LMFDB - Embedded newform 650.6.b.j.599.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$104.249482878$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 390x^{3} + 32400x^{2} - 135000x + 281250 $ x^6 - 2x^5 + 2x^4 - 390x^3 + 32400x^2 - 135000*x + 281250
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.5
Root			$2.14150 + 2.14150i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.j.599.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -3.71700i q^{3} -16.0000 q^{4} +14.8680 q^{6} +10.2576i q^{7} -64.0000i q^{8} +229.184 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -3.71700i q^{3} -16.0000 q^{4} +14.8680 q^{6} +10.2576i q^{7} -64.0000i q^{8} +229.184 q^{9} +197.274 q^{11} +59.4720i q^{12} +169.000i q^{13} -41.0305 q^{14} +256.000 q^{16} +949.722i q^{17} +916.736i q^{18} -2233.30 q^{19} +38.1276 q^{21} +789.095i q^{22} -367.911i q^{23} -237.888 q^{24} -676.000 q^{26} -1755.11i q^{27} -164.122i q^{28} +6602.16 q^{29} -9911.27 q^{31} +1024.00i q^{32} -733.267i q^{33} -3798.89 q^{34} -3666.94 q^{36} +9397.70i q^{37} -8933.22i q^{38} +628.173 q^{39} +20588.3 q^{41} +152.510i q^{42} -16341.8i q^{43} -3156.38 q^{44} +1471.64 q^{46} +13538.5i q^{47} -951.552i q^{48} +16701.8 q^{49} +3530.12 q^{51} -2704.00i q^{52} -35049.6i q^{53} +7020.43 q^{54} +656.488 q^{56} +8301.20i q^{57} +26408.6i q^{58} +6170.98 q^{59} +13746.1 q^{61} -39645.1i q^{62} +2350.88i q^{63} -4096.00 q^{64} +2933.07 q^{66} +62136.4i q^{67} -15195.6i q^{68} -1367.52 q^{69} -38355.6 q^{71} -14667.8i q^{72} +59967.3i q^{73} -37590.8 q^{74} +35732.9 q^{76} +2023.56i q^{77} +2512.69i q^{78} -89670.2 q^{79} +49167.9 q^{81} +82353.1i q^{82} -38134.8i q^{83} -610.042 q^{84} +65367.1 q^{86} -24540.2i q^{87} -12625.5i q^{88} +78278.2 q^{89} -1733.54 q^{91} +5886.57i q^{92} +36840.2i q^{93} -54154.1 q^{94} +3806.21 q^{96} -80567.1i q^{97} +66807.1i q^{98} +45212.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9}+O(q^{10}) $ 6 * q - 96 * q^4 + 176 * q^6 - 310 * q^9	$ 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9} + 96 q^{11} - 1872 q^{14} + 1536 q^{16} + 720 q^{19} + 13808 q^{21} - 2816 q^{24} - 4056 q^{26} + 6156 q^{29} - 10776 q^{31} + 12048 q^{34} + 4960 q^{36} + 7436 q^{39} - 31812 q^{41} - 1536 q^{44} + 18960 q^{46} - 15534 q^{49} - 67944 q^{51} - 81344 q^{54} + 29952 q^{56} + 155520 q^{59} + 35964 q^{61} - 24576 q^{64} + 180576 q^{66} + 63720 q^{69} - 9888 q^{71} - 202896 q^{74} - 11520 q^{76} - 84240 q^{79} + 293038 q^{81} - 220928 q^{84} + 314448 q^{86} - 39228 q^{89} - 79092 q^{91} - 142224 q^{94} + 45056 q^{96} - 700680 q^{99}+O(q^{100}) $ 6 * q - 96 * q^4 + 176 * q^6 - 310 * q^9 + 96 * q^11 - 1872 * q^14 + 1536 * q^16 + 720 * q^19 + 13808 * q^21 - 2816 * q^24 - 4056 * q^26 + 6156 * q^29 - 10776 * q^31 + 12048 * q^34 + 4960 * q^36 + 7436 * q^39 - 31812 * q^41 - 1536 * q^44 + 18960 * q^46 - 15534 * q^49 - 67944 * q^51 - 81344 * q^54 + 29952 * q^56 + 155520 * q^59 + 35964 * q^61 - 24576 * q^64 + 180576 * q^66 + 63720 * q^69 - 9888 * q^71 - 202896 * q^74 - 11520 * q^76 - 84240 * q^79 + 293038 * q^81 - 220928 * q^84 + 314448 * q^86 - 39228 * q^89 - 79092 * q^91 - 142224 * q^94 + 45056 * q^96 - 700680 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 3.71700i	− 0.238446i	−0.992868	−	0.119223i	$-0.961960\pi$
$3$	− 3.71700i	− 0.238446i	0.992868	−	0.119223i	$-0.0380403\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	14.8680	0.168607
$7$	10.2576i	0.0791228i	0.999217	+	0.0395614i	$0.0125961\pi$
$7$	10.2576i	0.0791228i	−0.999217	+	0.0395614i	$0.987404\pi$
$8$	− 64.0000i	− 0.353553i
$9$	229.184	0.943144
$10$	0	0
$11$	197.274	0.491572	0.245786	−	0.969324i	$-0.420954\pi$
$11$	197.274	0.491572	0.245786	+	0.969324i	$0.420954\pi$
$12$	59.4720i	0.119223i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−41.0305	−0.0559483
$15$	0	0
$16$	256.000	0.250000
$17$	949.722i	0.797029i	0.917162	+	0.398515i	$0.130474\pi$
$17$	949.722i	0.797029i	−0.917162	+	0.398515i	$0.869526\pi$
$18$	916.736i	0.666903i
$19$	−2233.30	−1.41927	−0.709633	−	0.704571i	$-0.751139\pi$
$19$	−2233.30	−1.41927	−0.709633	+	0.704571i	$0.751139\pi$
$20$	0	0
$21$	38.1276	0.0188665
$22$	789.095i	0.347594i
$23$	− 367.911i	− 0.145018i	−0.997368	−	0.0725092i	$-0.976899\pi$
$23$	− 367.911i	− 0.145018i	0.997368	−	0.0725092i	$-0.0231007\pi$
$24$	−237.888	−0.0843033
$25$	0	0
$26$	−676.000	−0.196116
$27$	− 1755.11i	− 0.463334i
$28$	− 164.122i	− 0.0395614i
$29$	6602.16	1.45778	0.728888	−	0.684633i	$-0.240037\pi$
$29$	6602.16	1.45778	0.728888	+	0.684633i	$0.240037\pi$
$30$	0	0
$31$	−9911.27	−1.85236	−0.926179	−	0.377084i	$-0.876927\pi$
$31$	−9911.27	−1.85236	−0.926179	+	0.377084i	$0.876927\pi$
$32$	1024.00i	0.176777i
$33$	− 733.267i	− 0.117213i
$34$	−3798.89	−0.563585
$35$	0	0
$36$	−3666.94	−0.471572
$37$	9397.70i	1.12854i	0.825590	+	0.564271i	$0.190842\pi$
$37$	9397.70i	1.12854i	−0.825590	+	0.564271i	$0.809158\pi$
$38$	− 8933.22i	− 1.00357i
$39$	628.173	0.0661330
$40$	0	0
$41$	20588.3	1.91276	0.956380	−	0.292126i	$-0.0943626\pi$
$41$	20588.3	1.91276	0.956380	+	0.292126i	$0.0943626\pi$
$42$	152.510i	0.0133406i
$43$	− 16341.8i	− 1.34781i	−0.738819	−	0.673904i	$-0.764616\pi$
$43$	− 16341.8i	− 1.34781i	0.738819	−	0.673904i	$-0.235384\pi$
$44$	−3156.38	−0.245786
$45$	0	0
$46$	1471.64	0.102543
$47$	13538.5i	0.893977i	0.894540	+	0.446989i	$0.147504\pi$
$47$	13538.5i	0.893977i	−0.894540	+	0.446989i	$0.852496\pi$
$48$	− 951.552i	− 0.0596114i
$49$	16701.8	0.993740
$50$	0	0
$51$	3530.12	0.190048
$52$	− 2704.00i	− 0.138675i
$53$	− 35049.6i	− 1.71393i	−0.515374	−	0.856965i	$-0.672347\pi$
$53$	− 35049.6i	− 1.71393i	0.515374	−	0.856965i	$-0.327653\pi$
$54$	7020.43	0.327627
$55$	0	0
$56$	656.488	0.0279741
$57$	8301.20i	0.338418i
$58$	26408.6i	1.03080i
$59$	6170.98	0.230794	0.115397	−	0.993319i	$-0.463186\pi$
