LMFDB - Embedded newform 656.6.a.h.1.5

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			1.5
Root			$4.25838$ 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)

$f(q)$	$=$	$q-5.25838 q^{3} -76.6066 q^{5} +175.490 q^{7} -215.349 q^{9} +O(q^{10})$	$q-5.25838 q^{3} -76.6066 q^{5} +175.490 q^{7} -215.349 q^{9} -693.182 q^{11} -730.584 q^{13} +402.827 q^{15} +1457.17 q^{17} -2701.01 q^{19} -922.791 q^{21} -4735.84 q^{23} +2743.57 q^{25} +2410.18 q^{27} -1352.71 q^{29} +8532.88 q^{31} +3645.02 q^{33} -13443.7 q^{35} -3725.56 q^{37} +3841.69 q^{39} -1681.00 q^{41} -9476.57 q^{43} +16497.2 q^{45} -19594.9 q^{47} +13989.6 q^{49} -7662.37 q^{51} -23071.7 q^{53} +53102.3 q^{55} +14202.9 q^{57} -17017.8 q^{59} -3320.14 q^{61} -37791.6 q^{63} +55967.5 q^{65} -32333.1 q^{67} +24902.9 q^{69} -41548.5 q^{71} +18792.8 q^{73} -14426.7 q^{75} -121646. q^{77} -83110.0 q^{79} +39656.3 q^{81} +82364.3 q^{83} -111629. q^{85} +7113.09 q^{87} -52490.4 q^{89} -128210. q^{91} -44869.2 q^{93} +206915. q^{95} +114884. q^{97} +149276. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.25838	−0.337325	−0.168663	−	0.985674i	$-0.553945\pi$
$3$	−5.25838	−0.337325	−0.168663	+	0.985674i	$0.553945\pi$
$4$	0	0
$5$	−76.6066	−1.37038	−0.685190	−	0.728364i	$-0.740281\pi$
$5$	−76.6066	−1.37038	−0.685190	+	0.728364i	$0.740281\pi$
$6$	0	0
$7$	175.490	1.35365	0.676825	−	0.736144i	$-0.263356\pi$
$7$	175.490	1.35365	0.676825	+	0.736144i	$0.263356\pi$
$8$	0	0
$9$	−215.349	−0.886212
$10$	0	0
$11$	−693.182	−1.72729	−0.863646	−	0.504098i	$-0.831825\pi$
$11$	−693.182	−1.72729	−0.863646	+	0.504098i	$0.831825\pi$
$12$	0	0
$13$	−730.584	−1.19898	−0.599490	−	0.800383i	$-0.704630\pi$
$13$	−730.584	−1.19898	−0.599490	+	0.800383i	$0.704630\pi$
$14$	0	0
$15$	402.827	0.462264
$16$	0	0
$17$	1457.17	1.22289	0.611447	−	0.791285i	$-0.290588\pi$
$17$	1457.17	1.22289	0.611447	+	0.791285i	$0.290588\pi$
$18$	0	0
$19$	−2701.01	−1.71649	−0.858246	−	0.513238i	$-0.828446\pi$
$19$	−2701.01	−1.71649	−0.858246	+	0.513238i	$0.828446\pi$
$20$	0	0
$21$	−922.791	−0.456620
$22$	0	0
$23$	−4735.84	−1.86671	−0.933357	−	0.358950i	$-0.883135\pi$
$23$	−4735.84	−1.86671	−0.933357	+	0.358950i	$0.883135\pi$
$24$	0	0
$25$	2743.57	0.877943
$26$	0	0
$27$	2410.18	0.636267
$28$	0	0
$29$	−1352.71	−0.298683	−0.149342	−	0.988786i	$-0.547715\pi$
$29$	−1352.71	−0.298683	−0.149342	+	0.988786i	$0.547715\pi$
$30$	0	0
$31$	8532.88	1.59475	0.797373	−	0.603487i	$-0.206222\pi$
$31$	8532.88	1.59475	0.797373	+	0.603487i	$0.206222\pi$
$32$	0	0
$33$	3645.02	0.582659
$34$	0	0
$35$	−13443.7	−1.85501
$36$	0	0
$37$	−3725.56	−0.447391	−0.223695	−	0.974659i	$-0.571812\pi$
$37$	−3725.56	−0.447391	−0.223695	+	0.974659i	$0.571812\pi$
$38$	0	0
$39$	3841.69	0.404446
$40$	0	0
$41$	−1681.00	−0.156174
$41$	−1681.00	−0.156174
$42$	0	0
$43$	−9476.57	−0.781592	−0.390796	−	0.920477i	$-0.627800\pi$
$43$	−9476.57	−0.781592	−0.390796	+	0.920477i	$0.627800\pi$
$44$	0	0
$45$	16497.2	1.21445
$46$	0	0
$47$	−19594.9	−1.29390	−0.646948	−	0.762534i	$-0.723955\pi$
$47$	−19594.9	−1.29390	−0.646948	+	0.762534i	$0.723955\pi$
$48$	0	0
$49$	13989.6	0.832366
$50$	0	0
$51$	−7662.37	−0.412513
$52$	0	0
$53$	−23071.7	−1.12821	−0.564105	−	0.825703i	$-0.690779\pi$
$53$	−23071.7	−1.12821	−0.564105	+	0.825703i	$0.690779\pi$
$54$	0	0
$55$	53102.3	2.36705
$56$	0	0
$57$	14202.9	0.579016
$58$	0	0
$59$	−17017.8	−0.636463	−0.318232	−	0.948013i	$-0.603089\pi$
$59$	−17017.8	−0.636463	−0.318232	+	0.948013i	$0.603089\pi$
$60$	0	0
$61$	−3320.14	−0.114244	−0.0571218	−	0.998367i	$-0.518192\pi$
$61$	−3320.14	−0.114244	−0.0571218	+	0.998367i	$0.518192\pi$
$62$	0	0
$63$	−37791.6	−1.19962
$64$	0	0
$65$	55967.5	1.64306
$66$	0	0
$67$	−32333.1	−0.879954	−0.439977	−	0.898009i	$-0.645013\pi$
$67$	−32333.1	−0.879954	−0.439977	+	0.898009i	$0.645013\pi$
$68$	0	0
$69$	24902.9	0.629690
$70$	0	0
$71$	−41548.5	−0.978160	−0.489080	−	0.872239i	$-0.662667\pi$
$71$	−41548.5	−0.978160	−0.489080	+	0.872239i	$0.662667\pi$
$72$	0	0
$73$	18792.8	0.412748	0.206374	−	0.978473i	$-0.433834\pi$
$73$	18792.8	0.412748	0.206374	+	0.978473i	$0.433834\pi$
$74$	0	0
$75$	−14426.7	−0.296152
$76$	0	0
$77$	−121646.	−2.33815
$78$	0	0
$79$	−83110.0	−1.49825	−0.749127	−	0.662426i	$-0.769527\pi$
$79$	−83110.0	−1.49825	−0.749127	+	0.662426i	$0.769527\pi$
$80$	0	0
$81$	39656.3	0.671583
$82$	0	0
$83$	82364.3	1.31233	0.656166	−	0.754616i	$-0.272177\pi$
$83$	82364.3	1.31233	0.656166	+	0.754616i	$0.272177\pi$
$84$	0	0
$85$	−111629.	−1.67583
$86$	0	0
$87$	7113.09	0.100753
$88$	0	0
