LMFDB - Embedded newform 400.6.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.47217.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 38x - 24 $ x^3 - x^2 - 38*x - 24
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.95752$ 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)

$f(q)$	$=$	$q-2.53448 q^{3} +247.370 q^{7} -236.576 q^{9} +O(q^{10})$	$q-2.53448 q^{3} +247.370 q^{7} -236.576 q^{9} -290.180 q^{11} -119.335 q^{13} +1100.71 q^{17} -1798.92 q^{19} -626.952 q^{21} -3056.46 q^{23} +1215.47 q^{27} +2565.53 q^{29} -1302.61 q^{31} +735.454 q^{33} -15014.8 q^{37} +302.452 q^{39} +13948.8 q^{41} -1779.13 q^{43} +3350.75 q^{47} +44384.7 q^{49} -2789.72 q^{51} +20874.1 q^{53} +4559.33 q^{57} -31645.3 q^{59} -34965.3 q^{61} -58521.8 q^{63} +38022.5 q^{67} +7746.52 q^{69} -69549.3 q^{71} -66299.4 q^{73} -71781.7 q^{77} -34853.9 q^{79} +54407.5 q^{81} -86638.0 q^{83} -6502.28 q^{87} +46305.1 q^{89} -29519.9 q^{91} +3301.43 q^{93} -16911.9 q^{97} +68649.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 70 q^{7} + 154 q^{9}+O(q^{10}) $ 3 * q + q^3 - 70 * q^7 + 154 * q^9	$ 3 q + q^{3} - 70 q^{7} + 154 q^{9} + 19 q^{11} + 196 q^{13} - 1223 q^{17} - 1221 q^{19} + 958 q^{21} - 2490 q^{23} + 4123 q^{27} + 11912 q^{29} - 7442 q^{31} + 6969 q^{33} - 14766 q^{37} - 27396 q^{39} + 3223 q^{41} - 41060 q^{43} + 29188 q^{47} + 66423 q^{49} - 43165 q^{51} + 12878 q^{53} - 77791 q^{57} - 64912 q^{59} + 22478 q^{61} - 112916 q^{63} + 26499 q^{67} + 178642 q^{69} - 86676 q^{71} - 8305 q^{73} - 106822 q^{77} - 21982 q^{79} - 71453 q^{81} - 213353 q^{83} + 14952 q^{87} + 182381 q^{89} - 149224 q^{91} - 196470 q^{93} - 76342 q^{97} + 149130 q^{99}+O(q^{100}) $ 3 * q + q^3 - 70 * q^7 + 154 * q^9 + 19 * q^11 + 196 * q^13 - 1223 * q^17 - 1221 * q^19 + 958 * q^21 - 2490 * q^23 + 4123 * q^27 + 11912 * q^29 - 7442 * q^31 + 6969 * q^33 - 14766 * q^37 - 27396 * q^39 + 3223 * q^41 - 41060 * q^43 + 29188 * q^47 + 66423 * q^49 - 43165 * q^51 + 12878 * q^53 - 77791 * q^57 - 64912 * q^59 + 22478 * q^61 - 112916 * q^63 + 26499 * q^67 + 178642 * q^69 - 86676 * q^71 - 8305 * q^73 - 106822 * q^77 - 21982 * q^79 - 71453 * q^81 - 213353 * q^83 + 14952 * q^87 + 182381 * q^89 - 149224 * q^91 - 196470 * q^93 - 76342 * q^97 + 149130 * 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
$3$	−2.53448	−0.162587	−0.0812934	−	0.996690i	$-0.525905\pi$
$3$	−2.53448	−0.162587	−0.0812934	+	0.996690i	$0.525905\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	247.370	1.90810	0.954050	−	0.299647i	$-0.0968689\pi$
$7$	247.370	1.90810	0.954050	+	0.299647i	$0.0968689\pi$
$8$	0	0
$9$	−236.576	−0.973566
$10$	0	0
$11$	−290.180	−0.723079	−0.361539	−	0.932357i	$-0.617749\pi$
$11$	−290.180	−0.723079	−0.361539	+	0.932357i	$0.617749\pi$
$12$	0	0
$13$	−119.335	−0.195844	−0.0979221	−	0.995194i	$-0.531220\pi$
$13$	−119.335	−0.195844	−0.0979221	+	0.995194i	$0.531220\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1100.71	0.923741	0.461871	−	0.886947i	$-0.347178\pi$
$17$	1100.71	0.923741	0.461871	+	0.886947i	$0.347178\pi$
$18$	0	0
$19$	−1798.92	−1.14322	−0.571609	−	0.820526i	$-0.693681\pi$
$19$	−1798.92	−1.14322	−0.571609	+	0.820526i	$0.693681\pi$
$20$	0	0
$21$	−626.952	−0.310232
$22$	0	0
$23$	−3056.46	−1.20476	−0.602378	−	0.798211i	$-0.705780\pi$
$23$	−3056.46	−1.20476	−0.602378	+	0.798211i	$0.705780\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1215.47	0.320876
$28$	0	0
$29$	2565.53	0.566477	0.283239	−	0.959049i	$-0.408591\pi$
$29$	2565.53	0.566477	0.283239	+	0.959049i	$0.408591\pi$
$30$	0	0
$31$	−1302.61	−0.243450	−0.121725	−	0.992564i	$-0.538843\pi$
$31$	−1302.61	−0.243450	−0.121725	+	0.992564i	$0.538843\pi$
$32$	0	0
$33$	735.454	0.117563
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−15014.8	−1.80308	−0.901542	−	0.432691i	$-0.857564\pi$
$37$	−15014.8	−1.80308	−0.901542	+	0.432691i	$0.857564\pi$
$38$	0	0
$39$	302.452	0.0318417
$40$	0	0
$41$	13948.8	1.29592	0.647961	−	0.761674i	$-0.275622\pi$
$41$	13948.8	1.29592	0.647961	+	0.761674i	$0.275622\pi$
$42$	0	0
$43$	−1779.13	−0.146736	−0.0733678	−	0.997305i	$-0.523375\pi$
$43$	−1779.13	−0.146736	−0.0733678	+	0.997305i	$0.523375\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3350.75	0.221257	0.110628	−	0.993862i	$-0.464714\pi$
$47$	3350.75	0.221257	0.110628	+	0.993862i	$0.464714\pi$
$48$	0	0
$49$	44384.7	2.64085
$50$	0	0
$51$	−2789.72	−0.150188
$52$	0	0
$53$	20874.1	1.02075	0.510373	−	0.859953i	$-0.329507\pi$
$53$	20874.1	1.02075	0.510373	+	0.859953i	$0.329507\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4559.33	0.185872
$58$	0	0
$59$	−31645.3	−1.18353	−0.591766	−	0.806110i	$-0.701569\pi$
$59$	−31645.3	−1.18353	−0.591766	+	0.806110i	$0.701569\pi$
$60$	0	0
$61$	−34965.3	−1.20313	−0.601565	−	0.798824i	$-0.705456\pi$
$61$	−34965.3	−1.20313	−0.601565	+	0.798824i	$0.705456\pi$
