LMFDB - Embedded newform 656.6.a.h.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [656,6,Mod(1,656)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(656, 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("656.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 656 = 2^{4} \cdot 41 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	656.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:	$105.211785797$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 1807 x^{8} - 1186 x^{7} + 1075622 x^{6} + 1575146 x^{5} - 242812142 x^{4} - 535064182 x^{3} + \cdots - 425549129499 $ x^10 - 1807x^8 - 1186x^7 + 1075622x^6 + 1575146x^5 - 242812142x^4 - 535064182x^3 + 19880651457x^2 + 41753967582x - 425549129499
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-28.8044$ of defining polynomial
Character	$\chi$	$=$	656.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.8044 q^{3} +33.0128 q^{5} +225.112 q^{7} +530.085 q^{9} +O(q^{10})$	$q+27.8044 q^{3} +33.0128 q^{5} +225.112 q^{7} +530.085 q^{9} +386.046 q^{11} -901.433 q^{13} +917.902 q^{15} -1148.95 q^{17} +1417.44 q^{19} +6259.09 q^{21} -2436.99 q^{23} -2035.15 q^{25} +7982.23 q^{27} +7379.46 q^{29} +7824.92 q^{31} +10733.8 q^{33} +7431.57 q^{35} +11802.9 q^{37} -25063.8 q^{39} -1681.00 q^{41} -1150.74 q^{43} +17499.6 q^{45} -11663.5 q^{47} +33868.2 q^{49} -31946.0 q^{51} +2693.32 q^{53} +12744.5 q^{55} +39411.1 q^{57} -47501.0 q^{59} -50485.8 q^{61} +119328. q^{63} -29758.8 q^{65} +46897.6 q^{67} -67759.2 q^{69} -28650.9 q^{71} -45401.7 q^{73} -56586.2 q^{75} +86903.4 q^{77} -23682.5 q^{79} +93130.6 q^{81} +24759.6 q^{83} -37930.2 q^{85} +205182. q^{87} +88260.8 q^{89} -202923. q^{91} +217567. q^{93} +46793.8 q^{95} -83679.1 q^{97} +204637. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 32 * q^5 - 88 * q^7 + 1194 * q^9	$ 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9} + 476 q^{11} - 456 q^{13} + 2 q^{15} + 1456 q^{17} - 2094 q^{19} + 7048 q^{21} - 7380 q^{23} + 15158 q^{25} - 9550 q^{27} + 9948 q^{29} + 840 q^{31} + 34828 q^{33} - 31214 q^{35} + 21780 q^{37} - 17832 q^{39} - 16810 q^{41} - 56636 q^{43} + 95584 q^{45} - 72666 q^{47} + 76574 q^{49} - 115660 q^{51} + 47528 q^{53} - 14182 q^{55} + 60356 q^{57} - 87380 q^{59} + 97364 q^{61} - 66998 q^{63} + 65716 q^{65} - 5724 q^{67} + 80692 q^{69} + 2834 q^{71} + 11228 q^{73} + 50282 q^{75} + 22400 q^{77} - 90094 q^{79} + 212530 q^{81} + 16132 q^{83} + 88840 q^{85} + 318756 q^{87} + 79872 q^{89} - 62004 q^{91} + 33652 q^{93} + 574026 q^{95} - 167548 q^{97} + 441774 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 32 * q^5 - 88 * q^7 + 1194 * q^9 + 476 * q^11 - 456 * q^13 + 2 * q^15 + 1456 * q^17 - 2094 * q^19 + 7048 * q^21 - 7380 * q^23 + 15158 * q^25 - 9550 * q^27 + 9948 * q^29 + 840 * q^31 + 34828 * q^33 - 31214 * q^35 + 21780 * q^37 - 17832 * q^39 - 16810 * q^41 - 56636 * q^43 + 95584 * q^45 - 72666 * q^47 + 76574 * q^49 - 115660 * q^51 + 47528 * q^53 - 14182 * q^55 + 60356 * q^57 - 87380 * q^59 + 97364 * q^61 - 66998 * q^63 + 65716 * q^65 - 5724 * q^67 + 80692 * q^69 + 2834 * q^71 + 11228 * q^73 + 50282 * q^75 + 22400 * q^77 - 90094 * q^79 + 212530 * q^81 + 16132 * q^83 + 88840 * q^85 + 318756 * q^87 + 79872 * q^89 - 62004 * q^91 + 33652 * q^93 + 574026 * q^95 - 167548 * q^97 + 441774 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	27.8044	1.78365	0.891827	−	0.452377i	$-0.149424\pi$
$3$	27.8044	1.78365	0.891827	+	0.452377i	$0.149424\pi$
$4$	0	0
$5$	33.0128	0.590552	0.295276	−	0.955412i	$-0.404588\pi$
$5$	33.0128	0.590552	0.295276	+	0.955412i	$0.404588\pi$
$6$	0	0
$7$	225.112	1.73641	0.868206	−	0.496204i	$-0.165273\pi$
$7$	225.112	1.73641	0.868206	+	0.496204i	$0.165273\pi$
$8$	0	0
$9$	530.085	2.18142
$10$	0	0
$11$	386.046	0.961961	0.480980	−	0.876731i	$-0.340281\pi$
$11$	386.046	0.961961	0.480980	+	0.876731i	$0.340281\pi$
$12$	0	0
$13$	−901.433	−1.47936	−0.739682	−	0.672957i	$-0.765024\pi$
$13$	−901.433	−1.47936	−0.739682	+	0.672957i	$0.765024\pi$
$14$	0	0
$15$	917.902	1.05334
$16$	0	0
$17$	−1148.95	−0.964228	−0.482114	−	0.876108i	$-0.660131\pi$
$17$	−1148.95	−0.964228	−0.482114	+	0.876108i	$0.660131\pi$
$18$	0	0
$19$	1417.44	0.900785	0.450392	−	0.892831i	$-0.351284\pi$
$19$	1417.44	0.900785	0.450392	+	0.892831i	$0.351284\pi$
$20$	0	0
$21$	6259.09	3.09716
$22$	0	0
$23$	−2436.99	−0.960583	−0.480291	−	0.877109i	$-0.659469\pi$
$23$	−2436.99	−0.960583	−0.480291	+	0.877109i	$0.659469\pi$
$24$	0	0
$25$	−2035.15	−0.651249
$26$	0	0
$27$	7982.23	2.10724
$28$	0	0
$29$	7379.46	1.62941	0.814704	−	0.579878i	$-0.196900\pi$
$29$	7379.46	1.62941	0.814704	+	0.579878i	$0.196900\pi$
$30$	0	0
$31$	7824.92	1.46243	0.731216	−	0.682146i	$-0.238953\pi$
$31$	7824.92	1.46243	0.731216	+	0.682146i	$0.238953\pi$
$32$	0	0
$33$	10733.8	1.71580
$34$	0	0
$35$	7431.57	1.02544
$36$	0	0
$37$	11802.9	1.41737	0.708685	−	0.705525i	$-0.249289\pi$
$37$	11802.9	1.41737	0.708685	+	0.705525i	$0.249289\pi$
$38$	0	0
$39$	−25063.8	−2.63867
$40$	0	0
$41$	−1681.00	−0.156174
