LMFDB - Embedded newform 882.6.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{130}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 130 $ x^2 - 130
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$11.4018$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} +21.0000 q^{5} +64.0000 q^{8} +84.0000 q^{10} -331.874 q^{11} +66.8211 q^{13} +256.000 q^{16} -240.537 q^{17} +441.979 q^{19} +336.000 q^{20} -1327.49 q^{22} +1071.37 q^{23} -2684.00 q^{25} +267.284 q^{26} -1791.75 q^{29} -5688.74 q^{31} +1024.00 q^{32} -962.147 q^{34} +11210.7 q^{37} +1767.92 q^{38} +1344.00 q^{40} -12077.9 q^{41} -9921.98 q^{43} -5309.98 q^{44} +4285.47 q^{46} +16869.2 q^{47} -10736.0 q^{50} +1069.14 q^{52} -5298.77 q^{53} -6969.35 q^{55} -7166.99 q^{58} -41384.8 q^{59} -21523.3 q^{61} -22754.9 q^{62} +4096.00 q^{64} +1403.24 q^{65} -26618.8 q^{67} -3848.59 q^{68} +58096.5 q^{71} -39987.5 q^{73} +44842.9 q^{74} +7071.66 q^{76} -43949.8 q^{79} +5376.00 q^{80} -48311.4 q^{82} -22421.4 q^{83} -5051.27 q^{85} -39687.9 q^{86} -21239.9 q^{88} -24062.0 q^{89} +17141.9 q^{92} +67477.0 q^{94} +9281.56 q^{95} +71896.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} + 42 q^{5} + 128 q^{8} + 168 q^{10} + 294 q^{11} - 140 q^{13} + 512 q^{16} - 1302 q^{17} - 1442 q^{19} + 672 q^{20} + 1176 q^{22} - 2646 q^{23} - 5368 q^{25} - 560 q^{26} - 1668 q^{29}+ \cdots - 43652 q^{97}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 + 42 * q^5 + 128 * q^8 + 168 * q^10 + 294 * q^11 - 140 * q^13 + 512 * q^16 - 1302 * q^17 - 1442 * q^19 + 672 * q^20 + 1176 * q^22 - 2646 * q^23 - 5368 * q^25 - 560 * q^26 - 1668 * q^29 - 14798 * q^31 + 2048 * q^32 - 5208 * q^34 + 5182 * q^37 - 5768 * q^38 + 2688 * q^40 + 5124 * q^41 - 4520 * q^43 + 4704 * q^44 - 10584 * q^46 + 14994 * q^47 - 21472 * q^50 - 2240 * q^52 - 24006 * q^53 + 6174 * q^55 - 6672 * q^58 - 38850 * q^59 - 23618 * q^61 - 59192 * q^62 + 8192 * q^64 - 2940 * q^65 + 32002 * q^67 - 20832 * q^68 + 89376 * q^71 - 47138 * q^73 + 20728 * q^74 - 23072 * q^76 - 40970 * q^79 + 10752 * q^80 + 20496 * q^82 - 68376 * q^83 - 27342 * q^85 - 18080 * q^86 + 18816 * q^88 - 123102 * q^89 - 42336 * q^92 + 59976 * q^94 - 30282 * q^95 - 43652 * q^97

Coefficient data

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

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

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	21.0000	0.375659	0.187830	−	0.982202i	$-0.439855\pi$
$5$	21.0000	0.375659	0.187830	+	0.982202i	$0.439855\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	0	0
$10$	84.0000	0.265631
$11$	−331.874	−0.826973	−0.413486	−	0.910510i	$-0.635689\pi$
$11$	−331.874	−0.826973	−0.413486	+	0.910510i	$0.635689\pi$
$12$	0	0
$13$	66.8211	0.109662	0.0548308	−	0.998496i	$-0.482538\pi$
$13$	66.8211	0.109662	0.0548308	+	0.998496i	$0.482538\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−240.537	−0.201864	−0.100932	−	0.994893i	$-0.532182\pi$
$17$	−240.537	−0.201864	−0.100932	+	0.994893i	$0.532182\pi$
$18$	0	0
$19$	441.979	0.280878	0.140439	−	0.990089i	$-0.455149\pi$
$19$	441.979	0.280878	0.140439	+	0.990089i	$0.455149\pi$
$20$	336.000	0.187830
$21$	0	0
$22$	−1327.49	−0.584758
$23$	1071.37	0.422298	0.211149	−	0.977454i	$-0.432279\pi$
$23$	1071.37	0.422298	0.211149	+	0.977454i	$0.432279\pi$
$24$	0	0
$25$	−2684.00	−0.858880
$26$	267.284	0.0775425
$27$	0	0
$28$	0	0
$29$	−1791.75	−0.395623	−0.197812	−	0.980240i	$-0.563383\pi$
$29$	−1791.75	−0.395623	−0.197812	+	0.980240i	$0.563383\pi$
$30$	0	0
$31$	−5688.74	−1.06319	−0.531596	−	0.846998i	$-0.678407\pi$
$31$	−5688.74	−1.06319	−0.531596	+	0.846998i	$0.678407\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−962.147	−0.142740
$35$	0	0
$36$	0	0
$37$	11210.7	1.34626	0.673131	−	0.739523i	$-0.264949\pi$
$37$	11210.7	1.34626	0.673131	+	0.739523i	$0.264949\pi$
$38$	1767.92	0.198611
$39$	0	0
$40$	1344.00	0.132816
$41$	−12077.9	−1.12210	−0.561048	−	0.827783i	$-0.689602\pi$
$41$	−12077.9	−1.12210	−0.561048	+	0.827783i	$0.689602\pi$
$42$	0	0
$43$	−9921.98	−0.818328	−0.409164	−	0.912461i	$-0.634180\pi$
$43$	−9921.98	−0.818328	−0.409164	+	0.912461i	$0.634180\pi$
$44$	−5309.98	−0.413486
$45$	0	0
$46$	4285.47	0.298610
$47$	16869.2	1.11391	0.556956	−	0.830542i	$-0.311969\pi$
$47$	16869.2	1.11391	0.556956	+	0.830542i	$0.311969\pi$
$48$	0	0
$49$	0	0
$50$	−10736.0	−0.607320
$51$	0	0
$52$	1069.14	0.0548308
$53$	−5298.77	−0.259111	−0.129555	−	0.991572i	$-0.541355\pi$
$53$	−5298.77	−0.259111	−0.129555	+	0.991572i	$0.541355\pi$
$54$	0	0
$55$	−6969.35	−0.310660
$56$	0	0
$57$	0	0
$58$	−7166.99	−0.279748
$59$	−41384.8	−1.54778	−0.773892	−	0.633317i	$-0.781693\pi$
$59$	−41384.8	−1.54778	−0.773892	+	0.633317i	$0.781693\pi$
$60$	0	0
$61$	−21523.3	−0.740601	−0.370300	−	0.928912i	$-0.620745\pi$
$61$	−21523.3	−0.740601	−0.370300	+	0.928912i	$0.620745\pi$
$62$	−22754.9	−0.751790
$63$	0	0
$64$	4096.00	0.125000
$65$	1403.24	0.0411954
$66$	0	0
$67$	−26618.8	−0.724437	−0.362219	−	0.932093i	$-0.617981\pi$
$67$	−26618.8	−0.724437	−0.362219	+	0.932093i	$0.617981\pi$