$62$	0	0
$63$	−58521.8	−1.85766
$64$	0	0
$65$	0	0
$66$	0	0
$67$	38022.5	1.03479	0.517397	−	0.855746i	$-0.326901\pi$
$67$	38022.5	1.03479	0.517397	+	0.855746i	$0.326901\pi$
$68$	0	0
$69$	7746.52	0.195877
$70$	0	0
$71$	−69549.3	−1.63737	−0.818685	−	0.574242i	$-0.805297\pi$
$71$	−69549.3	−1.63737	−0.818685	+	0.574242i	$0.805297\pi$
$72$	0	0
$73$	−66299.4	−1.45614	−0.728069	−	0.685504i	$-0.759582\pi$
$73$	−66299.4	−1.45614	−0.728069	+	0.685504i	$0.759582\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−71781.7	−1.37971
$78$	0	0
$79$	−34853.9	−0.628325	−0.314162	−	0.949369i	$-0.601724\pi$
$79$	−34853.9	−0.628325	−0.314162	+	0.949369i	$0.601724\pi$
$80$	0	0
$81$	54407.5	0.921395
$82$	0	0
$83$	−86638.0	−1.38043	−0.690213	−	0.723606i	$-0.742483\pi$
$83$	−86638.0	−1.38043	−0.690213	+	0.723606i	$0.742483\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6502.28	−0.0921017
$88$	0	0
$89$	46305.1	0.619660	0.309830	−	0.950792i	$-0.399728\pi$
$89$	46305.1	0.619660	0.309830	+	0.950792i	$0.399728\pi$
$90$	0	0
$91$	−29519.9	−0.373690
$92$	0	0
$93$	3301.43	0.0395817
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16911.9	−0.182501	−0.0912503	−	0.995828i	$-0.529086\pi$
$97$	−16911.9	−0.182501	−0.0912503	+	0.995828i	$0.529086\pi$
$98$	0	0
$99$	68649.7	0.703964
$100$	0	0
$101$	−151350.	−1.47631	−0.738155	−	0.674631i	$-0.764303\pi$
$101$	−151350.	−1.47631	−0.738155	+	0.674631i	$0.764303\pi$
$102$	0	0
$103$	−19867.7	−0.184525	−0.0922624	−	0.995735i	$-0.529410\pi$
$103$	−19867.7	−0.184525	−0.0922624	+	0.995735i	$0.529410\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−187685.	−1.58478	−0.792392	−	0.610012i	$-0.791165\pi$
$107$	−187685.	−1.58478	−0.792392	+	0.610012i	$0.791165\pi$
$108$	0	0
$109$	130806.	1.05454	0.527268	−	0.849699i	$-0.323217\pi$
$109$	130806.	1.05454	0.527268	+	0.849699i	$0.323217\pi$
$110$	0	0
$111$	38054.7	0.293158
$112$	0	0
$113$	−56375.6	−0.415332	−0.207666	−	0.978200i	$-0.566587\pi$
$113$	−56375.6	−0.415332	−0.207666	+	0.978200i	$0.566587\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	28231.9	0.190667
$118$	0	0
$119$	272282.	1.76259
$120$	0	0
$121$	−76846.6	−0.477157
$122$	0	0
$123$	−35353.0	−0.210700
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2888.85	0.0158933	0.00794667	−	0.999968i	$-0.497470\pi$
$127$	2888.85	0.0158933	0.00794667	+	0.999968i	$0.497470\pi$
$128$	0	0
$129$	4509.15	0.0238573
$130$	0	0
$131$	−212910.	−1.08397	−0.541985	−	0.840388i	$-0.682327\pi$
$131$	−212910.	−1.08397	−0.541985	+	0.840388i	$0.682327\pi$
$132$	0	0
$133$	−444999.	−2.18137
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−173546.	−0.789976	−0.394988	−	0.918686i	$-0.629251\pi$
$137$	−173546.	−0.789976	−0.394988	+	0.918686i	$0.629251\pi$
$138$	0	0
$139$	339914.	1.49222	0.746110	−	0.665823i	$-0.231919\pi$
$139$	339914.	1.49222	0.746110	+	0.665823i	$0.231919\pi$
$140$	0	0
$141$	−8492.38	−0.0359734
$142$	0	0
$143$	34628.7	0.141611
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−112492.	−0.429367
$148$	0	0
$149$	−354534.	−1.30825	−0.654127	−	0.756385i	$-0.726964\pi$
$149$	−354534.	−1.30825	−0.654127	+	0.756385i	$0.726964\pi$
$150$	0	0
$151$	1791.74	0.00639487	0.00319743	−	0.999995i	$-0.498982\pi$
$151$	1791.74	0.00639487	0.00319743	+	0.999995i	$0.498982\pi$
$152$	0	0
$153$	−260402.	−0.899323
$154$	0	0
$155$	0	0
$156$	0	0
$157$	511537.	1.65626	0.828129	−	0.560537i	$-0.189405\pi$
$157$	511537.	1.65626	0.828129	+	0.560537i	$0.189405\pi$
$158$	0	0
$159$	−52904.9	−0.165960
$160$	0	0
$161$	−756075.	−2.29879
$162$	0	0
$163$	−459676.	−1.35514	−0.677568	−	0.735460i	$-0.736966\pi$
$163$	−459676.	−1.35514	−0.677568	+	0.735460i	$0.736966\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−259189.	−0.719160	−0.359580	−	0.933114i	$-0.617080\pi$
$167$	−259189.	−0.719160	−0.359580	+	0.933114i	$0.617080\pi$
$168$	0	0
$169$	−357052.	−0.961645
$170$	0	0
$171$	425583.	1.11300
$172$	0	0
$173$	−374220.	−0.950631	−0.475315	−	0.879815i	$-0.657666\pi$
$173$	−374220.	−0.950631	−0.475315	+	0.879815i	$0.657666\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	80204.4	0.192427
$178$	0	0
$179$	−200493.	−0.467700	−0.233850	−	0.972273i	$-0.575132\pi$
$179$	−200493.	−0.467700	−0.233850	+	0.972273i	$0.575132\pi$
$180$	0	0
$181$	705255.	1.60011	0.800055	−	0.599927i	$-0.204804\pi$
$181$	705255.	1.60011	0.800055	+	0.599927i	$0.204804\pi$
$182$	0	0
$183$	88618.7	0.195613
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−319404.	−0.667938
$188$	0	0
$189$	300672.	0.612263
$190$	0	0
$191$	−491768.	−0.975386	−0.487693	−	0.873015i	$-0.662161\pi$
$191$	−491768.	−0.975386	−0.487693	+	0.873015i	$0.662161\pi$
$192$	0	0
$193$	362539.	0.700585	0.350293	−	0.936640i	$-0.386082\pi$
$193$	362539.	0.700585	0.350293	+	0.936640i	$0.386082\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−440774.	−0.809190	−0.404595	−	0.914496i	$-0.632588\pi$
