LMFDB - Embedded newform 234.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,6,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, 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("234.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	234.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:	$37.5298138362$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{849}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 212 $ x^2 - x - 212
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-14.0688$ of defining polynomial
Character	$\chi$	$=$	234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +16.0000 q^{4} +7.20641 q^{5} -53.6192 q^{7} -64.0000 q^{8} -28.8256 q^{10} -239.651 q^{11} -169.000 q^{13} +214.477 q^{14} +256.000 q^{16} +1973.88 q^{17} -373.872 q^{19} +115.303 q^{20} +958.605 q^{22} -51.4306 q^{23} -3073.07 q^{25} +676.000 q^{26} -857.908 q^{28} -4796.16 q^{29} +6906.01 q^{31} -1024.00 q^{32} -7895.50 q^{34} -386.402 q^{35} +11481.2 q^{37} +1495.49 q^{38} -461.210 q^{40} -12547.8 q^{41} +1156.07 q^{43} -3834.42 q^{44} +205.723 q^{46} +18644.9 q^{47} -13932.0 q^{49} +12292.3 q^{50} -2704.00 q^{52} -9318.90 q^{53} -1727.02 q^{55} +3431.63 q^{56} +19184.7 q^{58} +5066.63 q^{59} +54271.7 q^{61} -27624.0 q^{62} +4096.00 q^{64} -1217.88 q^{65} +40241.0 q^{67} +31582.0 q^{68} +1545.61 q^{70} +65236.5 q^{71} +68506.2 q^{73} -45924.7 q^{74} -5981.95 q^{76} +12849.9 q^{77} +10627.3 q^{79} +1844.84 q^{80} +50191.1 q^{82} +2357.98 q^{83} +14224.5 q^{85} -4624.27 q^{86} +15337.7 q^{88} +93620.0 q^{89} +9061.65 q^{91} -822.890 q^{92} -74579.5 q^{94} -2694.27 q^{95} -31195.8 q^{97} +55727.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 73 q^{5} + 155 q^{7} - 128 q^{8} + 292 q^{10} + 220 q^{11} - 338 q^{13} - 620 q^{14} + 512 q^{16} + 189 q^{17} - 2496 q^{19} - 1168 q^{20} - 880 q^{22} + 3044 q^{23} + 235 q^{25}+ \cdots - 51132 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 73 * q^5 + 155 * q^7 - 128 * q^8 + 292 * q^10 + 220 * q^11 - 338 * q^13 - 620 * q^14 + 512 * q^16 + 189 * q^17 - 2496 * q^19 - 1168 * q^20 - 880 * q^22 + 3044 * q^23 + 235 * q^25 + 1352 * q^26 + 2480 * q^28 - 1900 * q^29 + 2798 * q^31 - 2048 * q^32 - 756 * q^34 - 17119 * q^35 + 17805 * q^37 + 9984 * q^38 + 4672 * q^40 - 11634 * q^41 - 4069 * q^43 + 3520 * q^44 - 12176 * q^46 + 25489 * q^47 + 12783 * q^49 - 940 * q^50 - 5408 * q^52 + 4614 * q^53 - 38594 * q^55 - 9920 * q^56 + 7600 * q^58 + 23420 * q^59 + 96830 * q^61 - 11192 * q^62 + 8192 * q^64 + 12337 * q^65 + 72440 * q^67 + 3024 * q^68 + 68476 * q^70 + 36679 * q^71 + 47152 * q^73 - 71220 * q^74 - 39936 * q^76 + 108742 * q^77 + 52024 * q^79 - 18688 * q^80 + 46536 * q^82 + 37758 * q^83 + 157383 * q^85 + 16276 * q^86 - 14080 * q^88 + 105072 * q^89 - 26195 * q^91 + 48704 * q^92 - 101956 * q^94 + 167514 * q^95 + 104392 * q^97 - 51132 * q^98

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	7.20641	0.128912	0.0644561	−	0.997921i	$-0.479469\pi$
$5$	7.20641	0.128912	0.0644561	+	0.997921i	$0.479469\pi$
$6$	0	0
$7$	−53.6192	−0.413595	−0.206798	−	0.978384i	$-0.566304\pi$
$7$	−53.6192	−0.413595	−0.206798	+	0.978384i	$0.566304\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−28.8256	−0.0911546
$11$	−239.651	−0.597170	−0.298585	−	0.954383i	$-0.596515\pi$
$11$	−239.651	−0.597170	−0.298585	+	0.954383i	$0.596515\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	214.477	0.292456
$15$	0	0
$16$	256.000	0.250000
$17$	1973.88	1.65652	0.828261	−	0.560342i	$-0.189330\pi$
$17$	1973.88	1.65652	0.828261	+	0.560342i	$0.189330\pi$
$18$	0	0
$19$	−373.872	−0.237596	−0.118798	−	0.992918i	$-0.537904\pi$
$19$	−373.872	−0.237596	−0.118798	+	0.992918i	$0.537904\pi$
$20$	115.303	0.0644561
$21$	0	0
$22$	958.605	0.422263
$23$	−51.4306	−0.0202723	−0.0101361	−	0.999949i	$-0.503226\pi$
$23$	−51.4306	−0.0202723	−0.0101361	+	0.999949i	$0.503226\pi$
$24$	0	0
$25$	−3073.07	−0.983382
$26$	676.000	0.196116
$27$	0	0
$28$	−857.908	−0.206798
$29$	−4796.16	−1.05901	−0.529504	−	0.848308i	$-0.677622\pi$
$29$	−4796.16	−1.05901	−0.529504	+	0.848308i	$0.677622\pi$
$30$	0	0
$31$	6906.01	1.29069	0.645346	−	0.763890i	$-0.276713\pi$
$31$	6906.01	1.29069	0.645346	+	0.763890i	$0.276713\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−7895.50	−1.17134
$35$	−386.402	−0.0533174
$36$	0	0
$37$	11481.2	1.37874	0.689370	−	0.724410i	$-0.257888\pi$
$37$	11481.2	1.37874	0.689370	+	0.724410i	$0.257888\pi$
$38$	1495.49	0.168006
$39$	0	0
$40$	−461.210	−0.0455773
$41$	−12547.8	−1.16576	−0.582878	−	0.812560i	$-0.698073\pi$
$41$	−12547.8	−1.16576	−0.582878	+	0.812560i	$0.698073\pi$
$42$	0	0
$43$	1156.07	0.0953481	0.0476741	−	0.998863i	$-0.484819\pi$
$43$	1156.07	0.0953481	0.0476741	+	0.998863i	$0.484819\pi$
$44$	−3834.42	−0.298585
$45$	0	0
$46$	205.723	0.0143347
$47$	18644.9	1.23116	0.615580	−	0.788074i	$-0.288922\pi$
$47$	18644.9	1.23116	0.615580	+	0.788074i	$0.288922\pi$
$48$	0	0
$49$	−13932.0	−0.828939
$50$	12292.3	0.695356
$51$	0	0
$52$	−2704.00	−0.138675
$53$	−9318.90	−0.455696	−0.227848	−	0.973697i	$-0.573169\pi$
$53$	−9318.90	−0.455696	−0.227848	+	0.973697i	$0.573169\pi$
$54$	0	0
$55$	−1727.02	−0.0769825
$56$	3431.63	0.146228
$57$	0	0
$58$	19184.7	0.748831
$59$	5066.63	0.189491	0.0947456	−	0.995502i	$-0.469796\pi$
$59$	5066.63	0.189491	0.0947456	+	0.995502i	$0.469796\pi$
