LMFDB - Embedded newform 325.6.b.i.274.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.7
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.83767i q^{2} -12.7661i q^{3} -2.07840 q^{4} -74.5243 q^{6} -231.930i q^{7} -174.672i q^{8} +80.0268 q^{9} +O(q^{10})$	\(q-5.83767i q^{2} -12.7661i q^{3} -2.07840 q^{4} -74.5243 q^{6} -231.930i q^{7} -174.672i q^{8} +80.0268 q^{9} -223.383 q^{11} +26.5330i q^{12} +169.000i q^{13} -1353.93 q^{14} -1086.19 q^{16} -1329.26i q^{17} -467.170i q^{18} +1480.66 q^{19} -2960.84 q^{21} +1304.04i q^{22} -1324.23i q^{23} -2229.89 q^{24} +986.566 q^{26} -4123.79i q^{27} +482.042i q^{28} -1670.85 q^{29} +8484.76 q^{31} +751.294i q^{32} +2851.73i q^{33} -7759.78 q^{34} -166.327 q^{36} +4109.10i q^{37} -8643.62i q^{38} +2157.47 q^{39} -10014.5 q^{41} +17284.4i q^{42} +3443.28i q^{43} +464.279 q^{44} -7730.43 q^{46} -16459.1i q^{47} +13866.4i q^{48} -36984.6 q^{49} -16969.5 q^{51} -351.249i q^{52} +16934.3i q^{53} -24073.3 q^{54} -40511.8 q^{56} -18902.3i q^{57} +9753.89i q^{58} -11919.2 q^{59} +37419.0 q^{61} -49531.2i q^{62} -18560.6i q^{63} -30372.2 q^{64} +16647.5 q^{66} -7254.26i q^{67} +2762.73i q^{68} -16905.3 q^{69} +52098.4 q^{71} -13978.5i q^{72} +85653.0i q^{73} +23987.6 q^{74} -3077.40 q^{76} +51809.3i q^{77} -12594.6i q^{78} +45167.6 q^{79} -33198.2 q^{81} +58461.3i q^{82} +114443. i q^{83} +6153.80 q^{84} +20100.7 q^{86} +21330.3i q^{87} +39018.9i q^{88} -22697.0 q^{89} +39196.2 q^{91} +2752.28i q^{92} -108317. i q^{93} -96082.7 q^{94} +9591.09 q^{96} +107803. i q^{97} +215904. i q^{98} -17876.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 5.83767i	− 1.03196i	−0.856599	−	0.515982i	$-0.827427\pi$
$2$	− 5.83767i	− 1.03196i	0.856599	−	0.515982i	$-0.172573\pi$
$3$	− 12.7661i	− 0.818946i	−0.912322	−	0.409473i	$-0.865713\pi$
$3$	− 12.7661i	− 0.818946i	0.912322	−	0.409473i	$-0.134287\pi$
$4$	−2.07840	−0.0649499
$5$	0	0
$5$	0	0
$6$	−74.5243	−0.845122
$7$	− 231.930i	− 1.78901i	−0.447061	−	0.894503i	$-0.647529\pi$
$7$	− 231.930i	− 1.78901i	0.447061	−	0.894503i	$-0.352471\pi$
$8$	− 174.672i	− 0.964938i
$9$	80.0268	0.329328
$10$	0	0
$11$	−223.383	−0.556633	−0.278317	−	0.960489i	$-0.589776\pi$
$11$	−223.383	−0.556633	−0.278317	+	0.960489i	$0.589776\pi$
$12$	26.5330i	0.0531904i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−1353.93	−1.84619
$15$	0	0
$16$	−1086.19	−1.06073
$17$	− 1329.26i	− 1.11555i	−0.829993	−	0.557773i	$-0.811656\pi$
$17$	− 1329.26i	− 1.11555i	0.829993	−	0.557773i	$-0.188344\pi$
$18$	− 467.170i	− 0.339855i
$19$	1480.66	0.940962	0.470481	−	0.882410i	$-0.344080\pi$
$19$	1480.66	0.940962	0.470481	+	0.882410i	$0.344080\pi$
$20$	0	0
$21$	−2960.84	−1.46510
$22$	1304.04i	0.574425i
$23$	− 1324.23i	− 0.521969i	−0.965343	−	0.260984i	$-0.915953\pi$
$23$	− 1324.23i	− 0.521969i	0.965343	−	0.260984i	$-0.0840471\pi$
$24$	−2229.89	−0.790232
$25$	0	0
$26$	986.566	0.286215
$27$	− 4123.79i	− 1.08865i
$28$	482.042i	0.116196i
$29$	−1670.85	−0.368929	−0.184465	−	0.982839i	$-0.559055\pi$
$29$	−1670.85	−0.368929	−0.184465	+	0.982839i	$0.559055\pi$
$30$	0	0
$31$	8484.76	1.58575	0.792876	−	0.609383i	$-0.208583\pi$
$31$	8484.76	1.58575	0.792876	+	0.609383i	$0.208583\pi$
$32$	751.294i	0.129699i
$33$	2851.73i	0.455852i
$34$	−7759.78	−1.15120
$35$	0	0
$36$	−166.327	−0.0213898
$37$	4109.10i	0.493449i	0.969086	+	0.246725i	$0.0793543\pi$
$37$	4109.10i	0.493449i	−0.969086	+	0.246725i	$0.920646\pi$
$38$	− 8643.62i	− 0.971039i
$39$	2157.47	0.227135
$40$	0	0
$41$	−10014.5	−0.930399	−0.465200	−	0.885206i	$-0.654017\pi$
$41$	−10014.5	−0.930399	−0.465200	+	0.885206i	$0.654017\pi$
$42$	17284.4i	1.51193i
$43$	3443.28i	0.283989i	0.989867	+	0.141994i	$0.0453515\pi$
$43$	3443.28i	0.283989i	−0.989867	+	0.141994i	$0.954648\pi$
$44$	464.279	0.0361532
$45$	0	0
$46$	−7730.43	−0.538653
$47$	− 16459.1i	− 1.08683i	−0.839465	−	0.543414i	$-0.817131\pi$
$47$	− 16459.1i	− 1.08683i	0.839465	−	0.543414i	$-0.182869\pi$
$48$	13866.4i	0.868681i
$49$	−36984.6	−2.20055
$50$	0	0
$51$	−16969.5	−0.913572
$52$	− 351.249i	− 0.0180138i
$53$	16934.3i	0.828090i	0.910256	+	0.414045i	$0.135884\pi$
$53$	16934.3i	0.828090i	−0.910256	+	0.414045i	$0.864116\pi$
$54$	−24073.3	−1.12345
$55$	0	0
$56$	−40511.8	−1.72628
$57$	− 18902.3i	− 0.770596i
$58$	9753.89i	0.380722i
$59$	−11919.2	−0.445777	−0.222889	−	0.974844i	$-0.571549\pi$
$59$	−11919.2	−0.445777	−0.222889	+	0.974844i	$0.571549\pi$
$60$	0	0
$61$	37419.0	1.28756	0.643780	−	0.765210i	$-0.277365\pi$
$61$	37419.0	1.28756	0.643780	+	0.765210i	$0.277365\pi$
$62$	− 49531.2i	− 1.63644i
$63$	− 18560.6i	− 0.589170i
$64$	−30372.2	−0.926887
$65$	0	0
$66$	16647.5	0.470423
$67$	− 7254.26i	− 0.197427i	−0.995116	−	0.0987134i	$-0.968527\pi$
$67$	− 7254.26i	− 0.197427i	0.995116	−	0.0987134i	$-0.0314727\pi$
$68$	2762.73i	0.0724546i
$69$	−16905.3	−0.427464
$70$	0	0
$71$	52098.4	1.22653	0.613266	−	0.789877i	$-0.289855\pi$
$71$	52098.4	1.22653	0.613266	+	0.789877i	$0.289855\pi$
$72$	− 13978.5i	− 0.317781i
$73$	85653.0i	1.88120i	0.339513	+	0.940601i	$0.389738\pi$
$73$	85653.0i	1.88120i	−0.339513	+	0.940601i	$0.610262\pi$
$74$	23987.6	0.509222
$75$	0	0
$76$	−3077.40	−0.0611153
$77$	51809.3i	0.995820i
$78$	− 12594.6i	− 0.234395i
$79$	45167.6	0.814252	0.407126	−	0.913372i	$-0.366531\pi$
$79$	45167.6	0.814252	0.407126	+	0.913372i	$0.366531\pi$
$80$	0	0
$81$	−33198.2	−0.562215
$82$	58461.3i	0.960139i
$83$	114443.i	1.82345i	0.410802	+	0.911724i	$0.365249\pi$
