LMFDB - Embedded newform 441.6.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{19}, \sqrt{69})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 71x^{2} + 72x - 15 $ x^4 - 2x^3 - 71x^2 + 72*x - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.294413$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.71780 q^{2} +44.0000 q^{4} +99.6795 q^{5} +104.614 q^{8} +O(q^{10})$	$q+8.71780 q^{2} +44.0000 q^{4} +99.6795 q^{5} +104.614 q^{8} +868.986 q^{10} +374.865 q^{11} +868.986 q^{13} -496.000 q^{16} -1096.47 q^{17} -868.986 q^{19} +4385.90 q^{20} +3268.00 q^{22} +2972.77 q^{23} +6811.00 q^{25} +7575.64 q^{26} +4515.82 q^{29} -9558.84 q^{31} -7671.66 q^{32} -9558.84 q^{34} -5466.00 q^{37} -7575.64 q^{38} +10427.8 q^{40} +9868.27 q^{41} +12540.0 q^{43} +16494.1 q^{44} +25916.0 q^{46} -9967.95 q^{47} +59376.9 q^{50} +38235.4 q^{52} -15151.5 q^{53} +37366.4 q^{55} +39368.0 q^{58} +42662.8 q^{59} -47794.2 q^{61} -83332.1 q^{62} -51008.0 q^{64} +86620.0 q^{65} -29996.0 q^{67} -48244.9 q^{68} +61469.2 q^{71} -48663.2 q^{73} -47651.5 q^{74} -38235.4 q^{76} +80168.0 q^{79} -49441.0 q^{80} +86029.6 q^{82} +30701.3 q^{83} -109296. q^{85} +109321. q^{86} +39216.0 q^{88} -20833.0 q^{89} +130802. q^{92} -86898.6 q^{94} -86620.0 q^{95} -133824. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 176 q^{4}+O(q^{10}) $ 4 * q + 176 * q^4	$ 4 q + 176 q^{4} - 1984 q^{16} + 13072 q^{22} + 27244 q^{25} - 21864 q^{37} + 50160 q^{43} + 103664 q^{46} + 157472 q^{58} - 204032 q^{64} - 119984 q^{67} + 320672 q^{79} - 437184 q^{85} + 156864 q^{88}+O(q^{100}) $ 4 * q + 176 * q^4 - 1984 * q^16 + 13072 * q^22 + 27244 * q^25 - 21864 * q^37 + 50160 * q^43 + 103664 * q^46 + 157472 * q^58 - 204032 * q^64 - 119984 * q^67 + 320672 * q^79 - 437184 * q^85 + 156864 * q^88

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 $

$2$	8.71780	1.54110	0.770552	−	0.637377i	$-0.219981\pi$
$2$	8.71780	1.54110	0.770552	+	0.637377i	$0.219981\pi$
$3$	0	0
$3$	0	0
$4$	44.0000	1.37500
$5$	99.6795	1.78312	0.891560	−	0.452902i	$-0.149611\pi$
$5$	99.6795	1.78312	0.891560	+	0.452902i	$0.149611\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	104.614	0.577914
$9$	0	0
$10$	868.986	2.74797
$11$	374.865	0.934100	0.467050	−	0.884231i	$-0.345317\pi$
$11$	374.865	0.934100	0.467050	+	0.884231i	$0.345317\pi$
$12$	0	0
$13$	868.986	1.42611	0.713057	−	0.701106i	$-0.247310\pi$
$13$	868.986	1.42611	0.713057	+	0.701106i	$0.247310\pi$
$14$	0	0
$15$	0	0
$16$	−496.000	−0.484375
$17$	−1096.47	−0.920187	−0.460094	−	0.887870i	$-0.652184\pi$
$17$	−1096.47	−0.920187	−0.460094	+	0.887870i	$0.652184\pi$
$18$	0	0
$19$	−868.986	−0.552241	−0.276120	−	0.961123i	$-0.589049\pi$
$19$	−868.986	−0.552241	−0.276120	+	0.961123i	$0.589049\pi$
$20$	4385.90	2.45179
$21$	0	0
$22$	3268.00	1.43955
$23$	2972.77	1.17177	0.585884	−	0.810395i	$-0.300747\pi$
$23$	2972.77	1.17177	0.585884	+	0.810395i	$0.300747\pi$
$24$	0	0
$25$	6811.00	2.17952
$26$	7575.64	2.19779
$27$	0	0
$28$	0	0
$29$	4515.82	0.997107	0.498553	−	0.866859i	$-0.333865\pi$
$29$	4515.82	0.997107	0.498553	+	0.866859i	$0.333865\pi$
$30$	0	0
$31$	−9558.84	−1.78649	−0.893246	−	0.449568i	$-0.851578\pi$
$31$	−9558.84	−1.78649	−0.893246	+	0.449568i	$0.851578\pi$
$32$	−7671.66	−1.32439
$33$	0	0
$34$	−9558.84	−1.41810
$35$	0	0
$36$	0	0
$37$	−5466.00	−0.656395	−0.328198	−	0.944609i	$-0.606441\pi$
$37$	−5466.00	−0.656395	−0.328198	+	0.944609i	$0.606441\pi$
$38$	−7575.64	−0.851060
$39$	0	0
$40$	10427.8	1.03049
$41$	9868.27	0.916814	0.458407	−	0.888742i	$-0.348420\pi$
$41$	9868.27	0.916814	0.458407	+	0.888742i	$0.348420\pi$
$42$	0	0
$43$	12540.0	1.03425	0.517126	−	0.855909i	$-0.327002\pi$
$43$	12540.0	1.03425	0.517126	+	0.855909i	$0.327002\pi$
$44$	16494.1	1.28439
$45$	0	0
$46$	25916.0	1.80582
$47$	−9967.95	−0.658205	−0.329102	−	0.944294i	$-0.606746\pi$
$47$	−9967.95	−0.658205	−0.329102	+	0.944294i	$0.606746\pi$
$48$	0	0
$49$	0	0
$50$	59376.9	3.35887
$51$	0	0
$52$	38235.4	1.96091
$53$	−15151.5	−0.740912	−0.370456	−	0.928850i	$-0.620799\pi$
$53$	−15151.5	−0.740912	−0.370456	+	0.928850i	$0.620799\pi$
$54$	0	0
$55$	37366.4	1.66561
$56$	0	0
$57$	0	0
$58$	39368.0	1.53664
$59$	42662.8	1.59558	0.797792	−	0.602933i	$-0.206001\pi$
$59$	42662.8	1.59558	0.797792	+	0.602933i	$0.206001\pi$
$60$	0	0
$61$	−47794.2	−1.64456	−0.822282	−	0.569080i	$-0.807299\pi$
$61$	−47794.2	−1.64456	−0.822282	+	0.569080i	$0.807299\pi$
$62$	−83332.1	−2.75317
$63$	0	0
$64$	−51008.0	−1.55664
$65$	86620.0	2.54293
$66$	0	0
$67$	−29996.0	−0.816350	−0.408175	−	0.912904i	$-0.633835\pi$
$67$	−29996.0	−0.816350	−0.408175	+	0.912904i	$0.633835\pi$
$68$	−48244.9	−1.26526
$69$	0	0
$70$	0	0
$71$	61469.2	1.44714	0.723572	−	0.690249i	$-0.242499\pi$
$71$	61469.2	1.44714	0.723572	+	0.690249i	$0.242499\pi$
$72$	0	0
$73$	−48663.2	−1.06879	−0.534396	−	0.845234i	$-0.679461\pi$
