LMFDB - Embedded newform 1040.6.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,6,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1040.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1040.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:	$166.799172605$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 1252x^{5} + 1388x^{4} + 394896x^{3} - 722832x^{2} - 18679104x - 35596800 $ x^7 - 2x^6 - 1252x^5 + 1388x^4 + 394896x^3 - 722832x^2 - 18679104x - 35596800
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.57223$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.57223 q^{3} +25.0000 q^{5} +193.140 q^{7} -222.095 q^{9} -289.873 q^{11} +169.000 q^{13} -114.306 q^{15} -1813.23 q^{17} -1934.30 q^{19} -883.081 q^{21} -4946.34 q^{23} +625.000 q^{25} +2126.52 q^{27} +8657.48 q^{29} -2133.98 q^{31} +1325.37 q^{33} +4828.50 q^{35} +8823.83 q^{37} -772.707 q^{39} -7374.25 q^{41} +22164.4 q^{43} -5552.37 q^{45} +2806.61 q^{47} +20496.1 q^{49} +8290.50 q^{51} +20528.5 q^{53} -7246.83 q^{55} +8844.06 q^{57} -23362.5 q^{59} +4880.67 q^{61} -42895.4 q^{63} +4225.00 q^{65} +1610.81 q^{67} +22615.8 q^{69} -32885.3 q^{71} +42088.0 q^{73} -2857.64 q^{75} -55986.1 q^{77} +19996.4 q^{79} +44246.1 q^{81} -53725.6 q^{83} -45330.7 q^{85} -39584.0 q^{87} +91913.0 q^{89} +32640.7 q^{91} +9757.07 q^{93} -48357.5 q^{95} +47552.0 q^{97} +64379.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 175 q^{5} - 86 q^{7} + 807 q^{9} - 508 q^{11} + 1183 q^{13} + 50 q^{15} - 566 q^{17} - 164 q^{19} + 1816 q^{21} + 318 q^{23} + 4375 q^{25} + 2384 q^{27} + 6410 q^{29} - 6208 q^{31} + 10556 q^{33}+ \cdots - 59968 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 175 * q^5 - 86 * q^7 + 807 * q^9 - 508 * q^11 + 1183 * q^13 + 50 * q^15 - 566 * q^17 - 164 * q^19 + 1816 * q^21 + 318 * q^23 + 4375 * q^25 + 2384 * q^27 + 6410 * q^29 - 6208 * q^31 + 10556 * q^33 - 2150 * q^35 + 7870 * q^37 + 338 * q^39 + 24950 * q^41 - 7122 * q^43 + 20175 * q^45 + 3842 * q^47 + 34395 * q^49 - 41460 * q^51 + 36230 * q^53 - 12700 * q^55 + 90996 * q^57 - 48596 * q^59 + 126398 * q^61 - 85798 * q^63 + 29575 * q^65 - 58366 * q^67 + 127060 * q^69 - 160540 * q^71 + 56766 * q^73 + 1250 * q^75 + 226236 * q^77 - 146688 * q^79 + 149379 * q^81 - 139794 * q^83 - 14150 * q^85 - 182040 * q^87 + 290982 * q^89 - 14534 * q^91 + 78180 * q^93 - 4100 * q^95 + 206146 * q^97 - 59968 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−4.57223	−0.293309	−0.146654	−	0.989188i	$-0.546851\pi$
$3$	−4.57223	−0.293309	−0.146654	+	0.989188i	$0.546851\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	193.140	1.48980	0.744899	−	0.667177i	$-0.232498\pi$
$7$	193.140	1.48980	0.744899	+	0.667177i	$0.232498\pi$
$8$	0	0
$9$	−222.095	−0.913970
$10$	0	0
$11$	−289.873	−0.722314	−0.361157	−	0.932505i	$-0.617618\pi$
$11$	−289.873	−0.722314	−0.361157	+	0.932505i	$0.617618\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−114.306	−0.131172
$16$	0	0
$17$	−1813.23	−1.52170	−0.760852	−	0.648925i	$-0.775219\pi$
$17$	−1813.23	−1.52170	−0.760852	+	0.648925i	$0.775219\pi$
$18$	0	0
$19$	−1934.30	−1.22925	−0.614624	−	0.788820i	$-0.710692\pi$
$19$	−1934.30	−1.22925	−0.614624	+	0.788820i	$0.710692\pi$
$20$	0	0
$21$	−883.081	−0.436971
$22$	0	0
$23$	−4946.34	−1.94968	−0.974842	−	0.222897i	$-0.928449\pi$
$23$	−4946.34	−1.94968	−0.974842	+	0.222897i	$0.928449\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	2126.52	0.561384
$28$	0	0
$29$	8657.48	1.91160	0.955798	−	0.294023i	$-0.0949941\pi$
$29$	8657.48	1.91160	0.955798	+	0.294023i	$0.0949941\pi$
$30$	0	0
$31$	−2133.98	−0.398829	−0.199415	−	0.979915i	$-0.563904\pi$
$31$	−2133.98	−0.398829	−0.199415	+	0.979915i	$0.563904\pi$
$32$	0	0
$33$	1325.37	0.211861
$34$	0	0
$35$	4828.50	0.666258
$36$	0	0
$37$	8823.83	1.05963	0.529813	−	0.848114i	$-0.322262\pi$
$37$	8823.83	1.05963	0.529813	+	0.848114i	$0.322262\pi$
$38$	0	0
$39$	−772.707	−0.0813492
$40$	0	0
$41$	−7374.25	−0.685107	−0.342553	−	0.939498i	$-0.611292\pi$
$41$	−7374.25	−0.685107	−0.342553	+	0.939498i	$0.611292\pi$
$42$	0	0
$43$	22164.4	1.82803	0.914016	−	0.405677i	$-0.132964\pi$
$43$	22164.4	1.82803	0.914016	+	0.405677i	$0.132964\pi$
$44$	0	0
$45$	−5552.37	−0.408740
$46$	0	0
$47$	2806.61	0.185326	0.0926631	−	0.995698i	$-0.470462\pi$
$47$	2806.61	0.185326	0.0926631	+	0.995698i	$0.470462\pi$
$48$	0	0
$49$	20496.1	1.21950
$50$	0	0
$51$	8290.50	0.446329
$52$	0	0
$53$	20528.5	1.00385	0.501923	−	0.864912i	$-0.332626\pi$
$53$	20528.5	1.00385	0.501923	+	0.864912i	$0.332626\pi$
$54$	0	0
$55$	−7246.83	−0.323029
$56$	0	0
$57$	8844.06	0.360549
$58$	0	0
$59$	−23362.5	−0.873756	−0.436878	−	0.899521i	$-0.643916\pi$
$59$	−23362.5	−0.873756	−0.436878	+	0.899521i	$0.643916\pi$
$60$	0	0
$61$	4880.67	0.167940	0.0839701	−	0.996468i	$-0.473240\pi$
