LMFDB - Embedded newform 400.6.c.g.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.0000i q^{3} -142.000i q^{7} +122.000 q^{9} +O(q^{10})$	$q+11.0000i q^{3} -142.000i q^{7} +122.000 q^{9} -777.000 q^{11} +884.000i q^{13} +27.0000i q^{17} +1145.00 q^{19} +1562.00 q^{21} -1854.00i q^{23} +4015.00i q^{27} +4920.00 q^{29} -1802.00 q^{31} -8547.00i q^{33} -13178.0i q^{37} -9724.00 q^{39} -15123.0 q^{41} -7844.00i q^{43} -6732.00i q^{47} -3357.00 q^{49} -297.000 q^{51} +3414.00i q^{53} +12595.0i q^{57} +33960.0 q^{59} +47402.0 q^{61} -17324.0i q^{63} -13177.0i q^{67} +20394.0 q^{69} +7548.00 q^{71} -59821.0i q^{73} +110334. i q^{77} +75830.0 q^{79} -14519.0 q^{81} -46299.0i q^{83} +54120.0i q^{87} +30585.0 q^{89} +125528. q^{91} -19822.0i q^{93} -104018. i q^{97} -94794.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 244 q^{9}+O(q^{10}) $ 2 * q + 244 * q^9	$ 2 q + 244 q^{9} - 1554 q^{11} + 2290 q^{19} + 3124 q^{21} + 9840 q^{29} - 3604 q^{31} - 19448 q^{39} - 30246 q^{41} - 6714 q^{49} - 594 q^{51} + 67920 q^{59} + 94804 q^{61} + 40788 q^{69} + 15096 q^{71} + 151660 q^{79} - 29038 q^{81} + 61170 q^{89} + 251056 q^{91} - 189588 q^{99}+O(q^{100}) $ 2 * q + 244 * q^9 - 1554 * q^11 + 2290 * q^19 + 3124 * q^21 + 9840 * q^29 - 3604 * q^31 - 19448 * q^39 - 30246 * q^41 - 6714 * q^49 - 594 * q^51 + 67920 * q^59 + 94804 * q^61 + 40788 * q^69 + 15096 * q^71 + 151660 * q^79 - 29038 * q^81 + 61170 * q^89 + 251056 * q^91 - 189588 * 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
$3$	11.0000i	0.705650i	0.935689	+	0.352825i	$0.114779\pi$
$3$	11.0000i	0.705650i	−0.935689	+	0.352825i	$0.885221\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 142.000i	− 1.09533i	−0.836699	−	0.547663i	$-0.815518\pi$
$7$	− 142.000i	− 1.09533i	0.836699	−	0.547663i	$-0.184482\pi$
$8$	0	0
$9$	122.000	0.502058
$10$	0	0
$11$	−777.000	−1.93615	−0.968076	−	0.250658i	$-0.919353\pi$
$11$	−777.000	−1.93615	−0.968076	+	0.250658i	$0.919353\pi$
$12$	0	0
$13$	884.000i	1.45075i	0.688352	+	0.725377i	$0.258335\pi$
$13$	884.000i	1.45075i	−0.688352	+	0.725377i	$0.741665\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	27.0000i	0.0226590i	0.999936	+	0.0113295i	$0.00360637\pi$
$17$	27.0000i	0.0226590i	−0.999936	+	0.0113295i	$0.996394\pi$
$18$	0	0
$19$	1145.00	0.727648	0.363824	−	0.931468i	$-0.381471\pi$
$19$	1145.00	0.727648	0.363824	+	0.931468i	$0.381471\pi$
$20$	0	0
$21$	1562.00	0.772917
$22$	0	0
$23$	− 1854.00i	− 0.730786i	−0.930853	−	0.365393i	$-0.880935\pi$
$23$	− 1854.00i	− 0.730786i	0.930853	−	0.365393i	$-0.119065\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4015.00i	1.05993i
$28$	0	0
$29$	4920.00	1.08635	0.543175	−	0.839619i	$-0.317222\pi$
$29$	4920.00	1.08635	0.543175	+	0.839619i	$0.317222\pi$
$30$	0	0
$31$	−1802.00	−0.336783	−0.168392	−	0.985720i	$-0.553857\pi$
$31$	−1802.00	−0.336783	−0.168392	+	0.985720i	$0.553857\pi$
$32$	0	0
$33$	− 8547.00i	− 1.36625i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 13178.0i	− 1.58251i	−0.611489	−	0.791253i	$-0.709429\pi$
$37$	− 13178.0i	− 1.58251i	0.611489	−	0.791253i	$-0.290571\pi$
$38$	0	0
$39$	−9724.00	−1.02373
$40$	0	0
$41$	−15123.0	−1.40501	−0.702503	−	0.711681i	$-0.747934\pi$
$41$	−15123.0	−1.40501	−0.702503	+	0.711681i	$0.747934\pi$
$42$	0	0
$43$	− 7844.00i	− 0.646944i	−0.946238	−	0.323472i	$-0.895150\pi$
$43$	− 7844.00i	− 0.646944i	0.946238	−	0.323472i	$-0.104850\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6732.00i	− 0.444528i	−0.974986	−	0.222264i	$-0.928655\pi$
$47$	− 6732.00i	− 0.444528i	0.974986	−	0.222264i	$-0.0713447\pi$
$48$	0	0
$49$	−3357.00	−0.199738
$50$	0	0
$51$	−297.000	−0.0159894
$52$	0	0
$53$	3414.00i	0.166945i	0.996510	+	0.0834726i	$0.0266011\pi$
$53$	3414.00i	0.166945i	−0.996510	+	0.0834726i	$0.973399\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12595.0i	0.513465i
$58$	0	0
$59$	33960.0	1.27010	0.635050	−	0.772471i	$-0.280980\pi$
$59$	33960.0	1.27010	0.635050	+	0.772471i	$0.280980\pi$
$60$	0	0
$61$	47402.0	1.63107	0.815534	−	0.578709i	$-0.196443\pi$
$61$	47402.0	1.63107	0.815534	+	0.578709i	$0.196443\pi$
$62$	0	0
$63$	− 17324.0i	− 0.549917i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13177.0i	− 0.358616i	−0.983793	−	0.179308i	$-0.942614\pi$
$67$	− 13177.0i	− 0.358616i	0.983793	−	0.179308i	$-0.0573858\pi$
$68$	0	0
$69$	20394.0	0.515679
$70$	0	0
$71$	7548.00	0.177699	0.0888497	−	0.996045i	$-0.471681\pi$
$71$	7548.00	0.177699	0.0888497	+	0.996045i	$0.471681\pi$
$72$	0	0
$73$	− 59821.0i	− 1.31385i	−0.753955	−	0.656926i	$-0.771856\pi$
$73$	− 59821.0i	− 1.31385i	0.753955	−	0.656926i	$-0.228144\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	110334.i	2.12072i
$78$	0	0
$79$	75830.0	1.36702	0.683508	−	0.729943i	$-0.260454\pi$
$79$	75830.0	1.36702	0.683508	+	0.729943i	$0.260454\pi$
$80$	0	0
$81$	−14519.0	−0.245881
