# Properties

 Label 350.6.c.d.99.1 Level $350$ Weight $6$ Character 350.99 Analytic conductor $56.134$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 350.c (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: $$56.1343369345$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 350.99 Dual form 350.6.c.d.99.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000i q^{2} -10.0000i q^{3} -16.0000 q^{4} -40.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} +143.000 q^{9} +O(q^{10})$$ $$q-4.00000i q^{2} -10.0000i q^{3} -16.0000 q^{4} -40.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} +143.000 q^{9} -336.000 q^{11} +160.000i q^{12} -584.000i q^{13} +196.000 q^{14} +256.000 q^{16} -1458.00i q^{17} -572.000i q^{18} -470.000 q^{19} +490.000 q^{21} +1344.00i q^{22} +4200.00i q^{23} +640.000 q^{24} -2336.00 q^{26} -3860.00i q^{27} -784.000i q^{28} -4866.00 q^{29} -7372.00 q^{31} -1024.00i q^{32} +3360.00i q^{33} -5832.00 q^{34} -2288.00 q^{36} +14330.0i q^{37} +1880.00i q^{38} -5840.00 q^{39} +6222.00 q^{41} -1960.00i q^{42} -3704.00i q^{43} +5376.00 q^{44} +16800.0 q^{46} -1812.00i q^{47} -2560.00i q^{48} -2401.00 q^{49} -14580.0 q^{51} +9344.00i q^{52} +37242.0i q^{53} -15440.0 q^{54} -3136.00 q^{56} +4700.00i q^{57} +19464.0i q^{58} -34302.0 q^{59} +24476.0 q^{61} +29488.0i q^{62} +7007.00i q^{63} -4096.00 q^{64} +13440.0 q^{66} -17452.0i q^{67} +23328.0i q^{68} +42000.0 q^{69} +28224.0 q^{71} +9152.00i q^{72} -3602.00i q^{73} +57320.0 q^{74} +7520.00 q^{76} -16464.0i q^{77} +23360.0i q^{78} -42872.0 q^{79} -3851.00 q^{81} -24888.0i q^{82} +35202.0i q^{83} -7840.00 q^{84} -14816.0 q^{86} +48660.0i q^{87} -21504.0i q^{88} -26730.0 q^{89} +28616.0 q^{91} -67200.0i q^{92} +73720.0i q^{93} -7248.00 q^{94} -10240.0 q^{96} -16978.0i q^{97} +9604.00i q^{98} -48048.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 80 q^{6} + 286 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 80 * q^6 + 286 * q^9 $$2 q - 32 q^{4} - 80 q^{6} + 286 q^{9} - 672 q^{11} + 392 q^{14} + 512 q^{16} - 940 q^{19} + 980 q^{21} + 1280 q^{24} - 4672 q^{26} - 9732 q^{29} - 14744 q^{31} - 11664 q^{34} - 4576 q^{36} - 11680 q^{39} + 12444 q^{41} + 10752 q^{44} + 33600 q^{46} - 4802 q^{49} - 29160 q^{51} - 30880 q^{54} - 6272 q^{56} - 68604 q^{59} + 48952 q^{61} - 8192 q^{64} + 26880 q^{66} + 84000 q^{69} + 56448 q^{71} + 114640 q^{74} + 15040 q^{76} - 85744 q^{79} - 7702 q^{81} - 15680 q^{84} - 29632 q^{86} - 53460 q^{89} + 57232 q^{91} - 14496 q^{94} - 20480 q^{96} - 96096 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 - 80 * q^6 + 286 * q^9 - 672 * q^11 + 392 * q^14 + 512 * q^16 - 940 * q^19 + 980 * q^21 + 1280 * q^24 - 4672 * q^26 - 9732 * q^29 - 14744 * q^31 - 11664 * q^34 - 4576 * q^36 - 11680 * q^39 + 12444 * q^41 + 10752 * q^44 + 33600 * q^46 - 4802 * q^49 - 29160 * q^51 - 30880 * q^54 - 6272 * q^56 - 68604 * q^59 + 48952 * q^61 - 8192 * q^64 + 26880 * q^66 + 84000 * q^69 + 56448 * q^71 + 114640 * q^74 + 15040 * q^76 - 85744 * q^79 - 7702 * q^81 - 15680 * q^84 - 29632 * q^86 - 53460 * q^89 + 57232 * q^91 - 14496 * q^94 - 20480 * q^96 - 96096 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 4.00000i − 0.707107i
$$3$$ − 10.0000i − 0.641500i −0.947164 0.320750i $$-0.896065\pi$$
0.947164 0.320750i $$-0.103935\pi$$
$$4$$ −16.0000 −0.500000
$$5$$ 0 0
$$6$$ −40.0000 −0.453609
$$7$$ 49.0000i 0.377964i
$$8$$ 64.0000i 0.353553i
$$9$$ 143.000 0.588477
$$10$$ 0 0
$$11$$ −336.000 −0.837255 −0.418627 0.908158i $$-0.637489\pi$$
−0.418627 + 0.908158i $$0.637489\pi$$
$$12$$ 160.000i 0.320750i
$$13$$ − 584.000i − 0.958417i −0.877701 0.479208i $$-0.840924\pi$$
0.877701 0.479208i $$-0.159076\pi$$
$$14$$ 196.000 0.267261
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ − 1458.00i − 1.22359i −0.791017 0.611794i $$-0.790448\pi$$
0.791017 0.611794i $$-0.209552\pi$$
$$18$$ − 572.000i − 0.416116i
$$19$$ −470.000 −0.298685 −0.149343 0.988786i $$-0.547716\pi$$
−0.149343 + 0.988786i $$0.547716\pi$$
$$20$$ 0 0
$$21$$ 490.000 0.242464
$$22$$ 1344.00i 0.592028i
$$23$$ 4200.00i 1.65550i 0.561096 + 0.827751i $$0.310380\pi$$
−0.561096 + 0.827751i $$0.689620\pi$$
$$24$$ 640.000 0.226805
$$25$$ 0 0
$$26$$ −2336.00 −0.677703
$$27$$ − 3860.00i − 1.01901i
$$28$$ − 784.000i − 0.188982i
$$29$$ −4866.00 −1.07443 −0.537214 0.843446i $$-0.680523\pi$$
−0.537214 + 0.843446i $$0.680523\pi$$
$$30$$ 0 0
$$31$$ −7372.00 −1.37778 −0.688892 0.724864i $$-0.741903\pi$$
−0.688892 + 0.724864i $$0.741903\pi$$
$$32$$ − 1024.00i − 0.176777i
$$33$$ 3360.00i 0.537099i
$$34$$ −5832.00 −0.865207
$$35$$ 0 0
$$36$$ −2288.00 −0.294239
$$37$$ 14330.0i 1.72085i 0.509581 + 0.860423i $$0.329800\pi$$
−0.509581 + 0.860423i $$0.670200\pi$$
$$38$$ 1880.00i 0.211202i
$$39$$ −5840.00 −0.614825
$$40$$ 0 0
$$41$$ 6222.00 0.578057 0.289028 0.957321i $$-0.406668\pi$$
0.289028 + 0.957321i $$0.406668\pi$$
$$42$$ − 1960.00i − 0.171448i
$$43$$ − 3704.00i − 0.305492i −0.988265 0.152746i $$-0.951188\pi$$
0.988265 0.152746i $$-0.0488116\pi$$
$$44$$ 5376.00 0.418627
$$45$$ 0 0
$$46$$ 16800.0 1.17062
$$47$$ − 1812.00i − 0.119650i −0.998209 0.0598251i $$-0.980946\pi$$
0.998209 0.0598251i $$-0.0190543\pi$$
$$48$$ − 2560.00i − 0.160375i
$$49$$ −2401.00 −0.142857
$$50$$ 0 0