$197$	−440774.	−0.809190	−0.404595	+	0.914496i	$0.632588\pi$
$198$	0	0
$199$	985431.	1.76398	0.881990	−	0.471269i	$-0.156204\pi$
$199$	985431.	1.76398	0.881990	+	0.471269i	$0.156204\pi$
$200$	0	0
$201$	−96367.1	−0.168244
$202$	0	0
$203$	634634.	1.08090
$204$	0	0
$205$	0	0
$206$	0	0
$207$	723086.	1.17291
$208$	0	0
$209$	522012.	0.826637
$210$	0	0
$211$	−29389.2	−0.0454445	−0.0227222	−	0.999742i	$-0.507233\pi$
$211$	−29389.2	−0.0454445	−0.0227222	+	0.999742i	$0.507233\pi$
$212$	0	0
$213$	176271.	0.266215
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−322225.	−0.464526
$218$	0	0
$219$	168034.	0.236749
$220$	0	0
$221$	−131353.	−0.180909
$222$	0	0
$223$	−167815.	−0.225979	−0.112990	−	0.993596i	$-0.536043\pi$
$223$	−167815.	−0.225979	−0.112990	+	0.993596i	$0.536043\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	95840.2	0.123448	0.0617238	−	0.998093i	$-0.480340\pi$
$227$	95840.2	0.123448	0.0617238	+	0.998093i	$0.480340\pi$
$228$	0	0
$229$	557513.	0.702532	0.351266	−	0.936276i	$-0.385751\pi$
$229$	557513.	0.702532	0.351266	+	0.936276i	$0.385751\pi$
$230$	0	0
$231$	181929.	0.224322
$232$	0	0
$233$	914906.	1.10405	0.552023	−	0.833829i	$-0.313856\pi$
$233$	914906.	1.10405	0.552023	+	0.833829i	$0.313856\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	88336.5	0.102157
$238$	0	0
$239$	−1.45389e6	−1.64641	−0.823203	−	0.567747i	$-0.807815\pi$
$239$	−1.45389e6	−1.64641	−0.823203	+	0.567747i	$0.807815\pi$
$240$	0	0
$241$	1.06147e6	1.17724	0.588619	−	0.808411i	$-0.299672\pi$
$241$	1.06147e6	1.17724	0.588619	+	0.808411i	$0.299672\pi$
$242$	0	0
$243$	−433255.	−0.470682
$244$	0	0
$245$	0	0
$246$	0	0
$247$	214675.	0.223893
$248$	0	0
$249$	219582.	0.224439
$250$	0	0
$251$	985641.	0.987494	0.493747	−	0.869606i	$-0.335627\pi$
$251$	985641.	0.987494	0.493747	+	0.869606i	$0.335627\pi$
$252$	0	0
$253$	886922.	0.871133
$254$	0	0
$255$	0	0
$256$	0	0
$257$	317014.	0.299396	0.149698	−	0.988732i	$-0.452170\pi$
$257$	317014.	0.299396	0.149698	+	0.988732i	$0.452170\pi$
$258$	0	0
$259$	−3.71421e6	−3.44047
$260$	0	0
$261$	−606944.	−0.551503
$262$	0	0
$263$	50112.6	0.0446743	0.0223371	−	0.999750i	$-0.492889\pi$
$263$	50112.6	0.0446743	0.0223371	+	0.999750i	$0.492889\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−117359.	−0.100749
$268$	0	0
$269$	−1.28003e6	−1.07855	−0.539274	−	0.842130i	$-0.681301\pi$
$269$	−1.28003e6	−1.07855	−0.539274	+	0.842130i	$0.681301\pi$
$270$	0	0
$271$	−1.43785e6	−1.18929	−0.594646	−	0.803987i	$-0.702708\pi$
$271$	−1.43785e6	−1.18929	−0.594646	+	0.803987i	$0.702708\pi$
$272$	0	0
$273$	74817.5	0.0607571
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−583518.	−0.456935	−0.228468	−	0.973552i	$-0.573372\pi$
$277$	−583518.	−0.456935	−0.228468	+	0.973552i	$0.573372\pi$
$278$	0	0
$279$	308166.	0.237014
$280$	0	0
$281$	2.23261e6	1.68674	0.843368	−	0.537336i	$-0.180569\pi$
$281$	2.23261e6	1.68674	0.843368	+	0.537336i	$0.180569\pi$
$282$	0	0
$283$	1.30526e6	0.968792	0.484396	−	0.874849i	$-0.339039\pi$
$283$	1.30526e6	0.968792	0.484396	+	0.874849i	$0.339039\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.45052e6	2.47275
$288$	0	0
$289$	−208295.	−0.146702
$290$	0	0
$291$	42862.9	0.0296722
$292$	0	0
$293$	2.30501e6	1.56857	0.784284	−	0.620402i	$-0.213031\pi$
$293$	2.30501e6	1.56857	0.784284	+	0.620402i	$0.213031\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−352706.	−0.232018
$298$	0	0
$299$	364743.	0.235944
$300$	0	0
$301$	−440101.	−0.279986
$302$	0	0
$303$	383592.	0.240029
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−302646.	−0.183269	−0.0916346	−	0.995793i	$-0.529209\pi$
$307$	−302646.	−0.183269	−0.0916346	+	0.995793i	$0.529209\pi$
$308$	0	0
$309$	50354.2	0.0300013
$310$	0	0
$311$	1.41350e6	0.828696	0.414348	−	0.910119i	$-0.364010\pi$
$311$	1.41350e6	0.828696	0.414348	+	0.910119i	$0.364010\pi$
$312$	0	0
$313$	−924126.	−0.533176	−0.266588	−	0.963811i	$-0.585896\pi$
$313$	−924126.	−0.533176	−0.266588	+	0.963811i	$0.585896\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	113236.	0.0632903	0.0316451	−	0.999499i	$-0.489925\pi$
$317$	113236.	0.0632903	0.0316451	+	0.999499i	$0.489925\pi$
$318$	0	0
$319$	−744466.	−0.409608
$320$	0	0
$321$	475683.	0.257665
$322$	0	0
$323$	−1.98009e6	−1.05604
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−331524.	−0.171453
$328$	0	0
$329$	828872.	0.422180
$330$	0	0
$331$	896472.	0.449745	0.224873	−	0.974388i	$-0.427803\pi$
$331$	896472.	0.449745	0.224873	+	0.974388i	$0.427803\pi$
$332$	0	0
$333$	3.55215e6	1.75542
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.26530e6	1.56620	0.783102	−	0.621893i	$-0.213636\pi$
$337$	3.26530e6	1.56620	0.783102	+	0.621893i	$0.213636\pi$
$338$	0	0
$339$	142883.	0.0675274
$340$	0	0
$341$	377990.	0.176033
$342$	0	0
$343$	6.82189e6	3.13090
