LMFDB - Embedded newform 400.6.a.ba.1.3

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.1595208.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 20x^{2} + 33x - 3 $ x^4 - x^3 - 20x^2 + 33x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.98753$ 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)

$f(q)$	$=$	$q+4.69449 q^{3} +10.2635 q^{7} -220.962 q^{9} +O(q^{10})$	$q+4.69449 q^{3} +10.2635 q^{7} -220.962 q^{9} -596.423 q^{11} +420.629 q^{13} -974.149 q^{17} +380.528 q^{19} +48.1817 q^{21} +3543.51 q^{23} -2178.06 q^{27} +5440.89 q^{29} +3623.54 q^{31} -2799.90 q^{33} -1756.01 q^{37} +1974.64 q^{39} +263.984 q^{41} +14410.5 q^{43} +23464.8 q^{47} -16701.7 q^{49} -4573.13 q^{51} -33496.0 q^{53} +1786.38 q^{57} +2906.38 q^{59} +29431.9 q^{61} -2267.83 q^{63} +7163.34 q^{67} +16635.0 q^{69} +81353.2 q^{71} -55127.9 q^{73} -6121.36 q^{77} -16430.9 q^{79} +43468.8 q^{81} +116869. q^{83} +25542.2 q^{87} +99364.0 q^{89} +4317.11 q^{91} +17010.7 q^{93} -62987.8 q^{97} +131787. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9	$ 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9} + 368 q^{11} - 440 q^{13} + 672 q^{17} + 688 q^{19} + 992 q^{21} + 4492 q^{23} + 8152 q^{27} - 2936 q^{29} - 2112 q^{31} - 26864 q^{33} - 8792 q^{37} - 1504 q^{39} + 11800 q^{41} + 48276 q^{43} + 14724 q^{47} + 22500 q^{49} + 62400 q^{51} - 84296 q^{53} - 71024 q^{57} + 45840 q^{59} + 61928 q^{61} + 186292 q^{63} + 72700 q^{67} + 38368 q^{69} + 62816 q^{71} - 133072 q^{73} - 11440 q^{77} + 21632 q^{79} + 204836 q^{81} + 74660 q^{83} - 12472 q^{87} + 20952 q^{89} + 243808 q^{91} - 105600 q^{93} - 59456 q^{97} + 133424 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9 + 368 * q^11 - 440 * q^13 + 672 * q^17 + 688 * q^19 + 992 * q^21 + 4492 * q^23 + 8152 * q^27 - 2936 * q^29 - 2112 * q^31 - 26864 * q^33 - 8792 * q^37 - 1504 * q^39 + 11800 * q^41 + 48276 * q^43 + 14724 * q^47 + 22500 * q^49 + 62400 * q^51 - 84296 * q^53 - 71024 * q^57 + 45840 * q^59 + 61928 * q^61 + 186292 * q^63 + 72700 * q^67 + 38368 * q^69 + 62816 * q^71 - 133072 * q^73 - 11440 * q^77 + 21632 * q^79 + 204836 * q^81 + 74660 * q^83 - 12472 * q^87 + 20952 * q^89 + 243808 * q^91 - 105600 * q^93 - 59456 * q^97 + 133424 * 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
$3$	4.69449	0.301152	0.150576	−	0.988598i	$-0.451887\pi$
$3$	4.69449	0.301152	0.150576	+	0.988598i	$0.451887\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.2635	0.0791678	0.0395839	−	0.999216i	$-0.487397\pi$
$7$	10.2635	0.0791678	0.0395839	+	0.999216i	$0.487397\pi$
$8$	0	0
$9$	−220.962	−0.909308
$10$	0	0
$11$	−596.423	−1.48618	−0.743092	−	0.669189i	$-0.766642\pi$
$11$	−596.423	−1.48618	−0.743092	+	0.669189i	$0.766642\pi$
$12$	0	0
$13$	420.629	0.690305	0.345152	−	0.938547i	$-0.387827\pi$
$13$	420.629	0.690305	0.345152	+	0.938547i	$0.387827\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−974.149	−0.817529	−0.408764	−	0.912640i	$-0.634040\pi$
$17$	−974.149	−0.817529	−0.408764	+	0.912640i	$0.634040\pi$
$18$	0	0
$19$	380.528	0.241825	0.120913	−	0.992663i	$-0.461418\pi$
$19$	380.528	0.241825	0.120913	+	0.992663i	$0.461418\pi$
$20$	0	0
$21$	48.1817	0.0238415
$22$	0	0
$23$	3543.51	1.39673	0.698367	−	0.715740i	$-0.253910\pi$
$23$	3543.51	1.39673	0.698367	+	0.715740i	$0.253910\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2178.06	−0.574991
$28$	0	0
$29$	5440.89	1.20136	0.600682	−	0.799488i	$-0.294896\pi$
$29$	5440.89	1.20136	0.600682	+	0.799488i	$0.294896\pi$
$30$	0	0
$31$	3623.54	0.677219	0.338609	−	0.940927i	$-0.390043\pi$
$31$	3623.54	0.677219	0.338609	+	0.940927i	$0.390043\pi$
$32$	0	0
$33$	−2799.90	−0.447567
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1756.01	−0.210874	−0.105437	−	0.994426i	$-0.533624\pi$
$37$	−1756.01	−0.210874	−0.105437	+	0.994426i	$0.533624\pi$
$38$	0	0
$39$	1974.64	0.207886
$40$	0	0
$41$	263.984	0.0245255	0.0122628	−	0.999925i	$-0.496097\pi$
$41$	263.984	0.0245255	0.0122628	+	0.999925i	$0.496097\pi$
$42$	0	0
$43$	14410.5	1.18853	0.594263	−	0.804271i	$-0.297444\pi$
$43$	14410.5	1.18853	0.594263	+	0.804271i	$0.297444\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	23464.8	1.54943	0.774717	−	0.632308i	$-0.217892\pi$
$47$	23464.8	1.54943	0.774717	+	0.632308i	$0.217892\pi$
$48$	0	0
$49$	−16701.7	−0.993732
$50$	0	0
$51$	−4573.13	−0.246200
$52$	0	0
$53$	−33496.0	−1.63796	−0.818979	−	0.573823i	$-0.805460\pi$
$53$	−33496.0	−1.63796	−0.818979	+	0.573823i	$0.805460\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1786.38	0.0728261
$58$	0	0
$59$	2906.38	0.108698	0.0543492	−	0.998522i	$-0.482692\pi$
$59$	2906.38	0.108698	0.0543492	+	0.998522i	$0.482692\pi$
$60$	0	0
$61$	29431.9	1.01273	0.506366	−	0.862319i	$-0.330989\pi$
$61$	29431.9	1.01273	0.506366	+	0.862319i	$0.330989\pi$
$62$	0	0
$63$	−2267.83	−0.0719879
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7163.34	0.194952	0.0974762	−	0.995238i	$-0.468923\pi$