$68$	−3848.59	−0.100932
$69$	0	0
$70$	0	0
$71$	58096.5	1.36774	0.683870	−	0.729603i	$-0.260295\pi$
$71$	58096.5	1.36774	0.683870	+	0.729603i	$0.260295\pi$
$72$	0	0
$73$	−39987.5	−0.878248	−0.439124	−	0.898426i	$-0.644711\pi$
$73$	−39987.5	−0.878248	−0.439124	+	0.898426i	$0.644711\pi$
$74$	44842.9	0.951951
$75$	0	0
$76$	7071.66	0.140439
$77$	0	0
$78$	0	0
$79$	−43949.8	−0.792299	−0.396150	−	0.918186i	$-0.629654\pi$
$79$	−43949.8	−0.792299	−0.396150	+	0.918186i	$0.629654\pi$
$80$	5376.00	0.0939149
$81$	0	0
$82$	−48311.4	−0.793442
$83$	−22421.4	−0.357246	−0.178623	−	0.983918i	$-0.557164\pi$
$83$	−22421.4	−0.357246	−0.178623	+	0.983918i	$0.557164\pi$
$84$	0	0
$85$	−5051.27	−0.0758322
$86$	−39687.9	−0.578645
$87$	0	0
$88$	−21239.9	−0.292379
$89$	−24062.0	−0.322001	−0.161001	−	0.986954i	$-0.551472\pi$
$89$	−24062.0	−0.322001	−0.161001	+	0.986954i	$0.551472\pi$
$90$	0	0
$91$	0	0
$92$	17141.9	0.211149
$93$	0	0
$94$	67477.0	0.787655
$95$	9281.56	0.105514
$96$	0	0
$97$	71896.4	0.775850	0.387925	−	0.921691i	$-0.373192\pi$
$97$	71896.4	0.775850	0.387925	+	0.921691i	$0.373192\pi$
$98$	0	0
$99$	0	0
$100$	−42944.0	−0.429440
$101$	15865.5	0.154757	0.0773783	−	0.997002i	$-0.475345\pi$
$101$	15865.5	0.154757	0.0773783	+	0.997002i	$0.475345\pi$
$102$	0	0
$103$	−160574.	−1.49135	−0.745677	−	0.666307i	$-0.767874\pi$
$103$	−160574.	−1.49135	−0.745677	+	0.666307i	$0.767874\pi$
$104$	4276.55	0.0387713
$105$	0	0
$106$	−21195.1	−0.183219
$107$	205047.	1.73139	0.865693	−	0.500575i	$-0.166878\pi$
$107$	205047.	1.73139	0.865693	+	0.500575i	$0.166878\pi$
$108$	0	0
$109$	−112544.	−0.907313	−0.453657	−	0.891177i	$-0.649881\pi$
$109$	−112544.	−0.907313	−0.453657	+	0.891177i	$0.649881\pi$
$110$	−27877.4	−0.219670
$111$	0	0
$112$	0	0
$113$	118332.	0.871782	0.435891	−	0.900000i	$-0.356433\pi$
$113$	118332.	0.871782	0.435891	+	0.900000i	$0.356433\pi$
$114$	0	0
$115$	22498.7	0.158640
$116$	−28668.0	−0.197812
$117$	0	0
$118$	−165539.	−1.09445
$119$	0	0
$120$	0	0
$121$	−50910.9	−0.316116
$122$	−86093.2	−0.523684
$123$	0	0
$124$	−91019.8	−0.531596
$125$	−121989.	−0.698306
$126$	0	0
$127$	−245195.	−1.34897	−0.674485	−	0.738289i	$-0.735634\pi$
$127$	−245195.	−1.34897	−0.674485	+	0.738289i	$0.735634\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	5612.97	0.0291296
$131$	−69993.7	−0.356353	−0.178177	−	0.983999i	$-0.557020\pi$
$131$	−69993.7	−0.356353	−0.178177	+	0.983999i	$0.557020\pi$
$132$	0	0
$133$	0	0
$134$	−106475.	−0.512254
$135$	0	0
$136$	−15394.4	−0.0713698
$137$	187664.	0.854239	0.427119	−	0.904195i	$-0.359528\pi$
$137$	187664.	0.854239	0.427119	+	0.904195i	$0.359528\pi$
$138$	0	0
$139$	−78272.7	−0.343616	−0.171808	−	0.985130i	$-0.554961\pi$
$139$	−78272.7	−0.343616	−0.171808	+	0.985130i	$0.554961\pi$
$140$	0	0
$141$	0	0
$142$	232386.	0.967139
$143$	−22176.1	−0.0906872
$144$	0	0
$145$	−37626.7	−0.148620
$146$	−159950.	−0.621015
$147$	0	0
$148$	179372.	0.673131
$149$	92166.4	0.340100	0.170050	−	0.985435i	$-0.445607\pi$
$149$	92166.4	0.340100	0.170050	+	0.985435i	$0.445607\pi$
$150$	0	0
$151$	53416.0	0.190646	0.0953232	−	0.995446i	$-0.469612\pi$
$151$	53416.0	0.190646	0.0953232	+	0.995446i	$0.469612\pi$
$152$	28286.7	0.0993053
$153$	0	0
$154$	0	0
$155$	−119463.	−0.399398
$156$	0	0
$157$	−280501.	−0.908206	−0.454103	−	0.890949i	$-0.650040\pi$
$157$	−280501.	−0.908206	−0.454103	+	0.890949i	$0.650040\pi$
$158$	−175799.	−0.560240
$159$	0	0
$160$	21504.0	0.0664078
$161$	0	0
$162$	0	0
$163$	626810.	1.84785	0.923925	−	0.382574i	$-0.124962\pi$
$163$	626810.	1.84785	0.923925	+	0.382574i	$0.124962\pi$
$164$	−193246.	−0.561048
$165$	0	0
$166$	−89685.6	−0.252611
$167$	−293997.	−0.815741	−0.407870	−	0.913040i	$-0.633728\pi$
$167$	−293997.	−0.815741	−0.407870	+	0.913040i	$0.633728\pi$
$168$	0	0
$169$	−366828.	−0.987974
$170$	−20205.1	−0.0536215
$171$	0	0
$172$	−158752.	−0.409164
$173$	−714187.	−1.81425	−0.907124	−	0.420864i	$-0.861727\pi$
$173$	−714187.	−1.81425	−0.907124	+	0.420864i	$0.861727\pi$
$174$	0	0
$175$	0	0
$176$	−84959.7	−0.206743
$177$	0	0
$178$	−96248.1	−0.227689
$179$	−580050.	−1.35311	−0.676554	−	0.736393i	$-0.736528\pi$
$179$	−580050.	−1.35311	−0.676554	+	0.736393i	$0.736528\pi$
$180$	0	0
$181$	−308046.	−0.698907	−0.349454	−	0.936954i	$-0.613633\pi$
$181$	−308046.	−0.698907	−0.349454	+	0.936954i	$0.613633\pi$
$182$	0	0
$183$	0	0
$184$	68567.6	0.149305
$185$	235425.	0.505736
$186$	0	0
$187$	79827.8	0.166936
$188$	269908.	0.556956
$189$	0	0
$190$	37126.2	0.0746100
$191$	202789.	0.402218	0.201109	−	0.979569i	$-0.435545\pi$
$191$	202789.	0.402218	0.201109	+	0.979569i	$0.435545\pi$
$192$	0	0
$193$	605988.	1.17104	0.585519	−	0.810659i	$-0.300891\pi$
$193$	605988.	1.17104	0.585519	+	0.810659i	$0.300891\pi$
$194$	287586.	0.548609
$195$	0	0
$196$	0	0
$197$	310.819	0.000570614 0	0.000285307	−	1.00000i	$-0.499909\pi$
$197$	310.819	0.000570614 0	0.000285307		1.00000i	$0.499909\pi$
$198$	0	0
$199$	−409666.	−0.733327	−0.366664	−	0.930354i	$-0.619500\pi$
$199$	−409666.	−0.733327	−0.366664	+	0.930354i	$0.619500\pi$
$200$	−171776.	−0.303660
