991184i	$0.542299\pi$
$432$	0	0
$433$	−4.89035e6	−1.25349	−0.626744	−	0.779225i	$-0.715613\pi$
$433$	−4.89035e6	−1.25349	−0.626744	+	0.779225i	$0.715613\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	371212.	0.0929861
$438$	0	0
$439$	2.44464e6	0.605414	0.302707	−	0.953084i	$-0.402110\pi$
$439$	2.44464e6	0.605414	0.302707	+	0.953084i	$0.402110\pi$
$440$	0	0
$441$	2.12587e6	0.520525
$442$	0	0
$443$	3.84682e6	0.931308	0.465654	−	0.884967i	$-0.345819\pi$
$443$	3.84682e6	0.931308	0.465654	+	0.884967i	$0.345819\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.21906e6	−1.94560
$448$	0	0
$449$	6.57119e6	1.53825	0.769127	−	0.639096i	$-0.220691\pi$
$449$	6.57119e6	1.53825	0.769127	+	0.639096i	$0.220691\pi$
$450$	0	0
$451$	3.69019e6	0.854293
$452$	0	0
$453$	−4.36555e6	−0.999524
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.79203e6	−0.625359	−0.312679	−	0.949859i	$-0.601227\pi$
$457$	−2.79203e6	−0.625359	−0.312679	+	0.949859i	$0.601227\pi$
$458$	0	0
$459$	−1.56630e7	−3.47011
$460$	0	0
$461$	−272304.	−0.0596762	−0.0298381	−	0.999555i	$-0.509499\pi$
$461$	−272304.	−0.0596762	−0.0298381	+	0.999555i	$0.509499\pi$
$462$	0	0
$463$	−4.71209e6	−1.02155	−0.510777	−	0.859713i	$-0.670642\pi$
$463$	−4.71209e6	−1.02155	−0.510777	+	0.859713i	$0.670642\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.42758e6	−1.15163	−0.575816	−	0.817579i	$-0.695316\pi$
$467$	−5.42758e6	−1.15163	−0.575816	+	0.817579i	$0.695316\pi$
$468$	0	0
$469$	4.56636e6	0.958600
$470$	0	0
$471$	1.23098e7	2.55681
$472$	0	0
$473$	−1.48534e6	−0.305261
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.43624e7	−2.89023
$478$	0	0
$479$	−4.54900e6	−0.905893	−0.452947	−	0.891538i	$-0.649627\pi$
$479$	−4.54900e6	−0.905893	−0.452947	+	0.891538i	$0.649627\pi$
$480$	0	0
$481$	−2.97983e6	−0.587259
$482$	0	0
$483$	1.16660e7	2.27539
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.52219e6	−0.481897	−0.240949	−	0.970538i	$-0.577459\pi$
$487$	−2.52219e6	−0.481897	−0.240949	+	0.970538i	$0.577459\pi$
$488$	0	0
$489$	−1.43225e7	−2.70861
$490$	0	0
$491$	−4.33529e6	−0.811548	−0.405774	−	0.913973i	$-0.632998\pi$
$491$	−4.33529e6	−0.811548	−0.405774	+	0.913973i	$0.632998\pi$
$492$	0	0
$493$	3.73114e6	0.691392
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.46642e6	−0.992687
$498$	0	0
$499$	6.31959e6	1.13615	0.568077	−	0.822975i	$-0.307687\pi$
$499$	6.31959e6	1.13615	0.568077	+	0.822975i	$0.307687\pi$
$500$	0	0
$501$	8.79653e6	1.56573
$502$	0	0
$503$	−8.09276e6	−1.42619	−0.713094	−	0.701068i	$-0.752707\pi$
$503$	−8.09276e6	−1.42619	−0.713094	+	0.701068i	$0.752707\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.57898e7	2.72808
$508$	0	0