$60$	0	0
$61$	54271.7	1.86745	0.933724	−	0.357994i	$-0.116539\pi$
$61$	54271.7	1.86745	0.933724	+	0.357994i	$0.116539\pi$
$62$	−27624.0	−0.912657
$63$	0	0
$64$	4096.00	0.125000
$65$	−1217.88	−0.0357538
$66$	0	0
$67$	40241.0	1.09517	0.547585	−	0.836750i	$-0.315547\pi$
$67$	40241.0	1.09517	0.547585	+	0.836750i	$0.315547\pi$
$68$	31582.0	0.828261
$69$	0	0
$70$	1545.61	0.0377011
$71$	65236.5	1.53584	0.767918	−	0.640549i	$-0.221293\pi$
$71$	65236.5	1.53584	0.767918	+	0.640549i	$0.221293\pi$
$72$	0	0
$73$	68506.2	1.50461	0.752303	−	0.658818i	$-0.228943\pi$
$73$	68506.2	1.50461	0.752303	+	0.658818i	$0.228943\pi$
$74$	−45924.7	−0.974916
$75$	0	0
$76$	−5981.95	−0.118798
$77$	12849.9	0.246987
$78$	0	0
$79$	10627.3	0.191583	0.0957915	−	0.995401i	$-0.469462\pi$
$79$	10627.3	0.191583	0.0957915	+	0.995401i	$0.469462\pi$
$80$	1844.84	0.0322280
$81$	0	0
$82$	50191.1	0.824314
$83$	2357.98	0.0375703	0.0187851	−	0.999824i	$-0.494020\pi$
$83$	2357.98	0.0375703	0.0187851	+	0.999824i	$0.494020\pi$
$84$	0	0
$85$	14224.5	0.213546
$86$	−4624.27	−0.0674213
$87$	0	0
$88$	15337.7	0.211131
$89$	93620.0	1.25283	0.626417	−	0.779488i	$-0.284521\pi$
$89$	93620.0	1.25283	0.626417	+	0.779488i	$0.284521\pi$
$90$	0	0
$91$	9061.65	0.114711
$92$	−822.890	−0.0101361
$93$	0	0
$94$	−74579.5	−0.870562
$95$	−2694.27	−0.0306290
$96$	0	0
$97$	−31195.8	−0.336641	−0.168321	−	0.985732i	$-0.553834\pi$
$97$	−31195.8	−0.336641	−0.168321	+	0.985732i	$0.553834\pi$
$98$	55727.9	0.586148
$99$	0	0
$100$	−49169.1	−0.491691
$101$	86948.6	0.848124	0.424062	−	0.905633i	$-0.360604\pi$
$101$	86948.6	0.848124	0.424062	+	0.905633i	$0.360604\pi$
$102$	0	0
$103$	−43111.1	−0.400402	−0.200201	−	0.979755i	$-0.564159\pi$
$103$	−43111.1	−0.400402	−0.200201	+	0.979755i	$0.564159\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	37275.6	0.322226
$107$	175966.	1.48583	0.742913	−	0.669388i	$-0.233443\pi$
$107$	175966.	1.48583	0.742913	+	0.669388i	$0.233443\pi$
$108$	0	0
$109$	119578.	0.964020	0.482010	−	0.876166i	$-0.339907\pi$
$109$	119578.	0.964020	0.482010	+	0.876166i	$0.339907\pi$
$110$	6908.10	0.0544348
$111$	0	0
$112$	−13726.5	−0.103399
$113$	−47558.0	−0.350370	−0.175185	−	0.984536i	$-0.556052\pi$
$113$	−47558.0	−0.350370	−0.175185	+	0.984536i	$0.556052\pi$
$114$	0	0
$115$	−370.630	−0.00261334
$116$	−76738.6	−0.529504
$117$	0	0
$118$	−20266.5	−0.133990
$119$	−105838.	−0.685130
$120$	0	0
$121$	−103618.	−0.643388
$122$	−217087.	−1.32049
$123$	0	0
$124$	110496.	0.645346
$125$	−44665.8	−0.255682
$126$	0	0
$127$	−108774.	−0.598433	−0.299217	−	0.954185i	$-0.596725\pi$
$127$	−108774.	−0.598433	−0.299217	+	0.954185i	$0.596725\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	4871.53	0.0252817
$131$	340486.	1.73349	0.866744	−	0.498753i	$-0.166209\pi$
$131$	340486.	1.73349	0.866744	+	0.498753i	$0.166209\pi$
$132$	0	0
$133$	20046.7	0.0982685
$134$	−160964.	−0.774402
$135$	0	0
$136$	−126328.	−0.585669
$137$	−48143.4	−0.219147	−0.109574	−	0.993979i	$-0.534949\pi$
$137$	−48143.4	−0.219147	−0.109574	+	0.993979i	$0.534949\pi$
$138$	0	0
$139$	323803.	1.42149	0.710745	−	0.703450i	$-0.248358\pi$
$139$	323803.	1.42149	0.710745	+	0.703450i	$0.248358\pi$
$140$	−6182.43	−0.0266587
$141$	0	0
$142$	−260946.	−1.08600
$143$	40501.1	0.165625
$144$	0	0
$145$	−34563.1	−0.136519
$146$	−274025.	−1.06392
$147$	0	0
$148$	183699.	0.689370
$149$	−372945.	−1.37619	−0.688096	−	0.725620i	$-0.741553\pi$
$149$	−372945.	−1.37619	−0.688096	+	0.725620i	$0.741553\pi$
$150$	0	0
$151$	−159735.	−0.570110	−0.285055	−	0.958511i	$-0.592012\pi$
$151$	−159735.	−0.570110	−0.285055	+	0.958511i	$0.592012\pi$
$152$	23927.8	0.0840028
$153$	0	0
$154$	−51399.7	−0.174646
$155$	49767.5	0.166386
$156$	0	0
$157$	−484134.	−1.56753	−0.783765	−	0.621057i	$-0.786704\pi$
$157$	−484134.	−1.56753	−0.783765	+	0.621057i	$0.786704\pi$
$158$	−42509.4	−0.135470
$159$	0	0
$160$	−7379.36	−0.0227887
$161$	2757.67	0.00838451
$162$	0	0
$163$	−140172.	−0.413232	−0.206616	−	0.978422i	$-0.566245\pi$
$163$	−140172.	−0.413232	−0.206616	+	0.978422i	$0.566245\pi$
$164$	−200765.	−0.582878
$165$	0	0
$166$	−9431.91	−0.0265662
$167$	−218046.	−0.605002	−0.302501	−	0.953149i	$-0.597822\pi$
$167$	−218046.	−0.605002	−0.302501	+	0.953149i	$0.597822\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	−56898.2	−0.151000
$171$	0	0
$172$	18497.1	0.0476741
$173$	384852.	0.977638	0.488819	−	0.872385i	$-0.337428\pi$
$173$	384852.	0.977638	0.488819	+	0.872385i	$0.337428\pi$
$174$	0	0
$175$	164775.	0.406722
$176$	−61350.7	−0.149293
$177$	0	0
$178$	−374480.	−0.885888
$179$	−498273.	−1.16234	−0.581172	−	0.813781i	$-0.697405\pi$
$179$	−498273.	−1.16234	−0.581172	+	0.813781i	$0.697405\pi$
$180$	0	0
$181$	152201.	0.345320	0.172660	−	0.984981i	$-0.444764\pi$
$181$	152201.	0.345320	0.172660	+	0.984981i	$0.444764\pi$
$182$	−36246.6	−0.0811127
$183$	0	0
$184$	3291.56	0.00716733
$185$	82738.0	0.177736
$186$	0	0
$187$	−473042.	−0.989226
$188$	298318.	0.615580
$189$	0	0
$190$	10777.1	0.0216580
$191$	−600284.	−1.19062	−0.595310	−	0.803496i	$-0.702971\pi$
$191$	−600284.	−1.19062	−0.595310	+	0.803496i	$0.702971\pi$
$192$	0	0
$193$	350804.	0.677909	0.338955	−	0.940803i	$-0.389927\pi$
$193$	350804.	0.677909	0.338955	+	0.940803i	$0.389927\pi$
$194$	124783.	0.238041
$195$	0	0
$196$	−222912.	−0.414470