$59$	6170.98	0.230794	0.115397	+	0.993319i	$0.463186\pi$
$60$	0	0
$61$	13746.1	0.472993	0.236496	−	0.971632i	$-0.424001\pi$
$61$	13746.1	0.472993	0.236496	+	0.971632i	$0.424001\pi$
$62$	− 39645.1i	− 1.30981i
$63$	2350.88i	0.0746242i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	2933.07	0.0828824
$67$	62136.4i	1.69106i	0.533927	+	0.845530i	$0.320716\pi$
$67$	62136.4i	1.69106i	−0.533927	+	0.845530i	$0.679284\pi$
$68$	− 15195.6i	− 0.398515i
$69$	−1367.52	−0.0345790
$70$	0	0
$71$	−38355.6	−0.902989	−0.451495	−	0.892274i	$-0.649109\pi$
$71$	−38355.6	−0.902989	−0.451495	+	0.892274i	$0.649109\pi$
$72$	− 14667.8i	− 0.333452i
$73$	59967.3i	1.31707i	0.752552	+	0.658533i	$0.228823\pi$
$73$	59967.3i	1.31707i	−0.752552	+	0.658533i	$0.771177\pi$
$74$	−37590.8	−0.797999
$75$	0	0
$76$	35732.9	0.709633
$77$	2023.56i	0.0388946i
$78$	2512.69i	0.0467631i
$79$	−89670.2	−1.61652	−0.808259	−	0.588827i	$-0.799590\pi$
$79$	−89670.2	−1.61652	−0.808259	+	0.588827i	$0.799590\pi$
$80$	0	0
$81$	49167.9	0.832664
$82$	82353.1i	1.35253i
$83$	− 38134.8i	− 0.607611i	−0.952734	−	0.303806i	$-0.901743\pi$
$83$	− 38134.8i	− 0.607611i	0.952734	−	0.303806i	$-0.0982573\pi$
$84$	−610.042	−0.00943325
$85$	0	0
$86$	65367.1	0.953044
$87$	− 24540.2i	− 0.347601i
$88$	− 12625.5i	− 0.173797i
$89$	78278.2	1.04753	0.523764	−	0.851863i	$-0.324527\pi$
$89$	78278.2	1.04753	0.523764	+	0.851863i	$0.324527\pi$
$90$	0	0
$91$	−1733.54	−0.0219447
$92$	5886.57i	0.0725092i
$93$	36840.2i	0.441687i
$94$	−54154.1	−0.632137
$95$	0	0
$96$	3806.21	0.0421517
$97$	− 80567.1i	− 0.869417i	−0.900571	−	0.434709i	$-0.856851\pi$
$97$	− 80567.1i	− 0.869417i	0.900571	−	0.434709i	$-0.143149\pi$
$98$	66807.1i	0.702680i
$99$	45212.0	0.463623
$100$	0	0
$101$	−10770.8	−0.105062	−0.0525309	−	0.998619i	$-0.516729\pi$
$101$	−10770.8	−0.105062	−0.0525309	+	0.998619i	$0.516729\pi$
$102$	14120.5i	0.134384i
$103$	44993.4i	0.417884i	0.977928	+	0.208942i	$0.0670020\pi$
$103$	44993.4i	0.417884i	−0.977928	+	0.208942i	$0.932998\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	140198.	1.21193
$107$	194227.i	1.64003i	0.572345	+	0.820013i	$0.306034\pi$
$107$	194227.i	1.64003i	−0.572345	+	0.820013i	$0.693966\pi$
$108$	28081.7i	0.231667i
$109$	−216396.	−1.74455	−0.872275	−	0.489016i	$-0.837356\pi$
$109$	−216396.	−1.74455	−0.872275	+	0.489016i	$0.837356\pi$
$110$	0	0
$111$	34931.3	0.269096
$112$	2625.95i	0.0197807i
$113$	141318.i	1.04112i	0.853824	+	0.520562i	$0.174278\pi$
$113$	141318.i	1.04112i	−0.853824	+	0.520562i	$0.825722\pi$
$114$	−33204.8	−0.239298
$115$	0	0
$116$	−105635.	−0.728888
$117$	38732.1i	0.261581i
$118$	24683.9i	0.163196i
$119$	−9741.89	−0.0630632
$120$	0	0
$121$	−122134.	−0.758356
$122$	54984.3i	0.334456i
$123$	− 76526.7i	− 0.456089i
$124$	158580.	0.926179
$125$	0	0
$126$	−9403.53	−0.0527673
$127$	188285.i	1.03587i	0.855420	+	0.517935i	$0.173299\pi$
$127$	188285.i	1.03587i	−0.855420	+	0.517935i	$0.826701\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−60742.4	−0.321379
$130$	0	0
$131$	263916.	1.34365	0.671826	−	0.740709i	$-0.265510\pi$
$131$	263916.	1.34365	0.671826	+	0.740709i	$0.265510\pi$
$132$	11732.3i	0.0586067i
$133$	− 22908.4i	− 0.112296i
$134$	−248546.	−1.19576
$135$	0	0
$136$	60782.2	0.281792
$137$	337417.i	1.53591i	0.640504	+	0.767955i	$0.278725\pi$
$137$	337417.i	1.53591i	−0.640504	+	0.767955i	$0.721275\pi$
$138$	− 5470.10i	− 0.0244510i
$139$	86203.3	0.378431	0.189216	−	0.981936i	$-0.439405\pi$
$139$	86203.3	0.378431	0.189216	+	0.981936i	$0.439405\pi$
$140$	0	0
$141$	50322.7	0.213165
$142$	− 153422.i	− 0.638510i
$143$	33339.3i	0.136338i
$144$	58671.1	0.235786
$145$	0	0
$146$	−239869.	−0.931306
$147$	− 62080.5i	− 0.236953i
$148$	− 150363.i	− 0.564271i
$149$	290191.	1.07082	0.535412	−	0.844591i	$-0.320156\pi$
$149$	290191.	1.07082	0.535412	+	0.844591i	$0.320156\pi$
$150$	0	0
$151$	129889.	0.463585	0.231793	−	0.972765i	$-0.425541\pi$
$151$	129889.	0.463585	0.231793	+	0.972765i	$0.425541\pi$
$152$	142932.i	0.501787i
$153$	217661.i	0.751713i
$154$	−8094.24	−0.0275026
$155$	0	0
$156$	−10050.8	−0.0330665
$157$	46206.2i	0.149607i	0.997198	+	0.0748033i	$0.0238329\pi$
$157$	46206.2i	0.149607i	−0.997198	+	0.0748033i	$0.976167\pi$
$158$	− 358681.i	− 1.14305i
$159$	−130279.	−0.408679
$160$	0	0
$161$	3773.89	0.0114743
$162$	196672.i	0.588782i
$163$	379076.i	1.11753i	0.829328	+	0.558763i	$0.188724\pi$
$163$	379076.i	1.11753i	−0.829328	+	0.558763i	$0.811276\pi$
$164$	−329412.	−0.956380
$165$	0	0
$166$	152539.	0.429646
$167$	569176.i	1.57927i	0.613580	+	0.789633i	$0.289729\pi$
$167$	569176.i	1.57927i	−0.613580	+	0.789633i	$0.710271\pi$
$168$	− 2440.17i	− 0.00667031i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−511838.	−1.33857
$172$	261468.i	0.673904i
$173$	− 113968.i	− 0.289513i	−0.989467	−	0.144757i	$-0.953760\pi$
$173$	− 113968.i	− 0.289513i	0.989467	−	0.144757i	$-0.0462399\pi$
$174$	98160.9	0.245791
$175$	0	0
$176$	50502.1	0.122893
$177$	− 22937.5i	− 0.0550318i
$178$	313113.i	0.740714i
$179$	−320294.	−0.747165	−0.373582	−	0.927597i	$-0.621871\pi$
$179$	−320294.	−0.747165	−0.373582	+	0.927597i	$0.621871\pi$
$180$	0	0
$181$	451978.	1.02547	0.512733	−	0.858548i	$-0.328633\pi$
$181$	451978.	1.02547	0.512733	+	0.858548i	$0.328633\pi$
$182$	− 6934.15i	− 0.0155173i
$183$	− 51094.2i	− 0.112783i
$184$	−23546.3	−0.0512717
$185$	0	0
$186$	−147361.	−0.312320
$187$	187355.i	0.391798i
$188$	− 216616.i	− 0.446989i
$189$	18003.2	0.0366603
$190$	0	0
$191$	182918.	0.362805	0.181402	−	0.983409i	$-0.441936\pi$
$191$	182918.	0.362805	0.181402	+	0.983409i	$0.441936\pi$
$192$	15224.8i	0.0298057i
$193$	508696.i	0.983026i	0.870870	+	0.491513i	$0.163556\pi$
$193$	508696.i	0.983026i	−0.870870	+	0.491513i	$0.836444\pi$
$194$	322268.	0.614771
$195$	0	0
$196$	−267228.	−0.496870
$197$	− 111666.i	− 0.205000i	−0.994733	−	0.102500i	$-0.967316\pi$