$$51$$ −14580.0 −0.784932
$$52$$ 9344.00i 0.479208i
$$53$$ 37242.0i 1.82114i 0.413355 + 0.910570i $$0.364357\pi$$
−0.413355 + 0.910570i $$0.635643\pi$$
$$54$$ −15440.0 −0.720548
$$55$$ 0 0
$$56$$ −3136.00 −0.133631
$$57$$ 4700.00i 0.191607i
$$58$$ 19464.0i 0.759735i
$$59$$ −34302.0 −1.28289 −0.641445 0.767169i $$-0.721665\pi$$
−0.641445 + 0.767169i $$0.721665\pi$$
$$60$$ 0 0
$$61$$ 24476.0 0.842201 0.421101 0.907014i $$-0.361644\pi$$
0.421101 + 0.907014i $$0.361644\pi$$
$$62$$ 29488.0i 0.974240i
$$63$$ 7007.00i 0.222424i
$$64$$ −4096.00 −0.125000
$$65$$ 0 0
$$66$$ 13440.0 0.379786
$$67$$ − 17452.0i − 0.474961i −0.971392 0.237481i $$-0.923678\pi$$
0.971392 0.237481i $$-0.0763216\pi$$
$$68$$ 23328.0i 0.611794i
$$69$$ 42000.0 1.06201
$$70$$ 0 0
$$71$$ 28224.0 0.664466 0.332233 0.943197i $$-0.392198\pi$$
0.332233 + 0.943197i $$0.392198\pi$$
$$72$$ 9152.00i 0.208058i
$$73$$ − 3602.00i − 0.0791109i −0.999217 0.0395555i $$-0.987406\pi$$
0.999217 0.0395555i $$-0.0125942\pi$$
$$74$$ 57320.0 1.21682
$$75$$ 0 0
$$76$$ 7520.00 0.149343
$$77$$ − 16464.0i − 0.316453i
$$78$$ 23360.0i 0.434747i
$$79$$ −42872.0 −0.772869 −0.386435 0.922317i $$-0.626294\pi$$
−0.386435 + 0.922317i $$0.626294\pi$$
$$80$$ 0 0
$$81$$ −3851.00 −0.0652170
$$82$$ − 24888.0i − 0.408748i
$$83$$ 35202.0i 0.560883i 0.959871 + 0.280441i $$0.0904808\pi$$
−0.959871 + 0.280441i $$0.909519\pi$$
$$84$$ −7840.00 −0.121232
$$85$$ 0 0
$$86$$ −14816.0 −0.216015
$$87$$ 48660.0i 0.689246i
$$88$$ − 21504.0i − 0.296014i
$$89$$ −26730.0 −0.357704 −0.178852 0.983876i $$-0.557238\pi$$
−0.178852 + 0.983876i $$0.557238\pi$$
$$90$$ 0 0
$$91$$ 28616.0 0.362248
$$92$$ − 67200.0i − 0.827751i
$$93$$ 73720.0i 0.883849i
$$94$$ −7248.00 −0.0846055
$$95$$ 0 0
$$96$$ −10240.0 −0.113402
$$97$$ − 16978.0i − 0.183213i −0.995795 0.0916067i $$-0.970800\pi$$
0.995795 0.0916067i $$-0.0292003\pi$$
$$98$$ 9604.00i 0.101015i
$$99$$ −48048.0 −0.492705
$$100$$ 0 0
$$101$$ 99204.0 0.967667 0.483833 0.875160i $$-0.339244\pi$$
0.483833 + 0.875160i $$0.339244\pi$$
$$102$$ 58320.0i 0.555031i
$$103$$ 131644.i 1.22267i 0.791373 + 0.611333i $$0.209366\pi$$
−0.791373 + 0.611333i $$0.790634\pi$$
$$104$$ 37376.0 0.338852
$$105$$ 0 0
$$106$$ 148968. 1.28774
$$107$$ 48852.0i 0.412499i 0.978499 + 0.206250i $$0.0661259\pi$$
−0.978499 + 0.206250i $$0.933874\pi$$
$$108$$ 61760.0i 0.509504i
$$109$$ 56374.0 0.454478 0.227239 0.973839i $$-0.427030\pi$$
0.227239 + 0.973839i $$0.427030\pi$$
$$110$$ 0 0
$$111$$ 143300. 1.10392
$$112$$ 12544.0i 0.0944911i
$$113$$ − 8742.00i − 0.0644043i −0.999481 0.0322021i $$-0.989748\pi$$
0.999481 0.0322021i $$-0.0102520\pi$$
$$114$$ 18800.0 0.135486
$$115$$ 0 0
$$116$$ 77856.0 0.537214
$$117$$ − 83512.0i − 0.564007i
$$118$$ 137208.i 0.907140i
$$119$$ 71442.0 0.462473
$$120$$ 0 0
$$121$$ −48155.0 −0.299005
$$122$$ − 97904.0i − 0.595526i
$$123$$ − 62220.0i − 0.370823i
$$124$$ 117952. 0.688892
$$125$$ 0 0
$$126$$ 28028.0 0.157277
$$127$$ 315992.i 1.73847i 0.494401 + 0.869234i $$0.335388\pi$$
−0.494401 + 0.869234i $$0.664612\pi$$
$$128$$ 16384.0i 0.0883883i
$$129$$ −37040.0 −0.195973
$$130$$ 0 0
$$131$$ −24666.0 −0.125580 −0.0627900 0.998027i $$-0.520000\pi$$
−0.0627900 + 0.998027i $$0.520000\pi$$
$$132$$ − 53760.0i − 0.268550i
$$133$$ − 23030.0i − 0.112892i
$$134$$ −69808.0 −0.335848
$$135$$ 0 0
$$136$$ 93312.0 0.432604
$$137$$ 303234.i 1.38031i 0.723662 + 0.690155i $$0.242458\pi$$
−0.723662 + 0.690155i $$0.757542\pi$$
$$138$$ − 168000.i − 0.750951i
$$139$$ −250586. −1.10007 −0.550034 0.835142i $$-0.685385\pi$$
−0.550034 + 0.835142i $$0.685385\pi$$
$$140$$ 0 0
$$141$$ −18120.0 −0.0767557
$$142$$ − 112896.i − 0.469848i
$$143$$ 196224.i 0.802439i
$$144$$ 36608.0 0.147119
$$145$$ 0 0
$$146$$ −14408.0 −0.0559399
$$147$$ 24010.0i 0.0916429i
$$148$$ − 229280.i − 0.860423i
$$149$$ 60594.0 0.223596 0.111798 0.993731i $$-0.464339\pi$$
0.111798 + 0.993731i $$0.464339\pi$$
$$150$$ 0 0
$$151$$ 124448. 0.444166 0.222083 0.975028i $$-0.428714\pi$$
0.222083 + 0.975028i $$0.428714\pi$$
$$152$$ − 30080.0i − 0.105601i
$$153$$ − 208494.i − 0.720054i
$$154$$ −65856.0 −0.223766
$$155$$ 0 0
$$156$$ 93440.0 0.307412
$$157$$ 76040.0i 0.246203i 0.992394 + 0.123101i $$0.0392840\pi$$
−0.992394 + 0.123101i $$0.960716\pi$$
$$158$$ 171488.i 0.546501i
$$159$$ 372420. 1.16826
$$160$$ 0 0
$$161$$ −205800. −0.625721
$$162$$ 15404.0i 0.0461154i
$$163$$ − 124256.i − 0.366310i −0.983084 0.183155i $$-0.941369\pi$$
0.983084 0.183155i $$-0.0586310\pi$$
$$164$$ −99552.0 −0.289028
$$165$$ 0 0
$$166$$ 140808. 0.396604
$$167$$ − 72420.0i − 0.200940i −0.994940 0.100470i $$-0.967965\pi$$
0.994940 0.100470i $$-0.0320347\pi$$
$$168$$ 31360.0i 0.0857241i
$$169$$ 30237.0 0.0814370
$$170$$ 0 0
$$171$$ −67210.0 −0.175770
$$172$$ 59264.0i 0.152746i
$$173$$ 441552.i 1.12167i 0.827926 + 0.560837i $$0.189521\pi$$
−0.827926 + 0.560837i $$0.810479\pi$$
$$174$$ 194640. 0.487370
$$175$$ 0 0
$$176$$ −86016.0 −0.209314
$$177$$ 343020.i 0.822974i
$$178$$ 106920.i 0.252935i
$$179$$ 10692.0 0.0249417 0.0124709 0.999922i $$-0.496030\pi$$
0.0124709 + 0.999922i $$0.496030\pi$$
$$180$$ 0 0
$$181$$ −546064. −1.23893 −0.619465 0.785024i $$-0.712651\pi$$
−0.619465 + 0.785024i $$0.712651\pi$$
$$182$$ − 114464.i − 0.256148i
$$183$$ − 244760.i − 0.540272i
