LMFDB - Embedded newform 825.6.a.y.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + \cdots + 522579400 $ x^13 - 306x^11 - 206x^10 + 34574x^9 + 39928x^8 - 1788312x^7 - 2591628x^6 + 42852537x^5 + 63733360x^4 - 448113518x^3 - 549984598x^2 + 1518551280*x + 522579400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-4.09558$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.09558 q^{2} +9.00000 q^{3} -6.03505 q^{4} +45.8602 q^{6} +170.277 q^{7} -193.811 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+5.09558 q^{2} +9.00000 q^{3} -6.03505 q^{4} +45.8602 q^{6} +170.277 q^{7} -193.811 q^{8} +81.0000 q^{9} +121.000 q^{11} -54.3155 q^{12} +364.335 q^{13} +867.659 q^{14} -794.456 q^{16} +1307.63 q^{17} +412.742 q^{18} +619.655 q^{19} +1532.49 q^{21} +616.565 q^{22} -922.189 q^{23} -1744.30 q^{24} +1856.50 q^{26} +729.000 q^{27} -1027.63 q^{28} -3433.99 q^{29} +1479.05 q^{31} +2153.73 q^{32} +1089.00 q^{33} +6663.11 q^{34} -488.839 q^{36} +5289.91 q^{37} +3157.50 q^{38} +3279.02 q^{39} -7669.49 q^{41} +7808.93 q^{42} -17366.5 q^{43} -730.242 q^{44} -4699.09 q^{46} -6946.36 q^{47} -7150.11 q^{48} +12187.2 q^{49} +11768.6 q^{51} -2198.78 q^{52} +17379.4 q^{53} +3714.68 q^{54} -33001.5 q^{56} +5576.90 q^{57} -17498.2 q^{58} +34047.9 q^{59} +48554.5 q^{61} +7536.59 q^{62} +13792.4 q^{63} +36397.1 q^{64} +5549.09 q^{66} +34368.8 q^{67} -7891.59 q^{68} -8299.70 q^{69} +48448.3 q^{71} -15698.7 q^{72} -59439.7 q^{73} +26955.1 q^{74} -3739.65 q^{76} +20603.5 q^{77} +16708.5 q^{78} +3591.22 q^{79} +6561.00 q^{81} -39080.5 q^{82} +64310.2 q^{83} -9248.67 q^{84} -88492.6 q^{86} -30905.9 q^{87} -23451.1 q^{88} +26956.3 q^{89} +62037.9 q^{91} +5565.46 q^{92} +13311.4 q^{93} -35395.8 q^{94} +19383.5 q^{96} +16545.8 q^{97} +62100.8 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	5.09558	0.900780	0.450390	−	0.892832i	$-0.351285\pi$
$2$	5.09558	0.900780	0.450390	+	0.892832i	$0.351285\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−6.03505	−0.188595
$5$	0	0
$5$	0	0
$6$	45.8602	0.520066
$7$	170.277	1.31344	0.656720	−	0.754134i	$-0.271943\pi$
$7$	170.277	1.31344	0.656720	+	0.754134i	$0.271943\pi$
$8$	−193.811	−1.07066
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−54.3155	−0.108886
$13$	364.335	0.597920	0.298960	−	0.954266i	$-0.403360\pi$
$13$	364.335	0.597920	0.298960	+	0.954266i	$0.403360\pi$
$14$	867.659	1.18312
$15$	0	0
$16$	−794.456	−0.775836
$17$	1307.63	1.09739	0.548695	−	0.836023i	$-0.315125\pi$
$17$	1307.63	1.09739	0.548695	+	0.836023i	$0.315125\pi$
$18$	412.742	0.300260
$19$	619.655	0.393791	0.196896	−	0.980424i	$-0.436914\pi$
$19$	619.655	0.393791	0.196896	+	0.980424i	$0.436914\pi$
$20$	0	0
$21$	1532.49	0.758315
$22$	616.565	0.271595
$23$	−922.189	−0.363497	−0.181748	−	0.983345i	$-0.558176\pi$
$23$	−922.189	−0.363497	−0.181748	+	0.983345i	$0.558176\pi$
$24$	−1744.30	−0.618148
$25$	0	0
$26$	1856.50	0.538594
$27$	729.000	0.192450
$28$	−1027.63	−0.247709
$29$	−3433.99	−0.758235	−0.379118	−	0.925349i	$-0.623772\pi$
$29$	−3433.99	−0.758235	−0.379118	+	0.925349i	$0.623772\pi$
$30$	0	0
$31$	1479.05	0.276425	0.138212	−	0.990403i	$-0.455864\pi$
$31$	1479.05	0.276425	0.138212	+	0.990403i	$0.455864\pi$
$32$	2153.73	0.371805
$33$	1089.00	0.174078
$34$	6663.11	0.988507
$35$	0	0
$36$	−488.839	−0.0628651
$37$	5289.91	0.635249	0.317624	−	0.948217i	$-0.397115\pi$
$37$	5289.91	0.635249	0.317624	+	0.948217i	$0.397115\pi$
$38$	3157.50	0.354719
$39$	3279.02	0.345209
$40$	0	0
$41$	−7669.49	−0.712536	−0.356268	−	0.934384i	$-0.615951\pi$
$41$	−7669.49	−0.712536	−0.356268	+	0.934384i	$0.615951\pi$
$42$	7808.93	0.683075
$43$	−17366.5	−1.43233	−0.716164	−	0.697933i	$-0.754104\pi$
$43$	−17366.5	−1.43233	−0.716164	+	0.697933i	$0.754104\pi$
$44$	−730.242	−0.0568637
$45$	0	0
$46$	−4699.09	−0.327430
$47$	−6946.36	−0.458683	−0.229342	−	0.973346i	$-0.573657\pi$
$47$	−6946.36	−0.458683	−0.229342	+	0.973346i	$0.573657\pi$
$48$	−7150.11	−0.447929
$49$	12187.2	0.725126
$50$	0	0
$51$	11768.6	0.633578
$52$	−2198.78	−0.112765
$53$	17379.4	0.849857	0.424929	−	0.905227i	$-0.360299\pi$
$53$	17379.4	0.849857	0.424929	+	0.905227i	$0.360299\pi$
$54$	3714.68	0.173355
$55$	0	0
$56$	−33001.5	−1.40625
$57$	5576.90	0.227356
$58$	−17498.2	−0.683003
$59$	34047.9	1.27339	0.636693	−	0.771117i	$-0.280302\pi$
$59$	34047.9	1.27339	0.636693	+	0.771117i	$0.280302\pi$
$60$	0	0
$61$	48554.5	1.67073	0.835363	−	0.549699i	$-0.185258\pi$
$61$	48554.5	1.67073	0.835363	+	0.549699i	$0.185258\pi$
$62$	7536.59	0.248998
$63$	13792.4	0.437813
$64$	36397.1	1.11075
$65$	0	0
$66$	5549.09	0.156806
$67$	34368.8	0.935358	0.467679	−	0.883898i	$-0.345090\pi$
$67$	34368.8	0.935358	0.467679	+	0.883898i	$0.345090\pi$
$68$	−7891.59	−0.206963
$69$	−8299.70	−0.209865
$70$	0	0
$71$	48448.3	1.14060	0.570299	−	0.821437i	$-0.306827\pi$
$71$	48448.3	1.14060	0.570299	+	0.821437i	$0.306827\pi$
$72$	−15698.7	−0.356888
$73$	−59439.7	−1.30548	−0.652739	−	0.757583i	$-0.726380\pi$
$73$	−59439.7	−1.30548	−0.652739	+	0.757583i	$0.726380\pi$
$74$	26955.1	0.572219
$75$	0	0
$76$	−3739.65	−0.0742673
$77$	20603.5	0.396017
$78$	16708.5	0.310957
$79$	3591.22	0.0647402	0.0323701	−	0.999476i	$-0.489694\pi$
$79$	3591.22	0.0647402	0.0323701	+	0.999476i	$0.489694\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−39080.5	−0.641838