$197$	− 111666.i	− 0.205000i	0.994733	−	0.102500i	$-0.0326842\pi$
$198$	180848.i	0.327831i
$199$	−521218.	−0.933011	−0.466505	−	0.884518i	$-0.654487\pi$
$199$	−521218.	−0.933011	−0.466505	+	0.884518i	$0.654487\pi$
$200$	0	0
$201$	230961.	0.403226
$202$	− 43083.2i	− 0.0742899i
$203$	67722.4i	0.115343i
$204$	−56481.9	−0.0950241
$205$	0	0
$206$	−179974.	−0.295489
$207$	− 84319.2i	− 0.136773i
$208$	43264.0i	0.0693375i
$209$	−440572.	−0.697672
$210$	0	0
$211$	−543969.	−0.841140	−0.420570	−	0.907260i	$-0.638170\pi$
$211$	−543969.	−0.841140	−0.420570	+	0.907260i	$0.638170\pi$
$212$	560793.i	0.856965i
$213$	142568.i	0.215314i
$214$	−776909.	−1.15967
$215$	0	0
$216$	−112327.	−0.163813
$217$	− 101666.i	− 0.146564i
$218$	− 865585.i	− 1.23358i
$219$	222899.	0.314049
$220$	0	0
$221$	−160503.	−0.221056
$222$	139725.i	0.190279i
$223$	114372.i	0.154013i	0.997031	+	0.0770063i	$0.0245362\pi$
$223$	114372.i	0.154013i	−0.997031	+	0.0770063i	$0.975464\pi$
$224$	−10503.8	−0.0139871
$225$	0	0
$226$	−565274.	−0.736186
$227$	213250.i	0.274678i	0.990524	+	0.137339i	$0.0438550\pi$
$227$	213250.i	0.274678i	−0.990524	+	0.137339i	$0.956145\pi$
$228$	− 132819.i	− 0.169209i
$229$	−1.18502e6	−1.49326	−0.746631	−	0.665239i	$-0.768330\pi$
$229$	−1.18502e6	−1.49326	−0.746631	+	0.665239i	$0.768330\pi$
$230$	0	0
$231$	7521.58	0.00927425
$232$	− 422538.i	− 0.515402i
$233$	1.37128e6i	1.65476i	0.561641	+	0.827381i	$0.310170\pi$
$233$	1.37128e6i	1.65476i	−0.561641	+	0.827381i	$0.689830\pi$
$234$	−154928.	−0.184966
$235$	0	0
$236$	−98735.7	−0.115397
$237$	333304.i	0.385452i
$238$	− 38967.6i	− 0.0445924i
$239$	1.09024e6	1.23460	0.617301	−	0.786727i	$-0.288226\pi$
$239$	1.09024e6	1.23460	0.617301	+	0.786727i	$0.288226\pi$
$240$	0	0
$241$	84897.3	0.0941567	0.0470784	−	0.998891i	$-0.485009\pi$
$241$	84897.3	0.0941567	0.0470784	+	0.998891i	$0.485009\pi$
$242$	− 488536.i	− 0.536239i
$243$	− 609249.i	− 0.661879i
$244$	−219937.	−0.236496
$245$	0	0
$246$	306107.	0.322504
$247$	− 377429.i	− 0.393634i
$248$	634321.i	0.654907i
$249$	−141747.	−0.144882
$250$	0	0
$251$	425949.	0.426750	0.213375	−	0.976970i	$-0.431554\pi$
$251$	425949.	0.426750	0.213375	+	0.976970i	$0.431554\pi$
$252$	− 37614.1i	− 0.0373121i
$253$	− 72579.1i	− 0.0712870i
$254$	−753139.	−0.732471
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 250941.i	− 0.236995i	−0.992954	−	0.118497i	$-0.962192\pi$
$257$	− 250941.i	− 0.236995i	0.992954	−	0.118497i	$-0.0378077\pi$
$258$	− 242969.i	− 0.227249i
$259$	−96398.1	−0.0892933
$260$	0	0
$261$	1.51311e6	1.37489
$262$	1.05566e6i	0.950106i
$263$	− 1.48299e6i	− 1.32205i	−0.750363	−	0.661026i	$-0.770121\pi$
$263$	− 1.48299e6i	− 1.32205i	0.750363	−	0.661026i	$-0.229879\pi$
$264$	−46929.1	−0.0414412
$265$	0	0
$266$	91633.6	0.0794055
$267$	− 290960.i	− 0.249779i
$268$	− 994183.i	− 0.845530i
$269$	−398773.	−0.336004	−0.168002	−	0.985787i	$-0.553732\pi$
$269$	−398773.	−0.336004	−0.168002	+	0.985787i	$0.553732\pi$
$270$	0	0
$271$	182496.	0.150949	0.0754743	−	0.997148i	$-0.475953\pi$
$271$	182496.	0.150949	0.0754743	+	0.997148i	$0.475953\pi$
$272$	243129.i	0.199257i
$273$	6443.57i	0.00523262i
$274$	−1.34967e6	−1.08605
$275$	0	0
$276$	21880.4	0.0172895
$277$	79202.9i	0.0620215i	0.999519	+	0.0310107i	$0.00987260\pi$
$277$	79202.9i	0.0620215i	−0.999519	+	0.0310107i	$0.990127\pi$
$278$	344813.i	0.267591i
$279$	−2.27150e6	−1.74704
$280$	0	0
$281$	−1.51199e6	−1.14231	−0.571154	−	0.820843i	$-0.693504\pi$
$281$	−1.51199e6	−1.14231	−0.571154	+	0.820843i	$0.693504\pi$
$282$	201291.i	0.150730i
$283$	2.50781e6i	1.86135i	0.365848	+	0.930675i	$0.380779\pi$
$283$	2.50781e6i	1.86135i	−0.365848	+	0.930675i	$0.619221\pi$
$284$	613689.	0.451495
$285$	0	0
$286$	−133357.	−0.0964053
$287$	211187.i	0.151343i
$288$	234684.i	0.166726i
$289$	517885.	0.364744
$290$	0	0
$291$	−299468.	−0.207309
$292$	− 959477.i	− 0.658533i
$293$	1.75384e6i	1.19349i	0.802429	+	0.596747i	$0.203541\pi$
$293$	1.75384e6i	1.19349i	−0.802429	+	0.596747i	$0.796459\pi$
$294$	248322.	0.167551
$295$	0	0
$296$	601453.	0.399000
$297$	− 346237.i	− 0.227762i
$298$	1.16076e6i	0.757187i
$299$	62176.9	0.0402208
$300$	0	0
$301$	167628.	0.106642
$302$	519556.i	0.327804i
$303$	40035.1i	0.0250515i
$304$	−571726.	−0.354817
$305$	0	0
$306$	−870644.	−0.531541
$307$	1.55298e6i	0.940414i	0.882556	+	0.470207i	$0.155821\pi$
$307$	1.55298e6i	0.940414i	−0.882556	+	0.470207i	$0.844179\pi$
$308$	− 32377.0i	− 0.0194473i
$309$	167240.	0.0996426
$310$	0	0
$311$	−188125.	−0.110292	−0.0551462	−	0.998478i	$-0.517562\pi$
$311$	−188125.	−0.110292	−0.0551462	+	0.998478i	$0.517562\pi$
$312$	− 40203.1i	− 0.0233815i
$313$	846464.i	0.488369i	0.969729	+	0.244184i	$0.0785202\pi$
$313$	846464.i	0.488369i	−0.969729	+	0.244184i	$0.921480\pi$
$314$	−184825.	−0.105788
$315$	0	0
$316$	1.43472e6	0.808259
$317$	303332.i	0.169539i	0.996401	+	0.0847696i	$0.0270154\pi$
$317$	303332.i	0.169539i	−0.996401	+	0.0847696i	$0.972985\pi$
$318$	− 521117.i	− 0.288980i
$319$	1.30243e6	0.716603
$320$	0	0
$321$	721943.	0.391057
$322$	15095.6i	0.00811352i
$323$	− 2.12102e6i	− 1.13120i
$324$	−786687.	−0.416332
$325$	0	0
$326$	−1.51630e6	−0.790210
$327$	804345.i	0.415980i
$328$	− 1.31765e6i	− 0.676263i
$329$	−138873.	−0.0707340
$330$	0	0
$331$	−3.36224e6	−1.68678	−0.843390	−	0.537301i	$-0.819444\pi$
$331$	−3.36224e6	−1.68678	−0.843390	+	0.537301i	$0.819444\pi$
$332$	610156.i	0.303806i
$333$	2.15380e6i	1.06438i
$334$	−2.27670e6	−1.11671
$335$	0	0
$336$	9760.67	0.00471662
$337$	− 2.42001e6i	− 1.16076i	−0.814346	−	0.580380i	$-0.802904\pi$
$337$	− 2.42001e6i	− 1.16076i	0.814346	−	0.580380i	$-0.197096\pi$
$338$	− 114244.i	− 0.0543928i
$339$	525281.	0.248252
$340$	0	0
$341$	−1.95523e6	−0.910568
$342$	− 2.04735e6i	− 0.946514i
$343$	343720.i	0.157750i
$344$	−1.04587e6	−0.476522
$345$	0	0
$346$	455872.	0.204717