$73$	−48663.2	−1.06879	−0.534396	+	0.845234i	$0.679461\pi$
$74$	−47651.5	−1.01157
$75$	0	0
$76$	−38235.4	−0.759331
$77$	0	0
$78$	0	0
$79$	80168.0	1.44522	0.722609	−	0.691257i	$-0.242943\pi$
$79$	80168.0	1.44522	0.722609	+	0.691257i	$0.242943\pi$
$80$	−49441.0	−0.863699
$81$	0	0
$82$	86029.6	1.41291
$83$	30701.3	0.489172	0.244586	−	0.969628i	$-0.421348\pi$
$83$	30701.3	0.489172	0.244586	+	0.969628i	$0.421348\pi$
$84$	0	0
$85$	−109296.	−1.64080
$86$	109321.	1.59389
$87$	0	0
$88$	39216.0	0.539829
$89$	−20833.0	−0.278790	−0.139395	−	0.990237i	$-0.544516\pi$
$89$	−20833.0	−0.278790	−0.139395	+	0.990237i	$0.544516\pi$
$90$	0	0
$91$	0	0
$92$	130802.	1.61118
$93$	0	0
$94$	−86898.6	−1.01436
$95$	−86620.0	−0.984712
$96$	0	0
$97$	−133824.	−1.44412	−0.722061	−	0.691829i	$-0.756805\pi$
$97$	−133824.	−1.44412	−0.722061	+	0.691829i	$0.756805\pi$
$98$	0	0
$99$	0	0
$100$	299684.	2.99684
$101$	106358.	1.03745	0.518725	−	0.854941i	$-0.326407\pi$
$101$	106358.	1.03745	0.518725	+	0.854941i	$0.326407\pi$
$102$	0	0
$103$	47794.2	0.443897	0.221949	−	0.975058i	$-0.428758\pi$
$103$	47794.2	0.443897	0.221949	+	0.975058i	$0.428758\pi$
$104$	90907.7	0.824171
$105$	0	0
$106$	−132088.	−1.14182
$107$	−102983.	−0.869576	−0.434788	−	0.900533i	$-0.643177\pi$
$107$	−102983.	−0.869576	−0.434788	+	0.900533i	$0.643177\pi$
$108$	0	0
$109$	73854.0	0.595399	0.297699	−	0.954660i	$-0.403781\pi$
$109$	73854.0	0.595399	0.297699	+	0.954660i	$0.403781\pi$
$110$	325753.	2.56688
$111$	0	0
$112$	0	0
$113$	−29535.9	−0.217598	−0.108799	−	0.994064i	$-0.534700\pi$
$113$	−29535.9	−0.217598	−0.108799	+	0.994064i	$0.534700\pi$
$114$	0	0
$115$	296324.	2.08940
$116$	198696.	1.37102
$117$	0	0
$118$	371926.	2.45896
$119$	0	0
$120$	0	0
$121$	−20527.0	−0.127457
$122$	−416660.	−2.53444
$123$	0	0
$124$	−420589.	−2.45643
$125$	367419.	2.10323
$126$	0	0
$127$	175120.	0.963444	0.481722	−	0.876324i	$-0.340012\pi$
$127$	175120.	0.963444	0.481722	+	0.876324i	$0.340012\pi$
$128$	−199184.	−1.07456
$129$	0	0
$130$	755136.	3.91892
$131$	−144735.	−0.736876	−0.368438	−	0.929652i	$-0.620107\pi$
$131$	−144735.	−0.736876	−0.368438	+	0.929652i	$0.620107\pi$
$132$	0	0
$133$	0	0
$134$	−261499.	−1.25808
$135$	0	0
$136$	−114706.	−0.531789
$137$	−172229.	−0.783979	−0.391989	−	0.919970i	$-0.628213\pi$
$137$	−172229.	−0.783979	−0.391989	+	0.919970i	$0.628213\pi$
$138$	0	0
$139$	−19117.7	−0.0839263	−0.0419632	−	0.999119i	$-0.513361\pi$
$139$	−19117.7	−0.0839263	−0.0419632	+	0.999119i	$0.513361\pi$
$140$	0	0
$141$	0	0
$142$	535876.	2.23020
$143$	325753.	1.33213
$144$	0	0
$145$	450135.	1.77796
$146$	−424236.	−1.64712
$147$	0	0
$148$	−240504.	−0.902543
$149$	−9118.82	−0.0336491	−0.0168245	−	0.999858i	$-0.505356\pi$
$149$	−9118.82	−0.0336491	−0.0168245	+	0.999858i	$0.505356\pi$
$150$	0	0
$151$	310200.	1.10713	0.553566	−	0.832805i	$-0.313267\pi$
$151$	310200.	1.10713	0.553566	+	0.832805i	$0.313267\pi$
$152$	−90907.7	−0.319148
$153$	0	0
$154$	0	0
$155$	−952820.	−3.18553
$156$	0	0
$157$	−258089.	−0.835641	−0.417821	−	0.908530i	$-0.637206\pi$
$157$	−258089.	−0.835641	−0.417821	+	0.908530i	$0.637206\pi$
$158$	698888.	2.22723
$159$	0	0
$160$	−764707.	−2.36154
$161$	0	0
$162$	0	0
$163$	126556.	0.373090	0.186545	−	0.982446i	$-0.440271\pi$
$163$	126556.	0.373090	0.186545	+	0.982446i	$0.440271\pi$
$164$	434204.	1.26062
$165$	0	0
$166$	267648.	0.753864
$167$	−342100.	−0.949209	−0.474605	−	0.880199i	$-0.657409\pi$
$167$	−342100.	−0.949209	−0.474605	+	0.880199i	$0.657409\pi$
$168$	0	0
$169$	383843.	1.03380
$170$	−952820.	−2.52865
$171$	0	0
$172$	551760.	1.42210
$173$	−187497.	−0.476299	−0.238149	−	0.971229i	$-0.576541\pi$
$173$	−187497.	−0.476299	−0.238149	+	0.971229i	$0.576541\pi$
$174$	0	0
$175$	0	0
$176$	−185933.	−0.452455
$177$	0	0
$178$	−181618.	−0.429644
$179$	−127829.	−0.298193	−0.149096	−	0.988823i	$-0.547636\pi$
$179$	−127829.	−0.298193	−0.149096	+	0.988823i	$0.547636\pi$
$180$	0	0
$181$	219853.	0.498812	0.249406	−	0.968399i	$-0.419765\pi$
$181$	219853.	0.498812	0.249406	+	0.968399i	$0.419765\pi$
$182$	0	0
$183$	0	0
$184$	310992.	0.677181
$185$	−544848.	−1.17043
$186$	0	0
$187$	−411030.	−0.859547
$188$	−438590.	−0.905032
$189$	0	0
$190$	−755136.	−1.51754
$191$	−116130.	−0.230335	−0.115168	−	0.993346i	$-0.536741\pi$
$191$	−116130.	−0.230335	−0.115168	+	0.993346i	$0.536741\pi$
$192$	0	0
$193$	−306306.	−0.591919	−0.295959	−	0.955200i	$-0.595639\pi$
$193$	−306306.	−0.591919	−0.295959	+	0.955200i	$0.595639\pi$
$194$	−1.16665e6	−2.22554
$195$	0	0
$196$	0	0
$197$	−360202.	−0.661273	−0.330636	−	0.943758i	$-0.607263\pi$
$197$	−360202.	−0.661273	−0.330636	+	0.943758i	$0.607263\pi$
$198$	0	0
$199$	229412.	0.410661	0.205331	−	0.978693i	$-0.434173\pi$
$199$	229412.	0.410661	0.205331	+	0.978693i	$0.434173\pi$
$200$	712523.	1.25957
$201$	0	0
$202$	927208.	1.59882
$203$	0	0
$204$	0	0
$205$	983664.	1.63479
$206$	416660.	0.684091