$83$	64310.2	1.02467	0.512336	−	0.858785i	$-0.328780\pi$
$83$	64310.2	1.02467	0.512336	+	0.858785i	$0.328780\pi$
$84$	−9248.67	−0.143015
$85$	0	0
$86$	−88492.6	−1.29021
$87$	−30905.9	−0.437767
$88$	−23451.1	−0.322817
$89$	26956.3	0.360732	0.180366	−	0.983600i	$-0.442272\pi$
$89$	26956.3	0.360732	0.180366	+	0.983600i	$0.442272\pi$
$90$	0	0
$91$	62037.9	0.785332
$92$	5565.46	0.0685538
$93$	13311.4	0.159594
$94$	−35395.8	−0.413173
$95$	0	0
$96$	19383.5	0.214662
$97$	16545.8	0.178549	0.0892745	−	0.996007i	$-0.471545\pi$
$97$	16545.8	0.178549	0.0892745	+	0.996007i	$0.471545\pi$
$98$	62100.8	0.653179
$99$	9801.00	0.100504
$100$	0	0
$101$	137886.	1.34498	0.672492	−	0.740104i	$-0.265224\pi$
$101$	137886.	1.34498	0.672492	+	0.740104i	$0.265224\pi$
$102$	59968.0	0.570715
$103$	−63842.9	−0.592952	−0.296476	−	0.955040i	$-0.595812\pi$
$103$	−63842.9	−0.592952	−0.296476	+	0.955040i	$0.595812\pi$
$104$	−70612.1	−0.640171
$105$	0	0
$106$	88558.3	0.765534
$107$	−37586.9	−0.317378	−0.158689	−	0.987329i	$-0.550727\pi$
$107$	−37586.9	−0.317378	−0.158689	+	0.987329i	$0.550727\pi$
$108$	−4399.55	−0.0362952
$109$	−148487.	−1.19708	−0.598538	−	0.801095i	$-0.704251\pi$
$109$	−148487.	−1.19708	−0.598538	+	0.801095i	$0.704251\pi$
$110$	0	0
$111$	47609.2	0.366761
$112$	−135277.	−1.01901
$113$	189126.	1.39333	0.696666	−	0.717395i	$-0.254666\pi$
$113$	189126.	1.39333	0.696666	+	0.717395i	$0.254666\pi$
$114$	28417.5	0.204797
$115$	0	0
$116$	20724.3	0.143000
$117$	29511.2	0.199307
$118$	173494.	1.14704
$119$	222658.	1.44136
$120$	0	0
$121$	14641.0	0.0909091
$122$	247413.	1.50496
$123$	−69025.4	−0.411383
$124$	−8926.12	−0.0521325
$125$	0	0
$126$	70280.4	0.394374
$127$	−308080.	−1.69494	−0.847469	−	0.530846i	$-0.821874\pi$
$127$	−308080.	−1.69494	−0.847469	+	0.530846i	$0.821874\pi$
$128$	116545.	0.628737
$129$	−156299.	−0.826954
$130$	0	0
$131$	64949.6	0.330673	0.165336	−	0.986237i	$-0.447129\pi$
$131$	64949.6	0.330673	0.165336	+	0.986237i	$0.447129\pi$
$132$	−6572.17	−0.0328303
$133$	105513.	0.517222
$134$	175129.	0.842552
$135$	0	0
$136$	−253432.	−1.17493
$137$	−306828.	−1.39667	−0.698334	−	0.715772i	$-0.746075\pi$
$137$	−306828.	−1.39667	−0.698334	+	0.715772i	$0.746075\pi$
$138$	−42291.8	−0.189042
$139$	416315.	1.82762	0.913808	−	0.406147i	$-0.133128\pi$
$139$	416315.	1.82762	0.913808	+	0.406147i	$0.133128\pi$
$140$	0	0
$141$	−62517.3	−0.264821
$142$	246872.	1.02743
$143$	44084.6	0.180280
$144$	−64351.0	−0.258612
$145$	0	0
$146$	−302880.	−1.17595
$147$	109685.	0.418651
$148$	−31924.9	−0.119805
$149$	478891.	1.76714	0.883570	−	0.468300i	$-0.155133\pi$
$149$	478891.	1.76714	0.883570	+	0.468300i	$0.155133\pi$
$150$	0	0
$151$	251861.	0.898914	0.449457	−	0.893302i	$-0.351618\pi$
$151$	251861.	0.898914	0.449457	+	0.893302i	$0.351618\pi$
$152$	−120096.	−0.421618
$153$	105918.	0.365797
$154$	104987.	0.356724
$155$	0	0
$156$	−19789.1	−0.0651049
$157$	−35343.5	−0.114435	−0.0572177	−	0.998362i	$-0.518223\pi$
$157$	−35343.5	−0.114435	−0.0572177	+	0.998362i	$0.518223\pi$
$158$	18299.3	0.0583167
$159$	156415.	0.490665
$160$	0	0
$161$	−157027.	−0.477431
$162$	33432.1	0.100087
$163$	482364.	1.42202	0.711010	−	0.703182i	$-0.248238\pi$
$163$	482364.	1.42202	0.711010	+	0.703182i	$0.248238\pi$
$164$	46285.8	0.134381
$165$	0	0
$166$	327698.	0.923003
$167$	285420.	0.791941	0.395970	−	0.918263i	$-0.370408\pi$
$167$	285420.	0.791941	0.395970	+	0.918263i	$0.370408\pi$
$168$	−297013.	−0.811900
$169$	−238553.	−0.642492
$170$	0	0
$171$	50192.1	0.131264
$172$	104808.	0.270130
$173$	473489.	1.20280	0.601402	−	0.798947i	$-0.294609\pi$
$173$	473489.	1.20280	0.601402	+	0.798947i	$0.294609\pi$
$174$	−157484.	−0.394332
$175$	0	0
$176$	−96129.2	−0.233923
$177$	306431.	0.735190
$178$	137358.	0.324940
$179$	487786.	1.13788	0.568940	−	0.822379i	$-0.307354\pi$
$179$	487786.	1.13788	0.568940	+	0.822379i	$0.307354\pi$
$180$	0	0
$181$	−129180.	−0.293088	−0.146544	−	0.989204i	$-0.546815\pi$
$181$	−129180.	−0.293088	−0.146544	+	0.989204i	$0.546815\pi$
$182$	316119.	0.707411
$183$	436991.	0.964594
$184$	178730.	0.389182
$185$	0	0
$186$	67829.3	0.143759
$187$	158223.	0.330876
$188$	41921.7	0.0865056
$189$	124132.	0.252772
$190$	0	0
$191$	−386800.	−0.767191	−0.383595	−	0.923501i	$-0.625314\pi$
$191$	−386800.	−0.767191	−0.383595	+	0.923501i	$0.625314\pi$
$192$	327574.	0.641292
$193$	618444.	1.19511	0.597554	−	0.801828i	$-0.296139\pi$
$193$	618444.	1.19511	0.597554	+	0.801828i	$0.296139\pi$
$194$	84310.3	0.160833
$195$	0	0
$196$	−73550.3	−0.136755
$197$	−870016.	−1.59721	−0.798605	−	0.601856i	$-0.794428\pi$
$197$	−870016.	−1.59721	−0.798605	+	0.601856i	$0.794428\pi$
$198$	49941.8	0.0905318
$199$	−903963.	−1.61815	−0.809074	−	0.587707i	$-0.800031\pi$
$199$	−903963.	−1.61815	−0.809074	+	0.587707i	$0.800031\pi$
$200$	0	0
$201$	309320.	0.540029
$202$	702610.	1.21153
$203$	−584729.	−0.995897
$204$	−71024.3	−0.119490
$205$	0	0
$206$	−325317.	−0.534120
$207$	−74697.3	−0.121166
$208$	−289449.	−0.463888
$209$	74978.3	0.118733
$210$	0	0
$211$	414396.	0.640780	0.320390	−	0.947286i	$-0.396186\pi$
$211$	414396.	0.640780	0.320390	+	0.947286i	$0.396186\pi$
$212$	−104886.	−0.160279
$213$	436034.	0.658524
$214$	−191527.	−0.285888
$215$	0	0
$216$	−141288.	−0.206049