$347$	208658.i	0.0930274i	0.998918	+	0.0465137i	$0.0148111\pi$
$347$	208658.i	0.0930274i	−0.998918	+	0.0465137i	$0.985189\pi$
$348$	392644.i	0.173800i
$349$	2.56139e6	1.12567	0.562837	−	0.826568i	$-0.309710\pi$
$349$	2.56139e6	1.12567	0.562837	+	0.826568i	$0.309710\pi$
$350$	0	0
$351$	296613.	0.128506
$352$	202008.i	0.0868986i
$353$	− 2.22758e6i	− 0.951472i	−0.879588	−	0.475736i	$-0.842182\pi$
$353$	− 2.22758e6i	− 0.951472i	0.879588	−	0.475736i	$-0.157818\pi$
$354$	91750.2	0.0389134
$355$	0	0
$356$	−1.25245e6	−0.523764
$357$	36210.6i	0.0150372i
$358$	− 1.28118e6i	− 0.528325i
$359$	4.69711e6	1.92351	0.961756	−	0.273908i	$-0.0883164\pi$
$359$	4.69711e6	1.92351	0.961756	+	0.273908i	$0.0883164\pi$
$360$	0	0
$361$	2.51155e6	1.01432
$362$	1.80791e6i	0.725113i
$363$	453973.i	0.180827i
$364$	27736.6	0.0109724
$365$	0	0
$366$	204377.	0.0797497
$367$	3.22496e6i	1.24985i	0.780684	+	0.624926i	$0.214871\pi$
$367$	3.22496e6i	1.24985i	−0.780684	+	0.624926i	$0.785129\pi$
$368$	− 94185.1i	− 0.0362546i
$369$	4.71850e6	1.80401
$370$	0	0
$371$	359525.	0.135611
$372$	− 589443.i	− 0.220843i
$373$	− 3.09204e6i	− 1.15073i	−0.817898	−	0.575364i	$-0.804860\pi$
$373$	− 3.09204e6i	− 1.15073i	0.817898	−	0.575364i	$-0.195140\pi$
$374$	−749421.	−0.277043
$375$	0	0
$376$	866465.	0.316069
$377$	1.11576e6i	0.404314i
$378$	72013.0i	0.0259228i
$379$	3.63851e6	1.30115	0.650573	−	0.759444i	$-0.274529\pi$
$379$	3.63851e6	1.30115	0.650573	+	0.759444i	$0.274529\pi$
$380$	0	0
$381$	699854.	0.246999
$382$	731672.i	0.256542i
$383$	1.89242e6i	0.659204i	0.944120	+	0.329602i	$0.106915\pi$
$383$	1.89242e6i	0.659204i	−0.944120	+	0.329602i	$0.893085\pi$
$384$	−60899.4	−0.0210758
$385$	0	0
$386$	−2.03478e6	−0.695104
$387$	− 3.74527e6i	− 1.27118i
$388$	1.28907e6i	0.434709i
$389$	−79129.5	−0.0265133	−0.0132567	−	0.999912i	$-0.504220\pi$
$389$	−79129.5	−0.0265133	−0.0132567	+	0.999912i	$0.504220\pi$
$390$	0	0
$391$	349413.	0.115584
$392$	− 1.06891e6i	− 0.351340i
$393$	− 980975.i	− 0.320388i
$394$	446663.	0.144957
$395$	0	0
$396$	−723391.	−0.231812
$397$	1.82235e6i	0.580305i	0.956980	+	0.290153i	$0.0937061\pi$
$397$	1.82235e6i	0.580305i	−0.956980	+	0.290153i	$0.906294\pi$
$398$	− 2.08487e6i	− 0.659738i
$399$	−85150.6	−0.0267766
$400$	0	0
$401$	−117774.	−0.0365754	−0.0182877	−	0.999833i	$-0.505821\pi$
$401$	−117774.	−0.0365754	−0.0182877	+	0.999833i	$0.505821\pi$
$402$	923845.i	0.285124i
$403$	− 1.67500e6i	− 0.513752i
$404$	172333.	0.0525309
$405$	0	0
$406$	−270890.	−0.0815601
$407$	1.85392e6i	0.554760i
$408$	− 225928.i	− 0.0671922i
$409$	−3.73101e6	−1.10285	−0.551427	−	0.834223i	$-0.685917\pi$
$409$	−3.73101e6	−1.10285	−0.551427	+	0.834223i	$0.685917\pi$
$410$	0	0
$411$	1.25418e6	0.366231
$412$	− 719894.i	− 0.208942i
$413$	63299.6i	0.0182610i
$414$	337277.	0.0967132
$415$	0	0
$416$	−173056.	−0.0490290
$417$	− 320418.i	− 0.0902353i
$418$	− 1.76229e6i	− 0.493329i
$419$	2.69321e6	0.749438	0.374719	−	0.927138i	$-0.377739\pi$
$419$	2.69321e6	0.749438	0.374719	+	0.927138i	$0.377739\pi$
$420$	0	0
$421$	5.56789e6	1.53103	0.765517	−	0.643415i	$-0.222483\pi$
$421$	5.56789e6	1.53103	0.765517	+	0.643415i	$0.222483\pi$
$422$	− 2.17588e6i	− 0.594776i
$423$	3.10281e6i	0.843149i
$424$	−2.24317e6	−0.605966
$425$	0	0
$426$	−570271.	−0.152250
$427$	141002.i	0.0374245i
$428$	− 3.10763e6i	− 0.820013i
$429$	123922.	0.0325091
$430$	0	0
$431$	−3.75659e6	−0.974093	−0.487047	−	0.873376i	$-0.661926\pi$
$431$	−3.75659e6	−0.974093	−0.487047	+	0.873376i	$0.661926\pi$
$432$	− 449308.i	− 0.115834i
$433$	1.60978e6i	0.412615i	0.978487	+	0.206308i	$0.0661448\pi$
$433$	1.60978e6i	0.412615i	−0.978487	+	0.206308i	$0.933855\pi$
$434$	406664.	0.103636
$435$	0	0
$436$	3.46234e6	0.872275
$437$	821657.i	0.205820i
$438$	891595.i	0.222066i
$439$	1.92307e6	0.476248	0.238124	−	0.971235i	$-0.423468\pi$
$439$	1.92307e6	0.476248	0.238124	+	0.971235i	$0.423468\pi$
$440$	0	0
$441$	3.82778e6	0.937239
$442$	− 642012.i	− 0.156310i
$443$	− 1.43060e6i	− 0.346346i	−0.984891	−	0.173173i	$-0.944598\pi$
$443$	− 1.43060e6i	− 0.346346i	0.984891	−	0.173173i	$-0.0554019\pi$
$444$	−558900.	−0.134548
$445$	0	0
$446$	−457487.	−0.108903
$447$	− 1.07864e6i	− 0.255333i
$448$	− 42015.2i	− 0.00989035i
$449$	3.30270e6	0.773131	0.386565	−	0.922262i	$-0.373661\pi$
$449$	3.30270e6	0.773131	0.386565	+	0.922262i	$0.373661\pi$
$450$	0	0
$451$	4.06153e6	0.940260
$452$	− 2.26110e6i	− 0.520562i
$453$	− 482797.i	− 0.110540i
$454$	−853000.	−0.194227
$455$	0	0
$456$	531277.	0.119649
$457$	− 6.85658e6i	− 1.53574i	−0.640606	−	0.767869i	$-0.721317\pi$
$457$	− 6.85658e6i	− 1.53574i	0.640606	−	0.767869i	$-0.278683\pi$
$458$	− 4.74007e6i	− 1.05590i
$459$	1.66687e6	0.369291
$460$	0	0
$461$	8.33435e6	1.82650	0.913250	−	0.407400i	$-0.133565\pi$
$461$	8.33435e6	1.82650	0.913250	+	0.407400i	$0.133565\pi$
$462$	30086.3i	0.00655789i
$463$	− 2.87024e6i	− 0.622250i	−0.950369	−	0.311125i	$-0.899294\pi$
$463$	− 2.87024e6i	− 0.622250i	0.950369	−	0.311125i	$-0.100706\pi$
$464$	1.69015e6	0.364444
$465$	0	0
$466$	−5.48511e6	−1.17009
$467$	− 6.61814e6i	− 1.40425i	−0.712055	−	0.702124i	$-0.752235\pi$
$467$	− 6.61814e6i	− 1.40425i	0.712055	−	0.702124i	$-0.247765\pi$
$468$	− 619713.i	− 0.130790i
$469$	−637372.	−0.133801
$470$	0	0
$471$	171748.	0.0356731
$472$	− 394943.i	− 0.0815979i
$473$	− 3.22380e6i	− 0.662545i
$474$	−1.33322e6	−0.272556
$475$	0	0
$476$	155870.	0.0315316
$477$	− 8.03280e6i	− 1.61648i
$478$	4.36096e6i	0.872995i
$479$	1.47435e6	0.293603	0.146802	−	0.989166i	$-0.453102\pi$
$479$	1.47435e6	0.293603	0.146802	+	0.989166i	$0.453102\pi$
$480$	0	0
$481$	−1.58821e6	−0.313001
$482$	339589.i	0.0665789i
$483$	− 14027.6i	− 0.00273599i
$484$	1.95415e6	0.379178
$485$	0	0
$486$	2.43699e6	0.468019
$487$	− 2.63151e6i	− 0.502786i	−0.967885	−	0.251393i	$-0.919111\pi$