$201$	0	0
$202$	63461.9	0.109429
$203$	0	0
$204$	0	0
$205$	−253635.	−0.421526
$206$	−642294.	−1.05455
$207$	0	0
$208$	17106.2	0.0274154
$209$	−146681.	−0.232278
$210$	0	0
$211$	−441339.	−0.682442	−0.341221	−	0.939983i	$-0.610840\pi$
$211$	−441339.	−0.682442	−0.341221	+	0.939983i	$0.610840\pi$
$212$	−84780.3	−0.129555
$213$	0	0
$214$	820188.	1.22427
$215$	−208362.	−0.307412
$216$	0	0
$217$	0	0
$218$	−450177.	−0.641567
$219$	0	0
$220$	−111510.	−0.155330
$221$	−16072.9	−0.0221368
$222$	0	0
$223$	265133.	0.357028	0.178514	−	0.983937i	$-0.442871\pi$
$223$	265133.	0.357028	0.178514	+	0.983937i	$0.442871\pi$
$224$	0	0
$225$	0	0
$226$	473330.	0.616443
$227$	1.43044e6	1.84249	0.921244	−	0.388985i	$-0.127174\pi$
$227$	1.43044e6	1.84249	0.921244	+	0.388985i	$0.127174\pi$
$228$	0	0
$229$	−1.25805e6	−1.58529	−0.792646	−	0.609682i	$-0.791297\pi$
$229$	−1.25805e6	−1.58529	−0.792646	+	0.609682i	$0.791297\pi$
$230$	89994.9	0.112176
$231$	0	0
$232$	−114672.	−0.139874
$233$	−224723.	−0.271180	−0.135590	−	0.990765i	$-0.543293\pi$
$233$	−224723.	−0.271180	−0.135590	+	0.990765i	$0.543293\pi$
$234$	0	0
$235$	354254.	0.418452
$236$	−662156.	−0.773892
$237$	0	0
$238$	0	0
$239$	−1.28193e6	−1.45167	−0.725837	−	0.687867i	$-0.758547\pi$
$239$	−1.28193e6	−1.45167	−0.725837	+	0.687867i	$0.758547\pi$
$240$	0	0
$241$	−576125.	−0.638961	−0.319480	−	0.947593i	$-0.603508\pi$
$241$	−576125.	−0.638961	−0.319480	+	0.947593i	$0.603508\pi$
$242$	−203643.	−0.223528
$243$	0	0
$244$	−344373.	−0.370300
$245$	0	0
$246$	0	0
$247$	29533.5	0.0308015
$248$	−364079.	−0.375895
$249$	0	0
$250$	−487956.	−0.493777
$251$	609040.	0.610185	0.305092	−	0.952323i	$-0.401313\pi$
$251$	609040.	0.610185	0.305092	+	0.952323i	$0.401313\pi$
$252$	0	0
$253$	−355559.	−0.349229
$254$	−980780.	−0.953865
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.03067e6	−0.973389	−0.486695	−	0.873572i	$-0.661798\pi$
$257$	−1.03067e6	−0.973389	−0.486695	+	0.873572i	$0.661798\pi$
$258$	0	0
$259$	0	0
$260$	22451.9	0.0205977
$261$	0	0
$262$	−279975.	−0.251980
$263$	910585.	0.811767	0.405883	−	0.913925i	$-0.366964\pi$
$263$	910585.	0.811767	0.405883	+	0.913925i	$0.366964\pi$
$264$	0	0
$265$	−111274.	−0.0973374
$266$	0	0
$267$	0	0
$268$	−425900.	−0.362219
$269$	−1.42100e6	−1.19733	−0.598663	−	0.801001i	$-0.704301\pi$
$269$	−1.42100e6	−1.19733	−0.598663	+	0.801001i	$0.704301\pi$
$270$	0	0
$271$	−297530.	−0.246097	−0.123049	−	0.992401i	$-0.539267\pi$
$271$	−297530.	−0.246097	−0.123049	+	0.992401i	$0.539267\pi$
$272$	−61577.4	−0.0504661
$273$	0	0
$274$	750655.	0.604038
$275$	890749.	0.710270
$276$	0	0
$277$	849612.	0.665305	0.332653	−	0.943049i	$-0.392056\pi$
$277$	849612.	0.665305	0.332653	+	0.943049i	$0.392056\pi$
$278$	−313091.	−0.242973
$279$	0	0
$280$	0	0
$281$	7680.49	0.00580261	0.00290130	−	0.999996i	$-0.499076\pi$
$281$	7680.49	0.00580261	0.00290130	+	0.999996i	$0.499076\pi$
$282$	0	0
$283$	−985368.	−0.731362	−0.365681	−	0.930740i	$-0.619164\pi$
$283$	−985368.	−0.731362	−0.365681	+	0.930740i	$0.619164\pi$
$284$	929543.	0.683870
$285$	0	0
$286$	−88704.6	−0.0641255
$287$	0	0
$288$	0	0
$289$	−1.36200e6	−0.959251
$290$	−150507.	−0.105090
$291$	0	0
$292$	−639800.	−0.439124
$293$	2.16934e6	1.47625	0.738124	−	0.674665i	$-0.235712\pi$
$293$	2.16934e6	1.47625	0.738124	+	0.674665i	$0.235712\pi$
$294$	0	0
$295$	−869080.	−0.581440
$296$	717486.	0.475975
$297$	0	0
$298$	368666.	0.240487
$299$	71590.0	0.0463099
$300$	0	0
$301$	0	0
$302$	213664.	0.134807
$303$	0	0
$304$	113147.	0.0702195
$305$	−451989.	−0.278214
$306$	0	0
$307$	−1.39093e6	−0.842287	−0.421143	−	0.906994i	$-0.638371\pi$
$307$	−1.39093e6	−0.842287	−0.421143	+	0.906994i	$0.638371\pi$
$308$	0	0
$309$	0	0
$310$	−477854.	−0.282417
$311$	−2.03794e6	−1.19479	−0.597394	−	0.801948i	$-0.703797\pi$
$311$	−2.03794e6	−1.19479	−0.597394	+	0.801948i	$0.703797\pi$
$312$	0	0
$313$	−1.16958e6	−0.674789	−0.337395	−	0.941363i	$-0.609546\pi$
$313$	−1.16958e6	−0.674789	−0.337395	+	0.941363i	$0.609546\pi$
$314$	−1.12200e6	−0.642199
$315$	0	0
$316$	−703197.	−0.396150
$317$	−801137.	−0.447774	−0.223887	−	0.974615i	$-0.571875\pi$
$317$	−801137.	−0.447774	−0.223887	+	0.974615i	$0.571875\pi$
$318$	0	0
$319$	594634.	0.327170
$320$	86016.0	0.0469574
$321$	0	0
$322$	0	0
$323$	−106312.	−0.0566992
$324$	0	0
$325$	−179348.	−0.0941862
$326$	2.50724e6	1.30663
$327$	0	0
$328$	−772983.	−0.396721
$329$	0	0
$330$	0	0
$331$	2.64951e6	1.32922	0.664608	−	0.747192i	$-0.268598\pi$
$331$	2.64951e6	1.32922	0.664608	+	0.747192i	$0.268598\pi$
$332$	−358742.	−0.178623
$333$	0	0
$334$	−1.17599e6	−0.576816
$335$	−558994.	−0.272142
$336$	0	0
$337$	−3.40056e6	−1.63108	−0.815541	−	0.578699i	$-0.803560\pi$
$337$	−3.40056e6	−1.63108	−0.815541	+	0.578699i	$0.803560\pi$
$338$	−1.46731e6	−0.698603
$339$	0	0
$340$	−80820.4	−0.0379161
$341$	1.88794e6	0.879230
$342$	0	0
$343$	0	0
$344$	−635007.	−0.289322
$345$	0	0
$346$	−2.85675e6	−1.28287
$347$	617709.	0.275398	0.137699	−	0.990474i	$-0.456029\pi$
$347$	617709.	0.275398	0.137699	+	0.990474i	$0.456029\pi$
$348$	0	0