$$184$$ −268800. −0.585308
$$185$$ 0 0
$$186$$ 294880. 0.624975
$$187$$ 489888.i 1.02445i
$$188$$ 28992.0i 0.0598251i
$$189$$ 189140. 0.385149
$$190$$ 0 0
$$191$$ −575976. −1.14241 −0.571204 0.820808i $$-0.693523\pi$$
−0.571204 + 0.820808i $$0.693523\pi$$
$$192$$ 40960.0i 0.0801875i
$$193$$ 413938.i 0.799912i 0.916534 + 0.399956i $$0.130975\pi$$
−0.916534 + 0.399956i $$0.869025\pi$$
$$194$$ −67912.0 −0.129551
$$195$$ 0 0
$$196$$ 38416.0 0.0714286
$$197$$ − 494946.i − 0.908641i −0.890838 0.454320i $$-0.849882\pi$$
0.890838 0.454320i $$-0.150118\pi$$
$$198$$ 192192.i 0.348395i
$$199$$ −520364. −0.931482 −0.465741 0.884921i $$-0.654212\pi$$
−0.465741 + 0.884921i $$0.654212\pi$$
$$200$$ 0 0
$$201$$ −174520. −0.304688
$$202$$ − 396816.i − 0.684244i
$$203$$ − 238434.i − 0.406095i
$$204$$ 233280. 0.392466
$$205$$ 0 0
$$206$$ 526576. 0.864556
$$207$$ 600600.i 0.974225i
$$208$$ − 149504.i − 0.239604i
$$209$$ 157920. 0.250076
$$210$$ 0 0
$$211$$ 183284. 0.283412 0.141706 0.989909i $$-0.454741\pi$$
0.141706 + 0.989909i $$0.454741\pi$$
$$212$$ − 595872.i − 0.910570i
$$213$$ − 282240.i − 0.426255i
$$214$$ 195408. 0.291681
$$215$$ 0 0
$$216$$ 247040. 0.360274
$$217$$ − 361228.i − 0.520753i
$$218$$ − 225496.i − 0.321364i
$$219$$ −36020.0 −0.0507497
$$220$$ 0 0
$$221$$ −851472. −1.17271
$$222$$ − 573200.i − 0.780591i
$$223$$ 1.27746e6i 1.72023i 0.510100 + 0.860115i $$0.329608\pi$$
−0.510100 + 0.860115i $$0.670392\pi$$
$$224$$ 50176.0 0.0668153
$$225$$ 0 0
$$226$$ −34968.0 −0.0455407
$$227$$ − 1.28764e6i − 1.65856i −0.558835 0.829279i $$-0.688752\pi$$
0.558835 0.829279i $$-0.311248\pi$$
$$228$$ − 75200.0i − 0.0958034i
$$229$$ −350936. −0.442221 −0.221110 0.975249i $$-0.570968\pi$$
−0.221110 + 0.975249i $$0.570968\pi$$
$$230$$ 0 0
$$231$$ −164640. −0.203004
$$232$$ − 311424.i − 0.379867i
$$233$$ − 836154.i − 1.00901i −0.863408 0.504506i $$-0.831675\pi$$
0.863408 0.504506i $$-0.168325\pi$$
$$234$$ −334048. −0.398813
$$235$$ 0 0
$$236$$ 548832. 0.641445
$$237$$ 428720.i 0.495796i
$$238$$ − 285768.i − 0.327018i
$$239$$ −774336. −0.876869 −0.438434 0.898763i $$-0.644467\pi$$
−0.438434 + 0.898763i $$0.644467\pi$$
$$240$$ 0 0
$$241$$ −1.15285e6 −1.27859 −0.639293 0.768963i $$-0.720773\pi$$
−0.639293 + 0.768963i $$0.720773\pi$$
$$242$$ 192620.i 0.211428i
$$243$$ − 899470.i − 0.977172i
$$244$$ −391616. −0.421101
$$245$$ 0 0
$$246$$ −248880. −0.262212
$$247$$ 274480.i 0.286265i
$$248$$ − 471808.i − 0.487120i
$$249$$ 352020. 0.359806
$$250$$ 0 0
$$251$$ 1.35801e6 1.36056 0.680282 0.732951i $$-0.261858\pi$$
0.680282 + 0.732951i $$0.261858\pi$$
$$252$$ − 112112.i − 0.111212i
$$253$$ − 1.41120e6i − 1.38608i
$$254$$ 1.26397e6 1.22928
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ − 317742.i − 0.300083i −0.988680 0.150042i $$-0.952059\pi$$
0.988680 0.150042i $$-0.0479408\pi$$
$$258$$ 148160.i 0.138574i
$$259$$ −702170. −0.650418
$$260$$ 0 0
$$261$$ −695838. −0.632276
$$262$$ 98664.0i 0.0887985i
$$263$$ − 1.05101e6i − 0.936951i −0.883477 0.468475i $$-0.844804\pi$$
0.883477 0.468475i $$-0.155196\pi$$
$$264$$ −215040. −0.189893
$$265$$ 0 0
$$266$$ −92120.0 −0.0798270
$$267$$ 267300.i 0.229467i
$$268$$ 279232.i 0.237481i
$$269$$ −1.18958e6 −1.00234 −0.501169 0.865349i $$-0.667097\pi$$
−0.501169 + 0.865349i $$0.667097\pi$$
$$270$$ 0 0
$$271$$ −1.43008e6 −1.18287 −0.591435 0.806353i $$-0.701438\pi$$
−0.591435 + 0.806353i $$0.701438\pi$$
$$272$$ − 373248.i − 0.305897i
$$273$$ − 286160.i − 0.232382i
$$274$$ 1.21294e6 0.976026
$$275$$ 0 0
$$276$$ −672000. −0.531003
$$277$$ 63302.0i 0.0495699i 0.999693 + 0.0247849i $$0.00789010\pi$$
−0.999693 + 0.0247849i $$0.992110\pi$$
$$278$$ 1.00234e6i 0.777866i
$$279$$ −1.05420e6 −0.810795
$$280$$ 0 0
$$281$$ −496614. −0.375192 −0.187596 0.982246i $$-0.560070\pi$$
−0.187596 + 0.982246i $$0.560070\pi$$
$$282$$ 72480.0i 0.0542744i
$$283$$ 1.15842e6i 0.859803i 0.902876 + 0.429902i $$0.141452\pi$$
−0.902876 + 0.429902i $$0.858548\pi$$
$$284$$ −451584. −0.332233
$$285$$ 0 0
$$286$$ 784896. 0.567410
$$287$$ 304878.i 0.218485i
$$288$$ − 146432.i − 0.104029i
$$289$$ −705907. −0.497168
$$290$$ 0 0
$$291$$ −169780. −0.117531
$$292$$ 57632.0i 0.0395555i
$$293$$ − 1.43886e6i − 0.979151i −0.871961 0.489575i $$-0.837152\pi$$
0.871961 0.489575i $$-0.162848\pi$$
$$294$$ 96040.0 0.0648013
$$295$$ 0 0
$$296$$ −917120. −0.608411
$$297$$ 1.29696e6i 0.853170i
$$298$$ − 242376.i − 0.158106i
$$299$$ 2.45280e6 1.58666
$$300$$ 0 0
$$301$$ 181496. 0.115465
$$302$$ − 497792.i − 0.314073i
$$303$$ − 992040.i − 0.620758i
$$304$$ −120320. −0.0746713
$$305$$ 0 0
$$306$$ −833976. −0.509155
$$307$$ − 989098.i − 0.598954i −0.954104 0.299477i $$-0.903188\pi$$
0.954104 0.299477i $$-0.0968122\pi$$
$$308$$ 263424.i 0.158226i
$$309$$ 1.31644e6 0.784341
$$310$$ 0 0
$$311$$ −2.22050e6 −1.30182 −0.650909 0.759155i $$-0.725612\pi$$
−0.650909 + 0.759155i $$0.725612\pi$$
$$312$$ − 373760.i − 0.217373i
$$313$$ − 2.33008e6i − 1.34434i −0.740396 0.672171i $$-0.765362\pi$$
0.740396 0.672171i $$-0.234638\pi$$
$$314$$ 304160. 0.174092
$$315$$ 0 0
$$316$$ 685952. 0.386435
$$317$$ 427542.i 0.238963i 0.992836 + 0.119481i $$0.0381232\pi$$
−0.992836 + 0.119481i $$0.961877\pi$$
$$318$$ − 1.48968e6i − 0.826086i
$$319$$ 1.63498e6 0.899569
$$320$$ 0 0