$41$	−1681.00	−0.156174
$42$	0	0
$43$	−1150.74	−0.0949087	−0.0474543	−	0.998873i	$-0.515111\pi$
$43$	−1150.74	−0.0949087	−0.0474543	+	0.998873i	$0.515111\pi$
$44$	0	0
$45$	17499.6	1.28824
$46$	0	0
$47$	−11663.5	−0.770165	−0.385082	−	0.922882i	$-0.625827\pi$
$47$	−11663.5	−0.770165	−0.385082	+	0.922882i	$0.625827\pi$
$48$	0	0
$49$	33868.2	2.01513
$50$	0	0
$51$	−31946.0	−1.71985
$52$	0	0
$53$	2693.32	0.131704	0.0658519	−	0.997829i	$-0.479024\pi$
$53$	2693.32	0.131704	0.0658519	+	0.997829i	$0.479024\pi$
$54$	0	0
$55$	12744.5	0.568087
$56$	0	0
$57$	39411.1	1.60669
$58$	0	0
$59$	−47501.0	−1.77653	−0.888266	−	0.459330i	$-0.848090\pi$
$59$	−47501.0	−1.77653	−0.888266	+	0.459330i	$0.848090\pi$
$60$	0	0
$61$	−50485.8	−1.73718	−0.868589	−	0.495533i	$-0.834973\pi$
$61$	−50485.8	−1.73718	−0.868589	+	0.495533i	$0.834973\pi$
$62$	0	0
$63$	119328.	3.78784
$64$	0	0
$65$	−29758.8	−0.873640
$66$	0	0
$67$	46897.6	1.27633	0.638166	−	0.769899i	$-0.279693\pi$
$67$	46897.6	1.27633	0.638166	+	0.769899i	$0.279693\pi$
$68$	0	0
$69$	−67759.2	−1.71335
$70$	0	0
$71$	−28650.9	−0.674516	−0.337258	−	0.941412i	$-0.609499\pi$
$71$	−28650.9	−0.674516	−0.337258	+	0.941412i	$0.609499\pi$
$72$	0	0
$73$	−45401.7	−0.997161	−0.498581	−	0.866843i	$-0.666145\pi$
$73$	−45401.7	−0.997161	−0.498581	+	0.866843i	$0.666145\pi$
$74$	0	0
$75$	−56586.2	−1.16160
$76$	0	0
$77$	86903.4	1.67036
$78$	0	0
$79$	−23682.5	−0.426933	−0.213466	−	0.976950i	$-0.568475\pi$
$79$	−23682.5	−0.426933	−0.213466	+	0.976950i	$0.568475\pi$
$80$	0	0
$81$	93130.6	1.57717
$82$	0	0
$83$	24759.6	0.394502	0.197251	−	0.980353i	$-0.436799\pi$
$83$	24759.6	0.394502	0.197251	+	0.980353i	$0.436799\pi$
$84$	0	0
$85$	−37930.2	−0.569426
$86$	0	0
$87$	205182.	2.90630
$88$	0	0
$89$	88260.8	1.18112	0.590558	−	0.806995i	$-0.298908\pi$
$89$	88260.8	1.18112	0.590558	+	0.806995i	$0.298908\pi$
$90$	0	0
$91$	−202923.	−2.56878
$92$	0	0
$93$	217567.	2.60847
$94$	0	0
$95$	46793.8	0.531960
$96$	0	0
$97$	−83679.1	−0.903000	−0.451500	−	0.892271i	$-0.649111\pi$
$97$	−83679.1	−0.903000	−0.451500	+	0.892271i	$0.649111\pi$
$98$	0	0
$99$	204637.	2.09844
$100$	0	0
$101$	37197.9	0.362840	0.181420	−	0.983406i	$-0.441931\pi$
$101$	37197.9	0.362840	0.181420	+	0.983406i	$0.441931\pi$
$102$	0	0
$103$	88155.6	0.818760	0.409380	−	0.912364i	$-0.365745\pi$
$103$	88155.6	0.818760	0.409380	+	0.912364i	$0.365745\pi$
$104$	0	0
$105$	206630.	1.82903
$106$	0	0
$107$	42833.1	0.361676	0.180838	−	0.983513i	$-0.442119\pi$
$107$	42833.1	0.361676	0.180838	+	0.983513i	$0.442119\pi$
$108$	0	0
$109$	116395.	0.938357	0.469179	−	0.883103i	$-0.344550\pi$
$109$	116395.	0.938357	0.469179	+	0.883103i	$0.344550\pi$
$110$	0	0
$111$	328172.	2.52810
$112$	0	0
$113$	−143804.	−1.05943	−0.529717	−	0.848174i	$-0.677702\pi$
$113$	−143804.	−1.05943	−0.529717	+	0.848174i	$0.677702\pi$
$114$	0	0
$115$	−80452.1	−0.567274
$116$	0	0
$117$	−477836.	−3.22711
$118$	0	0
$119$	−258643.	−1.67430
$120$	0	0
$121$	−12019.5	−0.0746315
$122$	0	0
$123$	−46739.2	−0.278560
$124$	0	0
$125$	−170351.	−0.975148
$126$	0	0
$127$	−71491.1	−0.393317	−0.196659	−	0.980472i	$-0.563009\pi$
$127$	−71491.1	−0.393317	−0.196659	+	0.980472i	$0.563009\pi$
$128$	0	0
$129$	−31995.6	−0.169284
$130$	0	0
$131$	−208698.	−1.06253	−0.531263	−	0.847207i	$-0.678282\pi$
$131$	−208698.	−1.06253	−0.531263	+	0.847207i	$0.678282\pi$
$132$	0	0
$133$	319082.	1.56413
$134$	0	0
$135$	263516.	1.24444
$136$	0	0
$137$	76672.4	0.349010	0.174505	−	0.984656i	$-0.444167\pi$
$137$	76672.4	0.349010	0.174505	+	0.984656i	$0.444167\pi$
$138$	0	0
$139$	39498.4	0.173397	0.0866987	−	0.996235i	$-0.472368\pi$
$139$	39498.4	0.173397	0.0866987	+	0.996235i	$0.472368\pi$
$140$	0	0
$141$	−324296.	−1.37371
$142$	0	0
$143$	−347995.	−1.42309
$144$	0	0
$145$	243617.	0.962249
$146$	0	0
$147$	941686.	3.59429
$148$	0	0
$149$	201915.	0.745081	0.372541	−	0.928016i	$-0.378487\pi$
$149$	201915.	0.745081	0.372541	+	0.928016i	$0.378487\pi$
$150$	0	0
$151$	−142517.	−0.508657	−0.254329	−	0.967118i	$-0.581854\pi$
$151$	−142517.	−0.508657	−0.254329	+	0.967118i	$0.581854\pi$
$152$	0	0
$153$	−609043.	−2.10339
$154$	0	0
$155$	258323.	0.863642
$156$	0	0
$157$	−448564.	−1.45236	−0.726181	−	0.687503i	$-0.758707\pi$
$157$	−448564.	−1.45236	−0.726181	+	0.687503i	$0.758707\pi$
$158$	0	0
$159$	74886.2	0.234914
$160$	0	0
$161$	−548595.	−1.66797
$162$	0	0
$163$	−195919.	−0.577573	−0.288787	−	0.957393i	$-0.593252\pi$
$163$	−195919.	−0.577573	−0.288787	+	0.957393i	$0.593252\pi$
$164$	0	0
$165$	354353.	1.01327
$166$	0	0
$167$	−89084.7	−0.247179	−0.123590	−	0.992333i	$-0.539441\pi$
$167$	−89084.7	−0.247179	−0.123590	+	0.992333i	$0.539441\pi$
$168$	0	0
$169$	441288.	1.18852
$170$	0	0
$171$	751365.	1.96499
$172$	0	0
$173$	514692.	1.30747	0.653736	−	0.756723i	$-0.273201\pi$
$173$	514692.	1.30747	0.653736	+	0.756723i	$0.273201\pi$
$174$	0	0
$175$	−458136.	−1.13084
$176$	0	0
$177$	−1.32074e6	−3.16872
$178$	0	0
$179$	432605.	1.00916	0.504578	−	0.863366i	$-0.331648\pi$
$179$	432605.	1.00916	0.504578	+	0.863366i	$0.331648\pi$
$180$	0	0