$349$	2.70539e6	1.18896	0.594479	−	0.804111i	$-0.297358\pi$
$349$	2.70539e6	1.18896	0.594479	+	0.804111i	$0.297358\pi$
$350$	0	0
$351$	0	0
$352$	−339839.	−0.146189
$353$	3.18064e6	1.35856	0.679279	−	0.733880i	$-0.262293\pi$
$353$	3.18064e6	1.35856	0.679279	+	0.733880i	$0.262293\pi$
$354$	0	0
$355$	1.22003e6	0.513805
$356$	−384993.	−0.161001
$357$	0	0
$358$	−2.32020e6	−0.956792
$359$	4.45970e6	1.82629	0.913145	−	0.407635i	$-0.133646\pi$
$359$	4.45970e6	1.82629	0.913145	+	0.407635i	$0.133646\pi$
$360$	0	0
$361$	−2.28075e6	−0.921108
$362$	−1.23219e6	−0.494202
$363$	0	0
$364$	0	0
$365$	−839738.	−0.329922
$366$	0	0
$367$	461247.	0.178759	0.0893795	−	0.995998i	$-0.471512\pi$
$367$	461247.	0.178759	0.0893795	+	0.995998i	$0.471512\pi$
$368$	274270.	0.105575
$369$	0	0
$370$	941701.	0.357609
$371$	0	0
$372$	0	0
$373$	3.41954e6	1.27261	0.636305	−	0.771437i	$-0.280462\pi$
$373$	3.41954e6	1.27261	0.636305	+	0.771437i	$0.280462\pi$
$374$	319311.	0.118042
$375$	0	0
$376$	1.07963e6	0.393827
$377$	−119726.	−0.0433847
$378$	0	0
$379$	16355.6	0.00584882	0.00292441	−	0.999996i	$-0.499069\pi$
$379$	16355.6	0.00584882	0.00292441	+	0.999996i	$0.499069\pi$
$380$	148505.	0.0527572
$381$	0	0
$382$	811158.	0.284411
$383$	3.43643e6	1.19705	0.598523	−	0.801105i	$-0.295754\pi$
$383$	3.43643e6	1.19705	0.598523	+	0.801105i	$0.295754\pi$
$384$	0	0
$385$	0	0
$386$	2.42395e6	0.828048
$387$	0	0
$388$	1.15034e6	0.387925
$389$	4.81052e6	1.61183	0.805914	−	0.592033i	$-0.201675\pi$
$389$	4.81052e6	1.61183	0.805914	+	0.592033i	$0.201675\pi$
$390$	0	0
$391$	−257704.	−0.0852469
$392$	0	0
$393$	0	0
$394$	1243.28	0.000403485 0
$395$	−922946.	−0.297635
$396$	0	0
$397$	−3.29848e6	−1.05036	−0.525180	−	0.850991i	$-0.676002\pi$
$397$	−3.29848e6	−1.05036	−0.525180	+	0.850991i	$0.676002\pi$
$398$	−1.63867e6	−0.518540
$399$	0	0
$400$	−687104.	−0.214720
$401$	−136164.	−0.0422865	−0.0211432	−	0.999776i	$-0.506731\pi$
$401$	−136164.	−0.0422865	−0.0211432	+	0.999776i	$0.506731\pi$
$402$	0	0
$403$	−380127.	−0.116591
$404$	253847.	0.0773783
$405$	0	0
$406$	0	0
$407$	−3.72054e6	−1.11332
$408$	0	0
$409$	2.74869e6	0.812490	0.406245	−	0.913764i	$-0.366838\pi$
$409$	2.74869e6	0.812490	0.406245	+	0.913764i	$0.366838\pi$
$410$	−1.01454e6	−0.298064
$411$	0	0
$412$	−2.56918e6	−0.745677
$413$	0	0
$414$	0	0
$415$	−470849.	−0.134203
$416$	68424.8	0.0193856
$417$	0	0
$418$	−586725.	−0.164246
$419$	1.87219e6	0.520973	0.260486	−	0.965478i	$-0.416117\pi$
$419$	1.87219e6	0.520973	0.260486	+	0.965478i	$0.416117\pi$
$420$	0	0
$421$	655225.	0.180171	0.0900856	−	0.995934i	$-0.471286\pi$
$421$	655225.	0.180171	0.0900856	+	0.995934i	$0.471286\pi$
$422$	−1.76535e6	−0.482559
$423$	0	0
$424$	−339121.	−0.0916095
$425$	645601.	0.173377
$426$	0	0
$427$	0	0
$428$	3.28075e6	0.865693
$429$	0	0
$430$	−833446.	−0.217373
$431$	−6.37690e6	−1.65355	−0.826773	−	0.562535i	$-0.809826\pi$
$431$	−6.37690e6	−1.65355	−0.826773	+	0.562535i	$0.809826\pi$
$432$	0	0
$433$	4.80003e6	1.23034	0.615168	−	0.788396i	$-0.289088\pi$
$433$	4.80003e6	1.23034	0.615168	+	0.788396i	$0.289088\pi$
$434$	0	0
$435$	0	0
$436$	−1.80071e6	−0.453657
$437$	473522.	0.118614
$438$	0	0
$439$	−5.59508e6	−1.38562	−0.692811	−	0.721119i	$-0.743628\pi$
$439$	−5.59508e6	−1.38562	−0.692811	+	0.721119i	$0.743628\pi$
$440$	−446038.	−0.109835
$441$	0	0
$442$	−64291.7	−0.0156531
$443$	−4.14010e6	−1.00231	−0.501154	−	0.865358i	$-0.667091\pi$
$443$	−4.14010e6	−1.00231	−0.501154	+	0.865358i	$0.667091\pi$
$444$	0	0
$445$	−505303.	−0.120963
$446$	1.06053e6	0.252457
$447$	0	0
$448$	0	0
$449$	305966.	0.0716239	0.0358119	−	0.999359i	$-0.488598\pi$
$449$	305966.	0.0716239	0.0358119	+	0.999359i	$0.488598\pi$
$450$	0	0
$451$	4.00832e6	0.927943
$452$	1.89332e6	0.435891
$453$	0	0
$454$	5.72176e6	1.30284
$455$	0	0
$456$	0	0
$457$	892608.	0.199926	0.0999632	−	0.994991i	$-0.468127\pi$
$457$	892608.	0.199926	0.0999632	+	0.994991i	$0.468127\pi$
$458$	−5.03220e6	−1.12097
$459$	0	0
$460$	359980.	0.0793202
$461$	3.01465e6	0.660670	0.330335	−	0.943864i	$-0.392838\pi$
$461$	3.01465e6	0.660670	0.330335	+	0.943864i	$0.392838\pi$
$462$	0	0
$463$	3.45497e6	0.749017	0.374508	−	0.927224i	$-0.377812\pi$
$463$	3.45497e6	0.749017	0.374508	+	0.927224i	$0.377812\pi$
$464$	−458687.	−0.0989058
$465$	0	0
$466$	−898893.	−0.191753
$467$	−2.83776e6	−0.602120	−0.301060	−	0.953605i	$-0.597340\pi$
$467$	−2.83776e6	−0.602120	−0.301060	+	0.953605i	$0.597340\pi$
$468$	0	0
$469$	0	0
$470$	1.41702e6	0.295890
$471$	0	0
$472$	−2.64863e6	−0.547225
$473$	3.29284e6	0.676734
$474$	0	0
$475$	−1.18627e6	−0.241240
$476$	0	0
$477$	0	0
$478$	−5.12772e6	−1.02649
$479$	4.70442e6	0.936845	0.468422	−	0.883505i	$-0.344823\pi$
$479$	4.70442e6	0.936845	0.468422	+	0.883505i	$0.344823\pi$
$480$	0	0
$481$	749113.	0.147633
$482$	−2.30450e6	−0.451813
$483$	0	0
$484$	−814574.	−0.158058
$485$	1.50982e6	0.291455
$486$	0	0
$487$	4.16634e6	0.796036	0.398018	−	0.917378i	$-0.369698\pi$
$487$	4.16634e6	0.796036	0.398018	+	0.917378i	$0.369698\pi$
$488$	−1.37749e6	−0.261842
$489$	0	0
$490$	0	0
$491$	−1.57876e6	−0.295537	−0.147768	−	0.989022i	$-0.547209\pi$
