LMFDB - Embedded newform 495.6.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.34253.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 52x + 48 $ x^3 - x^2 - 52*x + 48
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$7.25531$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.25531 q^{2} +53.6607 q^{4} -25.0000 q^{5} +36.4478 q^{7} +200.476 q^{8} +O(q^{10})$	\(q+9.25531 q^{2} +53.6607 q^{4} -25.0000 q^{5} +36.4478 q^{7} +200.476 q^{8} -231.383 q^{10} +121.000 q^{11} -878.032 q^{13} +337.336 q^{14} +138.327 q^{16} -155.385 q^{17} -1932.65 q^{19} -1341.52 q^{20} +1119.89 q^{22} -1927.38 q^{23} +625.000 q^{25} -8126.46 q^{26} +1955.81 q^{28} +480.444 q^{29} +1759.49 q^{31} -5134.98 q^{32} -1438.14 q^{34} -911.195 q^{35} -1898.87 q^{37} -17887.3 q^{38} -5011.91 q^{40} +4500.03 q^{41} -4475.49 q^{43} +6492.94 q^{44} -17838.5 q^{46} +12371.2 q^{47} -15478.6 q^{49} +5784.57 q^{50} -47115.8 q^{52} -2145.12 q^{53} -3025.00 q^{55} +7306.92 q^{56} +4446.66 q^{58} +15857.9 q^{59} -36447.7 q^{61} +16284.6 q^{62} -51952.3 q^{64} +21950.8 q^{65} -15668.5 q^{67} -8338.07 q^{68} -8433.39 q^{70} -10689.5 q^{71} +12172.6 q^{73} -17574.6 q^{74} -103707. q^{76} +4410.19 q^{77} -87205.6 q^{79} -3458.17 q^{80} +41649.2 q^{82} -97230.6 q^{83} +3884.63 q^{85} -41422.0 q^{86} +24257.6 q^{88} -38639.2 q^{89} -32002.4 q^{91} -103424. q^{92} +114499. q^{94} +48316.2 q^{95} -36754.5 q^{97} -143259. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10}) $ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8	$ 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100}) $ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 - 175 * q^10 + 363 * q^11 - 654 * q^13 + 728 * q^14 - 415 * q^16 + 2366 * q^17 - 2872 * q^19 - 625 * q^20 + 847 * q^22 - 2272 * q^23 + 1875 * q^25 - 3422 * q^26 + 4592 * q^28 + 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 2506 * q^34 + 4300 * q^35 - 9126 * q^37 - 13076 * q^38 - 5775 * q^40 + 8758 * q^41 - 14672 * q^43 + 3025 * q^44 - 28768 * q^46 + 19392 * q^47 - 26629 * q^49 + 4375 * q^50 - 61506 * q^52 + 4598 * q^53 - 9075 * q^55 - 2688 * q^56 + 8550 * q^58 + 9348 * q^59 - 60078 * q^61 + 14096 * q^62 - 7087 * q^64 + 16350 * q^65 - 38468 * q^67 - 59778 * q^68 - 18200 * q^70 + 74032 * q^71 - 44442 * q^73 - 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 108116 * q^79 + 10375 * q^80 - 92230 * q^82 + 81892 * q^83 - 59150 * q^85 - 126412 * q^86 + 27951 * q^88 - 167342 * q^89 - 31832 * q^91 - 72960 * q^92 + 12728 * q^94 + 71800 * q^95 + 159702 * q^97 - 163121 * q^98

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 $

$2$	9.25531	1.63612	0.818061	−	0.575131i	$-0.195049\pi$
$2$	9.25531	1.63612	0.818061	+	0.575131i	$0.195049\pi$
$3$	0	0
$3$	0	0
$4$	53.6607	1.67690
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	36.4478	0.281142	0.140571	−	0.990071i	$-0.455106\pi$
$7$	36.4478	0.281142	0.140571	+	0.990071i	$0.455106\pi$
$8$	200.476	1.10749
$9$	0	0
$10$	−231.383	−0.731696
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−878.032	−1.44096	−0.720480	−	0.693476i	$-0.756079\pi$
$13$	−878.032	−1.44096	−0.720480	+	0.693476i	$0.756079\pi$
$14$	337.336	0.459983
$15$	0	0
$16$	138.327	0.135085
$17$	−155.385	−0.130403	−0.0652014	−	0.997872i	$-0.520769\pi$
$17$	−155.385	−0.130403	−0.0652014	+	0.997872i	$0.520769\pi$
$18$	0	0
$19$	−1932.65	−1.22820	−0.614100	−	0.789228i	$-0.710481\pi$
$19$	−1932.65	−1.22820	−0.614100	+	0.789228i	$0.710481\pi$
$20$	−1341.52	−0.749931
$21$	0	0
$22$	1119.89	0.493309
$23$	−1927.38	−0.759709	−0.379855	−	0.925046i	$-0.624026\pi$
$23$	−1927.38	−0.759709	−0.379855	+	0.925046i	$0.624026\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−8126.46	−2.35759
$27$	0	0
$28$	1955.81	0.471447
$29$	480.444	0.106084	0.0530418	−	0.998592i	$-0.483108\pi$
$29$	480.444	0.106084	0.0530418	+	0.998592i	$0.483108\pi$
$30$	0	0
$31$	1759.49	0.328838	0.164419	−	0.986391i	$-0.447425\pi$
$31$	1759.49	0.328838	0.164419	+	0.986391i	$0.447425\pi$
$32$	−5134.98	−0.886470
$33$	0	0
$34$	−1438.14	−0.213355
$35$	−911.195	−0.125731
$36$	0	0
$37$	−1898.87	−0.228029	−0.114015	−	0.993479i	$-0.536371\pi$
$37$	−1898.87	−0.228029	−0.114015	+	0.993479i	$0.536371\pi$
$38$	−17887.3	−2.00949
$39$	0	0
$40$	−5011.91	−0.495282
$41$	4500.03	0.418076	0.209038	−	0.977907i	$-0.432967\pi$
$41$	4500.03	0.418076	0.209038	+	0.977907i	$0.432967\pi$
$42$	0	0
$43$	−4475.49	−0.369121	−0.184561	−	0.982821i	$-0.559086\pi$
$43$	−4475.49	−0.369121	−0.184561	+	0.982821i	$0.559086\pi$
$44$	6492.94	0.505603
$45$	0	0
$46$	−17838.5	−1.24298
$47$	12371.2	0.816895	0.408448	−	0.912782i	$-0.366070\pi$
$47$	12371.2	0.816895	0.408448	+	0.912782i	$0.366070\pi$
$48$	0	0
$49$	−15478.6	−0.920959
$50$	5784.57	0.327224
$51$	0	0
$52$	−47115.8	−2.41634
$53$	−2145.12	−0.104897	−0.0524483	−	0.998624i	$-0.516702\pi$
$53$	−2145.12	−0.104897	−0.0524483	+	0.998624i	$0.516702\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	7306.92	0.311361
$57$	0	0
$58$	4446.66	0.173566
$59$	15857.9	0.593084	0.296542	−	0.955020i	$-0.404166\pi$
$59$	15857.9	0.593084	0.296542	+	0.955020i	$0.404166\pi$
$60$	0	0
$61$	−36447.7	−1.25414	−0.627070	−	0.778963i	$-0.715746\pi$
$61$	−36447.7	−1.25414	−0.627070	+	0.778963i	$0.715746\pi$