$181$	200372.	0.454611	0.227305	−	0.973824i	$-0.427008\pi$
$181$	200372.	0.454611	0.227305	+	0.973824i	$0.427008\pi$
$182$	0	0
$183$	−1.40373e6	−3.09852
$184$	0	0
$185$	389646.	0.837030
$186$	0	0
$187$	−443549.	−0.927550
$188$	0	0
$189$	1.79689e6	3.65904
$190$	0	0
$191$	81681.1	0.162009	0.0810043	−	0.996714i	$-0.474187\pi$
$191$	81681.1	0.162009	0.0810043	+	0.996714i	$0.474187\pi$
$192$	0	0
$193$	−53972.2	−0.104298	−0.0521491	−	0.998639i	$-0.516607\pi$
$193$	−53972.2	−0.104298	−0.0521491	+	0.998639i	$0.516607\pi$
$194$	0	0
$195$	−827427.	−1.55827
$196$	0	0
$197$	−825302.	−1.51512	−0.757561	−	0.652764i	$-0.773609\pi$
$197$	−825302.	−1.51512	−0.757561	+	0.652764i	$0.773609\pi$
$198$	0	0
$199$	149715.	0.267998	0.133999	−	0.990981i	$-0.457218\pi$
$199$	149715.	0.267998	0.133999	+	0.990981i	$0.457218\pi$
$200$	0	0
$201$	1.30396e6	2.27653
$202$	0	0
$203$	1.66120e6	2.82932
$204$	0	0
$205$	−55494.6	−0.0922287
$206$	0	0
$207$	−1.29181e6	−2.09543
$208$	0	0
$209$	547198.	0.866520
$210$	0	0
$211$	−388498.	−0.600734	−0.300367	−	0.953824i	$-0.597109\pi$
$211$	−388498.	−0.600734	−0.300367	+	0.953824i	$0.597109\pi$
$212$	0	0
$213$	−796621.	−1.20310
$214$	0	0
$215$	−37989.2	−0.0560485
$216$	0	0
$217$	1.76148e6	2.53938
$218$	0	0
$219$	−1.26237e6	−1.77859
$220$	0	0
$221$	1.03570e6	1.42644
$222$	0	0
$223$	1.39276e6	1.87549	0.937743	−	0.347329i	$-0.112911\pi$
$223$	1.39276e6	1.87549	0.937743	+	0.347329i	$0.112911\pi$
$224$	0	0
$225$	−1.07880e6	−1.42065
$226$	0	0
$227$	83677.6	0.107782	0.0538908	−	0.998547i	$-0.482838\pi$
$227$	83677.6	0.107782	0.0538908	+	0.998547i	$0.482838\pi$
$228$	0	0
$229$	1.41235e6	1.77972	0.889862	−	0.456229i	$-0.150800\pi$
$229$	1.41235e6	1.77972	0.889862	+	0.456229i	$0.150800\pi$
$230$	0	0
$231$	2.41630e6	2.97934
$232$	0	0
$233$	−101264.	−0.122198	−0.0610990	−	0.998132i	$-0.519461\pi$
$233$	−101264.	−0.122198	−0.0610990	+	0.998132i	$0.519461\pi$
$234$	0	0
$235$	−385045.	−0.454822
$236$	0	0
$237$	−658478.	−0.761500
$238$	0	0
$239$	1.02959e6	1.16592	0.582962	−	0.812500i	$-0.301894\pi$
$239$	1.02959e6	1.16592	0.582962	+	0.812500i	$0.301894\pi$
$240$	0	0
$241$	−509720.	−0.565313	−0.282656	−	0.959221i	$-0.591216\pi$
$241$	−509720.	−0.565313	−0.282656	+	0.959221i	$0.591216\pi$
$242$	0	0
$243$	649758.	0.705888
$244$	0	0
$245$	1.11809e6	1.19004
$246$	0	0
$247$	−1.27773e6	−1.33259
$248$	0	0
$249$	688427.	0.703654
$250$	0	0
$251$	−1.93969e6	−1.94333	−0.971666	−	0.236357i	$-0.924046\pi$
$251$	−1.93969e6	−1.94333	−0.971666	+	0.236357i	$0.924046\pi$
$252$	0	0
$253$	−940792.	−0.924043
$254$	0	0
$255$	−1.05463e6	−1.01566
$256$	0	0
$257$	−288478.	−0.272446	−0.136223	−	0.990678i	$-0.543496\pi$
$257$	−288478.	−0.272446	−0.136223	+	0.990678i	$0.543496\pi$
$258$	0	0
$259$	2.65696e6	2.46114
$260$	0	0
$261$	3.91174e6	3.55442
$262$	0	0
$263$	432228.	0.385322	0.192661	−	0.981265i	$-0.438288\pi$
$263$	432228.	0.385322	0.192661	+	0.981265i	$0.438288\pi$
$264$	0	0
$265$	88914.1	0.0777779
$266$	0	0
$267$	2.45404e6	2.10670
$268$	0	0
$269$	−211516.	−0.178223	−0.0891113	−	0.996022i	$-0.528403\pi$
$269$	−211516.	−0.178223	−0.0891113	+	0.996022i	$0.528403\pi$
$270$	0	0
$271$	−246156.	−0.203605	−0.101802	−	0.994805i	$-0.532461\pi$
$271$	−246156.	−0.203605	−0.101802	+	0.994805i	$0.532461\pi$
$272$	0	0
$273$	−5.64215e6	−4.58182
$274$	0	0
$275$	−785663.	−0.626476
$276$	0	0
$277$	−6851.70	−0.00536536	−0.00268268	−	0.999996i	$-0.500854\pi$
$277$	−6851.70	−0.00536536	−0.00268268	+	0.999996i	$0.500854\pi$
$278$	0	0
$279$	4.14787e6	3.19018
$280$	0	0
$281$	−2.32226e6	−1.75447	−0.877233	−	0.480065i	$-0.840613\pi$
$281$	−2.32226e6	−1.75447	−0.877233	+	0.480065i	$0.840613\pi$
$282$	0	0
$283$	−1.05095e6	−0.780038	−0.390019	−	0.920807i	$-0.627532\pi$
$283$	−1.05095e6	−0.780038	−0.390019	+	0.920807i	$0.627532\pi$
$284$	0	0
$285$	1.30107e6	0.948832
$286$	0	0
$287$	−378413.	−0.271182
$288$	0	0
$289$	−99764.7	−0.0702639
$290$	0	0
$291$	−2.32665e6	−1.61064
$292$	0	0
$293$	483079.	0.328737	0.164369	−	0.986399i	$-0.447441\pi$
$293$	483079.	0.328737	0.164369	+	0.986399i	$0.447441\pi$
$294$	0	0
$295$	−1.56814e6	−1.04913
$296$	0	0
$297$	3.08151e6	2.02709
$298$	0	0
$299$	2.19679e6	1.42105
$300$	0	0
$301$	−259045.	−0.164801
$302$	0	0
$303$	1.03426e6	0.647180
$304$	0	0
$305$	−1.66668e6	−1.02589
$306$	0	0
$307$	−422687.	−0.255960	−0.127980	−	0.991777i	$-0.540849\pi$
$307$	−422687.	−0.255960	−0.127980	+	0.991777i	$0.540849\pi$
$308$	0	0
$309$	2.45111e6	1.46038
$310$	0	0
$311$	−53709.3	−0.0314882	−0.0157441	−	0.999876i	$-0.505012\pi$
$311$	−53709.3	−0.0314882	−0.0157441	+	0.999876i	$0.505012\pi$
$312$	0	0
$313$	181021.	0.104440	0.0522201	−	0.998636i	$-0.483370\pi$
$313$	181021.	0.104440	0.0522201	+	0.998636i	$0.483370\pi$
$314$	0	0
$315$	3.93936e6	2.23692
$316$	0	0
$317$	3.09825e6	1.73169	0.865843	−	0.500317i	$-0.166783\pi$
$317$	3.09825e6	1.73169	0.865843	+	0.500317i	$0.166783\pi$
$318$	0	0
$319$	2.84881e6	1.56743
$320$	0	0
$321$	1.19095e6	0.645105
$322$	0	0
$323$	−1.62857e6	−0.868562
$324$	0	0
$325$	1.83455e6	0.963434
$326$	0	0