$217$	251847.	0.363068
$218$	−756626.	−1.07830
$219$	−534957.	−0.753718
$220$	0	0
$221$	476414.	0.656151
$222$	242596.	0.330371
$223$	526103.	0.708449	0.354224	−	0.935160i	$-0.384745\pi$
$223$	526103.	0.708449	0.354224	+	0.935160i	$0.384745\pi$
$224$	366729.	0.488344
$225$	0	0
$226$	963706.	1.25509
$227$	582127.	0.749813	0.374907	−	0.927063i	$-0.377675\pi$
$227$	582127.	0.749813	0.374907	+	0.927063i	$0.377675\pi$
$228$	−33656.9	−0.0428782
$229$	−59916.3	−0.0755016	−0.0377508	−	0.999287i	$-0.512019\pi$
$229$	−59916.3	−0.0755016	−0.0377508	+	0.999287i	$0.512019\pi$
$230$	0	0
$231$	185431.	0.228641
$232$	665544.	0.811814
$233$	−282811.	−0.341276	−0.170638	−	0.985334i	$-0.554583\pi$
$233$	−282811.	−0.341276	−0.170638	+	0.985334i	$0.554583\pi$
$234$	150377.	0.179531
$235$	0	0
$236$	−205481.	−0.240155
$237$	32321.0	0.0373778
$238$	1.13457e6	1.29834
$239$	−1.69607e6	−1.92065	−0.960327	−	0.278875i	$-0.910039\pi$
$239$	−1.69607e6	−1.92065	−0.960327	+	0.278875i	$0.910039\pi$
$240$	0	0
$241$	−1.04493e6	−1.15890	−0.579449	−	0.815008i	$-0.696732\pi$
$241$	−1.04493e6	−1.15890	−0.579449	+	0.815008i	$0.696732\pi$
$242$	74604.4	0.0818891
$243$	59049.0	0.0641500
$244$	−293029.	−0.315091
$245$	0	0
$246$	−351724.	−0.370565
$247$	225762.	0.235456
$248$	−286655.	−0.295958
$249$	578792.	0.591594
$250$	0	0
$251$	−1.92822e6	−1.93184	−0.965921	−	0.258837i	$-0.916661\pi$
$251$	−1.92822e6	−1.93184	−0.965921	+	0.258837i	$0.916661\pi$
$252$	−83238.0	−0.0825696
$253$	−111585.	−0.109598
$254$	−1.56984e6	−1.52677
$255$	0	0
$256$	−570842.	−0.544397
$257$	−686476.	−0.648324	−0.324162	−	0.946002i	$-0.605082\pi$
$257$	−686476.	−0.648324	−0.324162	+	0.946002i	$0.605082\pi$
$258$	−796434.	−0.744904
$259$	900748.	0.834361
$260$	0	0
$261$	−278153.	−0.252745
$262$	330956.	0.297863
$263$	507368.	0.452307	0.226154	−	0.974092i	$-0.427385\pi$
$263$	507368.	0.452307	0.226154	+	0.974092i	$0.427385\pi$
$264$	−211060.	−0.186378
$265$	0	0
$266$	537650.	0.465903
$267$	242606.	0.208269
$268$	−207418.	−0.176404
$269$	717068.	0.604198	0.302099	−	0.953277i	$-0.402313\pi$
$269$	717068.	0.604198	0.302099	+	0.953277i	$0.402313\pi$
$270$	0	0
$271$	−315074.	−0.260609	−0.130304	−	0.991474i	$-0.541595\pi$
$271$	−315074.	−0.260609	−0.130304	+	0.991474i	$0.541595\pi$
$272$	−1.03885e6	−0.851395
$273$	558341.	0.453412
$274$	−1.56347e6	−1.25809
$275$	0	0
$276$	50089.1	0.0395795
$277$	1.42881e6	1.11885	0.559427	−	0.828879i	$-0.311021\pi$
$277$	1.42881e6	1.11885	0.559427	+	0.828879i	$0.311021\pi$
$278$	2.12137e6	1.64628
$279$	119803.	0.0921416
$280$	0	0
$281$	432306.	0.326607	0.163303	−	0.986576i	$-0.447785\pi$
$281$	432306.	0.326607	0.163303	+	0.986576i	$0.447785\pi$
$282$	−318562.	−0.238545
$283$	−257624.	−0.191214	−0.0956071	−	0.995419i	$-0.530479\pi$
$283$	−257624.	−0.191214	−0.0956071	+	0.995419i	$0.530479\pi$
$284$	−292388.	−0.215112
$285$	0	0
$286$	224637.	0.162392
$287$	−1.30594e6	−0.935873
$288$	174452.	0.123935
$289$	290027.	0.204265
$290$	0	0
$291$	148912.	0.103085
$292$	358722.	0.246207
$293$	993168.	0.675855	0.337928	−	0.941172i	$-0.390274\pi$
$293$	993168.	0.675855	0.337928	+	0.941172i	$0.390274\pi$
$294$	558907.	0.377113
$295$	0	0
$296$	−1.02524e6	−0.680137
$297$	88209.0	0.0580259
$298$	2.44023e6	1.59180
$299$	−335986.	−0.217342
$300$	0	0
$301$	−2.95712e6	−1.88128
$302$	1.28338e6	0.809723
$303$	1.24098e6	0.776527
$304$	−492289.	−0.305518
$305$	0	0
$306$	539712.	0.329502
$307$	1.93257e6	1.17028	0.585139	−	0.810933i	$-0.301040\pi$
$307$	1.93257e6	1.17028	0.585139	+	0.810933i	$0.301040\pi$
$308$	−124343.	−0.0746870
$309$	−574587.	−0.342341
$310$	0	0
$311$	2.35905e6	1.38305	0.691523	−	0.722354i	$-0.256940\pi$
$311$	2.35905e6	1.38305	0.691523	+	0.722354i	$0.256940\pi$
$312$	−635509.	−0.369603
$313$	−133954.	−0.0772849	−0.0386425	−	0.999253i	$-0.512303\pi$
$313$	−133954.	−0.0772849	−0.0386425	+	0.999253i	$0.512303\pi$
$314$	−180096.	−0.103081
$315$	0	0
$316$	−21673.2	−0.0122097
$317$	−1.69094e6	−0.945102	−0.472551	−	0.881303i	$-0.656667\pi$
$317$	−1.69094e6	−0.945102	−0.472551	+	0.881303i	$0.656667\pi$
$318$	797025.	0.441981
$319$	−415513.	−0.228616
$320$	0	0
$321$	−338282.	−0.183238
$322$	−800146.	−0.430060
$323$	810277.	0.432143
$324$	−39596.0	−0.0209550
$325$	0	0
$326$	2.45792e6	1.28093
$327$	−1.33638e6	−0.691132
$328$	1.48643e6	0.762886
$329$	−1.18280e6	−0.602453
$330$	0	0
$331$	313289.	0.157172	0.0785860	−	0.996907i	$-0.474959\pi$
$331$	313289.	0.157172	0.0785860	+	0.996907i	$0.474959\pi$
$332$	−388115.	−0.193248
$333$	428482.	0.211750
$334$	1.45438e6	0.713364
$335$	0	0
$336$	−1.21750e6	−0.588328
$337$	−1.93465e6	−0.927957	−0.463979	−	0.885846i	$-0.653579\pi$
$337$	−1.93465e6	−0.927957	−0.463979	+	0.885846i	$0.653579\pi$
$338$	−1.21556e6	−0.578744
$339$	1.70213e6	0.804441
$340$	0	0
$341$	178964.	0.0833452
$342$	255758.	0.118240
$343$	−786647.	−0.361031
$344$	3.36582e6	1.53354
$345$	0	0
$346$	2.41270e6	1.08346
$347$	−360768.	−0.160844	−0.0804218	−	0.996761i	$-0.525627\pi$
$347$	−360768.	−0.160844	−0.0804218	+	0.996761i	$0.525627\pi$
$348$	186519.	0.0825609
$349$	−1.67266e6	−0.735095	−0.367547	−	0.930005i	$-0.619802\pi$
$349$	−1.67266e6	−0.735095	−0.367547	+	0.930005i	$0.619802\pi$
$350$	0	0
$351$	265600.	0.115070
$352$	260601.	0.112103