$82$	0	0
$83$	− 46299.0i	− 0.737694i	−0.929490	−	0.368847i	$-0.879753\pi$
$83$	− 46299.0i	− 0.737694i	0.929490	−	0.368847i	$-0.120247\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	54120.0i	0.766584i
$88$	0	0
$89$	30585.0	0.409292	0.204646	−	0.978836i	$-0.434396\pi$
$89$	30585.0	0.409292	0.204646	+	0.978836i	$0.434396\pi$
$90$	0	0
$91$	125528.	1.58905
$92$	0	0
$93$	− 19822.0i	− 0.237651i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 104018.i	− 1.12248i	−0.827653	−	0.561241i	$-0.810324\pi$
$97$	− 104018.i	− 1.12248i	0.827653	−	0.561241i	$-0.189676\pi$
$98$	0	0
$99$	−94794.0	−0.972060
$100$	0	0
$101$	−23898.0	−0.233109	−0.116554	−	0.993184i	$-0.537185\pi$
$101$	−23898.0	−0.233109	−0.116554	+	0.993184i	$0.537185\pi$
$102$	0	0
$103$	22636.0i	0.210236i	0.994460	+	0.105118i	$0.0335220\pi$
$103$	22636.0i	0.210236i	−0.994460	+	0.105118i	$0.966478\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	60633.0i	0.511976i	0.966680	+	0.255988i	$0.0824008\pi$
$107$	60633.0i	0.511976i	−0.966680	+	0.255988i	$0.917599\pi$
$108$	0	0
$109$	7090.00	0.0571584	0.0285792	−	0.999592i	$-0.490902\pi$
$109$	7090.00	0.0571584	0.0285792	+	0.999592i	$0.490902\pi$
$110$	0	0
$111$	144958.	1.11670
$112$	0	0
$113$	− 128841.i	− 0.949201i	−0.880201	−	0.474600i	$-0.842593\pi$
$113$	− 128841.i	− 0.949201i	0.880201	−	0.474600i	$-0.157407\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	107848.i	0.728362i
$118$	0	0
$119$	3834.00	0.0248190
$120$	0	0
$121$	442678.	2.74868
$122$	0	0
$123$	− 166353.i	− 0.991443i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	141338.i	0.777588i	0.921325	+	0.388794i	$0.127108\pi$
$127$	141338.i	0.777588i	−0.921325	+	0.388794i	$0.872892\pi$
$128$	0	0
$129$	86284.0	0.456516
$130$	0	0
$131$	−80052.0	−0.407562	−0.203781	−	0.979016i	$-0.565323\pi$
$131$	−80052.0	−0.407562	−0.203781	+	0.979016i	$0.565323\pi$
$132$	0	0
$133$	− 162590.i	− 0.797012i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 32253.0i	− 0.146814i	−0.997302	−	0.0734072i	$-0.976613\pi$
$137$	− 32253.0i	− 0.146814i	0.997302	−	0.0734072i	$-0.0233873\pi$
$138$	0	0
$139$	394865.	1.73345	0.866726	−	0.498785i	$-0.166220\pi$
$139$	394865.	1.73345	0.866726	+	0.498785i	$0.166220\pi$
$140$	0	0
$141$	74052.0	0.313682
$142$	0	0
$143$	− 686868.i	− 2.80888i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 36927.0i	− 0.140945i
$148$	0	0
$149$	491400.	1.81330	0.906650	−	0.421884i	$-0.138631\pi$
$149$	491400.	1.81330	0.906650	+	0.421884i	$0.138631\pi$
$150$	0	0
$151$	−200402.	−0.715253	−0.357626	−	0.933865i	$-0.616414\pi$
$151$	−200402.	−0.715253	−0.357626	+	0.933865i	$0.616414\pi$
$152$	0	0
$153$	3294.00i	0.0113761i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22942.0i	0.0742818i	0.999310	+	0.0371409i	$0.0118250\pi$
$157$	22942.0i	0.0742818i	−0.999310	+	0.0371409i	$0.988175\pi$
$158$	0	0
$159$	−37554.0	−0.117805
$160$	0	0
$161$	−263268.	−0.800448
$162$	0	0
$163$	336241.i	0.991246i	0.868538	+	0.495623i	$0.165060\pi$
$163$	336241.i	0.991246i	−0.868538	+	0.495623i	$0.834940\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	59748.0i	0.165780i	0.996559	+	0.0828900i	$0.0264150\pi$
$167$	59748.0i	0.165780i	−0.996559	+	0.0828900i	$0.973585\pi$
$168$	0	0
$169$	−410163.	−1.10469
$170$	0	0
$171$	139690.	0.365321
$172$	0	0
$173$	− 60696.0i	− 0.154186i	−0.997024	−	0.0770930i	$-0.975436\pi$
$173$	− 60696.0i	− 0.154186i	0.997024	−	0.0770930i	$-0.0245638\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	373560.i	0.896246i
$178$	0	0
$179$	−7995.00	−0.0186503	−0.00932515	−	0.999957i	$-0.502968\pi$
$179$	−7995.00	−0.0186503	−0.00932515	+	0.999957i	$0.502968\pi$
$180$	0	0
$181$	−454798.	−1.03186	−0.515932	−	0.856630i	$-0.672554\pi$
$181$	−454798.	−1.03186	−0.515932	+	0.856630i	$0.672554\pi$
$182$	0	0
$183$	521422.i	1.15096i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20979.0i	− 0.0438713i
$188$	0	0
$189$	570130.	1.16097
$190$	0	0
$191$	428298.	0.849499	0.424749	−	0.905311i	$-0.360362\pi$
$191$	428298.	0.849499	0.424749	+	0.905311i	$0.360362\pi$
$192$	0	0
$193$	− 835531.i	− 1.61462i	−0.590130	−	0.807308i	$-0.700924\pi$
$193$	− 835531.i	− 1.61462i	0.590130	−	0.807308i	$-0.299076\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 678318.i	− 1.24528i	−0.782508	−	0.622641i	$-0.786060\pi$
$197$	− 678318.i	− 1.24528i	0.782508	−	0.622641i	$-0.213940\pi$
$198$	0	0
$199$	−31900.0	−0.0571029	−0.0285514	−	0.999592i	$-0.509089\pi$
$199$	−31900.0	−0.0571029	−0.0285514	+	0.999592i	$0.509089\pi$
$200$	0	0
$201$	144947.	0.253057
$202$	0	0
$203$	− 698640.i	− 1.18991i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 226188.i	− 0.366897i
$208$	0	0
$209$	−889665.	−1.40884
$210$	0	0
$211$	423673.	0.655126	0.327563	−	0.944829i	$-0.393773\pi$
$211$	423673.	0.655126	0.327563	+	0.944829i	$0.393773\pi$
$212$	0	0
