LMFDB - Embedded newform 400.6.a.o.1.1

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{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.26209$ 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)

$f(q)$	$=$	$q-25.5242 q^{3} -131.048 q^{7} +408.483 q^{9} +O(q^{10})$	$q-25.5242 q^{3} -131.048 q^{7} +408.483 q^{9} -290.104 q^{11} +68.3868 q^{13} -310.644 q^{17} +2133.35 q^{19} +3344.90 q^{21} -873.145 q^{23} -4223.83 q^{27} -2580.97 q^{29} +9086.30 q^{31} +7404.67 q^{33} +3990.64 q^{37} -1745.52 q^{39} +16981.8 q^{41} +18017.7 q^{43} -24864.7 q^{47} +366.670 q^{49} +7928.93 q^{51} -7652.91 q^{53} -54451.9 q^{57} +9233.69 q^{59} +3326.17 q^{61} -53531.1 q^{63} +32340.7 q^{67} +22286.3 q^{69} +35885.9 q^{71} -26513.6 q^{73} +38017.7 q^{77} -71705.7 q^{79} +8548.28 q^{81} -39630.1 q^{83} +65877.1 q^{87} -117441. q^{89} -8961.98 q^{91} -231920. q^{93} +21878.3 q^{97} -118503. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{3} - 200 q^{7} + 196 q^{9}+O(q^{10}) $ 2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9	$ 2 q - 20 q^{3} - 200 q^{7} + 196 q^{9} + 196 q^{11} - 360 q^{13} + 1490 q^{17} + 3180 q^{19} + 2964 q^{21} - 1560 q^{23} - 6740 q^{27} - 3920 q^{29} + 1096 q^{31} + 10090 q^{33} + 2020 q^{37} - 4112 q^{39} + 27754 q^{41} + 3000 q^{43} - 25760 q^{47} - 11686 q^{49} + 17876 q^{51} - 26980 q^{53} - 48670 q^{57} - 11960 q^{59} - 24396 q^{61} - 38880 q^{63} + 40060 q^{67} + 18492 q^{69} + 87296 q^{71} - 70290 q^{73} + 4500 q^{77} - 65480 q^{79} + 46282 q^{81} - 92580 q^{83} + 58480 q^{87} - 72810 q^{89} + 20576 q^{91} - 276060 q^{93} - 126140 q^{97} - 221792 q^{99}+O(q^{100}) $ 2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9 + 196 * q^11 - 360 * q^13 + 1490 * q^17 + 3180 * q^19 + 2964 * q^21 - 1560 * q^23 - 6740 * q^27 - 3920 * q^29 + 1096 * q^31 + 10090 * q^33 + 2020 * q^37 - 4112 * q^39 + 27754 * q^41 + 3000 * q^43 - 25760 * q^47 - 11686 * q^49 + 17876 * q^51 - 26980 * q^53 - 48670 * q^57 - 11960 * q^59 - 24396 * q^61 - 38880 * q^63 + 40060 * q^67 + 18492 * q^69 + 87296 * q^71 - 70290 * q^73 + 4500 * q^77 - 65480 * q^79 + 46282 * q^81 - 92580 * q^83 + 58480 * q^87 - 72810 * q^89 + 20576 * q^91 - 276060 * q^93 - 126140 * q^97 - 221792 * 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
$3$	−25.5242	−1.63738	−0.818688	−	0.574238i	$-0.805298\pi$
$3$	−25.5242	−1.63738	−0.818688	+	0.574238i	$0.805298\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−131.048	−1.01085	−0.505425	−	0.862871i	$-0.668664\pi$
$7$	−131.048	−1.01085	−0.505425	+	0.862871i	$0.668664\pi$
$8$	0	0
$9$	408.483	1.68100
$10$	0	0
$11$	−290.104	−0.722891	−0.361445	−	0.932393i	$-0.617717\pi$
$11$	−290.104	−0.722891	−0.361445	+	0.932393i	$0.617717\pi$
$12$	0	0
$13$	68.3868	0.112231	0.0561156	−	0.998424i	$-0.482128\pi$
$13$	68.3868	0.112231	0.0561156	+	0.998424i	$0.482128\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−310.644	−0.260700	−0.130350	−	0.991468i	$-0.541610\pi$
$17$	−310.644	−0.260700	−0.130350	+	0.991468i	$0.541610\pi$
$18$	0	0
$19$	2133.35	1.35574	0.677871	−	0.735180i	$-0.262903\pi$
$19$	2133.35	1.35574	0.677871	+	0.735180i	$0.262903\pi$
$20$	0	0
$21$	3344.90	1.65514
$22$	0	0
$23$	−873.145	−0.344165	−0.172083	−	0.985083i	$-0.555050\pi$
$23$	−873.145	−0.344165	−0.172083	+	0.985083i	$0.555050\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4223.83	−1.11506
$28$	0	0
$29$	−2580.97	−0.569885	−0.284943	−	0.958545i	$-0.591975\pi$
$29$	−2580.97	−0.569885	−0.284943	+	0.958545i	$0.591975\pi$
$30$	0	0
$31$	9086.30	1.69818	0.849088	−	0.528252i	$-0.177152\pi$
$31$	9086.30	1.69818	0.849088	+	0.528252i	$0.177152\pi$
$32$	0	0
$33$	7404.67	1.18364
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3990.64	0.479224	0.239612	−	0.970869i	$-0.422980\pi$
$37$	3990.64	0.479224	0.239612	+	0.970869i	$0.422980\pi$
$38$	0	0
$39$	−1745.52	−0.183765
$40$	0	0
$41$	16981.8	1.57770	0.788851	−	0.614584i	$-0.210676\pi$
$41$	16981.8	1.57770	0.788851	+	0.614584i	$0.210676\pi$
$42$	0	0
$43$	18017.7	1.48603	0.743017	−	0.669273i	$-0.233394\pi$
$43$	18017.7	1.48603	0.743017	+	0.669273i	$0.233394\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−24864.7	−1.64187	−0.820933	−	0.571024i	$-0.806546\pi$
$47$	−24864.7	−1.64187	−0.820933	+	0.571024i	$0.806546\pi$
$48$	0	0
$49$	366.670	0.0218165
$50$	0	0
$51$	7928.93	0.426864
$52$	0	0
$53$	−7652.91	−0.374229	−0.187114	−	0.982338i	$-0.559913\pi$
$53$	−7652.91	−0.374229	−0.187114	+	0.982338i	$0.559913\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−54451.9	−2.21986
$58$	0	0
$59$	9233.69	0.345339	0.172669	−	0.984980i	$-0.444761\pi$
$59$	9233.69	0.345339	0.172669	+	0.984980i	$0.444761\pi$
$60$	0	0
$61$	3326.17	0.114451	0.0572256	−	0.998361i	$-0.481775\pi$
$61$	3326.17	0.114451	0.0572256	+	0.998361i	$0.481775\pi$
$62$	0	0
$63$	−53531.1	−1.69924
$64$	0	0
$65$	0	0
$66$	0	0