$83$	114443.i	1.82345i	−0.410802	+	0.911724i	$0.634751\pi$
$84$	6153.80	0.0951580
$85$	0	0
$86$	20100.7	0.293066
$87$	21330.3i	0.302133i
$88$	39018.9i	0.537116i
$89$	−22697.0	−0.303733	−0.151867	−	0.988401i	$-0.548528\pi$
$89$	−22697.0	−0.303733	−0.151867	+	0.988401i	$0.548528\pi$
$90$	0	0
$91$	39196.2	0.496181
$92$	2752.28i	0.0339018i
$93$	− 108317.i	− 1.29864i
$94$	−96082.7	−1.12157
$95$	0	0
$96$	9591.09	0.106216
$97$	107803.i	1.16333i	0.813429	+	0.581664i	$0.197598\pi$
$97$	107803.i	1.16333i	−0.813429	+	0.581664i	$0.802402\pi$
$98$	215904.i	2.27088i
$99$	−17876.6	−0.183315
$100$	0	0
$101$	196051.	1.91234	0.956172	−	0.292806i	$-0.0945889\pi$
$101$	196051.	1.91234	0.956172	+	0.292806i	$0.0945889\pi$
$102$	99062.1i	0.942773i
$103$	− 168848.i	− 1.56820i	−0.620633	−	0.784101i	$-0.713124\pi$
$103$	− 168848.i	− 1.56820i	0.620633	−	0.784101i	$-0.286876\pi$
$104$	29519.6	0.267626
$105$	0	0
$106$	98856.9	0.854559
$107$	69103.1i	0.583496i	0.956495	+	0.291748i	$0.0942369\pi$
$107$	69103.1i	0.583496i	−0.956495	+	0.291748i	$0.905763\pi$
$108$	8570.87i	0.0707075i
$109$	−31697.4	−0.255539	−0.127770	−	0.991804i	$-0.540782\pi$
$109$	−31697.4	−0.255539	−0.127770	+	0.991804i	$0.540782\pi$
$110$	0	0
$111$	52457.2	0.404108
$112$	251920.i	1.89766i
$113$	− 104371.i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$113$	− 104371.i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$114$	−110345.	−0.795228
$115$	0	0
$116$	3472.69	0.0239619
$117$	13524.5i	0.0913392i
$118$	69580.5i	0.460026i
$119$	−308295.	−1.99572
$120$	0	0
$121$	−111151.	−0.690160
$122$	− 218440.i	− 1.32872i
$123$	127846.i	0.761946i
$124$	−17634.7	−0.102994
$125$	0	0
$126$	−108351.	−0.608003
$127$	204212.i	1.12350i	0.827307	+	0.561749i	$0.189871\pi$
$127$	204212.i	1.12350i	−0.827307	+	0.561749i	$0.810129\pi$
$128$	201345.i	1.08621i
$129$	43957.3	0.232571
$130$	0	0
$131$	186798.	0.951027	0.475514	−	0.879708i	$-0.342262\pi$
$131$	186798.	0.951027	0.475514	+	0.879708i	$0.342262\pi$
$132$	− 5927.03i	− 0.0296075i
$133$	− 343410.i	− 1.68339i
$134$	−42348.0	−0.203737
$135$	0	0
$136$	−232185.	−1.07643
$137$	169640.i	0.772195i	0.922458	+	0.386098i	$0.126177\pi$
$137$	169640.i	0.772195i	−0.922458	+	0.386098i	$0.873823\pi$
$138$	98687.4i	0.441128i
$139$	−65860.1	−0.289125	−0.144562	−	0.989496i	$-0.546177\pi$
$139$	−65860.1	−0.289125	−0.144562	+	0.989496i	$0.546177\pi$
$140$	0	0
$141$	−210118.	−0.890053
$142$	− 304133.i	− 1.26574i
$143$	− 37751.8i	− 0.154382i
$144$	−86924.2	−0.349329
$145$	0	0
$146$	500014.	1.94133
$147$	472149.i	1.80213i
$148$	− 8540.34i	− 0.0320495i
$149$	191843.	0.707914	0.353957	−	0.935262i	$-0.384836\pi$
$149$	191843.	0.707914	0.353957	+	0.935262i	$0.384836\pi$
$150$	0	0
$151$	−252918.	−0.902687	−0.451343	−	0.892350i	$-0.649055\pi$
$151$	−252918.	−0.902687	−0.451343	+	0.892350i	$0.649055\pi$
$152$	− 258631.i	− 0.907970i
$153$	− 106376.i	− 0.367381i
$154$	302446.	1.02765
$155$	0	0
$156$	−4484.08	−0.0147524
$157$	− 246156.i	− 0.797007i	−0.917167	−	0.398503i	$-0.869530\pi$
$157$	− 246156.i	− 0.797007i	0.917167	−	0.398503i	$-0.130470\pi$
$158$	− 263673.i	− 0.840279i
$159$	216185.	0.678161
$160$	0	0
$161$	−307129.	−0.933806
$162$	193800.i	0.580185i
$163$	− 487736.i	− 1.43786i	−0.695084	−	0.718929i	$-0.744633\pi$
$163$	− 487736.i	− 1.43786i	0.695084	−	0.718929i	$-0.255367\pi$
$164$	20814.1	0.0604293
$165$	0	0
$166$	668080.	1.88173
$167$	336570.i	0.933866i	0.884293	+	0.466933i	$0.154641\pi$
$167$	336570.i	0.933866i	−0.884293	+	0.466933i	$0.845359\pi$
$168$	517178.i	1.41373i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	118493.	0.309885
$172$	− 7156.50i	− 0.0184450i
$173$	290715.i	0.738502i	0.929330	+	0.369251i	$0.120386\pi$
$173$	290715.i	0.738502i	−0.929330	+	0.369251i	$0.879614\pi$
$174$	124519.	0.311790
$175$	0	0
$176$	242636.	0.590438
$177$	152162.i	0.365067i
$178$	132497.i	0.313442i
$179$	−293142.	−0.683827	−0.341913	−	0.939732i	$-0.611075\pi$
$179$	−293142.	−0.683827	−0.341913	+	0.939732i	$0.611075\pi$
$180$	0	0
$181$	74419.3	0.168845	0.0844227	−	0.996430i	$-0.473095\pi$
$181$	74419.3	0.168845	0.0844227	+	0.996430i	$0.473095\pi$
$182$	− 228814.i	− 0.512041i
$183$	− 477695.i	− 1.05444i
$184$	−231307.	−0.503668
$185$	0	0
$186$	−632320.	−1.34015
$187$	296935.i	0.620950i
$188$	34208.5i	0.0705893i
$189$	−956431.	−1.94760
$190$	0	0
$191$	−464114.	−0.920537	−0.460269	−	0.887780i	$-0.652247\pi$
$191$	−464114.	−0.920537	−0.460269	+	0.887780i	$0.652247\pi$
$192$	387735.i	0.759070i
$193$	204016.i	0.394249i	0.980379	+	0.197124i	$0.0631602\pi$
$193$	204016.i	0.394249i	−0.980379	+	0.197124i	$0.936840\pi$
$194$	629319.	1.20051
$195$	0	0
$196$	76868.5	0.142925
$197$	− 792402.i	− 1.45472i	−0.686255	−	0.727361i	$-0.740747\pi$
$197$	− 792402.i	− 1.45472i	0.686255	−	0.727361i	$-0.259253\pi$
$198$	104358.i	0.189174i
$199$	635602.	1.13776	0.568882	−	0.822419i	$-0.307376\pi$
$199$	635602.	1.13776	0.568882	+	0.822419i	$0.307376\pi$
$200$	0	0
$201$	−92608.6	−0.161682
$202$	− 1.14448e6i	− 1.97347i
$203$	387521.i	0.660017i
$204$	35269.3	0.0593364
$205$	0	0
$206$	−985677.	−1.61833
$207$	− 105974.i	− 0.171899i
$208$	− 183566.i	− 0.294194i
$209$	−330755.	−0.523770
$210$	0	0
$211$	590802.	0.913557	0.456779	−	0.889580i	$-0.349003\pi$
$211$	590802.	0.913557	0.456779	+	0.889580i	$0.349003\pi$
$212$	− 35196.2i	− 0.0537843i
$213$	− 665094.i	− 1.00446i
$214$	403401.	0.602147
$215$	0	0
$216$	−720313.	−1.05048
$217$	− 1.96787e6i	− 2.83692i
$218$	185039.i	0.263707i