$207$	0	0
$208$	−431017.	−0.690774
$209$	−325753.	−0.515848
$210$	0	0
$211$	−1.21031e6	−1.87150	−0.935750	−	0.352664i	$-0.885276\pi$
$211$	−1.21031e6	−1.87150	−0.935750	+	0.352664i	$0.885276\pi$
$212$	−666667.	−1.01875
$213$	0	0
$214$	−897788.	−1.34011
$215$	1.24998e6	1.84420
$216$	0	0
$217$	0	0
$218$	643844.	0.917571
$219$	0	0
$220$	1.64412e6	2.29022
$221$	−952820.	−1.31229
$222$	0	0
$223$	745590.	1.00401	0.502005	−	0.864865i	$-0.332596\pi$
$223$	745590.	1.00401	0.502005	+	0.864865i	$0.332596\pi$
$224$	0	0
$225$	0	0
$226$	−257488.	−0.335340
$227$	140349.	0.180777	0.0903886	−	0.995907i	$-0.471189\pi$
$227$	140349.	0.180777	0.0903886	+	0.995907i	$0.471189\pi$
$228$	0	0
$229$	−774266.	−0.975667	−0.487833	−	0.872937i	$-0.662213\pi$
$229$	−774266.	−0.975667	−0.487833	+	0.872937i	$0.662213\pi$
$230$	2.58329e6	3.21999
$231$	0	0
$232$	472416.	0.576242
$233$	1.03550e6	1.24957	0.624785	−	0.780797i	$-0.285187\pi$
$233$	1.03550e6	1.24957	0.624785	+	0.780797i	$0.285187\pi$
$234$	0	0
$235$	−993600.	−1.17366
$236$	1.87716e6	2.19393
$237$	0	0
$238$	0	0
$239$	−1.11622e6	−1.26402	−0.632011	−	0.774960i	$-0.717770\pi$
$239$	−1.11622e6	−1.26402	−0.632011	+	0.774960i	$0.717770\pi$
$240$	0	0
$241$	−1.44425e6	−1.60177	−0.800887	−	0.598816i	$-0.795638\pi$
$241$	−1.44425e6	−1.60177	−0.800887	+	0.598816i	$0.795638\pi$
$242$	−178950.	−0.196424
$243$	0	0
$244$	−2.10295e6	−2.26128
$245$	0	0
$246$	0	0
$247$	−755136.	−0.787558
$248$	−999985.	−1.03244
$249$	0	0
$250$	3.20308e6	3.24129
$251$	−216105.	−0.216511	−0.108256	−	0.994123i	$-0.534527\pi$
$251$	−216105.	−0.216511	−0.108256	+	0.994123i	$0.534527\pi$
$252$	0	0
$253$	1.11439e6	1.09455
$254$	1.52666e6	1.48477
$255$	0	0
$256$	−104192.	−0.0993652
$257$	−1.28517e6	−1.21374	−0.606872	−	0.794800i	$-0.707576\pi$
$257$	−1.28517e6	−1.21374	−0.606872	+	0.794800i	$0.707576\pi$
$258$	0	0
$259$	0	0
$260$	3.81128e6	3.49653
$261$	0	0
$262$	−1.26177e6	−1.13560
$263$	−1.47794e6	−1.31755	−0.658774	−	0.752341i	$-0.728925\pi$
$263$	−1.47794e6	−1.31755	−0.658774	+	0.752341i	$0.728925\pi$
$264$	0	0
$265$	−1.51030e6	−1.32114
$266$	0	0
$267$	0	0
$268$	−1.31982e6	−1.12248
$269$	−2.28176e6	−1.92260	−0.961302	−	0.275497i	$-0.911158\pi$
$269$	−2.28176e6	−1.92260	−0.961302	+	0.275497i	$0.911158\pi$
$270$	0	0
$271$	−1.09058e6	−0.902055	−0.451028	−	0.892510i	$-0.648942\pi$
$271$	−1.09058e6	−0.902055	−0.451028	+	0.892510i	$0.648942\pi$
$272$	543851.	0.445716
$273$	0	0
$274$	−1.50146e6	−1.20819
$275$	2.55321e6	2.03589
$276$	0	0
$277$	−256234.	−0.200649	−0.100325	−	0.994955i	$-0.531988\pi$
$277$	−256234.	−0.200649	−0.100325	+	0.994955i	$0.531988\pi$
$278$	−166664.	−0.129339
$279$	0	0
$280$	0	0
$281$	889041.	0.671670	0.335835	−	0.941921i	$-0.390982\pi$
$281$	889041.	0.671670	0.335835	+	0.941921i	$0.390982\pi$
$282$	0	0
$283$	−496191.	−0.368284	−0.184142	−	0.982900i	$-0.558951\pi$
$283$	−496191.	−0.368284	−0.184142	+	0.982900i	$0.558951\pi$
$284$	2.70464e6	1.98982
$285$	0	0
$286$	2.83984e6	2.05296
$287$	0	0
$288$	0	0
$289$	−217601.	−0.153256
$290$	3.92418e6	2.74002
$291$	0	0
$292$	−2.14118e6	−1.46959
$293$	−731348.	−0.497686	−0.248843	−	0.968544i	$-0.580050\pi$
$293$	−731348.	−0.497686	−0.248843	+	0.968544i	$0.580050\pi$
$294$	0	0
$295$	4.25261e6	2.84512
$296$	−571818.	−0.379340
$297$	0	0
$298$	−79496.0	−0.0518567
$299$	2.58329e6	1.67107
$300$	0	0
$301$	0	0
$302$	2.70426e6	1.70621
$303$	0	0
$304$	431017.	0.267492
$305$	−4.76410e6	−2.93246
$306$	0	0
$307$	66911.9	0.0405189	0.0202594	−	0.999795i	$-0.493551\pi$
$307$	66911.9	0.0405189	0.0202594	+	0.999795i	$0.493551\pi$
$308$	0	0
$309$	0	0
$310$	−8.30650e6	−4.90923
$311$	−302627.	−0.177422	−0.0887108	−	0.996057i	$-0.528275\pi$
$311$	−302627.	−0.177422	−0.0887108	+	0.996057i	$0.528275\pi$
$312$	0	0
$313$	1.01324e6	0.584589	0.292294	−	0.956328i	$-0.405581\pi$
$313$	1.01324e6	0.584589	0.292294	+	0.956328i	$0.405581\pi$
$314$	−2.24997e6	−1.28781
$315$	0	0
$316$	3.52739e6	1.98717
$317$	638474.	0.356858	0.178429	−	0.983953i	$-0.442899\pi$
$317$	638474.	0.356858	0.178429	+	0.983953i	$0.442899\pi$
$318$	0	0
$319$	1.69282e6	0.931398
$320$	−5.08445e6	−2.77568
$321$	0	0
$322$	0	0
$323$	952820.	0.508165
$324$	0	0
$325$	5.91866e6	3.10824
$326$	1.10329e6	0.574970
$327$	0	0
$328$	1.03235e6	0.529840
$329$	0	0
$330$	0	0
$331$	2.63905e6	1.32397	0.661985	−	0.749517i	$-0.269714\pi$
$331$	2.63905e6	1.32397	0.661985	+	0.749517i	$0.269714\pi$
$332$	1.35086e6	0.672611
$333$	0	0
$334$	−2.98236e6	−1.46283
$335$	−2.98999e6	−1.45565
$336$	0	0
$337$	−455070.	−0.218275	−0.109137	−	0.994027i	$-0.534809\pi$
$337$	−455070.	−0.218275	−0.109137	+	0.994027i	$0.534809\pi$
$338$	3.34627e6	1.59319
$339$	0	0
$340$	−4.80902e6	−2.25611
$341$	−3.58328e6	−1.66876
$342$	0	0
$343$	0	0
$344$	1.31185e6	0.597709
$345$	0	0
$346$	−1.63456e6	−0.734025
$347$	−1.30756e6	−0.582957	−0.291479	−	0.956577i	$-0.594147\pi$