$487$	− 2.63151e6i	− 0.502786i	0.967885	−	0.251393i	$-0.0808886\pi$
$488$	− 879749.i	− 0.167228i
$489$	1.40903e6	0.266469
$490$	0	0
$491$	4.70461e6	0.880684	0.440342	−	0.897830i	$-0.354857\pi$
$491$	4.70461e6	0.880684	0.440342	+	0.897830i	$0.354857\pi$
$492$	1.22443e6i	0.228045i
$493$	6.27022e6i	1.16189i
$494$	1.50971e6	0.278341
$495$	0	0
$496$	−2.53728e6	−0.463089
$497$	− 393437.i	− 0.0714471i
$498$	− 566988.i	− 0.102447i
$499$	−409148.	−0.0735578	−0.0367789	−	0.999323i	$-0.511710\pi$
$499$	−409148.	−0.0735578	−0.0367789	+	0.999323i	$0.511710\pi$
$500$	0	0
$501$	2.11563e6	0.376569
$502$	1.70380e6i	0.301758i
$503$	9.62836e6i	1.69681i	0.529350	+	0.848404i	$0.322436\pi$
$503$	9.62836e6i	1.69681i	−0.529350	+	0.848404i	$0.677564\pi$
$504$	150456.	0.0263836
$505$	0	0
$506$	290317.	0.0504075
$507$	106161.i	0.0183420i
$508$	− 3.01255e6i	− 0.517935i
$509$	1.24824e6	0.213552	0.106776	−	0.994283i	$-0.465947\pi$
$509$	1.24824e6	0.213552	0.106776	+	0.994283i	$0.465947\pi$
$510$	0	0
$511$	−615122.	−0.104210
$512$	262144.i	0.0441942i
$513$	3.91969e6i	0.657595i
$514$	1.00376e6	0.167581
$515$	0	0
$516$	971878.	0.160689
$517$	2.67079e6i	0.439455i
$518$	− 385592.i	− 0.0631399i
$519$	−423620.	−0.0690332
$520$	0	0
$521$	−7.13050e6	−1.15087	−0.575434	−	0.817848i	$-0.695167\pi$
$521$	−7.13050e6	−1.15087	−0.575434	+	0.817848i	$0.695167\pi$
$522$	6.05243e6i	0.972196i
$523$	4.85487e6i	0.776110i	0.921636	+	0.388055i	$0.126853\pi$
$523$	4.85487e6i	0.776110i	−0.921636	+	0.388055i	$0.873147\pi$
$524$	−4.22265e6	−0.671826
$525$	0	0
$526$	5.93196e6	0.934833
$527$	− 9.41295e6i	− 1.47638i
$528$	− 187716.i	− 0.0293033i
$529$	6.30098e6	0.978970
$530$	0	0
$531$	1.41429e6	0.217672
$532$	366534.i	0.0561482i
$533$	3.47942e6i	0.530504i
$534$	1.16384e6	0.176620
$535$	0	0
$536$	3.97673e6	0.597880
$537$	1.19053e6i	0.178158i
$538$	− 1.59509e6i	− 0.237591i
$539$	3.29482e6	0.488495
$540$	0	0
$541$	−1.13451e7	−1.66653	−0.833266	−	0.552872i	$-0.813532\pi$
$541$	−1.13451e7	−1.66653	−0.833266	+	0.552872i	$0.813532\pi$
$542$	729983.i	0.106737i
$543$	− 1.68000e6i	− 0.244518i
$544$	−972516.	−0.140896
$545$	0	0
$546$	−25774.3	−0.00370002
$547$	− 401928.i	− 0.0574355i	−0.999588	−	0.0287177i	$-0.990858\pi$
$547$	− 401928.i	− 0.0574355i	0.999588	−	0.0287177i	$-0.00914240\pi$
$548$	− 5.39867e6i	− 0.767955i
$549$	3.15038e6	0.446100
$550$	0	0
$551$	−1.47446e7	−2.06897
$552$	87521.6i	0.0122255i
$553$	− 919803.i	− 0.127903i
$554$	−316812.	−0.0438558
$555$	0	0
$556$	−1.37925e6	−0.189216
$557$	− 7.58501e6i	− 1.03590i	−0.855411	−	0.517950i	$-0.826695\pi$
$557$	− 7.58501e6i	− 1.03590i	0.855411	−	0.517950i	$-0.173305\pi$
$558$	− 9.08601e6i	− 1.23534i
$559$	2.76176e6	0.373814
$560$	0	0
$561$	696400.	0.0934225
$562$	− 6.04796e6i	− 0.807733i
$563$	− 9.36555e6i	− 1.24527i	−0.782514	−	0.622633i	$-0.786063\pi$
$563$	− 9.36555e6i	− 1.24527i	0.782514	−	0.622633i	$-0.213937\pi$
$564$	−805163.	−0.106583
$565$	0	0
$566$	−1.00312e7	−1.31617
$567$	504346.i	0.0658827i
$568$	2.45476e6i	0.319255i
$569$	−1.36255e7	−1.76430	−0.882150	−	0.470969i	$-0.843904\pi$
$569$	−1.36255e7	−1.76430	−0.882150	+	0.470969i	$0.843904\pi$
$570$	0	0
$571$	1.29016e7	1.65598	0.827988	−	0.560746i	$-0.189485\pi$
$571$	1.29016e7	1.65598	0.827988	+	0.560746i	$0.189485\pi$
$572$	− 533428.i	− 0.0681688i
$573$	− 679907.i	− 0.0865093i
$574$	−844747.	−0.107016
$575$	0	0
$576$	−938737.	−0.117893
$577$	− 3.12690e6i	− 0.390998i	−0.980704	−	0.195499i	$-0.937367\pi$
$577$	− 3.12690e6i	− 0.390998i	0.980704	−	0.195499i	$-0.0626327\pi$
$578$	2.07154e6i	0.257913i
$579$	1.89082e6	0.234398
$580$	0	0
$581$	391172.	0.0480759
$582$	− 1.19787e6i	− 0.146590i
$583$	− 6.91436e6i	− 0.842521i
$584$	3.83791e6	0.465653
$585$	0	0
$586$	−7.01535e6	−0.843928
$587$	1.61598e7i	1.93571i	0.251511	+	0.967854i	$0.419072\pi$
$587$	1.61598e7i	1.93571i	−0.251511	+	0.967854i	$0.580928\pi$
$588$	993289.i	0.118476i
$589$	2.21349e7	2.62899
$590$	0	0
$591$	−415062.	−0.0488814
$592$	2.40581e6i	0.282135i
$593$	− 5.02852e6i	− 0.587223i	−0.955925	−	0.293612i	$-0.905143\pi$
$593$	− 5.02852e6i	− 0.587223i	0.955925	−	0.293612i	$-0.0948572\pi$
$594$	1.38495e6	0.161052
$595$	0	0
$596$	−4.64305e6	−0.535412
$597$	1.93737e6i	0.222472i
$598$	248708.i	0.0284404i
$599$	−6.83761e6	−0.778642	−0.389321	−	0.921102i	$-0.627290\pi$
$599$	−6.83761e6	−0.778642	−0.389321	+	0.921102i	$0.627290\pi$
$600$	0	0
$601$	−8.01912e6	−0.905608	−0.452804	−	0.891610i	$-0.649576\pi$
$601$	−8.01912e6	−0.905608	−0.452804	+	0.891610i	$0.649576\pi$
$602$	670511.i	0.0754075i
$603$	1.42407e7i	1.59491i
$604$	−2.07822e6	−0.231793
$605$	0	0
$606$	−160140.	−0.0177141
$607$	− 7.83186e6i	− 0.862766i	−0.902169	−	0.431383i	$-0.858026\pi$
$607$	− 7.83186e6i	− 0.862766i	0.902169	−	0.431383i	$-0.141974\pi$
$608$	− 2.28690e6i	− 0.250893i
$609$	251724.	0.0275031
$610$	0	0
$611$	−2.28801e6	−0.247945
$612$	− 3.48258e6i	− 0.375857i
$613$	− 1.57922e7i	− 1.69743i	−0.528849	−	0.848716i	$-0.677376\pi$
$613$	− 1.57922e7i	− 1.69743i	0.528849	−	0.848716i	$-0.322624\pi$
$614$	−6.21191e6	−0.664973
$615$	0	0
$616$	129508.	0.0137513
$617$	− 8.38747e6i	− 0.886989i	−0.896277	−	0.443494i	$-0.853739\pi$
$617$	− 8.38747e6i	− 0.886989i	0.896277	−	0.443494i	$-0.146261\pi$
$618$	668962.i	0.0704580i
$619$	−1.35269e6	−0.141896	−0.0709481	−	0.997480i	$-0.522602\pi$
$619$	−1.35269e6	−0.141896	−0.0709481	+	0.997480i	$0.522602\pi$
$620$	0	0
$621$	−645723.	−0.0671920
$622$	− 752500.i	− 0.0779884i
$623$	802948.i	0.0828834i
$624$	160812.	0.0165332
$625$	0	0
$626$	−3.38586e6	−0.345329
$627$	1.63761e6i	0.166357i
$628$	− 739299.i	− 0.0748033i
$629$	−8.92521e6	−0.899480
$630$	0	0
$631$	−7.20270e6	−0.720149	−0.360074	−	0.932924i	$-0.617249\pi$
$631$	−7.20270e6	−0.720149	−0.360074	+	0.932924i	$0.617249\pi$
$632$	5.73889e6i	0.571525i
$633$	2.02193e6i	0.200566i
$634$	−1.21333e6	−0.119882
$635$	0	0
$636$	2.08447e6	0.204340