$197$	321145.	0.589571	0.294785	−	0.955563i	$-0.404752\pi$
$197$	321145.	0.589571	0.294785	+	0.955563i	$0.404752\pi$
$198$	0	0
$199$	561416.	1.00497	0.502484	−	0.864586i	$-0.332420\pi$
$199$	561416.	1.00497	0.502484	+	0.864586i	$0.332420\pi$
$200$	196676.	0.347678
$201$	0	0
$202$	−347794.	−0.599714
$203$	257167.	0.438000
$204$	0	0
$205$	−90424.5	−0.150280
$206$	172444.	0.283127
$207$	0	0
$208$	−43264.0	−0.0693375
$209$	89598.9	0.141885
$210$	0	0
$211$	−751002.	−1.16127	−0.580637	−	0.814162i	$-0.697196\pi$
$211$	−751002.	−1.16127	−0.580637	+	0.814162i	$0.697196\pi$
$212$	−149102.	−0.227848
$213$	0	0
$214$	−703862.	−1.05064
$215$	8331.09	0.0122915
$216$	0	0
$217$	−370295.	−0.533824
$218$	−478313.	−0.681665
$219$	0	0
$220$	−27632.4	−0.0384912
$221$	−333585.	−0.459437
$222$	0	0
$223$	−598906.	−0.806485	−0.403243	−	0.915093i	$-0.632117\pi$
$223$	−598906.	−0.806485	−0.403243	+	0.915093i	$0.632117\pi$
$224$	54906.1	0.0731140
$225$	0	0
$226$	190232.	0.247749
$227$	1.32303e6	1.70414	0.852071	−	0.523426i	$-0.175346\pi$
$227$	1.32303e6	1.70414	0.852071	+	0.523426i	$0.175346\pi$
$228$	0	0
$229$	115752.	0.145861	0.0729307	−	0.997337i	$-0.476765\pi$
$229$	115752.	0.145861	0.0729307	+	0.997337i	$0.476765\pi$
$230$	1482.52	0.00184791
$231$	0	0
$232$	306954.	0.374416
$233$	−1.21925e6	−1.47131	−0.735655	−	0.677357i	$-0.763125\pi$
$233$	−1.21925e6	−1.47131	−0.735655	+	0.677357i	$0.763125\pi$
$234$	0	0
$235$	134362.	0.158711
$236$	81066.0	0.0947456
$237$	0	0
$238$	423351.	0.484460
$239$	1.66823e6	1.88913	0.944565	−	0.328323i	$-0.106484\pi$
$239$	1.66823e6	1.88913	0.944565	+	0.328323i	$0.106484\pi$
$240$	0	0
$241$	847710.	0.940167	0.470083	−	0.882622i	$-0.344224\pi$
$241$	847710.	0.940167	0.470083	+	0.882622i	$0.344224\pi$
$242$	414473.	0.454944
$243$	0	0
$244$	868347.	0.933724
$245$	−100400.	−0.106860
$246$	0	0
$247$	63184.3	0.0658972
$248$	−441984.	−0.456329
$249$	0	0
$250$	178663.	0.180794
$251$	1.09148e6	1.09353	0.546765	−	0.837286i	$-0.315859\pi$
$251$	1.09148e6	1.09353	0.546765	+	0.837286i	$0.315859\pi$
$252$	0	0
$253$	12325.4	0.0121060
$254$	435096.	0.423156
$255$	0	0
$256$	65536.0	0.0625000
$257$	10510.6	0.00992644	0.00496322	−	0.999988i	$-0.498420\pi$
$257$	10510.6	0.00992644	0.00496322	+	0.999988i	$0.498420\pi$
$258$	0	0
$259$	−615612.	−0.570240
$260$	−19486.1	−0.0178769
$261$	0	0
$262$	−1.36194e6	−1.22576
$263$	−1.66350e6	−1.48297	−0.741487	−	0.670967i	$-0.765879\pi$
$263$	−1.66350e6	−1.48297	−0.741487	+	0.670967i	$0.765879\pi$
$264$	0	0
$265$	−67155.8	−0.0587447
$266$	−80186.9	−0.0694863
$267$	0	0
$268$	643856.	0.547585
$269$	123097.	0.103721	0.0518606	−	0.998654i	$-0.483485\pi$
$269$	123097.	0.103721	0.0518606	+	0.998654i	$0.483485\pi$
$270$	0	0
$271$	2.14956e6	1.77798	0.888991	−	0.457925i	$-0.151407\pi$
$271$	2.14956e6	1.77798	0.888991	+	0.457925i	$0.151407\pi$
$272$	505312.	0.414131
$273$	0	0
$274$	192574.	0.154960
$275$	736465.	0.587246
$276$	0	0
$277$	818582.	0.641007	0.320503	−	0.947247i	$-0.396148\pi$
$277$	818582.	0.641007	0.320503	+	0.947247i	$0.396148\pi$
$278$	−1.29521e6	−1.00514
$279$	0	0
$280$	24729.7	0.0188506
$281$	−1.38003e6	−1.04261	−0.521305	−	0.853370i	$-0.674555\pi$
$281$	−1.38003e6	−1.04261	−0.521305	+	0.853370i	$0.674555\pi$
$282$	0	0
$283$	395823.	0.293788	0.146894	−	0.989152i	$-0.453072\pi$
$283$	395823.	0.293788	0.146894	+	0.989152i	$0.453072\pi$
$284$	1.04378e6	0.767918
$285$	0	0
$286$	−162004.	−0.117115
$287$	672803.	0.482151
$288$	0	0
$289$	2.47633e6	1.74407
$290$	138252.	0.0965334
$291$	0	0
$292$	1.09610e6	0.752303
$293$	−382789.	−0.260490	−0.130245	−	0.991482i	$-0.541576\pi$
$293$	−382789.	−0.260490	−0.130245	+	0.991482i	$0.541576\pi$
$294$	0	0
$295$	36512.2	0.0244277
$296$	−734795.	−0.487458
$297$	0	0
$298$	1.49178e6	0.973115
$299$	8691.78	0.00562252
$300$	0	0
$301$	−61987.4	−0.0394355
$302$	638941.	0.403129
$303$	0	0
$304$	−95711.2	−0.0593990
$305$	391104.	0.240737
$306$	0	0
$307$	−2.95976e6	−1.79230	−0.896150	−	0.443751i	$-0.853648\pi$
$307$	−2.95976e6	−1.79230	−0.896150	+	0.443751i	$0.853648\pi$
$308$	205599.	0.123493
$309$	0	0
$310$	−199070.	−0.117653
$311$	536900.	0.314770	0.157385	−	0.987537i	$-0.449694\pi$
$311$	536900.	0.314770	0.157385	+	0.987537i	$0.449694\pi$
$312$	0	0
$313$	−3.33301e6	−1.92299	−0.961493	−	0.274829i	$-0.911379\pi$
$313$	−3.33301e6	−1.92299	−0.961493	+	0.274829i	$0.911379\pi$
$314$	1.93653e6	1.10841
$315$	0	0
$316$	170038.	0.0957915
$317$	−1.50100e6	−0.838943	−0.419472	−	0.907768i	$-0.637785\pi$
$317$	−1.50100e6	−0.838943	−0.419472	+	0.907768i	$0.637785\pi$
$318$	0	0
$319$	1.14941e6	0.632407
$320$	29517.4	0.0161140
$321$	0	0
$322$	−11030.7	−0.00592875
$323$	−737977.	−0.393583
$324$	0	0
$325$	519348.	0.272741
$326$	560690.	0.292199
$327$	0	0
$328$	803058.	0.412157
$329$	−999723.	−0.509202
$330$	0	0
$331$	2.17404e6	1.09068	0.545341	−	0.838214i	$-0.316400\pi$
$331$	2.17404e6	1.09068	0.545341	+	0.838214i	$0.316400\pi$
$332$	37727.7	0.0187851
$333$	0	0
$334$	872184.	0.427801
$335$	289993.	0.141181
$336$	0	0
$337$	821494.	0.394030	0.197015	−	0.980400i	$-0.436875\pi$
$337$	821494.	0.394030	0.197015	+	0.980400i	$0.436875\pi$
$338$	−114244.	−0.0543928
$339$	0	0
$340$	227593.	0.106773
$341$	−1.65503e6	−0.770763
$342$	0	0
$343$	1.64820e6	0.756440
$344$	−73988.3	−0.0337107
$345$	0	0
$346$	−1.53941e6	−0.691295