$62$	16284.6	0.538020
$63$	0	0
$64$	−51952.3	−1.58546
$65$	21950.8	0.644417
$66$	0	0
$67$	−15668.5	−0.426424	−0.213212	−	0.977006i	$-0.568393\pi$
$67$	−15668.5	−0.426424	−0.213212	+	0.977006i	$0.568393\pi$
$68$	−8338.07	−0.218672
$69$	0	0
$70$	−8433.39	−0.205711
$71$	−10689.5	−0.251659	−0.125830	−	0.992052i	$-0.540159\pi$
$71$	−10689.5	−0.251659	−0.125830	+	0.992052i	$0.540159\pi$
$72$	0	0
$73$	12172.6	0.267347	0.133674	−	0.991025i	$-0.457323\pi$
$73$	12172.6	0.267347	0.133674	+	0.991025i	$0.457323\pi$
$74$	−17574.6	−0.373084
$75$	0	0
$76$	−103707.	−2.05956
$77$	4410.19	0.0847676
$78$	0	0
$79$	−87205.6	−1.57209	−0.786043	−	0.618171i	$-0.787874\pi$
$79$	−87205.6	−1.57209	−0.786043	+	0.618171i	$0.787874\pi$
$80$	−3458.17	−0.0604117
$81$	0	0
$82$	41649.2	0.684024
$83$	−97230.6	−1.54920	−0.774600	−	0.632451i	$-0.782049\pi$
$83$	−97230.6	−1.54920	−0.774600	+	0.632451i	$0.782049\pi$
$84$	0	0
$85$	3884.63	0.0583179
$86$	−41422.0	−0.603928
$87$	0	0
$88$	24257.6	0.333919
$89$	−38639.2	−0.517075	−0.258537	−	0.966001i	$-0.583241\pi$
$89$	−38639.2	−0.517075	−0.258537	+	0.966001i	$0.583241\pi$
$90$	0	0
$91$	−32002.4	−0.405115
$92$	−103424.	−1.27395
$93$	0	0
$94$	114499.	1.33654
$95$	48316.2	0.549268
$96$	0	0
$97$	−36754.5	−0.396626	−0.198313	−	0.980139i	$-0.563546\pi$
$97$	−36754.5	−0.396626	−0.198313	+	0.980139i	$0.563546\pi$
$98$	−143259.	−1.50680
$99$	0	0
$100$	33537.9	0.335379
$101$	185487.	1.80929	0.904647	−	0.426161i	$-0.140134\pi$
$101$	185487.	1.80929	0.904647	+	0.426161i	$0.140134\pi$
$102$	0	0
$103$	36890.7	0.342629	0.171315	−	0.985216i	$-0.445199\pi$
$103$	36890.7	0.342629	0.171315	+	0.985216i	$0.445199\pi$
$104$	−176025.	−1.59584
$105$	0	0
$106$	−19853.7	−0.171624
$107$	124996.	1.05545	0.527725	−	0.849415i	$-0.323045\pi$
$107$	124996.	1.05545	0.527725	+	0.849415i	$0.323045\pi$
$108$	0	0
$109$	−150975.	−1.21714	−0.608568	−	0.793502i	$-0.708256\pi$
$109$	−150975.	−1.21714	−0.608568	+	0.793502i	$0.708256\pi$
$110$	−27997.3	−0.220615
$111$	0	0
$112$	5041.71	0.0379780
$113$	−157970.	−1.16380	−0.581899	−	0.813261i	$-0.697690\pi$
$113$	−157970.	−1.16380	−0.581899	+	0.813261i	$0.697690\pi$
$114$	0	0
$115$	48184.5	0.339752
$116$	25781.0	0.177891
$117$	0	0
$118$	146770.	0.970359
$119$	−5663.45	−0.0366618
$120$	0	0
$121$	14641.0	0.0909091
$122$	−337335.	−2.05193
$123$	0	0
$124$	94415.4	0.551428
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	268814.	1.47891	0.739457	−	0.673204i	$-0.235082\pi$
$127$	268814.	1.47891	0.739457	+	0.673204i	$0.235082\pi$
$128$	−316515.	−1.70753
$129$	0	0
$130$	203161.	1.05435
$131$	366914.	1.86804	0.934020	−	0.357220i	$-0.116275\pi$
$131$	366914.	1.86804	0.934020	+	0.357220i	$0.116275\pi$
$132$	0	0
$133$	−70440.9	−0.345299
$134$	−145017.	−0.697682
$135$	0	0
$136$	−31151.0	−0.144419
$137$	182927.	0.832678	0.416339	−	0.909209i	$-0.363313\pi$
$137$	182927.	0.832678	0.416339	+	0.909209i	$0.363313\pi$
$138$	0	0
$139$	8429.85	0.0370069	0.0185035	−	0.999829i	$-0.494110\pi$
$139$	8429.85	0.0370069	0.0185035	+	0.999829i	$0.494110\pi$
$140$	−48895.4	−0.210837
$141$	0	0
$142$	−98934.8	−0.411745
$143$	−106242.	−0.434466
$144$	0	0
$145$	−12011.1	−0.0474420
$146$	112661.	0.437413
$147$	0	0
$148$	−101895.	−0.382381
$149$	−271810.	−1.00300	−0.501499	−	0.865158i	$-0.667218\pi$
$149$	−271810.	−1.00300	−0.501499	+	0.865158i	$0.667218\pi$
$150$	0	0
$151$	−236565.	−0.844322	−0.422161	−	0.906521i	$-0.638728\pi$
$151$	−236565.	−0.844322	−0.422161	+	0.906521i	$0.638728\pi$
$152$	−387450.	−1.36021
$153$	0	0
$154$	40817.6	0.138690
$155$	−43987.2	−0.147061
$156$	0	0
$157$	211824.	0.685846	0.342923	−	0.939364i	$-0.388583\pi$
$157$	211824.	0.685846	0.342923	+	0.939364i	$0.388583\pi$
$158$	−807114.	−2.57213
$159$	0	0
$160$	128375.	0.396441
$161$	−70248.7	−0.213586
$162$	0	0
$163$	−341315.	−1.00620	−0.503102	−	0.864227i	$-0.667808\pi$
$163$	−341315.	−1.00620	−0.503102	+	0.864227i	$0.667808\pi$
$164$	241475.	0.701071
$165$	0	0
$166$	−899899.	−2.53468
$167$	180548.	0.500958	0.250479	−	0.968122i	$-0.419412\pi$
$167$	180548.	0.500958	0.250479	+	0.968122i	$0.419412\pi$
$168$	0	0
$169$	399647.	1.07637
$170$	35953.4	0.0954153
$171$	0	0
$172$	−240158.	−0.618978
$173$	643322.	1.63423	0.817114	−	0.576476i	$-0.195572\pi$
$173$	643322.	1.63423	0.817114	+	0.576476i	$0.195572\pi$
$174$	0	0
$175$	22779.9	0.0562285
$176$	16737.5	0.0407296
$177$	0	0
$178$	−357618.	−0.845998
$179$	266922.	0.622662	0.311331	−	0.950302i	$-0.399225\pi$
$179$	266922.	0.622662	0.311331	+	0.950302i	$0.399225\pi$
$180$	0	0
$181$	281529.	0.638745	0.319372	−	0.947629i	$-0.396528\pi$
$181$	281529.	0.638745	0.319372	+	0.947629i	$0.396528\pi$
$182$	−296192.	−0.662818
$183$	0	0
$184$	−386393.	−0.841366
$185$	47471.7	0.101978
$186$	0	0
$187$	−18801.6	−0.0393179
$188$	663846.	1.36985
$189$	0	0
$190$	447182.	0.898669
$191$	−933723.	−1.85197	−0.925987	−	0.377556i	$-0.876765\pi$
$191$	−933723.	−1.85197	−0.925987	+	0.377556i	$0.876765\pi$
$192$	0	0
$193$	300124.	0.579973	0.289986	−	0.957031i	$-0.406349\pi$
$193$	300124.	0.579973	0.289986	+	0.957031i	$0.406349\pi$
$194$	−340174.	−0.648929
$195$	0	0
$196$	−830590.	−1.54435
$197$	944940.	1.73476	0.867378	−	0.497649i	$-0.165803\pi$
$197$	944940.	1.73476	0.867378	+	0.497649i	$0.165803\pi$