$219$	1.09345e6	1.54060
$220$	0	0
$221$	224645.	0.309397
$222$	− 306228.i	− 0.417025i
$223$	1.19025e6i	1.60279i	0.598139	+	0.801393i	$0.295907\pi$
$223$	1.19025e6i	1.60279i	−0.598139	+	0.801393i	$0.704093\pi$
$224$	174248.	0.232032
$225$	0	0
$226$	−609282.	−0.793500
$227$	− 925858.i	− 1.19256i	−0.802777	−	0.596279i	$-0.796645\pi$
$227$	− 925858.i	− 1.19256i	0.802777	−	0.596279i	$-0.203355\pi$
$228$	39286.4i	0.0500501i
$229$	−989943.	−1.24744	−0.623722	−	0.781646i	$-0.714380\pi$
$229$	−989943.	−1.24744	−0.623722	+	0.781646i	$0.714380\pi$
$230$	0	0
$231$	661403.	0.815523
$232$	291852.i	0.355994i
$233$	− 907234.i	− 1.09479i	−0.836876	−	0.547393i	$-0.815620\pi$
$233$	− 907234.i	− 1.09479i	0.836876	−	0.547393i	$-0.184380\pi$
$234$	78951.7	0.0942588
$235$	0	0
$236$	24772.9	0.0289532
$237$	− 576613.i	− 0.666828i
$238$	1.79973e6i	2.05951i
$239$	−702154.	−0.795129	−0.397564	−	0.917574i	$-0.630144\pi$
$239$	−702154.	−0.795129	−0.397564	+	0.917574i	$0.630144\pi$
$240$	0	0
$241$	−631777.	−0.700682	−0.350341	−	0.936622i	$-0.613934\pi$
$241$	−631777.	−0.700682	−0.350341	+	0.936622i	$0.613934\pi$
$242$	648862.i	0.712220i
$243$	− 578270.i	− 0.628224i
$244$	−77771.5	−0.0836269
$245$	0	0
$246$	746323.	0.786301
$247$	250232.i	0.260976i
$248$	− 1.48205e6i	− 1.53015i
$249$	1.46099e6	1.49331
$250$	0	0
$251$	655538.	0.656770	0.328385	−	0.944544i	$-0.393496\pi$
$251$	655538.	0.656770	0.328385	+	0.944544i	$0.393496\pi$
$252$	38576.3i	0.0382665i
$253$	295811.i	0.290545i
$254$	1.19212e6	1.15941
$255$	0	0
$256$	203471.	0.194045
$257$	− 1.90957e6i	− 1.80344i	−0.432320	−	0.901720i	$-0.642305\pi$
$257$	− 1.90957e6i	− 1.80344i	0.432320	−	0.901720i	$-0.357695\pi$
$258$	− 256608.i	− 0.240005i
$259$	953024.	0.882784
$260$	0	0
$261$	−133713.	−0.121499
$262$	− 1.09046e6i	− 0.981426i
$263$	− 1.00173e6i	− 0.893025i	−0.894777	−	0.446512i	$-0.852666\pi$
$263$	− 1.00173e6i	− 0.893025i	0.894777	−	0.446512i	$-0.147334\pi$
$264$	498119.	0.439869
$265$	0	0
$266$	−2.00471e6	−1.73719
$267$	289752.i	0.248741i
$268$	15077.2i	0.0128228i
$269$	298009.	0.251101	0.125551	−	0.992087i	$-0.459930\pi$
$269$	298009.	0.251101	0.125551	+	0.992087i	$0.459930\pi$
$270$	0	0
$271$	−505333.	−0.417979	−0.208990	−	0.977918i	$-0.567017\pi$
$271$	−505333.	−0.417979	−0.208990	+	0.977918i	$0.567017\pi$
$272$	1.44383e6i	1.18330i
$273$	− 500382.i	− 0.406345i
$274$	990303.	0.796878
$275$	0	0
$276$	35135.8	0.0277637
$277$	708331.i	0.554673i	0.960773	+	0.277337i	$0.0894517\pi$
$277$	708331.i	0.554673i	−0.960773	+	0.277337i	$0.910548\pi$
$278$	384469.i	0.298366i
$279$	679008.	0.522233
$280$	0	0
$281$	−1.37074e6	−1.03559	−0.517797	−	0.855503i	$-0.673248\pi$
$281$	−1.37074e6	−1.03559	−0.517797	+	0.855503i	$0.673248\pi$
$282$	1.22660e6i	0.918503i
$283$	− 1.46799e6i	− 1.08957i	−0.838574	−	0.544787i	$-0.816610\pi$
$283$	− 1.46799e6i	− 1.08957i	0.838574	−	0.544787i	$-0.183390\pi$
$284$	−108281.	−0.0796631
$285$	0	0
$286$	−220382.	−0.159317
$287$	2.32266e6i	1.66449i
$288$	60123.6i	0.0427134i
$289$	−347075.	−0.244444
$290$	0	0
$291$	1.37623e6	0.952702
$292$	− 178021.i	− 0.122184i
$293$	− 1.94491e6i	− 1.32352i	−0.749715	−	0.661760i	$-0.769810\pi$
$293$	− 1.94491e6i	− 1.32352i	0.749715	−	0.661760i	$-0.230190\pi$
$294$	2.75625e6	1.85973
$295$	0	0
$296$	717747.	0.476148
$297$	921186.i	0.605977i
$298$	− 1.11992e6i	− 0.730541i
$299$	223795.	0.144768
$300$	0	0
$301$	798600.	0.508058
$302$	1.47645e6i	0.931540i
$303$	− 2.50281e6i	− 1.56611i
$304$	−1.60828e6	−0.998108
$305$	0	0
$306$	−620990.	−0.379124
$307$	− 1.90204e6i	− 1.15179i	−0.817523	−	0.575896i	$-0.804653\pi$
$307$	− 1.90204e6i	− 1.15179i	0.817523	−	0.575896i	$-0.195347\pi$
$308$	− 107680.i	− 0.0646784i
$309$	−2.15553e6	−1.28427
$310$	0	0
$311$	1.59136e6	0.932970	0.466485	−	0.884529i	$-0.345520\pi$
$311$	1.59136e6	0.932970	0.466485	+	0.884529i	$0.345520\pi$
$312$	− 376851.i	− 0.219171i
$313$	3.09137e6i	1.78357i	0.452459	+	0.891785i	$0.350547\pi$
$313$	3.09137e6i	1.78357i	−0.452459	+	0.891785i	$0.649453\pi$
$314$	−1.43698e6	−0.822482
$315$	0	0
$316$	−93876.0	−0.0528856
$317$	− 953763.i	− 0.533080i	−0.963824	−	0.266540i	$-0.914120\pi$
$317$	− 953763.i	− 0.533080i	0.963824	−	0.266540i	$-0.0858804\pi$
$318$	− 1.26202e6i	− 0.699837i
$319$	373241.	0.205358
$320$	0	0
$321$	882177.	0.477852
$322$	1.79292e6i	0.963654i
$323$	− 1.96818e6i	− 1.04969i
$324$	68999.0	0.0365158
$325$	0	0
$326$	−2.84724e6	−1.48382
$327$	404653.i	0.209273i
$328$	1.74926e6i	0.897778i
$329$	−3.81736e6	−1.94434
$330$	0	0
$331$	−308119.	−0.154578	−0.0772890	−	0.997009i	$-0.524626\pi$
$331$	−308119.	−0.154578	−0.0772890	+	0.997009i	$0.524626\pi$
$332$	− 237858.i	− 0.118433i
$333$	328838.i	0.162507i
$334$	1.96479e6	0.963716
$335$	0	0
$336$	3.21603e6	1.55408
$337$	766929.i	0.367858i	0.982940	+	0.183929i	$0.0588817\pi$
$337$	766929.i	0.367858i	−0.982940	+	0.183929i	$0.941118\pi$
$338$	166730.i	0.0793819i
$339$	−1.33241e6	−0.629705
$340$	0	0
$341$	−1.89535e6	−0.882682
$342$	− 691721.i	− 0.319790i
$343$	4.67978e6i	2.14778i
$344$	601446.	0.274032
$345$	0	0
$346$	1.69710e6	0.762108
$347$	− 2.86866e6i	− 1.27895i	−0.768810	−	0.639477i	$-0.779151\pi$
$347$	− 2.86866e6i	− 1.27895i	0.768810	−	0.639477i	$-0.220849\pi$
$348$	− 44332.7i	− 0.0196235i
$349$	2.02182e6	0.888546	0.444273	−	0.895891i	$-0.353462\pi$
$349$	2.02182e6	0.888546	0.444273	+	0.895891i	$0.353462\pi$
$350$	0	0
$351$	696921.	0.301936
$352$	− 167827.i	− 0.0721945i