$347$	−1.75897e6	−0.784216	−0.392108	−	0.919919i	$-0.628254\pi$
$347$	−1.75897e6	−0.784216	−0.392108	+	0.919919i	$0.628254\pi$
$348$	0	0
$349$	−3.15359e6	−1.38593	−0.692965	−	0.720971i	$-0.743696\pi$
$349$	−3.15359e6	−1.38593	−0.692965	+	0.720971i	$0.743696\pi$
$350$	−659102.	−0.287596
$351$	0	0
$352$	245403.	0.105566
$353$	1.51456e6	0.646918	0.323459	−	0.946242i	$-0.395154\pi$
$353$	1.51456e6	0.646918	0.323459	+	0.946242i	$0.395154\pi$
$354$	0	0
$355$	470121.	0.197988
$356$	1.49792e6	0.626417
$357$	0	0
$358$	1.99309e6	0.821901
$359$	−1.68935e6	−0.691803	−0.345902	−	0.938271i	$-0.612427\pi$
$359$	−1.68935e6	−0.691803	−0.345902	+	0.938271i	$0.612427\pi$
$360$	0	0
$361$	−2.33632e6	−0.943548
$362$	−608805.	−0.244178
$363$	0	0
$364$	144986.	0.0573553
$365$	493683.	0.193962
$366$	0	0
$367$	−1.87184e6	−0.725443	−0.362722	−	0.931898i	$-0.618152\pi$
$367$	−1.87184e6	−0.725443	−0.362722	+	0.931898i	$0.618152\pi$
$368$	−13166.2	−0.00506807
$369$	0	0
$370$	−330952.	−0.125678
$371$	499672.	0.188474
$372$	0	0
$373$	628003.	0.233717	0.116858	−	0.993149i	$-0.462718\pi$
$373$	628003.	0.233717	0.116858	+	0.993149i	$0.462718\pi$
$374$	1.89217e6	0.699488
$375$	0	0
$376$	−1.19327e6	−0.435281
$377$	810552.	0.293716
$378$	0	0
$379$	478225.	0.171015	0.0855076	−	0.996338i	$-0.472749\pi$
$379$	478225.	0.171015	0.0855076	+	0.996338i	$0.472749\pi$
$380$	−43108.4	−0.0153145
$381$	0	0
$382$	2.40114e6	0.841895
$383$	4.58548e6	1.59731	0.798653	−	0.601791i	$-0.205546\pi$
$383$	4.58548e6	1.59731	0.798653	+	0.601791i	$0.205546\pi$
$384$	0	0
$385$	92601.7	0.0318396
$386$	−1.40322e6	−0.479354
$387$	0	0
$388$	−499133.	−0.168321
$389$	−2.62136e6	−0.878318	−0.439159	−	0.898409i	$-0.644724\pi$
$389$	−2.62136e6	−0.878318	−0.439159	+	0.898409i	$0.644724\pi$
$390$	0	0
$391$	−101518.	−0.0335815
$392$	891647.	0.293074
$393$	0	0
$394$	−1.28458e6	−0.416890
$395$	76585.0	0.0246974
$396$	0	0
$397$	961154.	0.306067	0.153034	−	0.988221i	$-0.451096\pi$
$397$	961154.	0.306067	0.153034	+	0.988221i	$0.451096\pi$
$398$	−2.24567e6	−0.710620
$399$	0	0
$400$	−786705.	−0.245845
$401$	4.00559e6	1.24396	0.621978	−	0.783034i	$-0.286329\pi$
$401$	4.00559e6	1.24396	0.621978	+	0.783034i	$0.286329\pi$
$402$	0	0
$403$	−1.16712e6	−0.357974
$404$	1.39118e6	0.424062
$405$	0	0
$406$	−1.02867e6	−0.309713
$407$	−2.75148e6	−0.823342
$408$	0	0
$409$	−3.12530e6	−0.923813	−0.461906	−	0.886929i	$-0.652834\pi$
$409$	−3.12530e6	−0.923813	−0.461906	+	0.886929i	$0.652834\pi$
$410$	361698.	0.106264
$411$	0	0
$412$	−689777.	−0.200201
$413$	−271669.	−0.0783726
$414$	0	0
$415$	16992.6	0.00484327
$416$	173056.	0.0490290
$417$	0	0
$418$	−358395.	−0.100328
$419$	3.34877e6	0.931859	0.465929	−	0.884822i	$-0.345720\pi$
$419$	3.34877e6	0.931859	0.465929	+	0.884822i	$0.345720\pi$
$420$	0	0
$421$	−3.01332e6	−0.828591	−0.414295	−	0.910143i	$-0.635972\pi$
$421$	−3.01332e6	−0.828591	−0.414295	+	0.910143i	$0.635972\pi$
$422$	3.00401e6	0.821145
$423$	0	0
$424$	596410.	0.161113
$425$	−6.06585e6	−1.62899
$426$	0	0
$427$	−2.91000e6	−0.772367
$428$	2.81545e6	0.742913
$429$	0	0
$430$	−33324.4	−0.00869142
$431$	−3.35224e6	−0.869245	−0.434623	−	0.900613i	$-0.643118\pi$
$431$	−3.35224e6	−0.869245	−0.434623	+	0.900613i	$0.643118\pi$
$432$	0	0
$433$	1.62188e6	0.415719	0.207859	−	0.978159i	$-0.433350\pi$
$433$	1.62188e6	0.415719	0.207859	+	0.978159i	$0.433350\pi$
$434$	1.48118e6	0.377471
$435$	0	0
$436$	1.91325e6	0.482010
$437$	19228.5	0.00481661
$438$	0	0
$439$	7.48948e6	1.85477	0.927386	−	0.374106i	$-0.122050\pi$
$439$	7.48948e6	1.85477	0.927386	+	0.374106i	$0.122050\pi$
$440$	110530.	0.0272174
$441$	0	0
$442$	1.33434e6	0.324871
$443$	−1.90156e6	−0.460364	−0.230182	−	0.973148i	$-0.573932\pi$
$443$	−1.90156e6	−0.460364	−0.230182	+	0.973148i	$0.573932\pi$
$444$	0	0
$445$	674664.	0.161506
$446$	2.39562e6	0.570271
$447$	0	0
$448$	−219624.	−0.0516994
$449$	−1.62580e6	−0.380585	−0.190293	−	0.981727i	$-0.560944\pi$
$449$	−1.62580e6	−0.380585	−0.190293	+	0.981727i	$0.560944\pi$
$450$	0	0
$451$	3.00709e6	0.696154
$452$	−760928.	−0.175185
$453$	0	0
$454$	−5.29213e6	−1.20501
$455$	65301.9	0.0147876
$456$	0	0
$457$	1.50323e6	0.336694	0.168347	−	0.985728i	$-0.446157\pi$
$457$	1.50323e6	0.336694	0.168347	+	0.985728i	$0.446157\pi$
$458$	−463009.	−0.103140
$459$	0	0
$460$	−5930.08	−0.00130667
$461$	−549546.	−0.120435	−0.0602173	−	0.998185i	$-0.519179\pi$
$461$	−549546.	−0.120435	−0.0602173	+	0.998185i	$0.519179\pi$
$462$	0	0
$463$	−3.34547e6	−0.725279	−0.362640	−	0.931929i	$-0.618124\pi$
$463$	−3.34547e6	−0.725279	−0.362640	+	0.931929i	$0.618124\pi$
$464$	−1.22782e6	−0.264752
$465$	0	0
$466$	4.87701e6	1.04037
$467$	−2.20487e6	−0.467833	−0.233916	−	0.972257i	$-0.575154\pi$
$467$	−2.20487e6	−0.467833	−0.233916	+	0.972257i	$0.575154\pi$
$468$	0	0
$469$	−2.15769e6	−0.452957
$470$	−537450.	−0.112226
$471$	0	0
$472$	−324264.	−0.0669952
$473$	−277053.	−0.0569390
$474$	0	0
$475$	1.14893e6	0.233647
$476$	−1.69340e6	−0.342565
$477$	0	0
$478$	−6.67293e6	−1.33582
$479$	655516.	0.130540	0.0652702	−	0.997868i	$-0.479209\pi$
$479$	655516.	0.130540	0.0652702	+	0.997868i	$0.479209\pi$
$480$	0	0
$481$	−1.94032e6	−0.382393
$482$	−3.39084e6	−0.664798
$483$	0	0
$484$	−1.65789e6	−0.321694
$485$	−224810.	−0.0433971
$486$	0	0
$487$	1.70140e6	0.325076	0.162538	−	0.986702i	$-0.448032\pi$