$89$	−52490.4	−0.702433	−0.351216	−	0.936294i	$-0.614232\pi$
$89$	−52490.4	−0.702433	−0.351216	+	0.936294i	$0.614232\pi$
$90$	0	0
$91$	−128210.	−1.62300
$92$	0	0
$93$	−44869.2	−0.537948
$94$	0	0
$95$	206915.	2.35225
$96$	0	0
$97$	114884.	1.23974	0.619872	−	0.784703i	$-0.287185\pi$
$97$	114884.	1.23974	0.619872	+	0.784703i	$0.287185\pi$
$98$	0	0
$99$	149276.	1.53075
$100$	0	0
$101$	82115.0	0.800975	0.400488	−	0.916302i	$-0.368841\pi$
$101$	82115.0	0.800975	0.400488	+	0.916302i	$0.368841\pi$
$102$	0	0
$103$	43352.9	0.402648	0.201324	−	0.979525i	$-0.435476\pi$
$103$	43352.9	0.402648	0.201324	+	0.979525i	$0.435476\pi$
$104$	0	0
$105$	70691.9	0.625743
$106$	0	0
$107$	50497.8	0.426396	0.213198	−	0.977009i	$-0.431612\pi$
$107$	50497.8	0.426396	0.213198	+	0.977009i	$0.431612\pi$
$108$	0	0
$109$	37667.4	0.303668	0.151834	−	0.988406i	$-0.451482\pi$
$109$	37667.4	0.303668	0.151834	+	0.988406i	$0.451482\pi$
$110$	0	0
$111$	19590.4	0.150916
$112$	0	0
$113$	−24856.1	−0.183120	−0.0915601	−	0.995800i	$-0.529185\pi$
$113$	−24856.1	−0.183120	−0.0915601	+	0.995800i	$0.529185\pi$
$114$	0	0
$115$	362797.	2.55811
$116$	0	0
$117$	157331.	1.06255
$118$	0	0
$119$	255719.	1.65537
$120$	0	0
$121$	319451.	1.98354
$122$	0	0
$123$	8839.34	0.0526814
$124$	0	0
$125$	29220.0	0.167265
$126$	0	0
$127$	101791.	0.560015	0.280008	−	0.959998i	$-0.409663\pi$
$127$	101791.	0.560015	0.280008	+	0.959998i	$0.409663\pi$
$128$	0	0
$129$	49831.4	0.263651
$130$	0	0
$131$	10479.4	0.0533528	0.0266764	−	0.999644i	$-0.491508\pi$
$131$	10479.4	0.0533528	0.0266764	+	0.999644i	$0.491508\pi$
$132$	0	0
$133$	−473999.	−2.32353
$134$	0	0
$135$	−184635.	−0.871928
$136$	0	0
$137$	−232043.	−1.05625	−0.528125	−	0.849167i	$-0.677105\pi$
$137$	−232043.	−1.05625	−0.528125	+	0.849167i	$0.677105\pi$
$138$	0	0
$139$	−60360.0	−0.264979	−0.132490	−	0.991184i	$-0.542297\pi$
$139$	−60360.0	−0.264979	−0.132490	+	0.991184i	$0.542297\pi$
$140$	0	0
$141$	103038.	0.436464
$142$	0	0
$143$	506428.	2.07099
$144$	0	0
$145$	103627.	0.409310
$146$	0	0
$147$	−73562.5	−0.280778
$148$	0	0
$149$	452881.	1.67116	0.835580	−	0.549368i	$-0.185132\pi$
$149$	452881.	1.67116	0.835580	+	0.549368i	$0.185132\pi$
$150$	0	0
$151$	−190342.	−0.679347	−0.339674	−	0.940543i	$-0.610317\pi$
$151$	−190342.	−0.679347	−0.339674	+	0.940543i	$0.610317\pi$
$152$	0	0
$153$	−313802.	−1.08374
$154$	0	0
$155$	−653675.	−2.18541
$156$	0	0
$157$	450034.	1.45712	0.728561	−	0.684981i	$-0.240189\pi$
$157$	450034.	1.45712	0.728561	+	0.684981i	$0.240189\pi$
$158$	0	0
$159$	121320.	0.380574
$160$	0	0
$161$	−831091.	−2.52688
$162$	0	0
$163$	295383.	0.870795	0.435398	−	0.900238i	$-0.356608\pi$
$163$	295383.	0.870795	0.435398	+	0.900238i	$0.356608\pi$
$164$	0	0
$165$	−279232.	−0.798465
$166$	0	0
$167$	−192584.	−0.534354	−0.267177	−	0.963648i	$-0.586091\pi$
$167$	−192584.	−0.534354	−0.267177	+	0.963648i	$0.586091\pi$
$168$	0	0
$169$	162460.	0.437551
$170$	0	0
$171$	581660.	1.52118
$172$	0	0
$173$	215682.	0.547896	0.273948	−	0.961745i	$-0.411670\pi$
$173$	215682.	0.547896	0.273948	+	0.961745i	$0.411670\pi$
$174$	0	0
$175$	481468.	1.18843
$176$	0	0
$177$	89486.1	0.214695
$178$	0	0
$179$	170607.	0.397984	0.198992	−	0.980001i	$-0.436233\pi$
$179$	170607.	0.397984	0.198992	+	0.980001i	$0.436233\pi$
$180$	0	0
$181$	−582931.	−1.32258	−0.661288	−	0.750132i	$-0.729990\pi$
$181$	−582931.	−1.32258	−0.661288	+	0.750132i	$0.729990\pi$
$182$	0	0
$183$	17458.5	0.0385372
$184$	0	0
$185$	285402.	0.613095
$186$	0	0
$187$	−1.01009e6	−2.11230
$188$	0	0
$189$	422961.	0.861282
$190$	0	0
$191$	−858405.	−1.70258	−0.851292	−	0.524692i	$-0.824181\pi$
$191$	−858405.	−1.70258	−0.851292	+	0.524692i	$0.824181\pi$
$192$	0	0
$193$	550286.	1.06340	0.531698	−	0.846934i	$-0.321554\pi$
$193$	550286.	1.06340	0.531698	+	0.846934i	$0.321554\pi$
$194$	0	0
$195$	−294299.	−0.554245
$196$	0	0
$197$	136107.	0.249870	0.124935	−	0.992165i	$-0.460128\pi$
$197$	136107.	0.249870	0.124935	+	0.992165i	$0.460128\pi$
$198$	0	0
$199$	233414.	0.417824	0.208912	−	0.977934i	$-0.433008\pi$
$199$	233414.	0.417824	0.208912	+	0.977934i	$0.433008\pi$
$200$	0	0
$201$	170020.	0.296831
$202$	0	0
$203$	−237387.	−0.404313
$204$	0	0
$205$	128776.	0.214017
$206$	0	0
$207$	1.01986e6	1.65430
$208$	0	0
$209$	1.87229e6	2.96488
$210$	0	0
$211$	−928703.	−1.43605	−0.718027	−	0.696016i	$-0.754954\pi$
$211$	−928703.	−1.43605	−0.718027	+	0.696016i	$0.754954\pi$
$212$	0	0
$213$	218478.	0.329958
$214$	0	0
$215$	725967.	1.07108
$216$	0	0
$217$	1.49743e6	2.15873
$218$	0	0
$219$	−98819.9	−0.139230
$220$	0	0
$221$	−1.06459e6	−1.46622