$491$	−1.57876e6	−0.295537	−0.147768	+	0.989022i	$0.547209\pi$
$492$	0	0
$493$	430981.	0.0798622
$494$	118134.	0.0217800
$495$	0	0
$496$	−1.45632e6	−0.265798
$497$	0	0
$498$	0	0
$499$	1.55712e6	0.279943	0.139972	−	0.990156i	$-0.455299\pi$
$499$	1.55712e6	0.279943	0.139972	+	0.990156i	$0.455299\pi$
$500$	−1.95182e6	−0.349153
$501$	0	0
$502$	2.43616e6	0.431466
$503$	1.19459e6	0.210523	0.105261	−	0.994445i	$-0.466432\pi$
$503$	1.19459e6	0.210523	0.105261	+	0.994445i	$0.466432\pi$
$504$	0	0
$505$	333175.	0.0581358
$506$	−1.42224e6	−0.246942
$507$	0	0
$508$	−3.92312e6	−0.674485
$509$	−3.16857e6	−0.542087	−0.271043	−	0.962567i	$-0.587369\pi$
$509$	−3.16857e6	−0.542087	−0.271043	+	0.962567i	$0.587369\pi$
$510$	0	0
$511$	0	0
$512$	262144.	0.0441942
$513$	0	0
$514$	−4.12268e6	−0.688290
$515$	−3.37204e6	−0.560241
$516$	0	0
$517$	−5.59846e6	−0.921175
$518$	0	0
$519$	0	0
$520$	89807.5	0.0145648
$521$	1.00658e6	0.162462	0.0812312	−	0.996695i	$-0.474115\pi$
$521$	1.00658e6	0.162462	0.0812312	+	0.996695i	$0.474115\pi$
$522$	0	0
$523$	8.78985e6	1.40517	0.702583	−	0.711602i	$-0.252030\pi$
$523$	8.78985e6	1.40517	0.702583	+	0.711602i	$0.252030\pi$
$524$	−1.11990e6	−0.178177
$525$	0	0
$526$	3.64234e6	0.574006
$527$	1.36835e6	0.214620
$528$	0	0
$529$	−5.28851e6	−0.821664
$530$	−445097.	−0.0688279
$531$	0	0
$532$	0	0
$533$	−807055.	−0.123051
$534$	0	0
$535$	4.30599e6	0.650412
$536$	−1.70360e6	−0.256127
$537$	0	0
$538$	−5.68399e6	−0.846638
$539$	0	0
$540$	0	0
$541$	−3.54864e6	−0.521277	−0.260639	−	0.965436i	$-0.583933\pi$
$541$	−3.54864e6	−0.521277	−0.260639	+	0.965436i	$0.583933\pi$
$542$	−1.19012e6	−0.174017
$543$	0	0
$544$	−246310.	−0.0356849
$545$	−2.36343e6	−0.340841
$546$	0	0
$547$	4.68179e6	0.669027	0.334513	−	0.942391i	$-0.391428\pi$
$547$	4.68179e6	0.669027	0.334513	+	0.942391i	$0.391428\pi$
$548$	3.00262e6	0.427119
$549$	0	0
$550$	3.56300e6	0.502237
$551$	−791915.	−0.111122
$552$	0	0
$553$	0	0
$554$	3.39845e6	0.470442
$555$	0	0
$556$	−1.25236e6	−0.171808
$557$	−5.79507e6	−0.791445	−0.395723	−	0.918370i	$-0.629506\pi$
$557$	−5.79507e6	−0.791445	−0.395723	+	0.918370i	$0.629506\pi$
$558$	0	0
$559$	−662997.	−0.0897392
$560$	0	0
$561$	0	0
$562$	30722.0	0.00410306
$563$	7.59703e6	1.01012	0.505060	−	0.863084i	$-0.331470\pi$
$563$	7.59703e6	1.01012	0.505060	+	0.863084i	$0.331470\pi$
$564$	0	0
$565$	2.48498e6	0.327493
$566$	−3.94147e6	−0.517151
$567$	0	0
$568$	3.71817e6	0.483569
$569$	9.81699e6	1.27115	0.635576	−	0.772038i	$-0.280763\pi$
$569$	9.81699e6	1.27115	0.635576	+	0.772038i	$0.280763\pi$
$570$	0	0
$571$	5.25888e6	0.674999	0.337499	−	0.941326i	$-0.390419\pi$
$571$	5.25888e6	0.674999	0.337499	+	0.941326i	$0.390419\pi$
$572$	−354818.	−0.0453436
$573$	0	0
$574$	0	0
$575$	−2.87555e6	−0.362703
$576$	0	0
$577$	8.63468e6	1.07971	0.539855	−	0.841758i	$-0.318479\pi$
$577$	8.63468e6	1.07971	0.539855	+	0.841758i	$0.318479\pi$
$578$	−5.44800e6	−0.678293
$579$	0	0
$580$	−602027.	−0.0743098
$581$	0	0
$582$	0	0
$583$	1.75852e6	0.214277
$584$	−2.55920e6	−0.310508
$585$	0	0
$586$	8.67737e6	1.04386
$587$	3.32014e6	0.397705	0.198852	−	0.980029i	$-0.436279\pi$
$587$	3.32014e6	0.397705	0.198852	+	0.980029i	$0.436279\pi$
$588$	0	0
$589$	−2.51430e6	−0.298627
$590$	−3.47632e6	−0.411140
$591$	0	0
$592$	2.86995e6	0.336565
$593$	6.40001e6	0.747384	0.373692	−	0.927553i	$-0.378092\pi$
$593$	6.40001e6	0.747384	0.373692	+	0.927553i	$0.378092\pi$
$594$	0	0
$595$	0	0
$596$	1.47466e6	0.170050
$597$	0	0
$598$	286360.	0.0327461
$599$	−7.49862e6	−0.853915	−0.426957	−	0.904272i	$-0.640415\pi$
$599$	−7.49862e6	−0.853915	−0.426957	+	0.904272i	$0.640415\pi$
$600$	0	0
$601$	−227052.	−0.0256412	−0.0128206	−	0.999918i	$-0.504081\pi$
$601$	−227052.	−0.0256412	−0.0128206	+	0.999918i	$0.504081\pi$
$602$	0	0
$603$	0	0
$604$	854656.	0.0953232
$605$	−1.06913e6	−0.118752
$606$	0	0
$607$	1.63037e7	1.79603	0.898015	−	0.439965i	$-0.145009\pi$
$607$	1.63037e7	1.79603	0.898015	+	0.439965i	$0.145009\pi$
$608$	452586.	0.0496527
$609$	0	0
$610$	−1.80796e6	−0.196727
$611$	1.12722e6	0.122153
$612$	0	0
$613$	−1.04062e7	−1.11852	−0.559259	−	0.828993i	$-0.688914\pi$
$613$	−1.04062e7	−1.11852	−0.559259	+	0.828993i	$0.688914\pi$
$614$	−5.56373e6	−0.595587
$615$	0	0
$616$	0	0
$617$	−4.74140e6	−0.501411	−0.250705	−	0.968063i	$-0.580663\pi$
$617$	−4.74140e6	−0.501411	−0.250705	+	0.968063i	$0.580663\pi$
$618$	0	0
$619$	−1.08534e7	−1.13851	−0.569256	−	0.822160i	$-0.692769\pi$
$619$	−1.08534e7	−1.13851	−0.569256	+	0.822160i	$0.692769\pi$
$620$	−1.91142e6	−0.199699
$621$	0	0
$622$	−8.15177e6	−0.844843
$623$	0	0
$624$	0	0
$625$	5.82573e6	0.596555
$626$	−4.67831e6	−0.477148
$627$	0	0
$628$	−4.48801e6	−0.454103
$629$	−2.69659e6	−0.271762
$630$	0	0
$631$	1.58978e7	1.58951	0.794757	−	0.606928i	$-0.207598\pi$
$631$	1.58978e7	1.58951	0.794757	+	0.606928i	$0.207598\pi$
$632$	−2.81279e6	−0.280120
$633$	0	0
$634$	−3.20455e6	−0.316624
$635$	−5.14909e6	−0.506753
$636$	0	0
$637$	0	0
$638$	2.37854e6	0.231344
$639$	0	0
$640$	344064.	0.0332039