$353$	268287.	0.114594	0.0572971	−	0.998357i	$-0.481752\pi$
$353$	268287.	0.114594	0.0572971	+	0.998357i	$0.481752\pi$
$354$	1.56144e6	0.662245
$355$	0	0
$356$	−162682.	−0.0680324
$357$	2.00392e6	0.832167
$358$	2.48555e6	1.02498
$359$	−2.33979e6	−0.958165	−0.479083	−	0.877770i	$-0.659031\pi$
$359$	−2.33979e6	−0.958165	−0.479083	+	0.877770i	$0.659031\pi$
$360$	0	0
$361$	−2.09213e6	−0.844928
$362$	−658247.	−0.264008
$363$	131769.	0.0524864
$364$	−374402.	−0.148110
$365$	0	0
$366$	2.22672e6	0.868887
$367$	1.87216e6	0.725569	0.362785	−	0.931873i	$-0.381826\pi$
$367$	1.87216e6	0.725569	0.362785	+	0.931873i	$0.381826\pi$
$368$	732639.	0.282014
$369$	−621229.	−0.237512
$370$	0	0
$371$	2.95931e6	1.11624
$372$	−80335.1	−0.0300987
$373$	−1.51377e6	−0.563363	−0.281681	−	0.959508i	$-0.590892\pi$
$373$	−1.51377e6	−0.563363	−0.281681	+	0.959508i	$0.590892\pi$
$374$	806236.	0.298046
$375$	0	0
$376$	1.34628e6	0.491095
$377$	−1.25112e6	−0.453364
$378$	632524.	0.227692
$379$	−2.85481e6	−1.02089	−0.510446	−	0.859910i	$-0.670520\pi$
$379$	−2.85481e6	−1.02089	−0.510446	+	0.859910i	$0.670520\pi$
$380$	0	0
$381$	−2.77272e6	−0.978572
$382$	−1.97097e6	−0.691070
$383$	3.23968e6	1.12851	0.564254	−	0.825601i	$-0.309164\pi$
$383$	3.23968e6	1.12851	0.564254	+	0.825601i	$0.309164\pi$
$384$	1.04891e6	0.363002
$385$	0	0
$386$	3.15133e6	1.07653
$387$	−1.40669e6	−0.477442
$388$	−99854.6	−0.0336735
$389$	−558246.	−0.187047	−0.0935237	−	0.995617i	$-0.529813\pi$
$389$	−558246.	−0.187047	−0.0935237	+	0.995617i	$0.529813\pi$
$390$	0	0
$391$	−1.20588e6	−0.398897
$392$	−2.36201e6	−0.776365
$393$	584547.	0.190914
$394$	−4.43324e6	−1.43873
$395$	0	0
$396$	−59149.6	−0.0189546
$397$	1.57535e6	0.501651	0.250826	−	0.968032i	$-0.419298\pi$
$397$	1.57535e6	0.501651	0.250826	+	0.968032i	$0.419298\pi$
$398$	−4.60622e6	−1.45760
$399$	949616.	0.298618
$400$	0	0
$401$	2.98866e6	0.928145	0.464073	−	0.885797i	$-0.346388\pi$
$401$	2.98866e6	0.928145	0.464073	+	0.885797i	$0.346388\pi$
$402$	1.57616e6	0.486448
$403$	538868.	0.165280
$404$	−832150.	−0.253658
$405$	0	0
$406$	−2.97953e6	−0.897084
$407$	640079.	0.191535
$408$	−2.28089e6	−0.678349
$409$	6.60301e6	1.95179	0.975897	−	0.218232i	$-0.0700291\pi$
$409$	6.60301e6	1.95179	0.975897	+	0.218232i	$0.0700291\pi$
$410$	0	0
$411$	−2.76145e6	−0.806367
$412$	385296.	0.111828
$413$	5.79757e6	1.67252
$414$	−380626.	−0.109143
$415$	0	0
$416$	784678.	0.222310
$417$	3.74683e6	1.05517
$418$	382058.	0.106952
$419$	−2.41796e6	−0.672842	−0.336421	−	0.941712i	$-0.609217\pi$
$419$	−2.41796e6	−0.672842	−0.336421	+	0.941712i	$0.609217\pi$
$420$	0	0
$421$	−315653.	−0.0867971	−0.0433986	−	0.999058i	$-0.513819\pi$
$421$	−315653.	−0.0867971	−0.0433986	+	0.999058i	$0.513819\pi$
$422$	2.11159e6	0.577202
$423$	−562656.	−0.152894
$424$	−3.36832e6	−0.909911
$425$	0	0
$426$	2.22185e6	0.593186
$427$	8.26771e6	2.19440
$428$	226839.	0.0598561
$429$	396761.	0.104084
$430$	0	0
$431$	−1.62617e6	−0.421669	−0.210835	−	0.977522i	$-0.567618\pi$
$431$	−1.62617e6	−0.421669	−0.210835	+	0.977522i	$0.567618\pi$
$432$	−579159.	−0.149310
$433$	−756561.	−0.193921	−0.0969603	−	0.995288i	$-0.530912\pi$
$433$	−756561.	−0.193921	−0.0969603	+	0.995288i	$0.530912\pi$
$434$	1.28331e6	0.327044
$435$	0	0
$436$	896126.	0.225763
$437$	−571439.	−0.143142
$438$	−2.72592e6	−0.678934
$439$	−3.70064e6	−0.916463	−0.458232	−	0.888833i	$-0.651517\pi$
$439$	−3.70064e6	−0.916463	−0.458232	+	0.888833i	$0.651517\pi$
$440$	0	0
$441$	987162.	0.241709
$442$	2.42761e6	0.591048
$443$	−6.39510e6	−1.54824	−0.774120	−	0.633039i	$-0.781807\pi$
$443$	−6.39510e6	−1.54824	−0.774120	+	0.633039i	$0.781807\pi$
$444$	−287324.	−0.0691694
$445$	0	0
$446$	2.68080e6	0.638157
$447$	4.31002e6	1.02026
$448$	6.19758e6	1.45891
$449$	−7.10566e6	−1.66337	−0.831684	−	0.555249i	$-0.812623\pi$
$449$	−7.10566e6	−1.66337	−0.831684	+	0.555249i	$0.812623\pi$
$450$	0	0
$451$	−928008.	−0.214838
$452$	−1.14138e6	−0.262776
$453$	2.26675e6	0.518988
$454$	2.96628e6	0.675417
$455$	0	0
$456$	−1.08086e6	−0.243421
$457$	−2.49947e6	−0.559832	−0.279916	−	0.960025i	$-0.590307\pi$
$457$	−2.49947e6	−0.559832	−0.279916	+	0.960025i	$0.590307\pi$
$458$	−305308.	−0.0680103
$459$	953259.	0.211193
$460$	0	0
$461$	−5.72233e6	−1.25407	−0.627033	−	0.778993i	$-0.715731\pi$
$461$	−5.72233e6	−1.25407	−0.627033	+	0.778993i	$0.715731\pi$
$462$	944881.	0.205955
$463$	−5.64981e6	−1.22485	−0.612423	−	0.790530i	$-0.709805\pi$
$463$	−5.64981e6	−1.22485	−0.612423	+	0.790530i	$0.709805\pi$
$464$	2.72815e6	0.588266
$465$	0	0
$466$	−1.44108e6	−0.307415
$467$	8.10977e6	1.72074	0.860372	−	0.509666i	$-0.170231\pi$
$467$	8.10977e6	1.72074	0.860372	+	0.509666i	$0.170231\pi$
$468$	−178101.	−0.0375883
$469$	5.85222e6	1.22854
$470$	0	0
$471$	−318092.	−0.0660693
$472$	−6.59885e6	−1.36337
$473$	−2.10135e6	−0.431863
$474$	164694.	0.0336691
$475$	0	0
$476$	−1.34375e6	−0.271833
$477$	1.40773e6	0.283286
$478$	−8.64247e6	−1.73009
$479$	5.82406e6	1.15981	0.579905	−	0.814684i	$-0.303090\pi$
$479$	5.82406e6	1.15981	0.579905	+	0.814684i	$0.303090\pi$
$480$	0	0
$481$	1.92730e6	0.379828
$482$	−5.32454e6	−1.04391
$483$	−1.41325e6	−0.275645
$484$	−88359.2	−0.0171450
$485$	0	0
$486$	300889.	0.0577851
$487$	−7.82723e6	−1.49550	−0.747749	−	0.663981i	$-0.768865\pi$