$$321$$ 488520. 0.264618
$$322$$ 823200.i 0.442452i
$$323$$ 685260.i 0.365468i
$$324$$ 61616.0 0.0326085
$$325$$ 0 0
$$326$$ −497024. −0.259020
$$327$$ − 563740.i − 0.291548i
$$328$$ 398208.i 0.204374i
$$329$$ 88788.0 0.0452235
$$330$$ 0 0
$$331$$ −396616. −0.198976 −0.0994879 0.995039i $$-0.531720\pi$$
−0.0994879 + 0.995039i $$0.531720\pi$$
$$332$$ − 563232.i − 0.280441i
$$333$$ 2.04919e6i 1.01268i
$$334$$ −289680. −0.142086
$$335$$ 0 0
$$336$$ 125440. 0.0606161
$$337$$ − 3.21819e6i − 1.54361i −0.635860 0.771805i $$-0.719354\pi$$
0.635860 0.771805i $$-0.280646\pi$$
$$338$$ − 120948.i − 0.0575847i
$$339$$ −87420.0 −0.0413154
$$340$$ 0 0
$$341$$ 2.47699e6 1.15356
$$342$$ 268840.i 0.124288i
$$343$$ − 117649.i − 0.0539949i
$$344$$ 237056. 0.108008
$$345$$ 0 0
$$346$$ 1.76621e6 0.793143
$$347$$ 2.78018e6i 1.23951i 0.784796 + 0.619755i $$0.212768\pi$$
−0.784796 + 0.619755i $$0.787232\pi$$
$$348$$ − 778560.i − 0.344623i
$$349$$ 338800. 0.148895 0.0744475 0.997225i $$-0.476281\pi$$
0.0744475 + 0.997225i $$0.476281\pi$$
$$350$$ 0 0
$$351$$ −2.25424e6 −0.976635
$$352$$ 344064.i 0.148007i
$$353$$ 362046.i 0.154642i 0.997006 + 0.0773209i $$0.0246366\pi$$
−0.997006 + 0.0773209i $$0.975363\pi$$
$$354$$ 1.37208e6 0.581931
$$355$$ 0 0
$$356$$ 427680. 0.178852
$$357$$ − 714420.i − 0.296676i
$$358$$ − 42768.0i − 0.0176365i
$$359$$ −876528. −0.358946 −0.179473 0.983763i $$-0.557439\pi$$
−0.179473 + 0.983763i $$0.557439\pi$$
$$360$$ 0 0
$$361$$ −2.25520e6 −0.910787
$$362$$ 2.18426e6i 0.876056i
$$363$$ 481550.i 0.191812i
$$364$$ −457856. −0.181124
$$365$$ 0 0
$$366$$ −979040. −0.382030
$$367$$ 2.98062e6i 1.15516i 0.816335 + 0.577578i $$0.196002\pi$$
−0.816335 + 0.577578i $$0.803998\pi$$
$$368$$ 1.07520e6i 0.413875i
$$369$$ 889746. 0.340173
$$370$$ 0 0
$$371$$ −1.82486e6 −0.688326
$$372$$ − 1.17952e6i − 0.441924i
$$373$$ − 3.91441e6i − 1.45678i −0.685162 0.728391i $$-0.740268\pi$$
0.685162 0.728391i $$-0.259732\pi$$
$$374$$ 1.95955e6 0.724399
$$375$$ 0 0
$$376$$ 115968. 0.0423027
$$377$$ 2.84174e6i 1.02975i
$$378$$ − 756560.i − 0.272342i
$$379$$ −3.60661e6 −1.28974 −0.644868 0.764294i $$-0.723088\pi$$
−0.644868 + 0.764294i $$0.723088\pi$$
$$380$$ 0 0
$$381$$ 3.15992e6 1.11523
$$382$$ 2.30390e6i 0.807804i
$$383$$ 2.66644e6i 0.928826i 0.885619 + 0.464413i $$0.153735\pi$$
−0.885619 + 0.464413i $$0.846265\pi$$
$$384$$ 163840. 0.0567012
$$385$$ 0 0
$$386$$ 1.65575e6 0.565623
$$387$$ − 529672.i − 0.179775i
$$388$$ 271648.i 0.0916067i
$$389$$ 213366. 0.0714910 0.0357455 0.999361i $$-0.488619\pi$$
0.0357455 + 0.999361i $$0.488619\pi$$
$$390$$ 0 0
$$391$$ 6.12360e6 2.02565
$$392$$ − 153664.i − 0.0505076i
$$393$$ 246660.i 0.0805596i
$$394$$ −1.97978e6 −0.642506
$$395$$ 0 0
$$396$$ 768768. 0.246353
$$397$$ − 4.09408e6i − 1.30371i −0.758345 0.651854i $$-0.773992\pi$$
0.758345 0.651854i $$-0.226008\pi$$
$$398$$ 2.08146e6i 0.658657i
$$399$$ −230300. −0.0724205
$$400$$ 0 0
$$401$$ 942366. 0.292657 0.146328 0.989236i $$-0.453254\pi$$
0.146328 + 0.989236i $$0.453254\pi$$
$$402$$ 698080.i 0.215447i
$$403$$ 4.30525e6i 1.32049i
$$404$$ −1.58726e6 −0.483833
$$405$$ 0 0
$$406$$ −953736. −0.287153
$$407$$ − 4.81488e6i − 1.44079i
$$408$$ − 933120.i − 0.277515i
$$409$$ 4.84561e6 1.43232 0.716160 0.697936i $$-0.245898\pi$$
0.716160 + 0.697936i $$0.245898\pi$$
$$410$$ 0 0
$$411$$ 3.03234e6 0.885469
$$412$$ − 2.10630e6i − 0.611333i
$$413$$ − 1.68080e6i − 0.484887i
$$414$$ 2.40240e6 0.688881
$$415$$ 0 0
$$416$$ −598016. −0.169426
$$417$$ 2.50586e6i 0.705694i
$$418$$ − 631680.i − 0.176830i
$$419$$ 1.73485e6 0.482754 0.241377 0.970431i $$-0.422401\pi$$
0.241377 + 0.970431i $$0.422401\pi$$
$$420$$ 0 0
$$421$$ −1.65145e6 −0.454109 −0.227055 0.973882i $$-0.572910\pi$$
−0.227055 + 0.973882i $$0.572910\pi$$
$$422$$ − 733136.i − 0.200403i
$$423$$ − 259116.i − 0.0704115i
$$424$$ −2.38349e6 −0.643870
$$425$$ 0 0
$$426$$ −1.12896e6 −0.301408
$$427$$ 1.19932e6i 0.318322i
$$428$$ − 781632.i − 0.206250i
$$429$$ 1.96224e6 0.514765
$$430$$ 0 0
$$431$$ 4.14360e6 1.07445 0.537223 0.843440i $$-0.319473\pi$$
0.537223 + 0.843440i $$0.319473\pi$$
$$432$$ − 988160.i − 0.254752i
$$433$$ 3.03966e6i 0.779121i 0.921001 + 0.389561i $$0.127373\pi$$
−0.921001 + 0.389561i $$0.872627\pi$$
$$434$$ −1.44491e6 −0.368228
$$435$$ 0 0
$$436$$ −901984. −0.227239
$$437$$ − 1.97400e6i − 0.494474i
$$438$$ 144080.i 0.0358855i
$$439$$ −2.54271e6 −0.629703 −0.314852 0.949141i $$-0.601955\pi$$
−0.314852 + 0.949141i $$0.601955\pi$$
$$440$$ 0 0
$$441$$ −343343. −0.0840682
$$442$$ 3.40589e6i 0.829229i
$$443$$ 2.43210e6i 0.588806i 0.955681 + 0.294403i $$0.0951208\pi$$
−0.955681 + 0.294403i $$0.904879\pi$$
$$444$$ −2.29280e6 −0.551961
$$445$$ 0 0
$$446$$ 5.10986e6 1.21639
$$447$$ − 605940.i − 0.143437i
$$448$$ − 200704.i − 0.0472456i
$$449$$ −1.82853e6 −0.428042 −0.214021 0.976829i $$-0.568656\pi$$
−0.214021 + 0.976829i $$0.568656\pi$$
$$450$$ 0 0
$$451$$ −2.09059e6 −0.483981
$$452$$ 139872.i 0.0322021i
$$453$$ − 1.24448e6i − 0.284933i
$$454$$ −5.15057e6 −1.17278
$$455$$ 0 0
$$456$$ −300800. −0.0677432
$$457$$ 1.58063e6i 0.354030i 0.984208 + 0.177015i $$0.0566441\pi$$
−0.984208 + 0.177015i $$0.943356\pi$$
$$458$$ 1.40374e6i 0.312697i
$$459$$ −5.62788e6 −1.24685
$$460$$ 0 0
$$461$$ 5.09604e6 1.11681 0.558407 0.829567i $$-0.311413\pi$$