$353$	− 2.28884e6i	− 0.977638i	−0.872385	−	0.488819i	$-0.837428\pi$
$353$	− 2.28884e6i	− 0.977638i	0.872385	−	0.488819i	$-0.162572\pi$
$354$	888271.	0.376736
$355$	0	0
$356$	47173.2	0.0197274
$357$	3.93573e6i	1.63439i
$358$	1.71127e6i	0.705684i
$359$	−989244.	−0.405105	−0.202552	−	0.979271i	$-0.564924\pi$
$359$	−989244.	−0.405105	−0.202552	+	0.979271i	$0.564924\pi$
$360$	0	0
$361$	−283739.	−0.114591
$362$	− 434435.i	− 0.174242i
$363$	1.41896e6i	0.565203i
$364$	−81465.2	−0.0322269
$365$	0	0
$366$	−2.78862e6	−1.08815
$367$	216471.i	0.0838948i	0.999120	+	0.0419474i	$0.0133562\pi$
$367$	216471.i	0.0838948i	−0.999120	+	0.0419474i	$0.986644\pi$
$368$	1.43837e6i	0.553669i
$369$	−801427.	−0.306407
$370$	0	0
$371$	3.92757e6	1.48146
$372$	225126.i	0.0843468i
$373$	− 1.42264e6i	− 0.529448i	−0.964324	−	0.264724i	$-0.914719\pi$
$373$	− 1.42264e6i	− 0.529448i	0.964324	−	0.264724i	$-0.0852808\pi$
$374$	1.73341e6	0.640798
$375$	0	0
$376$	−2.87495e6	−1.04872
$377$	− 282374.i	− 0.102323i
$378$	5.58333e6i	2.00985i
$379$	3.44946e6	1.23354	0.616770	−	0.787144i	$-0.288441\pi$
$379$	3.44946e6	1.23354	0.616770	+	0.787144i	$0.288441\pi$
$380$	0	0
$381$	2.60699e6	0.920084
$382$	2.70935e6i	0.949962i
$383$	− 1.77836e6i	− 0.619474i	−0.950822	−	0.309737i	$-0.899759\pi$
$383$	− 1.77836e6i	− 0.619474i	0.950822	−	0.309737i	$-0.100241\pi$
$384$	2.57038e6	0.889549
$385$	0	0
$386$	1.19098e6	0.406850
$387$	275555.i	0.0935255i
$388$	− 224057.i	− 0.0755579i
$389$	−3.65001e6	−1.22298	−0.611491	−	0.791251i	$-0.709430\pi$
$389$	−3.65001e6	−1.22298	−0.611491	+	0.791251i	$0.709430\pi$
$390$	0	0
$391$	−1.76025e6	−0.582280
$392$	6.46018e6i	2.12339i
$393$	− 2.38468e6i	− 0.778839i
$394$	−4.62578e6	−1.50122
$395$	0	0
$396$	37154.7	0.0119063
$397$	− 2.18400e6i	− 0.695465i	−0.937594	−	0.347733i	$-0.886952\pi$
$397$	− 2.18400e6i	− 0.695465i	0.937594	−	0.347733i	$-0.113048\pi$
$398$	− 3.71043e6i	− 1.17413i
$399$	−4.38401e6	−1.37860
$400$	0	0
$401$	3.39990e6	1.05586	0.527929	−	0.849289i	$-0.322969\pi$
$401$	3.39990e6	1.05586	0.527929	+	0.849289i	$0.322969\pi$
$402$	540619.i	0.166850i
$403$	1.43392e6i	0.439808i
$404$	−407472.	−0.124206
$405$	0	0
$406$	2.26222e6	0.681114
$407$	− 917905.i	− 0.274670i
$408$	2.96410e6i	0.881540i
$409$	4.95079e6	1.46341	0.731705	−	0.681622i	$-0.238725\pi$
$409$	4.95079e6	1.46341	0.731705	+	0.681622i	$0.238725\pi$
$410$	0	0
$411$	2.16564e6	0.632386
$412$	350932.i	0.101855i
$413$	2.76443e6i	0.797498i
$414$	−618641.	−0.177394
$415$	0	0
$416$	−126969.	−0.0359719
$417$	840776.i	0.236777i
$418$	1.93084e6i	0.540512i
$419$	−2.60913e6	−0.726040	−0.363020	−	0.931781i	$-0.618254\pi$
$419$	−2.60913e6	−0.726040	−0.363020	+	0.931781i	$0.618254\pi$
$420$	0	0
$421$	5.93872e6	1.63301	0.816503	−	0.577342i	$-0.195910\pi$
$421$	5.93872e6	1.63301	0.816503	+	0.577342i	$0.195910\pi$
$422$	− 3.44891e6i	− 0.942758i
$423$	− 1.31717e6i	− 0.357923i
$424$	2.95796e6	0.799056
$425$	0	0
$426$	−3.88260e6	−1.03657
$427$	− 8.67859e6i	− 2.30345i
$428$	− 143624.i	− 0.0378980i
$429$	−481943.	−0.126431
$430$	0	0
$431$	−5.74808e6	−1.49049	−0.745247	−	0.666789i	$-0.767668\pi$
$431$	−5.74808e6	−1.49049	−0.745247	+	0.666789i	$0.767668\pi$
$432$	4.47922e6i	1.15476i
$433$	− 2.59160e6i	− 0.664275i	−0.943231	−	0.332137i	$-0.892230\pi$
$433$	− 2.59160e6i	− 0.664275i	0.943231	−	0.332137i	$-0.107770\pi$
$434$	−1.14878e7	−2.92760
$435$	0	0
$436$	65879.8	0.0165972
$437$	− 1.96074e6i	− 0.491153i
$438$	− 6.38323e6i	− 1.58985i
$439$	765476.	0.189570	0.0947851	−	0.995498i	$-0.469784\pi$
$439$	765476.	0.189570	0.0947851	+	0.995498i	$0.469784\pi$
$440$	0	0
$441$	−2.95975e6	−0.724702
$442$	− 1.31140e6i	− 0.319286i
$443$	5.41714e6i	1.31148i	0.754988	+	0.655739i	$0.227643\pi$
$443$	5.41714e6i	1.31148i	−0.754988	+	0.655739i	$0.772357\pi$
$444$	−109027.	−0.0262468
$445$	0	0
$446$	6.94828e6	1.65402
$447$	− 2.44909e6i	− 0.579743i
$448$	7.04424e6i	1.65821i
$449$	1.21210e6	0.283742	0.141871	−	0.989885i	$-0.454688\pi$
$449$	1.21210e6	0.283742	0.141871	+	0.989885i	$0.454688\pi$
$450$	0	0
$451$	2.23707e6	0.517891
$452$	216924.i	0.0499414i
$453$	3.22877e6i	0.739251i
$454$	−5.40485e6	−1.23068
$455$	0	0
$456$	−3.30171e6	−0.743578
$457$	8.01067e6i	1.79423i	0.441796	+	0.897115i	$0.354341\pi$
$457$	8.01067e6i	1.79423i	−0.441796	+	0.897115i	$0.645659\pi$
$458$	5.77896e6i	1.28732i
$459$	−5.48159e6	−1.21444
$460$	0	0
$461$	−4.10263e6	−0.899105	−0.449552	−	0.893254i	$-0.648416\pi$
$461$	−4.10263e6	−0.899105	−0.449552	+	0.893254i	$0.648416\pi$
$462$	− 3.86105e6i	− 0.841590i
$463$	7.69493e6i	1.66822i	0.551601	+	0.834108i	$0.314017\pi$
$463$	7.69493e6i	1.66822i	−0.551601	+	0.834108i	$0.685983\pi$
$464$	1.81486e6	0.391335
$465$	0	0
$466$	−5.29613e6	−1.12978
$467$	1.97041e6i	0.418086i	0.977906	+	0.209043i	$0.0670348\pi$
$467$	1.97041e6i	0.418086i	−0.977906	+	0.209043i	$0.932965\pi$
$468$	− 28109.3i	− 0.00593247i
$469$	−1.68248e6	−0.353198
$470$	0	0
$471$	−3.14246e6	−0.652705
$472$	2.08196e6i	0.430147i
$473$	− 769171.i	− 0.158078i
$474$	−3.36608e6	−0.688143
$475$	0	0
$476$	640760.	0.129622
$477$	1.35520e6i	0.272713i
$478$	4.09894e6i	0.820544i
$479$	−5.97974e6	−1.19081	−0.595407	−	0.803424i	$-0.703009\pi$
$479$	−5.97974e6	−1.19081	−0.595407	+	0.803424i	$0.703009\pi$
$480$	0	0
$481$	−694438.	−0.136858
$482$	3.68810e6i	0.723079i
$483$	3.92084e6i	0.764736i
$484$	231016.	0.0448258
$485$	0	0
$486$	−3.37575e6	−0.648305
$487$	− 3.79919e6i	− 0.725887i	−0.931811	−	0.362943i	$-0.881772\pi$