$347$	−1.30756e6	−0.582957	−0.291479	+	0.956577i	$0.594147\pi$
$348$	0	0
$349$	1.05234e6	0.462480	0.231240	−	0.972897i	$-0.425722\pi$
$349$	1.05234e6	0.462480	0.231240	+	0.972897i	$0.425722\pi$
$350$	0	0
$351$	0	0
$352$	−2.87584e6	−1.23711
$353$	−1.17333e6	−0.501167	−0.250583	−	0.968095i	$-0.580622\pi$
$353$	−1.17333e6	−0.501167	−0.250583	+	0.968095i	$0.580622\pi$
$354$	0	0
$355$	6.12722e6	2.58043
$356$	−916653.	−0.383336
$357$	0	0
$358$	−1.11439e6	−0.459546
$359$	−2.98746e6	−1.22339	−0.611696	−	0.791093i	$-0.709512\pi$
$359$	−2.98746e6	−1.22339	−0.611696	+	0.791093i	$0.709512\pi$
$360$	0	0
$361$	−1.72096e6	−0.695030
$362$	1.91664e6	0.768721
$363$	0	0
$364$	0	0
$365$	−4.85072e6	−1.90579
$366$	0	0
$367$	−2.98236e6	−1.15583	−0.577916	−	0.816096i	$-0.696134\pi$
$367$	−2.98236e6	−1.15583	−0.577916	+	0.816096i	$0.696134\pi$
$368$	−1.47449e6	−0.567575
$369$	0	0
$370$	−4.74988e6	−1.80376
$371$	0	0
$372$	0	0
$373$	−1.58662e6	−0.590473	−0.295237	−	0.955424i	$-0.595398\pi$
$373$	−1.58662e6	−0.590473	−0.295237	+	0.955424i	$0.595398\pi$
$374$	−3.58328e6	−1.32465
$375$	0	0
$376$	−1.04278e6	−0.380386
$377$	3.92418e6	1.42199
$378$	0	0
$379$	1.27168e6	0.454756	0.227378	−	0.973807i	$-0.426985\pi$
$379$	1.27168e6	0.454756	0.227378	+	0.973807i	$0.426985\pi$
$380$	−3.81128e6	−1.35398
$381$	0	0
$382$	−1.01240e6	−0.354970
$383$	5.13589e6	1.78903	0.894517	−	0.447035i	$-0.147520\pi$
$383$	5.13589e6	1.78903	0.894517	+	0.447035i	$0.147520\pi$
$384$	0	0
$385$	0	0
$386$	−2.67031e6	−0.912208
$387$	0	0
$388$	−5.88825e6	−1.98567
$389$	4.92655e6	1.65070	0.825351	−	0.564619i	$-0.190977\pi$
$389$	4.92655e6	1.65070	0.825351	+	0.564619i	$0.190977\pi$
$390$	0	0
$391$	−3.25957e6	−1.07825
$392$	0	0
$393$	0	0
$394$	−3.14017e6	−1.01909
$395$	7.99111e6	2.57700
$396$	0	0
$397$	−219853.	−0.0700095	−0.0350047	−	0.999387i	$-0.511145\pi$
$397$	−219853.	−0.0700095	−0.0350047	+	0.999387i	$0.511145\pi$
$398$	1.99997e6	0.632872
$399$	0	0
$400$	−3.37826e6	−1.05570
$401$	3.18796e6	0.990038	0.495019	−	0.868882i	$-0.335161\pi$
$401$	3.18796e6	0.990038	0.495019	+	0.868882i	$0.335161\pi$
$402$	0	0
$403$	−8.30650e6	−2.54774
$404$	4.67975e6	1.42649
$405$	0	0
$406$	0	0
$407$	−2.04901e6	−0.613139
$408$	0	0
$409$	−2.54265e6	−0.751586	−0.375793	−	0.926704i	$-0.622630\pi$
$409$	−2.54265e6	−0.751586	−0.375793	+	0.926704i	$0.622630\pi$
$410$	8.57538e6	2.51938
$411$	0	0
$412$	2.10295e6	0.610358
$413$	0	0
$414$	0	0
$415$	3.06029e6	0.872252
$416$	−6.66656e6	−1.88873
$417$	0	0
$418$	−2.83984e6	−0.794976
$419$	3.57132e6	0.993787	0.496893	−	0.867812i	$-0.334474\pi$
$419$	3.57132e6	0.993787	0.496893	+	0.867812i	$0.334474\pi$
$420$	0	0
$421$	991750.	0.272707	0.136354	−	0.990660i	$-0.456462\pi$
$421$	991750.	0.272707	0.136354	+	0.990660i	$0.456462\pi$
$422$	−1.05512e7	−2.88417
$423$	0	0
$424$	−1.58506e6	−0.428184
$425$	−7.46809e6	−2.00557
$426$	0	0
$427$	0	0
$428$	−4.53127e6	−1.19567
$429$	0	0
$430$	1.08971e7	2.84210
$431$	−3.40602e6	−0.883189	−0.441595	−	0.897215i	$-0.645587\pi$
$431$	−3.40602e6	−0.883189	−0.441595	+	0.897215i	$0.645587\pi$
$432$	0	0
$433$	7.74266e6	1.98459	0.992294	−	0.123902i	$-0.0395409\pi$
$433$	7.74266e6	1.98459	0.992294	+	0.123902i	$0.0395409\pi$
$434$	0	0
$435$	0	0
$436$	3.24958e6	0.818673
$437$	−2.58329e6	−0.647098
$438$	0	0
$439$	−6.31753e6	−1.56454	−0.782268	−	0.622942i	$-0.785937\pi$
$439$	−6.31753e6	−1.56454	−0.782268	+	0.622942i	$0.785937\pi$
$440$	3.90903e6	0.962581
$441$	0	0
$442$	−8.30650e6	−2.02238
$443$	−485712.	−0.117590	−0.0587949	−	0.998270i	$-0.518726\pi$
$443$	−485712.	−0.117590	−0.0587949	+	0.998270i	$0.518726\pi$
$444$	0	0
$445$	−2.07662e6	−0.497116
$446$	6.49990e6	1.54728
$447$	0	0
$448$	0	0
$449$	−5.18374e6	−1.21347	−0.606733	−	0.794906i	$-0.707520\pi$
$449$	−5.18374e6	−1.21347	−0.606733	+	0.794906i	$0.707520\pi$
$450$	0	0
$451$	3.69927e6	0.856396
$452$	−1.29958e6	−0.299197
$453$	0	0
$454$	1.22353e6	0.278596
$455$	0	0
$456$	0	0
$457$	6.55690e6	1.46862	0.734308	−	0.678817i	$-0.237507\pi$
$457$	6.55690e6	1.46862	0.734308	+	0.678817i	$0.237507\pi$
$458$	−6.74990e6	−1.50360
$459$	0	0
$460$	1.30383e7	2.87293
$461$	4.15893e6	0.911442	0.455721	−	0.890123i	$-0.349381\pi$
$461$	4.15893e6	0.911442	0.455721	+	0.890123i	$0.349381\pi$
$462$	0	0
$463$	5.62854e6	1.22023	0.610117	−	0.792311i	$-0.291123\pi$
$463$	5.62854e6	1.22023	0.610117	+	0.792311i	$0.291123\pi$
$464$	−2.23985e6	−0.482973
$465$	0	0
$466$	9.02728e6	1.92572
$467$	4.20727e6	0.892706	0.446353	−	0.894857i	$-0.352723\pi$
$467$	4.20727e6	0.892706	0.446353	+	0.894857i	$0.352723\pi$
$468$	0	0
$469$	0	0
$470$	−8.66200e6	−1.80873
$471$	0	0
$472$	4.46311e6	0.922110
$473$	4.70081e6	0.966095
$474$	0	0
$475$	−5.91866e6	−1.20362
$476$	0	0
$477$	0	0
$478$	−9.73096e6	−1.94799
$479$	7.10954e6	1.41580	0.707901	−	0.706311i	$-0.249642\pi$
$479$	7.10954e6	1.41580	0.707901	+	0.706311i	$0.249642\pi$
$480$	0	0