$67$	32340.7	0.880161	0.440080	−	0.897958i	$-0.354950\pi$
$67$	32340.7	0.880161	0.440080	+	0.897958i	$0.354950\pi$
$68$	0	0
$69$	22286.3	0.563528
$70$	0	0
$71$	35885.9	0.844847	0.422424	−	0.906399i	$-0.361179\pi$
$71$	35885.9	0.844847	0.422424	+	0.906399i	$0.361179\pi$
$72$	0	0
$73$	−26513.6	−0.582319	−0.291159	−	0.956675i	$-0.594041\pi$
$73$	−26513.6	−0.582319	−0.291159	+	0.956675i	$0.594041\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	38017.7	0.730733
$78$	0	0
$79$	−71705.7	−1.29266	−0.646332	−	0.763056i	$-0.723698\pi$
$79$	−71705.7	−1.29266	−0.646332	+	0.763056i	$0.723698\pi$
$80$	0	0
$81$	8548.28	0.144766
$82$	0	0
$83$	−39630.1	−0.631437	−0.315719	−	0.948853i	$-0.602246\pi$
$83$	−39630.1	−0.631437	−0.315719	+	0.948853i	$0.602246\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	65877.1	0.933117
$88$	0	0
$89$	−117441.	−1.57161	−0.785806	−	0.618473i	$-0.787752\pi$
$89$	−117441.	−1.57161	−0.785806	+	0.618473i	$0.787752\pi$
$90$	0	0
$91$	−8961.98	−0.113449
$92$	0	0
$93$	−231920.	−2.78055
$94$	0	0
$95$	0	0
$96$	0	0
$97$	21878.3	0.236093	0.118047	−	0.993008i	$-0.462337\pi$
$97$	21878.3	0.236093	0.118047	+	0.993008i	$0.462337\pi$
$98$	0	0
$99$	−118503.	−1.21518
$100$	0	0
$101$	−75072.1	−0.732276	−0.366138	−	0.930561i	$-0.619320\pi$
$101$	−75072.1	−0.732276	−0.366138	+	0.930561i	$0.619320\pi$
$102$	0	0
$103$	−47928.6	−0.445145	−0.222573	−	0.974916i	$-0.571445\pi$
$103$	−47928.6	−0.445145	−0.222573	+	0.974916i	$0.571445\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−92012.3	−0.776938	−0.388469	−	0.921462i	$-0.626996\pi$
$107$	−92012.3	−0.776938	−0.388469	+	0.921462i	$0.626996\pi$
$108$	0	0
$109$	−10647.5	−0.0858387	−0.0429194	−	0.999079i	$-0.513666\pi$
$109$	−10647.5	−0.0858387	−0.0429194	+	0.999079i	$0.513666\pi$
$110$	0	0
$111$	−101858.	−0.784670
$112$	0	0
$113$	87373.9	0.643703	0.321852	−	0.946790i	$-0.395695\pi$
$113$	87373.9	0.643703	0.321852	+	0.946790i	$0.395695\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	27934.9	0.188661
$118$	0	0
$119$	40709.4	0.263528
$120$	0	0
$121$	−76890.5	−0.477429
$122$	0	0
$123$	−433447.	−2.58329
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−197379.	−1.08591	−0.542953	−	0.839763i	$-0.682694\pi$
$127$	−197379.	−1.08591	−0.542953	+	0.839763i	$0.682694\pi$
$128$	0	0
$129$	−459887.	−2.43320
$130$	0	0
$131$	118490.	0.603258	0.301629	−	0.953425i	$-0.402470\pi$
$131$	118490.	0.603258	0.301629	+	0.953425i	$0.402470\pi$
$132$	0	0
$133$	−279571.	−1.37045
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−302570.	−1.37728	−0.688642	−	0.725101i	$-0.741793\pi$
$137$	−302570.	−1.37728	−0.688642	+	0.725101i	$0.741793\pi$
$138$	0	0
$139$	−157190.	−0.690062	−0.345031	−	0.938591i	$-0.612132\pi$
$139$	−157190.	−0.690062	−0.345031	+	0.938591i	$0.612132\pi$
$140$	0	0
$141$	634650.	2.68835
$142$	0	0
$143$	−19839.3	−0.0811309
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−9358.95	−0.0357218
$148$	0	0
$149$	526340.	1.94223	0.971115	−	0.238612i	$-0.0766923\pi$
$149$	526340.	1.94223	0.971115	+	0.238612i	$0.0766923\pi$
$150$	0	0
$151$	−1849.08	−0.00659954	−0.00329977	−	0.999995i	$-0.501050\pi$
$151$	−1849.08	−0.00659954	−0.00329977	+	0.999995i	$0.501050\pi$
$152$	0	0
$153$	−126893.	−0.438237
$154$	0	0
$155$	0	0
$156$	0	0
$157$	343342.	1.11167	0.555837	−	0.831292i	$-0.312398\pi$
$157$	343342.	1.11167	0.555837	+	0.831292i	$0.312398\pi$
$158$	0	0
$159$	195334.	0.612753
$160$	0	0
$161$	114424.	0.347899
$162$	0	0
$163$	267463.	0.788487	0.394243	−	0.919006i	$-0.371007\pi$
$163$	267463.	0.788487	0.394243	+	0.919006i	$0.371007\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−122968.	−0.341193	−0.170596	−	0.985341i	$-0.554569\pi$
$167$	−122968.	−0.341193	−0.170596	+	0.985341i	$0.554569\pi$
$168$	0	0
$169$	−366616.	−0.987404
$170$	0	0
$171$	871437.	2.27901
$172$	0	0
$173$	288020.	0.731657	0.365829	−	0.930682i	$-0.380786\pi$
$173$	288020.	0.731657	0.365829	+	0.930682i	$0.380786\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−235682.	−0.565450
$178$	0	0
$179$	−246177.	−0.574268	−0.287134	−	0.957890i	$-0.592702\pi$
$179$	−246177.	−0.574268	−0.287134	+	0.957890i	$0.592702\pi$
$180$	0	0
$181$	433120.	0.982678	0.491339	−	0.870968i	$-0.336508\pi$
$181$	433120.	0.982678	0.491339	+	0.870968i	$0.336508\pi$
$182$	0	0
$183$	−84897.9	−0.187400
$184$	0	0
$185$	0	0
$186$	0	0
$187$	90119.1	0.188457
$188$	0	0
$189$	553526.	1.12715
$190$	0	0
$191$	701011.	1.39040	0.695202	−	0.718814i	$-0.255315\pi$
$191$	701011.	1.39040	0.695202	+	0.718814i	$0.255315\pi$
$192$	0	0
$193$	215730.	0.416887	0.208443	−	0.978034i	$-0.433160\pi$
$193$	215730.	0.416887	0.208443	+	0.978034i	$0.433160\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−700484.	−1.28598	−0.642988	−	0.765876i	$-0.722305\pi$
$197$	−700484.	−1.28598	−0.642988	+	0.765876i	$0.722305\pi$
$198$	0	0
$199$	−22097.5	−0.0395558	−0.0197779	−	0.999804i	$-0.506296\pi$