$67$	7163.34	0.194952	0.0974762	+	0.995238i	$0.468923\pi$
$68$	0	0
$69$	16635.0	0.420629
$70$	0	0
$71$	81353.2	1.91527	0.957633	−	0.287992i	$-0.0929878\pi$
$71$	81353.2	1.91527	0.957633	+	0.287992i	$0.0929878\pi$
$72$	0	0
$73$	−55127.9	−1.21078	−0.605388	−	0.795930i	$-0.706982\pi$
$73$	−55127.9	−1.21078	−0.605388	+	0.795930i	$0.706982\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6121.36	−0.117658
$78$	0	0
$79$	−16430.9	−0.296205	−0.148103	−	0.988972i	$-0.547317\pi$
$79$	−16430.9	−0.296205	−0.148103	+	0.988972i	$0.547317\pi$
$80$	0	0
$81$	43468.8	0.736148
$82$	0	0
$83$	116869.	1.86210	0.931050	−	0.364891i	$-0.118894\pi$
$83$	116869.	1.86210	0.931050	+	0.364891i	$0.118894\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	25542.2	0.361793
$88$	0	0
$89$	99364.0	1.32970	0.664851	−	0.746976i	$-0.268495\pi$
$89$	99364.0	1.32970	0.664851	+	0.746976i	$0.268495\pi$
$90$	0	0
$91$	4317.11	0.0546499
$92$	0	0
$93$	17010.7	0.203946
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−62987.8	−0.679715	−0.339858	−	0.940477i	$-0.610379\pi$
$97$	−62987.8	−0.679715	−0.339858	+	0.940477i	$0.610379\pi$
$98$	0	0
$99$	131787.	1.35140
$100$	0	0
$101$	40702.1	0.397021	0.198510	−	0.980099i	$-0.436390\pi$
$101$	40702.1	0.397021	0.198510	+	0.980099i	$0.436390\pi$
$102$	0	0
$103$	−108113.	−1.00412	−0.502058	−	0.864834i	$-0.667424\pi$
$103$	−108113.	−1.00412	−0.502058	+	0.864834i	$0.667424\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−198380.	−1.67509	−0.837546	−	0.546367i	$-0.816010\pi$
$107$	−198380.	−1.67509	−0.837546	+	0.546367i	$0.816010\pi$
$108$	0	0
$109$	−89150.3	−0.718715	−0.359357	−	0.933200i	$-0.617004\pi$
$109$	−89150.3	−0.718715	−0.359357	+	0.933200i	$0.617004\pi$
$110$	0	0
$111$	−8243.56	−0.0635049
$112$	0	0
$113$	165319.	1.21795	0.608973	−	0.793191i	$-0.291582\pi$
$113$	165319.	1.21795	0.608973	+	0.793191i	$0.291582\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−92943.0	−0.627700
$118$	0	0
$119$	−9998.14	−0.0647220
$120$	0	0
$121$	194669.	1.20874
$122$	0	0
$123$	1239.27	0.00738591
$124$	0	0
$125$	0	0
$126$	0	0
$127$	137238.	0.755031	0.377516	−	0.926003i	$-0.376778\pi$
$127$	137238.	0.755031	0.377516	+	0.926003i	$0.376778\pi$
$128$	0	0
$129$	67650.1	0.357927
$130$	0	0
$131$	355564.	1.81026	0.905128	−	0.425140i	$-0.139775\pi$
$131$	355564.	1.81026	0.905128	+	0.425140i	$0.139775\pi$
$132$	0	0
$133$	3905.53	0.0191448
$134$	0	0
$135$	0	0
$136$	0	0
$137$	232937.	1.06032	0.530160	−	0.847898i	$-0.322132\pi$
$137$	232937.	1.06032	0.530160	+	0.847898i	$0.322132\pi$
$138$	0	0
$139$	62526.2	0.274489	0.137245	−	0.990537i	$-0.456175\pi$
$139$	62526.2	0.274489	0.137245	+	0.990537i	$0.456175\pi$
$140$	0	0
$141$	110155.	0.466614
$142$	0	0
$143$	−250873.	−1.02592
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−78405.8	−0.299264
$148$	0	0
$149$	267151.	0.985804	0.492902	−	0.870085i	$-0.335936\pi$
$149$	267151.	0.985804	0.492902	+	0.870085i	$0.335936\pi$
$150$	0	0
$151$	329630.	1.17648	0.588240	−	0.808686i	$-0.299821\pi$
$151$	329630.	1.17648	0.588240	+	0.808686i	$0.299821\pi$
$152$	0	0
$153$	215250.	0.743385
$154$	0	0
$155$	0	0
$156$	0	0
$157$	400955.	1.29822	0.649108	−	0.760696i	$-0.275142\pi$
$157$	400955.	1.29822	0.649108	+	0.760696i	$0.275142\pi$
$158$	0	0
$159$	−157246.	−0.493274
$160$	0	0
$161$	36368.6	0.110576
$162$	0	0
$163$	−511411.	−1.50765	−0.753826	−	0.657074i	$-0.771794\pi$
$163$	−511411.	−1.50765	−0.753826	+	0.657074i	$0.771794\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−63650.1	−0.176607	−0.0883035	−	0.996094i	$-0.528145\pi$
$167$	−63650.1	−0.176607	−0.0883035	+	0.996094i	$0.528145\pi$
$168$	0	0
$169$	−194364.	−0.523479
$170$	0	0
$171$	−84082.0	−0.219894
$172$	0	0
$173$	33351.0	0.0847214	0.0423607	−	0.999102i	$-0.486512\pi$
$173$	33351.0	0.0847214	0.0423607	+	0.999102i	$0.486512\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13644.0	0.0327347
$178$	0	0
$179$	314739.	0.734207	0.367103	−	0.930180i	$-0.380350\pi$
$179$	314739.	0.734207	0.367103	+	0.930180i	$0.380350\pi$
$180$	0	0
$181$	415818.	0.943425	0.471712	−	0.881753i	$-0.343636\pi$
$181$	415818.	0.943425	0.471712	+	0.881753i	$0.343636\pi$
$182$	0	0
$183$	138168.	0.304986
$184$	0	0
$185$	0	0
$186$	0	0
$187$	581005.	1.21500
$188$	0	0
$189$	−22354.5	−0.0455208
$190$	0	0
$191$	−494250.	−0.980310	−0.490155	−	0.871635i	$-0.663060\pi$
$191$	−494250.	−0.980310	−0.490155	+	0.871635i	$0.663060\pi$
$192$	0	0
$193$	62426.1	0.120635	0.0603175	−	0.998179i	$-0.480789\pi$
$193$	62426.1	0.120635	0.0603175	+	0.998179i	$0.480789\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−513844.	−0.943334	−0.471667	−	0.881777i	$-0.656348\pi$
$197$	−513844.	−0.943334	−0.471667	+	0.881777i	$0.656348\pi$
$198$	0	0
$199$	29132.1	0.0521481	0.0260741	−	0.999660i	$-0.491699\pi$
$199$	29132.1	0.0521481	0.0260741	+	0.999660i	$0.491699\pi$
$200$	0	0