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.39262e6	−1.95839	−0.979197	−	0.202913i	$-0.934959\pi$
$347$	−4.39262e6	−1.95839	−0.979197	+	0.202913i	$0.934959\pi$
$348$	0	0
$349$	680341.	0.298994	0.149497	−	0.988762i	$-0.452235\pi$
$349$	680341.	0.298994	0.149497	+	0.988762i	$0.452235\pi$
$350$	0	0
$351$	−145049.	−0.0628416
$352$	0	0
$353$	423316.	0.180812	0.0904062	−	0.995905i	$-0.471183\pi$
$353$	423316.	0.180812	0.0904062	+	0.995905i	$0.471183\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−690092.	−0.286574
$358$	0	0
$359$	3.69571e6	1.51343	0.756715	−	0.653745i	$-0.226803\pi$
$359$	3.69571e6	1.51343	0.756715	+	0.653745i	$0.226803\pi$
$360$	0	0
$361$	760032.	0.306947
$362$	0	0
$363$	194766.	0.0775794
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.42747e6	0.553223	0.276612	−	0.960982i	$-0.410788\pi$
$367$	1.42747e6	0.553223	0.276612	+	0.960982i	$0.410788\pi$
$368$	0	0
$369$	−3.29997e6	−1.26166
$370$	0	0
$371$	5.16362e6	1.94769
$372$	0	0
$373$	−3.36477e6	−1.25223	−0.626113	−	0.779732i	$-0.715355\pi$
$373$	−3.36477e6	−1.25223	−0.626113	+	0.779732i	$0.715355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−306158.	−0.110941
$378$	0	0
$379$	168251.	0.0601671	0.0300835	−	0.999547i	$-0.490423\pi$
$379$	168251.	0.0601671	0.0300835	+	0.999547i	$0.490423\pi$
$380$	0	0
$381$	−7321.72	−0.00258405
$382$	0	0
$383$	−470292.	−0.163821	−0.0819106	−	0.996640i	$-0.526102\pi$
$383$	−470292.	−0.163821	−0.0819106	+	0.996640i	$0.526102\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	420899.	0.142857
$388$	0	0
$389$	−4.85362e6	−1.62627	−0.813133	−	0.582077i	$-0.802240\pi$
$389$	−4.85362e6	−1.62627	−0.813133	+	0.582077i	$0.802240\pi$
$390$	0	0
$391$	−3.36427e6	−1.11288
$392$	0	0
$393$	539615.	0.176239
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.36418e6	0.434405	0.217202	−	0.976127i	$-0.430307\pi$
$397$	1.36418e6	0.434405	0.217202	+	0.976127i	$0.430307\pi$
$398$	0	0
$399$	1.12784e6	0.354662
$400$	0	0
$401$	−1.99367e6	−0.619145	−0.309573	−	0.950876i	$-0.600186\pi$
$401$	−1.99367e6	−0.619145	−0.309573	+	0.950876i	$0.600186\pi$
$402$	0	0
$403$	155447.	0.0476782
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.35700e6	1.30377
$408$	0	0
$409$	−3.41895e6	−1.01061	−0.505305	−	0.862941i	$-0.668620\pi$
$409$	−3.41895e6	−1.01061	−0.505305	+	0.862941i	$0.668620\pi$
$410$	0	0
$411$	439849.	0.128440
$412$	0	0
$413$	−7.82810e6	−2.25830
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−861505.	−0.242615
$418$	0	0
$419$	−3.06851e6	−0.853871	−0.426935	−	0.904282i	$-0.640407\pi$
$419$	−3.06851e6	−0.853871	−0.426935	+	0.904282i	$0.640407\pi$
$420$	0	0
$421$	1.62229e6	0.446092	0.223046	−	0.974808i	$-0.428400\pi$
$421$	1.62229e6	0.446092	0.223046	+	0.974808i	$0.428400\pi$
$422$	0	0
$423$	−792707.	−0.215408
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.64935e6	−2.29569
$428$	0	0
$429$	−87765.6	−0.0230240
$430$	0	0
$431$	614867.	0.159437	0.0797183	−	0.996817i	$-0.474598\pi$
$431$	614867.	0.159437	0.0797183	+	0.996817i	$0.474598\pi$
$432$	0	0
$433$	−207106.	−0.0530851	−0.0265425	−	0.999648i	$-0.508450\pi$
$433$	−207106.	−0.0530851	−0.0265425	+	0.999648i	$0.508450\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.49834e6	1.37730
$438$	0	0
$439$	3.93815e6	0.975284	0.487642	−	0.873044i	$-0.337857\pi$
$439$	3.93815e6	0.975284	0.487642	+	0.873044i	$0.337857\pi$
$440$	0	0
$441$	−1.05004e7	−2.57104
$442$	0	0
$443$	−3.09207e6	−0.748584	−0.374292	−	0.927311i	$-0.622114\pi$
$443$	−3.09207e6	−0.748584	−0.374292	+	0.927311i	$0.622114\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	898558.	0.212705
$448$	0	0
$449$	2.16948e6	0.507855	0.253927	−	0.967223i	$-0.418278\pi$
$449$	2.16948e6	0.507855	0.253927	+	0.967223i	$0.418278\pi$
$450$	0	0
$451$	−4.04767e6	−0.937053
$452$	0	0
$453$	−4541.11	−0.00103972
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.38699e6	0.982598	0.491299	−	0.870991i	$-0.336522\pi$
$457$	4.38699e6	0.982598	0.491299	+	0.870991i	$0.336522\pi$
$458$	0	0
$459$	1.33788e6	0.296406
$460$	0	0
$461$	3.00490e6	0.658532	0.329266	−	0.944237i	$-0.393199\pi$
$461$	3.00490e6	0.658532	0.329266	+	0.944237i	$0.393199\pi$
$462$	0	0
$463$	−922183.	−0.199924	−0.0999619	−	0.994991i	$-0.531872\pi$
$463$	−922183.	−0.199924	−0.0999619	+	0.994991i	$0.531872\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.45571e6	−0.733239	−0.366620	−	0.930371i	$-0.619485\pi$
$467$	−3.45571e6	−0.733239	−0.366620	+	0.930371i	$0.619485\pi$
$468$	0	0
$469$	9.40561e6	1.97449
$470$	0	0
$471$	−1.29648e6	−0.269286
$472$	0	0
$473$	516266.	0.106101
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.93832e6	−0.993764
$478$	0	0
$479$	7.59868e6	1.51321	0.756605	−	0.653872i	$-0.226857\pi$
$479$	7.59868e6	1.51321	0.756605	+	0.653872i	$0.226857\pi$
$480$	0	0
$481$	1.79180e6	0.353123
$482$	0	0
$483$	1.91625e6	0.373753