$199$	−22097.5	−0.0395558	−0.0197779	+	0.999804i	$0.506296\pi$
$200$	0	0
$201$	−825469.	−1.44115
$202$	0	0
$203$	338231.	0.576068
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−356665.	−0.578542
$208$	0	0
$209$	−618893.	−0.980054
$210$	0	0
$211$	−910782.	−1.40834	−0.704172	−	0.710030i	$-0.748681\pi$
$211$	−910782.	−1.40834	−0.704172	+	0.710030i	$0.748681\pi$
$212$	0	0
$213$	−915958.	−1.38333
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.19074e6	−1.71660
$218$	0	0
$219$	676737.	0.953475
$220$	0	0
$221$	−21243.9	−0.0292587
$222$	0	0
$223$	−132745.	−0.178754	−0.0893768	−	0.995998i	$-0.528488\pi$
$223$	−132745.	−0.178754	−0.0893768	+	0.995998i	$0.528488\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	354321.	0.456386	0.228193	−	0.973616i	$-0.426718\pi$
$227$	354321.	0.456386	0.228193	+	0.973616i	$0.426718\pi$
$228$	0	0
$229$	366643.	0.462013	0.231007	−	0.972952i	$-0.425798\pi$
$229$	366643.	0.462013	0.231007	+	0.972952i	$0.425798\pi$
$230$	0	0
$231$	−970370.	−1.19649
$232$	0	0
$233$	1.02388e6	1.23555	0.617776	−	0.786355i	$-0.288034\pi$
$233$	1.02388e6	1.23555	0.617776	+	0.786355i	$0.288034\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.83023e6	2.11658
$238$	0	0
$239$	−1.19966e6	−1.35852	−0.679258	−	0.733899i	$-0.737698\pi$
$239$	−1.19966e6	−1.35852	−0.679258	+	0.733899i	$0.737698\pi$
$240$	0	0
$241$	−94967.5	−0.105325	−0.0526626	−	0.998612i	$-0.516771\pi$
$241$	−94967.5	−0.105325	−0.0526626	+	0.998612i	$0.516771\pi$
$242$	0	0
$243$	808203.	0.878021
$244$	0	0
$245$	0	0
$246$	0	0
$247$	145893.	0.152157
$248$	0	0
$249$	1.01153e6	1.03390
$250$	0	0
$251$	−418053.	−0.418839	−0.209419	−	0.977826i	$-0.567157\pi$
$251$	−418053.	−0.418839	−0.209419	+	0.977826i	$0.567157\pi$
$252$	0	0
$253$	253303.	0.248794
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.04586e6	1.93216	0.966079	−	0.258246i	$-0.0831444\pi$
$257$	2.04586e6	1.93216	0.966079	+	0.258246i	$0.0831444\pi$
$258$	0	0
$259$	−522967.	−0.484423
$260$	0	0
$261$	−1.05428e6	−0.957978
$262$	0	0
$263$	−1.64024e6	−1.46224	−0.731119	−	0.682250i	$-0.761002\pi$
$263$	−1.64024e6	−1.46224	−0.731119	+	0.682250i	$0.761002\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.99759e6	2.57332
$268$	0	0
$269$	−720582.	−0.607160	−0.303580	−	0.952806i	$-0.598182\pi$
$269$	−720582.	−0.607160	−0.303580	+	0.952806i	$0.598182\pi$
$270$	0	0
$271$	−1.14186e6	−0.944477	−0.472238	−	0.881471i	$-0.656554\pi$
$271$	−1.14186e6	−0.944477	−0.472238	+	0.881471i	$0.656554\pi$
$272$	0	0
$273$	228747.	0.185759
$274$	0	0
$275$	0	0
$276$	0	0
$277$	377028.	0.295239	0.147620	−	0.989044i	$-0.452839\pi$
$277$	377028.	0.295239	0.147620	+	0.989044i	$0.452839\pi$
$278$	0	0
$279$	3.71160e6	2.85464
$280$	0	0
$281$	−617249.	−0.466331	−0.233166	−	0.972437i	$-0.574908\pi$
$281$	−617249.	−0.466331	−0.233166	+	0.972437i	$0.574908\pi$
$282$	0	0
$283$	1.25311e6	0.930087	0.465044	−	0.885288i	$-0.346039\pi$
$283$	1.25311e6	0.930087	0.465044	+	0.885288i	$0.346039\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.22544e6	−1.59482
$288$	0	0
$289$	−1.32336e6	−0.932036
$290$	0	0
$291$	−558425.	−0.386574
$292$	0	0
$293$	818972.	0.557314	0.278657	−	0.960391i	$-0.410111\pi$
$293$	818972.	0.557314	0.278657	+	0.960391i	$0.410111\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.22535e6	0.806064
$298$	0	0
$299$	−59711.6	−0.0386261
$300$	0	0
$301$	−2.36119e6	−1.50216
$302$	0	0
$303$	1.91615e6	1.19901
$304$	0	0
$305$	0	0
$306$	0	0
$307$	136224.	0.0824915	0.0412458	−	0.999149i	$-0.486867\pi$
$307$	136224.	0.0824915	0.0412458	+	0.999149i	$0.486867\pi$
$308$	0	0
$309$	1.22334e6	0.728871
$310$	0	0
$311$	−2.62886e6	−1.54122	−0.770612	−	0.637304i	$-0.780050\pi$
$311$	−2.62886e6	−1.54122	−0.770612	+	0.637304i	$0.780050\pi$
$312$	0	0
$313$	−218161.	−0.125868	−0.0629341	−	0.998018i	$-0.520046\pi$
$313$	−218161.	−0.125868	−0.0629341	+	0.998018i	$0.520046\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.25865e6	−0.703491	−0.351745	−	0.936096i	$-0.614412\pi$
$317$	−1.25865e6	−0.703491	−0.351745	+	0.936096i	$0.614412\pi$
$318$	0	0
$319$	748750.	0.411965
$320$	0	0
$321$	2.34854e6	1.27214
$322$	0	0
$323$	−662711.	−0.353442
$324$	0	0
$325$	0	0
$326$	0	0
$327$	271770.	0.140550
$328$	0	0
$329$	3.25847e6	1.65968
$330$	0	0
$331$	3.21863e6	1.61473	0.807366	−	0.590051i	$-0.200892\pi$
$331$	3.21863e6	1.61473	0.807366	+	0.590051i	$0.200892\pi$
$332$	0	0
$333$	1.63011e6	0.805576
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.63574e6	−0.784585	−0.392293	−	0.919840i	$-0.628318\pi$
$337$	−1.63574e6	−0.784585	−0.392293	+	0.919840i	$0.628318\pi$
$338$	0	0
$339$	−2.23015e6	−1.05398
$340$	0	0
$341$	−2.63597e6	−1.22760
$342$	0	0
$343$	2.15448e6	0.988796
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.83815e6	−0.819514	−0.409757	−	0.912195i	$-0.634386\pi$
$347$	−1.83815e6	−0.819514	−0.409757	+	0.912195i	$0.634386\pi$