$201$	33628.2	0.0587102
$202$	0	0
$203$	55842.3	0.0951094
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−782980.	−1.27006
$208$	0	0
$209$	−226955.	−0.359397
$210$	0	0
$211$	330044.	0.510347	0.255173	−	0.966895i	$-0.417867\pi$
$211$	330044.	0.510347	0.255173	+	0.966895i	$0.417867\pi$
$212$	0	0
$213$	381912.	0.576785
$214$	0	0
$215$	0	0
$216$	0	0
$217$	37190.1	0.0536139
$218$	0	0
$219$	−258797.	−0.364627
$220$	0	0
$221$	−409756.	−0.564344
$222$	0	0
$223$	930113.	1.25249	0.626244	−	0.779627i	$-0.284591\pi$
$223$	930113.	1.25249	0.626244	+	0.779627i	$0.284591\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−37009.2	−0.0476699	−0.0238350	−	0.999716i	$-0.507588\pi$
$227$	−37009.2	−0.0476699	−0.0238350	+	0.999716i	$0.507588\pi$
$228$	0	0
$229$	506969.	0.638841	0.319421	−	0.947613i	$-0.396512\pi$
$229$	506969.	0.638841	0.319421	+	0.947613i	$0.396512\pi$
$230$	0	0
$231$	−28736.7	−0.0354329
$232$	0	0
$233$	−378560.	−0.456819	−0.228410	−	0.973565i	$-0.573353\pi$
$233$	−378560.	−0.456819	−0.228410	+	0.973565i	$0.573353\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−77134.5	−0.0892026
$238$	0	0
$239$	−105854.	−0.119870	−0.0599351	−	0.998202i	$-0.519089\pi$
$239$	−105854.	−0.119870	−0.0599351	+	0.998202i	$0.519089\pi$
$240$	0	0
$241$	−1.14897e6	−1.27429	−0.637144	−	0.770745i	$-0.719884\pi$
$241$	−1.14897e6	−1.27429	−0.637144	+	0.770745i	$0.719884\pi$
$242$	0	0
$243$	733333.	0.796683
$244$	0	0
$245$	0	0
$246$	0	0
$247$	160061.	0.166933
$248$	0	0
$249$	548639.	0.560775
$250$	0	0
$251$	−655956.	−0.657189	−0.328595	−	0.944471i	$-0.606575\pi$
$251$	−655956.	−0.657189	−0.328595	+	0.944471i	$0.606575\pi$
$252$	0	0
$253$	−2.11343e6	−2.07580
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−986720.	−0.931883	−0.465941	−	0.884816i	$-0.654284\pi$
$257$	−986720.	−0.931883	−0.465941	+	0.884816i	$0.654284\pi$
$258$	0	0
$259$	−18022.7	−0.0166944
$260$	0	0
$261$	−1.20223e6	−1.09241
$262$	0	0
$263$	865331.	0.771424	0.385712	−	0.922619i	$-0.373956\pi$
$263$	865331.	0.771424	0.385712	+	0.922619i	$0.373956\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	466463.	0.400442
$268$	0	0
$269$	−1.75595e6	−1.47956	−0.739780	−	0.672849i	$-0.765070\pi$
$269$	−1.75595e6	−1.47956	−0.739780	+	0.672849i	$0.765070\pi$
$270$	0	0
$271$	−1.07635e6	−0.890287	−0.445143	−	0.895459i	$-0.646847\pi$
$271$	−1.07635e6	−0.890287	−0.445143	+	0.895459i	$0.646847\pi$
$272$	0	0
$273$	20266.6	0.0164579
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−620129.	−0.485605	−0.242802	−	0.970076i	$-0.578067\pi$
$277$	−620129.	−0.485605	−0.242802	+	0.970076i	$0.578067\pi$
$278$	0	0
$279$	−800664.	−0.615800
$280$	0	0
$281$	884924.	0.668560	0.334280	−	0.942474i	$-0.391507\pi$
$281$	884924.	0.668560	0.334280	+	0.942474i	$0.391507\pi$
$282$	0	0
$283$	−1.06088e6	−0.787408	−0.393704	−	0.919237i	$-0.628806\pi$
$283$	−1.06088e6	−0.787408	−0.393704	+	0.919237i	$0.628806\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2709.39	0.00194163
$288$	0	0
$289$	−470891.	−0.331646
$290$	0	0
$291$	−295696.	−0.204697
$292$	0	0
$293$	585847.	0.398672	0.199336	−	0.979931i	$-0.436122\pi$
$293$	585847.	0.398672	0.199336	+	0.979931i	$0.436122\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.29905e6	0.854543
$298$	0	0
$299$	1.49050e6	0.964172
$300$	0	0
$301$	147902.	0.0940931
$302$	0	0
$303$	191076.	0.119563
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.33467e6	1.41377	0.706886	−	0.707328i	$-0.250099\pi$
$307$	2.33467e6	1.41377	0.706886	+	0.707328i	$0.250099\pi$
$308$	0	0
$309$	−507534.	−0.302391
$310$	0	0
$311$	−2.18712e6	−1.28225	−0.641123	−	0.767438i	$-0.721531\pi$
$311$	−2.18712e6	−1.28225	−0.641123	+	0.767438i	$0.721531\pi$
$312$	0	0
$313$	−2.76800e6	−1.59700	−0.798501	−	0.601993i	$-0.794374\pi$
$313$	−2.76800e6	−1.59700	−0.798501	+	0.601993i	$0.794374\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.71952e6	0.961077	0.480538	−	0.876974i	$-0.340441\pi$
$317$	1.71952e6	0.961077	0.480538	+	0.876974i	$0.340441\pi$
$318$	0	0
$319$	−3.24507e6	−1.78545
$320$	0	0
$321$	−931293.	−0.504457
$322$	0	0
$323$	−370691.	−0.197699
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−418515.	−0.216442
$328$	0	0
$329$	240830.	0.122665
$330$	0	0
$331$	1.24759e6	0.625895	0.312948	−	0.949770i	$-0.398684\pi$
$331$	1.24759e6	0.625895	0.312948	+	0.949770i	$0.398684\pi$
$332$	0	0
$333$	388011.	0.191749
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.96509e6	−0.942556	−0.471278	−	0.881985i	$-0.656207\pi$
$337$	−1.96509e6	−0.942556	−0.471278	+	0.881985i	$0.656207\pi$
$338$	0	0
$339$	776090.	0.366786
$340$	0	0
$341$	−2.16116e6	−1.00647
$342$	0	0
$343$	−343915.	−0.157839
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.27623e6	−1.46067	−0.730333	−	0.683091i	$-0.760635\pi$
$347$	−3.27623e6	−1.46067	−0.730333	+	0.683091i	$0.760635\pi$
$348$	0	0
$349$	101966.	0.0448117	0.0224058	−	0.999749i	$-0.492867\pi$