$198$	0	0
$199$	−1.00821e6	−1.80475	−0.902374	−	0.430954i	$-0.858177\pi$
$199$	−1.00821e6	−1.80475	−0.902374	+	0.430954i	$0.858177\pi$
$200$	125298.	0.221497
$201$	0	0
$202$	1.71674e6	2.96023
$203$	17511.1	0.0298246
$204$	0	0
$205$	−112501.	−0.186969
$206$	341435.	0.560583
$207$	0	0
$208$	−121455.	−0.194652
$209$	−233851.	−0.370316
$210$	0	0
$211$	497479.	0.769253	0.384626	−	0.923072i	$-0.374330\pi$
$211$	497479.	0.769253	0.384626	+	0.923072i	$0.374330\pi$
$212$	−115108.	−0.175901
$213$	0	0
$214$	1.15688e6	1.72684
$215$	111887.	0.165076
$216$	0	0
$217$	64129.5	0.0924504
$218$	−1.39732e6	−1.99138
$219$	0	0
$220$	−162324.	−0.226113
$221$	136433.	0.187905
$222$	0	0
$223$	1.14136e6	1.53695	0.768477	−	0.639878i	$-0.221015\pi$
$223$	1.14136e6	1.53695	0.768477	+	0.639878i	$0.221015\pi$
$224$	−187159.	−0.249224
$225$	0	0
$226$	−1.46206e6	−1.90411
$227$	−669451.	−0.862292	−0.431146	−	0.902282i	$-0.641891\pi$
$227$	−669451.	−0.862292	−0.431146	+	0.902282i	$0.641891\pi$
$228$	0	0
$229$	588061.	0.741026	0.370513	−	0.928827i	$-0.379182\pi$
$229$	588061.	0.741026	0.370513	+	0.928827i	$0.379182\pi$
$230$	445962.	0.555876
$231$	0	0
$232$	96317.6	0.117486
$233$	−199417.	−0.240642	−0.120321	−	0.992735i	$-0.538392\pi$
$233$	−199417.	−0.240642	−0.120321	+	0.992735i	$0.538392\pi$
$234$	0	0
$235$	−309279.	−0.365327
$236$	850947.	0.994541
$237$	0	0
$238$	−52416.9	−0.0599832
$239$	408055.	0.462088	0.231044	−	0.972943i	$-0.425786\pi$
$239$	408055.	0.462088	0.231044	+	0.972943i	$0.425786\pi$
$240$	0	0
$241$	1.24022e6	1.37548	0.687742	−	0.725956i	$-0.258602\pi$
$241$	1.24022e6	1.37548	0.687742	+	0.725956i	$0.258602\pi$
$242$	135507.	0.148738
$243$	0	0
$244$	−1.95581e6	−2.10306
$245$	386964.	0.411865
$246$	0	0
$247$	1.69693e6	1.76979
$248$	352736.	0.364183
$249$	0	0
$250$	−144614.	−0.146339
$251$	−30660.6	−0.0307183	−0.0153591	−	0.999882i	$-0.504889\pi$
$251$	−30660.6	−0.0307183	−0.0153591	+	0.999882i	$0.504889\pi$
$252$	0	0
$253$	−233213.	−0.229061
$254$	2.48796e6	2.41968
$255$	0	0
$256$	−1.26697e6	−1.20828
$257$	687971.	0.649737	0.324868	−	0.945759i	$-0.394680\pi$
$257$	687971.	0.649737	0.324868	+	0.945759i	$0.394680\pi$
$258$	0	0
$259$	−69209.6	−0.0641087
$260$	1.17790e6	1.08062
$261$	0	0
$262$	3.39590e6	3.05634
$263$	−1.70103e6	−1.51643	−0.758216	−	0.652004i	$-0.773929\pi$
$263$	−1.70103e6	−1.51643	−0.758216	+	0.652004i	$0.773929\pi$
$264$	0	0
$265$	53628.0	0.0469112
$266$	−651952.	−0.564952
$267$	0	0
$268$	−840785.	−0.715069
$269$	982644.	0.827972	0.413986	−	0.910283i	$-0.364136\pi$
$269$	982644.	0.827972	0.413986	+	0.910283i	$0.364136\pi$
$270$	0	0
$271$	−276206.	−0.228459	−0.114230	−	0.993454i	$-0.536440\pi$
$271$	−276206.	−0.228459	−0.114230	+	0.993454i	$0.536440\pi$
$272$	−21493.9	−0.0176154
$273$	0	0
$274$	1.69305e6	1.36236
$275$	75625.0	0.0603023
$276$	0	0
$277$	148776.	0.116502	0.0582509	−	0.998302i	$-0.481448\pi$
$277$	148776.	0.116502	0.0582509	+	0.998302i	$0.481448\pi$
$278$	78020.8	0.0605478
$279$	0	0
$280$	−182673.	−0.139245
$281$	−357772.	−0.270297	−0.135148	−	0.990825i	$-0.543151\pi$
$281$	−357772.	−0.270297	−0.135148	+	0.990825i	$0.543151\pi$
$282$	0	0
$283$	−492090.	−0.365240	−0.182620	−	0.983184i	$-0.558458\pi$
$283$	−492090.	−0.365240	−0.182620	+	0.983184i	$0.558458\pi$
$284$	−573607.	−0.422006
$285$	0	0
$286$	−983301.	−0.710839
$287$	164016.	0.117539
$288$	0	0
$289$	−1.39571e6	−0.982995
$290$	−111166.	−0.0776209
$291$	0	0
$292$	653189.	0.448314
$293$	−498120.	−0.338973	−0.169487	−	0.985532i	$-0.554211\pi$
$293$	−498120.	−0.338973	−0.169487	+	0.985532i	$0.554211\pi$
$294$	0	0
$295$	−396448.	−0.265235
$296$	−380678.	−0.252539
$297$	0	0
$298$	−2.51568e6	−1.64103
$299$	1.69230e6	1.09471
$300$	0	0
$301$	−163122.	−0.103776
$302$	−2.18948e6	−1.38141
$303$	0	0
$304$	−267337.	−0.165911
$305$	911194.	0.560869
$306$	0	0
$307$	−998760.	−0.604805	−0.302402	−	0.953180i	$-0.597789\pi$
$307$	−998760.	−0.604805	−0.302402	+	0.953180i	$0.597789\pi$
$308$	236654.	0.142147
$309$	0	0
$310$	−407115.	−0.240610
$311$	−1.88783e6	−1.10678	−0.553389	−	0.832923i	$-0.686666\pi$
$311$	−1.88783e6	−1.10678	−0.553389	+	0.832923i	$0.686666\pi$
$312$	0	0
$313$	−2.34787e6	−1.35461	−0.677303	−	0.735704i	$-0.736851\pi$
$313$	−2.34787e6	−1.35461	−0.677303	+	0.735704i	$0.736851\pi$
$314$	1.96050e6	1.12213
$315$	0	0
$316$	−4.67951e6	−2.63623
$317$	−562721.	−0.314518	−0.157259	−	0.987557i	$-0.550266\pi$
$317$	−562721.	−0.314518	−0.157259	+	0.987557i	$0.550266\pi$
$318$	0	0
$319$	58133.7	0.0319854
$320$	1.29881e6	0.709038
$321$	0	0
$322$	−650173.	−0.349454
$323$	300305.	0.160161
$324$	0	0
$325$	−548770.	−0.288192
$326$	−3.15897e6	−1.64627
$327$	0	0
$328$	902149.	0.463013
$329$	450902.	0.229664
$330$	0	0
$331$	−1.00593e6	−0.504657	−0.252328	−	0.967642i	$-0.581196\pi$
$331$	−1.00593e6	−0.504657	−0.252328	+	0.967642i	$0.581196\pi$
$332$	−5.21746e6	−2.59785
$333$	0	0
$334$	1.67103e6	0.819628
$335$	391714.	0.190703
$336$	0	0
$337$	−280192.	−0.134394	−0.0671971	−	0.997740i	$-0.521406\pi$
$337$	−280192.	−0.134394	−0.0671971	+	0.997740i	$0.521406\pi$
$338$	3.69886e6	1.76107
$339$	0	0
$340$	208452.	0.0977931
$341$	212898.	0.0991485
$342$	0	0
$343$	−1.17674e6	−0.540063
$344$	−897229.	−0.408796
$345$	0	0
