LMFDB - Embedded newform 400.6.c.n.49.4

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:	$4$
Coefficient field:	$\Q(i, \sqrt{241})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 121x^{2} + 3600 $ x^4 + 121*x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$7.26209i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.n.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.5242i q^{3} -131.048i q^{7} -408.483 q^{9} +O(q^{10})$	$q+25.5242i q^{3} -131.048i q^{7} -408.483 q^{9} -290.104 q^{11} -68.3868i q^{13} -310.644i q^{17} -2133.35 q^{19} +3344.90 q^{21} +873.145i q^{23} -4223.83i q^{27} +2580.97 q^{29} +9086.30 q^{31} -7404.67i q^{33} +3990.64i q^{37} +1745.52 q^{39} +16981.8 q^{41} -18017.7i q^{43} -24864.7i q^{47} -366.670 q^{49} +7928.93 q^{51} +7652.91i q^{53} -54451.9i q^{57} -9233.69 q^{59} +3326.17 q^{61} +53531.1i q^{63} +32340.7i q^{67} -22286.3 q^{69} +35885.9 q^{71} +26513.6i q^{73} +38017.7i q^{77} +71705.7 q^{79} +8548.28 q^{81} +39630.1i q^{83} +65877.1i q^{87} +117441. q^{89} -8961.98 q^{91} +231920. i q^{93} +21878.3i q^{97} +118503. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 392 q^{9}+O(q^{10}) $ 4 * q - 392 * q^9	$ 4 q - 392 q^{9} + 392 q^{11} - 6360 q^{19} + 5928 q^{21} + 7840 q^{29} + 2192 q^{31} + 8224 q^{39} + 55508 q^{41} + 23372 q^{49} + 35752 q^{51} + 23920 q^{59} - 48792 q^{61} - 36984 q^{69} + 174592 q^{71} + 130960 q^{79} + 92564 q^{81} + 145620 q^{89} + 41152 q^{91} + 443584 q^{99}+O(q^{100}) $ 4 * q - 392 * q^9 + 392 * q^11 - 6360 * q^19 + 5928 * q^21 + 7840 * q^29 + 2192 * q^31 + 8224 * q^39 + 55508 * q^41 + 23372 * q^49 + 35752 * q^51 + 23920 * q^59 - 48792 * q^61 - 36984 * q^69 + 174592 * q^71 + 130960 * q^79 + 92564 * q^81 + 145620 * q^89 + 41152 * q^91 + 443584 * 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
$3$	25.5242i	1.63738i	0.574238	+	0.818688i	$0.305298\pi$
$3$	25.5242i	1.63738i	−0.574238	+	0.818688i	$0.694702\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 131.048i	− 1.01085i	−0.862871	−	0.505425i	$-0.831336\pi$
$7$	− 131.048i	− 1.01085i	0.862871	−	0.505425i	$-0.168664\pi$
$8$	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
$13$	− 68.3868i	− 0.112231i	−0.998424	−	0.0561156i	$-0.982128\pi$
$13$	− 68.3868i	− 0.112231i	0.998424	−	0.0561156i	$-0.0178715\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 310.644i	− 0.260700i	−0.991468	−	0.130350i	$-0.958390\pi$
$17$	− 310.644i	− 0.260700i	0.991468	−	0.130350i	$-0.0416100\pi$
$18$	0	0
$19$	−2133.35	−1.35574	−0.677871	−	0.735180i	$-0.737097\pi$
$19$	−2133.35	−1.35574	−0.677871	+	0.735180i	$0.737097\pi$
$20$	0	0
$21$	3344.90	1.65514
$22$	0	0
$23$	873.145i	0.344165i	0.985083	+	0.172083i	$0.0550496\pi$
$23$	873.145i	0.344165i	−0.985083	+	0.172083i	$0.944950\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4223.83i	− 1.11506i
$28$	0	0
$29$	2580.97	0.569885	0.284943	−	0.958545i	$-0.408025\pi$
$29$	2580.97	0.569885	0.284943	+	0.958545i	$0.408025\pi$
$30$	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
$33$	− 7404.67i	− 1.18364i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3990.64i	0.479224i	0.970869	+	0.239612i	$0.0770202\pi$
$37$	3990.64i	0.479224i	−0.970869	+	0.239612i	$0.922980\pi$
$38$	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
$43$	− 18017.7i	− 1.48603i	−0.669273	−	0.743017i	$-0.733394\pi$
$43$	− 18017.7i	− 1.48603i	0.669273	−	0.743017i	$-0.266606\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 24864.7i	− 1.64187i	−0.571024	−	0.820933i	$-0.693454\pi$
$47$	− 24864.7i	− 1.64187i	0.571024	−	0.820933i	$-0.306546\pi$
$48$	0	0
$49$	−366.670	−0.0218165
$50$	0	0
$51$	7928.93	0.426864
$52$	0	0
$53$	7652.91i	0.374229i	0.982338	+	0.187114i	$0.0599135\pi$
$53$	7652.91i	0.374229i	−0.982338	+	0.187114i	$0.940087\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 54451.9i	− 2.21986i
$58$	0	0
$59$	−9233.69	−0.345339	−0.172669	−	0.984980i	$-0.555239\pi$
$59$	−9233.69	−0.345339	−0.172669	+	0.984980i	$0.555239\pi$
$60$	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
$63$	53531.1i	1.69924i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	32340.7i	0.880161i	0.897958	+	0.440080i	$0.145050\pi$
$67$	32340.7i	0.880161i	−0.897958	+	0.440080i	$0.854950\pi$