$61$	4880.67	0.167940	0.0839701	+	0.996468i	$0.473240\pi$
$62$	0	0
$63$	−42895.4	−1.36163
$64$	0	0
$65$	4225.00	0.124035
$66$	0	0
$67$	1610.81	0.0438386	0.0219193	−	0.999760i	$-0.493022\pi$
$67$	1610.81	0.0438386	0.0219193	+	0.999760i	$0.493022\pi$
$68$	0	0
$69$	22615.8	0.571859
$70$	0	0
$71$	−32885.3	−0.774205	−0.387102	−	0.922037i	$-0.626524\pi$
$71$	−32885.3	−0.774205	−0.387102	+	0.922037i	$0.626524\pi$
$72$	0	0
$73$	42088.0	0.924381	0.462190	−	0.886781i	$-0.347064\pi$
$73$	42088.0	0.924381	0.462190	+	0.886781i	$0.347064\pi$
$74$	0	0
$75$	−2857.64	−0.0586617
$76$	0	0
$77$	−55986.1	−1.07610
$78$	0	0
$79$	19996.4	0.360483	0.180242	−	0.983622i	$-0.442312\pi$
$79$	19996.4	0.360483	0.180242	+	0.983622i	$0.442312\pi$
$80$	0	0
$81$	44246.1	0.749311
$82$	0	0
$83$	−53725.6	−0.856024	−0.428012	−	0.903773i	$-0.640786\pi$
$83$	−53725.6	−0.856024	−0.428012	+	0.903773i	$0.640786\pi$
$84$	0	0
$85$	−45330.7	−0.680527
$86$	0	0
$87$	−39584.0	−0.560688
$88$	0	0
$89$	91913.0	1.22999	0.614996	−	0.788531i	$-0.289158\pi$
$89$	91913.0	1.22999	0.614996	+	0.788531i	$0.289158\pi$
$90$	0	0
$91$	32640.7	0.413196
$92$	0	0
$93$	9757.07	0.116980
$94$	0	0
$95$	−48357.5	−0.549736
$96$	0	0
$97$	47552.0	0.513145	0.256572	−	0.966525i	$-0.417407\pi$
$97$	47552.0	0.513145	0.256572	+	0.966525i	$0.417407\pi$
$98$	0	0
$99$	64379.3	0.660174
$100$	0	0
$101$	126491.	1.23384	0.616919	−	0.787027i	$-0.288381\pi$
$101$	126491.	1.23384	0.616919	+	0.787027i	$0.288381\pi$
$102$	0	0
$103$	120144.	1.11586	0.557928	−	0.829889i	$-0.311596\pi$
$103$	120144.	1.11586	0.557928	+	0.829889i	$0.311596\pi$
$104$	0	0
$105$	−22077.0	−0.195419
$106$	0	0
$107$	82869.1	0.699734	0.349867	−	0.936799i	$-0.386227\pi$
$107$	82869.1	0.699734	0.349867	+	0.936799i	$0.386227\pi$
$108$	0	0
$109$	−165073.	−1.33079	−0.665397	−	0.746490i	$-0.731738\pi$
$109$	−165073.	−1.33079	−0.665397	+	0.746490i	$0.731738\pi$
$110$	0	0
$111$	−40344.6	−0.310798
$112$	0	0
$113$	191436.	1.41035	0.705177	−	0.709032i	$-0.250868\pi$
$113$	191436.	1.41035	0.705177	+	0.709032i	$0.250868\pi$
$114$	0	0
$115$	−123658.	−0.871925
$116$	0	0
$117$	−37534.0	−0.253490
$118$	0	0
$119$	−350207.	−2.26703
$120$	0	0
$121$	−77024.6	−0.478262
$122$	0	0
$123$	33716.8	0.200948
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−66362.9	−0.365104	−0.182552	−	0.983196i	$-0.558436\pi$
$127$	−66362.9	−0.365104	−0.182552	+	0.983196i	$0.558436\pi$
$128$	0	0
$129$	−101341.	−0.536178
$130$	0	0
$131$	−74916.8	−0.381418	−0.190709	−	0.981647i	$-0.561079\pi$
$131$	−74916.8	−0.381418	−0.190709	+	0.981647i	$0.561079\pi$
$132$	0	0
$133$	−373591.	−1.83133
$134$	0	0
$135$	53163.0	0.251059
$136$	0	0
$137$	198208.	0.902235	0.451118	−	0.892464i	$-0.351025\pi$
$137$	198208.	0.902235	0.451118	+	0.892464i	$0.351025\pi$
$138$	0	0
$139$	−78716.9	−0.345566	−0.172783	−	0.984960i	$-0.555276\pi$
$139$	−78716.9	−0.345566	−0.172783	+	0.984960i	$0.555276\pi$
$140$	0	0
$141$	−12832.5	−0.0543578
$142$	0	0
$143$	−48988.6	−0.200334
$144$	0	0
$145$	216437.	0.854892
$146$	0	0
$147$	−93713.0	−0.357690
$148$	0	0
$149$	−467138.	−1.72377	−0.861885	−	0.507104i	$-0.830716\pi$
$149$	−467138.	−1.72377	−0.861885	+	0.507104i	$0.830716\pi$
$150$	0	0
$151$	128317.	0.457974	0.228987	−	0.973429i	$-0.426459\pi$
$151$	128317.	0.457974	0.228987	+	0.973429i	$0.426459\pi$
$152$	0	0
$153$	402708.	1.39079
$154$	0	0
$155$	−53349.6	−0.178362
$156$	0	0
$157$	402301.	1.30257	0.651287	−	0.758832i	$-0.274229\pi$
$157$	402301.	1.30257	0.651287	+	0.758832i	$0.274229\pi$
$158$	0	0
$159$	−93861.0	−0.294437
$160$	0	0
$161$	−955337.	−2.90464
$162$	0	0
$163$	464263.	1.36866	0.684330	−	0.729173i	$-0.260095\pi$
$163$	464263.	1.36866	0.684330	+	0.729173i	$0.260095\pi$
$164$	0	0
$165$	33134.2	0.0947472
$166$	0	0
$167$	78682.3	0.218316	0.109158	−	0.994024i	$-0.465185\pi$
$167$	78682.3	0.218316	0.109158	+	0.994024i	$0.465185\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	429598.	1.12350
$172$	0	0
$173$	−432227.	−1.09798	−0.548992	−	0.835827i	$-0.684988\pi$
$173$	−432227.	−1.09798	−0.548992	+	0.835827i	$0.684988\pi$
$174$	0	0
$175$	120713.	0.297960
$176$	0	0
$177$	106819.	0.256280
$178$	0	0
$179$	402481.	0.938886	0.469443	−	0.882963i	$-0.344455\pi$
$179$	402481.	0.938886	0.469443	+	0.882963i	$0.344455\pi$
$180$	0	0
$181$	110182.	0.249984	0.124992	−	0.992158i	$-0.460109\pi$
$181$	110182.	0.249984	0.124992	+	0.992158i	$0.460109\pi$
$182$	0	0
$183$	−22315.5	−0.0492583
$184$	0	0
$185$	220596.	0.473880
$186$	0	0
$187$	525606.	1.09915
$188$	0	0
$189$	410716.	0.836349
$190$	0	0
$191$	31198.5	0.0618801	0.0309400	−	0.999521i	$-0.490150\pi$
$191$	31198.5	0.0618801	0.0309400	+	0.999521i	$0.490150\pi$
$192$	0	0
$193$	511652.	0.988738	0.494369	−	0.869252i	$-0.335399\pi$
$193$	511652.	0.988738	0.494369	+	0.869252i	$0.335399\pi$
$194$	0	0
$195$	−19317.7	−0.0363805