$509$	2.00658e6	0.343291	0.171645	−	0.985159i	$-0.445092\pi$
$509$	2.00658e6	0.343291	0.171645	+	0.985159i	$0.445092\pi$
$510$	0	0
$511$	1.14258e6	0.193569
$512$	0	0
$513$	−1.75622e6	−0.294636
$514$	0	0
$515$	0	0
$516$	0	0
$517$	893748.	0.147058
$518$	0	0
$519$	4.04083e6	0.658495
$520$	0	0
$521$	2.22441e6	0.359021	0.179511	−	0.983756i	$-0.442549\pi$
$521$	2.22441e6	0.359021	0.179511	+	0.983756i	$0.442549\pi$
$522$	0	0
$523$	3.88746e6	0.621459	0.310729	−	0.950498i	$-0.399427\pi$
$523$	3.88746e6	0.621459	0.310729	+	0.950498i	$0.399427\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.70989e6	−0.268190
$528$	0	0
$529$	1.02135e6	0.158684
$530$	0	0
$531$	2.77597e7	4.27247
$532$	0	0
$533$	1.27621e7	1.94582
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.18650e7	1.77554
$538$	0	0
$539$	−866730.	−0.128503
$540$	0	0
$541$	8.65144e6	1.27085	0.635426	−	0.772162i	$-0.280824\pi$
$541$	8.65144e6	1.27085	0.635426	+	0.772162i	$0.280824\pi$
$542$	0	0
$543$	−1.13049e7	−1.64538
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.36845e6	0.767150	0.383575	−	0.923510i	$-0.374693\pi$
$547$	5.36845e6	0.767150	0.383575	+	0.923510i	$0.374693\pi$
$548$	0	0
$549$	−1.63421e7	−2.31407
$550$	0	0
$551$	418356.	0.0587039
$552$	0	0
$553$	−2.25855e6	−0.314063
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.41362e6	−0.875922	−0.437961	−	0.898994i	$-0.644299\pi$
$557$	−6.41362e6	−0.875922	−0.437961	+	0.898994i	$0.644299\pi$
$558$	0	0
$559$	−5.13686e6	−0.695293
$560$	0	0
$561$	1.00160e7	1.34365
$562$	0	0
$563$	−2.48424e6	−0.330311	−0.165155	−	0.986268i	$-0.552813\pi$
$563$	−2.48424e6	−0.330311	−0.165155	+	0.986268i	$0.552813\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.21642e7	−4.20161
$568$	0	0
$569$	5.11183e6	0.661905	0.330953	−	0.943647i	$-0.392630\pi$
$569$	5.11183e6	0.661905	0.330953	+	0.943647i	$0.392630\pi$
$570$	0	0
$571$	−1.13467e7	−1.45640	−0.728200	−	0.685365i	$-0.759643\pi$
$571$	−1.13467e7	−1.45640	−0.728200	+	0.685365i	$0.759643\pi$
$572$	0	0
$573$	−3.00135e7	−3.81883
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.90331e6	−0.237996	−0.118998	−	0.992895i	$-0.537968\pi$
$577$	−1.90331e6	−0.237996	−0.118998	+	0.992895i	$0.537968\pi$
$578$	0	0
$579$	−3.99661e6	−0.495445
$580$	0	0
$581$	−6.85428e6	−0.842406
$582$	0	0
$583$	5.85564e6	0.713515
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.64386e6	−0.316697	−0.158348	−	0.987383i	$-0.550617\pi$
$587$	−2.64386e6	−0.316697	−0.158348	+	0.987383i	$0.550617\pi$
$588$	0	0
$589$	−191723.	−0.0227712
$590$	0	0
$591$	−3.04627e7	−3.58757
$592$	0	0
$593$	3.02064e6	0.352746	0.176373	−	0.984323i	$-0.443563\pi$
$593$	3.02064e6	0.352746	0.176373	+	0.984323i	$0.443563\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.28670e7	2.62587