$68$	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
$73$	26513.6i	0.582319i	0.956675	+	0.291159i	$0.0940410\pi$
$73$	26513.6i	0.582319i	−0.956675	+	0.291159i	$0.905959\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	38017.7i	0.730733i
$78$	0	0
$79$	71705.7	1.29266	0.646332	−	0.763056i	$-0.276302\pi$
$79$	71705.7	1.29266	0.646332	+	0.763056i	$0.276302\pi$
$80$	0	0
$81$	8548.28	0.144766
$82$	0	0
$83$	39630.1i	0.631437i	0.948853	+	0.315719i	$0.102246\pi$
$83$	39630.1i	0.631437i	−0.948853	+	0.315719i	$0.897754\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	65877.1i	0.933117i
$88$	0	0
$89$	117441.	1.57161	0.785806	−	0.618473i	$-0.212248\pi$
$89$	117441.	1.57161	0.785806	+	0.618473i	$0.212248\pi$
$90$	0	0
$91$	−8961.98	−0.113449
$92$	0	0
$93$	231920.i	2.78055i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	21878.3i	0.236093i	0.993008	+	0.118047i	$0.0376633\pi$
$97$	21878.3i	0.236093i	−0.993008	+	0.118047i	$0.962337\pi$
$98$	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
$103$	47928.6i	0.445145i	0.974916	+	0.222573i	$0.0714455\pi$
$103$	47928.6i	0.445145i	−0.974916	+	0.222573i	$0.928555\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 92012.3i	− 0.776938i	−0.921462	−	0.388469i	$-0.873004\pi$
$107$	− 92012.3i	− 0.776938i	0.921462	−	0.388469i	$-0.126996\pi$
$108$	0	0
$109$	10647.5	0.0858387	0.0429194	−	0.999079i	$-0.486334\pi$
$109$	10647.5	0.0858387	0.0429194	+	0.999079i	$0.486334\pi$
$110$	0	0
$111$	−101858.	−0.784670
$112$	0	0
$113$	− 87373.9i	− 0.643703i	−0.946790	−	0.321852i	$-0.895695\pi$
$113$	− 87373.9i	− 0.643703i	0.946790	−	0.321852i	$-0.104305\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	27934.9i	0.188661i
$118$	0	0
$119$	−40709.4	−0.263528
$120$	0	0
$121$	−76890.5	−0.477429
$122$	0	0
$123$	433447.i	2.58329i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 197379.i	− 1.08591i	−0.839763	−	0.542953i	$-0.817306\pi$
$127$	− 197379.i	− 1.08591i	0.839763	−	0.542953i	$-0.182694\pi$
$128$	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.i	1.37045i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 302570.i	− 1.37728i	−0.725101	−	0.688642i	$-0.758207\pi$
$137$	− 302570.i	− 1.37728i	0.725101	−	0.688642i	$-0.241793\pi$
$138$	0	0
$139$	157190.	0.690062	0.345031	−	0.938591i	$-0.387868\pi$
$139$	157190.	0.690062	0.345031	+	0.938591i	$0.387868\pi$
$140$	0	0
$141$	634650.	2.68835
$142$	0	0
$143$	19839.3i	0.0811309i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 9358.95i	− 0.0357218i
$148$	0	0
$149$	−526340.	−1.94223	−0.971115	−	0.238612i	$-0.923308\pi$
$149$	−526340.	−1.94223	−0.971115	+	0.238612i	$0.923308\pi$
$150$	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.i	0.438237i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	343342.i	1.11167i	0.831292	+	0.555837i	$0.187602\pi$
$157$	343342.i	1.11167i	−0.831292	+	0.555837i	$0.812398\pi$
$158$	0	0
$159$	−195334.	−0.612753
$160$	0	0
$161$	114424.	0.347899
$162$	0	0
$163$	− 267463.i	− 0.788487i	−0.919006	−	0.394243i	$-0.871007\pi$
$163$	− 267463.i	− 0.788487i	0.919006	−	0.394243i	$-0.128993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 122968.i	− 0.341193i	−0.985341	−	0.170596i	$-0.945431\pi$
$167$	− 122968.i	− 0.341193i	0.985341	−	0.170596i	$-0.0545694\pi$
$168$	0	0
$169$	366616.	0.987404
$170$	0	0
$171$	871437.	2.27901
$172$	0	0
$173$	− 288020.i	− 0.731657i	−0.930682	−	0.365829i	$-0.880786\pi$
$173$	− 288020.i	− 0.731657i	0.930682	−	0.365829i	$-0.119214\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 235682.i	− 0.565450i
$178$	0	0
$179$	246177.	0.574268	0.287134	−	0.957890i	$-0.407298\pi$
$179$	246177.	0.574268	0.287134	+	0.957890i	$0.407298\pi$
$180$	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
$183$	84897.9i	0.187400i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	90119.1i	0.188457i
$188$	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.i	− 0.416887i	−0.978034	−	0.208443i	$-0.933160\pi$
$193$	− 215730.i	− 0.416887i	0.978034	−	0.208443i	$-0.0668397\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 700484.i	− 1.28598i	−0.765876	−	0.642988i	$-0.777695\pi$
$197$	− 700484.i	− 1.28598i	0.765876	−	0.642988i	$-0.222305\pi$
$198$	0	0
$199$	22097.5	0.0395558	0.0197779	−	0.999804i	$-0.493704\pi$