$487$	− 3.79919e6i	− 0.725887i	0.931811	−	0.362943i	$-0.118228\pi$
$488$	− 6.53607e6i	− 1.24242i
$489$	−6.22649e6	−1.17753
$490$	0	0
$491$	4.33342e6	0.811199	0.405599	−	0.914051i	$-0.367063\pi$
$491$	4.33342e6	0.811199	0.405599	+	0.914051i	$0.367063\pi$
$492$	− 265715.i	− 0.0494883i
$493$	2.22100e6i	0.411558i
$494$	1.46077e6	0.269318
$495$	0	0
$496$	−9.21605e6	−1.68206
$497$	− 1.20832e7i	− 2.19427i
$498$	− 8.52877e6i	− 1.54104i
$499$	2.99219e6	0.537945	0.268972	−	0.963148i	$-0.413316\pi$
$499$	2.99219e6	0.537945	0.268972	+	0.963148i	$0.413316\pi$
$500$	0	0
$501$	4.29669e6	0.764785
$502$	− 3.82681e6i	− 0.677763i
$503$	2.84092e6i	0.500656i	0.968161	+	0.250328i	$0.0805384\pi$
$503$	2.84092e6i	0.500656i	−0.968161	+	0.250328i	$0.919462\pi$
$504$	−3.24203e6	−0.568513
$505$	0	0
$506$	1.72685e6	0.299832
$507$	364613.i	0.0629958i
$508$	− 424434.i	− 0.0729711i
$509$	7.05718e6	1.20736	0.603680	−	0.797227i	$-0.293700\pi$
$509$	7.05718e6	1.20736	0.603680	+	0.797227i	$0.293700\pi$
$510$	0	0
$511$	1.98655e7	3.36548
$512$	5.25523e6i	0.885965i
$513$	− 6.10594e6i	− 1.02438i
$514$	−1.11474e7	−1.86109
$515$	0	0
$516$	−91360.6	−0.0151055
$517$	3.67668e6i	0.604964i
$518$	− 5.56344e6i	− 0.911002i
$519$	3.71129e6	0.604793
$520$	0	0
$521$	−6.86977e6	−1.10879	−0.554393	−	0.832255i	$-0.687050\pi$
$521$	−6.86977e6	−1.10879	−0.554393	+	0.832255i	$0.687050\pi$
$522$	780572.i	0.125382i
$523$	238445.i	0.0381183i	0.999818	+	0.0190592i	$0.00606709\pi$
$523$	238445.i	0.0381183i	−0.999818	+	0.0190592i	$0.993933\pi$
$524$	−388239.	−0.0617691
$525$	0	0
$526$	−5.84780e6	−0.921569
$527$	− 1.12784e7i	− 1.76898i
$528$	− 3.09752e6i	− 0.483537i
$529$	4.68275e6	0.727549
$530$	0	0
$531$	−953857.	−0.146807
$532$	713742.i	0.109336i
$533$	− 1.69245e6i	− 0.258046i
$534$	1.69147e6	0.256692
$535$	0	0
$536$	−1.26712e6	−0.190505
$537$	3.74228e6i	0.560017i
$538$	− 1.73968e6i	− 0.259127i
$539$	8.26173e6	1.22490
$540$	0	0
$541$	−2.33837e6	−0.343495	−0.171747	−	0.985141i	$-0.554941\pi$
$541$	−2.33837e6	−0.343495	−0.171747	+	0.985141i	$0.554941\pi$
$542$	2.94997e6i	0.431340i
$543$	− 950044.i	− 0.138275i
$544$	998665.	0.144685
$545$	0	0
$546$	−2.92107e6	−0.419334
$547$	1.02081e7i	1.45874i	0.684120	+	0.729370i	$0.260187\pi$
$547$	1.02081e7i	1.45874i	−0.684120	+	0.729370i	$0.739813\pi$
$548$	− 352579.i	− 0.0501540i
$549$	2.99452e6	0.424030
$550$	0	0
$551$	−2.47397e6	−0.347148
$552$	2.95289e6i	0.412476i
$553$	− 1.04757e7i	− 1.45670i
$554$	4.13501e6	0.572403
$555$	0	0
$556$	136883.	0.0187786
$557$	− 3.35038e6i	− 0.457568i	−0.973477	−	0.228784i	$-0.926525\pi$
$557$	− 3.35038e6i	− 0.457568i	0.973477	−	0.228784i	$-0.0734749\pi$
$558$	− 3.96382e6i	− 0.538925i
$559$	−581914.	−0.0787643
$560$	0	0
$561$	3.79069e6	0.508524
$562$	8.00193e6i	1.06870i
$563$	4.99891e6i	0.664667i	0.943162	+	0.332333i	$0.107836\pi$
$563$	4.99891e6i	0.664667i	−0.943162	+	0.332333i	$0.892164\pi$
$564$	436709.	0.0578088
$565$	0	0
$566$	−8.56964e6	−1.12440
$567$	7.69967e6i	1.00581i
$568$	− 9.10016e6i	− 1.18353i
$569$	−7.06035e6	−0.914209	−0.457105	−	0.889413i	$-0.651113\pi$
$569$	−7.06035e6	−0.914209	−0.457105	+	0.889413i	$0.651113\pi$
$570$	0	0
$571$	1.21226e7	1.55599	0.777994	−	0.628271i	$-0.216237\pi$
$571$	1.21226e7	1.55599	0.777994	+	0.628271i	$0.216237\pi$
$572$	78463.1i	0.0100271i
$573$	5.92493e6i	0.753870i
$574$	1.35589e7	1.71769
$575$	0	0
$576$	−2.43059e6	−0.305250
$577$	1.26694e7i	1.58422i	0.610380	+	0.792109i	$0.291017\pi$
$577$	1.26694e7i	1.58422i	−0.610380	+	0.792109i	$0.708983\pi$
$578$	2.02611e6i	0.252257i
$579$	2.60448e6	0.322868
$580$	0	0
$581$	2.65427e7	3.26216
$582$	− 8.03395e6i	− 0.983154i
$583$	− 3.78284e6i	− 0.460942i
$584$	1.49612e7	1.81524
$585$	0	0
$586$	−1.13538e7	−1.36583
$587$	1.25878e7i	1.50783i	0.656971	+	0.753916i	$0.271837\pi$
$587$	1.25878e7i	1.50783i	−0.656971	+	0.753916i	$0.728163\pi$
$588$	− 981311.i	− 0.117048i
$589$	1.25631e7	1.49213
$590$	0	0
$591$	−1.01159e7	−1.19134
$592$	− 4.46326e6i	− 0.523417i
$593$	− 1.42337e7i	− 1.66219i	−0.556130	−	0.831096i	$-0.687714\pi$
$593$	− 1.42337e7i	− 1.66219i	0.556130	−	0.831096i	$-0.312286\pi$
$594$	5.37758e6	0.625347
$595$	0	0
$596$	−398726.	−0.0459789
$597$	− 8.11415e6i	− 0.931767i
$598$	− 1.30644e6i	− 0.149395i
$599$	−3.07901e6	−0.350626	−0.175313	−	0.984513i	$-0.556094\pi$
$599$	−3.07901e6	−0.350626	−0.175313	+	0.984513i	$0.556094\pi$
$600$	0	0
$601$	4.32299e6	0.488200	0.244100	−	0.969750i	$-0.421507\pi$
$601$	4.32299e6	0.488200	0.244100	+	0.969750i	$0.421507\pi$
$602$	− 4.66197e6i	− 0.524298i
$603$	− 580535.i	− 0.0650182i
$604$	525663.	0.0586294
$605$	0	0
$606$	−1.46106e7	−1.61616
$607$	− 1.65670e7i	− 1.82503i	−0.409038	−	0.912517i	$-0.634136\pi$
$607$	− 1.65670e7i	− 1.82503i	0.409038	−	0.912517i	$-0.365864\pi$
$608$	1.11241e6i	0.122041i
$609$	4.94713e6	0.540518
$610$	0	0
$611$	2.78158e6	0.301432
$612$	221092.i	0.0238613i
$613$	7.82552e6i	0.841128i	0.907263	+	0.420564i	$0.138168\pi$
$613$	7.82552e6i	0.841128i	−0.907263	+	0.420564i	$0.861832\pi$
$614$	−1.11035e7	−1.18861
$615$	0	0
$616$	9.04966e6	0.960905
$617$	1.93061e6i	0.204165i	0.994776	+	0.102083i	$0.0325506\pi$
$617$	1.93061e6i	0.204165i	−0.994776	+	0.102083i	$0.967449\pi$
$618$	1.25833e7i	1.32532i
$619$	−1.06583e7	−1.11805	−0.559024	−	0.829151i	$-0.688824\pi$
$619$	−1.06583e7	−1.11805	−0.559024	+	0.829151i	$0.688824\pi$
$620$	0	0
$621$	−5.46086e6	−0.568240
$622$	− 9.28984e6i	− 0.962792i
$623$	5.26411e6i	0.543381i
$624$	−2.34342e6	−0.240929
$625$	0	0
$626$	1.80464e7	1.84058