$487$	−7.82723e6	−1.49550	−0.747749	+	0.663981i	$0.768865\pi$
$488$	−9.41039e6	−1.78878
$489$	4.34127e6	0.821004
$490$	0	0
$491$	−5.95082e6	−1.11397	−0.556985	−	0.830523i	$-0.688042\pi$
$491$	−5.95082e6	−1.11397	−0.556985	+	0.830523i	$0.688042\pi$
$492$	416572.	0.0775849
$493$	−4.49037e6	−0.832080
$494$	1.15039e6	0.212094
$495$	0	0
$496$	−1.17504e6	−0.214460
$497$	8.24962e6	1.49811
$498$	2.94928e6	0.532896
$499$	−1.80923e6	−0.325269	−0.162634	−	0.986686i	$-0.551999\pi$
$499$	−1.80923e6	−0.325269	−0.162634	+	0.986686i	$0.551999\pi$
$500$	0	0
$501$	2.56878e6	0.457227
$502$	−9.82539e6	−1.74016
$503$	−2.53156e6	−0.446137	−0.223068	−	0.974803i	$-0.571607\pi$
$503$	−2.53156e6	−0.446137	−0.223068	+	0.974803i	$0.571607\pi$
$504$	−2.67312e6	−0.468751
$505$	0	0
$506$	−568590.	−0.0987240
$507$	−2.14697e6	−0.370943
$508$	1.85928e6	0.319657
$509$	8.24104e6	1.40990	0.704948	−	0.709259i	$-0.250970\pi$
$509$	8.24104e6	1.40990	0.704948	+	0.709259i	$0.250970\pi$
$510$	0	0
$511$	−1.01212e7	−1.71467
$512$	−6.63821e6	−1.11912
$513$	451729.	0.0757852
$514$	−3.49799e6	−0.583998
$515$	0	0
$516$	943272.	0.155960
$517$	−840510.	−0.138298
$518$	4.58984e6	0.751576
$519$	4.26140e6	0.694439
$520$	0	0
$521$	5.59646e6	0.903273	0.451637	−	0.892202i	$-0.350840\pi$
$521$	5.59646e6	0.903273	0.451637	+	0.892202i	$0.350840\pi$
$522$	−1.41735e6	−0.227668
$523$	−1.34040e6	−0.214279	−0.107140	−	0.994244i	$-0.534169\pi$
$523$	−1.34040e6	−0.214279	−0.107140	+	0.994244i	$0.534169\pi$
$524$	−391974.	−0.0623634
$525$	0	0
$526$	2.58533e6	0.407429
$527$	1.93404e6	0.303346
$528$	−865163.	−0.135056
$529$	−5.58591e6	−0.867870
$530$	0	0
$531$	2.75788e6	0.424462
$532$	−636776.	−0.0975456
$533$	−2.79427e6	−0.426039
$534$	1.23622e6	0.187604
$535$	0	0
$536$	−6.66105e6	−1.00145
$537$	4.39007e6	0.656955
$538$	3.65388e6	0.544250
$539$	1.47465e6	0.218634
$540$	0	0
$541$	2.97220e6	0.436602	0.218301	−	0.975882i	$-0.429949\pi$
$541$	2.97220e6	0.436602	0.218301	+	0.975882i	$0.429949\pi$
$542$	−1.60548e6	−0.234751
$543$	−1.16262e6	−0.169215
$544$	2.81627e6	0.408015
$545$	0	0
$546$	2.84507e6	0.408424
$547$	−4.70253e6	−0.671991	−0.335995	−	0.941864i	$-0.609073\pi$
$547$	−4.70253e6	−0.671991	−0.335995	+	0.941864i	$0.609073\pi$
$548$	1.85172e6	0.263405
$549$	3.93292e6	0.556908
$550$	0	0
$551$	−2.12789e6	−0.298586
$552$	1.60857e6	0.224694
$553$	611501.	0.0850324
$554$	7.28059e6	1.00784
$555$	0	0
$556$	−2.51248e6	−0.344680
$557$	−5.61923e6	−0.767431	−0.383715	−	0.923451i	$-0.625356\pi$
$557$	−5.61923e6	−0.767431	−0.383715	+	0.923451i	$0.625356\pi$
$558$	610464.	0.0829993
$559$	−6.32724e6	−0.856417
$560$	0	0
$561$	1.42400e6	0.191031
$562$	2.20285e6	0.294201
$563$	−1.12518e7	−1.49607	−0.748036	−	0.663658i	$-0.769003\pi$
$563$	−1.12518e7	−1.49607	−0.748036	+	0.663658i	$0.769003\pi$
$564$	377295.	0.0499440
$565$	0	0
$566$	−1.31274e6	−0.172242
$567$	1.11719e6	0.145938
$568$	−9.38979e6	−1.22120
$569$	3.62036e6	0.468782	0.234391	−	0.972142i	$-0.424690\pi$
$569$	3.62036e6	0.468782	0.234391	+	0.972142i	$0.424690\pi$
$570$	0	0
$571$	−6.01771e6	−0.772397	−0.386199	−	0.922416i	$-0.626212\pi$
$571$	−6.01771e6	−0.772397	−0.386199	+	0.922416i	$0.626212\pi$
$572$	−266053.	−0.0339999
$573$	−3.48120e6	−0.442938
$574$	−6.65450e6	−0.843016
$575$	0	0
$576$	2.94816e6	0.370250
$577$	3.81655e6	0.477233	0.238617	−	0.971114i	$-0.423306\pi$
$577$	3.81655e6	0.477233	0.238617	+	0.971114i	$0.423306\pi$
$578$	1.47785e6	0.183998
$579$	5.56600e6	0.689996
$580$	0	0
$581$	1.09505e7	1.34584
$582$	758793.	0.0928572
$583$	2.10291e6	0.256242
$584$	1.15200e7	1.39773
$585$	0	0
$586$	5.06077e6	0.608797
$587$	4.77208e6	0.571627	0.285813	−	0.958285i	$-0.407736\pi$
$587$	4.77208e6	0.571627	0.285813	+	0.958285i	$0.407736\pi$
$588$	−661953.	−0.0789558
$589$	916498.	0.108854
$590$	0	0
$591$	−7.83015e6	−0.922149
$592$	−4.20260e6	−0.492849
$593$	1.97557e6	0.230704	0.115352	−	0.993325i	$-0.463200\pi$
$593$	1.97557e6	0.230704	0.115352	+	0.993325i	$0.463200\pi$
$594$	449476.	0.0522686
$595$	0	0
$596$	−2.89013e6	−0.333274
$597$	−8.13567e6	−0.934238
$598$	−1.71204e6	−0.195777
$599$	−7989.43	−0.000909806 0	−0.000454903	−	1.00000i	$-0.500145\pi$
$599$	−7989.43	−0.000909806 0	−0.000454903		1.00000i	$0.500145\pi$
$600$	0	0
$601$	1.62885e7	1.83948	0.919741	−	0.392526i	$-0.128399\pi$
$601$	1.62885e7	1.83948	0.919741	+	0.392526i	$0.128399\pi$
$602$	−1.50682e7	−1.69462
$603$	2.78388e6	0.311786
$604$	−1.51999e6	−0.169531
$605$	0	0
$606$	6.32349e6	0.699480
$607$	1.05514e7	1.16235	0.581176	−	0.813778i	$-0.302593\pi$
$607$	1.05514e7	1.16235	0.581176	+	0.813778i	$0.302593\pi$
$608$	1.33457e6	0.146414
$609$	−5.26256e6	−0.574981
$610$	0	0
$611$	−2.53081e6	−0.274256
$612$	−639219.	−0.0689876
$613$	1.45442e7	1.56329	0.781645	−	0.623723i	$-0.214381\pi$
$613$	1.45442e7	1.56329	0.781645	+	0.623723i	$0.214381\pi$
$614$	9.84757e6	1.05416
$615$	0	0
$616$	−3.99318e6	−0.424001
$617$	−1.74916e7	−1.84977	−0.924885	−	0.380246i	$-0.875839\pi$
$617$	−1.74916e7	−1.84977	−0.924885	+	0.380246i	$0.875839\pi$
$618$	−2.92785e6	−0.308374
$619$	−1.56180e7	−1.63832	−0.819160	−	0.573565i	$-0.805560\pi$
$619$	−1.56180e7	−1.63832	−0.819160	+	0.573565i	$0.805560\pi$
$620$	0	0
$621$	−672276.	−0.0699549
$622$	1.20208e7	1.24582
$623$	4.59003e6	0.473800
$624$	−2.60504e6	−0.267826
$625$	0	0