$199$	22097.5	0.0395558	0.0197779	+	0.999804i	$0.493704\pi$
$200$	0	0
$201$	−825469.	−1.44115
$202$	0	0
$203$	− 338231.i	− 0.576068i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 356665.i	− 0.578542i
$208$	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.i	1.38333i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.19074e6i	− 1.71660i
$218$	0	0
$219$	−676737.	−0.953475
$220$	0	0
$221$	−21243.9	−0.0292587
$222$	0	0
$223$	132745.i	0.178754i	0.995998	+	0.0893768i	$0.0284875\pi$
$223$	132745.i	0.178754i	−0.995998	+	0.0893768i	$0.971512\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	354321.i	0.456386i	0.973616	+	0.228193i	$0.0732817\pi$
$227$	354321.i	0.456386i	−0.973616	+	0.228193i	$0.926718\pi$
$228$	0	0
$229$	−366643.	−0.462013	−0.231007	−	0.972952i	$-0.574202\pi$
$229$	−366643.	−0.462013	−0.231007	+	0.972952i	$0.574202\pi$
$230$	0	0
$231$	−970370.	−1.19649
$232$	0	0
$233$	− 1.02388e6i	− 1.23555i	−0.786355	−	0.617776i	$-0.788034\pi$
$233$	− 1.02388e6i	− 1.23555i	0.786355	−	0.617776i	$-0.211966\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.83023e6i	2.11658i
$238$	0	0
$239$	1.19966e6	1.35852	0.679258	−	0.733899i	$-0.262302\pi$
$239$	1.19966e6	1.35852	0.679258	+	0.733899i	$0.262302\pi$
$240$	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.i	− 0.878021i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	145893.i	0.152157i
$248$	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.i	− 0.248794i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.04586e6i	1.93216i	0.258246	+	0.966079i	$0.416856\pi$
$257$	2.04586e6i	1.93216i	−0.258246	+	0.966079i	$0.583144\pi$
$258$	0	0
$259$	522967.	0.484423
$260$	0	0
$261$	−1.05428e6	−0.957978
$262$	0	0
$263$	1.64024e6i	1.46224i	0.682250	+	0.731119i	$0.261002\pi$
$263$	1.64024e6i	1.46224i	−0.682250	+	0.731119i	$0.738998\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.99759e6i	2.57332i
$268$	0	0
$269$	720582.	0.607160	0.303580	−	0.952806i	$-0.401818\pi$
$269$	720582.	0.607160	0.303580	+	0.952806i	$0.401818\pi$
$270$	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.i	− 0.185759i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	377028.i	0.295239i	0.989044	+	0.147620i	$0.0471612\pi$
$277$	377028.i	0.295239i	−0.989044	+	0.147620i	$0.952839\pi$
$278$	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
$283$	− 1.25311e6i	− 0.930087i	−0.885288	−	0.465044i	$-0.846039\pi$
$283$	− 1.25311e6i	− 0.930087i	0.885288	−	0.465044i	$-0.153961\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.22544e6i	− 1.59482i
$288$	0	0
$289$	1.32336e6	0.932036
$290$	0	0
$291$	−558425.	−0.386574
$292$	0	0
$293$	− 818972.i	− 0.557314i	−0.960391	−	0.278657i	$-0.910111\pi$
$293$	− 818972.i	− 0.557314i	0.960391	−	0.278657i	$-0.0898893\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.22535e6i	0.806064i
$298$	0	0
$299$	59711.6	0.0386261
$300$	0	0
$301$	−2.36119e6	−1.50216
$302$	0	0
$303$	− 1.91615e6i	− 1.19901i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	136224.i	0.0824915i	0.999149	+	0.0412458i	$0.0131327\pi$
$307$	136224.i	0.0824915i	−0.999149	+	0.0412458i	$0.986867\pi$
$308$	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.i	0.125868i	0.998018	+	0.0629341i	$0.0200458\pi$
$313$	218161.i	0.125868i	−0.998018	+	0.0629341i	$0.979954\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.25865e6i	− 0.703491i	−0.936096	−	0.351745i	$-0.885588\pi$
$317$	− 1.25865e6i	− 0.703491i	0.936096	−	0.351745i	$-0.114412\pi$
$318$	0	0
$319$	−748750.	−0.411965
$320$	0	0
$321$	2.34854e6	1.27214
$322$	0	0
$323$	662711.i	0.353442i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	271770.i	0.140550i
$328$	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
$333$	− 1.63011e6i	− 0.805576i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.63574e6i	− 0.784585i	−0.919840	−	0.392293i	$-0.871682\pi$
$337$	− 1.63574e6i	− 0.784585i	0.919840	−	0.392293i	$-0.128318\pi$
$338$	0	0
$339$	2.23015e6	1.05398
$340$	0	0
$341$	−2.63597e6	−1.22760
$342$	0	0
$343$	− 2.15448e6i	− 0.988796i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.83815e6i	− 0.819514i	−0.912195	−	0.409757i	$-0.865614\pi$
$347$	− 1.83815e6i	− 0.819514i	0.912195	−	0.409757i	$-0.134386\pi$
$348$	0	0