$196$	0	0
$197$	−521813.	−0.957964	−0.478982	−	0.877825i	$-0.658994\pi$
$197$	−521813.	−0.957964	−0.478982	+	0.877825i	$0.658994\pi$
$198$	0	0
$199$	44734.2	0.0800769	0.0400385	−	0.999198i	$-0.487252\pi$
$199$	44734.2	0.0800769	0.0400385	+	0.999198i	$0.487252\pi$
$200$	0	0
$201$	−7364.99	−0.0128583
$202$	0	0
$203$	1.67211e6	2.84789
$204$	0	0
$205$	−184356.	−0.306389
$206$	0	0
$207$	1.09856e6	1.78195
$208$	0	0
$209$	560701.	0.887904
$210$	0	0
$211$	−845530.	−1.30744	−0.653721	−	0.756735i	$-0.726793\pi$
$211$	−845530.	−1.30744	−0.653721	+	0.756735i	$0.726793\pi$
$212$	0	0
$213$	150359.	0.227081
$214$	0	0
$215$	554109.	0.817521
$216$	0	0
$217$	−412158.	−0.594175
$218$	0	0
$219$	−192436.	−0.271129
$220$	0	0
$221$	−306436.	−0.422045
$222$	0	0
$223$	1.14273e6	1.53880	0.769399	−	0.638769i	$-0.220556\pi$
$223$	1.14273e6	1.53880	0.769399	+	0.638769i	$0.220556\pi$
$224$	0	0
$225$	−138809.	−0.182794
$226$	0	0
$227$	−356171.	−0.458768	−0.229384	−	0.973336i	$-0.573671\pi$
$227$	−356171.	−0.458768	−0.229384	+	0.973336i	$0.573671\pi$
$228$	0	0
$229$	1.16047e6	1.46234	0.731168	−	0.682198i	$-0.238976\pi$
$229$	1.16047e6	1.46234	0.731168	+	0.682198i	$0.238976\pi$
$230$	0	0
$231$	255982.	0.315630
$232$	0	0
$233$	926173.	1.11764	0.558821	−	0.829288i	$-0.311254\pi$
$233$	926173.	1.11764	0.558821	+	0.829288i	$0.311254\pi$
$234$	0	0
$235$	70165.2	0.0828804
$236$	0	0
$237$	−91428.4	−0.105733
$238$	0	0
$239$	−1.30321e6	−1.47577	−0.737885	−	0.674927i	$-0.764175\pi$
$239$	−1.30321e6	−1.47577	−0.737885	+	0.674927i	$0.764175\pi$
$240$	0	0
$241$	537456.	0.596074	0.298037	−	0.954554i	$-0.403668\pi$
$241$	537456.	0.596074	0.298037	+	0.954554i	$0.403668\pi$
$242$	0	0
$243$	−719048.	−0.781164
$244$	0	0
$245$	512403.	0.545377
$246$	0	0
$247$	−326897.	−0.340932
$248$	0	0
$249$	245646.	0.251079
$250$	0	0
$251$	−305760.	−0.306335	−0.153167	−	0.988200i	$-0.548947\pi$
$251$	−305760.	−0.306335	−0.153167	+	0.988200i	$0.548947\pi$
$252$	0	0
$253$	1.43381e6	1.40828
$254$	0	0
$255$	207262.	0.199604
$256$	0	0
$257$	−465369.	−0.439506	−0.219753	−	0.975556i	$-0.570525\pi$
$257$	−465369.	−0.439506	−0.219753	+	0.975556i	$0.570525\pi$
$258$	0	0
$259$	1.70424e6	1.57863
$260$	0	0
$261$	−1.92278e6	−1.74714
$262$	0	0
$263$	1.37558e6	1.22630	0.613149	−	0.789967i	$-0.289902\pi$
$263$	1.37558e6	1.22630	0.613149	+	0.789967i	$0.289902\pi$
$264$	0	0
$265$	513212.	0.448934
$266$	0	0
$267$	−420248.	−0.360767
$268$	0	0
$269$	1.25766e6	1.05970	0.529851	−	0.848091i	$-0.322248\pi$
$269$	1.25766e6	1.05970	0.529851	+	0.848091i	$0.322248\pi$
$270$	0	0
$271$	1.98346e6	1.64059	0.820294	−	0.571943i	$-0.193810\pi$
$271$	1.98346e6	1.64059	0.820294	+	0.571943i	$0.193810\pi$
$272$	0	0
$273$	−149241.	−0.121194
$274$	0	0
$275$	−181171.	−0.144463
$276$	0	0
$277$	−1.83132e6	−1.43405	−0.717026	−	0.697047i	$-0.754497\pi$
$277$	−1.83132e6	−1.43405	−0.717026	+	0.697047i	$0.754497\pi$
$278$	0	0
$279$	473947.	0.364518
$280$	0	0
$281$	−716556.	−0.541358	−0.270679	−	0.962670i	$-0.587248\pi$
$281$	−716556.	−0.541358	−0.270679	+	0.962670i	$0.587248\pi$
$282$	0	0
$283$	955059.	0.708866	0.354433	−	0.935081i	$-0.384674\pi$
$283$	955059.	0.708866	0.354433	+	0.935081i	$0.384674\pi$
$284$	0	0
$285$	221102.	0.161243
$286$	0	0
$287$	−1.42426e6	−1.02067
$288$	0	0
$289$	1.86794e6	1.31558
$290$	0	0
$291$	−217419.	−0.150510
$292$	0	0
$293$	−1.67941e6	−1.14285	−0.571423	−	0.820655i	$-0.693609\pi$
$293$	−1.67941e6	−1.14285	−0.571423	+	0.820655i	$0.693609\pi$
$294$	0	0
$295$	−584063.	−0.390755
$296$	0	0
$297$	−616421.	−0.405496
$298$	0	0
$299$	−835931.	−0.540745
$300$	0	0
$301$	4.28083e6	2.72340
$302$	0	0
$303$	−578348.	−0.361895
$304$	0	0
$305$	122017.	0.0751051
$306$	0	0
$307$	−738204.	−0.447024	−0.223512	−	0.974701i	$-0.571752\pi$
$307$	−738204.	−0.447024	−0.223512	+	0.974701i	$0.571752\pi$
$308$	0	0
$309$	−549325.	−0.327291
$310$	0	0
$311$	−1.65404e6	−0.969715	−0.484857	−	0.874593i	$-0.661129\pi$
$311$	−1.65404e6	−0.969715	−0.484857	+	0.874593i	$0.661129\pi$
$312$	0	0
$313$	52478.5	0.0302775	0.0151388	−	0.999885i	$-0.495181\pi$
$313$	52478.5	0.0302775	0.0151388	+	0.999885i	$0.495181\pi$
$314$	0	0
$315$	−1.07239e6	−0.608940
$316$	0	0
$317$	341349.	0.190788	0.0953938	−	0.995440i	$-0.469589\pi$
$317$	341349.	0.190788	0.0953938	+	0.995440i	$0.469589\pi$
$318$	0	0
$319$	−2.50957e6	−1.38077
$320$	0	0
$321$	−378897.	−0.205238
$322$	0	0
$323$	3.50733e6	1.87055
$324$	0	0
$325$	105625.	0.0554700
$326$	0	0
$327$	754753.	0.390333
$328$	0	0
$329$	542069.	0.276099
$330$	0	0
$331$	−1.75708e6	−0.881500	−0.440750	−	0.897630i	$-0.645288\pi$
$331$	−1.75708e6	−0.881500	−0.440750	+	0.897630i	$0.645288\pi$
$332$	0	0
$333$	−1.95973e6	−0.968467
$334$	0	0
$335$	40270.2	0.0196052
$336$	0	0
$337$	3.52015e6	1.68844	0.844222	−	0.535993i	$-0.180063\pi$
$337$	3.52015e6	1.68844	0.844222	+	0.535993i	$0.180063\pi$
