LMFDB - Embedded newform 2205.4.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.4892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 4 $ x^3 - 11*x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.368173$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36817 q^{2} -6.12810 q^{4} +5.00000 q^{5} +19.3297 q^{8} +O(q^{10})$	$q-1.36817 q^{2} -6.12810 q^{4} +5.00000 q^{5} +19.3297 q^{8} -6.84087 q^{10} -9.07349 q^{11} +2.69663 q^{13} +22.5784 q^{16} +15.5050 q^{17} -46.4429 q^{19} -30.6405 q^{20} +12.4141 q^{22} -47.4618 q^{23} +25.0000 q^{25} -3.68946 q^{26} +8.21699 q^{29} +228.348 q^{31} -185.529 q^{32} -21.2135 q^{34} -334.179 q^{37} +63.5419 q^{38} +96.6485 q^{40} +35.3197 q^{41} +510.201 q^{43} +55.6033 q^{44} +64.9359 q^{46} -545.252 q^{47} -34.2043 q^{50} -16.5252 q^{52} +4.33412 q^{53} -45.3674 q^{55} -11.2423 q^{58} -791.890 q^{59} +215.724 q^{61} -312.420 q^{62} +73.2079 q^{64} +13.4832 q^{65} +402.221 q^{67} -95.0160 q^{68} +328.146 q^{71} +523.856 q^{73} +457.215 q^{74} +284.607 q^{76} -130.798 q^{79} +112.892 q^{80} -48.3235 q^{82} -507.390 q^{83} +77.5248 q^{85} -698.043 q^{86} -175.388 q^{88} +565.891 q^{89} +290.851 q^{92} +745.999 q^{94} -232.214 q^{95} +1439.25 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{4} + 15 q^{5} - 9 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + q^4 + 15 * q^5 - 9 * q^8	$ 3 q - 3 q^{2} + q^{4} + 15 q^{5} - 9 q^{8} - 15 q^{10} + q^{11} + 79 q^{13} - 79 q^{16} - 72 q^{17} + 29 q^{19} + 5 q^{20} + 143 q^{22} + 63 q^{23} + 75 q^{25} - 339 q^{26} - 220 q^{29} - 136 q^{31} + 155 q^{32} + 220 q^{34} + 43 q^{37} - 21 q^{38} - 45 q^{40} - 599 q^{41} + 170 q^{43} - 135 q^{44} + 265 q^{46} - 3 q^{47} - 75 q^{50} + 701 q^{52} - 331 q^{53} + 5 q^{55} - 472 q^{58} - 1520 q^{59} + 1160 q^{61} - 748 q^{62} + 17 q^{64} + 395 q^{65} + 806 q^{67} - 684 q^{68} + 406 q^{71} + 1192 q^{73} - 959 q^{74} + 591 q^{76} - 2590 q^{79} - 395 q^{80} + 1191 q^{82} - 508 q^{83} - 360 q^{85} + 742 q^{86} - 749 q^{88} + 42 q^{89} + 211 q^{92} + 1167 q^{94} + 145 q^{95} + 1020 q^{97}+O(q^{100}) $ 3 * q - 3 * q^2 + q^4 + 15 * q^5 - 9 * q^8 - 15 * q^10 + q^11 + 79 * q^13 - 79 * q^16 - 72 * q^17 + 29 * q^19 + 5 * q^20 + 143 * q^22 + 63 * q^23 + 75 * q^25 - 339 * q^26 - 220 * q^29 - 136 * q^31 + 155 * q^32 + 220 * q^34 + 43 * q^37 - 21 * q^38 - 45 * q^40 - 599 * q^41 + 170 * q^43 - 135 * q^44 + 265 * q^46 - 3 * q^47 - 75 * q^50 + 701 * q^52 - 331 * q^53 + 5 * q^55 - 472 * q^58 - 1520 * q^59 + 1160 * q^61 - 748 * q^62 + 17 * q^64 + 395 * q^65 + 806 * q^67 - 684 * q^68 + 406 * q^71 + 1192 * q^73 - 959 * q^74 + 591 * q^76 - 2590 * q^79 - 395 * q^80 + 1191 * q^82 - 508 * q^83 - 360 * q^85 + 742 * q^86 - 749 * q^88 + 42 * q^89 + 211 * q^92 + 1167 * q^94 + 145 * q^95 + 1020 * q^97

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 $

$2$	−1.36817	−0.483722	−0.241861	−	0.970311i	$-0.577758\pi$
$2$	−1.36817	−0.483722	−0.241861	+	0.970311i	$0.577758\pi$
$3$	0	0
$3$	0	0
$4$	−6.12810	−0.766013
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	19.3297	0.854260
$9$	0	0
$10$	−6.84087	−0.216327
$11$	−9.07349	−0.248705	−0.124353	−	0.992238i	$-0.539685\pi$
$11$	−9.07349	−0.248705	−0.124353	+	0.992238i	$0.539685\pi$
$12$	0	0
$13$	2.69663	0.0575316	0.0287658	−	0.999586i	$-0.490842\pi$
$13$	2.69663	0.0575316	0.0287658	+	0.999586i	$0.490842\pi$
$14$	0	0
$15$	0	0
$16$	22.5784	0.352788
$17$	15.5050	0.221206	0.110603	−	0.993865i	$-0.464722\pi$
$17$	15.5050	0.221206	0.110603	+	0.993865i	$0.464722\pi$
$18$	0	0
$19$	−46.4429	−0.560775	−0.280388	−	0.959887i	$-0.590463\pi$
$19$	−46.4429	−0.560775	−0.280388	+	0.959887i	$0.590463\pi$
$20$	−30.6405	−0.342571
$21$	0	0
$22$	12.4141	0.120304
$23$	−47.4618	−0.430281	−0.215140	−	0.976583i	$-0.569021\pi$
$23$	−47.4618	−0.430281	−0.215140	+	0.976583i	$0.569021\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−3.68946	−0.0278293
$27$	0	0
$28$	0	0
$29$	8.21699	0.0526158	0.0263079	−	0.999654i	$-0.491625\pi$
$29$	8.21699	0.0526158	0.0263079	+	0.999654i	$0.491625\pi$
$30$	0	0
$31$	228.348	1.32298	0.661492	−	0.749952i	$-0.269923\pi$
$31$	228.348	1.32298	0.661492	+	0.749952i	$0.269923\pi$
$32$	−185.529	−1.02491
$33$	0	0
$34$	−21.2135	−0.107002
$35$	0	0
$36$	0	0
$37$	−334.179	−1.48483	−0.742415	−	0.669940i	$-0.766320\pi$
$37$	−334.179	−1.48483	−0.742415	+	0.669940i	$0.766320\pi$
$38$	63.5419	0.271260
$39$	0	0
$40$	96.6485	0.382037
$41$	35.3197	0.134537	0.0672685	−	0.997735i	$-0.478572\pi$
$41$	35.3197	0.134537	0.0672685	+	0.997735i	$0.478572\pi$
$42$	0	0
$43$	510.201	1.80942	0.904708	−	0.426033i	$-0.140089\pi$
$43$	510.201	1.80942	0.904708	+	0.426033i	$0.140089\pi$
$44$	55.6033	0.190512
$45$	0	0
$46$	64.9359	0.208137
$47$	−545.252	−1.69219	−0.846097	−	0.533029i	$-0.821054\pi$
$47$	−545.252	−1.69219	−0.846097	+	0.533029i	$0.821054\pi$
$48$	0	0
$49$	0	0
$50$	−34.2043	−0.0967445
$51$	0	0
$52$	−16.5252	−0.0440700
$53$	4.33412	0.0112328	0.00561639	−	0.999984i	$-0.498212\pi$
$53$	4.33412	0.0112328	0.00561639	+	0.999984i	$0.498212\pi$