$349$	2.53806e6	1.11542	0.557710	−	0.830036i	$-0.311680\pi$
$349$	2.53806e6	1.11542	0.557710	+	0.830036i	$0.311680\pi$
$350$	0	0
$351$	−288854.	−0.125144
$352$	0	0
$353$	− 1.88471e6i	− 0.805023i	−0.915415	−	0.402511i	$-0.868137\pi$
$353$	− 1.88471e6i	− 0.805023i	0.915415	−	0.402511i	$-0.131863\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.03907e6i	− 0.431495i
$358$	0	0
$359$	305057.	0.124924	0.0624619	−	0.998047i	$-0.480105\pi$
$359$	305057.	0.124924	0.0624619	+	0.998047i	$0.480105\pi$
$360$	0	0
$361$	2.07507e6	0.838039
$362$	0	0
$363$	− 1.96257e6i	− 0.781731i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 727834.i	− 0.282077i	−0.990004	−	0.141038i	$-0.954956\pi$
$367$	− 727834.i	− 0.282077i	0.990004	−	0.141038i	$-0.0450441\pi$
$368$	0	0
$369$	−6.93680e6	−2.65212
$370$	0	0
$371$	1.00290e6	0.378289
$372$	0	0
$373$	− 4.77676e6i	− 1.77771i	−0.458188	−	0.888855i	$-0.651501\pi$
$373$	− 4.77676e6i	− 1.77771i	0.458188	−	0.888855i	$-0.348499\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 176504.i	− 0.0639590i
$378$	0	0
$379$	−701558.	−0.250880	−0.125440	−	0.992101i	$-0.540034\pi$
$379$	−701558.	−0.250880	−0.125440	+	0.992101i	$0.540034\pi$
$380$	0	0
$381$	5.03794e6	1.77804
$382$	0	0
$383$	− 4.01069e6i	− 1.39708i	−0.715570	−	0.698541i	$-0.753833\pi$
$383$	− 4.01069e6i	− 1.39708i	0.715570	−	0.698541i	$-0.246167\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.35994e6i	2.49803i
$388$	0	0
$389$	4.45952e6	1.49422	0.747108	−	0.664702i	$-0.231442\pi$
$389$	4.45952e6	1.49422	0.747108	+	0.664702i	$0.231442\pi$
$390$	0	0
$391$	271237.	0.0897237
$392$	0	0
$393$	3.02436e6i	0.987761i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3.36993e6i	− 1.07311i	−0.843865	−	0.536555i	$-0.819725\pi$
$397$	− 3.36993e6i	− 1.07311i	0.843865	−	0.536555i	$-0.180275\pi$
$398$	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.i	− 0.190588i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.15770e6i	− 0.346426i
$408$	0	0
$409$	998012.	0.295004	0.147502	−	0.989062i	$-0.452877\pi$
$409$	998012.	0.295004	0.147502	+	0.989062i	$0.452877\pi$
$410$	0	0
$411$	7.72284e6	2.25513
$412$	0	0
$413$	1.21006e6i	0.349085i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.01215e6i	1.12989i
$418$	0	0
$419$	5.53743e6	1.54090	0.770448	−	0.637503i	$-0.220033\pi$
$419$	5.53743e6	1.54090	0.770448	+	0.637503i	$0.220033\pi$
$420$	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
$423$	1.01568e7i	2.75998i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 435890.i	− 0.115693i
$428$	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
$433$	4.56166e6i	1.16924i	0.811308	+	0.584619i	$0.198756\pi$
$433$	4.56166e6i	1.16924i	−0.811308	+	0.584619i	$0.801244\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.86272e6i	− 0.466599i
$438$	0	0
$439$	2.92172e6	0.723565	0.361782	−	0.932263i	$-0.382168\pi$
$439$	2.92172e6	0.723565	0.361782	+	0.932263i	$0.382168\pi$
$440$	0	0
$441$	149779.	0.0366736
$442$	0	0
$443$	− 1.59752e6i	− 0.386756i	−0.981124	−	0.193378i	$-0.938056\pi$
$443$	− 1.59752e6i	− 0.386756i	0.981124	−	0.193378i	$-0.0619444\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.34344e7i	− 3.18016i
$448$	0	0
$449$	−3.11073e6	−0.728193	−0.364096	−	0.931361i	$-0.618622\pi$
$449$	−3.11073e6	−0.728193	−0.364096	+	0.931361i	$0.618622\pi$
$450$	0	0
$451$	−4.92650e6	−1.14051
$452$	0	0
$453$	− 47196.3i	− 0.0108059i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.47145e6i	1.44948i	0.689025	+	0.724738i	$0.258039\pi$
$457$	6.47145e6i	1.44948i	−0.689025	+	0.724738i	$0.741961\pi$
$458$	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
$463$	2.35489e6i	0.510526i	0.966872	+	0.255263i	$0.0821621\pi$
$463$	2.35489e6i	0.510526i	−0.966872	+	0.255263i	$0.917838\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.56027e6i	0.967606i	0.875177	+	0.483803i	$0.160745\pi$
$467$	4.56027e6i	0.967606i	−0.875177	+	0.483803i	$0.839255\pi$
$468$	0	0
$469$	4.23819e6	0.889710
$470$	0	0
$471$	−8.76351e6	−1.82023
$472$	0	0
$473$	5.22702e6i	1.07424i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.12609e6i	− 0.629079i
$478$	0	0
$479$	−1.88004e6	−0.374394	−0.187197	−	0.982322i	$-0.559940\pi$
$479$	−1.88004e6	−0.374394	−0.187197	+	0.982322i	$0.559940\pi$
$480$	0	0
$481$	272907.	0.0537839
$482$	0	0