$338$	0	0
$339$	−875290.	−0.413669
$340$	0	0
$341$	618585.	0.288080
$342$	0	0
$343$	712518.	0.327010
$344$	0	0
$345$	565395.	0.255743
$346$	0	0
$347$	2.59869e6	1.15859	0.579296	−	0.815117i	$-0.303327\pi$
$347$	2.59869e6	1.15859	0.579296	+	0.815117i	$0.303327\pi$
$348$	0	0
$349$	568423.	0.249809	0.124905	−	0.992169i	$-0.460138\pi$
$349$	568423.	0.249809	0.124905	+	0.992169i	$0.460138\pi$
$350$	0	0
$351$	359382.	0.155700
$352$	0	0
$353$	450591.	0.192463	0.0962313	−	0.995359i	$-0.469321\pi$
$353$	450591.	0.192463	0.0962313	+	0.995359i	$0.469321\pi$
$354$	0	0
$355$	−822132.	−0.346235
$356$	0	0
$357$	1.60123e6	0.664940
$358$	0	0
$359$	−3.62131e6	−1.48296	−0.741480	−	0.670975i	$-0.765876\pi$
$359$	−3.62131e6	−1.48296	−0.741480	+	0.670975i	$0.765876\pi$
$360$	0	0
$361$	1.26541e6	0.511051
$362$	0	0
$363$	352174.	0.140278
$364$	0	0
$365$	1.05220e6	0.413396
$366$	0	0
$367$	−383281.	−0.148543	−0.0742715	−	0.997238i	$-0.523663\pi$
$367$	−383281.	−0.148543	−0.0742715	+	0.997238i	$0.523663\pi$
$368$	0	0
$369$	1.63778e6	0.626167
$370$	0	0
$371$	3.96488e6	1.49553
$372$	0	0
$373$	2.25251e6	0.838292	0.419146	−	0.907919i	$-0.362329\pi$
$373$	2.25251e6	0.838292	0.419146	+	0.907919i	$0.362329\pi$
$374$	0	0
$375$	−71441.1	−0.0262343
$376$	0	0
$377$	1.46311e6	0.530182
$378$	0	0
$379$	−108337.	−0.0387417	−0.0193708	−	0.999812i	$-0.506166\pi$
$379$	−108337.	−0.0387417	−0.0193708	+	0.999812i	$0.506166\pi$
$380$	0	0
$381$	303427.	0.107088
$382$	0	0
$383$	5.25708e6	1.83125	0.915625	−	0.402034i	$-0.131697\pi$
$383$	5.25708e6	1.83125	0.915625	+	0.402034i	$0.131697\pi$
$384$	0	0
$385$	−1.39965e6	−0.481248
$386$	0	0
$387$	−4.92259e6	−1.67077
$388$	0	0
$389$	3.56120e6	1.19323	0.596613	−	0.802529i	$-0.296513\pi$
$389$	3.56120e6	1.19323	0.596613	+	0.802529i	$0.296513\pi$
$390$	0	0
$391$	8.96884e6	2.96684
$392$	0	0
$393$	342537.	0.111873
$394$	0	0
$395$	499911.	0.161213
$396$	0	0
$397$	−2.15566e6	−0.686444	−0.343222	−	0.939254i	$-0.611518\pi$
$397$	−2.15566e6	−0.686444	−0.343222	+	0.939254i	$0.611518\pi$
$398$	0	0
$399$	1.70814e6	0.537146
$400$	0	0
$401$	−717167.	−0.222720	−0.111360	−	0.993780i	$-0.535521\pi$
$401$	−717167.	−0.222720	−0.111360	+	0.993780i	$0.535521\pi$
$402$	0	0
$403$	−360643.	−0.110615
$404$	0	0
$405$	1.10615e6	0.335102
$406$	0	0
$407$	−2.55779e6	−0.765384
$408$	0	0
$409$	4.88152e6	1.44293	0.721467	−	0.692448i	$-0.243468\pi$
$409$	4.88152e6	1.44293	0.721467	+	0.692448i	$0.243468\pi$
$410$	0	0
$411$	−906253.	−0.264633
$412$	0	0
$413$	−4.51224e6	−1.30172
$414$	0	0
$415$	−1.34314e6	−0.382825
$416$	0	0
$417$	359912.	0.101358
$418$	0	0
$419$	4.77315e6	1.32822	0.664111	−	0.747634i	$-0.268810\pi$
$419$	4.77315e6	1.32822	0.664111	+	0.747634i	$0.268810\pi$
$420$	0	0
$421$	1.13836e6	0.313021	0.156511	−	0.987676i	$-0.449975\pi$
$421$	1.13836e6	0.313021	0.156511	+	0.987676i	$0.449975\pi$
$422$	0	0
$423$	−623333.	−0.169383
$424$	0	0
$425$	−1.13327e6	−0.304341
$426$	0	0
$427$	942653.	0.250197
$428$	0	0
$429$	223987.	0.0587597
$430$	0	0
$431$	594447.	0.154142	0.0770708	−	0.997026i	$-0.475443\pi$
$431$	594447.	0.154142	0.0770708	+	0.997026i	$0.475443\pi$
$432$	0	0
$433$	−289328.	−0.0741603	−0.0370801	−	0.999312i	$-0.511806\pi$
$433$	−289328.	−0.0741603	−0.0370801	+	0.999312i	$0.511806\pi$
$434$	0	0
$435$	−989599.	−0.250747
$436$	0	0
$437$	9.56770e6	2.39665
$438$	0	0
$439$	−7.28780e6	−1.80482	−0.902412	−	0.430873i	$-0.858206\pi$
$439$	−7.28780e6	−1.80482	−0.902412	+	0.430873i	$0.858206\pi$
$440$	0	0
$441$	−4.55208e6	−1.11459
$442$	0	0
$443$	−5.66527e6	−1.37155	−0.685774	−	0.727814i	$-0.740536\pi$
$443$	−5.66527e6	−1.37155	−0.685774	+	0.727814i	$0.740536\pi$
$444$	0	0
$445$	2.29783e6	0.550069
$446$	0	0
$447$	2.13586e6	0.505597
$448$	0	0
$449$	−4.91365e6	−1.15024	−0.575120	−	0.818069i	$-0.695045\pi$
$449$	−4.91365e6	−1.15024	−0.575120	+	0.818069i	$0.695045\pi$
$450$	0	0
$451$	2.13760e6	0.494862
$452$	0	0
$453$	−586693.	−0.134328
$454$	0	0
$455$	816017.	0.184787
$456$	0	0
$457$	−6.01249e6	−1.34668	−0.673339	−	0.739333i	$-0.735141\pi$
$457$	−6.01249e6	−1.34668	−0.673339	+	0.739333i	$0.735141\pi$
$458$	0	0
$459$	−3.85587e6	−0.854261
$460$	0	0
$461$	6.60229e6	1.44691	0.723456	−	0.690371i	$-0.242552\pi$
$461$	6.60229e6	1.44691	0.723456	+	0.690371i	$0.242552\pi$
$462$	0	0
$463$	−3.22312e6	−0.698753	−0.349376	−	0.936982i	$-0.613607\pi$
$463$	−3.22312e6	−0.698753	−0.349376	+	0.936982i	$0.613607\pi$
$464$	0	0
$465$	243927.	0.0523151
$466$	0	0
$467$	3.21943e6	0.683103	0.341552	−	0.939863i	$-0.389048\pi$
$467$	3.21943e6	0.683103	0.341552	+	0.939863i	$0.389048\pi$
$468$	0	0
$469$	311112.	0.0653107
$470$	0	0
$471$	−1.83941e6	−0.382056
$472$	0	0
$473$	−6.42485e6	−1.32041
$474$	0	0
$475$	−1.20894e6	−0.245850
$476$	0	0
$477$	−4.55927e6	−0.917486
$478$	0	0
$479$	8.37964e6	1.66873	0.834366	−	0.551211i	$-0.185834\pi$