$346$	5.95414e6	2.67380
$347$	−2.10913e6	−0.940328	−0.470164	−	0.882579i	$-0.655805\pi$
$347$	−2.10913e6	−0.940328	−0.470164	+	0.882579i	$0.655805\pi$
$348$	0	0
$349$	3.88469e6	1.70723	0.853617	−	0.520901i	$-0.174404\pi$
$349$	3.88469e6	1.70723	0.853617	+	0.520901i	$0.174404\pi$
$350$	210835.	0.0919967
$351$	0	0
$352$	−621333.	−0.267281
$353$	−1.35663e6	−0.579463	−0.289732	−	0.957108i	$-0.593566\pi$
$353$	−1.35663e6	−0.579463	−0.289732	+	0.957108i	$0.593566\pi$
$354$	0	0
$355$	267238.	0.112545
$356$	−2.07341e6	−0.867081
$357$	0	0
$358$	2.47045e6	1.01875
$359$	−3.93436e6	−1.61116	−0.805579	−	0.592488i	$-0.798146\pi$
$359$	−3.93436e6	−1.61116	−0.805579	+	0.592488i	$0.798146\pi$
$360$	0	0
$361$	1.25904e6	0.508476
$362$	2.60564e6	1.04506
$363$	0	0
$364$	−1.71727e6	−0.679336
$365$	−304315.	−0.119561
$366$	0	0
$367$	−2.82588e6	−1.09519	−0.547594	−	0.836744i	$-0.684456\pi$
$367$	−2.82588e6	−1.09519	−0.547594	+	0.836744i	$0.684456\pi$
$368$	−266608.	−0.102625
$369$	0	0
$370$	439365.	0.166848
$371$	−78184.9	−0.0294909
$372$	0	0
$373$	4.58790e6	1.70743	0.853713	−	0.520744i	$-0.174345\pi$
$373$	4.58790e6	1.70743	0.853713	+	0.520744i	$0.174345\pi$
$374$	−174015.	−0.0643290
$375$	0	0
$376$	2.48013e6	0.904699
$377$	−421845.	−0.152862
$378$	0	0
$379$	2.84827e6	1.01855	0.509277	−	0.860603i	$-0.329913\pi$
$379$	2.84827e6	1.01855	0.509277	+	0.860603i	$0.329913\pi$
$380$	2.59268e6	0.921065
$381$	0	0
$382$	−8.64190e6	−3.03006
$383$	2.78467e6	0.970013	0.485006	−	0.874511i	$-0.338817\pi$
$383$	2.78467e6	0.970013	0.485006	+	0.874511i	$0.338817\pi$
$384$	0	0
$385$	−110255.	−0.0379092
$386$	2.77774e6	0.948906
$387$	0	0
$388$	−1.97227e6	−0.665101
$389$	−3.95277e6	−1.32442	−0.662212	−	0.749316i	$-0.730382\pi$
$389$	−3.95277e6	−1.32442	−0.662212	+	0.749316i	$0.730382\pi$
$390$	0	0
$391$	299486.	0.0990682
$392$	−3.10308e6	−1.01995
$393$	0	0
$394$	8.74571e6	2.83827
$395$	2.18014e6	0.703059
$396$	0	0
$397$	−5.55351e6	−1.76844	−0.884221	−	0.467068i	$-0.845310\pi$
$397$	−5.55351e6	−1.76844	−0.884221	+	0.467068i	$0.845310\pi$
$398$	−9.33125e6	−2.95279
$399$	0	0
$400$	86454.2	0.0270169
$401$	279266.	0.0867277	0.0433639	−	0.999059i	$-0.486193\pi$
$401$	279266.	0.0867277	0.0433639	+	0.999059i	$0.486193\pi$
$402$	0	0
$403$	−1.54489e6	−0.473843
$404$	9.95334e6	3.03400
$405$	0	0
$406$	162071.	0.0487967
$407$	−229763.	−0.0687534
$408$	0	0
$409$	5.17128e6	1.52859	0.764293	−	0.644869i	$-0.223088\pi$
$409$	5.17128e6	1.52859	0.764293	+	0.644869i	$0.223088\pi$
$410$	−1.04123e6	−0.305905
$411$	0	0
$412$	1.97958e6	0.574554
$413$	577987.	0.166741
$414$	0	0
$415$	2.43076e6	0.692824
$416$	4.50868e6	1.27737
$417$	0	0
$418$	−2.16436e6	−0.605883
$419$	6.04152e6	1.68117	0.840585	−	0.541680i	$-0.182212\pi$
$419$	6.04152e6	1.68117	0.840585	+	0.541680i	$0.182212\pi$
$420$	0	0
$421$	−895386.	−0.246210	−0.123105	−	0.992394i	$-0.539285\pi$
$421$	−895386.	−0.246210	−0.123105	+	0.992394i	$0.539285\pi$
$422$	4.60432e6	1.25859
$423$	0	0
$424$	−430045.	−0.116171
$425$	−97115.7	−0.0260806
$426$	0	0
$427$	−1.32844e6	−0.352592
$428$	6.70738e6	1.76988
$429$	0	0
$430$	1.03555e6	0.270085
$431$	1.60867e6	0.417132	0.208566	−	0.978008i	$-0.433120\pi$
$431$	1.60867e6	0.417132	0.208566	+	0.978008i	$0.433120\pi$
$432$	0	0
$433$	1.86039e6	0.476853	0.238427	−	0.971161i	$-0.423368\pi$
$433$	1.86039e6	0.476853	0.238427	+	0.971161i	$0.423368\pi$
$434$	593538.	0.151260
$435$	0	0
$436$	−8.10142e6	−2.04101
$437$	3.72495e6	0.933075
$438$	0	0
$439$	2.75051e6	0.681164	0.340582	−	0.940215i	$-0.389376\pi$
$439$	2.75051e6	0.681164	0.340582	+	0.940215i	$0.389376\pi$
$440$	−606441.	−0.149333
$441$	0	0
$442$	1.26273e6	0.307436
$443$	−3.15804e6	−0.764555	−0.382278	−	0.924048i	$-0.624860\pi$
$443$	−3.15804e6	−0.764555	−0.382278	+	0.924048i	$0.624860\pi$
$444$	0	0
$445$	965981.	0.231243
$446$	1.05636e7	2.51464
$447$	0	0
$448$	−1.89355e6	−0.445740
$449$	127508.	0.0298484	0.0149242	−	0.999889i	$-0.495249\pi$
$449$	127508.	0.0298484	0.0149242	+	0.999889i	$0.495249\pi$
$450$	0	0
$451$	544504.	0.126055
$452$	−8.47675e6	−1.95157
$453$	0	0
$454$	−6.19598e6	−1.41082
$455$	800059.	0.181173
$456$	0	0
$457$	2.19209e6	0.490984	0.245492	−	0.969399i	$-0.421050\pi$
$457$	2.19209e6	0.490984	0.245492	+	0.969399i	$0.421050\pi$
$458$	5.44268e6	1.21241
$459$	0	0
$460$	2.58561e6	0.569729
$461$	−5.20229e6	−1.14010	−0.570050	−	0.821610i	$-0.693076\pi$
$461$	−5.20229e6	−1.14010	−0.570050	+	0.821610i	$0.693076\pi$
$462$	0	0
$463$	2.66624e6	0.578025	0.289013	−	0.957325i	$-0.406673\pi$
$463$	2.66624e6	0.578025	0.289013	+	0.957325i	$0.406673\pi$
$464$	66458.3	0.0143303
$465$	0	0
$466$	−1.84566e6	−0.393720
$467$	2.35578e6	0.499853	0.249926	−	0.968265i	$-0.419594\pi$
$467$	2.35578e6	0.499853	0.249926	+	0.968265i	$0.419594\pi$
$468$	0	0
$469$	−571084.	−0.119886
$470$	−2.86247e6	−0.597719
$471$	0	0
$472$	3.17914e6	0.656832
$473$	−541534.	−0.111294
$474$	0	0
$475$	−1.20791e6	−0.245640
$476$	−303905.	−0.0614780
$477$	0	0
$478$	3.77668e6	0.756032
$479$	4.78329e6	0.952551	0.476276	−	0.879296i	$-0.341986\pi$
$479$	4.78329e6	0.952551	0.476276	+	0.879296i	$0.341986\pi$
$480$	0	0
$481$	1.66727e6	0.328581
$482$	1.14786e7	2.25046
$483$	0	0
$484$	785646.	0.152445
$485$	918863.	0.177377
$486$	0	0