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.60362e6	0.879584	0.439792	−	0.898100i	$-0.355052\pi$
$487$	4.60362e6	0.879584	0.439792	+	0.898100i	$0.355052\pi$
$488$	0	0
$489$	1.16504e6	0.220327
$490$	0	0
$491$	6.01057e6	1.12515	0.562577	−	0.826745i	$-0.309810\pi$
$491$	6.01057e6	1.12515	0.562577	+	0.826745i	$0.309810\pi$
$492$	0	0
$493$	2.82391e6	0.523278
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.72044e7	−3.12427
$498$	0	0
$499$	−1.29512e6	−0.232841	−0.116421	−	0.993200i	$-0.537142\pi$
$499$	−1.29512e6	−0.232841	−0.116421	+	0.993200i	$0.537142\pi$
$500$	0	0
$501$	656908.	0.116926
$502$	0	0
$503$	−3.98765e6	−0.702744	−0.351372	−	0.936236i	$-0.614285\pi$
$503$	−3.98765e6	−0.702744	−0.351372	+	0.936236i	$0.614285\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	904940.	0.156351
$508$	0	0
$509$	479969.	0.0821142	0.0410571	−	0.999157i	$-0.486927\pi$
$509$	479969.	0.0821142	0.0410571	+	0.999157i	$0.486927\pi$
$510$	0	0
$511$	−1.64004e7	−2.77846
$512$	0	0
$513$	−2.18655e6	−0.366831
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−972319.	−0.159986
$518$	0	0
$519$	948452.	0.154560
$520$	0	0
$521$	−1.49769e6	−0.241729	−0.120864	−	0.992669i	$-0.538567\pi$
$521$	−1.49769e6	−0.241729	−0.120864	+	0.992669i	$0.538567\pi$
$522$	0	0
$523$	−7.57897e6	−1.21159	−0.605796	−	0.795620i	$-0.707145\pi$
$523$	−7.57897e6	−1.21159	−0.605796	+	0.795620i	$0.707145\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.43379e6	−0.224885
$528$	0	0
$529$	2.90559e6	0.451435
$530$	0	0
$531$	7.48654e6	1.15225
$532$	0	0
$533$	−1.66459e6	−0.253799
$534$	0	0
$535$	0	0
$536$	0	0
$537$	508146.	0.0760418
$538$	0	0
$539$	−1.28795e7	−1.90954
$540$	0	0
$541$	3.61980e6	0.531730	0.265865	−	0.964010i	$-0.414342\pi$
$541$	3.61980e6	0.531730	0.265865	+	0.964010i	$0.414342\pi$
$542$	0	0
$543$	−1.78745e6	−0.260157
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.13597e7	−1.62330	−0.811651	−	0.584143i	$-0.801431\pi$
$547$	−1.13597e7	−1.62330	−0.811651	+	0.584143i	$0.801431\pi$
$548$	0	0
$549$	8.27196e6	1.17133
$550$	0	0
$551$	−4.61520e6	−0.647607
$552$	0	0
$553$	−8.62180e6	−1.19891
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.82035e6	−0.248609	−0.124305	−	0.992244i	$-0.539670\pi$
$557$	−1.82035e6	−0.248609	−0.124305	+	0.992244i	$0.539670\pi$
$558$	0	0
$559$	212312.	0.0287373
$560$	0	0
$561$	809521.	0.108598
$562$	0	0
$563$	−1.21054e7	−1.60956	−0.804780	−	0.593573i	$-0.797717\pi$
$563$	−1.21054e7	−1.60956	−0.804780	+	0.593573i	$0.797717\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.34588e7	1.75811
$568$	0	0
$569$	9.71197e6	1.25755	0.628777	−	0.777585i	$-0.283556\pi$
$569$	9.71197e6	1.25755	0.628777	+	0.777585i	$0.283556\pi$
$570$	0	0
$571$	−7.72708e6	−0.991803	−0.495901	−	0.868379i	$-0.665162\pi$
$571$	−7.72708e6	−0.991803	−0.495901	+	0.868379i	$0.665162\pi$
$572$	0	0
$573$	1.24637e6	0.158585
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.68085e6	1.21053	0.605263	−	0.796026i	$-0.293068\pi$
$577$	9.68085e6	1.21053	0.605263	+	0.796026i	$0.293068\pi$
$578$	0	0
$579$	−918845.	−0.113906
$580$	0	0
$581$	−2.14316e7	−2.63399
$582$	0	0
$583$	−6.05724e6	−0.738080
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.28718e6	0.393757	0.196879	−	0.980428i	$-0.436920\pi$
$587$	3.28718e6	0.393757	0.196879	+	0.980428i	$0.436920\pi$
$588$	0	0
$589$	2.34329e6	0.278316
$590$	0	0
$591$	1.11713e6	0.131564
$592$	0	0
$593$	6.15399e6	0.718655	0.359327	−	0.933212i	$-0.383006\pi$
$593$	6.15399e6	0.718655	0.359327	+	0.933212i	$0.383006\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.49755e6	−0.286800
$598$	0	0
$599$	−7.73935e6	−0.881328	−0.440664	−	0.897672i	$-0.645257\pi$
$599$	−7.73935e6	−0.881328	−0.440664	+	0.897672i	$0.645257\pi$
$600$	0	0
$601$	1.13857e7	1.28580	0.642900	−	0.765950i	$-0.277731\pi$
$601$	1.13857e7	1.28580	0.642900	+	0.765950i	$0.277731\pi$
$602$	0	0
$603$	−8.99523e6	−1.00744
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.24328e6	0.908089	0.454044	−	0.890979i	$-0.349981\pi$
$607$	8.24328e6	0.908089	0.454044	+	0.890979i	$0.349981\pi$
$608$	0	0
$609$	−1.60847e6	−0.175739
$610$	0	0
$611$	−399862.	−0.0433319
$612$	0	0
$613$	−7.44252e6	−0.799961	−0.399980	−	0.916524i	$-0.630983\pi$
$613$	−7.44252e6	−0.799961	−0.399980	+	0.916524i	$0.630983\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.06327e7	−1.12443	−0.562214	−	0.826991i	$-0.690050\pi$
$617$	−1.06327e7	−1.12443	−0.562214	+	0.826991i	$0.690050\pi$
$618$	0	0
$619$	−7.90577e6	−0.829312	−0.414656	−	0.909978i	$-0.636098\pi$
$619$	−7.90577e6	−0.829312	−0.414656	+	0.909978i	$0.636098\pi$
$620$	0	0
$621$	−3.71505e6	−0.386576
$622$	0	0
$623$	1.14545e7	1.18237
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.32303e6	−0.134400
$628$	0	0
$629$	−1.65270e7	−1.66558
$630$	0	0
$631$	−6.26384e6	−0.626278	−0.313139	−	0.949707i	$-0.601381\pi$