$54$	0	0
$55$	−45.3674	−0.111224
$56$	0	0
$57$	0	0
$58$	−11.2423	−0.0254514
$59$	−791.890	−1.74738	−0.873689	−	0.486484i	$-0.838279\pi$
$59$	−791.890	−1.74738	−0.873689	+	0.486484i	$0.838279\pi$
$60$	0	0
$61$	215.724	0.452797	0.226399	−	0.974035i	$-0.427305\pi$
$61$	215.724	0.452797	0.226399	+	0.974035i	$0.427305\pi$
$62$	−312.420	−0.639957
$63$	0	0
$64$	73.2079	0.142984
$65$	13.4832	0.0257289
$66$	0	0
$67$	402.221	0.733420	0.366710	−	0.930335i	$-0.380484\pi$
$67$	402.221	0.733420	0.366710	+	0.930335i	$0.380484\pi$
$68$	−95.0160	−0.169447
$69$	0	0
$70$	0	0
$71$	328.146	0.548503	0.274252	−	0.961658i	$-0.411570\pi$
$71$	328.146	0.548503	0.274252	+	0.961658i	$0.411570\pi$
$72$	0	0
$73$	523.856	0.839900	0.419950	−	0.907547i	$-0.362048\pi$
$73$	523.856	0.839900	0.419950	+	0.907547i	$0.362048\pi$
$74$	457.215	0.718246
$75$	0	0
$76$	284.607	0.429561
$77$	0	0
$78$	0	0
$79$	−130.798	−0.186277	−0.0931387	−	0.995653i	$-0.529690\pi$
$79$	−130.798	−0.186277	−0.0931387	+	0.995653i	$0.529690\pi$
$80$	112.892	0.157772
$81$	0	0
$82$	−48.3235	−0.0650785
$83$	−507.390	−0.671003	−0.335502	−	0.942040i	$-0.608906\pi$
$83$	−507.390	−0.671003	−0.335502	+	0.942040i	$0.608906\pi$
$84$	0	0
$85$	77.5248	0.0989264
$86$	−698.043	−0.875254
$87$	0	0
$88$	−175.388	−0.212459
$89$	565.891	0.673982	0.336991	−	0.941508i	$-0.390591\pi$
$89$	565.891	0.673982	0.336991	+	0.941508i	$0.390591\pi$
$90$	0	0
$91$	0	0
$92$	290.851	0.329601
$93$	0	0
$94$	745.999	0.818552
$95$	−232.214	−0.250786
$96$	0	0
$97$	1439.25	1.50654	0.753269	−	0.657713i	$-0.228476\pi$
$97$	1439.25	1.50654	0.753269	+	0.657713i	$0.228476\pi$
$98$	0	0
$99$	0	0
$100$	−153.203	−0.153203
$101$	−1865.23	−1.83760	−0.918798	−	0.394728i	$-0.870839\pi$
$101$	−1865.23	−1.83760	−0.918798	+	0.394728i	$0.870839\pi$
$102$	0	0
$103$	−436.752	−0.417810	−0.208905	−	0.977936i	$-0.566990\pi$
$103$	−436.752	−0.417810	−0.208905	+	0.977936i	$0.566990\pi$
$104$	52.1251	0.0491470
$105$	0	0
$106$	−5.92983	−0.00543355
$107$	2054.87	1.85656	0.928279	−	0.371883i	$-0.121288\pi$
$107$	2054.87	1.85656	0.928279	+	0.371883i	$0.121288\pi$
$108$	0	0
$109$	321.240	0.282286	0.141143	−	0.989989i	$-0.454922\pi$
$109$	321.240	0.282286	0.141143	+	0.989989i	$0.454922\pi$
$110$	62.0705	0.0538017
$111$	0	0
$112$	0	0
$113$	−762.909	−0.635119	−0.317559	−	0.948238i	$-0.602863\pi$
$113$	−762.909	−0.635119	−0.317559	+	0.948238i	$0.602863\pi$
$114$	0	0
$115$	−237.309	−0.192428
$116$	−50.3546	−0.0403043
$117$	0	0
$118$	1083.44	0.845246
$119$	0	0
$120$	0	0
$121$	−1248.67	−0.938146
$122$	−295.148	−0.219028
$123$	0	0
$124$	−1399.34	−1.01342
$125$	125.000	0.0894427
$126$	0	0
$127$	−696.163	−0.486413	−0.243207	−	0.969974i	$-0.578199\pi$
$127$	−696.163	−0.486413	−0.243207	+	0.969974i	$0.578199\pi$
$128$	1384.07	0.955747
$129$	0	0
$130$	−18.4473	−0.0124457
$131$	−682.734	−0.455349	−0.227675	−	0.973737i	$-0.573112\pi$
$131$	−682.734	−0.455349	−0.227675	+	0.973737i	$0.573112\pi$
$132$	0	0
$133$	0	0
$134$	−550.309	−0.354772
$135$	0	0
$136$	299.706	0.188968
$137$	414.139	0.258265	0.129132	−	0.991627i	$-0.458781\pi$
$137$	414.139	0.258265	0.129132	+	0.991627i	$0.458781\pi$
$138$	0	0
$139$	1224.87	0.747428	0.373714	−	0.927544i	$-0.378084\pi$
$139$	1224.87	0.747428	0.373714	+	0.927544i	$0.378084\pi$
$140$	0	0
$141$	0	0
$142$	−448.960	−0.265323
$143$	−24.4679	−0.0143084
$144$	0	0
$145$	41.0850	0.0235305
$146$	−716.726	−0.406278
$147$	0	0
$148$	2047.88	1.13740
$149$	2375.94	1.30634	0.653169	−	0.757212i	$-0.273439\pi$
$149$	2375.94	1.30634	0.653169	+	0.757212i	$0.273439\pi$
$150$	0	0
$151$	2797.29	1.50755	0.753777	−	0.657131i	$-0.228230\pi$
$151$	2797.29	1.50755	0.753777	+	0.657131i	$0.228230\pi$
$152$	−897.727	−0.479048
$153$	0	0
$154$	0	0
$155$	1141.74	0.591657
$156$	0	0
$157$	−1332.66	−0.677439	−0.338720	−	0.940887i	$-0.609994\pi$
$157$	−1332.66	−0.677439	−0.338720	+	0.940887i	$0.609994\pi$
$158$	178.954	0.0901066
$159$	0	0
$160$	−927.644	−0.458354
$161$	0	0
$162$	0	0
$163$	−2436.70	−1.17090	−0.585451	−	0.810708i	$-0.699083\pi$
$163$	−2436.70	−1.17090	−0.585451	+	0.810708i	$0.699083\pi$
$164$	−216.443	−0.103057
$165$	0	0
$166$	694.197	0.324579
$167$	2395.37	1.10994	0.554968	−	0.831871i	$-0.312730\pi$
$167$	2395.37	1.10994	0.554968	+	0.831871i	$0.312730\pi$
$168$	0	0
$169$	−2189.73	−0.996690
$170$	−106.067	−0.0478529
$171$	0	0
$172$	−3126.56	−1.38604
$173$	2940.08	1.29208	0.646041	−	0.763303i	$-0.276424\pi$
$173$	2940.08	1.29208	0.646041	+	0.763303i	$0.276424\pi$
$174$	0	0
$175$	0	0
$176$	−204.865	−0.0877403
$177$	0	0
$178$	−774.238	−0.326020
$179$	−3131.30	−1.30751	−0.653754	−	0.756707i	$-0.726807\pi$
$179$	−3131.30	−1.30751	−0.653754	+	0.756707i	$0.726807\pi$
$180$	0	0
$181$	393.898	0.161758	0.0808791	−	0.996724i	$-0.474227\pi$
$181$	393.898	0.161758	0.0808791	+	0.996724i	$0.474227\pi$
$182$	0	0
$183$	0	0
$184$	−917.421	−0.367572
$185$	−1670.90	−0.664036
$186$	0	0
$187$	−140.684	−0.0550152
$188$	3341.36	1.29624
$189$	0	0
$190$	317.710	0.121311
$191$	−4749.23	−1.79917	−0.899586	−	0.436743i	$-0.856132\pi$