$487$	1.70140e6	0.325076	0.162538	+	0.986702i	$0.448032\pi$
$488$	−3.47339e6	−0.660243
$489$	0	0
$490$	401598.	0.0755616
$491$	−1.54564e6	−0.289338	−0.144669	−	0.989480i	$-0.546212\pi$
$491$	−1.54564e6	−0.289338	−0.144669	+	0.989480i	$0.546212\pi$
$492$	0	0
$493$	−9.46703e6	−1.75427
$494$	−252737.	−0.0465964
$495$	0	0
$496$	1.76794e6	0.322673
$497$	−3.49793e6	−0.635214
$498$	0	0
$499$	5.57437e6	1.00218	0.501089	−	0.865396i	$-0.332933\pi$
$499$	5.57437e6	1.00218	0.501089	+	0.865396i	$0.332933\pi$
$500$	−714653.	−0.127841
$501$	0	0
$502$	−4.36591e6	−0.773243
$503$	−318350.	−0.0561028	−0.0280514	−	0.999606i	$-0.508930\pi$
$503$	−318350.	−0.0561028	−0.0280514	+	0.999606i	$0.508930\pi$
$504$	0	0
$505$	626587.	0.109333
$506$	−49301.7	−0.00856023
$507$	0	0
$508$	−1.74038e6	−0.299217
$509$	−265194.	−0.0453701	−0.0226851	−	0.999743i	$-0.507221\pi$
$509$	−265194.	−0.0453701	−0.0226851	+	0.999743i	$0.507221\pi$
$510$	0	0
$511$	−3.67325e6	−0.622297
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−42042.3	−0.00701906
$515$	−310676.	−0.0516166
$516$	0	0
$517$	−4.46827e6	−0.735212
$518$	2.46245e6	0.403220
$519$	0	0
$520$	77944.5	0.0126409
$521$	−1.91132e6	−0.308489	−0.154245	−	0.988033i	$-0.549294\pi$
$521$	−1.91132e6	−0.308489	−0.154245	+	0.988033i	$0.549294\pi$
$522$	0	0
$523$	4.75910e6	0.760801	0.380400	−	0.924822i	$-0.375786\pi$
$523$	4.75910e6	0.760801	0.380400	+	0.924822i	$0.375786\pi$
$524$	5.44777e6	0.866744
$525$	0	0
$526$	6.65400e6	1.04862
$527$	1.36316e7	2.13806
$528$	0	0
$529$	−6.43370e6	−0.999589
$530$	268623.	0.0415388
$531$	0	0
$532$	320747.	0.0491342
$533$	2.12058e6	0.323322
$534$	0	0
$535$	1.26808e6	0.191541
$536$	−2.57542e6	−0.387201
$537$	0	0
$538$	−492389.	−0.0733420
$539$	3.33882e6	0.495018
$540$	0	0
$541$	−1.03919e7	−1.52652	−0.763261	−	0.646090i	$-0.776403\pi$
$541$	−1.03919e7	−1.52652	−0.763261	+	0.646090i	$0.776403\pi$
$542$	−8.59826e6	−1.25722
$543$	0	0
$544$	−2.02125e6	−0.292835
$545$	861730.	0.124274
$546$	0	0
$547$	−9.00790e6	−1.28723	−0.643614	−	0.765350i	$-0.722566\pi$
$547$	−9.00790e6	−1.28723	−0.643614	+	0.765350i	$0.722566\pi$
$548$	−770295.	−0.109574
$549$	0	0
$550$	−2.94586e6	−0.415246
$551$	1.79315e6	0.251616
$552$	0	0
$553$	−569830.	−0.0792378
$554$	−3.27433e6	−0.453260
$555$	0	0
$556$	5.18084e6	0.710745
$557$	−1.61153e6	−0.220090	−0.110045	−	0.993927i	$-0.535099\pi$
$557$	−1.61153e6	−0.220090	−0.110045	+	0.993927i	$0.535099\pi$
$558$	0	0
$559$	−195375.	−0.0264448
$560$	−98918.9	−0.0133294
$561$	0	0
$562$	5.52011e6	0.737237
$563$	5.73380e6	0.762380	0.381190	−	0.924497i	$-0.375514\pi$
$563$	5.73380e6	0.762380	0.381190	+	0.924497i	$0.375514\pi$
$564$	0	0
$565$	−342722.	−0.0451670
$566$	−1.58329e6	−0.207740
$567$	0	0
$568$	−4.17513e6	−0.543000
$569$	−4.81958e6	−0.624063	−0.312032	−	0.950072i	$-0.601009\pi$
$569$	−4.81958e6	−0.624063	−0.312032	+	0.950072i	$0.601009\pi$
$570$	0	0
$571$	7.86132e6	1.00903	0.504517	−	0.863402i	$-0.331671\pi$
$571$	7.86132e6	1.00903	0.504517	+	0.863402i	$0.331671\pi$
$572$	648017.	0.0828126
$573$	0	0
$574$	−2.69121e6	−0.340932
$575$	158050.	0.0199354
$576$	0	0
$577$	1.55342e7	1.94244	0.971221	−	0.238179i	$-0.0765504\pi$
$577$	1.55342e7	1.94244	0.971221	+	0.238179i	$0.0765504\pi$
$578$	−9.90531e6	−1.23324
$579$	0	0
$580$	−553010.	−0.0682594
$581$	−126433.	−0.0155389
$582$	0	0
$583$	2.23329e6	0.272128
$584$	−4.38440e6	−0.531958
$585$	0	0
$586$	1.53116e6	0.184194
$587$	1.84016e6	0.220425	0.110212	−	0.993908i	$-0.464847\pi$
$587$	1.84016e6	0.220425	0.110212	+	0.993908i	$0.464847\pi$
$588$	0	0
$589$	−2.58196e6	−0.306663
$590$	−146049.	−0.0172730
$591$	0	0
$592$	2.93918e6	0.344685
$593$	−1.11019e7	−1.29646	−0.648230	−	0.761445i	$-0.724490\pi$
$593$	−1.11019e7	−1.29646	−0.648230	+	0.761445i	$0.724490\pi$
$594$	0	0
$595$	−762709.	−0.0883215
$596$	−5.96712e6	−0.688096
$597$	0	0
$598$	−34767.1	−0.00397572
$599$	1.34553e7	1.53224	0.766122	−	0.642696i	$-0.222184\pi$
$599$	1.34553e7	1.53224	0.766122	+	0.642696i	$0.222184\pi$
$600$	0	0
$601$	1.49829e7	1.69204	0.846020	−	0.533151i	$-0.178992\pi$
$601$	1.49829e7	1.69204	0.846020	+	0.533151i	$0.178992\pi$
$602$	247950.	0.0278851
$603$	0	0
$604$	−2.55577e6	−0.285055
$605$	−746715.	−0.0829405
$606$	0	0
$607$	−1.18436e7	−1.30471	−0.652353	−	0.757915i	$-0.726218\pi$
$607$	−1.18436e7	−1.30471	−0.652353	+	0.757915i	$0.726218\pi$
$608$	382845.	0.0420014
$609$	0	0
$610$	−1.56441e6	−0.170227
$611$	−3.15098e6	−0.341462
$612$	0	0
$613$	6.44924e6	0.693198	0.346599	−	0.938013i	$-0.387336\pi$
$613$	6.44924e6	0.693198	0.346599	+	0.938013i	$0.387336\pi$
$614$	1.18390e7	1.26735
$615$	0	0
$616$	−822394.	−0.0873229
$617$	1.10487e7	1.16842	0.584210	−	0.811602i	$-0.301404\pi$
$617$	1.10487e7	1.16842	0.584210	+	0.811602i	$0.301404\pi$
$618$	0	0
$619$	7.71392e6	0.809186	0.404593	−	0.914497i	$-0.367413\pi$
$619$	7.71392e6	0.809186	0.404593	+	0.914497i	$0.367413\pi$
$620$	796280.	0.0831930
$621$	0	0
$622$	−2.14760e6	−0.222576
$623$	−5.01983e6	−0.518166
$624$	0	0
$625$	9.28146e6	0.950421
$626$	1.33321e7	1.35976
$627$	0	0
$628$	−7.74614e6	−0.783765
$629$	2.26624e7	2.28391
$630$	0	0
$631$	−9.79932e6	−0.979767	−0.489883	−	0.871788i	$-0.662961\pi$
$631$	−9.79932e6	−0.979767	−0.489883	+	0.871788i	$0.662961\pi$
$632$	−680150.	−0.0677348
$633$	0	0
$634$	6.00400e6	0.593222
$635$	−783870.	−0.0771453