$631$	−6.26384e6	−0.626278	−0.313139	+	0.949707i	$0.601381\pi$
$632$	0	0
$633$	74486.1	0.00738866
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5.29666e6	−0.517194
$638$	0	0
$639$	1.64537e7	1.59409
$640$	0	0
$641$	1.06859e7	1.02722	0.513611	−	0.858023i	$-0.328307\pi$
$641$	1.06859e7	1.02722	0.513611	+	0.858023i	$0.328307\pi$
$642$	0	0
$643$	1.10617e7	1.05510	0.527549	−	0.849524i	$-0.323111\pi$
$643$	1.10617e7	1.05510	0.527549	+	0.849524i	$0.323111\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.43619e6	0.134881	0.0674404	−	0.997723i	$-0.478517\pi$
$647$	1.43619e6	0.134881	0.0674404	+	0.997723i	$0.478517\pi$
$648$	0	0
$649$	9.18284e6	0.855786
$650$	0	0
$651$	816673.	0.0755258
$652$	0	0
$653$	7.84002e6	0.719506	0.359753	−	0.933048i	$-0.382861\pi$
$653$	7.84002e6	0.719506	0.359753	+	0.933048i	$0.382861\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.56849e7	1.41764
$658$	0	0
$659$	1.41636e7	1.27046	0.635230	−	0.772323i	$-0.280905\pi$
$659$	1.41636e7	1.27046	0.635230	+	0.772323i	$0.280905\pi$
$660$	0	0
$661$	8.42312e6	0.749842	0.374921	−	0.927057i	$-0.377670\pi$
$661$	8.42312e6	0.749842	0.374921	+	0.927057i	$0.377670\pi$
$662$	0	0
$663$	332912.	0.0294135
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.84144e6	−0.682466
$668$	0	0
$669$	425323.	0.0367412
$670$	0	0
$671$	1.01462e7	0.869957
$672$	0	0
$673$	−9.43547e6	−0.803019	−0.401509	−	0.915855i	$-0.631514\pi$
$673$	−9.43547e6	−0.803019	−0.401509	+	0.915855i	$0.631514\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.39970e6	−0.704356	−0.352178	−	0.935933i	$-0.614559\pi$
$677$	−8.39970e6	−0.704356	−0.352178	+	0.935933i	$0.614559\pi$
$678$	0	0
$679$	−4.18350e6	−0.348229
$680$	0	0
$681$	−242905.	−0.0200710
$682$	0	0
$683$	−4.92821e6	−0.404238	−0.202119	−	0.979361i	$-0.564783\pi$
$683$	−4.92821e6	−0.404238	−0.202119	+	0.979361i	$0.564783\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.41300e6	−0.114222
$688$	0	0
$689$	−2.49102e6	−0.199907
$690$	0	0
$691$	1.73031e7	1.37857	0.689285	−	0.724490i	$-0.257925\pi$
$691$	1.73031e7	1.37857	0.689285	+	0.724490i	$0.257925\pi$
$692$	0	0
$693$	1.69819e7	1.34323
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.53536e7	1.19710
$698$	0	0
$699$	−2.31881e6	−0.179503
$700$	0	0
$701$	−5.77146e6	−0.443599	−0.221800	−	0.975092i	$-0.571193\pi$
$701$	−5.77146e6	−0.443599	−0.221800	+	0.975092i	$0.571193\pi$
$702$	0	0
$703$	2.70105e7	2.06132
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.74393e7	−2.81695
$708$	0	0
$709$	7.29358e6	0.544911	0.272455	−	0.962168i	$-0.412164\pi$
$709$	7.29358e6	0.544911	0.272455	+	0.962168i	$0.412164\pi$
$710$	0	0
$711$	8.24562e6	0.611715
$712$	0	0
$713$	3.98136e6	0.293297
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.68485e6	0.267684
$718$	0	0
$719$	9.70735e6	0.700291	0.350145	−	0.936695i	$-0.386132\pi$
$719$	9.70735e6	0.700291	0.350145	+	0.936695i	$0.386132\pi$
$720$	0	0
$721$	−4.91467e6	−0.352092
$722$	0	0
$723$	−2.69026e6	−0.191403
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.73733e6	0.332428	0.166214	−	0.986090i	$-0.446846\pi$
$727$	4.73733e6	0.332428	0.166214	+	0.986090i	$0.446846\pi$
$728$	0	0
$729$	−1.21229e7	−0.844869
$730$	0	0
$731$	−1.95830e6	−0.135546
$732$	0	0
$733$	−2.81524e6	−0.193533	−0.0967667	−	0.995307i	$-0.530850\pi$
$733$	−2.81524e6	−0.193533	−0.0967667	+	0.995307i	$0.530850\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.10334e7	−0.748237
$738$	0	0
$739$	8.68260e6	0.584842	0.292421	−	0.956290i	$-0.405539\pi$
$739$	8.68260e6	0.584842	0.292421	+	0.956290i	$0.405539\pi$
$740$	0	0
$741$	−544089.	−0.0364019
$742$	0	0
$743$	−4.09845e6	−0.272363	−0.136181	−	0.990684i	$-0.543483\pi$
$743$	−4.09845e6	−0.272363	−0.136181	+	0.990684i	$0.543483\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.04965e7	1.34394
$748$	0	0
$749$	−4.64276e7	−3.02393
$750$	0	0
$751$	4.62828e6	0.299447	0.149723	−	0.988728i	$-0.452162\pi$
$751$	4.62828e6	0.299447	0.149723	+	0.988728i	$0.452162\pi$
$752$	0	0
$753$	−2.49808e6	−0.160553
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.87265e7	1.18773	0.593863	−	0.804566i	$-0.297602\pi$
$757$	1.87265e7	1.18773	0.593863	+	0.804566i	$0.297602\pi$
$758$	0	0
$759$	−2.24788e6	−0.141635
$760$	0	0
$761$	−5.03003e6	−0.314854	−0.157427	−	0.987531i	$-0.550320\pi$
$761$	−5.03003e6	−0.314854	−0.157427	+	0.987531i	$0.550320\pi$
$762$	0	0
$763$	3.23574e7	2.01216
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.77641e6	0.231788
$768$	0	0
$769$	−2.38788e7	−1.45612	−0.728060	−	0.685513i	$-0.759578\pi$
$769$	−2.38788e7	−1.45612	−0.728060	+	0.685513i	$0.759578\pi$
$770$	0	0
$771$	−803464.	−0.0486778
$772$	0	0
$773$	−1.69482e7	−1.02017	−0.510086	−	0.860123i	$-0.670387\pi$
$773$	−1.69482e7	−1.02017	−0.510086	+	0.860123i	$0.670387\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.41358e6	0.559374