0.558407 + 0.829567i $$0.311413\pi$$
$$462$$ 658560.i 0.143546i
$$463$$ 7.02338e6i 1.52263i 0.648384 + 0.761313i $$0.275445\pi$$
−0.648384 + 0.761313i $$0.724555\pi$$
$$464$$ −1.24570e6 −0.268607
$$465$$ 0 0
$$466$$ −3.34462e6 −0.713479
$$467$$ − 4.24845e6i − 0.901443i −0.892665 0.450722i $$-0.851167\pi$$
0.892665 0.450722i $$-0.148833\pi$$
$$468$$ 1.33619e6i 0.282003i
$$469$$ 855148. 0.179518
$$470$$ 0 0
$$471$$ 760400. 0.157939
$$472$$ − 2.19533e6i − 0.453570i
$$473$$ 1.24454e6i 0.255775i
$$474$$ 1.71488e6 0.350581
$$475$$ 0 0
$$476$$ −1.14307e6 −0.231236
$$477$$ 5.32561e6i 1.07170i
$$478$$ 3.09734e6i 0.620040i
$$479$$ −559284. −0.111377 −0.0556883 0.998448i $$-0.517735\pi$$
−0.0556883 + 0.998448i $$0.517735\pi$$
$$480$$ 0 0
$$481$$ 8.36872e6 1.64929
$$482$$ 4.61140e6i 0.904097i
$$483$$ 2.05800e6i 0.401400i
$$484$$ 770480. 0.149502
$$485$$ 0 0
$$486$$ −3.59788e6 −0.690965
$$487$$ − 1.32057e6i − 0.252312i −0.992010 0.126156i $$-0.959736\pi$$
0.992010 0.126156i $$-0.0402640\pi$$
$$488$$ 1.56646e6i 0.297763i
$$489$$ −1.24256e6 −0.234988
$$490$$ 0 0
$$491$$ 6.27193e6 1.17408 0.587040 0.809558i $$-0.300293\pi$$
0.587040 + 0.809558i $$0.300293\pi$$
$$492$$ 995520.i 0.185412i
$$493$$ 7.09463e6i 1.31466i
$$494$$ 1.09792e6 0.202420
$$495$$ 0 0
$$496$$ −1.88723e6 −0.344446
$$497$$ 1.38298e6i 0.251144i
$$498$$ − 1.40808e6i − 0.254422i
$$499$$ 3.93785e6 0.707959 0.353979 0.935253i $$-0.384828\pi$$
0.353979 + 0.935253i $$0.384828\pi$$
$$500$$ 0 0
$$501$$ −724200. −0.128903
$$502$$ − 5.43204e6i − 0.962063i
$$503$$ 7.59830e6i 1.33905i 0.742790 + 0.669525i $$0.233502\pi$$
−0.742790 + 0.669525i $$0.766498\pi$$
$$504$$ −448448. −0.0786386
$$505$$ 0 0
$$506$$ −5.64480e6 −0.980104
$$507$$ − 302370.i − 0.0522419i
$$508$$ − 5.05587e6i − 0.869234i
$$509$$ 7.82664e6 1.33900 0.669501 0.742812i $$-0.266508\pi$$
0.669501 + 0.742812i $$0.266508\pi$$
$$510$$ 0 0
$$511$$ 176498. 0.0299011
$$512$$ − 262144.i − 0.0441942i
$$513$$ 1.81420e6i 0.304363i
$$514$$ −1.27097e6 −0.212191
$$515$$ 0 0
$$516$$ 592640. 0.0979866
$$517$$ 608832.i 0.100178i
$$518$$ 2.80868e6i 0.459915i
$$519$$ 4.41552e6 0.719554
$$520$$ 0 0
$$521$$ 8.94454e6 1.44366 0.721828 0.692072i $$-0.243302\pi$$
0.721828 + 0.692072i $$0.243302\pi$$
$$522$$ 2.78335e6i 0.447087i
$$523$$ − 4.07481e6i − 0.651407i −0.945472 0.325704i $$-0.894399\pi$$
0.945472 0.325704i $$-0.105601\pi$$
$$524$$ 394656. 0.0627900
$$525$$ 0 0
$$526$$ −4.20403e6 −0.662524
$$527$$ 1.07484e7i 1.68584i
$$528$$ 860160.i 0.134275i
$$529$$ −1.12037e7 −1.74069
$$530$$ 0 0
$$531$$ −4.90519e6 −0.754952
$$532$$ 368480.i 0.0564462i
$$533$$ − 3.63365e6i − 0.554019i
$$534$$ 1.06920e6 0.162258
$$535$$ 0 0
$$536$$ 1.11693e6 0.167924
$$537$$ − 106920.i − 0.0160001i
$$538$$ 4.75834e6i 0.708760i
$$539$$ 806736. 0.119608
$$540$$ 0 0
$$541$$ −1.18676e7 −1.74329 −0.871644 0.490140i $$-0.836946\pi$$
−0.871644 + 0.490140i $$0.836946\pi$$
$$542$$ 5.72032e6i 0.836416i
$$543$$ 5.46064e6i 0.794775i
$$544$$ −1.49299e6 −0.216302
$$545$$ 0 0
$$546$$ −1.14464e6 −0.164319
$$547$$ − 5.37801e6i − 0.768516i −0.923226 0.384258i $$-0.874457\pi$$
0.923226 0.384258i $$-0.125543\pi$$
$$548$$ − 4.85174e6i − 0.690155i
$$549$$ 3.50007e6 0.495616
$$550$$ 0 0
$$551$$ 2.28702e6 0.320916
$$552$$ 2.68800e6i 0.375475i
$$553$$ − 2.10073e6i − 0.292117i
$$554$$ 253208. 0.0350512
$$555$$ 0 0
$$556$$ 4.00938e6 0.550034
$$557$$ − 5.64878e6i − 0.771466i −0.922611 0.385733i $$-0.873949\pi$$
0.922611 0.385733i $$-0.126051\pi$$
$$558$$ 4.21678e6i 0.573318i
$$559$$ −2.16314e6 −0.292789
$$560$$ 0 0
$$561$$ 4.89888e6 0.657188
$$562$$ 1.98646e6i 0.265301i
$$563$$ − 4.56407e6i − 0.606850i −0.952855 0.303425i $$-0.901870\pi$$
0.952855 0.303425i $$-0.0981303\pi$$
$$564$$ 289920. 0.0383778
$$565$$ 0 0
$$566$$ 4.63367e6 0.607973
$$567$$ − 188699.i − 0.0246497i
$$568$$ 1.80634e6i 0.234924i
$$569$$ −8.00165e6 −1.03609 −0.518047 0.855352i $$-0.673341\pi$$
−0.518047 + 0.855352i $$0.673341\pi$$
$$570$$ 0 0
$$571$$ −1.37164e7 −1.76055 −0.880275 0.474464i $$-0.842642\pi$$
−0.880275 + 0.474464i $$0.842642\pi$$
$$572$$ − 3.13958e6i − 0.401220i
$$573$$ 5.75976e6i 0.732855i
$$574$$ 1.21951e6 0.154492
$$575$$ 0 0
$$576$$ −585728. −0.0735597
$$577$$ 6.09797e6i 0.762510i 0.924470 + 0.381255i $$0.124508\pi$$
−0.924470 + 0.381255i $$0.875492\pi$$
$$578$$ 2.82363e6i 0.351551i
$$579$$ 4.13938e6 0.513144
$$580$$ 0 0
$$581$$ −1.72490e6 −0.211994
$$582$$ 679120.i 0.0831073i
$$583$$ − 1.25133e7i − 1.52476i
$$584$$ 230528. 0.0279699
$$585$$ 0 0
$$586$$ −5.75544e6 −0.692364
$$587$$ − 8.08462e6i − 0.968422i −0.874951 0.484211i $$-0.839107\pi$$
0.874951 0.484211i $$-0.160893\pi$$
$$588$$ − 384160.i − 0.0458214i
$$589$$ 3.46484e6 0.411524
$$590$$ 0 0
$$591$$ −4.94946e6 −0.582893
$$592$$ 3.66848e6i 0.430211i
$$593$$ − 1.41575e6i − 0.165330i −0.996577 0.0826649i $$-0.973657\pi$$
0.996577 0.0826649i $$-0.0263431\pi$$
$$594$$ 5.18784e6 0.603282
$$595$$ 0 0
$$596$$ −969504. −0.111798
$$597$$ 5.20364e6i 0.597546i
$$598$$ − 9.81120e6i − 1.12194i
$$599$$ −8.75460e6 −0.996941 −0.498470 0.866907i $$-0.666105\pi$$
−0.498470 + 0.866907i $$0.666105\pi$$
$$600$$ 0 0
$$601$$ 8.70276e6 0.982813 0.491407 0.870930i $$-0.336483\pi$$
0.491407 + 0.870930i $$0.336483\pi$$