$213$	83028.0i	0.125394i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	255884.i	0.368887i
$218$	0	0
$219$	658031.	0.927120
$220$	0	0
$221$	−23868.0	−0.0328727
$222$	0	0
$223$	− 398204.i	− 0.536221i	−0.963388	−	0.268110i	$-0.913601\pi$
$223$	− 398204.i	− 0.536221i	0.963388	−	0.268110i	$-0.0863992\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.25761e6i	− 1.61988i	−0.586515	−	0.809938i	$-0.699500\pi$
$227$	− 1.25761e6i	− 1.61988i	0.586515	−	0.809938i	$-0.300500\pi$
$228$	0	0
$229$	−203780.	−0.256787	−0.128393	−	0.991723i	$-0.540982\pi$
$229$	−203780.	−0.256787	−0.128393	+	0.991723i	$0.540982\pi$
$230$	0	0
$231$	−1.21367e6	−1.49648
$232$	0	0
$233$	823974.i	0.994314i	0.867661	+	0.497157i	$0.165623\pi$
$233$	823974.i	0.994314i	−0.867661	+	0.497157i	$0.834377\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	834130.i	0.964635i
$238$	0	0
$239$	−555960.	−0.629577	−0.314788	−	0.949162i	$-0.601934\pi$
$239$	−555960.	−0.629577	−0.314788	+	0.949162i	$0.601934\pi$
$240$	0	0
$241$	523577.	0.580681	0.290341	−	0.956923i	$-0.406231\pi$
$241$	523577.	0.580681	0.290341	+	0.956923i	$0.406231\pi$
$242$	0	0
$243$	815936.i	0.886422i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.01218e6i	1.05564i
$248$	0	0
$249$	509289.	0.520554
$250$	0	0
$251$	−113127.	−0.113340	−0.0566698	−	0.998393i	$-0.518048\pi$
$251$	−113127.	−0.113340	−0.0566698	+	0.998393i	$0.518048\pi$
$252$	0	0
$253$	1.44056e6i	1.41491i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 872958.i	− 0.824443i	−0.911084	−	0.412221i	$-0.864753\pi$
$257$	− 872958.i	− 0.824443i	0.911084	−	0.412221i	$-0.135247\pi$
$258$	0	0
$259$	−1.87128e6	−1.73336
$260$	0	0
$261$	600240.	0.545411
$262$	0	0
$263$	1.64647e6i	1.46779i	0.679264	+	0.733894i	$0.262299\pi$
$263$	1.64647e6i	1.46779i	−0.679264	+	0.733894i	$0.737701\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	336435.i	0.288817i
$268$	0	0
$269$	−1.78872e6	−1.50717	−0.753584	−	0.657352i	$-0.771677\pi$
$269$	−1.78872e6	−1.50717	−0.753584	+	0.657352i	$0.771677\pi$
$270$	0	0
$271$	−1.12140e6	−0.927552	−0.463776	−	0.885953i	$-0.653506\pi$
$271$	−1.12140e6	−0.927552	−0.463776	+	0.885953i	$0.653506\pi$
$272$	0	0
$273$	1.38081e6i	1.12131i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	598312.i	0.468520i	0.972174	+	0.234260i	$0.0752667\pi$
$277$	598312.i	0.468520i	−0.972174	+	0.234260i	$0.924733\pi$
$278$	0	0
$279$	−219844.	−0.169085
$280$	0	0
$281$	−1.53050e6	−1.15629	−0.578145	−	0.815934i	$-0.696223\pi$
$281$	−1.53050e6	−1.15629	−0.578145	+	0.815934i	$0.696223\pi$
$282$	0	0
$283$	1.79700e6i	1.33377i	0.745159	+	0.666887i	$0.232374\pi$
$283$	1.79700e6i	1.33377i	−0.745159	+	0.666887i	$0.767626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.14747e6i	1.53894i
$288$	0	0
$289$	1.41913e6	0.999487
$290$	0	0
$291$	1.14420e6	0.792079
$292$	0	0
$293$	754494.i	0.513437i	0.966486	+	0.256718i	$0.0826412\pi$
$293$	754494.i	0.513437i	−0.966486	+	0.256718i	$0.917359\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.11965e6i	− 2.05218i
$298$	0	0
$299$	1.63894e6	1.06019
$300$	0	0
$301$	−1.11385e6	−0.708614
$302$	0	0
$303$	− 262878.i	− 0.164493i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.96627e6i	− 1.19068i	−0.803472	−	0.595342i	$-0.797017\pi$
$307$	− 1.96627e6i	− 1.19068i	0.803472	−	0.595342i	$-0.202983\pi$
$308$	0	0
$309$	−248996.	−0.148353
$310$	0	0
$311$	599298.	0.351352	0.175676	−	0.984448i	$-0.443789\pi$
$311$	599298.	0.351352	0.175676	+	0.984448i	$0.443789\pi$
$312$	0	0
$313$	− 721366.i	− 0.416193i	−0.978108	−	0.208097i	$-0.933273\pi$
$313$	− 721366.i	− 0.416193i	0.978108	−	0.208097i	$-0.0667268\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 102348.i	− 0.0572046i	−0.999591	−	0.0286023i	$-0.990894\pi$
$317$	− 102348.i	− 0.0572046i	0.999591	−	0.0286023i	$-0.00910564\pi$
$318$	0	0
$319$	−3.82284e6	−2.10334
$320$	0	0
$321$	−666963.	−0.361276
$322$	0	0
$323$	30915.0i	0.0164878i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	77990.0i	0.0403338i
$328$	0	0
$329$	−955944.	−0.486903
$330$	0	0
$331$	−1.31048e6	−0.657445	−0.328722	−	0.944427i	$-0.606618\pi$
$331$	−1.31048e6	−0.657445	−0.328722	+	0.944427i	$0.606618\pi$
$332$	0	0
$333$	− 1.60772e6i	− 0.794509i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	804397.i	0.385830i	0.981215	+	0.192915i	$0.0617941\pi$
$337$	804397.i	0.385830i	−0.981215	+	0.192915i	$0.938206\pi$
$338$	0	0
$339$	1.41725e6	0.669804
$340$	0	0
$341$	1.40015e6	0.652063
$342$	0	0
$343$	− 1.90990e6i	− 0.876547i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.88321e6i	− 1.28544i	−0.766101	−	0.642720i	$-0.777806\pi$
$347$	− 2.88321e6i	− 1.28544i	0.766101	−	0.642720i	$-0.222194\pi$
$348$	0	0
$349$	−1.27355e6	−0.559696	−0.279848	−	0.960044i	$-0.590284\pi$
$349$	−1.27355e6	−0.559696	−0.279848	+	0.960044i	$0.590284\pi$
$350$	0	0
$351$	−3.54926e6	−1.53769
$352$	0	0