$481$	−4.74988e6	−0.936094
$482$	−1.25907e7	−2.46850
$483$	0	0
$484$	−903188.	−0.175253
$485$	−1.33395e7	−2.57504
$486$	0	0
$487$	7.03615e6	1.34435	0.672176	−	0.740392i	$-0.265360\pi$
$487$	7.03615e6	1.34435	0.672176	+	0.740392i	$0.265360\pi$
$488$	−4.99992e6	−0.950416
$489$	0	0
$490$	0	0
$491$	−1.08668e6	−0.203422	−0.101711	−	0.994814i	$-0.532432\pi$
$491$	−1.08668e6	−0.203422	−0.101711	+	0.994814i	$0.532432\pi$
$492$	0	0
$493$	−4.95148e6	−0.917525
$494$	−6.58312e6	−1.21371
$495$	0	0
$496$	4.74119e6	0.865332
$497$	0	0
$498$	0	0
$499$	3.05896e6	0.549948	0.274974	−	0.961452i	$-0.411331\pi$
$499$	3.05896e6	0.549948	0.274974	+	0.961452i	$0.411331\pi$
$500$	1.61664e7	2.89194
$501$	0	0
$502$	−1.88396e6	−0.333666
$503$	−6.23236e6	−1.09833	−0.549165	−	0.835714i	$-0.685054\pi$
$503$	−6.23236e6	−1.09833	−0.549165	+	0.835714i	$0.685054\pi$
$504$	0	0
$505$	1.06017e7	1.84990
$506$	9.71501e6	1.68681
$507$	0	0
$508$	7.70528e6	1.32474
$509$	−6.14574e6	−1.05143	−0.525714	−	0.850661i	$-0.676202\pi$
$509$	−6.14574e6	−1.05143	−0.525714	+	0.850661i	$0.676202\pi$
$510$	0	0
$511$	0	0
$512$	5.46557e6	0.921426
$513$	0	0
$514$	−1.12038e7	−1.87050
$515$	4.76410e6	0.791522
$516$	0	0
$517$	−3.73664e6	−0.614829
$518$	0	0
$519$	0	0
$520$	9.06163e6	1.46960
$521$	−6.78050e6	−1.09438	−0.547189	−	0.837009i	$-0.684302\pi$
$521$	−6.78050e6	−1.09438	−0.547189	+	0.837009i	$0.684302\pi$
$522$	0	0
$523$	−1.90134e6	−0.303952	−0.151976	−	0.988384i	$-0.548564\pi$
$523$	−1.90134e6	−0.303952	−0.151976	+	0.988384i	$0.548564\pi$
$524$	−6.36832e6	−1.01320
$525$	0	0
$526$	−1.28844e7	−2.03048
$527$	1.04810e7	1.64391
$528$	0	0
$529$	2.40101e6	0.373040
$530$	−1.31665e7	−2.03601
$531$	0	0
$532$	0	0
$533$	8.57538e6	1.30748
$534$	0	0
$535$	−1.02653e7	−1.55056
$536$	−3.13799e6	−0.471780
$537$	0	0
$538$	−1.98919e7	−2.96293
$539$	0	0
$540$	0	0
$541$	9.04950e6	1.32933	0.664663	−	0.747143i	$-0.268575\pi$
$541$	9.04950e6	1.32933	0.664663	+	0.747143i	$0.268575\pi$
$542$	−9.50743e6	−1.39016
$543$	0	0
$544$	8.41178e6	1.21868
$545$	7.36173e6	1.06167
$546$	0	0
$547$	1.91721e6	0.273969	0.136985	−	0.990573i	$-0.456259\pi$
$547$	1.91721e6	0.273969	0.136985	+	0.990573i	$0.456259\pi$
$548$	−7.57807e6	−1.07797
$549$	0	0
$550$	2.22583e7	3.13752
$551$	−3.92418e6	−0.550643
$552$	0	0
$553$	0	0
$554$	−2.23380e6	−0.309221
$555$	0	0
$556$	−841178.	−0.115399
$557$	2.62380e6	0.358337	0.179169	−	0.983818i	$-0.442659\pi$
$557$	2.62380e6	0.358337	0.179169	+	0.983818i	$0.442659\pi$
$558$	0	0
$559$	1.08971e7	1.47496
$560$	0	0
$561$	0	0
$562$	7.75048e6	1.03511
$563$	1.07498e7	1.42932	0.714662	−	0.699470i	$-0.246580\pi$
$563$	1.07498e7	1.42932	0.714662	+	0.699470i	$0.246580\pi$
$564$	0	0
$565$	−2.94412e6	−0.388003
$566$	−4.32569e6	−0.567563
$567$	0	0
$568$	6.43051e6	0.836324
$569$	914671.	0.118436	0.0592181	−	0.998245i	$-0.481139\pi$
$569$	914671.	0.118436	0.0592181	+	0.998245i	$0.481139\pi$
$570$	0	0
$571$	5.24616e6	0.673367	0.336683	−	0.941618i	$-0.390695\pi$
$571$	5.24616e6	0.673367	0.336683	+	0.941618i	$0.390695\pi$
$572$	1.43331e7	1.83168
$573$	0	0
$574$	0	0
$575$	2.02475e7	2.55389
$576$	0	0
$577$	1.15471e7	1.44389	0.721943	−	0.691953i	$-0.243249\pi$
$577$	1.15471e7	1.44389	0.721943	+	0.691953i	$0.243249\pi$
$578$	−1.89700e6	−0.236183
$579$	0	0
$580$	1.98059e7	2.44470
$581$	0	0
$582$	0	0
$583$	−5.67978e6	−0.692087
$584$	−5.09083e6	−0.617670
$585$	0	0
$586$	−6.37575e6	−0.766985
$587$	−5.01428e6	−0.600638	−0.300319	−	0.953839i	$-0.597093\pi$
$587$	−5.01428e6	−0.600638	−0.300319	+	0.953839i	$0.597093\pi$
$588$	0	0
$589$	8.30650e6	0.986574
$590$	3.70734e7	4.38462
$591$	0	0
$592$	2.71114e6	0.317941
$593$	4.82778e6	0.563781	0.281890	−	0.959447i	$-0.409039\pi$
$593$	4.82778e6	0.563781	0.281890	+	0.959447i	$0.409039\pi$
$594$	0	0
$595$	0	0
$596$	−401228.	−0.0462674
$597$	0	0
$598$	2.25206e7	2.57530
$599$	9.24522e6	1.05281	0.526405	−	0.850234i	$-0.323540\pi$
$599$	9.24522e6	1.05281	0.526405	+	0.850234i	$0.323540\pi$
$600$	0	0
$601$	−3.60455e6	−0.407066	−0.203533	−	0.979068i	$-0.565242\pi$
$601$	−3.60455e6	−0.407066	−0.203533	+	0.979068i	$0.565242\pi$
$602$	0	0
$603$	0	0
$604$	1.36488e7	1.52231
$605$	−2.04612e6	−0.227270
$606$	0	0
$607$	7.34988e6	0.809671	0.404835	−	0.914390i	$-0.367329\pi$
$607$	7.34988e6	0.809671	0.404835	+	0.914390i	$0.367329\pi$
$608$	6.66656e6	0.731380
$609$	0	0
$610$	−4.15325e7	−4.51922
$611$	−8.66200e6	−0.938675
$612$	0	0
$613$	−1.52374e7	−1.63780	−0.818898	−	0.573940i	$-0.805414\pi$
$613$	−1.52374e7	−1.63780	−0.818898	+	0.573940i	$0.805414\pi$
$614$	583324.	0.0624438
$615$	0	0
$616$	0	0
$617$	1.13748e7	1.20290	0.601450	−	0.798910i	$-0.294590\pi$
$617$	1.13748e7	1.20290	0.601450	+	0.798910i	$0.294590\pi$
$618$	0	0
$619$	1.50074e7	1.57427	0.787133	−	0.616783i	$-0.211564\pi$
$619$	1.50074e7	1.57427	0.787133	+	0.616783i	$0.211564\pi$
$620$	−4.19241e7	−4.38010
$621$	0	0