$598$	0	0
$599$	336944.	0.0383699	0.0191849	−	0.999816i	$-0.493893\pi$
$599$	336944.	0.0383699	0.0191849	+	0.999816i	$0.493893\pi$
$600$	0	0
$601$	1.06160e7	1.19887	0.599436	−	0.800422i	$-0.295391\pi$
$601$	1.06160e7	1.19887	0.599436	+	0.800422i	$0.295391\pi$
$602$	0	0
$603$	−2.16611e7	−2.42597
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.71997e6	0.960602	0.480301	−	0.877104i	$-0.340528\pi$
$607$	8.71997e6	0.960602	0.480301	+	0.877104i	$0.340528\pi$
$608$	0	0
$609$	1.31476e7	1.43649
$610$	0	0
$611$	3.09092e6	0.334954
$612$	0	0
$613$	1.32783e6	0.142722	0.0713611	−	0.997451i	$-0.477266\pi$
$613$	1.32783e6	0.142722	0.0713611	+	0.997451i	$0.477266\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.38634e6	0.886869	0.443435	−	0.896307i	$-0.353760\pi$
$617$	8.38634e6	0.886869	0.443435	+	0.896307i	$0.353760\pi$
$618$	0	0
$619$	−73772.8	−0.00773873	−0.00386936	−	0.999993i	$-0.501232\pi$
$619$	−73772.8	−0.00773873	−0.00386936	+	0.999993i	$0.501232\pi$
$620$	0	0
$621$	−3.52827e7	−3.67140
$622$	0	0
$623$	−1.26060e7	−1.30124
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.12304e6	0.114085
$628$	0	0
$629$	−3.82125e6	−0.385105
$630$	0	0
$631$	−1.03680e7	−1.03663	−0.518313	−	0.855191i	$-0.673440\pi$
$631$	−1.03680e7	−1.03663	−0.518313	+	0.855191i	$0.673440\pi$
$632$	0	0
$633$	−1.35070e7	−1.33983
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.99748e6	−0.292690
$638$	0	0
$639$	2.59306e7	2.51224
$640$	0	0
$641$	−2.77488e6	−0.266746	−0.133373	−	0.991066i	$-0.542581\pi$
$641$	−2.77488e6	−0.266746	−0.133373	+	0.991066i	$0.542581\pi$
$642$	0	0
$643$	−3.99310e6	−0.380875	−0.190438	−	0.981699i	$-0.560991\pi$
$643$	−3.99310e6	−0.380875	−0.190438	+	0.981699i	$0.560991\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.30149e6	−0.216147	−0.108073	−	0.994143i	$-0.534468\pi$
$647$	−2.30149e6	−0.216147	−0.108073	+	0.994143i	$0.534468\pi$
$648$	0	0
$649$	−1.13178e7	−1.05475
$650$	0	0
$651$	−6.02525e6	−0.557215
$652$	0	0
$653$	1.94361e7	1.78371	0.891857	−	0.452317i	$-0.149402\pi$
$653$	1.94361e7	1.78371	0.891857	+	0.452317i	$0.149402\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.41998e6	−0.489874
$658$	0	0
$659$	−1.05797e7	−0.948982	−0.474491	−	0.880260i	$-0.657368\pi$
$659$	−1.05797e7	−0.948982	−0.474491	+	0.880260i	$0.657368\pi$
$660$	0	0
$661$	1.01568e6	0.0904180	0.0452090	−	0.998978i	$-0.485605\pi$
$661$	1.01568e6	0.0904180	0.0452090	+	0.998978i	$0.485605\pi$
$662$	0	0
$663$	3.46390e7	3.06043
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.40482e6	0.731499
$668$	0	0
$669$	−3.89348e7	−3.36336
$670$	0	0
$671$	6.66276e6	0.571278
$672$	0	0