$626$	−682573.	−0.0696167
$627$	674805.	0.0685503
$628$	213300.	0.0215820
$629$	6.91722e6	0.697115
$630$	0	0
$631$	9.11463e6	0.911309	0.455655	−	0.890157i	$-0.349405\pi$
$631$	9.11463e6	0.911309	0.455655	+	0.890157i	$0.349405\pi$
$632$	−696016.	−0.0693149
$633$	3.72956e6	0.369955
$634$	−8.61630e6	−0.851329
$635$	0	0
$636$	−943972.	−0.0925372
$637$	4.44022e6	0.433567
$638$	−2.11728e6	−0.205933
$639$	3.92431e6	0.380199
$640$	0	0
$641$	1.47144e7	1.41448	0.707242	−	0.706971i	$-0.249939\pi$
$641$	1.47144e7	1.41448	0.707242	+	0.706971i	$0.249939\pi$
$642$	−1.72374e6	−0.165057
$643$	−1.99364e7	−1.90160	−0.950799	−	0.309807i	$-0.899735\pi$
$643$	−1.99364e7	−1.90160	−0.950799	+	0.309807i	$0.899735\pi$
$644$	947669.	0.0900413
$645$	0	0
$646$	4.12883e6	0.389266
$647$	−1.17295e7	−1.10159	−0.550794	−	0.834641i	$-0.685675\pi$
$647$	−1.17295e7	−1.10159	−0.550794	+	0.834641i	$0.685675\pi$
$648$	−1.27159e6	−0.118963
$649$	4.11980e6	0.383940
$650$	0	0
$651$	2.26662e6	0.209617
$652$	−2.91109e6	−0.268187
$653$	−1.62173e7	−1.48832	−0.744158	−	0.668004i	$-0.767149\pi$
$653$	−1.62173e7	−1.48832	−0.744158	+	0.668004i	$0.767149\pi$
$654$	−6.80964e6	−0.622558
$655$	0	0
$656$	6.09307e6	0.552811
$657$	−4.81462e6	−0.435159
$658$	−6.02708e6	−0.542678
$659$	−806061.	−0.0723027	−0.0361513	−	0.999346i	$-0.511510\pi$
$659$	−806061.	−0.0723027	−0.0361513	+	0.999346i	$0.511510\pi$
$660$	0	0
$661$	−1.54634e7	−1.37658	−0.688291	−	0.725435i	$-0.741639\pi$
$661$	−1.54634e7	−1.37658	−0.688291	+	0.725435i	$0.741639\pi$
$662$	1.59639e6	0.141577
$663$	4.28773e6	0.378829
$664$	−1.24640e7	−1.09708
$665$	0	0
$666$	2.18337e6	0.190740
$667$	3.16679e6	0.275616
$668$	−1.72252e6	−0.149356
$669$	4.73493e6	0.409023
$670$	0	0
$671$	5.87510e6	0.503743
$672$	3.30057e6	0.281945
$673$	−3.23414e6	−0.275246	−0.137623	−	0.990485i	$-0.543946\pi$
$673$	−3.23414e6	−0.275246	−0.137623	+	0.990485i	$0.543946\pi$
$674$	−9.85817e6	−0.835885
$675$	0	0
$676$	1.43968e6	0.121171
$677$	−8.24825e6	−0.691656	−0.345828	−	0.938298i	$-0.612402\pi$
$677$	−8.24825e6	−0.691656	−0.345828	+	0.938298i	$0.612402\pi$
$678$	8.67336e6	0.724624
$679$	2.81736e6	0.234514
$680$	0	0
$681$	5.23914e6	0.432905
$682$	911928.	0.0750757
$683$	1.23891e7	1.01622	0.508111	−	0.861291i	$-0.330344\pi$
$683$	1.23891e7	1.01622	0.508111	+	0.861291i	$0.330344\pi$
$684$	−302912.	−0.0247558
$685$	0	0
$686$	−4.00842e6	−0.325210
$687$	−539247.	−0.0435909
$688$	1.37970e7	1.11125
$689$	6.33194e6	0.508146
$690$	0	0
$691$	−1.75824e7	−1.40082	−0.700410	−	0.713741i	$-0.746999\pi$
$691$	−1.75824e7	−1.40082	−0.700410	+	0.713741i	$0.746999\pi$
$692$	−2.85753e6	−0.226843
$693$	1.66888e6	0.132006
$694$	−1.83832e6	−0.144885
$695$	0	0
$696$	5.98989e6	0.468701
$697$	−1.00288e7	−0.781930
$698$	−8.52316e6	−0.662158
$699$	−2.54530e6	−0.197036
$700$	0	0
$701$	−1.26176e7	−0.969797	−0.484898	−	0.874571i	$-0.661143\pi$
$701$	−1.26176e7	−0.969797	−0.484898	+	0.874571i	$0.661143\pi$
$702$	1.35339e6	0.103652
$703$	3.27792e6	0.250155
$704$	4.40405e6	0.334904
$705$	0	0
$706$	1.36708e6	0.103224
$707$	2.34788e7	1.76656
$708$	−1.84933e6	−0.138653
$709$	5.87588e6	0.438993	0.219496	−	0.975613i	$-0.429559\pi$
$709$	5.87588e6	0.438993	0.219496	+	0.975613i	$0.429559\pi$
$710$	0	0
$711$	290889.	0.0215801
$712$	−5.22441e6	−0.386222
$713$	−1.36396e6	−0.100479
$714$	1.02112e7	0.749600
$715$	0	0
$716$	−2.94381e6	−0.214599
$717$	−1.52646e7	−1.10889
$718$	−1.19226e7	−0.863096
$719$	−9.06132e6	−0.653686	−0.326843	−	0.945079i	$-0.605985\pi$
$719$	−9.06132e6	−0.653686	−0.326843	+	0.945079i	$0.605985\pi$
$720$	0	0
$721$	−1.08710e7	−0.778808
$722$	−1.06606e7	−0.761095
$723$	−9.40439e6	−0.669090
$724$	779608.	0.0552752
$725$	0	0
$726$	671440.	0.0472787
$727$	8.13363e6	0.570753	0.285376	−	0.958415i	$-0.407881\pi$
$727$	8.13363e6	0.570753	0.285376	+	0.958415i	$0.407881\pi$
$728$	−1.20236e7	−0.840826
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.27089e7	−1.57182
$732$	−2.63726e6	−0.181918
$733$	1.02344e7	0.703561	0.351780	−	0.936083i	$-0.385576\pi$
$733$	1.02344e7	0.703561	0.351780	+	0.936083i	$0.385576\pi$
$734$	9.53976e6	0.653578
$735$	0	0
$736$	−1.98614e6	−0.135150
$737$	4.15863e6	0.282021
$738$	−3.16552e6	−0.213946
$739$	−1.70091e7	−1.14570	−0.572850	−	0.819660i	$-0.694162\pi$
$739$	−1.70091e7	−1.14570	−0.572850	+	0.819660i	$0.694162\pi$
$740$	0	0
$741$	2.03186e6	0.135940
$742$	1.50794e7	1.00548
$743$	−1.55671e7	−1.03451	−0.517257	−	0.855830i	$-0.673047\pi$
$743$	−1.55671e7	−1.03451	−0.517257	+	0.855830i	$0.673047\pi$
$744$	−2.57989e6	−0.170871
$745$	0	0
$746$	−7.71354e6	−0.507466
$747$	5.20912e6	0.341557
$748$	−954882.	−0.0624016
$749$	−6.40018e6	−0.416857
$750$	0	0
$751$	−2.29900e7	−1.48744	−0.743719	−	0.668493i	$-0.766940\pi$
$751$	−2.29900e7	−1.48744	−0.743719	+	0.668493i	$0.766940\pi$
$752$	5.51858e6	0.355863
$753$	−1.73540e7	−1.11535
$754$	−6.37520e6	−0.408381
$755$	0	0
$756$	−749142.	−0.0476716
$757$	1.66179e7	1.05399	0.526994	−	0.849869i	$-0.323319\pi$
$757$	1.66179e7	1.05399	0.526994	+	0.849869i	$0.323319\pi$
$758$	−1.45469e7	−0.919599
$759$	−1.00426e6	−0.0632766
$760$	0	0
$761$	−2.43312e7	−1.52300	−0.761502	−	0.648162i	$-0.775538\pi$
$761$	−2.43312e7	−1.52300	−0.761502	+	0.648162i	$0.775538\pi$
$762$	−1.41286e7	−0.881478
$763$	−2.52839e7	−1.57229