$479$	8.37964e6	1.66873	0.834366	+	0.551211i	$0.185834\pi$
$480$	0	0
$481$	1.49123e6	0.293888
$482$	0	0
$483$	4.36802e6	0.851955
$484$	0	0
$485$	1.18880e6	0.229485
$486$	0	0
$487$	8.95783e6	1.71151	0.855757	−	0.517378i	$-0.173092\pi$
$487$	8.95783e6	1.71151	0.855757	+	0.517378i	$0.173092\pi$
$488$	0	0
$489$	−2.12272e6	−0.401440
$490$	0	0
$491$	−425627.	−0.0796756	−0.0398378	−	0.999206i	$-0.512684\pi$
$491$	−425627.	−0.0796756	−0.0398378	+	0.999206i	$0.512684\pi$
$492$	0	0
$493$	−1.56980e7	−2.90888
$494$	0	0
$495$	1.60948e6	0.295239
$496$	0	0
$497$	−6.35147e6	−1.15341
$498$	0	0
$499$	−1.12507e6	−0.202268	−0.101134	−	0.994873i	$-0.532247\pi$
$499$	−1.12507e6	−0.202268	−0.101134	+	0.994873i	$0.532247\pi$
$500$	0	0
$501$	−359753.	−0.0640340
$502$	0	0
$503$	4.07903e6	0.718847	0.359424	−	0.933175i	$-0.382973\pi$
$503$	4.07903e6	0.718847	0.359424	+	0.933175i	$0.382973\pi$
$504$	0	0
$505$	3.16229e6	0.551789
$506$	0	0
$507$	−130587.	−0.0225622
$508$	0	0
$509$	9.37759e6	1.60434	0.802170	−	0.597095i	$-0.203679\pi$
$509$	9.37759e6	1.60434	0.802170	+	0.597095i	$0.203679\pi$
$510$	0	0
$511$	8.12888e6	1.37714
$512$	0	0
$513$	−4.11333e6	−0.690080
$514$	0	0
$515$	3.00360e6	0.499026
$516$	0	0
$517$	−813560.	−0.133864
$518$	0	0
$519$	1.97624e6	0.322048
$520$	0	0
$521$	−1.20746e6	−0.194885	−0.0974423	−	0.995241i	$-0.531066\pi$
$521$	−1.20746e6	−0.194885	−0.0974423	+	0.995241i	$0.531066\pi$
$522$	0	0
$523$	−1.03495e7	−1.65450	−0.827250	−	0.561834i	$-0.810096\pi$
$523$	−1.03495e7	−1.65450	−0.827250	+	0.561834i	$0.810096\pi$
$524$	0	0
$525$	−551926.	−0.0873942
$526$	0	0
$527$	3.86940e6	0.606900
$528$	0	0
$529$	1.80299e7	2.80127
$530$	0	0
$531$	5.18870e6	0.798586
$532$	0	0
$533$	−1.24625e6	−0.190014
$534$	0	0
$535$	2.07173e6	0.312931
$536$	0	0
$537$	−1.84024e6	−0.275383
$538$	0	0
$539$	−5.94128e6	−0.880862
$540$	0	0
$541$	−9.73024e6	−1.42932	−0.714662	−	0.699470i	$-0.753419\pi$
$541$	−9.73024e6	−1.42932	−0.714662	+	0.699470i	$0.753419\pi$
$542$	0	0
$543$	−503776.	−0.0733226
$544$	0	0
$545$	−4.12683e6	−0.595149
$546$	0	0
$547$	640777.	0.0915669	0.0457835	−	0.998951i	$-0.485422\pi$
$547$	640777.	0.0915669	0.0457835	+	0.998951i	$0.485422\pi$
$548$	0	0
$549$	−1.08397e6	−0.153492
$550$	0	0
$551$	−1.67461e7	−2.34983
$552$	0	0
$553$	3.86212e6	0.537047
$554$	0	0
$555$	−1.00861e6	−0.138993
$556$	0	0
$557$	−6.38949e6	−0.872626	−0.436313	−	0.899795i	$-0.643716\pi$
$557$	−6.38949e6	−0.872626	−0.436313	+	0.899795i	$0.643716\pi$
$558$	0	0
$559$	3.74578e6	0.507005
$560$	0	0
$561$	−2.40319e6	−0.322390
$562$	0	0
$563$	−794793.	−0.105678	−0.0528388	−	0.998603i	$-0.516827\pi$
$563$	−794793.	−0.105678	−0.0528388	+	0.998603i	$0.516827\pi$
$564$	0	0
$565$	4.78590e6	0.630729
$566$	0	0
$567$	8.54569e6	1.11632
$568$	0	0
$569$	515557.	0.0667569	0.0333785	−	0.999443i	$-0.489373\pi$
$569$	515557.	0.0667569	0.0333785	+	0.999443i	$0.489373\pi$
$570$	0	0
$571$	7.21986e6	0.926698	0.463349	−	0.886176i	$-0.346648\pi$
$571$	7.21986e6	0.926698	0.463349	+	0.886176i	$0.346648\pi$
$572$	0	0
$573$	−142647.	−0.0181500
$574$	0	0
$575$	−3.09146e6	−0.389937
$576$	0	0
$577$	−9.48847e6	−1.18647	−0.593235	−	0.805030i	$-0.702149\pi$
$577$	−9.48847e6	−1.18647	−0.593235	+	0.805030i	$0.702149\pi$
$578$	0	0
$579$	−2.33939e6	−0.290006
$580$	0	0
$581$	−1.03766e7	−1.27530
$582$	0	0
$583$	−5.95066e6	−0.725093
$584$	0	0
$585$	−938350.	−0.113364
$586$	0	0
$587$	−4.21397e6	−0.504773	−0.252387	−	0.967626i	$-0.581215\pi$
$587$	−4.21397e6	−0.504773	−0.252387	+	0.967626i	$0.581215\pi$
$588$	0	0
$589$	4.12776e6	0.490260
$590$	0	0
$591$	2.38585e6	0.280979
$592$	0	0
$593$	−6.43958e6	−0.752006	−0.376003	−	0.926619i	$-0.622702\pi$
$593$	−6.43958e6	−0.752006	−0.376003	+	0.926619i	$0.622702\pi$
$594$	0	0
$595$	−8.75518e6	−1.01385
$596$	0	0
$597$	−204535.	−0.0234873
$598$	0	0
$599$	−1.06380e7	−1.21142	−0.605710	−	0.795686i	$-0.707111\pi$
$599$	−1.06380e7	−1.21142	−0.605710	+	0.795686i	$0.707111\pi$
$600$	0	0
$601$	1.48154e7	1.67311	0.836557	−	0.547879i	$-0.184565\pi$
$601$	1.48154e7	1.67311	0.836557	+	0.547879i	$0.184565\pi$
$602$	0	0
$603$	−357752.	−0.0400672
$604$	0	0
$605$	−1.92561e6	−0.213885
$606$	0	0
$607$	−1.52111e7	−1.67567	−0.837837	−	0.545921i	$-0.816180\pi$
$607$	−1.52111e7	−1.67567	−0.837837	+	0.545921i	$0.816180\pi$
$608$	0	0
$609$	−7.64526e6	−0.835312
$610$	0	0
$611$	474317.	0.0514003
$612$	0	0
$613$	1.08915e7	1.17068	0.585340	−	0.810788i	$-0.300961\pi$
$613$	1.08915e7	1.17068	0.585340	+	0.810788i	$0.300961\pi$
$614$	0	0
$615$	842919.	0.0898666
$616$	0	0
$617$	6.62264e6	0.700355	0.350177	−	0.936683i	$-0.386121\pi$
$617$	6.62264e6	0.700355	0.350177	+	0.936683i	$0.386121\pi$
$618$	0	0
$619$	1.82789e6	0.191745	0.0958723	−	0.995394i	$-0.469436\pi$
$619$	1.82789e6	0.191745	0.0958723	+	0.995394i	$0.469436\pi$
$620$	0	0