$327$	3.23629e6	1.67370
$328$	0	0
$329$	−2.62558e6	−1.33732
$330$	0	0
$331$	2.21225e6	1.10985	0.554925	−	0.831900i	$-0.312747\pi$
$331$	2.21225e6	1.10985	0.554925	+	0.831900i	$0.312747\pi$
$332$	0	0
$333$	6.25653e6	3.09188
$334$	0	0
$335$	1.54822e6	0.753740
$336$	0	0
$337$	−377573.	−0.181103	−0.0905517	−	0.995892i	$-0.528863\pi$
$337$	−377573.	−0.181103	−0.0905517	+	0.995892i	$0.528863\pi$
$338$	0	0
$339$	−3.99838e6	−1.88967
$340$	0	0
$341$	3.02078e6	1.40680
$342$	0	0
$343$	3.84068e6	1.76268
$344$	0	0
$345$	−2.23692e6	−1.01182
$346$	0	0
$347$	−1.65453e6	−0.737651	−0.368826	−	0.929499i	$-0.620240\pi$
$347$	−1.65453e6	−0.737651	−0.368826	+	0.929499i	$0.620240\pi$
$348$	0	0
$349$	−3.28081e6	−1.44184	−0.720920	−	0.693019i	$-0.756280\pi$
$349$	−3.28081e6	−1.44184	−0.720920	+	0.693019i	$0.756280\pi$
$350$	0	0
$351$	−7.19545e6	−3.11738
$352$	0	0
$353$	−3.23371e6	−1.38123	−0.690613	−	0.723225i	$-0.742659\pi$
$353$	−3.23371e6	−1.38123	−0.690613	+	0.723225i	$0.742659\pi$
$354$	0	0
$355$	−945847.	−0.398337
$356$	0	0
$357$	−7.19140e6	−2.98637
$358$	0	0
$359$	1.61594e6	0.661742	0.330871	−	0.943676i	$-0.392657\pi$
$359$	1.61594e6	0.661742	0.330871	+	0.943676i	$0.392657\pi$
$360$	0	0
$361$	−466959.	−0.188587
$362$	0	0
$363$	−334194.	−0.133117
$364$	0	0
$365$	−1.49884e6	−0.588875
$366$	0	0
$367$	−524142.	−0.203135	−0.101567	−	0.994829i	$-0.532386\pi$
$367$	−524142.	−0.203135	−0.101567	+	0.994829i	$0.532386\pi$
$368$	0	0
$369$	−891073.	−0.340681
$370$	0	0
$371$	606297.	0.228692
$372$	0	0
$373$	4.46340e6	1.66109	0.830546	−	0.556950i	$-0.188028\pi$
$373$	4.46340e6	1.66109	0.830546	+	0.556950i	$0.188028\pi$
$374$	0	0
$375$	−4.73652e6	−1.73933
$376$	0	0
$377$	−6.65209e6	−2.41049
$378$	0	0
$379$	−2.33026e6	−0.833311	−0.416655	−	0.909064i	$-0.636798\pi$
$379$	−2.33026e6	−0.833311	−0.416655	+	0.909064i	$0.636798\pi$
$380$	0	0
$381$	−1.98777e6	−0.701541
$382$	0	0
$383$	−4.05564e6	−1.41274	−0.706370	−	0.707843i	$-0.749668\pi$
$383$	−4.05564e6	−1.41274	−0.706370	+	0.707843i	$0.749668\pi$
$384$	0	0
$385$	2.86893e6	0.986434
$386$	0	0
$387$	−609990.	−0.207036
$388$	0	0
$389$	−5.31374e6	−1.78044	−0.890218	−	0.455534i	$-0.849448\pi$
$389$	−5.31374e6	−1.78044	−0.890218	+	0.455534i	$0.849448\pi$
$390$	0	0
$391$	2.79999e6	0.926221
$392$	0	0
$393$	−5.80272e6	−1.89518
$394$	0	0
$395$	−781826.	−0.252126
$396$	0	0
$397$	437449.	0.139300	0.0696501	−	0.997571i	$-0.477812\pi$
$397$	437449.	0.139300	0.0696501	+	0.997571i	$0.477812\pi$
$398$	0	0
$399$	8.87190e6	2.78987
$400$	0	0
$401$	1.89451e6	0.588349	0.294174	−	0.955752i	$-0.404955\pi$
$401$	1.89451e6	0.588349	0.294174	+	0.955752i	$0.404955\pi$
$402$	0	0
$403$	−7.05364e6	−2.16347
$404$	0	0
$405$	3.07450e6	0.931403
$406$	0	0
$407$	4.55645e6	1.36345
$408$	0	0
$409$	3.63202e6	1.07359	0.536796	−	0.843712i	$-0.319634\pi$
$409$	3.63202e6	1.07359	0.536796	+	0.843712i	$0.319634\pi$
$410$	0	0
$411$	2.13183e6	0.622513
$412$	0	0
$413$	−1.06930e7	−3.08479
$414$	0	0
$415$	817385.	0.232974
$416$	0	0
$417$	1.09823e6	0.309281
$418$	0	0
$419$	451051.	0.125514	0.0627568	−	0.998029i	$-0.480011\pi$
$419$	451051.	0.125514	0.0627568	+	0.998029i	$0.480011\pi$
$420$	0	0
$421$	3.90077e6	1.07262	0.536308	−	0.844022i	$-0.319818\pi$
$421$	3.90077e6	1.07262	0.536308	+	0.844022i	$0.319818\pi$
$422$	0	0
$423$	−6.18264e6	−1.68005
$424$	0	0
$425$	2.33829e6	0.627953
$426$	0	0
$427$	−1.13649e7	−3.01646
$428$	0	0
$429$	−9.67578e6	−2.53830
$430$	0	0
$431$	622521.	0.161421	0.0807107	−	0.996738i	$-0.474281\pi$
$431$	622521.	0.161421	0.0807107	+	0.996738i	$0.474281\pi$
$432$	0	0
$433$	1.06876e6	0.273942	0.136971	−	0.990575i	$-0.456263\pi$
$433$	1.06876e6	0.273942	0.136971	+	0.990575i	$0.456263\pi$
$434$	0	0
$435$	6.77362e6	1.71632
$436$	0	0
$437$	−3.45430e6	−0.865278
$438$	0	0
$439$	1.59116e6	0.394050	0.197025	−	0.980398i	$-0.436872\pi$
$439$	1.59116e6	0.394050	0.197025	+	0.980398i	$0.436872\pi$
$440$	0	0
$441$	1.79530e7	4.39584
$442$	0	0
$443$	−432368.	−0.104675	−0.0523377	−	0.998629i	$-0.516667\pi$
$443$	−432368.	−0.104675	−0.0523377	+	0.998629i	$0.516667\pi$
$444$	0	0
$445$	2.91374e6	0.697510
$446$	0	0
$447$	5.61413e6	1.32897
$448$	0	0
$449$	−1.37650e6	−0.322225	−0.161112	−	0.986936i	$-0.551508\pi$
$449$	−1.37650e6	−0.322225	−0.161112	+	0.986936i	$0.551508\pi$
$450$	0	0
$451$	−648943.	−0.150233
$452$	0	0
$453$	−3.96261e6	−0.907268
$454$	0	0
$455$	−6.69906e6	−1.51700
$456$	0	0
$457$	−5.07972e6	−1.13776	−0.568878	−	0.822422i	$-0.692622\pi$
$457$	−5.07972e6	−1.13776	−0.568878	+	0.822422i	$0.692622\pi$
$458$	0	0
$459$	−9.17121e6	−2.03186
$460$	0	0
$461$	8.82444e6	1.93390	0.966952	−	0.254959i	$-0.0820620\pi$
$461$	8.82444e6	1.93390	0.966952	+	0.254959i	$0.0820620\pi$
$462$	0	0
$463$	−5.54337e6	−1.20177	−0.600885	−	0.799335i	$-0.705185\pi$
$463$	−5.54337e6	−1.20177	−0.600885	+	0.799335i	$0.705185\pi$
$464$	0	0
$465$	7.18251e6	1.54044
$466$	0	0
$467$	1.88045e6	0.398996	0.199498	−	0.979898i	$-0.436069\pi$
$467$	1.88045e6	0.398996	0.199498	+	0.979898i	$0.436069\pi$
$468$	0	0
$469$	1.05572e7	2.21624