$353$	2.83061e6i	1.20905i	0.796587	+	0.604524i	$0.206637\pi$
$353$	2.83061e6i	1.20905i	−0.796587	+	0.604524i	$0.793363\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	42174.0i	0.0175136i
$358$	0	0
$359$	−981090.	−0.401766	−0.200883	−	0.979615i	$-0.564381\pi$
$359$	−981090.	−0.401766	−0.200883	+	0.979615i	$0.564381\pi$
$360$	0	0
$361$	−1.16507e6	−0.470528
$362$	0	0
$363$	4.86946e6i	1.93961i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.19105e6i	− 1.62427i	−0.583470	−	0.812134i	$-0.698306\pi$
$367$	− 4.19105e6i	− 1.62427i	0.583470	−	0.812134i	$-0.301694\pi$
$368$	0	0
$369$	−1.84501e6	−0.705394
$370$	0	0
$371$	484788.	0.182859
$372$	0	0
$373$	3.23455e6i	1.20377i	0.798584	+	0.601883i	$0.205583\pi$
$373$	3.23455e6i	1.20377i	−0.798584	+	0.601883i	$0.794417\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.34928e6i	1.57603i
$378$	0	0
$379$	1.39036e6	0.497196	0.248598	−	0.968607i	$-0.420030\pi$
$379$	1.39036e6	0.497196	0.248598	+	0.968607i	$0.420030\pi$
$380$	0	0
$381$	−1.55472e6	−0.548705
$382$	0	0
$383$	− 1.14197e6i	− 0.397795i	−0.980020	−	0.198897i	$-0.936264\pi$
$383$	− 1.14197e6i	− 0.397795i	0.980020	−	0.198897i	$-0.0637361\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 956968.i	− 0.324803i
$388$	0	0
$389$	−3.46299e6	−1.16032	−0.580159	−	0.814503i	$-0.697010\pi$
$389$	−3.46299e6	−1.16032	−0.580159	+	0.814503i	$0.697010\pi$
$390$	0	0
$391$	50058.0	0.0165589
$392$	0	0
$393$	− 880572.i	− 0.287596i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.94007e6i	− 1.89154i	−0.324839	−	0.945769i	$-0.605310\pi$
$397$	− 5.94007e6i	− 1.89154i	0.324839	−	0.945769i	$-0.394690\pi$
$398$	0	0
$399$	1.78849e6	0.562412
$400$	0	0
$401$	−2.27412e6	−0.706241	−0.353121	−	0.935578i	$-0.614879\pi$
$401$	−2.27412e6	−0.706241	−0.353121	+	0.935578i	$0.614879\pi$
$402$	0	0
$403$	− 1.59297e6i	− 0.488590i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.02393e7i	3.06397i
$408$	0	0
$409$	4.29552e6	1.26972	0.634859	−	0.772628i	$-0.281058\pi$
$409$	4.29552e6	1.26972	0.634859	+	0.772628i	$0.281058\pi$
$410$	0	0
$411$	354783.	0.103600
$412$	0	0
$413$	− 4.82232e6i	− 1.39117i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.34352e6i	1.22321i
$418$	0	0
$419$	1.79705e6	0.500062	0.250031	−	0.968238i	$-0.419559\pi$
$419$	1.79705e6	0.500062	0.250031	+	0.968238i	$0.419559\pi$
$420$	0	0
$421$	−257548.	−0.0708195	−0.0354098	−	0.999373i	$-0.511274\pi$
$421$	−257548.	−0.0708195	−0.0354098	+	0.999373i	$0.511274\pi$
$422$	0	0
$423$	− 821304.i	− 0.223179i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6.73108e6i	− 1.78655i
$428$	0	0
$429$	7.55555e6	1.98209
$430$	0	0
$431$	−2.22910e6	−0.578012	−0.289006	−	0.957327i	$-0.593325\pi$
$431$	−2.22910e6	−0.578012	−0.289006	+	0.957327i	$0.593325\pi$
$432$	0	0
$433$	− 4.20585e6i	− 1.07804i	−0.842294	−	0.539019i	$-0.818795\pi$
$433$	− 4.20585e6i	− 1.07804i	0.842294	−	0.539019i	$-0.181205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.12283e6i	− 0.531755i
$438$	0	0
$439$	352640.	0.0873314	0.0436657	−	0.999046i	$-0.486096\pi$
$439$	352640.	0.0873314	0.0436657	+	0.999046i	$0.486096\pi$
$440$	0	0
$441$	−409554.	−0.100280
$442$	0	0
$443$	− 1.28362e6i	− 0.310761i	−0.987855	−	0.155381i	$-0.950340\pi$
$443$	− 1.28362e6i	− 0.310761i	0.987855	−	0.155381i	$-0.0496604\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.40540e6i	1.27956i
$448$	0	0
$449$	2.10398e6	0.492521	0.246260	−	0.969204i	$-0.420798\pi$
$449$	2.10398e6	0.492521	0.246260	+	0.969204i	$0.420798\pi$
$450$	0	0
$451$	1.17506e7	2.72031
$452$	0	0
$453$	− 2.20442e6i	− 0.504719i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 825233.i	− 0.184836i	−0.995720	−	0.0924179i	$-0.970540\pi$
$457$	− 825233.i	− 0.184836i	0.995720	−	0.0924179i	$-0.0294596\pi$
$458$	0	0
$459$	−108405.	−0.0240169
$460$	0	0
$461$	4.50145e6	0.986507	0.493254	−	0.869886i	$-0.335807\pi$
$461$	4.50145e6	0.986507	0.493254	+	0.869886i	$0.335807\pi$
$462$	0	0
$463$	1.44212e6i	0.312642i	0.987706	+	0.156321i	$0.0499635\pi$
$463$	1.44212e6i	0.312642i	−0.987706	+	0.156321i	$0.950037\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	393348.i	0.0834612i	0.999129	+	0.0417306i	$0.0132871\pi$
$467$	393348.i	0.0834612i	−0.999129	+	0.0417306i	$0.986713\pi$
$468$	0	0
$469$	−1.87113e6	−0.392801
$470$	0	0
$471$	−252362.	−0.0524169
$472$	0	0
$473$	6.09479e6i	1.25258i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	416508.i	0.0838161i
$478$	0	0
$479$	−9.17697e6	−1.82751	−0.913757	−	0.406262i	$-0.866832\pi$
$479$	−9.17697e6	−1.82751	−0.913757	+	0.406262i	$0.866832\pi$
$480$	0	0
$481$	1.16494e7	2.29583
$482$	0	0
$483$	− 2.89595e6i	− 0.564837i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.60598e6i	1.26216i	0.775717	+	0.631080i	$0.217388\pi$
$487$	6.60598e6i	1.26216i	−0.775717	+	0.631080i	$0.782612\pi$
$488$	0	0
$489$	−3.69865e6	−0.699473