$$602$$ − 725984.i − 0.0816462i
$$603$$ − 2.49564e6i − 0.279504i
$$604$$ −1.99117e6 −0.222083
$$605$$ 0 0
$$606$$ −3.96816e6 −0.438942
$$607$$ − 1.69578e7i − 1.86809i −0.357157 0.934045i $$-0.616254\pi$$
0.357157 0.934045i $$-0.383746\pi$$
$$608$$ 481280.i 0.0528006i
$$609$$ −2.38434e6 −0.260510
$$610$$ 0 0
$$611$$ −1.05821e6 −0.114675
$$612$$ 3.33590e6i 0.360027i
$$613$$ − 1.76743e7i − 1.89973i −0.312658 0.949866i $$-0.601220\pi$$
0.312658 0.949866i $$-0.398780\pi$$
$$614$$ −3.95639e6 −0.423524
$$615$$ 0 0
$$616$$ 1.05370e6 0.111883
$$617$$ − 9.70636e6i − 1.02646i −0.858250 0.513232i $$-0.828448\pi$$
0.858250 0.513232i $$-0.171552\pi$$
$$618$$ − 5.26576e6i − 0.554613i
$$619$$ −1.48739e7 −1.56027 −0.780133 0.625613i $$-0.784849\pi$$
−0.780133 + 0.625613i $$0.784849\pi$$
$$620$$ 0 0
$$621$$ 1.62120e7 1.68697
$$622$$ 8.88202e6i 0.920525i
$$623$$ − 1.30977e6i − 0.135199i
$$624$$ −1.49504e6 −0.153706
$$625$$ 0 0
$$626$$ −9.32031e6 −0.950593
$$627$$ − 1.57920e6i − 0.160424i
$$628$$ − 1.21664e6i − 0.123101i
$$629$$ 2.08931e7 2.10561
$$630$$ 0 0
$$631$$ 1.26353e7 1.26331 0.631656 0.775248i $$-0.282375\pi$$
0.631656 + 0.775248i $$0.282375\pi$$
$$632$$ − 2.74381e6i − 0.273251i
$$633$$ − 1.83284e6i − 0.181809i
$$634$$ 1.71017e6 0.168972
$$635$$ 0 0
$$636$$ −5.95872e6 −0.584131
$$637$$ 1.40218e6i 0.136917i
$$638$$ − 6.53990e6i − 0.636092i
$$639$$ 4.03603e6 0.391023
$$640$$ 0 0
$$641$$ 6.23398e6 0.599267 0.299634 0.954054i $$-0.403136\pi$$
0.299634 + 0.954054i $$0.403136\pi$$
$$642$$ − 1.95408e6i − 0.187113i
$$643$$ − 1.06874e7i − 1.01940i −0.860352 0.509701i $$-0.829756\pi$$
0.860352 0.509701i $$-0.170244\pi$$
$$644$$ 3.29280e6 0.312860
$$645$$ 0 0
$$646$$ 2.74104e6 0.258425
$$647$$ 1.83258e7i 1.72109i 0.509376 + 0.860544i $$0.329876\pi$$
−0.509376 + 0.860544i $$0.670124\pi$$
$$648$$ − 246464.i − 0.0230577i
$$649$$ 1.15255e7 1.07411
$$650$$ 0 0
$$651$$ −3.61228e6 −0.334063
$$652$$ 1.98810e6i 0.183155i
$$653$$ 7.28857e6i 0.668897i 0.942414 + 0.334448i $$0.108550\pi$$
−0.942414 + 0.334448i $$0.891450\pi$$
$$654$$ −2.25496e6 −0.206155
$$655$$ 0 0
$$656$$ 1.59283e6 0.144514
$$657$$ − 515086.i − 0.0465550i
$$658$$ − 355152.i − 0.0319779i
$$659$$ −4.54337e6 −0.407534 −0.203767 0.979019i $$-0.565319\pi$$
−0.203767 + 0.979019i $$0.565319\pi$$
$$660$$ 0 0
$$661$$ −2.10021e7 −1.86964 −0.934821 0.355120i $$-0.884440\pi$$
−0.934821 + 0.355120i $$0.884440\pi$$
$$662$$ 1.58646e6i 0.140697i
$$663$$ 8.51472e6i 0.752292i
$$664$$ −2.25293e6 −0.198302
$$665$$ 0 0
$$666$$ 8.19676e6 0.716072
$$667$$ − 2.04372e7i − 1.77872i
$$668$$ 1.15872e6i 0.100470i
$$669$$ 1.27746e7 1.10353
$$670$$ 0 0
$$671$$ −8.22394e6 −0.705137
$$672$$ − 501760.i − 0.0428620i
$$673$$ − 3.46923e6i − 0.295253i −0.989043 0.147627i $$-0.952837\pi$$
0.989043 0.147627i $$-0.0471634\pi$$
$$674$$ −1.28728e7 −1.09150
$$675$$ 0 0
$$676$$ −483792. −0.0407185
$$677$$ − 1.80916e7i − 1.51707i −0.651631 0.758536i $$-0.725915\pi$$
0.651631 0.758536i $$-0.274085\pi$$
$$678$$ 349680.i 0.0292144i
$$679$$ 831922. 0.0692481
$$680$$ 0 0
$$681$$ −1.28764e7 −1.06397
$$682$$ − 9.90797e6i − 0.815687i
$$683$$ − 4.67752e6i − 0.383675i −0.981427 0.191838i $$-0.938555\pi$$
0.981427 0.191838i $$-0.0614447\pi$$
$$684$$ 1.07536e6 0.0878848
$$685$$ 0 0
$$686$$ −470596. −0.0381802
$$687$$ 3.50936e6i 0.283685i
$$688$$ − 948224.i − 0.0763730i
$$689$$ 2.17493e7 1.74541
$$690$$ 0 0
$$691$$ 1.68960e7 1.34614 0.673069 0.739579i $$-0.264976\pi$$
0.673069 + 0.739579i $$0.264976\pi$$
$$692$$ − 7.06483e6i − 0.560837i
$$693$$ − 2.35435e6i − 0.186225i
$$694$$ 1.11207e7 0.876466
$$695$$ 0 0
$$696$$ −3.11424e6 −0.243685
$$697$$ − 9.07168e6i − 0.707303i
$$698$$ − 1.35520e6i − 0.105285i
$$699$$ −8.36154e6 −0.647282
$$700$$ 0 0
$$701$$ 2.40964e6 0.185207 0.0926035 0.995703i $$-0.470481\pi$$
0.0926035 + 0.995703i $$0.470481\pi$$
$$702$$ 9.01696e6i 0.690585i
$$703$$ − 6.73510e6i − 0.513991i
$$704$$ 1.37626e6 0.104657
$$705$$ 0 0
$$706$$ 1.44818e6 0.109348
$$707$$ 4.86100e6i 0.365744i
$$708$$ − 5.48832e6i − 0.411487i
$$709$$ 5.77010e6 0.431090 0.215545 0.976494i $$-0.430847\pi$$
0.215545 + 0.976494i $$0.430847\pi$$
$$710$$ 0 0
$$711$$ −6.13070e6 −0.454816
$$712$$ − 1.71072e6i − 0.126468i
$$713$$ − 3.09624e7i − 2.28092i
$$714$$ −2.85768e6 −0.209782
$$715$$ 0 0
$$716$$ −171072. −0.0124709
$$717$$ 7.74336e6i 0.562512i
$$718$$ 3.50611e6i 0.253813i
$$719$$ 1.43716e7 1.03677 0.518385 0.855147i $$-0.326533\pi$$
0.518385 + 0.855147i $$0.326533\pi$$
$$720$$ 0 0
$$721$$ −6.45056e6 −0.462124
$$722$$ 9.02080e6i 0.644024i
$$723$$ 1.15285e7i 0.820214i
$$724$$ 8.73702e6 0.619465
$$725$$ 0 0
$$726$$ 1.92620e6 0.135631
$$727$$ − 1.40705e7i − 0.987353i −0.869646 0.493676i $$-0.835653\pi$$
0.869646 0.493676i $$-0.164347\pi$$
$$728$$ 1.83142e6i 0.128074i
$$729$$ −9.93049e6 −0.692073
$$730$$ 0 0
$$731$$ −5.40043e6 −0.373796
$$732$$ 3.91616e6i 0.270136i
$$733$$ 3.75000e6i 0.257793i 0.991658 + 0.128897i $$0.0411436\pi$$
−0.991658 + 0.128897i $$0.958856\pi$$
$$734$$ 1.19225e7 0.816819
$$735$$ 0 0
$$736$$ 4.30080e6 0.292654
$$737$$ 5.86387e6i 0.397664i
$$738$$ − 3.55898e6i − 0.240539i
$$739$$ −2.61318e7 −1.76019 −0.880093 0.474802i $$-0.842520\pi$$
−0.880093 + 0.474802i $$0.842520\pi$$
$$740$$ 0 0
$$741$$ 2.74480e6 0.183639
$$742$$ 7.29943e6i 0.486720i