$470$	0	0
$471$	−1.24721e7	−2.59051
$472$	0	0
$473$	−444238.	−0.0912984
$474$	0	0
$475$	−2.88471e6	−0.586635
$476$	0	0
$477$	1.42769e6	0.287301
$478$	0	0
$479$	−8.29933e6	−1.65274	−0.826370	−	0.563128i	$-0.809598\pi$
$479$	−8.29933e6	−1.65274	−0.826370	+	0.563128i	$0.809598\pi$
$480$	0	0
$481$	−1.06395e7	−2.09681
$482$	0	0
$483$	−1.52534e7	−2.97508
$484$	0	0
$485$	−2.76248e6	−0.533268
$486$	0	0
$487$	5.56785e6	1.06381	0.531906	−	0.846803i	$-0.321476\pi$
$487$	5.56785e6	1.06381	0.531906	+	0.846803i	$0.321476\pi$
$488$	0	0
$489$	−5.44741e6	−1.03019
$490$	0	0
$491$	−2.44509e6	−0.457710	−0.228855	−	0.973461i	$-0.573498\pi$
$491$	−2.44509e6	−0.457710	−0.228855	+	0.973461i	$0.573498\pi$
$492$	0	0
$493$	−8.47865e6	−1.57112
$494$	0	0
$495$	6.75566e6	1.23924
$496$	0	0
$497$	−6.44965e6	−1.17124
$498$	0	0
$499$	−6.31692e6	−1.13567	−0.567837	−	0.823141i	$-0.692220\pi$
$499$	−6.31692e6	−1.13567	−0.567837	+	0.823141i	$0.692220\pi$
$500$	0	0
$501$	−2.47695e6	−0.440882
$502$	0	0
$503$	1.31737e6	0.232160	0.116080	−	0.993240i	$-0.462967\pi$
$503$	1.31737e6	0.232160	0.116080	+	0.993240i	$0.462967\pi$
$504$	0	0
$505$	1.22801e6	0.214275
$506$	0	0
$507$	1.22697e7	2.11990
$508$	0	0
$509$	−6.21363e6	−1.06304	−0.531521	−	0.847045i	$-0.678379\pi$
$509$	−6.21363e6	−1.06304	−0.531521	+	0.847045i	$0.678379\pi$
$510$	0	0
$511$	−1.02205e7	−1.73148
$512$	0	0
$513$	1.13143e7	1.89817
$514$	0	0
$515$	2.91027e6	0.483520
$516$	0	0
$517$	−4.50264e6	−0.740868
$518$	0	0
$519$	1.43107e7	2.33208
$520$	0	0
$521$	6.38753e6	1.03095	0.515476	−	0.856904i	$-0.327615\pi$
$521$	6.38753e6	1.03095	0.515476	+	0.856904i	$0.327615\pi$
$522$	0	0
$523$	−814705.	−0.130241	−0.0651203	−	0.997877i	$-0.520743\pi$
$523$	−814705.	−0.130241	−0.0651203	+	0.997877i	$0.520743\pi$
$524$	0	0
$525$	−1.27382e7	−2.01702
$526$	0	0
$527$	−8.99046e6	−1.41012
$528$	0	0
$529$	−497406.	−0.0772808
$530$	0	0
$531$	−2.51796e7	−3.87536
$532$	0	0
$533$	1.51531e6	0.231038
$534$	0	0
$535$	1.41404e6	0.213588
$536$	0	0
$537$	1.20283e7	1.79999
$538$	0	0
$539$	1.30747e7	1.93847
$540$	0	0
$541$	−7.39808e6	−1.08674	−0.543370	−	0.839493i	$-0.682852\pi$
$541$	−7.39808e6	−1.08674	−0.543370	+	0.839493i	$0.682852\pi$
$542$	0	0
$543$	5.57121e6	0.810868
$544$	0	0
$545$	3.84253e6	0.554148
$546$	0	0
$547$	1.30116e6	0.185936	0.0929679	−	0.995669i	$-0.470365\pi$
$547$	1.30116e6	0.185936	0.0929679	+	0.995669i	$0.470365\pi$
$548$	0	0
$549$	−2.67618e7	−3.78952
$550$	0	0
$551$	1.04600e7	1.46775
$552$	0	0
$553$	−5.33120e6	−0.741331
$554$	0	0
$555$	1.08339e7	1.49297
$556$	0	0
$557$	1.16271e7	1.58794	0.793970	−	0.607957i	$-0.208011\pi$
$557$	1.16271e7	1.58794	0.793970	+	0.607957i	$0.208011\pi$
$558$	0	0
$559$	1.03731e6	0.140404
$560$	0	0
$561$	−1.23326e7	−1.65443
$562$	0	0
$563$	−1.33786e6	−0.177885	−0.0889423	−	0.996037i	$-0.528349\pi$
$563$	−1.33786e6	−0.177885	−0.0889423	+	0.996037i	$0.528349\pi$
$564$	0	0
$565$	−4.74737e6	−0.625651
$566$	0	0
$567$	2.09648e7	2.73862
$568$	0	0
$569$	1.13009e6	0.146329	0.0731646	−	0.997320i	$-0.476690\pi$
$569$	1.13009e6	0.146329	0.0731646	+	0.997320i	$0.476690\pi$
$570$	0	0
$571$	3.17480e6	0.407499	0.203750	−	0.979023i	$-0.434687\pi$
$571$	3.17480e6	0.407499	0.203750	+	0.979023i	$0.434687\pi$
$572$	0	0
$573$	2.27109e6	0.288967
$574$	0	0
$575$	4.95965e6	0.625578
$576$	0	0
$577$	1.35882e7	1.69912	0.849559	−	0.527493i	$-0.176868\pi$
$577$	1.35882e7	1.69912	0.849559	+	0.527493i	$0.176868\pi$
$578$	0	0
$579$	−1.50067e6	−0.186032
$580$	0	0
$581$	5.57368e6	0.685017
$582$	0	0
$583$	1.03975e6	0.126694
$584$	0	0
$585$	−1.57747e7	−1.90578
$586$	0	0
$587$	−9.96797e6	−1.19402	−0.597010	−	0.802234i	$-0.703645\pi$
$587$	−9.96797e6	−1.19402	−0.597010	+	0.802234i	$0.703645\pi$
$588$	0	0
$589$	1.10914e7	1.31734
$590$	0	0
$591$	−2.29470e7	−2.70245
$592$	0	0
$593$	−1.41338e7	−1.65053	−0.825264	−	0.564747i	$-0.808974\pi$
$593$	−1.41338e7	−1.65053	−0.825264	+	0.564747i	$0.808974\pi$
$594$	0	0
$595$	−8.53852e6	−0.988759
$596$	0	0
$597$	4.16273e6	0.478016
$598$	0	0
$599$	560596.	0.0638385	0.0319193	−	0.999490i	$-0.489838\pi$
$599$	560596.	0.0638385	0.0319193	+	0.999490i	$0.489838\pi$
$600$	0	0
$601$	1.08287e7	1.22290	0.611451	−	0.791282i	$-0.290586\pi$
$601$	1.08287e7	1.22290	0.611451	+	0.791282i	$0.290586\pi$
$602$	0	0
$603$	2.48597e7	2.78422
$604$	0	0
$605$	−396797.	−0.0440737
$606$	0	0
$607$	4.78906e6	0.527568	0.263784	−	0.964582i	$-0.415029\pi$
$607$	4.78906e6	0.527568	0.263784	+	0.964582i	$0.415029\pi$
$608$	0	0
$609$	4.61887e7	5.04653
$610$	0	0
$611$	1.05138e7	1.13935
$612$	0	0
$613$	552751.	0.0594125	0.0297063	−	0.999559i	$-0.490543\pi$
$613$	552751.	0.0594125	0.0297063	+	0.999559i	$0.490543\pi$
$614$	0	0
$615$	−1.54299e6	−0.164504
$616$	0	0
$617$	1.79810e7	1.90152	0.950761	−	0.309926i	$-0.100304\pi$
$617$	1.79810e7	1.90152	0.950761	+	0.309926i	$0.100304\pi$
$618$	0	0
$619$	−9.65428e6	−1.01273	−0.506365	−	0.862319i	$-0.669011\pi$
$619$	−9.65428e6	−1.01273	−0.506365	+	0.862319i	$0.669011\pi$
$620$	0	0
$621$	−1.94526e7	−2.02418
$622$	0	0
$623$	1.98685e7	2.05091