$627$	4.22245e6i	0.428939i
$628$	511610.i	0.0517655i
$629$	5.46207e6	0.550466
$630$	0	0
$631$	1.46286e7	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$631$	1.46286e7	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$632$	− 7.88953e6i	− 0.785703i
$633$	− 7.54224e6i	− 0.748154i
$634$	−5.56775e6	−0.550119
$635$	0	0
$636$	−449318.	−0.0440464
$637$	− 6.25039e6i	− 0.610321i
$638$	− 2.17886e6i	− 0.211922i
$639$	4.16927e6	0.403931
$640$	0	0
$641$	7.89850e6	0.759275	0.379638	−	0.925135i	$-0.376049\pi$
$641$	7.89850e6	0.759275	0.379638	+	0.925135i	$0.376049\pi$
$642$	− 5.14986e6i	− 0.493126i
$643$	9.26854e6i	0.884064i	0.896999	+	0.442032i	$0.145742\pi$
$643$	9.26854e6i	0.884064i	−0.896999	+	0.442032i	$0.854258\pi$
$644$	638336.	0.0606505
$645$	0	0
$646$	−1.14896e7	−1.08324
$647$	− 6.76009e6i	− 0.634880i	−0.948278	−	0.317440i	$-0.897177\pi$
$647$	− 6.76009e6i	− 0.634880i	0.948278	−	0.317440i	$-0.102823\pi$
$648$	5.79881e6i	0.542502i
$649$	2.66256e6	0.248134
$650$	0	0
$651$	−2.51220e7	−2.32328
$652$	1.01371e6i	0.0933887i
$653$	− 1.61068e7i	− 1.47818i	−0.673609	−	0.739088i	$-0.735257\pi$
$653$	− 1.61068e7i	− 1.47818i	0.673609	−	0.739088i	$-0.264743\pi$
$654$	2.36223e6	0.215962
$655$	0	0
$656$	1.08776e7	0.986904
$657$	6.85453e6i	0.619533i
$658$	2.22845e7i	2.00649i
$659$	9.81974e6	0.880818	0.440409	−	0.897797i	$-0.354833\pi$
$659$	9.81974e6	0.880818	0.440409	+	0.897797i	$0.354833\pi$
$660$	0	0
$661$	−1.14240e7	−1.01699	−0.508494	−	0.861066i	$-0.669797\pi$
$661$	−1.14240e7	−1.01699	−0.508494	+	0.861066i	$0.669797\pi$
$662$	1.79869e6i	0.159519i
$663$	− 2.86784e6i	− 0.253379i
$664$	1.99900e7	1.75952
$665$	0	0
$666$	1.91965e6	0.167701
$667$	2.21260e6i	0.192570i
$668$	− 699526.i	− 0.0606545i
$669$	1.51948e7	1.31259
$670$	0	0
$671$	−8.35878e6	−0.716699
$672$	− 2.22446e6i	− 0.190021i
$673$	− 1.41804e7i	− 1.20684i	−0.797422	−	0.603422i	$-0.793803\pi$
$673$	− 1.41804e7i	− 1.20684i	0.797422	−	0.603422i	$-0.206197\pi$
$674$	4.47708e6	0.379616
$675$	0	0
$676$	59361.0	0.00499614
$677$	− 1.05474e7i	− 0.884455i	−0.896903	−	0.442228i	$-0.854188\pi$
$677$	− 1.05474e7i	− 0.884455i	0.896903	−	0.442228i	$-0.145812\pi$
$678$	7.77815e6i	0.649833i
$679$	2.50028e7	2.08120
$680$	0	0
$681$	−1.18196e7	−0.976641
$682$	1.10644e7i	0.910896i
$683$	− 8.95552e6i	− 0.734580i	−0.930107	−	0.367290i	$-0.880286\pi$
$683$	− 8.95552e6i	− 0.734580i	0.930107	−	0.367290i	$-0.119714\pi$
$684$	−246274.	−0.0201270
$685$	0	0
$686$	2.73190e7	2.21644
$687$	1.26377e7i	1.02159i
$688$	− 3.74005e6i	− 0.301236i
$689$	−2.86190e6	−0.229671
$690$	0	0
$691$	−614647.	−0.0489700	−0.0244850	−	0.999700i	$-0.507795\pi$
$691$	−614647.	−0.0489700	−0.0244850	+	0.999700i	$0.507795\pi$
$692$	− 604220.i	− 0.0479656i
$693$	4.14613e6i	0.327952i
$694$	−1.67463e7	−1.31984
$695$	0	0
$696$	3.72581e6	0.291540
$697$	1.33119e7i	1.03790i
$698$	− 1.18027e7i	− 0.916947i
$699$	−1.15818e7	−0.896570
$700$	0	0
$701$	1.38233e6	0.106247	0.0531235	−	0.998588i	$-0.483082\pi$
$701$	1.38233e6	0.106247	0.0531235	+	0.998588i	$0.483082\pi$
$702$	− 4.06839e6i	− 0.311588i
$703$	6.08419e6i	0.464317i
$704$	6.78465e6	0.515936
$705$	0	0
$706$	−1.33615e7	−1.00889
$707$	− 4.54702e7i	− 3.42120i
$708$	− 316253.i	− 0.0237111i
$709$	−8.21660e6	−0.613870	−0.306935	−	0.951730i	$-0.599303\pi$
$709$	−8.21660e6	−0.613870	−0.306935	+	0.951730i	$0.599303\pi$
$710$	0	0
$711$	3.61461e6	0.268156
$712$	3.96453e6i	0.293084i
$713$	− 1.12358e7i	− 0.827713i
$714$	2.29755e7	1.68663
$715$	0	0
$716$	609266.	0.0444144
$717$	8.96376e6i	0.651167i
$718$	5.77488e6i	0.418054i
$719$	−2.34126e7	−1.68899	−0.844496	−	0.535562i	$-0.820100\pi$
$719$	−2.34126e7	−1.68899	−0.844496	+	0.535562i	$0.820100\pi$
$720$	0	0
$721$	−3.91609e7	−2.80552
$722$	1.65638e6i	0.118254i
$723$	8.06532e6i	0.573821i
$724$	−154673.	−0.0109665
$725$	0	0
$726$	8.28344e6	0.583269
$727$	1.33689e7i	0.938122i	0.883166	+	0.469061i	$0.155408\pi$
$727$	1.33689e7i	0.938122i	−0.883166	+	0.469061i	$0.844592\pi$
$728$	− 6.84649e6i	− 0.478784i
$729$	−1.54494e7	−1.07670
$730$	0	0
$731$	4.57701e6	0.316803
$732$	992839.i	0.0684859i
$733$	− 1.68737e7i	− 1.15998i	−0.814625	−	0.579988i	$-0.803057\pi$
$733$	− 1.68737e7i	− 1.15998i	0.814625	−	0.579988i	$-0.196943\pi$
$734$	1.26369e6	0.0865764
$735$	0	0
$736$	994888.	0.0676986
$737$	1.62048e6i	0.109894i
$738$	4.67847e6i	0.316201i
$739$	7.20516e6	0.485325	0.242663	−	0.970111i	$-0.421979\pi$
$739$	7.20516e6	0.485325	0.242663	+	0.970111i	$0.421979\pi$
$740$	0	0
$741$	3.19448e6	0.213725
$742$	− 2.29279e7i	− 1.52881i
$743$	− 2.32904e7i	− 1.54777i	−0.633328	−	0.773883i	$-0.718312\pi$
$743$	− 2.32904e7i	− 1.54777i	0.633328	−	0.773883i	$-0.281688\pi$
$744$	−1.89200e7	−1.25311
$745$	0	0
$746$	−8.30490e6	−0.546371
$747$	9.15849e6i	0.600513i
$748$	− 617147.i	− 0.0403306i
$749$	1.60271e7	1.04388
$750$	0	0
$751$	2.64679e7	1.71246	0.856228	−	0.516599i	$-0.172802\pi$
$751$	2.64679e7	1.71246	0.856228	+	0.516599i	$0.172802\pi$
$752$	1.78777e7i	1.15283i
$753$	− 8.36866e6i	− 0.537859i
$754$	−1.64841e6	−0.105593
$755$	0	0
$756$	1.98784e6	0.126496
$757$	− 1.34842e7i	− 0.855234i	−0.903960	−	0.427617i	$-0.859353\pi$
$757$	− 1.34842e7i	− 0.855234i	0.903960	−	0.427617i	$-0.140647\pi$
$758$	− 2.01368e7i	− 1.27297i
$759$	3.77636e6	0.237941
$760$	0	0
$761$	5.97967e6	0.374296	0.187148	−	0.982332i	$-0.440076\pi$
$761$	5.97967e6	0.374296	0.187148	+	0.982332i	$0.440076\pi$
$762$	− 1.52188e7i	− 0.949494i
$763$	7.35159e6i	0.457162i