$191$	−4749.23	−1.79917	−0.899586	+	0.436743i	$0.856132\pi$
$192$	0	0
$193$	−371.943	−0.138720	−0.0693602	−	0.997592i	$-0.522096\pi$
$193$	−371.943	−0.138720	−0.0693602	+	0.997592i	$0.522096\pi$
$194$	−1969.15	−0.728746
$195$	0	0
$196$	0	0
$197$	−732.895	−0.265059	−0.132529	−	0.991179i	$-0.542310\pi$
$197$	−732.895	−0.265059	−0.132529	+	0.991179i	$0.542310\pi$
$198$	0	0
$199$	−3117.09	−1.11038	−0.555188	−	0.831725i	$-0.687354\pi$
$199$	−3117.09	−1.11038	−0.555188	+	0.831725i	$0.687354\pi$
$200$	483.242	0.170852
$201$	0	0
$202$	2551.96	0.888886
$203$	0	0
$204$	0	0
$205$	176.599	0.0601667
$206$	597.552	0.202104
$207$	0	0
$208$	60.8858	0.0202965
$209$	421.399	0.139468
$210$	0	0
$211$	1368.10	0.446369	0.223184	−	0.974776i	$-0.428355\pi$
$211$	1368.10	0.446369	0.223184	+	0.974776i	$0.428355\pi$
$212$	−26.5599	−0.00860445
$213$	0	0
$214$	−2811.42	−0.898059
$215$	2551.00	0.809195
$216$	0	0
$217$	0	0
$218$	−439.512	−0.136548
$219$	0	0
$220$	278.016	0.0851993
$221$	41.8112	0.0127264
$222$	0	0
$223$	4161.15	1.24956	0.624779	−	0.780802i	$-0.285189\pi$
$223$	4161.15	1.24956	0.624779	+	0.780802i	$0.285189\pi$
$224$	0	0
$225$	0	0
$226$	1043.79	0.307221
$227$	−1834.99	−0.536530	−0.268265	−	0.963345i	$-0.586450\pi$
$227$	−1834.99	−0.536530	−0.268265	+	0.963345i	$0.586450\pi$
$228$	0	0
$229$	−1844.56	−0.532278	−0.266139	−	0.963935i	$-0.585748\pi$
$229$	−1844.56	−0.532278	−0.266139	+	0.963935i	$0.585748\pi$
$230$	324.680	0.0930815
$231$	0	0
$232$	158.832	0.0449475
$233$	−5346.58	−1.50329	−0.751644	−	0.659570i	$-0.770739\pi$
$233$	−5346.58	−1.50329	−0.751644	+	0.659570i	$0.770739\pi$
$234$	0	0
$235$	−2726.26	−0.756772
$236$	4852.78	1.33851
$237$	0	0
$238$	0	0
$239$	1406.13	0.380564	0.190282	−	0.981730i	$-0.439060\pi$
$239$	1406.13	0.380564	0.190282	+	0.981730i	$0.439060\pi$
$240$	0	0
$241$	−1800.27	−0.481185	−0.240593	−	0.970626i	$-0.577342\pi$
$241$	−1800.27	−0.481185	−0.240593	+	0.970626i	$0.577342\pi$
$242$	1708.40	0.453802
$243$	0	0
$244$	−1321.98	−0.346848
$245$	0	0
$246$	0	0
$247$	−125.239	−0.0322623
$248$	4413.90	1.13017
$249$	0	0
$250$	−171.022	−0.0432654
$251$	−2968.25	−0.746432	−0.373216	−	0.927744i	$-0.621745\pi$
$251$	−2968.25	−0.746432	−0.373216	+	0.927744i	$0.621745\pi$
$252$	0	0
$253$	430.644	0.107013
$254$	952.472	0.235289
$255$	0	0
$256$	−2479.31	−0.605300
$257$	−6383.78	−1.54945	−0.774726	−	0.632297i	$-0.782112\pi$
$257$	−6383.78	−1.54945	−0.774726	+	0.632297i	$0.782112\pi$
$258$	0	0
$259$	0	0
$260$	−82.6262	−0.0197087
$261$	0	0
$262$	934.098	0.220263
$263$	−7027.68	−1.64770	−0.823850	−	0.566808i	$-0.808178\pi$
$263$	−7027.68	−1.64770	−0.823850	+	0.566808i	$0.808178\pi$
$264$	0	0
$265$	21.6706	0.00502345
$266$	0	0
$267$	0	0
$268$	−2464.85	−0.561809
$269$	−4100.78	−0.929475	−0.464738	−	0.885448i	$-0.653851\pi$
$269$	−4100.78	−0.929475	−0.464738	+	0.885448i	$0.653851\pi$
$270$	0	0
$271$	8300.99	1.86070	0.930349	−	0.366676i	$-0.119504\pi$
$271$	8300.99	1.86070	0.930349	+	0.366676i	$0.119504\pi$
$272$	350.078	0.0780389
$273$	0	0
$274$	−566.614	−0.124928
$275$	−226.837	−0.0497411
$276$	0	0
$277$	−7315.61	−1.58683	−0.793416	−	0.608680i	$-0.791699\pi$
$277$	−7315.61	−1.58683	−0.793416	+	0.608680i	$0.791699\pi$
$278$	−1675.84	−0.361548
$279$	0	0
$280$	0	0
$281$	3004.11	0.637759	0.318880	−	0.947795i	$-0.396693\pi$
$281$	3004.11	0.637759	0.318880	+	0.947795i	$0.396693\pi$
$282$	0	0
$283$	−3658.45	−0.768453	−0.384226	−	0.923239i	$-0.625532\pi$
$283$	−3658.45	−0.768453	−0.384226	+	0.923239i	$0.625532\pi$
$284$	−2010.91	−0.420160
$285$	0	0
$286$	33.4763	0.00692131
$287$	0	0
$288$	0	0
$289$	−4672.60	−0.951068
$290$	−56.2113	−0.0113822
$291$	0	0
$292$	−3210.24	−0.643374
$293$	4069.16	0.811342	0.405671	−	0.914019i	$-0.367038\pi$
$293$	4069.16	0.811342	0.405671	+	0.914019i	$0.367038\pi$
$294$	0	0
$295$	−3959.45	−0.781451
$296$	−6459.58	−1.26843
$297$	0	0
$298$	−3250.69	−0.631905
$299$	−127.987	−0.0247548
$300$	0	0
$301$	0	0
$302$	−3827.18	−0.729237
$303$	0	0
$304$	−1048.61	−0.197835
$305$	1078.62	0.202497
$306$	0	0
$307$	−7902.82	−1.46918	−0.734589	−	0.678512i	$-0.762625\pi$
$307$	−7902.82	−1.46918	−0.734589	+	0.678512i	$0.762625\pi$
$308$	0	0
$309$	0	0
$310$	−1562.10	−0.286198
$311$	3031.83	0.552796	0.276398	−	0.961043i	$-0.410859\pi$
$311$	3031.83	0.552796	0.276398	+	0.961043i	$0.410859\pi$
$312$	0	0
$313$	328.926	0.0593993	0.0296996	−	0.999559i	$-0.490545\pi$
$313$	328.926	0.0593993	0.0296996	+	0.999559i	$0.490545\pi$
$314$	1823.31	0.327692
$315$	0	0
$316$	801.543	0.142691
$317$	−2871.25	−0.508724	−0.254362	−	0.967109i	$-0.581866\pi$
$317$	−2871.25	−0.508724	−0.254362	+	0.967109i	$0.581866\pi$
$318$	0	0
$319$	−74.5568	−0.0130858
$320$	366.040	0.0639445
$321$	0	0
$322$	0	0
$323$	−720.095	−0.124047
$324$	0	0
$325$	67.4158	0.0115063
$326$	3333.83	0.566392
$327$	0	0
$328$	682.719	0.114929
$329$	0	0
$330$	0	0
$331$	−9158.59	−1.52085	−0.760425	−	0.649425i	$-0.775010\pi$
$331$	−9158.59	−1.52085	−0.760425	+	0.649425i	$0.775010\pi$
$332$	3109.34	0.513997
$333$	0	0
$334$	−3277.28	−0.536901