$622$	−2.63824e6	−0.273425
$623$	0	0
$624$	0	0
$625$	1.53397e7	1.57079
$626$	8.83320e6	0.900911
$627$	0	0
$628$	−1.13559e7	−1.14901
$629$	5.99333e6	0.604006
$630$	0	0
$631$	−4.06654e6	−0.406586	−0.203293	−	0.979118i	$-0.565164\pi$
$631$	−4.06654e6	−0.406586	−0.203293	+	0.979118i	$0.565164\pi$
$632$	8.38666e6	0.835211
$633$	0	0
$634$	5.56609e6	0.549955
$635$	1.74559e7	1.71794
$636$	0	0
$637$	0	0
$638$	1.47577e7	1.43538
$639$	0	0
$640$	−1.98546e7	−1.91607
$641$	−9.79786e6	−0.941860	−0.470930	−	0.882171i	$-0.656082\pi$
$641$	−9.79786e6	−0.941860	−0.470930	+	0.882171i	$0.656082\pi$
$642$	0	0
$643$	−678678.	−0.0647346	−0.0323673	−	0.999476i	$-0.510305\pi$
$643$	−678678.	−0.0647346	−0.0323673	+	0.999476i	$0.510305\pi$
$644$	0	0
$645$	0	0
$646$	8.30650e6	0.783135
$647$	−1.51545e7	−1.42325	−0.711623	−	0.702561i	$-0.752040\pi$
$647$	−1.51545e7	−1.42325	−0.711623	+	0.702561i	$0.752040\pi$
$648$	0	0
$649$	1.59928e7	1.49044
$650$	5.15977e7	4.79013
$651$	0	0
$652$	5.56846e6	0.512999
$653$	6.64999e6	0.610292	0.305146	−	0.952306i	$-0.401295\pi$
$653$	6.64999e6	0.610292	0.305146	+	0.952306i	$0.401295\pi$
$654$	0	0
$655$	−1.44271e7	−1.31394
$656$	−4.89466e6	−0.444082
$657$	0	0
$658$	0	0
$659$	9.20368e6	0.825559	0.412780	−	0.910831i	$-0.364558\pi$
$659$	9.20368e6	0.825559	0.412780	+	0.910831i	$0.364558\pi$
$660$	0	0
$661$	−1.24361e7	−1.10708	−0.553540	−	0.832823i	$-0.686723\pi$
$661$	−1.24361e7	−1.10708	−0.553540	+	0.832823i	$0.686723\pi$
$662$	2.30067e7	2.04037
$663$	0	0
$664$	3.21177e6	0.282699
$665$	0	0
$666$	0	0
$667$	1.34245e7	1.16838
$668$	−1.50524e7	−1.30516
$669$	0	0
$670$	−2.60661e7	−2.24331
$671$	−1.79164e7	−1.53619
$672$	0	0
$673$	−9.53297e6	−0.811317	−0.405659	−	0.914025i	$-0.632958\pi$
$673$	−9.53297e6	−0.811317	−0.405659	+	0.914025i	$0.632958\pi$
$674$	−3.96721e6	−0.336384
$675$	0	0
$676$	1.68891e7	1.42148
$677$	−6.45494e6	−0.541279	−0.270639	−	0.962681i	$-0.587235\pi$
$677$	−6.45494e6	−0.541279	−0.270639	+	0.962681i	$0.587235\pi$
$678$	0	0
$679$	0	0
$680$	−1.14338e7	−0.948244
$681$	0	0
$682$	−3.12383e7	−2.57174
$683$	4.39884e6	0.360816	0.180408	−	0.983592i	$-0.442258\pi$
$683$	4.39884e6	0.360816	0.180408	+	0.983592i	$0.442258\pi$
$684$	0	0
$685$	−1.71677e7	−1.39793
$686$	0	0
$687$	0	0
$688$	−6.21984e6	−0.500966
$689$	−1.31665e7	−1.05663
$690$	0	0
$691$	2.30559e7	1.83691	0.918454	−	0.395528i	$-0.129438\pi$
$691$	2.30559e7	1.83691	0.918454	+	0.395528i	$0.129438\pi$
$692$	−8.24987e6	−0.654911
$693$	0	0
$694$	−1.13990e7	−0.898397
$695$	−1.90564e6	−0.149651
$696$	0	0
$697$	−1.08203e7	−0.843641
$698$	9.17410e6	0.712730
$699$	0	0
$700$	0	0
$701$	1.69755e7	1.30475	0.652374	−	0.757898i	$-0.273773\pi$
$701$	1.69755e7	1.30475	0.652374	+	0.757898i	$0.273773\pi$
$702$	0	0
$703$	4.74988e6	0.362488
$704$	−1.91211e7	−1.45406
$705$	0	0
$706$	−1.02288e7	−0.772350
$707$	0	0
$708$	0	0
$709$	3.61369e6	0.269982	0.134991	−	0.990847i	$-0.456899\pi$
$709$	3.61369e6	0.269982	0.134991	+	0.990847i	$0.456899\pi$
$710$	5.34158e7	3.97671
$711$	0	0
$712$	−2.17942e6	−0.161117
$713$	−2.84162e7	−2.09335
$714$	0	0
$715$	3.24708e7	2.37536
$716$	−5.62448e6	−0.410015
$717$	0	0
$718$	−2.60441e7	−1.88537
$719$	5.66778e6	0.408875	0.204437	−	0.978880i	$-0.434464\pi$
$719$	5.66778e6	0.408875	0.204437	+	0.978880i	$0.434464\pi$
$720$	0	0
$721$	0	0
$722$	−1.50030e7	−1.07111
$723$	0	0
$724$	9.67355e6	0.685866
$725$	3.07572e7	2.17321
$726$	0	0
$727$	1.72919e7	1.21341	0.606705	−	0.794927i	$-0.292491\pi$
$727$	1.72919e7	1.21341	0.606705	+	0.794927i	$0.292491\pi$
$728$	0	0
$729$	0	0
$730$	−4.22876e7	−2.93701
$731$	−1.37498e7	−0.951705
$732$	0	0
$733$	3.37427e6	0.231964	0.115982	−	0.993251i	$-0.462999\pi$
$733$	3.37427e6	0.231964	0.115982	+	0.993251i	$0.462999\pi$
$734$	−2.59996e7	−1.78126
$735$	0	0
$736$	−2.28061e7	−1.55187
$737$	−1.12445e7	−0.762553
$738$	0	0
$739$	2.16148e7	1.45593	0.727964	−	0.685616i	$-0.240467\pi$
$739$	2.16148e7	1.45593	0.727964	+	0.685616i	$0.240467\pi$
$740$	−2.39733e7	−1.60934
$741$	0	0
$742$	0	0
$743$	−1.51151e7	−1.00447	−0.502237	−	0.864730i	$-0.667490\pi$
$743$	−1.51151e7	−1.00447	−0.502237	+	0.864730i	$0.667490\pi$
$744$	0	0
$745$	−908959.	−0.0600003
$746$	−1.38318e7	−0.909981
$747$	0	0
$748$	−1.80853e7	−1.18188
$749$	0	0
$750$	0	0
$751$	2.45624e7	1.58917	0.794587	−	0.607151i	$-0.207688\pi$
$751$	2.45624e7	1.58917	0.794587	+	0.607151i	$0.207688\pi$
$752$	4.94410e6	0.318818
$753$	0	0
$754$	3.42102e7	2.19143
$755$	3.09206e7	1.97415
$756$	0	0
$757$	2.02916e6	0.128699	0.0643496	−	0.997927i	$-0.479503\pi$
$757$	2.02916e6	0.128699	0.0643496	+	0.997927i	$0.479503\pi$
$758$	1.10862e7	0.700826
$759$	0	0
$760$	−9.06163e6	−0.569079
$761$	−5.94179e6	−0.371926	−0.185963	−	0.982557i	$-0.559540\pi$
$761$	−5.94179e6	−0.371926	−0.185963	+	0.982557i	$0.559540\pi$
$762$	0	0
$763$	0	0
$764$	−5.10971e6	−0.316711
$765$	0	0
$766$	4.47736e7	2.75709