$348$	0	0
$349$	−2.53806e6	−1.11542	−0.557710	−	0.830036i	$-0.688320\pi$
$349$	−2.53806e6	−1.11542	−0.557710	+	0.830036i	$0.688320\pi$
$350$	0	0
$351$	−288854.	−0.125144
$352$	0	0
$353$	1.88471e6	0.805023	0.402511	−	0.915415i	$-0.368137\pi$
$353$	1.88471e6	0.805023	0.402511	+	0.915415i	$0.368137\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.03907e6	−0.431495
$358$	0	0
$359$	−305057.	−0.124924	−0.0624619	−	0.998047i	$-0.519895\pi$
$359$	−305057.	−0.124924	−0.0624619	+	0.998047i	$0.519895\pi$
$360$	0	0
$361$	2.07507e6	0.838039
$362$	0	0
$363$	1.96257e6	0.781731
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−727834.	−0.282077	−0.141038	−	0.990004i	$-0.545044\pi$
$367$	−727834.	−0.282077	−0.141038	+	0.990004i	$0.545044\pi$
$368$	0	0
$369$	6.93680e6	2.65212
$370$	0	0
$371$	1.00290e6	0.378289
$372$	0	0
$373$	4.77676e6	1.77771	0.888855	−	0.458188i	$-0.151501\pi$
$373$	4.77676e6	1.77771	0.888855	+	0.458188i	$0.151501\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−176504.	−0.0639590
$378$	0	0
$379$	701558.	0.250880	0.125440	−	0.992101i	$-0.459966\pi$
$379$	701558.	0.250880	0.125440	+	0.992101i	$0.459966\pi$
$380$	0	0
$381$	5.03794e6	1.77804
$382$	0	0
$383$	4.01069e6	1.39708	0.698541	−	0.715570i	$-0.253833\pi$
$383$	4.01069e6	1.39708	0.698541	+	0.715570i	$0.253833\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.35994e6	2.49803
$388$	0	0
$389$	−4.45952e6	−1.49422	−0.747108	−	0.664702i	$-0.768558\pi$
$389$	−4.45952e6	−1.49422	−0.747108	+	0.664702i	$0.768558\pi$
$390$	0	0
$391$	271237.	0.0897237
$392$	0	0
$393$	−3.02436e6	−0.987761
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.36993e6	−1.07311	−0.536555	−	0.843865i	$-0.680275\pi$
$397$	−3.36993e6	−1.07311	−0.536555	+	0.843865i	$0.680275\pi$
$398$	0	0
$399$	7.13583e6	2.24395
$400$	0	0
$401$	−3.00679e6	−0.933775	−0.466888	−	0.884317i	$-0.654625\pi$
$401$	−3.00679e6	−0.933775	−0.466888	+	0.884317i	$0.654625\pi$
$402$	0	0
$403$	621383.	0.190588
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.15770e6	−0.346426
$408$	0	0
$409$	−998012.	−0.295004	−0.147502	−	0.989062i	$-0.547123\pi$
$409$	−998012.	−0.295004	−0.147502	+	0.989062i	$0.547123\pi$
$410$	0	0
$411$	7.72284e6	2.25513
$412$	0	0
$413$	−1.21006e6	−0.349085
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.01215e6	1.12989
$418$	0	0
$419$	−5.53743e6	−1.54090	−0.770448	−	0.637503i	$-0.779967\pi$
$419$	−5.53743e6	−1.54090	−0.770448	+	0.637503i	$0.779967\pi$
$420$	0	0
$421$	1.98635e6	0.546198	0.273099	−	0.961986i	$-0.411951\pi$
$421$	1.98635e6	0.546198	0.273099	+	0.961986i	$0.411951\pi$
$422$	0	0
$423$	−1.01568e7	−2.75998
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−435890.	−0.115693
$428$	0	0
$429$	506382.	0.132842
$430$	0	0
$431$	−116512.	−0.0302118	−0.0151059	−	0.999886i	$-0.504809\pi$
$431$	−116512.	−0.0302118	−0.0151059	+	0.999886i	$0.504809\pi$
$432$	0	0
$433$	−4.56166e6	−1.16924	−0.584619	−	0.811308i	$-0.698756\pi$
$433$	−4.56166e6	−1.16924	−0.584619	+	0.811308i	$0.698756\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.86272e6	−0.466599
$438$	0	0
$439$	−2.92172e6	−0.723565	−0.361782	−	0.932263i	$-0.617832\pi$
$439$	−2.92172e6	−0.723565	−0.361782	+	0.932263i	$0.617832\pi$
$440$	0	0
$441$	149779.	0.0366736
$442$	0	0
$443$	1.59752e6	0.386756	0.193378	−	0.981124i	$-0.438056\pi$
$443$	1.59752e6	0.386756	0.193378	+	0.981124i	$0.438056\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.34344e7	−3.18016
$448$	0	0
$449$	3.11073e6	0.728193	0.364096	−	0.931361i	$-0.381378\pi$
$449$	3.11073e6	0.728193	0.364096	+	0.931361i	$0.381378\pi$
$450$	0	0
$451$	−4.92650e6	−1.14051
$452$	0	0
$453$	47196.3	0.0108059
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.47145e6	1.44948	0.724738	−	0.689025i	$-0.241961\pi$
$457$	6.47145e6	1.44948	0.724738	+	0.689025i	$0.241961\pi$
$458$	0	0
$459$	1.31211e6	0.290695
$460$	0	0
$461$	−5.47864e6	−1.20066	−0.600330	−	0.799752i	$-0.704964\pi$
$461$	−5.47864e6	−1.20066	−0.600330	+	0.799752i	$0.704964\pi$
$462$	0	0
$463$	−2.35489e6	−0.510526	−0.255263	−	0.966872i	$-0.582162\pi$
$463$	−2.35489e6	−0.510526	−0.255263	+	0.966872i	$0.582162\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.56027e6	0.967606	0.483803	−	0.875177i	$-0.339255\pi$
$467$	4.56027e6	0.967606	0.483803	+	0.875177i	$0.339255\pi$
$468$	0	0
$469$	−4.23819e6	−0.889710
$470$	0	0
$471$	−8.76351e6	−1.82023
$472$	0	0
$473$	−5.22702e6	−1.07424
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.12609e6	−0.629079
$478$	0	0
$479$	1.88004e6	0.374394	0.187197	−	0.982322i	$-0.440060\pi$
$479$	1.88004e6	0.374394	0.187197	+	0.982322i	$0.440060\pi$
$480$	0	0
$481$	272907.	0.0537839
$482$	0	0
$483$	−2.92058e6	−0.569642
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.69396e6	−0.323654	−0.161827	−	0.986819i	$-0.551739\pi$
$487$	−1.69396e6	−0.323654	−0.161827	+	0.986819i	$0.551739\pi$
$488$	0	0