$778$	0	0
$779$	−2.50929e7	−1.48152
$780$	0	0
$781$	2.01818e7	1.18395
$782$	0	0
$783$	3.11834e6	0.181769
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.60447e7	1.49893	0.749466	−	0.662043i	$-0.230310\pi$
$787$	2.60447e7	1.49893	0.749466	+	0.662043i	$0.230310\pi$
$788$	0	0
$789$	−127009.	−0.00726345
$790$	0	0
$791$	−1.39456e7	−0.792494
$792$	0	0
$793$	4.17259e6	0.235626
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.54203e7	1.41754	0.708769	−	0.705441i	$-0.249251\pi$
$797$	2.54203e7	1.41754	0.708769	+	0.705441i	$0.249251\pi$
$798$	0	0
$799$	3.68820e6	0.204384
$800$	0	0
$801$	−1.09547e7	−0.603280
$802$	0	0
$803$	1.92387e7	1.05290
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.24421e6	0.175358
$808$	0	0
$809$	−4.37704e6	−0.235131	−0.117565	−	0.993065i	$-0.537509\pi$
$809$	−4.37704e6	−0.235131	−0.117565	+	0.993065i	$0.537509\pi$
$810$	0	0
$811$	8.41629e6	0.449333	0.224667	−	0.974436i	$-0.427871\pi$
$811$	8.41629e6	0.449333	0.224667	+	0.974436i	$0.427871\pi$
$812$	0	0
$813$	3.64418e6	0.193363
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.20051e6	0.167751
$818$	0	0
$819$	6.98372e6	0.363812
$820$	0	0
$821$	1.43979e7	0.745488	0.372744	−	0.927934i	$-0.378417\pi$
$821$	1.43979e7	0.745488	0.372744	+	0.927934i	$0.378417\pi$
$822$	0	0
$823$	3.85762e6	0.198527	0.0992635	−	0.995061i	$-0.468351\pi$
$823$	3.85762e6	0.198527	0.0992635	+	0.995061i	$0.468351\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.10270e7	−0.560653	−0.280327	−	0.959905i	$-0.590443\pi$
$827$	−1.10270e7	−0.560653	−0.280327	+	0.959905i	$0.590443\pi$
$828$	0	0
$829$	1.64140e7	0.829523	0.414761	−	0.909930i	$-0.363865\pi$
$829$	1.64140e7	0.829523	0.414761	+	0.909930i	$0.363865\pi$
$830$	0	0
$831$	1.47891e6	0.0742916
$832$	0	0
$833$	4.88547e7	2.43946
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.58329e6	−0.0781171
$838$	0	0
$839$	−1.86354e7	−0.913974	−0.456987	−	0.889473i	$-0.651071\pi$
$839$	−1.86354e7	−0.913974	−0.456987	+	0.889473i	$0.651071\pi$
$840$	0	0
$841$	−1.39292e7	−0.679104
$842$	0	0
$843$	−5.65850e6	−0.274241
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.90095e7	−0.910464
$848$	0	0
$849$	−3.30815e6	−0.157513
$850$	0	0
$851$	4.58922e7	2.17228
$852$	0	0
$853$	2.28504e6	0.107528	0.0537639	−	0.998554i	$-0.482878\pi$
$853$	2.28504e6	0.107528	0.0537639	+	0.998554i	$0.482878\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.20029e7	−1.95356	−0.976781	−	0.214241i	$-0.931272\pi$
$857$	−4.20029e7	−1.95356	−0.976781	+	0.214241i	$0.931272\pi$
$858$	0	0
$859$	−6.39699e6	−0.295797	−0.147898	−	0.989003i	$-0.547251\pi$
$859$	−6.39699e6	−0.295797	−0.147898	+	0.989003i	$0.547251\pi$
$860$	0	0
$861$	−8.74526e6	−0.402036
$862$	0	0
$863$	8.50743e6	0.388840	0.194420	−	0.980918i	$-0.437717\pi$
$863$	8.50743e6	0.388840	0.194420	+	0.980918i	$0.437717\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	527920.	0.0238517
$868$	0	0
$869$	1.01139e7	0.454328
$870$	0	0
$871$	−4.53743e6	−0.202658
$872$	0	0
$873$	4.00097e6	0.177676
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.75823e7	1.21097	0.605483	−	0.795858i	$-0.292980\pi$
$877$	2.75823e7	1.21097	0.605483	+	0.795858i	$0.292980\pi$
$878$	0	0
$879$	−5.84199e6	−0.255028
$880$	0	0
$881$	3.10318e7	1.34700	0.673499	−	0.739188i	$-0.264791\pi$
$881$	3.10318e7	1.34700	0.673499	+	0.739188i	$0.264791\pi$
$882$	0	0
$883$	−1.68243e7	−0.726164	−0.363082	−	0.931757i	$-0.618276\pi$
$883$	−1.68243e7	−0.726164	−0.363082	+	0.931757i	$0.618276\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.41735e7	−0.604879	−0.302439	−	0.953169i	$-0.597801\pi$
$887$	−1.41735e7	−0.604879	−0.302439	+	0.953169i	$0.597801\pi$
$888$	0	0
$889$	714613.	0.0303261
$890$	0	0
$891$	−1.57880e7	−0.666241
$892$	0	0
$893$	−6.02774e6	−0.252945
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−924433.	−0.0383614
$898$	0	0
$899$	−3.34188e6	−0.137909
$900$	0	0
$901$	2.29763e7	0.942906
$902$	0	0
$903$	1.11543e6	0.0455220
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.35968e7	0.548807	0.274404	−	0.961615i	$-0.411520\pi$
$907$	1.35968e7	0.548807	0.274404	+	0.961615i	$0.411520\pi$
$908$	0	0
$909$	3.58057e7	1.43729
$910$	0	0
$911$	3.14453e7	1.25534	0.627668	−	0.778481i	$-0.284010\pi$
$911$	3.14453e7	1.25534	0.627668	+	0.778481i	$0.284010\pi$
$912$	0	0
$913$	2.51406e7	0.998157
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.26674e7	−2.06832
$918$	0	0
$919$	1.73140e7	0.676253	0.338127	−	0.941101i	$-0.390207\pi$
$919$	1.73140e7	0.676253	0.338127	+	0.941101i	$0.390207\pi$
$920$	0	0
$921$	767050.	0.0297971
$922$	0	0
$923$	8.29969e6	0.320669
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.70023e6	0.179647
$928$	0	0
$929$	1.19303e7	0.453535	0.226767	−	0.973949i	$-0.427184\pi$
$929$	1.19303e7	0.453535	0.226767	+	0.973949i	$0.427184\pi$
$930$	0	0
$931$	−7.98448e7	−3.01906