$483$	2.92058e6i	0.569642i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.69396e6i	− 0.323654i	−0.986819	−	0.161827i	$-0.948261\pi$
$487$	− 1.69396e6i	− 0.323654i	0.986819	−	0.161827i	$-0.0517386\pi$
$488$	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.i	− 0.148569i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.70279e6i	− 0.854013i
$498$	0	0
$499$	7.09934e6	1.27634	0.638170	−	0.769896i	$-0.279692\pi$
$499$	7.09934e6	1.27634	0.638170	+	0.769896i	$0.279692\pi$
$500$	0	0
$501$	3.13865e6	0.558661
$502$	0	0
$503$	9.24224e6i	1.62876i	0.580331	+	0.814381i	$0.302923\pi$
$503$	9.24224e6i	1.62876i	−0.580331	+	0.814381i	$0.697077\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.35758e6i	1.61675i
$508$	0	0
$509$	8.12506e6	1.39006	0.695028	−	0.718983i	$-0.255392\pi$
$509$	8.12506e6	1.39006	0.695028	+	0.718983i	$0.255392\pi$
$510$	0	0
$511$	3.47456e6	0.588637
$512$	0	0
$513$	9.01089e6i	1.51173i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.21335e6i	1.18689i
$518$	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
$523$	− 4.76222e6i	− 0.761299i	−0.924719	−	0.380649i	$-0.875700\pi$
$523$	− 4.76222e6i	− 0.761299i	0.924719	−	0.380649i	$-0.124300\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.82260e6i	− 0.442714i
$528$	0	0
$529$	5.67396e6	0.881550
$530$	0	0
$531$	3.77181e6	0.580515
$532$	0	0
$533$	− 1.16133e6i	− 0.177067i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.28346e6i	0.940292i
$538$	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
$543$	1.10550e7i	1.60901i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.74434e6i	0.820866i	0.911891	+	0.410433i	$0.134622\pi$
$547$	5.74434e6i	0.820866i	−0.911891	+	0.410433i	$0.865378\pi$
$548$	0	0
$549$	−1.35869e6	−0.192393
$550$	0	0
$551$	−5.50610e6	−0.772618
$552$	0	0
$553$	− 9.39691e6i	− 1.30669i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.29174e6i	− 0.995848i	−0.867221	−	0.497924i	$-0.834096\pi$
$557$	− 7.29174e6i	− 0.995848i	0.867221	−	0.497924i	$-0.165904\pi$
$558$	0	0
$559$	−1.23217e6	−0.166779
$560$	0	0
$561$	−2.30022e6	−0.308576
$562$	0	0
$563$	6.65348e6i	0.884663i	0.896852	+	0.442331i	$0.145849\pi$
$563$	6.65348e6i	0.884663i	−0.896852	+	0.442331i	$0.854151\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.12024e6i	− 0.146336i
$568$	0	0
$569$	−5.78715e6	−0.749349	−0.374675	−	0.927156i	$-0.622246\pi$
$569$	−5.78715e6	−0.749349	−0.374675	+	0.927156i	$0.622246\pi$
$570$	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
$573$	1.78927e7i	2.27662i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.02981e7i	− 1.28771i	−0.765149	−	0.643853i	$-0.777335\pi$
$577$	− 1.02981e7i	− 1.28771i	0.765149	−	0.643853i	$-0.222665\pi$
$578$	0	0
$579$	5.50634e6	0.682601
$580$	0	0
$581$	5.19346e6	0.638288
$582$	0	0
$583$	− 2.22014e6i	− 0.270526i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.30519e7i	− 1.56343i	−0.623636	−	0.781715i	$-0.714345\pi$
$587$	− 1.30519e7i	− 1.56343i	0.623636	−	0.781715i	$-0.285655\pi$
$588$	0	0
$589$	−1.93842e7	−2.30229
$590$	0	0
$591$	1.78793e7	2.10563
$592$	0	0
$593$	6.43920e6i	0.751961i	0.926628	+	0.375980i	$0.122694\pi$
$593$	6.43920e6i	0.751961i	−0.926628	+	0.375980i	$0.877306\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	564020.i	0.0647677i
$598$	0	0
$599$	−1.00760e7	−1.14741	−0.573707	−	0.819061i	$-0.694495\pi$
$599$	−1.00760e7	−1.14741	−0.573707	+	0.819061i	$0.694495\pi$
$600$	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
$603$	− 1.32106e7i	− 1.47955i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7.31039e6i	− 0.805321i	−0.915349	−	0.402660i	$-0.868086\pi$
$607$	− 7.31039e6i	− 0.805321i	0.915349	−	0.402660i	$-0.131914\pi$
$608$	0	0
$609$	8.63308e6	0.943241
$610$	0	0
$611$	−1.70041e6	−0.184269
$612$	0	0
$613$	1.31997e7i	1.41878i	0.704817	+	0.709389i	$0.251029\pi$
$613$	1.31997e7i	1.41878i	−0.704817	+	0.709389i	$0.748971\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.02423e7i	− 1.08314i	−0.840655	−	0.541570i	$-0.817830\pi$
$617$	− 1.02423e7i	− 1.08314i	0.840655	−	0.541570i	$-0.182170\pi$
$618$	0	0
$619$	1.05614e7	1.10788	0.553942	−	0.832555i	$-0.313123\pi$
$619$	1.05614e7	1.10788	0.553942	+	0.832555i	$0.313123\pi$
$620$	0	0
$621$	3.68802e6	0.383764
$622$	0	0