$767$	3.70734e7	2.27548
$768$	0	0
$769$	1.07928e6	0.0658140	0.0329070	−	0.999458i	$-0.489523\pi$
$769$	1.07928e6	0.0658140	0.0329070	+	0.999458i	$0.489523\pi$
$770$	0	0
$771$	0	0
$772$	−1.34775e7	−0.813889
$773$	−1.84656e6	−0.111151	−0.0555757	−	0.998454i	$-0.517699\pi$
$773$	−1.84656e6	−0.111151	−0.0555757	+	0.998454i	$0.517699\pi$
$774$	0	0
$775$	−6.51053e7	−3.89369
$776$	−1.39998e7	−0.834578
$777$	0	0
$778$	4.29487e7	2.54390
$779$	−8.57538e6	−0.506302
$780$	0	0
$781$	2.30427e7	1.35178
$782$	−2.84162e7	−1.66169
$783$	0	0
$784$	0	0
$785$	−2.57262e7	−1.49005
$786$	0	0
$787$	1.32764e7	0.764086	0.382043	−	0.924145i	$-0.375221\pi$
$787$	1.32764e7	0.764086	0.382043	+	0.924145i	$0.375221\pi$
$788$	−1.58489e7	−0.909250
$789$	0	0
$790$	6.96648e7	3.97142
$791$	0	0
$792$	0	0
$793$	−4.15325e7	−2.34534
$794$	−1.91664e6	−0.107892
$795$	0	0
$796$	1.00941e7	0.564659
$797$	−8.63793e6	−0.481686	−0.240843	−	0.970564i	$-0.577424\pi$
$797$	−8.63793e6	−0.481686	−0.240843	+	0.970564i	$0.577424\pi$
$798$	0	0
$799$	1.09296e7	0.605672
$800$	−5.22517e7	−2.88653
$801$	0	0
$802$	2.77920e7	1.52575
$803$	−1.82421e7	−0.998360
$804$	0	0
$805$	0	0
$806$	−7.24144e7	−3.92633
$807$	0	0
$808$	1.11265e7	0.599556
$809$	−2.39614e6	−0.128718	−0.0643592	−	0.997927i	$-0.520500\pi$
$809$	−2.39614e6	−0.128718	−0.0643592	+	0.997927i	$0.520500\pi$
$810$	0	0
$811$	−6.79547e6	−0.362800	−0.181400	−	0.983409i	$-0.558063\pi$
$811$	−6.79547e6	−0.362800	−0.181400	+	0.983409i	$0.558063\pi$
$812$	0	0
$813$	0	0
$814$	−1.78629e7	−0.944910
$815$	1.26150e7	0.665265
$816$	0	0
$817$	−1.08971e7	−0.571156
$818$	−2.21663e7	−1.15827
$819$	0	0
$820$	4.32812e7	2.24784
$821$	−2.64758e6	−0.137085	−0.0685426	−	0.997648i	$-0.521835\pi$
$821$	−2.64758e6	−0.137085	−0.0685426	+	0.997648i	$0.521835\pi$
$822$	0	0
$823$	−2.40725e7	−1.23886	−0.619430	−	0.785052i	$-0.712636\pi$
$823$	−2.40725e7	−1.23886	−0.619430	+	0.785052i	$0.712636\pi$
$824$	4.99992e6	0.256534
$825$	0	0
$826$	0	0
$827$	−1.55007e6	−0.0788110	−0.0394055	−	0.999223i	$-0.512546\pi$
$827$	−1.55007e6	−0.0788110	−0.0394055	+	0.999223i	$0.512546\pi$
$828$	0	0
$829$	−2.64111e7	−1.33475	−0.667375	−	0.744722i	$-0.732582\pi$
$829$	−2.64111e7	−1.33475	−0.667375	+	0.744722i	$0.732582\pi$
$830$	2.66790e7	1.34423
$831$	0	0
$832$	−4.43252e7	−2.21995
$833$	0	0
$834$	0	0
$835$	−3.41004e7	−1.69255
$836$	−1.43331e7	−0.709292
$837$	0	0
$838$	3.11340e7	1.53153
$839$	−1.47573e7	−0.723775	−0.361887	−	0.932222i	$-0.617868\pi$
$839$	−1.47573e7	−0.723775	−0.361887	+	0.932222i	$0.617868\pi$
$840$	0	0
$841$	−118525.	−0.00577856
$842$	8.64588e6	0.420270
$843$	0	0
$844$	−5.32536e7	−2.57331
$845$	3.82613e7	1.84339
$846$	0	0
$847$	0	0
$848$	7.51516e6	0.358879
$849$	0	0
$850$	−6.51053e7	−3.09079
$851$	−1.62492e7	−0.769143
$852$	0	0
$853$	6.12722e6	0.288331	0.144165	−	0.989554i	$-0.453950\pi$
$853$	6.12722e6	0.288331	0.144165	+	0.989554i	$0.453950\pi$
$854$	0	0
$855$	0	0
$856$	−1.07735e7	−0.502540
$857$	3.88119e7	1.80515	0.902574	−	0.430534i	$-0.141675\pi$
$857$	3.88119e7	1.80515	0.902574	+	0.430534i	$0.141675\pi$
$858$	0	0
$859$	−2.09339e6	−0.0967980	−0.0483990	−	0.998828i	$-0.515412\pi$
$859$	−2.09339e6	−0.0967980	−0.0483990	+	0.998828i	$0.515412\pi$
$860$	5.49992e7	2.53577
$861$	0	0
$862$	−2.96930e7	−1.36109
$863$	−4.08889e7	−1.86887	−0.934434	−	0.356137i	$-0.884094\pi$
$863$	−4.08889e7	−1.86887	−0.934434	+	0.356137i	$0.884094\pi$
$864$	0	0
$865$	−1.86896e7	−0.849298
$866$	6.74990e7	3.05846
$867$	0	0
$868$	0	0
$869$	3.00522e7	1.34998
$870$	0	0
$871$	−2.60661e7	−1.16421
$872$	7.72613e6	0.344089
$873$	0	0
$874$	−2.25206e7	−0.997245
$875$	0	0
$876$	0	0
$877$	1.65515e7	0.726670	0.363335	−	0.931659i	$-0.381638\pi$
$877$	1.65515e7	0.726670	0.363335	+	0.931659i	$0.381638\pi$
$878$	−5.50749e7	−2.41111
$879$	0	0
$880$	−1.85337e7	−0.806782
$881$	1.48157e7	0.643104	0.321552	−	0.946892i	$-0.395795\pi$
$881$	1.48157e7	0.643104	0.321552	+	0.946892i	$0.395795\pi$
$882$	0	0
$883$	2.52632e7	1.09040	0.545202	−	0.838305i	$-0.316453\pi$
$883$	2.52632e7	1.09040	0.545202	+	0.838305i	$0.316453\pi$
$884$	−4.19241e7	−1.80440
$885$	0	0
$886$	−4.23434e6	−0.181218
$887$	−3.23328e7	−1.37986	−0.689930	−	0.723877i	$-0.742359\pi$
$887$	−3.23328e7	−1.37986	−0.689930	+	0.723877i	$0.742359\pi$
$888$	0	0
$889$	0	0
$890$	−1.81036e7	−0.766107
$891$	0	0
$892$	3.28059e7	1.38051
$893$	8.66200e6	0.363488
$894$	0	0
$895$	−1.27419e7	−0.531714
$896$	0	0
$897$	0	0
$898$	−4.51908e7	−1.87008
$899$	−4.31660e7	−1.78132
$900$	0	0
$901$	1.66133e7	0.681778
$902$	3.22495e7	1.31980
$903$	0	0
$904$	−3.08986e6	−0.125753
$905$	2.19149e7	0.889442
$906$	0	0
$907$	1.10034e7	0.444129	0.222065	−	0.975032i	$-0.428720\pi$
$907$	1.10034e7	0.444129	0.222065	+	0.975032i	$0.428720\pi$
$908$	6.17534e6	0.248569
$909$	0	0
$910$	0	0
$911$	−4.12930e7	−1.64847	−0.824234	−	0.566249i	$-0.808394\pi$