$335$	2011.11	0.327996
$336$	0	0
$337$	−6200.00	−1.00218	−0.501091	−	0.865394i	$-0.667068\pi$
$337$	−6200.00	−1.00218	−0.501091	+	0.865394i	$0.667068\pi$
$338$	2995.93	0.482121
$339$	0	0
$340$	−475.080	−0.0757789
$341$	−2071.91	−0.329033
$342$	0	0
$343$	0	0
$344$	9862.02	1.54571
$345$	0	0
$346$	−4022.54	−0.625009
$347$	−3712.06	−0.574277	−0.287138	−	0.957889i	$-0.592704\pi$
$347$	−3712.06	−0.574277	−0.287138	+	0.957889i	$0.592704\pi$
$348$	0	0
$349$	619.951	0.0950865	0.0475433	−	0.998869i	$-0.484861\pi$
$349$	619.951	0.0950865	0.0475433	+	0.998869i	$0.484861\pi$
$350$	0	0
$351$	0	0
$352$	1683.39	0.254901
$353$	4175.57	0.629583	0.314792	−	0.949161i	$-0.398065\pi$
$353$	4175.57	0.629583	0.314792	+	0.949161i	$0.398065\pi$
$354$	0	0
$355$	1640.73	0.245298
$356$	−3467.84	−0.516279
$357$	0	0
$358$	4284.16	0.632471
$359$	−6211.84	−0.913227	−0.456614	−	0.889665i	$-0.650938\pi$
$359$	−6211.84	−0.913227	−0.456614	+	0.889665i	$0.650938\pi$
$360$	0	0
$361$	−4702.06	−0.685531
$362$	−538.921	−0.0782461
$363$	0	0
$364$	0	0
$365$	2619.28	0.375615
$366$	0	0
$367$	−4839.25	−0.688302	−0.344151	−	0.938914i	$-0.611833\pi$
$367$	−4839.25	−0.688302	−0.344151	+	0.938914i	$0.611833\pi$
$368$	−1071.61	−0.151798
$369$	0	0
$370$	2286.08	0.321209
$371$	0	0
$372$	0	0
$373$	11957.4	1.65987	0.829933	−	0.557863i	$-0.188378\pi$
$373$	11957.4	1.65987	0.829933	+	0.557863i	$0.188378\pi$
$374$	192.480	0.0266121
$375$	0	0
$376$	−10539.5	−1.44557
$377$	22.1582	0.00302707
$378$	0	0
$379$	−13037.0	−1.76692	−0.883462	−	0.468502i	$-0.844794\pi$
$379$	−13037.0	−1.76692	−0.883462	+	0.468502i	$0.844794\pi$
$380$	1423.03	0.192106
$381$	0	0
$382$	6497.76	0.870300
$383$	−4312.99	−0.575414	−0.287707	−	0.957718i	$-0.592893\pi$
$383$	−4312.99	−0.575414	−0.287707	+	0.957718i	$0.592893\pi$
$384$	0	0
$385$	0	0
$386$	508.882	0.0671021
$387$	0	0
$388$	−8819.89	−1.15403
$389$	8160.03	1.06357	0.531787	−	0.846878i	$-0.321521\pi$
$389$	8160.03	1.06357	0.531787	+	0.846878i	$0.321521\pi$
$390$	0	0
$391$	−735.893	−0.0951808
$392$	0	0
$393$	0	0
$394$	1002.73	0.128215
$395$	−653.990	−0.0833058
$396$	0	0
$397$	−2783.50	−0.351889	−0.175945	−	0.984400i	$-0.556298\pi$
$397$	−2783.50	−0.351889	−0.175945	+	0.984400i	$0.556298\pi$
$398$	4264.73	0.537114
$399$	0	0
$400$	564.461	0.0705576
$401$	1764.24	0.219705	0.109853	−	0.993948i	$-0.464962\pi$
$401$	1764.24	0.219705	0.109853	+	0.993948i	$0.464962\pi$
$402$	0	0
$403$	615.771	0.0761135
$404$	11430.3	1.40762
$405$	0	0
$406$	0	0
$407$	3032.17	0.369285
$408$	0	0
$409$	13570.8	1.64066	0.820332	−	0.571888i	$-0.193789\pi$
$409$	13570.8	1.64066	0.820332	+	0.571888i	$0.193789\pi$
$410$	−241.618	−0.0291040
$411$	0	0
$412$	2676.46	0.320048
$413$	0	0
$414$	0	0
$415$	−2536.95	−0.300082
$416$	−500.303	−0.0589648
$417$	0	0
$418$	−576.547	−0.0674637
$419$	12093.9	1.41008	0.705042	−	0.709166i	$-0.250928\pi$
$419$	12093.9	1.41008	0.705042	+	0.709166i	$0.250928\pi$
$420$	0	0
$421$	−6544.53	−0.757627	−0.378813	−	0.925473i	$-0.623668\pi$
$421$	−6544.53	−0.757627	−0.378813	+	0.925473i	$0.623668\pi$
$422$	−1871.80	−0.215919
$423$	0	0
$424$	83.7772	0.00959571
$425$	387.624	0.0442412
$426$	0	0
$427$	0	0
$428$	−12592.5	−1.42215
$429$	0	0
$430$	−3490.21	−0.391426
$431$	3580.10	0.400110	0.200055	−	0.979785i	$-0.435888\pi$
$431$	3580.10	0.400110	0.200055	+	0.979785i	$0.435888\pi$
$432$	0	0
$433$	14612.4	1.62178	0.810888	−	0.585202i	$-0.198985\pi$
$433$	14612.4	1.62178	0.810888	+	0.585202i	$0.198985\pi$
$434$	0	0
$435$	0	0
$436$	−1968.59	−0.216235
$437$	2204.26	0.241291
$438$	0	0
$439$	−11573.5	−1.25825	−0.629126	−	0.777303i	$-0.716587\pi$
$439$	−11573.5	−1.25825	−0.629126	+	0.777303i	$0.716587\pi$
$440$	−876.939	−0.0950146
$441$	0	0
$442$	−57.2049	−0.00615602
$443$	−16288.7	−1.74695	−0.873477	−	0.486865i	$-0.838140\pi$
$443$	−16288.7	−1.74695	−0.873477	+	0.486865i	$0.838140\pi$
$444$	0	0
$445$	2829.46	0.301414
$446$	−5693.18	−0.604439
$447$	0	0
$448$	0	0
$449$	−4951.02	−0.520385	−0.260193	−	0.965557i	$-0.583786\pi$
$449$	−4951.02	−0.520385	−0.260193	+	0.965557i	$0.583786\pi$
$450$	0	0
$451$	−320.473	−0.0334601
$452$	4675.18	0.486509
$453$	0	0
$454$	2510.58	0.259532
$455$	0	0
$456$	0	0
$457$	−6438.28	−0.659015	−0.329508	−	0.944153i	$-0.606883\pi$
$457$	−6438.28	−0.659015	−0.329508	+	0.944153i	$0.606883\pi$
$458$	2523.67	0.257475
$459$	0	0
$460$	1454.25	0.147402
$461$	−5615.41	−0.567323	−0.283661	−	0.958925i	$-0.591549\pi$
$461$	−5615.41	−0.567323	−0.283661	+	0.958925i	$0.591549\pi$
$462$	0	0
$463$	4355.36	0.437173	0.218586	−	0.975818i	$-0.429855\pi$
$463$	4355.36	0.437173	0.218586	+	0.975818i	$0.429855\pi$
$464$	185.527	0.0185622
$465$	0	0
$466$	7315.04	0.727173
$467$	−10550.8	−1.04547	−0.522735	−	0.852495i	$-0.675088\pi$
$467$	−10550.8	−1.04547	−0.522735	+	0.852495i	$0.675088\pi$
$468$	0	0
$469$	0	0
$470$	3729.99	0.366068
$471$	0	0
$472$	−15307.0	−1.49272
$473$	−4629.30	−0.450011
$474$	0	0
$475$	−1161.07	−0.112155
$476$	0	0
$477$	0	0
$478$	−1923.82	−0.184087
$479$	−10697.4	−1.02041	−0.510207	−	0.860051i	$-0.670431\pi$