$487$	−3.31515e6	−0.633405	−0.316702	−	0.948525i	$-0.602576\pi$
$487$	−3.31515e6	−0.633405	−0.316702	+	0.948525i	$0.602576\pi$
$488$	−7.30691e6	−1.38894
$489$	0	0
$490$	3.58147e6	0.673862
$491$	3.02276e6	0.565847	0.282924	−	0.959142i	$-0.408696\pi$
$491$	3.02276e6	0.565847	0.282924	+	0.959142i	$0.408696\pi$
$492$	0	0
$493$	−74653.9	−0.0138336
$494$	1.57056e7	2.89559
$495$	0	0
$496$	243384.	0.0444210
$497$	−389610.	−0.0707520
$498$	0	0
$499$	4.46530e6	0.802785	0.401392	−	0.915906i	$-0.368526\pi$
$499$	4.46530e6	0.802785	0.401392	+	0.915906i	$0.368526\pi$
$500$	−838448.	−0.149986
$501$	0	0
$502$	−283774.	−0.0502589
$503$	1.77528e6	0.312858	0.156429	−	0.987689i	$-0.450002\pi$
$503$	1.77528e6	0.312858	0.156429	+	0.987689i	$0.450002\pi$
$504$	0	0
$505$	−4.63717e6	−0.809141
$506$	−2.15846e6	−0.374772
$507$	0	0
$508$	1.44248e7	2.47999
$509$	485180.	0.0830058	0.0415029	−	0.999138i	$-0.486785\pi$
$509$	485180.	0.0830058	0.0415029	+	0.999138i	$0.486785\pi$
$510$	0	0
$511$	443664.	0.0751627
$512$	−1.59770e6	−0.269353
$513$	0	0
$514$	6.36739e6	1.06305
$515$	−922269.	−0.153228
$516$	0	0
$517$	1.49691e6	0.246303
$518$	−640556.	−0.104890
$519$	0	0
$520$	4.40061e6	0.713682
$521$	−9.92932e6	−1.60260	−0.801300	−	0.598262i	$-0.795858\pi$
$521$	−9.92932e6	−1.60260	−0.801300	+	0.598262i	$0.795858\pi$
$522$	0	0
$523$	5.04767e6	0.806931	0.403466	−	0.914995i	$-0.367805\pi$
$523$	5.04767e6	0.806931	0.403466	+	0.914995i	$0.367805\pi$
$524$	1.96889e7	3.13251
$525$	0	0
$526$	−1.57436e7	−2.48107
$527$	−273398.	−0.0428815
$528$	0	0
$529$	−2.72156e6	−0.422842
$530$	496343.	0.0767525
$531$	0	0
$532$	−3.77990e6	−0.579031
$533$	−3.95117e6	−0.602432
$534$	0	0
$535$	−3.12491e6	−0.472011
$536$	−3.14117e6	−0.472258
$537$	0	0
$538$	9.09467e6	1.35466
$539$	−1.87291e6	−0.277680
$540$	0	0
$541$	−4.46393e6	−0.655729	−0.327864	−	0.944725i	$-0.606329\pi$
$541$	−4.46393e6	−0.655729	−0.327864	+	0.944725i	$0.606329\pi$
$542$	−2.55637e6	−0.373788
$543$	0	0
$544$	797900.	0.115598
$545$	3.77438e6	0.544319
$546$	0	0
$547$	−6.62530e6	−0.946755	−0.473377	−	0.880860i	$-0.656965\pi$
$547$	−6.62530e6	−0.946755	−0.473377	+	0.880860i	$0.656965\pi$
$548$	9.81601e6	1.39632
$549$	0	0
$550$	699932.	0.0986619
$551$	−928530.	−0.130292
$552$	0	0
$553$	−3.17845e6	−0.441980
$554$	1.37696e6	0.190611
$555$	0	0
$556$	452351.	0.0620568
$557$	717580.	0.0980014	0.0490007	−	0.998799i	$-0.484396\pi$
$557$	717580.	0.0980014	0.0490007	+	0.998799i	$0.484396\pi$
$558$	0	0
$559$	3.92962e6	0.531889
$560$	−126043.	−0.0169843
$561$	0	0
$562$	−3.31129e6	−0.442238
$563$	−1.35097e7	−1.79629	−0.898143	−	0.439703i	$-0.855084\pi$
$563$	−1.35097e7	−1.79629	−0.898143	+	0.439703i	$0.855084\pi$
$564$	0	0
$565$	3.94924e6	0.520466
$566$	−4.55444e6	−0.597577
$567$	0	0
$568$	−2.14300e6	−0.278709
$569$	−1.35535e7	−1.75498	−0.877488	−	0.479598i	$-0.840782\pi$
$569$	−1.35535e7	−1.75498	−0.877488	+	0.479598i	$0.840782\pi$
$570$	0	0
$571$	7.82130e6	1.00390	0.501948	−	0.864898i	$-0.332617\pi$
$571$	7.82130e6	1.00390	0.501948	+	0.864898i	$0.332617\pi$
$572$	−5.70101e6	−0.728554
$573$	0	0
$574$	1.51802e6	0.192308
$575$	−1.20461e6	−0.151942
$576$	0	0
$577$	218443.	0.0273149	0.0136574	−	0.999907i	$-0.495653\pi$
$577$	218443.	0.0273149	0.0136574	+	0.999907i	$0.495653\pi$
$578$	−1.29177e7	−1.60830
$579$	0	0
$580$	−644524.	−0.0795553
$581$	−3.54384e6	−0.435546
$582$	0	0
$583$	−259559.	−0.0316275
$584$	2.44031e6	0.296083
$585$	0	0
$586$	−4.61026e6	−0.554602
$587$	5.46003e6	0.654033	0.327017	−	0.945019i	$-0.393957\pi$
$587$	5.46003e6	0.654033	0.327017	+	0.945019i	$0.393957\pi$
$588$	0	0
$589$	−3.40048e6	−0.403879
$590$	−3.66925e6	−0.433958
$591$	0	0
$592$	−262664.	−0.0308033
$593$	−1.41186e7	−1.64875	−0.824375	−	0.566044i	$-0.808473\pi$
$593$	−1.41186e7	−1.64875	−0.824375	+	0.566044i	$0.808473\pi$
$594$	0	0
$595$	141586.	0.0163956
$596$	−1.45855e7	−1.68192
$597$	0	0
$598$	1.56628e7	1.79108
$599$	−8.02044e6	−0.913338	−0.456669	−	0.889637i	$-0.650958\pi$
$599$	−8.02044e6	−0.913338	−0.456669	+	0.889637i	$0.650958\pi$
$600$	0	0
$601$	−1.20301e7	−1.35857	−0.679286	−	0.733874i	$-0.737710\pi$
$601$	−1.20301e7	−1.35857	−0.679286	+	0.733874i	$0.737710\pi$
$602$	−1.50974e6	−0.169790
$603$	0	0
$604$	−1.26942e7	−1.41584
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−1.58863e7	−1.75005	−0.875025	−	0.484078i	$-0.839155\pi$
$607$	−1.58863e7	−1.75005	−0.875025	+	0.484078i	$0.839155\pi$
$608$	9.92412e6	1.08876
$609$	0	0
$610$	8.43338e6	0.917650
$611$	−1.08623e7	−1.17711
$612$	0	0
$613$	1.27701e7	1.37260	0.686301	−	0.727318i	$-0.259233\pi$
$613$	1.27701e7	1.37260	0.686301	+	0.727318i	$0.259233\pi$
$614$	−9.24383e6	−0.989535
$615$	0	0
$616$	884137.	0.0938789
$617$	−5.32363e6	−0.562982	−0.281491	−	0.959564i	$-0.590829\pi$
$617$	−5.32363e6	−0.562982	−0.281491	+	0.959564i	$0.590829\pi$
$618$	0	0
$619$	−5.79882e6	−0.608293	−0.304147	−	0.952625i	$-0.598371\pi$
$619$	−5.79882e6	−0.608293	−0.304147	+	0.952625i	$0.598371\pi$
$620$	−2.36038e6	−0.246606
$621$	0	0
$622$	−1.74724e7	−1.81083
$623$	−1.40832e6	−0.145372
$624$	0	0
$625$	390625.	0.0400000
$626$	−2.17302e7	−2.21630
$627$	0	0
$628$	1.13666e7	1.15009
$629$	295056.	0.0297357
$630$	0	0
$631$	−8.25264e6	−0.825124	−0.412562	−	0.910929i	$-0.635366\pi$