$$743$$ 159072.i 0.0105711i 0.999986 + 0.00528557i $$0.00168246\pi$$
−0.999986 + 0.00528557i $$0.998318\pi$$
$$744$$ −4.71808e6 −0.312488
$$745$$ 0 0
$$746$$ −1.56577e7 −1.03010
$$747$$ 5.03389e6i 0.330067i
$$748$$ − 7.83821e6i − 0.512227i
$$749$$ −2.39375e6 −0.155910
$$750$$ 0 0
$$751$$ −2.65311e7 −1.71654 −0.858272 0.513196i $$-0.828461\pi$$
−0.858272 + 0.513196i $$0.828461\pi$$
$$752$$ − 463872.i − 0.0299126i
$$753$$ − 1.35801e7i − 0.872802i
$$754$$ 1.13670e7 0.728143
$$755$$ 0 0
$$756$$ −3.02624e6 −0.192575
$$757$$ − 1.52032e7i − 0.964260i −0.876100 0.482130i $$-0.839863\pi$$
0.876100 0.482130i $$-0.160137\pi$$
$$758$$ 1.44264e7i 0.911981i
$$759$$ −1.41120e7 −0.889169
$$760$$ 0 0
$$761$$ 4.71380e6 0.295059 0.147530 0.989058i $$-0.452868\pi$$
0.147530 + 0.989058i $$0.452868\pi$$
$$762$$ − 1.26397e7i − 0.788585i
$$763$$ 2.76233e6i 0.171776i
$$764$$ 9.21562e6 0.571204
$$765$$ 0 0
$$766$$ 1.06657e7 0.656779
$$767$$ 2.00324e7i 1.22954i
$$768$$ − 655360.i − 0.0400938i
$$769$$ 1.58977e6 0.0969434 0.0484717 0.998825i $$-0.484565\pi$$
0.0484717 + 0.998825i $$0.484565\pi$$
$$770$$ 0 0
$$771$$ −3.17742e6 −0.192504
$$772$$ − 6.62301e6i − 0.399956i
$$773$$ 9.69095e6i 0.583334i 0.956520 + 0.291667i $$0.0942100\pi$$
−0.956520 + 0.291667i $$0.905790\pi$$
$$774$$ −2.11869e6 −0.127120
$$775$$ 0 0
$$776$$ 1.08659e6 0.0647757
$$777$$ 7.02170e6i 0.417244i
$$778$$ − 853464.i − 0.0505518i
$$779$$ −2.92434e6 −0.172657
$$780$$ 0 0
$$781$$ −9.48326e6 −0.556327
$$782$$ − 2.44944e7i − 1.43235i
$$783$$ 1.87828e7i 1.09485i
$$784$$ −614656. −0.0357143
$$785$$ 0 0
$$786$$ 986640. 0.0569642
$$787$$ − 1.57170e6i − 0.0904549i −0.998977 0.0452275i $$-0.985599\pi$$
0.998977 0.0452275i $$-0.0144013\pi$$
$$788$$ 7.91914e6i 0.454320i
$$789$$ −1.05101e7 −0.601054
$$790$$ 0 0
$$791$$ 428358. 0.0243425
$$792$$ − 3.07507e6i − 0.174198i
$$793$$ − 1.42940e7i − 0.807180i
$$794$$ −1.63763e7 −0.921860
$$795$$ 0 0
$$796$$ 8.32582e6 0.465741
$$797$$ − 2.25298e6i − 0.125635i −0.998025 0.0628175i $$-0.979991\pi$$
0.998025 0.0628175i $$-0.0200086\pi$$
$$798$$ 921200.i 0.0512090i
$$799$$ −2.64190e6 −0.146403
$$800$$ 0 0
$$801$$ −3.82239e6 −0.210501
$$802$$ − 3.76946e6i − 0.206940i
$$803$$ 1.21027e6i 0.0662360i
$$804$$ 2.79232e6 0.152344
$$805$$ 0 0
$$806$$ 1.72210e7 0.933728
$$807$$ 1.18958e7i 0.643000i
$$808$$ 6.34906e6i 0.342122i
$$809$$ 2.37938e7 1.27818 0.639090 0.769132i $$-0.279311\pi$$
0.639090 + 0.769132i $$0.279311\pi$$
$$810$$ 0 0
$$811$$ 5.32300e6 0.284187 0.142093 0.989853i $$-0.454617\pi$$
0.142093 + 0.989853i $$0.454617\pi$$
$$812$$ 3.81494e6i 0.203048i
$$813$$ 1.43008e7i 0.758812i
$$814$$ −1.92595e7 −1.01879
$$815$$ 0 0
$$816$$ −3.73248e6 −0.196233
$$817$$ 1.74088e6i 0.0912460i
$$818$$ − 1.93824e7i − 1.01280i
$$819$$ 4.09209e6 0.213174
$$820$$ 0 0
$$821$$ 1.48802e7 0.770464 0.385232 0.922820i $$-0.374121\pi$$
0.385232 + 0.922820i $$0.374121\pi$$
$$822$$ − 1.21294e7i − 0.626121i
$$823$$ − 2.00601e7i − 1.03236i −0.856479 0.516182i $$-0.827353\pi$$
0.856479 0.516182i $$-0.172647\pi$$
$$824$$ −8.42522e6 −0.432278
$$825$$ 0 0
$$826$$ −6.72319e6 −0.342867
$$827$$ 1.21539e7i 0.617949i 0.951070 + 0.308975i $$0.0999858\pi$$
−0.951070 + 0.308975i $$0.900014\pi$$
$$828$$ − 9.60960e6i − 0.487113i
$$829$$ −3.21197e7 −1.62325 −0.811625 0.584179i $$-0.801417\pi$$
−0.811625 + 0.584179i $$0.801417\pi$$
$$830$$ 0 0
$$831$$ 633020. 0.0317991
$$832$$ 2.39206e6i 0.119802i
$$833$$ 3.50066e6i 0.174798i
$$834$$ 1.00234e7 0.499001
$$835$$ 0 0
$$836$$ −2.52672e6 −0.125038
$$837$$ 2.84559e7i 1.40397i
$$838$$ − 6.93938e6i − 0.341359i
$$839$$ 1.01320e6 0.0496922 0.0248461 0.999691i $$-0.492090\pi$$
0.0248461 + 0.999691i $$0.492090\pi$$
$$840$$ 0 0
$$841$$ 3.16681e6 0.154394
$$842$$ 6.60580e6i 0.321104i
$$843$$ 4.96614e6i 0.240686i
$$844$$ −2.93254e6 −0.141706
$$845$$ 0 0
$$846$$ −1.03646e6 −0.0497884
$$847$$ − 2.35960e6i − 0.113013i
$$848$$ 9.53395e6i 0.455285i
$$849$$ 1.15842e7 0.551564
$$850$$ 0 0
$$851$$ −6.01860e7 −2.84886
$$852$$ 4.51584e6i 0.213128i
$$853$$ − 234824.i − 0.0110502i −0.999985 0.00552510i $$-0.998241\pi$$
0.999985 0.00552510i $$-0.00175870\pi$$
$$854$$ 4.79730e6 0.225088
$$855$$ 0 0
$$856$$ −3.12653e6 −0.145840
$$857$$ 2.83802e7i 1.31997i 0.751279 + 0.659985i $$0.229437\pi$$
−0.751279 + 0.659985i $$0.770563\pi$$
$$858$$ − 7.84896e6i − 0.363994i
$$859$$ −4.00081e7 −1.84997 −0.924986 0.380001i $$-0.875924\pi$$
−0.924986 + 0.380001i $$0.875924\pi$$
$$860$$ 0 0
$$861$$ 3.04878e6 0.140158
$$862$$ − 1.65744e7i − 0.759748i
$$863$$ 2.08030e7i 0.950823i 0.879764 + 0.475411i $$0.157701\pi$$
−0.879764 + 0.475411i $$0.842299\pi$$
$$864$$ −3.95264e6 −0.180137
$$865$$ 0 0
$$866$$ 1.21586e7 0.550922
$$867$$ 7.05907e6i 0.318933i
$$868$$ 5.77965e6i 0.260377i
$$869$$ 1.44050e7 0.647088
$$870$$ 0 0
$$871$$ −1.01920e7 −0.455211
$$872$$ 3.60794e6i 0.160682i
$$873$$ − 2.42785e6i − 0.107817i
$$874$$ −7.89600e6 −0.349646
$$875$$ 0 0
$$876$$ 576320. 0.0253748
$$877$$ 3.03559e7i 1.33273i 0.745624 + 0.666367i $$0.232152\pi$$
−0.745624 + 0.666367i $$0.767848\pi$$
$$878$$ 1.01708e7i 0.445267i
$$879$$ −1.43886e7 −0.628125
$$880$$ 0 0
$$881$$ −2.58936e7 −1.12396 −0.561981 0.827150i $$-0.689961\pi$$
−0.561981 + 0.827150i $$0.689961\pi$$
$$882$$ 1.37337e6i 0.0594452i