$479$	−10697.4	−1.02041	−0.510207	+	0.860051i	$0.670431\pi$
$480$	0	0
$481$	−901.159	−0.0854247
$482$	2463.08	0.232760
$483$	0	0
$484$	7651.99	0.718631
$485$	7196.27	0.673744
$486$	0	0
$487$	2411.53	0.224388	0.112194	−	0.993686i	$-0.464212\pi$
$487$	2411.53	0.224388	0.112194	+	0.993686i	$0.464212\pi$
$488$	4169.88	0.386806
$489$	0	0
$490$	0	0
$491$	−20725.7	−1.90496	−0.952480	−	0.304601i	$-0.901477\pi$
$491$	−20725.7	−1.90496	−0.952480	+	0.304601i	$0.901477\pi$
$492$	0	0
$493$	127.404	0.0116389
$494$	171.349	0.0156060
$495$	0	0
$496$	5155.74	0.466733
$497$	0	0
$498$	0	0
$499$	−16507.1	−1.48088	−0.740442	−	0.672120i	$-0.765384\pi$
$499$	−16507.1	−1.48088	−0.740442	+	0.672120i	$0.765384\pi$
$500$	−766.013	−0.0685143
$501$	0	0
$502$	4061.09	0.361066
$503$	−7943.62	−0.704152	−0.352076	−	0.935971i	$-0.614524\pi$
$503$	−7943.62	−0.704152	−0.352076	+	0.935971i	$0.614524\pi$
$504$	0	0
$505$	−9326.14	−0.821798
$506$	−589.195	−0.0517647
$507$	0	0
$508$	4266.16	0.372599
$509$	−8134.56	−0.708365	−0.354183	−	0.935176i	$-0.615241\pi$
$509$	−8134.56	−0.708365	−0.354183	+	0.935176i	$0.615241\pi$
$510$	0	0
$511$	0	0
$512$	−7680.43	−0.662949
$513$	0	0
$514$	8734.12	0.749505
$515$	−2183.76	−0.186850
$516$	0	0
$517$	4947.34	0.420858
$518$	0	0
$519$	0	0
$520$	260.625	0.0219792
$521$	12838.3	1.07957	0.539785	−	0.841803i	$-0.318505\pi$
$521$	12838.3	1.07957	0.539785	+	0.841803i	$0.318505\pi$
$522$	0	0
$523$	8127.07	0.679487	0.339744	−	0.940518i	$-0.389660\pi$
$523$	8127.07	0.679487	0.339744	+	0.940518i	$0.389660\pi$
$524$	4183.86	0.348803
$525$	0	0
$526$	9615.08	0.797029
$527$	3540.53	0.292652
$528$	0	0
$529$	−9914.38	−0.814858
$530$	−29.6491	−0.00242996
$531$	0	0
$532$	0	0
$533$	95.2443	0.00774013
$534$	0	0
$535$	10274.4	0.830278
$536$	7774.82	0.626531
$537$	0	0
$538$	5610.57	0.449608
$539$	0	0
$540$	0	0
$541$	8403.45	0.667823	0.333912	−	0.942604i	$-0.391631\pi$
$541$	8403.45	0.667823	0.333912	+	0.942604i	$0.391631\pi$
$542$	−11357.2	−0.900061
$543$	0	0
$544$	−2876.62	−0.226717
$545$	1606.20	0.126242
$546$	0	0
$547$	15317.1	1.19728	0.598639	−	0.801019i	$-0.295709\pi$
$547$	15317.1	1.19728	0.598639	+	0.801019i	$0.295709\pi$
$548$	−2537.88	−0.197834
$549$	0	0
$550$	310.353	0.0240609
$551$	−381.621	−0.0295056
$552$	0	0
$553$	0	0
$554$	10009.0	0.767586
$555$	0	0
$556$	−7506.16	−0.572540
$557$	−19822.3	−1.50790	−0.753948	−	0.656934i	$-0.771853\pi$
$557$	−19822.3	−1.50790	−0.753948	+	0.656934i	$0.771853\pi$
$558$	0	0
$559$	1375.82	0.104099
$560$	0	0
$561$	0	0
$562$	−4110.15	−0.308498
$563$	10991.1	0.822773	0.411386	−	0.911461i	$-0.365045\pi$
$563$	10991.1	0.822773	0.411386	+	0.911461i	$0.365045\pi$
$564$	0	0
$565$	−3814.54	−0.284034
$566$	5005.39	0.371718
$567$	0	0
$568$	6342.95	0.468564
$569$	−1614.53	−0.118954	−0.0594768	−	0.998230i	$-0.518943\pi$
$569$	−1614.53	−0.118954	−0.0594768	+	0.998230i	$0.518943\pi$
$570$	0	0
$571$	−4264.17	−0.312522	−0.156261	−	0.987716i	$-0.549944\pi$
$571$	−4264.17	−0.312522	−0.156261	+	0.987716i	$0.549944\pi$
$572$	149.942	0.0109604
$573$	0	0
$574$	0	0
$575$	−1186.54	−0.0860562
$576$	0	0
$577$	−2540.18	−0.183274	−0.0916371	−	0.995792i	$-0.529210\pi$
$577$	−2540.18	−0.183274	−0.0916371	+	0.995792i	$0.529210\pi$
$578$	6392.92	0.460053
$579$	0	0
$580$	−251.773	−0.0180246
$581$	0	0
$582$	0	0
$583$	−39.3256	−0.00279365
$584$	10126.0	0.717493
$585$	0	0
$586$	−5567.32	−0.392464
$587$	−21188.9	−1.48988	−0.744940	−	0.667131i	$-0.767522\pi$
$587$	−21188.9	−1.48988	−0.744940	+	0.667131i	$0.767522\pi$
$588$	0	0
$589$	−10605.1	−0.741897
$590$	5417.22	0.378006
$591$	0	0
$592$	−7545.25	−0.523831
$593$	−10098.7	−0.699329	−0.349665	−	0.936875i	$-0.613704\pi$
$593$	−10098.7	−0.699329	−0.349665	+	0.936875i	$0.613704\pi$
$594$	0	0
$595$	0	0
$596$	−14560.0	−1.00067
$597$	0	0
$598$	175.108	0.0119744
$599$	−2080.47	−0.141913	−0.0709564	−	0.997479i	$-0.522605\pi$
$599$	−2080.47	−0.141913	−0.0709564	+	0.997479i	$0.522605\pi$
$600$	0	0
$601$	15577.1	1.05724	0.528621	−	0.848858i	$-0.322709\pi$
$601$	15577.1	1.05724	0.528621	+	0.848858i	$0.322709\pi$
$602$	0	0
$603$	0	0
$604$	−17142.1	−1.15481
$605$	−6243.36	−0.419551
$606$	0	0
$607$	−18533.6	−1.23930	−0.619650	−	0.784878i	$-0.712725\pi$
$607$	−18533.6	−1.23930	−0.619650	+	0.784878i	$0.712725\pi$
$608$	8616.49	0.574745
$609$	0	0
$610$	−1475.74	−0.0979523
$611$	−1470.34	−0.0973547
$612$	0	0
$613$	−12650.7	−0.833538	−0.416769	−	0.909012i	$-0.636838\pi$
$613$	−12650.7	−0.833538	−0.416769	+	0.909012i	$0.636838\pi$
$614$	10812.4	0.710675
$615$	0	0
$616$	0	0
$617$	795.348	0.0518955	0.0259477	−	0.999663i	$-0.491740\pi$
$617$	795.348	0.0518955	0.0259477	+	0.999663i	$0.491740\pi$
$618$	0	0
$619$	24655.7	1.60097	0.800483	−	0.599356i	$-0.204576\pi$
$619$	24655.7	1.60097	0.800483	+	0.599356i	$0.204576\pi$
$620$	−6996.70	−0.453217
$621$	0	0
$622$	−4148.07	−0.267400
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−450.027	−0.0287328
$627$	0	0
$628$	8166.68	0.518927
$629$	−5181.43	−0.328454
$630$	0	0
$631$	−12153.3	−0.766743	−0.383371	−	0.923594i	$-0.625237\pi$