$631$	−8.25264e6	−0.825124	−0.412562	+	0.910929i	$0.635366\pi$
$632$	−1.74826e7	−1.74106
$633$	0	0
$634$	−5.20816e6	−0.514590
$635$	−6.72036e6	−0.661391
$636$	0	0
$637$	1.35907e7	1.32707
$638$	538045.	0.0523320
$639$	0	0
$640$	7.91287e6	0.763632
$641$	−1.14111e7	−1.09694	−0.548471	−	0.836169i	$-0.684790\pi$
$641$	−1.14111e7	−1.09694	−0.548471	+	0.836169i	$0.684790\pi$
$642$	0	0
$643$	−3.26961e6	−0.311866	−0.155933	−	0.987768i	$-0.549838\pi$
$643$	−3.26961e6	−0.311866	−0.155933	+	0.987768i	$0.549838\pi$
$644$	−3.76959e6	−0.358162
$645$	0	0
$646$	2.77941e6	0.262043
$647$	9.95068e6	0.934527	0.467264	−	0.884118i	$-0.345240\pi$
$647$	9.95068e6	0.934527	0.467264	+	0.884118i	$0.345240\pi$
$648$	0	0
$649$	1.91881e6	0.178822
$650$	−5.07903e6	−0.471517
$651$	0	0
$652$	−1.83152e7	−1.68730
$653$	−1.52022e7	−1.39515	−0.697577	−	0.716509i	$-0.745739\pi$
$653$	−1.52022e7	−1.39515	−0.697577	+	0.716509i	$0.745739\pi$
$654$	0	0
$655$	−9.17285e6	−0.835413
$656$	622474.	0.0564757
$657$	0	0
$658$	4.17324e6	0.375758
$659$	9.63232e6	0.864007	0.432004	−	0.901872i	$-0.357807\pi$
$659$	9.63232e6	0.864007	0.432004	+	0.901872i	$0.357807\pi$
$660$	0	0
$661$	2.04631e7	1.82166	0.910832	−	0.412778i	$-0.135442\pi$
$661$	2.04631e7	1.82166	0.910832	+	0.412778i	$0.135442\pi$
$662$	−9.31016e6	−0.825681
$663$	0	0
$664$	−1.94924e7	−1.71572
$665$	1.76102e6	0.154422
$666$	0	0
$667$	−925997.	−0.0805926
$668$	9.68832e6	0.840054
$669$	0	0
$670$	3.62543e6	0.312013
$671$	−4.41018e6	−0.378137
$672$	0	0
$673$	1.57773e6	0.134275	0.0671376	−	0.997744i	$-0.478613\pi$
$673$	1.57773e6	0.134275	0.0671376	+	0.997744i	$0.478613\pi$
$674$	−2.59326e6	−0.219885
$675$	0	0
$676$	2.14454e7	1.80496
$677$	−6.92750e6	−0.580904	−0.290452	−	0.956890i	$-0.593806\pi$
$677$	−6.92750e6	−0.580904	−0.290452	+	0.956890i	$0.593806\pi$
$678$	0	0
$679$	−1.33962e6	−0.111509
$680$	778776.	0.0645862
$681$	0	0
$682$	1.97044e6	0.162219
$683$	277554.	0.0227665	0.0113833	−	0.999935i	$-0.496377\pi$
$683$	277554.	0.0227665	0.0113833	+	0.999935i	$0.496377\pi$
$684$	0	0
$685$	−4.57318e6	−0.372385
$686$	−1.08911e7	−0.883609
$687$	0	0
$688$	−619080.	−0.0498627
$689$	1.88348e6	0.151152
$690$	0	0
$691$	−2.12446e7	−1.69259	−0.846297	−	0.532712i	$-0.821173\pi$
$691$	−2.12446e7	−1.69259	−0.846297	+	0.532712i	$0.821173\pi$
$692$	3.45211e7	2.74043
$693$	0	0
$694$	−1.95206e7	−1.53849
$695$	−210746.	−0.0165500
$696$	0	0
$697$	−699238.	−0.0545184
$698$	3.59540e7	2.79324
$699$	0	0
$700$	1.22238e6	0.0942893
$701$	1.06971e7	0.822184	0.411092	−	0.911594i	$-0.365147\pi$
$701$	1.06971e7	0.822184	0.411092	+	0.911594i	$0.365147\pi$
$702$	0	0
$703$	3.66985e6	0.280066
$704$	−6.28623e6	−0.478034
$705$	0	0
$706$	−1.25561e7	−0.948072
$707$	6.76058e6	0.508669
$708$	0	0
$709$	6.52521e6	0.487505	0.243752	−	0.969838i	$-0.421622\pi$
$709$	6.52521e6	0.487505	0.243752	+	0.969838i	$0.421622\pi$
$710$	2.47337e6	0.184138
$711$	0	0
$712$	−7.74625e6	−0.572653
$713$	−3.39120e6	−0.249821
$714$	0	0
$715$	2.65605e6	0.194299
$716$	1.43232e7	1.04414
$717$	0	0
$718$	−3.64137e7	−2.63605
$719$	2.12413e7	1.53235	0.766176	−	0.642631i	$-0.222157\pi$
$719$	2.12413e7	1.53235	0.766176	+	0.642631i	$0.222157\pi$
$720$	0	0
$721$	1.34459e6	0.0963276
$722$	1.16528e7	0.831928
$723$	0	0
$724$	1.51071e7	1.07111
$725$	300278.	0.0212167
$726$	0	0
$727$	−1.06687e7	−0.748646	−0.374323	−	0.927298i	$-0.622125\pi$
$727$	−1.06687e7	−0.748646	−0.374323	+	0.927298i	$0.622125\pi$
$728$	−6.41571e6	−0.448659
$729$	0	0
$730$	−2.81652e6	−0.195617
$731$	695424.	0.0481345
$732$	0	0
$733$	−2.30788e7	−1.58655	−0.793274	−	0.608864i	$-0.791625\pi$
$733$	−2.30788e7	−1.58655	−0.793274	+	0.608864i	$0.791625\pi$
$734$	−2.61544e7	−1.79186
$735$	0	0
$736$	9.89705e6	0.673459
$737$	−1.89589e6	−0.128572
$738$	0	0
$739$	−2.27313e7	−1.53113	−0.765566	−	0.643358i	$-0.777541\pi$
$739$	−2.27313e7	−1.53113	−0.765566	+	0.643358i	$0.777541\pi$
$740$	2.54736e6	0.171006
$741$	0	0
$742$	−723625.	−0.0482507
$743$	1.49328e7	0.992360	0.496180	−	0.868220i	$-0.334736\pi$
$743$	1.49328e7	0.992360	0.496180	+	0.868220i	$0.334736\pi$
$744$	0	0
$745$	6.79525e6	0.448554
$746$	4.24624e7	2.79356
$747$	0	0
$748$	−1.00891e6	−0.0659321
$749$	4.55584e6	0.296732
$750$	0	0
$751$	−1.92964e7	−1.24846	−0.624232	−	0.781239i	$-0.714588\pi$
$751$	−1.92964e7	−1.24846	−0.624232	+	0.781239i	$0.714588\pi$
$752$	1.71126e6	0.110350
$753$	0	0
$754$	−3.90431e6	−0.250101
$755$	5.91412e6	0.377592
$756$	0	0
$757$	1.35397e7	0.858756	0.429378	−	0.903125i	$-0.358733\pi$
$757$	1.35397e7	0.858756	0.429378	+	0.903125i	$0.358733\pi$
$758$	2.63616e7	1.66648
$759$	0	0
$760$	9.68626e6	0.608306
$761$	−2.60669e7	−1.63165	−0.815825	−	0.578299i	$-0.803717\pi$
$761$	−2.60669e7	−1.63165	−0.815825	+	0.578299i	$0.803717\pi$
$762$	0	0
$763$	−5.50271e6	−0.342188
$764$	−5.01042e7	−3.10557
$765$	0	0
$766$	2.57730e7	1.58706
$767$	−1.39238e7	−0.854611
$768$	0	0
$769$	−1.65354e7	−1.00832	−0.504162	−	0.863609i	$-0.668199\pi$
$769$	−1.65354e7	−1.00832	−0.504162	+	0.863609i	$0.668199\pi$
$770$	−1.02044e6	−0.0620242
$771$	0	0
$772$	1.61049e7	0.972554
$773$	−8.95978e6	−0.539323	−0.269661	−	0.962955i	$-0.586912\pi$
$773$	−8.95978e6	−0.539323	−0.269661	+	0.962955i	$0.586912\pi$
$774$	0	0
$775$	1.09968e6	0.0657677