$222$	0	0
$223$	658031.	0.886103	0.443052	−	0.896496i	$-0.353896\pi$
$223$	658031.	0.886103	0.443052	+	0.896496i	$0.353896\pi$
$224$	0	0
$225$	−590826.	−0.778043
$226$	0	0
$227$	−623189.	−0.802704	−0.401352	−	0.915924i	$-0.631460\pi$
$227$	−623189.	−0.802704	−0.401352	+	0.915924i	$0.631460\pi$
$228$	0	0
$229$	64424.3	0.0811822	0.0405911	−	0.999176i	$-0.487076\pi$
$229$	64424.3	0.0811822	0.0405911	+	0.999176i	$0.487076\pi$
$230$	0	0
$231$	639662.	0.788716
$232$	0	0
$233$	−1.06558e6	−1.28587	−0.642935	−	0.765920i	$-0.722284\pi$
$233$	−1.06558e6	−1.28587	−0.642935	+	0.765920i	$0.722284\pi$
$234$	0	0
$235$	1.50110e6	1.77313
$236$	0	0
$237$	437024.	0.505399
$238$	0	0
$239$	−1.36562e6	−1.54645	−0.773223	−	0.634134i	$-0.781357\pi$
$239$	−1.36562e6	−1.54645	−0.773223	+	0.634134i	$0.781357\pi$
$240$	0	0
$241$	−5098.60	−0.00565468	−0.00282734	−	0.999996i	$-0.500900\pi$
$241$	−5098.60	−0.00565468	−0.00282734	+	0.999996i	$0.500900\pi$
$242$	0	0
$243$	−794201.	−0.862809
$244$	0	0
$245$	−1.07169e6	−1.14066
$246$	0	0
$247$	1.97331e6	2.05804
$248$	0	0
$249$	−433103.	−0.442683
$250$	0	0
$251$	619803.	0.620968	0.310484	−	0.950579i	$-0.399509\pi$
$251$	619803.	0.620968	0.310484	+	0.950579i	$0.399509\pi$
$252$	0	0
$253$	3.28280e6	3.22436
$254$	0	0
$255$	586988.	0.565300
$256$	0	0
$257$	−108794.	−0.102747	−0.0513737	−	0.998679i	$-0.516360\pi$
$257$	−108794.	−0.102747	−0.0513737	+	0.998679i	$0.516360\pi$
$258$	0	0
$259$	−653796.	−0.605610
$260$	0	0
$261$	291306.	0.264697
$262$	0	0
$263$	367033.	0.327202	0.163601	−	0.986527i	$-0.447689\pi$
$263$	367033.	0.327202	0.163601	+	0.986527i	$0.447689\pi$
$264$	0	0
$265$	1.76745e6	1.54608
$266$	0	0
$267$	276014.	0.236948
$268$	0	0
$269$	777113.	0.654792	0.327396	−	0.944887i	$-0.393829\pi$
$269$	777113.	0.654792	0.327396	+	0.944887i	$0.393829\pi$
$270$	0	0
$271$	984621.	0.814415	0.407207	−	0.913336i	$-0.366503\pi$
$271$	984621.	0.814415	0.407207	+	0.913336i	$0.366503\pi$
$272$	0	0
$273$	674176.	0.547478
$274$	0	0
$275$	−1.90180e6	−1.51646
$276$	0	0
$277$	2.19928e6	1.72219	0.861094	−	0.508446i	$-0.169780\pi$
$277$	2.19928e6	1.72219	0.861094	+	0.508446i	$0.169780\pi$
$278$	0	0
$279$	−1.83755e6	−1.41328
$280$	0	0
$281$	−2.00561e6	−1.51524	−0.757618	−	0.652698i	$-0.773637\pi$
$281$	−2.00561e6	−1.51524	−0.757618	+	0.652698i	$0.773637\pi$
$282$	0	0
$283$	2.31299e6	1.71675	0.858377	−	0.513019i	$-0.171473\pi$
$283$	2.31299e6	1.71675	0.858377	+	0.513019i	$0.171473\pi$
$284$	0	0
$285$	−1.08804e6	−0.793472
$286$	0	0
$287$	−294998.	−0.211404
$288$	0	0
$289$	703498.	0.495471
$290$	0	0
$291$	−604106.	−0.418197
$292$	0	0
$293$	2.37980e6	1.61946	0.809731	−	0.586801i	$-0.199613\pi$
$293$	2.37980e6	1.61946	0.809731	+	0.586801i	$0.199613\pi$
$294$	0	0
$295$	1.30368e6	0.872197
$296$	0	0
$297$	−1.67069e6	−1.09902
$298$	0	0
$299$	3.45993e6	2.23815
$300$	0	0
$301$	−1.66304e6	−1.05800
$302$	0	0
$303$	−431792.	−0.270189
$304$	0	0
$305$	254344.	0.156557
$306$	0	0
$307$	1.58796e6	0.961597	0.480798	−	0.876831i	$-0.340347\pi$
$307$	1.58796e6	0.961597	0.480798	+	0.876831i	$0.340347\pi$
$308$	0	0
$309$	−227966.	−0.135823
$310$	0	0
$311$	−1.23418e6	−0.723565	−0.361783	−	0.932263i	$-0.617832\pi$
$311$	−1.23418e6	−0.723565	−0.361783	+	0.932263i	$0.617832\pi$
$312$	0	0
$313$	−1.30169e6	−0.751014	−0.375507	−	0.926820i	$-0.622531\pi$
$313$	−1.30169e6	−0.751014	−0.375507	+	0.926820i	$0.622531\pi$
$314$	0	0
$315$	2.89508e6	1.64394
$316$	0	0
$317$	−1.01863e6	−0.569335	−0.284667	−	0.958626i	$-0.591883\pi$
$317$	−1.01863e6	−0.569335	−0.284667	+	0.958626i	$0.591883\pi$
$318$	0	0
$319$	937678.	0.515914
$320$	0	0
$321$	−265537.	−0.143834
$322$	0	0
$323$	−3.93584e6	−2.09909
$324$	0	0
$325$	−2.00441e6	−1.05263
$326$	0	0
$327$	−198070.	−0.102435
$328$	0	0
$329$	−3.43871e6	−1.75148
$330$	0	0
$331$	−3.31526e6	−1.66321	−0.831605	−	0.555368i	$-0.812577\pi$
$331$	−3.31526e6	−1.66321	−0.831605	+	0.555368i	$0.812577\pi$
$332$	0	0
$333$	802297.	0.396483
$334$	0	0
$335$	2.47693e6	1.20587
$336$	0	0
$337$	33920.1	0.0162698	0.00813490	−	0.999967i	$-0.497411\pi$
$337$	33920.1	0.0162698	0.00813490	+	0.999967i	$0.497411\pi$
$338$	0	0
$339$	130703.	0.0617711
$340$	0	0
$341$	−5.91485e6	−2.75459
$342$	0	0
$343$	−494428.	−0.226917
$344$	0	0
$345$	−1.90772e6	−0.862914
$346$	0	0
$347$	2.78072e6	1.23975	0.619874	−	0.784701i	$-0.287184\pi$
$347$	2.78072e6	1.23975	0.619874	+	0.784701i	$0.287184\pi$
$348$	0	0
$349$	1.58649e6	0.697225	0.348613	−	0.937267i	$-0.386653\pi$
$349$	1.58649e6	0.697225	0.348613	+	0.937267i	$0.386653\pi$
$350$	0	0
$351$	−1.76084e6	−0.762871