$631$	−12153.3	−0.766743	−0.383371	+	0.923594i	$0.625237\pi$
$632$	−2528.28	−0.159129
$633$	0	0
$634$	3928.37	0.246081
$635$	−3480.82	−0.217531
$636$	0	0
$637$	0	0
$638$	102.007	0.00632990
$639$	0	0
$640$	6920.34	0.427423
$641$	−10437.7	−0.643158	−0.321579	−	0.946883i	$-0.604214\pi$
$641$	−10437.7	−0.643158	−0.321579	+	0.946883i	$0.604214\pi$
$642$	0	0
$643$	−16184.5	−0.992621	−0.496311	−	0.868145i	$-0.665312\pi$
$643$	−16184.5	−0.992621	−0.496311	+	0.868145i	$0.665312\pi$
$644$	0	0
$645$	0	0
$646$	985.215	0.0600043
$647$	−26891.0	−1.63399	−0.816997	−	0.576642i	$-0.804363\pi$
$647$	−26891.0	−1.63399	−0.816997	+	0.576642i	$0.804363\pi$
$648$	0	0
$649$	7185.21	0.434583
$650$	−92.2365	−0.00556587
$651$	0	0
$652$	14932.4	0.896926
$653$	5184.33	0.310687	0.155343	−	0.987861i	$-0.450352\pi$
$653$	5184.33	0.310687	0.155343	+	0.987861i	$0.450352\pi$
$654$	0	0
$655$	−3413.67	−0.203638
$656$	797.464	0.0474630
$657$	0	0
$658$	0	0
$659$	30149.8	1.78220	0.891099	−	0.453809i	$-0.149935\pi$
$659$	30149.8	1.78220	0.891099	+	0.453809i	$0.149935\pi$
$660$	0	0
$661$	15182.4	0.893384	0.446692	−	0.894688i	$-0.352602\pi$
$661$	15182.4	0.893384	0.446692	+	0.894688i	$0.352602\pi$
$662$	12530.5	0.735670
$663$	0	0
$664$	−9807.69	−0.573211
$665$	0	0
$666$	0	0
$667$	−389.993	−0.0226396
$668$	−14679.1	−0.850226
$669$	0	0
$670$	−2751.54	−0.158659
$671$	−1957.37	−0.112613
$672$	0	0
$673$	12549.6	0.718799	0.359399	−	0.933184i	$-0.382982\pi$
$673$	12549.6	0.718799	0.359399	+	0.933184i	$0.382982\pi$
$674$	8482.67	0.484778
$675$	0	0
$676$	13418.9	0.763477
$677$	18743.3	1.06405	0.532026	−	0.846728i	$-0.321431\pi$
$677$	18743.3	1.06405	0.532026	+	0.846728i	$0.321431\pi$
$678$	0	0
$679$	0	0
$680$	1498.53	0.0845088
$681$	0	0
$682$	2834.74	0.159161
$683$	−25226.0	−1.41324	−0.706622	−	0.707591i	$-0.749782\pi$
$683$	−25226.0	−1.41324	−0.706622	+	0.707591i	$0.749782\pi$
$684$	0	0
$685$	2070.69	0.115499
$686$	0	0
$687$	0	0
$688$	11519.5	0.638340
$689$	11.6875	0.000646240 0
$690$	0	0
$691$	19872.5	1.09404	0.547022	−	0.837118i	$-0.315761\pi$
$691$	19872.5	1.09404	0.547022	+	0.837118i	$0.315761\pi$
$692$	−18017.1	−0.989751
$693$	0	0
$694$	5078.75	0.277790
$695$	6124.37	0.334260
$696$	0	0
$697$	547.631	0.0297604
$698$	−848.200	−0.0459955
$699$	0	0
$700$	0	0
$701$	−25586.8	−1.37860	−0.689301	−	0.724475i	$-0.742082\pi$
$701$	−25586.8	−1.37860	−0.689301	+	0.724475i	$0.742082\pi$
$702$	0	0
$703$	15520.2	0.832656
$704$	−664.251	−0.0355610
$705$	0	0
$706$	−5712.90	−0.304544
$707$	0	0
$708$	0	0
$709$	−23494.2	−1.24449	−0.622244	−	0.782824i	$-0.713779\pi$
$709$	−23494.2	−1.24449	−0.622244	+	0.782824i	$0.713779\pi$
$710$	−2244.80	−0.118656
$711$	0	0
$712$	10938.5	0.575755
$713$	−10837.8	−0.569255
$714$	0	0
$715$	−122.339	−0.00639893
$716$	19188.9	1.00157
$717$	0	0
$718$	8498.88	0.441748
$719$	14096.2	0.731156	0.365578	−	0.930781i	$-0.380871\pi$
$719$	14096.2	0.731156	0.365578	+	0.930781i	$0.380871\pi$
$720$	0	0
$721$	0	0
$722$	6433.23	0.331607
$723$	0	0
$724$	−2413.85	−0.123909
$725$	205.425	0.0105232
$726$	0	0
$727$	−25807.9	−1.31659	−0.658296	−	0.752759i	$-0.728722\pi$
$727$	−25807.9	−1.31659	−0.658296	+	0.752759i	$0.728722\pi$
$728$	0	0
$729$	0	0
$730$	−3583.63	−0.181693
$731$	7910.64	0.400254
$732$	0	0
$733$	25684.8	1.29425	0.647127	−	0.762382i	$-0.275970\pi$
$733$	25684.8	1.29425	0.647127	+	0.762382i	$0.275970\pi$
$734$	6620.93	0.332947
$735$	0	0
$736$	8805.52	0.441000
$737$	−3649.55	−0.182406
$738$	0	0
$739$	15928.6	0.792885	0.396443	−	0.918060i	$-0.370245\pi$
$739$	15928.6	0.792885	0.396443	+	0.918060i	$0.370245\pi$
$740$	10239.4	0.508660
$741$	0	0
$742$	0	0
$743$	9733.51	0.480603	0.240301	−	0.970698i	$-0.422754\pi$
$743$	9733.51	0.480603	0.240301	+	0.970698i	$0.422754\pi$
$744$	0	0
$745$	11879.7	0.584212
$746$	−16359.8	−0.802914
$747$	0	0
$748$	862.126	0.0421423
$749$	0	0
$750$	0	0
$751$	21805.5	1.05951	0.529756	−	0.848150i	$-0.322283\pi$
$751$	21805.5	1.05951	0.529756	+	0.848150i	$0.322283\pi$
$752$	−12310.9	−0.596986
$753$	0	0
$754$	−30.3163	−0.00146426
$755$	13986.5	0.674198
$756$	0	0
$757$	−5231.20	−0.251164	−0.125582	−	0.992083i	$-0.540080\pi$
$757$	−5231.20	−0.251164	−0.125582	+	0.992083i	$0.540080\pi$
$758$	17836.8	0.854701
$759$	0	0
$760$	−4488.63	−0.214237
$761$	−23557.2	−1.12214	−0.561069	−	0.827769i	$-0.689610\pi$
$761$	−23557.2	−1.12214	−0.561069	+	0.827769i	$0.689610\pi$
$762$	0	0
$763$	0	0
$764$	29103.7	1.37819
$765$	0	0
$766$	5900.92	0.278341
$767$	−2135.44	−0.100530
$768$	0	0
$769$	7441.48	0.348955	0.174478	−	0.984661i	$-0.444176\pi$
$769$	7441.48	0.348955	0.174478	+	0.984661i	$0.444176\pi$
$770$	0	0
$771$	0	0
$772$	2279.30	0.106262
$773$	16903.7	0.786523	0.393262	−	0.919427i	$-0.371347\pi$
$773$	16903.7	0.786523	0.393262	+	0.919427i	$0.371347\pi$
$774$	0	0
$775$	5708.70	0.264597
$776$	27820.3	1.28697
$777$	0	0
$778$	−11164.3	−0.514474
$779$	−1640.35	−0.0754450
$780$	0	0
$781$	−2977.43	−0.136416
$782$	1006.83	0.0460411
$783$	0	0
$784$	0	0
$785$	−6663.31	−0.302960
$786$	0	0