$636$	0	0
$637$	2.35450e6	0.229906
$638$	−4.59763e6	−0.447180
$639$	0	0
$640$	−118070.	−0.0113943
$641$	−1.53393e7	−1.47455	−0.737275	−	0.675593i	$-0.763888\pi$
$641$	−1.53393e7	−1.47455	−0.737275	+	0.675593i	$0.763888\pi$
$642$	0	0
$643$	1.45341e7	1.38631	0.693155	−	0.720789i	$-0.256220\pi$
$643$	1.45341e7	1.38631	0.693155	+	0.720789i	$0.256220\pi$
$644$	44122.7	0.00419226
$645$	0	0
$646$	2.95191e6	0.278305
$647$	1.26902e7	1.19181	0.595904	−	0.803055i	$-0.296794\pi$
$647$	1.26902e7	1.19181	0.595904	+	0.803055i	$0.296794\pi$
$648$	0	0
$649$	−1.21422e6	−0.113158
$650$	−2.07739e6	−0.192857
$651$	0	0
$652$	−2.24276e6	−0.206616
$653$	4.48209e6	0.411337	0.205669	−	0.978622i	$-0.434063\pi$
$653$	4.48209e6	0.411337	0.205669	+	0.978622i	$0.434063\pi$
$654$	0	0
$655$	2.45368e6	0.223468
$656$	−3.21223e6	−0.291439
$657$	0	0
$658$	3.99889e6	0.360060
$659$	1.76267e7	1.58109	0.790545	−	0.612403i	$-0.209797\pi$
$659$	1.76267e7	1.58109	0.790545	+	0.612403i	$0.209797\pi$
$660$	0	0
$661$	1.03733e7	0.923447	0.461724	−	0.887024i	$-0.347231\pi$
$661$	1.03733e7	0.923447	0.461724	+	0.887024i	$0.347231\pi$
$662$	−8.69617e6	−0.771229
$663$	0	0
$664$	−150911.	−0.0132831
$665$	144465.	0.0126680
$666$	0	0
$667$	246670.	0.0214685
$668$	−3.48874e6	−0.302501
$669$	0	0
$670$	−1.15997e6	−0.0998298
$671$	−1.30063e7	−1.11518
$672$	0	0
$673$	4.76273e6	0.405339	0.202669	−	0.979247i	$-0.435038\pi$
$673$	4.76273e6	0.405339	0.202669	+	0.979247i	$0.435038\pi$
$674$	−3.28598e6	−0.278622
$675$	0	0
$676$	456976.	0.0384615
$677$	−1.68783e7	−1.41533	−0.707665	−	0.706548i	$-0.750251\pi$
$677$	−1.68783e7	−1.41533	−0.707665	+	0.706548i	$0.750251\pi$
$678$	0	0
$679$	1.67270e6	0.139233
$680$	−910371.	−0.0754999
$681$	0	0
$682$	6.62013e6	0.545012
$683$	1.87331e7	1.53659	0.768294	−	0.640097i	$-0.221106\pi$
$683$	1.87331e7	1.53659	0.768294	+	0.640097i	$0.221106\pi$
$684$	0	0
$685$	−346941.	−0.0282507
$686$	−6.59280e6	−0.534884
$687$	0	0
$688$	295953.	0.0238370
$689$	1.57489e6	0.126387
$690$	0	0
$691$	2.12121e7	1.69000	0.845002	−	0.534763i	$-0.179599\pi$
$691$	2.12121e7	1.69000	0.845002	+	0.534763i	$0.179599\pi$
$692$	6.15763e6	0.488819
$693$	0	0
$694$	7.03590e6	0.554525
$695$	2.33345e6	0.183247
$696$	0	0
$697$	−2.47678e7	−1.93110
$698$	1.26143e7	0.980000
$699$	0	0
$700$	2.63641e6	0.203361
$701$	−2.61831e6	−0.201246	−0.100623	−	0.994925i	$-0.532084\pi$
$701$	−2.61831e6	−0.201246	−0.100623	+	0.994925i	$0.532084\pi$
$702$	0	0
$703$	−4.29249e6	−0.327583
$704$	−981612.	−0.0746463
$705$	0	0
$706$	−6.05824e6	−0.457440
$707$	−4.66212e6	−0.350780
$708$	0	0
$709$	1.41825e7	1.05959	0.529794	−	0.848127i	$-0.322269\pi$
$709$	1.41825e7	1.05959	0.529794	+	0.848127i	$0.322269\pi$
$710$	−1.88048e6	−0.139998
$711$	0	0
$712$	−5.99168e6	−0.442944
$713$	−355180.	−0.0261653
$714$	0	0
$715$	291867.	0.0213511
$716$	−7.97236e6	−0.581172
$717$	0	0
$718$	6.75739e6	0.489179
$719$	−1.39362e6	−0.100536	−0.0502681	−	0.998736i	$-0.516008\pi$
$719$	−1.39362e6	−0.100536	−0.0502681	+	0.998736i	$0.516008\pi$
$720$	0	0
$721$	2.31158e6	0.165604
$722$	9.34528e6	0.667189
$723$	0	0
$724$	2.43522e6	0.172660
$725$	1.47389e7	1.04141
$726$	0	0
$727$	5.67972e6	0.398557	0.199279	−	0.979943i	$-0.436140\pi$
$727$	5.67972e6	0.398557	0.199279	+	0.979943i	$0.436140\pi$
$728$	−579945.	−0.0405563
$729$	0	0
$730$	−1.97473e6	−0.137152
$731$	2.28193e6	0.157946
$732$	0	0
$733$	−1.55611e7	−1.06974	−0.534872	−	0.844933i	$-0.679640\pi$
$733$	−1.55611e7	−1.06974	−0.534872	+	0.844933i	$0.679640\pi$
$734$	7.48736e6	0.512966
$735$	0	0
$736$	52665.0	0.00358367
$737$	−9.64380e6	−0.654003
$738$	0	0
$739$	1.54351e7	1.03968	0.519839	−	0.854264i	$-0.325992\pi$
$739$	1.54351e7	1.03968	0.519839	+	0.854264i	$0.325992\pi$
$740$	1.32381e6	0.0888681
$741$	0	0
$742$	−1.99869e6	−0.133271
$743$	8.72110e6	0.579561	0.289781	−	0.957093i	$-0.406418\pi$
$743$	8.72110e6	0.579561	0.289781	+	0.957093i	$0.406418\pi$
$744$	0	0
$745$	−2.68759e6	−0.177408
$746$	−2.51201e6	−0.165263
$747$	0	0
$748$	−7.56867e6	−0.494613
$749$	−9.43514e6	−0.614531
$750$	0	0
$751$	1.07940e7	0.698365	0.349183	−	0.937055i	$-0.386459\pi$
$751$	1.07940e7	0.698365	0.349183	+	0.937055i	$0.386459\pi$
$752$	4.77309e6	0.307790
$753$	0	0
$754$	−3.24221e6	−0.207688
$755$	−1.15112e6	−0.0734941
$756$	0	0
$757$	1.87309e7	1.18801	0.594005	−	0.804462i	$-0.297546\pi$
$757$	1.87309e7	1.18801	0.594005	+	0.804462i	$0.297546\pi$
$758$	−1.91290e6	−0.120926
$759$	0	0
$760$	172433.	0.0108290
$761$	1.41785e7	0.887498	0.443749	−	0.896151i	$-0.353648\pi$
$761$	1.41785e7	0.887498	0.443749	+	0.896151i	$0.353648\pi$
$762$	0	0
$763$	−6.41170e6	−0.398714
$764$	−9.60454e6	−0.595310
$765$	0	0
$766$	−1.83419e7	−1.12947
$767$	−856260.	−0.0525554
$768$	0	0
$769$	−1.90780e7	−1.16337	−0.581683	−	0.813416i	$-0.697605\pi$
$769$	−1.90780e7	−1.16337	−0.581683	+	0.813416i	$0.697605\pi$
$770$	−370407.	−0.0225140
$771$	0	0
$772$	5.61287e6	0.338955
$773$	−1.31787e7	−0.793276	−0.396638	−	0.917975i	$-0.629823\pi$
$773$	−1.31787e7	−0.793276	−0.396638	+	0.917975i	$0.629823\pi$
$774$	0	0
$775$	−2.12226e7	−1.26924
$776$	1.99653e6	0.119021
$777$	0	0
$778$	1.04854e7	0.621065
$779$	4.69126e6	0.276979
$780$	0	0
$781$	−1.56340e7	−0.917155
$782$	406071.	0.0237457
$783$	0	0
$784$	−3.56659e6	−0.207235
$785$	−3.48886e6	−0.202074
$786$	0	0