$673$	−5.87188e6	−0.499735	−0.249867	−	0.968280i	$-0.580387\pi$
$673$	−5.87188e6	−0.499735	−0.249867	+	0.968280i	$0.580387\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.10597e7	−1.76596	−0.882978	−	0.469415i	$-0.844465\pi$
$677$	−2.10597e7	−1.76596	−0.882978	+	0.469415i	$0.844465\pi$
$678$	0	0
$679$	−8.03139e6	−0.668523
$680$	0	0
$681$	−3.31739e6	−0.274113
$682$	0	0
$683$	−2.05169e7	−1.68290	−0.841452	−	0.540331i	$-0.818299\pi$
$683$	−2.05169e7	−1.68290	−0.841452	+	0.540331i	$0.818299\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.05655e7	3.27918
$688$	0	0
$689$	2.02510e7	1.62517
$690$	0	0
$691$	1.61554e7	1.28713	0.643565	−	0.765392i	$-0.277455\pi$
$691$	1.61554e7	1.28713	0.643565	+	0.765392i	$0.277455\pi$
$692$	0	0
$693$	2.59050e7	2.04904
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.63657e7	1.27601
$698$	0	0
$699$	−2.98589e7	−2.31143
$700$	0	0
$701$	6.60014e6	0.507292	0.253646	−	0.967297i	$-0.418370\pi$
$701$	6.60014e6	0.507292	0.253646	+	0.967297i	$0.418370\pi$
$702$	0	0
$703$	−428460.	−0.0326980
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.52919e6	−0.716981
$708$	0	0
$709$	−2.10409e6	−0.157199	−0.0785995	−	0.996906i	$-0.525045\pi$
$709$	−2.10409e6	−0.157199	−0.0785995	+	0.996906i	$0.525045\pi$
$710$	0	0
$711$	1.07137e7	0.794814
$712$	0	0
$713$	−3.85173e6	−0.283748
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.37190e6	0.172305
$718$	0	0
$719$	1.86112e6	0.134262	0.0671309	−	0.997744i	$-0.478615\pi$
$719$	1.86112e6	0.134262	0.0671309	+	0.997744i	$0.478615\pi$
$720$	0	0
$721$	−8.73203e6	−0.625572
$722$	0	0
$723$	4.23369e7	3.01213
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.05408e7	−0.739671	−0.369836	−	0.929097i	$-0.620586\pi$
$727$	−1.05408e7	−0.739671	−0.369836	+	0.929097i	$0.620586\pi$
$728$	0	0
$729$	5.76863e7	4.02026
$730$	0	0
$731$	−6.58736e6	−0.455950
$732$	0	0
$733$	−1.47763e7	−1.01579	−0.507897	−	0.861418i	$-0.669577\pi$
$733$	−1.47763e7	−1.01579	−0.507897	+	0.861418i	$0.669577\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.83133e6	0.598904
$738$	0	0
$739$	3.19319e6	0.215087	0.107543	−	0.994200i	$-0.465702\pi$
$739$	3.19319e6	0.215087	0.107543	+	0.994200i	$0.465702\pi$
$740$	0	0
$741$	3.88391e6	0.259851
$742$	0	0
$743$	1.58587e7	1.05389	0.526945	−	0.849900i	$-0.323337\pi$
$743$	1.58587e7	1.05389	0.526945	+	0.849900i	$0.323337\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.25141e7	2.13192
$748$	0	0
$749$	1.11873e7	0.728653
$750$	0	0
$751$	−2.14341e7	−1.38677	−0.693387	−	0.720566i	$-0.743882\pi$
$751$	−2.14341e7	−1.38677	−0.693387	+	0.720566i	$0.743882\pi$
$752$	0	0
$753$	−5.49199e6	−0.352974
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.29941e7	−1.45840	−0.729201	−	0.684299i	$-0.760108\pi$