$$883$$ 1.88813e7i 0.814950i 0.913216 + 0.407475i $$0.133591\pi$$
−0.913216 + 0.407475i $$0.866409\pi$$
$$884$$ 1.36236e7 0.586354
$$885$$ 0 0
$$886$$ 9.72840e6 0.416349
$$887$$ − 2.34431e7i − 1.00048i −0.865888 0.500238i $$-0.833246\pi$$
0.865888 0.500238i $$-0.166754\pi$$
$$888$$ 9.17120e6i 0.390296i
$$889$$ −1.54836e7 −0.657079
$$890$$ 0 0
$$891$$ 1.29394e6 0.0546033
$$892$$ − 2.04394e7i − 0.860115i
$$893$$ 851640.i 0.0357378i
$$894$$ −2.42376e6 −0.101425
$$895$$ 0 0
$$896$$ −802816. −0.0334077
$$897$$ − 2.45280e7i − 1.01784i
$$898$$ 7.31412e6i 0.302671i
$$899$$ 3.58722e7 1.48033
$$900$$ 0 0
$$901$$ 5.42988e7 2.22833
$$902$$ 8.36237e6i 0.342226i
$$903$$ − 1.81496e6i − 0.0740709i
$$904$$ 559488. 0.0227703
$$905$$ 0 0
$$906$$ −4.97792e6 −0.201478
$$907$$ − 5.60873e6i − 0.226384i −0.993573 0.113192i $$-0.963892\pi$$
0.993573 0.113192i $$-0.0361076\pi$$
$$908$$ 2.06023e7i 0.829279i
$$909$$ 1.41862e7 0.569450
$$910$$ 0 0
$$911$$ 2.16215e7 0.863156 0.431578 0.902076i $$-0.357957\pi$$
0.431578 + 0.902076i $$0.357957\pi$$
$$912$$ 1.20320e6i 0.0479017i
$$913$$ − 1.18279e7i − 0.469602i
$$914$$ 6.32252e6 0.250337
$$915$$ 0 0
$$916$$ 5.61498e6 0.221110
$$917$$ − 1.20863e6i − 0.0474648i
$$918$$ 2.25115e7i 0.881654i
$$919$$ −4.51695e7 −1.76424 −0.882119 0.471028i $$-0.843883\pi$$
−0.882119 + 0.471028i $$0.843883\pi$$
$$920$$ 0 0
$$921$$ −9.89098e6 −0.384229
$$922$$ − 2.03842e7i − 0.789706i
$$923$$ − 1.64828e7i − 0.636835i
$$924$$ 2.63424e6 0.101502
$$925$$ 0 0
$$926$$ 2.80935e7 1.07666
$$927$$ 1.88251e7i 0.719512i
$$928$$ 4.98278e6i 0.189934i
$$929$$ 2.28729e7 0.869524 0.434762 0.900545i $$-0.356832\pi$$
0.434762 + 0.900545i $$0.356832\pi$$
$$930$$ 0 0
$$931$$ 1.12847e6 0.0426693
$$932$$ 1.33785e7i 0.504506i
$$933$$ 2.22050e7i 0.835117i
$$934$$ −1.69938e7 −0.637417
$$935$$ 0 0
$$936$$ 5.34477e6 0.199406
$$937$$ − 1.79616e7i − 0.668336i −0.942514 0.334168i $$-0.891545\pi$$
0.942514 0.334168i $$-0.108455\pi$$
$$938$$ − 3.42059e6i − 0.126939i
$$939$$ −2.33008e7 −0.862395
$$940$$ 0 0
$$941$$ −1.79697e7 −0.661558 −0.330779 0.943708i $$-0.607311\pi$$
−0.330779 + 0.943708i $$0.607311\pi$$
$$942$$ − 3.04160e6i − 0.111680i
$$943$$ 2.61324e7i 0.956974i
$$944$$ −8.78131e6 −0.320722
$$945$$ 0 0
$$946$$ 4.97818e6 0.180860
$$947$$ 4.32115e7i 1.56576i 0.622174 + 0.782879i $$0.286250\pi$$
−0.622174 + 0.782879i $$0.713750\pi$$
$$948$$ − 6.85952e6i − 0.247898i
$$949$$ −2.10357e6 −0.0758213
$$950$$ 0 0
$$951$$ 4.27542e6 0.153295
$$952$$ 4.57229e6i 0.163509i
$$953$$ 7.50965e6i 0.267848i 0.990992 + 0.133924i $$0.0427577\pi$$
−0.990992 + 0.133924i $$0.957242\pi$$
$$954$$ 2.13024e7 0.757806
$$955$$ 0 0
$$956$$ 1.23894e7 0.438434
$$957$$ − 1.63498e7i − 0.577074i
$$958$$ 2.23714e6i 0.0787551i
$$959$$ −1.48585e7 −0.521708
$$960$$ 0 0
$$961$$ 2.57172e7 0.898288
$$962$$ − 3.34749e7i − 1.16622i
$$963$$ 6.98584e6i 0.242746i
$$964$$ 1.84456e7 0.639293
$$965$$ 0 0
$$966$$ 8.23200e6 0.283833
$$967$$ − 1.69305e7i − 0.582242i −0.956686 0.291121i $$-0.905972\pi$$
0.956686 0.291121i $$-0.0940283\pi$$
$$968$$ − 3.08192e6i − 0.105714i
$$969$$ 6.85260e6 0.234448
$$970$$ 0 0
$$971$$ 2.86144e7 0.973949 0.486974 0.873416i $$-0.338101\pi$$
0.486974 + 0.873416i $$0.338101\pi$$
$$972$$ 1.43915e7i 0.488586i
$$973$$ − 1.22787e7i − 0.415787i
$$974$$ −5.28227e6 −0.178412
$$975$$ 0 0
$$976$$ 6.26586e6 0.210550
$$977$$ 3.69445e7i 1.23826i 0.785287 + 0.619132i $$0.212515\pi$$
−0.785287 + 0.619132i $$0.787485\pi$$
$$978$$ 4.97024e6i 0.166161i
$$979$$ 8.98128e6 0.299489
$$980$$ 0 0
$$981$$ 8.06148e6 0.267450
$$982$$ − 2.50877e7i − 0.830200i
$$983$$ 3.88787e7i 1.28330i 0.766998 + 0.641650i $$0.221750\pi$$
−0.766998 + 0.641650i $$0.778250\pi$$
$$984$$ 3.98208e6 0.131106
$$985$$ 0 0
$$986$$ 2.83785e7 0.929603
$$987$$ − 887880.i − 0.0290109i
$$988$$ − 4.39168e6i − 0.143133i
$$989$$ 1.55568e7 0.505743
$$990$$ 0 0
$$991$$ 2.49212e7 0.806092 0.403046 0.915180i $$-0.367951\pi$$
0.403046 + 0.915180i $$0.367951\pi$$
$$992$$ 7.54893e6i 0.243560i
$$993$$ 3.96616e6i 0.127643i
$$994$$ 5.53190e6 0.177586
$$995$$ 0 0
$$996$$ −5.63232e6 −0.179903
$$997$$ 1.01956e7i 0.324845i 0.986721 + 0.162422i $$0.0519307\pi$$
−0.986721 + 0.162422i $$0.948069\pi$$
$$998$$ − 1.57514e7i − 0.500603i
$$999$$ 5.53138e7 1.75356
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 350.6.c.d.99.1 2
5.2 odd 4 350.6.a.i.1.1 1
5.3 odd 4 14.6.a.a.1.1 1
5.4 even 2 inner 350.6.c.d.99.2 2
15.8 even 4 126.6.a.f.1.1 1
20.3 even 4 112.6.a.c.1.1 1
35.3 even 12 98.6.c.d.79.1 2
35.13 even 4 98.6.a.a.1.1 1
35.18 odd 12 98.6.c.c.79.1 2
35.23 odd 12 98.6.c.c.67.1 2
35.33 even 12 98.6.c.d.67.1 2
40.3 even 4 448.6.a.l.1.1 1
40.13 odd 4 448.6.a.e.1.1 1
60.23 odd 4 1008.6.a.b.1.1 1
105.83 odd 4 882.6.a.x.1.1 1
140.83 odd 4 784.6.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.a.a.1.1 1 5.3 odd 4
98.6.a.a.1.1 1 35.13 even 4
98.6.c.c.67.1 2 35.23 odd 12
98.6.c.c.79.1 2 35.18 odd 12
98.6.c.d.67.1 2 35.33 even 12
98.6.c.d.79.1 2 35.3 even 12
112.6.a.c.1.1 1 20.3 even 4
126.6.a.f.1.1 1 15.8 even 4
350.6.a.i.1.1 1 5.2 odd 4
350.6.c.d.99.1 2 1.1 even 1 trivial
350.6.c.d.99.2 2 5.4 even 2 inner
448.6.a.e.1.1 1 40.13 odd 4
448.6.a.l.1.1 1 40.3 even 4
784.6.a.i.1.1 1 140.83 odd 4
882.6.a.x.1.1 1 105.83 odd 4
1008.6.a.b.1.1 1 60.23 odd 4