$787$	13644.3	0.618003	0.309001	−	0.951062i	$-0.400005\pi$
$787$	13644.3	0.618003	0.309001	+	0.951062i	$0.400005\pi$
$788$	4491.25	0.203038
$789$	0	0
$790$	894.771	0.0402969
$791$	0	0
$792$	0	0
$793$	581.728	0.0260502
$794$	3808.32	0.170217
$795$	0	0
$796$	19101.9	0.850563
$797$	−35027.5	−1.55676	−0.778379	−	0.627795i	$-0.783958\pi$
$797$	−35027.5	−1.55676	−0.778379	+	0.627795i	$0.783958\pi$
$798$	0	0
$799$	−8454.10	−0.374324
$800$	−4638.22	−0.204982
$801$	0	0
$802$	−2413.78	−0.106276
$803$	−4753.20	−0.208888
$804$	0	0
$805$	0	0
$806$	−842.481	−0.0368178
$807$	0	0
$808$	−36054.3	−1.56978
$809$	40226.0	1.74817	0.874087	−	0.485770i	$-0.161461\pi$
$809$	40226.0	1.74817	0.874087	+	0.485770i	$0.161461\pi$
$810$	0	0
$811$	−16374.0	−0.708963	−0.354482	−	0.935063i	$-0.615343\pi$
$811$	−16374.0	−0.708963	−0.354482	+	0.935063i	$0.615343\pi$
$812$	0	0
$813$	0	0
$814$	−4148.54	−0.178632
$815$	−12183.5	−0.523644
$816$	0	0
$817$	−23695.2	−1.01468
$818$	−18567.2	−0.793625
$819$	0	0
$820$	−1082.21	−0.0460885
$821$	27603.5	1.17341	0.586705	−	0.809801i	$-0.300425\pi$
$821$	27603.5	1.17341	0.586705	+	0.809801i	$0.300425\pi$
$822$	0	0
$823$	8941.48	0.378713	0.189356	−	0.981908i	$-0.439360\pi$
$823$	8941.48	0.378713	0.189356	+	0.981908i	$0.439360\pi$
$824$	−8442.28	−0.356918
$825$	0	0
$826$	0	0
$827$	−36920.4	−1.55242	−0.776208	−	0.630476i	$-0.782860\pi$
$827$	−36920.4	−1.55242	−0.776208	+	0.630476i	$0.782860\pi$
$828$	0	0
$829$	−3222.22	−0.134997	−0.0674983	−	0.997719i	$-0.521502\pi$
$829$	−3222.22	−0.134997	−0.0674983	+	0.997719i	$0.521502\pi$
$830$	3470.99	0.145156
$831$	0	0
$832$	197.415	0.00822612
$833$	0	0
$834$	0	0
$835$	11976.9	0.496379
$836$	−2582.38	−0.106834
$837$	0	0
$838$	−16546.5	−0.682089
$839$	8534.92	0.351201	0.175601	−	0.984461i	$-0.443813\pi$
$839$	8534.92	0.351201	0.175601	+	0.984461i	$0.443813\pi$
$840$	0	0
$841$	−24321.5	−0.997232
$842$	8954.05	0.366481
$843$	0	0
$844$	−8383.85	−0.341924
$845$	−10948.6	−0.445733
$846$	0	0
$847$	0	0
$848$	97.8577	0.00396279
$849$	0	0
$850$	−530.337	−0.0214005
$851$	15860.7	0.638894
$852$	0	0
$853$	−12096.5	−0.485553	−0.242776	−	0.970082i	$-0.578058\pi$
$853$	−12096.5	−0.485553	−0.242776	+	0.970082i	$0.578058\pi$
$854$	0	0
$855$	0	0
$856$	39720.0	1.58598
$857$	−17873.6	−0.712427	−0.356213	−	0.934405i	$-0.615932\pi$
$857$	−17873.6	−0.712427	−0.356213	+	0.934405i	$0.615932\pi$
$858$	0	0
$859$	23738.0	0.942875	0.471438	−	0.881899i	$-0.343735\pi$
$859$	23738.0	0.942875	0.471438	+	0.881899i	$0.343735\pi$
$860$	−15632.8	−0.619854
$861$	0	0
$862$	−4898.20	−0.193542
$863$	−30271.5	−1.19404	−0.597019	−	0.802227i	$-0.703648\pi$
$863$	−30271.5	−1.19404	−0.597019	+	0.802227i	$0.703648\pi$
$864$	0	0
$865$	14700.4	0.577836
$866$	−19992.3	−0.784489
$867$	0	0
$868$	0	0
$869$	1186.79	0.0463282
$870$	0	0
$871$	1084.64	0.0421949
$872$	6209.47	0.241146
$873$	0	0
$874$	−3015.81	−0.116718
$875$	0	0
$876$	0	0
$877$	28444.7	1.09522	0.547611	−	0.836733i	$-0.315537\pi$
$877$	28444.7	1.09522	0.547611	+	0.836733i	$0.315537\pi$
$878$	15834.5	0.608645
$879$	0	0
$880$	−1024.33	−0.0392387
$881$	−6712.92	−0.256713	−0.128356	−	0.991728i	$-0.540970\pi$
$881$	−6712.92	−0.256713	−0.128356	+	0.991728i	$0.540970\pi$
$882$	0	0
$883$	21771.1	0.829735	0.414867	−	0.909882i	$-0.363828\pi$
$883$	21771.1	0.829735	0.414867	+	0.909882i	$0.363828\pi$
$884$	−256.223	−0.00974855
$885$	0	0
$886$	22285.8	0.845041
$887$	45189.5	1.71061	0.855307	−	0.518121i	$-0.173368\pi$
$887$	45189.5	1.71061	0.855307	+	0.518121i	$0.173368\pi$
$888$	0	0
$889$	0	0
$890$	−3871.19	−0.145801
$891$	0	0
$892$	−25500.0	−0.957177
$893$	25323.1	0.948941
$894$	0	0
$895$	−15656.5	−0.584736
$896$	0	0
$897$	0	0
$898$	6773.85	0.251722
$899$	1876.33	0.0696098
$900$	0	0
$901$	67.2004	0.00248476
$902$	438.463	0.0161854
$903$	0	0
$904$	−14746.8	−0.542556
$905$	1969.49	0.0723405
$906$	0	0
$907$	3677.44	0.134628	0.0673139	−	0.997732i	$-0.478557\pi$
$907$	3677.44	0.134628	0.0673139	+	0.997732i	$0.478557\pi$
$908$	11245.0	0.410989
$909$	0	0
$910$	0	0
$911$	45601.1	1.65843	0.829217	−	0.558927i	$-0.188787\pi$
$911$	45601.1	1.65843	0.829217	+	0.558927i	$0.188787\pi$
$912$	0	0
$913$	4603.80	0.166882
$914$	8808.68	0.318780
$915$	0	0
$916$	11303.6	0.407732
$917$	0	0
$918$	0	0
$919$	5418.51	0.194494	0.0972471	−	0.995260i	$-0.468996\pi$
$919$	5418.51	0.194494	0.0972471	+	0.995260i	$0.468996\pi$
$920$	−4587.11	−0.164383
$921$	0	0
$922$	7682.86	0.274427
$923$	884.888	0.0315563
$924$	0	0
$925$	−8354.48	−0.296966
$926$	−5958.89	−0.211470
$927$	0	0
$928$	−1524.49	−0.0539265
$929$	10022.6	0.353964	0.176982	−	0.984214i	$-0.443367\pi$
$929$	10022.6	0.353964	0.176982	+	0.984214i	$0.443367\pi$
$930$	0	0
$931$	0	0
$932$	32764.4	1.15154
$933$	0	0
$934$	14435.4	0.505717
$935$	−703.420	−0.0246035
$936$	0	0
$937$	25723.9	0.896865	0.448432	−	0.893817i	$-0.351982\pi$
$937$	25723.9	0.896865	0.448432	+	0.893817i	$0.351982\pi$
$938$	0	0
$939$	0	0
$940$	16706.8	0.579697
$941$	19300.0	0.668609	0.334305	−	0.942465i	$-0.391499\pi$
$941$	19300.0	0.668609	0.334305	+	0.942465i	$0.391499\pi$