$624$	0	0
$625$	736074.	0.0753740
$626$	0	0
$627$	1.52145e7	1.54557
$628$	0	0
$629$	−1.35609e7	−1.36667
$630$	0	0
$631$	−5.23646e6	−0.523558	−0.261779	−	0.965128i	$-0.584309\pi$
$631$	−5.23646e6	−0.523558	−0.261779	+	0.965128i	$0.584309\pi$
$632$	0	0
$633$	−1.08020e7	−1.07150
$634$	0	0
$635$	−2.36012e6	−0.232274
$636$	0	0
$637$	−3.05299e7	−2.98110
$638$	0	0
$639$	−1.51874e7	−1.47140
$640$	0	0
$641$	−5.16676e6	−0.496676	−0.248338	−	0.968673i	$-0.579884\pi$
$641$	−5.16676e6	−0.496676	−0.248338	+	0.968673i	$0.579884\pi$
$642$	0	0
$643$	−1.52900e7	−1.45841	−0.729207	−	0.684293i	$-0.760111\pi$
$643$	−1.52900e7	−1.45841	−0.729207	+	0.684293i	$0.760111\pi$
$644$	0	0
$645$	−1.05627e6	−0.0999711
$646$	0	0
$647$	−9.70061e6	−0.911042	−0.455521	−	0.890225i	$-0.650547\pi$
$647$	−9.70061e6	−0.911042	−0.455521	+	0.890225i	$0.650547\pi$
$648$	0	0
$649$	−1.83376e7	−1.70895
$650$	0	0
$651$	4.89769e7	4.52938
$652$	0	0
$653$	2.06182e7	1.89221	0.946104	−	0.323864i	$-0.104982\pi$
$653$	2.06182e7	1.89221	0.946104	+	0.323864i	$0.104982\pi$
$654$	0	0
$655$	−6.88971e6	−0.627477
$656$	0	0
$657$	−2.40668e7	−2.17523
$658$	0	0
$659$	1.01447e7	0.909968	0.454984	−	0.890500i	$-0.349645\pi$
$659$	1.01447e7	0.909968	0.454984	+	0.890500i	$0.349645\pi$
$660$	0	0
$661$	5.89873e6	0.525116	0.262558	−	0.964916i	$-0.415434\pi$
$661$	5.89873e6	0.525116	0.262558	+	0.964916i	$0.415434\pi$
$662$	0	0
$663$	2.87971e7	2.54428
$664$	0	0
$665$	1.05338e7	0.923701
$666$	0	0
$667$	−1.79837e7	−1.56518
$668$	0	0
$669$	3.87249e7	3.34522
$670$	0	0
$671$	−1.94898e7	−1.67110
$672$	0	0
$673$	7.19044e6	0.611953	0.305976	−	0.952039i	$-0.401017\pi$
$673$	7.19044e6	0.611953	0.305976	+	0.952039i	$0.401017\pi$
$674$	0	0
$675$	−1.62451e7	−1.37234
$676$	0	0
$677$	−1.72813e7	−1.44912	−0.724559	−	0.689213i	$-0.757956\pi$
$677$	−1.72813e7	−1.44912	−0.724559	+	0.689213i	$0.757956\pi$
$678$	0	0
$679$	−1.88371e7	−1.56798
$680$	0	0
$681$	2.32661e6	0.192245
$682$	0	0
$683$	−8.42624e6	−0.691165	−0.345583	−	0.938388i	$-0.612319\pi$
$683$	−8.42624e6	−0.691165	−0.345583	+	0.938388i	$0.612319\pi$
$684$	0	0
$685$	2.53117e6	0.206108
$686$	0	0
$687$	3.92695e7	3.17441
$688$	0	0
$689$	−2.42785e6	−0.194838
$690$	0	0
$691$	−1.30470e7	−1.03948	−0.519739	−	0.854325i	$-0.673971\pi$
$691$	−1.30470e7	−1.03948	−0.519739	+	0.854325i	$0.673971\pi$
$692$	0	0
$693$	4.60662e7	3.64376
$694$	0	0
$695$	1.30395e6	0.102400
$696$	0	0
$697$	1.93139e6	0.150587
$698$	0	0
$699$	−2.81558e6	−0.217959
$700$	0	0
$701$	1.67394e7	1.28660	0.643301	−	0.765613i	$-0.277564\pi$
$701$	1.67394e7	1.28660	0.643301	+	0.765613i	$0.277564\pi$
$702$	0	0
$703$	1.67299e7	1.27675
$704$	0	0
$705$	−1.07059e7	−0.811245
$706$	0	0
$707$	8.37367e6	0.630039
$708$	0	0
$709$	5.38084e6	0.402008	0.201004	−	0.979590i	$-0.435580\pi$
$709$	5.38084e6	0.402008	0.201004	+	0.979590i	$0.435580\pi$
$710$	0	0
$711$	−1.25537e7	−0.931320
$712$	0	0
$713$	−1.90693e7	−1.40479
$714$	0	0
$715$	−1.14883e7	−0.840408
$716$	0	0
$717$	2.86272e7	2.07960
$718$	0	0
$719$	5.01053e6	0.361461	0.180731	−	0.983533i	$-0.442154\pi$
$719$	5.01053e6	0.361461	0.180731	+	0.983533i	$0.442154\pi$
$720$	0	0
$721$	1.98448e7	1.42170
$722$	0	0
$723$	−1.41725e7	−1.00832
$724$	0	0
$725$	−1.50183e7	−1.06115
$726$	0	0
$727$	−1.55925e7	−1.09416	−0.547079	−	0.837081i	$-0.684260\pi$
$727$	−1.55925e7	−1.09416	−0.547079	+	0.837081i	$0.684260\pi$
$728$	0	0
$729$	−4.56459e6	−0.318114
$730$	0	0
$731$	1.32215e6	0.0915136
$732$	0	0
$733$	−1.08598e7	−0.746558	−0.373279	−	0.927719i	$-0.621767\pi$
$733$	−1.08598e7	−0.746558	−0.373279	+	0.927719i	$0.621767\pi$
$734$	0	0
$735$	3.10877e7	2.12261
$736$	0	0
$737$	1.81046e7	1.22778
$738$	0	0
$739$	−8.55357e6	−0.576151	−0.288076	−	0.957608i	$-0.593015\pi$
$739$	−8.55357e6	−0.576151	−0.288076	+	0.957608i	$0.593015\pi$
$740$	0	0
$741$	−3.55265e7	−2.37688
$742$	0	0
$743$	−2.37009e7	−1.57505	−0.787524	−	0.616284i	$-0.788637\pi$
$743$	−2.37009e7	−1.57505	−0.787524	+	0.616284i	$0.788637\pi$
$744$	0	0
$745$	6.66580e6	0.440009
$746$	0	0
$747$	1.31247e7	0.860574
$748$	0	0
$749$	9.64222e6	0.628018
$750$	0	0
$751$	2.01160e7	1.30149	0.650747	−	0.759295i	$-0.274456\pi$
$751$	2.01160e7	1.30149	0.650747	+	0.759295i	$0.274456\pi$
$752$	0	0
$753$	−5.39318e7	−3.46623
$754$	0	0
$755$	−4.70490e6	−0.300388
$756$	0	0
$757$	−1.88820e7	−1.19759	−0.598795	−	0.800902i	$-0.704354\pi$
$757$	−1.88820e7	−1.19759	−0.598795	+	0.800902i	$0.704354\pi$
$758$	0	0
$759$	−2.61582e7	−1.64817
$760$	0	0
$761$	−2.52164e7	−1.57842	−0.789208	−	0.614126i	$-0.789509\pi$
$761$	−2.52164e7	−1.57842	−0.789208	+	0.614126i	$0.789509\pi$
$762$	0	0
$763$	2.62019e7	1.62937
$764$	0	0
$765$	−2.01062e7	−1.24216
$766$	0	0
$767$	4.28190e7	2.62814
$768$	0	0
$769$	−7.74078e6	−0.472029	−0.236015	−	0.971750i	$-0.575841\pi$
$769$	−7.74078e6	−0.472029	−0.236015	+	0.971750i	$0.575841\pi$
$770$	0	0
$771$	−8.02096e6	−0.485948
$772$	0	0
$773$	1.40912e6	0.0848202	0.0424101	−	0.999100i	$-0.486496\pi$
$773$	1.40912e6	0.0848202	0.0424101	+	0.999100i	$0.486496\pi$
$774$	0	0
$775$	−1.59249e7	−0.952407