$623$	− 1.53905e7i	− 1.58866i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.57967e7i	1.60472i
$628$	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
$633$	− 2.32470e7i	− 2.30599i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	25075.4i	0.00244849i
$638$	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
$643$	1.79513e7i	1.71226i	0.516761	+	0.856130i	$0.327138\pi$
$643$	1.79513e7i	1.71226i	−0.516761	+	0.856130i	$0.672862\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.05470e7i	− 0.990534i	−0.868741	−	0.495267i	$-0.835070\pi$
$647$	− 1.05470e7i	− 0.990534i	0.868741	−	0.495267i	$-0.164930\pi$
$648$	0	0
$649$	2.67873e6	0.249642
$650$	0	0
$651$	3.03928e7	2.81072
$652$	0	0
$653$	− 1.00324e7i	− 0.920712i	−0.887734	−	0.460356i	$-0.847722\pi$
$653$	− 1.00324e7i	− 0.920712i	0.887734	−	0.460356i	$-0.152278\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.08304e7i	− 0.978879i
$658$	0	0
$659$	−8.99161e6	−0.806536	−0.403268	−	0.915082i	$-0.632126\pi$
$659$	−8.99161e6	−0.806536	−0.403268	+	0.915082i	$0.632126\pi$
$660$	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.i	− 0.0479074i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.25356e6i	0.196135i
$668$	0	0
$669$	−3.38820e6	−0.292687
$670$	0	0
$671$	−964938.	−0.0827357
$672$	0	0
$673$	− 1.53612e7i	− 1.30733i	−0.756783	−	0.653666i	$-0.773230\pi$
$673$	− 1.53612e7i	− 1.30733i	0.756783	−	0.653666i	$-0.226770\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.16026e7i	0.972934i	0.873699	+	0.486467i	$0.161715\pi$
$677$	1.16026e7i	0.972934i	−0.873699	+	0.486467i	$0.838285\pi$
$678$	0	0
$679$	2.86711e6	0.238655
$680$	0	0
$681$	−9.04375e6	−0.747275
$682$	0	0
$683$	− 1.20315e7i	− 0.986890i	−0.869777	−	0.493445i	$-0.835737\pi$
$683$	− 1.20315e7i	− 0.986890i	0.869777	−	0.493445i	$-0.164263\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 9.35825e6i	− 0.756490i
$688$	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
$693$	− 1.55296e7i	− 1.22836i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.27530e6i	− 0.411306i
$698$	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
$703$	− 8.51342e6i	− 0.649704i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.83807e6i	0.740221i
$708$	0	0
$709$	−5.90083e6	−0.440857	−0.220429	−	0.975403i	$-0.570746\pi$
$709$	−5.90083e6	−0.440857	−0.220429	+	0.975403i	$0.570746\pi$
$710$	0	0
$711$	−2.92906e7	−2.17297
$712$	0	0
$713$	7.93365e6i	0.584453i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.06204e7i	2.22440i
$718$	0	0
$719$	−1.36592e7	−0.985382	−0.492691	−	0.870204i	$-0.663987\pi$
$719$	−1.36592e7	−0.985382	−0.492691	+	0.870204i	$0.663987\pi$
$720$	0	0
$721$	6.28097e6	0.449975
$722$	0	0
$723$	− 2.42397e6i	− 0.172457i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.11594e7i	0.783079i	0.920161	+	0.391539i	$0.128057\pi$
$727$	1.11594e7i	0.783079i	−0.920161	+	0.391539i	$0.871943\pi$
$728$	0	0
$729$	2.27059e7	1.58242
$730$	0	0
$731$	−5.59710e6	−0.387409
$732$	0	0
$733$	1.52510e7i	1.04843i	0.851586	+	0.524215i	$0.175641\pi$
$733$	1.52510e7i	1.04843i	−0.851586	+	0.524215i	$0.824359\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.38217e6i	− 0.636260i
$738$	0	0
$739$	−1.11820e7	−0.753196	−0.376598	−	0.926377i	$-0.622906\pi$
$739$	−1.11820e7	−0.753196	−0.376598	+	0.926377i	$0.622906\pi$
$740$	0	0
$741$	−3.72379e6	−0.249138
$742$	0	0
$743$	7.71450e6i	0.512667i	0.966588	+	0.256334i	$0.0825146\pi$
$743$	7.71450e6i	0.512667i	−0.966588	+	0.256334i	$0.917485\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.61883e7i	− 1.06145i
$748$	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
$753$	− 1.06704e7i	− 0.685796i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.57267e7i	1.63171i	0.578254	+	0.815857i	$0.303734\pi$
$757$	2.57267e7i	1.63171i	−0.578254	+	0.815857i	$0.696266\pi$
$758$	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
$763$	− 1.39534e6i	− 0.0867700i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	631463.i	0.0387578i
$768$	0	0
$769$	5.57112e6	0.339724	0.169862	−	0.985468i	$-0.445668\pi$
$769$	5.57112e6	0.339724	0.169862	+	0.985468i	$0.445668\pi$
$770$	0	0
$771$	−5.22188e7	−3.16367
$772$	0	0
$773$	1.58230e7i	0.952447i	0.879324	+	0.476224i	$0.157995\pi$