$942$	0	0
$943$	−1676.34	−0.0578887
$944$	−17879.7	−0.616455
$945$	0	0
$946$	6333.68	0.217681
$947$	38253.3	1.31263	0.656317	−	0.754486i	$-0.272114\pi$
$947$	38253.3	1.31263	0.656317	+	0.754486i	$0.272114\pi$
$948$	0	0
$949$	1412.65	0.0483208
$950$	1588.55	0.0542519
$951$	0	0
$952$	0	0
$953$	−29645.9	−1.00769	−0.503843	−	0.863795i	$-0.668081\pi$
$953$	−29645.9	−1.00769	−0.503843	+	0.863795i	$0.668081\pi$
$954$	0	0
$955$	−23746.1	−0.804614
$956$	−8616.88	−0.291517
$957$	0	0
$958$	14636.0	0.493597
$959$	0	0
$960$	0	0
$961$	22351.8	0.750288
$962$	1232.94	0.0413218
$963$	0	0
$964$	11032.2	0.368594
$965$	−1859.71	−0.0620376
$966$	0	0
$967$	13170.0	0.437970	0.218985	−	0.975728i	$-0.429725\pi$
$967$	13170.0	0.437970	0.218985	+	0.975728i	$0.429725\pi$
$968$	−24136.4	−0.801420
$969$	0	0
$970$	−9845.74	−0.325905
$971$	33984.6	1.12319	0.561595	−	0.827413i	$-0.310188\pi$
$971$	33984.6	1.12319	0.561595	+	0.827413i	$0.310188\pi$
$972$	0	0
$973$	0	0
$974$	−3299.40	−0.108542
$975$	0	0
$976$	4870.71	0.159741
$977$	−28253.2	−0.925178	−0.462589	−	0.886573i	$-0.653079\pi$
$977$	−28253.2	−0.925178	−0.462589	+	0.886573i	$0.653079\pi$
$978$	0	0
$979$	−5134.61	−0.167623
$980$	0	0
$981$	0	0
$982$	28356.3	0.921472
$983$	38888.6	1.26180	0.630902	−	0.775863i	$-0.282685\pi$
$983$	38888.6	1.26180	0.630902	+	0.775863i	$0.282685\pi$
$984$	0	0
$985$	−3664.47	−0.118538
$986$	−174.311	−0.00563001
$987$	0	0
$988$	767.480	0.0247134
$989$	−24215.0	−0.778557
$990$	0	0
$991$	10093.5	0.323542	0.161771	−	0.986828i	$-0.448279\pi$
$991$	10093.5	0.323542	0.161771	+	0.986828i	$0.448279\pi$
$992$	−42365.1	−1.35594
$993$	0	0
$994$	0	0
$995$	−15585.5	−0.496576
$996$	0	0
$997$	−44247.3	−1.40554	−0.702771	−	0.711416i	$-0.748054\pi$
$997$	−44247.3	−1.40554	−0.702771	+	0.711416i	$0.748054\pi$
$998$	22584.6	0.716336
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bj.1.2		3
3.2	odd	2		735.4.a.s.1.2		3
7.3	odd	6		315.4.j.e.226.2		6
7.5	odd	6		315.4.j.e.46.2		6
7.6	odd	2		2205.4.a.bi.1.2		3
21.5	even	6		105.4.i.c.46.2	yes	6
21.17	even	6		105.4.i.c.16.2	✓	6
21.20	even	2		735.4.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.c.16.2	✓	6	21.17	even	6
105.4.i.c.46.2	yes	6	21.5	even	6
315.4.j.e.46.2		6	7.5	odd	6
315.4.j.e.226.2		6	7.3	odd	6
735.4.a.r.1.2		3	21.20	even	2
735.4.a.s.1.2		3	3.2	odd	2
2205.4.a.bi.1.2		3	7.6	odd	2
2205.4.a.bj.1.2		3	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-1.36817 q^{2} -6.12810 q^{4} +5.00000 q^{5} +19.3297 q^{8} +O(q^{10})\)	\(q-1.36817 q^{2} -6.12810 q^{4} +5.00000 q^{5} +19.3297 q^{8} -6.84087 q^{10} -9.07349 q^{11} +2.69663 q^{13} +22.5784 q^{16} +15.5050 q^{17} -46.4429 q^{19} -30.6405 q^{20} +12.4141 q^{22} -47.4618 q^{23} +25.0000 q^{25} -3.68946 q^{26} +8.21699 q^{29} +228.348 q^{31} -185.529 q^{32} -21.2135 q^{34} -334.179 q^{37} +63.5419 q^{38} +96.6485 q^{40} +35.3197 q^{41} +510.201 q^{43} +55.6033 q^{44} +64.9359 q^{46} -545.252 q^{47} -34.2043 q^{50} -16.5252 q^{52} +4.33412 q^{53} -45.3674 q^{55} -11.2423 q^{58} -791.890 q^{59} +215.724 q^{61} -312.420 q^{62} +73.2079 q^{64} +13.4832 q^{65} +402.221 q^{67} -95.0160 q^{68} +328.146 q^{71} +523.856 q^{73} +457.215 q^{74} +284.607 q^{76} -130.798 q^{79} +112.892 q^{80} -48.3235 q^{82} -507.390 q^{83} +77.5248 q^{85} -698.043 q^{86} -175.388 q^{88} +565.891 q^{89} +290.851 q^{92} +745.999 q^{94} -232.214 q^{95} +1439.25 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{4} + 15 q^{5} - 9 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + q^4 + 15 * q^5 - 9 * q^8	\( 3 q - 3 q^{2} + q^{4} + 15 q^{5} - 9 q^{8} - 15 q^{10} + q^{11} + 79 q^{13} - 79 q^{16} - 72 q^{17} + 29 q^{19} + 5 q^{20} + 143 q^{22} + 63 q^{23} + 75 q^{25} - 339 q^{26} - 220 q^{29} - 136 q^{31} + 155 q^{32} + 220 q^{34} + 43 q^{37} - 21 q^{38} - 45 q^{40} - 599 q^{41} + 170 q^{43} - 135 q^{44} + 265 q^{46} - 3 q^{47} - 75 q^{50} + 701 q^{52} - 331 q^{53} + 5 q^{55} - 472 q^{58} - 1520 q^{59} + 1160 q^{61} - 748 q^{62} + 17 q^{64} + 395 q^{65} + 806 q^{67} - 684 q^{68} + 406 q^{71} + 1192 q^{73} - 959 q^{74} + 591 q^{76} - 2590 q^{79} - 395 q^{80} + 1191 q^{82} - 508 q^{83} - 360 q^{85} + 742 q^{86} - 749 q^{88} + 42 q^{89} + 211 q^{92} + 1167 q^{94} + 145 q^{95} + 1020 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + q^4 + 15 * q^5 - 9 * q^8 - 15 * q^10 + q^11 + 79 * q^13 - 79 * q^16 - 72 * q^17 + 29 * q^19 + 5 * q^20 + 143 * q^22 + 63 * q^23 + 75 * q^25 - 339 * q^26 - 220 * q^29 - 136 * q^31 + 155 * q^32 + 220 * q^34 + 43 * q^37 - 21 * q^38 - 45 * q^40 - 599 * q^41 + 170 * q^43 - 135 * q^44 + 265 * q^46 - 3 * q^47 - 75 * q^50 + 701 * q^52 - 331 * q^53 + 5 * q^55 - 472 * q^58 - 1520 * q^59 + 1160 * q^61 - 748 * q^62 + 17 * q^64 + 395 * q^65 + 806 * q^67 - 684 * q^68 + 406 * q^71 + 1192 * q^73 - 959 * q^74 + 591 * q^76 - 2590 * q^79 - 395 * q^80 + 1191 * q^82 - 508 * q^83 - 360 * q^85 + 742 * q^86 - 749 * q^88 + 42 * q^89 + 211 * q^92 + 1167 * q^94 + 145 * q^95 + 1020 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more