$637$	2.82260e6i	0.275614i
$638$	5.20973e6i	0.506715i
$639$	−8.79048e6	−0.851649
$640$	0	0
$641$	2.63844e6	0.253631	0.126815	−	0.991926i	$-0.459524\pi$
$641$	2.63844e6	0.253631	0.126815	+	0.991926i	$0.459524\pi$
$642$	2.88777e6i	0.276519i
$643$	6.76792e6i	0.645547i	0.946476	+	0.322774i	$0.104615\pi$
$643$	6.76792e6i	0.645547i	−0.946476	+	0.322774i	$0.895385\pi$
$644$	−60382.2	−0.00573713
$645$	0	0
$646$	8.48408e6	0.799877
$647$	− 1.01251e7i	− 0.950907i	−0.879741	−	0.475453i	$-0.842284\pi$
$647$	− 1.01251e7i	− 0.950907i	0.879741	−	0.475453i	$-0.157716\pi$
$648$	− 3.14675e6i	− 0.294391i
$649$	1.21737e6	0.113452
$650$	0	0
$651$	−377893.	−0.0349475
$652$	− 6.06522e6i	− 0.558763i
$653$	− 8.17558e6i	− 0.750301i	−0.926964	−	0.375150i	$-0.877591\pi$
$653$	− 8.17558e6i	− 0.750301i	0.926964	−	0.375150i	$-0.122409\pi$
$654$	−3.21738e6	−0.294143
$655$	0	0
$656$	5.27060e6	0.478190
$657$	1.37435e7i	1.24218i
$658$	− 555492.i	− 0.0500165i
$659$	1.24909e7	1.12042	0.560211	−	0.828350i	$-0.310720\pi$
$659$	1.24909e7	1.12042	0.560211	+	0.828350i	$0.310720\pi$
$660$	0	0
$661$	5.43688e6	0.484001	0.242001	−	0.970276i	$-0.422196\pi$
$661$	5.43688e6	0.484001	0.242001	+	0.970276i	$0.422196\pi$
$662$	− 1.34490e7i	− 1.19273i
$663$	596590.i	0.0527099i
$664$	−2.44062e6	−0.214823
$665$	0	0
$666$	−8.61521e6	−0.752628
$667$	− 2.42900e6i	− 0.211404i
$668$	− 9.10681e6i	− 0.789633i
$669$	425120.	0.0367236
$670$	0	0
$671$	2.71174e6	0.232510
$672$	39042.7i	0.00333516i
$673$	9.60528e6i	0.817471i	0.912653	+	0.408736i	$0.134030\pi$
$673$	9.60528e6i	0.817471i	−0.912653	+	0.408736i	$0.865970\pi$
$674$	9.68004e6	0.820782
$675$	0	0
$676$	456976.	0.0384615
$677$	− 5.99107e6i	− 0.502380i	−0.967938	−	0.251190i	$-0.919178\pi$
$677$	− 5.99107e6i	− 0.502380i	0.967938	−	0.251190i	$-0.0808220\pi$
$678$	2.10112e6i	0.175541i
$679$	826427.	0.0687907
$680$	0	0
$681$	792650.	0.0654958
$682$	− 7.82093e6i	− 0.643869i
$683$	1.49302e7i	1.22465i	0.790605	+	0.612327i	$0.209766\pi$
$683$	1.49302e7i	1.22465i	−0.790605	+	0.612327i	$0.790234\pi$
$684$	8.18940e6	0.669286
$685$	0	0
$686$	−1.37488e6	−0.111546
$687$	4.40471e6i	0.356062i
$688$	− 4.18349e6i	− 0.336952i
$689$	5.92338e6	0.475359
$690$	0	0
$691$	−4.19350e6	−0.334104	−0.167052	−	0.985948i	$-0.553425\pi$
$691$	−4.19350e6	−0.334104	−0.167052	+	0.985948i	$0.553425\pi$
$692$	1.82349e6i	0.144757i
$693$	463767.i	0.0366832i
$694$	−834631.	−0.0657803
$695$	0	0
$696$	−1.57057e6	−0.122895
$697$	1.95531e7i	1.52453i
$698$	1.02456e7i	0.795972i
$699$	5.09704e6	0.394571
$700$	0	0
$701$	−1.88086e7	−1.44564	−0.722820	−	0.691036i	$-0.757155\pi$
$701$	−1.88086e7	−1.44564	−0.722820	+	0.691036i	$0.757155\pi$
$702$	1.18645e6i	0.0908673i
$703$	− 2.09879e7i	− 1.60170i
$704$	−808033.	−0.0614466
$705$	0	0
$706$	8.91031e6	0.672792
$707$	− 110483.i	− 0.00831278i
$708$	367001.i	0.0275159i
$709$	−1.59830e7	−1.19411	−0.597054	−	0.802201i	$-0.703662\pi$
$709$	−1.59830e7	−1.19411	−0.597054	+	0.802201i	$0.703662\pi$
$710$	0	0
$711$	−2.05510e7	−1.52461
$712$	− 5.00980e6i	− 0.370357i
$713$	3.64646e6i	0.268626i
$714$	−144843.	−0.0106329
$715$	0	0
$716$	5.12471e6	0.373582
$717$	− 4.05242e6i	− 0.294386i
$718$	1.87885e7i	1.36013i
$719$	−1.01853e7	−0.734771	−0.367386	−	0.930069i	$-0.619747\pi$
$719$	−1.01853e7	−0.734771	−0.367386	+	0.930069i	$0.619747\pi$
$720$	0	0
$721$	−461525.	−0.0330641
$722$	1.00462e7i	0.717231i
$723$	− 315564.i	− 0.0224513i
$724$	−7.23165e6	−0.512733
$725$	0	0
$726$	−1.81589e6	−0.127864
$727$	4.80166e6i	0.336942i	0.985707	+	0.168471i	$0.0538829\pi$
$727$	4.80166e6i	0.336942i	−0.985707	+	0.168471i	$0.946117\pi$
$728$	110946.i	0.00775863i
$729$	9.68323e6	0.674841
$730$	0	0
$731$	1.55201e7	1.07424
$732$	817507.i	0.0563915i
$733$	4.57724e6i	0.314662i	0.987546	+	0.157331i	$0.0502889\pi$
$733$	4.57724e6i	0.314662i	−0.987546	+	0.157331i	$0.949711\pi$
$734$	−1.28998e7	−0.883779
$735$	0	0
$736$	376741.	0.0256359
$737$	1.22579e7i	0.831279i
$738$	1.88740e7i	1.27563i
$739$	−2.54398e7	−1.71358	−0.856788	−	0.515669i	$-0.827543\pi$
$739$	−2.54398e7	−1.71358	−0.856788	+	0.515669i	$0.827543\pi$
$740$	0	0
$741$	−1.40290e6	−0.0938603
$742$	1.43810e6i	0.0958914i
$743$	− 1.06776e7i	− 0.709583i	−0.934946	−	0.354791i	$-0.884552\pi$
$743$	− 1.06776e7i	− 0.709583i	0.934946	−	0.354791i	$-0.115448\pi$
$744$	2.35777e6	0.156160
$745$	0	0
$746$	1.23681e7	0.813687
$747$	− 8.73987e6i	− 0.573065i
$748$	− 2.99768e6i	− 0.195899i
$749$	−1.99231e6	−0.129763
$750$	0	0
$751$	1.24759e7	0.807184	0.403592	−	0.914939i	$-0.367762\pi$
$751$	1.24759e7	0.807184	0.403592	+	0.914939i	$0.367762\pi$
$752$	3.46586e6i	0.223494i
$753$	− 1.58325e6i	− 0.101757i
$754$	−4.46306e6	−0.285893
$755$	0	0
$756$	−288052.	−0.0183302
$757$	− 1.89987e6i	− 0.120499i	−0.998183	−	0.0602495i	$-0.980810\pi$
$757$	− 1.89987e6i	− 0.120499i	0.998183	−	0.0602495i	$-0.0191896\pi$
$758$	1.45541e7i	0.920049i
$759$	−269777.	−0.0169981
$760$	0	0
$761$	1.16295e7	0.727949	0.363975	−	0.931409i	$-0.381420\pi$
$761$	1.16295e7	0.727949	0.363975	+	0.931409i	$0.381420\pi$
$762$	2.79942e6i	0.174655i
$763$	− 2.21971e6i	− 0.138034i
$764$	−2.92669e6	−0.181402
$765$	0	0
$766$	−7.56967e6	−0.466128
$767$	1.04290e6i	0.0640107i
$768$	− 243597.i	− 0.0149029i
$769$	2.05493e7	1.25309	0.626544	−	0.779386i	$-0.284469\pi$
$769$	2.05493e7	1.25309	0.626544	+	0.779386i	$0.284469\pi$
$770$	0	0
$771$	−932748.	−0.0565104
$772$	− 8.13913e6i	− 0.491513i
$773$	− 5.50647e6i	− 0.331455i	−0.986172	−	0.165727i	$-0.947003\pi$
$773$	− 5.50647e6i	− 0.331455i	0.986172	−	0.165727i	$-0.0529972\pi$
$774$	1.49811e7	0.898857
$775$	0	0
$776$	−5.15629e6	−0.307385
$777$	358312.i	0.0212916i
$778$	− 316518.i	− 0.0187478i
$779$	−4.59799e7	−2.71472
$780$	0	0
$781$	−7.56655e6	−0.443885
$782$	1.39765e6i	0.0817301i
$783$	− 1.15875e7i	− 0.675438i
$784$	4.27566e6	0.248435
$785$	0	0