$776$	0	0
$777$	7.38753e7	4.38982
$778$	0	0
$779$	−2.38272e6	−0.140679
$780$	0	0
$781$	−1.10606e7	−0.648858
$782$	0	0
$783$	5.89046e7	3.43356
$784$	0	0
$785$	−1.48084e7	−0.857695
$786$	0	0
$787$	−6.39739e6	−0.368185	−0.184092	−	0.982909i	$-0.558935\pi$
$787$	−6.39739e6	−0.368185	−0.184092	+	0.982909i	$0.558935\pi$
$788$	0	0
$789$	1.20178e7	0.687281
$790$	0	0
$791$	−3.23719e7	−1.83962
$792$	0	0
$793$	4.55095e7	2.56992
$794$	0	0
$795$	2.47220e6	0.138729
$796$	0	0
$797$	2.96000e7	1.65062	0.825308	−	0.564683i	$-0.191002\pi$
$797$	2.96000e7	1.65062	0.825308	+	0.564683i	$0.191002\pi$
$798$	0	0
$799$	1.34008e7	0.742615
$800$	0	0
$801$	4.67857e7	2.57651
$802$	0	0
$803$	−1.75272e7	−0.959230
$804$	0	0
$805$	−1.81107e7	−0.985020
$806$	0	0
$807$	−5.88108e6	−0.317887
$808$	0	0
$809$	1.67165e7	0.897993	0.448996	−	0.893534i	$-0.351782\pi$
$809$	1.67165e7	0.897993	0.448996	+	0.893534i	$0.351782\pi$
$810$	0	0
$811$	1.50423e7	0.803087	0.401544	−	0.915840i	$-0.368474\pi$
$811$	1.50423e7	0.803087	0.401544	+	0.915840i	$0.368474\pi$
$812$	0	0
$813$	−6.84423e6	−0.363160
$814$	0	0
$815$	−6.46784e6	−0.341087
$816$	0	0
$817$	−1.63111e6	−0.0854923
$818$	0	0
$819$	−1.07566e8	−5.60360
$820$	0	0
$821$	−1.76652e7	−0.914660	−0.457330	−	0.889297i	$-0.651194\pi$
$821$	−1.76652e7	−0.914660	−0.457330	+	0.889297i	$0.651194\pi$
$822$	0	0
$823$	2.00681e7	1.03278	0.516389	−	0.856354i	$-0.327276\pi$
$823$	2.00681e7	1.03278	0.516389	+	0.856354i	$0.327276\pi$
$824$	0	0
$825$	−2.18449e7	−1.11742
$826$	0	0
$827$	−3.23283e7	−1.64368	−0.821842	−	0.569715i	$-0.807054\pi$
$827$	−3.23283e7	−1.64368	−0.821842	+	0.569715i	$0.807054\pi$
$828$	0	0
$829$	−1.79696e7	−0.908140	−0.454070	−	0.890966i	$-0.650028\pi$
$829$	−1.79696e7	−0.908140	−0.454070	+	0.890966i	$0.650028\pi$
$830$	0	0
$831$	−190507.	−0.00956994
$832$	0	0
$833$	−3.89130e7	−1.94304
$834$	0	0
$835$	−2.94094e6	−0.145972
$836$	0	0
$837$	6.24603e7	3.08170
$838$	0	0
$839$	9.27963e6	0.455120	0.227560	−	0.973764i	$-0.426925\pi$
$839$	9.27963e6	0.455120	0.227560	+	0.973764i	$0.426925\pi$
$840$	0	0
$841$	3.39453e7	1.65497
$842$	0	0
$843$	−6.45691e7	−3.12936
$844$	0	0
$845$	1.45682e7	0.701880
$846$	0	0
$847$	−2.70572e6	−0.129591
$848$	0	0
$849$	−2.92210e7	−1.39132
$850$	0	0
$851$	−2.87635e7	−1.36150
$852$	0	0
$853$	−3.40818e7	−1.60380	−0.801900	−	0.597458i	$-0.796177\pi$
$853$	−3.40818e7	−1.60380	−0.801900	+	0.597458i	$0.796177\pi$
$854$	0	0
$855$	2.48047e7	1.16043
$856$	0	0
$857$	−4.53094e6	−0.210735	−0.105367	−	0.994433i	$-0.533602\pi$
$857$	−4.53094e6	−0.210735	−0.105367	+	0.994433i	$0.533602\pi$
$858$	0	0
$859$	3.27476e7	1.51424	0.757122	−	0.653273i	$-0.226605\pi$
$859$	3.27476e7	1.51424	0.757122	+	0.653273i	$0.226605\pi$
$860$	0	0
$861$	−1.05215e7	−0.483695
$862$	0	0
$863$	−4.18043e6	−0.191070	−0.0955352	−	0.995426i	$-0.530456\pi$
$863$	−4.18043e6	−0.191070	−0.0955352	+	0.995426i	$0.530456\pi$
$864$	0	0
$865$	1.69915e7	0.772130
$866$	0	0
$867$	−2.77390e6	−0.125326
$868$	0	0
$869$	−9.14253e6	−0.410693
$870$	0	0
$871$	−4.22750e7	−1.88816
$872$	0	0
$873$	−4.43570e7	−1.96982
$874$	0	0
$875$	−3.83480e7	−1.69326
$876$	0	0
$877$	2.15842e7	0.947624	0.473812	−	0.880626i	$-0.342878\pi$
$877$	2.15842e7	0.947624	0.473812	+	0.880626i	$0.342878\pi$
$878$	0	0
$879$	1.34317e7	0.586353
$880$	0	0
$881$	3.03382e7	1.31689	0.658445	−	0.752629i	$-0.271214\pi$
$881$	3.03382e7	1.31689	0.658445	+	0.752629i	$0.271214\pi$
$882$	0	0
$883$	3.71147e7	1.60193	0.800966	−	0.598710i	$-0.204320\pi$
$883$	3.71147e7	1.60193	0.800966	+	0.598710i	$0.204320\pi$
$884$	0	0
$885$	−4.36013e7	−1.87129
$886$	0	0
$887$	−2.31896e7	−0.989654	−0.494827	−	0.868991i	$-0.664769\pi$
$887$	−2.31896e7	−0.989654	−0.494827	+	0.868991i	$0.664769\pi$
$888$	0	0
$889$	−1.60935e7	−0.682960
$890$	0	0
$891$	3.59527e7	1.51718
$892$	0	0
$893$	−1.65323e7	−0.693753
$894$	0	0
$895$	1.42815e7	0.595959
$896$	0	0
$897$	6.10803e7	2.53466
$898$	0	0
$899$	5.77437e7	2.38290
$900$	0	0
$901$	−3.09450e6	−0.126992
$902$	0	0
$903$	−7.20259e6	−0.293947
$904$	0	0
$905$	6.61483e6	0.268471
$906$	0	0
$907$	5.77629e6	0.233147	0.116574	−	0.993182i	$-0.462809\pi$
$907$	5.77629e6	0.233147	0.116574	+	0.993182i	$0.462809\pi$
$908$	0	0
$909$	1.97180e7	0.791506
$910$	0	0
$911$	−1.70518e7	−0.680731	−0.340365	−	0.940293i	$-0.610551\pi$
$911$	−1.70518e7	−0.680731	−0.340365	+	0.940293i	$0.610551\pi$
$912$	0	0
$913$	9.55835e6	0.379495
$914$	0	0
$915$	−4.63410e7	−1.82984
$916$	0	0
$917$	−4.69803e7	−1.84498
$918$	0	0
$919$	1.34813e7	0.526555	0.263278	−	0.964720i	$-0.415196\pi$
$919$	1.34813e7	0.526555	0.263278	+	0.964720i	$0.415196\pi$
$920$	0	0
$921$	−1.17526e7	−0.456544
$922$	0	0
$923$	2.58269e7	0.997855
$924$	0	0
$925$	−2.40206e7	−0.923061
$926$	0	0
$927$	4.67300e7	1.78606
$928$	0	0
$929$	−4.73150e7	−1.79871	−0.899353	−	0.437224i	$-0.855962\pi$
$929$	−4.73150e7	−1.79871	−0.899353	+	0.437224i	$0.855962\pi$
$930$	0	0
$931$	4.80062e7	1.81519
$932$	0	0
$933$	−1.49335e6	−0.0561641
$934$	0	0
$935$	−1.46428e7	−0.547766
$936$	0	0
$937$	−3.81792e6	−0.142062	−0.0710310	−	0.997474i	$-0.522629\pi$