$776$	−7.36841e6	−0.439258
$777$	0	0
$778$	−3.65841e7	−2.16692
$779$	−8.69698e6	−0.513482
$780$	0	0
$781$	−1.29343e6	−0.0758781
$782$	2.77183e6	0.162088
$783$	0	0
$784$	−2.14110e6	−0.124407
$785$	−5.29561e6	−0.306720
$786$	0	0
$787$	−2.31723e7	−1.33362	−0.666812	−	0.745226i	$-0.732341\pi$
$787$	−2.31723e7	−1.33362	−0.666812	+	0.745226i	$0.732341\pi$
$788$	5.07061e7	2.90901
$789$	0	0
$790$	2.01779e7	1.15029
$791$	−5.75765e6	−0.327193
$792$	0	0
$793$	3.20023e7	1.80717
$794$	−5.13994e7	−2.89339
$795$	0	0
$796$	−5.41010e7	−3.02638
$797$	3.40811e7	1.90050	0.950250	−	0.311489i	$-0.100828\pi$
$797$	3.40811e7	1.90050	0.950250	+	0.311489i	$0.100828\pi$
$798$	0	0
$799$	−1.92230e6	−0.106525
$800$	−3.20936e6	−0.177294
$801$	0	0
$802$	2.58470e6	0.141897
$803$	1.47288e6	0.0806082
$804$	0	0
$805$	1.75622e6	0.0955188
$806$	−1.42984e7	−0.775265
$807$	0	0
$808$	3.71857e7	2.00377
$809$	−1.77957e6	−0.0955969	−0.0477985	−	0.998857i	$-0.515221\pi$
$809$	−1.77957e6	−0.0955969	−0.0477985	+	0.998857i	$0.515221\pi$
$810$	0	0
$811$	1.28099e7	0.683900	0.341950	−	0.939718i	$-0.388913\pi$
$811$	1.28099e7	0.683900	0.341950	+	0.939718i	$0.388913\pi$
$812$	939660.	0.0500127
$813$	0	0
$814$	−2.12653e6	−0.112489
$815$	8.53287e6	0.449988
$816$	0	0
$817$	8.64955e6	0.453355
$818$	4.78618e7	2.50095
$819$	0	0
$820$	−6.03687e6	−0.313528
$821$	−1.47980e7	−0.766205	−0.383102	−	0.923706i	$-0.625144\pi$
$821$	−1.47980e7	−0.766205	−0.383102	+	0.923706i	$0.625144\pi$
$822$	0	0
$823$	−1.17405e7	−0.604207	−0.302103	−	0.953275i	$-0.597689\pi$
$823$	−1.17405e7	−0.604207	−0.302103	+	0.953275i	$0.597689\pi$
$824$	7.39572e6	0.379457
$825$	0	0
$826$	5.34945e6	0.272809
$827$	−1.61763e7	−0.822461	−0.411231	−	0.911531i	$-0.634901\pi$
$827$	−1.61763e7	−0.822461	−0.411231	+	0.911531i	$0.634901\pi$
$828$	0	0
$829$	−1.56514e7	−0.790984	−0.395492	−	0.918470i	$-0.629426\pi$
$829$	−1.56514e7	−0.790984	−0.395492	+	0.918470i	$0.629426\pi$
$830$	2.24975e7	1.13354
$831$	0	0
$832$	4.56158e7	2.28458
$833$	2.40514e6	0.120096
$834$	0	0
$835$	−4.51370e6	−0.224035
$836$	−1.25486e7	−0.620982
$837$	0	0
$838$	5.59161e7	2.75060
$839$	−2.70692e7	−1.32761	−0.663806	−	0.747905i	$-0.731060\pi$
$839$	−2.70692e7	−1.32761	−0.663806	+	0.747905i	$0.731060\pi$
$840$	0	0
$841$	−2.02803e7	−0.988746
$842$	−8.28707e6	−0.402829
$843$	0	0
$844$	2.66951e7	1.28996
$845$	−9.99119e6	−0.481366
$846$	0	0
$847$	533632.	0.0255584
$848$	−296727.	−0.0141699
$849$	0	0
$850$	−898836.	−0.0426710
$851$	3.65984e6	0.173236
$852$	0	0
$853$	−2.43979e7	−1.14810	−0.574050	−	0.818820i	$-0.694629\pi$
$853$	−2.43979e7	−1.14810	−0.574050	+	0.818820i	$0.694629\pi$
$854$	−1.22951e7	−0.576884
$855$	0	0
$856$	2.50588e7	1.16889
$857$	−1.33841e7	−0.622496	−0.311248	−	0.950329i	$-0.600747\pi$
$857$	−1.33841e7	−0.622496	−0.311248	+	0.950329i	$0.600747\pi$
$858$	0	0
$859$	−2.60324e7	−1.20374	−0.601869	−	0.798595i	$-0.705577\pi$
$859$	−2.60324e7	−1.20374	−0.601869	+	0.798595i	$0.705577\pi$
$860$	6.00394e6	0.276815
$861$	0	0
$862$	1.48887e7	0.682479
$863$	2.45378e7	1.12152	0.560762	−	0.827977i	$-0.310508\pi$
$863$	2.45378e7	1.12152	0.560762	+	0.827977i	$0.310508\pi$
$864$	0	0
$865$	−1.60830e7	−0.730849
$866$	1.72185e7	0.780190
$867$	0	0
$868$	3.44123e6	0.155030
$869$	−1.05519e7	−0.474002
$870$	0	0
$871$	1.37575e7	0.614460
$872$	−3.02669e7	−1.34796
$873$	0	0
$874$	3.44755e7	1.52662
$875$	−569497.	−0.0251461
$876$	0	0
$877$	1.99034e7	0.873833	0.436917	−	0.899502i	$-0.356070\pi$
$877$	1.99034e7	0.873833	0.436917	+	0.899502i	$0.356070\pi$
$878$	2.54568e7	1.11447
$879$	0	0
$880$	−418438.	−0.0182148
$881$	3.90880e7	1.69669	0.848346	−	0.529442i	$-0.177599\pi$
$881$	3.90880e7	1.69669	0.848346	+	0.529442i	$0.177599\pi$
$882$	0	0
$883$	2.14490e7	0.925776	0.462888	−	0.886417i	$-0.346813\pi$
$883$	2.14490e7	0.925776	0.462888	+	0.886417i	$0.346813\pi$
$884$	7.32109e6	0.315098
$885$	0	0
$886$	−2.92287e7	−1.25091
$887$	−6.08624e6	−0.259741	−0.129870	−	0.991531i	$-0.541456\pi$
$887$	−6.08624e6	−0.259741	−0.129870	+	0.991531i	$0.541456\pi$
$888$	0	0
$889$	9.79769e6	0.415786
$890$	8.94045e6	0.378342
$891$	0	0
$892$	6.12462e7	2.57731
$893$	−2.39091e7	−1.00331
$894$	0	0
$895$	−6.67306e6	−0.278463
$896$	−1.15363e7	−0.480060
$897$	0	0
$898$	1.18012e6	0.0488356
$899$	845336.	0.0348843
$900$	0	0
$901$	333319.	0.0136788
$902$	5.03955e6	0.206241
$903$	0	0
$904$	−3.16691e7	−1.28889
$905$	−7.03823e6	−0.285655
$906$	0	0
$907$	−2.14459e7	−0.865618	−0.432809	−	0.901486i	$-0.642477\pi$
$907$	−2.14459e7	−0.865618	−0.432809	+	0.901486i	$0.642477\pi$
$908$	−3.59232e7	−1.44597
$909$	0	0
$910$	7.40479e6	0.296421
$911$	−1.03374e7	−0.412682	−0.206341	−	0.978480i	$-0.566156\pi$
$911$	−1.03374e7	−0.412682	−0.206341	+	0.978480i	$0.566156\pi$
$912$	0	0
$913$	−1.17649e7	−0.467102
$914$	2.02884e7	0.803310
$915$	0	0
$916$	3.15557e7	1.24262
$917$	1.33732e7	0.525185
$918$	0	0
$919$	−4.75142e7	−1.85581	−0.927907	−	0.372811i	$-0.878394\pi$
$919$	−4.75142e7	−1.85581	−0.927907	+	0.372811i	$0.878394\pi$
$920$	9.65984e6	0.376271
$921$	0	0
$922$	−4.81488e7	−1.86534
$923$	9.38575e6	0.362631
$924$	0	0
$925$	−1.18679e6	−0.0456059
$926$	2.46769e7	0.945720
$927$	0	0
$928$	−2.46707e6	−0.0940398
$929$	−6.08962e6	−0.231500	−0.115750	−	0.993278i	$-0.536927\pi$