$786$	3.92390e6	0.226549
$787$	− 1.66836e7i	− 0.960182i	−0.877219	−	0.480091i	$-0.840604\pi$
$787$	− 1.66836e7i	− 0.960182i	0.877219	−	0.480091i	$-0.159396\pi$
$788$	1.78665e6i	0.102500i
$789$	−5.51228e6	−0.315238
$790$	0	0
$791$	−1.44959e6	−0.0823767
$792$	− 2.89357e6i	− 0.163916i
$793$	2.32309e6i	0.131185i
$794$	−7.28942e6	−0.410338
$795$	0	0
$796$	8.33949e6	0.466505
$797$	1.53978e7i	0.858643i	0.903152	+	0.429321i	$0.141247\pi$
$797$	1.53978e7i	0.858643i	−0.903152	+	0.429321i	$0.858753\pi$
$798$	− 340602.i	− 0.0189339i
$799$	−1.28578e7	−0.712526
$800$	0	0
$801$	1.79401e7	0.987969
$802$	− 471097.i	− 0.0258627i
$803$	1.18300e7i	0.647434i
$804$	−3.69538e6	−0.201613
$805$	0	0
$806$	6.70002e6	0.363277
$807$	1.48224e6i	0.0801188i
$808$	689332.i	0.0371450i
$809$	2.01482e7	1.08234	0.541171	−	0.840912i	$-0.317981\pi$
$809$	2.01482e7	1.08234	0.541171	+	0.840912i	$0.317981\pi$
$810$	0	0
$811$	−3.88568e6	−0.207451	−0.103725	−	0.994606i	$-0.533076\pi$
$811$	−3.88568e6	−0.207451	−0.103725	+	0.994606i	$0.533076\pi$
$812$	− 1.08356e6i	− 0.0576717i
$813$	− 678337.i	− 0.0359931i
$814$	−7.41568e6	−0.392274
$815$	0	0
$816$	903710.	0.0475121
$817$	3.64961e7i	1.91290i
$818$	− 1.49240e7i	− 0.779836i
$819$	−397299.	−0.0206970
$820$	0	0
$821$	1.69007e7	0.875079	0.437539	−	0.899199i	$-0.355850\pi$
$821$	1.69007e7	0.875079	0.437539	+	0.899199i	$0.355850\pi$
$822$	5.01672e6i	0.258964i
$823$	166131.i	0.00854971i	0.999991	+	0.00427485i	$0.00136073\pi$
$823$	166131.i	0.00854971i	−0.999991	+	0.00427485i	$0.998639\pi$
$824$	2.87958e6	0.147744
$825$	0	0
$826$	−253198.	−0.0129125
$827$	1.64293e7i	0.835324i	0.908602	+	0.417662i	$0.137150\pi$
$827$	1.64293e7i	0.835324i	−0.908602	+	0.417662i	$0.862850\pi$
$828$	1.34911e6i	0.0683866i
$829$	−811068.	−0.0409894	−0.0204947	−	0.999790i	$-0.506524\pi$
$829$	−811068.	−0.0409894	−0.0204947	+	0.999790i	$0.506524\pi$
$830$	0	0
$831$	294397.	0.0147888
$832$	− 692224.i	− 0.0346688i
$833$	1.58621e7i	0.792040i
$834$	1.28167e6	0.0638060
$835$	0	0
$836$	7.04916e6	0.348836
$837$	1.73953e7i	0.858261i
$838$	1.07729e7i	0.529933i
$839$	−1.09462e7	−0.536855	−0.268427	−	0.963300i	$-0.586504\pi$
$839$	−1.09462e7	−0.536855	−0.268427	+	0.963300i	$0.586504\pi$
$840$	0	0
$841$	2.30773e7	1.12511
$842$	2.22715e7i	1.08261i
$843$	5.62007e6i	0.272378i
$844$	8.70351e6	0.420570
$845$	0	0
$846$	−1.24112e7	−0.596196
$847$	− 1.25281e6i	− 0.0600033i
$848$	− 8.97269e6i	− 0.428483i
$849$	9.32152e6	0.443831
$850$	0	0
$851$	3.45752e6	0.163659
$852$	− 2.28108e6i	− 0.107657i
$853$	1.24011e7i	0.583564i	0.956485	+	0.291782i	$0.0942482\pi$
$853$	1.24011e7i	0.583564i	−0.956485	+	0.291782i	$0.905752\pi$
$854$	−564009.	−0.0264631
$855$	0	0
$856$	1.24305e7	0.579837
$857$	− 1.78406e7i	− 0.829768i	−0.909874	−	0.414884i	$-0.863822\pi$
$857$	− 1.78406e7i	− 0.829768i	0.909874	−	0.414884i	$-0.136178\pi$
$858$	495688.i	0.0229874i
$859$	−1.43038e7	−0.661408	−0.330704	−	0.943735i	$-0.607286\pi$
$859$	−1.43038e7	−0.661408	−0.330704	+	0.943735i	$0.607286\pi$
$860$	0	0
$861$	784982.	0.0360871
$862$	− 1.50264e7i	− 0.688788i
$863$	2.13905e7i	0.977673i	0.872376	+	0.488836i	$0.162578\pi$
$863$	2.13905e7i	0.977673i	−0.872376	+	0.488836i	$0.837422\pi$
$864$	1.79723e6	0.0819067
$865$	0	0
$866$	−6.43910e6	−0.291763
$867$	− 1.92498e6i	− 0.0869717i
$868$	1.62666e6i	0.0732819i
$869$	−1.76896e7	−0.794636
$870$	0	0
$871$	−1.05011e7	−0.469016
$872$	1.38494e7i	0.616791i
$873$	− 1.84647e7i	− 0.819985i
$874$	−3.28663e6	−0.145536
$875$	0	0
$876$	−3.56638e6	−0.157024
$877$	3.28786e7i	1.44349i	0.692159	+	0.721745i	$0.256660\pi$
$877$	3.28786e7i	1.44349i	−0.692159	+	0.721745i	$0.743340\pi$
$878$	7.69227e6i	0.336758i
$879$	6.51902e6	0.284584
$880$	0	0
$881$	−2.63729e7	−1.14477	−0.572384	−	0.819986i	$-0.693981\pi$
$881$	−2.63729e7	−1.14477	−0.572384	+	0.819986i	$0.693981\pi$
$882$	1.53111e7i	0.662728i
$883$	− 1.89435e7i	− 0.817634i	−0.912616	−	0.408817i	$-0.865941\pi$
$883$	− 1.89435e7i	− 0.817634i	0.912616	−	0.408817i	$-0.134059\pi$
$884$	2.56805e6	0.110528
$885$	0	0
$886$	5.72241e6	0.244903
$887$	− 1.00174e7i	− 0.427509i	−0.976887	−	0.213755i	$-0.931431\pi$
$887$	− 1.00174e7i	− 0.427509i	0.976887	−	0.213755i	$-0.0685693\pi$
$888$	− 2.23560e6i	− 0.0951397i
$889$	−1.93135e6	−0.0819610
$890$	0	0
$891$	9.69955e6	0.409314
$892$	− 1.82995e6i	− 0.0770063i
$893$	− 3.02356e7i	− 1.26879i
$894$	4.31456e6	0.180548
$895$	0	0
$896$	168061.	0.00699353
$897$	− 231112.i	− 0.00959049i
$898$	1.32108e7i	0.546686i
$899$	−6.54357e7	−2.70032
$900$	0	0
$901$	3.32874e7	1.36605
$902$	1.62461e7i	0.664864i
$903$	− 623072.i	− 0.0254284i
$904$	9.04438e6	0.368093
$905$	0	0
$906$	1.93119e6	0.0781636
$907$	2.20644e7i	0.890581i	0.895386	+	0.445290i	$0.146900\pi$
$907$	2.20644e7i	0.890581i	−0.895386	+	0.445290i	$0.853100\pi$
$908$	− 3.41200e6i	− 0.137339i
$909$	−2.46850e6	−0.0990884
$910$	0	0
$911$	−2.02691e7	−0.809167	−0.404583	−	0.914501i	$-0.632583\pi$
$911$	−2.02691e7	−0.809167	−0.404583	+	0.914501i	$0.632583\pi$
$912$	2.12511e6i	0.0846045i
$913$	− 7.52299e6i	− 0.298685i
$914$	2.74263e7	1.08593
$915$	0	0
$916$	1.89603e7	0.746631
$917$	2.70715e6i	0.106314i
$918$	6.66746e6i	0.261128i
$919$	1.79004e7	0.699154	0.349577	−	0.936908i	$-0.386325\pi$
$919$	1.79004e7	0.699154	0.349577	+	0.936908i	$0.386325\pi$
$920$	0	0
$921$	5.77242e6	0.224238
$922$	3.33374e7i	1.29153i
$923$	− 6.48209e6i	− 0.250444i
$924$	−120345.	−0.00463713
$925$	0	0
$926$	1.14809e7	0.439997
$927$	1.03118e7i	0.394124i
$928$	6.76061e6i	0.257701i
$929$	−9.25686e6	−0.351904	−0.175952	−	0.984399i	$-0.556300\pi$
$929$	−9.25686e6	−0.351904	−0.175952	+	0.984399i	$0.556300\pi$
$930$	0	0
$931$	−3.73002e7	−1.41038
$932$	− 2.19404e7i	− 0.827381i
$933$	699260.i	0.0262987i
$934$	2.64726e7	0.992953
$935$	0	0
$936$	2.47885e6	0.0924828
$937$	8.47287e6i	0.315269i	0.987498	+	0.157635i	$0.0503868\pi$