$764$	964613.	0.0597888
$765$	0	0
$766$	−1.03815e7	−0.639275
$767$	− 2.01435e6i	− 0.123636i
$768$	− 2.59753e6i	− 0.158913i
$769$	−1.49826e7	−0.913634	−0.456817	−	0.889561i	$-0.651011\pi$
$769$	−1.49826e7	−0.913634	−0.456817	+	0.889561i	$0.651011\pi$
$770$	0	0
$771$	−2.43777e7	−1.47692
$772$	− 424025.i	− 0.0256064i
$773$	2.52693e7i	1.52106i	0.649305	+	0.760528i	$0.275060\pi$
$773$	2.52693e7i	1.52106i	−0.649305	+	0.760528i	$0.724940\pi$
$774$	1.60860e6	0.0965150
$775$	0	0
$776$	1.88302e7	1.12254
$777$	− 1.21664e7i	− 0.722952i
$778$	2.13076e7i	1.26207i
$779$	−1.48281e7	−0.875470
$780$	0	0
$781$	−1.16379e7	−0.682728
$782$	1.02758e7i	0.600892i
$783$	6.89025e6i	0.401634i
$784$	4.01722e7	2.33419
$785$	0	0
$786$	−1.39209e7	−0.803734
$787$	− 2.44969e7i	− 1.40985i	−0.709280	−	0.704927i	$-0.750980\pi$
$787$	− 2.44969e7i	− 1.40985i	0.709280	−	0.704927i	$-0.249020\pi$
$788$	1.64692e6i	0.0944840i
$789$	−1.27882e7	−0.731339
$790$	0	0
$791$	−2.42067e7	−1.37561
$792$	3.12256e6i	0.176888i
$793$	6.32381e6i	0.357105i
$794$	−1.27494e7	−0.717695
$795$	0	0
$796$	−1.32103e6	−0.0738976
$797$	1.45189e7i	0.809633i	0.914398	+	0.404817i	$0.132665\pi$
$797$	1.45189e7i	0.809633i	−0.914398	+	0.404817i	$0.867335\pi$
$798$	2.55924e7i	1.42267i
$799$	−2.18784e7	−1.21241
$800$	0	0
$801$	−1.81636e6	−0.100028
$802$	− 1.98475e7i	− 1.08961i
$803$	− 1.91335e7i	− 1.04714i
$804$	192477.	0.0105012
$805$	0	0
$806$	8.37078e6	0.453866
$807$	− 3.80441e6i	− 0.205638i
$808$	− 3.42447e7i	− 1.84529i
$809$	2.54949e6	0.136956	0.0684781	−	0.997653i	$-0.478186\pi$
$809$	2.54949e6	0.136956	0.0684781	+	0.997653i	$0.478186\pi$
$810$	0	0
$811$	−2.28468e7	−1.21976	−0.609880	−	0.792494i	$-0.708782\pi$
$811$	−2.28468e7	−1.21976	−0.609880	+	0.792494i	$0.708782\pi$
$812$	− 805422.i	− 0.0428680i
$813$	6.45114e6i	0.342302i
$814$	−5.35843e6	−0.283450
$815$	0	0
$816$	1.84320e7	0.969054
$817$	5.09833e6i	0.267223i
$818$	− 2.89011e7i	− 1.51019i
$819$	3.13674e6	0.163406
$820$	0	0
$821$	−1.97995e7	−1.02517	−0.512584	−	0.858637i	$-0.671312\pi$
$821$	−1.97995e7	−1.02517	−0.512584	+	0.858637i	$0.671312\pi$
$822$	− 1.26423e7i	− 0.652599i
$823$	2.65781e7i	1.36781i	0.729572	+	0.683904i	$0.239719\pi$
$823$	2.65781e7i	1.36781i	−0.729572	+	0.683904i	$0.760281\pi$
$824$	−2.94930e7	−1.51322
$825$	0	0
$826$	1.61378e7	0.822990
$827$	2.68344e7i	1.36436i	0.731186	+	0.682178i	$0.238967\pi$
$827$	2.68344e7i	1.36436i	−0.731186	+	0.682178i	$0.761033\pi$
$828$	220256.i	0.0111648i
$829$	−5.90654e6	−0.298502	−0.149251	−	0.988799i	$-0.547686\pi$
$829$	−5.90654e6	−0.298502	−0.149251	+	0.988799i	$0.547686\pi$
$830$	0	0
$831$	9.04263e6	0.454247
$832$	− 5.13291e6i	− 0.257072i
$833$	4.91621e7i	2.45481i
$834$	4.90817e6	0.244346
$835$	0	0
$836$	687440.	0.0340188
$837$	− 3.49894e7i	− 1.72632i
$838$	1.52312e7i	0.749247i
$839$	−2.72137e7	−1.33470	−0.667350	−	0.744745i	$-0.732571\pi$
$839$	−2.72137e7	−1.33470	−0.667350	+	0.744745i	$0.732571\pi$
$840$	0	0
$841$	−1.77194e7	−0.863891
$842$	− 3.46683e7i	− 1.68520i
$843$	1.74990e7i	0.848095i
$844$	−1.22792e6	−0.0593354
$845$	0	0
$846$	−7.68918e6	−0.369364
$847$	2.57792e7i	1.23470i
$848$	− 1.83938e7i	− 0.878381i
$849$	−1.87405e7	−0.892302
$850$	0	0
$851$	5.44140e6	0.257565
$852$	1.38233e6i	0.0652397i
$853$	2.41385e7i	1.13589i	0.823066	+	0.567946i	$0.192262\pi$
$853$	2.41385e7i	1.13589i	−0.823066	+	0.567946i	$0.807738\pi$
$854$	−5.06628e7	−2.37708
$855$	0	0
$856$	1.20704e7	0.563038
$857$	4.27143e7i	1.98665i	0.115351	+	0.993325i	$0.463201\pi$
$857$	4.27143e7i	1.98665i	−0.115351	+	0.993325i	$0.536799\pi$
$858$	2.81342e6i	0.130472i
$859$	1.60393e7	0.741657	0.370828	−	0.928701i	$-0.379074\pi$
$859$	1.60393e7	0.741657	0.370828	+	0.928701i	$0.379074\pi$
$860$	0	0
$861$	2.96513e7	1.36313
$862$	3.35554e7i	1.53814i
$863$	2.63477e7i	1.20425i	0.798403	+	0.602124i	$0.205679\pi$
$863$	2.63477e7i	1.20425i	−0.798403	+	0.602124i	$0.794321\pi$
$864$	3.09818e6	0.141196
$865$	0	0
$866$	−1.51289e7	−0.685508
$867$	4.43080e6i	0.200186i
$868$	4.09001e6i	0.184258i
$869$	−1.00897e7	−0.453240
$870$	0	0
$871$	1.22597e6	0.0547564
$872$	5.53667e6i	0.246580i
$873$	8.62713e6i	0.383117i
$874$	−1.14462e7	−0.506852
$875$	0	0
$876$	−2.27263e6	−0.100062
$877$	− 2.76379e6i	− 0.121341i	−0.998158	−	0.0606704i	$-0.980676\pi$
$877$	− 2.76379e6i	− 0.121341i	0.998158	−	0.0606704i	$-0.0193238\pi$
$878$	− 4.46860e6i	− 0.195630i
$879$	−2.48289e7	−1.08389
$880$	0	0
$881$	−2.52826e7	−1.09744	−0.548722	−	0.836005i	$-0.684886\pi$
$881$	−2.52826e7	−1.09744	−0.548722	+	0.836005i	$0.684886\pi$
$882$	1.72781e7i	0.747866i
$883$	− 2.46984e7i	− 1.06602i	−0.846108	−	0.533012i	$-0.821060\pi$
$883$	− 2.46984e7i	− 1.06602i	0.846108	−	0.533012i	$-0.178940\pi$
$884$	−466901.	−0.0200953
$885$	0	0
$886$	3.16235e7	1.35340
$887$	2.74349e7i	1.17083i	0.810734	+	0.585415i	$0.199068\pi$
$887$	2.74349e7i	1.17083i	−0.810734	+	0.585415i	$0.800932\pi$
$888$	− 9.16283e6i	− 0.389939i
$889$	4.73630e7	2.00995
$890$	0	0
$891$	7.41593e6	0.312947
$892$	− 2.47381e6i	− 0.104101i
$893$	− 2.43703e7i	− 1.02266i
$894$	−1.42970e7	−0.598274
$895$	0	0
$896$	4.66979e7	1.94324
$897$	− 2.85699e6i	− 0.118557i
$898$	− 7.07587e6i	− 0.292812i
$899$	−1.41768e7	−0.585030
$900$	0	0
$901$	2.25101e7	0.923773
$902$	− 1.30593e7i	− 0.534445i
$903$	− 1.01950e7i	− 0.416072i
$904$	−1.82307e7	−0.741962
$905$	0	0
$906$	1.88485e7	0.762881
$907$	3.72957e7i	1.50536i	0.658386	+	0.752681i	$0.271240\pi$
$907$	3.72957e7i	1.50536i	−0.658386	+	0.752681i	$0.728760\pi$