$911$	−4.12930e7	−1.64847	−0.824234	+	0.566249i	$0.808394\pi$
$912$	0	0
$913$	1.15088e7	0.456935
$914$	5.71617e7	2.26329
$915$	0	0
$916$	−3.40677e7	−1.34154
$917$	0	0
$918$	0	0
$919$	3.78206e6	0.147720	0.0738601	−	0.997269i	$-0.476468\pi$
$919$	3.78206e6	0.147720	0.0738601	+	0.997269i	$0.476468\pi$
$920$	3.09995e7	1.20750
$921$	0	0
$922$	3.62567e7	1.40463
$923$	5.34158e7	2.06379
$924$	0	0
$925$	−3.72289e7	−1.43063
$926$	4.90684e7	1.88051
$927$	0	0
$928$	−3.46438e7	−1.32055
$929$	1.47979e7	0.562550	0.281275	−	0.959627i	$-0.409243\pi$
$929$	1.47979e7	0.562550	0.281275	+	0.959627i	$0.409243\pi$
$930$	0	0
$931$	0	0
$932$	4.55620e7	1.71816
$933$	0	0
$934$	3.66781e7	1.37575
$935$	−4.09713e7	−1.53268
$936$	0	0
$937$	3.07412e7	1.14386	0.571929	−	0.820303i	$-0.306195\pi$
$937$	3.07412e7	1.14386	0.571929	+	0.820303i	$0.306195\pi$
$938$	0	0
$939$	0	0
$940$	−4.37184e7	−1.61378
$941$	2.56871e7	0.945673	0.472837	−	0.881150i	$-0.343230\pi$
$941$	2.56871e7	0.945673	0.472837	+	0.881150i	$0.343230\pi$
$942$	0	0
$943$	2.93361e7	1.07429
$944$	−2.11608e7	−0.772861
$945$	0	0
$946$	4.09807e7	1.48885
$947$	−6.25960e6	−0.226815	−0.113407	−	0.993549i	$-0.536177\pi$
$947$	−6.25960e6	−0.226815	−0.113407	+	0.993549i	$0.536177\pi$
$948$	0	0
$949$	−4.22876e7	−1.52422
$950$	−5.15977e7	−1.85490
$951$	0	0
$952$	0	0
$953$	−2.65561e7	−0.947180	−0.473590	−	0.880746i	$-0.657042\pi$
$953$	−2.65561e7	−0.947180	−0.473590	+	0.880746i	$0.657042\pi$
$954$	0	0
$955$	−1.15758e7	−0.410715
$956$	−4.91136e7	−1.73803
$957$	0	0
$958$	6.19795e7	2.18190
$959$	0	0
$960$	0	0
$961$	6.27423e7	2.19155
$962$	−4.14085e7	−1.44262
$963$	0	0
$964$	−6.35472e7	−2.20244
$965$	−3.05324e7	−1.05546
$966$	0	0
$967$	−3.33142e7	−1.14568	−0.572841	−	0.819667i	$-0.694159\pi$
$967$	−3.33142e7	−1.14568	−0.572841	+	0.819667i	$0.694159\pi$
$968$	−2.14740e6	−0.0736589
$969$	0	0
$970$	−1.16291e8	−3.96841
$971$	4.04061e6	0.137530	0.0687652	−	0.997633i	$-0.478094\pi$
$971$	4.04061e6	0.137530	0.0687652	+	0.997633i	$0.478094\pi$
$972$	0	0
$973$	0	0
$974$	6.13398e7	2.07178
$975$	0	0
$976$	2.37059e7	0.796586
$977$	−1.10096e7	−0.369008	−0.184504	−	0.982832i	$-0.559068\pi$
$977$	−1.10096e7	−0.369008	−0.184504	+	0.982832i	$0.559068\pi$
$978$	0	0
$979$	−7.80957e6	−0.260418
$980$	0	0
$981$	0	0
$982$	−9.47348e6	−0.313495
$983$	2.74119e7	0.904804	0.452402	−	0.891814i	$-0.350567\pi$
$983$	2.74119e7	0.904804	0.452402	+	0.891814i	$0.350567\pi$
$984$	0	0
$985$	−3.59047e7	−1.17913
$986$	−4.31660e7	−1.41400
$987$	0	0
$988$	−3.32260e7	−1.08289
$989$	3.72785e7	1.21190
$990$	0	0
$991$	2.29977e7	0.743875	0.371938	−	0.928258i	$-0.378694\pi$
$991$	2.29977e7	0.743875	0.371938	+	0.928258i	$0.378694\pi$
$992$	7.33322e7	2.36600
$993$	0	0
$994$	0	0
$995$	2.28677e7	0.732259
$996$	0	0
$997$	5.23347e7	1.66744	0.833722	−	0.552184i	$-0.186205\pi$
$997$	5.23347e7	1.66744	0.833722	+	0.552184i	$0.186205\pi$
$998$	2.66674e7	0.847527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.y.1.4	yes	4
3.2	odd	2	inner	441.6.a.y.1.1	✓	4
7.6	odd	2	inner	441.6.a.y.1.3	yes	4
21.20	even	2	inner	441.6.a.y.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.6.a.y.1.1	✓	4	3.2	odd	2	inner
441.6.a.y.1.2	yes	4	21.20	even	2	inner
441.6.a.y.1.3	yes	4	7.6	odd	2	inner
441.6.a.y.1.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+8.71780 q^{2} +44.0000 q^{4} +99.6795 q^{5} +104.614 q^{8} +O(q^{10})\)	\(q+8.71780 q^{2} +44.0000 q^{4} +99.6795 q^{5} +104.614 q^{8} +868.986 q^{10} +374.865 q^{11} +868.986 q^{13} -496.000 q^{16} -1096.47 q^{17} -868.986 q^{19} +4385.90 q^{20} +3268.00 q^{22} +2972.77 q^{23} +6811.00 q^{25} +7575.64 q^{26} +4515.82 q^{29} -9558.84 q^{31} -7671.66 q^{32} -9558.84 q^{34} -5466.00 q^{37} -7575.64 q^{38} +10427.8 q^{40} +9868.27 q^{41} +12540.0 q^{43} +16494.1 q^{44} +25916.0 q^{46} -9967.95 q^{47} +59376.9 q^{50} +38235.4 q^{52} -15151.5 q^{53} +37366.4 q^{55} +39368.0 q^{58} +42662.8 q^{59} -47794.2 q^{61} -83332.1 q^{62} -51008.0 q^{64} +86620.0 q^{65} -29996.0 q^{67} -48244.9 q^{68} +61469.2 q^{71} -48663.2 q^{73} -47651.5 q^{74} -38235.4 q^{76} +80168.0 q^{79} -49441.0 q^{80} +86029.6 q^{82} +30701.3 q^{83} -109296. q^{85} +109321. q^{86} +39216.0 q^{88} -20833.0 q^{89} +130802. q^{92} -86898.6 q^{94} -86620.0 q^{95} -133824. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 176 q^{4}+O(q^{10}) \) 4 * q + 176 * q^4	\( 4 q + 176 q^{4} - 1984 q^{16} + 13072 q^{22} + 27244 q^{25} - 21864 q^{37} + 50160 q^{43} + 103664 q^{46} + 157472 q^{58} - 204032 q^{64} - 119984 q^{67} + 320672 q^{79} - 437184 q^{85} + 156864 q^{88}+O(q^{100}) \) 4 * q + 176 * q^4 - 1984 * q^16 + 13072 * q^22 + 27244 * q^25 - 21864 * q^37 + 50160 * q^43 + 103664 * q^46 + 157472 * q^58 - 204032 * q^64 - 119984 * q^67 + 320672 * q^79 - 437184 * q^85 + 156864 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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