$757$	−2.29941e7	−1.45840	−0.729201	+	0.684299i	$0.760108\pi$
$758$	0	0
$759$	2.25621e7	1.42159
$760$	0	0
$761$	−1.59525e7	−0.998547	−0.499273	−	0.866445i	$-0.666400\pi$
$761$	−1.59525e7	−0.998547	−0.499273	+	0.866445i	$0.666400\pi$
$762$	0	0
$763$	2.50234e7	1.55609
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.91412e7	−2.40240
$768$	0	0
$769$	−1.41126e7	−0.860577	−0.430288	−	0.902691i	$-0.641588\pi$
$769$	−1.41126e7	−0.860577	−0.430288	+	0.902691i	$0.641588\pi$
$770$	0	0
$771$	8.09466e6	0.490414
$772$	0	0
$773$	9.05372e6	0.544977	0.272489	−	0.962159i	$-0.412153\pi$
$773$	9.05372e6	0.544977	0.272489	+	0.962159i	$0.412153\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.34652e7	−0.800127
$778$	0	0
$779$	1.83501e6	0.108342
$780$	0	0
$781$	−1.05721e7	−0.620200
$782$	0	0
$783$	−3.97636e7	−2.31783
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.33625e6	−0.192009	−0.0960046	−	0.995381i	$-0.530606\pi$
$787$	−3.33625e6	−0.192009	−0.0960046	+	0.995381i	$0.530606\pi$
$788$	0	0
$789$	3.36687e7	1.92546
$790$	0	0
$791$	2.78756e7	1.58410
$792$	0	0
$793$	2.30424e7	1.30120
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.58206e7	−0.882219	−0.441110	−	0.897453i	$-0.645415\pi$
$797$	−1.58206e7	−0.882219	−0.441110	+	0.897453i	$0.645415\pi$
$798$	0	0
$799$	3.96371e6	0.219652
$800$	0	0
$801$	5.97980e7	3.29310
$802$	0	0
$803$	2.20976e6	0.120936
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.75741e6	0.203098
$808$	0	0
$809$	1.69505e7	0.910564	0.455282	−	0.890347i	$-0.349538\pi$
$809$	1.69505e7	0.910564	0.455282	+	0.890347i	$0.349538\pi$
$810$	0	0
$811$	2.93880e6	0.156898	0.0784491	−	0.996918i	$-0.475003\pi$
$811$	2.93880e6	0.156898	0.0784491	+	0.996918i	$0.475003\pi$
$812$	0	0
$813$	5.47489e7	2.90502
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−738610.	−0.0387133
$818$	0	0
$819$	8.95895e7	4.66710
$820$	0	0
$821$	−3.13058e7	−1.62094	−0.810469	−	0.585781i	$-0.800788\pi$
$821$	−3.13058e7	−1.62094	−0.810469	+	0.585781i	$0.800788\pi$
$822$	0	0
$823$	1.19264e7	0.613774	0.306887	−	0.951746i	$-0.400713\pi$
$823$	1.19264e7	0.613774	0.306887	+	0.951746i	$0.400713\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.96149e7	−0.997292	−0.498646	−	0.866806i	$-0.666169\pi$
$827$	−1.96149e7	−0.997292	−0.498646	+	0.866806i	$0.666169\pi$
$828$	0	0
$829$	−3.33477e7	−1.68531	−0.842656	−	0.538453i	$-0.819009\pi$
$829$	−3.33477e7	−1.68531	−0.842656	+	0.538453i	$0.819009\pi$
$830$	0	0
$831$	−1.64445e7	−0.826073
$832$	0	0
$833$	−3.84389e6	−0.191937
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.82227e7	0.899082
$838$	0	0
$839$	1.78255e7	0.874254	0.437127	−	0.899400i	$-0.355996\pi$