$349$	101966.	0.0448117	0.0224058	+	0.999749i	$0.492867\pi$
$350$	0	0
$351$	−916157.	−0.396919
$352$	0	0
$353$	3.03342e6	1.29567	0.647837	−	0.761779i	$-0.275674\pi$
$353$	3.03342e6	1.29567	0.647837	+	0.761779i	$0.275674\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−46936.2	−0.0194911
$358$	0	0
$359$	−2.68388e6	−1.09908	−0.549538	−	0.835469i	$-0.685196\pi$
$359$	−2.68388e6	−1.09908	−0.549538	+	0.835469i	$0.685196\pi$
$360$	0	0
$361$	−2.33130e6	−0.941520
$362$	0	0
$363$	913874.	0.364015
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.17487e6	0.455329	0.227664	−	0.973740i	$-0.426891\pi$
$367$	1.17487e6	0.455329	0.227664	+	0.973740i	$0.426891\pi$
$368$	0	0
$369$	−58330.5	−0.0223013
$370$	0	0
$371$	−343784.	−0.129674
$372$	0	0
$373$	4.13661e6	1.53947	0.769737	−	0.638361i	$-0.220387\pi$
$373$	4.13661e6	1.53947	0.769737	+	0.638361i	$0.220387\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.28860e6	0.829308
$378$	0	0
$379$	4.02996e6	1.44113	0.720565	−	0.693387i	$-0.243883\pi$
$379$	4.02996e6	1.44113	0.720565	+	0.693387i	$0.243883\pi$
$380$	0	0
$381$	644262.	0.227379
$382$	0	0
$383$	−3.29187e6	−1.14669	−0.573345	−	0.819314i	$-0.694355\pi$
$383$	−3.29187e6	−1.14669	−0.573345	+	0.819314i	$0.694355\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.18418e6	−1.08074
$388$	0	0
$389$	3.23387e6	1.08355	0.541774	−	0.840524i	$-0.317753\pi$
$389$	3.23387e6	1.08355	0.541774	+	0.840524i	$0.317753\pi$
$390$	0	0
$391$	−3.45190e6	−1.14187
$392$	0	0
$393$	1.66919e6	0.545161
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.50766e6	−0.480095	−0.240047	−	0.970761i	$-0.577163\pi$
$397$	−1.50766e6	−0.480095	−0.240047	+	0.970761i	$0.577163\pi$
$398$	0	0
$399$	18334.5	0.00576549
$400$	0	0
$401$	−132613.	−0.0411837	−0.0205918	−	0.999788i	$-0.506555\pi$
$401$	−132613.	−0.0411837	−0.0205918	+	0.999788i	$0.506555\pi$
$402$	0	0
$403$	1.52417e6	0.467487
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.04732e6	0.313397
$408$	0	0
$409$	−4.30354e6	−1.27209	−0.636044	−	0.771653i	$-0.719430\pi$
$409$	−4.30354e6	−1.27209	−0.636044	+	0.771653i	$0.719430\pi$
$410$	0	0
$411$	1.09352e6	0.319317
$412$	0	0
$413$	29829.6	0.00860541
$414$	0	0
$415$	0	0
$416$	0	0
$417$	293529.	0.0826628
$418$	0	0
$419$	5.35654e6	1.49056	0.745280	−	0.666752i	$-0.232316\pi$
$419$	5.35654e6	1.49056	0.745280	+	0.666752i	$0.232316\pi$
$420$	0	0
$421$	2.95420e6	0.812335	0.406167	−	0.913799i	$-0.366865\pi$
$421$	2.95420e6	0.812335	0.406167	+	0.913799i	$0.366865\pi$
$422$	0	0
$423$	−5.18483e6	−1.40891
$424$	0	0
$425$	0	0
$426$	0	0
$427$	302074.	0.0801758
$428$	0	0
$429$	−1.17772e6	−0.308958
$430$	0	0
$431$	5.93581e6	1.53917	0.769586	−	0.638543i	$-0.220463\pi$
$431$	5.93581e6	1.53917	0.769586	+	0.638543i	$0.220463\pi$
$432$	0	0
$433$	2.72633e6	0.698810	0.349405	−	0.936972i	$-0.386384\pi$
$433$	2.72633e6	0.698810	0.349405	+	0.936972i	$0.386384\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.34840e6	0.337766
$438$	0	0
$439$	−6.39220e6	−1.58303	−0.791515	−	0.611150i	$-0.790707\pi$
$439$	−6.39220e6	−1.58303	−0.791515	+	0.611150i	$0.790707\pi$
$440$	0	0
$441$	3.69043e6	0.903609
$442$	0	0
$443$	−990808.	−0.239872	−0.119936	−	0.992782i	$-0.538269\pi$
$443$	−990808.	−0.239872	−0.119936	+	0.992782i	$0.538269\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.25414e6	0.296877
$448$	0	0
$449$	4.79185e6	1.12173	0.560864	−	0.827908i	$-0.310469\pi$
$449$	4.79185e6	1.12173	0.560864	+	0.827908i	$0.310469\pi$
$450$	0	0
$451$	−157446.	−0.0364495
$452$	0	0
$453$	1.54745e6	0.354299
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.50517e6	−0.337128	−0.168564	−	0.985691i	$-0.553913\pi$
$457$	−1.50517e6	−0.337128	−0.168564	+	0.985691i	$0.553913\pi$
$458$	0	0
$459$	2.12176e6	0.470072
$460$	0	0
$461$	1.28168e6	0.280883	0.140442	−	0.990089i	$-0.455148\pi$
$461$	1.28168e6	0.280883	0.140442	+	0.990089i	$0.455148\pi$
$462$	0	0
$463$	700105.	0.151779	0.0758893	−	0.997116i	$-0.475820\pi$
$463$	700105.	0.151779	0.0758893	+	0.997116i	$0.475820\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.89776e6	−0.402669	−0.201334	−	0.979523i	$-0.564528\pi$
$467$	−1.89776e6	−0.402669	−0.201334	+	0.979523i	$0.564528\pi$
$468$	0	0
$469$	73520.6	0.0154340
$470$	0	0
$471$	1.88228e6	0.390960
$472$	0	0
$473$	−8.59478e6	−1.76637
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.40133e6	1.48941
$478$	0	0
$479$	−6.88878e6	−1.37184	−0.685920	−	0.727677i	$-0.740600\pi$
$479$	−6.88878e6	−1.37184	−0.685920	+	0.727677i	$0.740600\pi$
$480$	0	0
$481$	−738628.	−0.145567
$482$	0	0
$483$	170732.	0.0333003
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.31559e6	0.633488	0.316744	−	0.948511i	$-0.397410\pi$
$487$	3.31559e6	0.633488	0.316744	+	0.948511i	$0.397410\pi$
$488$	0	0
$489$	−2.40081e6	−0.454032
$490$	0	0
$491$	6.55075e6	1.22627	0.613137	−	0.789977i	$-0.289908\pi$
$491$	6.55075e6	1.22627	0.613137	+	0.789977i	$0.289908\pi$