$490$	0	0
$491$	−38052.0	−0.00712318	−0.00356159	−	0.999994i	$-0.501134\pi$
$491$	−38052.0	−0.00712318	−0.00356159	+	0.999994i	$0.501134\pi$
$492$	0	0
$493$	132840.i	0.0246157i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.07182e6i	− 0.194639i
$498$	0	0
$499$	6.85670e6	1.23272	0.616359	−	0.787465i	$-0.288607\pi$
$499$	6.85670e6	1.23272	0.616359	+	0.787465i	$0.288607\pi$
$500$	0	0
$501$	−657228.	−0.116983
$502$	0	0
$503$	− 8.20016e6i	− 1.44512i	−0.691311	−	0.722558i	$-0.742966\pi$
$503$	− 8.20016e6i	− 1.44512i	0.691311	−	0.722558i	$-0.257034\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.51179e6i	− 0.779524i
$508$	0	0
$509$	−4.06581e6	−0.695589	−0.347794	−	0.937571i	$-0.613069\pi$
$509$	−4.06581e6	−0.695589	−0.347794	+	0.937571i	$0.613069\pi$
$510$	0	0
$511$	−8.49458e6	−1.43910
$512$	0	0
$513$	4.59718e6i	0.771254i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.23076e6i	0.860674i
$518$	0	0
$519$	667656.	0.108801
$520$	0	0
$521$	5.28408e6	0.852854	0.426427	−	0.904522i	$-0.359772\pi$
$521$	5.28408e6	0.852854	0.426427	+	0.904522i	$0.359772\pi$
$522$	0	0
$523$	− 2.53383e6i	− 0.405063i	−0.979276	−	0.202532i	$-0.935083\pi$
$523$	− 2.53383e6i	− 0.405063i	0.979276	−	0.202532i	$-0.0649169\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 48654.0i	− 0.00763119i
$528$	0	0
$529$	2.99903e6	0.465952
$530$	0	0
$531$	4.14312e6	0.637663
$532$	0	0
$533$	− 1.33687e7i	− 2.03832i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 87945.0i	− 0.0131606i
$538$	0	0
$539$	2.60839e6	0.386723
$540$	0	0
$541$	498752.	0.0732641	0.0366321	−	0.999329i	$-0.488337\pi$
$541$	498752.	0.0732641	0.0366321	+	0.999329i	$0.488337\pi$
$542$	0	0
$543$	− 5.00278e6i	− 0.728135i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.00269e6i	0.429084i	0.976715	+	0.214542i	$0.0688259\pi$
$547$	3.00269e6i	0.429084i	−0.976715	+	0.214542i	$0.931174\pi$
$548$	0	0
$549$	5.78304e6	0.818890
$550$	0	0
$551$	5.63340e6	0.790481
$552$	0	0
$553$	− 1.07679e7i	− 1.49733i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.27373e7i	1.73956i	0.493441	+	0.869779i	$0.335739\pi$
$557$	1.27373e7i	1.73956i	−0.493441	+	0.869779i	$0.664261\pi$
$558$	0	0
$559$	6.93410e6	0.938556
$560$	0	0
$561$	230769.	0.0309578
$562$	0	0
$563$	5.97082e6i	0.793894i	0.917841	+	0.396947i	$0.129930\pi$
$563$	5.97082e6i	0.793894i	−0.917841	+	0.396947i	$0.870070\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.06170e6i	0.269319i
$568$	0	0
$569$	9.26906e6	1.20020	0.600102	−	0.799924i	$-0.295127\pi$
$569$	9.26906e6	1.20020	0.600102	+	0.799924i	$0.295127\pi$
$570$	0	0
$571$	3.89535e6	0.499984	0.249992	−	0.968248i	$-0.419572\pi$
$571$	3.89535e6	0.499984	0.249992	+	0.968248i	$0.419572\pi$
$572$	0	0
$573$	4.71128e6i	0.599449i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 7.29416e6i	− 0.912086i	−0.889958	−	0.456043i	$-0.849266\pi$
$577$	− 7.29416e6i	− 0.912086i	0.889958	−	0.456043i	$-0.150734\pi$
$578$	0	0
$579$	9.19084e6	1.13935
$580$	0	0
$581$	−6.57446e6	−0.808015
$582$	0	0
$583$	− 2.65268e6i	− 0.323231i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.72820e6i	− 1.04551i	−0.852482	−	0.522756i	$-0.824904\pi$
$587$	− 8.72820e6i	− 1.04551i	0.852482	−	0.522756i	$-0.175096\pi$
$588$	0	0
$589$	−2.06329e6	−0.245060
$590$	0	0
$591$	7.46150e6	0.878734
$592$	0	0
$593$	− 1.30963e7i	− 1.52937i	−0.644407	−	0.764683i	$-0.722896\pi$
$593$	− 1.30963e7i	− 1.52937i	0.644407	−	0.764683i	$-0.277104\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 350900.i	− 0.0402947i
$598$	0	0
$599$	−1.30168e7	−1.48231	−0.741155	−	0.671334i	$-0.765721\pi$
$599$	−1.30168e7	−1.48231	−0.741155	+	0.671334i	$0.765721\pi$
$600$	0	0
$601$	−9.93997e6	−1.12253	−0.561266	−	0.827635i	$-0.689686\pi$
$601$	−9.93997e6	−1.12253	−0.561266	+	0.827635i	$0.689686\pi$
$602$	0	0
$603$	− 1.60759e6i	− 0.180046i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.56438e7i	1.72334i	0.507470	+	0.861670i	$0.330581\pi$
$607$	1.56438e7i	1.72334i	−0.507470	+	0.861670i	$0.669419\pi$
$608$	0	0
$609$	7.68504e6	0.839659
$610$	0	0
$611$	5.95109e6	0.644901
$612$	0	0
$613$	9.33793e6i	1.00369i	0.864958	+	0.501845i	$0.167345\pi$
$613$	9.33793e6i	1.00369i	−0.864958	+	0.501845i	$0.832655\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.06680e6i	0.535823i	0.963444	+	0.267911i	$0.0863334\pi$
$617$	5.06680e6i	0.535823i	−0.963444	+	0.267911i	$0.913667\pi$
$618$	0	0
$619$	1.37670e7	1.44415	0.722077	−	0.691813i	$-0.243188\pi$
$619$	1.37670e7	1.44415	0.722077	+	0.691813i	$0.243188\pi$
$620$	0	0
$621$	7.44381e6	0.774580
$622$	0	0
$623$	− 4.34307e6i	− 0.448308i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 9.78632e6i	− 0.994146i
$628$	0	0
$629$	355806.	0.0358580
$630$	0	0
$631$	−2.07060e6	−0.207025	−0.103513	−	0.994628i	$-0.533008\pi$
$631$	−2.07060e6	−0.207025	−0.103513	+	0.994628i	$0.533008\pi$
$632$	0	0