$641$	−1.11828e7	−1.07499	−0.537496	−	0.843267i	$-0.680629\pi$
$641$	−1.11828e7	−1.07499	−0.537496	+	0.843267i	$0.680629\pi$
$642$	0	0
$643$	1.55983e6	0.148782	0.0743909	−	0.997229i	$-0.476299\pi$
$643$	1.55983e6	0.148782	0.0743909	+	0.997229i	$0.476299\pi$
$644$	0	0
$645$	0	0
$646$	−425249.	−0.0400924
$647$	1.54824e7	1.45404	0.727020	−	0.686616i	$-0.240905\pi$
$647$	1.54824e7	1.45404	0.727020	+	0.686616i	$0.240905\pi$
$648$	0	0
$649$	1.37345e7	1.27998
$650$	−717391.	−0.0665997
$651$	0	0
$652$	1.00290e7	0.923925
$653$	−614752.	−0.0564179	−0.0282090	−	0.999602i	$-0.508980\pi$
$653$	−614752.	−0.0564179	−0.0282090	+	0.999602i	$0.508980\pi$
$654$	0	0
$655$	−1.46987e6	−0.133867
$656$	−3.09193e6	−0.280524
$657$	0	0
$658$	0	0
$659$	1.32697e7	1.19028	0.595139	−	0.803623i	$-0.297097\pi$
$659$	1.32697e7	1.19028	0.595139	+	0.803623i	$0.297097\pi$
$660$	0	0
$661$	5.03584e6	0.448299	0.224150	−	0.974555i	$-0.428040\pi$
$661$	5.03584e6	0.448299	0.224150	+	0.974555i	$0.428040\pi$
$662$	1.05980e7	0.939898
$663$	0	0
$664$	−1.43497e6	−0.126306
$665$	0	0
$666$	0	0
$667$	−1.91962e6	−0.167071
$668$	−4.70396e6	−0.407870
$669$	0	0
$670$	−2.23598e6	−0.192433
$671$	7.14301e6	0.612457
$672$	0	0
$673$	8.28068e6	0.704739	0.352369	−	0.935861i	$-0.385376\pi$
$673$	8.28068e6	0.704739	0.352369	+	0.935861i	$0.385376\pi$
$674$	−1.36022e7	−1.15335
$675$	0	0
$676$	−5.86925e6	−0.493987
$677$	1.52513e7	1.27889	0.639446	−	0.768836i	$-0.279164\pi$
$677$	1.52513e7	1.27889	0.639446	+	0.768836i	$0.279164\pi$
$678$	0	0
$679$	0	0
$680$	−323282.	−0.0268107
$681$	0	0
$682$	7.55177e6	0.621710
$683$	−7.88673e6	−0.646912	−0.323456	−	0.946243i	$-0.604845\pi$
$683$	−7.88673e6	−0.646912	−0.323456	+	0.946243i	$0.604845\pi$
$684$	0	0
$685$	3.94094e6	0.320903
$686$	0	0
$687$	0	0
$688$	−2.54003e6	−0.204582
$689$	−354069.	−0.0284145
$690$	0	0
$691$	−1.66190e6	−0.132407	−0.0662033	−	0.997806i	$-0.521089\pi$
$691$	−1.66190e6	−0.132407	−0.0662033	+	0.997806i	$0.521089\pi$
$692$	−1.14270e7	−0.907124
$693$	0	0
$694$	2.47084e6	0.194736
$695$	−1.64373e6	−0.129083
$696$	0	0
$697$	2.90517e6	0.226511
$698$	1.08216e7	0.840721
$699$	0	0
$700$	0	0
$701$	1.39364e7	1.07117	0.535583	−	0.844483i	$-0.320092\pi$
$701$	1.39364e7	1.07117	0.535583	+	0.844483i	$0.320092\pi$
$702$	0	0
$703$	4.95490e6	0.378135
$704$	−1.35935e6	−0.103372
$705$	0	0
$706$	1.27226e7	0.960646
$707$	0	0
$708$	0	0
$709$	−1.01874e7	−0.761108	−0.380554	−	0.924759i	$-0.624267\pi$
$709$	−1.01874e7	−0.761108	−0.380554	+	0.924759i	$0.624267\pi$
$710$	4.88010e6	0.363315
$711$	0	0
$712$	−1.53997e6	−0.113845
$713$	−6.09473e6	−0.448984
$714$	0	0
$715$	−465699.	−0.0340675
$716$	−9.28079e6	−0.676554
$717$	0	0
$718$	1.78388e7	1.29138
$719$	−1.39169e7	−1.00397	−0.501983	−	0.864877i	$-0.667396\pi$
$719$	−1.39169e7	−1.00397	−0.501983	+	0.864877i	$0.667396\pi$
$720$	0	0
$721$	0	0
$722$	−9.12301e6	−0.651321
$723$	0	0
$724$	−4.92874e6	−0.349454
$725$	4.80905e6	0.339793
$726$	0	0
$727$	3.42063e6	0.240033	0.120016	−	0.992772i	$-0.461705\pi$
$727$	3.42063e6	0.240033	0.120016	+	0.992772i	$0.461705\pi$
$728$	0	0
$729$	0	0
$730$	−3.35895e6	−0.233290
$731$	2.38660e6	0.165191
$732$	0	0
$733$	−1.03162e7	−0.709186	−0.354593	−	0.935021i	$-0.615381\pi$
$733$	−1.03162e7	−0.709186	−0.354593	+	0.935021i	$0.615381\pi$
$734$	1.84499e6	0.126402
$735$	0	0
$736$	1.09708e6	0.0746525
$737$	8.83406e6	0.599090
$738$	0	0
$739$	−1.03890e7	−0.699779	−0.349889	−	0.936791i	$-0.613781\pi$
$739$	−1.03890e7	−0.699779	−0.349889	+	0.936791i	$0.613781\pi$
$740$	3.76680e6	0.252868
$741$	0	0
$742$	0	0
$743$	1.04738e7	0.696035	0.348018	−	0.937488i	$-0.386855\pi$
$743$	1.04738e7	0.696035	0.348018	+	0.937488i	$0.386855\pi$
$744$	0	0
$745$	1.93550e6	0.127762
$746$	1.36782e7	0.899872
$747$	0	0
$748$	1.27725e6	0.0834681
$749$	0	0
$750$	0	0
$751$	−9.40445e6	−0.608462	−0.304231	−	0.952598i	$-0.598400\pi$
$751$	−9.40445e6	−0.608462	−0.304231	+	0.952598i	$0.598400\pi$
$752$	4.31853e6	0.278478
$753$	0	0
$754$	−478906.	−0.0306776
$755$	1.12174e6	0.0716181
$756$	0	0
$757$	−1.33677e7	−0.847848	−0.423924	−	0.905698i	$-0.639348\pi$
$757$	−1.33677e7	−0.847848	−0.423924	+	0.905698i	$0.639348\pi$
$758$	65422.4	0.00413574
$759$	0	0
$760$	594020.	0.0373050
$761$	2.22623e7	1.39350	0.696752	−	0.717312i	$-0.254628\pi$
$761$	2.22623e7	1.39350	0.696752	+	0.717312i	$0.254628\pi$
$762$	0	0
$763$	0	0
$764$	3.24463e6	0.201109
$765$	0	0
$766$	1.37457e7	0.846440
$767$	−2.76537e6	−0.169733
$768$	0	0
$769$	9.65833e6	0.588961	0.294480	−	0.955658i	$-0.404853\pi$
$769$	9.65833e6	0.588961	0.294480	+	0.955658i	$0.404853\pi$
$770$	0	0
$771$	0	0
$772$	9.69581e6	0.585519
$773$	5.10951e6	0.307561	0.153780	−	0.988105i	$-0.450855\pi$
$773$	5.10951e6	0.307561	0.153780	+	0.988105i	$0.450855\pi$
$774$	0	0
$775$	1.52686e7	0.913154
$776$	4.60137e6	0.274305
$777$	0	0
$778$	1.92421e7	1.13973
$779$	−5.33816e6	−0.315172
$780$	0	0
$781$	−1.92807e7	−1.13108
$782$	−1.03081e6	−0.0602787
$783$	0	0
$784$	0	0
$785$	−5.89051e6	−0.341176
$786$	0	0
$787$	2.51592e7	1.44797	0.723984	−	0.689816i	$-0.242309\pi$
$787$	2.51592e7	1.44797	0.723984	+	0.689816i	$0.242309\pi$