$352$	0	0
$353$	−1.80462e6	−0.770813	−0.385406	−	0.922747i	$-0.625939\pi$
$353$	−1.80462e6	−0.770813	−0.385406	+	0.922747i	$0.625939\pi$
$354$	0	0
$355$	3.18289e6	1.34045
$356$	0	0
$357$	−1.34467e6	−0.558398
$358$	0	0
$359$	−510422.	−0.209023	−0.104511	−	0.994524i	$-0.533328\pi$
$359$	−510422.	−0.209023	−0.104511	+	0.994524i	$0.533328\pi$
$360$	0	0
$361$	4.81934e6	1.94635
$362$	0	0
$363$	−1.67979e6	−0.669098
$364$	0	0
$365$	−1.43966e6	−0.565622
$366$	0	0
$367$	−1.73803e6	−0.673583	−0.336792	−	0.941579i	$-0.609342\pi$
$367$	−1.73803e6	−0.673583	−0.336792	+	0.941579i	$0.609342\pi$
$368$	0	0
$369$	362002.	0.138403
$370$	0	0
$371$	−4.04884e6	−1.52720
$372$	0	0
$373$	−3.52954e6	−1.31355	−0.656773	−	0.754088i	$-0.728079\pi$
$373$	−3.52954e6	−1.31355	−0.656773	+	0.754088i	$0.728079\pi$
$374$	0	0
$375$	−153650.	−0.0564227
$376$	0	0
$377$	988271.	0.358115
$378$	0	0
$379$	−2.18140e6	−0.780077	−0.390039	−	0.920798i	$-0.627538\pi$
$379$	−2.18140e6	−0.780077	−0.390039	+	0.920798i	$0.627538\pi$
$380$	0	0
$381$	−535255.	−0.188907
$382$	0	0
$383$	−3.70077e6	−1.28913	−0.644563	−	0.764551i	$-0.722961\pi$
$383$	−3.70077e6	−1.28913	−0.644563	+	0.764551i	$0.722961\pi$
$384$	0	0
$385$	9.31891e6	3.20415
$386$	0	0
$387$	2.04077e6	0.692656
$388$	0	0
$389$	1.32322e6	0.443363	0.221681	−	0.975119i	$-0.428846\pi$
$389$	1.32322e6	0.443363	0.221681	+	0.975119i	$0.428846\pi$
$390$	0	0
$391$	−6.90094e6	−2.28279
$392$	0	0
$393$	−55104.5	−0.0179972
$394$	0	0
$395$	6.36678e6	2.05318
$396$	0	0
$397$	−1.09555e6	−0.348863	−0.174432	−	0.984669i	$-0.555809\pi$
$397$	−1.09555e6	−0.348863	−0.174432	+	0.984669i	$0.555809\pi$
$398$	0	0
$399$	2.49247e6	0.783785
$400$	0	0
$401$	−3.63783e6	−1.12975	−0.564874	−	0.825177i	$-0.691075\pi$
$401$	−3.63783e6	−1.12975	−0.564874	+	0.825177i	$0.691075\pi$
$402$	0	0
$403$	−6.23399e6	−1.91207
$404$	0	0
$405$	−3.03793e6	−0.920324
$406$	0	0
$407$	2.58249e6	0.772775
$408$	0	0
$409$	−3.01310e6	−0.890646	−0.445323	−	0.895370i	$-0.646911\pi$
$409$	−3.01310e6	−0.890646	−0.445323	+	0.895370i	$0.646911\pi$
$410$	0	0
$411$	1.22017e6	0.356299
$412$	0	0
$413$	−2.98645e6	−0.861548
$414$	0	0
$415$	−6.30965e6	−1.79839
$416$	0	0
$417$	317396.	0.0893842
$418$	0	0
$419$	3.32631e6	0.925609	0.462805	−	0.886460i	$-0.346843\pi$
$419$	3.32631e6	0.925609	0.462805	+	0.886460i	$0.346843\pi$
$420$	0	0
$421$	4.66752e6	1.28346	0.641728	−	0.766932i	$-0.278218\pi$
$421$	4.66752e6	1.28346	0.641728	+	0.766932i	$0.278218\pi$
$422$	0	0
$423$	4.21976e6	1.14667
$424$	0	0
$425$	3.99786e6	1.07363
$426$	0	0
$427$	−582649.	−0.154646
$428$	0	0
$429$	−2.66299e6	−0.698596
$430$	0	0
$431$	−781341.	−0.202604	−0.101302	−	0.994856i	$-0.532301\pi$
$431$	−781341.	−0.202604	−0.101302	+	0.994856i	$0.532301\pi$
$432$	0	0
$433$	−5.85862e6	−1.50167	−0.750837	−	0.660488i	$-0.770350\pi$
$433$	−5.85862e6	−1.50167	−0.750837	+	0.660488i	$0.770350\pi$
$434$	0	0
$435$	−544909.	−0.138071
$436$	0	0
$437$	1.27915e7	3.20420
$438$	0	0
$439$	−5.52246e6	−1.36764	−0.683819	−	0.729651i	$-0.739682\pi$
$439$	−5.52246e6	−1.36764	−0.683819	+	0.729651i	$0.739682\pi$
$440$	0	0
$441$	−3.01265e6	−0.737653
$442$	0	0
$443$	1.19749e6	0.289910	0.144955	−	0.989438i	$-0.453696\pi$
$443$	1.19749e6	0.289910	0.144955	+	0.989438i	$0.453696\pi$
$444$	0	0
$445$	4.02111e6	0.962600
$446$	0	0
$447$	−2.38142e6	−0.563725
$448$	0	0
$449$	1.43673e6	0.336325	0.168162	−	0.985759i	$-0.446217\pi$
$449$	1.43673e6	0.336325	0.168162	+	0.985759i	$0.446217\pi$
$450$	0	0
$451$	1.16524e6	0.269758
$452$	0	0
$453$	1.00089e6	0.229161
$454$	0	0
$455$	9.82172e6	2.22412
$456$	0	0
$457$	4.61837e6	1.03442	0.517212	−	0.855857i	$-0.326970\pi$
$457$	4.61837e6	1.03442	0.517212	+	0.855857i	$0.326970\pi$
$458$	0	0
$459$	3.51204e6	0.778087
$460$	0	0
$461$	−703008.	−0.154066	−0.0770332	−	0.997029i	$-0.524545\pi$
$461$	−703008.	−0.154066	−0.0770332	+	0.997029i	$0.524545\pi$
$462$	0	0
$463$	4.40368e6	0.954692	0.477346	−	0.878715i	$-0.341599\pi$
$463$	4.40368e6	0.954692	0.477346	+	0.878715i	$0.341599\pi$
$464$	0	0
$465$	3.43727e6	0.737194
$466$	0	0
$467$	−4.37131e6	−0.927511	−0.463756	−	0.885963i	$-0.653498\pi$
$467$	−4.37131e6	−0.927511	−0.463756	+	0.885963i	$0.653498\pi$
$468$	0	0
$469$	−5.67412e6	−1.19115
$470$	0	0
$471$	−2.36645e6	−0.491524
$472$	0	0
$473$	6.56899e6	1.35004
$474$	0	0
$475$	−7.41041e6	−1.50698
$476$	0	0
$477$	4.96848e6	0.999833
$478$	0	0
$479$	3.22064e6	0.641363	0.320681	−	0.947187i	$-0.396088\pi$
$479$	3.22064e6	0.641363	0.320681	+	0.947187i	$0.396088\pi$
$480$	0	0
$481$	2.72183e6	0.536412
$482$	0	0
$483$	4.37019e6	0.852379