$773$	1.58230e7i	0.952447i	−0.879324	+	0.476224i	$0.842005\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.33483e7i	0.793183i
$778$	0	0
$779$	−3.62281e7	−2.13896
$780$	0	0
$781$	−1.04107e7	−0.610732
$782$	0	0
$783$	− 1.09016e7i	− 0.635454i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.31529e7i	0.756981i	0.925605	+	0.378491i	$0.123557\pi$
$787$	1.31529e7i	0.756981i	−0.925605	+	0.378491i	$0.876443\pi$
$788$	0	0
$789$	−4.18658e7	−2.39423
$790$	0	0
$791$	−1.14502e7	−0.650687
$792$	0	0
$793$	− 227466.i	− 0.0128450i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.58443e7i	− 1.44118i	−0.693361	−	0.720590i	$-0.743871\pi$
$797$	− 2.58443e7i	− 1.44118i	0.693361	−	0.720590i	$-0.256129\pi$
$798$	0	0
$799$	−7.72406e6	−0.428034
$800$	0	0
$801$	−4.79728e7	−2.64188
$802$	0	0
$803$	− 7.69170e6i	− 0.420953i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.83923e7i	0.994149i
$808$	0	0
$809$	−1.78857e7	−0.960804	−0.480402	−	0.877048i	$-0.659509\pi$
$809$	−1.78857e7	−0.960804	−0.480402	+	0.877048i	$0.659509\pi$
$810$	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
$813$	− 2.91451e7i	− 1.54646i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.84380e7i	2.01468i
$818$	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
$823$	2.13360e7i	1.09803i	0.835813	+	0.549015i	$0.184997\pi$
$823$	2.13360e7i	1.09803i	−0.835813	+	0.549015i	$0.815003\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.59813e6i	0.0812548i	0.999174	+	0.0406274i	$0.0129357\pi$
$827$	1.59813e6i	0.0812548i	−0.999174	+	0.0406274i	$0.987064\pi$
$828$	0	0
$829$	2.53923e7	1.28327	0.641633	−	0.767012i	$-0.278257\pi$
$829$	2.53923e7	1.28327	0.641633	+	0.767012i	$0.278257\pi$
$830$	0	0
$831$	−9.62333e6	−0.483418
$832$	0	0
$833$	113904.i	0.00568755i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.83790e7i	− 1.89356i
$838$	0	0
$839$	1.98528e7	0.973681	0.486841	−	0.873491i	$-0.338149\pi$
$839$	1.98528e7	0.973681	0.486841	+	0.873491i	$0.338149\pi$
$840$	0	0
$841$	−1.38498e7	−0.675231
$842$	0	0
$843$	− 1.57548e7i	− 0.763560i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.00764e7i	0.482609i
$848$	0	0
$849$	3.19846e7	1.52290
$850$	0	0
$851$	−3.48441e6	−0.164932
$852$	0	0
$853$	− 1.59794e7i	− 0.751948i	−0.926630	−	0.375974i	$-0.877308\pi$
$853$	− 1.59794e7i	− 0.751948i	0.926630	−	0.375974i	$-0.122692\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7.00157e6i	− 0.325644i	−0.986655	−	0.162822i	$-0.947940\pi$
$857$	− 7.00157e6i	− 0.325644i	0.986655	−	0.162822i	$-0.0520597\pi$
$858$	0	0
$859$	−7.28414e6	−0.336818	−0.168409	−	0.985717i	$-0.553863\pi$
$859$	−7.28414e6	−0.336818	−0.168409	+	0.985717i	$0.553863\pi$
$860$	0	0
$861$	5.68026e7	2.61132
$862$	0	0
$863$	1.76361e7i	0.806075i	0.915184	+	0.403037i	$0.132046\pi$
$863$	1.76361e7i	0.806075i	−0.915184	+	0.403037i	$0.867954\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.37776e7i	1.52609i
$868$	0	0
$869$	−2.08021e7	−0.934455
$870$	0	0
$871$	2.21167e6	0.0987816
$872$	0	0
$873$	− 8.93692e6i	− 0.396874i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.69004e7i	1.18102i	0.807029	+	0.590512i	$0.201074\pi$
$877$	2.69004e7i	1.18102i	−0.807029	+	0.590512i	$0.798926\pi$
$878$	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
$883$	− 3.22126e7i	− 1.39035i	−0.718840	−	0.695175i	$-0.755327\pi$
$883$	− 3.22126e7i	− 1.39035i	0.718840	−	0.695175i	$-0.244673\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.96139e6i	− 0.382443i	−0.981547	−	0.191221i	$-0.938755\pi$
$887$	− 8.96139e6i	− 0.382443i	0.981547	−	0.191221i	$-0.0612448\pi$
$888$	0	0
$889$	−2.58662e7	−1.09769
$890$	0	0
$891$	−2.47989e6	−0.104650
$892$	0	0
$893$	5.30449e7i	2.22595i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.52409e6i	0.0632454i
$898$	0	0
$899$	2.34514e7	0.967765
$900$	0	0
$901$	2.37733e6	0.0975613
$902$	0	0
$903$	− 6.02675e7i	− 2.45960i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.81689e6i	0.234786i	0.993086	+	0.117393i	$0.0374537\pi$
$907$	5.81689e6i	0.234786i	−0.993086	+	0.117393i	$0.962546\pi$
$908$	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
$913$	− 1.14969e7i	− 0.456460i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.55279e7i	− 0.609803i
$918$	0	0
$919$	−89962.4	−0.00351376	−0.00175688	−	0.999998i	$-0.500559\pi$