$621$	−1.05185e7	−1.09452
$622$	0	0
$623$	1.77521e7	1.83244
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−2.56366e6	−0.260430
$628$	0	0
$629$	−1.59996e7	−1.61244
$630$	0	0
$631$	2.45976e6	0.245935	0.122967	−	0.992411i	$-0.460759\pi$
$631$	2.45976e6	0.245935	0.122967	+	0.992411i	$0.460759\pi$
$632$	0	0
$633$	3.86596e6	0.383484
$634$	0	0
$635$	−1.65907e6	−0.163279
$636$	0	0
$637$	3.46385e6	0.338228
$638$	0	0
$639$	7.30365e6	0.707600
$640$	0	0
$641$	6.90052e6	0.663341	0.331670	−	0.943395i	$-0.392388\pi$
$641$	6.90052e6	0.663341	0.331670	+	0.943395i	$0.392388\pi$
$642$	0	0
$643$	6.14505e6	0.586135	0.293068	−	0.956092i	$-0.405324\pi$
$643$	6.14505e6	0.586135	0.293068	+	0.956092i	$0.405324\pi$
$644$	0	0
$645$	−2.53351e6	−0.239786
$646$	0	0
$647$	6.62103e6	0.621821	0.310910	−	0.950439i	$-0.399366\pi$
$647$	6.62103e6	0.621821	0.310910	+	0.950439i	$0.399366\pi$
$648$	0	0
$649$	6.77217e6	0.631126
$650$	0	0
$651$	1.88448e6	0.174277
$652$	0	0
$653$	−7.91365e6	−0.726263	−0.363131	−	0.931738i	$-0.618292\pi$
$653$	−7.91365e6	−0.726263	−0.363131	+	0.931738i	$0.618292\pi$
$654$	0	0
$655$	−1.87292e6	−0.170575
$656$	0	0
$657$	−9.34752e6	−0.844856
$658$	0	0
$659$	60778.0	0.00545171	0.00272586	−	0.999996i	$-0.499132\pi$
$659$	60778.0	0.00545171	0.00272586	+	0.999996i	$0.499132\pi$
$660$	0	0
$661$	1.60526e7	1.42903	0.714516	−	0.699619i	$-0.246647\pi$
$661$	1.60526e7	1.42903	0.714516	+	0.699619i	$0.246647\pi$
$662$	0	0
$663$	1.40109e6	0.123789
$664$	0	0
$665$	−9.33977e6	−0.818997
$666$	0	0
$667$	−4.28228e7	−3.72701
$668$	0	0
$669$	−5.22482e6	−0.451343
$670$	0	0
$671$	−1.41477e6	−0.121306
$672$	0	0
$673$	1.60234e7	1.36369	0.681846	−	0.731496i	$-0.261177\pi$
$673$	1.60234e7	1.36369	0.681846	+	0.731496i	$0.261177\pi$
$674$	0	0
$675$	1.32908e6	0.112277
$676$	0	0
$677$	1.38633e7	1.16251	0.581255	−	0.813722i	$-0.302562\pi$
$677$	1.38633e7	1.16251	0.581255	+	0.813722i	$0.302562\pi$
$678$	0	0
$679$	9.18421e6	0.764482
$680$	0	0
$681$	1.62849e6	0.134561
$682$	0	0
$683$	1.63727e7	1.34298	0.671488	−	0.741016i	$-0.265656\pi$
$683$	1.63727e7	1.34298	0.671488	+	0.741016i	$0.265656\pi$
$684$	0	0
$685$	4.95520e6	0.403492
$686$	0	0
$687$	−5.30596e6	−0.428916
$688$	0	0
$689$	3.46932e6	0.278417
$690$	0	0
$691$	1.81683e7	1.44750	0.723750	−	0.690062i	$-0.242417\pi$
$691$	1.81683e7	1.44750	0.723750	+	0.690062i	$0.242417\pi$
$692$	0	0
$693$	1.24342e7	0.983526
$694$	0	0
$695$	−1.96792e6	−0.154542
$696$	0	0
$697$	1.33712e7	1.04253
$698$	0	0
$699$	−4.23468e6	−0.327814
$700$	0	0
$701$	−1.57090e7	−1.20740	−0.603702	−	0.797210i	$-0.706308\pi$
$701$	−1.57090e7	−1.20740	−0.603702	+	0.797210i	$0.706308\pi$
$702$	0	0
$703$	−1.70679e7	−1.30254
$704$	0	0
$705$	−320811.	−0.0243095
$706$	0	0
$707$	2.44306e7	1.83817
$708$	0	0
$709$	−2.26506e7	−1.69225	−0.846125	−	0.532985i	$-0.821070\pi$
$709$	−2.26506e7	−1.69225	−0.846125	+	0.532985i	$0.821070\pi$
$710$	0	0
$711$	−4.44110e6	−0.329471
$712$	0	0
$713$	1.05554e7	0.777591
$714$	0	0
$715$	−1.22471e6	−0.0895921
$716$	0	0
$717$	5.95856e6	0.432856
$718$	0	0
$719$	−1.39425e7	−1.00582	−0.502909	−	0.864339i	$-0.667737\pi$
$719$	−1.39425e7	−1.00582	−0.502909	+	0.864339i	$0.667737\pi$
$720$	0	0
$721$	2.32046e7	1.66240
$722$	0	0
$723$	−2.45737e6	−0.174834
$724$	0	0
$725$	5.41092e6	0.382319
$726$	0	0
$727$	−2.82130e7	−1.97976	−0.989881	−	0.141903i	$-0.954678\pi$
$727$	−2.82130e7	−1.97976	−0.989881	+	0.141903i	$0.954678\pi$
$728$	0	0
$729$	−7.46414e6	−0.520189
$730$	0	0
$731$	−4.01890e7	−2.78173
$732$	0	0
$733$	−2.19905e7	−1.51174	−0.755868	−	0.654724i	$-0.772785\pi$
$733$	−2.19905e7	−1.51174	−0.755868	+	0.654724i	$0.772785\pi$
$734$	0	0
$735$	−2.34283e6	−0.159964
$736$	0	0
$737$	−466930.	−0.0316653
$738$	0	0
$739$	−7.09511e6	−0.477912	−0.238956	−	0.971030i	$-0.576805\pi$
$739$	−7.09511e6	−0.477912	−0.238956	+	0.971030i	$0.576805\pi$
$740$	0	0
$741$	1.49465e6	0.0999984
$742$	0	0
$743$	−1.37027e7	−0.910614	−0.455307	−	0.890334i	$-0.650471\pi$
$743$	−1.37027e7	−0.910614	−0.455307	+	0.890334i	$0.650471\pi$
$744$	0	0
$745$	−1.16784e7	−0.770893
$746$	0	0
$747$	1.19322e7	0.782380
$748$	0	0
$749$	1.60053e7	1.04246
$750$	0	0
$751$	2.18956e7	1.41664	0.708318	−	0.705894i	$-0.249454\pi$
$751$	2.18956e7	1.41664	0.708318	+	0.705894i	$0.249454\pi$
$752$	0	0
$753$	1.39800e6	0.0898507
$754$	0	0
$755$	3.20792e6	0.204812
$756$	0	0
$757$	3.74824e6	0.237732	0.118866	−	0.992910i	$-0.462074\pi$
$757$	3.74824e6	0.237732	0.118866	+	0.992910i	$0.462074\pi$
$758$	0	0
$759$	−6.55571e6	−0.413062
$760$	0	0
$761$	2.89635e6	0.181297	0.0906483	−	0.995883i	$-0.471106\pi$
$761$	2.89635e6	0.181297	0.0906483	+	0.995883i	$0.471106\pi$
$762$	0	0
$763$	−3.18823e7	−1.98261
$764$	0	0
$765$	1.00677e7	0.621981
$766$	0	0
$767$	−3.94827e6	−0.242336
$768$	0	0
$769$	−1.17597e7	−0.717101	−0.358550	−	0.933510i	$-0.616729\pi$