$787$	−2.47284e7	−1.42318	−0.711589	−	0.702596i	$-0.752024\pi$
$787$	−2.47284e7	−1.42318	−0.711589	+	0.702596i	$0.752024\pi$
$788$	5.13832e6	0.294785
$789$	0	0
$790$	−306340.	−0.0174637
$791$	2.55002e6	0.144911
$792$	0	0
$793$	−9.17191e6	−0.517937
$794$	−3.84461e6	−0.216422
$795$	0	0
$796$	8.98266e6	0.502484
$797$	2.11191e7	1.17769	0.588843	−	0.808248i	$-0.299584\pi$
$797$	2.11191e7	1.17769	0.588843	+	0.808248i	$0.299584\pi$
$798$	0	0
$799$	3.68026e7	2.03945
$800$	3.14682e6	0.173839
$801$	0	0
$802$	−1.60223e7	−0.879610
$803$	−1.64176e7	−0.898505
$804$	0	0
$805$	19872.9	0.00108087
$806$	4.66846e6	0.253126
$807$	0	0
$808$	−5.56471e6	−0.299857
$809$	−1.62945e7	−0.875326	−0.437663	−	0.899139i	$-0.644194\pi$
$809$	−1.62945e7	−0.875326	−0.437663	+	0.899139i	$0.644194\pi$
$810$	0	0
$811$	2.00401e7	1.06991	0.534954	−	0.844881i	$-0.320329\pi$
$811$	2.00401e7	1.06991	0.534954	+	0.844881i	$0.320329\pi$
$812$	4.11467e6	0.219000
$813$	0	0
$814$	1.10059e7	0.582191
$815$	−1.01014e6	−0.0532706
$816$	0	0
$817$	−432221.	−0.0226543
$818$	1.25012e7	0.653234
$819$	0	0
$820$	−1.44679e6	−0.0751400
$821$	−3.51296e7	−1.81893	−0.909463	−	0.415784i	$-0.863507\pi$
$821$	−3.51296e7	−1.81893	−0.909463	+	0.415784i	$0.863507\pi$
$822$	0	0
$823$	−2.51324e7	−1.29341	−0.646703	−	0.762742i	$-0.723853\pi$
$823$	−2.51324e7	−1.29341	−0.646703	+	0.762742i	$0.723853\pi$
$824$	2.75911e6	0.141563
$825$	0	0
$826$	1.08667e6	0.0554178
$827$	1.66576e7	0.846934	0.423467	−	0.905911i	$-0.360813\pi$
$827$	1.66576e7	0.846934	0.423467	+	0.905911i	$0.360813\pi$
$828$	0	0
$829$	−1.46404e7	−0.739887	−0.369944	−	0.929054i	$-0.620623\pi$
$829$	−1.46404e7	−0.739887	−0.369944	+	0.929054i	$0.620623\pi$
$830$	−67970.2	−0.00342471
$831$	0	0
$832$	−692224.	−0.0346688
$833$	−2.75000e7	−1.37316
$834$	0	0
$835$	−1.57133e6	−0.0779921
$836$	1.43358e6	0.0709425
$837$	0	0
$838$	−1.33951e7	−0.658924
$839$	−2.79895e7	−1.37275	−0.686374	−	0.727249i	$-0.740799\pi$
$839$	−2.79895e7	−1.37275	−0.686374	+	0.727249i	$0.740799\pi$
$840$	0	0
$841$	2.49204e6	0.121497
$842$	1.20533e7	0.585902
$843$	0	0
$844$	−1.20160e7	−0.580637
$845$	205822.	0.00991632
$846$	0	0
$847$	5.55593e6	0.266102
$848$	−2.38564e6	−0.113924
$849$	0	0
$850$	2.42634e7	1.15187
$851$	−590484.	−0.0279502
$852$	0	0
$853$	−2.22248e7	−1.04584	−0.522919	−	0.852382i	$-0.675157\pi$
$853$	−2.22248e7	−1.04584	−0.522919	+	0.852382i	$0.675157\pi$
$854$	1.16400e7	0.546146
$855$	0	0
$856$	−1.12618e7	−0.525319
$857$	−7.83995e6	−0.364637	−0.182319	−	0.983239i	$-0.558360\pi$
$857$	−7.83995e6	−0.364637	−0.182319	+	0.983239i	$0.558360\pi$
$858$	0	0
$859$	−3.08215e7	−1.42518	−0.712592	−	0.701579i	$-0.752479\pi$
$859$	−3.08215e7	−1.42518	−0.712592	+	0.701579i	$0.752479\pi$
$860$	133298.	0.00614576
$861$	0	0
$862$	1.34090e7	0.614649
$863$	4.53439e6	0.207249	0.103624	−	0.994617i	$-0.466956\pi$
$863$	4.53439e6	0.207249	0.103624	+	0.994617i	$0.466956\pi$
$864$	0	0
$865$	2.77340e6	0.126029
$866$	−6.48753e6	−0.293957
$867$	0	0
$868$	−5.92472e6	−0.266912
$869$	−2.54686e6	−0.114408
$870$	0	0
$871$	−6.80073e6	−0.303746
$872$	−7.65301e6	−0.340833
$873$	0	0
$874$	−76913.9	−0.00340586
$875$	2.39495e6	0.105749
$876$	0	0
$877$	3.31312e7	1.45458	0.727292	−	0.686328i	$-0.240779\pi$
$877$	3.31312e7	1.45458	0.727292	+	0.686328i	$0.240779\pi$
$878$	−2.99579e7	−1.31152
$879$	0	0
$880$	−442118.	−0.0192456
$881$	−6.36611e6	−0.276334	−0.138167	−	0.990409i	$-0.544121\pi$
$881$	−6.36611e6	−0.276334	−0.138167	+	0.990409i	$0.544121\pi$
$882$	0	0
$883$	−3.09630e7	−1.33641	−0.668207	−	0.743975i	$-0.732938\pi$
$883$	−3.09630e7	−1.33641	−0.668207	+	0.743975i	$0.732938\pi$
$884$	−5.33736e6	−0.229718
$885$	0	0
$886$	7.60626e6	0.325527
$887$	4.17692e7	1.78257	0.891287	−	0.453440i	$-0.149803\pi$
$887$	4.17692e7	1.78257	0.891287	+	0.453440i	$0.149803\pi$
$888$	0	0
$889$	5.83238e6	0.247509
$890$	−2.69866e6	−0.114202
$891$	0	0
$892$	−9.58250e6	−0.403243
$893$	−6.97079e6	−0.292519
$894$	0	0
$895$	−3.59075e6	−0.149840
$896$	878497.	0.0365570
$897$	0	0
$898$	6.50321e6	0.269114
$899$	−3.31223e7	−1.36685
$900$	0	0
$901$	−1.83944e7	−0.754871
$902$	−1.20284e7	−0.492255
$903$	0	0
$904$	3.04371e6	0.123875
$905$	1.09682e6	0.0445159
$906$	0	0
$907$	−4.51182e7	−1.82110	−0.910550	−	0.413399i	$-0.864342\pi$
$907$	−4.51182e7	−1.82110	−0.910550	+	0.413399i	$0.864342\pi$
$908$	2.11685e7	0.852071
$909$	0	0
$910$	−261208.	−0.0104564
$911$	−2.93224e6	−0.117059	−0.0585293	−	0.998286i	$-0.518641\pi$
$911$	−2.93224e6	−0.117059	−0.0585293	+	0.998286i	$0.518641\pi$
$912$	0	0
$913$	−565092.	−0.0224359
$914$	−6.01293e6	−0.238079
$915$	0	0
$916$	1.85203e6	0.0729307
$917$	−1.82566e7	−0.716962
$918$	0	0
$919$	1.06534e7	0.416101	0.208051	−	0.978118i	$-0.433288\pi$
$919$	1.06534e7	0.416101	0.208051	+	0.978118i	$0.433288\pi$
$920$	23720.3	0.000923956 0
$921$	0	0
$922$	2.19818e6	0.0851602
$923$	−1.10250e7	−0.425964
$924$	0	0
$925$	−3.52824e7	−1.35583
$926$	1.33819e7	0.512850
$927$	0	0
$928$	4.91127e6	0.187208
$929$	1.93436e7	0.735355	0.367677	−	0.929953i	$-0.380153\pi$
$929$	1.93436e7	0.735355	0.367677	+	0.929953i	$0.380153\pi$
$930$	0	0
$931$	5.20878e6	0.196952
$932$	−1.95081e7	−0.735655
$933$	0	0
$934$	8.81947e6	0.330808
$935$	−3.40893e6	−0.127523
$936$	0	0
$937$	−5.62445e6	−0.209282	−0.104641	−	0.994510i	$-0.533369\pi$
$937$	−5.62445e6	−0.209282	−0.104641	+	0.994510i	$0.533369\pi$