$932$	0	0
$933$	−3.58248e6	−0.134735
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.88245e7	−1.07254	−0.536270	−	0.844046i	$-0.680167\pi$
$937$	−2.88245e7	−1.07254	−0.536270	+	0.844046i	$0.680167\pi$
$938$	0	0
$939$	2.34218e6	0.0866873
$940$	0	0
$941$	3.86079e7	1.42135	0.710676	−	0.703519i	$-0.248389\pi$
$941$	3.86079e7	1.42135	0.710676	+	0.703519i	$0.248389\pi$
$942$	0	0
$943$	−4.26341e7	−1.56127
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.06877e7	1.83666	0.918328	−	0.395820i	$-0.129539\pi$
$947$	5.06877e7	1.83666	0.918328	+	0.395820i	$0.129539\pi$
$948$	0	0
$949$	7.91186e6	0.285176
$950$	0	0
$951$	−286994.	−0.0102902
$952$	0	0
$953$	−4.32695e7	−1.54330	−0.771649	−	0.636048i	$-0.780568\pi$
$953$	−4.32695e7	−1.54330	−0.771649	+	0.636048i	$0.780568\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.88683e6	0.0665967
$958$	0	0
$959$	−4.29301e7	−1.50735
$960$	0	0
$961$	−2.69324e7	−0.940732
$962$	0	0
$963$	4.44019e7	1.54289
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.77348e7	0.953804	0.476902	−	0.878956i	$-0.341760\pi$
$967$	2.77348e7	0.953804	0.476902	+	0.878956i	$0.341760\pi$
$968$	0	0
$969$	5.01850e6	0.171698
$970$	0	0
$971$	1.42588e7	0.485327	0.242664	−	0.970111i	$-0.421979\pi$
$971$	1.42588e7	0.485327	0.242664	+	0.970111i	$0.421979\pi$
$972$	0	0
$973$	8.40845e7	2.84730
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.74148e7	1.58919	0.794597	−	0.607137i	$-0.207682\pi$
$977$	4.74148e7	1.58919	0.794597	+	0.607137i	$0.207682\pi$
$978$	0	0
$979$	−1.34368e7	−0.448063
$980$	0	0
$981$	−3.09456e7	−1.02666
$982$	0	0
$983$	−4.76960e7	−1.57434	−0.787169	−	0.616738i	$-0.788454\pi$
$983$	−4.76960e7	−1.57434	−0.787169	+	0.616738i	$0.788454\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.10076e6	−0.0686409
$988$	0	0
$989$	5.43782e6	0.176780
$990$	0	0
$991$	−1.56020e7	−0.504657	−0.252328	−	0.967642i	$-0.581196\pi$
$991$	−1.56020e7	−0.504657	−0.252328	+	0.967642i	$0.581196\pi$
$992$	0	0
$993$	−2.27209e6	−0.0731226
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.28177e7	0.408388	0.204194	−	0.978930i	$-0.434543\pi$
$997$	1.28177e7	0.408388	0.204194	+	0.978930i	$0.434543\pi$
$998$	0	0
$999$	−1.82501e7	−0.578566

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.y.1.2		3
4.3	odd	2		200.6.a.h.1.2	✓	3
5.2	odd	4		400.6.c.p.49.4		6
5.3	odd	4		400.6.c.p.49.3		6
5.4	even	2		400.6.a.x.1.2		3
20.3	even	4		200.6.c.g.49.4		6
20.7	even	4		200.6.c.g.49.3		6
20.19	odd	2		200.6.a.i.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.a.h.1.2	✓	3	4.3	odd	2
200.6.a.i.1.2	yes	3	20.19	odd	2
200.6.c.g.49.3		6	20.7	even	4
200.6.c.g.49.4		6	20.3	even	4
400.6.a.x.1.2		3	5.4	even	2
400.6.a.y.1.2		3	1.1	even	1	trivial
400.6.c.p.49.3		6	5.3	odd	4
400.6.c.p.49.4		6	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.53448 q^{3} +247.370 q^{7} -236.576 q^{9} +O(q^{10})\)	\(q-2.53448 q^{3} +247.370 q^{7} -236.576 q^{9} -290.180 q^{11} -119.335 q^{13} +1100.71 q^{17} -1798.92 q^{19} -626.952 q^{21} -3056.46 q^{23} +1215.47 q^{27} +2565.53 q^{29} -1302.61 q^{31} +735.454 q^{33} -15014.8 q^{37} +302.452 q^{39} +13948.8 q^{41} -1779.13 q^{43} +3350.75 q^{47} +44384.7 q^{49} -2789.72 q^{51} +20874.1 q^{53} +4559.33 q^{57} -31645.3 q^{59} -34965.3 q^{61} -58521.8 q^{63} +38022.5 q^{67} +7746.52 q^{69} -69549.3 q^{71} -66299.4 q^{73} -71781.7 q^{77} -34853.9 q^{79} +54407.5 q^{81} -86638.0 q^{83} -6502.28 q^{87} +46305.1 q^{89} -29519.9 q^{91} +3301.43 q^{93} -16911.9 q^{97} +68649.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 70 q^{7} + 154 q^{9}+O(q^{10}) \) 3 * q + q^3 - 70 * q^7 + 154 * q^9	\( 3 q + q^{3} - 70 q^{7} + 154 q^{9} + 19 q^{11} + 196 q^{13} - 1223 q^{17} - 1221 q^{19} + 958 q^{21} - 2490 q^{23} + 4123 q^{27} + 11912 q^{29} - 7442 q^{31} + 6969 q^{33} - 14766 q^{37} - 27396 q^{39} + 3223 q^{41} - 41060 q^{43} + 29188 q^{47} + 66423 q^{49} - 43165 q^{51} + 12878 q^{53} - 77791 q^{57} - 64912 q^{59} + 22478 q^{61} - 112916 q^{63} + 26499 q^{67} + 178642 q^{69} - 86676 q^{71} - 8305 q^{73} - 106822 q^{77} - 21982 q^{79} - 71453 q^{81} - 213353 q^{83} + 14952 q^{87} + 182381 q^{89} - 149224 q^{91} - 196470 q^{93} - 76342 q^{97} + 149130 q^{99}+O(q^{100}) \) 3 * q + q^3 - 70 * q^7 + 154 * q^9 + 19 * q^11 + 196 * q^13 - 1223 * q^17 - 1221 * q^19 + 958 * q^21 - 2490 * q^23 + 4123 * q^27 + 11912 * q^29 - 7442 * q^31 + 6969 * q^33 - 14766 * q^37 - 27396 * q^39 + 3223 * q^41 - 41060 * q^43 + 29188 * q^47 + 66423 * q^49 - 43165 * q^51 + 12878 * q^53 - 77791 * q^57 - 64912 * q^59 + 22478 * q^61 - 112916 * q^63 + 26499 * q^67 + 178642 * q^69 - 86676 * q^71 - 8305 * q^73 - 106822 * q^77 - 21982 * q^79 - 71453 * q^81 - 213353 * q^83 + 14952 * q^87 + 182381 * q^89 - 149224 * q^91 - 196470 * q^93 - 76342 * q^97 + 149130 * q^99

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