$492$	0	0
$493$	−5.30024e6	−0.982150
$494$	0	0
$495$	0	0
$496$	0	0
$497$	834966.	0.151627
$498$	0	0
$499$	4.14654e6	0.745477	0.372738	−	0.927936i	$-0.378419\pi$
$499$	4.14654e6	0.745477	0.372738	+	0.927936i	$0.378419\pi$
$500$	0	0
$501$	−298805.	−0.0531855
$502$	0	0
$503$	386446.	0.0681035	0.0340517	−	0.999420i	$-0.489159\pi$
$503$	386446.	0.0681035	0.0340517	+	0.999420i	$0.489159\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−912440.	−0.157647
$508$	0	0
$509$	3.04725e6	0.521331	0.260665	−	0.965429i	$-0.416058\pi$
$509$	3.04725e6	0.521331	0.260665	+	0.965429i	$0.416058\pi$
$510$	0	0
$511$	−565802.	−0.0958545
$512$	0	0
$513$	−828813.	−0.139048
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.39950e7	−2.30274
$518$	0	0
$519$	156566.	0.0255140
$520$	0	0
$521$	8.88552e6	1.43413	0.717065	−	0.697007i	$-0.245485\pi$
$521$	8.88552e6	1.43413	0.717065	+	0.697007i	$0.245485\pi$
$522$	0	0
$523$	4.92458e6	0.787254	0.393627	−	0.919270i	$-0.371220\pi$
$523$	4.92458e6	0.787254	0.393627	+	0.919270i	$0.371220\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.52987e6	−0.553646
$528$	0	0
$529$	6.12010e6	0.950866
$530$	0	0
$531$	−642200.	−0.0988403
$532$	0	0
$533$	111040.	0.0169301
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.47754e6	0.221108
$538$	0	0
$539$	9.96126e6	1.47687
$540$	0	0
$541$	5.45948e6	0.801970	0.400985	−	0.916085i	$-0.368668\pi$
$541$	5.45948e6	0.801970	0.400985	+	0.916085i	$0.368668\pi$
$542$	0	0
$543$	1.95205e6	0.284114
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.70824e6	1.38731	0.693653	−	0.720309i	$-0.256000\pi$
$547$	9.70824e6	1.38731	0.693653	+	0.720309i	$0.256000\pi$
$548$	0	0
$549$	−6.50334e6	−0.920885
$550$	0	0
$551$	2.07041e6	0.290521
$552$	0	0
$553$	−168637.	−0.0234499
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.77260e6	0.651804	0.325902	−	0.945404i	$-0.394332\pi$
$557$	4.77260e6	0.651804	0.325902	+	0.945404i	$0.394332\pi$
$558$	0	0
$559$	6.06149e6	0.820446
$560$	0	0
$561$	2.72752e6	0.365899
$562$	0	0
$563$	2.16768e6	0.288220	0.144110	−	0.989562i	$-0.453968\pi$
$563$	2.16768e6	0.288220	0.144110	+	0.989562i	$0.453968\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	446140.	0.0582792
$568$	0	0
$569$	4.17864e6	0.541070	0.270535	−	0.962710i	$-0.412799\pi$
$569$	4.17864e6	0.541070	0.270535	+	0.962710i	$0.412799\pi$
$570$	0	0
$571$	7.89574e6	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$571$	7.89574e6	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$572$	0	0
$573$	−2.32025e6	−0.295222
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.34605e7	−1.68315	−0.841575	−	0.540141i	$-0.818371\pi$
$577$	−1.34605e7	−1.68315	−0.841575	+	0.540141i	$0.818371\pi$
$578$	0	0
$579$	293059.	0.0363294
$580$	0	0
$581$	1.19948e6	0.147418
$582$	0	0
$583$	1.99778e7	2.43431
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.43350e6	−0.171713	−0.0858564	−	0.996308i	$-0.527363\pi$
$587$	−1.43350e6	−0.171713	−0.0858564	+	0.996308i	$0.527363\pi$
$588$	0	0
$589$	1.37886e6	0.163769
$590$	0	0
$591$	−2.41223e6	−0.284087
$592$	0	0
$593$	1.38150e6	0.161330	0.0806649	−	0.996741i	$-0.474296\pi$
$593$	1.38150e6	0.161330	0.0806649	+	0.996741i	$0.474296\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	136760.	0.0157045
$598$	0	0
$599$	−6.53001e6	−0.743612	−0.371806	−	0.928310i	$-0.621261\pi$
$599$	−6.53001e6	−0.743612	−0.371806	+	0.928310i	$0.621261\pi$
$600$	0	0
$601$	−3.81467e6	−0.430795	−0.215398	−	0.976526i	$-0.569105\pi$
$601$	−3.81467e6	−0.430795	−0.215398	+	0.976526i	$0.569105\pi$
$602$	0	0
$603$	−1.58282e6	−0.177272
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.06018e7	−1.16791	−0.583954	−	0.811787i	$-0.698495\pi$
$607$	−1.06018e7	−1.16791	−0.583954	+	0.811787i	$0.698495\pi$
$608$	0	0
$609$	262151.	0.0286424
$610$	0	0
$611$	9.87000e6	1.06958
$612$	0	0
$613$	−5.80630e6	−0.624091	−0.312046	−	0.950067i	$-0.601014\pi$
$613$	−5.80630e6	−0.624091	−0.312046	+	0.950067i	$0.601014\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.80508e6	0.719648	0.359824	−	0.933020i	$-0.382837\pi$
$617$	6.80508e6	0.719648	0.359824	+	0.933020i	$0.382837\pi$
$618$	0	0
$619$	−852530.	−0.0894299	−0.0447150	−	0.999000i	$-0.514238\pi$
$619$	−852530.	−0.0894299	−0.0447150	+	0.999000i	$0.514238\pi$
$620$	0	0
$621$	−7.71798e6	−0.803110
$622$	0	0
$623$	1.01982e6	0.105270
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.06544e6	−0.108233
$628$	0	0
$629$	1.71061e6	0.172395
$630$	0	0
$631$	−8.01341e6	−0.801205	−0.400603	−	0.916252i	$-0.631199\pi$
$631$	−8.01341e6	−0.801205	−0.400603	+	0.916252i	$0.631199\pi$
$632$	0	0
$633$	1.54939e6	0.153692
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.02521e6	−0.685978
$638$	0	0
$639$	−1.79760e7	−1.74157
$640$	0	0
$641$	−3.68737e6	−0.354464	−0.177232	−	0.984169i	$-0.556714\pi$
$641$	−3.68737e6	−0.354464	−0.177232	+	0.984169i	$0.556714\pi$
$642$	0	0