$919$	−89962.4	−0.00351376	−0.00175688	+	0.999998i	$0.500559\pi$
$920$	0	0
$921$	−3.47702e6	−0.135070
$922$	0	0
$923$	− 2.45412e6i	− 0.0948183i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.95781e7i	− 0.748290i
$928$	0	0
$929$	−3.65192e7	−1.38830	−0.694149	−	0.719832i	$-0.744219\pi$
$929$	−3.65192e7	−1.38830	−0.694149	+	0.719832i	$0.744219\pi$
$930$	0	0
$931$	782234.	0.0295776
$932$	0	0
$933$	− 6.70994e7i	− 2.52357i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3.58659e7i	− 1.33454i	−0.744814	−	0.667272i	$-0.767462\pi$
$937$	− 3.58659e7i	− 1.33454i	0.744814	−	0.667272i	$-0.232538\pi$
$938$	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
$943$	1.48276e7i	0.542990i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.71846e7i	− 1.70972i	−0.518858	−	0.854861i	$-0.673643\pi$
$947$	− 4.71846e7i	− 1.70972i	0.518858	−	0.854861i	$-0.326357\pi$
$948$	0	0
$949$	1.81318e6	0.0653544
$950$	0	0
$951$	3.21261e7	1.15188
$952$	0	0
$953$	− 1.65226e6i	− 0.0589315i	−0.999566	−	0.0294657i	$-0.990619\pi$
$953$	− 1.65226e6i	− 0.0589315i	0.999566	−	0.0294657i	$-0.00938059\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.91112e7i	− 0.674541i
$958$	0	0
$959$	−3.96512e7	−1.39223
$960$	0	0
$961$	5.39316e7	1.88380
$962$	0	0
$963$	3.75855e7i	1.30603i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.23040e7i	1.11094i	0.831537	+	0.555470i	$0.187462\pi$
$967$	3.23040e7i	1.11094i	−0.831537	+	0.555470i	$0.812538\pi$
$968$	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
$973$	− 2.05995e7i	− 0.697549i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.58947e7i	0.867909i	0.900935	+	0.433954i	$0.142882\pi$
$977$	2.58947e7i	0.867909i	−0.900935	+	0.433954i	$0.857118\pi$
$978$	0	0
$979$	−3.40702e7	−1.13610
$980$	0	0
$981$	−4.34935e6	−0.144295
$982$	0	0
$983$	3.46040e7i	1.14220i	0.820880	+	0.571101i	$0.193483\pi$
$983$	3.46040e7i	1.14220i	−0.820880	+	0.571101i	$0.806517\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 8.31698e7i	− 2.71752i
$988$	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
$993$	8.21528e7i	2.64393i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.47350e7i	− 0.788086i	−0.919092	−	0.394043i	$-0.871076\pi$
$997$	− 2.47350e7i	− 0.788086i	0.919092	−	0.394043i	$-0.128924\pi$
$998$	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.c.n.49.4		4
4.3	odd	2		25.6.b.b.24.2		4
5.2	odd	4		400.6.a.w.1.2		2
5.3	odd	4		400.6.a.o.1.1		2
5.4	even	2	inner	400.6.c.n.49.1		4
12.11	even	2		225.6.b.i.199.3		4
20.3	even	4		25.6.a.d.1.1	yes	2
20.7	even	4		25.6.a.b.1.2	✓	2
20.19	odd	2		25.6.b.b.24.3		4
60.23	odd	4		225.6.a.l.1.2		2
60.47	odd	4		225.6.a.s.1.1		2
60.59	even	2		225.6.b.i.199.2		4

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

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)

\(f(q)\)	\(=\)	\(q+25.5242i q^{3} -131.048i q^{7} -408.483 q^{9} +O(q^{10})\)	\(q+25.5242i q^{3} -131.048i q^{7} -408.483 q^{9} -290.104 q^{11} -68.3868i q^{13} -310.644i q^{17} -2133.35 q^{19} +3344.90 q^{21} +873.145i q^{23} -4223.83i q^{27} +2580.97 q^{29} +9086.30 q^{31} -7404.67i q^{33} +3990.64i q^{37} +1745.52 q^{39} +16981.8 q^{41} -18017.7i q^{43} -24864.7i q^{47} -366.670 q^{49} +7928.93 q^{51} +7652.91i q^{53} -54451.9i q^{57} -9233.69 q^{59} +3326.17 q^{61} +53531.1i q^{63} +32340.7i q^{67} -22286.3 q^{69} +35885.9 q^{71} +26513.6i q^{73} +38017.7i q^{77} +71705.7 q^{79} +8548.28 q^{81} +39630.1i q^{83} +65877.1i q^{87} +117441. q^{89} -8961.98 q^{91} +231920. i q^{93} +21878.3i q^{97} +118503. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 392 q^{9}+O(q^{10}) \) 4 * q - 392 * q^9	\( 4 q - 392 q^{9} + 392 q^{11} - 6360 q^{19} + 5928 q^{21} + 7840 q^{29} + 2192 q^{31} + 8224 q^{39} + 55508 q^{41} + 23372 q^{49} + 35752 q^{51} + 23920 q^{59} - 48792 q^{61} - 36984 q^{69} + 174592 q^{71} + 130960 q^{79} + 92564 q^{81} + 145620 q^{89} + 41152 q^{91} + 443584 q^{99}+O(q^{100}) \) 4 * q - 392 * q^9 + 392 * q^11 - 6360 * q^19 + 5928 * q^21 + 7840 * q^29 + 2192 * q^31 + 8224 * q^39 + 55508 * q^41 + 23372 * q^49 + 35752 * q^51 + 23920 * q^59 - 48792 * q^61 - 36984 * q^69 + 174592 * q^71 + 130960 * q^79 + 92564 * q^81 + 145620 * q^89 + 41152 * q^91 + 443584 * q^99

Introduction

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