$489$	−6.82677e6	−1.29105
$490$	0	0
$491$	−1.48645e6	−0.278258	−0.139129	−	0.990274i	$-0.544430\pi$
$491$	−1.48645e6	−0.278258	−0.139129	+	0.990274i	$0.544430\pi$
$492$	0	0
$493$	801762.	0.148569
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.70279e6	−0.854013
$498$	0	0
$499$	−7.09934e6	−1.27634	−0.638170	−	0.769896i	$-0.720308\pi$
$499$	−7.09934e6	−1.27634	−0.638170	+	0.769896i	$0.720308\pi$
$500$	0	0
$501$	3.13865e6	0.558661
$502$	0	0
$503$	−9.24224e6	−1.62876	−0.814381	−	0.580331i	$-0.802923\pi$
$503$	−9.24224e6	−1.62876	−0.814381	+	0.580331i	$0.802923\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.35758e6	1.61675
$508$	0	0
$509$	−8.12506e6	−1.39006	−0.695028	−	0.718983i	$-0.744608\pi$
$509$	−8.12506e6	−1.39006	−0.695028	+	0.718983i	$0.744608\pi$
$510$	0	0
$511$	3.47456e6	0.588637
$512$	0	0
$513$	−9.01089e6	−1.51173
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.21335e6	1.18689
$518$	0	0
$519$	−7.35148e6	−1.19800
$520$	0	0
$521$	5.06245e6	0.817084	0.408542	−	0.912740i	$-0.366037\pi$
$521$	5.06245e6	0.817084	0.408542	+	0.912740i	$0.366037\pi$
$522$	0	0
$523$	4.76222e6	0.761299	0.380649	−	0.924719i	$-0.375700\pi$
$523$	4.76222e6	0.761299	0.380649	+	0.924719i	$0.375700\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.82260e6	−0.442714
$528$	0	0
$529$	−5.67396e6	−0.881550
$530$	0	0
$531$	3.77181e6	0.580515
$532$	0	0
$533$	1.16133e6	0.177067
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.28346e6	0.940292
$538$	0	0
$539$	−106373.	−0.0157709
$540$	0	0
$541$	−2.89920e6	−0.425877	−0.212939	−	0.977066i	$-0.568303\pi$
$541$	−2.89920e6	−0.425877	−0.212939	+	0.977066i	$0.568303\pi$
$542$	0	0
$543$	−1.10550e7	−1.60901
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.74434e6	0.820866	0.410433	−	0.911891i	$-0.365378\pi$
$547$	5.74434e6	0.820866	0.410433	+	0.911891i	$0.365378\pi$
$548$	0	0
$549$	1.35869e6	0.192393
$550$	0	0
$551$	−5.50610e6	−0.772618
$552$	0	0
$553$	9.39691e6	1.30669
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.29174e6	−0.995848	−0.497924	−	0.867221i	$-0.665904\pi$
$557$	−7.29174e6	−0.995848	−0.497924	+	0.867221i	$0.665904\pi$
$558$	0	0
$559$	1.23217e6	0.166779
$560$	0	0
$561$	−2.30022e6	−0.308576
$562$	0	0
$563$	−6.65348e6	−0.884663	−0.442331	−	0.896852i	$-0.645849\pi$
$563$	−6.65348e6	−0.884663	−0.442331	+	0.896852i	$0.645849\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.12024e6	−0.146336
$568$	0	0
$569$	5.78715e6	0.749349	0.374675	−	0.927156i	$-0.377754\pi$
$569$	5.78715e6	0.749349	0.374675	+	0.927156i	$0.377754\pi$
$570$	0	0
$571$	1.22059e7	1.56667	0.783336	−	0.621599i	$-0.213517\pi$
$571$	1.22059e7	1.56667	0.783336	+	0.621599i	$0.213517\pi$
$572$	0	0
$573$	−1.78927e7	−2.27662
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.02981e7	−1.28771	−0.643853	−	0.765149i	$-0.722665\pi$
$577$	−1.02981e7	−1.28771	−0.643853	+	0.765149i	$0.722665\pi$
$578$	0	0
$579$	−5.50634e6	−0.682601
$580$	0	0
$581$	5.19346e6	0.638288
$582$	0	0
$583$	2.22014e6	0.270526
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.30519e7	−1.56343	−0.781715	−	0.623636i	$-0.785655\pi$
$587$	−1.30519e7	−1.56343	−0.781715	+	0.623636i	$0.785655\pi$
$588$	0	0
$589$	1.93842e7	2.30229
$590$	0	0
$591$	1.78793e7	2.10563
$592$	0	0
$593$	−6.43920e6	−0.751961	−0.375980	−	0.926628i	$-0.622694\pi$
$593$	−6.43920e6	−0.751961	−0.375980	+	0.926628i	$0.622694\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	564020.	0.0647677
$598$	0	0
$599$	1.00760e7	1.14741	0.573707	−	0.819061i	$-0.305505\pi$
$599$	1.00760e7	1.14741	0.573707	+	0.819061i	$0.305505\pi$
$600$	0	0
$601$	1.57050e6	0.177358	0.0886791	−	0.996060i	$-0.471735\pi$
$601$	1.57050e6	0.177358	0.0886791	+	0.996060i	$0.471735\pi$
$602$	0	0
$603$	1.32106e7	1.47955
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.31039e6	−0.805321	−0.402660	−	0.915349i	$-0.631914\pi$
$607$	−7.31039e6	−0.805321	−0.402660	+	0.915349i	$0.631914\pi$
$608$	0	0
$609$	−8.63308e6	−0.943241
$610$	0	0
$611$	−1.70041e6	−0.184269
$612$	0	0
$613$	−1.31997e7	−1.41878	−0.709389	−	0.704817i	$-0.751029\pi$
$613$	−1.31997e7	−1.41878	−0.709389	+	0.704817i	$0.751029\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.02423e7	−1.08314	−0.541570	−	0.840655i	$-0.682170\pi$
$617$	−1.02423e7	−1.08314	−0.541570	+	0.840655i	$0.682170\pi$
$618$	0	0
$619$	−1.05614e7	−1.10788	−0.553942	−	0.832555i	$-0.686877\pi$
$619$	−1.05614e7	−1.10788	−0.553942	+	0.832555i	$0.686877\pi$
$620$	0	0
$621$	3.68802e6	0.383764
$622$	0	0
$623$	1.53905e7	1.58866
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.57967e7	1.60472
$628$	0	0
$629$	−1.23967e6	−0.124933
$630$	0	0
$631$	−1.90535e7	−1.90503	−0.952513	−	0.304497i	$-0.901512\pi$
$631$	−1.90535e7	−1.90503	−0.952513	+	0.304497i	$0.901512\pi$
$632$	0	0
$633$	2.32470e7	2.30599
$634$	0	0
$635$	0	0
$636$	0	0
$637$	25075.4	0.00244849
$638$	0	0
$639$	1.46588e7	1.42019