$937$	8.47287e6i	0.315269i	−0.987498	+	0.157635i	$0.949613\pi$
$938$	− 2.54949e6i	− 0.0946119i
$939$	3.14631e6	0.116449
$940$	0	0
$941$	−1.22548e7	−0.451161	−0.225580	−	0.974225i	$-0.572428\pi$
$941$	−1.22548e7	−0.451161	−0.225580	+	0.974225i	$0.572428\pi$
$942$	686994.i	0.0252247i
$943$	− 7.57465e6i	− 0.277385i
$944$	1.57977e6	0.0576984
$945$	0	0
$946$	1.28952e7	0.468490
$947$	− 2.12327e7i	− 0.769363i	−0.923049	−	0.384681i	$-0.874311\pi$
$947$	− 2.12327e7i	− 0.769363i	0.923049	−	0.384681i	$-0.125689\pi$
$948$	− 5.33287e6i	− 0.192726i
$949$	−1.01345e7	−0.365288
$950$	0	0
$951$	1.12749e6	0.0404259
$952$	623481.i	0.0222962i
$953$	− 2.31217e7i	− 0.824684i	−0.911029	−	0.412342i	$-0.864711\pi$
$953$	− 2.31217e7i	− 0.824684i	0.911029	−	0.412342i	$-0.135289\pi$
$954$	3.21312e7	1.14303
$955$	0	0
$956$	−1.74438e7	−0.617301
$957$	− 4.84114e6i	− 0.170871i
$958$	5.89739e6i	0.207609i
$959$	−3.46110e6	−0.121525
$960$	0	0
$961$	6.96040e7	2.43123
$962$	− 6.35285e6i	− 0.221325i
$963$	4.45137e7i	1.54678i
$964$	−1.35836e6	−0.0470784
$965$	0	0
$966$	56110.2	0.00193464
$967$	− 1.92947e7i	− 0.663548i	−0.943359	−	0.331774i	$-0.892353\pi$
$967$	− 1.92947e7i	− 0.663548i	0.943359	−	0.331774i	$-0.107647\pi$
$968$	7.81658e6i	0.268120i
$969$	−7.88383e6	−0.269729
$970$	0	0
$971$	1.63863e7	0.557743	0.278871	−	0.960328i	$-0.410040\pi$
$971$	1.63863e7	0.557743	0.278871	+	0.960328i	$0.410040\pi$
$972$	9.74798e6i	0.330940i
$973$	884241.i	0.0299425i
$974$	1.05260e7	0.355523
$975$	0	0
$976$	3.51900e6	0.118248
$977$	− 3.83317e6i	− 0.128476i	−0.997935	−	0.0642380i	$-0.979538\pi$
$977$	− 3.83317e6i	− 0.128476i	0.997935	−	0.0642380i	$-0.0204617\pi$
$978$	5.63611e6i	0.188422i
$979$	1.54422e7	0.514936
$980$	0	0
$981$	−4.95945e7	−1.64536
$982$	1.88185e7i	0.622738i
$983$	2.80937e7i	0.927310i	0.886016	+	0.463655i	$0.153462\pi$
$983$	2.80937e7i	0.927310i	−0.886016	+	0.463655i	$0.846538\pi$
$984$	−4.89771e6	−0.161252
$985$	0	0
$986$	−2.50809e7	−0.821581
$987$	516191.i	0.0168662i
$988$	6.03886e6i	0.196817i
$989$	−6.01231e6	−0.195457
$990$	0	0
$991$	−4.83597e7	−1.56423	−0.782113	−	0.623137i	$-0.785858\pi$
$991$	−4.83597e7	−1.56423	−0.782113	+	0.623137i	$0.785858\pi$
$992$	− 1.01491e7i	− 0.327454i
$993$	1.24974e7i	0.402206i
$994$	1.57375e6	0.0505207
$995$	0	0
$996$	2.26795e6	0.0724411
$997$	4.50212e7i	1.43443i	0.696852	+	0.717215i	$0.254584\pi$
$997$	4.50212e7i	1.43443i	−0.696852	+	0.717215i	$0.745416\pi$
$998$	− 1.63659e6i	− 0.0520132i
$999$	1.64940e7	0.522892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.b.j.599.5		6
5.2	odd	4		130.6.a.f.1.2	✓	3
5.3	odd	4		650.6.a.j.1.2		3
5.4	even	2	inner	650.6.b.j.599.2		6
20.7	even	4		1040.6.a.l.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.f.1.2	✓	3	5.2	odd	4
650.6.a.j.1.2		3	5.3	odd	4
650.6.b.j.599.2		6	5.4	even	2	inner
650.6.b.j.599.5		6	1.1	even	1	trivial
1040.6.a.l.1.2		3	20.7	even	4

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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -3.71700i q^{3} -16.0000 q^{4} +14.8680 q^{6} +10.2576i q^{7} -64.0000i q^{8} +229.184 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -3.71700i q^{3} -16.0000 q^{4} +14.8680 q^{6} +10.2576i q^{7} -64.0000i q^{8} +229.184 q^{9} +197.274 q^{11} +59.4720i q^{12} +169.000i q^{13} -41.0305 q^{14} +256.000 q^{16} +949.722i q^{17} +916.736i q^{18} -2233.30 q^{19} +38.1276 q^{21} +789.095i q^{22} -367.911i q^{23} -237.888 q^{24} -676.000 q^{26} -1755.11i q^{27} -164.122i q^{28} +6602.16 q^{29} -9911.27 q^{31} +1024.00i q^{32} -733.267i q^{33} -3798.89 q^{34} -3666.94 q^{36} +9397.70i q^{37} -8933.22i q^{38} +628.173 q^{39} +20588.3 q^{41} +152.510i q^{42} -16341.8i q^{43} -3156.38 q^{44} +1471.64 q^{46} +13538.5i q^{47} -951.552i q^{48} +16701.8 q^{49} +3530.12 q^{51} -2704.00i q^{52} -35049.6i q^{53} +7020.43 q^{54} +656.488 q^{56} +8301.20i q^{57} +26408.6i q^{58} +6170.98 q^{59} +13746.1 q^{61} -39645.1i q^{62} +2350.88i q^{63} -4096.00 q^{64} +2933.07 q^{66} +62136.4i q^{67} -15195.6i q^{68} -1367.52 q^{69} -38355.6 q^{71} -14667.8i q^{72} +59967.3i q^{73} -37590.8 q^{74} +35732.9 q^{76} +2023.56i q^{77} +2512.69i q^{78} -89670.2 q^{79} +49167.9 q^{81} +82353.1i q^{82} -38134.8i q^{83} -610.042 q^{84} +65367.1 q^{86} -24540.2i q^{87} -12625.5i q^{88} +78278.2 q^{89} -1733.54 q^{91} +5886.57i q^{92} +36840.2i q^{93} -54154.1 q^{94} +3806.21 q^{96} -80567.1i q^{97} +66807.1i q^{98} +45212.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9}+O(q^{10}) \) 6 * q - 96 * q^4 + 176 * q^6 - 310 * q^9	\( 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9} + 96 q^{11} - 1872 q^{14} + 1536 q^{16} + 720 q^{19} + 13808 q^{21} - 2816 q^{24} - 4056 q^{26} + 6156 q^{29} - 10776 q^{31} + 12048 q^{34} + 4960 q^{36} + 7436 q^{39} - 31812 q^{41} - 1536 q^{44} + 18960 q^{46} - 15534 q^{49} - 67944 q^{51} - 81344 q^{54} + 29952 q^{56} + 155520 q^{59} + 35964 q^{61} - 24576 q^{64} + 180576 q^{66} + 63720 q^{69} - 9888 q^{71} - 202896 q^{74} - 11520 q^{76} - 84240 q^{79} + 293038 q^{81} - 220928 q^{84} + 314448 q^{86} - 39228 q^{89} - 79092 q^{91} - 142224 q^{94} + 45056 q^{96} - 700680 q^{99}+O(q^{100}) \) 6 * q - 96 * q^4 + 176 * q^6 - 310 * q^9 + 96 * q^11 - 1872 * q^14 + 1536 * q^16 + 720 * q^19 + 13808 * q^21 - 2816 * q^24 - 4056 * q^26 + 6156 * q^29 - 10776 * q^31 + 12048 * q^34 + 4960 * q^36 + 7436 * q^39 - 31812 * q^41 - 1536 * q^44 + 18960 * q^46 - 15534 * q^49 - 67944 * q^51 - 81344 * q^54 + 29952 * q^56 + 155520 * q^59 + 35964 * q^61 - 24576 * q^64 + 180576 * q^66 + 63720 * q^69 - 9888 * q^71 - 202896 * q^74 - 11520 * q^76 - 84240 * q^79 + 293038 * q^81 - 220928 * q^84 + 314448 * q^86 - 39228 * q^89 - 79092 * q^91 - 142224 * q^94 + 45056 * q^96 - 700680 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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