$633$	4.66040e6i	0.462290i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.96759e6i	− 0.289771i
$638$	0	0
$639$	920856.	0.0892153
$640$	0	0
$641$	−1.79114e7	−1.72181	−0.860903	−	0.508768i	$-0.830101\pi$
$641$	−1.79114e7	−1.72181	−0.860903	+	0.508768i	$0.830101\pi$
$642$	0	0
$643$	1.71414e7i	1.63500i	0.575929	+	0.817500i	$0.304641\pi$
$643$	1.71414e7i	1.63500i	−0.575929	+	0.817500i	$0.695359\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.48773e6i	− 0.797133i	−0.917139	−	0.398567i	$-0.869508\pi$
$647$	− 8.48773e6i	− 0.797133i	0.917139	−	0.398567i	$-0.130492\pi$
$648$	0	0
$649$	−2.63869e7	−2.45910
$650$	0	0
$651$	−2.81472e6	−0.260306
$652$	0	0
$653$	− 2.45479e6i	− 0.225284i	−0.993636	−	0.112642i	$-0.964069\pi$
$653$	− 2.45479e6i	− 0.225284i	0.993636	−	0.112642i	$-0.0359313\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 7.29816e6i	− 0.659630i
$658$	0	0
$659$	−5.91557e6	−0.530619	−0.265309	−	0.964163i	$-0.585474\pi$
$659$	−5.91557e6	−0.530619	−0.265309	+	0.964163i	$0.585474\pi$
$660$	0	0
$661$	4.33095e6	0.385549	0.192775	−	0.981243i	$-0.438251\pi$
$661$	4.33095e6	0.385549	0.192775	+	0.981243i	$0.438251\pi$
$662$	0	0
$663$	− 262548.i	− 0.0231966i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.12168e6i	− 0.793890i
$668$	0	0
$669$	4.38024e6	0.378384
$670$	0	0
$671$	−3.68314e7	−3.15799
$672$	0	0
$673$	− 9.13985e6i	− 0.777860i	−0.921267	−	0.388930i	$-0.872845\pi$
$673$	− 9.13985e6i	− 0.777860i	0.921267	−	0.388930i	$-0.127155\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4.57229e6i	− 0.383409i	−0.981453	−	0.191704i	$-0.938599\pi$
$677$	− 4.57229e6i	− 0.383409i	0.981453	−	0.191704i	$-0.0614015\pi$
$678$	0	0
$679$	−1.47706e7	−1.22948
$680$	0	0
$681$	1.38337e7	1.14307
$682$	0	0
$683$	− 1.53221e7i	− 1.25681i	−0.777888	−	0.628403i	$-0.783709\pi$
$683$	− 1.53221e7i	− 1.25681i	0.777888	−	0.628403i	$-0.216291\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.24158e6i	− 0.181202i
$688$	0	0
$689$	−3.01798e6	−0.242196
$690$	0	0
$691$	−7.02548e6	−0.559733	−0.279866	−	0.960039i	$-0.590290\pi$
$691$	−7.02548e6	−0.559733	−0.279866	+	0.960039i	$0.590290\pi$
$692$	0	0
$693$	1.34607e7i	1.06472i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 408321.i	− 0.0318361i
$698$	0	0
$699$	−9.06371e6	−0.701638
$700$	0	0
$701$	7.91125e6	0.608065	0.304033	−	0.952662i	$-0.401667\pi$
$701$	7.91125e6	0.608065	0.304033	+	0.952662i	$0.401667\pi$
$702$	0	0
$703$	− 1.50888e7i	− 1.15151i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.39352e6i	0.255330i
$708$	0	0
$709$	1.54485e7	1.15418	0.577088	−	0.816682i	$-0.304189\pi$
$709$	1.54485e7	1.15418	0.577088	+	0.816682i	$0.304189\pi$
$710$	0	0
$711$	9.25126e6	0.686320
$712$	0	0
$713$	3.34091e6i	0.246116i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.11556e6i	− 0.444261i
$718$	0	0
$719$	2.30544e7	1.66315	0.831574	−	0.555414i	$-0.187440\pi$
$719$	2.30544e7	1.66315	0.831574	+	0.555414i	$0.187440\pi$
$720$	0	0
$721$	3.21431e6	0.230277
$722$	0	0
$723$	5.75935e6i	0.409758i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.62905e7i	1.14314i	0.820555	+	0.571568i	$0.193665\pi$
$727$	1.62905e7i	1.14314i	−0.820555	+	0.571568i	$0.806335\pi$
$728$	0	0
$729$	−1.25034e7	−0.871384
$730$	0	0
$731$	211788.	0.0146591
$732$	0	0
$733$	1.28279e7i	0.881853i	0.897543	+	0.440927i	$0.145350\pi$
$733$	1.28279e7i	0.881853i	−0.897543	+	0.440927i	$0.854650\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.02385e7i	0.694335i
$738$	0	0
$739$	−1.24535e7	−0.838840	−0.419420	−	0.907792i	$-0.637766\pi$
$739$	−1.24535e7	−0.838840	−0.419420	+	0.907792i	$0.637766\pi$
$740$	0	0
$741$	−1.11340e7	−0.744912
$742$	0	0
$743$	2.63247e7i	1.74941i	0.484656	+	0.874705i	$0.338945\pi$
$743$	2.63247e7i	1.74941i	−0.484656	+	0.874705i	$0.661055\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.64848e6i	− 0.370365i
$748$	0	0
$749$	8.60989e6	0.560780
$750$	0	0
$751$	1.74994e7	1.13220	0.566102	−	0.824335i	$-0.308451\pi$
$751$	1.74994e7	1.13220	0.566102	+	0.824335i	$0.308451\pi$
$752$	0	0
$753$	− 1.24440e6i	− 0.0799782i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.46381e6i	− 0.219692i	−0.993949	−	0.109846i	$-0.964964\pi$
$757$	− 3.46381e6i	− 0.219692i	0.993949	−	0.109846i	$-0.0350358\pi$
$758$	0	0
$759$	−1.58461e7	−0.998433
$760$	0	0
$761$	−1.26175e7	−0.789792	−0.394896	−	0.918726i	$-0.629219\pi$
$761$	−1.26175e7	−0.789792	−0.394896	+	0.918726i	$0.629219\pi$
$762$	0	0
$763$	− 1.00678e6i	− 0.0626070i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.00206e7i	1.84260i
$768$	0	0
$769$	−5.70804e6	−0.348074	−0.174037	−	0.984739i	$-0.555681\pi$
$769$	−5.70804e6	−0.348074	−0.174037	+	0.984739i	$0.555681\pi$
$770$	0	0
$771$	9.60254e6	0.581768
$772$	0	0
$773$	1.20827e7i	0.727303i	0.931535	+	0.363652i	$0.118470\pi$
$773$	1.20827e7i	0.727303i	−0.931535	+	0.363652i	$0.881530\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.05840e7i	− 1.22315i