$839$	1.78255e7	0.874254	0.437127	+	0.899400i	$0.355996\pi$
$840$	0	0
$841$	−1.10389e7	−0.538191
$842$	0	0
$843$	3.58006e7	1.73509
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.22017e7	0.584403
$848$	0	0
$849$	−3.02132e7	−1.43856
$850$	0	0
$851$	−8.60781e6	−0.407445
$852$	0	0
$853$	−2.37711e6	−0.111860	−0.0559302	−	0.998435i	$-0.517812\pi$
$853$	−2.37711e6	−0.111860	−0.0559302	+	0.998435i	$0.517812\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.68298e6	0.264316	0.132158	−	0.991229i	$-0.457809\pi$
$857$	5.68298e6	0.264316	0.132158	+	0.991229i	$0.457809\pi$
$858$	0	0
$859$	−2.02691e7	−0.937240	−0.468620	−	0.883400i	$-0.655249\pi$
$859$	−2.02691e7	−0.937240	−0.468620	+	0.883400i	$0.655249\pi$
$860$	0	0
$861$	5.76687e7	2.65114
$862$	0	0
$863$	5.83084e6	0.266504	0.133252	−	0.991082i	$-0.457458\pi$
$863$	5.83084e6	0.266504	0.133252	+	0.991082i	$0.457458\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.50668e6	0.0680725
$868$	0	0
$869$	−4.36803e6	−0.196217
$870$	0	0
$871$	3.05421e7	1.36412
$872$	0	0
$873$	3.80978e7	1.69186
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.35496e7	1.91199	0.955995	−	0.293383i	$-0.0947812\pi$
$877$	4.35496e7	1.91199	0.955995	+	0.293383i	$0.0947812\pi$
$878$	0	0
$879$	−7.75789e7	−3.38666
$880$	0	0
$881$	−3.38261e7	−1.46829	−0.734146	−	0.678992i	$-0.762417\pi$
$881$	−3.38261e7	−1.46829	−0.734146	+	0.678992i	$0.762417\pi$
$882$	0	0
$883$	1.15279e7	0.497562	0.248781	−	0.968560i	$-0.419970\pi$
$883$	1.15279e7	0.497562	0.248781	+	0.968560i	$0.419970\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.62136e7	−1.11871	−0.559354	−	0.828929i	$-0.688951\pi$
$887$	−2.62136e7	−1.11871	−0.559354	+	0.828929i	$0.688951\pi$
$888$	0	0
$889$	−1.07409e7	−0.455815
$890$	0	0
$891$	−6.22056e7	−2.62503
$892$	0	0
$893$	444432.	0.0186499
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.80283e7	3.23796
$898$	0	0
$899$	−4.34090e6	−0.179135
$900$	0	0
$901$	2.59694e7	1.06573
$902$	0	0
$903$	−2.32122e7	−0.947321
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.88118e7	−0.759296	−0.379648	−	0.925131i	$-0.623955\pi$
$907$	−1.88118e7	−0.759296	−0.379648	+	0.925131i	$0.623955\pi$
$908$	0	0
$909$	4.52029e7	1.81450
$910$	0	0
$911$	−2.51622e7	−1.00451	−0.502253	−	0.864721i	$-0.667495\pi$
$911$	−2.51622e7	−1.00451	−0.502253	+	0.864721i	$0.667495\pi$
$912$	0	0
$913$	−1.32562e7	−0.526310
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.20580e7	−1.65168
$918$	0	0
$919$	−2.96599e7	−1.15846	−0.579230	−	0.815164i	$-0.696647\pi$
$919$	−2.96599e7	−1.15846	−0.579230	+	0.815164i	$0.696647\pi$
$920$	0	0
$921$	−2.79768e7	−1.08680
$922$	0	0
$923$	−3.65622e7	−1.41263