$640$	0	0
$641$	8.56937e6	0.823766	0.411883	−	0.911237i	$-0.364871\pi$
$641$	8.56937e6	0.823766	0.411883	+	0.911237i	$0.364871\pi$
$642$	0	0
$643$	−1.79513e7	−1.71226	−0.856130	−	0.516761i	$-0.827138\pi$
$643$	−1.79513e7	−1.71226	−0.856130	+	0.516761i	$0.827138\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.05470e7	−0.990534	−0.495267	−	0.868741i	$-0.664930\pi$
$647$	−1.05470e7	−0.990534	−0.495267	+	0.868741i	$0.664930\pi$
$648$	0	0
$649$	−2.67873e6	−0.249642
$650$	0	0
$651$	3.03928e7	2.81072
$652$	0	0
$653$	1.00324e7	0.920712	0.460356	−	0.887734i	$-0.347722\pi$
$653$	1.00324e7	0.920712	0.460356	+	0.887734i	$0.347722\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.08304e7	−0.978879
$658$	0	0
$659$	8.99161e6	0.806536	0.403268	−	0.915082i	$-0.367874\pi$
$659$	8.99161e6	0.806536	0.403268	+	0.915082i	$0.367874\pi$
$660$	0	0
$661$	2.39297e6	0.213027	0.106513	−	0.994311i	$-0.466031\pi$
$661$	2.39297e6	0.213027	0.106513	+	0.994311i	$0.466031\pi$
$662$	0	0
$663$	542234.	0.0479074
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.25356e6	0.196135
$668$	0	0
$669$	3.38820e6	0.292687
$670$	0	0
$671$	−964938.	−0.0827357
$672$	0	0
$673$	1.53612e7	1.30733	0.653666	−	0.756783i	$-0.273230\pi$
$673$	1.53612e7	1.30733	0.653666	+	0.756783i	$0.273230\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.16026e7	0.972934	0.486467	−	0.873699i	$-0.338285\pi$
$677$	1.16026e7	0.972934	0.486467	+	0.873699i	$0.338285\pi$
$678$	0	0
$679$	−2.86711e6	−0.238655
$680$	0	0
$681$	−9.04375e6	−0.747275
$682$	0	0
$683$	1.20315e7	0.986890	0.493445	−	0.869777i	$-0.335737\pi$
$683$	1.20315e7	0.986890	0.493445	+	0.869777i	$0.335737\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−9.35825e6	−0.756490
$688$	0	0
$689$	−523358.	−0.0420001
$690$	0	0
$691$	−5.18616e6	−0.413191	−0.206595	−	0.978426i	$-0.566238\pi$
$691$	−5.18616e6	−0.413191	−0.206595	+	0.978426i	$0.566238\pi$
$692$	0	0
$693$	1.55296e7	1.22836
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−5.27530e6	−0.411306
$698$	0	0
$699$	−2.61338e7	−2.02306
$700$	0	0
$701$	6.00859e6	0.461825	0.230913	−	0.972974i	$-0.425829\pi$
$701$	6.00859e6	0.461825	0.230913	+	0.972974i	$0.425829\pi$
$702$	0	0
$703$	8.51342e6	0.649704
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.83807e6	0.740221
$708$	0	0
$709$	5.90083e6	0.440857	0.220429	−	0.975403i	$-0.429254\pi$
$709$	5.90083e6	0.440857	0.220429	+	0.975403i	$0.429254\pi$
$710$	0	0
$711$	−2.92906e7	−2.17297
$712$	0	0
$713$	−7.93365e6	−0.584453
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.06204e7	2.22440
$718$	0	0
$719$	1.36592e7	0.985382	0.492691	−	0.870204i	$-0.336013\pi$
$719$	1.36592e7	0.985382	0.492691	+	0.870204i	$0.336013\pi$
$720$	0	0
$721$	6.28097e6	0.449975
$722$	0	0
$723$	2.42397e6	0.172457
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.11594e7	0.783079	0.391539	−	0.920161i	$-0.371943\pi$
$727$	1.11594e7	0.783079	0.391539	+	0.920161i	$0.371943\pi$
$728$	0	0
$729$	−2.27059e7	−1.58242
$730$	0	0
$731$	−5.59710e6	−0.387409
$732$	0	0
$733$	−1.52510e7	−1.04843	−0.524215	−	0.851586i	$-0.675641\pi$
$733$	−1.52510e7	−1.04843	−0.524215	+	0.851586i	$0.675641\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.38217e6	−0.636260
$738$	0	0
$739$	1.11820e7	0.753196	0.376598	−	0.926377i	$-0.377094\pi$
$739$	1.11820e7	0.753196	0.376598	+	0.926377i	$0.377094\pi$
$740$	0	0
$741$	−3.72379e6	−0.249138
$742$	0	0
$743$	−7.71450e6	−0.512667	−0.256334	−	0.966588i	$-0.582515\pi$
$743$	−7.71450e6	−0.512667	−0.256334	+	0.966588i	$0.582515\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.61883e7	−1.06145
$748$	0	0
$749$	1.20581e7	0.785367
$750$	0	0
$751$	2.23973e7	1.44909	0.724545	−	0.689228i	$-0.242050\pi$
$751$	2.23973e7	1.44909	0.724545	+	0.689228i	$0.242050\pi$
$752$	0	0
$753$	1.06704e7	0.685796
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.57267e7	1.63171	0.815857	−	0.578254i	$-0.196266\pi$
$757$	2.57267e7	1.63171	0.815857	+	0.578254i	$0.196266\pi$
$758$	0	0
$759$	−6.46535e6	−0.407369
$760$	0	0
$761$	−1.48340e7	−0.928533	−0.464267	−	0.885696i	$-0.653682\pi$
$761$	−1.48340e7	−0.928533	−0.464267	+	0.885696i	$0.653682\pi$
$762$	0	0
$763$	1.39534e6	0.0867700
$764$	0	0
$765$	0	0
$766$	0	0
$767$	631463.	0.0387578
$768$	0	0
$769$	−5.57112e6	−0.339724	−0.169862	−	0.985468i	$-0.554332\pi$
$769$	−5.57112e6	−0.339724	−0.169862	+	0.985468i	$0.554332\pi$
$770$	0	0
$771$	−5.22188e7	−3.16367
$772$	0	0
$773$	−1.58230e7	−0.952447	−0.476224	−	0.879324i	$-0.657995\pi$
$773$	−1.58230e7	−0.952447	−0.476224	+	0.879324i	$0.657995\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.33483e7	0.793183
$778$	0	0
$779$	3.62281e7	2.13896
$780$	0	0
$781$	−1.04107e7	−0.610732
$782$	0	0
$783$	1.09016e7	0.635454
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.31529e7	0.756981	0.378491	−	0.925605i	$-0.376443\pi$