$484$	0	0
$485$	−8.80090e6	−1.69892
$486$	0	0
$487$	−1.02782e6	−0.196378	−0.0981890	−	0.995168i	$-0.531305\pi$
$487$	−1.02782e6	−0.196378	−0.0981890	+	0.995168i	$0.531305\pi$
$488$	0	0
$489$	−1.55323e6	−0.293741
$490$	0	0
$491$	−8.88208e6	−1.66269	−0.831344	−	0.555758i	$-0.812428\pi$
$491$	−8.88208e6	−1.66269	−0.831344	+	0.555758i	$0.812428\pi$
$492$	0	0
$493$	−1.97114e6	−0.365258
$494$	0	0
$495$	−1.14356e7	−2.09771
$496$	0	0
$497$	−7.29134e6	−1.32409
$498$	0	0
$499$	−9.50931e6	−1.70961	−0.854806	−	0.518947i	$-0.826324\pi$
$499$	−9.50931e6	−1.70961	−0.854806	+	0.518947i	$0.826324\pi$
$500$	0	0
$501$	1.01268e6	0.180251
$502$	0	0
$503$	218035.	0.0384243	0.0192122	−	0.999815i	$-0.493884\pi$
$503$	218035.	0.0384243	0.0192122	+	0.999815i	$0.493884\pi$
$504$	0	0
$505$	−6.29055e6	−1.09764
$506$	0	0
$507$	−854274.	−0.147597
$508$	0	0
$509$	−2.61808e6	−0.447908	−0.223954	−	0.974600i	$-0.571896\pi$
$509$	−2.61808e6	−0.447908	−0.223954	+	0.974600i	$0.571896\pi$
$510$	0	0
$511$	3.29795e6	0.558716
$512$	0	0
$513$	−6.50990e6	−1.09215
$514$	0	0
$515$	−3.32112e6	−0.551780
$516$	0	0
$517$	1.35829e7	2.23494
$518$	0	0
$519$	−1.13414e6	−0.184819
$520$	0	0
$521$	−5.74533e6	−0.927301	−0.463650	−	0.886018i	$-0.653461\pi$
$521$	−5.74533e6	−0.927301	−0.463650	+	0.886018i	$0.653461\pi$
$522$	0	0
$523$	4.66586e6	0.745895	0.372947	−	0.927852i	$-0.378347\pi$
$523$	4.66586e6	0.745895	0.372947	+	0.927852i	$0.378347\pi$
$524$	0	0
$525$	−2.53174e6	−0.400886
$526$	0	0
$527$	1.24339e7	1.95021
$528$	0	0
$529$	1.59919e7	2.48462
$530$	0	0
$531$	3.66477e6	0.564041
$532$	0	0
$533$	1.22811e6	0.187249
$534$	0	0
$535$	−3.86847e6	−0.584325
$536$	0	0
$537$	−897119.	−0.134250
$538$	0	0
$539$	−9.69733e6	−1.43774
$540$	0	0
$541$	−3.64584e6	−0.535556	−0.267778	−	0.963481i	$-0.586289\pi$
$541$	−3.64584e6	−0.535556	−0.267778	+	0.963481i	$0.586289\pi$
$542$	0	0
$543$	3.06527e6	0.446138
$544$	0	0
$545$	−2.88557e6	−0.416141
$546$	0	0
$547$	−2.18984e6	−0.312927	−0.156464	−	0.987684i	$-0.550009\pi$
$547$	−2.18984e6	−0.312927	−0.156464	+	0.987684i	$0.550009\pi$
$548$	0	0
$549$	714990.	0.101244
$550$	0	0
$551$	3.65369e6	0.512688
$552$	0	0
$553$	−1.45849e7	−2.02811
$554$	0	0
$555$	−1.50075e6	−0.206813
$556$	0	0
$557$	−1.94235e6	−0.265271	−0.132635	−	0.991165i	$-0.542344\pi$
$557$	−1.94235e6	−0.265271	−0.132635	+	0.991165i	$0.542344\pi$
$558$	0	0
$559$	6.92342e6	0.937112
$560$	0	0
$561$	5.31142e6	0.712531
$562$	0	0
$563$	−1.06333e7	−1.41384	−0.706918	−	0.707295i	$-0.749915\pi$
$563$	−1.06333e7	−1.41384	−0.706918	+	0.707295i	$0.749915\pi$
$564$	0	0
$565$	1.90414e6	0.250944
$566$	0	0
$567$	6.95926e6	0.909087
$568$	0	0
$569$	−1.29842e7	−1.68126	−0.840628	−	0.541613i	$-0.817814\pi$
$569$	−1.29842e7	−1.68126	−0.840628	+	0.541613i	$0.817814\pi$
$570$	0	0
$571$	2.04551e6	0.262549	0.131275	−	0.991346i	$-0.458093\pi$
$571$	2.04551e6	0.262549	0.131275	+	0.991346i	$0.458093\pi$
$572$	0	0
$573$	4.51382e6	0.574325
$574$	0	0
$575$	−1.29931e7	−1.63887
$576$	0	0
$577$	−906247.	−0.113320	−0.0566601	−	0.998394i	$-0.518045\pi$
$577$	−906247.	−0.113320	−0.0566601	+	0.998394i	$0.518045\pi$
$578$	0	0
$579$	−2.89361e6	−0.358711
$580$	0	0
$581$	1.44541e7	1.77644
$582$	0	0
$583$	1.59929e7	1.94875
$584$	0	0
$585$	−1.20526e7	−1.45610
$586$	0	0
$587$	−6.65881e6	−0.797630	−0.398815	−	0.917031i	$-0.630578\pi$
$587$	−6.65881e6	−0.797630	−0.398815	+	0.917031i	$0.630578\pi$
$588$	0	0
$589$	−2.30474e7	−2.73737
$590$	0	0
$591$	−715700.	−0.0842874
$592$	0	0
$593$	7.93324e6	0.926432	0.463216	−	0.886245i	$-0.346695\pi$
$593$	7.93324e6	0.926432	0.463216	+	0.886245i	$0.346695\pi$
$594$	0	0
$595$	−1.95897e7	−2.26849
$596$	0	0
$597$	−1.22738e6	−0.140943
$598$	0	0
$599$	7.31949e6	0.833516	0.416758	−	0.909017i	$-0.363166\pi$
$599$	7.31949e6	0.833516	0.416758	+	0.909017i	$0.363166\pi$
$600$	0	0
$601$	−387230.	−0.0437304	−0.0218652	−	0.999761i	$-0.506960\pi$
$601$	−387230.	−0.0437304	−0.0218652	+	0.999761i	$0.506960\pi$
$602$	0	0
$603$	6.96291e6	0.779826
$604$	0	0
$605$	−2.44720e7	−2.71820
$606$	0	0
$607$	−4.68107e6	−0.515672	−0.257836	−	0.966189i	$-0.583009\pi$
$607$	−4.68107e6	−0.515672	−0.257836	+	0.966189i	$0.583009\pi$
$608$	0	0
$609$	1.24827e6	0.136385
$610$	0	0
$611$	1.43157e7	1.55135
$612$	0	0
$613$	−2.75942e6	−0.296597	−0.148299	−	0.988943i	$-0.547380\pi$
$613$	−2.75942e6	−0.296597	−0.148299	+	0.988943i	$0.547380\pi$
$614$	0	0
$615$	−677152.	−0.0721935
$616$	0	0
$617$	−7.17797e6	−0.759082	−0.379541	−	0.925175i	$-0.623918\pi$
$617$	−7.17797e6	−0.759082	−0.379541	+	0.925175i	$0.623918\pi$