$764$	2.33436e6	0.144689
$765$	0	0
$766$	1.65080e7	1.01654
$767$	1.24049e7	0.761383
$768$	−5.13758e6	−0.314308
$769$	−1.81242e7	−1.10520	−0.552602	−	0.833445i	$-0.686365\pi$
$769$	−1.81242e7	−1.10520	−0.552602	+	0.833445i	$0.686365\pi$
$770$	0	0
$771$	−6.17828e6	−0.374310
$772$	−3.73235e6	−0.225392
$773$	2.02819e7	1.22084	0.610422	−	0.792077i	$-0.291000\pi$
$773$	2.02819e7	1.22084	0.610422	+	0.792077i	$0.291000\pi$
$774$	−7.16790e6	−0.430071
$775$	0	0
$776$	−3.20675e6	−0.191166
$777$	8.10673e6	0.481719
$778$	−2.84459e6	−0.168489
$779$	−4.75244e6	−0.280590
$780$	0	0
$781$	5.86224e6	0.343903
$782$	−6.14465e6	−0.359319
$783$	−2.50338e6	−0.145922
$784$	−9.68219e6	−0.562579
$785$	0	0
$786$	2.97860e6	0.171971
$787$	−1.37897e7	−0.793630	−0.396815	−	0.917899i	$-0.629885\pi$
$787$	−1.37897e7	−0.793630	−0.396815	+	0.917899i	$0.629885\pi$
$788$	5.25060e6	0.301226
$789$	4.56631e6	0.261140
$790$	0	0
$791$	3.22037e7	1.83006
$792$	−1.89954e6	−0.107606
$793$	1.76901e7	0.998960
$794$	8.02735e6	0.451878
$795$	0	0
$796$	5.45547e6	0.305175
$797$	3.04194e6	0.169631	0.0848155	−	0.996397i	$-0.472970\pi$
$797$	3.04194e6	0.169631	0.0848155	+	0.996397i	$0.472970\pi$
$798$	4.83885e6	0.268989
$799$	−9.08324e6	−0.503354
$800$	0	0
$801$	2.18346e6	0.120244
$802$	1.52290e7	0.836055
$803$	−7.19220e6	−0.393616
$804$	−1.86676e6	−0.101847
$805$	0	0
$806$	2.74585e6	0.148881
$807$	6.45361e6	0.348834
$808$	−2.67238e7	−1.44002
$809$	−6.70668e6	−0.360277	−0.180138	−	0.983641i	$-0.557655\pi$
$809$	−6.70668e6	−0.360277	−0.180138	+	0.983641i	$0.557655\pi$
$810$	0	0
$811$	−8.62214e6	−0.460323	−0.230162	−	0.973152i	$-0.573926\pi$
$811$	−8.62214e6	−0.460323	−0.230162	+	0.973152i	$0.573926\pi$
$812$	3.52887e6	0.187822
$813$	−2.83566e6	−0.150463
$814$	3.26157e6	0.172531
$815$	0	0
$816$	−9.34966e6	−0.491553
$817$	−1.07613e7	−0.564038
$818$	3.36462e7	1.75814
$819$	5.02507e6	0.261777
$820$	0	0
$821$	2.68361e7	1.38951	0.694754	−	0.719247i	$-0.255513\pi$
$821$	2.68361e7	1.38951	0.694754	+	0.719247i	$0.255513\pi$
$822$	−1.40712e7	−0.726359
$823$	1.23875e6	0.0637508	0.0318754	−	0.999492i	$-0.489852\pi$
$823$	1.23875e6	0.0637508	0.0318754	+	0.999492i	$0.489852\pi$
$824$	1.23734e7	0.634852
$825$	0	0
$826$	2.95420e7	1.50657
$827$	2.59198e7	1.31786	0.658929	−	0.752205i	$-0.271010\pi$
$827$	2.59198e7	1.31786	0.658929	+	0.752205i	$0.271010\pi$
$828$	450802.	0.0228513
$829$	−6.40573e6	−0.323729	−0.161865	−	0.986813i	$-0.551751\pi$
$829$	−6.40573e6	−0.323729	−0.161865	+	0.986813i	$0.551751\pi$
$830$	0	0
$831$	1.28593e7	0.645971
$832$	1.32607e7	0.664140
$833$	1.59363e7	0.795746
$834$	1.90923e7	0.950480
$835$	0	0
$836$	−452498.	−0.0223924
$837$	1.07822e6	0.0531980
$838$	−1.23209e7	−0.606083
$839$	−2.24119e7	−1.09919	−0.549597	−	0.835430i	$-0.685219\pi$
$839$	−2.24119e7	−1.09919	−0.549597	+	0.835430i	$0.685219\pi$
$840$	0	0
$841$	−8.71887e6	−0.425080
$842$	−1.60844e6	−0.0781851
$843$	3.89075e6	0.188566
$844$	−2.50090e6	−0.120848
$845$	0	0
$846$	−2.86706e6	−0.137724
$847$	2.49302e6	0.119404
$848$	−1.38072e7	−0.659350
$849$	−2.31862e6	−0.110398
$850$	0	0
$851$	−4.87829e6	−0.230911
$852$	−2.63149e6	−0.124195
$853$	−1.15120e7	−0.541725	−0.270862	−	0.962618i	$-0.587309\pi$
$853$	−1.15120e7	−0.541725	−0.270862	+	0.962618i	$0.587309\pi$
$854$	4.21288e7	1.97667
$855$	0	0
$856$	7.28474e6	0.339805
$857$	1.03784e7	0.482701	0.241350	−	0.970438i	$-0.422410\pi$
$857$	1.03784e7	0.482701	0.241350	+	0.970438i	$0.422410\pi$
$858$	2.02173e6	0.0937572
$859$	2.03041e7	0.938860	0.469430	−	0.882970i	$-0.344459\pi$
$859$	2.03041e7	0.938860	0.469430	+	0.882970i	$0.344459\pi$
$860$	0	0
$861$	−1.17534e7	−0.540327
$862$	−8.28627e6	−0.379831
$863$	3.21077e7	1.46752	0.733758	−	0.679411i	$-0.237765\pi$
$863$	3.21077e7	1.46752	0.733758	+	0.679411i	$0.237765\pi$
$864$	1.57007e6	0.0715539
$865$	0	0
$866$	−3.85512e6	−0.174680
$867$	2.61024e6	0.117932
$868$	−1.51991e6	−0.0684729
$869$	434537.	0.0195199
$870$	0	0
$871$	1.25218e7	0.559269
$872$	2.87783e7	1.28166
$873$	1.34021e6	0.0595164
$874$	−2.91182e6	−0.128939
$875$	0	0
$876$	3.22850e6	0.142148
$877$	−3.06108e7	−1.34393	−0.671964	−	0.740584i	$-0.734549\pi$
$877$	−3.06108e7	−1.34393	−0.671964	+	0.740584i	$0.734549\pi$
$878$	−1.88569e7	−0.825532
$879$	8.93851e6	0.390205
$880$	0	0
$881$	4.04714e7	1.75675	0.878373	−	0.477976i	$-0.158629\pi$
$881$	4.04714e7	1.75675	0.878373	+	0.477976i	$0.158629\pi$
$882$	5.03016e6	0.217726
$883$	3.01707e7	1.30222	0.651108	−	0.758985i	$-0.274304\pi$
$883$	3.01707e7	1.30222	0.651108	+	0.758985i	$0.274304\pi$
$884$	−2.87518e6	−0.123747
$885$	0	0
$886$	−3.25868e7	−1.39462
$887$	1.68446e7	0.718874	0.359437	−	0.933169i	$-0.382969\pi$
$887$	1.68446e7	0.718874	0.359437	+	0.933169i	$0.382969\pi$
$888$	−9.22716e6	−0.392677
$889$	−5.24588e7	−2.22620
$890$	0	0
$891$	793881.	0.0335013
$892$	−3.17506e6	−0.133610
$893$	−4.30435e6	−0.180626
$894$	2.19620e7	0.919028
$895$	0	0
$896$	1.98449e7	0.825809
$897$	−3.02387e6	−0.125482
$898$	−3.62075e7	−1.49833
$899$	−5.07902e6	−0.209595
$900$	0	0
$901$	2.27258e7	0.932625
$902$	−4.72874e6	−0.193521
$903$	−2.66141e7	−1.08616
$904$	−3.66546e7	−1.49179
$905$	0	0
$906$	1.15504e7	0.467494
$907$	−3.35783e7	−1.35532	−0.677659	−	0.735377i	$-0.737005\pi$
$907$	−3.35783e7	−1.35532	−0.677659	+	0.735377i	$0.737005\pi$