$778$	0	0
$779$	−1.73158e7	−1.02235
$780$	0	0
$781$	−5.86480e6	−0.344053
$782$	0	0
$783$	1.97538e7i	1.15145i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.37636e7i	− 0.792126i	−0.918223	−	0.396063i	$-0.870376\pi$
$787$	− 1.37636e7i	− 0.792126i	0.918223	−	0.396063i	$-0.129624\pi$
$788$	0	0
$789$	−1.81111e7	−1.03575
$790$	0	0
$791$	−1.82954e7	−1.03968
$792$	0	0
$793$	4.19034e7i	2.36628i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.77738e6i	0.489462i	0.969591	+	0.244731i	$0.0786997\pi$
$797$	8.77738e6i	0.489462i	−0.969591	+	0.244731i	$0.921300\pi$
$798$	0	0
$799$	181764.	0.0100726
$800$	0	0
$801$	3.73137e6	0.205488
$802$	0	0
$803$	4.64809e7i	2.54382i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.96759e7i	− 1.06353i
$808$	0	0
$809$	1.02046e7	0.548181	0.274091	−	0.961704i	$-0.411623\pi$
$809$	1.02046e7	0.548181	0.274091	+	0.961704i	$0.411623\pi$
$810$	0	0
$811$	1.17375e6	0.0626647	0.0313323	−	0.999509i	$-0.490025\pi$
$811$	1.17375e6	0.0626647	0.0313323	+	0.999509i	$0.490025\pi$
$812$	0	0
$813$	− 1.23354e7i	− 0.654527i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.98138e6i	− 0.470747i
$818$	0	0
$819$	1.53144e7	0.797794
$820$	0	0
$821$	−1.98062e7	−1.02552	−0.512759	−	0.858533i	$-0.671377\pi$
$821$	−1.98062e7	−1.02552	−0.512759	+	0.858533i	$0.671377\pi$
$822$	0	0
$823$	3.06722e7i	1.57850i	0.614070	+	0.789251i	$0.289531\pi$
$823$	3.06722e7i	1.57850i	−0.614070	+	0.789251i	$0.710469\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.55520e7i	1.29915i	0.760296	+	0.649577i	$0.225054\pi$
$827$	2.55520e7i	1.29915i	−0.760296	+	0.649577i	$0.774946\pi$
$828$	0	0
$829$	9.19402e6	0.464643	0.232321	−	0.972639i	$-0.425368\pi$
$829$	9.19402e6	0.464643	0.232321	+	0.972639i	$0.425368\pi$
$830$	0	0
$831$	−6.58143e6	−0.330611
$832$	0	0
$833$	− 90639.0i	− 0.00452588i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 7.23503e6i	− 0.356966i
$838$	0	0
$839$	−1.56910e7	−0.769564	−0.384782	−	0.923008i	$-0.625723\pi$
$839$	−1.56910e7	−0.769564	−0.384782	+	0.923008i	$0.625723\pi$
$840$	0	0
$841$	3.69525e6	0.180158
$842$	0	0
$843$	− 1.68355e7i	− 0.815937i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6.28603e7i	− 3.01070i
$848$	0	0
$849$	−1.97670e7	−0.941178
$850$	0	0
$851$	−2.44320e7	−1.15647
$852$	0	0
$853$	1.60111e6i	0.0753442i	0.999290	+	0.0376721i	$0.0119942\pi$
$853$	1.60111e6i	0.0753442i	−0.999290	+	0.0376721i	$0.988006\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.64613e7i	0.765616i	0.923828	+	0.382808i	$0.125043\pi$
$857$	1.64613e7i	0.765616i	−0.923828	+	0.382808i	$0.874957\pi$
$858$	0	0
$859$	1.96736e7	0.909705	0.454853	−	0.890567i	$-0.349692\pi$
$859$	1.96736e7	0.909705	0.454853	+	0.890567i	$0.349692\pi$
$860$	0	0
$861$	−2.36221e7	−1.08595
$862$	0	0
$863$	− 3.68068e7i	− 1.68229i	−0.540810	−	0.841145i	$-0.681882\pi$
$863$	− 3.68068e7i	− 1.68229i	0.540810	−	0.841145i	$-0.318118\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.56104e7i	0.705288i
$868$	0	0
$869$	−5.89199e7	−2.64675
$870$	0	0
$871$	1.16485e7	0.520264
$872$	0	0
$873$	− 1.26902e7i	− 0.563550i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.69596e7i	1.18363i	0.806075	+	0.591813i	$0.201588\pi$
$877$	2.69596e7i	1.18363i	−0.806075	+	0.591813i	$0.798412\pi$
$878$	0	0
$879$	−8.29943e6	−0.362307
$880$	0	0
$881$	3.47335e7	1.50768	0.753839	−	0.657059i	$-0.228200\pi$
$881$	3.47335e7	1.50768	0.753839	+	0.657059i	$0.228200\pi$
$882$	0	0
$883$	− 2.16187e7i	− 0.933101i	−0.884494	−	0.466551i	$-0.845497\pi$
$883$	− 2.16187e7i	− 0.933101i	0.884494	−	0.466551i	$-0.154503\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.48163e6i	0.191261i	0.995417	+	0.0956306i	$0.0304867\pi$
$887$	4.48163e6i	0.191261i	−0.995417	+	0.0956306i	$0.969513\pi$
$888$	0	0
$889$	2.00700e7	0.851712
$890$	0	0
$891$	1.12813e7	0.476062
$892$	0	0
$893$	− 7.70814e6i	− 0.323460i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.80283e7i	0.748124i
$898$	0	0
$899$	−8.86584e6	−0.365865
$900$	0	0
$901$	−92178.0	−0.00378282
$902$	0	0
$903$	− 1.22523e7i	− 0.500034i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.36639e7i	− 1.35877i	−0.733782	−	0.679385i	$-0.762247\pi$
$907$	− 3.36639e7i	− 1.35877i	0.733782	−	0.679385i	$-0.237753\pi$
$908$	0	0
$909$	−2.91556e6	−0.117034
$910$	0	0
$911$	1.03175e6	0.0411887	0.0205943	−	0.999788i	$-0.493444\pi$
$911$	1.03175e6	0.0411887	0.0205943	+	0.999788i	$0.493444\pi$
$912$	0	0
$913$	3.59743e7i	1.42829i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.13674e7i	0.446413i
$918$	0	0
$919$	4.10147e6	0.160196	0.0800978	−	0.996787i	$-0.474477\pi$
$919$	4.10147e6	0.160196	0.0800978	+	0.996787i	$0.474477\pi$
$920$	0	0
$921$	2.16289e7	0.840207
$922$	0	0
$923$	6.67243e6i	0.257798i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.76159e6i	0.105550i