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.14214e7	1.58316
$928$	0	0
$929$	−4.31730e7	−1.64124	−0.820622	−	0.571471i	$-0.806373\pi$
$929$	−4.31730e7	−1.64124	−0.820622	+	0.571471i	$0.806373\pi$
$930$	0	0
$931$	−430997.	−0.0162967
$932$	0	0
$933$	−7.43544e7	−2.79642
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.49826e6	−0.167377	−0.0836885	−	0.996492i	$-0.526670\pi$
$937$	−4.49826e6	−0.167377	−0.0836885	+	0.996492i	$0.526670\pi$
$938$	0	0
$939$	4.33239e7	1.60348
$940$	0	0
$941$	1.01968e7	0.375395	0.187697	−	0.982227i	$-0.439898\pi$
$941$	1.01968e7	0.375395	0.187697	+	0.982227i	$0.439898\pi$
$942$	0	0
$943$	3.68656e7	1.35003
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.08376e7	0.392696	0.196348	−	0.980534i	$-0.437092\pi$
$947$	1.08376e7	0.392696	0.196348	+	0.980534i	$0.437092\pi$
$948$	0	0
$949$	7.64218e6	0.275456
$950$	0	0
$951$	1.21177e7	0.434477
$952$	0	0
$953$	−4.39870e7	−1.56889	−0.784444	−	0.620199i	$-0.787052\pi$
$953$	−4.39870e7	−1.56889	−0.784444	+	0.620199i	$0.787052\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.54275e7	0.897478
$958$	0	0
$959$	8.08347e6	0.283826
$960$	0	0
$961$	−2.66398e7	−0.930514
$962$	0	0
$963$	−5.30683e7	−1.84404
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.72431e7	0.592992	0.296496	−	0.955034i	$-0.404182\pi$
$967$	1.72431e7	0.592992	0.296496	+	0.955034i	$0.404182\pi$
$968$	0	0
$969$	4.98062e6	0.170402
$970$	0	0
$971$	−5.11158e7	−1.73983	−0.869915	−	0.493201i	$-0.835827\pi$
$971$	−5.11158e7	−1.73983	−0.869915	+	0.493201i	$0.835827\pi$
$972$	0	0
$973$	4.20650e7	1.42442
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.28599e7	−1.77170	−0.885849	−	0.463974i	$-0.846423\pi$
$977$	−5.28599e7	−1.77170	−0.885849	+	0.463974i	$0.846423\pi$
$978$	0	0
$979$	−2.43800e7	−0.812973
$980$	0	0
$981$	−1.18702e8	−3.93808
$982$	0	0
$983$	1.40562e6	0.0463965	0.0231983	−	0.999731i	$-0.492615\pi$
$983$	1.40562e6	0.0463965	0.0231983	+	0.999731i	$0.492615\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.39671e7	0.456367
$988$	0	0
$989$	−1.48388e7	−0.482400
$990$	0	0
$991$	3.12457e6	0.101066	0.0505331	−	0.998722i	$-0.483908\pi$
$991$	3.12457e6	0.101066	0.0505331	+	0.998722i	$0.483908\pi$
$992$	0	0
$993$	1.01878e7	0.327875
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.07225e7	−0.341630	−0.170815	−	0.985303i	$-0.554640\pi$
$997$	−1.07225e7	−0.341630	−0.170815	+	0.985303i	$0.554640\pi$
$998$	0	0
$999$	4.07239e7	1.29103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.v.1.2		2
4.3	odd	2		100.6.a.c.1.1	✓	2
5.2	odd	4		400.6.c.m.49.1		4
5.3	odd	4		400.6.c.m.49.4		4
5.4	even	2		400.6.a.p.1.1		2
12.11	even	2		900.6.a.s.1.2		2
20.3	even	4		100.6.c.c.49.1		4