$643$	−9.97031e6	−0.951001	−0.475501	−	0.879715i	$-0.657733\pi$
$643$	−9.97031e6	−0.951001	−0.475501	+	0.879715i	$0.657733\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.73097e6	0.162566	0.0812830	−	0.996691i	$-0.474098\pi$
$647$	1.73097e6	0.162566	0.0812830	+	0.996691i	$0.474098\pi$
$648$	0	0
$649$	−1.73343e6	−0.161546
$650$	0	0
$651$	174588.	0.0161459
$652$	0	0
$653$	1.15248e6	0.105767	0.0528836	−	0.998601i	$-0.483159\pi$
$653$	1.15248e6	0.105767	0.0528836	+	0.998601i	$0.483159\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.21811e7	1.10097
$658$	0	0
$659$	1.53161e7	1.37384	0.686919	−	0.726734i	$-0.258963\pi$
$659$	1.53161e7	1.37384	0.686919	+	0.726734i	$0.258963\pi$
$660$	0	0
$661$	1.69450e6	0.150848	0.0754238	−	0.997152i	$-0.475969\pi$
$661$	1.69450e6	0.150848	0.0754238	+	0.997152i	$0.475969\pi$
$662$	0	0
$663$	−1.92359e6	−0.169953
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.92798e7	1.67799
$668$	0	0
$669$	4.36641e6	0.377189
$670$	0	0
$671$	−1.75539e7	−1.50511
$672$	0	0
$673$	−1.53955e6	−0.131026	−0.0655129	−	0.997852i	$-0.520868\pi$
$673$	−1.53955e6	−0.131026	−0.0655129	+	0.997852i	$0.520868\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.16833e7	0.979702	0.489851	−	0.871806i	$-0.337051\pi$
$677$	1.16833e7	0.979702	0.489851	+	0.871806i	$0.337051\pi$
$678$	0	0
$679$	−646473.	−0.0538116
$680$	0	0
$681$	−173739.	−0.0143559
$682$	0	0
$683$	−1.49762e7	−1.22843	−0.614215	−	0.789139i	$-0.710527\pi$
$683$	−1.49762e7	−1.22843	−0.614215	+	0.789139i	$0.710527\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.37996e6	0.192388
$688$	0	0
$689$	−1.40894e7	−1.13069
$690$	0	0
$691$	9.90115e6	0.788843	0.394422	−	0.918930i	$-0.370945\pi$
$691$	9.90115e6	0.788843	0.394422	+	0.918930i	$0.370945\pi$
$692$	0	0
$693$	1.35259e6	0.106987
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−257160.	−0.0200503
$698$	0	0
$699$	−1.77714e6	−0.137572
$700$	0	0
$701$	1.48933e7	1.14471	0.572357	−	0.820005i	$-0.306029\pi$
$701$	1.48933e7	1.14471	0.572357	+	0.820005i	$0.306029\pi$
$702$	0	0
$703$	−668209.	−0.0509946
$704$	0	0
$705$	0	0
$706$	0	0
$707$	417744.	0.0314313
$708$	0	0
$709$	3.63694e6	0.271719	0.135860	−	0.990728i	$-0.456620\pi$
$709$	3.63694e6	0.271719	0.135860	+	0.990728i	$0.456620\pi$
$710$	0	0
$711$	3.63059e6	0.269342
$712$	0	0
$713$	1.28400e7	0.945894
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−496929.	−0.0360991
$718$	0	0
$719$	1.17971e7	0.851044	0.425522	−	0.904948i	$-0.360091\pi$
$719$	1.17971e7	0.851044	0.425522	+	0.904948i	$0.360091\pi$
$720$	0	0
$721$	−1.10961e6	−0.0794936
$722$	0	0
$723$	−5.39384e6	−0.383754
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−7.53381e6	−0.528663	−0.264331	−	0.964432i	$-0.585151\pi$
$727$	−7.53381e6	−0.528663	−0.264331	+	0.964432i	$0.585151\pi$
$728$	0	0
$729$	−7.12030e6	−0.496226
$730$	0	0
$731$	−1.40380e7	−0.971655
$732$	0	0
$733$	1.36302e7	0.937003	0.468501	−	0.883463i	$-0.344794\pi$
$733$	1.36302e7	0.937003	0.468501	+	0.883463i	$0.344794\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.27238e6	−0.289735
$738$	0	0
$739$	−1.07655e7	−0.725141	−0.362571	−	0.931956i	$-0.618101\pi$
$739$	−1.07655e7	−0.725141	−0.362571	+	0.931956i	$0.618101\pi$
$740$	0	0
$741$	751405.	0.0502722
$742$	0	0
$743$	−2.53921e7	−1.68743	−0.843717	−	0.536788i	$-0.819637\pi$
$743$	−2.53921e7	−1.68743	−0.843717	+	0.536788i	$0.819637\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.58235e7	−1.69322
$748$	0	0
$749$	−2.03607e6	−0.132613
$750$	0	0
$751$	−8.06289e6	−0.521664	−0.260832	−	0.965384i	$-0.583997\pi$
$751$	−8.06289e6	−0.521664	−0.260832	+	0.965384i	$0.583997\pi$
$752$	0	0
$753$	−3.07938e6	−0.197914
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.08891e7	0.690639	0.345319	−	0.938485i	$-0.387771\pi$
$757$	1.08891e7	0.690639	0.345319	+	0.938485i	$0.387771\pi$
$758$	0	0
$759$	−9.92147e6	−0.625132
$760$	0	0
$761$	−1.07800e7	−0.674773	−0.337387	−	0.941366i	$-0.609543\pi$
$761$	−1.07800e7	−0.674773	−0.337387	+	0.941366i	$0.609543\pi$
$762$	0	0
$763$	−914990.	−0.0568991
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.22251e6	0.0750350
$768$	0	0
$769$	−8.24398e6	−0.502714	−0.251357	−	0.967894i	$-0.580877\pi$
$769$	−8.24398e6	−0.502714	−0.251357	+	0.967894i	$0.580877\pi$
$770$	0	0
$771$	−4.63215e6	−0.280638
$772$	0	0
$773$	3.76264e6	0.226487	0.113244	−	0.993567i	$-0.463876\pi$
$773$	3.76264e6	0.226487	0.113244	+	0.993567i	$0.463876\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−84607.5	−0.00502755
$778$	0	0
$779$	100453.	0.00593090
$780$	0	0
$781$	−4.85209e7	−2.84644
$782$	0	0
$783$	−1.18506e7	−0.690774
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.14314e7	0.657906	0.328953	−	0.944346i	$-0.393304\pi$
$787$	1.14314e7	0.657906	0.328953	+	0.944346i	$0.393304\pi$
$788$	0	0
$789$	4.06229e6	0.232315
$790$	0	0
$791$	1.69675e6	0.0964221
$792$	0	0
$793$	1.23799e7	0.699094