$928$	0	0
$929$	−7.71603e6	−0.293329	−0.146664	−	0.989186i	$-0.546854\pi$
$929$	−7.71603e6	−0.293329	−0.146664	+	0.989186i	$0.546854\pi$
$930$	0	0
$931$	−3.84376e6	−0.145339
$932$	0	0
$933$	6.59228e6i	0.247931i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.38458e7i	− 1.63147i	−0.578426	−	0.815735i	$-0.696333\pi$
$937$	− 4.38458e7i	− 1.63147i	0.578426	−	0.815735i	$-0.303667\pi$
$938$	0	0
$939$	7.93503e6	0.293687
$940$	0	0
$941$	−1.00215e7	−0.368944	−0.184472	−	0.982838i	$-0.559058\pi$
$941$	−1.00215e7	−0.368944	−0.184472	+	0.982838i	$0.559058\pi$
$942$	0	0
$943$	2.80380e7i	1.02676i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.79530e7i	1.01287i	0.862279	+	0.506434i	$0.169037\pi$
$947$	2.79530e7i	1.01287i	−0.862279	+	0.506434i	$0.830963\pi$
$948$	0	0
$949$	5.28818e7	1.90608
$950$	0	0
$951$	1.12583e6	0.0403665
$952$	0	0
$953$	− 2.31811e7i	− 0.826803i	−0.910549	−	0.413401i	$-0.864341\pi$
$953$	− 2.31811e7i	− 0.826803i	0.910549	−	0.413401i	$-0.135659\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 4.20512e7i	− 1.48422i
$958$	0	0
$959$	−4.57993e6	−0.160810
$960$	0	0
$961$	−2.53819e7	−0.886577
$962$	0	0
$963$	7.39723e6i	0.257041i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.58435e6i	− 0.0544861i	−0.999629	−	0.0272430i	$-0.991327\pi$
$967$	− 1.58435e6i	− 0.0544861i	0.999629	−	0.0272430i	$-0.00867280\pi$
$968$	0	0
$969$	−340065.	−0.0116346
$970$	0	0
$971$	−3.44552e7	−1.17275	−0.586376	−	0.810039i	$-0.699446\pi$
$971$	−3.44552e7	−1.17275	−0.586376	+	0.810039i	$0.699446\pi$
$972$	0	0
$973$	− 5.60708e7i	− 1.89869i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.93599e7i	0.984052i	0.870581	+	0.492026i	$0.163743\pi$
$977$	2.93599e7i	0.984052i	−0.870581	+	0.492026i	$0.836257\pi$
$978$	0	0
$979$	−2.37645e7	−0.792452
$980$	0	0
$981$	864980.	0.0286968
$982$	0	0
$983$	− 8.93957e6i	− 0.295075i	−0.989056	−	0.147538i	$-0.952865\pi$
$983$	− 8.93957e6i	− 0.295075i	0.989056	−	0.147538i	$-0.0471348\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.05154e7i	− 0.343583i
$988$	0	0
$989$	−1.45428e7	−0.472777
$990$	0	0
$991$	1.78899e7	0.578660	0.289330	−	0.957229i	$-0.406567\pi$
$991$	1.78899e7	0.578660	0.289330	+	0.957229i	$0.406567\pi$
$992$	0	0
$993$	− 1.44152e7i	− 0.463926i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 3.30517e7i	− 1.05307i	−0.850155	−	0.526533i	$-0.823492\pi$
$997$	− 3.30517e7i	− 1.05307i	0.850155	−	0.526533i	$-0.176508\pi$
$998$	0	0
$999$	5.29097e7	1.67734

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.g.49.2		2
4.3	odd	2		50.6.b.c.49.2		2
5.2	odd	4		400.6.a.j.1.1		1
5.3	odd	4		400.6.a.e.1.1		1
5.4	even	2	inner	400.6.c.g.49.1		2
12.11	even	2		450.6.c.a.199.1		2
20.3	even	4		50.6.a.f.1.1	yes	1
20.7	even	4		50.6.a.a.1.1	✓	1
20.19	odd	2		50.6.b.c.49.1		2
60.23	odd	4		450.6.a.j.1.1		1
60.47	odd	4		450.6.a.n.1.1		1
60.59	even	2		450.6.c.a.199.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.a.a.1.1	✓	1	20.7	even	4
50.6.a.f.1.1	yes	1	20.3	even	4
50.6.b.c.49.1		2	20.19	odd	2
50.6.b.c.49.2		2	4.3	odd	2
400.6.a.e.1.1		1	5.3	odd	4
400.6.a.j.1.1		1	5.2	odd	4
400.6.c.g.49.1		2	5.4	even	2	inner
400.6.c.g.49.2		2	1.1	even	1	trivial
450.6.a.j.1.1		1	60.23	odd	4
450.6.a.n.1.1		1	60.47	odd	4
450.6.c.a.199.1		2	12.11	even	2
450.6.c.a.199.2		2	60.59	even	2

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)

\(f(q)\)	\(=\)	\(q+11.0000i q^{3} -142.000i q^{7} +122.000 q^{9} +O(q^{10})\)	\(q+11.0000i q^{3} -142.000i q^{7} +122.000 q^{9} -777.000 q^{11} +884.000i q^{13} +27.0000i q^{17} +1145.00 q^{19} +1562.00 q^{21} -1854.00i q^{23} +4015.00i q^{27} +4920.00 q^{29} -1802.00 q^{31} -8547.00i q^{33} -13178.0i q^{37} -9724.00 q^{39} -15123.0 q^{41} -7844.00i q^{43} -6732.00i q^{47} -3357.00 q^{49} -297.000 q^{51} +3414.00i q^{53} +12595.0i q^{57} +33960.0 q^{59} +47402.0 q^{61} -17324.0i q^{63} -13177.0i q^{67} +20394.0 q^{69} +7548.00 q^{71} -59821.0i q^{73} +110334. i q^{77} +75830.0 q^{79} -14519.0 q^{81} -46299.0i q^{83} +54120.0i q^{87} +30585.0 q^{89} +125528. q^{91} -19822.0i q^{93} -104018. i q^{97} -94794.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 244 q^{9}+O(q^{10}) \) 2 * q + 244 * q^9	\( 2 q + 244 q^{9} - 1554 q^{11} + 2290 q^{19} + 3124 q^{21} + 9840 q^{29} - 3604 q^{31} - 19448 q^{39} - 30246 q^{41} - 6714 q^{49} - 594 q^{51} + 67920 q^{59} + 94804 q^{61} + 40788 q^{69} + 15096 q^{71} + 151660 q^{79} - 29038 q^{81} + 61170 q^{89} + 251056 q^{91} - 189588 q^{99}+O(q^{100}) \) 2 * q + 244 * q^9 - 1554 * q^11 + 2290 * q^19 + 3124 * q^21 + 9840 * q^29 - 3604 * q^31 - 19448 * q^39 - 30246 * q^41 - 6714 * q^49 - 594 * q^51 + 67920 * q^59 + 94804 * q^61 + 40788 * q^69 + 15096 * q^71 + 151660 * q^79 - 29038 * q^81 + 61170 * q^89 + 251056 * q^91 - 189588 * q^99

Introduction

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