$788$	4973.11	0.000285307 0
$789$	0	0
$790$	−3.69178e6	−0.210460
$791$	0	0
$792$	0	0
$793$	−1.43821e6	−0.0812155
$794$	−1.31939e7	−0.742716
$795$	0	0
$796$	−6.55466e6	−0.366664
$797$	3.35976e6	0.187354	0.0936768	−	0.995603i	$-0.470138\pi$
$797$	3.35976e6	0.187354	0.0936768	+	0.995603i	$0.470138\pi$
$798$	0	0
$799$	−4.05767e6	−0.224859
$800$	−2.74842e6	−0.151830
$801$	0	0
$802$	−544656.	−0.0299010
$803$	1.32708e7	0.726287
$804$	0	0
$805$	0	0
$806$	−1.52051e6	−0.0824426
$807$	0	0
$808$	1.01539e6	0.0547147
$809$	2.97050e7	1.59572	0.797862	−	0.602840i	$-0.205964\pi$
$809$	2.97050e7	1.59572	0.797862	+	0.602840i	$0.205964\pi$
$810$	0	0
$811$	1.26386e7	0.674757	0.337379	−	0.941369i	$-0.390460\pi$
$811$	1.26386e7	0.674757	0.337379	+	0.941369i	$0.390460\pi$
$812$	0	0
$813$	0	0
$814$	−1.48822e7	−0.787237
$815$	1.31630e7	0.694162
$816$	0	0
$817$	−4.38531e6	−0.229850
$818$	1.09948e7	0.574517
$819$	0	0
$820$	−4.05816e6	−0.210763
$821$	2.61760e7	1.35533	0.677666	−	0.735370i	$-0.262992\pi$
$821$	2.61760e7	1.35533	0.677666	+	0.735370i	$0.262992\pi$
$822$	0	0
$823$	−7.29843e6	−0.375604	−0.187802	−	0.982207i	$-0.560136\pi$
$823$	−7.29843e6	−0.375604	−0.187802	+	0.982207i	$0.560136\pi$
$824$	−1.02767e7	−0.527274
$825$	0	0
$826$	0	0
$827$	−1.18681e7	−0.603418	−0.301709	−	0.953400i	$-0.597557\pi$
$827$	−1.18681e7	−0.603418	−0.301709	+	0.953400i	$0.597557\pi$
$828$	0	0
$829$	−1.04338e7	−0.527298	−0.263649	−	0.964619i	$-0.584926\pi$
$829$	−1.04338e7	−0.527298	−0.263649	+	0.964619i	$0.584926\pi$
$830$	−1.88340e6	−0.0948957
$831$	0	0
$832$	273699.	0.0137077
$833$	0	0
$834$	0	0
$835$	−6.17394e6	−0.306441
$836$	−2.34690e6	−0.116139
$837$	0	0
$838$	7.48876e6	0.368383
$839$	2.91444e7	1.42939	0.714695	−	0.699436i	$-0.246565\pi$
$839$	2.91444e7	1.42939	0.714695	+	0.699436i	$0.246565\pi$
$840$	0	0
$841$	−1.73008e7	−0.843482
$842$	2.62090e6	0.127400
$843$	0	0
$844$	−7.06142e6	−0.341221
$845$	−7.70339e6	−0.371142
$846$	0	0
$847$	0	0
$848$	−1.35648e6	−0.0647777
$849$	0	0
$850$	2.58240e6	0.122596
$851$	1.20108e7	0.568524
$852$	0	0
$853$	2.63032e7	1.23776	0.618879	−	0.785487i	$-0.287587\pi$
$853$	2.63032e7	1.23776	0.618879	+	0.785487i	$0.287587\pi$
$854$	0	0
$855$	0	0
$856$	1.31230e7	0.612137
$857$	−3.43598e6	−0.159808	−0.0799040	−	0.996803i	$-0.525461\pi$
$857$	−3.43598e6	−0.159808	−0.0799040	+	0.996803i	$0.525461\pi$
$858$	0	0
$859$	1.09139e7	0.504659	0.252329	−	0.967641i	$-0.418803\pi$
$859$	1.09139e7	0.504659	0.252329	+	0.967641i	$0.418803\pi$
$860$	−3.33378e6	−0.153706
$861$	0	0
$862$	−2.55076e7	−1.16923
$863$	−2.19058e7	−1.00123	−0.500614	−	0.865671i	$-0.666892\pi$
$863$	−2.19058e7	−1.00123	−0.500614	+	0.865671i	$0.666892\pi$
$864$	0	0
$865$	−1.49979e7	−0.681539
$866$	1.92001e7	0.869979
$867$	0	0
$868$	0	0
$869$	1.45858e7	0.655210
$870$	0	0
$871$	−1.77869e6	−0.0794430
$872$	−7.20283e6	−0.320784
$873$	0	0
$874$	1.89409e6	0.0838729
$875$	0	0
$876$	0	0
$877$	6.46927e6	0.284025	0.142012	−	0.989865i	$-0.454643\pi$
$877$	6.46927e6	0.284025	0.142012	+	0.989865i	$0.454643\pi$
$878$	−2.23803e7	−0.979782
$879$	0	0
$880$	−1.78415e6	−0.0776650
$881$	−2.03983e7	−0.885430	−0.442715	−	0.896662i	$-0.645985\pi$
$881$	−2.03983e7	−0.885430	−0.442715	+	0.896662i	$0.645985\pi$
$882$	0	0
$883$	1.62381e7	0.700862	0.350431	−	0.936589i	$-0.386035\pi$
$883$	1.62381e7	0.700862	0.350431	+	0.936589i	$0.386035\pi$
$884$	−257167.	−0.0110684
$885$	0	0
$886$	−1.65604e7	−0.708739
$887$	6.79027e6	0.289786	0.144893	−	0.989447i	$-0.453716\pi$
$887$	6.79027e6	0.289786	0.144893	+	0.989447i	$0.453716\pi$
$888$	0	0
$889$	0	0
$890$	−2.02121e6	−0.0855336
$891$	0	0
$892$	4.24213e6	0.178514
$893$	7.45585e6	0.312873
$894$	0	0
$895$	−1.21810e7	−0.508308
$896$	0	0
$897$	0	0
$898$	1.22387e6	0.0506457
$899$	1.01928e7	0.420623
$900$	0	0
$901$	1.27455e6	0.0523052
$902$	1.60333e7	0.656155
$903$	0	0
$904$	7.57328e6	0.308221
$905$	−6.46897e6	−0.262551
$906$	0	0
$907$	3.42142e7	1.38098	0.690492	−	0.723340i	$-0.257394\pi$
$907$	3.42142e7	1.38098	0.690492	+	0.723340i	$0.257394\pi$
$908$	2.28870e7	0.921244
$909$	0	0
$910$	0	0
$911$	4.47390e6	0.178604	0.0893019	−	0.996005i	$-0.471536\pi$
$911$	4.47390e6	0.178604	0.0893019	+	0.996005i	$0.471536\pi$
$912$	0	0
$913$	7.44107e6	0.295433
$914$	3.57043e6	0.141369
$915$	0	0
$916$	−2.01288e7	−0.792646
$917$	0	0
$918$	0	0
$919$	−537900.	−0.0210094	−0.0105047	−	0.999945i	$-0.503344\pi$
$919$	−537900.	−0.0210094	−0.0105047	+	0.999945i	$0.503344\pi$
$920$	1.43992e6	0.0560878
$921$	0	0
$922$	1.20586e7	0.467164
$923$	3.88207e6	0.149989
$924$	0	0
$925$	−3.00896e7	−1.15628
$926$	1.38199e7	0.529635
$927$	0	0
$928$	−1.83475e6	−0.0699370
$929$	1.77241e7	0.673792	0.336896	−	0.941542i	$-0.390623\pi$
$929$	1.77241e7	0.673792	0.336896	+	0.941542i	$0.390623\pi$
$930$	0	0
$931$	0	0
$932$	−3.59557e6	−0.135590
$933$	0	0
$934$	−1.13510e7	−0.425763
$935$	1.67638e6	0.0627111
$936$	0	0
$937$	−3.32444e7	−1.23700	−0.618499	−	0.785786i	$-0.712259\pi$
$937$	−3.32444e7	−1.23700	−0.618499	+	0.785786i	$0.712259\pi$
$938$	0	0
$939$	0	0