$937$	−3.81792e6	−0.142062	−0.0710310	+	0.997474i	$0.522629\pi$
$938$	0	0
$939$	5.03318e6	0.186285
$940$	0	0
$941$	−2.79743e7	−1.02988	−0.514939	−	0.857227i	$-0.672185\pi$
$941$	−2.79743e7	−1.02988	−0.514939	+	0.857227i	$0.672185\pi$
$942$	0	0
$943$	4.09659e6	0.150018
$944$	0	0
$945$	5.93205e7	2.16085
$946$	0	0
$947$	−2.98435e7	−1.08137	−0.540685	−	0.841225i	$-0.681835\pi$
$947$	−2.98435e7	−1.08137	−0.540685	+	0.841225i	$0.681835\pi$
$948$	0	0
$949$	4.09266e7	1.47516
$950$	0	0
$951$	8.61451e7	3.08873
$952$	0	0
$953$	4.33581e6	0.154646	0.0773228	−	0.997006i	$-0.475363\pi$
$953$	4.33581e6	0.154646	0.0773228	+	0.997006i	$0.475363\pi$
$954$	0	0
$955$	2.69652e6	0.0956744
$956$	0	0
$957$	7.92095e7	2.79574
$958$	0	0
$959$	1.72599e7	0.606025
$960$	0	0
$961$	3.26002e7	1.13871
$962$	0	0
$963$	2.27052e7	0.788967
$964$	0	0
$965$	−1.78178e6	−0.0615935
$966$	0	0
$967$	1.62116e7	0.557520	0.278760	−	0.960361i	$-0.410077\pi$
$967$	1.62116e7	0.557520	0.278760	+	0.960361i	$0.410077\pi$
$968$	0	0
$969$	−4.52815e7	−1.54921
$970$	0	0
$971$	4.81971e7	1.64049	0.820243	−	0.572015i	$-0.193838\pi$
$971$	4.81971e7	1.64049	0.820243	+	0.572015i	$0.193838\pi$
$972$	0	0
$973$	8.89155e6	0.301089
$974$	0	0
$975$	5.10087e7	1.71843
$976$	0	0
$977$	1.81149e7	0.607156	0.303578	−	0.952807i	$-0.401819\pi$
$977$	1.81149e7	0.607156	0.303578	+	0.952807i	$0.401819\pi$
$978$	0	0
$979$	3.40727e7	1.13619
$980$	0	0
$981$	6.16993e7	2.04695
$982$	0	0
$983$	−1.77983e7	−0.587481	−0.293740	−	0.955885i	$-0.594900\pi$
$983$	−1.77983e7	−0.587481	−0.293740	+	0.955885i	$0.594900\pi$
$984$	0	0
$985$	−2.72456e7	−0.894758
$986$	0	0
$987$	−7.30028e7	−2.38532
$988$	0	0
$989$	2.80435e6	0.0911676
$990$	0	0
$991$	1.66882e7	0.539791	0.269895	−	0.962890i	$-0.413011\pi$
$991$	1.66882e7	0.539791	0.269895	+	0.962890i	$0.413011\pi$
$992$	0	0
$993$	6.15103e7	1.97959
$994$	0	0
$995$	4.94251e6	0.158267
$996$	0	0
$997$	−5.89661e7	−1.87873	−0.939365	−	0.342918i	$-0.888585\pi$
$997$	−5.89661e7	−1.87873	−0.939365	+	0.342918i	$0.888585\pi$
$998$	0	0
$999$	9.42133e7	2.98675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	656.6.a.h.1.10		10
4.3	odd	2		164.6.a.b.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.6.a.b.1.1	✓	10	4.3	odd	2
656.6.a.h.1.10		10	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:	\( N \)	\(=\)	\( 656 = 2^{4} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	656.a (trivial)

\(f(q)\)	\(=\)	\(q+27.8044 q^{3} +33.0128 q^{5} +225.112 q^{7} +530.085 q^{9} +O(q^{10})\)	\(q+27.8044 q^{3} +33.0128 q^{5} +225.112 q^{7} +530.085 q^{9} +386.046 q^{11} -901.433 q^{13} +917.902 q^{15} -1148.95 q^{17} +1417.44 q^{19} +6259.09 q^{21} -2436.99 q^{23} -2035.15 q^{25} +7982.23 q^{27} +7379.46 q^{29} +7824.92 q^{31} +10733.8 q^{33} +7431.57 q^{35} +11802.9 q^{37} -25063.8 q^{39} -1681.00 q^{41} -1150.74 q^{43} +17499.6 q^{45} -11663.5 q^{47} +33868.2 q^{49} -31946.0 q^{51} +2693.32 q^{53} +12744.5 q^{55} +39411.1 q^{57} -47501.0 q^{59} -50485.8 q^{61} +119328. q^{63} -29758.8 q^{65} +46897.6 q^{67} -67759.2 q^{69} -28650.9 q^{71} -45401.7 q^{73} -56586.2 q^{75} +86903.4 q^{77} -23682.5 q^{79} +93130.6 q^{81} +24759.6 q^{83} -37930.2 q^{85} +205182. q^{87} +88260.8 q^{89} -202923. q^{91} +217567. q^{93} +46793.8 q^{95} -83679.1 q^{97} +204637. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 32 * q^5 - 88 * q^7 + 1194 * q^9	\( 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9} + 476 q^{11} - 456 q^{13} + 2 q^{15} + 1456 q^{17} - 2094 q^{19} + 7048 q^{21} - 7380 q^{23} + 15158 q^{25} - 9550 q^{27} + 9948 q^{29} + 840 q^{31} + 34828 q^{33} - 31214 q^{35} + 21780 q^{37} - 17832 q^{39} - 16810 q^{41} - 56636 q^{43} + 95584 q^{45} - 72666 q^{47} + 76574 q^{49} - 115660 q^{51} + 47528 q^{53} - 14182 q^{55} + 60356 q^{57} - 87380 q^{59} + 97364 q^{61} - 66998 q^{63} + 65716 q^{65} - 5724 q^{67} + 80692 q^{69} + 2834 q^{71} + 11228 q^{73} + 50282 q^{75} + 22400 q^{77} - 90094 q^{79} + 212530 q^{81} + 16132 q^{83} + 88840 q^{85} + 318756 q^{87} + 79872 q^{89} - 62004 q^{91} + 33652 q^{93} + 574026 q^{95} - 167548 q^{97} + 441774 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 32 * q^5 - 88 * q^7 + 1194 * q^9 + 476 * q^11 - 456 * q^13 + 2 * q^15 + 1456 * q^17 - 2094 * q^19 + 7048 * q^21 - 7380 * q^23 + 15158 * q^25 - 9550 * q^27 + 9948 * q^29 + 840 * q^31 + 34828 * q^33 - 31214 * q^35 + 21780 * q^37 - 17832 * q^39 - 16810 * q^41 - 56636 * q^43 + 95584 * q^45 - 72666 * q^47 + 76574 * q^49 - 115660 * q^51 + 47528 * q^53 - 14182 * q^55 + 60356 * q^57 - 87380 * q^59 + 97364 * q^61 - 66998 * q^63 + 65716 * q^65 - 5724 * q^67 + 80692 * q^69 + 2834 * q^71 + 11228 * q^73 + 50282 * q^75 + 22400 * q^77 - 90094 * q^79 + 212530 * q^81 + 16132 * q^83 + 88840 * q^85 + 318756 * q^87 + 79872 * q^89 - 62004 * q^91 + 33652 * q^93 + 574026 * q^95 - 167548 * q^97 + 441774 * q^99

Introduction

L-functions

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Database

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