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.82556e7	−1.01801	−0.509003	−	0.860765i	$-0.669986\pi$
$797$	−1.82556e7	−1.01801	−0.509003	+	0.860765i	$0.669986\pi$
$798$	0	0
$799$	−2.28583e7	−1.26671
$800$	0	0
$801$	−2.19557e7	−1.20911
$802$	0	0
$803$	3.28795e7	1.79944
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.24331e6	−0.445572
$808$	0	0
$809$	2.82214e7	1.51603	0.758014	−	0.652238i	$-0.226170\pi$
$809$	2.82214e7	1.51603	0.758014	+	0.652238i	$0.226170\pi$
$810$	0	0
$811$	−2.56400e7	−1.36888	−0.684441	−	0.729068i	$-0.739954\pi$
$811$	−2.56400e7	−1.36888	−0.684441	+	0.729068i	$0.739954\pi$
$812$	0	0
$813$	−5.05291e6	−0.268111
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.48360e6	0.287416
$818$	0	0
$819$	−953916.	−0.0496936
$820$	0	0
$821$	−4.20567e6	−0.217759	−0.108880	−	0.994055i	$-0.534726\pi$
$821$	−4.20567e6	−0.217759	−0.108880	+	0.994055i	$0.534726\pi$
$822$	0	0
$823$	9.80010e6	0.504349	0.252174	−	0.967682i	$-0.418854\pi$
$823$	9.80010e6	0.504349	0.252174	+	0.967682i	$0.418854\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.45973e7	−1.25061	−0.625306	−	0.780379i	$-0.715026\pi$
$827$	−2.45973e7	−1.25061	−0.625306	+	0.780379i	$0.715026\pi$
$828$	0	0
$829$	−3.33070e7	−1.68325	−0.841627	−	0.540060i	$-0.818402\pi$
$829$	−3.33070e7	−1.68325	−0.841627	+	0.540060i	$0.818402\pi$
$830$	0	0
$831$	−2.91119e6	−0.146241
$832$	0	0
$833$	1.62699e7	0.812405
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.89230e6	−0.389395
$838$	0	0
$839$	2.31345e7	1.13463	0.567315	−	0.823501i	$-0.307982\pi$
$839$	2.31345e7	1.13463	0.567315	+	0.823501i	$0.307982\pi$
$840$	0	0
$841$	9.09212e6	0.443277
$842$	0	0
$843$	4.15427e6	0.201338
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.99798e6	0.0956936
$848$	0	0
$849$	−4.98028e6	−0.237129
$850$	0	0
$851$	−6.22243e6	−0.294534
$852$	0	0
$853$	−1.59506e7	−0.750592	−0.375296	−	0.926905i	$-0.622459\pi$
$853$	−1.59506e7	−0.750592	−0.375296	+	0.926905i	$0.622459\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.04392e7	−0.485529	−0.242764	−	0.970085i	$-0.578054\pi$
$857$	−1.04392e7	−0.485529	−0.242764	+	0.970085i	$0.578054\pi$
$858$	0	0
$859$	−4.94442e6	−0.228630	−0.114315	−	0.993445i	$-0.536467\pi$
$859$	−4.94442e6	−0.228630	−0.114315	+	0.993445i	$0.536467\pi$
$860$	0	0
$861$	12719.2	0.000584726 0
$862$	0	0
$863$	7.28636e6	0.333030	0.166515	−	0.986039i	$-0.446749\pi$
$863$	7.28636e6	0.333030	0.166515	+	0.986039i	$0.446749\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.21059e6	−0.0998759
$868$	0	0
$869$	9.79974e6	0.440215
$870$	0	0
$871$	3.01311e6	0.134577
$872$	0	0
$873$	1.39179e7	0.618070
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.14124e6	0.269623	0.134812	−	0.990871i	$-0.456957\pi$
$877$	6.14124e6	0.269623	0.134812	+	0.990871i	$0.456957\pi$
$878$	0	0
$879$	2.75025e6	0.120061
$880$	0	0
$881$	8.64342e6	0.375185	0.187593	−	0.982247i	$-0.439932\pi$
$881$	8.64342e6	0.375185	0.187593	+	0.982247i	$0.439932\pi$
$882$	0	0
$883$	−3.13575e7	−1.35344	−0.676720	−	0.736240i	$-0.736599\pi$
$883$	−3.13575e7	−1.35344	−0.676720	+	0.736240i	$0.736599\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.41228e7	1.02948	0.514741	−	0.857346i	$-0.327888\pi$
$887$	2.41228e7	1.02948	0.514741	+	0.857346i	$0.327888\pi$
$888$	0	0
$889$	1.40854e6	0.0597742
$890$	0	0
$891$	−2.59258e7	−1.09405
$892$	0	0
$893$	8.92902e6	0.374693
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.99715e6	0.290362
$898$	0	0
$899$	1.97153e7	0.813587
$900$	0	0
$901$	3.26301e7	1.33908
$902$	0	0
$903$	694324.	0.0283363
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.78230e7	0.719389	0.359694	−	0.933070i	$-0.382881\pi$
$907$	1.78230e7	0.719389	0.359694	+	0.933070i	$0.382881\pi$
$908$	0	0
$909$	−8.99361e6	−0.361014
$910$	0	0
$911$	2.55925e7	1.02169	0.510843	−	0.859674i	$-0.329333\pi$
$911$	2.55925e7	1.02169	0.510843	+	0.859674i	$0.329333\pi$
$912$	0	0
$913$	−6.97032e7	−2.76742
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.64932e6	0.143314
$918$	0	0
$919$	−3.28071e6	−0.128138	−0.0640691	−	0.997945i	$-0.520408\pi$
$919$	−3.28071e6	−0.128138	−0.0640691	+	0.997945i	$0.520408\pi$
$920$	0	0
$921$	1.09601e7	0.425760
$922$	0	0
$923$	3.42195e7	1.32212
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.38888e7	0.913050
$928$	0	0
$929$	1.34815e7	0.512506	0.256253	−	0.966610i	$-0.417512\pi$
$929$	1.34815e7	0.512506	0.256253	+	0.966610i	$0.417512\pi$
$930$	0	0
$931$	−6.35544e6	−0.240310
$932$	0	0
$933$	−1.02674e7	−0.386151
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.85534e7	−1.06245	−0.531225	−	0.847231i	$-0.678268\pi$
$937$	−2.85534e7	−1.06245	−0.531225	+	0.847231i	$0.678268\pi$
$938$	0	0
$939$	−1.29944e7	−0.480940
$940$	0	0
$941$	−2.61677e7	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$941$	−2.61677e7	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$942$	0	0
$943$	935431.	0.0342557