$940$	5.66807e6	0.209226
$941$	−2.78480e7	−1.02523	−0.512613	−	0.858620i	$-0.671322\pi$
$941$	−2.78480e7	−1.02523	−0.512613	+	0.858620i	$0.671322\pi$
$942$	0	0
$943$	−1.29398e7	−0.473859
$944$	−1.05945e7	−0.386946
$945$	0	0
$946$	1.31714e7	0.478523
$947$	2.30569e7	0.835459	0.417729	−	0.908571i	$-0.362826\pi$
$947$	2.30569e7	0.835459	0.417729	+	0.908571i	$0.362826\pi$
$948$	0	0
$949$	−2.67201e6	−0.0963102
$950$	−4.74509e6	−0.170583
$951$	0	0
$952$	0	0
$953$	−2.85536e7	−1.01843	−0.509213	−	0.860641i	$-0.670063\pi$
$953$	−2.85536e7	−1.01843	−0.509213	+	0.860641i	$0.670063\pi$
$954$	0	0
$955$	4.25858e6	0.151097
$956$	−2.05109e7	−0.725837
$957$	0	0
$958$	1.88177e7	0.662449
$959$	0	0
$960$	0	0
$961$	3.73258e6	0.130377
$962$	2.99645e6	0.104393
$963$	0	0
$964$	−9.21800e6	−0.319480
$965$	1.27257e7	0.439911
$966$	0	0
$967$	−3.45310e7	−1.18753	−0.593763	−	0.804640i	$-0.702358\pi$
$967$	−3.45310e7	−1.18753	−0.593763	+	0.804640i	$0.702358\pi$
$968$	−3.25830e6	−0.111764
$969$	0	0
$970$	6.03930e6	0.206090
$971$	−1.24933e7	−0.425236	−0.212618	−	0.977135i	$-0.568199\pi$
$971$	−1.24933e7	−0.425236	−0.212618	+	0.977135i	$0.568199\pi$
$972$	0	0
$973$	0	0
$974$	1.66654e7	0.562883
$975$	0	0
$976$	−5.50996e6	−0.185150
$977$	3.89440e7	1.30528	0.652641	−	0.757667i	$-0.273661\pi$
$977$	3.89440e7	1.30528	0.652641	+	0.757667i	$0.273661\pi$
$978$	0	0
$979$	7.98556e6	0.266286
$980$	0	0
$981$	0	0
$982$	−6.31503e6	−0.208976
$983$	−279901.	−0.00923889	−0.00461945	−	0.999989i	$-0.501470\pi$
$983$	−279901.	−0.00923889	−0.00461945	+	0.999989i	$0.501470\pi$
$984$	0	0
$985$	6527.20	0.000214356 0
$986$	1.72393e6	0.0564711
$987$	0	0
$988$	472536.	0.0154008
$989$	−1.06301e7	−0.345578
$990$	0	0
$991$	1.96343e7	0.635084	0.317542	−	0.948244i	$-0.397142\pi$
$991$	1.96343e7	0.635084	0.317542	+	0.948244i	$0.397142\pi$
$992$	−5.82527e6	−0.187948
$993$	0	0
$994$	0	0
$995$	−8.60299e6	−0.275481
$996$	0	0
$997$	−3.73374e7	−1.18961	−0.594807	−	0.803868i	$-0.702772\pi$
$997$	−3.73374e7	−1.18961	−0.594807	+	0.803868i	$0.702772\pi$
$998$	6.22847e6	0.197950
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bt.1.1		2
3.2	odd	2		98.6.a.c.1.1		2
7.2	even	3		126.6.g.e.109.1		4
7.4	even	3		126.6.g.e.37.1		4
7.6	odd	2		882.6.a.bl.1.1		2
12.11	even	2		784.6.a.bc.1.2		2
21.2	odd	6		14.6.c.b.11.2	yes	4
21.5	even	6		98.6.c.f.67.1		4
21.11	odd	6		14.6.c.b.9.2	✓	4
21.17	even	6		98.6.c.f.79.1		4
21.20	even	2		98.6.a.f.1.2		2
84.11	even	6		112.6.i.b.65.1		4
84.23	even	6		112.6.i.b.81.1		4
84.83	odd	2		784.6.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.2	✓	4	21.11	odd	6
14.6.c.b.11.2	yes	4	21.2	odd	6
98.6.a.c.1.1		2	3.2	odd	2
98.6.a.f.1.2		2	21.20	even	2
98.6.c.f.67.1		4	21.5	even	6
98.6.c.f.79.1		4	21.17	even	6
112.6.i.b.65.1		4	84.11	even	6
112.6.i.b.81.1		4	84.23	even	6
126.6.g.e.37.1		4	7.4	even	3
126.6.g.e.109.1		4	7.2	even	3
784.6.a.r.1.1		2	84.83	odd	2
784.6.a.bc.1.2		2	12.11	even	2
882.6.a.bl.1.1		2	7.6	odd	2
882.6.a.bt.1.1		2	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} +21.0000 q^{5} +64.0000 q^{8} +84.0000 q^{10} -331.874 q^{11} +66.8211 q^{13} +256.000 q^{16} -240.537 q^{17} +441.979 q^{19} +336.000 q^{20} -1327.49 q^{22} +1071.37 q^{23} -2684.00 q^{25} +267.284 q^{26} -1791.75 q^{29} -5688.74 q^{31} +1024.00 q^{32} -962.147 q^{34} +11210.7 q^{37} +1767.92 q^{38} +1344.00 q^{40} -12077.9 q^{41} -9921.98 q^{43} -5309.98 q^{44} +4285.47 q^{46} +16869.2 q^{47} -10736.0 q^{50} +1069.14 q^{52} -5298.77 q^{53} -6969.35 q^{55} -7166.99 q^{58} -41384.8 q^{59} -21523.3 q^{61} -22754.9 q^{62} +4096.00 q^{64} +1403.24 q^{65} -26618.8 q^{67} -3848.59 q^{68} +58096.5 q^{71} -39987.5 q^{73} +44842.9 q^{74} +7071.66 q^{76} -43949.8 q^{79} +5376.00 q^{80} -48311.4 q^{82} -22421.4 q^{83} -5051.27 q^{85} -39687.9 q^{86} -21239.9 q^{88} -24062.0 q^{89} +17141.9 q^{92} +67477.0 q^{94} +9281.56 q^{95} +71896.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} + 42 q^{5} + 128 q^{8} + 168 q^{10} + 294 q^{11} - 140 q^{13} + 512 q^{16} - 1302 q^{17} - 1442 q^{19} + 672 q^{20} + 1176 q^{22} - 2646 q^{23} - 5368 q^{25} - 560 q^{26} - 1668 q^{29}+ \cdots - 43652 q^{97}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 + 42 * q^5 + 128 * q^8 + 168 * q^10 + 294 * q^11 - 140 * q^13 + 512 * q^16 - 1302 * q^17 - 1442 * q^19 + 672 * q^20 + 1176 * q^22 - 2646 * q^23 - 5368 * q^25 - 560 * q^26 - 1668 * q^29 - 14798 * q^31 + 2048 * q^32 - 5208 * q^34 + 5182 * q^37 - 5768 * q^38 + 2688 * q^40 + 5124 * q^41 - 4520 * q^43 + 4704 * q^44 - 10584 * q^46 + 14994 * q^47 - 21472 * q^50 - 2240 * q^52 - 24006 * q^53 + 6174 * q^55 - 6672 * q^58 - 38850 * q^59 - 23618 * q^61 - 59192 * q^62 + 8192 * q^64 - 2940 * q^65 + 32002 * q^67 - 20832 * q^68 + 89376 * q^71 - 47138 * q^73 + 20728 * q^74 - 23072 * q^76 - 40970 * q^79 + 10752 * q^80 + 20496 * q^82 - 68376 * q^83 - 27342 * q^85 - 18080 * q^86 + 18816 * q^88 - 123102 * q^89 - 42336 * q^92 + 59976 * q^94 - 30282 * q^95 - 43652 * q^97

Introduction

L-functions

Modular forms

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