$938$	8.63076e6	0.320289
$939$	0	0
$940$	2.14980e6	0.0793557
$941$	2.18961e7	0.806106	0.403053	−	0.915177i	$-0.367949\pi$
$941$	2.18961e7	0.806106	0.403053	+	0.915177i	$0.367949\pi$
$942$	0	0
$943$	645341.	0.0236325
$944$	1.29706e6	0.0473728
$945$	0	0
$946$	1.10821e6	0.0402620
$947$	−931486.	−0.0337522	−0.0168761	−	0.999858i	$-0.505372\pi$
$947$	−931486.	−0.0337522	−0.0168761	+	0.999858i	$0.505372\pi$
$948$	0	0
$949$	−1.15775e7	−0.417302
$950$	−4.59573e6	−0.165214
$951$	0	0
$952$	6.77361e6	0.242230
$953$	−4.06914e7	−1.45135	−0.725673	−	0.688040i	$-0.758471\pi$
$953$	−4.06914e7	−1.45135	−0.725673	+	0.688040i	$0.758471\pi$
$954$	0	0
$955$	−4.32589e6	−0.153485
$956$	2.66917e7	0.944565
$957$	0	0
$958$	−2.62206e6	−0.0923059
$959$	2.58141e6	0.0906382
$960$	0	0
$961$	1.90638e7	0.665887
$962$	7.76128e6	0.270393
$963$	0	0
$964$	1.35634e7	0.470083
$965$	2.52804e6	0.0873907
$966$	0	0
$967$	1.43608e7	0.493869	0.246934	−	0.969032i	$-0.420577\pi$
$967$	1.43608e7	0.493869	0.246934	+	0.969032i	$0.420577\pi$
$968$	6.63157e6	0.227472
$969$	0	0
$970$	899239.	0.0306864
$971$	−4.38037e6	−0.149095	−0.0745475	−	0.997217i	$-0.523751\pi$
$971$	−4.38037e6	−0.149095	−0.0745475	+	0.997217i	$0.523751\pi$
$972$	0	0
$973$	−1.73621e7	−0.587921
$974$	−6.80562e6	−0.229864
$975$	0	0
$976$	1.38935e7	0.466862
$977$	−2.87093e7	−0.962247	−0.481123	−	0.876653i	$-0.659771\pi$
$977$	−2.87093e7	−0.962247	−0.481123	+	0.876653i	$0.659771\pi$
$978$	0	0
$979$	−2.24362e7	−0.748155
$980$	−1.60639e6	−0.0534301
$981$	0	0
$982$	6.18257e6	0.204593
$983$	−5.36798e7	−1.77185	−0.885925	−	0.463829i	$-0.846475\pi$
$983$	−5.36798e7	−1.77185	−0.885925	+	0.463829i	$0.846475\pi$
$984$	0	0
$985$	2.31430e6	0.0760028
$986$	3.78681e7	1.24046
$987$	0	0
$988$	1.01095e6	0.0329486
$989$	−59457.3	−0.00193292
$990$	0	0
$991$	2.18434e7	0.706539	0.353269	−	0.935522i	$-0.385070\pi$
$991$	2.18434e7	0.706539	0.353269	+	0.935522i	$0.385070\pi$
$992$	−7.07175e6	−0.228164
$993$	0	0
$994$	1.39917e7	0.449164
$995$	4.04579e6	0.129553
$996$	0	0
$997$	4.14924e7	1.32200	0.660999	−	0.750387i	$-0.270133\pi$
$997$	4.14924e7	1.32200	0.660999	+	0.750387i	$0.270133\pi$
$998$	−2.22975e7	−0.708646
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.6.a.h.1.2		2
3.2	odd	2		26.6.a.c.1.2	✓	2
12.11	even	2		208.6.a.g.1.1		2
15.2	even	4		650.6.b.h.599.3		4
15.8	even	4		650.6.b.h.599.2		4
15.14	odd	2		650.6.a.b.1.1		2
24.5	odd	2		832.6.a.k.1.1		2
24.11	even	2		832.6.a.m.1.2		2
39.5	even	4		338.6.b.b.337.2		4
39.8	even	4		338.6.b.b.337.4		4
39.38	odd	2		338.6.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.6.a.c.1.2	✓	2	3.2	odd	2
208.6.a.g.1.1		2	12.11	even	2
234.6.a.h.1.2		2	1.1	even	1	trivial
338.6.a.f.1.2		2	39.38	odd	2
338.6.b.b.337.2		4	39.5	even	4
338.6.b.b.337.4		4	39.8	even	4
650.6.a.b.1.1		2	15.14	odd	2
650.6.b.h.599.2		4	15.8	even	4
650.6.b.h.599.3		4	15.2	even	4
832.6.a.k.1.1		2	24.5	odd	2
832.6.a.m.1.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +7.20641 q^{5} -53.6192 q^{7} -64.0000 q^{8} -28.8256 q^{10} -239.651 q^{11} -169.000 q^{13} +214.477 q^{14} +256.000 q^{16} +1973.88 q^{17} -373.872 q^{19} +115.303 q^{20} +958.605 q^{22} -51.4306 q^{23} -3073.07 q^{25} +676.000 q^{26} -857.908 q^{28} -4796.16 q^{29} +6906.01 q^{31} -1024.00 q^{32} -7895.50 q^{34} -386.402 q^{35} +11481.2 q^{37} +1495.49 q^{38} -461.210 q^{40} -12547.8 q^{41} +1156.07 q^{43} -3834.42 q^{44} +205.723 q^{46} +18644.9 q^{47} -13932.0 q^{49} +12292.3 q^{50} -2704.00 q^{52} -9318.90 q^{53} -1727.02 q^{55} +3431.63 q^{56} +19184.7 q^{58} +5066.63 q^{59} +54271.7 q^{61} -27624.0 q^{62} +4096.00 q^{64} -1217.88 q^{65} +40241.0 q^{67} +31582.0 q^{68} +1545.61 q^{70} +65236.5 q^{71} +68506.2 q^{73} -45924.7 q^{74} -5981.95 q^{76} +12849.9 q^{77} +10627.3 q^{79} +1844.84 q^{80} +50191.1 q^{82} +2357.98 q^{83} +14224.5 q^{85} -4624.27 q^{86} +15337.7 q^{88} +93620.0 q^{89} +9061.65 q^{91} -822.890 q^{92} -74579.5 q^{94} -2694.27 q^{95} -31195.8 q^{97} +55727.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 73 q^{5} + 155 q^{7} - 128 q^{8} + 292 q^{10} + 220 q^{11} - 338 q^{13} - 620 q^{14} + 512 q^{16} + 189 q^{17} - 2496 q^{19} - 1168 q^{20} - 880 q^{22} + 3044 q^{23} + 235 q^{25}+ \cdots - 51132 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 73 * q^5 + 155 * q^7 - 128 * q^8 + 292 * q^10 + 220 * q^11 - 338 * q^13 - 620 * q^14 + 512 * q^16 + 189 * q^17 - 2496 * q^19 - 1168 * q^20 - 880 * q^22 + 3044 * q^23 + 235 * q^25 + 1352 * q^26 + 2480 * q^28 - 1900 * q^29 + 2798 * q^31 - 2048 * q^32 - 756 * q^34 - 17119 * q^35 + 17805 * q^37 + 9984 * q^38 + 4672 * q^40 - 11634 * q^41 - 4069 * q^43 + 3520 * q^44 - 12176 * q^46 + 25489 * q^47 + 12783 * q^49 - 940 * q^50 - 5408 * q^52 + 4614 * q^53 - 38594 * q^55 - 9920 * q^56 + 7600 * q^58 + 23420 * q^59 + 96830 * q^61 - 11192 * q^62 + 8192 * q^64 + 12337 * q^65 + 72440 * q^67 + 3024 * q^68 + 68476 * q^70 + 36679 * q^71 + 47152 * q^73 - 71220 * q^74 - 39936 * q^76 + 108742 * q^77 + 52024 * q^79 - 18688 * q^80 + 46536 * q^82 + 37758 * q^83 + 157383 * q^85 + 16276 * q^86 - 14080 * q^88 + 105072 * q^89 - 26195 * q^91 + 48704 * q^92 - 101956 * q^94 + 167514 * q^95 + 104392 * q^97 - 51132 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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