$769$	−1.17597e7	−0.717101	−0.358550	+	0.933510i	$0.616729\pi$
$770$	0	0
$771$	2.12778e6	0.128911
$772$	0	0
$773$	2.29593e7	1.38200	0.691002	−	0.722853i	$-0.257170\pi$
$773$	2.29593e7	1.38200	0.691002	+	0.722853i	$0.257170\pi$
$774$	0	0
$775$	−1.33374e6	−0.0797659
$776$	0	0
$777$	−7.79216e6	−0.463026
$778$	0	0
$779$	1.42640e7	0.842166
$780$	0	0
$781$	9.53256e6	0.559219
$782$	0	0
$783$	1.84103e7	1.07314
$784$	0	0
$785$	1.00575e7	0.582529
$786$	0	0
$787$	7.27898e6	0.418922	0.209461	−	0.977817i	$-0.432829\pi$
$787$	7.27898e6	0.418922	0.209461	+	0.977817i	$0.432829\pi$
$788$	0	0
$789$	−6.28946e6	−0.359684
$790$	0	0
$791$	3.69740e7	2.10114
$792$	0	0
$793$	824833.	0.0465782
$794$	0	0
$795$	−2.34653e6	−0.131676
$796$	0	0
$797$	−2.24859e6	−0.125391	−0.0626954	−	0.998033i	$-0.519970\pi$
$797$	−2.24859e6	−0.125391	−0.0626954	+	0.998033i	$0.519970\pi$
$798$	0	0
$799$	−5.08902e6	−0.282012
$800$	0	0
$801$	−2.04134e7	−1.12418
$802$	0	0
$803$	−1.22002e7	−0.667693
$804$	0	0
$805$	−2.38834e7	−1.29899
$806$	0	0
$807$	−5.75033e6	−0.310820
$808$	0	0
$809$	−3.73687e6	−0.200741	−0.100371	−	0.994950i	$-0.532003\pi$
$809$	−3.73687e6	−0.200741	−0.100371	+	0.994950i	$0.532003\pi$
$810$	0	0
$811$	−3.62597e7	−1.93585	−0.967926	−	0.251237i	$-0.919163\pi$
$811$	−3.62597e7	−1.93585	−0.967926	+	0.251237i	$0.919163\pi$
$812$	0	0
$813$	−9.06882e6	−0.481199
$814$	0	0
$815$	1.16066e7	0.612083
$816$	0	0
$817$	−4.28725e7	−2.24711
$818$	0	0
$819$	−7.24932e6	−0.377649
$820$	0	0
$821$	−1.51475e7	−0.784299	−0.392150	−	0.919901i	$-0.628268\pi$
$821$	−1.51475e7	−0.784299	−0.392150	+	0.919901i	$0.628268\pi$
$822$	0	0
$823$	−1.57234e7	−0.809183	−0.404592	−	0.914497i	$-0.632586\pi$
$823$	−1.57234e7	−0.809183	−0.404592	+	0.914497i	$0.632586\pi$
$824$	0	0
$825$	828354.	0.0423722
$826$	0	0
$827$	5.78012e6	0.293882	0.146941	−	0.989145i	$-0.453057\pi$
$827$	5.78012e6	0.293882	0.146941	+	0.989145i	$0.453057\pi$
$828$	0	0
$829$	−1.93183e7	−0.976297	−0.488149	−	0.872761i	$-0.662328\pi$
$829$	−1.93183e7	−0.976297	−0.488149	+	0.872761i	$0.662328\pi$
$830$	0	0
$831$	8.37321e6	0.420620
$832$	0	0
$833$	−3.71642e7	−1.85572
$834$	0	0
$835$	1.96706e6	0.0976339
$836$	0	0
$837$	−4.53796e6	−0.223896
$838$	0	0
$839$	6.32985e6	0.310448	0.155224	−	0.987879i	$-0.450390\pi$
$839$	6.32985e6	0.310448	0.155224	+	0.987879i	$0.450390\pi$
$840$	0	0
$841$	5.44407e7	2.65420
$842$	0	0
$843$	3.27626e6	0.158785
$844$	0	0
$845$	714025.	0.0344010
$846$	0	0
$847$	−1.48765e7	−0.712514
$848$	0	0
$849$	−4.36675e6	−0.207917
$850$	0	0
$851$	−4.36457e7	−2.06594
$852$	0	0
$853$	1.52040e7	0.715460	0.357730	−	0.933825i	$-0.383551\pi$
$853$	1.52040e7	0.715460	0.357730	+	0.933825i	$0.383551\pi$
$854$	0	0
$855$	1.07399e7	0.502443
$856$	0	0
$857$	−2.66908e7	−1.24139	−0.620697	−	0.784051i	$-0.713150\pi$
$857$	−2.66908e7	−1.24139	−0.620697	+	0.784051i	$0.713150\pi$
$858$	0	0
$859$	−2.97401e7	−1.37518	−0.687589	−	0.726100i	$-0.741331\pi$
$859$	−2.97401e7	−1.37518	−0.687589	+	0.726100i	$0.741331\pi$
$860$	0	0
$861$	6.51206e6	0.299372
$862$	0	0
$863$	−2.62439e6	−0.119950	−0.0599751	−	0.998200i	$-0.519102\pi$
$863$	−2.62439e6	−0.119950	−0.0599751	+	0.998200i	$0.519102\pi$
$864$	0	0
$865$	−1.08057e7	−0.491034
$866$	0	0
$867$	−8.54065e6	−0.385872
$868$	0	0
$869$	−5.79643e6	−0.260382
$870$	0	0
$871$	272227.	0.0121586
$872$	0	0
$873$	−1.05611e7	−0.468999
$874$	0	0
$875$	3.01782e6	0.133252
$876$	0	0
$877$	1.54222e7	0.677093	0.338547	−	0.940950i	$-0.390065\pi$
$877$	1.54222e7	0.677093	0.338547	+	0.940950i	$0.390065\pi$
$878$	0	0
$879$	7.67866e6	0.335207
$880$	0	0
$881$	1.54269e7	0.669637	0.334818	−	0.942283i	$-0.391325\pi$
$881$	1.54269e7	0.669637	0.334818	+	0.942283i	$0.391325\pi$
$882$	0	0
$883$	−7.88254e6	−0.340223	−0.170112	−	0.985425i	$-0.554413\pi$
$883$	−7.88254e6	−0.340223	−0.170112	+	0.985425i	$0.554413\pi$
$884$	0	0
$885$	2.67047e6	0.114612
$886$	0	0
$887$	1.52145e7	0.649306	0.324653	−	0.945833i	$-0.394753\pi$
$887$	1.52145e7	0.649306	0.324653	+	0.945833i	$0.394753\pi$
$888$	0	0
$889$	−1.28174e7	−0.543931
$890$	0	0
$891$	−1.28257e7	−0.541238
$892$	0	0
$893$	−5.42882e6	−0.227812
$894$	0	0
$895$	1.00620e7	0.419883
$896$	0	0
$897$	3.82207e6	0.158605
$898$	0	0
$899$	−1.84749e7	−0.762401
$900$	0	0
$901$	−3.72229e7	−1.52756
$902$	0	0
$903$	−1.95729e7	−0.798797
$904$	0	0
$905$	2.75454e6	0.111796
$906$	0	0
$907$	−1.83400e7	−0.740253	−0.370127	−	0.928981i	$-0.620686\pi$
$907$	−1.83400e7	−0.740253	−0.370127	+	0.928981i	$0.620686\pi$
$908$	0	0
$909$	−2.80931e7	−1.12769
$910$	0	0
$911$	4.69902e7	1.87591	0.937954	−	0.346761i	$-0.112718\pi$
$911$	4.69902e7	1.87591	0.937954	+	0.346761i	$0.112718\pi$
$912$	0	0
$913$	1.55736e7	0.618318
$914$	0	0
$915$	−557888.	−0.0220290
$916$	0	0
$917$	−1.44694e7	−0.568236
$918$	0	0
$919$	1.42824e7	0.557844	0.278922	−	0.960314i	$-0.410023\pi$