$787$	1.31529e7	0.756981	0.378491	+	0.925605i	$0.376443\pi$
$788$	0	0
$789$	4.18658e7	2.39423
$790$	0	0
$791$	−1.14502e7	−0.650687
$792$	0	0
$793$	227466.	0.0128450
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.58443e7	−1.44118	−0.720590	−	0.693361i	$-0.756129\pi$
$797$	−2.58443e7	−1.44118	−0.720590	+	0.693361i	$0.756129\pi$
$798$	0	0
$799$	7.72406e6	0.428034
$800$	0	0
$801$	−4.79728e7	−2.64188
$802$	0	0
$803$	7.69170e6	0.420953
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.83923e7	0.994149
$808$	0	0
$809$	1.78857e7	0.960804	0.480402	−	0.877048i	$-0.340491\pi$
$809$	1.78857e7	0.960804	0.480402	+	0.877048i	$0.340491\pi$
$810$	0	0
$811$	1.41608e7	0.756026	0.378013	−	0.925800i	$-0.376607\pi$
$811$	1.41608e7	0.756026	0.378013	+	0.925800i	$0.376607\pi$
$812$	0	0
$813$	2.91451e7	1.54646
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.84380e7	2.01468
$818$	0	0
$819$	−3.66082e6	−0.190708
$820$	0	0
$821$	−3.46248e7	−1.79279	−0.896394	−	0.443258i	$-0.853823\pi$
$821$	−3.46248e7	−1.79279	−0.896394	+	0.443258i	$0.853823\pi$
$822$	0	0
$823$	−2.13360e7	−1.09803	−0.549015	−	0.835813i	$-0.684997\pi$
$823$	−2.13360e7	−1.09803	−0.549015	+	0.835813i	$0.684997\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.59813e6	0.0812548	0.0406274	−	0.999174i	$-0.487064\pi$
$827$	1.59813e6	0.0812548	0.0406274	+	0.999174i	$0.487064\pi$
$828$	0	0
$829$	−2.53923e7	−1.28327	−0.641633	−	0.767012i	$-0.721743\pi$
$829$	−2.53923e7	−1.28327	−0.641633	+	0.767012i	$0.721743\pi$
$830$	0	0
$831$	−9.62333e6	−0.483418
$832$	0	0
$833$	−113904.	−0.00568755
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.83790e7	−1.89356
$838$	0	0
$839$	−1.98528e7	−0.973681	−0.486841	−	0.873491i	$-0.661851\pi$
$839$	−1.98528e7	−0.973681	−0.486841	+	0.873491i	$0.661851\pi$
$840$	0	0
$841$	−1.38498e7	−0.675231
$842$	0	0
$843$	1.57548e7	0.763560
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.00764e7	0.482609
$848$	0	0
$849$	−3.19846e7	−1.52290
$850$	0	0
$851$	−3.48441e6	−0.164932
$852$	0	0
$853$	1.59794e7	0.751948	0.375974	−	0.926630i	$-0.377308\pi$
$853$	1.59794e7	0.751948	0.375974	+	0.926630i	$0.377308\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.00157e6	−0.325644	−0.162822	−	0.986655i	$-0.552060\pi$
$857$	−7.00157e6	−0.325644	−0.162822	+	0.986655i	$0.552060\pi$
$858$	0	0
$859$	7.28414e6	0.336818	0.168409	−	0.985717i	$-0.446137\pi$
$859$	7.28414e6	0.336818	0.168409	+	0.985717i	$0.446137\pi$
$860$	0	0
$861$	5.68026e7	2.61132
$862$	0	0
$863$	−1.76361e7	−0.806075	−0.403037	−	0.915184i	$-0.632046\pi$
$863$	−1.76361e7	−0.806075	−0.403037	+	0.915184i	$0.632046\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.37776e7	1.52609
$868$	0	0
$869$	2.08021e7	0.934455
$870$	0	0
$871$	2.21167e6	0.0987816
$872$	0	0
$873$	8.93692e6	0.396874
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.69004e7	1.18102	0.590512	−	0.807029i	$-0.298926\pi$
$877$	2.69004e7	1.18102	0.590512	+	0.807029i	$0.298926\pi$
$878$	0	0
$879$	−2.09036e7	−0.912533
$880$	0	0
$881$	2.51911e7	1.09347	0.546735	−	0.837306i	$-0.315870\pi$
$881$	2.51911e7	1.09347	0.546735	+	0.837306i	$0.315870\pi$
$882$	0	0
$883$	3.22126e7	1.39035	0.695175	−	0.718840i	$-0.255327\pi$
$883$	3.22126e7	1.39035	0.695175	+	0.718840i	$0.255327\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.96139e6	−0.382443	−0.191221	−	0.981547i	$-0.561245\pi$
$887$	−8.96139e6	−0.382443	−0.191221	+	0.981547i	$0.561245\pi$
$888$	0	0
$889$	2.58662e7	1.09769
$890$	0	0
$891$	−2.47989e6	−0.104650
$892$	0	0
$893$	−5.30449e7	−2.22595
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.52409e6	0.0632454
$898$	0	0
$899$	−2.34514e7	−0.967765
$900$	0	0
$901$	2.37733e6	0.0975613
$902$	0	0
$903$	6.02675e7	2.45960
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.81689e6	0.234786	0.117393	−	0.993086i	$-0.462546\pi$
$907$	5.81689e6	0.234786	0.117393	+	0.993086i	$0.462546\pi$
$908$	0	0
$909$	−3.06657e7	−1.23096
$910$	0	0
$911$	1.96435e7	0.784192	0.392096	−	0.919924i	$-0.371750\pi$
$911$	1.96435e7	0.784192	0.392096	+	0.919924i	$0.371750\pi$
$912$	0	0
$913$	1.14969e7	0.456460
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.55279e7	−0.609803
$918$	0	0
$919$	89962.4	0.00351376	0.00175688	−	0.999998i	$-0.499441\pi$
$919$	89962.4	0.00351376	0.00175688	+	0.999998i	$0.499441\pi$
$920$	0	0
$921$	−3.47702e6	−0.135070
$922$	0	0
$923$	2.45412e6	0.0948183
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.95781e7	−0.748290
$928$	0	0
$929$	3.65192e7	1.38830	0.694149	−	0.719832i	$-0.255781\pi$
$929$	3.65192e7	1.38830	0.694149	+	0.719832i	$0.255781\pi$
$930$	0	0
$931$	782234.	0.0295776
$932$	0	0
$933$	6.70994e7	2.52357
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.58659e7	−1.33454	−0.667272	−	0.744814i	$-0.732538\pi$
$937$	−3.58659e7	−1.33454	−0.667272	+	0.744814i	$0.732538\pi$
$938$	0	0
$939$	5.56838e6	0.206094
$940$	0	0
$941$	−3.19693e7	−1.17695	−0.588476	−	0.808515i	$-0.700272\pi$