20.7	even	4		100.6.c.c.49.4		4
20.19	odd	2		100.6.a.e.1.2	yes	2
60.23	odd	4		900.6.d.m.649.1		4
60.47	odd	4		900.6.d.m.649.4		4
60.59	even	2		900.6.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.6.a.c.1.1	✓	2	4.3	odd	2
100.6.a.e.1.2	yes	2	20.19	odd	2
100.6.c.c.49.1		4	20.3	even	4
100.6.c.c.49.4		4	20.7	even	4
400.6.a.p.1.1		2	5.4	even	2
400.6.a.v.1.2		2	1.1	even	1	trivial
400.6.c.m.49.1		4	5.2	odd	4
400.6.c.m.49.4		4	5.3	odd	4
900.6.a.m.1.1		2	60.59	even	2
900.6.a.s.1.2		2	12.11	even	2
900.6.d.m.649.1		4	60.23	odd	4
900.6.d.m.649.4		4	60.47	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.a (trivial)

\(f(q)\)	\(=\)	\(q+30.2237 q^{3} -141.342 q^{7} +670.475 q^{9} +O(q^{10})\)	\(q+30.2237 q^{3} -141.342 q^{7} +670.475 q^{9} -273.356 q^{11} -945.370 q^{13} -1212.32 q^{17} -135.931 q^{19} -4271.90 q^{21} -2730.88 q^{23} +12919.9 q^{27} -3077.70 q^{29} +1410.44 q^{31} -8261.85 q^{33} +3152.03 q^{37} -28572.6 q^{39} -13499.5 q^{41} +5433.70 q^{43} -3269.53 q^{47} +3170.70 q^{49} -36640.7 q^{51} -21421.3 q^{53} -4108.35 q^{57} +41403.1 q^{59} -24373.9 q^{61} -94766.6 q^{63} -32307.0 q^{67} -82537.3 q^{69} +38675.0 q^{71} -8083.79 q^{73} +38636.8 q^{77} +15979.3 q^{79} +227562. q^{81} +48494.1 q^{83} -93019.6 q^{87} +89187.5 q^{89} +133621. q^{91} +42628.7 q^{93} +56822.2 q^{97} -183279. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{3} - 40 q^{7} + 532 q^{9}+O(q^{10}) \) 2 * q + 20 * q^3 - 40 * q^7 + 532 * q^9	\( 2 q + 20 q^{3} - 40 q^{7} + 532 q^{9} + 60 q^{11} - 920 q^{13} - 2910 q^{17} - 2092 q^{19} - 5308 q^{21} + 120 q^{23} + 16820 q^{27} + 3552 q^{29} + 8888 q^{31} - 11670 q^{33} - 12140 q^{37} - 28832 q^{39} - 12438 q^{41} + 1160 q^{43} - 1200 q^{47} - 3366 q^{49} - 19284 q^{51} - 26340 q^{53} + 15890 q^{57} + 36696 q^{59} + 19204 q^{61} - 108800 q^{63} - 90460 q^{67} - 111684 q^{69} - 2736 q^{71} - 12770 q^{73} + 72420 q^{77} + 16184 q^{79} + 221338 q^{81} - 30300 q^{83} - 160800 q^{87} - 47322 q^{89} + 136192 q^{91} - 33820 q^{93} + 2980 q^{97} - 229440 q^{99}+O(q^{100}) \) 2 * q + 20 * q^3 - 40 * q^7 + 532 * q^9 + 60 * q^11 - 920 * q^13 - 2910 * q^17 - 2092 * q^19 - 5308 * q^21 + 120 * q^23 + 16820 * q^27 + 3552 * q^29 + 8888 * q^31 - 11670 * q^33 - 12140 * q^37 - 28832 * q^39 - 12438 * q^41 + 1160 * q^43 - 1200 * q^47 - 3366 * q^49 - 19284 * q^51 - 26340 * q^53 + 15890 * q^57 + 36696 * q^59 + 19204 * q^61 - 108800 * q^63 - 90460 * q^67 - 111684 * q^69 - 2736 * q^71 - 12770 * q^73 + 72420 * q^77 + 16184 * q^79 + 221338 * q^81 - 30300 * q^83 - 160800 * q^87 - 47322 * q^89 + 136192 * q^91 - 33820 * q^93 + 2980 * q^97 - 229440 * q^99

Introduction

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