$618$	0	0
$619$	−3.24662e6	−0.340569	−0.170285	−	0.985395i	$-0.554469\pi$
$619$	−3.24662e6	−0.340569	−0.170285	+	0.985395i	$0.554469\pi$
$620$	0	0
$621$	−1.14142e7	−1.18773
$622$	0	0
$623$	−9.21151e6	−0.950847
$624$	0	0
$625$	−1.08121e7	−1.10716
$626$	0	0
$627$	−9.84522e6	−1.00013
$628$	0	0
$629$	−5.42878e6	−0.547112
$630$	0	0
$631$	4.71211e6	0.471131	0.235566	−	0.971858i	$-0.424306\pi$
$631$	4.71211e6	0.471131	0.235566	+	0.971858i	$0.424306\pi$
$632$	0	0
$633$	4.88347e6	0.484417
$634$	0	0
$635$	−7.79785e6	−0.767434
$636$	0	0
$637$	−1.02206e7	−0.997990
$638$	0	0
$639$	8.94746e6	0.866857
$640$	0	0
$641$	3.80372e6	0.365648	0.182824	−	0.983146i	$-0.441476\pi$
$641$	3.80372e6	0.365648	0.182824	+	0.983146i	$0.441476\pi$
$642$	0	0
$643$	8.87190e6	0.846232	0.423116	−	0.906076i	$-0.360936\pi$
$643$	8.87190e6	0.846232	0.423116	+	0.906076i	$0.360936\pi$
$644$	0	0
$645$	−3.81741e6	−0.361302
$646$	0	0
$647$	5.38788e6	0.506007	0.253004	−	0.967465i	$-0.418582\pi$
$647$	5.38788e6	0.506007	0.253004	+	0.967465i	$0.418582\pi$
$648$	0	0
$649$	1.17964e7	1.09936
$650$	0	0
$651$	−7.87407e6	−0.728193
$652$	0	0
$653$	6.92301e6	0.635348	0.317674	−	0.948200i	$-0.397098\pi$
$653$	6.92301e6	0.635348	0.317674	+	0.948200i	$0.397098\pi$
$654$	0	0
$655$	−802789.	−0.0731136
$656$	0	0
$657$	−4.04703e6	−0.365782
$658$	0	0
$659$	−1.50172e7	−1.34702	−0.673511	−	0.739177i	$-0.735214\pi$
$659$	−1.50172e7	−1.34702	−0.673511	+	0.739177i	$0.735214\pi$
$660$	0	0
$661$	−1.04818e7	−0.933113	−0.466556	−	0.884492i	$-0.654505\pi$
$661$	−1.04818e7	−0.933113	−0.466556	+	0.884492i	$0.654505\pi$
$662$	0	0
$663$	5.59801e6	0.494595
$664$	0	0
$665$	3.63114e7	3.18412
$666$	0	0
$667$	6.40624e6	0.557556
$668$	0	0
$669$	−3.46018e6	−0.298905
$670$	0	0
$671$	2.30146e6	0.197332
$672$	0	0
$673$	2.37971e6	0.202529	0.101264	−	0.994860i	$-0.467711\pi$
$673$	2.37971e6	0.202529	0.101264	+	0.994860i	$0.467711\pi$
$674$	0	0
$675$	6.61249e6	0.558606
$676$	0	0
$677$	4.40384e6	0.369283	0.184642	−	0.982806i	$-0.440888\pi$
$677$	4.40384e6	0.369283	0.184642	+	0.982806i	$0.440888\pi$
$678$	0	0
$679$	2.01610e7	1.67818
$680$	0	0
$681$	3.27697e6	0.270772
$682$	0	0
$683$	−3.56791e6	−0.292659	−0.146330	−	0.989236i	$-0.546746\pi$
$683$	−3.56791e6	−0.292659	−0.146330	+	0.989236i	$0.546746\pi$
$684$	0	0
$685$	1.77760e7	1.44746
$686$	0	0
$687$	−338768.	−0.0273848
$688$	0	0
$689$	1.68558e7	1.35270
$690$	0	0
$691$	541773.	0.0431641	0.0215820	−	0.999767i	$-0.493130\pi$
$691$	541773.	0.0431641	0.0215820	+	0.999767i	$0.493130\pi$
$692$	0	0
$693$	2.61965e7	2.07209
$694$	0	0
$695$	4.62397e6	0.363122
$696$	0	0
$697$	−2.44951e6	−0.190984
$698$	0	0
$699$	5.60324e6	0.433757
$700$	0	0
$701$	−8.15829e6	−0.627053	−0.313526	−	0.949579i	$-0.601510\pi$
$701$	−8.15829e6	−0.627053	−0.313526	+	0.949579i	$0.601510\pi$
$702$	0	0
$703$	1.00628e7	0.767943
$704$	0	0
$705$	−7.89336e6	−0.598121
$706$	0	0
$707$	1.44103e7	1.08424
$708$	0	0
$709$	1.55138e7	1.15905	0.579526	−	0.814954i	$-0.303238\pi$
$709$	1.55138e7	1.15905	0.579526	+	0.814954i	$0.303238\pi$
$710$	0	0
$711$	1.78977e7	1.32777
$712$	0	0
$713$	−4.04104e7	−2.97693
$714$	0	0
$715$	−3.87957e7	−2.83804
$716$	0	0
$717$	7.18095e6	0.521656
$718$	0	0
$719$	1.75750e7	1.26787	0.633933	−	0.773388i	$-0.281440\pi$
$719$	1.75750e7	1.26787	0.633933	+	0.773388i	$0.281440\pi$
$720$	0	0
$721$	7.60798e6	0.545044
$722$	0	0
$723$	26810.4	0.00190747
$724$	0	0
$725$	−3.71127e6	−0.262227
$726$	0	0
$727$	1.77097e6	0.124273	0.0621364	−	0.998068i	$-0.480209\pi$
$727$	1.77097e6	0.124273	0.0621364	+	0.998068i	$0.480209\pi$
$728$	0	0
$729$	−5.46027e6	−0.380536
$730$	0	0
$731$	−1.38090e7	−0.955804
$732$	0	0
$733$	6.17772e6	0.424686	0.212343	−	0.977195i	$-0.431891\pi$
$733$	6.17772e6	0.424686	0.212343	+	0.977195i	$0.431891\pi$
$734$	0	0
$735$	5.63537e6	0.384773
$736$	0	0
$737$	2.24127e7	1.51994
$738$	0	0
$739$	−1.21748e7	−0.820067	−0.410033	−	0.912071i	$-0.634483\pi$
$739$	−1.21748e7	−0.820067	−0.410033	+	0.912071i	$0.634483\pi$
$740$	0	0
$741$	−1.03764e7	−0.694228
$742$	0	0
$743$	1.30871e7	0.869701	0.434851	−	0.900503i	$-0.356801\pi$
$743$	1.30871e7	0.869701	0.434851	+	0.900503i	$0.356801\pi$
$744$	0	0
$745$	−3.46937e7	−2.29013
$746$	0	0
$747$	−1.77371e7	−1.16300
$748$	0	0
$749$	8.86184e6	0.577191
$750$	0	0
$751$	−2.60860e7	−1.68775	−0.843874	−	0.536542i	$-0.819730\pi$
$751$	−2.60860e7	−1.68775	−0.843874	+	0.536542i	$0.819730\pi$
$752$	0	0
$753$	−3.25916e6	−0.209468
$754$	0	0
$755$	1.45814e7	0.930964
$756$	0	0
$757$	2.82725e7	1.79318	0.896590	−	0.442862i	$-0.146037\pi$