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.17031e7	−0.786404	−0.393202	−	0.919452i	$-0.628633\pi$
$947$	−2.17031e7	−0.786404	−0.393202	+	0.919452i	$0.628633\pi$
$948$	0	0
$949$	−2.31884e7	−0.835805
$950$	0	0
$951$	8.07225e6	0.289430
$952$	0	0
$953$	−3.46596e7	−1.23621	−0.618104	−	0.786096i	$-0.712099\pi$
$953$	−3.46596e7	−1.23621	−0.618104	+	0.786096i	$0.712099\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.52340e7	−0.537691
$958$	0	0
$959$	2.39074e6	0.0839432
$960$	0	0
$961$	−1.54991e7	−0.541375
$962$	0	0
$963$	4.38344e7	1.52317
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.00486e7	1.37728	0.688639	−	0.725104i	$-0.258209\pi$
$967$	4.00486e7	1.37728	0.688639	+	0.725104i	$0.258209\pi$
$968$	0	0
$969$	−1.74020e6	−0.0595375
$970$	0	0
$971$	2.42657e7	0.825933	0.412967	−	0.910746i	$-0.364493\pi$
$971$	2.42657e7	0.825933	0.412967	+	0.910746i	$0.364493\pi$
$972$	0	0
$973$	641735.	0.0217307
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.73026e7	1.92060	0.960302	−	0.278963i	$-0.0899908\pi$
$977$	5.73026e7	1.92060	0.960302	+	0.278963i	$0.0899908\pi$
$978$	0	0
$979$	−5.92630e7	−1.97618
$980$	0	0
$981$	1.96988e7	0.653533
$982$	0	0
$983$	−9.13971e6	−0.301682	−0.150841	−	0.988558i	$-0.548198\pi$
$983$	−9.13971e6	−0.301682	−0.150841	+	0.988558i	$0.548198\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.13058e6	0.0369408
$988$	0	0
$989$	5.10638e7	1.66006
$990$	0	0
$991$	−2.80365e6	−0.0906858	−0.0453429	−	0.998971i	$-0.514438\pi$
$991$	−2.80365e6	−0.0906858	−0.0453429	+	0.998971i	$0.514438\pi$
$992$	0	0
$993$	5.85680e6	0.188489
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.72423e7	1.50520	0.752598	−	0.658481i	$-0.228801\pi$
$997$	4.72423e7	1.50520	0.752598	+	0.658481i	$0.228801\pi$
$998$	0	0
$999$	3.82470e6	0.121250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.ba.1.3		4
4.3	odd	2		200.6.a.j.1.2		4
5.2	odd	4		80.6.c.d.49.4		8
5.3	odd	4		80.6.c.d.49.5		8
5.4	even	2		400.6.a.z.1.2		4
15.2	even	4		720.6.f.n.289.4		8
15.8	even	4		720.6.f.n.289.3		8
20.3	even	4		40.6.c.a.9.4	✓	8
20.7	even	4		40.6.c.a.9.5	yes	8
20.19	odd	2		200.6.a.k.1.3		4
40.3	even	4		320.6.c.j.129.5		8
40.13	odd	4		320.6.c.i.129.4		8
40.27	even	4		320.6.c.j.129.4		8
40.37	odd	4		320.6.c.i.129.5		8
60.23	odd	4		360.6.f.b.289.3		8
60.47	odd	4		360.6.f.b.289.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.c.a.9.4	✓	8	20.3	even	4
40.6.c.a.9.5	yes	8	20.7	even	4
80.6.c.d.49.4		8	5.2	odd	4
80.6.c.d.49.5		8	5.3	odd	4
200.6.a.j.1.2		4	4.3	odd	2
200.6.a.k.1.3		4	20.19	odd	2
320.6.c.i.129.4		8	40.13	odd	4
320.6.c.i.129.5		8	40.37	odd	4
320.6.c.j.129.4		8	40.27	even	4
320.6.c.j.129.5		8	40.3	even	4
360.6.f.b.289.3		8	60.23	odd	4
360.6.f.b.289.4		8	60.47	odd	4
400.6.a.z.1.2		4	5.4	even	2
400.6.a.ba.1.3		4	1.1	even	1	trivial
720.6.f.n.289.3		8	15.8	even	4
720.6.f.n.289.4		8	15.2	even	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)

\(f(q)\)	\(=\)	\(q+4.69449 q^{3} +10.2635 q^{7} -220.962 q^{9} +O(q^{10})\)	\(q+4.69449 q^{3} +10.2635 q^{7} -220.962 q^{9} -596.423 q^{11} +420.629 q^{13} -974.149 q^{17} +380.528 q^{19} +48.1817 q^{21} +3543.51 q^{23} -2178.06 q^{27} +5440.89 q^{29} +3623.54 q^{31} -2799.90 q^{33} -1756.01 q^{37} +1974.64 q^{39} +263.984 q^{41} +14410.5 q^{43} +23464.8 q^{47} -16701.7 q^{49} -4573.13 q^{51} -33496.0 q^{53} +1786.38 q^{57} +2906.38 q^{59} +29431.9 q^{61} -2267.83 q^{63} +7163.34 q^{67} +16635.0 q^{69} +81353.2 q^{71} -55127.9 q^{73} -6121.36 q^{77} -16430.9 q^{79} +43468.8 q^{81} +116869. q^{83} +25542.2 q^{87} +99364.0 q^{89} +4317.11 q^{91} +17010.7 q^{93} -62987.8 q^{97} +131787. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9	\( 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9} + 368 q^{11} - 440 q^{13} + 672 q^{17} + 688 q^{19} + 992 q^{21} + 4492 q^{23} + 8152 q^{27} - 2936 q^{29} - 2112 q^{31} - 26864 q^{33} - 8792 q^{37} - 1504 q^{39} + 11800 q^{41} + 48276 q^{43} + 14724 q^{47} + 22500 q^{49} + 62400 q^{51} - 84296 q^{53} - 71024 q^{57} + 45840 q^{59} + 61928 q^{61} + 186292 q^{63} + 72700 q^{67} + 38368 q^{69} + 62816 q^{71} - 133072 q^{73} - 11440 q^{77} + 21632 q^{79} + 204836 q^{81} + 74660 q^{83} - 12472 q^{87} + 20952 q^{89} + 243808 q^{91} - 105600 q^{93} - 59456 q^{97} + 133424 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9 + 368 * q^11 - 440 * q^13 + 672 * q^17 + 688 * q^19 + 992 * q^21 + 4492 * q^23 + 8152 * q^27 - 2936 * q^29 - 2112 * q^31 - 26864 * q^33 - 8792 * q^37 - 1504 * q^39 + 11800 * q^41 + 48276 * q^43 + 14724 * q^47 + 22500 * q^49 + 62400 * q^51 - 84296 * q^53 - 71024 * q^57 + 45840 * q^59 + 61928 * q^61 + 186292 * q^63 + 72700 * q^67 + 38368 * q^69 + 62816 * q^71 - 133072 * q^73 - 11440 * q^77 + 21632 * q^79 + 204836 * q^81 + 74660 * q^83 - 12472 * q^87 + 20952 * q^89 + 243808 * q^91 - 105600 * q^93 - 59456 * q^97 + 133424 * q^99

Introduction

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