$908$	−3.51317e6	−0.141411
$909$	1.11688e7	0.448328
$910$	0	0
$911$	−1.06806e7	−0.426385	−0.213192	−	0.977010i	$-0.568386\pi$
$911$	−1.06806e7	−0.426385	−0.213192	+	0.977010i	$0.568386\pi$
$912$	−4.43060e6	−0.176391
$913$	7.78153e6	0.308950
$914$	−1.27363e7	−0.504285
$915$	0	0
$916$	361598.	0.0142393
$917$	1.10594e7	0.434319
$918$	4.85741e6	0.190238
$919$	−2.12108e7	−0.828453	−0.414226	−	0.910174i	$-0.635948\pi$
$919$	−2.12108e7	−0.828453	−0.414226	+	0.910174i	$0.635948\pi$
$920$	0	0
$921$	1.73931e7	0.675661
$922$	−2.91586e7	−1.12964
$923$	1.76514e7	0.681986
$924$	−1.11909e6	−0.0431206
$925$	0	0
$926$	−2.87891e7	−1.10332
$927$	−5.17128e6	−0.197651
$928$	−7.39587e6	−0.281916
$929$	4.20462e7	1.59841	0.799203	−	0.601061i	$-0.205255\pi$
$929$	4.20462e7	1.59841	0.799203	+	0.601061i	$0.205255\pi$
$930$	0	0
$931$	7.55186e6	0.285548
$932$	1.70678e6	0.0643631
$933$	2.12315e7	0.798503
$934$	4.13240e7	1.55001
$935$	0	0
$936$	−5.71958e6	−0.213390
$937$	−5.84174e6	−0.217367	−0.108683	−	0.994076i	$-0.534663\pi$
$937$	−5.84174e6	−0.217367	−0.108683	+	0.994076i	$0.534663\pi$
$938$	2.98204e7	1.10664
$939$	−1.20559e6	−0.0446205
$940$	0	0
$941$	−1.24009e7	−0.456540	−0.228270	−	0.973598i	$-0.573307\pi$
$941$	−1.24009e7	−0.456540	−0.228270	+	0.973598i	$0.573307\pi$
$942$	−1.62086e6	−0.0595139
$943$	7.07272e6	0.259004
$944$	−2.70496e7	−0.987940
$945$	0	0
$946$	−1.07076e7	−0.389013
$947$	5.45771e7	1.97759	0.988793	−	0.149293i	$-0.0476998\pi$
$947$	5.45771e7	1.97759	0.988793	+	0.149293i	$0.0476998\pi$
$948$	−195059.	−0.00704928
$949$	−2.16560e7	−0.780571
$950$	0	0
$951$	−1.52184e7	−0.545655
$952$	−4.31535e7	−1.54321
$953$	1.12765e7	0.402198	0.201099	−	0.979571i	$-0.435549\pi$
$953$	1.12765e7	0.402198	0.201099	+	0.979571i	$0.435549\pi$
$954$	7.17322e6	0.255178
$955$	0	0
$956$	1.02359e7	0.362227
$957$	−3.73961e6	−0.131992
$958$	2.96770e7	1.04473
$959$	−5.22456e7	−1.83444
$960$	0	0
$961$	−2.64416e7	−0.923589
$962$	9.82071e6	0.342141
$963$	−3.04454e6	−0.105793
$964$	6.30622e6	0.218563
$965$	0	0
$966$	−7.20131e6	−0.248295
$967$	−2.71697e6	−0.0934370	−0.0467185	−	0.998908i	$-0.514876\pi$
$967$	−2.71697e6	−0.0934370	−0.0467185	+	0.998908i	$0.514876\pi$
$968$	−2.83758e6	−0.0973330
$969$	7.29249e6	0.249498
$970$	0	0
$971$	−3.01426e6	−0.102597	−0.0512983	−	0.998683i	$-0.516336\pi$
$971$	−3.01426e6	−0.102597	−0.0512983	+	0.998683i	$0.516336\pi$
$972$	−356364.	−0.0120984
$973$	7.08888e7	2.40046
$974$	−3.98843e7	−1.34711
$975$	0	0
$976$	−3.85744e7	−1.29621
$977$	−1.75203e7	−0.587227	−0.293613	−	0.955924i	$-0.594858\pi$
$977$	−1.75203e7	−0.587227	−0.293613	+	0.955924i	$0.594858\pi$
$978$	2.21213e7	0.739544
$979$	3.26171e6	0.108765
$980$	0	0
$981$	−1.20274e7	−0.399025
$982$	−3.03229e7	−1.00344
$983$	2.19475e7	0.724439	0.362219	−	0.932093i	$-0.382019\pi$
$983$	2.19475e7	0.724439	0.362219	+	0.932093i	$0.382019\pi$
$984$	1.33779e7	0.440452
$985$	0	0
$986$	−2.28810e7	−0.749521
$987$	−1.06452e7	−0.347827
$988$	−1.36249e6	−0.0444059
$989$	1.60152e7	0.520646
$990$	0	0
$991$	−4.54381e7	−1.46972	−0.734862	−	0.678216i	$-0.762753\pi$
$991$	−4.54381e7	−1.46972	−0.734862	+	0.678216i	$0.762753\pi$
$992$	3.18546e6	0.102776
$993$	2.81960e6	0.0907433
$994$	4.20366e7	1.34946
$995$	0	0
$996$	−3.49304e6	−0.111572
$997$	−1.38771e7	−0.442141	−0.221070	−	0.975258i	$-0.570955\pi$
$997$	−1.38771e7	−0.442141	−0.221070	+	0.975258i	$0.570955\pi$
$998$	−9.21907e6	−0.292995
$999$	3.85634e6	0.122254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.y.1.9		13
5.2	odd	4		165.6.c.b.34.19	yes	26
5.3	odd	4		165.6.c.b.34.8	✓	26
5.4	even	2		825.6.a.v.1.5		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.8	✓	26	5.3	odd	4
165.6.c.b.34.19	yes	26	5.2	odd	4
825.6.a.v.1.5		13	5.4	even	2
825.6.a.y.1.9		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+5.09558 q^{2} +9.00000 q^{3} -6.03505 q^{4} +45.8602 q^{6} +170.277 q^{7} -193.811 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+5.09558 q^{2} +9.00000 q^{3} -6.03505 q^{4} +45.8602 q^{6} +170.277 q^{7} -193.811 q^{8} +81.0000 q^{9} +121.000 q^{11} -54.3155 q^{12} +364.335 q^{13} +867.659 q^{14} -794.456 q^{16} +1307.63 q^{17} +412.742 q^{18} +619.655 q^{19} +1532.49 q^{21} +616.565 q^{22} -922.189 q^{23} -1744.30 q^{24} +1856.50 q^{26} +729.000 q^{27} -1027.63 q^{28} -3433.99 q^{29} +1479.05 q^{31} +2153.73 q^{32} +1089.00 q^{33} +6663.11 q^{34} -488.839 q^{36} +5289.91 q^{37} +3157.50 q^{38} +3279.02 q^{39} -7669.49 q^{41} +7808.93 q^{42} -17366.5 q^{43} -730.242 q^{44} -4699.09 q^{46} -6946.36 q^{47} -7150.11 q^{48} +12187.2 q^{49} +11768.6 q^{51} -2198.78 q^{52} +17379.4 q^{53} +3714.68 q^{54} -33001.5 q^{56} +5576.90 q^{57} -17498.2 q^{58} +34047.9 q^{59} +48554.5 q^{61} +7536.59 q^{62} +13792.4 q^{63} +36397.1 q^{64} +5549.09 q^{66} +34368.8 q^{67} -7891.59 q^{68} -8299.70 q^{69} +48448.3 q^{71} -15698.7 q^{72} -59439.7 q^{73} +26955.1 q^{74} -3739.65 q^{76} +20603.5 q^{77} +16708.5 q^{78} +3591.22 q^{79} +6561.00 q^{81} -39080.5 q^{82} +64310.2 q^{83} -9248.67 q^{84} -88492.6 q^{86} -30905.9 q^{87} -23451.1 q^{88} +26956.3 q^{89} +62037.9 q^{91} +5565.46 q^{92} +13311.4 q^{93} -35395.8 q^{94} +19383.5 q^{96} +16545.8 q^{97} +62100.8 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more