$908$	1.92430e6i	0.0774565i
$909$	1.56893e7	0.629789
$910$	0	0
$911$	4.32029e7	1.72471	0.862356	−	0.506302i	$-0.168988\pi$
$911$	4.32029e7	1.72471	0.862356	+	0.506302i	$0.168988\pi$
$912$	2.05314e7i	0.817396i
$913$	− 2.55646e7i	− 1.01499i
$914$	4.67637e7	1.85158
$915$	0	0
$916$	2.05749e6	0.0810214
$917$	− 4.33240e7i	− 1.70139i
$918$	3.19997e7i	1.25326i
$919$	−2.00349e7	−0.782526	−0.391263	−	0.920279i	$-0.627962\pi$
$919$	−2.00349e7	−0.782526	−0.391263	+	0.920279i	$0.627962\pi$
$920$	0	0
$921$	−2.42816e7	−0.943254
$922$	2.39498e7i	0.927844i
$923$	8.80463e6i	0.340179i
$924$	−1.37466e6	−0.0529681
$925$	0	0
$926$	4.49205e7	1.72154
$927$	− 1.35123e7i	− 0.516453i
$928$	− 1.25530e6i	− 0.0478496i
$929$	2.77906e7	1.05647	0.528237	−	0.849097i	$-0.322853\pi$
$929$	2.77906e7	1.05647	0.528237	+	0.849097i	$0.322853\pi$
$930$	0	0
$931$	−5.47616e7	−2.07063
$932$	1.88559e6i	0.0711062i
$933$	− 2.03155e7i	− 0.764052i
$934$	1.15026e7	0.431449
$935$	0	0
$936$	2.36236e6	0.0881367
$937$	− 296766.i	− 0.0110425i	−0.999985	−	0.00552123i	$-0.998243\pi$
$937$	− 296766.i	− 0.0110425i	0.999985	−	0.00552123i	$-0.00175747\pi$
$938$	9.82177e6i	0.364488i
$939$	3.94647e7	1.46065
$940$	0	0
$941$	−1.24096e7	−0.456861	−0.228431	−	0.973560i	$-0.573359\pi$
$941$	−1.24096e7	−0.456861	−0.228431	+	0.973560i	$0.573359\pi$
$942$	1.83446e7i	0.673568i
$943$	1.32615e7i	0.485639i
$944$	1.29465e7	0.472850
$945$	0	0
$946$	−4.49017e6	−0.163130
$947$	2.07818e7i	0.753021i	0.926412	+	0.376511i	$0.122876\pi$
$947$	2.07818e7i	0.753021i	−0.926412	+	0.376511i	$0.877124\pi$
$948$	1.19843e6i	0.0433104i
$949$	−1.44754e7	−0.521752
$950$	0	0
$951$	−1.21758e7	−0.436563
$952$	5.38507e7i	1.92575i
$953$	− 4.37556e7i	− 1.56064i	−0.625383	−	0.780318i	$-0.715057\pi$
$953$	− 4.37556e7i	− 1.56064i	0.625383	−	0.780318i	$-0.284943\pi$
$954$	7.91119e6	0.281430
$955$	0	0
$956$	1.45935e6	0.0516435
$957$	− 4.76483e6i	− 0.168177i
$958$	3.49078e7i	1.22888i
$959$	3.93446e7	1.38146
$960$	0	0
$961$	4.33620e7	1.51461
$962$	4.05390e6i	0.141233i
$963$	5.53010e6i	0.192162i
$964$	1.31308e6	0.0455092
$965$	0	0
$966$	2.28886e7	0.789180
$967$	2.96578e7i	1.01994i	0.860193	+	0.509968i	$0.170343\pi$
$967$	2.96578e7i	1.01994i	−0.860193	+	0.509968i	$0.829657\pi$
$968$	1.94150e7i	0.665961i
$969$	−2.51260e7	−0.859636
$970$	0	0
$971$	2.43354e7	0.828304	0.414152	−	0.910208i	$-0.364078\pi$
$971$	2.43354e7	0.828304	0.414152	+	0.910208i	$0.364078\pi$
$972$	1.20187e6i	0.0408031i
$973$	1.52749e7i	0.517246i
$974$	−2.21784e7	−0.749089
$975$	0	0
$976$	−4.06441e7	−1.36576
$977$	− 1.45469e7i	− 0.487566i	−0.969830	−	0.243783i	$-0.921612\pi$
$977$	− 1.45469e7i	− 0.487566i	0.969830	−	0.243783i	$-0.0783884\pi$
$978$	3.63482e7i	1.21517i
$979$	5.07012e6	0.169068
$980$	0	0
$981$	−2.53664e6	−0.0841563
$982$	− 2.52971e7i	− 0.837128i
$983$	8.32905e6i	0.274924i	0.990507	+	0.137462i	$0.0438944\pi$
$983$	8.32905e6i	0.274924i	−0.990507	+	0.137462i	$0.956106\pi$
$984$	2.23312e7	0.735231
$985$	0	0
$986$	1.29655e7	0.424713
$987$	4.87327e7i	1.59231i
$988$	− 520081.i	− 0.0169503i
$989$	4.55970e6	0.148233
$990$	0	0
$991$	−1.94665e6	−0.0629657	−0.0314828	−	0.999504i	$-0.510023\pi$
$991$	−1.94665e6	−0.0629657	−0.0314828	+	0.999504i	$0.510023\pi$
$992$	6.37455e6i	0.205670i
$993$	3.93347e6i	0.126591i
$994$	−7.05377e7	−2.26441
$995$	0	0
$996$	−3.03651e6	−0.0969900
$997$	728335.i	0.0232056i	0.999933	+	0.0116028i	$0.00369337\pi$
$997$	728335.i	0.0232056i	−0.999933	+	0.0116028i	$0.996307\pi$
$998$	− 1.74674e7i	− 0.555140i
$999$	1.69451e7	0.537192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.7		22
5.2	odd	4		325.6.a.k.1.8	yes	11
5.3	odd	4		325.6.a.j.1.4	✓	11
5.4	even	2	inner	325.6.b.i.274.16		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.4	✓	11	5.3	odd	4
325.6.a.k.1.8	yes	11	5.2	odd	4
325.6.b.i.274.7		22	1.1	even	1	trivial
325.6.b.i.274.16		22	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.83767i q^{2} -12.7661i q^{3} -2.07840 q^{4} -74.5243 q^{6} -231.930i q^{7} -174.672i q^{8} +80.0268 q^{9} +O(q^{10})\)	\(q-5.83767i q^{2} -12.7661i q^{3} -2.07840 q^{4} -74.5243 q^{6} -231.930i q^{7} -174.672i q^{8} +80.0268 q^{9} -223.383 q^{11} +26.5330i q^{12} +169.000i q^{13} -1353.93 q^{14} -1086.19 q^{16} -1329.26i q^{17} -467.170i q^{18} +1480.66 q^{19} -2960.84 q^{21} +1304.04i q^{22} -1324.23i q^{23} -2229.89 q^{24} +986.566 q^{26} -4123.79i q^{27} +482.042i q^{28} -1670.85 q^{29} +8484.76 q^{31} +751.294i q^{32} +2851.73i q^{33} -7759.78 q^{34} -166.327 q^{36} +4109.10i q^{37} -8643.62i q^{38} +2157.47 q^{39} -10014.5 q^{41} +17284.4i q^{42} +3443.28i q^{43} +464.279 q^{44} -7730.43 q^{46} -16459.1i q^{47} +13866.4i q^{48} -36984.6 q^{49} -16969.5 q^{51} -351.249i q^{52} +16934.3i q^{53} -24073.3 q^{54} -40511.8 q^{56} -18902.3i q^{57} +9753.89i q^{58} -11919.2 q^{59} +37419.0 q^{61} -49531.2i q^{62} -18560.6i q^{63} -30372.2 q^{64} +16647.5 q^{66} -7254.26i q^{67} +2762.73i q^{68} -16905.3 q^{69} +52098.4 q^{71} -13978.5i q^{72} +85653.0i q^{73} +23987.6 q^{74} -3077.40 q^{76} +51809.3i q^{77} -12594.6i q^{78} +45167.6 q^{79} -33198.2 q^{81} +58461.3i q^{82} +114443. i q^{83} +6153.80 q^{84} +20100.7 q^{86} +21330.3i q^{87} +39018.9i q^{88} -22697.0 q^{89} +39196.2 q^{91} +2752.28i q^{92} -108317. i q^{93} -96082.7 q^{94} +9591.09 q^{96} +107803. i q^{97} +215904. i q^{98} -17876.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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