$757$	2.82725e7	1.79318	0.896590	+	0.442862i	$0.146037\pi$
$758$	0	0
$759$	−1.72622e7	−1.08766
$760$	0	0
$761$	6.29581e6	0.394085	0.197043	−	0.980395i	$-0.436866\pi$
$761$	6.29581e6	0.394085	0.197043	+	0.980395i	$0.436866\pi$
$762$	0	0
$763$	6.61024e6	0.411061
$764$	0	0
$765$	2.40393e7	1.48514
$766$	0	0
$767$	1.24329e7	0.763106
$768$	0	0
$769$	2.70954e7	1.65227	0.826133	−	0.563475i	$-0.190536\pi$
$769$	2.70954e7	1.65227	0.826133	+	0.563475i	$0.190536\pi$
$770$	0	0
$771$	572079.	0.0346593
$772$	0	0
$773$	−2.52998e7	−1.52289	−0.761445	−	0.648230i	$-0.775510\pi$
$773$	−2.52998e7	−1.52289	−0.761445	+	0.648230i	$0.775510\pi$
$774$	0	0
$775$	2.34106e7	1.40010
$776$	0	0
$777$	3.43791e6	0.204288
$778$	0	0
$779$	4.54039e6	0.268071
$780$	0	0
$781$	2.88007e7	1.68957
$782$	0	0
$783$	−3.26028e6	−0.190042
$784$	0	0
$785$	−3.44755e7	−1.99681
$786$	0	0
$787$	9.85448e6	0.567149	0.283574	−	0.958950i	$-0.408480\pi$
$787$	9.85448e6	0.567149	0.283574	+	0.958950i	$0.408480\pi$
$788$	0	0
$789$	−1.93000e6	−0.110374
$790$	0	0
$791$	−4.36198e6	−0.247881
$792$	0	0
$793$	2.42564e6	0.136976
$794$	0	0
$795$	−9.29390e6	−0.521531
$796$	0	0
$797$	−1.12833e6	−0.0629200	−0.0314600	−	0.999505i	$-0.510016\pi$
$797$	−1.12833e6	−0.0629200	−0.0314600	+	0.999505i	$0.510016\pi$
$798$	0	0
$799$	−2.85532e7	−1.58230
$800$	0	0
$801$	1.13038e7	0.622504
$802$	0	0
$803$	−1.30269e7	−0.712937
$804$	0	0
$805$	6.36670e7	3.46278
$806$	0	0
$807$	−4.08635e6	−0.220878
$808$	0	0
$809$	3.83865e6	0.206209	0.103104	−	0.994671i	$-0.467122\pi$
$809$	3.83865e6	0.206209	0.103104	+	0.994671i	$0.467122\pi$
$810$	0	0
$811$	−3.50511e7	−1.87132	−0.935662	−	0.352897i	$-0.885197\pi$
$811$	−3.50511e7	−1.87132	−0.935662	+	0.352897i	$0.885197\pi$
$812$	0	0
$813$	−5.17751e6	−0.274723
$814$	0	0
$815$	−2.26283e7	−1.19332
$816$	0	0
$817$	2.55963e7	1.34160
$818$	0	0
$819$	2.76099e7	1.43832
$820$	0	0
$821$	−2.73301e7	−1.41509	−0.707543	−	0.706670i	$-0.750197\pi$
$821$	−2.73301e7	−1.41509	−0.707543	+	0.706670i	$0.750197\pi$
$822$	0	0
$823$	2.58020e7	1.32787	0.663933	−	0.747792i	$-0.268886\pi$
$823$	2.58020e7	1.32787	0.663933	+	0.747792i	$0.268886\pi$
$824$	0	0
$825$	1.00004e7	0.511541
$826$	0	0
$827$	−1.15507e7	−0.587280	−0.293640	−	0.955916i	$-0.594867\pi$
$827$	−1.15507e7	−0.587280	−0.293640	+	0.955916i	$0.594867\pi$
$828$	0	0
$829$	4.06903e6	0.205639	0.102819	−	0.994700i	$-0.467214\pi$
$829$	4.06903e6	0.205639	0.102819	+	0.994700i	$0.467214\pi$
$830$	0	0
$831$	−1.15646e7	−0.580938
$832$	0	0
$833$	2.03852e7	1.01790
$834$	0	0
$835$	1.47532e7	0.732268
$836$	0	0
$837$	2.05657e7	1.01468
$838$	0	0
$839$	2.05209e7	1.00645	0.503224	−	0.864156i	$-0.332147\pi$
$839$	2.05209e7	1.00645	0.503224	+	0.864156i	$0.332147\pi$
$840$	0	0
$841$	−1.86813e7	−0.910788
$842$	0	0
$843$	1.05463e7	0.511127
$844$	0	0
$845$	−1.24455e7	−0.599611
$846$	0	0
$847$	5.60603e7	2.68502
$848$	0	0
$849$	−1.21626e7	−0.579104
$850$	0	0
$851$	1.76437e7	0.835150
$852$	0	0
$853$	−8.13962e6	−0.383029	−0.191514	−	0.981490i	$-0.561340\pi$
$853$	−8.13962e6	−0.383029	−0.191514	+	0.981490i	$0.561340\pi$
$854$	0	0
$855$	−4.45590e7	−2.08459
$856$	0	0
$857$	−2.41756e7	−1.12441	−0.562205	−	0.826998i	$-0.690047\pi$
$857$	−2.41756e7	−1.12441	−0.562205	+	0.826998i	$0.690047\pi$
$858$	0	0
$859$	1.51662e7	0.701285	0.350643	−	0.936509i	$-0.385963\pi$
$859$	1.51662e7	0.701285	0.350643	+	0.936509i	$0.385963\pi$
$860$	0	0
$861$	1.55121e6	0.0713121
$862$	0	0
$863$	1.85284e7	0.846857	0.423428	−	0.905930i	$-0.360827\pi$
$863$	1.85284e7	0.846857	0.423428	+	0.905930i	$0.360827\pi$
$864$	0	0
$865$	−1.65226e7	−0.750826
$866$	0	0
$867$	−3.69926e6	−0.167135
$868$	0	0
$869$	5.76104e7	2.58792
$870$	0	0
$871$	2.36220e7	1.05505
$872$	0	0
$873$	−2.47403e7	−1.09868
$874$	0	0
$875$	5.12780e6	0.226418
$876$	0	0
$877$	−2.23438e7	−0.980976	−0.490488	−	0.871448i	$-0.663181\pi$
$877$	−2.23438e7	−0.980976	−0.490488	+	0.871448i	$0.663181\pi$
$878$	0	0
$879$	−1.25139e7	−0.546285
$880$	0	0
$881$	4.29317e7	1.86354	0.931770	−	0.363050i	$-0.118265\pi$
$881$	4.29317e7	1.86354	0.931770	+	0.363050i	$0.118265\pi$
$882$	0	0
$883$	1.48609e7	0.641423	0.320712	−	0.947177i	$-0.396078\pi$
$883$	1.48609e7	0.641423	0.320712	+	0.947177i	$0.396078\pi$
$884$	0	0
$885$	−6.85522e6	−0.294214
$886$	0	0
$887$	3.03284e6	0.129432	0.0647159	−	0.997904i	$-0.479386\pi$
$887$	3.03284e6	0.129432	0.0647159	+	0.997904i	$0.479386\pi$
$888$	0	0
$889$	1.78632e7	0.758064
$890$	0	0
$891$	−2.74890e7	−1.16002
$892$	0	0
$893$	5.29261e7	2.22096
$894$	0	0
$895$	−1.30697e7	−0.545389
$896$	0	0
$897$	−1.81936e7	−0.754985
$898$	0	0