$919$	1.42824e7	0.557844	0.278922	+	0.960314i	$0.410023\pi$
$920$	0	0
$921$	3.37524e6	0.131116
$922$	0	0
$923$	−5.55761e6	−0.214726
$924$	0	0
$925$	5.51490e6	0.211925
$926$	0	0
$927$	−2.66833e7	−1.01986
$928$	0	0
$929$	−1.39002e7	−0.528424	−0.264212	−	0.964465i	$-0.585112\pi$
$929$	−1.39002e7	−0.528424	−0.264212	+	0.964465i	$0.585112\pi$
$930$	0	0
$931$	−3.96456e7	−1.49907
$932$	0	0
$933$	7.56263e6	0.284426
$934$	0	0
$935$	1.31402e7	0.491554
$936$	0	0
$937$	−1.81287e7	−0.674556	−0.337278	−	0.941405i	$-0.609506\pi$
$937$	−1.81287e7	−0.674556	−0.337278	+	0.941405i	$0.609506\pi$
$938$	0	0
$939$	−239944.	−0.00888066
$940$	0	0
$941$	3.21587e7	1.18392	0.591962	−	0.805966i	$-0.298353\pi$
$941$	3.21587e7	1.18392	0.591962	+	0.805966i	$0.298353\pi$
$942$	0	0
$943$	3.64755e7	1.33574
$944$	0	0
$945$	1.02679e7	0.374027
$946$	0	0
$947$	1.20017e7	0.434879	0.217440	−	0.976074i	$-0.430229\pi$
$947$	1.20017e7	0.434879	0.217440	+	0.976074i	$0.430229\pi$
$948$	0	0
$949$	7.11287e6	0.256377
$950$	0	0
$951$	−1.56072e6	−0.0559596
$952$	0	0
$953$	1.62560e7	0.579806	0.289903	−	0.957056i	$-0.406377\pi$
$953$	1.62560e7	0.579806	0.289903	+	0.957056i	$0.406377\pi$
$954$	0	0
$955$	779963.	0.0276736
$956$	0	0
$957$	1.14743e7	0.404993
$958$	0	0
$959$	3.82819e7	1.34415
$960$	0	0
$961$	−2.40753e7	−0.840935
$962$	0	0
$963$	−1.84048e7	−0.639536
$964$	0	0
$965$	1.27913e7	0.442177
$966$	0	0
$967$	3.35091e7	1.15238	0.576191	−	0.817315i	$-0.304539\pi$
$967$	3.35091e7	1.15238	0.576191	+	0.817315i	$0.304539\pi$
$968$	0	0
$969$	−1.60363e7	−0.548649
$970$	0	0
$971$	5.02968e7	1.71195	0.855977	−	0.517013i	$-0.172956\pi$
$971$	5.02968e7	1.71195	0.855977	+	0.517013i	$0.172956\pi$
$972$	0	0
$973$	−1.52034e7	−0.514824
$974$	0	0
$975$	−482942.	−0.0162698
$976$	0	0
$977$	−1.50963e7	−0.505982	−0.252991	−	0.967469i	$-0.581414\pi$
$977$	−1.50963e7	−0.505982	−0.252991	+	0.967469i	$0.581414\pi$
$978$	0	0
$979$	−2.66431e7	−0.888440
$980$	0	0
$981$	3.66619e7	1.21631
$982$	0	0
$983$	−3.25193e6	−0.107339	−0.0536694	−	0.998559i	$-0.517092\pi$
$983$	−3.25193e6	−0.107339	−0.0536694	+	0.998559i	$0.517092\pi$
$984$	0	0
$985$	−1.30453e7	−0.428415
$986$	0	0
$987$	−2.47846e6	−0.0809822
$988$	0	0
$989$	−1.09632e8	−3.56409
$990$	0	0
$991$	2.53501e6	0.0819964	0.0409982	−	0.999159i	$-0.486946\pi$
$991$	2.53501e6	0.0819964	0.0409982	+	0.999159i	$0.486946\pi$
$992$	0	0
$993$	8.03379e6	0.258552
$994$	0	0
$995$	1.11836e6	0.0358115
$996$	0	0
$997$	−3.45726e7	−1.10153	−0.550763	−	0.834662i	$-0.685663\pi$
$997$	−3.45726e7	−1.10153	−0.550763	+	0.834662i	$0.685663\pi$
$998$	0	0
$999$	1.87641e7	0.594858

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.v.1.3		7
4.3	odd	2		260.6.a.f.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.6.a.f.1.5	✓	7	4.3	odd	2
1040.6.a.v.1.3		7	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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q-4.57223 q^{3} +25.0000 q^{5} +193.140 q^{7} -222.095 q^{9} -289.873 q^{11} +169.000 q^{13} -114.306 q^{15} -1813.23 q^{17} -1934.30 q^{19} -883.081 q^{21} -4946.34 q^{23} +625.000 q^{25} +2126.52 q^{27} +8657.48 q^{29} -2133.98 q^{31} +1325.37 q^{33} +4828.50 q^{35} +8823.83 q^{37} -772.707 q^{39} -7374.25 q^{41} +22164.4 q^{43} -5552.37 q^{45} +2806.61 q^{47} +20496.1 q^{49} +8290.50 q^{51} +20528.5 q^{53} -7246.83 q^{55} +8844.06 q^{57} -23362.5 q^{59} +4880.67 q^{61} -42895.4 q^{63} +4225.00 q^{65} +1610.81 q^{67} +22615.8 q^{69} -32885.3 q^{71} +42088.0 q^{73} -2857.64 q^{75} -55986.1 q^{77} +19996.4 q^{79} +44246.1 q^{81} -53725.6 q^{83} -45330.7 q^{85} -39584.0 q^{87} +91913.0 q^{89} +32640.7 q^{91} +9757.07 q^{93} -48357.5 q^{95} +47552.0 q^{97} +64379.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 175 q^{5} - 86 q^{7} + 807 q^{9} - 508 q^{11} + 1183 q^{13} + 50 q^{15} - 566 q^{17} - 164 q^{19} + 1816 q^{21} + 318 q^{23} + 4375 q^{25} + 2384 q^{27} + 6410 q^{29} - 6208 q^{31} + 10556 q^{33}+ \cdots - 59968 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 175 * q^5 - 86 * q^7 + 807 * q^9 - 508 * q^11 + 1183 * q^13 + 50 * q^15 - 566 * q^17 - 164 * q^19 + 1816 * q^21 + 318 * q^23 + 4375 * q^25 + 2384 * q^27 + 6410 * q^29 - 6208 * q^31 + 10556 * q^33 - 2150 * q^35 + 7870 * q^37 + 338 * q^39 + 24950 * q^41 - 7122 * q^43 + 20175 * q^45 + 3842 * q^47 + 34395 * q^49 - 41460 * q^51 + 36230 * q^53 - 12700 * q^55 + 90996 * q^57 - 48596 * q^59 + 126398 * q^61 - 85798 * q^63 + 29575 * q^65 - 58366 * q^67 + 127060 * q^69 - 160540 * q^71 + 56766 * q^73 + 1250 * q^75 + 226236 * q^77 - 146688 * q^79 + 149379 * q^81 - 139794 * q^83 - 14150 * q^85 - 182040 * q^87 + 290982 * q^89 - 14534 * q^91 + 78180 * q^93 - 4100 * q^95 + 206146 * q^97 - 59968 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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