$929$	−6.08962e6	−0.231500	−0.115750	+	0.993278i	$0.536927\pi$
$930$	0	0
$931$	2.99146e7	1.13112
$932$	−1.07008e7	−0.403532
$933$	0	0
$934$	2.18034e7	0.817820
$935$	470040.	0.0175835
$936$	0	0
$937$	−2.73781e6	−0.101872	−0.0509360	−	0.998702i	$-0.516220\pi$
$937$	−2.73781e6	−0.101872	−0.0509360	+	0.998702i	$0.516220\pi$
$938$	−5.28556e6	−0.196148
$939$	0	0
$940$	−1.65961e7	−0.612615
$941$	−3.73849e7	−1.37633	−0.688165	−	0.725554i	$-0.741583\pi$
$941$	−3.73849e7	−1.37633	−0.688165	+	0.725554i	$0.741583\pi$
$942$	0	0
$943$	−8.67326e6	−0.317617
$944$	2.19358e6	0.0801166
$945$	0	0
$946$	−5.01206e6	−0.182091
$947$	4.08261e7	1.47932	0.739661	−	0.672979i	$-0.234986\pi$
$947$	4.08261e7	1.47932	0.739661	+	0.672979i	$0.234986\pi$
$948$	0	0
$949$	−1.06879e7	−0.385237
$950$	−1.11795e7	−0.401897
$951$	0	0
$952$	−1.13539e6	−0.0406024
$953$	4.60553e7	1.64266	0.821330	−	0.570453i	$-0.193232\pi$
$953$	4.60553e7	1.64266	0.821330	+	0.570453i	$0.193232\pi$
$954$	0	0
$955$	2.33431e7	0.828228
$956$	2.18965e7	0.774873
$957$	0	0
$958$	4.42709e7	1.55849
$959$	6.66730e6	0.234101
$960$	0	0
$961$	−2.55333e7	−0.891865
$962$	1.54311e7	0.537599
$963$	0	0
$964$	6.65509e7	2.30654
$965$	−7.50310e6	−0.259372
$966$	0	0
$967$	1.48162e7	0.509532	0.254766	−	0.967003i	$-0.418002\pi$
$967$	1.48162e7	0.509532	0.254766	+	0.967003i	$0.418002\pi$
$968$	2.93517e6	0.100680
$969$	0	0
$970$	8.50436e6	0.290210
$971$	3.27641e7	1.11519	0.557597	−	0.830112i	$-0.311723\pi$
$971$	3.27641e7	1.11519	0.557597	+	0.830112i	$0.311723\pi$
$972$	0	0
$973$	307250.	0.0104042
$974$	−3.06828e7	−1.03633
$975$	0	0
$976$	−5.04170e6	−0.169415
$977$	4.06772e6	0.136337	0.0681686	−	0.997674i	$-0.478284\pi$
$977$	4.06772e6	0.136337	0.0681686	+	0.997674i	$0.478284\pi$
$978$	0	0
$979$	−4.67535e6	−0.155904
$980$	2.07647e7	0.690655
$981$	0	0
$982$	2.79765e7	0.925795
$983$	−3.76095e7	−1.24141	−0.620704	−	0.784045i	$-0.713153\pi$
$983$	−3.76095e7	−1.24141	−0.620704	+	0.784045i	$0.713153\pi$
$984$	0	0
$985$	−2.36235e7	−0.775807
$986$	−690944.	−0.0226335
$987$	0	0
$988$	9.10583e7	2.96775
$989$	8.62596e6	0.280425
$990$	0	0
$991$	1.02888e7	0.332797	0.166399	−	0.986059i	$-0.446786\pi$
$991$	1.02888e7	0.332797	0.166399	+	0.986059i	$0.446786\pi$
$992$	−9.03495e6	−0.291505
$993$	0	0
$994$	−3.60596e6	−0.115759
$995$	2.52052e7	0.807108
$996$	0	0
$997$	4.37776e7	1.39481	0.697403	−	0.716680i	$-0.254339\pi$
$997$	4.37776e7	1.39481	0.697403	+	0.716680i	$0.254339\pi$
$998$	4.13277e7	1.31345
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.e.1.3		3
3.2	odd	2		165.6.a.a.1.1	✓	3
15.14	odd	2		825.6.a.j.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.a.1.1	✓	3	3.2	odd	2
495.6.a.e.1.3		3	1.1	even	1	trivial
825.6.a.j.1.3		3	15.14	odd	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+9.25531 q^{2} +53.6607 q^{4} -25.0000 q^{5} +36.4478 q^{7} +200.476 q^{8} +O(q^{10})\)	\(q+9.25531 q^{2} +53.6607 q^{4} -25.0000 q^{5} +36.4478 q^{7} +200.476 q^{8} -231.383 q^{10} +121.000 q^{11} -878.032 q^{13} +337.336 q^{14} +138.327 q^{16} -155.385 q^{17} -1932.65 q^{19} -1341.52 q^{20} +1119.89 q^{22} -1927.38 q^{23} +625.000 q^{25} -8126.46 q^{26} +1955.81 q^{28} +480.444 q^{29} +1759.49 q^{31} -5134.98 q^{32} -1438.14 q^{34} -911.195 q^{35} -1898.87 q^{37} -17887.3 q^{38} -5011.91 q^{40} +4500.03 q^{41} -4475.49 q^{43} +6492.94 q^{44} -17838.5 q^{46} +12371.2 q^{47} -15478.6 q^{49} +5784.57 q^{50} -47115.8 q^{52} -2145.12 q^{53} -3025.00 q^{55} +7306.92 q^{56} +4446.66 q^{58} +15857.9 q^{59} -36447.7 q^{61} +16284.6 q^{62} -51952.3 q^{64} +21950.8 q^{65} -15668.5 q^{67} -8338.07 q^{68} -8433.39 q^{70} -10689.5 q^{71} +12172.6 q^{73} -17574.6 q^{74} -103707. q^{76} +4410.19 q^{77} -87205.6 q^{79} -3458.17 q^{80} +41649.2 q^{82} -97230.6 q^{83} +3884.63 q^{85} -41422.0 q^{86} +24257.6 q^{88} -38639.2 q^{89} -32002.4 q^{91} -103424. q^{92} +114499. q^{94} +48316.2 q^{95} -36754.5 q^{97} -143259. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10}) \) 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8	\( 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100}) \) 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 - 175 * q^10 + 363 * q^11 - 654 * q^13 + 728 * q^14 - 415 * q^16 + 2366 * q^17 - 2872 * q^19 - 625 * q^20 + 847 * q^22 - 2272 * q^23 + 1875 * q^25 - 3422 * q^26 + 4592 * q^28 + 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 2506 * q^34 + 4300 * q^35 - 9126 * q^37 - 13076 * q^38 - 5775 * q^40 + 8758 * q^41 - 14672 * q^43 + 3025 * q^44 - 28768 * q^46 + 19392 * q^47 - 26629 * q^49 + 4375 * q^50 - 61506 * q^52 + 4598 * q^53 - 9075 * q^55 - 2688 * q^56 + 8550 * q^58 + 9348 * q^59 - 60078 * q^61 + 14096 * q^62 - 7087 * q^64 + 16350 * q^65 - 38468 * q^67 - 59778 * q^68 - 18200 * q^70 + 74032 * q^71 - 44442 * q^73 - 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 108116 * q^79 + 10375 * q^80 - 92230 * q^82 + 81892 * q^83 - 59150 * q^85 - 126412 * q^86 + 27951 * q^88 - 167342 * q^89 - 31832 * q^91 - 72960 * q^92 + 12728 * q^94 + 71800 * q^95 + 159702 * q^97 - 163121 * q^98

Introduction

L-functions

Modular forms

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Database

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