$941$	−3.19693e7	−1.17695	−0.588476	+	0.808515i	$0.700272\pi$
$942$	0	0
$943$	−1.48276e7	−0.542990
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.71846e7	−1.70972	−0.854861	−	0.518858i	$-0.826357\pi$
$947$	−4.71846e7	−1.70972	−0.854861	+	0.518858i	$0.826357\pi$
$948$	0	0
$949$	−1.81318e6	−0.0653544
$950$	0	0
$951$	3.21261e7	1.15188
$952$	0	0
$953$	1.65226e6	0.0589315	0.0294657	−	0.999566i	$-0.490619\pi$
$953$	1.65226e6	0.0589315	0.0294657	+	0.999566i	$0.490619\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.91112e7	−0.674541
$958$	0	0
$959$	3.96512e7	1.39223
$960$	0	0
$961$	5.39316e7	1.88380
$962$	0	0
$963$	−3.75855e7	−1.30603
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.23040e7	1.11094	0.555470	−	0.831537i	$-0.312538\pi$
$967$	3.23040e7	1.11094	0.555470	+	0.831537i	$0.312538\pi$
$968$	0	0
$969$	1.69151e7	0.578717
$970$	0	0
$971$	−1.15927e7	−0.394582	−0.197291	−	0.980345i	$-0.563214\pi$
$971$	−1.15927e7	−0.394582	−0.197291	+	0.980345i	$0.563214\pi$
$972$	0	0
$973$	2.05995e7	0.697549
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.58947e7	0.867909	0.433954	−	0.900935i	$-0.357118\pi$
$977$	2.58947e7	0.867909	0.433954	+	0.900935i	$0.357118\pi$
$978$	0	0
$979$	3.40702e7	1.13610
$980$	0	0
$981$	−4.34935e6	−0.144295
$982$	0	0
$983$	−3.46040e7	−1.14220	−0.571101	−	0.820880i	$-0.693483\pi$
$983$	−3.46040e7	−1.14220	−0.571101	+	0.820880i	$0.693483\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−8.31698e7	−2.71752
$988$	0	0
$989$	−1.57321e7	−0.511441
$990$	0	0
$991$	3.71464e7	1.20152	0.600762	−	0.799428i	$-0.294864\pi$
$991$	3.71464e7	1.20152	0.600762	+	0.799428i	$0.294864\pi$
$992$	0	0
$993$	−8.21528e7	−2.64393
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.47350e7	−0.788086	−0.394043	−	0.919092i	$-0.628924\pi$
$997$	−2.47350e7	−0.788086	−0.394043	+	0.919092i	$0.628924\pi$
$998$	0	0
$999$	−1.68558e7	−0.534362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.o.1.1		2
4.3	odd	2		25.6.a.d.1.1	yes	2
5.2	odd	4		400.6.c.n.49.4		4
5.3	odd	4		400.6.c.n.49.1		4
5.4	even	2		400.6.a.w.1.2		2
12.11	even	2		225.6.a.l.1.2		2
20.3	even	4		25.6.b.b.24.3		4
20.7	even	4		25.6.b.b.24.2		4
20.19	odd	2		25.6.a.b.1.2	✓	2
60.23	odd	4		225.6.b.i.199.2		4
60.47	odd	4		225.6.b.i.199.3		4
60.59	even	2		225.6.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.a.b.1.2	✓	2	20.19	odd	2
25.6.a.d.1.1	yes	2	4.3	odd	2
25.6.b.b.24.2		4	20.7	even	4
25.6.b.b.24.3		4	20.3	even	4
225.6.a.l.1.2		2	12.11	even	2
225.6.a.s.1.1		2	60.59	even	2
225.6.b.i.199.2		4	60.23	odd	4
225.6.b.i.199.3		4	60.47	odd	4
400.6.a.o.1.1		2	1.1	even	1	trivial
400.6.a.w.1.2		2	5.4	even	2
400.6.c.n.49.1		4	5.3	odd	4
400.6.c.n.49.4		4	5.2	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)

\(f(q)\)	\(=\)	\(q-25.5242 q^{3} -131.048 q^{7} +408.483 q^{9} +O(q^{10})\)	\(q-25.5242 q^{3} -131.048 q^{7} +408.483 q^{9} -290.104 q^{11} +68.3868 q^{13} -310.644 q^{17} +2133.35 q^{19} +3344.90 q^{21} -873.145 q^{23} -4223.83 q^{27} -2580.97 q^{29} +9086.30 q^{31} +7404.67 q^{33} +3990.64 q^{37} -1745.52 q^{39} +16981.8 q^{41} +18017.7 q^{43} -24864.7 q^{47} +366.670 q^{49} +7928.93 q^{51} -7652.91 q^{53} -54451.9 q^{57} +9233.69 q^{59} +3326.17 q^{61} -53531.1 q^{63} +32340.7 q^{67} +22286.3 q^{69} +35885.9 q^{71} -26513.6 q^{73} +38017.7 q^{77} -71705.7 q^{79} +8548.28 q^{81} -39630.1 q^{83} +65877.1 q^{87} -117441. q^{89} -8961.98 q^{91} -231920. q^{93} +21878.3 q^{97} -118503. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{3} - 200 q^{7} + 196 q^{9}+O(q^{10}) \) 2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9	\( 2 q - 20 q^{3} - 200 q^{7} + 196 q^{9} + 196 q^{11} - 360 q^{13} + 1490 q^{17} + 3180 q^{19} + 2964 q^{21} - 1560 q^{23} - 6740 q^{27} - 3920 q^{29} + 1096 q^{31} + 10090 q^{33} + 2020 q^{37} - 4112 q^{39} + 27754 q^{41} + 3000 q^{43} - 25760 q^{47} - 11686 q^{49} + 17876 q^{51} - 26980 q^{53} - 48670 q^{57} - 11960 q^{59} - 24396 q^{61} - 38880 q^{63} + 40060 q^{67} + 18492 q^{69} + 87296 q^{71} - 70290 q^{73} + 4500 q^{77} - 65480 q^{79} + 46282 q^{81} - 92580 q^{83} + 58480 q^{87} - 72810 q^{89} + 20576 q^{91} - 276060 q^{93} - 126140 q^{97} - 221792 q^{99}+O(q^{100}) \) 2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9 + 196 * q^11 - 360 * q^13 + 1490 * q^17 + 3180 * q^19 + 2964 * q^21 - 1560 * q^23 - 6740 * q^27 - 3920 * q^29 + 1096 * q^31 + 10090 * q^33 + 2020 * q^37 - 4112 * q^39 + 27754 * q^41 + 3000 * q^43 - 25760 * q^47 - 11686 * q^49 + 17876 * q^51 - 26980 * q^53 - 48670 * q^57 - 11960 * q^59 - 24396 * q^61 - 38880 * q^63 + 40060 * q^67 + 18492 * q^69 + 87296 * q^71 - 70290 * q^73 + 4500 * q^77 - 65480 * q^79 + 46282 * q^81 - 92580 * q^83 + 58480 * q^87 - 72810 * q^89 + 20576 * q^91 - 276060 * q^93 - 126140 * q^97 - 221792 * q^99

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