$899$	−1.15426e7	−0.476324
$900$	0	0
$901$	−3.36195e7	−1.37968
$902$	0	0
$903$	8.74489e6	0.356890
$904$	0	0
$905$	4.46564e7	1.81243
$906$	0	0
$907$	−1.13468e6	−0.0457991	−0.0228995	−	0.999738i	$-0.507290\pi$
$907$	−1.13468e6	−0.0457991	−0.0228995	+	0.999738i	$0.507290\pi$
$908$	0	0
$909$	−1.76834e7	−0.709833
$910$	0	0
$911$	3.77042e7	1.50520	0.752599	−	0.658479i	$-0.228800\pi$
$911$	3.77042e7	1.50520	0.752599	+	0.658479i	$0.228800\pi$
$912$	0	0
$913$	−5.70935e7	−2.26678
$914$	0	0
$915$	−1.33744e6	−0.0528107
$916$	0	0
$917$	1.83902e6	0.0722209
$918$	0	0
$919$	1.94383e7	0.759222	0.379611	−	0.925146i	$-0.376058\pi$
$919$	1.94383e7	0.759222	0.379611	+	0.925146i	$0.376058\pi$
$920$	0	0
$921$	−8.35009e6	−0.324371
$922$	0	0
$923$	3.03547e7	1.17279
$924$	0	0
$925$	−1.02213e7	−0.392783
$926$	0	0
$927$	−9.33602e6	−0.356831
$928$	0	0
$929$	−539525.	−0.0205103	−0.0102551	−	0.999947i	$-0.503264\pi$
$929$	−539525.	−0.0205103	−0.0102551	+	0.999947i	$0.503264\pi$
$930$	0	0
$931$	−3.77860e7	−1.42875
$932$	0	0
$933$	6.48979e6	0.244077
$934$	0	0
$935$	7.73793e7	2.89465
$936$	0	0
$937$	−3.09708e7	−1.15240	−0.576200	−	0.817309i	$-0.695465\pi$
$937$	−3.09708e7	−1.15240	−0.576200	+	0.817309i	$0.695465\pi$
$938$	0	0
$939$	6.84480e6	0.253336
$940$	0	0
$941$	−1.90998e7	−0.703160	−0.351580	−	0.936158i	$-0.614355\pi$
$941$	−1.90998e7	−0.703160	−0.351580	+	0.936158i	$0.614355\pi$
$942$	0	0
$943$	7.96095e6	0.291532
$944$	0	0
$945$	−3.24016e7	−1.18028
$946$	0	0
$947$	−4.07465e7	−1.47644	−0.738220	−	0.674560i	$-0.764333\pi$
$947$	−4.07465e7	−1.47644	−0.738220	+	0.674560i	$0.764333\pi$
$948$	0	0
$949$	−1.37297e7	−0.494877
$950$	0	0
$951$	5.35634e6	0.192051
$952$	0	0
$953$	1.60804e7	0.573541	0.286770	−	0.957999i	$-0.407418\pi$
$953$	1.60804e7	0.573541	0.286770	+	0.957999i	$0.407418\pi$
$954$	0	0
$955$	6.57595e7	2.33319
$956$	0	0
$957$	−4.93067e6	−0.174031
$958$	0	0
$959$	−4.07211e7	−1.42979
$960$	0	0
$961$	4.41810e7	1.54322
$962$	0	0
$963$	−1.08747e7	−0.377877
$964$	0	0
$965$	−4.21556e7	−1.45726
$966$	0	0
$967$	−4.83109e7	−1.66142	−0.830708	−	0.556708i	$-0.812064\pi$
$967$	−4.83109e7	−1.66142	−0.830708	+	0.556708i	$0.812064\pi$
$968$	0	0
$969$	2.06961e7	0.708076
$970$	0	0
$971$	−2.10254e7	−0.715642	−0.357821	−	0.933790i	$-0.616480\pi$
$971$	−2.10254e7	−0.715642	−0.357821	+	0.933790i	$0.616480\pi$
$972$	0	0
$973$	−1.05925e7	−0.358689
$974$	0	0
$975$	1.05399e7	0.355080
$976$	0	0
$977$	84838.1	0.00284351	0.00142175	−	0.999999i	$-0.499547\pi$
$977$	84838.1	0.00284351	0.00142175	+	0.999999i	$0.499547\pi$
$978$	0	0
$979$	3.63854e7	1.21331
$980$	0	0
$981$	−8.11166e6	−0.269115
$982$	0	0
$983$	−2.30016e7	−0.759233	−0.379616	−	0.925144i	$-0.623944\pi$
$983$	−2.30016e7	−0.759233	−0.379616	+	0.925144i	$0.623944\pi$
$984$	0	0
$985$	−1.04267e7	−0.342417
$986$	0	0
$987$	1.80820e7	0.590819
$988$	0	0
$989$	4.48795e7	1.45901
$990$	0	0
$991$	−3.03150e7	−0.980557	−0.490279	−	0.871566i	$-0.663105\pi$
$991$	−3.03150e7	−0.980557	−0.490279	+	0.871566i	$0.663105\pi$
$992$	0	0
$993$	1.74329e7	0.561043
$994$	0	0
$995$	−1.78810e7	−0.572578
$996$	0	0
$997$	2.07033e7	0.659631	0.329816	−	0.944045i	$-0.393013\pi$
$997$	2.07033e7	0.659631	0.329816	+	0.944045i	$0.393013\pi$
$998$	0	0
$999$	−8.97925e6	−0.284660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	656.6.a.h.1.5		10
4.3	odd	2		164.6.a.b.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.6.a.b.1.6	✓	10	4.3	odd	2
656.6.a.h.1.5		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)

\(f(q)\)	\(=\)	\(q-5.25838 q^{3} -76.6066 q^{5} +175.490 q^{7} -215.349 q^{9} +O(q^{10})\)	\(q-5.25838 q^{3} -76.6066 q^{5} +175.490 q^{7} -215.349 q^{9} -693.182 q^{11} -730.584 q^{13} +402.827 q^{15} +1457.17 q^{17} -2701.01 q^{19} -922.791 q^{21} -4735.84 q^{23} +2743.57 q^{25} +2410.18 q^{27} -1352.71 q^{29} +8532.88 q^{31} +3645.02 q^{33} -13443.7 q^{35} -3725.56 q^{37} +3841.69 q^{39} -1681.00 q^{41} -9476.57 q^{43} +16497.2 q^{45} -19594.9 q^{47} +13989.6 q^{49} -7662.37 q^{51} -23071.7 q^{53} +53102.3 q^{55} +14202.9 q^{57} -17017.8 q^{59} -3320.14 q^{61} -37791.6 q^{63} +55967.5 q^{65} -32333.1 q^{67} +24902.9 q^{69} -41548.5 q^{71} +18792.8 q^{73} -14426.7 q^{75} -121646. q^{77} -83110.0 q^{79} +39656.3 q^{81} +82364.3 q^{83} -111629. q^{85} +7113.09 q^{87} -52490.4 q^{89} -128210. q^{91} -44869.2 q^{93} +206915. q^{95} +114884. q^{97} +149276. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9}+O(q^{10}) \) 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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